Journal of Functional Analysis 258 (2010) 2865–2883 www.elsevier.com/locate/jfa
On the generalized self-similar singularities for the Euler and the Navier–Stokes equations Dongho Chae 1 Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 12 August 2008; accepted 3 February 2010 Available online 13 February 2010 Communicated by I. Rodnianski
Abstract In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier–Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier–Stokes equations depending on a free parameter. © 2010 Elsevier Inc. All rights reserved. Keywords: Euler equations; Navier–Stokes equations; Generalized self-similar singularities; A priori estimates
1. Self-similar singularities We are concerned on the following Euler equations for the homogeneous incompressible fluid flows in R3 : ⎧ ∂v ⎪ ⎪ + (v · ∇)v = −∇p, (x, t) ∈ R3 × (0, ∞), ⎨ ∂t (E) div v = 0, (x, t) ∈ R3 × (0, ∞), ⎪ ⎪ ⎩ v(x, 0) = v0 (x), x ∈ R3 , E-mail address:
[email protected]. 1 This research was done, while the author was visiting University of Chicago. The work is supported partially by NRF
Grant. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.006
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where v = (v1 , v2 , v3 ), vj = vj (x, t), j = 1, 2, 3, is the velocity of the flow, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity, satisfying div v0 = 0. The system (E) is first modeled by Euler in [13]. The local well-posedness of the Euler equations in H m (R3 ), m > 5/2, is established by Kato in [17], which says that given v0 ∈ H m (R3 ), there exists T ∈ (0, ∞] such that there exists a unique solution to (E), v ∈ C([0, T ); H m (R3 )). The finite time blowup problem of the local classical solution is known as one of the most important and difficult problems in partial differential equations (see e.g. [20,6,8,7,5] for graduate level texts and survey articles on the current status of the problem). We say a local in time classical solution v ∈ C([0, T ); H m (R3 )) blows up at T if lim supt→T v(t)H m = ∞ for all m > 5/2. The celebrated Beale–Kato–Majda criterion [1] states that the blow-up happens at T if and only if T
ω(t)
L∞
dt = ∞.
0
Although the original result of [1] is the blow-up criterion in the H m (R3 ) norm with m 3, it is easy to extend it to the case of m > 5/2. There are studies of geometric nature for the blowup criterion [9,7,12]. As another direction of studies of the blow-up problem mathematicians also consider various scenarios of singularities and study carefully their possibility of realization (see e.g. [10,11,2,3] for some of those studies). One of the purposes in this paper, especially in this section, is to study more deeply the notions related to the scenarios of the self-similar singularities in the Euler equations, the preliminary studies of which are done in [2,3]. We recall that system (E) has scaling property that if (v, p) is a solution of the system (E), then for any λ > 0 and α ∈ R the functions v λ,α (x, t) = λα v λx, λα+1 t ,
p λ,α (x, t) = λ2α p λx, λα+1 t
(1.1)
are also solutions of (E) with the initial data v0λ,α (x) = λα v0 (λx). In view of the scaling properties in (1.1), a natural self-similar blowing-up solution v(x, t) of (E) should be of the form, v(x, t) = p(x, t) =
1 α
(T − t) α+1 α+1 2α
(T − t) α+1
V
,
1
P
x (T − t) α+1 x
(1.2)
1
(T − t) α+1
(1.3)
for α = −1 and t sufficiently close to T . Substituting (1.2)–(1.3) into (E), we obtain the following stationary system:
αV + (y · ∇)V + (α + 1)(V · ∇)V = −∇P , div V = 0,
(1.4)
the Navier–Stokes equations version of which has been studied extensively after Leray’s pioneering paper [19,23,24,22,3,16]. Existence of solution of the system (1.4) is equivalent to the existence of solutions to the Euler equations of the form (1.2)–(1.3), which blows up in a self-similar fashion. Given (α, p) ∈ (−1, ∞) × (0, ∞], we say the blow-up is α-asymptotically
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self-similar in the sense of Lp if there exists V = V α ∈ W˙ 1,p (R3 ) such that the following convergence holds true:
1 · =0 ∇v(·, t) − ∇V lim (T − t) 1 t→T T −t (T − t) α+1 L∞ if p = ∞, while
1 · =0 1 ω(·, t) − T − t Ω (T − t) α+1 Lp
3 1− (α+1)p
lim (T − t)
t→T
if 0 < p < ∞, where and hereafter we denote Ω = curl V
and Ω = curl V .
The above limit function V ∈ Lp (R3 ) with Ω = 0 is called the blow-up profile. We observe that the self-similar blow-up given by (1.2)–(1.3) is trivial case of α-asymptotic self-similar blow-up with the blow-up profile given by the representing function V . We say a blow-up at T is of type I, if lim sup (T − t)∇v(t)L∞ < ∞. t→T
If the blow-up is not of type I, we say it is of type II. For the use of terminology, type I and type II blow-ups, we followed the literatures on the studies of the blow-up problem in the semilinear heat equations (see e.g. [21,15,14], and the references therein). The use of ∇v(t)L∞ rather than v(t)L∞ in our definition of types I and II is motivated by Beale–Kato–Majda’s blow-up criterion. Theorem 1.1. Let m > 5/2, and v ∈ C([0, T ); H m (R3 )) be a solution to (E) with v0 ∈ H m (R3 ), div v0 = 0. We set lim sup (T − t)∇v(t)L∞ := M(T ).
(1.5)
t→T
Then, either M(T ) 1, or the solution does not blow up at time T , which implies that M(T ) = 0. Hence, one can only have type I blow-up at time T if M(T ) 1. An immediate implication of the above theorem on the self-similar blow-up is the following. Corollary 1.1. There exists no self-similar blow-up for the solution of the 3D Euler equations with the blow-up profile V satisfying ∇V L∞ < 1. Proof. We just observe that if v(x, t) =
1
α
(T −t) α+1
V(
x 1
(T −t) α+1
(T − t)∇v(t)L∞ = ∇V L∞ ,
), then
∀t ∈ (0, T ).
2
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Proof of Theorem 1.1. It suffices to show that M(T ) < 1 implies non-blow-up at T , which, in turn, leads to M(T ) = 0, since ∇v(t)L∞ ∈ C([0, T ]) in this case. We suppose M(T ) < 1. Then, there exists t0 ∈ (0, T ) such that sup (T − t)∇v(t)L∞ := M0 < 1.
t0
Taking curl of the evolution part of (E), we have the vorticity equation, ∂ω + (v · ∇)ω = (ω · ∇)v. ∂t This, taking dot product with ξ = ω/|ω|, leads to ∂|ω| + (v · ∇)|ω| = (ξ · ∇)v · ξ |ω|. ∂t Integrating this over [t0 , t] along the particle trajectories {X(a, t)} defined by v(x, t), we have t
ω X(a, t), t = ω X(a, t0 ), t0 exp (ξ · ∇)v · ξ X(a, s), s ds ,
(1.6)
t0
from which we estimate ω(t)
L∞
ω(t0 )
t L∞
exp
∇v(τ )
L∞
dτ
t0
< ω(t0 )L∞ exp M0
T − t0 = ω(t0 )L∞ T −t
t
−1
(T − τ )
dτ
t0
M0 .
(1.7)
T Since M0 < 1, we have t0 ω(t)L∞ dt < ∞, and thanks to the Beale–Kato–Majda criterion there exists no blow-up at T , and we can continue our classical solution beyond T . 2 The following is our main theorem in this section. Theorem 1.2. Let a classical solution v ∈ C([0, T ); H m (R3 )) of the 3D Euler equations with initial data v0 ∈ H m (R3 ) ∩ W˙ 1,p (R3 ), div v0 = 0, ω0 = 0 blows up with type I. Let M = M(T ) be as in Theorem 1.1. Suppose (α, p) ∈ (−1, ∞) × (0, ∞] satisfies
3
.
M < 1 − (α + 1)p
(1.8)
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Then, there exists no α-asymptotically self-similar blow-up at t = T in the sense of Lp if ω0 ∈ Lp (R3 ). Hence, for any type I blow-up and for any α ∈ (−1, ∞) there exists p1 ∈ (0, ∞] such that it is not α-asymptotically self-similar in the sense of Lp1 . Remark 1.1. We note that the case p = ∞ of the above theorem follows from Theorem 1.1, which states that there is no singularity at all at t = T in this case. The above theorem can be regarded an improvement of the main theorem in [3], in the sense that we can consider the Lp convergence only to exclude nontrivial blow-up profile V , where p depends on M. Moreover, we 0 do not need to use the Besov space B˙ ∞,1 in the statement of the theorem, and the continuation principle of local solution in the Besov space in the proof. Proof of Theorem 1.2. We assume asymptotically self-similar blow-up happens at T . Let us introduce similarity variables defined by y=
x 1
(T − t) α+1
,
s=
T 1 log , α+1 T −t
and transformation of the unknowns (v, p) → (V , P ) according to v(x, t) =
1 (T − t)
α α+1
V (y, s),
p(x, t) =
1 2α
(T − t) α+1
P (y, s).
(1.9)
Substituting (v, p) into the (E) we obtain the equivalent evolution equation for (V , P ),
(E1 )
⎧ ⎪ ⎨ Vs + αV + (y · ∇)V + (α + 1)(V · ∇)V = −∇P , div V = 0, 1 ⎪ α ⎩ V (y, 0) = V (y) = T α+1 v T α+1 y . 0
0
Then the assumption of asymptotically self-similar singularity at T implies that there exists V = V α ∈ W˙ 1,p (R3 ) such that lim Ω(·, s) − Ω Lp = 0.
(1.10)
s→∞
Now the hypothesis (1.8) implies that there exists t0 ∈ (0, T ) such that
sup (T − t) ∇v(t) L∞ := M0 <
1 −
t0
3
. (α + 1)p
Taking Lp (R3 ) norm of (1.6), taking into account the following simple estimates, −∇v(·, t)L∞ (ξ · ∇)v · ξ(x, t) ∇v(·, t)L∞ , we obtain, for all p ∈ (0, ∞],
∀(x, t) ∈ R3 × [t0 , T ),
(1.11)
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t exp − ∇v(·, s)L∞ ds ω(t)Lp Lp
ω(t0 )
t0
t ω0 Lp exp
∇v(·, s)
L∞
ds ,
(1.12)
t0
where we use the fact that a → X(a, t) is a volume preserving map. From the fact t
∇v(·, s)
t L∞
ds M0
t0
−1
(T − τ ) t0
T −t , dτ = −M0 log T − t0
and ω(t)Lp = ω(t0 )Lp
T −t T − t0
3 (α+1)p −1
Ω(s)Lp , Ω(s0 )Lp
where we set
T 1 , log s0 = α+1 T − t0 we find that (1.12) leads us to
T −t T − t0
M0 +1−
3 (α+1)p
Ω(s)Lp Ω(s0 )Lp
T −t T − t0
−M0 +1−
3 (α+1)p
(1.13)
for all p ∈ (0, ∞]. Passing t → T , which is equivalent to s → ∞ in (1.13), we have from (1.10) Ω(s)Lp ΩLp = ∈ (0, ∞). s→∞ Ω(s0 )Lp Ω(s0 )Lp lim
By (1.11), M0 + 1 −
3 (α+1)p
< 0 or −M0 + 1 −
T −t lim t→T T − t0
3 (α+1)p
M0 +1−
(1.14)
> 0. In the former case we have
3 (α+1)p
= ∞,
(1.15)
while in the latter case
T −t lim t→T T − t0
−M0 +1−
3 (α+1)p
= 0.
(1.16)
Both of (1.15) and (1.16) contradict with (1.14). If the blow-up is of type I, and M(T ) < ∞, then one can always choose p1 ∈ (0, p0 ) so small that (1.8) is valid for p = p1 . With such p1 it is not α-asymptotically self-similar in Lp1 . 2
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For the self-similar blowing-up solution of the form (1.2)–(1.3) we observe that in order to be consistent with the energy conservation, v(t)L2 = v0 L2 for all t ∈ [0, T ), we need to fix α = 3/2. Since the self-similar blowing-up solution corresponds to a trivial convergence of the asymptotically self-similar blow-up, the following is immediate from Theorem 1.2. Corollary 1.2. Given p ∈ (0, ∞], there exists no self-similar blow-up with the blow-up profile V satisfying Ω ∈ Lp (R3 ) if ∇V
L∞
6
. < 1 − 5p
(1.17)
Remark 1.2. The above corollary implies that we can exclude self-similar singularity of the Euler equations under the assumption of Ω ∈ Lp (R3 ) if p satisfies the condition (1.17). The following is, in turn, immediate from the above corollary, which is essentially Theorem 1.1 in [2]. Note that here we do not need the diffeomorphism condition for the particle trajectory mapping generated by a local classical solution before the blow-up time, which was necessary in [2] in order to guarantee the existence of the back-to-label map. Corollary 1.3. There exists no self-similar blow-up with the blow-up profile V satisfying Ω ∈ Lp (R3 ) for all p ∈ (0, p0 ) for some p0 > 0. Proof. Suppose, on the contrary, there exists a self-similar blow-up with ∇V L∞ < ∞. Then, there exists p1 > 0 such that ∇V L∞
6
, < 1 − 5p
∀p ∈ (0, p1 ).
Due to Corollary 1.2 there should be no p ∈ (0, p1 ) such that curl V = Ω ∈ Lp (R3 ), which contradicts the hypothesis of the current corollary. 2 The following theorem is concerned on the possibility of type II asymptotically self-similar singularity of the Euler equations, for which the blow-up rate near the possible blow-up time T is ∇v(t)
L∞
∼
1 , (T − t)γ
γ > 1.
(1.18)
Theorem 1.3. Let v ∈ C([0, T ); H m (R3 )), m > 5/2, be local classical solution of the Euler equations. Suppose there exists γ > 1 and R1 > 0 such that the following convergence holds true: 3 γ lim (T − t)(α− 2 ) α+1 v(·, t) −
t→T
1 γα
(T − t) α+1
V
· (T − t)
γ α+1
L2 (BR1 )
= 0,
(1.19)
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where BR1 = {x ∈ R3 | |x| < R1 }. Then, the blow-up profile V ∈ L2loc (R3 ) is a weak solution of the following stationary Euler equations, (V · ∇)V = −∇P ,
div V = 0.
(1.20)
Remark 1.3. Eq. (1.20) seems to not have any immediate applicability in the Euler setting, but see its counterpart (1.27) and Theorem 1.4 below, where it is used to rule out type II asymptotically self-similar singularities for the Navier–Stokes equations. Proof of Theorem 1.3. We introduce a self-similar transform defined by v(x, t) =
1 (T − t)
αγ α+1
p(x, t) =
V (y, s),
1 2αγ
(T − t) α+1
(1.21)
P (y, s)
with y=
1 γ
(T − t) α+1
x,
T γ −1 1 s= −1 . (γ − 1)T γ −1 (T − t)γ −1
(1.22)
Substituting (v, p) in (1.21)–(1.22) into the (E), we have
(E2 )
⎧ γ α 1 ⎪ ⎪ ⎪ ⎨ − s(γ − 1) + T 1−γ α + 1 V + α + 1 (y · ∇)V = Vs + (V · ∇)V + ∇P , div V = 0, ⎪ ⎪ ⎪ γ αγ ⎩ V (y, 0) = V0 (y) = T α+1 v0 T α+1 y .
(1.23)
The hypothesis (1.19) is written as lim V (·, s) − V (·)L2 (B
s→∞
R(s)
= 0, )
R(s) = (γ − 1)s +
1 T γ −1
γ (α+1)(γ −1)
,
(1.24)
which implies that lim V (·, s) − V L2 (B
s→∞
R)
= 0,
∀R > 0,
(1.25)
where V (y, s) is defined by (1.21). Similarly to [16,3], we consider the scalar test function 1 ξ ∈ C01 (0, 1) with 0 ξ(s) ds = 0, and the vector test function φ = (φ1 , φ2 , φ3 ) ∈ C01 (R3 ) with div φ = 0. We multiply the first equation of (E2 ), in the dot product, by ξ(s − n)φ(y), and integrate it over R3 × [n, n + 1], and then we integrate by parts to obtain
D. Chae / Journal of Functional Analysis 258 (2010) 2865–2883
α + α+1
2873
1 g(s + n)ξ(s)V (y, s + n) · φ(y) dy ds 0 R3
1 − α+1
1 g(s + n)ξ(s)V (y, s + n) · (y · ∇)φ(y) dy ds 0 R3
1 =
ξs (s)φ(y) · V (y, s + n) dy ds 0
R3
1
ξ(s) V (y, s + n) · V (y, s + n) · ∇ φ(y) dy ds = 0,
+ 0 R3
where we set g(s) =
γ . s(γ − 1) + T 1−γ
1 1 Passing to the limit n → ∞ in this equation, using the facts 0 ξs (s) ds = 0, 0 ξ(s) ds = 0, V (·, s + n) → V in L2loc (R3 ), and finally g(s + n) → 0, we find that V ∈ L2loc (R3 ) satisfies V · (V · ∇)φ(y) dy = 0 R3
for all vector test function φ ∈ C01 (R3 ) with div φ = 0. On the other hand, we can pass s → ∞ directly in the weak formulation of the second equation of (E2 ) to have V · ∇ψ(y) dy = 0 R3
for all scalar test function ψ ∈ C01 (R3 ).
2
In the following theorem we rule out the possibility of the blow-up rate given by (1.18) in the setting of the Navier–Stokes equations. Theorem 1.4. Let p ∈ [3, ∞) and v ∈ C([0, T ); Lp (R3 )) be a local classical solution of the Navier–Stokes equations constructed by Kato [18]. Suppose there exist γ > 1 and V ∈ Lp (R3 ) such that the following convergence holds true: lim (T − t)
t→T
(p−3)γ 2p
(p−3)γ v(·, t) − (T − t)− 2p V
If the blow-up profile V belongs to H˙ 1 (R3 ), then V = 0.
· γ
(T − t) 2
= 0. Lp
(1.26)
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D. Chae / Journal of Functional Analysis 258 (2010) 2865–2883
Proof. Since the main part of the proof is essentially identical to that of Theorem 1.3, we will be brief. Introducing the self-similar variables of the form (1.21)–(1.23) with α = 12 , and substituting (v, p) into the Navier–Stokes equations, ∂v
+ (v · ∇)v = v − ∇p, ∂t div v = 0, v(x, 0) = v0 (x),
(NS)
we find that (V , P ) satisfies ⎧ γ ⎪ ⎪ ⎨ − 2s(γ − 1) + 2T 1−γ V + (y · ∇)V = Vs + (V · ∇)V − V + ∇P , div V = 0, ⎪ ⎪ γ γ ⎩ V (y, 0) = V0 (y) = T 2 v0 T 2 y .
The hypothesis (1.26) is now translated as lim V (·, s) − V (·)Lp = 0.
s→∞
Following exactly same argument as in the proof of Theorem 1.3, we can deduce that V is a stationary solution of the Navier–Stokes equations, namely there exists P such that (V · ∇)V = V − ∇P ,
div V = 0. (1.27) In the case V ∈ H˙ 1 ∩ Lp (R3 ), we easily obtain from (1.27) that R3 |∇V |2 dy = 0, which implies V = 0. 2 2. Generalized self-similar singularities Let us consider a classical solution to (E) v ∈ C([0, T ); H m (R3 )), m > 5/2, where we assume T ∈ (0, ∞] is the maximal time of existence of the classical solution. Let p(x, t) be the associated pressure. Let μ(·) ∈ C 1 ([0, T )) be a scalar function such that μ(t) > 0 for all t ∈ [0, T ). We transform from (v, p) to (V , P ) according to the formula, v(x, t) = μ(t)
α α+1
V μ(t)
1 α+1
t x,
μ(σ ) dσ ,
(2.1)
0
p(x, t) = μ(t)
2α α+1
P μ(t)
1 α+1
t x,
μ(σ ) dσ ,
(2.2)
0
where α ∈ (−1, ∞) as previously. This means that the space–time variables are transformed from (x, t) ∈ R3 × [0, T ) into (y, s) ∈ R3 × [0, ∞) as follows: y = μ(t)
1 α+1
t x,
s=
μ(σ ) dσ. 0
(2.3)
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Substituting (2.1)–(2.3) into the Euler equations, we obtain the equivalent equations satisfied by (V , P )
(E∗ )
⎧ μ (t) α 1 ⎪ ⎪ ⎪ ⎨ − μ(t)2 α + 1 V + α + 1 (y · ∇)V = Vs + (V · ∇)V + ∇P , div V = 0, ⎪ ⎪ ⎪ α 1 ⎩ V (y, 0) = V0 (y) = μ(0) α+1 v0 μ(0) α+1 y .
We note that the special cases μ(t) =
1 , T −t
μ(t) =
1 , (T − t)γ
γ > 1,
are considered in the previous section. Let us choose μ(t) = exp[±γ Then,
t
±γ α v(x, t) = exp α+1
t 0
∇v(τ )L∞ dτ ], γ 1.
∇v(τ )
L∞
dτ V (y, s),
(2.4)
0
±2γ α p(x, t) = exp α+1
t
∇v(τ )
L∞
dτ P (y, s)
(2.5)
0
with
±γ y = exp α+1 t s=
t
L∞
dτ x,
0
τ
exp ±γ 0
∇v(τ )
∇v(σ )
L∞
dσ dτ
(2.6)
0
respectively for the signs ±. Substituting (v, p) in (2.4)–(2.6) into the (E∗ ), we find that (E∗ ) becomes
(E± )
⎧ α 1 ⎪ ⎪ V+ (y · ∇)V = Vs + (V · ∇)V + ∇P , ⎨ ∓γ ∇V (s) L∞ α+1 α+1 ⎪ div V = 0, ⎪ ⎩ V (y, 0) = V0 (y) = v0 (y)
respectively for ±. Similar equations to the system (E± ), without the term involving (y · ∇)V , are introduced and studied in [4], where similarity type of transform with respect to only time variables was considered. The argument of the global/local well-posedness of the system (E± )
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respectively from the local well-posedness result of the Euler equations is as follows. We define S± =
T
τ
exp ±γ 0
∇v(σ )
L∞
dσ dτ.
0
Then, S ± is the maximal time of existence of classical solution for the system (E± ). Indeed, by the BKM criterion we have T
+
S =
τ exp γ ∇v(σ )L∞ dσ dτ
0
0
T
1 ω0 L∞
ω(τ )
L∞
dτ = ∞.
0
We also note the following integral invariant of the transform, T
∇v(t)
0
S ± ∇V ± (s) ∞ dt =
L
L∞
ds.
0
t Below we fix μ(t) := exp[ 0 ∇v(τ )L∞ dτ ]. We assume our local classical solution in H m (R3 ) blows up at T , and hence μ(T − 0) = T exp[ 0 ∇v(τ )L∞ dτ ] = ∞. Given (α, p) ∈ (−1, ∞) × (0, ∞], as previously, we say the blowup is α-asymptotically self-similar in the sense of Lp if there exists V = V α ∈ W˙ 1,p (R3 ) such that the following convergence holds true: 1 lim μ(t)−1 ∇v(·, t) − μ(t)∇V μ(t) α+1 (·) L∞ = 0
t→T
(2.7)
for p = ∞, and 3 −1+ (α+1)p
lim μ(t)
t→T
3 1− (α+1)p
ω(·, t) − μ(t)
1 Ω μ(t) α+1 (·) Lp = 0
(2.8)
for p ∈ (0, ∞). The above limiting function V with Ω = 0 is called the blow-up profile as previously. Proposition 2.1. Let α = 3/2. Then there exists no α-asymptotically self-similar blow-up in the sense of L∞ with the blow-up profile belonging to L2 (R3 ). Proof. Let us suppose that there exists V ∈ W˙ 1,∞ (R3 ) ∩ L2 (R3 ) such that (2.7) holds, then we will show that V = 0. In terms of the self-similar variables (2.7) is translated into lim ∇V (·, s) − ∇V L∞ = 0,
s→∞
where V is defined in (2.1). If ∇V L∞ = 0, then the condition V ∈ L2 (R3 ) implies that V = 0, and there is nothing to prove. Let us suppose ∇V L∞ > 0. As is done in the proof of Theorem 1.3 the equations satisfied V are easily shown to be
D. Chae / Journal of Functional Analysis 258 (2010) 2865–2883
⎧ ⎨ ⎩
−∇V L∞
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α 1 (y · ∇)V = (V · ∇)V + ∇P , V+ α+1 α+1
(2.9)
div V = 0
for a scalar function P . Taking L2 (R3 ) inner product of the first equation of (2.9) by V we obtain
3 ∇V L∞ α− V L2 = 0. α+1 2 Since ∇V L∞ = 0 and α = 32 , we have V L2 = 0, and V = 0.
2
Proposition 2.2. There exists no α-asymptotically self-similar blowing-up solution to (E) in the 3 sense of Lp if 0 < p < 2(α+1) . Proof. Suppose there exists α-asymptotically self-similar blow-up at T in the sense of Lp . Then, there exists Ω ∈ Lp (R3 ) such that, in terms of the self-similar variables introduced in (2.1)–(2.2), we have lim Ω(s)Lp = ΩLp < ∞.
(2.10)
s→∞
We represent the Lp norm of ω(t)Lp in terms of similarity variables to obtain ω(t)
Lp
Ω(s)
3 1− (α+1)p
= μ(t)
t Lp
μ(t) = exp
,
∇v(τ )
L∞
dτ .
(2.11)
0
Substituting this into the lower estimate part of (1.12), we have 3 −2+ (α+1)p
μ(t) If −2 +
3 (α+1)p
Ω(s)Lp . Ω0 Lp
(2.12)
> 0, then taking the limit t → T in the above inequality we obtain 3 −2+ (α+1)p
∞ = lim sup μ(t)
Ω0 Lp
t→T
lim sup Ω(s)Lp = ΩLp , s→∞
which is a contradiction to (2.10).
2
3. New a priori estimates One of the important advantages of the formulation of (E) in terms of (E± ) of the previous section is that after representing it by the vorticity formulation, the convection term is dominated by the linear term associated with ∓γ ∇V (s)L∞ (see (3.7) below), which enables us to derive new a priori estimates for ω(t)L∞ as follows.
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Theorem 3.1. Given m > 5/2 and v0 ∈ H m (R3 ) with div v0 = 0, let ω be the vorticity of the solution v ∈ C([0, T ); H m (R3 )) to the Euler equations (E). Then we have an upper estimate t ω0 L∞ exp[γ 0 ∇v(τ )L∞ dτ ] , t τ 1 + (γ − 1)ω0 L∞ 0 exp[γ 0 ∇v(σ )L∞ dσ ] dτ
(3.1)
t ω0 L∞ exp[−γ 0 ∇v(τ )L∞ dτ ] t τ 1 − (γ − 1)ω0 L∞ 0 exp[−γ 0 ∇v(σ )L∞ dσ ] dτ
(3.2)
ω(t) ∞ L and lower one ω(t)
L∞
for all γ 1 and t ∈ [0, T ). Moreover, the denominator of the right-hand side of (3.2) can be estimated from below as
t 1 − (γ − 1)ω0 L∞
τ
exp −γ 0
∇v(σ )
L∞
0
dσ dτ
1 , (1 + ω0 L∞ t)γ −1
(3.3)
which shows that the finite time blow-up does not follow from (3.2). Remark 3.1. We observe that for γ = 1, the estimates (3.1)–(3.2) reduce to the well-known ones in (1.12) with p = ∞. In this sense the above estimates seem to be a natural extension from the known ones, but their use is not clear at this point. Moreover, combining (3.1)–(3.2) together, we easily derive another new estimate,
t 0
sinh[γ cosh[γ
t 0t τ
∇v(τ )L∞ dτ ] ∇v(σ )L∞ dσ ] dτ
(γ − 1)ω0 L∞ .
(3.4)
Proof of Theorem 3.1. Below we denote V ± for the solutions of (E± ) respectively, and Ω ± = curl V ± . Note that V0± = v0 := V0 and Ω0± = ω0 := Ω0 . We will first derive the following estimates for the system (E± ): + Ω (s)
L∞
Ω0 L∞ , 1 + (γ − 1)sΩ0 L∞
(3.5)
− Ω (s)
L∞
Ω0 L∞ , 1 − (γ − 1)sΩ0 L∞
(3.6)
as long as V ± (s) ∈ H m (R3 ). Taking curl of the first equation of (E± ), we have ∓γ ∇V L∞ Ω −
1 (y · ∇)Ω = Ωs + (V · ∇)Ω − (Ω · ∇)V . α+1
(3.7)
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Multiplying Ξ = Ω/|Ω| on the both sides of (3.7), we deduce ∇V (s)L∞ (y · ∇)|Ω| α+1 = Ξ · ∇V · Ξ ∓ ∇V L∞ |Ω| −(γ − 1)∇V L∞ |Ω| ∓ (γ − 1)∇V L∞ |Ω| (γ − 1)∇V L∞ |Ω|
|Ω|s + (V · ∇)|Ω| ∓
for (E+ ), for (E− ),
(3.8)
since |Ξ · ∇V · Ξ | |∇V | ∇V L∞ . Given smooth solution V (y, s) of (E± ), we introduce the particle trajectories {Y± (a, s)} defined by ∇V (s)L∞ ∂Y (a, s) = V± Y (a, s), s ∓ Y (a, s); ∂s α+1
Y (a, 0) = a.
Recalling the estimate
∇V (s) ∞ Ω(s) ∞ Ω(y, s) , L L
∀y ∈ R3 ,
we can further estimate from (3.8) −(γ − 1)|Ω(Y (a, s), s)|2 ∂
Ω Y (a, s), s ∂s (γ − 1)|Ω(Y (a, s), s)|2
for (E+ ), for (E− ).
(3.9)
Solving these differential inequalities (3.9) along the particle trajectories, we obtain that
Ω Y (a, s), s
|Ω0 (a)| 1+(γ −1)s|Ω0 (a)| |Ω0 (a)| 1−(γ −1)s|Ω0 (a)|
for (E+ ),
(3.10)
for (E− ).
Writing the first inequality of (3.10) as
+
Ω Y (a, s), s
1 1 |Ω0 (a)|
+ (γ − 1)s
1 1 Ω0 L∞
+ (γ − 1)s
,
and then taking supremum over a ∈ R3 , which is equivalent to taking supremum over Y (a, s) ∈ R3 due to the fact that the mapping a → Y (a, s) is a diffeomorphism (although not volume preserving) on R3 as long as V ∈ C([0, S); H m (R3 )), we obtain (3.5). In order to derive (3.6) from the second inequality of (3.10), we first write − Ω (s)
L∞
Ω Y (a, s), s
1 1 |Ω0 (a)|
− (γ − 1)s
,
and then take supremum over a ∈ R3 . Finally, in order to obtain (3.1)–(3.2), we just change variables from (3.5)–(3.6) back to the original physical ones, using the fact
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Ω + (y, s) = exp −γ
t
∇v(τ )
L∞
dτ ω(x, t),
0
t s=
τ exp γ ∇v(σ )L∞ dσ dτ
0
0
for (3.1), while in order to deduce (3.2) from (3.6) we substitute t Ω (y, s) = exp γ ∇v(τ ) L∞ dτ ω(x, t), −
0
t
τ
exp −γ
s= 0
∇v(σ )
L∞
dσ dτ.
0
Now we can rewrite (3.2) as ω(t)
L∞
t τ 1 d log 1 − (γ − 1)ω0 L∞ exp −γ ∇v(σ )L∞ dσ dτ . − γ − 1 dt 0
0
Thus, t
∇v(τ )
t
L∞
0
dτ
ω(τ )
L∞
dτ
0
t τ 1 log 1 − (γ − 1)ω0 L∞ exp −γ ∇v(σ )L∞ dσ dτ . − γ −1 0
0
(3.11) Set t y(t) := 1 − (γ − 1)ω0 L∞
τ
exp −γ 0
∇v(σ )
L∞
dσ dτ.
0
We find further integrable structure in (3.11), which is γ
y (t) −(γ − 1)ω0 L∞ y(t) γ −1 . Solving this differential inequality, we obtain (3.3).
2
Similar method can also be applied to derive new a priori estimates for the 3D Navier–Stokes equations.
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Theorem 3.2. Given v0 ∈ H 1 (R3 ) with div v0 = 0, let ω be the vorticity of the classical solution v ∈ C([0, T ); H 1 (R3 )) ∩ C((0, T ); C ∞ (R3 )) to the Navier–Stokes equations (NS). Then, there exists an absolute constant C0 > 1 such that for all γ C0 the following enstrophy estimate holds true: ω(t)
L2
t ω0 L2 exp[ γ4 0 ω(τ )4L2 dτ ] . t τ 1 {1 + (γ − C0 )ω0 4L2 0 exp[γ 0 ω(σ )4L2 dσ ] dτ } 4
(3.12)
The denominator of (3.12) is estimated from below by t 1 + (γ
− C0 )ω0 4L2 0
τ 4 ω(σ ) L2 dσ dτ exp γ 0
1 (1 − C0 ω0 4L2 t)
(3.13)
γ −C0 C0
for all γ C0 . Proof. Let (v, p) be a classical solution of the Navier–Stokes equations, and ω be its vorticity. We transform from (v, p) to (V , P ) according to the formula, given by (2.1)–(2.3), where t 4 μ(t) = exp γ ω(τ )L2 dτ . 0
Substituting (2.1)–(2.3) with such μ(t) into (NS), we obtain the equivalent equations satisfied by (V , P )
(NS∗ )
⎧ −γ Ω(s)4L2 ⎪ ⎪ ⎨ V + (y · ∇)V = Vs + (V · ∇)V − V − ∇P , 2 ⎪ div V = 0, ⎪ ⎩ V (y, 0) = V0 (y) = v0 (y).
Operating curl on the evolution equations of (NS∗ ), we obtain −γ Ω(s)4L2 2Ω + (y · ∇)Ω = Ωs + (V · ∇)Ω − (Ω · ∇)V − Ω. 2
(3.14)
Taking L2 (R3 ) inner product of (3.14) by Ω, and integrating by part, we estimate 1 d γ Ω2L2 + ∇Ω2L2 + Ω6L2 = 2 ds 4
(Ω · ∇)V · Ω dy R3 3
3
ΩL3 ∇V L2 ΩL6 CΩL2 2 ∇ΩL2 2 ∇Ω2L2 +
C0 Ω6L2 4
(3.15)
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for an absolute constant C0 > 1, where we used the fact ΩL2 = ∇V L2 , the Sobolev imbedding, H˙ 1 (R3 ) → L6 (R3 ), the Gagliardo–Nirenberg inequality in R3 , 1
1
f L3 Cf L2 2 ∇f L2 2 , and Young’s inequality of the form ab a p /p + bq /q, 1/p + 1/q = 1. Absorbing the term ∇Ω2L2 to the left-hand side, we have from (3.15) d γ − C0 Ω2L2 − Ω6L2 . ds 2
(3.16)
Solving the differential inequality (3.16), we have Ω(s)
L2
Ω0 L2 1
[1 + (γ − C0 )sΩ0 4L2 ] 4
(3.17)
.
Transforming back to the original variables and functions, using the relations t s=
τ 4 exp γ ω(σ )L2 dσ dτ,
0
ω(t)
L2
= Ω(s)
0
γ exp L2 4
t
ω(τ )4 2 dτ , L
0
we obtain (3.12). Next, we observe (3.12) can be written as ω(t)4 2 L
τ t 4 d 1 4 ω(σ ) L2 dσ dτ , log 1 + (γ − C0 )ω0 L2 exp γ (γ − C0 ) dt 0
0
which, after integration over [0, t], leads to t
ω(τ )4 2 dτ L
0
τ t 4 1 4 ω(σ ) L2 dσ dτ log 1 + (γ − C0 )ω0 L2 exp γ (γ − C0 ) 0
0
(3.18) for all γ > C0 . Setting t y(t) := 1 + (γ
− C0 )ω0 4L2 0
τ 4 exp γ ω(σ )L2 dσ dτ, 0
D. Chae / Journal of Functional Analysis 258 (2010) 2865–2883
2883
we find that (3.18) can be written in the form of a differential inequality, γ
y (t) (γ − C0 )ω0 4L2 y(t) γ −C0 , which can be integrated to provide us with (3.13).
2
Acknowledgment The author would like to thank deeply to the anonymous referee for many invaluable suggestions and comments. References [1] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61–66. [2] D. Chae, Nonexistence of self-similar singularities for the 3D incompressible Euler equations, Comm. Math. Phys. 273 (1) (2007) 203–215. [3] D. Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations, Math. Ann. 338 (2) (2007) 435–449. [4] D. Chae, On the deformations of the incompressible Euler equations, Math. Z. 257 (3) (2007) 563–580. [5] D. Chae, Incompressible Euler equations: the blow-up problem and related results, in: Handbook of Differential Equations: Evolutionary Partial Differential Equations, vol. IV, Elsevier Science Ltd., 2008, pp. 1–55. [6] J.Y. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998. [7] P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36 (1994) 73–98. [8] P. Constantin, On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc. 44 (2007) 603–621. [9] P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Comm. Partial Differential Equations 21 (3–4) (1996) 559–571. [10] D. Córdoba, C. Fefferman, On the collapse of tubes carried by 3D incompressible flows, Comm. Math. Phys. 222 (2) (2001) 293–298. [11] D. Córdoba, C. Fefferman, R. De La Llave, On squirt singularities in hydrodynamics, SIAM J. Math. Anal. 36 (1) (2004) 204–213. [12] J. Deng, T.Y. Hou, X. Yu, Geometric and nonblowup of 3D incompressible Euler flow, Comm. Partial Differential Equations 30 (2005) 225–243. [13] L. Euler, Principes généraux du mouvement des fluides, Mémoires de l’Académie des Sciences de Berlin 11 (1755) 274–315. [14] Y. Giga, R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985) 297–319. [15] Y. Giga, R.V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36 (1) (1987) 1–40. [16] T.Y. Hou, R. Li, Nonexistence of locally self-similar blow-up for the 3D incompressible Navier–Stokes equations, Discrete Contin. Dyn. Syst. Ser. A 18 (4) (2007) 637–642. [17] T. Kato, Nonstationary flows of viscous and ideal fluids in R3 , J. Funct. Anal. 9 (1972) 296–305. [18] T. Kato, Strong Lp solutions of the Navier–Stokes equations in Rm with applications to weak solutions, Math. Z. 187 (1984) 471–480. [19] J. Leray, Essai sur le mouvement d’un fluide visqueux emplissant l’espace, Acta Math. 63 (1934) 193–248. [20] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Univ. Press, 2002. [21] H. Matano, F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. (2004) 1494–1541. [22] J.R. Miller, M. O’Leary, M. Schonbek, Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system, Math. Ann. 319 (4) (2001) 809–815. [23] J. Neˇcas, M. Ružiˇcka, V. Šverák, On Leray’s self-similar solutions of the Navier–Stokes equations, Acta Math. 176 (2) (1996) 283–294. [24] T.-P. Tsai, On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal. 143 (1) (1998) 29–51.
Journal of Functional Analysis 258 (2010) 2884–2909 www.elsevier.com/locate/jfa
Restriction estimates for some surfaces with vanishing curvatures ✩ Sanghyuk Lee a,∗ , Ana Vargas b a School of Mathematical Sciences, Seoul National University, Seoul 151-742, Republic of Korea b Department of Mathematics, University Autonoma de Madrid, 28049 Madrid, Spain
Received 18 June 2009; accepted 13 January 2010 Available online 11 February 2010 Communicated by J. Bourgain
Abstract We obtain bilinear restriction estimates for surfaces with vanishing curvatures. As application we also prove new linear restriction estimates for some class of conic surfaces. © 2010 Elsevier Inc. All rights reserved. Keywords: Restriction estimates; Conic surfaces
1. Introduction In this note we consider Fourier restriction estimates for some class of conic surfaces. It has been known that the curvature plays an important role in determining the boundedness of the restriction operators. Let S be a smooth compact surface in Rn+1 with the induced Lebesgue measure dσ . It is well known [3,7,11] that if k principal curvatures are nonzero at each point of the surface S, for q 2k+4 k f dσ Lq (Rn+1 ) Cf L2 (dσ ) . ✩
(1.1)
The first author was partially supported by the NRF of Korea grant (2009-0072531). The second author was partially supported by the grant MTM2007-60952 (Spain). * Corresponding author. E-mail addresses:
[email protected] (S. Lee),
[email protected] (A. Vargas). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.014
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The range on q is optimal as it can be easily seen using Knapp’s example. The natural conjecture is that the estimate f dσ Lq (Rn+1 ) Cf Lp (dσ ) 1 2k+2 holds if k+2 q k(1 − p ) and p > k . It was Bourgain [1] who first obtained a result beyond the sharp L2 –Lq restriction estimates when k 2. Recent development on the restriction problem has been made by considering a suitable bilinear version of the operator [5,8–10,12,13]. To be specific, let S1 , S2 be subsets of S with measures dσ1 , dσ2 . Let us consider the bilinear adjoint restriction estimate
f dσ1 g dσ2 Lp Cf L2 (dσ1 ) gL2 (dσ2 ) .
(1.2)
In this form of estimate one can impose additional conditions which specify the relative position of the two surfaces. One typical condition is transversality. For positively curved surfaces (e.g. the cone and the paraboloid) transversality makes it possible to get a wider range of boundedness than is allowed for the linear estimates. However, for the surfaces with positive and negative principal curvatures transversality is not enough to obtain such improvement. Actually one needs stronger (separation) conditions [5,12]. Unlike the case of linear estimate (1.1), the role of curvature in bilinear restriction estimates does not seem to be clearly understood. In fact, the sharp bilinear restriction estimates (1.2) for the cone and the paraboloid are valid for the same range of q except the endpoint even though the cone has only n − 1 nonzero principal curvatures. The estimate (1.2) has been studied mainly with surfaces with nonvanishing Gaussian curvature or one vanishing curvature. In this note we want to generalize the known bilinear restriction estimates to the surfaces having two or more vanishing curvatures. Let k n − 1 be an integer. We assume that S has k nonvanishing principal curvatures and n − k vanishing curvatures. In other words the surface S has n − k null directions along which the curvatures vanish. To be more precisely, let ±N(ξ ) ∈ Sn be the unite normal vector of S at ξ . By rotation and decomposition we may assume that |N(ξ ) − en+1 | 1/2. Definition 1.1. Suppose that S is a smooth compact surface with (possibly) boundary in Rn+1 . We say that S is of conic type with k nonvanishing curvatures if the following assumptions are satisfied: • The map dN : Tξ (S) → TN (ξ ) (Sn ) has k nonzero eigenvalues and n − k zero eigenvalues. • The nonzero eigenvalues have magnitude ∼ 1.1 We denote by Nξ (S) the span of the eigenvectors with zero eigenvalues of dN at ξ . We say that any nonzero vector v is in a null direction of S at ξ if v ∈ Nξ (S). Definition 1.2. Let S1 , S2 be subsets of a conic type surface S of k nonvanishing curvatures. We say that S1 and S2 are transversal if 1 For A, B > 0, A ∼ B means C −1 A B CA for some constant C > 0.
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N(ξ1 ) − N(ξ2 ) ∼ 1
(1.3)
for ξ1 ∈ S1 and ξ2 ∈ S2 . Let z1 , z2 be points in Rn+1 and let us denote the translated surfaces by Szj = Sj + zj , 2
j = 1, 2.
Then by the condition (1.3) we may assume that the intersection of two surfaces Sz1 and Sz2 is a smooth (n − 1)-dimensional immersed submanifold by dividing the surfaces into small pieces (if necessary), as long as the intersection is not empty. We denote the intersection by Πz1 ,z2 = Sz1 ∩ Sz2 . As it is well known, the dispersion of the normal vectors of a given surface accounts for the decay of Fourier transform of the surface measure, which was crucial in obtaining the linear restriction estimate (1.1). As it turns out, for the bilinear estimates the dispersion along the intersection Πz1 ,z2 is important. Roughly, the number of nonzero curvatures along Πz1 ,z2 takes the role that is played by the total number of nonzero curvatures in the linear estimates. (See Theorem 1.4 below.) This explains why the range of bilinear restriction estimates for the cone and paraboloid is essentially the same. Let us denote by Tξ (Πz1 ,z2 ) the tangent space of Πz1 ,z2 at ξ . Now we make an assumption on the surfaces S1 and S2 : dim Tξ (Πz1 ,z2 ) ⊕ Nξ1 (S1 ) = n, dim Tξ (Πz1 ,z2 ) ⊕ Nξ2 (S2 ) = n
(1.4)
for all ξ ∈ Πz1 ,z2 , ξ1 ∈ S1 and ξ2 ∈ S2 as long as Πz1 ,z2 = ∅. This is one of most important assumption which gives the maximal amount of dispersion of normal vectors along the intersection Πz1 ,z2 . Since the surface S has k nonvanishing principal curvatures, the maps Nzi : Szi → Sn have rank k. Here Nzj (ξ ) is the normal vector to Szi at ξ . Hence from the condition (1.4) one can j
easily see that for j = 1, 2, the map Nz1 ,z2 which is given by ξ ∈ Πz1 ,z2 :→ Nzj (ξ ) ∈ Sn j
is also of rank k. That is to say, the rank of dNz1 ,z2 is k, j = 1, 2. Finally we assume that / dN1z1 ,z2 Tξ (Πz1 ,z2 ) ⊕ span Nz1 (ξ1 ) , Nz2 (ξ2 ) ∈ / dN2z1 ,z2 Tξ (Πz1 ,z2 ) ⊕ span Nz2 (ξ2 ) Nz1 (ξ1 ) ∈
(1.5)
2 Obviously, multiplication of e−2π ixξ0 to f dσj does not have any effect on the estimate and the Fourier transform
of e−2π ixξ0 f dσj is supported in Sj + ξ0 . So, it is natural to consider conditions which are valid uniformly for the translated surfaces.
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for ξ ∈ Πz1 ,z2 , ξ1 ∈ Sz1 and ξ2 ∈ Sz2 as long as Πz1 ,z2 = ∅. For the positively curved surfaces (e.g. the cone, sphere, or paraboloid) this type of transversality can be obtained by the normal separation condition (1.3) but it is not the case for the surfaces with principal curvatures of different sings. This is actually the separation condition which was used to obtain the best possible bilinear restriction estimates for the hyperboloid [5,12]. Remark 1.3. The condition (1.5) is concerned with the transversality between the cone generated by the normal vectors from the intersection surface Πz1 ,z2 and the normal vectors to the opposite surface. More precisely, for j = 1, 2, let us set Γj = tNzj (ξ ): ξ ∈ Πz1 ,z2 , 1 |t| 2 . Then the condition (1.5) equivalently means that any normal vector N1 (N2 , resp.) of Sz1 (Sz2 , resp.) is transversal to Γ2 (Γ1 , resp.) because the tangent space of Γj is given by j dNz1 ,z2 (Tξ (Πz1 ,z2 )) ⊕ span{Nzj (ξ )}. The following is our main result: Theorem 1.4. Let 1 k n − 1. Suppose that S is a smooth compact surface of conic type in Rn+1 with k-nonvanishing curvatures. If the surfaces S1 , S2 ⊂ S satisfy the assumptions (1.3), (1.4) and (1.5), then for p > k+4 k+2 f dσ1 g dσ2 Lp Cf 2 g2 . This theorem is sharp in the sense that there are surfaces satisfying (1.3), (1.4) and (1.5) but the estimate fails for p < k+4 k+2 . See Remark 3.3 and the proof of Proposition 3.2. Remark 1.5. If one considers two transversal subsets of a cylinder in R3 satisfying (1.3), then the condition (1.5) is trivially satisfied. But Πz1 ,z2 is contained in a line which is parallel with the null direction. Hence (1.4) fails. As it can be easily seen, the best possible bilinear restrictionL2 estimate is L2 × L2 → L2 estimate. Also, considering the n-dimensional cylinder (ξ , ξ , 1 − |ξ |2 ), (ξ , ξ ) ∈ Rk × Rn−k and suitable transversal subsets S1 and S2 it is easy to see that (1.2) fails for p < k+3 k+1 even though the conditions (1.3) and (1.5) are satisfied. On the other hand, if we only assume the conditions (1.3) and (1.5) without (1.4), then one can show for 3 p > k+3 k+1 f dσ1 g dσ2 Lp Cf 2 g2 . But this is still better than the trivial L2 × L2 → L estimate (1.1).
k+2 k
estimate which follows from the linear
3 This can be shown by following the argument below. The only thing one has to observe is that the combinatorial 1
estimates in Lemma 2.3 lose an additional factor of R 2 without (1.4).
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The paper is organized as follows. In Section 2 we give the proof of Theorem 1.4. As application we consider some model surfaces of conic type in Section 3 and obtain new restriction estimates. 2. Proof of Theorem 1.4 This section is devoted to proving Theorem 1.4. The proof here is based on the induction on scale argument which was used to obtain the sharp bilinear restriction estimates [8,13] (also see [5,12]). However adaptations are needed to reflect the geometry of conic surfaces. By rotation, translation and breaking the surfaces S into surfaces of small diameter if necessary, we may assume that the surface S is given by the graph of a function. For a δ0 > 0 let φ be a smooth function such that φ : Q = [−δ0 , δ0 ]n → R and φ satisfies φ(0) = 0,
∇φ(0) = 0.
We may also assume that the surfaces S1 , S2 are given by the graphs of the function −φ over the set Q1 , Q2 ⊂ Q, respectively. That is to say, for j = 1, 2, Sj =
x, −φ(x) : x ∈ Qj
for some cubes Q1 , Q2 ⊂ Q. Then for j = 1, 2, let us define the extension operators Ej f (x, t) =
ei(x·ξ −tφ(ξ )) f (ξ ) dξ.
(2.1)
Qj
Since dσi is comparable to dξ , it is enough to show that for p >
k+4 k+2 ,
2 2
Ej fj C fj 2 . j =1
p
j =1
First, we decompose Ej f into a sum of wave packets. The wave packets have good localization properties in both Fourier transform side and (x, t)-space. Unlike the usual decomposition of an O(R −1 )-neighborhood of a conic surface [2,6,13], which takes into account the null directions, we use a more direct decomposition in spirit of [8] (also see [5]). It does not depend on the presence of null directions. 2.1. Wave packet decomposition at scale R For d > 0 and A ⊂ Rn , let us denote by A + O(d) the set x ∈ Rn : dist(x, A) < Cd , for some big constant C > 0. Let R 1. The wave packet decomposition at scale R makes the support of the functions be expanded by O(R −1/2 ) (see Lemma 2.1). So, we need to consider a little bit larger sets
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than Q1 , Q2 . For CR −1/2 < δ0 , let us set Vj = Qj + O( ). We define the spatial grid Y by Y = R 1/2 Zn and the frequency grids V1 , V2 , respectively by setting Vj = R −1/2 Zn ∩ Vj ,
j = 1, 2.
Let us set Wj = (y, v): (y, v) ∈ Y × Vj . For wj = (yj , vj ) ∈ Wj , we define the associated tube Twj by Twj = (x, t) ∈ Rn × R: |t| 2R, x − yj + t∇φ(vj ) R 1/2 . Obviously Tyj ,vj contains (yj , 0) and its major direction is parallel to (∇φ(vj ), 1) ∈ Rn × R, which is parallel to the normal vector of the surface Sj at (vj , φ(vj )). That is, ∇φ(vj ), 1 N(vj ,φ(vj )) .
(2.2)
The following is a modification of Lemma 4.1 in [8]. For a proof see [5]. Lemma 2.1 (Wave packet decomposition). Let φ be a smooth function defined on Q. Suppose that f1 , f2 are supported in Q1 , Q2 , respectively. If |t| 10R, we can write Ej fj as Ej fj (x, t) =
Cwj pwj (x, t),
(x, t) ∈ Rn × R,
wj ∈Wj
such that Cwj , pwj satisfy the following conditions: (P1) For j = 1, 2,
1/2 |Cwj |
2
Cfj 2 .
wj ∈Wj
(P2) For j = 1, 2, pwj = Ej pw j (·, 0) .
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(P3) If wj = (yj , vj ), then −1/2 . supp p wj (·, t) ⊂ ξ : ξ = vj + O R (P4) For any N > 0, |t| 10R,
|x − (yj + t∇φ(vj ))| −N −n/4 1+ . j ,vj (x, t) CN R R 1/2
py
In particular, if dist((x, t), Tyj ,vj ) R δ+1/2 , then py
j ,vj
(x, t) CR −100n .
(P5) For any S ⊂ Wj , 2 pwj (·, t) C#S. 2
wj ∈S
2.2. Induction on scale Since k 1, the Fourier transform of the surface measures have some decay at infinity. Hence, by the globalization Lemma 2.4 in [9], it is enough to show that for any α > 0 2
Ej fj i=1
k+4 L k+2 (Q(R))
CR α
2
fj 2 .
(2.3)
i=1
Here Q(R) is the cube which is centered at the origin and of side length R. Let us denote the estimates (2.3) by E ∗ (α). We may assume f1 2 = f2 2 = 1. Since |Ei fi (x, t)| Cf 2 , using the wave packet decomposition and the standard pigeonholing argument together with the property (P1) in Lemma 2.1 the proof of (2.3) reduces (modulo loss of (log R)2 in bounds4 ) to obtaining the estimate
w1 ∈W1 , w2 ∈W2
pw1 pw2
k+4
CR α |W1 |1/2 |W2 |1/2
(2.4)
L k+2 (Q(R))
for any subsets W1 ⊂ W1 and W2 ⊂ W2 whenever pwj is the L2 normalized wave packet satisfying (P2), (P3), (P4) and (P5). By rapid decay of pwj away from the associated tube Twj we may assume that Twj meets with Q(2R). The main part of the induction on scale argument is to establish the following implication: 4 Such loss is harmless due to the nature of the estimate.
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If (2.4) is valid for R 1, then for 0 < δ 1,
w1 ∈W1 , w2 ∈W2
pw1 pw2
CR α(1−δ)+cδ |W1 |1/2 |W2 |1/2
k+4
(2.5)
L k+2 (Q(R))
with c independent of R and δ. Hence we have the implication E ∗ (α) → E ∗ α(1 − δ) + cδ 5 for any α > 0. Choosing sufficiently small δ and iterating this estimates finitely many times one can show that R ∗ (α) is valid for any α > 0. Pigeonholing further we can specify some of the quantities involved. Let us partition Q(2R) into disjoint R 1/2 -cubes q and denote the collection of those cubes by Q(R). We classify the q, w1 and w2 using dyadic parameters. For each q ∈ Q let us set Wj (q) = wj ∈ Wj : R δ q ∩ Twj = ∅ . Here R δ q is the set having the same center as q and expanded to R δ times from q. We also set for each dyadic number μ1 , μ2 1, Qμ1 ,μ2 = q: Wj (q) ∼ μj , j = 1, 2 . For each wj , define Qμ1 ,μ2 (wj ) = q ∈ Qμ1 ,μ2 : R δ q ∩ wj = ∅ and for dyadic λ 1, we again set λ,μ1 ,μ2
Wj
= wj ∈ Wj : Qμ1 ,μ2 (wj ) ∼ λ .
Then by (P4) in Lemma 2.1 and pigeonholing, it is easy to see
w1 ∈W1 , w2 ∈W2
pw1 pw2
k+4
L k+2 (Q(R))
C(log R)
4
λj ,μ1 ,μ2
wj ∈Wj
, j =1,2
q∈Qμ1 ,μ2
χq pw1 pw2
k+4 L k+2 (Q(R))
+ O R −M
for some dyadic 1 λ1 , λ2 , μ1 , μ2 R C and large M. Hence the matter is reduced to showing 5 The losses of (log R)C are absorbed in R cδ .
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λj ,μ1 ,μ2
wj ∈Wj
χq pw1 pw2
q∈Qμ1 ,μ2
, j =1,2
k+4
L k+2 (Q(R))
C R α(1−δ)+cδ |W1 |1/2 |W2 |1/2 .
(2.6)
Now we fix the numbers λ1 , λ2 , μ1 , μ2 for the rest of this section. We partition Q(2R) into O(R (n+1)δ ) cubes b which are of side length R 1−δ and essentially disjoint. So, we get
q∈Qμ1 ,μ2
λj ,μ1 ,μ2 wj ∈Wj , j =1,2
λj ,μ1 ,μ2 wj ∈Wj , j =1,2
b
χq pw1 pw2
k+4
L k+2 (Q(R))
q∈Qμ1 ,μ2 , q⊂2b
χq pw1 pw2
k+4
.
L k+2 (b)
For each wj , let b(wj ) be the cube b which contains the maximal number of q ∈ Qμ1 ,μ2 (wj ). There may be many of such cubes but we simply choose one of them. We now define a relation λ ,μ ,μ between b and wj ∈ Wj j 1 2 by saying b wj
if b ∩ 10b(wj ) = ∅.
Since the number of b is O(R (n+1)δ ), it is obvious that q ∈ Qμ1 ,μ2 : wj ∩ R δ q = ∅, q ∩ 10b = ∅ R −cδ λj provided wj b. Then
λj ,μ1 ,μ2
b
wj ∈Wj
, j =1,2
q∈Qμ1 ,μ2 , q⊂2b
χq pw1 pw2
k+4
L k+2 (b)
I b + I b , b
where
I
I
b
b
=
=
w1 b, w2 b
w1 b, or w2 b
q∈Qμ1 ,μ2 , q⊂2b
χq pw1 pw2
q∈Qμ1 ,μ2 , q⊂2b
k+4
,
L k+2 (b)
χq pw1 pw2
k+4
L k+2 (b)
.
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To simplify the notation, in the summations to be in the set
λ ,μ ,μ W1 1 1 2
λ ,μ ,μ × W2 2 1 2 .
w1 b, w2 b ,
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w1 b, or w2 b , (w1 , w2 ) is assumed
Therefore we are reduced to showing
I b + I b C R α(1−δ)+cδ |W1 |1/2 |W2 |1/2 . b
From the induction hypothesis (2.3) and (P5), it follows that
I
b
CR
α(1−δ)
2
wj ∈ W λj ,μ1 ,μ2 : b wj 1/2 j i=1
since the side length of b is ∼ R 1−δ . By Cauchy–Schwarz’s inequality it follows that
I b CR α(1−δ)
b
2
1/2 wj ∈ W λj ,μ1 ,μ2 : b wj j b
i=1
CR
α(1−δ)
2
2
{b: b wj }
1/2
λj ,μ1 ,μ2
i=1
CR α(1−δ)
wj ∈Wj
|Wj |1/2 .
i=1
For the second inequality we use the fact that there are only O(1) cubes b with b wj for λ ,μ ,μ
wj ∈ Wj j 1 2 . The induction assumption is used to handle highly concentrating part and under proper relation this gives slightly improved bound because the considered cube is of size R 1−δ . It is in fact the major advantage of the induction scale argument. However, we have to trade off such easiness of improvement with a detailed analysis when we handle less concentrating part. Now we need to show I b CR cδ |W1 |1/2 |W2 |1/2 . It follows from interpolation between the easy L1 -estimate, I b CR|W1 |1/2 |W2 |1/26 and the L2 -estimate
w1 b, or w2 b
q∈Qμ1 ,μ2 , q⊂2b
2 χq pw1 pw2 2
(2.7)
L (b)
λ ,μ1 ,μ2
Here the summation is taken over wj ∈ Wj j 6 This follows from the trace lemma, or (P5).
CR −k/2+cδ |W1 ||W2 |.
, j = 1, 2. Hence it remains to show (2.7).
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2.3. L2 estimates over R 1/2 -cubes For given v1 ∈ V1 and v2 ∈ V2 , let us set z1 = v2 , φ v2 ,
z2 = v1 , φ(v1 )
and also set z1 ,z2 = Π
2 Sj + zj + O R −1/2 i=1
which also can be considered as Πz1 ,z2 + O(R −1/2 ). For Uj ⊂ Wj , j = 1, 2, let us set z ,z Π 1 2
Uj
z1 ,z2 . = wj = (yj , vj ) ∈ Uj : vj , φ(vj ) + zj ∈ Π
Lemma 2.2. Let q be an R 1/2 -cube and Uj ⊂ Wj (q), j = 1, 2. Then, 2 cδ −(n−1)/2 p p |U1 ||U2 | min sup U1 Πz1 ,z2 , sup U2 Πz1 ,z2 . w1 w2 R R 2 z ,z z ,z w1 ∈U1 w2 ∈U2
L
1
2
1
2
Proof. Let us write the left-hand side of the above inequality as w1 ∈U1 w2 ∈U2
Iw1 ,w2 ,
where Iw1 ,w2 =
pw1 pw2 ,
w2 ∈U2
w1 ∈U1
pw1 pw2 .
Fixing w1 ∈ U1 and w2 ∈ U2 , it is enough to show that |Iw1 ,w2 | R cδ R −(n−1)/2 min U1 Πz1 ,z2 , U2 Πz1 ,z2 . By symmetry we only need to show |Iw1 ,w | R cδ R −(n−1)/2 U1 Πz1 ,z2 2
(2.8)
with w1 = (y1 , v1 ), w2 = (y2 , v2 ), z1 = (v2 , φ(v2 )) and z2 = (v1 , φ(v1 )). We observe that the Fourier supports of the functions w2 ∈U2 pw1 pw2 , w ∈U1 pw1 pw2 are 1 contained in the sets S2 + z2 + O R −1/2 ,
S1 + z1 + O R −1/2 ,
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respectively. So it is obvious that pw1 pw2 , pw1 pw2 = 0 only if w1 = (y1 , v1 ) and w2 = (y2 , v2 ) satisfy z2 + v2 , φ(v2 ) ∈ Πz1 ,z2 + O R −1/2 , z1 + v1 , φ v1 ∈ Πz1 ,z2 + O R −1/2 , and z2 + v2 , φ(v2 ) = z1 + v1 , φ v1 + O R −1/2 .
(2.9)
Therefore, |Iw1 ,w2 | is bounded by
pw pw , pw pw . 1 2
1
{w2 ∈U2 : v1 +v2 =v1 +v2 +O(R −1/2 )}
{w1 ∈U1 Πz1 ,z2 }
2
Hence it is now sufficient to show that for fixed w1 , w2 , w1 ,
pw pw , pw pw R cδ R −(n−1)/2 . 1 2 1 2
{w2 ∈U2 : v2 =v1 +v2 −v1 +O(R −1/2 )}
Since w1 , w2 , w1 are given, there are at most O(1) possible v2 in the inner summation. Note that U2 ⊂ W2 (q). That is, all the tubes Tw2 are passing through R δ q. Hence, there are at most O(R cδ )–w2 = (y2 , v) satisfying v = v2 because y2 ∈ Y are R 1/2 -separated. Using (P4) and the transversality (1.3) between the tubes Tw1 and Tw2 , it is easy to see that |pw1 pw2 , pw1 pw2 | R −(n−1)/2 . Therefore, we get the desired estimate. 2 2.4. Proof of (2.7) λ ,μ1 ,μ2
It should be noticed that wj ∈ Wj j the left-hand side of (2.7) as
w1 b, or w2 b
=
even though it is not explicitly written out. We write
q∈Qμ1 ,μ2 , q⊂2b
q∈Qμ1 ,μ2 , q⊂2b
2 χq pw1 pw2 2
w1 b, or w2 b
L (b)
2 pw1 pw2 2
.
L (q)
Discarding harmless O(R −M ) terms and using Lemma 2.2, we see that the right-hand side of the above is bounded by
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CR −(n−1)/2+cδ ×
1
2
q∈Qμ1 ,μ2 , q⊂2b
λ ,μ ,μ ,b Π λ ,μ ,μ ,b Π sup W1 1 1 2 (q) z1 ,z2 + sup W2 2 1 2 (q) z1 ,z2 ,
z1 ,z2
λ ,μ1 ,μ2 ,b
where Wi i
λ1 ,μ1 ,μ2 λ2 ,μ1 ,μ2 W (q)W (q)
z1 ,z2
λ ,μ1 ,μ2
= {w ∈ Wi 1
λ ,μ1 ,μ2
: w b}. Since |Wj i
(q)| μj and
λj ,μ1 ,μ2 W (q) Cλj |Wj |, 7 j
q∈Qμ1 ,μ2
one can easily see that
λ1 ,μ1 ,μ2 λ2 ,μ1 ,μ2 W (q)W (q) C min λ1 μ2 |W1 |, λ2 μ1 |W2 | . 1
2
q∈Qμ1 ,μ2 , q⊂2b
Therefore, to show (2.7) it is enough to show that if q ∈ Qμ1 ,μ2 and q ⊂ 2b, then λ ,μ ,μ ,b Π n−1−k λ1 μ2 W1 1 1 2 (q) z1 ,z2 CR 2 +cδ |W2 |, λ ,μ ,μ ,b Π n−1−k λ2 μ1 W2 2 1 2 (q) z1 ,z2 CR 2 +cδ |W1 |.
(2.10)
By symmetry it is enough to show one of the estimates. In fact, to prove (2.10) we need only to show the following. Lemma 2.3. Let q0 ∈ Q(R) be contained in Q(0, R) and let Q be a subset of Q(R). Additionally, let Uj be subset of Wj for j = 1, 2. Suppose that w ∈ Wj : R δ q ∩ Tw = ∅ ∼ μj
for q ∈ Q,
(2.11)
and for j = 1, 2 and some λj , μj 1, q ∈ Q: R δ q ∩ Tw = ∅, dist(q, q0 ) 2R 1−δ R −cδ λj j
(2.12)
provided wj ∈ Uj . For each q ∈ Q let us set Uj (q) = wj ∈ Uj : R δ q ∩ Twj = ∅ . Then, there are constants C and c such that λj ,μ1 ,μ2 7 It can be shown by interchanging the order of summation. More precisely, use (q)| = q∈Qμ1 ,μ2 |Wj δ λj ,μ1 ,μ2 |{q: R Twj ∩ q = ∅}|. wj ∈Wj
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n−1−k z ,z +cδ |W2 | U1 (q0 )Π 2 1 2 CR , λ1 μ2 n−1−k z ,z +cδ |W1 | U2 (q0 )Π 2 1 2 CR λ2 μ1 with c independent of δ.
Proof. By symmetry it is enough to show the estimate for |U1 (q0 )Πz1 ,z2 |. To begin with, let us set (∇φ(v1 ), 1) z1 ,z2 + O R −1/2 . NΠz1 ,z2 = : v1 , φ1 (v1 ) + z1 ∈ Π |(∇φ(v1 ), 1)| We consider the set
Λ1 =
R Tw1 ∩ Q(0, 2R) \ Q q0 , R 1−δ . δ
w1 ∈U1 (q0 )Πz1 ,z2
From the definition of U1 (q0 )Πz1 ,z2 the associated tubes Tw1 have directions which are normal z1 ,z2 . More precisely, the normal directions of the tubes are contained vectors to S1 + z1 along Π z ,z Π 1 2 in the set N , and all the tubes Tw1 meet with R δ q0 . Hence one can see that Λ1 is contained 1/2+cδ -neighborhood of the conic set in an R Γ1 (R) = tN: N ∈ NΠz1 ,z2 , R 1−δ |t| 4R + q0 . 1
This is actually an isotropic dilation of the set Γ1 + (R − 2 ) by factor R. Since the surface S has k nonvanishing curvatures, and by (1.4), the normal map ξ ∈ Πz1 ,z2 :→ Nz1 (ξ ) ∈ Sn has rank k. Since Πz1 ,z2 has dimension n − 1, the tubes Tw1 , w1 ∈ U1 (q0 )Πz1 ,z2 can overlap at most n−1−k O(R cδ+ 2 ) over the set Λ. It can be more clearly seen using implicit function theorem. Hence we have
χR δ Tw CR cδ+
n−1−k 2
(2.13)
.
w∈U1 (q0 )Πz1 ,z2
By (2.12) we have λ1 U1 (q0 )Πz1 ,z2 CR cδ
q ∈ Q: R δ q ∩ Tw = ∅, dist(q, q0 ) R 1−δ .
w∈U1 (q0 )Πz1 ,z2
Hence it follows that n+1 λ1 U1 (q0 )Πz1 ,z2 CR cδ− 2
w∈U1 (q0 )Πz1 ,z2
By (2.13) we get
χR δ Tw Λ
q∈Q
χq dx dt.
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n+1 λ1 U1 (q0 )Πz1 ,z2 CR − 2
χR δ Tw
n−1−k 2
χq dx dt
q∈Q
w∈U1 (q0 )Πz1 ,z2
Λ
CR cδ+
{q ∈ Q: q ⊂ Λ1 }.
On the other hand it is not difficult to see that the condition (1.5) implies that the tube Tw2 meets the opposite tube cone Γ1 (R) transversally. Hence we see that Tw2 can intersect only O(R cδ ) number of q ⊂ Λ1 . (See Remark 1.3.) This means that the collections of tubes {Tw2 }w2 ∈W2 (q) are essentially disjoint along q ⊂ Λ1 (overlapping at most R cδ ) so that W2 (q) CR cδ |W2 |. q∈Λ1
Hence by (2.11) it follows that
μ2 {q ∈ Q: q ⊂ Λ1 } C
w ∈ W2 : R δ q ∩ Tw = ∅
q∈Q: q⊂Λ1
CR cδ |W2 |. Therefore we get the required inequality comparing the upper and lower bounds for |{q ∈ Q: q ⊂ Λ}|. 2 3. Applications to linear restriction estimates In this section we apply Theorem 1.4 to obtain linear restriction estimates for some surfaces with vanishing curvatures. It is also possible to obtain results for more general surfaces but instead we do it with some model surfaces to avoid unnecessary complications. Let 1 L n − 1 and let n1 , n2 , . . . , nL 2 be positive integers satisfying n1 + n2 + · · · + nL = n. For each 1 l L, let ηl ∈ Rnl −1 and ρ l ∈ [1, 2]. We set η = η1 , . . . , ηL ∈ Rn−L ,
ρ = ρ 1 , . . . , ρ L ∈ [1, 2]L .
Abusing notation, we sometimes denote (ρ 1 η1 , ρ 2 η2 , . . . , ρ L ηL ) by ρη = ρ 1 η1 , ρ 2 η2 , . . . , ρ L ηL when it is more convenient. Now, the angular variable θ = θ (ξ ) for ξ = (η, ρ) is defined by setting θ = θ 1, θ 2, . . . , θ L =
ηL η1 η2 , , . . . , ρL ρ1 ρ2
=
η ∈ [1, −1]n−L . ρ
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Using these notations, we write the variable ξ so that ξ = (η, ρ) = (ρθ, ρ). Let us set Q = (η, ρ): |η| 1, ρ ∈ [1, 2]L . Then we consider the smooth conic surface S=
ξ, −φ(ξ ) : ξ ∈ Q
with φ given by
φ(ξ ) =
L
±
l=1
L 2 |ηl |2 = ±ρ l θ l . l ρ
(3.1)
l=1
Here the sign can be chosen arbitrarily for each summand. Obviously S has L vanishing and n − L nonvanishing curvatures. Instead of dealing with the operator f → f dσ , we consider as before the equivalent operator Eφ given by Eφ f (x, t) =
ei(x·ξ −tφ(ξ )) f (ξ ) dξ.
Q
We define a mixed norm space by
f Lp Lqρ = θ
[1,−1]n−L
q> that
p/q
1
p
dθ
.
[1,2]L
Theorem 3.1. Let φ be given by (3.1). If 2n−2L+8 n−L+2
f (ρθ, ρ)q dρ
n−L+2 q
(n − L)(1 −
1 p)
and p 2, then, for
when n − L 2, and for q > 4 when n − L = 1, there is a constant C such
Eφ f q Cf Lp L2 . θ
ρ
This is obviously stronger than the usual Lp –Lq estimate when p > 2. An interpolation with the trivial estimate Eφ f ∞ Cf L1 L1 further strengthens the restriction estimate slightly. θ
ρ
(n − L)(1 − p1 ) By adapting Knapp’s example one can easily show that the condition n−L+2 q p is necessary. When the surface is the cone, the Lθ L2ρ → Lq estimates were obtained for
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the non-endpoint case [13].8 Theorem 3.1 also includes the endpoint case. We prove Theorem 3.1 by obtaining bilinear estimates with suitable separation. The following is what we need. Proposition 3.2. Let Q1 , Q2 be the cubes contained in Q and the extension operators E1 , E2 be defined by (2.1). Suppose that η1 η2 − ∼1 (3.2) ρ ρ2 1 (equivalently |θ1 − θ2 | ∼ 1) for each (η1 , ρ1 ) ∈ Q1 , (η2 , ρ2 ) ∈ Q2 . Then for q >
n−L+4 n−L+2 ,
2 2
Ej fj C fj 2 . j =1
q
j =1
Remark 3.3. Adapting the usual counterexample in [10], one can show that the estimate fails for q < n−L+4 n−L+2 . In fact, for 0 < 1 and i = 1, 2, let fi be the characteristic functions of the sets (η, ρ) ∈ Q: η1 1 + (−1)i 2 , ρ 1 − 1 , η∗ , where η = ((η1 )1 , η∗ ) ∈ R × Rn−L−1 . Let x = (y, z) ∈ Rn−L × RL , y = ((y 1 )1 , y ∗ ) ∈ R × Rn−L−1 and z = (z1 , z∗ ) ∈ R × RL−1 . Then the condition (3.2) is satisfied on the supports of f1 , f2 , and |Ef1 Ef2 (x, t)| ∼ 2(n−L+2) on the set {(x, t) = (y, z, t): |t| c −2 , |(y 1 )1 | c −2 , |y ∗ | c −1 , |z1 − t| c −1 , |z∗ | c} for some small c > 0. The bilinear L2 estimate implies 2(n−L+2) −(n−L+4)/q C (n−L+2) . Hence letting → 0, we see that q n−L+4 n−L+2 . Assuming Proposition 3.2 for the moment, we prove Theorem 3.1. The argument below is an adaptation of the usual argument which was used to derive linear estimates from bilinear estimates [10]. To obtain the endpoint estimates we also use a simple summation argument involving Lorentz spaces and real interpolation (see, for instance [4]). Proof of Theorem 3.1. Since f is supported in a fixed compact set, it is enough to show Theorem 3.1 at the endpoint (p, q) that satisfies
n−L+2 1 = (n − L) 1 − . (3.3) q p We consider separately the cases n − L 2 and n − L = 1 and first prove the case n − L 2. Since the other case can be handled similarly, we only give a brief remark at the end of proof. When n − L 2, due to the known L2 restriction estimate (1.1) [3,7,11] we may assume that p > 2 and q < 4. To exploit the separation condition we make a decomposition of Eφ f Eφ f . Let I = k of side 2−k . Here m is the [−1, 1]n−L . For each k ∈ Z+ we partition I into dyadic cubes Θm 8 That is, n−L+2 < (n − L)(1 − 1 ). q p
S. Lee, A. Vargas / Journal of Functional Analysis 258 (2010) 2884–2909
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k ∼ Θ k if Θ k and Θ k have index running over these cubes of side length 2−k . We write Θm m m m adjacent parents, but are not adjacent. As in [10], we use a Whitney type decomposition of I × I away from its diagonal D, so that
(I × I \ D) =
∞
k k Θm × Θm .
k ∼Θ k k=0 Θm m
We set fmk (η, ρ) = χΘmk
η f (η, ρ) = χΘmk (θ )f (ρθ, ρ). ρ
Then it follows that (Ef )(Ef ) =
E fmk E fmk .
(3.4)
k ∼Θ k k0 Θm m
Here we drop the subscript of Eφ and do so for the rest of the proof. Fixing k, we claim that for q > 2n−2L+8 n−L+2 and p 2 k k E f E f m m k ∼Θ k Θm m
C2
(n−L+2) 2k( (n−L) )−(n−L) p + q
f 2Lp L2 . θ
q/2
(3.5)
ρ
Once one has (3.5), then the desired bound can be shown by using a simple summation argun−L+2 ment. In fact, let us fix p0 , q0 satisfying q0 > 2n−2L+8 = (n − L)(1 − p10 ). n−L+2 , p0 > 2 and q0 Using (3.5), for q >
2n−2L+8 n−L+2
we have
k k E fm E fm k ∼Θ k Θm m
C2
2k(n−L+2)( q1 − q1 ) 0
q/2
f 2 p0
Lθ L2ρ
Let ∞K be an integer to be chosen later. Then using (3.4) and splitting the sum K+1 , we see that (x, t): (Ef Ef )(x, t) > λ (x, t):
(3.6)
. k
to
K
−∞ ,
K k k λ E fm E fm > 2 k −∞ k
+ (x, t):
Θm ∼Θm
∞
k ∼Θ k K+1 Θm
k k λ E fm E fm > . 2
m
Let us choose q1 , q2 such that 2n−2L+8 n−L+2 < q1 < q0 < q2 . Obviously, using triangle inequality and (3.6), by summation of geometric series we get
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K
k=−∞ Θ k ∼Θ k
∞
m
m
k=K+1 Θ k ∼Θ k m
k k E fm E fm
C2
1
0
f 2 p0
C2
2K(n−L+2)( q1 − q1 ) 2
,
Lθ L2ρ
q1 /2
k k E fm E fm
m
2K(n−L+2)( q1 − q1 )
0
f 2 p0
Lθ L2ρ
q2 /2
.
Hence by Chebyshev’s inequality and the above we have q1 (x, t): (Ef Ef )(x, t)> λ C2K(n−L+2)(1− q0 ) f q1p 0
Lθ L2ρ
+ C2
q
K(n−L+2)(1− q2 ) 0
f
λ−q1 /2
q2 λ−q2 /2 . p Lθ 0 L2ρ
Then by choosing K which optimizes the right-hand side of the above inequality we get Ef Lq0 ,∞ Cf Lp0 L2 . θ
ρ
Finally, this estimate is valid for p0 , q0 satisfying q0 > 2n−2L+8 n−L+2 , p0 > 2 and (3.3). Real interpolation among these estimates gives the desired because p0 q0 . Hence now it remains to show (3.5). k ∼ Θ k are supported in Observe that for a fixed k the Fourier transforms of E(fmk )E(fmk ), Θm m boundedly overlapping parallelepipeds. Hence, for 2 q 4 we have k k E fm E fm k ∼Θ k Θm m
C
q/2 2/q E f k E f k . m
q/2
m
q/2
(3.7)
k ∼Θ k Θm m
This follows from interpolation between L2 and trivial L1 estimates (see Lemma 6.1 in [10]). By k ∼ Θ k , then this, we are reduced to showing that if Θm m (n−L) (n−L+2) k k E f E f C22k( p + q )−(n−L) f k p 2 f k p 2 . m m L L m m L L q/2 θ
ρ
θ
ρ
In fact, putting this in the right-hand side of the above, we have k k E f E f m m k ∼Θ k Θm m
C2
(n−L+2) 2k( (n−L) )−(n−L) p + q
q/2
×
q/2 q/2 2/q f k p 2 f k p 2 . m L L m L L k ∼Θ k Θm m
θ
ρ
θ
ρ
(3.8)
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Therefore the desired inequality (3.5) follows if one can show that
q/2 q/2 2/q q/2 q/2 f k p 2 f k p 2 Cf Lp L2 f Lp L2 . m L L m L L k ∼Θ k Θm m
θ
ρ
θ
ρ
θ
ρ
θ
ρ
q By Cauchy–Schwarz’s inequality, the left-hand side is bounded by ( m fmk Lp L2 )2/q . It is θ
ρ
again bounded by f 2Lp L2 since q p and the angular projections of the supports of fmk are θ
ρ
disjoint. Now we proceed to show (3.8). Let Θ1 , Θ2 ⊂ [−1, 1]n−L be cubes of diam(Θ1 ), diam(Θ2 ) −k 2 , and dist(Θ1 , Θ2 ) 2−k . For j = 1, 2, let us set P Θj , 2−k = (η, ρ) ∈ Q: η = ρθ, θ ∈ Θj . For (3.8) we need show that Ef Egq/2 C2
(n−L+2) 2k( (n−L) )−(n−L) p + q
f Lp L2 gLp L2 θ
ρ
θ
(3.9)
ρ
whenever f , g are supported in P (Θ1 , 2−k ), P (Θ2 , 2−k ), respectively. Let θ0 = (θ01 , θ02 , . . . , θ0L ) ∈ Rn−L be the center of the smallest cube containing both of Θ1 , Θ2 . We write the phase part of the extension operator Ef as x · ξ − tφ(ξ ) = x˜ · η + x¯ · ρ − t
L l=1
|ηl |2 ± l ρ
where x = (x, ˜ x) ¯ ∈ Rn−L × RL . Let us set x˜ = (x˜ 1 , x˜ 2 , . . . , x˜ L ), x˜ l ∈ Rnl −1 and x¯ = 1 2 L (x¯ , x¯ , . . . , x¯ ), x¯ l ∈ R. The phase part is transformed to L 2 l l x˜ · θ0 + x¯ l ∓ t θ0l ρ l − t (x˜ ∓ 2tθ0 ) · η + l=1
L l=1
|ηl |2 ± l ρ
under the change of variables (η, ρ) → (η + ρθ0 , ρ) = η1 + θ01 ρ 1 , η2 + θ02 ρ 2 , . . . , ηL + θ0L ρ L , ρ . Note that this sends the parallelepipeds P (Θ1 , 2−k ), P (Θ2 , 2−k ) to P (Θ1∗ , 2−k ), P (Θ2∗ , 2−k ), respectively. Here Θ1∗ , Θ2∗ are cubes contained in [−2−k+1 , 2−k+1 ]n−L with dist(Θ1∗ , Θ2∗ ) 2−k . Performing the change of variables (η, ρ) → (η + ρθ0 , ρ) for both of the integrals E(f ), E(g), we see (Ef Eg)(x, t) = (E f˜E g)(T ˜ x, t),
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where f˜, g˜ are functions satisfying that f˜Lp L2 = f Lp L2 , g ˜ Lp L2 = gLp L2 , and θ
ρ
ρ
θ
θ
ρ
ρ
θ
2 2 T (x) = x˜ ∓ 2tθ, x˜ 1 · θ01 + x¯ 1 ∓ t θ01 , . . . , x˜ L · θ0L + x¯ L ∓ t θ0L . Since |det T | = 1, the matters are reduced to showing that Ef Egq/2 C2
(n−L+2) 2k( (n−L) )−(n−L) p + q
f Lp L2 gLp L2 θ
ρ
θ
ρ
whenever f , g are supported in P (Θ1∗ , 2−k ), P (Θ2∗ , 2−k ). Now we make an additional change of variables (η, ρ) → 2−k η, ρ for both of the integrals Ef , Eg to see we see ˜ x, ¯ 2−2k t (Ef Eg)(x, t) = 2−2k(n−L) (Efk Egk ) 2−k x, where fk , gk are functions satisfying that fk Lp L2 = 2k(n−L)/p f Lp L2 , θ
ρ
θ
ρ
gk Lp L2 = 2k(n−L)/p gLp L2 θ
ρ
θ
ρ
and the supports of fk , gk are contained in P (Θ1o , 12 ), P (Θ2o , 12 ), satisfying Θ1o , Θ2o ⊂ [−1, 1]n−L and dist(Θ1o , Θ2o ) 1. Then, after rescaling, it is sufficient to show that Efk Egk q/2 Cfk Lp L2 gk Lp L2 θ
ρ
θ
ρ
provide that fk , gk are supported in P (Θ1o , 12 ), P (Θ2o , 12 ), respectively. This follows from Proposition 3.2. Hence we get (3.5). This completes the proof of Theorem 3.1 for the case n − L 2. Now we turn to the remaining case n − L = 1. This condition implies that n = 2 and L = 1. We only need to show (3.5) in a neighborhood of the segment determined by 3/q + 1/p = 1, q > 4, because once this is obtained the rest of the argument works without modification. Since 4 < q, we need to replace (3.7) with k k E f E f m m k ∼Θ k Θm m
C q/2
k k (q/2) (2/q) E f E f
m
m
(q/2)
k ∼Θ k Θm m
which is valid for q 4. Then, using (3.9), one can get the desired estimate (3.5) as long as 2(q/2) p. There is a small neighborhood of the segment determined by 3/q + 1/p = 1, q > 4, where this condition is satisfied. 2
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Proof of Proposition 3.2. To prove Proposition 3.2, we use Theorem 1.4. It is enough to verify the conditions (1.3), (1.4) and (1.5) when the surface is given by (ξ, φ(ξ )) and (3.2) is satisfied. We only show it for the case φ(ξ ) =
L |ηl |2 l=1
ρl
.
It is not difficult to see that the same is valid for the other cases. The geometry of Πz1 ,z2 can be complicated and the conditions (1.3), (1.4) and (1.5) are not so easy to check directly in practical applications. So we simplify the picture by projecting Πz1 ,z2 to Rn . It generally makes some information be lost but in our example we can still get the sharp results. By finite decomposition we may assume that Q1 and Q2 are as small as we wish. More precisely, let (ρ1 θ1 , ρ1 ), (ρ2 θ2 , ρ2 ) ∈ Q. Then, we may assume that the surfaces are given as graphes of φ over the sets Q1 = (ρθ, ρ) ∈ Q: |θ − θ1 | < 0 , |ρ − ρ1 | < 0 , Q2 = (ρθ, ρ) ∈ Q: |θ − θ2 | < 0 , |ρ − ρ2 | < 0 , respectively, for some small 0 > 0. Hence by continuity, to verify the conditions (1.3), (1.4) and (1.5) it is enough to show them at each point (ρ1 θ1 , ρ1 ), (ρ2 θ2 , ρ2 ) ⊂ Q if 0 is sufficiently small. Let us denote by πz1 ,z2 the projection of Πz1 ,z2 to spatial space. That is, πz1 ,z2 = {v: (v, τ ) ∈ Πz1 ,z2 }. Let us write zi = (vi , τi ) for i = 1, 2. Since Szi = {(u, φ(u)) + zi : u ∈ Qi }, i = 1, 2, one can easily see πz1 ,z2 = v ∈ (Q1 + v1 ) ∩ (Q2 + v2 ): φ(v − v1 ) − φ(v − v2 ) = τ2 − τ1 . Hence πz1 ,z2 is an (n−1)-dimensional immersed surface as long as ∇φ(v −v1 )−∇φ(v −v2 ) = 0 for all v ∈ πz1 ,z2 . Obviously this condition is equivalent to (1.3). Now note that 2 2 ∇φ(ρθ, ρ) = 2θ 1 , 2θ 2 , . . . , 2θ L , −θ 1 , . . . , −θ L . So we have for ξ1 = (ρ1 θ1 , ρ1 ), ξ2 = (ρ2 θ2 , ρ2 ), 2 2 2 2 ∇φ(ξ1 ) − ∇φ(ξ2 ) = 2θ1 − 2θ2 , θ21 − θ11 , . . . , θ2L − θ1L . Hence, the condition (1.3) is satisfied by (3.2). Note that at the point ξ = (ρθ, ρ) the projected null directions9 are given the span of the vectors N 1 (θ ) = θ 1 , 0, . . . , 0, 1, 0, . . . , 0 , N 2 (θ ) = 0, θ 2 , 0, . . . , 0, 0, 1, 0, . . . , 0 , .. .
N L (θ ) = 0, . . . , 0, θ L , 0, . . . , 1 . 9 That is, (N j · ∇)2 φ = 0.
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The condition (1.4) is satisfied as long as at least one of the null directions from each surface is transversal to the tangent space of Πz1 ,z2 . Since the surface πz1 ,z2 has dimension n − 1, (1.4) is satisfied if one of the projected null directions N 1 (θ1 ), N 2 (θ1 ), . . . , N L (θ1 ) is transversal to the tangent space of πz1 ,z2 . On the other hand the normal vector to πz1 ,z2 is ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ) + O( 0 ). So it is enough to show that there are some i, k such that ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ), N i (θ1 ) ∼ 1, ∇φ(ρ1 θ1 , ρ1 ) − ∇φ(ρ2 θ2 , ρ2 ), N k (θ2 ) ∼ 1. By direct computation one can easily see that the right-hand side of the first expression is equal to |θ1i − θ2i |2 and that of the second is |θ1k − θ2k |2 . Due to the condition (3.2), there must be i, k (in fact, i = k) satisfying the above. Hence (1.4) follows. Now it remains to show (1.5). By Remark 1.3 we need to show that for j = 1, 2, the normal vectors Nj of surfaces Szj are transversal to the opposite tube cones Γ3−j . Instead of considering the sets Γ1 , Γ2 , we consider larger sets which are given by Γj = t ∇φ(v), 1 : v ∈ Qj , 2−1 |t| 2 . Obviously Γj ⊂ Γj . So, if the opposite normal Nj is transversal to Γ3−j , it is also transversal to Γ3−j . By the homogeneity of φ Γj = tΦ(θ ): θj ∈ Θj , 2−1 |t| 2 where Θj = { ρη : (η, ρ) ∈ Qj } and 2 2 Φ(θ ) = t 2θ 1 , 2θ 2 , . . . , 2θ L , −θ 1 , . . . , −θ L , 1 . We only consider the case j = 2 because the other case j = 1 is obvious from symmetry. From the above the tangent space of Γ1 at each point is spanned by the partial derivatives (∂(θ l )k Φ(θ1 ), 0), and (Φ(θ1 ), 1). Here (θ l )k denotes the k-th component of θ l . The direction of opposite tube is also given by (Φ(θ2 ), 1). So the transversality between N1 and Γ2 follows if one can show that these n − L + 1 vectors (∂(θ L )k Φ(θ1 ), 0), and Φ(θ1 ) − Φ(θ2 ) are linearly independent. Equivalently, we need to show that the n × (n − L + 1) matrix M = ∂θ Φ(θ1 ), Φ(θ1 ) − Φ(θ2 ) has a minor of size (n − L + 1) × (n − L + 1) with nonvanishing determinant. Here we write Φ(θ1 ), Φ(θ2 ) as column vectors and ∂θ Φ(θ1 ) is an n × (n − L) matrix. If one removes the last L − 1 rows from M except (n − L + l)-th, then we have the matrix l = 2In−L M v
u α
!
S. Lee, A. Vargas / Journal of Functional Analysis 258 (2010) 2884–2909
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where In−L is the identity matrix of (n − L) × (n − L), u = 2θ1 − 2θ2 , α = |θ2l |2 − |θ1l |2 and v = (0, . . . , 0, −2θ1l , 0, . . . , 0). By a simple computation one see l = 2n−L−1 2α − u, v = 2n−L θ l − θ l 2 . det M 1 2 l with nonzero determinant because Therefore there is an (n − L + 1) × (n − L + 1) minor M det M1 , . . . , det ML cannot be zero simultaneously by the condition (3.2). This completes the proof of Proposition 3.2. 2 Since the estimates are stable under a small smooth perturbation of the surface, it is possible to obtain the same results for other surfaces which are of similar type. For example, let us consider ψ(ξ ) =
L l ξ l=1
where ξ = (ξ 1 , . . . , ξ L ) ∈ Rn1 × · · · × RnL . Then let Eψ be the operator given by
ei(x·ξ −tφ(ξ )) f (ξ ) dξ.
Eψ f (x, t) = {|ξ l |∼1, 1lL}
We now split variable ξ l and write ξ l = (ηl , ρ l ) ∈ Rnl −1 × R. By a finite decomposition of Q and rotation in each ξ l one may assume Q = ξ = ξ 1 , . . . , ξ L : ηl 0 , ρ l ∼ 1, 1 l L with small 0 > 0. Then by the linear change of variables (x, t) → x˜ 1 − t x¯ 1 , x˜ 2 − t x¯ 2 , . . . , x˜ L − t x¯ L , t ˜ )= ψ is replaced by ψ(ξ ˜ )= ψ(ξ
L
l=1 |(η
l , ρ l )| − ρ l .
Then by Taylor’s expansion
L L l 4 l 4 1 l 2 1 |ηl |2 η θ . θ = + O ρ + O l 2 ρl 2 l=1
l=1
This is a small smooth perturbation of ψ . Hence the bilinear estimate can be similarly formulated for Eψ because all the required (1.3), (1.4) and (1.5) are stable under smooth perturbation. By this stability the bilinear → linear argument also works. Let r l , θ l be the spherical coordinates for ξ l such that ξ l = r l θ l and define a mixed norm
f Lp Lqr = θ
Sn1 −1 ×···×SnL −1
f (rθ )q dr
p/q
1
p
dθ
[1,2]L
where θ = (θ 1 , . . . , θ L ) and r = (r 1 , . . . , r L ). Then we get the following.
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Corollary 3.4. Let ψ be the function given as in the above. If then for q > that
2n−2L+8 n−L+2
n−L+2 q
(n−L)(1− p1 ) and p 2,
when n − L 2, and for q > 4 when n − L = 1, there is a constant C such
Eψ f q Cf Lp L2 . θ
r
Remark 3.5. Proposition 3.2 can be further generalized. Let A1 , . . . , AL be nonsingular symmetric matrices of (nl − 1) × (nl − 1). Then consider
φ(ξ ) =
L
ρ l Al θ l , θ l ,
l=1
where θ l = ηl /ρ l . Here we use the same notations which are used in Proposition 3.2. Then the same bilinear estimate holds under the conditions L l l A θ − θl , θl − θl ∼ 1 1 2 1 2 l
for (ρ1 θ1 , ρ1 ) ∈ Q1 , (ρ2 θ2 , ρ2 ) ∈ Q2 . This is a generalization of the bilinear restriction estimate for a conic surface which was studied in [5]. However, when Al has eigenvalues of different signs it is still in question whether the sharp linear estimates can be obtained from these bilinear estimates. References [1] J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991) 147–187. [2] J. Bourgain, Estimates for cone multipliers, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41–60. [3] A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981) 519–537. [4] S. Lee, Endpoint estimates for the circular maximal function, Proc. Amer. Math. Soc. 131 (5) (2003) 1433– 1442. [5] S. Lee, Bilinear restriction estimates for surfaces with curvatures of different signs, Trans. Amer. Math. Soc. 358 (8) (2006) 3511–3533. [6] G. Mockenhaupt, A note on the cone multiplier, Proc. Amer. Math. Soc. 117 (1) (1993) 145–152. [7] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (3) (1977) 705–714. [8] T. Tao, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003) 1359–1384. [9] T. Tao, A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal. 10 (2000) 185–215. [10] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjecture, J. Amer. Math. Soc. 11 (1998) 967–1000. [11] P. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975) 477–478. [12] A. Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (1) (2005) 97–111. [13] T. Wolff, A sharp cone restriction estimate, Ann. of Math. 153 (2001) 661–698.
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Further reading [14] B. Barcelo, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985) 321–333. [15] T. Tao, A. Vargas, A bilinear approach to cone multipliers. II. Application, Geom. Funct. Anal. 10 (2000) 216–258.
Journal of Functional Analysis 258 (2010) 2910–2936 www.elsevier.com/locate/jfa
Rough paths analysis of general Banach space-valued Wiener processes S. Dereich Philipps-Universität Marburg, Fb. 12 – Mathematik und Informatik, Hans-Meerwein-Straße, D-35032 Marburg, Germany Received 10 July 2009; accepted 14 January 2010 Available online 6 February 2010 Communicated by Paul Malliavin
Abstract In this article, we carry out a rough paths analysis for Banach space-valued Wiener processes. We show that most of the features of the classical Wiener process pertain to its rough path analog. To be more precise, the enhanced process has the same scaling properties and it satisfies a Fernique type theorem, a support theorem and a large deviation principle in the same Hölder topologies as the classical Wiener process does. Moreover, the canonical rough paths of finite dimensional approximating Wiener processes converge to the enhanced Wiener process. Finally, a new criterion for the existence of the enhanced Wiener process is provided which is based on compact embeddings. This criterion is particularly handy when analyzing Kunita flows by means of rough paths analysis which is the topic of a forthcoming article. © 2010 Elsevier Inc. All rights reserved. Keywords: Rough paths; Wiener process; Support theorem; Large deviation principle
1. Introduction The notion rough path was coined by Terry Lyons in 1994 [20]. The corresponding theory provides an extension of Young integrals to less regular driving signals. In the context of probability theory, it allows an alternative representation for solutions to Stratonovich differential equations as solutions to rough path differential equations (RDE). The power of the approach is that once the driving signal has been associated to a rough path, the solution can be written E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.018
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as continuous function (Itô map) of the rough path signal by Terry Lyons’ universal limit theorem [22]. In general irregular controls admit several extensions to rough paths. Nonetheless, in the context of stochastic analysis there is one canonical choice which is uniquely defined up to a null set. Since the null set does not depend on the choice of the RDE this allows one to pick a random RDE depending on the path itself and, in particular, the concept of filtrations becomes obsolete for the existence of solutions. As eluded by Ledoux, Qian, and Zhang [18] the theory of rough paths leads to natural proofs of support theorems (ST) and large deviation principles (LDP) since both properties behave nicely under an application of the continuous Itô map. Consequently, it suffices to prove a support theorem and a large deviation principle for the canonical rough path of the driving signal and then to infer the corresponding results for the solution of the SDE. This approach has been firstly carried out in [18] for the multi-dimensional Wiener process under the p-variation topology for p > 2. Later on, analogous results were proved under fine Hölder topologies by Friz and Victoir [10] (see also [12]). General Banach space-valued Wiener processes were embedded into the theory of rough paths by Ledoux et al. [17] in 2002. A series of articles by Inahama and Kawabi [14–16] followed which was mainly motivated by its applicability to heat kernel measures on loop spaces. Nowadays the theory of rough paths is well established and we refer the reader to the monographs [21,23], and [11] for a general account on the topic. Our results are manifold. First we establish a representation of the enhanced Wiener process as limit of finite dimensional enhanced Wiener processes. This Itô–Nisio type theorem implies that the enhanced Wiener process has the same scaling properties as the classical Wiener process. For finite dimensional Wiener processes there are various ways to define the canonical rough path (either as solution to a Stratonovich stochastic differential equation or via the limit of certain smooth approximations, see for instance [11]) and it is thus conceived as a universal object. Since we can freely approximate the enhanced infinite dimensional Wiener process by finite dimensional approximations, also the infinite dimensional canonical rough path can be seen as a universal object. We derive a support theorem and a large deviation principle in fine Hölder topologies similar as the one known for finite dimensional processes [10]. By doing so we extend results of [14] who analyzed the problem under p-variation topology. In general, the existence of the canonical rough path is not trivial (at least for projective tensor products), and Ledoux et al. provide a sufficient criterion in [17]. We relate their concept of finite dimensional approximation to entropy numbers of compact embeddings. Since these are known for various embeddings [7,8], we obtain a new sufficient criterion which can be easily verified in many cases. We phrase its implications in the case where the state space of the Wiener process is a Hölder–Zygmund space. Our main interest in the theory developed here is its applicability to stochastic flows generated by Kunita type SDEs. Indeed, we will establish a support theorem and a large deviation principle for Brownian flows in a forthcoming article [5]. We start with summarizing the results of the article. Here we introduce some notation rather in an informal way in order to enhance readability. All notation will be introduced in great detail at the end of this section. Results Let (V , | · |V ) be a separable Banach space, and let X = (Xt )t∈[0,1] denotes a V -valued Wiener process on a probability space (Ω, F , P). More explicitly, X is measurable with respect to the Borel sets of C([0, 1], V ) and satisfies, for 0 s < t 1,
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• Xt − Xs is independent of σ (Xw : w s) and • L((t − s)−1/2 (Xt − Xs )) = L(X1 ) is a centered Gaussian distribution. We denote by (H1 , | · |H1 ) and (H, | · |H ) the reproducing kernel Hilbert spaces of X1 and X = (Xt )t∈[0,1] . Note that H can be expressed in terms of H1 as · H=
2
ft dt: f ∈ L [0, 1], H1
,
0
where the integral is to be understood as Bochner integral. For a general account on Gaussian distributions we refer the reader to the books by Lifshits [19] and Bogachev [2]. We let ϕ : (0, 1] → (0, ∞) denote an increasing function with ϕ(δ) =∞ lim √ δ↓0 −δ log δ
(1)
and consider the geometric ϕ-Hölder rough path space GΩϕ (V), which will be rigorously introduced below. Moreover, X is assumed to possess a canonical rough path X in the sense of assumption (E), see Section 2. Theorem 1.1. X is almost surely an element of GΩϕ (V) and its range in that space is the closure of the lift of the reproducing kernel Hilbert space H of X into GΩϕ (V). Theorem 1.2. The family {Xε : ε > 0} with Xε being the canonical rough path of (ε · Xt )t∈[0,1] satisfies a LDP in GΩϕ (V) with good rate function J (h) =
1
2 2 |h|H
∞
if ∃h ∈ H with h = S(h), else,
where S denotes the canonical lift of H into GΩϕ (V). Suppose now that the reproducing kernel Hilbert space H1 of X1 is infinite dimensional and fix a complete orthonormal system (ei )i∈N of H1 . We represent X as the in C([0, 1], V ) almost sure limit Xt = lim
n→∞
n
(i)
ξt ei ,
(2)
i=1
(i)
with {(ξt )t∈[0,1] : i ∈ N} being an appropriate family of independent standard Wiener processes. (i) Theorem 1.3. Each X (n) = ( ni=1 ξt ei )t∈[0,1] possesses a canonical rough path X(n) , and one has almost sure convergence lim X(n) = X in GΩϕ (V).
n→∞
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Remark 1.4. The latter theorem can be proved in a more general setting. One can replace the assumption that (ei ) is a complete orthonormal system of the reproducing kernel Hilbert space by the assumption that, for any h ∈ H1 , n lim h, ei H1 ei = h.
n→∞
i=1
We proceed with a sufficient criterion for the existence property (E) for Hölder–Zygmund spaces. For an open and bounded set D, for n ∈ N0 , η ∈ (0, 1], and γ = n + η, we denote by γ C0 (D, Rd ) the set of n-times differentiable functions f : Rd → Rd whose derivatives are ηHölder continuous and satisfy the zero boundary condition f |D c = 0. The space is endowed with a canonical norm · C γ , see (14). Theorem 1.5. Let 0 < γ < γ¯ and D ⊂ D be bounded and open subsets of Rd with D¯ ⊂ D . γ¯ Let μ be a centered Gaussian measure on C0 (D, Rd ). Then there exists a separable and closed γ subset V ⊂ C0 (D , Rd ) and a V -valued Wiener process X = (Xt )t∈[0,1] satisfying the existence property (E) and L(X1 ) = μ. For instance, one may choose V as the closure of H1 . γ¯
Remark 1.6. In the theorem, the two Banach spaces C0 (D, Rd ) and C0 (D , Rd ) can be replaced by arbitrary Banach spaces V1 and V2 for which V1 is compactly embedded into V2 and for which the entropy numbers of the embedding decay at least at a polynomial order, see Section 5 for the details. In particular, one can use the results of Edmunds and Triebel [7,8] on embeddings of Sobolev and Besov spaces. γ
Let us summarize the implications in a language that does not incorporate rough paths. Let W denote a further Banach space, let f : W → L(V , W ) be a Lip(γ )-function for a γ > 2 in the sense of [23, Def. 1.21]. For a fixed absolutely continuous path g : [0, 1] → V with differential in L2 ([0, 1], V ), we consider the Young differential equation dyt = f (yt ) d[x + g]t ,
y0 = ξ.
(3)
By Picard’s theorem, the differential equation possesses a unique solution Ig (x) for any x ∈ BV(V ) (actually unique solutions exist under less restrictive assumptions). For a V -valued Wiener process X and n ∈ N, we denote by X(n) the dyadic interpolation of X with breakpoints Dn = [0, 1] ∩ (2−n Z). We call Y the Wong–Zakai solution of (3) for the control X, if {Ig (X(n)): n ∈ N} is a convergent sequence in Cϕ ([0, 1], W ) with limit Y . As is well known Wong–Zakai solutions are tightly related to stochastic differential equations in the Stratonovich sense: If the state space W is an M-type 2 Banach space and if dgt = v dt for some v ∈ V , then the Wong–Zakai solution solves the corresponding Stratonovich stochastic differential equation, see [3]. Theorem 1.7. Let X be a V -valued Wiener process with reproducing kernel Hilbert space H . If property (E) is valid, then the following is true:
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(I) X admits a Wong–Zakai solution Y of (3). (II) For n ∈ N, let X (n) be as in Theorem 1.3 and denote by Y (n) the corresponding Wong–Zakai solution of (3). Then {Y (n) : n ∈ N} converges almost surely to Y in Cϕ ([0, 1], W ). (III) The range of Y in Cϕ ([0, 1], W ) is the closure of Ig (H ). (IV) For ε > 0, we let Y ε denote the Wong–Zakai solution of (3) for the control ε · X. Then {Y ε : ε > 0} satisfies a large deviation principle in Cϕ ([0, 1], W ) with speed (ε 2 )ε>0 and good rate function
1 J (h) = inf x H : x ∈ H, Ig (x) = y . 2 Here and elsewhere, the infimum of the empty set is defined as ∞. Agenda The article is organized as follows. Sections 3 and 4 are concerned with the derivation of Theorems 1.1 and 1.2, respectively. Section 2 has rather preliminary character. Here, we prove a preliminary version of the representation provided by Theorem 1.3 (see Remark 3.3 for the extension to the stronger statement). Moreover, we derive a Fernique type theorem together with Lévy’s modulus of continuity. Section 5 is concerned with the proof of Theorem 1.5. Finally, we explain the implications of the theory of rough paths (Theorem 1.7) in Section 6. Notation We start with introducing the (for us) relevant notation of the theory of rough paths. For a profound background on the topic we refer the reader to the textbooks [23] and [11]. For two Banach spaces V1 and V2 we denote by V1 ⊗a V2 the algebraic tensor product of V1 and V2 . Moreover, we let V1 ⊗ V2 denote the projective tensor product, that is the completion of V1 ⊗a V2 under the projective tensor norm | · |V1 ⊗V2 given by |v|V1 ⊗V2 = inf
n
|fi |V1 |gi |V2 ,
i=1
where the infimum is taken over all representations v=
n
f i ⊗ gi
(n ∈ N, fi ∈ V1 , gi ∈ V2 for i = 1, . . . , n).
i=1
From now on V denotes a separable Banach space. For a (continuous) piecewise linear function x : [0, 1] → V , we may compute its iterated (Young) integrals t x1s,t
=
dxu ∈ V , s
x2s,t = s
dxu1 ⊗ dxu2 ∈ V ⊗2 ,
(4)
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and set xs,t = x1s,t + x2s,t ∈ V ⊕ V ⊗2 , where (s, t) ∈ := {(s , t ): 0 s t 1}. As is well known the iterated integrals satisfy the Chen condition: x1s,t = x1s,u + x1u,t
and x2s,t = x2s,u + x2u,t + x1s,u ⊗ x1u,t ,
for 0 s u t 1.
We need to consider the truncated tensor algebra. For ease of notation, we omit the real component and set V = V ⊕ V ⊗2 . It is endowed with the standard addition and the multiplication u ∗ v = u1 + v 1 + u 2 + v 2 + u 1 ⊗ v 1 = u + v + u 1 ⊗ v 1 . Using the convention u ⊗ v := u1 ⊗ v1 we also write u ∗ v = u + v + u ⊗ v. In terms of ∗ the Chen condition can be rewritten as xs,t = xs,u ∗ xu,t . We endow V and C( , V) with the norms |u|V := u1 V + u2 V ⊗V
and x ∞ = sup |xs,t |V , (s,t)∈
and call
Ω(V) = (xs,t )(s,t)∈ ∈ C( , V): x satisfies the Chen condition the set of multiplicative functionals on V. It is a closed subset of C( , V). In the following, we shall reserve the notation x = (xs,t )(s,t)∈ for V-valued multiplicative functionals. In the context of rough path theory we mainly work with the following homogeneous norm on V,
u = u1 V + u2 V ⊗2 . It enjoys the following properties, for u, v ∈ V, (i) (ii) (iii) (iv)
u = 0 ⇔ u = 0;
δt u = |t| u for the dilation operator δt u = tu1 + t 2 u2 (t ∈ R);
u = − u ;
u + v u + v .
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We consider rough path spaces with Hölder norm topologies: Let ϕ : (0, 1] → (0, ∞) be an increasing function, and set, for x ∈ C( , V),
x ϕ =
xs,t
. 0s
We denote by
Ωϕ (V) = x ∈ Ω(V): x ϕ < ∞ the set of ϕ-Hölder rough paths in V. It is equipped with the metric (x, y) → x − y ϕ . The set is nontrivial whenever lim infδ↓0 ϕ(δ)/δ > 0 which we assume in the following without further mentioning. Moreover, the set of geometric ϕ-Hölder rough paths is given by
GΩϕ (V) = S(x): x ∈ C [0, 1], V piecewise linear ⊂ Ωϕ (V), where S(x) = (xs,t )(s,t)∈ with xs,t as in (4), and the closure is taken with respect to · ϕ . Analogously, we denote by Cϕ ([0, 1], V ) the Hölder space induced by ϕ that is the space of all functions x : [0, 1] → V with finite Hölder norm
x ϕ = x ∞ +
|xs,t |V , ϕ(t − s) 0s
where we – as usual – denote xs,t = xt − xs and x ∞ = supt∈[0,1] |xt |V for functions x ∈ C([0, 1], V ). Moreover, we denote by BV(V ) the set of all functions x ∈ C([0, 1], V ) with finite bounded variation norm
x BV(V ) = x ∞ +
sup
n
0t0 <···
|xtl−1 ,tl |V .
Interpreting the integrals in (4) as Young integrals allows us to assign each path x ∈ BV(V ) to a path S(x) := x ∈ Ω(V). Furthermore, we call x ∈ BV(V ) absolutely continuous if it admits t x˙ ∈ L1 ([0, 1], V ) with x0,t = 0 x˙s ds. We will make use of the following three operators: The dilation operator, which appeared already above, is given by δt : V → V,
δt (u) = tu1 + t 2 u2 ,
for t ∈ R,
and the logarithm which is defined by log : V → V,
1 v → v − v ⊗ v. 2
Both operators naturally extend to continuous functions that map C( , V) into itself via δt (x) = (δt (xs,u ))(s,u)∈ and log x = (log xs,t )(s,t)∈ . Additionally, we consider the translation operator on the set of multiplicative functionals which is defined for f ∈ BV(V ) by
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Tf : Ω(V) → Ω(V), t Tf (x)s,t = xs,t + fs,t +
t xs,u ⊗ dfu +
s
t fs,u ⊗ dxu +
s
fs,u ⊗ dfu . s
All integrals are well defined Young integrals and restricting the translation operator to more regular paths allows to relax the assumption on f . Note that the definition of Tf is motivated by the following property: for x ∈ C([0, 1], V ) for which Γ (x) exists (in the sense that the limit converges in Ω(V)) and f ∈ BV(V ) one has Tf Γ (x) = Γ (x + f ).
(5)
Finally, we denote by πV : V → V and πV ⊗2 : V → V ⊗2 the projections onto the V - and respectively. In general, we use analogous notation for W -valued paths.
V ⊗2 -component,
2. The canonical rough path In this section, we introduce the canonical rough path (called enhanced Wiener process) associated to a Banach space-valued Wiener process. The main task will be to establish Lévy’s modulus of continuity together with a Fernique type theorem. As before we let X denote a Wiener process attaining values in a separable Banach space V . Let Dn = (2−n Z) ∩ [0, 1] (n ∈ N0 ) and denote by X(n) the linear interpolation of X with dyadic breakpoints Dn . Moreover, let X(n) = S(X(n)). The definition of the enhanced Wiener process relies on the following assumption: (E) There exists an Ω(V)-valued random element X such that lim X(n) = X in L1 P; C( , V) .
n→∞
A sufficient criterion for property (E) is provided in [17]: Definition 2.1. Let μ be a centered Gaussian measure on the Borel sets of the separable Banach space V . For a fixed tensor product norm | · |V ⊗V the measure μ is called exact, if for independent ˜ l (l ∈ N) and some constants C and α < 1 one has μ-distributed random elements Gl , G N ˜ l E Gl ⊗ G l=1
CN α
V ⊗V
for all N ∈ N. By [17, formula on p. 566] property (E) is satisfied if the measure μ is exact with respect to the projective tensor norm. From now on we assume that property (E) is satisfied without further mentioning. In general, we denote, for a path x ∈ C([0, 1], V ), Γ (x) = lim S x(n) , n→∞
(6)
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provided that the limit exists in C( , V). Here and elsewhere, we denote by x(n) the interpolation of x with dyadic breakpoints Dn . The proposition below allows us to choose X = Γ (X) as the canonical rough path (enhanced Wiener process) of X. Proposition 2.2. The family (log X(n))n∈N is a C( , V)-valued martingale and one has log X(n) = E[log X|Gn ], where Gn = σ (Xt : t ∈ Dn ) = σ (X(n)). In particular, lim X(n) = X,
almost surely.
n→∞
The proof is based on the following lemma. Lemma 2.3. Let f = (ft )t∈[0,1] be a BV(V )-valued random element such that almost surely f is a one-dimensional excursion, that is dim(span(im(f ))) = 1, almost surely. If the distribution L
of f is symmetric in the sense that f = −f and if E f 2BV(V ) < ∞, then for an arbitrary path x ∈ Ω(V) we have E log Tf (x) = log x. Proof. According to the definition of log and the shift operator we have (with f = S(f )) log Tf (x)s,t = log xs,t + log fs,t t t t t 1 + xs,u ⊗ dfu + fs,u ⊗ dxu − dxu ⊗ fs,u − dfu ⊗ xs,u . 2 s
s
s
s
Due to the symmetry of the distribution of f we have t E
t xs,u ⊗ dfu +
s
t fs,u ⊗ dxu −
s
t dxu ⊗ fs,u −
s
dfu ⊗ xs,u = 0. s
The expectation of the first component πV (log fs,t ) vanishes for the same reason. The second component of log fs,t is identically zero since f takes values in a one-dimensional subspace of V . Indeed, we can write ft = ξt · v for a V -valued random vector v and a real valued process (ξt ). Correspondingly, fs,t = ξs,t · v and therefore 1 πV ⊗V (log fs,t ) = 2
t fs,u ⊗ dfu − s
1 = 2
dfu ⊗ fs,u s
t
t ξs,u dξu −
s
t
dξu ξs,u v ⊗ v = 0.
s
2
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Proof of Proposition 2.2. Note that X(n) := X(n + 1) − X(n) can be written as the sum of 2n independent symmetric one-dimensional excursions that are also independent of the σ algebra Gn . Applying Lemma 2.3 2n -times then gives that E log X(n + 1)s,t |Gn = log X(n)s,t .
(7)
Hence, for general k ∈ N, we have E log X(n + k)s,t |Gn = log X(n)s,t so that by assumption (E) E[log X|Gn ] = log X(n). In particular, log X(n) converges almost surely to log X in C( , V) (see for instance [6, Proposition 5.3.20]). The inverse of log (that is x → (x + 12 x ⊗ x)(s,t)∈ ) is continuous on C( , V), and we also get that (X(n))n∈N converges to X. 2 Finite dimensional approximation Based on a complete orthonormal system (ei )i∈N of the reproducing kernel Hilbert space H1 , we establish a limit theorem for certain finite dimensional approximations to X. We represent X as the in C([0, 1], V ) almost sure limit Xt =
∞
(i)
ξt ei ,
i=1 (i)
where ξ (i) = (ξt )t∈[0,1] (i ∈ N) are independent Wiener processes. For m ∈ N denote X ∗ (m)t =
m
(i)
ξt ei .
i=1
Proposition 2.4. For X∗ (m) = Γ (X ∗ (m)) (m ∈ N) one has lim X∗ (m) = X,
m→∞
almost surely in Ω(V).
(8)
Remark 2.5. The theorem yields that, for the enhanced Wiener process X, L
Xs,t = δ√t−s (X0,1 ),
for (s, t) ∈ ,
where δ denotes the dilation operator. Indeed, this statement is true for Rd -valued Wiener processes and it can be easily extended via (8).
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Proof. We denote the n-th dyadic interpolation of X ∗ (m) by X ∗ (n, m) and let X∗ (n, m) = S(X ∗ (n, m)) and Gn,m = σ (X ∗ (n, m)). By Lemma A.3, one has, for n, m ∈ N, log X∗ (n, m)s,t − log X(n)s,t V n+2 ∗ X (n, m) − X(n) ∞ 1 + X ∗ (n, m)∞ + X(n)∞ . 2 Thus the Cauchy–Schwarz inequality implies together with the equivalence of moments of Gaussian measures that lim log X∗ (n, m) = log X(n),
m→∞
in L1 P, C( , V) .
As in the proof of Proposition 2.2 one verifies that (log X∗ (n, m))m∈N is a (Gn,m )m∈N -martingale for any fixed n ∈ N. Hence, log X∗ (n, m) = E log X(n)|Gn,m = E[log X|Gn,m ]. Recall that X ∗ (m) is a Wiener process for which log X∗ (m) = lim log X∗ (n, m) = lim E[log X|Gn,m ] = E[log X|G∞,m ], n→∞
where G∞,m = σ ( n∈N Gn,m ).
n→∞
in C( , V),
2
Lévy’s modulus of continuity In the rest of this section, we derive Lévy’s modulus of continuity for the enhanced Wiener process, that is X is almost surely an element of Ωφ (V), where φ : (0, 1] → (0, ∞) is a fixed increasing function with φ(δ) lim √ = 1. δ↓0 −δ ln δ Theorem 2.6. The canonical rough path X of the Wiener process X is almost surely an element of Ωφ (V) and one has 2
Eeα X φ < ∞ for some α > 0. Remark 2.7. Fernique’s theorem is also proven in [14] for p-variation topology. Moreover, finite dimensional analogs can be found in [9]. We remark that a Fernique type result is already known to hold for p-variation norm under the exactness assumption [14]. The proof of this assertion parallels a variant of the proof of the classical statement. Basically our approach relies on the isoperimetric inequality and the Garsia– Rodemich–Rumsey inequality and it is similar to the approach taken in [10].
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Lemma 2.8. For a multiplicative functional x ∈ Ω(V) and a function f ∈ BV(V ), one has Tf (x)s,t 2
sup
xu,v + f BV([s,t],V ) ,
for (s, t) ∈ ,
suvt
where f BV([s,t],V ) := supst0 <···
+
t fs,u ⊗ dxu +
s
xs,u ⊗ dfu + s
fs,u ⊗ dfu . s,t
Note that t xs,u ⊗ dfu
κ(s, t) f BV([s,t],V ) .
V ⊗V
s
The same estimate is valid for |
t s
fu ⊗ dxs,u |V ⊗V . Moreover, one has
fs,u ⊗ dfu s,t
V ⊗V
f 2BV([s,t],V )
so that π
V ⊗2
2 Tf (x)s,t V ⊗V x2s,t V ⊗V + κ(s, t) + f BV([s,t],V ) .
Finally, combining all estimates gives Tf (x)s,t xs,t + κ(s, t) + 2 f BV([s,t],V ) .
2
√ Corollary 2.9. For an increasing function ϕ : (0, 1] → (0, ∞) with α1 := infδ∈(0,1] ϕ(δ)/ δ > 0, one has for all x ∈ Ωϕ (V) and f absolutely continuous with f˙ ∈ L2 ([0, 1], V ), Tf (x) 2 α x ϕ + f˙ 2 L ([0,1],V ) ϕ and Tf (x)0,1 2
sup 0st1
xs,t + f˙ L2 ([0,1],V ) .
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Proof. The proof is an immediate consequence of Lemma 2.8 and the estimate
f BV([s,t],V )
√ t − s f˙ L2 ([0,1],V ) αϕ(t − s) f˙ L2 ([0,1],V ) .
2
Next, we apply the isoperimetric inequality for Gaussian measures together with the above estimates to infer the following lemma. Lemma 2.10. There exists α > 0 with Eeα X0,1 < ∞. 2
Moreover, if X is in Ωφ (V) with positive probability, then there exists β > 0 such that 2
Eeβ X φ < ∞. Proof. Fix δ > 0 sufficiently large such that P(X ∈ A) 1/2 for
A = x ∈ C [0, 1], V : Γ (x) exists and x ∞ δ . Recall that H is the reproducing kernel Hilbert space of X, and thus we get with the isoperimetric inequality that for r 0 P X ∈ A + BH (0, r) Φ(r), =:Ar
where Φ is the standard normal distribution function. By (5) and Corollary 2.9, we have for z = x + h ∈ Ar with x ∈ A and h ∈ BH (0, r), Γ (z)0,1 = Th Γ (x)
0,1
2[δ + σ r],
where σ is the norm of the canonical embedding of H1 into V . Hence, we can couple X0,1
with a standard normal random variable N such that X0,1 2[δ + σ (N ∨ 0)] and Fernique’s theorem implies the first assertion. The second assertion is proved analogously. 2 Proposition 2.11. One has X ∈ Ωφ (V), almost surely. This proposition implies together with Lemma 2.10 the assertion of Theorem 2.6. Proof. The proof is based on the Garsia–Rodemich–Rumsey Lemma [13] and since it is classical we only focus on the crucial facts that allow us to apply the argument. By Lemma A.1 in Appendix A, · possesses an equivalent norm ||| · ||| which satisfies the triangle inequality with respect to the group operation ∗. Moreover, one infers from the scaling property of the Wiener process (Remark 2.5) and Lemma 2.10 that
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Eeα(t−s)
−1 |||X
2 s,t |||
2923
= Eeα|||X0,1 ||| < ∞ 2
for an α > 0. The rest of the proof can be literally translated from classical proofs of that result (see for instance [10]). 2 3. Support theorem Again we assume the validity of property (E), and let ϕ : (0, 1] → (0, ∞) be an increasing function satisfying (1). The objective of this section is to prove the following support theorem for the enhanced Wiener process: Theorem 3.1. If X is a V -valued Wiener process satisfying assumption (E), then X is almost surely an element of GΩϕ (V) and its range (in GΩϕ (V)) is the closure of S(H ), where H is again the RKHS of X. Again we denote by ϕ, φ : (0, 1] → (0, ∞) an increasing functions with limδ↓0 For x ∈ C( , V) we set |||x|||ϕ =
sup 0s
|x1s,t |V |x2s,t |V ⊗2 + ϕ(t − s) ϕ(t − s)2
√ φ(δ) −δ log δ
= 1.
and we consider the space
Cϕ ( , V) = (xs,t )(s,t)∈ ∈ C( , V): |||x|||ϕ < ∞ endowed with the norm ||| · |||ϕ . It is a (non-separable) Banach space. Clearly, the distances · ϕ and ||| · |||ϕ generate the same topology on GΩϕ (V) ⊂ Cϕ ( , V). The proof of Theorem 3.1 relies on the following theorem. Theorem 3.2. One has almost sure convergence lim X(n) = X in GΩϕ (V).
n→∞
(9)
Remark 3.3. Theorem 3.2 is an extension of Proposition 2.2. With the same techniques we can strengthen the statement of Proposition 2.4 in order to get Theorem 1.3. The following criterion allows us to verify convergence in GΩϕ (V) (see [10] for a similar criterion in the finite dimensional setting). Lemma 3.4. Let x(n) (n ∈ N) and x be elements of Cφ ( , V). If • supn∈N |||x(n)|||φ < ∞ and • (x(n))n∈N converges to x in C( , V), then (x(n))n∈N converges in Cϕ ( , V) to x.
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Proof. Set η(t) := φ(t)/ϕ(t) (t ∈ [0, 1]) and note that for M = |||x|||φ ∨ supn∈N |||x(n)|||φ , x(n)1 − x1 2Mφ(t − s) = 2Mη(t − s)ϕ(t − s). s,t s,t V On the other hand, x(n)1 − x1 |||x(n) − x|||∞ ϕ(t − s), s,t s,t V ϕ(t − s) where ||| · |||∞ denotes the supremum norm |||x|||∞ = sup(s,t)∈ |xs,t |V . Recall that η(t) converges to 0 for t ↓ 0 so that for arbitrary ε > 0 there exists ζ ∈ (0, 1) with η(ζ ) ε/(2M). We apply the first estimate in the case where t − s ζ and apply the second estimate otherwise: x(n)1 − x1 ε ∨ |||x(n) − x|||∞ ϕ(t − s). s,t s,t ϕ(ζ ) Thus lim sup x(n)1 − x1 ϕ ε. n→∞
Analogously, one shows that lim supn→∞ |||x(n)2 − x2 |||ϕ ε so that the assertion follows by the triangle inequality and by noticing that ε > 0 is arbitrary. 2 From now on we denote for n ∈ N by Υn the canonical embedding of C(Dn , V ) into C([0, 1], V ) which linearly interpolates the values at the breakpoints Dn = (2−n Z) ∩ [0, 1]. With slight misuse of notation we sometimes also apply Υn on functions in C([0, 1], V ). Proof of Theorem 3.2. By Proposition 2.2, one has E[log X|Gn ] = log X(n) and we apply Jensen’s inequality [6, Thm. 5.1.15] to get log X(n) = E log X(n + 1)|Gn E log X(n + 1) |Gn φ φ φ E |||log X|||φ |Gn .
(10)
Hence, (|||log X(n)|||φ )n∈N∪{∞} is a submartingale with E log X(n) φ E|||log X|||φ E|||X|||φ + 2E|||X|||2φ < ∞. Here, we used the general inequality |||log x|||φ |||x|||φ + 2|||x|||2φ and Theorem 2.6. Consequently, the submartingale converges almost surely to a finite value so that supn∈N |||X(n)|||φ is almost surely finite, and the statement is an immediate consequence of Lemma 3.4. 2 ¯ Next, we consider X(−n) = X − X(n) for n ∈ N0 together with its enhanced process ¯ ¯ X(−n) = T−X(n) X = Γ X(n) .
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
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¯ Lemma 3.5. The sequence (log X(k)) k∈−N0 is a Cϕ ( , V)-valued (G¯k )-martingale, where G¯k = ¯ σ (X(k)) (k ∈ −N0 ). Moreover, one has ¯ lim X(k) = 0,
k→−∞
almost surely.
Proof. Set Xk = X(k + 1) − X(k) and G¯k = σ (X¯ k ). Notice that as in the proof of Lemma 3.2 the processes Xk are independent of G¯k and they can be written as a finite sum of onedimensional symmetric excursions so that ¯ ¯ + 1)|G¯k . log X(k) = E log X(k ¯ Hence, (log X(k)) k∈−N0 defines a Cϕ ( , V)-valued martingale. Applying a convergence theorem for reversed martingales (see for instance [6, p. 213]) we obtain almost sure convergence ¯ lim log X(k) = E[log X|G¯−∞ ],
k→−∞
in Cϕ ( , V), where G¯−∞ = k∈−N G¯k . Since G¯−∞ is a tail σ -field it only contains 0-1-events. Thus using the symmetry of log X the limit has to be zero. 2 We are now in a position to prove the support theorem. ¯ Proof of Theorem 3.1. Let X(n) and X(n) be as above. We use the standard notation for their enhanced processes and abridge Ωϕ = Ωϕ (V ). As a consequence of the standard support theorem and Lemma A.3 one has rangeΩϕ X(n) = rangeΩϕ S ◦ Υn (X) = S ◦ Υn (H )Ωϕ ⊂ S(H )Ωϕ . By Theorem 3.2, the family (X(n))n∈N converges almost surely in Ωϕ (V) to X so that rangeΩϕ (X) ⊂ S(H )Ωϕ . For the converse statement fix f ∈ H and let f = S(f ). Due to the continuity of Tf and T−f (see Lemma A.2) one has ∀ε > 0: P X ∈ BΩϕ (f, ε) > 0 if and only if ∀ε > 0:
P T−f X ∈ BΩϕ (0, ε) > 0.
Due to the Cameron Martin Theorem the latter statement is equivalent to ∀ε > 0:
P X ∈ BΩϕ (0, ε) > 0.
¯ On the other hand, we have X = TX(n) (X(n)), and by Corollary 2.9 there exists a constant c = c(ϕ) such that for any ε > 0:
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¯ ˙ P( X ϕ 2cε) P X(n)
ϕ ε, X(n)
L2 ([0,1],V ) ε ¯ ˙ P X(n)
ϕ ε P X(n)
L2 ([0,1],V ) ε . →1 (n→∞)
>0
¯ In the last step we have used the independence of X(n) and X(n).
2
4. Large deviations Let again X denote a V -valued Wiener process satisfying assumption (E) with enhanced process X. For ε > 0 let X ε = (Xtε )t∈[0,1] = (εXt )t∈[0,1] and recall that the family {X ε : ε > 0} satisfies a large deviation principle in C([0, 1], V ) with good rate function J (h) =
1
2 2 h H
if h ∈ H, else.
∞
The aim of this section is to prove: Theorem 4.1. The family {Xε : ε > 0} given by Xε = (δε (Xt ))t∈[0,1] = Γ (X ε ) satisfies a LDP in GΩϕ (V) with good rate function J (h) =
1
2 2 h H
∞
if ∃h ∈ H with h = S(h), else.
Similarly as in [17] and [10], the proof uses the concept of exponentially good approximations. It mainly relies on the isoperimetric inequality and the following estimate. Lemma 4.2. Let n ∈ N, and denote by h : [0, 1] → V an absolutely continuous function with h˙ ∈ L2 ([0, 1], V ). Set f = Υn (h) and g = h − f . There exists a universal constant C such that for x, y ∈ Ωϕ (V) and κ = x ϕ + y ϕ , one has Th (x) − Tf (y) 1 + 2 β x ϕ + y ϕ + 4βn h
˙ L2 ([0,1],V ) , ϕ βn where β = supδ∈(0,1]
√ δ ϕ(δ)
and βn = supδ∈(0,1∧21−n/2 ]
√ δ ϕ(δ)
→ 0 as n → ∞.
Proof. We denote x = x1 , y = y1 , f = Υn (h) and g = h − f . By Jensen’s inequality one ˙ L2 ([0,1],V ) and thus the triangle inequality gives that g
˙ L2 ([0,1],V ) has f˙ L2 ([0,1],V ) h
˙ L2 ([0,1],V ) . Moreover, 2 h
t Th (x)s,t − Tf (y)s,t = xs,t + S(f + g)s,t +
t (f + g)s,u ⊗ dxu +
s
t − ys,t − S(f )s,t −
t fs,u ⊗ dyu −
s
xs,u ⊗ d(f + g)u s
ys,t ⊗ dfu s
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
t = xs,t − ys,t + gs,t +
t hs,u ⊗ dgu +
s
gs,u ⊗ dfu s
t
t
+
fs,u ⊗ d(x − y)u +
gs,u ⊗ dxu
s
s
t
t
+
(x − y)s,u ⊗ dfu + s
xs,u ⊗ dgu .
(11)
s
We need to control the norms of each single term above. We start with | Clearly, t hs,u ⊗ dgu
V ⊗2
s
2927
t s
hs,u ⊗ dgu |V ⊗2 .
˙ 22 hs,· L∞ ([s,t],V ) g
˙ L1 ([s,t],V ) 2(t − s) h
. L ([s,t],V )
For t − s 2−n one can refine this estimate as follows. Let s t0 · · · tN t with {t0 , . . . , tN } = Dn ∩ [s, t], and observe that t hs,u ⊗ dgu s
V ⊗2
t0 hs,u ⊗ dgu s
+ V ⊗2
t + hs,u ⊗ dgu tN
ti+1 hs,u ⊗ dgu
N −1 i=0
ti
V ⊗2
. V ⊗2
˙ 22 Since t0 − s 2−n the first term is bounded by 2−n+1 h
. For v ∈ V one has L ([s,t0 ],V ) dgu = 0 so that ti+1 hs,u ⊗ dgu ti
V ⊗2
ti+1 = hti ,u ⊗ dgu ti
V ⊗2
˙ 22 2−n+1 h
L ([t ,t
i i+1 ],V )
.
Moreover, the remaining term is bounded by t hs,u ⊗ dgu tN
V ⊗2
˙ 22 hs,· L∞ ([s,t],V ) g
˙ L1 ([tN ,t],E) 2−n/2+1 h
. L ([0,1],V )
Combining the above estimates yields t hs,u ⊗ dgu s
V ⊗2
˙ 22 ˙ 22 2 2−n + 2−n/2 h
2−n/2+2 h
L ([0,1],V ) L ([0,1],V )
ti+1 ti
v⊗
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so that in general t hs,u ⊗ dgu s
V ⊗2
˙ 22 2 (t − s) ∧ 21−n/2 h
. L ([0,1],V )
Similarly, one finds the same estimate for
t s
gs,u ⊗ dfu and
! ˙ L2 ([0,1],V ) . |gs,t |V 2 (t − s) ∧ 21−n h
We proceed with the next term in (11): t fs,u ⊗ d(x − y)u s
√ ˙ L2 ([0,1],V ) ϕ(t − s) t − s x − y ϕ h
V ⊗2
√ ˙ L2 ([0,1],V ) , 2κϕ(t − s) t − s h
where κ := x ϕ + y ϕ . Analogously one √ finds that also the remaining three terms from (11) ˙ L2 ([0,1],V ) . We now combine the above estihave norm smaller or equal to 2κϕ(t − s) t − s h
mates: Th (x)s,t − Tf (y)s,t ! ˙ L2 ([0,1],V ) κϕ(t − s) + 2 (t − s) ∧ 21−n h
√ ˙ 22 ˙ L2 ([0,1],V ) . + 2 (t − s) ∧ 21−n/2 h
+ 2κϕ(t − s) t − s h
L ([0,1],V ) Next, we use that
!
(t − s) ∧ 21−n/2 βn ϕ(t − s) to conclude that
Th (x)s,t − Tf (y)s,t ˙ L2 ([0,1],V ) + 2 βn2 h
˙ 22 ˙ L2 ([0,1],V ) ϕ(t − s) κ + 2βn h
+ 2κβ h
L ([0,1],V ) β ˙ L2 ([0,1],V ) ϕ(t − s). κ + 4βn h
2 1+2 βn Proof of Theorem 4.1. We will use the concept of exponentially good approximations. Recall that by Lemma A.3, the map S ◦ Υn : C([0, 1], E) → GΩϕ (E) is continuous. Therefore, the processes Xε (n) = S(X ε (n)) (ε > 0) satisfy a large deviation principle with good rate function Jn (h) =
1
2 2 h H
∞
if ∃h ∈ H : h = S(h) and h = Υn (h), else.
It remains to show that the approximation is exponentially good (see [4, Thm. 4.2.23]), in the sense that for every δ > 0,
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
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lim lim sup ε 2 log P dϕ Xε , Xε (n) > δ = −∞
(12)
n→∞
ε↓0
and that for every α > 0,
lim sup dϕ S(x), S Υn (x) : x ∈ H, x H α = 0.
n→∞
(13)
We start with verifying (12). Recall that we can fix κ > 0 such that P(X ∈ A)
1 2
for # " A = x ∈ C [0, 1], V : x0 = 0, Γ (x) exists, Γ (x)ϕ + supΥn (x)ϕ κ . n∈N
With the isoperimetric inequality we conclude that P(X ∈ Ar ) Φ(r) for the sets Ar = A + BH (0, r)
(r 0).
Here Φ denotes again the standard normal distribution function. By Lemma 4.2 it follows that for z = x + h ∈ Ar (with x ∈ A and h ∈ BH (0, r)): z − z(n) = Th (x) − Tf x(n) C βn σ h H + κ(1 + β/βn ) , ϕ ϕ where β and βn → 0 are as in the lemma and σ is the norm of the canonical embedding of H1 into V . Hence, we can find a standard normal random variable N (on a possibly larger probability space) such that for any n ∈ N: X − X(n) C βn σ N+ + κ(1 + β/βn ) , ϕ where N+ = N ∨ 0. Now choose αn = 13 (C 2 σ 2 βn2 )−1 . Then limn→∞ αn = ∞ and 2 2 Cn := Eeαn X−X(n) ϕ E exp αn C 2 βn σ N+ + κ(1 + β/βn )
1 2 = E exp N+ + O(N+ ) < ∞. 3 By Chebychev’s inequality we get for η 0: 2 P X − X(n)ϕ η Cn e−αn η and hence lim sup η→∞
1 log P X − X(n)ϕ η −αn → −∞ as n → ∞. 2 η
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Now assertion (12) is a consequence of ε X − Xε (n) = δε (X) − δε X(n) = δε X − X(n) = ε X − X(n) . ϕ ϕ ϕ ϕ Finally note that (13) is an immediate consequence of Lemma 4.2. Indeed, for h ∈ H and f = Υn (h) one gets S(h) − S(f ) = Th (0) − Tf (0) 3βn h
˙ L2 ([0,1],E) 3σβn h
˙ H ϕ ϕ with limn→∞ βn = 0.
2
5. A sufficient criterion for exactness In this section we introduce a new sufficient criterion for the exactness of a Gaussian random vector attaining values in a Hölder–Zygmund space. For n ∈ N and for an n-times continuously differentiable function f : Rd → Rd , we set
f C n =
sup ∂ α f (x) ,
|α|n x∈R
d
where the sum is taken over all multiindices α with entries in {1, . . . , d} of length smaller or equal to n. Moreover, for γ = n + η with n ∈ N0 and η ∈ (0, 1], and an n-times continuously differentiable function f : Rd → Rd , we consider the Hölder–Zygmund norm of order γ
f C γ = f C n +
sup
|α|=n x=y
|∂ α f (x) − ∂ α f (y)| |x − y|η
(14)
γ
and we denote by C0 (D, Rd ) (D ⊂ Rd open) the set of n-times continuously differentiable functions satisfying f |D c ≡ 0
and f C γ < ∞.
It is endowed with the norm · C γ . Theorem 5.1. Let 0 γ < γ¯ and let D, D ⊂ Rd denote bounded open sets with D¯ ⊂ D . Every γ¯ centered Gaussian measure μ on C0 (D, Rd ) is exact and Bochner measurable when viewed as γ Gaussian measure on C0 (D , Rd ). The proof is based on the concept of finite dimensional approximation introduced in [17]. For a Banach space V (not necessarily separable) and a V -valued random vector Y , we denote the linear average Kolmogorov widths of Y by
n (Y ) = Vn (Y ) = inf E Y − Tn (Y ) V : Tn : V → V linear, rk(Tn ) n
(n ∈ N).
Lemma 5.2. Let Y be a V -valued Gaussian random vector and suppose that n (Y ) n−ε for some ε > 0. Then Y is exact and Bochner measurable in V .
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
2931
Proof. Since n (Y ) decays to zero Y is Bochner measurable. Without loss of generality we assume that V is infinite dimensional. Let G1 be a μ-distributed r.v. For fixed n ∈ N there exists a bounded operator Tn : V → V with n-dimensional range and E G1 − Tn (G1 ) 2n (Y ) =: ε(n). Set F1 := Tn (G1 ) and observe that there are n independent standard normals ξ11 , . . . , ξn1 and n vectors e1 , . . . , en ∈ V such that F1 =
n
ξi1 ei .
i=1
Moreover, set H1 = G1 − F1 . Then for each i = 1, . . . , n one has !
2/π|ei |V = E ξ 1 ei E|F1 | E|G1 | + E|H1 | C,
where C := C(Y ) := 21 (Y ) + E|Y | does only depend on the distribution of Y . ˜ l , F˜l , H˜ l , (ξ˜ l )i=1,...,n )l∈N denote independent Let now (Gl , Fl , Hl , (ξil )i=1,...,n )l2 and (G i 1 copies of (G1 , F1 , H1 , (ξi )i=1,...,n ). Then N
˜l = Gl ⊗ G
l=1
N
Fl ⊗ F˜l +
l=1
N
Fl ⊗ H˜ l +
l=1
N
Hl ⊗ F˜l +
l=1
N
Hl ⊗ H˜ l
(15)
l=1
and N E Fl ⊗ F˜l l=1
Using that E|
N
l ˜l l=1 ξi ξj |
V ⊗2
N n n E ξil ξ˜jl |ei ⊗ ej |V ⊗2 . i=1 j =1
l=1
√ N we arrive at
N E Fl ⊗ F˜l l=1
√ √ π d 2 N max |ei |2V C 2 d 2 N . i=1,...,n 2
(16)
NE|F1 |E|H˜ 1 | N ε(n)E|F1 | CN ε(n).
(17)
V ⊗2
On the other hand, N E Fl ⊗ H˜ l l=1
V ⊗2
The same estimate holds for E|
N
l=1 Hl
N ˜ E Hl ⊗ Hl l=1
⊗ F˜l |V ⊗2 . Finally, the last term in (15) is bounded by 2 N E|H1 | N ε(n)2 . V ⊗2
(18)
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When choosing n = n(N ) = N 1/(4+2ε) and letting n tend to infinity one obtains with (16), (17) and (18) that N ˜ l N (4+ε)/(4+2ε) , E Gl ⊗ G
as N → ∞,
l=1
which implies exactness.
2
In the forthcoming proof of Theorem 5.1, we use a result by Pisier [25] that provides an estimate for the average Kolmogorov width against entropy numbers of generating operators. For two Banach spaces E and V , and a compact operator ρ : E → V we define the n-th entropy number as
n−1 2$ BV (bj , ε) for some b1 , . . . , b2n−1 ∈ V . en (ρ) = inf ε > 0: u BE (0, 1) ⊂ j =1
Proof of Theorem 5.1. We denote by ρ the canonical embedding of the reproducing kernel γ¯ Hilbert space of Y into C0 (D, Rd ). By Pajor and Tomczak-Jaegermann [24], one has en (ρ) n−1/2 . The asymptotic behavior of the entropy numbers for general Besov embeddings were studied by γ¯ Edmunds and Triebel [7,8]. In particular, one has for the canonical embedding : C0 (D, Rd ) → γ d C0 (D , R ) that en () n−
γ¯ −γ d
.
Combining the above estimates with the general estimate ek+l−1 ( ◦ ρ) ek ()el (ρ) (k, l ∈ N0 ) gives 1
en ( ◦ ρ) n− 2 −
γ¯ −γ d
.
Now note that ◦ ρ generates the Gaussian random element Y on C0 (D , Rd ). In order to control (Y ) we use a result of Pisier (see [25, Thm. 9.1, p. 141]) combined with the duality of metric entropy found in [1]: γ
n (Y ) C1
k −1/2 (log k)ek ( ◦ ρ)
kC2 n
for two universal constants C1 , C2 > 0. Combining this estimate with the above result on the entropy numbers gives n (Y ) n−
γ¯ −γ d
log n.
2
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
2933
6. Consequences of Lyons’ universal limit theorem In this section, we use Lyons’ universal limit theorem together with our findings to derive Theorem 1.7. Let V and W denote two Banach spaces and let f : W → L(V , W ) a Lip(γ )function for a γ > 2 in the sense of [23, Def. 1.21]. As in the introduction, Ig denotes the solution operator for controls x ∈ BV(V ) to the differential equation dyt = f (yt ) d[x + g]t ,
y0 = ξ,
where g ∈ C([0, 1], V ) is an absolutely continuous function with g˙ ∈ L2 ([0, 1], V ). Next, fix a real p ∈ (2, 3 ∧ γ ) and an increasing function ϕ¯ : (0, 1] → (0, ∞) that is dominated log δ by ϕ and satisfies limδ↓0 −δϕ(δ) = 0 such that ϕ¯ p is convex. By Lemma A.4, there always exists ¯ an appropriate ϕ¯ and since Cϕ¯ ([0, 1], W ) is continuously embedded into Cϕ ([0, 1], W ), it suffices to prove Theorem 1.7 in Cϕ¯ ([0, 1], W ). By choice of ϕ¯ the space GΩϕ¯ (V) is continuously embedded into GΩp (V), where GΩp (V) denotes the space of all geometric rough paths induced by the p-variation norm. Hence, the universal limit theorem (see for instance [23, Thm. 5.3]) implies that the rough path differential equation dyt = f (yt ) dxt ,
y0 = idD
induces a continuous solution operator I : GΩϕ¯ (V) → GΩϕ¯ (W) (Itô map) and the following diagram commutes for any piecewise linear V -valued path x:
x
Tg
xg
I0
y ξ +πW (·)
S
x
I
y
Assuming assumption (E), the processes {X(n): n ∈ N} converge in GΩϕ¯ (V ) to X. By the continuity of Tg : GΩϕ¯ (V ) → GΩϕ¯ (V ) (Lemma A.2), I : GΩϕ¯ (V ) → GΩϕ¯ (W ) (Lyons’ universal limit theorem), and ξ + πW (·) : GΩϕ¯ (W ) → Cϕ¯ ([0, 1], W ), we conclude that {Y (n): n ∈ N} converges to ξ + πW ◦ I0 ◦ Tg (X) which is statement (I) of the theorem. Assertions (II)–(IV) are now immediate consequences of Theorems 1.3, 1.1, and 1.2, respectively. Appendix A. Preliminary results For u ∈ V we set |||u||| = inf
n
ui ,
(19)
i=1
where the infimum is taken over all representations u = arbitrary.
%n
i=1 ui
with n ∈ N and u1 , . . . , un ∈ V
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Lemma A.1. • ||| · ||| satisfies the triangle inequality with respect to ∗. • For t ∈ R and u ∈ V, one has |||δt u||| = |t||||u|||. • Moreover, · and ||| · ||| are equivalent: |||u||| u 2|||x|||. Proof. The proof of the first % two statements is straight forward and we only present the proof of the third statement. Let u = ni=1 ui . Then u1 = ni=1 u1i and |u1 |V ni=1 |u1i |V . Moreover, since u2 = ni=1 u2i + i<j u1i ⊗ u1j we get u 2
V ⊗V
& ' ' n 2 1 1 ( = ui + ui ⊗ uj i=1
& ' n ' ( u 2
i<j
)
i V ⊗V
i=1
+
V ⊗V
n 1 u i V i=1
n 1 u + u2 =
ui
i V i V ⊗V
n i=1
so that u 2|||u|||.
*2
i=1
2
Lemma A.2. Let f ∈ C([0, 1], V ) be an absolutely continuous with f˙ ∈ L2 ([0, 1], V ) and let √ > 0. Then the shift operator ϕ : (0, 1] → (0, ∞) be an increasing function with infδ∈(0,1] ϕ(δ) δ
Tf : Ωϕ (V) → Ωϕ (V) is continuous. Proof. For x, y ∈ Ωϕ (V), one has t Tf (x)s,t − Tf (y)s,t = xs,t − ys,t +
t fs,u ⊗ d(x − y)u +
s
(x − y)s,u ⊗ dfu , s
where x = πV (x) and y = πV (y). The process f is of bounded variation, and one has t fs,u ⊗ d(x − y)u s
f˙ L1 ([0,1],V ) V ⊗V
sup
|xu,v − yu,v |V .
suvt
√ Next, recall that f˙ L1 ([s,t],V ) t − s f˙ L2 ([0,1],V ) and that, by assumption, there exists a √ constant c = c(ϕ) with t − s c ϕ(t − s). Consequently,
S. Dereich / Journal of Functional Analysis 258 (2010) 2910–2936
t fs,u ⊗ d(x − y)u s
2935
c f˙ L2 ([0,1],V ) x − y ϕ ϕ(t − s)2 , V ⊗V
and ) t * fs,u ⊗ d(x − y)u
c f˙ L2 ([0,1],V ) x − y ϕ .
(s,t)∈ ϕ
s
t Analogously one finds the same estimate for ( s (x − y)s,u ⊗ dfu )(s,t)∈ ϕ so that Tf (x) − Tf (y) x − y ϕ + 2 c f˙ 2 L ([0,1],V ) x − y ϕ . ϕ
2
Lemma A.3. For n ∈ N, the maps S ◦ Υn : C(Dn , V ) → GΩϕ (V)
and
log ◦S ◦ Υn : C(Dn , V ) → Cϕ ( , V)
are continuous. Moreover, for x, y ∈ C([0, 1], V ), x = S ◦ Υn (x), and y = S ◦ Υn (y), one has |log xs,t − log ys,t |V 2n+1 x − y ∞ 1 + 2 x ∞ + 2 y ∞ (t − s),
for (s, t) ∈ .
The proof is straight-forward and therefore omitted. Lemma A.4. For any p > 2 and ϕ : (0, 1] → (0, ∞) increasing with exists a function ϕ¯ : (0, 1] → (0, ∞) such that ϕ¯ p is convex and ! Proof. Set φ(δ) =
√
−δ log δ ϕ(δ) ¯ ϕ(δ),
√ −δ log δ ϕ(δ) there
for δ ∈ (0, 1].
−δ log δ and define φm,u : (0, 1] → [0, ∞) for m ∈ N and u > 0 via
p φm,u (δ) =
mφ p (δ)
if δ u,
mφ p (u) + (mφ p ) (u)(δ − u)
otherwise.
As one easily verifies by taking derivatives, φ p is convex on an appropriate set (0, ε). Hence, p φm,u is convex provided that u ε. Moreover, one can check that, for every m ∈ N, there exists u(m) ∈ (0, ε) such that φm,u(m) ϕ. Consequently, taking ϕ¯ = supm∈N φm,u(m) finishes the proof. 2 References [1] S. Artstein, V. Milman, S.J. Szarek, Duality of metric entropy, Ann. of Math. (2) 159 (3) (2004) 1313–1328. [2] V.I. Bogachev, Gaussian Measures, Math. Surveys Monogr., vol. 62, American Mathematical Society, Providence, RI, 1998. [3] Z. Brze´zniak, A. Carroll, Approximations of the Wong–Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces, in: Séminaire de Probabilités XXXVII, in: Lecture Notes in Math., vol. 1832, Springer, Berlin, 2003, pp. 251–289.
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[4] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, second edition, Appl. Math. (New York), vol. 38, Springer, New York, 1998. [5] S. Dereich, G. Dimitroff, Large deviations and a support theorem for Brownian flows, in preparation. [6] G.A. Edgar, L. Sucheston, Stopping Times and Directed Processes, Encyclopedia Math. Appl., vol. 47, Cambridge University Press, Cambridge, 1992. [7] D.E. Edmunds, H. Triebel, Entropy numbers and approximation numbers in function spaces, Proc. Lond. Math. Soc. (3) 58 (1) (1989) 137–152. [8] D.E. Edmunds, H. Triebel, Entropy numbers and approximation numbers in function spaces, II, Proc. Lond. Math. Soc. (3) 64 (1) (1992) 153–169. [9] P. Friz, H. Oberhauser, Isoperimetry and rough path regularity, preprint. [10] P. Friz, N. Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 703–724. [11] P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, 2010. [12] P.K. Friz, Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm, in: Probability and Partial Differential Equations in Modern Applied Mathematics, in: IMA Vol. Math. Appl., vol. 140, Springer, New York, 2005, pp. 117–135. [13] A.M. Garsia, E. Rodemich, H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/1971) 565–578. [14] Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. Lond. Math. Soc. (2) 73 (3) (2006) 797–816. [15] Y. Inahama, H. Kawabi, Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths, J. Funct. Anal. 243 (1) (2007) 270–322. [16] Y. Inahama, H. Kawabi, On the Laplace-type asymptotics and the stochastic Taylor expansion for Itô functionals of Brownian rough paths, in: Proceedings of RIMS Workshop on Stochastic Analysis and Applications, in: RIMS Kôkyûroku Bessatsu, vol. B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 139–152. [17] M. Ledoux, T. Lyons, Z. Qian, Lévy area of Wiener processes in Banach spaces, Ann. Probab. 30 (2) (2002). [18] M. Ledoux, Z. Qian, T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl. 102 (2) (2002) 265–283. [19] M.A. Lifshits, Gaussian Random Functions, Math. Appl. (Dordrecht), vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. [20] T. Lyons, Differential equations driven by rough signals, I: An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451–464. [21] T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr., Oxford University Press, Oxford, 2002. [22] T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310. [23] T.L. Lyons, M.J. Caruana, T. Lévy, Differential equations driven by rough paths, in: Ecole d’Eté de Probabilités de Saint-Flour XXXIV – 2004, in: Lecture Notes in Math., vol. 1908, Springer, Berlin, 2007. [24] A. Pajor, N. Tomczak-Jaegermann, Remarques sur les nombres d’entropie d’un opérateur et de son transposé, C. R. Acad. Sci. Paris Sér. I Math. 301 (15) (1985) 743–746. [25] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge University Press, Cambridge, 1989.
Journal of Functional Analysis 258 (2010) 2937–2960 www.elsevier.com/locate/jfa
Quantum isometry groups of the Podles spheres Jyotishman Bhowmick 1 , Debashish Goswami ∗,2 Stat-Math Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700 108, India Received 18 July 2009; accepted 5 February 2010 Available online 13 February 2010 Communicated by Alain Connes
Abstract For μ ∈ (0, 1), c 0, we identify the quantum group SOμ (3) as the universal object in the category of compact quantum groups acting by ‘orientation and volume preserving isometries’ in the sense of 2 constructed Bhowmick and Goswami (2009) [4] on the natural spectral triple on the Podles sphere Sμ,c by Dabrowski, D’Andrea, Landi and Wagner (2007) in [9]. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantum isometry groups; Podles spheres; Spectral triples
1. Introduction In a series of articles initiated by [11] and followed by [3,4], we have formulated and studied a quantum group analogue of the group of Riemannian isometries of a classical or noncommutative manifold. This was motivated by previous work of a number of mathematicians including Wang, Banica, Bichon and others (see, e.g. [20,21,1,2,5,22] and the references therein), who have defined quantum automorphism and quantum isometry groups of finite spaces and finite dimensional algebras. Our theory of quantum isometry groups can be viewed as a natural generalization of such quantum automorphism or isometry groups of ‘finite’ or ‘discrete’ structures to the continuous or smooth set-up. Clearly, such a generalization is crucial to study the quantum * Corresponding author.
E-mail addresses:
[email protected] (J. Bhowmick),
[email protected] (D. Goswami). 1 The support from National Board of Higher Mathematics, India, is gratefully acknowledged. 2 Partially supported by a project on ‘Noncommutative Geometry and Quantum Groups’ funded by Indian National
Science Academy. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.009
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symmetries in noncommutative geometry a la Connes [6], and in particular, for a good understanding of quantum group equivariant spectral triples. The group of Riemannian isometries of a compact Riemannian manifold M can be viewed as the universal object in the category of all compact metrizable groups acting on M, with smooth and isometric action. Moreover, assume that the manifold has a spin structure (hence in particular orientable, so we can fix a choice of orientation) and D denotes the conventional Dirac operator acting as an unbounded self-adjoint operator on the Hilbert space H of square integrable spinors. Then, it can be proved that the action of a compact group G on the manifold lifts as a unitary ˜ which is topologically a 2-cover of G, see [7] and [8] representation (possibly of some group G for more details) on the Hilbert space H which commutes with D if and only if the action on the manifold is an orientation preserving isometric action. Therefore, to define the quantum analogue of the group of orientation-preserving Riemannian isometry group of a possibly noncommutative manifold given by a spectral triple (A∞ , H, D), it is reasonable to consider a category Q (D) of compact quantum groups having unitary (co-)representation, say U , on H, which commutes with D, and the action on B(H) obtained via conjugation by U maps A∞ into its weak closure. A universal object in this category, if it exists, should define the ‘quantum group of orientation preserving Riemannian isometries’ of the underlying spectral triple. Indeed (see [4]), if we consider a classical spectral triple, the subcategory of the category Q (D) consisting of groups has the classical group of orientation preserving isometries as the universal object, which justifies our definition of the quantum analogue. Unfortunately, if we consider quantum group actions, even in the finite dimensional (but with noncommutative A) situation the category Q (D) may often fail to have a universal object. It turns out, however, that if we fix any suitable faithful functional τR on B(H) (to be interpreted as the choice of a ‘volume form’) then there exists a universal object in the subcategory QR (D) of Q (D) obtained by restricting the object-class to the quantum group actions which also preserve the given functional. The subtle point to note here is that unlike the classical group actions on B(H) which always preserve the usual trace, a quantum group action may not do so. In fact, it was proved by one of the authors in [10] that given an object (Q, U ) of Q (D) (where Q is the compact quantum group and U denotes its unitary co-representation on H), we can find a suitable functional τR (which typically differs from the usual trace of B(H) and can have a nontrivial modularity) which is preserved by the action of Q. This makes it quite natural to work in the setting of twisted spectral data (as defined in [10]). It may also be mentioned that in [4] we have actually worked in slightly bigger category QR (D) of the so-called ‘quantum family of orientation and volume preserving isometries’ and deduced that the universal object in QR (D) exists and coincides with that of QR (D). It is very important to explicitly compute the (orientation and volume preserving) quantum group of isometries for as many examples as possible. This programme has been successfully carried out for a number of spectral triples, including classical spheres and tori as well as their Rieffel deformations. The aim of the present article is to identify SOμ (3) as the quantum group of 2 , orientation and volume preserving isometries for the spectral triples on the Podles spheres Sμ,c constructed by Dabrowski et al. in [9]. Let us mention here that although the quantum groups SOμ (3) are ‘deformations’ of the classical SO(3), these are not Rieffel deformations and so the results and techniques of [4] do not apply. Our characterization of SOμ (3) as the quantum isometry group of a noncommutative Riemannian manifold generalizes the classical description of the group SO(3) as the group of orientation preserving isometries of the usual Riemannian structure on the 2-sphere. It may be mentioned here that in a very recent article [19], P.M. Soltan has characterized SOμ (3) as the universal compact quantum group acting on the finite dimensional C ∗ -algebra M2 (C) such that the action
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preserves a functional ωμ defined in [19]. In the classical case, we have three equivalent descriptions of SO(3): (a) as a quotient of SU(2), (b) as the group of (orientation preserving) isometries of S 2 , and (c) as the automorphism group of M2 . In the quantum case the definition of SOμ (3) is an analogue of (a), so the characterization of SOμ (3) obtained in this paper as the quantum isometry group, together with Soltan’s characterization, completes the generalization of all three classical descriptions of SO(3). 2. Notations and preliminaries 2.1. Basics of the theory of compact quantum groups We begin by recalling the definition of compact quantum groups and their actions from [24,23]. A compact quantum group (to be abbreviated as CQG from now on) is given by a pair (S, ), where S is a unital separable C ∗ algebra equipped with a unital C ∗ -homomorphism : S → S ⊗ S (where ⊗ denotes the injective tensor product) satisfying (ai) ( ⊗ id) ◦ = (id ⊗ ) ◦ (co-associativity), and (aii) the linear spans of (S)(S ⊗ 1) and (S)(1 ⊗ S) are norm-dense in S ⊗ S. It is well known (see [24,23]) that there is a canonical dense ∗-subalgebra S0 of S, consisting of the matrix coefficients of the finite dimensional unitary (co)-representations (to be defined below) of S, and maps : S0 → C (co-unit) and κ : S0 → S0 (antipode) defined on S0 which make S0 a Hopf ∗-algebra. A CQG (S, ) is said to (co)-act on a unital C ∗ algebra B, if there is a unital C ∗ homomorphism (called an action) α : B → B ⊗ S satisfying the following: (bi) (α ⊗ id) ◦ α = (id ⊗ ) ◦ α, and (bii) the linear span of α(B)(1 ⊗ S) is norm-dense in B ⊗ S. For a Hilbert B-module E, (where B is a C ∗ algebra) we shall denote the set of adjointable B-linear maps on E by L(E). The norm-closure of the linear span of the finite-rank B-linear maps on E, to be called the set of compact operators on E, will be denoted by K(E). We note that L(E) = M(K(E)), where M(C) denotes the multiplier algebra of a C ∗ -algebra C. We shall also need the ‘leg-numbering’ notation: for an operator X in B(H1 ⊗ H2 ), X(12) and X(13) will denote the operators X ⊗ IH2 in B(H1 ⊗ H2 ⊗ H2 ), and Σ23 X12 Σ23 , respectively, where Σ23 is the unitary on H1 ⊗ H2 ⊗ H2 which flips the two copies of H2 . A unitary (co)-representation of a CQG (S, ) on a Hilbert space H is given by a unitary element U of M(K(H) ⊗ S) ≡ L(H ⊗ S) satisfying (id ⊗ )(U ) = U(12) U(13) . Given a unitary representation U we shall denote by αU the ∗-homomorphism αU (X) = U (X ⊗ 1S )U ∗ for X belonging to B(H). We shall sometimes identify U with the isometric map from the Hilbert space H to the Hilbert module H ⊗ S which sends a vector ξ of H to U (ξ ⊗ 1), and may even denote U (ξ ⊗ 1) by U ξ by a slight abuse of notation. We say that a (possibly unbounded) operator T on H commutes with U if T ⊗ I (with the natural domain) commutes with U . Such an operator will also be called U -equivariant or S-equivariant if U is understood.
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2.2. The quantum group of orientation preserving Riemannian isometries We briefly recall the definition of the quantum group of orientation preserving Riemannian isometries for a spectral triple (of compact type) (A∞ , H, D) as in [4]. We consider the category Q (A∞ , H, D) ≡ Q (D) whose objects (to be called orientation preserving isometries) are the triplets (S, , U ), where (S, ) is a CQG with a unitary representation U in H, satisfying the following: (i) U commutes with D, (ii) for every state ω on S, (id ⊗ ω) ◦ αU (a) belongs to (A∞ ) for all a in A∞ . The category Q (D) may not have a universal object in general, as pointed out in [4]. In case + (D), with the corresponding representathere is a universal object, we shall denote it by QISO + (D) generated tion U , say, and we denote by QISO+ (D) the Woronowicz subalgebra of QISO by the elements of the form ξ ⊗ 1, αU (a)(η ⊗ 1) , where ξ, η belong to H, a belongs to A∞ and + (D)-valued inner product of H ⊗ QISO + (D). The quantum group QISO+ (D)
·,· is the QISO will be called the quantum group of orientation-preserving Riemannian isometries of the spectral triple (A∞ , H, D). Although the category Q (D) may fail to have a universal object, we can always get a universal object in suitable subcategories which will be described now. Suppose that we are given an invertible positive (possibly unbounded) operator R on H which commutes with D. Then we consider the full subcategory QR (D) of Q (D) by restricting the object class to those (S, , U ) for which αU satisfies (τR ⊗ id)(αU (X)) = τR (X)1 for all X in the ∗-subalgebra generated by operators of the form |ξ η|, where ξ, η are eigenvectors of the operator D which by assumption has discrete spectrum, and τR (X) = Tr(RX) = η, Rξ for X = |ξ η|. We shall call the objects of QR (D) orientation and (R-twisted) volume preserving isometries. It is clear (see Remark 2.9 in [4]) that 2 when Re−tD is trace-class for some t > 0, the above condition is equivalent to the condition that 2 αU preserves the bounded normal functional Tr(· Re−tD ) on the whole of B(H). It is shown in + (D), and [4] that the category Q (D) always admits a universal object, to be denoted by QISO R
R
the Woronowicz subalgebra generated by { ξ ⊗ 1, αW (a)(η ⊗ 1) : ξ, η ∈ H, a ∈ A∞ } (where + (D) in H) will be denoted by QISO+ (D) and called W is the unitary representation of QISO R R the quantum group of orientation and (R-twisted) volume preserving Riemannian isometries of the spectral triple. 2.3. SU μ (2) and the Podles spheres Fix μ in (0, 1). The C ∗ algebra underlying the CQG SU μ (2) is defined as the universal unital C ∗ algebra generated by α, γ such that α ∗ α + γ ∗ γ = 1, αα ∗ + μ2 γ γ ∗ = 1, γ γ ∗ = γ ∗ γ , μγ α = αγ , μγ ∗ α = αγ ∗ . α −μγ ∗ The CQG structure is given by the following fundamental representation: γ α ∗ . The coproduct is defined by: (α) = α ⊗ α − μγ ∗ ⊗ γ , (γ ) = γ ⊗ α + α ∗ ⊗ γ .
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The Haar state of SU μ (2) will be denoted by h and the corresponding G.N.S. Hilbert space will be denoted by L2 (SU μ (2)). We will call the unital ∗-subalgebra of SU μ (2) (without any norm-closure) generated by α, γ the ‘co-ordinate Hopf ∗-algebra’ of SU μ (2) and denote it by O(SU μ (2)) as in [17]. We now recall the definition of the Podles sphere from [9] (see also the original article [15] by Podles). n −μ−n For c 0, let t in (0, 1] be given by c = t −1 − t. Let [n] ≡ [n]μ = μμ−μ −1 , n ∈ N. 2 is defined to be the universal unital C ∗ algebra generated by elements The Podles sphere Sμ,c x−1 , x0 , x1 satisfying the relations:
x−1 (x0 − t) = μ2 (x0 − t)x−1 , x1 (x0 − t) = μ−2 (x0 − t)x1 , −[2]x−1 x1 + μ2 x0 + t (x0 − t) = [2]2 (1 − t), −[2]x1 x−1 + μ−2 x0 + t (x0 − t) = [2]2 (1 − t). 2 is given by The involution on Sμ,c ∗ = −μ−1 x1 , x−1
x0∗ = x0 .
2 as defined above is the same as χ We note that Sμ,c q,α ,β in [13, p. 124] with q = μ, α = t, 2 −2 2 2 β = t + μ (μ + 1) (1 − t). 2 can be Thus, from the expressions of x−1 , x0 , x1 given in [13, p. 125], it follows that Sμ,c realized as a ∗-subalgebra of SU μ (2) by setting:
x−1 =
μα 2 + ρ(1 + μ2 )αγ − μ2 γ 2
, 1 μ(1 + μ2 ) 2 x0 = −μγ ∗ α + ρ 1 − 1 + μ2 γ ∗ γ − γ α ∗ , x1 =
where ρ 2 = Taking
μ2 γ ∗2
− ρμ(1 + μ2 )α ∗ γ ∗
− μα ∗2
1
(1 + μ2 ) 2
,
(1) (2) (3)
μ2 t 2 . (μ2 +1)2 (1−t)
A=
1 − t −1 x0 , 1 + μ2
− 1 B = μ 1 + μ2 2 t −1 x−1 ,
2 coincides with the original description given in [15], one obtains (see [9]) that the C ∗ algebra Sμ,c ∗ i.e., the universal C algebra generated by elements A and B satisfying the relations:
A∗ = A, B ∗ B = A − A2 + cI,
AB = μ−2 BA, BB ∗ = μ2 A − μ4 A2 + cI.
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2 ) the co-ordinate ∗-algebra of S 2 , i.e. the unital ∗-subalgebra generWe will denote by O(Sμ,c μ,c ated by A, B. We recall from [17] the Hopf ∗-algebra Uμ (su(2)) which is generated by elements F , E, K, K −1 with defining relations:
KK −1 = K −1 K = 1,
KE = μEK, F K = μKF, 2 −1 −1 EF − F E = μ − μ K − K −2
with involution E ∗ = F , K ∗ = K and comultiplication: (E) = E ⊗ K + K −1 ⊗ E,
(F ) = F ⊗ K + K −1 ⊗ F,
(K) = K ⊗ K.
The counit is given by (E) = (F ) = (K − 1) = 0 and antipode S(K) = K −1 , S(E) = −μE, S(F ) = −μ−1 F . There is a dual pairing .,. of Uμ (su(2)) and O(SU μ (2)), for which the nonzero values of the pairing among the generators are given below:
1 K ±1 , α ∗ = K ∓1 , α = μ± 2 ,
E, γ = F, −μγ ∗ = 1.
The left action and right action of Uμ (su(2)) on SU μ (2) are given by: f x = f, x(2) x(1) ,
x f = f, x(1) x(2) ,
x ∈ O SU μ (2) , f ∈ Uμ su(2) ,
where we have used the Sweedler notation (x) = x(1) ⊗ x(2) . The actions satisfy the following: (f x)∗ = S(f )∗ x ∗ , f xy = (f(1) x)(f(2) y),
(x f )∗ = x ∗ S(f )∗ , xy f = (x f(1) )(y f(2) ).
The action on generators is given by: E γ = α∗, E γ ∗ = E α ∗ = 0, E α = −μγ ∗ , F −μγ ∗ = α, F α∗ = γ , F α = F γ = 0, 1 1 1 1 K γ ∗ = μ 2 γ ∗, K γ = μ− 2 γ , K α∗ = μ 2 α∗; K α = μ− 2 α, γ E = α,
α ∗ E = −μγ ∗ ,
α E = γ ∗ E = 0,
α F = γ,
−μγ ∗ F = α ∗ ,
γ F = α ∗ F = 0,
1
α K = μ− 2 α,
1
γ ∗ K = μ− 2 γ ∗ ,
1
γ K = μ2 γ,
1
α∗ K = μ 2 α∗.
2 from [18] which we are going to need. We recall an alternative description of Sμ,c
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Let −1 1 1 Xc = μ 2 μ−1 − μ c− 2 1 − K 2 + EK + μF K,
c > 0,
X0 = 1 − K 2 . One has (Xc ) = 1 ⊗ Xc + Xc ⊗ K 2 . Moreover, we have the following [18, p. 9]: Theorem 2.1. We have, 2 O Sμ,c = x ∈ O SU μ (2) : x Xc = 0 . 2 ) is given by {Ak , Ak B l , Ak B ∗ m , k 0, l, m > 0}. A basis of the vector space O(Sμ,c 2 ) can be written as a finite linear combination of elements of the Thus, any element of O(Sμ,c l k k l k ∗ form A , A B , A B . Let ψ be the densely defined linear map on L2 (SU μ (2)) defined by ψ(x) = x Xc . 2 ⊆ Ker(ψ) where ψ is the closed extenLemma 2.2. The map ψ is closable and we have Sμ,c 2 denotes the Hilbert subspace generated by S 2 in L2 (SU (2)). Moreover, sion of ψ and Sμ,c μ μ,c 2 ) = O(SU (2)) ∩ Ker(ψ) = O(SU (2)) ∩ Ker(ψ). O(Sμ,c μ μ
Proof. From the expression of Xc , it is clear that O(SU μ (2)) ⊆ Dom(ψ ∗ ) implying that ψ is 2 )= closable, hence Ker(ψ) is closed. The lemma now follows from the observation that O(Sμ,c Ker(ψ) ⊆ Ker(ψ). 2 We end this subsection with a discussion on the CQG SOμ (3) as described in [16]. It is the universal unital C ∗ algebra generated by elements M, N , G, C, L satisfying: L∗ L = (I − N) I − μ−2 N ,
LL∗ = I − μ2 N I − μ4 N ,
M ∗M = N − N 2, CC ∗ = μ2 N − μ4 N 2 , CN = μ2 NC,
MM ∗ = μ2 N − μ4 N 2 , LN = μ4 N L,
LG = μ4 GL,
LG∗ = μ4 G∗ L,
M 2 = μ−1 LG,
C∗C = N − N 2,
GN = N G,
LM = μ2 ML,
G∗ G = GG∗ = N 2 ,
MN = μ2 N M,
MG = μ2 GM,
M ∗ L = μ−1 (I − N )C,
CM = MC, N ∗ = N.
This CQG can be identified with a Woronowicz subalgebra of SU μ (2) by taking: N = γ ∗γ ,
M = αγ ,
C = αγ ∗ ,
G = γ 2,
L = α2.
2 , i.e. the action obtained by restricting the coproduct of The canonical action of SU μ (2) on Sμ,c 2 SU μ (2) to the subalgebra Sμ,c , is actually a faithful action of SOμ (3). On the subspace spanned
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by {x−1 , x0 , x1 } this action is given by the following SOμ (3)-valued 3 × 3-matrix: ⎛
α2 1 Z := ⎝ (1 + μ−2 ) 2 αγ γ2
1
−μ(1 + μ−2 ) 2 αγ ∗ I − μ(μ + μ−1 )γ ∗ γ 1 (1 + μ−2 ) 2 γ α ∗
⎞ μ2 γ ∗2 1 −μ(1 + μ−2 ) 2 γ ∗ α ∗ ⎠ . α ∗2
3. Spectral triples on the Podles spheres and their quantum isometry groups 3.1. Description of the spectral triples 2 discussed in [9] (see also [17] for the case c = 0). We now recall the spectral triples on Sμ,c 1
Let s = −c− 2 λ− , λ± = For all j in 12 N,
1 2
1
± (c + 14 ) 2 .
uj = α ∗ − sγ ∗ α ∗ − μ−1 sγ ∗ . . . α ∗ − μ−2j +1 sγ ∗ , wj = (α − μsγ ) α − μ2 sγ . . . α − μ2j sγ , u−j = E 2j wj , u0 = w0 = 1, 1 1 1 y1 = 1 + μ−2 2 c 2 μ2 γ ∗2 − μγ ∗ α ∗ − μc 2 α ∗2 , −1 l Nkj = F l−k y1 l−|j | uj . l−|j |
l = N l F l−k (y uj ), l ∈ 12 N0 , j, k = −l, −l + 1, . . . , l. Define vk,j k,j 1 l : l = |N |, Let MN be the Hilbert subspace of L2 (SU μ (2)) with the orthonormal basis {vm,N |N| + 1, . . . , m = −l, . . . , l}. Set
H = M− 1 ⊕ M 1 , 2
2
2 on H by and define a representation π of Sμ,c l−1 l+1 l l = αi− (l, m; N )vm+i,N + αi0 (l, m; N )vm+i,N + αi+ (l, m; N )vm+i,N , π(xi )vm,N
where αi− , αi0 , αi+ are as defined in [9]. 2 ) with S 2 . We will often identify π(Sμ,c μ,c Finally by Proposition 7.2 of [9], the following Dirac operator D gives a spectral triple 2 ), H, D) which we are going to work with: (O(Sμ,c l l D vm,± 1 = (c1 l + c2 )v m,∓ 1 2
where c1 , c2 belong to R, c1 = 0.
2
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It is easy to see that the action of SU μ (2) on itself keeps the subspace H invariant and so induces a unitary representation, say U0 on H. We define a positive, unbounded operator R on H by R(v n 1 ) = μ−2i v n 1 . i,± 2
i,± 2
2 ) Proposition 3.1. αU0 preserves the R-twisted volume. In particular, for x belonging to π(Sμ,c −tD ), and h denotes the restriction and t > 0, we have h(x) = ττRR (x) (1) , where τR (x) := Tr(xRe 2 , which is the unique SU (2)-invariant state of the Haar state of SU μ (2) to the subalgebra Sμ,c μ 2 on Sμ,c . 2
Proof. It is enough to prove that τR is αU0 -invariant. Define R0 (v n
i,± 12
) = μ−2i∓1 v n
i,± 12
, and note
that it has been observed in [12] that Tr(R0 e−tD ) < ∞ (for all t > 0) and one has 2
(τR0 ⊗ id) U0 (x ⊗ 1)U0 ∗ = τR0 (x).1, for all x in B(H), where τR0 (x) = Tr(xR0 e−tD ). Let us denote by P 1 , P− 1 the projections onto the closed subspaces generated by {v l 1 } and 2
2
{v l
i, 2
2
−tD ). 1 } respectively. Moreover, let τ± be the functionals defined by τ± (x) = Tr(xR0 P± 1 e
i,− 2
2
2
We observe that R0 , e−tD and U0 commute with P± 1 so that for x belonging to B(H), 2
2
(τ± ⊗ id) αU0 (x) = (τR0 ⊗ id) αU0 (xP± 1 ) = τR0 (xP± 1 )1 = τ± (x)1, 2
2
i.e. τ± are αU0 -invariant. Moreover, since we have RP± 1 = μ± R0 P± 1 , the functional τR coin-
cides with μ−1 τ+ + μτ− , hence is αU0 -invariant.
2
2
2
Theorem 3.2. (SU μ (2), , U0 ) is an object in QR (D). Proof. The above spectral triple is equivariant with respect to this representation (see [9]) and it preserves τR by Proposition 3.1, which completes the proof. 2 We now note down some useful facts for later use. l and , we observe: Remark 3.3. Using the definition of vi,j
1. The eigenspaces of D corresponding to (c1 l + c2 ) and −(c1 l + c2 ) are span{v l vl 1 : m,− 2
−l m l} and
span{v l 1 m, 2
− vl 1 : m,− 2
−l m l} respectively.
m, 12
2. The eigenspace of |D| corresponding to the eigenvalue (c1 . 12 + c2 ) is span{α, γ , α ∗ , γ ∗ }. Remark 3.4. l−1 l+1 l l 1. π(A)vm,N belongs to Span{vm,N , vm,N , vm,N }, l−1 l+1 l l π(B)vm,N belongs to Span{vm−1,N , vm−1,N , vm−1,N }, l−1 l+1 l l ∗ π(B )vm,N belongs to Span{vm+1,N , vm+1,N , vm+1,N }.
+
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l−k l−k+1 l+k l 2. π(Ak )(vm,N ) belongs to Span{vm,N , vm,N , . . . , vm,N }.
l−(n +m −1)
l−m −n l ) belongs to Span{vm−n 3. π(Am B n )(vm,N ,N , vm−n ,N
l+n +m , . . . , vm−n ,N }.
l−s−r l−s−r+1 l+s+r l 4. π(Ar B ∗s )(vm,N ) belongs to Span{vm+s,N , vm+s,N , . . . , vm+s,N }.
˜ We shall now proceed to show that QISO+ R (D) is isomorphic with SOμ (3). Let (Q, U ) be an object in the category QR (D) of CQG s acting by orientation and R-twisted volume preserving isometries on this spectral triple and Q be the Woronowicz C ∗ subalgebra of Q˜ generated by 2 (where ·,· is the Q-valued ˜
(ξ ⊗ 1), αU (a)(η ⊗ 1) Q˜ , for ξ, η in H, a in Sμ,c inner product Q˜ ˜ We shall denote αU by φ from now on. of H ⊗ Q). The proof has two main steps: first, we prove that φ is ‘linear’, in the sense that it keeps the span of {1, A, B, B ∗ } invariant, and then we shall exploit the facts that φ is a ∗-homomorphism 2 , i.e. the restriction of the Haar state of and preserves the canonical volume form on Sμ,c SU μ (2). Remark 3.5. The first step does not make use of the fact that φ preserves the R-twisted volume, so linearity of the action follows for any object in the bigger category Q (D). 3.2. Linearity of the action For a vector v in H, we shall denote by Tv the map from B(H) to L2 (SU μ (2)) given by Tv (x) = xv ∈ H ⊂ L2 (SU μ (2)). It is clearly a continuous map with respect to the strong operator topology on B(H) and the Hilbert space topology of L2 (SU μ (2)). For an element a in SU μ (2), we consider the right multiplication Ra as a bounded linear map on L2 (SU μ (2)). Clearly the composition Ra Tv is a continuous linear map from B(H) (with the strong operator topology) to the Hilbert space L2 (SU μ (2)). We now define T = Rα ∗ T α + μ 2 R γ T γ ∗ . 2 , we have T (φ (x)) = φ (x) ≡ R (φ (x)) Lemma 3.6. For any state ω on Q˜ and x in Sμ,c ω ω 1 ω 2 ⊆ L2 (SU (2)), where φ (x) = (id ⊗ ω)(φ(x)). belonging to Sμ,c μ ω
Proof. It is clear from the definition of T (using αα ∗ + μ2 γ γ ∗ = 1) that T (x) = x ≡ R1 (x) for x 2 ⊂ B(H), where x in the right-hand side of the above denotes the identification of x ∈ S 2 in Sμ,c μ,c 2 , φ (x) belongs as a vector in L2 (SU μ (2)). Now, the lemma follows by noting that for x in Sμ,c ω 2 ) , which is the closure of S 2 in the strong operator topology, and the continuity of T to (Sμ,c μ,c in this topology discussed before. 2 Let l V l = Span vi,± 1 , −l i l , l l . 2
Since Span{v l
i,± 21
, −l i l} is the eigenspace of |D| corresponding to the eigenvalue c1 l + c2 ,
U and U ∗ must keep V l invariant for all l.
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Lemma 3.7. There is some finite dimensional subspace V of O(SU μ (2)) such that 1
1
Rα ∗ (φω (A)v 2
), Rγ (φω (A)v 2
j,± 12
j,± 12
˜ ) belong to V for all states ω on Q.
The same holds when A is replaced by B or B ∗ . Proof. We prove the result for A only, since a similar argument will work for B and B ∗ . 1
We have φ(A)(v 2 Now, U ∗ (v
j,± 12
1 2
j,± 12
1
⊗ 1) = U (π(A) ⊗ 1)U ∗ (v 2
j,± 12
⊗ 1).
1 ˜ and then using the definition of π as well as Re⊗ 1) belongs to V 2 ⊗ Q, 1
mark 3.4, we observe that (π(A) ⊗ 1)U ∗ (v 2
j,± 12
⊗ 1) belongs to Span{v l
j,± 12
: −l j l , 1
3 ˜ Again, U keeps V 32 ⊗ Q˜ invariant, so Rα ∗ (φω (A)v 2 1 ) belongs to l 32 } ⊗ Q˜ = V 2 ⊗ Q.
±2
1
3
3
Span{vα ∗ : v ∈ V 2 }. Similarly, Rγ (φω (A)(v 2 1 )) belongs to Span{vγ : v ∈ V 2 }. So, the lemma ±2
3
follows for A by taking V = Span{vα ∗ , vγ : v ∈ V 2 } ⊂ O(SU μ (2)). 1
Since α, γ ∗ belong to Span{v 2
j,± 12
2
}, we have the following immediate corollary:
Corollary 3.8. There is a finite dimensional subspace V of O(SU μ (2)) such that for every state ˜ we have T (φω (A)) belongs to V. A similar (hence for every bounded linear functional) ω on Q, ∗ conclusion holds for B and B as well. 2 )⊗ Proposition 3.9. φ(A), φ(B), φ(B ∗ ) belong to O(Sμ,c alg Q.
Proof. We give the proof for φ(A) only, the proof for B, B ∗ being similar. From Corollary 3.8 ˜ T (φω (A)) belongs to and Lemma 3.6 it follows that for every bounded linear functional ω on Q, 2 2 2 V ∩ Sμ,c ⊂ O(SU μ (2)) ∩ Ker(ψ) and hence V ∩ Sμ,c = V ∩ O(Sμ,c ), where V is the finite dimen2 ) is a finite dimensional subspace sional subspace mentioned in Corollary 3.8. Clearly, V ∩ O(Sμ,c 2 of O(Sμ,c ) implying that there must be finite m, say, such that for every ω, T (φω (A)) belongs to Span{Ak , Ak B l , Ak B ∗l : 0 k, l m}. Denote by W the (finite dimensional) subspace of B(H) spanned by {Ak , Ak B l , Ak B ∗l : 0 k, l m}. Since for every state (and hence for every ˜ we have T (φω (A)) = R1 (φω (A)) ≡ φω (A)1, it is clear that bounded linear functional) ω on Q, φω (A) belongs to W for every ω in Q˜ ∗ . Now, let us fix any faithful state ω on the separable unital C ∗ -algebra Q˜ and embed Q˜ in B(L2 (Q, ω)) ≡ B(K). Thus, we get a canonical embedding of ˜ in B(H ⊗K). Let us thus identify φ(A) as an element of B(H ⊗K), and then by choosL(H ⊗ Q) basis in K = L2 (ω), ing a countable family of elements {q1 , q2 , . . .} of Q˜ which is an orthonormal
∞ ij we can write φ(A) as a weakly convergent series of the form i,j =1 φ (A) ⊗ |qi qj |. But φ ij (A) = (id ⊗ ωij )(φ(A)), where ωij (·) = ω(qi∗ · qj ). Thus, φ ij (A) belongs to W for all i, j ,
and hence the sequence ni,j =1 φ ij (A) ⊗ |qi qj | ∈ W ⊗ B(K) converges weakly, and W being finite dimensional (hence weakly closed), the limit, i.e. φ(A), must belong
to W ⊗ B(K). In other words, if A1 , . . . , Ak denotes a basis of W, we can write φ(A) = ki=1 Ai ⊗ Bi for some Bi ∈ B(K). ˜ For any trace-class positive operator ρ in H, say We claim that each Bi must belong to Q. of the form ρ = j λj |ej ej |, where {e1 , e2 , . . .} is an orthonormal basis of H and λj 0,
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λj < ∞, let us denote by ψρ the normal functional on B(H) given by x → Tr(ρx), and it is ˜ given by ψ˜ ρ (X) = easy to see that it has a canonical extension ψ˜ ρ := (ψρ ⊗ id) on L(H ⊗ Q)
˜ ˜ λ
e ⊗ 1, X(e ⊗ 1) , where X belongs to L(H ⊗ Q) and
·,· denotes the Q-valued j j j j Q˜ Q˜ ˜ Clearly, ψ˜ ρ is a bounded linear map from L(H ⊗ Q) ˜ to Q. ˜ Now, inner product of H ⊗ Q. since A1 , . . . , Ak in the expression of φ(A) are linearly independent, we can choose trace class operators ρ1 , . . . , ρk such that ψρi (Ai ) = 1 and ψρi (Aj ) = 0 for j = i. Then, by applying ψ˜ ρi ˜ But by definition, Q is the Woronowicz subalgebra on φ(A) we conclude that Bi belongs to Q. 2 ), and hence ˜ of Q generated by ξ ⊗ 1, φ(x)(η ⊗ 1) Q˜ , with η, ξ belonging to H and x in O(Sμ,c it follows that Bi belongs to Q. 2 j
Proposition 3.10. φ keeps the span of 1, A, B, B ∗ invariant. Proof. We prove the result for φ(A) only, the proof for the other cases being quite similar. Using Proposition 3.9, we can write φ(A) as a finite sum of the form:
Ak ⊗ Qk +
Am B n ⊗ Rm ,n +
m ,n ,n =0
k0
Ar B ∗s ⊗ Rr,s .
r,s,s=0
l Let ξ = vm . 0 ,N0 l We have that U (ξ ) belongs to Span{vm,N , m = −l, . . . , l, N = ± 12 }. Let us write
U (ξ ⊗ 1) =
l l vm,N ⊗ q(m,N ),(m0 ,N0 ) ,
m=−l,...,l, N =± 12 l where q(m,N ),(m0 ,N0 ) belong to Q. Since αU preserves the R-twisted volume, we have:
m ,N
l l∗ q(m,N ),(m ,N ) q(m,N ),(m ,N ) = 1.
(4)
l It also follows that U (Aξ ) belongs to Span{vm,N , m = −l , . . . , l , l = l − 1, l, l + 1, N = ± 12 }. Recalling Remark 3.4, we have
φ(A)U (ξ ⊗ 1) =
l l Ak vm,N ⊗ Qk q(m,N ),(m0 ,N0 )
k,m=−l,...,l, N=± 12
+
l l Am B n vm,N ⊗ Rm ,n q(m,N ),(m0 ,N0 )
m ,n , n =0, m=−l,...,l, N=± 12
+
l l Ar B ∗s vm,N ⊗ Rr,s q(m,N ),(m0 ,N0 ) .
r,s, s=0, m=−l,...l, N =± 12
Let m0 denote the largest integer m such that there is a nonzero coefficient of Am B n , n 1 in the expression of φ(A). We claim that the coefficient of v l Rm0 ,n q(m,N ),(m0 ,N0 ) .
l−m0 −n m−n ,N
in φ(A)U (ξ ⊗ 1) is
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2949
l−m −n
Indeed, the term vm−n0 ,N can arise in three ways: it can come from a term of the form
l l l or Ak vm,N or Ar B ∗s vm.N for some m , n , k, r, s. Am B n vm,N In the first case, we must have l − m0 − n = l − m − n + t, 0 t 2m and m − n = m − n implying m = m0 + t, and since m0 is the largest integer such that Am0 B n appears in φ(A), we l−m −n
only have the possibility t = 0, i.e. vm−n0 ,N appears only in Am0 B n . In the second case, we have m − n = m implying n = 0 – a contradiction. In the last case, we have m − n = m + s so that −n = s which is only possible when n = s = 0 which is again a contradiction. It now follows from the above claim, using Remark 3.4 and comparing coefficients in the l equality U (Aξ ⊗ 1) = φ(A)U (ξ ⊗ 1), that Rm0 ,n q(m,N ),(m0 ,N0 ) = 0 for all n 1, for all m, N when m0 1. Now varying (m0 , N0 ), we conclude that the above holds for all (m0 , N0 ). Using (4), we conclude that Rm0 ,n
m ,N
l l∗ q(m,N ),(m ,N ) q(m,N ),(m ,N ) = 0 for all n 1,
that is, Rm0 ,n = 0 for all n 1 if m0 1. Proceeding by induction on m0 , we deduce Rm ,n = 0 for all m 1, n 1. = 0 for all r 1, s 1. Similarly, we have Qk = 0 for all k 2 and Rr,s Thus, φ(A) belongs to Span{1, A, B, B ∗ , B 2 , . . . , B n , B ∗2 , . . . , B ∗m }. But the coefficient of l−n vm−n ,N in φ(A)U (ξ ⊗ 1) is R0,n . Arguing as before, we conclude that R0,n = 0 for all n 2. In a similar way, we can prove R0,n 2 = 0 for all n 2. In view of the above, let us write: φ(A) = 1 ⊗ T1 + A ⊗ T2 + B ⊗ T3 + B ∗ ⊗ T4 ,
(5)
φ(B) = 1 ⊗ S1 + A ⊗ S2 + B ⊗ S3 + B ∗ ⊗ S4 ,
(6)
for some Ti , Si in Q. 3.3. Identification of SOμ (3) as the quantum isometry group In this subsection, we shall use the facts that φ is a ∗-homomorphism and it preserves the R-twisted volume to derive relations among Ti , Si in (5), (6). Lemma 3.11. 1 − T2 , 1 + μ2 −S2 S1 = . 1 + μ2
T1 =
Proof. We have the expressions of A and B in terms of the SU μ (2) elements from Eqs. (1), (2) and (3). From these, we note that h(A) = (1 + μ2 )−1 and h(B) = 0. By recalling Proposition 3.1, we use (h ⊗ id)φ(A) = h(A).1 and (h ⊗ id)φ(B) = h(B).1 to have the above two equations. 2
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Lemma 3.12. T1∗ = T1 ,
T2∗ = T2 ,
T4∗ = T3 .
Proof. It follows by comparing the coefficients of 1, A and B respectively in the equation φ(A∗ ) = φ(A). 2 Lemma 3.13. 2 2 S2∗ S2 + c 1 + μ2 S3∗ S3 + c 1 + μ2 S4∗ S4 2 2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T3 T3∗ − c 1 + μ2 T3∗ T3 + c 1 + μ2 .1, −2S2∗ S2 + 1 + μ2 S3∗ S3 + μ2 1 + μ2 S4∗ S4 = μ2 + 2T2 − 1 T2 − μ2 1 + μ2 T3 T3∗ − 1 + μ2 T3∗ T3 , S2∗ S2 − S3∗ S3 − μ4 S4∗ S4 = −T22 + μ4 T3 T3∗ + T3∗ T3 , S2∗ S4 + S3∗ S2 = − μ2 + T2 T3∗ + T3∗ (1 − T2 ), S2∗ S3
+ μ2 S4∗ S2 S4∗ S3
(7)
(8) (9) (10)
= −T2 T3 − μ T3 T2 ,
(11)
= −T32 .
(12)
2
Proof. It follows by comparing the coefficients of 1, A, A2 , B ∗ , AB and B 2 in the equation φ(B ∗ B) = φ(A) − φ(A2 ) + cφ(I ) and then using Lemmas 3.11 and 3.12. 2 Lemma 3.14. 2 2 −S2 (1 − T2 ) + c 1 + μ2 S3 T3∗ + c 1 + μ2 S4 T3 2 2 = −μ2 (1 − T2 )S2 + cμ2 1 + μ2 T3 S4 + cμ2 1 + μ2 T3∗ S3 , S2 − 2S2 T2 + 1 + μ2 μ2 S3 T3∗ + S4 T3 = μ2 S2 − 2μ2 T2 S2 + μ4 1 + μ2 T3 S4 + μ2 1 + μ2 T3∗ S3 ,
(13)
(14)
−S2 T3 + S3 (1 − T2 ) = −μ2 T3 S2 + μ2 (1 − T2 )S3 ,
(15)
−S2 T3∗
(16)
T3∗ S2 ,
+ S4 (1 − T2 ) = μ (1 − T2 )S4 − μ S 2 T3 + μ 2 S 3 T2 = μ 2 T2 S 3 + μ 2 T3 S 2 , 2
2
(17)
S 3 T3 = μ 2 T3 S 3 ,
(18)
S4 T3∗ = μ2 T3∗ S4 .
(19)
Proof. It follows by equating the coefficients of 1, A, B, B ∗ , AB, B 2 and B ∗ 2 in the equation φ(BA) = μ2 φ(AB) and then using Lemmas 3.11 and 3.12. 2
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−S2 S4∗ − S3 S2∗ = μ2 1 + μ2 T3 − μ4 (1 − T2 )T3 − μ4 T3 (1 − T2 ),
(20)
Lemma 3.15.
S2 S4∗ + μ2 S3 S2∗ = −μ4 T2 T3 − μ6 T3 T2 , S3 S4∗
= −μ
4
(21)
T32 .
(22)
Proof. The lemma is proved by equating the coefficient of B, AB, B 2 in the equation φ(BB ∗ ) = μ2 φ(A) − μ4 φ(A2 ) + cφ(I ) and then using Lemmas 3.11 and 3.12. 2 Now, we compute the antipode, say κ of Q. To begin with, we note that {x−1 , x0 , x1 } is a set of orthogonal vectors. Moreover, they have the same norm. The first assertion being easier, we prove below the second one. Lemma 3.16. ∗ x−1 = h x0∗ x0 = h x1∗ x1 h x−1 −1 2 = t 2 1 − μ2 1 − μ6 μ + t −1 μ4 + 2μ2 + 1 + t −μ4 − 2μ2 − 1 . ∗ x ∗ 2 −2 2 2 2 2 Proof. We have x−1 −1 = t μ (1 + μ )(A − A + cI ), x0 x0 = t (1 − 2(1 + μ )A + (1 + ∗ 2 2 2 2 2 2 4 2 μ ) A ), x1 x1 = t (1 + μ )(μ A − μ A + cI ). We recall from [18] that for all bounded Borel function f on σ (A), ∞ ∞ 2n 2n f λ + μ μ + γ− f λ− μ2n μ2n , h f (A) = γ+ n=0 1
n=0 1
where λ+ = 12 + (c + 14 ) 2 , λ− = 12 − (c + 14 ) 2 , γ+ = (1 − μ2 )λ+ (λ+ − λ− )−1 , γ− = (1 − μ2 )λ− (λ− − λ+ )−1 . ∗ x , x∗x , The lemma follows by applying this relation to the above expressions of x−1 −1 0 0 ∗ x1 x 1 . 2 , x , x is the normalized basis corresponding to {x , x , x }, then from (5) and (6) If x−1 −1 0 1 0 1 along with the fact that each of the vectors x−1 , x0 , x1 has the same norm, it follows that
− 1 φ x−1 = x−1 ⊗ S3 + x0 ⊗ −μ−1 1 + μ2 2 S2 + x1 ⊗ −μ−1 S4 , 1 1 φ x0 = x−1 ⊗ −μ 1 + μ2 2 T3 + x0 ⊗ T2 + x1 ⊗ 1 + μ2 2 T4 , − 1 φ x1 = x−1 ⊗ −μS4∗ + x0 ⊗ 1 + μ2 2 S2∗ + x1 ⊗ S3∗ . Since φ is kept invariant by the Haar state h of SU μ (2) and φ keeps the span of the or , x , x } invariant too, we get a unitary representation of the CQG Q on the thonormal set {x−1 0 1 span of {x−1 , x0 , x1 }. If we denote by Z the M3 (Q)-valued unitary corresponding to this unitary , x , x }, we get by using T = T ∗ from representation with respect to the ordered basis {x−1 4 0 1 3
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Lemma 3.12 the following: ⎛
S3 ⎜ √−S2 Z = ⎝ μ 1+μ2 −μ−1 S4
−μ 1 + μ2 T3
T2 1 + μ2 T3∗
⎞ −μS4∗ S2∗ ⎟ √ . 1+μ2 ⎠ S3∗
Recall that (see, for example, [14]), the antipode κ on the matrix elements of a finite dimensional unitary representation U α ≡ (uαpq ) is given by κ(uαpq ) = (uαqp )∗ . Thus, the antipode κ is given by: κ(T2 ) = T2 ,
κ(T3 ) =
κ(S3 ) = S3∗ ,
S2∗ , 2 μ (1 + μ2 )
κ(S4 ) = μ2 S4 ,
κ S2∗ = 1 + μ2 T3 ,
κ(S2 ) = μ2 1 + μ2 T3∗ , κ T3∗ =
κ S3∗ = S3 ,
S2 , 1 + μ2 κ S4∗ = μ−2 S4∗ .
Now we derive some more relations by applying the anti-homomorphism κ on the relations obtained earlier. Lemma 3.17. 3 2 2 −2μ4 1 + μ2 T3∗ T3 + μ2 1 + μ2 S3∗ S3 + μ4 1 + μ2 S4 S4∗ = μ2 1 + μ2 T2 μ2 + 2T2 − 1 − μ2 S2 S2∗ − S2∗ S2 , 4 2 2 μ4 1 + μ2 T3∗ T3 − μ2 1 + μ2 S3∗ S3 − μ6 1 + μ2 S4 S4∗ 2 = −μ2 1 + μ2 T22 + μ4 S2 S2∗ + S2∗ S2 , 2 2 μ2 1 + μ2 S4 T3 + μ2 1 + μ2 T3∗ S3 = −S2 μ2 + T2 + (1 − T2 )S2 , S4 S3 = −
(23)
(24) (25)
−S22 . μ2 (1 + μ2 )2
(26)
Proof. The relations follow by applying κ on (8), (9), (10) and (12) respectively.
2
Lemma 3.18. −μ2 (1 − T2 )T3∗ + cS2 S3∗ + cS2∗ S4 = −μ4 T3∗ (1 − T2 ) + cμ2 S4 S2∗ + cμ2 S3∗ S2 ,
(27)
S3 S2 = μ2 S2 S3 ,
(28)
S2 S4 = μ S4 S2 ,
(29)
2
−S2∗ T3∗ + (1 − T2 )S3∗ = −μ2 T3∗ S2∗ + μ2 S3∗ (1 − T2 ),
(30)
−S2 T3∗
(31)
+ (1 − T2 )S4 = μ S4 (1 − T2 ) − μ 2
2
T3∗ S2 .
Proof. The relations follow by applying κ on (13), (18), (19), (15) and (16) respectively.
2
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Lemma 3.19. μ2 S22 S3 S4 = − , (1 + μ2 )2 2 2 −μ2 1 + μ2 S4∗ T3∗ − μ2 1 + μ2 T3 S3∗ = μ2 1 + μ2 S2∗ − μ4 S2∗ (1 − T2 ) − μ4 (1 − T2 )S2∗ , 2 2 1 + μ2 S4∗ T3∗ + μ2 1 + μ2 T3 S3∗ = −μ2 S2∗ T2 − μ4 T2 S2∗ . Proof. The relations follow by applying κ on (22), (20) and (21) respectively.
(32)
(33) (34) 2
Remark 3.20. It follows from (26) and (32) that μ4 S4 S3 = S3 S4 . Lemma 3.21. S2∗ S2 = (1 − T2 ) μ2 + T2 . Proof. Subtracting the equation obtained by multiplying c(1 + μ2 ) with (8) from (7), we have 2 1 + 2c 1 + μ2 S2∗ S2 + c 1 + μ2 1 − μ2 S4∗ S4 = (1 − T2 ) μ2 + T2 − c 1 + μ2 μ2 + 2T2 − 1 T2 2 2 + c 1 + μ2 μ2 − 1 T3 T3∗ + c 1 + μ2 .1.
(35)
Again, by adding (7) with c(1 + μ2 )2 times (9) gives 2 2 1 + c 1 + μ2 S2∗ S2 + c 1 − μ4 1 + μ2 S4∗ S4 2 2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T22 + c 1 + μ2 μ4 − 1 T3 T3∗ + c 1 + μ2 .1.
(36)
Subtracting the equation obtained by multiplying (μ2 + 1) with (35) from (36) we obtain 2 − μ2 + c 1 + μ2 S2∗ S2 2 = (1 − T2 ) μ2 + T2 − c 1 + μ2 T22 2 2 − 1 + μ2 (1 − T2 ) μ2 + T2 − cμ2 1 + μ2 .1 + c 1 + μ2 μ2 + 2T2 − 1 T2 . The right-hand side can be seen to equal −(μ2 + c(1 + μ2 )2 )(1 − T2 )(μ2 + T2 ). Thus, S2∗ S2 = (1 − T2 )(μ2 + T2 ). 2 Lemma 3.22. 2 μ2 1 + μ2 T3∗ T3 = (1 − T2 ) μ2 + T2 , 2 1 + μ2 T3 T3∗ = (1 − T2 ) 1 + μ2 T2 , S2 S2∗ = μ2 (1 − T2 ) 1 + μ2 T2 .
(37) (38) (39)
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Proof. Applying κ on Lemma 3.21, we obtain (37). Unitarity of the matrix Z ((2, 2) position of the matrix Z ∗ Z) gives μ2 (1 + μ2 )T3∗ T3 + T22 + (1 + μ2 )T3 T3∗ = 1. Using (37) we deduce −(1 + μ2 )2 T3 T3∗ = (T2 − 1)(1 + μ2 T2 ). Thus we obtain (38). Applying κ on (38), we deduce (39). 2 Lemma 3.23. −2 S4∗ S4 = S4 S4∗ = 1 + μ2 μ2 (1 − T2 )2 . Proof. Adding (23) and (24), we have: 3 2 −μ4 1 + μ2 1 − μ2 T3∗ T3 + μ4 1 + μ2 1 − μ2 S4 S4∗ = −μ2 1 + μ2 1 − μ2 T2 (1 − T2 ) − μ2 1 − μ2 S2 S2∗ . Using μ2 = 1, we obtain, 3 2 −μ4 1 + μ2 T3∗ T3 + μ4 1 + μ2 S4 S4∗ = −μ2 1 + μ2 T2 (1 − T2 ) − μ2 S2 S2∗ . Now using (37) and (39), we reduce the above equation to 2 μ4 1 + μ2 S4 S4∗ = −μ2 (1 − T2 ) T2 + μ2 T2 + μ2 + μ4 T2 + μ2 1 + μ2 (1 − T2 ) μ2 + T2 = μ6 (1 − T2 )2 . Thus, S4 S4∗ = =
μ6 (1 − T2 )2 μ4 (1 + μ2 )2 μ2 (1 − T2 )2 . (1 + μ2 )2
μ2 (1 − T2 )2 . (1+μ2 )2 2 μ (1 − T2 )2 . 2 (1+μ2 )2
Applying κ, we have S4∗ S4 = Thus, S4∗ S4 = S4 S4∗ = Lemma 3.24.
2 μ2 1 + μ2 S3∗ S3 = μ2 + T2 μ2 1 + μ2 − (1 − T2 ) . Proof. Using Lemma 3.21 in (7), we have S3∗ S3 + T3∗ T3 + T3 T3∗ + S4∗ S4 = 1.
(40)
The lemma is derived by substituting the expressions of T3∗ T3 , T3 T3∗ and S4∗ S4 from (37), (38) and Lemma 3.23 in Eq. (40). 2
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Lemma 3.25. 2 1 + μ2 S3 S3∗ = 1 + μ2 T2 1 + μ2 − μ4 (1 − T2 ) . Proof. By unitarity of the matrix Z, in particular equating the (1, 1)-th entry of ZZ ∗ to 1 we get S3 S3∗ + μ2 (1 + μ2 )T3 T3∗ + μ2 S4∗ S4 = 1. Then the lemma follows by using (38) and Lemma 3.23 in the above equation. 2 Lemma 3.26. −S2∗ S3 = μ2 + T2 T3 . Proof. By applying the adjoint and then multiplying by μ2 on (10) we have μ2 S2∗ S3 + μ2 S4∗ S2 = −μ2 T3 (μ2 + T2 ) + μ2 (1 − T2 )T3 . Subtracting this from (11) we have (1 − μ2 )S2∗ S3 = −T2 T3 − μ2 T3 T2 + μ2 T3 (μ2 + T2 ) − μ2 (1 − T2 )T3 which implies −S2∗ S3 = (μ2 + T2 )T3 as μ2 = 1. 2 Lemma 3.27. S2 (1 − T2 ) = μ2 (1 − T2 )S2 . Proof. Applying κ to Lemma 3.26 and then taking adjoint, we have 2 μ2 1 + μ2 T3∗ S3 = − μ2 + T2 S2 .
(41)
Adding (33) and (34) and then taking adjoint, we get (by using μ2 = 1) 2 μ2 1 + μ2 T3 S4 = μ4 (1 − T2 )S2 .
(42)
Moreover, (25) gives 2 2 μ2 1 + μ2 S4 T3 = −S2 μ2 + T2 + (1 − T2 )S2 − μ2 1 + μ2 T3∗ S3 . Using (41), the right-hand side of this equation turns out to be S2 (1 − T2 ). Thus, 2 1 + μ2 S4 T3 = μ−2 S2 (1 − T2 ).
(43)
Again, application of adjoint to Eq. (33) gives: 2 2 μ2 1 + μ2 S3 T3∗ = −μ2 1 + μ2 T3 S4 − μ2 1 + μ2 S2 + μ4 (1 − T2 )S2 + μ4 S2 (1 − T2 ). Using (42), we get 2 1 + μ2 S3 T3∗ = −S2 1 + μ2 T2 .
(44)
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Using (41)–(44) to Eq. (14), we obtain: −1 −1 S2 − 2S2 T2 − 1 + μ2 μ2 S2 1 + μ2 T2 + μ−2 1 + μ2 S2 (1 − T2 ) −1 −1 2 μ + T2 S 2 . = μ2 S2 − 2μ2 T2 S2 + 1 + μ2 μ6 (1 − T2 )S2 − 1 + μ2 This gives μ2 1 + μ2 (S2 − S2 T2 ) − μ2 S2 − μ2 T2 S2 − μ2 1 + μ2 S2 T2 − μ2 T2 S2 − μ4 S2 − μ6 S2 T2 + S2 (1 − T2 ) − μ8 (S2 − T2 S2 ) + μ4 S2 + μ2 T2 S2 = 0. Thus, μ2 1 + μ2 S2 (1 − T2 ) − μ2 (1 − T2 )S2 + S2 (1 − T2 ) − μ2 (S2 − T2 S2 ) + μ6 S2 (1 − T2 ) − μ2 (1 − T2 )S2 − μ6 (1 − T2 )S2 + μ4 S2 (1 − T2 ) + μ2 S2 (1 − T2 ) − μ2 (1 − T2 )S2 = 0. On simplifying, (μ6 + 2μ4 + 2μ2 + 1)(S2 (1 − T2 ) − μ2 (1 − T2 )S2 ) = 0, which proves the lemma as 0 < μ < 1. 2 Lemma 3.28. T3 (1 − T2 ) = μ2 (1 − T2 )T3 ,
(45)
S3 S4∗ = μ4 S4∗ S3 .
(46)
Proof. Eq. (45) follows by applying κ on Lemma 3.27 and then taking adjoint. We have S4∗ S3 = −T32 from (12). On the other hand we have S3 S4∗ = −μ4 T32 from (22). Combining these two, we get (46). 2 Lemma 3.29. S 4 T2 = T2 S 4 . Proof. Subtracting (31) from (16) we get the required result.
2
Lemma 3.30. T3 S 2 = S 2 T3 . Proof. By applying adjoint on (30) and then subtracting it from (15) we obtain S2 T3 − T3 S2 = 0. 2 Lemma 3.31. S3 (1 − T2 ) = μ4 (1 − T2 )S3 .
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Proof. By adding (15) with (17) we obtain S3 (1 − T2 ) + μ2 S3 (T2 − 1) = μ2 μ2 − 1 T3 S2 . Thus, using μ2 = 1, S3 (1 − T2 ) = −μ2 T3 S2 .
(47)
Moreover, by taking adjoint of (30), we obtain μ2 (1 − T2 )S3 = μ2 S2 T3 − T3 S2 + S3 (1 − T2 ). Thus, μ4 (1 − T2 )S3 = μ4 S2 T3 − μ2 T3 S2 + μ2 S3 (1 − T2 ). Hence, to prove the lemma it suffices to prove: S3 (1 − T2 ) = μ4 S2 T3 − μ2 T3 S2 + μ2 S3 (1 − T2 ). After using T3 S2 = S2 T3 obtained from Lemma 3.30 we get this to be the same as (1 − μ2 )S3 (1 − T2 ) = μ2 (μ2 − 1)T3 S2 . This is equivalent to S3 (1 − T2 ) = −μ2 T3 S2 (as μ2 = 1) which follows from (47). 2 Proposition 3.32. The map SOμ (3) → Q sending M, L, G, N , C to −(1 + μ2 )−1 S2 , S3 , −μ−1 S4 , (1 + μ2 )−1 (1 − T2 ), μT3 respectively is a CQG homomorphism. Proof. It is enough to check that the map is ∗-homomorphic, since the coproducts on SOμ (3) and Q are determined in terms of the fundamental unitaries Z and Z respectively, and the map described in the statement of the proposition sends (ij )-th entry of Z to the (ij )-th entry of Z for all (ij ). Now, it can easily be checked that the proof of the homomorphic property of the given map reduces to verification of the relations on Q as derived in Lemmas 3.21–3.31 along with the following equations: S3 S4 = μ4 S4 S3 ,
(48)
S3 S2 = μ2 S2 S3 ,
(49)
S2 S4 = μ S4 S2 ,
(50)
2
S3 S4 = −
μ2
S2, 2 2 2 (1 + μ )
which follow from Remark 3.20, (28), (29), (32) respectively.
(51) 2
Theorem 3.33. We have the isomorphism: 2 ∼ QISO+ R O Sμ,c , H, D = SOμ (3).
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+ (D) as remarked before, and thus one gets a surjective Proof. SU μ (2) is an object in QISO R + (D) to SU (2) which clearly maps QISO+ (D) onto SO (3), identimorphism from QISO μ μ R R fying the latter as a quantum subgroup of QISO+ (D). Let us denote the surjective map from R + QISO+ (D) to SO (3) by Π . On the other hand, Proposition 3.32 implies that QISO μ R R (D) is a quantum subgroup of SOμ (3), and the corresponding surjective CQG morphism from SOμ (3) onto QISO+ R (D) is clearly seen to be the inverse of Π , thereby completing the proof. 2 Remark 3.34. Theorem 3.33 shows that for a fixed μ, the quantum isometry group QISO+ R (D) 2 of Sμ,c does not depend on c. This may appear somewhat surprising, but let us remark that in 2 are the classical situation (that is for μ = 1), c corresponds to the radius of the sphere and S1,c ∗ isomorphic as C algebras for all c 0. We refer the reader to [13, p. 126], for the details regarding this. Although in the noncommutative case, that is, when μ = 1, we do get non-isomorphic 2 for different choices of c, one may still think that the parameter c in some C ∗ -algebras Sμ,c sense determines the ‘radius’ of the noncommutative sphere, and thus one should get the same (quantum) isometry group for different choices of c. In view of the above, it seems impossible to ‘reconstruct’ the quantum homogeneous spaces 2 from the quantum isometry groups SO (3). In this context, it may be mentioned that for Sμ,c μ 2 are quantum homogeneous spaces corresponding to SO (3), only S 2 μ = 1, although all Sμ,c μ μ,0 arises as a quotient of SOμ (3) by a quantum subgroup (see [16] for more details). Thus, it is 2 from the quantum group SO (3). perhaps possible to somehow ‘reconstruct’ Sμ,0 μ + (D) 3.4. Existence of QISO For the above spectral triple, we have been unable to settle the issue of the existence of QISO+ (D) which is the universal object (if it exists) in the category Q (D) mentioned in Section 1. Nevertheless, we shall show that if a universal object in Q (D) exists, then QISO+ (D) must coincide with SOμ (3). + (D) exists, its induced action on S 2 , say α , must preserve the state h Lemma 3.35. If QISO 0 μ,c on the subspace spanned by {1, A, B, B ∗ , AB, AB ∗ , A2 , B 2 , B ∗ }. 2
Proof. Let W0 = C.1, W 1 = Span{1, A, B, B ∗ }, 2
W 3 = Span 1, A, B, B ∗ , AB, AB ∗ , A2 , B 2 , B ∗2 . 2
We note that the proof of Proposition 3.10 and the lemmas preceding it do not use the assumption that the action is R-twisted volume preserving, so the proof of Proposition 3.10 goes through verbatim implying that α0 keeps invariant the subspace spanned by {1, A, B, B ∗ } and hence it preserves W 3 as well. Let W 3 = W 1 ⊕ W be the orthogonal decomposition with respect to the 2
2
2
Haar state (say h0 ) of QISO+ (D). Since SOμ (3) is a sub-object of QISO+ (D), there is a CQG morphism π from QISO+ (D) onto SOμ (3) satisfying (id ⊗ π)α0 = , where is the SOμ (3) 2 . It follows from this that any QISO+ (D)-invariant subspace (in particular W ) is action on Sμ,c also SOμ (3)-invariant. On the other hand, it is easily seen that on W 3 , the SOμ (3)-action de2
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composes as W 1 ⊕ W (orthogonality with respect to h, the Haar state of SOμ (3)), where W 2 is a five-dimensional irreducible subspace. We claim that W = W , which will prove that the QISO+ (D)-action α0 has the same horthogonal decomposition as the SOμ (3)-action on W 3 , so preserves C.1 and its h-orthogonal 2 complements. This will prove that α0 preserves the Haar state h on W 3 . 2 We now prove the claim. Observe that V := W ∩ W is invariant under the SOμ (3)-action but due to the irreducibility of on the vector space W or W , it has to be zero or W = W . Now, dim(V) = 0 implies dim(W ) + dim(W ) = 5 + 5 > 9 = dim(W 3 ) which is a contradiction 2 unless W = W . 2 + (D) exists, then we must have that QISO+ (D) ∼ Theorem 3.36. If QISO = SOμ (3). Proof. In the proof of Lemma 3.35, it was noted that Proposition 3.10 follows under the assumption of the present theorem. To complete the proof of the theorem, we just need to observe that the other lemmas used for proving Theorem 3.33 require the conclusion of Lemma 3.35 as the only extra ingredient. 2 Let us conclude the article with brief explanation of the technical difficulties regarding the + (D). Let R be a positive, invertible operator commuting with D issue of existence of QISO
2 such that τR = τR and let φ denote the action of QISO+ R (D) on Sμ,c . The problem of existence + (D) is closely related to the question whether it is possible to identify QISO+ (D) as of QISO R
a quantum subgroup of SOμ (3) for a general R . By Theorem 3.36, a negative answer of this question will prove that Q (D) does not have a universal object. Now, as has been noted in Remark 3.5, φ is linear, that is, it keeps the span of {1, A, B, B ∗ } invariant and hence it is given by an expression similar to Eqs. (5) and (6) with Ti , Si replaced ∗ by some Ti , Si which generate QISO+ R (D) as a C -algebra. We can in principle write down all the relations satisfied by these generators, proceeding as in Section 3.3. These relations will be analogous to Eqs. (7)–(34), and in fact, the relations which make use of the homomorphism property only remain unchanged. However, the ones making use of the fact that φ preserves τR will change, since τR is in general different from τR . In particular, the expression of the antipode will change, which will affect all the relations starting from (23). We need to have a deeper and systematic understanding of the relations satisfied by Ti , Si for a general R , and possibly study their representations in concrete Hilbert spaces, to decide whether QISO+ R (D) is a quantum subgroup of SOμ (3) or not. We are not yet able to do this. + (D) exists, although we can identify QISO+ (D) with the wellMoreover, even if QISO + (D). If U denotes known quantum group SOμ (3), it is not so easy to explicitly compute QISO + (D), the fact that U commutes with D implies the unitary representation corresponding to QISO that U must preserve each of the two-dimensional eigenspaces span{v l and span{v l
m, 12
− vl
m,− 21
m, 21
+ vl
m,− 12
: m = ± 12 }
: m = ± 12 } of D. Suppose that (qij )i,j =1,2 and (rij )i,j =1,2 are the ma-
+ (D)) of U corresponding to these two spaces respectively. Then it trices (with entries in QISO + (D) will be generated by q , r ’s as well as the generators is clear that as a C ∗ algebra QISO ij
ij
Ti , Si of SOμ (3). However, the mutual relations among these generating elements have to be
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determined from the fact that U preserves each of the eigenspaces of D. In principle one gets infinitely many such relations which are quite complicated and it is not clear how to simplify them. References [1] T. Banica, Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (1) (2005) 27–51. [2] T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2) (2005) 243–280. [3] J. Bhowmick, D. Goswami, Quantum isometry groups: examples and computations, Comm. Math. Phys. 285 (2) (2009) 421–444, arXiv:0707.2648. [4] J. Bhowmick, D. Goswami, Quantum group of orientation preserving Riemannian isometries, J. Funct. Anal. 257 (8) (2009) 2530–2572. [5] J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (3) (2003) 665–673. [6] A. Connes, Noncommutative Geometry, Academic Press, London, New York, 1994. [7] Alain Connes, Michel Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Comm. Math. Phys. 230 (3) (2002) 539–579. [8] L. Dabrowski, Spinors and theta deformations, Russ. J. Math. Phys. 16 (3) (2009) 404–408. [9] L. Dabrowski, F. D’Andrea, G. Landi, E. Wagner, Dirac operators on all Podles quantum spheres, J. Noncommut. Geom. 1 (2007) 213–239. [10] D. Goswami, Twisted entire cyclic cohomology, JLO cocycles and equivariant spectral triples, Rev. Math. Phys. 16 (5) (2004) 583–602. [11] D. Goswami, Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys. 285 (1) (2009) 141–160, arXiv:0704.0041. [12] D. Goswami, Some noncommutative geometric aspects of SU q (2), arXiv:math-ph/0108003. [13] A. Klimyk, K. Schmudgen, Quantum Groups and Their Representations, Springer, New York, 1998. [14] A. Maes, A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wiskd. (4) 16 (1–2) (1998) 73–112. [15] P. Podles, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202. [16] P. Podles, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1995) 1–20. [17] K. Schmudgen, E. Wagner, Dirac operators and a twisted cyclic cocycle on the standard Podles’ quantum sphere, J. Reine Angew. Math. 574 (2004) 219–235. [18] K. Schmudgen, E. Wagner, Representation of cross product algebras of Podles quantum spheres, math.QA/0305309. [19] P.M. Soltan, Quantum SO(3) groups and quantum group actions on M2 , arXiv:0810.0398v1. [20] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [21] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998) 195–211. [22] S. Wang, Structure and isomorphism classification of compact quantum groups Au (Q) and Bu (Q), J. Operator Theory 48 (2002) 573–583. [23] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987) 613–665. [24] S.L. Woronowicz, Compact quantum groups, in: A. Connes, et al. (Eds.), Symétries Quantiques (Quantum Symmetries), Les Houches, 1995, Elsevier, Amsterdam, 1998, pp. 845–884.
Journal of Functional Analysis 258 (2010) 2961–2982 www.elsevier.com/locate/jfa
Similarity of analytic Toeplitz operators on the Bergman spaces ✩ Chunlan Jiang a , Dechao Zheng b,∗ a Department of Mathematics, Hebei Normal University, Shijianzhuang 050016, PR China b Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
Received 31 July 2009; accepted 16 September 2009 Available online 3 October 2009 Communicated by N. Kalton
Abstract In this paper we give a function theoretic similarity classification for Toeplitz operators on weighted Bergman spaces with symbol analytic on the closure of the unit disk. © 2009 Elsevier Inc. All rights reserved. Keywords: Bergman spaces; Toeplitz operator; Similarity; Blaschke product; Strongly irreducible
1. Introduction Let dA denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. For α > −1, the weighted Bergman space A2α is the space of analytic functions on D which are square-integrable with respect to the measure dAα (z) = (α + 1)(1 − |z|2 )α dA(z). For u ∈ L∞ (D, dA), the Toeplitz operator Tu with symbol u is the operator on A2α defined by Tu f = P (uf ); here P is the orthogonal projection from L2 (D, dAα ) onto A2α . Tg is called to be the analytic Toeplitz operator if g ∈ H ∞ (the set of bounded analytic functions on D). In this case, Tg is just the operator of multiplication by g on A2α . In this paper we study the similarity of Toeplitz operators with symbol analytic on the closure of the unit disk on A2α . On the Hardy ✩
The first author was partially supported by NSFC and the second author was partially supported by NSF.
* Corresponding author.
E-mail addresses:
[email protected] (C. Jiang),
[email protected] (D. Zheng). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.011
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space, Cowen showed that two Toeplitz operators with symbol analytic on the closure of the unit disk are similar if and only if they are unitarily equivalent [3]. However this is not true on the Bergman space. In [17], Sun showed that for two functions f and g analytic on the closure of the unit disk, the analytic Toeplitz operator Tf on the Bergman space A20 is unitarily equivalent to Tg if and only if there is an inner function χ of order one such that g = f ◦ χ . Also in [18], Sun and Yu showed that if the direct sum of two analytic Toeplitz operators is unitarily equivalent to an analytic Toeplitz operator, then they must be constants. So the Bergman space is rigid. But in [12], Jiang and Li showed that if f is a finite Blaschke product, then the analytic Toeplitz operator Tf is similar to the direct sum of finite copies of the Bergman shift Tz on the unweighted Bergman space A20 . In this paper, we will completely determine when two Toeplitz operators with symbol analytic on the closure of the unit disk are similar on the weighted Bergman spaces in terms of symbols, which is analogous to the result on the Hardy space [3]. While the Beurling theorem plays an important role on the Hardy space [3] and the Beurling theorem does not hold on the weighted Bergman spaces [8], we apply the general results [9–11] on similarity classification of the Cowen–Douglas classes to analytic Toeplitz operators. It was shown in [9,11] that two strongly irreducible members of Cowen–Douglas operator class Bn (Ω) [4] are similar if and only if the respective commutant algebras have isomorphic K0 groups and strongly irreducible decomposition operators give the similarity classification for Cowen–Douglas operator classes. As the adjoint of Toeplitz operators with symbol analytic on the closure of the unit disk is in the Cowen–Douglas operator classes, our main ideas are to identify the commutant of analytic Toeplitz operators as the commutant of Toeplitz operators with some finite Blaschke products and to use the strongly irreducible decomposition of analytic Toeplitz operators and K0 -groups of the commutants. The following theorem is our main result. It gives a function theoretic similarity classification of Toeplitz operators with symbol analytic on the closure of the unit disk. Theorem 1.1. Suppose that f and g are analytic on the closure of the unit disk D. Tf is similar to Tg on the weighted Bergman spaces A2α if and only if there are two finite Blaschke products B and B1 with the same order and a function h analytic on the closure of the unit disk such that f =h◦B
and g = h ◦ B1 .
2. Toeplitz operators with symbol as a finite Blashcke product A finite Blashcke product B is given by B=
n z − ak 1 − ak z
k=1
for n numbers {ak }nk=1 in the unit disk and some positive integer n. Here n is said to be the order of the Blaschke product B. In this section we will show that two Toeplitz operators with symbols as finite Blaschke products are similar on the weighted Bergman spaces if and only if their symbols have the same order. This was conjectured in [6] and proved in [12] on the unweighted Bergman space. For another proof, see [7]. Even for a very special Toeplitz operator Tzn on the weighted Bergman space A2α , the following result was established in [14] recently.
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Theorem 2.1. Let B be a Blaschke product with order n. Then TB on the weighted Bergman space A2α is similar to the direct sum nj=1 Tz on nj=1 A2α . We define the composition operator CB on A2α by CB (f )(z) = f B(z) for f ∈ A2α . Since B is a finite Blaschke product, the Nevanlinna counting function of B is equivalent to (1 − |z|2 ) on the unit disk. Thus CB is bounded below and hence has the closed range. For each k, define Mk =
Since T
1 1−ak z
f (B) : f ∈ A2α . 1 − ak z
is invertible, Mk is a closed subspace of A2α .
Assume that the n-th order Blaschke B has n distinct zeros. In [16], Stessin and Zhu showed that the Bergman space A20 is spanned by {M1 , . . . , Mn }. In [12], Jiang and Li showed that the Bergman space A20 is the Banach direct sum M1 M2 · · · Mn . The following theorem extends Jiang and He’s result to the weighted Bergman spaces, which immediately gives Theorem 2.1. Theorem 2.2. Let B be an n-th order Blaschke product with distinct zeros {ak }nk=1 in the unit disk. Then the weighted Bergman space A2α is the Banach direct sum of M1 , . . . , Mn , i.e., Aα = M1 M2 · · · Mn . Before going to the proof of the above theorems we need the following simple lemma. Lemma 2.3. Suppose that {ak }nk=1 are n distinct nonzero numbers in the unit disk. Then the following system ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 1−|a1 |2 1 1−a1 a2 1 1−a1 a3
1 1−a2 a1 1 1−|a2 |2 1 1−a2 a3
.. .
.. .
1 1−a1 an
1 1−a2 an
1 1−a3 a1 1 1−a3 a2 1 1−|a3 |2
.. .
1 1−a3 an
··· ··· ··· .. . ···
1 1−an a1 1 1−an a2 1 1−an a3
.. . 1 1−|an |2
⎞
⎛ ⎞ ⎛ ⎞ c1 0 ⎟ ⎟ ⎜ c2 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ c3 ⎟ = ⎜ 0 ⎟ ⎟⎜ . ⎟ ⎜ . ⎟ ⎟ ⎝ . ⎠ ⎝ .. ⎠ ⎠ . 0 cn
(2.1)
has only the trivial solution. Proof. Using row reductions and induction we obtain that the determinant of the coefficient matrix of system (2.1) equals
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1 1−|a |2 1 1 1−a1 a2 1 1−a1 a3 .. . 1 1−a1 an
1 1−a2 a1 1 1−|a2 |2 1 1−a2 a3
1 1−a3 a1 1 1−a3 a2 1 1−|a3 |2
.. .
.. .
1 1−a2 an
1 1−a3 an
1 1−|a1 |2 a (a1 −a2 ) (1−a11a2 )(1−|a 2 1| ) a1 (a1 −a3 ) = (1−a1 a3 )(1−|a1 |2 ) .. . a1 (a1 −an )
(1−a1 an )(1−|a1 |2 )
···
1 1−an a1 1 1−an a2 1 1−an a3
··· ··· .. . ···
.. . 1 1−|an |2
1 1−a2 a1 a2 (a1 −a2 ) (1−|a2 |2 )(1−a2 a1 ) a2 (a1 −a3 ) (1−a2 a3 )(1−a2 a1 )
1 1−a3 a1 a3 (a1 −a2 ) (1−a3 a2 )(1−a3 a1 ) a3 (a1 −a3 ) (1−|a3 |2 )(1−a3 a1 )
.. .
.. .
a2 (a1 −an ) (1−a2 an )(1−a2 a1 )
1 a1 (1−a1 a2 ) a1 − aj a1 1 (1−a1 a3 ) = 1 − a1 aj 1 − |a1 |2 .. j >1 . a1
(1−a1 an )
1 0 a1 − aj 1 0 = 1 − a1 aj . 1 − |a1 |2 j >1 .. 0
a3 (a1 −an ) (1−a3 an )(1−a3 a1 )
1 a3 (1−a3 a2 ) a3 (1−|a3 |2 )
.. .
.. .
a2 (1−a2 an )
a3 (1−a3 an )
.. .
an (a1 −an ) (1−|an |2 )(1−an a1 )
··· ···
1 an (1−an a2 ) an (1−an a3 )
··· .. . ···
.. . an (1−|an |2 )
−(a1 −a3 ) (1−a3 a2 )(1−a1 a2 ) −(a1 −a3 ) (1−|a3 |2 )(1−a1 a3 )
.. .
.. .
−(a1 −a2 ) (1−a2 an )(1−a1 an )
··· ···
1
−(a1 −a2 ) (1−|a2 |2 )(1−a1 a2 ) −(a1 −a2 ) (1−a2 a3 )(1−a1 a3 )
1−a2 an
n(n−1) 2
··· .. . ···
1
1
1 1−an a1 an (a1 −a2 ) (1−an a2 )(1−an a1 ) an (a1 −a3 ) (1−an a3 )(1−an a1 )
···
a2 (1−|a2 |2 ) a2 (1−a2 a3 )
1 1−|a2 |2 |a1 − aj |2 1−a12 a3 1 n−1 = (−1) 1 − |a1 |2 (1 − a1 aj )2 .. j >1 . 1
= (−1)
···
··· .. . ···
−(a1 −a3 ) (1−a3 an )(1−a1 an ) 1 1−a3 a2 1 1−|a3 |2
.. . 1 1−a3 an
··· ··· .. . ···
n |aj − ak |2 1 = 0. 2 (1 − aj ak )2 j =1 (1 − |aj | )
n
j =1 j
This gives that system (2.1) has only the trivial solution and hence c1 = c2 = · · · = cn = 0.
2
1 1−an a2 1 1−an a3
.. . 1 1−|an |2
1
−(a1 −an ) (1−an a2 )(1−a1 a2 ) −(a1 −an ) (1−an a3 )(1−a1 a3 )
.. .
−(a1 −an ) (1−|an |2 )(1−a1 an )
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From the above proof we immediately see 1 1−a1 b1 1 1−a1 b2 1 1−a b 1 3 . . . 1 1−a1 bn
= (−1)
1 1−a2 b1 1 1−a2 b2 1 1−a2 b3
1 1−a3 b1 1 1−a3 b2 1 1−a3 b3
.. .
.. .
1 1−a2 bn
1 1−a3 bn
n(n−1) 2
··· ··· ··· .. . ···
1 1−an b1 1 1−an b2 1 1−an b3
.. . 1 1−an bn
n (aj − ak )(bj − bk ) 1 . (1 − aj bk )2 j =1 (1 − aj bj )
n
(2.2)
j =1 j
The above formula will be used in the proof of Theorem 2.2. Proof of Theorem 2.2. Without loss of generality we may assume that B has n distinct zeros {ak }nk=1 with ak = 0 for each k. n that {Gm } is a Cauchy sequence in We will nshow that the sum k=1 Mk is closed. Suppose the sum k=1 Mk and converges to a function G in A2α . There are functions fkm in A2α such that
Gm (z) =
fnm (B(z)) f1m (B(z)) f2m (B(z)) + + ··· + . 1 − a1 z 1 − a2 z 1 − an z
(2.3)
To show that G is in nj=1 Mj , we need only to show that for each j , {fj m } is Cauchy in A2α . Since B is a finite Blaschke product, the Bochner theorem [21] gives that critical points of B in the closed unit disk are contained in a compact subset of the open unit disk. So we may assume that for each point z on the unit circle, there is an open neighborhood U (z) of z such that n (local) inverses ρj : U (z) → ρj (U (z)) of B −1 on U (z) (we have to emphasize that {ρj }nj=1 depend on U (z)) satisfying the following conditions: (1) (2) (3) (4) (5)
B(ρj (w)) = w for w ∈ U (z); each ρj is analytic on U (z); ρj (w) = ρk (w) for w in the closure of U (z) if j = k; each ρj (w) does not vanish on the closure of U (z); and ρj (U (z) ∩ D) ⊂ D for j = 1, . . . , n.
Substituting ρj into both sides of equality (2.3) by ρj and condition (1) give f1m (B(ρj (w))) f2m (B(ρj (w))) fnm (B(ρj (w))) + + ··· + Gm ρj (w) = 1 − a1 ρj (w) 1 − a2 ρj (w) 1 − an ρj (w) =
f2m (w) fnm (w) f1m (w) + + ··· + 1 − a1 ρj (w) 1 − a2 ρj (w) 1 − an ρj (w)
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for w ∈ U (z) ∩ D and each j = 1, . . . , n. We obtain the following system of equations: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 1−a1 ρ1 (w) 1 1−a1 ρ2 (w) 1 1−a1 ρ3 (w)
1 1−a2 ρ1 (w) 1 1−a2 ρ2 (w) 1 1−a2 ρ3 (w)
1 1−a3 ρ1 (w) 1 1−a3 ρ2 (w) 1 1−a3 ρ3 (w)
.. .
.. .
.. .
1 1−a1 ρn (w)
1 1−a2 ρn (w)
1 1−a3 ρn (w)
···
⎞
1 1−an ρ1 (w) 1 1−an ρ2 (w) 1 1−an ρ3 (w)
··· ··· .. . ···
⎛ ⎞ ⎞ ⎛ f1m (w) Gm (ρ1 (w)) ⎟ ⎟ ⎜ f2m (w) ⎟ ⎜ Gm (ρ2 (w)) ⎟ ⎟⎜ ⎟ ⎟ ⎜ ⎟ ⎜ f3m (w) ⎟ = ⎜ Gm (ρ3 (w)) ⎟ . ⎜ ⎟⎜ ⎟ ⎟ .. .. ⎟⎝ ⎠ ⎠ ⎝ . . ⎠ Gm (ρn (w)) fnm (w)
.. . 1 1−an ρn (w)
Using Cramer’s rule to solve the above system of equations, by conditions (1) and (5) and equality (2.2), we have that there are uniformly bounded functions Fj k (w) on U (z) such that fj m (w) =
n
Fj k (w)Gm ρk (w) .
k=1
Therefore for some positive constants C and Cz and any positive integers m and m , U (z) ∩ D
fj m (w) − fj m (w)2 dAα
C C
n
Gm ρj (w) − Gm ρj (w) 2 dAα
j =1 U (z)∩D n
Gm (λ) − Gm (λ)2
j =1 ρ (U (z)∩D) j
CCz
1 |ρj (ρj−1 (λ))|2
1 − |ρj−1 (λ)|2 1 − |λ|2
α dAα
Gm (w) − Gm (w)2 dAα .
D
The last inequality follows from condition (4) and 2 1 − |w|2 ≈ 1 − ρj (w) . This comes from condition (1) and the fact that 2 1 − |λ|2 ≈ 1 − B(λ) since 2 1 − |a1 |2 2 1 − B(λ) = 1 − |λ| |1 − a1 λ|2 2 1 − |a2 |2 + φa1 (λ) + ··· + |1 − a2 λ|2
n−1 2 1 − |an |2 φa (λ) . j |1 − an λ|2 j =1
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982
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Here φa (z) is the Mobius transform φa (z) =
z−a . 1 − az
Since the unit circle is compact, there are finitely many {U (zk )}lk=1 covering the unit circle. Thus there is 0 < r < 1 such that {w ∈ D: r < |w| < 1} is contained in lk=1 U (zj ) and so for any integers m and m ,
l fj m (w) − fj m (w)2 dAα
fj m (w) − fj m (w)2 dAα
k=1 U (z ) ∩ D k
r<|w|<1
lC
max Czi
1il
Gm (w) − Gm (w)2 dAα . D
Since χ{r<|w|<1} dAα is a reversed Carleson measure for A2α , by [15], there is a positive constant C2 such that for all f ∈ A2α ,
f (w)2 dAα C2
D
f (w)2 dAα .
r<|w|<1
Thus for any integers m and m ,
fj m (w) − fj m (w)2 dAα C2
D
fj m (w) − fj m (w)2 dAα
r<|w|<1
lCC2
2 Gm (w) − Gm (w) dAα . max Czi
1il
D
This implies that for each j , {fj m } is Cauchy in A2α since {Gm } is Cauchy in A2α . We may assume that fj m converges to a function fj in A2α and hence converges pointwise to fj . Taking the pointwise limit on both sides of (2.3) gives G(z) =
fn (B(z)) f1 (B(z)) f2 (B(z)) + + ··· + . 1 − a1 z 1 − a2 z 1 − an z
2 Therefore the sum nk=1 Mk is a closed n subspace of Aα . Next we will show that the sum k=1 Mk is a Banach direct sum M1 M2 · · · Mn on the weighted Bergman space A2α .
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To do so, let
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982 fk (B) 1−ak z
be functions in Mk such that n fk (B) =0 1 − ak z k=1
for some functions fk = easy calculations give
∞
m=0 ckm z
m
in the weighted Bergman space A2α . On the other hand,
n n ∞ fk (B) 1 = ckm B m 1 − ak z 1 − ak z k=1
k=1
m=0
∞ n
=
ckm
m=0 k=1
1 B m. 1 − ak z
Taking the inner product of both sides of the above equality with each reproducing kernel kaj of the weighted Bergman space A2α at aj and noting that the powers of B vanish at each aj , we have the following system of equations ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 1−|a1 |2 1 1−a1 a2 1 1−a1 a3
1 1−a2 a1 1 1−|a2 |2 1 1−a2 a3
.. .
.. .
1 1−a1 an
1 1−a2 an
···
1 1−a3 a1 1 1−a3 a2 1 1−|a3 |2
··· ··· .. . ···
.. .
1 1−a3 an
1 1−an a1 1 1−an a2 1 1−an a3
.. . 1 1−|an |2
⎞
⎛ ⎞ ⎛ ⎞ c10 0 ⎟ ⎟ ⎜ c20 ⎟ ⎜ 0 ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ c30 ⎟ = ⎜ 0 ⎟ . ⎟⎜ . ⎟ ⎜ . ⎟ ⎟ ⎝ . ⎠ ⎝ .. ⎠ . ⎠ 0 cn0
Lemma 2.3 gives that the above system has only the trivial solution and hence c10 = c20 = · · · = cn0 = 0. Let gk =
∞
ckj zj −1 .
j =1
Then {gk }nk=1 are functions in A2α such that fk = zgk and n fk (B) 0= 1 − ak z k=1
=
n Bgk (B) k=1
=B
1 − ak z
n gk (B) . 1 − ak z k=1
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So we obtain n gk (B) = 0. 1 − ak z k=1
Repeating the above argument gives c11 = c21 = · · · = cn1 = 0. By the induction we have that cij = 0 for any i, j to get fk (B) = 0. 1 − ak z This implies that 0 in the sum
n
k=1 Mk
has the unique decomposition n
0 = 0 + 0 + ··· + 0. Therefore the sum nj=1 Mk is a Banach direct sum M1 M2 · · · Mn . Next we will show that M1 M2 · · · Mn is dense in the weighted Bergman space A2α and hence equals A2α . Suppose that f is a function in the weighted Bergman space A2α but orthogonal to each Mk . Then f (z) D
1 dAα = 0 1 − zak
for each k. We claim f (z) D
1 dAα = 0 (1 − zak )m
for each k and all m 0. Assuming the claim, we have D
α f (z) 1 − |z|2
1 dA = 0 (1 − zak )m
(2.4)
for all m. This implies that the Bergman projection P˜ [f (z)(1 − |z|2 )α ] of the function f (z)(1 − |z|2 )α into the unweighted Bergman space A20 also equals zero at ak and all derivatives of P˜ [f (z)(1 − |z|2 )α ] equal zero at ak and so α ! = 0. P˜ f (z) 1 − |z|2
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Here P˜ is the Bergman projection from L2 (D, dA) onto the unweighted Bergman space A20 . Taking derivatives of P˜ [f (z)(1 − |z|2 )α ] and evaluating at 0 give
α f (z) 1 − |z|2 zm dA = 0
D
for all m 0. Thus
α f (z) 1 − |z|2 p(z) dA = 0
D
for any polynomials p(z) and hence
f (z)2 1 − |z|2 α dA = 0
D
as polynomials are dense in A2α . This gives that f equals 0. We are going to prove our claim by induction. We warn the reader that although having (2.4) for one point ak is enough, one needs it for all ak for the induction step. Clearly, our claim is true for m = 1. Assume that the claim is true for m K, i.e., 1 f (z) dAα = 0 (1 − zak )m D
for each k and all m K. For m = K + 1, we note that for each k, the rational function
BK 1−ak z
can
1 be written as a sum of partial fractions with the highest power of (1−a is K + 1 and the highest k z) 1 1 power of the rest term (1−aj z) is K for j = k. Thus the term (1−a z)K+1 is a linear combination k BK of terms { (1−a1j z)m }j =1,...,n; m=1,...,K and 1−a . So kz
f (z) D
This gives the claim.
1 dAα = 0. (1 − zak )K+1
2
For two bounded operators S and T , we say that T is similar to S if there is an invertible operator X such that T = XSX −1 . We use T ∼ S to denote that T is similar to S. Now we turn to the proof of Theorem 2.1. B−λ is the n-th order Blaschke product Proof of Theorem 2.1. Since for most λ in the unit disk, 1−λB with n distinct zeros in the unit disk, without loss of generality we may assume that B has n
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982
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distinct zeros {ak }nk=1 with ak = 0 for each k. Now we use the same notation as in the above proof. First we will show that on each Mk , TB is similar to Tz on the weighted Bergman space. To do this, let Yk = T 1 CB : A2α → Mk . Clearly, Yk is a bounded invertible operator from A2α onto Mk . Moreover,
1−ak z
Yk Tz f = Yk (zf ) Bf (B) 1 − ak z " # f (B) = TB 1 − ak z =
= TB Yk f for each f ∈ A2α . By Theorem 2.2, the weighed Bergman spaces A2α = M1 M2 · · · Mn . Thus we conclude that TB ∼
n $
Tz .
2
k=1
3. Some notation and lemmas In this section we introduce some notation and give a few lemmas which will be used in the proof of our main result in the last section. Let {Tf } denote the commutant of Tf , the set of bounded operators commuting with Tf on the Bergman space L2a . We need the following theorem to study the similarity of analytic Toeplitz operators. The version of the following theorem on the Hardy space was obtained in [20]. For more general results on the Hardy space, see [2] and [19]. The method in [20] also works on the weighted Bergman spaces. For the details of the proof of the following theorem, see the proof of theorem in [20, pp. 524–528] by replacing the reproducing kernel on the Hardy space by one on weighted Bergman spaces A2α . Theorem 3.1. If f is analytic on the closure of the unit disk, then there are a finite Blaschke product B and a function h analytic on the closure of unit disk such that f =h◦B and {Tf } = {TB } . Let T be a bounded operator on a Hilbert space H . An idempotent Q in the commutant {T } is said to be minimal if for every idempotent R in {T } , Q = R, whenever Ran R ⊂ Ran Q. Here Ran R denotes the range of the operator R.
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A bounded operator T on a Hilbert space is said to be strongly irreducible if there is no nontrivial idempotent operator in its commutant. In the other words, T is strongly irreducible if and only if XT X −1 is irreducible for each invertible operator X. Lemma 3.2. If Q is a minimal idempotent in {T } , then T |Ran Q is strongly irreducible. Proof. Let A = T |Ran Q . Let R be an idempotent in {A} . In other words, AR = RA, and Ran R ⊂ Ran Q. We write H as a Banach direct sum H = Ran Q Ran(I − Q) and extend R to H by ˜ + y) = Rx R(x if x ∈ Ran Q and y ∈ Ran(I − Q). Clearly, R˜ is an idempotent. For each x in Ran Q, T Rx = T |Ran Q Rx = ARx = RAx = RT |Ran Q x = RT x. For y ∈ Ran(I − Q), write y = (I − Q)z and then ˜ y = RT ˜ (I − Q)z RT ˜ − Q)T z = R(I =0 and ˜ = 0. T Ry This gives ˜ = RT ˜ y. T Ry
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982
2973
Thus we obtain ˜ = T R. ˜ RT So R˜ is in {T } . Since Q is a minimal idempotent in {T } , we conclude that R˜ = Q and hence R = Q. 2 Lemma 3.3. If f is analytic on the closure of the unit disk and B is a Blaschke product B with order n such that {Tf } = {TB } , then there is a function h bounded and analytic on the unit disk such that Tf ∼
n $
Th ,
{Th } = {Tz }
j =1
and Th is strongly irreducible. Proof. First we get a decomposition of Tf as a Banach direct sum of strongly irreducible operators. By Theorem 2.1, there is an invertible operator X from A2α to nj=1 A2α such that XTB X −1 =
n $
Tz .
j =1
Let Pi be the projection from
n
2 j =1 Aα
onto the i-th component. Clearly,
n $ j =1
Tz
= Tz . Ran Pi
Since {Tz } = {Tg : g ∈ H ∞ } is isomorphic to the Banach algebra H ∞ and H ∞ does not contain any nontrivialidempotents, Tz is strongly irreducible. Thus Pi is a minimal idempotent in the commutant { nj=1 Tz } . Let Qi = X −1 Pi X. So {Qi }ni=1 are n minimal idempotents in {Tf } with n
Qj = I.
j =1
This follows from the hypothesis: {Tf } = {TB } and the fact that {TB } contains exactly n minimal idempotents.
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Let Si = Tf |Ran Qi and Ti = XSi X −1 . We have XTf X −1 = X
n
Si X −1
i=1
=
n $
Ti .
i=1
The above direct sum follows from the fact that {Pi }ni=1 are orthogonal projections on Since %
n $
& Tz
n
2 j =1 Aα .
( ' = TF : F ∈ Mn H ∞ ,
j =1
we have Ti T z = T z T i . It is well known that there are functions f1 , . . . , fn in H ∞ such that Ti = Tfi for each i. This gives the decomposition of Tf as we desired: ⎛T XTf X
on
−1
⎜ ⎜ =⎜ ⎜ ⎝
f1
0 0 .. . 0
0 Tf2 0 .. . 0
0 0 Tf3 .. . 0
··· 0 ⎞ ··· 0 ⎟ ⎟ ··· 0 ⎟ .. .. ⎟ ⎠ . . · · · Tfn
n
2 j =1 Aα .
Next we need to show that f1 = f2 = · · · = fn . The above equalities follow from Tf1 = Tf2 = · · · = Tfn . To simplify the proof, we are going to show only that Tf1 = Tf2 since the same argument gives that Tfi = Tfj .
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982
2975
To do so, let ⎛
on
n
2 j =1 Aα .
⎛
Clearly, V is in {
0 ⎜I ⎜ ⎜0 ⎜. ⎝ ..
I 0 0 .. .
0 0 I .. .
0
0
0
⎞ 0 0⎟ ⎟ 0⎟ .. ⎟ .⎠
0 ⎜I ⎜ 0 V =⎜ ⎜. ⎝ ..
I 0 0 .. .
0 0 I .. .
··· ··· ··· .. .
0
0
0
··· I
n
j =1 Tz }
⎞⎛T 0 f1 0⎟⎜ 0 ⎟⎜ 0⎟⎜ 0 ⎜ . .. ⎟ . ⎠ ⎝ .. ··· I 0
··· ··· ··· .. .
0 Tf2 0 .. . 0
.
Easy computations give
0 0 Tf3 .. . 0
··· 0 ⎞ ⎛ 0 · · · 0 ⎟ ⎜ Tf1 ⎟ ⎜ ··· 0 ⎟ = ⎜ 0 ⎜ .. .. ⎟ ⎠ ⎝ .. . . . · · · Tfn 0
Tf2 0 0 .. . 0
0 0 Tf3 .. . 0
··· 0 ⎞ ··· 0 ⎟ ⎟ ··· 0 ⎟ .. .. ⎟ ⎠ . . · · · Tfn
Tf1 0 0 .. . 0
0 0 Tf3 .. . 0
··· 0 ⎞ ··· 0 ⎟ ⎟ ··· 0 ⎟. .. .. ⎟ ⎠ . . · · · Tfn
and ⎛T ⎜ ⎜ ⎜ ⎜ ⎝
f1
0 0 .. . 0
0 Tf2 0 .. . 0
0 0 Tf3 .. . 0
··· 0 ⎞⎛0 ··· 0 ⎟⎜I ⎟⎜ ··· 0 ⎟⎜0 ⎜ .. .. ⎟ ⎠ ⎝ ... . . 0 · · · Tfn
I 0 0 .. .
0 0 I .. .
0
0
⎞ ⎛ 0 0 0 ⎟ ⎜ Tf2 ⎟ ⎜ 0⎟ = ⎜ 0 ⎜ . .. ⎟ . ⎠ ⎝ .. ··· I 0
··· ··· ··· .. .
Since V
n $
Tfj =
j =1
n $
Tfj V ,
j =1
we have Tf1 = Tf2 to obtain XTf X −1 =
n $
Th
j
where h = f1 = f2 = · · · = fn . Moreover, since Pj is a minimal idempotent, by Lemma 3.2, Th is strongly irreducible. Finally we show that {Th } = {Tz } .
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Letting A be in {Th } , we define ⎛
⎞ 0 ··· 0 0 ··· 0 ⎟ ⎟ A ··· 0 ⎟. .. .. .. ⎟ . . .⎠ 0 ··· A
A 0 ⎜0 A ⎜ 0 0 U =⎜ ⎜. . ⎝ .. .. 0 0 Thus U is in the commutant %
n $
& Th
( ' = XTf X −1
j
and so A is an analytic Toeplitz operator. Hence ( ' {Th } = Tg : g ∈ H ∞ = {Tz } .
2
The decomposition of Tf in the above lemma is the so-called “strongly irreducible decomposition” of Tf . In fact the decomposition for the above Toeplitz operator is unique up to the similarity. To state the result in [1] and [13], we need some notation. Let B be a Banach algebra and Proj(B) be the set of all idempotents in B. Murray–von Neumann equivalence ∼a is introduced in Proj(B). Let e and e˜ be in Proj(B). We say that e ∼a e˜ if there are two elements x, y ∈ B such that xy = e,
yx = e. ˜
Let Proj(B) denote the equivalence classes of Proj(B) under Murray–von Neumann equivalence ∼a . Let M∞ (B) =
∞ )
Mn (B)
n=1
where Mn (B) is the algebra of n × n matrices with entries in B. Let * (B) = Proj M∞ (B) . The direct sum of two matrices gives a natural addition in M∞ (B) and hence induces an addition + in Proj(M∞ (B)) by [p] + [q] = [p ⊕ q] + where [p] denotes the equivalence class of p. ( (B), +) forms a semigroup and depends on B + only up to stable isomorphism. K0 (B) is the Grothendieck group of (B). Let A be a bounded operator on a Hilbert space H . We use H (n) to denote the direct sum of n copies of H and A(n) to denote the direct sum of n copies of A acting on H (n) .
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Let T be a bounded operator on a Hilbert space H and Q = {Qj : 1 j l} be a family of minimal idempotents such that l
Qj = I
j =1
and Qj Qi = 0 if j = i. We say that Q is a strongly irreducible decomposition of T . In fact, let Tj = T |Ran Qj . Then each Tj is strongly irreducible and T is similar to l $
Tj .
j =1
Under similarity, {T1 , . . . , Tn } is classified as equivalence classes {[Tj1 ], . . . , [Tjk ]}. Let ni be the number of elements in [Tji ]. Then T is similar to k $
(n )
Tji i .
i=1
If for any two strongly irreducible decompositions Q1 = {Qj : 1 j l1 } and Q2 = ˜ j : 1 j l2 } of T , we have that l1 = l2 and there are a permutation π of {1, 2, . . . , l1 } {Q and invertible bounded operators Xj from Ran Qj onto Ran Q˜ π(j ) such that Xj T |Ran Qj = T |Ran Q˜ π(j ) Xj , then we say that T has unique strongly irreducible decomposition up to similarity. We need the following theorem [1] and [13]. Theorem 3.4. Let T be a bounded operator on a Hilbert space H . The following are equivalent: (a) T is similar to
k i=1
(ni )
Ai
under the decomposition of the space
H=
k $
(ni )
Hi
,
i=1
where k and ni are finite, Ai is strongly irreducible and Ai is not similar to Aj if i = j . Moreover T (n) + has a unique strongly irreducible decomposition up to similarity. (b) The semigroup ({T } ) is isomorphic to the semigroup N(k) where N = {0, 1, 2, . . .} and the isomorphism φ sends
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[I ] → n1 e1 + n2 e2 + · · · + nk ek where {ei }ki=1 are the generators of N(k) and ni = 0. (n )
(n )
1 2 Remark. The above theorem + immediately gives that if T ∼ T1 ⊕ T1 , both T1 and T2 are strongly irreducible and ({T } ) is isomorphic to the semigroup N, then T1 is similar to T2 .
4. Proof of main result In this section we give the proof of Theorem 1.1. For a Fredholm operator T on a Hilbert space, we use index T to denote the Fredholm index: index T = dim ker T − dim ker T ∗ . Clearly, the product formula of the Fredholm index [5] gives index T = k index S if T = S (k) and both T and S are Fredholm. Proof of Theorem 1.1. Suppose that there are two finite Blaschke products B and B1 with the same order n and a function h analytic on the closure of the unit disk such that f = h ◦ B, g = h ◦ B1 . By Theorem 2.1, Tf is similar to Th(zn ) and Tg is similar to Th(zn ) . Thus Tf is similar to Tg . Conversely, suppose that Tf is similar to Tg . Since f and g are analytic functions on the closure of the unit disk, by Theorem 3.1, there are two finite Blaschke products B and B˜ and two functions h and h1 analytic on the closure of the unit disk such that f = h ◦ B,
˜ g = h1 ◦ B,
and {Tf } = {TB } ,
{Tg } = {TB˜ } .
˜ By Lemma 3.3, Tf has the following strongly Let n be the order of B and n1 the order of B. irreducible decomposition Tf ∼
n $ j =1
(n)
Th = Th
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2979
and Tg has the following strongly irreducible decomposition Tg ∼
n1 $ j =1
(n )
Th1 = Th1 1 .
First we show that Th ∼ Th1 . To do so, let T = Tf ⊕ Tg . Then T is similar to (n)
(n )
Th ⊕ Th1 1 (2n)
and hence it is similar to Th
. This means that there is an invertible operator Y such that Y T Y −1 = Th
(2n)
.
This implies ( ' (2n) ( ' = TF : F ∈ M2n H ∞ . Y {T } Y −1 = Th Thus the Banach algebra {T } is isomorphic to M2n (H ∞ ). By Theorem 6.11 in [13, p. 203] or Lemma 2.9 in [1, p. 248], we have * * ∞ M2n H ∞ = H = N. So * * M2n H ∞ = N. {T } = + The first equality follows from the fact that the semigroup (B) is invariant for Banach algebra isomorphisms. By the remark after Theorem 3.4, we get that Th is similar to Th1 . Next we show that n = n1 . Since we just proved that Th is similar to Th1 and Tf is similar to Tg , we have (n1 )
Tf ∼ Th
.
Noting that Tf −λ and Th−λ are Fredholm with nonzero index for some λ, by the Fredholm index product formula, we have (n) ! index Tf −λ = index Th−λ = n index Th−λ
and
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C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982 (n1 ) ! = n1 index Th−λ index Tf −λ = index Th−λ
to get that n = n1 . Thus B and B˜ have the same order. Finally we need to show that there is a Mobius transform χ such that h1 = h ◦ χ. Noting that {Th } = {Tz } and {Th1 } = {Tz } , as Th is similar to Th1 , we have that there is an invertible operator Z such that ZTh Z −1 = Th1 and Z{Th } Z −1 = {Th1 } . Thus Z{Tz } Z −1 = {Tz } . By the fact that ( ' {Tz } = TG : G ∈ H ∞ , we have ' ( ( ' Z TG : G ∈ H ∞ Z −1 = TG : G ∈ H ∞ . In particular, there is a function χ in H ∞ such that ZTz Z −1 = Tχ . The spectral picture of Tz forces χ to be a Mobius transform. Since h is analytic on the closure of the unit disk, we can write h as h=
∞
an z n
n=0
with ∞ j =0
for some r > 1. Thus
|an |r n < ∞
C. Jiang, D. Zheng / Journal of Functional Analysis 258 (2010) 2961–2982 ∞
2981
an Tzn = Th
n=0
in the norm topology. So ZTh Z −1 =
∞
n an ZTz Z −1
n=0
=
∞
an (Tχ )n
n=0
=
∞
a n Tχ n
n=0 = T ∞
n=0 an χ
n
.
Since ZTh Z −1 = Th1 , we have h1 =
∞
an χ n = h ◦ χ,
n=0
to obtain f = h ◦ B, ˜ = h ◦ B1 g = h1 ◦ B˜ = h ◦ (χ ◦ B) where B1 = χ ◦ B˜ is a Blaschke product with order n.
2
Acknowledgment We thank the referee for his many useful suggestions and pointing out Theorem 2.2 which is contained in the proof of Theorem 2.1 in the first version of this paper. References [1] Y. Cao, J. Fang, C. Jiang, K-groups of Banach algebras and strongly irreducible decompositions of operators, J. Operator Theory 48 (2002) 235–253. [2] C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978) 1–31. [3] C. Cowen, On equivalence of Toeplitz operators, J. Operator Theory 7 (1982) 167–172. [4] M. Cowen, R. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978) 187–261. [5] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, London, 1972. [6] R. Douglas, Operator theory and complex geometry, arXiv:0710.1880v1 [math.FA]. [7] K. Guo, H. Huang, On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces, J. Operator Theory, in press. [8] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Grad. Texts in Math., vol. 199, Springer-Verlag, New York, 2000. [9] C. Jiang, Similarity classification of Cowen–Douglas operators, Canad. J. Math. 56 (2004) 742–775. [10] C. Jiang, X. Guo, K. Ji, K-group and similarity classification of operators, J. Funct. Anal. 225 (2005) 167–192.
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[11] C. Jiang, K. Ji, Similarity classification of holomorphic curves, Adv. Math. 215 (2007) 446–468. [12] C. Jiang, Y. Li, The commutant and similarity invariant of analytic Toeplitz operators on Bergman space, Sci. China Ser. A 50 (2007) 651–664. [13] C. Jiang, Z. Wang, Structure of Hilbert Space Operators, World Scientific Publishing, Hackensack, NJ, 2006, x+248 pp. [14] Y. Li, On similarity of multiplication operator on weighted Bergman space, Integral Equations Operator Theory 63 (2009) 95–102. [15] D. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985) 85–111. [16] M. Stessin, K. Zhu, Generalized factorization in Hardy spaces and the commutant of Toeplitz operators, Canad. J. Math. 55 (2) (2003) 379–400. [17] S. Sun, On unitary equivalence of multiplication operators on Bergman space, Northeast. Math. J. 1 (1985) 213–222. [18] S. Sun, D. Yu, Super-isometrically dilatable operators, Sci. China 32 (1989) 1447–1457. [19] J. Thomson, The commutant of a class of analytic Toeplitz operators II, Indiana Univ. Math. J. 25 (1976) 793–800. [20] J. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977) 522–529. [21] J. Walsh, The Location of Critical Points, Amer. Math. Soc. Colloq. Publ., vol. 34, 1950.
Journal of Functional Analysis 258 (2010) 2983–3023 www.elsevier.com/locate/jfa
Infinitesimal non-crossing cumulants and free probability of type B Maxime Février a , Alexandru Nica b,∗,1 a Institut de Mathématiques de Toulouse, Equipe de Statistique et Probabilités, Université Paul Sabatier,
31062 Toulouse Cedex 09, France b Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Received 6 August 2009; accepted 21 October 2009 Available online 7 November 2009 Communicated by D. Voiculescu
Abstract Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003) 2263–2303]. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between “type B” and “infinitesimal” is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A1 , . . . , Ak in an infinitesimal noncommutative probability space (A, ϕ, ϕ ) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, ϕ, ϕ ), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A1 , . . . , Ak is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from “usual” free probability. We show that the lattices NC(B) (n) of non-crossing partitions of type B appear in the combinatorial study of (A, ϕ, ϕ ), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A1 , . . . , Ak in (A, ϕ)
* Corresponding author.
E-mail addresses:
[email protected] (M. Février),
[email protected] (A. Nica). 1 Research supported by a Discovery Grant from NSERC, Canada.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.010
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can be naturally upgraded to infinitesimal freeness in (A, ϕ, ϕ ), for a suitable choice of a “companion functional” ϕ : A → C. © 2009 Elsevier Inc. All rights reserved. Keywords: Free probability (of type B); Infinitesimal freeness; Infinitesimal non-crossing cumulants; Dual derivation system
1. Introduction 1.1. The framework of the paper This paper is concerned with a form of free independence for noncommutative random variables, which can be called “freeness of type B” or “infinitesimal freeness”, and occurs in relation to objects of the form (A, ϕ, ϕ ), where A is a unital algebra over C (1.1) and ϕ, ϕ : A → C are linear with ϕ(1A ) = 1, ϕ (1A ) = 0. The motivation for considering objects as in (1.1) is three-fold. (a) This framework generalizes the link-algebra associated to a noncommutative probability space of type B, in the sense introduced by Biane, Goodman and Nica [2]. One can thus take the point of view that (1.1) provides us with an enlarged framework for doing “free probability of type B”. This point of view is justified by the fact that lattices of non-crossing partitions of type B do indeed appear in the underlying combinatorics – see e.g. Theorem 6.4 below, concerning alternating products of infinitesimally free random variables. (b) It turns out to be beneficial to consolidate the functionals ϕ, ϕ from (1.1) into only one functional ϕ : A → G,
ϕ := ϕ + εϕ ,
(1.2)
where G denotes the two-dimensional Grassman algebra generated by an element ε which satisfies ε 2 = 0. Thus G is the extension of C defined as G = {α + εβ | α, β ∈ C},
(1.3)
with multiplication given by (α1 + εβ1 ) · (α2 + εβ2 ) = α1 α2 + ε(α1 β2 + β1 α2 ), and the structure from (1.1) could equivalently be treated as (A, ϕ ), where A is a unital algebra over C (1.4) and ϕ : A → G is C-linear with ϕ (1A ) = 1. The framework (1.4) was discussed in the PhD thesis of Oancea [6], under the name of “scarce2 G-probability space”. Specifically, Chapter 7 of [6] considers a concept of G-freeness for a family of unital subalgebras in a G-probability space, which is defined via a vanishing condition for mixed G-valued cumulants, and generalizes the concept of freeness of type B from [2]. 2 The adjective “scarce” is used in order to distinguish from the concept of “G-probability space” from operator-valued free probability, where one would require the functional ϕ to be G-linear.
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(c) The recent paper [1] by Belinschi and Shlyakhtenko discusses a concept of “infinitesimal distribution” (CX1 , . . . , Xk , μ, μ ) which is exactly as in (1.1), with CX1 , . . . , Xk denoting the algebra of polynomials in noncommuting indeterminates X1 , . . . , Xk . This remarkable paper brings forth the idea that interesting infinitesimal distributions arise when μ is the limit at 0 and μ is the derivative at 0 for a family of k-variable distributions (μt : CX1 , . . . , Xk → C)t∈T , where T is a set of real numbers having 0 as accumulation point. As we will show below, this ties in really nicely with the G-valued cumulant considerations mentioned in (b); indeed, one could say that [1] puts the ε from (1.3) in its right place – it is a sibling of the ε’s from calculus, only that instead of just having “ε 2 much smaller than ε” one goes for the radical requirement that ε 2 = 0. Upon consideration, it seems that what goes best with the framework from (1.1) is the “infinitesimal” terminology from (c), which is in particular adopted in the next definition. Throughout the paper some terminology inspired from (a) and (b) will also be used, in the places where it is suggestive to do so (e.g. when talking about “soul companions for ϕ” in Section 1.3 below). Definition 1.1. 1o A structure (A, ϕ, ϕ ) as in (1.1) will be called an infinitesimal noncommutative probability space (abbreviated as incps). 2o Let (A, ϕ, ϕ ) be an incps and let A1 , . . . , Ak be unital subalgebras of A. We will say that A1 , . . . , Ak are infinitesimally free with respect to (ϕ, ϕ ) when they satisfy the following condition: If i1 , . . . , in ∈ {1, . . . , k} are such that i1 = i2 , i2 = i3 , . . . , in−1 = in , and if a1 ∈ Ai1 , . . . , an ∈ Ain are such that ϕ(a1 ) = · · · = ϕ(an ) = 0, then ϕ(a1 · · · an ) = 0 and ⎧ ⎨ ϕ(a1 an )ϕ(a2 an−1 ) · · · ϕ(a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ), ϕ (a1 · · · an ) = ⎩ if n is odd and i1 = in , i2 = in−1 , . . . , i(n−1)/2 = i(n+3)/2 , 0, otherwise.
(1.5)
Recall that in the free probability literature it is customary to use the name noncommutative probability space for a pair (A, ϕ) where A is a unital algebra over C and ϕ : A → C is linear with ϕ(1A ) = 1. Thus the concept of infinitesimal noncommutative probability space is obtained by adding to (A, ϕ) another functional ϕ as in (1.1). It is also immediate that Definition 1.1.2o of infinitesimal freeness is obtained by adding the condition (1.5) to the “usual” definition for the freeness of A1 , . . . , Ak in (A, ϕ) (as appearing e.g. in [10, Definition 2.5.1]). Definition 1.1.2o is a reformulation of the concept with the same name from Definition 13 of [1]. The relations with [1,2] are discussed more precisely in Section 2 (cf. Remarks 2.8, 2.9). Section 2 also collects a few miscellaneous properties of infinitesimal freeness that follow directly from the definition. Most notable among them is that one can easily extend to infinitesimal framework the well-known free product construction of noncommutative probability spaces (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ), as presented e.g. in Lecture 6 of [5]. More precisely: if (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ) =: (A, ϕ) and if we are given linear functionals ϕi : Ai → C such that ϕi (1A ) = 0, 1 i k, then there exists a unique linear functional ϕ : A → C such that ϕ | Ai = ϕi , 1 i k, and such that A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ). (See Proposition 2.4 below.) The resulting incps (A, ϕ, ϕ ) can thus be taken, by definition, as the free product of (Ai , ϕi , ϕi ) for 1 i k.
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1.2. Non-crossing cumulants for (A, ϕ, ϕ ) An important tool in the combinatorics of free probability is the family of non-crossing cumulant functionals (κn : An → C)n1 associated to a noncommutative probability space (A, ϕ). These functionals were introduced in [9]; for a detailed presentation of their basic properties, see Lecture 11 of [5]. For every n 1, the multilinear functional κn : An → C is defined via a certain summation formula over the lattice NC(n) of non-crossing partitions of {1, . . . , n}. We will review the formula for a general κn in Section 3.2, here we only pick a special value of n that we use for illustration, e.g. n = 3. In this special case one has κ3 (a1 , a2 , a3 ) = ϕ(a1 a2 a3 ) − ϕ(a1 )ϕ(a2 a3 ) − ϕ(a2 )ϕ(a1 a3 ) − ϕ(a3 )ϕ(a1 a2 ) + 2ϕ(a1 )ϕ(a2 )ϕ(a3 ),
∀a1 , a2 , a3 ∈ A.
(1.6)
The expression on the right-hand side of (1.6) has 5 terms (premultiplied by integer coefficients3 such as 1, −1, or 2), corresponding to the fact that |NC(3)| = 5. Let now (A, ϕ, ϕ ) be an incps as in Definition 1.1. Then in addition to the non-crossing cumulant functionals κn : An → C associated to ϕ we will define another family of multilinear functionals (κn : An → C)n1 , which involve both ϕ and ϕ . For every n 1, the functional κn is obtained by taking a formal derivative in the formula for κn , where we postulate that the derivative of ϕ is ϕ and we invoke linearity and the Leibnitz rule for derivatives. For instance for n = 3 the term ϕ(a1 a2 a3 ) on the right-hand side of (1.6) is derivated into ϕ (a1 a2 a3 ), the term ϕ(a1 )ϕ(a2 a3 ) is derivated into ϕ (a1 )ϕ(a2 a3 ) + ϕ(a1 )ϕ (a2 a3 ), etc, yielding the formula for κ3 to be κ3 (a1 , a2 , a3 ) = ϕ (a1 a2 a3 ) − ϕ (a1 )ϕ(a2 a3 ) − ϕ(a1 )ϕ (a2 a3 ) − ϕ (a2 )ϕ(a1 a3 ) − ϕ(a2 )ϕ (a1 a3 ) − ϕ (a3 )ϕ(a1 a2 ) − ϕ(a3 )ϕ (a1 a2 ) + 2ϕ (a1 )ϕ(a2 )ϕ(a3 ) + 2ϕ(a1 )ϕ (a2 )ϕ(a3 ) + 2ϕ(a1 )ϕ(a2 )ϕ (a3 ).
(1.7)
We will refer to the functionals κn as infinitesimal non-crossing cumulants associated to (A, ϕ, ϕ ). The precise formula defining them appears in Definition 4.2 below. The passage from the formula for κn to the one for κn is related to a concept of dual derivation system on a space of multilinear functionals on A, which is discussed in Section 7 of the paper. The role of infinitesimal non-crossing cumulants in the study of infinitesimal freeness is described in the next theorem. Theorem 1.2. Let (A, ϕ, ϕ ) be an incps and let A1 , . . . , Ak be unital subalgebras of A. The following statements are equivalent: (1) A1 , . . . , Ak are infinitesimally free. (2) For every n 2, for every i1 , . . . , in ∈ {1, . . . , k} which are not all equal to each other, and for every a1 ∈ Ai1 , . . . , an ∈ Ain , one has that κn (a1 , . . . , an ) = κn (a1 , . . . , an ) = 0. 3 The meaning of these coefficients is that they are special values of the Möbius function of NC(3), as reviewed more precisely in Section 3.
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Theorem 1.2 provides an infinitesimal version for the basic result of Speicher which describes the usual freeness of A1 , . . . , Ak in (A, ϕ) in terms of the cumulants κn (cf. [5, Theorem 11.16]). In the remaining part of this subsection we point out some other interpretations of the formula defining κn (all corresponding to one or another of the points of view (a), (b), (c) listed at the beginning of Section 1.1). The easy verifications required by these alternative descriptions of κn are shown at the beginning of Section 4. First of all, one can consider, as in [1], the situation when ϕ, ϕ in (1.1) are obtained as the infinitesimal limit of a family of functionals {ϕt | t ∈ T }. Here T is a subset of R which has 0 as an accumulation point, every ϕt is linear with ϕt (1A ) = 1, and we have ϕ(a) = lim ϕt (a) t→0
and
ϕt (a) − ϕ(a) , t→0 t
ϕ (a) = lim
∀a ∈ A.
(1.8)
(Note that such families {ϕt | t ∈ T } can in fact always be found, e.g. by simply taking ϕt = ϕ + tϕ , t ∈ (0, ∞).) In such a situation, the formal derivative which leads from κn to κn turns d out to have the same effect as a “ dt ” derivative. Consequently, we get the alternative formula κn (a1 , . . . , an ) =
d (t) κn (a1 , . . . , an ) , dt t=0
(1.9)
(t)
where κn denotes the nth non-crossing cumulant functional of ϕt . Second of all, it is possible to take a direct combinatorial approach to the functionals κn , and identify precisely a set of non-crossing partitions which indexes the terms in the summation defining κn (a1 , . . . , an ). This set turns out to be
NCZ (B) (n) := τ ∈ NC(B) (n) τ has a zero-block ,
(1.10)
where NC(B) (n) is the lattice of non-crossing partitions of type B of {1, . . . , n} ∪ {−1, . . . , −n} (see Section 3.1 for a brief review of this). Hence in a terminology focused on types of noncrossing partitions, one could call the functionals κn and κn “non-crossing cumulants of type A and of type B”, respectively. The idea put forth here is that, in some sense, summations over NCZ (B) (n) appear as “derivatives for summation over NC(n)”. A more refined formula supporting this idea is shown in Proposition 7.6 below, in connection to the concept of dual derivation system. In the case n = 3 that we are using for illustration, the 10 terms appearing on the righthand side of (1.7) are indexed by the 10 partitions with zero-block in NC(B) (3). For example, the partitions corresponding to the first three terms and the last term from (1.7) are depicted in Fig. 1. The relation between a partition τ and the corresponding term is easy to follow: the zero-block Z of τ produces the ϕ ( · · · ) factor, and every pair V , −V of non-zero-blocks of τ produces a ϕ( · · · ) factor. Finally (third of all) one can also give a description of κn which corresponds to the “Gvalued” point of view appearing as (b) on the list from Section 1.1. This goes as follows. Let ϕ = ϕ + εϕ : A → G be as in (1.2), and consider the family of C-multilinear functionals ( κn : An → G)n1 defined by the same summation formula as for the usual non-crossing cumulant functionals (κn : An → C)n1 , only that now we use ϕ instead of ϕ in the summations. So, for example, for n = 3 we have
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Fig. 1. Some partitions in NCZ (B) (3).
κ3 (a1 , a2 , a3 ) = ϕ (a1 a2 a3 ) − ϕ (a1 ) ϕ (a2 a3 ) − ϕ (a2 ) ϕ (a1 a3 ) ϕ (a1 a2 ) + 2 ϕ (a1 ) ϕ (a2 ) ϕ (a3 ) ∈ G, − ϕ (a3 )
∀a1 , a2 , a3 ∈ A.
(1.11)
κn . It then turns out that the functional κn can be obtained by reading the ε-component of We take the opportunity to introduce here a piece of terminology from the literature on Grassman algebras (see e.g. [3, pp. 1–2]): the complex numbers α, β which give the two components of a Grassman number γ = α + εβ ∈ G will be called the body and respectively the soul of γ ; it will come in handy throughout the paper to denote them4 as α = Bo(γ ),
β = So(γ ).
(1.12)
This notation will also be used in connection to a G-valued function f defined on some set S – we define functions Bo f and So f from S to C by
(Bo f )(x) = Bo f (x) ,
(So f )(x) = So f (x) ,
∀x ∈ S.
(1.13)
Returning then to the functionals κn : An → G from the preceding paragraph, their connection to the κn (and also to the κn ) can be recorded as Bo κn = κn ,
So κn = κn ,
∀n 1.
(1.14)
4 Besides being amusing, “Bo” and “So” give a faithful analogue for the common notations “Re” and “Im” used when one introduces C as a 2-dimensional algebra over R.
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Due to (1.14), κn can be used as a simplifying tool in calculations with κn (in the sense that it may be easier to run the corresponding calculation with κn , in G, and only pick soul parts at the end of the calculation). In particular, this will be useful when proving Theorem 1.2, since the condition κn (a1 , . . . , an ) = κn (a1 , . . . , an ) = 0 from Theorem 1.2(2) amounts precisely to κn (a1 , . . . , an ) = 0. 1.3. Using derivations to find “soul companions” for a given ϕ When studying infinitesimal freeness it may be of interest to consider the situation where we have fixed a noncommutative probability space (A, ϕ) and a family A1 , . . . , Ak of unital subalgebras of A which are free in (A, ϕ). In this situation we can ask: how do we find interesting examples of functionals ϕ : A → C with ϕ (1A ) = 0 and such that A1 , . . . , Ak become infinitesimally free in (A, ϕ, ϕ )? A nice name for such functionals ϕ is suggested by the G-valued point of view described in Section 1.1: since ϕ and ϕ are the body part and respectively the soul part of the consolidated functional ϕ : A → G, one may say that we are looking for a suitable soul companion ϕ for the given “body functional” ϕ (and in reference to the given subalgebras A1 , . . . , Ak ). Let us note that the remark made at the end of Section 1.1 can be interpreted as a statement about soul companions. Indeed, this remark says that if (A, ϕ) is the free product of (A1 , ϕ1 ), . . . , (Ak , ϕk ), then a ϕ from the desired set of soul companions is parametrized precisely by a family of linear functionals ϕi : Ai → C such that ϕi (1A ) = 0, 1 i k. The point we follow here, with inspiration from [1], is that some interesting recipes to construct “soul companions” for a given ϕ : A → C arise from ideas pertaining to differentiability. This is intimately related to the fact that κn is a formal derivative for κn , hence to equations of the form dn (κn ) = κn ,
∀n 1,
where (dn )n1 is a dual derivation system on A. Indeed, suppose we are given a derivation D : A → A; then one has a natural dual derivation system associated to it, which acts by (dn f )(a1 , . . . , an ) =
n
f a1 , . . . , am−1 , D(am ), am+1 , . . . , an ,
(1.15)
m=1
for f : An → C multilinear and a1 , . . . , an ∈ A. By using the dn from (1.15), we obtain the following theorem. Theorem 1.3. Let (A, ϕ, ϕ ) be an incps, and let (κn )n1 , (κn )n1 be the families of non-crossing and respectively of infinitesimal non-crossing cumulant functionals associated to this incps. Suppose D : A → A is a derivation with the property that ϕ = ϕ ◦ D. Then for every n 1 and a1 , . . . , an ∈ A one has κn (a1 , . . . , an ) =
n
κn a1 , . . . , am−1 , D(am ), am+1 , . . . , an .
(1.16)
m=1
Moreover, when combined with Theorem 1.2, the formula for infinitesimal cumulants obtained in (1.16) has the following immediate consequence.
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Corollary 1.4. Let (A, ϕ) be a noncommutative probability space, and let A1 , . . . , Ak be unital subalgebras of A which are free in (A, ϕ). Suppose we found a derivation D : A → A such that D(Ai ) ⊆ Ai for every 1 i k. Then A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ), where ϕ = ϕ ◦ D. For comparison, let us also look at the parallel statement arising in connection to infinitesimal limits. This is essentially the same as Remark 15 from [1], and goes as follows. Proposition 1.5. Let (A, ϕ) be a noncommutative probability space, and let A1 , . . . , Ak be unital subalgebras of A which are free in (A, ϕ). Suppose we found a family of linear functionals (ϕt : A → C)t∈T with ϕt (1A ) = 1 for every t ∈ T (where T ⊆ R has 0 as an accumulation point), and such that the following conditions are fulfilled: (i) A1 , . . . , Ak are free in (A, ϕt ) for every t ∈ T . (ii) limt→0 ϕt (a) = ϕ(a), for every a ∈ A. (iii) The limit ϕ (a) := limt→0 (ϕt (a) − ϕ(a))/t exists, for every a ∈ A. Then A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ), where ϕ : A → C is defined by condition (iii). A natural example accompanying Proposition 1.5 comes in connection to -convolution powers of joint distributions of k-tuples (cf. Example 8.9 below). In Section 8 we also discuss a couple of natural situations when Corollary 1.4 applies (cf. Example 8.7). 1.4. Outline of the rest of the paper Besides the introduction, the paper has seven other sections. In Section 2 we collect some basic properties of infinitesimal freeness, and we discuss the relations between Definition 1.1 and the frameworks of [1,2]. Section 3 is a review of background concerning non-crossing partitions and non-crossing cumulants. In Section 4 we introduce the non-crossing infinitesimal cumulants, we verify the equivalence between their various alternative descriptions, and we prove Theorem 1.2. Sections 5 and 6 address the topic of alternating products of infinitesimally free random variables. Section 5 uses this topic to illustrate a “generic” method to obtain infinitesimal analogues for known results in usual free probability: one replaces C by G in the proof of the original result, then one takes the soul part in the G-valued statement that comes out. By using this method we obtain the infinitesimal versions of two important facts related to alternating products that were originally found in [4] – one of them is about compressions by free projections, the other concerns a method of constructing free families of free Poisson elements. In Section 6 we remember that the concept of incps has its origins in the considerations “of type B” from [2], and we look at how the essence of these considerations persists in the framework of the present paper. The main point of the section is that, when taking the soul part of the G-valued formulas for alternating products of infinitesimally free random variables, one does indeed obtain nice analogues of type B (with summations over NC(B) (n)) for the type A formulas. In particular, this offers another explanation for why the infinitesimal cumulant functional κn can be described by using a summation formula over NCZ (B) (n). In Section 7 we return to the point of view of treating κn as a derivative of the usual noncrossing cumulant functional κn , and we discuss the related concept of dual derivation system
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on a unital algebra A. Finally, Section 8 elaborates on the discussion about soul companions from the above Section 1.3. In particular, we show how the dual derivation system provided by a derivation D : A → A leads to the setting for infinitesimal freeness from Corollary 1.4. Section 8 (and the paper) concludes with a couple of examples related to the settings of Corollary 1.4 and of Proposition 1.5. 2. Basic properties of infinitesimal freeness In this section we collect some basic properties of infinitesimal freeness, and we discuss the relations between Definition 1.1 and the frameworks from [1,2]. Definition 2.1. Here are some standard variations of Definition 1.1. 1o The concept of infinitesimal freeness carries over to ∗-algebras. More precisely, we will use the name ∗-incps for an incps (A, ϕ, ϕ ) where A is a unital ∗-algebra and where (i) ϕ is positive definite, that is, ϕ(a ∗ a) 0, ∀a ∈ A; (ii) ϕ is selfadjoint, that is, ϕ (a ∗ ) = ϕ (a), ∀a ∈ A. 2o Another standard variation of the definitions is that infinitesimal freeness can be considered for arbitrary subsets of A (which don’t have to be subalgebras). So if (A, ϕ, ϕ ) is an incps (respectively a ∗-incps) and if X1 , . . . , Xk are subsets of A, then we will say that X1 , . . . , Xk are infinitesimally free (respectively infinitesimally ∗-free) when the unital subalgebras (respectively ∗-subalgebras) generated by X1 , . . . , Xk are so. Remark 2.2. Let (A, ϕ) be a noncommutative probability space and let A1 , . . . , Ak be unital subalgebras of A which are free in (A, ϕ). Let us denote M := Alg(A1 ∪ · · · ∪ Ak ) (the subalgebra of A generated by A1 ∪ · · · ∪ Ak ). It is easy to see (cf. Remark 2.5.2 in [10] or Examples 5.15 in [5]) that the way how ϕ acts on M can be reconstructed from the restrictions ϕ | Ai , 1 i k. The simplest illustration for how this works is provided by the formula ϕ(ab) = ϕ(a)ϕ(b),
∀a ∈ Ai1 , b ∈ Ai2 ,
with i1 = i2 ,
(2.1)
which is obtained by expanding the product and then collecting terms in the equation ϕ((a − ϕ(a)1A ) · (b − ϕ(b)1A )) = 0. A similar phenomenon turns out to take place when dealing with infinitesimal freeness: the way how ϕ acts on Alg(A1 ∪ · · · ∪ Ak ) can be reconstructed from the restrictions of ϕ and of ϕ to Ai , 1 i k. For example, the counterpart of Eq. (2.1) says that ϕ (ab) = ϕ (a)ϕ(b) + ϕ(a)ϕ (b),
∀a ∈ Ai1 , b ∈ Ai2 ,
where i1 = i2 .
(2.2)
This is obtained by expanding the product and then collecting terms in the equation ϕ ((a − ϕ(a)1A ) · (b − ϕ(b)1A )) = 0 (which is a particular case of Eq. (1.5)), and by taking into account that ϕ (1A ) = 0. We leave it as an easy exercise to the reader to verify that the similar calculation for an alternating product of 3 factors (which makes a more involved use of Eq. (1.5)) leads to the formula ϕ (a1 ba2 ) = ϕ (a1 a2 )ϕ(b) + ϕ(a1 a2 )ϕ (b),
for a1 , a2 ∈ Ai1 , b ∈ Ai2 ,
with i1 = i2 . (2.3)
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Remark 2.3. (Traciality.) Another well-known fact in usual free probability is that if the unital subalgebras A1 , . . . , Ak ⊆ A are free in (A, ϕ) and if ϕ | Ai is a trace for every 1 i k, then ϕ is a trace on Alg(A1 ∪ · · · ∪ Ak ). This too extends to the infinitesimal framework: if A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ) and if ϕ | Ai , ϕ | Ai are traces for every 1 i k, then ϕ and ϕ are traces on Alg(A1 ∪ · · · ∪ Ak ). Rather than writing an ad-hoc proof of this fact based directly on Definition 1.1, we find it more instructive to do this by using cumulants – see Proposition 4.11 below. We next move to describing the free product of infinitesimal noncommutative probability spaces announced at the end of Section 1.1. Proposition 2.4. Let (A1 , ϕ1 ), . . . , (Ak , ϕk ) be noncommutative probability spaces, and consider the free product (A, ϕ) = (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ) (as described e.g. in Lecture 6 of [5]). Suppose that for every 1 i k we are given a linear functional ϕi : Ai → C such that ϕi (1A ) = 0. Then there exists a unique linear functional ϕ : A → C such that ϕ | Ai = ϕi , 1 i k, and such that A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ). Proof. We start by reviewing a few basic facts and notations related to (A, ϕ). Each of A1 , . . . , Ak is identified as a unital subalgebra of A, such that ϕ | Ai = ϕi . For 1 i k we denote Aoi = {a ∈ Ai | ϕ(a) = 0}, and for every n 1 and 1 i1 , . . . , in k such that i1 = i2 , . . . , in−1 = in we put
Wi1 ,...,in := span a1 · · · an a1 ∈ Aoi1 , . . . , an ∈ Aoin .
(2.4)
It is known that Wi1 ,...,in is canonically isomorphic to the tensor product Aoi1 ⊗ · · · ⊗ Aoin , via the identification a1 · · · an a1 ⊗ · · · ⊗ an , for a1 ∈ Aoi1 , . . . , an ∈ Aoin . Moreover it is known that the spaces Wi1 ,...,in defined in (2.4) realize a direct sum decomposition of the kernel of ϕ. (See [5, pp. 81–84].) Due to the direct sum decomposition mentioned above, we may define the required functional ϕ by separately prescribing its behaviour at 1A and on each of the subspaces Wi1 ,...,in . We put ϕ (1A ) := 0. We also prescribe ϕ to be 0 on Wi1 ,...,in whenever n is even, and whenever n is odd but it is not true that im = in+1−m for all 1 m (n − 1)/2. Suppose next that n = 2m − 1, odd, and that the indices i1 , . . . , in are such that i1 = i2m−1 , i2 = i2m−2 , . . . , im−1 = im+1 . By using the identification Wi1 ,...,in Aoi1 ⊗ · · · ⊗ Aoin it is immediate that we can define a linear map on Wi1 ,...,in by the requirement that a1 · · · a2m−1 → ϕi1 (a1 a2m−1 )ϕi2 (a2 a2m−2 ) · · · ϕim−1 (am−1 am+1 ) · ϕim (am ), for every a1 ∈ Aoi1 , . . . , an ∈ Aoin ; we take this as the prescription for how ϕ is to act on Wi1 ,...,in . Directly from Definition 1.1 it is immediate that, with ϕ : A → C defined as in the preceding paragraph, A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ). The uniqueness of ϕ with this property is also immediate. 2 Definition 2.5. Let (A1 , ϕ1 , ϕ1 ), . . . , (Ak , ϕk , ϕk ) be infinitesimal noncommutative probability spaces. We define their free product to be (A, ϕ, ϕ ) where (A, ϕ) = (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ) and where ϕ : A → C is the functional provided by Proposition 2.4.
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Remark 2.6. In the context of Proposition 2.4, suppose that (A1 , ϕ1 ), . . . , (Ak , ϕk ) are ∗probability spaces. Then so is the free product (A, ϕ) (see [5, Theorem 6.13]). If moreover each of the functionals ϕi : Ai → C given in Proposition 2.4 is selfadjoint, then it is easily checked that the resulting functional ϕ : A → C is selfadjoint too. Hence if in Definition 2.5 each of (Ai , ϕi , ϕi ) is a ∗-incps, then the free product (A, ϕ, ϕ ) is a ∗-incps as well. Example 2.7. For an illustration of the above, we look at a simple example where the spaces Wi1 ,...,in are all 1-dimensional. Consider the k-fold free product group Z2 ∗ · · · ∗ Z2 and let ϕ be the canonical trace on the group algebra A := C[Z2 ∗ · · · ∗ Z2 ]. So A is a unital ∗-algebra freely generated by k unitaries u1 , . . . , uk of order 2, and has a linear basis B given by B = {1A } ∪ ui1 · · · uin
n 1, 1 i1 , . . . , in k, . with i = i , . . . , i 1 2 n−1 = in
(2.5)
The linear functional ϕ : A → C acts on the basis B by ϕ(1A ) = 1,
and ϕ(b) = 0,
∀b ∈ B \ {1A }.
It is easy to verify (see e.g. Lecture 6 in [5]) that we have (A, ϕ) = (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ), where for 1 i k we denote Ai = span{1A , ui } (2-dimensional ∗-subalgebra of A), and where ϕi := ϕ | Ai . The direct sum decomposition of A with respect to this free product structure simply has Wi1 ,...,in = 1-dimensional space spanned by ui1 · · · uin , for every n 1 and every alternating sequence i1 , . . . , in as described in (2.5). Now let ϕi : Ai → C be linear functionals such that ϕi (1A ) = 0, 1 i k. Clearly, these functionals are determined by the values ϕ1 (u1 ) =: α1 ,
...,
ϕk (uk ) =: αk .
The free product extension ϕ : A → C then acts by ϕ (ui1 · · · uin ) =
αim , 0,
if n = 2m − 1 and i1 = i2m−1 , . . . , im−1 = im+1 , otherwise.
(2.6)
Note that formula (2.6) looks particularly nice in the case when k = 2 – indeed, in this case the requirement that i1 = i2m−1 , . . . , im−1 = im+1 is automatically satisfied whenever n = 2m − 1 and i1 , . . . , in are as in (2.5). Remark 2.8. (Relation to [1]). Definition 13 of [1] introduces a concept of infinitesimal freeness for unital subalgebras A1 , A2 ⊆ A in an incps (A, μ, μ ). As explained there (immediately following to Definition 13), this amounts to two requirements: that A1 , A2 are free in (A, μ), and that they satisfy the following additional condition:
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μ p1 − μ(p1 )1A · · · pn − μ(pn )1A =
n
μ p1 − μ(p1 )1A · · · μ (pm ) · · · pn − μ(pn )1A
(2.7)
m=1
for p1 ∈ Ai1 , . . . , pn ∈ Ain , where i1 = i2 , . . . , in−1 = in . By denoting pm − μ(pm )1A =: qm and by taking into account that μ (qm ) = μ (pm ), 1 m n, one sees that condition (2.7) is equivalent to its particular case requesting that
μ (q1 · · · qn ) =
n
μ(q1 · · · qm−1 qm+1 · · · qn ) · μ (qm )
(2.8)
m=1
for q1 ∈ Ai1 , . . . , qn ∈ Ain , where i1 = i2 , . . . , in−1 = in and where μ(q1 ) = · · · = μ(qn ) = 0. But now, let A1 , A2 be unital subalgebras of A which are free in (A, μ). A standard calculation from usual free probability (see e.g. [5, Lemma 5.18 on p. 73]) says that, with q1 , . . . , qn as in (2.8), one has μ(q1 · · · qm−1 qm+1 · · · qn ) = 0 unless it is true that m − 1 = n − m and that im−1 = im+1 , im−2 = im+2 , . . . , i1 = in ; moreover, if the latter conditions are satisfied, then μ(q1 · · · qm−1 qm+1 · · · qn ) = μ(qm−1 qm+1 )μ(qm−2 qm+2 ) · · · μ(q1 qn ). This clearly implies that the sum on the right-hand side of (2.8) has at most one term which is different from 0; and moreover, when such a term exists, it is exactly as described in Eq. (1.5) of Definition 1.1. Hence, modulo an immediate reformulation, the concept of infinitesimal freeness from [1] is the same as the one used in this paper (which justifies the fact that we are calling it by the same name). Remark 2.9. (Relation to [2]). A noncommutative probability space of type B is defined in [2] as a system (A, ϕ, V, f, Φ), where (A, ϕ) is a noncommutative probability space, V is a complex vector space, f : V → C is a linear functional, and Φ : A × V × A → V is a two-sided action. We will write for short aξ b and respectively aξ , ξ b instead of Φ(a, ξ, b) and respectively Φ(a, ξ, 1A ), Φ(1A , ξ, b), for a, b ∈ A and ξ ∈ V. Let A1 , . . . , Ak be unital subalgebras of A and let V1 , . . . , Vk be linear subspaces of V, such that Vi is closed under the two-sided action of Ai , 1 i k. Definition 7.2 of [2] introduces a concept of what it means for (A1 , V1 ), . . . , (Ak , Vk ) to be free in (A, ϕ, V, f, Φ). This amounts to two requirements: that A1 , . . . , Ak are free in (A, ϕ), and that the following additional condition is satisfied: ⎧ ⎨ ϕ(a1 b1 ) · · · ϕ(an bn )f (ξ ), f (am . . . a1 ξ b1 . . . bn ) = if m = n and i1 = j1 , . . . , in = jn , ⎩ 0, otherwise,
(2.9)
holding for m, n 0 and a1 ∈ Ai1 , . . . , am ∈ Aim , b1 ∈ Aj1 , . . . , bn ∈ Ajn , ξ ∈ Vh , where any two consecutive indices among im , . . . , i1 , h, j1 , . . . , jn are different from each other, and where ϕ(am ) = · · · = ϕ(a1 ) = 0 = ϕ(b1 ) = · · · = ϕ(bn ).
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Now, to (A, ϕ, V, f, Φ) as above one associates a link-algebra, which is simply the direct product M = A × V endowed with the natural structure of complex vector space and with multiplication (a, ξ ) · (b, η) = (ab, aη + ξ b),
∀a, b ∈ A, ξ, η ∈ V.
(2.10)
If we define ψ, ψ : M → C by
ψ (a, ξ ) := ϕ(a),
ψ (a, ξ ) := f (ξ ),
∀(a, ξ ) ∈ M,
(2.11)
then (M, ψ, ψ ) becomes an incps. Let again A1 , . . . , Ak be unital subalgebras of A and V1 , . . . , Vk be linear subspaces of V such that Vi is closed under the two-sided action of Ai , 1 i k. Then M1 := A1 × V1 , . . . , Mk := Ak × Vk are unital subalgebras of the linkalgebra M, and we claim that ⎛
⎞ (A1 , V1 ), . . . , (Ak , Vk ) ⎝ are free in (A, ϕ, V, f, Φ), ⎠ in the sense of [2]
⎛ ⇔
⎞ M1 , . . . , Mk are inf. free ⎝ in (M, ψ, ψ ), in the ⎠ . sense of Definition 1.1
(2.12)
In order to prove the implication “⇐” in (2.12), we only have to write
f (am . . . a1 ξ b1 . . . bn ) = ψ (am , 0V ) · · · (a1 , 0V ) · (0A , ξ ) · (b1 , 0V ) · · · (bn , 0V ) and then invoke Eq. (1.5). For the implication “⇒”, consider some elements (a1 , ξ1 ) ∈ Mi1 , . . . , (an , ξn ) ∈ Min where i1 = i2 , . . . , in−1 = in and where ψ((a1 , ξ1 )) = · · · = ψ((an , ξn )) = 0 (which just means that ϕ(a1 ) = · · · = ϕ(an ) = 0). By using how the multiplication on M and how ψ are defined, we see that n
f (a1 · · · am−1 ξm am+1 · · · an ). ψ (a1 , ξ1 ) · · · (an , ξn ) =
(2.13)
m=1
But because of (2.9), at most one term in the sum on the right-hand side of (2.13) can be different from 0; moreover such a term can only occur for m = (n + 1)/2, if (n is odd and) i1 = i2m−1 , . . . , im−1 = im+1 . Finally, if the latter equalities of indices are satisfied, then the unique term left in the sum from (2.13) is ϕ(a1 a2m−1 ) · · · ϕ(am−1 am+1 )f (ξm ), and the conditions defining the infinitesimal freeness of M1 , . . . , Mk in (M, ψ, ψ ) follow. Hence, by focusing on the link-algebra, one can incorporate the freeness of type B from [2] into the framework of this paper. 3. Background on non-crossing partitions and non-crossing cumulants 3.1. Non-crossing partitions Notation 3.1. We will use the standard conventions of notation concerning non-crossing partitions (as they appear for instance in Lecture 9 of [5]). So for a positive integer n we denote by NC(n) the set of all non-crossing partitions of {1, . . . , n}. We will use the abbreviation “V ∈ π ” for “V is a block of π ”, and the number of blocks of π ∈ NC(n) will be denoted as |π|. On
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NC(n) we will consider the partial order given by reverse refinement; that is, for π, ρ ∈ NC(n) we write “π ρ” to mean that every block of ρ is a union of blocks of π . The minimal and maximal element of (NC(n), ) are denoted by 0n (the partition of {1, . . . , n} into n blocks of 1 element each) and respectively 1n (the partition of {1, . . . , n} into 1 block of n elements). It is easy to see that (NC(n), ) is a lattice, i.e. that every π, ρ ∈ NC(n) have a join (smallest common upper bound) and a meet (largest common lower bound), which will be denoted by π ∨ ρ and π ∧ ρ, respectively. Remark 3.2. A block W of a partition π ∈ NC(n) is called an interval-block if it is of the form W = [p, q] ∩ Z for some 1 p q n. Every non-crossing partition has interval-blocks, and it is actually easy to check that the following stronger statement holds: let π be in NC(n), let V be a block of π , and let i < j be two elements of V which are consecutive in V (in the sense that (i, j ) ∩ V = ∅). If j = i + 1 (hence the interval (i, j ) contains some integers) then there exists an interval-block W of π such that W ⊆ (i, j ). Notation 3.3. The lattice of non-crossing partitions of type B of 2n elements will be denoted by NC(B) (n). Following the paper of Reiner [8] where NC(B) (n) was introduced, it is customary to denote the 2n elements that are being partitioned as 1, . . . , n and −1, . . . , −n, taken in the order 1 < · · · < n < −1 < · · · < −n. So let NC(±n) denote the lattice of all non-crossing partitions of the ordered set {1, . . . , n} ∪ {−1, . . . , −n}. (NC(±n) is thus just a copy of NC(2n), where one puts different labels on some of the 2n points that are being partitioned.) Then NC(B) (n) consists of those partitions τ ∈ NC(±n) which have the symmetry property that (V is a block of τ )
⇒
(−V is a block of τ )
(with −V := {−v | v ∈ V } ⊆ {1, . . . , n} ∪ {−1, . . . , −n}). NC(B) (n) inherits from NC(±n) the partial order by reverse refinement, and is closed under the operations ∨, ∧, hence is a sublattice of NC(±n). Note also that NC(B) (n) contains the minimal and maximal elements of NC(B) (n), which will be denoted as 0±n and 1±n , respectively. A block Z of a partition τ ∈ NC(B) (n) is called a zero-block when it satisfies the condition Z = −Z. The set {τ ∈ NC(B) (n) | τ has zero-blocks} will be denoted by NCZ (B) (n). Due to the non-crossing property, it is immediate that every τ ∈ NCZ (B) (n) has exactly one zero-block (hence it is justified to talk about “the zero-block” of τ ). Remark 3.4 (Kreweras complementation). An important ingredient in the study of the lattice NC(n) is a special anti-automorphism Kr : NC(n) → NC(n), called the Kreweras complementation map (see [5, pp. 147–148]). Since NC(±n) NC(2n), one also has such a map Kr on NC(±n). (All occurrences of Kreweras complementation maps in this paper will be denoted in the same way, by “Kr”.) Moreover, the sublattice NC(B) (n) ⊆ NC(±n) turns out to be invariant under the Kr map of NC(±n), hence one can talk about the Kreweras complementation map on NC(B) (n) as well. It is easily checked that Kr : NC(B) (n) → NC(B) (n) maps the sets NCZ (B) (n) and NC(B) (n) \ NCZ (B) (n) bijectively onto each other (see Section 1.2 of [2]). Remark 3.5 (Absolute value map). Let Abs : {1, . . . , n} ∪ {−1, . . . , −n} → {1, . . . , n} denote the absolute value map sending ±i to i for 1 i n. In [2] it was observed that it makes sense to extend the concept of “absolute value” to non-crossing partitions. That is, for τ ∈ NC(B) (n) it makes sense to define Abs(τ ) ∈ NC(n) to be the partition of {1, . . . , n} into blocks of the form
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Abs(V ), V ∈ τ . Moreover, Section 1.4 of [2] puts into evidence the remarkable fact that the map Abs : NC(B) (n) → NC(n) so defined is an (n + 1)-to-1 map, and explains precisely how to find the n + 1 partitions in Abs−1 (π), for a given π ∈ NC(n). A part of this result which is important for the present paper is that for every π ∈ NC(n) and V ∈ π there exists a unique τ ∈ NCZ (B) (n) such that Abs(τ ) = π and such that the zero-block Z of τ has Abs(Z) = V . Clearly, this can be rephrased by saying that we have a bijection
⎧ (B) V ∈π ⎪ ⎨ NCZ (n) → (π, V ) π ∈ NC(n),
τ → Abs(τ ), Abs(Z) , ⎪ ⎩ where Z := the unique zero-block of τ .
(3.1)
Moreover, for every π ∈ NC(n), the n + 1 − |π| partitions in Abs−1 (π) that are not accounted by (3.1) are all from NC(B) (n) \ NCZ (B) (n), and are naturally indexed by the blocks of Kr(π). For the explanation of how this happens, we refer to the Remark on p. 2270 of [2]. Remark 3.6 (Möbius functions). We will use the notation “Möb(A) ” for the Möbius functions of the lattices NC(n). The value Möb(A) (π, ρ) for π ρ in NC(n) can be given explicitly, as a product of signed Catalan numbers (see [5, p. 163]). In the present paper we will not need the concrete values Möb(A) (π, ρ), but only the Möbius inversion formula; this says that if we have two families of vectors {fπ | π ∈ NC(n)} and {gπ | π ∈ NC(n)} in the same vector space over C, then the relations
gρ =
fπ ,
∀ρ ∈ NC(n)
(3.2)
π∈NC(n), πρ
are equivalent to fρ =
Möb(A) (π, ρ) · gπ ,
∀ρ ∈ NC(n).
(3.3)
π∈NC(n), πρ
We will use the notation “Möb(B) ” for the Möbius functions of the lattices NC(B) (n). The explicit values Möb(B) (σ, τ ) for σ τ in NC(B) (n) can be read off from the considerations in Section 3 of [8]. Here we will only need a simple connection between the types A and B, saying that
σ τ in NCZ (B) (n) ⇒ Möb(B) (σ, τ ) = Möb(A) Abs(σ ), Abs(τ ) .
(3.4)
For the proof of (3.4) one observes that Abs gives a poset isomorphism between the intervals [σ, τ ] ⊆ NC(B) (n) and [Abs(σ ), Abs(τ )] ⊆ NC(n), then uses the fact that the values Möb(B) (σ, τ ) and Möb(A) (Abs(σ ), Abs(τ )) only depend on the isomorphism classes (in the category of posets) of these intervals. 3.2. Non-crossing cumulants, in the usual C-valued setting The following notation for “restrictions of n-tuples” will be used throughout the whole paper.
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Notation 3.7. Let (a1 , . . . , an ) be an n-tuple of elements in a set A, and let V = {v1 , . . . , vm } be a non-empty subset of {1, . . . , n}, with v1 < · · · < vm . Then we denote (a1 , . . . , an ) | V := (av1 , . . . , avm ) ∈ Am .
(3.5)
Definition 3.8. Let (A, ϕ) be a noncommutative probability space. The multilinear functionals (κn : An → C)n1 defined by
κn (a1 , . . . , an ) =
Möb(A) (π, 1n ) ·
ϕ|V | (a1 , . . . , an ) V ,
V ∈π
π∈NC(n)
for n 1 and a1 , . . . , an ∈ A
(3.6)
(with ϕ|V | as in Remark 3.10 below) are called the non-crossing cumulant functionals associated to (A, ϕ). The importance of non-crossing cumulants for free probability theory comes from the following theorem, originally found in [9] (see also the detailed presentation in Lecture 11 of [5]). Theorem 3.9. Let (A, ϕ) be a noncommutative probability space and let A1 , . . . , Ak be unital subalgebras of A. The following statements are equivalent: (1) A1 , . . . , Ak are free. (2) For every n 2, for every i1 , . . . , in ∈ {1, . . . , k} which are not all equal to each other, and for every a1 ∈ Ai1 , . . . , an ∈ Ain , one has that κn (a1 , . . . , an ) = 0. Remark 3.10. Let (A, ϕ) be a noncommutative probability space. For every n 1 let us consider the multiplication map Multn : An → A,
Multn (a1 , . . . , an ) = a1 · · · an ,
(3.7)
and let us denote ϕn := ϕ ◦ Multn . The multilinear functionals (ϕn )n1 are called the moment functionals of (A, ϕ). For every n 1 and π ∈ NC(n) let us next define a multilinear functional ϕπ(A) : An → C by5 ϕπ(A) (a1 , . . . , an ) :=
ϕ|V | (a1 , . . . , an ) V ,
a1 , . . . , an ∈ A.
(3.8)
V ∈π
Then Definition 3.8 can be rephrased as saying that κn =
Möb(A) (π, 1n ) · ϕπ(A) ∈ Mn ,
(3.9)
π∈NC(n)
5 The superscript “(A)” is used in anticipation of the fact that some multilinear functionals ϕ (B) with τ ∈ NC(B) (n) τ
will appear in Section 6 of the paper.
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where Mn denotes the vector space of multilinear functionals from An to C. Moreover, if for (A) every π ∈ NC(n) we introduce (by analogy with (3.8)) a functional κπ ∈ Mn defined by
κπ(A) (a1 , . . . , an ) :=
κ|V | (a1 , . . . , an ) V ,
a1 , . . . , an ∈ A,
(3.10)
V ∈π
then it is not hard to see that the formula (3.9) for κn extends to
κρ(A) =
Möb(A) (π, ρ) · ϕπ(A) ,
∀ρ ∈ NC(n).
(3.11)
π∈NC(n), πρ (A)
(A)
Thus for a given n 1, the families of functionals {κπ | π ∈ NC(n)} and {ϕπ | π ∈ NC(n)} are exactly as in the above Remark 3.6. Eq. (3.11) and its equivalent counterpart which expresses (A) (A) ϕρ as the sum of the functionals {κπ | π ρ} go under the name of non-crossing momentcumulant formulas for (A, ϕ). 3.3. Non-crossing cumulants in the G-valued setting Remark 3.11. We will work with the Grassman algebra G from Section 1.1, and with the maps Bo, So : G → C defined in Section 1.2. It is immediate that the multiplication of G is commutative, and that the “body” map Bo : G → C is a homomorphism of unital algebras. Concerning how the “soul” map So behaves with respect to multiplication, we record the immediately verified formula So(γ1 · · · γn ) =
n
So(γi ) ·
Bo(γj ) ,
∀n 1, ∀γ1 , . . . , γn ∈ G.
(3.12)
j n, j =i
i=1
Notation 3.12. For the rest of this subsection we fix a pair (A, ϕ ) where A is a unital algebra over C and ϕ : A → G is C-linear with ϕ (1A ) = 1. In connection to this ϕ we will repeat all the constructions of functionals described in Remark 3.10, with the only difference that the range ϕ ◦ Multn : An → G, space of these functionals is now G. So for every n 1 we put ϕn = n where Multn : A → A is the same as in Equation (3.7). Then for every π ∈ NC(n) we define ϕπ : An → G by ϕπ (a1 , . . . , an ) :=
ϕ|V | (a1 , . . . , an ) V ,
a1 , . . . , an ∈ A.
(3.13)
V ∈π
This is followed by defining a family of cumulant functionals ( κn : An → G)n1 , where
κn =
Möb(A) (π, 1n ) · ϕπ ,
n 1.
(3.14)
π∈NC(n)
Finally, for every π ∈ NC(n) we define κπ : An → G by κπ (a1 , . . . , an ) :=
V ∈π
κ|V | (a1 , . . . , an ) V ,
a1 , . . . , an ∈ A.
(3.15)
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It is easily seen that, exactly as in the C-valued case from Remark 3.10, the families of funcϕπ | π ∈ NC(n)} are related by moment-cumulant formulas (i.e. tionals { κπ | π ∈ NC(n)} and { by summation formulas as shown in Eqs. (3.2), (3.3) of Remark 3.6). We only record here the special case of moment-cumulant formula which expresses ϕ1n as a sum of cumulant functionals, and thus says that ϕ (a1 · · · an ) = κπ (a1 , . . . , an ) ∈ G, ∀a1 , . . . , an ∈ A. (3.16) π∈NC(n)
Remark 3.13. A natural question concerning (A, ϕ ) is whether the analogue of Theorem 3.9 holds in this framework. As will be explained in detail in Remark 4.9 below, both conditions (1) and (2) from the statement of Theorem 3.9 can be faithfully transcribed in the context of (A, ϕ ), but then they are no longer equivalent to each other – the implication (2) ⇒ (1) still holds, but its converse does not. In the remaining part of this subsection we will point out two other facts from the theory of usual non-crossing cumulants where (unlike for Theorem 3.9) both the statement and the proof can be transcribed without any problems from usual C-valued framework to the G-valued framework of (A, ϕ ). Proposition 3.14. One has that κn (a1 , . . . , an ) = 0 whenever n 2, a1 , . . . , an ∈ A, and there exists 1 m n such that am ∈ C1A . Proof. This is the analogue of Proposition 11.15 in [5]. It is straightforward (left to the reader) to see that the proof shown on p. 182 of [5] goes without any changes to the G-valued framework. 2 Proposition 3.15. Let x1 , . . . , xs be in A and consider some products of the form a1 = x 1 · · · x s 1 ,
a2 = xs1 +1 · · · xs2 ,
...,
an = xsn−1 +1 · · · xsn ,
where 1 s1 < s2 < · · · < sn = s. Then κn (a1 , . . . , an ) =
κπ (x1 , . . . , xs ),
(3.17)
π∈NC(s) such that π∨θ=1s
where θ ∈ NC(s) is the partition with interval blocks {1, . . . , s1 }, {s1 + 1, . . . , s2 }, . . . , {sn−1 + 1, . . . , sn }. Proof. This is the analogue of Theorem 11.20 in [5], and the proof of this theorem (as shown on pp. 178–180 of [5]) goes without any changes to the G-valued framework. 2 4. Infinitesimal cumulants and the proof of Theorem 1.2 Notation 4.1. Throughout this whole section we fix an incps (A, ϕ, ϕ ). We will use the notation “κn ” for the non-crossing cumulant functionals associated to ϕ, as described in Section 3.2. Moreover, we will denote, same as in the introduction: ϕ = ϕ + εϕ : A → G
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and we will consider the family of non-crossing cumulant functionals ( κn : An → G)n1 which are associated to ϕ as in Section 3.3. Definition 4.2. For every n 1, consider the multilinear functional κn : An → C defined by the formula κn (a1 , . . . , an )
Möb(π, 1n )ϕ|V | (a1 , . . . , an ) V · ϕ|W | (a1 , . . . , an ) | W , (4.1) = π∈NC(n) V ∈π
W ∈π W =V
for a1 , . . . , an ∈ A. The functionals κn will be called infinitesimal non-crossing cumulant functionals associated to (A, ϕ, ϕ ). A moment’s thought shows that Eq. (4.1) is indeed obtained from the formula (3.6) defining κn , where one uses the formal derivation procedure announced in Section 1.2 of the introduction. We next make precise (in Propositions 4.3, 4.5 and Remark 4.4) the equivalence between Definition 4.2 and the other facets of κn that were mentioned in Section 1.2. Proposition 4.3. Suppose that ϕ, ϕ are the infinitesimal limit of a family {ϕt | t ∈ T }, in the sense (t) described in Eq. (1.8). Let us use the notation κn for the non-crossing cumulant functional of ϕt , for t ∈ T and n 1. Then for every n 1 and every a1 , . . . , an ∈ A one has that κn (a1 , . . . , an ) = lim κn(t) (a1 , . . . , an ), t→0
and κn (a1 , . . . , an ) =
d (t) κn (a1 , . . . , an ) . dt t=0
Proof. Fix n 1 and a1 , . . . , an ∈ A. For every t ∈ T we have that κn(t) (a1 , . . . , an ) =
π∈NC(n)
Möb(A) (π, 1n ) ·
ϕt (a1 , . . . , an ) V .
(4.2)
V ∈π
(t)
From (4.2) it is clear that limt→0 κn (a1 , . . . , an ) = κn (a1 , . . . , an ). Moreover, it is immediate that the function of t appearing on the right-hand side of (4.2) has a derivative at 0; and upon using linearity and the Leibnitz formula to compute this derivative, one obtains precisely the formula (4.1) that defined κn (a1 , . . . , an ). 2 Remark 4.4. As observed in Remark 3.5, the set {(π, V ) | π ∈ NC(n), V ∈ π} which indexes the sum on the right-hand side of Eq. (4.1) is the image of NCZ (B) (n) via the bijection (τ ∈ NCZ (B) (n) with zero-block Z) → (Abs(τ ), Abs(Z)). When τ and (π, V ) correspond to each other via this bijection, we have that Möb(B) (τ, 1±n ) = Möb(A) (π, 1n ) (cf. (3.4) in Remark 3.6); moreover, the rest of the product indexed by (π, V ) on the right-hand side of Eq. (4.1)
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(B)
is precisely equal to ϕτ (a1 , . . . , an ), where we anticipate here the notation ϕτ from Eq. (6.3). In conclusion, the change of variable from (V , π) to τ converts (4.1) into a summation formula which is a counterpart “of type B” for Eq. (3.9),
κn =
Möb(B) (τ, 1±n ) · ϕτ(B) .
(4.3)
τ ∈NCZ (B) (n)
It is easy to see that (4.3) is equivalent to a plain summation formula which writes ϕ (a1 · · · an ) in terms of cumulants (cf. Remark 6.5 below, where one also sees that the absence of terms indexed by partitions from NC(B) (n) \ NCZ (B) (n) is caused by the fact that ϕ (1A ) = 0). Proposition 4.5. For every n 1 one has that Bo κn = κn and So κn = κn . Proof. For the first statement we only have to take the body part on both sides of Eq. (3.14) and use the fact that Bo : G → C is a homomorphism of unital algebras. For the second statement we take soul parts in (3.14) and then use the multiplication formula (3.12). 2 We now go to Theorem 1.2. Note that, in view of Proposition 4.5, the equalities “κn (a1 , . . . , an ) = κn (a1 , . . . , an ) = 0” from condition (2) of Theorem 1.2 may be replaced with “ κn (a1 , . . . , an ) = 0”. We will prove Theorem 1.2 in this alternative form, which is stated below as Proposition 4.7. Lemma 4.6. Suppose that n is a positive integer and π is a partition in NC(n), such that the following two properties hold: (i) For every 1 i n − 1, the numbers i and i + 1 do not belong to the same block of π . (ii) π has at most one block of cardinality 1. Then n is odd, and π is the partition
{1, n}, {2, n − 1}, . . . , (n − 1)/2, (n + 3)/2 , (n + 1)/2 . Proof. Clearly, condition (i) implies that π cannot have interval-blocks V with |V | 2. By also taking (ii) and Remark 3.2 into account, we thus see that π has a unique interval-block Vo , of the form Vo = {p} for some 1 p n. Let V be a block of π , distinct from Vo . We claim that V ∩ [1, p) 1,
V ∩ (p, n] 1.
(4.4)
Indeed, assume for instance that we had |V ∩ [1, p)| 2. Then we could find i, j ∈ V such that i < j < p and (i, j ) ∩ V = ∅. Note that j = i + 1, due to condition (i); but then, as observed in Remark 3.2, the partition π must have an interval-block W ∩ (i, j ), in contradiction to the fact that the unique interval-block of π is Vo . For every block V = Vo of π it then follows that |V ∩ [1, p)| = |V ∩ (p, n]| = 1. Indeed, if in (4.4) one of the sets V ∩ [1, p), V ∩ (p, n] would be empty, then it would follow that |V | = 1 and hypothesis (ii) would be contradicted.
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The list of blocks of π which are distinct from Vo can thus be written in the form V1 = {i1 , j1 }, . . . , Vm = {im , jm }, where i1 < p < j1 , . . . , im < p < jm , and i1 < i2 < · · · < im .
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(4.5)
Observe that in (4.5) we must have j1 > j2 > · · · > jm . Indeed, if it was true that js < jt for some 1 s < t m, then it would follow that is < it < p < js < jt , and the blocks Vs , Vt would cross. Hence we have obtained i1 < · · · < im < p < jm < · · · < j1 ; together with (4.5), this implies that n = 2m + 1 and that π is precisely the partition indicated in the lemma. 2 Proposition 4.7. Let A1 , . . . , Ak be unital subalgebras of A. The following statements are equivalent: (1) A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ). (2) For every n 2, for every i1 , . . . , in ∈ {1, . . . , k} which are not all equal to each other, and κn (a1 , . . . , an ) = 0. for every a1 ∈ Ai1 , . . . , an ∈ Ain , one has that Proof. “(1) ⇒ (2)”. We prove the required statement about cumulants by induction on n. For the base case n = 2, consider elements a1 ∈ Ai1 and a2 ∈ Ai2 , where i1 = i2 . By using the formulas which define κ2 and κ2 and by invoking Eqs. (2.1) and (2.2) from Remark 2.2 we find that
κ2 (a1 , a2 ) = ϕ(a1 a2 ) − ϕ(a1 )ϕ(a2 ) = 0 and κ2 (a1 , a2 ) = ϕ (a1 a2 ) − ϕ (a1 )ϕ(a2 ) − ϕ(a1 )ϕ (a2 ) = 0,
hence κ2 (a1 , a2 ) = κ2 (a1 , a2 ) + εκ2 (a1 , a2 ) = 0. We now prove the induction step: assume that the vanishing of mixed cumulants is already proved for 1, 2, . . . , n − 1, where n 3. We consider elements a1 ∈ Ai1 , . . . , an ∈ Ain where κn (a1 , . . . , an ) = 0. By not all indices i1 , . . . , in are equal to each other, and we want to prove that invoking Proposition 3.14 we may replace every am with am − ϕ(am )1A , 1 m n, and therefore assume without loss of generality that ϕ(a1 ) = · · · = ϕ(an ) = 0. Observe that this implies ϕ (aq ) = (εϕ (ap )) · (εϕ (aq )) = 0, hence that ϕ (ap ) ϕ (ap aq ) − ϕ (ap ) ϕ (aq ) = ϕ (ap aq ), κ2 (ap , aq ) =
∀1 p < q n.
(4.6)
Another assumption that can be made without loss of generality is that im = im+1 , ∀1 m < n. Indeed, if there exists 1 m < n such that im = im+1 , then we invoke the special case of Proposition 3.15 which states that κn (a1 , . . . , an ) + κπ (a1 , . . . , an ). (4.7) κn−1 (a1 , . . . , am am+1 , . . . , an ) = π∈NC(n) with |π|=2 π separates m and m+1
The induction hypothesis gives us that the left-hand side and every term in the sum on the righthand side of Eq. (4.7) are equal to 0, and it follows that κn (a1 , . . . , an ) must be 0 as well. Hence for the rest of the proof of this induction step we will assume that ϕ(a1 ) = · · · = ϕ(an ) = 0 and that i1 = i2 , . . . , in−1 = in . This makes a1 , . . . , an be exactly as in Definition 1.1, so we get that ϕ(a1 · · · an ) = 0 and that ϕ (a1 · · · an ) is as described in Eq. (1.5). In terms of the functional ϕ , we have
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ϕ (a1 · · · an ) = εϕ (a1 · · · an ) ⎧ ⎨ εϕ(a1 an )ϕ(a2 an−1 ) · · · ϕ(a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ), = if n is odd and i1 = in , i2 = in−1 , . . . , i(n−1)/2 = i(n+3)/2 , ⎩ 0, otherwise.
(4.8)
Now let us consider the relation (3.16), written in the equivalent form κn (a1 , . . . , an ) = ϕ (a1 · · · an ) −
κπ (a1 , . . . , an ).
(4.9)
π∈NC(n), π=1n
Observe that if a partition π ∈ NC(n) has two distinct blocks {p}, {q} of cardinality one, then the term indexed by π on the right-hand side of (4.9) vanishes, because it contains the subproduct κ1 (aq ) = ϕ (ap ) ϕ (aq ) = 0. On the other hand if π ∈ NC(n) has a block V which contains κ1 (ap ) two consecutive numbers i and i + 1, then the term indexed by π on the right-hand side of (4.9) vanishes as well, due to the induction hypothesis. Hence the sum subtracted on the right-hand side of (4.9) can only get non-zero contributions from partitions π ∈ NC(n) which satisfy the hypotheses of Lemma 4.6; from the lemma it then follows that the sum in question is 0 for n even, and is equal to κ2 (a2 , an−1 ) · · · κ2 (a(n−1)/2 , a(n+3)/2 ) · κ1 (a(n+1)/2 ) κ2 (a1 , an )
(4.10)
for n odd. Let us focus for a moment on the quantity that appeared in (4.10). The vanishing of mixed cumulants of order 2 (which is part of our induction hypothesis) implies that this quantity vanishes unless i1 = in , i2 = in−1 , . . . , i(n−1)/2 = i(n+3)/2 . In the case that the latter equalities of indices hold, we can continue (4.10) with ϕ (a2 an−1 ) · · · ϕ (a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ) = ϕ (a1 an )
due to (4.6)
= εϕ(a1 an )ϕ(a2 an−1 ) · · · ϕ(a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ).
(4.11)
(The equality (4.11) holds because ϕ (a(n+1)/2 ) = εϕ (a(n+1)/2 ), and due to how the multiplication on G works.) So all in all, what we have obtained is that κn (a1 , . . . , an ) ⎧ ϕ (a1 · · · an ) − εϕ(a1 an )ϕ(a2 an−1 ) · · · ϕ(a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ), ⎨ = if n is odd and i1 = in , i2 = in−1 , . . . , i(n−1)/2 = i(n+3)/2 , ⎩ ϕ (a1 · · · an ), otherwise.
(4.12)
By comparing Eqs. (4.12) and (4.8) we see that, in all cases, we have κn (a1 , . . . , an ) = 0. This concludes the induction argument, and the proof of the implication (1) ⇒ (2) of the proposition. “(2) ⇒ (1)”. Consider indices i1 , . . . , in ∈ {1, . . . , k} and elements a1 ∈ Ai1 , . . . , an ∈ Ain such that i1 = i2 , . . . , in−1 = in and such that ϕ(a1 ) = · · · = ϕ(an ) = 0. We have to prove that ϕ(a1 · · · an ) = 0 and that ϕ (a1 · · · an ) is as described in formula (1.5) from Definition 1.1. To this
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end we consider the G-valued moment ϕ (a1 · · · an ) = ϕ(a1 · · · an ) + εϕ (a1 · · · an ), and write it in terms of G-valued cumulants as in Section 3.3: ϕ (a1 · · · an ) =
κ|V | (a1 , . . . , an ) V .
(4.13)
π∈NC(n) V ∈π
An argument very similar to the one used in the proof of the implication (1) ⇒ (2) above shows that the sum on the right-hand side of (4.13) can only get non-zero contributions from partitions π ∈ NC(n) which satisfy the hypotheses of Lemma 4.6. If n is even then there is no such partition, and we obtain ϕ (a1 · · · an ) = 0. If n is odd, then the sum in (4.13) reduces to only one term and we obtain that ϕ (a1 · · · an ) = κ2 (a1 , an ) κ2 (a2 , an−1 ) · · · κ2 (a(n−1)/2 , a(n+3)/2 ) · κ1 (a(n+1)/2 ).
(4.14)
Moreover, in the case when n is odd, the hypothesis that mixed cumulants vanish gives us that the right-hand side of (4.14) is equal to 0 unless we have i1 = in , . . . , i(n−1)/2 = i(n+3)/2 . And finally, if the latter equalities of indices hold, then the right-hand side of (4.14) gets converted into εϕ(a1 an )ϕ(a2 an−1 ) · · · ϕ(a(n−1)/2 a(n+3)/2 ) · ϕ (a(n+1)/2 ), by the same argument that led to (4.11) in the proof of the implication (1) ⇒ (2). The conclusion is that ϕ(a1 · · · an ) = 0 (in all cases), and that ϕ (a1 · · · an ) is as in Eq. (1.5), as required. 2 Corollary 4.8. Let X1 , . . . , Xk be subsets of A. The following statements are equivalent: (1) X1 , . . . , Xk are infinitesimally free in (A, ϕ, ϕ ). (2) For every n 2, for every i1 , . . . , in ∈ {1, . . . , k} which are not all equal to each other, and for every x1 ∈ Xi1 , . . . , xn ∈ Xin , one has that κn (x1 , . . . , xn ) = 0. Proof. This is a faithful copy of the proof giving the analogous result over C (cf. [5, Theorem 11.20]). For the reader’s convenience, we repeat here the highlights of the argument. Let Ai denote the unital subalgebra of A generated by Xi , 1 i k. The infinitesimal freeness of X1 , . . . , Xk is by definition equivalent to the one of A1 , . . . , Ak , hence to the fact that condition (2) from Proposition 4.7 holds. We must thus prove that ((2) in Proposition 4.7) is equivalent to ((2) in Corollary 4.8). The implication “⇒” is trivial. For “⇐” it suffices (by multilinearity of κn and Proposition 3.14) to prove that κn (a1 , . . . , an ) = 0 when a1 = x 1 · · · x s 1 ,
a2 = xs1 +1 · · · xs2 ,
...,
an = xsn−1 +1 · · · xsn
(4.15)
for n 2 and 1 s1 < s2 < · · · < sn , where x1 , . . . , xs1 ∈ Xi1 , xs1 +1 , . . . , xs2 ∈ Xi2 , . . . , xsn−1 +1 , . . . , xsn ∈ Xin , and where the indices i1 , . . . , in are not all equal to each other. But for a1 , . . . , an as in (4.15), Proposition 3.15 gives us the cumulant κn (a1 , . . . , an ) as a sum of cumulants κπ (x1 , . . . , xsn ); and a direct combinatorial analysis (exactly as on p. 186 of [5]) shows that all the latter cumulants vanish because of condition (2) from Corollary 4.8. 2 Remark 4.9. Since the functional ϕ : A → G and its associated cumulants κn play such a central role in the proof of Theorem 1.2, it is natural to ask: can’t one actually characterize infinitesimal freeness by the same kind of moment condition as in the definition of usual freeness, with the only
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modification that one now uses ϕ instead of ϕ? To be precise, consider the following condition which a family of unital subalgebras A1 , . . . , Ak ⊆ A may or may not satisfy: ⎧ ⎨ For every n 1 and 1 i1 , . . . , in k such that i1 = i2 , . . . , in−1 = in , and every a1 ∈ Ai1 , . . . , an ∈ Ain such that ϕ (a1 ) = · · · = ϕ (an ) = 0, (4.16) ⎩ one has that ϕ (a1 · · · an ) = 0. Isn’t then condition (4.16) equivalent to infinitesimal freeness? On the positive side it is immediate, directly from Definition 1.1, that (4.16) is indeed implied by infinitesimal freeness. However, the converse statement is not true: it may happen that (4.16) is satisfied and yet A1 , . . . , Ak are not infinitesimally free. What causes this to happen is that one cannot generally “center” an element a ∈ A with respect to ϕ (the scalars available are from C, and there may be no λ ∈ C such that ϕ (a − λ1A ) = 0). This limits the scope of condition (4.16), ϕ | Ai , and makes it insufficient for recomputing ϕ on Alg(A1 ∪ · · · ∪ Ak ) from the restrictions 1 i k. For a simple concrete example showing how (4.16) may fail to imply infinitesimal freeness, suppose we are in the situation from Example 2.7, with A = C[Z2 ∗ · · · ∗ Z2 ] and where A1 = span{1A , u1 }, . . . , Ak = span{1A , uk } are the k copies of C[Z2 ] canonically embedded into A. ϕ = ϕ + εϕ satisfies Suppose moreover that the linear functionals ϕ, ϕ : A → C are such that ϕ (1A ) = 1,
ϕ (u1 ) = · · · = ϕ (uk ) = ε.
(4.17)
Then, no matter how ϕ acts on words of length 2 made with u1 , . . . , uk , it will be true that ϕ ; this is due to the simple reason that the A1 , . . . , Ak satisfy condition (4.16) with respect to restrictions ϕ | Ai (1 i k) are one-to-one. But on the other hand, Remark 2.2 tells us that if ϕ is uniquely determined by (4.17); A1 , . . . , Ak are to be infinitesimally free in (A, ϕ, ϕ ), then for example, the formulas given for illustration in Eqs. (2.1), (2.2) imply that we must have ϕ (u1 ) ϕ (u2 ) = ε 2 = 0. Hence any choice of ϕ as in (4.17) and with ϕ (u1 u2 ) = 0 ϕ (u1 u2 ) = provides an example for how condition (4.16) does not imply infinitesimal freeness. We conclude this section by establishing the fact about traciality that was announced in Remark 2.3. Lemma 4.10. Let B be a unital subalgebra of A, and suppose that ϕ | B is a trace. Then κn (b1 , b2 , . . . , bn ) = κn (b2 , bn , . . . , b1 ),
∀n 2, b1 , . . . , bn ∈ B.
(4.18)
Proof. Let Γ be the cyclic permutation of {1, . . . , n} defined by Γ (1) = 2, . . . , Γ (n − 1) = n, Γ (n) = 1. It is easy to see (cf. [5, Exercise 9.41 on p. 153]) that Γ induces an automorphism of the lattice NC(n) which maps π = {V1 , . . . , Vp } ∈ NC(n) to Γ · π := {Γ (V1 ), . . . , Γ (Vp )}. κn (bΓ (1) , . . . , bΓ (n) ), Now let some b1 , . . . , bn ∈ B be given. The right-hand side of (4.18) is which is by definition equal to Möb(A) (π, 1n ) · ϕπ (bΓ (1) , . . . , bΓ (n) ). (4.19) π∈NC(n)
By taking into account the traciality of ϕ on B it is easily verified that ϕπ (bΓ (1) , . . . , bΓ (n) ) = ϕΓ ·π (b1 , . . . , bn ), ∀π ∈ NC(n). Since Möb(A) (Γ · π, 1n ) = Möb(A) (Γ · π, Γ · 1n ) =
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Möb(A) (π, 1n ), ∀π ∈ NC(n), it becomes clear that the change of variable Γ · π =: ρ will convert the sum from (4.19) into the one which defines κn (b1 , . . . , bn ). 2 Proposition 4.11. Let A1 , . . . , Ak be unital subalgebras of A that are infinitesimally free in (A, ϕ, ϕ ). If ϕ | Ai and ϕ | Ai are traces for every 1 i k, then ϕ and ϕ are traces on Alg(A1 ∪ · · · ∪ Ak ). Proof. The given hypothesis and the required conclusion can be rephrased by saying that ϕ is a trace on every Ai , and respectively that ϕ is a trace on Alg(A1 ∪ · · · ∪ Ak ). Clearly, the rephrased conclusion will follow if we prove that ϕ (x1 · · · xn−1 xn ) = ϕ (xn x1 · · · xn−1 )
(4.20)
for x1 ∈ Ai1 , . . . , xn ∈ Ain , with n 2 and 1 i1 , . . . , in k. Let us fix such n, i1 , . . . , in and x1 , . . . , xn . It is moreover convenient to denote y1 := xn , y2 := x1 , . . ., yn := xn−1 , so that (4.20) ϕ (y1 · · · yn ). takes the form ϕ (x1 · · · xn ) = Let πo be the partition of {1, . . . , n} defined by the requirement that for 1 p < q n we have (p, q in the same block of πo ) ⇔ ip = iq . The hypothesis that A1 , . . . , Ak are infinitesimally free and Proposition 4.7 imply that ϕ (x1 · · · xn ) =
κπ (x1 , . . . , xn ).
(4.21)
π∈NC(n) such that ππo
(Note that πo may not belong to NC(n), but the inequality π πo still makes sense, in reverse refinement order.) Now, by using Lemma 4.10 it is easily checked that for every π ∈ NC(n) such that π πo one has κπ (x1 , . . . , xn ) = κΓ ·π (y1 , . . . , yn ),
(4.22)
where “Γ · π ” has the same significance as in the proof of Lemma 4.10. If we combine (4.21) with (4.22) and then make the change of variable Γ · π =: ρ, we arrive to ϕ (x1 · · · xn ) =
κρ (y1 , . . . , yn ).
(4.23)
ρ∈NC(n) such that ρΓ ·πo
Finally, we invoke once more the infinitesimal freeness of A1 , . . . , Ak and Proposition 4.7, to conclude that the right-hand side of (4.23) is precisely the moment-cumulant expansion for ϕ (y1 · · · yn ). 2 5. Alternating products of infinitesimally free random variables In Proposition 4.7 we saw that infinitesimal freeness can be described as a vanishing condition for mixed G-valued cumulants. Because of this fact and because G is commutative, (which makes practically all calculations with non-crossing cumulants go without any change from C-valued to G-valued framework) we get a “generic method” for proving infinitesimal versions of various results presented in the monograph [5] – replace C by G in the proof of the original result, then
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take the soul part of what comes out. Note that the infinitesimal results so obtained do not have G in their statement, hence could also be attacked by using other approaches to infinitesimal freeness (in which case, however, proving them may be more than a straightforward routine). In this section we show how the generic method suggested above works when applied to the topic of alternating products of infinitesimally free random variables. In particular, we will obtain the infinitesimal versions for two important facts related to this topic, that were originally found in [4] – one of them is about compressions by free projections, the other concerns a method of constructing free families of free Poisson elements. Since the proofs of the G-valued formulas that we need are identical to those of their C-valued counterparts, we will not give them here, but we will merely indicate where in [5] can the C-valued proofs be exactly found. The starting point is provided by the following formulas, obtained by doing the C-to-G change in Theorem 14.4 of [5]. Proposition 5.1. Let (A, ϕ, ϕ ) be an incps and let A1 , A2 be unital subalgebras of A which are infinitesimally free. Consider the functional ϕ = ϕ + εϕ : A → G and the associated cumun lant functionals ( κn : A → G)n1 . Recall that for every n 1 and π ∈ NC(n) we also have functionals ϕπ , κπ : An → G, as defined in Notation 3.12. 1o For every a1 , . . . , an ∈ A1 and b1 , . . . , bn ∈ A2 one has that
ϕ (a1 b1 · · · an bn ) =
κπ (a1 , . . . , an ) · ϕKr(π) (b1 , . . . , bn ).
(5.1)
π∈NC(n)
2o For every a1 , . . . , an ∈ A1 and b1 , . . . , bn ∈ A2 one has that
κn (a1 b1 , . . . , an bn ) =
κπ (a1 , . . . , an ) · κKr(π) (b1 , . . . , bn ).
(5.2)
π∈NC(n)
We now start on the application to free compressions. Definition 5.2. Let (A, ϕ, ϕ ) be an incps, and let p ∈ A be an idempotent element such that ϕ(p) = 0. We denote ϕ(p) =: α and ϕ (p) = α . The compression of (A, ϕ, ϕ ) by p is then defined to be the incps (B, ψ, ψ ) where B := pAp = {b ∈ A | pb = b = bp}
(5.3)
and where ψ, ψ : B → C are defined by ψ(b) =
1 ϕ(b), α
ψ (b) =
1 α ϕ (b) − 2 ϕ(b), α α
b ∈ B.
(5.4)
Remark 5.3. 1o In the preceding definition, note that the Grassman number α := α + εα is 2 invertible in G, with inverse 1/ α = (1/α) − ε(α /α ). As a consequence, the two formulas given = ψ + εψ : B → G satisfying in (5.4) are equivalent to the consolidated functional ψ (b) = ψ
1 ϕ (b), α
∀b ∈ B.
(5.5)
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2o If in the preceding definition (A, ϕ, ϕ ) is a ∗-incps and p is a projection, then by using the relations p = p ∗ = p 2 we immediately infer that 0 < α 1 and α ∈ R. As a consequence, (B, ψ, ψ ) defined there is a ∗-incps as well. Theorem 5.4. Let (A, ϕ, ϕ ) be an incps. Let p ∈ A be an idempotent element such that ϕ(p) = 0. Denote ϕ(p) =: α, ϕ (p) =: α , and consider the compressed incps (B, ψ, ψ ) from Definition 5.2. For every n 1 let κn , κn : An → C and κ n , κ n : B n → C be the nth non-crossing cumulant and infinitesimal cumulant functional associated to (A, ϕ, ϕ ) and to (B, ψ, ψ ), respectively. Let X be a subset of A which is infinitesimally free from {p}. Then we have κ n (px1 p, . . . , pxn p) =
1 κn (αx1 , . . . , αxn ), α
∀n 1, x1 , . . . , xn ∈ X
(5.6)
and ⎧ ⎨ κ 1 (px1 p) = κ1 (x1 ),
∀x1 ∈ X , (n − 1)α ⎩ κ n (px1 p, . . . , pxn p) = κn (αx1 , . . . , αxn ), α2
∀n 2, x1 , . . . , xn ∈ X .
(5.7)
Proof. It is easily verified that Eqs. (5.6) and (5.7) are the body part and respectively the soul part for the formula κ n (px1 p, . . . , pxn p) = α n−1 · κn (x1 , . . . , xn ) ∈ G,
∀n 1, x1 , . . . , xn ∈ X ,
(5.8)
α = α + ε · α ). But the where the “tilde” notations have their usual meaning ( κ n = κ n + ε · κ n , latter formula is just the G-valued counterpart for Theorem 14.10 in [5]; its proof is obtained by faithfully doing the C-to-G transcription of the proof of that theorem from [5], with the minor change that the powers of α must be kept outside the cumulant functionals (one cannot write “ κn ( α x1 , . . . , α xn )”, since A is only a C-algebra). Note that the argument obtained in this way is indeed an application of Proposition 5.1, in the same way as Theorem 14.10 is an application of Theorem 14.4 in [5]. 2 Corollary 5.5. Let (A, ϕ, ϕ ) be an incps. Let p ∈ A be an idempotent element with ϕ(p) = 0, and consider the compressed incps (B, ψ, ψ ) defined as above. Let X1 , . . . , Xk be subsets of A such that {p}, X1 , . . . , Xk are infinitesimally free in (A, ϕ, ϕ ). Put Yi = pXi p ⊆ B, 1 i k. Then Y1 , . . . , Yk are infinitesimally free in (B, ψ, ψ ). Proof. This is an immediate consequence of Corollary 4.8, where the needed vanishing of mixed cumulants follows from the explicit formulas found in Theorem 5.4. 2 We now go to the construction of families of infinitesimally free Poisson elements. We will use the infinitesimal (a.k.a “type B”) versions of semicircular and of free Poisson elements that appeared in [7] in connection to limit theorems of type B, and are discussed in detail in Sections 4 and 5 of [1]. For the present paper it is most convenient to introduce these elements in terms of their infinitesimal cumulants, as stated in Definitions 5.6 and 5.8 below.
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Definition 5.6. Let (A, ϕ, ϕ ) be a ∗-incps. A selfadjoint element x ∈ A will be called infinitesimally semicircular when it satisfies κn (x, . . . , x) = κn (x, . . . , x) = 0,
∀n 3.
(5.9)
If in addition to that we also have κ1 (x) = 0,
κ2 (x, x) = 1,
(5.10)
then we will say that x is a standard infinitesimally semicircular element. Remark 5.7. 1o By using the multilinearity of κn , κn and Proposition 3.14, it is immediately seen that if x is infinitesimally semicircular then so is α(x − β1A ) for any α > 0 and β ∈ R. Moreover, leaving aside the trivial case when κ2 (x, x) = 0, one can always pick α and β so that α(x − β1A ) is standard. 2o Let x be standard infinitesimally semicircular in (A, ϕ, ϕ ). Then all moments ϕ(x n ) and ϕ (x n ) for n 1 are completely determined by the real parameters6 α1 , α2 defined by α1 := κ1 (x) = ϕ (x),
and α2 := κ2 (x, x) = ϕ x 2 .
(5.11)
It is in fact very easy to calculate what these moments are. Indeed, one can calculate the G-valued moments ϕ (x n ) = ϕ(x n ) + εϕ (x n ) by using the moment-cumulant formula (3.16), where one takes into account that κ1 (x) = εα1 ,
κ2 (x, x) = 1 + εα2 ,
and κn (x, . . . , x) = 0 for all n 3.
κπ (x, . . . , x) | π ∈ NC(n)} can get non-zero contributions The expansion of ϕ (x n ) in terms of { only from such partitions π where every block V of π has |V | 2 and where there is at most one block of π of cardinality 1 (the latter condition coming from the fact that ( κ1 (x))2 = 0). We distinguish two cases, depending on the parity of n. Case 1. n is even, n = 2m. We get a sum extending over non-crossing pairings in NC(n), which gives us
m
ϕ x 2m = Cm · 1 + εα2 = Cm · 1 + εmα2 , or in other words
ϕ x 2m = Cm ,
ϕ x 2m = α2 · (mCm ),
(5.12)
where Cm stands for the mth Catalan number. 6 Any two numbers α , α ∈ R can appear here. Indeed, Example 8.7 shows situations where one has α = 1, α = 0 1 2 1 2 and respectively α1 = 0, α2 = 2. One can rescale the functionals ϕ of these two special cases to get standard infinitesimal semicirculars x1 , x2 having any pairs of parameters α1 , 0 and respectively 0, α2 ; then due to Proposition 2.4 one √ may assume that x1 , x2 are infinitesimally free, and form the average (x1 + x2 )/ 2, which is standard infinitesimally semicircular with generic parameters in (5.11).
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Case 2. n is odd, n = 2m + 1. Here we get a sum extending over the partitions π ∈ NC(n) which have one block of 1 element and m blocks of 2 elements. There are (2m + 1)Cm such partitions; so we obtain
m , ϕ x 2m+1 = (2m + 1)Cm · εα1 1 + εα2 leading to
ϕ x 2m+1 = 0,
ϕ x 2m+1 = α1 · (2m + 1)Cm .
(5.13)
Definition 5.8. Let (A, ϕ, ϕ ) be a ∗-incps, and let λ, β , γ be real parameters, where λ > 0. A selfadjoint element y ∈ A will be called infinitesimally free Poisson of parameter λ and7 infinitesimal parameters β , γ when it has non-crossing cumulants given by κn (y, . . . , y) = λ, (5.14) κn (y, . . . , y) = β + nγ , ∀n 1. Theorem 5.9. Let (A, ϕ, ϕ ) be a ∗-incps. Let x ∈ A be a standard infinitesimally semicircular element, and let S be a subset of A which is infinitesimally free from {x}. Then for every n 1 and a1 , . . . , an ∈ S we have κn (xa1 x, . . . , xan x) = ϕ(a1 · · · an )
(5.15)
κn (xa1 x, . . . , xan x) = ϕ (a1 · · · an ) + nϕ x 2 · ϕ(a1 · · · an ).
(5.16)
and
Proof. Eqs. (5.15) and (5.16) are the body part and respectively the soul part for the formula
n κn (xa1 x, . . . , xan x) = κ2 (x, x) · ϕ (a1 · · · an ) ∈ G. (5.17) The proof of the latter formula is obtained by doing the C-to-G transcription either for the arguments used in Proposition 12.18 and Example 12.19 on pp. 207–208 of [5], or for the arguments in Propositions 17.20 and 17.21 on pp. 283–284 of [5]. 2 The ensuing construction of families of infinitesimally free Poisson elements is stated in the next corollary. Part 2o of the corollary has also appeared as Corollary 36 of [1]. Corollary 5.10. Let (A, ϕ, ϕ ) be a ∗-incps, and let x ∈ A be a standard infinitesimally semicircular element. Let e1 , . . . , ek ∈ A be projections such that ei ⊥ ej for 1 i < j k and such that {e1 , . . . , ek } is infinitesimally free from {x}. Then 1o The elements xe1 x, . . . , xek x form an infinitesimally free family in (A, ϕ, ϕ ). 2o For every 1 i k, xei x is infinitesimally free Poisson with parameter λi and infinitesimal parameters βi , γi given by λi = ϕ(ei ), βi = ϕ (ei ), γi = ϕ (x 2 ) · ϕ(ei ). 7 A more complete definition of these elements would also use a 4th parameter r > 0, and have each of λ, β , γ multiplied by r n in Eqs. (5.14). For the sake of simplicity, here we have set this additional parameter to r = 1.
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Proof. 1o This is an immediate consequence of Corollary 4.8, where the needed vanishing of mixed cumulants follows from the explicit formulas found in Theorem 5.9. 2o By putting a1 = · · · = an := ei in (5.15) and (5.16) we see that the cumulants of xei x have the form required in Definition 5.8, with parameters λi , βi , γi as stated. 2 6. Relations with the lattices NC(B) (n) In this section we remember that the concept of incps has its origins in the considerations “of type B” from [2], and we look at how the essence of these considerations persists in the framework of the present paper. The strategy of [2] was to study the type B analogue for an operation with power series called boxed convolution and denoted by . The focus on was motivated by the fact that it provides in some sense a “middle ground” between alternating products of free random variables and the structure of intervals in the lattices NC(n) (see discussion on pp. 2282-2283 of [2]). The key (B) (A) = G in the introduction point discovered in [2] (stated in the form of the equation of that paper) was that boxed convolution of type B can still be defined by the formulas from type A, provided that one uses scalars from G. For a detailed discussion on we refer the reader to Lecture 17 of [5]. What is important for us here is that the formula used to define (cf. Eq. (17.1) on p. 273 of [5]) has already made an appearance, in G-valued context, in Eqs. (5.1), (5.2) of the preceding section. So then, the present (B) (A) = G ” principle from [2] should just amount to the following fact: incarnation of the “ if one takes the soul parts of Eqs. (5.1) and (5.2), then summations over NC(B) (n) must arise. This is stated precisely in Theorem 6.4 below, which is actually an easy application of the fact that the absolute value map Abs : NC(B) (n) → NC(n) is an (n + 1)-to-1 cover. We start by introducing some notations that will be used in Theorem 6.4, namely the type B (A) (A) analogues for the functionals ϕπ and κπ from Section 3.2. Notation 6.1. Let (A, ϕ, ϕ ) be an incps and let (κn )n1 , (κn )n1 be the families of non-crossing and respectively of infinitesimal non-crossing cumulant functionals associated to this incps. For (B) every n 1 and τ ∈ NC(B) (n) we define a multilinear functional κτ : An → C, as follows. Case 1. If τ ∈ NCZ (B) (n), τ = {Z, V1 , −V1 , . . . , Vp , −Vp }, then we put
κτ(B) (a1 , . . . , an ) := κ |Z|/2 a1 , . . . , an ) Abs(Z) ×
p
κ|Vj | (a1 , . . . , an ) Abs(Vj ) ,
for a1 , . . . , an ∈ A.
(6.1)
j =1
Case 2. If τ ∈ NC(B) (n) \ NCZ (B) (n), τ = {V1 , −V1 , . . . , Vp , −Vp }, then we put κτ(B) (a1 , . . . , an ) :=
p
κ|Vj | (a1 , . . . , an ) Abs(Vj ) ,
a1 , . . . , an ∈ A.
(6.2)
j =1
Notation 6.2. Let (A, ϕ, ϕ ) be an incps. Consider the families of multilinear functionals (ϕn , ϕn : An → C)n1 defined by ϕn = ϕ ◦ Multn , ϕn = ϕ ◦ Multn , where Multn : An → A is the multiplication map, n 1 (same as used in Remark 3.10). Then for every n 1 and every
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(B)
τ ∈ NC(B) (n) we define a multilinear functional ϕτ : An → C by the same recipe as in Nota ) tion 6.1 (with discussion separated in 2 cases), where every occurrence of κm (respectively κm ). For example, the analogue of Case 1 is like this: for n 1 is replaced by ϕm (respectively ϕm (B) and for τ = {Z, V1 , −V1 , . . . , Vp , −Vp } in NCZ (B) (n) we define ϕτ : An → C by putting
a1 , . . . , an ) Abs(Z) · ϕ|Vj | (a1 , . . . , an ) Abs(Vj ) , p
ϕτ(B) (a1 , . . . , an ) := ϕ|Z|/2
(6.3)
j =1
for a1 , . . . , an ∈ A. Remark 6.3. 1o It is immediate that for τ ∈ NC(B) (n) \ NCZ (B) (n) one has (A)
(A)
κτ(B) = κAbs(τ ) ,
ϕτ(B) = ϕAbs(τ ) .
(6.4)
2o The functionals introduced in Notation 6.1 extend both families κn and κn . Indeed, we (B) (A) (B) have that κn = κ1±n and that κn = κ1n = κτ for every n 1 and any τ ∈ NC(B) (n) such that Abs(τ ) = 1n (e.g. τ = {{1, . . . , n}, {−1, . . . , −n}}). A similar remark holds in connection to the (B) functionals ϕτ – they extend both families ϕn and ϕn . Theorem 6.4. Let (A, ϕ, ϕ ) be an incps, and consider multilinear functionals on A as in Notations 6.1, 6.2. Let A1 , A2 be unital subalgebras of A which are infinitesimally free. Then for every a1 , . . . , an ∈ A1 and b1 , . . . , bn ∈ A2 one has
ϕ (a1 b1 · · · an bn ) =
(B)
κτ(B) (a1 , . . . , an ) · ϕKr(τ ) (b1 , . . . , bn )
(6.5)
τ ∈NC(B) (n)
and κn (a1 b1 , . . . , an bn ) =
(B)
κτ(B) (a1 , . . . , an ) · κKr(τ ) (b1 , . . . , bn ).
(6.6)
τ ∈NC(B) (n)
Proof. Consider the “tilde” notations from Proposition 5.1. Let π be a partition in NC(n), and let us look at the expression
So κπ (a1 , . . . , an ) κKr(π) (b1 , . . . , bn ) κ|V | (a1 , . . . , an ) V · = So V ∈π
κ|W | (b1 , . . . , bn ) | W .
W ∈Kr(π)
In view of the formula (3.12) describing the soul part of a product, the latter expression is equal to a sum of n + 1 terms, some of them indexed by the blocks V ∈ π , and the others indexed by the blocks W ∈ Kr(π). We leave it as a straightforward exercise to the reader to write these n + 1 terms explicitly, and verify that the natural correspondence to the n + 1 partitions in {τ ∈ NC(B) (n) | Abs(τ ) = π} leads to the formula
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So κπ (a1 , . . . , an ) κKr(π) (b1 , . . . , bn ) (B) κτ(B) (a1 , . . . , an ) · ϕKr(τ ) (b1 , . . . , bn ). =
(6.7)
τ ∈NC(B) (n)
such that Abs(τ )=π
(Note: the Kreweras complement Kr(τ ) from (6.7) is taken in the lattice NC(B) (n); we use here the fact that Abs(τ ) = π ⇒ Abs(Kr(τ )) = Kr(π) – cf. Lemma 1.4 in [2].) By summing over π ∈ NC(n) on both sides of (6.7), we obtain that
So right-hand side of Eq. (5.1) = right-hand side of Eq. (6.5) . Since the soul part of the left-hand side of Eq. (5.1) is ϕ (a1 b1 · · · an bn ), this proves that (6.5) holds. The verification of (6.6) is done in exactly the same way, by starting from Eq. (5.2) of Proposition 5.1. 2 Remark 6.5. If in the preceding theorem we make A1 = A and A2 = C1A , and if in Eq. (6.5) we take b1 = · · · = bn = 1A , then we obtain the formula κτ(B) (a1 , . . . , an ), ∀a1 , . . . , an ∈ A. (6.8) ϕ (a1 · · · an ) = τ ∈NCZ (B) (n)
The terms indexed by σ ∈ NC(B) (n) \ NCZ (B) (n) have disappeared in (6.8), due to the fact that ϕ (1A ) = 0. This formula was also noticed (via a direct argument from the definition of the G-valued functionals κn ) in Proposition 7.4.4 of [6]. 7. Dual derivation systems Notation 7.1. Let A be a unital algebra over C, and for every n 1 let Mn denote the vector space of multilinear functionals from An to C. If π = {V1 , . . . , Vp } is a partition in NC(n) where the blocks V1 , . . . , Vp are listed in increasing order of their minimal elements, then we define a multilinear map Jπ : M|V1 | × · · · × M|Vp | (f1 , . . . , fp ) → f ∈ Mn ,
(7.1)
where f (a1 , . . . , an ) :=
p
fj (a1 , . . . , an ) Vj ,
∀a1 , . . . , an ∈ A.
(7.2)
j =1
Remark 7.2. 1o For a concrete example of how Notation 7.1 works, say for instance that n = 6 and that π = {{1, 3, 6}, {2}, {4, 5}} ∈ NC(6). Then we have Jπ : M3 × M1 × M2 → M6 , and the fact that Jπ (f1 , f2 , f3 ) = f means that f (a1 , . . . , a6 ) = f1 (a1 , a3 , a6 ) · f2 (a2 ) · f3 (a4 , a5 ),
∀a1 , . . . , a6 ∈ A.
2o The formula (7.2) from the preceding notation is the same as those used to define the (A) (A) families of functionals {ϕπ | π ∈ NC(n)} and {κπ | π ∈ NC(n)} in Remark 3.10. Hence if
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(A, ϕ) is a noncommutative probability space and if (κn )n1 are the non-crossing cumulant functionals associated to ϕ, then for π = {V1 , . . . , Vp } ∈ NC(n) as in Notation 7.1 we get that Jπ (κ|V1 | , . . . , κ|Vp | ) = κπ(A) .
(7.3)
Likewise, for the same (A, ϕ) and π we get Jπ (ϕ|V1 | , . . . , ϕ|Vp | ) = ϕπ(A) ,
(7.4)
where ϕm = ϕ ◦ Multm : Am → C, m 1 (same as in Remark 3.10). 3o Let π = {V1 , . . . , Vp } ∈ NC(n) be as in Notation 7.1, and let 1 j p be such that Vj ∨
is an interval-block of π . Denote |Vj | =: m and let π ∈ NC(n − m) be the partition obtained by removing the block Vj out of π and by redenoting the elements of {1, . . . , n} \ Vj as 1, . . . , n − m, in increasing order. On the other hand, let us denote by γ ∈ NC(n) the partition of {1, . . . , n} into the two blocks Vj and {1, . . . , n} \ Vj . It is then immediate that for every f1 ∈ M|V1 | , . . . , fp ∈ M|Vp | we can write Jπ (f1 , . . . , fp ) = Jγ (g, fj )
where g := J ∨ (f1 , . . . fj −1 , fj +1 , . . . , fp ). π
(7.5)
Due to this observation and to the fact that every non-crossing partitions has interval-blocks, considerations about the multilinear functions Jπ from Notation 7.1 can sometimes be reduced (via an induction argument on |π|) to discussing the case when |π| = 2. Definition 7.3. Let Abe a unital algebra over C and let the spaces (Mn )n1 and the multilinear functions {Jπ | π ∈ ∞ n=1 NC(n)} be as in Notation 7.1. We will call dual derivation system a family of linear maps (dn : Dn → Mn )n1 where, for every n 1, Dn is a linear subspace of Mn , and where the following two conditions are satisfied. (i) Let π = {V1 , . . . , Vp } ∈ NC(n) be as in Notation 7.1. Then for every f1 ∈ D|V1 | , . . . , fp ∈ D|Vp | one has that Jπ (f1 , . . . , fp ) ∈ Dn and that
dn Jπ (f1 , . . . , fp ) = Jπ f1 , . . . , fj −1 , d|Vj | (fj ), fj +1 , . . . , fp . p
(7.6)
j =1
(ii) For every f ∈ D1 and every n 1 one has that f ◦ Multn ∈ Dn and that dn (f ◦ Multn ) = (d1 f ) ◦ Multn ,
(7.7)
where Multn : An → A is the multiplication map. Remark 7.4. 1o When verifying condition (i) in Definition 7.3, it suffices to check the particular case when |π| = 2. Indeed, the general case of Eq. (7.6) can then be obtained by induction on |π|, where one invokes the argument from (7.5). 2o In the setting of Definition 7.3, let us use the notation f × g for the functional obtained by “concatenating” f ∈ Mm and g ∈ Mn . So f × g ∈ Mm+n acts simply by (f × g)(a1 , . . . , am , b1 , . . . , bn ) = f (a1 , . . . , am )g(b1 , . . . , bn ),
∀a1 , . . . , am , b1 , . . . , bn ∈ A.
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Clearly one can write f × g = Jγ (f, g) where γ ∈ NC(m + n) is the partition with two blocks {1, . . . , m} and {m + 1, . . . , m + n}. By using Eq. (7.6) we thus obtain that
dm+n (f × g) = dm (f ) × g + f × dn (g) , ∀m, n 1, f ∈ Mm , g ∈ Mn . (7.8) So a dual derivation systemgives in particular a derivation on the algebra structure defined ∞ by using concatenation on n=1 Mn . Note however that Eq. (7.8) alone is not sufficient to ensure condition (i) from Definition 7.3 (since it cannot control Jπ for partitions such as π = {{1, 3}, {2}} ∈ NC(3)). Proposition 7.5. Let A be a unital algebra over C and let (dn : Dn → Mn )n1 be a dual derivation system on A. Let ϕ be a linear functional in D1 , and denote d1 (ϕ) =: ϕ . Consider the incps (A, ϕ, ϕ ), and let (κn )n1 , (κn )n1 be the families of non-crossing and respectively of infinitesimal non-crossing cumulant functionals associated to this incps. Then for every n 1 we have that κn ∈ Dn
and dn (κn ) = κn .
(7.9)
Proof. Denote as usual ϕn := ϕ ◦ Multn , ϕn := ϕ ◦ Multn , n 1. Since ϕ ∈ D1 , condition (ii) from Definition 7.3 implies that ϕn ∈ Dn and dn (ϕn ) = ϕn for every n 1. Now let π = {V1 , . . . , Vp } be a partition in NC(n), with V1 , . . . , Vp written in increasing order of their minimal elements. By using Eq. (7.4) from Remark 7.2 and condition (i) in Definition 7.3 we find that
Jπ ϕ|V1 | , . . . , ϕ|Vj −1 | , ϕ|V , ϕ|Vj +1 | , . . . , ϕ|Vp | dn ϕπ(A) = j| p
(7.10)
j =1
). (where the latter formula incorporates the fact that d|Vj | (ϕ|Vj | ) = ϕ|V j| We next consider the formula (3.9) which expresses a cumulant functional κn in terms of the (A) functionals {ϕπ | π ∈ NC(n)}. From this formula it follows that κn ∈ Dn and that p
Möb(π, 1n ) Jπ ϕ|V1 | , . . . , ϕ|Vj −1 | , ϕ|Vj | , ϕ|Vj +1 | , . . . , ϕ|Vp | . (7.11) dn (κn ) = π∈NC(n), π={V1 ,...,Vp }
j =1
It is immediate that on the right-hand side of (7.11) we have obtained precisely the sum over {(π, V ) | π ∈ NC(n), V block of π} which was used to introduce κn in Definition 4.2. 2 Proposition 7.6. Let (A, ϕ, ϕ ) be an incps, and consider the multilinear functionals ϕπ (π ∈ NC(n), n 1) which were introduced in Remark 3.10. Suppose that for every n 1 the set {ϕπ(A) | π ∈ NC(n)} is linearly independent in Mn ; let Dn denote its span, and let dn : Dn → Mn be the linear map defined by the requirement that dn (ϕπ(A) ) = ϕτ(B) , ∀π ∈ NC(n), (7.12) (A)
τ ∈NCZ (B) (n) such that Abs(τ )=π (B)
with ϕτ
as in Notation 6.2. Then (dn )n1 is a dual derivation system, and d1 (ϕ) = ϕ .
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Proof. It is obvious that the unique partition τ ∈ NCZ (B) (n) such that Abs(τ ) = 1n is τ = 1±n . (A) (B) Thus if we put π = 1n in Eq. (7.12) we obtain that dn (ϕ1n ) = ϕ1±n ; in other words, this means that dn (ϕ ◦ Multn ) = ϕ ◦ Multn ,
∀n 1.
(7.13)
The particular case n = 1 of (7.13) gives us that d1 (ϕ) = ϕ . Moreover, it becomes clear that dn (f ◦ Multn ) = (d1 f ) ◦ Multn ,
∀n 1 and f ∈ Cϕ;
since in this proposition we have D1 = Cϕ, we thus see that condition (ii) from Definition 7.3 is verified. The rest of the proof is devoted to verifying (i) from Definition 7.3. We fix a partition π = {V1 , . . . , Vp } ∈ NC(n) for which we will prove that Eq. (7.6) holds. Both sides of (7.6) behave multilinearly in the arguments f1 ∈ D|V1 | , . . . , fp ∈ D|Vp | ; hence, due to how D|V1 | , . . . , D|Vp | are defined, it suffices to prove the following statement: for every π1 ∈ NC(|V1 |), . . . , πp ∈ (A) (A) NC(|Vp |) we have that Jπ (ϕπ1 , . . . , ϕπp ) ∈ Dn and that
dn Jπ ϕπ(A) , . . . , ϕπ(A) p 1 =
p j =1
(A)
, ϕπj +1 , . . . , ϕπ(A) . Jπ ϕπ(A) , . . . , ϕπ(A) , d|Vj | ϕπ(A) p j 1 j −1
(7.14)
In what follows we fix some partitions π1 ∈ NC(|V1 |), . . . , πp ∈ NC(|Vp |), for which we will prove that this statement holds. Observe that, in view of how the maps d|Vj | are defined, on the right-hand side of (7.14) we have p
j =1 τ ∈NCZ (B) (n) such that Abs(τ )=πj
. Jπ ϕπ(A) , . . . , ϕπ(A) , ϕτ(B) , ϕπ(A) , . . . , ϕπ(A) p 1 j −1 j +1
But let us recall from Remark 3.5 that the partitions in {τ ∈ NCZ (B) (n) | Abs(τ ) = πj } are indexed by the set of blocks of πj . More precisely, for every 1 j p and V ∈ πj let us denote by τ (j, V ) the unique partition τ ∈ NCZ (B) (n) such that Abs(τ ) = πj and such that the zero-block Z of τ has Abs(Z) = V ; then the double sum written above for the right-hand side of Eq. (7.14) becomes p j =1 V ∈πj
(B) . Jπ ϕπ(A) , . . . , ϕπ(A) , ϕτ (j,V ) , ϕπ(A) , . . . , ϕπ(A) p 1 j −1 j +1
(7.15)
Now to the left-hand side of (7.14). For every 1 j p let πj be the partition of Vj obtained by transporting the blocks of πj via the unique order preserving bijection from π1 , . . . , πp form together a partition ρ ∈ NC(n) which refines π , {1, . . . , |Vj |} onto Vj . Then (A) (A) (A) and it is immediate that Jπ (ϕπ1 , . . . , ϕπp ) = ϕρ . In particular this shows of course that
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(A)
(A)
Jπ (ϕπ1 , . . . , ϕπp ) ∈ Dn . Moreover, by using how dn (ϕρ ) is defined, we obtain that the left (B) hand side of (7.14) is equal to W ∈ρ ϕσ (W ) , where for every W ∈ ρ we denote by σ (W ) the unique partition in NCZ (B) (n) such that Abs(σ (W )) = ρ and such that the zero-block Z of σ (W ) has Abs(Z) = W . Finally, we observe that the set of blocks of ρ is the disjoint union of the sets of blocks of πp , and is hence in natural bijection with {(j, V ) | 1 j p and V ∈ πj }. the partitions π1 , . . . , We leave it as a straightforward (though somewhat notationally tedious) exercise to the reader to verify that when W ∈ ρ corresponds to (j, V ) via this bijection, then the term indexed by (j, V ) (B) in (7.15) is precisely equal to ϕσ (W ) . Hence the double sum from (7.15) is identified term by term (B) to W ∈ρ ϕσ (W ) via the bijection W ↔ (j, V ), and the required formula (7.14) follows. 2 Remark 7.7. The linear independence hypothesis in Proposition 7.6 is necessary, otherwise we need some relations to be satisfied by ϕ and ϕ . Indeed, suppose for example that the set {ϕπ(A) | π ∈ NC(2)} is linearly dependent in M2 . It is immediately verified that this is equivalent to the fact that ϕ is a character of A (ϕ(ab) = ϕ(a)ϕ(b), ∀a, b ∈ A). Hence κ2 = 0, so if Proposition 7.6 is to work then we should have κ2 = d2 (κ2 ) = 0 as well, implying that ϕ satisfies the condition ϕ (ab) = ϕ(a)ϕ (b) + ϕ (a)ϕ(b), ∀a, b ∈ A. 8. Soul companions for a given ϕ In this section we elaborate on the facts announced in the Section 1.3 of the introduction. We start by recording some basic properties of the set of functionals ϕ which can appear as soul-companions for ϕ, when (A, ϕ) and A1 , . . . , Ak are given. Proposition 8.1. Let (A, ϕ) be a noncommutative probability space and let A1 , . . . , Ak be unital subalgebras of A which are freely independent in (A, ϕ). 1o The set of linear functionals ϕ linear, ϕ (1A ) = 0, and A1 , . . . , Ak F := ϕ : A → C are infinitesimally free in (A, ϕ, ϕ )
is a linear subspace of the dual of A. 2o Suppose that Alg(A1 ∪ · · · ∪ Ak ) = A, and consider the linear map F ϕ → ϕ A1 , . . . , ϕ Ak ∈ F1 × · · · × Fk ,
(8.1)
(8.2)
where F is as in (8.1) and where for 1 i k we denote Fi = {ϕ : Ai → C | ϕ linear, ϕ (1A ) = 0}. The map from (8.2) is one-to-one. Proof. 1o This is immediate from Definition 1.1, and specifically from the fact that ϕ makes a linear appearance on the right-hand side of Eq. (1.5). 2o Let ϕ ∈ F be such that ϕ | Ai = 0, ∀1 i k. Then from Eq. (1.5) it is immediate that ϕ (a1 · · · an ) = 0 for all choices of a1 , . . . , an ∈ A1 ∪ · · · ∪ Ak . The linear span of the products a1 · · · an formed in this way is the algebra generated by A1 ∪ · · · ∪ Ak , hence is all of A, and the conclusion that ϕ = 0 follows. 2
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Remark 8.2. In the framework of Proposition 8.1, the linear map (8.2) may not be surjective. For an example, consider the full Fock space over C2 ,
⊗n T = CΩ ⊕ C2 ⊕ C2 ⊗ C2 ⊕ · · · ⊕ C2 ⊕ ···, and let L1 , L2 ∈ B(T ) be the left-creation operators associated to the two vectors in the canonical orthonormal basis of C2 . Then L1 , L2 are isometries with mutually orthogonal ranges; this is recorded in algebraic form by the relations L∗1 L1 = L∗2 L2 = 1 (identity operator on T ),
L∗1 L2 = 0.
For i = 1, 2 let Ai denote the unital ∗-subalgebra of B(T ) generated by Li , and let A = Alg(A1 ∪ A2 ), the unital ∗-algebra generated by L1 and L2 together. It is well known (see e.g. [5, Lecture 7]) that A1 and A2 are free in (A, ϕ) where ϕ is the vacuum-state on A. Let ϕ2 : A2 → C be any linear functional such that ϕ2 (1A ) = 0 and ϕ2 (L2 ) = 1. Then there exists no linear functional ϕ : A → C such that ϕ | A2 = ϕ2 and such that A1 , A2 are infinitesimally free in (A, ϕ, ϕ ). Indeed, if such ϕ would exist then from Eq. (2.3) of Remark 2.2 it would follow that
ϕ L∗1 L2 L1 = ϕ L∗1 L1 ϕ (L2 ) + ϕ L∗1 L1 ϕ(L2 ) = 1 · 1 + 0 · 0 = 1, which is not possible, since L∗1 L2 L1 = 0. Remark 8.3. The example from the above remark shows that we can’t always extend a given system of functionals ϕi in order to get a soul companion ϕ for ϕ. But Proposition 2.4 gives us an important case when we are sure this is possible, namely the one when (A, ϕ) is the free product (A1 , ϕ1 ) ∗ · · · ∗ (Ak , ϕk ). In the remaining part of this section we will look at the two recipes for obtaining a soul companion that were stated in Corollary 1.4 and Proposition 1.5. For the first of them, we start by verifying that a derivation on A does indeed define a dual derivation system as indicated in Eq. (1.15). Proposition 8.4. Let A be a unital algebra over C and let D : A → A be a derivation. For every n 1 let Mn denote the space of multilinear functionals from An to C, and define dn : Mn → Mn by putting (dn f )(a1 , . . . , an ) :=
n
f a1 , . . . , am−1 , D(am ), am+1 , . . . , an ,
(8.3)
m=1
for f ∈ Mn and a1 , . . . , an ∈ A. Then (dn )n1 is a dual derivation system on A. Proof. We first do the immediate verification of condition (ii) from Definition 7.3. Let f be a functional in M1 , let n be a positive integer, and denote g = f ◦ Multn ∈ Mn . Then for every a1 , . . . , an ∈ A we have (dn g)(a1 , . . . , an ) =
n
f a1 · · · am−1 · D(am ) · am+1 · · · an = f D(a1 · · · an ) m=1
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(where at the first equality sign we used the definitions of dn and of g, and at the second equality sign we used the derivation property of D). Since d1 f is just f ◦ D, it is clear that we have obtained dn g = (d1 f ) ◦ Mn , as required. For the remaining part of the proof we fix π = {V1 , . . . , Vp } ∈ NC(n) and f1 ∈ M|V1 | , . . . , fp ∈ M|Vp | as in (i) of Definition 7.3, and we verify that the formula (7.6) holds. Denote f := Jπ (f1 , . . . , fp ) ∈ Mn . In the summation which defines dn f in Eq. (8.3) we group the terms by writing p j =1
f a1 , . . . , am−1 , D(am ), am+1 , . . . , an .
(8.4)
m∈Vj
It will clearly suffice to prove that, for every 1 j p, the term indexed by j in the sum (8.4) is equal to the term indexed by j on the right-hand side of (7.6). So then let us also fix a j , 1 j p. We write explicitly the block Vj of π as {v1 , . . . , vs } with v1 < · · · < vs . From the definition of f as Jπ (f1 , . . . , fp ) it is then immediate that for m = vr ∈ Vj we have
f a1 , . . . , am−1 , D(am ), am+1 , . . . , an
= fi (a1 , . . . , an ) Vi · fj av1 , . . . , avr−1 , D(avr ), avr+1 , . . . , avs .
(8.5)
1ip, i=j
When summing over 1 r s in (8.5), the sum on the right-hand side only affects the last factor of the product, which gets summed to (ds fj )(av1 , . . . , avs ). The result of this summation is hence that
f a1 , . . . , am−1 , D(am ), am+1 , . . . , an = Jπ f1 , . . . , fj −1 , d|Vj | (fj ), fj +1 , . . . , fp , m∈Vj
as required.
2
Corollary 8.5. Let (A, ϕ) be a noncommutative probability space, and let D : A → A be a derivation. Define ϕ := ϕ ◦ D. Let the non-crossing and the infinitesimal non-crossing cumulant functionals associated to (A, ϕ, ϕ ) be denoted by (κn )n1 and (κn )n1 , in the usual way. Then for every n 1 and a1 , . . . , an ∈ A one has κn (a1 , . . . , an ) =
n
κn a1 , . . . , am−1 , D(am ), am+1 , . . . , an .
(8.6)
m=1
Proof. This follows from Proposition 7.5, where we use the specific dual derivation system put into evidence in Proposition 8.4. 2 Corollary 8.6. In the notations of Corollary 8.5, let A1 , . . . , Ak be unital subalgebras of A which are freely independent with respect to ϕ, and such that D(Ai ) ⊆ Ai for 1 i k. Then A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ).
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Proof. We verify that condition (2) from Theorem 1.2 is satisfied. The vanishing of mixed cumulants κn follows from the hypothesis that A1 , . . . , Ak are free in (A, ϕ). But then the specific formula obtained for the infinitesimal cumulants κn in Corollary 8.5, together with the hypothesis that A1 , . . . , Ak are invariant under D, imply that the mixed infinitesimal cumulants κn vanish as well. 2 Example 8.7. Consider the situation where A is the algebra CX1 , . . . , Xk of noncommutative polynomials in k indeterminates. We will view A as a ∗-algebra, with ∗-operation uniquely determined by the requirement that each of X1 , . . . , Xk is selfadjoint. Consider the unital ∗subalgebras A1 , . . . , Ak ⊆ A where Ai = span{Xin | n 0}, 1 i k. We will look at two natural derivations on A that leave A1 , . . . , Ak invariant, and we will examine some examples of infinitesimal freeness given by these derivations. (a) Let D : A → A be the linear operator defined by putting D(1) = 0, D(Xi ) = 1 ∀1 i k, and D(Xi1 · · · Xin ) =
n
Xi1 · · · Xim−1 Xim+1 · · · Xin ,
∀n 2, ∀1 i1 , . . . , in k. (8.7)
m=1
It is immediate that D is a derivation on A, which is selfadjoint (in the sense that D(P ∗ ) = D(P )∗ , ∀P ∈ A). For every 1 i k we have that D(Ai ) ⊆ Ai and that D acts on Ai as the usual derivative (in the sense that D(P (Xi )) = P (Xi ), ∀P ∈ C[X]). Now let μ : A → C be a positive definite functional with μ(1) = 1 and such that A1 , . . . , Ak are free in (A, μ). Then Corollary 8.6 implies that A1 , . . . , Ak are infinitesimally free in the ∗-incps (A, μ, μ ), where μ := μ ◦ D. Note that in this special example we actually have κn (Xi1 , . . . , Xin ) = 0,
∀n 2, ∀1 i1 , . . . , in k;
(8.8)
this is an immediate consequence of formula (8.6), combined with the fact that a non-crossing cumulant vanishes when one of its arguments is a scalar. Eq. (8.8) gives in particular that κn (Xi , . . . , Xi ) = 0,
∀n 2 and 1 i k.
So if μ is defined such that every Xi has a standard semicircular distribution in (A, μ), then every Xi will become a standard infinitesimal semicircular element in (A, μ, μ ), where in Eq. (5.11) of Remark 5.7 we take α1 = 1, α2 = 0. (b) Let D# : A → A be the linear operator defined by putting D# (1) = 0 and D# (Xi1 · · · Xin ) = n Xi1 · · · Xin ,
∀n 1, ∀1 i1 , . . . , in k.
(8.9)
Then D# is a selfadjoint derivation, sometimes called “the number operator” on A. It is clear that D# leaves every Ai invariant, 1 i k. Hence if μ : A → C is as in part (a) above (such that A1 , . . . , Ak are free in (A, μ)), then Corollary 8.6 implies that A1 , . . . , Ak are infinitesimally free in the ∗-incps (A, μ, μ# ), where μ# := μ ◦ D# .
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Since D# (Xi ) = Xi for 1 i k, the formula (8.6) for infinitesimal non-crossing cumulants now gives κn (Xi1 , . . . , Xin ) = n · κn (Xi1 , . . . , Xin ),
∀n 1, ∀1 i1 , . . . , in k.
(8.10)
In the particular case when μ is such that every Xi is standard semicircular in (A, μ), it thus follows that every Xi becomes a standard infinitesimal semicircular element in (A, μ, μ# ), where we set the parameters from Eq. (5.11) to be α1 = 0 and α2 = 2. On the other hand, if μ is defined such that every Xi has a standard free Poisson distribution in (A, μ) (with κn (Xi , . . . , Xi ) = 1 for all n 1), then the Xi will become infinitesimal free Poisson elements in (A, μ, μ# ), in the sense of Definition 5.8 and where we take β = 0, γ = 1 in Eq. (5.14). We now move to the situation described in Proposition 1.5. Clearly, this is just an immediate consequence of Proposition 4.3. Corollary 8.8. In the notations of Proposition 4.3, suppose that A1 , . . . , Ak are unital subalgebras of A which are freely independent with respect to ϕt for every t ∈ T . Then A1 , . . . , Ak are infinitesimally free in (A, ϕ, ϕ ). Proof. Consider elements a1 ∈ Ai1 , . . . , an ∈ Ain where the indices i1 , . . . , in are not all equal (t) to each other. The freeness of A1 , . . . , Ak in (A, ϕt ) implies that κn (a1 , . . . , an ) = 0 for every (t) t ∈ T . The limit and derivative at 0 of the function t → κn (a1 , . . . , an ) must therefore vanish, which means (by Proposition 4.3) that κn (a1 , . . . , an ) = κn (a1 , . . . , an ) = 0. Hence condition (2) from Theorem 1.2 is satisfied, and the conclusion follows. 2 Example 8.9. Consider again the situation where A is the ∗-algebra CX1 , . . . , Xk , as in Example 8.7, and where μ : A → C is a positive definite functional with μ(1) = 1. Let (κn )n1 be the non-crossing cumulant functionals of μ, and let {κπ(A) | π ∈ ∞ n=1 NC(n)} be the extended family of multilinear functionals from Remark 3.10. For every t > 0, let μt : A → C be the linear functional defined by putting μt (1) = 1 and μt (Xi1 , . . . , Xin ) =
(t + 1)|π| · κπ (Xi1 , . . . , Xin ),
(8.11)
π∈NC(n)
for all n 1 and 1 i1 , . . . , in k. As is easily seen, μt is uniquely determined by the fact that (t) its non-crossing cumulant functionals (κn )n1 satisfy κn(t) (Xi1 , . . . , Xin ) = (t + 1) · κn (Xi1 , . . . , Xin ),
∀n 1, 1 i1 , . . . , in k.
(8.12)
Due to this fact, μt is called the “(t + 1)th convolution power of μ” with respect to the operation of free additive convolution – see pp. 231–233 of [5] for details. From (8.11) it is clear that the family {μt | t > 0} has infinitesimal limit (μ, μ ) at t = 0, where μ is the functional we started with, while μ is defined by putting μ (1) = 0 and μ (Xi1 · · · Xin ) =
π∈NC(n)
|π| · κπ (Xi1 , . . . , Xin ),
∀n 1, 1 i1 , . . . , in k.
(8.13)
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Note also that by using Eq. (8.12) and by invoking Proposition 4.3 we get κn (Xi1 , . . . , Xin ) = κn (Xi1 , . . . , Xin ),
∀n 1, 1 i1 , . . . , in k.
(8.14)
Now let A1 , . . . , Ak be the unital ∗-subalgebras of A that were also considered in Example 8.7, Ai = span{Xin | n 0} for 1 i k. Suppose that A1 , . . . , Ak are free in (A, μ). Then they are free in (A, μt ) for every t > 0; this follows from Eq. (8.12) and the description of freeness in terms of non-crossing cumulants (cf. [5, Theorem 11.20]), where we take into account that Ai is the unital algebra generated by Xi . Hence this is a situation where Corollary 8.8 applies, and we conclude that A1 , . . . , Ak are infinitesimally free in (A, μ, μ ). Note also that if Xi has a standard semicircular distribution in (A, μ), then Eq. (8.14) implies that Xi becomes an infinitesimal semicircular element in (A, μ, μ ), where the parameters α1 , α2 from Remark 5.7 are taken to be α1 = 0, α2 = 1. Likewise, if Xi is a standard free Poisson in (A, μ), then Eq. (8.14) implies that Xi becomes an infinitesimal free Poisson element in (A, μ, μ ), where the parameters β , γ from Definition 5.8 are taken to be β = 1, γ = 0. Acknowledgments We are grateful to Teodor Banica for very useful discussions during the incipient stage of this work. A. Nica would like to thank Mireille Capitaine and Muriel Casalis for inviting him to visit the Mathematics Institute of Université Paul Sabatier in Toulouse, where this work was started. Nica’s visit to Toulouse was supported in part by the project ANR-08-BLAN-0311-01 of ANR, France. References [1] S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at http://www.arxiv.org under reference arXiv:0903.2721. [2] P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003) 2263–2303. [3] B. DeWitt, Supermanifolds, second ed., Cambridge University Press, 1992. [4] A. Nica, R. Speicher, On the multiplication of free n-tuples of noncommutative random variables, with an appendix by Dan Voiculescu: Alternative proofs for the type II free Poisson variables and for the compression results, Amer. J. Math. 118 (1996) 799–837. [5] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser., vol. 335, Cambridge University Press, 2006. [6] I. Oancea, Posets of non-crossing partitions of type B and applications, University of Waterloo PhD thesis, October 2007, available online at uwspace.uwaterloo.ca/bitstream/10012/3402/1/BDTEZA.pdf. [7] M. Popa, Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B, preprint, 2007, available online at http://www.arxiv.org under reference arXiv:0709.0011. [8] V. Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997) 195–222. [9] R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann. 298 (1994) 611–628. [10] D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables, CRM Monogr. Ser., vol. 1, Amer. Math. Soc., 1992.
Journal of Functional Analysis 258 (2010) 3024–3047 www.elsevier.com/locate/jfa
Relations between some basic results derived from two kinds of topologies for a random locally convex module ✩ Tiexin Guo LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, PR China Received 10 August 2009; accepted 1 February 2010 Available online 11 February 2010 Communicated by Paul Malliavin Dedicated to Professor Berthold Schweizer on his 80th birthday
Abstract The purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies (namely the (ε, λ)-topology and the stronger locally L0 -convex topology) for a random locally convex module. First, we give an extremely simple proof of the known Hahn–Banach extension theorem for L0 -linear functions as well as its continuous variant. Then we give the relations between the hyperplane separation theorems in [D. Filipovi´c, M. Kupper, N. Vogelpoth, Separation and duality in locally L0 -convex modules, J. Funct. Anal. 256 (2009) 3996–4029] and a basic strict separation theorem in [T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794–3804]: in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. As applications of the fact, we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the (ε, λ)-topology are still valid under the locally L0 -convex topology, which considerably enriches financial applications of random normed modules. © 2010 Elsevier Inc. All rights reserved. Keywords: Random locally convex modules; (ε, λ)-topology; Locally L0 -convex topology; Random conjugate spaces; Hahn–Banach extension theorems; Hyperplane separation theorems; Countable concatenation property; Completeness
✩
Supported by NNSF No. 10871016. E-mail address:
[email protected].
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.002
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Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The random conjugate spaces of a random locally convex module under the (ε, λ)-topology and locally L0 -convex topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The (ε, λ)-topology and locally L0 -convex topology . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hahn–Banach extension theorems for random linear functionals . . . . . . . . . . . . . . . 2.3. Hahn–Banach extension theorems for L0 -linear functions . . . . . . . . . . . . . . . . . . . . 2.4. The Hahn–Banach extension theorems for continuous L0 -linear functions in random locally convex modules under the two kinds of topologies . . . . . . . . . . . . . . . . . . . 2.5. The relation between the random conjugate spaces of a random locally convex module under the two kinds of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Relations between the hyperplane separation theorems currently available . . . . . . . . . . . . . 3.1. A new countable concatenation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Relations between the hyperplane separation theorems currently available . . . . . . . . . 3.3. The hereditarily disjoint probability and more general forms of hyperplane separation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The relation between the two kinds of completeness of a random locally convex module under the two kinds of topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Representation theorems of random conjugate spaces of random normed modules under the locally L0 -convex topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Random metric theory originated from the theory of probabilistic metric spaces, cf. [13,14,28]. The original definition of a random normed space was presented in [28, p. 240] in which the random norm of a vector is a nonnegative random variable. A new version of a random normed space was presented in [11], in which the random norm of a vector is the equivalence class of a nonnegative random variable. Based on the new version, we further presented in [11] the elaborated definition of a random normed module, which was originally introduced in [7]. Random normed modules lead to the definitive notion of the random conjugate space of a random normed space, cf. [11], and in particular the theory of random normed modules together with their random conjugate spaces has obtained a systematic and deep development, cf. [8–10,15,16,18,19,21,23]. Subsequently, random locally convex modules were presented in [14] and deeply studied in [20,24] (called random seminormed modules in [14,20,24]) for the further development of the theory of random normed modules. Recently, using the theory of random conjugate spaces we established a basic strict separation theorem in random locally convex modules in [22, Theorem 3.1]. It should also be pointed out that random locally convex modules, including their special case—random normed modules, in all our previous papers, are endowed with a natural topology, called the (ε, λ)-topology. Motivated by financial applications, Filipovi´c, Kupper and Vogelpoth recently presented in [4] locally L0 -convex modules and in particular the locally L0 -convex topology, establishing their hyperplane separation theorems, cf. [4, Theorems 2.6 and 2.8], and some basic results on convex analysis over the modules, cf. [4, Theorems 3.7 and 3.8]. Besides, they also rediscovered the notions of random locally convex modules and random normed modules, and further utilized
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their nice notion of a gauge function to show that the theory of Hausdorff locally L0 -convex modules is equivalent to that of random locally convex modules endowed with the locally L0 convex topology, cf. [4, Theorem 2.4]. Thus our work forms one part of the theory of random locally convex modules—corresponding to the (ε, λ)-topology and Filipovi´c et al.’s work [4] forms the other part—corresponding to the locally L0 -convex topology. Although the locally L0 -convex topology is much stronger than the (ε, λ)-topology, there are many inherent relations between the above two parts. The purpose of this paper is to exhibit the relations. Since the theory of random conjugate spaces has played crucial roles in both the theory of random locally convex modules and their financial applications, this paper is focused on the relations between the basic results closely related to the theory of random conjugate spaces: in the process some interesting new results are also obtained. The remainder of this paper is organized as follows: Section 2 is devoted to proving that most of random locally convex modules including random normed modules under the two kinds of topologies have the same random conjugate spaces; in this section we also make use of the techniques obtained in our previous work of random linear functionals to give an extremely simple proof of the known Hahn–Banach extension theorem for L0 -linear functions as well as its continuous variant. Then in Section 3 we prove that our basic strict separation theorem implies Filipovi´c, Kupper and Vogelpoth’s hyperplane separation theorem II but is independent of their hyperplane separation theorem I; further based on the results in Section 2 we also give a general variant of either of their separation theorems, allowing a separation with an arbitrary probability rather than the only probability one, and in the process we obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. Based on the nice completeness relation, finally in Section 4 we give two important representation theorems of random conjugate spaces of random normed modules, which generalize and improve the corresponding results obtained in [4,27]. For the sake of convenience, let us first recapitulate the related terminology and notation. Throughout this paper, (Ω, F , P ) denotes a probability space, K the scalar field R of real numbers or C of complex numbers, R = [−∞, +∞], L0 (F , R) the set of extended real-valued F -measurable random variables on Ω, L0 (F , R) the set of equivalence classes of elements in L0 (F , R), L0 (F , K) the algebra of equivalence classes of K-valued F -measurable random variables on Ω under the ordinary scalar multiplication, addition and multiplication operations on equivalence classes. It is well known from [3] that L0 (F , R) is a complete lattice under the ordering : ξ η iff 0 ξ (ω) η0 (ω), for P -almost all ω in Ω (briefly, a.s.), where ξ 0 and η0 are arbitrarily chosen rep0 resentatives of ξ and η, respectively. Furthermore, every subset A of L (F , R) has a supremum, denoted by A, and an infimum, denoted {an | n ∈ N } and a by A, and there exist a sequence sequence {bn | n ∈ N } in A such that n1 an = A and n1 bn = A, where N denotes the set of positive integers. If, in addition, A is directed upwards (downwards), then the above {an } (resp. {bn }) can be chosen as nondecreasing (correspondingly, nonincreasing). Finally L0 (F , R), as a sublattice of L0 (F , R), is complete in the sense that every subset with an upper bound has a supremum. Besides, throughout this paper we distinguish random variables from their equivalence classes by means of symbols: for example, IA denotes the characteristic function of the F -measurable set A, then we use I˜A for its equivalence class. It is necessary when we apply the theory of lifting property [25,26] to random normed modules as in [13,14] and apply measurable selection theorems [31] and the theory of random normed modules to the theory of random operators, cf. [14,17,29]. Therefore, for a subset A of L0 (F , R), ess sup(A) and ess inf(A) denote its respec-
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tive essential supremum and infimum, they are still extended random variables; we also reserve ess sup(E) and ess inf(E) for the essential supremum and infimum of a subfamily E of F , respectively, as in [4]. Specially, L0+ (or L0+ (F , R)) = {ξ ∈ L0 (F , R) | ξ 0}, L0++ (or L0++ (F , R)) = {ξ ∈ L0 (F , R) | ξ > 0 on Ω}, where for A ∈ F , “ξ > η on A” means ξ 0 (ω) > η0 (ω) a.s. on A for any chosen representatives ξ 0 and η0 of ξ and η, respectively. As usual, ξ > η means ξ η and ξ = η. 2. The random conjugate spaces of a random locally convex module under the (ε, λ)-topology and locally L0 -convex topology In this section, we discuss the relation between the random conjugate spaces of a random locally convex module under the (ε, λ)-topology and locally L0 -convex topology, whose initial application yields the deep results on random reflexivity—Theorems 2.17 and 2.18. Besides, we also give an extremely concise proof of the known Hahn–Banach theorem for L0 -linear functions, in which Lemma 2.12 plays a key role. 2.1. The (ε, λ)-topology and locally L0 -convex topology To introduce the (ε, λ)-topology and locally L0 -convex topology, we first recall the following: Definition 2.1. (See [11,14,22].) An ordered pair (E, P) is called a random locally convex module over K with base (Ω, F , P ) if E is a left module over the algebra L0 (F , K) and P is a family of mappings from E to L0+ such that the following three axioms are satisfied: (i) {x | · ∈ P} = 0 iff x = θ (the null element of E); (ii) ξ · x = |ξ | · x, ∀ξ ∈ L0 (F , K), x ∈ E and · ∈ P; (iii) x + y x + y, ∀x, y ∈ E and · ∈ P. Furthermore, a mapping · : E → L0+ satisfying (ii) and (iii) is called an L0 -seminorm; in addition, if x = 0 also implies x = θ , then it is called an L0 -norm, in which case (E, · ) is called a random normed module (briefly, an RN module) over K with base (Ω, F , P ), and is a special case of a random locally convex module when P consists of a single L0 -norm · . Remark 2.2. In Definition 2.1, the terminologies “L0 -seminorm and norm” are adopted from [4] for simplicity. Since L0 (F , K) is not only a linear space but also a ring with unit element, a left module over the ring L0 (F , K) is automatically a left module over the algebra L0 (F , K), and hence the notion of an L0 -normed module in [4,27] is equivalent to that of an RN module introduced in [11]. In the sequel of this paper we continue to employ the terminology “random normed modules” in accordance with the one used in [8–19]. Following are two important examples of RN modules used in this paper. Let us first recall the known notion of a random variable with values in a normed space: let (B, · ) be a normed space over K. A mapping V : Ω → B is called an F -random element if V −1 (G) := {ω ∈ Ω | V (ω) ∈ G} ∈ F for each open subset G of B, further an F -random element with its range finite is called an F -simple random element. A mapping V : Ω → B is called an F -random variable if there exists a sequence {Vn | n ∈ N } of F -simple random elements such that Vn (ω) − V (ω) → 0 as
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n → ∞ for each ω ∈ Ω. Denote by L0 (F , B) the linear space of equivalence classes of B-valued F -random variables. Example 2.3. L0 (F , B) is an RN module over K with base (Ω, F , P ) if the L0 -norm · : L0 (F , B) → L0+ and the module multiplication · : L0 (F , K) × L0 (F , B) → L0 (F , B) are defined as follows: x = the equivalence class of x 0 , where x 0 is a representative of x, ξ · x = the equivalence class of ξ 0 · x 0 , where ξ 0 and x 0 are representatives of ξ in L0 (F , K) and x in L0 (F , B), respectively, and ξ 0 · x 0 is defined by (ξ 0 · x 0 )(ω) = ξ 0 (ω) · x 0 (ω), ∀ω ∈ Ω. Remark 2.4. Let B be the classical conjugate space of B, then in [8] we proved that L0 (F , B ) is isomorphically isometric to the random conjugate space of L0 (F , B) iff B has the Randon– Nikodým property with respect to (Ω, F , P ). p
Since the RN module LF (E) constructed in [4] is very important in the study of conditional risk measures, this motivates us to give the following more general Example 2.5. Example 2.5. Let (S, · ) be an RN module over K with base (Ω, E, P ) and F a sub-σ -algebra of E. For each 1 p +∞, define ||| · |||p : S → L0+ (F , R) = {ξ ∈ L0 (F , R) | ξ 0} as follows: for all x ∈ S, |||x|||p =
1
(E[xp | F ]) p , if p ∈ [1, +∞), {ξ ∈ L0+ (F , R) | ξ x}, if p = +∞,
where E[· | F ] denotes the conditional expectation, cf. [4,27]. p p Denote LF (S) = {x ∈ S | |||x|||p ∈ L0+ (F , R)}, then (LF (S), ||| · |||p ) is an RN module over q K with base (Ω, F , P ). Similarly, we can also have LF (S ∗ ) for all q ∈ [1, +∞]. When S = p p L0 (E, R), LF (S) is exactly LF (E) in [4,27]. From now on, for a random locally convex module (E, P) and for any finite subfamily Q ⊂ P, · Q always denotes the L0 -seminorm defined by xQ = {x | · ∈ Q}, ∀x ∈ E, unless stated otherwise. The following (ε, λ)-topology has its root in the theory of probabilistic metric spaces, see [28] for the related details. It is easy to see that the (ε, λ)-topology for L0 (F , B) is exactly the one of convergence in probability P , and it is frequently used in many other topics, cf. [1,29]. Proposition 2.6. (See [14,22].) Let (E, P) be a random locally convex module over K with base (Ω, F , P ). For any real numbers ε > 0, 0 < λ < 1 and any finite subfamily Q of P, let Nθ (Q, ε, λ) = {x ∈ E | P {ω ∈ Ω | xQ (ω) < ε} > 1 − λ} and Uθ = {Nθ (Q, ε, λ) | Q ⊂ P finite, ε > 0, 0 < λ < 1}, then Uθ is a local base at θ of some Hausdorff linear topology, called the (ε, λ)-topology induced by P. Further, we have the following statements: (1) L0 (F , K) is a topological algebra over K endowed with its (ε, λ)-topology, which is exactly the topology of convergence in probability P ;
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(2) E is a topological module over the topological algebra L0 (F , K) when E and L0 (F , K) are endowed with their respective (ε, λ)-topologies; (3) a net {xα , α ∈ ∧} in E converges in the (ε, λ)-topology to x ∈ E iff {xα − x, α ∈ ∧} converges in probability P to 0 for each · ∈ P. From now on, for all random locally convex modules, their (ε, λ)-topologies are denoted by Tε,λ unless there is a danger of confusion. Proposition 2.7. (See [4].) Let (E, P) be a random locally convex module over K with base (Ω, F , P ). For any ε ∈ L0++ and Q ⊂ P finite, let BQ (ε) = {x ∈ E | xQ ε} and Uθ = {BQ (ε) | Q ⊂ P finite, ε ∈ L0++ }. A set G ⊂ E is called Tc -open if for every x ∈ G there exists some BQ (ε) ∈ Uθ such that x + BQ (ε) ⊂ G. Let Tc be the family of Tc -open subsets, then Tc is a Hausdorff topology on E, called the locally L0 -convex topology induced by P. Further, the following statements are true: (1) L0 (F , K) is a topological ring endowed with its locally L0 -convex topology; (2) E is a topological module over the topological ring L0 (F , K) when E and L0 (F , K) are endowed with their respective locally L0 -convex topologies; (3) a net {xα , α ∈ ∧} in E converges in the locally L0 -convex topology to x ∈ E iff {xα − x, α ∈ ∧} converges in the locally L0 -convex topology of L0 (F , K) to θ for each · ∈ P. From now on, for all random locally convex modules, their locally L0 -convex topologies are denoted by Tc unless there is a possible confusion. Since Tc is not necessarily a linear topology as proved in [4], but (E, Tc ) is always a topological group with respect to the addition operation for any random locally convex module (E, P), and hence Tc -Cauchy nets and Tc -completeness are still well defined. Tc is called locally L0 -convex because it has a striking local base Uθ = {BQ (ε) | Q ⊂ P finite and ε ∈ L0++ }, each member U of which satisfies the following: (i) L0 -convex: ξ · x + (1 − ξ ) · y ∈ U for any x, y ∈ U and ξ ∈ L0+ such that 0 ξ 1; (ii) L0 -absorbent: there is ξ ∈ L0++ for each x ∈ E such that x ∈ ξ · U ; (iii) L0 -balanced: ξ · x ∈ U for any x ∈ U and any ξ ∈ L0 (F , K) such that |ξ | 1. Such an ordered pair (E, Tc ) that Tc possesses the above properties (i), (ii) and (iii) is called a locally L0 -convex module (see [4, Definition 2.2]); conversely, for every locally L0 -convex module (E, T ), T can also be induced by a family of L0 -seminorms on E as above, see [4, Theorem 2.4] for its proof. Thus the theory of Hausdorff locally L0 -convex modules amounts to the theory of random locally convex modules endowed with the locally L0 -convex topology. 2.2. Hahn–Banach extension theorems for random linear functionals Let us first recall from [13]: let X be a linear space over K, then a linear operator f from X to L0 (F , K) is called a random linear functional on X; a mapping p : X → L0+ is called a random seminorm on X if it satisfies the following: (1) p(αx) = |α|p(x), ∀α ∈ K and x ∈ X; (2) p(x + y) p(x) + p(y), ∀x, y ∈ X.
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Let X be a real linear space. A mapping p : X → L0 (F , R) is called a random sublinear functional if it satisfies the above (2) and the following: (3) p(αx) = αp(x), ∀α 0 and x ∈ X. The main results of the subsection are stated as follows: Theorem 2.8. Let E be a real linear space, M ⊂ E a linear subspace, f : M → L0 (F , R) a random linear functional and p : E → L0 (F , R) a random sublinear functional such that f (x) p(x), ∀x ∈ M. Then there exists a random linear functional g : E → L0 (F , R) such that g extends f and g(x) p(x), ∀x ∈ E. Theorem 2.9. Let E be a complex linear space, M ⊂ E a linear subspace, f : M → L0 (F , C) a random linear functional and p : E → L0+ a random seminorm such that |f (x)| p(x), ∀x ∈ M. Then there exists a random linear functional g : E → L0 (F , C) such that g extends f and |g(x)| p(x), ∀x ∈ E. The proofs of Theorems 2.8 and 2.9 are given in [5–7,13] but in an indirect way, see the survey paper [13] for details. In fact, Theorem 2.8 is known in [2,30] since R is, of course, an ordered ring, and its proof is only a copy of the classical Hahn–Banach extension theorem for a real linear functional by replacing the order-completeness of R with the one of L0 (F , R). But the proof of Theorem 2.9 is somewhat different from its classical prototype, since E is not an L0 (F , C)-module. We give a direct proof of Theorem 2.9 for the reader’s convenience, the idea of whose proof comes from [5]. Proof of Theorem 2.9. Let f1 : M → L0 (F , R) be the real part of f , then it is easy to see that f (x) = f1 (x) − if1 (ix), ∀x ∈ M. Then it is clear that f1 satisfies the following: f1 (x) p(x),
∀x ∈ M.
(2.1)
Regarding M and E as real linear spaces, then Theorem 2.8 yields a random linear functional g1 : E → L0 (F , R) such that g1 extends f1 and g1 satisfies: g1 (x) p(x),
∀x ∈ E.
(2.2)
Define g : E → L0 (F , C) by g(x) = g1 (x) − ig1 (ix), ∀x ∈ E, then it is easy to check that g is a random linear functional extending f ; we want to prove that g satisfies the following: g(x) p(x),
∀x ∈ E.
(2.3)
Given x ∈ E, let g10 (x), g 0 (x) and p 0 (x) be arbitrarily chosen representatives of g1 (x), g(x) and p(x), respectively. Then we have the following relations: g10 (x)(ω) p 0 (x)(ω), a.s., ∀x ∈ E, g10 (x)(ω) = Re g 0 (x)(ω) , a.s., ∀x ∈ E, g 0 (x + y)(ω) = g 0 (x)(ω) + g 0 (y)(ω),
a.s., ∀x, y ∈ E,
(2.4) (2.5) (2.6)
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g 0 (αx)(ω) = α · g 0 (x)(ω),
a.s., ∀α ∈ C and x ∈ E,
p 0 (αx)(ω) = |α| · p 0 (x)(ω),
a.s., ∀α ∈ C and x ∈ E.
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(2.7) (2.8)
It follows immediately from (2.4), (2.5), (2.7) and (2.8) that Re(α · g 0 (x)(ω)) = Re(g 0 (αx)(ω)) = g10 (αx)(ω) p 0 (αx)(ω) = |α| · p 0 (x)(ω), a.s., ∀α ∈ C and x ∈ E. Now, let x be an arbitrary but fixed element of E and {cn | n ∈ N } a countable subset dense in C. Then there exists An (x) ∈ F for each n ∈ N such that P (An (x)) = 1 and 0 Re(cn · g 0 (x)(ω)) |cn | · p (x)(ω), ∀ω ∈ An (x). Let A(x) = n∈N An (x). Then A(x) ∈ F , P (A(x)) = 1 and the following is also true: Re α · g 0 (x)(ω) |α| · p 0 (x)(ω),
∀ω ∈ A(x) and α ∈ C,
(2.9)
since {cn | n ∈ N} is dense in C. Given ω ∈ Ω, let θ (ω) be the principal argument of g 0 (x)(ω), then θ is a random variable and 0 |g (x)|(ω) = e−iθ(ω) · g 0 (x)(ω), ∀ω ∈ Ω. It follows immediately from (2.9) that |g 0 (x)|(ω) = Re(|g 0 (x)|(ω)) = Re(e−iθ(ω) · 0 g (x)(ω)) |e−iθ(ω) | · p 0 (x)(ω) = p 0 (x)(ω), ∀ω ∈ A(x). Thus, we have |g 0 (x)|(ω) p 0 (x)(ω), a.s., since P (A(x)) = 1. Namely, |g(x)| p(x), which comes from the definition of the ordering . Finally, since x is arbitrary, then (2.3) has been proved. 2 2.3. Hahn–Banach extension theorems for L0 -linear functions Let E be a left module over the algebra L0 (F , K), a module homomorphism f : E → is called an L0 (or L0 (F , K))-linear function. If K = R, then a mapping p : E → 0 L (F , R) is called an L0 -sublinear function if it satisfies the following: L0 (F , K)
(i) p(ξ x) = ξp(x), ∀ξ ∈ L0+ and x ∈ E; (ii) p(x + y) p(x) + p(y), ∀x, y ∈ E. The main results of this subsection are known, see [2,4,30], but we give them extremely concise proofs. They are stated as follows: Theorem 2.10. Let E be a left module over the algebra L0 (F , R), M ⊂ E an L0 (F , R)submodule, f : M → L0 (F , R) an L0 -linear function and p : E → L0 (F , R) an L0 -sublinear function such that f (x) p(x), ∀x ∈ M. Then there exists an L0 -linear function g : E → L0 (F , R) such that g extends f and g(x) p(x), ∀x ∈ E. Theorem 2.11. Let E be a left module over the algebra L0 (F , C), M ⊂ E an L0 (F , C)submodule, f : M → L0 (F , C) an L0 -linear function and p : E → L0+ an L0 -seminorm such that |f (x)| p(x), ∀x ∈ M. Then there exists an L0 -linear function g : E → L0 (F , C) such that g extends f and |g(x)| p(x), ∀x ∈ E. Theorem 2.11 follows immediately from Theorem 2.10, since E is a module and the method used to prove its classical prototype will still be feasible. The following simple lemma can lead to an extremely concise proof of Theorem 2.10.
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Lemma 2.12. Let E be a left module over the algebra L0 (F , R), f : E → L0 (F , R) a random linear functional and p : E → L0 (F , R) an L0 -sublinear function such that f (x) p(x), ∀x ∈ E. Then f is an L0 -linear function. If R is replaced by C and p is an L0 -seminorm such that |f (x)| p(x), ∀x ∈ E, then f is also an L0 -linear function. Proof. We only give the proof of the first part, since the second one is similar. Since f is linear, it suffices to prove that f (ξ x) = ξf (x), ∀ξ ∈ L0+ and x ∈ E. Let x ∈ E be fixed. For any ξ ∈ L0+ , since there exists a sequence {ξn | n ∈ N } of simple elements in L0+ such that {ξn | n ∈ N } converges a.s. to ξ in a nondecreasing way, and since |f (ξ x) − f (ξn x)| = |f [(ξ − ξn )x]| (ξ − ξn )(|p(x)| ∨ |p(−x)|), it also suffices to prove that f (I˜A x) = I˜A · f (x), ∀A ∈ F . Since −p(−I˜A x) f (I˜A x) p(I˜A x), namely −I˜A p(−x) f (I˜A x) I˜A p(x), we have ˜ IAc f (I˜A x) = 0, ∀A ∈ F , where Ac = Ω \ A. Then we have that f (I˜A x) = I˜A f (I˜A x) + I˜Ac f (I˜A x) = I˜A f (I˜A x) = I˜A f (I˜A x) + I˜A f (I˜Ac x) = I˜A f (I˜A x + I˜Ac x) = I˜A f (x). 2 We can now prove Theorem 2.10. Proof of Theorem 2.10. Applying Theorem 2.8 to f and p yields a random linear functional g : E → L0 (F , R) such that g extends f and g(x) p(x), ∀x ∈ E. Since E is a left module over the algebra L0 (F , R) and p is an L0 -sublinear function, Lemma 2.12 shows that g is again an L0 -linear function. 2 Remark 2.13. The idea of the proof of Theorem 2.10 comes from [7], which is also used in [20,24]. 2.4. The Hahn–Banach extension theorems for continuous L0 -linear functions in random locally convex modules under the two kinds of topologies The main result in this subsection is Theorem 2.15, its proof needs the characterizations of continuous L0 -linear functions, which are first given below. ∗ = Given a random locally convex module (E, P) over K with base (Ω, F , P ), let Eε,λ 0 0 {f : E → L (F , K) | f is a continuous module homomorphism from (E, Tε,λ ) to (L (F , K), Tε,λ )} and Ec∗ = {f : E → L0 (F , K) | f is a continuous module homomorphism from (E, Tc ) to (L0 (F , K), Tc )}, which are called the random conjugate spaces of (E, P) under Tε,λ and Tc , respectively. It should be noticed that Ec∗ was already used in [4] in anonymous way. It is easy to see that f ∈ Ec∗ iff there are ξ ∈ L0+ and Q ⊂ P finite such that |f (x)| ξ · xQ , ∀x ∈ E. In fact, let f ∈ Ec∗ , then there exists some BQ (ε) as in Proposition 2.7 such that f (BQ (ε)) ⊂ {ξ ∈ L0 (F , K) | |ξ | 1}, so that we can have |f ( xQε+1/n x)| 1, ∀x ∈ E and n ∈ N , which means that |f (x)| ξ xQ , ∀x ∈ E, where ξ = 1/ε, and the converse is obvious. Lemma 4.1 of [27] is a special case of this result when (E, P) is an RN module. ∗ , as shown in [24]: It is, however, not trivial to characterize an element in Eε,λ ∗ iff there are a countable partition {A | n ∈ N } of Ω Proposition 2.14. (See [24].) f ∈ Eε,λ n to F , a sequence {ξn | n ∈ N} in L0+ and a sequence {Qn | n ∈ N } of finite subfamilies of P such that |f (x)| n1 I˜An · ξn · xQn , ∀x ∈ E, in which case if, let ξ = n1 I˜An · ξn and x = n1 I˜An · xQn , ∀x ∈ E, then |f (x)| ξ · x, ∀x ∈ E.
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Theorem 2.15. Let (E, P) be a random locally convex module over K with base (Ω, F , P ) and M an L0 (F , K)-submodule of E. Then we have the following: (1) every continuous L0 -linear function f from (M, Tc ) to (L0 (F , K), Tc ) admits a continuous L0 -linear extension g from (E, Tc ) to (L0 (F , K), Tc ); (2) every continuous L0 -linear function f from (M, Tε,λ ) to (L0 (F , K), Tε,λ ) admits a continuous L0 -linear extension g from (E, Tε,λ ) to (L0 (F , K), Tε,λ ); (3) if (E, P) is an RN module (E, · ), then g in both (1) and (2) can be required to be such that g∗ = f ∗ , namely g is a random-norm preserving extension, where the definitions of f ∗ and g∗ are given in Definition 2.16 below. Proof. (1) Let PM = {·M | · ∈ P}, where ·M is the restriction of · to M, then (M, PM ) is still a random locally convex module. Since f is a continuous L0 -linear function from (M, Tc ) to (L0 (F , K), Tc ), there exist a finite subfamily Q of P and ξ ∈ L0+ such that |f (x)| ξ xQ , ∀x ∈ M. Then Theorems 2.10 and 2.11 jointly yield an L0 -linear function g : E → L0 (F , K) such that g extends f and |g(x)| ξ xQ , ∀x ∈ E. Of course, g satisfies the requirement of (1). (2) It follows immediately from Proposition 2.14 that there exist ξ ∈ L0+ , a countable partition {An | n ∈ N } of Ω to F and a sequence {Qn | n ∈ N } of finite subfamilies of P such that |f (x)| ξ x, ∀x ∈ M, where · : E → L0+ is given by x = n1 I˜An xQn , ∀x ∈ E. 0 Obviously, · is an L0 -seminorm, then Theorems 2.10 and 2.11 jointly yield an L -linear 0 function g : E → L (F , K) such that |g(x)| ξ x, ∀x ∈ E. Further, since n1 P (An ) = P (Ω) = 1, then P (An ) → 0 as n → ∞, and hence · is continuous from (E, Tε,λ ) to (L0+ , Tε,λ ), which implies that g is also continuous. (3) Let p(x) = f ∗ x, ∀x ∈ E, then Theorems 2.10 and 2.11 again jointly complete the proof. 2 2.5. The relation between the random conjugate spaces of a random locally convex module under the two kinds of topologies Given a random locally convex module (E, P) over K with base (Ω, F , P ), let us first recall from [4] that P is called having the countable concatenation property if n1 I˜An · · Qn still belongs to P for any countable partition {An | n ∈ N } of Ω to F and any sequence {Qn | n ∈ N } ∗ ⊃ E ∗ and further shows that the random of finite subfamilies of P. Proposition 2.14 implies Eε,λ c conjugate spaces of (E, P) under Tε,λ and Tc coincide if P has the countable concatenation property, in particular an RN module has the same random conjugate space under Tε,λ and Tc . Definition 2.16. (See [11].) Let (E, · ) be an RN module over K with base (Ω, F , P ) and ∗ = E ∗ . Define · ∗ : E ∗ → L0 by f ∗ = {ξ ∈ L0+ | |f (x)| ξ · x, ∀x ∈ E}, E ∗ := Eε,λ + c ∗ ∗ over K with base (Ω, F , P ), called the random conjugate then (E , · ) is an RN module space of (E, · ) (notice: f ∗ = {|f (x)| | x ∈ E and x 1}, cf. [19]). Let (E, · ) be an RN module, E ∗∗ denotes (E ∗ )∗ , the canonical embedding mapping J : E → E ∗∗ defined by (J x)(f ) = f (x), ∀x ∈ E and f ∈ E ∗ , is random-norm preserving. If J is surjective, then E is called random reflexive. In [9] we proved that E ∗ is Tε,λ -complete. Clearly, we will see from the discussion about completeness in Section 3.4 that if E is random reflexive, then E is complete under both Tc and Tε,λ , and has the countable concatenation property since E ∗∗ has these properties. The main
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results below of this subsection are essentially independent of a special choice of Tc and Tε,λ , which were established under Tε,λ . Since they are still valid under Tc , we only state them without proofs and mention of topologies. Theorem 2.17. (See [8].) L0 (F , B) (see Example 2.3) is random reflexive iff B is a reflexive Banach space. Theorem 2.18. (See [19, James theorem].) A complete RN module (S, · ) is random reflexive iff there exists p ∈ S(1) for each f ∈ E ∗ such that f (p) = f , where S(1) = {p ∈ S | p 1}. 3. Relations between the hyperplane separation theorems currently available 3.1. A new countable concatenation property Filipovi´c, Kupper and Vogelpoth introduced in [4] the two countable concatenation properties, one is relative to topology and the other is relative to the family of L0 -seminorms, in particular the two are essentially the same one for a random locally convex module (E, P) endowed with Tc . Thus neither of the two has anything to do with the L0 -module E itself. The main results in Sections 3 and 4 show that there is another kind of countable concatenation property, which is concerned with the L0 -module E itself and is very important for the theory of locally L0 -convex module. To introduce it, we give the following: First, we make the following convention that all the L0 (F , K)-modules E in the sequel of this paper satisfy the property: For any two elements x and y ∈ E, if there exists a countable partition {An | n ∈ N } of Ω to F such that I˜An x = I˜An y for each n ∈ N , then x = y, where I˜A x is called the A-stratification of x for any A ∈ F . It is easy to verify by definition that a random locally convex module satisfies the above property. Definition 3.1. Let E be a left module over the algebra L0 (F , K). Such a formal sum as ˜ n1 IAn xn for some countable partition {An | n ∈ N } of Ω to F and some sequence {xn | respect to {An | n ∈ N }. n ∈ N} in E, is called a countable concatenation of {xn | n ∈ N } with Furthermore a countable concatenation n1 I˜An xn is well defined or n1 I˜An xn ∈ E if there is x ∈ E such that I˜An x = I˜An xn , ∀n ∈ N . (Clearly, x is unique, in which case we write x = n1 I˜An xn .) A subset G of E is called having the countable concatenation property if ev ery countable concatenation n1 I˜An xn with xn ∈ G for each n ∈ N still belongs to G, namely ˜ ˜ n1 IAn xn is well defined and there exists x ∈ G such that x = n1 IAn xn . 0 Definition 3.2. Let E be a left module over the algebra L (F , K) and {Gn | n ∈ N } a sequence of subsets of E. The set of countable concatenations n1 I˜An xn with xn ∈ Gn for each n ∈ N is called the countable concatenation hull of the sequence {Gn | n ∈ N }, denoted by Hcc ({Gn | n ∈ N}). In particular when Gn = G for each n ∈ N , Hcc (G), denoting Hcc ({Gn | n ∈ N }), is called the countable concatenation hull of the subset G. (Clearly Hcc (G) ⊃ G, ∀G ⊂ E.)
Definition 3.1 leads to a notion of E having the countable concatenation property as a subset of itself, it is essentially different from the one introduced in [4]. To remove any possible confusions, when we talk of an L0 -module having the countable concatenation property we will always employ this one rather than that introduced in [4].
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Example 3.3 below, adopted from [4], exhibits that it is very necessary to introduce the notion of a well-defined countable concatenation. Example 3.3. For an L0 (F , K)-module E, M ⊂ E a subset of E, the L0 (F , K)-submodule generated by M is denoted by SpanL0 (M), namely SpanL0 (M) = { ni=1 ξi xi | ξi ∈ L0 (F , K), xi ∈ M, 1 i n and n ∈ N }. Take E = L0 (F , R), M = {I˜[1−2−(n−1) ,1−2−n ] | n ∈ N } and ∞ n ˜ F = SpanL0 (M), then it is easy to see that n=1 2 I[1−2−(n−1) ,1−2−n ] ∈ E \ F and the k n sequence { n=1 2 I˜[1−2−(n−1) ,1−2−n ] | k ∈ N } in F is not a Tc -Cauchy sequence but a T ε,λ -Cauchy sequence which has no limit in (F, Tε,λ ). Thus the countable concatenation ∞ n˜ n=1 2 I[1−2−(n−1) ,1−2−n ] is not well defined in both (F, Tc ) and (F, Tε,λ ) in any ways, namely neither of (F, Tc ) and (F, Tε,λ ) has the countable concatenation property. However, it is very easy to see that a Tε,λ -complete random locally convex module, the RN ∗ of any random locally module L0 (F , B) in Example 2.3 and the random conjugate space Eε,λ convex module E all have the countable concatenation property, our previous investigations [10,18,19,21,22] have shown that their countable concatenation property plays a crucial role in the development of the theory of random locally convex modules. The random normed modules p ϕ LF (E) and LF (E) in [4,27], which are important for the study of conditional risk measures, also have the countable concatenation property. 3.2. Relations between the hyperplane separation theorems currently available It is easy to see that Theorem 3.1 of [22] is independent of Theorem 2.6 of [4], the purpose of this subsection is to prove that Theorem 3.1 of [22] implies Theorem 2.8 of [4]. For this, we first restate Theorem 2.8 of [4] and an equivalent variant of Theorem 3.1 of [22] as Theorems 3.4 and 3.6 below, respectively, then we arrive at our aim by Theorem 3.10. Theorem 3.4. (See [4, Theorem 2.8, Hyperplane separation II].) Let (E, P) be a random locally convex module over R with base (Ω, F , P ) such that P has the countable concatenation property, x ∈ E and G ⊂ E a nonempty Tc -closed L0 -convex subset with the countable concatenation property. If x and G satisfy the following: I˜A {x} ∩ I˜A G = ∅ for all A ∈ F with P (A) > 0,
(3.10)
then there exist an f ∈ Ec∗ and ε ∈ L0++ such that f (x) > f (y) + ε
on Ω for all y ∈ G.
(3.11)
Remark 3.5. In Theorem 3.4, if R is replaced by C, then Theorem 3.4 is still true in the way: (Ref )(x) > (Ref )(y) + ε on Ω for all y ∈ G, where Ref : E → L0 (F , R) denotes the real part of f ; it is easy to see that f (x) = (Ref )(x) − i(Ref )(ix), ∀x ∈ E. ∗ (x, G) = For a random locally convex module (E, P), x ∈ E and G ⊂ E, let dQ {x − yQ | ∗ ∗ y ∈ G} for any finite subfamily Q of P and d (x, G) = {dQ (x, G) | Q ⊂ P finite}. If G is a nonempty Tε,λ -closed L0 -convex subset, then we proved in [22] that x ∈ G iff d ∗ (x, G) = 0.
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Theorem 3.6. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), / G, and ξ a chosen reprex ∈ E, G a nonempty Tε,λ -closed L0 -convex subset of E such that x ∈ ∗ such that sentative of d ∗ (x, G). Then there exists an f ∈ Eε,λ
(Ref )(y) y ∈ G ,
(3.12)
(Ref )(y) y ∈ G on [ξ > 0].
(3.13)
(Ref )(x) > and (Ref )(x) >
Theorem 3.7. Theorem 3.6 is equivalent to Theorem 3.1 of [22]. Proof. We only need to check that (3.12) and (3.13) are equivalent to (1) and (2) of Theorem 3.1 of [22], respectively. Let ξ , (Ref )(x) and {(Ref )(y) | y ∈ G} be the same ones as in Theorem 3.6, and let η and r be arbitrarily chosen representatives of (Ref )(x) and {(Ref )(y) | y ∈ G}, respectively. Then (1) and (2) of Theorem 3.1 of [22] are equivalent to the following (3.14) and (3.15), respectively:
(Ref )(y) y ∈ G , (Ref )(x) > P [η > r] [ξ > 0] = 0,
and
(3.14) (3.15)
where [η > r] = {ω ∈ Ω | η(ω) > r(ω)}, and [η > r] [ξ > 0] denotes the symmetric difference of [η > r] and [ξ > 0]. Clearly, (3.14) is just (3.12), and thus we only need to prove that (3.15) is equivalent to (3.13). (3.15) implies, of course, (3.13). On the other hand, (3.13) of Theorem 3.6 shows that [η > r] ⊃ [ξ > 0], a.s., we will prove [η > r] ⊂ [ξ > 0], a.s., as follows. Otherwise, let D = [η > r] \ [ξ > 0], then P (D) > 0. Thus ID (ω) · ξ(ω) = 0, a.s., namely I˜D · d ∗ (x, G) = 0, so that d ∗ (I˜D x, I˜D G) = 0, but from the following Lemma3.8, we have I˜D x ∈ ˜ ˜ ˜ ˜ the Tε,λ -closure of ID G, which means that ID (Ref )(x) = (Ref )(ID x) = {(Ref )(ID y) | y ∈ ˜ G} = ID · ( {(Ref )(y) | y ∈ G}), namely ID (ω)η(ω) = ID (ω)r(ω), a.s., which contradicts the fact that η(ω) > r(ω), a.s. on D. 2 In the sequel of this paper, for a subset G of a random locally convex module (E, P), Gε,λ deE, any finite subfamily Q of notes the Tε,λ -closure of G, and Gc the Tc -closure of G. For any x ∈ ∗ (x, G) = {ε ∈ L0++ | UQ,ε [x] ∩ G = P and ε ∈ L0++ , let UQ,ε [x] = {y ∈ E | x − yQ ε}, εQ ∗ ∗ ∅} and ε (x, G) = {εQ (x, G) | Q ⊂ P finite}. Lemma 3.8. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), x ∈ E and G a nonempty subset of E. Then we have the following: (1) d ∗ (x, G) = ε ∗ (x, G); (2) d ∗ (x, G) = d ∗ (x, Gε,λ ) = d ∗ (x, Gc ). If, in addition, G satisfies the following: I˜A y + I˜Ac z ∈ G
for all A ∈ F and all y, z ∈ G,
(3.16)
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then we have the following: (3) x ∈ Gε,λ iff d ∗ (x, G) = 0. ∗ (x, G) = ε ∗ (x, G) for each Q ⊂ P finite. By defiProof. (1) We only need to check that dQ Q ∗ (x, G) = ∗ (x, G) = nition, dQ {x − yQ | y ∈ G} and εQ {ε ∈ L0++ | UQ,ε [x] ∩ G = ∅}. If ε ∈ L0++ is such that U Q,ε [x] ∩ G = ∅, namely there exists y ∈ G such that x − yQ ε, ∗ (x, G), so that ε ∗ (x, G) d ∗ (x, G). In which means that ε {x − yQ | y ∈ G} = dQ Q Q the other direction, for any y ∈ G and n ∈ N , it is clear that x − yQ x − yQ + n1 , and x − yQ + n1 ∈ L0++ , if, let ε = x − yQ + n1 , then we have that y ∈ UQ,ε [x] ∩ G, of course, UQ,ε [x] ∩ G = ∅, and thus we have that ε = x − yQ + n1 {ε ∈ L0++ | UQ,ε [x] ∩ G = ∅}, ∗ (x, G), in turn d ∗ (x, G) = ∗ (x, G). ∀n ∈ N , so that x − yQ εQ {x − yQ | y ∈ G} εQ Q (2) is clear. (3) Lemma 2.2 of [22] shows that x ∈ F iff d ∗ (x, F ) = 0 for every Tε,λ -closed L0 -convex subset F of E, in which proof the only property (3.16) of an L0 -convex subset was used, and thus we have that x ∈ F iff d ∗ (x, F ) = 0 for every Tε,λ -closed subset F with the property (3.16). It is easy to see that Gε,λ also has the property (3.16) if G does. Applying the result to Gε,λ yields that x ∈ Gε,λ iff d ∗ (x, Gε,λ ) = 0, so that (2) has implied that x ∈ Gε,λ , iff d ∗ (x, G) = 0. 2
Perhaps, one would hope to have that x ∈ G iff d ∗ (x, G) = 0 for every Tc -closed subset G with the property (3.16), but it is not true since Tc is too strong (see the following Lemma 3.11)! Lemma 3.8 leads to the following equivalent variant Theorem 3.9 of Theorem 3.6, one only needs to notice that x ∈ / Gε,λ iff d ∗ (x, G) > 0 for every subset G with the property (3.16), and then applies Theorem 3.6 to x and Gε,λ . Theorem 3.9. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), x ∈ E ∗ such that and G an L0 -convex subset of E such that d ∗ (x, G) > 0. Then there exists an f ∈ Eε,λ (Ref )(x) >
(Ref )(y) y ∈ G ,
(3.17)
and (Ref )(x) >
(Ref )(y) y ∈ G
on [ξ > 0],
(3.18)
where ξ is an arbitrarily chosen representative of d ∗ (x, G). Theorem 3.10. Theorem 3.9 (equivalently, Theorem 3.6) implies Theorem 3.4. Proof. If x and G satisfy the conditions of Theorem 3.4, then the following Lemma 3.11 shows ∗ that d ∗ (x, G) > 0 on Ω, namely [ξ > 0] in Theorem 3.9 is just Ω, so there exists an f ∈ Eε,λ 1 such that (Ref )(x) > {(Ref )(y) | y ∈ G} on Ω by (3.18) of Theorem 3.9. Let ε = 2 ((Ref )(x) − {(Ref )(y) | y ∈ G}), then ε ∈ L0++ satisfies the following: (Ref )(x) >
(Ref )(y) y ∈ G + ε
on Ω,
(3.19)
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f , of course, satisfies the requirement of Theorem 3.4 if one notices that f is also in Ec∗ since ∗ by observing that P has the countable concatenation property. 2 Ec∗ = Eε,λ Lemma 3.11. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), x ∈ E and G ⊂ E a Tc -closed nonempty subset such that IˆA {x} ∩ I˜A G = ∅ for all A ∈ F with P (A) > 0 and such that G has the countable concatenation property. Then d ∗ (x, G) = ε ∗ (x, G) > 0 on Ω, namely ε ∗ (x, G) ∧ 1 ∈ L0++ . Proof. If it is not true that d ∗ (x, G) > 0 on Ω, then there is an A ∈ F such that P (A) > 0 and ∗ (x, G) = 0 for every finite subfamily Q of P. I˜A · d ∗ (x, G) = 0, so that I˜A · dQ We can, without loss of generality, assume that θ ∈ G (otherwise, by a translation). Since G has the countable concatenation property, it must have the property (3.16) so that {I˜A x − I˜A yQ | y ∈ G} is directed downwards for each Q ⊂ P finite, and it is also easy to see from the property (3.16) that I˜A G ⊂ G, so that we can easily prove that I˜A G is also Tc -closed. Now, for each fixed Q ⊂ P finite and each fixed α ∈ L0++ , we will prove that there is yQ,α ∈ G such that I˜A x − I˜A yQ,α Q α as follows. ∗ (x, G) = Since I˜A dQ {I˜A x − I˜A yQ | y ∈ G} = 0, then there exists a sequence {yn | n ∈ N} in G such that {I˜A x − I˜A yn Q | n ∈ N } converges to 0 in a nonincreasing way. Let εn = I˜A x − I˜A yn Q for each n ∈ N and choose a representative εn0 of εn for each n ∈ N 0 (ω) for each n ∈ N and ω ∈ Ω, and a representative α 0 of α such such that εn0 (ω) εn+1 0 0 that [εn α ] ↑ Ω, where [εn0 α 0 ] (denoted by En ) = {ω ∈ Ω | εn0 (ω) α 0 (ω)}. Again let An = En \ En−1 , n 1, where E0 = ∅, then {An | n ∈ N } forms a countable partition of Ω to F , so that yQ,α := n1 I˜An yn ∈ G and satisfies that I˜A x − I˜A yQ,α Q α. Finally, let Γ = F (P) × L0++ = {(Q, α) | Q ⊂ P finite and α ∈ L0++ }, where F (P) is the set of finite subfamilies of P. It is easy to see that Γ is directed upwards by the ordering: (Q1 , α1 ) (Q2 , α2 ) iff Q1 ⊂ Q2 and α2 α1 , so that {I˜A yQ,α , (Q, α) ∈ Γ } is a net in I˜A G which is convergent to I˜A x, and hence I˜A x ∈ I˜A G, which contradicts the fact that I˜F x ∈ / I˜F G for all F ∈ F with P (F ) > 0. 2 From the process of proof of Lemma 3.11, we can easily see that if G has the countable concatenation property and d ∗ (x, G) = 0 (at which time, take A = Ω) then x ∈ Gc , this yields a useful fact, namely Theorem 3.12 below, from which we have that a subset having the countable concatenation property has the same closure under Tc and Tε,λ , in particular it is Tc -closed iff it is Tε,λ -closed, which also derives a surprising fact on Tc -completeness, see Section 3.4. Theorem 3.12. Let (E, P) be a random locally convex module, x ∈ E and G ⊂ E a subset having the countable concatenation property. Then the following are equivalent: (1) x ∈ Gc ; (2) x ∈ Gε,λ ; (3) d ∗ (x, G) = 0. Theorem 3.6 (namely, Theorem 3.9) has been known for many years, which was mentioned in [21] without proof, where a special case of which was proved. We may say that Theorem 3.6 is general enough to meet all our needs under Tε,λ -topology, it implies Theorem 3.4 but is inde-
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pendent of Theorem 2.6 of [4]. It follows that we will generalize Theorems 2.6 and 2.8 of [4] to allow a separation with any probability rather than the merely probability 1. 0 Let E be a left module L (F , K), we make another convention: two count over the algebra able concatenations n1 I˜An xn and n1 I˜Bn yn are called equal if I˜Ai ∩Bj xi = I˜Ai ∩Bj yj , ∀i, j ∈ N . Theorem 3.13. Let E be a left module over the algebra L0 (F , K), M and G any two nonempty subsets of E such that I˜A M + I˜Ac M ⊂ M and I˜A G + I˜Ac G ⊂ G. If Hcc (M) ∩ Hcc (G) = ∅, then there exists an F -measurable subset H (M, G) unique a.s. such that the following are satisfied: (1) P (H (M, G)) > 0; (2) I˜A M ∩ I˜A G = ∅ for all A ∈ F , A ⊂ H (M, G) with P (A) > 0; (3) I˜A M ∩ I˜A G = ∅ for all A ∈ F , A ⊂ Ω \ H (M, G) with P (A) > 0. Proof. Let E = {A ∈ F | I˜A M ∩ I˜A G = ∅}. Then E is directed upwards: in fact, for any A and B ∈ E there exist x1 , x2 ∈ M and y1 , y2 ∈ G such that I˜A x1 = I˜A y1 and I˜B x2 = I˜B y2 . Since M and G are nonempty, take x0 in M and y0 in G, and let x = I˜A x1 + I˜B\A x2 + I˜(A∪B)c x0 and y = I˜A y1 + I˜B\A y2 + I˜(A∪B)c y0 , then I˜A∪B x = I˜A∪B y ∈ I˜A∪B M ∩ I˜A∪B G by noticing x ∈ M and y ∈ G. Define H (M, G) = Ω \ ess sup(E), then H (M, G), obviously, satisfies (2) and (3). We will verify that H (M, G) also has the property (1). In fact, if P (H (M, G)) = 0, then P (ess sup(E)) = 1. Let {Dn | n ∈ N} be a nondecreasing sequence of E such that Dn ↑ Ω, then there exist two sequences {xn | n ∈ N} in M and {yn | n ∈ N } in G such that I˜Dn xn = I˜Dn yn , ∀n ∈ N . Let An = Dn \ Dn−1 , ∀n 1, where D0 = ∅, then n1 I˜An xn = n1 I˜An yn ∈ Hcc (M) ∩ Hcc (G), which is a contradiction. 2 Definition 3.14. Let E, M and G be the same as in Theorem 3.13 such that Hcc (M) ∩ Hcc (G) = ∅, then H (M, G) is called the hereditarily disjoint stratification of H and M, and P (H (M, G)) is called the hereditarily disjoint probability of H and G. 3.3. The hereditarily disjoint probability and more general forms of hyperplane separation theorems First, we state the main results of this subsection—Theorems 3.15 and 3.16, whose proofs follow from Lemma 3.17. Please notice that Theorems 2.6 and 2.8 of [4] correspond to the special cases of Theorems 3.15 and 3.16 when H (M, G) = Ω and H ({x}, G) = Ω, respectively. Theorem 3.15. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), M and G two nonempty L0 -convex subsets such that G is Tc -open and Hcc (G) ∩ Hcc (M) = ∅. Then there exists an f ∈ Ec∗ such that for all x ∈ G and y ∈ M,
(3.20)
on H (M, G) for all x ∈ G and y ∈ M.
(3.21)
(Ref )(x) < (Ref )(y) and (Ref )(x) < (Ref )(y)
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In the following Theorem 3.16, we only need to notice that Hcc (G) = G and H ({x}, G) is just [ξ > 0] in Theorem 3.9. Theorem 3.16. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), such that P has the countable concatenation property, x ∈ E and G a nonempty Tc -closed L0 -convex subsets of E such that x ∈ / G and G has the countable concatenation property. Then there exists an f ∈ Ec∗ such that (Ref )(x) >
(Ref )(y) y ∈ G ,
(3.22)
and (Ref )(x) >
(Ref )(y) y ∈ G
on H {x}, G .
(3.23)
Lemma 3.17. Let (E, P) be a random locally convex module over K with base (Ω, F , P ), M a Tc -closed subset of E such that I˜A M + I˜Ac M ⊂ M, for all A ∈ F , and G a Tc -open subset of E such that I˜A G + I˜Ac G ⊂ G, for all A ∈ F . Then for each A ∈ F with P (A) > 0, I˜A M is relatively Tc -closed in I˜A E and I˜A G is relatively Tc -open in I˜A E. Proof. We can assume that θ ∈ G (otherwise, by a translation), then I˜A G ⊂ G, so that I˜A E ∩ G = I˜A G, which implies that I˜A G is relatively Tc -open in I˜A E. The proof of the fact that I˜A M is relatively Tc -closed in I˜A E is similar, so is omitted. 2 We can now prove Theorem 3.15. Proof of Theorem 3.15. Let Ω = H (M, G), F = Ω ∩ F = {Ω ∩ F | F ∈ F } and P : F → [0, 1] be defined by P (Ω ∩ F ) = P (Ω ∩ F )/P (Ω ), ∀F ∈ F . Take E = I˜Ω E, P = { · E | · ∈ P}, M = I˜Ω M, G = I˜Ω G and consider (E , P ) as a random locally convex module with base (Ω , F , P ). Then by Lemma 3.17 we have that M and G satisfy the condition of Theorem 2.6 of [4], so that there exists an f ∈ (E )∗c such that Ref (x) < Ref (y)
on Ω for all x ∈ G and y ∈ M .
(3.24)
By Theorem 2.15 f has an extension f ∈ Ec∗ . Now let f = I˜H (M,G) f , then f (x) = 0, ∀x ∈ I˜H (M,G)c E and f (x) = f (x), ∀x ∈ I˜H (M,G) E, so that f satisfies all the requirements of Theorem 3.15. 2 Proof of Theorem 3.16. It is completely similar to the proof of Theorem 3.15. (In fact, it can also be derived directly from Theorem 3.9.) 2 3.4. The relation between the two kinds of completeness of a random locally convex module under the two kinds of topologies The main result of this subsection is the following:
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Theorem 3.18. Let (E, P) be a random locally convex module over K with base (Ω, F , P ). Then E is Tc -complete if E is Tε,λ -complete. Furthermore, if E is Tc -complete and has the countable concatenation property, then E is also Tε,λ -complete. Proof of the first part of Theorem 3.18 is easy because the topological module (E, Tc ) has a local base Uθ = {BQ (ε) | Q ⊂ P finite and ε ∈ L0++ } as in Proposition 2.7 such that each BQ (ε) is Tε,λ -closed. However, the proof of the second part of it is a delicate matter since Tc is much stronger than Tε,λ , and it seems to be not an easy work for us to give a direct proof even for the case of an RN module. It is more interesting that we may give a clever proof of it by using Theorem 3.12 and the following Lemma 3.19. First, let us recall some terminologies as follows. Let (E, P) be a random locally convex module over K with base (Ω, F , P ). Two Tε,λ -Cauchy nets {xα , α ∈ Γ } and {yβ , β ∈ Λ} in E are called equivalent if {xα − yβ , (α, β) ∈ Γ × Λ} converges to 0 in probability P for each · ∈ P. Since the sum of {xα , α ∈ Γ } and {yβ , β ∈ Λ} is defined, as usual, to be {xα + yβ , (α, β) ∈ Γ × Λ}, which motivates us to do the following thing: if E has the countable concatenation property, {An | n ∈ N } is a countable partition of Ω to F and {{xαn , αn ∈ Γn } | n ∈ N } is a sequence of Cauchy nets in E, then we naturally define ˜ ˜ their
countable concatenation n1 IAn {xαn , αn ∈ Γn } = { n1 IAn xαn , (α1 , α2 , . . . , αn , . . .) ∈ Γ }, and it is easy to check that this is again a T -Cauchy net by noticing that P (An ) → 0 n ε,λ n1 as n → ∞, so that this countable concatenation is well defined. According to the same fact that P (An ) → 0 as n → ∞, we can verify that if {yβn : βn ∈ Λn } is equivalent to {xαn , αn ∈ Γn }
for each n ∈ N , then { n1 I˜An xαn , (α1 , α2 , . . . , αn , . . .) ∈ n1 Γn } is still equivalent to
{ n1 I˜An yβn , (β1 , β2 , . . . , βn , . . .) ∈ n1 Λn }. These observations lead to Lemma 3.19 below. Lemma 3.19. Let (E, P) be a random locally convex module over K with base (Ω, F , P ) such that E has the countable concatenation property. For a Tε,λ -Cauchy net {xα , α ∈ Γ }, [{xα , α ∈ Γ }] denotes its Tε,λ -equivalence class, the Tε,λ -equivalence class of a constant net with value x ∈ E is denoted by [x]. Let E˜ ε,λ = {[{xα , α ∈ Γ }] | {xα , α ∈ Γ } is a Tε,λ -Cauchy net}. The module operations are defined as follows: {xα , α ∈ Γ } + {yβ , β ∈ Λ} := xα + yβ , (α, β) ∈ Γ × Λ , ξ {xα , α ∈ Γ } = {ξ xα , α ∈ Γ } . Further, each · ∈ P induces an L0 -seminorm on E˜ ε,λ , still denoted by · , so that [{xα , α ∈ Γ }] = the limit of convergence in probability P of {xα , α ∈ Γ }. Then (E˜ ε,λ , P) is a Tε,λ -complete random locally convex module over K with base (Ω, F , P ) such that E˜ ε,λ still has the countable concatenation property, called the Tε,λ -completion of (E, P), further (E, P) is P-isometrically isomorphic with a dense submodule {[x] | x ∈ E} of E˜ ε,λ . Proof. From the proof of completion of a linear topological space, we can first have that (E˜ ε,λ , P) is Tε,λ -complete. Further, let
{{xαn , αn ∈ Γn } | n ∈ N } be a sequence of Tε,λ -Cauchy nets in E and define παn = xαn , ∀α ∈ n1 Γn and n ∈ N , then {xαn , αn ∈ Γn } is equivalent to {παn , α ∈ n1 Γn } for each n ∈ N . Thus for a countable partition {An | n ∈ N} of Ω to F we have that I˜Am [{ n1 I˜An xαn ,
(α1 , α2 , . . . , αn , . . .) ∈ n1 Γn }] = [{I˜Am παm , α ∈ n1 Γn }] (by the definition of the module
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multiplication) = I˜Am [{παm , α ∈ n1 Γn }] = I˜Am [{xαm , αm ∈ Γm }] for each m ∈ N , so
that m1 I˜Am [{xαm , αm ∈ Γm }] = [{ n1 I˜An xαn , (α1 , α2 , . . . , αn , . . .) ∈ n1 Γn }] ∈ E˜ ε,λ , namely E˜ ε,λ has the countable concatenation property. 2 Remark 3.20. Although every random locally convex module (E, P) admits a Tc -completion (E˜ c , P) and a Tε,λ -completion (E˜ ε,λ , P) such that (E, P) is P-isometrically isomorphic with a dense submodule of each of the latter two, respectively, but since Tc is so strong that a countable concatenation of a sequence of Tc -Cauchy nets is not necessarily well defined, so that we cannot give Lemma 3.19 for the Tc -topology. Lemma 3.19 is necessary since the countable concatenation property of E is reserved in E˜ ε,λ in a proper way when E and {[x] | x ∈ E} are identified. We can now prove Theorem 3.18. Proof of Theorem 3.18. Let (E˜ ε,λ , P) be the Tε,λ -completion of (E, P) as in Lemma 3.19 and regard E as a subset of E˜ ε,λ , then Theorem 3.12 shows that E c = E ε,λ = E˜ ε,λ . On the other hand, since E is Tc -complete, we always have that E c = E, so that E˜ ε,λ = E, hence E must be Tε,λ -complete. 2 Remark 3.21. Theorem 3.18 is a powerful result, for example, Kupper and Vogelpoth proved p ϕ in [27] that LF (E) and LF (E) are Tc -complete, then they must be Tε,λ -complete by the second part of Theorem 3.18, since they both have the countable concatenation property. On the other p hand, it is easy to verify that L1F (E) is Tc -complete, as to LF (E) when p > 1 they are, obviously, Tc -complete by the first part of Theorem 3.18 since they are all the random conjugate spaces of some RN modules and Tε,λ -complete. 4. Representation theorems of random conjugate spaces of random normed modules under the locally L0 -convex topology Since the (ε, λ)-topology is rarely a locally convex topology in the sense of traditional functional analysis, consequently, the theory of traditional conjugate spaces universally fails to serve for the deep development of RN modules under the (ε, λ)-topology, see [22] for details. It is under such a background that the theory of random conjugate spaces of random normed modules has been developed and has been being centered at our previous work, and in fact it is also the most difficult and deepest part of our previous work, cf. [8,10,16,19,23]. The locally L0 -convex topology has the nice convexity, but it is too strong, it is, certainly, also rather difficult to establish the corresponding results of [8,10,16,19,23] under the locally L0 convex topology in a direct way. Considering that an RN module has the same random conjugate space under the two kinds of topologies, even many results are independent of a special choice of the two kinds of topologies, we have established the corresponding Tc -variants—Theorems 2.17 and 2.18 of those deep results previously established in [8,19] under Tε,λ . The first main result of this section is to establish the Tc -variant Theorem 4.3 of the result in [23], based on Section 3.4 this will also become an easy matter in an indirect manner. The second main result of this section is Theorem 4.5 which generalizes and improves Theorem 4.5 of [27], in particular whose isometric argument is new and not easy. Corresponding to Definition 2.1, Definition 4.1 below is merely a restatement of the notion of a random inner product module introduced in [11].
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Definition 4.1. (See [11].) An ordered pair (E, ·,·) is called a random inner product module (briefly, an RIP module) over K with base (Ω, F , P ) if E is a left module over the algebra L0 (F , K) and ·,· is a mapping from E × E to L0 (F , K) such that the following are satisfied: (1) (2) (3) (4)
x, x ∈ L0+ , and x, x = 0 iff x = θ (the null vector of E); x, y = y, x, ∀x, y ∈ E where y, x denotes the complex conjugate of y, x; ξ x, y = ξ x, y, ∀ξ ∈ L0 (F , K), and x, y ∈ E; x + y, z = x, z + y, z, ∀x, y, z ∈ E,
where x, y is called the L0 -inner product between x and y. As in classical functional analysis, an RIP module (E, ·,·) can induce an RN module (E, · ) and one also has the Cauchy–Schwarz inequality: |x, y| xy, ∀x, y ∈ E. Example 4.2. Let (H, ·,·) be a Hilbert space over K. Denote by L0 (F , H ) the linear space of equivalence classes of H -valued F -random variables on Ω, which is a left module over the algebra L0 (F , K) under the module multiplication ξ x := the equivalence class of ξ 0 · x 0 defined by (ξ 0 · x 0 )(ω) = ξ 0 (ω) · x 0 (ω), ∀ω ∈ Ω, where ξ 0 and x 0 are arbitrarily chosen representatives of ξ ∈ L0 (F , K) and x ∈ L0 (F , H ). Then L0 (F , H ) becomes an RIP module over K with base (Ω, F , P ) under the L0 -inner product induced from the inner product ·,·, still denoted by ·,·, namely, for any x, y ∈ L0 (F , H ) with respective representatives x 0 , y 0 , we have x, y = the equivalence class of x 0 , y 0 defined by x 0 , y 0 (ω) = x 0 (ω), y 0 (ω), ∀ω ∈ Ω. It is easy to see that (L0 (F , H ), · ) is Tε,λ -complete, so that L0 (F , H ) is Tc -complete. Theorem 4.3. Let (E, ·,·) be a Tc -complete RIP module over K with base (Ω, F , P ) such that E has the countable concatenation property. Then for every f ∈ E ∗ there exists a unique π(f ) ∈ E such that f (x) = x, π(f ), ∀x ∈ E and such that f ∗ = π(f ). Finally the induced mapping π : E ∗ → E is a conjugate isomorphism, namely π(ξf + ηg) = ξ π(f ) + ηπ(g), ∀ξ, η ∈ L0 (F , K) and f, g ∈ E ∗ . Proof. By Theorem 3.18, we have that E is Tε,λ -complete. It follows immediately that the main result of [23] is just what we want to prove. 2 To state Theorem 4.5 below, we first give Proposition 4.4 below. Let 1 p ∞ and (S, · ) be an RN module over K with base (Ω, F , P ). Define · p : S → [0, +∞] by 1 xp = ( Ω xp dP ) p for p ∈ [1, +∞) and x∞ = the essential supremum of x, ∀x ∈ S, and denote Lp (S) = {x ∈ S | xp < +∞}, then (Lp (S), · p ) is a normed space, for all q, 1 q +∞ we can also have (Lq (S ∗ ), · q ) in a similar way. Proposition 4.4 below is essentially independent of a special choice of Tc and Tε,λ , which was first proved in [10] under Tε,λ , whose proof in English was given in [14]. Proposition 4.4. (See [10,14].) Let 1 p < +∞ and 1 < q +∞ be a pair of Hölder conjugate numbers. Then (Lq (S ∗ ), · q ) is isometrically isomorphic with the classical conjugate space of (Lp (S), · p ), denoted by (Lp (S)) , under the canonical mapping T : Lq (S ∗ ) → (Lp (S)) defined as Lq (S ∗ ), Tf (denoting T (f )) : Lp (S) → K is defined by follows. For each f ∈ p Tf (g) = Ω f (g) dP for all g ∈ L (S).
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Proposition 4.4 gives all representation theorems of the dual of Lebesgue–Bochner function spaces by taking S = L0 (F , B) (at which time Lp (S) is just Lp (F , B), the classical Lebesgue– Bochner function spaces, see [12] for more details). On the other hand, it provides the connection between the random conjugate space S ∗ and the classical conjugate space (Lp (S)) , and thus a powerful tool for the theory of random conjugate spaces, cf. [10,19,22]. In this paper, we will still employ it in the proof of Theorem 4.5. Theorem 4.5. Let 1 p < +∞ and 1 < q +∞ be a pair of Hölder conjugate numbers. q p The canonical mapping T : LF (S ∗ ) → (LF (S))∗ is surjective and random-norm preserving, q p where for each f ∈ LF (S ∗ ), Tf (denoting T (f )) : LF (S) → L0 (F , K) is defined by Tf (g) = p q p E[f (g) | F ] for all g ∈ LF (S), and LF (S ∗ ) and LF (S) are the same as in Example 2.5. For the sake of clearness, the proof of Theorem 4.5 is divided into the following two Lemmas 4.6 and 4.7, Lemma 4.6 shows that T is well defined and isometric (namely random-norm preserving) and Lemma 4.7 shows that T is surjective. Specially, we need to remind the readp q ers of noticing that ||| · |||p and ||| · |||q are the L0 -norms on LF (S) and on LF (S ∗ ), respectively, whereas · p and · q are norms. Lemma 4.6. T is well defined and isometric. Proof. For any fixed f ∈ LF (S ∗ ), we will first prove that Tf ∈ (LF (S))∗ and Tf = |||f |||q when p > 1 as follows. p For any g ∈ LF (S), Tf (g) = E[f (g) | F ], we have the following: q
p
Tf (g) E f (g) F E f · g F 1 |||f |||q · E gp F p = |||f |||q · |||g|||p .
(4.25)
This shows that Tf ∈ (LF (S))∗ and Tf |||f |||q , namely T is well defined. We remain to prove Tf = |||f |||q when p > 1. Let ξ be an arbitrary representative of |||f |||q and An = {ω ∈ Ω | n − 1 ξ(ω) < n} for each p n ∈ N . Then {An | n ∈ N} forms a countable partition of Ω to F . Observing Ω |||g|||p dP = p p p Ω E[g | F ] dP = Ω g dP , ∀g ∈ LF (S), thus we have the following relation: p
p Lp LF (S) = Lp (S).
(4.26)
Since p > 1, 1 < q < +∞, we also have the relation: q Lq L F S ∗ = L q S ∗ .
(4.27)
I˜An Tf = I˜An |||f |||q .
(4.28)
Now, we fix n and prove
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p Since I˜An Tf (g) = E[I˜An f (g) | F ] for all g ∈ LF (S), we have, of course, that I˜An Tf (g) = p p E[I˜An f (g) | F ] for all g ∈ Lp (LF (S)) = Lp (S). Further, since I˜A Lp (LF (S)) = I˜A Lp (S) ⊂ p p p L (LF (S)) = L (S) for all A ∈ F , we can have the following important relation:
I˜An I˜A Tf (g) = E I˜An I˜A f (g) F ,
(4.29)
p
for all A ∈ F and all g ∈ Lp (LF (S)) = Lp (S) Obviously, from (4.29), for all A ∈ F we can have the following relation:
(I˜An I˜A Tf )(g) dP =
Ω
(I˜An I˜A f )(g) dP ,
(4.30)
Ω
p
for all g ∈ Lp (LF (S)) = Lp (S). For each fixed A ∈ F , the left side of (4.30) defines a bounded linear functional on 1 1 p p L (LF (S)), whose norm is equal to ( Ω I˜An I˜A Tf q dP ) q = ( A (I˜An Tf )q dP ) q by app plying Proposition 4.4 to Lp (LF (S)). The same bounded linear functional is also a bounded p linear functional on L (S) defined by the right side of (4.30), whose norm is also equal to 1 ( A (I˜An f )q dP ) q . Consequently, A I˜An Tf q dP = A (I˜An f )q dP for all A ∈ F . Since I˜An Tf q ∈ L0+ (F , R), we have I˜An Tf q = E[(I˜An f )q | F ]. Again noticing An ∈ F , we can have 1
I˜An Tf = I˜An (E[f q | F ]) q , which is just (4.28). Since n1 An = Ω, Tf = |||f |||q . Finally, we consider the case of p = 1, for the sake of clearness, we use | · |∞ for the usual L∞ -norm on the Banach space L∞ (E, K) of equivalence classes of essentially bounded ∗ E-measurable K-valued functions on (Ω, E, P ). Then it is easy to see that L∞ (L∞ F (S )) = ∞ ∗ ∞ ∗ L (S ), in particular that f ∞ = ||||f |||∞ |∞ for all f ∈ L (S ). Since, we can, similarly to the case p > 1, have, by noticing I˜An f ∈ L∞ (S ∗ ), the following relation: |I˜A (I˜An Tf )|∞ = |(I˜An I˜A f )|∞ for all A ∈ F , but as stated above, the latter is just equal to |I˜A |||I˜An f |||∞ |∞ . Since I˜An Tf and |||I˜An f |||∞ (namely I˜An |||f |||∞ ) are both in L0+ (F , R), then we must have that I˜An Tf = I˜An |||f |||∞ , again since n1 An = Ω, we can have Tf = |||f |||∞ . 2 Lemma 4.7. T is surjective. Proof. Let F be an arbitrary element of (LF (S))∗ . We want to prove that there exists an f ∈ q LF (S ∗ ) such that F = Tf . Since F ∈ L0+ (F , R), letting ξ be a chosen representative of F and An = {ω ∈ Ω | n − 1 ξ(ω) < n} for each n ∈ N , then {An | n ∈ N } forms a countable partition of Ω to F . 1 1 p Since |F (g)| F |||g|||p = F (E[gp | F ]) p , |I˜An F (g)| n(E[gp | F ]) p , ∀g ∈ LF (S) and n ∈ N . 1 1 Now, we fix n and notice | Ω (I˜An F )(g) dP | n Ω (E[gp | F ]) p dP n( Ω gp dP ) p p for all inequality, then by Proposition 4.4 there exists fn ∈ Lq (S ∗ ) such g ∈ L (S) by the Hölder ˜ that Ω (IAn F )(g) dP = Ω fn (g) dP for all g ∈ Lp (S). p
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Since I˜A Lp (S) ⊂ Lp (S) for all A ∈ F , we have that A (I˜An F )(g) dP = Ω (I˜An F ) × (I˜A g) dP = Ω fn (I˜A g) dP = A fn (g) dP for all g ∈ Lp (S) and all A ∈ F , which yields the following important relation (by noticing I˜An F (g) ∈ L0 (F , K)): I˜An F (g) = E fn (g) F ,
(4.31)
p
for all g ∈ Lp (S) ≡ Lp (LF (S)). q p Since fn ∈ Lq (S ∗ ) ⊂ LF (S ∗ ), then Lemma 4.6 shows that Tfn ∈ (LF (S))∗ , and (4.31) just p p shows that I˜An F and Tfn are equal on Lp (LF (S)). Since I˜An F and Tfn are both in (LF (S))∗ = p p (LF (S))∗ε,λ , namely they are both continuous module homomorphism from (LF (S), Tε,λ ) to p p (L0 (F , K), Tε,λ ), and Lp (LF (S)) is Tε,λ -dense in LF (S) (cf. [19,22]), I˜An F = Tfn , namely we can have the following relation: I˜An F (g) = E fn (g) F ,
(4.32)
p
for all g ∈ LF (S). We have from (4.32) the following relation: I˜An F (g) = E I˜An fn (g) F ,
(4.33)
p
for all g ∈ LF (S). q Let f = n1 I˜An fn . Since LF (S ∗ ) has the countable concatenation property (or we can p directly define f (g) = n1 I˜An fn (g), ∀g ∈ LF (S) and verify that f is first in S ∗ and then q q f ∈ LF (S ∗ )), we have that f ∈ LF (S ∗ ). p Finally, (4.33) shows that F (g) = E[f (g) | F ] = Tf (g), ∀g ∈ LF (S). 2 Remark 4.8. Let H be a Hilbert space, then Theorem 4.3 shows that (L0 (E, H ))∗ = L0 (E, H ), in particular, (L0 (E, K))∗ = L0 (E, K), if we take S = L0 (E, R) in Theorem 4.5, then we have p q (LF (E))∗ = LF (E), which is just Theorem 4.5 of [27], so our Theorem 4.5 is surprisingly general. Besides, the proof of the isometric property of T of Theorem 4.5 is completely new since Theorem 4.5 of [27] does not involve any isometric arguments. Acknowledgments The author is greatly indebted to Professor Berthold Schweizer for ten years of invaluable suggestions and support to the author’s work. He is also greatly indebted to Professor Daniel W. Stroock for giving the author some invaluable directions to mathematics and generous financial support during a short stay at MIT from November of 2001 to March of 2002. Finally, he would like to sincerely thank Professors Shijian Yan, Yingming Liu, Kung-Ching Chang, Shunhua Sun, Mingzhu Yang, Peide Liu, Fuzhou Gong and Kunyu Guo for their advices and kindness given on some occasions.
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References [1] W. Brannath, W. Schachermayer, A bipolar theorem for subsets of L0+ (Ω, F , P ), in: Séminaire de Probabilités XXXIII, in: Lecture Notes in Math., vol. 1709, Springer, 1999, pp. 349–354. [2] W.W. Breckner, E. Scheiber, A Hahn–Banach extension theorem for linear mappings into ordered modules, Mathematica 19 (42) (1977) 13–27. [3] N. Dunford, J.T. Schwartz, Linear Operators (I), Interscience, New York, 1957. [4] D. Filipovi´c, M. Kupper, N. Vogelpoth, Separation and duality in locally L0 -convex modules, J. Funct. Anal. 256 (2009) 3996–4029. [5] T.X. Guo, The theory of probabilistic metric spaces with applications to random functional analysis, Master’s thesis, Xi’an Jiaotong University (China), 1989. [6] T.X. Guo, Random metric theory and its applications, PhD thesis, Xi’an Jiaotong University (China), 1992. [7] T.X. Guo, Extension theorems of continuous random linear operators on random domains, J. Math. Anal. Appl. 193 (1) (1995) 15–27. [8] T.X. Guo, The Radon–Nikodým property of conjugate spaces and the w∗ -equivalence theorem for w∗ -measurable functions, Sci. China Ser. A 39 (1996) 1034–1041. [9] T.X. Guo, Module homomorphisms on random normed modules, Chinese Northeast. Math. J. 12 (1996) 102–114. [10] T.X. Guo, A characterization for a complete random normed module to be random reflexive, J. Xiamen Univ. Natur. Sci. 36 (1997) 499–502. [11] T.X. Guo, Some basic theories of random normed linear spaces and random inner product spaces, Acta Anal. Funct. Appl. 1 (2) (1999) 160–184. [12] T.X. Guo, Representation theorems of the dual of Lebesgue–Bochner function spaces, Sci. China Ser. A 43 (2000) 234–243. [13] T.X. Guo, Survey of recent developments of random metric theory and its applications in China (I), Acta Anal. Funct. Appl. 3 (2001) 129–158. [14] T.X. Guo, Survey of recent developments of random metric theory and its applications in China (II), Acta Anal. Funct. Appl. 3 (2001) 208–230. [15] T.X. Guo, Several applications of the theory of random conjugate spaces to measurability problems, Sci. China Ser. A 50 (2007) 737–747. [16] T.X. Guo, The relation of Banach–Alaoglu theorem and Banach–Bourbaki–Kakutani–Šmulian theorem in complete random normed modules to stratification structure, Sci. China Ser. A 51 (2008) 1651–1663. [17] T.X. Guo, The theory of module homomorphisms in complete random inner product modules and its applications to Skorohod’s random operator theory, submitted for publication. [18] T.X. Guo, X.X. Chen, Random duality, Sci. China Ser. A 52 (2009) 2084–2098. [19] T.X. Guo, S.B. Li, The James theorem in complete random normed modules, J. Math. Anal. Appl. 308 (2005) 257–265. [20] T.X. Guo, S.L. Peng, A characterization for an L(μ, K)-topological module to admit enough canonical module homomorphisms, J. Math. Anal. Appl. 263 (2001) 580–599. [21] T.X. Guo, H.X. Xiao, A separation theorem in random normed modules, J. Xiamen Univ. Natur. Sci. 42 (2003) 270–274. [22] T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794–3804. [23] T.X. Guo, Z.Y. You, The Riesz’s representation theorem in complete random inner product modules and its applications, Chinese Ann. Math. Ser. A 17 (1996) 361–364. [24] T.X. Guo, L.H. Zhu, A characterization of continuous module homomorphisms on random seminormed modules and its applications, Acta Math. Sin. (Engl. Ser.) 19 (1) (2003) 201–208. [25] A. Ionescu Tulcea, C. Ionescu Tulcea, On the lifting property (I), J. Math. Anal. Appl. 3 (1961) 537–546. [26] A. Ionescu Tulcea, C. Ionescu Tulcea, On the lifting property (II), J. Math. Mech. 11 (5) (1962) 773–795. [27] M. Kupper, N. Vogelpoth, Complete L0 -normed modules and automatic continuity of monotone convex functions, Working paper Series No. 10, Vienna Institute of Finance, 2008. [28] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Elsevier/North-Holland, New York, 1983; reissued by Dover Publications, Mineola, New York, 2005. [29] A.V. Skorohod, Random Linear Operators, D. Reidel Publishing Company, Holland, 1984. [30] D. Vuza, The Hahn–Banach theorem for modules over ordered rings, Rev. Roumaine Math. Pures Appl. 9 (27) (1982) 989–995. [31] D.H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977) 859–903.
Journal of Functional Analysis 258 (2010) 3048–3081 www.elsevier.com/locate/jfa
Infinitely many solutions for the prescribed scalar curvature problem on SN Juncheng Wei a , Shusen Yan b,∗ a Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong b Department of Mathematics, The University of New England, Armidale, NSW 2351, Australia
Received 17 August 2009; accepted 13 December 2009 Available online 24 December 2009 Communicated by J. Coron
Abstract We consider the following prescribed scalar curvature problem on SN −SN u + u>0
N+2 N (N − 2) ˜ N−2 u = Ku 2
on SN ,
(∗)
where K˜ is positive and rotationally symmetric. We show that if K˜ has a local maximum point between the poles then Eq. (∗) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. © 2009 Elsevier Inc. All rights reserved. Keywords: Prescribed scalar curvature; Elliptic equation; Reduction argument
1. Introduction Consider the standard N -sphere (SN , g0 ), N 3. Let K˜ be a fixed smooth function. The prescribed curvature problem asks if one can find a conformally invariant metric g such that the ˜ The problem consists in solving the following equation on SN : scalar curvature becomes K. * Corresponding author.
E-mail addresses:
[email protected] (J. Wei),
[email protected] (S. Yan). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.008
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
−SN u +
N+2 N (N − 2) ˜ N−2 = 0 on SN , u − Ku 2
3049
(1.1)
u > 0. Problem (1.1) does not always admit a solution. A first necessary condition for the existence is that maxSN K˜ > 0, but there are also some obstructions, which are said of topological type. For example, a necessary condition is the following Kazdan–Warner condition:
2N
∇ K˜ · ∇xu N−2 = 0.
(1.2)
SN
The problem of determining which K˜ admits a solution to (1.1) has been studied extensively. See [1,3–11,13–21,23,24,30] and the references therein. Some existence results have been ob˜ see Chang and tained under some assumptions involving the Laplacian at the critical point of K, Yang [9], Bahri and Coron [3] and Schoen and Zhang [29] for the case N = 3, and Y. Li [19] for the case N 4. For example, in Bahri and Coron [3], it is assumed that N = 3, K˜ is a positive ˜ ˜ Morse function with K(x) = 0 if ∇ K(x) = 0, then if m(x) denotes the Morse index of the critical point x of K, (1.1) has a solution provided that
(−1)m(x) = −1.
(1.3)
˜ ˜ ∇ K=0, K(x)<0
The result has been extended to any SN , N 3 by Y. Li in [18,19]. Roughly, it is assumed that there exists β, N − 2 < β < N such that ˜ ) = K(ξ ˜ 0) + K(ξ
N
aj |ξj − ξ0,j |β + h.o.t.
(1.4)
j =1
N ˜ where aj = 0, N j =1 aj = 0. Let Σ = {ξ : ∇ K(ξ ) = 0, j =1 aj < 0} and i(ξ ) be the number of aj such that aj < 0. Then (1.1) has a solution provided
(−1)i(ξ ) = (−1)N .
(1.5)
ξ ∈Σ
By using the stereo-graphic projection, the prescribed scalar curvature problem (1.1) can be reduced to (1.6)
N+2
−u = K(y)u N−2 , u ∈ D 1,2 RN
u > 0, y ∈ RN ,
(1.6)
where D 1,2 (RN ) denotes the completion of C0∞ (RN ) under the norm RN |∇u|2 . Much less is known about the multiplicity of the solutions of (1.6). Amrosetti, Azorero and Peral [1], and Cao, Noussair and Yan [7] proved the existence of two or many solutions if K is a perturbation of the constant, i.e.
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K˜ = K0 + εh(x),
0 < ε 1.
(1.7)
On the other hand, Y. Li proved in [16] that (1.6) has infinitely many solutions if K(x) is periodic, while similar result was obtained in [30] if K(x) has a sequence of strict local maximum points tending to infinity. Note that this condition for K(x) at the infinity implies that the corresponding function K˜ defined on SN has a singularity at the south pole. In this paper, we consider the simplest case, i.e., K˜ is rotationally symmetric, K = K(r), r = |y|. It follows from the Pohozaev identity (1.2) that (1.6) has no solution if K (r) has fixed sign. Thus we assume that K is positive and not monotone. On the other hand, Bianchi [5] showed that any solution of (1.6) is radially symmetric if there is an r0 > 0, such that K(r) is non-increasing in (0, r0 ], and non-decreasing in [r0 , +∞). Moreover, in [6], it was proved that (1.6) has no solutions for some function K(r), which is non-increasing in (0, 1], and nondecreasing in [1, +∞). Therefore, we see that to obtain a solution for (1.6), it is natural to assume that K(r) has a local maximum at r0 > 0. The purpose of this paper is to answer the following two questions: Q1. Does the existence of a local maximum of K guarantee the existence of a solution to (1.6)? Q2. Are there non-radially symmetric solutions to (1.6)? (Question Q2 has been asked by Bianchi [5].) The aim of this paper is to show that if K(r) has a local maximum at r0 > 0, then (1.6) has infinitely many non-radial solutions. This answers Q1 and Q2 affirmatively. As far as we know, we believe our result is the first on the existence of infinitely many solution for (1.6). We assume that K(r) satisfies the following condition: (K): There is a constant r0 > 0, such that K(r) = K(r0 ) − c0 |r − r0 |m + O |r − r0 |m+θ ,
r ∈ (r0 − δ, r0 + δ),
where c0 > 0, θ > 0 are some constants, and the constant m satisfies m ∈ [2, N − 2). Without loss of generality, we assume that K(r0 ) = 1. Our main result in this paper can be stated as follows: Theorem 1.1. Suppose that N 5. If K(r) satisfies (K), then problem (1.6) has infinitely many non-radial solutions. Remark 1.2. The condition (1.3) (or (1.5)) is a global one while our condition (K) is local. Remark 1.3. Theorem 1.1 shows that the condition in [5] is optimal. We shall prove Theorem 1.1 by constructing solutions with large number of bubbles lying near the sphere |y| = r0 . So the energy of these solutions can be made arbitrary large and the distance between different bubbles can be made arbitrary small. When N = 3, 4, we know that the energy of the solutions to (1.6) is
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uniformly bounded and the distance between bubbles is uniformly bounded from below. See [29] (for N = 3) and Theorem 0.10 of [19] (for N = 4). On the other hand, if K(y) = K(y0 ) + O(|y − y0 |m ) where m ∈ [N − 2, N ) for N 5, the energy of solutions is also be bounded. See [19]. So our assumptions on N and m are almost optimal in the construction of the solutions in this paper. Remark 1.4. The radial symmetry can be replaced by the following weaker symmetry assumption: after suitably rotating the coordinate system, (K1) K(y) = K(y , y ) = K(|y |, |yN0 +1 |, . . . , |yN |), where y = (y , y ) ∈ R2 × RN −2 , (K2) K(y) = K(y0 ) − c0 |y − y0 |m + O(|y − y0 |m+θ ), |y | ∈ (|y0 | − δ, |y0 | + δ), |y | δ, where y0 = (y0 , 0). Remark 1.5. Theorem 1.1 exhibits a new phenomena for the prescribed scalar curvature problem. It suggests that if the critical points of K are not isolated, new solutions to (1.6) may bifurcate. We formulate the following conjecture in the general case. Conjecture. Assume that the set {K(x) = maxx∈RN K(x)} is an l-dimensional smooth manifold without boundary, where 1 l N − 1. The problem (1.6) admits infinitely many positive solutions. Before we close this introduction, let us outline the main idea in the proof of Theorem 1.1. Let us fix a positive integer k k0 , where k0 is a large integer, which is to be determined later. Set N−2
μ = k N−2−m , to be the scaling parameter. y − N−2 2 u( ), we find that (1.6) becomes Let 2∗ = N2N −2 . Using the transformation u(y) → μ μ ⎧
⎪ ⎨ −u = K |y| u2∗ −1 , μ ⎪ ⎩ u ∈ D 1,2 RN .
u > 0, y ∈ RN ,
It is well known that the functions N−2 Ux,Λ (y) = N (N − 2) 4
Λ 2 1 + Λ |y − x|2
N−2 2
are the only solutions to the problem N+2
−u = u N−2 ,
u > 0 in RN .
,
μ > 0, x ∈ RN ,
(1.8)
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Let y = (y , y ), y ∈ R2 , y ∈ RN −2 . Define Hs =
u: u ∈ D 1,2 RN , u is even in yh , h = 2, . . . , N,
2πj 2πj , r sin θ + , y . u r cos θ, r sin θ, y = u r cos θ + k k
Let
2(j − 1)π 2(j − 1)π xj = r cos , r sin ,0 , k k
j = 1, . . . , k,
where 0 is the zero vector in RN −2 , and let k N−2 Wr,Λ (y) = N (N − 2) 4 j =1
Λ
N−2 2
(1 + Λ2 |y − xj |2 )
N−2 2
.
In this paper, we always assume that 1 1 r ∈ r0 μ − ¯ , r0 μ + ¯ , μθ μθ
for some small θ¯ > 0,
and L 0 Λ L1 ,
for some constants L1 > L0 > 0.
Theorem 1.1 is a direct consequence of the following result: Theorem 1.6. Suppose that N 5. If K(r) satisfies (K), then there is an integer k0 > 0, such that for any integer k k0 , (1.8) has a solution uk of the form uk = Wrk ,Λk (y) + ωk , where ωk ∈ Hs , and as k → +∞, ω L∞ → 0, rk ∈ [r0 μ −
1 , r0 μ + 1θ¯ ], μθ¯ μ
and L0 Λk L1 .
We will use the techniques in the singularly perturbed elliptic problems to prove Theorem 1.6. We know that there is always a small parameter in a singularly perturbed elliptic problem. In Theorem 1.6, we only know that the number of the bubbles in the solutions are large. This suggests that although there is no parameter in (1.6), we can use k, the number of the bubbles of the solutions, as the parameter in the construction of bubbles solutions for (1.6). This is the new idea of this paper. This is partly motivated by recent paper of Lin, Ni and Wei [22] where they constructed multiple spikes to a singularly perturbed problem. There they allowed the number of spikes to depend on the small parameter. The main difficulty in constructing solution with k-bubbles is that we need to obtain a better control of the error terms. Since the number of the bubbles is large, it is very hard to carry out the reduction procedure by using the standard norm as in [2,25]. Noting that the maximum norm will
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not be affected by the number of the bubbles, we will carry out the reduction procedure in a space with weighted maximum norm. Similar weighted maximum norm has been used in [12,26–28]. But the estimates in the reduction procedure in this paper are much more complicated than those in [12,26–28], because the number of the bubbles is large. 2. Finite-dimensional reduction In this section, we perform a finite-dimensional reduction. Let
k
u ∗ = sup
y∈RN
j =1
−1
1 (1 + |y − xj |)
N−2 2 +τ
u(y),
(2.1)
and f ∗∗ = sup
y∈RN
k j =1
−1
1 (1 + |y − xj |)
N+2 2 +τ
f (y),
(2.2)
where τ = 1 + η¯ and η¯ > 0 is small. Let Zi,1 =
∂Uxi ,Λ , ∂r
Zi,2 =
∂Uxi ,Λ . ∂Λ
Consider ⎧
k 2 ∗ ⎪ ∗ −2 |y| ⎪ 2∗ −2 ⎪ cj Ux2i ,Λ Zi,j , ⎪ ⎨ −φk − 2 − 1 K μ Wr,Λ φk = hk + ⎪ ⎪ φk ∈ Hs , ⎪ ⎪ ⎩ 2∗ −2 Uxi ,Λ Zi,l , φk = 0 i = 1, . . . , k, l = 1, 2, for some numbers ci , where u, v =
RN
j =1
i=1
in RN , (2.3)
uv.
Lemma 2.1. Assume that φk solves (2.3) for h = hk . If hk ∗∗ goes to zero as k goes to infinity, so does φk ∗ . Proof. We argue by contradiction. Suppose that there are k → +∞, h = hk , Λk ∈ [L1 , L2 ], rk ∈ [r0 μ − 1θ¯ , r0 μ + 1θ¯ ], and φk solving (2.3) for h = hk , Λ = Λk , r = rk , with hk ∗∗ → 0, μ
μ
and φk ∗ c > 0. We may assume that φk ∗ = 1. For simplicity, we drop the subscript k.
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We rewrite (2.3) as φ(y) = 2∗ − 1
RN
1 |z| 2∗ −2 Wr,Λ K φ(z) dz μ |z − y|N −2
k 2 1 2∗ −2 cj Zi,j (z)Uxi ,Λ (z) dz. h(z) + |z − y|N −2
+
j =1
RN
(2.4)
i=1
Using Lemma B.3, we have
∗ 1 |z| 2∗ −2 2 −1 W K φ(z) dz r,Λ N −2 μ |z − y| RN
C φ ∗ RN
C φ ∗
1 1 2∗ −2 Wr,Λ dz N−2 N −2 |z − y| (1 + |z − xj |) 2 +τ k
j =1
k j =1
1 (1 + |y − xj |)
N−2 2 +τ +θ
(2.5)
.
It follows from Lemma B.2 that k 1 1 1 C h h(z) dz dz ∗∗ N+2 N −2 N −2 +τ |z − y| |z − y| j =1 (1 + |z − xj |) 2 N N R
R
C h ∗∗
k j =1
1 (1 + |y − xj |)
N−2 2 +τ
(2.6)
,
and k k 1 1 1 2∗ −2 C Z (z)U (z) dz dz i,l xi ,Λ |z − y|N −2 |z − y|N −2 (1 + |z − xi |)N +2 i=1
RN
i=1
C
k i=1
RN
1 (1 + |y − xi |)
N−2 2 +τ
.
(2.7)
Next, we estimate cl , l = 1, 2. Multiplying (2.3) by Z1,l and integrating, we see that ct satisfies
∗ −2 |y| 2∗ −2 Wr,Λ Ux2i ,Λ Zi,t , Z1,l ct = −φ − 2∗ − 1 K φ, Z1,l − h, Z1,l . μ
2 k t=1 i=1
(2.8)
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
3055
It follows from Lemma B.1 that h, Z1,l C h ∗∗
RN
k 1 1 dz N+2 +τ (1 + |z − x1 |)N −2 j =1 (1 + |z − xj |) 2
C h ∗∗ . On the other hand,
|z| 2∗ −2 −φ − 2 − 1 K Wr,Λ φ, Z1,l μ
|z| 2∗ −2 = −Z1,l − 2∗ − 1 K Wr,Λ Z1,l , φ μ
∗ |z| 2∗ −2 Wr,Λ Z1,l , φ = 2 −1 1−K μ 2∗ −2 |z| = φ ∗ O K μ − 1Wr,Λ (z)
∗
RN
k 1 1 dz . × N−2 (1 + |z − x1 |)N −2 2 +τ (1 + |z − x |) j j =1 Similar to the proof of Lemma B.3, we obtain √ ||z|−μr0 | μ
C √ μ
k 1 1 K |z| − 1W 2∗ −2 (z) dz r,Λ N−2 +τ μ (1 + |z − x1 |)N −2 j =1 (1 + |z − xj |) 2 ∗
2 −2 Wr,Λ (z)
RN
k 1 1 dz N−2 N −2 +τ (1 + |z − x1 |) j =1 (1 + |z − xj |) 2
C √ , μ and √ ||z|−μr0 | μ
C σ μ
RN
C , μσ
k 1 1 K |y| − 1W 2∗ −2 (z) dz N−2 r,Λ +τ μ (1 + |z − x1 |)N −2 j =1 (1 + |z − xj |) 2 ∗
2 −2 Wr,Λ (z)
k 1 1 dz N−2 N −2 +τ −2σ (1 + |z − x1 |) j =1 (1 + |z − xj |) 2
(2.9)
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
since if ||z| − μr0 |
√ μ, then
|z| − |x1 | |z| − μr0 − |x1 | − μr0 √μ − 1 1 √μ. 2 μθ¯ Thus, k 1 1 C K |z| − 1W 2∗ −2 (z) dz σ , N−2 r,Λ +τ μ μ (1 + |z − x1 |)N −2 j =1 (1 + |z − xj |) 2 N
R
which, together with (2.9), gives
|z| 1 2∗ −2 . Wr,Λ −φ − 2∗ − 1 K φ, Z1,l = φ ∗ O μ μσ
(2.10)
But there is a constant c¯ > 0, k
∗ −2 Ux2i ,Λ Zi,t , Z1,l = c¯ + o(1) δtl .
i=1
Thus we obtain from (2.8) that
cl = O
1 . φ + h ∗ ∗∗ μσ
(2.11)
So, k
j =1 φ ∗ hk ∗∗ + k
1 N−2 +τ +θ
(1+|y−xj |) 2 1 N−2 +τ j =1 (1+|y−xj |) 2
.
(2.12)
Since φ ∗ = 1, we obtain from (2.12) that there is R > 0, such that φ(y)
BR (xi )
a > 0,
(2.13)
¯ for some i. But φ(y) = φ(y − xi ) converges uniformly in any compact set to a solution u of 2∗ −2 −u − 2∗ − 1 U0,Λ u = 0,
in RN ,
(2.14)
for some Λ ∈ [L1 , L2 ], and u is perpendicular to the kernel of (2.14). So, u = 0. This is a contradiction to (2.13). 2 From Lemma 2.1, using the same argument as in the proof of Proposition 4.1 in [12], we can prove the following result:
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
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Proposition 2.2. There exists k0 > 0 and a constant C > 0, independent of k, such that for all k k0 and all h ∈ L∞ (RN ), problem (2.3) has a unique solution φ ≡ Lk (h). Besides, Lk (h) C h ∗∗ , ∗
|cl | C h ∗∗ .
(2.15)
Now, we consider ⎧
k 2 ⎪ ∗ −2 |y| ⎪ 2∗ −1 ⎪ (Wr,Λ + φ) + ct Ux2i ,Λ Zi,t , ⎪ −(Wr,Λ + φ) = K ⎨ μ ⎪ φk ∈ Hs , ⎪ ⎪ ⎪ ⎩ 2∗ −2 Uxi ,Λ Zi,l , φk = 0,
t=1
in RN ,
i=1
(2.16)
i = 1, . . . , k, l = 1, 2.
We have Proposition 2.3. There is an integer k0 > 0, such that for each k k0 , L0 Λ L1 , |r − μr0 | 1θ¯ , where θ¯ > 0 is a fixed small constant, (2.16) has a unique solution φ = φ(r, Λ), μ satisfying φ ∗ C
m +σ 1 2 , μ
|ct | C
m +σ 1 2 , μ
where σ > 0 is a small constant. Rewrite (2.16) as ⎧
k 2 ⎪ ∗ ∗ −2 |y| ⎪ 2∗ −2 ⎪ −φ − 2 W − 1 K φ = N (φ) + l + c Ux2i ,Λ Zi,t , ⎪ k i r,Λ ⎨ μ ⎪ φ ∈ Hs , ⎪ ⎪ ⎪ ⎩ 2∗ −2 Uxi ,Λ Zi,l , φ = 0,
t=1
i=1
in RN , (2.17)
i = 1, . . . , k, l = 1, 2,
where
2∗ −2 |y| ∗ 2∗ −1 N (φ) = K (Wr,Λ + φ)2 −1 − Wr,Λ − 2∗ − 1 Wr,Λ φ , μ and k ∗ −1 |y| 2∗ −1 Wr,Λ − Ux2j ,Λ . lk = K μ
j =1
In order to use the contraction mapping theorem to prove that (2.17) is uniquely solvable in the set that φ ∗ is small, we need to estimate N (φ) and lk .
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Lemma 2.4. If N 5, then N (φ)
∗∗
∗ −1,2)
C φ ∗min(2
.
Proof. We have N (φ)
∗ −1
C|φ|2
N 6;
,
1 3
7 3
CWr,Λ φ 2 + C|φ| ,
N = 5.
Firstly, we consider N 6. Using the Hölder inequality, we obtain N(φ) C φ 2∗ −1
k
∗
j =1
(1 + |y − xj |)
k
2∗ −1
C φ ∗
j =1
j =1
N−2 2 +τ
1 (1 + |y − xj |)
k
∗ −1
C φ ∗2
2∗ −1
1
N+2 2 +τ
1 (1 + |y − xj |)
N+2 2 +τ
k j =1
1 (1 + |y − xj |)τ
4 N−2
(2.18)
,
since k j =1
1 C C+ C. τ (1 + |y − xj |) |x1 − xj |τ k
j =2
Thus, the result follows. It remains to prove the result for N = 5. Similar to (2.18), we have N(φ) C φ 2
k
∗ i=1 7 3
+ φ ∗
1 (1 + |y − xi |)3
3
1
j =1
(1 + |y − xj |) 2 +τ
7
1
i=1
(1 + |y − xi |) 2 +τ
3
3
i=1
7 3
1 (1 + |y − xi |)
k
1
i=1
(1 + |y − xi |) 2 +τ
7
2
k
3
k
k 7 C φ 2∗ + φ ∗3 C φ 2∗
1
.
3 2 +τ
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
3059
Thus, for N = 5, N (φ)
∗∗
C φ 2∗ .
2
Next, we estimate lk . Lemma 2.5. Assume that ||x1 | − μr0 | 1θ¯ , where θ¯ > 0 is a fixed small constant. μ If N 5, then there is a small σ > 0, such that
m +σ 1 2 lk ∗∗ C . μ Proof. Define xj π y 2 N −2 Ωj = y: y = y , y ∈ R × R , cos . , |y | |xj | k We have
|y| lk = K μ
k k ∗ ∗ |y| 2 −1 2 −1 2∗ −1 −1 Uxj ,Λ + Uxj ,Λ K Wr,Λ − μ j =1
j =1
=: J1 + J2 . From the symmetry, we can assume that y ∈ Ω1 . Then, |y − xj | |y − x1 |,
∀y ∈ Ω1 .
Thus, |J1 | C
k 1 1 (1 + |y − x1 |)4 (1 + |y − xj |)N −2 j =2
+C
k j =2
1 (1 + |y − xj |)N −2
2∗ −1 .
Using Lemma B.1, for any 0 < α min(4, N − 2), we obtain k 1 1 (1 + |y − x1 |)4 (1 + |y − xj |)N −2 j =2
C
k j =2
1 1 1 + (1 + |y − x1 |)N +2−α (1 + |y − xj |)N +2−α |xj − x1 |α
(2.19)
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
C
k 1 kα 1 1 C . |xj − x1 |α (1 + |y − x1 |)N +2−α (1 + |y − x1 |)N +2−α μα
(2.20)
j =2
Since τ < 2, we can choose α >
N −2 2
with N + 2 − α
N +2 2
+ τ . Thus
+σ k 1 2 1 1 1 C . N+2 4 N −2 (1 + |y − x1 |) (1 + |y − xj |) (1 + |y − x1 |) 2 +τ μ m
j =2
On the other hand, for y ∈ Ω1 , using Lemma B.1 again, 1 1 1 N−2 N−2 N −2 (1 + |y − xj |) (1 + |y − x1 |) 2 (1 + |y − xj |) 2
C N−2
N−2
|xj − x1 | 2 − N+2 τ
1 1 × + N−2 N−2 N−2 N−2 (1 + |y − x1 |) 2 + N+2 τ (1 + |y − xj |) 2 + N+2 τ 1
C |xj − x1 |
N−2 N−2 2 − N+2 τ
If N 5, τ = 1 + η¯ and η¯ > 0 is small, then k j =2
(1 + |y − x1 |)
N −2 2
−
N −2 N +2 τ
N−2 N−2 2 + N+2 τ
.
> 1. Thus
N−2 − N−2 τ 2 N+2 1 1 k C , N−2 N−2 N −2 μ (1 + |y − xj |) (1 + |y − x1 |) 2 + N+2 τ
which, gives
k j =2
1 (1 + |y − xj |)N −2
2∗ −1
N+2 −τ 2 1 k C . N+2 μ (1 + |y − x1 |) 2 +τ
Thus, we have proved J1 ∗∗ C
m +σ 1 2 . μ
Now, we estimate J2 . For y ∈ Ω1 , and j > 1, using Lemma B.1, we have ∗
−1 Ux2j ,Λ (y) C
C
1
1 (1 + |y − x1 |)
N+2 2
(1 + |y − xj |) 1
1 (1 + |y − x1 |)
N+2 2
N+2 2 +τ
|x1 − xj |
N+2 2 −τ
,
(2.21)
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
3061
which implies k k 1 |y| 1 2∗ −1 K − 1 Uxj ,Λ C N+2 N+2 μ (1 + |y − x1 |) 2 +τ j =2 |x1 − xj | 2 −τ j =2
N+2 −τ 2 k C . N+2 +τ μ (1 + |y − x1 |) 2 1
(2.22)
For y ∈ Ω1 and ||y| − μr0 | δμ, where δ > 0 is a fixed constant, then |y| − |x1 | |y| − μr0 − |x1 | − μr0 1 δμ. 2 As a result,
2∗ −1 |y| U x1 ,Λ K μ − 1 C
1
1
(1 + |y − x1 |)
N+2 2 +τ
μ
N+2 2 −τ
.
(2.23)
If y ∈ Ω1 and ||y| − μr0 | δμ, then m K |y| − 1 C |y| − r0 C |y| − |x1 |m + |x1 | − μr0 m m μ μ μ m C C m |y| − |x1 | + , μ μm+θ¯ and |y| − |x1 | |y| − μr0 + μr0 − |x1 | 2δμ. As a result, ||y| − |x1 ||m 1 m μ (1 + |y − x1 |)N +2 since
N +2 2
−τ −
m 2
m
C m
μ 2 +σ C
||y| − |x1 || 2 +σ (1 + |y − x1 |)N +2 1
1
m
N+2 2 +τ
m
N+2 2 +τ
μ 2 +σ (1 + |y − x1 |) 1 C μ 2 +σ (1 + |y − x1 |)
−σ
N +2 2
2∗ −1 |y| U K − 1 x1 ,Λ μ
−τ − C μ
m 2 +σ
N −2 2
(1 + |y − x1 |)
N+2 m 2 −τ − 2 −σ
,
− σ > 0. Thus, we obtain 1
(1 + |y − x1 |)
N+2 2 +τ
,
|y| − μr0 δμ.
(2.24)
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Combining (2.22), (2.23) and (2.24), we reach J2 ∗∗ C
m +σ 1 2 . μ
2
Now, we are ready to prove Proposition 2.3. Proof of Proposition 2.3. Let us recall that N−2
μ = k N−m−2 . Let
m N 1 2 2∗−2 E = u: u ∈ C R ∩ Hs , u ∗ , Uxi ,Λ Zi,l φ = 0, i = 1, . . . , k, l = 1, 2 . μ RN
Then, (2.17) is equivalent to φ = A(φ) =: Lk N (φ) + Lk (lk ), where Lk is defined in Proposition 2.2. We will prove that A is a contraction map from E to E. In fact, A(φ) C N (φ) + C lk ∗∗ ∗ ∗∗
m +σ 1 2 ∗ C φ ∗min(2 −1,2) + C μ
m min(2∗ −1,2)
m +σ 1 2 1 2 C +C μ μ
m +σ m 1 2 1 2 C . μ μ Thus, A maps E to E. On the other hand, A(φ1 ) − A(φ2 ) = Lk N (φ1 ) − Lk N (φ2 ) C N (φ1 ) − N (φ2 ) . ∗ ∗ ∗∗ If N 6, then N (t) C|t|2∗ −2 . As a result,
(2.25)
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
N (φ1 ) − N (φ2 ) C |φ1 |2∗ −2 + |φ2 |2∗ −2 |φ1 − φ2 | k 2∗ −1 1 2∗ −2 2∗ −2 C φ1 ∗ φ1 − φ2 ∗ + φ2 ∗ N−2 +τ j =1 (1 + |y − xj |) 2 1 ∗ ∗ C φ1 ∗2 −2 + φ2 ∗2 −2 φ1 − φ2 ∗ φ1 − φ2 ∗ . 2 Thus, A is a contraction map. The case N = 5 can be discussed in a similar way. It follows from the contraction mapping theorem that there is a unique φ ∈ E, such that φ = A(φ). Moreover, it follows from Proposition 2.2 that φ ∗ C
m +σ 1 2 . μ
Finally, the estimate of ct comes from (2.15). See also (2.11).
2
3. Proof of Theorem 1.6 Let F (r, Λ) = I (Wr,Λ + φ), where r = |x1 |, φ is the function obtained in Proposition 2.3, and I (u) =
1 2
|Du|2 − RN
1 2∗
K
|y| ∗ |u|2 . μ
RN
Proposition 3.1. We have
F (r, Λ) = I (Wr,Λ ) + O
=k A+ −
k j =2
B1 m Λ μm
k
μm+σ +
B2 μr0 m−2 Λ μm
2 − |x1 |
3 B3 1 1 , +O + m μr0 − |x1 | μm+σ μ ΛN −2 |x1 − xj |N −2
where σ > 0 is a fixed constant, Bi > 0, i = 1, 2, 3, is some constant.
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Proof. Since
I (Wr,Λ + φ), φ = 0,
∀φ ∈ E,
there is t ∈ (0, 1) such that 1 F (r, Λ) = I (Wr,Λ ) − D 2 I (Wr + tφ)(φ, φ) 2
∗ 1 |y| 2 2∗ −2 2 |Dφ| − 2 − 1 K (Wr,Λ + tφ) φ = I (Wr,Λ ) − 2 μ = I (Wr,Λ ) + −
1 2
RN 2∗ − 1
K
2
|y| ∗ ∗ (Wr + tφ)2 −2 − Wr2 −2 φ 2 μ
RN
N (φ) + lk φ
RN
= I (Wr,Λ ) + O
2∗ |φ| + N (φ)|φ| + |lk ||φ| .
RN
But
N(φ)|φ| + |lk ||φ| C N (φ)
∗∗
+ lk ∗∗ φ ∗
RN
×
k
RN
j =1
k
1 (1 + |y − xj |)
N+2 2 +τ
i=1
1 (1 + |y − xi |)
N−2 2 +τ
Using Lemma B.1 k j =1
k
1 (1 + |y − xj |)
=
N+2 2 +τ
i=1
1 (1 + |y − xi |)
k
N−2 2 +τ
1 1 1 + N+2 N−2 +τ (1 + |y − xj |)N +2τ (1 + |y − xi |) 2 +τ j =1 j =1 i=j (1 + |y − xj |) 2
k j =1
C
k
k k 1 1 1 + C (1 + |y − xj |)N +τ |xj − x1 |τ (1 + |y − xj |)N +2τ
k j =1
j =1
1 , (1 + |y − xj |)N +τ
since τ > 1. Thus, we obtain
j =2
.
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
N (φ)|φ| + |lk ||φ| Ck N (φ)
3065
m+σ 1 |φ + l Ck . k ∗∗ ∗ ∗∗ μ
RN
On the other hand,
2∗
k
2∗
|φ|
C φ ∗
RN
j =1
RN
2∗
1 (1 + |y − xj |)
.
N−2 2 +τ
But using Lemma B.1, if y ∈ Ω1 , k j =2
1 (1 + |y − xj |)
N−2 2 +τ
k j =2
C C
1
1
(1 + |y − x1 |)
N−2 1 4 +2τ
1 (1 + |y − x1 |)
N−2 1 2 + 2 η¯
1 (1 + |y − x1 |)
N−2 1 2 + 2 η¯
(1 + |y − xj |)
N−2 1 4 +2τ
k
1
j =2
|xj − x1 |τ − 2 η¯
1
.
Thus,
k j =1
2∗
1 (1 + |y − xj |)
N−2 2 +τ
C ∗ 1 η¯ 2
(1 + |y − x1 |)N +2
,
y ∈ Ω1 .
Thus, k RN
j =1
2∗
1 (1 + |y − xj |)
N−2 2 +τ
Ck.
So, we have proved
∗
∗
|φ|2 Ck φ 2∗ Ck
m+σ 1 . μ
2
RN
Proposition 3.2. We have k ∂F (r, Λ) B3 (N − 2) B1 m = k − m+1 m + ∂Λ Λ μ ΛN −1 |x1 − xj |N −2 j =2
2 1 1 +O + m μr0 − |x1 | , μm+σ μ
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
where σ > 0 is a fixed constant. Proof. We have ∂φ ∂Wr,Λ ∂F (r, Λ) = I (Wr,Λ + φ), + ∂Λ ∂Λ ∂Λ 2 k ∂Wr,Λ ∂φ 2∗ −2 + . = I (Wr,Λ + φ), cl Uxi ,Λ Zi,l , ∂Λ ∂Λ l=1 i=1
But
∗
−2 Ux2i ,Λ Zi,l ,
∗ −2 ∂(Ux2i ,Λ Zi,l ) ∂φ =− ,φ . ∂Λ ∂Λ
Thus, using Proposition 2.3, k k ∗ 1 ∂φ 2 −2 cl Uxi ,Λ Zi,l , C|cl | φ ∗ ∂Λ (1 + |y − xi |)N +2 i=1
RN
×
k j =1
i=1
1 (1 + |y − xj |)
N−2 2 +τ
C . μm+σ
On the other hand, D(Wr,Λ + φ)D
∂Wr,Λ = ∂Λ
RN
DWr,Λ D
∂Wr,Λ , ∂Λ
RN
and
∂Wr,Λ |y| ∗ (Wr,Λ + φ)2 −1 K μ ∂Λ
RN
=
∗ |y| |y| 2∗ −1 ∂Wr,Λ 2∗ −2 ∂Wr,Λ 2∗ Wr,Λ + 2 −1 Wr,Λ φ+O K K |φ| . μ ∂Λ μ ∂Λ
RN
RN
RN
Moreover, from φ ∈ E, RN
k |y| |y| 2∗ −2 ∂Wr,Λ 2∗ −2 ∂Wr,Λ 2∗ −2 ∂Uxj ,Λ Wr,Λ φ= K − K Uxj ,Λ Wr,Λ φ μ ∂Λ μ ∂Λ ∂Λ
RN
j =1
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
+
k ∗ −2 ∂Uxj ,Λ |y| K − 1 Ux2j ,Λ φ μ ∂Λ j =1
RN
|y| K μ
=k Ω1
+k
2∗ −2 ∂Wr,Λ Wr,Λ
∂Λ
−
k j =1
∗ −2 ∂Uxj ,Λ Ux2j ,Λ
∗ −2 ∂Ux1 ,Λ |y| K − 1 Ux21 ,Λ φ, μ ∂Λ
RN
k |y| 2∗ −2 ∂Wr,Λ 2∗ −2 ∂Uxj ,Λ − Uxj ,Λ Wr,Λ φ K μ ∂Λ ∂Λ j =1
Ω1
C
2∗ −2
Ux1 ,Λ
Ω1
C μm+σ
k
Uxj ,Λ +
j =2
k j =2
2∗ −1
Uxj ,Λ |φ|
,
and |y| 2∗ −2 ∂Ux1 ,Λ K − 1 U φ x1 ,Λ μ ∂Λ RN
√ ||y|−μr0 | μ
+
|y| 2∗ −2 ∂Ux1 ,Λ K − 1 Ux1 ,Λ φ μ ∂Λ
√ ||y|−μr0 | μ
|y| 2∗ −2 ∂Ux1 ,Λ K − 1 Ux1 ,Λ φ μ ∂Λ
C . μm+σ
Thus, we have proved
∂F (r, Λ) ∂I (Wr,Λ ) 1 , = +O ∂Λ ∂Λ μm+σ and the result follows from Proposition A.2.
2
Since |xj − x1 | = 2|x1 | sin we have
3067
(j − 1)π , k
j = 2, . . . , k,
∂Λ
φ
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081 k
k 1 1 1 = N −2 N −2 (j −1)π |x − x1 | (2|x1 |) )N −2 j =2 j j =2 (sin k ⎧ k2 ⎪ 1 1 1 ⎪ ⎨ (2|x1 |)N−2 j =2 (sin (j −1)π )N−2 + (2|x1 |)N−2 , if k is even; k = [ k ] ⎪ 1 ⎪ ⎩ (2|x 1|)N−2 j 2=2 if k is old. (j −1)π N−2 , (sin
1
k
)
But
0
sin (j −1)π k (j −1)π k
c ,
k . j = 2, . . . , 2
So, there is a constant B4 > 0, such that k j =2
1 B4 k N −2 k . = +O |xj − x1 |N −2 |x1 |N −2 |x1 |N −2
Thus, we obtain
F (r, Λ) = k A +
B1 B2 + (μr0 − r)2 Λm μm Λm−2 μm
B4 k N −2 1 1 k 3 , − N −2 N −2 + O + m |μr0 − r| + N −2 μm+σ μ Λ r r
and
∂F (r, Λ) B1 m B4 (N − 2)k N −2 = k − m+1 m + ∂Λ Λ μ ΛN −1 r N −2
1 1 k 2 . +O + |μr − r| + 0 μm+σ μm r N −2 Let Λ0 be the solution of −
B4 (N − 2) B1 m + = 0. Λm+1 ΛN −1 r0N −2
Then
Λ0 =
B4 (N − 2) B1 mr0N −2
1 N−2−m
.
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
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Define 1 1 1 1 D = (r, Λ): r ∈ μr0 − ¯ , μr0 + ¯ , Λ ∈ Λ0 − 3 ¯ , Λ0 + 3 ¯ , μθ μθ μ2θ μ2θ where θ¯ > 0 is a small constant. For any (r, Λ) ∈ D, we have
1 r . = r0 + O μ μ1+θ¯ Thus,
1 . r N −2 = μN −2 r0N −2 + O μ1+θ¯ So, 1 B1 B4 B2 F (r, Λ) = k A + − + (μr0 − r)2 Λm ΛN −2 r0N −2 μm Λm−2 μm
1 1 k 3 , (r, Λ) ∈ D, +O + m |μr0 − r| + N −2 μm+σ μ μ
(3.1)
and
B1 m B4 (N − 2) 1 − m+1 + Λ ΛN −1 r0N −2 μm
1 1 k 2 , +O + |μr − r| + 0 μm+σ μm μN −2
∂F (r, Λ) =k ∂Λ
(r, Λ) ∈ D.
Now, we define F¯ (r, Λ) = −F (r, Λ),
(r, Λ) ∈ D.
Let α2 = k(−A + η),
1 B1 B4 1 , α1 = k −A − − − 5 ¯ −2 N −2 μm Λm ΛN r0 μm+ 2 θ 0 0
where η > 0 is a small constant. Let F¯ α = (r, Λ) ∈ D, F¯ (r, Λ) α .
(3.2)
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Consider ⎧ dr ⎪ ⎪ = −Dr F¯ , ⎪ ⎪ ⎨ dt dΛ ⎪ = −DΛ F¯ , ⎪ ⎪ dt ⎪ ⎩ (r, Λ) ∈ F α2 .
t > 0; t > 0;
Then Proposition 3.3. The flow (r(t), Λ(t)) does not leave D before it reaches F α1 . Proof. If Λ = Λ0 +
1 3 θ¯
μ2
, noting that |r − μr0 |
1 , μθ¯
we obtain from (3.2) that
1 1 ∂ F¯ (r, Λ) > 0. =k c +O 3 ¯ ∂Λ μm+2θ¯ μm+ 2 θ So, the flow does not leave D. Similarly, if Λ = Λ0 − 13 θ¯ , then we obtain from (3.2) that μ2
1 1 ∂ F¯ (r, Λ) +O < 0. = k −c 3 ¯ ∂Λ μm+2θ¯ μm+ 2 θ So, the flow does not leave D. Suppose now |r − μr0 | = 1θ¯ . Since |Λ − Λ0 | μ
1 3 θ¯
, we see
μ2
B1 B4 B1 B4 − = m − N −2 N −2 + O |Λ − Λ0 |2 m N −2 Λ Λ0 ΛN −2 r0 Λ0 r0
1 B1 B4 = m − N −2 N −2 + O . Λ0 Λ0 r0 μ3θ¯ So, using (3.1), we obtain
1 B1 B4 B2 1 2 ¯ F (r, Λ) = k −A − − N −2 N −2 − (μr0 − r) + O Λm μm Λm−2 Λ0 r0 μm μm+3θ¯ 0 0
B1 1 B4 B2 1 k −A − − − + O −2 N −2 μm Λm ΛN r0 Λm−2 μm+2θ¯ μm+3θ¯ 0 0 0 < α1 .
2
(3.3)
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
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Proof of Theorem 1.6. We will prove that F¯ , and thus F , has a critical point in D. Define Γ = h: h(r, Λ) = h1 (r, Λ), h2 (r, Λ) ∈ D, (r, Λ) ∈ D, h(r, Λ) = (r, Λ), if |r − μr0 | =
. ¯
1
μθ
Let c = inf max F¯ h(r, Λ) . h∈Γ (r,Λ)∈D
We claim that c is a critical value of F¯ . To prove this, we need to prove (i) α1 < c < α2 ; (ii) sup|r−μr0 |= 1 F¯ (h(r, Λ)) < α1 , ∀h ∈ Γ. μθ¯
To prove (ii), let h ∈ Γ . Then for any r¯ with |¯r − μr0 | = ˜ Thus, by (3.3), some Λ.
1 , μθ¯
˜ for we have h(¯r , Λ) = (¯r , Λ)
˜ < α1 . F¯ h(r, Λ) = F¯ (¯r , Λ) Now we prove (i). It is easy to see that c < α2 . For any h = (h1 , h2 ) ∈ Γ . Then h1 (r, Λ) = r, if |r − μr0 | =
1 . μθ¯
Define
h˜ 1 (r) = h1 (r, Λ0 ). Then h˜ 1 (r) = r, if |r − μr0 | =
1 . μθ¯
So, there is an r¯ ∈ (μr0 −
1 , μr0 μθ¯
+
1 ), μθ¯
such that
h˜ 1 (¯r ) = μr0 . Let Λ¯ = h2 (¯r , Λ0 ). Then from (3.1) ¯ max F¯ h(r, Λ) F¯ h(¯r , Λ0 ) = F¯ (μr0 , Λ)
(r,Λ)∈D
1 B1 B4 1 k = k −A − − +O + N −2 μm+σ μ Λ¯ m Λ¯ N −2 r0N −2 μm
B1 1 1 B4 = k −A − − N −2 N −2 +O 2 > α1 . m m m+3 θ¯ Λ0 μ Λ0 r0 μ
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Acknowledgments The first author is supported by an Earmarked Grant from RGC of Hong Kong. The second author is partially supported by ARC. Appendix A. Energy expansion In all of Appendixes A and B, we always assume that
2(j − 1)π 2(j − 1)π , r sin ,0 , xj = r cos k k where 0 is the zero vector in RN −2 , and r ∈ [r0 μ − Let recall that
j = 1, . . . , k,
1 , r0 μ + 1θ¯ ] μθ¯ μ
for some small θ¯ > 0.
N−2
μ = k N−2−m ,
|y| 1 1 ∗ I (u) = |Du|2 − ∗ K |u|2 , 2 2 μ RN
RN
N−2 Uxj ,Λ (y) = N (N − 2) 4
Λ
N−2 2
(1 + Λ2 |y − xj |2 )
N−2 2
,
and k N−2 Wr,Λ (y) = N (N − 2) 4 j =1
Λ
N−2 2
(1 + Λ2 |y − xj |2 )
N−2 2
.
In this section, we will calculate I (Wr ). Proposition A.1. We have
I (Wr,Λ ) = k A + −
k j =2
B1 m Λ μm
+
B2 (μr0 m−2 Λ μm
− r)2
B3 1 1 3 , + O + |μr − r| 0 μm+σ μm ΛN −2 |x1 − xj |N −2
where Bi , i = 1, 2, 3, is some positive constant, A > 0 is a constant, and r = |x1 |. Proof. By using the symmetry, we have |DWr,Λ |2 = RN
k k j =1 i=1
RN
∗
−1 Ux2j ,Λ Uxi ,Λ
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
2∗ U0,1
=k
+
2∗ U0,1
=k
+
k j =2
RN
k i=2
RN
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∗ −1 Ux21 ,Λ Uxi ,Λ
RN
B0 +O ΛN −2 |x1 − xj |N −2
k j =2
1 |x1 − xj |N −2+σ
.
Let xj π y , cos . Ωj = y: y = y , y ∈ R2 × RN −2 , |y | |xj | k Then, |y| |y| ∗ 2∗ |Wr,Λ | = k K |Wr,Λ |2 K μ μ
RN
Ω1
k ∗ −1 |y| |y| 2∗ ∗ Ux1 ,Λ − 2 K K Ux21 ,Λ Uxi ,Λ μ μ
=k Ω1
i=2
Ω1
2∗ /2 Ux1 ,Λ
+O
k
2∗ /2 Uxi ,Λ
.
i=2
Ω1
Note that for y ∈ Ω1 , |y −xi | |y −x1 |. Using Lemma B.1, we find that for any α ∈ (1, N 2−2 ), k
Uxi ,Λ C
i=2
k i=2
1
1
(1 + |y − x1 |)
N−2 2
(1 + |y − xi |)
N−2 2
α k k 1 1 1 C . |xi − x1 |α μ (1 + |y − x1 |)N −2−α (1 + |y − x1 |)N −2−α i=2
As a result, Ω1
2∗ /2 Ux1 ,Λ
k i=2
2∗ /2 Uxi ,Λ
Nα 1 k N−2 . N N + N−2 (N −2−α) μ (1 + |y − x |) 1 Ω 1
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
If we take the constant α with max(1, (N −2) ) < α < N − 2, then N 2
2∗ /2 Ux1 ,Λ
k
2∗ /2 =O
Uxi ,Λ
i=2
Ω1
N −2+σ k . μ
On the other hand, it is easy to show
K
k k k ∗ −1 ∗ −1 |y| |y| 2∗ −1 K −1 Ux1 ,Λ Uxi ,Λ = Ux21 ,Λ Uxi ,Λ + Ux21 ,Λ Uxi ,Λ μ μ i=2
Ω1
Ω1 i=2
=
k j =2
i=2
Ω1
B0 +O N −2 Λ |x1 − xj |N −2
N −2+σ k . μ
Finally,
m ∗ |y| c0 2∗ 2∗ Ux1 ,Λ = Ux1 ,Λ − m |y| − μr0 Ux21 ,Λ K μ μ
Ω1
Ω1
Ω1
+ O μ−m−θ =
=
x1 ,Λ
Ω1 2∗ U0,1
c0 − m μ
RN
|y| − μr0 m+θ U 2∗
|y| − μr0 m U 2∗
x1 ,Λ
+O
μm+θ
Ω1 2∗ U0,1
c0 − m μ
RN
1
|y − x1 | − μr0 m U 2∗ + O 0,Λ
1
μm+θ
.
RN
But 1 μm
|y − x1 | − μr0 m U 2∗ C 0,Λ
RN \B|x1 |/2 (0)
|y|m 2∗ + 1 U0,Λ m μ
RN \B|x1 |/2 (0)
C . μN
On the other hand, if y ∈ B|x1 |/2 (0), y = (y1 , y ∗ ), y ∗ = (y2 , . . . , yN ), then |x1 | − y1 > 0. So,
|x1 | 2
|y |2 |y − x1 | = |x1 | − y1 + O |x1 | − y1 As a result,
|y |2 = |x1 | − y1 + O . |x1 |
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
m
2 |y − x1 | − μr0 m = |x1 | − y1 + O |y| − μr0 |x |
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1
|y|2 = |y1 | + m|y1 | y1 μr0 − |x1 | + O |x1 |
2 2 1 |y| m−2 μr0 − |x1 | + O + m(m − 1)|y1 | 2 |x1 |
2 2+σ |y| . + O μr0 − |x1 | + O |x1 | m
m−2
Thus, using
∗
2 |y1 |m−2 y1 U0,Λ = 0, B|x1 |/2 (0)
we obtain
|y − x1 | − μr0 m U 2∗
0,Λ
B|x1 |/2 (0)
=
1 2∗ |y1 |m U0,Λ + m(m − 1) 2
RN
2 2∗ μr0 − |x1 | |y1 |m−2 U0,Λ
RN
2+σ 1 . + O μr0 − |x1 | + μ
So, we have proved
K
|y| c0 ∗ ∗ 2∗ |Wr,Λ |2 = k |U0,1 |2 − m m |y1 |m U0,1 μ Λ μ
RN
RN
−
RN
1 m(m − 1) Λm−2 μm 2 c0
2 2∗ μr0 − |x1 | |y1 |m−2 U0,1
RN
+2
∗
k j =2
We also need to calculate
∂I (Wr ) ∂Λ .
B0 1 . +O μm+σ ΛN −2 |x1 − xj |N −2
2
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
Proposition A.2. We have k ∂I (Wr,Λ ) B3 (N − 2) mB1 = k − m+1 m + N −1 ∂Λ Λ μ Λ |x1 − xj |N −2 j =2
+O
2 1 + m μr0 − |x1 | μ
1 μm+σ
,
where Bi , i = 1, 2, 3, is same positive constant in Proposition A.1. Proof. The proof of this proposition is similar to the proof of Proposition A.1. So we just sketch it. We have k ∗ ∂I (Wr,Λ ) |y| 2∗ −2 ∂Ux1 ,Λ 2∗ −1 ∂Wr,Λ =k 2 −1 Uxi ,Λ − K Wr,Λ Ux1 ,Λ . ∂Λ ∂Λ μ ∂Λ i=2
RN
Ω1
It is easy to check that for y ∈ Ω1 , k 2∗ /2 k ∂ ∗ /2 ∗ ∗ ∗ 2 −1 2 − Ux21 ,Λ − 2∗ Ux21 ,Λ Uxi ,Λ CUx1 ,Λ Uxi ,Λ . Wr,Λ ∂Λ i=2
i=2
Thus, k 2∗ /2 k ∂ ∂ 2∗ 2∗ /2 2∗ −1 2∗ ∗ ∂ +2 Uxi ,Λ + O Ux1 ,Λ Uxi ,Λ . W = U Ux1 ,Λ ∂Λ r,Λ ∂Λ x1 ,Λ ∂Λ i=2
i=2
As a result, we have 2∗
K
Ω1
=
|y| 2∗ −1 ∂Wr Wr,Λ μ ∂Λ
k |y| ∂ 2∗ |y| ∂ 2∗ −1 ∗ U K +2 K Uxi ,Λ Ux1 ,Λ μ ∂Λ x1 ,Λ μ ∂Λ
Ω1
i=2
Ω1
+O
2∗ /2 Ux1 ,Λ
i=2
Ω1
So, we obtain the desired result.
k
2
2∗ /2 Uxi ,Λ
.
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
3077
Appendix B. Basic estimates For each fixed i and j , i = j , consider the following function gij (y) =
1 1 , (1 + |y − xj |)α (1 + |y − xi |)β
(B.1)
where α 1 and β 1 are two constants. Lemma B.1. For any constant 0 < σ min(α, β), there is a constant C > 0, such that gij (y)
C |xi − xj |σ
1 1 . + (1 + |y − xi |)α+β−σ (1 + |y − xj |)α+β−σ
Proof. Let dij = |xi − xj |. If y ∈ B 1 dij (xi ), then 2
1 |y − xj | |xj − xi |, 2
1 |y − xj | |y − xi |, 2
which gives gij (y)
1 C , |xi − xj |σ (1 + |y − xi |)α+β−σ
y ∈ B 1 dij (xi ).
1 C , σ |xi − xj | (1 + |y − xj |)α+β−σ
y ∈ B 1 dij (xj ).
2
Similarly, we can prove gij (y)
2
Now we consider y ∈ RN \ (B 1 dij (xi ) ∪ B 1 dij (xj )). Then we have 2
1 |y − xi | |xj − xi |, 2
2
1 |y − xj | |xj − xi |. 2
If |y − xi | 2|xi − xj |, then 1 |y − xj | |y − xi | − |xi − xj | |y − xi |. 2 As a result, gij (y)
1 C C1 , (1 + |y − xi |)α+β |xi − xj |σ (1 + |y − xi |)α+β−σ
because |y − xi | 12 |xj − xi |.
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
If |y − xi | 2|xi − xj |, then gij (y)
C 1 1 C , α β σ (1 + |y − xi |) |xi − xj | |xi − xj | (1 + |y − xi |)α+β−σ
because |y − xj | 12 |xj − xi |.
2
Lemma B.2. For any constant 0 < σ < N − 2, there is a constant C > 0, such that RN
1 1 C dz . N −2 2+σ (1 + |y|)σ |y − z| (1 + |z|)
Proof. The result is well known. For the sake of completeness, we give the proof. We just need to obtain the estimate for |y| 2. Let d = 12 |y|. Then, we have Bd (0)
1 1 C dz N −2 |y − z|N −2 (1 + |z|)2+σ d
C d N −2
Bd (0)
1 dz (1 + |z|)2+σ
d N −2−σ
C , dσ
and Bd (y)
1 1 C dz 2+σ N −2 2+σ |y − z| (1 + |z|) d
Bd (y)
1 C dz σ . N −2 d |z − y|
Suppose that z ∈ RN \ (Bd (0) ∪ Bd (y)). Then 1 |z − y| |y|, 2
1 |z| |y|. 2
If |z| 2|y|, then |z − y| |z| − |y| 12 |z|. As a result, 1 1 C N −2 . |y − z|N −2 (1 + |z|)2+σ |z| (1 + |z|)2+σ If |z| 2|y|, then 1 1 C C1 N −2 N −2 . N −2 2+σ 2+σ |y − z| (1 + |z|) |y| (1 + |z|) |z| (1 + |z|)2+σ Thus, we have proved that 1 1 C N −2 , N −2 2+σ |y − z| (1 + |z|) |z| (1 + |z|)2+σ which, give
z ∈ RN \ Bd (0) ∪ Bd (y) ,
J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
RN \(Bd (0)∪Bd (y))
1 1 C dz σ . N −2 2+σ d |y − z| (1 + |z|)
2
Let recall that k N−2 Wr,Λ (y) = N (N − 2) 4 j =1
Λ
N−2 2
(1 + Λ2 |y − xj |2 )
N−2 2
.
Lemma B.3. Suppose that N 5 and τ ∈ (0, 2). Then there is a small θ > 0, such that RN
4 1 1 N−2 W (z) dz r,Λ N−2 |y − z|N −2 (1 + |z − xj |) 2 +τ
k
j =1
C
k j =1
1 (1 + |y − xj |)
N−2 2 +τ +θ
.
Proof. Recall that xj π y Ωj = y: y = y , y ∈ R2 × RN −2 , cos . , |y | |xj | k Note that for any τ1 k j =2
N −2−m N −2 ,
k 1 Ck τ1 1 |xj − x1 |τ1 μτ1 j τ1
Ck τ1 ln k
C, μ τ1 Ck μτ1 C,
j =2
τ1 1; τ1 1,
N−2
since μ = k N−2−m . For z ∈ Ω1 , we have |z − xj | |z − x1 |. Using Lemma B.1, we obtain k
k 1 1 1 N−2 (1 + |z − xj |)N −2 (1 + |z − x1 |) N−2 2 j =2 j =2 (1 + |z − xj |) 2 k 1 C |xj − x1 |τ1 (1 + |z − x1 |)N −2−τ1 j =2
C . (1 + |z − x1 |)N −2−τ1
Thus, 4
N−2 Wr,Λ (z)
C 4τ1
(1 + |z − x1 |)4− N−2
.
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J. Wei, S. Yan / Journal of Functional Analysis 258 (2010) 3048–3081
As a result, for z ∈ Ω1 , using Lemma B.1 again, we find 4 N−2 (z) Wr,Λ
k j =1
1 (1 + |z − xj |)
N−2 2 +τ
C (1 + |z − x1 |)
+
4τ1 N+6 2 +τ − N−2
C (1 + |z − x1 |)
N+6 (N+2)τ1 2 − N−2 +τ
k
C (1 + |z − x1 |)
4τ1 N+6 2 − N−2 −τ1 +τ
j =2
1 |xj − x1 |τ1
.
So, we obtain Ω1
4 1 1 N−2 Wr,Λ (z) dz N−2 N −2 |y − z| (1 + |z − xj |) 2 +τ
k
Ω1
j =1
1 C C dz , (N+2)τ1 N+2 (N+2)τ1 |y − z|N −2 (1 + |z − x |) N+6 − +τ 2 N−2 2 − N−2 +τ (1 + |y − x |) 1 1
which gives Ω
4 1 1 N−2 Wr,Λ (z) dz N−2 N −2 |y − z| (1 + |z − xj |) 2 +τ
k
j =1
=
k i=1 Ω i
(N +2)τ1 N −2
k
j =1
k i=1
since 2 −
4 1 1 N−2 Wr,Λ (z) dz N−2 N −2 |y − z| (1 + |z − xj |) 2 +τ
C (1 + |y − xi |)
> 0.
N+2 (N+2)τ1 2 − N−2 +τ
k i=1
C (1 + |y − xi |)
N−2 2 +θ+τ
,
2
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Journal of Functional Analysis 258 (2010) 3082–3096 www.elsevier.com/locate/jfa
Induction automorphe pour GL(n, C) Guy Henniart a,b,∗ a Univ. Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay Cedex F-91405, France b CNRS, Orsay Cedex F-91405, France
Reçu le 24 août 2009 ; accepté le 16 décembre 2009 Disponible sur Internet le 31 décembre 2009 Communiqué par P. Delorme
Résumé Pour F = R ou C, les classes d’isomorphisme de (g, K)-modules irréductibles pour GL(n, F ) sont paramétrées par des représentations de dimension n du groupe de Weil WF de F . Comme le groupe de Weil de C est d’indice 2 dans celui de R, on peut induire à WR une représentation de WC . Cela donne une opération d’« induction automorphe » qui à un (g, K)-module irréducible τ pour GL(n, C) associe un (g, K)-module irréductible π = τ C/R pour GL(2n, R). Dans cet article, nous prouvons que si τ est unitaire et générique, alors π est déterminé par τ via une identité de caractères entièrement analogue à l’identité qui intervient dans la théorie de l’induction automorphe pour les corps p-adiques, due à R. Herb et l’auteur. Cela complète ainsi la théorie de l’induction automorphe pour les représentations locales et globales de GL(n) sur un corps de nombres. © 2009 Elsevier Inc. Tous droits réservés. Abstract For F = R or C, isomorphism classes of irreducible (g, K)-modules for GL(n, F ) are parametrized by n-dimensional representations of the Weil group WF of F . We can induce to WR a representation of WC , which has index 2 in WR . That gives a process of “automorphic induction” which to an irreducible (g, K)module τ for GL(n, C) associates an irreducible (g, K)-module π = τ C/R for GL(2n, R). In the present paper we show that if τ is unitary and generic then π is determined by τ , up to isomorphism, via a character identity entirely analogous to the character identity occurring in the automorphic induction process for p-adic fields. This completes the theory of automorphic induction for local and global representations of GL(n) over number fields. © 2009 Elsevier Inc. Tous droits réservés.
* Adresse pour correspondance : Univ. Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay Cedex F-91405, France. Adresse e-mail :
[email protected].
0022-1236/$ – see front matter © 2009 Elsevier Inc. Tous droits réservés. doi:10.1016/j.jfa.2009.12.013
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Mots-clés : Correspondance de Langlands ; Induction automorphe ; Représentations de groupes de Lie ; (g, K)-modules ; Modèles de Whittaker Keywords: Langlands correspondence; Automorphic induction; Lie group representation; (g, K)-module; Whittaker model
1. Introduction 1.1. La théorie des formes automorphes amène à s’intéresser aux représentations unitaires de GL(n, R) et GL(n, C) et, par suite des travaux de Harish–Chandra, aux (g, K)-modules pour ces groupes. Langlands [16] a classifié les (g, K)-modules irréductibles pour GL(n, F ), quand F est un corps local commutatif archimédien, en termes de représentations de dimension n du groupe de Weil WF de F . On peut induire à WR une représentation de dimension n de WC , et on obtient une représentation de WR de dimension 2n, puisque WC est d’indice 2 dans WR . Par la correspondance de Langlands, on obtient ainsi une opération d’induction automorphe qui à un (g, K)-module irréductible τ pour GL(n, C) associe une classe d’isomorphisme de (g, K)modules irréductibles π = τ C/R pour GL(2n, R), qui ne dépend que de la classe d’isomorphisme de τ . Dans cet article, nous nous intéressons surtout aux (g, K)-modules irréductibles qui peuvent apparaître comme composants aux places archimédiennes de représentations automorphes cuspidales des groupes GL(n) sur les corps de nombres. On sait que ces composants sont unitaires et génériques, au sens où ils possèdent un modèle de Whittaker [19]. Nous prouvons ici que si τ est unitaire et générique, alors π = τ C/R est déterminé par τ via une identité de caractères (ci-après Théorème 1). 1.2. Précisons. La représentation τ possède un caractère χτ , qui est une fonction analytique sur l’ensemble des éléments (semisimples) réguliers de GL(n, C), et qui détermine τ à isomorphisme près. La représentation π = τ C/R , par sa définition, est isomorphe à sa tordue επ par le caractère ε d’ordre 2 de GL(2n, R). On peut donc choisir un opérateur d’entrelacement A entre επ et π , ce qui donne une distribution qui, à une fonction ϕ de classe C ∞ , à support compact, dans l’ensemble des éléments (semisimples) réguliers de GL(2n, R), associe la trace de l’opérateur π(ϕ) ◦ A ; cette distribution est représentée par une fonction analytique χ(π,A) . Par ailleurs, on dispose de « facteurs de transfert » qui dépendent de données auxiliaires. Nous utilisons ici les facteurs de transfert de [9,10], où l’on s’est donné une base du R-espace vectoriel Cn , d’où un plongement de H = GL(n, C) dans G = GL(2n, R). On obtient une fonction sur l’ouvert de H formé des éléments qui sont (semisimples) réguliers dans G. Théorème 1. Soit τ un (g, K)-module irréductible pour GL(n, C), qu’on suppose unitaire et générique. Soit π l’induite automorphe de τ , et choisissons un isomorphisme A de επ sur π . Alors il existe une constante non nulle c = c(τ, A) telle que (i) pour tout élément γ de H régulier dans G on ait : 1/2 DG (γ )1/2 χ(π,A) (γ ) = c(τ, A) ε x −1 x −1 γ x DH x −1 γ x C χτ x −1 γ x ; R x∈Xγ
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(ii) pour γ régulier dans G mais non conjugué à un élément de H , χ(π,A) (γ ) = 0. Dans la formule plus haut, les facteurs DG et DH sont les facteurs discriminants usuels, les valeurs absolues sont normalisées, et Xγ est obtenu en choisissant, pour chaque classe de conjugaison C dans H contenue dans la classe de conjugaison de γ dans G, un élément x de G tel que x −1 γ x appartienne à C. Ce théorème est l’exact pendant du résultat principal de [9], où la théorie de l’induction automorphe est établie pour les extensions cycliques de corps p-adiques [9, 3.10 et 3.11]. Que cette identité détermine π si τ est donnée vient de l’indépendance linéaire des caractères pour SL(2n, R). 1.3. B. Lemaire et l’auteur ont démontré [10], dans le cas de l’induction automorphe pour les corps p-adiques, qu’en un sens précis la constante c(τ, A) ne dépend pas de τ , et nous établissons ici l’analogue de ce résultat dans le cas archimédien. Plus précisément, fixons un caractère non trivial ψ du groupe additif R. On peut alors normaliser l’opérateur A en imposant que pour toute fonctionnelle de Whittaker λ, relative à ψ, sur l’espace de la représentation π , on ait λ ◦ A = λ (on voit aussitôt que si τ est unitaire et générique, il en est de même de π ). On obtient alors dans le théorème 1 une constante c(τ, A) qu’on note plutôt c(τ, ψ). Théorème 2. La constante c(τ, ψ) ne dépend que de ψ et pas de τ . On obtient ainsi un facteur c(ψ) qui ne dépend que de ψ et des données fixées pour définir les facteurs de transfert ; notre méthode permet de calculer c(ψ) par réduction au cas de GL(1, C), qui était pour l’essentiel connu de Labesse et Langlands au début des années 1970 [15]. Alors que je rédigeais cet article, j’ai appris que Hiraga et Ichino savent démontrer que si l’on choisit les facteurs de transfert « canoniques » de Kottwitz et Shelstad [14], qui font aussi intervenir un choix de ψ , alors la constante c(ψ) vaut toujours 1. Je renvoie à leur travail en cours de rédaction pour la comparaison explicite des facteurs de transfert. 1.4. Nous prouvons les théorèmes 1 et 2 ensemble. Notre méthode est purement locale et ne fait pas intervenir les représentations automorphes. A la section 2, nous donnons quelques rappels sur les représentations de GL(n, R) et GL(n, C). Dans la section 3, nous définissons l’induction automorphe pour GL(n), dans le cadre légèrement plus général d’une algèbre cyclique E sur un corps local commutatif archimédien F , et nous énonçons dans ce cadre notre résultat principal (Théorème 3.7 ci-dessous). La démonstration occupe la section 4. Le cas où l’algèbre E/F est déployée est facile, et ne figure que parce qu’on le rencontre nécessairement dans les situations globales. Le cas où E/F n’est pas déployée se ramène en fait, par les formules de caractères pour l’induction parabolique, au cas où n = 1 et où E/F est isomorphe à C/R, cas qui est bien connu depuis le début des années 1970 [15]. Dans la dernière section, nous indiquons comment notre résultat pour les algèbres cycliques permet d’étendre les théorèmes de [8] au cas des places archimédiennes : si E/F est une extension cyclique de corps de nombres, τ une représentation automorphe de GL(m, AE ), induite de cuspidale unitaire, et π l’induite automorphe de τ , alors pour chaque place archimédienne v de F , πv est l’induite automorphe de τv , l’induction étant prise pour l’algèbre cyclique E ⊗F Fv sur Fv ; ainsi τv et πv vérifient l’identité de caractères correspondante.
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1.5. Cet article est écrit alors que je suis en délégation au CNRS, affecté à l’Institut de Mathématiques de Jussieu ; je remercie ces deux institutions pour leur accueil. Je tiens à remercier également, pour des conversations et échanges de courriels, I. Badulescu, K. Hiraga, J.-P. Labesse, B. Lemaire, C. Mœglin, D. Renard et M. Tadi´c. R) et GL(n,C C) 2. Représentations de GL(n,R 2.1. Cette section est consacrée à des rappels sur les représentations de GL(n, F ), où F est un corps local commutatif archimédien. Ainsi F est isomorphe à R, l’isomorphisme étant alors unique, ou à C auquel cas on a deux isomorphismes possibles, se déduisant l’un de l’autre par l’action sur C de la conjugaison complexe. Si F est isomorphe à C, le groupe de Weil WF de F n’est autre que le groupe F × . Si F est isomorphe à R, on doit d’abord choisir une clôture algébrique F de F et le groupe de Weil WF de F est en fait relatif à ce choix de clôture algébrique : c’est l’extension de Gal(F /F ) par F × , où l’élément non trivial de Gal(F /F ) agit de la façon naturelle sur F × et se relève dans WF en un élément c de carré −1 dans F × . On munit WF de sa topologie naturelle. Deux choix différents de la clôture algébrique F donnent des groupes naturellement isomorphes. Il existe un homomorphisme de groupes N de WF dans F × qui coïncide avec la norme de F à F sur le sous-groupe F × , et prend la valeur −1 en c ; cet homomorphisme induit un isomorphisme de l’abélianisé de WF sur F × . 2.2. D’après Langlands [16], les classes d’isomorphisme de (g, K)-modules irréductibles pour GL(n, F ) sont paramétrées par les représentations continues semisimples de dimension n de WF , à isomorphisme près : à une telle représentation σ de WF , on associe une classe d’isomorphisme π(σ ) de (g, K)-modules irréductibles pour GL(n, F ). Noter que cette paramétrisation se comporte bien si l’on change la clôture algébrique F de F choisie. Pour n = 1, la correspondance σ → π(σ ) est l’identité sur l’ensemble des caractères de F × si F est isomorphe à C, tandis que si F est isomorphe à R, elle est caractérisée par π(σ ) ◦ N = σ pour tout caractère σ de WF . 2.3. On s’intéresse ici essentiellement aux représentations unitaires irréductibles (par quoi nous entendons toujours topologiquement irréductibles) de GL(n, F ) et aux (g, K)-modules sous-jacents. On sait que ces (g, K)-modules sont exactement les (g, K)-modules unitarisables et que deux représentations unitaires irréductibles de GL(n, F ) sont isomorphes si et seulement si les (g, K)-modules correspondants le sont. D. Vogan a classifié [23] les représentations unitaires irréductibles de GL(n, F ). Nous utilisons ici une présentation du résultat parallèle à la classification de Tadi´c pour les corps locaux non archimédiens [20] ; grâce à la preuve de la conjecture de Kirillov par Baruch [4], on peut d’ailleurs maintenant donner une démonstration analogue dans les cas archimédiens et non archimédiens [21] voir aussi la présentation de [3]. Rappelons cette classification du dual unitaire de GL(n, F ). On considère à la fois tous les groupes GL(n, F ) pour n entier strictement positif. Les briques de base sont les représentations « de la série discrète ». Si F est isomorphe à C, elles n’existent que pour n = 1, ce sont les caractères unitaires de GL(1, F ) = F × . Si F est isomorphe à R, il faut ajouter aux caractères unitaires de GL(1, R) les représentations de la série discrète de GL(2, R). Ces dernières sont celles qui correspondent, par la correspondance de Langlands, aux représentations continues irréductibles et unitaires, de dimension 2, de WF . Plus précisément, une représentation continue irréductible de dimension 2 de WF est forcément induite à partir du sous-groupe WF = F × de
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WF ; si χ est un caractère de F × , son induite σ (χ) à WF est irréductible exactement quand χ est régulier, c’est-à-dire distinct de son conjugué χ c par l’action de la conjugaison de F sur F ; on a σ (χ) σ (χ c ) et si χ est un caractère de F × distinct de χ et χ c , σ (χ ) n’est pas isomorphe à σ (χ). Les représentations de la série discrète pour GL(2, F ) s’obtiennent quand χ est un caractère unitaire de F × . 2.4. Notons D l’ensemble des représentations de la série discrète : ce sont des classes de représentations unitaires irréductibles de GL(1, F ) ou, si F est isomorphe à R, de GL(2, F ). Pour δ dans D et n entier strictement positif, on note u(δ, n) le quotient de Langlands de l’induite n−1 n−1 1−n parabolique ν 2 δ × ν 2 −1 δ × · · · × ν 2 δ, où ν est le caractère donné par la valeur absolue normalisée de F . (Ici et dans toute la suite, le signe × indique une induction parabolique normalisée ; pour fixer les idées, nous prenons toujours le sous-groupe parabolique formé de matrices triangulaires supérieures.) Alors u(δ, n) est une représentation unitaire irréductible, et pour tout nombre réel α strictement entre 0 et 1/2, l’induite parabolique π(δ, n; α) = ν α u(δ, n) × ν −α u(δ, n) est également unitaire irréductible (c’est la « série complémentaire »). Notons B l’ensemble des représentations u(δ, n) et π(δ, n; α) ainsi obtenues. La classification de Vogan s’exprime ainsi : (i) Pour k entier positif et π1 , . . . , πk dans B, π1 × · · · × πk est unitaire irréductible et sa classe d’isomorphisme ne dépend pas de l’ordre des πi . (ii) Une représentation unitaire irréductible de GL(n, F ) est de la forme π1 × · · · × πk où k est bien déterminé, et où les πi dans B sont bien déterminés à l’ordre près. 2.5. Nous nous intéressons en fait surtout aux représentations unitaires irréductibles qui sont génériques au sens de Shalika [19, §3], voir aussi [13, §6]. On fixe un caractère additif non trivial ψ de F ; si Nn désigne le groupe des matrices triangulaires supérieures unipotentes dans GL(n, F ), on définit le caractère θ = θψ de Nn par la formule θ (uij ) = ψ
n−1
ui,i+1
pour (uij ) ∈ Nn .
i=1
Si π est une représentation unitaire (ou plus généralement admissible) de GL(n, F ) dans un espace de Hilbert H, on note H∞ l’ensemble des vecteurs C ∞ de H ; il est canoniquement muni d’une structure d’espace de Fréchet [13, §6.1], et π est dite générique s’il existe une forme linéaire continue λ non nulle qui vérifie λ π(u)x = θψ (u)λ(x)
pour x dans H∞ et u ∈ Nn .
D’ailleurs, l’existence de λ ne dépend pas du choix de ψ non trivial, et λ est alors unique à multiplication par un scalaire près [19, Thm. 3.1]. On dit que λ est une fonctionnelle de Whittaker sur π (relative à ψ). On sait aussi que si π est composant en une place archimédienne d’une représentation automorphe cuspidale unitaire de GL(n) sur un corps de nombres, alors π est unitaire générique [19, §5, Corollary]. 2.6. D’après Vogan [22, p. 98], il découle des résultats de Kostant [13] qu’une représentation unitaire irréductible de GL(n, F ) est générique si et seulement si le (g, K)-module sous-jacent est
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« gros » i.e. de dimension de Gelfand–Kirillov maximale (Vogan dit « large » et Kostant « maximally faithfull »). Mais en fait Kostant ne traite que des groupes de Lie semisimples. Pour la commodité du lecteur, nous déduisons l’assertion ci-dessus du Thm. 6.7.2 de [13]. 2.7. Supposons d’abord F isomorphe à R ; on identifie même F à R, ce qui, pour notre question, est sans danger. On note G le sous-groupe de GL(n, R) formé des matrices de déterminant 1 ou −1 ; ainsi GL(n, R) est produit de G et du sous-groupe central formé des matrices scalaires à coefficients strictement positifs. Si π est une représentation unitaire irréductible de GL(n, R), sa restriction à G est irréductible, et la généricité de π se teste sur cette restriction. De même le (g, K)-module sous-jacent à π donne un (g, K)-module irréductible pour G, et la dimension de Gelfand–Kirillov, qui est basée sur l’action de l’algèbre de Lie, se teste également sur la restriction à G, de sorte que π est « grosse » si et seulement si sa restriction à G l’est. Suivant les constructions précédant le Thm. 6.7.2 de [13], on introduit le groupe F∞ des matrices diagonales dans SL(n, C) normalisant l’algèbre de Lie s de SL(n, R) et induisant sur λ un automorphisme d’ordre 1 ou 2 ; on voit que F∞ est formé des matrices diagonales diag(λε1 , . . . , λεn ) avec εi = ±1 pour i = 1, . . . , n et λn ε1 · · · εn = 1. Ces matrices normalisent G, et d’ailleurs l’action par conjugaison de diag(λε1 , . . . , λεn ) est la même que celle de diag(ε1 , . . . , εn ), qui appartient à G. On voit donc qu’avec les notations de [13], si π est une représentation unitaire irréductible de G sur un espace = H, et le Thm. 6.7.2 de [13] dit bien que les deux conditions suivantes de Hilbert H, on a H sont équivalentes : (i) π est « gros » ; (ii) π est générique. Le théorème assure de plus que l’espace des fonctionnelles de Whittaker pour π est de dimension 1. 2.8. On peut alors se rapporter à la classification donnée par Vogan [22] des représentations « grosses ». Le Thm. 6.2 (a) ⇔ (f ) de [22] donne qu’une représentation unitaire irréductible π de GL(n, R) est générique si et seulement si π est induite parabolique irréductible d’une représentation limite de séries discrètes ; pour GL(n, R) cela signifie simplement induite parabolique irréductible d’une représentation de la série discrète. Ecrivant π comme produit d’éléments de B par la classification du dual unitaire (2.4), cela signifie que les u(δ, n) pour n > 1 n’apparaissent pas. Autrement dit, π est générique si et seulement si elle est isomorphe à un produit π1 × · · · × πk où pour i = 1, . . . , k, πi est dans D ou de la forme π(δ, 1; α) pour un élément δ de D et un nombre réel α strictement entre 0 et 1/2. C’est bien la classification, utilisée en général sans commentaire, par exemple par Mœglin et Waldspurger dans [18]. 2.9. Si maintenant F est isomorphe à C, on peut supposer F = C. On reprend alors le même raisonnement que plus haut, en plus simple. On prend pour G le groupe SL(n, C), auquel les résultats de Kostant s’appliquent. Une représentation unitaire irréductible de GL(n, C) reste irréductible par restriction à SL(n, C), et reste générique si elle l’était au départ. Le groupe noté F∞ par Kostant agit alors trivialement par conjugaison sur G et ses représentations, de sorte que le Thm. 6.7.2 de [13] dit encore qu’une représentation unitaire irréductible de G est générique si et seulement si elle est grosse. Cela se transfère aussitôt à GL(n, C) et par le même Thm. 6.2
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de [22], les représentations unitaires irréductibles de GL(n, C) sont celles qui sont de la forme π1 × · · · × πk où pour i = 1, . . . , k, πi est dans D (i.e. πi est un caractère unitaire de C× ) ou de la forme π(δ, 1; α), δ un caractère unitaire de C× et α un nombre réel strictement entre 0 et 1/2. 3. L’induction automorphe 3.1. Considérons maintenant l’extension quadratique C/R. Via la correspondance de Langlands, les (g, K)-modules irréductibles pour GL(n, C) correspondent aux représentations continues semisimples de dimension n de WF : en fait toute telle représentation de WF est somme de caractères χ1 , . . . , χn de C× , et on lui associe le quotient de Langlands de la série principale attachée à χ1 , . . . , χn . Les représentations unitaires irréductibles de GL(n, C) sont obtenues pour des choix de χi imposés par la classification rappelée à la section 2. Les représentations unitaires irréductibles génériques de GL(n, C) sont obtenues pour des choix encore plus restreints, et en ce cas d’ailleurs la série principale correspondante est irréductible. Plus précisément, les représentations unitaires irréductibles génériques de GL(n, C) sont celles dont la représentation de WC correspondante est somme de représentations de la forme suivante : (i) un caractère unitaire ; (ii) une somme ν α χ ⊕ ν −α χ où χ est un caractère unitaire de C× et α strictement entre 0 et 1/2. De la même façon, les représentations unitaires irréductibles génériques de GL(n, R) sont celles dont la représentation de WR correspondante est somme de représentations de la forme suivante : (i) un caractère unitaire ; (ii) une représentation irréductible unitaire de dimension 2 ; (iii) une somme ν α ρ ⊕ ν −α ρ où ρ est comme dans (i) ou (ii) et α strictement entre 0 et 1/2. 3.2. On peut induire de WC à WR une représentation continue semisimple de dimension n de WC , et on obtient une représentation continue semisimple de dimension 2n de WR . Cela donne une opération d’induction σ → σ C/R , bien définie sur les classes d’isomorphisme, et on la transporte en une opération d’induction dite « automorphe » des classes d’isomorphisme de (g, K)-modules irréductibles pour GL(n, C) vers les classes d’isomorphisme de (g, K)-modules irréductibles pour GL(2n, R). Vu la classification rappelée plus haut, il est clair que cette opération d’induction automorphe envoie un (g, K)-module sous-jacent à une représentation unitaire irréductible générique de GL(n, C) vers le (g, K)-module sous-jacent à une représentation unitaire irréductible générique de GL(n, R). On obtient donc une opération d’induction automorphe π → π C/R sur les classes d’isomorphisme de représentations unitaires irréductibles génériques de GL(n, C). 3.3. Bien sûr, une représentation continue semisimple de dimension n de WR donne par restriction une représentation continue semisimple de dimension n de WC . Cela donne des opérations dites de changement de base, des (g, K)-modules irréductibles pour GL(n, R) vers les (g, K)-modules irréductibles pour GL(n, C), et aussi des représentations unitaires irréductibles génériques de GL(n, R) vers les représentations unitaires irréductibles génériques de GL(n, C). Il découle de la partie archimédienne des travaux d’Arthur et Clozel [2, Chap. 1, §7] que si
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π est une représentation unitaire irréductible générique de GL(n, R), et Π la représentation de GL(n, C) obtenue de π par changement de base, alors π et Π vérifient des identités de caractères « à la Shintani » qui déterminent Π à partir de π . Ce que nous voulons établir ici, les théorèmes 1 et 2 de l’introduction, est l’analogue des identités à la Shintani. 3.4. Par transport de structure, les n◦ 3.1 à 3.3 sont valables pour une extension quadratique quelconque de corps locaux archimédiens. En fait, pour les applications globales, on a besoin d’une situation plus générale, que nous introduisons maintenant. On fixe un entier strictement positif d, un groupe cyclique Γ à d éléments, et un générateur σ de Γ . On suppose donnée une F -algèbre cyclique E de groupe Γ , où F est un corps local commutatif archimédien. Dans cette situation, E est un produit de corps E1 × · · · × Er où r est un diviseur de d, d = rd1 , et où les Ei sont des extensions cycliques de degré d1 de F ; on peut choisir les notations de sorte que σ agisse par permutation cyclique sur les facteurs, et que E apparaisse comme E1r , σ agissant sur E1r par σ (x1 , . . . , xr ) = (σ1 xr , x1 , . . . , xr−1 ), où σ1 est un générateur de Gal(E1 /F ). Le groupe NE/F (E × ) est égal à NE1 /F (E1× ), qui est F × si E1 = F et est formé des éléments positifs de F × si E1 est quadratique sur F (auquel cas E1 /F est isomorphe à C/R) ; nous notons ε le caractère de F × de noyau NE/F (E × ) : ainsi ε = 1 si E1 = F et ε est le caractère signe sinon. 3.5. Fixons un entier strictement positif m, et posons n = md. Alors GL(m, E) s’identifie à GL(m, E1 )r , sur lequel σ agit de la façon indiquée plus haut. Une représentation unitaire ir ···⊗ τr où pour i = 1, . . . , r, τi est réductible τ de GL(m, E) est un produit tensoriel τ = τ1 ⊗ une représentation unitaire irréductible de GL(m, E1 ) ; de plus τ est générique si et seulement si chacun des τi est générique. ···⊗ τr une représentation unitaire irréductible générique de GL(m, E). Notant Soit τ = τ1 ⊗ E /F τi 1 l’induite automorphe de τi (c’est tout simplement τi si E1 = F ), l’induite automorphe E /F E /F τ E/F de τ est par définition τ1 1 × · · · × τr 1 , une représentation unitaire irréductible générique de GL(n, F ). On vérifie immédiatement qu’on peut caractériser π = τ E/F de la façon suivante : la tordue επ de π par le caractère ε ◦ det de GL(n, F ) est isomorphe à π (bien sûr cette condition est vide pour E1 = F ), et le changement de base de π pour l’algèbre cyclique E/F est l’induite d−1 parabolique τ × τ σ × · · · × τ σ , une représentation de GL(n, E) = GL(n, E1 )d : il suffit de traduire cette assertion en termes de représentations de WF et WE . Cette caractérisation montre que le théorème 6 de [8] est valable pour des corps locaux archimédiens. 3.6. Avant de donner notre théorème principal, qui donne l’identité de caractères reliant τ et π = τ E/F quand τ est comme en 3.5, il nous faut rappeler la normalisation de l’opérateur d’entrelacement A déjà apparu dans l’introduction, et un peu de la définition des facteurs de transfert, d’après [9,10]. Soit π une représentation unitaire irréductible générique de GL(n, R), ou plus généralement une représentation admissible irréductible générique de GL(n, R) sur un espace de Hilbert. On suppose επ isomorphe à π ; on peut donc choisir un isomorphisme A de επ sur π : il stabilise l’espace des vecteurs C ∞ de π . Si λ est une fonctionnelle de Whittaker pour π (relative au caractère ψ de R fixé), λ ◦ A est encore une fonctionnelle de Whittaker pour π , et on peut donc normaliser A en imposant l’égalité λ ◦ A = λ pour toute telle fonctionnelle de Whittaker. Bien sûr, si ε = 1, A est l’identité.
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On note A(ψ) l’opérateur d’entrelacement ainsi normalisé entre επ et π . On dispose alors de la distribution f → tr(π(f ) ◦ A(ψ)) sur l’ouvert GL(n, F )reg de GL(n, F ) formé des éléments (semisimples) réguliers. Si ε = 1, cette distribution est représentée par une fonction analytique sur GL(n, F )reg , localement L1 sur GL(n, R) [12, §10], le caractère χπ de π . On pose alors χ(π,A(ψ)) = χπ . Vérifions qu’on a les mêmes propriétés si ε = 1 (ce qui implique que n = md est pair et que E1 /F est isomorphe à C/R). Le noyau de ε ◦ det est le sous-groupe G+ de GL(n, F ) formé des matrices à déterminant strictement positif, c’est-à-dire le produit de SL(n, F ) par le sous-groupe central des matrices scalaires à coefficients strictement positifs. Comme επ est isomorphe à π , on voit par la théorie de Clifford que la restriction à G+ de π est somme de deux représentations irréductibles non isomorphes. Appliquant le théorème 6.7.2 de [13], comme en 2.7, au groupe SL(n, F ), on voit qu’un seul de ces composants est générique pour le choix de ψ (attention, comme il s’agit du groupe SL(n, F ), le choix fixé de ψ est ici important). Notons π + le composant générique pour ψ – il faudrait le noter π + (ψ) –, et π − l’autre composant. L’opérateur A(ψ) coïncide alors avec l’identité sur l’espace de π + et avec x → −x sur l’espace de π − . Notant χπ + et χπ − les caractères de π + et π − , qui sont des fonctions analytiques sur GL(n, F )reg ∩ G+ , localement intégrables sur G+ , on voit aussitôt que la distribution f → tr(π(f ) ◦ A(ψ) )) sur GL(n, F )reg est représentée par la fonction χπ + − χπ − , qu’on note, conformément à l’introduction, χ(π,A(ψ)) . 3.7. Il nous faut aussi introduire les facteurs de transfert , pour cela nous suivons [9], [10, §2]. On fixe une base de E m comme espace vectoriel sur F , ce qui donne un plongement de GL(m, E), qu’on note désormais H , dans GL(n, F ), noté G. On fixe également un élément e de E × tel que σ e = (−1)m(d−1) e. On dispose alors [10, 2.1 et 2.2, où le caractère ici noté ε est noté κ] de la fonction sur H ∩ Greg ; si ε = 1, est identiquement 1 ; si ε = 1, est à valeurs dans {±1} et vérifie (x −1 hx) = ε(x)(h) pour h ∈ H ∩ Greg et x ∈ G tel que x −1 hx appartienne à H ∩ Greg . Le théorème suivant est notre résultat principal. Dans le cas de l’algèbre cyclique E/F = C/R, il redonne les théorèmes 1 et 2 de l’introduction. Théorème 3. Il existe une racine de l’unité c(ψ) dans C qui vérifie la propriété suivante : Soit τ une représentation unitaire irréductible générique de H , et soit π = τ E/F l’induite automorphe de τ . Alors on a (i) si γ ∈ H ∩ Greg DG (γ )1/2 χ(π,A(ψ)) (γ ) F 1/2 = c(ψ) ε ◦ det x −1 x −1 γ x DH x −1 γ x E χτ x −1 γ x ; x∈X(γ )
(ii) si γ ∈ Greg n’est pas conjugué à un élément de H , χ(π,A(ψ)) (γ ) = 0. Dans la formule (ii), X(γ ) est choisi comme dans l’introduction : pour chaque classe de H -conjugaison C dans la classe de G-conjugaison de γ , on choisit un élément x de G tel
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que x −1 γ x soit dans C. Le facteur DG est le facteur discriminant habituel, | |F est la valeur absolue normalisée de F . Le facteur DH est aussi le facteur discriminant standard ; si l’on identifie H à GL(m, E1 )d , DH donne le discriminant usuel sur chaque facteur GL(m, E1 ), et pour
x = (x1 , . . . , xd ) dans E1d , on a posé |x|E = di=1 |xi |E1 , où | |E1 est la valeur absolue normalisée sur E1 . 3.8. Remarquons qu’on a, pour x dans G et γ dans Greg , χ(π,A(ψ)) x −1 γ x = ε ◦ det(x)χ(π,A(ψ) (γ ) ; c’est clair si ε = 1, et si ε = 1 cela découle du fait que la conjuguée de π + par x ∈ G est π + si x est dans G+ , et π − sinon. Il s’ensuit que les propriétés (i) et (ii) du théorème déterminent la fonction χ(π,A(ψ)) sur Greg , en termes de χτ . On sait par ailleurs [12, Thm. 10.6] que les caractères de représentations admissibles irréductibles deux à deux non isomorphes de G+ sont linéairement indépendants. On en déduit que, τ étant fixée, π est l’unique représentation admissible irréductible de G, à équivalence infinitésimale près, qui soit isomorphe à επ et vérifie les identités (i) et (ii) du théorème : on peut même pour cette caractérisation affaiblir l’identité (i) en prenant au lieu de c(ψ) un nombre complexe non nul pouvant dépendre de τ et π . La section suivante donne la démonstration du théorème. Comme nous l’avons dit dans l’introduction, Hiraga et Ichino savent maintenant prouver qu’avec la normalisation de Kottwitz et Shelstad [14] pour les facteurs de transferts, c(ψ) vaut en fait toujours 1. 3.9. Comme dans la théorie de l’induction automorphe pour les corps locaux non archimédiens, le théorème 3.7 peut se traduire en termes de fonctions concordantes [10, 2.4], ce qui peut être utile quand on a besoin de la formule des traces. 4. Preuve du théorème principal 4.1. Commençons par le cas où ε = 1, de sorte que E1 = F et que E s’identifie à l’algèbre cyclique déployée F d . En ce cas, nous l’avons dit, est identiquement 1. L’identité à prouver ne dépend pas du choix de plongement de H dans G, de sorte qu’on peut voir H comme le sous-groupe de Levi diagonal de G, de blocs diagonaux de taille m × m. Si τ est une représentation unitaire irréductible générique de H , produit tensoriel des représentations τ1 , . . . , τd de GL(m, F ), alors π = τ E/F est l’induite parabolique τ1 × · · · × τd . Les identités à prouver sont alors : (i) χπ (γ ) = 0 si γ ∈ Greg n’est pas conjugué à un élément de H ; 1/2 1/2 (ii) |DG (γ )|F χπ (γ ) = x∈X(γ ) |DH (x 1 γ x)|E χτ (x −1 γ x) pour γ dans H ∩ Greg . Mais il s’agit là simplement des formules décrivant le caractère d’une représentation induite parabolique, dues à Hirai [11] pour un groupe semisimple, à Wolf [24] un peu plus généralement ; voir [7, Théorème 5.7] pour un énoncé couvrant notre cas ; pour passer de la formulation de Hecht et Schmid à la nôtre, il suffit de remarquer qu’un élément de H régulier dans G n’appartient qu’à un sous-groupe de Cartan de G ; il serait agréable et important d’avoir une preuve de [7] parallèle à celle de Clozel [5] dans le cas non archimédien.
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4.2. Traitons ensuite le cas où m vaut 1 et où E est une extension quadratique de F , de sorte qu’on peut identifier E/F à C/R si l’on veut. Le résultat est alors essentiellement connu depuis le début des années 1970, et découle des investigations de Labesse et Langlands sur la Lindiscernabilité pour SL(2) [15, §2], voir aussi [17, §7]. Le théorème en ce cas est explicitement écrit et discuté dans [14, §5.3] : il suffit de remarquer que notre facteur de transfert ne diffère de celui utilisé par Kottwitz et Shelstad que par une racine de l’unité ne dépendant pas de τ . Remarquons aussi que dans le cas présent, le résultat du théorème est valable si l’on part de n’importe quel caractère τ de E × , et qu’on prend pour π la représentation τ E/F : cette représentation π est irréductible générique, isomorphe à επ , de sorte qu’on peut bien définir A(ψ) et χ(π,A(ψ)) . En fait, il existe un caractère η de F × tel que τ = τ/η ◦ NE/F soit unitaire et, posant π = τ E/F on a π = (η ◦ det) ⊗ π ; les identités de caractères (i) et (ii) reliant τ et π découlent immédiatement de celles reliant τ et π . 4.3. Passons maintenant au cas général. Si l’on change de plongement de H dans G, le membre de gauche des identités à prouver est multiplié par un signe ±1 ne dépendant pas de τ , tandis que le membre de droite est inchangé. On pourra donc choisir le plongement de façon appropriée. A ce stade, il est commode de poser d = 2r, d’identifier l’extension E1 /F à C/R et E à l’algèbre cyclique Cr sur R, où σ agit par σ (x1 , . . . , xr ) = (x r , x1 , . . . , xr−1 ). Si (e1 , . . . , em ) est la base canonique de Cm comme C-espace vectoriel, on peut choisir (e1 , ie1 , e2 , ie2 , . . . , em , iem ) comme base de Cm sur R, ce qui donne un plongement de GL(m, C) dans GL(2m, R). On choisit la base sur R de E m = (Cr )m de sorte que H = GL(m, E) soit identifié au sous-groupe GL(m, C)r du sous-groupe de Levi diagonal GL(2m, R)r de GL(n, R), donné par le sous-groupe GL(m, C) de GL(2m, R) dans chaque bloc diagonal. 4.4. La représentation unitaire irréductible générique τ de H est un produit tensoriel ···⊗ τr où pour i = 1, . . . , r, τi est une représentation unitaire irréductible générique de τ1 ⊗ C/R GL(m, C). Posant πi = τi , l’induite automorphe π de τ , pour l’algèbre cyclique E/F , est par définition l’induite parabolique π1 × · · · × πr . En fait, écrivant chaque τi comme une série principale, π apparaît comme une induite parabolique ρ1 × · · · × ρmr , où chaque ρj est une représentation admissible irréductible générique de GL(2, R) sur un espace de Hilbert, induite automorphe d’un caractère (pas forcément unitaire) ηj de C× . Les représentations τi sont obtenues en prenant les caractères ηj successivement m à m et en formant la série principale correspondante ; on a la recette correspondante pour les πi à partir des ρj . Notons M le sous-groupe de Levi de G, formé des blocs diagonaux de taille 2 × 2 ; l’intersection H ∩ M est le tore maximal diagonal T = C×mr de H . 4.5. Il convient d’identifier les composants π + et π − et de calculer χπ + − χπ − en termes de quantités analogues pour les ρj . Chaque ρj , par restriction au sous-groupe GL(2, R)+ de GL(2, R) formé des matrices à déterminant positif, se casse en deux représentations irréductibles inéquivalentes. Seule l’une d’elles est générique pour ψ, on la note ρj+ , et on note ρj− l’autre ···⊗ ρmr de M = GL(2, R)nr . composante irréductible. Considérons la représentation ρ = ρ1 ⊗ mr + + ++ Notons M le sous-groupe M ∩ G de M, et M le sous-groupe GL(2, R)+ de M + . La ε1 εmr ++ mr restriction de ρ à M a exactement 2 composants irréductibles distincts ρ1 ⊗ · · · ⊗ ρmr , ± où les εi sont des signes ±1 indiquant le choix du composant ρj . La restriction de ρ à M + se εmr ···⊗ ρmr casse en deux morceaux irréductibles, le morceau ρ + somme des composants ρ1ε1 ⊗
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avec Πεi = 1, et le morceau ρ − somme des autres composants. Il est clair que ρ + est induite de M ++ à M + , à partir de n’importe lequel de ses composants, et il en est de même de ρ − . En particulier, le caractère de ρ + et celui de ρ − s’annulent hors de M ++ et pour γ = (γ1 , . . . , γmr ) dans M ++ , régulier dans M, on a χρ + (γ ) =
mr j =1
χρ εj (γj ), j
où la somme porte sur les signes εj de produit 1, et χρ − (γ ) est la somme analogue sur les signes de produit −1 ; on en tire l’égalité (χρ + − χρ − )(γ ) =
mr
(γρ + − χρ − )(γj ).
j =1
j
j
4.6. Notons que la notation choisie pour ρ + et ρ − est cohérente avec nos conventions pour les représentations génériques. Par le lemme 14.11 de [1] (citant [6]) appliqué au groupe SL(n, R) et à son sous-groupe de Levi M ∩ SL(n, R), le composant π + de ρ1 × · · · × ρmr qui est générique relativement à ψ est l’induite parabolique, de M + à G+ , de la représentation ρ + ; de même π − est l’induite parabolique de ρ − . On peut alors appliquer à π + et π − les formules de [7, Thm. 5.7]. On trouve que pour γ dans Greg ∩ G+ , on a DG (γ )1/2 χπ ± (γ ) = DM (δ)1/2 χρ ± (δ) R
R
δ
où la somme porte sur les éléments δ de M (à conjugaison près dans M) conjugués à γ dans G. Par la formule finale de 4.5, on en tire, toujours pour γ dans Greg ∩ G+ , mr DG (γ )1/2 (χπ + − χπ − )(γ ) = D2 (δj )1/2 (χ R
où D2 est (δ1 , . . . , δmr ).
le
δ j =1
facteur
discriminant
pour
R
GL(2, R)
ρj+
et
− χρ − )(δj ) j
où
on
a
écrit
δ =
4.7. Les termes individuels du membre de droite dans la formule ci-dessus se calculent par le cas déjà établi du théorème. On voit en particulier que (χπ + − χπ − )(γ ) s’annule si γ n’est pas conjugué dans G à un élément de T , ce qui donne déjà l’assertion (ii) du théorème 3.7. Supposons maintenant γ dans H ∩ Greg ; il est alors conjugué dans H à un élément de T , et pour prouver l’assertion (i) du théorème 3.7, on peut aussi bien prendre γ , et les δ qui apparaissent dans les sommes ci-dessus, dans T . Ces éléments δ sont alors les éléments de T = (C× )mr dont les coordonnées sont une permutation de celles de γ . D’après le cas déjà traité de l’induction automorphe, de C× à GL(2, R), il existe une constante c2 (ψ) telle que, notant δj la j ème composante de δ pour j entier entre 1 et mr, on ait D2 (δj )1/2 (χ R
ρj+
− χρ − )(δj ) = c2 (ψ)2 (δj ) ηj (δj ) + ηj (δ j ) j
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dans cette formule, le facteur 2 est celui choisi pour le transfert de C× à GL(2, R), et la formule est valable car il est immédiat que les seuls éléments de C× conjugués à δj dans GL(2, R) sont δj et δj , et que d’autre part on a ε ◦ det(x) = −1 si δj = x −1 δj x, et 2 (δj ) = −2 (δj ). Finalement, pour γ = (γ1 , . . . , γmr ) dans T , régulier dans G, on obtient mr DG (γ )1/2 (χπ + − χπ − )(γ ) = c2 (ψ)mr 2 (δj ) ηj (δj ) + ηj (δ j ) R
δ j =1
la somme portant sur les δ dans T conjugués de γ dans H . 4.8. Il convient d’établir une expression comparable pour le membre de droite de (i). Notons A la quantité en facteur de c(ψ) dans cette formule. Pour chaque élément α = x −1 γ x apparaissant dans A, on peut supposer α pris dans T , et on a mr DH (α)1/2 χτ (α) = ηj (βj ) E
β j =1
où la somme porte sur les β = (β1 , . . . , βmr ) dans T conjugués dans H à α. D’autre part, les facteurs (x −1 γ x) et (β) dans T conjugués à γ dans G
étant égaux, A est somme sur les β −1 −1 de termes c(ψ)εβ (β) mr j =1 ηj (βj ), où εβ est le signe ε ◦ det(x ) quand on écrit β = x γ x avec x dans G. Ces éléments β de T sont obtenus en permutant les coordonnées de γ et en changeant certaines d’entre elles en leur conjuguée complexe, le signe εβ valant (−1)a(β) si le nombre de passages au conjugué complexe est a(β). Remarquant que change également alors par (−1)a(β) , on trouve A=
mr ηj (δj ) + ηj (δj ) , (δ) j =1
δ
où cette fois la somme porte sur les δ dans T conjugués à γ dans H . Mais on prouve, exactement comme dans [10, §3.6] que la restriction à T de est à un signe ξ près, la fonction δ →
mr
2 (δj ).
j =1
On a donc A=ξ
mr
2 (δj ) ηj (δj ) + ηj (δj ) .
δ j =1
Comparant avec la formule obtenue en 4.7, on obtient la partie (i) du Théorème 3.7, avec c(ψ) = ξ −1 c2 (ψ)mr , qui est bien une racine de l’unité dans C.
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5. Applications globales 5.1. Dans cette dernière section, très courte, nous indiquons comment les résultats du présent article complètent ceux de [8] sur l’induction automorphe globale pour GL(n) sur les corps de nombres. Soit F un corps de nombres, et E une extension cyclique de F . Notons d le degré de E sur F , Γ le groupe de Galois de E sur F , et fixons un générateur σ de Γ . Soit m un entier strictement × × positif, et posons n = md. Enfin fixons un caractère ε de A× F de noyau F NE/F (AE ). L’induction automorphe de [8] associe à une représentation automorphe τ de GL(m, AE ), induite de cuspidale unitaire, une représentation automorphe π = τ E/F de GL(n, AF ), également induite de cuspidale unitaire. La représentation π est caractérisée par les propriétés suivantes : (i) π est isomorphe à sa tordue επ par ε ◦ det ; d−1 (ii) le changement de base de π à E est l’induite parabolique τ × τ σ × · · · × τ σ , une représentation automorphe de GL(n, AE ). Comme τ et π sont induites de cuspidale unitaire, leurs composants locaux sont unitaires et génériques. 5.2. Soit v une place archimédienne de F . Posons Ev = E ⊗F Fv ; c’est une algèbre cyclique sur Fv , de groupe Γ . Le composant πv de π en v est isomorphe à sa tordue εv πv par le caractère εv ◦ det de GL(n, Fv ) et, par la compatibilité des changements de base locaux et d−1 globaux [2, Chap. III], le changement de base de πv à Ev est τv × τvσ × · · · × τvσ . La caractérisation donnée en 3.5 de l’induction automorphe archimédienne – qui donne l’équivalent, dans le cas archimédien, du Théorème 6 de [8] – entraîne que πv est l’induite automorphe de τv , pour l’algèbre cyclique Ev /Fv , de sorte que le Théorème 5 de [8] est encore valable pour les places archimédiennes de F . Nous pouvons énoncer : Théorème 4. Soit τ une représentation automorphe de GL(m, AE ), induite de cuspidale unitaire, et soit π = τ E/F l’induite automorphe de τ . Alors pour toute place v de τ , πv est l’induite automorphe de τv , pour l’algèbre cyclique Ev /Fv . Remarque. Cette dernière assertion équivaut au fait que πv est l’induite parabolique π1 × · · · × πe , où w1 , . . . , we sont les places de E au-dessus de v et πi l’induite automorphe de τwi , pour l’extension cyclique Ewi /Fwi . Si v est une place infinie, cela découle directement de la définition même de l’induction automorphe archimédienne. Si v est une place finie, cela découle des résultats de [10]. Références [1] J. Adams, D. Barbasch, D. Vogan, The Langlands Classification and Irreducible Characters for Real Reductive Group, Progr. Math., vol. 104, Birkhäuser, 1992. [2] J. Arthur, L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud., vol. 120, Princeton University Press, Princeton, 1989. [3] I. Badulescu, D. Renard, Unitary dual of GL(n) at Archimedean places and global Jacquet–Langlands correspondence, prépublication, 2009. [4] E.M. Baruch, A proof of Kirillov’s conjecture, Ann. of Math. 158 (2003) 207–252.
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[5] L. Clozel, Théorème d’Atiyah-Bott pour les variétés p-adiques et caractères des groupes réductifs, Mém. SMF 15 (1984) 39–54. [6] M. Hashizume, Whittaker models for real reductive groups, Jpn. J. Math. 5 (1979) 349–401. [7] H. Hecht, W. Schmid, Characters, asymptotics and n-homology of Harish–Chandra modules, Acta Math. 151 (1983) 49–151. [8] G. Henniart, Induction automorphe pour GL(n) sur les corps de nombres, prépublication, 2008. [9] G. Henniart, R. Herb, Automorphic induction for GL(n) (over local non-archimedean fields), Duke Math. J. 78 (1995) 131–192. [10] G. Henniart, B. Lemaire, Formules de caractères pour l’induction automorphe, prépublication, 2008. [11] T. Hirai, The characters of some induced representations of semisimple Lie groups, J. Math. Kyoto Univ. 8 (1968) 313–363. [12] A. Knapp, Representation Theory of Semisimple Groups, an Overview Based on Examples, Princeton Math. Ser., vol. 36, Princeton University Press, Princeton, 1986. [13] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978) 101–184. [14] R. Kottwitz, D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999). [15] J.-P. Labesse, R.P. Langlands, L-indistinguishability for SL2 , Canad. J. Math. 31 (1979) 726–785. [16] R.P. Langlands, On the classification of irreducible representations of real algebraic groups, in: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, in: Math. Surveys Monogr., vol. 31, Amer. Math. Soc., 1989, pp. 101–170. [17] R.P. Langlands, Base Change for GL(2), Ann. of Math. Stud., vol. 96, Princeton University Press, Princeton, 1980. [18] C. Mœglin, J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. Ecole Norm. Sup. 22 (1989) 605–674. [19] J. Shalika, The multiplicity one theorem for GL(n), Ann. of Math. 100 (1974) 171–193. [20] M. Tadi´c, Classification of unitary representations in irreducible representations of general linear group (nonarchimedean case), Ann. Sci. Ecole Norm. Sup. 19 (1986) 335–382. [21] M. Tadi´c, GL(n, C)∧ and GL(n, R)∧ , Contemp. Math., Amer. Math. Soc., 2009. [22] D. Vogan, Gelfand–Kirillov dimension for Harish–Chandra modules, Invent. Math. 48 (1978) 75–98. [23] D. Vogan, The unitary dual of GL(n) over an Archimedean field, Invent. Math. 83 (1986) 449–505. [24] J. Wolf, Unitary Representations on Partially Holomorphic Cohomology Spaces, Mem. Amer. Math. Soc., vol. 138, 1974.
Journal of Functional Analysis 258 (2010) 3097–3116 www.elsevier.com/locate/jfa
Ergodicity of non-equilibrium Glauber dynamics in continuum Yuri Kondratiev a,b , Oleksandr Kutoviy a,b,∗ , Robert Minlos c a Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany b Research Center BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany c IITP, Russian Academy of Sciences, Russia, Moscow
Received 27 August 2009; accepted 11 September 2009 Available online 11 November 2009 Communicated by Paul Malliavin
Abstract We study asymptotic properties of the continuous Glauber dynamics with unbounded death and constant birth rates. In particular, an information about the location of the spectrum for the symbol of the Markov generator is obtained. The latter fact is used for the proof of the ergodicity of this process. We show that the speed of convergence to the equilibrium is exponential. © 2009 Elsevier Inc. All rights reserved. Keywords: Configuration space; Continuous Glauber dynamics; Ergodicity of Markov process
1. Introduction The continuous Glauber type dynamics may be characterized as a birth-and-death Markov processes on the configuration space (CS) in continuum with the given grand canonical Gibbs equilibrium states as symmetrizing measures. The Markov generators of these processes are related with the (non-local) Dirichlet forms of Gibbs measures. The latter fact gives the possibility to construct the so-called equilibrium Glauber dynamics associated with these forms (see e.g. [10]). There are many possibilities to choose a particular form of the square field expression inside of the Dirichlet forms. Each of them leads to the corresponding birth and death intensi* Corresponding author at: Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany.
E-mail address:
[email protected] (O. Kutoviy). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.005
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ties in the generator of the Glauber dynamics. For example, one may consider a constant death rate, which means that all information about the interaction in the system is included in the birth rate (G+ Glauber dynamics). Another extremal possibility is to take a constant birth rate and to assure the reversibility of the dynamics via proper death coefficient (G− Glauber dynamics). In the case of G+ equilibrium dynamics and a positive interaction potential, there exists a spectral gap for the generator in the high temperature and low density regime [2,10,17]. This gives the exponential L2 ergodicity for the corresponding Markov semigroup. Coming to the problem of the non-equilibrium Glauber dynamics, we note that only recently were constructed Markov processes with some classes of initial distributions (i.e., Markov functions) for G+ [9] and G− [8] dynamics. These papers use a constructive approach to the dual Kolmogorov equation describing the evolution of the initial distributions in the Glauber stochastic dynamics. Let us stress that in the infinite particle case, the class of admissible initial conditions should be considered as an essential parameter in the study of dynamical properties. Depending on the initial conditions, stochastic dynamics may have very different types of behavior including the possibility to explode in a finite time. Roughly speaking, a choice of initial states defines the level of deviation from the equilibrium dynamics. In the present paper we analyze ergodic properties of such non-equilibrium random evolutions. Namely, we consider the case of G− stochastic dynamics for interaction potentials which satisfy stability and strong integrability conditions (but admit a possible negative part). The main result concerning the ergodicity is stated in Theorem 3.2. We show that under natural restrictions on the parameters of the system the time evolution for a class of initial measures converges to the invariant Gibbs measure. More precisely, we need to consider the equilibrium state in the high temperature and low density regime to assure the uniqueness of the limiting Gibbs measure. Then, depending on these parameters, we define explicitly the set of admissible initial states. Let us stress that this set forms a ball (in a proper metric) in the space of all probability measures on the CS. This ball includes the invariant Gibbs measure as well as all probability measures on the CS with the common Ruelle bounds. The convergence of measures on the CS is defined in the sense of their correlation functions convergence. We show the exponential rate of such convergence in the Ruelle type norm on correlation functions. 2. Foundations We consider the Euclidian space Rd . By B(Rd ) we denote the family of all Borel sets in Rd . Bb (Rd ) denotes the system of all sets in B(Rd ) which are bounded. The space of n-point configurations is (n)
Γ0
(n) = Γ0,Rd := η ⊂ Rd |η| = n ,
n ∈ N0 := N ∪ {0},
where |A| denotes the cardinality of the set A. (n) (n) (n) The space ΓΛ = Γ0,Λ for Λ ∈ Bb (Rd ) is defined analogously to the space Γ0 . As a set, (n)
Γ0
is equivalent to the symmetrization of n n Rd = (x1 , . . . , xn ) ∈ Rd xk = xl if k = l ,
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d )n /S , where S is the permutation group of {1, . . . , n}. Hence, one can introduce i.e., to (R n n (n) (n) the corresponding topology and Borel σ -algebra, which we denote by O(Γ0 ) and B(Γ0 ), respectively. The space of finite configurations Γ0 :=
(n)
Γ0
n∈N0
is equipped with the topology O(Γ0 ) of disjoint union. Let B(Γ0 ) denote the corresponding Borel σ -algebra. B ∈ B(Γ0 ) is called bounded if there exists Λ ∈ Bb (Rd ) and N ∈ N such that B ⊂ NA set(n) n=0 ΓΛ . The configuration space Γ := γ ⊂ Rd |γ ∩ Λ| < ∞ for all Λ ∈ Bb Rd is equipped with the vague topology O(Γ ). It is a Polish space (see e.g. [7]). B(Γ ) denotes the corresponding Borel σ -algebra. The filtration on Γ with a base set Λ ∈ Bb (Rd ) is given by BΛ (Γ ) := σ NΛ Λ ∈ Bb Rd , Λ ⊂ Λ , where NΛ : Γ0 → N0 is such that NΛ (η) := |η ∩ Λ|. For short we write ηΛ := η ∩ Λ. (n) For every Λ ∈ Bb (Rd ) the projection pΛ : Γ → ΓΛ := n0 ΓΛ is defined as pΛ (γ ) := γΛ . One can show that Γ is the projective limit of the spaces {ΓΛ }Λ∈Bb (Rd ) w.r.t. these projections. In the sequel we will use the following classes of function on Γ0 : • L0 (Γ0 ) – the set of all measurable functions on Γ0 ; • L0ls (Γ0 ) – the set of measurable functions with local support, i.e., G ∈ L0ls (Γ0 ) if there exists Λ ∈ Bb (Rd ) such that G Γ0 \ΓΛ = 0; • L0bs (Γ0 ) – the set of measurable functions with bounded support, i.e., G ∈ L0bs (Γ0 ) if there exists Λ ∈ Bb (Rd ) and N ∈ N such that G Γ \ N Γ (n) = 0; 0 n=0 Λ • B(Γ0 ) – the set of bounded measurable functions; • Bbs (Γ0 ) – the set of bounded functions with bounded support. Remark 2.1. Any function G : Γ0 → R can be presented as the collection of functions G = (G(n) , n 0), G(n) : Γ0(n) → R due to the structure of the space Γ0 . On Γ we consider the set of cylinder functions F L0 (Γ ), i.e., the set of all measurable functions F ∈ L0 (Γ ) which are measurable w.r.t. BΛ (Γ ) for some Λ ∈ Bb (Rd ). These functions are characterized by the following relation: F (γ ) = F ΓΛ (γΛ ).
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Those cylinder functions which are measurable w.r.t. BΛ (Γ ) for fixed Λ ∈ Bb (Rd ) we will denote by F L0 (Γ, BΛ (Γ )). Next we would like to describe some facts from the harmonic analysis on the configuration spaces based on [6]. The following mapping between functions on Γ0 and functions on Γ plays the key role in our further considerations: KG(γ ) :=
G(ξ ),
G ∈ L0ls (Γ0 ), γ ∈ Γ,
ξ γ
see e.g. [12,13]. The summation in the latter expression is taken over all finite subconfigurations ξ of γ , which is denoted by the symbol ξ γ . The K-transform is linear, positivity preserving, and invertible, with K −1 F (η) :=
(−1)|η\ξ | F (ξ ),
F ∈ F L0 (Γ ), η ∈ Γ0 .
(1)
ξ ⊂η
The map K, as well as the map K −1 , can be extended to more wide classes of functions. For details and further properties of the map K see, e.g. [6]. One can introduce a convolution : L0 (Γ0 ) × L0 (Γ0 ) → L0 (Γ0 ) (G1 , G2 ) → (G1 G2 )(η)
:=
G1 (ξ1 ∪ ξ2 )G2 (ξ2 ∪ ξ3 ),
(2)
(ξ1 ,ξ2 ,ξ3 )∈P∅3 (η)
where P∅3 (η) denotes the set of all partitions (ξ1 , ξ2 , ξ3 ) of η in 3 parts, i.e., all triples (ξ1 , ξ2 , ξ3 ) with ξi ⊂ η, ξi ∩ ξj = ∅ if i = j , and ξ1 ∪ ξ2 ∪ ξ3 = η. It has the property that for G1 , G2 ∈ L0ls (Γ0 ) K(G1 G2 ) = KG1 · KG2 . Due to this convolution we can interpret the K-transform as the Fourier transform in configuration space analysis, see also [1]. Let M1 (Γ ) be the set of all probability measures on B(Γ ), and let M1fm (Γ ) be the set of all probability measures μ which have finite local moments of all orders, i.e.
|γΛ |n μ(dγ ) < +∞ Γ
for all Λ ∈ Bb (Rd ) and n ∈ N0 . A measure ρ on (Γ0 , B(Γ0 )) is called locally finite if ρ(A) < ∞ for all bounded sets A from B(Γ0 ). The set of such measures is denoted by Mlf (Γ0 ).
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A measure ρ ∈ Mlf (Γ0 ) is called positive definite if
(G G)(η) ρ(dη) 0,
∀G ∈ Bbs (Γ0 ),
Γ0
where G is the complex conjugate of G. A measure ρ is called normalized if and only if ρ({∅}) = 1. One can define a transform K ∗ : M1fm (Γ ) → Mlf (Γ0 ), which is dual to the K-transform, i.e., for every μ ∈ M1fm (Γ ), G ∈ Bbs (Γ0 ) we have
KG(γ ) μ(dγ ) =
Γ
G(η)(K ∗ μ)(dη).
Γ0
The measure ρμ := K ∗ μ is called the correlation measure of μ. As it is shown in [6], for μ ∈ M1fm (Γ ) and any G ∈ L1 (Γ0 , ρμ ) the series KG(γ ) :=
G(η)
(3)
ηγ
is μ-a.s. absolutely convergent. Furthermore, KG ∈ L1 (Γ, μ) and
G(η) ρμ (dη) = Γ0
(KG)(γ ) μ(dγ ).
(4)
Γ
Fix a non-atomic and locally finite measure σ on (Rd , B(Rd )). For any n ∈ N the product d )n and hence on Γ (n) . The measure σ ⊗n can be considered by restriction as a measure on (R 0 (n) measure on Γ0 we denote by σ (n) . The Lebesgue–Poisson measure λzσ on Γ0 is defined as λzσ :=
∞ n z n=0
n!
σ (n) .
Here z > 0 is the so-called activity parameter. The restriction of λzσ to ΓΛ will be also denoted by λzσ . We write λz instead of λzσ , if the measure σ is considered to be fixed. The Poisson measure πzσ on (Γ, B(Γ )) is given as the projective limit of the family of meaΛ} Λ Λ −zσ (Λ) λ . sures {πzσ zσ Λ∈Bb (Rd ) , where πzσ is the measure on ΓΛ defined by πzσ := e −1 A measure μ ∈ M1fm (Γ ) is called locally absolutely continuous w.r.t. πzσ iff μΛ := μ ◦ pΛ −1 Λ d is absolutely continuous with respect to πzσ = πzσ ◦ pΛ for all Λ ∈ Bb (R ). In this case, ρμ := K ∗ μ is absolutely continuous w.r.t. λzσ . Let kμ : Γ0 → R+ be the corresponding Radon– Nikodym derivative, i.e. kμ (η) :=
dρμ (η), dλzσ
η ∈ Γ0 .
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Remark 2.2. Let σ be the Lebesgue measure. Then, the functions
kμ(n) (x1 , . . . , xn ) :=
n kμ(n) : Rd → R+ , kμ ({x1 , . . . , xn }),
d )n , if (x1 , . . . , xn ) ∈ (R
0,
otherwise
(5)
are the well-known correlation functions in statistical physics, see e.g. [15,16]. For technical purposes we also recall the following result: Lemma 2.1. Let n ∈ N, n 2, and z > 0 be given. Then
... Γ0
G(η1 ∪ · · · ∪ ηn )H (η1 , . . . , ηn ) dλzσ (η1 ) . . . dλzσ (ηn )
Γ0
=
G(η) Γ0
H (η1 , . . . , ηn ) dλzσ (η)
(η1 ,...,ηn )∈P∅n (η)
for all measurable functions G : Γ0 → R and H : Γ0 × · · · × Γ0 → R with respect to which both sides of the equality make sense. Here P∅n (η) denotes the set of all ordered partitions of η in n parts, which may be empty. This lemma is known in the literature as Minlos lemma (cf., [11,14]), and it will be crucial for calculations in many places below. 3. Glauber dynamics with competition We study the special class of the Glauber dynamics on Γ with the birth rate equal to a constant (see [8,9]) and the death rate equal to some unbounded function. In applications, this death rate may be considered as a competition between particles (or individuals) of a system. 3.1. Potential and Gibbs measures on configuration spaces A pair potential is a Borel, even function φ : Rd → R ∪ {+∞}. We assume that φ satisfies the following standard conditions, known from statistical physics: (S) (Stability). There exists B > 0 such that, for any η ∈ Γ0 , |η| 2 E(η) :=
φ(x − y) −B|η|.
{x,y}⊂η
Note that the stability condition implies that the potential φ is bounded from below. Namely, φ(x) −2B,
∀x ∈ Rd .
(6)
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(SI) (Strong Integrability). For any β > 0,
Cst (β) :=
1 − exp βφ(x) dx < ∞.
Rd
Throughout the paper we assume that the conditions (S) and (SI) are satisfied. For γ ∈ Γ and x ∈ Rd \ γ we define the relative energy of interaction as follows: E(x, γ ) :=
y∈γ
φ(x − y),
+∞,
if y∈γ |φ(x − y)| < ∞, otherwise.
A probability measure μ on (Γ, B(Γ )) is called a Gibbs measure with parameters β > 0 and z > 0 if it satisfies
Γ
F (γ , x) μ(dγ ) =
x∈γ
e−βE(x,γ ) F (γ ∪ x, x)z dx μ(dγ )
(7)
Γ Rd
for any measurable function F : Γ × Rd → [0, +∞]. Note that any fixed γ ∈ Γ as a set in Rd has zero Lebesgue measure, so that the expression E(x, γ ) on the right-hand side of (7) is almost surely well-defined. The set of all Gibbs measures, which correspond to the potential φ, activity parameter z > 0, and inverse temperature β > 0, will be denoted by G(φ, z, β). For the fixed potential φ we will write G(z, β) instead of G(φ, z, β). Remark 3.1. It is well known (see e.g. [14,15]) that if φ is stable and
C(β) :=
1 − e−βφ(x) dx < z−1 e−1−2Bβ
Rd
then the class G(φ, z, β) is non-empty. Moreover, under these conditions, for any μ ∈ G(φ, z, β) the corresponding correlation function kμ satisfies the following bound kμ (η) const C(β)−|η| ,
η ∈ Γ0 .
3.2. Generator of Glauber type dynamics The mechanism of the evolution of configurations in Γ we describe by some formally given generator. The action of such generator in the case of Glauber type dynamics has the following form (see e.g. [8]) (LF )(γ ) :=
x∈γ
eβE(x,γ \x) Dx− F (γ ) +
Dx+ F (γ ) dx,
Rd
where Dx− F (γ ) = F (γ \ x) − F (γ ) and Dx+ F (γ ) = F (γ ∪ x) − F (γ ).
(8)
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Remark 3.2. It is known that a Gibbs measure μ ∈ G(, β) is reversible with respect to the Markov process associated with L (i.e. the operator L is symmetric in L2 (Γ, μ)). Remark 3.3. In the case where E(x, γ ) is given by a potential with a non-trivial positive part, the death rate of the operator L will be unbounded. In the considered model, the death rate reflects a competition between points of the configuration. In the spatial ecology models such a situation is related with the density dependent mortality notion (see e.g. [4]). 3.3. Spectral properties of the symbol Let us consider the operator L on those functions from F L0 (Γ, BΛ (Γ )) for which (8) is well-defined. One can easily check that this operator has the Markov property (it satisfies the maximum principle for the generators of Markov semigroups). Therefore, one may think about this operator as a Markov pre-generator. As it was shown in [8], the following result holds: Proposition 3.1. The image of L under the K-transform (or symbol of L) on functions G ∈ Bbs (Γ0 ) is given by Lˆ := K −1 LK = L0 + L1 + L2 , where L0 G(η) := −A(η)G(η), L1 G(η) := −
G(ξ )
ξ ⊂η,ξ =η
A(η) =
x∈η y∈η\x
eβφ(x−y) − 1 ;
eβφ(x−y)
x∈ξ y∈ξ \x
L2 G(η) :=
eβφ(x−y) ;
y∈η\ξ
G(η ∪ x) dx,
> 0.
Rd
ˆ D(L)) ˆ in the Banach space In the present paper, we study the operator (L, LC := L1 Γ0 , C |η| λ(dη) with the domain ˆ := {G ∈ LC | L0 G ∈ LC }, D(L) where C > 0 and λ := λ1 is the Lebesgue–Poisson measure with intensity z = 1. Remark 3.4. In our recent paper [8] we have shown that for any triplet of positive constants C, , and β which satisfies 2eCst (β)C + 2e2Bβ C −1 < 3 ˆ D(L)) ˆ is a generator of a holomorphic semigroup Tt in LC . Moreover, there the symbol (L, exists a non-equilibrium Markov process which corresponds to L.
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In order to analyze the ergodic properties of the constructed non-equilibrium random evoluˆ For this purpose, we have to decompose the tion we investigate the spectrum of the operator L. space LC . Namely, we set L0 := {G ∈ LC | G = cψ0 , c ∈ R},
ψ0 (η) =
1, 0,
η = ∅, η = ∅,
(9)
and L1 := G ∈ LC G(∅) = 0 . Then, one can easily see that any G ∈ LC can be presented as the sum of functions from L0 and L1 . Moreover, such decomposition is unique due to the structure of functions on Γ0 (see Remark 2.1). The decomposition corresponding to G ∈ LC will be denoted by (G0 , G1 ), where G0 ∈ L0 , G1 ∈ L1 . As result, LC = L0 + L1
(10)
and the operator Lˆ can be represented in the form Lˆ :=
L00 L10
L01 L11
,
where L00 : L0 → L0 , L01 : L1 → L0 , L10 : L0 → L1 , and L11 : L1 → L1 are the parts ˆ D(L)) ˆ corresponding to the decomposition of LC . of the operator (L, Simple calculations show that L00 = 0, L10 = 0, and
ˆ L01 G(η) = ψ0 (η) G(1) (x) dx, L11 G(η) = 1 − ψ0 (η) LG(η), Rd
for any G = (G(n) , n 0) ∈ D(L01 ) = D(L11 ), where ˆ . D(L11 ) := G1 ∈ L1 ∃G0 ∈ L0 : (G0 , G1 ) ∈ D(L) Remark 3.5. Since L10 = 0, the space L0 is invariant for the operator L11 . Moreover, it is clear ˆ and, as result, D(L11 ) coincides with D(L) ˆ ∩ L1 . that L0 ⊂ D(L) The main result concerning the ergodicity will be stated in terms of correlation functions convergence. For this reason we introduce also the Banach space KC := k : Γ0 → R k · C −|·| ∈ L∞ (Γ0 , λ) , which is dual to LC with respect to the duality defined as
G, k := G(η)k(η) λ(dη). Γ0
C > 0,
(11)
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Remark 3.6. The evolution of correlation functions in the space KC corresponding to the nonequilibrium process with generator L was constructed in [8]. Analogously to the decomposition of LC we get the following representation for the space KC : KC = K0 + K1 ,
(12)
K0 := {k ∈ KC | k = cψ0 , c ∈ R},
(13)
where
and K1 := k ∈ KC k(∅) = 0 . Then, according to this decomposition the adjoint operator to Lˆ on KC has form Lˆ ∗ :=
L∗00 L∗10
L∗01 L∗11
.
Since the operator Lˆ is closed and densely defined in LC , the corresponding adjoint operator Lˆ ∗ in KC , with respect to the duality (11), has the same spectrum as Lˆ (see e.g. [5]). The spectral properties of the operator Lˆ are described in the following theorem. Theorem 3.1. Let 1 D = D(β, , C) := exp e2βB CCst (β) + 2e2Bβ C −1 < 1 + √ . 2 Then the following statements are fulfilled: 1. The point z = 0 is an eigenvalue of the operator Lˆ which corresponds to the eigenvector ψ0 . 2. There exists z0 > 0 such that I1 = {z ∈ C: Re z > −z0 } \ {0} and 3π I2 = z ∈ C: |arg z| < \ {0} 4 ˆ belong to the resolvent set of the operator L.
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Proof. At the beginning we will show that any z = u + iw, u 0, w ∈ R belongs to the resolvent set of the operator L11 acting in the space L1 . In order to do this it is enough to show existence of the following resolvents in the space L1 : −1 11 L0 − (u + iw)1 and −1 −1 11 1 + L11 L1 + L11 , 0 − (u + iw)1 2 where L11 i is the part of the operator L11 corresponding to the operator Li , i = 0, 1, 2. The existence of (L11 − (u + iw)1)−1 in L1 follows then from the identity
−1 11 −1 −1 11 −1 L1 + L11 L0 − (u + iw)1 . = 1 + L11 L11 − (u + iw)1 0 − (u + iw)1 2
−1 exists in the space L The first resolvent (L11 1 since 0 − (u + iw)1)
A(η) > 0,
∀η = ∅.
It is also clear that the second resolvent −1 −1 11 1 + L11 L1 + L11 0 − (u + iw)1 2 exists in L1 if 11 L − (u + iw)1 −1 L11 + L11 < 1, 0
1
2
where · := · L1 throughout the proof of this theorem. Below we verify the latter bound. Let η = ∅, then for G ∈ L1 11 −1 L0 − (u + iw)1 L11 1 G(η) βφ(x−y) βφ(x−y) − 1) ξ ⊂η,ξ =η G(ξ ) x∈ξ y∈ξ \x e y∈η\ξ (e =− . βφ(x−y) + u + iw x∈η y∈η\x e As result 11 L − (u + iw)1 −1 L11 G(η) 0 1 βE(x,ξ \x) −1 G(ξ ) A (η) e B(η \ ξ, x) − G(η) A(η) , ξ ⊂η
where B(η, x) :=
y∈η |e
βφ(x−y)
x∈ξ
− 1|. Then, using the Minlos lemma
η = ∅,
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11 L − (u + iw)1 −1 L11 G(η)C |η| λ(dη) 0
1
Γ0 \{∅}
λ(dξ1 )
Γ0 \{∅}
λ(dξ2 )
|G(η)|
−
x∈ξ1
eβE(x,ξ1 \x) B(ξ2 , x)C |ξ1 | C |ξ2 | eβE(x,ξ1 \x) e−βm|ξ2 |
βφ(x−y) y∈η\x e C |η| λ(dη), βφ(x−y) y∈η\x e
x∈η
x∈η
Γ0 \{∅}
x∈ξ1
Γ0
|G(ξ1 )|
(14)
where m := − min {0, min φ} 2B (see (6)). For the last bound we have used the following inequality:
eβφ(x−y)
x∈ξ1 ∪ξ2 y∈ξ1 ∪ξ2 \{x}
eβφ(x−y)
x∈ξ1 y∈ξ1 \x
eβφ(x−y)
y∈ξ2
eβφ(x−y) e−βm|ξ2 | .
x∈ξ1 y∈ξ1 \x
The estimate (14) gives
11 L − (u + iw)1 −1 L11 G(η)C |η| λ(dη) G exp eβm CCst (β) − G 0
1
Γ0 \{∅}
= exp eβm CCst (β) − 1 G.
It is clear now that 11 L − (u + iw)1 −1 L11 exp e2βB CCst (β) − 1. 0
1
−1 11 Next we estimate the norm of (L11 0 − (u + iw)1) L2 in L1 . The simple inequality
11 L − (u + iw)1 −1 L11 G(η) A−1 (η) 0
2
G(η ∪ x) dx,
η = ∅,
Rd
yields
11 L − (u + iw)1 −1 L11 G(η)C |η| λ(dη) 0
Γ0 \{∅}
e
2
2Bβ Γ0 \{∅}
1 |η|
Rd
= e2Bβ (1)
G(η ∪ x)C |η| dx λ(dη)
Γ0 \(Γ0 ∪{∅})
2e2Bβ |η| C |η|−1 G(η) λ(dη) G, |η| − 1 C
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where B > 0 is the constant from the stability condition for φ. In the estimate above we have also used the following inequality 1 βφ(x−y) 2β E(η) e−2βB . e exp |η| x∈η |η| y∈η\x
Therefore, 11 L − (u + iw)1 −1 L11 2e2Bβ C −1 0
2
and finally by assumption of the theorem 11 L − (u + iw)1 −1 L11 + L11 exp e2βB CCst (β) + 2e2Bβ C −1 − 1 0
1
2
= D − 1 < 1. The latter fact implies L11 − (u + iw)1 −1 −1 11 −1 11 −1 = 1 + L11 L1 + L11 L0 − (u + iw)1 0 − (u + iw)1 2 −1 , (2 − D)−1 L11 0 − (u + iw)1
(15)
but 11 L − (u + iw)1 −1 0
1 1
((min|ξ |1 A(ξ ) + u)2 + w 2 ) 2 − 1 2 2 min |ξ |e−2βB + u + w 2 . |ξ |1
As result we have shown that {z ∈ C | Re z 0} is a subset of the resolvent set of L11 . Moreover, for any z ∈ C: Re z 0, z = 0 1 L11 − (u + iw)1 −1 (2 − D)−1 min e−4βB + w 2 − 2 , 1 , |z| where z = u + iw. Now, let 0 > u > −z0 , z0 = (2 − D)e−2βB , w ∈ R. Since L11 − (u + iw)1 −1 = 1 − (L11 − iw1)−1 u −1 (L11 − iw1)−1 we have −1 −4βB − 1 L11 − (u + iw)1 −1 (2 − D)−1 1 − |u| e + w2 2 . z0
(16)
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Therefore, I1 is a subset of the resolvent set of L11 . Next we note that imaginary and real parts of each z = u + iw ∈ I2 , with u < 0 satisfy the following relation |w| > |u|. This inequality implies 1 A(η) + u + iw > A(η) + u 2 + u2 2 A(η) √ . 2 Using this bound and the same arguments as before, one can show that I2 belongs to the resolvent π set of L11 as well. Moreover, for each z = u + iw ∈ C, such that 3π 4 > |arg z| > 2 , z = 0 − 1 2 √ 2 L11 − (u + iw)1 −1 1 − 2(D − 1) −1 min A(ξ ) + u + w 2 |ξ |1
√ √ (1 − 2(D − 1))−1 2(1 − 2(D − 1))−1 < |w| |w| + |u| √ 2(1 − 2(D − 1))−1 . |z|
(17)
Now we are ready to prove the facts claimed in the theorem. It is clear that the vector ψ0 is an eigenvector of the operator Lˆ with the eigenvalue 0. Suppose that there exists another ˆ ψ¯ 0 ∈ L0 , ψ¯ 1 ∈ L1 , which corresponds to the eigenvalue 0. eigenvector ψ¯ = (ψ¯ 0 , ψ¯ 1 ) ∈ D(L), Then the following should be true: L00 ψ¯ 0 + L01 ψ¯ 1 = 0, L11 ψ¯ 1 = 0. The last equality implies that ψ¯1 = 0, and hence ψ¯ = const ψ¯ 0 . That proves the first statement of the theorem. Let z ∈ I1 ∪ I2 , z = 0 be arbitrary and fixed. To prove the second statement of the theorem, we consider the following equation (Lˆ − z1)R = G,
ˆ G ∈ LC . R ∈ D(L),
It can also be rewritten in the form (L00 − z1)R0 + L01 R1 = G0 , (L11 − z1)R1 = G1 , where R = (R0 , R1 ), G = (G0 , G1 ), and R0 , G0 ∈ L0 , R1 , G1 ∈ L1 . The solution to the second equation exists and it is unique: R1 = (L11 − z1)−1 G1 .
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Therefore, R0 =
1 L01 (L11 − z1)−1 G1 − G0 , z
which means that the resolvent of the operator Lˆ is defined for all z ∈ I1 ∪ I2 , z = 0.
2
Remark 3.7. It is easy to see that, under conditions of Theorem 3.1, for any ε > 0 there exists 0 < ω = ω(ε) < π2 such that π π Sect −z0 + ε, + ω := z ∈ C arg {z + z0 − ε} + ω , 2 2
where z0 = (2 − D)e−2βB is a subset of I1 ∪ I2 . Moreover, there exists M = M(ε) > 0 such that (L11 − z1)−1 M(ε) |z|
(18)
for all z ∈ Sect(−z0 + ε, π2 + ω) \ {0}. For the readers convenience we recall below some facts well known from the functional analysis which we use in the sequel (see e.g. [3]): Definition 3.1. A closed linear operator (A, D(A)) with dense domain D(A) in a Banach space X is called sectorial (of angle ω) if there exists 0 < ω π2 such that the sector Sect
π π + ω := z ∈ C |arg z| < + ω \ {0} 2 2
is contained in the resolvent set ρ(A), and if for each ε ∈ (0, ω) there exists Mε 1 such that (A − z1)−1 Mε , |z|
for all 0 = z ∈ Sect
π +ω−ε . 2
Theorem 3.2. Let (A, D(A)) be a sectorial operator of angle ω. Then A is the infinitesimal generator of a C0 semigroup T (t) satisfying T (t) C for some constant C. Moreover, T (t) =
1 2πi
eζ t (ζ 1 − A)−1 dζ
(19)
Υ
where Υ is any piecewise smooth curve in Sect( π2 + ω) running from ∞e−iθ to ∞eiθ for θ < π2 + ω. The integral (19) converges for t > 0 in the uniform operator topology.
π 2
<
The direct application of this theorem to the operator (L11 , D(L11 )) and Remark 3.7 imply the following proposition:
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Proposition 3.2. Let D(β, , C) < 1 +
√1 . 2
Then, (L11 , D(L11 )) is the infinitesimal generator of a uniformly bounded holomorphic semigroup T˜t . Moreover, the semigroup T˜t can be given in the form (19).
4. Ergodicity The existence of the semigroup T˜t corresponding to L11 in L1 proved in this paper using the spectral properties of Lˆ can also be derived from the existence of Tt in LC , obtained in our recent paper [8]. Below we show how to do this. The condition which ensures the existence of Tt corresponding to Lˆ has form (see e.g. [8]): 3 exp CCst (β) + e2Bβ C −1 < . 2 Suppose that it is satisfied. The space L0 is invariant with respect to the semigroup Tt . As result, ˆ L in this there exist the factor semigroup Tt /L0 on the factor space LC /L0 and the operator L/ 0 space, the action of which is given by ˆ ˆ L )[G] := [LG], (L/ 0
(20)
ˆ is the where [G] ∈ LC /L0 is the equivalence class corresponding to G ∈ LC . Since L0 ⊂ D(L) ˆ ˆ ˆ invariant subspace for L, the definition (20) is correct. Moreover, (L/L0 , D(L)/L0 ) turns out to be the generator for Tt /L0 . Identifying the space LC /L0 with L1 one can easily see that ˆ L corresponds to the operator L11 in the sense of similarity, i.e. there exists an the operator L/ 0 (isometric) isomorphism J such that ˆ L . J −1 L11 J = L/ 0 Moreover, the latter isomorphism determines also the semigroup on L1 : T˜t = J (Tt /L0 )J −1 . Its generator is exactly (L11 , D(L11 )) by the construction. In addition, for each t 0 the operator T˜t coincides with (Tt )11 , which is the part of the operator Tt , corresponding to the decomposition (10), acting in the space L1 . Remark 4.1. Let (Tt∗ )t0 be the semigroup adjoint to (Tt )t0 with respect to the duality (11). It easy to see that the subspace K1 is invariant for the operators Tt∗ , t 0. Moreover, (Tt∗ )11 (the part of Tt∗ acting in K1 with respect to the decomposition (12)) coincides with Tt∗ |K1 (restriction of Tt∗ to the invariant subspace K1 ). In the sequel we will consider only those parameters β, , C which satisfy the following conditions 1 D(β, , C) = exp e2βB CCst (β) + 2e2Bβ C −1 < 1 + √ 2
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and 3 exp CCst (β) + e2Bβ C −1 < . 2 The class of all such parameters we denote by P. Remark 4.2. The conditions above are satisfied if, for example, 3 D(β, , C) < . 2 In the next theorem we study the properties of the semigroup (Tt∗ )t0 on the invariant subspace K1 . Theorem 4.1. Assume (β, , C) ∈ P. Then, for any ε > 0 there exists C(ε) > 0 such that ∗ T
t K1
e−(z0 −ε)t C(ε),
where z0 = (2 − D)e−2βB . Proof. Let ε > 0 be arbitrary and fixed. According to Remark 3.7, there exists 0 < ω = ω(ε) < π π 2 such that Sect(−z0 + ε, 2 + ω) is a subset of the resolvent set of L11 . Moreover, there exists M = M(ε) > 0 such that (L11 − z1)−1 M(ε) |z|
(21)
for all z ∈ Sect(−z0 + ε, π2 + ω) \ {0}. Due to Proposition 3.2, for any moment of time t 0 1 T˜t = 2πi
eζ t (ζ 1 − L11 )−1 dζ,
(22)
Υ0
where Υ0 is a piecewise smooth curve in Sect( π2 + ω) running from ∞e−i( 2 +θ) to ∞ei( 2 +θ) for 0 < θ < ω. Since the integrand in the formula (22) is analytic in the region containing Sect(−z0 +ε, π2 +ω), the corresponding integral is by the Cauchy’s integral theorem independent of the particular choice of Υ0 . Hence, we may change Υ0 by the contour Υ−z0 +ε := Υ0 − z0 + ε, i.e.
1 T˜t = eζ t (ζ 1 − L11 )−1 dζ. 2πi π
π
Υ−z0 +ε
The change of variables ζ → (ζ + z0 − ε) in the last integral leads us to the following equality 1 T˜t = e−(z0 −ε)t 2πi
Υ0
−1 eζ t (ζ − z0 + ε)1 − L11 dζ.
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Using the estimate (21) we get M(ε) T˜t e−(z0 −ε)t π
∞
−1 π e−ρt sin θ ρei( 2 +θ) − z0 + ε dρ.
0
One may easily check that the last integral is finite. Therefore, for t 0 ∗ T
t K1
e−(z0 −ε)t C(ε),
where we have used the fact that T˜t = (Tt∗ )11 = Tt∗ |K1 .
(23) 2
Now, given an initial correlation function k0 from the class KC , we want to study the behavior of kt := Tt∗ k0 for t > 0. More precisely, we would like to show in the last part of the work that kt converges to the invariant correlation function as t → 0. Toward this end we should adjust the initial correlation functions with the possible parameters of the considered system. The corresponding result is described in the theorem below. Theorem 4.2. Let the initial measure μ0 ∈ M1 (Γ ) has the correlation function k0 , which belongs to KC for some C > 0. Assume that (β, , C) ∈ P and, additionally, e2βB C −1 exp e2βB CCst (β) < 1.
(24)
Denote kμ the correlation function corresponding to μ ∈ G(, β). Then, for any ε > 0 there exists C(ε) > 0 such that ∗ T k0 − kμ t
KC
e−(z0 −ε)t C(ε)k0 − kμ KC ,
where z0 = (2 − D)e−2βB . Proof. Let us clarify the sense of each condition assumed in the theorem. The assumption (β, , C) ∈ P ensures existence of the semigroup Tt∗ in the Banach space KC . Moreover, according to Theorem 4.1, it implies the following bound ∗ T
t K1
e−(z0 −ε)t C(ε).
The condition (24) gives the existence of μ ∈ G(, β) and the following bound for the corresponding correlation function kμ (η) C |η| ,
η ∈ Γ0 .
(25)
Since μ is invariant measure for the operator L, using the properties of the K-transform, it is not difficult to see that Tt∗ kμ = kμ .
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Denote rt := kt − kμ , where kt := Tt∗ k0 and k0 ∈ KC . Since k0 (∅) = 1 and kμ (∅) = 1, we have r0 (∅) = 0. Hence, the condition (24) implies that r0 is a function from K1 . Indeed, |r0 | = |k0 − kμ | const C |η| . Finally, using Theorem 4.1 we have kt − kμ KC = Tt∗ r0 K e−(z0 −ε)t C(ε)k0 − kμ KC , C
that concludes the proof of the theorem.
2
Remark 4.3. For any fixed C > 0 there exist small > 0 and β > 0 such that the conditions of Theorem 4.2 are fulfilled. Acknowledgments The authors are very thankful to Michael Röckner for several fruitful discussions. The financial support of the DFG through the SFB 701 (Bielefeld University) and GermanUkrainian Projects 436 UKR 113/94 and UKR 113/97 is gratefully acknowledged. Yu. Kondratiev and O. Kutoviy gratefully acknowledge the financial support of FCT through POCI and PTDC/MAT/67965/2006. R. Minlos gratefully acknowledges the financial support of RFFI No. 06-01-00449 and CRDF research funds N RM1-2085. References [1] Yu.M. Berezansky, Yu.G. Kondratiev, T. Kuna, E. Lytvynov, On a spectral representation for correlation measures in configuration space analysis, Methods Funct. Anal. Topology 5 (4) (1999) 87–100. [2] L. Bertini, N. Cancrini, F. Cesi, The spectral gap for a Glauber-type dynamics in a continuous gas, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 91–108. [3] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., Springer, 1999. [4] D.L. Finkelshtein, Yu.G. Kondratiev, O. Kutoviy, Individual based model with competition in spatial ecology, SIAM J. Math. Anal. 41 (1) (2009) 297–317. [5] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/Heidelberg/New York, 1966. [6] Yu.G. Kondratiev, T. Kuna, Harmonic analysis on configuration space I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2) (2002) 201–233. [7] Yu.G. Kondratiev, O.V. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (7) (2006) 774–783. [8] Yu.G. Kondratiev, O.V. Kutoviy, R.A. Minlos, On non-equilibrium stochastic dynamics for interacting particle systems in continuum, J. Funct. Anal. 255 (7) (2007) 200–227. [9] Yu.G. Kondratiev, O.V. Kutoviy, E. Zhizhina, Nonequilibrium Glauber-type dynamics in continuum, J. Math. Phys. 47 (11) (2006), 17 pp. [10] Yu.G. Kondratiev, E. Lytvynov, Glauber dynamics of continuous particle systems, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 685–702. [11] Yu.G. Kondratiev, R. Minlos, E. Zhizhina, One-particle subspaces of the generator of Glauber dynamics of continuous particle systems, Rev. Math. Phys. 16 (9) (2004) 1–42.
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[12] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Ration. Mech. Anal. 59 (1975) 219–239. [13] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II, Arch. Ration. Mech. Anal. 59 (1975) 241–256. [14] R.A. Minlos, Lectures on statistical physics, Uspekhi Mat. Nauk (1) (1968) 133–190 (in Russian). [15] D. Ruelle, Statistical Mechanics, Benjamin, New York, 1969. [16] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970) 127–159. [17] L. Wu, Estimate of spectral gap for continuous gas, Ann. Inst. H. Poincaré Probab. Statist. 40 (4) (2004) 387–409.
Journal of Functional Analysis 258 (2010) 3117–3133 www.elsevier.com/locate/jfa
On ideals in the bidual of the Fourier algebra and related algebras M. Filali a , M. Neufang b,∗,1 , M. Sangani Monfared c,1 a Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland b School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada c Department of Mathematics and Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada
Received 28 August 2009; accepted 19 December 2009 Available online 6 January 2010 Communicated by N. Kalton
Abstract Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G × H ) the Fourier algebra of G × H , and UC2 (G × H ) the space of uniformly continuous functionals in VN(G × H ) = A(G × H )∗ . We use weak factorization of b(G) operators in the group von Neumann algebra VN(G × H ) to prove that there exist at least 22 left ideals b(G) of dimensions at least 22 in A(G × H )∗∗ and in UC2 (G × H )∗ . We show that every nontrivial right b(G) ideal in A(G × H )∗∗ and in UC2 (G × H )∗ has dimension at least 22 . © 2010 Elsevier Inc. All rights reserved. Keywords: Amenable groups; Compact nonmetrizable groups; Group von Neumann algebra; Factorization; Double dual of Fourier algebra; Dual of uniformly continuous functionals; Unitary representations; One-sided ideals; Cancellable sets
1. Introduction The study of the ideal structure of the double dual of the group algebra was initiated by Civin and Yood [3]. It was proved by Filali and Pym [7] that for a noncompact, locally compact * Corresponding author.
E-mail addresses:
[email protected] (M. Filali),
[email protected] (M. Neufang),
[email protected] (M. Sangani Monfared). 1 The authors were partially supported by NSERC. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.011
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group K, with compact covering number κ(K), every nontrivial right ideal in LUC(K)∗ has diκ(K) mension at least 22 . The case of the dimension of right ideals in L(K)∗∗ was however left κ(K) open in [7]. Recently, Filali and Salmi in [8] proved that there exist 22 left ideals in L1 (K)∗∗ ∗ and in LUC(K) with trivial pairwise intersections, and that the dimension of every nontrivκ(K) ial right ideal in L1 (K)∗∗ is at least 22 . The method used by Filali and Salmi in obtaining these results relies on factorization of elements of L∞ (K) (adapted from an earlier result by Neufang [17]) and on the existence of right cancellable points in L1 (G)∗∗ . In Filali, Neufang and Sangani Monfared [6], the authors proved a factorization theorem for operators in the group von Neumann algebra of certain amenable groups, and applied the factorization theorem to determine the topological centre of the Fourier algebra as well as to prove the automatic continuity of certain module homomorphisms on the group von Neumann algebra. We should also mention that, in 1981, Cecchini and Zappa [1] conjectured that for all amenable groups, the centre of UC2 (G)∗ is equal to B(G). Recent results of Losert [16] showed that this conjecture is not true in general. However, as we showed in [6, Corollary 3.14], the Cecchini–Zappa conjecture is in fact true for compact groups whose local weight has uncountable cofinality. In this paper, we shall study further applications of our factorization theorem by obtaining analogous results of [7] and [8] on the number and dimension of ideals for the double dual of the Fourier algebra and the dual of the space of uniformly continuous functionals. If G is a locally compact group, the local weight of G denoted by b(G), is the smallest cardinality of an open base at the identity element e of G. Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality, and let H be an arbitrary amenable locally compact group. Let A(G × H ) be the Fourier algebra of G × H , and UC2 (G × H ) be the space of uniformly continuous functionals in VN(G × H ). In Theorem 4.3, we show that b(G) every nontrivial right ideal in A(G × H )∗∗ and in UC2 (G × H )∗ has dimension at least 22 . b(G) In Theorem 4.4, we show that there exist at least 22 left ideals of dimensions greater than b(G) or equal to 22 in A(G × H )∗∗ and in UC2 (G × H )∗ . The final result of our paper concerns the number of topological invariant means (TIMs) in A(G × H )∗∗ . The cardinality of the set of TIMs in the bidual of the Fourier algebra was determined by Chou [2], for the case of metrizable groups, and by Hu [12], in the general case. In Theorem 4.8, we approach this problem by a b(G) different method than those used by the above authors, and we show that there are at least 22 linearly independent TIMs in VN(G × H )∗ . We hope that this alternative method can eventually be applied to the general case of nonmetrizable locally compact groups. In obtaining our results, we rely on the construction of right cancellable sets in VN(G × H )∗ and UC2 (G×H )∗ (Corollary 3.2), and we use a modified version of the factorization of operators in VN(G × H ), which we obtained in [6]. We note that, in the spirit of duality between the group algebra and the Fourier algebra, the compact covering number κ in the results of [7] and [8], is replaced with the local weight b in our results. 2. Preliminaries Let K be a locally compact group. For detailed properties of the Fourier algebra A(K), the group von Neumann algebra VN(K) = A(K)∗ , and the Fourier–Stieltjes algebra B(K) we refer our readers to [5]. The space A(K)∗∗ is a Banach algebra under the left Arens product 2 which is defined by the following relations:
M. Filali et al. / Journal of Functional Analysis 258 (2010) 3117–3133
3119
Φ 2 Ψ, T = Φ, Ψ · T , Ψ · T , u = Ψ, T · u, T · u, v = T , uv, where Φ, Ψ ∈ A(K)∗∗ , T ∈ VN(K) and u, v ∈ A(K). In this paper we shall also consider the space of uniformly continuous functionals on K, defined by UC2 (K) = VN(K) · A(K)
·
;
in the litthe closure is automatic if K is amenable. This space is often denoted by UC(K) erature, but we shall use the notation UC2 (K) to avoid any possible confusion arising from to denote the set of irreducible unitary representation on K (our nothe common usage of K tation also emphasizes that one can define a p-version of this space as a subset of PM p (K) for 1 < p < ∞). If K is abelian, then via the transpose of the Fourier transform, UC2 (K) can be identified with the Banach algebra of bounded uniformly continuous functions on the dual It is known, and is easy to verify, that UC2 (K)◦ — the polar of UC2 (K) — is a closed group K. two sided ideal in (A(K)∗∗ , 2), and hence UC2 (K)∗ = (A(K)∗∗ , 2)/UC2 (K)◦ is a quotient algebra of (A(K)∗∗ , 2). We denote this quotient product on UC2 (K)∗ by 2 as well. We also note that, VN(K) is a left Banach UC2 (K)∗ -module with the action defined by the relations: ξ T , u = ξ, T · u
ξ ∈ UC2 (K)∗ , T ∈ VN(K), u ∈ A(K) .
It follows that A(K)∗∗ has a natural right Banach UC2 (K)∗ -module structure defined by Φ ξ, T = Φ, ξ T
Φ ∈ A(K)∗∗ , ξ ∈ UC2 (K)∗ , T ∈ VN(K) .
It is easily verified that ξ T = ξ · T,
Φ ξ = Φ 2 ξ,
where ξ is any extension of ξ to a continuous linear functional on VN(K). For more information on these spaces see Lau [13,14]. Throughout this paper, unless otherwise stated, G is a compact nonmetrizable group such that its local weight b(G) has uncountable cofinality, and H is an amenable locally compact group. to denote the family of all equivalence classes of irreducible, (strongly) continuous We write G unitary representations of G. Since by Hewitt and Ross [10, Theorem 28.2 and Remark 28.58(b)], = b(G), we may write G = {Vγ : γ < b(G)}, with V0 the trivial one-dimensional represen|G| tation defined by V0 (t) = 1 (t ∈ G). For every 0 < α < b(G), we define 0 γ < α}. α = {Vγ ∈ G: G Let (Nα )0αb(G) be a decreasing family of compact normal subgroups of G (N0 = G and Nb(G) = {e}) whose existence was shown in [15,12]. In Filali, Neufang and Sangani Monfared [6], the authors showed the existence of two families of irreducible unitary representations α = (u˜ α )α
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α = n where n and n are independent of α. In an extension of Uα , (iii) dim Uα = n and dim U addition, it was shown that the coordinate functions u˜ αij satisfy a certain orthogonality condition. We state this orthogonality result for our later purposes. Lemma 2.1. (See [6].) Let G be a compact nonmetrizable group such that b(G) has uncountable be arbitrary and let V be the conjugate representation of V . Let α be large cofinality. Let V ∈ G α . Then if β = α, 1 i, k n , 1 j, l n and 1 p , q dim V , we enough so that V , V ∈ G have β u˜ αij (t)u˜ kl (t)v p q (t) dλβ (t) = 0. (1) Nβ
3. Right cancellation in VN(G × H )∗ and UC2 (G × H )∗ Throughout this section, G, H , Nα , u˜ αij , n and n are as stated in Section 2. For each α > 0, let λα be the normalized Haar measure on Nα . As in [15,12] we consider the family (Pα )0α
We define u˜ αij = u˜ αij ⊗ 1H ∈ B(G × H ), α = Pα ⊗ I ∈ VN(G × H ). P ˇ We let b(G) be equipped with the discrete topology and β(b(G)) be the Stone–Cech compactification of b(G) (cf. [11]). Let C(b(G)) be the set of all cofinal ultrafilters on b(G). Recall that an ultrafilter a on b(G) is cofinal if every set I in a is cofinal in b(G), that is, sup I = b(G) (cf. [19]). By a result of van Douwen [19] C b(G) = 22b(G) .
(3)
For each 1 k, l n, the bounded map b(G) −→ UC2 (G × H )∗ ,
α −→ u˜ αkl ,
has a continuous extensions to β(b(G)) denoted by θkl : β b(G) −→ UC2 (G × H )∗ ,
a −→ ξkla .
Since a ∈ I for every I ∈ a, it follows that the cluster point ξkla of (˜uαkl )α∈I does not depend on I , for every I ∈ a. We write ξkla for any Hahn–Banach extension of ξkla to an element in ∗ VN(G × H ) .
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Given any Q ∈ VN(G × H ), we define the operators a,I Pklr (Q) =
β 1 β β β u˜ kr · Pβ u˜ r1 · Q u˜ 1l · P 6 n
∗ w -limit ,
(4)
β∈I
a,I Pkl (Q) =
n
a,I Pklr (Q).
(5)
r=1
The w ∗ -convergence of the sum in the first equation is shown in [6, Lemma 4.4]. When there a,I a,I a,I a,I and Pkl in place of Pklr (Q) and Pkl (Q), respectively. is no fear of confusion, we write Pklr Now we are prepared to state the following factorization theorem necessary for this paper. Theorem 3.1. (i) Given Q ∈ VN(G × H ), a ∈ C(b(G)) and I ∈ a, we have n n
a,I ξkla · Pkl =
k=1 l=1
n n
a,I ξkla Pkl = Q.
k=1 l=1
(ii) If b ∈ C(b(G)) is such that there exists J ∈ b with J ∩ I = ∅, then ξklb Pka,I
l = 0
k, k , l, l ∈ {1, . . . , n} .
Proof. Statement (i) for the case that I = b(G) was proved in [6, Lemma 4.5]. However an inspection of the proof of that lemma shows that b(G) can be replaced with any cofinal set I in b(G), and hence (i) follows by essentially the same proof which we do not repeat for briefness. (ii) We show that the action ξklb Pka,I
l r on every element of A(G × H ) is zero. Let v ∈ A(G) and w ∈ A(H ). Since G is compact, we may assume that for s ∈ G, v(s) = vij (s) = dim V = d, and {e1 , . . . , ed } is the standard basis of Cd . Let (QF ), (V (s)e j |ei ), where V ∈ G, i i QF = i∈F Q1 ⊗ Q2 (where F is a finite subset of N), be a net in VN(G × H ) converging in w ∗ -topology to Q. Then, for every 1 r n , following the calculations leading to Eq. (33) in [6], we have
ξklb Pka,I
l r , vij ⊗ w =
β 1 β u˜ β · Q u˜ β · P β , (vij ⊗ w)˜uαkl u˜ k r · P lim r1 1l 6 n α∈J β∈I
=
β 1 β u˜ β · Qi ⊗ Qi u˜ β · P β , (vij ⊗ w)˜uαkl u˜ k r · P lim lim 1 2 r1 1l 6 F n α∈J β∈I
=
i∈F
β β β
1 u˜ k r · Pβ u˜ r1 · Qi1 u˜ 1l · Pβ , vij u˜ αkl Qi2 , w lim lim 6 F n α∈J β∈I
i∈F
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=
d n n d β
1 u˜ r1 · Qi1 , u˜ αqq vpp Qi2 , w lim lim 6 F n α∈J
×
p=1 q=1 p =1 q =1 β∈I
i∈F
β
u˜ k r (t)u˜ αkq (t)vip (t) dλβ (t)
Nβ
β
u˜ 1l (x)u˜ αq l (x)vp j (x) dλβ (x).
Nβ
However by Lemma 2.1, for large enough α,
β
u˜ 1l (x)u˜ αq l (x)vp j (x) dλβ (x) = 0. Nβ
Hence, it follows that ξklb Pka,I
l r = 0 for every r, and therefore the lemma follows.
2
The next result concerns the right cancellable sets in VN(G × H )∗ and UC2 (G × H )∗ . It can be considered as the dual version of the results in [7,8] concerning the right cancellable points in L1 (G)∗∗ and LUC(G)∗ . We call a set X in an algebra A right cancellable if for every nonzero a ∈ A, ab = 0 for some b ∈ X. When X = {b} is a singleton, then the set X is right cancellable if and only if the right multiplication operation by b is injective. The matrix ( ξkla ) may not be right cancellable in the matrix algebra Mn (VN(G × H )∗ ). Similarly, the entries ξkla may not be right cancellable points in VN(G × H )∗ . However, we have the following result. Corollary 3.2. Let a ∈ C(b(G)). For Ψ ∈ VN(G × H )∗ , let [Ψ ] be the n × n diagonal matrix whose diagonal entries are equal to Ψ . Then (i) the linear map VN(G × H )∗ −→ Mn VN(G × H )∗ , a ξkla n×n Ψ −→ [Ψ ] ξkl n×n = Ψ 2 is injective; (ii) { ξkla : k, l = 1, . . . , n} is a right cancellable set in VN(G × H )∗ ; (iii) {ξkla : k, l = 1, . . . , n} is a right cancellable set in UC2 (G × H )∗ . Proof. (i) The map is clearly linear. Let Ψ ∈ VN(G × H )∗ be nonzero and pick Q ∈ VN(G × H ) a,I ξkla · Pkl , and therefore such that Ψ, Q = 0. Let I ∈ a. Then by Theorem 3.1 , Q = nk=1 nl=1 n n
Ψ
k=1 l=1
a,I 2 ξkla , Pkl =
Ψ,
n n
a,I ξkla · Pkl
= Ψ, Q = 0.
k=1 l=1
This shows that Ψ 2 ξkla must be nonzero for some k, l ∈ {1, . . . , n}. (ii) This statement is an immediate consequence of (i).
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(iii) Let η ∈ UC2 (G × H )∗ be nonzero and pick v · Q ∈ UC2 (G × H ) such that η, v · Q = 0. Arguing as in (i) and using the identities Q=
n n
a,I a,I = v · ξkla Pkl , ξkla v · Pkl
a,I ξkla Pkl ,
k=1 l=1
we can write n n
η 2 ξkla , v
a,I = · Pkl
η, v ·
k=1 l=1
n n
ξkla
a,I Pkl
= η, v · Q = 0.
k=1 l=1
Thus η 2 ξkla must be nonzero for some k, l ∈ {1, . . . , n}.
2
We end this section with an interesting result which shows that each column of the matrix (ξkla )n×n gives rise to an injective map C b(G) −→ Mn×1 UC2 (G × H )∗ ,
a −→ ξkla 1kn .
Theorem 3.3. Let a1 and a2 be two distinct elements of C(b(G)). Then given any 1 l n, there exists 1 k n such that ξkla1 = ξka 2l , for all k , l ∈ {1, 2, . . . , n}. Proof. Let I1 and I2 be two disjoint subsets of b(G) such that a1 ∈ I1 and a2 ∈ I2 , the closure being taken in β(b(G)). Consider the operator Pkla1 ,I1 =
β u˜ kl · P β
∗ w -limit .
(6)
β∈I1
It follows from [6, Lemmas 3.7, 4.2, 4.3] that Pkla1 ,I1 ∈ VN(G × H ). To prove our result it suffices to show that given l ∈ {1, . . . , n}, we have n
ξkla1 · Pkla1 ,I1 =
k=1
n k=1
ξka 2l Pkla1 ,I1 = 0
ξkla1 Pkla1 ,I1 = n2 I ⊗ I,
(7)
k, k , l ∈ {1, . . . , n} ,
(8)
where I ⊗ I is the identity operator in VN(G × H ). Since then, there must exist some 1 k n such that ξkla1 Pkla1 ,I1 = 0, while simultaneously, ξka 2l Pkla1 ,I1 = 0 for all k , l , proving that ξkla1 = ξka 2l . Write ξkla1 = limα∈I1 u˜ αkl . Then ξkla1 Pkla1 ,I1 = w ∗ - lim u˜ αkl · Pkla1 ,I1 = w ∗ - lim α∈I1
= w ∗ - lim
α∈I1
α∈I1
β∈I1
β u˜ αkl u˜ kl · Pβ
⊗ I.
β∈I1
β β u˜ αkl · u˜ kl · P
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Now given vij ⊗ w ∈ A(G × H ), where as in the proof of Theorem 3.1, vij (·) = (V (·)ej |ei ) for we have some V ∈ G,
a1 β (9) u˜ αkl u˜ kl · Pβ , vij I, w. ξkl Pkla1 ,I1 , vij ⊗ w = lim α∈I1
β∈I1
α . We can Without loss of generality we can assume that α is large enough that so that V , V ∈ G then write
β β β Pβ , vij u˜ αkl u˜ kl = u˜ αkl u˜ kl · Pβ , vij = Pβ vij u˜ αkl u˜ kl ˇ(e) β∈I1
β∈I1
β∈I1
β = vij u˜ αkl u˜ kl ˇ t −1 e dλβ (t) β∈I1 N
=
β∈I1 N
(Lemma 2.1)
β
=
β
u˜ αkl (t)u˜ kl (t)vij (t) dλβ (t)
β
u˜ αkl (t)u˜ αkl (t)vij (t) dλα (t)
(10)
Nα
= u˜ αkl u˜ αkl · Pα , vij .
(11)
α that for each k = n + 1, . . . , n , we have It follows from our choice of the representations U α u˜ kl (t) = 0 if t ∈ Nα (see Eq. (4) of [6]). Thus, it follows from (10) and (11) that
n
n α α α α u˜ kl u˜ kl · Pα , vij = u˜ kl u˜ kl · Pα , vij . k=1
k=1
α is a unitary representation, we have Consequently using (9) and (11) and the fact that U n
n
a1 a1 ,I1 α α ξkl Pkl , vij ⊗ w = lim u˜ kl u˜ kl · Pα , vij I, w α∈I1
k=1
= lim
α∈I1
k=1
n
u˜ αkl u˜ αkl
· Pα , vij I, w
k=1
= lim n2 Pα , vij I, w α∈I1
by (2)
= n2 I, w lim Pα (vij )ˇ(e) α∈I1 = n2 I, w lim vij (t) dλα (t) α∈I1 Nα
= n2 I, wvij (e) = n2 I ⊗ I, vij ⊗ w, proving (7).
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Next let ξka 2l = limα∈I2 u˜ αk l . Then following the calculations leading to (9), we have β ξka 2l Pkla1 ,I1 , vij ⊗ w = lim u˜ αk l u˜ kl · Pβ , vij I, w.
α∈I2
(12)
β∈I1
Now repeating the calculations leading to (10), and applying Lemma 2.1, we obtain
β u˜ αk l u˜ kl · Pβ , vij =
β∈I1
β∈I1 N
β
u˜ αk l (t)u˜ kl (t)vij (t) dλβ (t) = 0,
(13)
β
α . Our claim in (8) follows from (13) and (12). if α is large enough so that V , V ∈ G
2
Following (3), Corollary 3.2, and Theorem 3.3, we can state Corollary 3.4. Let A denote either of the algebras VN(G × H ) or UC2 (G × H ). Then (i) the number of matrices (ξkl )n×n in the matrix algebra Mn (A∗ ), for which the map Ψ −→ b(G) (Ψ 2 ξkl )n×n , A∗ −→ Mn (A∗ ) is injective, is at least 22 ; b(G) (ii) the number of sets in A∗ which are right cancellable in A∗ is at least 22 . 4. Number and dimension of ideals in A(G × H )∗∗ and UC2 (G × H )∗ We keep the notation introduced in the previous two sections. It is shown by Filali and Salmi [8, Theorem 5] that for a noncompact, locally compact κ(K) group K, there exist 22 left ideals in L1 (K)∗∗ and in LUC(K)∗ with trivial pairwise intersections, and that the dimension of every nontrivial right ideal in L1 (K)∗∗ and in LUC(K)∗ is κ(K) at least 22 [8, Theorem 6] (see also [7]). In this section we prove the dual of these results for the Banach algebras UC2 (G × H )∗ and A(G × H )∗∗ . We start by determining the dimension of right ideals. Lemma 4.1. Let Φ be a nonzero element of VN(G × H )∗ , and [Φ] be the n × n diagonal matrix with Φ on the diagonal. Then, the dimensions of the right ideals generated by [Φ] in the matrix b(G) algebras Mn (VN(G × H )∗ ) and Mn (UC2 (G × H )∗ ) are at least 22 . Proof. We give the proof for the ideal generated in Mn (UC2 (G × H )∗ ); the proof of the other case is similar. By (3), it suffices to show that the set a [Φ] ξkl : a ∈ C b(G) is linearly independent. Let Q ∈ VN(G × H ) be such that Φ, Q = 1. Suppose that for some scalars λi and distinct cofinal ultrafilters ai , i = 1, . . . , m, we have m i=1
λi [Φ] ξklai = 0.
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It follows that m
a
λi Φ ξkli = 0 for all 1 k, l n.
i=1
Let Ii , i = 1, . . . , m, be pairwise disjoint members of ai . Let 1 j m be fixed, and consider a ,I the operators Pklj j . Then by Theorem 3.1, 0=
n m
a ,I a λi Φ ξkli , Pklj j
k,l=1 i=1
=
n m
a ,I λi Φ, ξklai Pklj j
k,l=1 i=1
= λj Φ,
n
a ξklj
a ,I Pklj j
k,l=1
= λj Φ, Q = λj , as required.
2
We omit the proof of the following easy lemma. Lemma 4.2. Let X, Y , Z be vector spaces over C and X = Y × Z. If dim X κ for an infinite cardinal κ, then dim Y κ or dim Z κ. Theorem 4.3. Every nontrivial right ideal in A(G × H )∗∗ and in UC2 (G × H )∗ has dimension b(G) at least 22 . Proof. We give the proof for right ideals in UC2 (G × H )∗ . Let R be a nontrivial right ideal in UC2 (G × H )∗ and pick any nonzero element Φ in R. It follows from Lemmas 4.1 and 4.2 b(G) that there exist k, l ∈ {1, 2, . . . , n} and a subset S of C(b(G)) with cardinality 22 such that {Φ ξkla : a ∈ S} is a linearly independent subset of R. 2 As a corollary of Theorem 4.3 it follows immediately that both A(G×H )∗∗ and UC2 (G×H )∗ b(G) have dimensions greater than or equal to 22 . Next we turn our attention to left ideals. Nontrivial left ideals in A(G × H )∗∗ and in UC2 (G × H )∗ can be of finite dimensions of course, as one can easily generate such ideals using topological invariant means (TIMs). More generally, if K is a nondiscrete locally compact b(K) group, then by a result of Hu [12, Theorem 5.9], there are 22 TIMs in A(K)∗∗ (for the metrizable case see Chou [2, Theorem 3.3]). Now if Ψ is a TIM, then for every Φ ∈ A(K)∗∗ and every Q ∈ VN(K), we have
Φ 2 Ψ, Q = Φ, Ψ · Q = Φ, Ψ, QI = Φ(I)Ψ, Q,
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where I is the identity operator in VN(K). Thus, Φ 2Ψ = Φ(I)Ψ . In other words, A(K)∗∗ 2Ψ = b(K) CΨ , and therefore there are at least 22 left ideals of dimension one in A(K)∗∗ . a In the next theorem, using the set {ξkl : 1 k, l n, a ∈ C(b(G))}, we show that left ideals of b(G) can be produced in both A(G × H )∗∗ and UC2 (G × H )∗ . infinite dimensions up to at least 22 b(G)
Theorem 4.4. There exists at least 22 b(G) of dimension at least 22 .
left ideals in A(G × H )∗∗ and in UC2 (G × H )∗ , each
Proof. Let a1 , a2 be two distinct elements of C(b(G)) and let I1 and I2 be two disjoint subsets of b(G) such that a1 ∈ I1 and a2 ∈ I2 . Consider the two closed left ideals J1 and J2 in VN(G × H )∗ , generated, respectively, by the sets VN(G × H )∗ ξkla1 : 1 k, l n ,
VN(G × H )∗ ξkla2 : 1 k, l n .
Let Ψ ∈ VN(G × H )∗ be nonzero and pick Q ∈ VN(G × H ) such that Ψ, Q = 0. It follows from Theorem 3.1 that
n n n n
a1 a1 ,I1 a1 a1 ,I1 Ψ ξkl , Pkl = Ψ, ξkl Pkl = Ψ, Q = 0, (14) k=1 l=1
k=1 l=1
while for all Φ ∈ VN(G × H )∗ and all 1 k, k , l, l n,
1 1 Φ ξkla2 , Pka 1l,I = Φ, ξkla2 Pka 1l,I = Φ, 0 = 0.
(15)
Therefore, for some 1 k0 , l0 n, we have Ψ ξka01l0 , Pka01l,I0 1 = 0, and thus Ψ ξka01l0 ∈ / J2 , b(G)
showing that J1 ⊂ J2 . Similarly one can show that J2 ⊂ J1 . Since there are 22 cofinal ultrab(G) filters in β(b(G)) it follows that VN(G × H )∗ has at least 22 closed left ideals. Similarly, if η ∈ UC2 (G × H )∗ is nonzero, pick Q ∈ VN(G × H ) and v ∈ A(G × H ) such that η, v · Q = 0. Then, once again, n n
η 2 ξkla1 , v
a1 ,I1 = · Pkl
η, v ·
k=1 l=1
n n
ξkla1
a1 ,I1 Pkl
= η, v · Q = 0,
k=1 l=1
while for all η ∈ UC2 (G × H )∗ and all 1 k, k , l, l n,
1 1 η 2 ξkla2 , v · Pka 1l,I = η , v · ξkla2 Pka 1l,I = η , 0 = 0.
As in the case of VN(G × H )∗ , it follows that the two closed ideals in UC2 (G × H )∗ generated, respectively, by the sets UC2 (G × H )∗ 2 ξkla1 : 1 k, l n b(G)
are distinct. Since there are 22 b(G) has 22 closed left ideals.
and UC2 (G × H )∗ 2 ξkla2 : 1 k, l n ,
cofinal ultrafilters in β(b(G)), this shows that UC2 (G × H )∗
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Next we show that the dimension of each of these left ideals is at least 22 cofinal ultrafilter on b(G). By Corollary 3.2(i), the map VN(G × H )∗ −→ Mn VN(G × H )∗ ,
. Let a be a fixed
Φ −→ [Φ] ξkla ,
is a linear isomorphism onto its image. Therefore, by Theorem 4.3, the dimension of the linear b(G) subspace {[Φ](ξkla ): Φ ∈ VN(G × H )∗ } of Mn (VN(G × H )∗ ) is at least 22 . Therefore by Lemma 4.2, for at least one pair 1 k, l n, the linear space {Φ ξkla : Φ ∈ VN(G × H )∗ } is of b(G) dimension at least 22 . A fortiori the dimension of each of left ideal VN(G × H )∗ ξkla : 1 k, l n b(G)
is at least 22 . The case of the left ideals in UC2 (G × H )∗ can be argued in a similar way.
2
Our next result shows that we can locate the left ideals of Theorem 4.4 within VN(G × H )∗ and UC2 (G × H )∗ . Suppose that PF 2 (G × H ) is the norm closed subspace of VN(G × H ), generated by the operators λ2 (f ⊗ g), where f ⊗ g ∈ L1 (G × H ), and λ2 is the left regular representation of L1 (G × H ) on L2 (G × H ). Theorem 4.5. Let a ∈ C(b(G)). Then the left ideals UC2 (G × H )∗ 2 ξkla : 1 k, l n ,
VN(G × H )∗ ξkla : 1 k, l n
are contained in the polar of PF 2 (G × H ) in the spaces UC2 (G × H )∗ and VN(G × H )∗ , respectively. Proof. Consider the first ideal. We need only to check that ξkla annihilates PF 2 (G × H ), since it can be easily checked that the polar of PF 2 (G × H ) in UC2 (G × H ), that is, PF 2 (G × H )◦ , is a w ∗ -closed left ideal in UC2 (G × H )∗ (that the polar is in fact a two sided ideal is proved in [9,4]). So let f ⊗ g ∈ L1 (G × H ) = L1 (G) ⊗γ L1 (H ). Since C(G) ⊗ C(H ) is dense in L1 (G) ⊗ L1 (H ), which itself is dense in the projective tensor product L1 (G) ⊗γ L1 (H ), we may suppose that both f and g are continuous functions. Since
ξkla , λ2 (f ⊗ g) = lim u˜ αkl ⊗ 1H , λ2 (f ) ⊗ λ2 (g) α = lim u˜ αkl (x)f (x) dx g(y) dy, α
G
H
it suffices to show that the limit of the first integral is equal to zero. By Hewitt and Ross [10, Theorem 27.39(ii)], we may assume that φ is in fact a trigonometric polynomial. Without loss of generality we may assume further that φ is a coordinate function of an irreducible unitary representation Vγ of G. Then we may choose α large enough so that α Vγ (since by [6, Lemma 3.3(iii)] we can ensure the two representations have inequivalent U
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restrictions to Nα ). By the orthogonality relations between coordinate functions of irreducible unitary representations we have u˜ αkl (x)φ(x) dx = 0,
lim α
G
as we wanted to show. Our claim for the left ideals VN(G × H )∗ {ξkla : 1 k, l n} also follows from the above proved fact that ξkla annihilates PF 2 (G × H ), since once again,
Ψ ∈ VN(G × H )∗ : Ψ |PF2 (G×H ) = 0
is a w ∗ -closed left ideal in VN(G × H )∗ .
2
Remark 4.6. It follows from the above theorem that the left ideals VN(G × H )∗ {ξkla : 1 k, l n} have trivial intersection with the image of A(G × H ) in VN(G × H )∗ , since nonzero elements of A(G × H ) do not annihilate PF 2 (G × H ). Let T be a linearly independent set of TIMs of cardinality κ in VN(G × H )∗ . Then lin VN(G × H )∗ 2 T = lin(T ) (closure in the norm topology) is a left ideal in VN(G×H )∗ of dimension at least κ. Each element of this left ideal is clearly topologically invariant. We check that these left ideals are different from those in VN(G × H )∗ , given in Theorem 4.5. Let u ∈ A(G × H ) be a nonzero function such that u(e) = 0 (where e denotes the identity of G × H ), then u 2 lin(T ) = {0}, while by Corollary 3.2, u2 ξkla is nonzero for some 1 k, l n, and therefore u2VN(G×H ) {ξkla : 1 k, l n} is nonzero. Note that if u(e) = 1, then the left ideal lin(T ) is strictly contained in the two sided ideal u 2 A(G × H )∗∗ . By our Theorem 4.3, these latter ideals have dimensions greater than or equal b(G) to 22 . As the last result in this paper, we give another application of our factorization Theorem 3.1. Using the cluster points ξkla we give an alternative proof for the fact that VN(G × H )∗ has at least b(G) linearly independent topological invariant means (TIMs). For a different and more general 22 approach to this problem see Chou [2] and Hu [12]. We thank Z. Hu for kindly pointing out to us that a careful analysis of the main results in the last two references does indeed show that TIMs constructed in these papers are linearly independent. We start with the following lemma. Lemma 4.7. Let Ψ be a TIM, a ∈ C(b(G)), and I ∈ a. Let I ⊗ I be the identity operator in a,I VN(G × H ) and Pkl (I ⊗ I) be defined as in (5). If 1 k, l n, and k = l, then a,I (I ⊗ I) = 0. Ψ, ξkla Pkl
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Proof. Recall that by (4),
a,I Pkl (I ⊗ I) =
n
a,I Pklr (I ⊗ I);
r=1
therefore it suffices to show that for every 1 r n , a,I (I ⊗ I) = 0. Ψ, ξkla Pklr
If r = 1, then a,I Pklr (I ⊗ I) =
β 1 β β β u˜ kr · Pβ u˜ r1 · I ⊗ I u˜ 1l · P n6 β∈I
β 1 β β β = 0, = 6 u˜ kr · Pβ u˜ r1 (e)I ⊗ I u˜ 1l · P n
(16)
β∈I
β β (e) is the n × n identity matrix. since u˜ r1 (e) = 0, as U β So it remains to consider the case r = 1. Then u˜ 11 (e) = 1, and therefore we have
β 1
a,I β (I ⊗ I) u˜ β · P β Ψ, ξkla Pkl1 Ψ, ξkla u˜ k1 · P (I ⊗ I) = 6 1l n
β∈I
β 1
β Ψ, ξkla u˜ kl · P = 6 n β∈I
1
β β Ψ, u˜ kl · ξkla P = 6 n β∈I
β
= u˜ kl (e)
1
β = 0, Ψ, ξkla P 6 n β∈I
where to obtain the second identity, we used the relation β β u˜ β · P β = u˜ β · P β u˜ k1 · P kl 1l proved in [6, Lemma 4.2], and in the last line of the calculations, we have used the facts that Ψ β is a TIM and u˜ kl (e) = 0. 2 b(G)
Theorem 4.8. There are at least 22
linearly independent TIMs in VN(G × H )∗ .
α are not normalized, Proof. For the purpose of this proof we assume that our representations U α so that u˜ kl have norm less than or equal to 1 in A(G × H ) (with this assumption, the factorization formula in Theorem 3.1(i) will be affected by a multiplicative constant on the right-hand side, which is irrelevant for our purposes). Let Ψ be a TIM in VN(G × H )∗ , whose existence was
M. Filali et al. / Journal of Functional Analysis 258 (2010) 3117–3133
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shown, for example, in [18]. If a is a cofinal ultrafilter and 1 k n, then we show below that a is also a TIM. In fact, for every u ⊗ v ∈ A(G × H ), and every Q ∈ VN(G × H ), Ψ ξkk
a a Ψ ξkk , (u ⊗ v) · Q = Ψ, ξkk (u ⊗ v) · T
a Q = Ψ, (u ⊗ v) · ξkk
a ,Q . = I ⊗ I, u ⊗ v Ψ ξkk
Furthermore, using the fact that Ψ (I ⊗ I) = 1, we have
a a Ψ ξkk , I ⊗ I = Ψ, ξkk · (I ⊗ I) a (I ⊗ I)Ψ (I ⊗ I) = ξkk
= lim u˜ αkk , I ⊗ I α
= lim u˜ αkk ⊗ 1H , I ⊗ I α
= 1H , I lim u˜ αkk (e) = 1. α
a belongs to the closed unit ball of Since ˜uαkk = u˜ αkk 1 for all α, it follows that ξkk a a a is ∗ UC2 (G × H ) , and hence Ψ ξkk Ψ ξkk 1. Thus we have shown that Ψ ξkk a TIM. Next, let a1 and a2 be distinct cofinal ultrafilters. We take disjoint subsets I1 and I2 of b(G) a1 ,I1 a1 ,I1 such that a1 ∈ I1 and a2 ∈ I2 . Define the operators Pkl = Pkl (I ⊗ I) as in (5). Then by Theorem 3.1, for all 1 k, l, k , l n, we have
I⊗I=
n n
a1 ,I1 ξkla1 Pkl
1 but ξkla2 Pka 1l,I = 0,
k=1 l=1
α , the first equality holds up a nonzero multiplicative where, by our current assumptions on U constant. Now using Lemma 4.7, we can write 1 = Ψ, I ⊗ I = Ψ,
n n
ξkla1
a1 ,I1 Pkl
k=1 l=1
=
n
n
a1 a1 ,I1 a1 a1 ,I1 Ψ, ξkk = Ψ ξkk . Pkk , Pkk
k=1
k=1
It follows that for at least one k0 , we have 0. Ψ ξka01k0 , Pka01k,I01 =
(17)
On the other hand, by Theorem 3.1(ii), for all 1 k, l, k , l n,
1 1 Ψ ξkla2 , Pka 1l,I = Ψ, ξkla2 Pka 1l,I = Ψ, 0 = 0.
(18)
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Thus Ψ ξka01k0 = Ψ ξkla2 for all 1 k, l n. We can repeat the same argument for any other a ∈ C(b(G)) such that a = a1 . Since there are 22 cofinal ultrafilters, we can find a subset C of C(b(G)), with the same cardinality, and an index 1 k1 n, such that Ψ ξka11k1 = Ψ ξkla for all 1 k, l n, and all a ∈ C . Replacing C(b(G)) with C , a transfinite induction will complete the proof. It remains to show that the TIMs we obtained above, are linearly independent. Suppose that b(G)
n j =1
a
λj Ψ ξkjjkj = 0,
where λ1 , . . . , λn are complex numbers. For each 1 j n, let Ij ∈ aj be such that Ii ∩ Ij = ∅ a ,I if i = j . Then, for a given i, we may act both sides of the above identity on the operator Pkiiki i =
Pkaiik,Ii i (I ⊗ I), and use (18) to obtain 0=
n j =1
a λj Ψ ξkjjkj , Pkaiik,Ii i = λi Ψ ξkaiiki , Pkaiik,Ii i .
Since by (17), Ψ ξkaiiki , Pkaiik,Ii i = 0, it follows that λi = 0, which is what we wanted to show. 2 Acknowledgments We would like to thank Zhiguo Hu and the anonymous referee for their valuable comments and suggestions. The first author wishes to thank Pekka Salmi for comments and useful discussions. References [1] C. Cecchini, A. Zappa, Some results on the center of an algebra of operators on VN(G) for the Heisenberg group, Canad. J. Math. 33 (1981) 1469–1486. [2] C. Chou, Topological invariant means on the von Neumann algebra VN(G), Trans. Amer. Math. Soc. 273 (1982) 207–229. [3] P. Civin, B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961) 847– 870. [4] A. Derighetti, M. Filali, M. Sangani Monfared, On the ideal structure of some Banach algebras related to convolution operators on Lp (G), J. Funct. Anal. 215 (2004) 341–365. [5] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. [6] M. Filali, M. Neufang, M. Sangani Monfared, Weak factorization of operators in the group von Neumann algebra of certain amenable groups and applications, preprint, 2009. [7] M. Filali, J.S. Pym, Right cancellation in LUC-compactification of a locally compact group, Bull. Lond. Math. Soc. 35 (2003) 128–134. [8] M. Filali, P. Salmi, One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras, J. Lond. Math. Soc. 75 (2007) 35–46. [9] F. Ghahramani, A.T. Lau, V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990) 273–283. [10] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 2, Springer-Verlag, New York, 1970. ˇ [11] N. Hindman, D. Strauss, Algebra in the Stone–Cech Compactification, Theory and Applications, Walter de Gruyter, New York, 1998.
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[12] Z. Hu, On the set of topologically invariant means on the von Neumann algebra VN(G), Illinois J. Math. 39 (1995) 463–490. [13] A.T.-M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986) 273–283. [14] A.T.-M. Lau, Uniformly continuous functionals on Banach algebras, Colloq. Math. 51 (1987) 195–205. [15] A.T.-M. Lau, V. Losert, The C ∗ -algebra generated by the operators with compact support on a locally compact group, J. Funct. Anal. 112 (1993) 1–30. [16] V. Losert, Further results on the centre of the bidual of Fourier algebras, Lecture at the Conference “Harmonic Analysis, Operator Algebras and Representations”, CIRM, 08/11/2008. [17] M. Neufang, A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis, Arch. Math. 82 (2004) 164–171. [18] P.F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972) 285–291. [19] E.K. van Douwen, The number of cofinal ultrafilters, Topology Appl. 39 (1991) 61–63.
Journal of Functional Analysis 258 (2010) 3134–3164 www.elsevier.com/locate/jfa
Lower bounds for densities of Asian type stochastic differential equations Vlad Bally a , Arturo Kohatsu-Higa b,c,∗,1 a Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, University Paris-Est-Marne-la-Vallée,
Cité Descartes, 5, Bld Descartes, Champs-sur-Marne, F-77454 Marne-la Vallée Cedex 2, France b Osaka University, Graduate School of Engineering Sciences, Machikaneyama cho 1–3, Osaka 560-8531, Japan c Japan Science and Technology Agency, Japan
Received 29 August 2009; accepted 30 October 2009 Available online 18 November 2009 Communicated by Paul Malliavin
Abstract We obtain lower bounds for densities of solutions of certain hypoelliptic two-dimensional stochastic differential equations where one of the components is the Lebesgue integral of the other. These results are non-trivial extensions of previous work of the authors. In particular, these type of equations are linked to the so-called Asian option set-up. © 2009 Elsevier Inc. All rights reserved. Keywords: Lower bounds; Density function; Asian type sde’s; Malliavin calculus
1. Introduction We consider the bi-dimensional diffusion process solution of the equation t Xt1
=x +
t σ (Xs ) dWs +
1
0
t b1 (Xs ) ds,
Xt2
=x +
0
2
b2 (Xs ) ds,
t ∈ [0, T ].
0
* Corresponding author at: Osaka University, Graduate School of Engineering Sciences, Machikaneyama cho 1–3, Osaka 560-8531, Japan. E-mail addresses:
[email protected] (V. Bally),
[email protected] (A. Kohatsu-Higa). 1 Supported by grants of the Japanese government.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.027
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We assume that the coefficients σ, b1 and b2 are five times differentiable and have bounded derivatives but the functions themselves do not need to be bounded. Also notice that we have just one Brownian motion W driving the process X = (X 1 , X 2 ) so the ellipticity assumption fails at any point, and the strong Hörmander condition (based on the coefficients of the Brownian motion only) fails as well. But we assume that σ (x) = (σ (x), 0) and [σ , b](x) span R2 (here [.,.] denotes the Lie bracket) so that the weak Hörmander’s condition holds true. Therefore the law of XT (x) is absolutely continuous with respect to the Lebesgue measure and has a continuous density pT (x, y). Our aim is to give lower bounds for pT (x, y). Such lower bounds have already been obtained by C.L. Fefferman and A. Sanchez-Calle in [13] and [6], and by S. Kusuoka and D. Stroock in [8]. But the results in those papers do not cover the case of the weak Hörmander’s condition. On the same theme, S. Polidoro, A. Pascucci and U. Boscain in [12,11,4] give an analytical approach to the problem discussed here. The lower bounds in those articles are analogues with the ones obtained here. Their framework is a little bit more general because they consider a diffusion process which has the same structure as ours but they work in an arbitrary dimension (we think that our approach works also in arbitrary dimension but this would ask some technical effort). On the other hand the framework here is more general that the analytical approach because in these articles, the representative example is b2 (x) = x 1 (or either a polynomial) with σ bounded. Our approach is a probabilistic one based on the results in A. Kohatsu-Higa [7] and V. Bally [2]. Other methods can be found in P. Malliavin and E. Nualart [9] and F. Delarue and S. Menozzi [5]. We consider here the 2-dimensional case in order not to obscure the crucial arguments. We believe that similar arguments should be used in the general d-dimensional case. Also note that our elliptic hypothesis on σ will be only local (see (8)) while in the other articles mentioned above this is not the case. Let us consider a fixed control φ ∈ L2 ([0, T ]) and define the corresponding skeleton i xt = xti (φ) t xt1
=x +
t σ (xs )φs ds +
1
0
t
bˆ1 (xs ) ds,
xt2
=x + 2
0
b2 (xs ) ds,
(1)
0
where bˆ1 (x) = b1 (x) − 12 (σ ∂1 σ )(x). It is well known (see [1]) that the support of the diffusion process Xt is concentrated on the curves of the previous form. So a necessary condition for pT (x, y) > 0 is that there exists a control φ such that xT1 (φ) = y (this condition is not sufficient: see [1] and [3]). So in a first stage we assume that such a control exists and we prove that (under appropriate assumptions on σ and b2 , see (8), (9) and (24)) we have
T
pT (x, y) C1 exp −C2
φt2 Ht2 dt
,
0
where C1 and C2 are constants which depend on the bounds of the coefficients and H is an explicit function (see Theorem 14 and the comments right before Lemma 13). Once this result is proved a second problem appears: how to find the control φ and how T to compute 0 φt2 dt. We solve this problem in two specific situations. First of all we assume that |σ (z)| ε > 0 and |∂1 b2 (z)| ε > 0. We call this case the “elliptic” case because the first
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component satisfies an ellipticity condition (although the whole diffusion does not). Then we prove (see Theorem 17) that |y 2 − x 2 − b2 (x)T |2 |y 1 − x 1 |2 . + pT (x, y) C1 exp −C2 T + T T3 This estimate is in the good range: if we consider the case σ = 1, b1 = 0 and b2 (x) = x 1 then XT is a Gaussian vector and the density of the law of this vector has upper and lower bounds of this form. The estimates in [4,11,12] are of the same spirit. But in the estimates obtained under the strong Hörmander condition in [6,8,13] the term T −3 × |y 2 − x 2 − b2 (x)T |2 does not appear. As another representative example, we consider the “linear case”. That is, t Xt1
=x +
t σ¯ (Xs )Xs1 dWs
1
0
+
b¯1 (Xs )Xs1 ds,
0
t Xt2
=x + 2
b2 (Xs ) ds. 0
This framework is interesting in mathematical finance because it represents the model used for Asian type options. Under similar hypothesis we obtain a lower bound for the density (see Example 20). Note that, one can easily prove from the results in Yor [14] that in the particular case that σ¯ (x) = 2, b¯1 (x) = 2 and b2 (x) = x1 then limy 2 →∞ y 2 log(|y 2 |2 pT (x, y)) < 0. This result seems to indicate that the bound obtained here (see Example 20 and also [12]) is not optimal for this case. The result of M. Yor is far more exact but it uses strongly the particular structure of the example while here the spirit is to develop a somewhat general theory. Throughout the text we denote by | · | the Euclidean norm on the corresponding d-dimensional space Rd . Given two positive definite invertible matrices A and B, we say that A B if A − B is a positive definite matrix. B([a, b]) stands for the Borelian sets on the interval [a, b]. We use the following directional derivative notation ∂σ b = σ1 ∂1 b + σ2 ∂2 b for a 2-dimensional 2 2 2 2 vector function σ = (σ 1 , σ2 ) : R → R and a function b : R → R . Also we define Lb := 2 1 2 i i,j =1 σi σj ∂i ∂j b + i=1 b ∂i b. 2 2. Evolution sequences In this section we recall the framework and some results from [2]. We consider a probability space (Ω, F , P ) with a filtration Ft , t 0, and a one-dimensional Brownian motion Wt , t 0. In order to define an “elliptic evolution sequence” we consider the following objects. We give a time grid 0 = t0 < t1 < · · · < tN = T and we define δk = tk − tk−1 . We consider also a space grid yk ∈ Rd , k = 0, . . . , N. Moreover, we consider a sequence of positive definite and invertible deterministic (d × d)-dimensional matrices Mk , k = 1, . . . , N. To these matrices we associate the
norms |x|k = Mk−1 x, x, x ∈ Rd , and we fix a sequence of numbers Hk 1, k = 1, . . . , N, such that for k = 2, . . . , N |x|k Hk |x|k−1 .
(2)
Moreover we fix a sequence of numbers ρk ∈ (0, 1), k = 1, . . . , N, such that |yk − yk−1 |k
ρk . 4
(3)
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We consider now a sequence of d-dimensional random variables Fk , k = 0, . . . , N, such that Fk is Ftk measurable and for k = 1, . . . , N tk Fk = Fk−1 +
hk (s, ω) dWs + Rk .
(4)
tk−1
Here hk : [tk−1 , tk ] × Ω → R d , k = 1, . . . , N, are B([tk−1 , tk ]) × Ftk−1 measurable and tk tk |hk (s, ω)|2 ds] < ∞. So, conditionally to Ftk−1 , Jk := tk−1 hk (s, ω) dWs is a centered E[ tk−1 Gaussian random variable with covariance matrix tk C (Jk ) = ij
j
hik (s, ω)hk (s, ω) ds,
i, j ∈ {1, . . . , d}.
tk−1
Rk is a remainder which has to be small with respect to the Gaussian part. In order to express this assumption we need the following norms on the Wiener space. For a d-dimensional functional F : Ω → Rd which is m times differentiable in Malliavin sense (we refer to [10] and [2] for notations and definitions) we define m 1/p
Etk−1 F k,m,p = Etk−1 |F |p + i=1
2 s1 ,...,si F ds1 · · · dsi
i D
p/2 1/p .
[tk−1 ,tk ]i
Here Etk−1 designs the conditional expectation with respect to Ftk−1 . Notice that F k,m,p was denoted by F tk−1 ,δk ,m,p in [2]. Finally we define the sets for k = 1, . . . , N ρi Ak = ω ∈ Ω: |Fi−1 − yi |i , i = 1, . . . , k ∈ Ftk−1 . 2 In the following definition we will use two universal constants pd = 22(d+2) p ∗ (d),
and Cd = c∗ (d)μ(d + 1)(2π)d/2 43(d+3) (d + 1)d+3 . 3
Here p ∗ (d) and c∗ (d) are given in Proposition 3 of [2] and μ(d + 1) is given in Eq. (2) of [2]. They are universal constants depending on d only. In particular, that is the case of the constants Cd and pd . We assume without loss of generality that Cd 1. We are now able to give our definition. Definition 1. The sequence Fk , k = 0, . . . , N, is called an elliptic evolution sequence if there exist some constants ak 1, k = 0, . . . , N, such that for every k = 1, . . . , N and every ω ∈ Ak one has i) ii)
1 Mk , ak 2 −1/2
8(d+1)2 + 12 ak−1 (ρk +ρk−1 ) −1 M 8 R C a e . k d k k k,d+2,p
ak Mk C(Jk )
d
(5)
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In this case, we define N N N
2
1 1 1 Hk 1/2 + θ = ln 8 (2πd) + ln(ak ) + ln ak−1 . 2N N ρk 8dN k=1
k=2
(6)
k=2
A slight modification of the main result from [2] gives the result that we will use in this paper. For some details of this proof see Appendix A. Theorem 2. Let Fk , k = 0, . . . , N, be an elliptic evolution sequence such that the law of FN is absolutely continuous with respect to the Lebesgue measure on Rd and has a continuous density p. Then p(yN )
1 × e−N ×d×θ . √ 4(2π)d/2 det MN
(7)
3. General framework We consider the two-dimensional diffusion process t Xt = x +
t σ (Xs ) dWs +
0
b(Xs ) ds 0
with σ, b : R2 → R2 five times differentiable functions. And W is an one-dimensional Brownian motion. We denote by bˆ = (bˆ1 , bˆ2 ) = b − 12 ∂σ σ the drift in the equation of X when written with a Stratonovich integral and by L its infinitesimal generator. We will also use the skeleton associated to this equation. That is, for a control φ ∈ L2 [0, T ] we define xt = xt (φ) = (xt1 , xt2 ) as the solution of the equation t xt = x +
t σ (xs )φs ds +
0
ˆ s ) ds. b(x
0
We have to precise our hypothesis. The first one concerns the non-degeneracy. We assume that for a fixed control function φ, (H1 (φ))
i)
σ2 (x) = 0,
∀x ∈ R2 .
ii)
There exist a function εφ : [0, ∞) → (0, ∞) such that min σ1 (xt ), ∂σ b2 (xt ) > εφ (t), t 0.
(8)
Remark. Under the present setting condition ii) is implied by the weak Hörmander condition. From now until Section 4 we assume that the control function φ is fixed. We also assume that we have the following upper bounds. For a multi-index α = (α1 , . . . , αp ) ∈ {1, 2}p , p ∈ N, we denote its length by |α| = p and ∂α = ∂α1 · · · ∂αp . We assume that for a
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3139
fixed control function φ, there exist a function Cφ : [0, ∞) → (0, ∞) and a constant C∗ 1 such that for any ε ∈ [0, 1] and f = σ, b, ∂σ b, ∂σ σ, σ + ε∂σ b, Lb we have f (xt ) Cφ (t), ∀t 0, ii) ∂α f (x) C∗ , ∀x ∈ R2 , 1 |α| 5, i)
(H2 (φ))
iii)
Cφ (t) 1 εφ (t).
(9)
Notice that we assume that the derivatives of the coefficients are bounded but the coefficients themselves may have linear growth (the bound in (H2 (φ)).i) concerns xt , t 0 only). Note that the bounds like in (H2 (φ)).i) are also valid for the functions ∂σ b and σ + ε∂σ b uniformly for ε ∈ [0, 1] as well as its derivatives up to order 5 if we only assume |σ (xt )| + |b(xt )| Cφ (t). In fact, we have that ∂σ b(xt ) Cφ (t)C∗ , (σ + ε∂σ b)(xt ) Cφ (t)(1 + εC∗ ), ∂σ σ (xt ) Cφ (t)C∗ . We have preferred this presentation which may be redundant as far as bounds are concerned. But this setting avoids writing cumbersome constants. A similar remark is valid about the hypothesis |∂α ∂σ b(x)| C∗ in (H2 (φ)).ii). 3.1. Evolution sequence We start this subsection with a description of the main goal in the first part of the article. We consider a time grid 0 = t−1 = t0 < t1 < · · · < tN with tk+1 − tk 1 and we denote for k = 1, . . . , N δk = tk − tk−1 ,
Ck =
sup
tk−2 ttk
Cφ (t),
εk = min εφ (tk−1 ), εφ (tk−2 ) .
Note that due to (H2 (φ)) we have that εk 1 Ck . Moreover we consider a sequence of numbers ρk 1, k = 1, . . . , N, such that tk φt2 dt . ρk := max δk ,
(10)
tk−1
Define the following quantities
2
ak = 196 Ck εk−1 ,
3/2
Hk = 6Uk
Ck εk−1
and
Uk = max{δk , δk−1 }/ min{δk , δk−1 } for k = 1, . . . , N. We assume that there exists μ 1 such that μfk+1 fk μ−1 fk+1 for fk = εk , Ck and all k. Furthermore, we will prove in Lemma 10 that there exists a constant C = C(p, T , C∗ ) such that for u ∈ [tk−1 , tk ] and |X(ti−1 ) − x(ti−1 )|i ρ2i for all i = 1, . . . , k
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Xu − Xtk−1 k,4,p CCk ρk , Xu k,4,p CCk + |xtk−1 |. Let C ∗ ≡ C ∗ (μ) be a positive constant such that
145 −1 49 2 ∗ 2 2 ∗ − 2 (C (μ)) (μ +1) 2 C (μ) min CCd 196 e , ,1 . 9C∗ 49
Such a constant exists because x → xe 2 (1+μ is zero at x = 0. Moreover, denote
σ k = σ x(tk−1 ) ,
2 )2 x 2
(11)
is an increasing continuous function which
bk = ∂σ b x(tk−1 )
and ck = σ k + bk
δk 2
and we consider the matrices Nk =
δk
ck,1 ck,2
bk,1 √δk
12 δk √ bk,2 12
,
Mk = N k × N ∗k .
Finally we consider Fk := X(tk ) − x(tk ),
k = 1, . . . , N,
and we notice that, if x = X0 and y = x(tN ) then ptN (x, y) = pFN (0), where pFN is the density of FN . The goal in the first part of the article is to prove the following lower bound result for ptN (x, y). Theorem 3. We suppose that (Hi (φ)), i = 1, 2, hold and suppose that we have
148 . ρk C ∗ εk Ck−1 Then Fk : k = 1, . . . , N is an elliptic evolution sequence with parameters Mk , ak , Hk , ρk and consequently we have ptN (x, y)
1 × e−2N θ √ 8π det MN
with θ defined in (6). The proof is given through a sequence of lemmas. The basic decomposition that leads to (4) is given by the following lemma. We take yk = 0. In particular the relation (3) trivially holds true.
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Lemma 4. With the previous definitions, we have that (4) is satisfied with Fk − Fk−1 = Jk + Rk , where Jk = σ (Xtk−1 ) k + ∂σ b(Xtk−1 ) k , tk
k := W (tk ) − W (tk−1 ),
k :=
(tk − s) dWs
tk−1
and Rk :=
6
Rk,i
i=1
tk =
σ (Xs ) − σ (Xtk−1 ) dWs +
tk−1
tk
∂σ b(Xs ) − ∂σ b(Xtk−1 ) (tk − s) dWs
tk−1
tk
tk +
Lb(Xs )(tk − s) ds −
tk−1
tk −
1 σ (xs )φs + ∂σ σ (xs ) ds 2
tk−1
b(xs ) − b(xtk−1 ) ds + b(Xtk−1 ) − b(xtk−1 ) δk .
tk−1
Proof. It is enough to note that tk b(Xs ) ds tk−1
tk = b(Xtk−1 )δk +
b(Xs ) − b(Xtk−1 ) ds
tk−1
tk s = b(Xtk−1 )δk +
∂σ b(Xr ) dWr +
tk−1
tk−1
Lb(Xr ) dr ds
tk−1
tk = b(Xtk−1 )δk + tk−1
s
s ∂σ b(Xr )(tk − r) dWr + tk−1
Lb(Xr )(tk − r) dr,
(12)
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tk ∂σ b(Xr )(tk − r) dWr tk−1
tk = ∂σ b(Xtk−1 )
tk (tk − r) dWr +
tk−1
∂σ b(Xr ) − ∂σ b(Xtk−1 ) (tk − r) dWr .
tk−1
Furthermore, tk xtk − xtk−1 = b(xtk−1 )δk +
tk
1 σ (xs )φs − ∂σ σ (xs ) ds + b(xs ) − b(xtk−1 ) ds. 2
tk−1
(13)
tk−1
So we have the above decomposition if we note that Fk − Fk−1 = (Xtk − Xtk−1 ) − (xtk − xtk−1 ).
2
3.2. Covariance matrices For x, y ∈ R2 we define [x, y] = x, y ⊥ = x1 y2 − x2 y1 . Notice that [x, y] = 0, y = 0 implies that x and y are colinear. Also one has that [x, y] |x||y|. We denote for k = 1, . . . , N σk = σ (Xtk−1 ),
bk = ∂σ b(Xtk−1 ),
ck = σ k + bk
δk , 2
dk = [ck , bk ] = [σk , bk ]. The context will determine the difference between the vector valued random variables bk and σk as defined above and the functions bk and σk , k = 1, 2, used in (1). We will denote the components of the above random vector by fk = (fk,1 , fk,2 ) for f = b, c, d, σ . We compute for k = 1, . . . , N : 2 2 2 2 δk , i = 1, 2, Etk−1 Jk,i = δk ck,i + bk,i 12 δk2 . Etk−1 [Jk,1 × Jk,2 ] = δk ck,1 ck,2 + bk,1 bk,2 12 So the conditional covariance matrix of the random vector Jk is C(Jk ) = Nk × Nk∗ ,
V. Bally, A. Kohatsu-Higa / Journal of Functional Analysis 258 (2010) 3134–3164
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where Nk :=
δk
ck,1
bk,1 ×
ck,2
bk,2 ×
√δk 12 √δk 12
and Nk∗ denotes the transpose of Nk . We define now the deterministic matrix Mk which will control the covariance matrix C(Jk ). The idea is to replace the random variable Xtk−1 by the corresponding deterministic point xtk−1 . We denote for k = 1, . . . , N d k = [ck , bk ] = [σ k , bk ]. Notice that if d¯k = 0 then N¯ k is invertible and we have N −1 k =
√ 12 bk,2 × 3/2
√δk 12
−bk,1 ×
−ck,2
δk d k
√δk 12
ck,1
.
Note that under hypotheses (8) and (9), we have |bk | Ck , |σ k | Ck , |ck | Ck and εk2 |d k | |σ k ||bk | Ck2 . 3.3. Properties of the norm | · |k As stated in Section 2, we define for k = 1, . . . , N 2 |x|2k = Mk−1 x, x = N −1 k x . Therefore |x|2k
=
1
12 2 [x, bk ] + 2 [x, ck ] . δk 2
δk d 2k
Similarly
C(Jk )−1 x, x =
1 12 2 2 [x, b . ] + [x, c ] k k δk dk2 δk2
We will use the following two inequalities of norms. Lemma 5. We have for k = 1, . . . , N |x|2
2 δk √ |bk | + |ck | |x|2k δk , 12
(14)
and for x = (x1 , x2 ) |x|2k 24Ck2
√ |(x1 √δk , 2x2 )|2 2
δk3 d 2k
.
(15)
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Proof. Let α, β ∈ R2 . We solve the system of equations y1 = [x, α], y2 = [x, β] and we obtain x=
1 1 (y1 β1 − y2 α1 , y1 β2 − y2 α2 ) = (y1 β − y2 α). [β, α] [β, α]
It follows that [β, α]2 |x|2 = |y1 β|2 + |y2 α|2 − 2y1 y2 α, β |y1 β|2 + |y2 α|2 + 2|y1 β||y2 α|
2
2
= |y1 β| + |y2 α| |y|2 |β| + |α| . So we have |x|2 Using (16) with α = bk and β =
|x| 2
d 2k
(16)
√
12 δk c k
( √δk |bk | + |ck |)2 12
2 [x, α]2 + [x, β]2
|β| + |α| . [β, α]2 we obtain
2 δk 12 2 [x, bk ] + 2 [x, ck ] = √ |bk | + |ck | |x|2k δk . δk 12 2
Now we proceed with the proof of (15). Let vki = (ck,i , bk,i √δk ), i = 1, 2. Then clearly, 12
2 2 2 1 2 2 1 2 |x|2k = 12δk−3 d −2 . k x1 vk − 2x1 x2 vk , vk + x2 vk Then the result follows because (here, we use that σ2 ≡ 0) 2 2 2 v C 2 δk , k k 2 1 2 2 v 2C . k
And this yields the result.
k
2
3.4. Proof of the property (2) We start this subsection establishing a general matrix inequality that will help us prove the property (2). First, we give a preparatory lemma from linear algebra. Its proof is left to the reader. Lemma 6. Assume that for positive definite, symmetric and invertible d × d matrices A and B we have that x Ax x Bx for all x ∈ Rd . Then x A−1 x x B −1 x for all x ∈ Rd .
α γ Lemma 7. Assume that for the matrices Ii = 0i βii , i = 1, 2, there exist positive constants θ > 0, ν > 0 such that the following inequalities are satisfied
V. Bally, A. Kohatsu-Higa / Journal of Functional Analysis 258 (2010) 3134–3164
|α1 | θ |α2 |,
|β1 | θ |β2 |,
|γ1 | ν|β1 |,
|γ2 | ν|β2 |.
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Then |I1 x| θ (1 + 2ν)|I2 x| for all x ∈ R2 . Proof. We consider the auxiliary matrices Ji =
0 βi
αi 0
Ki =
,
0 0
γi 0
and we notice that |K1 x| = |γ1 x2 | νθ |β2 x2 | νθ |I2 x|
and |K2 x| = |γ2 x2 | ν|β2 x2 | ν|I2 x|.
Then
|I1 x| |J1 x| + |K1 x| θ |J2 x| + |K1 x| θ |I2 x| + |K2 x| + |K1 x| θ (1 + 2ν)|I2 x|.
2
Now we apply the above lemma to our particular case to obtain the property (2). Lemma 8. Assume hypotheses (H1 (φ)) and (H2 (φ)). Furthermore assume ρk2
2εk , 9C∗ Ck
(17)
where ρk is given by (10). Then |x|k Hk |x|k−1
(18)
with 3/2
Hk = 6Uk
Ck εk−1 ,
Uk = max{δk , δk−1 }/ min{δk , δk−1 }. Proof. First note that Lemma 6 gives us the following sequence of statements |N k−1 x| Hk |N k x|
⇒
Mk−1 x, x Hk2 Mk x, x
⇒
|x|k Hk |x|k−1 .
Furthermore, note that Nk =
δk I¯k Q,
where (here we use that σ2 ≡ 0) I¯k :=
σ k,1 0
b¯k,1 δk , b¯k,2 δk
Q=
1
0
1 2
√1 12
.
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Therefore (18) will follow if we prove that Uk |I¯k−1 x| Hk |I¯k x|. 1/2
(19)
To obtain this result, we will apply Lemma 7 with α1 = σ k−1,1 , β1 = b¯k−1,2 δk−1 , γ1 = b¯k−1,1 δk−1 and α2 = σ k,1 , β2 = b¯k,2 δk , γ2 = b¯k,1 δk . To prove the required inequalities in Lemma 7, we need to consider following properties. 1. From (13), we have |xtk − xtk−1 | 92 Ck ρk2 . In fact, |xtk − xtk−1 | tk tk
1 σ (xs )φs − ∂σ σ (xs ) ds + = b(xtk−1 )δk + b(xs ) − b(xtk−1 ) ds 2 tk−1
Ck
7 δk + 2
tk
tk−1
|φs | ds Ck
7 δ k + δk ρ k . 2
tk−1
2. Using (H2 (φ)) we have that for f· = σ¯ ·,i , b¯·,i , i = 1, 2, then 9 |fk − fk−1 | C∗ Ck ρk2 . 2 3. Similarly, from (H1 (φ)) and (H2 (φ)), we have for f· = σ¯ ·,1 , b¯·,2 , that |fk | εk and 9 fk−1 2 −1 f − 1 2 C∗ Ck ρk εk . k Using (17), we obtain that 9 2 −1 |fk−1 | 1 + C∗ Ck ρk εk |fk | 2|fk |. 2
Therefore, this yields |σ k−1,1 | 2|σ k,1 |, |b¯k−1,2 δk−1 | 2Uk |b¯k,2 δk |. Since |bk−1,2 | ∧ |bk,2 | εk and |bk−1,1 | ∧ |bk,1 | Ck we also obtain |b¯k−1,1 δk−1 | Ck εk−1 |b¯k−1,2 δk−1 |, |b¯k,1 δk | Ck εk−1 |b¯k,2 δk |. Therefore we apply Lemma 7 with θ = 2Uk and ν = Ck εk−1 . This gives
(20)
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|I¯k−1 x| 2Uk 1 + 2Ck εk−1 |I¯k x|. From here (19) follows and therefore we obtain (18).
2
3.5. Proof of property (5) In a similar fashion we now prove that (5), ak Mk C(Jk )
1 ak Mk ,
is satisfied. We recall that
ρi Ak = ω ∈ Ω: |Xti−1 − xti−1 |i , i = 1, . . . , k . 2 Lemma 9. Assume hypotheses (H1 (φ)) and (H2 (φ)) and (17). Then the matrix inequality ak Mk C(Jk ) a1k Mk , k = 1, . . . , N, is satisfied on Ak with
2
ak = 196 Ck εk−1 . Proof. In order to prove that C(Jk ) ak Mk , it is enough to prove that |Nk x|
√ ak |N¯ k x|.
As in the proof of Lemma 8, we will apply Lemma 7 with α1 = σk,1 , α2 = σ¯ k,1 , β1 = bk,2 , β2 = b¯k,2 , γ1 = bk,1 and γ2 = b¯k,1 . In order to prove the inequalities required in Lemma 7, we prove the following properties. 1. We prove first that |Xtk−1 − xtk−1 | Ck ρk2 for ω ∈ Ak . As ω ∈ Ak we know that |Xtk−1 − xtk−1 |k = |Fk−1 |k ρ2k and using (14) and (9) we have that |Xtk−1 − xtk−1 |
δk |b¯k | ρk 1/2 δ + |c¯k | √ 2 k 12
< Ck ρk2 .
(21)
2. Using (9), we have that |f¯k − fk | C∗ |Xtk−1 − xtk−1 | C∗ Ck ρk2 , for f· = b·,i , σ·,i , i = 1, 2. Furthermore, |fk | Ck (1 + C∗ Ck ρk2 ). 3. On Ak , we have that |σk,1 | |σ¯ k,1 | − C∗ Ck ρk2 εk − C∗ Ck ρk2 Since |bk,2 | εk we obtain 1 |bk,2 | εk − C∗ Ck ρk2 εk . 2 4. Let f· = σ·,1 , b·,2 , then fk − 1 C∗ Ck ρ 2 ε −1 2 k k f¯ 9 k
εk . 2
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or equivalently
11 |fk | 1 + C∗ Ck ρk2 εk−1 |f¯k | |f¯k |. 9 Using (17) and Ck εk−1 1, we have that the inequalities in Lemma 7 are satisfied with θ = 2 and ν = 3Ck εk−1 . Finally one has that
|Ik x| 2 1 + 6Ck εk−1 |I¯k x|. Although the constants change slightly, the other inequality is obtained exactly in the same way. We still have θ = 2 and ν = 3Ck εk−1 . 2 3.6. The remainder We will now give estimations for the remainder. Before doing so we give some estimates needed in the next lemma. The proofs are standard and can be obtained by suitable modifications of similar statements in [2] or [7]. In this subsection, C ≡ C(p, T , C∗ ) denotes a constant bigger than C∗ that depends exclusively on T , C∗ and an integer p 2 chosen independently of the other parameters. Lemma 10. Suppose that the hypotheses (H1 (φ)) and (H2 (φ)) hold true. For u ∈ [tk−1 , tk ] and ω ∈ Ak there exists a positive deterministic constant C ≡ C(p, T , C∗ ), which is increasing in T , such that Xu − Xtk−1 k,4,p CCk ρk , Xu k,4,p CCk + |xtk−1 |. Proof. It is enough to consider the following decomposition
Xt − xt = Xtk−1 − xtk−1 +
5
Ii (t)
i=1
with t I1 (t) =
t
σ (Xs ) − σ (xs ) dWs ,
I2 (t) =
tk
tk
t I3 (t) =
t σ (xs ) dWs ,
tk
b(Xs ) − b(xs ) ds,
I4 (t) =
t b(xs ) ds,
tk
I5 (t) = − tk
ˆ s ) ds. σ (xs )φs + b(x
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We have the following estimates 1 1 2−p
t
I1 (t)k,0,p C∗ δk
1/p p Xs − xs k,0,p ds
,
tk 1− 1 I2 (t)k,0,p C∗ δk p
t
1/p p Xs − xs k,0,p ds
tk
√ and Ii (t)k,0,p 2Ck ( δk + ρk ), i = 3, 4, 5. Since ω ∈ Ak we also have |Xtk−1 − xtk−1 | Ck ρk2 (see 1. in the proof of Lemma 9). We conclude that p Xt − xt k,0,p
C
p Ck (
t δk + ρk )p + (2C∗ )p
p Xs − xs k,0,p ds
tk
with C a universal constant√ which depends on p only. Then using Gronwall’s lemma we obp p tain Xt − xt k,0,p CCk ( δk + ρk )p where C is a constant that only depends on p, C∗ and (increasingly in ) T . Using (21) and (20), we obtain Xu − Xtk−1 k,0,p Xu − xu k,0,p + |xu − xtk−1 | + Ck ρk2 11 CCk ( δk + ρk ) + Ck ρk2 2 CCk ρk .
(22)
Similarly, Xu − Xtk−1 k,4,p CCk ρk , Xu k,4,p CCk + |xtk−1 |.
2
Lemma 11. Suppose that the hypotheses (H1 (φ)) and (H2 (φ)) hold true. For any integer p 2 there exists some universal constant C = C(C∗ , T , p), which is increasing in T , such that for ω ∈ Ak −1/2 M Rk k
C k,4,p
Ck3 ρk εk2
.
Proof. We use the definition for Rk in (12) and also Lemma 5 together with estimates of the conditional Sobolev norms of Rk,i . Recall that d k εk2 and ck,2 = bk,2 δ2k (because σ2 = 0). So for every random vector V = (V 1 , V 2 ) which is four times differentiable in Malliavin sense we have √ −1/2 3Ck 4 1 V 2 M V 1 . V + k k,4,p k,4,p k,4,p 1/2 δk δk εk2
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Using Lemma 10, (10), (20), (21) and (14), we obtain 1 R
k,1 k,4,p
2
i R
k,2 k,4,p
1 1/2 + Rk,4 CCk ρk δk k,4,p
i i
3/2 CCk ρk δk + Rk,5 + Rk,6 k,4,p k,4,p
i=1
i R
k,3 k,4,p
CCk2 δk2 .
2 = R 2 = 0 and so the result follows. Since σ2 = 0 we have Rk,1 k,4
2
The actual value of the constant C = C(C∗ , T , p) > 1 appearing in Lemma 11 may change from one line to the next, but it always remains a constant that depends only on C∗ , T and p. Due to the increasing property of C(C∗ , T , p) in T , we will consider in what follows a time T > T . So that, our estimates are not valid for times T going to infinity. Now we verify that condition ii) in Definition 1 is satisfied. That is, we prove Theorem 3. For the reader’s convenience, we restate it here. Theorem 12. Suppose that (H1 (φ)) and (H2 (φ)). Furthermore, assume that there exists a constant μ 1, independent of k such that μfk+1 fk μ−1 fk+1 for fk = εk , Ck and all k. There 145 49 ∗ 2 2 exists a positive constant C ∗ (μ) such that C ∗ (μ) min{(CCd 196 2 )−1 e− 2 (C (μ)) (μ +1) , 2 9C∗ , 1}
for which we assume 148 εk ρk C ∗ (μ) Ck
(23)
then Fn , n N is an elliptic evolution sequence with
2
ak = 196 Ck εk−1 and 3/2
Hk = 6Uk
Ck εk−1
with Uk = max{δk , δk−1 }/ min{δk , δk−1 } for k = 1, . . . , N where θ is defined as in (6). Furthermore, we obtain pT (x, y)
1 × e−2N θ . √ 8π det MN
Proof. The existence of the constant C ∗ (μ) follows from the increasing property of the function 2 2 f (x) = xe196(μ +1)x for x 0. Then for p = 148. Then we have
V. Bally, A. Kohatsu-Higa / Journal of Functional Analysis 258 (2010) 3134–3164
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1
2 49
2 1 1 12 −1 2 ak−1 (ρk + ρk−1 )2 = ak−1 Ck−1 εk−1 ρk + ak−1 ρk−1 ρk−1 + μ2 Ck εk−1 ρk 8 8 2
p−1
p−1 2 49 ∗ 2
−1 C (μ) Ck−1 εk−1 + μ2 Ck−1 εk . 2
2 49 C ∗ (μ)2 1 + μ2 . 2
Furthermore using Lemma 11, we have −1/2 M Rk k
k,4,p
C
Ck3 ρk εk2
p−3 εk 1 CC (μ) 49 ∗ 2 2 2 −1 2 145 Ck Cd (196(Ck εk ) ) 2 e 2 (C (μ)) (1+μ ) ∗
145 ak−1 (ρk +ρk−1 )2
−1 8 Cd a k 2 e ,
and therefore condition ii) in Definition 1 is satisfied and furthermore due to the choice of C ∗ (μ), we also have that Lemma 8 and Theorem 2 can be applied. 2 4. Construction of an evolution sequence associated to a given skeleton Up to now we supposed that the time grid tk , k ∈ N, was given. Now we will give a specific construction of the skeleton φ and the associated time grid. We introduce the following class of functions. For μ 1 and h∗ > 0, define the set of functions Λμ,h∗ as the functions f : (0, ∞) → R which verify f (s) μf (t)
if |t − s| h∗ .
The control φ to be used in the examples will belong to this class. This class is useful in order to simplify the lower bounds for the density of X. We make the following supplementary assumption. (H3 (φ)) We assume that εφ , Cφ , εφ Cφ−1 ∈ Λμ,h∗ for some μ 1 and h∗ > 0. Moreover we assume that there exists a control φ such that |φt | |φt | and φ ∈ Λμ,h∗ for some μ 1. We define εφ (t) 148 hφ (t) := min h∗ , C ∗ (μ) , Cφ (t)
(24)
where C ∗ is the universal constant given in (H2 (φ)).ii) and C∗ is given in (9). Note that under hypothesis (H3 (φ)), hφ ∈ Λμ∗ ,h∗ for μ∗ = μ296 and that due to (23) we have that hφ (t) < 1. This function will serve to define the grid. We construct now the time grid: we put t0 = 0 and if tk is given we denote hk = hφ (tk ) and we define tk+1 by
t
tk+1 = (tk + hk ) ∧ inf t > tk :
φ 2s ds tk
> hk .
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t t Notice that δk := tk+1 − tk hk and tkk+1 φs2 ds tkk+1 φ 2s ds hk , therefore ρk = hk (see (10)) which together with (24) will allow the application of Theorem 3. Notice also that we may choose φ = φ and this seems to be the natural choice, but in some cases it may be simpler to choose another φ for which μ is simpler to compute. For example if we know that |φ(t)| Q then we may take φ(t) = Q and then μ = 1. Lemma 13. Suppose that (H1 (φ)), (H2 (φ)) and (H3 (φ)) hold true. Then (Fk )k∈N is an elliptic evolution sequence. We define NT := sup{k: tk T }. Then NT μ
∗
T
1 + φ 2t dt. hφ (t)
(25)
0
Then for all non-negative integers a, b and c and any strictly increasing positive function f , we have N T −1 k=0
f
Cka
μ
εkb hck
∗
T
∗ c Cφ (t)a 1 + φ 2t a+b f μ dt. μ hφ (t) εφ (t)b hφ (t)
(26)
0
Furthermore, Uk μ∗ μ4 . Proof. Using the definition of NT , we have that T
T 1 + φ 2t dt hφ (t)
N
tk
k=1tk−1
0
1 + φ 2t dt. hφ (t)
(27)
Since hφ ∈ Λμ∗ ,h∗ and tk − tk−1 hk−1 h∗ we have tk tk−1
1 1 + φ 2t dt ∗ hφ (t) μ hk−1
tk
1 + φ 2t dt
tk−1
1 1 hk−1 = ∗ . μ∗ hk−1 μ
Applying this to (27), it follows that T
NT 1 + φ 2t dt ∗ hφ (t) μ
0
and (25) is proved. The proof of (26) is analogue. From here, it follows in particular that NT is finite. The estimate for Uk is a consequence of the following inequality: δk μ∗ μ4 δk−1 . In order to prove it, we assume first that δk−1 = hk−1 = hφ (tk−1 ). Since hφ (tk−1 ) (μ∗ )−1 hφ (tk ) = (μ∗ )−1 hk (μ∗ )−1 δk , ourinequality is proved. Assume now by contradiction that δk−1 < hk−1 tk and μ∗ μ4 δk−1 < δk . Then tk−1 φ 2s ds = hk−1 = hφ (tk−1 ). Furthermore,
V. Bally, A. Kohatsu-Higa / Journal of Functional Analysis 258 (2010) 3134–3164 tk +μ∗ μ4 δk−1
φ 2s ds μ−2 φ(tk )2 × μ∗ μ4 δk−1 μ∗
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tk−1+δk−1
φ 2s ds = μ∗ hφ (tk−1 )
tk
tk−1
hφ (tk ) = hk . This leads to a contradiction and therefore it proves that μ∗ μ4 δk−1 δk . For the reverse inequality note that if δk = hk then we have as before that δk−1 hk−1 μ∗ hk = μ∗ δk . Then assume tk+1 ∗ 4 by contradiction that δk < hk and δk−1 > μ μ δk then tk φ 2s ds = hk and ∗ 4 tk−1 +μ μ δk
φ 2s ds
μ
−2
∗ 4
φ(tk ) × μ μ δk μ 2
tk−1
∗
t k +δk
φ 2s ds = μ∗ hk hk−1
tk
which again leads to a contradiction.
2
Theorem 14. Suppose that the hypotheses (H1 (φ)), (H2 (φ)) and (H3 (φ)) hold true (see (8), (9) and (24)). Let y = xT , where xt , t 0, is the skeleton associated to the control φ. Then T
Cφ (T ) 220
Cφ (t) 150 ∗ ∗ 2 1 + φt exp −K2 μ, μ dt , pT (x, y) K1 μ, μ εφ (T ) εφ (t) 0
where √ 3 , ∗ 4π(μ )3/2 μ2
9
5/2 6 4
−1 2μ∗ ln 2 × 21π 1/2 μ∗ + 150 + 49μ4 . μ¯ μ h∗ ∧ C ∗ (μ) K2 μ, μ∗ = ∗ h∗ ∧ C (μ)
K1 μ, μ∗ =
Proof. We have using the properties of the functions in the class Λ that (see the discussion after (24)) 3 det MNT δN |cNT |2 |bNT |2 /12 T
3 μ∗ μ4 h3φ (T )Cφ4 (T )/12.
Furthermore using the definitions for Hk , ak and ρk together with (6) and (7) as well as from Theorem 12 and Lemma 13, we obtain that 1/2 NT NT −1
a Hk 1 + NT |θ | NT ln 27 π 1/2 + ln k ak ρk 16 k=1
k=1
NT N 3/2 T −1
2
(Ck εk−1 )2 Uk + 49 Ck εk−1 = NT ln 29 × 21π 1/2 + ln hk k=1
k=1
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μ
∗
T
1 + φ 2t hφ (t)
1 Cμ, μ¯
0
Cφ (t)2 + ln εφ (t)2 hφ (t)
2 4 Cφ (t) dt + 49μ εφ (t)2
1 = ln(29 × 21π 1/2 (μ∗ )5/2 μ with Cμ, ¯ 6 μ4 ). From here it follows that for Cμ2 = 2 + 49μ4 μ¯
p(yN )
√ 3 3/2
4π(μ∗ )3/2 μ2 hφ (T )Cφ2 (T ) × exp −2μ∗
T
1 + φ 2t hφ (t)
0
2 C (t) 1 φ 1 dt . Cμ, + Cμ2 μ¯ + ln hφ (t) εφ (t)2
˜ Furthermore if we define C(μ) = h∗ ∧ C ∗ (μ) 1, we have that εφ (t) 148 ˜ hφ (t) C(μ) . Cφ (t) Therefore we have that √ p(yN )
3Cφ220 (T )
4π(μ∗ )3/2 μ2 εφ222 (T ) × exp −2μ
∗
T 0
Therefore the result follows.
1 + φ 2t Cφ (t) 150 1 2 ˜ Cμ,μ¯ − ln C(μ) + 148 + Cμ dt . ˜ εφ (t) C(μ)
2
5. Construction of skeletons In this section we consider the construction of skeletons satisfying the equation t xt1
=x + 1
t
σ (xs )φs + b1 (xs ) ds,
xt2
0
=x + 2
b2 (xs ) ds.
(28)
0
We fix x, y ∈ R2 , T > 0 and we want to construct control functions φ that generate solutions to the above ordinary differential equation (28) xt = xt (φ), with x0 (φ) = x = (x 1 , x 2 ), xT (φ) = y = (y 1 , y 2 ). First, consider any differentiable function zt , 0 t T , which verifies T z0 = b2 (x),
zT = b2 (y),
zt dt = y 2 − x 2 . 0
(29)
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Then we define t xt2
=x + 2
zs ds
(30)
0
and we denote f (t, a) = b2 (a, xt2 ). We suppose that b2 is once differentiable and ! min inf ∂1 b2 (x), inf σ (x) ε0 > 0. x∈R2
x∈R2
(31)
Although this restriction is stronger than (8), we will relax it in the next subsection. By (31) we have ∂a f (t, a) = ∂1 b2 (a, xt2 ) ε0 > 0 so the function a → f (t, a) is invertible for every t 0. We denote by g(t, a) = f −1 (t, a) the corresponding inverse function. Therefore, we have that f (t, g(t, a)) = g(t, f (t, a)) = a. Then we define xt1 = g(t, zt ), φt =
∂t zt − (b2 ∂2 b2 )(xt ) − (b1 ∂1 b2 )(xt ) ∂t zt − ∂b b2 (xt ) = . (σ ∂1 b2 )(xt ) ∂σ b2 (xt )
(32) (33)
We sometimes denote by xt (z), φt (z) the functions constructed in this way. Lemma 15. We suppose that (31) holds and that z satisfies (29) for some fixed x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ R2 and T > 0. Then the function xt = (xt1 , xt2 ) defined by (30) and (32) satisfies that x0 = x = (x 1 , x 2 ), xT = y = (y 1 , y 2 ) and Eq. (28) holds with the control φt = φt (z) defined by (33). Proof. First, it is clear that
x0 = f −1 0, b2 (x) , x 2 = x
and xT = f −1 T , b2 (y) , xT2 = y.
Next, we have ∂a f (t, a) = ∂1 b2 (a, xt2 ) and ∂t f (t, a) = ∂2 b2 (a, xt2 )∂t xt2 = ∂2 b2 (a, xt2 )zt . Taking derivatives with respect to t in the equality f (t, g(t, a)) = a we obtain
0 = ∂t f t, g(t, a) = (∂t f ) t, g(t, a) + (∂a f ) t, g(t, a) ∂t g(t, a) which gives ∂t g(t, a) = −
∂2 b2 (g(t, a), xt2 )zt (∂t f )(t, g(t, a)) =− . (∂a f )(t, g(t, a)) ∂1 b2 (g(t, a), xt2 )
And by the inverse function theorem, we have that ∂a g(t, a) =
1 1 = . (∂a f )(t, g(t, a)) ∂1 b2 (g(t, a), xt2 )
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We compute now
∂t xt1 = ∂t g(t, zt ) = (∂t g)(t, zt ) + (∂a g)(t, zt )∂t zt =
∂t zt − (b2 ∂2 b2 )(xt ) ∂t zt − ∂2 b2 (g(t, zt ), xt2 )zt = 2 ∂1 b2 (xt ) ∂1 b2 (g(t, zt ), xt )
the last equality being a consequence of g(t, zt ) = xt1 and zt = b2 (xt ). Eq. (28) reads ∂t xt1 = φt σ (xt ) + b1 (xt ) which, in view of the previous equality, amounts to φt σ (xt ) + b1 (xt ) = And this is true by the definition of φ.
∂t zt − (b2 ∂2 b2 )(xt ) . ∂1 b2 (xt )
2
From now on, we assume global ellipticity and boundedness (that is, (31) and (34) below). Nevertheless, we will see later, that the core of the argument in the general case is in the following proof. Lemma 16. Assume (31) and for ε ∈ (0, 1) i) ∂α f (x) C∗ ∀x ∈ R2 for f = σ, b, ∂σ b, ∂b b2 , ∂σ σ, σ + ε∂σ b, Lb and |α| 5.
(34)
Then there exists a function z∗ generating a control φ = φ(z∗ ) which gives a solution xt = (xt1 , xt2 ) of (30) and (32) which satisfies that x0 = x = (x 1 , x 2 ), xT = y = (y 1 , y 2 ) and Eq. (28) holds with T φt2 dt 0
T
64 |y 2 − x 2 − b2 (x)T |2 |b2 (y) − b2 (x)|2 2 C , T + + ∗ T T3 ε04
T Cφ (t)a 1 + φt2 , ε ) φt2 dt dt C(C f ∗ 0 hφ (t) εφ (t)b hφ (t)c
0
0
for any increasing positive function f. Proof. We construct now a function z which verifies (29) and which is linear on [0, T /2] and on [T /2, T ]. We fix q ∈ R and we ask that zT /2 = q. Since z0 = b2 (x) and zT = b2 (y) we have T
b2 (x) + b2 (y) T q+ . zt dt = 2 2
0
We solve the equation y 2 − x 2 =
T 0
q=
zt dt and we find y 2 − x 2 b2 (x) + b2 (y) − . T /2 2
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And we denote by zt∗ the function corresponding to this value of q. Furthermore, ∂t zt∗ =
q − b2 (x) b2 (y) − q 1(0,T /2) (t) + 1(T /2,T ) (t). T /2 T /2
Therefore ∗ 4|y 2 − x 2 − b2 (x)T | 3|b2 (y) − b2 (x)| ∂t z . + t T T2 From here it follows that ∗ ∂t zt∗ − ∂b b2 (xt ) φt z = ∂ b (x ) σ 2 t 1 4|y 2 − x 2 − b2 (x)T | 3|b2 (y) − b2 (x)| 2 + C + ∗ . T T2 ε0 The second estimate follows by noting that εφ (t) = ε0 , Cφ (t) = C∗ and hφ (t) = C(1, C∗ ).
2
Therefore we obtain the following result by direct application of Lemma 16 to Theorem 14. Theorem 17. Assume the conditions (31) and (34) then the density of X is bounded below as follows: pT (x, y)
1 |y 2 − x 2 − b2 (x)T |2 |b2 (y) − b2 (x)|2 exp −C(C∗ , ε0 ) T + . + C(C∗ , ε0 ) T T3
The above result may seem restrictive due to condition (31). Nevertheless we claim that this assumption is needed just to define the path z∗ . Finding a similar path in non-uniform elliptic cases is possible. Similarly, one could replace condition (34) by a localized version on the range of x(φ(z∗ )). This is in part the objective of the next subsection. 5.1. An optimal version of the lower bound result So far, we have only used a particular skeleton in order to obtain the lower bounds for the densities of X. One could decide to optimize over the possible paths satisfying certain conditions. This is, in general, a procedure that will not lead to explicit results and can only be treated on a case by case basis. Nevertheless one can leave the optimization procedure unsolved and obtain a lower bound. This is done in this subsection. For this, we consider as before Eqs. (28). We fix x = (x 1 , x 2 ) and we assume that σ (x) ∧ ∂1 b2 (x) > 0.
(35)
The component xt1 (z) is constructed in the following lemma. Lemma 18. Assume (35). Let z : [0, ∞) → R be a deterministic differentiable function such that t z0 = b2 (x). Let xt2 (z) ∈ R2 , t ∈ [0, ∞), be defined as xt2 (z) = x 2 + 0 zs ds. Then we have
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A. There exist a unique time τ ∈ (0, ∞] and a unique continuous curve xt1 ∈ R, t ∈ [0, τ ), such that x01 = x 1 and
i) b2 xt1 , xt2 (z) = zt , 0 t < τ,
ii) ∂1 b2 xt1 , xt2 (z) > 0, 0 t < τ,
iii) lim ∂1 b2 xt1 , xt2 (z) = 0, if τ < ∞. t↑τ
We denote by τb (z) the above time and by xt1 (z) the above curve. We also denote τσ (z) = inf{t τb (z): |σ (xt1 (z))| = 0} and τ (z) = τb (z) ∧ τσ (z). B. Moreover the unique curve xt1 (z), t ∈ [0, τ ), constructed in A. together with xt2 (z), t ∈ [0, τ ), is the unique solution of (28) with the control φ given by φt (z) =
∂t zt − ∂b b2 (xt (z)) . ∂σ b2 (xt (z))
Proof. A. Step 1. We prove that there exist τ > 0 and a continuous curve xt1 ∈ R, t ∈ [0, τ ), such that i) and ii) hold true. Define f (t, a) := b2 (a, xt2 (z)) and δ := 12 |∂1 b2 (x)|. By assumption (35), we have that |∂a f (0, x 1 )| = |∂1 b2 (x)| > 0. Therefore by a continuity argument, there exists η > 0 such that for 0 t < η and |a − x 1 | < η we have |∂a f (t, a)| > δ. Therefore for each 0 t < η, the function f (t, .) : Bη (x 1 ) → Ut =: f (t, Bη (x 1 )) is a bijection. Moreover, since |∂a f (t, a)| > δ for a ∈ Bη (x 1 ), we may find r(δ) > 0 (which depends on δ but not on t) such that Br(δ) (f (t, x 1 )) ⊂ Ut . Next, we define τ = η ∧ inf{t: |f (t, x 1 ) − zt | r(δ)}. We have z0 = b2 (x) = f (0, x 1 ) and by continuity τ > 0. For every t < τ , we have that zt ∈ Br(δ) (f (t, x 1 )) so we may define xt1 = g(t, zt ), where g(t, .) : Br(δ) (f (t, x 1 )) → Bη (x 1 ) is the inverse of f (t, .). By the definition of xt1 we have b2 (xt1 , xt2 (z)) = f (t, xt1 ) = f (t, g(t, zt )) = zt . And for t τ η we have xt1 ∈ Bη (x 1 ) so that |∂a f (t, xt1 )| > δ > 0. Therefore, the properties i) and ii) are satisfied. Finally, for t = 0 we have x01 = g(0, z0 ) = g(0, f (0, x 1 )) = x 1 . Step 2. In this step we prove the uniqueness of xt1 for t τ satisfying i) and ii) above. For this, consider another continuous curve yt , t τ such that y0 = x 1 and b2 (yt , xt1 (z)) = zt and |∂1 b2 (yt , xt1 (z))| > 0 hold for t τ . Let θ = inf{t: yt = xt1 }. As before, note that there exists η > 0 such that for all t < η and for all a such that |a −x 1 | < η we have that |∂a f (t, a)| > δ. Define μX := inf{t: xt ∈ / Bη (x 1 )} and similarly μY = inf{t: yt ∈ / 1 Bη (x )}. If μX < θ then due to the bijection property of f it follows that xt = yt up to τ. Otherwise, assume that μX > θ then μY > θ and then there ε > 0 such that yt , xt1 ∈ Bη (x 1 ) for θ t θ + ε. And using i) we have f (t, yt ) = f (t, xt1 ) = zt . Since f (t, .) is injective on Bη (x 1 ) we get yt = xt1 , which is in contradiction with the definition of θ. Step 3. Let τ = sup{τ : ∃xt1 , t τ which satisfies i), ii)}. Using the uniqueness property proved at Step 2 we may construct a path {xt1 , t < τ } which satisfies i) and ii). Suppose that τ < ∞ and let us prove by contradiction that limt↑τ ∂1 b2 (xt1 , xt2 (z)) = 0.
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In fact, otherwise there exist δ > 0 and a sequence tn ↑ τ such that |∂1 b2 (xt1n , xt2n (z))| > δ. We fix n and we come back to the reasoning from Step 1 with x replaced by xn = (xt1n , xt2n (z)). Since |∂1 b2 (xn )| > δ, we may find η > 0 such that for tn t tn + η and |a − xn1 | < η we have |∂1 b2 (a, xn2 )| > 12 δ. Notice that η depends on the Lipschitz constant of ∂1 b2 and on suptτ |zt | but not on n. We recall that f (t, a) = b2 (a, xt2 (z)) and the function f (t, .) : Bη (xn1 ) → Ut =: f (t, Bη (xn1 )), tn t tn + η is a bijection. Moreover, since |∂a f (t, a)| > δ/2 for a ∈ Bη (xn1 ) and tn t tn + η, we may find r(δ) > 0 (which depends on δ but not on t) such that Br(δ) (f (t, xn1 )) ⊂ Ut . Then we define τn = η ∧ inf{t > tn : |f (t, xn1 ) − zt | r(δ)}. We have ztn = b2 (xn ) = f (tn , xn1 ) so τn > 0. Furthermore, for every tn < t < τn we have zt ∈ Br(δ) (f (t, xn1 )) so we may define xt1 = g(t, zt ) where g is the inverse of f. Notice that τn − tn does not depend on n. It depends on the modulus of continuity of zt , on the Lipschitz continuity constant of ∂1 b2 and on suptτ |zt | but not on n. So we have obtained an extension of xt1 on (tn , τn ]. Since τn − tn does not depend on n and tn ↑ τ we obtain an extension of xt1 beyond τ. And this is in contradiction with the definition of τ. So iii) is proved. Suppose now that there exist τ and x t , t < τ which satisfy i), ii), iii). By the uniqueness argument given in Step 1, for every t < τ ∧ τ we have x t = xt1 . In particular, limt↑τ ∧τ |∂1 b2 (xt1 )| = 0. And then ii) implies that τ = τ . So we have uniqueness of the pair (τ, {xt1 , t < τ }). B. We recall that f (t, a) = b2 (a, xt2 (z)) and xt1 (z) = g(t, zt ) where g(t, ·) is the inverse of f (t, ·). Fix t0 < τ (z), and we know that |∂a f (t0 , xt10 (z))| = |∂1 b2 (xt0 (z))| = δ for some δ > 0. So there exists r > 0 such that for every t ∈ (t0 − r, t0 + r) the function f (t, .) : Br (xt10 (z)) → Ut = f (t, Br (xt10 (z))) is invertible and g(t, .) : Ut → Br (xt10 (z)) is its inverse. So for every t ∈ (t0 − r, t0 + r) and a ∈ Ut we have f (t, g(t, a)) = a and for every a ∈ Br (xt10 (z)) we have g(t, f (t, a)) = a. Finally, ∂a f (t, a) = ∂1 b2 (a, xt2 (z)) and ∂t f (t, a) = ∂2 b2 (a, xt2 (z))∂t xt2 (z) = ∂2 b2 (a, xt2 (z))zt . From here, the exact argument (in localized version) that was used in the proof of Lemma 15 applies. 2 So far, we have constructed z and then defined x. We now prove that the reverse procedure is also possible. For this, we define the following sets. For x, y ∈ R2 and T > 0 and we define CT (x, y) to be the class of the differentiable functions z : [0, ∞) such that τ (z) > T and T z0 = b2 (x),
zT = b2 (y)
zt dt = y 2 − x 2 .
and 0
Let ΛT (x, y) be the set of derivatives ∂t xt2 (φ), t ∈ [0, T ] such that there exists xt1 (φ) so that for x(φ) = (x 1 (φ), x 2 (φ)) it satisfies x0 (φ) = x, xT (φ) = y and it solves Eqs. (28). Lemma 19. ΛT (x, y) = CT (x, y). t Proof. Let z ∈ CT (x, y). As in the previous lemma define xt2 = x 2 + 0 zs ds and using xt1 as in the statement of the previous lemma we obtain that xt = (xt1 , xt2 ) ∈ ΛT (x, y). Similarly, if x ∈ ΛT (x, y) then one defines zt = ∂t xt2 (φ) and all the needed properties follow straightforwardly. 2
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Using the previous results we can study the following examples. Example 20 (The Asian case). Let σ (x) = x1 , b1 (x) = 0 and b2 (x) = 0 for x = (x1 , x2 ). In this case, note that ∂1 b2 = 1 therefore τ = ∞. As in the previous section we consider as z∗ , a linear function on [0, T /2] joining x 1 and q and then joining q and y 1 on [T /2, T ]. Note that the path z∗ stays away from zero. To make the arguments simple, assume that q=
y2 − x2 x1 + y1 − > max x 1 , y 1 > 0. T /2 2
Then we have that 1 1 ∗ φt z max q − x , q − y ≡ φ. ¯ x 1 T /2 y 1 T /2 Therefore in order to apply Theorem 14, note that εφ (t) = min{x 1 , y 1 }, C∗ = 1, Cφ (t) = 2q so then we obtain pT (x, y) K1
2q min{x 1 , y 1 }
220
exp −K2 T 1 + φ 2
2q min{x 1 , y 1 }
150 ,
where √ K1 =
3 , 4π
9
−1 2 ln 2 × 21π 1/2 T ∧ C ∗ (1) + 199 , T ∧ C ∗ (1) "
145 −1 2 ∗ 2 C ∗ (1) min CCd 196 2 . e−49(C (1)) , 9 K2 =
From here one can obtain the following result lim inf
y 2 →∞
ln((y 2 )−220 pT (x, y)) −C x 1 , y 1 , x 2 , T , 2 152 (y )
where C(x 1 , y 1 , x 2 , T ) is a positive constant. Similar results can be easily obtained. For example, the above result is also valid for the case y 2 = αy 1 for α > T . Example 21. Consider the case σ = 1, b1 = 0 and b2 (x) is a smooth increasing function such √ −1/2 that b2 (x) = x1 for x1 1. In this case, ∂1 b2 (x) = 12 x1 . As before we consider the path z : [0,√T ] → (1, +∞) to be a piecewise linear function as in the proof of Lemma 16 such that 1 1 z0 = x , zT = y , zT /2 = q > 1 with T zt dt = y 2 − x 2 . 0
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1 1 Therefore τ = ∞. For √ the sake of the argument we assume that min{x , y } > 1 and q = √ 1+ y1 x 1 2 2 > 1. In this case, we have T /2 (y − x ) − 2
T
−1 −2 220 150 pT (x, y) K1 (μ) 2 zT 1 + (2zt ∂t zt )2 2−1 zt−2 exp −K2 (μ) dt , 0
where √ 3 K1 (μ) = , 4πμ446
−1 2μ296 9 ln 2 × 21π 1/2 μ744 μ¯ 6 T ∧ C ∗ (μ) + 150 + 49μ4 , ∗ T ∧ C (μ) √ √ 1 q− y 1 max{| q−T /2x |, | T /2 |} μ¯ = μ , √ √ 1 q− y 1 min{| q−T /2x |, | T /2 |} √ max{ x 1 , y 1 , q} μ= . √ min{ x 1 , y 1 , q}
K2 (μ) =
√ −1 so we have a degeneracy at infinity. ThereNote that for |x1 | > 1, we have ∂σ b2 (x) = (2 |x|) t 1 2 2 fore xt = (zt ) , φt = 2zt ∂t zt and x2 (t) = x + 0 zs ds. We then have that εφ (t) = (2zt )−1 , Cφ (t) = zt ,
148
, hφ (t) = min h∗ , C ∗ (μ) 2−1 zt−2
where we have that zt , zt−2 ∈ Λμ,h∗ for h∗ = T . Furthermore we also set |φt | = |2zt ∂t zt | = φ¯ t ∈ Λμ,h ¯ ∗ . As before, one can use the lower bound for the density to prove that
ln((y 2 )446 pt (x, y)) −C x 1 , y 1 , x 2 , T . 2 4 2 (y ) y →∞
lim inf
Remark. A variety of similar situations can be treated with the above arguments. For example, in the case that σ = 1, b1 = 0 and b2 (x) is a smooth function such that for |x1 | big enough b2 (x) = e−|x1 | . It is clear from the above argument that if we assume that min{x 1 , y 1 } > 1, a similar argument as the one above can be used to obtain a lower bound for the density in this case. We now give a lower bound in optimal form. For z ∈ CT (x, y) we denote
εz (t) = ∂1 b2 xt (z) ∧ σ xt (z) ∧ 1 > 0,
Cz (t) = 1 + |σ | + |b| + |∂σ σ | + |∂σ b| + |Lb| xt (z) .
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We also fix μ > 1 > h and we define the class CT ,μ,h (x, y) = z ∈ CT (x, y): εz , Cz , εz Cz−1 ∈ Λμ,h . Following (24), we define εz (t) 148 hz (t) := min h∗ , C ∗ (μ) , Cz (t) φt (z) =
∂t zt − ∂b b2 (xt (z)) . ∂σ b2 (xt (z))
So, for z ∈ CT ,μ,h (x, y), the hypothesis (H2 (φ)).i) holds with φ = φ(z), (H1 (φ)) holds with εφ = ε(z) and (H3 (φ)) holds with φ¯ = φ, μ and h∗ = 1. Then as an immediate consequence of Theorem 14 we obtain Theorem 22. Suppose that the (H2 (φ)).ii) (see (9)) is satisfied and CT (x, y) = ∅ for some μ > 1 > h. Then Cz (T ) 220 K1 (μ) εz (T ) μ>1>h z∈CT (x,y)
pT (x, y) sup
sup
T ∂t zt − ∂b b2 (xt (z)) 2 Cz (t) 150 1+ × exp −K2 (μ) dt . ∂σ b2 (xt (z)) εz (t) 0
Here, K1 (μ) =
√ 3 , 4πμ448
K2 (μ) =
2μ296 9 ln 2 × 21π 1/2 μ¯ 6 μ744 C ∗ (μ)−1 + 150 + 49μ4 . ∗ C (μ)
In this theorem one may use μ¯ = maxt∈[0,T ] |φt (z)|. Appendix A. Proof of Theorem 2 We recall the reader that the notation appearing in this proof corresponds to the one in [2]. We use that framework with M˜ k =
1 ak Mk ,
H˜ k =
1/2
ak Hk 1/2
ak−1
, a˜ k = ak2 with the set A˜ k redefined as
ρ˜i A˜ k = ω ∈ Ω: |Fi−1 − yi |ı˜ , i = 1, . . . , k . 2 Here the norm |x|ı˜ = M˜ i−1 x, x1/2 and ρ˜i = ai ρi . Then the proofs in [2] apply exactly chang˜ z) by ρ˜ and ing the constant 1 which appears in the definition of the hypothesis (H1 , a, ˜ A, 1/2
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(ρ+ ˜ c) ˜2 η ∈ (0, c˜ M˜ ) which gives the same result except that e2 is changed by e 2 . This factor is not considered in the definition of Cd above as it was the case in [2]. Therefore in this set˜ z) is ting the constant e2 in [2] starts to depend on ρ˜ and c. ˜ Similarly, the condition (H2 , a, ˜ A, changed to −1/2 M˜ R t,δ,d+2,p d
1 (ρ+ ˜ c) ˜2 2
a˜ 4(d+1) Cd e 2
.
Proposition 8 becomes 1
pη (z)(ω) 4e
(2π a) ˜ d/2 det M˜
(ρ+ ˜ c) ˜2 2
˜ and some η ∈ (0, c˜ M˜ ). Then Section 2.3 applies for ω ∈ A ⊂ {ω: V (ω) − zM −1 ρ} exactly with the following changes. 1. The estimate in Corollary 9 becomes (where c˜k = ρ˜˜k ) 8Hk
P (A˜ k )
P (A˜ k−1 )ρ˜kd 8d+1 H˜ kd e
(c˜k +ρ˜k−1 /2)2 2
(2dak−1 π)d/2
under the condition that −1/2 M˜ Rk t k
k−1 ,δk ,d+2,pd
1
4(d+1)2
Cd a˜ k
e
(c˜k +ρ˜k−1 /2)2 2
.
2. This estimate naturally leads to the estimate in Theorem 15 with pFN (xN )
e−N dθ 4(2π)d/2 det M˜ N
with ˜ N N N
1 1 1 Hk ρ˜k−1 2 + c˜k + ln(a˜ k ) + ln . θ = ln 82 (2πd)1/2 + 2N N ρ˜k 8dN 2 k=1
k=2
k=2
Finally replacing all the above parameters changes and the inequality c˜k + ρ˜k−1 = a 1/2 ρk−1 + ρk a 1/2 ρk−1 + ρk k−1 k−1 2 2 8Hk 2 one obtains the result.
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References [1] S. Aida, S. Kusuoka, D. Stroock, On the support of Wiener functionals, in: K.D. Elworthy, N. Ikeda (Eds.), Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, in: Pitman Res. Notes Math. Ser., vol. 284, Longman Scient. Tech., 1993, pp. 3–34. [2] V. Bally, Lower bounds for the density of the law of locally elliptic Ito processes, Ann. Probab. 34 (2006) 2406– 2440. [3] G. Ben-Arous, R. Leandre, Decroissance exponentille du noyau de la chaleur sur la diagonale (II), Probab. Theory Related Fields 90 (1991) 377–402. [4] U. Boscain, S. Polidoro, Gaussian estimates for hypoelliptic operators via optimal control, Rend. Lincei Mat. Appol. 18 (2007) 333–342. [5] F. Delarue, S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations, 2009, in preparation. [6] C.L. Fefferman, A. Sanchez-Calle, Fundamental solutions of second order subelliptic operators, Ann. of Math. (2) 124 (1986) 247–272. [7] A. Kohatsu-Higa, Lower bounds for densities of uniform elliptic random variables on Wiener space, Probab. Theory Related Fields 126 (2003) 421–457. [8] S. Kusuoka, D. Stroock, Applications of the Malliavin calculus, part III, J. Fac. Sci. Univ Tokyo Sect. 1A Math. 34 (1987) 391–442. [9] P. Malliavin, E. Nualart, Density minoration of a strongly non-degenerated random variable, J. Funct. Anal. 256 (2009) 4197–4214. [10] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin, 1995. [11] A. Pascucci, S. Polidoro, Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators, Trans. Amer. Math. Soc. 358 (2006) 4873–4893. [12] S. Polidoro, A global lower bound for the fundamental solution of a Kolomgorov–Fokker–Planck equation, Arch. Ration. Math. Anal. 137 (1997) 321–340. [13] A. Sanchez-Calle, Fundamental solutions and geometry of the sum of square of vector fields, Invent. Math. 78 (1) (1984) 143–160. [14] M. Yor, On some exponential functionals of Brownian motion, Adv. in Appl. Probab. 24 (1992) 509–531.
Journal of Functional Analysis 258 (2010) 3165–3194 www.elsevier.com/locate/jfa
Existence and multiplicity results for a fourth order mean field equation ✩ Mohamed Ben Ayed a , Mohameden Ould Ahmedou b,∗ a Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia b Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Received 1 September 2009; accepted 9 January 2010 Available online 8 February 2010 Communicated by J. Coron
Abstract In this article we consider the following fourth order mean field equation on smooth domain Ω R4 : Keu u Ω Ke
2 u =
in Ω,
u = u = 0 on ∂Ω, where ∈ R and 0 < K ∈ C 2 (Ω). Through a refined blow up analysis, we characterize the critical points at infinity of the associated variational problem and compute their contribution of the difference of topology between the level sets of the associated Euler–Lagrange functional. We then use topological and dynamical methods to prove some existence and multiplicity results. © 2010 Elsevier Inc. All rights reserved. Keywords: Fourth order nonlinear elliptic equation; Critical point at infinity; Morse theory; Topological methods
✩
The research of M. Ould Ahmedou has been partly supported by SFB TR 71.
* Corresponding author.
E-mail addresses:
[email protected] (M. Ben Ayed),
[email protected] (M. Ould Ahmedou). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.009
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1. Introduction and statement of main results In this paper we consider the following fourth order mean field equation:
u
2 u = KeKeu
in Ω,
u = u = 0
on ∂Ω,
Ω
(1.1)
where ∈ R, 0 < K ∈ C 2 (Ω) and Ω is a bounded and smooth domain of R4 . In dimension 2 the analogue problem
u
−u = KeKeu
in Ω,
u=0
on ∂Ω,
Ω
(1.2)
where Ω is a bounded smooth domain in R2 and K ∈ C 2 (Ω) has been extensively studied, see [9,14,13,19,20] and the references therein. Our interest in (1.1) grew up, in particular from its resemblance to the prescribed Q-curvature problem on 4-dimensional riemannian manifold (M 4 , g). Indeed on such a manifold the Paneitz operator is defined as: Pg4 ϕ
= 2g ϕ
− divg
2 Rg g − 2 Ricg dϕ, 3
where Rg and Ricg denote respectively the scalar and Ricci curvature. The Paneitz operator gives rise to a fourth order curvature: the Q-curvature defined as: Q :=
1 −R + R 2 − 3|Ric|2 . 12
Now since under conformal change of metrics g = e2w g, there holds Pg4 w + 2Qg = 2Qg e4w ,
(1.3)
the following natural question arises: Does there exist a metric g˜ conformally equivalent to g such that Qg˜ = K? In view of the above transformation law, this amounts to solve the following nonlinear fourth order equation: Pg4 w + 2Qg = 2Ke4w
in M 4 .
(1.4)
The above equation has been during the last decades extensively studied. See the works [2,7,10,12,11,15–17,25,26] and references therein. Coming back to our fourth order mean field Eq. (1.1), we point out that it has a variational structure, indeed its solutions are in one-to-one correspondence with the critical points of the following functional defined on H := H 2 (Ω) ∩ H01 (Ω) by J (u) := 1/2 Ω
|u|2 dx − Log Keu dx . Ω
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The analytic features of Eq. (1.1) and of its associated Euler–Lagrange functional depend strongly on some critical values of the parameter . Indeed depending on wether is a multiple of 64π 2 or not, the noncompactness of the variational problem and therefore the way of finding critical points of the functional J on H change drastically. Therefore we will first focus on the case = 64mπ 2 , m ∈ N. Theorem 1.1. Assume that ∈ (64mπ 2 , 64(m + 1)π 2 ); m ∈ N∗ and that Ω is not contractible, then the problem (1.1) has at least one solution. Furthermore, for generic K’s, there holds
1 # solutions of (1.1) 1 − χ(Ω) · · · m − χ(Ω) . m! Remark 1.2. Under the stronger condition that χ(Ω) 0, where χ(Ω) denotes the Euler characteristic of Ω, the above theorem has been proved in [22]. Their proof which is drastically different from ours, uses a topological degree argument. We observe that, the embedding of S2 in R4 has a positive Euler characteristic but nontrivial topology. We point out that a main ingredient in the proof of Theorem 1.1 is the fact that for = 64mπ 2 , m ∈ N∗ , although the functional does not satisfy the Palais Smale Condition, the change of topology between any finite level sets is due only to the existence of critical points of the functional J and from another part the topology of the sublevel set J −L := {u ∈ H; J (u) < −L} for very large L is homotopically equivalent to the set of formal barycenter Bm (K) := K m ×σm m−1 , where K Ω is a compact subset of Ω and m−1 denotes the standard simplex. We recall that the role of topology of formal barycenters in noncompact variational problems involving exponential nonlinearities was first discovered by Djadli and Malchiodi (see [16]). Now we address the case = 64π 2 . To state our results in this case, we need to introduce the following notation. Let G4 (a, .) be the Green’s function of 2 under Navier boundary conditions and H4 (a, .) its regular part and set fK : Ω → R defined by fK (y) := Log K(y) − 32π 2 H4 (y, y).
(1.5)
We say that the function K satisfies the condition (C0 ) if fK has only nondegenerate critical points and at each critical point of fK there holds that (Log K)(y) − 64π 2 1 H (y, y) = 0, where 1 H (y, y) denotes the Laplacian of the function H (y, .). We set
K− := y ∈ Ω; ∇fK (y) = 0 and (Log K)(y) − 64π 2 1 H (y, y) < 0 . Now we are ready to state our next result:
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Theorem 1.3. Let = 64π 2 and assume that K satisfies the condition (C0 ). If
(−1)morse(fK ,q) = 1,
q∈K−
then the problem (1.1) has at least one solution. Here morse(fK , q) denotes the Morse index of fK at the critical point q. Furthermore for generic K’s the number of solutions of (1.1) is lower bounded by morse(fK ,q) 1 − (−1) . q∈K−
In view of the above results, one may think about the situation where the total sum equals 1 but a partial one is not equal 1. A natural question arises: Is it possible in this case to use such an information to derive an existence result? In the following theorem we give a partial result to this question. Theorem 1.4. Let = 64π 2 and K be a function satisfying (C0 ). Assume that there exists k ∈ N∗ such that: ∀q ∈ K− ;
(1)
ι(q) = k,
where ι(q) := 4 − morse(fK , q) is the coindex of q. (2) (−1)morse(fK ,q) = 1. q∈K− ; ι(q)
Then the problem (1.1) has at least one solution w, whose generalized Morse index is less than or equals k. Recall that the generalized Morse index of a solution w is the dimension of the space of nonpositivity of the associated linearized operator Kew ϕ L(ϕ) = 2 ϕ − . w Ω Ke Moreover for generic K’s, the number of solutions having their Morse indices less than or equals k is lower bounded by 1 −
q∈K− ; ι(q)
.
morse(fK ,q)
(−1)
The proofs of Theorems 1.1 and 1.3 rely on a careful analysis of the loss of compactness of the associated variational problem when = 64π 2 . As a byproduct of such analysis, we have the following existence result:
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Theorem 1.5. Let = 64π 2 and 0 < K ∈ C 2 (Ω). There exists a constant c0 depending on Ω such that if K(y) |∇K(y)|2 < c0 , − max x∈Ω K(y) K(y) then the problem (1.1) admits a minimal solution. Now we consider the case = 64mπ 2 , m 2. In this case the functional is neither upper bounded or lower bounded and the analysis of the lack of compactness is more delicate. To state our results in this case, we introduce the following notation. We set F K : Ω m \ Fm (Ω) → R, where Fm (Ω) := {(x1 , . . . , xm ): there exist i, j such that xi = xj } denotes the thick diagonal. The function F K is defined as: F K (y1 , . . . , ym ) :=
m
Log K(yi ) − 32π 2 H4 (yi , yi ) + 64π 2
G4 (yi , yj ) .
(1.6)
j =i
i=1
For q := (q1 , . . . , qm ), let q
Fi (x) := eLog K(x)−64π
2 (H
4 (qi ,x)−
j =i
G4 (qj ,x))
.
We say that K satisfies the condition (C1 ) if the critical points of F K are nondegenerate and at every critical point q := (q1 , . . . , qm ) there holds l(q) :=
q m F (qi ) i = 0. q Fi (qi ) i=1
We set
F∞ := q = (q1 , . . . , qm ) a critical point of F K such that l(q) < 0 . To each point q ∈ F∞ we associate an index: i : F∞ → N defined by i(q) := 5m − 1 − morse F K , q , where morse(F K , q) denotes the Morse index of F K at its critical point q. Now we state our main results in the case = 64mπ 2 ; m 2.
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Theorem 1.6. Assume that = 64mπ 2 ; m 2 and let 0 < K ∈ C 2 (Ω) satisfying the condition (C1 ). If
(−1)i(q) =
q∈F∞
1 −χ(Ω) + 1 −χ(Ω) + 2 · · · −χ(Ω) + m − 1 . (m − 1)!
Then the problem (1.1) has at least one solution. Furthermore, for generic K’s, there holds
# solutions of (1.1)
1 i(q) 1 − χ(Ω) · · · m − 1 − χ(Ω) − (−1) . (m − 1)! q∈F∞
The remainder of this paper is organized as follows: We set up our notation in Section 2 and expand the associated Euler–Lagrange functional J near potential neighborhood of critical points at infinity then prove a deformation lemma in Section 4. In Section 5 we expand the gradient of J near its potential end points and in Section 6 we perform a Morse type reduction near potential neighborhood at infinity while Section 7 is devoted to the proof of our existence results. Finally we collect in Appendix 8 some useful computations. 2. Notation In this article we will use the following notation: On H := H 2 (Ω) ∩ H01 (Ω), we define the following norm
u :=
|u|2
2
Ω
to which we associate the inner product u, v =
uv
for u, v ∈ H.
Ω
Furthermore, for a ∈ Ω and λ > 0, we define on R4 the following function: δa,λ (x) := Log
λ4 (1 + λ2 |x − a|2 )4
K(a) − Log , c0
where c0 := 6 × 64.
Observe that this function satisfies 2 δa,λ = K(a)eδa,λ
in R4 .
Let P δa,λ be the unique function satisfying the following equation
2 P δa,λ = K(a)eδa,λ
in Ω,
P δa,λ = P δa,λ = 0
on ∂Ω.
(2.1)
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For x ∈ Ω, let G4 (x, .) be the Green’s function of 2 under Navier boundary condition and H4 (x, .) its regular part. That is 1 1 − H4 (x, y). Log G4 (x, y) = |x − y| 8π 2 For x ∈ Ω, let G2 (x, .) be the Green’s function of under Dirichlet boundary condition and H2 (x, .) its regular part. That is G2 (x, y) =
1 1 − H2 (x, y). 2 4π |x − y|2
Remark 2.1. In the sequel, the function F K will be defined by (1.6) for m 2 and by (1.5) for m = 1. 3. Expansion of the functional in potential neighborhoods of infinity Let p ∈ N∗ , ε > 0 and let V (p, ε) be a neighborhood of potential critical points at infinity defined as p C1 V (p, ε) := αi P δai ,λi + w: for each i, |αi − 1| 2 , λi ε −1 , λi i=1 λi dist(ai , ∂Ω) η, and < C1 , |ai − aj | 2η for i = j , λj where C1 is a large positive constant and η is a fixed positive constant. Following the ideas of Bahri and Coron [5], for u ∈ V (p, ε) and ε small, the following minimization problem has, up to permutation, only one solution. p αi P δai ,λi . min u − αi >0; ai ∈Ω; λi >0
(3.1)
i=1
Hence every u ∈ V (p, ε) can be written as u=
p
αi P δai ,λi + w,
(3.2)
i=1
where w satisfies ∂P δai ,λi = 0, w, P δai ,λi = w, ∂λi
∂P δai ,λi w, = 0 and w < ε. ∂ai
Remark that, from the definition of V (p, ε), we have Bη (ai ) := B(ai , η) ⊂ Ω
for each i
and Bη (ai ) ∩ Bη (aj ) = ∅ for i = j.
(3.3)
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In this section, we give an asymptotic expansion of the functional J in V (p, ε) and we start with the following lemma: Lemma 3.1. Let u :=
p
i=1 αi P δai ,λi
∈ V (p, ε). On Bi := Bη (ai ) there holds
λ8i
Keu =
FiA (x) (1 + λ2i |x − ai |2 )4 Log2 λ A 2 2 × 1 + Bi (x) + (αi − 1) 8 Log λi − 4 Log 1 + λi |x − ai | + O λ4 λ8i Log λ , = FiA (x) 1 + O 2 2 4 λ2 (1 + λi |x − ai | )
where, if p 2, A = (a1 , . . . , ap ), FiA (x) := eLog(K(x))−64π
2 (H
4 (ai ,x)−
j =i
G4 (aj ,x))
,
(αj − 1)G4 (aj , x) BiA (x) := −64π 2 (αi − 1)H4 (ai , x) − + 16π 2
j =i
H2 (ai , x) G2 (aj , x) , − 2 λ2i λ i i=j
and if p = 1, then A = a1 , F1A (x) := eLog(K(x))−64π
2H
4 (ai ,x)
,
B1A (x) := −64π 2 (α1 − 1)H4 (a1 , x) + 16π 2
H2 (a1 , x) . λ21
Proof. The lemma follows easily from Lemma 8.1 and the fact that for each j , we have |αj − 1| C1 /λ2j (see the definition of V (p, ε)). 2 Using the above lemma, we can expand the integral part of the functional J . In fact, we have Lemma 3.2. Let u :=
p
Ke dx = u
Ω
i=1 αi P δai ,λi
p 2 π i=1
6
∈ V (p, ε), then we have
λ4i FiA (ai ) +
π2 4 A π2 2 λi Fi (ai )BiA (ai ) + λ FiA (ai ) 6 24 i
5 4π 2 4 A 2 (αi − 1)λi Fi (ai ) Log λi − + O Log λ . + 3 12
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3173
Proof. For simplicity, in all the proofs, we will write Fi and Bi instead of FiA and BiA respectively. We notice first that, since the function u in Ω \ ( Bi ) is bounded, we get Ω\(
Keu dx = O(1). Bi )
Now using Lemma 3.1, we derive:
Keu dx = Bi
Bi
λ8i Fi (x) (1 + λ2i |x
− ai |2 )4
+ (αi − 1) Bi
1 + Bi (x) + O Log2 λ
λ8i Fi (x) (1 + λ2i |x − ai |2 )4
8 Log λi − 4 Log 1 + λ2i |x − ai |2 .
Now we remark that, on Bi , Fi is a C ∞ function and we have Fi C ∞ (Bi ) is bounded. Thus, expanding around ai , we obtain
Keu dx = λ4i Fi (ai ) 1 + Bi (ai )
R4
Bi
dy 1 + O (1 + |y|2 )4 λ4
1 + (Fi + Fi Bi )(ai )λ2i 8
R4
|y|2 dy 1 + O Log2 λ + O 2 4 2 (1 + |y| ) λ
4 + (αi − 1) 8λi Log λi Fi (ai ) R4
dy 1 +O 4 2 4 (1 + |y| ) λ
2 Log(1 + |y|2 ) dy Log λ 4 + O λi Log λi . − 4λi Fi (ai ) +O (1 + |y|2 )4 λ4 R4
Finally using easy computations (to obtain the values of the integrals) and the fact that for each j , we have |αj − 1| C1 /λ2j , the lemma follows. 2 In the following, we will show that the w-part in the parametrization of functions in V (p, ε) does not play in the lack of compactness. More precisely we perform a finite dimensional reduction of the functional in such potential neighborhoods of infinity. Indeed we have Proposition 3.3. Let u =
p
i=1 αi P δai ,λi
J (u) = J
p i=1
where
+ w ∈ V (p, ε). Then we have that
αi P δai ,λi
1 − f (w) + Q(w) + o w 2 , 2
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α Pδ
Ke j aj ,λj w
f (v) := Ω αj P δaj ,λj Ω Ke and
α Pδ
Ke j aj ,λj w 2
Q(w) := w 2 − Ω . αj P δaj ,λj Ke Ω Moreover Q is a positive definite quadratic form and f satisfies: for every γ ∈ (0, 1) there exists a constant C(γ ) such that f (w) C(γ ) w
|∇FiA (ai )| Log λ . + λ1−γ λ2−γ
Before giving the proof of Proposition 3.3, we need the following lemma: Lemma 3.4. Let u =
p
i=1 αi P δai ,λi
+ w ∈ V (p, ε).
(i) Let β 1. For every γ ∈ (0, 1), there exists a constant C(γ ) such that
Keu−w w β C(γ )λγ +4 w β ,
Ω
Keu−w ew − 1 − w − w 2 /2 = o λ4 w 2 .
(ii) Ω
Proof. Observe that as u − w is bounded in Ω \ ( Bi ), it follows:
Ω\(
It remains the integral on
Keu−w |w|β C Ω
Bi )
Bi . From Lemma 8.1, it follows that
Keu−w |w|β C
Bi
|w|β C w β .
Bi
λ8i (1 + λ2i |x − ai |2 )4
Cλ8i Bi γ −4
Cλ8i λi
|w|β
1 2 (1 + λi |x − ai |2 )4
4 4−γ
C(γ ) w β ,
4−γ 4
w
β 4
Lγ
(3.4) 4
where we have used the continuity of the embedding H → L γ (Ω).
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3175
Now to prove the second statement we argue as follows: Let t0 be a small positive constant. Using (i) of Lemma 3.4, for every γ ∈ (0, 1), there exists C(γ ) such that
Ke
u−w
Ω∩{|w|
w2 e −1−w− dx C Keu−w |w|3 dx C(γ )λ4+γ w 3 . 2 w
Ω
Furthermore, if |w| t0 , there exists a constant C > 0 such that 2 |w| 2 |w|2 w e − 1 − w − w 2 /2 Ce|w| Ce|w|(1−32π w 2 ) e32π w 2
Ce
t0 (1−32π 2
t0 )
w 2
e
32π 2
|w|2
w 2
.
Now using Moser–Trudinger Inequality [1] we derive that
Ke
u−w
Ω∩{|w|t0 }
w2 e −1−w− dx Cλ8 2
w
Ω∩{|w|t0 }
Cλ8 e
t0 (1−32π 2
2 w e − 1 − w − w dx 2 t0 )
w 2
.
Since w is very small, we get
e
t0 (1−32π 2
t0 )
w 2
ce
−32π 2
t02
w 2
c w 10 .
Finally, using the fact that w Cλ−2 (see the definition of V (p, ε)), the lemma follows. Proof of Proposition 3.3. First, using the orthogonality of w to P δai ,λi , we derive that 1 1 J (u) = u − w 2 + w 2 2 2 u−w 2 u−w w 2 1 + w + w /2 + Ke e − 1 − w − w /2 . − Log Ke Using Lemmas 3.2 and 3.4, we derive that Log Keu−w 1 + w + w 2 /2 + Keu−w ew − 1 − w − w 2 /2
= Log Hence
Ke
u−w
Keu−w w Keu−w w 2 Keu−w w 2 1 + + − + o w 2 . u−w u−w u−w 2 Ke Ke 2 Ke
2
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Keu−w w 1 Keu−w w 2 1 2 J (u) = J (u − w) − + −
w
u−w 2 2 Ke Keu−w Keu−w w 2 1 − + o w 2 . u−w 2 Ke To complete the proof, it remains to estimate |f (w)|. Observe that Keu−w w =
p i=1 B
Ω
i
=
p
Log λ w + O F (x) 1 + O i λ2 (1 + λ2i |x − ai |2 )4
λ4i Fi (ai )
i=1
+O Bi
Bi
λ4i w (1 + λ2i |x
− ai |2 )4
Bi
Log λ + 2 2 2 4 λ (1 + λi |x − ai | )
Bi
Ω\(
Bi )
λ8i (x − ai )w
+ ∇Fi (ai )
λ8i |x − ai |2 |w|
|w|
λ8i
(1 + λ2i |x − ai |2 )4
λ8i |w| (1 + λ2i |x − ai |2 )4
+ w .
Since w is orthogonal to P δi , it holds
P δai ,λi w = c0
λ4i
2
Ω
Ω
(1 + λ2i |x − ai |2 )4
w = 0,
16 3 3 4 λ4i w λi c c w
4 w . (1 + λ2 |x − a |2 )4 2 2 λ 1 + λ |x − a | i i i i Bi
Ω\Bi
As in the proof of (3.4), for every γ ∈ (0, 1), there holds Keu−w w = O
∇Fi (ai ) λ3+γ + λ2+γ Log λ w .
Ω
Now using Lemma 3.2, the expansion claimed in Proposition 3.3 follows. Now we can proceed as in [3, pp. 65–68] to prove that the quadratic form Q is positive definite. Proposition 3.3 is thereby proved. 2 Now it follows from the above estimate on |f (w)| and the fact that the quadratic form Q is positive definite that: Corollary 3.5. Let u :=
p
i=1 αi P δai ,λi
∈ V (p, ε). Then there exists a unique w such that
J (u + w) = min J (u + w); u + w ∈ V (p, ε) . Furthermore, for every γ > 0 there exists a constant C(γ ) such that
M. Ben Ayed, M. Ould Ahmedou / Journal of Functional Analysis 258 (2010) 3165–3194
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|∇FiA (ai )| 1 ,
w C(γ ) + 1−γ 2−γ λi λi where A = (a1 , . . . , ap ) and FiA is defined in Lemma 3.1. Corollary 3.6. Let u := w − w → v such that
p
i=1 αi P δai ,λi
∈ V (p, ε). Then there exists a change of variable
1 J (u + w) = J (u + w) + ∂ 2 J (u + w)(v, v). 2 Using the above estimate (Corollary 3.5), we derive the following expansion. Proposition 3.7. Let u :=
p
i=1 αi P δai ,λi
J (u + w) = 4 × 64π 2
p
∈ V (p, ε). Then we have
αi2 Log λi
i=1
− 64π 2
p i=1
5 2 αi + 32π 2 αi2 H4 (ai , ai ) − αi αj G4 (ai , aj ) 3 j =i
p 1 1 1 1 H (a , a ) − G (a , a ) + 2 i i 2 i j 2 λ2i λ2i λ2j j =i i=1 p π2 4 A − Log λi Fi (ai ) 6 + 16 × 64π 2
i=1
p 1 − p λ4i FiA (ai )BiA (ai ) + λ2i FiA (ai ) 4 A 4 i=1 λi Fi (ai ) i=1 Log λ 5 + w 2 . f (w) + 8(αi − 1)λ4i FiA (ai ) Log λi − +O + 12 λ4
Proof. The proof follows immediately from Lemmas 3.2, 8.3, Proposition 3.3 and Remark 8.4. 2 Corollary 3.8. Let u :=
p
i=1 αi P δai ,λi
∈ V (p, ε). There holds
J (u) = 4 64pπ 2 − Log λ1 + O(1). Proof. Since u ∈ V (p, ε) then there exists a constant C1 such that C1−1
λi C1 λj
∀i, j,
and |αi − 1|
C1 λ2
∀i.
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Therefore Log λi = Log λ1 + O(1)
and
αi2
1 . =p+O λ
It follows then from Proposition 3.7 that J (u) = 4 × 64π 2 p Log λ1 − 4 Log λ1 + O(1). Thus the corollary follows.
2
As an immediate result we have Corollary 3.9. Assume that = 64π 2 m with m 1. (1) There exists L > 0 such that ∀u ∈ V (m, ε),
−L J (u) L.
(2) For each b > L, we can choose ε > 0 small enough such that ∀p > m, ∀u ∈ V (p, ε),
J (u) > b.
(3) Assume that m 2. For each a < −L, we can choose ε > 0 small enough such that ∀p < m, ∀u ∈ V (p, ε),
J (u) < a.
4. Deformation lemma Lemma 4.1. Let a, b ∈ R such that a < b and there is no critical value of J in [a, b]. (1) If = 64mπ 2 , m ∈ N∗ , then J a is a retract by deformation of J b . (2) If = 64mπ 2 , m ∈ N∗ then there are two possibilities J a is a retract by deformation of J b , or J b retracts by deformation onto J a ∪ σ, where σ ⊂ V (m, ε) and for c ∈ R, J c := {u ∈ H: J (u) < c}. Proof. We first point out that it follows from an abstract result of [18,23] that given a < b ∈ R, regarding the difference of topology between the sublevel sets J a and J b there are only two possibilities: or J a is a retract by deformation of J b or there exists a sequence k and a sequence of solutions uk of
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⎧ u ⎨ 2 u = Ke k k k uk Ω Ke dx ⎩ uk = uk = 0
in Ω,
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(4.1)
on ∂Ω,
such that k →
and a J (uk ) b.
Now observe that if uk were bounded, it would converge to a solution of the problem (1.1) which contradicts our assumption that in [a, b], there is no critical value of J . Therefore the sequence uk should blow up and it follows from the blow up analysis of Lin and Wei [21] that k → 64mπ 2
and uk ∈ V (p, ε)
for some p ∈ N∗ .
It follows then from Corollary 3.9 that p = m and = 64mπ 2 . As a consequence, in case where = 64mπ 2 , we have for every a < b ∈ R, J a is a retract by deformation of J b and if = 64mπ 2 and J a is not homotopically equivalent to J b then J b J a ∪ σ, where σ ⊂ V (m, ε) and means “retracts by deformation”.
2
Corollary 4.2. Let = 64mπ 2 , m ∈ N∗ and assume that (1.1) has a finite number l ∈ N of solutions. Thus there exists a large positive constant L1 such that: H retracts by deformation onto J L1 . Proof. Let w1 , . . . , wl be all the solutions of (1.1) with l ∈ N (if there exist). Thus there exists ˜ where L is defined in L˜ such that J (wi ) L˜ for each i l. Now let b > a > L1 := max(L, L) Corollary 3.9. By Lemma 4.1, we get that J b J a . Hence our corollary follows. 2 5. Expansion of the gradient near its potential end points Proposition 5.1. Let = 64mπ 2 , m 1 and u = P δi := P δai ,λi
m
i=1 αi P δai ,λi
and γi := 1 −
∈ V (m, ε). Setting
mλ4i FiA (ai ) π 2 . u 6 Ω Ke
There holds 1 ∂P δi −2π 2 λ4i ∇FiA (ai ) 64π 2 ∂H4 (ai , ai ) ∇J (u), = γi − u λi ∂ai 3 λi λi ∂a Ω Ke 64π 2 ∂G4 (ai , aj ) 1 + γj (if m 2) + O 2 . λi ∂a λ j =i
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Proof. Using Lemmas 3.1, 8.1 and expanding around ai , we derive that
1 ∂P δi Ke = λi ∂ai
Bi
Bi
λ8i Fi (x)
u
(1 + λ2i |x − ai |2 )4
λ8i
1 1 + (αi − 1) Log +O 2 2 2 4 λ (1 + λi |x − ai | )
λi (x − ai ) 64π 2 ∂H4 (ai , x) 1 × 8 − +O 3 2 2 λ ∂a 1 + λi |x − ai | λi i i
= 8λ3i
π2 π 2 ∂H4 (ai , ai ) ∇Fi (ai ) − 64π 2 λ3i Fi (ai ) + O λ2 . 12 6 ∂a
Now for i = j (if m 2) there holds Keu
1 ∂P δi = λi ∂ai
Bj
Bj
(1 + λ2j |x − aj |2 )4
1+O
Log λ λ2
64π 2 ∂G4 (ai , x) 1 × +O 3 λi ∂a λ
=
π 2 ∂G4 (ai , aj ) 64π 2 4 + O(λ Log λ). λj Fj (aj ) λi 6 ∂a
Finally using Lemma 8.2, the proposition follows. Proposition 5.2. Let = 64mπ 2 , m 1 and u =
λ8j Fj (x)
∂P δi ∇J (u), λi ∂λi
2
p
i=1 αi P δi
∈ V (m, ε). There holds
8π 2 Fi (ai ) 7 + 2 = 4 × 64π γi 1 − 2 H2 (ai , ai ) + Bi (ai ) + 8(αi − 1) Log λi − 24 λi λi Fi (ai ) 2
40 × 64π 2 (αi − 1) − 32 × 64π 2 (αi − 1) Log λi 3 64π 4 Log λ 2 Fi (ai ) . + 32 × 2 − 32π 2 γj G2 (ai , aj ) (if m 2) + O λ4 λi Fi (ai ) λi j =i − 4 × 64π 2 Bi (ai ) +
Proof. From one part we have
∂P δi Ke λi = ∂λi
u
Bi
Bi
λ8i (1 + λ2i |x
− ai
|2 )4
Fi (x) 1 + Bi (x) + (αi − 1)
× 8 Log λi − 4 Log 1 + λ2i |x − ai |2 8 32π 2 Log λ × − 2 H2 (ai , x) + O λ4 1 + λ2i |x − ai |2 λi
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π2 π2 = 8 1 + 8(αi − 1) Log λi λ4i Fi (ai ) + λ2i Fi (ai ) 12 12 2 π π2 + 8λ4i Fi (ai )Bi (ai ) − 32π 2 λ2i Fi (ai )H2 (ai , ai ) 12 6 2 7π + O(Log λ). − 32(αi − 1)λ4i Fi (ai ) 144 On the other hand for j = i
∂P δi Ke λi = ∂λi
u
Bj
Bj
=
λ8j Fj (x) (1 + λ2j |x − aj |2 )4
32π 2 Log λ G2 (ai , x) + O λ4 λ2i
32π 2 π2 4 + O(Log λ). G (a , a )λ F (a ) 2 i j j j j 6 λ2i
The proof follows from Lemma 8.2, the definition of γi and the above estimates.
2
Observe that the above proposition can be written as: Corollary 5.3. Let = 64mπ 2 , m 1 and u =
p
i=1 αi P δi
∈ V (m, ε). There holds
FiA (ai ) ∂P δi 5 2 A = 4 × 64π γi − Bi (ai ) − 8(αi − 1) Log λi − − 2 A ∇J (u), λi ∂λi 12 8λi Fi (ai ) m Log λ Log λ + |γj | . +O λ4 λ2 j =1
From Lemma 3.2, for m = 1, it is easy to get that γ1 = B1 (a1 ) +
Log(λ1 ) 1 F1 (a1 ) 5 , + O + 8(α − 1) Log(λ ) − 1 1 4 λ21 F1 (a1 ) 12 λ41
and therefore Corollary 5.3 can be written as follows: Corollary 5.4. Let = 64π 2 and u = α1 P δa1 ,λ1 ∈ V (1, ε). Then we have A Log2 (λ) ∂P δ1 2 F1 (a1 ) ∇J (u), λ1 = 32π 2 A . +O ∂λ1 λ4 λ1 F1 (a1 ) For m 2, we can obtain a similar result. Indeed: Corollary 5.5. Let u =
m
i=1 αi P δai ,λi
∈ V (m, ε) with m 2 and assume that
|γj | C
Log λ λ2
for each j.
(5.1)
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(i) For each j , we have mλ4j FjA (aj ) Log λ =1+O .
4 A λ2 λi Fi (ai ) F A (ai ) ∂P δi Log2 λ i 2 +O = 32π . ∇J (u), λi ∂λi λ4 λ2i FiA (ai )
(ii)
Proof. Claim (i) follows immediately from the assumption on γj and Lemma 3.2. Now using claim (i), we can improve Lemma 3.2 and it becomes
Ke = u
Ω
π2 4 1 1 Fi (ai ) λi Fi (ai ) 1 + Bi (ai ) + 6 m 4m λ2i Fi (ai )
+
Log2 (λ) 5 8 . (αi − 1) Log λi − +O m 12 λ4
Hence, using the definition of Proposition 5.6. Let u =
γj , claim (ii) follows immediately from Corollary 5.3.
p
i=1 αi P δi
2
∈ V (m, ε) and = 64mπ 2
∂P δi 5 2 ∇J (u), P δi = 2 Log λi − − 16π H4 (ai , ai ) ∇J (u), λi 6 ∂λi 2 F A (ai ) ∂P δj − 64π 2 Log λi 2 iA + 64π 2 G4 (ai , aj ) ∇J (u), λj ∂λj λi Fi (ai ) j =i 1 Log λ |γj | . + 8 × 64π 2 (αi − 1) Log λi + O 2 + 2 λ λ
!
"
Proof.
Ke P δi = u
Bi
Bi
λ8i Fi (x) (1 + λ2i |x
− ai
|2 )4
1 + Bi (x) + (αi − 1)
× 8 Log λi − 4 Log 1 + λ2i |x − ai |2 1 × 8 Log λi − 4 Log 1 + λ2i |x − ai |2 − 64π 2 H4 (ai , x) + O 2 λ
π2 π2 = 8 Log λi 1 + 8(αi − 1) Log λi λ4i Fi (ai ) + λ2i Log λi Fi (ai ) 6 3 2 4 5π − 4 1 + 8(αi − 1) Log λi λi Fi (ai ) 36 π2 − 64π 2 1 + 8(αi − 1) Log λi λ4i Fi (ai )H4 (ai , ai ) 6
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5π 2 π2 + 8 Log λi λ4i Fi (ai )Bi (ai ) + O λ2 36 6 2
5 π 4 2 λ Fi (ai ) 1 + 8(αi − 1) Log λi 2 Log λi − − 16π H4 (ai , ai ) = 6 i 6 − 32(αi − 1) Log λi λ4i Fi (ai )
π2 5π 2 − 32(αi − 1) Log λi λ4i Fi (ai ) 3 36 2 π + O λ2 . + 8 Log λi λ4i Fi (ai )Bi (ai ) 6 + Fi (ai )λ2i Log(λi )
Furthermore for j = i
Keu P δi =
Bj
(1 + 8(αj − 1) Log λj ) (1 + λ2j |x − aj |2 )4
Bj
λ8j Fj (x) 64π 2 G4 (ai , x) + O λ2
π2 + O λ2 . = 1 + 8(αj − 1) Log λj 64π 2 G4 (ai , x) λ4i Fi (ai ) 6 It follows then from the above estimates and Lemma 8.3 that " ! 5 ∇J (u), P δi = 4 × 64π 2 2 Log λi − − 16π 2 H4 (ai , ai ) 6 2 mλ4i Fi (ai ) π6 1 + 8(αi − 1) Log λi + (αi − 1)8 × 64π 2 Log λi × 1− u Ω Ke 2 mλ4j Fj (aj ) π6 2 2 1 + 8(αj − 1) Log λj + 64π G4 (ai , aj ) 1 − u Ω Ke
j =i
π2 5π 2 2 4 λ − 32(α Log λ F (a ) − 1)λ Log λ F (a ) − i i i i i i i i i u 3 36 Ω Ke 2 1 π +O 2 . + 8 Log λi Bi (ai )λ4i Fi (ai ) 6 λ Recall that mλ4i Fi (ai )π 2 /6 = 1 − γi , u Ω Ke it follows 5 2 ∇J (u), P δi = 4 × 64π 2 Log λi − − 16π H4 (ai , ai ) 6 × γi 1 + 8(αi − 1) Log λi − 8(αi − 1) Log λi 2 G4 (ai , aj ) γj 1 + 8(αj − 1) Log λj − 8(αj − 1) Log λj + 64π 2
!
"
2
j =i
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Fi (ai ) λ2i Fi (ai ) 80 1 + 8 × 64π 2 (αi − 1) Log λi + 64π 2 (1 − γi )(αi − 1) Log(λi ) + O 2 . 3 λ
− 8 × 64π 2 (1 − γi ) Log λi Bi (ai ) − 2 × 64π 2 (1 − γi ) Log λi
Finally, using Proposition 5.2 our proposition follows.
2
6. Morse lemma at infinity Using the above estimate, we are now able to construct a pseudogradient near neighborhood of potential critical points at infinity. The analysis of the end points of such a pseudogradient is easier than the genuine gradient flow. In order to construct such a pseudogradient we have to divide V (m, ε) in different regions, to construct an appropriate pseudogradient in each region and then glue up through convex combinations. Proposition 6.1. Let = 64mπ 2 with m 2 (resp. m = 1) and assume that the function K satisfies the condition (C1 ) (resp. (C0 )). Then there exists a pseudogradient W defined in V (m, ε) and satisfying the following properties:
There exists a constant C independent of u = m i=1 αi P δi such that (1)
(2)
m 1 |∇Fi (ai )| −∇J (u), W C + + |αi − 1| . λi λ2i i=1
!
"
m 1 ∂w(W ) |∇Fi (ai )| C −∇J (u + w), W + + + |α − 1| . i ∂(α, λ, a) λi λ2i i=1
(3) |W | is bounded and the only region where the maximum of the λi ’s increases along the flow lines of W is: (a1 , . . . , am ) is near a critical point q := (q1 , . . . , qm ) of F K (resp. fK ) with l(q) :=
q m F (qi ) i <0 q Fi (qi ) i=1
resp. q1 ∈ K− .
Proof. We divide the set V (m, ε) onto four subsets and in each one we will define a vector field. The required pseudogradient will be a convex combination of all them. First, we will focus on the case m 2. Let C and c be two large positive constants. We set
V1 := u ∈ V (m, ε): ∃i such that γi > C Log λi /λ2i ,
V2 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i and ∃j s.t. ∇Fj (aj ) > c/λj ,
V3 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i ; ∇Fi (ai ) < 2c/λi and l(q) > 0 ,
V4 := u ∈ V (m, ε): ∀i, |γi | < 2C Log λi /λ2i ; ∇Fi (ai ) < 2c/λi and l(q) < 0 , where γi is defined in Proposition 5.1.
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Remark. From Lemma 3.2, it is easy to see that m i=1
Log λ . γi = O λ2
(6.1)
Therefore, if u ∈ / V1 , we derive that γi > −(3/2)C Log λi /λ2i for each i and thus u ∈ V2 ∪ V3 ∪ V4 . For u =
αi P δi ∈ V1 , we define Y1 := −
i∈F1
λi
where F1 := i: γi > C Log λi /λ2i .
∂P δi , ∂λi
From Corollary 5.3, we get m m " Log λi −∇J (u), Y1 c γi c + c |αi − 1|. λ2i i∈F i=1 i=1
!
(6.2)
1
In V1 , we define W1 := Y1 + Wa ,
with Wa :=
m ∇F (ai ) 1 ∂P δi ξ λi ∇F (ai ) , |∇F (ai )| λi ∂ai
(6.3)
i=1
where ξ is a C ∞ positive function satisfying ξ(t) = 0 if t μ and ξ(t) = 1 if t 2μ where μ is a small positive constant. From Proposition 5.1 and (6.2), we derive that !
m m m " Log λi |∇F (ai )| + c |α − 1| + c . − ∇J (u), W1 c i 2 λi λ i i=1 i=1 i=1
(6.4)
For u = αi P δi ∈ V2 , we use the vector field Wa defined in (6.3). Note that, since u ∈ V2 , there exists at least one index i such that ξ(ai ) 1 (since μ is small with respect to c). Now, using Proposition 5.1, we derive " −∇J (u), Wa c
!
i: ξ(ai )1
c
|∇F (ai )| 1 +O 2 λi λ
(6.5)
m m m 1 |∇F (ai )| + c |α − 1| + c . i 2 λi λ i=1 i i=1 i=1
For u = αi P δi ∈ V3 , it is easy to get that the m-tuple (a1 , . . . , am ) is close to a critical point q := (q1 , . . . , qm ) of F K with l(q) > 0. In this case we define W3 := −
m i=1
λi
∂P δi , ∂λi
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and from Corollary 5.5 we derive that m ! " Log(λ) F (ai ) 2 −∇J (u), W3 = 32π . +O λ4 λ2 F (ai ) i=1 i
(6.6)
Now using claim (i) of Corollary 5.5, we get λ4i F (ai ) = λ4j F (aj ) + O λ2 Log(λ)
for each i, j,
which implies that m m Log(λ) 1 F (ai ) F (ai ) . + O = √ √ λ4 λ2 F (ai ) λ21 F (a1 ) i=1 F (ai ) i=1 i Thus, (6.6) becomes !
" −∇J (u), W3 c c
l(A) √ F (a1 )
λ21 m i=1
m m 1 |∇F (ai )| + c |α − 1| + c . i λi λ2i i=1 i=1
Finally, for u = αi P δi ∈ V4 , the m-tuple (a1 , . . . , am ) is close to a critical point q := (q1 , . . . , qm ) of F K with l(q) < 0, in this case we will increase the variables λi ’s and we move the concentration points ai ’s and the variables αi ’s by defining
(αi − 1) 2 Wαi , (αi − 1) W4 := μWa − 4W3 − ξ λ1 | (αi − 1)| where μ is a small positive constant, Wa and ξ are defined in (6.3) and Wαi :=
∂P δj 1 64π 2 P δi − G4 (ai , aj )λj Log(λi ) Log(λi ) ∂λj − 2−
j =i
16π 2 ∂P δi 5 − H4 (ai , ai ) λi . 6 Log(λi ) Log(λi ) ∂λi
Using (6.6) and Propositions 5.1, 5.6, we derive that m m m " 1 |∇F (ai )| −∇J (u), W4 c + c |α − 1| + c . i 2 λi λ i=1 i i=1 i=1
!
Now, we define the pseudogradient W by a convex combination of W1 , . . . , W4 and therefore claim (1) follows. Concerning claim (2), it follows as in [8] since the norm of w 2 is small with respect to the lower bound of claim (1). Finally, claim (3) follows from the definitions of Wi ’s
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and the fact that λi ∂P δi /∂λi and λ−1 i ∂P δi /∂ai are bounded. This completes the proof of our proposition in the case where m 2. Now, for m = 1, the proof is similar to the one done for m 2 and it is more easy since for example V1 = ∅ (from (5.1)). 2 As a consequence of Proposition 6.1 we are able to identify the critical points at infinity of J . Indeed we have Corollary 6.2. Let = 64mπ 2 , m 2 (resp. m = 1). The critical points at infinity of J are m
P δqi ,∞ ,
i=1
such that: q := (q1 , . . . , qm ) is a critical point of F K (resp. fK ) with qi = qj if i = j and l(q) :=
q m F (qi ) i < 0 resp. q1 ∈ K− . q Fi (qi ) i=1
Furthermore the energy level of such a critical point at infinity (q1 , . . . , qm )∞ denoted C∞ (q1 , . . . , qm )∞ is given by C∞ (q1 , . . . , qm )∞ = −
m 640mπ 2 π2 − 64mπ 2 Log m − 64π 2 Log K(qi ) 6 6
+
2 1 64π 2 2
m
i=1
H4 (qi , qi ) −
i=1
G4 (qi , qj ) (if m 2) .
j =i
Moreover the Morse index of such a critical point at infinity (q1 , . . . , qm )∞ is given by 5m − 1 − morse F K , (q1 , . . . , qm )
resp. 4 − morse(fK , q1 ) .
7. Proof of the main results First of all we point out that, just like for usual critical points, it is associated to each critical point at infinity x∞ of J stable and unstable manifolds Ws∞ (x∞ ) and Wu∞ (x∞ ), see [4]. These manifolds can be easily described once a finite dimensional reduction like the one we performed in Section 3 is established. The stable manifold is, indeed defined to be the set of points attracted by the asymptote while the unstable one is a shadow object, which is the limit of Wu (xλ ), xλ being critical point of the reduced problem and Wu (xλ ) its associated unstable manifold. Proof of Theorem 1.1. We prove the theorem by contradiction, therefore, we assume that Eq. (1.1) does not have solution. Let K Ω be a compact subset of Ω such that Ω retracts by deformation on K and denote by Bm (K) := K m ×σ m m−1 ,
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the set of formal barycenters in K, where m−1 := (α1 , . . . , αm ); αi 0,
m
αi = m .
i=1
It follows from [16,24] that J −L is homotopically equivalent to Bm (K) for some compact subset K Ω which is a retract by deformation of Ω itself. Therefore, it follows from the fact that Ω is not contractible that J −L is not contractible. From another part, according to Lemma 4.1, for L large enough we have that the whole space H retracts by deformation onto J L . Moreover, if (1.1) has no solution, then J L retracts by deformation on J −L and therefore H, which is contractible, retracts by deformation on J −L which is not contractible. A contradiction! To prove the multiplicity part of the statement, we observe that, it follows from Sard–Smale Theorem that for generic K’s, the solutions of (1.1) are all nondegenerate, in the sense that, the associated linearized operator does not admit zero as eigenvalue. Now in case Eq. (1.1) has infinitely many solutions, we are done, otherwise, there exists L1 1 such that all solutions are in the sublevel J L . We choose it to be larger than L in Lemma 4.2. It 1 follows then that J L is contractible. Therefore it follows from the Euler–Poincaré Theorem that
1 = χ J −L +
(−1)m(w) ,
(7.1)
w;∇J (w)=0
where m(w) denotes the Morse index of the solution w. It follows from [16,24] that J −L is homotopically equivalent to Bm (K) for some compact subset K Ω which is a retract by deformation of Ω itself. Therefore 1 −χ(Ω) + 1 (· · ·) −χ(Ω) + m . χ J −L = χ Bm (K) = 1 − m! Hence the lower bound on the number of solutions follows from (7.1) and (7.2).
(7.2) 2
Proof of Theorem 1.3. First, we remark that, for = 64π 2 , the functional J is lower bounded. We prove the existence result by contradiction. Therefore, we assume that Eq. (1.1) does not have solution. We recall, from Lemma 4.2, that there exists a large L > 0 such that H J L. Therefore J L is contractible. Now thanks to Lemma 4.1, we can compute the Euler–Poincaré characteristic of J L using the pseudogradient constructed in Proposition 6.1, whose “zeros”, under the assumption that (1.1) has no solution, are the critical points at infinity of J . It follows then from a theorem of Bahri and Rabinowitz [6] that JL
# {w∞ : critical point at infinity}
Wu (w∞ ).
(7.3)
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These critical points at infinity, according to Corollary 6.2, are in one-to-one correspondence to the elements of the set K− . It follows then from the Euler–Poincaré Theorem and the assumption of the theorem that (−1)morse(fK ,q) = 1 χ JL = q∈K−
which contradicts the fact that J L is contractible. Moreover we observe that for generic K’s, Eq. (1.1) admits only nondegenerate solutions. Now using Lemmas 4.1, 4.2 and a theorem of Bahri and Rabinowitz [6], we derive that #
JL
#
Wu (w∞ ) ∪
{w∞ : critical point at infinity}
Wu (w).
{w: critical point}
Now using the Euler–Poincaré Theorem, we derive that 1=
q∈K−
Our result follows.
(−1)morse(fK ,q) +
(−1)morse(J,w) .
w: critical point of J
2
Proof of Theorem 1.4. We set #
X∞ :=
Wu∞ (q∞ ),
{q∈K− ; ι(q)
where q∞ denotes the critical point at infinity associated to q ∈ K− and Wu∞ (q∞ ) its associated unstable manifold. Observe that X∞ is a stratified set of top dimension k − 1, which is contractible in H. Let U denote the image of such a contraction. To prove the first part, arguing by contradiction, we assume that (1.1) has no solution. Using the pseudogradient constructed in Proposition 6.1, we can deform U . By tranversality arguments, we can assume that such a deformation avoids all critical points at infinity of Morse index greater than or equals k + 1. Note that, from the assumption (1) of the theorem, there is no critical point at infinity with Morse index k. It follows then from a theorem of Bahri and Rabinowitz [6] that #
U
Wu∞ (q∞ ) = X∞ .
{q∈K− ; ι(q)
Hence from the Euler–Poincaré Theorem and the assumption of Theorem 1.4 we get 1 = χ(U ) =
(−1)morse(fK ,q) = 1,
q∈K− ; ι(q)
which is a contradiction. Hence the existence part. Regarding the multiplicity result, we observe that for generic K’s the functional J admits only nondegenerate critical points. Hence the set U will be deformed onto
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U
#
Wu∞ (q∞ ) ∪
{q∈K− ; ι(q)
#
Wu (w),
{w∈Ck }
where Ck denotes the set of the critical points of Morse index less than or equals k, which are dominated by U . Finally, if the cardinal of Ck is finite (if not, we are done), using the Euler–Poincaré Theorem, the proof follows. 2 Proof of Theorem 1.5. Recall that it follows from Corollary 6.2 that the critical points at infinity of J64π 2 are in one-to-one correspondence with the elements of the set
K− := y ∈ Ω; ∇fK (y) = 0 and (Log K)(y) − 64π 2 1 H4 (y, y) < 0 . Now observe that (Log K) =
K |∇K|2 − K K2
and since ⎧ ⎨ 1 H4 (., y) = 0 1 1 ⎩ 1 H4 (., y) = − 4π 2 |x − a|2
in Ω, on ∂Ω,
(7.4)
it follows from the maximum principle that 1 H4 (., y) < 0
in Ω.
Hence if we choose c0 := min −64π 2 1 H4 (., y) y∈Ω
then K− = ∅. In particular every minimizing sequence has a converging subsequence and therefore the global minimum of J64π 2 is achieved. 2 Proof of Theorem 1.6. For the existence result, arguing by contradiction, we assume that Eq. (1.1) has no solution. Therefore using Lemmas 4.1 and 4.2, there exists L > 0 large enough, such that H J L J −L ∪ σ, where σ ⊂ V (m, ε) for some small ε > 0. Recall that, according to Corollary 6.2, the critical points at infinity of J64mπ 2 ; m 2 are in one-to-one correspondence with the elements of the set F∞ , defined as
F∞ := q = (q1 , . . . , qm ) a critical point of F K ; such that l(q) < 0 .
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Therefore 1 = χ JL = (−1)ι(q) + χ(J−L ).
(7.5)
q∈F∞
Moreover, it follows from [16,24] that J −L is homotopically equivalent to Bm−1 (K) for some K Ω a compact subset of Ω, which is a retract by deformation of Ω. Now it is well known, see [14,24,21] that χ Bm−1 (K) = 1 −
1 −χ(Ω) + 1 (· · ·) −χ(Ω) + m − 1 . (m − 1)!
It follows then, under the assumption that Eq. (1.1) has no solution for = 64mπ 2 , m 2: 1=
(−1)ι(q) + 1 −
q∈F∞
1 −χ(Ω) + 1 (· · ·) −χ(Ω) + m − 1 . (m − 1)!
A contradiction with the assumption of Theorem 1.6. Concerning the multiplicity result, assume that (1.1) has a finite number of solutions (if not, we are done) and for generic K’s, these solutions are nondegenerate. Arguing as in the proof of Theorem 1.3, (7.5) becomes 1 = χ J L2 = (−1)ι(q) + q∈F∞
(−1)morse(J,w) + χ(J−L2 ),
{w: criticalpoint}
where L2 satisfies: L2 L and |J (w)| < L2 for each critical point w. (L is defined in Corollary 3.9.) Finally, using the Euler–Poincaré Theorem, the proof follows. 2 8. Appendix In this appendix, the concentration points are assumed to be in a compact set of Ω and the concentration speeds are of the same order and large enough. Furthermore, for sake of simplicity, O(1/λα ) designs some quantities like O( 1/λαi ). Lemma 8.1. Let η > 0 such that Bη (a) := B(a, η) ⊂ Ω. (1) On Bη (a) there holds 16π 2 Log(λ) − 64π H4 (x, a) + 2 H2 (x, a) + O , λ λ4 ∂P δa,λ 8 32π 2 Log(λ) λ , = − 2 H2 (x, a) + O ∂λ 1 + λ2 |x − a|2 λ λ4 8λ(x − a) Log(λ) 64π 2 ∂H4 (a, x) 16π 2 ∂H2 (a, x) . = + 3 +O − λ ∂a ∂a 1 + λ2 |x − a|2 λ λ4
P δa,λ = Log
1 ∂P δa,λ λ ∂a
λ8 (1 + λ2 |x − a|2 )4
2
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(2) On Ω \ Bη (a) there holds 16π 2 Log(λ) , G (x, a) + O 2 λ2 λ4 ∂P δa,λ 32π 2 Log(λ) λ , = 2 G2 (x, a) + O λ λ λ4 64π 2 ∂G4 (a, x) 16π 2 ∂G2 (a, x) Log(λ) . = − 3 +O λ ∂a ∂a λ λ4
P δa,λ = 64π 2 G4 (a, x) −
1 ∂P δa,λ λ ∂a Lemma 8.2.
∂P δa,λ 64π 4 Log(λ) 2 P δa,λ , λ , = 4 × 64π − 32 × 2 H2 (a, a) + O ∂λ λ λ4 2 1 ∂H4 1 ∂P δai ,λi 1 P δai ,λi , = − 64π 2 (ai , ai ) + O 3 , λi ∂ai λi ∂a λ
where ∂H4 /∂a denotes the derivative with respect to the first variable. For i = j and λi |ai − aj | large enough, there holds 64π 4 ∂P δai ,λi Log(λ) = 32 × 2 G2 (ai , aj ) + O , P δaj ,λj , λi ∂λi λi λ4i 1 ∂P δai ,λi 1 2 2 1 ∂G4 (ai , aj ) P δaj ,λj , = 64π +O 3 . λi ∂ai λi ∂ai λ Lemma 8.3. |P δa,λ |2 dx = 8 × 64π 2 Log λ − Ω
− 16 × 64π 4
2 640π 2 − 64π 2 H4 (a, a) 3
1 H4 (a, a) 64π 4 Log λ , + 16 H (a, a) + O 2 λ2 λ2 λ4
where 1 H4 denotes the Laplacian of H4 with respect to the first variable. For i = j and λi |ai − aj | large enough, there holds 2 π4 P δai ,λi , P δaj ,λj = 64π 2 G4 (ai , aj ) + 16 × 64 2 1 G4 (ai , aj ) λi π4 Log λ , − 16 × 64 2 G2 (ai , aj ) + O λ4 λj where 1 G4 denotes the Laplacian of G4 with respect to the first variable.
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Remark 8.4. It is easy to see that −1 H4 (x, a) = H2 (x, a)
and −1 G4 (x, a) = G2 (x, a).
Acknowledgments Part of this work has been written while the first author enjoyed the support of the collaborative research group SFB TR 71 and the hospitality of the University of Tübingen. He would like, in particular to acknowledge the excellent working conditions. References [1] David R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (2) (1988) 385–398. [2] Adimurthi, F. Robert, M. Struwe, Concentration phenomena for Liouville’s equation in dimension four, J. Eur. Math. Soc. (JEMS) 8 (2) (2006) 171–180. [3] A. Bahri, Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, vol. 182, Longman Scientific & Technical, Harlow, New York, 1989, copublished in the United States with John Wiley & Sons. [4] A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension, A celebration of John F. Nash, Jr., Duke Math. J. 81 (2) (1996) 323–466. [5] A. Bahri, J.-M. Coron, The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal. 95 (1) (1991) 106–172. [6] A. Bahri, P. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (6) (1991) 561–649. [7] S. Baraket, M. Dammak, T. Ouni, F. Pacard, Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (6) (2007) 875–895. [8] Ben Ayed, Chen, Chtioui, Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J. 84 (1996) 633–667. [9] H. Brézis, F. Merle, Uniform estimates and blow-up behavior for solutions of −u = V (x)eu in two dimensions, Comm. Partial Differential Equations 16 (1991) 1223–1253. [10] A. Chang, Non-Linear Elliptic Equations in Conformal Geometry, Zur. Lect. Adv. Math., European Mathematical Society (EMS), Zürich, 2004. [11] A. Chang, Paul Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 (1) (1995) 171–212. [12] A. Chang, P. Yang, Non-linear partial differential equations in conformal geometry, in: Proceedings of the International Congress of Mathematicians, vol. I, Beijing, 2002, Higher Ed. Press, Beijing, 2002, pp. 189–207. [13] C.-C. Chen, C.-S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math. 55 (6) (2002) 728–771. [14] C.-C. Chen, C.-S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56 (12) (2003) 1667–1727. [15] M. Clapp, C. Munoz, M. Musso, Singular limits for the bi-Laplacian operator with exponential nonlinearity in R4 , Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (5) (2008) 1015–1041. [16] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Ann. of Math. (2) 168 (3) (2008) 813–858. [17] O. Druet, F. Robert, Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc. 134 (3) (2006) 897–908. [18] J. Horak, M. Lucia, A Minmax theorem in the presence of unbounded Palais–Smale sequences, Israel J. Math. 172 (2009) 125–143. [19] Yan Yan Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (2) (1999) 421–444. [20] Y. Li, I. Shafrir, Blow-up analysis for solutions of −u = V eu in dimension two, Indiana Univ. Math. J. 43 (1994) 1255–1270. [21] Chang-Shou Lin, Juncheng Wei, Sharp estimates for bubbling solutions of a fourth order mean field equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (4) (2007) 599–630.
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[22] C.-S. Lin, L.-P. Wang, Juncheng Wei, Topological degree for solutions of a fourth order mean field equation, preprint. [23] M. Lucia, A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal. 30 (2007) 113–138. [24] A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations 13 (11–12) (2008) 1109–1129. [25] F. Robert, Quantization effects for a fourth-order equation of exponential growth in dimension 4, Proc. Roy. Soc. Edinburgh Sect. A 137 (3) (2007) 531–553. [26] F. Robert, Juncheng Wei, Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition, Indiana Univ. Math. J. 57 (5) (2008) 2039–2060.
Journal of Functional Analysis 258 (2010) 3195–3225 www.elsevier.com/locate/jfa
Operator algebras of functions ✩ Meghna Mittal ∗ , Vern I. Paulsen Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA Received 2 September 2009; accepted 5 January 2010 Available online 8 February 2010 Communicated by K. Ball
Abstract We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler’s theory of commuting contractions and the Arveson–Drury–Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices. © 2010 Elsevier Inc. All rights reserved. Keywords: Reproducing kernel Hilbert spaces; Agler type factorization results; Operator algebras
1. Introduction A concrete operator algebra A is just a subalgebra of B(H), the bounded operators on a Hilbert space H. The operator norm on B(H) gives rise to a norm on A. Moreover, the identification Mn (A) ∼ = A ⊗ Mn ⊆ B(H ⊗ Cn ) ∼ = B(Hn ) endows the matrices over A with a family of norms in a natural way, where Mn denotes the algebra of n × n matrices. It is common practice to identify two operator algebras A and B as being the “same” if and only if there exists an algebra isomorphism π : A → B that is not only an isometry, but which also preserves all the ✩
This research was supported in part by NSF grant DMS-0600191.
* Corresponding author.
E-mail addresses:
[email protected] (M. Mittal),
[email protected] (V.I. Paulsen). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.006
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matrix norms, that is such that (π(ai,j ))Mn (B) = (ai,j )Mn (A) , for every n and every element (ai,j ) ∈ Mn (A). Such a map π is called a completely isometric isomorphism. In [15] an abstract characterization of operator algebras, in the above sense, was given and since that time a theory of such algebras has evolved. For more details on the abstract theory of operator algebras, see [13,25] or [29]. In this note we present a theory for a special class of abstract abelian operator algebras which contains many important examples arising in function theoretic operator theory, including the Schur–Agler and the Arveson–Drury–Popescu algebras. This theory will allow us to answer certain types of questions about such algebras in a unified manner. We will prove that our hypotheses give an abstract characterization of operator algebras that are completely isometrically isomorphic to multiplier algebras of vector-valued reproducing kernel Hilbert spaces. This result can be viewed as expanding on the Agler–McCarthy concept of realizable algebras [3]. Our results will show that under certain mild hypotheses, operator algebra norms which are defined by taking the supremum of certain families of operators on Hilbert spaces of arbitrary dimensions can be obtained by restricting the family of operators to finite-dimensional Hilbert spaces. Thus, in a certain sense, which will be explained later, our results give conditions that guarantee that an algebra is residually finite-dimensional. Finally, we apply our results to study “quantized function theories” on various domains. Our work in this direction should be compared to that of Ambrozie and Timotin [5], Ball and Bolotnikov [9] and Kalyuzhnyi-Verbovetzkii [22]. These authors obtain the same type of factorization theorem via unitary colligation methods as we obtain via operator algebra methods, but they assume somewhat different (and in many cases stronger) hypotheses. We, also, obtain a bit more information about the algebras themselves, including the fact that in many cases their norms can be obtained as supremums over matrices instead of operators. We now give the relevant definitions. Recall that given any set X the set of all complex-valued functions on X is an algebra over the field of complex numbers. Definition 1.1. We call A an operator algebra of functions on a set X provided: (1) A is a subalgebra of the algebra of functions on X, (2) A separates the points of X and contains the constant functions, (3) for each n, Mn (A) is equipped with a norm · Mn (A) , such that the set of norms satisfy the BRS axioms [15] to be an abstract operator algebra, (4) for each x ∈ X, the evaluation functional, πx : A → C, given by πx (f ) = f (x) is bounded. A few remarks and observations are in order. First note that if A is an operator algebra of functions on X and B ⊆ A is any subalgebra, which contains the constant functions and still separates points, then B, equipped with the norms that Mn (B) inherits as a subspace of Mn (A) is still an operator algebra of functions. The basic example of an operator algebra of functions is ∞ (X), the algebra of all bounded functions on X. If for (fi,j ) ∈ Mn (∞ (X)) we set (fi,j ) M
n (
∞ (X))
= (fi,j )∞ ≡ sup fi,j (x) M : x ∈ X , n
where · Mn is the norm on Mn obtained via the identification Mn = B(Cn ), then it readily follows that properties (1)–(4) of the above definition are satisfied. Thus, ∞ (X) is an operator algebra of functions on X in our sense and any subalgebra of ∞ (X) that contains the constants
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and separates points will be an operator algebra of functions on X when equipped with the subspace norms. Proposition 1.2. Let A be an operator algebra of functions on X, then A ⊆ ∞ (X), and for every n and every (fi,j ) ∈ Mn (A), we have (fij )∞ (fij )Mn (A) . Proof. Since πx : A → C is bounded and the norm is sub-multiplicative, we have that for any f ∈ A, |f (x)|n = |πx (f n )| πx f n πx f n . Taking the n-th root of each side of this inequality and letting n → +∞, yields |f (x)| f , and hence, f ∈ ∞ (X). Note also that πx = 1. Finally, since every bounded, linear functional on an operator space is completely bounded and the norm and the cb-norm are equal, we have that πx cb = πx = 1. Thus, for (fi,j ) ∈ Mn (A), we have (fi,j (x))Mn = (πx (fi,j )) (fi,j )Mn (A) . 2 Given an operator algebra A of functions on a set X and F = {x1 , . . . , xk } a set of k distinct points in X, we set IF = {f ∈ A: f (x) = 0 for all x ∈ F }. Note that for each n, Mn (IF ) = {f ∈ Mn (A): f (x) = 0 for all x ∈ F }. The quotient space A/IF has a natural set of matrix norms given by defining (fi,j + IF ) = inf{(fi,j + gi,j )Mn (A) : gi,j ∈ IF }. Alternatively, this is the norm on Mn (A/IF ) that comes via the identification, Mn (A/IF ) = Mn (A)/Mn (IF ), where the latter space is given its quotient norm. It is easily checked that this family of matrix norms satisfies the BRS conditions and so gives A/IF the structure of an abstract operator algebra. We let πF (f ) = f + IF denote the quotient map πF : A → A/IF so that for each n, πF(n) : Mn (A) → Mn (A/IF ) ∼ = Mn (A)/Mn (IF ). Since A is an algebra which separates points on X and contains constant functions, it follows that there exist functions f1 , . . . , fk ∈ A, such that fi (xj ) = δi,j , where δi,j denotes the Dirac delta function. If we set Ej = πF (fj ), then it is easily seen that whenever f ∈ A and f (xi ) = λi , i = 1, . . . , k, then πF (f ) = λ1 E1 +· · ·+λk Ek . Moreover, Ei Ej = δi,j Ei , and E1 +· · ·+Ek = 1, where 1 denotes the identity of the algebra A/IF . Thus, A/IF = span{E1 , . . . , Ek }, is a unital algebra spanned by k commuting idempotents. Such algebras were called k-idempotent operator algebras in [26] and we will use a number of results from that paper. Definition 1.3. An operator algebra of functions A on a set X, is called a local operator algebra of functions if it satisfies (n) supπF (fij ) = (fij )
for all (fij ) ∈ Mn (A) and for every n
F
where the supremum is taken over all finite subsets of X. The following result shows that every operator algebra of functions can be re-normed so that it becomes local. Proposition 1.4. Let A be an operator algebra of functions on X, let AL = A and define a family of matrix norms on AL , by setting (fi,j )Mn (AL ) = supF (πF (fi,j ))Mn (A/IF ) , where the supremum is taken over all finite subsets of X. Then AL is a local operator algebra of functions on X and the identity map, id : A → AL , is completely contractive.
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Proof. It is clear from the definition of the norms on AL that the identity map is completely contractive and it is readily checked that AL is an operator algebra of functions on X. Let π˜ F : AL → AL /IF , denote the quotient map, so that π˜ F (f ) = inf{f +gL : g ∈ IF } inf{f + g: g ∈ IF } = πF (f ), since f + gL f + g. We claim that for any f ∈ A, and any finite subset F ⊆ X, we have that πF (f ) = π˜ F (f ). To see the other inequality note that for g ∈ IF , and G ⊆ X a finite set, we have f + gL = supG πG (f + g) πF (f + g) = πF (f ). Hence, π˜ F (f ) πF (f ), and equality follows. A similar calculation shows that (π˜ F (fi,j )) = (πF (fi,j )), for any matrix of functions. Now it easily follows that AL is local, since sup π˜ F (fi,j ) = sup πF (fi,j ) = (fi,j )M (A ) . n L F
2
F
Definition 1.5. Given an operator algebra of functions A on X we say that f : X → C is a BPW limit of A if there exists a uniformly bounded net (fλ )λ ∈ A that converges pointwise on X to f . We let A˜ denote the set of BPW limits of functions in A. We say that A is BPW complete, if ˜ A = A. ˜ we set Given (fi,j ) ∈ Mn (A), (fi,j )
Mn (A˜ )
λ λ C , = inf C: fi,j (x) = lim fi,j (x) and fi,j λ
λ ∈ A. with fi,j
˜ It is also easIt is easily checked that for each n, the above formula defines a norm on Mn (A). ily checked that a matrix-valued function, (fi,j ) : X → Mn is the pointwise limit of a uniformly λ ) ∈ M (A) if and only if f bounded net (fi,j n i,j ∈ A˜ for every i, j . ˜ Then Lemma 1.6. Let A be an operator algebra of functions on X and let (fi,j ) ∈ Mn (A). (fi,j )
Mn (A˜ )
F = inf C: for each finite F ⊆ X there exists gi,j ∈ A with F F (fi,j |F ) = gi,j F , gi,j C .
Proof. The collection of finite subsets of X determines a directed set, ordered by inclusion. If F ) satisfying the conditions of the right-hand set, we choose for each finite set F , functions (gi,j then these functions define a net that converges BPW to (fi,j ) and hence, the right-hand side is λ ) that converges pointwise to (f ) and satisfies larger than the left. Conversely, given a net (fi,j i,j λ ) C and any finite set F = {x , . . . , x }, choose functions in A such that f (x ) = δ . (fi,j 1 k i j i,j λ (x )), then (g λ ) = (f λ ) + Aλ f + · · · + Aλ f ∈ M (A) and If we let Aλl = (fi,j (xl )) − (fi,j l n i,j i,j k k 1 1 λ ) (f λ ) + Aλ f + · · · + Aλ f . is equal to (fi,j ) on F . Moreover, (gi,j M 1 M k A A n n i,j k 1 Thus, given > 0, since the functions f1 , . . . , fk depend only on F , we may choose λ so that λ ) < C + . This shows the other inequality. 2 (gi,j Lemma 1.7. Let A be an operator algebra of functions on the set X, then A˜ equipped with the ˜ given in Definition 1.5 is an operator algebra. collection of norms on Mn (A)
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Proof. It is clear from the definition of A˜ that it is an algebra. Thus, it is enough to check that the axioms of BRS are satisfied by the algebra A˜ equipped with the matrix norms given in Definition 1.5. ˜ then for > 0 there exists If L and M are scalar matrices of appropriate sizes and G ∈ Mn (A), Gλ ∈ Mn (A) such that limλ Gλ (x) = G(x) for all x ∈ X and supλ Gλ Mn (A) GMn (A˜ ) + . Since A is an operator space, LGλ M ∈ Mn (A) and LGλ MMn (A) LGλ Mn (A) M. Note that it follows that LGMMn (A˜ ) LGMn (A˜ ) M, since LGλ M → LGM pointwise and supλ LGλ MMn (A) L(GMn (A˜ ) + )M for any > 0. ˜ then for every > 0 there exist Gλ , Hλ ∈ Mn (A) such that limλ Gλ (x) = If G, H ∈ Mn (A), G(x) and limλ Hλ (x) = H (x) for x ∈ X. Also, we have that supλ Gλ Mn (A) GMn (A˜ ) + and supλ Hλ Mn (A) H Mn (A˜ ) + . Let L = GH and Lλ = Gλ Hλ . Since A is matrix normed algebra, Lλ ∈ Mn (A) and Lλ Mn (A) Gλ Mn (A) Hλ Mn (A) for every λ. This implies that limλ Lλ (x) = L(x) and that LMn (A˜ ) sup Lλ Mn (A) sup Gλ Mn (A) sup Hλ Mn (A) . λ
λ
λ
This yields LMn (A˜ ) GMn (A˜ ) H Mn (A˜ ) , and so the multiplication is completely contractive. ˜ and H ∈ Mm (A). ˜ Given > 0 Finally, to see that the L∞ conditions are met, let G ∈ Mn (A) there exist Gλ ∈ Mn (A) and Hλ ∈ Mm (A) such that limλ Gλ (x) = G(x), limλ Hλ (x) = H (x) and supλ Gλ Mn (A) GMn (A˜ ) + , supλ Hλ Mn (A) H Mn (A˜ ) + . Note that Gλ ⊕ Hλ ∈ Mn+m (A) and Gλ ⊕ Hλ = max{Gλ Mn (A) , Hλ Mn (A) } for every λ ˜ and which implies that G ⊕ H ∈ Mn+m (A), G ⊕ H Mn+m (A˜ ) sup Gλ ⊕ Hλ = sup max Gλ Mn (A) , Hλ Mm (A) λ
λ
= max sup Gλ Mn (A) , supH λ M (A) m λ λ max GMn (A˜ ) + , H Mm (A˜ ) + . This shows that G ⊕ H Mn+m (A˜ ) max{GMn (A˜ ) , H Mm (A˜ ) }, and so the L∞ condition follows. This completes the proof of the result. 2 Lemma 1.8. If A is an operator algebra of functions on the set X, then A˜ equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (fi,j ) ∈ Mn (A), (fi,j )Mn (A˜ ) = (fi,j )Mn (AL ) . Proof. It is clear from the definition of the norms on A˜ that the identity map from A to A˜ is completely contractive and thus A ⊆ A˜ as sets. This shows that A˜ separates points of X and contains the constant functions. ˜ and > 0, then there exists a net (f λ ) ∈ Mn (A) such that limλ (f λ (x)) = Let (fij ) ∈ Mn (A) ij ij (fij (x)) for each x ∈ X and supλ (fijλ )Mn (A) (fij )Mn (A˜ ) + . Since A is an operator al-
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gebra of functions on the set X, we have that (fijλ )∞ (fijλ )Mn (A) . Thus, supλ (fijλ )∞ (fij )Mn (A˜ ) + . Fix z ∈ X, then fij (z) = lim f λ (z) sup f λ ij
λ
λ
ij
∞
(fij )M (A˜ ) + . n
By letting → 0 and taking the supremum over z ∈ X, we get that (fij )∞ (fij )Mn (A˜ ) . Hence, A˜ is an operator algebra of functions on the set X. ˜ f |F ≡ 0} and let (fij ) ∈ Mn (A). ˜ Then, clearly supF (fij + I˜F ) Set I˜F = {f ∈ A: Mn (A˜ /I˜F ) ˜ (fij ) ˜ . To see the other inequality, assume that supF (fij + IF ) < 1. Then for every fiM n ( A)
˜ such that (hF )|F = (f F )|F and supF hF 1. Fix a nite F ⊆ X there exists (hFij ) ∈ Mn (A) ij ij ij ˜ Then for all finite F ⊆ X there exists (k F ) ∈ Mn (A) such that set F ⊆ X and (hF ) ∈ Mn (A).
ij
ij
(kijF )|F = (hFij )|F and supF kijF 1. In particular, let F = F then (kijF )|F = (hFij )|F = (fij )|F and supF kijF 1. Hence, (fij )Mn (A˜ ) 1, and (fij )Mn (A˜ ) supF (fij + I˜F )Mn (A˜ /I˜ ) . F ˜ Finally, given that (fij ) ∈ Mn (A), (fij ) ˜ = supF (fij + IF ) ˜ ˜ . Note that for M n (A )
M n ( A/ I F )
any F ⊆ X we have (fij + I˜F )Mn (A˜ /I˜ ) (fij + IF )Mn (A/IF ) , since IF ⊆ I˜F . We claim F that for any (fij ) ∈ Mn (A), and for any finite subset F ⊆ X, we have that (fij + IF ) = (fij + I˜F ). To see the other inequality, let (gij ) ∈ Mn (I˜F ). Then for > 0 and G ⊆ X, we may G G choose (hG ij ) ∈ Mn (A) such that (hij )|G = (fij + gij )|G and supG (hij ) (fij + gij ) + . Hence, (fij + IF ) = (hFij + IF ) (hFij ) (fij + gij ) + . Since > 0 was arbitrary, the equality follows. Now it is clear that, (fij ) and so the result follows.
Mn (A˜ )
= sup (fij + IF ) = (fij )M (A ) , n L F
2
Corollary 1.9. If A is a BPW complete operator algebra then AL = A˜ completely isometrically. Proof. Since A is BPW complete, A = A˜ as sets. But by Lemma 1.6, the norm defined on AL ˜ 2 agrees with the norm defined on A. Remark 1.10. In the view of above corollary, we denote the norm on A˜ by · L . Lemma 1.11. If A is an operator algebra of functions on X, then Ball(AL ) is BPW dense in ˜ and A˜ is BPW complete, i.e., A˜˜ = A. ˜ Ball(A) Proof. It can be easily checked that the statement is equivalent to showing that AL is BPW dense ˜ since the other containment follows immediately by ˜ We’ll only prove that AL BPW ⊆ A, in A. ˜ the definition of A. Let {fλ } be a net in AL such that fλ → f pointwise and supλ fλ AL < C. Then for fixed F ⊆ X and > 0, there exists λF such that |fλF (z) − f (z)| < for z ∈ F . Also since supλ fλ < C, there exists gλF ∈ IF such that fλF + gλF < C. Note that the function
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˜ Thus, AL is hF = fλF + gλF ∈ A satisfies hF A < C, and hF → f pointwise. Hence, f ∈ A. ˜ BPW dense in A. Finally, a similar argument yields that A˜ is BPW complete. 2 All the above lemmas can be summarized as the following theorem. Theorem 1.12. If A is an operator algebra of functions on X, then A˜ is a BPW complete local operator algebra of functions on X which contains AL completely isometrically as a BPW dense subalgebra. Definition 1.13. Given an operator algebra of functions A on X, we call A˜ the BPW completion of A. We now present a few examples to illustrate these concepts. We will delay the main family of examples to a later section. Example 1.14. If A is a uniform algebra, then there exists a compact, Hausdorff space X, such that A can be represented as a subalgebra of C(X) that separates points. If we endow A with the matrix-normed structure that it inherits as a subalgebra of C(X), namely, (fi,j ) = (fi,j )∞ ≡ sup{(fi,j )Mn : x ∈ X}, then A is a local operator algebra of functions on X. Indeed, to achieve the norm, it is sufficient to take the supremum over all finite subsets consisting of one point. In this case the BPW completion A˜ is completely isometrically isomorphic to the subalgebra of ∞ (X) consisting of functions that are bounded, pointwise limits of functions in A. Example 1.15. Let A = A(D) ⊆ C(D− ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying Mn (A(D)) ⊆ Mn (C(D− )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it’s usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D− . If we regard it as a local operator algebra of functions on D− , then A(D) A(D). n To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f (z) → χ{1} , the characteristic function of the singleton {1}. However, if we regard A(D) as a local operator algebra of functions on D, then its BPW = H ∞ (D), the bounded analytic functions on the disk, with its usual operator completion A(D) structure. Example 1.16. Let X = D, 0 < < 1 and A = {f |X : f ∈ H ∞ (D)}. If we endow A with the matrix-normed structure on H ∞ (D), then A is an operator algebra of functions on X. Also, it ˜ Indeed, if F = can be verified that A is a local operator algebra of functions and that A = A. (fij ) ∈ Mn (A) with (fij + IY )∞ < 1 for all finite subset Y ⊆ X, then there exists HY ∈ Mn (A) such that HY ∞ 1 and HY → F pointwise on X. Note by Montel’s theorem there will exist a subnet HY and G ∈ Mn (H ∞ (D)) such that G∞ 1 and HY → G uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, F Mn (A) 1 and so A is a local operator algebra. A similar argument shows that if f is a BPW limit on X, then there exists g ∈ H ∞ (D) such that g|X = f , and so A is BPW complete. By Lemma 1.11, A˜ = A completely isometrically.
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∞ Example 1.17. Let = H (D) but endowed with a new norm. Fix b > 1, and set F = A max{F ∞ , F ( 00 0b )}, F ∈ Mn (A). It can be easily verified that A is a BPW complete operator algebra of functions. However, we also claim that A is local. To prove this we proceed by contradiction. Suppose there exists F = (fij ) ∈ Mn (H ∞ (D)) such that F > 1 > c, where c = supY (fij + IY ). In this case, F = F ( 00 0b ), since (fij + IY ) = F (λ) when Y = {λ}. Let = 1−c 4b and Y = {0, } ⊆ D, then ∃G ∈ Mn (H ∞ (D)) such that G|Y = 0 and F + G < 1+c 2 . Thus, we z− ∞ can write BY (z) = 1−¯ z , so that we can write G(z) = zBY (z)H (z), for some H ∈ Mn (H ). It follows that H ∞ < 2, since G∞ < 2. We now consider
0 b 0 b 0 b + G (F + G) F 1< 0 0 0 0 0 0
1+c 0 bG (0) = 1 + c + bBY (0)H (0) + 0 0 2 2
1+c 1+c 1−c + 2b = + 2b = 1, 2 2 4b
which is a contradiction. Example 1.18. This is an example of a non-local algebra that arises from boundary behavior. 1 )}, Let A = A(D) equipped with the family of matrix norms F = max{F ∞ , F ( 10 −1 F ∈ Mn (A). Note that when F (1) = F (−1), then F = F ∞ . It is easy to check that A is an operator algebra of on the set D. Also, it can be verified that A is not local. To see this, functions 1 > 1. Fix α > 0, such that 1 + 2α < z. For each Y = {z1 , z2 , . . . , zn }, note that z = 10 −1 z−zi we define BY (z) = ni=1 ( 1− z¯i z ) and choose h ∈ A such that h(1) = −BY (1), h(−1) = BY (−1), and h∞ 2. Let g(z) = z + BY (z)h(z)α, then g ∈ A, g(1) = g(−1) and g|Y = z|Y . Hence, πY (z) = πY (g) g = g∞ 1 + 2α < z. Thus, since α was arbitrary, supY ⊆D πY (z) = 1 < z and hence A is not local. Example 1.19. This example shows that one can easily build non-local algebras by adding “values” outside of the set X. Let A be the algebra of polynomials regarded as functions on the set X = D. Then A endowed with the matrix-normed structure as (pij ) = max{(pij )∞ , (pij (2))}, is an operator algebra of functions on the set X. To see that A is not local, let p ∈ A be such that p∞ < |p(2)|. For each finite subset Y = {z1 , . . . , zn } of X, let |p(2)|−p∞ > 0. Note that hY (z) = ni=1 (z − zi ) and gY (z) = p(z) − αhY (z)p(2), where α = 2|p(2)|h Y ∞ gY (1 − α)|p(2)| and gY |Y = p|Y . Hence, πY (p) = πY (gY ) gY (1 − α)|p(2)| < p. It follows that A is not local. Finally, be BPW complete. For example, if we take
observe that in this case A cannot 1 for z ∈ D and pn < f , which implies that pn = 13 ni=0 ( 3z )i ∈ A then pn (z) → f (z) = 3−z ˜ AL A. Example 1.20. It is still an open problem as to whether or not every unital contractive, homomorphism ρ : H ∞ (D) → B(H) is completely contractive. For a recent discussion of this problem see [27]. Let’s assume that ρ is a contractive homomorphism that is not completely contractive.
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Let B = H ∞ (D), but endow it with the family of matrix-norms given by, (fi,j ) = max (fi,j ) , ρ(fi,j ) . ∞ Note that |f | = f ∞ , for f ∈ B. It is easily checked that B is a BPW complete operator algebra of functions on D. However, since every contractive homomorphism of A(D) is completely contractive, we have that for (fi,j ) ∈ Mn (A(D)), |(fi,j )| = (fi,j )∞ . If Y = {x1 , . . . , xk } is a finite subset of D and F = (fi,j ) ∈ Mn (B), then there is G = (gi,j ) ∈ Mn (A(D)), such that F (x) = G(x) for all x ∈ Y , (n) (n) and G∞ = F ∞ . Hence, πY (F ) F ∞ . Thus, supY πY (F ) = F ∞ . It follows that B is not local and that B˜ = BL = H ∞ (D), with its usual supremum norm operator algebra structure. In particular, if there does exist a contractive but not completely contractive representation of H ∞ (D), then we have constructed an example of a non-local BPW complete operator algebra of functions on D. 2. A characterization of local operator algebras of functions The main goal of this section is to prove that every BPW complete local operator algebra of functions is completely isometrically isomorphic to the algebra of multipliers on a reproducing kernel Hilbert space of vector-valued functions. Moreover, we will show that every such algebra is a dual operator algebra in the precise sense of [13]. We will then prove that for such BPW algebras, weak∗ -convergence and BPW convergence coincide on bounded balls. First we need to recall a few basic facts and some terminology from the theory of vectorvalued reproducing kernel Hilbert spaces. Given a set X and a Hilbert space H, then by a reproducing kernel Hilbert space of H-valued functions on X, we mean a vector space L of H-valued functions on X that is equipped with a norm and an inner product that makes it a Hilbert space and which has the property that for every x ∈ X, the evaluation map Ex : L → H, is a bounded, linear map. Recall that given a Hilbert space H, a matrix of operators, T = (Ti,j ) ∈ Mk (B(H)) is regarded as an operator on the Hilbert space H(k) ≡ H ⊗ Ck , which is the direct sum of k copies of H. A function K : X × X → B(H), where H is a Hilbert space, is called a positive definite operator-valued function on X, provided that for every finite set of (distinct) points {x1 , . . . , xk } in X, the operator-valued matrix, (K(xi , xj )) is positive semidefinite. Given a reproducing kernel Hilbert space of H-valued functions, if we set K(x, y) = Ex Ey∗ , then K is positive definite and is called the reproducing kernel of L. There is a converse to this fact, generally called Moore’s theorem, which states that given any positive definite operator-valued function K : X × X → B(H), then there exists a unique reproducing kernel Hilbert space of H-valued functions on X, such that K(x, y) = Ex Ey∗ . We will denote this space by L(K, H). Given v, w ∈ H, we let v ⊗ w ∗ : H → H denote the rank one operator given by (v ⊗ w ∗ )(h) = h, wv. A function g : X → H belongs to L(K, H) if and only if there exists a constant C > 0 such that the function C 2 K(x, y) − g(x) ⊗ g(y)∗ is positive definite. In which case the norm of g is the least such constant. Finally, given any reproducing kernel Hilbert space L of H-valued functions with reproducing kernel K, a function
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f : X → C is called a (scalar) multiplier provided that for every g ∈ L, the function f g ∈ L. In this case it follows by an application of the closed graph theorem that the map Mf : L → L, defined by Mf (g) = f g, is a bounded, linear map. The set of all multipliers is denoted M(K) and is easily seen to be an algebra of functions on X and a subalgebra of B(L). The reader can find proofs of the above facts in [16,4]. Also, we refer to the fundamental work of Pedrick [28] for further treatment of vector-valued reproducing kernel Hilbert spaces. Another good source is [3]. Lemma 2.1. Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak∗ -closed subalgebra. Proof. It is enough to show that the unit ball is weak∗ -closed by the Krein–Smulian theorem. So let {Mfλ } be a net of multipliers in the unit ball of B(L) that converges in the weak∗ -topology to an operator T . We must show that T is a multiplier. Let x ∈ X be fixed and assume that there exists g ∈ L, with g(x) = h = 0. Then T g, Ex∗ hL = limλ Mfλ g, Ex∗ hL = limλ Ex (Mfλ g), hH = limλ fλ (x)h2 . This shows that at every such x the net {fλ (x)} converges to some value. Set f (x) equal to this limit and for all other x’s set f (x) = 0. We claim that f is a multiplier and that T = Mf . Note that if g(x) = 0 for every g ∈ L, then Ex = Ex∗ = 0. Thus, we have that for any g ∈ L and any h ∈ H, Ex (T g), hH = limλ Ex (Mfλ g), hH = limλ fλ (x)g(x), hH = f (x)g(x), hH . Since this holds for every h ∈ H, we have that Ex (T g) = f (x)g(x), and so T = Mf and f is a multiplier. 2 Every weak∗ -closed subspace V ⊆ B(H) has a predual and it is the operator space dual of this predual. Also, if an abstract operator algebra is the dual of an operator space, then it can be represented completely isometrically and weak∗ -continuously as a weak∗ -closed subalgebra of the bounded operators on some Hilbert space. For this reason an operator algebra that has a predual as an operator space is called a dual operator algebra. See the book of [13] for the proofs of these facts. Thus, in summary, the above lemma shows that every multiplier algebra is a dual operator algebra in the sense of [13]. Theorem 2.2. Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X. Proof. The multiplier norm of a given matrix-valued function F = (fi,j ) ∈ Mn (M(K)) is the least constant C such that 2 C In − F (xi )F (xj )∗ ⊗ K(xi , xj ) 0, for all sets of finitely many points, Y = {x1 , . . . , xk } ⊆ X. Applying this fact to a set consisting of a single point, we have that (C 2 In − F (x)F (x)∗ ) ⊗ K(x, x) 0, and it follows that C 2 In − F (x)F (x)∗ 0. Thus, F (x) C = F and we have that point evaluations are completely contractive on M(K). Since M(K) contains the constants and separates points by hypothesis, it is an operator algebra of functions on X.
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Suppose that M(K) was not local, then there would exist F ∈ Mn (M(K)), and a real num(n) ber C, such that supY πY < C < F . Then for each finite set Y = {x1 , . . . , xk } we could choose G ∈ Mn (M(K)), with G < C, and G(x) = F (x), for every x ∈ Y . But then we would have that ((C 2 In − F (xi )F (xj )∗ ) ⊗ K(xi , xj )) = ((C 2 In − G(xi )G(xj )∗ ) ⊗ K(xi , xj )) 0, and since Y was arbitrary, F C, a contradiction. Thus, M(K) is local. Finally, assume that fλ ∈ M(K), is a net in M(K), with fλ C, and limλ fλ (x) = f (x), pointwise. If g ∈ L with gL = M, then ∗ (MC)2 K(x, y) − fλ (x)g(x) ⊗ fλ (y)g(y) is positive definite. By taking pointwise limits, we obtain that (MC)2 K(x, y) − f (x)g(x) ⊗ (f (y)g(y))∗ is positive definite. From the earlier characterization of functions in L and their norms in a reproducing kernel Hilbert space, this implies that f g ∈ L, with f gL MC. Hence, f ∈ M(K) with Mf C. Thus, M(K) is BPW complete. 2 In general, M(K) need not separate points on X. In fact, it is possible that L does not separate points and if g(x1 ) = g(x2 ), for every g ∈ L, then necessarily f (x1 ) = f (x2 ) for every f ∈ M(K). Following [26], we call C a k-idempotent operator algebra, provided that there are k operators, {E1 , . . . , Ek } on some Hilbert space H, such that Ei Ej = Ej Ei = δi,j Ei , I = E1 + · · · + Ek and C = span{E1 , . . . , Ek }. Proposition 2.3. Let C = span{E1 , . . . , Ek } be a k-idempotent operator algebra on the Hilbert space H, let Y = {x1 , . . . , xk } be a set of k distinct points and define K : Y × Y → B(H) by K(xi , xj ) = Ei Ej∗ . Then K is positive definite and C is completely isometrically isomorphic to M(K) via the map that sends a1 E1 + · · · + ak Ek to the multiplier f (xi ) = ai . Proof. It is easily checked that K is positive definite. We first prove that the map is an isometry. Given B = ki=1 ai ⊗ Ei ∈ C, let f : Y → C be defined by f (xi ) = ai . We have that f ∈ M(K) with f C if and only if P = ((C 2 −f (xi )f (xj )∗ )K(xi , xj )) is positive semidef inite in B(H(k) ). Let v = e1 ⊗ v1 + · · · + ek ⊗ vk ∈ H(k) , let h = kj =1 Ej∗ vj and note that
Ej∗ h = Ej∗ vj . Finally, set h = ki=1 hi . Thus, P v, v =
k 2 C − ai a¯j Ei Ej∗ vj , vi i,j =1
=
k 2 C − ai a¯j Ej∗ h, Ei∗ h = C 2 h2 − B ∗ h, B ∗ h i,j =1
2 = C 2 h2 − B ∗ h . Hence, B C implies that P is positive and so Mf B.
For the converse, given any h let v = kj =1 ej ⊗ Ej∗ h, and note that P v, v 0, implies that ∗ B h C, and so B Mf . The proof of the complete isometry is similar but notationally cumbersome. 2
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Theorem 2.4. Let A be an operator algebra of functions on the set X then there exist a Hilbert space, H and a positive definite function K : X × X → B(H) such that M(K) = A˜ completely isometrically. Proof. Let Y be a finite subset of X. Since A/IY is a |Y |-idempotent operator algebra, by the above lemma, there exists a vector valued kernel KY such that A/IY = M(KY ) completely isometrically. Define KY (x, y) when (x, y) ∈ Y × Y, Y (x, y) = K 0 when (x, y) ∈ / Y ×Y
Y , where the direct sum is over all finite subsets of X. K and set K = Y It is easily checked that K is positive definite. Let f ∈ Mn (M(K)) with Mf 1, which is equivalent to ((In − f (x)f (y)∗ ) ⊗ K(x, y)) being positive definite. This is in turn equivalent to ((In − f (x)f (y)∗ ) ⊗ KY (x, y)) being positive definite for every finite subset Y of X. This last condition is equivalent to the existence for each such Y of some fY ∈ Mn (A) such that πY (fY ) 1 and fY = f on Y . The net of functions {fY } then converges BPW to f . Hence, f ∈ A˜ with f L 1. ˜ isometrically, for every n, and the result follows. 2 This proves that Mn (M(K)) = Mn (A) Corollary 2.5. Every BPW complete local operator algebra of functions is a dual operator algebra. Proof. In this case we have that A = A˜ = M(K) completely isometrically. By Lemma 2.1, this latter algebra is a dual operator algebra. 2 The above theorem gives a weak∗ -topology to a local operator algebra of functions A by using the identification A ⊆ A˜ = M(K) and taking the weak∗ -topology of M(K). The following proposition proves that convergence of bounded nets in this weak∗ -topology on A is same as BPW convergence. Proposition 2.6. Let A be a local operator algebra of functions on the set X. Then the net (fλ )λ ∈ Ball(A) converges in the weak∗ -topology if and only if it converges pointwise on X. Proof. Let L denote the reproducing kernel Hilbert space of H-valued functions on X with kernel K for which A˜ = M(K). Recall that if Ex : L → H, is the linear map given by evaluation at x, then K(x, y) = Ex Ey∗ . Also, if v ∈ H, and h ∈ L, then h, Ex∗ vL = h(x), vH . First assume that the net (fλ )λ ∈ Ball(A) converges to f in the weak∗ -topology. Using the identification of A˜ = M(K), we have that the operators Mfλ of multiplication by fλ , converge in the weak∗ -topology of B(L) to Mf . Then for any x ∈ X, h ∈ L, v ∈ H, we have that fλ (x) h(x), v H = f (x)h(x), v H = Mfλ h, Ex∗ v L → Mf h, Ex∗ v L = f (x) h(x), v H . Thus, if there is a vector in H and a vector in L such that h(x), vH = 0, then we have that fλ (x) → f (x). It is readily seen that such vectors exist if and only if Ex = 0, or equivalently,
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K(x, x) = 0. But this follows from the construction of K as a direct sum of positive definite functions over all finite subsets of X. For fixed x ∈ X and the one element subset Y0 = {x}, we have that the 1-idempotent algebra A/IY0 = 0 and so KY0 (x, x) = 0, which is one term in the direct sum for K(x, x). Conversely, assume that fλ < K, for all λ and fλ → f pointwise on X. We must prove that Mfλ → Mf in the weak∗ -topology on B(L). But since this is a bounded net of operators, it will be enough to show convergence in the weak operator topology and arbitrary vectors can be replaced by vectors from a spanning set. Thus, it will be enough to show that for v1 , v2 ∈ H and x1 , x2 ∈ X, we have that Mfλ Ex∗1 v1 , Ex∗2 v2 L → Mf Ex∗1 v1 , Ex∗2 v2 L . But we have,
Mfλ Ex∗1 v1 , Ex∗2 v2 L = Ex2 Mfλ Ex∗1 v1 , v2 H = fλ (x2 ) K(x2 , x1 )v1 , v2 H → f (x2 ) K(x2 , x1 )v1 , v2 H = Mf Ex∗1 v1 , Ex∗2 v2 L ,
and the result follows.
2
Corollary 2.7. The ball of a local operator algebra of functions is weak∗ -dense in the ball of its BPW completion. 3. Residually finite-dimensional operator algebras A C∗ -algebra B is called residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations, that is, of ∗-homomorphisms into matrix algebras. Since every ∗-homomorphism of a C∗ -algebra is completely contractive, a C∗ -algebra B is RFD if and only if for all n, and for every (bi,j ) ∈ Mn (B), we have that (bi,j )Mn (B) = sup{(π(bi,j ))} where the suprema is taken over all ∗-homomorphisms, π : B → Mk with k arbitrary. Residually finite-dimensional C∗ -algebras have been studied in [21,12,17,6]. Moreover, for C∗ -algebras, a homomorphism is a ∗ -homomorphism if and only if it is completely contractive. Thus, the following definition gives us a natural way of extending the notion of RFD to operator algebras. Definition 3.1. An operator algebra, B is called RFD if for every n and for every (bi,j ) ∈ Mn (B), (bi,j ) = sup{(π(bi,j ))}, where the suprema is taken over all completely contractive homomorphisms, π : B → Mk with k arbitrary. A dual operator algebra B is called weak∗ -RFD if this last equality holds when the completely contractive homomorphisms are also required to be weak∗ -continuous. The following result is implicitly contained in [26], but the precise statement that we shall need does not appear there. Thus, we refer the reader to [26] to be able to fully understand the proof since we have used some of the definitions and results from [26] without stating them. Lemma 3.2. Let B = span{F1 , . . . , Fk } be a concrete k-idempotent operator algebra. Then S(B ∗ B) is a Schur ideal affiliated with B, i.e., B = A(S(B ∗ B)) completely isometrically.
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Proof. From Corollary 3.3 of [26] we have that the Schur ideal S(B ∗ B) is non-trivial and bounded. Thus, we can define the algebra A(S(B ∗ B)) = span{E1 , . . . , Ek }, where Ei =
n
Q∈Sn−1 (B∗ B )
Q1/2 (In ⊗ Eii )Q−1/2
is the idempotent operator that lives on n Q∈S −1 Mk (Mn ). By using Theorem 3.2 of [26] n we get that S(A(S(B ∗ B))A(S(B ∗ B))∗ ) = S(B ∗ B). This further implies that ∗ A S B∗ B A S B∗ B = B∗ B completely order isomorphically under the map which sends Ei∗ Ej to Fi∗ Fj . Finally, by restricting the same map to A we get a map which sends Ei to Fi completely isometrically. Hence, the result follows. 2 Theorem 3.3. Every k-idempotent operator algebra is weak∗ -RFD. Proof. Let A be an abstract k-idempotent operator algebra. Note that A is a dual operator algebra since it is a finite-dimensional operator algebra. From this it follows that there exist a Hilbert space, H and a weak∗ -continuous completely isometric homomorphism, π : A → B(H). Note that B = π(A) is a concrete k-idempotent algebra generated by the idempotents, B = span{F1 , F2 , . . . , Fk } contained in B(H). Thus, from the above lemma B = A(S(B ∗ B)) completely isometrically. Q For each n ∈ N and Q ∈ Sn−1 , we define πn : B → Mk (Mn ) via πnQ (Fi ) = Q1/2 (In ⊗ Eii )Q−1/2 . Assume for the moment that we have proven that πn is a weak∗ -continuous completely contracQ tive homomorphism. Then for every (bij ) ∈ Mk (B) we must have that supn,Q∈S −1 (πn (bij )) = n (bij ), and hence (bij ) = sup{(ρ(bij ))} where the supremum is taken over all weak∗ continuous completely contractive homomorphisms ρ : B → Mm with m arbitrary. Since π : A → B is a complete isometry and weak∗ -continuous, this would imply the result for A by composition. Q Thus, it remains to show that πn is a weak∗ -continuous completely contractive homomorphism on B. Note that it is easy to check that it is a completely contractive homomorphism and it is completely isometric by the proof of the previous lemma. Q Finally, the fact that πn is weak∗ -continuous follows from the fact that B is finite-dimensional ∗ so that the weak -topology and the norm topology are equal. 2 Q
Theorem 3.4. Every BPW complete local operator algebra of functions is weak∗ -RFD. Proof. Let A be a BPW complete local operator algebra of functions on the set X and let F be a finite subset of X, so that A/IF is an |F |-idempotent operator algebra. It follows from the above lemma that A/IF is weak∗ -RFD, i.e., for ([fij ]) ∈ Mk (A/IF ) we have ([fij ]) = sup{(ρ([fij ]))} where the supremum is taken over all weak∗ -continuous completely contractive homomorphisms ρ from A/IF into matrix algebras.
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Let (fij ) ∈ Mk (A), then (fij )Mk (A) = supF ([fij ]) since A is local. Recall, that the weak∗ -topology on A requires all the quotient maps of the form πF : A → A/IF , πF (f ) = [f ] to be weak∗ -continuous. Thus, for each finite subset F ⊆ X, πF is a weak∗ -continuous completely contractive homomorphism. The result now follows by considering the composition of the weak∗ -continuous quotient maps with the weak∗ -continuous finite-dimensional representations of each quotient algebra. 2 Corollary 3.5. Every local operator algebra of functions is RFD. Proof. This follows immediately from the fact that every local operator algebra is completely isometrically contained in a BPW complete local operator algebra. 2 4. Quantized function theory on domains Whenever one replaces scalar variables by operator variables in a problem or definition, then this process is often referred to as quantization. It is in this sense that we would like to quantize the function theory on a family of complex domains. In some sense this process has already been carried out for balls in the work of Drury [20], Popescu [30], Arveson [7], and Davidson and Pitts [18] and for polydisks in the work of Agler [1,2], and Ball and Trent [10]. Our work is closely related to the idea of “quantizing” other domains defined by inequalities that occurs in the work of Ambrozie and Timotin [5], Ball and Bolotnikov [9], and Kalyuzhnyi-Verbovetzkii [22], but the terminology is our own. We approach these same ideas via operator algebra methods. We will show that in many cases this process yields local operator algebras of functions to which the results of the earlier sections can be applied. We begin by defining a family of open sets for which our techniques will apply. Definition 4.1. Let G ⊆ CN be an open set. If there exists a set of matrix-valued functions, Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I , whose components are analytic functions on G, and satisfy Fk (z) < 1, k ∈ I , then we call G an analytically presented domain and we call the set of functions R = {Fk : G → Mmk ,nk : k ∈ I } an analytic presentation of G. The subalgebra A of the algebra of bounded analytic functions on G generated by the component functions {fk,i,j : 1 i mk , 1 j nk , k ∈ I } and the constant function is called the algebra of the presentation. We say that R is a separating analytic presentation provided that the algebra A separates points on G. Remark 4.2. An analytic presentation of G by a finite set of matrix-valued functions, Fk : G → Mmk ,nk , 1 k K, can always be replaced by the single block-diagonal matrix-valued function, F (z) = F1 (z) ⊕ · · · ⊕ FK (z) into Mm,n with m = m1 + · · · + mK , n = n1 + · · · + nK and we will sometimes do this to simplify proofs. But it is often convenient to think in terms of the set, especially since this will explain the sums that occur in Agler’s factorization formula. Definition 4.3. Let G ⊆ CN be an analytically presented domain with presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I }, let A be the algebra of the presentation and let H be a Hilbert space. A homomorphism π : A → B(H) is called an admissible representation provided that (π(fk,i,j )) 1 in Mmk ,nk (B(H)) = B(Hnk , Hmk ), for every k ∈ I . We call the homomorphism π an admissible strict representation when these inequalities are all strictly less than 1. Given (gi,j ) ∈ Mn (A) we set (gi,j )u = sup{(π(gi,j ))}, where the supremum is
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taken over all admissible representations π of A. We let (gi,j )u0 denote the supremum that is obtained when we restrict to admissible strict representations. The theory of [5,9] studies domains defined as above with the additional restrictions that the set of defining functions is a finite set of polynomials. However, they do not need their polynomials to separate points, while we shall shortly assume that our presentations are separating, in order to invoke the results of the previous sections. This latter assumption can, generally, be dropped in our theory, but it requires some additional argument. There are several other places where our results and definitions given below differ from theirs. So while our results extend their results in many cases, in other cases we are using different definitions and direct comparisons of the results are not so clear. Proposition 4.4. Let G have a separating analytical presentation and let A be the algebra of the presentation. Then A endowed with either of the family of norms · u or · u0 is an operator algebra of functions on G. Proof. It is clear that it is an operator algebra and by definition it is an algebra of functions on G. It follows from the hypotheses that it separates points of G. Finally, for every λ = (λ1 , . . . , λN ) ∈ G, we have a representation of A on the one-dimensional Hilbert space given by πλ (f ) = f (λ). Hence, |f (λ)| f u and so A is an operator algebra of functions on G. 2 It will be convenient to say that matrices, A1 , . . . , Am are of compatible sizes if the product, A1 · · · Am exists, that is, provided that each Ai is an ni × ni+1 matrix. Given an analytically presented domain G, we include one extra function, F1 which denotes the constant function 1. By an admissible block-diagonal matrix over G we mean a blockdiagonal matrix-valued function of the form D(z) = diag(Fk1 , . . . , Fkm ) where ki ∈ I ∪ {1} for 1 i m. Thus, we are allowing blocks of 1’s in D(z). Finally, given a matrix B we let B (q) = diag(B, . . . , B) denote the block-diagonal matrix that repeats B q times. Theorem 4.5. Let G be an analytically presented domain with presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I }, let A be the algebra of the presentation and let P = (pij ) ∈ Mm,n (A), where m, n are arbitrary. Then the following are equivalent: (i) P u < 1, (ii) there exist an integer l, matrices of scalars Cj , 1 j l with Cj < 1 and admissible block-diagonal matrices Dj (z), 1 j l, which are of compatible sizes and are such that P (z) = C1 D1 (z) · · · Cl Dl (z), (iii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 , Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is a finite set, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +
(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗
k∈K
where rk = qk mk and z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ G.
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Proof. Although we will not logically need it, we first show that (ii) implies (i), since this is the easiest implication and helps to illustrate some ideas. Note that if π : A → B(H) is any admissible representation, then the norm of π of any admissible block-diagonal matrix is at most 1. Thus, if P has the form of (ii), then for any admissible π , we will have (π(pi,j )) expressed as a product of scalar matrices and operator matrices all of norm at most one and hence, (π(Pi,j )) C1 · · · Cl < 1. Thus, P u C1 · · · Cl < 1. We now prove that (i) implies (ii). The ideas of the proof are similar to [25, Corollary 18.2], [14, Corollary 2.11] and [23, Theorem 1] and use in an essential way the abstract characterization of operator algebras. For each m, n ∈ N, one proves that P m,n := inf{C1 · · · Cl }, defines a norm on Mm,n (A), where the infimum is taken over all l and all ways to factor P (z) = C1 D1 (zi1 ) · · · Cl Dl (zil ) as a product of matrices of compatible sizes with scalar matrices Cj , 1 j l and admissible block-diagonal matrices Dj , 1 j l. Moreover, one can verify that Mm,n (A) with this family { · m,n }m,n of norms satisfies the axioms for an abstract unital operator algebra as given in [15] and hence by the Blecher–Ruan– Sinclair representation theorem [15] (see also [25]) there exist a Hilbert space H and a unital completely isometric isomorphism π : A → B(H). Thus, for every m, n ∈ N and for every P = (pij ) ∈ Mm,n (A), we have that P m,n = (π(pij )). However, π (mk ,nk ) (Fk ) = (π(fk,i,j )) 1 for 1 i K, and so, π is an admissible representation. Thus, P m,n = (π(pij )) P u . Hence, if P u < 1, then P m,n < 1 which implies that such a factorization exists. This completes the proof that (i) implies (ii). We will now prove that (ii) implies (iii). Suppose that P has a factorization as in (ii). Let K ⊆ I be the finite subset of all indices that appear in the block-diagonal matrices appearing in the factorization of P . We will use induction on l to prove that (iii) holds. First, assume that l = 1 so that P (z) = C1 D1 (z). Then, ∗ Im − P (z)P (w)∗ = Im − C1 D1 (z) C1 D1 (w) = Im − C1 C1∗ + C1 I − D1 (z)D1 (w)∗ C1∗ . Since D1 (z) is an admissible block-diagonal matrix the (i, i)-th block-diagonal entry of I − D1 (z)D1 (w)∗ is I − Fki (z)Fki (w)∗ for some finite collection, ki . Let Ek be the diagonal matrix that has 1’s wherever Fk appears (so Ek = 0 when there is no Fk term in D1 ). Hence, C1 I − D1 (z)D1 (w)∗ C1∗ = C1 Ek I − Fk (z)Fk (w)∗ Ek C1∗ . k
Therefore, gathering terms for common values of i, Im − P (z)P (w)∗ = R0 +
Pk I − Fk (z)Fk (w)∗ Pk∗ ,
k∈K
where R0 = Im − C1 C1∗ is a positive, invertible matrix and Pi is, in this case a constant. Thus, the form (iii) holds, when l = 1. We now assume that the form (iii) holds for any R(z) that has a factorization of length at most l − 1, and assume that
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P (z) = C1 D1 (z) · · · Dl−1 (z)Cl Dl (z) = C1 D1 (z)R(z), where R(z) has a factorization of length l − 1. Note that a sum of expressions such as on the right-hand side of (iii) is again such an expression. This follows by using the fact that given any two expressions A(z), B(z), we can write A(z)A(w)∗ + B(z)B(w)∗ = C(z)C(w)∗ where C(z) = (A(z), B(z)). Thus, it will be sufficient to show that Im − P (z)P (w)∗ is a sum of expressions as above. To this end we have that, Im − P (z)P (w)∗ = Im − C1 D1 (z)D1 (w)∗ C1∗ + C1 D1 (z) I − R(z)R(w)∗ D1 (w)∗ C1∗ . The first term of the above equation is of the form as on the right-hand side of (iii) by case l = 1. Also, the quantity (I − R(z)R(w)∗ ) = R0 R0∗ + R0 (z)R0 (w)∗ + k∈K Rk (z)(I − Fk (z)Fk (w)∗ )(qk ) Rk (w)∗ by the inductive hypothesis. Hence, C1 D1 (z) I − R(z)R(w)∗ D1 (w)∗ C1∗ ∗ ∗ = C1 D1 (z)R0 C1 D(w)R0 + C1 D1 (z)R0 (z) C1 D1 (w)R0 (w) (q ) ∗ C1 D1 (z)Rk (z) I − Fk (z)Fk (w)∗ k C1 D1 (w)Rk (w) . + k∈K
Thus, we have expressed (I − P (z)P (w)∗ ) as a sum of two terms both of which can be written in the form desired. Using again our remark that the sum of two such expressions is again such an expression, we have the required form. Finally, we will prove (iii) implies (i). Let π : A → B(H) be an admissible representation and let P = (pi,j ) ∈ Mm,n (A) have a factorization as in (iii). To avoid far too many superscripts we simplify π (m,n) to Π . Now observe that Im − Π(P )Π(P )∗ = Π(R) + Π(P0 )Π(P0 )∗ (q ) ∗ + Π(Pk ) I − Π(Fk )Π(Fk )∗ k Π(Pk ) . k∈K
Clearly the first two terms of the sum are positive. But since π is an admissible representation, Π(Fk ) 1 and hence, (I − Π(Fk )Π(Fk )∗ ) 0. Hence, each term on the right-hand side of the above inequality is positive and since R is strictly positive, say R δIm for some scalar ∗ δ > 0, we have that Im − Π(P √ )Π(P ) δIm . Therefore, Π(P ) 1 − δ. Thus, since π was an arbitrary admissible representation, √ P u 1 − δ < 1, which proves (i). 2 When we require the functions in the presentation to be row vector-valued, then the above theory simplifies somewhat and begins to look more familiar. Let G be an analytically presented domain with presentation Fk : G → M1,nk , k ∈ I . We identify M1,n with the Hilbert space Cn so
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that 1 − Fk (z)Fk (w)∗ = 1 − Fk (z), Fk (w), where the inner product is in Cn . In this case we shall say that G is presented by vector-valued functions. Corollary 4.6. Let G be presented by vector-valued functions, Fk = (fk,j ) : G → M1,nk , k ∈ I , let A be the algebra of the presentation and let P = (pij ) ∈ Mm,n (A). Then the following are equivalent: (i) P u < 1, (ii) there exist an integer l, matrices of scalars Cj , 1 j l with Cj < 1 and admissible block-diagonal matrices Dj (z), 1 j l, which are of compatible sizes and are such that P (z) = C1 D1 (z) · · · Cl Dl (z), (iii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 ∈ Mm,r0 (A), Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is finite, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +
1 − Fk (z), Fk (w) Pk (z)Pk (w)∗ k∈K
where z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ G. The following result gives us a Nevanlinna-type result for the algebra of the presentation. Theorem 4.7. Let Y be a finite subset of an analytically presented domain G with separating analytic presentation Fk = (fk,i,j ) : G → Mmk ,nk , k ∈ I , let A be the algebra of the presentation and let P be an Mm,n -valued function defined on a finite subset Y = {x1 , . . . , xl } of G. Then the following are equivalent: (i) there exists P˜ ∈ Mmn (A) such that P˜ |Y = P and P˜ u < 1, (ii) there exist a positive, invertible matrix R ∈ Mm and matrices P0 ∈ Mm,r0 (A), Pk ∈ Mm,rk (A), k ∈ K, where K ⊆ I is a finite set, such that Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +
(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗
k∈K
where rk = qk mk and z = (z1 , . . . , zN ), w = (w1 , . . . , wN ) ∈ Y . Proof. Note that (i) ⇒ (ii) follows immediately as a corollary of Theorem 4.5. Thus it only remains to show that (ii) ⇒ (i). Since A is an operator algebra of functions, therefore, A/IY is a finite-dimensional operator algebra of idempotents and A/IY = span{E1 , . . . , El } where l = |Y |. Thus there exist a Hilbert space HY and a completely isometric representation π of A/IY . By Theorem 2.4, there exists a kernel KY such that π(A/IY ) = M(KY ) completely isometrically under the map ρ : π(A/IY ) → M(KY ) which sends π(B) to Mf , where B = li=1 ai π(Ei ) and f : Y → C is a function defined by f (xi ) = ai . Note that I − Fk (xi )Fk (xj ) ⊗ KY (xi , xj ) = I − π(Fk + IY )(xi )π(Fk + IY )(xj )∗ ⊗ KY (xi , xj ) ij 0
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since π(Fk + IY ) Fk u 1 for all k ∈ I . From this it follows that Im − π(P + IY ) (xi ) π(P + IY ) (xj )∗ ⊗ KY (xi , xj ) ij 0. Using that R > 0, we get that π(P + IY ) < 1. This shows that there exists P˜ ∈ A such that P˜ |Y = P and P˜ u < 1. This completes the proof. 2 We now turn towards defining quantized versions of the bounded analytic functions on these domains. For this we need to recall that the joint Taylor spectrum [31] of a commuting N -tuple of operators T = (T1 , . . . , TN ), is a compact set, σ (T ) ⊆ CN and that there is an analytic functional calculus [32,33] defined for any function that is holomorphic in a neighborhood of σ (T ). Definition 4.8. Let G ⊆ CN be an analytically presented domain, with presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. We define the quantized version of G to be the collection of all commuting N -tuples of operators, Q(G) = T = (T1 , T2 , . . . , TN ) ∈ B(H): σ (T ) ⊆ G and Fk (T ) 1, ∀k ∈ I , where H is an arbitrary Hilbert space. We set Q0,0 (G) = T = (T1 , T2 , . . . , TN ) ∈ Mn : σ (T ) ⊆ G and Fk (T ) 1, ∀k ∈ I , where n is an arbitrary positive integer. Note that if we identify a point (λ1 , . . . , λN ) ∈ CN with an N -tuple of commuting operators on a one-dimensional Hilbert space, then we have that G ⊆ Q(G). If T = (T1 , . . . , TN ) ∈ Q(G), is a commuting N -tuple of operators on the Hilbert space H, then since the joint Taylor spectrum of T is contained in G, we have that if f is analytic on G, then there is an operator f (T ) defined and the map π : Hol(G) → B(H) is a homomorphism, where Hol(G) denotes the algebra of analytic functions on G [33]. Definition 4.9. Let G ⊆ CN be an analytically presented domain, with presentation R = ∞ (G) to be the set of functions f ∈ Hol(G), such that {Fk : G− → Mmk ,nk , k ∈ I }. We define HR ∞ (G)), we set (f ) = f R ≡ sup{f (T ): T ∈ Q(G)} is finite. Given (fi,j ) ∈ Mn (HR i,j R sup{(fi,j (T )): T ∈ Q(G)}. ∞ (G) = A, ˜ completely isometrically and whether We are interested in determining when HR or not the · R norm is attained on the smaller set Q0,0 (G). ∞ (G) ⊆ H ∞ (G), and f Note that since each point in G ⊆ Q(G), we have that HR ∞ ∞ f R . Also, we have that A ⊆ HR (G) and for (fi,j ) ∈ Mn (A), (fi,j )R (fi,j )u0 ∞ (G) might not even be isometric. (fi,j )u . Thus, the inclusion of A into HR
Theorem 4.10. Let G be an analytically presented domain with a separating presentation R = {Fk : G → Mmk ,nk : k ∈ I }, let A be the algebra of the presentation and let A˜ be the BPWcompletion of A. Then ∞ (G), completely isometrically, (i) A˜ = HR
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∞ (G) is a local weak∗ -RFD dual operator algebra. (ii) HR
˜ with f L < 1. Then there exists a net of functions, fλ ∈ Mn (A), Proof. Let f ∈ Mn (A), such that fλ u < 1 and limλ fλ (z) = f (z) for every z ∈ G. Since fλ ∞ < 1, by Montel’s theorem, there is a subsequence {fn } of this net that converges to f uniformly on compact sets. Hence, if T ∈ Q(G), then limn f (T ) − fn (T ) = 0 and so f (T ) supn fn (T ) 1. ∞ (G)), with f 1. This proves that A˜ ⊆ H ∞ (G) and that Thus, we have that f ∈ Mn (HR R R f L f R . ∞ (G)) with g < 1. Given any finite set Y = {y , . . . , y } ⊆ G, Conversely, let g ∈ Mn (HR 1 t R let A/IY = span{E1 , . . . , Et } be the corresponding t-idempotent algebra and let πY : A → A/IY denote the quotient map. Write yi = (yi,1 , . . . , yi,N ), 1 i t and let Tj = y1,j E1 +· · ·+yt,j Et , 1 j N so that T = (T1 , . . . , TN ) is a commuting N -tuple of operators with σ (T ) = Y . For k ∈ I , we have that Fk (T ) = Fk (y1 ) ⊗ E1 + · · · + Fk (yt ) ⊗ Et = πY (Fk ) Fk u = 1. Thus, T ∈ Q(G), and so, g(T ) = g(y1 ) ⊗ E1 + · · · + g(yt ) ⊗ Et gR < 1. Since A separates points, we may pick f ∈ Mn (A) such that f = g on Y . Hence, πY (f ) = f (T ) = g(T ) and πY (f ) < 1. Thus, we may pick fY ∈ Mn (A), such that πY (fY ) = πY (f ) and fY u < 1. This net of functions, {fY } converges to g pointwise and is bounded. Therefore, ˜ and gL 1. This proves that H ∞ (G) ⊆ A˜ and that gL gR . g ∈ Mn (A) R ∞ (G) and the two matrix norms are equal for matrices of all sizes. The rest of Thus, A˜ = HR the conclusions follows from the results on BPW-completions. 2 ∞ (G), f = sup{π(f )} Remark 4.11. The above result yields that for every f ∈ HR R where the supremum is taken over all finite-dimensional weak∗ -continuous representations, ∞ (G) → M with n arbitrary. For many examples, we can show that π : HR n
f R = sup f (T ): T ∈ Q00 (G) ∞ (G). Also, we can verify these hypotheses are met for most of the algebras given for any f ∈ HR in the example section. In particular, for Examples 5.1, 5.2, 5.3, 5.4, 5.6, 5.7, and 5.8. It would be interesting to know if this can be done in general. ∞ (G). In particular, we wish We now seek other characterizations of the functions in HR to obtain analogues of Agler’s factorization theorem and of the results in [5,9]. By Theorem 2.4, if we are given an analytically presented domain G ⊆ CN , with presentation R = {Fk : G → Mmk ,nk , k ∈ I }, then there exist a Hilbert space H and a positive definite function, K : G × G → B(H) such that A˜ = M(K). We shall denote any kernel satisfying this property by KR .
Definition 4.12. Let G ⊆ CN be an analytically presented domain, with presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. We shall call a function H : G × G → Mm an R-limit, provided that H is the pointwise limit of a net of functions Hλ : G × G → Mm of the form given by Theorem 4.5(iii).
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Corollary 4.13. Let G ⊆ CN be an analytically presented domain, with a separating presentation R = {Fk : G− → Mmk ,nk , k ∈ I }. Then the following are equivalent: ∞ (G)) and f 1, (i) f ∈ Mm (HR R (ii) (Im − f (z)f (w)∗ ) ⊗ KR (z, w) is positive definite, (iii) Im − f (z)f (w)∗ is an R-limit.
In the case when the presentation contains only finitely many functions we can say considerably more about R-limits. Proposition 4.14. Let G be an analytically presented domain with a finite presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , 1 k K}. For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form, Im − P (z)P (w)∗ = R + P0 (z)P0 (w)∗ +
K
(q ) Pk (z) I − Fk (z)Fk (w)∗ k Pk (w)∗ ,
k=1
then Pk (z) C for all k ∈ I and for all z ∈ S. Proof. By the continuity of the functions, there is a constant δ > 0, such that Fk (z) 1 − δ, for all k ∈ I and for all z ∈ S. Thus, we have that I − Fk (z)Fk (z)∗ δI , for all k ∈ I and for all z ∈ S. Also, we have that (q ) Im Im − P (z)P (z)∗ Pk (z) I − Fk (z)Fk (z)∗ k Pk (z)∗ δPk (z)Pk (z)∗ . This shows that Pk (z) 1/δ for all k ∈ I and for all z ∈ S.
2
The proof of the following result is essentially contained in [9, Lemma 3.3]. Proposition 4.15. Let G be a bounded domain in CN and let F = (fi,j ) : G → Mm,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → Mp is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions Pλ ∈ Mp,rλ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H (z, w) is the pointwise limit of Hλ (z, w) = Pλ (z)(Im − F (z)F (w)∗ )(qλ ) Pλ (w)∗ where rλ = qλ mk , then there exist a Hilbert space H and an analytic function, R : G → B(H ⊗ CM , Cp ) such that H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ IH ]R(w)∗ . Proof. We identify (Im − F (z)F (w)∗ )(qλ ) = (Im − F (z)F (w)∗ ) ⊗ ICqλ , and the p × mqλ matrixvalued function Pλ as an analytic function, Pλ : G → B(Cm ⊗ Cqλ , Cp ). Writing Cm ⊗ Cqλ = Cqλ ⊕ · · · ⊕ C qλ (m times) allows us to write Pλ (z) = [P1λ (z), . . . , Pmλ (z)] where each Piλ (z) be the (1, n) vectors that represent the rows of the is p × qλ . Also, if we let f1 (z), . . . , fm (z) ∗ matrix F , then we have that F (z)F (w)∗ = m i,j =1 fi (z)fj (w) Ei,j . Finally, we have that Hλ (z, w) =
m i=1
Piλ (z)Piλ (w)∗ −
m i,j =1
fi (z)fj (w)∗ Piλ (z)Pjλ (w).
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Let Kλ (z, w) = (Piλ (z)Pjλ (w)∗ ), so that Kλ : G×G → Mm (Mp ) = B(Cm ⊗Cp ), is a positive definite function that is analytic in z and co-analytic in w. By dropping to a subnet, if necessary, we may assume that Kλ converges uniformly on compact subsets of G to K = (Ki,j ) : G × G → Mm (Mp ). Note that this implies that Piλ (z)Pjλ (w)∗ → Ki,j (z, w) for all i, j and that K is a positive definite function that is analytic in z and coanalytic in w. The positive definite function K gives rise to a reproducing kernel Hilbert space H of analytic Cm ⊗ Cp -valued functions on G. If we let E(z) : H → Cm ⊗ Cp , be the evaluation functional, then K(z, w) = E(z)E(w)∗ and E : G → B(H, Cm ⊗ Cp ) is analytic. Identifying Cm ⊗ Cp = Cp ⊕ · · · ⊕ Cp (m times), yields analytic functions, Ei : G → B(H, Cp ), i = 1, . . . , m, such that (Ki,j (z, w)) = K(z, w) = E(z)E(w)∗ = (Ei (z)Ej (w)∗ ). Define an analytic map R : G → B(H ⊗ Cm , Cp ) by identifying H ⊗ Cm = H ⊕ · · · ⊕ H (m times) and setting R(z)(h1 ⊕ · · · ⊕ hm ) = E1 (z)h1 + · · · + Em (z)hm . Thus, we have that R(z) Im − F (z)F (w)∗ ⊗ IH R(w)∗ =
m
Ei (z)Ei (w)∗ −
=
Ki,i (z, w) −
λ
m
fi (z)fj (w)∗ Ki,j (z, w)
i,j =1
i=1
= lim
fi (z)fj (w)∗ Ei (z)Ej (w)∗
i,j =1
i=1 m
m
m
Piλ (z)Piλ (w)∗ −
fi (z)fj (w)∗ Piλ (z)Pjλ (w)∗ = H (z, w),
i,j =1
i=1
and the proof is complete.
m
2
Remark 4.16. Conversely, any function that can be written in the form H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ IH ]R(w)∗ can be expressed as a limit of a net as above by considering the directed set of all finite-dimensional subspaces of H and for each finite-dimensional subspace setting HF (z, w) = RF (z)[(Im − F (z)F (w)∗ ) ⊗ IF ]RF (w)∗ , where RF (z) = R(z)(PF ⊗ Im ) with PF the orthogonal projection onto F . Definition 4.17. We shall refer to a function H : G × G → Mm (Mp ) that can be expressed as H (z, w) = R(z)[(Im − F (z)F (w)∗ ) ⊗ H]R(w)∗ for some Hilbert space H and some analytic function R : G → B(H ⊗ Cm , Cp ), as an F -limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {Fk = (fk,i,j ) : G → Mmk ,nk , 1 k K}, let f = (fij ) be an Mm,n -valued function defined on G. Then the following are equivalent: ∞ (G)) and f 1, (1) f ∈ Mmn (HR R (2) there exist an analytic operator-valued function R0 : G → B(H0 , Cm ) and Fk -limits, Hk : G × G → Mm , such that ∗
∗
I − f (z)f (w) = R0 (z)R0 (w) +
K k=1
Hk (z, w),
∀z, w ∈ G,
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(3) there exist Fk -limits, Hk (z, w), such that I − f (z)f (w)∗ =
K
Hk (z, w),
∀z, w ∈ G.
k=1 ∞ (G). Let us first assume that f ∈ M (A) Proof. Recall that A˜ = HR mn ˜ and f Mmn (A) ˜ < 1. Then for each finite set Y , there exists fY ∈ Mmm (A) such that fY converges to f pointwise and fY fY u 1. We may assume that fY u < 1 by replacing fY by 1+1/|Y | , where |Y | denotes the cardinality of the set Y . Thus by Theorem 4.5 there exist a positive, invertible matrix R Y ∈ Mm and matrices Y Pk ∈ Mm,rkY (A), 0 k K, such that
Im − fY (z)fY (w)∗ = R Y + P0Y (z)P0Y (w)∗ +
K
(q ) PkY (z) I − Fk (z)Fk (w)∗ kY PkY (w)∗
k=1
where rkY = qkY mk and z, w ∈ G. If we define a map F0 : G → Mm0 ,n0 as the zero map then the above factorization can be written as Im − fY (z)fY (w)∗ = R Y +
K
(q ) PkY (z) I − Fk (z)Fk (w)∗ kY PkY (w)∗
k=0
where rkY = qkY mk and z, w ∈ G. Note that the net R Y is uniformly bounded above by 1, thus there exists R ∈ Mm and a subnet R Ys which converges to R. Finally, since the net fY converges to f pointwise we have that the net K
(q ) PkY (z) I − Fk (z)Fk (w)∗ kY PkY (w)∗
k=1
converges pointwise on G. Also note that for each k, {PkY } is a net of vector-valued holomorphic functions and is uniformly bounded on compact subsets of G by Proposition 4.14. Thus by Proposition 4.15 there exist Fk -limits for each 0 k K, that is, there exist K + 1 Hilbert spaces Hk and K + 1 analytic function, Rk : G → B(Hk ⊗ CM , Cp ) such that H (z, w) = Rk (z)[(Im − Fk (z)Fk (w)∗ ) ⊗ IHk ]Rk (w)∗ and the corresponding subnet of the net
kK
K Y ∗ (qkY ) P Y (w)∗ converges to k=0 Pk (z)(I − Fk (z)Fk (w) ) k=0 Hk (z, w) for all z, w ∈ G. This k completes the proof that (1) implies (2). To show the converse, assume that there exist an analytic operator-valued function R0 : G → M p spaces B(H0 , Cm ) and K analytic functions, Rk : G → B(H
Kk ⊗ C , C ) on some Hilbert ∗ ∗ Hk such that I − f (z)f (w) = R0 (z)R0 (w) + k=1 Rk (z)(I − Fk (z)Fk (w)∗ )(qk ) Rk (w)∗ for z, w ∈ G. ˜ By using Theorem 2.4 there exists a vector-valued kernel K such that Mn (M(K)) = Mn (A) ∗ completely isometrically for every n. It is easy to see that ((I − f (z)f (w) ) ⊗ K(z, w)) 0 for z, w ∈ G. This is equivalent to f ∈ Mm (M(K)) and Mf 1 which in turn is equivalent to (1). Thus, (1) and (2) are equivalent.
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Clearly, (3) implies (2). The argument for why (2) implies (3) is contained in [9] and we recall it. If we fix any k0 , then since Fk0 (z) < 1 on G, we have that |fk0 ,1,1 (z)|2 + · · · + |fk0 ,1,m (z)|2 < 1 on G. From this it follows that H (z, w) = (1 − fk0 ,1,1 (z)f¯k0 ,1,1 (w) − · · · − fk0 ,1,m (z)f¯k0 ,1,m (w)) is an Fk0 -limit and that H −1 (z, w) is positive definite. Now we have that R0 (z)R0 (w)∗ H −1 (z, w) is positive definite and so we may write, R0 (z)R0 (w)∗ H −1 (z, w) = G0 (z)G0 (w)∗ and we have that R0 (z)R0 (w)∗ = G0 (z)H (z, w)G(w)∗ . This shows that R0 (z)R0 (w)∗ is an Fk -limit and so it may be absorbed into the sum. 2 5. Examples and applications In this section we present a few examples to illustrate the above definitions and results. Example 5.1. Let G = DN be the polydisk which has a presentation given by the coordinate functions Fi (z) = zi , 1 i N . Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T1 , . . . , TN ) on a Hilbert space. Given a matrix of polynomials, (pi,j )u = sup (pi,j (T1 , . . . , TN )) where the supremum is taken over all N -tuples of commuting contractions. This is the norm considered by Agler in [1], which is sometimes called the Schur–Agler norm [23]. Our Q(DN ) = {T = (T1 , . . . , TN ): σ (T ) ⊆ DN and Ti 1}. Note that if we replace such a T by rT = (rT1 , . . . , rTN ) then rTi < 1, rT ∈ QR (DN ) and taking suprema over all T ∈ QR (DN ) will be the same as taking a suprema over this smaller set. Thus, the algebra ∞ (DN ) consists of those analytic functions f such that HR f R = sup f (T1 , . . . , TN ): Ti < 1, i = 1, . . . , N < +∞. Our result that this is a weak∗ -RFD algebra shows that this supremum is also attained by considering commuting N -tuples of matrices satisfying Ti < 1, i = 1, . . . , N . By Theorem 4.18 for ∞ (DN )), we have that f 1 if and only if f ∈ Mm,n (HR R Im − f (z)f (w)∗ =
N (1 − zi w¯ i )Ki (z, w), i=1
for some analytic-coanalytic positive definite functions, Ki : DN × DN → Mm . Example 5.2. Let G = BN denote the unit Euclidean ball in CN . If we let F1 (z) = (z1 , . . . , zN ) : BN → M1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , Tn ) such that T1 T1∗ + · · · + TN TN∗ I , which is commonly called a row contraction and an admissible strict representation is given when T1 T1∗ + · · · + TN TN∗ < I , which is generally referred to as a strict row contraction. In this case one can again easily see that · u = · u0 by ∞ (B ) if and only if using the same r < 1 argument as in the last example and that f ∈ HR N f R = sup f (T ): T1 T1∗ + · · · + TN TN∗ < I < +∞.
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These are the norms on polynomials considered by Drury [20], Popescu [30], Arveson [7], and Davidson and Pitts [18]. Again our weak∗ -RFD result shows that f R is attained by taking the supremum over commuting N -tuples of matrices satisfying T1 T1∗ + · · · + TN TN∗ < I . ∞ (B )) that f 1 if and only if By Theorem 4.18 we will have for f ∈ Mm,n (HR N R Im − f (z)f (w)∗ = 1 − z, w K(z, w), where K : BN × BN → Mm is an analytic-coanalytic positive definite function. Example 5.3. Let G = BN as above and let F1 (z) = (z1 , . . . , zN )t : BN → MN,1 . Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , TN ) such that (T1 , . . . , TN )t 1, i.e., such that T1∗ T1 + · · · + TN∗ TN I , which is generally referred to as a ∞ (B ) will be defined by taking suprema over all column contraction. This time the norm on HR N strict column contractions and we will have that f R 1 if and only if Im − f (z)f (w)∗ = R1 (z) IN − (zi w¯j ) ⊗ IH R1 (w)∗ for some R1 : BN → B(Cm , H), analytic. Again, the weak∗ -RFD result shows that f R is attained by taking the supremum over matrices that form strict column contractions. Example 5.4. Let G = BN as above, let F1 (z) = (z1 , . . . , zN ) : BN → M1,N and F2 (z) = (z1 , . . . , zN )t : BN → MN,1 . Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N -tuple of commuting operators (T1 , . . . , TN ) such that T1 T1∗ + · · · + TN TN∗ I and T1∗ T1 + · · · + TN∗ TN I , ∞ (B ) is defined that is, which is both a row and column contraction. This time the norm on HR N as the supremum over all commuting N -tuples that are both strict row and column contractions and again this is attained by restricting to commuting N -tuples of matrices that are strict row and ∞ (B )) with f 1 if and only if column contractions. We will have that f ∈ Mm,n (HR N R Im − f (z)f (w)∗ = 1 − z, w K1 (z, w) + R1 (z) IN − (zi w¯j ) ⊗ IH R1 (w)∗ , where K1 and R1 are as before. The last three examples illustrate that it is possible to have multiple rational representations of G, all with the same algebra, but which give rise to (possibly) different operator algebra norms on A. Thus, the operator algebra norm depends not just on G, but also on the particular presentation of G that one has chosen. We have surpressed this dependence on R to keep our notation simplified. Example 5.5. Let G = D be the open unit disk in the complex plane and let F1 (z) = z2 , F2 (z) = z3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function so that A is the span of the monomials, {1, zn : n 2}. Also, A separates the points of G. In this case a (strict) admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict) contractions,
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A = π(z2 ), B = π(z3 ), satisfying A3 = B 2 . Again, it is easy to see that · u = · u0 . On the other hand Q(D) = T : σ (T ) ⊆ D and T 2 1, T 3 1 ∞ (D) is defined by and it can be seen that HR
f R = sup f (T ): T 2 < 1, T 3 < 1 < +∞. ∞ (D)) and f 1 if and only if In this case we have that f ∈ Mm,n (HR R
Im − f (z)f (w)∗ = 1 − z2 w¯ 2 K1 (z, w) + 1 − z3 w¯ 3 K2 (z, w). However, our weak∗ -RFD result only guarantees that f R is attained by taking the supremum all finite-dimensional representations π such that π(z2 ) = A and π(z3 ) = B are commuting strict contractions satisfying A3 = B 2 . However, given such a pair there is, in general, no single matrix T such that T 2 = A and T 3 = B. So our results do not guarantee, that f R is attained by taking the supremum over all matrices T satisfying T 2 < 1 and T 3 < 1. Example 5.6. Let L = {z ∈ C: |z − a| < 1, |z − b| < 1}, where |a − b| < 1, then the functions f1 (z) = z − a, f2 (z) = z − b give a polynomial presentation of this “lens”. The algebra of this presentation is again the algebra of polynomials. An admissible representation of this algebra is defined by choosing any operator satisfying T − aI 1 and T − bI 1, with strict inequalities for the admissible strict representations. In this case we easily see that · u = · u0 , since given any operator T satisfying T − aI 1 and T − bI 1, and r < 1, Sr = rT + (1 − r)(a + b) corresponds to the admissible strict representations and for any matrix of polynomials (pi,j (T )) = limr→1 (pij (Sr )). This algebra with this norm was studied in [11]. Their work shows that this norm is completely boundedly equivalent to the usual supremum ∞ (L) = H ∞ (L), as sets, but the norms are different. norm and consequently, HR ∞ (L)) and f 1 if and only if Our results imply that f ∈ Mm,n (HR R Im − f (z)f (w)∗ = 1 − (z − a)(w − b) K1 (z, w) + 1 − (z − b)(w − b) K2 (z, w). Since the coordinate function z belongs to the algebra A, our weak∗ -RFD results again show that f R is attained by choosing matrices satisfying T − aI < 1, T − bI < 1. Example 5.7. Let G = {(zi,j ) ∈ MM,N : (zi,j ) < 1} and let F : G → MM,N be the identity map F (z) = (zi,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the MN variables {zi,j }. An admissible representation of this algebra is given by any choice of MN commuting operators {Ti,j } on a Hilbert space H, such that (Ti,j ) 1 in MM,N (B(H)) and as above, one can show that · R is achieved by taking suprema over all commuting MN -tuples of matrices for which (Ti,j ) < 1. We have that f ∈ ∞ (G)) and f 1 if and only if Mm,n (HR R Im − f (z)f (w)∗ = R1 (z) IM − (zi,j )(wi,j )∗ ⊗ IH R1 (w)∗ , for some appropriately chosen R1 .
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All of the above examples are also covered by the theory of [5,9], except that their definition of the norm is slightly different and the fact that the suprema are attained over matrices rather than operators, i.e., the weak∗ -RFD consequences, seem to be new. We address the difference between their definition of the norm and ours in a later remark. We now turn to some examples that are not covered by these other theories. Example 5.8. Let 0 < r < 1 be fixed and let Ar = {z ∈ C: r < |z| < 1} be an annulus. Then this has a rational presentation given by F1 (z) = z and F2 (z) = rz−1 , and the algebra of this presentation is just the Laurent polynomials. Admissible representations of this algebra are given by selecting any invertible operator T satisfying T 1 and T −1 r −1 . Admissible strict representations are given by invertible operators satisfying T < 1 and T −1 < r −1 . The al∞ (A ) is also introduced in [3] where it is called the Douglas–Paulsen gebra that we denote HR r algebra. It is no longer quite so clear that · u = · u0 . However, this algebra with these norms is studied by the first author in [24] and among other results the equality of these norms was shown. Consequently, f R is attained by taking the supremum over matrices T satisfying T < 1 and T −1 < r −1 . The formula for the norm is given by f R 1 if and only if ¯ 1 (z, w) + 1 − r 2 z−1 w −1 K2 (z, w). Im − f (z)f (w)∗ = (1 − zw)K The scalar version of this formula is also shown in [3]. Douglas and the second author showed in [19] that · u is completely boundedly equivalent to the usual supremum norm, but that the two norms are not equal. In fact, they exhibit an explicit function for which the norms are different. Since the norms are equivalent, it fol∞ (A ) = H ∞ (A ) as sets. Badea, Beckermann and Crouzeix [8] show that not only lows that HR r r are the norms equivalent, but that there is a universal constant C, independent of r, such that f ∞ f R Cf ∞ . Example 5.9. Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C: |φ(z)| < 1} and Q(G) = {T : σ (T ) ⊆ G and φ(T ) 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ, regarded as a subalgebra of the algebra of analytic functions on G. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that · u = ∞ (G) if and only if f ∈ Hol(G) and · u0 and that f ∈ HR f R = sup f (T ): T ∈ QR (G) < +∞. Our results imply that f R 1 if and only if 1 − f (z)f (w)∗ = 1 − φ(z)φ(w)∗ K1 (z, w). In particular, if we take φ(z) = z−1 z+1 then it maps the half plane H = {z: Re(z) > 0} to the unit disk. For this particular φ, we have that QR = {T : σ (T ) ⊆ H and Re(T ) 0}.
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Example 5.10. Similarly, if we let G = {z ∈ CN : |φi (z)| < 1, i = 1, . . . , N } where φi (z) = zi −1 zi +1 , then G is an intersection of half planes and QR consists of all commuting N -tuples of operators, (T1 , . . . , TN ) such that σ (Ti ) ⊆ H and Re(Ti ) 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D. KalyuzhnyiVerbovetzkii in [22]. Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes Hθ . Each half plane can be expressed as {z: |Fθ (z)| < 1} for some family of linear fractional maps. If we let R = {Fθ }, then QR (G) = {T : σ (T ) ⊆ G and Fθ (T ) 1, ∀θ }. Moreover, each inequality Fθ (T ) 1 can be re-written as an operator inequality for the real z part of some translate and rotation of T . For example, when G = D, we may take Fθ (z) = z−2e iθ , iθ for 0 θ < 2π . In this case, one checks that Fθ (T ) 1 if and only if Re(e T ) I . Thus, it follows that Q(D) = T : σ (T ) ⊆ D and w(T ) 1 , ∞ (D) becomes the “universal” operator where w(T ) denotes the numerical radius of T . Thus, HR algebra that one obtains by substituting an operator of numerical radius less than one for the variable z and we have a quite different quantization of the unit disk. Our results give a formula for this norm, but only in terms of R-limits, so further work would need to be done to make it explicit.
Example 5.12. There is a second way that one can quantize many convex sets. Let G = {z: |z − ak | < rk , k ∈ I } ⊆ C be an open, bounded convex set that can be expressed as an intersection of a possibly infinite set of open disks. For example, the open unit square cannot be expressed as such an intersection, but any convex set with a smooth boundary with uniformly bounded curvature can be expressed in such a fashion. Then G has a rational presentation given by Fk (z) = rk−1 (z − ak ), k ∈ I the algebra of the presentation is just the polynomial algebra and an admissible representation is given by selecting any operator T satisfying, T − ak I rk , k ∈ I . Thus, we again a factorization result, but only in terms of R-limits. The above definitions allow one to consider many other examples. For example, one could fix 0 < r < 1 and let G = {z ∈ BN : r < |z1 |}, with rational presentation f1 (z) = (z1 , . . . , zN ) ∈ M1,N , and f2 (z) = rz1−1 . An admissible representation would then correspond to a commuting row contraction with T1 invertible and T1−1 r −1 . We now compare and contrast some of our hypotheses with those of [5,9]. Remark 5.13. Let G = {z ∈ CN : Fk (z) < 1, k = 1, . . . , K} where the Fk ’s are matrix-valued polynomials defined on G. Then for f ∈ Hol(G), [5,9] really study a norm given by f s = sup{f (T )} where the supremum is taken over all commuting N -tuples of operators T with Fk (T ) < 1, for 1 k K. We wish to contrast this norm with our f R . In [5] it is shown that the hypotheses Fk (T ) < 1, k = 1, . . . , K implies that σ (T ) ⊆ G. Thus, we have that f s f R . In fact, we have that f s = f R . This can be seen by the fact that they obtain identical factorization theorems to ours. This can also be seen directly in some cases where the algebra A contains the polynomials and when it can be seen that · R is attained by taking the supremum over matrices (see Remark 4.11). Indeed, if f R is attained as the supremum over
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commuting N -tuples of finite matrices T = (T1 , . . . , TN ) satisfying σ (T ) ⊆ G and Fk (T ) 1 then such an N -tuple of commuting matrices, can be conjugated by a unitary to be simultaneously put in upper triangular form. Now it is easily argued that the strictly upper triangular entries can be shrunk slightly so that one obtains new N -tuples T = (T1, , . . . , TN, ) satisfying, Fk (T ) < 1, k = 1, . . . , K and Ti − Ti, < . But we do not have a simple direct argument that works in all cases. Remark 5.14. We do not know how generally it is the case that · u is a local norm. That is, we do not know if f u = f R for f ∈ Mn (A). In particular, we do not know if this is the case for Example 5.5. In this case, the algebra of the presentation is A = span{zn : n 0, n = 1}. If we write a polynomial p ∈ A in terms of its even and odd decomposition, p = pe + po , then pe (z) = q(z2 ) and po = z3 r(z2 ) for some polynomials q, r. In this case it is easily seen that pu = sup q(A) + Br(A): A 1, B 1, AB = BA, A3 = B 2 , while pL = pR = sup p(T ): T 2 1, T 3 1 . References [1] J. Agler, Some interpolation theorems of Nevanlinna–Pick type, preprint, 1988. [2] J. Agler, On the representation of certain holomorphic function defined on a polydisk, in: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, in: Oper. Theory Adv. Appl., vol. 48, 1990, pp. 47–66. [3] J. Agler, J. McCarthy, Pick Interpolation and Hilbert Function Spaces, Grad. Stud. Math., vol. 44, American Mathematical Society, Providence, RI, 2002. [4] D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur functions, operator colligation, and reproducing Kernel Pontryagin spaces, in: Oper. Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997, pp. 1–13. [5] C.G. Ambrozie, D. Timotin, A von Neumann type inequality for certain domains in Cn , Proc. Amer. Math. Soc. 131 (2003) 859–869. [6] R.J. Archbold, On residually finite-dimensional C∗ -algebras, Proc. Amer. Math. Soc. 123 (9) (1995) 2935–2937. [7] W. Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (2) (1998) 159–228. [8] C. Badea, B. Beckermann, M. Crouzeix, Intersections of several disks of the Riemann sphere as K-spectral sets, preprint. [9] J.A. Ball, V. Bolotnikov, Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining functions in Cn , J. Funct. Anal. 213 (1) (2004) 45–87. [10] J.A. Ball, T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna–Pick interpolation in several variables, J. Funct. Anal. 197 (1998) 1–61. [11] B. Beckermann, M. Crouzeix, A lenticular version of a von Neumann inequality, Arch. Math. 86 (2006) 352–355. [12] B. Blackadar, Operator algebras: Theory of C∗ -algebras and von Neumann algebras, in: Operator Algebras and Non-commutative Geometry III, in: Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [13] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules – an Operator Space Approach, London Math. Soc. Monogr., vol. 30, Oxford University Press, 2004. [14] D.P. Blecher, V.I. Paulsen, Explicit construction of universal operator algebras and an application to polynomial factorization, Proc. Amer. Math. Soc. 112 (1991) 839–850. [15] D.P. Blecher, Z.-J. Ruan, A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990) 188–201. [16] J. Burbea, P. Masani, Banach and Hilbert spaces of vector-valued functions, in: Their General Theory and Applications to Holomorphy, in: Res. Notes Math., vol. 90, Pitman Advanced Publishing Program, Boston/London/ Melbourne, 1984. [17] M. Dadarlat, Residually finite dimensional C∗ algebras, in: Operator Algebras and Operator Theory: International Conference on Operator Algebras and Operator Theory, By Liming Ge Shanghai, China, July 4–9, 1997, pp. 45–50.
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[18] K.R. Davidson, D. Pitts, Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (3) (1998) 321–337. [19] R.G. Douglas, V.I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged) 50 (1– 2) (1986) 143–157. [20] S.W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (3) (1978) 300–304. [21] K.R. Goodearl, P. Menal, Free and residually finite-dimensional C∗ -algebras, J. Funct. Anal. 90 (2) (1990) 391–410. [22] D. Kalyuzhnyi-Verbovetzkii, On the Bessmertnyi class of homogeneous positive holomorphic functions of several variables, in: J.A. Ball, J.W. Helton, M. Klaus, L. Rodman (Eds.), Current Trends in Operator Theory and Its Applications, vol. OT 149, Birkhäuser-Verlag, Basel, 2004, pp. 255–289. [23] S. Lata, M. Mittal, V.I. Paulsen, An operator algebraic proof of Agler’s factorization theorem, Proc. Amer. Math. Soc. 137 (11) (2009) 3741–3748. [24] M. Mittal, PhD thesis, University of Houston. [25] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, 2002. [26] V.I. Paulsen, Operator algebras of idempotents, J. Funct. Anal. 181 (2001) 209–226. [27] V.I. Paulsen, M. Raghupathi, Representations of logmodular algebras, Trans. Amer. Math. Soc., in press. [28] G.B. Pedrick, Theory of reproducing kernels in Hilbert spaces of vector-valued functions, Univ. of Kansas Tech. Rep., vol. 19, Lawrence, 1957. [29] G. Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge University Press, 2003. [30] G. Popescu, Interpolation problems in several variables, J. Math. Anal. Appl. 227 (1) (1998) 227–250 (English summary). [31] J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191. [32] J.L. Taylor, The analytic functional calculus for several commuting contractions, Acta Math. 125 (1970) 1–38. [33] J.L. Taylor, Analytic Functional Calculus and Spectral Decompositions, Math. Appl., vol. 1, D. Reidel Publishing Co. Dordrecht, Holland, 1982.
Journal of Functional Analysis 258 (2010) 3227–3240 www.elsevier.com/locate/jfa
Endpoint Strichartz estimates for the magnetic Schrödinger equation Piero D’Ancona a , Luca Fanelli b,∗ , Luis Vega b , Nicola Visciglia c a SAPIENZA – Università di Roma, Dipartimento di Matematica, Piazzale A. Moro 2, I-00185 Roma, Italy b Universidad del Pais Vasco, Departamento de Matemáticas, Apartado 644, 48080, Bilbao, Spain c Universitá di Pisa, Dipartimento di Matematica, Largo B. Pontecorvo 5, 56100 Pisa, Italy
Received 24 January 2009; accepted 4 February 2010 Available online 19 February 2010 Communicated by I. Rodnianski
Abstract We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n 3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials. © 2010 Elsevier Inc. All rights reserved. Keywords: Strichartz estimates; Dispersive equations; Schrödinger equation; Magnetic potential
1. Introduction Recent research on linear and nonlinear dispersive equations is largely focused on measuring precisely the rate of decay of solutions. Indeed, decay and Strichartz estimates are one of the central tools of the theory, with immediate applications to local and global well posedness, existence of low regularity solutions, and scattering. This point of view includes most fundamental equations of physics like the Schrödinger, Klein–Gordon, wave and Dirac equations. Strichartz estimates appeared in [29]; the basic framework for this study was laid out in the two papers * Corresponding author.
E-mail addresses:
[email protected] (P. D’Ancona),
[email protected] (L. Fanelli),
[email protected] (L. Vega),
[email protected] (N. Visciglia). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.007
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[12,18], which examined in an exhaustive way the case of constant coefficient, unperturbed equations. This leads naturally to the possible extensions to equations perturbed with electromagnetic potentials or with variable coefficients; a general theory of dispersive properties for such equations is still under construction and very actively researched. In the present paper we shall focus on the time dependent Schrödinger equation i∂t u(t, x) = H u(t, x),
u(0, x) = ϕ(x),
x ∈ Rn , n 3,
(1.1)
associated with the electromagnetic Schrödinger operator H := −∇A2 + V (x),
∇A := ∇ − iA(x)
(1.2)
where A = (A1 , . . . , An ) : Rn → Rn , V : Rn → R. We recall that in the unperturbed case A ≡ 0, V ≡ 0, dispersive properties are best expressed in terms of the mixed norms on R1+n Lp Lq := Lp Rt ; Lq Rnx as follows: for every n 3, it e ϕ
p q
Lt Lx
cn ϕL2 ,
provided the couple (p, q) satisfies the admissibility condition n n 2 = − , p 2 q
2 p ∞.
(1.3)
These estimates are usually referred to as Strichartz estimates. Our main goal is to find sufficient conditions on the potentials A, V such that Strichartz estimates are true for the perturbed equation (1.1). In the purely electric case A ≡ 0 the literature is extensive and almost complete; we may cite among many others the papers [1,2,13,14,23]. It is now clear that in this case the decay V (x) ∼ 1/|x|2 is critical for the validity of Strichartz estimates; suitable counterexamples were constructed in [15]. In the magnetic case A = 0, the Coulomb decay |A| ∼ 1/|x| is likely to be critical, however no explicit counterexamples are available at the time. An intense research is ongoing concerning Strichartz estimates for the magnetic Schrödinger equation, see e.g. [5,7,8, 11,21]; see also [22] for a more general class of first order perturbations. Due to the perturbative techniques used in the above mentioned papers, an assumption concerning absence of zero-energy resonances for the perturbed operator H is typically required in order to preserve the dispersion. In the case A ≡ 0 it was shown in [2] how this abstract condition can be dispensed with, by directly proving some weak dispersive estimates (also called Morawetz or smoothing estimates) via multipliers methods. Here we shall give a very short proof of Strichartz estimates for the magnetic Schrödinger equation with potentials of almost Coulomb decay, based uniquely on the weak dispersive estimates proved in [10]. The leading theme is that direct multiplier techniques allow to avoid, under suitable repulsivity conditions on V and nontrapping conditions on A (see also [9]), the presence of nondispersive components, and to preserve Strichartz estimates.
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We begin by introducing some notations. Regarding as usual the potential A as a 1-form, we define the corresponding magnetic field as the 2-form B = dA, which can be identified with the anti-symmetric gradient of A: B ∈ Mn×n ,
B = DA − (DA)t ,
(1.4)
where (DA)ij = ∂i Aj , (DA)tij = (DA)j i . In dimension 3, B is uniquely determined by the vector field curl A via the vector product Bv = curl A × v,
∀v ∈ R3 .
(1.5)
We define the trapping component of B as Bτ (x) =
x B(x); |x|
(1.6)
when n = 3 this reduces to Bτ (x) =
x × curl A(x), |x|
n = 3,
(1.7)
thus we see that Bτ is a tangential vector. The trapping component may be interpreted as an obstruction to the dispersion of solutions; some explicit examples of potentials A with Bτ = 0 in dimension 3 are given in [9,10]. Moreover, by ∂r V = ∇V ·
x , |x|
we denote the radial derivative of V , and we decompose it into its positive and negative part ∂r V = (∂r V )+ − (∂r V )− . The positive part (∂r V )+ also represents an obstruction to dispersion, and indeed we shall require it to be small in a suitable sense. To ensure good spectral properties of the operator we shall also assume that the negative part V− is not too large in the sense of the Kato norm: Definition 1.1. Let n 3. A measurable function V (x) is said to be in the Kato class Kn provided lim sup
r↓0 x∈Rn |x−y|r
|V (y)| dy = 0. |x − y|n−2
We shall usually omit the reference to the space dimension and write simply K instead of Kn . The Kato norm is defined as |V (y)| dy. V K = sup n |x − y|n−2 x∈R
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A last notation we shall need is the radial-tangential norm p f Lp L∞ (S ) r r
∞ :=
sup |f |p dr.
|x|=r
0
In our results we always assume that the operators H and −A := −(∇ −iA)2 are self-adjoint and positive on L2 , in order to ensure the existence of the propagator eitH and of the powers H s via the spectral theorem. There are several sufficient conditions for self-adjointness and positivity, which can be expressed in terms of the local integrability properties of the coefficients (see the standard references [4,19]); here we prefer to leave this as an abstract assumption. Our main result is the following: 1 (Rn \ {0}), assume the operators = −(∇ − iA)2 Theorem 1.1. Let n 3. Given A, V ∈ Cloc A and H = −A + V are self-adjoint and positive on L2 . Moreover assume that n
V− K <
π2 Γ ( n2 − 1)
(1.8)
and j ∈Z
2j sup |A| + x∈Cj
22j sup |V | < ∞,
(1.9)
x∈Cj
j ∈Z
where Cj = {x: 2j |x| 2j +1 } and the Coulomb gauge condition div A = 0.
(1.10)
Finally, when n = 3, we assume that for some M > 0 (M + 12 )2 3 2 1 |x| 2 Bτ 2 ∞ + (2M + 1)|x|2 (∂r V )+ L1 L∞ (S ) < , L L (S ) r r r r M 2
(1.11)
while for n 4 we assume that 2 |x| Bτ (x)2 ∞ + 2|x|3 (∂r V )+ (x) ∞ < 2 (n − 1)(n − 3). L L 3
(1.12)
Then, for any Schrödinger admissible couple (p, q), the following Strichartz estimates hold: itH e ϕ p q Cϕ 2 , L L L
2 n n = − , p 2 q
p 2,
p = 2 if n = 3.
(1.13)
In dimension n = 3, we have the endpoint estimate |D| 12 eitH ϕ
L2 L6
1 H 4 ϕ L2 .
(1.14)
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Remark 1.1. By optimizing condition (1.11) with respect to M, we can rephrase the condition as follows: writing for short 3 2 α = |x| 2 Bτ L2 L∞ (S ) , r
r
β = |x|2 (∂r V )+ L1 L∞ (S ) , r
r
we can rewrite it in the following equivalent form: √ √ √ √ α + α + 2β α + α + 2β √ 1 α+β < . √ 2 2 α + 2β Notice that when Bτ ≡ 0 the condition reduces to β < 1/2, and when (∂r V )+ ≡ 0 the condition reduces to α < 1/4. 1 (Rn \ {0}) is actually Remark 1.2. Let us remark that the regularity assumption A, V ∈ Cloc stronger than what we really require. For the validity of the theorem, we just need to give meaning to inequalities (1.11), (1.12).
Remark 1.3. Assumptions (1.10), (1.11), and (1.12) imply the weak dispersion of the propagator eitH (see Theorems 1.9, 1.10, assumptions (1.24), and (1.27) in [10]). Actually assumption (1.24) in [10] seems to be stronger than (1.11), but reading carefully the proof of Theorem 1.9 in [10] it is clear that the real assumption is our (1.11) (see inequality (3.14) in [10]). The strict inequality in (1.11), (1.12) is essential, in order to dispose of the weighted L2 -estimate in the above mentioned Theorems by [10] (see also inequality (3.5) below). Remark 1.4. We recall that, usually, suitable spectral conditions must be required for the dispersion to hold, see e.g. [7,8] where resonances at zero are excluded. In our case, such conditions are implied by the smallness assumptions (1.11), (1.12) which can be checked easily in concrete examples. The derivation of Strichartz estimates from the weak dispersive ones turns out to be remarkably simple if working on the half derivative |D|1/2 u, see Section 3 for details. As a drawback, the final estimates are expressed in terms of fractional Sobolev spaces generated by the perturbed magnetic operator −A . Thus, in order to revert to standard Strichartz norms as in (1.13), we need suitable bounds for the perturbed Sobolev norms in terms of the standard ones. This is provided by the following theorem, which we think is of independent interest. Theorem 1.2. Let n 3. Given A ∈ L2loc (Rn ; Rn ), V : Rn → R, assume the operators A = −(∇ − iA)2 and H = −A + V are self-adjoint and positive on L2 . Moreover, assume that V+ is of Kato class, V− satisfies n
V− K <
π2 , Γ ( n2 − 1)
(1.15)
and n
|A|2 − i∇ · A + V ∈ L 2 ,∞ ,
A ∈ Ln,∞ .
(1.16)
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Then the following estimate holds: 1 H 4 f q Cq |D| 12 f q , L L
1 < q < 2n,
n 3.
(1.17)
4 < q < 4, 3
n 3.
(1.18)
In addition we have the reverse estimate 1 H 4 f
Lq
1 cq |D| 2 f Lq ,
2. Proof of Theorem 1.2 We start with the proof of Theorem 1.2, divided into several steps. First we need to prove that the heat kernel associated with the operator H is well behaved under quite general assumptions: Proposition 2.1. Consider the self-adjoint operator H = −(∇ − iA(x))2 + V (x) on L2 (Rn ), n 3. Assume that A ∈ L2loc (Rn , Rn ), moreover the positive and negative parts V± of V satisfy V+ is of Kato class,
(2.1)
V− K < cn = π n/2 /Γ (n/2 − 1).
(2.2)
Then e−tH is an integral operator and its heat kernel pt (x, y) satisfies the pointwise estimate pt (x, y)
(2πt)−n/2 2 e−|x−y| /(8t) . 1 − V− K /cn
(2.3)
Proof. We recall Simon’s diamagnetic pointwise inequality (see e.g. Theorem B.13.2 in [27]), which holds under weaker assumptions than ours: for any test function g(x), t[(∇−iA(x))2 −V ] e g et (−V ) |g|. Notice that by choosing a delta sequence g of (positive) test functions, this implies an analogous pointwise inequality for the corresponding heat kernels. Now we can apply the second part of Proposition 5.1 in [6] which gives precisely estimate (2.3) for the heat kernel of e−t (−V ) under (2.1), (2.2). 2 The second tool we shall use is a weak type estimate for imaginary powers of self-adjoint operators, defined in the sense of spectral theory. This follows easily from the previous heat kernel bound and the techniques of Sikora and Wright (see [26]): Proposition 2.2. Let H be as in Proposition 2.1, and assume in addition that H 0. Then for all y ∈ R the imaginary powers H iy satisfy the (1, 1) weak type estimate iy H
L1 →L1,∞
C(1 + |y|)n/2 .
(2.4)
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Proof. By Theorem 3 in [25] we obtain immediately√ that our heat kernel bound (2.3) implies the finite speed of propagation for the wave kernel cos(t H ), in the sense of [25,26], i.e., √ cos(t H )φ, ψ L2 = 0 for all φ, ψ ∈ L2 with support in B(ξ1 , x1 ), B(ξ2 , x2 ) respectively, provided |t| < 2−1/2 (|x1 − x2 | − ξ1 − ξ2 ). Then we are in a position to apply Theorem 2 from [26] which gives the required bound. 2 We are ready to prove the first part of Theorem 1.2. Proof of (1.17). We shall use the Stein–Weiss interpolation theorem applied to the analytic family of operators Tz = H z · (−)−z . Here H z is defined by spectral theory while (−)−z e.g. by the Fourier transform. Writing z = x + iy, we can decompose Tz = H iy H x (−)−x (−)−iy ,
y ∈ R, x ∈ [0, 1].
The operators H iy and (−)iy are obviously bounded on L2 . On the side z = 0 the operator reduces to the composition of pure imaginary powers Tiy = H iy (−)−iy and by the weak type estimate (2.4) we obtain immediately by interpolation that H iy , and hence Tiy is bounded on Lp for all 1 < p < ∞: n/2 Tz f Lp C 1 + |y| f Lp
for z = 0, 1 < p < ∞.
(2.5)
Next we consider the case z = 1. We start by proving the estimate Hf Lr Cf Lr ,
n 1
For f ∈ Cc∞ (Rn ) we can write Hf = −f − 2iA · ∇f + |A|2 − i∇ · A + V f. We have then by Hölder’s inequality in Lorentz spaces and assumption (1.16) A · ∇f Lr CALn,∞ ∇f
nr
L n−r ,r
1 r < n,
and using the precised Sobolev embedding g
nr
L n−r ,r
C∇gLr,r = ∇gLr
(2.6)
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(and the boundedness of Riesz operators, which rules out the case r = 1) we obtain A · ∇f Lr Cf Lr ,
1 < r < n.
In a similar way, 2 |A| − i∇ · A + V f
C |A|2 − i∇ · A + V Ln/2,∞ f
Lr
nr ,r L n−2r
,
n 1r < , 2
and again by the Sobolev embedding f
nr ,r
L n−2r
Cf Lr
we conclude that 2 |A| − i∇ · A + V f
Lr
n 1
Cf Lr ,
Summing up we obtain (2.6). Combining (2.6) with the Lr -boundedness of the purely imaginary powers H iy and −iy , we get T1+iy : Lr → Lr ,
n 1
(2.7)
Interpolating (2.7) with (2.5) we obtain T1/4 f Lp Cf Lp for 1 3 1 = + , q 4p 4r which concludes the proof.
1 < p < ∞, 1 < r <
n 2
⇒
1 < q < 2n
2
We pass now to the proof of the reverse estimate (1.18). We shall need the following lemma: Lemma 2.1. Assume that A ∈ L2loc Rn ,
V− K < 4π n/2 /Γ (n/2 − 1).
(2.8)
Then for some constant a < 1 the following inequality holds: V− |f |2 dx a∇A f 2L2 .
(2.9)
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Proof. The proof follows a standard argument. We begin by showing that V− |f |2 dx a∇f 2L2 ,
(2.10)
for some a < 1. This can be restated as 1/2 1/2 V− (−)−1/2 f, V− (−)−1/2 f af 2 ,
a < 1,
i.e. we must prove that the operator T = V− (−)−1/2 is bounded on L2 with norm smaller than one. Equivalently, we must prove that the operator 1/2
T T ∗ = V− (−)−1 V− 1/2
1/2
satisfies 1/2 V (−)−1 V 1/2 f 2 bf 2 , − −
b < 1.
Writing explicitly the kernel of (−)−1 , we are reduced to prove I=
2 1/2 V− (x) |V− (y)| f (y) dy dx k 2 bf 2 n |x − y|n−2
where kn = 4π n/2 /Γ (n/2 − 1), b < 1. Now by Cauchy–Schwartz I
V− (x)
|V− (y)| dy |x − y|n−2
|f (y)|2 dy dx |x − y|n−2
which gives I V− K
2 |V− (x)| f (y) dy dx = V− 2K f 2 n−2 |x − y|
and this proves (2.10) under the smallness assumption (2.8). Applying the same computation to the function |f | instead of f , we deduce from (2.10) that
2 V− |f |2 dx a ∇|f |L2 .
Since A ∈ L2loc , we can apply the diamagnetic inequality ∇|f | |∇A f |, (see e.g. [20]) to obtain (2.9).
2
a.e. in Rn
(2.11)
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Proof of (1.18). We begin by proving the L2 inequality (−)1/2 f
L2
∇f L2 C H 1/2 f L2 .
(2.12)
We can write, with the notation ∇A = ∇ − iA, 1/2 2 H f = (Hf, f ) = − ∇ 2 f, f +
V |f |2 = ∇A f 2L2 +
A
V |f |2
and this implies 1/2 2 H f ∇A f 2 2 −
L
V− |f |2 .
Thus by Lemma 2.1 we have for some a < 1 1/2 2 H f (1 − a)∇A f 2 2 L
so that, in order to prove (2.12), it is sufficient to prove the inequality ∇f L2 C∇A f L2 .
(2.13)
Now, using as in the first half of the proof the Hölder inequality and the Sobolev embedding in Lorentz spaces, we can write A|f |2 2 + L
V |f |2 C |A|2 + V Ln/2,∞ |f |
2n ,2 L n−2
C ∇|f |L2
by assumption (1.16). Then, by the diamagnetic inequality ∇|f | |∇A f | we obtain Af 2L2
+
V |f |2 C∇A f 2L2 .
(2.14)
Moreover we have (∇ − iA)f 2 |∇f | − |Af | 2
⇒
|∇f |2 2|∇A f |2 + 2|Af |2
which implies ∇f 2L2 2∇A f 2L2 + 2Af 2L2 and combining this with (2.14) we get ∇f 2L2 C∇A f 2L2 − 2
V |f |2 C∇A f 2L2 + 2
V− |f |2 .
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Using again Lemma 2.1 we finally arrive at (2.13), so that the claimed estimate (2.12) is proved. Now we can use again the Stein–Weiss interpolation theorem, applied to the analytic family of operators Tz = (−)z H −z with z in the range 0 z 1/2. Writing z = x + iy we have Tz = (−)iy (−)x H −x H −iy ,
y ∈ R, x ∈ [0, 1/2].
The operators H −iy and (−)iy are bounded on L2 , while estimate (2.12) proves that (−)1/2 H −1/2 is also bounded on L2 . This shows that Tz f L2 Cf L2
for z = 1/2.
(2.15)
On the side z = 0 the operator reduces to the composition of pure imaginary powers Tiy = (−)iy H −iy and arguing as in the proof of (2.5) we get that Tiy is bounded on Lp for all 1 < p < ∞: n/2 Tz f Lp C 1 + |y| f Lp
for z = 0, 1 < p < ∞.
(2.16)
Then by the Stein–Weiss interpolation theorem we obtain as above T1/4 f Lq Cf Lq with 1 1 1 = + , q 4 2p
1
⇒
4 < q < 4. 3
2
3. Proof of Strichartz estimates Let us first recall some well-known facts about the free propagator. First of all, the free Strichartz estimates for T (t) = eit , its dual operator and the operator T T ∗ are it e ϕ
Lp Lq
CϕL2 ,
e−is F (s, ·) ds t i(t−s) F (s, ·) ds e 0
L2
CF Lp L q ,
CF Lp L q , Lp Lq
(3.1)
(3.2)
(3.3)
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for all Schrödinger admissible couples (p, q), (
p ,
q ) satisfying 2 n n = − , p 2 q
p 2,
with p = 2 if n = 2 (see [12,18]). Moreover, we recall the following estimate: t 1 i(t−s) F (s, ·) ds |D| 2 e 0
j
2 2 Fj L2 L2 ,
(3.4)
j ∈Z
Lp Lq
for any admissible couple (p, q) as above, where F=
supp Fj ⊂ 2j |x| 2j +1 ,
Fj ,
x ∈ [0, t].
j ∈Z
Estimate (3.4) was proved in [24] first; actually it follows by mixing the free Strichartz estimates for T (t) with the dual of the local smoothing estimates which were proved independently by [3,17,28] and [30]. In the paper [24] the endpoint estimate for p = 2 is not proved (and indeed it predates the Keel–Tao paper [18]). The endpoint case p = 2 in dimension n 3 is a consequence of Lemma 3 in [16]. Finally, we need to recall the local smoothing estimates for the magnetic propagator eitH proved in [10] under assumptions less restrictive than the ones of the present paper: we have 1 sup R R>0
|x|R
∇A eitH ϕ dx dt + sup 1 2 R>0 R
itH 2 e ϕ dσR dt (−A ) 14 ϕ 2 2 , (3.5) L
|x|=R
where the constant in the inequality only depends on Bτ and (∂r V )+ . We are now ready to prove Theorem 1.1. Since div A = 0, we can expand H as follows: H = − + 2iA · ∇A − |A|2 + V .
(3.6)
As a consequence, by the Duhamel formula we can write t eitH ϕ = eit ϕ +
ei(t−s) R(x, D)eitH ϕ ds,
(3.7)
0
where the perturbative operator R(x, D) is given by R(x, D) = 2iA · ∇A − |A|2 + V .
(3.8)
By (3.1) and (3.4) we have |D| 12 e−itH ϕ
Lp Lq
j 1 C |D| 2 ϕ L2 + 2 2 χj R(x, D)eitH ϕ L2 L2 , j ∈Z
(3.9)
P. D’Ancona et al. / Journal of Functional Analysis 258 (2010) 3227–3240
3239
where χj is the characteristic function of the ring 2j |x| 2j +1 . For the first term at the RHS of (3.9), by (1.18) we have |D| 12 ϕ
1 C H 4 ϕ L2 .
L2
(3.10)
On the other hand, we can split the second term as follows j ∈Z
j 2 2 χj R(x, D)eitH ϕ L2 L2
j ∈Z
j 2 2 χj A · ∇A eitH ϕ L2 L2 + χj V − |A|2 eitH ϕ L2 L2 = I + II.
(3.11)
By Hölder inequality, assumption (1.9) and the smoothing estimates (3.5) we have I
j ∈Z
II
1 2 sup |A| · sup R>0 R |x|∼2j
∇A eitH ϕ 2 dx dt
j
1 2
1 (−A ) 4 ϕ L2 , (3.12)
|x|R
j ∈Z
1 2 itH 2 1 e ϕ dσR dt 22j sup |V | + |A|2 · sup 2 R>0 R |x|∼2j |x|=R
1 2 (−A ) 4 ϕ 2 .
(3.13)
L
Now we remark that all the assumptions of Theorem 1.2 are satisfied. Indeed, we know that A ∈ L2loc ; moreover, assumption (1.9) implies that |A| 1/|x| and |V | 1/|x|2 , hence (1.16) is satisfied. Thus by Theorem 1.2 (which holds also in the special case V ≡ 0) we get (−A ) 14 ϕ
L2
Cϕ
H˙
1 2
1 H 4 ϕ L2 .
Collecting (3.11), (3.12) and (3.13) we obtain j ∈Z
1 j 2 2 χj R(x, D)eitH ϕ L2 L2 H 4 ϕ L2
(3.14)
and by (3.9), (3.10) and (3.14) we deduce |D| 12 e−itH ϕ
Lp Lq
1 H 4 ϕ L2 ,
(3.15)
for any admissible couple (p, q); notice that this includes also the 3D endpoint estimate (1.14). In order to conclude the proof, it is now sufficient to use estimate (1.17) which gives 1 −itH H 4 e ϕ
Lp Lq
1 H 4 ϕ L2 ,
(3.16)
and commuting H 1/4 with the flow eitH we obtain (1.13). However, in dimension 3 (1.17) does not cover the endpoint q = 6 and we are left with (3.15).
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References [1] J.A. Barceló, A. Ruiz, L. Vega, Some dispersive estimates for Schrödinger equations with repulsive potentials, J. Funct. Anal. 236 (2006) 1–24. [2] N. Burq, F. Planchon, J. Stalker, S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (6) (2004) 1665–1680. [3] P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. (1988) 413–439. [4] H.L. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts Monogr. Phys., Springer Verlag, Berlin, Heidelberg, New York, 1987. [5] P. D’Ancona, L. Fanelli, Strichartz and smoothing estimates for dispersive equations with magnetic potentials, Comm. Partial Differential Equations 33 (2008) 1082–1112. [6] P. D’Ancona, V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal. 227 (2005) 30–77. [7] M.B. Erdo˘gan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Math. 21 (2009) 687–722. [8] M.B. Erdo˘gan, M. Goldberg, W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R3 , J. Eur. Math. Soc. (JEMS) 10 (2008) 507–532. [9] L. Fanelli, Non-trapping magnetic fields and Morrey–Campanato estimates for Schrödinger operators, J. Math. Anal. Appl. 357 (2009) 1–14. [10] L. Fanelli, L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann. 344 (2009) 249–278. [11] V. Georgiev, A. Stefanov, M. Tarulli, Smoothing – Strichartz estimates for the Schrödinger equation with small magnetic potential, Discrete Contin. Dyn. Syst. Ser. A 17 (2007) 771–786. [12] J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1) (1995) 50–68. [13] M. Goldberg, Dispersive estimates for the three-dimensional Schrödinger equation with rough potential, Amer. J. Math. 128 (2006) 731–750. [14] M. Goldberg, W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys. 251 (2004) 157–178. [15] M. Goldberg, L. Vega, N. Visciglia, Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, Int. Math. Res. Not. 2006 (2006), article ID 13927. [16] A.D. Ionescu, C. Kenig, Well-posedness and local smoothing of solutions of Schrödinger equations, Math. Res. Lett. 12 (2005) 193–205. [17] T. Kato, K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989) 481–496. [18] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (5) (1998) 955–980. [19] H. Leinfelder, C. Simader, Schrödinger operators with singular magnetic vector potentials, Math. Z. 176 (1981) 1–19. [20] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math., vol. 14, 2001. [21] J. Marzuola, J. Metcalfe, D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal. 255 (2008) 1497–1553. [22] L. Robbiano, C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Mem. Soc. Math. Fr. (N.S.) (2005) 101–102. [23] I. Rodnianski, W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (3) (2004) 451–513. [24] A. Ruiz, L. Vega, On local regularity of Schrödinger equations, Int. Math. Res. Not. IMRN 1 (1993) 13–27. [25] A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford Ser. (2) 47 (187) (1996) 371–382. [26] A. Sikora, J. Wright, Imaginary powers of Laplace operators, Proc. Amer. Math. Soc. 129 (2000) 1745–1754. [27] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982) 447–526. [28] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math. J. 55 (1987) 699–715. [29] R. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977) 705–774. [30] L. Vega, The Schrödinger equation: pointwise convergence to the initial date, Proc. Amer. Math. Soc. 102 (1988) 874–878.
Journal of Functional Analysis 258 (2010) 3241–3265 www.elsevier.com/locate/jfa
Branching laws for discrete Wallach points Stéphane Merigon a,∗,1 , Henrik Seppänen b,2 a Fachbereich Mathematik, AG AGF, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt,
Germany b Universität Paderborn, Fakultät für Elektrotechnik, Informatik und Mathematik, Institut für Mathematik,
Warburger Str. 100, 33098 Paderborn, Germany Received 9 September 2009; accepted 31 January 2010 Available online 11 February 2010 Communicated by P. Delorme
Abstract We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain V ⊕ iΩ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of GL(Ω). Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density of the Plancherel measure involves quotients of Γ -functions and the c-function for a symmetric cone of smaller rank. © 2010 Elsevier Inc. All rights reserved. Keywords: Lie group; Holomorphic discrete series; Branching law; Symmetric tube domains; Jordan algebras; Spherical functions; Plancherel theorem
1. Introduction Let G be the identity component of the group of biholomorphisms of an irreducible bounded symmetric domain D. The scalar holomorphic discrete series of G can be realised in the space of holomorphic functions on this domain. By reproducing kernel techniques, M. Vergne and * Corresponding author.
E-mail addresses:
[email protected] (S. Merigon),
[email protected] (H. Seppänen). 1 The author was supported by the fellowship “Bourse Lavoisier” from the Ministry for Foreign and European affairs. 2 The author was supported by a Post Doctoral Fellowship from the Swedish Research Council.
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.025
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H. Rossi [27] have shown (see also [3,28,5]) that it has an analytic continuation as a family of (projective) irreducible unitary representations πα of G, parametrised by the so-called Wallach set. Let r be the rank of the domain and d its characteristic number (cf. next section for a definition). Then the Wallach set is the union of the half-line α > (r − 1) d2 and a discrete part consisting of r points l d2 , l = 0, . . . , r − 1. When α > p − 1, where p is the genus of D, the representation spaces are weighted Bergman spaces (cf. for example [1, p. 382] for more details and a definition of the genus). Let τ be an antilinear involution of D. Then D := Dτ is a totally geodesic submanifold, hence a Riemannian symmetric space, and G := Gτ contains its group of displacements. Such a domain is called a real bounded symmetric domain. When one restricts an irreducible unitary representation of a group to a subgroup, the representation need not to be irreducible anymore, and the decomposition into irreducibles is called a branching law. In our context two branching problems have been extensively studied: the de π α and the restriction of πα to symmetric composition of the tensor product representation πα ⊗ subgroups G = Gτ where τ is an antilinear involution of D. A formula for the first problem and for α > p − 1 was given without proof by Berezin for classical domains in [4]. H. Upmeier and A. Unterberger extended it to all domains and gave a Jordan theoretic proof [24]. The second problem was solved (for the same parameters) by G. Zhang and (independently) by G. van Dijk and M. Pevzner [30,26], and also by Y. Neretin for classical groups [15]. Those two problems are in fact similar. The restriction map from D to D (resp. from D × D to D) gives rise to the Berezin transform on D (resp. D), which is a kernel operator. The solution then consists in computing the spectral symbol of the Berezin transform, or, if one prefers, in computing the spherical Fourier transform of the Berezin kernel. In [29] and in [26, Section 5] the problem of decompos π α+l where l ∈ N is also solved, by the same method. A similar problem is also studied ing πα ⊗ in [6]. For arbitrary parameters, those problems are more complicated, and no general method seems to apply. In [18] the tensor product problem for G = SU(2, 2) is solved for any parameter. In [31] π d is decomposed for any G (π d is called the minimal holomorphic the representation π d ⊗ 2 2 2 representation). In [17], Y. Neretin solves the restriction problem from U (r, s) to O(r, s) (r s) for any parameter by analytic continuation of the result for large parameters. If r = s the support of the Plancherel formula remains the same for all α > r − 1 (here d = 2) but when s − r is sufficiently large new pieces appear when α crosses p − 1 = 2(r + s) − 1 and the situation gets worse as α approaches to (r − 1) d2 , as he had already explained in [16]. For points in the discrete Wallach set, the situation is not clear. In his thesis the second author manages to decompose the restriction of SO(2, n) to SO(1, n) for any parameter [22], as well as the restriction of the minimal holomorphic representation of SU(p, q) to SO(p, q) [21], and the minimal holomorphic representation of Sp(n, R) (resp. SU(n, n)) to GL+ (n, R) (resp. to GL(n, C)) [23]. Assume that D is of tube type, i.e. that D is biholomorphic to the tube domain TΩ over the symmetric cone Ω. Then the inverse image of Ω is a real bounded symmetric domain. In this paper, generalising [23], we establish, for any parameter in the discrete Wallach set, the branching rule for the restriction of the associated representation of G = G0 (TΩ ) to G = GL0 (Ω). We use the model by Rossi and Vergne which realises the representation given by the l-th point in the Wallach set as L2 (∂l Ω, μl ), where ∂l Ω is the set of positive semidefinite elements in ∂Ω of rank l, and μl is a relatively G-invariant measure on ∂l Ω. A key observation is that for any x in ∂l Ω, the function g → ν (g ∗ x) on G, where ν is the power function of the Jordan algebra (cf. (2) for the definition), transforms like a function in a certain parabolically induced representation. A naive approach to construct an intertwining operator from L2 (∂l Ω, μl ) into a direct
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3243
sum of parabolically induced representations would then to weight the functions above by compactly supported smooth functions, i.e., to consider mappings f → ∂l Ω f (x)ν (g ∗ x) dμl (x), for f in C0∞ (∂l Ω). It will become clear that this approach is in fact fruitful. However, there are two problems that have to be dealt with. First of all, it is not obvious that the natural target spaces are unitarisable. Secondly, and more importantly, the integrals above need not converge for the suitable choice of parameters ν. However, as we shall see, both these problems can be solved. The paper is organised as follows. In Section 2 we recall some facts about Jordan algebras and symmetric cones that will be needed in the paper. In Section 3 we prove an identity between the restriction of a spherical function of the cone Ω to a cone of lower rank in its boundary and the corresponding spherical function of the lower rank cone. In Section 4 we define a class of irreducible unitary spherical representations that provides target spaces for the integral operators discussed above. These are constructed using the Levi decomposition of the group G by twisting parabolically induced unitary representations of the semisimple factor of G by a certain character. In Section 5 we construct the intertwining operator as an analytic continuation of the integral operator above. After this has been taken care of, a polar decomposition for the measure μl due to J. Arazy and H. Upmeier [1] will allow to express the restriction of the intertwining operator to K-invariant vectors in terms of the Fourier transform for a cone of rank l. Using this identification, the inversion formula for the Fourier transform can be used to prove the Plancherel theorem for the branching problem. In Appendix A we provide a framework for certain restrictions of distributions to submanifolds which will be useful for giving an analytic continuation for the integral that should give an intertwining operator. It should be pointed out that the standard theory for restricting distributions (e.g. [8, Corollary 8.2.7]) does not apply to our situation since the condition on the wave front set for the distribution is not satisfied. Instead we have to use restrictions based on extensions of test functions in such a way that they are constant in certain directions from the submanifold (cf. Appendix A). We finally want to mention that branching problems related to holomorphic involutions of D have also been studied in [20,11,2,19]. 2. Jordan theoretic preliminaries Let V be a Euclidean Jordan algebra. It is a commutative real algebra with unit element e such that the multiplication operator L(x) satisfies [L(x), L(x 2 )] = 0, and provided with a scalar product for which L(x) is symmetric. An element is invertible if its quadratic representation P (x) = 2L(x)2 − L(x 2 ) is so. The cone of invertible squares to be denoted Ω is a symmetric cone: it is homogeneous under the identity component, G, of the Lie group GL(Ω) = {g ∈ GL(V ) | gΩ = Ω}, and it is self-dual. It follows that the involution θ (g) := (g ∗ )−1 (where g ∗ is the adjoint of g with respect to the scalar product of V ) preserves G (which is hence reductive). The stabiliser K = Ge of e coincides with the identity component of the group Aut(V ) of automorphisms of V and with the fixed points of G under the involution θ , and hence is compact. Thus Ω is a Riemannian symmetric space. The tube TΩ = V ⊕ iΩ over Ω in the complexification of V is a Hermitian symmetric space of the non-compact type, diffeomorphic via the Cayley transform to a (tube type) bounded symmetric domain. Any element of GL(Ω), when extended complex-linearly, preserves TΩ . In this fashion G is seen as a subgroup of the identity component G of the group of biholomorphisms of TΩ .
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We assume that V is simple. Then there exists a positive integer r, called the rank of V , such that any family of mutually orthogonal minimal idempotents has r elements. Such a family is called a Jordan frame. Let g=k⊕p be the Cartan decomposition of the Lie algebra g of G. Then the map x → L(x) yields an isomorphism V → p. The subspace generated by a Jordan frame (more precisely by the associated multiplication operators) is a maximal abelian subspace of p and conversely, any maximal abelian subspace of p determines (up to permutation) a Jordan frame. From now we fix a choice of a Jordan frame (c1 , . . . , cr ) and let a = L(cj ), j = 1, . . . , r and A = exp a. Any x in V can be written x=k
λj cj ,
1j r
where k ∈ K and the λj are real numbers, and the family (λ1 , . . . , λr ) is unique up to permutation (its members are called the eigenvalues of x). Then x belongs to Ω if and only if for all 1 j r, λj > 0, and this spectral decomposition corresponds to the KAK decomposition of G, the Acomponent in the decomposition being unique up to conjugation by an element of the Weyl group W = Sr of G (cf. [10, Chapter VII.3]). The rank of x is defined to be the number of its nonzero eigenvalues. There exist on V a K-invariant polynomial function (x) (the determinant) and a K-invariant linear function tr(x) (the trace) that satisfy (x) =
r
λj
and
tr(x) =
j =1
r
λj .
j =1
The determinant defines the character (g) := (ge)
(1)
of the group G. A Jordan frame gives rise to the Peirce decomposition. Since multiplications by orthogonal idempotents commute, the space V decomposes into a direct sum of joint eigenspaces for the (symmetric) operators (L(cj ))j =1,...,r . The eigenvalues of L(c) when c is an idempotent, belong to {0, 12 , 1}. Let us denote by V (c, α) the eigenspace corresponding to the value α. The decomposition into joint eigenspaces is then given by V=
1ij r
where
Vij ,
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Vii = V (ci , 1) ∩
3245
V (cj , 0),
j =i
and when i = j ,
1 1 Vij = V ci , V (cj , 0). ∩ V cj , ∩ 2 2 k ∈{i,j / }
We have Vii = Rci and the Vij all have the same dimension d, called the degree of the Jordan algebra. We can now describe the roots of (g, a). Let (δj )j =1,...,r be the dual basis of (L(cj ))j =1,...,r in a∗ . Then the roots are αij± = ±
δj − δi , 2
1 i < j r,
and the corresponding root spaces are g+ ij = {a ei | a ∈ Vij }, g− ij = {a ej | a ∈ Vij }, where x y = L(L(x)y) + [L(x), L(y)]. Let N be the nilpotent subgroup
N = exp
g+ ij .
1i<j r
Then G has the Iwasawa decomposition G = N AK. For any idempotent c, the projection on V (1, c) is P (c), and V (1, c) is a Jordan subalgebra, hence a Euclidean Jordan algebra with neutral element c (note that it is simple with rank the one of c). We denote by Ω1 (c) its symmetric cone. In particular for ej =
l
ck ,
k=1
we set V (l) = V (1, el )
and Ω (l) = Ω1 (el ),
and also note G(l) the identity component of G(Ω (l) ), K (l) = Gel and (l) the determinant of V (l) . The principal minors of V are then defined by the formula
(j ) (x) := (j ) P (ej )(x) . Then x is in Ω if and only if for all 1 j r, (j ) (x) > 0. Let ν ∈ Cr and set for x in Ω, ν −ν3
2 1 −ν2 (x)(2) ν (x) = ν(1)
ν
−ν
r r−1 r (x) . . . (r−1) (x)ν(r) (x).
(2)
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∗ . Then if a(g) is the projection of Using the basis (δj ), we can identify ν with an element of aC g on A in the Iwasawa decomposition,
ν (gx) = eν log a(g) ν (x).
(3)
The action of G on the boundary ∂Ω of Ω has r − 1 orbits, which may be parametrised by the rank of their elements. We denote by ∂l Ω the orbit of rank l elements, i.e., ∂l Ω = Gel . There exists on ∂l Ω a unique (up to constant) relatively G-invariant measure μl , which transforms according to ld
dμl (gx) = 2 (g) dμl (x). The Hilbert space associated to the Wallach point l d2 is, up to renormalisation, isometric to L2 (∂l Ω, μl ) [5, Theorem X.III.4], and the representation of G in this picture is then given by ld π l (g)f = 4 (g)f g ∗ · .
(4)
The measures μl were constructed by M. Lassalle [14] and can also be obtain as Riesz distributions, thanks to S. Gindikin’s theorem [5, VII.3]. A major tool for our purpose will be the Arazy–Upmeier polar decomposition of μl [1, Theorem 2.7]. Let Πl = K.el be the set of idempotents of rank l. Then ∂l Ω is the disjoint union ∂l Ω =
Ω(u).
u∈Πl
Since elements of G permute the faces of Ω (which are of the form Ω(u) for idempotents u), an action is induced on Πl , such that the preceding equality defines a G-equivariant fibration ∂l Ω → Πl . For any f in the space C0∞ (∂l Ω) of smooth functions with compact support on ∂l Ω,
f dμl = ∂l Ω
dk
K
rd
2 (l) (x)f (kx) d∗(l) x,
(5)
Ω (l) (l)
where the Haar measure on K is normalised and d∗ x is the unique (up to constant) G(l) -invariant measure on Ω (l) . The set ∂l Ω is not a submanifold of V . However let Vl be the (open) set of elements in V with rank greater or equal than l. To any l-element subset Il ⊂ {1, . . . , r} one can associate the idempotent eIl = j ∈Il cj and the minor Il (x) := (P (eIl )x + e − eIl ). Then
S. Merigon, H. Seppänen / Journal of Functional Analysis 258 (2010) 3241–3265
Vl =
3247
x ∈ V Il (x) = 0 ,
Il ⊂{1,...,r}
and [14, Propositions 3 and 7] show that ∂l Ω is a (closed) submanifold of Vl . 3. An identity between spherical functions The spherical functions on Ω may be defined for ν in a∗C by the formula Φν (x) =
ν (kx) dk. K
When ν satisfies ν1 · · · νr 0, a property that we will denote by ν 0, the generalised power function ν and the spherical function Φν extend continuously to Ω. Now let al = L(cj ), j = 1, . . . , l
(6)
for 1 l r − 1 and assume that ν belongs to (al C )∗ , i.e. that its (r − l) last coordinates van(l) ish. Then ν also defines a spherical function Φν of Ω (l) . Let α be a real number. When ν appears in the argument of an object related to V (l) , we will use the convention that ν + α := (ν1 + α, . . . , νl + α). Recall that Ω (l) ⊂ ∂l Ω. Theorem 3.1. Let ν in al ∗C such that ν 0. Then for all x in Ω (l) , Φν (x) = γν(l) Φν(l) (x),
where γν(l) =
ΓΩ (l) ( rd 2 )ΓΩ (l) (ν + ΓΩ (l) ( ld2 )ΓΩ (l) (ν +
ld 2) . rd 2 )
Here ΓΩ (l) is the Gindikin Gamma function for the cone Ω (l) , ΓΩ (l) (ν) = (2π)
l(l−1)d 4
l j =1
d Γ νj − (j − 1) . 2
The theorem is proved in the case ν ∈ Nl in [1, Proposition 3.7]. We use this result and the following lemma, which is based on Blaschke’s theorem (see [12, Lemma A.1] for a detailed proof). Lemma 3.2. Let f be a holomorphic function defined on the right half-plane {z ∈ C | z > 0}. If f is bounded and f (n) = 0 for n ∈ N, then f is identically zero. Proof of Theorem 3.1. Let us set zj = νj − νj +1 , j = 1, . . . , l − 1, zl = νl , so that zj 0 and νj = lk=j zk . Let x ∈ Ω (l) and let (l)
(l)
F (z1 , . . . , zl ) = Φν(z1 ,...,zl ) (x) − γν(z1 ,...,zl ) Φν(z1 ,...,zl ) (x).
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Let us fix zj = mj ∈ N, j = 2, . . . , l. If b a > 0, one can see by Stirling’s formula that is bounded on the right half-plane. It follows that the function z →
Γ (z + Γ (z +
l
k=2 mk
+
k=2 mk
+
l
Γ (z+a) Γ (z+b)
ld 2) rd 2 )
(l) is bounded and hence also z → γν(z,m . Now 2 ,...,ml )
Φν(z,m
2 ,...,ml ) (x)
z (kx)m2 (kx) . . . ml (kx) dk (2) (l) (1)
K
(z) 2
ml sup m (kx) . . . (kx) sup (kx) , (1) (2) (l) K
K
and (z) (l) m2
ml Φ sup (1) (kx) . ν(z,m2 ,...,ml ) (x) sup (2) (kx) . . . (l) (kx) K (l)
K (l)
Let δ > supK (1) (kx) supK (l) (1) (kx). Then the holomorphic function f (z) = F (z, m2 , . . . , ml )δ −z is bounded and vanishes on N, hence on the right half-plane, i.e., for every z ∈ C with z > 0 and mj ∈ N, F (z, m2 , . . . , ml ) = 0. By the same argument one shows that for every z1 ∈ C with z1 > 0 and mj ∈ N, the map z → F (z1 , z, m3 , . . . , ml ) vanishes identically, and the proof follows by induction. 2 4. A series of spherical unitary representations In this section we introduce a family of spherical unitary representations that will occur in the decomposition of L2 (∂l Ω) under the action of G. For 1 l r − 1 let nl =
g− ij .
li<j
Note that it is a (nilpotent) Lie algebra and that (cf. (6)) al ⊕ RL(e) =
ker αij± .
li<j
The closed subgroup ZG (al ) = ZG (al ⊕ RL(e)) normalises N l = exp nl hence Ql = ZG (al )N l
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is a subgroup of G. Moreover ZG (al ) ∩ N l = {id} so this decomposition is a semidirect product. The Lie algebra of Ql , ql = zg (al ) ⊕ nl = m ⊕ a ⊕
g± ij ⊕
l
g− ij ,
li<j
where m = zk (a), is a parabolic subalgebra of g, and since Ql is the normaliser of ql in G [10, 7.83], it is a closed subgroup of G (the parabolic subgroup associated to ql ). It is also the stabiliser of a flag of idempotents (e1 , e2 , . . . , el ). Note also that since V (L(cj ), 1) = Rcj , we have ZG (al )cj = R+ cj ,
j = 1, . . . , l.
(7)
Lemma 4.1. Let Al = exp al and Ml =
l
ZG (al )cj .
j =1
Then the multiplication map Ml × Al × N l → Ql is a diffeomorphism. Proof. It is clear from (7) that the product map Ml × Al → ZG (al ) is a smooth bijection hence it is a diffeomorphism. 2 Note that the decomposition in the preceding lemma is not exactly the Langlands decomposition of Ql . However, it is more adapted to our purpose. We will let al (q) denote the Al -component of q ∈ Ql in the preceding decomposition. For ν ∈ (al C )∗ , let 1 ⊗ eν ⊗ 1 be the character of Ql defined by (1 ⊗ eν ⊗ 1)(q) = eν log al (q) , and let us denote by C(G, Ql , 1 ⊗ eν ⊗ 1) the Fréchet space of continuous complex valued functions on G that are Ql -equivariant with respect to 1 ⊗ eν ⊗ 1, i.e.,
C G, Ql , 1 ⊗ eν ⊗ 1 = f ∈ C(G) ∀q ∈ Ql , f (gq) = e−ν log al (q) f (g) . ν The induced representation IndG Ql (1 ⊗ e ⊗ 1) is the left regular representation of G on C(G, Ql , 1 ⊗ eν ⊗ 1). We will know determine values of ν ∈ (al C )∗ for which the representation
ν − ld4 IndG Ql 1 ⊗ e ⊗ 1 ⊗ (cf. (1)) can be made unitary and irreducible. The group G admits the Levi decomposition G = G × R+
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where the semisimple part G is the kernel of the character . Then K is a maximal compact subgroup of G and the Lie algebra g of G has Cartan decomposition g = k ⊕ p with p = {L(x) ∈ p | tr x = 0}, and a = a ∩ p is maximal abelian in p . Let al
=
1j l
L(e − el ) = R L(cj ) − ker αij± , r −l li<j
r−1
ml = m ⊕
j =l+1
L(e − el ) ⊕ R L(cj ) − g± ij . r −l l
Then ql = ml ⊕ al ⊕ nl , is a parabolic subalgebra of G . The corresponding parabolic subgroup Ql admits the Langlands decomposition Ql = Ml Al N l , where Al = exp al and (cf. [10, Chapter VII, Propositions 7.25, 7.27 and 7.82])
Ml = ZK al exp ml ∩ p , whose Lie algebra is ml . For ν ∈ (al C )∗ the induced representation
1 ⊗ eν ⊗ 1 IndG Q l
is defined in the same way as for G. Lemma 4.2. (i) Ql = Ql × R+ , (ii) Ml ⊂ Ml . Proof. To prove (i) we observe that since R+ ⊂ Ql , we can write Ql = Q × R+ , for some subgroup Q ⊂ G . Since Ql (resp. Ql ) is the normaliser of ql (resp. ql ) in G (resp. G ), the inclusion Ql ⊂ Q is obvious and the converse follows from the fact that Ad(G) preserves g . Since X.cj = 0 for X ∈ ml and 1 j l, the assertion (ii) will follow from ZK (al ) = ZK (al ). Let k ∈ ZK (al ), i.e., for j = 1, . . . , l,
L(e − el ) L(e − kel ) L(e − el ) = L(kcj ) − = L(cj ) − . k. L(cj ) − r −l r −l r −l By summing over j one obtains L(kel ) −
l l L(e − kel ) = L(el ) − L(e − el ), r −l r −l
(8)
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and hence L(kel ) = L(el ). By (8) we then have L(kcj ) = L(cj ) for j = 1, . . . , l, i.e., k ∈ ZK (al ). 2 s ∗ l) Let us denote by (δjs )j =1,...,l the dual basis of (L(cj ) − L(e−e r−l )j =1,...,l in (al C ) . By δj → δj we define an isomorphism (al C )∗ (al C )∗ , ν → ν s . Let mν be the character of R+ defined by
mν (ζ ) = ζ −
rld 4 + 1j νj
(9)
.
Proposition 4.3.
s ν − ld4 IndG 1 ⊗ eν ⊗ 1 ⊗ mν . IndG Ql 1 ⊗ e ⊗ 1 ⊗ Q l
Proof. Let us denote by al (q ) the Al -component of q in the Langlands decomposition of Ql . Then for all q = q ξ ∈ Ql = Ql × R+ , eν log al (q) = ξ Indeed, if q = m e
λj (L(cj )−
L(e−el ) r−l )
νj ν s log al (q )
e
(10)
.
n is the Langlands decomposition of q in Ql , then
l q = m e(− r−l +log ξ )L(e−el ) e λj L(cj )+log ξ L(el ) n is the Langlands decomposition of q in Ql . If f ∈ C(G, Ql , 1 ⊗ eν ⊗ 1), then by (10), its res striction f to G belongs to the space C(G , Ql , 1 ⊗ eν ⊗ 1). Conversely, if f ∈ C(G , Ql , s 1 ⊗ eν ⊗ 1), one obtains, again by (10), a function f ∈ C(G, Ql , 1 ⊗ eν ⊗ 1) by setting f (g ζ ) := s − j νj ζ f (g ) and we obtain thereby a bijection C(G, Ql , 1 ⊗ eν ⊗ 1) C(G , Ql , 1 ⊗ eν ⊗ 1). Now the operator
s T : C G, Ql , 1 ⊗ eν ⊗ 1 ⊗ C → C G , Ql , 1 ⊗ eν ⊗ 1 ⊗ C, f ⊗ z → f ⊗ z, ν intertwines the actions of G. Indeed, if f ∈ C(G, Ql , 1 ⊗ e ⊗ 1) and g = g ζ ∈ G = G × R+ ν −1 −1 −1 −1 j then f (g ·) = f (g ζ ·) = ζ f (g ·), and hence
ld ld T f g −1 · ⊗ − 4 (g)z = f g −1 · ⊗ − 4 (g)z
rld = ζ νj f g −1 · ⊗ ζ − 4 z
rld = f g −1 · ⊗ ζ − 4 + νj z. ld
2
−4 ν The representation IndG extends to a continuous representation (denoted Ql (1 ⊗ e ⊗ 1) ⊗ by the same symbol) on the Hilbert completion of C(G, Ql , 1 ⊗ eν ⊗ 1) ⊗ C with respect to 2 |f ⊗ z|2 = f (k) dk|z|2 . K
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This Hilbert space is the tensor product of the Hilbert space completion of C(G, Ql , 1 ⊗ eν ⊗ 1) with respect to the L2 (K)-inner product and C. Lemma 4.4. For any m ∈ Ml and any x ∈ V , (j ) (mx) = (j ) (x),
1 j l.
Proof. Let m ∈ Ml . Then for j = 1, . . . , l, m commutes with L(ej ), and m ∈ Aut V (j ) , so
(j ) (mx) = (j ) P (cj )(mx) = (j ) mP (cj )(x) = (j ) P (cj )(x) = (j ) (x).
2
Proposition 4.5. The map g → −ν (g ∗ e) is a (norm one) K-invariant vector in C(G, Ql , 1 ⊗ eν ⊗ 1). Proof. Let q ∈ Ql and let q = man ∈ Ml Al N l . Then for g ∈ G, one has, since n∗ ∈ N and a ∗ = a,
−ν (gq)∗ e = −ν n∗ a ∗ m∗ g ∗ e = e−ν log(a) −ν m∗ g ∗ e and because of Lemma 4.4,
−ν (gq)∗ e = e−ν log(a) −ν g ∗ e .
2
Let ρl =
l d d δi − δ j = (r + 1 − 2j )δj 2 2 4 j =1
li<j
be the half sum of the negative al ⊕ RL(e)-restricted roots (counted with multiplicities). Theorem 4.6. For almost every λ ∈ a∗l (with respect to Lebesgue measure), the representation ld
−4 ν IndG with ν = iλ + ρ l + Ql (1 ⊗ e ⊗ 1) ⊗ tion.
ld 4
Proof. First, let us remark that if ν = iλ + ρ l +
is an irreducible unitary spherical representa-
ld 4
with λ ∈ a∗l , then since
l(l + 1) rld d l(r + 1 + l) − 2 =i λj + − rl = i λj , νj − 4 4 2
the character mν (cf. (9)) is unitary. We now claim that (ρ l + al -restricted roots (counted with multiplicities). Indeed,
ld s 4)
is the half sum of the negative
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d δi − δj L(e − el ) L(ek ) − 2 2 r −l li<j
d δi − δj L(e − el ) d − = (r + 1 − 2k) + 4 2 2 r −l il<j
1 d d (−δj ) − L(ck ) = (r + 1 − 2k) + 4 4 r −l il<j
ld ld d = ρl + = (r + 1 − 2k) + 4 4 4
s
k>l
L(e − el ) L(ek ) − . r −l
The theorem now follows from Proposition 4.3 and Bruhat’s theorem [25, Theorem 2.6]. Let us note ρ l = ρ l +
ld 4.
2
We now set
ld iν+ρ l ⊗ 1 ⊗ − 4 , (πν , Hν ) := IndG Ql 1 ⊗ e and
vν := −(iν+ρ l ) (·)∗ e ⊗ 1. Let us compute the positive definite spherical function associated to πν , that is,
Φ(g) = πν (g)vν , vν
ν
=
∗ ld −(iν+ρ l ) g −1 k e dk − 4 (g).
K
Since (g ∗ )−1 e = θ (g)e = (ge)−1 [5, Theorem III.5.3], Φ can be written, as a function on Ω, Φ(x) =
ld ld −(iν+ρ l ) x −1 dk − 4 (x) = Φ−(iν+ρ l ) x −1 − 4 (x),
K
and since Φ−(iν+ρ l ) (x −1 ) = Φiν+ρ l +2ρ (x) with ρ=
r
(2j − r − 1)δj
j =1
[5, Theorem XIV.3.1(iv)], Φ(x) = Φiν+ηl +ρ (x)
where ηl =
r d (2j − l − r − 1)δj . 4 j =l+1
Recall [5, Theorem XIV.3.1(iii)] that Φν +ρ = Φν+ρ if and only if ν = wν for w ∈ W . Hence the representations πλ and πλ , with λ, λ in a∗l , are equivalent if and only if iλ + ηl = w(iλ + ηl ). Since ηl is real it follows that πλ is equivalent to πλ if and only if λ = wλ with w ∈ Wl := Sl .
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5. The intertwining operator and the Plancherel formula Let ν ∈ Cl such that (−(iν + ρ l )) 0. Then for f ∈ C0∞ (∂l Ω) and g ∈ G, the formula Tν f (g) =
f (x)−(iν+ρ l ) g ∗ x dμl (x)
(11)
∂l Ω
defines a continuous function on G. Moreover, it follows from (3) and Lemma 4.4 that
Tν f ∈ C G, Ql , 1 ⊗ eiν+ρ l ⊗ 1 .
We will also view Tν as an operator with values in C(G, Ql , 1 ⊗ eiν+ρ l ⊗ 1) ⊗ C (in the obvious way), and hence in Hν . Lemma 5.1. For (−(iν + ρ l )) 0, the operator Tν : C0∞ (∂l Ω) → Hν intertwines π l and πν . Proof. Let us set ν = −(iν + ρ l ). Let h ∈ G. Then
ld 4 (h)f h∗ x ν g ∗ x dμl (x)
Tν (h.f )(g) = ∂Ωl
− ld4
=
(h)
−1 f (x)ν g ∗ h∗ x dμl (x)
∂Ωl − ld4
=
(h)
∗ f (x)ν h−1 g x dμl (x),
∂Ωl
i.e.,
ld Tν (h.f )(g) = − 4 (h)Tν (f ) h−1 g .
2
(12)
Since ρ l = d2 lj =1 (r + l + 1 − 2j )δj , we do not have ((−iλ + ρ l )) 0 when λ ∈ a∗l , and the integral (11) does not converge. This means that the integral has to be interpreted in a suitable sense using analytic continuation in the parameter ν. For this we recall that when ν ∈ Cl , the Riesz distribution Rν+ ld on V can be defined as the analytic continuation of the following 2 integral
Rν+ ld (F ) = ΓΩ (l) 2
ld ν+ 2
−1 F (x)ν (x) dμl (x), ∂l Ω
F ∈ S(V ),
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where S(V ) is the Schwartz space of V , and that it has support in ∂l Ω (cf. [9, Theorems 5.1 and 5.2], where the integral is actually defined over Ol = {x ∈ ∂l Ω | (l) (x) = 0}, but μl (∂l Ω \ Ol ) = 0). The restriction (denoted by the same symbol) to the open set Vl is then a distribution with support in the submanifold ∂l Ω. We can therefore consider the vertical restrictions Rν+ ld |∂l Ω (cf. Appendix A). Since Rν+ ld is a measure with support on ∂l Ω for ν 0, 2 2 these restrictions do not depend on the choice of a tubular neighbourhood (cf. Proposition A.9). For (−(iν + ρ l )) 0, we have
−1
ld ld − 2 (g)(R−(iν+ρ )+ ld )∂ Ω f g ∗ · . Tν f (g) = ΓΩ (l) − iν + ρ l + l l 2 2 Hence, we can let the right-hand side define an analytic continuation of the integrals Tν f (g). It is defined on the complement Z of the set of poles of the meromorphic function ΓΩ (l) (−(iν + ρ l )). Since for fixed f ∈ C0∞ (∂l Ω) the map g → f ((g ∗ )−1 ·) is continuous, the function g → Tν (f )(g) is continuous. Proposition 5.2. For any ν ∈ Z and f ∈ C0∞ (∂l Ω),
Tν f ∈ C G, Ql , 1 ⊗ eiν+ρ l ⊗ 1 , and the operator Tν : C0∞ (∂l Ω) → Hν intertwines π l and πν . Proof. The equation describing the Ql -equivariance as well as Eq. (12) are analytic in the parameter ν. Hence they hold by analytic continuation since they hold on the open set where (−(iν + ρ l )) > 0. 2 We now recall, in order to fix the notations, the definition of the spherical Fourier transform on Ω (l) . If f is a continuous function with compact support on Ω (l) which is K (l) -invariant, its spherical Fourier transform is fˆ(ν) =
(l)
f (x)Φ−ν+ρ (l) (x) d∗(l) x,
Ω (l)
where ν ∈ (al C )∗ and ρ (l) = d4 lj =1 (2j − l − 1)δj . Since f has compact support, the function fˆ is holomorphic on (al C )∗ . For latter use we also recall the inversion formula for (l) f ∈ C0∞ (Ω (l) )K and λ ∈ a∗l (cf. [5, Theorem XIV.5.3] and [7, Chapter III, Theorem 7.4]): (l)
f (x) = c0
a∗l
(l) fˆ(iλ)Φiλ+ρ (l) (x)
dλ , |c(l) (λ)|2
(13)
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where dλ is the Lebesgue measure on a∗l Rl , c(l) (λ) is the Harish-Chandra c-function for Ω (l) , and c0(l) is a positive constant. Now let f ∈ C0∞ (∂l Ω)K , and observe that Ω (l) , being a fibre of ∂l Ω → Πl , is closed in ∂l Ω, and hence f |Ω (l) (sometimes still denoted by f ) has compact support in Ω (l) . Moreover, since any k ∈ K (l) extends to an element of K, the function f |Ω (l) is K (l) -invariant. Proposition 5.3. If f ∈ C0∞ (∂l Ω) is K-invariant then (cf. Theorem 3.1)
rd (l) ˆ Tν f (g) = γ−(iν+ρ −(iν+ρ l ) g ∗ e . f iν − ) l 4
(14)
Proof. Let us set ν = −(iν + ρ l ) in the following. Again, by analytic continuation, it suffices to prove the equality for (ν ) 0. Since f and μl are K-invariant one has for all k in K, Tν f (g) =
f k −1 x ν g ∗ x dμl (x) =
∂Ωl
f (x)ν g ∗ kx dμl (x).
∂Ωl
Hence Tν f (g) =
Tν f (g) dk =
K
∗ f (x) ν g kx dk dμl (x).
∂Ωl
K
Writing g ∗ = th, t ∈ N A, h ∈ K, we have ν (thkx) = ν (hkx)ν (te) = ν (hkx)ν g ∗ e , and using the left invariance of the Haar measure of K,
ν g ∗ kx dk =
K
ν (hkx) dk ν g ∗ e
K
=
ν (kx) dk ν g ∗ e
K
= Φν (x)ν g ∗ e , hence, Tν f (g) =
f (x)Φν (x) dμl (x) ν g ∗ e .
Ω (l)
Now Upmeier and Arazy’s polar decomposition (5) for μl yields Tν f (g) = Ω (l)
rd 2 f (x)Φν (x)(l) (x) d∗(l) x ν g ∗ e ,
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and by Theorem 3.1, (l) Tν f (g) = γν
rd (l) 2 f (x)Φν (x)(l) (x) d∗(l) x ν g ∗ e
Ω (l)
= γν(l)
f (x)Φ (l)
ν + rd 2
Ω (l)
ˆ = γν(l) f Since −ρ l + ρ (l) −
rd 2
−ν + ρ
(x) d∗(l) x ν g ∗ e
rd ν g ∗ e . − 2
(l)
= − rd 4 , we eventually get
(l) Tν f (g) = γ−(iν+ρ ) fˆ l
rd −(iν+ρ l ) g ∗ e . iν − 4
2
In the following we set,
rd (l) ˆ , f (ν) = γ−(iν+ρ ) f iν − l 4 and we note that it defines a meromorphic function whose poles are those of ΓΩ (−(iν +ρ l )+ ld2 ). The following lemma will be used in the proof of the Plancherel formula. Lemma 5.4 (Inversion formula). Let f ∈ C0∞ (∂l Ω)K and let λ ∈ a∗l . Then for x ∈ Ω (l) , (l) f (x) = c0
a∗l
(l) f(λ)Φ
iλ+ρ (l) − rd 4
−1 (l) (x) γ−(iλ+ρ ) l
dλ |c(l) (λ)|2
.
Proof. We have
rd (l)
−1 rd 4 = f(λ) γ−(iλ+ρ ) , f (l) (iλ) = fˆ iλ − l 4 rd 4 hence the inversion formula (13) applied to the function f (l) gives the desired identity.
(15) 2
Recall the notations from the end of Section 4. Let p be the measure on a∗l /Wl defined by (l)
dp(λ) =
c0
(l) (l) 2 |γ−(iλ+ρ ) c (λ)|
dλ.
(16)
l
Consider the direct integral Hilbert spaces a∗ Hλ dp(λ) with inner product ((vλ ), (wλ )) := l a∗l vλ , wλ λ dp(λ), where ·,·λ denotes the inner product on Hλ . We now state the main result of the article.
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Theorem 5.5 (Plancherel theorem). There exists an isomorphism of unitary representations
T : π l , L2 (∂l Ω)
πλ dp(λ),
a∗l
Hλ dp(λ) ,
a∗l
such that for every f ∈ C0∞ (∂l Ω), (Tf )λ = Tλ f . Proof. First we prove that for any K-invariant function f in C0∞ (∂l Ω),
f (x)2 dμl (x) =
f(λ)2 dp(λ).
(17)
a∗l
∂l Ω
For this purpose we use the polar decomposition (5) for μl and the inversion formula of Lemma 5.4. Then f (x)2 dμl (x) ∂l Ω
=
rd f (x)2 2 (x) d (l) x ∗
(l)
Ω (l)
=
rd
(l)
(l) f(λ)Φ
2 f (x)(l) (x)c0
a∗l
Ω (l)
=
(l) f(λ) γ−(iλ+ρ ) −1
= a∗l
= a∗l
(l)
(x) γ−(iλ+ρ ) −1 l
dλ |c(l) (λ)|2
rd
(l)
4 f (x)(l) (x)Φ−iλ+ρ (l) (x) d∗(l) x
l
a∗l
iλ+ρ (l) − rd 4
Ω (l)
d∗(l) x
(l)
c0 dλ |c(l) (λ)|2
(l) rd
(l) c0 dλ 4 (iλ) (l) f(λ) γ−(iλ+ρ ) −1 f (l) l |c (λ)|2
f(λ)2
(l)
c0 dλ (l)
|γ−(iλ+ρ ) c(λ)|2
.
l
In the last equality we have used again the formula (15). The next step is to prove that for a dense subset of functions f in C0∞ (∂l Ω), the identity ∂l Ω
f (x)2 dμl (x) =
|Tλ f |2λ dp(λ) a∗l
holds. Recall that L1 (G) is a Banach ∗-algebra when equipped with convolution as multiplication, and ϕ ∗ (g) := ϕ(g −1 ). Let L1 (G)# denote the commutative closed subalgebra of left and right
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K-invariant functions in L1 (G) (recall that (G, K) is a Gelfand pair since it is a Riemannian symmetric pair, cf. [7, Chapter IV, Theorem 3.1]). There is a natural projection L1 (G) → L1 (G)# ,
ϕ k1−1 · k2 dk1 dk2 .
ϕ → ϕ := #
K K
For a unitary representation (τ, H ) of G, there is a ∗-representation (also denoted by τ ) of L1 (G) on H given by τ (ϕ)v :=
ϕ(g)τ (g)v dg,
v∈H .
G
The representations of K and L1 (G) are related by
τ (k1 )τ (ϕ)τ (k2 ) = τ ϕ k1−1 · k2−1 ,
ϕ ∈ L1 (G), k1 , k2 ∈ K.
(18)
The subspace H K of K-invariants is invariant under L1 (G)# . From (18), it follows that for any ϕ ∈ L1 (G), and u, v ∈ H K ,
τ (ϕ)u, v = τ ϕ # u, v .
(19)
Let ξ be the K-invariant cyclic vector in L2 (∂l Ω). We claim that there exists a sequence ∞ K 2 {ξn }∞ n=1 ⊆ C0 (∂l Ω) , such that ξn → ξ in L (∂l Ω). To see this, we can first choose a se∞ ∞ quence {ζn }n=1 ⊆ C0 (∂l Ω) that converges to ξ . Next, observe that the orthogonal projection P : L2 (∂l Ω) → L2 (∂l Ω)# is given by f → K f (k −1 ·) dk. Then P (f ) is smooth if f is smooth. Moreover, supp f is contained in the image of the map K × supp f → ∂l Ω, (k, x) → kx. It follows that P (C0∞ (∂l Ω)) ⊆ C0∞ (∂l Ω)K . Hence, the claim holds with ξn := P (ζn ). The subspace H0 := π l (f )ξn f ∈ C0∞ (G), n ∈ N is then dense in L2 (∂l Ω). For ϕ ∈ C0∞ (G), n ∈ N, we have, by (19) and (17),
π l (ϕ)ξn , π l (ϕ)ξn
L2 (∂l Ω)
= π l ϕ ∗ ∗ ϕ ξn , ξn L2 (∂ Ω) l l ∗
# = π ϕ ∗ ϕ ξn , ξn L2 (∂ Ω) l l ∗
# Tλ π ϕ ∗ ϕ ξn , Tλ (ξn ) λ dp(λ) = a∗l
=
# πλ ϕ ∗ ∗ ϕ Tλ (ξn ), Tλ (ξn ) λ dp(λ)
a∗l
= a∗l
πλ ϕ ∗ ∗ ϕ Tλ (ξn ), Tλ (ξn ) λ dp(λ)
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=
πλ (ϕ)Tλ (ξn ), π(ϕ)Tλ (ξn ) λ dp(λ)
a∗l
=
Tλ π l (ϕ)ξn , Tλ π l (ϕ)ξn λ dp(λ).
a∗l
Hence, the operator T defined on H0 by T (π l (ϕ)ξn ) = (πλ (ϕ)Tλ (ξn )) extends uniquely to a G-equivariant isometric operator 2 T : L (∂l Ω) → Hλ dp(λ). a∗l
It now only remains to prove the surjectivity of T . Assume therefore that there exists a vector (ηλ ) which is orthogonal to the image of T . Then for all ϕ in L1 (G) and h ∈ L1 (G)# , πλ (ϕ ∗ h)(T ξ )λ , ηλ λ dp(λ) = 0, a∗l
i.e.,
ˇ h(λ) πλ (ϕ)(T ξ )λ , ηλ λ dp(λ) = 0,
a∗l
ˇ where h(λ) is the Gelfand transform of h restricted to a∗l . Recall that the character space of L1 (G)# can be parametrised by the quotient of a subset of (aC )∗ modulo the Weyl group (cf. [7, Chapter IV, Theorems 3.3 and 4.3]). Since the image of L1 (G)# under the Gelfand transform separates points in this space, it follows from the Stone–Weierstrass theorem that the functions hˇ are dense in the space of continuous functions on a∗l which are invariant under the action of Wl . Hence πλ (f )(T ξ )λ , ηλ λ = 0 p-almost everywhere. By separability of L1 (G), there is a set U with p(a∗l \ U ) = 0 such that for all f in L1 (G) and λ ∈ U , πλ (f )(T ξ )λ , ηλ λ = 0. By cyclicity of (T ξ )λ (note that (T ξ )λ is nonzero p-almost everywhere), ηλ is zero p-almost everywhere. 2 Remark 5.6. We want to point out that it is actually not necessary to prove the analytic continuation of Tν (and hence to use the theory of Riesz distributions) to derive the decomposition of π l (however, the natural operator T above is then replaced by an abstract one). Indeed, by the Cartan–Helgason theorem [7, Chapter III, Lemma 3.6] we have HλK = Cvλ when λ ∈ a∗l , and hence we can set f → f(λ)vλ , and by (17) we thus obtain an operator T : L2 (∂l Ω)K → a∗ HλK dp(λ). Assume that we can l prove that T intertwines the respective actions of C0∞ (G)# on L2 (∂l Ω)K and a∗ HλK dp(λ). Tλ : C0∞ (∂l Ω)K → HλK ,
Then for ϕ ∈ C0∞ (G)# ,
l
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π (ϕ)ξ, ξ = T π (ϕ)ξ, T ξ = l
l
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πλ (ϕ)(T ξ )λ , (T ξ )λ λ dp(λ)
a∗l
2 ϕ(λ) ˇ (T ξ )λ λ dp(λ),
= a∗l
ˇ where ϕˇ is defined by πλ (ϕ)vλ = ϕ(λ)v λ . Here ξ is again a K-invariant cyclic vector in L2 (∂l Ω)K . The proof of [22, Theorem 10] shows that the decomposition of π l then follows. We now prove the intertwining property. It is equivalent to the equality l (ϕ)f (λ) = f(λ)ϕ(λ). π ˇ
(20)
Let ν ∈ Cl . Then for f ∈ C0∞ (∂l Ω) and ϕ ∈ C0∞ (G)# we have, with ν = −(iν + ρ l ),
πν (ϕ) ν (·)∗ e ⊗ 1 (g) =
−1 ld ϕ(h)ν g ∗ h∗ e ⊗ − 4 (he) dh
G
=
− ld4
ϕ(h) G
=
K
∗ −1
− ld4 ϕ(h) (he) ν k h e dk dh ν g ∗ e ⊗ 1
G
=
∗ ∗ −1 (he) ν g k h e dk ⊗ 1 dh
K
−1
ld ϕ(h)− 4 (he)Φν h∗ e dh ν g ∗ e ⊗ 1 ,
G
i.e., πν (ϕ)vν = ϕ(ν)v ˇ ν, where ϕ(ν) ˇ is holomorphic on Cl . If (−(iν + ρ l )) 0, then, by Lemma 5.1, the operator Tν intertwines the actions of C0∞ (G)# , and hence l (ϕ)f (ν) = f(ν)ϕ(ν). π ˇ
Thus (20) follows by analytic continuation. Acknowledgment The authors would like to thank Karl-Hermann Neeb for enlightening discussions and comments that led to substantial improvement of the presentation.
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Appendix A. Restrictions of distributions Let X be a smooth n-dimensional manifold. Let D(X) denote the space of compactly supported smooth functions on X, i.e., the test functions on X. For any chart (V , φ), compact subset := φ(V ), and N ∈ N, consider the seminorm K ⊆V pV ,K,N (f ) :=
supx∈K D α f ◦ φ −1 (x)
(A.1)
α∈Nn , |α|N
on the space of smooth functions on X. Here D α :=
∂ |α| α ∂x1 1 ···∂xnαn
for α = (α1 , . . . , αn ), and |α| =
|α1 | + · · · + |αn |. For a compact set K ⊆ X, let D(K) be the space of smooth functions with support in K equipped with the topology induced by the above seminorms. We recall that a distribution on X is a continuous functional on D(X) = K D(K) equipped with the inductive limit topology. We let D (X) denote the space of distributions on X. Let {Ui } be an open cover of X. Then a linear functional on D(X) is continuous if and only if its restriction to every D(Ui ) is continuous. We will now construct restrictions to a closed submanifold Y of distributions that have support on Y . To have a well-defined notion of restriction one cannot permit arbitrary extensions to X of test functions on Y . Instead, we will require the extension to be locally constant along some predescribed direction. This can be made precise using tubular neighbourhoods. Definition A.1. Let X be a smooth n-dimensional manifold, and let Y be a k-dimensional submanifold. A tubular neighbourhood of Y in X consists of a smooth vector bundle π : E → Y , an open neighbourhood Z of the image, ζE (Y ), of the zero section in E, and a diffeomorphism f : Z → O onto an open set O ⊆ X containing Y , such that the diagram Z f ζE j
Y
X
commutes. Here j : Y → X is the inclusion map. Remark A.2. Any closed submanifold Y of X admits a tubular neighbourhood (cf. [13, Chapter IV, F, Theorem 9]). Any splitting of the tangent bundle of X over Y , T (X)|Y = T (Y ) ⊕ E, gives such a vector bundle E. In particular, given a Riemannian metric on X, E can be chosen as the orthogonal complement to T (Y ) in T (X)|Y . Since the concepts we are dealing with are of a local nature we can without loss of generality assume that X = Z itself is a tubular neighbourhood of Y . Definition A.3. A function f on X is said to be locally vertically constant around Y , l.v.c., if for any x ∈ Y , there exists an open neighbourhood Wx of x in X such that for y ∈ Wx , f (y) = f (π(y)). Moreover, if g is a function on Y , and f |Y = g, f is called an l.v.c. extension of g.
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Lemma A.4. (i) Any test function ϕ on Y admits an l.v.c. extension ϕ ∈ D(X). (ii) An l.v.c. function f that vanishes on Y vanishes on some neighbourhood of Y . Proof. Since supp ϕ is compact, it can be covered by finitely many open neighbourhoods O1 , . . . , ON diffeomorphic to products Ui × Vi ⊂ Rk ⊕ Rn−k , where Ui is open in Rk and Vi is to the projection onto the an open neighbourhood of 0 in Rn−k , in such a way that π corresponds π(O first coordinate. Let (ψi ) be a smooth partition of unity on U = N i ) subordinate to the i=1 cover π(Oi ), i = 1, . . . , N , and let ϕi = ϕψi . Then each ϕi can be identified with a test function on Ui , and by multiplying this with a test functionon Vi which is 1 on some neighbourhood of 0, ϕ= ϕi is an l.v.c. extension of ϕ. This proves (i). we obtain an l.v.c. extension ϕi of ϕi . Then For (ii) just observe that if f is an l.v.c. function that vanishes on Y , then Y is in the complement of the support of f . 2 Assume now that u ∈ D (X) has support on the submanifold Y . Then l.v.c. test functions that vanish on a neighbourhood of Y are in the kernel of u, and the preceding lemma enables us to make the following definition. Definition A.5. Let u ∈ D (X) be a distribution with support on Y . The vertical restriction, u|Y , of u to Y is the unique distribution on Y that satisfies u|Y (ϕ|Y ) = u(ϕ), for any ϕ ∈ D(X) which is l.v.c. around Y . To see that the functional u|Y really is a distribution on Y , it suffices to verify the continuity for test functions with support in trivialising open sets. In this case the verification is straightforward using the l.v.c. extension from the proof of Lemma A.4. Remark A.6. Note that the vertical restriction depends on the choice of tubular neighbourhood, i.e., on a choice of complement E in the splitting of vector bundles in Remark A.2. However, when u is a measure on X with support on Y , the vertical restriction u|Y is u itself, now viewed as a distribution on Y . We now consider holomorphic families of distributions and their properties under restriction. Definition A.7. Let Ω ⊆ Cm be an open set, and let {uz }z∈Ω be a family of distributions on a smooth manifold X. Then this family is called a holomorphic family of distributions if the map z → uz (ϕ) is holomorphic on Ω for every ϕ ∈ D(X). Remark A.8. It follows immediately from Definition A.5 that if {uz }z∈Ω is a holomorphic family of distributions with support on Y , then the family {uz |Y }z∈Ω is a holomorphic family of distributions on Y . Proposition A.9. Let Ω ⊆ Cn be open and connected, and let {uz }z∈Ω be a holomorphic family of distributions on X with support on Y . Assume that there exists an open subset U ⊆ Ω, such
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that uz is a measure with support on Y for z ∈ U . Then the whole family {uz |Y }z∈Ω is independent of E. References [1] J. Arazy, H. Upmeier, Boundary measures for symmetric domains and integral formulas for the discrete Wallach points, Integral Equations Operator Theory 47 (4) (2003) 375–434. [2] S. Ben Saïd, Weighted Bergman spaces on bounded symmetric domains, Pacific J. Math. 206 (1) (2002) 39– 68. [3] F.A. Berezin, Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (2) (1975) 363–402, 472. [4] F.A. Berezin, The connection between covariant and contravariant symbols of operators on classical complex symmetric spaces, Dokl. Akad. Nauk SSSR 241 (1) (1978) 15–17. [5] J. Faraut, A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., The Clarendon Press/Oxford University Press/Oxford Science Publications, New York, 1994. [6] J. Faraut, M. Pevzner, Berezin kernels and analysis on Makarevich spaces, Indag. Math. (N.S.) 16 (3–4) (2005) 461–486. [7] S. Helgason, Groups and geometric analysis, in: Integral Geometry, Invariant Differential Operators, and Spherical Functions, in: Pure Appl. Math., vol. 113, Academic Press, Orlando, FL, 1984. [8] L. Hörmander, The analysis of linear partial differential operators. I, in: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin, 1983. [9] H. Ishi, Positive Riesz distributions on homogeneous cones, J. Math. Soc. Japan 52 (1) (2000) 161–186. [10] A.W. Knapp, Lie Groups Beyond an Introduction, second ed., Progr. Math., vol. 140, Birkhäuser Boston, Boston, MA, 2002. [11] T. Kobayashi, Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1) (1998) 100–135. [12] B. Krötz, Formal dimension for semisimple symmetric spaces, Compos. Math. 125 (2) (2001) 155–191. [13] S. Lang, Differential Manifolds, Addison–Wesley Publishing, Reading, MA/London/Don Mills, ON, 1972. [14] M. Lassalle, Algèbre de Jordan et ensemble de Wallach, Invent. Math. 89 (2) (1987) 375–393. [15] Y.A. Neretin, Matrix analogues of the B-function, and the Plancherel formula for Berezin kernel representations, Mat. Sb. 191 (5) (2000) 67–100. [16] Y.A. Neretin, On the separation of spectra in the analysis of Berezin kernels, Funktsional. Anal. i Prilozhen. 34 (3) (2000) 49–62, 96. [17] Y.A. Neretin, Plancherel formula for Berezin deformation of L2 on Riemannian symmetric space, J. Funct. Anal. 189 (2) (2002) 336–408. [18] B. Ørsted, G. Zhang, Tensor products of analytic continuations of holomorphic discrete series, Canad. J. Math. 49 (6) (1997) 1224–1241. [19] L. Peng, G. Zhang, Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal. 210 (1) (2004) 171–192. [20] J. Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math. 31 (4) (1979) 836– 844. [21] H. Seppänen, Branching laws for minimal holomorphic representations, J. Funct. Anal. 251 (1) (2007) 174–209. [22] H. Seppänen, Branching of some holomorphic representations of SO(2, n), J. Lie Theory 17 (1) (2007) 191–227. [23] H. Seppänen, Tube domains and restrictions of minimal representations, Internat. J. Math. 19 (10) (2008) 1247– 1268. [24] A. Unterberger, H. Upmeier, The Berezin transform and invariant differential operators, Comm. Math. Phys. 164 (3) (1994) 563–597. [25] E.P. van den Ban, Induced representations and the Langlands classification, in: Representation Theory and Automorphic Forms, Edinburgh, 1996, in: Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 123–155. [26] G. van Dijk, M. Pevzner, Berezin kernels of tube domains, J. Funct. Anal. 181 (2) (2001) 189–208. [27] M. Vergne, H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1–2) (1976) 1–59. [28] N.R. Wallach, The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc. 251 (1–17) (1979) 19–37.
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[29] G. Zhang, Berezin transform on line bundles over bounded symmetric domains, J. Lie Theory 10 (1) (2000) 111– 126. [30] G. Zhang, Berezin transform on real bounded symmetric domains, Trans. Amer. Math. Soc. 353 (9) (2001) 3769– 3787 (electronic). [31] G. Zhang, Tensor products of minimal holomorphic representations, Represent. Theory 5 (2001) 164–190 (electronic).
Journal of Functional Analysis 258 (2010) 3266–3294 www.elsevier.com/locate/jfa
Hierarchy of Hamilton equations on Banach Lie–Poisson spaces related to restricted Grassmannian Tomasz Goli´nski ∗ , Anatol Odzijewicz Institute of Mathematics, University in Białystok, Lipowa 41, 15-424 Białystok, Poland Received 10 September 2009; accepted 14 January 2010 Available online 2 February 2010 Communicated by D. Voiculescu
Abstract We consider the Banach Lie–Poisson space iR ⊕ U L1res and its complexification C ⊕ L1res , where the first one of them contains the restricted Grassmannian Grres as a symplectic leaf. Using the Magri method we define an involutive family of Hamiltonians on these Banach Lie–Poisson spaces. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases we give the explicit solutions. © 2010 Elsevier Inc. All rights reserved. Keywords: Integrable systems; Banach Lie–Poisson spaces; Restricted Grassmannian; Banach Lie groups and algebras
1. Introduction The foundation of Banach Poisson differential geometry was developed in [11]. This theory, in particular, gives us a possibility to formulate geometrically and analytically rigorous language for the theory of infinite dimensional Hamiltonian systems. A special place in this theory is occupied by the Banach Lie–Poisson spaces. Recall that by definition b is a Banach Lie–Poisson space if its dual b∗ is a Banach Lie algebra such that ad∗x b ⊂ b ⊂ b∗∗ for x ∈ b∗ , where ad∗x : b∗∗ → b∗∗ is dual to the adjoint representation adx := [x, ·] : b∗ → b∗ . * Corresponding author.
E-mail addresses:
[email protected] (T. Goli´nski),
[email protected] (A. Odzijewicz). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.019
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Many infinite dimensional physical systems can be considered as systems on some Banach Lie–Poisson space b in the Hamilton way d ρ = − ad∗Dh(ρ) ρ, dt
(1.1)
where ρ ∈ b and h ∈ C ∞ (b), e.g. see [13]. The first aim of this paper is to investigate Banach Lie–Poisson spaces related to the restricted Grassmannian Grres , see [2,12]. The restricted Grassmannian Grres has its own long story as one of the most important infinite dimensional Kähler manifolds in mathematical physics. It is a set of Hilbert subspaces W ⊂ H of a polarized Hilbert space H = H+ ⊕ H− such that the projectors P+ : W → H+ and P− : W → H− are Fredholm and Hilbert–Schmidt operators respectively. The geometry of Grres and its symmetry group play an important role in the quantum field theory [14,22,23,25,26], the loop group theory [15,21], and the integration of the KdV and KP hierarchies [8,17,20]. The second aim is to define and investigate a hierarchy of the Hamilton equations on the Banach Lie–Poisson space iR ⊕ UL1res and on its complexification C ⊕ L1res . This hierarchy is obtained from the involutive system of Hamiltonians constructed on iR ⊕ UL1res and C ⊕ L1res by the Magri method [7]. The pair of coupled operator Ricatti equations, see (3.41), belongs to this hierarchy. As we show in Example 4.3 the finite dimensional version of the hierarchy provides the non-trivial example of an integrable Hamiltonian system. In our considerations we use the functional analytical methods as well as Banach differential geometric methods. All the results we have obtained are also valid in finite dimension case. Since the hierarchy consists of Hamilton equations, the flows preserve symplectic leaves of iR ⊕ UL1res and C ⊕ L1res . In particular, they preserve Grres , which is one of the symplectic leaves of iR ⊕ UL1res , see [2]. We show in Example 4.1 that after restriction to Grres the flows linearize in natural complex coordinates. The central place of the paper is occupied by Section 3, where (using the Magri method) we construct the infinite hierarchy under consideration. We also discuss its various realizations, see (3.20), (3.19), (3.38), (3.52) and (3.54). In Section 2 we prepare the material necessary for the application of Magri method, i.e. we res,0 of GLres,0 , the find explicit formulas for coadjoint representation of central extension GL 1 Poisson bracket and Casimirs of the Banach Lie–Poisson spaces C ⊕ Lres and iR ⊕ UL1res . Finally Section 4 gives formulas for solutions in some particular cases. We also include in the paper two appendices, where we present the Magri method and the theory of extensions of Banach Lie–Poisson spaces. 2. Banach Lie–Poisson spaces related to restricted Grassmannian We investigate the extensions of Banach Lie groups, Banach Lie algebras and Banach Lie– Poisson spaces related to the restricted Grassmannian Grres . One of the aims of this section is to obtain explicit formulas for adjoint and coadjoint actions of constructed Banach Lie groups and Banach Lie algebras. To this end we apply the methods described in Appendix A. Before that let us recall the definitions of objects we are going to use and fix the notation. For more information see [15,21,26].
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2.1. Preliminary definitions and notation Let us consider a complex separable Hilbert space with a fixed decomposition into two orthogonal Hilbert subspaces H = H+ ⊕ H− .
(2.1)
Let P+ and P− denote the orthogonal projectors onto H+ and H− respectively. We assume that in general both Hilbert subspaces are infinite dimensional. However we also admit the case when one (or both) of them is finite dimensional. In this case many analytical problems are considerably simplified. In what follows we omit the symbols H and H± in the notation for various operator algebras and groups and put the subscript ± if we mean that the operators act in H± . In this way, for example instead of L2 (H) or L2 (H+ ) we write L2 or L2+ . In order to simplify our notation we use the block decomposition A++ 0 , 0 0 0 0 P− AP+ = , A−+ 0
P+ AP+ =
0 A+− , 0 0 0 0 P− AP− = 0 A−−
P+ AP− =
(2.2)
and we identify the operators A++ : H+ → H+ , A−− : H− → H− , A−+ : H+ → H− and A+− : H− → H+ with P+ AP+ , P− AP− , P− AP+ , P+ AP− respectively when there is no risk of confusion. By Lp we denote the Schatten classes of operators acting in H equipped with the norm · p . The Lp spaces are ideals in associative algebra L∞ of bounded operators in H. In particular L1 denotes the ideal of trace-class operators and L2 is the ideal of Hilbert–Schmidt operators. By L0 ⊂ L∞ one denotes the ideal of compact operators, which is · ∞ -norm closure Lp = L0 of any Lp ideal, see [4,18]. Let GL∞ be the Banach Lie group of invertible bounded operators in H. By UL∞ ⊂ GL∞ we denote the real Banach Lie group of the unitary operators and its Lie algebra is denoted by UL∞ . By GL1+ we denote the group of invertible operators on H+ which have a determinant (i.e. they differ from identity by a trace-class operator) and by SL1+ —its subgroup which consists of operators with the determinant equal to 1. See [5,16] for the definition and properties of the determinant for this case. The unitary restricted group ULres is defined as ULres := u ∈ UL∞ [u, P+ ] ∈ L2 . (2.3) It possesses the Banach Lie group structure given by the embedding ULres u → (u, u−+ ) ∈ UL∞ × L2+− .
(2.4)
This structure is not compatible with Banach Lie group structure of UL∞ . The Banach Lie algebra of ULres is (2.5) ULres := x ∈ L∞ x + = −x, [x, P+ ] ∈ L2
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xres := x++ ∞ + x−− ∞ + x−+ 2 + x+− 2 ,
(2.6)
with the norm
where x + is the operator adjoint to x. Note that the topology of ULres is strictly stronger than the operator topology on L∞ . C The complexifications of ULres and ULres are ULC res = GLres and ULres = Lres respectively, where (2.7) GLres = g ∈ GL∞ [g, P+ ] ∈ L2 and Lres = x ∈ L∞ [x, P+ ] ∈ L2 .
(2.8)
By definition the restricted Grassmannian Grres consists of Hilbert subspaces W ⊂ H such that: i) the projection P+ restricted to W is a Fredholm operator; ii) the projection P− restricted to W is a Hilbert–Schmidt operator; see e.g. [15,26]. The restricted Grassmannian is a Hilbert manifold modelled on the Hilbert space L2+− . The groups ULres and GLres act on it transitively. In this way the tangent space to the restricted Grassmannian in the point H+ can be described as follows ∞ (2.9) TH+ Grres ∼ = ULres / UL∞ + × UL− . Both ULres and GLres are disconnected and their connected components are ULres,k and GLres,k , where g ∈ ULres,k and g ∈ GLres,k iff the Fredholm index ind g++ of the upper left block g++ of the operator g is equal to k, see [3,15]. The maximal connected subgroups ULres,0 and GLres,0 will be of special interest. In a similar fashion, connected components of the restricted Grassmannian Grres are the sets Grres,k consisting of elements of Grres such that the index of the orthogonal projection P+ restricted to that element is equal to k. Let us note that ULres,0 acts transitively on Grres,0 . 2.2. Extensions of GLres,0 The central object in the following construction is the group E defined as 1 E := (q, A) ∈ GL∞ + × GLres,0 A++ − q ∈ L+
(2.10)
with pairwise multiplication. The topology and Banach manifold structure on E is given by the embedding (q, A) → (A++ − q, A) ∈ L1+ × GLres , see [15,26].
(2.11)
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Let us consider the Banach Lie group extensions presented in the following commutative diagram: {1}
{1}
C×
{1}
ι
res,0 GL
det
{1}
GL1+
{1}
SL1+
{1}
{1}
π
GLres,0 id
δ ι1
ι1
{1}
E
π2
SL1+ × {1}
GLres,0
π2
{1}
(2.12)
{1}
{1}
The map ι1 is defined by ι1 (q) := (q, 1) and the map π2 is a projection onto the second 1 component of the Cartesian product GL∞ + × GLres,0 . Thus ι1 (SL+ ) is a normal subgroup of E res,0 is defined as the quotient group and the group GL res,0 := E/ι1 SL1+ . GL
(2.13)
The maps ι and π in upper row of diagram (2.12) are given as quotients of ι1 and π2 respectively. res,0 . In this way all rows and columns in The map δ is the quotient map E → E/ι1 (SL1+ ) = GL diagram (2.12) are exact sequences of Banach Lie groups. Using the approach described in Appendix A we define a local section (A.2) of the bundle GL1+ → E → GLres,0 by σ (A) := (A++ , A)
(2.14)
for A ∈ GLres,0 such that A++ is invertible. The Banach Lie group E can be locally identified with GL1+ ×Φ,Ω GLres,0 through the isomorphism Ψ : GL1+ ×Φ,Ω GLres,0 → E given in the properly chosen neighborhood of identity by Ψ (n, A) = (nA++ , A).
(2.15)
The maps Φ : GLres,0 → Aut GL1+ and Ω : GLres,0 × GLres,0 → GL1+ defined in the general case by (A.7) and (A.8) in this case can be expressed locally as follows:
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Φ(A)(n) = A++ nA−1 ++ ,
(2.16)
Ω(A1 , A2 ) = A1++ A2++ (A1 A2 )−1 ++
(2.17)
for n ∈ GL1+ , A, A1 , A2 ∈ GLres,0 such that A++ , A1++ , A2++ and (A1 A2 )++ are invertible. The map Φ(A) descends to the trivial automorphism of C× . Thus the Banach Lie group GLres,0 can be identified with C× ×id,Ω˜ GLres,0 for Ω˜ := det ◦ Ω
(2.18)
and it is a central extension of GLres,0 . 2.3. Extensions of Lres The Banach Lie algebra counterpart of diagram (2.12) is the following: {0}
{0}
{0}
C
C ⊕ Lres
Lres
{0}
Tr1
Tr
{0}
L1+
{0}
S L1+
{0}
ι1
L1+ ⊕ Lres
ι1
S L1+ ⊕ {0}
{0}
id π2
π2
Lres
{0}
(2.19)
{0}
{0}
where S L1+ := ρ ∈ L1+ Tr ρ = 0
(2.20)
is the Banach Lie algebra of the group SL1+ . The Banach Lie algebra (L1+ ⊕ Lres )/(S L1+ ⊕ {0}) res,0 is naturally identified by DΨ (1, 1) with C ⊕ Lres . The map Tr1 is of the quotient group GL given by taking trace of the first component of (ρ, X) ∈ L1+ ⊕ Lres . The direct sums in diagram (2.19) are understood as direct sums of Banach spaces. Similarly as in the group case, all rows and columns in diagram (2.19) are exact sequences of Banach Lie algebras.
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By using formula (A.14) with functions (2.16) and (2.17) we obtain a local formula for the adjoint action Ad(n,A) (ρ, X) = nA++ (ρ + X++ )(A++ )−1 n−1 − AXA−1 ++ , AXA−1
(2.21)
for (n, A) in some open set in GL1+ ×Φ,Ω GLres,0 and (ρ, X) ∈ L1+ ⊕ Lres . From (A.16) and (A.17) we get that ϕ(X) := [X++ , ·],
(2.22)
ω(X, Y ) := −X+− Y−+ + Y+− X−+ .
(2.23)
Bracket (A.15) for ϕ and ω given by (2.22) and (2.23) assumes the form
(ρ, X), ρ , Y = ρ, ρ + X++ , ρ − [Y++ , ρ] − X+− Y−+ + Y+− X−+ , [X, Y ]
(2.24)
where (ρ, X), (ρ , Y ) ∈ L1+ ⊕ Lres . This Banach Lie algebra was presented in [12] (up to the sign conventions) as an example of extensions of Banach Lie algebras. The structure of Banach Lie algebra on C ⊕ Lres is given by the function ϕ(X) ˜ ≡0
(2.25)
ω(X, ˜ Y ) = −s(X, Y ) = − Tr(X+− Y−+ − Y+− X−+ ),
(2.26)
and the cocycle
where s(X, Y ) is called the Schwinger term, see [19,26]. Thus the adjoint representation of the Lie group C× ×id,Ω˜ GLres,0 on C ⊕ Lres is given by Ad(γ ,A) (λ, X) = λ + Tr P+ − A−1 P+ A , AXA−1
(2.27)
for (γ , A) ∈ C× ×id,Ω˜ GLres,0 , (λ, X) ∈ C ⊕ Lres . Moreover, the Lie bracket for (λ, X), (λ , Y ) ∈ C ⊕ Lres is the following
(λ, X), λ , Y = −s(X, Y ), [X, Y ] .
(2.28)
Let us note that formula (A.14) allows one to express Ad only locally. However the right-hand side of formula (2.27) defines some global representation of C× ×id,Ω˜ GLres,0 , which coincides with Ad on an open neighborhood of (1, 1). However every open neighborhood of the unit element in Banach Lie group generates a connected component, and C× ×id,Ω˜ GLres,0 is connected. Thus the formula (2.27) is valid for all (γ , A) ∈ C× ×id,Ω˜ GLres,0 .
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2.4. Extensions of complex Banach Lie–Poisson space L1res In order to find a Banach space predual to the Banach Lie algebra Lres , we define the Banach space L1res := μ ∈ Lres μ++ ∈ L1+ , μ−− ∈ L1−
(2.29)
μ∗ := μ++ 1 + μ−− 1 + μ−+ 2 + μ+− 2 .
(2.30)
with the norm
Moreover we define the restricted trace Trres : L1res → C by Trres μ := Tr(μ++ + μ−− ),
(2.31)
for μ ∈ L1res . The domain of Trres is larger than L1 since L1 ⊂ Lres . However for trace-class operators the restricted trace Trres coincides with the standard trace Tr. The properties of the restricted trace are similar to the properties of the standard trace but one needs to replace L∞ with Lres . Proposition 2.1. The Banach space L1res is an ideal (in the sense of commutative algebras) in the Banach space Lres . Moreover for μ ∈ L1res , ν ∈ Lres we have Trres (μν) = Trres (νμ).
(2.32)
Proof. The conclusion that L1res is an ideal follows from the fact that L1 and L2 are ideals in L∞ and a product of two operators from L2 is trace-class. To prove formula (2.32) we expand its left-hand side Trres (μν) = Tr(μ++ ν++ + μ+− ν−+ + μ−+ ν+− + μ−− ν−− ).
(2.33)
By assumptions of the proposition, we conclude that each term in the right-hand side is a traceclass operator. Since for A ∈ L∞ and B ∈ L1 or for A, B ∈ L2 one has Tr(AB) = Tr(BA),
(2.34)
we conclude that Trres (μν) = Trres (νμ).
2
(2.35)
As a corollary to this proposition we get that for g ∈ GLres and μ ∈ L1res , the operator μg −1 belongs to L1res and Trres gμg −1 = Trres (μ).
(2.36)
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Using the pairing between μ ∈ L1res , A ∈ Lres given by μ, A := Trres (μA) = Tr(μ++ A++ ) + Tr(μ+− A−+ ) + Tr(μ−+ A+− ) + Tr(μ−− A−− ),
(2.37)
we conclude that (L1res )∗ ∼ = Lres , i.e. the Banach space L1res is predual of Lres . This duality can be found in [2,12]. Proposition 2.2. Space L1res with norm · ∗ is Banach ∗-algebra. Proof. The only point yet to be shown is the inequality μρ∗ μ∗ ρ∗
(2.38)
for μ, ρ ∈ L1res . In order to prove it we observe that μρ∗ = (μρ)++ 1 + (μρ)+− 2 + (μρ)−+ 2 + (μρ)−− 1 = μ++ ρ++ + μ+− ρ−+ 1 + μ++ ρ+− + μ+− ρ−− 2 + μ−+ ρ++ + μ−− ρ−+ 2 + μ−− ρ−− + μ−+ ρ+− 1 .
(2.39)
Next, applying the following inequalities ρ2 ρ1 ,
(2.40)
ρμ1 ρ∞ μ1 ρ1 μ1
(2.41)
for ρ, μ ∈ L1 and the inequalities ρμ1 ρ2 μ2 ,
(2.42)
ρμ2 ρ∞ μ2 ρ2 μ2
(2.43)
for ρ, μ ∈ L2 , we obtain μρ∗ μ++ 1 ρ++ 1 + μ+− 2 ρ−+ 2 + μ++ 1 ρ+− 2 + μ+− 2 ρ−− 2 + μ−+ 2 ρ++ 1 + μ−− 1 ρ−+ 2 + μ−− 1 ρ−− 1 + μ−+ 2 ρ+− 2 μ∗ ρ∗
(2.44)
The latter inequality in (2.44) follows directly from (2.30).
2
The Banach space predual to extended Banach Lie algebra L1+ ⊕ Lres is L0+ ⊕ L1res with natural component-wise pairing
(A, μ), (ρ, X) = Tr(Aρ) + Trres (μX).
(2.45)
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It follows from the fact that the Banach space dual to the ideal of compact operators L0+ is L1+ , see [24]. Analogously, the predual of C ⊕ Lres is C ⊕ L1res with the pairing given by
(γ , μ), (λ, X) = γ λ + Trres (μX), (2.46) for μ ∈ L1res , X ∈ Lres , γ , λ ∈ C. 1 ∗ The map Tr∗ : C → L∞ + dual to Tr : L+ → C is given by Tr (λ) = λ1. Since the ideal of 0 1 compact operators L+ is the Banach space predual to L+ and Tr∗ does not take values in L0+ , we conclude that Tr∗ cannot be restricted to predual spaces. Therefore only horizontal exact sequences in diagram (2.19) have their predual counterparts {0} {0}
L1res L1res
π2∗
π2∗
C ⊕ L1res L0+ ⊕ L1res
ι∗1
ι∗1
C L0+
{0} {0}
(2.47) (2.48)
where the map π2∗ is an injection into the second argument and the map ι∗1 is the projection onto the first component of the respective direct sums. It follows from (A.25) and from (2.16), (2.17) that the coadjoint representation of the Banach Lie group GL1+ ×Φ,Ω GLres,0 on the predual Banach space L0+ ⊕ L1res is given by Ad∗(n,A) (τ, μ) = (A++ )−1 n−1 τ nA++ , (A++ )−1 n−1 τ nA++ − A−1 τ A + A−1 μA (2.49) for (n, A) ∈ GL1+ ×Φ,Ω GLres,0 and (τ, μ) ∈ L0+ ⊕ L1res . Similarly, the coadjoint representation of C× ×id,Ω˜ GLres,0 on C ⊕ L1res is the following Ad∗(λ,A) (γ , μ) = γ , A−1 μA + γ P+ − A−1 P+ A (2.50) where (λ, A) ∈ C× ×id,Ω˜ GLres,0 and (γ , μ) ∈ C ⊕ L1res . Let us also note that these coadjoint representations preserve the Banach subspaces L0+ ⊕ L1res ⊂ (L1+ ⊕ Lres )∗ and C ⊕ L1res ⊂ C ⊕ L∗res respectively Ad∗(n,A) L0+ ⊕ L1res ⊂ L0+ ⊕ L1res , Ad∗(λ,A) C ⊕ L1res ⊂ C ⊕ L1res .
(2.51) (2.52)
The coadjoint representation of the Banach Lie algebra L1+ ⊕ Lres on its predual L0+ ⊕ L1res is the following (2.53) ad∗(ρ,X) (τ, μ) = [−ρ, τ ] − [X++ , τ ], −[X, μ] − [ρ, τ ] − τ X+− + X−+ τ where (ρ, X) ∈ L1+ ⊕ Lres , (τ, μ) ∈ L0+ ⊕ L1res . The coadjoint representation of the Banach Lie algebra C ⊕ Lres on C ⊕ L1res is given by ad∗(λ,X) (γ , μ) = 0, −[X, μ] − γ (X+− − X−+ ) (2.54) where (λ, X) ∈ C ⊕ Lres , (γ , μ) ∈ C ⊕ L1res .
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We observe that the conditions (A.26) are satisfied for both extensions, thus the Banach spaces L0+ ⊕ L1res and C ⊕ L1res are Banach Lie–Poisson spaces. The Poisson bracket for F, G ∈ C ∞ (C ⊕ L1res ) is obtained from the general formula (A.27) and is given by
{F, G}(γ , μ) = μ, D2 F (γ , μ), D2 G(γ , μ) − γ s D2 F (γ , μ), D2 G(γ , μ) ,
(2.55)
where D2 denotes the partial Fréchet derivative with respect to the second variable. 2.5. Extensions of real Banach Lie–Poisson space UL1res In previous subsections we considered the extensions of the complex linear restricted group GLres,0 , its Banach Lie algebra Lres and the complex Banach Lie–Poisson space L1res , which is Banach predual of Lres . As we mentioned above they are complexifications of ULres,0 , ULres and (ULres )∗ ∼ = UL1res := μ ∈ L1res μ+ = −μ
(2.56)
respectively. The pairing between elements of ULres and UL1res as in the complex case is given by (2.37). By a construction similar to those for GLres,0 , we obtain the central extension of ULres,0 by U (1) {1}
U (1)
res,0 UL
ULres,0
{1} .
(2.57)
Namely we define the real Banach Lie group U E := (A, q) ∈ E A ∈ ULres,0 , q ∈ UL∞ + ,
(2.58)
res,0 in (2.57) is defined as which complexification U E C is E . The group UL res,0 := U E/ ι1 SU L1+ , UL
(2.59)
where SU L1+ := SL1+ ∩ UL∞ and ι1 (q) := (q, 1). Restricting the Schwinger term (2.26) to ULres we obtain the central extension {0}
iR
iR ⊕ ULres
ULres
{0}
(2.60)
of the real Banach Lie algebra ULres . The exact sequence of Banach Lie–Poisson spaces predual to (2.60) is {0}
iR
iR ⊕ UL1res
UL1res
{0} .
(2.61)
The complexification of (2.61) gives (2.47). All expressions obtained above, including the ones for the Poisson bracket (2.55) and coadjoint representation (2.50), (2.54) are valid for the real case if one assumes that γ = −γ and μ+ = −μ.
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3. Hierarchy of Hamilton equations on Banach Lie–Poisson spaces C ⊕ L1res and R ⊕ U L1res iR In this section we use the Magri method (see [7]), to introduce the hierarchy of the Hamiltonian systems on the Banach Lie–Poisson spaces C ⊕ L1res and iR ⊕ UL1res , which were investigated in Section 2. Short description of Magri method is presented in Appendix B. To this end we define for any k ∈ N the function I k (γ , μ) := Trres (μ − γ P+ )k+1 − (−γ )k (μ − γ P+ )
(3.1)
on C ⊕ L1res . Note that the expression under Trres is a polynomial in variable μ without a free term, thus from Proposition 2.1 it follows that the function I k is well defined. We observe that I k is invariant with respect to the coadjoint representation (2.50) I k Ad∗[n,A] (γ , μ) = I k γ , A−1 μA + γ P+ − A−1 P+ A k+1 A = Trres A−1 μ − γ P+ + γ AP+ A−1 − γ AP+ A−1 − A−1 (−γ )k μ − γ P+ + γ AP+ A−1 − γ AP+ A−1 A = Trres (μ − γ P+ )k+1 − (−γ )k (μ − γ P+ ) = I k (γ , μ).
(3.2)
Thus the functions I k , k ∈ N, are Casimirs
I k, · = 0
(3.3)
for Poisson bracket (2.55). Note that the coordinate function γ is a Casimir too. Observing that Poisson brackets for F, G ∈ C ∞ (C ⊕ L1res ) given by
{F, G}1 (γ , μ) := μ, D2 F (γ , μ), D2 G(γ , μ)
(3.4)
{F, G}2 (γ , μ) := −γ s D2 F (γ , μ), D2 G(γ , μ)
(3.5)
and by
are compatible, we introduce a Poisson pencil {F, G}ε (γ , μ) := {F, G}1 (γ , μ) + ε{F, G}2 (γ , μ)
= μ, D2 F (γ , μ), D2 G(γ , μ) − εγ s D2 F (γ , μ), D2 G(γ , μ)
(3.6)
on C ⊕ L1res . Compatibility of {·,·}1 and {·,·}2 follows from the fact that the Poisson tensor for {·,·}2 is constant with respect to the variable μ and {·,·}1 depends only on the derivations with respect to the variable μ. Due to the latter equality in (3.6) the Casimirs for {·,·}ε are: Iεk (γ , μ) = Trres (μ − εγ P+ )k+1 − (−εγ )k (μ − εγ P+ )
(3.7)
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where k ∈ N. According to Magri method, we expand these Casimirs with respect to the parameter −εγ Iεk (γ , μ) =
k−1 (−εγ )n Trres Wnk+1 (μ) + (−εγ )k Trres Wkk+1 (μ) − μ ,
(3.8)
n=0
where the operators Wnk (μ) are polynomials in operator arguments μ and P+ defined by the equality (μ + λP+ )k =
k
λn Wnk (μ),
λ ∈ R.
(3.9)
n=0
In this way from (B.7) it follows that we obtain a family hkn (γ , μ) = γ n Trres Wnk+1 (μ), 0 n k − 1, hkk (γ , μ) = γ k Trres Wkk+1 − μ
(3.10a) (3.10b)
of Hamiltonians in involution
hkn , hlm
ε
=0
(3.11)
with respect to the brackets {·,·}ε for ε ∈ R. In the particular case ε = 1 they are in involution with respect to the bracket {·,·} given by (2.55). Let us now investigate the infinite system of the Hamilton equations on the Banach Lie– Poisson space C ⊕ L1res ∂ (γ , μ) = − ad∗(D hk (γ ,μ),D hk (γ ,μ)) (γ , μ) 1 n 2 n ∂tnk
(3.12)
defined by the hierarchy of Hamiltonians hkn , k ∈ N, n = 0, 1, . . . , k. Using the explicit form of coadjoint action (2.54) one sees that observe that Eq. (3.12) take the form ∂ γ = 0, ∂tnk
∂ μ = − μ, D2 hkn (γ , μ) + γ P+ D2 hkn (γ , μ)P− − P− D2 hkn (γ , μ)P+ . ∂tnk
(3.13a) (3.13b)
In (3.13) the real parameter tnk parametrizes the Hamiltonian flow generated by hkn . In order to compute D2 hkn (γ , μ) we apply the partial derivative operator D2 to both sides of equality (3.8). Since D2 Iεk (γ , μ) = (k + 1)(μ − εγ P+ )k − (−εγ )k 1
(3.14)
D2 hkn (γ , μ) = (k + 1)γ n Wnk (μ)
(3.15a)
we get that
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for 0 n k − 1, and D2 hkk (γ , μ) = γ k (k + 1)Wkk (μ) − 1 .
(3.15b)
Substituting (3.15) into (3.13) we obtain
∂ μ = −(k + 1)γ n μ − γ P+ , Wnk (μ) k ∂tn
(3.16)
for 0 n k. From (3.9) we the following recurrence rules k (μ)P+ , Wnk+1 (μ) = Wnk (μ)μ + Wn−1 k Wnk+1 (μ) = μWnk (μ) + P+ Wn−1 (μ)
(3.17)
which yield the commutation relation
k μ, Wnk (μ) + P+ , Wn−1 (μ) = 0,
(3.18)
k := 0. Using (3.18) we can express the Hamilton equations where 0 n k and we put W−1 (3.16) in the following two ways
∂ k μ = −(k + 1)γ n μ, Wnk (μ) + γ Wn+1 (μ) , k ∂tn
∂ k μ = (k + 1)γ n P+ , γ Wnk (μ) + Wn−1 (μ) , k ∂tn
(3.19) (3.20)
where 0 n k. Rewriting (3.20) in the block form (2.2) we obtain ∂ μ++ = 0, ∂tnk
∂ μ−− = 0 ∂tnk
(3.21)
and ⎧ ∂ k n k ⎪ ⎪ ⎨ ∂t k μ+− = (k + 1)γ P+ γ Wn (μ) + Wn−1 (μ) P− , n
⎪ ∂ ⎪ k (μ) P . ⎩ μ−+ = −(k + 1)γ n P− γ Wnk (μ) + Wn−1 + k ∂tn
(3.22)
Let us observe that Eq. (3.19) is a Hamilton equation for the Poisson bracket {·,·}1 and the Hamiltonian hkn + hkn+1 , while Eq. (3.20) is a Hamilton equation for the Poisson bracket {·,·}2 and the Hamiltonian hkn + hkn−1 . From (3.21) we conclude that diagonal blocks μ++ and μ−− are invariants for all Hamiltonian flows under consideration. The symplectic leaves for C ⊕ L1res and iR ⊕ UL1res with Poisson bracket {·,·}2 are the affine spaces obtained by shifting the vector spaces L2+− ⊕L2−+ or L2+− respectively by diagonal blocks μ++ and μ−− . This fact explain why we have obtained additional integrals of motion (3.21).
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Eqs. (3.16) and (3.19) are in the Lax form. In the first case it is an equation on μ − γ P+ , while it is on μ in the other. Let us calculate several operators Wnk (μ). We can do this by iterating the recurrence (3.17). The result is: Wkk = P+ ,
(3.23)
k 2, (3.24) k = μ2 P+ + μP+ μ + P+ μ2 + (k − 3) P+ μ2 P+ + P+ μP+ μ + μP+ μP+ Wk−2
k Wk−1
= μP+ + P+ μ + (k − 2)P+ μP+ ,
+
(k − 3)(k − 4) P+ μP+ μP+ , 2
k 4,
(3.25)
.. . W1k = P+ μk−1 + μP+ μk−2 + · · · + μk−1 P+ ,
(3.26)
W0k = μk .
(3.27)
It is obvious that the Hamiltonians hkn are functionally interdependent and it implies the interdependence of tnk -flows given by (3.22). The above formulas suggest the introduction of the homogeneous polynomials 1
Hnl (μ) :=
i0 ,i1 ,...il =0 i0 +···+il =n
i
i
P+0 μP+i1 μ . . . μP+l
(3.28)
of the degree l ∈ N in the operator variable μ ∈ L1res , where n l + 1. These polynomials are linearly independent and they satisfy the recurrences l+1 l (μ) = P+ μHnl (μ) + μHn+1 (μ) Hn+1
(3.29a)
l+1 l (μ) = P+ μHl+1 (μ) Hl+2
(3.29b)
for n l, l ∈ N and
for l ∈ N. Proposition 3.1. Polynomials Wnk are linear combinations of the homogeneous polynomials Hnl k Wk−l (μ) =
l+1
max 0, pnl (k) Hnl (μ)
(3.30)
n=1
for l < k and W0k (μ) = H0k (μ), where pnl ∈ Rn−1 [x] are polynomials of degree n − 1 that are defined by the recurrences:
(3.31)
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l pn+1 (k) =
k−1
max 0, pnl (i) ,
3281
(3.32)
i=l+1
pnl+1 (k) = pnl (k − 1)
(3.33)
with initial condition p1l (k) = 1. Proof. We prove formula (3.30) by induction with respect to l. From recurrence (3.32) we infer that p21 (k) = k − 2 and thus from (3.23) and (3.24) we see that formula (3.30) is satisfied for l = 0 and l = 1. From (3.17) we conclude that k−1 k (μ) = μWk−l (μ) + P+ μ Wk−l
k−l
k−i−1 Wk−l−i (μ).
(3.34)
i=1 k assuming that (3.30) is satisfied for l − 1 and obtain We apply (3.34) to Wk−l
k Wk−l (μ) = μ
l
max 0, pnl−1 (k − 1) Hnl−1 (μ)
n=1
+ P+ μ
k−l−1 l
max 0, pnl−1 (k − i − 1) H l−1 (μ) + P+ μH0l−1 (μ).
(3.35)
i=1 n=1
Changing the order of summation and using recurrences (3.32) and (3.33) we get
k Wk−l (μ) = μ
l−1
l−1 l−1 max 0, pn+1 (k − 1) Hn+1 (μ)
n=0
+ P+ μ
l
l−1 max 0, pn+1 (k − 1) Hnl−1 (μ) + P+ μH0l−1 (μ).
(3.36)
n=1
By rearranging terms in the sums and applying (3.29) we end up with (3.30). Direct check shows that relations (3.32) and (3.33) are compatible. The fact that deg pnl = n − 1 follows from (3.32). 2 From Proposition 3.1 we conclude: Corollary 3.2. k }∞ i) The dimension of the complex vector space spanned by {Wk−l k=l+1 is equal to l + 1 and l+1 l {Hn }n=1 is a basis of this space; k , where ii) Using (3.30) one can express Hnl , 0 n l + 1 as a finite linear combination of Wk−l k l + 1.
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k }∞ Proof. i) From (3.30) it follows that all elements of the set {Wk−l k=l+1 are linear combinations l+1 l l of {Hn }n=1 . Let us also note that the polynomials pn assume positive values pnl (k) > 0 for k large enough and that the set {pnl }l+1 n=1 spans an (l + 1)-dimensional vector space. This concludes the proof. ii) This statement is a consequence of i). 2
Introducing new variables τnl ∈ R through the linear combination k tk−l
= (k − 1)γ
k−l
l+1
max 0, pnl (k) τnl ,
(3.37)
n=1
we rewrite the hierarchy (3.16) in the equivalent form
∂ μ = μ − γ P+ , Hnl (μ) l ∂τn
(3.38)
where l ∈ N and n = 1, . . . , l + 1. Let us write out explicitly several equations from hierarchy (3.38). For H0k and H1k in block notation (2.2) we obtain ⎧ k ∂ ⎪ ⎪ μ = −γ μ +− , +− ⎪ k ⎨ ∂τ 0 k ⎪ ∂ ⎪ ⎪ ⎩ k μ−+ = γ μ −+ ∂τ0
(3.39)
⎧ k+1 i k−i ∂ ⎪ ⎪ − γ k−1 , ⎪ i=0 μ ++ μ +− +− ⎨ ∂τ k μ+− = − μ 1 k+1 ⎪ ∂ ⎪ ⎪ + γ ki=1 μi −+ μk−i ++ ⎩ k μ−+ = μ −+ ∂τ1
(3.40)
and
respectively. For k = 1 and k = 2 we get from (3.39) linear equations and for k = 3 we obtain from (3.39) a pair of coupled operator Ricatti-type equations ⎧ ∂ ⎪ 2μ 2 , ⎪ μ = −γ (μ ) + μ μ μ + μ μ μ + μ (μ ) +− ++ +− +− −+ +− ++ +− −− +− −− ⎪ 3 ⎨ ∂τ 0 ⎪ ∂ ⎪ 2 2 ⎪ ⎩ 3 μ−+ = γ μ−+ (μ++ ) + μ−− μ−+ μ++ + μ−+ μ+− μ−+ + (μ−− ) μ−+ . ∂τ0
(3.41)
Let us recall that blocks μ++ and μ−− are constant with respect to all flows. Moreover if we assume that μ++ = 0 or μ−− = 0 then Eqs. (3.38) become linear. After certain modifications we can also consider the hierarchy of Eqs. (3.38) on the real Banach Lie–Poisson iR ⊕ UL1res . To this end we have to modify the Hamiltonians hkn is such a way
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that they will take real values when restricted to iR ⊕ UL1res . In consequence we obtain the following equations
∂ μ = i l+1 μ − γ P+ , Hnl (μ) ∂τnl
(3.42)
where μ ∈ UL1res , γ ∈ iR. Now we express the Hamiltonian hierarchy (3.16) in a more compact and elegant form. To this end let us define the “generating” Hamiltonian for the Hamiltonians (3.10) hκ,λ (γ , μ) :=
∞ k=1
1 κk λn hkn (γ , μ), k+1 k+1
(3.43)
n=0
where κ, λ ∈ R. In order to show that the series of functions (3.43) is convergent on some nonempty open subset of C ⊕ L1res we observe that k+1
λn hkn (γ , μ) = Trres (μ + γ λP+ )k+1 − (γ λ)k (μ + γ λP+ ) .
(3.44)
n=0
Equality (3.44) follows from (3.10) and (3.9). Next, let us prove the following lemma. Lemma 3.3. One has (μ + βP+ )k+1 − β k (μ + βP+ ) μ∗ + |β| k+1 − |β|k μ∗ + |β| , ∗
(3.45)
where β ∈ C and μ ∈ L1res . Proof. We expand the left-hand side and apply the triangle inequality and Proposition 2.2. Moreover we note that νP+ ∗ ν∗ for ν ∈ L1res . In this way we get k+1 k+1 (μ + βP+ )k+1 − β k (μ + βP+ ) μi∗ |β|k−i+1 − |β|k μ∗ . (3.46) ∗ i i=1
By adding and subtracting the term |β|k+1 and collecting terms we obtain the right-hand side. 2 From this lemma and Proposition 2.2 we conclude: Proposition 3.4. One has 1 log(1 − κλγ ) hκ,λ (γ , μ) = Trres − log 1 − κ(μ + γ λP+ ) + (μ + γ λP+ ) κ κλγ
(3.47)
and {hκ,λ , hκ ,λ } = 0
(3.48)
for |κ|(μ∗ + |λγ |) < 1 and |κ |(μ∗ + |λ γ |) < 1, where the Poisson bracket in (3.48) is given by (2.55).
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Now we find the explicit form of the Hamilton equation ∂ (γ , μ) = − ad∗Dhκ,λ (γ ,μ) (γ , μ) ∂tκ,λ
(3.49)
generated by Hamiltonian (3.43). Using (2.54) we obtain ∂ γ = 0, ∂tκ,λ
(3.50a)
∂ μ = − μ, D2 hκ,λ (γ , μ) + γ P+ D2 hκ,λ (γ , μ)P− − P− D2 hκ,λ (γ , μ)P+ . ∂tκ,λ
(3.50b)
Since −1 log(1 − κλγ ) D2 hκ,λ (γ , μ) = 1 − κ(μ + λγ P+ ) + κλγ
(3.51)
∂ x = −α P+ , (1 − x)−1 , ∂tκ,λ
(3.52)
x := κ(μ + λγ P+ ),
(3.53)
we get
where
and α := κ(1 + λ)γ . Replacing x in (3.52) by y := (1 − x)−1 we get the hierarchy of equations ∂ y = α[y, yP+ y], ∂tκ,λ
λ, κ ∈ C
(3.54)
equivalent to the hierarchy (3.16). In this paper we don’t intend to address the problem of finding general solutions for the considered Hamiltonian systems but in the next section we will present several examples of solutions. Let us also observe that if H+ is finite dimensional, then the operator μ − γ P+ has a discrete res,0 coincide with spectrum. Formula (2.50) shows that orbits of coadjoint action of group GL ∞ orbits of standard coadjoint action of GLres,0 ⊂ GL shifted by γ P+ . Thus one can use the spectrum spec(μ − γ P+ ) to distinguish partially symplectic leaves of the Banach Lie–Poisson space iR ⊕ UL1res , i.e. if spec(μ1 − γ P+ ) = spec(μ2 − γ P+ ) then they belong to different symplectic leaves. If the dimension of H+ is infinite then the shift μ − γ P+ of the operator μ ∈ L1res by the operator γ P+ is not an element of L1res , but of Lres . However from Weyl’s criterion (see [16]) we deduce that the set spec(μ − γ P+ ) \ {0, −γ } is discrete. Therefore if H+ is infinite dimensional, one can use also elements of spec(μ − γ P+ ) for the partial indexation of the symplectic leaves. The problem of description of these leaves is complicated, see [2] for more information.
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4. Examples of solutions In this section we present several examples of explicit solutions to Eqs. (3.38) in some particular cases. Example 4.1 (Restricted Grassmannian). The connected component Grres,0 of the restricted res,0 in the Grassmannian Grres can be identified with the coadjoint orbit Oγ of the group UL 1 Banach Lie–Poisson space iR ⊕ ULres generated from point (γ , 0), see [2]. Namely from (2.50) we see that Ad∗(λ,g) (γ , 0) = γ , γ P+ − g −1 P+ g
(4.1)
for (λ, g) ∈ U (1) ×id,Ω˜ ULres,0 and γ ∈ iR. This suggests to define the map ιγ : Grres,0 → Oγ in the following way ιγ (W ) := γ , γ (P+ − PW ) .
(4.2)
Since ULres,0 acts transitively on Grres,0 , we see that ιγ maps Grres,0 bijectively on Oγ . Let us introduce homogeneous coordinates on some open subset in Grres . To this end we fix a basis {|n}, n ∈ Z in H, such that |n for n < 0 spans H− and for n 0 spans H+ . Let us fix a basis w1 , w2 , . . . in a subspace W ∈ Grres and put the coefficients of wk in the basis {|n}n∈Z in the matrix form α := n | wk n∈Z,k∈N , (4.3) β where α, β are blocks obtained for n 0 and n < 0 respectively. Let us consider a subspace W ∈ Grres such that there exists an orthonormal basis {wk }k∈N such that α is invertible. Then we define z := βα −1 .
(4.4)
Definition of z is independent of the choice of the basis {wk }k∈N . The matrix of the projector PW is n|PW k n,k∈Z =
α + + α β . β
(4.5)
Thus ιγ (W ) takes the following form ιγ (W ) =
(1 + z+ z)−1 − 1 z(1 + z+ z)−1
(1 + z+ z)−1 z+ z(1 + z+ z)−1 z+
,
(4.6)
where we consider z as an operator z : H+ → H− . Hamilton equations (3.13) can be written in terms of z. Due to (3.21) we note that z+ z is constant. Thus any polynomial of ιγ (W ) has only constant and linear terms in z and Eqs. (3.13) are linear in homogeneous coordinates on Oγ .
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Example 4.2 (Vector case). Let us consider a particular case of the equations given above. We assume that dim H+ = 1. We introduce the block notation for elements μ ∈ L1res μ=
a w
v+ A
(4.7)
,
where a ∈ C, A ∈ L∞ (H− ), v, w ∈ L2 (C, H− ) ∼ = 2 . We consider Eqs. (3.38) as non-linear equations for two vectors v, w coupled by interaction depending on constants a and A. Nonlinear behavior is due to the terms of the type v | Al w. First of all let us remark that in general all functions hkl are linear combinations of funck1 k2 kn tions Trres μk = hk−1 0 (γ , μ) and Trres (μ P+ μ P+ · · · + P+ μ P+ ). However due to the fact that dim H+ = 1 we have k hkn (γ , μ) 1 h 1 (γ , μ) Trres μk1 P+ μk2 P+ · · · + P+ μkn P+ = n 1 ... 1 . γ k1 + 1 kn + 1
(4.8)
Thus all integrals of motion hkl are functionally dependent on hk0 and hk1 . Therefore one has only two independent families of Hamilton equations, i.e. (3.39) and (3.40). In order to solve these families we note that k μ −+ = Mk (γ , μ)w, k μ +− = v + Mk (γ , μ),
(4.9) (4.10)
where Mk (γ , μ) :=
hk−2 (γ , μ) h2 (γ , μ) k−3 hk−1 1 (γ , μ) + 1 A + ··· + 1 A + aAk−2 + Ak−1 γk γ (k − 1) 3γ
(4.11)
is a time independent operator. Thus Eqs. (3.39) take the form ⎧ ∂ ⎪ + ⎪ ⎪ ⎨ ∂τ k v = −γ Mk γ , μ v, 0 ∂ ⎪ ⎪ ⎪ ⎩ k w = γ Mk (γ , μ)w. ∂τ0
(4.12)
In this way we have reduced system (3.39) to a linear system. Thus its solution is ∞ k 2 3 + Mk γ , μ (0, 0, . . .) τ0 v(0, 0, . . .), v τ0 , τ0 , . . . = exp −γ
(4.13)
k=2
∞ 2 3 k w τ0 , τ0 , . . . = exp γ Mk γ , μ(0, 0, . . .) τ0 w(0, 0, . . .) k=2
where v(0, 0, . . .), w(0, 0, . . .) ∈ 2 , A ∈ L∞ (H− ), a ∈ C are initial conditions.
(4.14)
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In the case of Eqs. (3.40) we get ⎧ k−1 j h (γ , μ+ ) ⎪ ∂ ⎪ 1 ⎪ Mk−j γ , μ+ + Mk+1 γ , μ+ v, v=− ⎪ ⎪ k ⎪ j +1 ⎨ ∂τ1 j =1 ⎪ k−1 k−j ⎪ h1 (γ , μ) ⎪ ∂ ⎪ ⎪ Mj (γ , μ) + Mk+1 (γ , μ) w. ⎪ ⎩ ∂τ k w = k−j +1 1
(4.15)
j =1
These equation are also linear and their solution can be obtained by exponentiation. Example 4.3 (4-dimensional case). In this example we solve equation (3.41) in the iR ⊕ UL1res case assuming dim H+ = dim H− = 2. We will use the following notation A Z , (4.16) γ = iχ, μ=i Z+ D where χ ∈ R and A = A+ , D = D + , Z ∈ Mat2×2 (C). Substituting (4.16) into (3.41) we obtain d A = 0, dt
d D=0 dt
(4.17)
and d Z = −iχ A2 Z + ZD 2 + AZD + ZZ + Z , dt d + Z = iχ Z + A2 + D 2 Z + + DZ + A + Z + ZZ + . dt
(4.18) (4.19)
Let us note that Eqs. (4.18) do not change their form with respect to the transformation A → U AU + , D → V DV + , Z → V ZU + , where U U + = 1 and V V + = 1. So without loss of the generality we can assume d1 0 a b a1 0 , D= , Z= , (4.20) A= c d 0 a2 0 d2 where a1 , a2 , d1 , d2 ∈ R are constants and a, b, c, d are complex-valued functions of t ∈ R. From (4.18) we obtain d a = iχ a12 + a1 d1 + d12 + |a|2 + |b|2 + |c|2 a + iχbcd, dt d b = iχ a12 + a1 d2 + d22 + |a|2 + |b|2 + |d|2 b + iχacd, dt d c = iχ a22 + a2 d1 + d12 + |a|2 + |c|2 + |d|2 c + iχabd, dt d c = iχ a22 + a2 d2 + d22 + |b|2 + |c|2 + |d|2 c + iχ abc. dt
(4.21)
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Now we consider the generic case, i.e. a1 = a2 and d1 = d2 . In order to solve this system of equations we calculate explicitly the integrals of motion h10 (μ) = Trres μ2 , h20 (μ) = Trres μ3 , h21 (μ) = γ Tr(μ2 P+ ), h31 (μ) = γ Tr(μ3 P+ ) and h30 (μ) = Trres μ4 . From that we conclude that |a|2 + |b|2 =: p 2 = const, |a|2 + |c|2 =: q 2 = const, |c|2 + |d|2 =: r 2 = const, |b|2 + |d|2 =: s 2 = const,
(4.22)
|a|2 a1 d1 + |b|2 a1 d2 + |c|2 a2 d1 + |d|2 a2 d2 + |a|2 |c|2 + |b|2 |d|2 + 2 Re(abcd) =: = const,
(4.23)
where p 2 + r 2 = q 2 + s 2 . Using (4.22) and (4.21) we find d 2 d d d |a| = − |b|2 = − |c|2 = |d|2 = 2χ Im(abcd). dt dt dt dt
(4.24)
Now from (4.23) and (4.24) we obtain the following equation d x = ± w(x) dt
(4.25)
w(x) := 4x p 2 − x q 2 − x r 2 − q 2 − x − v 2 (x)
(4.26)
v(x) := x(a1 − a2 )(d1 − d2 ) − x p 2 − x − q 2 − x r 2 − q 2 + x − a1 d2 p 2 − a2 d1 q 2 − a2 d2 r 2 − q 2 .
(4.27)
on the function x := |a|2 , where
and
Since w is a polynomial of the fourth degree, this equation is solved by an elliptic integral of the first kind t=
dx . √ w(x)
(4.28)
This allows us to express x(t) as an elliptic function of time parameter t. By (4.22) we may calculate |b|2 , |c|2 and |d|2 in terms of x(t). In order to find the functions a(t), b(t), c(t) and d(t) we substitute their polar decompositions a = |a|eiα , b = |b|eiβ , c = |c|eiγ and d = |d|eiδ into Eqs. (4.21). In this way we obtain
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d v(x) α = χ a12 + a1 d1 + d12 + p 2 + q 2 − x + , dt 2x d v(x) 2 2 2 2 2 β = χ a1 + a1 d2 + d 2 + p + r − q + x + , dt 2(p 2 − x) v(x) d , γ = χ a22 + a2 d1 + d12 + r 2 + x + dt 2(q 2 − x) d v(x) δ = χ a22 + a2 d2 + d22 + p 2 + r 2 − x + . dt 2(r 2 − q 2 + x)
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(4.29)
Since the right-hand sides of Eqs. (4.29) are known, we can find α(t), β(t), γ (t) and δ(t) by integration. In this way we have solved equations (4.18) in quadratures. The Hamiltonian system solved in this example have applications for example in the nonlinear optics, see [6]. Acknowledgments The authors would like to thank A.B. Tumpach, D. Belti¸taˇ and V. Dragovic for many valuable remarks and comments. This work is partially supported by Polish Government grant 1 P03A 001 29. The authors would like to thank Banach Center for their hospitality during the workshop “Banach Lie–Poisson spaces and integrable systems” (5–10 August 2008, B¸edlewo). Appendix A. Extensions of Banach Lie groups and related Banach Lie–Poisson spaces Let us present an abbreviated description of extensions of Banach Lie groups, Banach Lie algebras and Banach Lie–Poisson spaces associated with them. Some of the results given below can be found in papers [1,9,10,12]. Our main aim is to compute the formulas for the adjoint and coadjoint actions of the extended Banach Lie group. A.1. Extensions of Banach Lie groups Let us consider an exact sequence of Banach Lie groups {eN }
N
ι
G
π
H
{eH } .
(A.1)
We assume that N → G → H is a smooth principal bundle, i.e. the maps ι and π are smooth and there exists a smooth local section σ : U → G, where U ⊂ H is an open neighborhood of identity. Additionally we impose on σ the normalization condition σ (eH ) = eG . One can extend σ to a global section σ : H −→ G,
(A.2)
but in general such extension will not be smooth. Let us define the map Ψ : N × H −→ G by Ψ (n, h) := ι(n)σ (h).
(A.3)
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Since G/N ∼ = H , we get that for g ∈ G there exists a unique n ∈ N such that g = ι(n)σ (π(g)). Thus Ψ is a locally smooth bijection with the inverse Ψ −1 : G −→ N × H given by −1 Ψ −1 (g) := ι−1 g σ π(g) , σ π(g) . Using Ψ one defines the multiplication on N × H by (n1 , h1 ) · (n2 , h2 ) := Ψ −1 Ψ (n1 , h1 )Ψ (n2 , h2 )
(A.4)
(A.5)
and can express it as follows (n1 , h1 ) · (n2 , h2 ) = n1 Φ(h1 )(n2 )Ω(h1 , h2 ), h1 h2 ,
(A.6)
where maps Φ : H → Aut(N ) and Ω : H × H → N are defined by Φ(h)(n) := ι−1 σ (h)ι(n)σ (h)−1 , Ω(h1 , h2 ) := ι−1 σ (h1 )σ (h2 )σ (h1 h2 )−1 .
(A.7) (A.8)
Let us denote by Φ the map Φ : H × N (h, n) → Φ(h)(n) ∈ N.
(A.9)
One has the following properties of Φ and Ω: Φ(eH ) = id,
(A.10a)
Ω(eH , h) = Ω(h, eH ) = eN , Ω(h1 , h2 )Ω(h1 h2 , h3 ) = Φ(h1 ) Ω(h2 , h3 ) Ω(h1 , h2 h3 ),
(A.10b)
Ω(h1 , h2 )Φ(h1 h2 )(n) = Φ(h1 ) ◦ Φ(h2 )(n)Ω(h1 , h2 ).
(A.10d)
(A.10c)
Forgetting about definitions (A.7), (A.8) we can consider Φ and Ω as abstract maps satisfying conditions (A.10). We assume that the map Φ is smooth on U × N and Ω is smooth on some neighborhood of (eH , eH ). Moreover we have to assume that the map H x → Ω(h, x)Ω(hxh−1 , h)−1 is smooth for all h ∈ H on a neighborhood of eH (if H is connected then this condition is automatically satisfied). Under these conditions there exists on N × H a structure of Banach Lie group defined by (A.6), see [10]. One denotes this Banach Lie group by N ×Φ,Ω H . We get that the inverse of (n, h) in N ×Φ,Ω H is given by −1 (n, h)−1 = Ω h−1 , h Φ h−1 n−1 , h−1
(A.11)
and the inner automorphism I(n,h) (m, g) := (n, h) · (m, g) · (n, h)−1 can be expressed in terms of Φ and Ω as −1 I(n,h) (m, g) = nΦ(h)(m)Ω(h, g)Ω hgh−1 , h Φ hgh−1 n−1 , hgh−1 . (A.12) Now, let us pass to the extensions of related Banach Lie algebras.
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A.2. Extensions of Banach Lie algebras We will denote the Banach Lie algebras of G, H , N by g, h, n respectively. Taking derivatives of the maps in (A.1) we obtain the exact sequence of Banach Lie algebras {0}
n
Dι(eN )
g
Dπ(eG )
h
{0} .
(A.13)
The derivative DΨ −1 (eG ) : g → n ⊕ h of Ψ −1 at the point eG allows us to identify the Banach space g with n ⊕ h. The adjoint representation of N ×Φ,Ω H on g ∼ = n ⊕ h can be locally computed from (A.12) and it is the following Ad(n,h) (ζ, η) = Adn D2 Φ(h, eN )(ζ ) + D2 Ω(h, eH )(η) − D1 Ω(eH , h)(Adh η) + DLn n−1 ◦ D1 Φ eH , n−1 ◦ Adh (η), Adh η (A.14) for (n, h) ∈ N ×Φ,Ω U , (η, ζ ) ∈ n ⊕ h, where Φ is defined by (A.9), and Di denotes partial derivative with respect to the ith argument. We denote by Ln the left group action Ln m = nm, n, m ∈ N , on itself. Differentiating (A.14) we obtain the formula for the Lie bracket
(ζ, η), (ν, ξ ) := [ζ, ν] + ϕ(η)(ν) − ϕ(ξ )(ζ ) + ω(η, ξ ), [η, ξ ] ,
(A.15)
for (ζ, η), (ν, ξ ) ∈ n ⊕ h, where ϕ : h → Aut(n) is the linear continuous map and ω : h × h → n is the continuous bilinear skew symmetric map defined by Φ and Ω as follows: ϕ(η)(ζ ) := D1 D2 Φ(eH , eN )(η, ζ ),
(A.16)
ω(η, ξ ) := D1 D2 Ω(eH , eH )(η, ξ ) − D1 D2 Ω(eH , eH )(ξ, η).
(A.17)
In these formulas D1 D2 is the second mixed partial derivative. The maps ϕ and ω satisfy the following infinitesimal version of conditions (A.10):
ω η, η , η
+ ω η , η
, η + ω η
, η , η − ϕ(η) ω η , η
− ϕ η ω η
, η − ϕ η
ω η, η = 0,
(A.18)
adω(η,η ) +ϕ η, η − ϕ(η), ϕ η = 0
(A.19)
and
for all η, η , η
∈ h. If we forget about the underlying Banach Lie groups and consider their Banach Lie algebras only, then the maps ϕ and ω satisfying conditions (A.18)–(A.19) with additional smoothness conditions, define the structure of Banach Lie algebra on n ⊕ h, see [1,12].
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A.3. Extensions of Banach Lie–Poisson spaces According to [11] the Banach Lie–Poisson space is a Banach space b such that its dual b∗ is Banach Lie algebra with the property ad∗x b ⊂ b ⊂ b∗∗
(A.20)
for all x ∈ b∗ . This property allows us to define the Poisson bracket on b
{f, g}(b) = Df (b), Dg(b) , b ,
(A.21)
where Df (b), Dg(b) ∈ b∗ are Fréchet derivatives at point b ∈ b. The bracket makes the Banach space b a Banach Poisson space in the sense of [11]. Let us assume that Banach Lie algebras n, h and g possess predual Banach spaces n∗ , h∗ and g∗ satisfying condition (A.20). We also assume that maps (Dι(eN ))∗ , (Dπ(eG ))∗ (h∗ ) dual to Dι(eN ) and Dπ(eG ) preserve predual spaces, i.e. ∗ ∗ Dι(eN ) (g∗ ) ⊂ n∗ , Dπ(eG ) (h∗ ) ⊂ g∗ . (A.22) In that situation one obtains the exact sequence of predual Banach spaces {0}
h∗
(Dπ(eG ))∗
g∗
(Dι(eG ))∗
n∗
{0} ,
(A.23)
see Lemma 3.7 in [12]. We can identify g∗ with n∗ ⊕ h∗ by the map dual to the derivative DΨ −1 (eG ) at the point eG . This identification allows us to compute coadjoint actions of N ×Φ,Ω H and n ⊕ h on n∗ ⊕ h∗ as follows: ∗ ∗ ∗ Ad∗(n,h) (τ, μ) = D2 Φ(h, eN ) Ad∗n τ, D2 Ω(h, eH ) Ad∗n − Ad∗h D1 Ω(eH , h) Ad∗n ∗ τ + Ad∗h μ , + Ad∗h D1 Φ eH , n−1 (A.24) where (n, h) ∈ N ×Φ,Ω U , (τ, μ) ∈ n∗ ⊕ h∗ and ∗ ∗ ∗ ad∗(ζ,η) (τ, μ) = ad∗ζ τ + ϕ(η) τ, − ϕ( · )(ζ ) τ + ω(η, ·) τ + ad∗η μ
(A.25)
for (ζ, η) ∈ n ⊕ h, (τ, μ) ∈ n∗ ⊕ h∗ . The coadjoint representation (A.25) satisfies condition (A.20) if and only if ∗ ∗ ∗ ϕ(η) (n∗ ) ⊂ n∗ , ϕ(·)(ζ ) (n∗ ) ⊂ h∗ , ω(η, ·) (n∗ ) ⊂ h∗ .
(A.26)
So, under these conditions the Banach space n ⊕ h is a Banach Lie–Poisson space. Using definition (A.21) and Lie bracket (A.15) we obtain the Poisson bracket on n∗ ⊕ h∗ :
{f, g}(τ, μ) = [D1 f, D1 g] + ϕ(D2 f )(D1 g) − ϕ(D2 g)(D1 f ) + ω(D2 f, D2 g), τ
+ [D2 f, D2 g], μ (A.27) for f, g ∈ C ∞ (n∗ ⊕ h∗ ).
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Further investigation of extensions of Lie groups and Lie algebras is beyond the scope of this paper, so for more information we refer to [1,9,10,12]. Appendix B. Magri method We briefly recall the Magri method of constructing integrals of motion in involution. For more details see e.g. [7]. Let us consider a pencil of compatible Poisson brackets {·,·}ε := {·,·}1 + ε{·,·}2
(B.1)
where ε ∈ R. Compatibility of Poisson brackets means that {·,·}ε is also a Poisson bracket for any parameter ε. Let Iεk be a family of Casimirs for Poisson bracket {·,·}ε indexed by k ∈ N, i.e. k Iε , · ε = 0.
(B.2)
Assuming that Iεk depends analytically on the parameter ε one expands the equality (B.2) and computes the coefficients in front of ε n . Thus one obtains that {hk0 , ·}1 = 0 and
hkl , ·
1
= hkl+1 , · 2 ,
l = 0, 1, . . .
(B.3)
where hkl are defined by =
Iεk
∞
hkl ε l .
(B.4)
l=0
Due to relation (B.3), the sequence {hkl }l∈N∪{0} is called a Magri chain. By using (B.3) twice one gets that
hkl , hkn
1
= hkl−1 , hkn+1 1 .
(B.5)
Next, by iterating this procedure one concludes that k k
hl , hn 1 = hk0 , hkn+l 1 = 0.
(B.6)
Thus functions hkl are in involution k k
hl , hn ε = hkl , hkn 1 = hkl , hkn 2 = 0 for all Poisson brackets under consideration.
(B.7)
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References [1] D. Alekseevsky, P.W. Michor, W. Ruppert, Extensions of Lie algebras, ESI preprint 881. [2] D. Belti¸taˇ , T.S. Ratiu, A.B. Tumpach, The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (2007) 138–168. [3] A.L. Carey, C.A. Hurst, D.M. O’Brien, Automorphisms of the canonical anticommutation relations and index theory, J. Funct. Anal. 48 (1982) 360–393. [4] P. de la Harpe, Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space, SpringerVerlag, 1972. [5] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. 1, Birkhäuser, 1990. [6] D.D. Holm, Geometric Mechanics, Part I: Dynamics and Symmetry, Imperial College Press, 2008. [7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156–1162. [8] T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press, 2000. [9] K.-H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier 52 (2002) 1365–1442. [10] K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier 56 (2006) 209–271. [11] A. Odzijewicz, T.S. Ratiu, Banach Lie–Poisson spaces and reduction, Comm. Math. Phys. 243 (2003) 1–54. [12] A. Odzijewicz, T.S. Ratiu, Extensions of Banach Lie–Poisson spaces, J. Funct. Anal. 217 (2004) 103–125. [13] A. Odzijewicz, T.S. Ratiu, Induced and coinduced Banach Lie–Poisson spaces and integrability, J. Funct. Anal. 255 (2008) 1225–1272. [14] R.T. Powers, E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970) 1–33. [15] A. Pressley, G. Segal, Loop Groups, Oxford Math. Monogr., Clarendon Press, Oxford, 1986. [16] M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, 1978. [17] M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Lect. Notes in Num. Appl. Anal. 5 (1982) 259–271. [18] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, 1960. [19] J. Schwinger, Field theory commutators, Phys. Rev. Lett. 3 (6) (1959) 296–297. [20] G.B. Segal, G. Wilson, Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985) 5–65. [21] A. Sergeev, Kähler Geometry of Loop Spaces, MCNMO, Moscow, 2001 (in Russian). [22] D. Shale, W.F. Stinespring, States on the Clifford algebra, Ann. of Math. 80 (1964) 365–381. [23] M. Spera, T. Wurzbacher, Determinants, Pfaffians, and quasi-free representations of the CAR algebra, Rev. Math. Phys. 10 (5) (1998) 705–721. [24] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, 1979. [25] B. Thaller, The Dirac Equation, Springer, 1992. [26] T. Wurzbacher, Fermionic second quantization and the geometry of the restricted Grassmannian, in: A. Huckleberry, T. Wurzbacher (Eds.), Infinite Dimensional Kähler Manifolds, in: DMV Seminar, vol. 31, Birkhäuser, Basel, 2001.
Journal of Functional Analysis 258 (2010) 3295–3318 www.elsevier.com/locate/jfa
Lane–Emden systems with negative exponents ✩ Marius Ghergu School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland Received 15 September 2009; accepted 3 February 2010 Available online 12 February 2010 Communicated by J. Coron
Abstract We study the elliptic system ⎧ ⎨ −u = u−p v −q −v = u−r v −s ⎩ u=v=0
in Ω, in Ω, on ∂Ω,
in a bounded domain Ω ⊂ RN (N 1) with a smooth boundary, p, s 0 and q, r > 0. We investigate the existence, non-existence, and uniqueness of C 2 (Ω) ∩ C(Ω) solutions in terms of p, q, r and s. A necessary and sufficient condition for the C 1 -regularity of solutions up to the boundary is also obtained. © 2010 Elsevier Inc. All rights reserved. Keywords: Lane–Emden equation; Elliptic system; Negative exponent; Boundary behavior
1. Introduction In this paper we study the elliptic system ⎧ ⎨ −u = u−p v −q , u > 0 −v = u−r v −s , v > 0 ⎩ u=v=0
in Ω, in Ω, on ∂Ω,
(1)
✩ This research was partially supported by the ESF Research Network “Harmonic and Complex Analysis and Applications” (HCAA). E-mail address:
[email protected].
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.003
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where Ω ⊂ RN (N 1) is a bounded domain with C 2 -boundary, p, s 0 and q, r > 0. By solution of (1) we understand a pair (u, v) with u, v ∈ C 2 (Ω) ∩ C(Ω) such that u, v > 0 in Ω and satisfies (1) pointwise. The first motivation for the study of system (1) comes from the so-called Lane–Emden equation (see [5,8,15]) −u = up
in BR (0), R > 0,
(2)
subject to Dirichlet boundary condition. In astrophysics, the exponent p is called polytropic index and positive radially symmetric solutions of (2) are used to describe the structure of the polytropic stars (we refer the interested reader to the book by Chandrasekhar [2] for an account on the above equation as well as for various mathematical techniques to describe the behavior of the solution to Lane–Emden equation). Systems of type (1) with p, s 0 and q, s < 0 have received considerably attention in the last decade (see, e.g., [1,3,6,7,16,18–23] and the references therein). It has been shown that for such range of exponents system (1) has a rich mathematical structure. Various techniques such as moving plane method, Pohozaev-type identities, rescaling arguments have been developed and suitably adapted to deal with (1) in this case. Recently, there has been some interest in systems of type (1) where not all the exponents are negative. In [10–12] the system (1) is considered under the hypothesis p, r < 0 < q, s. This corresponds to the singular Gierer–Meinhardt system arising in the molecular biology. In [9] the authors provide a nice sub and supersolution device that applies to general systems both in cooperative and non-cooperative setting. This method was then used to discuss singular counterpart of some well-known models such as Gierer–Meinhardt, Lotka–Voltera or predator-prey systems. In this paper, we shall be concerned with system (1) in case p, s 0 and q, r > 0. This corresponds to the prototype equation (2) in which the polytropic index p is negative. For such range of exponents, the above mentioned methods do not apply; another difficulties in dealing with system (1) come from the non-cooperative character of our system and from the lack of a variational structure. In turn, our approach relies on the boundary behavior of solutions to (2) (with p < 0) or more generally, to singular elliptic problems of the type
−u = k δ(x) u−p , u=0
u > 0 in Ω, on ∂Ω,
(3)
where δ(x) = dist(x, ∂Ω),
x ∈ Ω,
and k : (0, ∞) → (0, ∞) is a decreasing function such that limt0 k(t) = ∞. The approach we adopt in this paper can be used to study more general systems in the form
−Lu = f (x, u, v), u > 0 in Ω, −Lv = g(x, u, v), v > 0 in Ω, u=v=0 on ∂Ω,
where L is a second order differential operator not necessarily in divergence form and f (x, u, v) = k1 (x)u−p v −q ,
g(x, u, v) = k2 (x)u−r v −s ,
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or f (x, u, v) = k11 (x)u−p + k12 (x)v −q ,
g(x, u, v) = k21 (x)u−r + k22 (x)v −s ,
with ki , kij : Ω → (0, ∞) (i, j = 1, 2) continuous functions that behave like −a
δ(x)
b
log
A δ(x)
near ∂Ω,
(4)
for some A, a > 0 and b ∈ R. Our first result concerning the study of (1) is the following. Theorem 1.1 (Non-existence). Let p, s 0, q, r > 0 be such that one of the following conditions holds: 2−q (i) r min{1, 1+p } 2;
(ii) q min{1, 2−r 1+s } 2; (iii) p > max{1, r − 1}, 2r > (1 − s)(1 + p) and q(1 + p − r) > (1 + p)(1 + s); (iv) s > max{1, q − 1}, 2q > (1 − p)(1 + s) and r(1 + s − q) > (1 + p)(1 + s). Then the system (1) has no solutions. Remark that condition (i) in Theorem 1.1 restricts the range of the exponent q to the interval (0, 2) while in (iii) the exponent q can take any value greater than 2, provided we adjust the other three exponents p, r, s accordingly. The same remark applies for the exponent r from the above conditions (ii) and (iv). The existence of solutions to (1) is obtained under the following assumption on the exponents p, q, r, s: (1 + p)(1 + s) − qr > 0.
(5)
We also introduce the quantities 2−r , α = p + q min 1, 1+s
2−q β = r + s min 1, . 1+p
The above values of α and β are related to the boundary behavior of the solution to the singular elliptic problem (3) as explained in Proposition 2.6 below. Our existence result is as follows. Theorem 1.2 (Existence). Let p, s 0, q, r > 0 satisfy (5) and one of the following conditions: (i) α 1 and r < 2; (ii) β 1 and q < 2; (iii) p, s 1 and q, r < 2. Then, the system (1) has at least one solution.
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The proof of the existence is based on the Schauder’s fixed point theorem in a suitable chosen closed convex subset of C(Ω) × C(Ω) that contains all the functions having a certain rate of decay expressed in terms of the distance function δ(x) up to the boundary of Ω. From Theorem 1.1(i)–(ii) and Theorem 1.2(i)–(ii) we have the following necessary and sufficient conditions for the existence of solutions to (1): Corollary 1.3. Let p, s 0, q, r > 0 satisfy (5). (i) Assume p + q 1. Then system (1) has solutions if and only if r < 2; (ii) Assume r + s 1. Then system (1) has solutions if and only if q < 2. A particular feature of system (1) is that it does no posses C 2 (Ω) solutions. Indeed, due to the fact that q, r < 0 and to the homogeneous Dirichlet boundary condition imposed on u and v we have that u−p v −q and u−r v −s are unbounded around ∂Ω, so there are no C 2 (Ω) solutions of (1). In turn, C 2 (Ω) ∩ C 1 (Ω) may exist and our next result provides necessary and sufficient conditions in terms of p, q, r and s for the existence of such solutions. Theorem 1.4 (C 1 -regularity). Let p, s 0, q, r > 0 satisfy (5). Then: (i) System (1) has a solution (u, v) with u ∈ C 1 (Ω) if and only if α < 1 and r < 2; (ii) System (1) has a solution (u, v) with v ∈ C 1 (Ω) if and only if β < 1 and q < 2; (iii) System (1) has a solution (u, v) with u, v ∈ C 1 (Ω) if and only if p + q < 1 and r + s < 1. Another feature of system (1) is that under some conditions on p, q, r, s it has a unique solution (see Theorem 1.5 below). This is a striking difference between our setting and the case p, s 0 and q, r < 0 largely investigated in the literature so far, where the uniqueness does not seem to occur. In our framework, the uniqueness is achieved from the boundary behavior of solution to (1) deduced from the study of the prototype model (3). Theorem 1.5 (Uniqueness). Let p, s 0, q, r > 0 satisfy (5) and one of the following conditions: (i) p + q < 1 and r < 2; (ii) r + s < 1 and q < 2. Then, the system (1) has a unique solution. The rest of the paper is organized as follows. In Section 2 we obtain some useful properties related to the boundary behavior of the solution to (3). Sections 3–6 are devoted to the proofs of the above results. 2. Preliminary results In this section we collect some old and new results concerning problems of type (3). Note that the method of sub and supersolutions is also valid in the singular framework as explained in [13, Theorem 1.2.3]. Our first result is a straightforward comparison principle between subsolutions and supersolutions for singular elliptic equations.
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Proposition 2.1. Let p 0 and φ : Ω → (0, ∞) be a continuous function. If u is a subsolution and u is a supersolution of
−u = φ(x)u−p , u=0
u > 0 in Ω, on ∂Ω,
then u u in Ω. Proof. If p = 0 the result follows directly from the maximum principle. Let now p > 0. Assume by contradiction that the set ω := {x ∈ Ω: u(x) < u(x)} is not empty and let w := u − u. Then, w achieves its maximum on Ω at a point that belongs to ω. At that point, say x0 , we have
0 −w(x0 ) φ(x0 ) u(x0 )−p − u(x0 )−p < 0, which is a contradiction. Therefore, ω = ∅, that is, u u in Ω.
2
Proposition 2.2. Let u ∈ C 2 (Ω) ∩ C(Ω) be such that u = 0 on ∂Ω and 0 −u cδ(x)−a
in Ω,
where 0 < a < 2 and c > 0. Then, u ∈ C 0,γ (Ω) for some 0 < γ < 1. Furthermore, if 0 < a < 1, then u ∈ C 1,1−a (Ω). Proof. Let G denote Green’s function for the negative Laplace operator. Thus, for all x ∈ Ω we have u(x) = − G(x, y)u(y) dy. Ω
Let x1 , x2 ∈ Ω. Then u(x1 ) − u(x2 ) −
G(x1 , y) − G(x2 , y)u(y) dy
Ω
c
G(x1 , y) − Gx (x2 , y)δ(y)−a dy.
Ω
Next, using the method in [14, Theorem 1.1] we have u(x1 ) − u(x2 ) C|x1 − x2 |γ
for some 0 < γ < 1.
Hence u ∈ C 0,γ (Ω). Assume now 0 < a < 1. Then, Gx (x, y)u(y) dy
∇u(x) = − Ω
for all x ∈ Ω,
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and ∇u(x1 ) − ∇u(x2 ) −
Gx (x1 , y) − Gx (x2 , y)u(y) dy
Ω
c
Gx (x1 , y) − Gx (x2 , y)δ(y)−a dy.
Ω
The same technique as in [14, Theorem 1.1] yields ∇u(x1 ) − ∇u(x2 ) C|x1 − x2 |1−a Therefore u ∈ C 1,1−a (Ω).
for all x1 , x2 ∈ Ω.
2
Proposition 2.3. Let (u, v) be a solution of system (1). Then, there exists a constant c > 0 such that u(x) cδ(x)
and v(x) cδ(x)
in Ω.
(6)
Proof. Let w be the solution of
−w = 1, w=0
w>0
in Ω, on ∂Ω.
(7)
Using the smoothness of ∂Ω, we have w ∈ C 2 (Ω) and by Hopf’s boundary point lemma (see [17]), there exists c0 > 0 such that w(x) c0 δ(x) in Ω. Since −u C = −(Cw) in Ω, for some constant C > 0, by standard maximum principle we deduce u(x) Cw(x) cδ(x) in Ω and similarly v(x) cδ(x) in Ω, where c > 0 is a positive constant. 2 Let (λ1 , ϕ1 ) be the first eigenvalue/eigenfunction of − in Ω. It is well known that λ1 > 0 and ϕ1 ∈ C 2 (Ω) has constant sign in Ω. Further, using the smoothness of Ω and normalizing ϕ1 with a suitable constant, we can assume c0 δ(x) ϕ1 (x) δ(x)
in Ω,
(8)
1 for some 0 < c0 < 1. By Hopf’s boundary point lemma we have ∂ϕ ∂n < 0 on ∂Ω, where n is the outer unit normal vector at ∂Ω. Hence, there exists ω Ω and c > 0 such that
|∇ϕ1 | > c
in Ω \ ω.
(9)
Theorem 2.4. Let p 0, A > diam(Ω) and k : (0, A) → (0, ∞) be a decreasing function such that A tk(t) dt = ∞. 0
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Then, the inequality
−u k δ(x) u−p , u=0
u > 0 in Ω, on ∂Ω,
(10)
has no solutions u ∈ C 2 (Ω) ∩ C(Ω). Proof. Suppose by contradiction that there exists a solution u0 of (10). For any 0 < ε < A − diam(Ω) we consider the perturbed problem
−u = k δ(x) + ε (u + ε)−p , u=0
u > 0 in Ω, on ∂Ω.
(11)
Then, u = u0 is a supersolution of (11). Also, if w is the solution of problem (7) it is easy to see that u = cw is a subsolution of (11) provided c > 0 is small enough. Further, by Proposition 2.1 it follows that u u in Ω. Thus, by the sub and supersolution method we deduce that problem (11) has a solution uε ∈ C 2 (Ω) such that cw uε u0
in Ω.
(12)
Multiplying with ϕ1 in (11) and then integrating over Ω we find
uε ϕ1 dx =
λ1 Ω
k δ(x) + ε (uε + ε)−p ϕ1 dx.
Ω
Using (12) we obtain M := λ1
u0 ϕ1 dx λ1
Ω
uε ϕ1 dx Ω
k δ(x) + ε (u0 + ε)−p ϕ1 dx,
ω
for all ω Ω. Passing to the limit with ε → 0 in the above inequality and using (8) we find M
−p −p k δ(x) u0 ϕ1 dx c0 u0 ∞
ω
k δ(x) δ(x) dx.
ω
Since ω Ω was arbitrary, we deduce
k δ(x) δ(x) dx < ∞.
Ω
Using the smoothness of ∂Ω, the above condition yields assumption on k. Hence, (10) has no solutions. 2
A 0
tk(t) dt < ∞, which contradicts our
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A direct consequence of Theorem 2.4 is the following result. Corollary 2.5. Let p 0 and q 2. Then, there are no functions u ∈ C 2 (Ω) ∩ C(Ω) such that
−u δ(x)−q u−p , u=0
u > 0 in Ω, on ∂Ω.
Proposition 2.6. Let p 0 and 0 < q < 2. There exists c > 0 and A > diam(Ω) such that any supersolution u of
−u = δ(x)−q u−p , u=0
u > 0 in Ω, on ∂Ω,
(13)
satisfies: (i) u(x) cδ(x) in Ω, if p + q < 1; 1
A (ii) u(x) cδ(x) log 1+p ( δ(x) ) in Ω if p + q = 1; 2−q
(iii) u(x) cδ(x) 1+p in Ω, if p + q > 1. A similar result holds for subsolutions of (13). Proof. If p > 0 then the result follows from Theorem 3.5 in [4] (see also [13, Section 9]). If p = 0 we proceed as in [4, Theorem 3.5], namely, for m > 0 we show that the function ⎧ mϕ (x) 1
if q < 1, ⎪ ⎨ A if q = 1, A > diam(Ω), u(x) = mϕ1 (x) log ϕ1 (x) ⎪ ⎩ mϕ1 (x)2−q if q > 1, satisfies −u δ(x)−q in Ω. Thus, the estimates in Proposition 2.6 follow from (8) and the maximum principle. 2 Theorem 2.7. Let 0 < a < 1, A > diam(Ω), p 0 and q > 0 be such that p + q = 1. Then, the problem
−u = δ(x)−q log−a u=0
A u−p , δ(x)
u > 0 in Ω,
(14)
on ∂Ω,
has a unique solution u which satisfies 1−a
c1 δ(x) log 1+p for some c1 , c2 > 0.
A δ(x)
1−a
u(x) c2 δ(x) log 1+p
A δ(x)
in Ω,
(15)
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3303
Proof. Let w(x) = ϕ1 (x) logb where b =
1−a 1+p
A , ϕ1 (x)
x ∈ Ω,
∈ (0, 1). A straightforward computation yields
A A + b |∇ϕ1 |2 − λ1 ϕ12 ϕ1−1 logb−1 ϕ1 (x) ϕ1 (x)
A + b(1 − b)|∇ϕ1 |2 ϕ1−1 logb−2 in Ω. ϕ1 (x)
−w = λ1 ϕ1 logb
Using (9) we can find C1 , C2 > 0 such that C1 ϕ1−1 logb−1
A ϕ1 (x)
−w C2 ϕ1−1 logb−1
A ϕ1 (x)
in Ω,
that is, −q
C1 ϕ1 log−a
A A −q w −p −w C2 ϕ1 log−a w −p ϕ1 (x) ϕ1 (x)
in Ω.
We now deduce that u = mw and u = Mw are respectively subsolution and supersolution of (14) for suitable 0 < m < 1 < M. Hence, the problem (14) has a solution u ∈ C 2 (Ω) ∩ C(Ω) such that 1−a
mϕ1 log 1+p
A ϕ1 (x)
1−a
u M log 1+p
A ϕ1 (x)
in Ω.
(16)
The uniqueness follows from Proposition 2.1 while the boundary behavior of u follows from (16) and (8). This finishes the proof. 2 Corollary 2.8. Let C > 0 and a, A, p, q be as in Theorem 2.7. Then, there exists c > 0 such that any solution u of
−q
−u Cδ(x)
−a
log
u=0
A u−p , δ(x)
u > 0 in Ω, on ∂Ω,
satisfies u(x) cδ(x) log
1−a 1+p
A δ(x)
in Ω.
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Proposition 2.9. Let A > 3 diam(Ω) and C > 0. There exists c > 0 such that any solution u ∈ C 2 (Ω) ∩ C(Ω) of
−u Cδ
−1
(x) log
−1
u=0
A , δ(x)
u > 0 in Ω, on ∂Ω,
satisfies
A u(x) cδ(x) log log δ(x)
in Ω.
(17)
Proof. Let
A , w(x) = ϕ1 (x) log log ϕ1 (x)
x ∈ Ω.
An easy computation yields
|∇ϕ1 |2 − λ1 ϕ12 |∇ϕ1 |2 A −w = λ1 ϕ1 log log + + ϕ1 (x) ϕ1 log( ϕ1A(x) ) ϕ1 log2 ( ϕ1A(x) )
c0
in Ω,
ϕ1 log( ϕ1A(x) )
for some c0 > 0. Using (8) we can find m > 0 small enough such that −(mw)
C A δ(x) log( δ(x) )
in Ω.
Now by maximum principle we deduce u mw in Ω and by (8) we obtain that u satisfies the estimate (17). 2 Theorem 2.10. Let p 0, A > diam(Ω) and a ∈ R. Then, problem
−u = δ(x)−2 log−a u=0
A u−p , δ(x)
u > 0 in Ω,
(18)
on ∂Ω,
has solutions if and only if a > 1. Furthermore, if a > 1 then (21) has a unique solution u and there exist c1 , c2 > 0 such that c1 log
1−a 1+p
A δ(x)
u(x) c2 log
1−a 1+p
A δ(x)
in Ω.
(19)
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3305
Proof. Fix B > A be such that the function k : (0, B) → R, k(t) = t −2 log−a ( Bt ) is decreasing on (0, A). Then, any solution u of (18) satisfies −u ck δ(x) u−p , u > 0 in Ω, u=0 on ∂Ω, A where c > 0. By virtue of Theorem 2.4 we deduce 0 tk(t) dt < ∞, that is, a > 1. For a > 1, let
B b , x ∈ Ω, w(x) = log ϕ1 (x) where b =
1−a 1+p
< 0. It is easy to see that
B ϕ1 (x)
B in Ω. − b(b − 1)|∇ϕ1 |2 ϕ1−2 logb−2 ϕ1 (x)
−w = −b |∇ϕ1 |2 + λ1 ϕ12 ϕ1−2 logb−1
Choosing B > 0 large enough, we may assume
B log 2(1 − b) ϕ1 (x)
in Ω.
(20)
Therefore, from (9) and (20) there exist C1 , C2 > 0 such that
B B −2 −2 b−1 b−1 −w C2 ϕ1 log in Ω, C1 ϕ1 log ϕ1 (x) ϕ1 (x) that is, C1 ϕ1−2 log−a
B B w −p −w C2 ϕ1−2 log−a w −p ϕ1 (x) ϕ1 (x)
in Ω.
As before, from (8) it follows that u = mw and u = Mw are respectively subsolution and supersolution of (18) provided m > 0 is small and M > 1 is large enough. The rest of the proof is the same as for Theorem 2.7. 2 Corollary 2.11. Let C > 0, p 0, A > diam(Ω) and a > 1. Then, there exists c > 0 such that any solution u ∈ C 2 (Ω) ∩ C(Ω) of
A −2 −a u−p , u > 0 in Ω, −u Cδ(x) log (21) ϕ1 (x) u=0 on ∂Ω, satisfies u(x) c log
1−a 1+p
A δ(x)
in Ω.
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3. Proof of Theorem 1.1 Since the system (1) is invariant under the transform (u, v, p, q, r, s) → (v, u, s, r, q, p), we only need to prove (i) and (iii). (i) Assume that there exists (u, v) a solution of system (1). Note that from (i) we have 0 < q < 2. Also, using Proposition 2.3, we can find c > 0 such that (6) holds. Case 1: p + q < 1. From our hypothesis (i) we deduce r 2. Using the estimates (6) in the first equation of the system (1) we find
−u c1 δ(x)−q u−p , u=0
u>0
in Ω, on ∂Ω,
(22)
for some c1 > 0. From Proposition 2.6(i) we now deduce u(x) c2 δ(x) in Ω, for some c2 > 0. Using this last estimate in the second equation of (1) we find
−v c3 δ(x)−r v −s , u=0
v > 0 in Ω, on ∂Ω,
(23)
where c3 > 0. According to Corollary 2.5, this is impossible, since r 2. Case 2: p + q > 1. From hypothesis (i) we also have r(2−q) 1+p 2. In the same manner as above, u satisfies (22). Thus, by Proposition 2.6(iii), there exists c4 > 0 such that 2−q
u(x) c4 δ(x) 1+p
in Ω.
Using this estimate in the second equation of system (1) we obtain
− r(2−q) 1+p v −s ,
−v c5 δ(x) u=0
v > 0 in Ω, on ∂Ω,
for some c5 > 0, which is impossible in view of Corollary 2.5, since
r(2−q) 1+p
2.
Case 3: p + q = 1. From (i) it follows that r 2. As in the previous two cases, we easily find that u is a solution of (22), for some c1 > 0. Using Proposition 2.6(ii), there exists c6 > 0 such that u(x) c6 δ(x) log
1 1+p
A δ(x)
in Ω,
for some A > 3 diam(Ω). Using this estimate in the second equation of (1) we obtain
−v c7 δ(x)−r log u=0
r − 1+p
A v −s , δ(x)
v > 0 in Ω,
where c7 is a positive constant. From Theorem 2.4 it follows that
on ∂Ω,
(24)
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
1 t
1−r
r − 1+p
log
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A dt < ∞. t
0
Since r 2, the above integral condition implies r = 2. Now, using (24) (with r = 2) and Corollary 2.11, there exists c8 > 0 such that p−1 (1+p)(1+s)
v(x) c8 log
A δ(x)
in Ω.
(25)
Using the estimate (25) in the first equation of system (1) we deduce
−u c9 log
q(1−p) (1+p)(1+s)
u=0
A u−p , δ(x)
u > 0 in Ω,
(26)
on ∂Ω,
for some c9 > 0. Fix 0 < a < 1 − p. Then, from (26) we can find a constant c10 > 0 such that u satisfies −u c10 δ(x)−a u−p , u > 0 in Ω, u=0 on ∂Ω. By Proposition 2.6(i) (since a + p < 1) we derive u(x) c11 δ(x) in Ω, where c11 > 0. Using this last estimate in the second equation of (1) we finally obtain (note that r = 2):
−v c12 δ(x)−2 v −s , v=0
v>0
in Ω, on ∂Ω,
which is impossible according to Corollary 2.5. Therefore, the system (1) has no solutions. (iii) Suppose that the system (1) has a solution (u, v) and let M = maxx∈Ω v. From the first equation of (1) we have
−u c1 u−p , u=0
u > 0 in Ω, on ∂Ω, 2
where c1 = M −q > 0. Using Proposition 2.6(iii) there exists c2 > 0 such that u(x) c2 δ(x) 1+p in Ω. Combining this estimate with the second equation of (1) we find
Since
2r 1+p
2r − 1+p v −s ,
−v c3 δ(x) v=0
v > 0 in Ω, on ∂Ω.
+ s > 1, again by Proposition 2.6(iii) we obtain that the function v satisfies 2(1+p−r)
v(x) c4 δ(x) (1+p)(1+s)
in Ω,
for some c4 > 0. Using the above estimate in the first equation of (1) we find c5 > 0 such that
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2q(1+p−r) − (1+p)(1+s) u−p ,
−u c5 δ(x) u=0
u>0
in Ω, on ∂Ω,
which contradicts Corollary 2.5 since q(1 + p − r) > (1 + p)(1 + s). Thus, the system (1) has no solutions. This ends the proof of Theorem 1.1. 4. Proof of Theorem 1.2 (i) We divide the proof into six cases according to the boundary behavior of singular elliptic problems of type (3), as described in Proposition 2.6. Case 1: r + s > 1 and α = p + 1 < c2 such that:
q(2−r) 1+s
< 1. By Proposition 2.6(i) and (iii) there exist 0 < c1 <
• Any subsolution u and any supersolution u of the problem
−u = δ(x)− u=0
q(2−r) 1+s
u−p ,
u > 0 in Ω, on ∂Ω,
(27)
satisfy u(x) c1 δ(x)
and u(x) c2 δ(x)
in Ω.
(28)
• Any subsolution v and any supersolution v of the problem
−v = δ(x)−r v −s , v=0
v > 0 in Ω, on ∂Ω,
(29)
satisfy 2−r
v(x) c1 δ(x) 1+s
2−r
and v(x) c2 δ(x) 1+s
in Ω.
(30)
We fix 0 < m1 < 1 < M1 and 0 < m2 < 1 < M2 such that q
r
M11+s m2 c1 < c2 M1 m21+p ,
(31)
and q
r
M21+p m1 c1 < c2 M2 m11+s .
(32)
Note that the above choice of mi , Mi (i = 1, 2) is possible in view of (5). Set A = (u, v) ∈ C(Ω) × C(Ω):
m1 δ(x) u(x) M1 δ(x) 2−r
in Ω 2−r
m2 δ(x) 1+s v(x) M2 δ(x) 1+s
in Ω
.
For any (u, v) ∈ A, we consider (T u, T v) the unique solution of the decoupled system
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
⎧ ⎨ −(T u) = v −q (T u)−p , T u > 0 −(T v) = u−r (T v)−s , T v > 0 ⎩ Tu=Tv =0
in Ω, in Ω, on ∂Ω,
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(33)
and define F : A → C(Ω) × C(Ω)
by F (u, v) = (T u, T v) for any (u, v) ∈ A.
(34)
Thus, the existence of a solution to system (1) follows once we prove that F has a fixed point in A. To this aim, we shall prove that F satisfies the conditions: F (A) ⊆ A,
F is compact and continuous.
Then, by Schauder’s fixed point theorem we deduce that F has a fixed point in A, which, by standard elliptic estimates, is a classical solution to (1). Step 1: F (A) ⊆ A. Let (u, v) ∈ A. From 2−r
v(x) M2 δ(x) 1+s
in Ω,
it follows that T u satisfies q(2−r) −q −(T u) M2 δ(x)− 1+s (T u)−p , Tu=0
Tu>0
in Ω, on ∂Ω.
q
Thus, u := M21+p T u is a supersolution of (27). By (28) and (32) we obtain q − 1+p
T u = M2
q − 1+p
u c 1 M2
δ(x) m1 δ(x)
in Ω.
2−r
From v(x) m2 δ(x) 1+s in Ω and the definition of T u we deduce that q(2−r) −q −(T u) m2 δ(x)− 1+s (T u)−p , T u > 0 in Ω, Tu=0 on ∂Ω. q
Thus, u := m21+p T u is a subsolution of problem (27). Hence, from (28) and (31) we obtain q − 1+p
T u = m2
q − 1+p
u c2 m2
δ(x) M1 δ(x)
in Ω.
We have proved that T u satisfies m1 δ(x) T u M1 δ(x)
in Ω.
In a similar manner, using the definition of A and the properties of the sub and supersolutions of problem (29) we show that T v satisfies 2−r
2−r
m2 δ(x) 1+s T v M2 δ(x) 1+s Thus, (T u, T v) ∈ A for all (u, v) ∈ A, that is, F (A) ⊆ A.
in Ω.
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Step 2: F is compact and continuous. Let (u, v) ∈ A. Since F (u, v) ∈ A, one can find 0 < a < 2 such that 0 −(T u), −(T v) cδ(x)−a
in Ω,
for some positive constant c > 0. Using Proposition 2.2 we now deduce T u, T v ∈ C 0,γ (Ω) (0 < γ < 1). Since the embedding C 0,γ (Ω) → C(Ω) is compact, it follows that F is also compact. It remains to prove that F is continuous. To this aim, let {(un , vn )} ⊂ A be such that un → u and vn → v in C(Ω) as n → ∞. Using the fact that F is compact, there exists (U, V ) ∈ A such that up to a subsequence we have T un → U,
T vn → V
in C(Ω) as n → ∞.
On the other hand, by standard elliptic estimates, the sequences {T un } and {T vn } are bounded in C 2,β (ω) (0 < β < 1) for any smooth open set ω Ω. Therefore, up to a diagonally subsequence, we have T un → U,
T vn → V
in C 2 (ω) as n → ∞,
for any smooth open set ω Ω. Passing to the limit in the definition of T un and T vn we find that (U, V ) satisfies
−U = v −q U −p , U > 0 in Ω, −V = u−r V −s , V > 0 in Ω, U =V =0 on ∂Ω.
By uniqueness of (33), it follows that T u = U and T v = V . Hence T un → T u,
T vn → T v
in C(Ω) as n → ∞.
This proves that F is continuous. We are now in a position to apply the Schauder’s fixed point theorem. Thus, there exists (u, v) ∈ A such that F (u, v) = (u, v), that is, T u = u and T v = v. By standard elliptic estimates, it follows that (u, v) is a solution of system (1). The remaining five cases will be considered in a similar way. Due to the different boundary behavior of solutions described in Proposition 2.6, the set A and the constants c1 , c2 have to be modified accordingly. We shall point out the way we choose these constants in order to apply the Schauder’s fixed point theorem. Case 2: r + s = 1 and α = p + q < 1. According to Proposition 2.6(i)–(ii) there exist 0 < a < 1 and 0 < c1 < 1 < c2 such that: • Any subsolution u of the problem
−u = δ(x)−q u−p , u=0
u > 0 in Ω, on ∂Ω,
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
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satisfies u(x) c2 δ(x)
in Ω.
• Any supersolution u of the problem −u = δ(x)−q(1−a) u−p , u=0
u > 0 in Ω, on ∂Ω,
satisfies u(x) c1 δ(x)
in Ω.
• Any subsolution v and any supersolution v of problem (29) satisfy the estimates v(x) c2 δ(x)1−a
and v(x) c1 δ(x)
in Ω.
We now define m1 δ(x) u(x) M1 δ(x) A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)1−a
in Ω , in Ω
where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31), (32) and
a m2 diam(Ω) < M2 .
(35)
We next define the operator F in the same way as in Case 1 by (33) and (34). The fact that F (A) ⊆ A and that F is continuous and compact follows in the same manner. Case 3: r + s < 1 and α = p + q < 1. In the same manner we define m1 δ(x) u(x) M1 δ(x) A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)
in Ω , in Ω
where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31)–(32) for suitable constants c1 and c2 . Case 4: r + s < 1 and α = p + q = 1. The approach is the same as in Case 2 above if we interchange u with v in the initial system (1). Case 5: r + s > 1 and α = p + q = 1. Let 0 < a < 1 be fixed such that ar + s > 1. From Proposition 2.6(i), (iii), there exist 0 < c1 < 1 < c2 such that: • Any subsolution u of the problem
−u = δ(x)− u=0
q(2−ar) 1+s
u−p ,
u > 0 in Ω, on ∂Ω,
satisfies u(x) c2 δ(x)a
in Ω.
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• Any supersolution u of the problem
−u = δ(x)− u=0
q(2−r) 1+s
u−p ,
u > 0 in Ω, on ∂Ω,
satisfies u(x) c1 δ(x)
in Ω.
• Any subsolution v of problem (29) satisfies 2−r
v(x) c2 δ(x) 1+s
in Ω.
• Any supersolution v of the problem
−v = δ(x)−ar v −s , v=0
v > 0 in Ω, on ∂Ω,
satisfies 2−ar
v(x) c1 δ(x) 1+s We now define A = (u, v) ∈ C(Ω) × C(Ω):
in Ω.
m1 δ(x) u(x) M1 δ(x)a 2−ar
in Ω 2−r
m2 δ(x) 1+s v(x) M2 δ(x) 1+s
in Ω
,
where 0 < mi < 1 < Mi (i = 1, 2) satisfy (31)–(32) in which the constants c1 , c2 are those given above and
1−a < M1 , m1 diam(Ω)
r(1−a) m2 diam(Ω) 1+s < M2 .
Case 6: r + s = 1 and α = p + q = 1. We proceed in the same manner as above by considering m1 δ(x) u(x) M1 δ(x)1−a A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x) v(x) M2 δ(x)1−a
in Ω , in Ω
where 0 < a < 1 is a fixed constant and mi , Mi (i = 1, 2) satisfy (31)–(32) for suitable c1 , c2 > 0 and
a mi diam(Ω) < Mi ,
i = 1, 2.
(iii) Let a=
2(1 + s − q) , (1 + p)(1 + s) − qr
b=
2(1 + p − r) . (1 + p)(1 + s) − qr
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
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Then (1 + p)a + bq = 2,
ar + (1 + s)b = 2.
(36)
Since p + bq > 1 and s + ar > 1, from Proposition 2.6(iii) and (36) above we can find 0 < c1 < 1 < c2 such that: • Any subsolution u and any supersolution u of the problem
−u = δ(x)−bq u−p , u=0
u > 0 in Ω, on ∂Ω,
satisfy u(x) c1 δ(x)a
and u(x) c2 δ(x)a
in Ω.
• Any subsolution v and any supersolution v of the problem
−v = δ(x)−ar v −s , v=0
v > 0 in Ω, on ∂Ω,
satisfy v(x) c1 δ(x)b
and v(x) c2 δ(x)b
in Ω.
As before, we now define m1 δ(x)a u(x) M1 δ(x)a A = (u, v) ∈ C(Ω) × C(Ω): m2 δ(x)b v(x) M2 δ(x)b
in Ω , in Ω
where 0 < m1 < 1 < M1 and 0 < m2 < 1 < M2 satisfy (31)–(32). This concludes the proof of Theorem 1.2. 5. Proof of Theorem 1.4 (i) Assume first that the system (1) has a solution (u, v) with u ∈ C 1 (Ω). Then, there exists c > 0 such that u(x) cδ(x) in Ω. Using this fact in the second equation of (1), we derive that v satisfies the elliptic inequality (23) for some c3 > 0. By Corollary 2.5 this entails r < 2. In order to prove that α < 1 we argue by contradiction. Suppose that α 1 and we divide our argument into three cases. Case 1: r + s > 1. Then, α = p + q(2−r) 1+s 1. From Proposition 2.3 we have u(x) cδ(x) in Ω, for some c > 0. Then v satisfies
−v c1 δ(x)−r v −s , u=0
v > 0 in Ω, on ∂Ω,
(37)
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where c1 > 0. Since r < 2, from Proposition 2.6(iii) we find v(x) c2 δ(x) 1+s in Ω, for some c2 > 0. Using this estimate in the first equation of system (1) we deduce
where c3 > 0. Now, if q(2−r) 1+s
−u c3 δ(x)− u=0
q(2−r) 1+s
q(2−r) 1+s
u−p ,
u > 0 in Ω, on ∂Ω,
(38)
2, from Corollary 2.5 the above inequality is impossible. Assume
next that < 2. If α > 1, from (8), (38) and Proposition 2.6(iii) we find u(x) c4 δ(x)τ c4 ϕ1 (x)τ
in Ω,
(39)
where τ=
2−
q(2−r) 1+s
1+p
∈ (0, 1)
and c4 > 0.
Fix x0 ∈ ∂Ω and let n be the outer unit normal vector on ∂Ω at x0 . Using (39) and the fact that 0 < τ < 1 we have ∂u u(x0 + tn) − u(x0 ) (x0 ) = lim t 0 ∂n t c4 lim
t 0
ϕ1 (x0 + tn) − ϕ1 (x0 ) τ −1 ϕ1 (x0 + tn) t
∂ϕ1 (x0 ) lim ϕ1τ −1 (x0 + tn) t 0 ∂n = −∞. = c4
Hence, u ∈ / C 1 (Ω). If α = 1 we proceed in the same manner. From (38) and Proposition 2.6(ii) we deduce 1
u(x) c5 δ(x) log 1+p
A δ(x)
where c5 , c6 > 0. As before, we obtain
1
c6 ϕ1 (x) log 1+p
∂u ∂n (x0 ) = −∞, x0
A ϕ1 (x)
in Ω,
∈ ∂Ω, which contradicts u ∈ C 1 (Ω).
Case 2: r + s < 1. Then, α = p + q 1. As in Case 1, v fulfills (37) and by Proposition 2.6(i) we find v(x) c7 δ(x) in Ω, for some c7 > 0. Thus, u satisfies
−u c8 δ(x)−q u−p , u=0
u>0
in Ω, on ∂Ω,
where c8 > 0. From Corollary 2.5 it follows q < 2. Since α = p + q 1, it follows that u satisfies either the estimate (ii) (if p + q = 1) or the estimate (iii) (if p + q > 1) in Proposition 2.6. Proceeding in the same way as before we derive that the outer unit normal derivative of u on ∂Ω is −∞, which is impossible.
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
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Case 3: r + s = 1. This also yields α = p + q 1. As before v satisfies (37) and by Proposition 2.6(ii) we deduce v(x) c9 δ(x) log
1 1+s
A δ(x)
in Ω,
where c9 > 0. It follows that u satisfies
q
−u c10 δ(x)−q log− 1+s
u=0
A u−p , δ(x)
u > 0 in Ω,
(40)
on ∂Ω,
where c10 > 0. If q − b 2 the above inequality is impossible in the light of Corollary 2.5. Assume next that q − b < 2. If α = p + q > 1, we fix 0 < b < min{q, p + q − 1} and from (40) we have that u satisfies −u c11 δ(x)−(q−b) u−p , u > 0 in Ω, u=0 on ∂Ω, for some c11 > 0. Now, since p + q − b > 1, from Proposition 2.6(iii) we find u(x) c12 δ(x)
2−(q−b) 1+p
in Ω,
where c12 > 0. Since 0 < 2−(q−b) 1+p < 1, we obtain as before that the normal derivative of u on ∂Ω is infinite which is impossible. It remains to consider the case α = p + q = 1, that is, p + q = r + s = 1. First, if q < 1 + s, that is, q = 1 and s = 0, by (40) and Corollary 2.8 we deduce 1+s−q
u(x) c13 δ(x) log (1+p)(1+s)
A δ(x)
in Ω,
for some c13 > 0. Proceeding as before we obtain ∂u ∂n = −∞ on ∂Ω, which is impossible. If q = 1 and s = 0 then we apply Proposition 2.9 to obtain
A in Ω, u(x) c14 δ(x) log log δ(x) where c14 > 0. This also leads us to the same contradiction ∂u ∂n = −∞ on ∂Ω. Thus, we have proved that if the system (1) has a solution (u, v) with u ∈ C 1 (Ω) then α < 1 and r < 2. Conversely, assume now that α < 1 and r < 2. By Theorem 1.2(i) (Cases 1, 2 and 3) there exists a solution (u, v) of (1) such that u(x) cδ(x)
in Ω,
and v(x) cδ(x)
in Ω, if r + s 1,
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or 2−r
v(x) cδ(x) 1+s
in Ω, if r + s > 1,
for some c > 0. Using the above estimates we find −u = u−p v −q Cδ(x)−α
in Ω,
for some C > 0. By Proposition 2.2, we now deduce u ∈ C 1,1−α (Ω). The proof of (ii) is similar. (iii) Assume first that the system (1) has a solution (u, v) with u, v ∈ C 1 (Ω). Then, there exists c > 0 such that v(x) cδ(x) in Ω. Using this estimate in the first equation of (1) we find that −u Cδ(x)−q u−p , u > 0 in Ω, u=0 on ∂Ω, where C is a positive constant. If p + q 1, then we combine the result in Proposition 2.6(ii)– / C 1 (Ω). Thus, p + q < 1 (iii) with the techniques used above to deduce ∂u ∂n = −∞ on ∂Ω, so u ∈ and in a similar way we obtain r + s < 1. Assume now that p + q < 1 and r + s < 1. By Theorem 1.2(i) (Case 3) we have that (1) has a solution (u, v) such that u(x), v(x) cδ(x) in Ω, for some c > 0. This yields −u Cδ(x)−(p+q)
in Ω,
−(r+s)
in Ω,
−v Cδ(x)
where C > 0. Now Proposition 2.2 implies u, v ∈ C 1 (Ω). This concludes the proof. 6. Proof of Theorem 1.5 We shall prove only (i); the case (ii) follows in the same manner. Let (u1 , v1 ) and (u2 , v2 ) be two solutions of system (1). Using Proposition 2.3 there exists c1 > 0 such that ui (x), vi (x) c1 δ(x)
in Ω, i = 1, 2.
(41)
Hence, ui satisfies
−p
−ui c2 δ(x)−q ui , ui = 0
ui > 0 in Ω, on ∂Ω,
for some c2 > 0. By Proposition 2.6(i) and (41) there exists 0 < c < 1 such that 1 cδ(x) ui (x) δ(x) c
in Ω, i = 1, 2.
This means that we can find a constant C > 1 such that Cu1 u2 and Cu2 u1 in Ω.
(42)
M. Ghergu / Journal of Functional Analysis 258 (2010) 3295–3318
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We claim that u1 u2 in Ω. Supposing the contrary, let M = inf{A > 1: Au1 u2 in Ω}. By our assumption, we have M > 1. From Mu1 u2 in Ω, it follows that −s −r −r −s −v2 = u−r 2 v2 M u1 v2
in Ω.
r
Therefore v1 is a solution and M 1+s v2 is a supersolution of
−s −w = u−r 1 w , w=0
w>0
in Ω, on ∂Ω.
By Proposition 2.1 we obtain r
v1 M 1+s v2
in Ω.
The above estimate yields −p −q
qr
−p −q
−u1 = u1 v1 M − 1+s u1 v2
in Ω.
qr
It follows that u2 is a solution and M (1+p)(1+s) u1 is a supersolution of
−q
−w = v2 w −p , w=0
w > 0 in Ω, on ∂Ω.
By Proposition 2.1 we now deduce qr
M (1+p)(1+s) u1 u2
in Ω.
qr Since M > 1 and (1+p)(1+s) < 1, the above inequality contradicts the minimality of M. Hence, u1 u2 in Ω. Similarly we deduce u1 u2 in Ω, so u1 ≡ u2 which also yields v1 ≡ v2 . Therefore, the system has a unique solution. This completes the proof of Theorem 1.4.
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[8] R.H. Fowler, Further studies of Emden’s and similar differential equations, Q. J. Math. (Oxford Ser.) 2 (1931) 259–288. [9] J. Hernández, F.J. Mancebo, J.M. Vega, Positive solutions for singular semilinear elliptic systems, Adv. Differential Equations 13 (2008) 857–880. [10] M. Ghergu, Steady-state solutions for Gierer–Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc. 361 (2009) 3953–3976. [11] M. Ghergu, V. R˘adulescu, On a class of singular Gierer–Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris 344 (2007) 163–168. [12] M. Ghergu, V. R˘adulescu, A singular Gierer–Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 1215–1234. [13] M. Ghergu, V. R˘adulescu, Singular Elliptic Equations: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl., vol. 37, Oxford University Press, 2008. [14] C. Gui, F. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 1021–1029. [15] J.H. Lane, On the theoretical temperature of the sun under hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Amer. J. Sci. 50 (1869) 57–74. [16] M. Naito, H. Usami, Existence of nonoscillatory solutions to second-order elliptic systems of Emden–Fowler type, Indiana Univ. Math. J. 55 (2006) 317–340. [17] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, NJ, 1967. [18] P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49–81. [19] W. Reichel, H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations 161 (2000) 219–243. [20] J. Serrin, H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Semin. Mat. Univ. Modena XLVI (1998) 369–380. [21] J. Serrin, H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differential Integral Equations 9 (1996) 635–653. [22] Ph. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math. 221 (2009) 1409–1427. [23] H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their applications, Math. Ann. 323 (2002) 713–735.
Journal of Functional Analysis 258 (2010) 3319–3346 www.elsevier.com/locate/jfa
Joint extensions in families of contractive commuting operator tuples ✩ Stefan Richter ∗ , Carl Sundberg Department of Mathematics, University of Tennessee, Knoxville, TN 37996, United States Received 15 September 2009; accepted 10 January 2010 Available online 2 February 2010 Communicated by N. Kalton
Abstract In this paper we systematically study extension questions in families of commuting operator tuples that are associated with the unit ball in Cd . © 2010 Elsevier Inc. All rights reserved. Keywords: Extension; Dilation; Spherical contraction; Row contraction; Spherical isometry
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Spherical isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Extremality of the adjoint of the d-shift . . . . . . . . . . . . . . . . 4. A proposition about tuples of the type S ∗ ⊕ V . . . . . . . . . . . 5. Finite rank extensions of spherical contractions and isometries 6. Extensions of spherical contractions . . . . . . . . . . . . . . . . . . 7. Rank one extensions of row contractions . . . . . . . . . . . . . . . 8. Extensions of row contractions . . . . . . . . . . . . . . . . . . . . . . 9. An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Work of the authors was supported by the National Science Foundation, grant DMS-0556051.
* Corresponding author.
E-mail addresses:
[email protected] (S. Richter),
[email protected] (C. Sundberg). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.011
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1. Introduction It is fair to say that the Sz.-Nagy dilation theorem is of central importance for the theory of contraction operators on Hilbert spaces. One version of this theorem states that every contraction on a Hilbert space can be extended to a co-isometric operator acting on a larger Hilbert space. Because of the known structure of the co-isometric operators, this means that one can use the function theory of the Hardy space of the unit disc to study arbitrary contractions. Partial extensions of Sz.-Nagy’s theorem are available for the study of tuples of operators. The best known result is Ando’s theorem which says that for any pair of commuting contraction operators S and T acting on a Hilbert space H, there is a pair U, V of commuting co-isometric operators acting on a larger space K ⊇ H such that U extends S and V extends T [2]. It is also known that a direct analogue of Ando’s theorem fails for three or more commuting contractions. Ando’s theorem relates the study of commuting contractions to function theory on the bidisc, while it remains an open problem to find an effective model for three or more commuting contractions. The spherical contractions and the row contractions are collections of operator tuples which have been studied recently and which can be associated with function theory in the unit ball of Cd . A convenient way to approach many such theorems is through J. Agler’s model theory (see [1]). In this note we will present some examples of this model theory for the multivariable context. The following definition is from [1]. We will assume that all our Hilbert spaces are separable. Definition 1.1. Let d 1. A family is a collection F of d-tuples T = (T1 , . . . , Td ) of Hilbert space operators, Ti ∈ B(H), such that: (a) F is bounded, i.e. there exists c > 0 such that for all T = (T1 , . . . , Td ) ∈ F we have Ti c for all i = 1, . . . , d, (b) F is preserved under restrictions to invariant subspaces, i.e. whenever T ∈ F and M ⊆ H such that Ti M ⊆ M for all i, then T |M ∈ F , (c) F is preserved under direct sums, i.e. whenever Tn ∈ F is a sequence of tuples, then n Tn ∈ F , (d) F is preserved under unital ∗-representations, i.e. if π : B(H) → B(K) is a ∗-homomorphism with π(I ) = I and if T = (T1 , . . . , Td ) ∈ F , then π(T ) = (π(T1 ), . . . , π(Td )) ∈ F . For d = 1 some examples are given by the families of contractions, isometries, subnormal contractions, and hyponormal contractions. For d > 1 we will be interested only in families which consist of commuting tuples of operators. The family of commuting contractions has already been mentioned. A tuple (T1 , . . . , Td ) is called a spherical isometry if di=1 Ti x2 = x2 for every x ∈ H. It is immediately clear that the collection of spherical isometries satisfies (a), (b) and (c) of Definition 1.1. Furthermore (d) follows as well, because (T1 , . . . , Td ) is a spherical isometry if and only if di=1 Ti∗ Ti = I . We will write Fsi to denote the family of commuting spherical isometries. The spherical contractions Fsc are those commuting d-tuples T = (T1 , . . ∗. , Td ) of Hilbert consists of the space operators satisfying dj =1 Tj∗ Tj I . The collection of adjoint tuples Fsc row contractions Frc . They satisfy
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
d 2 d Tj x j xj 2 j =1
3321
for all x1 , . . . , xd
j =1
in the Hilbert space. As for the spherical isometries it is easy to check that both Fsc and Frc = ∗ form a family. Fsc Suppose T is an operator tuple acting on a Hilbert space H and R is a tuple acting on K. We will write R T if R is an extension of T , i.e. if H ⊆ K is a subspace which is invariant for each Ri , and if Ti = Ri |H for all i. In this case we will call dim K H the rank of the extension. If R = T ⊕ B for some operator tuple B, then R is called a trivial extension of T . Definition 1.2. Let F be a family. An operator tuple T ∈ F acting on H is called an extremal for F if T has only trivial extensions in F , i.e. whenever R ∈ F satisfies R T , then H reduces R. We shall write ext(F ) for the extremals of the family F . Theorem (J. Agler). If F is a family and if T ∈ F , then T can be extended to a tuple S ∈ ext(F ). The theorem is stated for families of single operators in [1], but it is mentioned there that the result also holds in the multivariable context. For a proof we refer the reader to [14] or the unpublished note [6]. Thus it is an important question to identify the extremals of families of interest. We note that it is easy to see that the extremals for the family of contractions are the co-isometric operators, the extremals for the isometric operators are the unitary operators, and extremals for the subnormal contractions are the normal contractions. It is unknown what the extremals for the hyponormal contractions are (see [13]). Next we discuss some examples for d > 1. Ando’s theorem can be used to show that the pairs of two commuting co-isometric operators are extremal for the pairs of commuting contractions. Alternatively, one can use a one-step extension as in the proof of the commutant lifting theorem (see [20, p. 65]) to identify the extremals. In this case Ando’s theorem follows from the above theorem of Agler’s. It is an open problem to identify the extremals for the d-tuples of commuting contractions if d > 2. On the other hand the extremals for the family of commuting isometries are easily identified as the tuples of commuting unitary operators. The resulting extension theorem is due to Ito [17] and Brehmer [9]. In this paper we will discuss extremals of families that are associated with the unit ball in Cd . A particular emphasis is placed on identifying which spatial and spectral properties of an operator tuple allow nontrivial extensions. A tuple U = (U1 , . . . , Ud ) of commuting operators is called spherical unitary if di=1 Ui∗ Ui = I and each Ui is a normal operator. Our first theorem is the following. Theorem 1.3. Let Fsi be the family of commuting spherical isometries, then ext(Fsi ) equals the collection of commuting spherical unitaries. The resulting extension theorem says that commuting spherical isometries are jointly subnormal and it is due to Athavale [7].
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We now turn to spherical and row contractions. An important example of a row contraction is the d-shift Mz = (Mz1 , . . . , Mzd ) acting on the Drury–Arveson space Hd2 . Hd2 is the reproducing kernel Hilbert space defined by the kernel kλ (z) =
1 , 1 − z, λ
λ, z ∈ Bd , z, λ =
d
zi λi .
i=1
Hd2 consists of analytic functions in Bd , and for d > 1 it is properly contained in the classical 1 Hardy space H 2 (∂Bd ), which has reproducing kernel (1− z,λ ) d. Since Mz∗i kλ = λi kλ it follows that
d
Mzi Mz∗i kλ
i=1
(z) =
z, λ = kλ (z) − 1. 1 − z, λ
This implies that d
Mzi Mz∗i = I − 1 ⊗ 1 I,
(1.1)
i=1
thus Mz∗ is a spherical contraction and Mz is a row contraction. We say that S is a direct sum of d-shifts if S = (S1 , . . . , Sd ), Si = Mzi ⊗ I ∈ B(Hd2 ⊗ C) for some Hilbert space C. Theorem 1.4. Let Fsc be the family of commuting spherical contractions, and let T = (T1 , . . . , Td ) be a commuting operator tuple. Then the following are equivalent: (i) T ∈ ext(Fsc ), (ii) T = S ∗ ⊕ U , where U is spherical unitary and S is a direct sum of d-shifts, d Ti∗ Ti = P = a projection, (iii) (a) i=1 d ∗ (b) i=1 Ti Ti I , (c) if x1 , . . . , xd ∈ H with Ti xj = Tj xi , then there is an x ∈ H with xi = Ti x for all i. When we write T = S ∗ ⊕ U , we want to include the possibility that one of the summands is absent. Note that (iii)(c) says that the Koszul complex for T is exact at Λ1 (H) (Section 3 contains a short summary of elementary facts about the Koszul complex). The resulting extension theorem (i.e. that any R ∈ Fsc has an extension T of the type as in (ii)) had been known and is due to Müller–Vasilescu [18] and to Arveson [4]. Arveson also proved that the adjoint of the d-shift is an extremal spherical contraction (see [4, pp. 205, 206]). Among other things his proofs are based on his earlier results [3] and an analysis of the C ∗ -algebra generated by the d-shift and the identity operator. We note that the extremality of S ∗ also follows directly from Agler’s theorem once the implication (i) ⇒ (ii) of Theorem 1.4 has been established. Indeed, by Agler’s theorem the zero tuple 0 = (0, . . . , 0) acting on a nonzero space extends to an extremal spherical contraction. By (i) ⇒ (ii) there must be an extremal of the type S ∗ ⊕ U
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such that 0 = S ∗ ⊕ U |M where S is a direct sum of d-shifts, U is a spherical unitary tuple, and M is invariant for S ∗ ⊕ U . If the direct summand S ∗ were absent, then 0 = U |M would have to be a spherical isometry, which is absurd. Thus S ∗ ⊕ U is extremal and definitely has a d-shift as a direct summand. Now it is easy to verify that if X ⊕ Y is extremal for a family F , then both X and Y have to be extremal for F also. Hence the adjoint of the d-shift must be extremal for Fsc . For this paper we decided to present yet another proof of the extremality of S ∗ , this one based on spatial properties of S = Mz as it acts on Hd2 (Section 3). When we apply Theorem 1.4 to the tuple of adjoints we obtain the following corollary. Corollary 1.5 (Wold-decomposition). A d-tuple of commuting operators is of the form T = S ⊕U if and only if d Ti Ti∗ is a projection, (a) i=1 d ∗ (b) i=1 Ti Ti I , and d (c) whenever x1 , . . . , xd ∈ H with there is an antisymmetric matrix i=1 Ti xi = 0, then {yij }1i,j d with entries yij ∈ H such that xi = dj =1 Tj yij for each i (i.e. the Koszul complex for T is exact at Λd−1 (H)). We will give a short proof at the end of Section 6. Note that when d = 1 and condition (a) is satisfied, then T is a partial isometry. In this case each of the conditions (b) or (c) implies that T is 1–1 and thus T must be an isometry. For d > 1 neither conditions (a) and (b) nor conditions (a) and (c) alone will imply that T = S ⊕ U . In fact if T = (Mz∗ , H 2 (∂Bd )) then T ∗ is a spherical isometry, thus it satisfies (a). Furthermore, it is well known that the Koszul complex for (Mz∗ , H 2 (∂Bd )) is exact at all stages except at the first one, so (c) is satisfied provided d > 1 (see e.g. Section 2 of [15]), but (b) is not satisfied. In order to exhibit an example which satisfies (a) and (b) but not (c) we let M0 = {f ∈ Hd2 : f (0) = 0} and T = S|M0 , where S is the d-shift acting on Hd2 . It is clear that T satis fies (b), because S does. Furthermore, di=1 Si Si∗ = P = I − 1 ⊗ 1 is the projection from Hd2 d onto M0 (see (1.1)). Then i=1 Ti Ti∗ = di=1 P Si P Si∗ P = P ( di=1 Si Si∗ − Si (1 ⊗ 1)Si∗ )P = P − di=1 zi ⊗ zi and this is a projection, because zi = 1 for all i and zi ⊥ zj for all i = j (see Eq. (1.2) below). Thus, T satisfies (a). We will now show that for d > 1 Tdoes not satisfy (c). Let f1 (z) = z2 , f2 (z) = −z1 , and f3 = · · · = fd = 0. Then fi ∈ M0 and di=1 Ti fi = 0. If z2 = f1 (z) = dj =1 zj g1j for some g1j ∈ M0 , then we take a partial derivative with respect to z2 and obtain 1 = g12 (z) + dj =1 zj ∂z∂ 2 g1j (z). Evaluating at z = 0 we conclude 1 = g12 (0), but this contradicts g12 ∈ M0 . For the family of row contractions we have partial results. Theorem 1.6. Let Frc be the family of commuting row contractions. Let T ∈ Frc and write D∗ = (I − di=1 Ti Ti∗ )1/2 . (i) If D∗ = 0, then T ∈ ext(Frc ). / ext(Frc ). (ii) If D∗ is onto, then T ∈ / ext(Frc ) if and only if there are x1 , . . . , xd ∈ di=1 ker Ti∗ (iii) If D∗ is a projection, then T ∈ d with i=1 xi 2 > 0 and Ti xj = Tj xi for all i, j .
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(iv) If D∗ has rank one, i.e. if D∗ = u ⊗ u for some u = 0, then T ∈ ext(Frc ) if and only if dim span{u, T1 u, . . . , Td u} 3. If d = 1, then part (i) of Theorem 1.6 describes all extremals (the co-isometric operators). For d > 1 the d-shift is an example of an extremal with D∗ = 0. For the d-shift one verifies that D∗ is a projection of rank 1 (see Eq. (1.1)), so its extremality can be derived either from part (iii) or part (iv) of Theorem 1.6. We will see that in all of the above cases, when T is not extremal, then T actually has a nontrivial rank 1 extension in Frc . If S = (Mz , Hd2 ) is the d-shift, and if M Hd2 is invariant for S, then T = PM⊥ S|M⊥ ∈ Frc and D∗ has rank 1. Because of this one can use Theorem 1.6 to verify the following corollary (see Corollary 8.4). Corollary 1.7. Let Frc be the family of commuting row contractions. If M = Hd2 is an invariant subspace for the d-shift S = (Mz , Hd2 ), and if L= a+
d
bi zi : a, b1 , . . . , bd ∈ C
i=1
denotes the collection of polynomials of degree less than or equal to one, then T = PM⊥ S|M⊥ ∈ ext(Frc ) if and only if dim M ∩ L < d − 1. Under the hypothesis of the corollary one easily checks that D∗2 = ϕ ⊗ ϕ, where ϕ = PM⊥ 1 (see the proof of Corollary 8.4). Thus D∗ is a projection if and only if 1 ∈ M⊥ , and the corollary can be used to exhibit many examples of extremal row contractions whose defect operators are not projections. For example, if d = 2 and λi = (λi1 , λi2 ), i = 1, 2, 3 are three distinct points in B2 , then we can let M = f ∈ H22 : f (λ1 ) = f (λ2 ) = f (λ3 ) = 0 . In this case T = (T1 , T2 ) acts on the 3-dimensional space M⊥ = span{kλ1 , kλ2 , kλ3 }, and by the corollary T is extremal if and only if M ∩ L = {0}. From this one deduces with a little bit of elementary algebra that T is extremal if and only if (λ31 − λ11 )(λ22 − λ12 ) = (λ21 − λ11 )(λ32 − λ12 ). Hence there are extremal row contractions on finite dimensional spaces that are not spherical unitaries. We also note that the above examples of extremals where the defect operator is not a projection show that part (iii) of Theorem 1.6 does not cover all extremals. This is in contrast to the family Fsc where for all extremals the defect operator D = (I − di=1 Ti∗ Ti )1/2 must be a projection (see Theorem 1.4). If d = 1 and if F is either the family of contractions or the family of isometries, then any nonextremal operator T ∈ F has a nontrivial rank one extension in F . This is well known and easy to see (compare Lemma 7.2). If d > 1, then for each of the families Fsc , Fsi and Frc there
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is a difference between extremals and operator tuples that allow nontrivial finite rank or rank one extensions. We shall show in Corollary 5.4 that a spherical isometry V has no nontrivial rank one extensions in Fsi if and only if the Koszul complex for V −b is exact at Λ1 for all b ∈ Bd . We will review definitions and elementary properties of the Koszul complex in Section 3. Furthermore, in Theorem 6.1 we will show that if T ∈ Fsc , then T has no nontrivial finite rank extensions if and only if T has no nontrivial rank one extensions and this happens if and only if T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry with no nontrivial rank one extension. For the row contractions we only have the following technical condition, and we note that all extension results of Theorem 1.6 follow from this result, i.e. if either D∗ is onto, or if T ∈ Frc is nonextremal and D∗ is a projection or a rank one operator, then T has a nontrivial rank 1 extension in Frc . Theorem 1.8. Let Frc be the family of commuting row contractions, and let T ∈ Frc . The following are equivalent: (a) T has a nontrivial rank 1 extension in Frc , (b) T has a nontrivial finite rank extension in Frc , (c) there are a1 , . . . , ad ∈ ran D∗ , i ai > 0, and b = (b1 , . . . , bd ) ∈ Bd such that (Ti − bi )aj = (Tj − bj )ai for all i, j . In Section 9 we will present an example of a nonextremal commuting row contraction which has no nontrivial finite dimensional extensions. The remainder of the paper is structured as follows. In Section 2 we will prove Theorem 1.3 and we will see that spherical unitaries are extremal spherical contractions. Section 3 contains a proof that the adjoint of the d-shift is an extremal among the spherical contractions. A basic proposition about spherical contractions of the form S ∗ ⊕ V , where S is a sum of d-shifts and V is a spherical isometry will be presented and proved in Section 4. Section 5 contains a theorem characterizing the spherical isometries that have nontrivial rank one extensions (Corollary 5.4) and it also has some preliminary results about rank one and finite rank extensions of spherical contractions. Theorem 6.1 characterizes spherical contractions with nontrivial finite rank extensions and Corollaries 6.2 and 6.3 are Theorem 1.4 and Corollary 1.5. In Section 7 we give our results about finite rank extensions of row contractions and in Section 8 we present our main results about extremals of Frc . At various places throughout the paper we will use multinomial notation. If α ∈ Nd0 , then
|α|! = α! . If z ∈ Cd and if α = (α1 , . . . , αd ), |α| = α1 + · · · + αd , α! = α1 · . . . · αd !, and |α| α α α α α T = (T1 , . . . , Td ) ∈ B(H)d , then zα = z1 1 · . . . · zd d and T α = T1 1 . . . Td d . Furthermore, we will use ej = (0, . . . , 0, 1, 0, . . . , 0) where the 1 is in the j th spot. The reproducing kernel for the Drury–Arveson space Hd2 satisfies kλ (z) =
∞ ∞ |α| 1 n α α z λ = zα λα . =
z, λ n = α α 1 − z, λ d n=0
n=0 |α|=n
α∈N0
Since we also have |α| |β| kλ (z) = kλ , kz = z β λα w α , w β H 2 α β d α∈Nd0 β∈Nd0
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it follows that α β 1 w , w H 2 = δαβ |α| , d
where δαβ = 0 if α = β,
From this one deduces that for f ∈ Hol(Bd ), f (z) = f 2H 2 = d
d
i=1 zi z
α 2
=
α∈Nd0
fˆ(α)zα one has
|fˆ(α)|2 α! fˆ(α)2 .
|α| = |α|! d d α
α∈N0
If α ∈ N0 d , then
δαβ = 1 if α = β.
α
α! |α|+d |α|! |α|+1 , d
(1.2)
α∈N0
hence it follows that
zi f 2 f 2
(1.3)
i=1
for all f ∈ Hd2 . Furthermore, one calculates that for α ∈ Nd0 and 1 i d Mz∗i zα =
αi α−ei z |α|
whenever αi > 0,
(1.4)
and Mz∗i zα = 0 if αi = 0. 2. Spherical isometries In this section we will prove Theorem 1.3 and part of the proof of (ii) ⇒ (i) of Theorem 1.4. The fact that extremals of the spherical isometries must be jointly normal follows easily from the arguments of Attele and Lubin [8], who presented an alternate proof of Athavale’s theorem. In fact, let T = (T1 , . . . , Td ) be a commuting spherical isometry acting on H and assume that T1 is not normal. We must show that T is not extremal, i.e. we have to construct a commuting spherical isometry S that extends T nontrivially. By Corollary 6 of [8] T1 is a subnormal contraction. Thus we can let S1 ∈ B(K) be the minimal normal extension of T1 . Since we assumed that T1 is not normal, it is clear that any extension of T of the form S = (S1 , S2 , . . . , Sd ) will be nontrivial. In order to define S2 , . . . , Sd we use the standard extensions n n k ∗ Si S1 xk = S1∗ k Ti xk , i = 2, . . . , d, x0 , . . . , xn ∈ H, k=0
k=0
see the proof of Proposition 7 of [8]. Since S1 is normal it is easy to verify that n 2 n 2 d S1∗ k Ti xk = S1∗ k xk . i=1 k=0
k=0
This implies that S2 , . . . , Sd are well defined and extend to K and S = (S1 , S2 , . . . , Sd ) forms a spherical isometry. Finally, we see that for all 1 i, j d
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
Sj Si
n
S1∗ k xk = Sj
k=0
n
3327
S1∗ k Ti xk
k=0
=
n
S1∗ k Tj Ti xk
k=0
= Si Sj
n
S1∗ k xk .
k=0
Thus S forms a tuple of commuting operators, and this proves that the extremals of the spherical isometries must be commuting normals. For later reference we make some simple observations about extremals. Lemma 2.1. Let F and G be families. (a) Let U ∈ ext(F ) and V ∈ F . If R ∈ F with R U ⊕ V , then R = U ⊕ R for some R V , R ∈ F . (b) Finite or infinite direct sums of extremals are extremal. (c) If F ⊆ G, then ext(G) ∩ F ⊆ ext(F ). Proof. (a) is obvious and it easily implies (b) for finite direct sums. In order to prove (b) for infinite direct sums let Un ∈ ext(F ) ∩ B(Hn )d , V = U1 ⊕ U2 ⊕ · · · and R ∈ F ∩ B(K)d with R V . For n ∈ N we let Pn be the projection from K onto H1 ⊕ · · · ⊕ Hn . By the finite case we have Ri Pn = Pn Ri for all i and n. The sequence Pn converges in the strong operator topology to P , the projection from K onto H1 ⊕ H2 ⊕ · · · . It follows that P commutes with R. Hence V must be extremal. (c) is immediate. 2 Theorem 2.2. Commuting spherical unitary tuples are extremal for the families of commuting spherical contractions and commuting spherical isometries. Proof. Let U = (U1 , . . . , Ud ) be a commuting spherical unitary tuple. By Lemma 2.1(c) it suffices to show that U is extremal for the commuting spherical contractions. Thus let S U be a commuting spherical contraction. Then for each 1 i d we have Si =
Ui 0
Ai Bi
∈ B(H ⊕ K)
with Ui Aj + Ai Bj = Uj Ai + Aj Bi
(2.1)
for all 1 i, j d and d i=1
Si∗ Si
=
d
I
∗ i=1 Ai Ui
d
∗ i=1 Ui Ai ∗ ∗ i=1 Ai Ai + Bi Bi
d
I 0
0 . I
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It follows that
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
d
∗ i=1 Ui Ai
= 0 and d
A∗i Ai + Bi∗ Bi I.
(2.2)
i=1
Let j ∈ {1, . . . , d}. We shall establish Lemma 2.2 by showing that Aj = 0. By hypothesis each Ui is normal and Ui Uj = Uj Ui . Hence it follows from Fuglede’s theorem [19] that Ui∗ Uj = Uj Ui∗ . We now apply Ui∗ on the left in Eq. (2.1), sum in i, and obtain d
Ui∗ Ui Aj +
i=1
d
Ui∗ Ai Bj =
i=1
d
Ui∗ Uj Ai +
i=1
d
Ui∗ Aj Bi .
i=1
Since di=1 Ui∗ Ui = I and di=1 Ui∗ Ai = 0 this implies Aj = di=1 Ui∗ Aj Bi . Since U is a spherical contraction it follows that U ∗ = (U1∗ , . . . , Ud∗ ) is a row contraction. Hence for x ∈ K, x 1 we have d 2 d Aj x2 = Ui∗ Aj Bi x Aj Bi x2 i=1
Aj 2
i=1
d
Bi x2 .
i=1
By Eq. (2.2) this implies Aj x Aj 2
2
x − 2
d
Ai x
2
Aj 2 1 − Aj x2 .
i=1
We now rearrange the terms to get Aj x2 (1 + Aj 2 ) Aj 2 and after taking the sup over x 1 we obtain Aj 2 (1 + Aj 2 ) Aj 2 which implies that Aj = 0. 2 3. Extremality of the adjoint of the d-shift Let T = (T1 , . . . , Td ) be a commuting tuple of operators on a Hilbert space H. We will now define the Koszul complex of T . We will follow [5]. For more information of a general type on the Koszul complex and its relationship to invertible and Fredholm tuples, the reader is also referred to [10] and [21]. Let Λ = Λ[e] = Λd [e] be the exterior algebra generated by the d symbols e1 , . . . , ed , along with the identity e0 defined by e0 ∧ ξ = ξ for all ξ . Then Λ is the algebra of forms in e1 , . . . , ed with complex coefficients, subject to the anticommutative property ei ∧ej +ej ∧ei = 0 (1 i, j d). In fact, we can make Λ into a 2d -dimensional Hilbert space with orthonormal basis {e0 } ∪ ei1 ∧ · · · ∧ eik ij ∈ {1, . . . , d}, i1 < i2 < · · · < ik .
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
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For each i = 0, 1, . . . , d let Ei : Λ → Λ be given by Ei ξ = ei ∧ ξ . E0 is thus the identity on Λ. For i = 1, . . . , d the Ei are called the creation operators and they satisfy the following anticommutation relations Ei Ej + Ej Ei = 0 and Ei∗ Ej + Ej Ei∗ = δij E0 . Let Λ(H) := H ⊗C Λ and define ∂T : Λ(H) → Λ(H) by ∂T :=
d
Ti ⊗ E i .
i=1
It follows easily from the anticommutation relationships that ∂T2 = 0. Thus, the Koszul complex of the tuple T can be defined by ∂T ,0
∂T ,1
∂T ,d−1
K(T ) : 0 → Λ0 (H) −−→ Λ1 (H) −−→ · · · −−−−→ Λd (H) → 0 where Λp (H) is the collection of p forms in Λ(H) and ∂T ,p := ∂T |Λp (H). For purposes of notation we also define Λ−1 (H) = 0 and ∂T ,−1 and ∂T ,d to be the zero maps at the two ends of the complex. The identity ∂T2 = 0 implies that for each p = 0, 1, . . . , d ran ∂T ,p−1 ⊆ ker ∂T ,p and ran ∂T∗ ,p ⊆ ker ∂T∗ ,p−1 , and one says that the Koszul complex K(T ) is exact at Λp (H), if ran ∂T ,p−1 = ker ∂T ,p . In particular, if K(T ) is exact at Λp (H), then ran ∂T ,p−1 must be closed, hence ∂T ,p−1 ∂T∗ ,p−1 is 1–1 and onto when restricted to ran ∂T ,p−1 = (ker ∂T∗ ,p−1 )⊥ . Furthermore, in this case one also has that ran ∂T∗ ,p is dense in ker ∂T∗ ,p−1 . It follows that the operator DT ,p : Λp (H) → Λp (H),
DT ,p = ∂T ,p−1 ∂T∗ ,p−1 + ∂T∗ ,p ∂T ,p
(3.1)
is 1–1 and has dense range whenever K(T ) is exact at Λp (H), and DT ,p is invertible if it is also known that ran ∂T ,p , or what is the same, ran ∂T∗ ,p is closed. In order to relate properties of the Koszul complex for T with the Koszul complex for T ∗ we define the Hodge ∗-operator (see [16] for more information on this topic). For p = 0, 1, . . . , d we have dim Λp = dim Λd−p and ∗ establishes a conjugate linear isomorphism between Λp and Λd−p that is compatible with ∂. Indeed, ∗(ei1 ∧ ei2 ∧ · · · ∧ eip ) = (−1)ε ej1 ∧ ej2 ∧ · · · ∧ ejd−p where {ei1 , ei1 , . . . , eip , ej1 , . . . , ejd−p } = {e1 , . . . , ed } and ε ∈ {0, 1} is chosen so that η ∧ ∗ω =
η, ω e1 ∧ · · · ∧ ed for all p-forms η, ω ∈ Λp . If η ∈ Λp−1 and ω ∈ Λp , then
η ∧ ∗Ei∗ ω = η, Ei∗ ω e1 ∧ · · · ∧ ed = Ei η, ω e1 ∧ · · · ∧ ed = Ei η ∧ ∗ω = ei ∧ η ∧ ∗ω = (−1)p−1 η ∧ ei ∧ ∗ω = (−1)p−1 η ∧ Ei (∗ω).
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Thus for x ∈ H and ω ∈ Λp we have (I ⊗ ∗)∂T∗ ,p−1 (x ⊗ ω) =
d
Ti∗ x ⊗ ∗Ei∗ ω
i=1
= (−1)p−1
d
Ti∗ x ⊗ Ei (∗ω)
i=1
= (−1)p−1 ∂T ∗ ,d−p (I ⊗ ∗)(x ⊗ ω). Now note that elementary functional analysis results imply that for any p ran ∂T ,p is closed if and only if ran ∂T∗ ,p is closed, and by the above this happens if and only if ran ∂T ∗ ,d−(p+1) is closed. Thus, if it is known for some p that ran ∂T ,p is closed and K(T ) is exact at Λp , then K(T ∗ ) is exact at Λd−p . It is known that if α > 0 and Kα is the Hilbert space of analytic functions on Bd with reproducing kernel kλ (z) = (1 − z, λ )−α , then the Koszul complex for S = (Mz , Kα ) is exact at all stages p = 0, . . . , d − 1 and at the last stage we have dim ker ∂S,d / ran ∂S,d−1 = 1 (see Proposition 2.6 of [15]). Thus, ran ∂S,p is closed for all p. From this and the above remarks about the Hodge ∗-operator it follows that for T = S ∗ we have ran ∂T ,p is closed for all p and K(T ) is exact at Λp (H) for each p 1. In the following let T be a commuting d-tuple of operators on H. We will later take T so that (T ∗ , H) = (Mz , Hd2 ) is the d-shift. Suppose that we have an extension R ∈ B(H ⊕ K)d of T . Then Ti Ai . Ri = 0 Bi Define ∂T : Λ(H) → Λ(H),
∂T =
d
Ti ⊗ E i ,
i=1
∂A : Λ(K) → Λ(H),
∂A =
d
Ai ⊗ Ei ,
i=1
∂B : Λ(K) → Λ(K),
∂B =
d i=1
Lemma 3.1. If R is a tuple of commuting operators, then ∂T ∂A + ∂A ∂B = 0. Proof. Since Λ(H ⊕ K) = Λ(H) ⊕ Λ(K) we can write ∂R =
∂T 0
∂A ∂B
.
Bi ⊗ Ei .
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
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The lemma follows easily, because the expression equals the (1, 2)-entry of the matrix for ∂R2 = 0. 2 Proposition 3.2. Let d 2 and let T = (T1 , . . . , Td ) ∈ B(H)d be a commuting tuple of operators which is graded in the sense that there is a decomposition of H as a direct sum of mutually orthogonal subspaces, H = H0 ⊕ H1 ⊕ · · · such that Tj (H0 ) = (0) and Tj (Hn ) ⊆ Hn−1 for all n 1 and all 1 j d. Assume that the Koszul complex K(T ) is exact at Λp (H) for p = 1 and p = 2. If R ∈ B(H ⊕ K)d is a commuting extension of T of the form Ri = and if
d
∗ j =1 Aj Tj
Ti 0
Ai Bi
,
i = 1, . . . , d,
= 0, then Ai = 0 for all i = 1, . . . , d.
Proof. We start by noting that in terms of the Koszul complex the hypothesis dj =1 A∗j Tj = 0 ∗ ∂ ∗ can be restated as ∂A,0 T ,0 = 0 and that we have to show that ∂A,0 = 0. p p+1 Since Tj (Hn ) ⊆ Hn−1 we see that ∂T ,p (Λ (Hn )) ⊆ Λ (Hn−1 ) for all n 1. Furthermore, one easily checks that for all n 0 and all 1 j d we have Tj∗ (Hn ) ⊆ Hn+1 . Thus for each p the selfadjoint operators ∂T ,p−1 ∂T∗ ,p−1 and DT ,p (see Eq. (3.1)) leave Λp (Hn ) invariant, and the hypothesis implies that
DT ,1 Λ1 (Hn ) = Λ1 (Hn )
(3.2)
and that
DT ,2 Λ2 (Hn )
is dense in Λ2 (Hn )
(3.3)
for each n 0. Define C = ∂T ,1 ∂A,0 , then ∗ C ∗ ∂T ,1 = ∂A,0 DT ,1 ,
(3.4)
∗ ∂ ∗ because ∂A,0 T ,0 ∂T ,0 = 0. We also have
∗ ∗ ∂T∗ ,1 ∂T ,1 ∂T∗ ,1 + ∂T∗ ,2 ∂T ,2 = ∂A,0 ∂T∗ ,1 ∂T ,1 ∂T∗ ,1 C ∗ DT ,2 = ∂A,0 ∗ ∗ = −∂B,0 ∂A,1 ∂T ,1 ∂T∗ ,1 ,
(3.5)
by Lemma 3.1 and because ∂T∗ ,1 ∂T∗ ,2 = 0. We shall now show inductively that
1 ∗ Λ (Hn ) = (0) ∂A,0
and C ∗ Λ2 (Hn ) = (0)
for each n 0.
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We start by applying (3.4) to Λ1 (H0 ). Since ∂T ,1 (Λ1 (H0 )) = 0 and DT ,1 (Λ1 (H0 )) = ∗ (Λ1 (H )) = 0. This implies that for each i we have A∗ (H ) = 0, thus Λ1 (H0 ) we have ∂A,0 0 0 i ∗ ∂ ∗ 2 ∗ ∂A,1 T ,1 ∂T ,1 (Λ (H0 )) = 0. In light of (3.5) and (3.3) this means that C = 0 on a dense subset of Λ2 (H0 ), hence C ∗ (Λ2 (H0 )) = (0). ∗ (Λ1 (H )) = (0) and C ∗ (Λ2 (H )) = 0. Then Next suppose that for some n 0 we have ∂A,0 n n 1 2 since ∂T ,1 (Λ (Hn+1 )) ⊆ (Λ (Hn )) we can use (3.4) and the induction hypothesis to see that ∗ D ∗ ∗ 1 1 0 = ∂A,0 T ,1 (Λ (Hn+1 )) = ∂A,0 (Λ (Hn+1 )). Thus, for each i we have Ai (Hn+1 ) = 0 and so
∗ ∂T ,1 ∂T∗ ,1 Λ2 (Hn+1 ) = 0. ∂A,1 In light of (3.5) and (3.3) this means that C ∗ = 0 on a dense subset of Λ2 (Hn+1 ), hence C ∗ (Λ2 (Hn+1 )) = (0). 2 Theorem 3.3. If S = (Mz , Hd2 ), then S ∗ is extremal for the family of spherical contractions. Proof. Set T = S ∗ and let R ∈ B(Hd2 ⊕ K)d be a commuting spherical contraction which extends T . Then R is of the form R = (R1 , . . . , Rd ), Ti Ai . Ri = 0 Bi We shall use Proposition 3.2 to show that each Ai equals 0. To this end let Hn be the homogeneous polynomials of degree n, so that Hd2 = H0 ⊕ H1 ⊕ · · · , Tj (H0 ) = (0) and Tj (Hn ) ⊆ Hn−1 for all n 1 and all 1 j d (see (1.4)). We noted earlier in this section that the Koszul com∗ plex d K(T∗ ) = K(S ) is exact at stages 1 and 2. Thus by Proposition 3.2 it suffices to show that j =1 Aj Tj = 0. The condition I − di=1 Ri∗ Ri 0 implies that for all f ∈ Hd2 and y ∈ K we have f − 2
d i=1
Ti f − 2 Re 2
d
A∗i Ti f, y
+ y2 −
i=1
d
Ai y2 + Bi y2 0.
(3.6)
i=1
Now we recall that I − di=1 Ti∗ Ti = I − di=1 Si Si∗ equals the projection onto H0 (the con stants, see Eq. (1.1)). Thus, if f ⊥ H0 , then (3.6) implies that di=1 A∗i Ti f = di=1 A∗i Si∗ f = 0. If f ∈ H0 , then Si∗ f = 0 for all 1 i d, hence di=1 A∗i Ti = 0. 2 4. A proposition about tuples of the type S ∗ ⊕ V In this section we shall prove the following proposition which will easily imply one of the equivalent statements of Theorem 1.4. Proposition 4.1. Let T ∈ B(H)d be a commuting operator tuple which satisfies the following two conditions: d ∗ (a) i=1 Ti Ti is a projection, and (b) if x1 , . . . , xd ∈ H with Ti xj = Tj xi for all i, j , then there is an x ∈ H with xi = Ti x for all i.
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
3333
Then T is unitarily equivalent to S ∗ ⊕ V , where S is a direct sum of d-shifts, and V is a spherical isometry. Note that for d = 1 condition (b) says that T is surjective, while condition (a) implies that T is a partial isometry. Thus T must be a co-isometry, and if T ∗ = S ⊕ V ∗ , then V ∗ must be isometric. This means that for d = 1 the operator V in the proposition is automatically unitary. If d > 1, then the operator tuple T = (Mz , H 2 (∂Bd )) provides an example of a d-tuple that satisfies conditions (a) and (b) of the proposition, but is not a spherical unitary tuple. Let E0 = di=1 ker Ti . Inductively define a sequence of positive operators by
P0 = I
and PN +1 =
d
Ti∗ PN Ti
for N = 0, 1, . . . .
(4.1)
i=1
One verifies that for N 1 PN =
N T ∗α T α . α
|α|=N
Hence it follows that ker PN = |α|=N ker T α , and E0 = ker P1 . Note that for N 1 we have PN − PN +1 = di=1 Ti∗ (PN −1 − PN )Ti . Hence part (a) of the hypothesis and an induction argument imply that {PN }N ∈N is a nonincreasing sequence of positive operators which thus converges strongly to a positive operator P . Our first step will be to show that P and each PN are projections, and that Ti P = P Ti for all 1 i d. This means that M = ran P reduces each Ti and we will see that T |M is a spherical isometry and that T ∗ |M⊥ is unitarily equivalent to the d-shift acting on Hd2 (E0 ). Lemma 4.2. Let T ∈ B(H)d be as in Proposition 4.1. Then for each N ∈ N the operator PN is a projection such that Ti PN = PN −1 Ti for all i ∈ {1, . . . , d}. Proof. We will start by using induction on N ∈ N to show that Ti PN = PN −1 Ti for all i ∈ {1, . . . , d}. The hypothesis (a) of Proposition 4.1 implies that P1 is a projection, hence ran(I − P1 ) = ker P1 = di=1 ker Ti . This implies Ti (I − P1 ) = 0 for all i ∈ {1, . . . , d}. Thus P0 Ti = Ti = Ti P1 and the statement is true for N = 1. Next suppose that N > 1 and that Ti PN −1 = PN −2 Ti for all i ∈ {1, . . . , d}. For x ∈ H and j ∈ {1, . . . , d} set zj = PN −1 Tj x. Then for all i and j we have Ti zj = Ti PN −1 Tj x = PN −2 Ti Tj x = PN −2 Tj Ti x = Tj zi . Thus the hypothesis (b) of Proposition 4.1 implies that there exists y ∈ H such that PN −1 Tj x = zj = Tj y for all j . Then for all i we have Ti PN x = Ti
d j =1
Tj∗ PN −1 Tj x = Ti
d j =1
Tj∗ Tj y = Ti P1 y = Ti y = PN −1 Ti x.
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Thus Ti PN = PN −1 Ti for all N ∈ N and i ∈ {1, . . . , d}. For N > 1 we can iterate this to obtain Ti Tj PN = Ti PN −1 Tj = PN −2 Ti Tj for all i, j . Continuing the same way and using that P0 = I we see that for all N ∈ N and all multiindices α ∈ Nd0 with |α| = N we have T α PN = T α . Thus PN2 =
N N T ∗α T α PN = T ∗α T α = PN , α α |α|=N
|α|=N
2
which shows that each PN is a projection.
Lemma 4.3. Let T be a commuting operator tuple on H which satisfies condition (b) of Proposition 4.1, and let N ∈ N. Suppose that for each α ∈ Nd with |α| = N we are given an element xα ∈ H. Then there is an x ∈ H such that xα = T α x for all |α| = N if and only if Ti xβ+ej = Tj xβ+ei
for all 1 i, j d and all β ∈ Nd0 with |β| = N − 1.
(4.2)
Proof. It is clear that if xα = T α x for all |α| = N , then Ti xβ+ej = Tj xβ+ei for all 1 i, j d and all β ∈ Nd0 with |β| = N − 1. We will use induction on N to verify the sufficiency of condition (4.2). For N = 1 this is just the hypothesis of the lemma. Suppose that the lemma holds for N 1, and suppose that {xα }|α|=N +1 satisfies (4.2) with N + 1 instead of N . Let |β| = N . For i = 1, . . . , d set zi = xβ+ei . Then we have Tj zi = Ti zj for all i and j . Thus by our hypothesis on T there exists xβ ∈ H with xβ+ei = zi = Ti xβ for all i. The collection {xβ }|β|=N satisfies (4.2). Indeed, let |γ | = N − 1 and 1 i, j d, then Tj xγ +ei = x(γ +ei )+ej = x(γ +ej )+ei = Ti xγ +ej . Hence the induction hypothesis and the construction imply that there is an x ∈ H such that xβ = T β x and xβ+ei = Ti xβ = T β+ei for all |β| = N and 1 i d. Thus, xα = T α x for all |α| = N + 1. 2 Lemma 4.4. Let T ∈ B(H)d be as in Proposition 4.1 and let N ∈ N. Then for all x ∈ E0 and all α, β ∈ Nd0 with |α| = |β| = N we have
N α
N T α T ∗β x = δαβ x, β
where δαβ = 1 if α = β and δαβ = 0 otherwise. Of course, the definition of E0 then immediately implies that T γ T ∗β x = 0 for all x ∈ E0 and β, γ ∈ Nd0 with |γ | > N = |β|. Proof. Define the column operator T (N ) : H → T
(N )
x=
|α|=N
N Tα α
H by
. |α|=N
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
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Then T
(N ) ∗
T
(N )
N T ∗α T α = PN . = α |α|=N
∗
∗
Lemma 4.2 implies that T (N ) T (N ) is a projection and hence it follows that T (N ) T (N ) is the orthogonal projection onto ran T (N ) . d Now let x ∈ E0 , fix β ∈ N0 with |β| = N , and define a column vector z = {xα }|α|=N ∈ |α|=N H by xα = 0 if α = β and xβ = x. Then Ti xγ +ej = 0 = Tj xγ +ei for all 1 i, j d and all |γ | = N − 1. Thus it follows from Lemma 4.3 that there is a w ∈ H such that xα = T α w for all |α| = N and hence z ∈ ran T (N ) and z=T
(N )
T
(N ) ∗
z=
N α
The lemma now follows from the definition of z.
N α ∗β T T x β
. |α|=N
2
Proof of Proposition 4.1. From Lemma 4.2 and the remarks preceding it we know that the sequence {PN }N∈N forms a decreasing sequence of projections. Let P denote the strong limit of this sequence. Then P is a projection and the assertion PN −1 Ti = Ti PN of Lemma 4.2 implies that P Ti = Ti P for all 1 i d. Thus M = ran P reduces T and the identity P = di=1 Ti∗ P Ti shows that T |M is a spherical isometry. Let S denote the d-shift acting on Hd2 (E0 ), then S is unitarily equivalent to Mz ⊗ I acting on Hd2 ⊗ E0 . We will show that Mz ⊗ I is unitarily equivalent to T ∗ |M⊥ . Since E0 ⊆ M⊥ we can define a linear transformation U : Hd2 ⊗ E0 → M⊥ by setting U (p ⊗ x) = p(T ∗ )x for every polynomial p and x ∈ E0 . Note that for x, y ∈ E0 and α, β ∈ Nd0 Lemma 4.4 implies
Thus if p(z) = have
ˆ α p(α)z
α
|α| α
|β| ∗α T x, T ∗β y = δαβ x, y . β
and q(z) =
β
β are polynomials, then for all x, y ∈ E we q(β)z ˆ 0
∗
∗α p T x, q T ∗ y = p(α) ˆ q(β) ˆ T x, T ∗β y α,β
=
p(α) ˆ q(α) ˆ
|α| x, y = p, q H 2 x, y , α
d
α
where the identity for the Hd2 -inner product follows from (1.2). This implies that 2 ∗
∗ U = p T x T xj = p ⊗ x , p
pi , pj H 2 xi , xj j j i i j d j
i,j
i,j
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S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
2 = pj ⊗ x j
,
Hd2 ⊗E0
j
thus U extends to be an isometric operator on Hd2 ⊗ E0 . We shall now finish the proof by showing that U has dense range. Let [E0 ]T ∗ denote the smallest common invariant subspace for T1∗ , . . . , Td∗ that contains E0 . We have to show that [E0 ]T ∗ = M⊥ = ran(I − P ). For k 0 we set Qk = Pk − Pk+1 , then each Qk is a projection and I − P = limN →∞ I − PN = ∞ k=0 Qk . Note that ran Q0 = E0 . Thus if we define Ek = ran Qk then we must show that for each k 0 we have Ek ⊆ [E0 ]T ∗ . This is trivially true for k = 0. Thus assume that Ek ⊆ [E0 ]T ∗ for some k 0. Then for x ∈ H we have Qk+1 x =
d
Ti∗ Qk Ti x ∈
i=1
d
Ti∗ Ek ⊆ [E0 ]T ∗ .
i=1
Hence the density of ran U in M⊥ follows by induction.
2
5. Finite rank extensions of spherical contractions and isometries We start out with a trivial lemma that will be used repeatedly and without further mention. Lemma 5.1. Let T be a commuting d-tuple of operators acting on a Hilbert space H and let R = (R1 , . . . , Rd ) be a nontrivial rank one extension of T acting on H ⊕ C i.e. Ti Ai Ri = , 0 Bi where Ai 1 = εxi and Bi 1 = bi for some ε > 0, x1 , . . . , xd ∈ H, di=1 xi 2 = 1, and b = (b1 , . . . , bd ) ∈ Cd . Then R is a commuting d-tuple if and only if for all i, j we have (Ti − bi )xj = (Tj − bj )xi . The following two lemmas are only preliminary results. A more definitive result for spherical contractions will be presented in Theorem 6.1, the result about spherical isometries will follow in Corollary 5.4. family of commuting spherical contractions, let T ∈ Fsc ∩ B(H)d , Lemma 5.2. Let F scd be the ∗ and let D = (I − i=1 Ti Ti )1/2 . Then T has a nontrivial rank one extension in Fsc if and only if there exist b = (b1 , . . . , bd ) ∈ Cd , x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| d< 1, ∗and (iv) i=1 Ti xi ∈ ran D. Proof. Let R be a commuting rank 1 extension of T as in Lemma 5.1. Let x ∈ H and y ∈ C and calculate
S. Richter, C. Sundberg / Journal of Functional Analysis 258 (2010) 3319–3346
3337
2 d d Ri x = Ti x + εxi y2 + |b|2 |y|2 y i=1
i=1
=
d i=1
Ti x + 2ε Re y x, 2
d
Ti∗ xi
+ ε
2
i=1
d
xi + |b| 2
2
|y|2 .
i=1
We now set x0 = di=1 Ti∗ xi and recall di=1 xi 2 = 1. Then we see that R is a spherical contraction if and only if for all x ∈ H and y ∈ C we have
2ε Re y x, x0 + ε 2 + |b|2 |y|2 Dx2 + |y|2 .
(5.1)
Assume now that R is a spherical contraction for some ε > 0, then clearly |b| < 1. By changing the argument of y if necessary, it follows from (5.1) that 2ε|y|| x, x0 | Dx2 + |y|2 for all x ∈ H and y ∈ C. Thinking of this as a quadratic inequality in |y| we conclude that ε 2 | x, x0 |2 Dx2 for all x ∈ H. By the Douglas lemma [12] di=1 Ti∗ xi = x0 ∈ ran D. This proves the necessity of the four conditions. Conversely assume (i)–(iv). In particular, x0 = Dz0 for some z0 ∈ H. Then
2ε Re y x, x0 + ε 2 + |b|2 |y|2 2ε y Dx, z0 + ε 2 + |b|2 |y|2
εDx2 z0 2 + ε + ε 2 + |b|2 |y|2 , which will be Dx2 + |y|2 for sufficiently small ε > 0.
2
Lemma 5.3. Let Fsi be the family of commuting spherical isometries and let T ∈ Fsi ∩ B(H)d . Then the following are equivalent: (a) T has a nontrivial rank one extension in Fsi , (b) T has a nontrivial rank one extension in Fsc , (c) there exist b = (b1 , . . . , bd ) ∈ Cd and x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| d< 1, ∗and (iv) i=1 Ti xi = 0. Proof. (i) ⇒ (ii) is trivial and (ii) ⇒ (iii) follows immediately from Lemma 5.2, because D = 0. The implication (iii) ⇒ (i) follows from the proof of Lemma 5.2. Indeed since D = 0 and x0 = 0 we can take ε = 1 − |b|2 to obtain equality in (5.1). 2 Using the definition as given e.g. in Section 3 one checks that the Koszul complex for a commuting operator tuple R = (R1 , . . . , Rd ) acting on H is exact at Λ1 (H) if and only if whenever x1 , . . . , xd ∈ H are such that Ri xj = Rj xi for all i, j , then there is an x ∈ H such that xi = Ri x for all i. Corollary 5.4. Let T ∈ B(H)d be a commuting spherical isometry. Then the following are equivalent:
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(i) T has a nontrivial rank one extension in Fsi , (ii) T has a nontrivial rank one extension in Fsc , (iii) there exists b ∈ Bd such that the Koszul complex for T − b is not exact at Λ1 (H). For example, if Mz = (Mz , H 2 (∂D)) is the unilateral shift, then T = (Mz , 0) is a commuting spherical isometry. One easily checks that for (b1 , b2 ) ∈ B2 the Koszul complex for T − b is exact at Λ1 (H 2 (∂D)) if and only if b2 = 0, and since Mz has a nontrivial rank 1 extension it is of course clear that T has a nontrivial rank one extension. On the other hand, if d > 1 and if T = Mz = (Mz1 , . . . , Mzd ) acting on H 2 (∂Bd ), then it is known that the Koszul complex for Mz − b is exact at Λ1 (H 2 (∂Bd )) for all b ∈ Bd (see e.g. Proposition 2.6 of [15]). Thus Mz does not have a nontrivial rank one extension in Fsi . Proof. We have already seen the equivalence of (i) and (ii). To prove (i) ⇒ (iii) suppose that T has a nontrivial rank one extension in Fsi , then there exist a b ∈ Bd and x1 , . . . , xd ∈ H such that (i)–(iv) of Lemma 5.3(c) are satisfied. We will show that the Koszul complex for T − b is not exact at Λ1 (H). If it were exact, then by (ii) and the definition of exactness at Λ1 (H) there is an x ∈ H such that xi = (Ti − bi )x for all i. By (i) we have x = 0, and by (iv) we have 0=
d i=1
Ti∗ xi =
d
Ti∗ (Ti − bi )x = x −
i=1
d
Ti∗ bi x.
i=1
Thus x2 = di=1 Ti∗ bi x2 di=1 bi x2 = |b|2 x2 , because the adjoints of spherical isometries must be row contractions. Since x = 0 we conclude |b| 1 which is a contradiction. Hence the Koszul complex for T − b cannot be exact at Λ1 (H). We now prove (iii) ⇒ (i). Suppose that b ∈ Bd such that the Koszul complex for T − b is not exact at Λ1 (H). First we will assume that b = 0. Since T is a spherical isometry, the range of ∂0 : H → H ⊕ · · · ⊕ H, x → (T1 x, . . . , Td x) must be closed. Thus if the Koszul complex is not exact at Λ1 (H) H ⊕ · · · ⊕ H, then there is (x1 , . . . , xd ) ⊥ ran ∂0 such that di=1 xi 2 = 1 and Ti xj = Tj xi for all i, j . Then di=1 Ti∗ xi = ∂0∗ (x1 , . . . , xd ) = 0, so (i)–(iv) of Lemma 5.3(c) are satisfied with b = 0 and hence T has a nontrivial rank one extension in Fsi . If b = 0, then we consider a ball automorphism ϕb that takes b to 0. As in the paragraph preceding Lemma 2.4 of [15] we can define S = ϕb (T ). Then one checks that S is a commuting spherical isometry. Thus it is clear that T has a nontrivial rank one extension in Fsi if and only if S has a nontrivial rank one extension in Fsi . By Lemma 2.4 of [15] the Koszul complex for S is isomorphic to the Koszul complex for T − b. Hence the result follows from the case b = 0. 2 If F is a family and if an operator tuple T ∈ F has a nontrivial finite rank extension R ∈ F acting on H ⊕ K, then the compressions of the Ri to K will have a common eigenvector x0 and R = R|(H ⊕ Cx0 ) will be a rank one extension of T in F . However, it may happen that R is a trivial extension of T . For such situations the following lemma is useful. Lemma 5.5. Let H1 , H2 , H3 be Hilbert spaces, let K = H1 ⊕H2 ⊕H3 , and let R = (R1 , . . . , Rd ) be an operator tuple acting on K with matrix representation of the form Ri =
Ti 0 0
0 Bi 0
Ai Ci Di
.
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If R is a commuting spherical contraction (resp. commuting row contraction), then S = (S1 , . . . , Sd ), Si =
Ti 0
Ai Di
is a commuting spherical contraction (resp. commuting row contraction). Proof. Write H13 = H1 ⊕ H3 and note that S = PH13 R|H13 . The contractiveness assertions thus follow immediately. The commutativity follows from the special form of R. For all i, j we have Si Sj = PH13 Ri PH13 Rj |H13 = PH13 Ri Rj |H13 − PH13 Ri PH2 Rj |H13 = PH13 Ri Rj |H13
since PH13 Ri PH2 = 0
= PH13 Rj Ri |H13 = Sj Si .
2
Corollary 5.6. Let F = Fsc or F = Frc and let T ∈ F . Then T has a nontrivial finite rank extension in F if and only if T has a nontrivial rank one extension in F . Proof. Suppose T acts on a Hilbert space H and let R be a nontrivial finite rank extension of T in F acting on H ⊕ K with 1 < dim K < ∞. Let B = PK R|K be the compression of R to K. Then B is a commuting tuple of linear transformations on a finite dimensional space, thus the transformations Bi will have a common eigenvector x0 = 0. Then either R = R|(H ⊕ Cx0 ) is a nontrivial rank one extension of T in F or R is of the form as in Lemma 5.5 with H1 = H and H2 = Cx0 . In the latter case we can use the lemma to get a nontrivial extension R of T acting on H ⊕ K with R ∈ F and dim K = dim K − 1. Thus the result follows by an induction argument. 2 6. Extensions of spherical contractions Theorem 6.1. Let T be a commuting spherical contraction. Then the following are equivalent: (i) T has only trivial rank one extensions in Fsc , (ii) T has only trivial finite rank extensions in Fsc , (iii) T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry such that for all b ∈ Bd the Koszul complex for V − b is exact at Λ1 , d ∗ (iv) (a) i=1 Ti Ti is a projection, and (b) for all b ∈ Bd the Koszul complex for T − b is exact at Λ1 . Proof. (i) and (ii) are equivalent by Corollary 5.6. (iii) ⇒ (i): If R is a rank one extension of T = S ∗ ⊕ V in Fsc , then since S ∗ is extremal in Fsc Lemma 2.1(a) implies that R = S ∗ ⊕ R for some R ∈ Fsc with R V . Then clearly R is a rank one extension of V and (i) follows from the hypothesis and the equivalence of (ii) and (iii) of Corollary 5.4.
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(iv) ⇒ (iii): By Proposition 4.1 the conditions (iv)(a) and (b) with b = 0 imply that T is of the form T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry. The Koszul complex of S ∗ ⊕ V splits into a direct sum of Koszul complexes. Thus it is clear that (iv)(b) implies that for each b ∈ Bd the Koszul complex for V − b is exact at Λ1 . (i) ⇒ (iv): We will show the contrapositive. First suppose that (iv)(a) is not satisfied, i.e. d ∗ will use Lemma 5.2 with b = 0. i=1 Ti Ti is not a projection. We If E is the spectral measure for di=1 Ti∗ Ti , then there are real numbers r, s such that 0 < r < s < 1 and such that Q = E([r, s]) = 0. Let x0 ∈ ran Q, x0 = 0 and set xi = Ti x0 . Then d
xi = 2
i=1
d
1 Ti x0 =
t d Et x0 , x0 rx0 2 = 0.
2
i=1
0
Thus by scaling x0 we may assume that di=1 xi 2 = 1, and we have (i), (ii), and (iii) of s Lemma 5.2. Furthermore, since s < 1 we have Q = D r √ 1 dE, so ran Q ⊆ ran D. Hence 1−t
d
Ti∗ xi =
i=1
d
Ti∗ Ti x0 =
i=1
d
Ti∗ Ti Qx0 = Q
i=1
d
Ti∗ Ti x0 ∈ ran D
i=1
and (iv) of Lemma 5.2 is also satisfied. Thus T must have a nontrivial rank one extension in Fsc i.e. condition (i) of Theorem 6.1 does not hold. Next suppose that di=1 Ti∗ Ti = P1 is a projection, but that the Koszul complex for T is not exact at Λ1 . This implies that the column operator T (1) : H → Hd defined by ⎛
⎞ T1 x . T (1) x = ⎝ .. ⎠ Td x ∗
satisfies P1 = T (1) T (1) and hence T (1) is a partial isometry and in particular has closed range. x1 . / ran T (1) . Furthermore, there exist x1 , . . . , xd ∈ H such that Ti xj = Tj xi for all i, j , but .. ∈ xd x1 . Since ran T (1) is closed we may assume that .. ⊥ ran T (1) and di=1 xi 2 = 1. But this xd x1 d ∗ .. ∗ (1) or i=1 Ti xi = 0 ∈ ran D. Thus again we can use Lemma 5.2 to means that . ∈ ker T xd
see that condition (i) of Theorem 6.1 does not hold. Finally we suppose that di=1 Ti∗ Ti is a projection, that the Koszul complex for T is exact at Λ1 , but that there is a b ∈ Bd , b = 0 such that the Koszul complex for T − b is not exact at Λ1 . Then Proposition 4.1 implies that T = S ∗ ⊕ V , where S is a direct sum of d-shifts and V is a spherical isometry. Since the Koszul complex of S ∗ − b is exact at Λ1 , it follows that the Koszul complex for V − b cannot be exact at Λ1 . Thus in this case it follows from Corollary 5.4 that T would have a nontrivial rank one extension in Fsc . This concludes the proof of (iv) ⇒ (i). 2 Corollary 6.2. Theorem 1.4 holds.
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Proof. The implication (ii) ⇒ (i) follows from Theorems 3.3 and 2.2 and Lemma 2.1. We now show (iii) ⇒ (ii). If T satisfies the conditions (a), (b), and (c) of part (iii) of Theorem 1.4, then by Proposition 4.1 T is unitarily equivalent to S ∗ ⊕ V , where S is a direct sum of d-shifts and V = (V1 , . . . , Vd ) is a spherical isometry. We will see that V is a spherical unitary. Condition (b) implies that 0
d
Vi Vi∗ − I =
i=1
d
Vi Vi∗ − Vi∗ Vi .
i=1
We already mentioned that each operator in a spherical isometric tuple must be subnormal (also see Theorem 1.3), thus for each i we have Vi Vi∗ − Vi∗ Vi 0. Hence Vi Vi∗ = Vi∗ Vi for all 1 i d. Finally we prove (i) ⇒ (iii). If T is extremal then it has no nontrivial rank one extension, hence conditions (iii)(a) and (c) follow from the equivalence of (i) and (iv) in Theorem 6.1. Then it follows from Proposition 4.1 that T = S ∗ ⊕ V for a spherical isometry V . Since T is extremal Theorem 2.2 implies that V must in fact be a spherical unitary tuple. Then T ∗ = S ⊕ U for some spherical unitary tuple U and a direct sum of d-shifts S. Condition (iii)(b) follows easily (see (1.3)). 2 Corollary 6.3. Corollary 1.5 holds. Proof. Let T = S ⊕ U , where S is a direct sum of d-shifts and U is a spherical unitary tuple. It follows from (1.1) and (1.3) that T satisfies conditions (a) and (b) of Corollary 1.5. Furthermore, in Section 3 we mentioned that the Koszul complex for the d-shift is exact at Λp (H) for all p with 1 p d − 1. The Taylor spectrum of any spherical unitary tuple U must be contained in ∂Bd . Since such U is normal this can easily be deduced from Proposition 7.2 of [11]. Hence the Koszul complex of T = S ⊕ U is exact at Λd−1 (H), i.e. (c) of Corollary 1.5 holds as well. Conversely suppose that T satisfies (a), (b), and (c) of Corollary 1.5. Then the adjoint tuple T ∗ satisfies (iii)(a) and (iii)(b) of Theorem 1.4. Since di=1 Ti Ti∗ is a projection the operator H → Hd defined by x → (T1∗ x, . . . , Td∗ x) is a partial isometry and thus has closed range. This operator is unitarily equivalent to ∂T ∗ ,0 , hence ∂T ∗ ,0 has closed range in Λ1 (H). Thus, as the hypothesis (c) is that K(T ) is exact at Λd−1 (H) the discussion about the Hodge ∗-operator at the beginning of Section 3 implies that K(T ∗ ) is exact at Λ1 (H), i.e. T ∗ satisfies (iii)(c) of Theorem 1.4. Hence Theorem 1.4 implies that T ∗ = S ∗ ⊕ U , where S is a direct sum of d-shifts and U (and hence U ∗ ) is a spherical unitary tuple. This concludes the proof of the corollary. 2 7. Rank one extensions of row contractions of commuting row contractions, Next we consider row contractions. Let Frc denote the family let T = (T1 , . . . , Td ) ∈ Frc ∩ B(H)d and write D∗ = (I − di=1 Ti Ti∗ )1/2 for the defect operator. If R ∈ B(H ⊕ K) is an operator tuple that extends T , then we will use the notation Ri =
Ti 0
Ai Bi
,
i = 1, . . . , d.
(7.1)
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Note that R will be a commuting tuple if and only if Ti Aj − Tj Ai = Aj Bi − Ai Bj
and Bi Bj = Bj Bi
for all i, j.
(7.2)
Lemma 7.1. Let T be a commuting row contraction. (a) If b = (b1 , . . . , bd ) ∈ Cd , |b| = 1, then ker(I − di=1 bi Ti ) ⊆ ker D∗ . (b) If R is a row contraction that extends T , then di=1 Ai A∗i D∗2 and for each i we have ran Ai ⊆ ran D∗ . Proof. (a) Let |b| = 1 and let x ∈ ker(I −
d
i=1 bi Ti ).
Then
2 2 d d ∗ x = x, b i Ti x = bi Ti x, x 4
i=1
|b|2
i=1
d
∗ 2
T x x2 = x2 − D∗ x2 x2 . i
i=1
This implies D∗ x = 0. (b) Recall that R is a row contraction if and only if R ∗ is a spherical contraction, i.e. for all x ∈ H, y ∈ K we have d ∗ x 2 2 2 R i y x + y . i=1
A short calculation shows that this happens if and only if d ∗ A x + B ∗ y 2 D∗ x2 + y2 . i
i
(7.3)
i=1
In particular, we see that if R is a row contraction, then di=1 Ai A∗i D∗2 and hence for each i we must have ran Ai ⊆ ran D∗ (by the Douglas lemma [12]). This proves (b). 2 Lemma 7.2.Let Frc be the family of commuting row contractions, let T ∈ Frc ∩ B(H)d , and let D∗ = (I − di=1 Ti Ti∗ )1/2 . Then T has a nontrivial rank one extension in Frc , if and only if there exist b = (b1 , . . . , bd ) ∈ Cd , x1 , . . . , xd ∈ H such that d 2 (i) i=1 xi = 1, (ii) (Ti − bi )xj = (Tj − bj )xi for all i, j , (iii) |b| < 1, and (iv) xi ∈ ran D∗ for each i.
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3343
Proof. First assume that we are given b ∈ Cd and x1 , . . . , xd ∈ H such that (i)–(iv) are satisfied. For any ε > 0 we define a rank one extension R of T as in Lemma 5.1. By (i) and (ii) it be nontrivial and commutative. Thus by (7.3) R will be a row contraction if and only if will d |ε x, xi + bi y|2 D∗ x2 + |y|2 for all x ∈ H, y ∈ C. i=1 Since each xi ∈ ran D∗ , there are zi ∈ H such that xi = D∗ zi . Then for x ∈ H and y ∈ C we have d d d ε x, xi + bi y 2 ε 2 D∗ x, zi 2 + 2ε D∗ x, zi |bi ||y| + |b|2 |y|2 i=1
i=1
ε
2
d
i=1
! d ! " 2 zi D∗ x + 2ε D∗ x zi |b||y| + |b|2 |y|2 2
2
i=1
i=1
d
ε2 + ε zi 2 D∗ x2 + (1 + ε)|b|2 |y|2 D∗ x2 + |y|2 , i=1
whenever ε is sufficiently small. Thus T has a nontrivial rank one extension in Frc . Conversely, assume that T has a nontrivial rank one extension R in Frc . Then R can be written as in Lemma 5.1. Thus we have ε > 0, b ∈ Cd and x1 , . . . , xd ∈ H satisfying (i) and (ii). Furthermore, a calculation similar to what was done in the first part of the proof shows that since R is a row contraction we must have d d d ε x, xi + bi y 2 = ε 2 x, xi 2 + 2ε Re
x, xi bi y + |b|2 |y|2 i=1
i=1
i=1
D∗ x + |y| 2
2
for all x ∈ H and all y ∈ C. By taking y = 0 we see that the Douglas lemma [12] implies that (iv) must be satisfied, and by taking x = 0 it follows that |b| 1. We will be done if we can rule out the possibility that |b| = 1. Note that R ∗ is a nontrivial extension in Fsc of the tuple of scalars b : C → C. Hence Theorem 2.2 implies |b| < 1. A somewhat |b| = 1, then more direct argument goes as follows: If the above inequality implies that Re di=1 x, xi bi y = 0 for all x and y. Hence di=1 bi xi = 0. Now we multiply (ii) by bi and sum in i to obtain ( di=1 bi Ti − I )xj = (Tj − bj ) di=1 bi xi = 0 for each j . Thus each xj ∈ ker(I − di=1 bi Ti ) and hence by Lemma 7.1(a) xj ∈ ker D∗ . This contradicts (i) and (iv), which is already known to hold. 2 8. Extensions of row contractions In this section we shall prove Theorems 1.6 and 1.8 and Corollary 1.7. Proposition 8.1. (a) If D∗ = 0, then T ∈ ext(Frc ). (b) If D∗ is onto, then T has a rank one extension in ext(Frc ).
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(c) If D∗ is a projection, then the following are equivalent: (i) T has a nontrivial rank one extension in Frc , (ii) T ∈ / ext(Frc ), (iii) there are x1 , . . . , xd ∈ dj =1 ker Tj∗ with di=1 xi 2 > 0 and Ti xj = Tj xi for all i, j . Proof. (a) follows directly from Lemma 7.1(b). (b) Clearly the zero tuple, T = (0, . . . , 0), is not extremal. Thus assume that D∗ is onto and one of the Ti ’s is not zero. Then we can set b = 0 and choose x ∈ H such that the hypothesis of Lemma 7.2 is satisfied with xi = Ti x. (c) (iii) ⇒ (i) follows directly from Lemma 7.2 with b = 0. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): We assume that D∗ is a projection and that we have a nontrivial extension in Frc . Then with the notation as in (7.1) we set xi = Ai x, where x is chosen so that di=1 xi 2 > 0. Lemma 7.1(b) implies that for all k we have ran Ak ⊆ ran D∗ . Thus x1 , . . . , xd ∈ ran D∗ . Furthermore, since D∗ is a projection we have d #
ran Ak ⊆ ran D∗ =
d $
ker Tj∗
=
j =1
k=1
d #
⊥ ran Tj
.
j =1
Thus, commutativity implies that for all i and j Ti xj − Tj xi = Ti Aj x − Tj Ai x = Aj Bi x − Ai Bj x ∈
d #
ran Ak ∩
k=1
This establishes (iii).
d #
ran Tj = (0).
j =1
2
Proposition 8.2. If T ∈ Frc and if there is a u ∈ ran D∗ , u = 1, such that dim span{u, T1 u, . . . , Td u} 2, then T has a nontrivial rank one extension in Frc . Proof. The hypothesis implies that there is v ∈ H, v ⊥ u and α = (α1 , . . . , αd ), β = (β1 , . . . , βd ) ∈ Cd , β = 0 such that Ti u = αi u + βi v for i = 1, . . . , d. Indeed, if dim span{u, T1 u, . . . , Td u} = 2, then we can find such a unit vector v satisfying this, while if dim span{u, T1 u, . . . , Td u} = 1 we take v = 0 and any β = 0. We set γ = Pβ ⊥ α = α − cβ, where c = α,β . Then β, γ = 0 and α, γ = α, Pβ ⊥ α = |γ |2 . |β|2
βi The conclusion will follow from Lemma 7.2 with xi = |β| u and b = γ . Conditions (i) and (iv) are obvious from the definition of the xi . In order to verify (iii) we calculate
(Ti − γi )xj =
βi βj (v − cu) = (Tj − γj )xi |β|
for all i, j . Finally, di=1 γi Ti u = α, γ u + β, γ v = |γ |2 u. Since T is a row contraction this implies d 2 d |γ | = Ti (γi u) γi u2 = |γ |2 . 4
i=1
Hence |γ | 1.
i=1
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If |γ | = 1, then (I − di=1 γi Ti )u = 0, thus Lemma 7.1(a) implies that u ∈ ker D∗ . But since u ∈ ran D∗ this would mean u = 0, which is impossible. Hence |γ | < 1. 2 We shall now prove part (iv) of Theorem 1.6. Theorem 8.3. Let T be a commuting row contraction with D∗ = u ⊗ u for some u ∈ H, u = 0. Then the following are equivalent: (i) T ∈ ext(Frc ), (ii) T has only trivial rank one extensions in Frc , (iii) dim span{u, T1 u, . . . , Td u} 3. Proof. (i) ⇒ (ii) is trivial and (ii) ⇒ (iii) follows directly from Proposition 8.2. (iii) ⇒ (i): Suppose that dim span{u, T1 u, . . . , Td u} 3. Since u = 0 we may without loss of generality assume that the set {u, T1 u, T2 u} is linearly independent. Let R be an extension of T in Frc and assume each Ri is of the form as in (7.1). We must show that each Ai = 0. Since D∗ = u ⊗ u Lemma 7.1(b) implies the existence of x1 , . . . , xd ∈ K such that Ai x =
x, xi u for each i. Then commutativity (see (7.2)) implies for all x ∈ K and all i, j
x, xj Ti u − x, xi Tj u = Bi x, xj u − Bj x, xi u = x, Bi∗ xj − Bj∗ xi u.
(8.1)
Take i = 1 and j = 2. Then the linear independence of {u, T1 u, T2 u} shows that x, x1 =
x, x2 = 0 for all x ∈ K. Thus x1 = x2 = 0. Next consider (8.1) with i = 1 and j > 2. Since x1 = 0 we get
x, xj T1 u = x, B1∗ xj u. Again linear independence implies xj = 0. Thus Aj = 0 for all j , and T must be extremal.
2
Corollary 8.4. Corollary 1.7 holds. Proof. Write P = PM⊥ for the projection of Hd2 onto M⊥ . Recall that if S denotes the d-shift on Hd2 , then I − di=1 Si Si∗ = 1 ⊗ 1 is the projection onto the constants. For i = 1, . . . , d we have Ti = P Si |M⊥ , hence D∗2
= IM⊥ −
d
Ti Ti∗
=P I −
i=1
d
Si Si∗
P = ϕ ⊗ ϕ,
i=1
/ M, thus ϕ = 0 and rank D∗ = 1. where ϕ = P 1. Since we are assuming M = Hd2 we have 1 ∈ Let α0 , . . . , αd ∈ C, then α0 ϕ +
d i=1
αi Ti ϕ = 0
⇔
α0 +
d
αi zi ∈ M.
i=1
This implies that span{ϕ, T1 ϕ, . . . , Td ϕ} is isomorphic to L/M ∩ L. But dim L/M ∩ L = d + 1 − dim(M ∩ L). Thus Corollary 8.4 follows from Theorem 8.3. 2
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9. An example Let S = (Mz , Mw ) be the 2-shift on H22 and let M = {f ∈ H22 : f (z, 0) = 0}. M is invariant for S, thus T = S|M is a nonextremal row contraction. We claim that T has no nontrivial finite rank extensions in Frc . Note that the linear span of monomials of the form zn w m , n 0, m > 0 is dense in M and one computes % 1 n D∗2 zn w m = n+1 z w if m = 1, 0 if m > 1. From this one easily sees that there are no nonzero f, g ∈ ran D∗ and b = (b1 , b2 ) ∈ B2 such that (z − b1 )f = (w − b2 )g. Hence Theorem 1.8 applies to show the claim. References [1] Jim Agler, An abstract approach to model theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, in: Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 1–23. [2] T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90. [3] William Arveson, Subalgebras of C ∗ -algebras. II, Acta Math. 128 (3–4) (1972) 271–308. [4] William Arveson, Subalgebras of C ∗ -algebras. III. Multivariable operator theory, Acta Math. 181 (2) (1998) 159– 228. [5] William Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189 (1) (2002) 53–79. [6] William Arveson, Notes on the unique extension property, unpublished manuscript, see http://math.berkeley.edu/~ arveson/texfiles.html, 2003. [7] Ameer Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (2) (1990) 339–350. [8] K.R.M. Attele, A.R. Lubin, Dilations commutant lifting for jointly isometric operators—a geometric approach, J. Funct. Anal. 140 (2) (1996) 300–311. [9] S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961) 106– 111. [10] Raul E. Curto, Fredholm invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1) (1981) 129–159. [11] Raúl E. Curto, Applications of several complex variables to multiparameter spectral theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, in: Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 25–90. [12] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966) 413–415. [13] Michael A. Dritschel, Scott McCullough, Model theory for hyponormal contractions, Integral Equations Operator Theory 36 (2) (2000) 182–192. [14] Michael A. Dritschel, Scott A. McCullough, Boundary representations for families of representations of operator algebras and spaces, J. Operator Theory 53 (1) (2005) 159–167. [15] Jim Gleason, Stefan Richter, Carl Sundberg, On the index of invariant subspaces in spaces of analytic functions of several complex variables, J. Reine Angew. Math. 587 (2005) 49–76. [16] Phillip Griffiths, Joseph Harris, Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley & Sons Inc., New York, 1994, reprint of the 1978 original. [17] Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958) 1–15. [18] V. Müller, F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (4) (1993) 979–989. [19] Walter Rudin, Functional Analysis, second edition, Internat. Ser. Pure Appl. Math., McGraw–Hill Inc., New York, 1991. [20] Béla Sz.-Nagy, Ciprian Foia¸s, Harmonic Analysis of Operators on Hilbert Space, translated from the French and revised, North-Holland Publishing Co., Amsterdam, 1970. [21] Joseph L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191.
Journal of Functional Analysis 258 (2010) 3347–3361 www.elsevier.com/locate/jfa
On Schechter’s linking theorems W. Zou a,∗,1 , S. Li b,2 a Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China b Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China
Received 16 September 2009; accepted 6 November 2009 Available online 26 November 2009 Communicated by L. Gross
Abstract The existence theorems of critical points due to M. Schechter [M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995) 1603–1626] are developed to the case of sign-changing critical points. A new relationship between the linking without value-splitting property and the sign-changing (weak) Palais– Smale sequence is established. The new abstract theorems are applied to find nodal solutions to elliptic equations under rather weak hypotheses. © 2009 Elsevier Inc. All rights reserved. Keywords: Schechter’s linking; Sign-changing solution; PS-sequence; Value-splitting
1. Introduction A fundamental method of finding critical points of a C1 functional on a Banach space E consists of finding two subsets A and B of E such that A links B in the sense of Schechter [17] (or Rabinowitz [11]) and satisfy the value-splitting condition a0 := sup G b0 := inf G. A
B
* Corresponding author.
E-mail addresses:
[email protected] (W. Zou),
[email protected] (S. Li). 1 Supported by NSFC and the program of the Ministry of Education in China for NCET. 2 Supported by NSFC.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.005
(1.1)
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In [17] it was shown the existence of a sequence {uk } in E with G(uk ) → a b0 and G (uk ) → 0. This kind sequence is called Palais–Smale sequence ((PS) sequence, for short). The (PS) sequence usually can deduce a critical point (a solution of the equations) in case that additional hypotheses are imposed. However, if (1.1) is violated, the existence of the (PS) sequence is not assured. In Schechter’s paper [14], he considered this situation in which there is a sequence {Ak } in E such that Ak links B for each k and (1.1) is violated. He got the following alternative: Either (1) a (PS) sequence exists, or (2) there exists a sequence {uk } such that G(uk ) → a,
uk → ∞,
G(uk ) → 0, uk θ
G (uk ) → 0, uk θ−1
where θ 1 is a constant. The usefulness of this alternative theorem lies in the fact that the kind of arguments which are used in showing the boundedness of the (PS) sequences can be used to eliminate option (2) from the alternative. By this way, the existence of critical points can also be deduced. The abstract theorems of [14] are successfully applied to solve the existence of solutions to a series of semilinear elliptic equations with very weak assumptions, where the standard linking methods are usually ineffective. However, when ones want to study the existence of sign-changing (nodal) solutions to elliptic equations, Schechter’s theorems in [14] fail. The aim of the present paper is to develop Schechter’s theory and establish new theorems where (1.1) is not needed and sign-changing critical points can be obtained. Roughly speaking, I introduce some new linking sets A and B. If (1.1) is violated, we either get a sequence such that {uk } is sign-changing,
G(uk ) → a
and G (uk ) → 0,
or show that there exists a sequence {uk } such that {uk } is sign-changing,
G(uk ) → 0 and uk β
G (uk ) → 0, uk β−1
where β 1. The new saddle point alternative theorem may also produce the sign-changing critical points. Moreover, in this paper, a series of new theorems on the existence of sign-changing (PS) (or weak) sequence for various linkings will be established when (1.1) does not hold. Even if (1.1) holds, we shall build new theorems on the existence of sign-changing weak (PS) sequences. We illustrate this method with several applications to semilinear elliptic boundary value problems. 2. On Schechter’s linking theorems Let E be a Hilbert space endowed with an inner product ·,· and the associated norm · . Suppose that there is another norm · ∗ of E such that u∗ C∗ u for all u ∈ E, here C∗ > 0 is a constant. Moreover, we assume that un − u∗ ∗ → 0 whenever un u∗ weakly in (E, · ). For example, in the Sobolev space H01 (Ω), we may choose · ∗ = · p , the usual Lp -norm (see [12,21,22]). A functional G ∈ C1 (E, R) maps bounded sets to bounded sets. Let K := {u ∈ E: G (u) = 0}, E˜ := E\K, K[a, b] := {u ∈ K: G(u) ∈ [a, b]}, Ga := {u ∈ E: G(u) a}. As in
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Schechter [14–16], let ±P be a positive (negative) cone of E. Define a class of contractions of E as follows: ⎫ ⎧ Γ (·,·) ∈ C [0, 1] × E, E , Γ (0, ·) = id, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for each t ∈ [0, 1), Γ (t, ·) is a homeomorphism of E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 onto itself and Γ (t, ·) is continuous on [0, 1) × E; ⎬ . (2.1) Φ := Γ : ⎪ ⎪ there exists an x0 ∈ E such that Γ (1, x) = x0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for each x ∈ E and that Γ (t, x) → x0 as t → 1 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ uniformly on bounded subsets of E Obviously, Γ (t, u) = (1 − t)u ∈ Φ. Let A ⊂ E be a bounded subset of E. Definition 2.1. (See [17,19].) A subset A of E is linked to a subset B of E if A ∩ B = ∅ and, for every Γ ∈ Φ, there is a t ∈ [0, 1] such that Γ (t, A) ∩ B = ∅. Since Φ = ∅ and A is bounded, we may choose a β > 0 such that ΦA,β := Γ ∈ Φ: Γ [0, 1], A ⊂ Gβ = ∅.
(2.2)
Definition 2.2. Subset A of E is called ΦA,β -linked to a subset B of E if A ∩ B = ∅ and, for every Γ ∈ ΦA,β , there is a t ∈ [0, 1] such that Γ (t, A) ∩ B = ∅. Obviously, if A links B in the sense of Definition 2.1, then we may find a β > 0 such that A is ΦA,β -linked to B. The following linking is different from the classical ones in Rabinowitz [11]. Proposition 2.1. Let E = M ⊕ Y , where M, Y are closed subspaces and Y is of finite dimensional. Choose R > 0, ρ > 0 and y0 ∈ M\{0} with yR0 = ρ and y0 ∗ = 1. Set A := u = v + sy0 : v ∈ Y, s 0, u = R ∪ u ∈ Y : u R , B := u ∈ M: u∗ = ρ , then A links B in the sense of Definition 2.1 and hence, of Definition 2.2 for some β > 0. Remark 2.1. The choice of B is essential since in applications, dist(B ∩ Gβ , ±P) > 0 and this plays an important role in getting sign-changing solutions (see [12]). However, in classical case, B = {u ∈ M: u = ρ} and dist(B, ±P) = 0 in applications, hence we can not be applied directly to get sign-changing critical points. Proof. We first let Q = {sy0 + v: v ∈ Y, s 0, sy0 + v R}. Then A = ∂Q in RN +1 . Let u = v + w with v ∈ Y , w ∈ M, we define F u = v + w∗ y0 . Then F |Q = id and B = F −1 (ρy0 ). We can apply Proposition 2.6.2 of Schechter [17] to conclude that A links B in the sense of Definition 2.1. Hence, A is ΦA,β -linked to B for some β > 0. 2 As in [12,22], we make the following assumption. (A1 ) B ∩ Gc = ∅ ⇒ dist(B ∩ Gc , ±P) > 0.
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In most applications, (A1 ) can be satisfied readily. Note that if B = {u0 } and u0 ∈ / ±P, then (A1 ) is satisfied. For μ0 > 0, define Dμ0 := {u ∈ E: dist(u, P) < μ0 }. Set D := Dμ0 ∪ (−Dμ0 ), Sβ = E\D. Noting (A1 ) and B ∩ Gβ = ∅, we may assume μ0 small enough so that B ∩ Gβ ⊂ Sβ . Define a0 := sup G, A∩Sβ
b0 := inf G, B
a :=
inf
sup
Γ ∈ΦA,β Γ ([0,1],A)∩Sβ
G(u).
(2.3)
If A ∩ Sβ = ∅, we set a0 = −∞. Note that a is well defined if A is ΦA,β -linked to B. Further, we make the following hypotheses. (A2 ) Assume G is of the form G (u) = u − G(u), where G : E → E is a continuous operator and G(±Dμ0 ) ⊂ ±Dμ for some μ ∈ (0, μ0 ); K ∩ (E\{−P ∪ P}) ⊂ Sβ . Theorem 2.1. Assume (A1 )–(A2 ) and A is ΦA,β -linked to B. Assume that a0 b0 . Then a ∈ [b0 , β] and there exists a sequence {uk } such that {uk } ∈ Sβ ,
G(uk ) → a,
G (uk ) → 0.
(2.4)
If a = b0 , then we may require that dist(uk , B) → 0. Proof. Obviously, a ∈ [b0 , β]. We first consider the case of a > b0 . √ If (2.4) were false, there would exist a positive constant ε¯ ∈ (0, 1/8) such that G (u) 8¯ε for u ∈ T1 := {u ∈ Sβ : |G(u) − a| 3δ} for some δ ∈ (0, ε¯ /2) ∩ (0, a − b0 ). Note that T1 = ∅. Let V0 : E˜ → E be a ˜ ⊂ (±Dμ ) for some pseudo-gradient vector field of G with V0 (u) = u−L0 (u) and L0 (±Dμ0 ∩ E) V0 (u) μ ∈ (0, μ0 ). The construction of such V0 and L0 can be found in [22]. Let Y0 (u) := 1+V . 0 (u) Then Y0 (u) is locally Lipschitz continuous and G (u), Y0 (u) ε¯ on T1 . Let T2 := {u ∈ Sβ : dist(u,T2 ) |G(u) − a| 2δ}, T3 := {u ∈ Sβ : |G(u) − a| δ}. Define ξ(u) = dist(u,T . We remark 3 )+dist(u,T2 )
(t,u) that T3 = ∅. If T2 = ∅, we set ξ ≡ 1. Consider the Cauchy problem: dωdt = −ξ(ω)Y0 (ω), ω(0, u) = u ∈ E. Then ω(t, ±Dμ0 ) ⊂ ±Dμ0 for all t 0 (cf. [20,22]); and d G(ω(t, u))/dt / T3 for some t1 ∈ [0, 1], then ei−¯ε ξ(ω(t, u)) if ω(t, u) ∈ Sβ . Let u ∈ Ga+δ . If ω(t1 , u) ∈ / Sβ (hence, ω(t, u) ∈ D for all t t1 ). ther G(ω(1, u)) G(ω(t1 , u)) < a − δ or ω(t1 , u) ∈ On the other hand, if σ (t, u) ∈ T3 for all t ∈ [0, 1], then G(ω(1, u)) a − δ. Summing up, we have ω(1, Ga+δ ) ⊂ Ga−δ ∪ D. By the definition of a, we have a Γ ∈ ΦA,β such that Γ ([0, 1], A) ⊂ Ga+δ ∪ D. Set Γ1 (s, u) = ω(2s, u) if s ∈ [0, 1/2] and = ω(1, Γ (2s − 1, u)) for all s ∈ [1/2, 1]. Then Γ1 ∈ ΦA,β . It is easy to check that Γ1 (s, A) ⊂ Ga−δ ∪ D, which contradicts the definition of a. Next we consider the case of a = b0 . Here we have to construct different vector field and need a careful analysis of √ the flow. If it were not true, there would exist positive numbers ε, δ, T such that G (u) 8ε for u ∈ T4 := {u ∈ Sβ : dist(u, B) 4T , |G(u)−a| < 3δ} for some δ < T ε/2. Let T5 := {u ∈ Sβ : dist(u, B) 3T , |G(u)−a| < 2δ}, T6 := {u ∈ Sβ : dist(u, B) 2T , |G(u) − a| δ}. Then we observe that T6 = ∅. Define dist(u,E\T5 ) ζ (u) = dist(u,E\T . Consider the similar Cauchy problem as above, where ξ is replaced 5 )+dist(u,T6 ) by ζ and we get a new flow ω satisfying ω(t, ±Dμ0 ) ⊂ ±Dμ0 for all t 0. Take any u ∈ Ga+δ , / T6 , then either ω(t1 , u) ∈ D or ω(t1 , u) < a − δ if there exists a t1 T such that ω(t1 , u) ∈ or dist(ω(t1 , u), B) > 2T . For the last case, since ω(t, u) − ω(s, u) |t − s|, we see that dist(ω(t, u), B) > T for all t ∈ [0, T ]. On the other hand, if ω(t, u) ∈ T6 for all t ∈ [0, T ],
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then G(ω(T , u)) a − δ. Summing up, we see that ω(T , Ga+δ ) ∩ B ∩ Sβ = ∅. We claim that ω([0, T ], A) ∩ B ∩ Sβ = ∅. Otherwise, we have an ω(t, u) ∈ B ∩ Sβ , u ∈ A ∩ Sβ . Note that
t G(ω(t, u)) a0 − ε 0 ζ (ω(s, u)) ds and G(ω(t, u)) b0 , hence ζ (ω(s, u)) = 0 for all s ∈ [0, t] hence ω(s, u) ∈ E\T5 , that is, either G(ω(s, u)) < a −δ for all s ∈ [0, t] or dist(ω(s, u), B) > 2T for all s ∈ [0, t] or ω(s, u) ∈ D for all s ∈ [0, t]. These are absurd. Then our claim is correct. By the definition of a, we have a Γ ∈ ΦA,β such that Γ ([0, 1], A) ⊂ Ga+δ ∪ D. Define Γ1 such that Γ1 = ω(2sT , u) for s ∈ [0, 1/2] and = ω(T , Γ (2s − 1, u)) if s ∈ [1/2, 1]. We have Γ1 ∈ ΦA,β . But Γ1 ([0, 1], A) ∩ B ∩ Sβ = ∅, also a contradiction. 2 Remark 2.2. If we replace (A1 ) by “B ⊂ Sβ ”, then Theorem 2.1 is still true (see [20,22]). Lemma 2.1. Assume (A2 ). If for any Γ ∈ ΦA,β , there holds / A, G(v) a0 = ∅, SΓ := v = Γ (s, u): s ∈ (0, 1], u ∈ A, v ∈ Sβ , v ∈ then there is a sequence {uk } ∈ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. We set B0 := Γ ∈ΦA,β SΓ . Then A ∩ B0 = ∅, B0 ⊂ Sβ and A is ΦA,β -linked to B0 . In particular a0 infB0 G. By Theorem 2.1 and Remark 2.2, we get the conclusion. 2 Lemma 2.2. Assume (A2 ) and either (A1 ) or B ⊂ Sβ . If a0 = a, then there is a sequence {uk } ∈ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. Note that A ∩ Sβ ⊂ Γ ([0, 1], A) ∩ Sβ , we see that a > a0 . Hence, for any Γ ∈ ΦA,β , there exists a u ∈ A, s ∈ (0, 1] such that G(Γ (s, u)) > a0 , Γ (s, u) ∈ Sβ , hence SΓ = ∅. By the above Lemma 2.1, we get the conclusion. 2 Theorem 2.2. Let A, a0 be as above. Then the sufficient and necessary condition of SΓ = ∅ for all Γ ∈ ΦA,β is there exists a B0 ⊂ Sβ such that A is ΦA,β -linked to B0 and a0 infB0 G. Proof. If SΓ = ∅, then B0 = Γ ∈ΦA,β SΓ is what we want. If SΓ = ∅ for some Γ ∈ ΦA,β , then for any B0 ⊂ Sβ with A ∩ B0 = ∅ and a0 infB0 G, we must have Γ ([0, 1], A) ∩ B0 = ∅, a contradiction. 2 Theorem 2.3. If for each Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G is attained at w ∈ / A, then there is a sequence {uk } ⊂ Sβ such that G(uk ) → a and G (uk ) → 0 as k → ∞. Proof. For any Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G a0 . If for each Γ ∈ ΦA,β , supΓ ([0,1],A)∩Sβ G is attained at w ∈ / A, then w = Γ (s, e) ∈ Sβ , G(w) a0 . Hence, SΓ = ∅. By Lemma 2.1, we get the conclusion. 2 ˜ B). Set A˜ := {u ∈ A ∩ Sβ : G(u) > b0 }. Then A˜ = ∅ if and only if a0 b0 . Let α := dist(A, If A˜ = ∅, we take α = ∞. Let φ(t) be a positive nondecreasing function on [0, ∞) such that
T φ(t)2 dt for some T α. a0 − b0 < 14 0 1+φ(t)
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Theorem 2.4. Assume (A1 )–(A2 ). If A is ΦA,β -linked to B and b0 ∈ R, then for each ε > 0 there is a uε ∈ E such that uε ∈ Sβ ,
b0 − ε G(uε ) a + ε,
G (uε ) < φ dist(uε , B) .
Proof. By Lemma 2.2, we may assume a = a0 . By negation, we assume that there is an ε > 0 such that ε < φ(dist(u, B)) G (u) for all u ∈ Q := {u ∈ Sβ : b0 − 3ε G(u) a + 3ε}. We note that Q = ∅ for such an ε. We may choose ε < 1 small enough and let T < α such
T φ 2 (t) φ 2 (0) that a − b0 + ε < 14 0 1+φ(t) dt and ε < 4(1+φ(0)) T . Let Q0 := {u ∈ Q: either b0 − 2ε G(u) or G(u) a + 2ε}, Q1 := {u ∈ Q: b0 − ε G(u) a + ε}, and η(u) =
dist(u,Q0 ) dist(u,Q1 )+dist(u,Q0 ) . Then
Q1 = ∅. If Q0 = ∅, we let η ≡ 1. Consider dσ dt = −η(σ )Y0 (σ ), σ (0, u) = u. Then σ (t, u) − u =
t − 0 η(σ )Y0 (σ ) dt, σ ([0, ∞), ±Dμ0 ) ⊂ ±Dμ0 (cf. [22]) and dist(u, B) − t dist σ (t, u), B dist(u, B) + t Further,
d G(σ (t,u)) dt
∀u ∈ E, t 0.
(2.5)
2
φ (dist(σ,B)) − 14 η(σ ) 1+φ(dist(σ,B)) if σ (t, u) ∈ Sβ and
1 G σ (t, u) G(u) − 4
t 0
φ 2 (dist(σ (s, u), B)) ds η σ (s, u) 1 + φ(dist(σ (s, u), B))
if σ (t, u) ∈ Sβ .
Assume for the moment that σ (T , A) ∩ Sβ = ∅ and let W1 := {u ∈ A: σ (T , u) ∈ Sβ }. Then for each u ∈ W1 , we have that σ (t, u) ∈ Sβ for all t ∈ [0, T ]. In particular, Sβ ∩ A = ∅. Now we consider u ∈ W1 . If there is a t1 ∈ [0, T ] such that σ (t1 , u) ∈ / Q1 , then G(σ (T , u)) G(σ (t1 , u)) < b0 − ε. On the other hand, if σ (t, u) ∈ Q1 for all t ∈ [0, T ], then 1 G σ (T , u) G(u) − 4
T 0
1 G(u) − 4
T 0
φ 2 (dist(σ (s, u), B)) ds 1 + φ(dist(σ (s, u), B)) φ 2 (dist(u, B) − s) ds. 1 + φ(dist(u, B) − s)
˜ B) = α > T , then If u ∈ A˜ ∩ W1 , then dist(u, B) dist(A, 1 G σ (T , u) a − 4
dist(u,B)
dist(u,B)−T
φ 2 (τ ) 1 dτ a − 1 + φ(τ ) 4
T 0
φ 2 (τ ) dτ. 1 + φ(τ )
˜ then G(u) b0 , then G(σ (T , u)) < b0 − ε. It implies that G(σ (T , u)) < b0 − ε. If u ∈ W1 \A, Let A1 = σ (T , A), a10 = supA1 ∩Sβ G. If A1 ∩ Sβ = ∅, i.e., W1 = ∅, we set a10 = −∞. Then
W. Zou, S. Li / Journal of Functional Analysis 258 (2010) 3347–3361
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a10 < b0 . Denote ΦA1 ,β := {Γ ∈ Φ: Γ ([0, 1], A1 ) ⊂ Gβ }. We claim ΦA1 ,β = ∅. Indeed, take Γ ∈ ΦA,β . Set Γ1 (s, u) =
σ (2sT , u)−1 , Γ (2s − 1, σ (T , u)−1 ),
s ∈ [0, 1/2], s ∈ (1/2, 1].
Then Γ1 ∈ Φ and G(σ (T − 2sT , A)), G Γ1 s, σ (T , A) = G(Γ (2s − 1, A)),
s ∈ [0, 1/2], s ∈ (1/2, 1].
Thus supΓ1 ([0,1],A1 ) G supΓ ([0,1],A) G β. It follows that Γ1 ∈ ΦA1 ,β . Define a1 := infΓ ∈ΦA1 ,β supΓ ([0,1],A1 )∩Sβ G. To show that a1 is well defined, we have to show that A1 is ΦA1 ,β -linked to B. To this aim, we first prove σ ([0, T ], A) ∩ B = ∅. Recall A ⊂ Gβ , then σ ([0, T ], A) ⊂ Gβ . If it were not true, then there would be σ (t0 , u0 ) ∈ B ∩ Gβ ⊂ Sβ for some ˜ by (2.5), we see that dist(σ (t0 , u0 ), B) t0 ∈ [0, T ] and u0 ∈ A (hence, u0 ∈ Sβ ). If u0 ∈ A, α − t0 α − T > 0, a contradiction. Then we must have u0 ∈ A ∩ Sβ \A˜ and then G(u0 ) b0 , ¯ 0 . But then hence G(σ (t0 , u0 )) < b0 . Unless η(σ (s, u0 )) = 0 for all s ∈ [0, t0 ], i.e., σ (s, u0 ) ∈ Q G(σ (t0 , u0 )) b0 − 2ε. Hence G(σ (t0 , u0 )) < b0 , a contradiction. Let Γ be any map in ΦA1 ,β . Set σ (2sT , u), s ∈ [0, 1/2], Γ1 (s, u) = Γ (2s − 1, σ (T , u)), s ∈ (1/2, 1]. Then Γ1 ([0, 1], A) ⊂ Gβ . Then Γ1 ∈ ΦA,β . Since A is ΦA,β -linked to B, there is an s1 ∈ [0, 1] such that Γ1 (s1 , A) ∩ B = ∅. Since σ (2sT , A) ∩ B = ∅ for all s ∈ [0, 1/2], we must have s1 > 1/2. Hence, Γ (2s1 − 1, σ (T , A)) ∩ B = ∅. Then, A1 is ΦA1 ,β -linked to B. Thus, a1 is well defined. Next, we show that a10 < b0 a1 a. Let Γ ∈ ΦA,β . Set Γ1 (s, u) =
σ (2sT , u)−1 , Γ (2s − 1, σ (T , u)−1 ),
s ∈ [0, 1/2], s ∈ (1/2, 1].
Then Γ1 ∈ Φ and G(σ (T − 2sT , u), G Γ1 s, σ (T , u) = G(Γ (2s − 1, u)),
s ∈ [0, 1/2], s ∈ (1/2, 1].
Thus Γ1 ∈ ΦA1 ,β and a1 supΓ1 ([0,1],A1 )∩Sβ G supΓ ([0,1],A)∩Sβ G. It follows that a1 a. Obviously, we have known b0 a1 . Finally, by the previous arguments, we have a10 < b0 . We now apply Theorem 2.1 to A1 and B, there exists a sequence {uk } such that {uk } ⊂ Sβ , G(uk ) → a1 and G (uk ) → 0. If a1 = b0 , then we may require that dist(uk , B) → 0. However, this contradicts the assumptions at the beginning of the proof. 2 Let {Ak , Bk } be a sequence of pairs of subsets of E, where each Ak is bounded. Assume that there is a βk > 0 such that (A3 ) ΦAk ,βk := {Γ ∈ Φ: Γ ([0, 1], Ak ) ⊂ Gβk } = ∅, ∀k ∈ N. Moreover, Ak is ΦAk ,βk -linked to Bk and Bk ∩ Gβk ⊂ Sβk .
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Let ak,0 := supAk ∩Sβ G and bk,0 := infBk G. Define ak = infΓ ∈ΦAk ,βk supΓ ([0,1],Ak )∩Sβ G. Set k k A˜ k = {u ∈ Ak ∩ Sβ : G(u) > bk,0 } and αk = dist(A˜ k , Bk ). Theorem 2.5. Assume (A2 )–(A3 ) and bk,0 ∈ R. Suppose that αk → ∞ and there exists a θ 0 ak,0 −bk,0 such that (α θ+1 → 0 as k → ∞. Then there exists a sequence {uk } such that uk ∈ Sβk ⊂ k) E\(−P ∪ P) and that bk,0 −
G (uk ) → 0, (1 + dist(uk , Bk ))θ
1 1 G(uk ) ak + , k k
k → ∞.
Proof. Let h(t) := 4(θ + 1)(ak,0 − bk,0 )(1 + t) /(αk ) θ
Then we may check that
αk 0
φ 2 (t) 1+φ(t)
θ+1
,
φ(t) :=
h(t) +
h2 (t) + 4h(t) . 2
dt > 4(ak,0 − bk,0 ). We may assume that bk,0 < ak,0 for
each k. We now apply Theorem 2.4 for each k there is a uk ∈ E such that uk ∈ Sβk , bk,0 − G(uk ) ak + k1 and G (uk ) < φ(dist(uk , Bk )), which implies the conclusion. 2
1 k
An immediate consequence of Theorem 2.5 is the following corollary. Corollary 2.1. Assume that αk → ∞ and there exist a ρ1 > −∞ and ρ2 < ∞ such that bk,0 ρ1 , ak,0 ρ2 . Then there exists a sequence {uk } such that uk ∈ Sβk and that G(uk ) → c ∈ [ρ1 , ∞), G (uk ) → 0, k → ∞. Let E, M, Y, A and B be as in Proposition 2.1. Theorem 2.6. Assume (A1 )–(A2 ). Assume there is a constant c0 such that G(u) c0 for all u ∈ Y and G(u) c0 for all u ∈ {u ∈ M: u∗ = ρ}. Further, G(sy0 + v) Ck for s 0, v ∈ Y and sy0 + v = k, here y0 ∈ M with y0 ∗ = 1 and ρ > 0. If there is a θ 0 such that k limk→∞ kCθ+1 = 0, then there exists a sequence {um } such that um ∈⊂ E\(−P ∪ P) and that G(um ) → c ∈ [c0 , ∞],
G (um ) → 0, (1 + um )θ
m → ∞.
Proof. Let Ak := {sw0 + v = u: s 0, v ∈ Y, u = k} ∪ {u ∈ Y : u k} and B := Bk := {u ∈ M: u∗ = ρ}. Then A˜ k ⊂ {sw0 + v = u: s 0, v ∈ Y, u = k}. Note that sw0 + v − u∗ CY sw0 + v − ρ = CY k − ρ, where CY > 0 is a constant depending on Y only. Then we observe that αk = dist(A˜ k , Bk ) (CY k − ρ)/C∗ → ∞. On the other hand, we may assume a −b that ak,0 > bk,0 (otherwise, we are done!), then k,0θ+1k,0 → 0. Then by Theorem 2.5, we get the conclusion.
2
αk
In particular, if Ck is uniformly bounded (this is indeed true for superlinear cases), then we have
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Theorem 2.7. In Theorem 2.6, if Ck < c∗ for all k > 0, then there exists a sequence {um } such that um ∈ E\(−P ∪ P) and that G(um ) → c ∈ [c0 , c∗ ], G (um ) → 0, m → ∞. Next we consider the existence of sign-changing Cerami’s sequence. Theorem 2.8. Assume (A1 )–(A2 ). Assume that A is bounded and A is ΦA,β -linked to B and ∞ a0 b0 . Let φ(t) be a positive nonincreasing function on (0, ∞) such that 1 φ(t) dt = ∞. Then there exists a sequence {uk } such that {uk } ⊂ Sβ ,
G(uk ) → a,
G (uk ) → 0. φ(uk )
Proof. We note that if the theorem were false, there would be an ε > 0 and a φ(t) satisfying G (u) φ(u) when u ∈ Q := {u ∈ Sβ : |G(u) − a| 3ε} = ∅. Assume b0 < a and 3ε < a − b0 . Let Q0 := {u ∈ Sβ : |G(u) − a| 2ε} and Q1 := {u ∈ Sβ : |G(u) − a| (t,u) dist(u,Q2 ) ε}, Q2 := E\Q0 , η(u) = dist(u,Q . Consider d γdt = −η(γ )Y (γ ), γ (0, u) = u, 1 )+dist(u,Q2 )
) where Y (u) = V0 (u)/V0 (u), V0 is as that in the proof of Theorem 2.1. Then d G(γ dt 1 − 4 η(γ (t, u))φ(u + t) and γ ([0, ∞), ±Dμ0 ) ⊂ ±Dμ0 . By the definition of a, there is a Γ ∈ ΦA,β such that G(Γ ([0, 1], A) ∩ Sβ ) < a + ε. Set ξ := sup{Γ (s, u): s ∈ [0, 1], u ∈ A}.
ξ +T Choose a T > 0 such that 8ε < ξ φ(t) dt. Let W := {v: v = Γ (s, u) ∈ Sβ for some s ∈ [0, 1], u ∈ A}. Next, we only consider those v ∈ W satisfying γ (T , v) ∈ Sβ . If there is a t0 T such that γ (t0 , v) ∈ / Q1 , then we must have G(γ (T , v)) < a − ε. If γ (t, v) ∈ Q1 for
T all t ∈ [0, T ], then we have G(γ (T , v)) a + ε − 14 0 φ(ξ + t) dt < a − ε. Summing up, we see that G(γ [T , Γ ([0, 1], A) ∩ Sβ ] ∩ Sβ ) < a − ε. Now we define Γ ∗ (t, u) = γ (2tT , u) if t ∈ [0, 1/2] and Γ ∗ (t, u) = γ (T , Γ (2t − 1, u)) if t ∈ [1/2, 1]. Then Γ ∗ ∈ Φ. Obviously, G(Γ ∗ ([0, 1], A) ∩ Sβ ) a − ε and G(Γ ∗ ([0, 1], A)) β. This contradicts the definition of a. Next we consider the case of b0 = a. For each u ∈ A ∩ Sβ , we note that G(γ (t, u)) a0 −
1 t 4 0 η(γ (s, u))φ(γ (s, u)) ds. We show that γ (t, A) ∩ B ∩ Sβ = ∅ for all t 0. Otherwise, there is a γ (t, u) ∈ B ∩ Sβ with u ∈ A ∩ Sβ . Then G(γ (s, u)) b0 for all s ∈ [0, t], hence η(γ (s, u)) = 0 which implies that γ (s, u) ∈ Q2 and then G(γ (s, u)) a − ε = b0 − ε and thus γ (s, u) ∈ / B for all s ∈ [0, t]. Similar to the first case, γ (T , Γ ([0, 1], A) ∩ Sβ ) ∩ B ∩ Sβ = ∅. Let Γ ∗ be defined as in the first case, we know that Γ ∗ ([0, 1], A) ∩ B = ∅, this contradicts the fact that A is ΦA,β -linked to B. 2
We have the following immediate consequence on the existence of the sign-changing Cerami’s sequence. Theorem 2.9 (Sign-changing Cerami’s sequence). Assume (A1 )–(A2 ). Suppose that A is bounded and A is ΦA,β -linked to B, a0 b0 . Then there exists a sequence {uk } such that {uk } ⊂ Sβ ,
G(uk ) → a,
1 + uk G (uk ) → 0.
Let a0 , b0 , a, A˜ and α be defined as before. We shall establish the following alternative theorems.
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Theorem 2.10. Assume (A1 )–(A2 ). Suppose that A is bounded and A is ΦA,β -linked to B. Assume a, b0 ∈ R and let φ be a positive nondecreasing function on [0, +∞) such that a0 − b0 <
1 T 4 0 φ(t) dt for some T ∈ (0, α). Then the following alternative holds: Either (a) there is a sequence {uk } such that {uk } ⊂ Sβ ,
G(uk ) → c ∈ [b0 , a],
G (uk ) → 0;
or (b) for each ε > 0 there is a uε ∈ Sβ such that b0 − ε G(uε ) a + ε,
˜ T + ε, G (uε ) < φ dist(uε , B) . dist(uε , A)
Proof. By Theorem 2.1 and Lemma 2.2, we may assume that a = a0 and b0 < a0 . Hence, A˜ = ∅ and α > 0. If option (b) were false, there would be an ε > 0 such that φ(dist(u, B)) G (u) for all u in the set ˜ T + 3ε . Q := u ∈ Sβ : b0 − 3ε G(u) a + 3ε, dist(u, A)
(2.6)
We remark that Q is nonempty as long as ε small enough. We assume that a − b0 + ε <
T 1 0 φ(t) dt, ε < 4 φ(0)T . Let ˜ T + 2ε , Q0 := u ∈ Q: b0 − 2ε G(u) a + 2ε and dist(u, A) ˜ T +ε , Q1 := u ∈ Q: b0 − ε G(u) a + ε and dist(u, A) Q2 := E\Q0 , η(u) = dist(u, Q2 )/ dist(u, Q1 ) + dist(u, Q2 ) . Consider dσ dt = −η(σ )Y (σ ) with σ (0, u) = u, where Y comes from the proof of Theorem 2.8. Then σ ([0, ∞), ±Dμ0 ) ⊂ ±Dμ0 . dist(u, B) − t dist σ (t, u), B dist(u, B) + t;
∀u ∈ E, t 0,
(2.7)
and d G(σdt(t,u)) − 14 η(σ (t, u))G (σ ) − 14 η(σ )φ(dist(σ, B)). It follows that G(σ (t, u))
t G(u) − 14 0 η(σ (s, u))φ(dist(σ (s, u), B)) ds. Now we consider D := {u ∈ A: σ (T , u) ∈ Sβ }. ˜ If there is a t1 ∈ [0, T ] Then for each u ∈ D, σ (t, u) ∈ Sβ for all t ∈ [0, T ]. Suppose that u ∈ A. such that σ (t1 , u) ∈ / Q1 , then G(σ (T , u)) < b0 − ε. On the other hand, if σ (t, u) ∈ Q1 for all T t ∈ [0, T ], then we have G(σ (T , u)) G(u) − 14 0 φ(dist(σ (t, u), B)) dt. Hence 1 G σ (T , u) a − 4
T
1 φ dist(u, B) − t dt a − 4
0
a−
1 4
φ(s) ds dist(u,B)−T
T φ(s) ds < b0 − ε. 0
dist(u,B)
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˜ we also have G(σ (T , u)) b0 − 1 φ(0)T < b0 − ε. Let A1 = σ (T , A) and If u ∈ A ∩ Sβ \A, 4 a1,0 = supA1 ∩Sβ G, a1,0 = −∞ if D = ∅. As in the proof of Theorem 2.4, ΦA1 ,β = ∅. Define d := infΓ ∈ΦA1 ,β supΓ ([0,1],A1 ])∩Sβ G. We now show that A1 is ΦA1 ,β links B. First as in the proof of Theorem 2.4, we know that σ (t, A) does not intersect B for any t ∈ [0, T ]. Now we let Γ be any map in ΦA1 ,β and let Γ ∗ (t, u) = σ (2tT , u) for t ∈ [0, 1/2] and Γ ∗ (t, u) = Γ (2t − 1, σ (T , u)) for all s ∈ [1/2, 1]. Then Γ ∗ ∈ Φ. Since A is ΦA,β -linked to B, we may find a t1 ∈ [0, T ] such that Γ ∗ (t1 , A) ∩ B = ∅. Since σ (2tT , A) ∩ B = ∅ for [0, 1/2], we must have Γ (2t1 − 1, σ (T , A)) ∩ B = ∅ showing that A1 is ΦA1 ,β links B. Next we show that a1,0 < b0 d a. Define Γ1 (t, u) = σ (2tT , u)−1 for t ∈ [0, 1/2] and Γ1 (t, u) = Γ (2t −1, σ (T , u)−1 ) for all t ∈ [1/2, 1]. Then Γ1 ∈ Φ and G(Γ1 (t, σ (T , u))) G(u) if t ∈ [0, 1/2] and G(Γ (2t − 1, u)) for all t ∈ (1/2, 1]. This show that d a. Finally, a1,0 < b0 d are obvious. Applying Theorem 2.1 to A1 , B and d we get the conclusion (a). 2 Corollary 2.2. Assume (A2 ). Let {Ak , Bk } be a sequence of pairs of subsets of E satisfying (A3 ) ak,0 −bk,0 such that αk := dist(A˜ k , Bk ) → ∞ and lim supk→∞ (α θ+1 0, where θ 0 is a constant. k) Then either (1) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [infk bk,0 , ∞) with G (um ) → 0 or (2) there is a sequence {uk } ∈ E\(−P ∪ P) such that bk,0 − k1 G(uk ) ak,0 + k1 , dist(uk , Bk ) → ∞ and G(uk ) − bk,0 = 0, k→∞ (dist(uk , Bk ))θ+1 lim
G (uk ) = 0. k→∞ (dist(uk , Bk ))θ lim
Proof. Let εk → 0 such that ak,0 − bk,0 < εk (αk )θ+1 . Set φ(t) = (θ + 1)2θ+1 εk t θ , Tk = αk /2.
T Then ak,0 − bk,0 < 0 k φ(t) dt. If option (1) is not true, we find a uk ∈ E\(−P ∪ P) such that bk,0 − k1 G(uk ) ak,0 + k1 and dist(uk , A˜ k ) Tk + k1 and G (uk ) < φ(dist(uk , Bk )). These imply the conclusion of the current corollary. 2 Furthermore, we have Theorem 2.11. Let E, M, Y, A and B be as in Proposition 2.1. Assume (A1 )–(A2 ). Assume that G(u) c0 for all u ∈ Y , G(u) c0 for all u ∈ {u ∈ M: u∗ = ρ} and G(tu0 + u) mk for all k t 0, u ∈ Y with tu0 + u = k > ρ, where u0 ∈ M with u0 ∗ = 1. If lim supk→∞ kmθ+1 0 for some θ 0, then either (1) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [c0 , ∞) with G (um ) → 0 or (2) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [c0 , ∞] and G(um ) = 0, m→∞ um θ+1 lim
G (um ) = 0, m→∞ um θ lim
um → ∞.
Proof. Let Ak := {w = su0 + u: s 0, u ∈ Y, w = k} ∪ {u ∈ Y : u k} and Bk := B := {u ∈ M: u∗ = ρ}. Then Ak links Bk with respect to Φ and A˜ k ⊂ {w = su0 + u: s 0, u ∈ Y,
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w = k}. Then similar to the proof of Theorem 2.6, αk := dist(A˜ k , B) → ∞. Apply Corollary 2.2. 2 Theorem 2.12. Assume (A2 ) and lim supR→∞ (−P ∪ P), either
supu=R G R θ+1
0 for some θ 0, then for any w0 ∈ /
(1) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞) with G (um ) → 0 or (2) there is a sequence {um } ∈ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞] and G(um ) = 0, m→∞ um θ+1 lim
G (um ) = 0, m→∞ um θ lim
um → ∞.
Proof. Let Ak := {u: u = k} and Bk := B := {w0 }. Then Ak links Bk with respect to Φ (cf. [17]). Then similarly, using Corollary 2.2 we get the conclusion. 2 Remark 2.3. Assume (A2 ) and lim supR→∞ G (u)
G(u) limu→∞ u θ+1
supu=R G R θ+1
0 for some θ 0 and either
0 or limu→∞ uθ = 0. Then for any w0 ∈ = / (−P ∪ P), there is a sequence {um } ⊂ E\(−P ∪ P) such that G(um ) → c ∈ [G(w0 ), ∞) with G (um ) → 0. If moreover, for any such a (PS) sequence {um }, it converges to a sign-changing critical point, then G has infinitely many sign-changing critical points, where G is not necessarily even. Remark 2.4. The above theorems and corollaries can be traced back to the results due to Martin Schechter [14,17], where there is no any information on the sign-changing property of the critical point or of the (PS) sequence. Also some other earlier results on sign-changing solutions have been obtained in [1,6,12,21,22], etc. 3. Applications In this section, we just give two theorems to illustrate the efficacy of the abstract theorems in Section 2. We will observe that it is very convenient to get sign-changing solutions under very weak hypotheses. We believe the abstract theorems may have more deep applications. Consider the sign-changing solutions to the following Dirichlet boundary value problem
−u = f (x, u), u = 0,
in Ω, on ∂Ω,
(3.1)
where Ω ⊂ RN is a bounded domain with the smooth boundary ∂Ω and finite measure meas Ω := |Ω|. Let E := H01 (Ω) be the usual Sobolev space endowed with the inner prod uct u, v := Ω (u · v) dx for u, v ∈ E and the norm u := u, u1/2 . Let 0 < λ1 < · · · < λm < · · · denote the distinct Dirichlet eigenvalues of − on Ω with zero boundary value. Then each λm has finite multiplicity. The principal eigenvalue λ1 is simple with a positive eigenfunction ϕ1 , and the eigenfunctions ϕm corresponding to λm (m 2) are sign-changing. Let Nm denote the eigenspace of λm . Then dim Nm < ∞. We fix m and let Em := N1 ⊕ · · · ⊕ Nm . We first consider the following assumptions:
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(B1 ) f is a Carathèodory function and f (x, t)t 0 for (x, t) ∈ Ω × R; limt→0 f (x,t) = 0 unit formly for x ∈ Ω.
t (B2 ) C(1 + |t|2 ) 2F (x, t) λm t 2 − κ0 for all x ∈ Ω, t ∈ R, where F (x, t) = 0 f (x, s) ds, C, κ0 > 0 is a constant. (x,t)t > 0 a.e. on Ω. (B3 ) lim inf|t|→∞ 2F (x,t)−f |t| The above assumptions include the following cases:
f (x, t)/t → a f (x, t)/t → b
a.e. x ∈ Ω as t → −∞, a.e. x ∈ Ω as t → ∞.
The existence of solutions of (3.1) is deeply related to the equation −u = bu+ − au− , where u± = max{±u, 0}. Conventionally, the set Σ := (a, b) ∈ R2 : −u = bu+ − au− has nontrivial solutions is called the Fuˇcík spectrum of − (see S. Fuˇcík [2] and M. Schechter [13]). It plays a key role in most results on this aspect. However, so far no complete description of Σ has been found. Only partial answers were given. It was shown in M. Schechter [13] that in the square (λl−1 , λl+1 )2 there are decreasing curves Cl1 , Cl2 (which may or may not coincide) passing through the point (λl , λl ) such that all points above or below both curves in the square (the so-called type (I) region) are not in Σ, while points on the curves are in Σ . Usually, the status of points between the curves (referred to as type (II) region, if the curves do not coincide) is unknown. However, it was shown in T. Gallouët, O. Kavian [3] that when λ is a simple eigenvalue, then points of type (II) region are not in Σ. On the other hand, C.A. Margulies, W. Margulies [7] have shown that there are boundary value problems for which many curves in Σ emanate from a point (λl , λl ) when λl is a multiple eigenvalue. Certainly, these curves are contained in the region (II). We refer the readers to [8–10] for further developments. Recently, related applications also can be seen in [5,4] (and the references cited therein). Their results heavily rely on Σ . Our results in this current paper are independent of Σ. By our assumption, F (x, u) λ41 |u|2 + CF |u|p , ∀x ∈ Ω, u ∈ R, where p ∈ (2, 2∗ ). On the other hand, we have a constant Cp > 0 such that u2p Cp u2 for u ∈ E. Set S0 := {u ∈ ⊥ : u = ρ} for ρ := ( 1 )1/(p−2) . Ek−1 p 8CF Cp Theorem 3.1. Assume (B1 )–(B3 ). If κ0 ( 8Cp1CF )2/(p−2) 4Cp1|Ω| , then Eq. (3.1) has a signchanging solution. Proof. For u ∈ Em−1 we see that G(u)
κ0 |Ω| 2 .
For any u ∈ S0 , we have that
1 p G(u) u2p − CF up 4Cp
1 8Cp CF
2/(p−2)
1 . 8Cp
We choose u0 ∈ Nm with u0 ∗ := u0 p = 1 and set mk = sup{G(tu0 + u): t 0, u ∈ Em−1 , tu0 + u = k}. Then lim supk→∞ mkk 0. Let P := {u ∈ E: u(x) 0 for a.e. x ∈ Ω}. Then P is the positive cone of E. We define Dμ0 := {u ∈ E: dist(u, P) < μ0 }. Then by (B1 ), we may check that conditions (A1 )–(A2 ) are satisfied (see for example [22]). By Theorem 2.11, either
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(1) there is a sequence {uk } ∈ E\(−P ∪ P) such that G(uk ) → c ∈ [ κ0 |Ω| 2 , ∞) with G (uk ) → 0 or (2) there is a sequence {uk } ∈ E\(−P ∪ P) such that G(uk ) → c ∈ [ κ0 |Ω| 2 , ∞] and
lim
k→∞
G(uk ) = 0, uk
lim G (uk ) = 0,
k→∞
uk → ∞.
Assume the second alternative is true. Let u¯ k = uk /uk , then u¯ k → u¯ strongly in L2 . By (B3 ) we know that u¯ ≡ 0. But by (B2 ) we may observe that u¯ ≡ 0. This eliminate the case of (2). Hence, we just have case (1). By standard arguments, we know that {uk } has a convergent subsequence whose limit is still sign-changing (by (B1 )) and its energy is κ0 |Ω| 2 . 2 We define the following numbers (cf. M. Schechter [18,17]): αk = max u2 : u ∈ Ek , u 0, u2 = 1 λk , 2 γk (a) = sup u2 − a u− 2 : u ∈ Ek , u+ 2 = 1 λ1 , 2 Γk (a) = inf u2 − a u− 2 : u ∈ Ek⊥ , u+ 2 = 1 . Then α1 = λ1 = γ1 (a) for any a ∈ R. Moreover, (1) (2) (3) (4) (5)
γk (a), Γk (a) are continuous and decreasing, if k 2, αk < λk , if k 2 and a > αk , then γk (a) < ∞ and the supremum is attained, for each k, the infimum in the definition of Γk (a) is attained, if a λk+1 , then Γk (a) λk+1 .
Theorem 3.2. Assume (B1 ). Assume there are numbers m > αk−1 , m1 > αk , p > 2 and β < λk such that 2 2 m t − + γk−1 (m) t + − W1 (x) 2F (x, t) βt 2 + C0 |t|p + W2 (x) and 2 2 m1 t − + γk (m1 ) t + 2F (x, t) + W3 (x) for all x ∈ Ω, t ∈ R. Further, f (x, t) = β+ (x), t→+∞ t
f (x, t) = β− (x), t→−∞ t
lim
lim
uniformly for x ∈ Ω and β± ∈ L∞ (Ω). Assume moreover, B1 + B2 <
1 C0
2 p−2
p/(p−2) 1 β 1− , 2Cp λk
where Bj = Ω Wj (x) dx. If the equation −u = β+ u+ − β− u− has the only solution u ≡ 0, then (3.1) has a sign-changing solution.
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Proof. For any u ∈ Ek−1 we have that G(u) B1 /2 and for any u ∈ Ek we see that G(u) ⊥ , we have that B3 /2. For any u ∈ Ek−1 p/(p−2) 2 1 1 p−2 1 β B2 G(u) 1− := C1 − 2 C0 2Cp λk 2 for up := u∗ = ( 2C01Cp (1 − λβk ))1/(p−2) := ρ. Let y0 ∈ Nk with y0 p = 1, then we have G(u) B1 /2 for all u ∈ Ek−1 ; G(u + sy0 ) B3 /2 for all s ∈ R and u ∈ Ek−1 ; G(u) C1 > ⊥ with w = ρ. By Theorem 2.6, there is a sign-changing sequence {u } ∈ B1 /2 for all u ∈ Ek−1 ∗ k E\(−P ∪ P) and a c ∈ R such that G(uk ) → c ∈ [C1 , B3 /2], G (uk ) → 0. Since the equation −u = β+ u+ − β− u− has the only solution u ≡ 0, it is easy to see that {uk } is bounded, hence it has a convergent subsequence whose limit is still sign-changing (by (B1 )). 2 Acknowledgment The authors thank so much the anonymous referee for his/her generous and kind suggestions. References [1] T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001) 117–152. ˇ [2] S. Fuˇcík, Boundary value problems with jumping nonlinearities, Casopis Pˇest. Mat. 101 (1976) 69–87. [3] T. Gallouët, O. Kavian, Résultats d’existence et de non existence pour certains problèms demi linéaires à l’infini, Ann. Fac. Sci. Toulouse Math. 3 (1981) 201–246. [4] C. Li, S. Li, Z. Liu, Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems, Calc. Var. Partial Differential Equations 32 (2008) 237–251. [5] C. Li, S. Li, Z. Liu, J. Pan, On the Fuck spectrum, J. Differential Equations 244 (2008) 2498–2528. [6] S. Li, Z.Q. Wang, Ljusternik–Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc. 354 (2002) 3207–3227. [7] C.A. Margulies, W. Margulies, An example of the Fuˇcík spectrum, Nonlinear Anal. TMA 29 (1997) 1373–1378. [8] K. Perera, On the Fuˇcík spectrum of the p-Laplacian, NoDEA Nonlinear Differential Equations Appl. 11 (2) (2004) 259–270. [9] K. Perera, M. Schechter, Computation of critical groups in Fuˇcík resonance problems, J. Math. Anal. Appl. 279 (2003) 317–325. [10] K. Perera, M. Schechter, Double resonance problems with respect to the Fuˇcík spectrum, Indiana Univ. Math. J. 52 (2003) 1–17. [11] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., 1986. [12] M. Ramos, H. Tavares, W. Zou, A Bahri–Lions theorem revisted, Adv. Math. 222 (2009) 2173–2195. [13] M. Schechter, The Fuˇcík spectrum, Indiana Univ. Math. J. 43 (1994) 1139–1157. [14] M. Schechter, The saddle point alternative, Amer. J. Math. 117 (1995) 1603–1626. [15] M. Schechter, Critical point theory with weak-to-weak linking, Comm. Pure Appl. Math. 51 (1998) 1247–1254. [16] M. Schechter, Infinite-dimensional linking, Duke Math. J. 94 (1998) 573–595. [17] M. Schechter, Linking Methods in Critical Point Theory, Birkhäuser, Boston, 1999. [18] M. Schechter, Sandwich pairs in critical point theory, Trans. Amer. Math. Soc. 360 (2008) 2811–2823. [19] M. Schechter, Minimax Systems and Critical Point Theory, Birkhäuser, 2009. [20] M. Schecher, W. Zou, Sign-changing critical points of linking type theorems, Trans. Amer. Math. Soc. 358 (2006) 5293–5318. [21] M. Schechter, W. Zou, On the Brézis–Nirenberg problem, Arch. Ration. Mech. Anal., in press. [22] W. Zou, Sign-changing Critical Points Theory, Springer, New York, 2008.
Journal of Functional Analysis 258 (2010) 3362–3375 www.elsevier.com/locate/jfa
On cocycle twisting of compact quantum groups Kenny De Commer 1 Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium Received 16 September 2009; accepted 4 November 2009 Available online 12 November 2009 Communicated by D. Voiculescu
Abstract We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C∗ -algebra of the quantum group changes. © 2009 Elsevier Inc. All rights reserved. Keywords: Compact quantum group; von Neumann algebraic quantum group; Cocycle twisting; Comonoidal W∗ -Morita equivalence
0. Introduction In the seventies, Kac and Vainerman [11], and independently Enock and Schwartz [6], introduced the notion of (what was called by the latter) a Kac algebra, based on the fundamental work of Kac concerning ring groups in the sixties [10]. Such Kac algebras, which are von Neumann algebras M with a coproduct : M → M ⊗ M satisfying certain conditions, can naturally be made into a category, containing as a full sub-category the category of all locally compact groups, but allowing a duality functor which extends the Pontryagin duality functor on the sub-category of all abelian locally compact groups. Moreover, Kac algebras with a commutative underlying von Neumann algebra arise precisely from locally compact groups, by passing to the L ∞ -space of the latter with respect to the left (or right) Haar measure, and with dual to the multiplication in the group. E-mail address:
[email protected]. 1 Research Assistant of the Research Foundation, Flanders (FWO, Vlaanderen).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.003
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However, these Kac algebras do not cover all interesting examples of what could be called ‘locally compact quantum groups’. In the eighties, Woronowicz introduced ‘compact matrix quantum groups’ [20], which are to be seen as quantum versions of compact Lie groups. He also constructed in that paper certain compact matrix quantum groups SU q (2), which are deformations of the classical SU(2)-group by some parameter q ∈ R with 0 < |q| < 1. These quantum groups do not fit into the Kac algebra framework. The reason for this is that the antipode of these quantum groups is no longer a ∗ -preserving anti-automorphism, but some unbounded operator on the associated C∗ -algebra of the quantum group. A satisfactory theory, covering both the compact quantum groups, the Kac algebras and some isolated examples, was obtained in 2000, when Kustermans and Vaes introduced their C∗ -algebraic quantum groups [12]. In a follow-up paper, they also introduced von Neumann algebraic quantum groups [13], and proved that the C∗ -algebra approach and the von Neumann algebra approach were just different ways to look at the same structure (in that one can pass from the von Neumann algebra setup to a (reduced or universal) C∗ -algebraic setup and back). We also remark that in [18], an slightly alternative approach to von Neumann algebraic quantum groups was presented. In this paper, we will be mainly using the von Neumann algebraic approach, which simply asks for the existence of a coproduct and invariant weights on a von Neumann algebra (see Definition 1.2). An interesting and important part of the theory consists in finding construction methods for von Neumann algebraic quantum groups. For example, in [1] the double product construction was worked out, while in [16] the bicrossed product construction was treated. In [4], we developed another construction method, namely the generalized twisting of a von Neumann algebraic quantum group (by a Galois object for its dual). This covers in particular the twisting by unitary 2-cocycles, special situations of which had been considered by Enock and Vainerman in [7], and by Fima and Vainerman in [8]. When we have two von Neumann algebraic quantum groups, one of which is obtained from the other by the above generalized twisting construction, we call them comonoidally W∗ -Morita equivalent. The reason for this name is simple: the underlying von Neumann algebras of two such quantum groups are Morita equivalent (in the sense of Rieffel [14]), with the equivalence ‘respecting the coproduct structure’. One can also show then that the representation categories of their associated universal C∗ -algebraic quantum groups are unitarily comonoidally equivalent (so that they have the ‘same’ tensor category of ∗ -representations). The special case of cocycle twisting corresponds to those comonoidal Morita equivalences whose underlying Morita equivalence is (isomorphic to) the identity. It was shown in [5] that comonoidal W∗ -Morita equivalence provides a genuine equivalence relation between von Neumann algebraic quantum groups. It is then a natural question to find out which properties are preserved by this equivalence relation. In [2], it was shown for example that the discreteness of a quantum group is preserved, while its amenability is not. It also follows from the results of that paper that ‘being discrete and Kac’ is not preserved. In the general setting of von Neumann algebraic quantum groups, we showed in [5] that the scaling constant is an invariant for comonoidal W∗ -Morita equivalence. In this article, we will show in a very concrete way that the notion of ‘being compact’ is not an invariant. This implies in particular that the representation category of a locally compact quantum group, as a monoidal W∗ -category, does not necessarily remember the topology of the quantum group. In [5], we have also shown, by more general methods, that ‘being compact and Kac’ is an invariant. This article is divided into two sections. In the first section, we recall some notions concerning compact quantum groups, von Neumann algebraic quantum groups and the twisting by unitary
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2-cocycles. In the second section, we recall the definition of Woronowicz’s SU q (2) quantum groups. We then consider an infinite tensor product of these compact quantum groups over varying q’s, with the condition that the q’s go to zero sufficiently fast. Taking a limit of appropriate coboundaries, we obtain a 2-cocycle Ω on this infinite product quantum group which is no longer a coboundary. By giving an explicit formula for a non-finite but semi-finite invariant weight, we conclude that the Ω-twisted quantum group is no longer compact. Remarks on notation: When A is a set, we denote by ι the identity map A → A. We will need the following tensor products: we denote by the algebraic tensor product of two vector spaces over C, by ⊗min the minimal tensor product of two C∗ -algebras, and by ⊗ the spatial tensor product between von Neumann algebras or Hilbert spaces. We will further use the following notations concerning weights on von Neumann algebras. If M is a von Neumann algebra, and ϕ : M + → [0, +∞] an nsf (= normal semi-finite faithful) weight, we denote by Nϕ ⊆ M the σ -weakly dense left ideal of square integrable elements: Nϕ = x ∈ M ϕ x ∗ x < ∞ . We denote by L 2 (M, ϕ) the Hilbert space completion of Nϕ with respect to the inner product x, y = ϕ y ∗ x , and by Λϕ the canonical embedding Nϕ → L 2 (M, ϕ). We remark that Λϕ is (σ -strong)-(norm) closed. We denote Mϕ+ for the space of elements x ∈ M + for which ϕ(x) < ∞, and Mϕ for the complex linear span of Mϕ+ . One can show then that Mϕ = Nϕ∗ · Nϕ , i.e. any element x of Mϕ can be written as ni=1 xi∗ yi with xi , yi ∈ Nϕ . As far as applicable, we use the same notation when considering, more generally, operator valued weights. We also remark here that if M1 and M2 are von Neumann algebras, and ϕ an nsf weight on M2 , we can make sense of (ι ⊗ ϕ) as an nsf operator valued weight from M1 ⊗ M2 to M1 in a natural way: if x ∈ (M1 ⊗ M2 )+ , we let (ι ⊗ ϕ)(x) be the element ω ∈ (M1 )+ ∗ → ϕ (ω ⊗ ι)(x) ∈ [0, +∞] in the extended positive cone of M1 . For more information concerning the theory of weights and operator valued weights, we refer to the first chapters of [15]. 1. Compact and von Neumann algebraic quantum groups As we mentioned in the Introduction, S.L. Woronowicz developed the notion of a compact matrix quantum group in [20] (there called compact matrix pseudogroup). Later, he also introduced the more general notion of a compact quantum group [22]. Definition 1.1. A compact quantum group consists of a couple (A, ), where A is a unital C∗ -algebra, and a unital ∗ -homomorphism A → A ⊗min A such that 1. the map is coassociative: ( ⊗ ι) = (ι ⊗ ),
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and 2. the linear subspaces (A)(1 ⊗ A) :=
(ai )(1 ⊗ bi ) ai , bi ∈ A
i
and (A)(A ⊗ 1) :=
(ai )(bi ⊗ 1) ai , bi ∈ A
i
are norm-dense in A ⊗min A. The compact quantum group is called a compact matrix quantum group if there exist n ∈ N0 and a unitary u = ni,j =1 uij ⊗ eij ∈ A ⊗ Mn (C), such that (uij ) = nk=1 uik ⊗ ukj and such that the uij generate A as a unital C∗ -algebra. Such a unitary is then called a fundamental unitary corepresentation. It is not difficult to show that any compact quantum group (A, ) with A commutative is of the form (C(G), ) for some compact group G, and dual to the group multiplication. Moreover, (A, ) will then be a compact matrix quantum group iff G is a compact Lie group. It is common practice to denote, by analogy, also a general compact quantum group (A, ) as (C(G), ), although this notation is now of course purely formal, since there is no underlying object G. In [22] (and [17] for the non-separable case), it is proven that to any compact quantum group (C(G), ) one can associate a unique state ϕ satisfying (ι ⊗ ϕ)(a) = ϕ(a)1 = (ϕ ⊗ ι)(a),
∀a ∈ C(G).
This state is called the invariant state of the compact quantum group. It is also proven there that with any compact quantum group, one can associate a Hopf ∗ -algebra (Pol(G), ) such that Pol(G) ⊆ C(G) is a norm-dense sub-∗ -algebra, the comultiplication being the restriction of the comultiplication on C(G). The invariant state is then faithful on Pol(G). Conversely, any Hopf ∗ -algebra (Pol(G), ) possessing an invariant state can be completed to a compact quantum group in essentially two ways. The first construction gives the associated reduced compact quantum group. Its underlying C∗ -algebra Cr (G) is given as the closure of the image of the GNS-representation of Pol(G) with respect to the invariant state. The second construction gives the associated universal compact quantum group. Its underlying C∗ -algebra Cu (G) is now the universal C∗ -envelope of Pol(G) (which can be shown to exist). For coamenable compact quantum groups, which are those compact quantum groups possessing both a bounded counit and a faithful invariant state, the reduced and universal construction for the underlying Hopf ∗ -algebra both coincide with the original compact quantum group, so that in this case, one only has to specify the Hopf ∗ -algebra to determine completely the associated C∗ -algebraic structure. With any compact quantum group C(G), one can also associate a von Neumann algebra, which we will denote as L ∞ (G). It is the σ -weak closure of the image of Pol(G) under the GNS-representation for ϕ. Then can be completed to a normal unital ∗ -homomorphism
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: L ∞ (G) → L ∞ (G) ⊗ L ∞ (G). This makes (L ∞ (G), ) into a von Neumann algebraic quantum group, whose definition we now present. Definition 1.2. (See [13].) A von Neumann algebraic quantum group is a couple (M, ), consisting of a von Neumann algebra M and a normal unital ∗ -homomorphism : M → M ⊗ M, such that 1. the map is coassociative: ( ⊗ ι) = (ι ⊗ ), and 2. there exist normal, semi-finite, faithful (nsf) weights ϕ and ψ on M such that, for any state ω ∈ M∗ , we have ϕ (ω ⊗ ι)(x) = ϕ(x),
∀x ∈ Mϕ+
ψ (ι ⊗ ω)(x) = ψ(x),
∀x ∈ Mψ+ .
and
The weights appearing in this definition turn out to be unique (up to multiplication with a positive non-zero scalar), and are called respectively the left and right invariant weights. The von Neumann algebraic quantum groups (L ∞ (G), ) arising from compact quantum groups can be characterized as those von Neumann algebraic quantum groups (M, ) which have a left invariant normal state. One can also show that from such a (L ∞ (G), ), a σ -weakly dense Hopf ∗ -subalgebra (Pol(G), ) can be reconstructed, providing a one-to-one correspondence between von Neumann algebraic quantum groups with an invariant normal state and Hopf ∗ -algebras with an invariant state. We now introduce the notion of a unitary 2-cocycle. Definition 1.3. Let (M, ) be a von Neumann algebraic quantum group. A unitary 2-cocycle for (M, ) is a unitary element Ω ∈ M ⊗ M satisfying the 2-cocycle equation (Ω ⊗ 1)( ⊗ ι)(Ω) = (1 ⊗ Ω)(ι ⊗ )(Ω). It is then easily seen that if (M, ) is a von Neumann algebraic quantum group, and Ω a unitary 2-cocycle for it, we can define a new coproduct Ω on M by putting Ω (x) := Ω(x)Ω ∗ ,
∀x ∈ M.
(1)
The coassociativity of Ω then follows precisely from the 2-cocycle equation. A non-trivial result from [4] states that (M, Ω ) in fact possesses invariant nsf weights, so that it is a von Neumann algebraic quantum group. The construction of these weights is rather complicated, and relies on some deep theorems from non-commutative integration theory. However, in the concrete example which we develop in the next section, we will be able to construct the weights on our cocycle twisted quantum group in a fairly straightforward way.
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2. Twisting does not preserve compactness The compact quantum groups which we will need will be constructed using the ‘twisted SU(2)’ groups from [20] (see also [21]). We recall their definition. Definition-Proposition 2.1. Let q be a real number with 0 < |q| 1. Define Pol(SU q (2)) as the unital ∗ -algebra, generated (as a unital ∗ -algebra) by two generators a and b satisfying the relations
a ∗ a + b∗ b = 1, ab = qba, aa ∗ + q 2 bb∗ = 1, a ∗ b = q −1 ba ∗ , bb∗ = b∗ b.
Then there exists a Hopf ∗ -algebra structure (Pol(SU q (2)), ) on Pol(SU q (2)) which satisfies
(a) = a ⊗ a − qb∗ ⊗ b, (b) = b ⊗ a + a ∗ ⊗ b.
Moreover, this Hopf ∗ -algebra possesses an invariant state ϕ, and has a unique completion to a compact matrix quantum group (C(SU q (2)), ). We will always use a and b to denote the generators of C(SU q (2)). Later on however, when we will be working with a sequence of SU qn (2)’s, we will index these generators by the corresponding index n of qn . We will follow the same convention for the comultiplication, the invariant state, and the other special elements in C(SU q (2)) which we will later introduce. The following proposition gives us more information about how the underlying C∗ -algebra and the associated invariant state of C(SU q (2)) look like. Proposition 2.2. (See [21].) Let q be a real number with 0 < |q| < 1. Let H be the Hilbert space l 2 (N) ⊗ l 2 (Z), whose canonical basis elements we denote as ξn,k (and with the convention ξn,k = 0 when n < 0). Then there exists a faithful unital ∗ -representation of C(SU q (2)) on H , determined by
π(a)ξn,k = 1 − q 2n ξn−1,k , π(b)ξn,k = q n ξn,k+1 .
The invariant state ϕ on C(SU q (2)) is given by 2n ϕ(x) = 1 − q 2 q π(x)ξn,0 , ξn,0 . n∈N
We now introduce special elements which will be of importance later on. Define, in the notation of the previous proposition, matrix units fmn on l 2 (N)⊗l 2 (Z), by putting fmn ξr,k = δr,n ξm,k , with δ the Kronecker delta, and where m, n take values in N. It is clear then that each fmn is in the unital C∗ -algebra generated by π(a). Hence emn := π −1 (fmn ) is an element of C(SU q (2)). We define elements p, p , w ∈ C(SU q (2)) by the following formulas:
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p := e00 ,
(2)
p := e11 ,
(3)
w := e01 + e12 + e20 +
∞
ekk .
(4)
k=3
Thus, with respect to the matrix units emn , the element w is the unitary ⎛
0 ⎜0 w=⎝ 1 0
1 0 0 0
⎞ 0 0 1 0⎟ ⎠. 0 0 0 I
We need to form infinite tensor products of the above quantum groups (see [19] for more detailed information). Let q := (qn )n∈N0 be a sequence of reals satisfying 0 < |qn | < 1. Then we can form an inductive sequence n Pol SU qk (2) , k k=1
of Hopf ∗ -algebras, by tensoring with 1 to the right at each step. We denote the inductive limit as (Pol(SU q (2)), q ). It is easy to see that this is again a Hopf ∗ -algebra with an invariant state ϕq , which is the pointwise limit of the functionals nk=1 ϕk . The associated compact quantum group will then again be coamenable, with underlying C∗ -algebra C(SU q (2)) the universal infinite tensor product of the C(SU qk (2)). This will equal the reduced C∗ -tensor product with respect to the states ϕk , by nuclearity of the C(SU qk (2)) (see the appendix of [21]). We will show now that those (C(SU q (2)), q ) for which q is square summable possess the property enunciated in the abstract, namely: they possess a unitary 2-cocycle which allows to twist them into a non-compact quantum group. Therefore, we now fix some q satisfying the property of square summability. We need to show some properties of L ∞ (SU q (2)). We begin with some well-known general lemmas. Lemma 2.3. Let M ⊆ B(H ) be a von Neumann algebra, and ξ ∈ H a separating vector for M. Let xn be a bounded sequence in M for which xn ξ converges to a vector η. Then xn converges in the σ -strong topology. Lemma 2.4. (See [9, Proposition 1.ı].) Let (Hn , ξn ) be a sequence of Hilbert spaces with distinguished unit vectors, and let (H , ξ ) be their tensor product. Let ηn ∈ Hn be non-zero infinite n ∞ vectors with
η
1. Then ( η ) ⊗ ( ξ n k k ) ∈ H converges to a non-zero vector k=1 k=n+1 ∞ |1 − ηk , ξk | < ∞. k=1 ηk if Recall the special element p (defined at (2)), for which we will now also use index notation. ∞ Lemma 2.5. The sequence ( nk=1 p k ) ⊗ 1 converges σ -strongly in L (SU q (2)) to an operator ∞ ∞ k=1 pk , while the sequence 1 ⊗ ( k=n+1 pk ) converges σ -strongly to 1.
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∞ Proof. Note first that2 the GNS-construction of L (SU q (2)) with respect to ϕq can be iden(L (SU (2)), ξ ), with ξ the separating and tified with ∞ qk k k k=1 cyclic vector associated to ϕk . Combining Lemma 2.3 and Lemma 2.4, we only have to see if (1 − ϕk (pk )) < ∞ to have the first statement of the lemma. Since ϕk (pk ) = 1 − qk2 , this follows by the assumption that qk is a square summable sequence. n ∞ Then we can of course also make sense of x ⊗( ∞ k=n+1 pk ), for any x ∈ k=1 L (SU qk (2)). Since
n 2 ∞ ∞ ∞ ξqk ⊗ pk ξqk ξqk = 1 − ϕqk (pk ) − k=1
k=n+1
k=1
k=n+1
=1−
∞ 1 − qk2 , k=n+1
2 the second statement follows from the convergence of ∞ k=1 (1−qk ) to a non-zero number, which in turn follows easily from the square summability of the qn . 2 ∞ Corollary 2.6. Denote En (x) = sn xsn , where sn = 1 ⊗ ( ∞ k=n+1 pk ) and x ∈ L (SU q (2)). Then En (x) converges to x in the σ -strong topology. Recall the special elements w defined by the formula (4). We again denote this element, when we regard it inside some C(SU qn (2)), as wn . Lemma 2.7. Let 0 < |q| < 1. Let ξ be the cyclic separating vector in the GNS-construction for L ∞ (SU q (2)) w.r.t. the invariant state ϕ. Then ∗ w − a ∗ ξ 3q 2 . Proof. In the matrix representation introduced just after Proposition 2.2, it is easy to calculate that (w − a) w ∗ − a ∗ ⎛ (1 − 1 − q 2 )2 ⎜ 0 ⎜ ⎜ 0 ⎜ =⎜ ⎜ 0 ⎜ ⎜ 0 ⎝ .. . Since 1 −
0 (1 − 1 − q 4 )2 0 0
0 0 2 − q6 − 1 − q6
0 .. .
0 .. .
0 0 − 1 − q6 2 − q8 − 1 − q8 .. .
0 0 0
− 1 − q8 2 − q 10 .. .
⎞ ... ...⎟ ⎟ ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .
√ 1 − c c for 0 c 1, we have
ϕ (w − a) w ∗ − a ∗ 2 2 = 1 − q2 1 − 1 − q2 + q2 1 − 1 − q4 + q4 2 − q6 + q6 2 − q8 + · · ·
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2 1 − q 2 q 4 + q 10 + q 4 + q 6 + · · · 9q 4 . So (w ∗ − a ∗ )ξ 3q 2 .
2
∞ ∗ ∞ Theorem 2.8. The σ -strong∗ limit n=1 ((wn ⊗ wn )n (wn )) exists in L (SU q (2)) ⊗ ∞ ∞ L (SU q (2)), and determines a unitary 2-cocycle Ω for L (SU q (2)). Proof. Remark that (L ∞ (SU q (2)) ⊗ L ∞ (SU q (2)), ϕq ⊗ ϕq ) can be identified with
∞ ∞ ∞ ∞ L SU qk (2) ⊗ L SU qk (2) , (ϕk ⊗ ϕk ) . k=1
k=1
Then by Lemmas 2.3 and 2.4, it is enough to prove that ∞ 1 − (ϕn ⊗ ϕn ) (wn ⊗ wn )n w ∗ < ∞ n
n=1
to know that nk=1 ((wk ⊗ wk )k (wk∗ )) converges in the σ -strong∗ -topology. Denoting by ξk the GNS vector associated with ϕk , we estimate ∞ 1 − (ϕn ⊗ ϕn ) (wn ⊗ wn )n w ∗ n
n=1
∞ ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + (ϕn ⊗ ϕn ) (an ⊗ an ) n a ∗ − n w ∗ n
n=1
+
n
n
n=1
∞ (ϕn ⊗ ϕn ) (an ⊗ an − wn ⊗ wn )n w ∗ n
n=1
∞ ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + (ϕn ⊗ ϕn ) n (an − wn )(an − wn )∗ 1/2 n
n=1
+
n=1
∞
(ϕn ⊗ ϕn ) (an ⊗ an − wn ⊗ wn )(an ⊗ an − wn ⊗ wn )∗ 1/2
n=1
=
∞
1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ n
n=1
+
∞ ∞ (an − wn )∗ ξn + (an ⊗ an − wn ⊗ wn )∗ (ξn ⊗ ξn ) n=1
n=1
∞
∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ + 3 (an − wn )∗ ξn . n
n=1
n=1
K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375
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By the previous lemma and the square summability of the qn , we have ∞ (an − wn )∗ ξn < ∞. n=1
So we only have to compute if ∞ 1 − (ϕn ⊗ ϕn ) (an ⊗ an )n a ∗ < ∞. n
n=1
Now an easy calculation shows that (ϕn ⊗ ϕn ) (an ⊗ an )n an∗ = Since 1 − Thus
1 (1+qn2 )2
1 . (1 + qn2 )2
2qn2 , we can again conclude convergence by square summability of the qn .
Ω=
∞ (wn ⊗ wn )n wn∗ n=1
is a well-defined unitary as a σ -strong∗ limit of unitaries. Since multiplication is jointly continuous on the group of unitaries with the σ -strong∗ topology, Ω will satisfy the 2-cocycle identity since each nk=1 ((wk ⊗ wk )k (wk∗ )) does. 2 Lemma 2.9. Let 0 < |q| < 1, and take w ∈ C(SU q (2)) as defined by the formula (4). Then the invariant state ϕ for C(SU q (2)) satisfies ϕ q −2 ϕ w ∗ · w . Proof. This follows from a straightforward computation, using the concrete form of ϕ as in Proposition 2.2. 2 Hence we can form on L ∞ (SU q (2)) the normal faithful weight ϕΩ := lim
n→∞
n k=1
qk−2
n
∞ ∗ ϕk wk · wk ⊗ ϕk ,
k=1
k=n+1
the limit being taken pointwise on elements of L ∞ (SU q (2))+ . This is a well-defined normal faithful weight, since it is an increasing sequence of normal, faithful, positive functionals. It is clear that ϕΩ is not finite. Proposition 2.10. The weight ϕΩ on L ∞ (SU q (2)) is semi-finite.
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K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375
Proof. We prove that the projections 1 ⊗ ( ∞ k=n+1 pk ), introduced in Lemma 2.5, are integrable with respect to ϕΩ . By Corollary 2.6, this will prove the proposition. 2 2 But wn∗ p n wn = pn , using also the notation of (3). Since ϕn (pn ) = qn (1 − qn ), the integrability p ) follows. 2 of all 1 ⊗ ( ∞ k=n+1 k We now end by proving that ϕΩ is an invariant nsf weight for the couple (L ∞ (SU q (2)), Ω ), where Ω was introduced in Theorem 2.8 and ϕΩ just before Lemma 2.10, and where Ω is the twisted coproduct defined by (1). Theorem 2.11. The nsf weight ϕΩ is a left and right invariant nsf weight for (L ∞ (SU q (2)), Ω ). Proof. We only prove left invariance, since the proof for right invariance follows by symmetry. Denote, for n ∈ N0 ,
Pn = x ⊗
∞
pk
n ∞ L SU qk (2) , x∈
k=n+1
k=1
still using the notation (2), and denote P = n∈N0 Pn . By the proof of Lemma 2.10, we know that P consists of integrable elements for ϕΩ . We first show that also Ω (P) ⊆ M(ι⊗ϕΩ ) , and that (ι ⊗ ϕΩ ) Ω (y) = ϕΩ (y)1 for y in P. Choose n ∈ N0 , and choose x = nk=1 xk ∈ nk=1 L ∞ (SU qk (2)) with all xk positive. Put y =x⊗( ∞ k=n+1 pk ). Then (ι ⊗ ϕΩ )(Ω (y)) is an element in the extended positive cone of L ∞ (SU q (2)). As such, it is the pointwise supremum of the elements zm =
n k=1
αk ⊗
n+m k=n+1
∞
βk ⊗
γk ,
k=n+m+1
regarded as semi-linear functionals on L ∞ (SU q (2))+ ∗ , where αk = qk−2 ι ⊗ ϕk wk∗ · wk (wk ⊗ wk )k wk∗ xk wk wk∗ ⊗ wk∗ , βk = qk−2 ι ⊗ ϕk wk∗ · wk (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ and γk = (ι ⊗ ϕk ) (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ , and where the infinite tensor product is to be seen as a σ -strong limit. We can simplify αk and βk to respectively qk−2 ϕk (wk∗ xk wk )1 and qk−2 ϕk (wk∗ pk wk )1, by invariance of ϕk , while γk satisfies γk qk−2 ϕk wk∗ pk wk 1
K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375
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by Lemma 2.9 and left invariance of ϕk . This implies that zm ϕΩ (y)1. Since zm is increasing, and the invariant state ϕq on L ∞ (SU q (2)) is a faithful normal state, we only have to prove that ϕq (zm ) → ϕΩ (y) to conclude that (ι ⊗ ϕΩ )(Ω (y)) = ϕΩ (y)1. But it is easily seen that ϕq (zm ) converges to ϕΩ (y) if we can show that ∞
ϕk (γk ) − −−−→ 1. m→∞
k=n+m+1
This is equivalent with proving that ∞
(ϕk ⊗ ϕk ) (wk ⊗ wk )k wk∗ pk wk wk∗ ⊗ wk∗ = 0.
1
Since the left hand side equals (ϕq ⊗ ϕq )(Ωq (s)Ω ∗ ), with s = ∞ k=1 pk , the above product is indeed non-zero, by faithfulness of ϕq . One then easily concludes that, since any y ∈ P is a linear combination of elements of the above form, we have Ω (y) ∈ Mι⊗ϕΩ for y ∈ P, with (ι ⊗ ϕΩ )Ω (y) = ϕΩ (y). Next we prove that P is a σ -strong-norm core for the GNS-map ΛϕΩ associated with the nsf ϕΩ for weight ϕΩ . For this, it is enough to prove that sn = 1 ⊗ ( ∞ k=n+1 pk ) is invariant under σt any t ∈ R, since then, using Corollary 2.6, we can conclude that, whenever y ∈ NϕΩ , we will have sn ysn → y in the σ -strong topology and ΛϕΩ (sn ysn ) = sn JϕΩ sn JϕΩ ΛϕΩ (y) → ΛϕΩ (y) in the norm topology (where JϕΩ is the modular conjugation on L 2 (M, ϕΩ )). ϕ By formula (A1.4) of [20] (or by a direct verification), we have that σt k (ak ) = |q|2it ak for ∗ any t ∈ R. Hence ak ak is in the centralizer of ϕk , and so the same is true of pk and pk . Since sn ∈ MϕΩ , we will have xsn , sn x ∈ MϕΩ for x ∈ NϕΩ ∩ Nϕ∗Ω , and then ϕΩ (sn x) = lim
n+m
m→∞
qk−2
k=1
= lim
m→∞
ϕk (pk ·)
qk−2
k=1
= lim
m→∞
⊗
ϕk (pk ·)
qk−2
k=1 ∞
k=n+m+1
= ϕΩ (xsn ).
n+m ∗ ∗ ϕk wk · wk ⊗ ϕk pk wk · wk k=n+1
k=n+m+1
n+m
n
(x)
k=1
∞
⊗
k=n+1
k=n+m+1
n+m
n+m ∗ ∗ ϕk wk · wk ⊗ ϕk wk pk · wk
k=1
∞
⊗
n
n
(x) n+m ∗ ∗ ϕk wk · wk ⊗ ϕk wk · wk pk
k=1
k=n+1
ϕk (·pk )
(x)
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K. De Commer / Journal of Functional Analysis 258 (2010) 3362–3375 ϕ
From this, the equality σt Ω (sn ) = sn for any t ∈ R follows (for example by Theorem VIII.2.6 of [15]). We can now conclude the proof. By left invariance of ϕΩ on P, we can introduce an isometry ∗ on L 2 (SU (2), ϕ ) ⊗ L 2 (SU (2), ϕ ) by putting WΩ q Ω q Ω ∗ WΩ ΛϕΩ (x) ⊗ ΛϕΩ (y) = (ΛϕΩ ⊗ ΛϕΩ ) Ω (y)(x ⊗ 1) for x ∈ NϕΩ and y ∈ P. Then by the core-property of P, we conclude that for x ∈ NϕΩ and any y ∈ NϕΩ , we have Ω (y)(x ⊗ 1) square integrable for ϕΩ ⊗ ϕΩ , with ∗ WΩ ΛϕΩ (x) ⊗ ΛϕΩ (y) = (ΛϕΩ ⊗ ΛϕΩ ) Ω (y)(x ⊗ 1) . Hence ϕΩ (ω ⊗ ι)Ω (y) = ϕΩ (y) for y ∈ Mϕ+Ω and ω a state of the form ·ΛϕΩ (x), ΛϕΩ (x) with x ∈ NϕΩ . Hence the elements (ι⊗ϕΩ )(Ω (y)) and ϕΩ (y)1 in the extended cone of M are equal on a normdense subset of M∗+ . Since the latter element is bounded, the same is true of the former by lower-semi-continuity, and then their equality everywhere follows. 2 Since the C∗ -algebra underlying a non-compact von Neumann algebraic quantum group is non-unital, we obtain the following corollary. Corollary 2.12. There exist a von Neumann algebraic quantum group (M, ) and a unitary 2-cocycle Ω ∈ M ⊗ M, such that the reduced (resp. universal) C∗ -algebra associated to (M, ) is not isomorphic to the reduced (resp. universal) C∗ -algebra associated to (M, Ω ), the Ω-twisted von Neumann algebraic quantum group. Remark 1. Note that the compact quantum group L ∞ (SU q (2)) is not a compact matrix quantum group. It would therefore be interesting to see if one can also twist compact matrix quantum groups into non-compact locally compact quantum groups. By the results of [5], this is closely related to the question whether a compact matrix quantum group can act ergodically on an infinite-dimensional type I -factor. Added in proof: in the meantime, the author succeeded in proving that this phenomenon is indeed also possible for compact matrix quantum groups, in fact even for the compact quantum group SU q (2). He intends to treat this example in detail in a future paper. Remark 2. It follows from the results of [3] that if (C(G), ) is a compact quantum group, and Ω a unitary 2-cocycle inside Pol(G) Pol(G), then the cocycle twisted von Neumann algebraic quantum group is again compact, with Pol(G) as the ∗ -algebra underlying the associated Hopf ∗ -algebra. Hence also the associated C∗ -algebras remain unaltered. Acknowledgments I would like to thank my thesis advisor Alfons Van Daele for his unwavering support of my work, and Stefaan Vaes for valuable suggestions concerning the presentation of the results in this article.
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References [1] S. Baaj, S. Vaes, Double crossed products of locally compact quantum groups, J. Inst. Math. Jussieu 4 (2005) 135–173. [2] J. Bichon, A. De Rijdt, S. Vaes, Ergodic coactions with large quantum multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006) 703–728. [3] K. De Commer, Galois objects for algebraic quantum groups, J. Algebra 321 (6) (2009) 1746–1785. [4] K. De Commer, Galois objects and cocycle twisting for locally compact quantum groups, J. Operator Theory, submitted for publication. [5] K. De Commer, Galois coactions for algebraic and locally compact quantum groups, PhD thesis, K.U. Leuven, 2009. [6] M. Enock, J.-M. Schwartz, Une dualité dans les algèbres de von Neumann, Supp. Bull. Soc. Math. France Mémoire 44 (1975) 1–144. [7] M. Enock, L. Vainerman, Deformation of a Kac algebra by an abelian subgroup, Comm. Math. Phys. 178 (3) (1996) 571–596. [8] P. Fima, L. Vainerman, Twisting and Rieffel’s deformation of locally compact quantum groups. Deformation of the Haar measure, Comm. Math. Phys. 286 (3) (2009) 1011–1050. [9] A. Guichardet, Produits tensoriels infinis et représentations des relations d’anti-commutation, Ann. Sci. Ecole Norm. Sup. 83 (1966) 1–52. [10] G. Kac, Ring groups and the principle of duality, I, II, Trans. Moscow Math. Soc. (1963) 291–339; (1965) 94–126. [11] G. Kac, L. Vainerman, Nonunimodular ring groups and Hopf–von Neumann algebras, Math. USSR Sb. 23 (1974) 185–214. [12] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup. 33 (6) (2000) 837–934. [13] J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (1) (2003) 68–92. [14] M. Rieffel, Morita equivalence for C∗ -algebras and W∗ -algebras, J. Pure Appl. Algebra 5 (1974) 51–96. [15] M. Takesaki, Theory of Operator Algebras II, Springer, Berlin, 2003. [16] S. Vaes, L. Vainerman, Extensions of locally compact quantum groups and the bicrossed product construction, Adv. Math. 175 (1) (2003) 1–101. [17] A. Van Daele, The Haar measure on a compact quantum group, Proc. Amer. Math. Soc. 123 (1995) 3125–3128. [18] A. Van Daele, Locally compact quantum groups. A von Neumann algebra approach, arXiv:math.OA/0602212. [19] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (3) (1995) 671–692. [20] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987) 613–665. [21] S.L. Woronowicz, Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987) 117–181. [22] S.L. Woronowicz, Compact quantum groups, in: Symétries Quantiques, Les Houches, 1995, North-Holland, 1998, pp. 845–884.
Journal of Functional Analysis 258 (2010) 3376–3387 www.elsevier.com/locate/jfa
Ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space near BMO−1 Tsuyoshi Yoneda Institute for Mathematics and its Applications, University of Minnesota, 114 Lind Hall 207 Church Street S.E., Minneapolis, MN 55455, USA Received 18 September 2009; accepted 2 February 2010 Available online 11 February 2010 Communicated by J. Bourgain
Abstract The ill-posedness of the 3D-Navier–Stokes equations in a generalized Besov space which is smaller than −1 (q > 2) is considered. In 2008, Bourgain–Pavlovi´c proved that the 3D-Navier–Stokes equation is B∞,q
−1 ill-posed in B∞,∞ by showing norm inflation phenomena of the solution for some initial data. On the other −1 hand, in 2008, Germain proved that the flow map is not C 2 in the space B∞,q for q > 2. However he did not treat ill-posed problem in such spaces. Thus our result is an extension of these previous results. © 2010 Elsevier Inc. All rights reserved.
Keywords: Navier–Stokes equations; Ill-posedness; Besov spaces
1. Introduction We consider the nonstationary incompressible Navier–Stokes equations in R3 : ⎧ ⎨ ∂u − u + (u · ∇)u + ∇p = 0, div u = 0 in x ∈ R3 , t ∈ (0, T ), ∂t ⎩ u|t=0 = u0 ,
(1)
where u = u(t) = (u1 (x, t), u2 (x, t), u3 (x, t)) and p = p(t) = p(x, t) denote the velocity vector field and the pressure of fluid at the point (x, t) ∈ R3 × (0, T ), respectively, while u0 = (u10 (x), u20 (x), u30 (x)) is a given initial velocity vector field. E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.005
T. Yoneda / Journal of Functional Analysis 258 (2010) 3376–3387
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In this paper we are concerned with the ill-posedness of the Cauchy problem for (1). More precisely for a given function space X = X(R3 ) we say that the Cauchy problem is wellposed in X if there exists a space Y ⊂ C([0, T ), X) such that for all u0 ∈ X there exists a unique solution u ∈ Y for (1), and the flow map u0 → u = Φ(u0 ) is continuous from X to C([0, T ), X). Also we say that the Cauchy problem is ill-posed in X if it is not. The classical results on the existence theorem of the mild solution were shown by Kato [9] and Giga and Miyakawa [7]. Making use of the iteration procedure, they constructed a global solution in the class C([0, ∞); Ln (Rn )) ∩ C((0, ∞); Lp (Rn )) for n < p ∞, when an initial data u0 is small enough in Ln (Rn ). To construct a solution in more general classes of initial data is very important problem. Giga and Miyakawa [8], Kato [10] and Taylor [20] proved the well-posedness in certain Morrey spaces. Cannone [3] and Kozono and Yamazaki [13] investigated this problem in Besov spaces. In particular, Koch and Tataru [12] obtained the global solvability for (1), when the initial data u0 is small enough in BMO−1 . BMO−1 includes above function spaces and it has been considered as the largest space of initial data (see Lemarié-Rieusset [14]). On the other hand, Montgomery-Smith [16] introduced an equation similar to Navier–Stokes equation and proved −1 , which is larger than BMO−1 . In 2008, Bourgain and ill-posedness in the Besov space B∞,∞ −1 by showing norm inflation phenomena of the Pavlovi´c [2] showed that (1) is ill-posed in B∞,∞ solution for some initial data. More precisely, they proved that for any δ > 0 there exists an initial data u0 with u0 B −1 < δ such that the corresponding solution u satisfies u(t)B −1 > 1/δ for ∞,∞ ∞,∞ some t < δ. This shows that the flow map Φ is not continuous. On the other hand, Germain [4] −1 for q > 2. However he did not proved that the flow map is not C 2 in the Besov spaces B∞,q treat ill-posed problem in such spaces. The purpose of the present paper is to show ill-posedness of 3D-Navier–Stokes equations in a generalized Besov space V which is strictly smaller than −1 (q > 2). Thus our result is an extension of both Bourgain–Pavlovi´ B∞,q c’s and Germain’s results. Now we give a sketch of the proof. First we introduce a generalized Besov space V which is −1 (q > 2). The idea is proposed by [18]. Second, we introduce initial data which smaller than B∞,q is composed by a sum of r cosine functions. The idea of setting of the initial data is proposed by [2] and [4]. We take a lacunary frequency set, and the norm of initial data in V is controlled by r. Third, we extract an inflation term from second approximation. Fourth, we estimate the remainder term y. The remainder term satisfies certain integral equation composed by first and second approximations including an inflation term. We also control the remainder term by r. Since we set refined initial data from Bourgain–Pavlovi´c’s setting, we can get better estimate of second approximation than their estimate. According to their setting of initial data, using BMO−1 norm to estimate remainder term y is important. Since we got better estimate of second approximation, we can use the bilinear estimate of a class of bounded uniformly continuous functions (equipped with the L∞ norm). 2. Preliminaries Before presenting our results, we define the Besov spaces, a generalized Besov space which −1 (q > 2), the bilinear operator, j -th approximation and the remainder is smaller than B˙ ∞,q term. Now, we recall the Littlewood–Paley decomposition ψ, ϕj ∈ S, j = 0, 1, . . . , such that supp ψˆ ⊂ |ξ | 5/6 ,
supp ϕˆ ⊂ 3/5 |ξ | 5/3 ,
ϕj (x) = 2nj ϕ 2j x ,
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T. Yoneda / Journal of Functional Analysis 258 (2010) 3376–3387
ˆ )+ 1 = ψ(ξ
∞
ϕˆ j (ξ )
ξ ∈ Rn ,
j =0 ∞
1=
ξ ∈ Rn \ {0} ,
ϕˆj (ξ )
(2)
j =−∞
where fˆ denotes the Fourier transform of f . Definition 1. (See Besov space cf. [1,19].) The inhomogeneous and homogeneous Besov spaces s and B s are defined as follows: ˙ p,q Bp,q s s Bp,q ≡ f ∈ S ; f Bp,q <∞ ,
s B˙ p,q ≡ f ∈ S ; f B˙ s < ∞ , p,q
where
s f Bp,q = ψ ∗ f p +
f B˙ s = p,q
∞ js 2 ϕj ∗ f q p j =0
∞ js 2 ϕj ∗ f q p
1/q ,
1/q
j =−∞
for s ∈ R, 1 p, q ∞. Note that
f ∈ S ; f B˙ s < ∞, f = p,q
∞
ϕj ∗ f in S
s ∼ /P, = B˙ p,q
(3)
j =−∞
holds if s < n/p,
or
s = n/p and q = 1.
(4)
For details, see [13]. Here, P denotes the set of all polynomials. Hence, when s, p, and q satisfy (4), we may modify the definition of homogeneous Besov space as
∞ s
ϕj ∗ f in S . (5) B˙ p,q ≡ f ∈ S ; f B˙ s < ∞, f = p,q
j =−∞
s when s, p, and q satisfy (4). Then, if s, p, and q Hereinafter we use (5) as the definition of B˙ p,q s satisfy (4), B˙ p,q is a Banach space and
f B˙ s = 0 if and only if p,q
f = 0 in S .
−1 −1 We now define a generalized Besov space V which is bigger than B∞,2 but smaller than B∞,q with q > 2. The idea of the definition is based on [18].
T. Yoneda / Journal of Functional Analysis 258 (2010) 3376–3387
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Definition 2 (A generalized Besov space V ). s,α denotes the set of all f ∈ S /P for which the norm (1) B˙ ∞,∞
α js s,α = sup 2 |j | + 1 ϕj ∗ f ∞ < ∞. f B˙ ∞,∞
(6)
j ∈Z
−1,1/2 −1 is the set of all f ∈ S /P that is written as a sum f = f1 + f2 , where (2) V := B˙ ∞,∞ + B˙ ∞,2 −1,1/2 −1 . The norm is defined by f1 ∈ B˙ ∞,∞ and f2 ∈ B˙ ∞,2
f V := f B˙ −1,1/2 +B˙ −1 = ∞,∞
∞,2
inf
f =f1 +f2
f1 B˙ −1,1/2 + f2 B˙ −1 ∞,∞
(7)
∞,2
−1,1/2 −1 for f ∈ B˙ ∞,∞ + B˙ ∞2 , where f1 , f2 run over all admissible representation f = f1 + f2 −1,1/2 −1 ˙ with f1 ∈ B∞,∞ and f2 ∈ B˙ ∞,2 .
Let us investigate the above generalized Besov space. The following proposition is also based on [18]. Proposition 3. −1 −1 for all q > 2. ⊂ V ⊂ B˙ ∞,q (1) B˙ ∞,2
−1,1/2 −1 and B˙ ∞,∞ . (2) There is no inclusion relationship between B˙ ∞,2
Proof of Proposition 3. (1) is easy to check from the definition of the norm. So let us consider (2). Let δz be the Dirac delta function massed at z ∈ R3 . Define f=
∞
aj δ2j
j =1
for {aj }∞ j =1 ⊂ R.
(8)
Then we have f B˙ −1
∞,2
So if we take aj =
j √2 j +1
2−2j |aj |2
1 2
,
j ∈Z
f B˙ −1,1/2 sup 2−j j + 1|aj |. ∞,∞
for j 0, aj = 0 for j < 0, then f B˙ −1,1/2 1, f B˙ −1 = ∞. ∞,∞
−1,1/2 −1 Therefore, B˙ ∞,2 is not included in B˙ ∞,∞ .
Let δj k be Kronecker’s delta. For fixed k ∈ N, if we take aj = j < 0, then we have f B˙ −1,1/2 1, f B˙ −1 = −1 in B˙ ∞,2 .
∞,∞
2
(9)
j ∈Z
∞,2
1 k.
δ 2j √j k j +1
∞,2
for j 0, aj = 0 for −1,1/2
Since k is arbitrary, B˙ ∞,∞ is not included
Now we define the bilinear operator and j -th approximation of the solution to the Navier– Stokes equations.
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T. Yoneda / Journal of Functional Analysis 258 (2010) 3376–3387
Definition 4 (Bilinear operator). For w1 , w2 ∈ (L1 (0, T : L∞ ))3 with w1 ⊗ w2 ∈ (L1 (0, T : L∞ ))3×3 , let t B(w1 , w2 ) :=
∇ · e(t−τ ) P w1 (τ ) ⊗ w2 (τ ) dτ,
0
where P is the Helmholtz projection. The bilinear estimate is important to consider ill-posed problem. According to [2], they used the bilinear estimate in BMO−1 provided by Koch and Tataru [12]. In this paper we use the following estimate. See [5,15] for example (for elementary proof, see [6]). Proposition 5. There exists a constant c > 0 such that t ∇e Pf
∞
ct −1/2 f ∞
for t > 0, f ∈ L∞ .
Corollary 6. Let w1 , w2 ∈ (L1 (0, T : L∞ ))3 be such that w1 ⊗ w2 ∈ (L1 (0, T : L∞ ))3×3 . Then we have the following estimate: B(w1 , w2 )
t
∞
0
C w1 (τ ) w2 (τ ) dτ. ∞ ∞ (t − τ )1/2
Now we define j -th approximation of the solution u, and the remainder term y. Definition 7. Let ⎧ u1 = u1 (t) := et u0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ uj = uj (t) :=
B(uk1 , uk2 )
for j 2,
k1 +k2 =j, 1k1 ,k2 j −1
⎪ ⎪ ⎪ ⎪ uk . y = y(t) := ⎪ ⎪ ⎩ k3
For example, u2 = B(u1 , u1 ) and u3 = B(u1 , u2 ) + B(u2 , u1 ). Remark 8. By a formal calculation, we see that u = u1 + u2 + y. Moreover, the remainder term satisfies the following integral equation: y = B(y, y) + B(y, u2 + u1 ) + B(u2 + u1 , y) + B(u2 , u2 ) + B(u2 , u1 ) + B(u1 , u2 ) (10) on (0, ∞) with the initial condition y(0) = 0. Throughout this paper, we only treat periodic functions, since the following embedding inequality is necessary for (13). Let BUC be the space of all bounded uniformly continuous functions equipped with the L∞ -norm.
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Remark 9. A fundamental observation shows that for L > 0, if an initial data u0 ∈ BUC is a periodic function in [0, L)3 and its mean value is zero, then the solution u(t) ∈ BUC (t > 0) to Eq. (1) is also periodic in [0, L)3 and its mean value is also zero. Moreover we have the following embedding inequality: u(t) CL u(t) (11) V ∞ for t > 0. According to the above remark, homogeneous and inhomogeneous type function spaces are equivalent in this paper. We always denote by C > 0 universal constants unless no confusion occurs. 3. Main result Recall that BUC is the space of all bounded uniformly continuous functions equipped with the L∞ -norm. The main result is as follows. Theorem 10. For any δ > 0, there exist real-valued initial data u0 ∈ BUC with div u0 = 0 in S
and u0 V < δ such that the corresponding solution u exists in C([0, T ] : BUC) (T < δ) with u(T )V > 1/δ. −1 or F˙ −1 (Triebel–Lizorkin spaces) Remark 11. The same result is true if we replace V by B˙ ∞,q ∞,q for q > 2. The proof is quite similar to the case of V . Thus we omit its detail.
Remark 12. The solution (u, p) is unique in L∞ ((0, T ) × R3 ) × L1loc ((0, T ) : L1,φ ), where L1,φ is generalized Campanato spaces which include BMO (see [11,17]). Proof of Theorem 10. First, we set initial data which bring norm inflation phenomena. In order to set initial data, we need several definitions. For sufficiently small > 0, let Γ1 , Γ2 : N → R be such that Γ1 (m) :=
m
s −1 ,
1− 2
Γ2 (m) := Γ1
(m)
s=1
for m ∈ N. We take sufficiently small C1 = C1 ( ) > 0 and sufficiently large C2 = C2 ( ) > 0, and let us take Γ3 : N → N satisfying C1 Γ13 (m) Γ3 (m) C2 Γ13 (m) for m ∈ N. Let us set coefficients v 0 := (0, 0, 1) and v 1 := (0, 1, 0), and let {ks0 }rs=1 ⊂ N and {ks1 }rs=1 ⊂ N be frequency sets satisfying the following property, ks0 := 23s T −1/2 a0 ,
ks1 := ks0 + a1
(s = 1, . . . , r),
where a0 = (1, 0, 0), a1 = (0, 0, 1) and T = (1/Γ32 (r)) for r ∈ N. The constant T with r 1 will be the existence time. To obtain the embedding inequality (11), we require Γ3 to be integervalued.
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0 0 2 −1 , 1 |k 0 | |k 1 | 2|k 0 | for Remark 13. It is easy to see that |ks0 | 4 s−1 s s s s =1 |ks |, |k1 | = 64T s = 1, . . . , r, and ⎧ 0 0 a · v = 0, ⎪ ⎪ ⎨ 1 0 a · v = 1, (12) 0 1 ⎪ ⎪ ⎩ a · v = 0, a 1 · v 1 = 0. If the space dimension is two, we cannot obtain the above properties. Thus, ill-posed result is obtained only for the space dimension n 3. We set the initial data as follows: u0 (x) :=
r 1 0 −1/2 0 ks s v cos ks0 · x + v 1 cos ks1 · x . Γ2 (r) s=1
Remark 14. We state properties of the initial data. • In [2], the coefficients of v 0 and v 1 depend on s. • A direct calculation yields div u0 = 0. • Since {ks0 }rs=1 and {ks1 }rs=1 are lacunary, and homogeneous and inhomogeneous type function spaces are equivalent, we see u0 V u0 B˙ −1,1/2 = sup 2−j j + 1ϕj ∗ u0 ∞ ∞,∞
= sup 2 j ∈N
−j
j ∈Z
j + 1ϕj ∗ u0 ∞
√ r j + 1 |ks0 | ϕj ∗ cos k 0 · x + ϕj ∗ cos k 1 · x s s ∞ ∞ j 1/2 2 s s=1 √ j +1 1 sup 1/2 → 0 as r → ∞. Γ2 (r) j ∈N j
1 sup Γ2 (r) j ∈N
Our strategy is to decompose second approximation u2 as Bourgain and Pavlovi´c [2] did. Since r 1 u2 = B(u1 , u1 ) = 2 Γ2 (r) s,s =1
t
σ e(t−τ ) PUs,s
(τ, x) dτ,
σ ∈{0,1}3 0
where σ ,σ2 ,σ3
σ 1 Us,s
(τ, x) = Us,s
(τ, x) 0 0 −1/2 −(|k σ1 |2 +|k σ2 |2 )τ s
e s := (1/2)ks ks ss × v σ1 sin ksσ1 + (−1)σ3 ksσ 2 · x v σ2 · ksσ1 + (−1)σ3 ksσ 2 ,
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we can decompose u2 as u2 = u02 + u12 + u˜ 2 , where ⎧ t r ⎪ ⎪ ⎪ (σ1 ,σ2 ,0) 0 2 ⎪ u2 := 1/Γ2 (r) e(t−τ ) PUs,s (τ, x) dτ, ⎪ ⎪ ⎪ ⎪ s=1 σ ,σ ∈{0,1} ⎪ 1 2 0 ⎪ ⎪ ⎪ ⎪ t ⎪ r ⎨ (σ1 ,σ2 ,1) 1 2 u := 1/Γ2 (r) e(t−τ ) PUs,s (τ, x) dτ, ⎪ 2 ⎪ s=1 σ ,σ ∈{0,1} ⎪ 1 2 0 ⎪ ⎪ ⎪ ⎪ ⎪ t r ⎪ ⎪ ⎪ σ 2 ⎪ u˜ := 1/Γ2 (r) e(t−τ ) P Usσ ,s (τ, x) + Us,s ⎪
(τ, x) dτ. ⎪ ⎩ 2
s=1 s <s σ ∈{0,1}3 0
We note u12 is an inflation term. Remark 15. By (12), we see that 1 for (σ1 , σ2 ) = (1, 0), v σ2 · ksσ1 + (−1)σ3 ksσ 2 = 0 otherwise. Lemma 16. We have the following key estimates of u12 , u02 , u˜ 2 , u1 and y. • The estimate of the inflation term u12 . We have the following inequalities: 1 u (t) 2
∞
C
Γ1 (r) = CΓ1 (r) Γ22 (r)
for t > 0, 1 u (t) C Γ1 (r) = CΓ (r) 2 1 V Γ22 (r) for T /64 t 1 with sufficiently large r. • The estimate of the first approximation and another part of the second approximation (exclude the inflation term). We have the following inequalities: ⎧ u0 (t) C , ⎪ ⎪ 2 ⎪ ∞ ⎪ Γ22 (r) ⎪ ⎪ ⎪ ⎨ u˜ 2 (t) C , ∞ ⎪ Γ22 (r) ⎪ ⎪ ⎪ ⎪ ⎪ C ⎪ ⎩ et u0 ∞ Γ2 (r)t 1/2
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for 0 < t < T = (1/Γ32 (r)) and t e u0
7 1 Γ3 (r) − = Γ1 2 2 (r) Γ2 (r)
∞
for T /64 < t < T .
• The estimate of the remainder term. 3
−1
Let ρ1 (r) := Γ1 (r)/Γ23 (r) = Γ1 2 2 (r) and ρ2 (r) := Γ12 (r)/(Γ24 (r)Γ3 (r)) CΓ1− (r). (Note that ρ1 (r), ρ2 (r) → 0 as r → ∞.) There exists the remainder term y ∈ C([0, T ] : BUC), y(0) = 0 satisfying the following estimate: y(t) 2C ρ1 (r) + ρ2 (r) ∞ for 0 < t < T = 1/Γ32 (r) with sufficiently large r. We postpone to prove the above lemma. We now prove the main theorem. By the embedding inequality (11) and Lemma 16, we have the following estimate: u(t) u1 (t) − u0 (t) − u˜ 2 (t) − et u0 − y(t) 2 2 V V ∞ ∞ ∞ ∞ CΓ1 (r) −
C Γ22 (r)
−
Γ3 (r) − 2C ρ1 (r) + ρ2 (r) CΓ1 (r). Γ2 (r)
for T /64 < t < T (T = 1/Γ32 (r)) with sufficiently large r. This is the desired estimate.
(13) 2
4. Proof of the key lemma In this section, we prove Lemma 16. First we estimate the inflation term u12 . It is easy to see that 001 111 ≡ 0 and Us,s ≡ 0 for s = 1, . . . , r. Us,s 011 ≡ 0 for s = 1, . . . , r and Since v 1 · a1 = 0 and v 0 · a1 = 1, we also see that Us,s Pv 1 (v 0 · a1 ) sin(a1 · x) = v 1 sin x3 . Thus we have the following equality:
1 Γ22 (r) s=1 r
u12 (x, t) =
=
t (1,0,1) e(t−τ ) PUs,s (τ, x) dτ 0
r 1 −1 0 2 s ks Γ22 (r) s=1
t
e−(t−τ )|a1 | e−(|ks | 2
1 2 +|k 0 |2 )τ s
dτ Pv 1 v 0 · a1 sin(a1 · x).
0
We now estimate lower and upper bound of u12 . By Remark 13, we have 0 2 k
t
s
0
e−(t−τ )|a1 | e−(|ks | 2
1 2 +|k 0 |2 )τ s
2 1 − e−(|ks1 |2 +|ks0 |2 −1)t 0 2 e−t 1 − e−2|k1 | t C dτ = e−t ks0 0 2 1 2 |ks | + |ks | − 1
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for t ∈ [T /64, 1] = [1/|k10 |2 , 1]. Since we easily estimate sin x3 in V , we have 1 u (t) C Γ1 (r) 2 V Γ22 (r) for t ∈ [T /64, 1] = [1/|k10 |2 , 1] with sufficiently large r. We also have u12 (t)∞ B(Γ1 (r)/ Γ22 (r)) for some constant B > C and t > 0. Thus we complete the estimate of the inflation term. Next we consider u˜ 2 ∞ , u02 ∞ and et u0 ∞ . However, we only calculate u˜ 2 ∞ , since the calculations of u02 ∞ and et u0 ∞ are easy. Let t Js,s :=
σ1 2 σ σ σ | +|ks 2 |2 )τ −|ks 1 +(−1)σ3 ks 2 |2 (t−τ )
e−(|ks
dτ
0
=e
σ
σ
−|ks 1 +(−1)σ3 ks 2 |2 t
σ3 σ1 σ2 e2(−1) ks ·ks t − 1 . 2(−1)σ3 ksσ1 · ksσ 2
A direct calculation shows that σ u˜ 2
r 2 −1/2 0 0 ss ks ks |Js,s |. Γ22 (r) s=1 s <s
By Remark 15, we only need to consider just two cases, the case σ = (σ1 , σ2 , σ3 ) = (1, 0, 1) and the case σ = (σ1 , σ2 , σ3 ) = (1, 0, 0). If σ3 = 1, the function
e
σ σ 2(−1)σ3 ks 1 ·k 2 t s −1 σ σ 2(−1)σ3 ks 1 ·ks 2 t
is uniformly
bounded with respect to t > 0. But if σ3 = 0, it is not. Thus we need to distinguish the cases σ3 = 0 and σ3 = 1. The case σ = (1, 0, 1). Since s (s − 1) < s, we see that 2 2
−ksσ1 + (−1)σ3 ksσ 2 = −23s T −1/2 a0 − 23s T −1/2 a0 − a1 2 2
− 23s − 23s T −1/2 a0 − 23s − 23(s−1) T −1/2 a0 49 0 2 ks . =− 64 Thus we have |Js,s | Cte−C|ks | t for t > 0. 0 2
The case σ = (1, 0, 0). A direct calculation yields Js,s = e Thus we have
σ
2
σ
σ
−|ks 1 +ksσ |2 t 2ks 1 ·ks 2 t
e
σ1
σ2 t
1 − e−2ks ·ks
2ksσ1 · ksσ 2
σ1 2 σ | +|ks 2 |2 )t
Cte−(|ks
Cte−|ks | t . 0 2
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|u˜ 2 |
r C −1/2 0 0 −C|ks0 |2 t ks ks te ss Γ22 (r) s=1 s <s
C 0 2 −C|ks0 |2 t C ks te 2 , 2 Γ2 (r) s=1 Γ2 (r) r
0 0 2 −C|ks0 |2 t C j j where we used ∞ j ∈Z 2 exp(−2 ) < ∞ for t > 0 and 4 s <s |ks | s=1 t|ks | e |ks0 |. Thus we complete the estimate of the term u˜ 2 . Now we consider the remainder term y which satisfies (10). Recall that ρ1 (r) = 3
−1
Γ1 (r)/Γ23 (r) = Γ1 2 2 (r) and ρ2 (r) = Γ12 (r)/(Γ24 (r)Γ3 (r)) CΓ1− (r). Let ρ3 (r) := Γ1 (r)/(Γ22 (r)Γ3 (r)) CΓ1−2 (r). Note that ρ1 (r) → 0, ρ2 (r) → 0 and ρ3 (r) → 0 as r → ∞. Also we recall that T = 1/Γ32 (r). By the above estimates of u1 ∞ , u2 ∞ and Corollary 6, and since the worst term to estimate in B(u1 , u2 ) or B(u2 , u1 ) is B(u12 , u1 ), we have the following inequality: B u1 (t), u2 (t) + B u2 (t), u1 (t) C B u1 (t), u1 (t) Cρ1 (r). 2 ∞ ∞ ∞ Since the worst term to estimate in B(u2 , u2 ) is B(u12 , u12 ) and B(y, u2 ) is B(y, u12 ), we have 1 B u (t), u1 (t) Cρ2 (r) C 2 2 ∞ ∞
B u2 (t), u2 (t) and B u1 (t) + u2 (t), y(t)
∞
+ B y(t), u1 (t) + u2 (t) ∞ C B y(t), u12 (t) ∞ Cρ3 (r) sup y(τ ) ∞ 0<τ T
for 0 < t < T = 1/Γ32 (r) with sufficiently large r. Therefore we have sup y(t) ∞ Cρ3 (r) + sup y(t) ∞ sup y(t) ∞ + C ρ1 (r) + ρ2 (r) .
0
0
(14)
0
By an absorbing argument, we have the following a priori bound, sup y(t) ∞ 2C ρ1 (r) + ρ2 (r) 0
for sufficiently large r in order to satisfy Cρ3 (r) + 2C(ρ1 (r) + ρ2 (r)) 1/2. By an iteration procedure and the above a priori bound, we obtain existence of y(t) in C([0, T ); BUC). Acknowledgments The author is deeply grateful to Professor Dongho Chae for encouragement and valuable suggestions. This paper developed during a stay of the author at the Department of Mathematics at Sungkyunkwan University. The author would like to thank the Department of Mathematics for support and hospitality. The author is also grateful for Professors Yoshikazu Giga, Kyungkeun
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Kang, Jihoon Lee and Hideyuki Miura for their interest and expository comments. The author also would like to thank Professors Okihiro Sawada and Nataša Pavlovi´c for discussion about our refined initial data. Finally, the author would like to thank Professor Yoshihiro Sawano and the referee for advices about a generalized Besov space. References [1] J. Bergh, J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin–New York–Heidelberg, 1976. [2] J. Bourgain, Jean N. Pavlovi´c, Ill-posedness of the Navier–Stokes equations in a critical space in 3D, J. Funct. Anal. 255 (2008) 2233–2247. [3] M. Cannone, A generalization of a theorem by Kato on Navier–Stokes equations, Rev. Mat. Iberoamericana 13 (1997) 515–541. [4] P. Germain, The second iterate for the Navier–Stokes equation, J. Funct. Anal. 255 (2008) 2248–2264. [5] Y. Giga, K. Inui, S. Matsui, On the Cauchy problem for the Navier–Stokes equations with nondecaying initial data, in: Advances in Fluid Dynamics, in: Quad. Mat., vol. 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999, pp. 27–68. [6] Y. Giga, H. Jo, A. Mahalov, T. Yoneda, On time analyticity of the Navier–Stokes equations in a rotating frame with spatially almost periodic data, Phys. D 237 (2008) 1422–1428. [7] Y. Giga, T. Miyakawa, Solutions in Lr of the Navier–Stokes initial value problem, Arch. Ration. Mech. Anal. 89 (1985) 267–281. [8] Y. Giga, T. Miyakawa, Navier–Stokes flow in R3 with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations 14 (1989) 577–618. [9] T. Kato, Storing Lp -solutions of the Navier–Stokes equation in Rm with applications to weak solutions, Math. Z. 187 (1984) 471–480. [10] T. Kato, Strong solutions of the Navier–Stokes equation in Morrey spaces, Bull. Braz. Math. Soc. (N.S.) 22 (1992) 127–155. [11] J. Kato, The uniqueness of nondecaying solutions for the Navier–Stokes equations, Arch. Ration. Mech. Anal. 169 (2003) 159–175. [12] H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001) 22–35. [13] H. Kozono, M. Yamazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994) 959–1014. [14] P.G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Res. Notes Math., Chapman and Hall/CRC, Boca Raton, FL, 2002. [15] Y. Maekawa, Y. Terasawa, The Navier–Stokes equations with initial data in uniformly local Lp spaces, Differential Integral Equations 19 (2006) 369–400. [16] S. Montgomery-Smith, Finite time blow up for a Navier–Stokes like equation, Proc. Amer. Math. Soc. 129 (2001) 3025–3029. [17] E. Nakai, T. Yoneda, Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier–Stokes equations, preprint. [18] Y. Sawano, personal communication. [19] Y. Taniuchi, T. Tashiro, T. Yoneda, On the two-dimensional Euler equations with spatially almost periodic initial data, J. Math. Fluid Mech., in press. [20] M. Taylor, Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations, Comm. Partial Differential Equations 17 (1992) 1407–1456.
Journal of Functional Analysis 258 (2010) 3388–3412 www.elsevier.com/locate/jfa
Quantization of abelian varieties: Distributional sections and the transition from Kähler to real polarizations Thomas Baier, José M. Mourão ∗ , João P. Nunes Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Received 21 September 2009; accepted 26 January 2010 Available online 4 February 2010 Communicated by L. Gross
Abstract We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil–Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all non-negative polarizations. The Blattner–Kostant–Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of Sp(2g, R). Moreover, these maps are shown to be unitary. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantization; Abelian varieties; Theta functions; Bohr–Sommerfeld fibers
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries on the prequantum line bundle and its sections . 2.1. The prequantum line bundle L . . . . . . . . . . . . . . . . 2.2. Spaces of sections of Lk . . . . . . . . . . . . . . . . . . . . 2.3. Invariant complex structures and theta functions . . . . 2.4. Invariant polarizations . . . . . . . . . . . . . . . . . . . . . .
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* Corresponding author.
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[email protected] (J.M. Mourão),
[email protected] (J.P. Nunes). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.023
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2.5. The action of the real symplectic group . . . . . . . . A distributional construction of the quantum bundle . . . . 3.1. The weak equations of covariant constancy . . . . . . 3.2. The half-form correction . . . . . . . . . . . . . . . . . . 3.3. The Weil representation of the metaplectic group . . 3.4. The extended quantum Hilbert bundle . . . . . . . . . 4. The BKS pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The BKS pairing on the extended quantum bundle . 4.2. Further properties of the BKS pairing . . . . . . . . . 5. Tropical theta divisors . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.
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1. Introduction Abelian varieties provide a very rich example of interplay between ideas of algebraic geometry and of geometric quantization. A general problem of great interest in the geometric quantization of a large class of symplectic manifolds is the dependence of quantization on the complex structure [3,16]. (See also [2].) As is well known, the holomorphic quantization of a symplectic torus, once it is equipped with the structure of abelian variety, produces the space of theta functions. In this case, quantizations in different complex structures yield spaces of theta functions which are naturally unitarily equivalent. This equivalence between holomorphic quantization spaces has been obtained in [3] by means of the parallel transport induced by a heat equation. See also [7,8] for an intrinsically finite-dimensional approach. In this work, we consider the geometric quantization of the standard symplectic torus of dimension 2g, (T2g , ω), in holomorphic and real non-negative invariant polarizations and for any level k ∈ N. Due to the group structure of T2g , and consequent triviality of the tangent bundle, the space of invariant polarizations is canonically identified with the non-negative Lagrangian Grassmannian of a fixed symplectic vector space, C2g . Thus, one obtains some similarities with the geometric quantization of a vector space [3,21]. For instance, below we construct a Blattner–Kostant–Sternberg (BKS) pairing, between quantization spaces associated with invariant non-negative polarizations, which is transitive, once the half-form correction is included. The main novel point in our approach is the treatment of the equations of covariant constancy of wave functions, for real and holomorphic polarizations, from a unified analytical stand point, considering the operators of covariant derivation to act on distributional sections of the prequantum line bundle. For real polarizations this yields, as expected, distributional sections supported on Bohr–Sommerfeld fibers; still, it does not always produce the same result as the more traditional cohomological wave function approach to quantization of [26], as can be seen in the case of toric manifolds comparing [4] and [15]. For previous related work on the geometric quantization of abelian varieties see [2], where, in particular, it is explicitly shown that the quantization spaces for any two invariant real polarizations are isomorphic. In the language of geometric quantization, the space of invariant complex structures corresponds to the space of positive Lagrangian subspaces, which is usually identified with Siegel upper half-space Hg . To allow the study of the dependence of quantization on the complex structure, including the case of degenerate complex structures on the boundary of Hg , it is convenient
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to consider also another chart that uses the closed Siegel disc, which we denote by Dg . In this way, we obtain a global parameter also for the non-positive definite part (that becomes identified with ∂Dg ) of the Lagrangian Grassmannian. A crucial contribution for the nice behavior of quantization (or rather of the relation between any two quantizations) on the boundary of the upper half-space is given by the half-form correction. As in many other examples in geometric quantization [29], the half-form is necessary for the unitarity of the pairing maps relating quantizations in different polarizations. Symmetry groups play an important role in this work. On one hand, the BKS pairing between different invariant polarizations is intimately related with the action of Sp(2g, R) on Dg . This action does not represent a geometric symmetry of the manifold we are quantizing, but has, instead, an analytical flavor connected to the natural representation of the metaplectic group on L2 (Rg ). Also, one of the most interesting feature of the geometric quantization of T2g , that is not present in the linear case R2g (see [21]), besides the appearance of distributional sections of quite different nature over different degenerate Lagrangian subspaces, is the interplay with natural invariance groups of certain data of the prequantization of the torus. Namely, there are natural geometric actions of the integer symplectic group, Sp(2g, Z), and of the group (Z/kZ)2g on T2g that leave the holonomies of the prequantum line bundle representing kω invariant. While the latter gives rise to the finite Heisenberg group when lifted to the line bundle, the former gives origin, for non-degenerate complex structures, to the classical algebro-geometric thetatransformation formula (see, for example, [6,24,19]). We will address some of these issues in [5]. In [22], Manoliu shows (for even level k) that the BKS pairing maps relating two reducible real polarizations is unitary if one takes into account the half-form correction. In this work, where we consider arbitrary level k, we obtain an analogous result for all holomorphic polarizations, while the real polarizations are included as limiting cases on the boundary of the space of complex structures. Abelian varieties, therefore, give one more family of symplectic manifolds, in addition to noncompact complex Lie groups (see [14,9]), where half-form corrected holomorphic quantizations in different complex structures, including real polarizations as degenerate cases, can be related by unitary BKS pairing maps [25,29]. The paper is organized as follows. After some preliminaries in Section 2, we use the Sp(2g, R) action in Section 3 to give a unified analytical description of half-form quantization in complex, real, or mixed polarizations, resulting in Theorem 3.10. In Section 4, we show that the BKS pairing maps are unitary and transitive. Section 5 contains a brief illustration of how tropical geometry can be seen to emerge as the complex structure degenerates. 2. Preliminaries on the prequantum line bundle and its sections 2.1. The prequantum line bundle L Let (T2g = R2g /Z2g , ω) be the standard even-dimensional torus, with g periodic coordinates (x, y) with x, y ∈ Rg , and the invariant symplectic form given by ω = i=1 dyi ∧ dxi . Consider a prequantization of T2g given by the line bundle L, representing the cohomology class of the symplectic form, with Hermitian structure and a compatible connection with curvature −2πiω. L is defined by L = R2g ×Z2g C, where the Z2g action is λ · (u, ζ ) = u + λ, α(λ)e−πiω(u,λ) ζ ,
u=
x ∈ R2g , λ ∈ Z2g , y
(1)
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where α : Z2g → {±1} ⊆ U (1) is the so-called “canonical” semi-character g
α(λ) = (−1)
j =1 λj λg+j
and ω is identified with symplectic bilinear form g 0 t ω(u, v) = (ug+j vj − uj vg+j ) = u I j =1
−I v. 0
The Hermitian structure and connection are defined to be h (u, ζ ), u, ζ = ζ ζ
(2)
and g (yj dxj − xj dyj ),
∇s = ds − πis
(3)
j =1
respectively. Here and below, global sections of L (for example smooth ones, s ∈ Γ ∞ (L)) will be identified with sections of the (trivialized) pull-back of L to R2g , that is, with functions on R2g that satisfy the appropriate quasi-periodicity conditions, s(u + λ) = α(λ)e−πiω(u,λ) s(u),
λ ∈ Z2g .
Since the trivialization of the bundle is unitary and the connection 1-form purely imaginary, the connection is Hermitian. 2.2. Spaces of sections of Lk If s is a smooth section of Lk , then t
ekπi xy s(x, y) is periodic in x and hence admits a Fourier expansion. This observation leads to an isomorphism given by the Weil–Brezin expansion (also known as Zak expansion) [10] given by Γ ∞ Lk →
S Rg
l∈(Z/kZ)g
s → (s)l (y) := [0,1]g
with inverse
l kπit x(y+ l ) −2πit lx g k e e s x, y + d x k
(4)
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S R g → Γ ∞ Lk
l∈(Z/kZ)g
t (s)l l∈(Z/kZ)g → s(x, y) := e−kπi xy
l 2πit (km+l)x e (s)l y − m − , k g
(5)
l∈(Z/kZ)g m∈Z
where 0 lj < k. We will use the round bracket notation (s)l for the Weil–Brezin coefficients of a section s throughout the paper. The Weil–Brezin map is an isomorphism between topological vector spaces (of smooth sections and Schwartz functions, respectively), hence it extends to an isomorphism between the dual spaces, that is between the space of distributional sections of the bundle Lk and a product of k g copies ofthe space of tempered distributions S (Rg ). This map in turn restricts to a unitary isomorphism l∈(Z/kZ)g L2 (Rg ) ↔ ΓL2 (Lk ), with
s, s =
(s)l , s l =
l∈(Z/kZ)g
(s)l s l .
(6)
l∈(Z/kZ)gRg
2.3. Invariant complex structures and theta functions Invariant complex structures on T2g are determined by their restriction to the tangent space at any point and hence can be parametrized by matrices in Siegel upper half-space Hg (see, for instance, [19]). The complex structure on T2g can be described by its lift to the universal cover R2g : for any Ω ∈ Hg , let ΛΩ be the lattice Zg ⊕ ΩZg ⊂ Cg , so that the torus is equipped with the structure of an Abelian variety via the smooth isomorphism φΩ : T2g → XΩ := Cg /ΛΩ induced by (x, y ∈ Rg ) R2g u = (x, y) → zΩ := x − Ωy ∈ Cg .
(7)
(The choice of the sign here, as well as for the symplectic form ω was made so that the action of the symplectic group turns out to be the expected one in all the coordinates). In the coordinates x, y, the complex structure takes the form JΩ =
−Ω1 Ω2−1
Ω1 Ω2−1 Ω1 + Ω2
−Ω2−1
Ω2−1 Ω1
,
Ω = Ω1 + iΩ2 ,
(8)
but we will need this only in the last section. The corresponding holomorphic structure on (L, ∇) is given by the action of the lattice ΛΩ on Cg × C obtained by combining (1), (7) and the isomorphism R2g × C (u, ζ ) = (x, y), ζ → x − Ωy, eπiFΩ (zΩ ,zΩ ) ζ ∈ Cg × C. We will denote this holomorphic line bundle by LΩ := Cg ×ΛΩ C. Here FΩ is the bilinear form FΩ (z, w) = −t z(Im Ω)−1 Im w or, in terms of real coordinates, z = x − Ωy, w = u − Ωv,
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FΩ (x − Ωy, u − Ωv) = t (x − Ωy)v = t xv − t yΩv. The quasi-periodicity condition for functions on Cg to define holomorphic sections of LΩ takes the well-known form corresponding to classical theta functions, λ ∈ ΛΩ .
ϑ(z + λ) = α(λ)e2πiFΩ (z,λ)+πiFΩ (λ,λ) ϑ(z),
It is well known (see, for example, [19,24]) that a basis of H 0 (XΩ , LkΩ ) is given by l k (kz, kΩ) ϑ , 0 l∈Zg /kZg
(9)
which corresponds to the sections l l −kπiFΩ (x−Ωy,x−Ωy) ϑΩ (x, y) := e ϑ k (kx = e−kπi xy t
0
t
l
− kΩy, kΩ) l
t
l
ekπi (y−m− k )Ω(y−m− k )+k2πi (m+ k )x
(10)
m∈Zg
of Lk . Note that the Weil–Brezin coefficients of these sections are the Gaussians l t ϑΩ l (y) = δll ekπi yΩy . 2.4. Invariant polarizations Geometric quantization (for example, of the symplectic torus) can be performed not only using the Kähler structures from the last paragraph, but using the more general notion of a polarization. Here, we describe all invariant polarizations on T2g . Let LT2g → T2g be the bundle of non-negative Lagrangian subspaces of the complexified tangent bundle on the symplectic manifold (T2g , ω). A (non-negative) polarization, in the sense of geometric quantization (see, for instance, [29]), is a section of LT2g that provides an involutive distribution on T2g , Pol T2g ⊂ Γ ∞ LT2g . In the present case, LT2g is canonically trivial LT2g ∼ = T2g × LC2g (where LC2g denotes the Grassmannian of non-negative Lagrangian subspaces in C2g equipped with the standard symplectic form), since T T2g is (canonically) trivial. A polarization P, therefore, is simply a function P : T2g → LC2g , and it is invariant if this function is constant, PolT
2g
2g T ∼ = LC2g .
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The complex structures from the previous section, parametrized by matrices Ω ∈ Hg in Siegel upper half-space via the coordinates zΩ = x − Ωy, correspond to the positive (or Kähler) polar2g izations L+ C2g in PolT (T2g ),
∂ ∂ ∂ PΩ = span = span Ω lk + . j ∂xl ∂yk k=1,...,g ∂zΩ j =1,...,g l Remark 2.1. Notice that the positive Lagrangian Grassmannian bundle L+ T2g ∼ = T2g × Hg carries also a natural complex structure: equipping every fiber T2g × {Ω} with the complex structure from the isomorphism T2g ∼ = Cg /ΛΩ gives the universal bundle of Abelian varieties (with a marked basis of the first homology) over Siegel upper half-space. This coordinate chart Hg → PolT (T2g ) is not convenient for the description of all genuine non-negative polarizations (namely, only those transverse to the polarization spanned by the ∂x∂ j directions appear in the closure of Hg as a space of matrices). The convenient substitute (see, for instance, [20]) for it is the closed Siegel disc Dg , which is the closure of the image of Hg under the Cayley transform 2g
˘ g, Hg Ω → τ = (i − Ω)(i + Ω)−1 ∈ D
(11)
with inverse Ω = i(1 − τ )(1 + τ )−1 . The Siegel disc is a global chart of LC2g , and the parametrization of invariant polarizations now reads
∂ ∂ −i(1 − τ )lk + (1 + τ )lk . Pτ = span ∂xl ∂yl k=1,...,g
(12)
l
Throughout the paper, τ will always denote a point in the closed Siegel disc, while Ω denotes a point in the upper half-space. Remark 2.2. A polarization is real if and only if τ is unitary. Following [29], we will call a real polarization reducible, if its space of leaves is a Hausdorff manifold (or equivalently, in our case, if the leaves are compact). Note that this happens if and only if τ is unitary and its entries lie in Q[i]. Reducible polarizations and the quantizations defined by them (whose elements are supported on leaves along which ∇ has trivial holonomy, the so-called Bohr–Sommerfeld leaves) have been studied in the present context using geometric methods (see [22] and also earlier work ´ by Sniatycki [26]), and will also be investigated in more detail below. Both reducible and nonreducible polarizations are also considered in [2]. 2.5. The action of the real symplectic group Below, we will consider an action of the real symplectic group Sp(2g, R) on the Lagrangian Grassmannian (see [29]), which is intimately related to the construction of the half-form (or metaplectic) correction and the Blattner–Kostant–Sternberg pairing. On the upper half-space chart on the positive Lagrangian Grassmannian, the action is given by the usual fractional linear transformation
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M(Ω) = (AΩ + B)(CΩ + D)−1 ,
M=
A C
B ∈ Sp(2g, R), D
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(13)
and it is transitive on Hg . It extends to all of LC2g (or Dg ), where it becomes transitive on each stratum; in terms of the parameter τ and setting τ := M(τ ), the transformation is characterized by the equation τ + 1 (A + iB) + τ − 1 (iC − D) τ = τ + 1 (A − iB) + τ − 1 (iC + D).
(14)
In particular, the action of the integer symplectic group Sp(2g, Z) by symplectomorphism of T2g T
2g
x x
→ M ∈ T2g , y y
induces an action on polarizations via push-forward given by M∗ PΩ = PM(Ω) . Remark 2.3. If we write this action on the symplectic coordinates (x, y) in terms of the complex ones, we recover the usual transformation zΩ → zΩ = t (CΩ + D)−1 zΩ . The metaplectic group (discussed in more detail in Section 3.3 below) is defined as the connected two-fold covering group Mp(2g, R) → Sp(2g, R). It cannot be realized as a matrix group. A convenient description for our purpose (see [28,12]) involves an open subset U ⊂ Sp(2g, R) that generates the group in two steps, i.e. U 2 = Sp(2g, R); U is parametrized by triples (P , L, Q) of g × g matrices, with L invertible, P and Q symmetric, via
P L−1 (P , L, Q) → L−1
P L−1 Q − t L . L−1 Q
The usefulness of this subset U for the description of the metaplectic group lies in the fact that in Mp(2g, Z) consists of two disjoint diffeomorphic copies of U . This gives a its pre-image U workable description of the metaplectic group, as we will see below. 3. A distributional construction of the quantum bundle In order to treat quantizations using complex, mixed and real polarizations in a uniform way, it is convenient to widen our perspective on the prequantum Hilbert space ΓL2 (Lk ) of squareintegrable sections, and consider instead the whole Gelfand triplet Γ ∞ Lk ⊆ ΓL2 Lk ⊆ Γ −∞ Lk . Once we take into account the factors arising from the half-form correction, the pairing on the prequantum Hilbert space ΓL2 and its extension to the Gelfand triplet will, by construction, provide the BKS pairing, as we will see in the next section. The fact that the pairing splits nicely
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into a “prequantum factor” and a “half-form factor”, as in the vector space case, is ultimately a consequence of the group structure on (T2g , ω). Before studying the pairing, we will define the quantum bundle (or rather, interpret the usual definition, as in [29] or [20]) in the setting just outlined. Remark 3.1. Similar analytic behavior occurs in the quantization of a flat symplectic vector space, where in particular it also turns out to be necessary to include half-form correction to obtain continuous behavior of the BKS pairing up to the boundary [21]. Also, in the context of toric varieties, a similar convergence to distributional sections occurs when holomorphic polarizations degenerate to the toric real polarization, with fibers the compact Lagrangian tori [4]. Examining the set of differential equations attached to a choice of polarization that singles out the subspaces of covariantly constant sections, we will see that they actually admit a natural weak variant that coincides with the usual equations for square-integrable sections in the case of a complex polarization, but with the advantage that there is a (non-zero) space of solutions also for mixed and real polarizations. It is our aim, in this section, to show that this definition is the natural one from the point of view of the transition from complex to mixed or real polarizations. This is achieved in Theorem 3.10 below. 3.1. The weak equations of covariant constancy Given an invariant polarization P, a smooth, at first, local section s is covariantly constant along P if ∀ξ ∈ P:
∇ξ s = ds · ξ − kπis(x, y)
(yj dxj − xj dyj ) · ξ = 0.
(15)
Note that if the polarization is given by a complex structure, these are just the Cauchy–Riemann equations. In any case, every ξ ∈ P defines a continuous linear operator ∇ξ : Γ ∞ (Lk ) → Γ ∞ (Lk ), and the quantum space associated with a complex polarization P is given by the intersection of the kernels of these operators. If the polarization has real directions (and the manifold we are quantizing is compact), there are no non-zero global smooth solutions to Eqs. (15), since any such solution would have to be supported on Bohr–Sommerfeld leaves: smooth sections can be restricted to leaves, and nontrivial holonomy along any non-contractible loop in a leaf forbids the existence of non-zero horizontal sections. It is, therefore, natural to consider the weak version of the operators (15) acting on distributional sections of Lk . Distributional sections in the quantization of abelian varieties appeared also in [2]. As already mentioned in the introduction, in the case of toric manifolds this approach has the advantage of including contributions from all, even singular, Bohr–Sommerfeld fibers [4]; thus, the dimension of the quantization space in the real toric polarization equals the dimension obtained from Kähler polarization. Remark 3.2. Locally, the neighborhood of a Bohr–Sommerfeld fiber of T2g is equivariantly symplectomorphic to a neighborhood of a non-singular Bohr–Sommerfeld fiber of a toric manifold. As seen in [4], each Bohr–Sommerfeld fiber will be the support of a one-dimensional space of polarized distributional sections. Moreover, the explicit form of these sections is described in [4].
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For an open subset U ⊂ T2g , consider first the natural injection ι : Γ ∞ Lk U → Γ −∞ Lk U = Γc∞ Lk U ωn s → ιs(φ) = sφ . n! U
Then, the definition should make the following diagram commute, Γ ∞ (Lk )
ι
Γ −∞ (Lk ) ∇ξ
∇ξ
Γ ∞ (Lk )
ι
Γ −∞ (Lk )
so that the operator ∇ξ extends the operator ∇ξ to distributional sections. Explicitly, on any open set U , for any smooth section s ∈ Γc∞ (Lk |U ) and for any test section φ ∈ Γc∞ (L−k |U ) with compact support and smooth section ξ ∈ Γ ∞ (P|U ) of the polarization on U , ∇ξ ιs (φ) =
(∇ξ s)φ dg x dg y U
=
U
=−
ds · ξ − kπis
(yj dxj − xj dyj ) · ξ φ dg x dg y
s div ξ φ + dφ · ξ + kπiφ (yj dxj − xj dyj ) · ξ dg x dg y
U
=−
s div ξ φ + ∇ξ−1 φ dg x dg y,
U
where ∇ −1 stands for the connection on the inverse bundle L−k induced by ∇. It is, therefore, natural to define the operation of covariant differentiation of distributional sections, ∀σ ∈ Γ −∞ (Lk |U ), ∀ξ ∈ Γ ∞ (P|U ), ∇ξ σ (φ) := −σ div ξ φ + ∇ξ−1 φ ,
∀φ ∈ Γc∞ L−k U ,
(16)
and to define the quantum space associated with the polarization as the intersections of the kernels of the operators ∇ξ : Γ −∞ (Lk ) → Γ −∞ (Lk ) for ξ ∈ Γ ∞ (P), as before. For Kähler polarizations, regularity of the Cauchy–Riemann equations (that are the equations of covariant constancy in this case) guarantees that this definition is conservative, that is, one does not find distributional solutions to them which are not holomorphic functions. On regularity of the Cauchy–Riemann equations see, for instance, Chapter 6 in [13] (the argument given there for the case of distributions on C extends to Cn ), or [18].
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3.2. The half-form correction To define the half-form correction for the torus T2g we have been considering, let us first recall its definition for the case of a symplectic vector space (V , ω) of real dimension 2g (see [29,21]). As before, we can for simplicity parametrize polarizations (non-negative Lagrangian subspaces P ⊂ VC ) by the Siegel disc Dg . Over it, one considers the canonical line bundle K ⊂ Dg × Λg VC∗ → Dg , where each fiber is given by the space of g-forms that vanish upon contraction with the conjugate of any vector in the polarization. We will use the same letter K for the pull-back of this bundle to Hg . The (indefinite) pairing on the space of g-forms given by
η, η
(kω)g g!
= 2−g ig (−1)
g(g−1) 2
η ∧ η ,
(17)
induces a pairing between any two fibers of this fibration. Notice that this pairing is invariant under the natural action of the real symplectic group Sp(V , ω) ∼ = Sp(2g, R). Any Ω ∈ Hg defines a positive polarization by specifying a complex coordinate zΩ = x − Ωy 1 ∧ · · · ∧ dzg is a g-form generating the line of K over Ω. A short on V , and dg zΩ := dzΩ Ω calculation shows that the pairing comparing the two different fibers over Ω and Ω is, then, 1 Ω − Ω , dg zΩ , dg zΩ = det 2ki
(18)
and in particular we see that the pairing is positive definite over Hg . The half-form (or metaplectic) correction consists in the choice of a square root of K. Since K is trivial, this looks like a trivial operation, but the behavior of the inner product on the square root is an essential analytic ingredient for the BKS pairing. We take advantage of the fact that there is a natural way of defining a specific branch of the square root of a determinant as in (18), given by a Gaussian integral
1 t ξ(Ω−Ω )ξ t ξ Ωξ −2 t ξ Ωξ 1 det Ω −Ω := e−π 2ki dg ξ = eπi 2k , eπi 2k L2 (Rg ) . 2ki Rg
√ Keeping a traditional, though possibly slightly misleading, notation dg zΩ for the generator of the complex line of the half-form bundle over Ω ∈ Hg , we define the pairing via the embedding 1 t ξ Ωξ α dg zΩ → eπi 2k , α
α ∈ C \ {0},
(19)
or, more explicitly (and with the sign determined by the Gaussian integral)
1 2 1 Ω − Ω dg zΩ , dg zΩ = det . 2ki
(20)
We will see below that this point of view is also very convenient for the description of the action of the metaplectic group on half-forms.
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This definition works only over the positive Lagrangian Grassmannian; we need to rescale the trivializing section to describe the half-form bundle over the whole of the non-negative Lagrangian Grassmannian. The canonical line determined by the polarization Pτ is generated by the g-form g (1 + τ ) dx − i(1 − τ ) dy = det(1 + τ ) dg zΩ(τ ) , d (x, y)τ = g
where the last equality holds whenever dg zΩ(τ ) is defined. The half-form bundle is then described by any of these two trivializations, C dg zΩ(.) ∼ = C dg (x, y). → Dg , where the isomorphism is given by − 1 dg zΩ(τ ) = det(1 + τ ) 2 dg (x, y)τ ,
(21)
and the branch of the square root is fixed by demanding it to be 1 for τ = 0. Let us now address the half-form correction for T2g . Recall from Section 2.4 that the bundle of non-negative Lagrangian subspaces LT2g of the complexified tangent bundle is canonically trivialized, LT2g ∼ = T2g × LC2g . The canonical bundle K → LT2g , generated over any point 2g (x, y, τ ) ∈ T × Dg by dg (x, y)τ is again topologically trivial. It is clear that K restricted to T2g × {Ω} ∼ = XΩ is the usual canonical bundle KΩ → XΩ of holomorphic g-forms. As before (17), the canonical bundle K comes with a natural Hermitian structure hK determined by the Liouville form. As Hermitian bundle, the half-form bundle δ → LT2g is just the pull-back δ
√ C dg (x, y).
LT2g ∼ = T2g × Dg
Dg
A connection on δ is determined as follows. K is equipped with a natural partial connection ∇ part given pointwise at ((x, y), τ ) by the Lie derivative of complexified g-forms on T2g at (x, y) along the directions of Pτ . In fact, as K is trivial, ∇ part extends to the relative trivial connection ∇ triv of the family LT2g → Dg , corresponding to covariant derivatives along the directions of the fibers XΩ . In particular, the section (x, y) → dg (x, y)τ is parallel relative to this connection. on the half-form bundle δ is that it satisfies The condition imposed on the connection ∇ Leibniz’s rule . ⊗ μ + μ ⊗ ∇μ ∇ triv μ ⊗ μ = (∇μ) is determined by the holonomies along 2g generators Thus, the choice of connection ∇ 2g of H1 (T , R), subject to the condition that they square to 1. The Hermitian structure on δ is obtained by taking a consistent square root of the Hermitian structure on K.
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√ is the trivial Note that dg zΩ is a parallel section (along the base T2g , as above) of δ only if ∇ and an invariant holomorphic polarization PΩ is equivalent connection. Choosing the pair (δ, ∇) to fixing a specific point χ of order at most two in Pic0 (XΩ ), that is, a half-characteristic. If χ is not zero (and does not represent the trivial holomorphic bundle), there are no global holomorphic √ sections; in particular, dg zΩ , being a smooth trivialization, cannot be parallel along the anti∗ δ gives, then, the holomorphic directions. Solving equations of covariant constancy on Lk ⊗ PΩ 0 k space of holomorphic sections H (L ⊗ χ), i.e. leads to theta functions with half-characteristic. corresponding In the remainder of the paper, for simplicity, we will use the correction (δ, ∇) to χ = 1 ∈ Pic0 (XΩ ). All the results extend to other choices for χ . 3.3. The Weil representation of the metaplectic group In order to study solutions of the equations of covariant constancy for polarizations on the boundary of Dg , and in view of the Weil–Brezin expansion, it is convenient to use a natural action of the metaplectic group on L2 (Rg ). Recall from [28] (see also [12,10,11]) that the metaplectic group Mp(2g, R), the connected two-fold cover of Sp(2g, R), can be constructed as a group of unitary operators on the Hilbert space L2 (Rg ), as follows. Considering real g × g matrices P , L, Q with P and Q symmetric and L invertible, and an integer m mod 4 indexing a choice of square root im of sign det L, consider the integral operator S(P , L, Q)m : S(Rg ) → S(Rg ) given by g
g
S(P , L, Q)m f (u) := i− 2 +m k 2
|det L|
t
t t
t
ekπi( uP u−2 u Lv+ vQv) f (v) dg v,
πi
(22)
where we use the notation iφ := e 2 φ for any real number φ. These operators are continuous on S(Rg ), therefore also on S (Rg ), and are unitary isomorphisms when restricted to L2 (Rg ). The metaplectic group is the group of unitary operators generated by all operators of this form. The covering maps each generator S(P , L, Q)m to the symplectic matrix specified by (P , L, Q). Note that this projection differs from the one in [12] by an automorphism of Sp(2g, R). Any element in Mp(2g, R) can be represented by the product of two operators of this form. We then have Lemma 3.3. Let Ω ∈ Hg ; then t S(P , L, Q)m ekπi vΩv (u) = im
|det L| det(Ω + Q)
1 2
t
ekπi uΩ u ,
where Ω ∈ Hg is given by Ω = P − t L(Ω + Q)−1 L =
P L−1 L−1
P L−1 Q − t L (Ω). L−1 Q
Proof. Applying the definitions, g
g
S(P , L, Q)m ekπi vΩv (u) = i− 2 +m k 2 t
|det L|
t
t t
t
ekπi( uP u−2 u Lv+ v(Ω+Q)v) dg v
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=i
− g2 +m
k
g 2
|det L|
πg det −kπi(Ω + Q)
× ekπi uP u e−kπi u L(Ω+Q) t
from which the assertion follows.
3401
t t
−1 Lu
,
2
√ Remark 3.4. Note that the sign of the square root in the statement det(Ω + Q) is determined by evaluation of the Gaussian integral in the proof. We will not need to specify it explicitly. The action of Sp(2g, R) on the Lagrangian Grassmannian, given by Ω → M(Ω), lifts to K, where one obtains the ordinary pull-back of g-forms ∗ −1 g M d zΩ = det(CΩ + D) dg zΩ ,
(23)
for M ∈ Sp(2g, R) as in (13). By the construction of the half-form bundle δ and the identification of the Hermitian metric (19) on it, the unitary representation of the metaplectic group that lifts (23) is then given by S(P , L, Q)m
1 dg zΩ = i−m |det L|− 2 det(Ω + Q) dg zΩ ,
(24)
where S(P , L, Q)m is any generator of Mp(2g, R) and Ω = M(Ω), with M = (P , L, Q). From this follows immediately Proposition 3.5. The product of the action of the metaplectic group Mp(2g, R) √ on Γ −∞ (Lk ) with −∞ k the natural action on δ descends to an action of Sp(2g, R) on Γ (L ) ⊗ dg (x, y)τ → Dg that lifts the symplectic action on the Siegel disc. 3.4. The extended quantum Hilbert bundle As explained above, since we consider real polarizations as limits of holomorphic polarizations, and since for the real polarizations there are no smooth solutions of the equations of covariant constancy, we use the weak version of these equations and look for solutions in the space of distributional sections Γ −∞ (Lk ). This is particularly simple to do in terms of the Weil–Brezin expansion in (4), and making use of the parameter τ ∈ Dg for the polarization in Section 2.4. τ be the space of distributional sections σ ∈ Γ −∞ (Lk ) that are solutions of the weak Let Q equations of covariant constancy (16) with respect to the polarization Pτ . These spaces naturally form a bundle of vector subspaces in the bundle of distributional sections, ⊂ Γ −∞ Lk × Dg → Dg . Q Lemma 3.6. Under the identification of global sections of Γ −∞ (Lk ) with elements in S (Rg )g τ become indevia the Weil–Brezin transform, the equations of covariant constancy defining Q g pendent for each of the g components S (R ), and are identical on all of them.
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Explicitly, using the first-order differential operators Ξτ : S (Rg ) → S (Rg )g defined by Ξτ = (τ + 1)
∂ − 2kπ(τ − 1)y, ∂y
τ ∈ Dg ,
(25)
we have τ σ ∈Q
⇔
∀l = 1, . . . , g:
(σ )l ∈ Ker Ξτ .
Proof. Since all the operators involved are continuous, it suffices to check the identity on the dense subspace of smooth sections s ∈ Γ ∞ (Lk ). Using (3) and (5) we obtain (∇ξ s)l (y) = t β
∂ (s)l (y) + k2πit αy(s)l (y), ∂y
∂ ∂ where ξ = t α ∂x + t β ∂y is any constant vector field. From (12) we immediately get the lemma. Note that we get the same equation for each Weil–Brezin coefficient separately. 2
Definition 3.7. The quantum Hilbert bundle Q over the space of invariant polarizations of (T2g , ω) is defined as ⊗ C dg (x, y). → Dg , Q := Q √ g d (x, y). → Dg is the half-form bundle. To simplify the notation we will write Qτ = where √ C τ dg (x, y)τ . Q Recall that due to regularity of the Cauchy–Riemann equations the elements of the quantum ˘g ∼ Hilbert space QΩ over a point Ω in the interior D = Hg of the Lagrangian Grassmannian k are given by the holomorphic sections of√L , with the holomorphic structure determined by Ω, tensored with the half-form correction C dg zΩ , QΩ ∼ = H 0 XΩ , LkΩ dg zΩ . Definition 3.7 extends the bundle of quantizations over the Siegel upper half-space to the boundary of the Siegel disc Dg , proceeding as depicted in the following diagram: √ Γ ∞ (Lk ) dg z.
Γ −∞ (Lk ) dg (x, y)τ (.)
∪
∪
Q
Q
˘g Hg ∼ =D
Dg
The quantum bundle is naturally viewed as a sub-bundle of the trivial (infinite-dimensional) bundle of smooth (on the left part of the diagram) or distributional (on the right part of the
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diagram) sections of the half-form corrected prequantum bundle. By abuse of notation, we will usually not distinguish the spaces of smooth sections from the distributions they naturally define by integrating against Liouville measure. τ ) is independent of τ ∈ Dg . (ReFirst, we prove that the dimension of the spaces Qτ (or Q covering a result also in [2].) Lemma 3.8. For all τ ∈ Dg with det(1 + τ ) = 0, dim Qτ = k g . τ of Γ −∞ (Lk ) is spanned by the sections with Weil– Explicitly, the corresponding subspace Q Brezin coefficients l 1 t ϑτ l := δl,l det(1 + τ )− 2 ekπi yΩ(τ )y ∈ S Rg ,
(26)
where the branch of the square root is the same one as in (21). Proof. Note that the Weil–Brezin coefficients in the statement of the lemma are well-defined elements in S (Rg ) even when the imaginary part of Ω is only positive semi-definite, and that the dependence on τ is continuous in this case. By Lemma 3.6 we have to identify the kernel of the operators Ξτ . Since (1 + τ ) is invertible and
∂ ∂ − 2kπiΩy , Ξτ = (τ + 1) − 2kπ(τ − 1)y = (τ + 1) ∂y ∂y the kernel of Ξτ on distributions is one-dimensional and is spanned by the Gaussian in the statement of the lemma (see e.g. the proof of Theorem 7.6.1 in [17]). 2 In order to treat the points where det(1 + τ ) = 0, we now show that the Sp(2g, R) action preserves the kernels of the operators Ξτ . Consider the operators Ξτ ⊗ 1 acting on S (Rg ) ⊗ δ, where δ is the half-form bundle. Recall that, from Proposition 3.5, the action of the metaplectic group Mp(2g, R) on Γ −∞ (Lk ) ⊗ δ actually descends to an action of Sp(2g, R). Note also that the group Sp(2g, R) acts diagonally on the g factors of S (Rg ) in the Weil–Brezin expansion. In fact, we now show that the action of Sp(2g, R) lifts to the Hilbert quantum bundle Q. Proposition 3.9. Let, as above, M = (P , L, Q) ∈ Sp(2g, R) be a matrix with g × g block entries A, B, C, D, and τ = M(τ ); furthermore, consider the matrix Xτ :=
1 τ + 1 (A + iB) + τ − 1 (iC − D) 2
acting on a vector of g elements of S (Rg ). Then (Ξτ ⊗ 1) ◦ M = Xτ ◦ M ◦ (Ξτ ⊗ 1).
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In particular, the symplectic transformation M preserves the kernels of the family of operators Ξt , which define the bundle Q. Proof. To shorten the expressions, set t
t t
t
e(u, v) := ekπi( uP u−2 u Lv+ vQv) ; and calculate, using an arbitrary lifting Mm ∈ Mp(2g, Z) and Proposition 3.5, (Ξτ ⊗ 1 ◦ M) f (u) dg (x, y)τ
∂Mm f − 2kπ τ − 1 uMm f (u) Mm dg (x, y)τ = τ +1 ∂u
∂e(u, v) τ +1 − 2kπ τ − 1 ue(u, v) f (v) dg v dg (x, y)τ = ∂u = τ + 1 2kπi P u − t Lv − 2kπ τ − 1 u e(u, v)f (v) dg v dg (x, y)τ . Using the fact that ue(u, v) = L−1 Qve(u, v) −
∂e(u, v) 1 L−1 2kπi ∂v
and integrating by parts, we find (Ξτ ⊗ 1 ◦ M) f (u) dg (x, y)τ
i τ + 1 P L−1 Qve(u, v) − = 2kπ
∂e(u, v) 1 L−1 − t Lve(u, v) 2kπi ∂v
∂e(u, v) 1 L−1 f (v) dg v dg (x, y)τ − τ − 1 L−1 Qve(u, v) − 2kπi ∂v
∂f 1 i τ + 1 P L−1 Qvf (v) + L−1 − t Lvf (v) = 2kπ 2kπi ∂v
∂f 1 L−1 e(u, v) dg v dg (x, y)τ − τ − 1 L−1 Qvf (v) + 2kπi ∂v ∂f τ +1 A+i τ −1 C + 2kπ i τ + 1 B − τ − 1 D vf (v) = ∂v × e(u, v) dg v dg (x, y)τ .
Therefore, to complete the proof it suffices to show that τ + 1 A + i τ − 1 C = Xτ (1 + τ ), i τ + 1 B − τ − 1 D = Xτ (1 − τ ),
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but these two equations are equivalent to the following τ + 1 (A + iB) + τ − 1 (iC − D) = 2Xτ , τ + 1 (A − iB) + τ − 1 (iC + D) = 2Xτ τ, the first of which is the definition of Xτ , and the second one is precisely (14).
2
Now, we can use the metaplectic group action to show that the rank of the bundle is constant ˘ g, over all of the Siegel disc Dg . Consider the following section of Q → Hg ∼ =D l σΩl := ϑΩ
dg zΩ = ϑτl (Ω)
dg (x, y)τ (Ω) ∈ H 0 XΩ , LkΩ dg zΩ ,
where accordingly l ϑτl = det(1 + τ )− 2 ϑΩ(τ ), 1
l defined by (10) for Ω ∈ H and where the branch of the square root is the natural one, with ϑΩ g as in (21). Putting things together, we obtain
Theorem 3.10. Each section ϑτl extends continuously to a map Dg → Γ −∞ (Lk ). In particular, l τ → {στ }l∈Zg /kZg is a set of global sections of Q, which is therefore trivialized and of rank k g . Furthermore, the elements of the trivializing frame {στl }l∈Zg /kZg for Q are invariant under the Sp(2g, R) action. Proof. From Proposition 5.4.7 in [29], for any τ ∈ ∂Dg , there exists a symplectic transformation M ∈ Sp(2g, R) such that M(τ ) does not have eigenvalue −1. Since M −1 acts continuously on Q by the previous proposition, the assertion follows. The invariance of the sections στl under the Sp(2g, R) action is an immediate consequence of Lemma 3.3 and of (24). 2 Remark 3.11. For the case of reducible polarizations, the dimension k g coincides with the one ´ obtained by considering Sniatycki’s quantization procedure [26,22] where one considers smooth sections of the restriction of Lk to Bohr–Sommerfeld fibers. In fact, solutions of the weak equations of covariant constancy must be supported along fibers of trivial holonomy, and polarized sections are linear combinations of Dirac delta distributions with phase variation along the Bohr– Sommerfeld fibers. Example 3.12. For the case of the horizontal polarization, given by τ = −1, the basis l } {σ−1 l∈(Z/kZ)g consists of distributional sections, each supported on a single compact Bohr– Sommerfeld fiber of the vertical polarization: this follows immediately from the equations of covariant constancy written in terms of the Weil–Brezin coefficients. Since τ = −1, the operator Ξ−1 in (25) is just multiplication by 4kπy, and its kernel is generated by the Dirac delta l (each of distribution supported at y = 0. Therefore, from (5), the corresponding sections σ−1 which has a single non-zero Weil–Brezin coefficient) will be supported on the points with the components of y being congruent to kl . These points define the k g Bohr–Sommerfeld fibers of
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the vertical polarization. Since each basis element is supported on a single Bohr–Sommerfeld l } fiber, {σ−1 l∈(Z/kZ)g is a so-called Bohr–Sommerfeld basis [27] for Q−1 . In order to describe Bohr–Sommerfeld basis for other reducible polarizations, it is convenient to study a natural geometric action of the integer symplectic group Sp(2g, Z). For M ∈ Sp(2g, Z) and τ a reducible polarization, this action transforms Bohr–Sommerfeld leaves of the polarization τ into Bohr–Sommerfeld leaves of the reducible polarization M(τ ). We will describe and study some properties of this action, and of its lift to the quantum bundle, in a forthcoming paper [5]. In fact, that study amounts to a study of the algebro-geometric theta transformation formula from a symplectic point of view. In the following section, we study the Hilbert space structure on the fibers of the bundle Q → Dg , and how they are related for different fibers. 4. The BKS pairing 4.1. The BKS pairing on the extended quantum bundle By the very construction of the quantum bundle Q, over the interior of the Siegel disc the BKS pairing is already implemented as the product of the pairing of square integrable sections with the pairing of half-forms. Explicitly, one has Theorem 4.1. For Ω, Ω ∈ Hg , the BKS pairing is given by
l ϑΩ ⊗
l dg zΩ , ϑΩ ⊗
dg zΩ
BKS
g
= 2− 2 k −g δl,l .
(27)
l , ϑ l in the prequantum Hilbert space; from the Proof. We have to calculate the pairing ϑΩ Ω unitary isomorphism (6) between the space of square integrable sections of Lk and (L2 (Rg ))k that restricts to the Weil–Brezin expansion on smooth sections, and the Gaussians in (26) it is clear that this gives
− 1 l l −g 2, ϑΩ , ϑΩ = δl,l (ki) 2 det Ω − Ω
which, combined with (20) proves this result.
(28)
2
Note that from Theorem 3.10, the corresponding BKS pairing map BΩ,Ω : QΩ → QΩ , defined by BΩ,Ω σ, σ Q = σ, σ BKS , Ω
∀σ ∈ QΩ , σ ∈ QΩ ,
corresponds to the action on Weil–Brezin coefficients of the element of Sp(2g, R) relating Ω and Ω . It follows that the family of indexed basis {σΩl }l∈Zg /kZg is parallel with respect to the family of pairing maps.
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Moreover, from Theorem 3.10, the BKS pairing can be extended continuously to the boundary of Dg , so that, ∀τ, τ ∈ Dg , we can define unitary BKS pairing maps Bτ,τ : Qτ → Qτ l Bτ,τ στ = στl , ∀l ∈ Zg /kZg .
(29)
˘ g , as in TheRemark 4.2. The pairing between elements of Qτ and Qτ , for τ ∈ Dg and τ ∈ D orem 4.1 is still given by evaluating the distributional sections in Qτ on the conjugate of the smooth sections in Qτ and multiplying with the factor that arises from half-form correction. In this situation, for σ ∈ Qτ and σ ∈ Qτ ,
σ, σ
BKS
= σ σ = σ σ (1),
where in the last term the distribution σ σ is evaluated at the constant function 1. Therefore, we have Corollary 4.3. The family of indexed bases τ → στl l∈Zg /kZg ⊆ Qτ ,
τ ∈ Dg ,
is parallel with respect to the transitive family of unitary pairing maps Bτ,τ : Qτ → Qτ , τ, τ ∈ Dg , defined by
Bτ,τ σ, σ Q = σ, σ BKS , τ
∀σ ∈ Qτ , σ ∈ Qτ .
These results show that symplectic tori also provide examples of symplectic manifolds where quantizations in different polarizations can be related by transitive unitary BKS pairing maps, if the half-form correction is included. Remark 4.4. From a different but related perspective, the unitary maps BΩ,Ω for Ω, Ω ∈ Hg can also be realized as coherent state transformations associated to different Kähler structures on the complex Lie group (C∗ )g . (See [7] and also [14,9].) Remark 4.5. These results extend straightforwardly to non-principally polarized abelian varieties. An interesting application is then to the moduli space of rank n semistable (degree zero) vector bundles on an elliptic curve EΩ , for Ω ∈ H1 . This moduli space is isomorphic to Pn−1 and can be realized as a quotient of the (non-principally polarized) abelian variety M = EΩ ⊗ Λˇ R by the Weyl group of SLn (C), Pn−1 ∼ = M/W , where Λˇ R is the corresponding co-root lattice. Nonabelian theta functions in genus one are then realized as W anti-invariant theta functions on M. The unitary equivalence between spaces of non-abelian theta functions, in genus one, associated with different complex structures studied in [3,8], can therefore be formulated equivalently in terms of BKS pairing maps.
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4.2. Further properties of the BKS pairing Recall that the subgroup of translations of Xτ preserving the holomorphic structure on Lk has a central extension, given by the “finite” Heisenberg group Hk which is given by Hk =
λ, (a, b) λ ∈ U (1), (a, b) ∈ (Z/kZ)2g ,
with the group law πi λ, (a, b) λ , a , b = λλ e k (ab −ba ) , a + a , b + b . Hk acts naturally on H 0 (Lk ) ∼ = Qτ by
b a . λ, (a, b) σ (x, y) = λeπiω((x,y),(a,b)) σ x − , y − k k This well-known natural algebro-geometric irreducible unitary representation of Hk on Qτ , ˘ g (see for example Section 6.4 of [6], or [24]) is which is unique up to isomorphism, for τ ∈ D given in the parallel basis by λ, (a, 0) στl = λe−2πila/k στl , λ, (0, b) στl = λστl+b . It is then natural to consider this representation for all τ ∈ Dg , including degenerate points on ∂Dg . Remark 4.6. From Corollary 4.3 it follows immediately that the unitary BKS pairing maps Bτ τ : Qτ → Qτ intertwine the canonical representations of Hk . We will now show that the BKS pairing between transverse real polarizations is, as expected, given by an intersection pairing. Let τ ∈ ∂Dg define a real reducible polarization Pτ and let BS ⊂ T2g be the union of its Bohr–Sommerfeld fibers. Recall from Remark 3.2 that if σ ∈ Qτ then the support of σ is contained in BS. Proposition 4.7. Let τ, τ ∈ ∂Dg be such that Pτ , Pτ are real, reducible and transverse. If σ ∈ Qτ , σ ∈ Qτ then the pairing σ, σ BKS is obtained by evaluating a distribution supported in BS ∩ BS on the constant function 1. Proof. From Remark 3.2 and [4], one has that σ is a linear combination of smooth phases multiplying codimension g Dirac delta distributions supported in BS, and similarly for σ . Since, Pτ and Pτ are transverse, Theorem 8.2.4 and Example 8.2.11 of [17] guarantee that the product σ σ gives a well-defined distribution on T2g , supported in BS ∩ BS . From the continuity of the product (see Theorem 8.2.4 in [17]) and Remark 4.2, it is clear that σ σ (1) = σ, σ BKS . 2
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Remark 4.8. In [22], Manoliu has considered the quantization of the standard symplectic torus in real reducible polarizations at even level k. She introduces geometrically an intersection pairing between quantization spaces for two reducible (transverse) polarizations via the intersection points of the respective Bohr–Sommerfeld leaves. After including the half-form correction these pairings are unitary. Remark 4.9. Another particularly interesting point on ∂Dg is the one corresponding to the vertical polarization, τ = 1. In this case, the period matrix Ω = 0 and the theta functions become linear combinations of Dirac delta distributions supported on Bohr–Sommerfeld fibers. Remarkably, the linear combination coefficients are precisely given by the modular transformation Smatrix for characters of the affine Lie algebra u(1)k [7]. Thus, in terms of Bohr–Sommerfeld basis, the BKS pairing map B−1,1 is exactly represented by this modular transformation matrix. 5. Tropical theta divisors Tropical geometry is known to appear in the description of degeneration data of boundary points in the compactification of the moduli space of polarized abelian varieties [1]. In [23], the authors consider tropical curves, their Jacobians as well as tropical theta functions and tropical theta divisors. Tropical geometry is also known to arise when one degenerates Kähler metrics. In this section, we will comment briefly on the appearance of tropical geometry as one takes the complex structure of T2g to the boundary of Dg . Recall that tropical geometry is the algebraic geometry of curves over the tropical semi-field R ∪ {−∞}, where the operations are defined by x ⊕ y = max{x, y} and x y = x + y, for x, y ∈ R ∪ {−∞}. The evaluation of a polynomial in n variables over this ring defines a piecewise linear map from (R ∪ {−∞})n → R ∪ {−∞}. The associated tropical affine variety is then given by the set of points where this function is not smooth, which is a piecewise linear set. Let us now describe several examples of metric degeneration of Kähler structure at a real reducible polarization. Namely, for computational simplicity, we will choose the point τ = −1. We will see that the outcome depends considerably on the path in Dg through which τ approaches the point τ = −1 ∈ ∂Dg . From the expression (8) for the complex structure defined by Ω ∈ Hg , it is clear that the corresponding Kähler metric γΩ = ω(., JΩ .) is given by the matrix γΩ =
Ω2−1
−Ω2−1 Ω1
−Ω1 Ω2−1
Ω1 Ω2−1 Ω1 + Ω2
.
(30)
Example 5.1. For the standard hyperbolic metric in Hg invariant under Sp(2g, R), the geodesics rays going to the point at infinity corresponding to τ = −1 are given by Ω(s) = B t A + iAe2sΛt A,
s > 0,
where Λ is a positive diagonal matrix, A ∈ GLn (R) and B t A is symmetric. Then, from (30), γΩ(s) =
t
A−1 e−2sΛ A−1 −Be−2sΛ A−1
−t A−1 e−2sΛt B . Be−2sΛt B + Ae2sΛt A
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We see that, after rescaling the metric appropriately, as s → ∞ at least g dimensions collapse. In fact, the number of surviving dimensions equals the multiplicity of the largest eigenvalue of Λ. Let us therefore consider the case when Λ = λ1. Then the rescaled metrics e−2sλ γΩ(s) converge in the Gromov–Hausdorff sense, 2g −2sλ T ,e γΩ(s) → Tg , At A . One may ask how theta divisors behave under this metric degeneration. For the sake of simplicity, consider for instance the theta divisor V (ϑΩ ) for level k = 1. Let u = e2sλ . The absolute values of the terms of the series defining ϑΩ(s) are am (y) = e−πu (y−m)A A(y−m) , t
t
m ∈ Zg .
Call m ∈ Zg a lattice neighbor of y if ∀m ∈ Zg \ {m}:
y − m At A y − m At A .
As At A is positive definite, it is easy to see that for any point y0 which does not have at least two distinct lattice neighbors, the theta function cannot equal zero for s large enough and any value of x, i.e. y0 ∈ / μ V (ϑΩ(s) ) , where μ stands for the group-valued moment map T2g (x, y) → μ(x, y) = y ∈ Tg . The theta divisor then approaches the tropical theta divisor of [1,23] (see Section 5.2 of [23]), defined by the non-smoothness locus of the functions
maxg −t (y − m)At A(y − m)
m∈Z
or
maxg −t mAt Am + 2t yAt Am .
m∈Z
Note that this set depends only on the limit metric and on the location of the lattice points which define the Bohr–Sommerfeld fiber. This behavior of theta divisors as one approaches the boundary of Dg is consistent with the fact that the theta function approaches a distributional section supported on the Bohr–Sommerfeld fiber, so that its zeroes are away from this fiber. In general, the limiting behavior of the theta divisor will not be related to the limit metric in such a simple manner as above. Example 5.2. Let s > 0. Considering a half-line of complex structures Ω + s Ω˙ with Ω, Ω˙ ∈ Hg and s → ∞, the rescaled metrics 1s γΩ+s Ω˙ converge in the Gromov–Hausdorff sense,
1 T2g , γΩ+s Ω˙ → Tg , Ω˙ 1 Ω˙ 2−1 Ω˙ 1 + Ω˙ 2 . s
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Note that the point on the boundary of Dg we are approaching (or the real polarization) is still the same, τ = −1. A computation analogous to the one in Example 5.1 shows that the limit of the theta divisors is the same as above for the metric Ω˙ 2 . This coincides with the limit metric only in the case Ω˙ 1 = 0, which is when the half-line is a (reparametrized) geodesic. Example 5.3. For higher level k > 1, a particular tropical theta divisor is given in an analogous way by the non-smoothness locus of
y → maxg −t mΩ˙ 2 m + 2k t y Ω˙ 2 m . m∈Z
This piecewise linear object, which is equidistant from Bohr–Sommerfeld fibers, is obtained by degeneration of the divisor of the theta function l∈Zg /kZg στl , which is invariant by the subgroup Zg /kZg ⊂ Hk generated by the elements of the form (1, (0, b)). (See Section 4.2.) Similar objects, the geometric quantization amoebas of [4], appear in the geometric quantization of toric manifolds, where also such a particular tropical divisor, equidistant from Bohr–Sommerfeld points in the polytope, (asymptotically) selects a particular degenerating holomorphic section. Acknowledgments We wish to thank C. Florentino for useful discussions, J. Drumond Silva for help with the proof of Proposition 4.7 and C. Baier for help with the literature. We also wish to thank the referee for important suggestions. Partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems, by Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER and by the Projects POCI/MAT/58549/2004 and PPCDT/MAT/58549/2004. T.B. is also supported by the fellowship SFRH/BD/22479/2005 of Fundação para a Ciência e a Tecnologia. References [1] V. Alexeev, I. Nakamura, On Mumford’s construction of degenerating abelian varieties, Tohoku Math. J. 51 (1999) 399–420. [2] J.E. Andersen, Jones–Witten theory and the Thurston boundary of Teichmüller spaces, University of Oxford PhD thesis, 1992; J.E. Andersen, Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations, Comm. Math. Phys. 183 (1997) 401–421. [3] S. Axelrod, D. Della Pietra, E. Witten, Geometric quantization of Chern–Simons gauge theory, J. Differential Geom. 33 (1991) 787–902. [4] T. Baier, C. Florentino, J. Mourão, J.P. Nunes, Large complex structure limits, quantization and compact tropical amoebas on toric varieties, arXiv:0806.0606. [5] T. Baier, J. Mourão, J.P. Nunes, The theta transformation formula from a symplectic point of view, in preparation. [6] C. Birkenhake, H. Lange, Complex Abelian Varieties, Springer-Verlag, Berlin, 1992. [7] C. Florentino, J. Mourão, J.P. Nunes, Coherent state transforms and abelian varieties, J. Funct. Anal. 192 (2002) 410–424; C. Florentino, J. Mourão, J.P. Nunes, Coherent state transforms and theta functions, Proc. Steklov Inst. Math. 246 (2004) 283–302. [8] C. Florentino, J. Mourão, J.P. Nunes, Coherent state transforms and vector bundles on elliptic curves, J. Funct. Anal. 204 (2003) 355–398.
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Journal of Functional Analysis 258 (2010) 3413–3451 www.elsevier.com/locate/jfa
On type II0 E0-semigroups induced by boundary weight doubles Christopher Jankowski ∗,1 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA Received 24 September 2009; accepted 8 December 2009 Available online 24 December 2009 Communicated by D. Voiculescu
Abstract Powers has shown that each spatial E0 -semigroup can be obtained from the boundary weight map of a CP-flow acting on B(K ⊗ L2 (0, ∞)) for some separable Hilbert space K. In this paper, we define boundary weight maps through boundary weight doubles (φ, ν), where φ : Mn (C) → Mn (C) is a q-positive map and ν is a boundary weight over L2 (0, ∞). These doubles induce CP-flows over K for 1 < dim(K) < ∞ which then minimally dilate to E0 -semigroups by a theorem of Bhat. Through this construction, we obtain uncountably many mutually non-cocycle conjugate E0 -semigroups for each n > 1, n ∈ N. © 2009 Elsevier Inc. All rights reserved. Keywords: E0 -semigroup; CP-flow; Completely positive map
1. Introduction Let H be a separable Hilbert space, denoting its inner product by the symbol ( , ) which is conjugate-linear in its first coordinate and linear in its second. A result of Wigner in [16] shows that every weakly continuous one-parameter group of ∗-automorphisms {αt }t∈R of B(H ) is implemented by a strongly continuous unitary group {Ut }t∈R in that αt (A) = Ut AUt∗ for all A ∈ B(H ) and t ∈ R. This leads us to pursue the more general task of classifying all suitable semigroups of ∗-endomorphisms of B(H ): * Fax: +972 8 647 7648.
E-mail address:
[email protected]. 1 Present address: Department of Mathematics, Ben-Gurion University of the Negev, PO Box 653, Be’er Sheva 84105,
Israel. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.005
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Definition 1.1. We say a family {αt }t0 of ∗-endomorphisms of B(H ) is an E0 -semigroup if: 1. αs+t = αs ◦ αt for all s, t 0, and α0 (A) = A for all A ∈ B(H ). 2. For each f, g ∈ H and A ∈ B(H ), the inner product (f, αt (A)g) is continuous in t. 3. αt (I ) = I for all t 0 (in other words, α is unital). We have two different notions of what it means for two E0 -semigroups to be the same, namely conjugacy and cocycle conjugacy, the latter of which arises from Alain Connes’ definition of outer conjugacy. Definition 1.2. Let α and β be E0 -semigroups on B(H1 ) and B(H2 ), respectively. We say that α and β are conjugate if there is a ∗-isomorphism θ from B(H1 ) onto B(H2 ) such that θ ◦ αt = βt ◦ θ for all t 0. We say that α and β are cocycle conjugate if α is conjugate to β , where β is an E0 -semigroup on B(H2 ) satisfying the following condition: For some strongly continuous family of unitaries U = {Ut : t 0} acting on H2 and satisfying Ut+s = Ut βt (Us ) for all s, t 0, we have βt (A) = Ut βt (A)Ut∗ for all A ∈ B(H2 ) and t 0. Such a family of unitaries is called a unitary cocycle for β. E0 -semigroups are divided into three types based upon the existence, and structure of, their units. More specifically, let α be an E0 -semigroup on B(H ). A unit for α is a strongly continuous semigroup of bounded operators U = {U (t): t 0} such that αt (A)U (t) = U (t)A for all A ∈ B(H ). Let Uα be the set of all units for α. We say α is spatial if Uα = ∅, while we say that α is completely spatial if, for each t 0, the closed linear span of the set {U1 (t1 ) · · · Un (tn )f : f ∈ H, ti = t} is H . If an E0 -semigroup α is completely spatial, we say it is ti 0 and Ui ∈ Uα ∀i, of type I. If α is spatial but is not completely spatial, we say α is of type II. If α has no units, we say it is of type III. If α is of type I or II, we may further assign an integer n ∈ Z0 ∪ {∞} to α, in which case we say α is of type In or IIn . We call n the index of α. It was initially defined in different ways in [12] and [2], and the connection between these definitions was explored in [14]. The index of α is the dimension of a particular Hilbert space associated to its units, and it is perhaps the most fundamental cocycle conjugacy invariant for spatial E0 -semigroups. Arveson showed in [2] that the type I E0 -semigroups are entirely classified (up to cocycle conjugacy) by their index: the type I0 E0 -semigroups are semigroups of ∗-automorphisms, while for n ∈ N ∪ {∞}, every type In E0 -semigroup is cocycle conjugate to the CAR flow of rank n. However, at the present time, we do not have such a classification for those of type II or III. The first type II and type III examples were constructed by Powers in [11] and [13]. Through Arveson’s theory of product systems, Tsirelson became the first to exhibit uncountably many mutually non-cocycle conjugate E0 -semigroups of types II and III (see [15]). A dilation theorem of Bhat in [3] shows that every unital CP-flow α can be dilated to an E0 -semigroup, and that there is a minimal dilation α d of α which is unique up to conjugacy. Using Bhat’s result, Powers proved in [8] that every spatial E0 -semigroup can be obtained from the boundary weight map of a CP-flow over a separable Hilbert space K. In [9], he constructed spatial E0 -semigroups using boundary weights over K when dim(K) = 1 and then began to investigate the case when dim(K) = 2. Our goal is to use boundary weight maps to induce unital CP-flows over K for 1 < dim(K) < ∞ and to classify their minimal dilations to E0 -semigroups up to cocycle conjugacy. To do so, we define a natural boundary weight map ρ → ω(ρ) using a unital completely positive map φ
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and a normalized boundary weight ν over L2 (0, ∞). The necessary and sufficient condition that this map induces a unital CP-flow α is that φ satisfies a definition of q-positive analogous to that from [8] (see Definition 3.1 and Proposition 3.2), in which case we say that α is the CP-flow induced by the boundary weight double (φ, ν). We develop a comparison theory for boundary weight doubles (φ, ν) and (ψ, ν) (φ and ψ unital) √ in the case √ that ν is a normalized unbounded boundary weight over L2 (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ), finding that the doubles induce cocycle conjugate E0 -semigroups if and only if there is a hyper maximal q-corner from φ to ψ (see Definition 4.4 and Proposition 4.6). The problem of determining hyper maximal q-corners from φ to ψ becomes much easier if we focus on a particular class of q-positive maps, called the q-pure maps, which have the least possible q-subordinates (Definition 4.2). Given a q-positive map φ acting on Mn (C) and a unitary U ∈ Mn (C), we can form a new map φU by φU (A) = U ∗ φ(U AU ∗ )U . We describe the order isomorphism between the q-subordinates of φ and those of φU , which in turn leads to the existence of a hyper maximal q-corner from φ to φU if φ is unital and q-pure (Proposition 4.5). With this result in mind, we begin the task of classifying the unital q-pure maps. We find that the rank one unital q-pure maps φ : Mn (C) → Mn (C) are precisely the maps φ(A) = ρ(A)I for faithful states ρ on Mn (C) (Proposition 5.2). That these maps give us an enormous class of mutually non-cocycle conjugate E0 -semigroups in one of our main results (Theorem 5.4). Furthermore, for n > 1, none of the E0 -semigroups constructed from boundary weight doubles satisfying the conditions of Theorem 5.4 are cocycle conjugate to any of the E0 -semigroups obtained from one-dimensional boundary weights by Powers in [9] (Corollary 5.5). We turn our attention to the unital q-pure maps that are invertible. These maps are best understood through their (conditionally negative) inverses. In Theorem 6.11, we find a necessary and sufficient condition for an invertible unital map φ on Mn (C) to √ be q-pure. In √ this case, however, if ν is a normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ), then the E0 -semigroup induced by the boundary weight double (φ, ν) is entirely determined by ν. This E0 -semigroup is the one induced by ν in the sense of [9]. 2. Background 2.1. Completely positive maps Let φ : U → B be a linear map between C ∗ -algebras. For each n ∈ N, define φn : Mn (U) → Mn (B) by ⎛
A11 . φn ⎝ .. An1
··· .. . ···
⎞ ⎛ φ(A11 ) A1n .. ⎠ ⎝ .. = . . Ann
φ(An1 )
··· .. . ···
⎞ φ(A1n ) .. ⎠. . φ(Ann )
We say that φ is completely positive if φn is positive for all n ∈ N. A linear map φ : B(H1 ) → B(H2 ) is completely positive if and only if for all A1 , . . . , An ∈ B(H1 ), f1 , . . . , fn ∈ H2 , and n ∈ N, we have n fi , φ A∗i Aj fj 0. i,j =1
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Stinespring’s theorem asserts that if U is a unital C ∗ -algebra and φ : U → B(H ) is a unital completely positive map, then φ dilates to a ∗-homomorphism in that there is a Hilbert space K, a ∗-homomorphism π : U → B(K), and an isometry V : H → K such that φ(A) = V ∗ π(A)V for all A ∈ U. From the work of Choi [4] and Arveson [1], we know that a normal linear map φ : B(H1 ) → B(H2 ) is completely positive if and only if it can be written in the form φ(A) =
n
Si ASi∗
i=1
for some n ∈ N ∪ {∞} and maps Si : H1 → H2 which are linearly independent over 2 (N) in rn the sense that if i=1 zi Si = 0 for a sequence {zi }ri=1 ∈ 2 (N), then zi = 0 for all i. With these hypotheses satisfied, the number n is unique. We will use the above conditions for complete positivity interchangeably. 2.2. Conditionally negative maps Wesay a self-adjoint linear map ψ : B(K) → B(K) is conditionally negative if, whenm ever i=1 Ai fi = 0 for A1 , . . . , Am ∈ B(K), f1 , . . . , fm ∈ K, and m ∈ N, we have m ∗ n (f i=1 i , ψ(Ai Aj )fj ) 0. If K = C , then from the literature (see, for example, Theorem 3.1 of [10]) we know that ψ has the form ψ(A) = sA + Y A + AY ∗ −
p
λi Si ASi∗ ,
i=1
where s ∈ R, tr(Y ) = 0, and for all i and j we have λi > 0, tr(Si ) = 0 and tr(Si∗ Sj ) = nδij , where p n2 is independent of the maps Si . This form for ψ is unique in the sense that if ψ is written in the form ∗
ψ(A) = tA + ZA + AZ −
p
μi Ti ATi∗ ,
i=1
where t ∈ R, tr(Z) = 0, and for all i and j we have μi > 0, tr(Ti ) = 0, and tr(Ti∗ Tj ) = nδij , p p μi Ti ATi∗ for all A ∈ Mn (C). Indeed, let {vk }nk=1 then s = t, Z = Y , and i=1 λi Si ASi∗ = i=1 √ n be any orthonormal basis for C , let hk = vk / n for each k, let f ∈ Cn be arbitrary, and for k = 1, . . . , n, define Ak ∈ Mn (C) by Ak = f h∗k . Using the trace conditions, we find n k=1
ψ(Ak )hk =
n n n hk , Y ∗ hk f (hk , hk )sf + (hk , hk )Yf + k=1
−
k=1
n
p
k=1
i=1
λi hk , Si∗ hk Si f
k=1
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
= sf + Yf + 0 −
n p i=1
= sf + Yf −
p
λi hk , Si∗ hk Si f
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k=1
λi (0)Si f = sf + Yf.
i=1
An analogous computation shows that nk=1 ψ(Ak )hk = tf + Zf . Since f ∈ Cn was arbitrary, we conclude (t − s)I = Y − Z. Therefore, tr((t − s)I ) = tr(Y − Z) = 0, so t = s and Y = Z. p p Consequently, i=1 λi Si ASi∗ = i=1 μi Ti ATi∗ for all A ∈ Mn (C). 2.3. CP-flows and Bhat’s theorem Let K be a separable Hilbert space and let H = K ⊗ L2 (0, ∞). We identify H with the space of K-valued measurable functions on (0, ∞) which are square integrable. Under this identification, the inner product on H is L2 ((0, ∞); K),
∞ (f, g) = f (x), g(x) dx. 0
Let U = {Ut }t0 be the right shift semigroup on H , so for all t 0 and f ∈ H we have (Ut f )(x) = f (x − t) for x > t and (Ut f )(x) = 0 otherwise. Let Λ : B(K) → B(H ) be the map defined by (Λ(A)f )(x) = e−x Af (x) for all A ∈ B(K), f ∈ H . Definition 2.1. Assume the above notation. A strongly continuous semigroup α = {αt : t 0} of completely positive contractions of B(H ) into itself is a CP-flow if αt (A)Ut = Ut A for all A ∈ B(H ). A theorem of Bhat in [3] allows us to generate E0 -semigroups from unital CP-flows, and, more generally, from strongly continuous completely positive semigroups of unital maps on B(H ), called CP-semigroups. We give a reformulation of Bhat’s theorem (see Theorem 2.1 of [9]): Theorem 2.2. Suppose α is a unital CP-semigroup of B(H1 ). Then there is an E0 -semigroup α d of B(H2 ) and an isometry W : H1 → H2 such that αt (A) = W ∗ αtd W AW ∗ W and αt (W W ∗ ) W W ∗ for all t > 0. If the projection E = W W ∗ is minimal in that the closed linear span of the vectors αtd1 (EA1 E) · · · αtdn (EAn E)Ef for f ∈ K, Ai ∈ B(H1 ) and ti 0 for all i = 1, 2, . . . , n and n = 1, 2, . . . is H2 , then α d is unique up to conjugacy.
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In [8], Powers showed that every spatial E0 -semigroup acting on B(H) (for H a separable Hilbert space) is cocycle conjugate to an E0 -semigroup which is a CP-flow, and that every CPflow over K arises from a boundary weight map over H = K ⊗ L2 (0, ∞). The boundary weight map ρ → ω(ρ) of a CP-flow α associates to every ρ ∈ B(K)∗ a boundary weight, that is, a linear functional ω(ρ) acting on the null boundary algebra A(H ) =
IH − Λ(IK )B(H ) IH − Λ(IK )
which is normal in the following sense: If we define a linear functional (ρ) on B(H ) by
(ρ)(A) = ω(ρ) IH − Λ(IK )A IH − Λ(IK ) , then (ρ) ∈ B(H )∗ . If ω(ρ)(IH − Λ(IK )) = ρ(IK ) for all ρ ∈ B(K)∗ , then α is unital. For the sake of neatness, we will omit the subscripts H and K from the previous sentence ∞ when they are clear. Let δ be the generator of α, and define Γ : B(H ) → B(H ) by Γ (A) = 0 e−t Ut AUt∗ . ∞ The resolvent Rα := (I − δ)−1 of α satisfies Rα (A) = 0 e−t αt (A) dt for all A ∈ B(H ). Its associated predual map Rˆ α is given by ˆ +η Rˆ α (η) = Γˆ ω(Λη)
(1)
for all η ∈ B(H )∗ . A CP-flow α over K is entirely determined by a set of normal completely positive contractions π # = {πt# : t > 0} from B(H ) into B(K), called the generalized boundary representation of α. Its relationship to the boundary weight map is as follows. For each t > 0, denote by πˆ t : B(K)∗ → B(H )∗ the predual map induced by πt# . For the truncated boundary weight maps ρ → ωt (ρ) ∈ B(H )∗ defined by ωt (ρ)(A) = ω(ρ) Ut Ut∗ AUt Ut∗ ,
(2)
ˆ t )−1 and ωt = πˆ t (I − Λˆ πˆ t )−1 for all t > 0. The maps {π # }b>0 have we have πˆ t = ωt (I + Λω b # a σ -strong limit π0 as b → 0 for each A ∈ t>0 Ut B(H )Ut∗ , called the normal spine of α. If α is unital, then the index of α d as an E0 -semigroup is equal to the rank of π0# as a completely positive map (Theorem 4.49 of [8]). Having seen that every CP-flow has an associated boundary weight map, we would like to approach the situation from the opposite direction. More specifically, under what conditions is a map ρ → ω(ρ) from B(K)∗ to weights acting on A(H ) the boundary weight map of a CP-flow over K? Powers has found the answer (see Theorem 3.3 of [9]): Theorem 2.3. If ρ → ω(ρ) is a completely positive mapping from B(K)∗ into weights on B(H ) satisfying ω(ρ)(I − Λ(IK )) ρ(IK ) for all positive ρ ∈ B(K)∗ , and if the maps πˆ t := ωt (I + ˆ t )−1 are completely positive contractions from B(K)∗ into B(H )∗ for all t > 0, then ρ → Λω ω(ρ) is the boundary weight map of a CP-flow over K. The CP-flow is unital if and only if ω(ρ)(I − Λ(IK )) = ρ(IK ) for all ρ ∈ B(K)∗ .
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If dim(K) = 1, the boundary weight map is just c ∈ C → ω(c) = cω(1), so we may view our boundary weight map as a single positive boundary weight ω := ω(1) acting on A(L2 (0, ∞)). Since the functional defined on B(H ) by (A) = ω
I − Λ(1)A I − Λ(1)
is positive and normal, it has the form (A) = n∈N∪{∞} , so vectors {fk }k=1 ω
n
k=1 (fk , Afk )
for some mutually orthogonal
n
I − Λ(1)A I − Λ(1) = (fk , Afk ) k=1
for all A ∈ B(H ). If ω is normalized (that is, ω(I − Λ(1)) = 1), then nk=1 fk 2 = 1. In [9], 2 Powers induced E0 -semigroups using normalized boundary weights √ over L (0,√∞). The type d induced by a normalized boundary weight ω( I − Λ(1)A I − Λ(1) ) = of E -semigroup α n 0 k=1 (fk , Afk ) depends on whether ω is bounded in the sense that for some r > 0 we have |ω(B)| rB for all B ∈ A(H ). Results from [8] imply that α d is of type In if ω is bounded and of type II0 if ω is unbounded. If ω is unbounded, then both ωt (I ) and ωt (Λ(1)) approach infinity as t approaches√zero. We will√focus on normalized unbounded boundary weights over L2 (0, ∞) of the form ω( I − Λ(1)A I − Λ(1) ) = (f, Af ). We note that, as discussed in detail in [7], such boundary weights are not normal weights. If α and β are CP-flows, we say that α β if αt − βt is completely positive for all t 0. The subordinates of a CP-flow are entirely determined by the subordinates of its generalized boundary representation (see Theorem 3.4 of [9]): Theorem 2.4. Let α and β be CP-flows over K with generalized boundary representations π # = {πt# } and ξ # = {ξt# }, respectively. Then β is subordinate to α if and only if πt# − ξt# is completely positive for all t > 0. Given two unital CP-flows α and β, it is natural to ask when their minimally dilated E0 semigroups are cocycle conjugate. The following definition from [8] provides us with a key: Definition 2.5. Let α and β be CP-flows over K1 and K2 , respectively, where H1 = K1 ⊗ L2 (0, ∞) and H2 = K2 ⊗ L2 (0, ∞). We say that a family of linear maps γ = {γt : t 0} from B(H2 , H1 ) into itself is a flow corner from α to β if the family of maps Θ = {Θt : t 0} defined by Θt
A11 A21
A12 A22
=
αt (A11 ) γt∗ (A21 )
γt (A12 ) βt (A22 )
is a CP-flow over K1 ⊕ K2 . If γ is a flow corner from α to β, we consider subordinates Θ of Θ that are CP-flows of the form Θt
A11 A21
A12 A22
:=
αt (A11 ) γt∗ (A21 )
γt (A12 ) . βt (A22 )
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We say that γ is a hyper maximal flow corner from α to β if, for every such subordinate Θ of Θ, we have α = α and β = β . Our results will involve type II0 E0 -semigroups. These are spatial E0 -semigroups which are not semigroups of ∗-automorphisms and have only one unit V = {Vt }t0 up to scaling by etλ for λ ∈ C. In the case that unital CP-flows α and β minimally dilate to type II0 E0 -semigroups, we have a necessary and sufficient condition for α d and β d to be cocycle conjugate (Theorem 4.56 of [8]): Theorem 2.6. Suppose α and β are unital CP-flows over K1 and K2 and α d and β d are their minimal dilations to E0 -semigroups. Suppose γ is a hyper maximal flow corner from α to β. Then α d and β d are cocycle conjugate. Conversely, if α d is a type II0 and α d and β d are cocycle conjugate, then there is a hyper maximal flow corner from α to β. We will later use this theorem to determine a necessary and sufficient condition for some of the E0 -semigroups we construct to be cocycle conjugate (see Definition 4.4 and Proposition 4.6). 3. Our boundary weight map Recall that a completely positive linear map φ can have negative eigenvalues. Moreover, even if I + tφ is invertible for a given t, it does not necessarily follow that φ(I + tφ)−1 is completely positive. In our boundary weight construction, we will require a special kind of completely positive map: Definition 3.1. A linear map φ : Mn (C) → Mn (C) is q-positive if φ has no negative eigenvalues and φ(I + tφ)−1 is completely positive for all t 0. Henceforth, we naturally identify a finite-dimensional Hilbert space K with Cn and B(K ⊗ with Mn (B(L2 (0, ∞))). Under these identifications, the right shift t units on K ⊗ 2 L (0, ∞) is the matrix whose ij th entry is δij Vt for Vt the right shift on L2 (0, ∞). The map Λn×n : B(K) → B(K ⊗ L2 (0, ∞)) sends an n × n matrix B = (bij ) ∈ Mn (C) to the matrix Λn×n (B) whose ij th entry is bij Λ(1) ∈ B(L2 (0, ∞)). The null boundary algebra A(H ) is simply Mn (A(L2 (0, ∞))). Given a boundary weight ν over L2 (0, ∞), we write Ων,n×k for the map that sends an n × k matrix A = (Aij ) ∈ Mn×k (A(L2 (0, ∞))) to the matrix Ων,n×k (A) ∈ Mn×k (C) whose ij th entry is ν(Aij ). We will suppress the integers n and k when they are clear, writing the above maps as Ων and Λ. In the proposition and corollary that follow, we show how to construct a CP-flow using a q-positive map φ : Mn (C) → Mn (C), a normalized boundary weight ν over L2 (0, ∞), and the map Ων := Ων,n×n : A(H ) → Mn (C). The map Ων is completely positive since ν is positive. L2 (0, ∞))
Proposition 3.2. Let H = Cn ⊗ L2 (0, ∞). Let φ : Mn (C) → Mn (C) be a unital completely positive map with no negative eigenvalues, and let ν be a normalized unbounded boundary weight over L2 (0, ∞). Then the map ρ → ω(ρ) from Mn (C)∗ into boundary weights on A(H ) defined by ω(ρ)(A) = ρ φ Ων (A)
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ˆ t )−1 define normal completely is completely positive. Furthermore, the maps πˆ t := ωt (I + Λω positive contractions πt# of B(H ) into Mn (C) for all t > 0 if and only if φ is q-positive. Proof. The map ρ → ω(ρ) is completely positive since it is the composition of two completely positive maps. Before proving either direction, we let st = νt (Λ(1)) for all t > 0 and prove the equality πˆ t (ρ) = ρ φ(I + st φ)−1 Ωνt
(3)
for all ρ ∈ Mn (C)∗ . Denoting by Ut the right shift on H for every t > 0, we claim that (I + ˆ t )−1 = (I + st φ) ˆ −1 . Indeed, for arbitrary t > 0, B ∈ Mn (C), and ρ ∈ Mn (C)∗ , we have Λω ˆ t (ρ)(B) = ρ φ Ων Ut Ut∗ Λ(B)Ut Ut∗ = ρ φ Ωνt Λ(B) = st ρ φ(B) , Λω ˆ t = st φˆ and (I + Λω ˆ t )−1 = (I + st φ) ˆ −1 . hence Λω For any t > 0 and A ∈ B(H ), we have ˆ t )−1 (ρ)(A) = (I + Λω ˆ t )−1 (ρ) φ Ωνt (A) πˆ t (ρ)(A) = ωt (I + Λω ˆ −1 (ρ) φ Ωνt (A) = ρ (I + st φ)−1 φ Ωνt (A) = (I + st φ) = ρ φ(I + st φ)−1 Ωνt (A) , establishing (3). Assume the hypotheses of the backward direction and let t > 0. By construction, πˆ t maps Mn (C)∗ into B(H )∗ . It is also a contraction, since for all ρ ∈ Mn (C)∗ we have πˆ t (ρ) = ρ φ(I + st φ)−1 Ων ρφ(I + st φ)−1 Ων t t = ρφ(I + st φ)−1 Ωνt (I ) = ρφ(I + st φ)−1 νt (I )ICn νt (I ) νt (I ) n = ρ I 1 + s C = ρ 1 + ν (Λ(1)) ρ, t t where the last inequality follows from the fact that νt I − Λ(1) ν I − Λ(1) = 1. Therefore, for every t > 0, πˆ t defines a normal contraction πt# from B(H ) into Mn (C) satisfying πˆ t (ρ) = ρ ◦ πt# for all ρ ∈ Mn (C)∗ . From Eq. (3) we see πt# = φ(I + st φ)−1 Ωνt , so πt# is the composition of completely positive maps and is thus completely positive for all t > 0. Now assume the hypotheses of the forward direction. By unboundedness of ν, the (monotonically decreasing) values {st }t>0 form a set equal to either (0, ∞) or [0, ∞). Choose any t > 0 such that st > 0. Let T ∈ B(H ) be the matrix with ij th entry (1/νt (I ))I , and let κt : Mn (C) → B(H ) be the map that sends B = (bij ) ∈ Mn (C) to the matrix κt (B) ∈ B(H ) whose ij th entry is (bij /νt (I ))I . We note that κt is the Schur product B → B · T , which is completely positive since T is positive. For all B ∈ Mn (C), we have φ(I + st φ)−1 (B) = πt# κt (B) ,
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so φ(I + st φ)−1 is the composition of completely positive maps and is thus completely positive. As noted above, the values {st }t>0 span (0, ∞), so φ is q-positive. 2 Corollary 3.3. The map ρ → ω(ρ) in Proposition 3.2 is the boundary weight map of a unital CP-flow α over Cn , and the Bhat minimal dilation α d of α is a type II0 E0 -semigroup. Proof. The first claim of the corollary follows immediately from Theorem 2.3 and Proposition 3.2 since ω(ρ) I − Λ(ICn ) = ρ φ(ICn ) = ρ(ICn )
(4)
for all ρ ∈ Mn (C)∗ . For the second assertion, we note that by Theorem 4.49 of [8], the index of α d is equal to the rank of the normal spine π0# of α, where π0# is the σ -strong limit of the maps {πb# }b>0 for each A ∈ t>0 Ut B(H )Ut∗ . Fix t > 0, and let A ∈ Ut B(H )Ut∗ . From formula (3), −1 Ωνb (A) . πb# (A) = φ I + νb Λ(1) φ For all b < t we have Ωνb (A) = Ωνt (A) < ∞. Since νb (Λ(1)) → ∞ as b → 0, we conclude limb→0 πb# (A) = 0, hence π0# = 0 and the index of α is zero. However, α d is not completely spatial since α is not derived from the zero boundary weight map (see Lemma 4.37 and Theorem 4.52 of [8]), so α d is of type II0 . 2 Given a q-positive φ : Mn (C) → Mn (C) and a normalized unbounded boundary weight ν over L2 (0, ∞), we call (φ, ν) a boundary weight double. As we have seen, if φ is unital then the boundary weight double naturally defines a boundary weight map through the construction of Proposition 3.2, inducing a type II0 E0 -semigroup α d which is unique up to conjugacy by Theorem 2.2. We should note that it is not necessary for φ to be unital in order for the boundary weight double to induce a CP-flow: If φ is any q-positive contraction such that νt (I )φ(I + νt (Λ(1))φ)−1 1 for all t > 0, then the arguments given in the proofs of Proposition 3.2 and Corollary 3.3 show that the boundary weight double (φ, ν) induces a CP-flow α. However, if φ is not unital, then by Eq. (4) and Theorem 2.3, neither is α. Motivated by [8], we make the following definition: Definition 3.4. Suppose α : B(H1 ) → B(K1 ) and β : B(H2 ) → B(K2 ) are normal and completely positive. Write each A ∈ B(H1 ⊕ H2 ) as A = (Aij ), where Aij ∈ B(Hj , Hi ) for each i, j = 1, 2. We say a linear map γ : B(H2 , H1 ) → B(K2 , K1 ) is a corner from α to β if ψ : B(H1 ⊕ H2 ) → B(K1 ⊕ K2 ) defined by ψ
A11 A21
A12 A22
=
α(A11 ) γ ∗ (A21 )
γ (A12 ) β(A22 )
is a normal completely positive map. We will repeatedly use the following lemma, which gives us the form of any corner between normal completely positive contractions of finite index. We believe that this result is already present in the literature, but we present a proof here for the sake of completeness:
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
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Lemma 3.5. Let H1 , H2 , K1 , and K2 be separable Hilbert spaces. Let α : B(H1 ) → B(K1 ) and β : B(H2 ) → B(K2 ) be normal completely positive contractions of the form α(A11 ) =
n
Si A11 Si∗ ,
β(A22 ) =
p
Tj A22 Tj∗ ,
j =1
i=1 p
where n, p ∈ N and the sets of maps {Si }ni=1 and {Tj }j =1 are both linearly independent. A linear map γ : B(H2 , H1 ) → B(K2 , K1 ) is a corner from α to β if and only if for all A12 ∈ B(H2 , H1 ) we have γ (A12 ) =
cij Si A12 Tj∗ ,
i,j
where C = (cij ) ∈ Mn×p (C) is any matrix such that C 1. Proof. For the backward direction, let C = (cij ) ∈ M n×p (C) be any contraction, and define a linear map γ : B(H2 , H1 ) → B(K2 , K1 ) by γ (A) = i,j cij Si ATj∗ . We need to show that the map L
A11 A21
A12 A22
=
α(A11 ) γ ∗ (A21 )
γ (A12 ) β(A22 )
is normal and completely positive. To prove this, we first assume that n p and note that by Polar Decomposition we may write Cn×p = Vn×p Tp×p , where Vn×p is a partial isometry of ∗ D rank p and T is positive. Unitarily diagonalizing T we see Cn×p = Vn×p Wp×p p×p Wp×p . ∗ We may easily add columns to Vn×p Wp×p to form a unitary matrix in Mn (C), which we call U ∗ . Defining D˜ = (dij ) ∈ Mn×p (C) to be the matrix obtained from D by adding n − p rows of ∗ D, so C ∗ ˜ zeroes, we see U ∗ D˜ = Vn×p Wp×p n×p = U DW p×p and ∗ ˜ U Cn×p Wp×p = D.
In other words,
cij uki w j =
i,j
δk dk if k p . 0 if k > p
Next, define {Si }ni=1 : H1 → K1 and {Tj }j =1 : H2 → K2 by p
Si
=
n
uik Sk ,
k=1
so Si =
n
k=1 uki Sk
and Tj =
p
=1 w j Tj
Tj
=
p =1
for all i and j .
w j Tj ,
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C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
Since U and W are unitary, it follows that D = C 1 and that the maps {Si }ni=1 are p linearly independent, as are the maps {Tj }j =1 . We observe that for any A11 ∈ B(H1 ) and A22 ∈ B(H2 ), n
Si A11 Si∗ =
i=1
n
∗ Si A11 Si
p
and
Tj A22 Tj∗ =
j =1
i=1
p
∗ Tj A22 Tj .
j =1
Finally, for any A12 ∈ B(H2 , H1 ), we use our above computations to find that i,j
∗ ∗ cij uki w j Sk A12 T = cij uki w j Sk A12 T
cij Si A12 Tj∗ =
i,j,k,
=
(kp),
∗ cij uki w j Sk A12 T
(k>p),
i,j
+
=
k,
∗ T
cij uki w j Sk A
i,j
i,j
∗ ∗ Tk + 0 = dkk Sk A Tk . p
dkk Sk A12
k=1
kp
We have shown that n S A11 (Si )∗ L(A) = p i=1 i ∗ i=1 dii Ti A21 (Si )
p
∗ i=1 dii Si A12 (Ti ) p ∗ i=1 Ti A22 (Ti )
for all A=
A11 A21
A12 A22
∈ B(H1 ⊕ H2 ).
For each i = 1, . . . , p, define Zi : H1 ⊕ H2 → K1 ⊕ K2 by Zi =
dii Si 0
0 Ti
,
so L(A) =
p i=1
Zi AZi∗
p (1 − |dii |2 )Si A11 Si∗ + 0 i=1
0 0
n Si A11 Si∗ + 0 i=p+1
0 . 0
Since D 1, the line above shows that L is the sum of two normal completely positive maps and is thus normal and completely positive. Therefore, γ is a corner from α to β. If, on the other hand, n < p, then the same argument we just used shows that γ ∗ is a corner from β to α, which is equivalent to showing that γ is a corner from α to β. For the forward direction, suppose that γ is a corner from α to β, so the map Υ : B(H1 ⊕ H2 ) → B(K1 ⊕ K2 ) defined by
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
Υ
A12 A22
A11 A21
n
∗ i=1 Si A11 Si γ ∗ (A21 )
=
p
3425
γ (A12 )
∗ j =1 Tj A22 Tj
is normal and completely positive. Therefore, for some q ∈ N ∪ {∞} and maps Yi : H1 ⊕ H2 → K1 ⊕ K2 for i = 1, 2, . . . , linearly independent over 2 (N), we have ˜ = Υ (A)
q
˜ i∗ Yi AY
i=1
for all A˜ ∈ B(H1 ⊕ H2 ). For i = 1, 2, let Ei ∈ B(H1 ⊕ H2 ) be projection onto Hi , and let Fi ∈ B(K1 ⊕ K2 ) be projection onto Ki . Since α and β are contractions we have Υ (E1 ) F1 and Υ (E2 ) F2 , so Yi Ej Yi∗ Fj for each i and j . It follows that each Yi , i = 1, . . . , q, can be written in the form Yi =
S˜i 0
0 T˜i
for some S˜i ∈ B(H1 , K K2 ). 1 ) and T˜i ∈ B(H2 , q Note that α(A11 ) = ni=1 Si A11 Si∗ = i=1 S˜i A11 S˜i∗ for all A11 ∈ B(H1 ). For each S˜i , define a completely positive map Li by Li (A) = S˜i AS˜i∗ for A ∈ B(H1 ). Since α − Li is completely positive, it follows from the work of Arveson in [1] that S˜i can be written as S˜i =
n
rij Sj
j =1
for some complex coefficients {rij }nj=1 . The same argument shows that for each T˜i we have T˜i =
p
bij Tj
j =1 p
q
for some coefficients {bij }j =1 . It now follows from linear independence of the maps {Yi }i=1 that q n + p. Let R = (rij ) ∈ Mq×n (C) and B = (bij ) ∈ Mq×p (C), and let A ∈ B(H1 ). We calculate n
Si ASi∗
=
i=1
q
S˜i AS˜i∗
i=1
=
n
=
n q i=1
j,k=1
q i=1
rij rik Sj ASk∗
j,k=1
rij rik Sj ASk∗ .
(5)
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C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
q Let M = R T (R T )∗ ∈ Mn (C), so its j kth entry is mj k = i=1 rij rik . Unitarily diagonalizing M as U MU ∗ = D for some diagonal D and defining maps {Si }ni=1 by Si = nk=1 uik Sk , we see that Eq. (5) and the same linear algebra technique from the proof of the backward direction yield n
Si ASi∗ =
n
i=1
n
Si ASi∗ =
i=1
mj k Sj ASk∗ =
j,k=1
n
dii Si ASi∗ .
i=1
Therefore D = I and consequently M = I , hence R = 1. An identical argument shows that B = 1. Let A11 A12 ∈ B(H1 ⊕ H2 ) A˜ = A21 A22 be arbitrary. Let C = (cj k ) ∈ Mn×p (C) be the matrix C = (B ∗ R)T , noting that C 1. ˜ = q Yi AY ˜ ∗ yields A straightforward computation of Υ (A) i i=1 γ (A12 ) =
q
Si A12 Ti∗
=
i=1
=
q j,k
n q i=1
aij Sj A12
j =1
aij bik Sj A12 Tk∗ =
p
bik Tk∗
k=1
cj k Sj A12 Tk∗ ,
j,k
i=1
hence γ is of the form claimed.
2
4. Comparison theory for q-positive maps Just as in the general study of various classes of linear operators, it is natural to impose, and examine, an order structure for q-positive maps. If φ and ψ are q-positive maps acting on Mn (C), we say that φ q-dominates ψ (and write φ q ψ ) if φ(I + tφ)−1 − ψ(I + tψ)−1 is completely positive for all t 0. We would like to find the q-positive maps with the least complicated structure of q-subordinates. That last statement is not as simple as it seems. We might think to define a q-positive map φ to be “q-pure” if φ q ψ q 0 implies ψ = λφ for some λ ∈ [0, 1], but there exist q-positive maps φ such that for every λ ∈ (0, 1) we have φ q λφ. One such example is the Schur map φ on M2 (C) given by φ
a11 a21
a12 a22
=
a11 ( 1−i 2 )a21
( 1+i 2 )a12 a22
.
As it turns out, every q-positive map is guaranteed to have a one-parameter family of q-subordinates of a particular form: Proposition 4.1. Let φ q 0. For each s 0, let φ (s) = φ(I + sφ)−1 . Then φ (s) q 0 for all s 0. Furthermore, the set {φ (s) }s0 is a monotonically decreasing family of q-subordinates of φ, in the sense that φ (s1 ) q φ (s2 ) if s1 s2 .
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
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Proof. For all s 0 and t 0, we have −1 −1 = φ(I + sφ)−1 I + tφ(I + sφ)−1 φ (s) I + tφ (s) −1 = φ I + tφ(I + sφ) (I + sφ) −1 = φ I + (s + t)φ , which is completely positive by q-positivity of φ. Therefore, φ (s) q 0 for all s 0. To prove that φ (s1 ) q φ (s2 ) if s1 s2 , we let t 0 be arbitrary and examine the map −1 −1 Φ := φ (s1 ) I + tφ (s1 ) − φ (s2 ) I + tφ (s2 ) . Letting t1 = s1 + t and t2 = s2 + t, we make the following observations: −1 φ (sj ) I + tφ (sj ) = φ (tj ) for j = 1, 2, φ (t1 ) − φ (t2 ) = (I + t2 φ)−1 (I + t2 φ)φ − φ(I + t1 φ) (I + t1 φ)−1 .
(6) (7)
Eqs. (6) and (7) give us Φ = (I + t2 φ)−1 (I + t2 φ)φ − φ(I + t1 φ) (I + t1 φ)−1 = (I + t2 φ)−1 (t2 − t1 )φ 2 (I + t1 φ)−1 = (t2 − t1 ) φ(I + t2 φ)−1 φ(I + t1 φ)−1 . The last line is a non-negative multiple of a composition of completely positive maps and is thus completely positive. We conclude that φ (s1 ) q φ (s2 ) . 2 We now have the correct notion of what it means to be q-pure: Definition 4.2. Let φ : Mn (C) → Mn (C) be unital and q-positive. We say that φ is q-pure if its set of q-subordinates is precisely {0} ∪ {φ (s) }s0 . Lemma 4.3. Let ν be a normalized unbounded boundary weight over L2 (0, ∞) of the form ν
I − Λ(1)B I − Λ(1) = (f, Bf ).
Let φ : Mn (C) → Mn (C) be a q-positive contraction such that νt (I )φ(I + νt (Λ(1))φ)−1 1 for all t > 0, and let α be the CP-flow derived from the boundary weight double (φ, ν), with boundary generalized representation π = {πt# }t>0 . Let β be any CP-flow over Cn , with generalized boundary representation ξ # = {ξt# }t>0 and boundary weight map ρ → η(ρ). Then α β if and only if β is induced by the boundary weight double (ψ, ν), where ψ : Mn (C) → Mn (C) is a q-positive map satisfying φ q ψ.
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Proof. As before, for each t > 0 we let st = νt (Λ(1)). Assume the hypotheses of the backward direction. Then ξt# = ψ(I + st ψ)−1 Ωνt , and the direction now follows from Theorem 2.4 since the line below is completely positive for all t > 0: πt# − ξt# = φ(I + st φ)−1 − ψ(I + st ψ)−1 Ωνt . Now assume the hypotheses of the forward direction. Recall that by construction of ν, the set {st }t>0 is decreasing. If st > 0 for all t > 0 we define P = ∞. Otherwise, we define P to be the smallest positive number such that sP = 0. Fix any t0 ∈ (0, P ). Notationally, write each g ∈ H := Cn ⊗ L2 (0, ∞) in its components as g(x) = (g1 (x), . . . , gn (x)), and write ft0 for the function Vt0 Vt∗0 f ∈ L2 (0, ∞), where Vt0 is the right shift t0 units on L2 (0, ∞). Let Ut0 be the right shift t0 units on H. Under our identifications, Ut0 Ut∗0 is the diagonal matrix in Mn (B(L2 (0, ∞))) with iith entries Vt0 Vt∗0 . Define S : H → Cn by Sg = (ft0 , g1 ), . . . , (ft0 , gn ) , noting that Ωνt0 (A) = SAS ∗ for all A ∈ B(H ). Since φ(I + st0 φ)−1 is completely positive, ∗ we know it has the form φ(I + st0 φ)−1 (M) = m i=1 Ri MRi for some R1 , . . . , Rm ∈ Mn (C). Therefore, πt#0 (A) =
m −1 Ωνt0 (A) = φ(I + st0 φ) Ri SAS ∗ Ri∗ . i=1
The map ξt#0 is a subordinate of πt#0 , so from Arveson’s work in metric operator spaces in [1], we know that ξt#0 has the form ξt#0 (A) =
m
cij Ri SAS ∗ Rj∗ ,
i,j =1
for some complex numbers {cij }. Let Lt0 be the map Lt0 (M) = i,j cij Ri MRj∗ , noting that ξt#0 (A) = Lt0 (SAS ∗ ) = Lt0 (Ωνt0 (A)) for all A ∈ B(H ). Defining ψt0 : Mn (C) → Mn (C) by ψt0 = (I − ξt#0 Λ)−1 Lt0 , we find that for arbitrary A ∈ B(H ) and A´ ∈ Mn (C), −1 # ξt0 (A) ηt0 (ρ)(A) = ξˆt0 (I − Λˆ ξˆt0 )−1 (ρ)(A) = ρ I − ξt#0 Λ −1 = ρ I − ξt#0 Λ Lt0 Ωνt0 (A) = ρ ψt0 (Ωνt0 A)
(8)
´ ´ = ηt0 (ρ) Λ(A) ´ = ρ ψt0 Ωνt Λ(A) ´ , ˆ t0 (ρ)(A) = st0 ρ ψt0 (A) Λη 0
(9)
and
ˆ t0 = st0 ψˆ t0 . so Λη ˆ t0 )−1 , we find Using formulas (8) and (9) and the fact that ξˆt0 = ηt0 (I + Λη
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
3429
ˆ t0 )−1 (ρ) = (I + Λη ˆ t0 )−1 (ρ) (ψt0 Ωνt ) ρ ξt#0 = ξˆt0 (ρ) = ηt0 (I + Λη 0 = (I + st0 ψˆ t0 )−1 (ρ) (ψt0 Ωνt0 ) = ρ (I + st0 ψt0 )−1 ψt0 Ωνt0 = ρ ψt0 (I + st0 ψt0 )−1 Ωνt0 for all ρ ∈ Mn (C)∗ , hence ξt#0 = ψt0 (I + st0 ψt0 )−1 Ωνt0 . We now show that the maps {ψt }t>0 are constant on the interval (0, P ). Let t ∈ [t0 , P ) be arbitrary. For each A´ = (aij ) ∈ Mn (C), let A ∈ B(H ) be the matrix with ij th entry (aij /νt (I ))Vt Vt∗ . Let ρ ∈ Mn (C)∗ . Straightforward computations using formula (2) yield Ωt0 (A) = Ωt (A) = A´ and ηt0 (ρ)(A) = ηt (ρ)(A). Combining these equalities gives us ´ = ρ ψt0 Ωνt (A) = ηt0 (ρ)(A) ρ ψt0 (A) 0 ´ . = ηt (ρ)(A) = ρ ψt Ωνt (A) = ρ ψt (A) Since the above formula holds for every A´ ∈ Mn (C) and ρ ∈ Mn (C)∗ , we have ψt0 = ψt . But both t0 ∈ (0, P ) and t ∈ [t0 , P ) were chosen arbitrarily, so the previous sentence shows that ψt = ψt0 for all t ∈ (0, P ). Letting ψ = ψt0 , we have ξt# = ψ(I + st ψ)−1 Ωνt
(10)
for all t ∈ (0, P ). Defining κt as in the proof of Proposition 3.2, we observe that ψ(I + st ψ)−1 = ξt# κt for all t ∈ (0, P ), where the right hand side is completely positive by hypothesis. Since every t ∈ (0, ∞) can be written as t = st for some t ∈ (0, P ), it follows that ψ(I + tψ)−1 is completely positive for all t > 0. Furthermore, ψ(I + st ψ)−1 → ψ in norm as t → ∞, hence ψ q 0. Similarly, since πt# − ξt# is completely positive for all t > 0 by assumption, it follows from our formula φ(I + st φ)−1 − ψ(I + st ψ)−1 = πt# − ξt# κt that φ(I + st φ)−1 − ψ(I + st ψ)−1 is completely positive for all t > 0, and so its norm limit (as t → ∞) φ − ψ is completely positive. Therefore, φ q ψ . Finally, since the CP-flow β is entirely determined by its generalized boundary representation ξ # , which itself is determined by any sequence {ξt#n } with tn tending to 0 (see the remarks preceding Theorem 4.29 of [8]), it follows from (10) that β is induced by the boundary weight double (ψ, ν). 2 In a manner analogous to that used by Powers in [9] and [8], we define the terms q-corner and hyper maximal q-corner: Definition 4.4. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be q-positive maps. A corner γ : Mn×k (C) → Mn×k (C) from φ to ψ is said to be a q-corner from φ to ψ if the map Υ
An×n Ck×n
Bn×k Dk×k
=
φ(An×n ) γ ∗ (Ck×n )
γ (Bn×k ) ψ(Dk×k )
is q-positive. A q-corner γ is called hyper maximal if, whenever
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C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
Υ q Υ =
φ γ∗
γ ψ
q 0,
we have Υ = Υ . Proposition 4.5. For any q-positive φ : Mn (C) → Mn (C) and unitary U ∈ Mn (C), define a map φU by φU (A) = U ∗ φ U AU ∗ U. 1. The map φU is q-positive, and there is an order isomorphism between q-positive maps β such that φ q β and q-positive maps βU such that φU βU . In particular, φ is q-pure if and only if φU is q-pure. 2. If φ is unital and q-pure, then there is a hyper maximal q-corner from φ to φU . Proof. To prove the first assertion, we define a completely positive map ζ on Mn (C) by ζ (A) = U ∗ AU , noting that ζ −1 is also completely positive. For every t 0 and A ∈ Mn (C), we find that (I + tφU )−1 (A) = U ∗ (I + tφ)−1 (U AU ∗ )U and φU (I + tφU )−1 (A) = U ∗ φ U U ∗ (I + tφ)−1 U AU ∗ U U ∗ U = U ∗ φ(I + tφ)−1 U AU ∗ U = ζ ◦ φ(I + tφ)−1 ◦ ζ −1 (A),
(11)
so φU q 0. Given any q-positive map β such that φ q β, define βU by βU (A) = U ∗ β(U AU ∗ )U . Then βU is q-positive by (11), and for each t 0 we have φU (I + tφU )−1 − βU (I + tβU )−1 = ζ ◦ φ(I + tφ)−1 − β(I + tβ)−1 ◦ ζ −1 , hence φU q βU . Of course, since φ = (φU )U ∗ , the argument just used gives an identical correspondence between q-subordinates α of φU and q-subordinates αU ∗ of φ. Our first assertion now follows. To prove the second statement, we define γ : Mn (C) → Mn (C) by γ (A) = φ(AU ∗ )U . By Lemma 3.5, γ is a corner from φ to φU , so the map Θ
A11 A21
A12 A22
=
φ(A11 ) γ ∗ (A21 )
γ (A12 ) φU (A22 )
is completely positive. We calculate γ (I + tγ )−1 (A) = φ(I + tφ)−1 (AU ∗ )U , so for each t 0 and A˜ = (Aij ) ∈ M2n (C), we have ˜ = Θ(I + tΘ)−1 (A)
φ(I + tφ)−1 (A11 ) U ∗ φ(I + tφ)−1 (U A21 )
φ(I + tφ)−1 (A12 U ∗ )U φU (I + tφU )−1 (A22 )
.
This shows that γ (I + tγ )−1 is a corner from φ(I + tφ)−1 to φU (I + tφU )−1 for all t 0, so γ is a q-corner. Finally, if
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
Θ
A11 A21
A12 A22
=
α(A11 ) γ ∗ (A21 )
γ (A12 ) β(A22 )
3431
is q-positive and Θ q Θ , then since φ and φU are q-pure we have α = φ(I + tφ)−1 for some t 0 and β = φU (I + sφU )−1 for some s 0. Complete positivity of Θ implies that Θ
I U∗
U I
=
1 1+t I U∗
U 1 1+s I
so s = t = 0 and Θ = Θ , hence γ is hyper maximal.
0,
2
We have arrived at the key result of the section, which tells us that, under certain conditions, the problem of determining whether two E0 -semigroups induced by boundary weight doubles are cocycle conjugate can be reduced to the much simpler problem of finding hyper maximal q-corners between q-positive maps: Proposition√4.6. Let ν be√a normalized unbounded boundary weight over L2 (0, ∞) which has the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Let φ and ψ be unital q-positive maps on Mn (C) and Mk (C), respectively, and induce CP-flows α and β through the boundary weight doubles (φ, ν) and (ψ, ν). Then α d and β d are cocycle conjugate if and only if there is a hyper maximal q-corner from φ to ψ . Proof. Let N = n + k. For the forward direction, suppose α d and β d are cocycle conjugate. Since α d and β d are of type II0 , we know from Theorem 2.6 that there is a hyper maximal flow corner σ from α to β, with associated CP-flow Θ=
α σ∗
σ β
.
Let Π # = {Πt# }, π # = {πt# }, and ξ # = {ξt# } be the generalized boundary representations for Θ, α, and β, respectively. Define st = νt (Λ(1)) for all positive t, so for each t > 0 there is some Zt such that Πt# =
πt# Z∗t
Zt ξt#
=
φ(I + st φ)−1 ◦ Ωνt ,n×n Z∗t
Zt ψ(I + st ψ)−1 ◦ Ωνt ,k×k
.
Since each Zt is a corner from φ(I + st φ)−1 ◦ Ωνt ,n×n to ψ(I + st φ)−1 ◦ Ωνt ,k×k , we have Zt = Lt ◦ Ωνt ,n×k for some Lt . Define Bt for each t > 0 by Bt =
φ(I + st φ)−1 L∗t
Lt ψ(I + st ψ)−1
.
We observe that Πt# = Bt ◦ Ωνt ,N ×N for all t > 0, whereby the same argument given in the proof of Lemma 4.3 shows that each Bt has the form Bt = Wt (I + st Wt )−1 for some
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Wt : Mn (C) → Mn (C) and that the maps Wt are independent of t. Therefore, for some γ : Mn×k (C) → Mn×k (C), we have Zt = γ (I + st γ )−1 ◦ Ωνt ,n×k for all t > 0. Define κt,N ×N : MN (C) → B(H ) as in Proposition 3.2. Letting ϑ=
φ γ∗
γ ψ
,
we observe for each t that ϑ(I + st ϑ)−1 = Πt# ◦ κt,N ×N is the composition of completely positive maps and is thus completely positive, hence ϑ q 0. Suppose that for some map ϑ we have
ϑ q ϑ =
φ γ∗
γ ψ
q 0.
As in Proposition 3.2, the boundary weight map ρ ∈ MN (C)∗ → L(ρ) defined by L(ρ)(C) = ρ(ϑ (Ων,N×N (C))) induces a CP-flow Θ over CN , where for some CP-flows α over Cn and β over Ck , we have
Θ =
α σ∗
σ β
.
By Lemma 4.3, we have Θ Θ since ϑ q ϑ . But Θ is a hyper maximal flow corner, so Θ = Θ . Our formulas for the generalized boundary representations imply that φ(I + tφ)−1 = φ (I + tφ )−1 and ψ(I + tψ)−1 = ψ (I + tψ )−1 for all t > 0, hence φ = φ and ψ = ψ . We conclude that γ is a hyper maximal q-corner. For the backward direction, suppose there is a hyper maximal q-corner γ from φ to ψ , so the map Υ : MN (C) → MN (C) defined by Υ
An×n Ck×n
Bn×k Dk×k
=
φ(An×n ) γ ∗ (Ck×n )
γ (Bn×k ) ψ(Dk×k )
is q-positive. By Proposition 3.2, the boundary weight map ρ ∈ MN (C)∗ → Ξ (ρ) defined by Ξ (ρ)(A) = ρ Υ Ων,N×N (A) is the boundary weight map of a CP-flow θ over CN , where for some Σ we have θ=
α Σ∗
Σ β
.
Let θ =
α Σ∗
Σ β
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be any CP-flow such that θ θ . Letting Zt = γ (I + st γ )−1 ◦ Ωνt ,n×k for all t > 0, we see the generalized boundary representations Π # = {Πt# } and Π = {Πt } for θ and θ satisfy Πt# =
πt# Zt∗
Zt ξt#
Πt =
πt Zt∗
Zt ξt
for all t > 0. Lemma 4.3 implies that for some φ and ψ with φ q φ q 0 and ψ q ψ q 0 we have πt = φ (I + st φ )−1 ◦ Ωνt ,n×n and ξt = ψ (I + st ψ )−1 ◦ Ωνt ,k×k for all t > 0. Defining Υ : MN (C) → MN (C) by Υ
An×n Ck×n
Bn×k Dk×k
=
φ (An×n ) γ ∗ (Ck×n )
γ (Bn×k ) , ψ (Dk×k )
we observe that Πt ◦ κνt ,N ×N = Υ (I + st Υ )−1 for all st > 0, hence γ is a q-corner from φ to ψ . Hyper maximality of γ implies φ = φ and ψ = ψ , thus θ = θ . Therefore, σ is a hyper maximal flow corner from α to β, so α d and β d are cocycle conjugate by Theorem 2.6. 2 5. E0 -semigroups obtained from rank one unital q-pure maps Any unital linear map φ : Mn (C) → Mn (C) of rank one is of the form φ(A) = τ (A)I for some linear functional τ . If φ is positive, then τ is positive and τ (I ) = 1, so τ is a state. On the other hand, given any state ρ, the map φ defined by φ(A) = ρ(A)I is unital and completely positive. Furthermore, φ is q-positive since φ(I + tφ)−1 = (1/(1 + t))φ for all t > 0. The rank one unital q-positive maps are therefore precisely the maps A → ρ(A)I for states ρ. The goal of this section is to determine when such maps are q-pure, and then to determine when the E0 -semigroups induced by (φ, ν) and (ψ, ν) are cocycle conjugate, where φ and ψ are√rank one unital √ q-pure maps and ν is a normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) (Theorem 5.4). We also obtain a partial result for comparing E0 -semigroups induced by (φ, ν) and (ψ, μ) for rank one unital q-pure maps φ and ψ and any normalized unbounded boundary weights ν and μ over L2 (0, ∞) (Corollary 5.5). We begin with a lemma: Lemma 5.1. Let ρ be a faithful state on Mn (C), and define a unital q-positive map φ : Mn (C) → Mn (C) by φ(A) = ρ(A)I . For any non-zero positive linear functional τ on Mn (C) and non-zero positive operator C ∈ Mn (C), define ψτ,C : Mn (C) → Mn (C) by ψτ,C (A) = τ (A)C. Then ψτ,C is q-positive, and φ q ψτ,C if and only if ψτ,C = λφ for some λ ∈ (0, 1]. Proof. Note that for all A ∈ Mn (C) and t 0, we have (I + tψτ,C )−1 (A) = A − tτ (A)/(1 + tτ (C))C, so ψτ,C (I + tψτ,C )−1 (A) =
τ (A) C, 1 + tτ (C)
(12)
hence ψτ,C is q-positive. It follows from (12) that φ(I + tφ)−1 (A) = (ρ(A)/(1 + t))I for all A ∈ Mn (C). Assume the hypotheses of the forward direction. Since φ q ψτ,C , we have
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τ (A)C ρ(A)I 1+t 1 + tτ (C)
(13)
for all t 0 and A 0. This is impossible if τ (C) = 0, so we may assume τ (C) = 0. Letting t → ∞ in (13) yields ρ(A)I
τ (A)C τ (C)
(14)
for all A 0. Setting A = C in (14), we see ρ(C)I − C 0, yet ρ ρ(C)I − C = ρ(C) − ρ(C) = 0, hence C = ρ(C)I by faithfulness of ρ. Rewriting (14) as ρ(A)I
τ (A) τ (A) ρ(C)I = I τ (ρ(C)I ) τ
for all A 0, we see that ρ − τ/τ is a positive linear functional. Therefore, ρ − τ = ρ(I ) − τ (I ) = 1 − 1 = 0, τ τ hence τ = τ ρ. Setting t = 0 and A = I in (13) gives us τ = τ (I ) = λ/ρ(C) for some λ ∈ (0, 1]. Therefore, ψτ,C (A) = τ (A)C = τ ρ(A)ρ(C)I = λρ(A)I = λφ(A) for all A ∈ Mn (C), proving the forward direction. The backward direction follows from Proposition 4.1 since λφ = φ (−1+1/λ) for every λ ∈ (0, 1]. 2 Remark. Let ψ : Mn (C) → Mn (C) be a non-zero q-positive contraction such that the maps Lψt := tψ(I + tψ)−1 satisfy Lψt < 1 for all t > 0. By compactness of the unit ball of B(Mn (C)), the maps Lψt have some norm limit as t → ∞. This limit is unique: Pick any orthonormal basis with respect to the trace inner product (A, B) = tr(A∗ B) of Mn (C), and let Mt be the n2 × n2 matrix of Lψt with respect to this basis. From the cofactor formula for (I + tψ)−1 , we know that the ij th entry of Mt is a rational function rij (t). Uniqueness of limt→∞ Lψt now follows from the fact that each rij (t) has a unique limit as t → ∞. We call this limit Lψ . Noting that tψ = Lψt (I − Lψt )−1 = Lψt + L2ψt + · · · for each t > 0, we claim that Lψ fixes a positive element T of norm one. To prove this, we first observe for each k ∈ N and t > 0 that
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∞ (Lψ )kn (I ) tψ = t ψ(I ) Lψt (I ) + · · · + (Lψt )k−1 (I ) + k t n=1
< (k − 1) + k
∞
(Lψ )k (I )n , t
n=1
hence 1 = lim (Lψt )k (I ) = (Lψ )k (I ). t→∞
Therefore, all elements of the sequence {Tk }k∈N defined by Tk = (Lψ )k (I ) satisfy Tk 0 and Tk = 1. Since Tk − Tk+1 = (Lψ )k (I − T1 ) 0 for all k, the sequence {Tk }k∈N is monotonically decreasing and therefore has a positive norm limit T with T = 1. Finally, Lψ fixes T since Lψ (T ) = limk→∞ Lk+1 ψ (I ) = T . The information at hand suffices in showing that a large class of maps is q-pure: Proposition 5.2. Let ρ be a state on Mn (C), and define a q-positive map φ on Mn (C) by φ(A) = ρ(A)I . Then φ is q-pure if and only if ρ is faithful. Proof. For the forward direction, we prove the contrapositive. If ρ is not faithful, then for some k < n and mutually orthogonal vectors f1 , . . . , fk with ki=1 fi 2 = 1, we have ρ(A) = k i=1 (fi , Afi ) for all A ∈ Mn (C). Let P be the projection onto the k-dimensional subspace of Cn spanned by the vectors f1 , . . . , fk , and define a q-positive map ψ : Mn (C) → Mn (C) by ψ(A) = ρ(A)P . For each t 0 and A ∈ Mn (C), we find (t) φ − ψ (t) (A) =
1 1 φ(A) − ψ(A) = ρ(A)(I − P ), 1+t 1+t
so φ q ψ . Obviously, ψ = φ (s) for any s 0, so φ is not q-pure. To prove the backward direction, suppose φ q ψ q 0 for some ψ = 0, and form Lψ and Lφ . Since Lφt = (t/(1 + t))φ for each t > 0, we have Lφ = φ. The map Lφt − Lψt is completely positive for all t, so by taking its limit as t → ∞ we see φ − Lψ is completely positive. By the remarks preceding this proposition, we know that Lψ fixes a positive T with T = 1. But (φ − Lψ )(T ) = ρ(T )I − T 0, so ρ(T ) = 1, hence T = I by faithfulness of ρ. By complete positivity of φ − Lψ , we have φ − Lψ = φ(I ) − Lψ (I ) = 0, so φ = Lψ . Therefore, I I I +ψ = lim φ + ψ − Lψt +ψ 0 = lim (φ − Lψt ) t→∞ t→∞ t t t φ I φ = lim + φψ − tψ(I + tψ)−1 +ψ = lim + φψ − ψ t→∞ t t→∞ t t = φψ − ψ.
(15)
Letting τ be the positive linear functional τ = ρ ◦ ψ , we conclude from (15) that ψ(A) = ρ(ψ(A))I = τ (A)I for all A ∈ Mn (C). Lemma 5.1 implies that ψ = λφ = φ (−1+1/λ) for some λ ∈ (0, 1]. 2
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To prove the main result of the section, we need the following: Lemma 5.3. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be rank one unital q-pure maps, and let ν and μ be normalized unbounded boundary weights over L2 (0, ∞). If the boundary weight doubles (φ, ν) and (ψ, μ) induce cocycle conjugate E0 -semigroups α d and β d , then there is a corner γ from φ to ψ such that γ = 1. Proof. By construction, α d and β d are type II0 E0 -semigroups. If they are cocycle conjugate, then by Theorem 2.6, there is a hyper maximal flow corner σ from α to β with associated CPflow α σ . Θ= σ∗ β Let H1 = Cn ⊗ L2 (0, ∞), let H2 = Ck ⊗ L2 (0, ∞), and let H = Cn+k ⊗ L2 (0, ∞). Write the boundary representation Π = {Πt# } for Θ as Πt#
=
1 1+νt (Λ(1)) φ ◦ Ωνt ,n×n Z∗t
Zt 1 1+μt (Λ(1)) ψ
◦ Ωμt ,k×k
for some maps {Zt }t>0 . Let ρ11 → ω(ρ11 ) and ρ22 → η(ρ22 ) denote the boundary weight maps for α and β, respectively. Let ρ → Ξ (ρ) be the boundary weight map for Θ, so for some map ρ12 → (ρ12 ) from Mn×k (C)∗ to weights on B(H2 , H1 ) we have Ξ
ρ11 ρ21
ρ12 ρ22
=
ω(ρ11 ) ∗ (ρ21 )
(ρ12 ) . η(ρ22 )
Denote by Ut the right shift t units on H . For every A = (Aij ) ∈ t>0 Ut B(H )Ut∗ and bounded family of functionals {ρ(t) = (ρij (t))}t>0 in Mn+k (C)∗ , we observe that the argument used in Corollary 3.3 to show that π0# = ξ0# = 0 implies ˆ t )−1 ρ11 (t) (A11 ) = lim ηt (I + Λη ˆ t )−1 ρ22 (t) (A22 ) = 0, lim ωt (I + Λω
t→0
t→0
so by complete positivity of the generalized boundary representation, we have ˆ t )−1 ρ12 (t) (A12 ) = 0. lim t (I + Λ
t→0
(16)
We claim that ρ12 → (ρ12 ) is unbounded. If is bounded, then for each ρ12 ∈ Mn×k (C)∗ , ˆ t )(ρ12 ) is bounded, and it follows from (16) that the family ρ12 (t) := (I + Λ lim t (ρ12 )(A12 ) = 0
t→0
(17)
for each A12 ∈ t>0 Wt B(H2 , H1 )Xt∗ , where Wt and Xt are the right shift t units on H1 and H2 , respectively. Let A12 ∈ t>0 Wt B(H2 , H1 )Xt∗ , so A12 = Ws BXs∗ for some s > 0 and B ∈ B(H2 , H1 ). For all b < s, we have
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b (ρ12 )(A12 ) = b (ρ12 ) Ws BXs∗ = (ρ12 ) Wb Wb∗ Ws BXs∗ Xb Xb∗ ∗ = (ρ12 ) Wb Ws−b BXs−b Xb∗ = (ρ12 ) Ws BXs∗ = (ρ12 )(A12 ). Therefore, by Eq. (17) we have (ρ12 )(A12 ) = 0. Let A ∈ B(H2 , H1 ), ρ12 ∈ Mn×k (C)∗ , and t > 0 be arbitrary. From above we have t (ρ12 )(A) = (ρ12 ) Wt AXt∗ = 0, hence t ≡ 0 for all t > 0. We conclude from uniqueness of the generalized boundary representation that ρ12 → (ρ12 ) is the zero map. The boundary weight map ρ → Ξ (ρ) defined by Ξ
ρ11 ρ21
ρ12 ρ22
=
0 0
ω(ρ11 ) 0
gives rise to the CP-flow Θ =
α σ∗
σ β
,
where β is the non-unital CP-flow βt (A22 ) = Xt A22 Xt∗ . Trivially, Θ = Θ and Θ Θ , contradicting hyper maximality of σ . Therefore, the map ρ12 → (ρ12 ) is unbounded. Since Πt# is a contraction for every t > 0, so is Zt , hence the map Zt ◦ Λ : Mn×k (C) → Mn×k (C) is a contraction for each t > 0. A compactness argument shows that Ztn ◦ Λ has a norm limit γ for some sequence {tn } tending to zero, where γ 1. From unboundedness of and the formula t = Zˆ t (I − Λˆ Zˆ t )−1 for all t > 0, it follows that I − γ is not invertible, so γ 1, hence γ = 1. We claim that γ is a corner from φ to ψ. Indeed, for the family of completely positive maps {Rt }t>0 defined by Rt = Πt# ◦ Λ, we have
lim Rtn = lim
n→∞
n→∞
νtn (Λ(1)) 1+νtn (Λ(1)) φ
Ztn ◦ Λ
(Ztn ◦ Λ)∗
μtn (Λ(1)) 1+μtn (Λ(1)) ψ
=
φ γ∗
γ ψ
.
2
If ν is a √normalized unbounded boundary weight over L2 (0, ∞) of the form √ ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) and if φ : Mn (C) → Mn (C) is unital and q-pure, we know from Propositions 4.5 and 4.6 that the condition ψ = φU is sufficient for the boundary weight doubles (φ, ν) and (ψ, ν) to induce cocycle conjugate E0 -semigroups. In the case that φ is a rank one unital q-pure map, this condition is also necessary: Theorem 5.4. Let φ1 : Mn (C) → Mn (C) and φ2 : Mk (C) → Mk (C) be rank one unital 2 q-pure √ maps. Let √ ν be a normalized unbounded boundary weight over L (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Then the boundary weight doubles (φ1 , ν) and (φ2 , ν) induce cocycle conjugate E0 semigroups if and only if n = k and φ2 = (φ1 )U for some unitary U ∈ Mn (C).
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Proof. The backward direction follows immediately from Propositions 4.5 and 4.6. Assume the hypotheses of the forward direction. Since φ1 and φ2 are rank one, unital, and q-pure, there exist faithful states ρ1 on Mn (C) and ρ2 on Mk (C) such that φ1 (M) = ρ1 (M)In×n and φ2 (B) = ρ2 (B)Ik×k for all M ∈ Mn (C), B ∈ Mk (C). By Lemma 5.3, there is a corner γ from φ1 to φ2 such that γ = 1. Therefore, for some A0 ∈ Mn×k (C) of norm one and unit vectors f0 ∈ Cn and g0 ∈ Ck , we have |(f0 , γ (A0 )g0 )| = 1. Define ω ∈ Mn×k (C)∗ by ω(A) = (f0 , γ (A)g0 ), noting that ω = |ω(A0 )| = 1. We claim that the map ψ˜ : Mn+k (C) → M2 (C) defined by ψ˜
A11 A21
A12 A22
=
ρ1 (A11 ) ω∗ (A21 )
ω(A12 ) ρ2 (A22 )
is completely positive. To see this, let {F˜i } i=1 be arbitrary vectors in C2 , writing each F˜i as F˜i =
λ1i λ2i
for some complex numbers {λ1i } i=1 and {λ2i } i=1 . Since the map ψ : Mn+k (C) → Mn+k (C) defined by ψ
A11 A21
A12 A22
=
ρ1 (A11 )I γ ∗ (A21 )
γ (A12 ) ρ2 (A22 )I
is completely positive by assumption, we know that for any A1 , . . . , A ∈ Mn+k (C) and the vectors λ1i f0 ∈ Cn+k , i = 1, . . . , k, Fi = λ2i g0 we have Fi , ψ A∗i Aj Fj 0. i,j =1
However, for each i and j we find that Fi , ψ A∗i Aj Fj Cn+k = λ1i λ1j ρ1 A∗i Aj 11 + λ1i λ2j ω A∗i Aj 12 ∗ + λ2i λj 1 ω A∗i Aj 21 + λ2i λ2j ρ2 A∗i Aj 22 = F˜i , ψ˜ A∗i Aj F˜j C2 . Therefore, for all ∈ N, A1 , . . . , A ∈ Mn+k (C), and F˜1 , . . . , F˜ ∈ C2 , we have ˜ ˜ ∗ ˜ ˜ i,j =1 (Fi , ψ(Ai Aj )Fj ) 0, so ψ : M2n (C) → M2 (C) is completely positive. Since ρ1 and ρ2 are positive linear functionals (hence completely positive maps), ω is a corner from ρ1 to ρ2 . monotonically By faithfulness of ρ1 and ρ2 , there exist increasing sequences of strictly positive numbers {λi }ni=1 and {μj }kj =1 with ni=1 λ2i = kj =1 μ2j = 1, along with orthonor mal sets of vectors {fi }ni=1 and {gj }kj =1 , such that ρ1 (M) = ni=1 λ2i (fi , Mfi ) and ρ2 (B) =
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
3439
k
for all M ∈ Mn (C), B ∈ Mk (C). Given A ∈ Mn×k (C), let A˜ be the ma˜ = A. Let Dλ and Dμ be the diagonal trix whose j ith entry is (fi , Agj ), observing that A matrices whose iith entries are λi and μi , respectively, for all i, and let Dλ2 and Dμ2 be the diagonal matrices whose iith entries are λ2i and μ2i , respectively, observing that Dλ2 = (Dλ )2 and Dμ2 = (Dμ )2 . By Proposition 3.5, ω has the form 2 j =1 μj (gj , Bgj )
ω(A) =
˜ λ ) = tr CDμ Dλ A˜ ∗ ∗ cij λi μj (fi , Agj ) = tr(CDμ AD
i,j
for some C = (cij ) ∈ Mn×k (C) such that C 1. By the Cauchy–Schwartz inequality for the inner product (B, A) = tr(AB ∗ ) on Mn×k (C), we have 2 ∗ 2 2 1 = ω(A0 ) = tr CDμ Dλ A˜ ∗0 = CDμ , Dλ A˜ ∗0 (CDμ , CDμ ) Dλ A˜ ∗0 , Dλ A˜ ∗0 = tr Dμ C ∗ CDμ tr Dλ A˜ ∗0 A˜ 0 Dλ tr(Dμ2 Ik )tr(Dλ2 In ) 1 ∗ 1 = 1.
(18)
Since equality holds in all the inequalities above, we have mCDμ = Dλ A˜ ∗0 for some m ∈ C. It follows from (18) that |m| = 1 since CDμ tr = Dλ A˜ ∗0 tr = 1. Furthermore, since equality holds in (18) and the trace map is faithful, we have C ∗ C = Ik and A˜ ∗0 A˜ 0 = In . But C ∈ Mn×k (C) and A˜ ∗0 ∈ Mn×k (C), so n = k, hence C and A˜ 0 are unitary. Writing Dλ = mCDμ A˜ 0 = (mC A˜ 0 )(A˜ ∗0 Dμ A˜ 0 ), we observe that mC A˜ 0 is unitary and A˜ ∗0 Dμ A˜ 0 is positive. Uniqueness of the right Polar Decomposition for the invertible matrix Dλ implies Dλ = A˜ ∗0 Dμ A˜ 0 . Since the diagonal entries in Dλ and Dμ are listed in increasing order, it follows that Dλ = Dμ , hence ρ2 is of the form ρ2 (M) = ni=1 λ2i (gi , Mgi ). Defining a unitary U ∈ Mn (C) by letting Ugi = fi for all i and extending linearly, we observe that ρ2 (M) =
n i=1
n λ2i U ∗ fi , MU ∗ fi = λ2i fi , U MU ∗ fi = ρ1 U MU ∗ i=1
for all M ∈ Mn (C). In other words, φ2 = (φ1 )U .
2
In [9], Powers constructed E0 -semigroups using boundary weights over L2 (0, ∞). It is routine to check that in our notation, these are the E0 -semigroups arising from the boundary weight doubles (ıC , η), where ıC is the identity map on C and η is any boundary weight over L2 (0, ∞). Corollary 5.5. Let φ : Mn (C) → Mn (C) and ψ : Mk (C) → Mk (C) be unital rank one q-pure maps, and let ν and η be normalized unbounded boundary weights over L2 (0, ∞). Denote by α d
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and β d the Bhat minimal dilations of the CP-flows induced by the boundary weight doubles (φ, ν) and (ψ, μ), respectively. If n = k, then α d and β d are not cocycle conjugate. In particular, if n = 1, then α d is not cocycle conjugate to the E0 -semigroup induced by (ıC , μ). Proof. From the proof of Theorem 5.4, we know that every corner γ from φ to ψ satisfies γ < 1 since n = k. The result now follows from Lemma 5.3. 2 6. Invertible unital q-pure maps Now that we have classified the unital q-pure maps on Mn (C) of rank one, we explore the unital q-pure maps φ which are invertible. In a stark contrast to the√rank one case, √ we find that for a given normalized unbounded boundary weight of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) on L2 (0, ∞), the doubles (φ, ν) and (ψ, ν) always induce cocycle conjugate E0 semigroups if φ and ψ are unital invertible q-pure maps on Mn (C) and Mk (C), respectively. The following proposition gives us a bijective correspondence between invertible unital q-positive maps φ : Mn (C) → Mn (C) and unital conditionally negative maps Ψ : Mn (C) → Mn (C): Proposition 6.1. If φ : Mn (C) → Mn (C) is an invertible unital q-positive map, then φ −1 is conditionally negative. On the other hand, if Ψ : Mn (C) → Mn (C) is a unital conditionally negative map, then Ψ is invertible and Ψ −1 is q-positive. Proof. Let ψ = φ −1 . Since φ is self-adjoint, so is ψ , and the first statement of the proposition now follows from the fact that for large positive t we have −1
tφ(I + tφ)
= tψ
−1
ψ −1 ψ ψ2 −1 −1 −1 I + tψ = t (ψ + tI ) = I + =I − + − ···. t t t
To prove the second statement, let Ψ : Mn (C) → Mn (C) be any unital conditionally negative map. Since Ψ is conditionally negative, it follows from a result of Evans and Lewis in [5] that −sΨ = e−sΨ (I ) = e−s I = e−s for e−sΨ is completely positive ∞for all s 0. Therefore, e all s 0, and the integral 0 e−sΨ ds converges. Observing that (d/ds)(−e−sΨ ) = Ψ e−sΨ , we find that
∞ Ψ
e
−sΨ
0
∞
ds =
s Ψ e−sΨ ds = lim −e−sΨ 0 = I, s→∞
0
∞ so Ψ is invertible and Ψ −1 = 0 e−sψ ds. Since Ψ −1 is the integral of completely positive maps, it is completely positive. Furthermore, we find that tI + Ψ is invertible for every t > 0 and that Ψ −1 q 0, since the following holds for all t > 0: ∞ e 0
−st −sΨ
e
∞ ds = 0
−1 e−s(tI +Ψ ) ds = (tI + Ψ )−1 = Ψ −1 I + tΨ −1 .
2
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Examining the inverse of a unital invertible q-positive map φ is the key to finding the invertible q-subordinates of φ, as we find in the following proposition and corollary: Proposition 6.2. Let φ1 : Mn (C) → Mn (C) be an invertible unital q-positive map, and let ψ1 = φ1−1 . Suppose ψ2 : Mn (C) → Mn (C) is conditionally negative and ψ2 − ψ1 is completely positive. Then ψ2 is invertible, and φ2 := (ψ2 )−1 satisfies φ1 q φ2 q 0. Proof. Assume the hypotheses of the proposition, and let s > 0 be arbitrary. Define a function f 1 on R by f (t) = e−tsψ1 e(t−1)sψ2 . The equality below is f (1) − f (0) = 0 f (t) dt:
e
−sψ1
−e
−sψ2
1
se−tsψ1 (ψ2 − ψ1 )e(t−1)sψ2 dt.
= 0
The inside of the integral above is the composition of completely positive maps, so e−sψ1 − e−sψ2 is completely positive. This implies e−sψ1 (I ) − e−sψ2 (I ) 0, so −sψ −sψ e 2 = e 2 (I ) e−sψ1 (I ) = e−s (I ) = e−s . ∞ Now the argument given in the previous proposition shows that 0 e−sψ2 ds converges and is equal to ψ2−1 . Letting φ2 = ψ2−1 , we observe that φ1 q φ2 since the quantity below is completely positive for every t 0: −1
φ1 (I + tφ1 )
−1
− φ2 (I + tφ2 )
∞ =
e−st e−sψ1 − e−sψ2 ds.
2
0
Corollary 6.3. Let φ1 : Mn (C) → Mn (C) be an invertible unital q-positive map, and let φ2 : Mn (C) → Mn (C) be linear and invertible. Then φ1 q φ2 q 0 if and only if φ2−1 is conditionally negative and φ2−1 − φ1−1 is completely positive. Proof. The backward direction follows from Proposition 6.2. Assume the hypotheses of the forward direction and let ψ1 = φ1−1 and ψ2 = φ2−1 . Since φ2 is self-adjoint, so is ψ2 . For sufficiently large positive t we have ψ2 −1 ψ2 ψ22 + 2 − ··· tφ2 (I + tφ2 )−1 = I + =I − t t t and 2 ψ2 − ψ12 ψ23 − ψ13 −1 −1 t φ1 (I + tφ1 ) − φ2 (I + tφ2 ) = ψ2 − ψ1 + + ··· . − t t2 2
The first equation shows that φ2−1 is conditionally negative, while the second shows that φ2−1 − φ1−1 is completely positive. 2
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Now that we know how to find all invertible q-subordinates of an invertible unital q-positive map φ, we ask if there can be any other q-subordinates of φ. We will find that the answer is no (see Proposition 6.9). Proving this will require the use of some machinery (notably Lemma 6.8), which we now build. Definition 6.4. For every φ : Mn (C) → Mn (C) and I + (1 − )φ.
∈ [0, 1], we define a map φ by φ =
If φ is q-positive, then φ is invertible for all ∈ (0, 1]. In the lemmas that follow, we make frequent use of the fact that for all t 0 we have tφ(I + tφ)−1 = I − (I + tφ)−1 .
(19)
We present a quick consequence of (19) for all a 0 and b 0: a(I + btφ)−1 = aI − abtφ(I + btφ)−1 .
(20)
Lemma 6.5. Let φ : Mn (C) → Mn (C) be completely positive. If φ k q 0 for some monotonically decreasing sequence { k } of positive real numbers tending to 0, then φ q 0. Proof. Assume the hypotheses of the lemma. Let k be arbitrary. Since φ k q 0, we know I − (I + tφ k )−1 is completely positive for all t 0. Noting that −1 I − (I + tφ )−1 = I − (1 + t I ) + (1 − )tφ =I − and substituting t = t (1 −
k )/(1 + t k ),
1 1+t
t (1 − ) −1 I+ φ 1+t
we see
I − (I + tφ )−1 = I −
1+
1 k
1− k +t
−1 I + t φ . t k
Varying t throughout [0, ∞), we find that the above equation is completely positive for all t ∈ [0, −1 + 1/ k ). Of course, for any t ∈ [0, −1 + 1/ k ), we have t ∈ [0, −1 + 1/ ) for all k by monotonicity of the sequence { n }. Therefore, we may repeat the same argument to conclude that for any t ∈ [0, −1 + 1/ k ), the map I−
1+
1
1− +t
−1 I + t φ t
is completely positive for all k. Now fix any t > 0, so t ∈ (0, −1 + 1/ k ) for some k ∈ N. A straightforward computation shows that the sequence {cn } defined by cn = n /(1 − n + t n ) monotonically decreases to 0. From the previous paragraph, we know that the map I−
−1 1 I + t φ 1 + c t
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is completely positive for all k. Since cn ↓ 0 it follows that −1 I − I + t φ is completely positive. In other words, t φ(I + t φ)−1 is completely positive. Since t > 0 was chosen arbitrarily and φ is completely positive, the lemma follows. 2 Lemma 6.6. If φ : Mn (C) → Mn (C) and φ q 0, then φ q 0 for all ∈ [0, 1). Proof. Suppose that φ q 0, and let ∈ [0, 1) be arbitrary. For each t > 0, we apply formula (20) to a = 1/(1 + t ) and b = t (1 − )/(1 + t ) to find t (1 − ) −1 I+ φ 1+t t (1 − ) 1 t (1 − ) −1 I+ = 1− φ I + , φ 1+t 1+t (1 + t )2
I − (I + tφ )−1 = I −
1 1+t
where both terms on the last line are completely positive by assumption. Furthermore, φ is completely positive, hence φ q 0. 2 Corollary 6.7. Let φ : Mn (C) → Mn (C) be a completely positive map. Then φ q 0 if and only if φ q 0 for all ∈ (0, 1). Lemma 6.8. Let φ : Mn (C) → Mn (C) and ψ : Mn (C) → Mn (C) be q-positive maps. Then φ q ψ if and only if φ q ψ for all ∈ (0, 1). Proof. For any ∈ (0, 1) we have φ − ψ = (φ − ψ), so φ − ψ is completely positive if and only if φ − ψ is completely positive for all ∈ (0, 1). For all t > 0 we have −1 −1 −1 −1 = I + t ψ − ψ I + t ψ − I + t φ , t φ I + t φ
(21)
and for all t > 0 we have t φ (I + tφ )−1 − ψ (I + tψ )−1 = I − (I + tφ )−1 − I − (I + tψ )−1 −1 t (1 − ) 1 t (1 − ) −1 . I+ ψ φ = − I+ 1+t 1+t 1+t
(22)
Assume the hypotheses of the forward direction. Showing that φ q ψ for all ∈ (0, 1) is equivalent to proving that (22) is completely positive for every t ∈ (0, ∞) and ∈ (0, 1). But this follows from complete positivity of (21) since t (1 − )/(1 + t ) ∈ (0, ∞) for every ∈ (0, 1) and t ∈ (0, ∞). Now assume the hypotheses of the backward direction. Any t ∈ (0, ∞) can be written as t (1 − )/(1 + t ) for some ∈ (0, 1) and t ∈ (0, ∞), so complete positivity of (22) for all such and t implies that (21) is completely positive for all t > 0, hence φ q ψ . 2 We are now in a position to prove what is perhaps the most striking result of the section:
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Proposition 6.9. Let ξ : Mn (C) → Mn (C) be an invertible unital q-positive map. If φ : Mn (C) → Mn (C) is q-positive and ξ q φ, then φ is either invertible or identically zero. Proof. For every ∈ (0, 1), form ξ and φ as in Definition 6.4, and let ψ := (φ )−1 . By Lemma 6.8 we have ξ q φ for each , so ψ is conditionally negative and ψ − (ξ )−1 is completely positive by Corollary 6.3. We first examine the case when the norms ψ remain bounded as → 0. More precisely, suppose that for all sufficiently small we have ψ < r for some r > 0. By compactness of the closed unit ball of radius r in B(Mn (C)), there is a decreasing sequence { k }k∈N converging to 0 such that {ψ k }k∈N has a (bounded) norm limit ψ as k → ∞. Noting that I − φψ = φ k ψ k − φψ = (φ k − φ)(ψ k − ψ) + φ(ψ k − ψ) + (φ k − φ)ψ and then applying the triangle inequality, we find that I − φψ = φ k ψ k − φψ φ k − φψ k − ψ + φψ k − ψ + φ k − φψ for all k ∈ N. But φ and ψ are bounded maps while ψ k → ψ in norm and φ k → φ in norm, so the above equation tends to 0 as k → ∞. We conclude that φψ = I . Similarly ψφ = I , hence φ is invertible and ψ = φ −1 . If the first case does not hold, then for some decreasing sequence { k } tending to zero, the norms {ψ k }k∈N form an unbounded sequence. For each k ∈ N, we write −1
(ξ k )
(A) = sk A + Yk A + AYk∗
−
mk
Ski ASk∗i
i=1
and ψ k (A) = tk A + Zk A + AZk∗ −
k
Tki ATk∗i ,
i=1
where mk , k n2 , sk ∈ R, tk ∈ R, tr(Yk ) = tr(Zk ) = 0, tr(Ski ) = 0 and tr(Sk∗i Skj ) is non-zero if and only if i = j (i, j mk ), and tr(Tki ) = 0 and tr(Tk∗i Tkj ) is non-zero if and only if i = j (i, j k ). Since ψ k − (ξ k )−1 is completely positive for all k ∈ N, we know that for each k, there pk pk , and maps {Xki }i=1 with tr(Xki ) = 0, such that for all exist pk n2 , complex numbers {xki }i=1 A ∈ Mn (C), k ψ k − (ξ k )−1 (A) = (Xki + xki I )A(Xki + xki I )∗
p
i=1
=
pk i=1
|xki |
2
A+
pk i=1
xki Xki A + A
pk i=1
∗ xki Xki
+
pk
Xki AXk∗i .
i=1
(23)
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Simultaneously, for all A ∈ Mn (C) we have ψ k − (ξ k )−1 (A) = (tk − sk )A + (Zk − Yk )A + A(Zk − Yk )∗
m k k ∗ ∗ + Ski ASki − Tki ATki . i=1
(24)
i=1
We claim that m p k k Xki AXk∗i Ski ASk∗i i=1
(25)
i=1
√ for all k ∈ N. To prove this, we let {vj }nj=1 be any orthonormal basis for Cn , let hj = vj / n for each i, let f ∈ Cn be arbitrary, and define maps Aj for j = 1, . . . , n by Aj = f h∗j . Using the trace conditions on the maps Yk , Zk , {Tki }, {Ski }, and {Xki }, we find that n ψ k − (ξ k )−1 (Aj )hj = (tk − sk )f + (Zk − Yk )f j =1
=
pk
|xki |
f+
2
i=1
pk
xki Xki f.
i=1
Since f was arbitrary, it follows that
tk − sk −
pk
|xki |
2
I=
i=1
pk
xki Xki
− (Zk − Yk ).
i=1
Taking the trace of both sides yields
0 = tr
pk
xki Xki
− (Zk − Yk ) = tr
tk − sk −
i=1
so tk − sk =
pk
2 i=1 |xki |
|xki |2 I ,
i=1
pk
and Zk − Yk = pk
i=1 xki Xki .
Xki AXk∗i
=
i=1
Therefore, the map A →
pk
mk
Ski ASk∗i
Formulas (23) and (24) now imply that −
i=1
mk
∗ i=1 Ski ASki
−
k
Tki ATk∗i
.
i=1
k
∗ i=1 Tki ATki
is completely positive, and
m m p k k k k Xki Xk∗i = Ski Sk∗i − Tki Tk∗i Ski Sk∗i , i=1
establishing (25).
i=1
i=1
i=1
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We now show that there exists some M ∈ N such that Xki M
(26)
for all k ∈ N and i ∈ {1, . . . , pk }. To do this, we first note that since the sequence of invertible maps {ξ k }k∈N converges in norm to the invertible map ξ , the sequence {(ξ k )−1 }k∈N converges in norm to ξ −1 . Write ξ −1 in the form ξ −1 (A) = sA + Y A + AY ∗ −
m
Si ASi∗ ,
i=1
where m n2 , s ∈ R, tr(Y ) = 0, and for all i and j , tr(Si ) = 0 and tr(Si Sj∗ ) is non-zero if and only if i = j . Let f ∈ Cn be arbitrary, and define vectors {hj }nj=1 and maps {Aj }nj=1 ex actly as we did earlier in the proof. Then nj=1 (ξ k )−1 (Aj )hj = sk f + Yk f for all k ∈ N n and j =1 ξ −1 (Aj )hj = sf + Yf . Since (ξ k )−1 converges to ξ −1 as k → ∞, we see that (sk − s)f + (Yk − Y )f converges to 0 as k → ∞. But f was arbitrary, so lim (sk − s)I + Yk − Y = 0.
k→∞
The limit of the trace of the above equation must also be zero, so sk converges to s and consequently Yk converges to Y . This implies that not only are the sequences of complex numbers {Yk }∞ {Wk }∞ {sk }∞ k=1 and maps k=1 both bounded, but that the sequence of linear maps k=1 demk ∗ ∗ fined by Wk (A) = i=1 Ski ASki is bounded and converges to the map W (A) = m i=1 Si ASi . Choose M ∈ N so that M 2 n2 supk∈N {Wk }. For every k ∈ N and i ∈ {1, . . . , mk }, we have Ski 2 Wk M 2 /n2 . Combining this fact with (25), we find that for every k ∈ N and i ∈ {1, . . . , pk }, p m m k k k ∗ ∗ ∗ Xki Xki Ski Ski Ski 2 Xki = Xki Xki i=1 i=1 i=1 2 2 2 n max Ski : i = 1, . . . , mk M , 2
proving (26). Since ψ k → ∞ as k → ∞ while (ξ k )−1 → (ξ )−1 < ∞, there is a sequence of maps {A k } of norm one such that (ψ k − (ξ k )−1 )(A k ) → ∞ as k → ∞. However, we also have p
p k k −1 2 ψ − (ξ ) (A ) = |xki | A k + xki Xki A k k k k i=1 i=1
p ∗ pk k +A k xki Xki + Xki A k Xk∗i i=1
pk i=1
|xki |2 + 2M
i=1
pk i=1
|xki | + pk M 2 .
(27)
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We note that
pk i=1
2 |xki |
pk i=1
pk pk |xki |)2 ( i=1 |xki |)2 ( i=1 |xki | pk n2 2
(28)
pk for all k. For each k, let λk = i=1 |xki |, noting that λk → ∞ as k → ∞ since Eq. (27) tends to infinity as k → ∞. Let A ∈ Mn (C) be any matrix such that A = 1, and let C = supk∈N (ξ k )−1 < ∞. Using the reverse triangle inequality and (28), we find that for each k ∈ N, ψ (A) ψ − (ξ )−1 (A) − (ξ )−1 (A) k k k k
λ2k − 2Mλk − n2 M 2 − C. n2
(29)
Since limk→∞ λk = ∞, Eq. (29) tends to infinity as k → ∞. For all k large enough that (29) is positive, we have φ k =
1 1 , 2 2 inf{ψ k (A): A = 1} λk /n − 2Mλk − n2 M 2 − C
so limk→∞ φ k = 0. But the sequence {φ k }∞ k=1 converges to φ in norm, hence φ ≡ 0.
2
Proposition 6.10. An invertible unital linear map φ : Mn (C) → Mn (C) is q-pure if and only if φ −1 is of the form φ −1 (A) = A + Y A + AY ∗ for some Y = −Y ∗ ∈ Mn (C) such that tr(Y ) = 0. Proof. Let ψ = φ −1 . Assume the hypotheses of the forward direction. Write ψ(A) = sA + Y A + AY ∗ −
k
λi Xi AXi∗ ,
i=1
where s ∈ R, tr(Y ) = 0, and for each i and j we have λi 0, tr(Xi ) = 0, and tr(Xi∗ Xj ) = nδij . Defining ψ : Mn (C) → Mn (C) by ψ (A) = sA + Y A + AY ∗ , we conditionally negative, and ψ − ψ is completely positive since (ψ − ψ)(A) = knote that ψ is ∗ −1 j =1 λj Xj AXj for all A. By Lemma 6.2, it follows that ψ is invertible and that φ := (ψ ) satisfies φ q φ q 0. Since φ is q-pure, there is some t0 0 such that φ = φ (t0 ) , hence
−1 −1 −1 −1 −1 ψ = φ I + t0 ψ −1 = φ(I + t0 φ)−1 = ψ = (t0 I + ψ)−1 = t0 I + ψ.
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Therefore, for all A ∈ Mn (C) we have ψ (A) = ψ(A) +
k
λj Xj AXj∗ = ψ(A) + t0 A,
j =1
so the map L : A → λj Xj AXj∗ satisfies L = t0 I . We repeat a familiar argument: Let f ∈ Cn be √ arbitrary, choose an orthonormal basis {vk }nk=1 of Cn , define hk = vk / n for neach k, and form the maps {X } imply that {Ak }nk=1 by Ak = f h∗k . The trace conditions for j k=1 L(Ak )hk = 0. However, since L = t0 I , we must also have nk=1 L(Ak )hk = t0 f . From arbitrariness of f , we conclude t0 = 0. Therefore, ψ has the form ψ(A) = sA + Y A + AY ∗ . Since ψ(I ) = I = sI + Y + Y ∗ and tr(Y ) = 0, we have s = 1 and consequently Y = −Y ∗ . Now assume the hypotheses of the backward direction. Note that ψ is conditionally negative and unital, hence φ is q-positive by Proposition 6.1. Let Φ be any non-zero q-positive map such that φ q Φ, so by Corollary 6.3 and Proposition 6.9, Φ is invertible and Ψ := (Φ)−1 is a conditionally negative map such that Ψ − ψ is completely positive. Write Ψ in the form Ψ (A) = s A + ZA + AZ ∗ −
m
μi Ti ATi∗ ,
i=1
where s ∈ R and for all i and j , μi > 0, tr(Ti ) = 0, and tr(Ti∗ Tj ) = nδij . Writing C = Z − Y , we have m ∗ μi Ti ATi∗ . (Ψ − ψ)(A) = s − 1 A + CA + AC − i=1
By a familiar argument, complete positivity of Ψ − ψ and the trace conditions for the above maps imply that s 1, C = 0, and Ti = 0 for all i. Therefore Ψ = ψ + (s − 1)I , so Φ = Ψ −1 = φ (s −1) . We conclude that φ is q-pure. 2 Let the matrices {ej k }nj,k=1 denote the standard basis for Mn (C), writing each A = (aj k ) ∈ Mn (C) as A = j,k aj k ej k . The following theorem classifies all unital invertible q-pure maps on Mn (C): Theorem 6.11. An invertible unital linear map φ : Mn (C) → Mn (C) is q-pure if and only if for some unitary U ∈ Mn (C), the map φU is the Schur map ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k φU (aj k ej k ) = aj k ej k ⎪ aj k ⎩ e 1−i(λj −λk ) j k
if j < k, if j = k, if j > k
for all A = (aj k ) ∈ Mn (C) and j, k = 1, . . . , n, where λ1 , . . . , λn ∈ R and λ1 + · · · + λn = 0. Proof. Assume the hypotheses of the forward direction. By the previous proposition, ψ := φ −1 has the form ψ(A) = A + Y˜ A + AY˜ ∗ for some Y˜ ∈ Mn (C) with Y˜ = −Y˜ ∗ and tr(Y˜ ) = 0. Let B = −i Y˜ , so B = B ∗ . Defining Y := (1/2)I + Y˜ = (1/2)I + iB, we find ψ(A) = Y A + AY ∗
C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451
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for all A ∈ Mn (C). Since B is self-adjoint, there is some unitary U ∈ Mn (C) such that U ∗ BU is a diagonal matrix D. For each k ∈ {1, . . . , n} let λk ∈ R be the kk entry of D. Note that since tr(B) = 0 we have nk=1 λk = 0, and that U ∗ Y U is the diagonal matrix M whose kk entry is 1/2 + iλk . Defining a map ψU by ψU (A) = U ∗ ψ(U AU ∗ )U for all A ∈ Mn (C), we find that ψU (A) = U ∗ Y U AU ∗ + U AU ∗ Y ∗ U ∗ = U ∗ Y U A + A U ∗ Y U = MA + AM ∗ . A quick calculation shows that this is just the Schur map ⎧ ⎨ (1 + i(λj − λk ))aj k ej k ψU (aj k ej k ) = aj k ej k ⎩ (1 − i(λj − λk ))aj k ej k
if j < k, if j = k, if j > k,
and so (ψU )−1 has the form ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k (ψU )−1 (aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k
if j < k, if j = k, if j > k.
It is straightforward to verify that (ψU )−1 is the map φU (A) = U ∗ φ(U AU ∗ )U . Assume the hypotheses of the backward direction. Let T be the diagonal matrix whose kkth entry is λk for every k = 1, . . . , n. We observe that tr(T ) = 0 and T = T ∗ . Now let C = iT , and let T˜ = (1/2)I + C. We routinely verify that C = −C ∗ and tr(C) = 0, and that (φU )−1 satisfies (φU )−1 (A) = T˜ A + AT˜ ∗ = A + CA + AC ∗ for all A ∈ Mn (C). Proposition 6.10 implies that φU is q-pure, whereby φ is q-pure by Proposition 4.5. 2 As it turns out, boundary weight doubles (φ, ν) for invertible unital q-pure maps φ : 2 M√ n (C) → Mn (C) √ and normalized unbounded boundary weights ν over L (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ) give us nothing new in terms of E0 -semigroups: Theorem 6.12. Let φ : Mn (C) → Mn (C) be unital, invertible, and√q-pure, and √ let ν be a normalized unbounded boundary weight over L2 (0, ∞) of the form ν( I − Λ(1)B I − Λ(1) ) = (f, Bf ). Then (φ, ν) and (ıC , ν) induce cocycle conjugate E0 -semigroups. Proof. By Theorem 6.11 and Propositions 4.5 and 4.6, we may assume that φ is the Schur map ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k φ(aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k
if j < k, if j = k, if j > k
for some λ1 , . . . , λn ∈ R with nk=1 λk = 0. By Proposition 4.6, it suffices to find a hyper maximal q-corner from φ to ıC . For this, define γ : Mn×1 (C) → Mn×1 (C) by
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C. Jankowski / Journal of Functional Analysis 258 (2010) 3413–3451 1 ⎞ ⎛ 1+iλ b1 ⎞ b1 1 1 ⎟ ⎜ b2 ⎟ ⎜ ⎜ 1+iλ2 b2 ⎟ ⎟ = γ⎜ ⎜ ⎟. .. ⎝ ... ⎠ ⎝ ⎠ . bn 1 1+iλn bn
⎛
Now define Υ : Mn+1 (C) → Mn+1 (C) by Υ
An×n C1×n
Bn×1 a
=
φ(An×n ) γ ∗ (C1×n )
γ (Bn×1 ) . a
Letting λn+1 = 0, we observe that Υ is the Schur map satisfying ⎧ aj k ⎪ ⎨ 1+i(λj −λk ) ej k Υ (aj k ej k ) = aj k ej k ⎪ aj k ⎩ 1−i(λj −λk ) ej k
if j < k, if j = k, if j > k
n for all j, k = 1, . . . , n + 1 and A = (aj k ) ∈ Mn (C). Since n+1 i=1 λk = 0, it follows i=1 λk = from Theorem 6.11 that Υ is q-positive (in fact, q-pure), hence γ is a q-corner from φ to ıC . Now suppose that Υ q Υ q 0 for some Υ of the form Υ
An×n C1×n
Bn×1 a
=
φ (An×n ) γ ∗ (C1×n )
γ (Bn×1 ) . ı (a)
Since Υ is q-pure and Υ is not the zero map, we know that Υ = Υ (t) for some t 0, and a quick calculation gives us Υ
An×n C1×n
Bn×1 a
=
φ (t) (An×n ) (γ ∗ )(t) (C1×n )
γ (t) (Bn×1 ) . 1 1+t (a)
By inspecting the two formulas for Υ we see γ = γ (t) . But γ (t) has the form 1 ⎞ ⎛ 1+t+iλ b1 ⎞ b1 1 1 ⎜ b2 ⎟ ⎜ b ⎟ ⎟ ⎜ 1+t+iλ2 2 ⎟ γ (t) ⎜ ⎟, . ⎝ ... ⎠ = ⎜ ⎠ ⎝ .. bn 1 1+t+iλn bn
⎛
hence t = 0. Therefore, Υ = Υ , and we conclude the q-corner γ is hyper maximal.
2
In conclusion, we approach the broader question of simply finding all unital q-pure maps φ : Mn (C) → Mn (C), as they provide us with the simplest way to construct E0 -semigroups through boundary weight doubles. We believe that all q-pure maps are invertible or have rank one. For n = 2, we find in [6] that this conjecture holds: There is no unital q-pure map φ : M2 (C) → M2 (C) of rank 2, and there is no unital q-positive map φ : M2 (C) → M2 (C) of rank 3. It seems that for n = 3, the key to classifying unital q-pure maps is through investigation of the limits Lφ = limt→∞ tφ(I + tφ)−1 , though the situation becomes very complicated if n > 3.
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Acknowledgments The author very gratefully thanks his thesis advisor, Robert Powers, for his boundless enthusiasm, constant encouragement, and guidance in research. His help in the author’s thesis work has been indispensable. The author would also like to thank Geoff Price for proofreading an earlier draft of the paper and making suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.B. Arveson, The index of a quantum dynamical semigroup, J. Funct. Anal. 146 (1997) 557–588. W.B. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (409) (1989). B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. Amer. Math. Soc. 348 (2) (1996) 561–583. M. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975) 285–290. D.E. Evans, J.T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A 24 (1977). C. Jankowski, Unital q-pure maps on M2 (C), in preparation. D. Markiewicz, R.T. Powers, Local unitary cocycles of E0 -semigroups, J. Funct. Anal. 256 (5) (2009) 1511–1543. R.T. Powers, Continous spatial semigroups of completely positive maps of B(H ), New York J. Math. 9 (2003) 165–269. R.T. Powers, Construction of E0 -semigroups of B(H) from CP-flows, in: Advances in Quantum Dynamics, in: Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 57–97. R.T. Powers, Induction of semigroups of endomorphisms of B(H) from completely positive semigroups of (n × n) matrix algebras, Internat. J. Math. 10 (7) (1999) 773–790. R.T. Powers, New examples of continuous spatial semigroups of ∗-endomorphisms of B(H), Internat. J. Math. 10 (2) (1999) 215–288. R.T. Powers, An index theory for semigroups of ∗-endomorphisms of B(H) and type II1 factors, Canad. J. Math. 40 (1988) 86–114. R.T. Powers, A nonspatial continous semigroup of ∗-endomorphisms of B(H), Publ. Res. Inst. Math. Sci. 23 (6) (1987) 1053–1069. R.T. Powers, G. Price, Continuous spatial semigroups of ∗-endomorphisms of B(H), Trans. Amer. Math. Soc. 321 (1990) 347–361. B. Tsirelson, Non-isomorphic product systems, in: Advances in Quantum Dynamics, in: Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 273–328. E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. 40 (1939) 149–204.
Journal of Functional Analysis 258 (2010) 3452–3468 www.elsevier.com/locate/jfa
A renorming in some Banach spaces with applications to fixed point theory Carlos A. Hernandez Linares, Maria A. Japon ∗ Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain Received 24 September 2009; accepted 23 October 2009 Available online 5 November 2009 Communicated by N. Kalton
Abstract We consider a Banach space X endowed with a linear topology τ and a family of seminorms {Rk (·)} which satisfy some special conditions. We define an equivalent norm ||| · ||| on X such that if C is a convex bounded closed subset of (X, ||| · |||) which is τ -relatively sequentially compact, then every nonexpansive mapping T : C → C has a fixed point. As a consequence, we prove that, if G is a separable compact group, its Fourier–Stieltjes algebra B(G) can be renormed to satisfy the FPP. In case that G = T, we recover P.K. Lin’s renorming in the sequence space 1 . Moreover, we give new norms in 1 with the FPP, we find new classes of nonreflexive Banach spaces with the FPP and we give a sufficient condition so that a nonreflexive subspace of L1 (μ) can be renormed to have the FPP. © 2009 Elsevier Inc. All rights reserved. Keywords: Fixed point theory; Renorming theory; Nonexpansive mappings; Fourier algebras; Fourier–Stieltjes algebra; Topology of convergence locally in measure
1. Introduction Let (X, · ) be a Banach space and C a convex closed bounded subset of X. A mapping T : C → C is called nonexpansive if for any x, y ∈ C we have T x − T y x − y. A point x ∈ C is a fixed point of T if T x = x. It is clear that Banach’s Contraction Principle does not * Corresponding author.
E-mail addresses:
[email protected] (C.A.H. Linares),
[email protected] (M.A. Japon). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.025
C.A.H. Linares, M.A. Japon / Journal of Functional Analysis 258 (2010) 3452–3468
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extend to the setting of nonexpansive mappings. However, some positive results concerning the existence of fixed points for this class of mappings where given in 1965 by F.E. Browder [5] and D. Göhde [13] for uniformly convex Banach spaces and by W. Kirk [16] for reflexive Banach spaces with normal structure. Since then, many authors have studied the problem of the existence of fixed points for nonexpansive mappings and many positive results have been found (see for instance [12,17] and the references therein). It is usually said that a Banach space X has the fixed point property (FPP) if every nonexpansive mapping defined from a closed convex bounded subset onto itself has a fixed point. It is well known that the geometry of the Banach space plays a fundamental role to assure the FPP. In fact, Kirk’s result [16] means that a reflexive Banach space with normal structure has the FPP. Many other geometric properties are known to imply the FPP for reflexive Banach spaces (uniform Kadec Klee property, uniform Opial condition, existence of a monotone unconditional basis, etc.). Moreover the classical nonreflexive Banach spaces 1 , c0 , L1 do not have the FPP (in fact L1 does not satisfy a stronger condition called the weakly fixed point property [2]). For a long time, it was an open question whether all Banach spaces with the FPP were reflexive. In 2008, P.K. Lin [20] found the first known nonreflexive Banach space with the FPP. In fact, the Banach space given by P.K. Lin was the sequence space 1 endowed with an equivalent norm to the usual one. His result raises the question: can any Banach space be renormed to have the FPP? This is not, in general, the case because the Banach spaces 1 (Γ ) and c0 (Γ ), if Γ is uncountable, and the Banach space ∞ cannot be renormed to have the FPP [8]. A positive partial answer was given by T. Domínguez Benavides [6], who proved that every reflexive Banach space can be renormed to have the FPP. This leads to the following question: Which type of nonreflexive Banach spaces can be renormed to have the FPP? In this paper we find some classes of nonreflexive Banach spaces which under an equivalent renorming satisfy the FPP. Our techniques are inspired by those of P.K. Lin’s paper [20] but our applications go beyond the sequence space 1 as we will illustrate with many examples. As particular cases, we will recover P.K. Lin’s result and we will find new renormings in 1 with the FPP. Moreover, we will renorm the Fourier–Stieltjes algebra of a separable compact group to have the FPP. Notice that if G is locally compact, its Fourier–Stieltjes algebra B(G) has the FPP if and only if G is finite [18]. We also find new classes of nonreflexive Banach spaces with the FPP which are nonisomorphic to any subspace of 1 . Finally, we will apply our results to the particular case of subspaces of L1 (μ) for a σ -finite measure. It is known that a closed subspace X of L1 (μ) has the FPP if and only if X is reflexive [23,7]. Nevertheless, we will show that some nonreflexive subspaces of L1 (μ) can still be renormed to have the FPP. This paper is organized as follows: Section 2 is dedicated to the necessary fixed point background and we establish a technical lemma which is basic in our proofs. In Section 3 we will state our main Theorem and we will introduce the first applications: we are able to find new renormings in 1 with the FPP, we renorm B(G) with the FPP if G is a separable compact group and we give new examples of nonreflexive Banach spaces with the FPP that are nonisomorphic to any subspace of 1 . Section 4 is dedicated to the proof of the main Theorem. Finally, in Section 5, we will apply the main Theorem to closed subspaces of L1 (μ) when (Ω, Σ, μ) is a σ -finite measure space. We obtain a sufficient condition to assure that a nonreflexive subspace X of L1 (μ) can be renormed to have the FPP (it is known that, with the usual norm, X fails to have this property [7]). We finish the paper by introducing some examples of nonreflexive subspaces of L1 (μ) which can be renormed to have the FPP.
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2. Fixed point background Let C be a closed convex bounded subset of a Banach space (X, · ) and T : C → C a nonexpansive mapping. Fix any x0 ∈ C. A direct application of the Banach’s Contraction Principle to the sequence of mappings Tn : C → C defined by 1 1 T x; Tn x = x 0 + 1 − n n provides a sequence {xn }n ⊂ C, where xn is the unique point of Tn , such that lim xn − T xn = 0. n
Such sequences are called approximated fixed point sequences (a.f.p.s.). Moreover, if d > 0 and the set D = x ∈ C: lim sup xn − x d n
is nonempty, it is easy to check that D is convex, closed and a T -invariant subset of C. Hence, we can find another a.f.p.s. in D. As an application of Cantor’s Intersection Theorem, we can prove the following: Lemma 1. Let (X, · ) be a Banach space and C a convex, closed, bounded subset of X. Let T : C → C be a nonexpansive mapping and suppose that T is fixed point free. Then there exist some a > 0 and a convex closed T -invariant subset D of C such that for each approximated fixed point sequence (xn ) in D and for any z ∈ D lim sup xn − z a. n
Proof. If the statement is false there exists an a.f.p.s. (xn1 ) in C and z1 ∈ C such that 1 lim supxn1 − z1 < . 2 n Hence 1 D1 = z ∈ C: lim supxn1 − z 2 n is a nonempty, convex, closed, T -invariant subset of C. With the same argument, we deduce the existence of an approximated fixed point sequence (xn2 ) in D1 and z2 ∈ D1 such that 1 lim supxn2 − z2 < 2 . 2 n
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Hence the set 2 1 D2 = z ∈ D1 : lim sup xn − z 2 2 n is again a nonempty, convex, closed, T -invariant subset of D1 . In this way we construct a decreasing sequence (Dn ) of convex closed bounded T -invariant 1 . By the Cantor’s Intersection Theorem, n Dn is a subsets of C such that diam(Dn ) 2n−1 singleton. Since each Dn is T -invariant this point has to be a fixed point of T . Thus, we have obtained a contradiction since T is fixed point free. 2 Remark. Notice that if X is endowed with a topology τ such that every bounded sequence has a τ -convergent subsequence, then the conditions of Lemma 1 also imply that inf lim sup xn − x: (xn ) ⊂ D, (xn ) a.f.p.s. xn → x in τ > 0. n
Indeed, applying the triangular inequality, for every (xn ) ⊂ D a.f.p.s. such that (xn ) converges to x in the topology τ , we have lim sup xn − x n
1 a lim sup lim sup xn − xm . 2 m 2 n
3. Main result and first examples In this section we state the main result of this paper. As a consequence, we obtain the renorming given in [20] in the sequence space 1 , which provided the first known nonreflexive Banach space with the FPP. Also we will give new equivalent norms on 1 with the FPP and we will obtain new classes of nonreflexive Banach spaces with the FPP. In particular, we will prove that the Fourier–Stieltjes algebra B(G) of a separable compact group can be renormed to have the FPP. Notice that B(G) itself has the FPP if and only if G is finite [18] (Theorem 5.8). More applications of the main Theorem will be studied in the last section. Let (X, · ) be a Banach space endowed with a linear topology τ . Assume that there exists a family of seminorms Rk : X → [0, +∞) (k 1) that satisfy the following properties: (I) R1 (x) = x while for k 2, Rk (x) x for all x ∈ X. (II) limk Rk (x) = 0 for all x ∈ X. (III) If xn → 0 in τ and is norm-bounded, then for all k 1 lim sup Rk (xn ) = lim sup xn . n
n
(IV) If xn → 0 in τ , is norm-bounded and x ∈ X, then lim sup Rk (xn + x) = lim sup Rk (xn ) + Rk (x) n
for all k 1.
n
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Then we can state the following: Theorem 1. Let {γk }k ⊂ (0, 1) be any nondecreasing sequence such that limk γk = 1 and define |||x||| = sup γk Rk (x);
x ∈ X.
k1
Then ||| · ||| is an equivalent norm on X such that (X, ||| · |||) satisfies the following property: for every nonempty closed convex bounded subset C which is τ -relatively sequentially compact and for every T : C → C nonexpansive, there exists a fixed point. That ||| · ||| is an equivalent norm on (X, · ) is clear. In fact, γ1 x |||x||| x for all x ∈ X. We will prove Theorem 1 in the next section. Now we give several families of Banach spaces where our results can be applied. Example 1. A first application of Theorem 1 is a generalization of P.K. Lin’s example given in [20], where he proves that if γk = 8k /(1 + 8k ), then the renorming on 1 given by ∞ 8k
|||x||| = sup x e n n k k 1+8 n=k
has the FPP. Notice that Lin’s result can be derived from Theorem 1 defining the seminorms Rk (x) = ∞ n=k xn en and τ the weak-star topology associated to the duality σ (1 , c0 ). Since the unit ball is weak-star compact and c0 is separable, every closed convex bounded subset is σ (1 , c0 )-sequentially compact. Also, we obtain the renorming given in [10] where P.K. Lin’s result is generalized by using any nondecreasing sequence (γk ) ∈ (0, 1) with limk γk = 1 and γ1 > 2/3. Notice that in our approach the condition γ1 > 2/3 can be dropped. Example 2. If we consider again the sequence space 1 and change the family of seminorms, then we can obtain new renormings on 1 with the FPP. For instance, let p > 1 and for k 2 define ∞
x(n) + Rk (x) = n=2k
2k−1
1/p |xn |
p
n=k
and R1 (x) = x1 . It is easy to check that {Rk (·)}k is a family of seminorms that verify properties (I), (II), (III) and (IV), so 1 with the norm generated by the seminorms {Rk (·)}k satisfies the FPP. Corollary 1. Let {Xn }n be a sequence of finite dimensional Banach spaces and consider X = ⊕1
Xn = x = (xn )n : xn ∈ Xn , x = xn Xn < ∞ .
n
Then X can be renormed to have the FPP.
n
C.A.H. Linares, M.A. Japon / Journal of Functional Analysis 258 (2010) 3452–3468
Proof. Define the seminorms Rk (x) = the predual of X is
nk xn Xn
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and let τ be the weak star topology where
E = x = (xn ): xn ∈ Xn , lim xn Xn = 0, x = sup xn Xn . n
It is not difficult to check that the family {Rk (·)}k satisfies properties (I), (II), (III) and (IV). Using the renorming given in Theorem 1, the Banach space (X, ||| · |||) has the FPP. 2 A first application of Corollary 1 is the following example: Example 3. Consider Xn = np for some 1 < p +∞ and X = ⊕1
np .
n
Then we obtain a nonreflexive Banach space that can be renormed to have the FPP and that is not isomorphic to any subspace of 1 . Indeed, p is finitely representable in X and the type and the cotype of X is equal to the type and the cotype of p respectively. Notice that for every 1 < p +∞, either the type or the cotype of p is different from that of 1 , since type(p ) = min{2, p} and cotype(p ) = max{2, p} (see [22], p. 73). Thus, X is not isomorphic to any subspace of 1 and we obtain new classes of nonreflexive Banach spaces with the FPP. For the definitions of the Fourier algebra A(G) and the Fourier–Stieltjes algebra B(G) of a locally compact group see [9] or [19] and the references therein. When G is compact, B(G) = A(G). Another application of Corollary 1 is the following. Corollary 2. Let G be a separable compact group and B(G) its Fourier–Stieltjes algebra. Then B(G) can be renormed to have the FPP. Proof. Using the arguments in the proof of Lemma 3.1 of [19], and having in mind that B(G) is norm separable when G is a separable compact group [14, Corollary 6.9], the Banach space B(G) can be written as B(G) = ⊕1
T(Hn )
n
where Hn is a finite dimensional Hilbert space and T(Hn ) is the trass class operators on Hn . Applying Corollary 1 we obtain a renorming of B(G) with the FPP. 2 In the particular case that G = T, the circle group, B(G) is isometric to 1 (Z) via Bochner’s Theorem. Thus, Corollary 2 includes the sequence space 1 and the renorming given by P.K. Lin [20]. Also, recall that B(G) with its usual norm has the FPP if and only if G is finite [18].
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4. Proof of the main result Before proving Theorem 1 we prove two technical lemmas: Lemma 2. Let X be a Banach space endowed with a linear topology τ and a family of seminorms {Rk (·)k } satisfying properties (I), (II), (III) and (IV) stated above. Define the ||| · ||| norm as in Theorem 1 and let (xn ), (yn ) be two bounded sequences in X. Then the following statements are satisfied: (1) If xn → 0 in τ , then lim sup |||xn ||| = lim sup xn . n
n
(2) If xn → x and yn → y in τ then lim sup lim sup |||xn − ym ||| lim sup |||xn − x||| + lim sup |||ym − y|||. m
n
n
m
Proof. (1) For every k 1, using the definition of the ||| · ||| norm and property (III), we have lim sup xn lim sup |||xn ||| γk lim sup Rk (xn ) = γk lim sup xn . n
n
n
n
Taking limit as k goes to infinity we deduce (1). (2) By property (IV) we have lim sup lim sup Rk (xn − ym ) = lim sup lim sup Rk (xn − x) + Rk (x − ym ) m
n
m
n
= lim sup Rk (xn − x) + lim sup Rk (ym − y) + Rk (x − y), n
m
for every k 1. Then, using again the definition of ||| · ||| and property (III), lim sup lim sup |||xn − ym ||| γk lim sup Rk (xn − x) + lim sup Rk (ym − y) + Rk (x − y) m
n
n
m
γk lim sup |||xn − x||| + lim sup |||ym − y||| . n
m
Taking limit as k goes to infinity we get the desired result.
2
The following lemma is the key for the arguments in the proof of Theorem 1: Lemma 3. Consider the Banach space (X, ||| · |||) and let C and T be as in Theorem 1. If T is fixed point free we find D as in Lemma 1. Let K be any closed convex T -invariant subset of D and denote ρ = inf lim sup |||xn − x|||: (xn ) ⊂ K is an a.f.p.s. and xn → x in τ > 0. n
C.A.H. Linares, M.A. Japon / Journal of Functional Analysis 258 (2010) 3452–3468
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Then for every a.f.p.s. (xn ) ⊂ K which is τ -convergent and for every z ∈ K we have lim sup |||xn − z||| 2ρ. n
Proof. Assume that there exist a τ -convergent approximate fixed point sequence (xn ) in K and z ∈ K such that r = lim sup |||xn − z||| < 2ρ. n
Then K1 = w ∈ K: lim sup |||xn − w||| r n
is a nonempty, convex, closed, bounded T -invariant subset of K. Choose an approximate fixed τ point sequence (yn ) in K1 such that yn → y. Denote by x the τ -limit of the sequence (xn ). Then by (2) of Lemma 2, we have r lim sup lim sup |||xn − ym ||| m
n
lim sup |||xn − x||| + lim sup |||yn − y||| n
n
ρ + ρ = 2ρ, which is a contradiction.
2
Now we prove Theorem 1. Proof. Assume the contrary, that T has no fixed point. Let D be as in the conclusion of Lemma 1. Define τ c = inf lim sup |||xn − x|||: (xn ) ⊂ D is an a.f.p.s. and xn → x n
which is greater than zero by the remark made after Lemma 1. Without loss generality we can assume that c = 1. Take 0 < 1 < 1/2 and an a.f.p.s. (xn ) ⊂ D τ such that xn → x and lim supn |||xn − x||| < 1 + 1 . Again, by translation, we can assume that x = 0. Let us consider now K = z ∈ D: lim sup |||xn − z||| 2 + 2 1 . n
The set K is closed, convex, T -invariant and nonempty. Indeed, we can find n0 such that xn ∈ K for all n n0 . Define
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τ ρ = inf lim sup |||yn − y|||: (yn ) ⊂ K is an a.f.p.s. and yn → y . n
It is clear that 1 ρ lim supn |||xn ||| < 1 + 1 . We are going to find an a.f.p.s. (yn ) ⊂ K and z ∈ K such that lim sup |||yn − z||| < 2ρ n
and then we obtain a contradiction according to Lemma 3. τ Notice the following: If (yn ) ⊂ K is an a.f.p.s. and yn → y, then for all k, 2 + 2 1 lim sup lim sup |||xn − ym ||| = lim sup lim sup xn − (ym − y) − y m
n
m
n
γk lim sup lim sup Rk xn − (ym − y) − y m
n
= γk lim sup Rk (xn ) + lim sup Rk (ym − y) + Rk (y) n
m
= γk lim sup |||xn ||| + lim sup |||ym − y||| + Rk (y) γk 2 + Rk (y) . n
m
τ
Consequently, if (yn ) ⊂ K is an a.f.p.s. and yn → y, we have Rk (y) 2
1 + 1 −1 . γk
Let p := 1 + 1 + 2
1 + 1 − 1 > ρ, γ1
δ ∈ ( 1 , 1/2),
0 < 2 < ρ − 2δ.
Since, by Lemma 2(1), lim supn xn = lim supn |||xn ||| < 1 + 1 , we can find x ∈ K such that x < 1 + 1 . Also there exists m ∈ N such that if k m Rk (x) < 2
by property (II)
and 1 + 1 < γk 1+δ
since lim γk = 1 . k
We take λ ∈ (0, 1) such that λ<
ρ(1 − γm ) . γm (p − ρ)
Since (2 − λ)ρ + λ( 2 + 2δ) = 2ρ − λ ρ − (2δ + 2 ) < 2ρ
C.A.H. Linares, M.A. Japon / Journal of Functional Analysis 258 (2010) 3452–3468
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and γm (2 − λ)ρ + λp < ρ(1 + γm ) < 2ρ, we can find 3 > 0 such that (i)
(2 − λ)(ρ + 3 ) + λ( 2 + 2δ) < 2ρ
and (ii)
γm (2 − λ)(ρ + 3 ) + λp < 2ρ. τ
Take (yn ) ⊂ K to be an a.f.p.s. such that yn → y and lim sup yn − y = lim sup |||yn − y||| < ρ + 3 n
using Lemma 2(1) .
n
There exists s ∈ N such that yN − y < ρ + 3 for all N s and define z = (1 − λ)ys + λx which belongs to K because K is convex. Let us prove that lim supn |||yn − z||| < 2ρ. In order to do this, we will prove that there exists M > 0 such that for all k and N s we have γk Rk (yN − z) < M < 2ρ. We split the proof into two cases: Case 1: k m: γk Rk (yN − z) = γk Rk yN − (1 − λ)ys − λx Rk yN − y − (1 − λ)(ys − y) − λ(x − y) Rk (yN − y) + (1 − λ)Rk (ys − y) + λRk (x − y) yN − y + (1 − λ)ys − y + λRk (x − y) (ρ + 3 ) + (1 − λ)(ρ + 3 ) + λ Rk (x) + Rk (y) (2 − λ)(ρ + 3 ) + λ 2 + Rk (y) 1 + 1 (2 − λ)(ρ + 3 ) + λ 2 + 2 −1 γk < (2 − λ)(ρ + 3 ) + λ( 2 + 2δ) < 2ρ
by (i).
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Case 2: k m: γk Rk (yN − z) γm Rk yN − (1 − λ)ys − λx γm Rk yN − y − (1 − λ)(ys − y) − λ(x − y) γm Rk (yN − y) + (1 − λ)Rk (ys − y) + λRk (x − y) γm (ρ + 3 ) + (1 − λ)(ρ + 3 ) + λ Rk (x) + Rk (y) γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + Rk (y) 1 + 1 −1 < 2ρ γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + 2 γ1
by (ii).
Take M = max (2 − λ)(ρ + 3 ) + λ( 2 + 2δ), 1 + 1 γm (2 − λ)(ρ + 3 ) + λ 1 + 1 + 2 −1 . γ1 Then, for all N s, |||yN − z||| < M < 2ρ. Thus lim supn |||yn − z||| < 2ρ and this finishes the proof. 2 5. Applications to the Lebesgue function space L1 (μ) In this section we are going to consider the Banach space L1 (μ), where (Σ, Ω, μ) is a σ -finite measure space and we will apply Theorem 1 to this space. As a consequence we will obtain new results about renorming and FPP for nonreflexive subspaces of L1 (μ). In order to do that we will define a family of seminorms {Rk (·)}k1 which satisfies properties (I), (II), (III) and (IV) stated in Section 3. We denote by · the usual norm on L1 (μ), that is x = |x| dμ, for all x ∈ L1 (μ) Ω
x for all x ∈ L1 (μ). and R1 (x) = Let Ω = n Ωn be such that μ(Ωn ) < +∞ for all n ∈ N and denote Ak = kn=1 Ωn . For k 2 define the seminorms 1 + xχAck . Rk (x) = sup |x| dμ: μ(E) < (1) k E∩Ak
Let τ be the topology of locally convergence in measure, which is given by the metric d(x, y) =
∞
1 1 |x − y| dμ; n 2 μ(Ωn ) 1 + |x − y| n=1
Ωn
x, y ∈ L1 (μ).
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This topology is related to the convergence almost everywhere in the following way: every sequence that converges almost everywhere also converges locally in measure to the same function. Moreover, if a sequence converges locally in measure, then it has a subsequence that converges almost everywhere [15], pp. 157–158. Fix any nondecreasing sequence (γk )k ⊂ (0, 1) such that limk γk = 1 and define the equivalent norm on L1 (μ) as |||x||| = sup γk Rk (x). k
Now we have all the ingredients to state the following: Theorem 2. The seminorms {Rk (·)}k defined above satisfy properties (I), (II), (III) and (IV) stated in Section 3. Thus the following holds: Let C be a convex bounded closed subset of L1 (μ) and T : C → C a ||| · |||-nonexpansive mapping. If every sequence in C has a subsequence which is almost everywhere convergent in L1 (μ), then T has a fixed point. Remark 1. Notice that the above fixed point result does not hold for the usual norm in L1 (μ). Indeed, consider (hn ) a disjointly supported normalized sequence in L1 (μ). Let C=
tn hn ; tn 0,
n
tn = 1 = co(hn ).
n
The set C is closed convex bounded and every sequence in C has a convergent subsequence almost everywhere. Indeed, consider a sequence (fk ) ⊂ C. Then fk = n tn (k)hn where tn (k) 0 and n tn (k) = 1 for every k. Define t k = (tn (k))n . The sequence (t k )k belongs to the unit ball of 1 which is σ (1 , c0 )-compact. So there exists a subsequence, denoted again by t k , which is σ (1 , c0 )-convergent to some t = (tn )n belonging to unit ball of 1 . Now we can easily check that (fk ) is pointwise convergent to f = n tn hn (notice that f is not, in general, in C). Define T : C → C by T
n
tn hn =
tn hn+1 .
n
It is easy to check that T is · -nonexpansive and has no fixed point in C. Remark 2. If the measure space is not σ -finite and we consider the convergence almost everywhere in L1 (μ), we cannot renorm the space L1 (μ) so that Theorem 2 remains true. Indeed, in this case 1 (Γ ) is contained isometrically in L1 (μ) for some uncountable set Γ and every sequence in a bounded subset of 1 (Γ ) has a pointwise convergent subsequence. So if Theorem 2 holds then we would have a renorming in 1 (Γ ) with the FPP. This is impossible since every renorming of 1 (Γ ) contains an asymptotically isometric copy of 1 and then it fails the FPP [8]. Before proving Theorem 2 we give a simpler definition of the ||| · ||| norm in two special cases: when μ is finite and when μ is purely atomic.
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Remark 3. Assume that the measure μ is finite. Consider Ak = Ω for all k. Then Ack = ∅ and xχAck = 0 for all x ∈ L1 (μ). Therefore 1 Rk (x) = sup |x| dμ: μ(E) < k E
for all k ∈ N. In this case, the topology of the convergence locally in measure is given by the metric d(x, y) = Ω
|x − y| dμ; 1 + |x − y|
x, y ∈ L1 (μ),
and the convergence with respect to this topology is the convergence in measure. Remark 4. Assume now that Ω = N and μ is the counting measure defined on the subsets of N. Then the space L1 (μ) becomes the sequence space 1 . Denote Ωk = {n} and Ak = {1, . . . , n}. Then Rk (x) = xχAck =
∞
x(n) . n=k+1
In this case we recover again the ||| · ||| norm defined by P.K. Lin in [20] for the particular case γk = 8k /(1 + 8k ). Notice that the convergence almost everywhere in 1 is the pointwise convergence and every bounded sequence in 1 has a pointwise convergent subsequence because the unit ball of 1 is σ (1 , c0 )-compact. In general, if the measure space is σ -finite and purely atomic, L1 (μ) is isometric to 1 . Thus by Theorem 2, it can be renormed to have the FPP. Now, to prove Theorem 2 we only have to check that properties (I), (II), (III), (IV) are fulfilled for the family of seminorms Rk (·) defined on L1 (μ). Proof. To simplify the notation we let Sk (x) := sup
|x| dμ: μ(E) <
1 k
E
for x ∈ L1 (μ), therefore Rk (x) = Sk (xχAk ) + xχAck . (I) Using the absolute continuity of the norm and that limk xχAck = 0 we can check that limk Rk (x) = 0 for all x ∈ L1 (μ). (II) We have Sk (xχAk ) xχAk . Therefore Rk (x) xχAk + xχAck = x for every x ∈ L1 (μ).
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(III) Fix k 1 and let (xn ) be a sequence convergent to the null function locally in measure. Assume the contrary, that is, lim supn Rk (xn ) < lim supn xn . We can take a sequence, again denoted by (xn ), such that limn xn , limn Sk (xn χAk ), limn xn χAk and limn xn χAck exist, (xn ) converges to the null function almost everywhere and limn Rk (xn ) < limn xn . Let us prove that limn Sk (xn χAk ) = limn xn χAk : Using Egoroff’s Theorem there exists a measurable set E ⊂ Ak with μ(E) < 1/k and such that xn → 0 uniformly on Ak \ E. In particular xn χAk \E → 0 in norm and limn xn χAk = limn xn χE . Therefore lim xn χAk lim Sk (xn χAk ) lim xn χE = lim xn χAk . n
n
n
n
Now lim Rk (xn ) lim Sk (xn χAk ) + lim xn χAck n
n
n
= lim xn χAk + lim xn χAck n
n
= lim xn n
which is a contradiction and property (III) holds. (IV) If k = 1, since R1 (·) = · and using [4] we obtain lim sup xn + x = lim sup xn + x. n
n
Assume that k 2. Suppose by contradiction that property (IV) does not hold. We recall the following lemma for finite measure spaces [1]: Let (Ω, σ, μ) be a finite measure space and (hn ) be a bounded sequence in L1 (μ) converging to the null function in measure. Then there exists a subsequence (hnl ) and a sequence of pairwise disjoint measurable sets (El ) such that lim hnl − hnl χEl = 0. l
In particular, for all k 1, liml Sk (hnl − hnl χEl ) = 0. Using the above result we can take a subsequence, again denoted by (xn ), such that there exists a sequence (hn ) of measurable functions defined in Ak which is disjointly supported, limn Sk ((xn − hn )χAk ) = 0 and lim supn Rk (xn + x) < lim supn Rk (xn ) + Rk (x) for some x ∈ L1 (μ). Therefore lim sup Rk (xn + x) = lim sup Sk (xn + x)χAk + lim sup(xn + x)χAck n
n
n
n
n
= lim sup Sk (hn + x)χAk + lim sup xn χAck + xχAck .
Let us prove that lim supn Sk ((hn + x)χAk ) = lim supn Sk (hn χAk ) + Sk (xχAk ): Denote by En ⊂ Ak the support of the function hn and let > 0. By the definition of Sk (·), there exists a measurable set A ⊂ Ak with μ(A) < k1 such that
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|x| dμ Sk (xχAk ) − . A
Since
n μ(En ) μ(Ak ) < +∞ there exists n0
such that μ(A) +
nn0
μ(En ) < k1 . Therefore
lim sup Sk (hn + x) lim sup n
n
A∪ nn En
0
= lim sup n
A∪ nn En
lim sup n
|hn + x| dμ |hn | dμ +
0
A∪
|hn | dμ +
En
nn0
|x| dμ En
|x| dμ A
lim sup hn χAk + Sk (xχAk ) −
n
= lim sup Sk (hn χAk ) + Sk (xχAk ) − . n
Since is arbitrary we obtain the desired equality. Therefore: lim sup Rk (xn + x) = lim sup Sk (hn χAk ) + Sk (xχAk ) + lim sup xn χAck + xχAck n
n
n
= lim sup Sk (xn χAk ) + Sk (xχAk ) + lim sup xn χAck + xχAck n
n
= lim sup Rk (xn ) + Rk (x) n
and we obtain (IV).
2
Although it is well known that the space L1 (μ) does not have the FPP (it does not satisfies the weak fixed point property w-FPP [2]), in 1980, B. Maurey [23] proved that all reflexive subspaces of L1 (μ) do have the FPP. In 1997 P. Dowling and C. Lennard [7] proved that the converse holds, that is, a subspace X of L1 (μ) has the FPP if and only if X is reflexive. This leads us to the following question: Can a nonreflexive subspace of L1 (μ) be renormed to have the FPP? Theorem 2 lets us give a partial answer to this question: Corollary 3. Let X be a closed subspace of L1 (μ). If the unit ball of X is relatively sequentially compact for the topology of the convergence locally in measure, then (X, ||| · |||) has the FPP. Corollary 4. Let X be a closed subspace of L1 (μ). If X is a dual space such that the topology of the convergence locally in measure coincides with the weak star topology on the unit ball of X, then (X, ||| · |||) has the FPP. Notice that this is the case of the sequence space 1 . Here we present another example.
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Example 4. Let D denote the open unit disc. The Bergman space La (D) is defined as the subspace of L1 (D) of all analytic functions on D. This space is a dual space and for bounded sequences weak* convergence is equivalent to uniform convergence on compact sets [24]. This shows that the weak* topology is finer than the topology of convergence in measure on the unit ball of La (D) and consequently these two topologies coincide for BLa (D) . Thus the Bergman space endowed with the ||| · ||| norm given in this section has the FPP. Notice that, from P.K. Lin’s paper [20], it is deduced that the Bergman space can be renormed to have the FPP. Indeed, J. Lindenstrauss, A. Pelczynski [21] proved that the Bergman space and the sequence space 1 are isomorphic, although they did not give an explicit definition of the isomorphism. In fact, it turns out to be a difficult problem to find a system of functions which is a basis in La (D) equivalent to the unit vector basis in 1 (see Notes and Remarks in Chapter III.A of [25] and the references therein). However, using Theorem 2 we can give explicitly the renorming on the Bergman space with the FPP. The following example satisfies the hypothesis of Corollary 3 but does not fit in the scope of Corollary 4: Example 5. In [11], Théorème 7, we can find an example of a subspace X of L1 [0, 1] such that its unit ball BX is compact for the topology of convergence in measure but not locally convex for this topology. Then, by Corollary 3, (X, ||| · |||) has the FPP. To finish we show another example of a Banach space which can be renormed to satisfy the FPP: Example 6. In [3] we can find an example of a Banach space E contained in L1 , over a probability space, and such that E fails to have the Radon–Nikodym property and every L1 -bounded sequence in E has a subsequence converging in measure. Applying again Theorem 2, we deduce that E can be renormed to have the FPP and by the failure of the Radon–Nikodym property, E is not isomorphic to any subspace of 1 . Acknowledgments The authors would like to thank Professors T. Domínguez Benavides, B. Sims and A.T.-M. Lau for their valuable suggestions in the preparation of this paper. References [1] F. Albiac, N.J. Kalton, Topics in Banach Space Theory, Grad. Texts in Math., vol. 233, Springer-Verlag, New York, 2006. [2] D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [3] J. Bourgain, H.P. Rosenthal, Martingales valued in certain subspaces of L1 , Israel J. Math. 37 (1–2) (1980) 54–75. [4] M. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (3) (1993) 486–490. [5] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965) 1041– 1044. [6] T. Domínguez Benavides, A renorming of some nonseparable Banach spaces with the Fixed Point Property, J. Math. Anal. Appl. 350 (2) (2009) 525–530. [7] P.N. Dowling, C.J. Lennard, Every nonreflexive subspace of L1 [0, 1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (2) (1997) 443–446.
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[8] P.N. Dowling, C.J. Lennard, B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (1996) 653–662. [9] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. [10] H. Fetter, personal communication. [11] G. Godefroy, N.J. Kalton, D. Li, Propriété d’approximation métrique inconditionnelle et sous-espaces de L1 dont la boule est compacte en measure, C. R. Acad. Sci. Paris Sér. I 320 (1995) 1069–1073. [12] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. [13] D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965) 251–258. [14] C. Graham, A.T.-M. Lau, M. Leinert, Separable translation-invariant subspaces of M(G) and other dual spaces on locally compact groups, Colloq. Math. 55 (1) (1988) 131–145. [15] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1965. [16] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1964) 1004–1006. [17] W.A. Kirk, B. Sims, Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001. [18] A.T.-M. Lau, M. Leinert, Fixed point property and the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 360 (12) (2008) 6389–6402. [19] A.T.-M. Lau, P.F. Mah, Fixed point property for Banach algebras associated to locally compact groups, J. Funct. Anal. 258 (2) (2010) 357–372. [20] P.K. Lin, There is an equivalent norm on 1 that has the fixed point property, Nonlinear Anal. 68 (8) (2008) 2303– 2308. [21] J. Lindenstrauss, A. Pelczynski, Contributions of the theory of the classical Banach spaces, J. Funct. Anal. 8 (2) (1971) 225–249. [22] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Springer-Verlag, 1979. [23] B. Maurey, Points fixes des contractions sur un convexe ferme de L1 , in Seminaire d’Analyse Fonctionnelle, 1980– 1981, Ecole Polytechnique. [24] M. Nowak, Compact Hankel operators with conjugate analytic symbols, Rend. Circ. Mat. Palermo (2) XLVII (1998) 363–374. [25] P. Wojtaszczyk, Banach Spaces of Analysts, Cambridge Stud. Adv. Math., vol. 25, 1991.
Journal of Functional Analysis 258 (2010) 3469–3491 www.elsevier.com/locate/jfa
Local solvability for a class of evolution equations ✩ Ferruccio Colombini a , Paulo D. Cordaro b,∗ , Ludovico Pernazza c a Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy b Departamento de Matemática Aplicada, IME–USP, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil c Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
Received 29 September 2009; accepted 8 December 2009 Available online 12 January 2010 Communicated by Paul Malliavin
Abstract Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L2 and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg–Treves condition (ψ) which is suitable to our study. © 2009 Published by Elsevier Inc. Keywords: Local solvability; Linear PDE; Evolution equations
1. Introduction In 1971 Y. Kannai [7] presented an example of a linear partial differential operator which is hypoelliptic but not locally solvable. More precisely, in R2 , where the coordinates are written as (y, t), Kannai considered the operator
K=
✩
∂2 ∂ +t 2 ∂t ∂y
This research project was partially supported by CNPq and Fapesp.
* Corresponding author.
E-mail addresses:
[email protected] (F. Colombini),
[email protected] (P.D. Cordaro),
[email protected] (L. Pernazza). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.12.004
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and showed that K is hypoelliptic in any open set Ω ⊆ R2 and is not locally solvable near any point of the form (y0 , 0).1 Since the transpose of a hypoelliptic operator is locally solvable (cf. [10, Theorem 52.2]) it follows in particular that the operator K = −t
∂2 ∂ − , 2 ∂t ∂y
= ±1,
is locally solvable, say, near the origin if and only if = −1. A natural extension of this statement was proved in [2]. In this work the authors considered real operators in the form B = −t d
m j,k=1
m ∂ ∂ ∂ ∂ cj k (y, t) − − hj (y, t) + g(y, t), ∂yj ∂yk ∂t ∂yj
= ±1,
()
j =1
m+1 and in which d ∈ N, the coefficients m are smooth in an open neighborhood of the origin in R the quadratic form ξ → j,k=1 cj k (y, t)ξj ξk is positive definite everywhere. The main result in [2] states that when d is even then B is always locally solvable near the origin whereas when d is odd then B is locally solvable near the origin if and only if = −1. Furthermore, if either d is even or if d is odd and = −1 then the solvability of B occurs in the L2 sense. Following the work of R. Beals and C. Fefferman [1] we can account for this result in an invariant formulation. For this we consider real, smooth operators defined in an open neighborhood of the origin in Rn of the form
Q=−
n j,k=1
n ∂ ∂ ∂ d ϕ(x) aj k (x) − bj (x) + c(x), ∂xj ∂xk ∂xj j =1
where d ∈ N and the following conditions are imposed: ϕ(0) = 0 and dϕ = 0 on ϕ −1 (0); n . ξ → A(x)(ξ ) = aj k (x)ξj ξk
is nonnegative for every x ∈ Ω;
(i) (ii)
j,k=1
A(0) has rank n − 1;
(iii)
A(x)(dϕ) = 0 for all x ∈ Ω; . ϑ= bk ∂ϕ/∂xk (0) = 0.
(iv) (v)
k
Notice that, thanks to (iv), the function k bk ∂ϕ/∂xk is invariantly defined on ϕ −1 {0}. In particular, the sign of ϑ is invariantly defined even after multiplication of Q by a nonvanishing smooth function. √ 2s(∂/∂s + ∂ 2 /∂y 2 ) and consequently K is indeed hypoelliptic and locally solvable near any point (y0 , t0 ) with t0 = 0. 1 In the new variable s = t 2 /2 we can write K =
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If we choose the coordinates (x1 , . . . , xn ) in such a way that ϕ = xn then conditions (ii), (iii) and (iv) imply that aj n = anj = 0 for all j = 1, . . . , n and that ξ → n−1 j,k aj k (x)ξj ξk is positive . definite everywhere. Consequently, if we write yj = xj , j = 1, . . . , m = n − 1, t = xn we easily conclude that a nonvanishing multiple of Q has the form (), where equals the sign of ϑ . We can then state: Theorem 1.1. If d is even then Q is locally solvable near the origin. If d is odd then Q is locally solvable near the origin if and only if ϑ < 0. Furthermore, if either d is even or if d is odd and ϑ < 0 then the solvability of Q occurs in the L2 sense.2 Hence, for this class of operators, the local solvability is dictated by the natural generalization of the so-called Nirenberg–Treves condition (ψ) (cf. [8,2]), which in this case reads n ∂ bj (x) sgn ϕ d 0 as a measure. ∂xj j =1
Here sgn(τ ) denotes the sign function, which is defined by sgn(τ ) = 1 if τ > 0, sgn(τ ) = −1 if τ < 0 and sgn(0) = 0. In the present work we shall deepen our analysis for linear partial differential operators in the form P = X ∗ f X − Y + g,
(1)
where now X and Y are real-valued, smooth vector fields defined in an open, connected neighborhood Ω of the origin in Rn , X ∗ denotes the adjoint of X, f is a real-valued, real-analytic function defined in Ω and g is a real-valued, smooth function also defined in Ω. We assume that f does not vanish identically and also that f (0) = 0,
Y = 0 in Ω.
(2)
For operators in the form (1) we have A(x)(ξ ) = −σX (x, ξ )2 , where σX denotes the symbol of X, and then (ii) is always satisfied (recall that σX is purely imaginary). Noticing furthermore that (iv) is equivalent to Xϕ = 0, (iii) is equivalent to n = 2 and X(0) = 0 and (v) is equivalent to Y ϕ(0) = 0, we conclude Corollary 1.1. Suppose that n = 2 and assume that f = ϕ d , for some d ∈ N, where ϕ satisfies (i). Suppose also that X(0) = 0, Y ϕ(0) = 0 and Xϕ = 0. Then the same conclusion as in Theorem 1.1 holds for P . Our goal in the present work is to extend Corollary 1.1 under hypotheses much less restrictive than (i), (iii), (iv) and (v). Although the characterization of the local solvability of P seems to us a rather difficult problem in full generality, we have been able to prove fairly general results that we believe bring a significant understanding to the question. 2 The weak solvability of Q when d = 1 and ϑ < 0 also follows from [1, Theorem 2] in conjunction with [10, Theorem 52.2].
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We summarize the contents of the paper. After a preliminary section where we prove that any partial derivative of the function sgn(f ) is indeed a Radon measure (a result where the assumption of real-analiticity of f plays a crucial role) and study the main properties of such measures, in Section 3 we tackle the study of the L2 -solvability for the operator P . We prove that P is always L2 -solvable near the origin when either f does not change sign or else when Y sgn(f ) 0 as a measure (a property that will be called condition (ψ) in this work) and a weak version of (v) is satisfied. Conversely, in Section 4, we assume that P is real-analytic and prove the necessity of condition (ψ) for the weak solvability of P now under a suitable version of condition (iv). We emphasize that the necessity and the sufficiency of condition (ψ) are proved under different geometrical assumptions. In Section 6 we analyze this gap by presenting the study of a particular but illustrative example. Section 5, in which we relax the assumptions on P , extends the contents of [2, Section 2.2]. We now assume that P is only smooth and prove, by the so-called method of concatenations [4], the local solvability of P when Xf (0) = 0. Here condition (ψ ) is irrelevant and this result is important to illustrate the connection between condition (ψ ) and property (iv) (needless to recall that A(x)(df ) = −σX (x, df )2 = (Xf )2 ). A related example is discussed in Section 6.2. Finally, the authors take the opportunity to express their gratitude to François Treves for his interest in this work and his constant encouragement. 2. Geometrical preliminaries Let f be a real-analytic real-valued function defined in an open neighborhood Ω of the origin in Rn . Assume that f does not vanish identically and that f (0) = 0. Proposition 2.1. Let L be a real vector field on Ω with C 1 coefficients. Then the distribution . μ[L; f ] = L{sgn(f )} is a real Radon measure. Moreover μ[L; f ] = 0 if f does not change sign and f μ[L; f ] = 0, that is, the support of μ[L; f ] is contained in V , the zero set of f . For the proof we recall the following result (cf. [9]): Lemma 2.1. Let p : E → X be a real-analytic morphism between real-analytic spaces such that all fibers p −1 {x} have dimension equal to one. Let also F : E → R be a real-analytic function. Then, given compact sets K2 ⊂ X, K1 ⊂ E there is N = N (K1 , K2 ) such that, for all x ∈ K2 , the restriction of F to p −1 {x} ∩ K1 changes sign at most N times. Proof of Proposition 2.1. We can assume L = ∂/∂x1 . Write the coordinates as x = (x1 , x ), x = (x2 , . . . , xn ) and consider a neighborhood of the origin of the form U = I × B, where I (resp. B) is an open interval (resp. ball) centered at the origin in R (resp. Rn−1 ). Let also K1 (resp. K2 ) be a compact subset of I (resp. B) and set also K = K1 × K2 ⊂ U . If ϕ ∈ Cc∞ (K) then ∂ ∂ϕ x1 , x dx1 dx . ; f (ϕ) = − sgn(f ) x1 , x μ ∂x1 ∂x1
K2
K1
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Applying Lemma 2.1 with the choices p : U → B, p(x1 , x ) = x and F = f we obtain
μ ∂ ; f (ϕ) 2N |K2 | ∂x 1
sup f −1 {0}∩K
|ϕ|.
This inequality shows that μ[∂/∂x1 ; f ] is indeed a (real) Radon measure satisfying the stated properties. 2 We continue the discussion of the measure μ[L; f ]. By contracting Ω about the origin we can assume that df does not vanish in the complement of V . Thanks to this property we can consider the distributions δ(f − σ ), where δ is the Dirac distribution and σ = 0 is real and close to zero. We have (Lf )δ(f − σ ) (ϕ) =
f −1 {σ }
Lf ϕ dHn−1 , |∇f |
ϕ ∈ Cc∞ (Ω),
(3)
where Hn−1 denotes the (n − 1)-Hausdorff measure on f −1 {σ } (cf. [6, Theorem 6.1.5]). Applying a result in [5, Section 1] we obtain, for K ⊂ Ω compact, (Lf )δ(f − σ ) (ϕ) C sup |ϕ|,
ϕ ∈ Cc∞ (K),
(4)
where C > 0 depends on K but not on σ . This shows that the family {(Lf )δ(f − σ )}, σ = 0 small, is bounded in D 0 (Ω), the space of distributions of order 0 defined in Ω. Furthermore, since sgn(f − σ ) → sgn(f ) pointwise in Ω \ V as σ → 0, by the dominated convergence theorem, we obtain sgn(f − σ ) → sgn(f ) in D (Ω) and hence (4) and the argument in the proof of [6, Theorem 2.1.9] show that . μ = μ[L; f ] = lim 2(Lf )δ(f − σ ) in D 0 (Ω).
(5)
σ →0
We further consider the positive measures μ+ = lim 2(Lf )+ δ(f − σ ), σ →0
μ− = lim 2(Lf )− δ(f − σ ), σ →0
|μ| = lim 2|Lf |δ(f − σ ). σ →0
Notice that μ = μ+ − μ− and that |μ| = μ+ + μ− (in other words, (μ+ , μ− ) is the Hahn decomposition of the signed measure μ). We shall now prove the following result. Proposition 2.2. Let F be a compact, subanalytic subset of V of dimension n − 2. Then |μ|(F ) = 0. Proof. Denote by dF the distance function to F and by F ε the set of points x such that dF (x) ε. It is enough to show that |μ|(F ε ) → 0 as ε → 0. Letting ψ(τ ) be a real-valued,
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continuous function on R, such that ψ(τ ) = 1 for 0 τ 1, ψ(τ ) = 0 for τ 2, we have (with an abuse of notation) |μ| F ε |μ| ψ(dF /ε) = 2 lim
σ →0 f −1 {σ }
|Lf | ψ(dF /ε) dHn−1 . |∇f |
Since clearly we have f −1 {σ }
|Lf | n−1 ψ(dF /ε) dH Hn−1 f −1 {σ } ∩ F 2ε , |∇f |
it suffices to show that Hn−1 f −1 {σ } ∩ F 2ε = O(ε)
(6)
uniformly in σ . For this we shall use an integral-geometric estimate [3, 3.2.27]. For each j = 1, . . . , n let pj denote the orthogonal projection pj (x1 , . . . , xn ) = (x1 , . . . , xˆj , . . . , xn ) and let also
H0 f −1 {σ } ∩ F 2ε ∩ pj−1 {w} dw.
aj = Rn−1
Then we have
n
1/2 aj2
n Hn−1 f −1 {σ } ∩ F 2ε aj .
j =1
(7)
j =1
It then suffices to show that aj = O(ε) uniformly in σ , for all j = 1, . . . , n. For simplicity we assume j = 1. Consider the map G : K → Rn , G = (f, p1 ), where K is a compact subanalytic subset of Ω that contains all sets F 2ε , with ε > 0 small. Applying a result in [5, Section 1] we obtain a bound sup H0 G−1 {x} : G−1 (x) is finite < ∞. This means . M = sup H0 f −1 {σ } ∩ K ∩ p1−1 {w} : f −1 {σ } ∩ K ∩ p1−1 {w} is finite < ∞. Applying (7) we obtain a1 Hn−1 f −1 {σ } ∩ F 2ε
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and consequently H0 (f −1 {σ } ∩ F 2ε ∩ p1−1 {w}) < ∞ for w ∈ Bσ,ε , where p1 (F 2ε ) \ Bσ,ε has (n − 1)-dimensional Lebesgue measure equal to zero. But then H0 (f −1 {σ } ∩ F 2ε ∩ p1−1 {w}) M for w ∈ Bσ,ε and so a1 MLn−1 p1 F 2ε . Finally we have p1 (F 2ε ) ⊂ {w: d(w, p1 (F )) 2ε}. Since p1 (F ) is a compact subanalytic subset of Rn−1 and since F has dimension n − 2 it follows that dim p1 (F ) n − 2 and consequently Ln−1 ({w: d(w, p1 (F )) 2ε}) = O(ε) (cf. [3, 3.2.34] in conjunction with [5, Section 3]). 2 It follows from Proposition 2.2 that |μ| = 0 if dim V n − 2. Moreover if we denote by Rn−1 (V ) the set of all points p ∈ V for which V is an (n − 1)-dimensional manifold in a neighborhood of p, and if we set Sn−1 (V ) = V \ Rn−1 (V ), then |μ|(Sn−1 (V )) = 0. Thus we assume dim V = n − 1 and compute the expression of μ± on Rn−1 (V ). We first remark that if we start with Ω sufficiently contracted around the origin, we can decompose Rn−1 (V ) as a finite, disjoint union of connected, real-analytic (n − 1)-dimensional manifolds Nι , each of them containing the origin in its closure (each one of these manifolds is referred to as an (n − 1)-dimensional stratum of V ). For each ι we can take an open connected set Uι containing Nι as a closed set, with Uι ∩ V = Nι , and a real-analytic function ρι defined on Uι such that Nι is defined by ρι = 0 and dρι = 0 on Nι . On Uι we can write f = fι• ριkι , with kι ∈ N and fι• not identically zero in Nι . Notice that either fι• 0 or fι• 0 in Uι and that fι• = 0 on Uι \ Nι . We have μ|Uι = 0 if kι is even. On the other hand, if kι is odd we have sgn(f ) = γ (f, ρι ) sgn(ρι ), where γ (f, ρι ) = 1 if fι• 0 and γ (f, ρι ) = −1 if fι• 0. We have μ|Uι = 2γ (f, ρι )
Lρι dHn−1 |∇ρι |
where now Hn−1 denotes the Hausdorff measure on Nι . We then obtain μ+ |Uι =
2(γ (f, ρι )Lρι )+ dHn−1 , |∇ρι | |μ||Uι =
μ− |Uι =
2(γ (f, ρι )Lρι )− dHn−1 , |∇ρι |
2|Lρι | dHn−1 . |∇ρι |
We shall now take a closer look at the closure (in Ω) of the set where f changes sign: . V0 = Nι .
(8)
kι odd
Notice that V0 is a semianalytic subset of Ω which has dimension n − 1 when f changes sign (otherwise V0 is empty) and that μ[L; f ] is supported in V0 for any C 1 vector field L. Furthermore, if x ∈ / V0 then it is easily seen that f is either 0 or 0 in a neighborhood of x. After a suitable redefinition of the function sgn(f ) on V \ V0 we obtain a new function sgn (f ) which is continuous on Ω \ V0 and satisfies
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sgn (f ) = sgn(f )
−1 sgn (f ) {0} = V0 .
on (Ω \ V ) ∪ V0 ,
(9)
In particular sgn(f ) = sgn (f ) a.e. In order to prepare for the final result of this section we need a simple lemma. Lemma 2.2. Let U be an open and convex subset of Rn and let s : U → {−1, 0, 1} be such that s −1 {−1} and s −1 {1} are open subsets of U . If D denotes the distance function to the closed set . Z = s −1 {0} and if λ = sD then sgn(λ) = s and λ(x) − λ(y) |x − y|,
x, y ∈ U.
(10)
Proof. That sgn(λ) = s is obvious. For the other statement we first point out that (10) is certainly true if s(x) and s(y) have the same sign or else if either s(x) = 0 or s(y) = 0. Suppose then that s(x) = 1 and s(y) = −1. We consider the segment [x, y] joining x to y. Since s −1 {−1} and s −1 {1} are open subsets of U by connectness it follows that [x, y] ∩ Z is nonempty. Let p the nearest point to x belonging to [x, y] ∩ Z and let pj ∈ [x, p[, pj → p. Then s(pj ) = 1 for all j and thus λ(x) − λ(pj ) = D(x) − D(pj ) |x − pj | |x − p|. Likewise let q be the nearest point to y belonging to [x, y] ∩ Z and let qj ∈ ]q, y], qj → q. By the same argument λ(qj ) − λ(y) |q − y| and consequently λ(x) − λ(y) |x − p| + |q − y| + λ(pj ) − λ(qj ) |x − y| + λ(pj ) − λ(qj ). Letting j → ∞ and noticing that λ(pj ), λ(qj ) → 0 we obtain (10).
2
We can now state Proposition 2.3. Let L be a real, smooth vector field in Ω with no critical points and assume that L is transversal to V0 in the following sense: H1 x ∈ γ : D0 (x) ε = O(ε)
(11)
uniformly for an arbitrary orbit γ of L (here D0 denotes the distance function to V0 and H1 denotes the one-dimensional Hausdorff measure on the orbits of L). Then there is a Lipschitz continuous solution to the equation Lh = sgn(f ) in some neighborhood of the origin. . Proof. By Lemma 2.2 the function λ = sgn (f )D0 satisfies (10). We fix a smooth function ψ : R → R satisfying ψ(τ ) = sgn(τ ) for |τ | 1 and set, for ε > 0, ψε (τ ) = ψ(τ/ε). Then ψε (λ) −→ sgn (f ), as ε → 0+ , in D (U ), by the dominated convergence theorem.
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Thanks to the Ascoli–Arzelà theorem, in order to conclude the proof it suffices to show that, in some open and convex neighborhood U of the origin, we can solve Lhε = ψε (λ), where {hε } is a family of Lipschitz continuous functions defined on U satisfying, for some constant C > 0, sup |hε | + sup |∇hε | C, U
ε > 0.
U
After a smooth change of variables we can assume that L = ∂/∂x1 . We shall write x = (x1 , x ), x = (x2 , . . . , xn ) and work in an open neighborhood of the origin of the form U = ]−a, a[ × B Ω, where a > 0 and B is an open ball centered at the origin in Rn−1 . Our transversality hypothesis reads L1 x1 ∈ ]−a, a[: D0 x1 , x ε = O(ε)
(12)
uniformly for x ∈ B. Here now L1 denotes the Lebesgue measure in R. Thus if we form x1 hε (x) =
ψε λ σ, x dσ
−a
we have sup |hε | 2a sup |ψ|, U
∂hε sup |ψ|, sup U ∂x1
and, moreover, (10) and (12) give a ∂hε 1 ψ λ σ, x /ε dσ C supψ ∂x (x) ε k −a
for almost all x ∈ U , all k = 2, . . . , n and some constant C. The proof is complete.
2
Corollary 2.1. Under the same hypotheses as in Proposition 2.3, there is a sequence {hk } ⊂ Cc∞ (Rn ) which converges uniformly in Rn and satisfies the following properties: (1) {hk } is bounded in Cc1 (Rn ); (2) Lhk → sgn (f ) pointwise a.e. in some neighborhood W of the origin in Rn . Proof. Let h be as in the conclusion of Proposition 2.3. After cutting off h near the origin we can take a neighborhood W of the origin in Rn and a compactly supported and Lipschitz continuous function h˜ in Rn such that h˜ = h in W . If {ρε }ε>0 denotes a family of compactly supported . smooth mollifiers and if we set hε = ρε h˜ then {hε }ε>0 is bounded in Cc1 (Rn ), hε → h˜ as + n + p n ˜ ε → 0 uniformly in R and ∂hε /∂xj → ∂ h/∂x j as ε → 0 in L (R ), for all 1 p < ∞ . + and all j = 1, . . . , n. Hence for a conveniently chosen sequence εk → 0 the functions hk = hεk satisfy the required properties. 2
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3. Sufficient conditions for L2 -solvability We recall that we are dealing with a linear partial differential operator in the form P = X ∗ f X − Y + g, where X and Y are real-valued, smooth vector fields defined in an open, connected neighborhood Ω of the origin in Rn , f is a real-valued, real-analytic function defined in Ω and g is a real-valued, smooth function also defined in Ω. We assume that f does not vanish identically and also that f (0) = 0,
Y = 0 in Ω.
The operator P is said to be locally solvable near the origin if there is an open neighborhood U ⊂ Ω of the origin for which P D (U ) ⊃ Cc∞ (U ). A stronger concept is that of L2 -solvability: P is said to be L2 -solvable in the open set U ⊂ Ω if P L2 (U ) ⊃ L2 (U ). In the sequel we shall make use of the following well-known statement: P is L2 -solvable in U if and only if there is a constant C > 0 such that u C P ∗ u,
u ∈ Cc∞ (U ),
where the norm is the L2 -norm and P ∗ denotes the (formal) L2 -adjoint of P . We now start the study of the solvability of P , by first stating a fundamental identity. Proposition 3.1. The following identity holds, for all u ∈ Cc∞ (Ω) and all w ∈ C ∞ (Ω): 1 P ∗ u, wu = (wf Xu, Xu) + 2
P w + g + div(Y ) w |u|2 .
(13)
Here (·,·) denotes the inner product in L2 . Before the proof we first point out that for any real, smooth vector field Z and any real-valued, smooth function ψ we have 2(Zu, ψu) =
∗ 2 Z ψ |u| .
(14)
Proof of Proposition 3.1. It suffices to compute P ∗ u, wu = (wf Xu, Xu) + Xu, (f Xw)u + (Y u, wu) + g + div(Y ) u, wu and observe that (14) yields (Y u, wu) =
1 2
∗ 2 Y w |u| ,
From this (13) follows immediately.
2
1 Xu, (f Xw)u = 2
∗ X (f Xw) |u|2 .
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Remark 3.1. By taking limits it is clear that (13) remains true for w ∈ D (Ω), where now the integrals must be interpretated as duality brackets between Cc∞ (Ω) and D (Ω). Let now φ be a smooth function on Ω, also real-valued. If Mφ denotes the operator “multiplication” by exp{φ} then we have . Pφ = M−φ P Mφ = X ∗ f X − Yφ + gφ
(15)
Yφ = Y + 2f (Xφ)X
(16)
gφ = g − Y φ − (Xf )(Xφ) − f div(X)Xφ + X 2 (φ) + (Xφ)2 .
(17)
where
and
All these formulas follow from direct computation. We will now prove the main result of this section. For this, and following Nirenberg and Treves [8], we introduce the following definition: Definition 3.1. We shall say that P satisfies condition (ψ) if μ[Y ; f ] 0 as a measure.
(18)
Remark 3.2. It is easily seen that (ψ) is invariant under real-analytic changes of coordinates. On the other hand, if h is a real-valued, real-analytic function defined in Ω and nowhere vanishing, then hP = X ∗ (hf )X − hY + f (Xh)X + hg. Since μ hY − f (Xh)X; hf = |h|μ[Y ; f ] − sgn(h)f (Xh)μ[X; f ] = |h|μ[Y ; f ], it follows that (ψ) is also invariant after multiplication of P by a real-analytic factor that never vanishes. Theorem 3.1. Assume that P satisfies condition (ψ) and that Y is transversal to V0 in the sense of Proposition 2.3. Then P is L2 -solvable in some neighborhood of the origin. Proof. We apply (13) with Pφ substituted for P and w = sgn(f ). Since Pφ sgn(f ) = −μ[Y, f ] + gφ sgn(f ) (13) and condition (ψ ) imply Pφ∗ u, sgn(f )u
1 gφ + div(Yφ ) sgn(f )|u|2 . 2
(19)
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Now we have div(Yφ ) = div(Y ) + 2f X(φ) div(X) + 2f ∇(Xφ) · X + 2X(f )X(φ) and then we obtain gφ +
1 1 div(Yφ ) = g + div(Y ) − Y (φ) + f ∇(Xφ) · X − X 2 (φ) − (Xφ)2 . 2 2
Now we take a smooth solution v to the equation Y v = g + (1/2) div(Y ) and set φk = v − hk , where {hk } is the sequence given by Corollary 2.1 when L is replaced by Y . We can of course assume that v is also defined in V , the neighborhood of the origin described in the conclusion of Corollary 2.1. We then apply (19) after replacing φ by φk . Since gφ k +
1 div(Yφk ) = Y hk + f ∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 2
we derive, for u ∈ Cc∞ (V ), Pφ∗k u, sgn(f )u
(Y hk ) sgn(f )|u| − 2
|f |∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 |u|2 .
We then observe that the following property holds: given ε > 0 there is an open neighborhood Vε ⊂ V of the origin such that
|f | ∇(Xφk ) · X − X 2 (φk ) − (Xφk )2 |u|2 −εu2 ,
u ∈ Cc∞ (Vε ), k ∈ N.
Indeed, since f vanishes at the origin, thanks to Corollary 2.1(1) it suffices to show that φ → ∇(Xφ) · X − X 2 (φ) is a first order operator. If we write X = ∇(Xφ) =
aj ∂j we have
(∂j φ)∇(aj ) + aj ∇(∂j φ)
which gives (∂j φ)∇(aj ) · X + aj ∇(∂j φ) · X = aj X(∂j φ) + first order terms =X aj ∂j φ + first order terms
∇(Xφ) · X =
= X 2 (φ) + first order terms, hence our claim.
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Choosing ε = 1/2 we obtain, in some neighborhood W of the origin, Pφ∗k u, sgn(f )u
(Y hk ) sgn(f ) − 1/2 |u|2 ,
u ∈ Cc∞ (W ).
Since the Cauchy–Schwarz inequality and Corollary 2.1(1) give Pφ∗k u, sgn(f )u C P ∗ M−φk uM−φk u, where C > 0 is independent of k, we get
e2φk (Y hk ) sgn(f ) − 1/2 |u|2 CP ∗ uu,
u ∈ Cc∞ (W ).
Letting k → ∞, and taking advantage of the properties (1) and (2) in Corollary 2.1, we finally obtain, with a new constant C > 0, u C P ∗ u, The proof of Theorem 3.1 is complete.
u ∈ Cc∞ (W ).
2
Remark 3.3. It is not difficult to see that Theorem 3.1 remains true even if we replace g by a properly supported pseudo-differential operator of order 0. The argument in the proof requires only small modifications. As a consequence we derive our first positive result. Corollary 3.1. If f keeps constant sign in a neighborhood of the origin (in particular, if the germ of V at the origin has dimension n − 2) then P is L2 -solvable in some neighborhood of the origin. Proof. Indeed, in such situation we have μ[Y ; f ] = 0 and V0 = ∅.
2
As a further consequence we state a generalization of Corollary 1.1: Corollary 3.2. Assume f = ϕ d , where d is odd and ϕ is real-analytic. If Y ϕ < 0 then P is L2 -solvable in some neighborhood of the origin. Proof. Since dϕ(0) = 0 it follows that Y is transversal to V = V0 . Moreover, since also sgn(f ) = sgn(ϕ), we have μ[Y ; f ] = μ[Y ; ϕ] =
Yϕ dHn−1 , |∇ϕ|
where Hn−1 denotes the (n − 1)-dimensional Hausdorff measure on the regular set V . But then condition (ψ) holds and the result follows. 2
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4. A partial converse to Theorem 3.1 If (ψ) is not satisfied in an open neighborhood U ⊂ Ω of the origin then there is ϕ ∈ Cc (U ), ϕ 0 such that μ(ϕ) = μ[Y, f ](ϕ) > 0. By Proposition 2.2 we have
ϕ dμ =
V
ϕ dμ =
ι
Rn−1 (V )
ϕ dμ > 0
Nι
and then there must exist an (n − 1)-dimensional stratum Nι such that ϕYρι dHn−1 > 0. ϕ dμ = γ (f, ρι ) |∇ρι | Nι
Nι
If we now assume that condition (ψ) is not satisfied in any neighborhood of the origin then the preceding argument allows us to assert the existence of an (n − 1)-dimensional stratum Nι and of a sequence of nonnegative functions {ϕj } ⊂ Cc (U ) such that supp ϕj → {0} and γ (f, ρι ) Nι
ϕj Yρι dHn−1 > 0, |∇ρι |
∀j.
We define Aι as being the open subset of Nι formed by all points p ∈ Nι such that Y (p) is transversal to Nι and f changes sign at p from − to + along the orbit of Y through p. Of course we have Aι = p ∈ Nι : γ (f, ρι )Yρι (p) > 0 . We can then state Lemma 4.1. If condition (ψ) is not satisfied in any neighborhood of the origin then there is an (n − 1)-dimensional stratum Nι of V such that Aι contains the origin in its closure. We are now in a position to prove a partial converse to Theorem 3.1. Theorem 4.1. Assume that P is real-analytic and that condition (ψ) is not satisfied in any neighborhood of the origin and let Nι be an (n − 1)-dimensional stratum of V such that Aι contains the origin in its closure. If X|Nι is tangent to Nι and not identically zero then P is not solvable near the origin. Proof. Since {p ∈ Nι : X(p) = 0}, being the complement of a proper analytic subset of Nι , is dense in Nι , any arbitrary open neighborhood U of the origin contains a point p0 ∈ Nι such that γ (f, ρι )Yρι (p0 ) > 0,
fι• (p0 ) = 0,
X(p0 ) = 0.
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We then choose local coordinates (x1 , . . . , xn ) centered at p0 and defined on an open ball B0 in such a way that we can write on B0 : B0 ∩ Nι = B0 ∩ V = {x1 = 0},
f = x12+1 f • ,
Y = ∂/∂x1 ,
f • > 0.
(20)
We then write X=
aj (x)∂xj .
j 1
Since X is tangent to S, we must have a1 (0, x ) = 0 for every x = (x2 , . . . , xn ) near the origin; then we can write a1 = x1 a˜ 1 . Let us now take a real-analytic function v = exp(h), where h solves Xh = −a˜ 1 . This is possible since X does not vanish near p0 . Then Xv + a˜ 1 v = 0 and the change of variables y1 = x1 v(x),
yj = xj ,
j = 2, . . . , n
takes now ∂x1 into (v + x1 vx1 )∂y1 and ∂xj into ∂yj + x1 vxj ∂y1 . The vector field X then becomes
X = a1 (v + x1 vx1 ) + x1 aj vxj ∂x1 + aj ∂yj j 2
= x1 [a˜ 1 v + Xv]∂y1 + =
j 2
aj ∂yj
j 2
aj ∂yj
j 2
and thus P=
j 2
αj ∂yj
∗ f αj ∂yj − β∂y1 + γ . j 2
Dividing by β, whose value at the origin is equal to one, we then have that ∗ • 1 f γ P = y12+1 αj ∂yj αj ∂yj − ∂y1 + bj ∂yj + , β vβ β j 2
j 2
j 2
which is not solvable near p0 thanks to [2, Theorem 3.1].3 It then follows that P is not solvable in U , and this concludes the proof of Theorem 4.1. 2 Again we derive a particular case in order to compare it with Corollary 1.1. 3 What is needed here is indeed a slightly more general version of this result. An analysis of the proof of Theorem 3.1 in [2] shows that, when = 1, there is nonsolvability with the only assumption that there exists ξ ∈ Rn \ {0} such that Q(0, 0)(ξ, ξ ) > 0.
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Corollary 4.1. Assume that P is real-analytic and that f = ϕ d , where d is odd and ϕ is realanalytic. If X(0) = 0, X(ϕ) = O(ϕ) and Y ϕ > 0 then P is not solvable near the origin. Indeed, as in the proof of Corollary 3.2, the condition Y (ϕ) > 0 implies that (ψ) is not satisfied on any neighborhood of the origin. 5. Solvability when (ψ) is not necessarily satisfied In this section we prove Theorem 5.1 below. Since its proof only requires that P be smooth, this result generalizes the contents in [2, Section 2.2]. Theorem 5.1. Assume that X, Y, f and g are smooth in Ω. If X(f )(0) = 0 then P is locally solvable near the origin. Proof. We can choose, thanks to our hypotheses, local coordinates (y, t), y = (y1 , . . . , ym ), n = m + 1, such that f (y, t) = t and X = ∂/∂t. The key ingredient in the proof is the so-called “method of concatenations”. It is convenient to write P in the form P = Dt tDt + iB, where B = B(y, t; Dy , Dt ) is a differential operator of order 1 with real principal part. . We set Λ = (1 + Dt2 )1/2 . Making use of the elementary fact that for χ(Dt ), a pseudodifferential operator in R with symbol χ(τ ), we have t, χ(Dt ) = iχ (Dt ),
(21)
then −1/2 Dt tDt Λ = ΛDt tDt + iDt3 1 + Dt2 and thus, setting T = Dt2 (1 + Dt2 )−1 , we can write P Λ = ΛP1 , with P1 = Dt tDt + iT Dt + iΛ−1 BΛ. Iterating the process we can write P Λk = Λk Pk , where Pk = Dt tDt + ikT Dt + iΛ−k BΛk .
(22)
Since the local solvability of P is equivalent to that of Pk it suffices to show that if k is sufficiently large then Pk is L2 -solvable in a small neighborhood U of the origin. We shall assume that U = W × ]−δ, δ[, where W is an open neighborhood of the origin in Rm , and that the coefficients of B are defined in an open set that contains the closure of U .
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Let u ∈ Cc∞ (U ). We write Pk∗ u, tu = (Dt tDt u, tu) − k(iT Dt u, tu) − iΛk B ∗ Λ−k u, tu .
(23)
Now, an elementary computation gives 2(Dt tDt u, tu) = 2tDt u2 − u2
(24)
2 2(−iT Dt u, tu) = i T Dt (tu) − tT Dt (u) , u = ψ(Dt )u ,
(25)
and we also have
where ψ(Dt )2 = i[T Dt , t]. By the remark above ψ(τ ) = τ
(3 + τ 2 )1/2 . (1 + τ 2 )
Lemma 5.1. If δ > 0 is small enough then there is c > 0 such that ψ(Dt )u2 cu2 ,
u ∈ Cc∞ (U ).
(26)
Proof. We can write ψ(Dt ) = Dt ψ1 (Dt ) = ψ1 (Dt )Dt where, for every s, ψ1 (Dt ) defines isomorphisms ψ1 (Dt ) : H s (R) → H s+1 (R). Denoting by | · |s the norm in H s (R), it follows the existence of c1 > 0 such that ψ(Dt )v c1 |Dt v|−1 , 0
v ∈ S(R).
On the other hand, the ellipticity of Dt on R implies that, if δ > 0 and c2 > 0 are small enough, |Dt v|−1 c2 |v|0 ,
v ∈ Cc∞ ] − δ, δ[ .
Hence ψ(Dt )v c1 c2 |v|0 , 0
v ∈ Cc∞ ] − δ, δ[ .
Applying this inequality to v = u(y, ·), where u ∈ Cc∞ (U ), and integrating in y ∈ Rm , imply (26). 2 From (23), (24), (25) and (26) we derive 2 Pk∗ u, tu (ck − 1)u2 − 2 iΛk B ∗ Λ−k u, tu . Let us now write B=
m j =1
aj (y, t)Dyj + a0 (y, t)Dt + g(y, t),
(27)
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where aj are real-valued for j = 0, 1, . . . , m. After multiplication by a cut-off function we can assume that aj and b are defined in the whole Rn and have compact support. In order to study the last term in the right-hand side of (27) we first notice that iΛk Dyj aj Λ−k u, tu = (iDyj Sk,j u, u), where Sk,j = Λk (taj + M)Λ−k and M is any real constant4 . Next select M > 0 such that taj + M 1. With such a choice we can write Sk,j = Λk (taj + M)1/2 Λ−k 1/2
and consequently 1/2 1/2 1/2 1/2 iΛk Dyj aj Λ−k u, tu = iDyj Sk,j u, Sk,j u + i Dyj , Sk,j Sk,j u, u 1/2 1/2 = i Dyj , Sk,j Sk,j u, u . Notice that the operator Sk,j (y, t; Dt ) is a pseudo-differential operator in t of order 0, depending 1/2 1/2 smoothly in y, and thus bounded in L2 , and that the same is true for [Dyj , Sk,j ]Sk,j . Moreover, the pseudo-differential calculus gives 1/2 1/2 Dyj , Sk,j Sk,j = hj (y, t) + Rj,k (y, t; Dt ), where hj (y, t) is a smooth function and Rj,k (y, t; Dt ) is a pseudo-differential operator in t of order −1, depending smoothly in y. Hence there are constants C > 0 (which is independent of k and j ) and Ck > 0 (which is independent of j ) such that k iΛ Dy aj Λ−k u, tu Cu2 + Ck u0,−1 u, j
(28)
where · 0,−1 denotes the norm in L2 (Rm , H −1 (R)). Next we have 2 iΛk Dt a0 Λ−k u, tu = i tΛk Dt a0 Λ−k − Λ−k a0 Dt Λk t u, u . By the pseudo-differential calculus, tΛk Dt a0 Λ−k − Λ−k a0 Dt Λk t is a pseudo-differential operator in Dt of order zero which can be written as
∂a0 k −k −k k itΛ Dt a0 Λ − iΛ a0 Dt Λ t = t − a0 + tRk,0 (y, t; Dt ) + Rk,−1 (y, t; Dt ), ∂t where Rk,0 (y, t; Dt ) and Rk,−1 (y, t; Dt ) are pseudo-differential operators in Dt of order zero and −1 respectively, depending on k. Hence increasing the constants C > 0 and Ck > 0 if necessary we can write k iΛ Dt a0 Λ−k u, tu Cu2 + Ck δu2 + u0,−1 u . 4 Here we use the elementary fact that (iD u, u) = 0 for every u ∈ S. yj
(29)
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Finally, by a very similar argument, we can write k −k iΛ gΛ u, tu Ck δu2 ,
(30)
where the constants Ck > 0 are again appropriately increased if necessary. Putting together (28), (29) and (30) and giving ε > 0 arbitrary we obtain, with new constants C > 0 and Ck > 0 and δ > 0 appropriately small, 2 iΛk B ∗ Λ−k u, tu Cu2 + εCk u2 ,
u ∈ Cc∞ (U ).
(31)
Inserting (31) in (27) gives 2 Pk∗ u, tu (ck − 1 − C − εCk )u2 .
(32)
Choosing k such that ck − 1 − C 1 and then ε 1/(2Ck ), (32) finally yields 4 Pk∗ u, tu u2 ,
u ∈ Cc∞ (U ),
(33)
for some δ > 0 chosen conveniently small. Hence, (33) together with the Cauchy–Schwarz inequality implies the L2 solvability of Pk . The proof of Theorem 5.1 is now complete. 2 6. Final remarks 1. As already mentioned in the Introduction the necessity and the sufficiency of condition (ψ) are proved under different hypotheses. Since filling this gap seems to be a quite difficult question, we limit ourselves to a much more modest task. We assume that P has smooth coefficients and work under the condition X(0) = 0
and Yf > 0 on V
(34)
(recall that V denotes the zero set of the function f ). We provide an answer to a very particular case. Theorem 6.1. Assume that n = 2 and that condition (34) holds. Then P is locally solvable at the origin if and only if Xf (0) = 0. We sketch the proof. Thanks to Theorem 5.1 it suffices to show that P is not locally solvable near the origin when Xf (0) = 0. We start by selecting local coordinates (y, t) near the origin such that Y = ∂/∂y. We then have fy (0, 0) = 0,
Yf (0, 0) > 0.
If we write Y = b(y, t)∂y + c(y, t)∂t
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we must have c(0, 0)ft (0, 0) > 0 and then replacing P by −P /c allows us to reduce the proof of Theorem 6.1 to the proof of nonsolvability for the operator P = f (y, t)∂y2 + ∂t + b(y, t)∂y + g(y, t), where fy (0, 0) = 0, ft (0, 0) > 0. After a factorization we write P in the form P = t − γ (y) q(y, t)∂y2 + ∂t + b(y, t)∂y + g(y, t),
(35)
with γ (0) = γ (0) = 0.
q(0, 0) > 0,
(36)
Notice that the case γ ≡ 0 follows from Corollary 4.1 (and is also a direct consequence of Theorem 3.1 in [2]). In the sequel we show that a slight modification of the arguments in [2, Theorem 3.1], allows us to complete the proof of Theorem 6.1. We first observe that P ∗ = t − γ (y) q(y, t)∂y2 − ∂t + b• (y, t)∂y + g • (y, t) for conveniently defined real-analytic functions b• and g • . We then look for quasi-solutions of P ∗ u = 0 of the form uλ (y, t) = eiλΦ(y,t) K(y, t). We have e−iλΦ(y,t) P ∗ uλ = −Kt − λΦt K + b• (Ky + iλΦy K) + g • K + t − γ (y) q(y, t) Kyy − 2iλΦy Ky + iλΦyy K − λ2 Φy2 K
(37)
and consequently, after performing the change of variables z = λ5/2 y,
s = λ3 t,
we see from (37) that we must have −3 λ s − γ˜ (z) q(z, ˜ s) λ5 K˜ zz − 2iλ6 Φ˜ z K˜ z + iλ6 Φ˜ zz K˜ − λ7 Φ˜ z2 K˜ − λ3 K˜ s − λ4 Φ˜ s K˜ + b˜ • λ5/2 K˜ z + iλ7/2 Φ˜ z K˜ + g˜ • K˜ = 0,
(38)
˜ K˜ etc. to denote the functions in the new variables. Furthermore, as in where we used Φ, [2, p. 116], we shall take K˜ to be a formal series of the form ˜ s) = K(z,
j 0
kj (z, s)λ−j/2 .
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Thus, after expanding the coefficients γ˜ , q, ˜ b˜ • , g˜ • as power series in (λ−5/2 z, λ−3 s), and observing that (36) gives γ˜ =
∞
γ (j ) (0)λ−5j/2 zj /j !
j =2
we see that the term in (38) with biggest power in λ is given by (−s q(0, ˜ 0)Φ˜ z2 − iΦs )λ4 . From this we obtain the eikonal equation −s q(0, ˜ 0)Φ˜ z2 − iΦs = 0, which we solve under the initial condition Φ(z, 0) = iz + z2 . It is worth mentioning that this Cauchy problem is exactly the same as that arising in [2, p. 116], where it is proved that, near the origin, its solution has imaginary part of the form Φ(z, s) = z2 +
2 q(0, 0) 2 s + O (z, s) . 2
From now on the argument can be completed, with only minor modifications, as in [2, pp. 116–119], for the coefficients kj can recursively be found by solving the usual “transport equations”. The details are left to the interested reader. 2. Finally we discuss a natural question that can be raised after Theorem 5.1. Assume that V is a regular hypersurface. Does the transversality of X to V near the origin imply local solvability? The answer is negative in general, as it can be seen by the following example. Consider the operator in R2 in coordinates (y, t): M,a = ∂t∗ t 3 ∂t − ∂t + a∂y , where = ±1 and a ∈ R. Since this operator is locally L2 -solvable at the origin when = −1 (cf. Theorem 3.1) we limit ourselves to the case = 1. We have Proposition 6.1. M1,a is locally solvable near the origin if and only if a = 0. Proof. We first observe that M1,0 = −∂t t 3 ∂t + 1 . =L and that, by (14), 2 L∗ u, u = 2u2 + 2 t 3 u, ut = 2u2 − 3 if the diameter of U in the t-direction is
t 2 |u|2 u2 ,
u ∈ Cc∞ (U )
√ 3/3. Hence M1,0 is solvable near the origin.
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We now show that M1,a is not solvable near the origin for a a = 0. After scaling the y variable we can reduce to the case when a = 1. We proceed as in the proof of Theorem 6.1, that is, we ∗ u = 0 of the form start by finding formal solutions to the equation M1,1 λ uλ (y, t) = eiλφ(y,t) K(y, t). A simple computation then gives e−iλφ M ∗ uλ = −t 3 Ktt + 2iλφt Kt − (λφt )2 K + iλφtt K + 1 − 3t 2 {iλφt K + Kt } − iλφy K − Ky = 0. We now make the change of variables s = λ1/2 t, z = λ1/2 y. In the new variables we must then have 3/2 ˜ + K˜ s − K˜ z − 3is 2 φ˜ s K˜ − is 3 φ˜ ss K˜ − 2is 3 φ˜ s K˜ s λ1/2 i φ˜ s − i φ˜ z + s 3 φ˜ s2 Kλ − s 3 K˜ ss + 3s 2 K˜ s λ−1/2 = 0, ˜ K˜ denote the functions in the new variables. If we choose a real-analytic funcwhere again φ, ˜ tion φ φ˜ s − φ˜ z − is 3 φ˜ s2 = 0
(39)
and take K˜ of the form ˜ s) = K(z,
kj (z, s)λ−j/2
j 0 ∗ u = 0 can be found by solving recursively the it is easily seen that a formal solution to M1,1 λ equations
Lkj = 0, Lkj = s 3 ∂s2 kj −2
j = 0, 1,
+ 3s ∂s kj −2 , 2
(40) j 2,
(41)
where we have written L = 1 − 2is 3 φ˜ s ∂s − ∂z − i 3s 2 φ˜ s + s 3 φ˜ ss . Equations (40) are solved under the conditions kj (0, s) = 1 (j = 0, 1) whereas equations (41) ˜ s) = under the conditions kj (0, s) = 0, j 2. As far as (39) is concerned we write φ(z, j and solve it under the condition φ (z)s j 0 j ˜ 0) = φ0 (z) = z + iz2 . φ(z, After computing the first terms explicitly φ1 (z) = 1 + 2iz,
φ2 (z) = i,
φ3 (z) = 0,
φ4 (z) = i(1 + 2iz)2 /4,
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we conclude that ˜ s) = (z + s)2 + s 4 /4 − z2 s 4 + O s 5 , φ(z, which shows that φ˜ has a local strict minimum at the origin. At this point we can refer again to [2, pp. 116–119], since the conclusion of the argument is now completely standard. 2 References [1] R. Beals, C. Fefferman, On hypoellipticity of second order equations, Comm. Partial Differential Equations 1 (1976) 73–85. [2] F. Colombini, L. Pernazza, F. Treves, Solvability and nonsolvability of second-order evolution equations, in: Hyperbolic Problems and Related Topics, in: Grad. Ser. Anal., Int. Press, Somerville, MA, 2003, pp. 111–120. [3] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss., vol. 153, Springer-Verlag, 1996 (reprint of the 1969 edition). [4] A. Gilioli, F. Treves, An example in the solvability theory of linear PDE’s, Amer. J. Math. 96 (1974) 367–385. [5] R. Hardt, Some analytic bounds for subanalytic sets, in: R. Brockett, R. Millman, H. Sussmann (Eds.), Differential Geometric Control Theory, in: Progr. Math., vol. 27, Birkhäuser, 1983, pp. 259–267. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Grundlehren Math. Wiss., vol. 256, SpringerVerlag, 1983. [7] Y. Kannai, An unsolvable hypoelliptic differential operator, Israel J. Math. 9 (1971) 306–315. [8] L. Nirenberg, F. Treves, On local solvability of linear partial differential equations I, Necessary conditions, Comm. Pure Appl. Math. 23 (1970) 1–38. [9] B. Teissier, Appendice: sur trois questions de finitude en géométrie analytique réelle, Acta Math. 51 (1983) 39–48. [10] F. Treves, Topological Vector Spaces, Distributions and Kernels, Pure Appl. Math., vol. 25, Academic Press, 1967.
Journal of Functional Analysis 258 (2010) 3492–3516 www.elsevier.com/locate/jfa
Cameron–Martin formula for the σ -finite measure unifying Brownian penalisations Kouji Yano Graduate School of Science, Kobe University, Kobe, Japan Received 30 September 2009; accepted 25 November 2009 Available online 9 December 2009 Communicated by Paul Malliavin
Abstract Quasi-invariance under translation is established for the σ -finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ -finite measure on C(R+ , R) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic)], plays a key role. © 2009 Elsevier Inc. All rights reserved. Keywords: Cameron–Martin formula; Quasi-invariance; Penalisation; Wiener integral
1. Introduction Let Ω = C([0, ∞) → R). Let (Xt : t 0) denote the coordinate process and set F∞ = σ (Xt : t 0). We consider the following σ -finite measure on (Ω, F∞ ): ∞ W = 0
du Π (u) • R √ 2πu
where Π (u) • R is given as follows: E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.021
(1.1)
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(i) Π (u) denotes the law of the Brownian bridge from 0 to 0 of length u; (ii) R denotes the law of the symmetrized 3-dimensional Bessel process; (iii) Π (u) • R denotes the concatenation of Π (u) and R. This measure W has been introduced by Najnudel, Roynette and Yor [11,12] so that it unifies various Brownian penalisations. The Brownian penalisations can be explained roughly as follows (we will discuss details in Section 2): For a “good” family {Γt (X)} of non-negative F∞ functionals such that Γt (X) → Γ (X) as t → ∞, it holds that πt (1.2) W Fs (X)Γt (X) −→ W Fs (X)Γ (X) t→∞ 2 for any bounded Fs -measurable functional Fs (X). The purpose of this paper is to establish quasi-invariance of W under h-translation when h belongs to the Cameron–Martin type space:
t h ∈ Ω: ht =
f (s) ds for some f ∈ L (ds) ∩ L (ds) . 2
1
(1.3)
0
Now we state our main theorem. t Theorem 1.1. Suppose that ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then, for any nonnegative F∞ -measurable functional F (X), it holds that W F (X + h) = W F (X)E(f ; X)
(1.4)
where ∞ E(f ; X) = exp
1 f (s) dXs − 2
0
∞ 2
f (s) ds .
(1.5)
0
Theorem 1.1 will be proved in Section 4. ∞ Theorem 1.1 involves Wiener integral, i.e., the stochastic integral 0 f (s) dXs of a deterministic function f . (To avoid confusion, we give the following remark: In [3,4], the Wiener integral means the integral with respect to the Wiener measure.) The author has proved in his recent work [18] that this Wiener integral is well defined if f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), i.e., ∞ ∞ f (s) 2 ds + f (s) 0
0
ds √ < ∞. 1+ s
(1.6)
Note the obvious inclusion: L1 (ds) ⊂ L1 ( 1+ds√s ). We will discuss details in Section 3. One may t conjecture that Theorem 1.1 is valid for ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), but we have not succeeded at this point. We give several remarks which help us to understand Theorem 1.1 deeply.
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1◦ ). Rephrasing the main theorem. Let g(X) denote the last exit time from 0 for X: g(X) = sup{u 0: Xu = 0}.
(1.7)
For u 0, let θu X denote the shifted process: (θu X)s = Xu+s , s 0. Then the definition (1.1) says that the measure W can be described as follows: (i) W (g(X) ∈ du) = √du ; 2πu (ii) For (Lebesgue) a.e. u ∈ [0, ∞), it holds that, given g(X) = u, (iia) (Xs : s u) is a Brownian bridge from 0 to 0 of length u; (iib) ((θu X)s : s 0) is a symmetrized 3-dimensional Bessel process. In the same manner as this, Theorem 1.1 can be rephrased as the following corollary. We write Th∗ W for the image measure of X + h under W . For u ∈ [0, ∞), we define u Eu (f ; X) = exp 0
Corollary 1.2. Suppose that ht = Th∗ W
t 0
1 f (s) dXs − 2
u 2
f (s) ds .
(1.8)
0
f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then it holds that ∞
=
du ρ f (u)Π (u),f • R f (·+u)
(1.9)
0
where
1 ρ f (u) = √ Π (u) Eu (f ; ·) R E f (· + u); · , 2πu Π (u),f (dX) = R f (·+u) (dX) =
(1.10)
Eu (f ; X)Π (u) (dX) , Π (u) [Eu (f ; ·)]
(1.11)
E(f (· + u); X)R(dX) . R[E(f (· + u); ·)]
(1.12)
In other words, the law of the process X + h under W may be described as follows: (i ) W (g(X + h) ∈ du) = ρ f (u) du; (ii ) For a.e. u ∈ [0, ∞), it holds that, given g(X + h) = u, (iia ) (Xs + hs : s u) has law Π (u),f ; (iib ) ((θu (X + h))s : s 0) has law R f (·+u) . 2◦ ). Sketch of the proof. We will divide the proof of Theorem 1.1 into the following steps: Step 1. W [F (X + h·∧T )] = W [F (X)ET (f ; X)] for 0 < T < ∞; Step 2. W [F (X)ET (f ; X)] → W [F (X)E(f ; X)] as T → ∞;
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Step 3. W [F (X + h·∧T )] → W [F (X + h)] as T → ∞. Note that, in Steps 2 and 3, we will confine ourselves to certain particular classes of test functions F . One may think that Step 1 should be immediate from the following rough argument using (1.2): For any “good” Fs -measurable functional Fs (X), W Fs (X + h·∧T )Γ (X + h·∧T ) = lim
t→∞
πt W Fs (X + h·∧T )Γt (X + h·∧T ) 2
πt W Fs (X)ET (f ; X)Γt (X) = lim t→∞ 2 = W Fs (X)ET (f ; X)Γ (X) .
(1.13) (1.14) (1.15)
This observation, however, should be justified carefully, because the functional ET (f ; X) is not bounded. We shall utilize Markov property for {(Xt ), W } (see Section 2.4 for the details): W FT (X)G(θT X) = W FT (X)WXT G(·)
(1.16)
where Wx is the image measure of x + X under W (dX). The identity (1.16) suggests, in a way, that {Wx : x ∈ R} is a family of exit laws whose transition up to finite time is the Brownian motion, while the Markov property of the Brownian motion asserts that W FT (X)G(θT X) = W FT (X)WXT G(·) .
(1.17)
This makes a remarkable contrast with Itô’s excursion law n (see [8]), which satisfies the Markov property: n FT (X)G(θT X) = n FT (X)WX0 T G(·)
(1.18)
where {(Xt ), (Wx0 )} denotes the Brownian motion killed upon hitting the origin. In other words, n produces a family of entrance laws whose transition after positive time is the killed Brownian motion. ∞ We remark again that the Wiener integral 0 f (s) dXs is not Gaussian. In order to prove necessary estimates involving Wiener integrals in Step 2, we utilize the theory of Wiener integrals for centered Bessel processes, which is due to Funaki, Hariya and Yor [5]. For the 3-dimensional Bessel process {(Xt ), Ra+ } starting from a 0, we define t(a) = Xt − Ra+ [Xt ] X
(1.19)
t ), Ra+ } the centered Bessel process. We shall apply, to the convex function and call {(X |x| ψ(x) = (e − 1)2 , the following theorem, which was proved by Funaki, Hariya and Yor [5] via Brascamp–Lieb inequality [2], and from which we derive our necessary estimates. (a)
Theorem 1.3. (See [5].) For any f ∈ L2 (ds) and any non-negative convex function ψ on R, it holds that
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Ra+
∞
∞
(a) t f (t) dX f (t) dXt . ψ W ψ 0
(1.20)
0
For the proof of Theorem 1.3, see [5, Proposition 4.1]. 3◦ ). Comparison with the Brownian case. Let us recall the well-known Cameron–Martin formula for Brownian motion (see [3,4]). Let W stand for the Wiener measure on Ω with W (X0 = 0) = 1. t It is well known that, if ht = 0 f (s) ds with f ∈ L2 (ds), W F (X + h) = W F (X)E(f ; X)
(1.21)
for any non-negative F∞ -measurable functional F (X). It is also well known that, if h ∈ / H , the to W (dX). image measure of X + h under W (dX) is mutually singular on F ∞ t It is immediate from (1.21) that, if ht = 0 f (s) ds with f ∈ L2loc (ds), then W Ft (X + h) = W Ft (X)Et (f ; X)
(1.22)
for any non-negative Ft -measurable functional Ft (X) where t Et (f ; X) = exp
1 f (s) dXs − 2
0
t
f (s)2 ds .
(1.23)
0
Now we give some remarks about comparison between the two cases of W and W . (i) Let f ∈ L2 (ds). As a corollary of (1.21), we see that W [E(f ; X)] < ∞ and, consequently, that W [E(f ; X)p ] < ∞ for any p 1. This shows that, if F (X) ∈ Lp (W (dX)) for some p > 1, then F (X + h) ∈ L1 (W (dX)). Let f ∈ L2 (ds) ∩ L1 (ds). In the case of W , however, we see immediately by taking F ≡ 1 in (1.4) that W E(f ; X) = ∞,
(1.24)
which we should always keep in mind. Now the following question arises: W E(f ; X)Γ (X) < ∞
(1.25)
holds for what functional Γ (X)? The problem is that we do not know the distribution of the ∞ Wiener integral 0 f (s) dXs under W ; in fact, it is no longer Gaussian! In Theorem 4.2, we will appeal to a certain penalisation result and establish (1.25) for Feynman–Kac functionals Γ (X), the class of which we shall introduce in Section 2.2. (ii) In the Brownian case, we have the following criterion: The h-translation of W is quasi/ L2 (ds), respectively. invariant or singular with respect to W according as h ∈ L2 (ds) or h ∈ In the case of W , however, we do not know what happens on W (dX) when h ∈ / H or when f∈ / L1 (ds).
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(iii) Let f ∈ L2loc (ds). In the Brownian case, we have the quasi-invariance (1.22) on each Ft . In the case of W (dX), however, we find a drastically different situation (see Theorem 2.5): For any non-negative Ft -measurable functional Ft (X), W Ft (X + h) = W Ft (X) = 0 or ∞
(1.26)
according as W (Ft (X) = 0) = 1 or W (Ft (X) = 0) < 1. 4◦ ). Integration by parts formulae. From the Cameron–Martin theorem (1.21) in the Brownian case, we immediately obtain the following integration by parts formula: ∞ W ∇h F (X) = W F (X) f (s) dXs
(1.27)
0
t
for ht = 0 f (s) ds with f ∈ L2 (ds) and for any good functional F (X), where ∇ denotes the Gross–Sobolev–Malliavin derivative (see, e.g., [17]). In the case of W , from Theorem 1.1, we may expect the following integration by parts formula: ∞ W ∂h F (X) = W F (X) f (s) dXs
(1.28)
0
t for ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds) and for any good functional F (X), where ∂h is in the Gâteaux sense. We have not succeeded in finding a reasonable class of functionals F for which both sides of (1.28) make sense and coincide. Let us give a remark about 3-dimensional Bessel bridge of length u from 0 to 0, which we denote by {(Xs : s ∈ [0, u]), R +,(u) }. Although we do not have the Cameron–Martin formula for the bridge, there is a remarkable result due to Zambotti [20,21] that the following integration by parts formula holds:
R
+,(1)
∞ +,(1) ∂h F (X) = R F (X) f (s) dXs + (BC)
(1.29)
0
t for ht = 0 f (s) ds with f satisfying a certain regularity condition and for any good functional F (X), where ∂h is in the Gâteaux sense and where 1
(BC) = − 0
du hu 2πu3 (1 − u)3
+,(u)
R • R +,(1−u) F (·) .
(1.30)
The remainder term (BC) may describe the boundary contribution. Indeed, the measure R +,(1) is supported on the set of non-negative continuous paths on [0, 1], while the measure R +,(u) • R +,(1−u) is supported on the subset of paths which hit 0 once and only once; the latter set may be regarded in a certain sense as the boundary of the former. See also Bonaccorsi and
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Zambotti [1], Zambotti [22], Hariya [7] and Funaki and Ishitani [6] for similar results about integration by parts formulae. The organization of this paper is as follows. In Section 2, we recall several results of Brownian penalisations. In Section 3, we study Wiener integrals for the processes considered. Section 4 is devoted to the proofs of our main theorems. 2. Brownian penalisations 2.1. Notations Let X = (Xt : t 0) denote the coordinate process of the space Ω = C([0, ∞); R) of continuous functions from [0, ∞) to R. Let Ft = σ (Xs : s t) for 0 < t < ∞ and F∞ = σ ( t Ft ). For 0 < u < ∞, we write X (u) = (Xt : 0 t u) and Ω (u) = C([0, u]; R). 1◦ ). Brownian motion. For a ∈ R, we denote by Wa the Wiener measure on Ω with Wa (X0 = a) = 1. We simply write W for W0 . 2◦ ). Brownian bridge. We denote by Π (u) the law on Ω (u) of the Brownian bridge: Π (u) (·) = W (·|Xu = 0).
(2.1)
The process X (u) under Π (u) is a centered Gaussian process with covariance Π (u) [Xs Xt ] = s − st/u for 0 s t u. As a realization of {X (u) , Π (u) }, we may take s Bs − Bu : s ∈ [0, u] . u
(2.2)
3◦ ). 3-Dimensional Bessel process. For a 0, we denote by Ra+ the law √ on Ω of the 3-dimensional Bessel process starting from a, i.e., the law of the process ( Zt ) where (Zt ) is the unique strong solution to the stochastic differential equation dZt = 2 |Zt | dβt + 3 dt,
Z0 = a 2
(2.3)
with (βt ) a one-dimensional standard Brownian motion. Under Ra+ , the process X satisfies dXt = dBt +
1 dt, Xt
X0 = a
(2.4)
with {(Bt ), Ra+ } a one-dimensional standard Brownian motion. − For a > 0, we denote by R−a the law on Ω of (−Xt ) under Ra+ . We define Ra = and
Ra+ Ra−
if a > 0, if a < 0
(2.5)
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R = R0 =
R0+ + R0− ; 2
3499
(2.6)
in other words, R is the law on Ω of (εXt ) under the product measure P (dε) ⊗ R0+ (dX) where P (ε = 1) = P (ε = −1) = 1/2. 4◦ ). The σ -finite measure W . For u > 0 and for two processes X (u) = (Xt : 0 t u) and Y = (Yt : t 0), we define the concatenation X (u) • Y as ⎧ ⎨ Xt (u)
X • Y t = Yt−u ⎩ Xu
if 0 t < u, if t u and Xu = Y0 , if t u and Xu = Y0 .
(2.7)
We define the concatenation Π (u) • R as the law of X (u) • Y under the product measure Π (u) (dX (u) ) ⊗ R(dY ). Then we define ∞ W = 0
du Π (u) • R. √ 2πu
(2.8)
For x ∈ R, we define Wx as the image measure of x + X under W (dX); in other words, Wx F (X) = W F (x + X)
(2.9)
for any non-negative F∞ -measurable functional F (X). 5◦ ). Random times. For a ∈ R, we denote the first hitting time of a by τa (X) = inf{t > 0: Xt = a}.
(2.10)
We denote the last exit time from 0 by g(X) = sup{t 0: Xt = 0}.
(2.11)
2.2. Feynman–Kac penalisations y
Let Lt (X) denote the local time by time t of level y: For Wx (dX)-a.e. X, it holds that
t 1A (Xs ) ds = 0
y
Lt (X) dy,
A ∈ B(R), t 0.
(2.12)
A
For a non-negative Borel measure V on R and a process (Xt ) under W (dX), we write Kt (V ; X) = exp − Lxt (X) V (dx) R
(2.13)
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and x K(V ; X) = exp − L∞ (X) V (dx) .
(2.14)
R
The following theorem is due to Roynette, Vallois and Yor [14], [15, Theorem 4.1] and [16, Theorem 2.1]. Theorem 2.1. (See [16, Theorem 2.1].) Let V be a non-negative Borel measure on R and suppose that
0< 1 + |x| V (dx) < ∞. (2.15) R
Then the following statements hold: (i) ϕV (x) := limt→∞ πt 2 Wx [Kt (V ; X)] and the limit exists in R+ ; (ii) ϕV is the unique solution of the Sturm–Liouville equation ϕV (x) = 2ϕV (x) V (dx)
(2.16)
in the sense of distributions (see, e.g., [13, Appendix §8]) subject to the boundary condition: lim ϕ (x) = −1 x→−∞ V
and
lim ϕ (x) = 1; x→∞ V
(2.17)
(iii) For any 0 < s < ∞ and any bounded Fs -measurable functional Fs (X), Wx [Fs (X)Kt (V ; X)] ϕV (Xs ) → Wx Fs (X) Ks (V ; X) as t → ∞; Wx [Kt (V ; X)] ϕV (X0 )
(2.18)
(Xs ) (iv) (Ms(V ) (X) := ϕϕVV (X Ks (V ; X): s 0) is a (Wx , (Fs ))-martingale which converges a.s. to 0 0) as s → ∞; (V ) (v) Under the probability measure Wx on F∞ induced by the relation
Wx(V ) Fs (X) = Wx Fs (X)Ms(V ) (X) ,
(2.19)
the process (Xt ) solves the stochastic differential equation t Xt = x + Bt +
ϕV (Xs ) ds ϕV
(2.20)
0
where (Bt ) is a (Wx(V ) , (Ft ))-Brownian motion starting from 0; in particular, the process (Xt ) is a transient diffusion which admits the following function γV (x) as its scale function:
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
x γV (x) = 0
3501
dy . ϕV (y)2
(2.21)
Remark 2.2. By (ii) of Theorem 2.1, we see that the function ϕV also enjoys the following properties: (V )
(vi) ϕV (x) ∼ |x| as x → ∞. This suggests that the process {(Xt ), (Wx )} behaves like 3dimensional Bessel process when the value of |Xt | is large. (vii) infx∈R ϕV (x) > 0. This shows that the origin is regular for itself. Example 2.3. (A key example for [14].) Suppose that V = λδ0 with some λ > 0 where δ0 denotes the Dirac measure at 0. That is,
Kt (λδ0 ; X) = exp −λL0t (X) .
(2.22)
Then we can solve Eqs. (2.16)–(2.17) and consequently we obtain 1 + |x|, λ
(λδ ) Mt 0 (X) = 1 + λ|Xt | exp −λL0t (X) ϕλδ0 (x) =
(2.23) (2.24)
and t Xt = x + Bt + 0
sgn(Xs ) 1 λ
+ |Xs |
ds
(V )
under Wx .
(2.25)
2.3. The universal σ -finite measure Najnudel, Roynette and Yor [11,12] introduced the measure W on F∞ defined by (2.8) to give a global view on the Brownian penalisations. It unifies the Feynman–Kac penalisations in the sense of the following theorem, which is due to Najnudel, Roynette and Yor [12, Theorem 1.1.2 and Theorem 1.1.6]; see also Yano, Yano and Yor [19, Theorem 8.1]. See also Najnudel and Nikeghbali [9,10] for careful treatment of augmentation of filtrations. Theorem 2.4. (See [12].) Let x ∈ R and let V be a non-negative measure on R satisfying (2.15). Then it holds that Wx Zt (X)K(V ; X) = Wx Zt (X)ϕV (Xt )Kt (V ; X)
(2.26)
for any t 0 and any non-negative Ft -measurable functional Zt (X), where K(V ; X) has been defined as (2.14). Consequently, it holds that ϕV (x) = Wx K(V ; X) and that
(2.27)
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Wx(V ) (dX) =
1 K(V ; X) Wx (dX) ϕV (x)
on F∞ .
(2.28)
The following theorem can be found in [12, p. 6, point v) and Theorem 1.1.6]; see also [19, Theorem 5.1]. Theorem 2.5. (See [12].) The following statements hold: (i) W (g(X) ∈ du) =
√du 2πu
on [0, ∞).
In particular, W is σ -finite on F∞ ; (ii) For A ∈ Ft with 0 < t < ∞,
W (A) =
0 if W (A) = 0, ∞ if W (A) > 0.
In particular, W is not σ -finite on Ft . We give the proof for completeness of this paper. Proof. Claim (i) is obvious by definition (1.1) of W . Let us prove claim (ii). Let 0 < t < ∞. Suppose that A ∈ Ft and W (A) = 0. Then we have W [1A K(δ0 ; X)] = 0 by (2.26), which implies that W (A) = 0. Suppose in turn that A ∈ Ft and W (A) > 0. For λ > 0, we apply (2.26) for V = λδ0 and we have 1 1 0 −λL0∞ −λL0t W 1A e−λLt . + |Xt | e = W 1A (2.29) W (A) W 1A e λ λ Letting λ → 0+, we obtain, by the monotone convergence theorem, that W [1A e−λLt ] → W (A) > 0, and consequently, that W (A) = ∞. 2 0
We also need the following property. Proposition 2.6. For x ∈ R, it holds that
Wx τ0 (X) = ∞ = |x|.
(2.30)
Proof. By symmetry, we have only to prove the claim for x 0. Let V = δ0 and F (X) = 1{τ0 (X)=∞} . Note that L0∞ (X) = 0 if τ0 (X) = ∞. Hence it follows from Example 2.3 and Theorem 2.4 that
Wx τ0 (X) = ∞ = ϕδ0 (x)Wx(δ0 ) τ0 (X) = ∞ . (2.31) Since ϕδ0 (x) = 1 + x and since γδ0 (x) =
x 1+x ,
we have
γδ (x) − γδ0 (0) = x. Wx τ0 (X) = ∞ = (1 + x) · 0 γδ0 (∞) − γδ0 (0) The proof is complete.
2
(2.32)
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2.4. Markov property of {(Xt ), (Ft ), (Wx )} We may say that {(Xt ), (Ft ), (Wx )} possesses Markov property in the following sense. Theorem 2.7. (See [11,12].) Let x ∈ R and t 0. Let F be a non-negative F∞ -measurable functional. Then it holds that Wx Zt (X)F (θt X) = Wx Zt (X)WXt F (·)
(2.33)
for any non-negative Ft -measurable functional Zt (X). Moreover, the constant time t in (2.33) may be replaced by any finite (Ft )-stopping time τ . Proof. Let V be as in Theorem 2.1. Then we have Wx Zt (X)Kt (V ; X)F (θt X)K(V ; θt X)
by the multiplicativity property of K(V ; X) = Wx Zt (X)F (θt X)K(V ; X)
by (2.28) = ϕV (x)Wx(V ) Zt (X)F (θt X) (V ) (V ) = ϕV (x)Wx(V ) Zt (X)WXt F (·) by the Markov property of W· ϕV (Xt )
(V ) Kt (V ; X) by (2.19) = ϕV (x)Wx Zt (X)WXt F (·) · ϕV (X0 )
= Wx Zt (X)Kt (V ; X)WXt F (·)K(V ; ·) by (2.28) .
(2.34) (2.35) (2.36) (2.37) (2.38) (2.39)
Taking V = λδ0 and letting λ → 0+, we obtain (2.33) by the monotone convergence theorem. In the same way, we can prove (2.33) also in the case where the constant time t is replaced by a finite stopping time τ . 2 Since the measure Wx has infinite total mass, we cannot consider conditional expectation in the usual sense. But, by the help of Theorem 2.7, we can introduce a counterpart in the following sense. Corollary 2.8. (See [11,12]; see also [19].) Let x ∈ R and t 0. Let F be a F∞ -measurable functional which is in L1 (Wx ). Then there exists a unique {(Ft ), Wx }-martingale Mt [F ; X] such that Wx Zt (X)F (X) = Wx Zt (X)Mt [F ; X]
(2.40)
for any bounded Ft -measurable functional Zt (X). Moreover, it is given as Mt [F ; X] = Ω
WXt (dY )F X (t) • Y ,
Wx (dX)-a.s.
(2.41)
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Remark 2.9. If F ∈ L1 (Wx ), then the family of the conditional expectations {Wx [F |Ft ]: t 0} is a uniformly integrable martingale. In contrast with this fact, if F ∈ L1 (Wx ), the martingale {Mt [F ; X]: t 0} under Wx converges to 0 as t → ∞, and consequently, it is not uniformly integrable. Remark 2.10. Since Mt is an operator from L1 (Wx ) to L1 (Wx ), we do not have a counterpart of the tower property for the usual conditional expectation. Example 2.11. Let V be a non-negative measure on R satisfying (2.15). Then (iv) and (v) of Theorem 2.1 may be rewritten as Mt K(V ; ·); X = ϕV (Xt )Kt (V ; X).
(2.42)
From this and from Remark 2.2, we see that Mt K(V ; ·); X ∈ Lp (W )
for any p 1.
(2.43)
In particular, formula (2.24) may be rewritten as Mt K(λδ0 ; ·); X =
1 + |Xt | Kt (λδ0 ; X). λ
(2.44)
3. Wiener integrals Let S denote the set of all step functions f on [0, ∞) of the form: f (t) =
n
ck 1[tk−1 ,tk ) (t),
t 0
(3.1)
k=1
with n ∈ N, ck ∈ R (k = 1, . . . , n) and 0 = t0 < t1 < · · · < tn < ∞. Note that S is dense in L2 (ds). For a function f ∈ S and a process X, we define ∞ f (t) dXt =
n
ck (Xtk − Xtk−1 ).
(3.2)
k=1
0
∞
∞ If 0 f (t) dXt can be defined as the limit in some sense of 0 fn (t) dXt for an approximating sequence {fn } of f , then we will call it Wiener integral of f for the process X. We have the following facts: If a sequence {fn } ⊂ S approximates f in L2 (ds), then it holds that ∞
∞ fn (s) dXs −→
f (s) dXs
n→∞
0
and that, for any u > 0,
0
in W -probability
(3.3)
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
u
3505
u fn (s) dXs −→
0
in Π (u) -probability.
f (s) dXs
n→∞
(3.4)
0
3.1. Wiener integral for 3-dimensional Bessel process Let pt (x) denote the density of the Brownian semigroup: pt (x) = √
2 x , exp − 2t 2πt 1
t > 0, x ∈ R.
(3.5)
Let a 0 be fixed. It is well known (see, e.g., [13, §VI.3]) that, for t > 0 and x > 0, x
a {pt (x − a) − pt (x 2x 2 t pt (x) dx,
Ra+ (Xt ∈ dx) =
+ a)} dx, a > 0, a = 0.
(3.6)
From this formula, it is straightforward that, for t > 0 and x > 0, φa (t) := Ra+
1 Xt
⎧ a ⎨ a1 −a pt (x) dx, = ⎩ 2pt (0) = 2 , πt
a > 0, a = 0.
(3.7)
Since pt (x) pt (0), it is obvious by definition that φa (t) φ0 (t),
a > 0, t > 0.
(3.8)
Note that φa (t) has the following asymptotics as t → 0+:
1/a φa (t) ∼ √ 2/(πt)
if a > 0, if a = 0.
(3.9)
By the stochastic differential equation (2.4), we see that Ra+ [Xt ] = a
t +
Ra+
t 1 ds = a + φa (s) ds, Xs
0
(3.10)
0
Now the following lemma is obvious. Lemma 3.1. Let f ∈ L2 (ds) ∩ L1 (φa (s) ds). Then, according to the stochastic differential equation (2.4), the Wiener integral may be defined as ∞
∞ f (s) dXs =
0
∞ f (s) dBs +
0
f (s) ds. Xs
0
If a sequence {fn } ⊂ S approximates f both in L2 (ds) and in L1 (φa (s) ds), i.e.,
(3.11)
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∞ ∞ fn (s) − f (s) 2 ds + fn (s) − f (s) φa (s) ds −→ 0, n→∞
0
(3.12)
0
then it holds that ∞
∞ fn (s) dXs −→
0
in Ra+ -probability.
f (s) dXs
n→∞
(3.13)
0
Following Funaki, Hariya and Yor [5], we may propose another way of constructing the Wiener integral. We define s(a) = Xs − Ra+ [Xs ] X
(3.14)
s for X s(0) . By applying s(a) ), Ra+ } the centered Bessel process. We simply write X and we call {(X Theorem 1.3 with ψ(x) = x 2 , we obtain the following fact: If a sequence {fn } ⊂ S approximates f in L2 (ds), then it holds that ∞
s(a) −→ fn (s) dX
∞
n→∞
0
s(a) f (s) dX
in Ra+ -probability.
(3.15)
0
We then obtain the following lemma. Lemma 3.2. Let f ∈ L2 (ds) ∩ L1 (φa (s) ds). Then it holds that ∞
∞ f (s) dXs =
0
s(a) f (s) dX
0
∞ +
f (s)φa (s) ds
Ra+ -a.s.
(3.16)
0
3.2. Wiener integral for X under W Define L1+ (W ) = G : Ω → R+ , F -measurable, W (G = 0) = 0, W [G] < ∞ .
(3.17)
For G ∈ L1+ (W ), we define a probability measure W G on (Ω, F ) by W G (A) =
W [1A G] , W [G]
A ∈ F.
(3.18)
We recall the following notion of convergence. Proposition 3.3. Let Z, Z1 , Z2 , . . . be F∞ -measurable functionals. Then the following statements are equivalent:
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
(i) (ii) (iii) (iv)
3507
For any ε > 0 and any A ∈ F with W (A) < ∞, it holds that W (A ∩ {|Zn − Z| ε}) → 0. Zn → Z in W G -probability for some G ∈ L1+ (W ). Zn → Z in W G -probability for any G ∈ L1+ (W ). One can extract, from an arbitrary subsequence, a further subsequence {n(k): k = 1, 2, . . .} along which Zn(k) → Z W -a.e.
If one (and hence all) of the above statements holds, then we say that Zn → Z
locally in W -measure.
(3.19)
For the proof of Proposition 3.3, see, e.g., [18]. Wiener integral for X under W (dX) may be defined with the help of the following theorem. Theorem 3.4. (See [18].) Let f ∈ L2 (ds) ∩ L1 ( 1+ds√s ). Suppose that a sequence {fn } ⊂ S approximates f both in L2 (ds) and in L1 ( 1+ds√s ), i.e., ∞ ∞ fn (s) − f (s) 2 ds + fn (s) − f (s) 0
0
ds √ −→ 0. 1 + s n→∞
(3.20)
(Note that this condition is strictly weaker than the condition (3.12).) Then it holds that ∞
∞ fn (s) dXs −→
f (s) dXs
n→∞
0
locally in W -measure.
(3.21)
0
Moreover, there exists a functional J (f ; u, X) measurable with respect to the product σ -field B([0, ∞)) ⊗ F∞ such that ∞
f (s) dXs = J f ; g(X), X W -a.e.
(3.22)
0
and that it holds du-a.e. that
J f ; u, X (u) • Y =
u
∞ f (s) dXs +
0
f (s + u) dYs
(3.23)
0
is valid a.e. with respect to Π (u) (dX (u) ) ⊗ R(dY ). The following lemma allows us to use the same notation for Wiener integrals under W (dX) and W (dX). Let us temporarily write I W (f ; X) (resp. I W (f ; X)) for the Wiener integral I (f ; X) under W (dX) (resp. W (dX)).
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Lemma 3.5. Suppose that there exist F ∈ L1 (W ) and G ∈ L1 (W ) such that W H (X)F (X) = W H (X)G(X)
(3.24)
holds for any bounded measurable functional H (X). Then, for any f ∈ L2 (ds) ∩ L1 ( 1+ds√s ), it holds that
W ϕ I W (f ; X) H (X)F (X) = W ϕ I W (f ; X) H (X)G(X)
(3.25)
for any bounded Borel function ϕ on R. Proof. This is obvious by Theorem 3.4 and by the dominated convergence theorem.
2
3.3. Integrability lemma For later use, we need the following lemma. Lemma 3.6. Let f ∈ L1 (ds). Define ∞ ∞ ds ds f (s + t) √ = f (s) √ f (t) = , s s −t
t > 0.
(3.26)
t
0
Then the following statements hold: (i) For any a > 0, it holds that a
√ f (t) dt 2 a
0
∞ f (s) ds;
(3.27)
0
(ii) There exists a sequence t (n) → ∞ such that f (t (n)) → 0. Proof. (i) Let a > 0. Then we have a
a f (t) dt =
0
a dt
0
a = 0
a 0
t
f (s) √ ds + s−t
ds f (s)
s 0
a
dt + √ s −t
√ f (s) (2 s) ds +
∞ a
dt 0
∞
∞ f (s) √ ds s −t
(3.28)
a
ds f (s)
a
ds f (s)
a 0
a 0
dt √ s−t
dt √ a−t
(3.29)
(3.30)
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
√ 2 a
∞ f (s) ds.
3509
(3.31)
0
(ii) Let 0 < a < b < ∞. Then we have (b − a) 1 inf f (t) √ √ t: t>a b b
b f (t) dt 2 a
∞ f (s) ds.
(3.32)
0
√ Since (b − a)/ b → ∞ as b → ∞ with a fixed, we see that inft: t>a f (t) = 0 for any a > 0. This implies that lim inf f (t) = 0.
(3.33)
t→∞
The proof is now complete.
2
4. Cameron–Martin formula For a function ht = x ∈ R, we write
t 0
f (s) ds with f ∈ L2 (ds) ∩ L1 ( 1+ds√s ) and a process (Xs ) under Wx for t
Et (f ; X) = exp
1 f (s) dXs − 2
0
t f (s)2 ds
(4.1)
0
and ∞ E(f ; X) = exp 0
1 f (s) dXs − 2
∞ 2
f (s) ds .
(4.2)
0
In what follows, let V be a non-negative Borel measure satisfying (2.15). 4.1. The first step t Proposition 4.1. Let ht = 0 f (s) ds with f ∈ L2 (ds) and let T > 0. Then, for any non-negative F∞ -measurable functional F (X), it holds that W F (X + h·∧T ) = W F (X)ET (f ; X) .
(4.3)
If, moreover, MT [F ; X] ∈ Lp (W ) for some p > 1, then F (X + h·∧T ) ∈ L1 (W ). Proof. Let t T be fixed. By the multiplicativity property of K(δ0 ; ·) and since h(·+t)∧T = hT , we have K(δ0 ; X + h·∧T ) = Kt (δ0 ; X + h·∧T )K(δ0 ; θt X + hT ).
(4.4)
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Let Gt (X) be a non-negative Ft -measurable functional. Then, by the Markov property (2.41), by (2.9) and by (2.23), we have Mt K(δ0 ; · + h·∧T ); X = Kt (δ0 ; X + h·∧T )WXt K(δ0 ; X + hT ) = Kt (δ0 ; X + h·∧T )WXt +hT K(δ0 ; X)
= Kt (δ0 ; X + h·∧T ) 1 + |Xt + hT | .
(4.5) (4.6) (4.7)
Hence we obtain W Gt (X + h·∧T )K(δ0 ; X + h·∧T )
= W Gt (X + h·∧T )Kt (δ0 ; X + h·∧T ) 1 + |Xt + hT | .
(4.8)
By the Cameron–Martin formula (1.21), by formula (2.44), and then by the Markov property (2.33), we have
(4.8) = W Gt (X)Kt (δ0 ; X) 1 + |Xt | ET (f ; X) = W Gt (X)Mt K(δ0 ; ·); X ET (f ; X) = W Gt (X)K(δ0 ; X)ET (f ; X) .
(4.9) (4.10) (4.11)
Since t T is arbitrary, we see that W G(X + h·∧T )K(δ0 ; X + h·∧T ) = W G(X)K(δ0 ; X)ET (f ; X)
(4.12)
holds for any non-negative F∞ -measurable functional G(X). Replacing the functional G(X) by F (X)K(δ0 ; X)−1 , we obtain (4.3). Suppose that MT [F ; X] ∈ Lp (W ) for some p > 1. Since ET (h; X) is FT -measurable, we have W F (X)ET (f ; X) = W MT [F ; X]ET (f ; X) 1/p 1/q W MT [F ; X]p W ET (f ; X)q <∞ where q is the conjugate exponent to p: (1/p) + (1/q) = 1. The proof is now complete.
(4.13) (4.14) 2
4.2. Integrability under W , when weighed by Feynman–Kac functionals We need the following theorem. t Theorem 4.2. Let ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let V be as in Theorem 2.1 and set CV = infx∈R ϕV (x) > 0. Then it holds that W K(V ; X)E(f ; X) ϕV (0) exp
1 f L1 (ds) . CV
(4.15)
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Proof. By Theorem 2.4, we have 1 W K(V ; X)E(f ; X) = W (V ) E(f ; X) . ϕV (0)
(4.16)
By (v) of Theorem 2.1, we see that (4.16) = W
(V )
∞ E(f ; B) exp
ϕ f (s) V (Xs ) ds ϕV
(4.17)
0
where {(Bt ), W (V ) } is a Brownian motion. Since |ϕV (x)| 1 and ϕV (x) CV for any x ∈ R, we have (4.17) W
(V )
∞ 1 f (s) ds . E(f ; B) exp CV
(4.18)
0
Since W (V ) [E(f ; B)] = 1, we obtain the desired inequality.
2
4.3. The second step We utilize the following lemma. Lemma 4.3. Let ht = that
t 0
f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then, for any 0 < s < ∞, it holds W Et (f ; X)e−g(X) ; g(X) > t −→ 0.
(4.19)
t→∞
Proof. By the Markov property (2.33), we see that W Et (f ; X)e−g(X) ; g(X) > t = W Et (f ; X)e−t WXt e−g(X) ; τ0 (X) < ∞ .
(4.20)
By the strong Markov property (2.33), we see, for any x ∈ R, that
Wx e
−g(X)
; τ0 (X) < ∞ = Wx e−τ0 (X) W0 e−g(X)
∞ 0
du −u 1 e =√ . √ 2πu 2
(4.21)
Hence we obtain 1 1 (4.20) √ e−t W Et (f ; X) = √ e−t −→ 0. t→∞ 2 2 The proof is now complete.
2
(4.22)
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Lemma 4.4. Let ht = holds that
t 0
f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let V be as in Theorem 2.1. Then it
W E(f ; X)K(V ; X)e−g(X) ; g(X) > t −→ 0. t→∞
(4.23)
Proof. Since W [E(f ; X)K(V ; X)] < ∞ by Theorem 4.2. The desired conclusion is now obvious by the dominated convergence theorem. 2 Lemma 4.5. Let ht =
t 0
f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Set ∞ ds f (t) = f (s + t) √ , s 0
! ! σt = !f (· + t)! =
∞
t > 0,
(4.24)
1/2 2
f (s) ds
,
t > 0,
(4.25)
t
and set 2 1 E(t) = E exp σt |N | + cf (t) + σt2 − 1 , 2 where N stands for the standard Gaussian variable and c =
t >0
√ 2/π . Then it holds that
2
Ra E f (· + t); · − 1 E(t) for any t > 0 and any a ∈ R. Proof. Let us write f, g =
∞ 0
(4.26)
(4.27)
f1 (s)f2 (s) ds for f1 , f2 ∈ L2 (ds). Note that
∞ # 1 2 "
(a) s + f (· + t), φa − σt f (s + t) dX E f (· + t); X = exp under Ra+ . 2
(4.28)
0
Since |eb − 1| e|b| − 1 for any b ∈ R, we have 2 ∞ 2
# " 1 E f (· + t); · − 1 exp f (s + t) dX s(a) + f (· + t), φa − σt2 − 1 . 2
(4.29)
0
Since, for any constant b ∈ R, ψ(x) = (e|x+b| − 1)2 is a convex function, we may apply Theorem 1.3 and obtain Ra+ Since
2
# " E f (· + t); · − 1 2 E exp σt N + f (· + t), φa − 1 σ 2 − 1 . t 2
(4.30)
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
# " # " f (· + t), φa f (· + t) , φ0 = cf (t),
t 0
(4.31)
2
we obtain the desired result. Lemma 4.6. Let ht = t (n) → ∞ such that
3513
f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Then there exists a sequence
W e−g(X) K(V ; X) E(f ; X) − Et (n) (f ; X) → 0.
(4.32)
Proof. By Lemmas 4.3 and 4.4, it suffices to prove that W e−g(X) K(V ; X) E(f ; X) − Et (f ; X) ; g(X) t
(4.33)
converges to 0 along some sequence t = t (n) → ∞. By the multiplicativity:
E(f ; X) = Et (f ; X)E f (· + t); θt X ,
(4.34)
(4.33) = W e−g(X) K(V ; X)Et (f ; X) E f (· + t); θt X − 1 ; g(X) t .
(4.35)
we have
By the Schwarz inequality, (4.35) is dominated by A1/2 B 1/2 where A = W K(V ; X)2 Et (f ; X)2
(4.36)
2
B = W e−2g(X) E f (· + t); θt X − 1 ; g(X) t .
(4.37)
and
By Theorem 4.2, we see that
A W K(2V ; X)E(2f 1[0,t) ; X) exp f 2L2 (ds) 2 2 f L1 (ds) . ϕ2V (0) exp f L2 (ds) + C2V
(4.38) (4.39)
By Lemma 4.5, we see that t B= 0
t = 0
2
du −2u (u) Π • R E f (· + t); θt X − 1 e √ 2πu
(4.40)
2
du −2u e R RXt−u E f (· + t); · − 1 √ 2πu
(4.41)
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∞ E(t) 0
du −2u e . √ 2πu
(4.42)
Therefore we see that (4.33) is dominated by E(t) up to a multiplicative constant. The proof is now completed by (ii) of Lemma 3.6. 2 4.4. The third step In what follows, we take and utilize a non-negative, bounded, continuous function v0 on R such that v0 (x) = 1 for |x| 2 and v0 (x) = 0 for |x| 3. We write v1 = 1[−1,1] . We set V0 (dx) = v0 (x) dx and V1 (dx) = v1 (x) dx. For any V , we write Γ (V ; X) = e−g(X) K(V ; X). Lemma 4.7. Let ht =
t 0
(4.43)
f (s) ds with f ∈ L1 (ds). Suppose that ∞ f (s) ds 1
(4.44)
T
for some 0 < T < ∞. Then it holds that K(V0 ; X + h·∧t ) K(V1 ; X + h·∧T ),
t T.
(4.45)
Proof. Note that we have |ht − hT | 1 for any t T . If s T satisfies |Xs + hT | 1, then we have |Xs + hs∧t | 2. Hence we have ∞
∞ v0 (Xs + hs∧t ) ds
0
This completes the proof.
v1 (Xs + hs∧T ) ds,
t T.
(4.46)
0
2
t Lemma 4.8. Let ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let 0 < r < ∞ and let Gr (X) be a non-negative, bounded, continuous Fr -measurable functional. Then it holds that W Gr (X + h·∧t )Γ (V0 ; X + h·∧t ) −→ W Gr (X + h)Γ (V0 ; X + h) . t→∞
(4.47)
Proof. Note that g(X + h·∧t ) → g(X + h) as t → ∞, because h·∧t → h uniformly. By the continuity assumptions on Gr and v, we have Gr (X + h·∧t )Γ (V0 ; X + h·∧t ) → Gr (X + h)Γ (V0 ; X + h)
(4.48)
for W (dX)-almost every path X. Since Gr (X) is bounded, it suffices to find Z ∈ L1 (W ) such that Γ (V0 ; X + h·∧t ) Z(X), W -a.e. for any large t; in fact, we may obtain (4.47) by the dominated convergence theorem.
K. Yano / Journal of Functional Analysis 258 (2010) 3492–3516
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Since f ∈ L1 (ds), we may take T > 0 such that (4.44) holds. By Lemma 4.7, we have (4.45), and hence we have Γ (V0 ; X + h·∧t ) K(V1 ; X + h·∧T ),
t T.
(4.49)
Since MT [K(V1 ; ·); X] ∈ L2 (W ) by (2.43), we see, by Proposition 4.1, that K(V1 ; X + h·∧T ) ∈ L1 (W ). Therefore this functional K(V1 ; X + h·∧T ) is as desired.
(4.50)
2
4.5. Proof of Theorem 1.1 We now proceed to prove Theorem 1.1. t Proof of Theorem 1.1. Let ht = 0 f (s) ds with f ∈ L2 (ds) ∩ L1 (ds). Let 0 < s < ∞ and let Gs (X) be a non-negative, bounded, continuous Fs -measurable functional. Let T > 0. Then, by Proposition 4.1, we have W Gs (X + h·∧T )Γ (V0 ; X + h·∧T ) = W Gs (X)Γ (V0 ; X)ET (f ; X) .
(4.51)
By Lemma 4.8, we have W Gs (X + h·∧T )Γ (V0 ; X + h·∧T ) −→ W Gs (X + h)Γ (V0 ; X + h) . T →∞
(4.52)
By Lemma 4.6, we have W Gs (X)Γ (V0 ; X)ET (f ; X) → W Gs (X)Γ (V0 ; X)E(f ; X)
(4.53)
along some sequence T = t (n) → ∞. Thus, taking the limit as T = t (n) → ∞ in both sides of (4.51), we obtain W Gs (X + h)Γ (V0 ; X + h) = W Gs (X)Γ (V0 ; X)E(f ; X) .
(4.54)
Hence we obtain W G(X + h)Γ (V0 ; X + h) = W G(X)Γ (V0 ; X)E(f ; X)
(4.55)
for any non-negative F∞ -measurable functional G(X). Replacing G(X) by F (X)Γ (V0 ; X)−1 , we obtain the desired conclusion. 2 Acknowledgments The author would like to thank Professors Marc Yor, Tadahisa Funaki, Shinzo Watanabe and Ichiro Shigekawa for their useful comments which helped improve this paper.
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References [1] S. Bonaccorsi, L. Zambotti, Integration by parts on the Brownian meander, Proc. Amer. Math. Soc. 132 (3) (2004) 875–883 (electronic). [2] H.J. Brascamp, E.H. Lieb, On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (4) (1976) 366–389. [3] R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2) 45 (1944) 386–396. [4] R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under a general class of linear transformations, Trans. Amer. Math. Soc. 58 (1945) 184–219. [5] T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic). [6] T. Funaki, K. Ishitani, Integration by parts formulae for Wiener measures on a path space between two curves, Probab. Theory Related Fields 137 (3–4) (2007) 289–321. [7] Y. Hariya, Integration by parts formulae for Wiener measures restricted to subsets in Rd , J. Funct. Anal. 239 (2) (2006) 594–610. [8] K. Itô, Poisson point processes attached to Markov processes, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, California, 1970/1971, in: Probab. Theory, vol. III, University California Press, Berkeley, CA, 1972, pp. 225–239. [9] J. Najnudel, A. Nikeghbali, On some universal sigma finite measures and some extensions of Doob’s optional stopping theorem, preprint, arXiv:0906.1782, 2009. [10] J. Najnudel, A. Nikeghbali, A new kind of augmentation of filtrations, preprint, arXiv:0910.4959, 2009. [11] J. Najnudel, B. Roynette, M. Yor, A remarkable σ -finite measure on C(R+ , R) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466. [12] J. Najnudel, B. Roynette, M. Yor, A Global View of Brownian Penalisations, MSJ Mem., vol. 19, Mathematical Society of Japan, Tokyo, 2009. [13] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Grundlehren Math. Wiss., vol. 293, Springer-Verlag, Berlin, 1999. [14] B. Roynette, P. Vallois, M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I, Studia Sci. Math. Hungar. 43 (2) (2006) 171–246. [15] B. Roynette, P. Vallois, M. Yor, Some penalisations of the Wiener measure, Jpn. J. Math. 1 (1) (2006) 263–290. [16] B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Math., vol. 1969, Springer, Berlin, 2009. [17] A.S. Üstünel, An Introduction to Analysis on Wiener Space, Lecture Notes in Math., vol. 1610, Springer-Verlag, Berlin, 1995. [18] K. Yano, Wiener integral for the coordinate process under the σ -finite measure unifying Brownian penalisations, preprint, arXiv:0909.5130, 2009. [19] K. Yano, Y. Yano, M. Yor, Penalising symmetric stable Lévy paths, J. Math. Soc. Japan 61 (3) (2009) 757–798. [20] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields 123 (4) (2002) 579–600. [21] L. Zambotti, Integration by parts on δ-Bessel bridges, δ > 3 and related SPDEs, Ann. Probab. 31 (1) (2003) 323– 348. [22] L. Zambotti, Integration by parts on the law of the reflecting Brownian motion, J. Funct. Anal. 223 (1) (2005) 147–178.
Journal of Functional Analysis 258 (2010) 3517–3542 www.elsevier.com/locate/jfa
Gradient estimates for the heat equation under the Ricci flow Mihai Bailesteanu a , Xiaodong Cao a , Artem Pulemotov a,b,∗ a Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853-4201, USA b Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637-1514, USA
Received 10 October 2009; accepted 8 December 2009 Available online 21 December 2009 Communicated by L. Gross
Abstract The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li–Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M. © 2009 Elsevier Inc. All rights reserved. Keywords: Ricci flow; Heat equation; Li–Yau inequality; Harnack inequality; Manifold with boundary
1. Introduction The paper deals with a manifold M evolving under the Ricci flow and with positive solutions to the heat equation on M. We establish a series of gradient estimates for such solutions including several Li–Yau-type inequalities. First, we study the case where M is a complete manifold without boundary. Our results contain estimates of both local and global nature. Second, we look at the situation where M is compact and has nonempty boundary ∂M. We impose the condition that ∂M remain convex and umbilic at all times. Our arguments then yield two global estimates. * Corresponding author at: Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637-1514, USA. E-mail addresses:
[email protected] (M. Bailesteanu),
[email protected] (X. Cao),
[email protected] (A. Pulemotov).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.003
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Suppose M is a manifold without boundary. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow ∂ g(x, t) = −2 Ric(x, t), ∂t
x ∈ M, t ∈ [0, T ].
(1.1)
We assume its curvature remains uniformly bounded for all t ∈ [0, T ]. Consider a positive function u(x, t) defined on M × [0, T ]. In Section 2, we assume u(x, t) solves the equation −
∂ u(x, t) = 0, ∂t
x ∈ M, t ∈ [0, T ].
(1.2)
The symbol here stands for the Laplacian given by g(x, t). It is important to emphasize that depends on the parameter t. Thus, we look at the Ricci flow (1.1) combined with the heat equation (1.2). Note that formula (1.1) provides us with additional information about the coefficients of the operator appearing in (1.2) but is itself fully independent of (1.2). To learn about the history, the intuitive meaning, the technical aspects, and the applications of the Ricci flow, one should refer to the many quality books on the subject such as, for example, [12,35,24,9,10]. Problem (1.1) combined with (1.2) admits a simple interpretation in terms of the process of heat conduction. More specifically, one may think of the manifold M with the initial metric g(x, 0) as an object having the temperature distribution u(x, 0). Suppose we let M evolve under the Ricci flow and simultaneously let the heat spread on M. Then the solution u(x, t) will represent the temperature of M at the point x at time t. The work [2] provides a probabilistic interpretation of (1.1)–(1.2). In particular, it constructs a Brownian motion related to u(x, t). The study of system (1.1)–(1.2) arose from R. Hamilton’s paper [16]. The original idea in [16] was to investigate the Ricci flow combined with the heat flow of harmonic maps. The system we examine in Section 2 may be viewed as a special case. The idea to consider the Ricci flow combined with the heat flow of harmonic maps was further exploited in [30,31] for the purposes of regularizing non-smooth Riemannian metrics. We point out, without a deeper explanation, that looking at the two evolutions together leads to interesting simplifications in the analysis. After its conception in [16], the study of (1.1)–(1.2) was pursued in [14,25,38,2,6]. A large amount of work was done to understand several problems that are similar to (1.1)–(1.2) in one way or another. The list of relevant references includes but is not limited to [38,5,6] and [10, Chapter 16]. For instance, there are substantial results concerning the Ricci flow combined with the conjugate heat equation. The connection of this problem to (1.1)–(1.2) is beyond superficial. Q. Zhang used a gradient estimate for (1.1)–(1.2) to prove a Gaussian bound for the conjugate heat equation in [38]. The results of the present paper may have analogous applications. System (1.1)–(1.2) could serve as a model for researching the Ricci flow combined with the heat flow of harmonic maps. There are other geometric evolutions for which (1.1)–(1.2) plays the same role. One example is the Ricci Yang–Mills flow; see [19,33,37]. The analysis of this evolution is technically complicated. Its properties are not yet well understood. We expect that investigating the simpler model case of system (1.1)–(1.2) will provide insight on the behavior of the Ricci Yang–Mills flow. We also speculate that the results of the present paper may aid in proving relevant existence theorems; cf. [1,26] and also [27,7]. The scalar curvature of a surface which evolves under the Ricci flow satisfies the heat equation with a potential on that surface. In the same spirit, we expect to find geometric quantities on M that obey (1.1)–(1.2). The gradient estimates in this paper would then lead to new knowledge
M. Bailesteanu et al. / Journal of Functional Analysis 258 (2010) 3517–3542
3519
about the behavior of the metric g(x, t) under the Ricci flow. In particular, we believe that our results will be helpful in classifying ancient solutions of (1.1). L. Ni’s work [25] offers yet another way to use the Ricci flow combined with the heat equation to study the evolution of g(x, t). Section 2.2 discusses space-only gradient estimates for system (1.1)–(1.2). The first predecessor of these results was obtained by R. Hamilton in the paper [15]. It applies to the case where M is a closed manifold, the metric g(x, t) is independent of t, and Eq. (1.1) is not in the picture. New versions of R. Hamilton’s result were proposed in [32,38,5,6]. The beginning of Section 2 describes them thoroughly. For related work done by probabilistic methods, one should consult [18, Chapter 5] and [2]. Theorem 2.2 states a space-only gradient estimate for (1.1)–(1.2). It is a result of local nature. Section 2.3 deals with space–time gradient estimates for (1.1)–(1.2). Our results resemble the Li–Yau inequalities from the paper [22]; see also [28, Chapter IV]. More precisely, the solution u(x, t) of Eq. (1.2) satisfies |∇u|2 ut n , − u 2t u2
x ∈ M, t ∈ (0, T ],
(1.3)
if M is a closed manifold with nonnegative Ricci curvature, the metric g(x, t) does not depend on t, and (1.1) is not assumed. Here, ∇ stands for the gradient, the subscript t denotes the derivative in t, and n is the dimension of M. This result goes back to [22] and constitutes the simplest Li–Yau inequality. It opened new possibilities for the comparison of the values of solutions of (1.2) at different points and led to important Gaussian bounds in heat kernel analysis. Integrating the above estimate along a space–time curve yields a Harnack inequality. A precursory form of (1.3) appeared in [3]. Many variants of (1.3) now exist in the literature; see, e.g., [20,11,4,21]. R. Hamilton proved one in [15] which further extended our ability to compare the values of solutions of (1.2). Li–Yau inequalities served as prototypes for many estimates connected to geometric flows. The list of relevant references includes but is not limited to [8,17,10]. In particular, the Li–Yau-type inequality for the Ricci flow became one of the central tools in classifying ancient solutions to the flow as detailed in [12, Chapter 9]. Analogous results played a significant part in the study of Kähler manifolds; see [9, Chapter 2]. Our Theorems 2.7 and 2.9 establish space–time gradient estimates for (1.1)–(1.2). As an application, we lay down two Harnack inequalities for (1.1)–(1.2). They help compare the values of a solution at different points. We are also hopeful that the techniques in Section 2.3 will lead to the discovery of new informative Li– Yau-type inequalities related to the Ricci flow and other geometric flows. Our investigation of (1.1)–(1.2) would then be a model for the proof of such inequalities. In Section 3, we consider the case where M is a compact manifold and ∂M = ∅. We impose the boundary condition on the Ricci flow (1.1) by demanding that the second fundamental form II(x, t) of the boundary with respect to g(x, t) satisfy II(x, t) = λ(t)g(x, t),
x ∈ ∂M, t ∈ [0, T ],
(1.4)
for some nonnegative function λ(t) defined on [0, T ]. Thus, ∂M must remain convex and umbilic1 for all t ∈ [0, T ]. We then assume u(x, t) solves the heat equation (1.2) and satisfies the 1 There is ambiguity in the literature as to the use of the term “umbilic” in this context. See the discussion in [13].
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Neumann boundary condition ∂ u(x, t) = 0, ∂ν
x ∈ ∂M, t ∈ [0, T ].
(1.5)
∂ The outward unit normal ∂ν is determined by the metric g(x, t) and, therefore, depends on the parameter t. The Ricci flow on manifolds with boundary is not yet deeply understood. We remind the reader that Eq. (1.1) fails to be strictly parabolic. As a consequence, it is not even clear how to impose the boundary conditions on (1.1) to obtain a well-posed problem. Progress in this direction was made by Y. Shen in the paper [29]. He proposed to consider the Ricci flow on a manifold with boundary assuming formula (1.4) holds with λ(t) identically equal to a constant. Furthermore, he managed to prove the short-time existence of solutions to the flow in this case. The work [13] continues the investigation of problem (1.1) subject to (1.4) with λ(t) equal to a constant. It also contains a complete set of references on the subject. In the present paper, we consider a more general situation by allowing λ(t) to depend on the parameter t nontrivially. Note that Y. Shen’s method of proving the short-time existence applies to this case, as well. Section 3.1 ponders on the geometric meaning of the function λ(t). We explain why it is beneficial to let λ(t) depend on t. The discussion is rather informal. Section 3.2 provides gradient estimates for system (1.1)–(1.2) subject to the boundary conditions (1.4)–(1.5). Theorems 3.1 and 3.4 state versions of inequalities from Theorems 2.4 and 2.9. Related work was done in [22,36,4,26]. Note that Theorem 3.1 appears to be new in the case where ∂M is nonempty even if g(x, t) is independent of t (see Remark 3.3 for the details). At the same time, the proof is not particularly complicated. Theorems 3.1 and 3.4 are likely to have applications similar to those of Theorems 2.4 and 2.9. We hope that the material in Section 3 will help shed light on the behavior of the Ricci flow on manifolds with boundary. Last but not least, our results may serve as a model for the investigation of problems similar to (1.1)–(1.2)–(1.4)–(1.5). For example, it is natural to look at the Ricci flow subject to (1.4) combined with the conjugate heat equation. As we previously explained, such problems were actively studied on manifolds without boundary, but the case where ∂M is nonempty remains unexplored.
Note. After this paper was completed, we became aware that space–time gradient estimates for (1.1)–(1.2) were researched independently by Shiping Liu in [23], and Jun Sun in [34]. The results of those works are not identical to ours. 2. Manifolds without boundary Our goal is to investigate the Ricci flow combined with the heat equation. The present section establishes space-only and space–time gradient estimates in this context. 2.1. The setup Suppose M is a connected, oriented, smooth, n-dimensional manifold without boundary. Some of the results in this section, but not all of them, concern the case where M is compact.
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Given T > 0, assume (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow ∂ g(x, t) = −2 Ric(x, t), ∂t
x ∈ M, t ∈ [0, T ].
(2.1)
Suppose a smooth positive function u : M × [0, T ] → R satisfies the heat equation ∂ u(x, t) = 0, − ∂t
x ∈ M, t ∈ [0, T ].
(2.2)
Here, stands for the Laplacian given by g(x, t). In what follows, we will use the notation ∇ and | · | for the gradient and the norm with respect to g(x, t). It is clear that , ∇, and | · | all depend on t ∈ [0, T ]. We will write XY for the scalar product of the vectors X and Y with respect to g(x, t). Section 2.2 offers space-only gradient estimates for u(x, t). These results require that u(x, t) be a bounded function. A local space-only gradient estimate for solutions of (2.2) was originally proved in the paper [32] in the situation where g(x, t) did not depend on t ∈ [0, T ] and (2.1) was not in the picture. It was further generalized in [38] to hold in the case of the backward Ricci flow combined with the heat equation. Our Theorem 2.2 constitutes a version of this result for u(x, t). A global space-only gradient estimate for solutions of (2.2) was originally established in [15] with g(x, t) independent of t ∈ [0, T ] and (2.1) not assumed. It is now known to hold in the cases of both the backward Ricci flow and the Ricci flow combined with the heat equation; see [38,5,6]. We restate it in Theorem 2.4 for the completeness of our exposition. Section 2.3 contains Li–Yau-type estimates for (2.1)–(2.2). As applications, we obtain two Harnack inequalities. The results in this section prevail, with obvious modifications, if the function u(x, t) is defined on M × (0, T ] instead of M × [0, T ]. In order to see this, it suffices to replace u(x, t) and g(x, t) with u(x, t + ) and g(x, t + ) for a sufficiently small > 0, apply the corresponding formula, and then let go to 0. We thus justify, for example, the application of the theorems in Section 2.3 to heat-kernel-type functions. Two more pieces of notation should be introduced at this point. Let us fix x0 ∈ M and ρ > 0. We write dist(χ, x0 , t) for the distance between χ ∈ M and x0 with respect to the metric g(x, t). The notation Bρ,T stands for the set {(χ, t) ∈ M × [0, T ] | dist(χ, x0 , t) < ρ}. We point out that Theorems 2.2 and 2.7 still hold if u(x, t) is defined on Bρ,T instead of M × [0, T ] and satisfies the heat equation in Bρ,T . The proofs in this section will often involve local computations. Therefore, we assume a coordinate system {x1 , . . . , xn } is fixed in a neighborhood of every point x ∈ M. The notation Rij refers to the corresponding components of the Ricci tensor. In order to facilitate the computations, we often implicitly assume that {x1 , . . . , xn } are normal coordinates at x ∈ M with respect to the appropriate metric. We use the standard shorthand: Given a real-valued function f on the ∂f manifold M, the notation fi stands for ∂x , the notation fij refers to the Hessian of f applied to i ∂ ∂ and , and f is the third covariant derivative applied to ∂x∂ i , ∂x∂ j , and ∂x∂ k . The subscript t ij k ∂xi ∂xj designates the differentiation in t ∈ [0, T ]. The proofs of Theorems 2.2 and 2.7 will involve a cut-off function on Bρ,T . The construction of this function will rely on the basic analytical result stated in the following lemma. This result is well known. For example, it was previously used in the proofs of Theorems 2.3 and 3.1 in [38]; see also [28, Chapter IV] and [32].
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Lemma 2.1. Given τ ∈ (0, T ], there exists a smooth function Ψ¯ : [0, ∞) × [0, T ] → R satisfying the following requirements: 1. The support of Ψ¯ (r, t) is a subset of [0, ρ] × [0, T ], and 0 Ψ¯ (r, t) 1 in [0, ρ] × [0, T ]. ¯ 2. The equalities Ψ¯ (r, t) = 1 and ∂∂rΨ (r, t) = 0 hold in [0, ρ2 ] × [τ, T ] and [0, ρ2 ] × [0, T ], respectively. 1
¯ ¯ ¯ 3. The estimate | ∂∂tΨ | C Ψτ 2 is satisfied on [0, ∞) × [0, T ] for some C¯ > 0, and Ψ¯ (r, 0) = 0 for all r ∈ [0, ∞). 2 ¯ ¯a ¯a ¯ 4. The inequalities − CaρΨ ∂∂rΨ 0 and | ∂∂rΨ2 | CaρΨ2 hold on [0, ∞) × [0, T ] for every a ∈ (0, 1) with some constant Ca dependent on a.
2.2. Space-only gradient estimates Let us begin by stating the local space-only gradient estimate. Theorem 2.2. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Assume that |Ric(x, t)| k for some k > 0 and all (x, t) ∈ Bρ,T . Suppose u : M × [0, T ] → R is a smooth positive function solving the heat equation (2.2). If u(x, t) A for some A > 0 and all (x, t) ∈ Bρ,T , then there exists a constant C that depends only on the dimension of M and satisfies √ 1 A |∇u| 1 1 + log C +√ + k u ρ u t
(2.3)
for all (x, t) ∈ B ρ ,T with t = 0. 2
We will now establish a lemma of computational character. It will play a significant part in the proof of Theorem 2.2. Lemma 2.3. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). Assume that |∇f |2 . Then the inequality u(x, t) 1 for all (x, t) ∈ Bρ,T . Let f = log u and w = (1−f )2
∂ 2f − w ∇f ∇w + 2(1 − f )w 2 ∂t 1−f
holds in Bρ,T . Proof. A direct computation demonstrates that n 2f 2 fi fij fj fi fj fij ∂ ij w= − +8 −4 ∂t (1 − f )2 (1 − f )3 (1 − f )2 i,j =1
+6
|∇f |4 |∇f |4 − 2 (1 − f )4 (1 − f )3
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and
4
n fi fij fj |∇f |4 ∇f ∇w − 4 = 2 (1 − f ) (1 − f )3 (1 − f )4
i,j =1
at every point (x, t) ∈ Bρ,T ; cf. [32,38]. Using these formulas, we conclude that n 2f 2 fi fij fj fi fj fij ∂ ij − w= +4 −4 ∂t (1 − f )2 (1 − f )3 (1 − f )2 i,j =1
|∇f |4 |∇f |4 ∇f ∇w −2 +2 4 (1 − f ) (1 − f ) (1 − f )3 2 n fi fj fij + =2 1−f (1 − f )2 +2
i,j =1
+2 at (x, t) ∈ Bρ,T .
|∇f |4 ∇f ∇w +2 − 2∇f ∇w (1 − f ) (1 − f )3
2f ∇f ∇w + 2(1 − f )w 2 1−f
2
The preparations required to prove Theorem 2.2 are now completed. Note that we will also make use of arguments from the paper [38]. Proof of Theorem 2.2. Without loss of generality, we can assume A = 1. If this is not the case, one should just carry out the proof replacing u(x, t) with u(x,t) A . Let us pick a number τ ∈ (0, T ] and fix a function Ψ¯ (r, t) satisfying the conditions of Lemma 2.1. We will establish (2.3) at (x, τ ) for all x such that dist(x, x0 , τ ) < ρ2 . Because τ is chosen arbitrarily, the assertion of the theorem will immediately follow. Define Ψ : M × [0, T ] → R by the formula Ψ (x, t) = Ψ¯ dist(x, x0 , t), t . It is easy to see that Ψ (x, t) is supported in the closure of Bρ,T . This function is smooth at (x , t ) ∈ M × [0, T ] whenever x = x0 and x is not in the cut locus of x0 with respect to the |∇f |2 metric g(x, t ). We will employ the notation f = log u and w = (1−f introduced in Lemma 2.3. )2 2f It will also be convenient for us to write β instead of − 1−f ∇f . Our strategy is to estimate
( − ∂t∂ )(Ψ w) and scrutinize the produced formula at a point where Ψ w attains its maximum. The desired result will then follow.
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We use Lemma 2.3 to conclude that ∂ − (Ψ w) Ψ −β∇w + 2(1 − f )w 2 + (Ψ )w + 2∇Ψ ∇w − Ψt w ∂t in the portion of Bρ,T where Ψ (x, t) is smooth. This implies 2 ∂ (Ψ w) −β∇(Ψ w) + ∇Ψ ∇(Ψ w) + 2Ψ (1 − f )w 2 − ∂t Ψ + wβ∇Ψ − 2
|∇Ψ |2 w + (Ψ )w − Ψt w. Ψ
(2.4)
The latter inequality holds in the part of Bρ,T where Ψ (x, t) is smooth and nonzero. Now let (x1 , t1 ) be a maximum point for Ψ w in the closure of Bρ,T . If (Ψ w)(x1 , t1 ) is equal to 0, then (Ψ w)(x, τ ) = w(x, τ ) = 0 for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . This yields ∇u(x, τ ) = 0, and the estimate (2.3) becomes obvious at (x, τ ). Thus, it suffices to consider the case where (Ψ w)(x1 , t1 ) > 0. In particular, (x1 , t1 ) must be in Bρ,T , and t1 must be strictly positive. A standard argument due to E. Calabi (see, for example, [28, p. 21]) enables us to assume that Ψ (x, t) is smooth at (x1 , t1 ). Because (x1 , t1 ) is a maximum point, the formulas (Ψ w)(x1 , t1 ) 0, ∇(Ψ w)(x1 , t1 ) = 0, and (Ψ w)t (x1 , t1 ) 0 hold true. Together with (2.4), they yield 2Ψ (1 − f )w 2 −wβ∇Ψ + 2
|∇Ψ |2 w − (Ψ )w + Ψt w Ψ
(2.5)
at (x1 , t1 ). We will now estimate every term in the right-hand side. This will lead us to the desired result. A series of computations implies that |wβ∇Ψ | Ψ (1 − f )w 2 +
c1 f 4 , ρ 4 (1 − f )3
|∇Ψ |2 1 c1 w Ψ w2 + 4 , Ψ 8 ρ 1 c1 −(Ψ )w Ψ w 2 + 4 + c1 k 2 8 ρ at (x1 , t1 ) for some constant c1 > 0; see [32,38]. Here, we have used the inequality for the weighted arithmetic mean and the weighted geometric mean, as well as the properties of the function Ψ¯ (r, t) given by Lemma 2.1. Our next mission is to find a suitable bound for (Ψt w)(x1 , t1 ). It is clear that (Ψt w)(x1 , t1 ) =
∂ Ψ¯ dist(x1 , x0 , t1 ), t1 w(x1 , t1 ) ∂t ∂ ∂ Ψ¯ dist(x1 , x0 , t1 ), t1 dist(x1 , x0 , t1 ) w(x1 , t1 ). + ∂r ∂t
(2.6)
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We also observe that ∂ Ψ¯ 1 c2 2 ∂t dist(x1 , x0 , t1 ), t1 w(x1 , t1 ) 16 Ψ w (x1 , t1 ) + τ 2 for a positive constant c2 . Because the function Ψ¯ (r, t) satisfies the conditions listed in Lemma 2.1, the inequality ∂ Ψ¯ C 12 1 ∂r dist(x1 , x0 , t1 ), t1 ρ Ψ 2 (x1 , t1 )
(2.7)
holds with C 1 > 0. It remains to estimate the derivative of the distance. Utilizing the assumptions 2 of the theorem, we conclude that ∂ dist(x1 , x0 , t1 ) sup ∂t
dist(x 1 ,x0 ,t1 )
Ric d ζ (s), d ζ (s) ds ds ds
0
k dist(x1 , x0 , t1 ) kρ.
(2.8)
In this particular formula, Ric designates the Ricci curvature of g(x, t1 ). The supremum is taken over all the minimal geodesics ζ (s), with respect to g(x, t1 ), that connect x0 to x1 and are parametrized by arclength; see, e.g., [12, proof of Lemma 8.28]. It now becomes clear that Ψt w
1 1 c2 1 c2 Ψ w 2 + 2 + C 1 kwΨ 2 Ψ w 2 + 2 + c3 k 2 2 16 8 τ τ
at (x1 , t1 ) for some c3 > 0. We have thus found estimates for every term in the right-hand side of (2.5). We will combine them all, and the assertion of the theorem will shortly follow. Given the preceding considerations, formula (2.5) implies Ψ (1 − f )w 2
c4 f 4 4 ρ (1 − f )3
1 c4 c4 + Ψ w 2 + 4 + 2 + c4 k 2 2 ρ τ
at the point (x1 , t1 ). The constant c4 here equals max{3c1 , c2 , c1 + c3 }. Since f (x, t) 0 and f4 1, we can conclude that (1−f )4 Ψ w2
c4 f 4 4 ρ (1 − f )4
Ψ 2 w2 Ψ w2
1 c4 c4 + Ψ w 2 + 4 + 2 + c4 k 2 , 2 ρ τ
4c4 2c4 + 2 + 2c4 k 2 ρ4 τ
at (x1 , t1 ). Because Ψ (x, τ ) = 1 when dist(x, x0 , τ ) < ρ2 , the estimate
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w(x, τ ) = (Ψ w)(x, τ ) (Ψ w)(x1 , t1 )
C2 C2 + + C2k τ ρ2
√ holds with C = 2 c4 for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . Recalling the definition of w(x, t) and the fact that τ ∈ (0, T ] was chosen arbitrarily, we obtain the inequality √ |∇f (x, t)| 1 1 C +√ + k 1 − f (x, t) ρ t for (x, t) ∈ B ρ ,T provided t = 0. The assertion of the theorem follows by an elementary compu2 tation. 2 Our next step is to assume M is compact and state a global gradient estimate for the function u(x, t). This result was previously established in [38,6]. We restate it here for the completeness of our exposition. Moreover, we believe it is appropriate to present the proof, which is quite short. A computation from this proof will be used in Section 3. Theorem 2.4. (See Q. Zhang [38], X. Cao and R. Hamilton [6].) Suppose the manifold M is compact, and let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (2.1). Assume a smooth positive function u : M × [0, T ] → R satisfies the heat equation (2.2). Then the estimate |∇u| u
A 1 log , t u
x ∈ M, t ∈ (0, T ],
(2.9)
holds with A = supM u(x, 0). Remark 2.5. The maximum principle implies that A is actually equal to supM×[0,T ] u(x, t). This explains why the right-hand side of (2.9) is well defined. A Proof of Theorem 2.4. Consider the function P = t |∇u| u − u log u on the set M × [0, T ]. It is clear that P (x, 0) is nonpositive for every x ∈ M. A computation shows that 2
∂ ∂ |∇u|2 − P =t − ∂t ∂t u n ui uj 2 t uij − =2 0, u u
x ∈ M, t ∈ [0, T ].
i,j =1
In accordance with the maximum principle, this implies P (x, t) is nonpositive for all (x, t) ∈ M × [0, T ]. The desired assertion follows immediately. 2 2.3. Space–time gradient estimates This subsection establishes Li–Yau-type inequalities for system (2.1)–(2.2). We will obtain a local and a global estimate. The following lemma will be important to our considerations; cf. Lemma 1 in [28, Chapter IV]. It will also reoccur in Section 3.
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Lemma 2.6. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Assume that −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ Bρ,T . Suppose u : M × [0, T ] → R is a smooth positive function satisfying the heat equation (2.2). Given α 1, define f = log u and F = t (|∇f |2 − αft ). The estimate 2 ∂ 2aαt − F −2∇f ∇F + |∇f |2 − ft − |∇f |2 − αft ∂t n αtn max k12 , k22 , (x, t) ∈ Bρ,T , − 2k1 αt|∇f |2 − (2.10) 2b holds for any a, b > 0 such that a + b = α1 . Proof. We begin by finding a convenient bound on F . Observe that
n 2 F = t 2 fij + 2fj fj ii − α(ft ) , x ∈ M, t ∈ [0, T ]. i,j =1
Our assumption on the Ricci curvature of M implies the inequality n
fj fj ii =
i,j =1
n
(fj fiij + Rij fi fj )
i,j =1
= ∇f ∇(f ) + Ric(∇f, ∇f ) ∇f ∇(f ) − k1 |∇f |2 at an arbitrary point (x, t) ∈ Bρ,T . Using (2.1), we can show that (ft ) = (f )t − 2
n
Rij fij .
i,j =1
Consequently, the estimate
n 2 2 F t 2 fij + 2αRij fij + 2∇f ∇(f ) − 2k1 |∇f | − α(f )t i,j =1
holds at (x, t) ∈ Bρ,T . Our next step is to find a suitable bound on those terms in the right-hand side that involve fij . We do so by completing the square. More specifically, observe that n n 2 fij + αRij fij = (aα + bα)fij2 + αRij fij i,j =1
i,j =1
=
n √ Rij 2 α aαfij2 + α b fij + √ − Rij2 4b 2 b i,j =1
n α 2 2 aαfij − Rij 4b i,j =1
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at (x, t) ∈ Bρ,T for any a, b > 0 such that a + b = α1 . Employing the standard inequality n i,j =1
fij2
(f )2 n
and the assumptions of the lemma, we obtain the estimate n 2 aα αn (f )2 − max k12 , k22 , fij + αRij fij n 4b
(x, t) ∈ Bρ,T .
i,j =1
It is easy to conclude that
2 2 2aα αn 2 2 F t (f ) + 2∇f ∇(f ) − 2k1 |∇f | − α(f )t − max k1 , k2 n 2b 2 2aαt ft − |∇f |2 + 2t∇f ∇ ft − |∇f |2 = n αtn − 2k1 t|∇f |2 − αt ft − |∇f |2 t − max k12 , k22 (2.11) 2b in the set Bρ,T . Formula (2.11) provides us with a convenient bound on F . Let us now include the derivative of F in t ∈ [0, T ] into our considerations. One easily computes ∂F = |∇f |2 − αft + t |∇f |2 − αft t . ∂t Subtracting this from (2.11), we see that the inequality 2 2aαt ∂ F ft − |∇f |2 + 2t∇f ∇ ft − |∇f |2 − 2k1 t|∇f |2 − ∂t n αtn max k12 , k22 − |∇f |2 − αft + (α − 1)t |∇f |2 t − 2b holds in the set Bρ,T . In order to arrive to (2.10) from here, we need to estimate (|∇f |2 )t . The Ricci flow equation (2.1) and the assumptions of the lemma imply |∇f |2 t = 2∇f ∇(ft ) + 2 Ric(∇f, ∇f ) 2∇f ∇(ft ) − 2k1 |∇f |2 at (x, t) ∈ Bρ,T . As a consequence, −
2 2aαt ∂ F ft − |∇f |2 − |∇f |2 − αft ∂t n αtn max k12 , k22 − 2t∇f ∇ |∇f |2 − αft − 2k1 αt|∇f |2 − 2b
in Bρ,T . The desired assertion follows immediately.
2
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With Lemma 2.6 at hand, we are ready to establish the local space–time gradient estimate. We will also make use of arguments from the proof of Theorem 4.2 in [28, Chapter IV]. Recall that n designates the dimension of M. Theorem 2.7. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Suppose −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ Bρ,T . Consider a smooth positive function u : M × [0, T ] → R solving the heat equation (2.2). There exists a constant C
that depends only on the dimension of M and satisfies the estimate |∇u|2 α2 1 ut nk1 α 3
2 C + max{k + − α α , k } + 1 2 2 2 u α−1 u ρ (α − 1) t
(2.12)
for all α > 1 and all (x, t) ∈ B ρ ,T with t = 0. 2
Proof. We preserve the notation f = log u and F = t (|∇f |2 − αft ) from Lemma 2.6. Our strategy in this proof will be similar to that in the proof of Theorem 2.2. The role of the function w(x, t) now goes to the function F (x, t). Let us pick τ ∈ (0, T ] and fix Ψ¯ (r, t) satisfying the conditions of Lemma 2.1. Define Ψ : M × [0, T ] → R by setting Ψ (x, t) = Ψ¯ dist(x, x0 , t), t . We will establish (2.12) at (x, τ ) for x ∈ M such that dist(x, x0 , τ ) < ρ2 . This will complete the proof. Our plan is to estimate ( ∂t∂ − )(Ψ F ) and analyze the result at a point where the function Ψ F attains its maximum. The required conclusion will follow therefrom. Lemma 2.6 and some straightforward computations imply
∂ − (Ψ F ) −2∇f ∇(Ψ F ) + 2F ∇f ∇Ψ ∂t 2 2aαt + |∇f |2 − ft − |∇f |2 − αft Ψ n αtn ¯ 2 − 2k1 αt|∇f |2 + k Ψ 2b + (Ψ )F + 2
∇Ψ |∇Ψ |2 ∂Ψ ∇(Ψ F ) − 2 F− F Ψ Ψ ∂t
(2.13)
with k¯ = max{k1 , k2 }. This inequality holds in the part of Bρ,T where Ψ (x, t) is smooth and strictly positive. Let (x1 , t1 ) be a maximum point for the function Ψ F in the set {(x, t) ∈ M × [0, τ ] | dist(x, x0 , t) ρ}. We may assume (Ψ F )(x1 , t1 ) > 0 without loss of generality. Indeed, if this is not the case, then F (x, τ ) 0 and (2.12) is evident at (x, τ ) whenever dist(x, x0 , τ ) < ρ2 . We may also assume that Ψ (x, t) is smooth at (x1 , t1 ) due to a standard trick explained, for example, in [28, p. 21]. Since (x1 , t1 ) is a maximum point, the formulas (Ψ F )(x1 , t1 ) 0, ∇(Ψ F )(x1 , t1 ) = 0, and (Ψ F )t (x1 , t1 ) 0 hold true. Combined with (2.13), they yield
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0 2F ∇f ∇Ψ 2 2aαt1 αt1 n ¯ 2 |∇f |2 − ft − |∇f |2 − αft − 2k1 αt1 |∇f |2 − + k Ψ n 2b + (Ψ )F − 2
|∇Ψ |2 ∂Ψ F− F Ψ ∂t
(2.14)
at (x1 , t1 ). We will now use (2.14) to show that a certain quadratic expression in Ψ F is nonpositive. The desired result will then follow. Let us recall Lemma 2.1 and apply the Laplacian comparison theorem to conclude that C 21 |∇Ψ |2 − 22 , − Ψ ρ Ψ −
C1
2
ρ2
1
−
C1 Ψ 2 2
ρ
1 d1 d1 Ψ 2 (n − 1) k1 coth( k1 ρ) − 2 − k1 ρ ρ
at the point (x1 , t1 ) with d1 a positive constant depending on n. There exists C¯ > 0 such that the inequality ¯ 12 1 CΨ ∂Ψ ¯ 2 − − C 1 kΨ − 2 ∂t τ holds true; cf. (2.6), (2.7), and (2.8). Using these observations along with (2.14), we find the estimate 0 −2F |∇f ||∇Ψ | 2 2aαt1 αt1 n ¯ 2 2 2 2 |∇f | − ft − |∇f | − αft − 2k1 αt1 |∇f | − + k Ψ n 2b 1 1 1 1 Ψ2 Ψ2 ¯ 2 F + d2 − 2 − − kΨ k1 − ρ τ ρ ¯ If one further multiplies by tΨ and makes at (x1 , t1 ). Here, d2 is equal to max{3d1 , C 1 , 3C 21 , C}. 2
2
a few elementary manipulations, one will obtain 3
0 −2t1 F
C1 Ψ 2 2
ρ
|∇f |
2 2t12 n2 α ¯ 2 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ n 4b √ 1 k1 1 ¯ − − k (Ψ F ) − Ψ F + d 2 t1 − 2 − ρ τ ρ
+
(2.15)
at (x1 , t1 ). Our next step is to estimate the first two terms in the right-hand side. In order to do so, we need a few auxiliary pieces of notation.
M. Bailesteanu et al. / Journal of Functional Analysis 258 (2010) 3517–3542 1
Define y = Ψ |∇f |2 and z = Ψft . It is clear that y 2 (y − αz) = yields
3531
3
Ψ 2 F |∇f | t
when t = 0, which
2 n2 α ¯ 2 2 2t 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ ρ n 4b n2 α ¯ 2 2 nC 12 1 2t 2 aα(y − z)2 − nk1 αy − k Ψ − y 2 (y − αz) . n 4b ρ 3
−2tF
C1 Ψ 2 2
|∇f | +
Let us observe that (y − z)2 =
1 (α − 1)2 2 2(α − 1) 2 (y − αz) + y + y(y − αz) α2 α2 α2
and plug this into the previous estimate. Regrouping the terms and applying the inequality κ1 v 2 − κ2
κ2 v − 4κ21 valid for κ1 , κ2 > 0 and v ∈ R, we obtain 2 n2 α ¯ 2 2 2t 2 aα Ψ |∇f |2 − Ψft − nk1 αΨ 2 |∇f |2 − k Ψ ρ n 4b 2 2 3 2 2 2 n k1 α 2t n d2 α n2 α ¯ 2 2 2t a − . (y − αz)2 − (y − αz) + Ψ k n α n 8aρ 2 (α − 1) 4b 4a(α − 1)2 3
−2tF
C1 Ψ 2 2
|∇f | +
Because t (y − αz) = Ψ F by definition, (2.15) now implies 0
2a α nd2 t1 ρ2 (Ψ F )2 + − 2 + 1 + ρ k¯ + + ρ 2 k¯ − 1 (Ψ F ) nα a(α − 1) τ ρ
nk12 α 3 αn 2 ¯ 2 2 t k Ψ t12 − 2 2b 1 2a(α − 1) α ρ2 2a d 3 t1 2 2¯ (Ψ F ) + − 2 + + ρ k − 1 (Ψ F ) nα a(α − 1) τ ρ −
−
nk12 α 3 αn 2 ¯ 2 2 t k Ψ t12 − 2 2b 1 2a(α − 1)
at (x1 , t1 ) with d3 = 4nd2 . The expression in the last two lines is a polynomial in Ψ F of degree 2. Consequently, in accordance with the quadratic formula,
α nα d3 t1 a ¯ k1 α ρ2 2¯ ΨF + +ρ k +1+ t1 + t1 kΨ 2a ρ 2 a(α − 1) τ α−1 b at (x1 , t1 ). We will now use this conclusion to obtain a bound on F (x, τ ) for an appropriate range of x ∈ M. Recall that Ψ (x, τ ) = 1 whenever dist(x, x0 , τ ) < ρ2 . Besides, (x1 , t1 ) is a maximum point for Ψ F in the set {(x, t) ∈ M × [0, τ ] | dist(x, x0 , t) ρ}. Hence
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F (x, τ ) = (Ψ F )(x, τ ) (Ψ F )(x1 , t1 )
α ρ2 nk1 α 2 ατ nk¯ 1 nα nαd3 τ 2¯ + +ρ k + + τ+ τ 2a 2a(α − 1) 2 ab 2aρ 2 a(α − 1) for all x ∈ M such that dist(x, x0 , τ ) < ρ2 . Since τ ∈ (0, T ] was chosen arbitrarily, this formula implies α ρ2 αd4 + + ρ 2 k¯ |∇f |2 − αft (x, t) 2 t aρ a(α − 1)
αnk¯ 1 nk1 α 2 + , (x, t) ∈ B ρ ,T , + 2 2a(α − 1) 2 ab 1 with d4 = max{nd3 , n} as long as t = 0. If we set a = 2α , note that b = α1 − a, and define the
constant C appropriately, estimate (2.12) will follow by a straightforward computation. 2 1 Remark 2.8. The value 2α for the parameter a in the proof of the theorem might not be optimal. It is not unlikely that a different a will lead to a sharper estimate.
Let us now consider the case where the manifold M is compact. We will present a global estimate on u(x, t) demanding that the Ricci curvature of M be nonnegative. A related inequality for (2.1)–(2.2) may be found in [14]. Theorem 2.9. Suppose the manifold M is compact and (M, g(x, t))t∈[0,T ] is a solution to the Ricci flow (2.1). Assume that 0 Ric(x, t) kg(x, t) for some k > 0 and all (x, t) ∈ M × [0, T ]. Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). The estimate n |∇u|2 ut kn + − u 2t u2
(2.16)
holds for all (x, t) ∈ M × (0, T ]. Proof. As before, we write f instead of log u. It will be convenient for us to denote F1 = t (|∇f |2 − ft ). Fix τ ∈ (0, T ] and choose a point (x0 , t0 ) ∈ M × [0, τ ] where F1 attains its maximum on M × [0, τ ]. Our first step is to show that n F1 (x0 , t0 ) t0 kn + . 2
(2.17)
The assertion of the theorem will follow therefrom. If t0 = 0, then F1 (x, t0 ) is equal to 0 for every x ∈ M and estimate (2.17) becomes evident. Consequently, we can assume t0 > 0 without loss of generality. Lemma 2.6 and our conditions on the Ricci curvature of M imply the inequality 2a F12 F1 t0 n ∂ k2 − − − F1 −2∇f ∇F1 + ∂t n t0 t0 2(1 − a)
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for all a ∈ (0, 1) at the point (x0 , t0 ). Now recall that F1 attains its maximum at (x0 , t0 ). This tells us that F1 (x0 , t0 ) 0, ∂t∂ F1 (x0 , t0 ) 0, and ∇F1 (x0 , t0 ) = 0. In consequence, the estimate 2a F12 F1 t0 n k2 0 − − n t0 t0 2(1 − a) holds at (x0 , t0 ), and the quadratic formula yields n F1 (x0 , t0 ) 1+ 4a
4at02 2 k . 1+ 1−a
1+kt0 . The expression in the right-hand side is minimized in a ∈ (0, 1) when a is equal to 1+2kt 0 Plugging this value of a into the above inequality, we arrive at (2.17). Only a simple argument is now needed to complete the proof. The fact that (x0 , t0 ) is a maximum point for F1 on M × [0, τ ] enables us to conclude that
F1 (x, τ ) F1 (x0 , t0 ) t0 kn +
n n τ kn + 2 2
for all x ∈ M. Therefore, the estimate n |∇u|2 ut kn + − u 2τ u2 holds at (x, τ ). Because the number τ ∈ (0, T ] can be chosen arbitrarily, the assertion of the theorem follows. 2 Our last goal in this section is to state two Harnack inequalities for (2.1)–(2.2). These may be viewed as applications of Theorems 2.7 and 2.9; cf., for example, [28, Chapter IV]. One can find other Harnack inequalities for (2.1)–(2.2) in the papers [14,25]. We first introduce a piece of notation. Given x1 , x2 ∈ M and t1 , t2 ∈ (0, T ) satisfying t1 < t2 , define 2 t2 d Γ (x1 , t1 , x2 , t2 ) = inf γ (t) dt. dt t1
The infimum is taken over the set Θ(x1 , t1 , x2 , t2 ) of all the smooth paths γ : [t1 , t2 ] → M that connect x1 to x2 . We remind the reader that the norm | · | depends on t. Let us now present a lemma. It will be the key to the proof of our results. Lemma 2.10. Suppose (M, g(x, t))t∈[0,T ] is a complete solution to the Ricci flow (2.1). Let u : M × [0, T ] → R be a smooth positive function satisfying the heat equation (2.2). Define f = log u and assume that 1 ∂f A3 |∇f |2 − A2 − , ∂t A1 t
x ∈ M, t ∈ (0, T ],
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for some A1 , A2 , A3 > 0. Then the inequality − A3 A1 t2 A1 A2 u(x2 , t2 ) u(x1 , t1 ) exp − Γ (x1 , t1 , x2 , t2 ) − (t2 − t1 ) t1 4 A1 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) such that t1 < t2 . Proof. The method we use is rather traditional; see, for example, [28, Chapter IV] and [6]. Consider a path γ (t) ∈ Θ(x1 , t1 , x2 , t2 ). We begin by computing d d ∂ f γ (t), t = ∇f γ (t), t γ (t) + f γ (t), s s=t dt dt ∂s d 2 1 A3 ∇f γ (t), t − A2 − −∇f γ (t), t γ (t) + dt A1 t 2 A1 d 1 A3 A2 + , t ∈ [t1 , t2 ]. − γ (t) − 4 dt A1 t κ2
The last step is a consequence of the inequality κ1 v 2 − κ2 v − 4κ21 valid for κ1 , κ2 > 0 and v ∈ R. The above implies t2 f (x2 , t2 ) − f (x1 , t1 ) =
d f γ (t), t dt dt
t1
A1 − 4
2 t2 d γ (t) dt − A2 (t2 − t1 ) − A3 ln t2 . dt A1 A1 t1
t1
The assertion of the lemma follows by exponentiating.
2
We are ready to formulate our Harnack inequalities for (2.1)–(2.2). The first one applies on noncompact manifolds. The second one does not, but it provides a more explicit estimate. Theorem 2.11. Let (M, g(x, t))t∈[0,T ] be a complete solution to the Ricci flow (2.1). Assume that −k1 g(x, t) Ric(x, t) k2 g(x, t) for some k1 , k2 > 0 and all (x, t) ∈ M × [0, T ]. Suppose a smooth positive function u : M × [0, T ] → R satisfies the heat equation (2.2). Given α > 1, the estimate −C α t2 u(x2 , t2 ) u(x1 , t1 ) t1 α nk1 α 2
(t2 − t1 ) × exp − Γ (x1 , t1 , x2 , t2 ) − C α max{k1 , k2 } + 4 α−1 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) such that t1 < t2 . The constant C
comes from Theorem 2.7.
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Proof. Letting ρ go to infinity in (2.12), we conclude that nk1 α 3 ut 1 |∇u|2 C α 2
2 − − C α max{k1 , k2 } + u α t α−1 u2 on M × (0, T ]. The desired assertion is now a consequence of Lemma 2.10.
2
Theorem 2.12. Suppose M is compact and (M, g(x, t))t∈[0,T ] is a solution to the Ricci flow (2.1). Assume that 0 Ric(x, t) kg(x, t) for some k > 0 and all (x, t) ∈ M × [0, T ]. Consider a smooth positive function u : M × [0, T ] → R satisfying the heat equation (2.2). The estimate − n 2 t2 1 u(x2 , t2 ) u(x1 , t1 ) exp − Γ (x1 , t1 , x2 , t2 ) − kn(t2 − t1 ) t1 4 holds for all (x1 , t1 ) ∈ M × (0, T ) and (x2 , t2 ) ∈ M × (0, T ) as long as t1 < t2 . Proof. Theorem 2.9 implies ut |∇u|2 n − kn − , 2 u 2t u
x ∈ M, t ∈ (0, T ].
One may now use Lemma 2.10 to complete the proof.
2
3. Manifolds with boundary This section considers a compact manifold with boundary evolving under the Ricci flow and offers heat equation estimates on this manifold. We will present variants of Theorems 2.4 and 2.9. The proofs are largely based on the Hopf maximum principle. 3.1. The Ricci flow Suppose M is a compact, connected, oriented, smooth manifold with nonempty boundary ∂M. Consider a Riemannian metric g(x, t) on M that evolves under the Ricci flow. The parameter t runs through the interval [0, T ]. We investigate the case where the boundary ∂M remains umbilic for all t ∈ [0, T ]. More precisely, given a smooth nonnegative function λ(t) on [0, T ], we assume that (M, g(x, t))t∈[0,T ] is a solution to the problem ∂ g(x, t) = −2 Ric(x, t), x ∈ M, t ∈ [0, T ], ∂t II(x, t) = λ(t)g(x, t), x ∈ ∂M, t ∈ [0, T ].
(3.1)
In the second line, g(x, t) is understood to be restricted to the tangent bundle of ∂M. The notation II(x, t) here stands for the second fundamental form of ∂M with respect to g(x, t). That is, ∂ II(X, Y ) = DX Y ∂ν
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Fig. 1. The Ricci flow (3.1) with λ(t) = λ0 after the normalization.
Fig. 2. The Ricci flow (3.1) with λ(t) = λ1 (t) before the normalization.
if X and Y are tangent to the boundary at the same point. The letter D refers to the Levi-Civita ∂ is the outward unit normal vector field on ∂M with connection corresponding to g(x, t), and ∂ν respect to g(x, t). We should explain that problem (3.1) has different geometric meanings for different choices of the function λ(t). E.g., let us assume that λ(t) is equal to the same constant λ0 for all t ∈ [0, T ]. The papers [29,13] discuss this case in detail. Theorem 3 in [13] suggests that the Ricci flow (3.1), if normalized so as to preserve the volume of M, takes a sufficiently well-behaved Riemannian metric on M to a metric with totally geodesic boundary. An example of such an evolution is shown in Fig. 1. By letting λ(t) be a nontrivial function of t, we allow our results to include several cases which are, in a sense, more natural than the one just described. For instance, suppose we apply the Ricci flow to the sphere S in Fig. 1. The manifold M will then evolve along with S. This evolution will be described by Eqs. (3.1) with λ(t) equal to some nonconstant function λ1 (t). We provide an illustration in Fig. 2. Let us normalize the Ricci flow on the sphere S so as to preserve the volume of S. It is well known that S will then remain unchanged for all t. Analogously, we can normalize the Ricci flow (3.1) with λ(t) = λ1 (t) so as to preserve the volume of M. This will allow a better comparison with the situation shown in Fig. 1. After such a normalization, the flow will keep M unchanged for all t. 3.2. Gradient estimates Let us recollect some notation. The operator is the Laplacian given by the metric g(x, t). We write ∇ and | · | for the gradient and the norm with respect to g(x, t). Our attention will be centered round the heat equation ∂ u(x, t) = 0, x ∈ M, t ∈ [0, T ], (3.2) − ∂t
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with the Neumann boundary condition ∂ u(x, t) = 0, ∂ν
x ∈ ∂M, t ∈ [0, T ].
(3.3)
The results in this section still hold, with obvious modifications, if the solution u(x, t) is only defined on M × (0, T ]. In this case, one just has to replace u(x, t) and g(x, t) with u(x, t + ) and g(x, t + ) for a sufficiently small > 0, apply the corresponding theorem, and then let go to 0. Our first result is a space-only estimate. It is analogous to (2.9). Theorem 3.1. Let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (3.1). Suppose u(x, t) : M × [0, T ] → R is a smooth positive function satisfying the heat equation (3.2) with the Neumann boundary condition (3.3). Then the estimate |∇u| u
A 1 log , t u
x ∈ M, t ∈ (0, T ],
(3.4)
holds with A = supM u(x, 0). Remark 3.2. Using the strong maximum principle and the Hopf maximum principle, one can show that A is actually equal to supM×[0,T ] u(x, t). Consequently, the right-hand side of (3.4) is well defined. We emphasize that the Laplacian , the normal vector field | · | appearing above depend on the parameter t ∈ [0, T ].
∂ ∂ν ,
the gradient ∇, and the norm
A Proof of Theorem 3.1. Introduce the function P = t |∇u| u − u log u . One may repeat the computation from the proof of Theorem 2.4 and conclude that 2
∂ − P 0 ∂t
(3.5)
for all (x, t) ∈ M × (0, T ]. Employing this inequality, we will demonstrate that P must be nonpositive. The assertion of the theorem will immediately follow. Fix τ ∈ (0, T ]. Let us prove that the function P is nonpositive on M × [0, τ ]. If P attains its largest value at the point (x, 0) for some x ∈ M, then P is less than or equal to −u log Au computed at (x, 0). In this case, P must be nonpositive. Suppose this function attains its largest value at the point (x, t) for some x in the interior of M and some t in the interval (0, τ ]. We then use estimate (3.5) and the strong maximum principle. They imply P must also assume its largest value at (x, 0). As a consequence, P is nonpositive. Thus, we only have to consider the situation where this function has no maxima on M × [0, τ ] away from ∂M × (0, τ ]. Unless this is the case, P cannot become strictly greater than 0 anywhere. Let (x0 , t0 ) ∈ ∂M × (0, τ ] be a point where the function P attains its largest value on M × [0, τ ]. The Hopf maximum principle tells us that the inequality ∂ P (x0 , t0 ) > 0 ∂ν
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holds true. But the Neumann boundary condition (3.3) and the second line of (3.1) imply ∂ ∂ ∂ |∇u|2 ∂ A ∂ 2 1 P =t |∇u| −t 2 u− u log + u ∂ν ∂ν u ∂ν u ∂ν u ∂ν 1 t t ∂ |∇u|2 = 2 D ∂ (∇u) ∇u = −2 II(∇u, ∇u) =t ∂ν u u ∂ν u t = −2 λ(t)|∇u|2 0 u for all (x, t) ∈ ∂M × [0, τ ] (related computations appear in [26] and [28, Chapter IV]). Consequently, P must have a maximum on M × [0, τ ] away from ∂M × (0, τ ]. We conclude that P is nonpositive on M × [0, τ ]. Since the number τ ∈ (0, T ] can be chosen arbitrarily, the same assertion holds on M × [0, T ]. The theorem follows at once. 2 Remark 3.3. Consider the case where the metric g(x, t) does not depend on t and Eqs. (3.1) are not assumed. Suppose the Ricci curvature of M is nonnegative and ∂M is convex in the sense that the second fundamental form of ∂M is nonnegative definite. Then the solution u(x, t) of problem (3.2)–(3.3) satisfies (3.4). This fact can be established by the same argument we used to prove the theorem. The computation leading to (3.5) in this case may be found in [15]. Our next estimate is similar to (2.16). Henceforth, the subscript t denotes the derivative in t. The number n is the dimension of the manifold M. Theorem 3.4. Let (M, g(x, t))t∈[0,T ] be a solution to the Ricci flow (3.1). Consider a smooth positive function u(x, t) : M × [0, T ] → R satisfying the heat equation (3.2) with the Neumann boundary condition (3.3). If 0 Ric(x, t) kg(x, t) for a fixed k > 0 and all (x, t) ∈ M × [0, T ], then the estimate n |∇u|2 ut kn + − u 2t u2
(3.6)
holds for all (x, t) ∈ M × (0, T ]. Remark 3.5. We will make use of Lemma 2.6 in the arguments below. The proof of this lemma relies on local computations. Therefore, it prevails on manifolds with boundary. Proof of Theorem 3.4. Fix τ ∈ (0, T ]. Introduce the functions f = log u and F1 = t (|∇f |2 − ft ). Let us pick a point (x0 , t0 ) ∈ M × [0, τ ] where F1 attains its maximum on M × [0, τ ]. We will demonstrate that the inequality F1 (x0 , t0 ) t0 kn +
n 2
(3.7)
holds true. The assertion of the theorem will follow therefrom. If t0 = 0, then F1 (x, t0 ) = 0 for every x ∈ M and estimate (3.7) is evident. Consequently, we assume t0 > 0. In accordance with Lemma 2.6 and our conditions on the Ricci curvature of M,
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the inequality −
∂ 2a F12 F1 t0 n F1 −2∇f ∇F1 + k2 − − ∂t n t0 t0 2(1 − a)
holds for all a ∈ (0, 1) at the point (x0 , t0 ). Setting a = and using the quadratic formula, we see that
1+kt0 1+2kt0
like in the proof of Theorem 2.9
∂ nkt0 (1 + 2kt0 ) n − F1 + 2∇f ∇F1 F1 + F1 − t0 kn − . ∂t 2(1 + kt0 ) 2
(3.8)
If (3.7) fails to hold, then the right-hand side of (3.8) must be strictly positive. We will now show this is impossible. Suppose x0 lies in the interior of M. The fact that (x0 , t0 ) is a maximum point then yields F1 (x0 , t0 ) 0, ∂t∂ F1 (x0 , t0 ) 0, and ∇F1 (x0 , t0 ) = 0. Hence the right-hand side of (3.8) cannot be strictly positive. Suppose now x0 lies in the boundary of M. If the right-hand side of (3.8) is indeed positive, then the Hopf maximum principle tells us that the inequality ∂ F1 > 0 ∂ν
(3.9)
holds at (x0 , t0 ). We will make a computation to show this cannot be the case. Fix a system {y1 , . . . , yn } of local coordinates in a neighborhood U of the point x0 demanding that U ∩ ∂M = {x ∈ U | yn (x) = 0}. We write gij and Rij for the corresponding components of the metric and the Ricci tensor. Clearly, they depend on the parameter t. Without loss of generality, assume ∂y∂ 1 , . . . , ∂y∂n−1 are all orthogonal to ∂y∂ n on the boundary with respect to g(x, t0 ). It is easy to see that g in ∂ ∂ =− 1 ∂ν (g nn ) 2 ∂yi n
(3.10)
i=1
in U ∩ ∂M. Here, g ij are the components of the matrix inverse to (gij )ni,j =1 . ∂ The Neumann boundary condition (3.3) implies ∂ν f = 0. Utilizing this fact, we obtain ∂ ∂ ∂ ∂ 2 F1 = t |∇f | − ft = t 2 D ∂ (∇f ) ∇f − ft ∂ν ∂ν ∂ν ∂ν ∂ν ∂ ∂ ∂ ∂ f− = t −2 II(∇f, ∇f ) + f ∂t ∂ν ∂t ∂ν ∂ ∂ f . = t −2 II(∇f, ∇f ) + ∂t ∂ν For related computations, see [26] and [28, Chapter IV]. According to (3.10) and the first formula in (3.1), the equality
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n ∂ nn ∂ in nn 12 ∂ 1 ∂ ∂ ∂t g in g = − nn g − g 1 ∂t ∂ν g ∂t ∂y i 2(g nn ) 2 i=1 n 2Rj l g j i g nl Rj l g j n g nl g in ∂ =− − 1 1 ∂yi (g nn ) 2 g nn (g nn ) 2 i,j,l=1
holds in U ∩ ∂M. A calculation based on the Codazzi equation and the second line in (3.1) then implies ∂ ∂ ∂ = Rnn g nn ∂t ∂ν ∂ν near x0 at time t0 . Here, we make use of that fact that ∂y∂ 1 , . . . , ∂y∂n−1 are orthogonal to boundary with respect to g(x, t0 ). Combining the above equalities, we conclude that
∂ ∂yn
on the
∂ ∂ F1 = t0 −2 II(∇f, ∇f ) + Rnn g nn f ∂ν ∂ν = −2t0 II(∇f, ∇f ) = −2t0 λ(t0 )|∇f |2 0 at the point (x0 , t0 ). But this contradicts (3.9). Thus, the right-hand side of (3.8) cannot be strictly positive, and our assumption that (3.7) failed to hold must have been false. Because (x0 , t0 ) is a maximum point for F1 on M × [0, τ ], it is easy to see that F1 (x, τ ) F1 (x0 , t0 ) t0 kn +
n n τ kn + 2 2
for any x ∈ M. Consequently, n |∇u|2 ut kn + − 2 u 2τ u at (x, τ ). Since the number τ ∈ (0, T ] can be chosen arbitrarily, this yields the assertion of the theorem. 2 Acknowledgments Xiaodong Cao wishes to thank Professor Qi Zhang for useful discussions. X.C.’s research is partially supported by NSF grant DMS 0904432. Artem Pulemotov is grateful to Professor Leonard Gross for helpful conversations. A large portion of this paper was written when A.P. was a graduate student at Cornell University. At that time, he was partially supported by Professor Alfred Schatz’s NSF grant DMS 0612599. References [1] M. Arnaudon, R.O. Bauer, A. Thalmaier, A probabilistic approach to the Yang–Mills heat equation, J. Math. Pures Appl. 81 (2002) 143–166. [2] M. Arnaudon, K.A. Coulibaly, A. Thalmaier, Brownian motion with respect to a metric depending on time: definition, existence and applications to Ricci flow, C. R. Math. Acad. Sci. Paris 346 (2008) 773–778.
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Journal of Functional Analysis 258 (2010) 3543–3591 www.elsevier.com/locate/jfa
Dynamics of stochastic 2D Navier–Stokes equations Salah Mohammed a,∗,1 , Tusheng Zhang b a Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA b Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England,
United Kingdom Received 11 October 2009; accepted 9 November 2009 Available online 18 November 2009 Communicated by Paul Malliavin
Abstract In this paper, we study the dynamics of a two-dimensional stochastic Navier–Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2D Navier–Stokes equation generate a perfect and locally compacting C 1,1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier–Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space. © 2009 Elsevier Inc. All rights reserved. Keywords: Stochastic Navier–Stokes equation; Cocycle; Lyapunov exponents; Stable manifolds; Invariant manifolds
1. Introduction Two-dimensional stochastic Navier–Stokes equations (SNSE’s) are often used to describe the time evolution of an incompressible fluid in a smooth bounded planar domain. In this article, we characterize the long-time asymptotics of the following two-dimensional stochastic Navier–Stokes equation with Dirichlet boundary conditions: * Corresponding author.
E-mail addresses:
[email protected] (S. Mohammed),
[email protected] (T. Zhang). 1 The research of this author is supported by NSF award DMS-0705970.
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.007
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
du − νu dt + (u · ∇)u dt + ∇p dt = γ u dt +
∞
σk u(t) dWk (t),
k=1
(div u)(t, x) = 0,
x ∈ D, t > 0,
⎫ ⎪ ⎪ t > 0, ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
u(t, x) = 0, x ∈ ∂D, t > 0, u(0, x) = u0 (x), x ∈ D,
(1.1)
where D is a bounded domain in R2 with smooth boundary ∂D, u(t, x) ∈ R2 denotes the velocity field at time t and position x ∈ D, p(t, x) denotes the pressure field, and ν > 0 the viscosity coefficient. Moreover, the random force field is driven by independent one-dimensional standard Brownian motions Wk , k 1, and a deterministic linear drift term γ u dt with a fixed parameter γ . The Brownian motions Wk , k 1, are defined on a complete filtered Wiener space F , (Ft )t0 , P ). We assume that the noise parameters σk , k 1, are such that ∞ (Ω, 2 < ∞. σ k=1 k To formulate the dynamics of the above stochastic Navier–Stokes equation, we introduce the following standard spaces: Consider the Hilbert space
V := v ∈ H01 D, R2 : ∇ · v = 0 a.e. in D , with the norm vV :=
1 2
|∇v| dx 2
D
and inner product ·,·. Denote by H the closure of V in the L2 -norm |v|H :=
1 |v|2 dx
2
.
D
The inner product on H will be denoted by ·,·. Denote by PH the Helmholtz–Hodge projection from the Hilbert space L2 (D, R2 ) onto H . Define the (Stokes) operator A in H by the formula Au := −νPH u,
u ∈ H 2 D, R2 ∩ V ,
and the nonlinear operator B by
B(u, v) := PH (u · ∇)v , whenever u, v are such that (u · ∇v) belongs to the space L2 . We will often use the short notation B(u) := B(u, u). By applying the operator PH to each term of the above stochastic Navier–Stokes equation (SNSE), we can rewrite the equation in the following abstract form:
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 ∞
du(t) + Au(t) dt + B u(t) dt = γ u(t) dt + σk u(t) dWk (t)
3545
(1.2)
k=1
in L2 (0, T ; V ) with the initial condition u(0) = u0 ∈ H,
(1.3)
where V is the dual of V . Finally, and for the remainder of the article, we will adopt the following convention: Definition 1.1 (Perfection). A family of propositions {P (ω): ω ∈ Ω} is said to hold perfectly in ω if there is a sure event Ω ∗ ∈ F such that θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R and P (ω) is true for every ω ∈ Ω ∗ . There is a large amount of literature on the stochastic Navier–Stokes equation. We will only refer to some of it. A good reference for stochastic Navier–Stokes equations driven by additive noise is the book [4] and the references therein; see also [5]. The existence and uniqueness of solutions of stochastic 2D Navier–Stokes equations with multiplicative noise were obtained in [9,17]. Ergodic properties, invariant measures, asymptotic compactness and absorbing sets of stochastic 2D Navier–Stokes equations are studied in [8,12,11,2]. The results in [12] address important aspects of the ergodic theory and invariant measures for 2D stochastic Navier–Stokes equations with (additive) periodic random “kicks”. The small noise large deviation of the stochastic 2D Navier–Stokes equations was established in [17] and the large deviation of occupation measures was considered in [10]. Related results on the dynamics of semilinear stochastic partial differential equations are given in [6,7,3]. The purpose of this paper is to study the dynamics of the two-dimensional stochastic Navier– Stokes equation (1.1) driven by multiplicative noise. In particular, we will establish the following: • Existence of a perfect locally compacting C 1,1 cocycle (semiflow) generated by all solutions of the stochastic Navier–Stokes equation; • Long-time asymptotics for the stochastic semiflow given by a countable non-random Lyapunov spectrum of the linearized cocycle at the equilibrium (viz. stationary solution); • Existence of countable families of C 1,1 local and global flow-invariant submanifolds through the equilibrium (when γ = 0); • Sufficient conditions for hyperbolicity of the equilibrium; viz. existence of flow-invariant local stable/unstable manifolds in the neighborhood of the equilibrium; • Sufficient conditions on the parameters ν, γ , σi , i 1, and the geometry of the domain D to guarantee uniqueness of the (zero) equilibrium. We believe that it is possible to modify the arguments in this article so as to cover the case of additive noise, white in time and sufficiently smooth in space. 2. Preliminaries Let us identify the Hilbert space H in Section 1 with its dual H . We then consider the stochastic Navier–Stokes equation (1.1) in the framework of the Gelfand triple: V ⊂H ∼ = H ⊂ V
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
where V is the dual of V . Thus, we may consider the Stokes operator A as a bounded linear map from V into V . Moreover, we also denote by ·,· : V × V → R, the canonical bilinear pairing between V and V . Hence, using integration by parts, we have Au, w = ν
2
∂i uj ∂i wj dx = νu, w
(2.1)
i,j =1 D
for u = (u1 , u2 ) ∈ V , w = (w1 , w2 ) ∈ V . Define the real-valued trilinear form b on H × H × H by setting b(u, v, w) :=
2
(2.2)
ui ∂i vj wj dx,
i,j D
whenever the integral in (2.2) makes sense. In particular, if u, v, w ∈ V , then 2 b(u, v, w) = B(u, v), w = (u · ∇)v, w =
ui ∂i vj wj dx.
i,j D
Using integration by parts, it is easy to see that b(u, v, w) = −b(u, w, v),
(2.3)
b(u, v, v) = 0
(2.4)
for all u, v, w ∈ V . Thus,
for all u, v ∈ V . Throughout the paper, we will denote various generic positive constants by the same letter c, although the constants may differ from line to line. We now list some well-known estimates for b which will be used frequently in the sequel (see [19,15] for example): b(u, v, w) cuV · vV · wV , u, v, w ∈ V , b(u, v, w) c|u|H · vV · |Aw|H , u ∈ H, v ∈ V , w ∈ D(A), b(u, v, w) cuV · |v|H · |Aw|H , u ∈ V , v ∈ H, w ∈ D(A), 1 1 1 1 b(u, v, w) 2u 2 · |u| 2 · w 2 · |w| 2 · vV , u, v, w ∈ V . V H V H
(2.5) (2.6) (2.7) (2.8)
Moreover, combining (2.3) and (2.8), we obtain B(u, w) = sup b(u, w, v) = sup b(u, v, w) V vV 1 1 2
vV 1
1 2
1 2
1
2uV · |u|H · wV · |w|H2 for all u, w ∈ V .
(2.9)
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3547
3. Existence of the cocycle In this section, we will show that strong solutions of the stochastic NSE generate a Fréchet C 1,1 locally compacting cocycle (viz. stochastic semiflow) u : R+ × H × Ω → H on the Hilbert space H . Our approach is to use a variational technique which transforms the SNSE into a random NSE that we then analyze using a combination of Galerkin approximations and a priori estimates (cf. [19,15]). Consider the SNSE ⎧ ∞ ⎪ ⎨ du(t, f ) + Au(t, f ) dt + B u(t, f ) dt = γ u(t, f ) dt + σ u(t, f ) dW (t), t > 0, k
⎪ ⎩
k
k=1
u(0, f ) = f ∈ H. (3.1)
It is known that for each f ∈ H , the SNSE (3.1) admits a unique strong solution u(·, f ) ∈ L2 (Ω; C([0, T ]; H )) ∩ L2 (Ω × [0, T ]; V ) [1]. Writing (3.1) in integral form, we have
t
t
u(t, f ) = f −
Au(s, f ) ds − 0
+
B u(s, f ) ds + γ
0
∞ t
t u(s, f ) ds 0
(3.2)
σk u(s, f ) dWk (s),
k=1 0
for all t ∈ [0, T ]. Let Q : [0, ∞) × Ω → R be the solution of the one-dimensional linear stochastic ordinary differential equation dQ(t) = γ Q(t) dt +
∞
σk Q(t) dWk (t),
⎫ ⎪ t 0, ⎬
k=1
Q(0) = 1.
⎪ ⎭
(3.3)
t 0.
(3.4)
By Itô’s formula, we have Q(t) = exp
∞ k=1
∞ t 2 σk Wk (t) − σk + γ t , 2 k=1
This implies that EQ∞ < ∞, where Q∞ ≡ Q(·, ω)∞ := sup Q(t, ω), 0tT
ω ∈ Ω,
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for any finite positive T . Define v(t, f ) := u(t, f )Q−1 (t),
t 0.
(3.5)
Applying Itô’s formula to the relation u(t, f ) = v(t, f )Q(t), t 0, and using (3.3), it is easy to see that v(t) ≡ v(t, f ) satisfies the random NSE
dv(t) = −Av(t) dt − Q(t)B v(t) dt, v(0) = f ∈ H.
t 0,
(3.6)
Our next proposition gives a priori bounds on solutions of the random NSE (3.6). Proposition 3.1. For f ∈ H and ω ∈ Ω, let v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) be a solution of (3.6) on [0, T ] for some T > 0. Then for each ω ∈ Ω and any f ∈ H , the following estimates hold sup v(t, f, ω)H |f |H
(3.7)
v(t, f, ω)2 dt 1 |f |2 . V 2ν H
(3.8)
0tT
and
T 0
Moreover, for each ω ∈ Ω, the map H f → v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) is Lipschitz on bounded sets in H . Proof. Let f ∈ H and v(t) ≡ v(t, f, ω), t ∈ [0, T ], be a solution of (3.6). We fix and suppress ω ∈ Ω throughout this proof. Employing the divergence free condition, B(v), v = 0, we obtain v(t, f )2 = |f |2 − 2 H H
t
Av(s, f ), v(s, f ) ds − 2
0
Q(s) B v(s, f ) , v(s, f ) ds
0
t = |f |2H − 2
t
Av(s, f ), v(s, f ) ds
0
t = |f |2H − 2ν 0
for all t ∈ [0, T ]. Hence,
v(s, f )2 ds V
(3.9)
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
v(t, f )2 + 2ν H
t
v(s, f )2 ds = |f |2 , H V
3549
t ∈ [0, T ].
0
This immediately gives (3.7) and (3.8). It remains to prove the last assertion of the proposition. Let f, g ∈ H , and t ∈ [0, T ] for the rest of the proof. Using the identity b(u, v, v) = 0,
u, v ∈ V ,
and the chain rule we obtain v(t, f ) − v(t, g)2 = |f − g|2 − 2 H H
t
A v(s, f ) − v(s, g) , v(s, f ) − v(s, g) ds
0
t −2
Q(s) B v(s, f ) − B v(s, g) , v(s, f ) − v(s, g) ds
0
t = |f
− g|2H
− 2ν
v(s, f ) − v(s, g)2 ds V
0
t −2
Q(s) b v(s, f ), v(s, f ), v(s, f ) − v(s, g)
0
− b v(s, g), v(s, g), v(s, f ) − v(s, g) ds
t = |f
− g|2H
− 2ν
v(s, f ) − v(s, g)2 ds V
0
t −2
Q(s)b v(s, f ) − v(s, g), v(s, g), v(s, f ) − v(s, g) ds.
0
(3.10) Thus, we have v(t, f ) − v(t, g)2 |f − g|2 − 2ν H H
t
v(s, f ) − v(s, g)2 ds V
0
t + 4Q∞ 0
v(s, f ) − v(s, g) v(s, g) v(s, f ) − v(s, g) ds V V H
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t |f
− g|2H
−ν
v(s, f ) − v(s, g)2 ds V
0
+4
Q2∞ ν
t
v(s, g)2 v(s, f ) − v(s, g)2 ds. V H
(3.11)
0
Applying Gronwall’s lemma (Lemma 5.1) to the above inequality and using (3.8), we get v(t, f ) − v(t, g)2 + ν H
t
v(s, f ) − v(s, g)2 ds V
0
Q2∞ |f − g|2H exp 4 ν
T
v(s, g)2 ds V
0
2 |f − g|2H exp 2 Q2∞ |g|2H . ν
(3.12)
This implies that, for each ω ∈ Ω, the solution map
H f → v(·, f, ω) ∈ C [0, T ], H ∩ L2 [0, T ], V (when it exists) is Lipschitz on bounded sets in H .
2
Our next proposition proves the existence of a unique strong global solution to the random NSE (3.6). Proposition 3.2. Let f ∈ H , ω ∈ Ω. Then for each T > 0, there exists a unique solution v(·, f, ω) ∈ C([0, T ], H ) ∩ L2 ([0, T ], V ) to Eq. (3.6). Furthermore, the solution map R+ × H × Ω (t, f, ω) → v(t, f, ω) ∈ H is jointly measurable, and for each f the process v(·, f, ·) : R+ × Ω → H is (Ft )t0 -adapted. Proof. The proof is based on Galerkin approximations coupled with a priori estimates (cf. [19,18]). Let f ∈ H , fix and suppress ω ∈ Ω. We use Galerkin approximations to prove existence of a solution to the random NSE
⎫ dv(t) = −Av(t) dt − Q(t)B v(t) dt, t > 0, ⎪ ⎬
(3.13) v(0) = f ∈ L2 D, R2 = H, ⎪ ⎭ v(t)|∂D = 0, t > 0. Let {ei }∞ i=1 be a complete orthonormal basis of H that consists of eigenvectors of the operator −A under Dirichlet boundary conditions with corresponding eigenvalues {μi }∞ i=1 ; that is A(ei ) = −μi ei , ei |∂D = 0, i 1. Let Hn denote the n-dimensional subspace of H spanned by {e1 , e2 , . . . , en }. Define fn ∈ Hn by
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
fn :=
n
3551
f, ej ej .
j =1
Clearly, the sequence {fn }∞ n=1 converges to f in H . Now for every integer n 1, we seek a solution vn of the random NSE
dvn (t) = −Avn (t) dt − Q(t)B vn (t) dt, vn (0) = fn , vn (t)|∂D = 0, t > 0,
⎫ t > 0, ⎪ ⎬ ⎪ ⎭
(3.13n )
such that vn (t) :=
n
gj n (t)ej ,
t 0,
j =1
for appropriate choice of the real-valued random processes gj n . We will show that the Fourier coefficients gj n (t) solve a system of random ordinary differential equations with locally Lipschitz coefficients. To see this, we proceed as follows. Since vn satisfies the NSE (3.13n ), then for each 1 j n, we have dgj n (t) = d vn (t), ej
= − Avn (t), ej dt − Q(t) vn (t) · ∇ vn (t), ej dt
= − vn (t), Aej dt − Q(t) vn (t) · ∇ vn (t), ej dt
= μj gj n (t) dt − Q(t) vn (t) · ∇ vn (t), ej dt
(3.14)
for all t > 0. Consider
vn (t) · ∇ vn (t) =
n i=1
=
n
gin (t)(ei · ∇)
n
gkn (t)ek
k=1
gin (t)gkn (t)(ei · ∇)(ek ).
i,k=1
Hence n
vn (t) · ∇ vn (t), ej = gin (t)gkn (t) (ei · ∇)(ek ), ej i,k=1
=
n
gin (t)gkn (t)b(ei , ek , ej ),
1 j n.
(3.15)
i,k=1
Substituting (3.15) into (3.14) gives the following random system of ode’s for gj n (t), 1 j n,
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
dgj n (t) = μj gj n (t) dt − Q(t)
n
gin (t)gkn (t)b(ei , ek , ej ) dt,
⎫ ⎪ ⎪ t > 0, ⎬ ⎪ ⎪ ⎭
i,k=1
gj n (0) = f, ej .
(3.16)
The vector fields in (3.16) are locally Lipschitz and so the system (3.16) of random ode’s admits a unique local solution defined on a local time interval [0, T0 ), where T0 is possibly random. Hence the system (3.13n ) has a unique local solution defined on [0, T0 ). Since Q is jointly measurable and (Ft )t0 -adapted, then so are the gj n ’s. To show that T0 = ∞ a.s., we first derive a priori estimates on vn (or the gj n ). Suppose T0 ≡ T0 (ω) < ∞ for some ω ∈ Ω. Multiply both sides of (3.13n ) by vn (t), integrate over D and use the relation
B vn (t) , vn (t) H = 0,
n 1, t ∈ [0, T0 ),
to obtain 2 d vn (t)H = −2 Avn (t), vn (t) H dt,
t ∈ [0, T0 ).
Therefore,
2 d vn (t) + 2ν
∇vn (t)2 dt = 0,
0 < t < T0 .
D
Hence, vn (t)2 − vn (0)2 + 2ν H
t
vn (s)2 ds = 0, V
0 < t < T0 ;
0
and so vn (t)2 + 2ν H
t
vn (s)2 ds = |fn |2 |f |2 , H H V
0
for all n 1 and all t ∈ (0, T0 ). In particular, 2 sup vn (t)H |f |2H
(3.17)
vn (s)2 ds 1 |f |2 V 2ν H
(3.18)
0t
and
t 0
for all t ∈ [0, T0 ) and all n 1.
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3553
Use the estimate B(u1 , u2 )
V
1/2
1/2
1/2
1/2
2u1 V · |u1 |H · u2 V · |u2 |H ,
u1 , u2 ∈ V ,
(3.19)
to get B(u)
V
2uV · |u|H ,
u ∈ V.
(3.20)
We now view (3.13n ) as an integral equation in V :
t vn (t) = fn −
t Avn (s) ds −
Q(s)B vn (s) ds,
0 t < T0 .
(3.21)
2Q∞ · vn (s)H · vn (s)V 2Q∞ · |f |H · vn (s)V , 0 s < T0 .
(3.22)
0
0
Consider
Q(s)B vn (s)
V
In order to show that
T0 lim vn (t) = fn −
t→T0−
T0 Avn (s) ds −
0
Q(s)B vn (s) ds,
(3.23)
0
it is sufficient to prove that the map
t [0, T0 ) t → θ (t) :=
Q(s)B vn (s) ds ∈ V
(3.24)
0
is uniformly continuous. To do this, let 0 t1 < t2 < T0 . Using (3.22), Hölder’s inequality and (3.18), we get θ (t2 ) − θ (t1 )
V
t2
= Q(s)B vn (s) ds t1
t2
V
Q(s)B vn (s) ds V
t1
t2 2Q∞ · |f |H · t1
vn (s) ds V
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t2 2Q∞ · |f |H ·
vn (s)2 ds V
1/2 · (t2 − t1 )1/2
t1
1 2Q∞ · |f |H · √ · |f |H · (t2 − t1 )1/2 2ν 2 2 = |f | · Q∞ (t2 − t1 )1/2 . ν H
(3.25)
The above inequality implies that the map (3.24) is uniformly 12 -Hölder continuous on [0, T0 ). Since vn (s) ∈ Hn , the n-dimensional linear span of {ei }ni=1 , and Hn is invariant under A, then it follows from (3.17) that Avn (s)
V
Avn (s)H A|Hn L(Hn ) · vn (s)H A|Hn L(Hn ) · |f |H
(3.26)
for all s ∈ [0, T0 ) and all n 1. In the above inequality, A|Hn ∈ L(Hn ) is the restriction of A to Hn . Hence (3.23) holds for all n 1. Define
T0 vn (T0 ) := lim vn (t) = fn − t→T0−
T0 Avn (s) ds −
0
Q(s)B vn (s) ds,
n 1.
0
By local existence, we get a local solution vn : [T0 , T0 + ) × Ω → R2 of the NSE
dvn (t) = −Avn (t) dt − Q(t)B vn (t) dt,
T0 < t < T0 + ,
vn (t)|t=T0 = vn (T0 ) ∈ H, for some > 0. This contradicts the maximality of T0 . Hence T0 = ∞ a.s. We next show that the sequence {vn }∞ n=1 converges to a weak solution v of the random NSE (3.13). As before, we view Eq. (3.13n ) as an equation in V :
dv n (t) = −Av n (t) − Q(t)B v n (t) . dt Therefore, using (2.9), (3.17), (3.18) and the fact that A|V : V → V is continuous linear, we have
T n 2
T
T dv (t)
dt 2 Av n (t)2 dt + 2Q∞ B v n (t) 2 dt dt
V V V 0
0
T C 0
0
n 2 v (t) dt + C V
T 0
n 2 n 2 v (t) v (t) dt V H
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
C |f |2H 1 + |f |2H , 2ν
3555
(3.27)
where C is a positive random constant independent of f . Since the embedding V → H is compact, by the proof of Theorem 2.1 in [19, pp. 111–113], it follows from (3.17), (3.18) and (3.27) that there exists a subsequence v nk (t), k 1, and v(t) such that v nk (·) → v(·) in the weak star topology of C([0, T ], H ), v nk (·) → v(·) weakly in L2 ([0, T ], V ) and moreover v nk (·) → v(·) strongly in L2 ([0, T ], H ) as k → ∞. For w ∈ Hm , if nk m we have
v (t), w = v nk (0), w − nk
t
t
Av (s), w ds + nk
0
= v nk (0), w −
t
Q(s)b v nk (s), w, v nk (s) ds
0
Av nk (s), w ds
0
+
2 t
Q(s)vink (s)(∂i wj )vjnk (s) dx ds
(3.28)
i,j =1 0 D
for all t ∈ [0, T ]. Letting k → ∞ in the above relation, we obtain
t
v(t), w = f, w −
Av(s), w ds +
t
Av(s), w ds −
= (f, w) − 0
Q(s)vi (s)(∂i wj )vj (s) dx ds
i,j =1 0 D
0
t
2 t
Q(s) B v(s) , w ds
(3.29)
0
for all t ∈ [0, T ] and all m 1. Since m is arbitrary and ∞ m=1 Hm is dense in V , it follows that v is a solution to Eq. (3.6). Uniqueness follows by setting f = g in (3.12). Since the Galerkin approximations vn are jointly measurable and (Ft )t0 -adapted, it follows that the limiting process v must have the same measurability properties. 2 Our next result addresses the issue of local compactness of the solution map H f → v(t, f, ω) ∈ H for t > 0, ω ∈ Ω. Proposition 3.3. For t > 0 and ω ∈ Ω, the solution map H f → v(t, f, ω) ∈ H of (3.6) sends bounded sets into relatively compact sets in H . Proof. Fix ω ∈ Ω throughout this proof. √ First, we show that the map [0, T ] t → tv(t, f, ω) ∈ V is L∞ , and provide a bound for v in L∞ ([0, T ], V ). Let f ∈ V . Then by an argument similar to the proof of Theorem 3.10
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
in [18, p. 314], one can show that v(·, f, ω) ∈ L2 ([0, T ], H 2 (D)) ∩ L∞ ([0, T ], V ) and the following energy equation holds:
d v(t, f, ω)2 + 2Av(t, f, ω)2 = −Q(t) B v(t, f, ω) , Av(t, f, ω) V H dt
(3.30)
for all t ∈ (0, T ). Consequently, 2
d t v(t, f, ω)V dt 2 2
= −2t Av(t, f, ω)H − tQ(t) B v(t, f, ω) , Av(t, f, ω) + v(t, f, ω)V 2 1 2 3 −2t Av(t, f, ω)H + ctQ(t)v(t, f, ω)H2 v(t, f, ω)V Av(t, f, ω)H2 + v(t, f, ω)V 2 2 4 ctQ(t)4 v(t, f, ω) v(t, f, ω) + v(t, f, ω) , t ∈ (0, T ], (3.31) H
V
V
where the following Young’s inequality “with ”: ab
3 4/3 4/3 a4 b , + 4 4 4
a, b 0,
and the fact that 1 1 B(u) c|u| 2 u|Au| 2 , H H H
u ∈ V,
have been used. By Gronwall’s inequality it follows from (3.31) that √ ·v(·, f, ω)2 ∞
L ([0,T ],V )
T
T v(t, f, ω)2 dt exp cQ4 v(t, f, ω)2 v(t, f, ω)2 dt ∞ V H V
0
0
1 2 4 1 4 |f |H exp cQ∞ |f |H . 2ν 2ν
(3.32)
Since the right side of (3.32) depends only on the H -norm of f , by a limiting procedure it is easy to see that (3.32) also holds for all f ∈ H . Fix t > 0, f ∈ H . Then, for 0 < δ < t, v(t, f, ω) = Tδ v(t − δ, f, ω) +
t
Tt−s B v(s, f, ω) ds.
(3.33)
t−δ
Suppose δk 0, 0 < δk < t, and let {fn }∞ n=1 ⊂ H be a bounded sequence; i.e., there exists M > 0 such that |fn | M for all n 1.
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3557
∞ Claim. There exists a subsequence {f˜n }∞ n=1 of {fn }n=1 such that for each k 1, the sequence ∞ ˜ {Tδk [v(t − δk , fn , ω)]}n=1 converges in H .
Proof. We use a diagonalization argument. The set {v(t − δ1 , fn , ω): n 1} is bounded in H because |v(t − δ1 , fn , ω)|H |fn |H M for all n 1. So by compactness of Tδ1 : H → H , the sequence {Tδ1 [v(t − δ1 , fn , ω)]}∞ n=1 has a convergent subsequence. Therefore, there is a ∞ such that the sequence {T [v(t − δ , f 1 , ω)]}∞ converges. subsequence {fn1 }∞ of {f } n n=1 δ1 1 n n=1 n=1 Similarly, by compactness of the map H f → Tδ2 v(t − δ2 , f, ω) ∈ H ∞ 1 ∞ 2 there is a subsequence {fn2 }∞ n=1 ⊂ {fn }n=1 such that {Tδ2 [v(t − δ2 , fn , ω)]}n=1 converges in H . ∞ By induction, there are subsequences {fnk }n=1 , k 1, such that {Tδk [v(t − δk , fnk , ω)]}∞ n=1 con∞ ∞ k+1 k n ˜ verges and {fn }n=1 ⊂ {fn }n=1 for each k 1. Let fn := fn , n 1, be the diagonal subse∞ ˜ quence of {fn }∞ n=1 . Then the sequence {Tδk [v(t − δk , fn , ω)]}n=1 converges in H for each k 1. This proves the claim. 2
We will now show that the map H f → v(t, f, ω) ∈ H is compact i.e., takes bounded sets in H into relatively compact sets. It is sufficient to show that for the bounded sequence {fn }∞ n=1 ⊂ H ∞ ∞ ˜ ˜ there exists a subsequence {fn }n=1 such that {v(t, fn , ω)}n=1 converges. Pick the subsequence {f˜n } ⊂ {fn } as in the claim with each sequence {Tδk [v(t − δk , f˜n , ω)]}∞ n=1 convergent. Consider v(t, f˜n , ω) − v(t, f˜m , ω) Tδ v(t − δk , f˜n , ω) − Tδ v(t − δk , f˜m , ω) k k H H
t
Tt−s B v(s, f˜n , ω) − Tt−s B v(s, f˜m , ω) ds H
+ t−δk
Tδk v(t − δk , f˜n , ω) − Tδk v(t − δk , f˜m , ω) H 1 + 2CM √ δk , t − δk for all k 1 and all m, n 1, where the following estimate has been used
Tt−s B v(s, f˜n , ω) − Tt−s B v(s, f˜m , ω) 1 C√ t −s
B v(s, f˜n , ω)
V
H
˜
+ B v(s, fm , ω) V
1 v(s, f˜n , ω)H v(s, f˜n , ω)V + v(s, f˜m , ω)H v(s, f˜m , ω)V
C√ t −s √ 1 sup v(s, f˜n , ω)H sup sv(s, f˜n , ω)V
C√ √ t − s s 0sT 0sT √ + sup v(s, f˜m , ω) sup sv(s, f˜m , ω)
0sT
H
0sT
V
(3.34)
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
CM (ω) √
1 √ , t −s s
because of (3.32). For fixed k 1, the claim implies lim sup Tδk v(t − δk , f˜n , ω) − Tδk v(t − δk , f˜m , ω) H = 0.
m,n→∞
Take lim supm,n→∞ in (3.34) (with k 1 fixed) and use the above relation to get √ δk ˜ ˜ lim sup v(t, fn , ω) − v(t, fm , ω) H 2CM √ , t − δk m,n→∞
(3.35)
for all k 1. Now let k → ∞ in (3.35) to obtain √ δk ˜ ˜ lim sup v(t, fn , ω) − v(t, fm , ω) H 2CM lim √ = 0. k→∞ t − δk m,n→∞ Therefore, {v(t, f˜n , ω)}∞ n=1 is a Cauchy sequence and hence converges in H . This proves compactness of the map H f → v(t, f, ω) ∈ H . 2 Theorem 3.1. The solution map H f → v(t, f, ω) ∈ H is C 1,1 for ω ∈ Ω and all t 0, and has bounded Fréchet derivatives on bounded sets in H . Furthermore, the Fréchet derivative Dv(t, f, ω)(·) : H → H is compact for any t > 0, f ∈ H and ω ∈ Ω. Proof. Let f, g ∈ H . Fix and suppress ω ∈ Ω in this proof. Consider the following random integral equation
t z(t, f )(g) = g −
Az(s, f )(g) ds − 0
t −
t
Q(s) z(s, f )(g) · ∇ v(s, f ) ds
0
Q(s) v(s, f ) · ∇ z(s, f )(g) ds,
t ∈ [0, T ].
(3.36)
0
The existence and uniqueness of the solution z(t, f )(g) of (3.36) can be proved similarly as for Eq. (3.6), using Galerkin approximations (cf. proof of Proposition 3.2). Furthermore, uniqueness of the solution to (3.36) implies that the solution z(t, f )(g) is linear in g. We will now derive some useful estimates for the solution z(t, f )(g) of (3.36). Using the chain rule in (3.36), we obtain
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
z(t, f )(g)2 = |g|2 − 2ν H
t
3559
z(s, f )(g)2 ds V
0
t −2
Q(s)b z(s, f )(g), v(s, f ), z(s, f )(g) ds
0
t −2
Q(s)b v(s, f ), z(s, f )(g), z(s, f )(g) ds
0
t |g|2H − 2ν
z(s, f )(g)2 ds V
0
t
z(s, f )(g) v(s, f ) z(s, f )(g) ds H V V
+ cQ∞ 0
t |g|2H
− 2ν
z(s, f )(g)2 ds + cQ∞
V
0
t
z(s, f )(g)2 ds V
0
+ cQ∞ −1
t
z(s, f )(g)2 v(s, f )2 ds H V
0
t = |g|2H
−ν
z(s, f )(g)2 ds + cQ∞ −1 V
0
t
v(s, f )2 z(s, f )(g)2 ds V H
0
+ cQ∞ − ν
t
z(s, f )(g)2 ds, V
t ∈ [0, T ],
(3.37)
0
where we have used the following “Young inequality with ”: ab a 2 + −1 b2 ,
a, b > 0,
for any > 0. Now in (3.37), choose sufficiently small (and random) such that cQ∞ < ν. So (3.37) implies z(t, f )(g)2 |g|2 − ν H H
t 0
for all t ∈ [0, T ].
z(s, f )(g)2 ds + cQ ˜ ∞ V
t 0
v(s, f )2 z(s, f )(g)2 ds V H
3560
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
By Gronwall’s lemma (Lemma 5.1), the above inequality gives 2 sup z(t, f )(g)H + ν
0tT
T 0
T z(s, f )(g)2 ds |g|2 exp cQ2 v(s, f )2 ds ∞ H V V 0
1 |g|2H exp cQ2∞ |f |2H , 2ν
(3.38)
where (3.8) has been used for the last inequality. Since z(t, f )(g) is linear in g, (3.38) implies that z(t, f )(·) ∈ L(H ) for each t ∈ [0, T ], and z(·, f )(·) ∈ L(H, L2 ([0, T ], V )). Furthermore, 1 2 1 2 ˜ |f | . sup z(t, f ) L(H ) exp cQ ∞ 2 2ν H 0tT
(3.39)
Next we will show that the map H f → v(t, f, ω) ∈ H has a continuous Fréchet derivative given by Dv(t, f, ω) = z(t, f, ω)(·). To this end, it suffices to prove that v(t, f + hg, ω) − v(t, f, ω) lim sup − z(t, f )(g) = 0 h→0 h |g|H 1
(3.40)
H
and the map H f → z(t, f, ω) ∈ L(H ) is continuous. First, we prove lim sup
h→0 |g|H 1
2 sup v(t, f + hg) − v(t, f )H + ν
0tT
T
v(s, f + hg) − v(s, f )2 ds = 0. V
0
(3.41) Using the equations satisfied by v(t, f ) and v(t, f + hg), the chain rule and “Young’s inequality with ”, it follows that v(t, f + hg) − v(t, f )2 H
t =h
2
|g|2H
− 2ν 0
v(t, f + hg) − v(t, f )2 ds V
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t −2
3561
Q(s)b v(s, f + hg) − v(s, f ), v(s, f ), v(s, f + hg) − v(s, f ) ds
0
t h
2
|g|2H
− 2ν
v(t, f + hg) − v(t, f )2 ds V
0
t + cQ∞
v(s, f + hg) − v(s, f ) v(s, f ) v(s, f + hg) − v(s, f ) ds H V V
0
t h
2
|g|2H
−ν
v(t, f + hg) − v(t, f )2 ds V
0
t + cQ2∞
v(s, f + hg) − v(s, f )2 v(s, f )2 ds H V
(3.42)
0
for all t ∈ [0, T ]. By Gronwall’s inequality, we deduce that 2 sup v(t, f + hg) − v(t, f )H + ν
T
0tT
v(s, f + hg) − v(s, f )2 ds V
0
h
2
|g|2H
exp
t
cQ2∞
v(s, f )2 ds V
(3.43)
0
for all f, g ∈ H and h ∈ R. This immediately implies (3.41). Set v(t, f + hg, ω) − v(t, f, ω) , h X(t, f, g, h) = U (t, f, g, h) − z(t, f )(g),
U (t, f, g, h) =
for t ∈ [0, T ] and h ∈ R\{0}. Then,
t X(t, f, g, h) = −
t AX(s, f, g, h) ds −
0
t −
Q(s) v(s, f ) · ∇ X(s, f, g, h) ds
0
Q(s) X(s, f, g, h) · ∇ v(s, f + hg) ds
0
t + 0
Q(s) z(s, f )(g) · ∇ v(s, f ) − v(s, f + hg) ds,
(3.44)
3562
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
for all t ∈ [0, T ]. By the chain rule, X(t, f, g, h)2 = −2ν H
t
X(s, f, g, h)2 ds V
0
t −2
Q(s)b X(s, f, g, h), v(s, f + hg), X(s, f, g, h) ds
0
t +2
Q(s)b z(s, f )(g), v(s, f ) − v(s, f + hg), X(s, f, g, h) ds,
0
(3.45) where b(u, v, v) = 0 has been used. Hence, X(t, f, g, h)2 −2ν H
t
X(s, f, g, h)2 ds V
0
t +c
Q(s)X(s, f, g, h)H v(s, f + hg)V X(s, f, g, h)V ds
0
t +c
1 1 1 Q(s)z(s, f )(g)V2 z(s, f )(g)H2 v(s, f + hg) − v(s, f )V2
0
1 × v(s, f + hg) − v(s, f )H2 X(s, f, g, h)V ds,
(3.46)
for t ∈ [0, T ]. We next prove the following estimate X(t, f, g, h)2 −ν H
t 0
t +c
X(s, f, g, h)2 ds + c V
t
2 2 Q(s)X(s, f, g, h)H v(s, f + hg)V ds
0
2 2 Q(s)z(s, f )(g)V v(s, f + hg) − v(s, f )H ds
0
t +c
2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds,
0
for t ∈ [0, T ]. Use (3.46) and “Young’s inequality with ” to see that
(3.47)
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
X(t, f, g, h)2 −2ν H
t
3563
X(s, f, g, h)2 ds V
0
t +c
Q(s)X(s, f, g, h)H v(s, f + hg)V X(s, f, g, h)V ds
0
t +c
1/2 1/2 1/2 Q(s)z(s, f )(g)V z(s, f )(g)H v(s, f + hg) − v(s, f )V
0
1/2 × v(s, f + hg) − v(s, f )H X(s, f, g, h)V ds
t −2ν
X(s, f, g, h)2 ds V
0
+ c
−1
t
2 2 Q(s)X(s, f, g, h)H v(s, f + hg)V ds
0
t + cQ∞
X(s, f, g, h)2 ds + cQ∞ V
0
+ −1 c
t
t
X(s, f, g, h)2 ds V
0
Q(s)z(s, f )(g)V z(s, f )(g)H v(s, f + hg) − v(s, f )V
0
× v(s, f + hg) − v(s, f )H ds
t −2ν
X(s, f, g, h)2 ds V
0
+ c
−1
t
2 2 Q(s)X(s, f, g, h)H v(s, f + hg)V ds
0
t + cQ∞ 0
1 + −1 c 2
t
X(s, f, g, h)2 ds + cQ∞ V
t
X(s, f, g, h)2 ds V
0
2 2 Q(s)z(s, f )(g)V v(s, f + hg) − v(s, f )H ds
0
1 + −1 c 2
t 0
2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds,
(3.48)
3564
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
for t ∈ [0, T ]. Now choose small enough such that 2 cQ∞ < ν. This gives X(t, f, g, h)2 −ν H
t
X(s, f, g, h)2 ds V
0
+ c
−1
t
2 2 Q(s)X(s, f, g, h)H v(s, f + hg)V ds
0
1 + −1 c 2
t
2 2 Q(s)z(s, f )(g)V v(s, f + hg) − v(s, f )H ds
0
1 + −1 c 2
t
2 2 Q(s)z(s, f )(g)H v(s, f + hg) − v(s, f )V ds
(3.49)
0
for all t ∈ [0, T ], which implies (3.47). By (3.47) and Gronwall’s inequality (Lemma 5.1), it follows that 2 sup X(t, f, g, h)H + ν
0tT
T
X(s, f, g, h)2 ds V
0
! c Q∞
+ Q∞
2 sup v(t, f + hg) − v(t, f )H
0tT
z(s, f )(g)2 ds V
0
2 sup z(t, f )(g)H
0tT
T
T
v(t, f + hg) − v(s, f )2 ds V
"
0
T
× exp cQ∞
v(s, f + hg)2 ds V
(3.50)
0
for all f, g ∈ H and h ∈ R\{0}. By virtue of (3.41) and (3.38), (3.50) implies that lim sup
h→0 |g|H 1
2 sup X(t, f, g, h)H + ν
0tT
T
X(s, f, g, h)2 ds = 0. V
(3.51)
0
The equality (3.40) follows immediately from the above relation. To complete the proof that the map H f → v(t, f ) ∈ H is C 1,1 (Fréchet), observe first that the Gateaux derivative z(t, f ) ∈ L(H ). So for the map H f → v(t, f ) ∈ H to be Fréchet continuously differentiable, it is sufficient to prove that the map H f → z(t, f ) ∈ L(H ) is Lipschitz continuous on bounded sets.
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3565
In what follows, let g, f1 , f2 ∈ H be such that |fi |H M, i = 1, 2, |g|H 1 and t ∈ [0, T ]. From (3.36), we obtain z(t, f1 )(g) − z(t, f2 )(g)
t =−
A z(s, f1 )(g) − z(s, f2 )(g) ds −
0
t
Q(s) z(s, f1 )(g) · ∇ v(s, f1 ) ds
0
t
Q(s) z(s, f2 )(g) · ∇ v(s, f2 ) ds −
+ 0
t
Q(s) v(s, f1 ) · ∇ z(s, f1 )(g) ds
0
t
Q(s) v(s, f2 ) · ∇ z(s, f2 )(g) ds
+ 0
t =−
A z(s, f1 )(g) − z(s, f2 )(g) ds
0
t −
Q(s) z(s, f1 )(g) − z(s, f2 )(g) · ∇ v(s, f1 ) ds
0
t −
Q(s) z(s, f2 )(g) · ∇ v(s, f1 ) − v(s, f2 ) ds
0
t −
Q(s) v(s, f1 ) · ∇ z(s, f1 )(g) − z(s, f2 )(g) ds
0
t −
Q(s) v(s, f1 ) − v(s, f2 ) · ∇ z(s, f2 )(g) ds.
(3.52)
0
Differentiating both sides of (3.52) with respect to t, taking inner products of the resulting differential equation with z(t, f1 )(g) − z(t, f2 )(g), using the chain rule and integrating over t gives z(t, f1 )(g) − z(t, f2 )(g)2 H
t = −2ν
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
t −2 0
Q(s)b z(s, f1 )(g) − z(s, f2 )(g), v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g) ds
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t −2
Q(s)b z(s, f2 )(g), v(s, f1 ) − v(s, f2 ), z(s, f1 )(g) − z(s, f2 )(g) ds
0
t −2
Q(s)b v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds
0
t −2
Q(s)b v(s, f1 ) − v(s, f2 ), z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds
0
t = −2ν
z(s, f1 )(g) − z(s, f2 )(g)2 ds + I1 + I2 + I3 , V
(3.53)
0
where we have used the fact that b(u, w, w) = 0, and where
t I1 := −2
Q(s)b z(s, f1 )(g) − z(s, f2 )(g), v(s, f1 ), z(s, f1 )(g) − z(s, f2 )(g) ds,
(3.54)
Q(s)b z(s, f2 )(g), v(s, f1 ) − v(s, f2 ), z(s, f1 )(g) − z(s, f2 )(g) ds,
(3.55)
Q(s)b v(s, f1 ) − v(s, f2 ), z(s, f2 )(g), z(s, f1 )(g) − z(s, f2 )(g) ds.
(3.56)
0
t I2 := −2 0
t I3 := −2 0
We now estimate each of the terms on the right-hand side of (3.54), (3.55) and (3.56). Using “Young’s inequality with ” in (3.54), we obtain
t |I1 | : 4Q∞
v(s, f1 ) z(s, f1 )(g) − z(s, f2 )(g) z(s, f1 )(g) − z(s, f2 )(g) ds V H V
0
t 4 Q∞
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
+ 4
−1
t Q∞
v(s, f1 )2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H
(3.57)
0
From (3.55), (3.38), “Young’s inequality with ”, Hölder’s inequality and (3.43), it follows that
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t |I2 | : 4Q∞
3567
z(s, f2 )(g)1/2 z(s, f2 )(g)1/2 v(s, f1 ) − v(s, f2 ) V
H
V
0
1/2 1/2 × z(s, f1 )(g) − z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)H ds
t 2cQ∞
z(s, f2 )(g) v(s, f1 ) − v(s, f2 )2 ds V H
0
t + 2cQ∞
z(s, f1 )(g) − z(s, f2 )(g) z(s, f2 )(g) V V
0
× z(s, f1 )(g) − z(s, f2 )(g)H ds t 2 v(s, f1 ) − v(s, f2 ) ds 2cQ∞ sup z(s, f2 )(g)H · V 0sT
t + 2cQ∞
0
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
+ 2c
−1
t Q∞
z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds V H
0
$ # # $ 1 1 1 2cQ∞ |g|2H exp cQ2∞ |f2 |2H √ |f1 − f2 |H exp Q2∞ c 2ν ν ν
t + 2cQ∞
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
+ 2c
−1
t Q∞
z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H
(3.58)
0
In (3.56), we use Hölder inequality and “Young’s inequality with ”, together with (3.38), (3.43) and (3.8), to get the following estimates
t |I3 | 4Q∞
v(s, f1 ) − v(s, f2 )1/2 v(s, f1 ) − v(s, f2 )1/2 V
H
0
1/2 1/2 × z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)V z(s, f1 )(g) − z(s, f2 )(g)H ds
t 2cQ∞ 0
v(s, f1 ) − v(s, f2 ) v(s, f1 ) − v(s, f2 ) z(s, f2 )(g) ds V H V
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t + 2Q∞
z(s, f1 )(g) − z(s, f2 )(g)
V
0
× z(s, f1 )(g) − z(s, f2 )(g)H z(s, f2 )(g)V ds t 2cQ∞
z(s, f2 )(g)2 ds V
1/2 t ·
0
v(s, f1 ) − v(s, f2 )2 ds V
1/2
0
× sup v(s, f1 ) − v(s, f2 )H + 2 Q∞ 0sT
t
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
+ 2 −1 Q∞
t
z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds V H
0
$ $ # # 1 1 2 1 2 2 2 2cQ∞ |g|H exp cQ∞ |f2 |H √ |f1 − f2 |H exp cQ∞ 2ν ν ν
t + 2 Q∞
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
+ 2
−1
t Q∞
z(s, f2 )(g)2 z(s, f1 )(g) − z(s, f2 )(g)2 ds. V H
(3.59)
0
Choose > 0 sufficiently small such that 6 Q∞ + 2c Q∞ < ν.
(3.60)
Using (3.57), (3.58), (3.59), and (3.60) in (3.53), we get z(t, f1 )(g) − z(t, f2 )(g)2 + ν H
t
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
t c|f1 − f2 |2H
+c
v(s, f1 )2 + z(s, f2 )(g)2 V V
0
2 × z(s, f1 )(g) − z(s, f2 )(g)H ds,
(3.61)
for all f1 , f2 , g ∈ H such that |fi |H M, i = 1, 2, |g|H 1, with c a random constant (dependent on M, ν and T ). Applying Gronwall’s lemma (Lemma 5.1) to (3.61) and using (3.8) and (3.38), we get
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
z(t, f1 )(g) − z(t, f2 )(g)2 + ν H
t
3569
z(s, f1 )(g) − z(s, f2 )(g)2 ds V
0
t c|f1 − f2 |2H exp
v(s, f1 )2 + z(s, f2 )(g)2 ds V V
0
& %
|g|2 1 c|f1 − f2 |2H exp |f1 |2H + |f2 |2H × H exp cQ2∞ |f2 |2H 2ν ν & % 1 2 1 2 2 2 2M · exp cQ∞ M c|f1 − f2 |H exp 2ν ν c|f1 − f2 |2H .
(3.62)
Therefore, z(t, f1 ) − z(t, f2 )
1/2
L(H )
c|f1 − f2 |H
(3.63)
for all f1 , f2 ∈ H with |f1 |H M, |f2 |H M and all t ∈ [0, T ]. This proves that the map H f → v(t, f, ω) ∈ H is C 1 for each ω ∈ Ω and each t ∈ [0, T ]. Furthermore, its Fréchet derivative H f → Dv(t, f, ω) = z(t, f, ω) ∈ L(H ) is Lipschitz continuous on bounded sets in H . The compactness of the Fréchet derivative Dv(t, f, ω); H → H , t > 0, follows immediately from the fact that the map H f → v(t, f, ω) ∈ H , t > 0, is C 1 and carries bounded sets into relatively compact ones (Proposition 3.3). See the proof of Theorem 3.1 and Lemma 3.1 in [13, Part I]. This completes the proof of Theorem 3.1. 2 We are now ready to state the main result in this section. Theorem 3.2 (The cocycle). Let u(t, f, ·) be the unique global solution of the stochastic Navier– Stokes equation (3.1) for t 0 and f ∈ H . Denote by θ : R+ × Ω → Ω the standard Brownian shift θ (t, ω)(s) := ω(t + s) − ω(t),
t, s 0, ω ∈ Ω,
(3.64)
on Wiener space (Ω, F , P ). Then there is a version u : R+ × H × Ω → H of the solution of (3.1) with the following properties: (i) The map u : R+ × H × Ω → H is jointly measurable, and for each f ∈ H , the process u(·, f, ·) : R+ × Ω → H is (Ft )t0 -adapted. (ii) For each t > 0 and ω ∈ Ω, the map u(t, ·, ω) : H → H takes bounded sets into relatively compact sets. (iii) (u, θ ) is a C 1,1 perfect cocycle; viz.
u t2 , u(t1 , f, ω), θ (t1 , ω) = u(t1 + t2 , f, ω) for all t1 , t2 0, f ∈ H , ω ∈ Ω.
(3.65)
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(iv) For each (t, f, ω) ∈ R+ × H × Ω, the Fréchet derivative Du(t, f, ω) ∈ L(H ) of the map u(t, ·, ω) is compact linear, and the map R+ × H × Ω → L(H ), (t, f, ω) → Du(t, f, ω) is strongly measurable. (v) For fixed ρ, a > 0, E log+
sup
0t1 ,t2 a |f |H ρ
u t2 , f, θ (t1 , ·) + Du t2 , f, θ (t1 , ·) < ∞. H L(H )
(3.66)
Proof. To prove assertion (i) of the theorem, define the required version u : R+ × H × Ω → H by setting u(t, f, ω) := Q(t, ω)v(t, f, ω),
t 0, ω ∈ Ω, f ∈ H.
(3.67)
Note first that Q is jointly measurable and (Ft )t0 -adapted. In view of Proposition 3.2, it follows from (3.67) that u satisfies assertion (i). Assertion (ii) of the theorem follows immediately from (3.67) and Proposition 3.3. Next, we establish the perfect cocycle property (iii). To see this, observe that Q has the cocycle property
Q(t1 + t2 , ω) = Q t2 , θ (t1 , ω) Q(t1 , ω),
t1 , t2 0, ω ∈ Ω.
(3.68)
This follows directly from (3.4). Thus, (3.65) will follow if we prove the following identity
v(t1 + t2 , f, ω) = Q(t1 , ω)−1 v t2 , Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)
(3.69)
for t1 , t2 0, ω ∈ Ω, f ∈ H . Indeed, assume that (3.69) holds. Fix ω ∈ Ω and t1 0 throughout this proof. Then, for t 0, we have
u t, u(t1 , f, ω), θ (t1 , ω) = Q t, θ (t1 , ω) v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)
= Q(t1 + t, ω)Q(t1 , ω)−1 v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) = Q(t1 + t, ω)v(t1 + t, f, ω) = u(t1 + t, f, ω). Hence the perfect cocycle property (3.65) holds. We now show (3.69). To do this, define the processes
z(t, ω) := Q(t1 , ω)−1 v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) , z(t, ω) := v(t + t1 , f, ω),
(3.70)
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for all t 0 and all ω ∈ Ω. Shifting the time-variable t by t1 in the integral equation for v, it is easy to see that z(t, ω) satisfies the following equation
t z(t, ω) = v(t1 , f, ω) −
t Az(s, ω) ds −
0
Q(t1 + s, ω)B z(s, ω) ds
(3.71)
0
for all t 0, ω ∈ Ω. On the other hand,
v t, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω)
t = Q(t1 , ω)v(t1 , f, ω) −
Av s, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) ds
0
t −
Q s, θ (t1 , ω) B v s, Q(t1 , ω)v(t1 , f, ω), θ (t1 , ω) ds
(3.72)
0
for all t 0, ω ∈ Ω. Multiplying the above equation by Q(t1 , ω)−1 and using the bilinear property of B, we easily see that
t z(t, ω) = v(t1 , f, ω) −
t Az(s, ω) ds −
0
Q(t1 + s, ω)B z(s, ω) ds
(3.73)
0
for all t 0, ω ∈ Ω. Subtract z(t, ω) from z(t, ω) and use a similar calculation as for (3.12) to deduce that z(t, ω) = z(t, ω) for all t 0 and ω ∈ Ω. This proves the identity (3.69), and so the cocycle property (iii) holds for the solution map u : R+ × H × Ω → H of the SNSE (3.1). Assertion (iv) of the theorem is a consequence of Theorem 3.1, Proposition 3.2 and relation (3.67). (The strong measurability of Du(t, f, ω) follows from Eq. (3.40).) Let us now prove the integrability estimate in assertion (v) of the theorem. Let 0 t1 , t2 a and f ∈ H with |f |H ρ. It follows from (3.7) that
u t2 , f, θ (t1 , ω) = Q t2 , θ (t1 , ω) v t2 , f, θ (t1 , ω) H H
Q t2 , θ (t1 , ω) |f |H = Q(t1 + t2 , ω)Q−1 (t1 , ω)|f |H ρQ∞ Q−1 ∞ ,
(3.74)
where Q−1 ∞ := sup0t2a Q−1 (t). Using (3.39) we have
Du t2 , f, θ (t1 , ω)
L(H )
= Q t2 , θ (t1 , ω) Dv t2 , f, θ (t1 , ω) L(H ) $ # −1 2 −1 2 1 2 |f |H Q∞ Q exp cQ∞ Q ∞ 2ν ∞ # $ 2 1 Q∞ Q−1 ∞ exp cQ2∞ Q−1 ∞ ρ 2 . 2ν
(3.75)
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Combining (3.74) and (3.75), we get E log+
sup
0t1 ,t2 a |f |H ρ
u t2 , f, θ (t1 , ·) + E log+ H
sup
0t1 ,t2 a |f |H ρ
Du t2 , f, θ (t1 , ·)
L(H )
& %
2 1 E 2 log+ Q∞ Q−1 ∞ + cQ2∞ Q−1 ∞ ρ 2 + log+ ρ < ∞. 2ν (3.76) The above relation implies the integrability condition (3.66). The proof of Theorem 3.2 is now complete. 2 It is easy to see from (3.76) and (3.63) that the following stronger estimate holds E log+
sup
0t1 ,t2 a
u t2 , ·, θ (t1 , ·)
C 1,1
< ∞,
where · C 1,1 denotes the C 1,1 norm on the ball B(0, ρ) in H . 4. The multiplicative ergodic theory Our objective in this section is to characterize the local behavior of solutions of the SNSE (3.1) near an equilibrium or a stationary point/solution. We next describe the concepts of equilibrium or a stationary point for the SNSE (3.1). Definition 4.1 (Equilibrium/stationary point). An F -measurable random variable Y : Ω → H is said be an equilibrium or a stationary random point for the cocycle (u, θ ) if
u t, Y (ω), ω = Y θ (t, ω)
(4.1)
perfectly in ω ∈ Ω for all t ∈ R+ . A trivial equilibrium or stationary solution of the SNSE (3.1) is u(t, 0, ω) ≡ 0 corresponding to the zero initial function f ≡ 0 ∈ H . 4.1. Dynamics near a general equilibrium In order to analyze the dynamics of the SNSE (3.1) near a general equilibrium or stationary point Y : Ω → H , we linearize the C 1,1 cocycle u : R+ × H × Ω → H at Y . This gives a linear cocycle of Fréchet derivatives Du(t, Y (ω), ω) ∈ L(H ) satisfying the following random equations Du(t, Y ) = Q(t, ·)Dv(t, Y ), and
t 0,
(4.2)
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
t Dv(t, Y )(g) = g −
t ADv(s, Y )(g) ds −
0
t −
3573
Q(s) Dv(s, Y )(g) · ∇ v(s, Y ) ds
0
Q(s) v(s, f ) · ∇ Dv(s, Y )(g) ds,
t 0, g ∈ H.
(4.3)
0
We next apply the Oseledec–Ruelle spectral theorem to the compact linear cocycle (Du(t, Y (ω), ω), θ (t, ω)), t 0, ω ∈ Ω ([16, Theorem 2.1.1], [14]). This gives Theorem 4.1 (The Lyapunov spectrum: General equilibrium). Let (u(t, ·, ω), θ (t, ω)) be the C 1,1 cocycle on H generated by the stochastic Navier–Stokes equation (3.1). Suppose that Y : Ω → H is a stationary random point for the cocycle (u, θ ) of the SNSE (3.1) with E log+ |Y | < ∞. Then the following limit Λ(ω) := lim
∗
1/2t Du t, Y (ω), ω ◦ Du t, Y (ω), ω
t→∞
(4.4)
exists in the uniform operator norm in L(H ), perfectly in ω. The Oseledec operator Λ(ω) in (4.4) is compact, self-adjoint and non-negative with discrete non-random spectrum eλ1 > eλ2 > eλ3 > · · · > eλn > · · · .
(4.5)
The Lyapunov exponents {λn }∞ n=1 correspond to values of the limit
1 log Du t, Y (ω), ω (g)H ∈ {λn }∞ n=1 t→∞ t lim
for any g ∈ H , perfectly in ω. Each eigenvalue eλj has a fixed finite multiplicity mj with a corresponding finite-dimensional eigenspace Fj (ω) such that mj := dim Fj (ω), j 1, ω ∈ Ω. If we set ! E1 (ω) := H,
En (ω) :=
n−1 '
"⊥ Fj (ω)
,
n > 1,
j =1
then for each n 1, codim En (ω) =
n−1
j =1 mj
< ∞, and the following assertions are true:
En (ω) ⊂ En−1 (ω) ⊂ · · · ⊂ E2 (ω) ⊂ E1 (ω) = H,
1 lim log Du t, Y (ω), ω (g)H = λn t→∞ t
n > 1; (4.6)
for g ∈ En (ω)\En+1 (ω);
1 log Du t, Y (ω), ω L(H ) = λ1 ; t→∞ t lim
and
(4.7)
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Du t, Y (ω), ω En (ω) ⊆ En θ (t, ω)
(4.8)
for all t 0, perfectly in ω ∈ Ω, for all n 1. Proof. Recall the Oseledec integrability condition E log+
sup
0t1 ,t2 a
Du t2 , Y θ (t1 , ·) , θ (t1 , ·)
L(H )
<∞
(4.9)
for any 0 < a < ∞, which follows directly from (3.66) in Theorem 3.2. Using the above integrability condition and the Ruelle–Oseledec theorem [14, Theorem 2.1.1], there is a random family of compact self-adjoint positive operators Λ(ω) ∈ L(H ), defined perfectly in ω, and satisfies Λ(ω) := lim
∗
1/2t Du t, Y (ω), ω ◦ Du t, Y (ω), ω .
t→∞
(4.10)
The above almost sure limit exists in the uniform operator norm in L(H ), perfectly in ω. The operator Λ(ω) has a discrete non-random spectrum eλ1 > eλ2 > eλ3 > · · · > eλn > · · ·
(4.11)
due to the ergodicity of the Brownian shift θ . The assertions (4.7) and (4.8) of the theorem follow from the Oseledec–Ruelle spectral theorem [14, Theorem 2.1.1]. 2 Remark. If the cocycle is linearized at the zero equilibrium Y ≡ 0, the Oseledec–Ruelle operator Λ(ω) is non-random. Consequently, the Oseledec spaces {En : n 1} are also non-random. This will be shown later in the section. Definition 4.2 (Hyperbolicity). A stationary point Y : Ω → H for the SNSE (3.1) is hyperbolic if the linearized cocycle (Du(t, Y (ω), ω), θ (t, ω)) has a non-vanishing Lyapunov spectrum: λi = 0 for all i 1. Theorem 4.2 below is a consequence of the nonlinear multiplicative ergodic theorem [14, Theorem 2.2.1]. It describes the saddle-point behavior of the random flow of the SNSE (3.1) in the neighborhood of any equilibrium. ¯ For any ρ > 0 and any f ∈ H , we will denote by B(f, ρ) the closed ball in H center f and radius ρ. Theorem 4.2 (The local stable manifold theorem: General equilibrium). Assume that Y : Ω → H is a hyperbolic stationary random point for the cocycle (u, θ ) of the SNSE (3.1) with E log+ |Y | < ∞. Denote by {· · · < λi+1 < λi < · · · < λ2 < λ1 } the Lyapunov spectrum of the linearized cocycle (Du(t, Y (ω), ω), θ (t, ω), t 0) as given in Theorem 4.1. Define i0 := min{i: λi < 0}. Fix 1 ∈ (0, −λi0 ) and 2 ∈ (0, λi0 −1 ). Then there exist (i) a sure event Ω ∗ ∈ F with θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R;
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3575
(ii) F -measurable random variables ρi , βi : Ω ∗ → (0, 1), βi > ρi > 0, i = 1, 2, such that for each ω ∈ Ω ∗ , the following is true: ¯ (ω), ρ1 (ω)) and B(Y ¯ (ω), ρ2 (ω)) (resp.) There are C 1,1 submanifolds S(ω), U(ω) of B(Y with the following properties: ¯ (ω), ρ1 (ω)) such that (a) For λi0 > −∞, S(ω) is the set of all f ∈ B(Y
u(n, f, ω) − Y θ (n, ω) β1 (ω) exp (λi + 1 )n 0 H ¯ (ω), ρ1 (ω)) such for all integers n 0. If λi0 = −∞, then S(ω) is the set of all f ∈ B(Y that
u(n, f, ω) − Y θ (n, ω) β1 (ω)eλn H for all integers n 0 and any λ ∈ (−∞, 0). Furthermore, lim sup t→∞
1 log u(t, f, ω) − Y θ (t, ω) H λi0 t
(4.12)
for all f ∈ S(ω). The stable subspace S 0 (ω) of the linearized cocycle (Du(t, Y (ω), ·), θ (t, ·)) is tangent at Y (ω) to the submanifold S(ω), viz. TY (ω) S(ω) = S 0 (ω). In partici −1 ular, codim S(ω) = codim S 0 (ω) = j0=1 dim Fj (ω) is fixed and finite. (b)
$& % # |u(t, f1 , ω) − u(t, f2 , ω)|H 1 lim sup log sup : f1 = f2 , f1 , f2 ∈ S(ω) λi0 . |f1 − f2 |H t→∞ t
(c) (Cocycle-invariance of the stable manifolds): There exists τ1 (ω) 0 such that
u(t, ·, ω) S(ω) ⊆ S θ (t, ω)
(4.13)
for all t τ1 (ω). Also
Du t, Y (ω), ω S 0 (ω) ⊆ S 0 θ (t, ω) ,
t 0.
(4.14)
¯ (ω), ρ2 (ω)) with the property that there is a discrete(d) U(ω) is the set of all f ∈ B(Y time “history” process y(·, ω) : {−n: n 0} → H such that y(0, ω) = f and for each integer n 1, one has u(1, y(−n, ω), θ (−n, ω)) = y(−(n − 1), ω) and
y(−n, ω) − Y θ (−n, ω) β2 (ω) exp −(λi −1 − 2 )n . 0 H ¯ (ω), ρ2 (ω)) with the property that there is If λi0 −1 = ∞, U(ω) is the set of all f ∈ B(Y a discrete-time “history” process y(·, ω) : {−n: n 0} → H such that y(0, ω) = f and for each integer n 1,
y(−n, ω) − Y θ (−n, ω) β2 (ω) exp{−λn}, H
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for any λ ∈ (0, ∞). Furthermore, for each f ∈ U(ω), there is a unique continuoustime “history” process also denoted by y(·, ω) : (−∞, 0] → H such that y(0, ω) = f , u(t, y(s, ω), θ (s, ω)) = y(t + s, ω) for all s 0, 0 t −s, and lim sup t→∞
1 log y(−t, ω) − Y θ (−t, ω) H −λi0 −1 . t
Each unstable subspace U 0 (ω) of the linearized cocycle (Du(t, Y (·), ·), θ (t, ·)) is tangent at Y (ω) to U(ω), viz. TY (ω) U(ω) = U 0 (ω). In particular,
dim U(ω) =
i 0 −1
dim Fj (ω)
j =1
is finite and non-random. (e) Let y(·, fi , ω), i = 1, 2, be the history processes associated with fi = y(0, fi , ω) ∈ U(ω), i = 1, 2. Then % # $& |y(−t, f1 , ω) − y(−t, f2 , ω)|H 1 lim sup log sup : f1 = f2 , fi ∈ U(ω), i = 1, 2 |f1 − f2 |H t→∞ t −λi0 −1 . (f) (Cocycle-invariance of the unstable manifolds): There exists τ2 (ω) 0 such that
U(ω) ⊆ u t, ·, θ (−t, ω) U θ (−t, ω)
(4.15)
for all t τ2 (ω). Also
Du t, ·, θ (−t, ω) U 0 θ (−t, ω) = U 0 (ω),
t 0;
and the restriction
Du t, ·, θ (−t, ω) U 0 θ (−t, ω) : U 0 θ (−t, ω) → U 0 (ω),
t 0,
is a linear homeomorphism onto. (g) The submanifolds U(ω) and S(ω) are transversal, viz. H = TY (ω) U(ω) ⊕ TY (ω) S(ω). We will only give an outline of the proof of Theorem 4.2. Full details of the proof may be obtained by adapting the arguments in [13,14].
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An outline of the proof of Theorem 4.2. • Develop perfect continuous-time versions of Kingman’s subadditive ergodic theorem as well as the ergodic theorem [14, Lemma 2.3.1(ii), (iii)]. The linearized cocycle (Du(t, Y (ω), ·), θ (t, ·)) at the equilibrium Y can be shown to satisfy the hypotheses of these perfect ergodic theorems. As a consequence of the perfect ergodic theorems, one obtains stable/unstable subspaces for the linearized cocycle, which will constitute tangent spaces to the local stable and unstable manifolds of the nonlinear cocycle (u, θ ). • We use hyperbolicity of the equilibrium Y , the continuous-time integrability condition (3.66) on the cocycle and perfect versions of the ergodic and subadditive ergodic theorems to show the existence of local stable/unstable manifolds for the discrete cocycle (u(n, ·, ω), θ (n, ω)) near Y (ω) (cf. [16, Theorems 5.1 and 6.1]). These manifolds are random objects and are perfectly defined for ω ∈ Ω. Using interpolation between discrete times and the (continuoustime) integrability condition (3.66), it can be shown that the above manifolds for the discretetime cocycle (u(n, ·, ω), θ (n, ω)), n 1, also serve as perfectly defined local stable/unstable manifolds for the continuous-time cocycle (u(t, ·, ω), θ (t, ω)), t 0, near the equilibrium Y (see [13,14,16]). • Again, by using the integrability condition (3.66) on the nonlinear cocycle and its Fréchet derivatives, it is possible to control the excursions of the continuous-time cocycle (u(t, ·, ω), θ (t, ω)), t 0, between discrete times. In view of the perfect subadditive ergodic theorem, these estimates show that the local stable manifolds are asymptotically invariant under the nonlinear cocycle. The asymptotic invariance of the unstable manifolds is obtained via the concept of a stochastic history process for the cocycle. The existence of a stochastic history process is needed because the (locally compact) cocycle is not invertible. This completes the outline of the proof of Theorem 4.2.
2
We next discuss the behavior of the cocycle (u, θ ) near the zero equilibrium. 4.2. Dynamics near the zero equilibrium For the rest of the article, we will focus on the dynamics of the SNSE (3.1) relative to its zero equilibrium Y ≡ 0. In this special case, we are able to express the Lyapunov spectrum of the linearized cocycle (Du(t, 0, ω), θ (t, ω)) explicitly in terms of the parameters ν, γ , σi , i 1, in the SNSE (3.1). Let {Tt }t0 be the strongly continuous semigroup of the operator −A = ν with a Dirichlet boundary condition on ∂D. The operator −A has a discrete spectrum of eigenvalues {μn : n 1} and a complete orthonormal system of corresponding eigenfunctions {en : n 1}. We assume μ1 < μ2 < · · · < μn < · · · . Let Fn0 be the finite-dimensional eigenspace of −A corresponding to the eigenvalue μn for n 1. Next, we linearize the cocycle u : R+ × H × Ω → H at the zero equilibrium. To do this, put Y ≡ 0 in Eq. (4.3) to obtain
t Dv(t, 0, ω)(g) = g −
ADv(s, 0, ω)(g) ds,
(4.16)
0
for all g ∈ H , t ∈ [0, T ] and ω ∈ Ω. This implies that Dv(t, 0, ω) = Tt , t 0, ω ∈ Ω, and thus
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Du(t, 0, ω) = Q(t, ω)Dv(t, 0, ω) = Q(t, ω)Tt ,
t 0, ω ∈ Ω.
(4.17)
The next result is a special case (Y ≡ 0) of the Oseledec–Ruelle spectral theorem (Theorem 4.1). Theorem 4.3 (The Lyapunov spectrum: Zero equilibrium). Let (u(t, ·, ω), θ (t, ω)) be the C 1,1 cocycle on H generated by the stochastic Navier–Stokes equation (3.1). Then all the assertions of Theorem 4.1 hold perfectly in ω ∈ Ω subject to the following: (i) Y ≡ 0;
∞
(ii) λn = λ0n := −μn + γ −
1 2 σk , 2
n 1;
k=1
1
(iii) The Oseledec–Ruelle operator is deterministic and given by Λ0 (ω) = eγ − 2 (iv) The Oseledec spaces are deterministic and given by ! E10
:= H,
En0
:=
n−1 '
∞
2 k=1 σk
T1 ;
"⊥ Fj0
,
n > 1.
j =1
(v) The Lyapunov exponents satisfy the relations ∞ 1 1 2 log Du(t, 0, ω)(g)H = −μn + γ − σk t→∞ t 2
lim
k=1
0 ; and for g ∈ En0 \En+1 ∞ 1 1 2 log Du(t, 0, ω)L(H ) = −μ1 + γ − σk . t→∞ t 2
lim
k=1
Proof. In order to evaluate the Lyapunov spectrum {λ0n : n 1} of the linearized cocycle (Du(t, 0, ω), θ (t, ω)), we first compute the Oseledec–Ruelle operators Λ0 (ω) associated with this cocycle. To do this, use relation (4.17) to obtain Λ0 (ω) := lim
∗ 1/2t Du(t, 0, ω) ◦ Du(t, 0, ω)
t→∞
"1/2t ∞
∗ 2 2σk Wk (t) − σk t = lim exp 2γ t + Tt ◦ Tt !
t→∞
k=1
1/2t Wk (t) σk2 σk − = lim exp γ + lim T ∗ ◦ Tt t→∞ t→∞ t t 2 k=1 ∞
1/2t 1 2 = exp γ − σk lim Tt∗ ◦ Tt . t→∞ 2 ∞
k=1
(4.18)
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Now, it is easy to see that ∗
Tt ◦ Tt (en ) = exp{2μn t}en for all n 1. Therefore,
Tt∗ ◦ Tt
1/2t
= T1
(4.19)
for all t > 0. By (4.18) and (4.19),assertion (iii) of the theorem holds. In particular, the Oseledec– ∞ 1 2 Ruelle operator Λ0 (ω) = eγ − 2 k=1 σk T1 is non-random. Consequently, the Oseledec spaces {En : n 1} are also non-random. Assertions (iv) and (v) of the theorem follow directly from Theorem 4.1. 2 The next corollary is an immediate consequence of the above theorem: It gives necessary and sufficient conditions for hyperbolicity of the zero equilibrium Y ≡ 0. Corollary 4.3.1 (Hyperbolicity of the zero equilibrium). In the SNSE (3.1), the zero equilibrium is hyperbolic if and only if the following conditions hold 2 (i) −μ1 + γ − 12 ∞ k=1 σk > 0; 1 ∞ (ii) −μn + γ − 2 k=1 σk2 = 0 for all n 2. Theorem 4.4 below is a consequence of Theorem 4.2. It describes saddle-point behavior of the random flow of the SNSE (3.1) in the neighborhood of the zero equilibrium. Theorem 4.4 (The local stable manifold theorem: Zero equilibrium). In the SNSE (3.1), assume that the zero equilibrium is hyperbolic. Then all the assertions of the local stable manifold theorem (4.2) hold under the same choice of parameters as in Theorem 4.3. Our next result gives sufficient conditions on the parameters of the SNSE (3.1) to guarantee that the zero equilibrium is its only stationary point. Theorem 4.5 (Uniqueness of the stationary solution). Suppose that ∞
μ1 + γ +
1 2 σk < 0. 2
(4.20)
k=1
Then the zero equilibrium Y ≡ 0 is the only equilibrium (stationary point) of the SNSE (3.1). Proof. Assume that the SNSE (3.1) admits a non-zero stationary solution u0 (t). By stationarity, a := E[|u0 (t)|2H ] > 0 and b := E[u0 (t)2V ] > 0 are independent of t. Suppose t > s > 0. Then from (3.1) and Ito’s formula, we have
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u0 (t)2 = u0 (s)2 − 2ν H H
t
∞ u0 (r)2 dr + 2 V
t
2 σk u0 (r)H dWk (r)
k=1 s
s
+ 2γ
t
∞ u0 (r)2 dr + H
t
2 σk2 u0 (r)H dr.
(4.21)
k=1 s
s
Taking expectations on both sides of the above identity, we obtain a = a − 2ν(t − s)b +
∞
σk2 (t − s)a + 2γ (t − s)a.
(4.22)
k=1
Hence ! 2νb =
∞
" σk2
+ 2γ a.
(4.23)
k=1
Combining the above equality with the Poincare inequality: a
νb , −μ1
(4.24)
it follows that ∞
−μ1 γ +
1 2 σk . 2
(4.25)
k=1
This proves the theorem.
2
We conclude this section by stating the Local and Global Invariant Manifold Theorems for the SNSE (3.1) when γ = 0 (Theorems 4.6 and 4.7 below). The Local Invariant Manifold Theorem (Theorem 4.6) characterizes the almost sure asymptotic stability of the random flow of the SNSE (3.1) in the neighborhood of the zero equilibrium, in the special case when the linear drift vanishes (γ = 0). On the other hand, the Global Invariant Manifold Theorem (Theorem 4.7) gives a random cocycle-invariant foliation of the energy space H . The leaves of the foliation are 2 characterized by the Lyapunov exponents {λ0i = −μi + γ − 12 ∞ k=1 σk : i 1} of the linearized cocycle (Du(t, 0, ω), θ (t, ω)). Theorem 4.6(Local invariant manifolds). Consider the SNSE (3.1) with γ = 0. Fix 1 ∈ 2 (0, −μ1 + 12 ∞ k=1 σk ). Then there exist (i) a sure event Ω ∗ ∈ F with θ (t, ·)(Ω ∗ ) = Ω ∗ for all t ∈ R; (ii) F -measurable random variables ρi , βi : Ω ∗ → (0, 1), βi > ρi ρi+1 > 0, i 1, such that for each ω ∈ Ω ∗ , the following is true: ¯ ρi (ω)) with the following properties: There are C 1,1 submanifolds Si (ω), i 1, of B(0,
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
3581
¯ ρi (ω)) such that (a) Si (ω) is the set of all f ∈ B(0, u(n, f, ω) βi (ω) exp H
∞ 1 2 μi − σ k + 1 n 2 k=1
for all integers n 0. Furthermore, lim sup t→∞
∞ 1 1 2 σk log u(t, f, ω)H μi − t 2
(4.26)
k=1
for all f ∈ Si (ω). Each Oseledec space Ei0 of the linearized cocycle (Du(t, 0, ·), θ (t, ·)) is tangent at 0 to the submanifold Si (ω), viz. T0 Si (ω) = Ei0 . In particular, codim Si (ω)= 0 codim Ei0 = i−1 j =1 dim Fj ( fixed and finite). (b) % # $& |u(t, f1 , ω) − u(t, f2 , ω)|H 1 : f1 = f2 , f1 , f2 ∈ Si (ω) lim sup log sup |f1 − f2 |H t→∞ t ∞
μi −
1 2 σk . 2 k=1
(c) (Cocycle-invariance): There exists τi (ω) 0 such that
u(t, ·, ω) Si (ω) ⊆ Si θ (t, ω)
(4.27)
Du(t, 0, ω) Ei0 ⊆ Ei0 ,
(4.28)
for all t τi (ω). Also t 0.
2 ∗ ∗ ∗ Proof. Let 1 ∈ (0, −μ1 + 12 ∞ k=1 σk ). Then there exist Ω ∈ F such that θ (t, ·)(Ω ) = Ω for ∗ all t ∈ R, and F -measurable random variables ρi , βi : Ω → (0, 1) such that βi (ω) > ρi (ω) > 0, ¯ ρi (ω)) such that and C 1,1 local stable submanifolds Si (ω) ⊂ B(0,
1 ∞ 2 Si (ω) := f ∈ B¯ 0, ρi (ω) : u(n, f, ω)H βi (ω)e(μi − 2 k=1 σk + 1 )n for all n 1 . (4.29) Furthermore, ∞ 1 1 2 lim sup log u(t, f, ω) H μi − σk 2 t→∞ t k=1
for all f ∈ Si (ω).
(4.30)
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Also, T0 Si (ω) = Ei0 , the Oseledec space for the linearized cocycle (Du(t, 0, ω), θ (t, ω)) cor 2 responding to the Lyapunov exponent λi := μi − 12 ∞ k=1 σk , i 1. Following the argument in [13,16], the random variables ρi (ω), βi (ω) may be selected such that
ρi (ω)e(λi + 1 )t ρi θ (t, ω)
(4.31)
βi (ω)e(λi + 1 )t βi θ (t, ω)
(4.32)
and
for all t 0 and ω ∈ Ω ∗ . We now show that there exists τi (ω) > 0 such that
u(t, ·, ω) Si (ω) ⊆ Si θ (t, ω)
(4.33)
for all t τi (ω) and all ω ∈ Ω ∗ . Let f ∈ Si (ω), t 0 and let n 0 be any integer. Then (by the cocycle property),
u n, u(t, f, ω), θ (t, ω) = u(n + t, f, ω) . H H
(4.34)
From [13,16], we have 1 lim sup log t→∞ t
% sup
f ∈Si (ω) f =0
|u(t, f, ω)|H |f |H
& λi .
From the above estimate, for any ∈ (0, 1 ), there exists N0 = N0 ( ) > 0 such that % 1 sup log tN t
sup
f ∈Si (ω) f =0
|u(t, f, ω)|H |f |H
&
λi + ,
for all N N0 . Thus sup
f ∈Si (ω) f =0
|u(t, f, ω)|H
e(λi + )t , |f |H
for all t N0 . Define
βi (ω) := sup
sup
0tN0 f ∈Si (ω) f =0
|u(t, f, ω)|H −(λi + )N0 ·e . |f |H
Therefore, sup
f ∈Si (ω) f =0
|u(t, f, ω)|H
βi (ω) · e(λi + )t |f |H
(4.35)
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3583
for all t 0. Since f ∈ Si (ω) ⊂ B(0, 1), then |f |H 1 and u(t, f, ω) β (ω)e(λi + )t i H
(4.36)
for all t 0. Therefore, from (4.34) and (4.36),
u n, u(t, f, ω), θ (t, ω) β (ω)e(λi + )(n+t) i H
βi (ω)e(λi + )t · e(λi + 1 )n
(4.37)
for t 0 and all integers n 0. Since < 1 , it follows that
β (ω)e(λi + )t βi (ω) ( − 1 )t ·e = lim = 0. lim i t→∞ βi (ω)e(λi + 1 )t t→∞ βi (ω) Hence there exists τ˜i (ω) > 0 so that
βi (ω)e(λi + )t βi (ω)e(λi + 1 )t ,
(4.38)
for all t τ˜i (ω). By (4.37), (4.38) and (4.32), we get
u n, u(t, f, ω), θ (t, ω) βi (ω)e(λi + 1 )t · e(λi + 1 )n H
βi θ (t, ω) e(λi + 1 )n
(4.39)
for all t τ˜i (ω) and all n 1. Again, because < 1 , we have
βi (ω)e(λi + )t β (ω)e(λi + )t lim i = 0. t→∞ ρi (θ (t, ω)) t→∞ ρi (ω)e(λi + 1 )t lim
(4.40)
Therefore, there exists τ˜˜ i (ω) > 0 such that
βi (ω)e(λi + )t ρi θ (t, ω)
(4.41)
¯ ρi (θ (t, ω))) for all t τ˜˜ i (ω). for all t τ˜˜ i (ω). Hence u(t, f, ω) ∈ B(0, ˜ ¯ ρi (θ (t, ω))) and satisfies (4.39) for all Set τi (ω) := τ˜i (ω) ∨ τ˜ i (ω). Then u(t, f, ω) ∈ B(0, n 0 and all t τi (ω). By definition of Si (θ (t, ω)), it follows that u(t, f, ω) ∈ Si (θ (t, ω)) for all t τi (ω). Thus u(t, ·, ω)(Si (ω)) ⊆ Si (θ (t, ω)) for all t τi (ω). Note that τi (ω), τ˜i (ω), τ˜˜ i (ω)
are all independent of f ∈ Si (ω) because βi (ω), βi (ω) and ρi (ω) are independent of f ∈ Si (ω). This completes the proof of asymptotic invariance of Si (ω), i 1. 2 Our final result (Theorem 4.7 below) gives the existence of a global invariant flag for the cocycle (u, θ ). The foliation is induced by the Lyapunov spectrum {λi }∞ i=1 of the linearized cocycle.
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Theorem 4.7 (Global invariant flag). Consider the SNSE (3.1) with γ = 0. Define the random family of sets {Mi (ω): ω ∈ Ω ∗ , i 1} by $ # 1 Mi (ω) := f ∈ H : lim log u(t, f, ω) H λi t→∞ t
(4.42)
for i 1, ω ∈ Ω ∗ . For fixed i 1, ω ∈ Ω ∗ , define the sequence {Sin (ω)}∞ n=1 , inductively by: Si1 (ω) := Si (ω), u(n, ·, ω)−1 [Si (θ (n, ω))], n Si (ω) := Sin−1 (ω),
(4.43) if
Sin−1 (ω) ⊆ u(n, ·, ω)−1 [Si (θ (n, ω))],
otherwise,
(4.44)
for all n 2. In (4.43) and (4.44), the Si (ω) are the local invariant C 1,1 Hilbert submanifolds of H constructed in Theorem 4.6. Then the following is true for each i 1 and ω ∈ Ω ∗ : (i) The sets {Mi (ω): ω ∈ Ω ∗ , i 1} are cocycle-invariant:
u(t, ·, ω) Mi (ω) ⊆ Mi θ (t, ω)
(4.45)
for all t 0. (ii) Sin (ω) ⊆ Sin+1 (ω) for all n 1, and Mi (ω) =
∞ (
Sin (ω),
i 1,
(4.46)
n=1
( perfectly in ω). (iii) Mi+1 (ω) ⊆ Mi (ω). (iv) For any f ∈ Mi (ω)\Mi+1 (ω), 1 log u(t, f, ω)H ∈ (λi+1 , λi ]. t→∞ t lim
(4.47)
Proof. Fix ω ∈ Ω ∗ , where Ω ∗ is defined as in Theorem 4.6. (i) To prove the cocycle invariance property (4.45), let f ∈ Mi (ω) and t1 > 0. Then by definition (4.42) of Mi (ω), we have 1 log u(t, f, ω)H λi . t→∞ t lim
(4.48)
By the cocycle property of (u, θ ), we have
1 1 log u t, u(t1 , f, ω), θ (t1 , ω) H = lim log u(t + t1 , f, ω)H t→∞ t t→∞ t 1 t + t1 = lim log u(t + t1 , f, ω)H · lim t→∞ t + t1 t→∞ t lim
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591
= lim
1
t→∞ t
3585
log u(t, f, ω)H
λi . The above inequality implies that u(t1 , f, ω) ∈ Mi (θ (t1 , ω)). Hence u(t1 , ·, ω)(Mi (ω)) ⊆ Mi (θ (t1 , ω)) and so (4.45) holds for all t 0. (ii) To prove assertion (ii) of the theorem, observe first that (4.44) implies that Sin (ω) ⊆ n+1 Si (ω) for all n 1. Next, we show that Sin (ω) ⊂ Mi (ω)
(4.49)
for all n 1. We prove (4.49) by induction on n 1. Let f ∈ Si1 (ω) = Si (ω). By Theorem 4.6 and assertion (4.26), it follows that 1 log u(t, f, ω)H λi t→∞ t lim
(4.50)
perfectly in ω. Therefore, f ∈ Mi (ω). Hence Si1 (ω) = Si (ω) ⊂ Mi (ω). Assume, by induction, that Sik (ω) ⊂ Mi (ω) for all 1 k n. If Sin+1 (ω) u(n + 1, ·, ω)−1 [Si (θ (n + 1, ω))], then Sin+1 (ω) = Sin (ω) ⊂ Mi (ω), by inductive hypothesis. Otherwise, Sin+1 (ω) = u(n + 1, ·, ω)−1 [Si (θ (n + 1, ω))]. Let f ∈ Sin+1 (ω) = u(n+1, ·, ω)−1 [Si (θ (n+1, ω))]. Then by the cocycle property and the definition of Si (θ (n + 1, ω)), it follows that
u n + n + 1, f, ω βi θ (n + 1, ω) en λi H
(4.51)
for all n 1. This implies that
1 u n + n + 1, f, ω λi . log n →∞ n
lim
Hence lim
n
→∞
1 1 log u n + n + 1, f, ω H log u n
, f, ω H = lim
n n →∞ n + n + 1
n
1 · lim log u n + n + 1, f, ω H = lim
n →∞ n + n + 1 n →∞ n λi .
Therefore, f ∈ Mi (ω), and Sin+1 (ω) ⊂ Mi (ω). So, by induction, it follows that Sin (ω) ⊂ Mi (ω) for all n 1. Thus
(4.52)
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S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 ∞ (
Sin (ω) ⊆ Mi (ω).
(4.53)
n=1
In order to prove the converse inclusion Mi (ω) ⊆
∞ (
Sin (ω),
(4.54)
n=1
we establish the following: Claim. There exist an increasing (random) sequence of integers nk ↑ ∞ such that
−1 k Sin (ω) = u nk , ·, ω Si θ n, ω for all k 1. Proof. Define n1 := inf{n > 1: Sin−1 (ω) ⊆ u(n, ·, ω)−1 [Si (θ (n, ω))]}. Then 1 −1
Sin
−1 1
Si θ n , ω , (ω) ⊆ u n1 , ·, ω
and by definition (4.44),
Sin1 (ω) = u(n1 , ·, ω)−1 Si θ (n1 , ω) .
(4.55)
Furthermore, Sin−1 (ω) u(n, ·, ω)−1 [Si (θ (n, ω))] for all 1 < n < n1 , and so by definition (4.44), Sin (ω) = Sin−1 (ω) = Sin−2 (ω) = · · · = Si1 (ω) = Si (ω) for all 1 < n < n1 . In particular, 1 −1
Si (ω) = Sin
−1 1
Si θ n , ω . (ω) ⊆ u n1 , ·, ω
Therefore,
u n1 , ·, ω Si (ω) ⊆ Si θ n1 , ω . Hence
n1 = inf n > 1: u(n, ·, ω) Si (ω) ⊆ Si θ (n, ω) .
(4.56)
Since Si (ω) is asymptotically cocycle invariant (Theorem 4.6(c), (4.27)), it follows from (4.56) that 1 < n1 < ∞. Next, define n2 > n1 by
n2 := inf n > n1 : Sin−1 (ω) ⊆ u(n, ·, ω)−1 Si θ (n, ω) . As before, the definition (4.44) implies that
(4.57)
S. Mohammed, T. Zhang / Journal of Functional Analysis 258 (2010) 3543–3591 2 −1
Sin
1 +1
(ω) = Sin
−1 1
Si θ n , ω (ω) = u n1 , ·, ω
3587
(4.58)
and
−1 2
2 Si θ n , ω . Sin (ω) = u n2 , ·, ω
(4.59)
Since 2 −1
Sin
−1 2
Si θ n , ω , (ω) ⊆ u n2 , ·, ω
(4.60)
it follows from (4.58) that
−1 1
−1 2
Si θ n , ω ⊆ u n2 , ·, ω Si θ n , ω . u n1 , ·, ω Therefore,
−1 1
−1 1
2 Si θ n , ω ⊆ Si θ n2 , ω . ◦ u n , ·, ω u n , ·, ω
(4.61)
Using the cocycle property (Theorem 3.2(iii)), (4.61) implies
u n2 − n1 , ·, θ n1 , ω Si θ n1 , ω ⊆ Si θ n2 − n1 , θ n1 , ω .
(4.62)
By the asymptotic cocycle invariance of Si (θ (n1 , ω)), it follows from (4.62) that n1 < n2 < k ∞. Hence by induction, there exists an increasing sequence of integers {nk }∞ k=1 such that n ↑ ∞ as k → ∞ and
−1 k
k Si θ n , ω Sin (ω) = u nk , ·, ω for all integers k 1. This completes the proof of our claim.
(4.63)
2
We now proceed to prove the inclusion (4.54). Let f ∈ Mi (ω). Then by definition of Mi (ω), we have 1 log u(t, f, ω)H λi . t→∞ t lim
(4.64)
Fix 1 ∈ (0, −λ1 ) as in Theorem 4.6. Let 0 < < 1 . Then, using (4.64), there exists a positive integer n0 such that 1 log u(t, f, ω)H < λi +
tn t
sup for all n n0 . In particular,
u(n, f, ω) < en(λi + ) H for all n n0 . Define
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K(ω) := max u(k, f, ω)H . 1k
Therefore, u(n, f, ω) K(ω)en(λi + ) H
(4.65)
for all n 1. Pick m0 sufficiently large such that
K(ω)en( − 1 ) βi θ (n, ω)
(4.66)
for all n m0 . Let n m0 , n 0. Using the cocycle property and (4.65), we obtain
u n , u(n, f, ω), θ (n, ω) = u n + n, f, ω H H
K(ω)e(n+n )(λi + )
K(ω)en( − 1 ) · en (λi + 1 ) .
(4.67)
Pick m1 m0 and sufficiently large such that
K(ω)en( − 1 ) βi θ (n, ω)
(4.68)
for all n m1 . From (4.67) and (4.68), we get
u n , u(n, f, ω), θ (n, ω) βi θ (n, ω) en (λi + 1 ) H
(4.69)
for all n 0 and n m1 . Since u(n, f, ω) → 0 as n → ∞, then there exists m2 > 0 such that ¯ ρi (ω)) for all n m2 . Thus (4.69) implies that u(n, f, ω) ∈ B(0,
u(n, f, ω) ∈ Si θ (n, ω) for all n max(m1 , m2 ); i.e. f ∈ u(n, ·, ω)−1 [Si (θ (n, ω))], for all n max(m1 , m2 ). Now pick nk k k −1 k k sufficiently large such ∞that nn max(m1 , m2 ) and f ∈ u(n , ·, ω) [Si (θ (n , ω))] = Si (ω). This proves that f ∈ n=1 Si (ω); and so the inclusion (4.54) holds. The proof of assertion (ii) of the theorem is complete. Assertions (iii) and (iv) of the theorem follow directly from the definition (4.42) of the flag Mi (ω), i 1. 2 Remark. It is not clear if the Mi (ω) in Theorem 4.7 are C 1,1 immersed submanifolds in H . This would require transversality of the global semiflow u(n, ·, ω) and the local stable manifold Si (θ (n, ω)).
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5. Appendix The following version of Gronwall’s lemma is used throughout Section 3. Lemma 5.1. Suppose α : [0, T ] → R is C 1 and h, ψ : [0, T ] → R0 are continuous. Assume that
t α(t) +
t ψ(s) ds α(0) +
h(s)α(s) ds
0
0
t
t
(5.1)
for all t ∈ [0, T ]. Then ψ(s) ds α(0) exp
α(t) +
h(s) ds
0
(5.2)
0
for all t ∈ [0, T ]. Proof. Suppose (5.1) holds. Since α is C 1 , then (5.1) implies
t
t h(s)α(s) ds −
0
t
α (s) ds − 0
ψ(s) ds 0 0
for all t ∈ [0, T ]. Hence
t
h(s)α(s) − α (s) − ψ(s) ds 0
(5.3)
0
for all t ∈ [0, T ]. The above relation implies h(t)α(t) − α (t) − ψ(t) 0 for all t ∈ [0, T ]; i.e., α (t) − h(t)α(t) −ψ(t)
(5.4)
for) all t ∈ [0, T ]. We now multiply both sides of (5.4) by the “integrating factor” μ(t) := t e− 0 h(s) ds . This gives d μ(t)α(t) −μ(t)ψ(t) dt for all t ∈ [0, T ]. Integrating both sides of (5.5), we get
(5.5)
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t μ(t)α(t) − α(0) −
μ(s)ψ(s) ds 0
for all t ∈ [0, T ]. Therefore, −1
t
α(t) + μ(t)
μ(s)ψ(s) ds α(0)μ(t)−1
(5.6)
0
for all t ∈ [0, T ]. So α(t) + e
t
)t
0 h(s) ds
e−
)s 0
h(u)du
ψ(s) ds α(0)e
)t 0
h(s) ds
;
0
i.e.,
t α(t) +
e
)t s
h(u)du
ψ(s) ds α(0)e
)t 0
h(s) ds
(5.7)
0
for all t ∈ [0, T ]. Since h(u) 0 for all u ∈ [0, T ], (5.7) implies that
t α(t) +
t ψ(s) ds α(t) +
0
for all t ∈ [0, T ]. Therefore (5.2) holds.
e
)t s
h(u)du
ψ(s) ds α(0)e
)t 0
h(s) ds
0
2
References [1] Z. Brze´zniak, M. Capinski, F. Flandoli, Stochastic Navier–Stokes equations with multiplicative noise, Stoch. Anal. Appl. 10 (1992) 523–532. [2] Z. Brze´zniak, Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains, Trans. Amer. Math. Soc. 358 (12) (2006) 5587–5629. [3] T. Caraballo, P.E. Kloeden, B. Schmalfuss, Exponentially stable stationary solution for stochastic evolution equations and their perturbation, Appl. Math. Optim. 50 (2004) 183–207. [4] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [5] G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. [6] J. Duan, K. Lu, B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab. 31 (2003) 2109–2135. [7] J. Duan, K. Lu, B. Schmalfuss, Stable and unstable manifolds for stochastic partial differential equations, J. Dynam. Differential Equations 16 (4) (2004) 949–972. [8] F. Flandoli, Stochastic Flows for Nonlinear Second-Order Parabolic SPDE’s, Stoch. Monogr., vol. 9, Gordon and Breach Science Publishers, Yverdon, 1995. [9] F. Flandoli, Stochastic differential equations in fluid dynamics, Rend. Sem. Mat. Fis. Milano 66 (1996) 121–148. [10] M. Gourcy, A large deviation principle for 2D stochastic Navier–Stokes equation, Stochastic Process. Appl. 117 (7) (2007) 904–927.
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[11] M. Hairer, J.C. Mattingly, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Ann. of Math. (2) 164 (3) (2006) 993–1032. [12] N. Masmoudi, L.-S. Young, Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs, Comm. Math. Phys. 227 (2002) 461–481. [13] S.-E.A. Mohammed, M.K.R. Scheutzow, The stable manifold theorem for nonlinear stochastic systems with memory, Part I: Existence of the semiflow, J. Funct. Anal. 205 (2003) 271–305, Part II: The local stable manifold theorem J. Funct. Anal. 206 (2004) 253–306. [14] S.-E.A. Mohammed, T.S. Zhang, H.Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Part 1: The stochastic semiflow, Part 2: Existence of stable and unstable manifolds, Mem. Amer. Math. Soc. 196 (2008) 105. [15] J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2001. [16] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. 115 (1982) 243–290. [17] S.S. Sritharan, P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (11) (2006) 1636–1659. [18] R. Temam, Navier–Stokes Equations, North-Holland Amsterdam, New York, Oxford, 1979. [19] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988.
Journal of Functional Analysis 258 (2010) 3593–3603 www.elsevier.com/locate/jfa
The role of BMOA in the boundedness of weighted composition operators ✩ Eva A. Gallardo-Gutiérrez a,∗ , Jonathan R. Partington b a Departamento de Matemáticas, Universidad de Zaragoza e IUMA, Plaza San Francisco s/n, 50009 Zaragoza, Spain b School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
Received 22 April 2009; accepted 27 February 2010
Communicated by K. Ball
Abstract Boundedness (resp. compactness) of weighted composition operators Wh,ϕ acting on the classical Hardy space H2 as Wh,ϕ f = h(f ◦ ϕ) are characterized in terms of a Nevanlinna counting function associated to the symbols h and ϕ whenever h ∈ BMOA (resp. h ∈ VMOA). Analogous results are given for Hp spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space). © 2010 Elsevier Inc. All rights reserved. Keywords: Weighted composition operators; BMOA and VMOA; Nevanlinna counting functions
1. Introduction Let D denote the open unit disk of the complex plane, T its boundary and m the normalized arc-length measure on T. Let Hp , 1 p < ∞ be the classical Hardy space, that is, the space of holomorphic functions f on D for which the norm
✩
The authors are partially supported by Plan Nacional I+D grant No. MTM2007-61446 and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, Ref. DGA E-64. * Corresponding author. E-mail addresses:
[email protected] (E.A. Gallardo-Gutiérrez),
[email protected] (J.R. Partington). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.019
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f p =
sup 0r<1
1/p f (rζ )p dm(ζ )
T
is finite. The space consisting of bounded analytic functions on D will be denoted by H∞ . Given two analytic functions h and ϕ on D such that ϕ(D) ⊂ D, it is possible to define for those functions f ∈ Hp for which it makes sense the linear map Wh,ϕ defined by Wh,ϕ f (z) = h(z)f ϕ(z) .
(1)
If f belongs to the domain of Wh,ϕ , which be will denoted by D(Wh,ϕ ), the linear map in (1) is called a weighted composition operator. Hence, the key question is: When does a weighted composition operator take Hp boundedly into itself? As a consequence of the Littlewood Subordination Theorem [12], it is straightforward that the condition h ∈ H∞ always suffices for boundedness of Wh,ϕ . By considering the image of the constant functions, it is clear that h ∈ Hp is a necessary condition. Of course, if ϕ is a constant function, h ∈ Hp is also sufficient. Throughout this paper, we will assume that ϕ is a non-constant function. Note that the most interesting feature in this sense is that there exist symbols h (unbounded) and ϕ ∈ H∞ with ϕ(D) ⊆ D such that Wh,ϕ is a bounded operator in Hp . The pioneering works on weighted composition operators can be traced back to the sixties. Indeed, de Leeuw showed that the isometries in the Hardy space H1 are weighted composition operators, while Forelli obtained the same result for the Hardy spaces Hp when 1 < p < ∞, p = 2 (see [11,6]). Recently, there have been an increasing interest in studying weighted composition operators acting on different spaces of analytic functions. For instance, boundedness and compactness of weighted composition operators on Hardy spaces have been studied in [2] in terms of the pullback of Carleson measures. Our approach to the question of when a weighted composition operator Wh,ϕ is bounded (resp. compact) on Hp is completely different, and it involves a condition related to a Nevanlinna Counting Function associated to the symbols h and ϕ. As a particular instance, we will show that the space BMOA (resp. VMOA) will play a prominent role in the characterization of boundedness and compactness of weighted composition operators in H2 . Furthermore, as we will see, BMOA (resp. VMOA) can be characterized in terms of the boundedness (resp. compactness) of weighted composition operators. Let us point out that, in the particular case that h is the constant function 1, our condition on compactness agrees with that one proved by Shapiro in his well-known work [14] (see also [3,15]). Finally, we characterize boundedness and compactness of weighted composition operators on the scale of weighted Bergman spaces, where in the particular instance of the Hilbert ones, BMOA is replaced by the classical Bloch space B (and VMOA by the little Bloch space B0 ).
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A word about notation. Throughout the paper, for any set E and any integrable function f respect to a measure μ, we will denote the integral mean of f over E by f dμ =
1 μ(E)
E
f dμ. E
In addition, we will denote a ≈ b whenever there exist two positive universal constants c and C, such that cb a Cb. Further, for the sake of simplicity, C will always denote an independent constant, which can be different from one display to another. 2. Weighted composition operators on the Hardy space In this section we characterize boundedness and compactness of weighted composition operators on the classical Hardy spaces Hp , for 1 p < ∞. Firstly, we will state and prove the result in H2 , where the classical space BMOA plays a prominent role. Later, we will indicate the pertinent changes for the corresponding result in Hp . Before going further, recall that the space BMO consists, by definition, of all functions f ∈ L1 (T; m) satisfying f ∗ = sup z∈D
f (ζ ) − f ∗ Pz Pz (ζ ) dm(ζ ) < ∞,
T
where Pz (ζ ) denotes the Poisson kernel at z, i.e., Pz (ζ ) =
1 − |z|2 |ζ − z|2
(ζ ∈ T),
and f ∗ Pz stands for T f (ζ )Pz (ζ ) dm(ζ ). Alternative characterizations of BMO, as well as a systematic treatment of the subject, can be found in Garnett’s book [9, Chapter VI]. The analytic subspace BMOA is defined by BMOA = BMO ∩ H1 , and one has H∞ ⊂ BMOA ⊂
Hp .
0
The space VMO is the subspace of BMO consisting of functions of vanishing mean oscillation; that is, such that f ∗ =
f (ζ ) − f ∗ Pz Pz (ζ ) dm(ζ ) → 0,
T
as |z| → 1. Then VMOA = BMOA ∩ VMO. In order to state our main result, recall that a Carleson disc in D centered at ζ ∈ ∂D of radius r is given by D(ζ, r) = {z ∈ D: |z − ζ | < r}. A positive Borel measure μ on D is a Carleson measure, if
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μ(D(ζ, r)) < ∞, r ζ ∈T, 0
and a vanishing Carleson measure, if μ(D(ζ, r)) → 0, r ζ ∈T sup
as r → 0. Theorem 2.1. Let h ∈ H2 and ϕ ∈ H∞ such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) h ∈ BMOA; (b) Wh,ϕ is a bounded operator on H2 ;
2 1 h(z) log (c) sup dA(u) = O(r), |z| ζ ∈T D(ζ,r)
ϕ(z)=u
where dA denotes normalized Lebesgue measure on D. Then any two of these properties implies the third. Proof. Assume first that h ∈ BMOA and Wh,ϕ is bounded on H2 and let us fix a Carleson disc D(ζ0 , r0 ) in D, centered at ζ0 ∈ ∂D of radius r0 . Let us consider w0 = (1 − r0 )ζ0 . Then, if kw0 denotes the normalized reproducing kernel in H2 at w0 ∈ D, that is, kw0 (z) =
(1 − |w0 |2 )1/2 1 − w0 z
(z ∈ D),
it is clear that Wh,ϕ kw0 2 is bounded by Wh,ϕ . By the Littlewood–Paley identity [15, p. 51], we have 2 2 Wh,ϕ kw0 22 = h(0) kw0 ϕ(0) + 2
D
h(kw ◦ ϕ) (z)2 log 1 dA(z) Wh,ϕ 2 . 0 |z|
Thus, D
h (z)kw ϕ(z) + h(z)k ϕ(z) ϕ (z)2 log 1 dA(z) C. w0 0 |z|
We write this as the sum of three terms, namely,
(2)
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T1 = T2 = D
D
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h (z)kw ϕ(z) 2 log 1 dA(z), 0 |z|
h(z)k ϕ(z) ϕ (z)2 log 1 dA(z), w0 |z|
1
ϕ(z) ϕ (z) log dA(z), h (z)kw0 ϕ(z) h(z)kw 0 |z|
T3 = 2 Re D
and of course T32 4T1 T2 by the Cauchy–Schwarz inequality. 1 dA(z) is a Carleson measure [1, p. 146], and so Now h ∈ BMOA, so that |h (z)|2 log |z| T1 C
iθ 2 dθ = CCϕ kw 2 CCϕ 2 , k w ϕ e 0 0 2
T
so T1 remains bounded. By the elementary identity x2 x1 + x2 + x1 for norms, we see that T2 also remains bounded, i.e., D
h(z)k ϕ(z) ϕ (z)2 log 1 dA(z) C. w0 |z|
Changing variables to u = ϕ(z) in T2 (see [3, Chapter 2]), we have
k (u)2 w0
h(z)2 log 1 dA(u) C. |z| ϕ(z)=u
D
In particular the same holds for the integral over the Carleson disc D(ζ0 , r0 ), giving D(ζ0 ,r0 )
1 − |w0 |2 |1 − w0 u|4
h(z)2 log 1 dA(u) C. |z| ϕ(z)=u
Now, for u ∈ D(ζ0 , r0 ), one has |1 − w0 u| ≈ 1 − |w0 | = r0 . Hence,
h(z)2 log 1 dA(u) Cr0 , |z|
D(ζ0 ,r0 )
ϕ(z)=u
and therefore,
h(z)2 log 1 dA(u) = O(r0 ), |z|
D(ζ0 ,r0 )
which proves (c).
ϕ(z)=u
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Next, suppose that (c) holds and h ∈ BMOA. To prove that Wh,ϕ is bounded, it is enough to show that Wh,ϕ kw 2 is uniformly bounded in w, by [10, Thm. 3.3]. By the Littlewood–Paley identity above, it is enough to show that both integrals D
h(z)k ϕ(z) ϕ (z)2 log 1 dA(z), w |z|
(3)
and D
h (z)kw ϕ(z) 2 log 1 dA(z), |z|
(4)
remain uniformly bounded in w. Note that (4) is bounded because h ∈ BMOA, and therefore 1 |h (z)|2 log |z| dA(z) is a Carleson measure. So, we need to deal just with (3). Let w ∈ D and consider the Carleson disc centered at ζw = w/|w| of radius rw = 1 − |w|. For n 1, let us consider the sets n (w) = u ∈ D: 2n−1 rw |ζw − u| < 2n rw . Upon changing variables in (3), we deduce D
h(z)k ϕ(z) ϕ (z)2 log 1 dA(z) = w |z|
=
k (u)2 w
h(z)2 log 1 dA(u) |z| ϕ(z)=u
n (w)
∞
k (u)2 w
n=1 (w) n
≈
h(z)2 log 1 dA(u) |z| ϕ(z)=u
∞
1 − |w|2 n=1
(2n rw )4 n (w)
h(z)2 log 1 dA(u), |z| ϕ(z)=u
where the third line follows since for any u ∈ n (w) one has |1 − wu| ≈ 2n rw . Now, since n (w) ⊂ D(ζw , 2n rw ), one deduces that the above series is bounded by ∞
n=1
1 2n 2 rw
D(ζw ,2n rw )
h(z)2 log 1 dA(u). |z| ϕ(z)=u
This along with the assumption (c), yields that the integral in (3) remains uniformly bounded in w. Hence, Wh,ϕ kw 2 is uniformly bounded in w, and therefore Wh,ϕ is a bounded operator. Finally, let us assume that (c) holds and Wh,ϕ is bounded. Proving that h ∈ BMOA is equiva1 lent to showing that |h (z)|2 log |z| dA(z) is a Carleson measure. Note that for this latter task, it suffices to test it on normalized reproducing kernels [13].
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Now, taking into account the fact that Wh,ϕ is bounded if and only if Wh,ϕ kw 2 is uniformly bounded in w (see [10, Thm. 3.3]), and proceeding in a similar way as before, the desired conclusion follows. 2 An analogous result holds for compactness. Theorem 2.2. Let h ∈ H2 and ϕ ∈ H∞ such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) h ∈ VMOA; (b) Wh,ϕ is a compact operator on H2 (D);
h(z)2 log 1 dA(u) = o(r) (c) sup |z| ζ ∈T D(ζ,r)
as r → 0.
ϕ(z)=u
Then any two of these properties implies the third. Proof. The proof follows the lines of the proof of Theorem 2.1, bearing in mind that Wh,ϕ is compact if and only if Wh,ψ kw 2 → 0 as |w| → 1 (see [10]) along with the fact that h ∈ VMOA 1 is equivalent to the condition that |h (z)|2 log |z| dA(z) is a vanishing Carleson measure. 2 Example 2.3. Consider the singular inner function ϕ defined by z+1 ϕ(z) = exp z−1 for z ∈ D, and the weight function h defined by h(z) = log(1 − z), which is well known to lie in BMOA but not H∞ . We may use Theorem 2.1 to test the boundedness of the operator Wh,ϕ on H2 . Indeed, a little computation shows that for u ∈ D(ξ, r) the solutions zn to ϕ(z) = u satisfy 2 log(1 − zn )2 log 1 = O r(log n) |zn | n2 and therefore, condition (c) in Theorem 2.1 holds. Remark 2.4. Some of the ideas in Theorem 2.1 and Theorem 2.2 are connected to the boundedness and compactness of composition operators on Hardy spaces on Lavrentiev domains in [7]. Weighted composition operators on Hp spaces. In general, the boundedness or compactness of a weighted composition operator Wh,ϕ acting on a Hardy space Hp depends on p. The next results are the corresponding ones for Hp spaces. Theorem 2.5. Let h ∈ Hp , 1 p < ∞, and ϕ ∈ H∞ such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) |h (z)|2 |h(z)|p−2 dA(z) is a Carleson measure;
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(b) Wh,ϕ is a bounded operator on Hp ;
p 1 h(z) log dA(u) = O(r). (c) sup |z| ζ ∈T D(ζ,r)
ϕ(z)=u
Then any two of these properties implies the third. Theorem 2.6. Let h ∈ Hp , 1 p < ∞, and ϕ ∈ H∞ such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) |h (z)|2 |h(z)|p−2 dA(z) is a vanishing Carleson measure; (b) Wh,ϕ is a compact operator on Hp ;
p 1 h(z) log dA(u) = o(r) (c) sup |z| ζ ∈T
as r → 0.
ϕ(z)=u
D(ζ,r)
Then any two of these properties implies the third. The proof goes as for H2 , just considering the identity (see [18] or [17], for instance) p p 2 p f Hp = f (0) + 2
D
f (z)p−2 f (z)2 log 1 dA(z) |z|
as well as the test functions: ka (z) =
1 − |a|2 (1 − az)2
1/p (a ∈ D).
We refer to [4] where it is proved that boundedness and compactness of weighted composition operators on the Hardy spaces Hp can be tested on these functions. Remark 2.7. Note that if h has no zeros, then |h (z)|2 |h(z)|p−2 = |∇hp/2 |, and therefore condition (a) in Theorem 2.5 is hp/2 ∈ BMOA (resp. in Theorem 2.6 is hp/2 ∈ VMOA). 3. Weighted composition operators on weighted Bergman spaces The methods of Section 2 can be applied with minor modifications to the study of weighted p composition operators on a weighted Bergman space Aα , which is defined for 1 p < ∞ and α > −1 as the space of analytic functions in D such that
p
f Ap := α
f (z)p 1 − |z|2 α dA(z) < ∞.
D
Clearly, one may replace the factor (1 − |z|2 ) by log(1/|z|), obtaining an equivalent norm, and it is sometimes convenient to do so.
E.A. Gallardo-Gutiérrez, J.R. Partington / Journal of Functional Analysis 258 (2010) 3593–3603
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The Bloch space B is the space of analytic functions f defined on D and satisfying f B := sup 1 − |z|2 f (z) < ∞. z∈D
Finally the little Bloch space B0 is the subspace consisting of those functions in B such that 1 − |z|2 f (z) → 0
as |z| → 1.
We have two results which are the direct analogues of Theorems 2.1 and 2.2. Theorem 3.1. Let α > −1, let h ∈ A2α and ϕ ∈ H∞ be such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) h ∈ B; (b) Wh,ϕ is a bounded operator on A2α ; (c)
α+2
h(z)2 log 1 dA(u) = O r α+2 . sup |z| ζ ∈T D(ζ,r)
ϕ(z)=u
Then any two of these properties implies the third. Theorem 3.2. Let α > −1, let h ∈ A2α and ϕ ∈ H∞ be such that ϕ(D) ⊆ D. Consider the following three properties that h may possess: (a) h ∈ B0 ; (b) Wh,ϕ is a compact operator on A2α ; (c)
α+2
h(z)2 log 1 dA(u) = o r α+2 as r → 0. sup |z| ζ ∈T D(ζ,r)
ϕ(z)=u
Then any two of these properties implies the third. Proof. The proof now follows the lines of the proofs of Theorems 2.1 and 2.2, with the following minor modifications: • The Bergman norm can be equivalently expressed by f A2 2
α
2 ≈ f (0) +
D
α+2 2 f (z) log 1 dA(z), |z|
a formula given in [16]. • One replaces boundedness on H2 by boundedness on A2α , and uses the result of [8, Prop. 3.1] (the case α = 0 was given originally in [10]) that this is equivalent to the condition that
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Wh,ψ kα,w A2α remains uniformly bounded independently of w ∈ D, where kα,w denotes the normalized reproducing kernel of A2α , which is given by kα,w (z) =
(1 − |w|2 )(α+2)/2 . (1 − wz)α+2
• One replaces BMOA by the Bloch space; equivalently, the condition that 2 2 h (z) log 1 dA(z) |z| be a Carleson measure for the space A2 : this means (see [5, p. 61]) that μ(D(ζ, r)) < ∞. r2 ζ ∈T, r>0 sup
• The analogous modifications are made for Theorem 3.2, where we require the little Bloch space and the vanishing Carleson measures for A2 . 2 p
A final remark. Further results in the same fashion as Theorem 3.1 may be derived for Aα using the formula p p f Ap ≈ f (0) +
α
D
α+2 f (z)p−2 f (z)2 log 1 dA(z), |z|
which is given in [16] as well as the test functions (p) kα,w (z) =
(1 − |w|2 )(α+2)/2 (1 − wz)α+2
1/p (w ∈ D).
References [1] M. Andersson, Topics in Complex Analysis, Springer, New York, 1996. [2] M.D. Contreras, A.G. Hernández-Díaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (1) (2001) 224–233. [3] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995. ˘ ckovi´c, R. Zhao, Weighted composition operators between different weighted Bergman spaces and different [4] Z. Cu˘ Hardy spaces, Illinois J. Math. 51 (2) (2007) 479–498. [5] P. Duren, A. Schuster, Bergman Spaces, Math. Surveys Monogr., vol. 100, Amer. Math. Soc., Providence, RI, 2004. [6] F. Forelli, The isometries of H p , Canad. J. Math. 16 (1964) 721–728. [7] E.A. Gallardo-Gutiérrez, M.J. González, A. Nicolau, Composition operators on Hardy spaces on Lavrentiev domains, Trans. Amer. Math. Soc. 360 (1) (2008) 395–410. [8] E.A. Gallardo-Gutiérrez, R. Kumar, J.R. Partington, Boundedness, compactness and Schatten-class membership of weighted composition operators, 2009, submitted for publication. [9] J.B. Garnett, Bounded Analytic Functions, revised first ed., Springer-Verlag, New York, 2007. [10] Z. Harper, Applications of the discrete Weiss conjecture in operator theory, Integral Equations Operator Theory 54 (1) (2006) 69–88. [11] K. Hoffman, Banach Spaces of Analytic Functions, Dover Publications, 1988.
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[12] [13] [14] [15] [16]
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J.E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2) 23 (1925) 481–519. N.K. Nikol’skii, Treatise on the Shift Operator, Springer-Verlag, Berlin, 1986. J.H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (2) (1987) 375–404. J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (6) (1996) 2331–2348. [17] C.S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (2) (1986) 459–468. [18] S. Yamashita, Criteria for functions to be of Hardy class H p , Proc. Amer. Math. Soc. 75 (1) (1979) 69–72.
Journal of Functional Analysis 258 (2010) 3604–3661 www.elsevier.com/locate/jfa
The disintegration of the Lebesgue measure on the faces of a convex function L. Caravenna, S. Daneri ∗ SISSA, via Beirut 4, 34151 Trieste, Italy Received 18 May 2009; accepted 28 January 2010
Communicated by C. Villani
Abstract We consider the disintegration of the Lebesgue measure on the graph of a convex function f : Rn → R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure on the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green–Gauss formula for these directions holds on special sets. © 2010 Elsevier Inc. All rights reserved. Keywords: Disintegration of measures; Faces of a convex function; Conditional measures; Hausdorff dimension; Absolute continuity; Divergence formula
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Outline of the article . . . . . . . . . . . . . . . . . . . . . An abstract disintegration theorem . . . . . . . . . . . . . . . . Statement of the main theorem . . . . . . . . . . . . . . . . . . . 3.1. Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Absolute continuity of the conditional probabilities The explicit disintegration . . . . . . . . . . . . . . . . . . . . . .
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E-mail addresses:
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[email protected] (S. Daneri). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.024
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4.1. A disintegration technique . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Measurability of the directions of the k-dimensional faces . . . . 4.3. Partition into model sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. An absolute continuity estimate . . . . . . . . . . . . . . . . . . . . . . 4.5. Properties of the density function . . . . . . . . . . . . . . . . . . . . . 4.6. The disintegration on model sets . . . . . . . . . . . . . . . . . . . . . . 4.7. The global disintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. A divergence formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Vector fields parallel to the faces . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Study on model sets . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Global version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The currents of k-faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Recalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Divergence of the current of k-faces on model sets . . . 5.2.3. Divergence of the current of k-faces in the whole space 6. Table of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper we deal with the explicit disintegration of the Lebesgue measure on the graph of a convex function w.r.t. the partition given by its faces. As the graph of a convex function naturally supports the Lebesgue measure, its faces, being convex, have a well defined linear dimension and therefore they naturally support a proper dimensional Hausdorff measure. Our main result is that the conditional measures induced by the disintegration are equivalent to the Hausdorff measure on the faces on which they are concentrated.
Theorem 1. Let f : Rn → R be a convex function and let H n (graph f ) be the Hausdorff measure on its graph. Define a face of f as the convex set obtained by the intersection of its graph with a supporting hyperplane and consider the partition of the graph of f into the relative interiors of the faces {Fα }α∈A . Then, the Lebesgue measure on the graph of the convex function admits a unique disintegration
H n (graph f ) =
λα dm(α) A
w.r.t. this partition. The conditional measure λα is concentrated on the relative interior of the face Fα and it is equivalent to H k Fα , where k is the linear dimension of Fα .
This apparently intuitive fact does not always hold. Indeed, on one hand existence and uniqueness of a disintegration are obtained by classical theorems, thanks to measurability conditions satisfied by the faces. Nevertheless, even for a partition given by a Borel measurable collection of segments in R3 (1-dimensional convex sets), if any Lipschitz regularity of the directions of
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these segments is not known, it may happen that the conditional measures induced by the disintegration of the Lebesgue measure are Dirac deltas (see the couterexamples in [14,3]). Also in our case, up to our knowledge, the directions of the faces of a convex function are just Borel measurable. Therefore, our result, other than answering a quite natural question, enriches the regularity properties of the faces of a convex function, which have been intensively studied for example in [9,13,14,16,4,19]. As a byproduct, we recover the Lebesgue negligibility of the set of relative boundary points of the faces, which was first obtained in [15]. Our result is also interesting for possible applications. Indeed, the disintegration theorem is an effective tool in dimensional reduction arguments, where it may be essential to have an explicit expression for the conditional measures. In particular, the problem of the absolute continuity of the conditional measures w.r.t. a partition given by affine sets arose naturally in a work by Sudakov [21]. In absence of any Lipschitz regularity for the directions of the faces, the proof of our theorem does not rely on Area or Coarea formula, which in several situations allows to obtain in one step both the existence and the absolute continuity of the disintegration (in applications to optimal mass transport problem, see for example [22,11,5]). The basis of the technique we use was first presented in order to solve a variational problem in [7] and it has been successfully applied to the existence of optimal transport maps for strictly convex norms in [8]. Just to give an idea of how this technique works, focus on a collection of 1-dimensional faces C which are transversal to a fixed hyperplane H0 = {x ∈ Rn : x · e = 0} and such that the projection of each face on the line spanned by the fixed vector e contains the interval [h− , h+ ], with h− < 0 < h+ . Indeed, we will obtain the disintegration of the Lebesgue measure on the k-dimensional faces, with k > 1, from a reduction argument to this case. First, we slice C with the family of affine hyperplanes Ht = {x · e = t}, where t ∈ [h− , h+ ], which are parallel to H0 . In this way, by Fubini–Tonelli theorem, the Lebesgue measure L n of C can be recovered by integrating the (n − 1)-dimensional Hausdorff measures of the sections of C ∩ Ht over the segment [h− , h+ ] which parameterizes the parallel hyperplanes. Then, as the faces in C are transversal to H0 , one can see each point in C ∩ Ht as the image of a map σ t defined on C ∩ H0 which couples the points lying on the same face. Suppose that the (n − 1)-dimensional Hausdorff measure H n−1 (C ∩ Ht ) is absolutely continuous w.r.t. the pushforward measure σ#t (H n−1 (C ∩ H0 )) with Radon–Nikodym derivative α t . Then we can reduce each integral over the section C ∩ Ht to an integral over the section C ∩ H0 : dL n = H n−1 (C ∩ Ht ) dt = α t σ t (z) dH n−1 (z) dt.
[h− ,h+ ]
C
[h− ,h+ ] C ∩H0
Exchanging the order of the last iterated integrals, we obtain the following: n dL = α t σ t (z) dt dH n−1 (z). C −
+
C ∩H0 [h− ,h+ ]
Since the sets {σ [h ,h ] (z)}z∈C ∩H0 are exactly the elements of our partition, i.e. the 1dimensional faces of f , the last equality provides the explicit disintegration we are looking − + for: in particular, the conditional measure concentrated on σ [h ,h ] (z) is absolutely continuous − + w.r.t. H 1 σ [h ,h ] (z).
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The core of the proof is then to show that H n−1
(C ∩ Ht ) σ#t H n−1 (C ∩ H0) .
We prove this fact as a consequence of the following quantitative estimate: for all 0 t h+ and S ⊂ C ∩ H0 , H n−1 σ t (S)
t − h− −h−
n−1 H n−1 (S).
(1.1)
This fundamental estimate, as in [7,8], is proved approximating the 1-dimensional faces with a sequence of finitely many cones with vertex in C ∩ Hh− and basis in C ∩ Ht . At this step of the technique, the construction of such approximating sequence heavily depends on the nature of the partition one has to deal with. In this case, our main task is to find the suitable cones relying on the fact that we are approximating the faces of a convex function. One can also derive an estimate symmetric to the above one, showing that σ#t (H n−1 (C ∩ H0 )) is absolutely continuous w.r.t. H n−1 (C ∩ Ht ): as a consequence, α t is strictly positive and therefore the conditional measures are not only absolutely continuous w.r.t. the proper Hausdorff measure, but equivalent to it. The fundamental estimate (1.1) implies moreover a Lipschitz continuity and BV regularity of α t (z) w.r.t. t: this yields an improvement of the regularity of the partition that now we are going to describe. Consider a vector field v which at each point x is parallel to the face through that point x. If we restrict the vector field to an open Lipschitz set Ω which does not contain points in the relative boundaries of the faces, then we prove that its distributional divergence is the sum of two terms: an absolutely continuous measure, and an (n − 1)-rectifiable measure representing the flux of v through the boundary of Ω. The density (div v)a.c. of the absolutely continuous part is related to the density of the conditional measures defined by the disintegration above. In the case of the set C previously considered, if the vector field is such that v · e = 1, the expression of the density of the absolutely continuous part of the divergence satisfies
∂t α t = (div v)a.c. α t . Up to our knowledge, no piecewise BV regularity of the vector field v of the directions of the faces is known. Therefore, it is a remarkable fact that a divergence formula holds. The divergence of the whole vector field v is the limit, in the sense of distributions, of the sequence of measures which are the divergence of truncations of v on the elements {K }∈N of a suitable partition of Rn . However, in general, it fails to be a measure. In the last part, we change point of view: instead of looking at vector fields constrained to the faces of the convex function, we describe the faces as an (n + 1)-uple of currents, the k-th one corresponding to the family of k-dimensional faces, for k = 0, . . . , n. The regularity results obtained for the vector fields can be rewritten as regularity results for these currents. More precisely, we prove that they are locally flat chains. When truncated on a set Ω as above, they are locally normal, and we give an explicit formula for their border; the (n + 1)-uple of currents is the limit, in the flat norm, of the truncations on the elements of a partition.
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An application of this kind of further regularity is presented in Section 8 of [7]. Given a vector field v constrained to live on the faces of f , the divergence formula we obtain allows to reduce the transport equation div ρv = g to a PDE on the faces of the convex function. We do not pursue this issue in the paper. 1.1. Outline of the article In the following we describe the structure of the paper. In Section 2 we first give the definition of disintegration of a measure consistent with a fixed partition of the ambient space. Then, we report an abstract disintegration theorem which guarantees the existence and uniqueness of the disintegration under quite general assumptions on the σ -algebra of the ambient space and on the partition; under these hypotheses, the conditional probabilities given by the disintegration are concentrated on the sets of the partition. In Section 3, after giving the basic definitions and notations we will be working with, we apply the disintegration theorem recalled in Section 2 and get the existence and uniqueness of the disintegration of the Lebesgue measure on the faces of a convex function. For notational convenience, we work with the projections of the faces on Rn and we neglect the set where the convex function is not differentiable. In Section 3.2 we state our main theorem on the equivalence between the conditional probabilities and the k-Hausdorff measure on the k-dimensional faces where they are concentrated. As the conditional measures on the 0-dimensional and n-dimensional faces are already determined (they must be respectively given by Dirac deltas and by the H n -measure on the corresponding faces), we focus on the disintegration of the Lebesgue measure on the k-dimensional faces for k = 1, . . . , n − 1. In Section 4 we prove the explicit disintegration theorem. In Section 4.1 we explain the first idea of our disintegration technique, which consists in the reduction to countably many model sets like C and in the application to these sets of the Fubini– Tonelli technique, which has been briefly sketched in the introduction. In Section 4.2 we address the Borel measurability of the multivalued function D which assigns to each point x ∈ Rn the directions of the face passing through x. This is needed in the following in order to reduce the ambient space into countably many model sets. In Section 4.3 we define the partition of Rn into the model sets (called D-cylinders) on which we will first prove our disintegration theorem. When k = 1, the k-dimensional faces are partitioned into sets like C . When k > 1, each model set C k is defined taking a collection of k-dimensional faces which are transversal to a fixed (n − k)-dimensional affine plane (as, e.g., H = {x ∈ Rn : x · e1 = · · · = x · ek = 0}) and considering the points of these faces whose projection on the perpendicular k-plane (H ⊥ = {x ∈ Rn : x · ek+1 = · · · = x · en = 0}) is contained + in a fixed rectangle (as, e.g., {x ∈ Rn : x · ek+1 = · · · = x · en = 0, x · ei ∈ [h− i , hi ] for i = ± 1, . . . , k and hi ∈ R}). Section 4.4 is devoted to the proof of the quantitative estimate (1.1). Actually, in Lemma 4.7 we prove that an estimate like (1.1) holds for the pushforward of the H n−k -dimensional measure on the sections of a model set C k which are obtained cutting it with transversal (n − k)dimensional affine planes. The core of the proof, which is the construction of a suitable sequence of approximating cones for the 1-dimensional faces of f , is contained in Lemma 4.14.
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In Section 4.5 we study some regularity properties of the Radon–Nikodym derivative α t ; they will be used in Section 5 to study the regularity of the divergence of vector fields parallel to the faces. In Section 4.6 we prove the explicit disintegration theorem on the model sets C k of the partition. In Section 4.7 we collect the disintegrations obtained on the model set and we obtain the global result, namely the disintegration of the Lebesgue measure on all Rn . Section 5 deals with the divergence of the directions of the faces, with two equivalent approaches. In Section 5.1 we consider the divergence of any vector field which at each point x is parallel to the face of f through x. In Section 5.1.1 we truncate this vector field to k-dimensional D-cylinders. The divergence of these truncated vector fields turns out to be a Radon measure. The density of the absolutely continuous part of this distributional divergence involves the density α defined in (4.67). The precise statement is given in Corollary 5.4. In Section 5.1.2 we consider the vector field v in the whole Rn . Its divergence is the limit, in the sense of distributions, of finite sums of the Radon measures corresponding to the divergence of the truncations of v on a family of D-cylinders as above, which constitute a partition of Rn . In general, the divergence of v fails to be a measure. In Section 5.2 we consider the k-dimensional currents associated to k-faces, where each kface is thought as a k-covector field, for k = 0, . . . , n. We rephrase the results of Section 5.1 in this formalism. Section 5.2.1 is devoted to recalls on tensors and currents in order to fix the notation. In Section 5.2.2 we fix the attention on the current associated to a k-vector field that gives the directions of the k-faces on C k and vanishes elsewhere. The border of this current is the sum of two currents, both representable by integration. One is the integral on C k of the divergence of the k-vector field truncated to C k and it is again related to α. The other one is concentrated on an (n − 1)-dimensional set dC k (see definition (4.23)) and arises from the truncation of the faces to the D-cylinder. The statement is given in Lemma 5.9. In Section 5.2.3 we consider the (n + 1)-uple of currents associated to the faces of f , the k-th one acting on k-forms on Rn . By means of a partition of Rn into D-cylinders as above, we recover each of them as the limit, in the flat norm, of the normal currents defined as truncations of this (n + 1)-uple to the elements of the partition. The last section contains a long Table of Notations, for the reader’s convenience. 2. An abstract disintegration theorem A disintegration of a measure over a partition of the space on which it is defined is a way to write that measure as a “weighted sum” of probability measures which are possibly concentrated on the elements of the partition. Let (X, Σ, μ) be a measure space (which will be called the ambient space of the disintegration), i.e. Σ is a σ -algebra of subsets of X and μ is a measure with finite total variation on Σ, and let {Xα }α∈A ⊂ X be a partition of X. After defining the following equivalence relation on X x∼y
⇔
∃α ∈ A: x, y ∈ Xα ,
we make the identification A = X/∼ and we denote by p the quotient map p : x ∈ X → [x] ∈ A.
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Moreover, we endow the quotient space A with the measure space structure given by the largest σ -algebra that makes p measurable, i.e. A := F ⊂ A: p −1 (F ) ∈ Σ , and by the pushforward measure ν = p# μ, defined as ν(F ) := μ(p −1 (F )). Definition 2.1 (Disintegration). A disintegration of μ consistent with the partition {Xα }α∈A is a family {μα }α∈A of probability measures on X such that 1. ∀E ∈ Σ , α → μα (E) is ν-measurable; 2. μ = μα dν, i.e. μ E ∩ p −1 (F ) =
μα (E) dν(α),
∀E ∈ Σ, F ∈ A .
(2.1)
F
The disintegration is unique if the measures μα are uniquely determined for ν-a.e. α ∈ A. The disintegration is strongly consistent with p if μα (X\Xα ) = 0 for ν-a.e. α ∈ A. The measures μα are also called conditional probabilities of μ w.r.t. ν. Remark 2.2. When a disintegration exists, formula (2.1) can be extended by Beppo Levi’s theorem to measurable functions f : X → R as f dμ = f dμα dν(α). The existence and uniqueness of a disintegration can be obtained under very weak assumptions which concern only the ambient space. Nevertheless, in order to have the strong consistency we need structural assumptions also on the quotient measure algebra, otherwise in general μα (Xα ) = 1 (i.e. the disintegration is consistent but not strongly consistent). The more general result of existence of a disintegration which is consistent with a given partition is contained in [18], while a weak sufficient condition in order that a consistent disintegration is also strongly consistent is given in [12]. In the following we recall an abstract disintegration theorem, in the form presented in [6]. It guarantees, under suitable assumptions on the ambient and on the quotient measure spaces, the existence, uniqueness and strong consistency of a disintegration. Before stating it, we recall that a measure space (X, Σ) is countably-generated if Σ coincides with the σ -algebra generated by a sequence of measurable sets {Bn }n∈N ⊂ Σ . Theorem 2.3. Let (X, Σ) be a countably-generated measure space and let μ be a measure on X with finite total variation. Then, given a partition {Xα }α∈A of X, there exists a unique consistent disintegration {μα }α∈A . Moreover, if there exists an injective measurable map from (A, A ) to (R, B(R)), where B(R) is the Borel σ -algebra on R, the disintegration is strongly consistent with p. Remark 2.4. If the total variation of μ is not finite, a disintegration of μ consistent with a given partition as defined in (2.1) in general does not exists, even under the assumptions on the ambient
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and on the quotient space made in Theorem 2.3 (take for example X = Rn , Σ = B(Rn ), μ = L n and Xα = {x: x · e = α}, where e is a fixed vector in Rn and α ∈ R). Nevertheless, if μ is σ -finite and (X, Σ), (A, A ) satisfy the hypotheses of Theorem 2.3, as soon as we replace the possibly infinite-valued measure ν = p# μ with an equivalent σ -finite measure m on (A, A ), we can find a family of σ -finite measures {μ˜ α }α∈A on X such that μ = μ˜ α dm(α) (2.2) and μ˜ α (X\Xα ) = 0 for m-a.e. α ∈ A.
(2.3)
For example, we can take m = p# θ , where θ is a finite measure equivalent to μ. We recall that two measures μ1 and μ2 are equivalent if and only if μ1 μ2
and μ2 μ1 .
(2.4)
Moreover, if λ and {λ˜ α }α∈A satisfy (2.2) and (2.3) as well as m and {μ˜ α }α∈A , with m and λ equivalent to p# μ (and therefore to each other), then the following relation holds λ˜ α =
dm (α)μ˜ α , dλ
where dm dλ is the Radon–Nikodym derivative of m w.r.t. λ. In the following, whenever μ is a σ -finite measure with infinite total variation, by disintegration of μ strongly consistent with a given partition we will mean any family of σ -finite measures {μ˜ α }α∈A which satisfies the above properties; in fact, whenever μ has finite total variation we will keep the definition of disintegration given in (2.1). Finally, we recall that any disintegration of a σ -finite measure μ can be recovered by the disintegrations of the finite measures {μ Kn }n∈N , where {Kn }n∈N ⊂ X is a partition of X into sets of finite μ-measure.
3. Statement of the main theorem In this section, after setting the notation and some basic definitions, we apply Theorem 2.3 to get the existence, uniqueness and strong consistency of the disintegration of the Lebesgue measure on the faces of a convex function. Then, we give a rigorous formulation of the problem we are going to deal with and we state our main theorem. 3.1. Setting Let us consider the ambient space
Rn , B Rn , L n
K ,
where L n is the Lebesgue measure on Rn , B(Rn ) is the Borel σ -algebra, K is any set of finite Lebesgue measure and L n K is the restriction of the Lebesgue measure to the set K. Indeed,
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by Remark 2.4, the disintegration of the Lebesgue measure w.r.t. a given partition is determined by the disintegrations of the Lebesgue measure restricted to finite measure sets. Then, let f : Rn → R be a convex function. We recall that the subdifferential of f at a point x ∈ Rn is the set ∂ − f (x) of all r ∈ Rn such that f (w) − f (x) r · (w − x),
∀w ∈ Rn .
From the basic theory of convex functions, as f is real-valued and is defined on all Rn , ∂ − f (x) = ∅ for all x ∈ Rn and it consists of a single point if and only if f is differentiable at x. Moreover, in that case, ∂ − f (x) = {∇f (x)}, where ∇f (x) is the differential of f at the point x. We denote by dom ∇f a σ -compact set where f is differentiable and such that Rn \ dom ∇f is Lebesgue negligible. The σ -compactness is just a secondary technical requirement for simplifying measurability arguments. ∇f : dom ∇f → R denotes the differential map and Im ∇f the image of dom ∇f under the differential map. The partition of Rn on which we want to decompose the Lebesgue measure is given by the sets ∇f −1 (y) = x ∈ Rn : ∇f (x) = y ,
y ∈ Im ∇f,
along with the set Σ 1 (f ) = Rn \ dom ∇f . By the convexity of f , we can moreover assume w.l.o.g. that the intersection of ∇f −1 (y) with dom ∇f is convex. Since ∇f is a Borel map and Σ 1 (f ) is an L n -negligible Borel set (see e.g. [2,1]), we can assume that the quotient map p of Definition 2.1 is given by ∇f and that the quotient space is given by (Im ∇f, B(Im ∇f )), which is measurably included in (Rn , B(Rn )). Then, this partition satisfies the hypotheses of Theorem 2.3 and there exists a family {μy }y∈Im ∇f of probability measures on Rn such that L
n
K B ∩ ∇f
−1
(A) =
μy (B) d∇f# L n
K (y),
∀A, B ∈ B Rn .
A
In the following we give a formal definition of face of a convex function and we relate this object to the sets ∇f −1 (y) of our partition. Definition 3.1. A tangent hyperplane to the graph of a convex function f : Rn → R is a subset of Rn+1 of the form Hy = where x ∈ ∇f −1 (y).
z, hy (z) : z ∈ Rn , hy (z) = f (x) + y · (z − x) ,
(3.1)
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We note that, by convexity, the above definition is independent of x ∈ ∇f −1 (y). Definition 3.2. A face of a convex function f : Rn → R is a set of the form Hy ∩ graph f|dom ∇f .
(3.2)
It is easy to check that, ∀y ∈ Im ∇f and ∀z such that (z, f (z)) ∈ Hy ∩ graph f|dom ∇f , we have that y = ∇f (z). If we denote by πRn : Rn+1 → Rn the projection map on the first n coordinates, one can see that, for all y ∈ Im ∇f , ∇f −1 (y) = πRn (Hy ∩ graph f|dom ∇f ). For notational convenience, the set ∇f −1 (y) will be denoted as Fy and called projected face. We also write Fyk instead of Fy whenever we want to emphasize the fact that the latter has dimension k, for k = 0, . . . , n (where the dimension of a convex set C is the dimension of its affine hull aff(C)) and we denote the set of k-dimensional projected faces as
Fk =
(3.3)
Fy .
{y: dim(Fy )=k}
3.2. Absolute continuity of the conditional probabilities Since the measure we are disintegrating (L n ) has the same Hausdorff dimension as the space on which it is concentrated (Rn ) and since the sets of the partition on which the conditional probabilities are concentrated have a well defined linear dimension, we address the problem of whether this absolute continuity property of the initial measure is still satisfied by the conditional probabilities produced by the disintegration: we want to see if dim(Fy ) = k
⇒
μy H k
Fy .
(3.4)
The answer to this question is not trivial. Indeed, when n 3 one can construct sets of full Lebesgue measure in Rn and Borel partitions of those sets into convex sets such that the conditional probabilities of the corresponding disintegration do not satisfy property (3.4) for k = 1 (see e.g. [4]). However, we show that the absolute continuity property is preserved by the disintegration w.r.t. the partition given by the faces of a convex function. Our main result is the following: Theorem 3.3. Let {μy }y∈Im ∇f be the family of probability measures on Rn such that L
n
K B ∩ ∇f
−1
(A) =
μy (B) d∇f# L n
K (y),
∀A, B ∈ B Rn .
(3.5)
A
Then, for ∇f# (L n K)-a.e. y ∈ Im ∇f , the conditional probability μy is equivalent to the k-dimensional Hausdorff measure H k restricted to Fyk ∩ K, i.e.
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μy H k
k Fy ∩ K
and H k
k Fy ∩ K μ y .
(3.6)
Remark 3.4. The result for k = 0, n is trivial. Indeed, for all y such that Fy ∩ K = ∅ and dim(Fy ) = 0 we must put μy = δ{Fy } , where δx0 is the Dirac mass supported in x0 , whereas L nF if dim(Fy ∩ K) = n we have that μy = |L nFyy | . Remark 3.5. Since the map id × f : Rn → Rn+1 , x → x, f (x) is locally Lipschitz and preserves the Hausdorff dimension of sets, Theorem 3.3 holds also for the disintegration of the n-dimensional Lebesgue measure over the partition of the graph of f given by the faces defined in (3.2). We have chosen to deal with the disintegration of the Lebesgue measure over the projections of the faces on Rn only for notational convenience. Theorem 3.3 will be proved in Section 4.7, where we provide also an explicit expression for the conditional probabilities. If we knew some Lipschitz regularity for the field of directions of the faces of a convex function, we could try to apply the Area or Coarea formula in order to obtain within a single step the disintegration of the Lebesgue measure and the absolute continuity property (3.6). However, such regularity is presently not known and for this reason we have to follow a different approach. 4. The explicit disintegration 4.1. A disintegration technique In this paragraph we give an outline of the technique we use in order to prove Theorem 3.3. This kind of strategy was first used in order to disintegrate the Lebesgue measure on a collection of disjoint segments in [7], and then in [8]. For simplicity, we focus on the disintegration of the Lebesgue measure on the 1-dimensional faces and, in the end, we give an idea of how we will extend this technique in order to prove the absolute continuity of the conditional probabilities on the faces of higher dimension. The disintegration on model sets: Fubini–Tonelli theorem and absolute continuity estimates on affine planes which are transversal to the faces. First of all, let us suppose that the projected 1-dimensional faces of f are given by a collection of disjoint segments C whose projection on a fixed direction e ∈ Sn−1 is equal to a segment [h− e, h+ e] with h− < 0 < h+ , more precisely C=
a(z), b(z) ,
(4.1)
z∈Zt
where Zt is a compact subset of an affine hyperplane of the form {x · e = t} for some t ∈ R and a(z) · e = h− , b(z) · e = h+ . Any set of the form (4.1) will be called a model set (see also Fig. 1).
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Fig. 1. A model set of 1-dimensional projected faces. Given a subset Z = Z0 of the hyperplane {x · e = 0}, the above model set is made of the 1-dimensional faces of f passing through some z ∈ Z0 , truncated between {x · e = h− }, {x · e = h+ } and projected on Rn .
Then, we want to find the conditional probabilities of the disintegration of the Lebesgue measure on the segments which are contained in the model set C and see if they are absolutely continuous w.r.t. the H 1 -measure. The idea of the proof is to obtain the required disintegration by a Fubini–Tonelli argument, that reverts the problem of absolute continuity w.r.t. H 1 of the conditional probabilities on the projected 1-dimensional faces to the absolute continuity w.r.t. H n−1 of the pushforward by the flow induced by the directions of the faces of the H n−1 -measure on transversal hyperplanes. First of all, we cut the set C with the affine hyperplanes which are perpendicular to the segment [h− e, h+ e], we apply Fubini–Tonelli theorem and we get h+
ϕ(x) dL (x) = n
C
h−
ϕ dH n−1 dt
∀ϕ ∈ Cc0 Rn .
(4.2)
{x·e=t}∩C
Then we observe the following: for every s, t ∈ [h− , h+ ], the points of {x · e = t} ∩ C are in bijective correspondence with the points of the section {x · e = s} ∩ C and a bijection is obtained by pairing the points that belong to the same segment [a(z), b(z)], for some z ∈ Zt . For example, a map which sends the transversal section Z = {x · e = 0} ∩ C into the section Zt = {x · e = t} ∩ C (for any t ∈ [h− , h+ ]) is given by σ t : Z → σ t (Z) = {x · e = t} ∩ C , z → z + t
ve (z) = {x · e = t} ∩ a(z), b(z) , |ve (z) · e|
b(z)−a(z) where [a(z), b(z)] is the segment of C passing through the point z and ve (z) = |b(z)−a(z)| . Therefore, as soon as we fix a transversal section of C , say e.g. Z = {x · e = 0} ∩ C , we can try to rewrite the inner integral in the r.h.s. of (4.2) as an integral of the function ϕ ◦ σ t w.r.t. the H n−1 -measure on the fixed section Z. This can be done if
t −1 n−1 σ # H
σ t (Z)
H n−1
Z.
(4.3)
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Indeed,
ϕ(y) dH n−1 (y) = σ t (Z)
−1 ϕ σ t (z) d σ t # H n−1
σ t (Z) (z)
(4.4)
Z
and if (4.3) is satisfied for all t ∈ [h− , h+ ], then h+ (4.2) =
ϕ σ t (z) α(t, z) dH n−1 (z) dt,
h− Z
n−1 where α(t, z) is the Radon–Nikodym derivative of (σ t )−1 σ t (Z)) w.r.t. H n−1 Z. # (H Having turned the r.h.s. of (4.2) into an iterated integral over a product space isomorphic to Z + [h− e, h+ e], the final step consists in applying Fubini–Tonelli theorem again so as to exchange the order of the integrals and get
h+ ϕ(x) dL n (x) = ϕ σ t (z) α(t, z) dt dH n−1 (z).
C
Z
(4.5)
h−
This final step can be done if α is Borel-measurable and locally integrable in (t, z). By the uniqueness of the disintegration stated in Theorem 2.3 we have that dμz (t) =
α(t, z) dH 1 [a(z), b(z)](t) h+ h− α(s, z) ds
for H n−1 -a.e. z ∈ Z.
(4.6)
The same reasoning can be applied to the case k > 1. Indeed, let us consider a collection C k of faces whose projection on a certain k-plane e1 , . . . , ek is given by a rectangle kk-dimensional − + + [h e , h e ], with h− i i i=1 i i i < 0 < hi for all i = 1, . . . , k (see Fig. 2, p. 3617). Then, as soon as we fix an affine (n − k)-dimensional plane which is perpendicular to the k-plane e1 , . . . , ek , as for example H k = ki=1 {x · ei = 0}, and we denote by πe1 ,...,ek : Rn → e1 , . . . , ek the projection map on the k-plane e1 , . . . , ek , the k-dimensional faces in C k can be parameterized with the map σ te (z) = z + t
ve (z) , |πe1 ,...,ek (ve (z))|
(4.7)
where z ∈ Z k = H k ∩ C k , e is a unit vector in the k-plane e1 , . . . , ek , t ∈ R satisfies te · ei ∈ + [h− i , hi ] for all i = 1, . . . , k and ve (z) is the unit direction contained in the face passing through πe1 ,...,ek (ve (z)) |πe1 ,...,ek (ve (z))| = e. C k with affine hyperplanes which are perpendicular to
z which is such that
If we cut the set ei for i = 1, . . . , k and apply k-times the Fubini–Tonelli theorem, the main point is again to show that, for every e and t as above, te −1 n−k σ # H
σ te Zk
H n−k
Zk
(4.8)
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Fig. 2. Sheaf sets and D-cylinders (Definitions 4.3, 4.5). Roughly, a sheaf set Z k is a collection of relative interiors of kplane. faces of f , projected on Rn , which intersect exactly at one point some set Z k contained in an (n − k)-dimensional k −1 − + k k A D-cylinder C k is the intersection of a sheaf set with πe ,...,e (C ), for some rectangle C = i=1 [ti ei , ti ei ], 1
k
where {e1 , . . . , en } constitutes an orthonormal basis of Rn . Such sets Z k are called sections, while the k-plane e1 , . . . , ek is the axis of C k .
and, after this, that the Radon–Nikodym derivative between the above measures satisfies proper measurability and integrability conditions. Then, to prove Theorem 3.3 on model sets that are, up to translations and rotations, like the set C k , it is sufficient to prove (4.8) and some weak properties of the related density function, such as Borel-measurability and local integrability. Actually, the properties of this function will follow immediately from our proof of (4.8), which is given in a stronger form in Lemma 4.31. Partition of Rn into model sets and the global disintegration theorem. In the next section we show that the set F k defined in (3.3), for k = 1, . . . , n − 1, can be partitioned, up to a negligible set, into a countable collection of Borel-measurable model sets like C k . After proving the disintegration theorem on the model sets we will see how to glue the “local” results in order to obtain a global disintegration theorem for the Lebesgue measure over the whole faces of the convex function (restricted to a set of finite L n -measure). 4.2. Measurability of the directions of the k-dimensional faces The aim of this subsection is to show that the set of the projected k-dimensional faces of a convex function f can be parameterized by an L n -measurable (and multivalued) map. This will allow us to decompose Rn into a countable family of Borel model sets on which to prove Theorem 3.3. First of all we give the following definition, which generalizes Definition 3.1. Definition 4.1. A supporting hyperplane to the graph of a convex function f : Rn → R is an affine hyperplane in Rn+1 of the form H = w ∈ Rn+1 : w · b = β ,
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where b = 0, w · b β for all w ∈ epi f = {(x, t) ∈ Rn × R: t f (x)} and w · b = β for at least one w ∈ epi f . As f is defined and real-valued on all Rn , every supporting hyperplane is of the form Hy =
z, hy (z) : z ∈ Rn , hy (z) = f (x) + y · (z − x) ,
(4.9)
for some y ∈ ∂ − f (x). Whenever y ∈ Im ∇f , Hy is a tangent hyperplane to the graph of f according to Definition 3.1. Then we define the map x → P(x) = z ∈ Rn : ∃y ∈ ∂ − f (x) such that f (z) − f (x) = y · (z − x) . By definition, P(x) = Moreover, the map
y∈∂ − f (x) πRn (Hy
(4.10)
∩ graph f ).
dom ∇f x → R(x) := P(x) ∩ dom ∇f, gives precisely the set Fy of our partition that passes through the point x. As the disintegration over the 0-dimensional faces is trivial, we will restrict our attention to the set T = x ∈ dom ∇f : R(x) = {x} . For all such points there is at least one segment [w, z] ⊂ R(x) such that w = z. We can also define the multivalued map giving the unit directions contained in the faces passing through the set T, that is
T x → D(x) =
z−x : z ∈ R(x), z = x . |z − x|
(4.11)
We recall that a multivalued map is defined to be Borel measurable if the inverse image of any open set is Borel. The measurability of the above maps is proved in the following lemma: Lemma 4.2. The graph of the multivalued function P is a closed set in Rn × Rn . As a consequence, P, R and D are Borel measurable multivalued maps and T is a Borel set. Proof. The closedness of the graph of P follows immediately from the continuity of f and from the upper-semicontinuity of its subdifferential. Then, the graph of P is σ -compact in Rn × Rn and, due to the continuity of the projections from Rn × Rn to Rn , it follows that the map P is Borel. Moreover, since we chose dom ∇f to be σ -compact, also the graph of R is σ -compact, thus R is a Borel map. The same reasoning that is made for the map P can be applied to the multifunction R\id (where id denotes the identity map id(x) := x), thus giving the measurability of the set T, since
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T = π graph(R\id) ∩ dom ∇f, where π : Rn × Rn → Rn denotes the projection on the first n coordinates. The measurability of D follows by the continuity of the map Rn × Rn (x, z) → the diagonal. 2
z−x |z−x|
out of
4.3. Partition into model sets First of all, we introduce some preliminary notation. If C ⊂ Rd is a convex set and aff(C) is its affine hull, we denote by ri(C) the relative interior of C, which is the interior of C in the topology of aff(C), and by rb(C) its relative boundary, which is the boundary of C in aff(C). In order to find a countable partition of F k into model sets like the set C k which was defined in Section 4.1, we have to neglect the points that lie on the relative boundary of the k-dimensional faces. More precisely, from now onwards we look for the disintegration of the Lebesgue measure over the sets Ey = ri(Fy ),
y ∈ Im ∇f.
(4.12)
As we did for the sets Fy , we set Eyk = Ey
if dim(Ey ) = k
and
Ek =
Eyk .
(4.13)
{y∈Im ∇f : dim(Ey )=k}
This restriction will not affect the characterization of the conditional probabilities because, as we will prove in Lemma 4.19, the set
T\
n
Ek
k=1
is Lebesgue negligible. Now we can start to build the partition of E k into model sets. Definition 4.3. For all k = 1, . . . , n, we call sheaf set a σ -compact subset of E k of the form Zk=
z∈Z k
ri R(z) ,
(4.14)
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where Z k is a σ -compact subset of E k which is contained in an affine (n − k)-plane in Rn and is such that ri R(z) ∩ Z k = {z},
∀z ∈ Z k .
(4.15)
We call sections of Z k all the sets Y k that satisfy the same properties of Z k in the definition. A subsheaf of a sheaf set Z k is a sheaf set W k of the form Wk=
ri R(w) ,
w∈W k
where W k is a σ -compact subset of a section of the sheaf set Z k . Similarly to Lemma 2.6 in [8], we prove that the set E k can be covered with countably many disjoint sets of the form (4.14). First of all, let us take a dense sequence {Vi }i∈N ⊂ G(k, n), where G(k, n) is the compact set of all the k-planes in Rn passing through the origin, and fix, ∀i ∈ N, an orthonormal set {ei1 , . . . , eik } in Rn such that Vi = ei1 , . . . , eik .
(4.16)
Denote by Sn−1 ∩ V the k-dimensional unit sphere of a k-plane V ⊂ Rn w.r.t. the Euclidean norm and by πi = πVi : Rn → Vi the projection map on the k-plane Vi . For every fixed 0 < ε < 1 the following sets form a disjoint covering of the k-dimensional unit spheres in Rn : Sk−1 = Sn−1 ∩ V : V ∈ G(k, n), i
inf
x∈Sn−1 ∩V
i−1
πi (x) 1 − ε \ Sk−1 , j
j =1
i = 1, . . . , I,
(4.17)
where I ∈ N depends on the ε we have chosen. In order to determine a countable partition of E k into sheaf sets we consider the k-dimensional rectangles in the k-planes (4.16) whose boundary points have dyadic coordinates. For all l = (l1 , . . . , lk ),
m = (m1 , . . . , mk ) ∈ Zk
with lj < mj ∀j = 1, . . . , k
(4.18)
k and for all distinct i1 , . . . , ik ⊂ {1, . . . , I }, p ∈ N, let Ciplm be the rectangle
k = 2−p Ciplm
k
[lj eij , mj eij ].
(4.19)
j =1
Lemma 4.4. The following sets are sheaf sets covering E k : for i = 1, . . . , I, p ∈ N, and S ⊂ Zk take
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k ZipS = x ∈ E k : D(x) ⊂ Sk−1 and S ⊂ Zk is the maximal set such that i
k Cipl(l+1)
⊂ πi ri R(x) .
(4.20)
l∈S
Moreover, a disjoint family of sheaf sets that cover E k is obtained in the following way: in case k as above, whereas for all p > 1 we take a set Z k if and only p = 1 we consider all the sets ZipS ipS k k does not contain any rectangle of the form Cip if the set l∈S Cipl(l+1) l(l+1) for every p < p. As soon as a nonempty sheaf set Z k belongs to this partition, it will be denoted by Z¯ k . ipS
ipS
For the proof of this lemma we refer to the analogous Lemma 2.6 in [8]. Then, we can refine the partition into sheaf sets by cutting them with sections which are perpendicular to fixed k-planes. Definition 4.5. (See Fig. 2.) A k-dimensional D-cylinder is a σ -compact set of the form k −1 C , C k = Z k ∩ πe 1 ,...,ek
(4.21)
where Z k is a k-dimensional sheaf set, e1 , . . . , ek is any fixed k-dimensional subspace which is perpendicular to a section of Z k and C k is a rectangle in e1 , . . . , ek of the form
Ck =
k − ti ei , ti+ ei , i=1
with −∞ < ti− < ti+ < +∞ for all i = 1, . . . , k, such that C k ⊂ πe1 ,...,ek ri R(z)
k −1 ∀z ∈ Z k ∩ πe C . 1 ,...,ek
(4.22)
We set C k = C k (Z k , C k ) when we want to refer explicitly to a sheaf set Z k and to a rectangle C k that can be taken in the definition of C k . The k-plane e1 , . . . , ek is called the axis of the D-cylinder and every set Z k of the form −1 (w), C k ∩ πe 1 ,...,ek
for some w ∈ ri C k
is called a section of the D-cylinder. We also define the border of C k transversal to D and its outer unit normal as k −1 rb C , dC k = C k ∩ πe ,...,e k 1
k −1 nˆ |dC k (x) = outer unit normal to πe C at x, for H n−1 -a.e. x ∈ dC k . 1 ,...,ek
(4.23)
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Lemma 4.6. The set E k can be covered by the D-cylinders k k C k ZipS , , Cipl(l+1)
(4.24)
k , Ck where S ⊂ Zk , l ∈ S and ZipS ipl(l+1) are the sets defined in (4.19), (4.20). Moreover, there exists a countable covering of E k with D-cylinders of the form (4.24) such that
k k k k k πi C k ZipS ∩ C k Zipk S , Cip ⊂ rb Cipl(l+1) ∩ rb Cip , Cipl(l+1) l (l +1) l (l +1)
(4.25)
for any couple of D-cylinders which belong to this countable family (if i = i , it follows from the k , Ck k k k definition of sheaf set that C k (ZipS ipl(l+1) ) ∩ C (Zi p S , Ci p l (l +1) ) must be empty). Proof. If x ∈ E k , then x ∈ Eyk for y = ∇f (x). By definition Eyk is the relative interior of a convex k-dimensional set. Moreover, by construction, its projection on some k-plane Vi is an open, k . In particuconvex set and therefore it can be covered by rectangles of dyadic coordinates Ciplm k k k k lar, the projection of x on Vi belongs to some Ciplm and this means that x ∈ C (ZipS , Cip(+1) ) k for some . This proves that E can be covered by the D-cylinders defined in (4.24). Our aim is then to construct a countable covering of E k with D-cylinders which satisfy property (4.25). k which belongs to the countable partition of E k First of all, let us fix a nonempty sheaf set Z¯ipS given in Lemma 4.4. In the following we will determine the D-cylinders of the countable covering which are conk ; the others can be selected in the same way starting from a different sheaf set of tained in Z¯ipS the partition given in Lemma 4.4. Then, the D-cylinders that we are going to choose are of the form C k Z k ˆ , Ck i pˆ S
i pˆ ˆl(ˆl+1)
,
k . where Z k ˆ is a subsheaf of the sheaf set Z¯ipS i pˆ S
The construction is done by induction on the natural number pˆ which determines the diamk on the axis obtained projecting the D-cylinders contained in Z¯ipS eter of the squares C k ˆ ˆ i pˆ l(l+1) ei1 , . . . , eik . Then, as the induction step increases, the diameter of the k-dimensional rectangles associated to the D-cylinders that we are going to add to our countable partition will be smaller and smaller (see Fig. 3). k is a nonempty element of the partition defined By definition (4.20) and by the fact that Z¯ipS in Lemma 4.4, the smallest natural number pˆ such that there exists a k-dimensional rectangle of k ) is exactly p; then, w.l.o.g., we can assume in which is contained in πi (Z¯ipS the form C k ˆ ˆ i pˆ l(l+1) our induction argument that p = 1. For all pˆ ∈ N, we call Cylpˆ the collection of the D-cylinders which have been chosen up to step p. ˆ When pˆ = 1 we set k k Cyl1 = C k Z¯i1S : l∈S . , Ci1l(l+1) Now, let us suppose to have determined the collection of D-cylinders Cylpˆ for some pˆ ∈ N.
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Fig. 3. Partition of E k into D-cylinders (Lemma 4.6).
Then, we define k k : Zi(kp+1) Cylp+1 = Cylpˆ ∪ C k = C k Zi(kp+1) , C is a subsheaf of Z¯ipS and ˆ ˜ ˜ ˜ ˆ S ˆ S˜ i(p+1) ˆ l(l+1) k k k C k C k Zipk S , Cip Zip S , Cik p l (l +1) ∈ Cylpˆ . l (l +1) for all C
2 (4.26)
As we did in (4.7), any k-dimensional D-cylinder C k = C k (Z k , C k ) can be parameterized in the following way: if we fix w ∈ ri(C k ), then −1 C k = σ w+te (z): z ∈ Z k = πe (w) ∩ C k , e ∈ Sn−1 ∩ e1 , . . . , ek 1 ,...,ek and t ∈ R is such that (w + te) · ej ∈ tj− , tj+ ∀j = 1, . . . , k ,
(4.27)
where σ w+te (z) = z + t
ve (z) , |πe1 ,...,ek (ve (z))|
πe
(4.28)
,...,e (ve (z))
and ve (z) ∈ D(z) is the unit vector such that |πe1 ,...,ek (ve (z))| = e. k 1 We observe that, according to our notation, w+te −1 σ = σ (w+te)−te .
(4.29)
4.4. An absolute continuity estimate According to the strategy outlined in Section 4.1, in order to prove Theorem 3.3 for the disintegration of the Lebesgue measure on the D-cylinders we have to show that, for every D-cylinder
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C k parameterized as in (4.27) w+te −1 n−k H σ #
σ w+te Zk
H n−k
Zk .
(4.30)
This will allow us to make a change of variables from the measure space (σ w+te (Z k ), H n−k (σ w+te (Z k ))) to (Z k , αH n−k Z k ), where α is an integrable function w.r.t. H n−k Z k (see Section 4.1). It is clear that the domain of the parameter t, which can be interpreted as a time parameter for a flow σ w+te that moves points along the k-dimensional projected faces of a convex function, depends on the section Z k which has been chosen for the parameterization of C k and on the direction e. Then, if e1 , . . . , ek is the axis of a D-cylinder C k , for every w ∈ ri(C k ) and for every e ∈ n−1 S ∩ e1 , . . . , ek , we define the numbers
h− (w, e) = inf t ∈ R: w + te ∈ C k ,
h+ (w, e) = sup t ∈ R: w + te ∈ C k .
We observe that, since w ∈ ri(C k ), h− (w, e) < 0 < h+ (w, e). We obtain (4.30) in Corollary 4.15 as a consequence of the following fundamental lemma. Lemma 4.7 (Absolutely continuous pushforward). Let C k be a k-dimensional D-cylinder parameterized as in (4.27). Then, for all S ⊂ Z k the following estimate holds:
h+ (w, e) − t h+ (w, e) − s
n−k
H n−k σ w+se (S) H n−k σ w+te (S)
t − h− (w, e) s − h− (w, e)
n−k
H n−k σ w+se (S) ,
(4.31)
where h− (w, e) < s t < h+ (w, e). Moreover, if s = h− (w, e) the left inequality in (4.31) still holds and if t = h+ (w, e) the right one. Lemma 4.7 will be proven at p. 3634. The idea to prove this lemma, as in [7] and [8], is to get the estimate (4.31) for the flow σjw+te induced by simpler vector fields {vj }j ∈N and then to show that they approximate the initial vector field ve in such a way that the inequalities in (4.31) pass to the limit. The main problem in our proof is then to find a suitable sequence of vector fields {vj }j ∈N that approximate, in a certain region, the geometry of the projected k-dimensional faces of a convex function in the direction e, which is described by the vector field ve . For the construction of this family of vector fields we strongly rely on the fact that the sets on which we want to disintegrate the Lebesgue measure are, other than disjoint, the projections of the k-dimensional faces of a convex function. For simplicity, we first prove the estimate (4.31) for 1-dimensional D-cylinders. In this case, if e is the axis of a 1-dimensional D-cylinder C , there are only two possible directions ±e that can be chosen to parameterize it. Up to translations by a multiple of the same vector, we can assume that w = 0 and s = 0. Moreover, since the choice of −e instead of e in
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the definition of the parameterization map (4.28) simply reverses the order of s and t in (4.31), in order to prove (4.31) it is sufficient to show that, for all 0 t h+ and for all S ⊂ σ t (Z), H n−1 (S)
t − h− −h−
n−1
−1 H n−1 σ t (S) ,
(4.32)
where σ t = σ 0+te and h± = h± (0, e). In our construction we first approximate the 1-dimensional faces that lie on the graph of f restricted to the given D-cylinder and then we get the approximating vector fields {vj }j ∈N simply projecting the directions of those approximations on the first n coordinates. Before giving the details we recall and introduce some useful notation: Sn−1 = x ∈ Rn : x = 1 ; e ∈ Sn−1 a fixed vector; Hs := x ∈ Rn : x · e = s , where s ∈ h− , h+ and h− , h+ ∈ R: h− < 0 < h+ ; Bn−1 R (x) = z ∈ H{x·e} : z − x R ; Z is the σ -compact section of the 1-dimensional D-cylinder C which is contained in H0 ; ve (x) = D(x), πe ve (x) = ve (x) · e e; C = σ t (z): z ∈ Z, t ∈ h− , h+ , Ct =
σ t (z) = z + t
ve (z) ; |πe (ve (z))|
Hs ∩ C ;
s∈[h− ,t]
lt (x) = R(x) ∩ Ct , ∀x ∈ Ct ; ∀x ∈ Rn , x˜ := x, f (x) ∈ Rn+1
and ∀A ⊂ Rn ,
A˜ := graph f|A .
Moreover, we recall the following definitions: Definition 4.8. The convex envelope of a set of points X ⊂ Rn is the smallest convex set conv(X) that contains X. The following characterization holds: conv(X) =
J j =1
λj xj : xj ∈ X, 0 λj 1,
J
λj = 1, J ∈ N .
(4.33)
j =1
Definition 4.9. The graph of a compact convex set C ⊂ Rn+1 , that we denote by graph(C), is the graph of the function g : πRn (C) → R which is defined by g(x) = min t ∈ R: (x, t) ∈ C .
(4.34)
Definition 4.10. A supporting k-plane to the graph of a convex function f : Rn → R is an affine k-dimensional subspace of a supporting hyperplane to the graph of f (see Definition 4.1) whose intersection with graph f is nonempty.
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Definition 4.11. An R-face of a convex set C ⊂ Rd is a convex subset C of C such that every closed segment in C with a relative interior point in C has both endpoints in C . The 0-dimensional R-faces of a convex set are also called extreme points and the set of all extreme points in a convex set C will be denoted by ext(C). The definition of R-face corresponds to the definition of face of a convex set given in [20]. We also recall the following propositions, for which we refer to Section 18 of [20]. Proposition 4.12. Let C = conv(D), where D is a set of points in Rd , and let C be a nonempty R-face of C. Then C = conv(D ), where D consists of the points in D which belong to C . Proposition 4.13. Let C be a bounded closed convex set. Then C = conv(ext(C)). The key to get the fundamental estimate (4.32) is contained in the following lemma: Lemma 4.14 (Construction of regular approximating vector fields). For all 0 t h+ , there exists a sequence of H n−1 -measurable vector fields t vj j ∈N ,
vjt : σ t (Z) → Sn−1
such that 1. 2.
vjt converges H n−1 -a.e. to ve on σ t (Z); t − h− n−1 n−1 t −1 n−1 σv t (S) H (S) , H j −h−
(4.35) ∀S ⊂ σ t (Z),
where σvt t is the flow map associated to the vector field vjt .
(4.36)
j
Indeed, if we have such a sequence of vector fields, the proof of the estimate (4.32) follows as in [8]. Proof. Step 1. Preliminary considerations. First of all, let us fix t ∈ [0, h+ ]. Partitioning, if necessary, C into a countable collection of sets, we can assume that σ t (Z) and − n−1 h h− σ (Z) are bounded, with σ t (Z) ⊂ Bn−1 R1 (x1 ) ⊂ Ht and σ (Z) ⊂ BR2 (x2 ) ⊂ Hh− . Then, if n−1 we call Kt the convex envelope of Bn−1 R1 (x1 ) ∪ BR2 (x2 ), the function f|Kt is uniformly Lipschitz with a certain Lipschitz constant Lf . Step 2. Construction of approximating functions (see Fig. 4). Now we define a sequence of functions {fj }j ∈N whose 1-dimensional faces approximate, in a certain sense, the pieces of the 1-dimensional faces of f which are contained in Ct . The directions of a properly chosen subcollection of the 1-dimensional faces of fj will give, when projected on the first n coordinates, the approximate vector field vjt . h− ˜ First of all, take a sequence {y˜i }i∈N ⊂ σ˜ (Z) such that the collection of segments {lt (yi )}i∈N ˜ is dense in y∈σ h− (Z) lt (y).
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Fig. 4. Illustration of a vector field approximating the 1-dimensional faces of f (Lemma 4.14). One can see in the picture the graph of f4 , which is the convex envelope of {y˜i }i=1,...,4 and f |H . The faces of fj connect H n−1 -a.e. point of t Ht to a single point among the {y˜i }i=1,...,j , while the remaining points of Ht correspond to some convex envelope conv({y˜i } )—here represented by the segments [y˜i , y˜i+1 ]. The region where the map Dt4 , giving the directions of the faces of f4 , is multivalued is basically the projection of the k-faces of f4 with k > 1. If x belongs to this region, i.e. Dt4 (x) consists of the directions of a ‘planar’ face of f4 , intersecting the affine hull of this ‘planar’ face with Ht × R one finds a supporting hyperplane for f|H passing through the point (x, f (x)). These supporting hyperplanes are represented by t
the tangent lines in the right side of the picture. The intersection of σ t (Z) ⊂ Ht with any supporting plane to the graph of f |H must contain just one point, otherwise D would be multivalued at some point of σ t (Z). t
For all j ∈ N, let Cj be the convex envelope of the set j
{y˜i }i=1 ∪ graph f|
(4.37)
Bn−1 R1 (x1 )
and call fj : πRn (Cj ) → R the function whose graph is the graph of the convex set Cj . j j = graph(conv({y˜i }i=1 )). We note that πRn (Cj )∩Hh− = conv({yi }i=1 ) and graph fj | j conv({yi }i=1 )
We claim that the graph of fj is made of segments that connect the points of j (indeed, by convexity and by the fact that graph(conv({y˜i }i=1 )) to the graph of f| n−1 BR
1
(x1 )
y˜i = (yi , f (yi )), fj = f on Bn−1 R1 (x1 )). In order to prove this, we first observe that, by definition, all segments of this kind are contained in the set Cj . On the other hand, by (4.33), all the points in Cj are of the form w=
J i=1
λi wi ,
(4.38)
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where
J
i=1 λi
j
= 1, 0 λi 1 and wi ∈ {y˜i }i=1 ∪ graph f|
Bn−1 R1 (x1 )
w = αz + (1 − α)r,
. In particular, we can write
j where 0 α 1, z ∈ conv {y˜i }i=1 and r ∈ epi f|
Bn−1 R1 (x1 )
Moreover, if we take two points z ∈ graph(conv({y˜i }i=1 )), r ∈ graph f| j
πRn (z ) = πRn (z)
and
πRn (r ) = πRn (r),
(4.39)
.
Bn−1 R1 (x1 )
such that
we have that the point
w = αz + (1 − α)r
(4.40) j
belongs to Cj , lies on a segment which connects graph(conv({y˜i }i=1 )) to graph f|
Bn−1 R1 (x1 )
and its
(n + 1)-coordinate is less than the (n + 1)-coordinate of w. j The graph of fj contains also all the pieces of 1-dimensional faces {l˜t (yi )}i=1 , since by construction it contains their endpoints and it lies over the graph of f|π n (Cj ) . R
Step 3. Construction of approximating vector fields (see Fig. 4). j Among all the segments in the graph of fj that connect the points of graph(conv({y˜i }i=1 )) j to the graph of f| n−1 we select those of the form [x, ˜ y˜k ], where x ∈ σ t (Z), yk ∈ {yi }i=1 , and BR
1
(x1 )
we show that for H n−1 -a.e. x ∈ σ t (Z) there exists only one segment within this class which passes through x. ˜ The approximating vector field will be given by the projection on the first n coordinates of the directions of these segments. First of all, we claim that for all x ∈ Bn−1 R1 (x1 ) the graph of fj contains at least a segment of the form [x, ˜ y˜i ] for some i ∈ {1, . . . , j }. Indeed, we show that if x˜ is the endpoint of a segment of the form [x, ˜ (y, fj (y))] where y j j / ext(conv({yi }i=1 )), then there are at least two segments belongs to conv({yi }i=1 ) but (y, fj (y)) ∈ j j of the form [x, ˜ y˜k ] with y˜k ∈ ext(conv({y˜i }i=1 )) ⊂ {y˜i }i=1 (here we assume that j 2). ˜ (y, fj (y))) and a supIn order to prove this, take a point (z, fj (z)) in the open segment (x, porting hyperplane H (z) to the graph of fj that contains that point. By definition, H (z) contains the whole segment [x, ˜ (y, fj (y))] and the set H (z) ∩ (Hh− × R) is a supporting hyperplane to j the set graph(conv({y˜i }i=1 )) that contains the point (y, fj (y)). j j Now, take the smallest R-face C of conv({y˜i }i=1 ) which is contained in graph(conv({y˜i }i=1 )) and contains the point (y, fj (y)), that is given by the intersection of all R-faces which contain (y, fj (y)). j By Propositions 4.12 and 4.13, C = conv[ext(conv({y˜i }i=1 )) ∩ C] and as (y, fj (y)) does not j j belong to ext(conv({y˜i }i=1 )), dim(C) 1 and the set ext(conv({y˜i }i=1 )) ∩ C contains at least two points y˜k , y˜l . In particular, since both C and x˜ belong to H (z) ∩ graph fj , by definition of supporting hy˜ y˜k ], [x, ˜ y˜l ] and our claim is perplane we have that the graph of fj contains the segments [x, proved. n Now, for each j ∈ N, we define the (possibly multivalued) map Dtj : Bn−1 R1 (x1 ) → R as follows:
yi − x : [x, ˜ y˜i ] ⊂ graph fj (4.41) Dtj : x → |yi − x|
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and we prove that the set t Bj := σ t (Z) ∩ x ∈ Bn−1 R1 (x1 ): Dj (x) is multivalued
(4.42)
is H n−1 -negligible, ∀j ∈ N. Thus, if we neglect the set B = j ∈N Bj , we can define our approximating vector field as Dtj (x) = vjt (x) ,
∀x ∈ σ t (Z)\B, ∀j ∈ N.
(4.43)
In order to show that H n−1 (Bj ) = 0 we first prove that, for H n−1 -a.e. x ∈ Bn−1 R1 (x1 ), whent ever Dj (x) contains the directions of two segments, fj must be linear on their convex envelope. ˜ y˜ik ], where ik ∈ {1, . . . , j } and Indeed, suppose that the graph of fj contains two segments [x, k = 1, 2, and consider two points (zk , fj (zk )) ⊂ [x, ˜ y˜ik ] such that z1 = x + se + a1 v1 ,
s ∈ [h− − t, 0), v1 ∈ H0 ;
z2 = x + se + a2 v2 ,
s ∈ [h− − t, 0), v2 ∈ H0 .
(4.44)
As fj is linear on [x, yik ], we have that fj (zk ) = fj (x) + rk · (se + ak vk ),
(4.45)
where rk ∈ ∂ − fj (x), k = 1, 2. Moreover, since πH0 ∂ − fj (x) = ∂ − f|
Bn−1 R1 (x1 )
and the set where ∂ − f|
Bn−1 R1 (x1 )
(x)
(4.46)
is multivalued is H n−2 -rectifiable (see for e.g. [23,1]), we have
that, for H n−1 -a.e. x ∈ Bn−1 R1 (x1 ) r · v = ∇(f|
Bn−1 R1 (x1 )
Then, if we put w = ∇(f|
Bn−1 R1 (x1 )
)(x) · v,
∀r ∈ ∂ − fj (x), ∀v ∈ H0 .
(4.47)
)(x), (4.45) becomes
fj (zk ) = fj (x) + rk · se + w · ak vk .
(4.48)
If zλ = (1 − λ)z1 + λz2 , we have that fj (zλ ) (1 − λ)fj (z1 ) + λfj (z2 ) (4.48) = fj (x) + s (1 − λ)r1 + λr2 · e + w · (1 − λ)a1 v1 + λa2 v2 .
(4.49)
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As ((1 − λ)r1 + λr2 ) ∈ ∂ − fj (x), we also obtain that fj (zλ ) fj (x) + s (1 − λ)r1 + λr2 · e + (1 − λ)r1 + λr2 · (1 − λ)a1 v1 + λa2 v2 = fj (x) + s (1 − λ)r1 + λr2 · e + w · (1 − λ)a1 v1 + λa2 v2 (4.48)
= (1 − λ)fj (z1 ) + λfj (z2 ).
(4.50)
Thus, we have that fj ((1 − λ)z1 + λz2 ) = (1 − λ)fj (z1 ) + λfj (z2 ) and our claim is proved. In particular, there exists a supporting hyperplane to the graph of fj which contains the affine hull of the convex envelope of {[x, ˜ y˜ik ]}k=1,2 and then this affine hull must intersect Ht × R into which is parallel to the segment [y˜i1 , y˜i2 ]. a supporting line to the graph of f| n−1 BR
1
(x1 )
Thus, if all the supporting lines to the graph of f|
Bn−1 R1 (x1 )
which are parallel to a segment
[y˜k , y˜m ] (with k, m ∈ {1, . . . , j }, k = m) are parameterized as lk,m + w,
(4.51)
where lk,m is the linear subspace of Rn+1 which is parallel to [y˜k , y˜m ] and w ∈ Wk,m ⊂ Ht × R is perpendicular to lk,m , we have that Bj = σ (Z) ∩ t
π (lk,m + w) .
(4.52)
Rn
k,m∈{1,...,j } w∈Wk,m k<m
By this characterization of the set Bj and by Fubini theorem on Ht w.r.t. the partition given by the lines which are parallel to πRn (lk,m ) for every k and m, in order to show that H n−1 (Bj ) = 0 it is sufficient to prove that, ∀w ∈ Wk,m , H n−1 σ t (Z) ∩ πRn (lk,m + w) = 0.
(4.53)
Finally, (4.53) follows from the fact that a supporting line to the graph of f|
Bn−1 R1 (x1 )
cannot
contain two distinct points of σ˜ t (Z), because otherwise they would be contained in a higher dimensional face of the graph of f contradicting the definition of σ˜ t (Z). Then, the vector field defined in (4.43) is defined H n−1 -a.e. Step 4. Convergence of the approximating vector fields. Here we prove the convergence property of the vector field defined in (4.43) as stated in (4.35). This result is obtained as a consequence of the uniform convergence of the approximating functions fj to the function fˆ which is the graph of the set Cˆ = conv l˜t (yi ) i∈N .
(4.54)
ˆ First of all we observe that, since Cj C, ˆ dom fj = πRn (Cj ) dom fˆ = πRn (C) and fj (x) fˆ(x) where fj (x) is defined ∀j j0 such that x ∈ πRn (Cj0 ).
ˆ , ∀x ∈ ri πRn (C)
(4.55)
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In order to prove that fj (x) fˆ(x) uniformly, we show that the functions fj are uniformly Lipschitz on their domain, with uniformly bounded Lipschitz constants. We recall that the graph of fj is made of segments that connect the points of graph f| n−1 BR
j
1
(x1 )
to the points of graph(conv({y˜i }i=1 )). In order to find an upper bound for the incremental ratios between points z, w ∈ dom fj , we distinguish two cases. j Case 1: [z, w] ⊂ [x, yk ], where x ∈ Bn−1 ˜ y˜k ] ⊂ graph fj . R1 (x1 ), yk ∈ {yi }i=1 and [x, In this case we have that |fj (z) − fj (w)| |fj (x) − fj (yk )| |f (x) − f (yk )| = = Lf , |z − w| |x − yk | |x − yk |
(4.56)
where Lf is the Lipschitz constant of f on Kt . Case 2: Otherwise we observe that, since fj is convex, fj (z) − fj (w)
sup
r∈∂ − fj (z)∪∂ − fj (w)
r · (z − w) .
(4.57)
Let then r ∈ ∂ − fj (z) ∪ ∂ − fj (w) be a maximizer of the r.h.s. of (4.57) and let us suppose, without loss of generality, that r ∈ ∂ − fj (z). If x ∈ Bn−1 ˜ ⊂ R1 (x1 ) is such that (z, fj (z)) ⊂ [(y, fj (y)), x] j
graph fj for some y ∈ conv({yi }i=1 ), we have the following unique decomposition x−z + γj (z, w)q, w − z = βj (z, w) |x − z|
(4.58)
where q ∈ Sn−1 ∩ H0 and βj (z, w), γj (z, w) ∈ R. Then, x−z r · (w − z) = βj (z, w) r · + γj (z, w)(r · q). |x − z|
(4.59)
The first scalar product in (4.59) can be estimated as in Case 1. As for the second term, we note that the supporting hyperplane to the graph of fj given by the graph of the affine function h(p) = fj (z) + r · (p − z) contains the segment [(z, fj (z)), x] ˜ and its intersection with the hyperplane Ht × R is given by a supporting hyperplane to the graph of f| n−1 which contains the point x. ˜ BR
1
(x1 )
Moreover, as q ∈ H0 , we have that r · q = πH0 (r) · q, and we know that πH0 (r) ∈ ∂ − f|
Bn−1 R1 (x1 )
(x).
By definition of subdifferential, for all s ∈ ∂ − f| x − λq ∈ Bn−1 R1 (x1 ),
(4.60)
Bn−1 R1 (x1 )
(x) and for all λ > 0 such that x + λq,
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f (x + λq) − f (x) f (x) − f (x − λq) s ·q λ λ
(4.61)
and so the term |r · q| is bounded from above by the Lipschitz constant of f . As the scalar products βj (z, w), γj (z, w) are uniformly bounded w.r.t. j on dom fj ⊂ dom fˆ, we conclude that the functions {fj }j ∈N are uniformly Lipschitz on the sets {dom fj }j ∈N and ˆ their Lipschitz constants are uniformly bounded by some positive constant L. ˆ ˆ If we call fj a Lipschitz extension of fj to the set dom f which has the same Lipschitz constant (Mac Shane lemma), by Ascoli–Arzelá theorem we have that fˆj → fˆ uniformly on dom fˆ. Now we prove that, for H n−1 -a.e. x ∈ σ t (Z)\B, vjt (x) → ve (x). j
Given a point x ∈ σ t (Z)\B, we call y˜j (x) , where j ∈ N, the unique point y˜k ∈ {y˜i }i=1 such that vjt (x) =
yk − x . |yk − x|
By compactness of graph(conv({y˜i }i∈N )), there is a subsequence {jn }n∈N ⊂ N such that y˜jn (x) → yˆ ∈ graph f, hence vjt n (x) → vˆ =
yˆ − x . |yˆ − x|
As the functions fj converge to fˆ uniformly, the point yˆ and the whole segment [x, ˜ y] ˆ belong to the graph of fˆ. ˜ y] ˆ which belong to the graph of fˆ and pass through So, there are two segments l˜t (x) and [x, the point x. ˜ Since fˆ| n−1 = f| n−1 , we can apply the same reasoning we made in order to prove that BR
1
(x1 )
BR
1
(x1 )
the set (4.42) was H n−1 -negligible to conclude that the set ˆ σ t (Z) ∩ x ∈ Bn−1 R1 (x1 ): ∃ more than two segments in the graph of f that connect x˜ to a point of graph conv {y˜i }i∈N has zero H n−1 -measure. Then, [x, ˜ y] ˆ = l˜t (x) and vˆ = ve (x) for H n−1 -a.e. x ∈ σ t (Z), so that property (4.35) is proved. Step 5. Proof of the estimate (4.36) (see Fig. 5). The estimate for the map σvt t induced by the approximating vector fields vjt follows as in [7] j
and [8] from the fact that the collection of segments with directions given by vjt and endpoints in −
j
dom vjt , σ h (Z) form a finite union of cones with bases in dom vjt and vertex in {yi }i=1 .
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
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Fig. 5. The vector field ve (that one can see in the picture between {x · e = t} and {x · e = h+ }) is approximated by directions of approximating cones (in the picture one can see the first approximating cone between {x · e = h− } and {x · e = t}). At the same time, Z is approximated by the pushforward of σ t (Z) with the approximating vector field: compare the blue area (outer) with the yellow one (inner). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Indeed, if we define the sets
yi − x , Ωij = x ∈ σ t (Z): Dtj (x) = vjt (x) and vjt (x) = |yi − x|
j ∈ N, i = 1, . . . , j, (4.62)
for all S ⊂ σ t (Z)\B we have that H n−1 (S) =
j
H n−1 (S ∩ Ωij )
i=1
and t −1 n−1 t −1 σv t σv t (S) = H (S ∩ Ωij ) . j
H
n−1
j
j
i=1
Then it is sufficient to prove (4.36) when the vector field vjt is defined as vjt (x) =
yi − x . |yi − x|
After these preliminary considerations, (4.36) follows from the fact that the set
−1 σvst ◦ σvt t (S)
s∈[h− ,t]
j
(4.63)
j
is a cone with basis S ⊂ Ht and vertex yi ∈ Hh− and (σvt t )−1 (S) is the intersection of this cone with the hyperplane H0 .
2
j
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Proof of Lemma 4.7. Given a k-dimensional D-cylinder C k parameterized as in (4.27), the collection of segments
σ w+te (z): t ∈ h− (w, e), h+ (w, e)
(4.64)
z∈Z k
is a 1-dimensional D-cylinder of the convex function f restricted to the (n − k + 1)-dimensional set −1 πe w + te: t ∈ h− (w, e), h+ (w, e) . 1 ,...,ek
(4.65)
Then, as in Lemma 4.14, we can construct a sequence of approximating vector fields also for the directions of the segments (4.64). The only difference with respect to the approximation of the 1-dimensional faces of f is that the domain of the approximating vector fields will be a subset −1 of an (n − k)-dimensional affine plane of the form πe (w) and so the measure involved in 1 ,...,ek n−k n−1 the estimate (4.32) will be H instead of H . Finally, we pass to the limit as with the approximating vector fields given in Lemma 4.14 and we obtain the fundamental estimate (4.31) for the k-dimensional D-cylinders. 2 4.5. Properties of the density function In this subsection, we show that the quantitative estimates of Lemma 4.14 allow not only to derive the absolute continuity of the pushforward with σ w+te , but also to find regularity estimates on the density function. This regularity properties will be used in Section 5. Corollary 4.15. Let C k be a k-dimensional D-cylinder parameterized as in (4.27) and let σ w+se (Z k ), σ w+te (Z k ) be two sections of C k with s and t as in (4.31). Then, if we put s = w + se and t = w + te, we have that t−|s−t|e
H n−k
σ#
σ t Zk
H n−k
σ s Zk
(4.66)
and by the Radon–Nikodym theorem there exists a function α(t, s, ·) which is H n−k -a.e. defined on σ s (Z k ) and is such that t−|s−t|e
σ#
H n−k
σ t Zk
= α(t, s, ·)H n−k
σ s Zk .
(4.67)
Proof. Without loss of generality we can assume that s = 0. If H n−k (A) = 0 for some A ⊂ Z k , by definition of pushforward of a measure we have that w+te −1 n−k H σ #
σ w+te Zk
(A) = H n−k σ w+te (A)
(4.68)
and taking s = 0 in (4.31) we find that H n−k (A) = 0 implies that H n−k (σ w+te (A)) = 0.
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Remark 4.16. The function α = α(t, s, y) defined in (4.67) is measurable w.r.t. y and, for H n−k a.e. y ∈ σ w+te (Z k ), we have that −1 . α s, t, y = α t, s, σ t−|s−t|e y
(4.69)
Moreover, from Lemma 4.7 we immediately get the uniform bounds:
h+ (t, e) − u h+ (t, e) u − h− (t, e) −h− (t, e)
n−k
α(t + ue, t, ·)
n−k α(t + ue, t, ·)
u − h− (t, e) −h− (t, e) h+ (t, e) − u h+ (t, e)
n−k n−k
if u ∈ 0, h+ (t, e) , if u ∈ h− (t, e), 0 .
(4.70)
We conclude this section with the following proposition: Proposition 4.17. Let C k (Z k , C k ) be a k-dimensional D-cylinder parameterized as in (4.27) and assume without loss of generality that w = πe1 ,...,ek (Z k ) = 0. Then, the function α(t, 0, z) defined in (4.67) is locally Lipschitz in t ∈ ri(C k ) (and so jointly measurable in (t, z)). Moreover, for H n−k -a.e. y ∈ σ t (Z) the following estimates hold: 1. Derivative estimate d n−k α(t + ue, t, y) α(t + ue, t, y) − + h (t, e) − u du n−k α(t + ue, t, y); u − h− (t, e)
(4.71)
2. Integral estimate
|h+ (t, e) − u| |h+ (t, e)|
n−k (−1)1{u<0} α(t + ue, t, y)(−1)1{u<0}
|h− (t, e) − u| |h− (t, e)|
n−k (−1)1{u<0} ;
(4.72)
3. Total variation estimate h+ (t,e) h− (t,e)
+ − n−k d |h+ − h− |n−k α(t + ue, 0, z) du 2α(t, 0, z) |h − h | + − 1 , (4.73) du |h+ |n−k |h− |n−k
where h+ , h− stand for h+ (t, e), h− (t, e). Proof. Lipschitz regularity estimate. First we prove the local Lipschitz regularity of α(t, 0, z) w.r.t. t ∈ ri(C k ).
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Given s, t ∈ C k , we set e =
s−t |s−t| .
As s
t
σ s−|s| |s| = σ t−|t| |t| ◦ σ s−|s−t|e , then s s−|s| |s|
H n−k
σ#
σ s(Z)
t−|t| |t|t s−|s−t|e σ# H n−k
σ s(Z) t−|t| α(s, t, y) · H n−k σ t (Z) = σ# = α(t, 0, z)α s, t, σ t (z) H n−kZ.
= σ#
t |t|
(4.74)
By definition of α it follows that α(s, 0, z) − α(t, 0, z) = α(t, 0, z) α s, t, σ t (z) − 1 .
(4.75)
Now we want to estimate the term [α(s, t, σ t (z)) − 1] with the lenght |s − t| times a constant which is locally bounded w.r.t. t. In order to do this, we proceed as in the Corollary 2.19 of [8] using the estimate
h+ (t, e) − u2 h+ (t, e) − u1
n−k
H n−k σ t+u1 e (S) H n−k σ t+u2 e (S)
u2 − h− (t, e) u1 − h− (t, e)
n−k
H n−k σ t+u1 e (S) ,
(4.76)
which holds ∀h− (t, e) < u1 u2 < h+ (t, e) and ∀S ⊂ σ t (Z). Indeed, (4.76) can be rewritten in the following way:
h+ (t, e) − u2 h+ (t, e) − u1
n−k
S
α(t + u2 e, t, y) dH n−k (y)
S
α(t + u1 e, t, y) dH n−k (y)
u2 − h− (t, e) u1 − h− (t, e)
n−k α(t + u1 e, t, y) dH n−k (y).
(4.77)
S
Therefore, there is a dense sequence {ui }i∈N in (h− (t, e), h+ (t, e)) such that for H n−k -a.e. y ∈ S and for all ui uj , i, j ∈ N, the following inequalities hold
h+ (t, e) − uj h+ (t, e) − ui
n−k
− 1 α(t + ui e, t, y)
α(t + uj e, t, y) − α(t + ui e, t, y) uj − h− (t, e) n−k − 1 α(t + ui e, t, y). ui − h− (t, e)
(4.78)
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
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Thanks to the uniform bounds (4.70), for all y ∈ σ t (Z) such that (4.78) holds, the function α(t + ·e, t, y) is locally Lipschitz on {ui }i∈N and for every [a, b] ⊂ (h− (t, e), h+ (t, e)) the Lipschitz constants of α on {ui }i∈N ∩ [a, b] are uniformly bounded w.r.t. y. Then, on every compact interval [a, b] ⊂ (h− (t, e), h+ (t, e)) there exists a Lipschitz extension α(t ˜ + ·e, t, y) of α(t + ·e, t, y) which has the same Lipschitz constant. By the dominated convergence theorem, whenever {ujn }n∈N ⊂ {uj }j ∈N converges to some u ∈ [a, b] we have
α(t + ujn e, t, y) dH
n−k
(y) →
S
α(t ˜ + ue, t, y) dH n−k (y),
∀S ⊂ σ t (Z).
S
However, the integral estimate (4.77) implies that
α(t + ujn e, t, y) dH n−k (y) →
S
α(t + ue, t, y) dH n−k (y), S
so that the Lipschitz extension α˜ is an L1 (H n−k ) representative of the original density α for all u ∈ [a, b]. Repeating the same reasoning for an increasing sequence of compact intervals {[an , bn ]}n∈N that converge to (h− (t, e), h+ (t, e)), we can assume that the density function α(t + ue, t, y) is locally Lipschitz in u with a Lipschitz constant that depends continuously on t and on e. Then, by (4.75), the local Lipschitz regularity in t of the function α(t, 0, z) is proved. Derivative estimate. If we derive w.r.t. u the pointwise estimate (4.78) (which holds for all u ∈ (h− (t, e), h+ (t, e)) by the first part of the proof) we obtain the derivative estimate (4.71). Integral estimate (4.71) implies the monotonicity of the following quantities: α(t + ue, t, y) d 0, du (h+ (t, e) − u)n−k
d α(t + ue, t, y) 0. du (u − h− (t, e))n−k
Integrating the above inequalities from u ∈ (h− (t, e), h+ (t, e)) to 0 we obtain (4.72). Total variation estimate. In order to prove (4.73) we proceed as in Corollary 2.19 of [8]. 0 d α(t + ue, 0, z) du du
h− (t,e)
d α(t + ue, 0, z) du du
d { du α(t+ue,0,z)>0}∩{u∈(h− (t,e),0)}
0 + h− (t,e)
(n − k)α(t + ue, 0, z) du |h+ (t, e) − u|
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0
d α(t + ue, 0, z) du du
h− (t,e)
0
(n − k)α(t + ue, 0, z) du |h+ (t, e) − u|
+2 h− (t,e)
0 α(t, 0, z) + 2 h− (t,e)
(n − k)α(t + ue, 0, z) du. |h+ (t, e) − u|
(4.79)
From (4.75) we know that α(t + ue, 0, z) = α(t, 0, z) α(t + ue, t, σ t (z)). Moreover, since u < 0, α t + ue, t, σ t (z)
(4.72)
|h+ (t, e) − u| |h+ (t, e)|
n−k .
If we substitute this inequality in (4.79) we find that 0 (4.79) α(t, 0, z) + 2α(t, 0, z) h− (t,e)
= −α(t, 0, z) + 2α(t, 0, z)
(n − k)|h+ (t, e) − u|n−k−1 du |h+ (t, e)|n−k
|h+ (t, e) − h− (t, e)|n−k . |h+ (t, e)|n−k
Adding the symmetric estimate on (0, h+ (t, e)) we obtain (4.73).
(4.80)
2
4.6. The disintegration on model sets Now we conclude the proof of Theorem 3.3 on the model sets, giving also an explicit formula for the conditional probabilities. We consider a k-dimensional D-cylinder C k = C k (Z k , C k ) parameterized as in (4.27) and ± we assume, without loss of generality, that πe1 ,...,ek (Z k ) = 0 ∈ Rn . We also set h± j = h (0, ej ), ∀j = 1, . . . , k, and we omit the point w = 0 in the notation for the map (4.28). Theorem 4.18. Let C k be a k-dimensional D-cylinder parameterized as in (4.27). Then, ∀ϕ ∈ 1 (Rn ), Lloc +
ϕ dL n = Ck
+
hk
h1
... Z k h− k
α(t1 e1 + · · · + tk ek , 0, z) ϕ σ (t1 e1 +···+tk ek ) (z) dt1 . . . dtk dH n−k (z).
h− 1
(4.81)
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3639
Then, as (Z k , B(Z k )) is isomorphic to the quotient space determined by the map ∇f on C k and by the uniqueness of the disintegration, the conditional probabilities of the disintegration of the Lebesgue measure on the pieces of k-dimensional faces of f which are contained in C k are given by μz (dt1 . . . dtk ) =
α(t1 e1 + · · · + tk ek , 0, z) H k [ri(R(z)) ∩ C k ](dt1 . . . dtk ) , h+k h+1 − ... − α(s1 e1 + · · · + sk ek , 0, z) ds1 . . . dsk h h k
(4.82)
1
for H n−k -a.e. z ∈ Z k . Proof. We proceed using the disintegration technique which was presented in Section 4.1: +
+
hk ϕ(x) dL n (x)
=
h1
h− k
Ck
ϕ dH n−k dt1 . . . dtk
C k ∩{x·ek =tk }∩···∩{x·e1 =t1 } h− 1
+
+
hk =
(4.66)
...
h1 α(tk ek , 0, z) . . .
... h− k
h− 1
Zk
· α t1 e1 + · · · + tk ek , t2 e2 + · · · + tk ek , σ (t2 e2 +···+tk ek ) (z) · ϕ σ (t1 e1 +···+tk ek ) (z) dH n−k (z) dt1 . . . dtk +
+
hk =
(4.74)
h1 α(t1 e1 + · · · + tk ek , 0, z)
... h− k
Zk h− 1
· ϕ σ (t1 e1 +···+tk ek ) (z) dH n−1 (z) dt1 . . . dtk +
+
hk =
(4.70) Prop. 4.17 Z k h− k
h1
...
α(t1 e1 + · · · + tk ek , 0, z)
h− 1
· ϕ σ (t1 e1 +···+tk ek ) (z) dt1 . . . dtk dH n−1 (z).
2
(4.83)
4.7. The global disintegration In this section we prove Theorem 3.3, concerning the disintegration of the Lebesgue measure (restricted to a set of finite Lebesgue measure K ⊂ Rn ) on the whole k-dimensional faces of a convex function. The idea is to put side by side the disintegrations on the model D-cylinders which belong to the countable family defined in Lemma 4.6, so as to obtain a global disintegration. What will remain apart will be set T \ nk=1 E k , projection of those points which do not belong to the relative interior of any face. Nevertheless, the following lemma ensures that this set is L n -negligible. Indeed, the union of the relative boundaries of the n-dimensional faces has
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zero Lebesgue measure by convexity and by the fact that the n-dimensional faces of f are at most countable. For faces of dimension k, with 1 < k < n, the proof is by contradiction: one considers a Lebesgue point of suitable subsets of y (Fyk \ Eyk ) and applies the fundamental estimate (4.31) in order to show that the complementary is too big. Eq. (4.84) below was first proved using a different technique in [15], where it was shown that the union of the relative boundaries of the R-faces (see Definition 4.11) of an n-dimensional convex body C which have dimension at least 1 has zero H n−1 -measure. Lemma 4.19. The set of points which do not belong to the relative interior of any face is L n negligible: L
n
T\
n
E
= 0,
k
where E k =
ri Fyk .
(4.84)
y
k=1
Proof. Consider any n-dimensional face Fyn . Being convex, it has nonempty interior. As a consequence, since two different faces cannot intersect, there are at most countably many ndimensional faces {Fyni }i∈N ; moreover, by convexity, each Fyni has an L n -negligible boundary. Thus Ln
rb Fyni = 0.
i
Since T ⊂
n
k=1 F
k,
the thesis is reduced in showing that, for 0 < k < n, L n F k \ E k = 0.
(4.85)
We provide a countable covering of F k \E k with measurable sets Ap,e,V , choosing the following parameters: p ∈ N0 = N ∪ {0}, a k-dimensional subspace V in a dense sequence in G(k, n) and a unit direction e belonging to a dense sequence in Sn−1 ∩ V ; this direction e is identified by n−1 ∩ e , . . . , e , for 1 i < · · · < i n, which vary in a sequence dense its projections i1 ik 1 k on S in {0} ∪ 1i1 <···
d∈D(x)
k −p+1 πV F∇f e + 2−p Sn−1 ∩ V . (x) ⊃ conv πV (x) ∪ πV (x) + 2
(4.86) (4.87)
The measurability of each Ap,e,V can be deduced as follows. The set defined in (4.86) is exactly k the complementary in T \ ri(F∇f (x) ) of 1 D−1 ◦ πV−1 V ∩ ri √ Bn . 2
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3641
Moreover, (4.87) is equivalent to πV R(x) − x ⊃ conv 2−p+1 e + 2−p Sn−1 ∩ V . Since R and D are measurable (Lemma 4.2), then the Borel measurability of Ap,e,V follows. As a consequence, if (4.85) does not hold, then there exists a subset Ap,e,V of F k \E k with positive Lebesgue measure. Up to rescaling, one can assume w.l.o.g. that p = 0, V = e1 , . . . , ek , where {e1 , . . . , en } is an orthonormal basis of Rn , and e = e1 . Moreover, we will denote Ap,e,V simply with A. In the following we show the absurd. Before reaching the contradiction L n (A) = 0, we need the following remarks. First of all we notice that, for 0 h 3 and t ∈ πV (A), one can prove the fundamental estimate H
n−k
t+he σ (S)
3−h 3
n−k H n−k (S)
∀S ⊂ A ∩ πV−1 (t)
(4.88)
exactly as in Lemma 4.7, with the approximating vector field given in Step 3, p. 3628. Indeed, the (n − k + 1)-plane πV−1 (Re) cuts the face of each z ∈ A ∩ πV−1 (t) into exactly one line l; this line has projection on V containing at least [t, t + 3e]. Notice moreover that, by (4.87), each point x ∈ l, with πV (x) ∈ ri([t, t + 3e]), is a point in the relative interior of the face. In particular, it does not belong to A. Let us now prove the claim, assuming by contradiction that L n (A) > 0 (see also Fig. 6). Fix any ε > 0 small enough. W.l.o.g. one can suppose the origin to be a Lebesgue point of A. Therefore, by definition of Lebesgue point, for any ε > 0 sufficiently small there exists r¯ (ε) s.t. for every 0 < r < r¯ (ε) < 1, L n A ∩ [0, r]n 1 − ε 2 r n . By Fubini theorem, there exists T ⊂
k
i=1 [0, rei ],
with H k (T ) > (1 − ε)r k , such that
H n−k A ∩ πV−1 (t) ∩ [0, r]n (1 − ε)r n−k Moreover, there is a set Q ⊂ [0, re], with H 1 (Q) > (1 −
for all t ∈ T .
(4.89)
√ ε)r, such that
√ −1 (q) > (1 − ε )r k−1 H k−1 T ∩ πe
for q ∈ Q.
(4.90)
−1 Consider two points q, s := q + 2εre ∈ Q, and take t ∈ T ∩ πe (q). By the fundamental estimate (4.88), one has
H n−k σ t+2εre (St,r ) (1 − ε)n−k H n−k (St,r )
where St,r := A ∩ πV−1 (t) ∩ [0, r]n .
Furthermore, condition (4.86) implies that x + 2εre − σ t+2εre (x) 2εr for each x ∈ A ∩ πV−1 (t). Moving points of πV−1 (t) ∩ [0, r]n by means of the map σ t+2εre , they can therefore reach
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Fig. 6. Illustration of the construction in the proof of Lemma 4.19. A is the set of points on the relative boundary of the k-faces of f , projected on Rn , having directions close to V = e1 , . . . , ek and such that, for each point x ∈ A, k k πV (F∇f (x) ) contains a fixed k-cone centered at x with direction e1 . T is a subset of the square i=1 [0, rei ] such that, for every t ∈ T , πV−1 (t) ∩ A is ‘big’. Finally, q, s = q + 2εre1 are points on [0, re1 ] such that the intersection of T with
−1 −1 the affine hyperplanes πe (q), πe (s) is ‘big’. The absurd arises from the following. Due to the fundamental estimate, 1
1
−1 −1 (q), one finds points in the complementary of T . Since T ∩ πe (q) translating by 2εre1 the points in the set T ∩ πe 1 1 −1 −1 was ‘big’, then T \ πe (s) should be ‘big’, contradicting the fact that T ∩ πe (s) is ‘big’. 1
1
only the square πV−1 (t + 2εre) ∩ [−2εr, (1 + 2ε)r]n . Notice that for ε small, since our proof is needed for n 3 and k 1, n−k H n−k −2εr, (1 + 2ε)r \ [0, r]n−k = (1 + 4ε)n−k r n−k − r n−k 4(n − k)εr n−k + o(ε) < n2n εr n−k . As a consequence, the portion which stays inside πV−1 (t + 2εre) ∩ [0, r]n can be estimated as follows: H n−k σ t+2εre (St,r ) ∩ [0, r]n H n−k σ t+2εre (St,r ) − n2n εr n−k . As noticed before, condition (4.87) implies that the points σ t+2εre (St,r ) ∩ [0, r]n belong to the complementary of A. By the above inequalities we obtain then H n−k Ac ∩ πV−1 (t + 2εre) ∩ [0, r]n H n−k σ t+2εre (St,r ) ∩ [0, r]n (1 − ε)n−k H n−k (St,r ) − n2n εr n−k
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
3643
(4.89)
(1 − ε)n−k+1 r n−k − n2n εr n−k 1 n−k . r 2
−1 The last estimate shows that, for each t ∈ T ∩ πe (q), the point t + 2εre does not satisfy the −1 (q)) + 2εre lies in the complementary of T . In particular inequality in (4.89): thus (T ∩ πe
−1 −1 H k−1 T ∩ πe (s) < r k−1 − H k−1 T ∩ πe (q) . However, by construction both t and s belong to Q. This yields the contradiction, by definition of Q: 1 k−1 (4.90) k−1 −1 −1 (4.90) 1 k−1 r r . T ∩ πe < H (s) < r k−1 − H k−1 T ∩ πe (t) < 2 2
2
Proof of Theorem 3.3. As we observed in Remark 3.4, it is sufficient to prove the theorem for the disintegration of the Lebesgue measure on the set F k when k ∈ {1, . . . , n − 1}. Thanks to Lemma 4.19, we can further restrict the disintegration to the set E k defined in (4.13); moreover, by (4.25), for all k = 1, . . . , n − 1 there exists an L n -negligible set N k such that E k \N k =
Cjk \dCjk ,
j ∈N
where {Cjk }j ∈N is the countable collection of k-dimensional D-cylinders covering E k which was constructed in Lemma 4.6, so that the sets Cˆjk := Cjk \dCjk are disjoint. The fundamental observation is the following:
j ∈N
Cˆjk =
j ∈N y∈Im ∇f|
k Ey,j =
y∈Im ∇f|
Ek
Ek
k Ey,j =
y∈Im ∇f|
j ∈N
Eyk \N k ,
(4.91)
Ek
k = E k ∩ Cˆk . where Ey,j y j For all j ∈ N, we set
k Yj = y ∈ Im ∇f|E k : Ey,j = ∅ = ∇f Cˆjk ,
(4.92)
we denote by pj : Cˆjk → Yj the quotient map corresponding to the partition Cˆjk =
y∈Im ∇f|
k Ey,j = Ek
k Ey,j
y∈Yj
and we set νj = pj # (L n (Cˆjk ∩ K)). Since the quotient space (Yj , B(Yj )) is isomorphic to (Zjk , B(Zjk )), where Zjk is a section of Cjk , by Theorem 4.18 we have that
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L
n
k Cj ∩ K Ej ∩ pj−1 (Fj ) =
j
μy (Ej ) dνj (y), Fj
∀Ej ∈ B Cjk , Fj ∈ B(Yj ),
(4.93)
j
k ∩ K) for ν -a.e. y ∈ Y . where μy is equivalent to H k (Ey,j j j n k Moreover, for every E ∈ B(R ) ∩ E there exist sets Ej ∈ B(Cjk ) such that
E=
Ej
j ∈N
and for all F ∈ B(Y ), where Y = F=
j ∈N Yj
= Im ∇f|E k , setting F := F ∩ Yj we have that
and ∇f −1 (F ) =
Fj
j ∈N
pj−1 (Fj ).
j ∈N
Then, Ln
K E ∩ ∇f −1(F )
=
+∞
Ln
j =1
=
(4.93)
=
+∞ j =1F
Cjk Ej ∩ pj−1(Fj ) j
μy (Ej ) dνj (y)
j
+∞
j
1Fj (y)μy (Ej ) dνj (y)
j =1Y
j
=
+∞
j
1Fj (y)μy (Ej )fj (y) dν(y),
(4.94)
j =1 Y
where fj is the Radon–Nikodym derivative of νj w.r.t. the measure ν = ∇f# (L n K) on Y . Since, as we proved in Section 3.1, there exists a unique disintegration {μy }y∈Im ∇f| k such E that L n K E ∩ ∇f −1 (F ) = μy (E) dν(y) for all E ∈ B Rn , F ∈ B(Y ),
F
we conclude that the last term in (4.94) converges and μy =
+∞ j =1
so that the theorem is proved.
2
j
fj (y) μy
for ν-a.e. y ∈ Y,
(4.95)
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
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5. A divergence formula The previous section led to a definition of a function α, on any D-cylinder C k = C k (Z k , C k ), as the Radon–Nikodym derivative in (4.67). In the present section we find that on C k the function α satisfies the system of ODEs ∂t α t = πe1 ,...,ek (x), 0, x −
k
x · ei vi (x)
i=1
= (div v )a.c. (x)α πe1 ,...,ek (x), 0, x −
k
x · ei vi (x)
i=1
for = 1, . . . , k, where we assume w.l.o.g. that 0 ∈ C k , e1 , . . . , ek is the axis of C k , vi (x) is the vector field −1 x → 1C k (x) D(x) ∩ πe (e ) 1 ,...,ek i and (div vi )a.c. (x) is the density of the absolutely continuous part of the divergence of vi , that we prove to be a measure. This is a consequence of the absolute continuity of the conditional measures of the disintegration, stated in Theorem 4.18, and of the regularity estimates on the density α proved in Proposition 4.17. Notice that even the fact that the divergence of vi is a measure is not trivial, since the vector field is just Borel. Heuristically, the ODEs above can be formally derived as follows. In Section 4 we saw that C k is the image of the product space C k + Z k , where Z k = C k ∩ −1 πe1 ,...,ek (0) is a section of C k , under the change of variable Φ(t + z) = z +
k
ti vi (z) = σ t (z)
for all t =
i=1
k
ti e i ∈ C k , z =
i=1
n
zi ei ∈ Z k .
(5.1)
i=k+1
In Theorem 4.18 we found that the weak Jacobian of this change of variable is defined, and given by J(t + z) = α(t, 0, z).
(5.2)
From (5.1) one finds that, if vi was smooth instead of only Borel, this Jacobian would be ! J(t + z) = det
[vj · ei ] i=1,...,n
j =1,...,k
! k " t ∂zj v (z) · ei + δi,j =1
" .
(5.3)
i=1,...,n j =k+1,...,n
Moreover, by direct computations with Cramer rule and the multilinearity of the determinant, starting from (5.2)-(5.3) one would prove the relation
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−1 J(t + z), ∂t J(t + z) = trace J v (z) J Φ(t + z) where J g denotes the Jacobian matrix of a function g. By the Lipschitz regularity of α w.r.t. the {ti }ki=1 variables given in Proposition 4.17, one could then expect that ∂t α(t, 0, z) =
n j =1
−1 ∂xj vi Φ (x) · ej x=Φ(t+z) α(t, 0, z).
(5.4)
Notice that j ∂xj (vi (Φ −1 (x)) · ej )|x=Φ(t+z) is the pointwise divergence of the vector field vi (Φ −1 (x)) evaluated at x = Φ(t + z). In this article, we denote it with (div(vi ◦ Φ −1 ))a.c. . Finally, given a regular domain Ω ⊂ Rn , by the Green–Gauss–Stokes formula one should have div vi ◦ Φ −1 a.c. dL n (x) = vi Φ −1 (x) · nˆ dH n−1 (x), (5.5) Ω
∂Ω
where nˆ is the outer normal to the boundary of Ω. The analogue of formulas (5.4) and (5.5) is the additional regularity we prove in this section, in a weak context, for vector fields parallel to the faces and for the current of k-faces. Actually, for simplicity of notations we will continue working with the projection of the faces on Rn instead of with the faces themselves. We give now the idea of the proof, in the case of 1-dimensional faces. −1 (0). Fix the attention on a 1-dimensional D-cylinder C with axis e and section Z = C ∩ πe Consider the distributional divergence of the vector field v giving pointwise on C the direction of projected faces, normalized with v · e = 1, and vanishing elsewhere. The explicit disintegration in Theorem 4.18 decomposes integrals on C to integrals first on the projected faces, with the additional density factor α, then on Z. By means of it, one then reduces the integral − C ∇ϕ · v, defining the distributional divergence, to the following integrals on the projected faces: −
∇ϕ x = z + t1 v(z) · v(z)α(t, 0, z) dH 1 (t)
where z varies in Z.
[h− e,h+ e]
Since α is Lipschitz in t and ∇ϕ(x = σ w+t1 e (z)) · v(z) = ∂t1 (ϕ ◦ σ w+t (z)), by integrating by parts one arrives to t=h+ e ϕ ◦ σ w+t (z)∂t1 α(t, 0, z) dH 1 (t) − ϕ ◦ σ w+t (z)α(t, 0, z) t=h− e . [h− e,h+ e]
Applying again the disintegration theorem in the other direction, by the invertibility of α, one comes back to integrals on the D-cylinder, where in the first addend ϕ is now integrated with the factor ∂t1 α/α. An argument of this kind yields an explicit representation of the distributional divergence of the truncation to C k of a vector field v parallel at each point x to the projected face through x. This divergence is a Radon measure: the absolutely continuous part is basically given by (5.4)
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
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and, as in (5.5), there is an additional singular term representing the flux of v through dC k , the border of C k transversal to D, which we have already defined as k −1 rb C . dC k = C k ∩ πe 1 ,...,ek
(5.6)
−1 (C k ). We also remind that nˆ |dC k denotes the outer unit normal to πe 1 ,...,ek k As C are not regular sets, but just σ -compact, there is a loss of regularity for the divergence of v in the whole Rn . In general, the distributional divergence of v will be just a series of measures.
5.1. Vector fields parallel to the faces In the present subsection, we study the regularity of a vector field parallel, at each point, to the corresponding face through that point. 5.1.1. Study on model sets As a preliminary step, fix the attention on the D-cylinder C k = C k Z k , Ck . One can assume w.l.o.g. that the axis of C k is identified by vectors {e1 , . . . , ek } which are the first k coordinate vectors of Rn and that C k is the square
Ck =
k
[−ei , ei ].
i=1 −1 Denote with Z k the section Z k ∩ πe (0). We also assume w.l.o.g. that C k is bounded. 1 ,...,ek
Definition 5.1 (Coordinate vector fields). We define on Rn k-coordinate vector fields for C k as follows:
vi (x) =
0 v ∈ D(x)
if x ∈ / C k, such that πe1 ,...,ek v = ei if x ∈ C k .
The k-coordinate vector fields are a basis for the module on the algebra of measurable functions from Rn to R constituted by the vector fields with values in D(x) at each point x ∈ C k , and vanishing elsewhere. Consider the distributional divergence of vi , denoted by div vi . As a consequence of the absolute continuity of the pushforward with σ , and by the regularity of the density α, one gains more regularity for the divergence. Let us fix a notation. Given any vector field v : Rn → Rn whose distributional divergence is a Radon measure, we will denote by (div v)a.c. the density of the absolutely continuous part of the measure div v.
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L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
Lemma 5.2. The distribution div vi is a Radon measure. Its absolutely continuous part has density ∂ti α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div vi )a.c. (x) = 1C k (x). α(πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) Its singular part is H n−1
(5.7)
(C k ∩ {x · ei = −1}) − H n−1 (C k ∩ {x · ei = 1}).
n Proof. Consider any test function ϕ ∈ C∞ c (R ) and apply the explicit disintegration of Theorem 4.18:
div vi , ϕ := −
∇ϕ(x) · vi (x) dL n (x)
Ck
=−
α(t, 0, z) ∇ϕ σ t (z) · vi (z) dH k (t) dH n−k (z),
(5.8)
Zk C k
where we used that vi is constant on the faces, i.e. vi (z) = vi (σ t (z)). Being σ t (z) = z + k i=1 ti vi (z), one has ∇x ϕ x = σ t (z) · vi (z) = ∇x ϕ x = σ t (z) · ∂ti σ t (z) = ∂ti ϕ σ t (z) . The inner integral is thus
∇ϕ σ t (z) · vi (z)α(t, 0, z) dH k (t) =
Ck
∂ti ϕ σ t z α(t, 0, z) dH k (t).
Ck
Since Proposition 4.17 ensures that α is Lipschitz in t, for t ∈ C k , one can integrate by parts: Ck
∂ti ϕ σ t (z) α(t, 0, z) dH k (t) = −
Ck
ϕ σ t (z) ∂ti α(t, 0, z) dH k (t)
+
ϕ σ t (z) α(t, 0, z) dH k−1 (t)
C k ∩{ti =1}
−
ϕ σ t (z) α(t, 0, z) dH k−1 (t).
C k ∩{ti =−1}
Substitute now the last formula in the first expression (5.8). Recall moreover the definition of α in (4.67), as a Radon–Nikodym derivative of a push-forward measure, and its invertibility and Lipschitz estimates (Remark 4.16, Proposition 4.17), among which in particular the L1 estimate on the function ∂ti α/α. Then, pushing the measure from t = 0 to a generic t, one comes back to the integral on the D-cylinder
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
div vi , ϕ =
3649
ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) dH n−k (z)
Zk C k
−
ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)
Z k C k ∩{ti =1}
ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)
+ Z k C k ∩{ti =−1}
=
ϕ(x)(div vi )a.c. (x) dL (x) − n
Ck
ϕ(x) dH n−1 (x)
C k ∩{x·ei =1}
+
ϕ(x) dH n−1 (x),
C k ∩{x·ei =−1}
where (div vi )a.c. 1C k is the function proved by the last formula. 2
∂ti α α
precisely written in the statement. Thus our thesis is
Remark 5.3. Consider a function λ ∈ L1 (C k ; R) constant on each face, meaning that λ(σ t (z)) = λ(z) for t ∈ C k and z ∈ Z k . One can regard this λ as a function of ∇f (x). Then the same statement of Lemma 5.2 applies to the vector field λvi , but the divergence is clearly div(λvi ) = λ div vi . The proof is the same, observing that div(λvi ), ϕ
:=
−
∇ϕ(x) · λ(x)vi (x) dL n (x)
Ck Th. 4.18
=
−
λ(z)∇ϕ σ t (z) · vi (z)α(t, 0, z) dH k (t) dH n−k (z)
Zk C k
=
−
λ(z)∂ti ϕ σ t (z) α(t, 0, z) dH k (t) dH n−k (z)
Zk C k
=
Zk C k
λ(z)ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) dH n−k (z)
−
λ(z)ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)
Z k C k ∩{ti =1}
+ Z k C k ∩{ti =−1}
λ(z)ϕ σ t (z) α(t, 0, z) dH k−1 (t) dH n−k (z)
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L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661 Th. 4.18
=
ϕ(x)λ(x)(div vi )a.c. (x) dL (x) − n
Ck
ϕ(x)λ(x) dH n−1 (x)
C k ∩{x·ei =1}
+
ϕ(x)λ(x) dH n−1 (x).
C k ∩{x·e
(5.9)
i =−1}
Suitably adapting the integration by parts in the above equality (5.9) with
λ σ t (z) ∂ti ϕ σ t (z) α(t, 0, z) dH k (t)
Ck
=− Ck
λ σ t (z) ϕ σ t (z) ∂ti α(t, 0, z) dH k (t) −
∂ti λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k (t)
Ck
λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k−1 (t)
+ C k ∩{ti =1}
−
λ σ t (z) ϕ σ t (z) α(t, 0, z) dH k−1 (t)
C k ∩{ti =−1}
one finds moreover that for all λ ∈ L1 (Rn ; R) continuously differentiable along vi with integrable directional derivative ∂vi λ, the following relation holds: div(λvi ) = λ div vi + ∂vi λ L n .
(5.10)
Notice that in (5.10) there is the addend λH n−1 (C k ∩ {x · ei = 1}), hidden in the term λ div vi , which would make no sense for a general λ ∈ L1 (Rn ; R). Now we prove that the restriction to k k n−k -a.e. z ∈ Z k , C k ∩ {x · ei = 1} of each representative of λ which is C1 (F∇f (z) ∩ C ), for H 1 k identifies the same function in L (C ∩ {x · ei = 1}). ˜ λˆ of the L1 -class of λ can differ only on an L n -negligible Indeed, any two representatives λ, set N . By the explicit disintegration in Theorem 4.18, and using moreover Fubini theorem for reducing the integral on C k to integrals on lines parallel to ei , one has that the intersection of N with each of the lines on the projected faces with projection on e1 , . . . , ek parallel to ei is almost always negligible: H 1 N ∩ q + vi (q) = 0
for q ∈ C k ∩ {x · ei = 0} \ M, with H n−1 (M)=0.
˜ λˆ are continuously differentiable along vi , one can have that N ∩ (q + vi (q)) = ∅ for Since λ, all q ∈ C k ∩ {x · ei = 0} \ M. As a consequence N ∩ {x · ei = t} is a subset of τ tei (M), where τ tei is the map moving along each projected face with tvi : C k ∩ {x · ei = 0} q → τ tei (q) := q + tvi = σ πe1 ,...,ek (q)+te1 (q).
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
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By the pushforward formula (4.67), denoting wq := πe1 ,...,ek (q) and zq := πek+1 ,...,en (q), H n−1
te te τ i (S) = α(wq , wq + tei , zq )τ# i H n−1 (q)
S
for S ⊂ C k ∩ {x · e1 = 0} and t ∈ [−1, 1]. Therefore, as H n−1 (M) = 0, one has that λ˜ and λˆ identify the same integrable function on each section of C k perpendicular to ei , showing that the measure λH n−1 ({x · ei = 1}) is well defined. Actually, the same argument as above should be used in (5.9) in order to show that λ(z) is integrable on Z k , so that one can separate the three integrals as we did. Indeed, being constant on each face by assumption, the restriction of λ to a section is trivially well defined, since it associates to a point the value of λ corresponding to the face of that point, but the integrability w.r.t. H n−1 on each slice is a consequence of the pushforward estimate.
As a direct consequence of (5.10), by linearity, one gets a divergence formula for any sufficiently regular vector field which, at each point of C k , is parallel to the corresponding projected face of f , and vanishes elsewhere. The precise statement is given in the following: Corollary 5.4. Consider any vector field v = ki=1 λi vi with λi ∈ L1 (C k ; R) continuously differentiable along vi , with directional derivative ∂vi λi integrable on C k . Then the divergence of v is a Radon measure and for every ϕ ∈ C1c (Rn ), div v, ϕ =
ϕ(x)(div v)a.c. (x) dL (x) − n
Ck
ϕ(x) v(x) · n(x) ˆ dH n−1 (x),
(5.11)
dC k
where dC k , the border of C k transversal to D, and n, ˆ the outer unit normal, are defined in formula (5.6). In particular, the density (div v)a.c. (x) of the absolutely continuous part of the divergence vanishes out of C k , while for x ∈ C k one has the expression ∂ti α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div v)a.c. (x) = λi (x) α(πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) i=1 k
+
k
∂vi λi (x).
(5.12)
i=1
Remark 5.5. The result is essentially based on the application of the integration by parts formula when the integral on C k is reduced, by our explicit disintegration theorem, to integrals on C k : this is why we assume the C1 regularity of the λi , w.r.t. the directions of the k-face passing through each point of C k . Such regularity could be further weakened, however we do not pursue this issue here. As a consequence, one can easily extend the statement of the previous corollary −1 to sets of the form CΩk = F k ∩ πe (Ω), for an open set Ω ⊂ e1 , . . . , ek with piecewise 1 ,...,ek −1 (rb(Ω)). Lipschitz boundary, defining dCΩk := F k ∩ πe 1 ,...,ek
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5.1.2. Global version We study now the distributional divergence of an integrable vector field v on T, as we did in Section 5.1.1 for such a vector field truncated on D-cylinders. Corollary 5.6. Consider a vector field v ∈ L1 (T; Rn ) such that v(x) ∈ D(x) for x ∈ Rn , where we define D(x) = {0} for x ∈ / T. Suppose moreover that the restriction to every face Ey , for y ∈ Im ∇f , is continuously differentiable with integrable derivatives. Then, for every ϕ ∈ C1c (Rn ) one can write div v, ϕ = lim
→∞
ϕ(x) div(1Ci v) a.c. (x) dL n (x) −
i=1 C i
ϕ(x)v(x) · nˆ i (x) dH
n−1
(x) ,
dCi
(5.13) where {C }∈N is the countable partition of T in D-cylinders given in Lemma 4.6, while (div(1Ci v))a.c. is the one of Corollary 5.4 and dCi , nˆ i are defined in formula (5.6). Remark 5.7. By construction of the partition, each of the second integrals in the r.h.s. of (5.13) appears two times in the series, with opposite sign. Intuitively, the finite sum of these border terms is the integral on a perimeter which tends to the singular set of points in the relative boundary of projected k-faces. Remark 5.8. Suppose that div v is a Radon measure. Then Corollary 5.6 implies that 1C k (div v)a.c. ≡ div(1C k v) a.c. . Proof of Corollary 5.6. The partition of nk=1 E k into the sets {C }∈N is given by Lemma 4.6, as stated in the corollary. Moreover, Lemma 4.19 shows that the set T \ nk=1 Ek is Lebesgue negligible. Therefore, by dominated convergence theorem one finds that
v(x) · ∇ϕ(x) dL (x) = − lim
div v, ϕ = −
n
→∞
T
i=1 C
v(x) · ∇ϕ(x) dL n (x).
i
The addends in the r.h.s. are, by definition, the distributional divergence of the vector fields v1Ci applied to ϕ. In particular, by Corollary 5.4, they are equal to −
v(x) · ∇ϕ(x) dL n (x)
Ci
=−
1Ci (x)v(x) · ∇ϕ(x) dL n (x)
Rn Cor. 5.4
=
Rn
ϕ(x)1Ci (x) div(1Ci v) a.c. (x) dL n (x) +
dCik
ϕ(x)v(x) · nˆ i (x) dH n−1 (x)
L. Caravenna, S. Daneri / Journal of Functional Analysis 258 (2010) 3604–3661
=
3653
ϕ(x)(div v)a.c. (x) dL (x) +
ϕ(x)v(x) · nˆ i (x) dH n−1 (x),
n
Ci
dCik
proving the thesis.
2
5.2. The currents of k-faces In the present subsection, we change point of view. Instead of looking at vector fields constrained to the faces of f , we regard the k-dimensional faces of f as a k-dimensional current. We establish that this current is a locally flat chain, providing a sequence of normal currents converging to it in the flat norm.The border of these normal currents has formally the same representation one would have in a smooth setting. Before proving it, we devote Section 5.2.1 to recalls on this argument, in order to fix the notations. They are taken mainly from Chapter 4 of [17] and Sections 1.5.1, 4.1 of [10]. 5.2.1. Recalls Let {e1 , . . . , en } be a basis of Rn . The wedge product between vectors is multilinear and alternating, i.e.:
n
λi ei ∧ u1 ∧ · · · ∧ um =
i=1
n
λi (ei ∧ u1 ∧ · · · ∧ um ),
m ∈ N, λ1 , . . . , λn ∈ R,
i=1
u0 ∧ · · · ∧ ui ∧ · · · ∧ um = (−1)i ui ∧ u0 ∧ · · · ∧ u#i ∧ · · · ∧ um ,
0 < i m, u0 , . . . , um ∈ Rn ,
where the element under the hat is missing. The space of all linear combinations of ei1 ...im := ei1 ∧ · · · ∧ eim : i1 < · · · < im in {1, . . . , n} is the space of m-vectors, denoted by Λm Rn . The space Λ0 R is just R. Λm Rn has the inner product given by ei1 ...im · ej1 ...jm =
m
δi k j k
where δij =
k=1
1 if i = j , 0 otherwise.
The induced norm is denoted by · . An m-vector field is a map ξ : Rn → Λm Rn . The dual Hilbert space to Λm Rn , denoted by Λm Rn , is the space of m-covectors. The element dual to ei1 ...im is denoted by dxi1 ...im . A differential m-form is a map ω : Rn → Λm Rn . We denote by ·, · the duality pairing between m-vectors and m-covectors. Moreover, the same symbol denotes in this paper the bilinear pairing, which is a map Λp Rn × Λq Rn → Λp−q Rn for p > q and Λp Rn × Λq Rn → Λq−p Rn for q > p whose non-vanishing images on a basis are dxi1 ...i = dxi1 ...i ∧ dxi+1 ...i+m , ei+1 ...i+m , ei+1 ...i+m = dxi1 ...i , ei1 ...i ∧ ei+1 ...i+m ,
if p = + m > m = q, if p = < + m = q.
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Consider any differential m-form ω=
ωi1 ...im dxi1 ...im
i1 ...im
which is differentiable. The exterior derivative dω of ω is the differential (m + 1)-form dω =
n ∂ωi
1 ...im
∂xj
i1 ...im j =1
dxj ∧ dxi1 ...im .
If ω ∈ Ci (Rn ; Λm Rn ), the i-th exterior derivative is denoted by d i ω. Consider any m-vector field ξ=
ξi1 ...im ei1 ...im
which is differentiable. The pointwise divergence (div ξ )a.c. of ξ is the (m − 1)-vector field (div ξ )a.c. :=
n ∂ξi i1 ...im j =1
1 ...im
∂xj
dxj , ei1 ...im .
Consider the space D m of C∞ -differential m-form with compact support. The topology is generated by the seminorms i νK (φ) =
sup
x∈K, 0j i
j d φ(x)
with K compact subset of Rn , i ∈ N.
The dual space to D m , endowed with the weak topology, is called the space of m-dimensional currents and it is denoted by Dm . The support of a current T ∈ Dm is the smallest closed set K ⊂ Rn such that T (ω) = 0 whenever ω ∈ D m vanishes out of K. The mass of a current T ∈ Dm is defined as M(T ) = sup T (ω): ω ∈ D m , sup ω(x) 1 . x∈Rn
The flat norm of a current T ∈ Dm is defined as F(T ) = sup T (ω): ω ∈ D m , sup ω(x) 1, sup dω(x) 1 . x∈Rn
x∈Rn
An m-dimensional current T ∈ Dm is representable by integration, and we denote it by T = μ ∧ ξ , if there exists a Radon measure μ over Rn and a μ-locally integrable m-vector field ξ such that T (ω) = ω, ξ dμ ∀ω ∈ D m . Rn
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If m 1, the boundary of an m-dimensional current T is defined as (∂T )(ω) = T (dω) whenever ω ∈ D m−1 .
∂T ∈ Dm−1 ,
If either m = 0, or both T and ∂T are representable by integration, then we will call T locally normal. If T is locally normal and compactly supported, then T is called normal. The F-closure, in Dm , of the normal currents is the space of locally flat chains. Its subspace of currents with finite mass is the M-closure, in Dm , of the normal currents. To each L n -measurable m-vector field ξ such that ξ is locally integrable there corresponds the current L n ∧ ξ ∈ Dm (Rn ). If ξ is of class C1 , then this current is locally normal and the divergence of ξ is related to the boundary of the corresponding current by −∂ L n ∧ ξ = L n ∧ div ξ. ˆ Moreover, if Ω is an open set with C1 boundary, nˆ is its outer unit normal and d nˆ the dual of n, then ∂ L n ∧ (1Ω ξ ) = − L n
Ω
∧ div ξ + H n−1
∂Ω
∧ d n, ˆ ξ .
(5.14)
In the next subsection, we are going to find the analogue of the Green–Gauss formula (5.14) for the k-dimensional current associated to k-faces, restricted to D-cylinders. It will involve the distributional divergence of a less regular k-vector field (see below). 5.2.2. Divergence of the current of k-faces on model sets As a preliminary study, restrict again the attention to a D-cylinder as in Section 5.1.1, and keep the notation we had there. The k-faces, restricted to C k , define a k-vector field ξ = v1 ∧ · · · ∧ vk . In general, this vector field does not enjoy much regularity. Nevertheless, as a consequence of the study of Section 4, already exploited in Section 5.1 with a different formalism, one can find a representation of ∂(L n ∧ ξ ) like the one in a regular setting given by (5.14). This involves the density α of the push-forward with σ which was studied before, see (4.67) and (5.11). Lemma 5.9. Consider a function λ∈ L1 (C k ) such that it is continuously differentiable on each face and assume C k bounded. Then, the k-dimensional current (L n ∧ λξ ) is normal and the following formula holds ∂ L n ∧ λξ = −L n ∧ (div λξ )a.c. + H n−1
dC k
∧ d n, ˆ λξ ,
where dC k , nˆ are defined in (5.6), d nˆ is the differential 1-form at each point dual to the vector field n, ˆ and (div λξ )a.c. is defined here as (div λξ )a.c. :=
k (−1)i+1 (div λvi )a.c. v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk i=1
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where the functions (div λvi )a.c. are given in (5.7): ∂t α(t = πe1 ,...,ek (x), 0, x − ki=1 x · ei vi (x)) (div λvi )a.c. (x) = λ(x) i + ∂vi λ(x) 1C k (x). k α(πe1 ,...,ek (x), 0, x − i=1 x · ei vi (x)) Proof. Actually, reducing to computations in coordinates, this follows from Corollary 5.4 in Section 5.1.1. One has to verify the equality of the two currents on a basis. For simplicity, consider first ω = φ dx2 ∧ · · · ∧ dxk . with φ ∈ C1 (Rn ). Then dω = ∂x1 φ dx1 ∧ · · · ∧ dxk +
∂xi φ dxi ∧ dx2 ∧ · · · ∧ dxk ,
i=k+1
dω, ξ = ∇φ · v1
n
ω, (div λξ )a.c. = (div λv1 )a.c. φ,
ω, d n, ˆ ξ = φ nˆ · e1
and the thesis reduces exactly to Lemma 5.2 and Remark 5.3:
∂ L n ∧ λξ (ω)
:=
dω, λξ dL n Ck
Lemma 5.2
=
−
Ck
=:
ω, d n, ˆ λξ dH n−1
ω, (div λξ )a.c. dL + n
dC k
−L n ∧ (div λξ )a.c. + H n−1
dC k
∧ d n, ˆ λξ .
(5.15)
Consider then $i ∧ · · · ∧ dxk . ω = φ dx1 ∧ · · · ∧ dx The equality analogous to (5.15) can be deduced again by Lemma 5.2, as above, observing that the following formulas hold: dω, ξ = (−1)i+1 ∇φ · vi ,
ω, (div λξ )a.c. = (−1)i+1 (div λvi )a.c. φ, ω, d n, ˆ ξ = (−1)i+1 φ nˆ · ei .
Let us show the equality more in general. By a direct computation, one can verify that v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk =
k−1
h=0 k
i
i
k−1 sgn σ vσh+1 (h+1) . . . vσ (k−1) eσ (1)...σ (h)ih+1 ...ik−1 ,
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j
where vi is the j -th component of vi , S(1 . . . ıˆ . . . k) denotes the group of permutations of the integers {1, . . . , ıˆ, . . . , k}, with i missing, sgn σ is 1 if the permutation is even, −1 otherwise, and k−1 ij i ik−1 vσh+1 j =h+1 vσ (j ) , which is 1 if h + 1 > k − 1. (h+1) . . . vσ (k−1) denotes the product On the other hand, consider now a (k − 1)-form ω = φ dxi1 ...ih ∧ dxih+1 ...ik−1 , where 1 i1 < · · · < ih k, and k < ih+1 < · · · < ik−1 n. Then, again by direct computation, dω, ξ =
n (−1)jl +1 l=1
=
σ ∈S(1...k) σ (1)=i1 ,...,σ (h)=ih
i ik−1 ∂xl φ vlσ (jl ) vσh+1 (h+1) . . . vσ (k−1) i
i
k−1 (∇φ · vσ (1) ) sgn σ vσh+1 (h+2) . . . vσ (k) ,
σ ∈S(1...k) σ (2)=i1 ,...,σ (h+1)=ih
where jl ∈ N is such that, setting l = ijl , one has i1 < · · · < ijl < · · · < ik . Moreover,
ω, (div λξ )a.c. = φ
k (−1)i+1 (div λvi )a.c.
i
i
k−1 sgn σ vσh+1 (h+1) . . . vσ (k−1)
σ ∈S(1...ˆı ...k) σ (1)=i1 ,...,σ (h)=ih
i=1
=
i ik−1 φ · (div λvσ (1) )a.c. sgn σ vσh+1 (h+2) . . . vσ (k)
σ ∈S(1...k) σ (2)=i1 ,...,σ (h+1)=ih
and finally k (−1)i+1 (nˆ · ei )ω, v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk ω, d n, ˆ ξ = i=1
=
i
i
k−1 (φ nˆ · vσ (1) ) sgn σ vσh+1 (h+2) . . . vσ (k) .
σ ∈S(1...k) σ (2)=i1 ,...,σ (h+1)=ih
Therefore the thesis reduces to Corollary 5.4, being each vij constant on each face.
2
5.2.3. Divergence of the current of k-faces in the whole space In the previous section, we considered a k-dimensional current L n ∧ ξ identified by the restriction to a D-cylinder C k of the k-faces of f , projected on Rn . We established a formula analogous to (5.14) for the border of this current, which is representable by integration w.r.t. the measures L n C k and H n−1 dC k . In particular, when C k is bounded it is a normal current. Moreover, we have related the density of the absolutely continuous part to the function α by
k k i=1 x · ei vi (x)) i+1 ∂ti α(t = πe1 ,...,ek (x), 0, x − · (−1) (div ξ )a.c. = k α(πe1 ,...,ek (x), 0, i=1 x − x · ei vi (x)) i=1 1C k (x)v1 ∧ · · · ∧ vˆ i ∧ · · · ∧ vk .
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We observe now that the partition of Rn into the sets {F k }nk=1 and the remaining set that we call now F˜ 0 define an (n + 1)-uple of currents. The elements of this (n + 1)-uple are described by the following statement, which rephrases Corollary 5.6 in this setting. Corollary 5.10. Let {Ck }∈N be a countable partition of E k in D-cylinders as in Lemma 4.6 and, up to a refinement of the partition, assume moreover that the D-cylinders are bounded. Consider a k-vector field ξk ∈ L1 (Rn ; Λk Rn ) corresponding, at each point x ∈ E k , to the kplane D(x), and vanishing elsewhere. Assume moreover that it is continuously differentiable if k restricted to any set E∇f (x) , with locally integrable derivatives; more precisely, we assume that k 1 w +t 1 k n ξk ◦ σ (z) belongs to LH n−k (z) (Z ; C (C ; Λk R )) for each . Then, the k-dimensional current L n ∧ ξk is a locally flat chain, since it is the limit in the flat norm of normal currents: indeed, for k > 0 one has ∂ L n ∧ ξk = F- lim −L n ∧ div(1C k ξk ) a.c. + H n−1
i
dCik
∧ d nˆ i , ξk ,
i=1
where (div(1C k ξk ))a.c. is the one of Lemma 5.9, dCik and nˆ i are defined in formula (5.6), and i d nˆ i is the dual of nˆ i . Finally, notice that the current L n ∧ξk is itself locally normal if restricted to the interior of E k . However, in general E k can have empty interior. If ∂(L n ∧ ξk ) is representable by integration, then the density of its absolutely continuous part w.r.t. L n , at any point x ∈ Ck , is given by (div(1C k ξk ))a.c. (x).
6. Table of notations The following table collects some of the notations in the article. B(Rd ) Borel sets in Rd Ld d-dimensional Lebesgue measure d-dimensional Hausdorff outer measure Hd (X, Σ, μ) Σ = σ -algebra of subsets of X and μ = measure on Σ, i.e. μ : Σ → [0, +∞], μ(∅) = 0 and μ is countably additive on disjoint sets of Σ 1 (μ) (locally) integrable functions (w.r.t. μ) L(loc) ∞ L(loc) (locally) essentially bounded functions k C(c) k-times continuously differentiable functions (with compact support) 1A 1A (x) = 1 if x ∈ A, 1A (x) = 0 otherwise restriction of a measure μ to a set A μ A μ = μα dν disintegration of μ, see Definition 2.1 μν μ(A) = 0 whenever ν(A) = 0 (absolute continuity of a measure μ w.r.t. ν) equivalent μ is equivalent to ν if μ ν and ν μ separated two sets A and B sets are separated if each is disjoint from the other’s closure
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v·w · Sn−1 , Bn G(k, n) πL perpendicular ·,· v1 , . . . , vk aff(A) conv(A) dim(A) ri(C) rb(C) R-face extreme points ext(C) dom g graph g epi g ∇g ∂ −g g|a g|ab g|A f dom ∇f Im ∇f face of f k-face of f Fy Fyk Ey , Eyk Ek , F k P(x) R(x) T D Zk Zk
3659
Euclidean scalar product in Rn Euclidean norm in Rn {x ∈ Rn : x = 1}, {x ∈ Rn : x 1} Grassmaniann of k-dimensional vector spaces in Rn orthogonal projection from Rd to the affine plane L ⊂ Rd a set A is perpendicular to an affine plane H of Rd if ∃w ∈ H s.t. πH (A) = w pairing, see Section 5.2.1 linear span of vectors {v1 , . . . , vk } in Rn affine hull of A, the smallest affine plane containing A convex envelope of A, the smallest convex set containing A linear dimension of aff(A) relative interior of C, the interior of C w.r.t. the topology of aff(C) relative boundary of C, the boundary of C w.r.t. the topology of aff(C) see Definition 4.11 zero-dimensional R-faces extreme points of a convex set C the domain of a function g {(x, g(x)): x ∈ dom g} (graph) {(x, t): x ∈ dom g, t g(x)} (epigraph) gradient of g subdifferential of g, see p. 3612 evaluation of g at the point a the difference g(b) − g(a) the restriction of g to a subset A of dom g a fixed convex function Rn → R a fixed σ -compact set where f is differentiable, see Section 3.1 {∇f (x): x ∈ dom ∇f }, see Section 3.1 intersection of graph f|dom ∇f with a tangent hyperplane k-dimensional face of f ∇f −1 (y) = {x ∈ dom ∇f : ∇f (x) = y} Fy when dim(Fy ) = k, k = 0, . . . , n the sets, respectively, ri(Fy ) and ri(Fyk ) the sets, respectively, y Eyk and y Fyk see (4.10) F∇f (x) , for every x ∈ dom ∇f {x ∈ dom ∇f : R(x) = {x}} multivalued map of unit directions of the faces, see (4.11) sheaf set, see Definition 4.3 section of a sheaf set, see Definition 4.3
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[v, w] k [v , w ] Πi=1 i i
segment that connects v to w, i.e. {(1 − λ)v + λw: λ ∈ [0, 1]} k-dimensional rectangle in Rn with sides {[vi , wi ]}ki=1 , equal to the convex envelope of {vi , wi }ki=1 k-dimensional D-cylinder C k , see Definition 4.5 border of C k transversal to D and outer unit normal, see (5.6) a map which parameterizes a D-cylinder C k (Z k , C k ), see (4.28) σ te = σ 0+te , where e ∈ Sn−1 , t ∈ R if we write t = te with e a unit direction, then σ t = σ 0+te see (4.67) 1 (Rn ; Rn ), its divergence is the distribution C1 (Rn ) ϕ → − v · ∇ϕ if v ∈ Lloc c see p. 3647 and formula (5.12) see Definition 5.1 see formula (5.7)
C k (Z k , C k ) dC k , nˆ |dC k σ w+te σ te σt α(t, s, x) div v (div v)a.c. vi (div vi )a.c.
We avoid to recall here the notation on tensors and currents, which is the subject matter of Section 5.2.1. Acknowledgments The authors thank Professor Stefano Bianchini for having proposed the problem and for his unfailing support and guidance. References [1] G. Alberti, L. Ambrosio, A geometrical approach to monotone functions in R n , Math. Z. 230 (2) (1999) 259–316. [2] G. Alberti, L. Ambrosio, P. Cannarsa, On the singularities of convex functions, Manuscripta Math. 76 (3–4) (1992) 421–435. [3] G. Alberti, B. Kirchheim, D. Preiss, personal communication in [4]. [4] L. Ambrosio, B. Kirchheim, A. Pratelli, Existence of optimal transport maps for crystalline norms, Duke Math. J. 125 (2) (2004) 207–241. [5] L. Ambrosio, A. Pratelli, Existence and stability results in the L1 theory of optimal transportation, in: Optimal Transportation and Applications, Martina Franca, 2001, in: Lecture Notes in Math., vol. 1813, Springer-Verlag, Berlin, 2003, pp. 123–160. [6] S. Bianchini, L. Caravenna, On the extremality, uniqueness and optimality of transference plans, Bull. Inst. Math. Acad. Sin. (N.S.) 4 (4) (2009) 353–454. [7] S. Bianchini, M. Gloyer, On the Euler Lagrange equation for a variational problem: the general case II, Math. Z., doi:10.1007/s00209-009-0547-2, in press. [8] L. Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Z., doi:10.1007/s00209-010-0677-6, in press. [9] G. Ewald, D.G. Larman, C.A. Rogers, The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space, Mathematika 17 (1970) 1–20. [10] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969. [11] M. Feldman, R. McCann, Monge’s transport problem on a Riemannian manifold, Trans. Amer. Math. Soc. 354 (2002) 1667–1697. [12] J. Hoffmann-Jørgensen, Existence of conditional probabilities, Math. Scand. 28 (1971) 257–264. [13] V. Klee, M. Martin, Semicontinuity of the face-function of a convex set, Comment. Math. Helv. 46 (1971) 1–12. [14] D.G. Larman, A compact set of disjoint line segments in R3 whose end set has positive measure, Mathematika 18 (1971) 112–125. [15] D.G. Larman, On a conjecture of Klee and Martin for convex bodies, Proc. Lond. Math. Soc. (3) 23 (1971) 668–682.
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[16] D.G. Larman, C.A. Rogers, Increasing paths on the one-skeleton of a convex body and the directions of line segments on the boundary of a convex body, Proc. Lond. Math. Soc. (3) 23 (1971) 683–698. [17] F. Morgan, Geometric Measure Theory: A Beginner’s Guide, Academic Press Inc., 2000. [18] J.K. Pachl, Disintegration and compact measures, Math. Scand. 43 (1) (1978/1979) 157–168. [19] D. Pavlica, L. Zajíˇcek, On the directions of segments and r-dimensional balls on a convex surface, J. Convex Anal. 14 (1) (2007) 149–167. [20] R.T. Rockafellar, Convex Analysis, Princeton Math. Ser., vol. 28, Princeton University Press, Princeton, NJ, 1970. [21] V.N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math. 2 (1979) 1–178; Number in Russian series statements 141 (1976). [22] N.S. Trudinger, X.J. Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations 13 (2001) 19–31. [23] L. Zajíˇcek, On the points of multiplicity of monotone operators, Comment. Math. Univ. Carolin. 19 (1) (1978) 179–189.
Journal of Functional Analysis 258 (2010) 3662–3674 www.elsevier.com/locate/jfa
A note about proving non-Γ under a finite non-microstates free Fisher information assumption ✩ Yoann Dabrowski a,b,∗ a Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, United States b Laboratoire d’Informatique, Institut Gaspard Monge, Université Paris-Est, 5 bd Descartes,
Champs-sur-Marne F-77454, Marne-la-Vallée cedex 2, France Received 12 July 2009; accepted 8 February 2010 Available online 24 February 2010 Communicated by D. Voiculescu
Abstract We prove that if X1 , . . . , Xn (n > 1) are self-adjoints in a W ∗ -probability space with finite nonmicrostates free Fisher information, then the von Neumann algebra W ∗ (X1 , . . . , Xn ) they generate doesn’t have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy. © 2010 Elsevier Inc. All rights reserved. Keywords: Property Γ ; Free Fisher information; Non-microstates free entropy
0. Introduction In a fundamental series of papers, Voiculescu introduced analogs of entropy and Fisher information in the context of free probability theory. A first microstates free entropy χ(X1 , . . . , Xn ) is defined as a normalized limit of the volume of sets of microstates i.e. matricial approximants (in moments) of the n-tuple of self-adjoints Xi in a (tracial) W ∗ -probability space M. The study of this entropy proved useful for the understanding of the von Neumann algebras W ∗ (X1 , . . . , Xn ) ⊂ M generated by X1 , . . . , Xn , e.g. this leads to the absence of Cartan subalge✩
Research supported in part by NSF grant DMS-0555680.
* Address for correspondence: Department of Mathematics, University of California at Los Angeles, Los Angeles, CA
90095, United States. E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.010
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bras [16] and primeness [6] of free group factors or more generally von Neumann algebras generated by tuples with χ(X1 , . . . , Xn ) > −∞. Voiculescu also extended in [16] the well-known fact that free group factors doesn’t have property Γ to this class of von Neumann algebras generated by tuples with finite microstates free entropy. Starting from a definition recalled later of free Fisher information [17], Voiculescu also defined a non-microstates free entropy χ ∗ (X1 , . . . , Xn ) with up to now less applications to von Neumann algebras. This entropy is known by the fundamental work [1] to be greater than the previous microstates entropy, and believed to be equal (at least modulo Connes’ embedding conjecture), so that the question naturally arises of proving the above applications to von Neumann algebras for χ ∗ (X1 , . . . , Xn ) > −∞. For more details, we also refer the reader to the survey [18] for a list of properties as well as applications of free entropies in the theory of von Neumann algebras. The aim of this note is to prove the easiest result in that direction i.e. under the assumption that the free Fisher Information Φ ∗ (X1 , . . . , Xn ) < ∞ (an assumption stronger than χ ∗ (X1 , . . . , Xn ) > −∞ by a logarithmic Sobolev inequality of [17]), we intend to prove that W ∗ (X1 , . . . , Xn ) doesn’t have property Γ (especially is not amenable) (cf. [16] for the corresponding result in the case of microstates free entropy). Let us note that this especially implies that for any X1 , . . . , Xn , in a W ∗ -probability space, and S1 , . . . , Sn free semicircular elements free with X1 , . . . , Xn , then W ∗ (X1 + tS1 , . . . , Xn + tSn ) doesn’t have property Γ (a result not known, to the best of our knowledge, at least when W ∗ (X1 , . . . , Xn ) is not known to satisfy Connes’ embedding conjecture into an ultrapower of the hyperfinite II 1 factor). We will also prove factoriality under finiteness of non-microstates entropy, especially proving the same kind of degenerate convexity as the one of microstates entropy, i.e. all non-extremal states have χ ∗ (X1 , . . . , Xn ) = −∞. More precisely, let us recall that Φ ∗ (X1 , . . . , Xn ) is defined (in [17]) thanks to Hilbert– Schmidt-valued derivations, the so-called partial free difference quotients δi := ∂Xi : CX1 ,...,Xˆ i ,...,Xn : CX1 , . . . , Xn → H S L2 W ∗ (X1 , . . . , Xn ) , δi (Xj ) := δij 1 ⊗ 1, 2 ∗ H S L W (X1 , . . . , Xn ) L2 W ∗ (X1 , . . . , Xn ), τ ⊗ L2 W ∗ (X1 , . . . , Xn ), τ . If Φ ∗ (X1 , . . . , Xn ) = ni=1 δi∗ 1 ⊗ 1 22 < ∞, these derivations are closable, thanks to a result of Voiculescu. And, having in mind of proving first factoriality, if an element, say Z, of the center of W ∗ (X1 , . . . , Xn ) were in the domain of δi , we would write 0 = δi ([Z, Xj ]) = [δi (Z), Xj ] for j = i thanks to Leibniz rule and center property. And thus we would obtain that δi (Z), seen as a Hilbert–Schmidt operator, thus a compact operator, commutes with a diffuse operator, and thus is zero. A free Poincaré inequality (due to Voiculescu [19] and recalled later) would imply our result, that is Z is a scalar times the unit of the von Neumann algebra. At that point, we have thus to remove the domain assumption assumed valid on the element Z in the center. In Section 1, we prove factoriality under a slightly more general assumption for Fisher information relative to a subalgebra B. We will then, in Section 2, using a variant of Free Poincaré inequality and new boundedness results of (unbounded) dual systems, show our main result according to which W ∗ (X1 , . . . , Xn ) does not have property Γ . Let us mention that a previous preprint version of this paper used deeply the notion of L2 -rigidity introduced in [12] to get the same result under a supplementary non-amenability assumption. Here, we thus get non-amenability as a byproduct. Moreover, Section 3 applies the same tools to prove factoriality
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under finite non-microstates entropy. We also give a corresponding quantitative inequality in terms of one variant of non-microstates free entropy dimension. 1. Factoriality under finite Fisher information Let us fix some notations (close to those of [12]). We consider M a finite von Neumann algebra with normal faithful tracial state τ , and H an M–M-bimodule. D(δ) a weakly dense ∗-subalgebra of M. We suppose here that δ : D(δ) → H is a real closable derivation (real means δ(x), yδ(z) = δ(z∗ )y ∗ , δ(x ∗ )). = δ ∗ δ¯ the corresponding generator of a completely Dirichlet form, as proved in [14] (see this paper for the non-commutative definition of a Dirichlet form, here the Dirichlet form is E(x) = δ(x), δ(x), D(E) = D(1/2 ), completely means that ⊗ In is also the generator of a Dirichlet form on Mn (M)). Let us introduce a deformation of resolvent maps (a multiple of a so-called strongly continuous contraction resolvent, cf. e.g. [10] for the terminology) ηα = α(α + )−1 , which are unital, tracial (τ ◦ ηα = τ ), positive, completely positive maps, and moreover contractions on L2 (M, τ ) and normal contractions on M, such that
x − ηα (x) 2 x and x − ηα (x) 2 →α→∞ 0 (as recalled e.g. in Proposition 2.5 of [2]). We will also consider φt = e−t the semigroup of generator −. Let us recall two relations of the resolvent maps (see [10] for the first and [12] for the second, the integrals are understood as pointwise Riemann integral): ∞ ∀α > 0,
ηα = α
e−αt φt dt,
0
∀α > 0,
ζα := ηα1/2
=π
−1
∞ 0
t −1/2 ηα(1+t)/t dt. 1+t
¯ and Range(ηα1/2 ) = D(1/2 ) = D(δ) ¯ so that The point is that Range(ηα ) = D() ⊂ D(δ) ¯δ ◦ ζα is bounded (remark that this way to pre-compose with ηα1/2 to extend a map to the whole space is usual in classical Dirichlet form theory (especially in the relation with Malliavin calculus), in that way, for instance, the gradient operator of Malliavin calculus is extended to a distribution valued operator (after post-composition with another operator)). We now prove the first theorem of that note: Theorem 1. Let (M, τ ) a tracial W ∗ -probability space (i.e. M a von Neumann algebra with τ a faithful tracial normal state). Let (X1 , . . . , Xn ) an n-tuple (of self-adjoints, n 2) such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞, then W = W ∗ (X1 , . . . , Xn ) is a factor. Proof. Let δi = ∂Xi : CX1 ,...,Xˆ i ,...,Xn following the notation of Voiculescu for the non-commutative difference quotient. We see δi : CX1 , . . . , Xn → H S(L2 (W )) L2 (W, τ ) ⊗ L2 (W, τ ). First, thanks to a result of Voiculescu, Φ ∗ (X1 , . . . , Xn ) < ∞ implies that all the derivations δi are closable as unbounded operators L2 (W, τ ) → H S(L2 (W )) and they are even real closable derivations. But let us now fix i and consider Y ∈ CX1 , . . . , Xˆ i , . . . , Xn . By definition, we have δi Y = 0, so that if i = δi∗ δ¯i , we have especially i Y = 0 (and Y ∈ D(i )).
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Using a complete positivity argument (or an easy differential equation argument on the corresponding semigroup) one can easily show that ζα,i (ZY ) = ζα,i (Z)Y and ζα,i (Y Z) = Y ζα,i (Z). Thus, ζα,i ([Z, Y ]) = [ζα,i (Z), Y ], and if we note δ˜α,i = α −1/2 δi ◦ ζα (a bounded map as already noted), we have, using Leibniz rule and δ(Y ) = 0: δ˜α,i [Z, Y ] = δ˜α,i (Z), Y . Consequently, if Z is in the center of W ∗ (X1 , . . . , Xn ), we have proved [δ˜α,i (Z), Y ] = 0. But now, if Y = Xj (j = i), Y is diffuse (inasmuch as Φ ∗ (X1 , . . . , Xn ) < ∞ implies χ ∗ (X1 ) + · · · + χ ∗ (Xn ) χ ∗ (X1 , . . . , Xn ) > −∞, and if ξ ∈ L2 (W ) were an eigenvector of Xj with eigenvalue λ, the projector on ξ in B(L2 (W )) were not zero, implying the spectral projection 1Xj =λ to be not zero, and by faithfulness τ (1Xj =λ ) = 0 a contradiction, since χ ∗ (Xj ) > −∞ implies that the distribution of Xj has no point masses). But now, a Hilbert–Schmidt (thus compact) operator commuting with a diffuse one is zero (using the spectral theorem for compact operators, the diffuse one should have an eigenvector!). We have eventually proved δ˜α,i (Z) = 0 for all i (and all α > 0) as soon as Z is in the center of ∗ W (X1 , . . . , Xn ) and thus, by closability, knowing Z − ζα,i (Z) 2 →α→∞ 0, we obtain the fact that Z ∈ D(δ¯i ) and δ¯i (Z) = 0. Then, we conclude with the following lemma, due to Voiculescu (unpublished [19]). 2 Lemma 2 (Free Poincaré inequality). (See [19].) Consider δi the partial free difference quotient with respect to X1 , . . . , Xn , and Y a self-adjoint variable in the domain of all the operators δ¯i (as unbounded operators L2 (W ∗ (X1 , . . . , Xn )) → H S(L2 (W ∗ (X1 , . . . , Xn )))), then, there exists a positive constant C depending on the Xi but not on Y such that:
C
n j =1
δ¯j Y HS Y − τ (Y ) 2 .
We refer the reader to [19] for a proof, but we note that the key tool is the following remark, that for a polynomial Y = P (X1 , . . . , Xn ) ∈ CX1 , . . . , Xn , we verify immediately by linearity and monomial case that: n (δj P )(Xj ⊗ 1) − (1 ⊗ Xj )(δj P ) = P ⊗ 1 − 1 ⊗ P .
(1)
j =1
Remark 3. As Jesse Peterson pointed out to us, after reading an earlier version of this note, once we have shown δ¯1 (Z) = 0 (using only commutation with X2 , . . . , Xn ), we can conclude by writing down 0 = δ¯1 ([Z, X1 ]) = [δ¯1 (Z), X1 ] + [Z, 1 ⊗ 1] = [Z, 1 ⊗ 1] and conclude taking the
. 2 norm. (We have used our original proof inasmuch as a variant of free Poincaré inequality will be essential in the next part. But somehow, the following result is the only one not provable under weakened assumptions in what follows. The counterpart of the powerfulness of free Poincaré like technique being its non-applicability in the case Φ ∗ (X : B) < ∞, up to now.) We have thus also proved the following result:
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Theorem 4. Let X a self-adjoint element in M a tracial W ∗ -probability space, and B a subalgebra of M, algebraically free with X and containing a diffuse element. Suppose moreover that the free Fisher information of X relative to B: Φ ∗ (X : B) < ∞, then W ∗ (X, B) is a factor. Let us end this part with a corollary. Consider, e.g. as in [7], the full (universal) free-product C ∗ -algebra C([−R, R])N and note T the space of tracial states on this C ∗ -algebra. It is (elementarily known to be) a compact convex set for the weak-∗ topology. It is moreover known by the reduction theory for von Neumann algebras that this is a Choquet simplex. It is known (see the beginning of the proof of Theorem 3.1.18 of [13] using mainly Proposition 3.1.10 and the discussion before the theorem) that factorial states (i.e. states for which bicommutants in the GNS construction give factors) are exactly extreme points of this convex set. We will show that T is a Poulsen simplex (see [9]), i.e. a metrizable Choquet simplex in which the extreme points form a dense set. Since, as a weak-∗ compact of the dual of a separable space, T is metrizable, we have merely to prove the last statement about density of the set of extreme points. Let us prove this in the following: Corollary 5. The set of tracial states T on C = C([−R, R])N is a Poulsen simplex. Proof. To conclude the proof, consider thus a tracial state τ on C, consider Xi , the function f0 (t) = t in the i-th copy of C([−R, R]), the usual self-adjoint generators (as a C ∗ -algebra) of C. Let W the weak closure of πτ (C), the GNS construction associated to τ , we always note τ the associated (faithful normal tracial) state on W . We can consider the von Neumann algebra M generated by W and a free family of semicircular elements M = W ∗ (W, {Si }), and get another faithful tracial state on M (the last one noted τ , cf. [5] for faithfulness). Consider ∗ i +tSi ) Yi,t = R(X R+2t . By Corollary 3.9 in [17], Φ (Y1,t , . . . , YN,t ) < ∞ and thus by our Theorem 1, ∗ W (Y1,t , . . . , YN,t ) ⊂ M is a factor. But since Yi,t R, we have a ∗-homomorphism C → M sending Xi to Yi,t (e.g. Proposition 2.1 in [7]). This defines by composition with the state on M, a state τt on C. Since the state considered on M is faithful, the kernel of the ∗-homomorphism is nothing but the ideal of elements with τt (Z ∗ Z) = 0 by which we quotient C in the GNS construction for τt on C, we thus get a ∗-isomorphism, from this quotient on its image which preserves the trace, and thus, L2 (W ∗ (Y1,t , . . . , YN,t ), τ ) is isomorphic to L2 (C, τt ) (by the induced map). And a step further we get the ∗-isomorphism between W ∗ (Y1,t , . . . , YN,t ) and πτt (C). Thus, τt is a factorial, thus an extremal tracial state in T . Now, to get weak-∗ convergence of τ1/n to τ in T , and thus the claimed density, we have merely to consider convergence on monomials on which a usual trick shows the concluding inequality:
τ (Xi . . . Xi ) − τt (Xi . . . Xi ) = τ (Xi . . . Xi ) − τ (Yi ,t . . . Yi ,t )
p p p p 1 1 1 1 R(Xi + tSi ) − X R p−1 p sup i R + 2t i R p−1 p
4Rt . R + 2t
2
2. Non-Γ In the preceding part, we used the semigroup and resolvent maps associated to a derivation δi = ∂Xi : CX1 ,...,Xˆ i ,...,Xn . The drawback is that, if we doesn’t have exactly a commutator equal to
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zero, as this is the case when we want to prove non-Γ , we cannot move the estimate on the commutator (giving an estimate on the Hilbert–Schmidt operator), to an estimate on Z − τ (Z) using something like free Poincaré inequality, inasmuch as we have not the same resolvent maps for different derivations δi . We have thus searched to move (somehow) the preceding reasoning in case we consider δ := (δ1 , . . . , δn ), the resolvent map associated to it ηα (Z), and then δi ◦ ηα (Z). This was our approach in a previous version of this paper where we assumed non-amenability and used then an L2 -rigidity technique to conclude. However, working a little bit more from the following variant of free Poincaré inequality will be much more efficient, enabling us to prove non-Γ without any other assumption, and thus proving non-amenability instead of assuming it. 2.1. Two preliminaries First in order to get our inequality, we will use the following general result about derivations in von Neumann algebras, which can be thought of as a “Kaplansky’s density theorem for derivations” (the proof also confirms this), which is of independent interest and really likely known to specialists but for which we have not found any reference. Proposition 6. Let δ a symmetric derivation defined on a weakly dense ∗-algebra D(δ) of the tracial W ∗ -probability space (M, τ ), closable as an operator D(δ) ⊂ L2 (M, τ ) → H, H an involutive M–M Hilbert W ∗ -bimodule (with isometric involution, as usual we assume σ -weak continuity of both actions). Then the following properties are equivalent: ¯ ∩ M is a ∗-algebra on which δ| ¯ D(δ)∩M (i) D(δ) is a (symmetric) derivation. ¯ ¯ (ii) For any Z ∈ D(δ) ∩ M, there exists a sequence Zn ∈ D(δ) with Zn Z , Zn − Z 2 , ¯
δ(Zn ) − δ(Z)
2 → 0. Proof. The fact that (ii) implies (i) is clear since taking Zn , Yn for Z and Y , as in (ii), Zn Yn − ZY 2 → 0, and for any ξ ∈ H, we get successively ξ(Yn∗ − Y ∗ ) H → 0, by coincidence of L2 and σ -∗-strong topologies on bounded sets in M and σ -weak (thus σ -∗-strong) continuity of the action, and thus δ(Zn ), ξ Yn∗ → δ(Z), ξ Y ∗ , which gives at the end weak convergence of ¯ ) + δ(Z)Y ¯ ¯ we get ZY ∈ D(δ) ¯ ∩M , and by (weak) closability of the graph of δ, δ(Zn Yn ) to Z δ(Y with the derivation property. The proof of the converse follows verbatim the proof of Kaplansky’s density theorem. We may first assume that δ : D(δ) → L2 (M) is closed as a derivation M → L2 (M), since it is closable and then obtaining Zn in this enlarged D(δ) is harmless. We can also assume Z 1. Consider X = (1 + (1 − ZZ ∗ )1/2 )−1 Z ∈ M, then X ∈ D(δ) by closability of any closed derivation on a C ∗ -algebra by C 1 -functional calculus. Look at f (x) = 2(1 + xx ∗ )−1 x = 2x(1 + x ∗ x)−1 so that ¯ n − X) 2 → 0. Then consider Zn = Z = f (X). Take Xn ∈ D(δ) converging to X in L2 with δ(X 2 f (Xn ) so that Zn 1 and Zn converge to Z in L (even if this is not a consequence of ∗-strong continuity of f since we don’t know whether Xn converge to X ∗-strongly since it is not bounded as a sequence in M, the proof is however standard, like in the proof of Kaplansky’s density ¯ D(δ)∩M theorem, see bellow for an example for the derivative of f ). Since, by hypothesis, δ and δ| ¯ are derivations, closed seen as derivation M → H, we get δ(Zn ) = 2δ(Xn )(1 + Xn∗ Xn )−1 − ¯ using appropriate series 2Xn (1 + Xn∗ Xn )−1 δ(Xn∗ Xn )(1 + Xn∗ Xn )−1 and the analog for δ(Z), expansions. Now the boundedness as sequences in M of Xn (1 + Xn∗ Xn )−1 Xn∗ , (1 + Xn∗ Xn )−1 ¯ and Xn (1 + Xn∗ Xn )−1 shows that it suffices to show the convergence of 2δ(X)(1 + Xn∗ Xn )−1 −
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¯ ∗ )Xn + Xn∗ δ(X))(1 ¯ 2Xn (1 + Xn∗ Xn )−1 (δ(X + Xn∗ Xn )−1 . Likewise, by coincidence of L2 and σ -∗-strong topologies on bounded sets in M and σ -∗-strong continuity of the action, it suffices to show convergence in L2 (M) of Xn (1 + Xn∗ Xn )−1 Xn∗ , (1 + Xn∗ Xn )−1 and Xn (1 + Xn∗ Xn )−1 . Let us for instance prove the first one, let us write: −1 −1 Xn 1 + Xn∗ Xn Xn∗ − X 1 + X ∗ X X ∗ −1 ∗ −1 ∗ Xn − X ∗ + Xn 1 + Xn∗ Xn X − Xn∗ X = Xn 1 + Xn∗ Xn −1 −1 + Xn∗ (X − Xn ) 1 + X ∗ X X ∗ + (Xn − X) 1 + X ∗ X X ∗ . Since for each term, both sides of X − Xn or its adjoint are bounded by functional calculus, this concludes. 2 Remark 7. Let us note that in order to apply the previous proposition to the free difference quotient in case of finite Fisher information, we prove (i) using Proposition 3.4 in [4] to get ¯ ∩ M is an algebra (knowing that D(δ) ¯ is the domain of a Dirichlet form, as already recalled, D(δ) thanks to [14]), and then, for instance use the formula for δ ∗ given by Corollary 4.3 in [17] to show δ¯ is closable as an operator valued in L1 (M ⊗ M), and we prove there the derivation property, deducing it for the derivation valued in L2 as a consequence. Second, we recall for the reader convenience some results about bounded and unbounded dual systems in the sense of Voiculescu and Shlyakhtenko respectively. Even if we will not use their results explicitly, this will enable to express some assumptions and results in terms of these standard objects. Let us recall the following result of [15] (deduced from Theorem 1 and its proof), δi the i-th partial difference quotient as earlier. Proposition 8. (See [15].) δi∗ 1 ⊗ 1 exists (in L2 (W ), W = W ∗ (X1 , . . . , Xn )) if and only if there exists a closable unbounded operator Yi : L2 (W ) → L2 (W ) with CX1 , . . . , Xn ⊂ D(Yi ), Yi 1 = 0, 1 ∈ D(Yi∗ ) such that [Yi , Xj ] = δi (Xj ). Moreover, necessarily such a Yi = 1 ⊗ τ ◦ δi (or is an extension of it beyond CX1 , . . . , Xn ). Then Corollary 1 of the same paper noticed that Y˜i = 12 (Yi − Yi∗ ) is an anti-symmetric closable dual system. Moreover the proof of Theorem 1 also shows that Yi∗ X = Xδi∗ 1 ⊗ 1 − Yi X (also a consequence of Corollary 4.3 in [17] in the free difference quotient case we are interested in here), so that Y˜i (X) = Yi (X) − 12 Xδi∗ 1 ⊗ 1 i.e. Y˜i 1 = − 12 δi∗ 1 ⊗ 1. Moreover, it is easily seen that such a Y˜i gives in inverting the above relation to get a Yi similar to the one in the previous proposition, we will thus be later interested in bounded dual systems in the sense of Voiculescu verifying this relation for the specific relation they have with the canonical dual system of the previous proposition (for which we will get latter e.g. nice boundedness properties). 2.2. A mixed Poincaré-non-Γ (in)equality Our main tool will be a lemma based on the same argument as free Poincaré inequality. After proving it, we develop several consequences under (more or less) stronger assumptions for further use.
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Lemma 9. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of selfadjoints, n 2 in order to have a non-trivial result) such that the microstates free Fisher ¯ (δ the free difference quoinformation Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ W ∗ (X1 , . . . , Xn ) ∩ D(δ) tient), then we have the following equality: n 2 2(n − 1) Z − τ (Z) 2 = [Z, Xi ], Z, (Xi ) i=1
+ 2 (τ ⊗ 1 − 1 ⊗ τ ) δ¯i (Z) , [Z, Xi ] .
Proof. It suffices to show the result for Z ∈ CX1 , . . . , Xn (using Proposition 6). Inasmuch as δi is a derivation, we have δi [Z, Xi ] = [δi (Z), Xi ] + [Z, 1 ⊗ 1] and we have already noticed that [Z, 1 ⊗ 1] 2HS = 2 Z − τ (Z) 22 . Everything will be based on the equality on which is based free Poincaré inequality. Let us compute [Z, 1 ⊗ 1] 2HS = δi [Z, Xi ] − [δi (Z), Xi ], [Z, 1 ⊗ 1]: δi (Z), Xi , [Z, 1 ⊗ 1] = δi (Z), [Z, 1 ⊗ 1], Xi = δi (Z), Z, [1 ⊗ 1, Xi ] − δi (Z), 1 ⊗ 1, [Z, Xi ] . At that point, we notice that: Z, [1 ⊗ 1, Xi ] = [Z, 1 ⊗ Xi − Xi ⊗ 1] = Z ⊗ Xi − ZXi ⊗ 1 − 1 ⊗ Xi Z + Xi ⊗ Z = (1 ⊗ Xi )[Z, 1 ⊗ 1] − [Z, 1 ⊗ 1](Xi ⊗ 1). But now, we have in fact written an “inner” commutant of Xi and [Z, 1 ⊗ 1] (i.e. a commutant with the action of the von Neumann algebra on M ⊗ M on the side of the tensor product not on the outer side, remark that the preceding equation is just commutation of the two actions after writing [1 ⊗ 1, Xi ] in terms of an inner commutant). We will merely now use that the scalar product of Hilbert Schmidt operators is compatible with this inner commutant (which is nothing but an extension of traciality of τ ⊗ τ on M ⊗ M):
n δi (Z), Z, [1 ⊗ 1, Xi ] = (1 ⊗ Xi )δi (Z) − δi (Z)(Xi ⊗ 1), [Z, 1 ⊗ 1]
n i=1
i=1
= (1 ⊗ Z − Z ⊗ 1), [Z, 1 ⊗ 1] 2 = − [Z, 1 ⊗ 1] .
We have used Eq. (1) on which is based the proof of free Poincaré inequality. Thus, we have obtained: n n 2 δi (Z), Xi , [Z, 1 ⊗ 1] = − [Z, 1 ⊗ 1] − δi (Z), 1 ⊗ 1, [Z, Xi ] . i=1
i=1
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We have now to compute δi [Z, Xi ] , [Z, 1 ⊗ 1] = Z ∗ , δi [Z, Xi ] , 1 ⊗ 1 = δi Z ∗ , [Z, Xi ] , 1 ⊗ 1 − δi Z ∗ , [Z, Xi ] , 1 ⊗ 1 = [Z, Xi ], Z, (Xi ) + δi Z ∗ , 1 ⊗ 1, Z ∗ , Xi . We can now conclude using that 1 ⊗ τ (δi (Z ∗ )) = τ ⊗ 1(δi (Z))∗ : n 2 2 n [Z, 1 ⊗ 1] HS = [Z, 1 ⊗ 1] + [Z, Xi ], Z, (Xi ) i=1 n +2 δi (Z), 1 ⊗ 1, [Z, Xi ] .
2
i=1
For our purpose, the following lemma is only an intermediary step to the next lemma, but, as the remark after it shows, it can have an independent interest. Lemma 10. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple of n 2 self-adjoints such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ ¯ then the following inequality holds: W ∗ (X1 , . . . , Xn ) ∩ D(δ),
(1 ⊗ τ ) δ¯i (Z) − Z(Xi ) 2 Z(Xi ) 2 + δ¯i Z ∗ Z , 1 ⊗ (Xi ) . 2
2
Thus, if we assume moreover we have second order conjugate variables J2,j = J2 (Xj : CX1 , . . . , Xˆ j , . . . , Xn ) (in L1 (M, τ ), as defined in [17, Definition 3.1]). Then the following inequality holds:
(1 ⊗ τ ) δ¯i (Z) 2 Z(Xi ) + Z ∗ Z, J2,i 1/2 . 2 2 As a consequence, (1 ⊗ τ ) ◦ δ¯i extends as a bounded map M → L2 (M, τ ). Remark 11. If we assume moreover we have bounded first and second order conjugate variables (i.e. (Xi ), J2,i ∈ M or more generally bounded conjugate variable and dual system Y˜i in the sense of Voiculescu), then the previous lemma shows that (1 ⊗ τ ) ◦ δ¯i extends as a bounded map on L2 (M) and moreover the inequality above implies that X1 , . . . , Xn is a non-Γ ¯ in the sense of [11]. As a consequence of Corollary 3.3 in [11] this shows that set (for D(δ)) ∗ W (X1 , . . . , Xn ) doesn’t have property (T). A study under less restrictive assumptions will need new investigations, but we can already note (using well-known results of [17]) that this implies that for any X1 , . . . , Xn , if S1 , . . . , Sn is a free semicircular system free with X1 , . . . , Xn , then W ∗ (X1 + S1 , . . . , Xn + Sn ) doesn’t have property (T) (for any > 0). This result was proved in [8] assuming moreover W ∗ (X1 , . . . , Xn ) embeddable in R ω . Proof. The only non-trivial statement is the first one (using the previous lemma for proving consequences). Moreover we can assume Z ∈ CX1 , . . . , Xn as usual (take the limit in the fifth
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¯ ∩ M). Let us compute (using line below using Proposition 6 and then compute with Z ∈ D(δ) the formula for δi∗ , Corollary 4.3 in [17], and coassociativity in the third line): (1 ⊗ τ ) δ¯i (Z) 2 = (1 ⊗ τ ) δ¯i (Z) ⊗ 1, δ¯i (Z) 2 = (1 ⊗ τ ) δ¯i (Z) (Xi ), Z − (1 ⊗ τ )δ¯i (1 ⊗ τ ) δ¯i (Z) , Z = (1 ⊗ τ ) δ¯i (Z) (Xi ), Z − (1 ⊗ τ ⊗ τ ) 1 ⊗ δ¯i ◦ δ¯i (Z) , Z = (1 ⊗ τ ) δ¯i (Z) (Xi ) − (1 ⊗ τ ) δ¯i (Z)(Xi ) , Z = δ¯i (Z), Z (Xi ), 1 ⊗ 1 = δ¯i (Z), Z(Xi ) ⊗ 1 − δ¯i Z ∗ Z − δ¯i Z ∗ Z, 1 ⊗ (Xi ) = δ¯i (Z), Z(Xi ) ⊗ 1 + δ¯i Z ∗ , 1 ⊗ (Xi )Z ∗ − δ¯i Z ∗ Z , 1 ⊗ (Xi ) . Now note we can use (1 ⊗ τ )(δ¯i (Z))∗ = (τ ⊗ 1)(δ¯i (Z ∗ )) (using δ¯i is a real derivation), to conclude: (1 ⊗ τ ) δ¯i (Z) − Z(Xi ) 2 = Z(Xi ) 2 − δ¯i Z ∗ Z , 1 ⊗ (Xi ) . 2
2
2
Lemma 12. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple of n 2 self-adjoints such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞. Let Z ∈ ¯ a self-adjoint or a unitary, then we have the following inequality: W ∗ (X1 , . . . , Xn ) ∩ D(δ) (1 ⊗ τ ) δ¯i (Z) − Z(Xi ) (Xi ) Z . 2 2 As a consequence, (1 ⊗ τ ) ◦ δ¯i extends as a bounded map M → L2 (M, τ ) and: n 2 1 (n − 1) Z − τ (Z) 2 − [Z, Xi ], Z, (Xi ) +2 [Z, Xi ] 2 (Xi ) 2 Z . 2 i=1
Proof. Take Z of norm less than 1 ( Z < 1). If it is self-adjoint, we can write Z as a half ¯ using stability by C 1 functional calculus (e.g. sum of two unitaries in W ∗ (X1 , . . . , Xn ) ∩ D(δ) Lemma 7.2 in [2]), we have only to prove the inequality for any unitary U . This follows at once from the previous lemma. The second statement is a direct consequence. 2 2.3. The main result Now Lemma 12 contains immediately the non-Γ result we wanted: Theorem 13. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of self-adjoints, n 2) such that the microstates free Fisher information Φ ∗ (X1 , . . . , Xn ) < ∞, then W ∗ (X1 , . . . , Xn ) doesn’t have property Γ , i.e. all central sequences Zm (i.e. bounded in W ∗ (X1 , . . . , Xn ) and such that ∀Y ∈ W ∗ (X1 , . . . , Xn ) [Zm , Y ] 2 → 0) are trivial: Zm − τ (Zm ) 2 → 0. As a consequence, W ∗ (X1 , . . . , Xn ) is not amenable.
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3. Factoriality under finite non-microstates entropy We will now prove factoriality under the weaker assumption χ ∗ (X1 , . . . , Xn ) > −∞ as another consequence of Lemma 12. In this part, we thus let X1 , . . . , Xn , n 2, √ self-adjoints, S1 , . . . , Sn , a free semicircular system, free with X1 , . . . , Xn . Let Yjt = Xj + tSj and Et the (trace preserving) conditional expectation onto W ∗ (Y1t , . . . , Ynt ) (seen as a sub-von Neumann algebra of W ∗ ({Xi , Si })). Then recall that the non-microstates entropy is defined by the following integral:
1 χ (X1 , . . . , Xn ) = 2 ∗
∞ 0
√ √ n n ∗ − Φ (X1 + tS1 , . . . , Xn + tSn ) dt + log 2πe. 1+t 2
√ √ It is readily seen that if lim inft→0 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ) = 0 we have necessarily χ ∗ (X1 , . . . , Xn ) = −∞, we will thus use the assumption in that way. More generally, a variant √ of (X , . . . , X ) = n − lim inf ∗ (X + tS , free entropy dimension was defined in [3] by δ tΦ 1 n t→0 1 1 √ . . . , Xn + tSn ), we will thus express our result in function of this entropy dimension. We can now prove our claimed result: Theorem 14. Let (M, τ ) a tracial W ∗ -probability space. Let (X1 , . . . , Xn ) an n-tuple (of selfadjoints, n 2) then the following inequality holds for any central self-adjoint Z: Z − τ (Z) 2 2 n − δ (X1 , . . . , Xn ) Z 2 . 2 n−1
√ √ As a consequence, if n − δ ∗ (X1 , . . . , Xn ) := lim inft→0 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ) = 0, then W ∗ (X1 , . . . , Xn ) is a factor. ∗ As another example, if δ ∗ (X1 , . . . , Xn ) > n+1 2 , W (X1 , . . . , Xn ) has no central projection of trace one half, especially, doesn’t have diffuse center. Proof. Let Z in the center of W ∗ (X1 , . . . , Xn ), with Z 1 and apply Lemma 12 to Et (Z) ∈ W ∗ (Y1t , . . . , Ynt ) to get: 2 (n − 1) Et (Z) − τ Et (Z) 2
n
−
1 Et (Z), Yit , Et (Z), Yit +2 Et (Z), Yit 2 Yit 2 Z
2
−
1 1 1 Et Z, Yit , Et (Z), √ Et (Si ) + 2 Et Z, Yit 2 (S ) E √ t i Z
2 t t 2
−
1 Et [Z, Si ] , Et (Z), Et (Si ) +2 Et [Z, Si ] 2 Et (Si ) 2 Z , 2
i=1
=
n i=1
=
n i=1
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2 (n − 1) Et (Z) − τ Et (Z) 2
n
−
i=1
1 Et Z − Et (Z), Si , Et (Z), Et (Si ) 2
1 Et (Z), Et (Si ) , Et (Z), Et (Si ) 2 + 2 Et Z − Et (Z), Si 2 + Et (Z), Et (Si ) 2 Et (Si ) 2 Z
−
n i=1
2 1 12 Z − Et (Z) 2 Et (Si ) 2 Z − Et (Z), Et (Si ) 2 2
2 2 + Et (Z), Et (Si ) 2 + Et (Si ) 2 Z
n 1 Et (Z), Et (Si ) 2 + Et (Si ) Z 2 12n Z − Et (Z) 2 Z + 2 2 2 i=1
n Et (Si ) 2 . 12n Z − Et (Z) 2 Z + 2 Z 2 2 i=1
We used at the second line the result of [17] about the conjugate variable in the algebra generated by Yit : (Yit ) = √1t Et (Si ). We also used conditional expectation property and then at line 3 that Z commutes with Xi . In the fourth line we used Z = Z − Et (Z) + Et (Z) and then we only compute using Si = 2, Si 2 = 1 and arithmetico geometric inequality. At the end, we thus get using the definition of free Fisher information and the result of [17] above: 2 (n − 1) Et (Z) − τ Et (Z) 2 12n Z − Et (Z) 2 Z
√ √ + 2 Z 2 tΦ ∗ (X1 + tS1 , . . . , Xn + tSn ). It is thus sufficient to notice that Et (Z) − Z 2 goes to 0 with t to get the inequality stated by taking a lim inf. Et (Z) − Z 2 → 0 follows from Kaplansky’s density theorem, and from the remark that for P a non-commutative polynomial Et (P (X1 , . . . , Xn )) − P (X1 , . . . , Xn ) 2
P (Y1t , . . . , Ynt ) − P (X1 , . . . , Xn ) 2 . The consequences are trivial: for instance for the second, apply the inequality to Z = 1 − 2P the corresponding central self-adjoint unitary of trace 0 if P a central projection of trace 1/2. 2 Acknowledgments The author would like to thank Professor Dan Voiculescu for allowing him to use the argument of his free Poincaré inequality in another context, Professor Dimitri Shlyakhtenko for showing him references [14,2,12], and Professor Jesse Peterson for pointing out to his attention his result Corollary 3.3 in [11]. The author would also like to thank D. Shlyakhtenko, J. Peterson and P. Biane for useful discussions and comments.
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References [1] P. Biane, A. Guionnet, M. Capitaine, Large deviation bounds for matrix brownian motion, Invent. Math. 152 (2003) 433–459. [2] F. Cipriani, J.-L. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Anal. 201 (2003) 78–120. [3] A. Connes, D. Shlyakhtenko, L2 -homology for von Neumann algebras, J. Reine Angew. Math. 586 (2005) 125–168. [4] E. Davis, J.M. Lindsay, Non-commutative symmetric Markov semigroups, Math. Z. 210 (1992) 379–411. [5] K. Dykema, D. Voiculescu, A. Nica, Free Random Variables, CRM Monogr. Ser., AMS, 1992. [6] L. Ge, Applications of free entropy to finite von Neumann algebras, II, Ann. of Math. 147 (1998) 143–157. [7] F. Hiaï, Free analog of pressure and its Legendre transform, Comm. Math. Phys. 255 (1) (2005) 229–252. [8] K. Jung, D. Shlyakhtenko, All generating sets of all property T von Neumann algebras have free entropy dimension 1, preprint, arXiv:math.OA/0603669, 2006. [9] J. Lindenstrauss, G. Olsen, Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1) (1978) 91–114. [10] Z. Ma, M. Röckner, Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, Berlin, 1992. [11] J. Peterson, A 1-cohomology characterization of property (T) in von Neumann algebras, preprint, arXiv:math.OA/ 0409527, 2004. [12] J. Peterson, L2 -rigidity in von Neumann algebras, preprint, arXiv:math.OA/0605033, 2006. [13] S. Sakai, C ∗ -Algebras and W ∗ -Algebras, Springer, 1971. [14] J.-L. Sauvageot, Strong Feller semigroups on C ∗ -algebras, J. Operator Theory 42 (1999) 83–102–120. [15] D. Shlyakhtenko, Remarks on free entropy dimension, arXiv:math.OA/0504062, 2005. [16] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, III: The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1) (1996) 172–199. [17] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, V: Non commutative Hilbert Transforms, Invent. Math. 132 (1998) 189–227. [18] D. Voiculescu, Free entropy, Bull. Lond. Math. Soc. 34 (3) (2002) 257–278. [19] D. Voiculescu, A free Poincaré inequality.
Journal of Functional Analysis 258 (2010) 3675–3724 www.elsevier.com/locate/jfa
Functions of operators under perturbations of class Sp ✩ A.B. Aleksandrov a , V.V. Peller b,∗ a St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Received 29 August 2009; accepted 16 February 2010 Available online 26 February 2010 Communicated by N. Kalton
Abstract This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα (R) and 1 < p < ∞, the operator f (A) − f (B) belongs to S p/α , whenever A and B are self-adjoint operators with A − B ∈ S p . We also obtain sharp estimates for the Schatten–von Neumann norms f (A) − f (B)S p/α in terms of A − BS p and establish similar results for other operator ideals. We also estimate Schatten– m−j m f (A + j K). We prove that analogous von Neumann norms of higher order differences m j j =0 (−1) results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) − f (B) to belong to S q under the assumption that A − B ∈ S p . We also obtain Schatten–von Neumann estimates for quasicommutators f (A)R − Rf (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A − K) − 2f (A) + f (A + K). © 2010 Elsevier Inc. All rights reserved. Keywords: Operator ideals; Schatten–von Neumann classes; Self-adjoint operators; Unitary operators; Contractions; Perturbations; Functions of operators
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3676 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3679
✩
The first author is partially supported by RFBR grant 08-01-00358-a and by Russian Federation presidential grant NSh-2409.2008.1; the second author is partially supported by NSF grant DMS 0700995 and by ARC grant. * Corresponding author. E-mail address:
[email protected] (V.V. Peller). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.011
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3. Ideals of operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Multiple operator integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Self-adjoint operators. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . 6. Unitary operators. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . 7. The case of contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Finite rank perturbations and necessary conditions. Unitary operators . . . . 9. Finite rank perturbations and necessary conditions. Self-adjoint operators . 10. Spectral shift function for second order differences . . . . . . . . . . . . . . . . 11. Commutators and quasicommutators . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction This paper is a continuation of our paper [2]. In [2] we obtained sharp estimates for the norms of f (A) − f (B) in terms of the norm of A − B for various classes of functions f . Here A and B are self-adjoint operators on Hilbert space and f is a function on the real line R. We also obtained in [2] sharp estimates for the norms of higher order differences m m def K f (A) = (−1)m−j j =0
m f (A + j K), j
(1.1)
where A and K are self-adjoint operators. Similar results were obtained in [2] for functions of unitary operators and for functions of contractions. In this paper we are going to obtain sharp estimates for the Schatten–von Neumann norms of first order differences f (A) − f (B) and higher order differences (m K f )(A) for functions f that belong to a Hölder–Zygmund class Λα (R), 0 < α < ∞, (see Section 2 for the definition of these spaces). In particular we study the question, under which conditions on f the operator f (A) − f (B) (or (m K f )(A)) belongs to the Schatten–von Neumann class S q , whenever A − B (or K) belongs to S p . We also obtain related results for more general ideals of operators on Hilbert space (see Section 3 for the introduction to operator ideals on Hilbert space). In connection with the Lifshits–Krein trace formula, M.G. Krein asked in [16] the question whether f (A) − f (B) ∈ S 1 , whenever f is a Lipschitz function (i.e., |f (x) − f (y)| const |x − y|, x, y ∈ R) and A − B ∈ S 1 . Functions f satisfying this property are called trace class perturbations preserving. Farforovskaya constructed in [13] an example that shows that the answer to the Krein question is negative. Later in [27] and [29] necessary conditions and sufficient conditions for f to be trace class perturbations preserving were found. It was shown in [27] and [29] that if f belongs to the Besov 1 (R) (see Section 2), then f is trace class perturbations preserving. On the other hand, space B∞1 it was shown in [27] that if f is trace class perturbations preserving, then it belongs to the Besov space B11 (R) locally. This necessary condition also proves that a Lipschitz function does not have to be trace class perturbations preserving. Moreover, in [27] and [29] a stronger necessary
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condition was also found. Note that a function is trace class perturbations preserving if and only if it is operator Lipschitz (see [27] and [18]). We also mention here the paper [28], in which analogs of the above results were obtained for perturbations of class S p with p ∈ (0, 1). On the other hand, Birman and Solomyak in [9] proved that a Lipschitz function f must preserve Hilbert–Schmidt class perturbations: f (A) − f (B) ∈ S 2 , whenever A − B ∈ S 2 and f (A) − f (B)
S2
sup x=y
|f (x) − f (y)| A − BS 2 . |x − y|
To prove that result, Birman and Solomyak developed in [7,8], and [9] their beautiful theory of double operator integrals and established a formula for f (A) − f (B) in terms of double operator integrals (see Section 4). Note also that the paper [18] studies functions that preserve perturbations belonging to operator ideals. We mention here two recent results. In [22] it was proved that if f is a Lipschitz function and rank(A − B) < ∞, then f (A) − f (B) belongs the weak space S 1,∞ (see Section 3 for the definition). It was also shown in [22] that if A − B ∈ S 1 , then f (A) − f (B) belongs to the ideal S Ω , i.e., n sj f (A) − f (B) const log(2 + n) j =0
(here sj is the j th singular value). This allowed the authors of [22] to deduce that for p 1 and ε > 0, the operator f (A) − f (B) belongs to S p+ε , whenever f is a Lipschitz function and A − B ∈ Sp . The epsilon was removed later in [34] in the case 1 < p < ∞. It was shown in [34] that for p ∈ (1, ∞), the operator f (A) − f (B) belongs to S p , whenever A − B ∈ S p and f is a Lipschitz function. Note that similar results also hold for functions on the unit circle T and unitary operators. It was shown in [6] that if A and B are positive self-adjoint operators and I is a normed ideal of operators on Hilbert space with majorization property, then for α ∈ (0, 1), the following inequality holds: α A − B α |A − B|α . I I In this paper we study the problem under which conditions on a function f and a (quasi)normed ideal I of operators on Hilbert space the following inequality holds: f (A) − f (B) const|A − B|α . I I In Section 5 of this paper among other results we show that if f belongs to the Hölder class Λα , 0 < α < 1, and 1 < p < ∞, then f (A) − f (B) ∈ S p/α and f (A) − f (B)
S p/α
constf Λα (R) A − BαS p .
On the other hand, this is not true for p = 1 (a counter-example is given in Section 9). Nevertheless, for p = 1, under the assumptions that f ∈ Λα (R) and A − B ∈ S 1 , we
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prove that f (A) − f (B) belongs to the weak space S 1/α,∞ . To make the conclusion that f (B) − f (A) ∈ S 1/α under the assumption that A − B ∈ S 1 , we need the stronger condition: α . We also obtain similar results for other ideals of operators f belongs to the Besov space B∞1 on Hilbert space. In particular, we show that for every p ∈ (1, ∞) and every l 0, the following inequality holds l l 1/α p p p/α sj f (A) − f (B) sj (A − B) , constf Λα (R) j =0
j =0
where the constant does not depend on l. We also establish in Section 5 similar results for higher order differences (m K f )(A) and functions f ∈ Λα (R) with α ∈ [m − 1, m). In Section 6 we obtain analogs of the result of Section 5 for functions on T and unitary operators, while in Section 7 we establish similar results for functions analytic in the unit disk and contractions. In Section 8 we obtain refinements of some results of Section 6 in the case of finite rank perturbations of unitary operators. We also give some necessary conditions on a function f for f (U ) − f (V ) to belong to S q , whenever U − V ∈ S p . Analogs of the results of Section 8 for self-adjoint operators are given in Section 9. In Section 10 we consider the problem of evaluating the trace of f (A − K) − 2f (A) + 2 (R). We f (A + K) under the assumptions that K ∈ S 2 and f belongs to the Besov class B∞1 introduce a spectral shift function ς associated with the pair (A, K) and establish the following trace formula:
trace f (A − K) − 2f (A) + f (A + K) = f (x)ς(x) dx. R
We also show that similar results hold in the case of unitary operators. The final Section 11 is devoted to estimates of commutators and quasicommutators in the norm of Schatten–von Neumann classes (as well as in the norms of more general operator ideals). We consider a bounded operator R, self-adjoint operators A and B and for a function f ∈ Λα (R), we prove that f (A)R − Rf (B) ∈ S p/α , whenever p > 1 and AR − RB ∈ S p . We also obtain norm estimates for f (A)R − Rf (B) that are similar to the estimates obtained in Section 5 for first order differences f (A) − f (B). In Section 2 we give a brief introduction to Besov spaces and, in particular, we discuss Hölder– Zygmund classes Λα (R), 0 < α < ∞. In Section 3 we introduce quasinormed ideals of operators on Hilbert space and define the upper Boyd index of a quasinormed ideal. In Section 4 we give an introduction to double and multiple operator integrals which will be used in the paper to obtain desired estimates. We also define multiple operator integrals with respect to semi-spectral measures. Note that in this paper we give detailed proofs in the case of bounded self-adjoint operators and explain briefly that the main results also hold in the case of unbounded self-adjoint operators. We are going to consider in detail the case of unbounded self-adjoint operators in [3]. Note also that we are going to consider separately in [4] similar problems for perturbations of dissipative operators and improve earlier results of [20]. The main results of this paper have been announced without proofs in [1].
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2. Besov spaces The purpose of this section is to give a brief introduction to Besov spaces that play an important role in problems of perturbation theory. We start with Besov spaces on the unit circle. s of functions (or distributions) on T can be Let 0 < p, q ∞ and s ∈ R. The Besov class Bpq defined in the following way. Let w be an infinitely differentiable function on R such that 1 supp w ⊂ , 2 , 2
x and w(x) = 1 − w for x ∈ [1, 2]. 2
w 0,
(2.1)
Consider the trigonometric polynomials Wn , and Wn defined by k Wn (z) = w n zk , 2
n 1,
W0 (z) = z¯ + 1 + z,
and Wn (z) = Wn (z),
n 0.
k∈Z
Then for each distribution f on T, f=
f ∗ Wn +
n0
f ∗ Wn .
n1
s consists of functions (in the case s > 0) or distributions f on T such that The Besov class Bpq
ns 2 f ∗ Wn p ∈ q L n1
and
ns q 2 f ∗ W p n L n1 ∈ .
(2.2)
Besov classes admit many other descriptions. In particular, if s > 0 and s > 1/p − 1, the space s admits the following characterization. A function f ∈ Lp belongs to B s , s > 0, if and only Bpq pq if
T
q
nτ f Lp dm(τ ) < ∞ for q < ∞ |1 − τ |1+sq
and nτ f Lp < ∞ for q = ∞, s τ =1 |1 − τ | sup
(2.3)
where m is normalized Lebesgue measure on T, n is an integer greater than s, and τ , τ ∈ T, is the difference operator: (τ f )(ζ ) = f (τ ζ ) − f (ζ ),
ζ ∈ T.
s . We use the notation Bps for Bpp def
α form the Hölder–Zygmund scale. If 0 < α < 1, then f ∈ Λ if and The spaces Λα = B∞ α only if
f (ζ ) − f (τ ) const|ζ − τ |α ,
ζ, τ ∈ T,
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while f ∈ Λ1 if and only if f is continuous and f (ζ τ ) − 2f (ζ ) + f (ζ τ¯ ) const|1 − τ |,
ζ, τ ∈ T.
By (2.3), α > 0, f ∈ Λα if and only if f is continuous and def f Λα = sup |1 − τ |−α nτ f L∞ < ∞, τ
where n is the positive integer such that n − 1 α < n. Note that the f Λα is equivalent to f − fˆ(0) ∗ W0 ∞ + sup 2nα f ∗ Wn ∞ + f ∗ W ∞ , n L L L n1
where for a function or a distribution f on T, fˆ(n) is the nth Fourier coefficient of f . It is easy to see from the definition of Besov classes that the Riesz projection P+ , P+ f =
fˆ(n)zn ,
n0 def
s . Functions (or distributions) in (B s ) = P B s admit a natural extension is bounded on Bpq + pq pq + s ) admit the to analytic functions in the unit disk D. It is well known that the functions in (Bpq + following description:
s f ∈ Bpq +
1 ⇐⇒
q (1 − r)q(n−s)−1 fr(n) p dr < ∞,
q < ∞,
0
and s f ∈ Bp∞ +
⇐⇒
sup (1 − r)n−s fr(n) p < ∞,
0
def
where fr (ζ ) = f (rζ ) and n is a nonnegative integer greater than s. Let us proceed now to Besov spaces on the real line. We consider homogeneous Besov spaces s (R) of functions (distributions) on R. We use the same function w as in (2.1) and define the Bpq
functions Wn and Wn on R by x F Wn (x) = w n , 2
F Wn (x) = F Wn (−x),
where F is the Fourier transform:
(F f )(t) = R
f (x)e−ixt dx,
f ∈ L1 .
n ∈ Z,
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With every tempered distribution f ∈ S (R) we associate a sequences {fn }n∈Z , def
fn = f ∗ Wn + f ∗ Wn . s (R) as the set of all f ∈ S (R) such that Initially we define the (homogeneous) Besov class B˙ pq
ns 2 fn Lp n∈Z ∈ q (Z).
(2.4)
s (R) contains all polynomials. Moreover, the distribuAccording to this definition, the space B˙ pq tion f is defined by the sequence {fn }n∈Z uniquely up to a polynomial. It is easy to see that the series n0 fn converges in S (R). However, the series n<0 fn can diverge in general. It is (r) easy to prove that the series n<0 fn converges uniformly on R for each nonnegative integer (r) r > s − 1/p if q > 1 and the series n<0 fn converges uniformly, whenever r s − 1/p if q 1. s (R). We say that a distribuNow we can define the modified (homogeneous) Besov class Bpq (r) s (R) if {2ns f p } q (r) = tion f belongs to Bpq n L n∈Z ∈ (Z) and f n∈Z fn in the space S (R), where r is the minimal nonnegative integer such that r > s − 1/p in the case q > 1 and r is the minimal nonnegative integer such that r s − 1/p in the case q 1. Now the function f is determined uniquely by the sequence {fn }n∈Z up to a polynomial of degree less that r, and a s (R) if and only if deg ϕ < r. polynomial ϕ belongs to Bpq
We use the same notation Wn and Wn for functions on T and on R. This will not lead to a confusion. s (R) admit equivalent definitions that are similar to those discussed above in Besov spaces Bpq def
the case of Besov spaces of functions on T. In particular, the Hölder–Zygmund classes Λα (R) = α (R), α > 0, can be described as the classes of continuous functions f on R such that B∞ m f (x) const|t|α , t
t ∈ R,
where the difference operator t is defined by (t f )(x) = f (x + t) − f (x),
x ∈ R,
and m is the integer such that m − 1 α < m. As in the case of functions on the unit circle, we consider the following (semi)norm on Λα (R): sup 2nα f ∗ Wn L∞ + f ∗ Wn L∞ ,
f ∈ Λα (R).
n∈Z
We refer the reader to [24] and [30] for more detailed information on Besov spaces. 3. Ideals of operators on Hilbert space In this section we give a brief introduction to quasinormed ideals of operators on Hilbert space. First we recall the definition of quasinormed vector spaces.
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Let X be a vector space. A functional · : X → [0, ∞) is called a quasinorm on X if (i) x = 0 if and only if x = 0; (ii) αx = |α| · x, for every x ∈ X and α ∈ C; (iii) there exists a positive number c such that x + y c(x + y) for every x and y in X. We say that a sequence {xj }j 1 of vectors of a quasinormed space X converges to x ∈ X if limj →∞ xj − x = 0. It is well known that there exists a translation invariant metric on X which induces an equivalent topology on X. A quasinormed space is called quasi-Banach if it is complete. To proceed to operator ideals on Hilbert space, we also recall the definition of singular values of bounded linear operators on Hilbert space. Let T be a bounded linear operator. The singular values sj (T ), j 0, are defined by
sj (T ) = inf T − R: rank R j . Clearly, s0 (T ) = T and T is compact if and only if sj (T ) → 0 as j → ∞. For a bounded operator T on Hilbert space we also introduce the sequence {σn (T )}n0 defined by def
σn (T ) =
1 sj (T ). n+1 n
(3.1)
j =0
Definition. Let H be a Hilbert space and let I be a nonzero linear manifold in the set B(H ) of bounded linear operators on H that is equipped with a quasi-norm · I that makes I a quasi-Banach space. We say that I is a quasinormed ideal if for every A and B in B(H ) and T ∈ I, AT B ∈ I
and AT BI A · B · T I .
(3.2)
A quasinormed ideal I is called a normed ideal if · I is a norm. Note that we do not require that I = B(H ). It is easy to see that if T1 and T2 are operators in a quasinormed ideal I and sj (T1 ) = sj (T2 ) for j 0, then T1 I = T2 I . Thus there exists a function Ψ = ΨI defined on the set of nonincreasing sequences of nonnegative real numbers with values in [0, ∞] such that T ∈ I if and only if Ψ (s0 (T ), s1 (T ), s2 (T ), . . .) < ∞ and T I = Ψ s0 (T ), s1 (T ), s2 (T ), . . . ,
T ∈ I.
If T is an operator from a Hilbert space H1 to a Hilbert space H2 , we say that T belongs to I if Ψ (s0 (T ), s1 (T ), s2 (T ), . . .) < ∞. For a quasinormed ideal I and a positive number p, we define the quasinormed ideal I{p} by
p/2 ∈I , I{p} = T : T ∗ T
p/2 1/p def . T I{p} = T ∗ T I
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If T is an operator on a Hilbert space H and d is a positive integer, we denote by [T ]d the operator on the orthogonal sum dj =1 Tj , where Tj = T , 1 j d. It is easy to see that sn [T ]d = s[n/d] (T ),
n 0,
where [x] denotes the largest integer that is less than or equal to x. We denote by βI,d the quasinorm of the transformer T → [T ]d on I. Clearly, the sequence {βI,d }d1 is nondecreasing and submultiplicative, i.e., βI,d1 d2 βI,d1 βI,d2 . It is well known that the last inequality implies that lim
d→∞
log βI,d log βI,d = inf . d2 log d log d
(3.3)
An analog of (3.3) for submultiplicative functions on (0, ∞) is proved in [17], Ch. 2, Theorem 1.3. To reduce the case of sequences to the case of functions, one can proceed as follows. Suppose that {βn }n1 is a nondecreasing submultiplicative sequence such that β1 = 1. We can define the function v on (0, ∞) by v(t) = min{βn : n t}. Then v(n) = βn and to prove (3.3), it suffices to apply Theorem 1.3 of Ch. 2 of [17] to the function v. Definition. If I is a quasinormed ideal, the number def
βI = lim
d→∞
log βI,d log βI,d = inf d2 log d log d
is called the upper Boyd index of I. It is easy to see that βI 1 for an arbitrary normed ideal I. It is also clear that βI < 1 if and only if limd→∞ d −1 βI,d = 0. Note that the upper Boyd index does not change if we replace the initial quasinorm in the quasinormed ideal with an equivalent one that also satisfies (3.2). It is also easy to see that βI{p} = p −1 βI . Theorem 3.1 below is known to experts. Its analog for rearrangement invariant spaces can be found in [17], Ch. 2, Theorem 6.6 (see also [19], Theorem 2(i)). A similar method can be used to prove Theorem 3.1. We give a proof here for reader’s convenience. Theorem 3.1. Let I be a quasinormed ideal. The following are equivalent: (i) βI < 1; (ii) for every nonincreasing sequence {sn }0 of nonnegative numbers, ΨI {σn }n0 const ΨI {sn }n0 , def
where σn = (1 + n)−1
n
j =0 sj .
(3.4)
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In the proof of Theorem 3.1 we are going to use an elementary fact that if n1 xn is a series of vectors in a quasi-Banach space X such that xn const γ n for some γ < 1, then the series converges in X. This is obvious if cγ < 1, where c is the constant in the definition of quasinorms. In the general case we can partition the series n1 xn in several series, after which each resulting series satisfies the above assumption. Proof of Theorem 3.1. Let us first show that (i) ⇒ (ii). Suppose that βI < δ < 1. Then there exists C > 0 such that βI,d Cδ d for all positive d. Let {sn }n0 be a nonincreasing sequence of positive numbers such that Ψ ({sn }n0 ) < ∞. Let {ej }j 0 be an orthonormal basis in a Hilbert space H . For k 0, we consider the operator Ak ∈ B(H ) defined by Ak ej = s[2−k j ] ej , j 0. It is easy to see that Ak is unitarily equivalent to the operator [A0 ]2k and Ak I C2δk A0 I = C2δk Ψ {sn }n0 . −k It follows that the series A = ∞ in I and AI cΨ ({sn }n0 ), where c is k=0 2 ∞Ak converges a positive number. Clearly, sn (A) = k=0 2−k s[ nk ] . 2 We have n
sj = sn +
j =0
1+[log2 n] [2−k+1 n]−1
sj
j =[2−k n]
k=1
2−k+1 n − 2−k n s[2−k n]
1+[log2 n]
sn +
k=1 ∞ 2−k n + 1 s[2−k n] sn + 3n 2−k s[2−k n]
1+[log2 n]
sn +
k=1
3(n + 1)
∞
k=1
2−k s[2−k n] .
k=0
Hence, σn 3sn (A), n 0, and so ΨI {σn }n0 3Ψ σn (A) n0 = 3AI 3cΨI {sn }n0 . Let us prove now that (ii) ⇒ (i). Let {sn }n0 be a nonincreasing sequence of nonnegative numbers. Put n n n k n 1 1 1 1 1 ξn = σn = sj = sj . n+1 n+1 k+1 n+1 k+1 def
k=0
k=0
j =0
j =0
k=j
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For an arbitrary positive integer d, we have n [n/d] s[n/d] 1 ξn n+1 k+1 j =0 k=j n s[n/d] 1 [n/d] + 1 n+1 k+1 k=[n/d]
([n/d] + 1)s[n/d] n+1
n [n/d]
dx x +1
s[n/d] n+2 log d [n/d] + 1 log d s[n/d] . d This together with inequality (3.4) applied twice yields ΨI {s[n/d] }n0
d d ΨI {ξn }n0 const ΨI {sn }n0 log d log d
for d 2. Thus βI,d < d for sufficiently large d, and so βI < 1.
2
Remark. Suppose that I is a normed ideal and let C I be the best possible constant in inequality (3.4). It is easy to see from the proof of Theorem 3.1 that CI 3
∞
2−k βI,2k .
(3.5)
k=0
Let S p , 0 < p < ∞, be the Schatten–von Neumann class of operators T on Hilbert space such that p 1/p def T S p = sj (T ) . j 0
This is a normed ideal for p 1. We denote by S p,∞ , 0 < p < ∞, the ideal that consists of operators T on Hilbert space such that p 1/p def . T S p,∞ = sup (1 + j ) sj (T ) j 0
The quasinorm · p,∞ is not a norm, but it is equivalent to a norm if p > 1. It is easy to see that βS p = βS p,∞ =
1 , p
0 < p < ∞.
Thus S p and S p,∞ satisfy the hypotheses of Theorem 3.1 for p > 1.
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It follows easily from (3.5) that for p > 1, −1 C S p 3 1 − 21/p−1 . Suppose now that I is a quasinormed ideal of operators on Hilbert space. With a nonnegative integer l we associate the ideal (l) I that consists of all bounded linear operators on Hilbert space and is equipped with the norm Ψ(l) I (s0 , s1 , s2 , . . .) = Ψ (s0 , s1 , . . . , sl , 0, 0, . . .). It is easy to see that for every bounded operator T ,
T (l) I = sup RT I : R 1, rank R l + 1
= sup T RI : R 1, rank R l + 1 . The following fact is obvious. Lemma 3.2. Let I be a quasinormed ideal. Then for all l 0, C (l) I C I . def (l)
Note that if I = S p , p 1, then S lp = linear operators equipped with the norm
def
T S l = p
S p is the normed ideal that consists of all bounded
l p sj (T )
1/p .
j =0
It is well known that · S l is a norm for p 1 (see [10]). p We need the following well-known inequality: T1 T2 S l T1 S l T2 S l , r
p
(3.6)
q
where T1 and T2 bounded operator on Hilbert space and 1/p + 1/q = 1/r. Inequality (3.6) can be deduced from the corresponding inequality for S p norms. Indeed, let R be an operator of rank l such that T1 T2 S l = T1 T2 RS r . There exists an orthogonal projection P of rank l such r that T1 T2 RS r = P T1 T2 RS r . Then T1 T2 S l = P T1 T2 RS r P T1 S p T2 RS q T1 S l T2 S l . r
p
q
We refer the reader to [14] and [10] for further information on singular values and normed ideals of operators on Hilbert space.
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4. Multiple operator integrals 4.1. Multiple operator integrals with respect to spectral measures In this subsection we review some aspects of the theory of double and multiple operator integrals. Double operator integrals appeared in the paper [12] by Daletskii and S.G. Krein. In that paper the authors obtained the following formula d f (A + tK) − f (A) = t=0 dt
(Df )(x, y) dEA (x) K dEA (y)
for a function f of class C 2 (R), and bounded self-adjoint operators A and K (EA stands for the spectral measure of A). Here we use the notation def
(Df )(x, y) =
f (x) − f (y) , x −y
(Df )(x, x) = f (x), def
x = y,
x, y ∈ R.
However, the beautiful theory of double operator integrals was developed later by Birman and Solomyak in [7–9], see also their survey [11]. We are not going to define double operator integrals Φ(x, y) dE1 (x) Q dE2 (y) in the case of Hilbert–Schmidt operators Q that was the starting point in [7–9]. We use the approach based on (integral) projective tensor products. In the case of bounded or trace class operators Q this approach is equivalent to the approach of Birman and Solomyak, see [27]. Let (X , E1 ) and (Y , E2 ) be spaces with spectral measures E1 and E2 on Hilbert spaces H1 and H2 . Suppose that a function Φ on X × Y belongs to the projective tensor product ˆ L∞ (E2 ) of L∞ (E1 ) and L∞ (E2 ), i.e., Φ admits a representation L∞ (E1 ) ⊗ Φ(x, y) =
ϕn (x)ψn (y),
(4.1)
ϕn L∞ ψn L∞ < ∞.
(4.2)
n0
where ϕn ∈ L∞ (E1 ), ψn ∈ L∞ (E2 ), and n0
Then for an arbitrary bounded linear operator Q : H2 → H1 we put
def
Φ(x, y) dE1 (x) Q dE2 (y) = X Y
ϕn dE1 Q ψn dE2 .
n0 X
Y
ˆ ∞ (E2 ), its norm in L∞ (E1 ) ⊗ ˆ Note that if Φ belongs to the projective tensor product L∞ (E1 ) ⊗L ∞ L (E2 ) is, by definition, the infimum of the left-hand side of (4.2) over all representations (4.1). It is easy to see that
Φ ∞ Φ(x, y) dE (x) Q dE (y) 1 2 ˆ ∞ (E2 ) Q. L (E1 )⊗L X Y
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ˆ L∞ (E2 ), then Moreover, if Q belongs to a normed ideal I and Φ ∈ L∞ (E1 ) ⊗
Φ(x, y) dE1 (x) Q dE2 (y) ∈ I X Y
and
Φ(x, y) dE1 (x) Q dE2 (y) ΦL∞ (E1 )⊗L ˆ ∞ (E2 ) QI . I
X Y
We can enlarge the class of functions Φ, for which double operator integrals can be defined by considering integral projective tensor products. We do this in the case of multiple operator integrals. This approach for multiple operator integrals was given in [32]. To simplify the notation, we consider here the case of triple operator integrals; the case of arbitrary multiple operator integrals can be treated in the same way. Let (X , E1 ), (Y , E2 ), and (Z, E3 ) be spaces with spectral measures E1 , E2 , and E3 on Hilbert spaces H1 , H2 , and H3 . Suppose that Φ belongs to the integral projective tensor product ˆ i L∞ (E2 ) ⊗ ˆ i L∞ (E3 ), i.e., Φ admits a representation L∞ (E1 ) ⊗
Φ(x, y, z) =
ϕ(x, ω)ψ(y, ω)χ(z, ω) dσ (ω),
(4.3)
Ω
where (Ω, σ ) is a measure space with a σ -finite measure, ϕ is a measurable function on X × Ω, ψ is a measurable function on Y × Ω, χ is a measurable function on Z × Ω, and
ϕ(·, ω)
L∞ (E1 )
ψ(·, ω)
L∞ (E2 )
χ(·, ω)
L∞ (E3 )
dσ (ω) < ∞.
(4.4)
Ω
Suppose now that T1 is a bounded linear operator from H2 to H1 and T2 is a bounded linear ˆ i L∞ (E2 ) ⊗ ˆ i L∞ (E3 ) of the form (4.3), operator from H3 to H2 . For a function Φ in L∞ (E1 ) ⊗ we put
Φ(x, y, z) dE1 (x)T1 dE2 (y)T2 dE3 (z) X Y Z def
=
Ω
X
ϕ(x, ω) dE1 (x) T1 ψ(y, ω) dE2 (y) T2 χ(z, ω) dE3 (z) dσ (ω). Y
(4.5)
Z
It was shown in [32] (see also [5] for a different proof) that the above definition does not depend on the choice of a representation (4.3). The norm ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ is defined as the infimum of the left-hand side of (4.4) over all representations (4.3).
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It is easy to see that the following inequality holds
Φ(x, y, z) dE1 (x) T1 dE2 (y) T2 dE3 (z) ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ T1 · T2 . X Y Z
In particular, the triple operator integral on the left-hand side of (4.5) can be defined if Φ ˆ L∞ (E2 ) ⊗ ˆ L∞ (E3 ). It is easy to see that if belongs to the projective tensor product L∞ (E1 ) ⊗ T1 ∈ S p and T2 ∈ S q , and 1/p + 1/q 1, then the triple operator integral (4.5) belongs to S r and
Φ(x, y, z) dE1 (x) T1 dE2 (y) T2 dE3 (z) ΦL∞ ⊗ˆ i L∞ ⊗ˆ i L∞ T1 S p · T2 S q , Sr
X Y Z
where 1/r = 1/p + 1/q. Note tat similar inequalities hold for multiple operator integrals and for S lp norms. Recall that multiple operator integrals were considered earlier in [23] and [36]. However, in those papers the class of functions Φ for which the left-hand side of (4.5) was defined is much narrower than in the definition given above. It follows from the results of Birman and Solomyak [9] that if A is a self-adjoint operator (not necessarily bounded), K is a bounded self-adjoint operator, and f is a continuously differentiable ˆ i L∞ (EA ), then function on R such that Df ∈ L∞ (EA+K ) ⊗
f (A + K) − f (A) =
(Df )(x, y) dEA+K (x) K dEA (y)
(4.6)
R×R
and if K belongs to a normed ideal I, then f (A + K) − f (A) ∈ I and f (A + K) − f (A) constDf ∞ ˆ ∞ (EA ) KI . L (EA+K )⊗L I In case I = S p or I = S lp , p 1, the last inequality admits the following improvement: f (A + K) − f (A) constDf ∞ ˆ i L∞ (EA ) KI . L (EA+K )⊗ I
(4.7)
Similar results also hold for unitary operators, in which case we have to integrate the divided difference Df of a function f on the unit circle with respect to the spectral measures of the corresponding operator integrals. It was shown in [27] that if f is a trigonometric polynomial of degree d, then Df C(T)⊗C(T) const df L∞ . ˆ
(4.8)
On the other hand, it was shown in [29] that if f is a bounded function on R whose Fourier transform is supported on [−σ, σ ] (in other words, f is an entire function of exponential type at ˆ i L∞ and most σ that is bounded on R), then Df ∈ L∞ ⊗ Df L∞ ⊗ˆ i L∞ const σ f L∞ (R) .
(4.9)
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In this paper we are going to integrate divided differences of higher orders to estimate the norms of higher order operator differences (1.1). For a function f on R, the divided difference Dk f of order k is defined inductively as follows: def
D0 f = f ; if k 1, then in the case when x1 , x2 , . . . , xk+1 are distinct points in R, k−1 f )(x , . . . , x k−1 f )(x , . . . , x k def (D 1 k−1 , xk ) − (D 1 k−1 , xk+1 ) D f (x1 , . . . , xk+1 ) = xk − xk+1
(the definition does not depend on the order of the variables). Clearly, Df = D1 f. If f ∈ C k (R), then Dk f extends by continuity to a function defined for all points x1 , x2 , . . . , xk+1 . It can be shown that n+1 n+1 k−1 n f (xk ) (xk − xj )−1 (xk − xj )−1 . D f (x1 , . . . , xn+1 ) = j =1
k=1
j =k+1
Similarly, one can define the divided difference of order k for functions on T. It was shown in [32] that if f is a trigonometric polynomial of degree d, then k D f
ˆ ⊗C(T) ˆ C(T)⊗···
const d k f L∞ .
(4.10)
It was also shown in [32] that if f is an entire function of exponential type at most σ and is bounded on R, then k D f
ˆ i ···⊗ ˆ i L∞ L∞ ⊗
const σ k f L∞ (R) .
(4.11)
4.2. Multiple operator integrals with respect to semi-spectral measures Let H be a Hilbert space and let (X , B) be a measurable space. A map E from B to the algebra B(H ) of all bounded operators on H is called a semi-spectral measure if E () 0,
∈ B,
E (∅) = 0 and E (X ) = I,
and for a sequence {j }j 1 of disjoint sets in B, E
∞ j =1
j
= lim
N →∞
N
E (j )
in the weak operator topology.
j =1
Multiple operator integrals with respect to semi-spectral measures were defined in [33] (see also [28]); the definition is based on integral projective tensor products of L∞ spaces.
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If T is a contraction on a Hilbert space H , then by the Sz.-Nagy dilation theorem (see [37]), T has a unitary dilation, i.e., there exist a Hilbert space K such that H ⊂ K and a unitary operator U on K such that T n = PH U n |H ,
n 0,
(4.12)
where PH is the orthogonal projection onto H . Let EU be the spectral measure of U . Consider the operator set function E defined on the Borel subsets of the unit circle T by E () = PH EU ()|H ,
⊂ T.
Then E is a semi-spectral measure. It follows immediately from (4.12) that
Tn =
ζ n dE (ζ ) = PH
T
ζ n dEU (ζ )|H ,
n 0.
(4.13)
T
Such a semi-spectral measure E is called a semi-spectral measure of T. Note that it is not unique. To have uniqueness, we can consider a minimal unitary dilation U of T , which is unique up to an isomorphism (see [37]). The following analog of the Birman–Solomyak formula holds:
f (R) − f (T ) =
(Df )(ζ, τ ) dER (ζ ) (R − T ) dET (τ ). T×T
Here T and R contractions on Hilbert space, ET and ER are their semi-spectral measures, and f 1 ) (see [28] and [33]). is an analytic function in D of class (B∞1 + 5. Self-adjoint operators. Sufficient conditions Recall that for l 0 and p > 0, the normed ideal S lp consists of all bounded linear operators equipped with the norm def
T S l = p
l p sj (T )
1/p .
j =0
Theorem 5.1. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα (R), and for arbitrary self-adjoint operators A and B on Hilbert space with bounded A − B, the following inequality holds: sj f (A) − f (B) cf Λα (R) (1 + j )−α/p A − BαS l
p
for every j l.
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Proof. Put fn = f ∗ Wn + f ∗ Wn , n ∈ Z, and fix an integer N . We have by (4.7) and (4.9), N fn (A) − fn (B)
S lp
n=−∞
N fn (A) − fn (B)
n=−∞
const
N
S lp
2n fn L∞ A − BS l
p
n=−∞ N
constf Λα (R)
2n(1−α) A − BS l
p
n=−∞
const 2N (1−α) f Λα (R) A − BS l . p
On the other hand, 2 f (A) − f (B) fn L∞ n n n>N
n>N
constf Λα (R)
2−nα
n>N
const 2
−N α
f Λα (R) .
Put def
RN =
N fn (A) − fn (B)
def
and QN =
n=−∞
fn (A) − fn (B) . n>N
Clearly, for j l, sj f (A) − f (B) sj (RN ) + QN (1 + j )−1/p f (A) − f (B)S l + QN p −1/p N (1−α) const (1 + j ) 2 f Λα (R) A − BS l + 2−N α f Λα (R) . p
To obtain the desired estimate, it suffices to choose the number N so that 2−N < (1 + j )−1/p A − BS l 2−N +1 . p
2
Theorem 5.2. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R) and arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ S 1 , the operator f (A) − f (B) belongs to S 1/α,∞ and the following inequality holds: f (A) − f (B)
S 1/α,∞
cf Λα (R) A − BαS 1 .
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Proof. This is an immediate consequence of Theorem 5.1 in the case p = 1.
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2
Note that the assumptions of Theorem 5.2 do not imply that f (A) − f (B) ∈ S 1/α . In Section 9 we obtain a necessary condition on f for f (A) − f (B) ∈ S 1/α whenever A − B ∈ S 1 . The following result ensures that the assumption that A − B ∈ S 1 implies that f (A) − f (B) ∈ S 1/α under a slightly more restrictive condition on f . Theorem 5.3. Let 0 < α 1. Then there exists a positive number c > 0 such that for every α (R) and arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ S , f ∈ B∞1 1 the operator f (A) − f (B) belongs to S 1/α and the following inequality holds: f (A) − f (B)
S 1/α
α α (R) A − B . cf B∞1 S1
Note that in the case α = 1 this was proved earlier in [29].
Proof of Theorem 5.3. Put fn = f ∗ Wn + f ∗ Wn . Clearly, fn is trace class perturbations preserving and it is easy to see that fn (A) − fn (B) Since f (A) − f (B) =
S 1/α
α 1−α fn (A) − fn (B)S fn (A) − fn (B) . 1
n∈Z (fn (A) − fn (B)),
(5.1)
it suffices to prove that
fn (A) − fn (B)
S 1/α
< ∞.
n∈Z
We have by (5.1) and (4.7), fn (A) − fn (B) n∈Z
S 1/α
fn (A) − fn (B)α · fn (A) − fn (B)1−α S 1
n∈Z
const
1−α α 2nα fn αL∞ · 21−α fn L ∞ A − BS 1
n∈Z
const
2nα fn L∞ A − BαS 1
n∈Z α α (R) A − B . constf B∞1 S1
2
Theorem 5.4. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R) and arbitrary self-adjoint operators A and B on Hilbert space with bounded A − B, the following inequality holds: 1/α 1/α cf Λα (R) σj (A − B), sj f (A) − f (B) Proof. It suffices to apply Theorem 5.1 with l = j and p = 1.
j 0.
2
Now we are in a position to obtain a general result in the case f ∈ Λα (R) and A − B ∈ I for an arbitrary quasinormed ideal I with upper Boyd index less than 1.
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Theorem 5.5. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα (R), for an arbitrary quasinormed ideal I with βI < 1, and for arbitrary self-adjoint operators A and B on Hilbert space with A − B ∈ I, the operator |f (A) − f (B)|1/α belongs to I and the following inequality holds: f (A) − f (B) 1/α cC I f 1/α A − BI . Λα (R) I Proof. Theorem 3.1 implies that ΨI σn (A − B) n0 constA − BI , and Theorem 5.4 implies that f (A) − f (B) 1/α cf 1/α ΨI σn (A − B) . Λα (R) I n0
2
We can reformulate Theorem 5.5 in the following way. Theorem 5.6. Under the hypothesis of Theorem 5.5, the operator f (A) − f (B) belongs to I{1/α} and f (A) − f (B)
I{1/α}
cα C αI f Λα (R) A − BαI .
We deduce now some more consequences of Theorem 5.5. Theorem 5.7. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , and arbitrary self-adjoint operators A and B with bounded A − B, the following inequality holds: l l 1/α p p p/α sj f (A) − f (B) sj (A − B) . cf Λα (R) j =0
j =0
Proof. The result immediately follows from Theorem 5.5 and Lemma 3.2.
2
Theorem 5.8. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα (R) and for arbitrary self-adjoint operators A and B with A − B ∈ S p , the operator f (A) − f (B) belongs to S p/α and the following inequality holds: f (A) − f (B)
S p/α
cf Λα (R) A − BαS p .
Proof. The result is an immediate consequence of Theorem 5.7.
2
Suppose now that m−1 α < m and f ∈ Λα (R). For a self-adjoint operator A and a bounded self-adjoint operator K, we consider the finite difference m m def K f (A) = (−1)m−j j =0
m f (A + j K). j
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In the case when A is unbounded, by the right-hand side we mean the following operator m m−j m fn (A + j K), (−1) j n∈Z j =0
where as usual, fn = f ∗Wn +f ∗Wn . It has been proved in [2] that under the above assumptions, m m−j m (−1) fn (A + j K) < ∞. j n∈Z j =0
(We refer the reader to [3], where the situation with unbounded A will be discussed in detail.) We are going to use the following representation for (m K f )(A) in terms of multiple operator integrals: m K f (A)
m = m! · · · D f (x1 , . . . , xm+1 ) dEA (x1 )K dEA+K (x2 )K · · · K dEA+mK (xm+1 ), (5.2) m+1 m (R). Forwhere A is a self-adjoint operator, K is a bounded self-adjoint operator, and f ∈ B∞1 mula (5.2) was obtained in [2]. It follows from (5.2), (4.11), and (3.6) that if p m 1, l 0, and f is an entire function of exponential type at most σ that is bounded on R, then
m f (A) K
S lp
m
const σ m f L∞ Km . Sl
(5.3)
p
Moreover, the constant in (5.3) does not depend on p. Theorem 5.9. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ Λα (R), and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: −α/p sj m KαS l K f (A) cf Λα (R) (1 + j )
p
for j l. Proof. As in the proof of Theorem 5.1, we put def
RN =
m K fn (A) nN
def
and QN =
m K fn (A). n>N
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It follows from (5.3) that RN S l
p/m
const 2mn fn L∞ Km Sl
p
nN
Km f Λα (R) Sl p
2
2(m−α)n
nN
(m−α)N
f Λα (R) Km . Sl p
On the other hand, it is easy to see that QN const
fn L∞ f Λα (R)
n>N
2−nα 2−αN f Λα (R) .
n>N
Hence, sj m K fn (A) sj (RN ) + QN (1 + j )−m/p RN S l + QN p/m −αN constf Λα (R) (1 + j )−m/p 2(m−α)N Km . + 2 l S p
To complete the proof, it suffices to choose N such that 2−N < (1 + j )−1/p KS l 2−N +1 . p
2
The following result is an immediate consequence from Theorem 5.9. Theorem 5.10. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), and for an arbitrary self-adjoint operator A and an arbitrary self-adjoint m operator K of class S m , the operator (m K f )(A) belongs to S α ,∞ and the following inequality holds: m f (A) K
S m ,∞ α
cf Λα (R) KαS m .
As in the case 0 < α < 1 (see Theorem 5.3), we are going to improve the conclusion of Theorem 5.10 under a slightly more restrictive assumption on f . Note that in the following theorem α is allowed to be equal to m. Theorem 5.11. Let α > 0 and m − 1 α m. There exists a positive number c such that for α (R), and for an arbitrary self-adjoint operator A and an arbitrary self-adjoint every f ∈ B∞1 m operator K of class S m , the operator (m K f )(A) belongs to S α and the following inequality holds: m f (A) K
Sm α
α α (R) K cf B∞1 Sm .
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Proof. Clearly, m fn (A) K
Sm α
α/m m 1−α/m fn (A) m . K fn (A) S K 1
By (5.3), m fn (A) K
S1
const 2mn fn L∞ Km Sm .
Thus m fn (A) K
Sm
α
n∈Z
m fn (A)α/m m fn (A)1−α/m K K S 1
n∈Z
const
1−α/m
α/m
2αn fn L∞ KαS m fn L∞
n∈Z
constKαS m
2αn fn L∞
n∈Z α α (R) K constf B∞1 Sm .
2
Recall that for a bounded linear operator T the numbers, σj (T ) are defined by (3.1). Theorem 5.12. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: m/α m/α sj m cf Λα (R) σj |K|m , K f (A)
j 0.
Proof. The result follows immediately from Theorem 5.9 in the case j = l and p = m.
2
Theorem 5.13. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα (R), every quasinormed ideal I with βI < m−1 , and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: m 1/α 1/α f (A) cC 1/m f Λα (R) KI . K I I{1/m} Proof. Clearly, |K|m ∈ I{1/m} and βI{1/m} = mβI < 1. Therefore, by Theorem 5.12, m m/α f (A)
I{1/m}
K
which implies the result.
2
m/α cC I{1/m} f Λα (R) |K|m I{1/m}
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Theorem 5.14. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , and for arbitrary self-adjoint operator A and bounded self-adjoint operator K, the following inequality holds: l l m 1/α p p p/α sj K f (A) sj (K) . cf Λα (R) j =0
j =0
Proof. The result follows from Theorem 5.13 and Lemma 3.2.
2
The last theorem of this section is an immediate consequence of Theorem 5.14. Theorem 5.15. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), for an arbitrary self-adjoint operator A, and an arbitrary selfadjoint operator K of class S p , the following inequality holds: m f (A) K
S p/α
cf Λα (R) KαS p .
6. Unitary operators. Sufficient conditions In this section we are going to obtain analogs of the results of the previous section for functions of unitary operators. In the case of first order differences we can use the Birman–Solomyak formula for functions of unitary operators and the proofs are the same as in the case of functions of self-adjoint operators. However, in the case of higher order differences, formulae that express a difference of order m involves not only multiple operator integrals of multiplicity m + 1, but also multiple operator integrals of lower multiplicities, see [2]. This makes proofs more complicated than in the self-adjoint case. We start with first order differences. If U and V are unitary operators, then by the Birman– Solomyak formula,
f (U ) − f (V ) = T×T
f (ζ ) − f (τ ) dEU (ζ ) (U − V ) dEV (τ ), ζ −τ
(6.1)
ˆ L∞ . Here EU and EV are the spectral whenever the divided difference Df belongs to L∞ ⊗ 1 . measures of U and V . Recall that it was shown in [27] that (6.1) holds if f ∈ B∞1 It follows from (4.8) that if I is a normed ideal, U −V ∈ I and f is a trigonometric polynomial of degree d, then f (U ) − f (V ) ∈ I and f (U ) − f (V ) const df L∞ U − V I . I
(6.2)
Moreover, the constant does not depend on I. Theorem 6.1. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα , and for arbitrary unitary operators U and V on Hilbert space, the following inequality holds:
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sj f (U ) − f (V ) cf Λα (1 + j )−α/p U − V αS l
p
for every j l. Theorem 6.2. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα and arbitrary unitary operators U and V on Hilbert space with U −V ∈ S 1 , the operator f (U ) − f (V ) belongs to S 1/α,∞ and the following inequality holds: f (U ) − f (V )
S 1/α,∞
cf Λα U − V αS 1 .
As in the self-adjoint case, the assumptions of Theorem 6.2 do not imply that f (U ) − f (V ) ∈ S 1/α . In Section 8 we obtain a necessary condition on f for f (U ) − f (V ) ∈ S 1/α , whenever U − V ∈ S1. Theorem 6.3. Let 0 < α 1. Then there exists a positive number c > 0 such that for every α and arbitrary unitary operators U and V on Hilbert space with U − V ∈ S , the f ∈ B∞1 1 operator f (U ) − f (V ) belongs to S 1/α and the following inequality holds: f (U ) − f (V )
S 1/α
α α U − V . cf B∞1 S1
Note that in the case α = 1 this was proved earlier in [27]. Theorem 6.4. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα and arbitrary unitary operators U and V on Hilbert space, the following inequality holds: 1/α 1/α cf Λα σj (U − V ), sj f (U ) − f (V )
j 0.
Recall that the numbers σj (U − V ) are defined in (3.1). Theorem 6.5. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every f ∈ Λα , for an arbitrary quasinormed ideal I with βI < 1, and for arbitrary unitary operators U and V on Hilbert space with U − V ∈ I, the operator |f (U ) − f (V )|1/α belongs to I and the following inequality holds: f (U ) − f (V ) 1/α cC I f 1/α U − V I . Λα (R) I Theorem 6.6. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα , every l ∈ Z+ , and arbitrary unitary operators U and V , the following inequality holds: l l 1/α p p p/α sj f (U ) − f (V ) sj (U − V ) . cf Λα j =0
j =0
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Theorem 6.7. Let 0 < α < 1 and 1 < p < ∞. Then there exists a positive number c such that for every f ∈ Λα and for arbitrary unitary operators U and V with U − V ∈ S p , the operator f (U ) − f (V ) belongs to S p/α and the following inequality holds: f (U ) − f (V )
S p/α
cf Λα U − V αS p .
The proofs of the above results are almost the same as in the self-adjoint case. The only difference is that we have to use (6.2) instead of the corresponding inequality for self-adjoint operators. We proceed now to higher order differences. Let U be a unitary operator and A a self-adjoint operator. We are going to study properties of the following higher order differences m k m f eikA U . (−1) k
(6.3)
k=0
As we have already mentioned in the introduction to this section, such finite differences can be expressed as a linear combination of multiple operator integrals of multiplicity at most m + 1. We refer the reader to [2], Theorem 5.2. For simplicity, we state the formula in the case m = 3. 2 . Let U , U , and U be unitary operators. Then Let f ∈ B∞1 1 2 3 f (U1 ) − 2f (U2 ) + f (U3 )
2 =2 D f (ζ, τ, υ) dE1 (ζ ) (U1 − U2 ) dE2 (τ ) (U2 − U3 ) dE3 (υ)
+ (Df )(ζ, τ ) dE1 (ζ ) (U1 − 2U2 + U3 ) dE3 (τ ).
(6.4)
Let U1 = U , U2 = eiA U , and U3 = e2iA U . Lemma 6.8. Let I be a normed ideal such that I{1/2} is also a normed ideal. If f is a trigonometric polynomial of degree d and A ∈ I, then f (U ) − 2f (eiA U ) + f (e2iA U ) ∈ I{1/2} and f (U ) − 2f eiA U + f e2iA U
I{1/2}
const · d 2 f L∞ A2I .
Moreover, the constant does not depend on I. Proof. Let U1 = U , U2 = eiA U , and U3 = e2iA U . By (4.10), we have
2 D f (ζ, τ, υ) dE1 (ζ ) (U1 − U2 ) dE2 (τ ) (U2 − U3 ) dE3 (υ) const · d f L∞ U1 − U2 I U2 − U3 I . 2
Clearly, U1 − U2 I = U2 − U3 I = I − eiA I constAI .
I{1/2}
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On the other hand, by (4.8),
(Df )(ζ, τ ) dE1 (ζ ) (U1 − 2U2 + U3 ) dE3 (τ )
I{1/2}
3701
const · dU1 − 2U2 + U3 I{1/2}
and 2 U1 − 2U2 + U3 I{1/2} = I − eiA I{1/2} constA2I . The result follows now from (6.4).
2
In the general case, for a trigonometric polynomial f of degree d, the following inequality holds: m m f eikA U (−1)k const · d m f L∞ Am (6.5) I, k {1/m} k=0
I
whenever I is a normed ideal such that I{1/m} is also a normed ideal. This follows from an analog of formula (6.4) for higher order differences, see [2], Theorem 5.2. We state the remaining results in this section without proofs. The proofs are practically the same as in the self-adjoint case. The only difference is that instead of inequality (5.3), one has to use inequality (6.5) with I = S lp , p m. Theorem 6.9. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ Λα , and for arbitrary unitary operator U self-adjoint operator A, the following inequality holds: m ikA k m f e U cf Λα (1 + j )−α/p AαS l (−1) sj k p k=0
for j l. Theorem 6.10. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , and for an arbitrary unitary operator U and an arbitrary self-adjoint operator A of class S m , the operator (6.3) belongs to S mα ,∞ and the following inequality holds: m ikA k m f e U (−1) k k=0
cf Λα AαS m .
S m ,∞ α
Theorem 6.11. Let α > 0 and m − 1 α m. There exists a positive number c such that for α , and for an arbitrary unitary operator U and an arbitrary self-adjoint operator every f ∈ B∞1 A of class S m , the operator (6.3) belongs to S mα and the following inequality holds: m ikA k m f e U (−1) k k=0
α α A cf B∞1 Sm .
Sm α
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Theorem 6.12. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: m/α m m m/α f eikA U sj (−1)k cf Λα σj |A|m , k
j 0.
k=0
Theorem 6.13. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ Λα , every quasinormed ideal I with βI < m−1 , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: m 1/α m 1/m 1/α f eikA U cC I{1/m} f Λα AI . (−1)k k k=0
I
Theorem 6.14. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα , every l ∈ Z+ , and for arbitrary unitary operator U and bounded self-adjoint operator A, the following inequality holds: 1/α p m l l p m p/α f eikA U sj (A) . (−1)k cf Λα sj k j =0
j =0
k=0
Theorem 6.15. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λα , for an arbitrary unitary operator U , and an arbitrary self-adjoint operator A of class S p , the following inequality holds: m m f eikA U (−1)k k k=0
cf Λα AαS p .
S p/α
7. The case of contractions In this section we obtain analogs of the results of Sections 5 and 6 for contractions. To obtain desired estimates, we use multiple operator integrals with respect to semi-spectral measures. Suppose that T and R are contractions on Hilbert space and f is a function in the diskalgebra CA (i.e., f is analytic in D and continuous in clos D). We are going to study properties of differences m k=0
(−1)k
k m f T + (R − T ) , k m
m 1.
(7.1)
In particular, when m = 1, we obtain first order differences f (T ) − f (R). In this section we are not going to state separately results for first order differences. They can be obtained from the general results by putting m = 1.
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It was shown in [2] that
k m f T + (R − T ) (−1) k m k=0
m m! D f (ζ1 , . . . , ζm+1 ) dE1 (ζ1 )(R − T ) · · · (R − T ) dEm+1 (ζm+1 ), (7.2) = m ··· m
m
k
m+1
where Ek is a semi-spectral measure of T + mk (R − T ). Suppose now that I is a normed ideal such that I{1/m} is also a normed ideal. It follows from (7.2) and (4.11) that for an arbitrary trigonometric polynomial f of degree d, m k k m f T + (R − T ) (−1) k m k=0
const · d m f L∞ T − Rm I,
(7.3)
I{1/m}
where the constant can depend only on m. We state the results without proofs. The proofs are almost the same as in the self-adjoint case. The only difference is that to estimate higher order differences, we should use inequality (7.3). Theorem 7.1. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every l 0, p ∈ [m, ∞), f ∈ (Λα )+ , and for arbitrary contractions T and R on Hilbert space, the following inequality holds: sj
m k k m f T + (R − T ) (−1) cf Λα (1 + j )−α/p T − RαS l k p m k=0
for j l. Theorem 7.2. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , and for arbitrary contractions T and R with T − R ∈ S m , the operator (7.1) belongs to S mα ,∞ and the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0
cf Λα T − RαS m .
S m ,∞ α
Theorem 7.3. Let α > 0 and m − 1 α m. There exists a positive number c such that for α ) , and for arbitrary contractions T and R with T − R ∈ S , the operator (7.1) every f ∈ (B∞1 + m belongs to S mα and the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0
α α T − R cf B∞1 Sm .
Sm α
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Theorem 7.4. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , and for arbitrary contractions T and R, the following inequality holds: m m/α k m/α k m f T + (R − T ) sj (−1) cf Λα σj |T − R|m , k m
j 0.
k=0
Theorem 7.5. Let α > 0 and m − 1 α < m. There exists a positive number c such that for every f ∈ (Λα )+ , every quasinormed ideal I with βI < m−1 , and for arbitrary contractions T and R, the following inequality holds: m 1/α k 1/m 1/α k m f T + (R − T ) cC I{1/m} f Λα T − RI . (−1) k m k=0
I
Theorem 7.6. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ (Λα )+ , every l ∈ Z+ , and for arbitrary contractions T and R, the following inequality holds: m 1/α p l l p k m p/α f T + (R − T ) sj (T − R) . (−1)k cf Λα sj k m j =0
j =0
k=0
Theorem 7.7. Let α > 0, m − 1 α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ (Λα )+ , for arbitrary contractions T and R with T − R ∈ S p , the following inequality holds: m k m f T + (R − T ) (−1)k k m k=0
cf Λα T − RαS p . S p/α
8. Finite rank perturbations and necessary conditions. Unitary operators In this sections we study the case of finite rank perturbations of unitary operators. We also obtain some necessary conditions. In particular we show that the assumptions that rank(U − V ) = 1 and f ∈ Λα , 0 < α < 1, do not imply that f (U ) − f (V ) ∈ S 1/α . Let us introduce the notion of Hankel operators. For ϕ ∈ L∞ (T), the Hankel operator Hϕ def
from the Hardy class H 2 to H−2 = L2 H 2 is defined by Hϕ g = P− ϕg,
g ∈ H 2,
where P− is the orthogonal projection from L2 onto H−2 . Note that the operator Hϕ has Hankel matrix def Γϕ = ϕ(−j ˆ − k) j 1, k0 with respect to the orthonormal bases {zk }k0 and {¯zj }j 1 of H 2 and H−2 .
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We need the following description of Hankel operators of class S p that was obtained in [25] for p 1 and [26] and [35] for p < 1 (see also [30], Ch. 6): Hϕ ∈ S p
⇐⇒
1/p
P− ϕ ∈ Bp ,
0 < p < ∞.
(8.1)
The following result gives us a necessary condition on f for the assumption U − V ∈ S 1 to imply that f (U ) − f (V ) ∈ S 1/α . Theorem 8.1. Suppose that 0 < p < ∞. Let f be a continuous function on T such that f (U ) − f (V ) ∈ S p , whenever U and V are unitary operators with rank(U − V ) = 1. Then 1/p f ∈ Bp . Proof. Consider the operators U and V on the space L2 (T) with respect to normalized Lebesgue measure on T defined by Uf = z¯ f
and Vf = z¯ f − 2(f, 1)¯z,
f ∈ L2 .
It is easy to see that both U and V are unitary operators and rank(V − U ) = 1. It is also easy to verify that for n 0, ⎧ j −n ⎪ ⎨z , n j V z = −zj −n , ⎪ ⎩ j −n z ,
j n, 0 j < n, j < 0.
It follows that for f ∈ C(T), we have f (V ) − f (U ) zj , zk = fˆ(n) V n zj , zk − zj −n , zk + fˆ(n) V n zj , zk − zj −n , zk n>0
⎧ ⎪ ⎨ fˆ(j − k), = −2 fˆ(j − k), ⎪ ⎩ 0,
n<0
j 0, k < 0, j < 0, k 0, otherwise.
If f (U ) − f (V ) ∈ S p , it follows that the operators on 2 with Hankel matrices
fˆ(j + k) j 0, k1
and
fˆ(−j − k) j 0, k1 1/p
belong to S p . It follows now from (8.1) that f ∈ Bp .
2
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Remark. Recall that Theorem 6.2 says that under the assumptions U − V ∈ S 1 and f ∈ Λα , 0 < α < 1, the operator f (U ) − f (V ) belongs to S 1 ,∞ . On the other hand, Theorem 6.3 shows α α implies that f (U ) − f (V ) ∈ S that the slightly stronger condition f ∈ B∞1 1/α . However, the above theorem tells us that even under the much stronger assumption rank(U − V ) = 1 the α . This follows condition f ∈ Λα does not imply that f (U ) − f (V ) ∈ S 1/α . Indeed, Λα ⊂ B1/α from the fact that
k
ak z 2 ∈ Λ α
⇐⇒
2αk ak
k0
∈ ∞
(8.2)
k0
∈ 1/α .
(8.3)
k0
and from the fact that
k
α ak z2 ∈ B1/α
⇐⇒
2αk ak
k0
Both (8.2) and (8.3) follows easily from (2.2). Note that the proof of Theorem 8.1 shows that if U and V are the unitary operators constructed in the proof of Theorem 8.1 and I is a quasinormed ideal, then f (U ) − f (V ) ∈ I if and only if both Hf and Hf¯ belong to I. The following result is closely related to Theorem 6.2, it shows that if we replace the assumption U − V ∈ S 1 with the stronger assumption rank(U − V ) < +∞, we can obtain the same conclusion for all α > 0. Theorem 8.2. Let 0 < α < ∞ and let U and V be unitary operators such that rank(U − V ) < +∞. Then f (U ) − f (V ) ∈ S 1 ,∞ for every function f ∈ Λα (T). α
Proof. Let m be a positive integer and let f ∈ Λα . By Bernstein’s theorem, we can represent f in the form f = f1 + f2 , where f1 is a trigonometric polynomial of degree at most m and f2 L∞ const m−α (this can be deduced easily from (2.2)). It is easy to see that Um − V m =
m−1
U j (U − V )V m−1−j .
j =0
Hence, m Range f1 (U ) − f1 (V ) ⊂ Range U j (U − V ) + Range U −j U ∗ − V ∗ , j =1
and so rank f1 (U ) − f1 (V ) 2m rank(U − V ), while f2 (U ) − f2 (V ) 2f2 L∞ const m−α . It follows that s2m rank(U −V ) f (U ) − f (V ) const m−α .
2
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We can compare Theorem 8.2 with the following result obtained in [28]: if 0 < p 1, and U 1/p and V are unitary operators such that U − V ∈ S p , then f (U ) − f (V ) ∈ S p for every f ∈ B∞p . The following result allows us to estimate the singular values of Hankel operators with symbols in Λα . Lemma 8.3. Let 0 < α < ∞. Then there exists a positive number c such that for every f ∈ Λα (T), the following inequality holds: sm (Hf ) cf Λα (1 + m)−α . Proof. We can represent f in the form f = f1 + f2 , where f1 is a trigonometrical polynomial of degree at most m and f2 const(1 + m)−α . Then rank Hf1 m and Hf2 const(1 + m)−α which implies the result. 2 The following theorem shows that Theorems 8.2 and 6.2 cannot be improved. Theorem 8.4. Let α > 0. There exist unitary operators U and V and a real function h in Λα such that rank(U − V ) = 1 and sm h(U ) − h(V ) (1 + m)−α ,
m 0.
Proof. Let U and V be the unitary operators defined in the proof of Theorem 8.1. Consider the function g defined by ∞ def −αn 4n
g(ζ ) =
4
ζ
n + ζ¯ 4 ,
ζ ∈ T.
(8.4)
n=1
It follows easily from (2.2) that g ∈ Λα (T). By Lemma 8.3, sm (Hg ) const(1 + m)−α , m 0. Let us obtain a lower estimate for sm (Hg ). Consider the matrix Γg of the Hankel operator Hg with respect to the standard orthonormal bases:
Γg = g(−j ˆ − k) j 1, k0 = g(j ˆ + k) j 1, k0 . Let n 1. Define the 3 · 4n−1 × 3 · 4n−1 matrix Tn by
Tn = gˆ j + k + 4n−1 + 1 0j, k<3·4n−1 . Clearly, 4αn Tn is an orthogonal matrix. Hence, Tn − R 4−αn for every 3 · 4n−1 × 3 · 4n−1 matrix with rank R < 3 · 4n−1 . The matrix Tn can be considered as a submatrix of Γg . Hence Γg − R 4−αn for every infinite matrix R with rank R < 3 · 4n−1 . Thus, sj (Γg ) 4−αn for j < 3 · 4n−1 . To complete the proof, it suffices to take h = cg for a sufficiently large number c. 2
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In Section 6 we have obtained sufficient conditions on a function f on T for the condition U − V ∈ S p to imply that f (U ) − f (V ) ∈ S q for certain p and q. We are going to obtain here necessary conditions and consider other values p and q. We denote by U(S p , S q ) the set of all continuous functions f on T such that f (U ) − f (V ) ∈ S q , whenever U and V are unitary operators such that U − V ∈ S p . We also denote by Uc (S p , S q ) the set of all continuous functions f on T such that f (U ) − f (V ) ∈ S q , whenever U and V are commuting unitary operators such that U − V ∈ S p . Obviously, both U(S p , S q ) and Uc (S p , S q ) contain the set of constant functions. We say that U(S p , S q ) (or Uc (S p , S q )) is trivial if it contains no other functions. Recall that the space Lip of Lipschitz functions on T is defined as the space of functions f such that def
f Lip = sup ζ =τ
|f (ζ ) − f (τ )| < ∞. |ζ − τ |
Theorem 8.5. Let 0 < p, q < +∞. Then ! Uc (S p , S q ) =
Λp/q ,
p < q,
Lip,
p = q.
The space Uc (S p , S q ) is trivial if p > q. Proof. It is easy to see that f ∈ Uc (S p , S q ) if and only if for every two sequences {ζn } and {τn } in T,
|ζn − τn |p < ∞
⇒
f (ζn ) − f (τn ) q < ∞.
(8.5)
Clearly, the condition |f (ζ ) − f (ξ )| const|ζ − ξ |p/q implies (8.5). Consider the modulus of continuity ωf associated with f :
def ωf (δ) = sup f (x) − f (y) : |x − y| < δ ,
δ > 0.
Condition (8.5) obviously implies that ωf (δ) < ∞ for some δ > 0, and so it is finite for all δ > 0. We have to prove that (8.5) implies that ωf (δ) const · δ p/q . Assume the contrary. Then there exist two sequences {ζn } and {τn } in T such that ζn = τn for all n, lim |ζn − τn |p = 0 and
n→∞
|f (ζn ) − f (τn )|q = ∞. n→∞ |ζn − τn |p lim
Now the result is a consequence of the following elementary fact: If {αk } and {βk } are sequences of positive numbers such that limk→∞ βk = 0 and −1 lim k→∞ αk βk = +∞, then there exists a sequence {nk } of nonnegative integers such that nk βk < +∞ and nk αk = +∞. 2
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Corollary 8.6. Let 0 < p, q < +∞. Then ! U(S p , S q ) ⊂
Λp/q , Lip,
p < q, p = q.
The space U(S p , S q ) is trivial if p > q. It has been shown recently in [34] that if f is a Lipschitz function on R and 1 < p < ∞, then f (A) − f (B)S p constA − BS p , whenever A and B are self-adjoint operators such that A − B ∈ Sp . Remark 1. The Potapov–Sukochev theorem implies an analogous result for unitary operators: if U and V are unitary operators such that U − V ∈ S p , 1 < p < ∞, and f is a Lipschitz function on T, then f (U ) − f (V ) ∈ S p and f (U ) − f (V )S p constU − V S p . Indeed, it is easy to reduce the general case to the case when the support of f is contained in an arc I with m(I ) = 1/4. Without loss of generality, we may assume that −I = I¯. Then f (ζ ) + f (ζ¯ ) = 2g(ζ + ζ¯ )
and f (ζ ) − f (ζ¯ ) = 2(ζ − ζ¯ )h(ζ + ζ¯ ),
ζ ∈ T,
for some functions g, h ∈ Lip(R) ∩ L∞ (R). Hence f (ζ ) = g(ζ + ζ¯ ) + ζ h(ζ + ζ¯ ) − ζ¯ h(ζ + ζ¯ ). It remains to apply the Potapov–Sukochev theorem to the self-adjoint operators U + U ∗ , V + V ∗ , and the functions g, h. We are going to use this analog of the Potapov–Sukochev theorem for unitary operators in the following result. Theorem 8.7. Let 1 < q p < +∞. Then ! U(S p , S q ) =
Λp/q , Lip,
p < q, p = q.
Proof. By Corollary 8.6, it suffices to show that Λp/q ⊂ U(S p , S q ) for p < q and Lip ⊂ U(S p , S q ) for p = q. The fact that Λp/q ⊂ U(S p , S q ) for q < p is a consequence of Theorem 6.7. The inclusion Lip ⊂ U(S p , S q ) for q = p is the analog of the Potapov–Sukochev theorem mentioned above. 2 Remark 2. There exists a function f of class Lip such that f ∈ / U(S p , S q ) for any p > 0 and q ∈ (0, 1]. Indeed, if U and V are the unitary operators constructed in the proof of Theorem 8.1, then rank(U − V ) = 1 and f (U ) − f (V ) ∈ S 1 if and only if f ∈ B11 . It suffices to take a Lipschitz function f that does not belong to B11 . Remark 3. Let α > 0. There exists a function f in Λα such that f ∈ / U(S p , S q ) for any p > 0 and q ∈ (0, 1/α]. Indeed, it suffices to consider the unitary operators U and V constructed in the α . proof of Theorem 8.1 and take a function f ∈ Λα that does not belong to B1/α
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Theorem 8.8. Let 0 < p, q < +∞. Then Λp/q ⊂ U(S p , S q ) if and only if 1 < p < q. Proof. If 1 < p, q < +∞ or p > q, the result follows from Corollary 8.6 and Theorem 8.7. On the other hand, if p q and p 1, then Λp/q ⊂ U(S p , S q ) by Remark 2. 2 Theorem 8.9. Let 0 < p, q < +∞. Then Lip ⊂ U(S p , S q ) if and only if 1 < p q or p 1 < q. Proof. As in the proof of Theorem 8.8, it suffices to consider the case p 1. It was shown in [22] that Lip ⊂ U(S 1 , S q ) ⊂ U(S p , S q ) if p 1 < q. It remains to apply Remark 1. 2 Now we are going to obtain a quantitative refinement of Corollary 8.6. Let f ∈ C(T). Put
def Ωf,p,q (δ) = sup f (U ) − f (V )S : U − V S p δ, U, V are unitary operators . q
Lemma 8.10. Let U1 and U2 be a unitary operators with U1 − U2 ∈ S p . Then there exists a unitary operator V such that U1 − V S p
πU1 − U2 S p
and U2 − V S p
4
πU1 − U2 S p 4
.
Proof. Clearly, there exists a self-adjoint operator A such that exp(iA) = U1−1 U2 and A π . Note that π|eiθ − 1| 2|θ | for |θ | π . Hence, AS p π2 U1 − U2 S p . It remains to put V = U1 exp( 2i A). 2 Corollary 8.11. Let 0 < q < +∞. Then there exists a positive number cq such that for every p ∈ (0, ∞), Ωf,p,q (2δ) cq Ωf,p,q (δ),
δ > 0.
Lemma 8.12. Let 0 < p, q < ∞ and let f ∈ U(S p , S q ). Then Ωf,p,q n1/p δ n1/q Ωf,p,q (δ) for every positive integer n. Proof. The result is trivial if Ωf,p,q (δ) = 0 or Ωf,p,q (δ) = ∞. Suppose now that p 0 < Ωf,p,q (δ) < ∞. Fix ε ∈ (0, 1). Let U and V be unitary operators such that U − V S δ and f (U ) − f (V )S q (1 − ε)Ωf,p,q (δ). Put U = nj=1 U and V = nj=1 V (the orthogonal sum of n copies of U and V ). Clearly, U − VS p δn1/p , and f (U) − f (V)
Sq
(1 − ε)n1/q Ωf,p,q (δ)
Hence, Ωf,p,q (n1/p δ) (1 − ε)n1/q Ωf,p,q (δ) for every ε ∈ (0, 1).
2
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Theorem 8.13. Let 0 < p, q < ∞ and let f ∈ U(S p , S q ). Then Ωf,p,q (δ) < +∞ for all δ > 0 and Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) = inf sup = lim , p/q p/q p/q δ→∞ δ→0 δ>0 δ δ δ δ p/q δ>0 lim
where both limits exist in [0, +∞]. In particular, if f is a nonconstant function, then Ωf,p,q (δ) c1 δ p/q for every δ ∈ (0, 1] and Ωf,p,q (δ) c2 δ p/q for every δ ∈ [1, ∞), where c1 and c2 are positive numbers. Proof. Since Ωf,p,q is nondecreasing, Corollary 8.11 implies that either Ωf,p,q (δ) is finite for all δ > 0 or Ωf,p,q (δ) = ∞ for all δ > 0. The latter is impossible because we would be able to find sequences of unitary operators {Uj } and {Vj } such that "
(Uj − Vj ) ∈ S p ,
but
j
" f (Uj ) − f (Vj ) ∈ / Sq . j
Hence, Ωf,p,q (δ) < +∞ for all δ > 0. We can find a sequence {δj }∞ j =1 of positive numbers such −p/q
that δj → 0 and limj →∞ δj
def
Ωf,p,q (δj ) = lim supδ→0 δ −p/q Ωf,p,q (δ) = a. Fix ε ∈ (0, 1). −p/q
Ωf,p,q (δj ) (1 − ε)a for all j > N . Then there exists a positive integer N such that δj Lemma 8.12 implies Ωf,p,q (n1/p δj ) (1 − ε)a(n1/p δj )p/q for all j > N and n > 0. Hence, Ωf,p,q (δ) (1 − ε)aδ p/q for all δ > 0 and ε ∈ (0, 1). Thus Ωf,p,q (δ) aδ p/q for all δ > 0 and limδ→0 δ −p/q Ωf,p,q (δ) = a. In the same way we can prove that Ωf,p,q (δ) Ωf,p,q (δ) = lim . p/q δ→∞ δ δ p/q δ>0
sup
2
9. Finite rank perturbations and necessary conditions. Self-adjoint operators We are going to obtain in this section analogs of the results of the previous section in the case of self-adjoint operators. We obtain estimates for f (A) − f (B) in the case when rank(A − B) < ∞. We also obtain some necessary conditions. In particular, we show that f (A) − f (B) does not have to belong to S 1/α under the assumptions rank(A − B) = 1 and f ∈ Λα (R). However, there is a distinction between the case of unitary operators and the case of selfadjoint operators. To describe the class of functions f on R, for which f (A) − f (B) ∈ S q , whenever A − B ∈ S p , we have to introduce the space Λα of functions on R that satisfy the Hölder condition of order α uniformly on all intervals of length 1. We are going to deal in this section with Hankel operators on the Hardy class H 2 (C+ ) of functions analytic in the upper half-plane C+ . Recall that the space L2 (R) can be represented as L2 (R) = H 2 (C+ ) ⊕ H 2 (C− ), where H 2 (C− ) is the Hardy class of functions analytic in the lower half-plane C− . We denote by P + and P − the orthogonal projections onto H 2 (C+ ) and H 2 (C− ). For a function ϕ in L∞ (R), the Hankel operator Hϕ : H 2 (C+ ) → H 2 (C− ) is defined by def
Hϕ g = P − ϕg,
g ∈ H 2 (C+ ).
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As in the case of Hankel operators on the Hardy class H 2 of functions analytic in D, the Hankel operators Hϕ of class S p can be described in terms of Besov spaces: Hϕ ∈ S p
⇐⇒
1/p
P− ϕ ∈ Bp (R),
0 < p < ∞,
(9.1)
where the operator P− on L∞ (R) is defined by def P− ϕ = P− (ϕ ◦ ω) ◦ ω−1 ,
ϕ ∈ L∞ (R),
def
and ω(ζ ) = i(1 + ζ )(1 − ζ )−1 , ζ ∈ T. This was proved in [25] for p 1, and in [27] and [35] for 0 < p < 1, see also [30], Ch. 6. Note also that by Kronecker’s theorem, Hϕ has finite rank if and only if P− ϕ is a rational function (see [30], Ch. 1). Recall that the Hilbert transform H is defined on L2 (R) by H g = −ig+ + ig− , where we use def
def
the notation g+ = P + g and g− = P − g. Theorem 9.1. Let A and B be bounded self-adjoint operators on Hilbert space such that rank(A − B) < ∞ and let α > 0. Then f (A) − f (B) ∈ S 1 ,∞ for every function f in Λα (R). α
Proof. Consider the Cayley transforms of A and B: U = (A − iI )(A + iI )−1
and V = (B − iI )(B + iI )−1 .
It is well known that U and V are unitary operators. Moreover, it is easy to see that rank(U − V ) < ∞. Indeed, (A − iI )(A + iI )−1 − (B − iI )(B + iI )−1 = 2i (B + iI )−1 − (A + iI )−1 = 2i(A + iI )−1 (A − B)(B + iI )−1 , and so rank(U − V ) rank(A − B). Without loss of generality, we may assume that f has compact support. Otherwise, we can multiply f by an infinitely smooth function with compact support that is equal to 1 on an interval containing the spectra of A and B. Consider the function h on T defined by h(ζ ) = f (−i(ζ + i)(ζ − i)−1 ). Obviously, h ∈ Λα . By Theorem 8.2, h(U ) − h(V ) ∈ S 1/α,∞ . It remains to observe that h(U ) = f (A) and h(V ) = f (B). 2 In Section 5 we have proved Theorem 5.2 that says that the condition f ∈ Λα (R), 0 < α < 1, implies that f (A) − f (B) ∈ S 1/α,∞ , whenever A − B ∈ S 1 . On the other hand, by Theorem 5.3, α (R), 0 < α < 1, implies that f (A) − f (B) ∈ S the stronger condition f ∈ B∞1 1/α , whenever A−B ∈ S 1 . The following result gives a necessary condition on f for the assumption A−B ∈ S 1 to imply that f (A) − f (B) ∈ S 1/α . It shows that the condition f ∈ Λα (R) does not ensure that f (A) − f (B) ∈ S 1/α even under the much stronger assumption that A − B has finite rank.
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Theorem 9.2. Let f be a continuous function on R and let p > 0. Suppose that f (A) − f (B) ∈ S p , whenever A and B are bounded self-adjoint operators such that 1/p rank(A − B) < ∞. Then f ◦ h ∈ Bp (R) for every rational function h that is real on R and has no pole at ∞. Proof. Let ϕ ∈ L∞ (R) and let Mϕ denote multiplication by ϕ. For g ∈ L2 (R), we have Mϕ g − H −1 Mϕ H g = ϕg + H ϕ(H g) = ϕ − (ϕg+ )+ + (ϕg− )+ + (ϕg+ )− − (ϕg− )− = ϕ(g+ + g− ) − (ϕg+ )+ + (ϕg− )+ + (ϕg+ )− − (ϕg− )− = 2(ϕg+ )− + 2(ϕg− )+ = 2Hϕ g+ + 2Hϕ∗ g− .
(9.2)
Hence, by (9.1), Mϕ − H −1 Mϕ H ∈ S p if and only if ϕ ∈ Bp (R). Moreover, by Kronecker’s theorem, rank(Mϕ − H −1 H ϕ H ) < +∞ if and only if ϕ is a rational function. Suppose now that h is a rational function that takes real values on R and has no pole at ∞. Define the bounded self-adjoint operators A and B by 1/p
def
A = Mh ,
B = H −1 Mh H . def
and
By (9.2), rank(A − B) < ∞. Again, by (9.2) with ϕ = f ◦ h, the f (A) − f (B) = Mf ◦h − H −1 Mf ◦h H 1/p
belongs to S p if and only if f ◦ h ∈ Bp (R).
2 1/p
Note that the conclusion of Theorem 9.2 implies that f belongs to Bp (R) locally, i.e., the 1/p restriction of f to an arbitrary finite interval can be extended to a function of class Bp (R). Now we are going to show that Theorem 5.2 cannot be improved even under the assumption that rank(A − B) = 1. Denote by L2e (R) the set of even functions in L2 (R) and by L2o (R) the set of odd functions in 2 L (R). Clearly, L2 (R) = L2e (R) ⊕ L2o (R). Let ϕ be an even function in L∞ (R). Then L2e (R) and L2o (R) are invariant subspaces of the operators Mϕ and H −1 Mϕ H . The orthogonal projections Pe and Pe onto L2e (R) and L2o (R) are given by (Pe g)(x) =
1 g(x) + g(−x) 2
and (Po g)(x) =
1 g(x) − g(−x) . 2
Lemma 9.3. Let ϕ(x) = (x 2 + 1)−1 , x ∈ R. Then (H ϕ)(x) = x(x 2 + 1)−1 and Mϕ f − H −1 Mϕ H f =
1 1 (f, ϕ)ϕ − (f, H ϕ)H ϕ. π π
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In particular, # Mϕ f − H
−1
Mϕ H f =
Proof. It is easy to see that ϕ+ (x) = x ∈ R. Hence,
1 π (f, ϕ)ϕ, − π1 (f, H ϕ)H ϕ,
i 2(x+i) ,
f is even, f is odd.
i ϕ− (x) = − 2(x−i) , and (H ϕ)(x) = x(x 2 + 1)−1 ,
Mϕ f − H −1 Mϕ H f = ϕf + H ϕH f = 2(ϕf+ )− + 2(ϕf− )+ = 2(ϕ− f+ )− + 2(ϕ+ f− )+ f− (−i) f+ f− f+ (i) +i = −i +i = −i x −i − x+i + x−i x +i 1 1 1 1 1 1 f, − f, =− 2π x + i x − i 2π x −i x +i 2 2 (f, ϕ+ )ϕ− + (f, ϕ− )ϕ+ π π 1 1 (f, ϕ + iH ϕ)(ϕ − iH ϕ) + (f, ϕ − iH ϕ)(ϕ + iH ϕ) = 2π 2π 1 1 = (f, ϕ)ϕ − (f, H ϕ)H ϕ. 2 π π =
Corollary 9.4. Let ϕ(x) = (x 2 + 1)−1 , x ∈ R. Then rank(Mϕ − H −1 Mϕ H ) = 2, rank(Pe (Mϕ − H −1 Mϕ H )Pe ) = 1 and rank(Po (Mϕ − H −1 Mϕ H )Po ) = 1. Lemma 9.5. Let ϕ be an even function in L∞ (R). Then √ sn Mϕ − H −1 Mϕ H Pe 2sn (Hϕ ) and √ sn Mϕ − H −1 Mϕ H Po 2sn (Hϕ ). Proof. Note that P − (Mϕ − H −1 Mϕ H )|H 2 (C+ ) = 2Hϕ . It remains to observe that isometrically from L2e (R) onto H 2 (C+ ) and from L2o (R) onto H 2 (C+ ). 2
√ 2P + acts
Lemma 9.6. There exists a function ρ ∈ C ∞ (T) such that ρ(ζ ) + ρ(iζ ) = 1, ρ(ζ ) = ρ(ζ¯ ) for all ζ ∈ T, and ρ vanishes in a neighborhood of the set {−1, 1}. Proof. Fix a function ψ ∈ C ∞ (T) such that ψ vanishes in a neighborhood of the set {−1, 1}, ψ 0, and ψ(ζ ) + ψ(iζ ) > 0 for all ζ ∈ T. Put ρ0 (ζ ) =
ψ(ζ ) + ψ(−ζ ) . ψ(ζ ) + ψ(iζ ) + ψ(−ζ ) + ψ(−iζ )
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Clearly, ρ0 vanishes in a neighborhood of the set {−1, 1}, ρ0 0, and ρ0 (ζ ) + ρ0 (iζ ) = 1 for all ζ ∈ T. It remains to put ρ(ζ ) = 12 (ρ0 (ζ ) + ρ0 (ζ¯ )). 2 In what follows we fix such a function ρ. Lemma 9.7. Let g be a function in Λα such that g(iζ ) = g(ζ ) for all ζ ∈ T. Suppose that infn0 (n + 1)α sn (Hg ) > 0. Then infn0 (n + 1)α sn (Hρg ) > 0. Proof. Fix a positive p such that αp < 1. Clearly, there exists a positive number c1 such that p p Hg S n c1 n1−αp for all n 0. Note that Hρ(z)g(z) S np = Hρ(iz)g(z) S np . Hence, Hρg S n p
p
1 1−αp for all n 0. By Lemma 8.3, there exists a positive number c such that H p 2 ρg S n 2 c1 n p p p c2 n1−αp for all n 1. Hence, there exists an integer M such that Hρg Mn − Hρg S n n1−αp S
Hρg Thus (sn (Hρg ))p
p
p
for all n 1. Note that
1 −αp M−1 n
p S Mn p
p p − Hρg S n (M − 1)n sn (Hρg ) .
for all n 1.
p
2
Lemma 9.8. There exists a real function g0 ∈ Λα that vanishes in a neighborhood of the set {−1, 1} and such that g0 (ζ ) = g0 (ζ¯ ), ζ ∈ T, and sn (Hg0 ) (n + 1)−α for all n 0. def
Proof. Let g is the function given by (8.4). We can put g0 = Cρg for a sufficiently large number C. 2 Theorem 9.9. Let α > 0. Let ϕ(x) = (x 2 + 1)−1 . Consider the operators A and B on L2e (R) defined by Ag = H −1 Mϕ H g and Bg = ϕg. Then (i) rank(A − B) = 1, (ii) there exists a real bounded function f ∈ Λα (R) such that sn f (A) − f (B) (n + 1)−α ,
n 0.
Proof. The equality rank(A − B) = 1 is a consequence of Corollary 9.4. Let g0 be the function obtained in Lemma 9.8. It is easy to see that there exists a real bounded function f ∈ Λα (R) such that f (ϕ(x)) = g0 ( x−i x+i ). It is well known (see [30], Ch. 1, Section 8) that Hf ◦ϕ can be obtained from Hg0 by multiplying on the left and on the right by unitary operators. Hence, by Lemma 9.5, √ √ √ sn f (B) − f (A) 2sn (Hf ◦ϕ ) = 2sn (Hg0 ) 2(n + 1)−α .
2
Remark. The same result holds if we consider operators A and B on L2o (R) defined in the same way. In Section 5 we have obtained sufficient conditions on a function f on R for the condition A − B ∈ S p to imply that f (A) − f (B) ∈ S q for certain p and q. We are going to obtain here necessary conditions and consider other values p and q.
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As in the case of functions on T, we consider the space Lip(R) of Lipschitz functions on R such that $ ! |f (x) − f (y)| def f Lip(R) = sup : x, y ∈ R, x = y < +∞. |x − y| For α ∈ (0, 1], we denote by Λα the set of all functions defined on R and satisfying the Hölder condition of the order α uniformly on all intervals of a fixed length: f Λα
$ |f (x) − f (y)| = sup : x, y ∈ R, x = y, |x − y| 1 < +∞. |x − y|α
def
!
Clearly, f ∈ Λα if and only if ωf (δ) const δ α for δ ∈ (0, 1]. Note that Λ1 = Lip(R). Lemma 9.10. Let 0 < α < 1. Then Λα = Λα (R) + Lip(R). Proof. The inclusion Λα (R) + Lip(R) ⊂ Λα is evident. Let f ∈ Λα . We can consider the piecewise linear function f0 such that f0 (n) = f (n) and f0 |[n, n + 1] is linear for all n ∈ Z. Clearly, f0 ∈ Lip(R) and f − f0 ∈ Λα (R). 2 Denote by SA(S p , S q ) the set of all continuous functions f on R such that f (B)−f (A) ∈ S q , whenever A and B are self-adjoint operators such that B − A ∈ S p . We also denote by SAc (S p , S q ) the set of all continuous functions f on R such that f (B) − f (A) ∈ S q , whenever A and B are commuting self-adjoint operators such that B − A ∈ Sp . As in the case of unitary operators we say that the class SA(S p , S q ) (or SAc (S p , S q )) is trivial if it contains only constant functions. Theorem 9.11. Let 0 < p, q < +∞. Then SAc (S p , S q ) = Λp/q for p q and the space SAc (S p , S q ) is trivial for p > q. Proof. To prove the inclusion Λp/q ⊂ SAc (S p , S q ), it suffices to observe that f ∈ SAc (S p , S q ) if and only if for every two sequences {xn } and {yn } in R
|xn − yn |p < +∞
⇒
f (xn ) − f (yn ) q < +∞.
(9.3)
Condition (9.3) easily implies that ωf (δ) < +∞ for some δ > 0, and so for all δ > 0. To complete the proof, we have to prove that (9.3) implies that ωf (δ) Cδ p/q for δ ∈ (0, 1]. This can be done in exactly the same way as in the case of unitary operators, see the proof of Theorem 8.5. 2 The following result is an immediate consequence of Theorem 9.11. Theorem 9.12. Let 0 < p, q < +∞. Then SA(S p , S q ) ⊂ Λp/q for p q and SA(S p , S q ) is trivial for p > q.
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Theorem 9.13. Let 1 < p q < +∞. Then SA(S p , S q ) = Λp/q . Proof. In view of Theorem 9.12, we have to prove that Λp/q ⊂ SA(S p , S q ). In the case p = q this was proved by Potapov and Sukochev [34]. Suppose now that p < q. By Lemma 9.10, it is sufficient to verify that Lip(R) ⊂ SA(S p , S q ) and Λp/q (R) ⊂ SA(S p , S q ). The first inclusion follows from the results of [34] as well as from the results of [22]. Indeed, Lip(R) ⊂ SA(S p , S p ) ⊂ SA(S p , S q ). The inclusion Λp/q (R) ⊂ SA(S p , S q ) follows from Theorem 5.8. 2 Theorem 9.14. If 0 < p, q 1, then Lip(R) ⊂ SA(S p , S q ). If 0 < α < 1, 0 < p 1, and 0 < q 1/α, then Λα (R) ⊂ SA(S p , S q ). Proof. The result follows from Theorem 9.2. Indeed, there exists a function in Lip(R) which does not belong to B11 (R) locally, and for each α ∈ (0, 1) there exists a function in Λα (R) that α (R) locally. 2 does not belong to B1/a Theorem 9.15. Let 0 < q, p < +∞. Then Λp/q (R) ⊂ SA(S p , S q ) if and only if 1 < p < q. Proof. If 1 < q, p < +∞ or q < p, the result follows from Theorems 9.13 and 9.12. If p q and p 1, then Λp/q (R) ⊂ SA(S p , S q ) by Theorem 9.14. 2 Theorem 9.16. Let 0 < p, q < +∞. Then Lip(R) ⊂ SA(S p , S q ) if and only if 1 < p q or p 1 < q. Proof. In the same way as in the proof of Theorem 9.15, we see that it suffices to consider the case p 1. From the results of [22] or the results of [34] it follows that Lip(R) ⊂ SA(S 1 , S q ) ⊂ SA(S p , S q ) if p 1 < q. The converse follows from Theorem 9.14. 2 Now we are going to obtain a quantitative refinement of Theorem 9.12. Let f ∈ C(R). Put
def Ωf,p,q (δ) = sup f (A) − f (B)S : A − BS p δ, A, B are self-adjoint operators . q
It is easy to see that given q > 0, there exists a positive number cq such that Ωf,p,q (2δ) cq Ωf,p,q (δ). Theorem 9.17. Let 0 < p, q < ∞ and let f ∈ SA(S p , S q ). Then Ωf,p,q (δ) < +∞ for all δ > 0 and Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) Ωf,p,q (δ) = inf sup = lim , p/q p/q p/q δ→+∞ δ→0 δ>0 δ δ δ δ p/q δ>0 lim
(both limits exist and take values in [0, ∞]). In particular, if f is a nonconstant function, then Ωf,p,q (δ) c1 δ p/q for every δ ∈ (0, 1] and Ωf,p,q (δ) c2 δ p/q for every δ ∈ [1, +∞), where c1 and c2 are positive numbers. The proof of Theorem 9.17 is the same as that of Theorem 8.13.
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Theorem 9.18. Let f ∈ C(R) and p ∈ [1, +∞). Then either Ωf,p,p (δ) = +∞ for all δ > 0 or Ωf,p,p is a linear function. Proof. If f ∈ / SA(S p , S p ), then Ωf,p,p (δ) = +∞ for all δ > 0. Suppose now that f ∈ SA(S p , S p ). By the analog of Lemma 8.12 for self-adjoint operators (it is easy to see that it holds for self-adjoint operators), Ωf,p,p (nδ) nΩf,p,p (δ) for all positive integer n. On the other hand, clearly, Ωf,p,p (nδ) nΩf,p,p (δ) for all positive integer n. Hence, Ωf,p,p is a linear function. 2 10. Spectral shift function for second order differences In this section we obtain trace formulae for second order differences in the case of self-adjoint operators and unitary operators. By Theorem 5.11, if A is a self-adjoint operator, K is a self-adjoint operator of class S 2 and 2 (R), then f (A + K) − 2f (A) + f (A − K) ∈ S . We are going to obtain a formula for f ∈ B∞1 1 the trace of this operator. In this section m denotes Lebesgue measure on the real line. Theorem 10.1. Let A be a self-adjoint operator and K a self-adjoint operator of class S 2 . Then 2 (R), there exists a unique function ς ∈ L1 (R) such that for every f ∈ B∞1 trace f (A + K) − 2f (A) + f (A − K) =
f (x)ς(x) dm(x).
(10.1)
R
Moreover, ς(x) 0, x ∈ R. Definition. The function ς satisfying (10.1) is called the second order spectral shift function associated with the pair (A, K). We are going to use the spectral shift function of Koplienko. Koplienko proved in [15] that with each pair of a self-adjoint operator A and a self-adjoint operator K of class S 2 , there exists a function η ∈ L1 (R) such that η 0 and for every rational function f with poles off R, the following trace formula holds
d = f (x)η(x) dm(x). trace f (A + K) − f (A) − f (A + tK) t=0 dt
(10.2)
R
The function η is called the Koplienko spectral shift function associated with the pair (A, K). 2 (R). Note that Note that later in [31] it was proved that trace formula (10.2) holds for f ∈ B∞1 the derivative d f (A + tK) t=0 dt
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1 (R) (see [29] and [32]) and does not have to exist under the exists under the condition f ∈ B∞1 2 2 (R), by condition f ∈ B∞1 (R). However, in the case f ∈ B∞1
f (A + K) − f (A) −
d f (A + tK) t=0 dt
we can understand n∈Z
d fn (A + K) − fn (A) − fn (A + tK) , t=0 dt def
and the series converges absolutely, see [31]. Here, as usual, fn = f ∗ Wn + f ∗ Wn . Proof of Theorem 10.1. Let η1 be the spectral shift function associated with the pair (A, K) and let η2 be the spectral shift function associated with the pair (A, −K). We have
d = f (x)η1 (x) dm(x) trace f (A + K) − f (A) − f (A + tK) t=0 dt R
and
d = f (x)η2 (x) dm(x) trace f (A − K) − f (A) − f (A − tK) t=0 dt R
2 (R). Taking the sum, we obtain for f ∈ B∞1
trace f (A + K) − 2f (A) + f (A − K) =
f (x) η1 (x) + η2 (x) dm(x).
R def
It remains to put ς = η1 + η2 . Uniqueness is obvious. 2 We proceed now to the case of unitary operators. Suppose that U is a unitary operator and V 2 , then is a unitary operator such that I − V ∈ S 2 . It follows from Theorem 6.11 that if f ∈ B∞1 ∗ f (V U ) − 2f (U ) + 2(V U ) ∈ S 1 . We are going to obtain a trace formula for this operator. Theorem 10.2. Let U be a unitary operator and let V be a unitary operator such that I −V ∈ S 2 . Then there exists an integrable function ς on T such that trace f (V U ) − 2f (U ) + 2 V ∗ U =
T
f ς dm.
(10.3)
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It is easy to see that ς is determined by (10.3) modulo a constant. It is called a second order spectral shift function associated with the pair (U, V ). We are going to use a trace formula of Neidhardt [21], which is an analog of the Koplienko trace formula for unitary operators. Suppose that U and V be unitary operators such that U − V ∈ S 2 . Then V can be represented as V = eiA U , where A is a self-adjoint operator of class S 2 whose spectrum σ (A) is a subset of (−π, π]. It was shown in [21] that one can associate with the pair (U, V ) a function η in L1 (T) (a Neidhardt spectral shift function) such that if the second derivative f of a function f has absolutely converging Fourier series, then
d isA = f η dm. f e U trace f (V ) − f (U ) − s=0 ds
(10.4)
T
2 . Later it was shown in [31] that formula (10.4) holds for an arbitrary function f in B∞1
Proof of Theorem 10.2. We can represent V as V = eiA , where A is a self-adjoint operator of def
class S 2 such that the spectrum σ (A) of A is a subset of (−π, π]. Let V1 = V U . Clearly, V1 is a def
unitary operator and U − V1 ∈ S 2 . We have V1 = eiA U . Let V2 = V ∗ U . Then U − V2 ∈ S 2 and V2 = e−iA U . Let η1 be the Neidhardt spectral shift function associated with (U, V1 ) and let η2 be the Neidhardt spectral shift function associated with (U, V2 ). We have
d isA = f η1 dm f e U trace f (V1 ) − f (U ) − s=0 ds T
and
d −isA f e U trace f (V2 ) − f (U ) − = f η2 dm s=0 ds T
2 . Taking the sum, we obtain for f ∈ B∞1
trace f (V U ) − 2f (U ) + 2 V ∗ U = trace f (V1 ) − 2f (U ) + f (V2 )
= f (η1 + η2 ) dm. T def
It remains to put ς = η1 + η2 .
2
11. Commutators and quasicommutators In this section we obtain estimates for the norm of quasicommutators f (A)R − Rf (B) in terms of AR − RB for self-adjoint operators A and B and a bounded operator R. We assume for simplicity that A and B are bounded. However, we obtain estimates that do not depend on
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the norms of A and B. In [3] we will consider the case of not necessarily bounded operators A and B. Note that in the special case A = B, this problem turns into the problem of estimating the norm of commutators f (A)R − Rf (A) in terms of AR − RA. On the other hand, in the special case R = I the problem turns into the problem of estimating f (A) − f (B) in terms A − B. Similar results can be obtained for unitary operators and for contractions. Birman and Solomyak (see [11]) discovered the following formula
f (A)R − Rf (B) =
f (x) − f (y) dEA (x) (AR − RB) dEB (y), x−y
(11.1)
whenever f is a function, for which Df is a Schur multiplier of class M(EA , EB ) (see Section 4). Theorem 11.1. Let 0 < α < 1. There exists a positive number c > 0 such that for every l 0, p ∈ [1, ∞), f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R, the following inequality holds: sj f (A)R − Rf (B) cf Λα (R) (1 + j )−α/p R1−α AR − RBαS l
p
for every j l.
Proof. Clearly, we may assume that R = 0. As usual, fn = f ∗ Wn + f ∗ Wn , n ∈ Z. Fix an integer N . We have by (11.1) and (4.9), N fn (A)R − Rfn (B) n=−∞
S lp
N fn (A)R − Rfn (B) n=−∞
const
N
S lp
2n fn L∞ AR − RBS l
p
n=−∞
const 2N (1−α) f Λα (R) AR − RBS l . p
On the other hand, fn (A)R − Rfn (B) 2R fn L∞ n>N
n>N
constf Λα (R) R
2−nα
n>N
const 2−N α f Λα (R) R.
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Put N fn (A)R − Rfn (B)
def
XN =
def
and YN =
n=−∞
fn (A)R − Rfn (B) . n>N
Clearly, for j l, −1 sj f (A)R − Rf (B) sj (XN ) + YN (1 + j ) p AR − RBS l + YN p
−1 constf Λα (R) (1 + j ) p 2N (1−α) AR − RBS l + 2−N α R . p
To obtain the desired estimate, it suffices to choose the number N so that 2−N < (1 + j )−1/p AR − RBS l R−1 2−N +1 . p
2
The proofs of the remaining results of this section are the same as those of the results of Section 5 for first order differences. Theorem 11.2. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ S 1 and an arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S 1/α,∞ and the following inequality holds: f (A)R − Rf (B)
S 1/α,∞
cf Λα (R) R1−α AR − RBαS 1 .
Theorem 11.3. Let 0 < α 1. There exists a positive number c > 0 such that for every f ∈ α (R), for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ S and an B∞1 1 arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S 1/α and the following inequality holds: f (A)R − Rf (B)
S 1/α
1−α α (R) R cf B∞1 AR − RBαS 1 .
Theorem 11.4. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R on Hilbert space, the following inequality holds: 1/α 1−α 1/α sj f (A)R − Rf (B) cf Λα (R) R α σj (AR − RB),
j 0.
Theorem 11.5. Let 0 < α < 1. There exists a positive number c > 0 such that for every f ∈ Λα (R), for an arbitrary quasinormed ideal I with βI < 1, for arbitrary bounded self-adjoint operators A and B with AR − RB ∈ I, the operator |f (A)R − Rf (B)|1/α belongs to I and the following inequality holds: f (A)R − Rf (B) 1/α cC I f 1/α R 1−α α AR − RBI . Λα (R) I
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Theorem 11.6. Let 0 < α < 1 and 1 < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), every l ∈ Z+ , for arbitrary bounded self-adjoint operators A and B and an arbitrary bounded operator R, the following inequality holds: l l 1/α p p 1−α p/α sj f (A)R − Rf (B) sj (AR − RB) . cf Λα (R) Rp α j =0
j =0
Theorem 11.7. Let 0 < α < 1 and 1 < p < ∞. There exists a positive number c such that for every f ∈ Λα (R), for arbitrary bounded self-adjoint operators A and B, and for an arbitrary bounded operator R, the operator f (A)R − Rf (B) belongs to S p/α and the following inequality holds: f (A)R − Rf (B)
S p/α
cf Λα (R) R1−α AR − RBαS p .
References [1] A.B. Aleksandrov, V.V. Peller, Functions of perturbed operators, C. R. Math. Acad. Sci. Paris Sér. I 347 (2009) 483–488. [2] A.B. Aleksandrov, V.V. Peller, Operator Hölder–Zygmund functions, Adv. Math. (2010), doi:10.1016/j.aim. 2009.12.018, in press. [3] A.B. Aleksandrov, V.V. Peller, Functions of perturbed unbounded self-adjoint operators, in preparation. [4] A.B. Aleksandrov, V.V. Peller, Functions of perturbed dissipative operators, in preparation. [5] N.A. Azamov, A.L. Carey, P.G. Dodds, F.A. Sukochev, Operator integrals, spectral shift and spectral flow, Canad. J. Math. 61 (2009) 241–263. [6] M.S. Birman, L.S. Koplienko, M.Z. Solomyak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 154 (3) (1975) 3–10 (in Russian); English translation: Soviet Math. (Iz. VUZ) 19 (3) (1975) 1–6. [7] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals, in: Problems Math. Phys., vol. 1, Leningrad. Univ., 1966, pp. 33–67 (in Russian); English translation in: Topics Math. Phys., vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967, pp. 25–54. [8] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals. II, in: Problems Math. Phys., vol. 2, Leningrad. Univ., 1967, pp. 26–60 (in Russian); English translation in: Topics Math. Phys., vol. 2, Consultants Bureau Plenum Publishing Corporation, New York, 1968, pp. 19–46. [9] M.S. Birman, M.Z. Solomyak, Double Stieltjes operator integrals. III, in: Problems Math. Phys., vol. 6, Leningrad. Univ., 1973, pp. 27–53 (in Russian). [10] M.Sh. Birman, M.Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Math. Appl. (Sov. Ser.), D. Reidel Publishing Co., Dordrecht, 1987. [11] M.S. Birman, M.Z. Solomyak, Double operator integrals in Hilbert space, Integral Equations Operator Theory 47 (2003) 131–168. [12] Yu.L. Daletskii, S.G. Krein, Integration and differentiation of functions of Hermitian operators and application to the theory of perturbations, Trudy Sem. Funktsion. Anal. Voronezh. Gos. Univ. 1 (1956) 81–105 (in Russian). [13] Yu.B. Farforovskaya, An example of a Lipschitzian function of selfadjoint operators that yields a nonnuclear increase under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972) 146–153 (in Russian). [14] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Nauka, Moscow, 1965; English translation: Amer. Math. Soc., Providence, RI, 1969. [15] L.S. Koplienko, The trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (5) (1984) 62–71 (in Russian); English translation: Sib. Math. J. 25 (1984) 735–743. [16] M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian). [17] S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English translation: Transl. Math. Monogr., vol. 54, American Mathematical Society, Providence, RI, 1982.
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[18] E. Kissin, V.S. Shulman, Classes of operator-smooth functions. III. Stable functions and Fuglede ideals, Proc. Edinb. Math. Soc. (2) 48 (1) (2005) 175–197. [19] S.J. Montgomery-Smith, The Hardy operator and Boyd indices, in: Interaction Between Functional Analysis, Harmonic Analysis, and Probability, Columbia, MO, 1994, in: Lect. Notes Pure Appl. Math., vol. 175, Dekker, New York, 1996, pp. 359–364. [20] S.N. Naboko, Estimates in operator classes for the difference of functions from the Pick class of accretive operators, Funktsional. Anal. i Prilozhen. 24 (3) (1990) 26–35 (in Russian); English translation: Funct. Anal. Appl. 24 (1990) 187–195. [21] H. Neidhardt, Spectral shift function and Hilbert–Schmidt perturbation: extensions of some work of L.S. Koplienko, Math. Nachr. 138 (1988) 7–25. [22] F.L. Nazarov, V.V. Peller, Lipschitz functions of perturbed operators, C. R. Math. Acad. Sci. Paris Sér. I 347 (2009) 857–862. [23] B.S. Pavlov, On multiple operator integrals, in: Linear Operators and Operator Equations, in: Problems Math. Anal., vol. 2, Izdat. Leningrad. Univ., Leningrad, 1969, pp. 99–122 (in Russian). [24] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, 1976. [25] V.V. Peller, Hankel operators of class Sp and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Mat. Sb. 113 (1980) 538–581; English translation: Math. USSR Sb. 41 (1982) 443–479. [26] V.V. Peller, A description of Hankel operators of class Sp for p > 0, an investigation of the rate of rational approximation, and other applications, Mat. Sb. 122 (1983) 481–510; English translation: Math. USSR Sb. 50 (1985) 465–494. [27] V.V. Peller, Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funktsional. Anal. i Prilozhen. 19 (2) (1985) 37–51 (in Russian); English translation: Funct. Anal. Appl. 19 (1985) 111–123. [28] V.V. Peller, For which f does A − B ∈ Sp imply that f (A) − f (B) ∈ Sp ?, in: Oper. Theory Adv. Appl., vol. 24, Birkhäuser, Basel, 1987, pp. 289–294. [29] V.V. Peller, Hankel operators in the perturbation theory of unbounded self-adjoint operators, in: Analysis and Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529–544. [30] V.V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003. [31] V.V. Peller, An extension of the Koplienko–Neidhardt trace formulae, J. Funct. Anal. 221 (2005) 456–481. [32] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006) 515–544. [33] V.V. Peller, Differentiability of functions of contractions, in: Linear and Complex Analysis, in: Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 109–131. [34] D. Potapov, F. Sukochev, Operator-Lipschitz functions in Schatten–von Neumann classes, arXiv:0904.4095, 2009. [35] S. Semmes, Trace ideal criteria for Hankel operators and applications to Besov classes, Integral Equations Operator Theory 7 (1984) 241–281. [36] V.V. Sten’kin, Multiple operator integrals, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (79) (1977) 102–115 (in Russian); English translation: Soviet Math. (Iz. VUZ) 21 (4) (1977) 88–99. [37] B.Sz.- Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, Akadémiai Kiadó, Budapest, 1970.
Journal of Functional Analysis 258 (2010) 3725–3757 www.elsevier.com/locate/jfa
Characterization of the critical Sobolev space on the optimal singularity at the origin Sei Nagayasu a , Hidemitsu Wadade b,∗ a Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan b Department of Mathematics, Taida Institute for Mathematical Sciences, National Taiwan University,
Taipei 106, Taiwan Received 10 September 2009; accepted 25 February 2010 Available online 10 March 2010 Communicated by H. Brezis
Abstract In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the n
critical Sobolev space H p Nirenberg type inequality:
,p
(Rn ), 1 < p < ∞. With this purpose, we shall show the weighted Gagliardo–
1 uLq (Rn ; dx ) C n−s |x|s
1 + 1 q
p
(n−s)p 1 n 1− (n−s)p nq nq q p uLp (Rn ) (−) 2p uLp (Rn ) ,
(GN)
where C depends only on n and p. Here, 0 s < n and p˜ q < ∞ with some p˜ ∈ (p, ∞) determined only n p < ∞, we can prove the growth orders for s as by n and p. Additionally, in the case n 2 and n−1 s ↑ n and for q as q → ∞ are both optimal. (GN) allows us to prove the Trudinger type estimate with the homogeneous weight. Furthermore, it is obvious that (GN) cannot hold with the weight |x|n itself. However, 1 )r |x|n at the origin, we cover this critical weight. with a help of the logarithmic weight of the type (log |x| Simultaneously, we shall give the minimal exponent r = q+p p so that the continuous embedding can hold. © 2010 Elsevier Inc. All rights reserved.
Keywords: Sobolev embedding theorem; Gagliardo–Nirenberg type inequality; Trudinger type inequality; Caffarelli–Kohn–Nirenberg inequality
* Corresponding author.
E-mail addresses:
[email protected] (S. Nagayasu),
[email protected] (H. Wadade). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.015
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1. Introduction and main results n
,p
In the present paper, we give some characterization of the functions in H p (Rn ) with n ∈ N and 1 < p < ∞ called the critical Sobolev space in the sense that the continuous embedding n n ,p ,p H p (Rn ) → Lq (Rn ) holds for all p q < ∞, but H p (Rn ) ⊂ L∞ (Rn ) which implies that n ,p n n H p (R ) possibly has a singularity at some point. Indeed, at least in the case n 2 and n−1 1 τ 1 p < ∞, a compactly supported function such as [log( |x| )] with 0 < τ < p at the origin implies the failure of the embedding in the case q = ∞, see Lemma 2.6 in Section 2. More precisely, Ozawa [11] gave the Gagliardo–Nirenberg type estimate of the following type: p 1 1− pq n uLq (Rn ) Cq p uLq p (Rn ) (−) 2p uLp (R n) n
(1.1)
p holds for all u ∈ H p (Rn ) and p q < ∞, where C depends only on n and p, and p := p−1 denotes the Hölder conjugate exponent of p. The inequality (1.1) was originally obtained by n Ogawa [9] and Ogawa and Ozawa [10] in the case p = 2, i.e., H 2 ,2 (Rn ). Moreover, we refer n ,p to Kozono and Wadade [6] which treats the marginal case of (1.1) as p → ∞ in H p (Rn ). In fact, the functions having bounded mean oscillation BMO can be expressed as the limit case of n n ,p H p (Rn ) with p = ∞ in some sense, and [6] proved (1.1) with (−) 2p uLp (Rn ) replaced by uBMO . In addition, Wadade [18] is also a generalization of (1.1) in terms of the Besov and the Triebel–Lizorkin spaces. Our purpose in this article is to generalize (1.1) with the weighted Lebesgue space. In general, dx ) as the function space endowed for a measurable weight function w(x), we define Lq (Rn ; w(x) with the norm: ,p
uLq (Rn ;
dx w(x) )
:= Rn
u(x)q dx w(x)
1 q
for 1 < q < ∞.
We shall show the following inequality with the homogeneous weight w(x) = |x|s : Theorem 1.1. Let n ∈ N and 1 < p < ∞. Then there exist positive constants p˜ ∈ (p, ∞) and C which both depend only on n and p such that the inequality
1 uLq (Rn ; dxs ) C |x| n−s n
1 + 1 q
p
1 1−θ n q p uθLp (Rn ) (−) 2p uLp (Rn )
,p
holds for all u ∈ H p (Rn ), 0 s < n and p˜ q < ∞, where θ := 1
1
(n−s)p nq
(1.2)
∈ (0, 1). Further1
1 q + p p < ∞, the growth orders ( n−s ) as s ↑ n and q p as q → ∞ are more, if n 2 and 1 1 1 1 1 1 −ε 1 q + p 1 q + p −ε both optimal in the sense that we cannot replace ( n−s ) and q p by ( n−s ) and q p n n−1
for any small ε > 0, respectively.
Remark 1.2. (i) If we do not pay attention to the growth orders of s and q, the inequality (1.2) itself is shown by Caffarelli, Kohn and Nirenberg [1] with the first order derivative, i.e., pn = 1. However, we aim to obtain the optimal growth orders of s and q, and in fact we can prove that
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
3727
n those orders are optimal in the case n 2 and n−1 p < ∞. Unfortunately, we do not know n because of some the optimality in the cases n = 1 and 1 < p < ∞, or n 2 and 1 < p < n−1 technical reason, see Lemma 2.6 in Section 2. Moreover, we shall prove a weighted Trudinger 1
type estimate as an effect of this growth order q p as q → ∞, which will be stated below. (ii) The exponent p˜ actually can be chosen as p˜ := max{p + 1, p + 1, n + 1}. This restriction for the range of q will be used to prove Lemma 2.4. As stated in Remark 1.2(i), we can prove a weighted Trudinger type estimate as an application of Theorem 1.1: Corollary 1.3. Let n ∈ N, 1 < p < ∞, and define the function Φn,p by Φn,p (t) := exp t −
j 0 −1 j t j =0
j!
for t ∈ R with j0 := min j ∈ N; p j p˜ ,
where p˜ ∈ (p, ∞) is the positive constant given by Theorem 1.1. Then there exist two positive constants α and β which depend only on n and p such that Rn n
,p
(n−s)p p dx
β n u Φn,p α(n − s)u(x) p (Rn ) L |x|s n−s
n
holds for all u ∈ H p (Rn ) with (−) 2p uLp (Rn ) 1 and 0 s < n. Remark 1.4. The procedure to get the Trudinger type estimate from the Gagliardo–Nirenberg type estimate was originally seen in [9,10,12,11]. Especially, [12] clarified the relationship between the positive constants in the Trudinger and the Gagliardo–Nirenberg type estimates with the exact formula, which shows these two inequalities are actually equivalent each other. Next, we shall state the result which deals with the critical weight s = n in Theorem 1.1. Obviously, the inequality (1.1) cannot hold with the weight |x|n itself. However, with a help of the logarithmic weight, we shall show the following inequality: Theorem 1.5. Let n ∈ N, 1 < p < r < ∞ and p q (r − 1)p . Then there exists a positive constant C which depends only on n, p, q and r such that uLq (Rn ; n
dx wr (x) )
Cu
n ,p
Hp
(Rn )
(1.3)
,p
holds for all u ∈ H p (Rn ), where the weight function wr (x) is given by r 1 wr (x) := log e + |x|n . |x|
(1.4)
n Furthermore, if n 2 and n−1 p < ∞, the bound (r − 1)p is sharp in the sense that the inequality (1.3) no longer holds provided q > (r − 1)p .
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
Remark 1.6. (i) There are more general results of such embeddings in case of Besov and Triebel– Lizorkin spaces including the Sobolev scale, cf. [5] and [7,8], but restricted to Muckenhoupt weights or so-called admissible weights, the former allows the weight to have a local singularity, while the latter is some class of smooth functions. We emphasize that these classes of weight functions do not cover the above limiting situation. Indeed, it is well known that the weight w1r as in (1.4) no longer belongs to even the class of Muckenhoupt weights. (ii) Since there exists an upper bound (r − 1)p with respect to q so that the inequality (1.3) holds, we cannot deduce the Trudinger type estimate from the inequality (1.3) unlike the case with the subcritical weight |x|s with 0 s < n. We additionally note that the critical exponent q = (r − 1)p comes from the following computation:
{|x|< 12 }
q 1 q p 1 dx log =∞ |x| wr (x)
provided that q (r − 1)p . Here, note that the marginal case q = (r − 1)p is included in the above observation. However, we shall overcome this difficulty to get (1.3) by using the generalized Young inequality by O’Neil [13], see Theorem B in Section 2. Finally let us describe the organization of this article. Section 2 is devoted to prepare the several lemmas for the proof of main theorems, and we shall show our theorems in Section 3. 2. Preliminaries This section is devoted to prepare several lemmas for the proof of main theorems. First, let us introduce the higher-dimensional Hardy inequality proved by Drábek, Heinig and Kufner [2]: Theorem A. (i) Let U1 and V1 be non-negative weight functions in Rn , and 1 < p q < ∞. Then the inequality
q
f (y) dy
Rn
1 q
U1 (x) dx
C1
{|y|<|x|}
1 f (x)p V1 (x) dx
p
Rn
holds for all f 0 a.e. in Rn if and only if A1 := sup R>0
1 q U1 (x) dx
{|x|>R}
−(p −1)
V1 (x)
1 dx
p
< ∞.
{|x|
Moreover, the constant C1 can be taken as
1 1 C1 = p p p q A1 . (ii) Let U2 and V2 be non-negative weight functions in Rn , and 1 < p q < ∞. Then the inequality
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
q
f (y) dy
1 q
U2 (x) dx
C2
{|y|>|x|}
Rn
3729
1
p
p
f (x) V2 (x) dx Rn
holds for all f 0 a.e. in Rn if and only if
1 q
A2 := sup
−(p −1)
U2 (x) dx
R>0
V2 (x)
{|x|
1 dx
p
< ∞.
{|x|>R}
Moreover, the constant C2 can be taken as
1 1 C2 = p p p q A2 . By scaling and changing a variable, we have the following variant of Theorem A: Theorem A . (i) Let U1 and V1 be non-negative weight functions in Rn , and 1 < p q < ∞. Then the inequality
q
f (y) dy
Rn
1 q
U1 (x) dx
C˜ 1
{2|y|<|x|}
1
p
p
f (x) V1 (x) dx Rn
holds for all f 0 a.e. in Rn if and only if A˜ 1 := sup
1 q
−(p −1)
U1 (x) dx
R>0
V1 (x)
1 dx
p
< ∞.
{|x|
{|x|>2R}
Moreover, the constant C˜ 1 can be taken as
1 1 C˜ 1 = p p p q A˜ 1 . (ii) Let U2 and V2 be non-negative weight functions in Rn , and 1 < p q < ∞. Then the inequality
q
f (y) dy
Rn
1 q
U2 (x) dx
C˜ 2
{|y|>2|x|}
1 f (x)p V2 (x) dx
p
Rn
holds for all f 0 a.e. in Rn if and only if A˜ 2 := sup
R>0
1 q
U2 (x) dx {|x|
−(p −1)
V2 (x) {|x|>2R}
1 dx
p
< ∞.
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
Moreover, the constant C˜ 2 can be taken as
1 1 C˜ 2 = p p p q A˜ 2 . In what follows, C denotes a positive constant which may vary from line to line. We shall show key lemmas by applying Theorem A below. The idea of this procedure was inspired by Rakotondratsimba [14,15], who proved the weighted Young inequalities for convolutions towards the functions behaving like the Riesz potential |x|−(n−α) with 0 < α < n. However, we need to consider not only the Riesz potential but also ϕ, ψ and the Bessel potential Gα which are defined below, and for the purpose to get exact growth orders concerning s and q, we investigate these individual kernels precisely. Lemmas 2.1–2.4 will be used to prove Theorem 1.1 in Section 3. Lemma 2.1. Let n ∈ N, 1 < p < ∞ and ϕ(x) = e−π|x| . Then there exists a positive constant C which depends only on n and p such that 2
1
1 ϕ ∗ uLq (Rn ; dxs ) C |x| n−s
q
uLp (Rn )
(2.1)
holds for all u ∈ Lp (Rn ), 0 s < n and p q < ∞. Proof. Obviously, it is enough to show the inequality (2.1) for non-negative functions. First, we decompose the integral into three parts:
dx (ϕ ∗ u)(x) 3q |x|s
q
Rn
dx |x|s
ϕ(x − y)u(y) dy
q
Rn
{|y|< |x| 2 }
q
+
ϕ(x − y)u(y) dy
Rn
{ |x| 2 |y|2|x|}
q
+
ϕ(x − y)u(y) dy
Rn
{|y|>2|x|}
dx |x|s
dx |x|s
=: 3 (S1 + S2 + S3 ). q
We first estimate S1 . Note that |y| < ϕ(x − y)u(y) dy {|y|< |x| 2 }
|x| 2
implies
|x| 2
sup ϕ(z) { |x| 2 <|z|}
< |x − y|. Hence, we see
u(y) dy = e
− π|x| 4
{|y|< |x| 2 }
q
S1
u(y) dy Rn
{|y|< |x| 2 }
e−
u(y) dy. {|y|< |x| 2 }
Thus we have
2
πq|x|2 4
|x|−s dx.
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
3731
To apply Theorem A (i), we need to check the following condition:
e
− πq|x| 4
2
−s
|x|
1 q
1
dx
p
dx
A˜ 1
(2.2)
{|x|
{2R<|x|}
holds for all R > 0, where A˜ 1 is independent of R. Indeed, once (2.2) has been established, the Hardy inequality yields that 1
1 1 S1q p p p q A˜ 1 uLp (Rn ) C A˜ 1 uLp (Rn ) , 1
where C is independent of p and q since p q and sup1
e
− πq|x| 4
2
1 q
−s
|x|
dx
{2R<|x|}
1 q
e
− πqR|x| 2
e
− πqR|x| − πqR|x| 4 4
−s
|x|
dx
{2R<|x|}
=
e
−s
|x|
1 q
dx
{2R<|x|}
e−
πR 2 2
e−
πqR|x| 4
|x|−s dx
1 q
{2R<|x|}
e−
πR 2 2
e−
π|x| 4
1 q
dx
Ce−
πR 2 2
.
{2<|x|}
By using the above estimate, we obtain
e
− πq|x| 4
2
−s
|x|
1 q
1
dx
dx
p
{|x|
{2R<|x|}
for all R 1. Case 2. We assume 0 < R < 1. In this case, we have
e
{2R<|x|}
− πq|x| 4
2
−s
|x|
1 q
dx
e Rn
− πq|x| 4
2
−s
|x|
1 q
dx
Ce−
πR 2 2
n
R p C
(2.3)
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
e
− πq|x| 4
2
−s
|x|
1
q
+
dx
e
{|x|<1} −s
|x|
1 q
dx
+
e
{|x|<1}
C
2
−s
|x|
1 q
dx
{|x|1}
− πq|x| 4
− π|x| 4
1
2
q
dx
{|x|1}
1 n−s
1 q
+ C.
Thus we have
e
− πq|x| 4
2
1 q
−s
|x|
1
dx
p
dx {|x|
{2R<|x|}
1 C n−s
1 q
(2.4)
1
1 q for all 0 < R < 1. Therefore, combining (2.3) with (2.4), we can take A˜ 1 = C( n−s ) .
Next, we estimate S3 in the similar way as S1 . Note that |y| > 2|x| implies Hence, we see
ϕ(x − y)u(y) dy {|y|>2|x|}
{ |y| 2 <|z|}
{|y|>2|x|}
sup ϕ(z) u(y) dy =
e−
π|y|2 4
|y| 2
< |x − y|.
u(y) dy.
{|y|>2|x|}
Hence, we have
q
S3
h(y) dy Rn
|x|−s dx,
where h(y) := e−
π|y|2 4
u(y).
{|y|>2|x|}
To apply Theorem A (ii), we need to check the following condition:
−s
|x|
1 q
πp|x|2 −(p −1) e 4 dx
dx
{|x|
1 p
A˜ 2
(2.5)
{2R<|x|}
holds for all R > 0, where A˜ 2 is independent of R. Indeed, once (2.5) has been established, the Hardy inequality yields that
1 1 S3 p p p q A˜ 2 1 q
p
h(x) e
πp|x|2 4
1
p
dx
Rn
To check the condition (2.5), we distinguish two cases:
C A˜ 2 uLp (Rn ) .
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
3733
Case 1. We assume R 1. Then we have
−s
|x|
1 q
dx
{|x|
1 C n−s
1 q
R
1 C n−s
n−s q
1 q
Rn.
On the other hand, we see
e
− πp 4|x|
1
2
p
dx
{2R<|x|}
1
e
− πp 2R|x|
p
dx
πR 2
=
e
{2R<|x|}
e−
e
e−
πp R|x| 4
1 p
dx
e−
πR 2 2
{2R<|x|}
Ce
dx
p
{2R<|x|}
2
2 − πR 2
1
− πp 4R|x| − πp 4R|x|
e−
π|x| 4
1 dx
p
{2<|x|}
.
Thus by using above estimates, we obtain
−s
|x|
1
q
dx
e
{|x|
− πp 4|x|
1
2
dx
p
{2R<|x|}
1 C n−s
1 q
n − πR 2
R e
2
1 C n−s
1 q
(2.6)
for all R 1. Case 2. We assume 0 < R < 1. In this case, we see
−s
|x|
1
q
dx
e
{|x|
− πp 4|x|
1
2
p
dx
{2R<|x|}
|x|−s dx
1 q
{|x|<1}
e−
πp |x|2 4
1 dx
p
C
Rn
1 n−s
1 q
(2.7)
1
1 q for all 0 < R < 1. Thus combining (2.6) with (2.7), we can take A˜ 2 = C( n−s ) . k k+1 imply that 2k−1 Finally, we estimate S2 . Note that |x| 2 |y| 2|x| and 2 |x| < 2 nq |y| < 2k+2 , and take r := n−s ∈ [q, ∞). Then by the Hölder inequality and the Young inequality, we see q ϕ(x − y)u(y) dy |x|−s dx S2 = k∈Z
k∈Z
{2k |x|<2k+1 }
2
−ks
{ |x| 2 |y|2|x|}
q
ϕ(x − y)u(y) dy {2k |x|<2k+1 }
{ |x| 2 |y|2|x|}
dx
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
2
−ks
μ 2 |x| < 2 k
k+1
1− q
{2k |x|<2k+1 }
2
−ks+kn(1− qr )
k∈Z
=C
r ϕ(x − y)u(y)χ{2k−1 |·|<2k+2 } (y) dy
Rn
Rn q
q
ϕ ∗ uχ{2k−1 |·|<2k+2 } Lr (Rn ) CϕLr˜ (Rn )
C
q r
dx
{ |x| 2 |y|2|x|}
k∈Z q
r
ϕ(x − y)u(y) dy
k∈Z
C
r
q r
dx q
uχ{2k−1 |·|<2k+2 } Lp (Rn )
k∈Z
k∈Z
q
p
u(x) dx
p
C
q
{2k−1 |x|<2k+2 }
q
p
p
u(x) dx
k∈Z k−1 {2 |x|<2k+2 }
q
= C q uLp (Rn ) , where r˜ ∈ [1, ∞) is defined by 1r + 1 = 1r˜ + p1 , and μ denotes the Lebesgue measure. In the above estimates, we used the fact that max1˜r ∞ ϕLr˜ (Rn ) < ∞ since ϕ ∈ S(Rn ). Thus we finish the proof. 2 We proceed to prove the following lemma: Lemma 2.2. Let n ∈ N, 0 < α < n and define the function ψ by ψ(x) :=
−(n−α) 2 |x| − |x − y|−(n−α) e−π|y| dy
for x ∈ Rn \ {0}.
Rn
Then there exists a positive constant C which depends only on n and α such that ψ satisfies ψ(x) C min |x|−(n−α) , |x|−(n−α+1)
for all x ∈ Rn \ {0}.
Proof. We first decompose ψ into three integrals:
−(n−α) 2 |x| − |x − y|−(n−α) e−π|y| dy
ψ(x) = {|y| |x| 2 }
−(n−α) 2 |x| − |x − y|−(n−α) e−π|y| dy
+ { |x| 2 <|y|<2|x|}
+ {|y|2|x|}
−(n−α) 2 |x| − |x − y|−(n−α) e−π|y| dy =: ψ1 (x) + ψ2 (x) + ψ3 (x).
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757 |x| 2 ,
First, we estimate ψ1 . For |y|
3735
we see
1 −(n−α) d |x| |x − τy|−(n−α) dτ − |x − y|−(n−α) = dτ 0
1
|x − τy|−(n−α+1) dτ
(n − α)|y| 0
1 x y −(n−α+1) dτ − τ |x| |x|
−(n−α+1)
= (n − α)|y||x|
0
−(n−α+1) 1 1 − τ |y| dτ |x|
−(n−α+1)
(n − α)|y||x|
0
1 τ −(n−α+1) 1− dτ 2
−(n−α+1)
(n − α)|y||x|
0 −(n−α+1)
= C|y||x|
.
Thus on one hand, we see −(n−α+1)
ψ1 (x) C|x|
|y|e
−π|y|2
−(n−α+1)
dy C|x|
|y|e−π|y| dy = C|x|−(n−α+1) , 2
Rn
{|y| |x| 2 }
and on the other hand, we obtain ψ1 (x) C|x|−(n−α+1)
C|x|−(n−α)
|y|e−π|y| dy C|x|−(n−α) 2
{|y| |x| 2 }
e−π|y| dy 2
{|y| |x| 2 }
e−π|y| dy = C|x|−(n−α) . 2
Rn
Next, we estimate ψ2 as follows: ψ2 (x) |x|−(n−α)
e−π|y| dy + 2
{ |x| 2 <|y|<2|x|}
|x|−(n−α) e−
π|x|2 4
{|y|<2|x|}
|x − y|−(n−α) e−π|y| dy 2
{ |x| 2 <|y|<2|x|}
dy + e−
π|x|2 4
{ |x| 2 <|y|<2|x|}
|x − y|−(n−α) dy
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
C|x|α e−
π|x|2 4
+ e−
π|x|2 4
|z|−(n−α) dz = C|x|α e−
π|x|2 4
{|z|<3|x|}
C min |x|−(n−α) , |x|−(n−α+1) . Finally, we estimate ψ3 .
ψ3 (x) |x|−(n−α)
e−π|y| dy + 2
{|y|2|x|}
|x − y|−(n−α) e−π|y| dy =: ψ31 (x) + ψ32 (x). 2
{|y|2|x|}
On one hand, we see −(n−α)
ψ31 (x) |x|
e−π|y| dy = C|x|−(n−α) , 2
Rn
and on the other hand, for |x| 1 we have −(n−α)
ψ31 (x) |x|
e
{|y|2|x|}
|x|−(2n−α) e−2π|x|
2
−2π|x||y|
−(n−α) −2π|x|2
dy |x|
e
e−π|x||y| dy
{|y|2|x|}
e−π|z| dz C|x|−(n−α+1) .
Rn
Next, note that |y| 2|x| implies |x − y| |x|. Then we see
ψ32 (x) |x|−(n−α)
e−π|y| dy = ψ31 (x). 2
{|y|2|x|}
Thus the estimate of ψ32 can be reduced to the estimate of ψ31 (x), and we finish the proof.
2
We can get the following lemma by applying Lemma 2.2: Lemma 2.3. Let n ∈ N, 1 < p < ∞ and set the function ψ as in Lemma 2.2 with α = pn . Then there exists a positive constant C which depends only on n and p such that the estimate ψLr (Rn ) C holds for all
np n+p
1 1 + p − r (n + p )r − np
< r < p .
Proof. By using Lemma 2.2, we see that ψrLr (Rn )
= {|x|<1}
ψ(x) dx + r
{|x|1}
ψ(x)r dx
1 r
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
− nr p
C
|x|
−( pn +1)r
dx +
{|x|<1}
|x|
3737
dx
{|x|1}
1 1 . + C p − r (n + p )r − np
2
Furthermore, by applying Lemmas 2.2 and 2.3, we prove the following: Lemma 2.4. Let n ∈ N, 1 < p < ∞ and set the function ψ as in Lemma 2.2 with α = pn . Then there exist positive constants p˜ ∈ (p, ∞) and C which both depend only on n and p such that the inequality
1 ψ ∗ uLq (Rn : dxs ) C |x| n−s
1 + 1 q
p
1
q p uLp (Rn )
holds for all u ∈ Lp (Rn ), 0 s < n and p˜ q < ∞. We prove Lemma 2.4 in the similar way to the proof of Lemma 2.1. However, it should be noted that the function ψ has a singularity at the origin which is a major difference between ψ and ϕ ∈ S(Rn ). Proof of Lemma 2.4. We may assume u is non-negative, and we take p˜ := max{p + 1, p + 1, n + 1} q < ∞. The integral is decomposed into three parts:
−s
(ψ ∗ u)(x) |x| q
dx 3
q
ψ(x − y)u(y) dy
q
Rn
Rn
|x|−s dx
{|y|< |x| 2 }
q
ψ(x − y)u(y) dy
+ Rn
{ |x| 2 |y|2|x|}
q
ψ(x − y)u(y) dy
+ Rn
|x|−s dx
|x|−s dx
{|y|>2|x|}
=: 3q (T1 + T2 + T3 ). First, we estimate T1 . Note that |y| <
q
T1
u(y) dy Rn
|x| 2
implies
|x| 2
< |x − y|. Hence, we see
˜ q |x|−s dx, ψ(x)
˜ where ψ(x) := sup ψ(z).
{|y|< |x| 2 }
To apply Theorem A (i), we need to check the following condition:
{|z|> |x| 2 }
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
˜ ψ(x) |x|
−s
q
1 q
1
dx
p
dx
A˜ 1
(2.8)
{|x|
{2R<|x|}
holds for all R > 0. Indeed, once (2.8) has been established, the Hardy inequality yields 1
T1q C A˜ 1 uLp (Rn ) . Note that Lemma 2.2 shows −n −( n +1) ˜ ψ(x) C min |x| p , |x| p
for all x ∈ Rn \ {0}.
(2.9)
We distinguish two cases: Case 1. We assume R 1. By using the latter estimate of (2.9), we see
˜ q |x|−s dx ψ(x)
1
q
−( pn +1)q−s
C
|x|
{2R<|x|}
1 q
dx
{2R<|x|}
C
( pn
1 + 1)q − (n − s)
( pn
1 + 1)q − n
C
1 q
R
1 q
R
− pn + n−s q −1
− pn + n−s q −1
CR
− pn + n−s q −1
,
where we used q n + 1 to get a constant C independent of s and q. Thus since q n + 1 and R 1, the above estimate yields that
˜ q |x|−s dx ψ(x)
1 q
1 dx
p
CR
n−s q −1
C
for all R 1.
{|x|
{2R<|x|}
Case 2. We assume 0 < R < 1. In this case, we have
˜ ψ(x) |x| q
−s
1 q
dx
{2R<|x|}
˜ ψ(x) |x| q
−s
1 q
dx
{2R<|x|<2}
=: B1 + B2 . By using q n + 1 and the latter estimate of (2.9), we see B2 C
−( pn +1)q−s
|x| {|x|2}
1 q
+
˜ ψ(x) |x| q
{|x|2}
−s
1 q
dx
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
C
1 −[( n +1)q−(n−s)] 2 p + 1)q − (n − s)
( pn
1 q
C
( pn
1 + 1)q − n
3739
1 q
C,
and by q p + 1 and the former estimate of (2.9), we obtain
− nq −s p
B1 C
|x|
1 q
dx
C
R
nq p
{2R<|x|<2}
1 C nq p − n
1 q
R
− pn + n−s q
CR
−[ nq −(n−s)] p
−1 − (n − s)
− pn + n−s q
1 q
for all 0 < R < 1.
Thus we get
˜ q |x|−s dx ψ(x)
1 q
1 p
dx
n
n−s C R q + R p C
for all 0 < R < 1.
{|x|
{2R<|x|}
As a consequence, we can take A˜ 1 = C which depends only on n and p. Next, we estimate T3 . Note that 2|x| < |y| implies
q
T3
h(y) dy
|y| 2
|x|−s dx,
< |x − y|. Then we see
˜ where h(y) := ψ(y)u(y).
{|y|>2|x|}
Rn
To apply Theorem A (ii), we need to check the following condition:
|x|−s dx
1
q
{|x|
˜ −p −(p −1) dx ψ(x)
1 p
A˜ 2
(2.10)
{2R<|x|}
holds for all R > 0 with some A˜ 2 independent of R. Indeed, once (2.10) has been established, the Hardy inequality yields 1 q
T3
1 1 p p p q A˜ 2
˜ h(x) ψ(x) p
−p
1
p
dx
C A˜ 2 uLp (Rn ) .
Rn
We distinguish two cases: Case 1. We assume R 1. In this case, by q n + 1 and the latter estimate of (2.9), we have
−s
|x| {|x|
1 q
dx {2R<|x|}
˜ p dx ψ(x)
1 p
1 C n−s
1 q
R
n−s q
−( pn +1)p
|x| {2R<|x|}
1 dx
p
3740
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
1 q n−s 1 −1 =C R q n−s 1 q 1 C for all R 1. n−s
Case 2. We assume 0 < R < 1. In this case, we see
˜ p dx ψ(x)
1 p
˜ p dx ψ(x)
{2R<|x|}
1 p
˜ p dx ψ(x)
+
{2R<|x|<2}
1 p
=: B˜ 1 + B˜ 2 .
{|x|2}
The latter estimate of (2.9) yields B˜ 2 < ∞, and the former estimate of (2.9) shows B˜ 1 C
−n
|x|
1 p
dx
{2R<|x|<2}
1 1 p = C log R
for all 0 < R < 1.
Thus we get
|x|−s dx
1 q
{|x|
1
˜ p dx ψ(x)
p
C
{2R<|x|}
1 n−s
1 q
R
n−s q
1 1 p 1 + log R
(2.11)
for all 0 < R < 1. Elementary calculus gives 1
− q 1 p max g(R) := max R log = g e p (n−s) {0
(2.12)
Combining (2.11) with (2.12) yields
−s
|x| dx {|x|
1
q
˜ p dx ψ(x)
{2R<|x|} 1
1
1 p
1 C n−s
1 + 1 q
p
1
q p ,
1
1 q + p p ) q . and then we can take A˜ 2 = C( n−s k k+1 imply that 2k−1 Finally, we estimate T2 . Note that |x| 2 |y| 2|x| and 2 |x| < 2 nq k+2 |y| < 2 , and take r := n−s ∈ [q, ∞). Then by the Hölder inequality and the Young inequality, we see
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
T2 =
k∈Z
{2k |x|<2k+1 }
2
−ks
|x|−s dx
{ |x| 2 |y|2|x|}
q
ψ(x − y)u(y) dy
q
2−ks+kn(1− r )
r ψ(x − y)u(y)χ{2k−1 |·|<2k+2 } (y) dy
Rn
Rn q
q
ψ ∗ uχ{2k−1 |·|<2k+2 } Lr (Rn ) CψLr˜ (Rn )
k∈Z
1 r
+1=
0 s < n, we see max{1,
+ p1 , i.e.,
1 r˜
np n+p } < r˜
q r
dx q
uχ{2k−1 |·|<2k+2 } Lp (Rn )
k∈Z
q
CψLr˜ (Rn ) uLp (Rn ) , where r˜ is defined by
dx
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
k∈Z
=C
q
ψ(x − y)u(y) dy
k∈Z
C
3741
1 r˜
=
1 p
+
n−s nq .
Since q max{p + 1, n + 1} and
< p . By Lemma 2.3, we easily see that
q ψLr˜ (Rn ) C n−s
1 p
.
Therefore, we get 1 q
T2 which finishes the proof.
q C n−s
1 p
uLp (Rn ) ,
2
Next, for the proof of Theorem 1.5, we prepare several tools. First, we recall the Bessel potential Gα (x) with 0 < α < n defined by Gα (x) :=
1 (4π)
α 2
∞
( α2 ) 0
t−
n−α 2 −1
e−
π|x|2 t t − 4π
dt
for x ∈ Rn ,
where denotes the Gamma function. By virtue of the identity (I − )− 2 f = Gα ∗ f for f ∈ S (Rn ), Theorem 1.5 can be changed into the following equivalent form: α
Theorem 1.5 . Let n ∈ N, 1 < p < r < ∞, p q (r − 1)p , and let wr (x) be the weight function as in Theorem 1.5. Then there exists a positive constant C which depends only on n, p, q and r such that G pn ∗ f Lq (Rn ; holds for all f ∈ Lp (Rn ).
dx wr (x) )
Cf Lp (Rn )
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
For the treatment of the marginal case q = (r − 1)p in Theorem 1.5 , we need to recall the p weak Lebesgue space Lw (Rn ) for 1 < p < ∞ and the generalized Young inequality: We say p f ∈ Lw (Rn ) if the following norm is finite, i.e., 1
f Lpw (Rn ) := sup μ x ∈ Rn ; f (x) > λ p λ, λ>0
where μ denotes the Lebesgue measure. Then O’Neil [13] proved the following inequality which generalizes the usual Young inequality: Theorem B. Let n ∈ N and 1 < p < q < ∞. Then there exists a positive constant C which depends only on n, p and q such that f ∗ gLq (Rn ) Cf Lrw (Rn ) gLp (Rn ) , where the exponent r ∈ (1, ∞) is determined by 1 +
1 q
=
1 r
(2.13)
+ p1 .
Actually, O’Neil [13] proved more general inequality than (2.13) in terms of Lorentz spaces p p Lq (Rn ) with 1 < p < ∞ and 1 q ∞. However, since it is well known that L∞ (Rn ) = p n Lw (R ), we obtain Theorem B as a particular case of the result in [13]. Furthermore, we establish the decay estimate for Gα (x), which is essentially shown in Stein [16]. However, we shall include the verification for the sake of completeness. Lemma 2.5. Let n ∈ N and 0 < α < n. Then there exists a positive constant C which depends only on n and α such that Gα (x)
C|x|−(n−α) Ce−|x|
for x ∈ Rn \ {0}, for x ∈ Rn with |x| 1.
Proof. For x ∈ Rn \ {0}, changing a variable yields ∞ Gα (x) C
t−
n−α 2 −1
e−
π|x|2 t
dt = C|x|−(n−α)
0
∞
τ−
n−α 2 −1
π
e− τ dτ = C|x|−(n−α) ,
0
which proves the former decay estimate in Lemma 2.5. Next assume |x| 1. First, we obtain 1
2
t 0
π|x| t − n−α 2 −1 − t − 4π
e
1 dt
2
t
π|x| π|x| − n−α 2 −1 − 2t − 2t
e
2
dt e
− π|x| 2
0
e−
2
1
t−
n−α 2 −1
e−
π|x|2 2t
dt
0 π|x|2 2
1 0
t−
n−α 2 −1
π
e− 2t dt = Ce−
π|x|2 2
Ce−|x| .
(2.14)
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
Moreover, elementary calculus shows that the function e− t = 2π|x|, and then we have ∞
2
t
π|x| t − n−α 2 −1 − t − 4π
e
dt e
−|x|
∞
1
t−
π|x|2 t t − 4π
n−α 2 −1
3743
for t > 0 has a maximum at
dt = Ce−|x| .
(2.15)
1
Combining (2.14) with (2.15), we get the latter estimate in Lemma 2.5.
2
In the end of this section, we prove a lemma which is necessary for the proof of the optimality of both Theorem 1.1 and Theorem 1.5. We first define 1 τ η |x| vτ (x) := log |x|
for x ∈ Rn \ {0},
(2.16)
where we take η ∈ C ∞ ([0, ∞)) satisfying the followings: (i)
0 η(t) 1
1 η(t) ≡ 1 for 0 t ; 6 1 η(t) ≡ 0 for t . 5
for t ∈ [0, ∞); (iii)
(ii)
We remark that the following lemma can be understood as the explicit version of the extremal function studied in [3, Theorem 2.7.1] and [4, Theorem 2.1]. Lemma 2.6. Let n 2 and the estimate:
n n−1
n
,p
p < ∞. Then vτ ∈ H p (Rn ) holds for any τ ∈ (0, p1 ) with
vτ
H
n p ,p (Rn )
C
1 1 p
−τ
1
p
,
where a positive constant C depends only on n and p. Proof. We first remark that the direct computation yields the derivative estimates of vτ : for any l ∈ N, there exists cl depending only on l such that β
∂ vτ (x) cl v˜τ,l |x|
holds for 1 |β| l, τ ∈ (0, 1) and x ∈ Rn \ {0},
x
where v˜τ,l (t) := t −l (log 1t )τ −1 χ[0, 1 ] (t), t ∈ (0, ∞). We note that the function v˜τ,l is non4 increasing on (0, ∞) for l 1 and τ ∈ (0, 1). In this proof, C denotes a positive constant which depends only on n and p. It is easy to show vτ Lp (Rn ) C for all τ ∈ (0, 1). Now let pn = m+α, where m is a non-negative integer and α ∈ [0, 1). In the case of α = 0, we can prove β ∂ vτ p n C x L (R )
1 1 p
−τ
1
p
for 1 |β| m and 0 < τ <
1 , p
3744
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
by directly estimating the Lp (Rn )-norm of the derivatives of vτ . Then hereafter we assume that n α ∈ (0, 1). We have 0 m n − 2 by the assumption p n−1 and α = 0. We prove this lemma n
,p
by applying the characterization of H p (Rn ) in [17, §1.7, §2.1]. Thus it is enough to show that Rn
Rn
β
β
|(∂x vτ )(x + y) − (∂x vτ )(x)| dy |y|n+α
for |β| m and 0 < τ <
p dx
C 1 p
−τ
1 , p
(2.17)
because we already obtained vτ Lp (Rn ) C for all 0 < τ < 1. Then we focus on the estimate of the integrand of (2.17). We first divide the integrand into three parts as follows:
β
β
|(∂x vτ )(x + y) − (∂x vτ )(x)| dy |y|n+α
J (x) := Rn
1
β ∇∂ vτ (x + ty) dt |y|−n−α+1 dy x
Rn 0
1 C
v˜τ,m+1 |x + ty| dt |y|−n−α+1 dy
Rn 0
1
C
v˜τ,m+1 |x + ty| dt |y|−n−α+1 dy
0 {|y|< |x| 2 }
1
+
v˜τ,m+1 |x + ty| dt |y|−n−α+1 dy
0 { |x| 2 |y|2|x|}
1
+
v˜τ,m+1 |x + ty| dt |y|−n−α+1 dy
{|y|>2|x|} 0
=: C J1 (x) + J2 (x) + J3 (x) . Since we have |x + ty|
1
{|y|< |x| 2 }
0
J1 (x)
|x| 2
for any |y| <
v˜τ,m+1
|x| 2
and 0 t 1, we estimate J1 as follows:
|x| 2 τ −1 − pn −n−α+1 log dt|y| dy = C|x| χ[0, 1 ] |x| . 2 2 |x|
Next, we estimate J2 . By changing a variable z = x + ty, we have
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
1
−n−α+1
3745
v˜τ,m+1 |x + ty| dy dt
J2 (x) C|x|
0 { |x| |y|2|x|} 2
1
−n−α+1
v˜τ,m+1 |z| dz t −n dt
= C|x|
0 { t|x| |z−x|2t|x|} 2
1
−n−α+1
= C|x|
{ |x| 2 |z−x|2|x|} 2|z−x| |x|
+ {|z−x|< |x| 2 }
t −n dt v˜ τ,m+1 |z| dz
|z−x| 2|x|
t −n dt v˜ τ,m+1 |z| dz
|z−x| 2|x|
=: C|x|−n−α+1 J21 (x) + J22 (x) . Note that |x| 2 |z − x| 2|x| implies m n − 2, we can estimate J21 as
|z−x| 2|x|
v˜τ,m+1 |z| dz C
J21 (x) C
1 4
and |z| 3|x|. Then by using the condition
v˜τ,m+1 |z| dz
{|z|3|x|}
{ |x| 2 |z−x|2|x|}
∞ ⎧ C τ −1 e−σ dσ C ⎪ ⎨ (n−m−1)τ (n−m−1) log 4 σ τ −1 = ∞ C ⎪ τ −1 e−σ dσ C|x|n−m−1 log 1 ⎩ (n−m−1) σ τ (n−m−1) log 1 3|x| 3|x|
if |x| >
1 12 ,
if |x|
1 12 ,
where we used the following claim: Claim. The estimate
∞ t
σ τ −1 e−σ dσ t τ −1 e−t holds for any t > 0 and 0 < τ < 1.
Indeed, this claim is shown as ∞ σ
τ −1 −σ
e
dσ = t
∞
τ −1 −t
− (1 − τ )
e
t
σ τ −2 e−σ dσ t τ −1 e−t .
t
On the other hand, we can estimate J22 as J22 (x) C|x|n−1 {|z−x| |x| 2 }
1 v˜τ,m+1 |z| dz n−1 |z − x|
3746
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
C|x|n−1 v˜τ,m+1
|x| 2
{|z−x| |x| 2 }
1 dz |z − x|n−1
2 τ −1 = C|x|n−m−1 log χ[0, 1 ] |x| 2 |x| |x| since |z| |x| 2 holds for |z − x| 2 . Lastly, we estimate J3 . By changing a variable z = ty, we divide the integral into two parts as follows:
1
v˜τ,m+1 |x + z| |z|−n−α+1 t α−1 dz dt
J3 (x) = 0 {|z|>2t|x|}
1
=
t α−1 dt v˜τ,m+1 |x + z| |z|−n−α+1 dz
{|z|>2|x|} 0 |z|
2|x|
+
t α−1 dt v˜τ,m+1 |x + z| |z|−n−α+1 dz
{|z|2|x|} 0
=: J31 (x) + J32 (x). We now estimate J31 . We first remark that we have J31 (x) ≡ 0 for |x| > 14 since |x + z| > 14 holds for |z| > 2|x| and |x| > 14 . Then we consider |x| 14 . Since we have |x + z| > |z| 2 for any |z| > 2|x|, we have J31 (x) =
1 α
{|z|>2|x|} n p
C = n τ (p)
n p
1 v˜τ,m+1 |x + z| |z|−n−α+1 dz α
1 log |x|
− pn
σ τ −1 eσ dσ C|x|
v˜τ,m+1 {|z|>2|x|}
|z| |z|−n−α+1 dz 2
1 τ −1 log |x|
log 4
for |x| 14 , where we used the following claim: Claim. Fix a > 0. Then there exists a positive constant Ca depending only on a such that t a
holds for any t > a and 0 < τ < 1.
σ τ −1 eσ dσ Ca t τ −1 et
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
Indeed, this claim is shown as follows. First, it is easy to show t 2. Hence we may assume a < 2. Then we have 2
σ τ −1 eσ dσ Ca
t 2
3747
σ τ −1 eσ dσ 2t τ −1 et for
and t τ −1 et (1 − τ )τ −1 e1−τ 1 for t > 0 and 0 < τ < 1,
a
where a positive constant Ca depends only on a. Therefore, this claim is true. Now we estimate J32 . We divide it into two parts as follows:
J32 (x) = C|x|−α
v˜τ,m+1 |x + z| |z|−n+1 dz
{|z|2|x|}
= C|x|−α
v˜τ,m+1 |x + z| |z|−n+1 dz
{|z|< |x| 2 }
+
v˜τ,m+1 |x + z| |z|−n+1 dz
{ |x| 2 |z|2|x|}
=: C|x|−α J321 (x) + J322 (x) . Since |z| <
|x| 2
yields |x + z| >
|x| 2 ,
J321 (x)
v˜τ,m+1
{|z|< |x| 2 }
we have
|x| 2 τ −1 −n+1 −m log |z| dz = C|x| χ[0, 1 ] |x| . 2 2 |x|
On the other hand, we remark that |x + z| 3|x| holds for |z| 2|x|. Hence, we have −n+1
J322 (x) C|x|
v˜τ,m+1 |x + z| dz C|x|−n+1
v˜τ,m+1 |y| dy
{|y|3|x|}
{ |x| 2 |z|2|x|}
C|x|−n+1
if |x| >
1 τ −1 ) C|x|−m (log 3|x|
if |x|
1 12 , 1 12
since m n − 2 in the same way as the estimate of J21 . Summing up, we obtain
1 τ −1 −n log J (x) C |x| p χ[0, 1 ] |x| + |x|−n−α+1 χ( 1 ,∞) |x| . 4l 12 l|x| 1 l= 2 ,1,3
Therefore, we have
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
J
Lp (Rn )
−n
1 τ −1 p log C χ[0, 1 ] | · | | · | 4l l| · | l= 12 ,1,3
Lp (Rn )
+ C | · |−n−α+1 χ( 1 ,∞) | · | Lp (Rn ) 12
1 4
=C
1 ∞ 1 p p 1 p(τ −1) 1 log dt +C t −p(n−m−1)−1 dt t t
0
1 12
C
1
1 1 p
since m n − 2 and 0 < τ <
p
−τ
1 p ,
which is the desired estimate.
2
3. Proof of main theorems In this section, we prove Theorem 1.1 and Theorem 1.5 by using lemmas in Section 2. Proof of Theorem 1.1. We first prove the optimality of the growth orders with respect to s and q, which is easily seen by applying Lemma 2.6. Indeed, let vτ be the function defined in (2.16) for 0 < τ < p1 . Then the direct computation yields vτ Lq (Rn ; dxs ) |x|
{|x| 16 }
τ q 1 1 +τ q 1 dx q 1 log =O qτ s |x| |x| n−s n
(3.1)
,p
as s ↑ n and q → ∞ for all τ ∈ (0, p1 ). Since vτ ∈ H p (Rn ) for all τ ∈ (0, p1 ) if n 2 and n n−1 p < ∞ by Lemma 2.6, (3.1) clearly implies the growth orders of s and q are both optimal. Thus we proceed to the proof of the affirmative part of Theorem 1.1. We may assume u ∈ n ,p S(Rn ) since S(Rn ) is dense in H p (Rn ). In what follows, F and F −1 denote the Fourier and the Fourier inverse transforms, respectively. Then for any K > 0, the function u can be decomposed into two parts such as u(x) =
e2πix·ξ (F u)(ξ ) dξ Rn
=
e Rn
2πix·ξ
ξ (F u)(ξ )ϕ K
dξ + Rn
e
2πix·ξ
ξ dξ (F u)(ξ ) 1 − ϕ K
=: u1 (x) + u2 (x), where ϕ is the function as in Lemma 2.1. We first estimate the integral of u1 . Since F ϕ = ϕ, we have
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
3749
· (x) = K n ϕ(Kx) =: ϕK (x). F −1 ϕ K Then note that u1 (x) = ϕK ∗ u(x). Since we have the scaling (ϕK ∗ u)( Kx ) = ϕ ∗ (u( K· ))(x), Lemma 2.1 yields
u1 (x)q |x|−s dx
1 q
ϕK ∗ u(x)q |x|−s dx
=
Rn
1 q
Rn
=K
− n−s q
q 1 q (ϕK ∗ u) y |y|−s dy K Rn
q 1 q · − n−s −s q (y) |y| dy =K ϕ∗ u K Rn
C
1 n−s
1 =C n−s
1 q
K 1 q
· u K p n L (R )
− n−s q n
Kp
− n−s q
uLp (Rn )
(3.2)
for all K > 0. Next, we estimate the integral of u2 . For any K > 0, the function u2 can be rewritten as u2 (x) =
e2πix·ξ Rn
n n 1 − ϕ( Kξ )
p (F u)(ξ ) dξ = ψ ˜ K ∗ (−) 2p u(x), n 2π|ξ | (2π|ξ |) p
where
− n · · −n −n (x) = C |x| p − | · | p ∗ F −1 ϕ (x) ψ˜ K (x) := F −1 2π| · | p 1 − ϕ K K
− n
−n 2 −n −n |x| p − |x − y| p e−π|Ky| dy, (3.3) = C |x| p − | · | p ∗ ϕK (x) = CK n Rn
where the last equality follows from K n Moreover, we have the scaling such as
e−π|Ky| dy = 2
Rn
Rn
e−π|y| dy = 1 for all K > 0.
n n
x · −n ψ˜ K ∗ (−) 2p u = K p ψ˜ 1 ∗ (−) 2p u (x) K K Thus by (3.3), (3.4) and Lemma 2.4, we have
2
for all K > 0.
(3.4)
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
u2 (x)q |x|−s dx
1 q
n ψ˜ k ∗ (−) 2p u(x)q |x|−s dx
=
Rn
1 q
Rn
=K
− n−s q
q 1 q
n ψ˜ K ∗ (−) 2p u x |x|−s dx K Rn
=K
− pn − n−s q
1 q q n
ψ˜ 1 ∗ (−) 2p u · |x|−s dx (x) K Rn
CK
− pn − n−s q
Rn
C =C
1 n−s 1 n−s
1 + 1 q
p
1 + 1 q
p
q 1 q
n · −s 2p (x) |x| dx ψ ∗ (−) u K
n · 2p u (−) K Lp (Rn )
1
− pn − n−s q
1
− n−s q
q p K q p K
n (−) 2p uLp (Rn )
(3.5)
for all K > 0, where ψ is the function as in Lemma 2.4, and we used |ψ˜ 1 | Cψ . By combining (3.2) with (3.5), we have
1 uLq (Rn ; dxs ) C |x| n−s
1 + 1 q
p
1
n n−s n − − n−s q p K p q uLp (Rn ) + K q (−) 2p uLp (Rn )
for all K > 0. In the end, in order to optimize the right-hand side with respect to K, we especially take K as K :=
n
(−) 2p uLp (Rn ) uLp (Rn )
p n
,
which provides the desired interpolation inequality, and we finish the proof.
2
Next, we shall prove Theorem 1.5. Proof of Theorem 1.5. First, we prove the optimality of the bound (r − 1)p , which is shown by Lemma 2.6 again. Indeed, it is easy to see that if q > (r − 1)p , taking τ close enough to p1 shows that vτ Lq (Rn ; n
,p
dx wr (x) )
= ∞.
(3.6)
n On the other hand, vτ ∈ H p (Rn ) for all τ ∈ (0, p1 ) if n 2 and n−1 p < ∞ by Lemma 2.6. Thus (3.6) implies the optimality of the bound (r − 1)p in that case.
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Thus we proceed to the proof of the affirmative part of Theorem 1.5. As we discussed in Section 2, it suffices to prove Theorem 1.5 . We may assume the function f is non-negative, and we first decompose the integral into three parts: Rn
dx 3q (G ∗ f )(x) wr (x)
q
dx wr (x)
G (x − y)f (y) dy
q
n p
n p
Rn
{|y|< |x| 2 }
q
G pn (x − y)f (y) dy
+ Rn
{ |x| 2 |y|2|x|}
Rn
q
G pn (x − y)f (y) dy
+ {|y|>2|x|}
dx wr (x)
dx wr (x)
=: 3 (U1 + U2 + U3 ). q
We first investigate U1 . Note that G pn (x) is radial function and non-increasing with respect to |x|. Moreover, |y| <
|x| 2
implies that
|x| 2
< |x − y|. Thus we see q
U1
f (y)dy Rn
{ |x| 2 <|z|}
{|y|< |x| 2 }
q
=
f (y) dy Rn
q sup G pn (z)
{|y|< |x| 2 }
dx wr (x)
q x dx . G pn 2 wr (x)
To apply Theorem A (i), we need to check the following condition:
q 1 q x dx G 2 wr (x)
1 dx
n p
{2R<|x|}
p
A˜ 1
{|x|
holds for all R > 0. Indeed, once the above estimate has been established, the Hardy inequality yields 1
1 1 U1q p p p q A˜ 1 f Lp (Rn ) .
We distinguish two cases: Case 1. We assume R 1. In this case, by the latter estimate in Lemma 2.5, we have {2R<|x|}
q x dx C G pn 2 wr (x)
{2R<|x|}
e−
q|x| 2
dx [log(e +
1 r n |x| )] |x|
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
C
e
q|x| − q|x| 4 − 4
dx Ce
{2R<|x|}
− qR 2
e−
q|x| 4
dx = Ce−
qR 2
.
{2<|x|}
Thus we have for any R 1,
q 1 q x dx G pn 2 wr (x)
1 dx
p
n
R
Ce− 2 R p C.
{|x|
{2R<|x|}
Case 2. We assume 0 < R < 1. In this case, we see q x dx = G 2 wr (x)
q x dx + G 2 wr (x)
n p
{2R<|x|}
n p
{2R<|x|<2}
{|x|2}
q x dx . G 2 wr (x) n p
By the latter estimate in Lemma 2.5, the second term is integrable, and by the former estimate in Lemma 2.5, the first term can be estimated as follows:
q x dx G 2 wr (x)
n p
{2R<|x|<2}
q x dx G C 2 |x|n
n p
{2R<|x|<2}
CR
− nq p
− nq −n p
|x|
{2R<|x|<2}
.
Thus we obtain for any 0 < R < 1,
q 1 q x dx G 2 wr (x)
1 dx
n p
{2R<|x|}
p
−n n C R p + 1 R p C.
{|x|
Next, we estimate U3 . Note that 2|x| < |y| implies
U3
G pn Rn
{|y|>2|x|}
|y| 2
< |x − y|. Thus we see
q y dx f (y) dy . 2 wr (x)
To apply Theorem A (ii), we need to check the following condition: {|x|
We distinguish two cases:
dx wr (x)
1 q
p 1 p x G dx A˜ 2 . 2 n p
{2R<|x|}
dx
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
3753
Case 1. We assume R 1. By Lemma 2.5, we have
p x G dx C 2
n p
{2R<|x|}
e−
p |x| 2
e−
p |x| p |x| 4 − 4
dx
{2R<|x|}
=C
dx Ce−
{2R<|x|}
p R 2
e−
p |x| 4
dx = Ce−
p R 2
.
{2<|x|}
Furthermore, we see {|x|
dx = wr (x)
dx + wr (x)
{|x|< 12 }
dx , wr (x)
{ 21 |x|
and it is easy to see that the first term is integrable since r > 1. The second term will be estimated as
dx wr (x)
{ 12 |x|
{ 12 |x|
dx C(1 + log R). |x|n
Combining the above estimates, we have for any R 1, {|x|
dx wr (x)
1 q
{2R<|x|}
p 1 1 p R x G pn dx C 1 + (log R) q e− 2 C. 2
Case 2. We assume 0 < R < 1, which is a crucial case where we use the condition q (r − 1)p . First, Lemma 2.5 yields G pn {2R<|x|}
p x dx = 2
G pn
{2R<|x|<2}
p x dx + 2
1 . C 1 + log R
{|x|2}
Moreover, it is easy to see that {|x|
1 −(r−1) dx C log e + . wr (x) R
Thus combining above two estimates shows that
G pn
p x dx 2
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S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
{|x|
dx wr (x)
1
q
{2R<|x|}
p 1 1 r−1 p p x 1 1 − q 1 + log G pn dx C log e + 2 R R C
since r > 1 and − r−1 q +
1 p
0, i.e., q (r − 1)p .
Finally, we estimate U2 . We first write U2 as U2 =
k∈Z
q
G pn (x − y)f (y) dy
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
dx . wr (x)
1 r Since w˜ r (x) := [log( |x| )] |x|n is non-decreasing with respect to |x| near the origin, there exists k0 ∈ Z with k0 −3 such that w˜ r (x) is non-decreasing in |x| ∈ (0, 2k0 +1 ). We decompose U2 with k0 :
U2 =
k0
q
G (x − y)f (y) dy n p
k=−∞
+
{2k |x|<2k+1 }
{ |x| 2 |y|2|x|}
∞ k=k0 +1
dx wr (x) q
G pn (x − y)f (y) dy
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
dx =: U21 + U22 . wr (x)
We first investigate U22 which is easier to estimate compared to U21 . Note that |x| 2 |y| 2|x| and 2k |x| < 2k+1 imply 2k−1 |y| < 2k+2 . Then by the Young inequality, we see U22 C
k=k0 +1
∞
q
G pn (x − y)f (y) dy
{2k |x|<2k+1 }
q
=C
k=k0 +1
∞
q
CG pn ∗ f χ{2k−1 |·|<2k+2 } Lq (Rn ) CG pn Lr˜ (Rn ) ∞
dx
{ |x| 2 |y|2|x|}
q
p
f (x) dx
p
C
k=k0 +1
{2k−1 |x|<2k+2 }
k∈Z
q
f χ{2k−1 |·|<2k+2 } Lp (Rn ) q
p
p
f (x) dx {2k−1 |x|<2k+2 }
q
= Cf Lp (Rn ) , where the exponent r˜ ∈ [1, ∞) is determined by 1 +
1 q
=
1 r˜
+ p1 , and in the above estimate, we
used G pn ∈ Lr˜ (Rn ) which is easily seen by using Lemma 2.5.
Next, we estimate U21 . Recall that w˜ r (x) is non-decreasing in |x| ∈ (0, 2k0 +1 ), and note that |y| 2|x| implies |x| |x−y| 3 . Thus by Lemma 2.5, we have
S. Nagayasu, H. Wadade / Journal of Functional Analysis 258 (2010) 3725–3757
U21 C
k=−∞
=C
|x| 2
f (y) dy
− pn
f (y)
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
q
− pn
|x − y|
1
w˜ r (x) q
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
k0
− pn
|x − y|
f (y) 1
k=−∞
Here, note that
|x − y|
k0 k=−∞
C
k0
q w˜ r ( x−y 3 )
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
3755
dx w˜ r (x)
q dy
dx q
dy
dx.
|y| 2|x| and 2k |x| < 2k+1 with k k0 yield and |x − y| 3|x| < 3 · 2k0 +1
2k−1 |y| < 2k+2
3 4
since k0 −3.
Then we further keep evaluating U21 : U21 C
k0
q
f (y) r
k=−∞
C
k0
1 [log( |x−y| )] q |x − y|
{ |x| 2 |y|2|x|}
{2k |x|<2k+1 }
q
W ∗ f χ{2k−1 |·|<2k+2 } Lq (Rn ) ,
n n q + p
where W (x) :=
k=−∞
dy
dx
χB 3 (0) (x) 4
n
n
+ 1 [log( |x| )] q |x| q p r
.
Now we distinguish two cases: Case 1. We assume p q < (r − 1)p . In this case, the Young inequality yields k0
q U21 CW Lr˜ (Rn )
q
q
q
f χ{2k−1 |·|<2k+2 } Lp (Rn ) CW Lr˜ (Rn ) f Lp (Rn ) ,
k=−∞
where the exponent r˜ ∈ [1, ∞) is determined by 1 + q1 = 1r˜ + p1 . To complete the above estimate,
we check W ∈ Lr˜ (Rn ) below. Note that ( qn + variable, we see W rL˜ r˜ (Rn )
= B 3 (0) 4
since
rp p +q
> 1, i.e., q < (r − 1)p .
n r p )˜
= n and
∞
dx 1 [log( |x| )]
rp p +q
r r˜ q
=C |x|n
log( 43 )
=
rp p +q .
dt rp
t p +q
Thus by changing a
<∞
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Case 2. We assume q = (r − 1)p . In this case, one sees that W ∈ / Lr˜ (Rn ) but W ∈ Lrw˜ (Rn ) with p r˜ = r ∈ (1, ∞). Indeed, we have χB 3 (0) (x)
W (x) =
4
1 [log( |x| )]
r p
|x|
nr p
C |x|
nr p
=: W˜ (x)
for all x ∈ Rn \ {0},
and we can easily observe that W˜ ∈ Lrw˜ (Rn ) from the definition of the weak Lebesgue space. Then by the above observation and Theorem B, we have
q
U21 CW Lr˜ (Rn ) w
k0
q q q q f χ{2k−1 |·|<2k+2 } Lp (Rn ) CW˜ Lr˜ (Rn ) f Lp (Rn ) = Cf Lp (Rn ) . w
k=−∞
Thus we finish the proof.
2
We end this section with the proof of Corollary 1.3 which is an immediate consequence of Theorem 1.1: Proof of Corollary 1.3. Let p˜ and C be positive constants depending only on n and p given by n ,p Theorem 1.1, and let α > 0 which will be chosen small enough later. Then for any u ∈ H p (Rn ) n with (−) 2p uLp (Rn ) 1, the Taylor expansion yields Rn
∞ p dx
[α(n − s)]j p j u p j n dx . Φn,p α(n − s) u(x) = L (R ; |x|s ) |x|s j! j =j0
˜ and we get Furthermore, we apply Theorem 1.1 for each norm uLp j (Rn ; dxs ) with p j p, |x|
Rn
∞ (n−s)p (αC p p j )j p dx
1 Φn,p α(n − s)u(x) uLpn(Rn ) . s |x| n−s j!
In the end, we take α =
j =1
1 2C p p e
so that β =
∞
j =1
j j ( 2e ) j!
< ∞.
2
Acknowledgments The authors would like to express their gratitude to the referee for his/her valuable comments. In addition, the first author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, and the second author is supported by the Fujukai Foundation.
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References [1] L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (3) (1984) 259–275. [2] P. Drábek, H. Heinig, A. Kufner, Higher-dimensional Hardy inequality, in: General Inequalities 7, in: Internat. Ser. Numer. Math., vol. 123, Birkhäuser, Basel, 1997, pp. 3–16. [3] D.E. Edmunds, H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996. [4] D.E. Edmunds, H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr. 207 (1999) 79–92. [5] D.D. Haroske, L. Skrzypczak, Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights I, Rev. Mat. Complut. 21 (1) (2008) 135–177. [6] H. Kozono, H. Wadade, Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO, Math. Z. 259 (4) (2008) 935–950. [7] Th. Kühn, H.G. Leopold, W. Sickel, L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces II, Proc. Edinb. Math. Soc. (2) 49 (2) (2006) 331–359. [8] Th. Kühn, H.G. Leopold, W. Sickel, L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type, Math. Z. 255 (1) (2007) 1–15. [9] T. Ogawa, A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equation, Nonlinear Anal. 14 (9) (1990) 765–769. [10] T. Ogawa, T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl. 155 (2) (1991) 531–540. [11] T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (2) (1995) 259–269. [12] T. Ozawa, Characterization of Trudinger’s inequality, J. Inequal. Appl. 1 (4) (1997) 369–374. [13] R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963) 129–142. [14] Y. Rakotondratsimba, Weighted Lp –Lq inequalities for the fractional integral operator when 1 < q < p < ∞, SUT J. Math. 34 (2) (1998) 209–222. [15] Y. Rakotondratsimba, Weighted Young inequalities for convolutions, Southeast Asian Bull. Math. 26 (2) (2002) 77–99. [16] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970. [17] R.S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967) 1031–1060. [18] H. Wadade, Remarks on the Gagliardo–Nirenberg type inequality in the Besov and the Triebel–Lizorkin spaces in the limiting case, J. Fourier Anal. Appl. 15 (6) (2009) 857–870.
Journal of Functional Analysis 258 (2010) 3758–3774 www.elsevier.com/locate/jfa
Duality on gradient estimates and Wasserstein controls Kazumasa Kuwada ∗,1,2 Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Received 13 October 2009; accepted 11 January 2010 Available online 8 February 2010 Communicated by C. Villani
Abstract We establish a duality between Lp -Wasserstein control and Lq -gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance. © 2010 Elsevier Inc. All rights reserved. Keywords: Wasserstein distance; Gradient estimate; Subelliptic diffusion; Hamilton–Jacobi semigroup
1. Introduction There are several ways to formulate a quantitative estimate on rate of convergence to equilibrium. By means of functional inequalities, an Lq -gradient estimate for a heat semigroup Pt |∇Pt f |(x) e−kt Pt |∇f |q (x)1/q
(1.1)
has been known to be a very powerful tool. It implies several functional inequalities such as Poincaré inequalities (when q = 2) and logarithmic Sobolev inequalities (when q = 1), which * Fax: +49 228 73 6716.
E-mail address:
[email protected]. 1 Partially supported by the JSPS fellowship for research abroad. 2 Permanent address: Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan.
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.010
K. Kuwada / Journal of Functional Analysis 258 (2010) 3758–3774
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quantify convergence rates (see [2,4,5,21] and the references therein). As a different approach to this problem, F. Otto [30] discussed a contraction of Lp -Wasserstein distance dpW (μt , νt ) e−kt dpW (μ0 , ν0 )
(1.2)
for two (linear or nonlinear) diffusions μt , νt of masses when p = 2. His heuristic observation based on the geometry of the L2 -Wasserstein space has been a source of enormous developments in the theory of optimal transport (see [37] and the references therein). To investigate a relation between these formulations makes a connection between different approaches and hence it is an interesting problem. M.-K. von Renesse and K.-T. Sturm [31] unified several formulations of this kind for linear heat equation on a complete Riemannian manifold. As a consequence of their work, (1.1) or (1.2) is shown to be equivalent to the presence of a lower Ricci curvature bound by k (it also holds for k < 0). But, in a more general framework, such a sort of duality has been known only when p = 1 and q = ∞, which is the weakest form for (1.1) and (1.2) both. The main result of this paper extends the duality to that between an Lq -gradient estimate and an Lp -Wasserstein control for p, q ∈ [1, ∞] with p −1 + q −1 = 1 beyond the case of a heat flow on a complete Riemannian manifold (see Theorem 2.2 for the precise statement). We should emphasize that our duality does not require any other kind of curvature conditions. An L∞ -Wasserstein control has been used in the literature as a tool to show L1 -gradient estimate in a coupling method for stochastic processes (for instance, see [38] and the references therein). In the case of heat flows in a complete Riemannian manifolds, any construction of a coupling which carries L∞ -Wasserstein control relies on lower Ricci curvature bounds. In fact, such an argument was used in von Renesse and Sturm’s work. As a result, their proof employs a lower Ricci curvature bound to deduce Wasserstein controls from gradient estimates. Our result enables us to derive Wasserstein controls directly from gradient estimates. Such an implication is not known even in the case of heat flows on a Riemannian manifold. Furthermore, this is a great advantage under the lack of an appropriate notion of lower curvature bounds. Our work is strongly motivated by recent development on gradient estimates on a Lie group endowed with a sub-Riemannian structure [5,8,12,13,22,27]. To explain a consequence of our duality, we deal with the 3-dimensional Heisenberg group here. It is the simplest example of spaces possessing a non-Riemannian sub-Riemannian structure like a flat Euclidean space in Riemannian geometry. But, unlike Euclidean spaces, some results [12,17] indicate that the “Ricci curvature” should be regarded as being unbounded from below (in a generalized sense). Nevertheless, Lq -gradient estimates hold for q ∈ [1, ∞] with a constant K > 1 instead of e−kt in (1.1) [5,12,13,22]. We can apply our duality to this case to obtain the corresponding Lp -Wasserstein control for any p ∈ [1, ∞]. In the theory of optimal transport on the Heisenberg group, an L2 Wasserstein control for the heat flow would be important (cf. [18]). In probabilistic point of view, the heat flow is described by motions of a pair of the 2-dimensional Euclidean Brownian motion and the associated Lévy stochastic area. Our L∞ -Wasserstein control means the existence of a coupling of two particles so that the distance between them at time t is controlled by the initial distance almost surely. It is sometimes a complicated issue to construct a “well-behaved” coupling in the absence of curvature bounds. Especially, see [9,20] for works on a successful coupling on the Heisenberg group and its extension. Note that our formulation also fits with studying a heat semigroup under backward (super-)Ricci flow, in which case Wasserstein contractions with respect to a time-dependent distance function is shown recently [3,26].
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The notion of lower Ricci curvature bound has been extended in many ways. Although our result does not need those notions, they should be related since (1.1) and (1.2) are analytic and probabilistic characterizations of a lower Ricci curvature bound respectively. Here we review two extensions and observe how these are connected with our result. In an analytic way, D. Bakry and M. Émery [6] (see also [2] and the references therein) extend the notion of lower Ricci curvature bound to Γ2 -criterion or curvature-dimension condition. In an abstract framework where it works, a Γ2 -criterion is equivalent to an L1 -gradient estimate. Note that their notion of gradient is different from ours. But, once these two notions coincide, a Γ2 -criterion becomes equivalent to L∞ -Wasserstein control with the aid of our result. In a sufficiently regular case as diffusions on a manifold, such an equivalence is well known. Our result possibly provides an extension of this equivalence. In connection with the theory of optimal transport, convexities of entropy functionals are proposed by J. Lott, C. Villani and K.-T. Sturm [24,35] as a natural extension of lower Ricci curvature bound. Under this condition, the existence of a heat flow and an L2 -Wasserstein control follow in some cases beyond Riemannian manifolds [29,33] (see [14,37] for the case on a Riemannian manifold). With the aid of Theorem 8 in [33], we can apply our duality to show an L2 -gradient estimate for the heat semigroup. The idea of the proof of our main theorem is simple. The implication from a Wasserstein control to the corresponding gradient estimate is just a slight modification of existing arguments. The converse is based on the Kantorovich duality. If p = 1, the Kantorovich duality becomes the Kantorovich–Rubinstein formula and the problem becomes much simpler. In the case p > 1, we employ a general theory of Hamilton–Jacobi semigroup developed in [7,23] to analyze the variational formula. When p = ∞, we use an approximation of p by finite numbers because we are no longer able to apply the Kantorovich duality directly. Note that no semigroup property for heat semigroups is required in the proof. With keeping such a generality, our duality is sufficiently sharp in the sense that the control rate does not change when we obtain one estimate from the other, like the same e−kt appears in (1.1) and (1.2) both. The organization of this paper is as follows. In the next section, we introduce our framework and state our main theorem. We review the notion of Wasserstein distance and gradient there. Our main theorem is shown in Section 3. For the proof, we show basic properties of Wasserstein distances and summarize recent results on Hamilton–Jacobi semigroup there. In Section 4, we consider a heat flow on a sub-Riemannian manifold and apply our main theorem to these cases. 2. Framework and the main result Let (X, d) be a complete, separable, proper, length metric space. Here, we say that d is a length metric if, for every x, y ∈ X, d(x, y) equals infimum of the length of a curve joining x and y. Properness means that all closed metric balls in X of finite radii are compact. Under these assumptions, there exists a curve joining x and y whose length realizes d(x, y) for each x, y (see [10], for instance). We call it minimal geodesic. Let d˜ be a continuous distance function on X, possibly different from d. Assume that for any x, y ∈ X, there is a minimal geodesic with ˜ respect to d˜ joining x and y. We call such a curve “d-minimal geodesic”. For two probability measures μ and ν on X, we denote the space of all couplings of μ and ν by Π(μ, ν). That is, π ∈ Π(μ, ν) means that π is a probability measure on X × X satisfying π(A × X) = μ(A) and π(X × A) = ν(A) for each Borel set A. For p ∈ [1, ∞] and a measurable function ρ : X × X → [0, ∞), we define ρpW (μ, ν) by ρpW (μ, ν) := inf ρLp (π) π ∈ Π(μ, ν) .
(2.1)
K. Kuwada / Journal of Functional Analysis 258 (2010) 3758–3774
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˜ If dpW (μ, ν) < ∞, then there always exists a We are interested in the case ρ = d and ρ = d. minimizer of the infimum on the right-hand side in (2.1). In addition, dpW satisfies all properties of distance function on the space of probability measures though it may take the value +∞. The same are also true for d˜pW . These facts are well known for p ∈ [1, ∞) and we can show it similarly even when p = ∞. It is sometimes reasonable to restrict dpW on all probability measures having finite p-th moments in order to ensure dpW (μ, ν) < ∞. But, in this paper, we do not adopt such a restriction. Note that, when p < ∞, we usually call the restriction of dpW the Lp Wasserstein distance. See [36] for more details and a proof of these facts. Let Cb (X) be the space of bounded continuous functions on X equipped with the supremum norm. Let CL (X) be the collection of all Lipschitz continuous functions on X and Cb,L (X) := Cb (X) ∩ CL (X). Note that, if we merely say “Lipschitz”, it means “Lipschitz with respect to d”. ˜ we use the expression “d-Lipschitz”. ˜ For Lipschitz continuity with respect to d, For a measurable function f on X and x ∈ X, we define |∇d f |(x) by f (x) − f (y) . r↓0 0
|∇d f |(x) := lim
sup
We set ∇d f ∞ := supx∈X |∇d f |(x). Note that ∇d f ∞ < ∞ holds if and only if f ∈ CL (X). In addition, for f ∈ CL (X), f (x) − f (y) . ∇d f ∞ = sup d(x, y) x =y
(2.2)
For a pair of measurable functions f and g on X, we say that g is an upper gradient of f if, for each rectifiable curve γ : [0, l] → X parametrized with the arc-length, we have f γ (l) − f γ (0)
l
g γ (s) ds.
0
We will use the following fact as a basic tool. Lemma 2.1. (See [11, Proposition 1.11], [16, Proposition 10.2].) For f ∈ CL (X), |∇d f | is an upper gradient of f . ˜ All the properties described above for |∇d f |, including We also use the same notations for d. Lemma 2.1, are also true for |∇d˜ f |. Set P(X) be the space of all probability measures on X equipped with the topology of weak convergence. Let (Px )x∈X be a family of elements in P(X). Assume that x → Px is continuous as a map from X to P(X). Then (Px )x∈X defines a bounded linear operator P on Cb (X) by Pf (x) := X f (y)Px (dy). Let P ∗ be the adjoint operator of P . Note that P ∗ (P(X)) ⊂ P(X) holds. For describing our main theorem, we state the following conditions:
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Assumption 1. There exists a positive Radon measure v on X such that (i) (X, d, v) enjoys the local volume doubling condition. That is, there are constants D, R1 > 0 such that v(B2r (x)) Dv(Br (x)) holds for all x ∈ X and r ∈ (0, R1 ). (ii) (X, d, v) supports a (1, p0 )-local Poincaré inequality for some p0 1. That is, for every R > 0, there are constants λ 1 and CP > 0 such that, for any f ∈ L1loc (v) and any upper gradient g of f ,
|f − fx,r | dv CP r Br (x)
g p0 dv
1/p0 (2.3)
Bλr (x)
holds for every x ∈ X and r ∈ (0, R), where fx,r := v(Br (x))−1 Br (x) f dv. (iii) Px is absolutely continuous with respect to v for all x ∈ X; Px (dy) = Px (y) v(dy). In addition, the density Px (y) is continuous with respect to x for v-almost every y ∈ X. Now we are in turn to state our main theorem. Theorem 2.2. Suppose that Assumption 1 holds. Then, for any p ∈ [1, ∞], the following are equivalent: (i) For all μ, ν ∈ P(X), dpW P ∗ μ, P ∗ ν d˜pW (μ, ν).
(Cp )
(ii) When p > 1, for all f ∈ Cb,L (X) and x ∈ X, |∇d˜ Pf |(x) P |∇d f |q (x)1/q ,
(Gq )
where q is the Hölder conjugate of p; 1/p + 1/q = 1. When p = 1, for all f ∈ Cb,L (X), ∇d˜ Pf ∞ ∇d f ∞ .
(G∞ )
Remark 2.3. We give several remarks on Assumption 1 and Theorem 2.2. (i) If Assumption 1(i) holds, then Assumption 1(ii) follows once we obtain (2.3) with p0 = 1 for some R > 0 by a well-known argument. See [32, Lemma 5.3.1], for instance. The same is true for a (2, 2)-Poincaré inequality, which yield a (1, 2)-Poincaré inequality. (ii) It is shown in [11] that, under Assumption 1(i)–(ii), |∇d f | coincides with an Lp0 -minimal generalized upper gradient gf for those f for which gf is well defined. This fact itself is not used in this article. But, it will be helpful when we apply our main theorem to more concrete problems. In fact, the notion of minimal generalized upper gradients is regarded as a sort of weak derivative in the theory of Sobolev spaces. We can identify these two notions on Euclidean spaces or Riemannian manifolds. (iii) Assumption 1 is used only when we show the implication (Gq ) ⇒ (Cp ) for p ∈ (1, ∞]. Thus the rest holds true without Assumption 1. We need Assumption 1(i)–(ii) only for employing a property of Hamilton–Jacobi semigroups. To make these facts clear, in the rest of this paper, we will mention Assumption 1 when we require it.
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(iv) The duality between (1.1) and (1.2) is resumed by choosing P = Pt and d˜ = e−kt d. The case d˜ is essentially different from d naturally occurs if we consider a heat flow under a backward (super-)Ricci flow (see [3,26]). (v) Obviously (Gq ) implies (Gq ) for q, q ∈ [1, ∞] with q < q by the Hölder inequality. The dual implication (Cp ) ⇒ (Cp ) for p, p ∈ [1, ∞] with p > p also holds true without using the equivalence in Theorem 2.2 (see Corollary 3.4 below). For a heat flow on a Riemannian manifold (i.e. P = Pt and d˜ = e−kt d), if (Cp ) or (Gq ) holds for some p ∈ [1, ∞], then (Cp ) and (Gq ) hold for any p ∈ [1, ∞]. At this moment, it is not clear that what condition guarantees such an “Lp -independence”. 3. Proof of Theorem 2.2 We begin with showing the implication (Cp ) ⇒ (Gq ). Proposition 3.1. Suppose (Cp ) for p ∈ [1, ∞]. Then (Gq ) holds for q ∈ [1, ∞] with p −1 + q −1 = 1. Proof. For x, y ∈ X, take πxy ∈ Π(Px , Py ) such that dLp (πxy ) = dpW (Px , Py ). Since Pz = ˜ y). For f ∈ Cb,L (X), P ∗ δz for z ∈ X, (Cp ) yields dpW (Px , Py ) d˜pW (δx , δy ) = d(x, f (z) − f (w) πxy (dz dw). Pf (x) − Pf (y) = f dPx − f dPy X
X
X×X
(i) The case p = 1: (2.2) together with (C1 ) implies f (z) − f (w) πxy (dz dw) ∇d f ∞ d W (Px , Py ) ∇d f ∞ d(x, ˜ y). 1 X×X
˜ y) and by taking supremum in x = y, the Hence, by dividing the above inequalities by d(x, conclusion follows. (ii) The case p ∈ (1, ∞): Let us define Gr : X → R by f (z) − f (w) . sup Gr (z) := d(z, w) w∈Br (z)\{z} ˜ y)1/(2q) . The Hölder inequality and the Chebyshev inequality yield Set r := d(x, f (z) − f (w) πxy (dz dw) X×X
f (z) − f (w) = d(z, w) 1{0
+ X×X
f (z) − f (w)1{d(z,w)>r} πxy (dz dw)
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1/q 2f ∞ dLp (πxy ) f (z) − f (w) q dLp (πxy ) + d(z, w) 1{0
Gr Lq (Px ) dpW (Px , Py ) +
2f ∞ dpW (Px , Py )p rp
˜ y) + 2f ∞ d(x, ˜ y)1+(p−1)/2 . Gr Lq (Px ) d(x, Here the last inequality follows from (Cp ). Since limy→x r = 0, limy→x Gr (z) = |∇d f |(z) holds. By virtue of |Gr (z)| ∇d f ∞ , we can apply the dominated convergence theorem to obtain ˜ y) and limy→x Gr Lq (Px ) = |∇d f |Lq (Px ) . Thus, by dividing the above inequalities by d(x, by tending y → x, the conclusion follows. ˜ y) for πxy -a.e. (z, w). Hence we have (iii) The case p = ∞: (C∞ ) implies d(z, w) d(x,
f (z) − f (w) πxy (dz dw) d(x, ˜ y)G ˜
d(x,y) L1 (Px ) .
X×X
Thus the proof will be completed by following a similar argument as above.
2
For the converse implication, first we show two auxiliary lemmas concerning to Wasserstein distances. The first one will be used to deal with L∞ -Wasserstein distance. Lemma 3.2. Let ρ : X × X → [0, ∞) be a continuous function. Then W lim ρ W (μ, ν) = ρ∞ (μ, ν) p→∞ p
for any μ, ν ∈ P(X). Proof. Note that ρpW (μ, ν) is increasing in p by the Hölder inequality. Hence C := limp→∞ ρpW (μ, ν) ∈ [0, ∞] exists. Take πn ∈ Π(μ, ν) for n ∈ N such that ρnW (μ, ν) = ρLn (πn ) holds. Since πn ∈ Π(μ, ν), (πn )n∈N is tight. Thus there exists a convergent subsequence (πnk )k∈N of (πn )n∈N . We denote the limit of πnk by π∞ . Take R > 0 and n ∈ N arbitrary. Since ρ ∧ R ∈ Cb (X × X), we have ρ ∧ RLn (π∞ ) = lim ρ ∧ RLn (πnk ) lim ρLnk (πnk ) = C. k→∞
k→∞
Here the inequality follows from the Hölder inequality for sufficiently large k. Thus, as W (μ, ν) = ∞. R → ∞ and n → ∞, we obtain ρL∞ (π∞ ) C. Thus the assertion holds if ρ∞ W When ρ∞ (μ, ν) < ∞, we can take π ∈ Π(μ, ν) such that ρL∞ (π) < ∞. Then ρpW (μ, ν) W (μ, ν) and hence the conρLp (π) ρL∞ (π) . Thus C ρL∞ (π) holds. It yields C ρ∞ clusion holds. 2 The next one is useful to reduce the problem in a simpler case. Lemma 3.3. If (Cp ) holds for any pair of Dirac measures, then (Cp ) holds for any μ, ν ∈ P(X).
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Although this is probably well known for experts at least when p ∈ [1, ∞), we give a proof for completeness. Proof of Lemma 3.3. First we consider the case p < ∞. Given μ, ν ∈ P(X), take π ∈ Π(μ, ν) ˜ Lp (π) = d˜pW (μ, ν). We may assume d˜pW (μ, ν) < ∞ without loss of generality. so that d For x, y ∈ X, take Px,y ∈ Π(Px , Py ) so that dLp (Px,y ) = dpW (Px , Py ). By Corollary 5.22 of [37], we can choose {Px,y }x,y∈X so that the map (x, y) → Px,y is measurable. Define π˜ ∈ Π(P ∗ μ, P ∗ ν) by π˜ (A) := X×X Px,y (A) π(dx dy). Then (Cp ) for Dirac measures implies dpW
∗ P μ, P ∗ ν dLp (π˜ ) =
1/p
p dLp (Px,y ) π(dx dy)
X×X
˜ Lp (π) = d˜pW (μ, ν). d Thus the assertion holds. When p = ∞, (C∞ ) for Dirac measures implies (Cp ) for Dirac measures for any 1 p < ∞. Thus we obtain (Cp ) for any μ, ν ∈ P(X). Hence applying Lemma 3.2 for ρ = d and ρ = d˜ yields the conclusion. 2 By the Hölder inequality, (Cp ) for Dirac measures yields (Cp ) for Dirac measures if p < p. Thus we obtain the following as a by-product of Lemma 3.3. Corollary 3.4. (Cp ) implies (Cp ) for any p, p ∈ [1, ∞] with p > p . Next we introduce the notion and some properties of Hamilton–Jacobi semigroup, which plays an essential role in the sequel. Let L : [0, ∞) → [0, ∞) be a convex superlinear function with L(0) = 0. Note that L is continuous and increasing. We denote the Legendre conjugate of L by L∗ : [0, ∞) → [0, ∞), which is given by L∗ (z) = supw0 [wz − L(w)]. For f ∈ Cb (X) and t > 0, we define a function Qt f on X by
d(x, y) . Qt f (x) := inf f (y) + tL y∈X t For convenience, we write Q0 f := f . We call Qt the Hamilton–Jacobi semigroup associated with L. Several basic properties of Qt f in an abstract framework are studied in [7,23]. In [23], they assumed X to be compact and L(s) = s 2 . In [7], they assumed f ∈ CL (X). Among them, the following are all we need in this paper. Lemma 3.5. (See [7, Theorem 2.5], [23, Theorem 2.5].) (i) (ii) (iii) (iv)
infy∈X f (y) Qt f (x) f (x). In particular, Qt f ∈ Cb (X). Qt (Qs f ) = Qt+s f . Qt f (x) is nonincreasing in t and limt↓0 Qt f (x) = f (x). Set u(t, x) = Qt f (x). If f ∈ CL (X), then u ∈ CL ((0, ∞) × X). Moreover, sup
s =t y =x
|u(t, x) − u(s, y)| ∇d f ∞ ∨ L∗ ∇d f ∞ . |t − s| + d(x, y)
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(v) Suppose Assumption 1(i)–(ii) holds. Then, for t > 0 and v-a.e. x ∈ X, Qt f satisfies the Hamilton–Jacobi equation associated with L∗ : lim s↓0
Qt+s f (x) − Qt f (x) = −L∗ |∇d Qt f |(x) . s
We do not use Lemma 3.5(ii) in the sequel. But, it explains why we call Qt “semigroup” well. Note that Lemma 3.5(v) is shown in [7,23] for the subgradient norm instead of the gradient norm |∇d f |. Since these two notions coincide v-almost everywhere in this case (see [23, Remark 2.27]), Lemma 3.5(v) is still valid. Finally, we review the Kantorovich duality (see [36, Theorem 1.3] or [37, Theorem 5.10], for example). For μ, ν ∈ X and 1 p < ∞, the following duality holds: ⎧ ⎨
⎫ f, g ∈ Cb (X), ⎬ dpW (μ, ν)p = sup g dμ − f dν g(y) − f (x) d(x, y)p , ⎩ ⎭ for any x, y ∈ X X X ∗ f dμ − f dν , = sup f ∈Cb (X)
X
(3.1)
X
where f ∗ (y) := infx∈X [f (x) + d(x, y)p ]. In particular, when p = 1, (3.1) is written as follows: d1W (μ, ν) =
sup
f ∈CL (X) ∇f ∞ 1 M
f dμ −
f dν .
(3.2)
M
This is the so-called Kantorovich–Rubinstein formula (see [36, Theorem 1.14] or [37, Particular Case 5.16]). Remark 3.6. An observation on the proof in [37] tells us that the latter supremum in (3.1) can be approximated by elements in Cb,L (X). Actually, in that proof, there appears a sequence of pairs of functions φk , ψk ∈ Cb (X) approximating the former supremum in (3.1) by taking f = ψk , g = φk . We can easily verify ψk ∈ Cb,L (X) and that (ψk )k∈N also approximates the latter supremum in (3.1). Moreover, we can assume that each element of approximating sequence has a compact support without loss of generality, thanks to the tightness of μ, ν and the properness of X. Now we are in position to complete the proof of Theorem 2.2. Proposition 3.7. Suppose that Assumption 1 holds. Then (Gq ) implies (Cp ) for p, q ∈ [1, ∞] with p −1 + q −1 = 1. ˜ Proof. By virtue of Lemma 3.3, it suffices to show (Cp ) for μ = δx , ν = δy , x = y. Take a dminimal geodesic γ : [0, 1] → X from y to x, which is re-parametrized to have a constant speed. ˜ s , γt ) = |s − t|d(x, ˜ y). Note that, by (Gq ), Pf is d-Lipschitz ˜ Here “constant speed” means d(γ continuous if f ∈ CL (X).
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(i) The case p = 1: The Kantorovich–Rubinstein formula (3.2) yields d1W (Px , Py ) =
Pf (x) − Pf (y) .
sup
f ∈CL (X) ∇d f ∞ 1
(3.3)
For f ∈ CL (X), we can apply Lemma 2.1 to Pf . Thus (G∞ ) yields Pf (x) − Pf (y)
˜ d(x,y)
˜ y). |∇d˜ Pf |(γs ) ds ∇d f ∞ d(x,
0
Combining this estimate with (3.3), the conclusion follows. (ii) The case 1 < p < ∞: Let Qt be the Hamilton–Jacobi semigroup associated with L(s) := p −1 s p . Note that its Legendre conjugate L∗ is computed as L∗ (s) = q −1 s q . By (3.1) and Remark 3.6, we have dpW (Px , Py )p =
∗ P f (x) − Pf (y)
sup
f ∈Cb,L (X)
=p
P Q1 f (x) − Pf (y) .
sup
f ∈Cb,L (X)
(3.4)
To obtain an integral expression of the term in the above supremum (see (3.5) below), we give some estimates. (Gq ) and Lemma 3.5(i) and (iv) yield |∇d˜ P Qs f |(z) |∇d Qs f |Lq (P ) ∇d f ∞ ∨ L∗ ∇d f ∞ z
for s 0 and z ∈ X. Thus Lemmas 2.1 and 3.5(iv) imply P Qt+s f (γt+s ) − P Qs f (γs ) P Qt+s f (γt+s ) − P Qt+s f (γs ) t t Qt+s f − Qs f + dPγs t X t+s ˜ y) d(x, |∇d˜ P Qt+s f |(γu ) du t s
Qt+s f − Qs f dPγ + s t X
˜ y) ∇d f ∞ ∨ L∗ ∇d f ∞ 1 + d(x,
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for s 0. It means that P Qs f (γs ) is Lipschitz continuous as a function of s ∈ [0, 1]. Hence there exists a derivative ∂s (P Qs f (γs )) for a.e. s ∈ [0, 1] and we have 1 P Q1 f (x) − Pf (y) =
∂s P Qs f (γs ) ds.
(3.5)
0
Let s ∈ (0, 1) be a point where P Qs f (γs ) is differentiable. It implies P Qs+t f (γs+t ) − P Qs f (γs ) ∂s P Qs f (γs ) = lim . t↓0 t
(3.6)
We have P Qs+t f (γs+t ) − P Qs f (γs ) = t
Qs+t f − Qs f dPγs+t t
X
+
P Qs f (γs+t ) − P Qs f (γs ) . t
(3.7)
By Lemma 2.1 together with (Gq ), s+t ˜ y) 1/q P Qs f (γs+t ) − P Qs f (γs ) d(x, P |∇d Qs f |q (γu ) du. t t
(3.8)
s
By virtue of Assumption 1(iii), the Fatou lemma together with the boundedness of |∇d Qt f | implies that (P |∇d Qs f |q )(γu ) is upper semi-continuous in u. Thus (3.8) yields lim sup t↓0
P Qs f (γs+t ) − P Qs f (γs ) ˜ y)|∇d Qs f | q d(x, . L (Pγs ) t
(3.9)
For the first term in (3.7), Lemma 3.5(iii) implies the integrand is nonpositive. Thanks to Assumption 1(i)–(ii), Lemma 3.5(v) is applicable to the integrand. Thus the Fatou lemma together with Assumption 1(iii) yields lim sup t↓0
X
Qt+s f − Qs f dPγs+t = lim sup t t↓0
X
lim sup X
t↓0
=− X
Qt+s f (z) − Qs f (z) Pγs+t (z) v(dz) t Qt+s f (z) − Qs f (z) Pγs+t (z) v(dz) t
L∗ |∇d Qs f |(z) Pγs (z) v(dz).
(3.10)
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Combining (3.7), (3.9) and (3.10) with (3.5) and (3.6), 1 P Q1 f (x) − Pf (y)
˜ y)|∇d Qs f | d(x,
Lq (Pγs )
− L∗ |∇d Qs f |Lq (P
γs )
ds
0
˜ y) , L d(x, where the second inequality comes from the definition of L∗ as the Legendre conjugate. Substituting this estimate into (3.4), we obtain the desired estimate. (iii) The case p = ∞: Since (Gq ) holds with q = 1, the Hölder inequality implies (Gq ) for any q > 1. Thus we obtain (Cp ) for any 1 p < ∞. Therefore, by virtue of Lemma 3.2, the conclusion follows by tending p to ∞ in (Cp ). 2 Remark 3.8. Our duality between Lp and Lq can be extended to a similar one between Orlicz norms. In fact, there are Hölder-type inequalities (see [1], for instance) which will be used in the implication (i) ⇒ (ii). For the converse, all properties of Hamilton–Jacobi semigroup we will use in the proof still hold in such a generality. Remark 3.9. If (Cp ) holds with p > 1, then we obtain the following slightly stronger version of (G∞ ); for any f ∈ Cb,L (X) and x ∈ X, |∇d˜ Pf |(x) |∇d f |L∞ (P ) . (G∞ ) x
As we have seen in the proof of Proposition 3.7, a weaker condition (G∞ ) is sufficient to obtain (C1 ). At this moment, the author does not know any example that (Cp ) holds only for p = 1 and (G∞ ) fails. 4. Applications In a class of sub-Riemannian manifolds, Lq -gradient estimates of a subelliptic heat semigroup is shown recently by analytic methods. In these cases, we can obtain the corresponding Lp Wasserstein control via Theorem 2.2 though their notion of gradient looks different from ours. To explain how we deal with it, we will demonstrate a general framework of sub-Riemannian geometry generated by a family of vector fields. We refer to [16,28,34] for details. Throughout this section, we assume X to be a finite dimensional, σ -compact, connected, smooth differentiable manifold. Consider a family of vector fields {X1 , . . . , Xn } on X. We assume that {Xi (x)}ni=1 is linearly independent on Tx X for all x ∈ X and that {Xi }ni=1 satisfies the Hörmander condition. The latter one means that there exists a number m such that the family of vector fields generated by {Xi }ni=1 and their commutators up to the length m spans Tx X for each x ∈ X. Let H ⊂ T X be the subbundle generated by {Xi }ni=1 ; Hx := Span{X1 (x), . . . , Xn (x)}. We define a metric on H such that {Xi (x)}ni=1 becomes an orthonormal basis of Hx for x ∈ X. We are interested in the case H = T X. Associated with this metric, we define a function d on X × X as follows. We say a piecewise smooth curve γ : [0, l] → X horizontal if γ˙ (t) ∈ Hγ (t) for every t where γ is differentiable. For x, y ∈ X, we define d(x, y) by l d(x, y) := inf 0
γ˙ (t)
γ : [0, l] → X horizontal curve, . Hγ (t) dt γ (0) = x, γ (l) = y
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By the Chow theorem, the Hörmander condition ensures that d(x, y) < ∞ for x, y ∈ X. As a result, the function d : X × X → [0, ∞) becomes a distance. It is called the Carnot–Caratheodory distance. Note that the topology determined by d coincides with the original one on X. We assume that (X, d) is complete. Let v be a Borel measure on X such that its restriction on each local coordinate has a smooth density with respect to the Lebesgue measure associated with the coordinate. Let H := ni=1 Xi∗ Xi be the sub-Laplacian associated with {Xi }ni=1 and v. Here Xi∗ is the adjoint operator of Xi with respect to v. By the completeness of d, H is essentially self-adjoint (see [34]). Take the self-adjoint extension of H (also denoted by H ) and consider the associated heat semigroup Pt = exp(tH /2). By the hypoellipticity of H , Pt has a smooth density function with respect to v. In particular, Pt becomes a Feller semigroup. We assume that Pt is conservative, i.e. Pt 1 = 1. For a smooth function f : X → R, we define the carré du champ operator Γ (f ) : X → R by Γ (f )(x) := ni=1 |Xi f (x)|2 . An Lq -gradient estimate for Pt associated with Γ is formulated as follows; given q ∈ [1, ∞), there exists Kq (t) > 0 for each t > 0 such that, for any f ∈ Cc∞ (X), 1/q Γ (Pt f )(x)1/2 Kq (t) Pt Γ (f )q/2 (x) ,
(4.1)
where Cc∞ (X) is the set of all smooth functions f : X → R with compact support. As we see in the following, (4.1) implies our gradient estimate. Proposition 4.1. Eq. (4.1) for f ∈ Cc∞ (X) implies (Gq ) for P = Pt , d˜ = Kq (t)d and any f ∈ CL (X) with compact support. Proof. First we extend (4.1) for f ∈ Cb,L (X). By virtue of Corollary 11.8 of [16], for f ∈ Cb,L (X), the distributional derivatives {Xi f }ni=1 are represented as a bounded functions and |Γf |1/2 ∇d f ∞ holds v-almost everywhere. Moreover, Theorem 11.7 of [16] implies |Γf |1/2 gf for any upper gradient gf . In particular, Lemma 2.1 implies |Γf |1/2 |∇d f |. Though they discussed the case that X is an open subset of a Euclidean space in [16], we can extend it to our case with the aid of a partition of unity. By a mollifier argument together with use of a partition of unity again, we can take a sequence fk ∈ Cc∞ (X) such that fk → f and Γfk → Γf almost surely (cf. [16, Theorem 11.9]). Thus (4.1) holds for any f ∈ Cb,L (X) with compact support. Note that |Γf |1/2 is an upper gradient if f ∈ C ∞ (X) (see [16, Proposition 11.6], for instance). Since Pt f ∈ C ∞ (X) in our case, for a minimal geodesic γ joining x and y, d(x,y)
1/2 Γ (Pt f ) γ (s) ds
Pt f (x) − Pt f (y) 0
d(x,y)
1/q Pt Γ (f )q/2 γ (s) ds
Kq (t) 0
d(x,y)
1/q Pt |∇d f |q γ (s) ds.
Kq (t) 0
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Hence the conclusion follows by dividing the above inequality by d(x, y) and by letting y → x. 2 Remark 4.2. If we suppose Assumption 1(i)–(ii) holds in Proposition 4.1, then Theorem 6.1 of [11] asserts that the minimal generalized upper gradient of f coincides with |∇f | almost everywhere. Since the first part of the proof of Proposition 4.1 implies that |Γf |1/2 is the minimal generalized upper gradient for f ∈ CL (X) with compact support, the proof can be completed there in this case. As far as the author knows, (4.1) is established in the following cases: • The case q = 1 with K1 (t) ≡ K for some K > 0 on groups of type H [13] (including the Heisenberg group of arbitrary dimension, see [5,22] also). • The case q > 1 on an arbitrary Lie group [27]. Especially, Kp (t) ≡ Kp for some Kp > 0 if it is nilpotent. • The case q > 1 with Kq (t) = Kq e−t for some Kq > 0 on SU(2) [8]. In all these cases, v is chosen to be a right-invariant Haar measure and hence the associated subLaplacian is of the form H = ni=1 Xi2 . All conditions in Assumption 1 hold in these cases. For (iii), we have already observed. By the homogeneity of the space, we can reduce the assertion in the case of a Euclidean domain (see Remark 2.3 also). Thus (i) and (ii) with p0 = 1 follow from Theorems 11.19 and 11.21 of [16]. Note that (4.1) is shown on a wider class of functions than Cc∞ (X) in some cases. But it is not necessary for our purpose. Combining Proposition 4.1 with Theorem 2.2 in these cases, we obtain (Cp ) for P = Pt and d˜ = Kq (t)d. Though f is restricted to have a compact support in Proposition 4.1, it is sufficient to show (Cp ) (see Remark 3.6). The following simple examples explain a probabilistic meaning of these consequences. Example 4.3. The 3-dimensional Heisenberg group is realized on R3 with the multiplication defined by
1 (x, y, z) · x , y , z := x + x , y + y , z + z + xy − yx . 2 The Lebesgue measure v on R3 is a bi-invariant Haar measure. Let us define left-invariant vector fields X and Y by X :=
y ∂ ∂ − , ∂x 2 ∂z
Y :=
∂ x ∂ + . ∂y 2 ∂z
Set H := Span{X, Y }. Then the diffusion process {Bxt }t0 associated with H /2 = (X 2 + Y 2 )/2 starting at x = (x, y, z) ∈ R3 is given by Bxt
= x
(1) + Wt , y
(2) + Wt , z +
1 2
t 0
(1) (2) (2) (1) x + Ws dWs − y + Ws dWs ,
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K. Kuwada / Journal of Functional Analysis 258 (2010) 3758–3774 (1)
(2)
where (Wt , Wt ) is a Brownian motion on R2 . It means that the diffusion process associated with H /2 is given by the 2-dimensional Euclidean Brownian motion and the associated Lévy stochastic area. The corresponding heat semigroup is given by Pt f (x) = E[f (Bxt )] for f ∈ Cb (X). In this framework, (4.1) for q = 1, P = Pt and K1 (t) ≡ K is shown in [5,22]. Thus ¯ xt , B¯ yt ) of we obtain (C∞ ). It means that, for each t > 0 and x, y ∈ R3 , there exists a coupling (B y Bxt and Bt such that y d B¯ xt , B¯ t K d(x, y)
(4.2)
holds almost surely. Here d is the Carnot–Caratheodory distance associated with H. In this case, it is known that d is equivalent to the so-called Korányi distance. That is, there exist constants C1 , C2 > 0 such that, for any x = (x, y, z), y = (x , y , z ) ∈ R3 , C1 d(x, y)
x − x
2
2 2 2 1/4 1 + y − y + z − z + xy − yx 2
C2 d(x, y). Thus (4.2) is also interpreted in terms of the Korányi distance. y
Remark 4.4. In Example 4.3, (C∞ ) provides only a coupling of Bxt and Bt for each fixed t > 0. ¯ yt )t0 When X is a Riemannian manifold, (C∞ ) holds if and only if there exists a coupling (B¯ xt , B y x of two Brownian motions (Bt )t0 and (Bt )t0 starting from x and y respectively such that (4.2) holds for every t 0 with K = e−kt almost surely (see [31], for instance). In Example 4.3, it is not clear whether a similar result holds or not. Actually, in Riemannian case, the fact that the constant e−kt is multiplicative in t 0 plays a prominent role to construct a coupling of Brownian motions from a control of their infinitesimal motions. As observed in [12], we cannot expect such a multiplicativity in the case of Example 4.3. Example 4.5. On Rn × Rn(n−1)/2 , we introduce a structure of nilpotent Lie group of step 2. We index elements in Rn(n−1)/2 as that in upper triangle matrices. For x, y ∈ Rn × Rn(n−1)/2 written as x = (xi )ni=1 ; (zij )1i<j n ,
y=
n xi i=1 ; zij 1i<j n ,
we define x · y by
1 n . x · y := xi + xi i=1 ; zij + zij + xi xj − xj xi 2 1i<j n As in Example 4.3, the Lebesgue measure v on Rn × Rn(n−1)/2 becomes a bi-invariant Haar measure. Let us define left-invariant vector fields {Xi }ni=1 by Xi :=
xj ∂ xj ∂ ∂ − + . ∂xi 2 ∂zj i 2 ∂zij i<j n
1j
K. Kuwada / Journal of Functional Analysis 258 (2010) 3758–3774
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Set H := Span{Xi }ni=1 . The diffusion process {Bxt }t0 associated with the sub-Laplacian H /2 = ni=1 Xi2 /2 starting at x = ((xi )ni=1 ; (zij )1i<j n ) ∈ Rn × Rn(n−1)/2 is given by Bxt
t 1 (j ) (j ) (i) n (i) (i) xi + Ws dWs − xj + Ws dWs = xi + Wt i=1 ; zij + 2 0
. 1i<j n
We can easily verify that this group is of type H only if n = 1 (see Corollary 1 of [19], for example). But it is still in the framework of [27]. Thus, for each p ∈ [1, ∞), there is a constant y y Kp > 0 such that, for any pair x, y ∈ Rn × Rn(n−1)/2 , there is a coupling (B¯ xt , B¯ t ) of Bxt and Bt satisfying y p 1/p E d B¯ xt , B¯ t Kp d(x, y).
(4.3)
Finally, we give a remark that a different kind of coupling of this process is studied by Kendall [20]. He showed the existence of a successful coupling. As mentioned there, studying a coupling of this process has a possibility of a future application to rough path theory [15,25]. References [1] R. Adams, J. Fournier, Sobolev Spaces, 2nd edition, Pure Appl. Math. (Amst.), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. [2] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panor. Synthèses, vol. 10, Société Mathématique de France, Paris, 2000. [3] M. Arnaudon, K. Coulibaly, A. Thalmaier, Horizontal diffusion in C 1 -path space, in: Séminaire de Probabilités, in: Lecture Notes in Math., 2009, in press, arXiv:0904.2762. [4] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, in: New Trends in Stochastic Analysis, Charingworth, 1994, World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. [5] D. Bakry, F. Baudoin, M. Bonnefont, D. Chafaï, On gradient bounds for the heat kernel on the Heisenberg group, J. Funct. Anal. 255 (8) (2008) 1905–1938. [6] D. Bakry, M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math. 299 (15) (1984) 775–778. [7] Z. Balogh, A. Engoulatov, L. Hunziker, O. Maasalo, Functional inequalities and Hamilton–Jacobi equations in geodesic spaces, preprint, arXiv:0906.0476. [8] F. Baudoin, M. Bonnefont, The subelliptic heat kernel on SU(2): Representations, asymptotics and gradient bounds, Math. Z. 263 (3) (2009) 647–672. [9] G. Ben Arous, M. Cranston, W. Kendall, Coupling constructions for hypoelliptic diffusions: two examples, in: Stochastic Analysis, Ithaca, NY, 1993, in: Proc. Sympos. Pure Math., vol. 57, American Mathematical Society, Providence, RI, 1995, pp. 193–212. [10] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Grad. Stud. Math., vol. 33, American Mathematical Society, Providence, RI, 2001. [11] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (3) (1999) 428–517. [12] B. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (5) (2005) 340–365. [13] N. Eldredge, Gradient estimates for the subelliptic heat kernel on H-type groups, J. Funct. Anal. 258 (2) (2010) 504–533. [14] M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space, Ann. Inst. H. Poincaré, in press. [15] P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge University Press, in press. [16] P. Hajłasz, P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (688) (2000) 1–101.
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[17] N. Juillet, Geometric inequalities and generalized Ricci bound on the Heisenberg group, Int. Math. Res. Not. IMRN 2009 (13) (2009) 2347–2373. [18] N. Juillet, Diffusion by optimal transport on Heisenberg groups, preprint. [19] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1) (1980) 147–153. [20] W.S. Kendall, Coupling all the Lévy stochastic areas of multidimensional Brownian motion, Ann. Probab. 35 (3) (2007) 935–953. [21] M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6) 9 (2) (2000) 305–366. [22] H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236 (2) (2006) 369–394. [23] J. Lott, C. Villani, Hamilton–Jacobi semigroup on length spaces and applications, J. Math. Pures Appl. (9) 88 (3) (2007) 219–229. [24] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (3) (2009) 903–991. [25] T. Lyons, Z. Qian, System Control and Rough Paths, Oxford University Press, 2002. [26] R. McCann, P. Topping, Ricci flow, entropy and optimal transportation, Amer. J. Math., in press. [27] T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, Stochastic Process. Appl. 118 (3) (2008) 368–388. [28] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., vol. 91, American Mathematical Society, Providence, RI, 2002. [29] S.-i. Ohta, Gradient flows on Wasserstein spaces over compact Alexandrov spaces, Amer. J. Math. 131 (2) (2009) 475–516. [30] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (1–2) (2001) 101–174. [31] M.-K. von Renesse, K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math. 58 (7) (2005) 923–940. [32] L. Saloff-Coste, Aspects of Sobolev-type Inequalities, London Math. Soc. Lecture Note Ser., vol. 289, Cambridge University Press, Cambridge, 2002. [33] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Acad. Sci. Paris 345 (3) (2007) 151–154. [34] R.S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (2) (1986) 221–263. [35] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (1) (2006) 65–131. [36] C. Villani, Topics in Optimal Transportations, Grad. Stud. Math., vol. 58, American Mathematical Society, Providence, RI, 2003. [37] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer-Verlag, 2008. [38] F.-Y. Wang, On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. Theory Related Fields 108 (1) (1997) 87–101.
Journal of Functional Analysis 258 (2010) 3775–3791 www.elsevier.com/locate/jfa
Cumulants on the Wiener space Ivan Nourdin a , Giovanni Peccati b,c,∗ a Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, Boîte courrier 188, 4 Place Jussieu,
75252 Paris Cedex 5, France b Equipe Modal’X, Université Paris Ouest – Nanterre la Défense, 200 Avenue de la République, 92000 Nanterre, France c LSTA, Université Paris VI, France
Received 13 October 2009; accepted 26 October 2009 Available online 7 November 2009 Communicated by Paul Malliavin
Abstract We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual “graph/diagram computations” adopted in most of the probabilistic literature. © 2009 Elsevier Inc. All rights reserved. Keywords: Cumulants; Diagram formulae; Gaussian processes; Malliavin calculus; Ornstein–Uhlenbeck semigroup
1. Introduction The integration by parts formula of Malliavin calculus, combining derivative operators and anticipative integrals into a flexible tool for computing and assessing mathematical expectations, is a cornerstone of modern stochastic analysis. The scope of its applications, ranging e.g. from density estimates for solutions of stochastic differential equations to concentration inequalities, from anticipative stochastic calculus to “Greeks” computations in mathematical finance, is vividly described in the three classic monographs by Malliavin [10], Janson [9] and Nualart [20]. In recent years, infinite-dimensional integration by parts techniques have found another fertile ground for applications, that is, limit theorems and (more generally) probabilistic approxima* Corresponding author at: Equipe Modal’X, Université Paris Ouest – Nanterre la Défense, 200 Avenue de la République, 92000 Nanterre, France. E-mail addresses:
[email protected] (I. Nourdin),
[email protected] (G. Peccati).
0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.024
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tions. The starting point of this active line of research is the paper [21], where the authors use Malliavin calculus in order to refine some criteria for asymptotic normality on a fixed Wiener chaos, originally proved in [22,24] (see also [12] for some non-central version of these results). Another important step appears in [13], where integration by parts on Wiener space is combined with the so-called Stein’s method for probabilistic approximations (see e.g. [6,25]), thus yielding explicit upper bounds in the normal and gamma approximations of the law of functionals of Gaussian fields. The techniques introduced in [13] have led to several applications and generalizations, for instance: in [14] one can find applications to Edgeworth expansions and reversed Berry–Esseen inequalities; [18] contains results for multivariate normal approximations; [1] focuses on further developments in the multivariate case, in relation with quasi-sure analysis; [16] deals with infinite-dimensional second order Poincaré inequalities; in [19], one can find new explicit expressions for the density of functionals of Gaussian field as well as new concentration inequalities (see also [3] for some applications in mathematical statistics); in [17], the results of [13] are combined with Lindeberg-type invariance principles in order to deduce universality results for homogeneous sums (these findings are further applied in [15] to random matrix theory). The aim of this note is to develop yet another striking application of the infinite-dimensional integration by parts formula of Malliavin calculus, namely the computation of cumulants for general functionals of a given Gaussian field. As discussed below, our techniques make a crucial use of a recursive formula for moments (see relation (2.2) below), which is the starting point of some well-known computations performed by Barbour in [2] in connection with the Stein’s method for normal approximations. As such, the techniques developed in the forthcoming sections can be seen as further ramifications of the findings of [13]. The main achievement of the present work is a recursive formula for cumulants (see (4.19)), based on a repeated use of integration by parts. Note that cumulants of order greater than two are not linear operators (for instance, the second cumulant coincides with the variance): however, our formula (4.19) implies that cumulants of regular functionals of Gaussian fields can be always represented as the mathematical expectation of some recursively defined random variable. We shall prove in Section 5 that this implies a new compact representation for cumulants of random variables belonging to a fixed Wiener chaos. We claim that this result may replace the classic “diagram computations” adopted in most of the probabilistic literature (see e.g. [4,5,8], as well as [23] for a general discussion of related combinatorial results). The paper is organized as follows. In Section 2 we state and prove some useful recursive formulae for moments. Section 3 contains basic concepts and results related to Malliavin calculus. Section 4 is devoted to our main statements about cumulants on Wiener space. Finally, in Section 5 we specialize our results to random variables contained in a fixed Wiener chaos. From now on, every random object is defined on a common suitable probability space (Ω, F , P ). 2. Moment expansions The starting point of our analysis (Proposition 2.2) is a well-known recursive relation involving moments and cumulants. As already discussed, this result is the seed (as well as a special case) of some remarkable formulae by A.D. Barbour [2, Lemma 1 and Corollary 1], providing Edgeworth-type expansions for smooth functions of random variables with finite moments. Since we only need Barbour’s results in the special case of polynomial transformations, and for the sake
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of completeness, we choose to provide a self-contained presentation in this simpler setting. See also Rotar’ [26] for further extensions of Barbour’s findings. Definition 2.1 (Cumulants). Let F be a real-valued random variable such that E|F |m < ∞ for some integer m 1, and define φF (t) = E(eitF ), t ∈ R, to be the characteristic function of F . Then, for j = 1, . . . , m, the j th cumulant of F , denoted by κj (F ), is given by κj (F ) = (−i)j
dj log φF (t)|t=0 . dt j
(2.1)
For instance, κ1 (F ) = E(F ), κ2 (F ) = E(F 2 ) − E(F )2 = Var(F ), κ3 (F ) = E(F 3 ) − 3E(F 2 )E(F ) + 2E(F )3 , and so on. The following relation is exploited throughout the paper. Proposition 2.2. Fix m = 0, 1, 2, . . . , and suppose that E|F |m+1 < ∞. Then m m κs+1 (F )E F m−s . E F m+1 = s
(2.2)
s=0
Proof. By Leibniz rule, one has that d m+1 E F m+1 = (−i)m+1 m+1 φF (t)|t=0 dt m d m+1 d log φF (t) φF (t) = (−i) m dt dt t=0
m s+1 m−s s+1 m d m−s d log φF (t) × (−i) = (−i) φF (t) m−s s+1 s dt dt s=0
with
d0 dt 0
equal to the identity operator. This yields the desired conclusion.
, t=0
2
Finally, observe that (2.2) can be rewritten as m κs+1 (F ) E F m+1 = m(m − 1) · · · (m − s + 1)E F m−s , s! s=0
implying (by linearity) that, for F as in Proposition 2.2 and for every polynomial f : R → R of degree at most m 1, s s m ∞ d d κs+1 (F ) κs+1 (F ) E Ff (F ) = E E f (F ) = f (F ) . s! dx s s! dx s s=0
s=0
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Remark 2.3. In [2, Corollary 1], one can find sufficient conditions ensuring that the infinite expansion s ∞ κs+1 (F ) d E Ff (F ) = f (F ) E s! dx s s=0
holds for some infinitely differentiable function f which is not necessarily a polynomial. 3. Malliavin operators and Gaussian analysis We shall now present the basic elements of Gaussian analysis and Malliavin calculus that are used in this paper. The reader is referred to [9,10,20] for any unexplained definition or result. Let H be a real separable Hilbert space. For any q 1, let H⊗q be the qth tensor power of H and denote by Hq the associated qth symmetric tensor power. We write X = {X(h), h ∈ H} to indicate an isonormal Gaussian process over H, defined on some probability space (Ω, F , P ). This means that X is a centered Gaussian family, whose covariance is given by the relation E[X(h)X(g)] = h, g H . We also assume that F = σ (X), that is, F is generated by X. For every q 1, let Hq be the qth Wiener chaos of X, defined as the closed linear subspace of L2 (Ω, F , P ) generated by the family {Hq (X(h)), h ∈ H, h H = 1}, where Hq is the qth Hermite polynomial given by Hq (x) = (−1)q e
x2 2
d q − x2 e 2 . dx q
We write by convention H0 = R. For any q 1, the mapping Iq (h⊗q ) = q!Hq (X(h)) can be extended to a linear isometry between the symmetric tensor product Hq (equipped with the √ modified norm q! · H⊗q ) and the qth Wiener chaos Hq . For q = 0, we write I0 (c) = c, c ∈ R. It is well known that L2 (Ω) := L2 (Ω, F , P ) can be decomposed into the infinite orthogonal sum of the spaces Hq . It follows that any square integrable random variable F ∈ L2 (Ω) admits the following (Wiener–Itô) chaotic expansion F=
∞
Iq (fq ),
(3.3)
q=0
where f0 = E[F ], and the fq ∈ Hq , q 1, are uniquely determined by F . For every q 0, we denote by Jq the orthogonal projection operator on the qth Wiener chaos. In particular, if F ∈ L2 (Ω) is as in (3.3), then Jq F = Iq (fq ) for every q 0. Let {ek , k 1} be a complete orthonormal system in H. Given f ∈ Hp and g ∈ Hq , for every r = 0, . . . , p ∧ q, the contraction of f and g of order r is the element of H⊗(p+q−2r) defined by f ⊗r g =
∞
f, ei1 ⊗ · · · ⊗ eir H⊗r ⊗ g, ei1 ⊗ · · · ⊗ eir H⊗r .
(3.4)
i1 ,...,ir =1
Notice that the definition of f ⊗r g does not depend on the particular choice of {ek , k 1}, and r g ∈ H(p+q−2r) . that f ⊗r g is not necessarily symmetric; we denote its symmetrization by f ⊗
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Moreover, f ⊗0 g = f ⊗ g equals the tensor product of f and g while, for p = q, f ⊗q g = f, g H⊗q . It can also be shown that the following multiplication formula holds: if f ∈ Hp and g ∈ Hq , then Ip (f )Iq (g) =
p∧q r=0
p q r g). r! Ip+q−2r (f ⊗ r r
(3.5)
We now introduce some basic elements of the Malliavin calculus with respect to the isonormal Gaussian process X. Let S be the set of all cylindrical random variables of the form F = g X(φ1 ), . . . , X(φn ) , (3.6) where n 1, g : Rn → R is an infinitely differentiable function such that its partial derivatives have polynomial growth, and φi ∈ H, i = 1, . . . , n. The Malliavin derivative of F with respect to X is the element of L2 (Ω, H) defined as DF =
n ∂g X(φ1 ), . . . , X(φn ) φi . ∂xi i=1
In particular, DX(h) = h for every h ∈ H. By iteration, one can define the mth derivative D m F , which is an element of L2 (Ω, Hm ), for every m 2. For m 1 and p 1, Dm,p denotes the closure of S with respect to the norm · m,p , defined by the relation m p
p E D i F H⊗i .
F m,p = E |F |p + i=1
One also writes D∞ = m1 p1 Dm,p . The Malliavin derivative D obeys the following chain rule. If ϕ : Rn → R is continuously differentiable with bounded partial derivatives and if F = (F1 , . . . , Fn ) is a vector of elements of D1,2 , then ϕ(F ) ∈ D1,2 and Dϕ(F ) =
n ∂ϕ (F )DFi . ∂xi i=1
2 Note also that a random variable F as in (3.3) is in D1,2 if and only if ∞ q=1 q Jq F L2 (Ω) < ∞ ∞ and, in this case, E( DF 2H ) = q=1 q Jq F 2L2 (Ω) . If H = L2 (A, A , μ) (with μ non-atomic), then the derivative of a random variable F as in (3.3) can be identified with the element of L2 (A × Ω) given by Dx F =
∞
qIq−1 fq (·, x) ,
x ∈ A.
(3.7)
q=1
We denote by δ the adjoint of the operator D, also called the divergence operator. A random element u ∈ L2 (Ω, H) belongs to the domain of δ, noted Dom δ, if and only if it verifies |EDF, u H | cu F L2 (Ω) for any F ∈ D1,2 , where cu is a constant depending only on u. If
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u ∈ Dom δ, then the random variable δ(u) is defined by the duality relationship (called integration by parts formula) E F δ(u) = EDF, u H ,
(3.8)
which holds for every F ∈ D1,2 . The family (Pt , t 0) of operators is defined through the projection operators Jq as Pt =
∞
e−qt Jq ,
(3.9)
q=0
and is called the Ornstein–Uhlenbeck semigroup. Assume that the process X , which stands for an independent copy of X, is such that X and X are defined on the product probability space (Ω × Ω , F ⊗ F , P × P ). Given a random variable F ∈ D1,2 , we can regard it as a measurable mapping from RH to R, determined P ◦ X −1 -almost surely. Then, for any t 0, we have the so-called Mehler’s formula: Pt F = E F e−t X + 1 − e−2t X ,
(3.10)
where E denotes the mathematical expectation with respect to the probability P . By means of p this formula, it is immediate to prove that ∞Pt is a contraction operator on L (Ω), for all p 1. The operator L is defined as L = q=0 −qJq , and it can be proven to be the infinitesimal generator of the Ornstein–Uhlenbeck semigroup (Pt )t0 . The domain of L is Dom L = F ∈ L (Ω): 2
∞
q
2
Jq F 2L2 (Ω)
< ∞ = D2,2 .
q=1
There is an important relation between the operators D, δ and L. A random variable F belongs to D2,2 if and only if F ∈ Dom(δD) (i.e. F ∈ D1,2 and DF ∈ Dom δ) and, in this case, δDF = −LF.
(3.11)
1 −1 is called the For any F ∈ L2 (Ω), we define L−1 F = ∞ q=0 − q Jq (F ). The operator L pseudo-inverse of L. Indeed, for any F ∈ L2 (Ω), we have that L−1 F ∈ Dom L = D2,2 , and LL−1 F = F − E(F ).
(3.12)
We now present two useful lemmas, that we will need throughout the sequel. The first statement exploits the two fundamental relations (3.11) and (3.12). Note that these relations have been extensively applied in [13], in the context of the normal approximation of functionals of Gaussian fields by means of Stein’s method. Lemma 3.1. Suppose that F ∈ D1,2 and G ∈ L2 (Ω). Then, L−1 G ∈ D2,2 and we have: E(F G) = E(F )E(G) + E DF, −DL−1 G H .
(3.13)
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Proof. By (3.11) and (3.12), E(F G) − E(F )E(G) = E F G − E(G) = E F × LL−1 G = E F δ −DL−1 G , and the result is obtained by using the integration by parts formula (3.8).
2
Remark 3.2. Observe that DF, −DL−1 G H is not necessarily square-integrable, albeit it is by construction in L1 (Ω). On the other hand, we have
DF, −DL
−1
GH=
∞
e−t DF, Pt DG H dt,
(3.14)
0
see indeed identity (3.46) in [13]. The next result is a consequence of the previous Lemma 3.1. Proposition 3.3. Fix p 2, and assume that F ∈ L4p−4 (Ω) ∩ D1,4 . Then, F p ∈ D1,2 and DF p = pF p−1 DF . Moreover, for any G ∈ L2 (Ω), E F p G = E F p E(G) + pE F p−1 DF, −DL−1 G H .
(3.15)
4. A recursive formula for cumulants The aim of this section is to deduce from formula (2.2) a recursive relation for cumulants of sufficiently regular functionals of the isonormal process X. To do this, we need to (recursively) introduce some further notation. Definition 4.1. Let F ∈ D1,2 . We define Γ0 (F ) = F and Γ1 (F ) = DF, −DL−1 F H . If, for j 1, the random variable Γj (F ) is a well-defined element of L2 (Ω), we set Γj +1 (F ) = DF, −DL−1 Γj (F ) H . Observe that the definition of Γj +1 (F ) is well given, since (as already observed in general) the square-integrability of Γj (F ) implies that L−1 Γj (F ) ∈ Dom L = D2,2 ⊂ D1,2 . The following statement provides sufficient conditions on F , ensuring that the random variable Γj (F ) is well defined. Lemma 4.2. j −1
1. Fix an integers j 1, and let F, G ∈ Dj,2 . Then DF, −DL−1 G H ∈ Dj −1,2 , where we set by convention (but consistently!) D0,1 = L1 (Ω). j 2. Fix an integer j 1, and let F ∈ Dj,2 . Then, for all k = 1, . . . , j , we have that Γk (F ) is a j −k well-defined element of Dj −k,2 ; in particular, one has that Γk (F ) ∈ L1 (Ω) and that the quantity E[Γk (F )] is well defined and finite. 3. If F ∈ D∞ (in particular if F equals a finite sum of multiple integrals), then Γj (F ) ∈ D∞ for every j 0. j
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Proof. Without loss of generality, we assume during all the proof that H has the form L2 (A, A , μ), where (A, A ) is a measurable space and μ is a σ -finite measure with no atoms. 1. Let k ∈ {0, . . . , j − 1}. Using Leibniz rule for D (see e.g. [20, Exercice 1.2.13]), one has that −D k DF, −DL−1 G H = D k Da F Da L−1 G μ(da) A
=
k k l=0
l
D l Da L−1 G μ(da), D k−l (Da F )⊗
(4.16)
A
the usual symmetric tensor product. Note that, to deduce (4.16), it is sufficient to conwith ⊗ sider random variables F, G that have the form (3.6) (for which the formula is evident, since it basically boils down to the original Leibniz rule for differential calculus), and then to apply a standard approximation argument. From (4.16), one therefore deduces that k D DF, −DL−1 G ⊗k H H
k k k−l l −1 D (Da F )⊗D Da L G μ(da) ⊗k l H l=0
k l=0
k l=0
=
k l
A
k−l D (Da F )
H⊗(k−l)
l D Da L−1 G
H⊗l
μ(da)
A
2 2 k D k−l (Da F )H⊗(k−l) μ(da) D l Da L−1 G H⊗l μ(da) l A
A
k k D k−l+1 F ⊗(k−l+1) D l+1 L−1 G ⊗(l+1) . H H l
(4.17)
l=0
By mimicking the arguments used in the proof of Proposition 3.1 in [16] (see also (3.14)), it is possible to prove that −D
l+1 −1
L
∞ G=
e−(l+1)t Pt D l+1 G dt.
0
Consequently, for any real p 1, p
E D l+1 L−1 G ⊗(l+1) E
∞ e
H
−(l+1)t
Pt D
l+1
G
p H⊗(l+1)
dt
0
1 (l + 1)p−1
∞ 0
p
e−(l+1)t E Pt D l+1 GH⊗(l+1) dt
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l+1 p 1 D G ⊗(l+1) E H (l + 1)p−1 =
∞
3783
e−(l+1)t dt
0
p
1 E D l+1 GH⊗(l+1) , p (l + 1)
(4.18)
where, to get the last inequality, we used the contraction property of Pt on Lp (Ω). Finally, by combining (4.17) with (4.18) through the Cauchy–Schwarz inequality on the one hand, and by the assumptions on F and G on the other hand, we immediately get that D k DF, −DL−1 G H H⊗k j −1 belongs to L2 (Ω) for all k = 0, . . . , j − 1, yielding the announced result. j 2. Fix j 1 and F ∈ Dj,2 . The proof is achieved by recursion on k. For k = 1, the desired property is true, due to Point 1 applied to G = F . Now, assume that the desired property is true for k ( j − 1), and let us prove that it also holds for k + 1. Indeed, we have that j j −k (assumption on F ) and that Γk+1 (F ) = DF, −DL−1 Γk (F ) H , that F ∈ Dj,2 ⊂ Dj −k,2 j −k Γk (F ) ∈ Dj −k,2 (recurrence assumption). Point 1 yields the desired conclusion. 3. The proof is immediately obtained by a repeated application of Point 2. 2 We will now provide a new characterization of cumulants for functionals of Gaussian processes: it is the fundamental tool yielding the main results of the paper. Note that, due to m Lemma 4.2 (Point 2), the following statement applies in particular to random variables in Dm,2 (m 2). Theorem 4.3. Fix an integer m 2, and suppose that: (i) the random variable F is an element of L4m−4 (Ω) ∩ D1,4 , (ii) for every j m − 1, the random variable Γj (F ) is in L2 (Ω). Then, for every s m,
κs+1 (F ) = s!E Γs (F ) .
(4.19)
Proof. The proof is achieved by recursion on s. First observe that κ1 (F ) = E(F ) = E[Γ0 (F )], so that the claim is proved for s = 0. Now suppose that (4.19) holds for every s = 0, . . . , l, where l m − 1. According to (2.2), we have that l−1 l κs+1 (F )E F l−s + κl+1 (F ). E F l+1 = s
(4.20)
s=0
On the other hand, by applying (3.15) to the case p = l and G = F , we deduce that E F l+1 = E F l E(F ) + lE F l−1 DF, −DL−1 F H = E F l κ1 (F ) + lE F l−1 Γ1 (F ) . By the recurrence assumption, and by applying again (3.15) to the case p = l −1 and G = Γ1 (F ), one deduces therefore that E F l+1 = E F l κ1 (F ) + lE F l−1 Γ1 (F ) = κ1 (F )E F l + κ2 (F )lE F l−1 + l(l − 1)E F l−2 Γ2 (F ) .
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Iterating this procedure yields eventually l−1
l κs+1 (F )E F l−s + l!E Γl (F ) , E F l+1 = s s=0
so that one deduces from (4.20) that relation (4.19) holds for s = l + 1. The proof is concluded. 2 Remark 4.4. Suppose that F = Iq (f ), where q 2 and f ∈ Hq . Then, L−1 F = −q −1 F , and consequently
E[Γs (F )] = E DF, −DL−1 Γs−1 (F ) H = E F Γs−1 (F )
1 = E −DL−1 F, DΓs−1 (F ) H = E DF, DΓs−1 (F ) H . q
(4.21)
Several applications of formula (4.19) (and, implicitly, of (4.21)) are detailed in the next section. 5. Cumulants on Wiener chaos 5.1. General statement We now focus on the computation of cumulants associated to random variables belonging to a fixed Wiener chaos, that is, random variables having the form of a multiple Wiener–Itô integral. Our main findings are collected in the following statement, providing a quite compact representation for cumulants associated with multiple integrals of arbitrary orders. In the forthcoming Section 5.2, we will compare our results with the classic diagram formulae that are customarily used in the probabilistic literature. Theorem 5.1. Let q 2, and assume that F = Iq (f ), where f ∈ Hq . Denote by κs (F ), s 1, the cumulants of F . We have κ1 (F ) = 0, κ2 (F ) = q! f 2H⊗q and, for every s 3, κs (F ) = q!(s − 1)! cq (r1 , . . . , rs−2 ) r2 f . . . ⊗ rs−3 f ⊗ rs−2 f, f ⊗q , r1 f ) ⊗ × . . . (f ⊗ H where the sum runs over all collections of integers r1 , . . . , rs−2 such that: (i) (ii) (iii) (iv)
(5.22)
1 r1 , . . . , rs−2 q; r1 + · · · + rs−2 = (s−2)q 2 ; (s−2)q r1 < q, r1 + r2 < 3q , 2 . . . , r1 + · · · + rs−3 < 2 ; r2 2q − 2r1 , . . . , rs−2 (s − 2)q − 2r1 − · · · − 2rs−3 ;
and where the combinatorial constants cq (r1 , . . . , rs−2 ) are recursively defined by the relations q −1 2 cq (r) = q(r − 1)! , r −1
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and, for a 2, aq − 2r1 − · · · − 2ra−1 − 1 q − 1 cq (r1 , . . . , ra ) = q(ra − 1)! cq (r1 , . . . , ra−1 ). ra − 1 ra − 1 Remark 5.2. 1. If sq is odd, then κs (F ) = 0, see indeed condition (ii). 2. If q = 2 and F = I2 (f ), f ∈ H2 , then the only possible integers r1 , . . . , rs−2 verifying (i)–(iv) in the previous statement are r1 = · · · = rs−2 = 1. On the other hand, we immediately compute that c2 (1) = 2, c2 (1, 1) = 4, c2 (1, 1, 1) = 8, and so on. Therefore, κs (F ) = 2s−1 (s − 1)! . . . (f ⊗1 f ) . . . f ⊗1 f, f H⊗2 , and we recover the classical expression of the cumulants of a double integral (as used e.g. in [7] or [14]). 3. If q 2 and F = Iq (f ), f ∈ Hq , then (5.22) for s = 4 reads r f ) ⊗ q−r f, f ⊗q cq (r, q − r) (f ⊗ κ4 Iq (f ) = 6q! H q−1 r=1
=
q−1 3 2 q 4 r f ) ⊗q−r f, f ⊗q rr! (2q − 2r)! (f ⊗ H r q r=1
q−1 3 2 q 4 r f, f ⊗r f ⊗(2q−2r) = rr! (2q − 2r)! f ⊗ H r q r=1
3 = q
q−1 r=1
4 q r f 2 ⊗(2q−2r) , rr! (2q − 2r)! f ⊗ H r 2
(5.23)
and we recover the expression for κ4 (F ) deduced in [17, Section 3.1] by a different route. Formula (5.23) should be compared with the following identity, first established in [22, p. 183]: for every q 2 and every f ∈ Hq , q!4
f ⊗r f 2H⊗(2q−2r) r!2 (q − r)!2 r=1 2q − 2r 2 +
f ⊗r f H⊗(2q−2r) . q −r
κ4 Iq (f ) = q−1
(5.24)
Note that it is in principle much more difficult to deal with (5.24), since it involves both symmetrized and non-symmetrized contractions. Proof of Theorem 5.1. Let us first show that the following formula (5.25) is in order: for any s 2,
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Γs−1 (F ) =
[(s−1)q−2r1 −···−2rs−2 ]∧q
q
...
r1 =1
cq (r1 , . . . , rs−1 )1{r1
rs−1 =1
× Isq−2r1 −···−2rs−1
1 +···+rs−2 <
(s−1)q 2 }
r1 f ) ⊗ r2 f . . . f ⊗ rs−1 f . . . . (f ⊗
(5.25)
We shall prove (5.25) by induction, assuming without loss of generality that H has the form L2 (A, A , μ), where (A, A ) is a measurable space and μ is a σ -finite measure without atoms. When s = 2, identity (5.25) simply reads Γ1 (F ) =
q
r f ). cq (r)I2q−2r (f ⊗
(5.26)
r=1
Let us prove (5.26) by means of the multiplication formula (3.5), see also [21] for similar computations. We have 1 Γ1 (F ) = DF, −DL−1 F H = DF 2H = q q =q
q−1 r=0
=q
q−1 r=0
=q
q r=1
2 Iq−1 f (·, a) μ(da)
A
q −1 2 r! I2q−2−2r f (·, a) ⊗r f (·, a) μ(da) r A
q −1 2 r+1 f ) r! I2q−2−2r (f ⊗ r q −1 2 r f ), (r − 1)! I2q−2r (f ⊗ r −1
thus yielding (5.26). Assume now that (5.25) holds for Γs−1 (F ), and let us prove that it continues to hold for Γs (F ). We have, still using the multiplication formula (3.5) and proceeding as above, Γs (F ) = DF, −DL−1 Γs−1 F H =
q
[(s−1)q−2r1 −···−2rs−2 ]∧q
...
r1 =1
rs−1 =1
qcq (r1 , . . . , rs−1 )1{r1
1 +···+rs−2 <
(s−1)q 2 }
r1 f ) ⊗ r2 f . . . f ⊗ rs−1 f × 1{r1 +···+rs−1 < sq } Iq−1 (f ), Isq−2r1 −···−2rs−1 −1 . . . (f ⊗ H 2
=
q r1 =1
[(s−1)q−2r1 −···−2rs−2 ]∧q [sq−2r1 −···−2rs−1 ]∧q
...
rs−1 =1
rs =1
q −1 1{r1
cq (r1 , . . . , rs−1 ) × q(rs − 1)!
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which is the desired formula for Γs (F ). The proof of (5.25) for all s 1 is thus finished. Now, let us take the expectation on both sides of (5.25). We get
κs (F ) = (s − 1)!E Γs−1 (F ) = (s − 1)!
q
[(s−1)q−2r1 −···−2rs−2 ]∧q
...
r1 =1
rs−1 =1
× 1{r1 +···+rs−1 = sq } × 2
cq (r1 , . . . , rs−1 )1{r1
1 +···+rs−2 <
(s−1)q 2 }
r1 f ) ⊗ r2 f . . . f ⊗ rs−1 f. . . . (f ⊗
Observe that, if 2r1 + · · · + 2rs−1 = sq and rs−1 (s − 1)q − 2r1 − · · · − 2rs−2 then 2rs−1 = q + (s − 1)q − 2r1 − · · · − 2rs−2 q + rs−1 , so that rs−1 q. Therefore, κs (F ) = (s − 1)!
q
[(s−2)q−2r1 −···−2rs−3 ]∧q
r1 =1
× 1{r
...
(s−2)q 1 +···+rs−2 = 2 }
rs−2 =1
cq (r1 , . . . , rs−2 , q)1{r1
1 +···+rs−3 <
(s−2)q 2 }
r1 f ) ⊗ r2 f . . . f ⊗ rs−2 f, f ⊗q , . . . (f ⊗ H
which is the announced result, since cq (r1 , . . . , rs−2 , q) = q!cq (r1 , . . . , rs−2 ).
2
5.2. Combinatorial expression of cumulants We now provide a classic combinatorial representation of cumulants of the type κs (F ), in the case where: (i) s 2, (ii) q 2, (iii) sq is even, (iv) F = Iq (f ) (with f ∈ Hq ) and (v) H = L2 (A, A , μ) is a non-atomic measure space. Assumptions (i)–(v) will be in order throughout this section. As explained e.g. in [23], one can equivalently express cumulants of chaotic random variables by using diagrams or graphs: here, we choose to adopt the (somewhat simpler) representation in terms of graphs. See [23, Section 4] for an explicit connection between graphs and cumulants; see [11] for some striking application of graph counting to the computation of cumulants of non-linear functionals of spherical Gaussian fields; see [4,5,8] for classic examples of how to use diagram enumeration to deduce CLTs for Gaussian subordinated fields. Definition 5.3. For s 2, consider the set [s] = {1, . . . , s} of the first s integers. For q 2, we denote by K(s, q) the class of all non-oriented graphs γ over [s] satisfying the following properties: – γ does not contain edges of the type (j, j ), j = 1, . . . , s, that is, no edges of γ connect a vertex with itself. One can equivalently say that γ “has no loops”. – Multiple edges are allowed, that is, an edge can appear k 2 times into γ ; in this case, we say that k is the multiplicity of the edge. Also, if i, j are connected by an edge of multiplicity k, we say that i and j are connected k times. – Every vertex appears in exactly q edges (counting multiplicities). – γ is connected. If sq is odd, then K(s, q) is empty. If sq is even, then each γ ∈ K(s, q) contains exactly sq/2 edges (counting multiplicities). For instance: an element of K(3, 2) is γ = {(1, 2), (2, 3), (3, 1)};
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an element of K(3, 4) is γ = {(1, 2), (1, 2), (2, 3), (2, 3), (1, 3), (1, 3)} (note that each edge has multiplicity 2). Definition 5.4. Given q, s 2 such that sq is even, and γ ∈ K(s, q), we define the constant w(γ ) as follows. (a) For every j = 1, . . . , s, consider a generic vector L(j ) = (l(j, 1), . . . , l(j, q)) of q distinct objects. Write {L(j )} for the set of the components of L(j ). (b) Starting from γ , build a matching m(γ ) over L := j =1,...,s {L(j )} as follows. Enumerate the vertices v1 , . . . , vsq/2 of γ . If v1 links i1 and j1 and has multiplicity k1 , then pick k1 elements of L(i1 ) and k1 elements of L(j1 ) and build a matching between the two k1 -sets. If v2 links i2 and j2 and has multiplicity k2 , then pick k2 elements of L(i2 ) (among those not already chosen at the previous step, whenever i2 equals i1 or j1 ) and k2 elements of L(j2 ) (among those not already chosen at the previous step, whenever j2 equals i1 or j1 ) and build a matching between the two k2 -sets. Repeat the operation up to the step sq/2. (c) Define Sq as the group of all permutations of [q]. For every σ ∈ Sq , define the vector Lσ (j ) = (lσ (j, 1), . . . , lσ (j, q)), where lσ (j, p) = l(j, σ (p)), p = 1, . . . , q. (s) (s) (d) Define Sq as the sth product group of Sq , that is, Sq is the collection of all s-vectors of the type σ = (σ1 , . . . , σs ), where σj ∈ Sq , j = 1, . . . , s, endowed with the usual product group structure. (s) (e) For every σ = (σ1 , . . . , σs ) ∈ Sq , build a new matching mσ (γ ) over L by repeating the same operation as at point (b), with the vector Lσj (j ) replacing L(j ) for every j = 1, . . . , s. (s)
(f) Define an equivalence relation over Sq by writing σ ∼γ π whenever mσ (γ ) = mπ (γ ). Let (s) (s) Sq /∼γ be the quotient of Sq with respect to ∼γ . (s) (g) Define w(γ ) to be the cardinality of Sq /∼γ . For instance, one can prove that w(γ ) = 2s−1 for every s 2 and every γ ∈ K(s, 2). Also, w(γ ) = q! for every q 2 and every γ ∈ K(2, q). Definition 5.5. For q 2, let f ∈ Hq , that is, f is a symmetric element of L2 (Aq , A q , μq ). Fix γ ∈ K(s, q), where s 2 and sq is even. Starting from f and γ , we define a function (a1 , . . . , asq/2 ) → fγ (a1 , . . . , asq/2 ), of sq/2 variables, as follows: (i) juxtapose s copies of f , and (ii) if the vertices j and l are linked by r edges, then identify r variables in the argument of the j th copy of f with r variables in the argument of the lth copy. By symmetry, the position of the identified r variables is immaterial. Also, by connectedness, one has necessarily r < q. The resulting function fγ is a (not necessarily symmetric) element of L1 A(sq/2) , A (sq/2) , μ(sq/2) .
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For instance, if γ = {(1, 2), (2, 3), (3, 1)} ∈ K(3, 2), then fγ (x, y, z) = f (x, y)f (y, z)f (z, x). If γ = {(1, 2), (1, 2), (2, 3), (2, 3), (1, 3), (1, 3)} ∈ K(3, 4), then fγ (t, u, v, x, y, z) = f (t, u, v, x)f (t, u, y, z)f (y, z, v, x). We now turn to the main statement of this section, relating graphs and cumulants. The first part is classic (see e.g. [23] for a proof), and shows how to use the functions fγ to compute the cumulants of the random variable F = Iq (f ). The second part of the statement makes use of (5.22), and shows indeed that Theorem 5.1 implicitly provides a compact representation of well-known combinatorial expressions. Proposition 5.6. Let q 2 and assume that F = Iq (f ), where f ∈ Hq . Assume the integer s 2 is such that sq is even. Then, w(γ ) fγ (a1 , . . . , asq/2 ) μ(da1 ) · · · μ(dasq/2 ). (5.27) κs (F ) = γ ∈K(s,q)
A(sq/2)
As a consequence, by using (5.22), one deduces the identity (s − 1)!q! =
r1 f ) ⊗ r2 f . . . f ⊗ rs−2 f, f ⊗q cq (r1 , . . . , rs−2 ) . . . (f ⊗ H w(γ ) fγ (a1 , . . . , asq/2 ) μ(da1 ) · · · μ(dasq/2 ).
γ ∈K(s,q)
(5.28) (5.29)
A(sq/2)
where runs over all collections of integers r1 , . . . , rs−2 verifying the conditions pinpointed in the statement of Theorem 5.1. Remark 5.7. Being based on a sum over the whole set K(s, q), formula (5.27) is of course more compact than (5.22). However, since it does not contain any hint about how one should enumerate the elements of K(s, q), expression (5.27) is much harder to compute and asses. On the other hand, (5.22) is based on recursive relations and inner products of contractions, so that one could in principle compute κs (F ) by implementing an explicit algorithm. 5.3. CLTs on Wiener chaos We conclude the paper by providing a new proof (based on our new formula (5.22)) of the following result, first proved in [22] and yielding a necessary and sufficient condition for CLTs on a fixed chaos. Theorem 5.8. [See [22].] Fix an integer q 2, and let (Fn )n1 be a sequence of the form Fn = Iq (fn ), with fn ∈ Hq such that E[Fn2 ] = q! fn 2H⊗q = 1 for all n 1. Then, as n → ∞, r fn H⊗(2q−2r) → 0 for all r = 1, . . . , q − 1. we have Fn → N(0, 1) in law if and only if fn ⊗ Proof. Observe that κ1 (Fn ) = 0 and κ2 (Fn ) = 1. For κs (Fn ), s 3, we consider the expression (5.22). Let r1 , . . . , rs−2 be some integers such that (i)–(iv) in Theorem 5.1 are sat r h H⊗(p+q−2r) g ⊗r isfied. Using Cauchy–Schwarz inequality and then successively g ⊗ h H⊗(p+q−2r) g H⊗p h H⊗q whenever g ∈ Hp , h ∈ Hq and r = 1, . . . , p ∧ q, we get that
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. . . (fn ⊗ r1 fn ) ⊗ r2 fn . . . fn ) ⊗ rs−2 fn , fn ⊗q H r1 fn ) ⊗ rs−2 fn H⊗q fn H⊗q r 2 fn . . . fn ⊗ . . . (fn ⊗ r1 fn H⊗(2q−2r1 ) fn s−2 fn ⊗ H⊗q s
r1 fn H⊗(2q−2r1 ) . = (q!)1− 2 fn ⊗
(5.30)
r fn H⊗(2q−2r) → 0 for all r = 1, . . . , q − 1, and fix an integer s 3. By Assume now that fn ⊗ combining (5.22) with (5.30), we get that κs (Fn ) → 0 as n → ∞. Hence, applying the method of moments or cumulants, we get that Fn → N (0, 1) in law. Conversely, assume that Fn → N ∼ N(0, 1) in law. Since the sequence (Fn ) lives inside the qth chaos, and because E[Fn2 ] = 1 for all n, we have that, for every p 1, supn1 E[|Fn |p ] < ∞ (see e.g. Janson [9, Ch. V]). This implies immediately that κ4 (Fn ) = E[Fn4 ] − 3 → E[N 4 ] − 3 = 0. Hence, identity (5.23) allows r fn H⊗(2q−2r) → 0 for all r = 1, . . . , q − 1. 2 to conclude that fn ⊗ Acknowledgments Part of this paper has been written while we were visiting the Department of Statistics of Purdue University, in the occasion of the workshop “Stochastic Analysis at Purdue”, from September 28 to October 1, 2009. We heartily thank Professor Frederi G. Viens for his warm hospitality and generous support. References [1] H. Airault, P. Malliavin, F.G. Viens, Stokes formula on the Wiener space and n-dimensional Nourdin–Peccati analysis, J. Funct. Anal. 258 (5) (2010) 1763–1783. [2] A.D. Barbour, Asymptotic expansions based on smooth functions in the central limit theorem, Probab. Theory Related Fields 72 (2) (1986) 289–303. [3] J.-C. Breton, I. Nourdin, G. Peccati, Exact confidence intervals for the Hurst parameter of a fractional Brownian motion, Electron. J. Stat. 3 (2009) 416–425 (electronic). [4] P. Breuer, P. Major, Central limit theorems for non-linear functionals of Gaussian fields, J. Multivariate Anal. 13 (1983) 425–441. [5] D. Chambers, E. Slud, Central limit theorems for nonlinear functionals of stationary Gaussian processes, Probab. Theory Related Fields 80 (1989) 323–349. [6] L.H.Y. Chen, Q.-M. Shao, Stein’s method for normal approximation, in: A.D. Barbour, L.H.Y. Chen (Eds.), An Introduction to Stein’s Method, in: Lecture Notes Ser., vol. 4, Institute for Mathematical Sciences, National University of Singapore, Singapore University Press, World Scientific, 2005, pp. 1–59. [7] R. Fox, M.S. Taqqu, Multiple stochastic integrals with dependent integrators, J. Multivariate Anal. 21 (1) (1987) 105–127. [8] L. Giraitis, D. Surgailis, CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. Verw. Gebiete 70 (1985) 191–212. [9] S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, Cambridge, 1997. [10] P. Malliavin, Stochastic Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1997. [11] D. Marinucci, A central limit theorem and higher order results for the angular bispectrum, Probab. Theory Related Fields 141 (2007) 389–409. [12] I. Nourdin, G. Peccati, Non-central convergence of multiple integrals, Ann. Probab. 37 (4) (2007) 1412–1426. [13] I. Nourdin, G. Peccati, Stein’s method on Wiener chaos, Probab. Theory Related Fields 145 (1) (2009) 75–118. [14] I. Nourdin, G. Peccati, Stein’s method and exact Berry–Esséen asymptotics for functionals of Gaussian fields, Ann. Probab. (2009), in press. [15] I. Nourdin, G. Peccati, Universal Gaussian fluctuations of non-Hermitian matrix ensembles, preprint, 2008. [16] I. Nourdin, G. Peccati, G. Reinert, Second order Poincaré inequalities and CLTs on Wiener space, J. Func. Anal. 257 (2009) 593–609.
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[17] I. Nourdin, G. Peccati, G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos, preprint, 2009. [18] I. Nourdin, G. Peccati, A. Réveillac, Multivariate normal approximation using Stein’s method and Malliavin calculus, Ann. Inst. H. Poincaré Probab. Statist. (2009), in press. [19] I. Nourdin, F.G. Viens, Density estimates and concentration inequalities with Malliavin calculus, Electron. J. Probab. 14 (2009) 2287–2309. [20] D. Nualart, The Malliavin calculus and related topics of Probability and Its Applications, Springer-Verlag, Berlin, 2006. [21] D. Nualart, S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic Process. Appl. 118 (4) (2008) 614–628. [22] D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (1) (2005) 177–193. [23] G. Peccati, M.S. Taqqu, Moments, cumulants and diagram formulae for non-linear functionals of random measures (survey), preprint, 2008. [24] G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: Séminaire de Probabilités XXXVIII, in: Lecture Notes in Math., vol. 1857, Springer-Verlag, Berlin, Heidelberg, New York, 2005, pp. 247– 262. [25] G. Reinert, Three general approaches to Stein’s method, in: A.D. Barbour, L.H.Y. Chen (Eds.), An Introduction to Stein’s Method, in: Lecture Notes Ser., vol. 4, Institute for Mathematical Sciences, National University of Singapore, Singapore University Press, World Scientific, 2005, pp. 183–221. [26] V. Rotar’, Stein’s method, Edeworth’s expansions and a formula of Barbour, in: A.D. Barbour, L.H.Y. Chen (Eds.), An Introduction to Stein’s Method, in: Lecture Notes Ser., vol. 4, Institute for Mathematical Sciences, National University of Singapore, Singapore University Press, World Scientific, 2005, pp. 59–84.
Journal of Functional Analysis 258 (2010) 3792–3800 www.elsevier.com/locate/jfa
3-Manifold group and finite decomposition complexity Qinggang Ren 1 Fudan University, School of Mathematical Sciences, 220 Handan Road, Shanghai 200433, China Received 14 October 2009; accepted 25 January 2010 Available online 1 February 2010 Communicated by Alain Connes
Abstract In this paper, we prove that the fundamental group of an orientable compact 3-manifold has finite decomposition complexity if Thurston’s Hyperbolization Conjecture is true. © 2010 Elsevier Inc. All rights reserved. Keywords: 3-Manifold; Finite decomposition complexity; Hyperbolization Conjecture
1. Introduction Inspired by the property of finite asymptotic dimension of Gromov [4], E. Guentner, R. Tessera and G. Yu introduced the concept of finite decomposition complexity for metric spaces in [7]. And Guentner, Tessera and Yu proved the stable Borel conjecture for closed aspherical manifold M whose fundamental group has finite decomposition complexity. Roughly speaking, a metric space has finite decomposition complexity when there is an algorithm to decompose a space into nice pieces in a certain way. We know that any countable group can be given a proper length metric (if G is a finitely generated group, then the word length metric is such an example). Groups having finite decomposition complexity are a large collection including all countable linear groups, all countable subgroups of an almost connected Lie group, all hyperbolic groups and all elementary amenable groups. In addition, the collection of the groups having finite decomposition complexity is closed under the formation of subgroups, products, extensions, amalgamated E-mail address:
[email protected]. 1 Supported by NNSFC (No. 10731020), SFC (No. 10731020), Shanghai Pujiang Program (08PJ14006), Shanghai
Shuguang Program (07SG38) and the Ministry of Education, PR China. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.020
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free products, HNN-extensions and direct limits [7]. There is no example of a group without finite decomposition complexity other than Gromov’s random groups [5,6] at the moment. In this paper, we prove that the fundamental group of an orientable compact 3-manifold (possible with boundary) has finite decomposition complexity provided Thurston’s Hyperbolization Conjecture is true. There are many important works on this conjecture, see [2,15,18,20–22]. Theorem 1.1. Suppose Thurston’s Hyperbolization Conjecture is true, let G be the fundamental group of an orientable compact 3-manifold M (possible with boundary), then G has finite decomposition complexity. The author would like to express his gratefulness to Professor Guoliang Yu and Professor Xiaoman Chen. 2. Finite decomposition complexity In this section, we will recall the definition of finite decomposition complexity introduced in [7]. Recall that a (countable) collection of subspaces {Zi } of a metric space Z is r-disjoint for some r 0 if for all i = j , we have d(Zi , Zj ) r. Let Z=
Zi
r -disjoint
denote that Z is the union of subspaces Zi with {Zi } r-disjoint. A family of metric spaces {Zi } is bounded if there is a uniform bound on the diameter of the individual Zi : sup diam(Zi ) < ∞. Definition 2.1. A metric space X has finite asymptotic dimension if there exists a d ∈ N such that for every r > 0, the space X can be written as a union of d + 1 families of subspaces, each of which is an r-disjoint union of uniformly bounded subsets: X=
d
Xi ,
Xi =
Xij .
r -disjoint
i=0
Example 2.2. Hyperbolic groups have finite asymptotic dimensions, in particular, free group Fn of n generators has finite asymptotic dimension [23,25,26]. In the same spirit, the definition of finite decomposition complexity can be formulated. Definition 2.3. A metric family X is (d, r)-decomposable over a metric family Y if every X ∈ X admits a decomposition X=
d
Xi ,
Xi =
Xij ,
r -disjoint
i=0 d,r
where each Xij ∈ Y. And denote by X −−→ Y if X is (d, r)-decomposable over Y.
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Roughly speaking, a metric family has finite decomposition complexity if it can be decomposed, through a finite number of applications of the decomposability relation with d = 0 or 1, into a bounded family. Definition 2.4. For each ordinal α, define a collection of families of metric spaces according to the following prescription: (1) Let D0 be the collection of bounded families: D0 = {X : X is bounded}. (2) If α is an ordinal greater than 0, let Dα be the collection of families of metric spaces defined by d,r
Dα = {X : ∃0 d 1, ∀r > 0, ∃Y ∈ Dβ for some β < α such that X −−→ Y}. A metric family X has finite decomposition complexity if there exists an ordinal α such that X ∈ Dα . Any proper metric space (recall that a metric space is said to be proper if any bounded closed ball is compact) with finite asymptotic dimension clearly has finite decomposition complexity. A proper metric space X is a member of Dn for some n ∈ N precisely when X has finite asymptotic dimension. This follows from a theorem of Dranishnikov and Zarichnyi stating that any proper metric space with finite asymptotic dimension admits a coarse embedding into the product space of finitely many locally finite trees [3]. 3. The fundamental group of a surface In this section, we prove that the fundamental group of a surface has finite decomposition complexity. Here by a surface S we mean a connected compact 2-manifold, with or without boundary. Let Fn denote the free group of n generators, then Fn is a tree and has finite decomposition complexity [23]. We first show that: Theorem 3.1. Let S be a closed surface, i.e., a connected compact 2-manifold without boundary, then G = π1 (S) has finite decomposition complexity. Proof. By the classification of the closed surfaces [19], any closed surface is homeomorphic to a sphere, or a sphere with handles, or a sphere with crosscaps. Let S 2 denote the 2-sphere, π1 (S 2 ) = {e} is trivial so has finite decomposition complexity. Recall that a handle T is obtained from a torus T 2 = S 1 × S 1 by removing a disk D, we have π1 (T ) = F2 . Let Hn be a surface obtained by attaching n handles to S 2 , then by Van Kampen theorem, we have π1 (H1 ) = π1 (T ) ∗π1 (S 1 ) π1 S 2 \ D = F2 ∗Z {e} and we know the set of groups having finite decomposition complexity is closed under taking amalgamated free products [7], so π1 (H ) has finite decomposition complexity. For H2 , we have
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π1 (H2 ) = π1 (H1 \ D) ∗π1 (S 1 ) π1 (T ) = π1 S 2 \ D D ∗π1 (S 1 ) π1 (T ) ∗π1 (S 1 ) π1 (T ) = F1 ∗Z F2 ∗Z F2 . Thus π1 (H2 ) has finite decomposition complexity. By induction on n, we know that the fundamental group of the surface homeomorphic to a sphere with n handles has finite decomposition complexity. The same argument applies to the surface which is homeomorphic to a sphere with crosscaps just by replacing handles with Möbius bands. 2 For surfaces with boundary, we have: Theorem 3.2. Let S be a surface with boundary, then G = π1 (S) has finite decomposition complexity. Proof. We also know that any surface with boundary is homeomorphic to a sphere with holes, or a sphere with handles and holes, or a sphere with crosscaps and holes [19]. Let S˜ be a sphere ˜ = Fk−1 . Let H˜ be the surface obtained by attaching a handle T to S. ˜ with k holes, then π1 (S) ˜ ˜ ˜ By Van Kampen theorem, we have π1 (H ) = π1 (S \ D) ∗π1 (S 1 ) π1 (T ) = Fk ∗Z F2 , so H has finite decomposition complexity. By induction on n, we have that the fundamental group of the surface which is homeomorphic to a sphere with n handles and holes has finite decomposition complexity. The same argument applies to the surface which is homeomorphic to a sphere with crosscaps and holes just by replacing handles with Möbius bands. 2 4. Haken manifold In this section, we introduce the definition of Haken manifold and prove that its fundamental group has finite decomposition complexity. Let M be a 3-manifold with or without boundary. Definition 4.1. A surface Σ in M here means a compact connect 2-dimensional manifold (possible with boundary ∂Σ). Σ is called properly embedded in M if ∂Σ = Σ ∩ ∂M or Σ is embedded into ∂M. Definition 4.2. A surface Σ in M is called 2-sided if it is embedded in ∂M or it admits a tubular neighborhood in M which is a trivial line bundle, see [17]. Definition 4.3. A surface is called incompressible if it is 2-sided, not diffeomorphic to a 2sphere nor a disk and if for each disk D in M with D ∩ Σ = ∂D there is a disk D ⊂ Σ such that ∂D = ∂D . Then the inclusion Σ → M induces a monomorphism π1 (Σ) → π1 (M), see [8]. Definition 4.4. A 3-manifold M is called P 2 -irreducible if it is irreducible (recall a 3-manifold M is called irreducible if every embedded 2-sphere in M bounds a 3-ball) and if it contains no embedded 2-sided real projective plane. A compact connected 3-manifold M is called Haken if it is P 2 -irreducible and contains a properly embedded 2-sided incompressible surface (M is oriented then M is Haken if it is irreducible and contains a properly embedded incompressible oriented surface) [17].
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We have the following lemma to recognize a Haken manifold [9]. Lemma 4.5. If a compact connected 3-manifold M is orientable and ∂M = ∅ and ∂M doesn’t consist of a collection of 2-spheres, then H1 (M, Z) is infinite. If M is also irreducible, then M is Haken. Now we prove the finite decomposition complexity for Haken manifolds. Theorem 4.6. The fundamental group of a Haken manifold has finite decomposition complexity. Proof. If M is Haken, there exists a so-called hierarchy [1,9] M = M0 ⊃ M1 ⊃ M2 ⊃ · · · ⊃ M n , i.e., a finite sequence of 3-manifolds where each connect component of Mn is a 3-ball and Mi+1 is obtained from Mi by cutting it open along an incompressible surface Fi → Mi . n is called the length of the hierarchy. , we have Mi+1 If Fi separates Mi into disjoint pieces Mi+1 ∗π1 (Fi ) π1 Mi+1 . π1 (Mi ) = π1 Mi+1 If Fi doesn’t separate Mi then the two embeddings of Fi → ∂Mi+1 induce two monomorphisms τ, θ : π1 (Fi ) → π1 (Mi+1 ), then π1 (Mi ) = HNN π1 (Mi+1 ), π1 (Fi ), τ, θ . In Section 3, we know that the finite decomposition complexity is true for the fundamental group of the surface Fi . As Mn has trivial fundamental group at each component and finite decomposition complexity is closed under taking HNN-extensions or amalgamated free products, by induction on the length of the hierarchy, we know that the fundamental group of a Haken manifold has finite decomposition complexity. 2 5. Seifert manifold and hyperbolizable manifold In this section, we introduce the definition of Seifert manifold and hyperbolizable manifold. Now suppose that a 3-manifold M is connected and S is a sphere in M such that M \ S has two components, M1 and M2 . Let Mi be obtained from Mi by filling in its boundary sphere corresponding to S with a ball. In this situation we say M is the connected sum M1 # M2 . We remark that Mi is uniquely determined by Mi since any two ways of filling in a ball B 3 differ by a diffeomorphism of ∂B 3 , and any diffeomorphism of ∂B 3 extends to a diffeomorphism of B 3 . This last fact follows from the stronger assertion that any diffeomorphism of S 2 is isotopic to either the identity or a reflection (orientation-reversing), and each of these two diffeomorphisms extends over a ball. See [8]. Definition 5.1. A 3-manifold M is called prime if M can be expressed as a connected sum M # M then either M or M is diffeomorphism to S 3 . For example, S 3 and S 1 × S 2 are prime 3-manifolds.
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Lemma 5.2. If an orientable prime manifold M is not irreducible then it is S 1 × S 2 . Proof. Cf. [8] or [9].
2
Definition 5.3. A compact 3-manifold is called Seifert if it admits a foliation by circles. From [12], we have a deep result for Seifert manifolds. Theorem 5.4. A prime compact 3-manifold M with infinite fundamental group π1 (M) is Seifert if and only if π1 (M) contains an infinite cyclic normal group. There exists a short exact sequence of groups 1 → Z → π1 (M) → Γ → 1 where Γ is a discrete subgroup of the following Lie groups: O(3),
R2 O(2),
SO(2, 1).
Since any countable discrete group of linear groups O(3), R2 O(2), SO(2, 1) has finite decomposition complexity [7] and finite decomposition complexity is closed under taking group extensions, for manifolds which have finite fundamental group clearly have finite decomposition complexity, thus the fundamental group of a prime Seifert manifold has finite decomposition complexity. We have the following theorem: Theorem 5.5. The fundamental group of a prime Seifert manifold has finite decomposition complexity. Recall that an n-dimensional manifold is called hyperbolizable if its geometric interior M \ ∂M admits a complete Riemannian metric for which the sectional curvature is constant with value −1. In this case, π1 (M) ∼ = π1 (M \ ∂M) is isomorphic to a discrete subgroup of Lie group SO(n, 1) [14]. So the fundamental group of a hyperbolizable manifold has finite decomposition complexity [7]. 6. Kneser decomposition and JSJ-decomposition In this section, we will introduce Kneser decomposition and JSJ-decomposition to decompose a 3-manifold into nice pieces. Recall that the capped off manifold M˜ of a manifold M means that for every sphere S 2 in ∂M, the boundary of M, we fill a 3-ball. This does not change the fundamental group. So we can assume M does not have an S 2 as a boundary component. Prime manifolds are important because we can decompose a manifold into prime manifolds. We have the following decomposition theorem [9,16]. Theorem 6.1 (Kneser prime decomposition). Each compact connected 3-manifold can be expressed as a connected sum of finite numbers of prime compact connected 3-manifolds M = M1 # M2 # · · · # Mq . If M is orientable, then each Mi is orientable and some Mi is diffeomorphic ˜ S 2 , the to S 1 × S 2 . If M is not orientable, then some Mi is diffeomorphic to S 1 × S 2 or S 1 × 2 1 nonorientable S bundle over S .
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By the Van Kampen theorem, π1 (M) = π1 (M1 ) ∗ π1 (M2 ) ∗ · · · ∗ π1 (Mq ). Now, we define JSJ-decomposition for the orientable irreducible manifolds established by Jaco [10] and Shalen [11] and Johannson [13], also known as Torus Decomposition. Recall that a properly embedded surface Σ ⊂ M is called ∂-parallel if it is isotopic, fixing ∂Σ, to a subsurface of ∂M [8]. An irreducible manifold M is called atoroidal if every incompressible torus is ∂-parallel. Theorem 6.2 (Jaco–Shalen–Johannson decomposition). For any irreducible closed connected orientable 3-manifold M, there exists a minimal finite family {Tj }j ∈J (possible empty) of embedded disjoint incompressible 2-sided closed 2-tori that separates M into a finite set of pieces {Mk }k∈K of irreducible compact connected orientable 3-manifolds, each of which is Seifert or atoroidal, possible both. Using JSJ-decomposition, we can express the fundamental group π1 (M) of an irreducible connected orientable 3-manifold M as a graph of groups G. Recall that a graph of groups G is a non-empty graph GG = (EG , VG ) (possible with loops) equipped with two families of groups {Ge }e∈EG and {Gv }v∈VG parameterized by the edge set EG and vertex set VG , respectively, and there is a family {τe,v : Ge → Gv | v ∈ e}e∈EG of injective group homomorphisms, one for each pair (e, v) ∈ EG × VG consisting of an edge and an adjacent vertex. The groups in {Ge }e∈EG and {Gv }v∈VG are called the edge groups and the vertex groups, respectively, of G. If GG is a finite connected graph, the fundamental group π1 (G) is a group defined, up to isomorphism, by a finite induction process mixing the group Gv and Ge using the incidence relation of GG and maps τe,v via amalgamated free product and HNN-extensions. The group π1 (G) acts on the graph GG with, up to isomorphism, vertex-stabilizers {Gv }v∈VG and edge-stabilizers {Ge }e∈EG . So we know that π1 (G) has finite decomposition complexity if all its vertex groups and edge groups have finite decomposition complexity. For a given irreducible connected orientable 3-manifold M, we apply the JSJ-decomposition. It turns out there is a graph of groups G with EG = J and VG = K. For j ∈ J and k ∈ K, Gj = π1 (Tj ) ∼ = Z2 , Gk = π1 (Mk ) and τj,k := π1 (incl : Ti → Mk ), with the incidence relation dictated by combinatorial configuration of separating family of tori. In addition, π1 (M) ∼ = π1 (G). 7. Proof of the main theorem We are now in the position to prove the main theorem: Theorem 7.1. Suppose that Thurston’s Hyperbolization Conjecture is true, let G be the fundamental group of an orientable compact 3-manifold M (possible with boundary), then G has finite decomposition complexity. Proof. Given such a manifold M, we may further assume it is connected and it is capped off. We first apply Kneser’s prime decomposition to express π1 (M) = π1 (M1 ) ∗ π1 (M2 ) ∗ · · · ∗ π1 (Mn ) with some Mi diffeomorphic to S 1 × S 2 and others irreducible. And π1 (S 1 × S 2 ) = Z has finite decomposition complexity. We can further assume M is irreducible. After the capped off, if ∂M = ∅ and π1 (M) is infinite, then M is Haken. Otherwise M is closed, we apply JSJ-decomposition to express π1 (M) = π1 (G). And we know its edge group is π1 (Tj ) = Z2 has finite decomposition complexity. We only concerned on the vertex groups, the fundamental group of pieces, say N .
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(1) If N is Seifert this is the result of Section 5. (2) If π1 (N ) is finite, this is obvious. (3) If N is Haken, this is the result of Section 4. (4) If N is non-Seifert non-Haken and has infinite fundamental group, then ∂N = ∅ otherwise N is Haken by Lemma 4.5, so N should be hyperbolizable by the Thurston’s Hyperbolization Conjecture [24]. So N has finite decomposition complexity by Section 5. Thus all the edge groups and vertex groups of π1 (M) = π1 (G) have finite decomposition complexity, then π1 (M) has the finite decomposition complexity, we finish the proof. 2 Remark 7.2. Using the orientable double cover, the conclusion is also true for nonorientable compact 3-manifold. References [1] C. Béguin, H. Bettaieb, A. Valette, K-theory for C ∗ -algebra of one relator groups, J. K-Theory 16 (3) (1999) 277– 298. [2] H. Cao, X. Zhu, A complete proof of the Poincaré and geometrization conjectures-application of the Hamilton– Perelman theory of the Ricci flow, Asian J. Math. 10 (2) (2006) 165–492. [3] A. Dranishnikov, M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2–3) (2004) 203– 225. [4] M. Gromov, Asymptotic invariants of infinite groups, in: Geometric Group Theory, vol. 2, Sussex, 1991, in: London Math. Soc. Lecture Note Ser., vol. 182, Cambridge University Press, Cambridge, 1993, pp. 1–295. [5] M. Gromov, Spaces and questions, in: GAFA 2000, Tel Aviv, 1999, Geom. Funct. Anal. (2000) 118–161, special volume, Part I. [6] M. Gromov, Random walks in random groups, Geom. Funct. Anal. 13 (1) (2003) 73–146. [7] E. Guentner, R. Tessera, G. Yu, Decomposition complexity and stable Borel conjecture, preprint. [8] A. Hatcher, Notes on basic 3-manifold topology, available on line at http://www.math.cornell.edu/~hatcher/. [9] J. Hempel, 3-Manifolds, Ann. of Math. Stud., vol. 86, Princeton University Press, 1976. [10] W. Jaco, Finitely presented subgroups of three manifolds, Invent. Math. 13 (1971) 335–346. [11] W. Jaco, P. Shalen, Seifert fibered spaces in 3-manifolds, Mem. Amer. Math. Soc. 21 (220) (1979), 192 pp. [12] M. Jankin, W.D. Neumann, Lectures on Seifert Manifolds, Brandeis Lecture Notes, vol. 2, Brandeis University, 1983. [13] K. Johannson, Homotopy Equivalence of 3-Manifolds with Boundaries, Lecture Notes in Math., vol. 761, Springer, 1979. [14] M. Kapovich, Hyperbolic Manifolds and Discrete Groups, Progr. Math., vol. 183, Birkhäuser, 2001. [15] B. Kleiner, J. Lott, Notes on Perelman’s paper, arXiv:math.DG/0605667. [16] H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein. 38 (1929) 248–260. [17] M. Matthey, H. Oyono-Oyono, W. Pitsch, Homotopy invariance of higher signatures and 3-manifold groups, Bull. Soc. Math. France 136 (1) (2008) 1–25. [18] J.W. Morgan, Gang Tian, Ricci flow and the Poincaré conjecture, arXiv:math.DG/0607607v2. [19] J. Munkres, Topology, second ed., Prentice Hall, Upper Saddle River, NJ, 2000. [20] G. Perelmann, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159v1, November 11, 2002, preprint. [21] G. Perelmann, Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109v1, March 10, 2003, preprint. [22] G. Perelmann, Finite extinction time to the solutions to the Ricci flow on certain three manifolds, arXiv:math.DG/ 0307245, July 17, 2003, preprint. [23] J. Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc. 133 (9) (2005) 2489–2490, no. 28, 39 (electronic). [24] W.P. Thurston, Three-Dimensional Geometry and Topology, vol. 1, Princeton Math. Ser., vol. 35, Princeton University Press, 1997.
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[25] G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (2) (1998) 325–355. [26] G. Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (1) (2000) 201–240.
Journal of Functional Analysis 258 (2010) 3801–3817 www.elsevier.com/locate/jfa
Fixed point properties of semigroups of nonexpansive mappings Narcisse Randrianantoanina Department of Mathematics, Miami University, Oxford, OH 45056, USA Received 19 October 2009; accepted 25 January 2010 Available online 1 February 2010 Communicated by K. Ball
Abstract We prove that if H is a Hilbert space then the Schatten (trace) class operators on H has the weak∗ fixed point property for left reversible semigroups. This answered positively a problem raised by A.T.-M. Lau. We also prove that if M is a finite von Neumann algebra then any nonempty bounded convex subset of the non-commutative L1 -space associated to M that is compact for the measure topology has the fixed point property for left reversible semigroups. © 2010 Elsevier Inc. All rights reserved. Keywords: Non-commutative L1 -space; Weak∗ fixed point property; Nonexpansive mapping; Left reversible semigroup
1. Introduction Let X be a Banach space and K be a nonempty bounded closed convex subset of X. We say that the subset K has the fixed point property if every nonexpansive mapping T : K → K (i.e., T x − T y x − y for all x, y ∈ K) admits a fixed point. The Banach space X is said to have the weak fixed point property if every weakly compact convex subset of X has the fixed point property. If X is a dual Banach space then it is said to have the weak∗ fixed point property if every weak∗ compact convex subset of X has the fixed point property. The purpose of the present paper is to study stronger versions of the fixed point properties defined above that are related to common fixed points of some types of semigroups of nonexpansive mappings which we now formally define: E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.022
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Let S be a semitopological semigroup, i.e., S is a semigroup equipped with a Hausdorff topology such that for each a ∈ S, the mappings s → as and s → sa are continuous. Such semitopological semigroup S is called left reversible if for any a, b ∈ S, one has aS ∩ bS = ∅. It is clear that abelian semitopological semigroups and semitopological groups are left reversible. Other examples of left reversible semigroups and connections with the notion of left amenability can be found for instance in [10,12–14,17,23]. For a topological space K, an action of S on K is a map ψ : S × K → K, denoted by ψ(s, k) = s(k) or sk for s ∈ S and k ∈ K, that satisfies s1 s2 (k) = s1 (s2 k) for all s1 , s2 ∈ S, and k ∈ K. We also assume that ψ is separately continuous, i.e., for every s0 ∈ S and k0 ∈ K, the maps s → sk0 (S → K) and k → s0 k (K → K) are continuous. The following notion of fixed point property for Banach spaces isolates the primary topic of this paper: Definition 1.1. A Banach space X is said to have the weak fixed point property for left reversible semigroups if whenever S is a left reversible semigroup and K is a nonempty weakly compact convex subset of X for which the action of S on K (with the norm topology) is separately continuously and nonexpansive then S has a common fixed point. If X is a dual Banach space then a similar notion of weak∗ fixed point property for left reversible semigroups can be defined using nonempty weak∗ compact convex subsets of X in the preceding definition. Common fixed point theorems for semigroups of nonexpansive mappings have been extensively studied by several authors (see for instance [12] and the references contained therein). It is clear that the weak fixed point property for left regular semigroups (respectively, weak∗ fixed point property for left reversible semigroups) is stronger than the usual weak fixed point (respectively, weak∗ fixed point property). Indeed, whenever T : K → K is a nonexpansive mapping, one can consider the semigroup S = (N, +) with the action of S on K is given by the mappings {T n : n ∈ N} so that any common fixed point of the semigroup S is clearly a fixed point of T and vice versa. For the converse, we have a general result due to Bruck [3] which states that if a Banach space has the weak fixed point property then it has the weak fixed point property for abelian semigroups and Lim proved in [21] that any Banach space with the weak normal structure has the weak fixed point property for left reversible semigroups. Unfortunately, any corresponding general statement is still unknown for weak∗ normal structure and weak∗ fixed point property for left reversible semigroups (even for commuting semigroups). In this paper we are mainly concerned with various aspects of fixed point properties related to actions of left reversible semigroups on subsets of certain classes of non-commutative L1 -spaces. We recall first a result of Lennard [18] which states that if H is a Hilbert space then the ideal of trace class operators on H has the uniform Kadec–Klee property for the weak∗ topology. It then follows from general fixed point theory that it has the weak∗ normal structure and therefore it has the weak∗ fixed point property. The main motivation for the present paper comes from a question of whether or not the ideal of trace class operators on H has the weak∗ fixed property for left reversible semigroups. Such question was raised by Lau in [12, Problem 5] (see also the recent article [14, Open Problem 6.3]). We obtain that the answer to the above question is positive. We also provide necessary and sufficient conditions for Fourier–Stieltjes algebra and Fourier algebra of any given locally compact group to have the weak fixed point property for left reversible semigroups. These improve recent results of Lau and Mah in [14]. Our notation and terminology are standard as may be found in the books [8] for fixed point theory, [5] for Banach spaces, and [29] for operator algebras.
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The paper is organized as follows. In Section 2, we establish an inequality involving sequences on certain types of non-commutative L1 -spaces which is crucial for our approach. In Section 3, we prove our main result concerning the weak∗ fixed point property for left reversible semigroups for spaces of trace class operators on Hilbert spaces. Section 4 is devoted to fixed point properties of Fourier–Stieltjes algebras and Fourier algebras of locally compact groups, providing in particular generalizations of results from [14, Section 5]. In Section 5, we explore extensions of our method to the study of fixed point property for left reversible semigroups acting on subsets of non-commutative L1 -spaces (associated with finite von Neumann algebras) which are compact with respect to the topology of convergence in measure. 2. Preliminary results Throughout, M denotes a von Neumann algebra. The identity of M will be denoted by 1, and Mproj stands for the set of all (self adjoint) projections in M. As customary, for every ϕ ∈ M∗ and x ∈ M, xϕ (respectively, ϕx) denotes the (normal) functional y → ϕ(yx) (respectively, y → ϕ(xy)). Our first result will be used as an intermediate step toward the key technical inequality that can be considered as the primary tool for our approach. Lemma 2.1. Assume that {Dα : α ∈ Λ} is a decreasing net of bounded subsets of M∗ and (ξm )m1 is a bounded and disjointly supported sequence in M∗ (in the sense that there is a sequence of mutually disjoint sequence (em )m1 in Mproj such that for every m 1, ξm = em ξm em ), then lim sup ψ: ψ ∈ Dα + lim ξm = lim lim sup ξm − ψ: ψ ∈ Dα . α
m→∞
m→∞ α
As observed in [14, Remark 6.1], Lemma 2.1 cannot be extended to any arbitrary sequence in M∗ . In Proposition 2.4 below, we will provide a weaker version of Lemma 2.1 that is suitable for general bounded sequences that are not necessarily disjoint. For the proof of Lemma 2.1, we will make use of the following two facts which can be found in [27, Proposition 2.13]. Sublemma 2.2. If ξ ∈ M∗ and e ∈ Mproj then (i) ξ eξ e + (1 − e)ξ(1 − e); (ii) ξ e ξ 1/2 e|ξ |e1/2 , where |ξ | is the modulus of ξ . Proof of Lemma 2.1. Our proof should be compared with the argument used in the proof of [14, Lemma 3.1]. First, we observe that one inequality is an immediate consequence of triangle inequality. Indeed, for each ψ ∈ Dα and m 1, ψ + ξm ξm − ψ. Hence for every α and m 1, sup ψ: ψ ∈ Dα + ξm sup ξm − ψ: ψ ∈ Dα .
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Taking the limit on α, we get that for every m 1, lim sup ψ: ψ ∈ Dα + ξm lim sup ξm − ψ: ψ ∈ Dα . α
α
Taking limits over m provides the desired inequality. To prove the converse inequality, we may assume without loss of generality that {Dα : α ∈ Λ} is a decreasing sequence of bounded subsets of M∗ which we will now denote by {Dn : n 1}. For n ∈ N, we can choose ψn ∈ Dn such that: lim ψn = lim sup ψ: ψ ∈ Dn
n→∞
n→∞
and both limm→∞ ξm := q and limm→∞ limn→∞ ξm − ψn := r exist. Thus it suffices to show the following inequality: lim ψn r − q.
n→∞
(2.1.1)
Assume to the contrary that there exists p > 0 so that limn→∞ ψn = r − q + p. Fix ε > 0 so that p − 3ε > 0. We claim that there exist k N1 < N2 < · · · < Nk < · · · and m1 < m2 < · · · < mk < · · · sequences in N such that if πk = m j =1 ej , then for every n Nk , (πk − πk−1 )ψn (πk − πk−1 ) + ψn (πk − πk−1 ) + (πk − πk−1 )ψn p − 3ε. (2.1.2) To establish (2.1.2), we proceed inductively. Set m0 = 0 and choose m1 1 so that ξm1 q − ε
and
lim ξm1 − ψn < r + ε.
n→∞
Then N1 can be chosen so that for every n N1 , ξm1 − ψn < r + ε
and ψn > r − q + p − ε.
Observe that ξm1 is supported by em1 so that (1 − π1 )ξm1 = ξm1 (1 − π1 ) = 0. Using this fact with Sublemma 2.2(i), we have for each n N1 , r + ε > ξm1 − ψn π1 (ξm1 − ψn )π1 + (1 − π1 )(ξm1 − ψn )(1 − π1 ) = ξm1 − π1 ψn π1 + (1 − π1 )ψn (1 − π1 ). Let vn = (1−π1 )ψn (1−π1 ) = ψn −ψn π1 −π1 ψn (1−π1 ) then ψn = vn +ψn π1 +π1 ψn (1−π1 ) and therefore ψn (1 − π1 )ψn (1 − π1 ) + ψn π1 + π1 ψn . We deduce that
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r + ε > ξm1 − π1 ψn π1 + ψn − ψn π1 − π1 ψn , that is, r + ε > (q − ε) + (r − q + p − ε) − π1 ψn π1 − ψn π1 − π1 ψn which implies that for every n N1 , π1 ψn π1 + ψn π1 + π1 ψn p − 3ε. Now assume that N1 < N2 < · · · < Nk−1 and m1 < m2 < · · · < mk−1 have been chosen. First, choose mk > mk−1 with ξmk q − ε
and
lim ξmk − ψn < r + ε.
n→∞
Then as above, Nk > Nk−1 can be chosen so that for every n Nk , ξmk − ψn < r + ε
and ψn > r − q + p − ε.
Note that ξmk is supported by πk − πk−1 . Writing 1 = (πk − πk−1 ) + [1 − (πk − πk−1 )], it follows as above that r + ε > ξmk − ψn (πk − πk−1 )(ξmk − ψn )(πk − πk−1 ) + 1 − (πk − πk−1 ) (ξmk − ψn ) 1 − (πk − πk−1 ) = ξmk − (πk − πk−1 )ψn (πk − πk−1 ) + 1 − (πk − πk−1 ) ψn 1 − (πk − πk−1 ) . Now we repeat the argument used in the above estimate with the projection πk − πk−1 in place of π1 to deduce that (πk − πk−1 )ψn (πk − πk−1 ) + ψn (πk − πk−1 ) + (πk − πk−1 )ψn p − 3ε. This complete the induction and proves (2.1.2). Now for k ∈ N and n Nk , we use Sublemma 2.2(ii) with ξ = ψn and e = πk − πk−1 to deduce that ψn (πk − πk−1 ) ψn 1/2 . (πk − πk−1 )|ψn |(πk − πk−1 )1/2
1/2 ψn 1/2 . (πk − πk−1 ) |ψn | + ψn∗ (πk − πk−1 ) . Similarly, taking adjoints leads to
(πk − πk−1 )ψn ψn 1/2 . (πk − πk−1 ) ψ ∗ (πk − πk−1 )1/2 n
1/2 ψn 1/2 . (πk − πk−1 ) |ψn | + ψn∗ (πk − πk−1 ) .
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Combining with the trivial fact that (πk −πk−1 )ψn (πk −πk−1 ) (πk −πk−1 )ψn , we deduce from (2.1.2) that for every n Nk ,
1/2 3ψn 1/2 . (πk − πk−1 ) |ψn | + ψn∗ (πk − πk−1 ) p − 3ε. That is,
(p − 3ε)2 ψn . (πk − πk−1 ) |ψn | + ψn∗ (πk − πk−1 ) . 9
(2.1.3)
Taking the sum over j ∈ [1, k], we deduce from Sublemma 2.2(i) that for every n Nk ,
k(p − 3ε)2 , ψn . |ψn | + ψn∗ 9 which gives that √ (p − 3ε) k ψn . √ 3 2 This is in contradiction with the fact that the sequence (ψn )n1 is in D1 and therefore is bounded. This proves (2.1.1) and completes the proof. 2 The following subsequence decomposition principle is needed in the sequel. It is often refers to as the (non-commutative) Kadec–Pełczy´nski’s decomposition. We only state the version we need and refer the reader to [26–28] for more general versions. Theorem 2.3. (See [26].) Let (φn )n1 be a bounded sequence in M∗ . Then there exist a sub(1) (2) sequence (φnk )k1 of (φn )n1 , two bounded sequences (φk )k1 and (φk )k1 in M∗ , and a mutually disjoint sequence of projections (ek )k1 in Mproj such that: (1)
(2)
(i) φnk = φk + φk for all k 1; (ii) {φk(1) : k 1} is a relatively weakly compact subset of M∗ with the property that (1) ek φk ek = 0 for all k 1; (iii) the sequence (φk(2) )k1 is disjointly supported with ek φk(2) ek = φk(2) for all k 1. We now come to the principal estimate needed for our main result. It can be viewed as an extension of Lemma 2.1 to general bounded sequences in preduals of some special von Neumann algebras. Proposition 2.4. Assume that M∗ is a dual Banach space such that every bounded disjointly supported sequence in M∗ converges to zero for the weak∗ topology. If {Dα : α ∈ Λ} is a decreasing net of bounded subsets of M∗ and (φm )m1 is a weak∗ convergent sequence with weak∗ limit φ, then there exists a sequence (νm )m1 with νm ∈ conv{φm , φm+1 , . . .} for all m 1, that satisfies:
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lim sup φ − ψ: ψ ∈ Dα + lim νm − φ α m→∞ lim lim sup φm − ψ: ψ ∈ Dα . m→∞ α
Proof. First, we apply Theorem 2.3 to the sequence (φm )m1 : there exist a subsequence of (1) (φm )m1 (which we will assume to be (φm )m1 itself), bounded sequences (φm )m1 and (2) (φm )m1 in M∗ and a mutually disjoint sequence of projections (em )m1 in Mproj such that: (i) (ii) (iii) (iv)
(1)
(2)
φm = φm + φm for all m 1; (1) {φm : m 1} is a relatively weakly compact subset of M∗ ; (2) (2) (2) the sequence (φm )∞ m=1 is such that em φm em = φm for all m 1; (by taking subsequences if necessary) we may also assume that both limits (2) limm→∞ φm − φ and limm→∞ φm exist. (2)
Observe that since (φm )m1 is a bounded disjointly supported sequence, by assumption, it con(1) (1) verges weak∗ to zero. Hence, (φm )n1 converges weak∗ to φ. But since {φm : m 1} is a (1) relatively weakly compact subset of M∗ and φ is the unique weak cluster point of (φm )m1 , (1) it follows that (φm )m1 converges weakly to φ. Choose a sequence of convex combinations (1) (1) (νm )m1 of (φm )m1 so that: (1) lim νm − φ = 0.
(2.1.4)
m→∞
βm (1) (1) For each m 1, write νm = j =α λ φ with 1 α1 < β1 < α2 < β2 < · · · , λj ∈ [0, 1] for m j j βm all j 1, and j =αm λj = 1 for all m 1. For m 1 set (2) νm :=
βm
(2)
λj φj
and νm :=
j =αm
βm
λj φj
j =αm
so that for m 1, (1) (2) νm = νm + νm .
(2.1.5)
Apply Lemma 2.1 to the decreasing net of bounded subsets {Dα − φ: α ∈ Λ} (here Dα − φ = (2) {ψ − φ: ψ ∈ Dα }) and the mutually disjointly supported bounded sequence (νm )m1 to obtain that (2) lim sup ψ − φ: ψ ∈ Dα + lim νm α m→∞ (2) lim lim sup νm + φ − ψ : ψ ∈ Dα . m→∞ α
(2)
(2)
(2.1.6)
Since limm→∞ φm exists and (νm )m1 consists of block convex combinations of the se(2) (2) (2) (1) quence (φm )m1 , it follows that limm→∞ νm exists. Moreover, as νm = νm − νm =
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(1)
(2)
(νm −φ)+(φ −νm ) and limm→∞ νm −φ = 0, we have limm→∞ νm −φ limm→∞ νm and therefore it can be deduced from (2.1.6) that lim sup φ − ψ: ψ ∈ Dα + lim νm − φ α m→∞ (2) lim lim sup νm + φ − ψ : ψ ∈ Dα .
(2.1.7)
m→∞ α
Similarly, for every m 1, α ∈ Λ, and ψ ∈ Dα , (2) ν + φ − ψ νm − ψ + ν (1) − φ m
m
βm
(1) λi φi − ψ + νm − φ .
i=αm
It follows that for every m 1 and α ∈ Λ, βm (2) (1) sup νm + φ − ψ : ψ ∈ Dα λi sup φi − ψ: ψ ∈ Dα + νm − φ , i=αm
and taking the limit on α, we get that for every m 1, (2) lim sup νm + φ − ψ : ψ ∈ Dα α
βm
(1) λi lim sup φi − ψ: ψ ∈ Dα + νm − φ ,
i=αm
α
which in turn implies that (2) lim lim sup νm + φ − ψ : ψ ∈ Dα
m→∞ α
lim
m→∞
βm i=αm
λi lim sup φi − ψ: ψ ∈ Dα α
lim lim sup φm − ψ: ψ ∈ Dα . m→∞ α
We get the desired inequality by combining this last estimate with (2.1.7).
2
Remark 2.5. There are examples of class of von Neumann algebras N for which N∗ is a dual Banach space but does not satisfy the assumption of Proposition 2.4. For instance, if N = B(l 2 )∗∗ then N∗ = B(l 2 )∗ is a dual Banach space. Since weak∗ convergent sequences in B(l 2 )∗ are weakly convergent (see [25]), a bounded disjointly supported sequence (xn )n1 in B(l 2 )∗ converges to zero for the weak∗ topology if and only if limn→∞ xn = 0. We do not know however if the assumption in the statement of Proposition 2.4 can be removed.
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3. Weak∗ fixed point properties for dual non-commutative L1 -spaces Recall some general concepts from fixed point theory. Let X be a Banach space, C be a nonempty subset of X, and {Dα : α ∈ Λ} be a decreasing net of bounded subsets of X. For each x ∈ C and α ∈ Λ, we set rα (x) := sup x − y: y ∈ Dα and r(x) := lim rα (x) = inf rα (x). α
α
We recall that the asymptotic radius of {Dα : α ∈ Λ} with respect to C is the quantity r := inf r(x): x ∈ C and the asymptotic center of {Dα : α ∈ Λ} with respect to C is the subset (possibly empty) of C defined by AC {Dα : α ∈ Λ} = x ∈ C: r(x) = r . It is clear that if C is convex then AC({Dα : α ∈ Λ}) is also a convex subset of X. The following result can be viewed as the principal result of this section. Theorem 3.1. Assume that M∗ is a dual Banach space with the property that every bounded disjointly supported sequence in M∗ converges to zero for the weak∗ topology. If C is a nonempty weak∗ sequentially compact and convex subset of M∗ , and {Dα : α ∈ Λ} is a decreasing net of bounded subsets of M∗ , then AC({Dα : α ∈ Λ}) (with respect to C) is a nonempty weakly compact subset of M∗ . As expected, the proof depends heavily on Proposition 2.4 which using the terminology introduced above can be restated in the following form: Lemma 3.2. Assume that M∗ satisfies the assumptions of Theorem 3.1. If (φm )m1 is a weak∗ convergent sequence in M∗ with weak∗ limit φ, then there exists a sequence (νm )m1 with νm ∈ conv{φm , φm+1 , . . .} that satisfies r(φ) + lim νm − φ lim r(φm ). m→∞
m→∞
We will also make use of the following well-known fact: Lemma 3.3. Let E be a real or complex Banach space. A bounded subset A of E is relatively weakly compact if and only if for any given sequence (xn ) in A, there exists a sequence (yn ) with yn ∈ conv{xn , xn+1 , . . .} that converges weakly.
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Proof of Theorem 3.1. Let r be the asymptotic radius of {Dα : α ∈ Λ} with respect to C and K = {x ∈ C: r(x) = r} its asymptotic center. We will verify first that K is nonempty. For each s > r, set Ks := {x ∈ C: r(x) s}. Claim 1. For every s > r, the set Ks is weak∗ sequentially closed. To verify this claim, we observe that if (φj )j 1 is a sequence in Ks with (φj )j 1 converges weak∗ to φ, then by Lemma 3.2, there exists a sequence (νm )m1 with νm ∈ conv{φm , φm+1 , . . .} for every m 1 and such that r(φ)+limm→∞ νm −φ limj →∞ r(φj ). It follows immediately since r(φj ) s for all j 1 that r(φ) s that is φ ∈ Ks . This proves the claim. (s) For each s > r, set r (s) := inf{r(x): x ∈ C ∩ Ks } and K0 := {x ∈ C ∩ Ks : r(x) = r (s) }. From the definition of Ks , we clearly have r r (s) < s. Claim 2. For every s > r, there exists xs ∈ C ∩ Ks with r(xs ) = r (s) . Indeed, for s > r and j 1, there exists yj ∈ C ∩ Ks so that r (s) r(yj ) < r (s) + 1/j . Taking a subsequence if necessary, we may assume that (yj )j 1 converges weak∗ to xs ∈ C ∩ Ks (this is possible since C is weak∗ sequentially compact and Ks is weak∗ sequentially closed). As above, we have r(xs ) limj →∞ r(yj ) r (s) which verifies the claim. Fix a sequence of numbers (sj )j 1 with sj > r for all j and limj →∞ sj = r (it follows that limj →∞ r (sj ) = r) and consider the corresponding sequence (xsj )j 1 in C obtained from Claim 2 above (i.e., r(xsj ) = r (sj ) for all j 1). Again by passing to a subsequence if necessary, we may assume that (xsj )j 1 converges weak∗ to some x ∈ C. It follows as above that r(x) limj →∞ r(xsj ) = limj →∞ r (sj ) = r. This proves that x ∈ K and therefore K = ∅. We will now prove that K is weakly compact. For this, let (φj )j 1 be a sequence in K with (φj )j 1 converges weak∗ to φ. Since K ⊂ Ks (for all s > r) and Ks is weak∗ sequentially closed (see Claim 1 above), we have φ ∈ s>r Ks and by the definition of the Ks ’s, r(φ) r, i.e., φ ∈ K. Now, by Lemma 3.2, there exists a sequence (νm )m1 with νm ∈ conv{φm , φm+1 , . . .} for every m 1 and such that r(φ) + lim νm − φ lim r(φj ). m→∞
j →∞
From the definition of K, we have r(φ) = r(φj ) = r for all j 1. We deduce that limm→∞ νm − φ = 0. We then conclude from Lemma 3.3 that K is a relatively weakly compact convex subset of M∗ . But since it is already norm closed, we have that K is weakly compact. The proof is complete. 2 Remark 3.4. For the case where the predual of the dual Banach space M∗ is separable, the proof above can be streamlined. In fact, for this special case, any weak∗ closed bounded subset of M∗ is a compact metric space and therefore one can deduce from Lemma 2.4 that the realvalued function r(·) is weak∗ lower semicontinuous. With this observation, it follows that for every s > 0, the bounded subset Ks is weak∗ compact. Another consequence is that Claim 2 becomes trivial. Indeed, it follows immediately from the general fact that any lower semicontinuous function defined on a compact set attains its minimum.
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We will apply Theorem 3.1 to ideals of trace class operators on Hilbert spaces. Let H be a Hilbert space. We equip the (semifinite) von Neumann algebra B(H) with its usual trace tr. It is well known that the trace class T (H) can be realized as the predual of the semifinite von Neumann algebra (B(H), tr), i.e., T (H) = L1 (B(H), tr). Moreover, T (H) is a dual Banach space with its predual being the C ∗ -algebra K(H) of all compact operators on H. We assume that the next lemma is well known. It shows that the assumptions of Theorem 3.1 are satisfied by T (H). We include a short argument for completeness. Lemma 3.5. Let (γn )n1 be a bounded and disjointly supported sequence in T (H). Then (γn )n1 converges to zero for the weak∗ topology. Proof. Let (en )n1 be a sequence of mutually orthogonal projections in B(H) so that γn = en γn en for all n 1. If h ∈ H, we have n1 en h2 h2 . In particular lim en h = 0,
n→∞
i.e., limn→∞ en = 0 for the strong operator topology. Fix an operator a ∈ K(H). Since a is compact, limn→∞ en a(h) = 0 uniformly on all h in the unit ball of H. Therefore, limn→∞ en aen = 0. Now for a ∈ K(H) and n 1, | γn , a| = | en γn en , a| = | γn , en aen | γn . en aen . Thus, we get that limn→∞ γn , a = 0, proving that (γn ) converges weak∗ to zero. 2 Our next result solved positively a question raised by Lau in [12]. Theorem 3.6. Let H be a (not necessarily separable) Hilbert space. The space T (H) of trace class operators on H has the weak∗ fixed point property for left reversible semigroups. Proof. Let S be a left reversible semitopological semigroup and C be a nonempty weak∗ compact convex subset of T (H) for which the action of S on (C, · ) is separately continuous and nonexpansive. The crucial part here (when H is nonseparable) is that C is weak∗ sequentially compact. Indeed, since T (H) has the Radon–Nikodým property, it follows from [9, Corollary 2] (see also [5, Theorem 6, p. 230]) that the unit ball of T (H) is weak∗ sequentially compact and since C is norm bounded it is also weak∗ sequentially compact. The set S is directed by a b if and only if aS ⊆ bS. Fix u ∈ C and set for a ∈ S, Da := aS(u). Then {Da : a ∈ S} is a decreasing net of subsets of C. Denote by K the asymptotic center of {Da : a ∈ S} with respect to C. By Theorem 3.1, K is a nonempty weakly compact convex subset of C. We claim that K is S-invariant. The proof of this claim follows exactly the argument used in [14, Theorem 4.1] but we include the details for completeness. Let x ∈ K, s ∈ S, and fix ε > 0. If we denote by r the asymptotic radius of {Da : a ∈ S} with respect to C, then by the definition of K, we have r(x) = r. That is, lim ra (x) = lim sup x − y; y ∈ Da = r. a
a
Hence there exists a0 ∈ S so that sup{x − y; y ∈ Da0 } < r + ε. In particular, a0 S(u) ⊂ Da0 ⊂ B(x, r + ε)
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where B(x, r +ε) is the ball centered at x and with radius r +ε. Now since s ∈ S is nonexpansive, it follows that sa0 S(u) ⊂ B s(x), r + ε . Taking the (norm) closure, we have Dsa0 ⊂ B(s(x), r + ε) and this is equivalent to rsa0 s(x) = sup s(x) − y ; y ∈ Dsa0 r + ε. This shows that r(s(x)) r + ε. Since ε is arbitrary, we conclude that r(s(x)) r proving that s(x) ∈ K. To conclude the proof of the theorem, we recall from [18] that T (H) has the uniform Kadec– Klee property for the weak∗ topology and therefore it satisfies the weak∗ normal structure. A fortiori, T (H) has the weak normal structure and by a general result of Lim [20, Theorem 3], it has the weak fixed point property for left reversible semigroups. Now as K is a nonempty weakly compact convex of T (H) and the action of S on K is separately continuous and nonexpansive, we can conclude that S has a common fixed point on K and therefore on C. This completes the proof. 2 The argument above can easily be adapted to prove the following: Corollary 3.7. If {Hρ ; ρ ∈ I } is a collection of Hilbert spaces then ( ρ T (Hρ ))l 1 (with its natural predual ( ρ K(Hρ ))c0 ) has the weak∗ fixed point property for left reversible semigroups. In the next result, we explore possible extensions of Theorem 3.6 to the context of scattered C ∗ -algebras. We refer the reader to [4,11] for the definition, examples, and basic properties of scattered C ∗ -algebras. Corollary 3.8. If A is a dual scattered C ∗ -algebra then A∗ has the weak∗ fixed point property for left reversible semigroups. exists Proof. If A is a dual C ∗ -algebra then there
a collection of Hilbert spaces {Hρ ; ρ ∈ I } so that A is ∗-isomorphic to the C ∗ -algebra ( ρ K(Hρ ))c0 (see for instance [29, p. 157]). The conclusion then follows immediately from Corollary 3.7. 2 The preceding result naturally leads to the following general problem: Question 3.9. If A is a scattered C ∗ -algebra. Does A∗ have the weak∗ fixed point property for left reversible semigroups? One should note that according C ∗ -algebra then A∗ is iso to [31, p. 101], if A is a scattered ∗ morphically isometric to ( ρ T (Hρ ))l 1 and therefore A has the weak fixed point property for left reversible semigroups. However, in general, such isometry is not necessarily weak∗ continuous.
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4. Weak fixed point properties for semigroups In this section we consider weak fixed point properties for concrete spaces related to group C ∗ -algebras. Let G be a locally compact group with a fixed left Haar measure m. We define C ∗ (G), the group C ∗ -algebra of G, to be the completion of L1 (G) with respect to the norm f ∗ = sup πf , where the supremum is taken over all nondegenerate ∗-representations π of L1 (G) as a ∗-algebra of bounded linear operators on a Hilbert space. Let C(G) be the Banach space of bounded continuous complex-valued functions on G with the usual supremum norm. Denote the set of continuous positive definite functions on G by P (G), and the set of continuous functions on G with compact support by C00 (G). Define the Fourier–Stieltjes algebra of G, denoted by B(G), to be the linear span of P (G). Then B(G) is a Banach algebra when equipped with the norm defined by:
1
φ = sup f (t)φ(t) dm(t) : f ∈ L (G), f ∗ 1 ,
φ ∈ B(G).
The Fourier algebra of G, denoted by A(G), is defined to be the closed linear span of P (G) ∩ C00 (G). Clearly, A(G) = B(G) if G is compact. Let λ be the left regular representation of G, i.e., for every f ∈ L1 (G), λ(f ) is the bounded operator in B(L2 (G)) defined by λ(f )(h) = f ∗ h (the convolution of f with h). Then denote by VN(G) the closure of 2 that C ∗ (G)∗ = B(G) {λ(f ): f ∈ L1 (G)} in the weak operator topology in B(L (G)). It is known ∗ where the duality is given by f, φ = f (t)φ(t) dm(t). Also, A(G) = VN(G) (we refer to [7] for more details). Various fixed point properties of the spaces C ∗ (G), A(G), and B(G) have been extensively studied by several authors (see for instance [13,15,16,30]). A locally compact group G is said to be an [AU]-group if the von Neumann algebra generated by every continuous unitary representation of G is atomic. It is an [AR]-group if the von Neumann algebra VN(G) is atomic. It is known from [30, Theorem 4.1 and Theorem 4.2] that the classes of groups for which A(G) and B(G) have the Radon–Nikodým property are precisely the [AR]-groups and [AU]-groups, respectively. Since A(G) ⊂ B(G), it is clear that [AU] ⊂ [AR]. Moreover, compact groups are [AU]-groups. Our results for this section provide full characterizations of Fourier–Stieltjes algebras and Fourier algebras with weak fixed point properties. They generalize [14, Proposition 5.1 and Proposition 5.2]. Theorem 4.1. Let G be a locally compact group. The following are equivalent: (a) B(G) has the weak fixed point property; (b) B(G) has the weak fixed point property for left reversible semigroups; (c) G is an [AU]-group. Proof. Since (b) ⇒ (a) is trivial and (c) ⇒ (a) is already contained in [14, Proposition 5.1], we only need to verify that (a) ⇒ (b) and (c). Our proof is based on the well-known result of Alspach [1] that L1 [0, 1] fails the weak fixed point property. We denote by R the type II1 hyperfinite factor and τR its usual trace (see [29] for formal definition and details). Consider the von Neumann algebra B(G)∗ = C ∗ (G)∗∗ . We observe first that if B(G) has the weak fixed
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point property then B(G)∗ is of type I. Indeed, according to [22], if B(G)∗ is not of type I von Neumann algebra then L1 (R, τR ) embeds isometrically into B(G). But clearly as L1 [0, 1] embeds isometrically into L1 (R, τR ), it follows that L1 [0, 1] embeds isometrically into B(G) which is a contradiction. Therefore we may write, ∗
B(G) =
, Aα ⊗ B(Hα ) l∞
α
where Aα = L∞ (Ωα , μα ) and B(Hα ) is the space of all bounded operators on the Hilbert space Hα . Moreover, since B(G) is the unique predual of its dual, we deduce that, B(G) =
. L1 Ωα , μα , T (Hα ) l1
α
Next, we note that for each index α, L1 (Ωα , μα ) embeds isometrically into B(G) and therefore has the weak fixed point property. Thus, (Ωα , μα ) is an atomic measure space, i.e., there is a set Γα so that L1 (Ωα , μα ) is isometric to l 1 (Γα ) (see for instance [13, Proposition 3.8]). Consequently, we can further state that, B(G) =
α
l 1 Γα , T (Hα ) . l1
Since l 1 -sums of Banach spaces with the Radon–Nikodým also have the Radon–Nikodým property, we can conclude that B(G) has the Radon–Nikodým property. This proves that G is an [AU]-group. On the other hand, since B(G) is isometric to a subspace of T (H ) for H = ( α [l 2 (Γα ) ⊗2 Hα ])l 2 , it follows that B(G) has the weak fixed point property for left reversible semigroups. The proof is complete. 2 In the argument above, we observe that once we get that B(G)∗ is atomic, we can also deduce immediately from [11, Theorem 2.2] that C ∗ (G) is a scattered C ∗ -algebra, but we prefer to use the above argument since it can easily be adapted to the case of VN(G) which we now formally state: Remark 4.2. Applying the argument used in the proof of Theorem 4.1 above to the von Neumann algebra VN(G), we obtain the corresponding statement for the Fourier algebra: if G is a locally compact group then the following are equivalent: (a) A(G) has the weak fixed point property; (b) A(G) has the weak fixed point property for left reversible semigroups; (c) G is an [AR]-group. Remark 4.3. We do not know if the three statements listed in Theorem 4.1 are equivalent to the property that B(G) has the weak∗ fixed point property for left reversible semigroups. The proof of Theorem 4.1 shows that if G is an [AU]-group then B(G) is isometric to a dual Banach space with the weak∗ fixed point property for left reversible semigroups but in general such isometry may not be a weak∗ isomorphism.
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5. Remarks on convergence in measure In this section, we consider the topology of convergence in measure for measurable operators. Assume that M is finite equipped with a normal tracial state τ and is a subspace of B(H ) for a given Hilbert space H . A closed and densely defined operator a on H is said to be affiliated with M if ua = au for all unitary operator u in the commutant M of M. A closed and densely defined operator x, affiliated with M, is said to be τ -measurable if for every ε > 0, there exists p ∈ Mproj such that p(H ) ⊆ dom(x), τ (1 − p) < ε, and xp ∈ M. The set of all τ -measurable operators on H will be denoted by L0 (M, τ ). The set L0 (M, τ ) is a ∗-algebra with respect to the strong sum, the strong product, and the adjoint operation. The topology on L0 (M, τ ) defined by the system of neighborhoods of zero: for ε > 0, δ > 0, V (ε, δ) = x ∈ L0 (M, τ ): xe ε for some e ∈ Mproj with τ (1 − e) δ , is called the measure topology. It is now well known that L0 (M, τ ) (with the measure topology) is a complete metrizable topological ∗-algebra and M is dense in L0 (M, τ ) (see for instance [24] that if M is the abelian von Neumann algebra L∞ (μ) and τ (f ) = for more details). We remark 0 f dμ, then the space L (M, τ ) coincides exactly with the classical space L0 (μ) of all μmeasurable functions with the usual topology of convergence in measure. We recall that the non-commutative L1 -space associated with the couple (M, τ ), denoted by L1 (M, τ ), can be realized as the space of all operators x ∈ L0 (M, τ ) such that x1 = τ (|x|) < ∞. Our result in this section can be viewed as a non-commutative extension of [2,19] although they only considered single nonexpansive mappings. Theorem 5.1. Let C be a nonempty closed bounded and convex subset of L1 (M, τ ). If C is compact for the measure topology then C has the fixed point property for left reversible semigroups. The key technical tool for this result is a companion of Proposition 2.4 for the measure topology. The crucial observation here is that taking convex combinations are no longer needed for this case. Proposition 5.2. Let {Dα : α ∈ Λ} be a decreasing net of bounded subsets of L1 (M, τ ) and (φm )m1 be a bounded sequence in L1 (M, τ ) that converges to φ for the measure topology, then lim sup φ − ψ: ψ ∈ Dα + lim φm − φ α m→∞ = lim lim sup φm − ψ: ψ ∈ Dα . m→∞ α
Sketch of the proof. We observe first that Lemma 2.1 is independent of any topology other than the L1 -norm. We also note that as in Lemma 2.1, one inequality follows directly from triangle inequality. For the non-trivial inequality, we proceed as in the proof of Proposition 2.4. Fix a bounded sequence (φm )m1 in L1 (M, τ ) that converges to φ for the measure topology. Apply Theorem 2.3 to (φm )m1 : there exist a further subsequence (which again we will assume to (1) (2) be (φm )m1 itself), bounded sequences (φm )m1 and (φm )m1 in L1 (M, τ ) and a mutually disjoint sequence of projections (em )m1 in Mproj such that:
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N. Randrianantoanina / Journal of Functional Analysis 258 (2010) 3801–3817 (1)
(2)
(i) φm = φm + φm for all m 1; (1) (ii) {φm : m 1} is a relatively weakly compact subset of L1 (M, τ ); (2) (2) (2) (iii) the sequence (φm )m1 is such that em φm em = φm for all m 1. We note that since M is finite, any bounded mutually disjointly supported sequence in L1 (M, τ ) (2) converges to zero for the measure topology. In particular, (φm )m1 converges to zero for the (1) measure topology and therefore (φm )n1 converges to φ for the measure topology. But since (1) {φm : m 1} is relatively weakly compact, it follows (see for instance [27, Proposition 2.11]) (1) that (φm )m1 converges to φ for the L1 -norm. From here on, the remaining part of the argument follows verbatim (but without the use of convex combinations) the proof of Proposition 2.4 so we leave the details to the interested reader. 2 Now we sketch the proof of Theorem 5.1. Let C be a nonempty subset of L1 (M, τ ) that is and convex. If {Dα : α ∈ Λ} is a decreasing net of bounded subset of L1 (M, τ ), then one can apply the same argument used in the proof of Theorem 3.1 (with Proposition 5.2 in place of Proposition 2.4) to prove that the set K = AC({Dα : α ∈ Λ}) (with respect to C) is nonempty. The crucial difference is that since we don’t need to take convex combinations in Proposition 5.2, the argument actually gives that K is also compact for the L1 -norm. It follows that K has the fixed point property for left reversible semigroups. This fact combined with the proof of Theorem 3.6 gives the conclusion of Theorem 5.1. L0 -compact
Remark 5.3. As a consequence of [6, Theorem 2.7], it is known that if E is a symmetric Banach function space on the interval [0, 1] that satisfies a lower q-estimate with constant 1 for some q 1, then the corresponding non-commutative space E(M, τ ) has the fixed point property for the measure topology. We do not know if Theorem 5.1 can be extended to these general noncommutative spaces. References [1] D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [2] M. Besbes, Points fixes et théorèmes ergodiques dans les espaces L1 (E), Studia Math. 103 (1992) 79–97. [3] R.E. Bruck Jr., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53 (1974) 59–71. [4] C.-H. Chu, A note on scattered C ∗ -algebras and the Radon–Nikodým property, J. London Math. Soc. (2) 24 (1981) 533–536. [5] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math., vol. 92, Springer-Verlag, New York, 1984. [6] P.G. Dodds, T.K. Dodds, P.N. Dowling, C.J. Lennard, F.A. Sukochev, A uniform Kadec–Klee property for symmetric operator spaces, Math. Proc. Cambridge Philos. Soc. 118 (1995) 487–502. [7] P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964) 181–236. [8] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28, Cambridge University Press, Cambridge, 1990. [9] J. Hagler, W.B. Johnson, On Banach spaces whose dual balls are not weak∗ sequentially compact, Israel J. Math. 28 (1977) 325–330. [10] R.D. Holmes, A.T.-M. Lau, Nonexpansive actions of topological semigroups and fixed points, J. London Math. Soc. 5 (1972) 330–336. [11] H.E. Jensen, Scattered C ∗ -algebras, Math. Scand. 41 (1977) 308–314. [12] A.T.-M. Lau, Fixed point property for reversible semigroup of non-expansive mappings on weak∗ -compact convex sets, in: Fixed Point Theory Appl., vol. 3, Nova Sci. Publ., Huntington, NY, 2002, pp. 167–172. [13] A.T.-M. Lau, M. Leinert, Fixed point property and the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc. 360 (2008) 6389–6402.
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[14] A.T.-M. Lau, P. Mah, Fixed point property for Banach algebras associated to locally compact groups, J. Funct. Anal. 258 (2010) 357–372. [15] A.T.-M. Lau, P. Mah, A. Ülger, Fixed point property and normal structure for Banach spaces associated to locally compact groups, Proc. Amer. Math. Soc. 125 (1997) 2021–2027. [16] A.T.-M. Lau, A. Ülger, Some geometric properties on the Fourier and Fourier–Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993) 321–359. [17] A.T.-M. Lau, Y. Zhang, Fixed point properties of semi-groups of nonexpansive mappings, J. Funct. Anal. 254 (2008) 2534–2554. [18] C. Lennard, C1 is uniformly Kadec–Klee, Proc. Amer. Math. Soc. 109 (1990) 71–77. [19] C. Lennard, A new convexity property that implies a fixed point property for L1 , Studia Math. 100 (1991) 95–108. [20] T.C. Lim, Characterizations of normal structure, Proc. Amer. Math. Soc. 43 (1974) 313–319. [21] T.C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980) 135–143. [22] J.L. Marcolino Nhany, La stabilité des espaces Lp non-commutatifs, Math. Scand. 81 (1997) 212–218. [23] T. Mitchell, Fixed points of reversible semigroups of nonexpansive mappings, Kõdai Math. Sem. Rep. 22 (1970) 322–323. [24] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116. [25] H. Pfitzner, Weak compactness in the dual of a C ∗ -algebra is determined commutatively, Math. Ann. 298 (1994) 349–371. [26] N. Randrianantoanina, Kadec–Pełczy´nski decomposition for Haagerup Lp -spaces, Math. Proc. Cambridge Philos. Soc. 132 (2002) 137–154. [27] N. Randrianantoanina, Sequences in non-commutative Lp -spaces, J. Operator Theory 48 (2002) 255–272. [28] Y. Raynaud, Q. Xu, On subspaces of non-commutative Lp -spaces, J. Funct. Anal. 203 (2003) 149–196. [29] M. Takesaki, Theory of Operator Algebras. I, Springer-Verlag, New York, 1979. [30] K.F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983) 183–190. [31] J. Tomiyama, A characterization of C ∗ -algebras whose conjugate spaces are separable, Tôhoku Math. J. (2) 15 (1963) 96–102.
Journal of Functional Analysis 258 (2010) 3818–3840 www.elsevier.com/locate/jfa
BMO estimates for the H ∞(Bn) Corona problem Serban ¸ Costea a , Eric T. Sawyer a,1 , Brett D. Wick b,∗,2 a McMaster University, Department of Mathematics and Statistics, 1280 Main Street West, Hamilton,
Ontario L8S 4K1, Canada b Georgia Institute of Technology, School of Mathematics, 686 Cherry Street, Atlanta, GA 30332-1060, USA
Received 20 October 2009; accepted 24 December 2009 Available online 13 January 2010 Communicated by J. Bourgain
Abstract
We study the H ∞ (Bn ) Corona problem N j =1 fj gj = h and show it is always possible to find solutions f that belong to BMOA(Bn ) for any n > 1, including infinitely many generators N . This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space H ∞ · BMOA with N = ∞, while the latter result obtains BMOA(Bn ) solutions for just N = 2 generators with h = 1. Our method of proof is to solve ∂-problems and to exploit the connection between BMO functions and Carleson measures for H 2 (Bn ). Key to this is the exact structure of the kernels that solve the ∂ equation for (0, q) forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov–Sobolev spaces is also given. © 2009 Elsevier Inc. All rights reserved. Keywords: Corona problem; Carleson measures; BMO; Besov–Sobolev spaces
1. Introduction In 1962 Lennart Carleson demonstrated in [3] the absence of a corona in the maximal ideal ∞ space of H ∞ (D) by showing that if {gj }N j =1 is a finite set of functions in H (D) satisfying * Corresponding author.
E-mail addresses:
[email protected] (S. ¸ Costea),
[email protected] (E.T. Sawyer),
[email protected] (B.D. Wick). 1 Research supported in part by a grant from the National Science and Engineering Research Council of Canada. 2 Research supported in part by National Science Foundation DMS Grant # 0752703. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.015
S¸ . Costea et al. / Journal of Functional Analysis 258 (2010) 3818–3840 N gj (z) δ > 0,
z ∈ D,
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(1.1)
j =1 ∞ then there are functions {fj }N j =1 in H (D) with N
fj (z)gj (z) = 1,
z ∈ D and
j =1
N
fj ∞ C.
(1.2)
j =1
Later, Hörmander noted a connection between the Corona problem and the Koszul complex, and in the late 1970’s Tom Wolff gave a simplified proof using the theory of the ∂ equation and Green’s theorem (see [6]). This proof has since served as a model for proving corona type theorems for other Banach algebras. While there is a large literature on such corona theorems in one complex dimension (see e.g. [8]), progress in higher dimensions has been limited. Indeed, apart from the simple cases in which the maximal ideal space of the algebra can be identified with a compact subset of Cn , no corona theorem has been proved in higher dimensions until the recent work of the authors [5] on the Drury-Arveson Hardy space multipliers. Instead, partial results have been obtained, which we will discuss more below. We of course have the analogous question in several complex variables when we consider H ∞ (Bn ). The Corona problem for the Banach algebra H ∞ (Bn ) is to show that if g1 , . . . , gN ∈ H ∞ (Bn ) satisfy N gj (z) 1,
∀z ∈ Bn ,
j =1
N ∞ then the ideal generated by {gj }N j =1 fj (z)gj (z) = 1 for all j =1 is all of H (Bn ), equivalently z ∈ Bn for some f1 , . . . , fN ∈ H ∞ (Bn ). This famous problem has remained open for n > 1 since Lennart Carleson proved the n = 1 dimensional case in 1962, but there are some partial results. Most notably, there is the classical result of Varopoulos where BMOA(Bn ) estimates were obtained for solutions f to the Bézout equation f1 g1 + f2 g2 = 1 [11]. The restriction to just N = 2 generators provides some algebraic simplifications to the problem. Note also that the more general equation f1 g1 + f2 g2 = h,
h ∈ H ∞,
can then be solved for f ∈ H ∞ · BMOA. Over two decades later, the case 2 N ∞ was studied by Andersson and Carlsson [1] in 2000 who obtained H ∞ · BMOA solutions f to the infinite Bézout equation ∞ i=1 fi gi = 1, and hence also to the more general equation ∞
fi gi = h,
h ∈ H ∞.
(1.3)
i=1
To see that H ∞ · BMOA is strictly larger than BMOA, recall that the multiplier algebra of BMOA is a proper subspace of H ∞ satisfying a vanishing Carleson condition (see e.g. Theorem 6.2 in [1]).
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Our proof uses the methods of [5], that in turn generalize the integration by parts and estimates of Ortega and Fabrega [9]. Key to these new estimates are the almost invariant holomorphic derivatives from Arcozzi, Rochberg and Sawyer [2]. Consequently our proof can be used to handle any number of generators N with no additional difficulty and always yields BMOA(Bn ) solutions f to (1.3). See [1] for further references to related material. This leads to the main result of this paper in which we obtain BMOA(Bn ) solutions to the H ∞ (Bn ) Corona Problem (1.3) with infinitely many generators. ∞ 2 Theorem 1. There is a constant Cn,δ such that given g = (gi )∞ i=1 ∈ H (Bn ; ) satisfying
1
∞ gj (z)2 δ 2 > 0,
z ∈ Bn ,
(1.4)
j =1
there is for each h ∈ H ∞ (Bn ) a vector-valued function f ∈ BMOA(Bn ; 2 ) satisfying f BMOA(Bn ;2 ) Cn,δ hH ∞ (Bn ) , ∞
fj (z)gj (z) = h(z),
z ∈ Bn .
(1.5)
j =1
This theorem can be generalized to hold for the multiplier algebras MBpσ (Bn ) of the Besov– Sobolev spaces Bpσ (Bn ) in place of the multiplier algebra H ∞ (Bn ) of the classical Hardy space n
H 2 (Bn ) = B22 (Bn ). See Theorem 5 in the final section below. Our method of proof uses the notation and techniques from [5]. However, for the convenience of the reader, this paper is written so that it is mostly self-contained. 2. Preliminaries We begin by collecting all the relevant facts that will be necessary to prove Theorem 1. While many of these facts may be known to experts, we collect them all in one location for convenience. 2.1. BMO and Carleson measures In this subsection we recall the well-known connection between BMO functions on the boundary ∂Bn of the ball and Carleson measures for H 2 (Bn ). First, we define the space CM(Bn ). This is the collection of functions on the unit ball Bn such that 2 Sζ |h(z)| dλn (z) < ∞, hCM(Bn ) ≡ sup (1 − |ζ |)n ζ ∈Bn and for ζ ∈ Bn \ {0} the Carleson tent Sζ is defined by 1 1 − |ζ | . Sζ = z ∈ Bn : > |1 − ζ z| 2
(2.1)
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In classical language h ∈ CM(Bn ) if and only if the measure dμh (z) = |h(z)|2 dλn (z) is a Carleson measure for H 2 (Bn ); i.e. H 2 (Bn ) ⊂ L2 (dμh ). Also, recall the space BMO(∂Bn ), which is the collection of functions that are in L2 (∂Bn ) such that 1 |b − bQδ (η) |2 dσ (ζ ) < ∞, b2BMO(∂Bn ) ≡ sup |Q (η)| δ Qδ (η)⊂∂Bn Qδ (η)
where Qδ (η) is the non-isotropic ball of radius δ > 0 and center η in ∂Bn and bQδ (η) =
1 |Qδ (η)|
b(ξ ) dσ (ξ ). Qδ (η)
One then defines BMOA(Bn ) = BMO(∂Bn ) ∩ H 2 (Bn ). Finally, we define vector-valued versions BMO(∂Bn ; 2 ), H 2 (Bn ; 2 ), BMOA(Bn ; 2 ) and CM(Bn ; 2 ) in the usual way. A well-known fact connecting the spaces BMOA(Bn ; 2 ) and CM(Bn ) is the following: Lemma 1. For g ∈ H 2 (Bn ; 2 ) we have
n +1 cgBMOA(Bn ;2 ) 1 − |z|2 2 g (z) CM(B
n ;
2)
CgBMOA(Bn ;2 ) .
(2.2)
Proof. The scalar version of Lemma 1 is proved in Theorem 5.14 of [12]. The proof given there extends to the 2 -valued case in a routine manner with a couple of possible exceptions which we now address. A key step in proving the first inequality in (2.2) is: f (ζ )g(ζ ) dσ (ζ ) ∂Bn
1 −2n = Rf (z)Rg(z)|z| dV (z) log |z| Bn
Bn
1 1 2 2 2 −2n |Rf (z)|2 −2n 1 1 f (z) Rg(z) |z| |z| dV (z) dV (z) , log log |f (z)| |z| |z| Bn
followed by the two inequalities Bn
|Rf (z)|2 −2n 1 |z| dV (z) Cf H 1 (Bn ;2 ) , log |f (z)| |z|
(2.3)
and Bn
n f (z)Rg(z)2 |z|−2n log 1 dV 1 − |z|2 2 +1 g (z)
CM(Bn ;2 ) f H 1 (Bn ;2 ) . |z|
(2.4)
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A crucial equality used in the proof of (2.3) in the scalar case when n = 1 is f H 1 (D) = D
1 |f (z)|2 log dV (z), |f (z)| |z|
which in turn follows from Green’s theorem and the identity |f (z)|2
f (z) = . |f (z)| 2 When f = {fk }∞ k=1 is -valued, this identity becomes
−1 1 ∂ ∂ ∂ 1 f (z)f (z) 2 = 4 f (z) f (z) · f (z)
f (z) = 4 ∂z ∂z ∂z 2 −3 −1
1 = 2 f (z) f (z) · f (z) − f (z) f (z) · f (z) f (z) · f (z) 2 2 2 1 2 2
f (z) f (z) − , f (z), f (z) = 2 |f (z)|3 which by the Cauchy–Schwarz inequality yields the approximation |f (z)|2
f (z) ≈ . |f (z)| This approximation and Green’s theorem lead to 1 2π
iθ f e dθ = f 1 ≈ H (D)
T
D
1 |f (z)|2 log dV (z) |f (z)| |z|
(2.5)
in the 2 -valued case when n = 1. Now we consider the case of dimension n > 1 and apply (2.5) to each slice function fζ (z) with ζ ∈ ∂Bn . The result when f (0) = 0 is f H 1 (Bn ) = c
iθ fζ e dθ dσ (ζ )
∂Bn T
≈
|f (z)|2 ζ
∂Bn D
=C Bn
|fζ (z)|
log
1 dV (z) dσ (ζ ) |z|
|Rf (w)|2 1 |w|−2n log dV (w), |f (w)| |w|
since fζ (z) = z−1 Rf (zζ ) and dV (z) dσ (ζ ) = r dr dθ dσ (ζ ) = Cr 2−2n dV (rζ ) dθ . This proves (2.3).
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3823
The inequality (2.4) is the 2 -valued version of the Hörmander–Carleson Theorem when p = 1. The scalar case is proved in Theorem 5.9 of [12] using the theory of the invariant Poisson integral P together with the subharmonic inequality (Corollary 4.5 in [12]): p p f (z) 2 P |f | 2 (z),
z ∈ Bn .
The subharmonic inequality extends to 2 -valued f (z) by noting that p p p p f (z) 2 = sup f (z), v 2 sup P f, v 2 (z) P |f | 2 (z), |v|1
|v|1
and then the proof of (2.4) is completed using the scalar theory of P as in [12]. The arguments in [12] now complete the proof of (2.2). 2 We will also need the following slight generalization of the special case p = 2 and σ = n2 of the multilinear estimate in Proposition 3 of [5] (the scalar case is Theorem 3.5 in [9]). Note that n
B22 (Bn ) = H 2 (Bn ) and Mg
n
n
B22 (Bn )→B22 (Bn ;2 )
= gH ∞ (Bn ;2 ) .
Lemma 2. Suppose that M, m 1 and α = (α0 , . . . , αM ) ∈ ZM+1 with |α| = m. For g1 , . . . , + gM ∈ H ∞ (Bn ; 2 ) and h ∈ H 2 (Bn ) we have
n 2 2 2 1 − |z|2 Y α1 g1 (z) . . . Y αM gM (z) Y α0 h (z) dλn (z)
Bn
Cn,M,m
M j =1
gj 2H ∞ (B ;2 ) n
h2H 2 (B ) . n
(2.6)
Here Y m is the vector of all differential operators of the form X1 X2 . . . Xm where each Xi is either (1 − |z|2 )I , (1 − |z|2 )R or D. The operator I is the identity, the operator R is the radial derivative, and the operator D = 1 − |z|2 Pz ∇ + 1 − |z|2 Qz ∇ is an almost invariant derivative defined in [5]. The iteration X1 X2 . . . Xm is not a composition of operators, but as in [5] one fixes the coefficients, then composes the frozen operators, and then unfreezes the coefficients. Finally, the generalization in (2.6) is that the multiplier functions gj need not be the same function, as they were in Proposition 3 of [5]. However, the proof given in [5] applies to different gj as well (the scalar case in [9] is for different gj ). We observe that the proof is actually simplified due to the fact that for s n2 , Mg B s (Bn )→B s (Bn ;2 ) = gH ∞ (Bn ;2 ) . 2
2
Using the geometric characterization of Carleson measures for H 2 (Bn ; 2 ), Lemma 1 says that g ∈ BMOA(Bn ; 2 ) if and only if the measure μm g associated to g by 2 n 2 2 m dμm Y g(z) dλn (z), g (z) ≡ 1 − |z|
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2 is a Carleson measure for H 2 (Bn ; 2 ); i.e. H 2 (Bn ; 2 ) ⊂ L2 (μm g ; ), for some (equivalently all) 2 2 m 1. On the other hand, g ∈ H (Bn ; ) if and only if g is holomorphic and the measure μm g is finite.
Remark 1. We note in passing that CM(Bn ) ⊂ L2 (λn ). Moreover, L2 (λn ) is Möbius-invariant and so the functions g in L2 (λn ) satisfy nothing better than the growth estimate Sζ |g|2 dλn g2L2 (λ ) , while the functions g in CM(Bn ) satisfy the restrictive growth estimate n
n |g|2 dλn g2CM(Bn ) 1 − |ζ | .
Sζ
2.2. The Koszul complex Here we briefly review the algebra behind the Koszul complex as presented for example in [7] in the finite dimensional setting. A more detailed treatment in that setting can be found in Section 5.5.3 of [10]. Fix h holomorphic as in (1.5). Now if g = (gj )∞ j =1 satisfies 2 δ 2 > 0, let |g | |g|2 = ∞ j =1 j Ω01
g = 2= |g|
gj |g|2
∞ j =1
∞ = Ω01 (j ) j =1 ,
which we view as a 1-tensor (in 2 = C∞ ) of (0, 0)-forms with components Ω01 (j ) = = Ω01 h satisfies Mg f
gj . |g|2
Then
f = f · g = h, but in general fails to be holomorphic. The Koszul complex provides a scheme which we now recall for solving a sequence of ∂ equations that result in a correction term Λg Γ02 that when subtracted from f above yields a holomorphic solution to the second line in (1.5). See below. g 1 ∞ The 1-tensor of (0, 1)-forms ∂Ω0 = (∂ |g|j2 )∞ j =1 = (∂Ω0 (j ))j =1 is given by ∂Ω01 (j ) = ∂
∞ |g|2 ∂gj − gj ∂|g|2 gj 1 = = gk {gk ∂gj − gj ∂gk } |g|2 |g|4 |g|4 k=1
and can be written as ∂Ω01
= Λg Ω12
≡
∞ k=1
∞ Ω12 (j, k)gk
, j =1
where the antisymmetric 2-tensor Ω12 of (0, 1)-forms is given by Ω12
2 ∞ {gk ∂gj − gj ∂gk } ∞ = Ω1 (j, k) j,k=1 = |g|4 j,k=1
and Λg Ω12 denotes its contraction by the vector g in the final variable.
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We can repeat this process and by induction we have q+1
∂Ωq
q+2
= Λg Ωq+1 ,
0 q n,
(2.7)
q+1
where Ωq is an alternating (q + 1)-tensor of (0, q)-forms. Recall that h is holomorphic. When q = n we have that Ωnn+1 h is ∂-closed and this allows us to solve a chain of ∂ equations q
q
q+1
∂Γq−2 = Ωq−1 h − Λg Γq−1 , q
for alternating q-tensors Γq−2 of (0, q − 2)-forms, using the ameliorated Charpentier solution 0,q
operators Cn,s defined in Theorem 4 below (note that our notation suppresses the dependence of Γ on h). With the convention that Γnn+2 ≡ 0 we have q+1 q+2 = 0, ∂ Ωq h − Λg Γq q+1
0 q n, q+1
∂Γq−1 = Ωq
q+2
h − Λg Γq
,
1 q n.
(2.8)
Now f ≡ Ω01 h − Λg Γ02 is holomorphic by the first line in (2.8) with q = 0, and since Γ02 is antisymmetric, we compute that Λg Γ02 · g = Γ02 (g, g) = 0 and Mg f = f · g = Ω01 h · g − Λg Γ02 · g = h − 0 = h. Thus f = (fi )∞ i=1 is a vector of holomorphic functions satisfying the second line in (1.5). The first line in (1.5) is the subject of the remaining sections of the paper. 2.2.1. Wedge products and factorization of the Koszul complex Here we record the remarkable factorization of the Koszul complex in Andersson and Carlsson [1]. To describe the factorization we introduce an exterior algebra structure on 2 = C∞ . Let {e1 , e2 , . . .} be the usual basis in C∞ , and for an increasing multi-index I = (i1 , . . . , i ) of integers in N, define eI = ei 1 ∧ ei 2 ∧ · · · ∧ ei , where we use ∧ to denote the wedge product in the exterior algebra Λ∗ (C∞ ) of C∞ , as well as for the wedge product on forms in Cn . Note that {eI : |I | = r} is a basis for the alternating ∞ r-tensors on C . in Cn , If f = |I |=r fI eI is an alternating r-tensor on C∞ with values that are (0, k)-forms ∞ n which may be viewed as a member of the exterior algebra of C ⊗ C , and if g = |J |=s gJ eJ is an alternating s-tensor on C∞ with values that are (0, )-forms in Cn , then as in [1] we define the wedge product f ∧ g in the exterior algebra of C∞ ⊗ Cn to be the alternating (r + s)-tensor on C∞ with values that are (0, k + )-forms in Cn given by
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f ∧g=
∧
f I eI
|I |=r
|J |=s
=
(fI ∧ gJ )(eI ∧ eJ )
|I |=r,|J |=s
=
gJ e J
|K|=r+s
±
f I ∧ gJ e K .
(2.9)
I +J =K
Note that we simply write the exterior product of an element from Λ∗ (C∞ ) with an element from Λ∗ (Cn ) as juxtaposition, without writing an explicit wedge symbol. This should cause no ∗ n confusion since the basis we use in Λ∗ (C∞ ) is {ei }∞ i=1 , while the basis we use in Λ (C ) is n {dzj , d zj }j =1 , quite different in both appearance and interpretation. In terms of this notation we then have the following factorization in Theorem 3.1 of Andersson and Carlsson [1]: Ω01
∧
1 = Ω 0
∞ ∞ ∂gk gk0 1 i ek0 ∧ eki = − Ω +1 , 2 2 +1 |g| |g|
k0 =1
i=1
(2.10)
ki =1
i=1
where
Ω01
=
gi |g|2
∞
1 = and Ω 0
i=1
∂gi |g|2
∞ . i=1
The factorization in [1] is proved in the finite dimensional case, but this extends to the infinite dimensional case by continuity. Since the 2 norm is quasi-multiplicative on wedge products by Lemma 5.1 in [1] we have 1 2 +1 2 , Ω C Ω 1 2 Ω 0
0
0 n,
(2.11)
where the constant C depends only on the number of factors in the wedge product, and not on the underlying dimension of the vector space (which is infinite for 2 = C∞ ). 2.3. Charpentier’s solution operators In Theorem I.1 on p. 127 of [4], Charpentier proves the following formula for (0, q)-forms: Theorem 2. For q 0 and all forms f (ξ ) ∈ C 1 (Bn ) of degree (0, q + 1), we have for z ∈ Bn : f (z) = Cq Bn 0,q
0,q+1
∂f (ξ ) ∧ Cn
(ξ, z) + cq ∂ z
0,q f (ξ ) ∧ Cn (ξ, z) .
Bn
Here Cn (ξ, z) is an (n, n − q − 1)-form in ξ on the ball and a (0, q)-form in z on the ball that will be recalled below. Using Theorem 2, we can solve ∂ z u = f for a ∂-closed (0, q + 1)-form f
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3827
as follows. Set
0,q
f (ξ ) ∧ Cn (ξ, z).
u(z) ≡ cq Bn
Taking ∂ z of this we see from Theorem 2 and ∂f = 0 that
∂ z u = cq ∂ z
0,q f (ξ ) ∧ Cn (ξ, z)
= f (z).
Bn 0,q
The actual structure of the kernels Cn (ξ, z) is very important for our proof. The case of q = 0 is given in [4], and additional properties of the kernels of general (p, q) were illustrated. In [5] 0,q we explicitly compute the kernels Cn (ξ, z). Before we give the structure of the kernels, first we introduce some notation. q Notation 1. Let ωn (z) = nj=1 dzj . For n a positive integer and 0 q n − 1 let Pn denote the collection of all permutations ν on {1, . . . , n} that map to {iν , Jν , Lν } where Jν is an increasing multi-index with card(Jν ) = n − q − 1 and card(Lν ) = q. Let ν ≡ sgn(ν) ∈ {−1, 1} denote the signature of the permutation ν. Note that the number of increasing multi-indices of length n − q − 1 is
n! (q+1)!(n−q−1)! ,
while
n! q!(n−q)! . Since we are only allowed certain
the number of increasing multi-indices of length q are combinations of Jν and Lν (they must have disjoint intersection and they must be increasing q multi-indices), it is straightforward to see that the total number of permutations in Pn that we n! are considering is (n−q−1)!q! . Denote by : Cn × Cn → [0, ∞) the map
(w, z) ≡ |1 − wz|2 − 1 − |w|2 1 − |z|2 . We remark that it is possible to view (z, w) in many other ways due to the symmetry of the unit ball Bn . For example we will later use 2
(w, z) = |1 − wz|2 ϕw (z) .
(2.12)
See [5] for the additional representations of this function. It is convenient to isolate the following factor common to all summands in the formula: q
Φn (w, z) ≡
(1 − wz)n−1−q (1 − |w|2 )q ,
(w, z)n
0 q n − 1.
(2.13)
Theorem 3. Let n be a positive integer and suppose that 0 q n − 1. Then 0,q
Cn (w, z) =
q
ν∈Pn
q
(−1)q Φn (w, z) sgn(ν)(wiν − ziν )
j ∈Jν
dwj
l∈Lν
dzl
ωn (w).
(2.14)
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The proof of this theorem is a long computation that can be found in [5]. We also need the following ameliorations of the Charpentier solution operators. These are 0,q obtained by treating the solution operators Cn (w, z) with w, z ∈ Cn as actually being a function with w, z ∈ Cs with s > n. One then can integrate out the extra variables to obtain the following result. Theorem 4. Suppose that s > n and 0 q n − 1. Then we have
0,q 0,q Cn,s (w, z) = Cn (w, z) q
= Φn,s (w, z)
1 − |w|2 1 − wz
s−n n−q−1
cj,n,s
j =0
(1 − |w|2 )(1 − |z|2 ) |1 − wz|2
j
c (−1)μ(k,J ) (zk − wk ) dzJ ∧ dw (J ∪{k}) ∧ ωn (w). |J |=q k ∈J /
The interested reader can find this theorem in [5]. In order to establish appropriate inequalities for the Charpentier solution operators, we will ∂m m k need to control terms of the form (z − w)α ∂w α F (w), Dz (w, z) and D{(1 − wz) } inside the integral for T as given in the integration by parts formula in Lemma 4 below. We collect the necessary estimates in the following proposition. Proposition 1. For z, w ∈ Bn and m ∈ N, we have the following three crucial estimates:
√ m
(w, z) m m (z − w)α ∂ F (w) C , D F (w) ∂w α 1 − |w|2
m = |α|.
Dz (w, z) C 1 − |z|2 (w, z) 12 + (w, z) , ! 1 − |z|2 R (w, z) C 1 − |z|2 (w, z), m D (1 − wz)k C|1 − wz|k z 1 − |z|2 m R m (1 − wz)k C|1 − wz|k
1 − |z|2 |1 − wz| 1 − |z|2 |1 − wz|
(2.15)
(2.16)
m 2
, m .
(2.17)
Lemma 3. Let b > −1. For Ψ ∈ C(Bn ) ∩ C ∞ (Bn ) we have Bn
b 1 − |w|2 Ψ (w) dV (w) =
b+m m 1 − |w|2 Rb Ψ (w) dV (w).
Bn
When estimating the solution operators in the space CM(Bn ) the following lemma will play an important role. Lemma 4. Suppose that s > n and 0 q n − 1. For all m 0 and smooth (0, q + 1)-forms η in Bn we have the formula
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0,q
Cn,s η(z) =
m−1
m
ck,n,s Sn,s Dk η [Z](z) + c,n,s Φn,s D η (z),
3829
q
k=0
(2.18)
=0
have kernels given by where the ameliorated operators Sn,s and Φn,s
1 − |w|2 s−n−1 (1 − |w|2 )s−n−1 1 Sn,s (w, z) = cn,s = cn,s , (1 − wz)s 1 − wz (1 − wz)n+1
n−−1 (1 − |w|2 )(1 − |z|2 ) j 1 − |w|2 s−n (w, z) = Φn (w, z) cj,n,s . Φn,s 1 − wz |1 − wz|2 j =0
We have included these theorems so that this paper would be mostly self-contained. Remark 2. The proof of Lemma 4 can be found in [5, pp. 18 and 19], and the proof for the case of the nonameliorated operators Sn and Φn can be found on pp. 64–66. However, in the latter proof we only considered the two cases m = 0 and 1. The reader can find the cases m 2 treated in the first version of the paper [5] on the arXiv website. 3. Carleson measures and Schur’s lemma Key to the proof of Theorem 1 will be the knowledge that certain positive operators are bounded on CM(Bn ). In particular, these operators will be connected with the Charpentier solution operators. Lemma 5. Let a, b, c ∈ R. Then the operator Ta,b,c h(z) = Bn
√ (1 − |z|2 )a (1 − |w|2 )b ( (w, z))c h(w) dV (w) |1 − wz|n+1+a+b+c
(3.1)
is bounded on CM(Bn ) if c > −2n and
− 2a < −n < 2(b + 1).
(3.2)
We remind the reader that in [5] it is shown that the operator Ta,b,c is bounded on L2 (λn ) if and only if (3.2) holds. Note that since Ta,b,c is a positive operator, Minkowski’s inequality yields Ta,b,c hi (z)
∞ Ta,b,c {hi }∞ i=1 (z), i=1
z ∈ Bn ,
∞ and it follows that the extension Ta,b,c {hi }∞ i=1 = {Ta,b,c hi }i=1 is bounded on
CM Bn ; 2 = h ∈ L2 λn ; 2 : |h| ∈ CM(Bn ) if and only if Ta,b,c is bounded on CM(Bn ).
(3.3)
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Proof. Fix ζ ∈ Bn \ {0} and let
δ = 1 − |ζ |
and N = log2
1 . 1 − |ζ |
For 0 k N set 1 − 2k δ ζ k ζk = 1 − 2 δ = ζ. |ζ | 1−δ Then ζ0 = ζ and ζk lies on the real line through ζ and is 2k times as far from the boundary as is ζ : 1 − |ζk | = 2k δ. For a positive function h ∈ CM(Bn ) define h1 ≡ χSζ1 h, hk ≡ χSζk \Sζk−1 h,
2 k N.
Then 1
|Ta,b,c h| dλn 2
2
ChCM(Bn ) +
N
1 |Ta,b,c hk | dλn 2
2
.
k=1 S ζ
Sζ
Since Ta,b,c is bounded on L2 (λn ) by [5], we have
|Ta,b,c hk |2 dλn C
|hk |2 dλn = C
Bn
Sζ
|h|2 dλ
Sζk
n Ch2CM(Bn ) 1 − |ζk | n = C2kn h2CM(Bn ) 1 − |ζ | , which is an adequate estimate for k bounded. For k large we claim that z ∈ Sζ and w ∈ Sζk \ Sζk−1 imply |1 − wz| ≈ 2k δ
! and ϕw (z) ≈ 1 and
(w, z) ≈ 2k δ.
(3.4)
The second equivalence is obvious by Möbius invariance, and the third equivalence follows from the first two and the formula for (w, z) in (2.12). We now prove the first equivalence in (3.4), for which it suffices to prove that c2k δ |1 − wζ | C2k δ. 1
(3.5)
Indeed, since d(w, z) = |1 − wz| 2 satisfies the triangle inequality on the ball Bn , and since |1 − zζ | < 2δ by (2.1), we have from (3.5) that
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3831
! ! √ 2δ + C2k δ C2k+2 δ, ! ! √ 1 |1 − wz| 2 c2k δ − 2δ c2k−2 δ, 1
|1 − wz| 2
for k large enough. So to complete the proof of the claim (3.4), we must demonstrate (3.5). However, (3.5) clearly holds for ζ bounded away from the boundary ∂Bn , and so we may suppose that 0 < δ 14 . Now from (2.1) we have 1 − |ζk | |1 − ζk w|
>
1 2
and
1 − |ζk−1 |
1 , 2 |1 − ζk−1 w|
which yields k 1 − 1 − 2 δ ζ w < 2k+1 δ 1−δ
1 − 2k−1 δ and 1 − ζ w 2k δ. 1−δ
Now we conclude from the Euclidean triangle inequality that 1 − 2k δ 1 − 2k δ |1 − ζ w| 1 − |ζ w| ζ w + 1 − 1−δ 1−δ
(2k − 1)δ (2k − 1) k+1 k+1 δ 2k+2 δ, 2 2 δ+ + 3 1−δ 4 as well as 1 − 2k−1 δ 1 − 2k−1 δ |ζ w| |1 − ζ w| 1 − ζ w − 1 − 1−δ 1−δ
(2k−1 − 1)δ (2k−1 − 1) δ 2k−2 δ, 2k − 2k δ − 3 1−δ 4 provided 0 < δ 14 . This completes the proof of (3.5), and hence (3.4). We thus have for k large enough, say k K, |Ta,b,c hk |2 dλn Sζ
=
Sζ Sζk \Sζk−1
C
2 c (1 − |z|2 )a (1 − |w|2 )b (w, z) 2 dλn (z) h(w) dV (w) |1 − wz|n+1+a+b + c
Sζ Sζk \Sζk−1
2 (1 − |z|2 )a (1 − |w|2 )b (2k δ)c dλn (z) h(w) dV (w) k n+1+a+b+c (2 δ)
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−2(n+1+a+b) C 2k δ
2a−n−1 1 − |z|2 dV (z)
Sζ
2 2 b+n+1 1 − |w| h(w) dλn (w) .
× Sζk \Sζk−1
Now by Hölder’s inequality,
2 b+n+1 1 − |w|2 h(w) dλn (w)
Sζk \Sζk−1
2 2(b+n+1) 1 − |w| dλn (w)
Sζk \Sζk−1
h(w)2 dλn (w)
Sζk \Sζk−1
2(b+n+1) n 1 − |w|2 dλn (w) h2CM(Bn ) 2k δ
Sζk \Sζk−1
n 2(b+n+1) C 2k δ h2CM(Bn ) 2k δ 2b+3n+2 = C 2k δ h2CM(Bn ) , provided 2(b + n + 1) > n, i.e. 2(b + 1) > −n. Indeed, to obtain the estimate
2(b+n+1) 2(b+n+1) 1 − |w|2 dλn (w) C 2k δ ,
Sζk \Sζk−1
we decompose the annulus Sζk \ Sζk−1 into a union of unit radius Bergman balls Bj whose Euclidean distance from the boundary is approximately 2− 2k δ:
Sζk \ Sζk−1 =
A ∞ " "
Bj .
=0 j =1
Since A |Bj | n |Sζ | n k k = 2k δ , A 2− 2k δ = − k 2 δ 2 2 δ j =1
we have the estimate A 2n . Thus we compute that
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1 − |w|2
2(b+n+1)
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dλn (w)
Sζk \Sζk−1
=
A A ∞ ∞ 2(b+n+1) − k 2(b+n+1) 1 − |w|2 2 2 δ dλn (w) ≈ =0 j =1 Bj
=0 j =1
∞ − 2(b+n+1) n 2(b+n+1) 2(b+n+1) 2 2k δ 2 C 2k δ , =0
since 2(b + n + 1) > n. We also compute that
2a−n−1 1 − |z|2 dV (z) Cδ n
Sζ
δ t 2a−n−1 dt Cδ n δ 2a−n = Cδ 2a , 0
so that altogether we have
−2(n+1+a+b) 2a k 2b+3n+2 δ 2 δ |Ta,b,c hk |2 dλn C 2k δ h2CM(Bn )
Sζ
= Ch2CM(Bn ) 2k(n−2a) δ n . Summing we obtain
1 |Ta,b,c h| dλn 2
Sζ
2
∞
1 |Ta,b,c hk | dλn 2
2
k=1 S ζ ∞ n n n C2Kn hCM(Bn ) 1 − |ζ | 2 + ChCM(Bn ) 2k( 2 −a) δ 2
n ChCM(Bn ) 1 − |ζ | 2
k=K
provided a > n2 . The two requirements 2(b + 1) > −n and a > n2 are precisely the conditions on a and b in (3.2), and this completes the proof of Lemma 5. 2 4. Proof of Theorem 1 To prove Theorem 1 we follow the argument in [5]. We obtain from the Koszul complex a function f = Ω01 h − Λg Γ02 ∈ H (Bn ) that solves (1.5) where Γ02 is an antisymmetric 2-tensor of (0, 0)-forms that solves ∂Γ02 = Ω12 h − Λg Γ13 ,
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and inductively where Γq
is an alternating (q + 2)-tensor of (0, q)-forms that solves q+2
∂Γq
q+2
q+3
= Ωq+1 h − Λg Γq+1 ,
up to q = n − 1 (since Γnn+2 = 0 and the (0, n)-form Ωnn+1 is ∂-closed). Using the Charpentier 0,q solution operators Cn,s on (0, q + 1)-forms we have f = Ω01 h − Λg Γ02 0,0 0,0 0,0 = Ω01 h − Λg Cn,s Ω12 h + Λg Cn,s Λ C 0,1 Ω23 h − Λg Cn,s Λ C 0,1 Λ C 0,2 Ω34 h + · · · 1 1 g n,s2 1 g n,s2 g n,s3 0,0 0,n−1 n+1 . . . Λg Cn,s Ωn h + (−1)n Λg Cn,s n 1
≡ F 0 + F 1 + · · · + F n.
(4.1)
The goal is then to establish f = Ω01 h − Λg Γ02 ∈ BMOA Bn ; 2 , which we accomplish, through an application of Lemmas 1 and 2, by showing that for m = 1,
n
1 − |z|2 2 1 − |z|2 ∂ F μ (z)
∂z
CM(Bn ;2 )
Cn,δ (g)hH ∞ (Bn ) ,
0 μ n.
(4.2)
It is useful at this point to recall the analogous inequality from [5] with L2 (λn ; 2 ) and H 2 (Bn ) in place of CM(Bn ; 2 ) and H ∞ (Bn ) respectively:
n
1 − |z|2 2 1 − |z|2 ∂ F μ (z)
Cn,δ (g)hH 2 (Bn ) ,
2 ∂z L (λn ;2 )
0 μ n.
In [5] we constructed integers 1 = m0 < m1 < m2 < · · · < mn and used (3.3) and the boundedness of the operators Ta,b,c on L2 (λn ) for a, b, c satisfying (3.2) in order to prove
n n μ+1
1 − |z|2 2 +1 ∂ F μ (z)
Cn,δ 1 − |z|2 2 X mμ Ωμ h (z) L2 (λ ;2 ) ,
n ∂z L2 (λn ;2 )
(4.3)
μ #1 μ+1 μ+1 Dgi ∞ #1 where Ωμ = Ω01 ∧ i=1 Ω = Ω01 ∧ 0 and Ω0 = ( |g|2 )i=1 . Recall from (2.10) that Ωμ μ 1 μ+1 μ+1 ∂gi ∞ 1 is obtained from Ωμ by replacing i=1 Ω0 where Ω0 = ( |g|2 )i=1 , and so the form Ωμ each occurrence of ∂ with D. We then went on to prove in (8.10) of [5] that
n μ+1
1 − |z|2 2 X mμ Ω
μ h (z)
L2 (λn ;2 )
using the multilinear inequality in Lemma 2.
m +μ
Cn,δ gHμ∞ (B
n ;
2)
hH 2 (Bn ) ,
(4.4)
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3835
We can now prove
n
1 − |z|2 2 +1 ∂ X m0 F μ (z)
∂z
CM(Bn ;2 )
n μ+1 Cn,δ 1 − |z|2 2 X mμ Ωμ h (z) CM(B
n ;
2)
(4.5)
by following verbatim the argument in [5] used to prove (4.3), but using the boundedness of Ta,b,c on CM(Bn ; 2 ) rather than on L2 (λn ; 2 ). Recall from (3.3) that the boundedness of Ta,b,c on CM(Bn ; 2 ) is equivalent to the boundedness of Ta,b,c on the scalar space CM(Bn ). The routine verification of these assertions are left to the reader. Finally, we prove
n μ+1
1 − |z|2 2 X mμ Ω
μ h (z)
CM(Bn ;2 )
m +μ
Cn,δ gHμ∞ (B
n ;
2)
hH ∞ (Bn ) ,
(4.6)
using Lemma 2 and a slight variant of the argument used to prove (8.10) in [5]. Following the argument at the top of page 41 in [5], the Liebniz formula yields #1 μ μ+1 X m Ωμ h = X m Ω01 ∧ Ω 0 h α 1 α #1 α X 0 Ω0 ∧ X j Ω0 X μ+1 h .
=
μ+2
α∈Z+
μ
j =1
: |α|=m
Since dν is a Carleson measure if and only if
ϕ(z)2 dν(z) Cϕ2
H 2 (Bn )
,
ϕ ∈ H 2 (Bn ),
Bn
it thus suffices to show that Bn
2 μ 2 2 n 1 − |z|2 X α0 Ω01 ∧ X αj +1 Ω01 X αμ+1 h ϕ(z) j =1
2(m +μ) 2 2 2 hH ∞ (Bn ) ϕH 2 (Bn ) , n ; )
Cn,δ gH ∞μ(B
for all ϕ ∈ H 2 (Bn ). Now we recall (8.12) and (8.14) from [5]: X k Ω01 = X k and
g |g|2
=
k c X k− g X |g|−2 , =0
(4.7)
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−2 2 X |g|
−4−2
cα |g|
1α1 α2 ···αM : α1 +α2 +···+αM =
∞ M 2 α m X g i . m=1
(4.8)
i=1
Thus we see that it suffices to prove
2 n 2 2 2 1 − |z|2 Y α1 g (z) . . . Y αM g (z) Y α0 h (z) ϕ(z) dλn (z)
Bn
Cn,δ g2M H ∞ (B
n ;
2)
h2H ∞ (Bn ) ϕ2H 2 (B ) , n
and this latter inequality follows easily from Lemma 2 with appropriate choices of function. Altogether this yields (4.2) and completes the proof of Theorem 1. Remark 3. We comment briefly on how we obtain BMO, as opposed to H ∞ · BMOA, estimates for solutions to the Bezout equation (1.3). In [1] Andersson and Carlsson obtain H ∞ · BMOA estimates for solutions f to (1.2) with constants independent of dimension by establishing inequalities of the form
K (Λg K)μ−1 Ω μ+1 h
μ
BMOA
− 1 C 1 − |z|2 2 (Λg K)μ−1 Ωnn+1 h CM ,
(4.9)
and
k−1
1 − |z|2 2 (Λg K)μ−1−k Ω n+1 h
n
CM
k C 1 − |z|2 2 (Λg K)μ−2−k Ωnn+1 h CM ,
(4.10)
and finally
μ
1 − |z|2 2 −1 Ω n+1 h
n
CM
C(g)h2H ∞ ,
(4.11)
where K is a certain solution operator to the ∂-equation on strictly pseudoconvex domains (see (4.6), (4.5) and (5.3) in [1]). Iterating these inequalities yields
K (Λg K)μ−1 Ω μ+1 h
μ
BMOA
C(g)h2H ∞ ,
and then applying the final contraction Λg results in the H ∞ · BMOA estimate. These methods yield the best known estimates in terms of the positive parameter δ in (1.4), and yield estimates independent of the number of generators N . On the other hand, after using Lemma 1 to reduce matters to
n
1 − |z|2 2 +1 ∂ Λg C 0,0 . . . Λg C 0,μ−1 Ω μ+1 h (z)
C(g)h2H ∞ , n,s1 μ,sμ μ
∂z CM we follow [5] in using the integration by parts in Lemma 4 together with the estimates in Proposition 1 to reduce matters to an inequality of the form
n μ+1
2
T1 T2 . . . Tμ 1 − |z|2 2 X m Ω
μ h CM C(g)hH ∞ ,
S¸ . Costea et al. / Journal of Functional Analysis 258 (2010) 3818–3840
3837
μ+1 where the Ti are operators of the type Ta,b,c in (3.1), and where Ωμ is the form obtained from μ+1 Ωμ by replacing each occurrence of ∂ with the “larger” D. Then Lemma 5 reduces matters to proving
n μ+1
1 − |z|2 2 X m Ω
μ h
CM
C(g)h2H ∞ ,
which finally follows from the multilinear inequality in Lemma 2. It is in this way that we avoid having to multiply a BMOA solution by a bounded holomorphic function. 5. A generalization In [5] the corona theorem (including the semi-infinite matrix case) was established for the multiplier algebras MBpσ (Bn ) of Bpσ (Bn ) when p = 2 and 0 σ 12 . However, the Corona problem remains open for all of the remaining multiplier algebras MBpσ (Bn ) . In this section we consider two weaker assertions and prove the weakest one. Recall that Bpσ (Bn ; 2 ) can be characterized as consisting of those f ∈ H (Bn ; 2 ) such that p σ dμf (z) ≡ 1 − |z|2 Y m f (z) dλn (z) is a finite measure. Moreover, Proposition 3 in [5] shows that the multiplier space MBpσ (Bn )→Bpσ (Bn ;2 ) satisfies the containment MBpσ (Bn )→Bpσ (Bn ;2 ) ⊂ f ∈ H ∞ Bn ; 2 : μf is a Carleson measure for Bpσ (Bn ) .
(5.1)
Remark 4. Theorem 3.7 of [9] shows that equality actually holds in the scalar-valued version of (5.1). The argument there can be extended to prove equality in (5.1) itself, but as we do not need this result, we do not pursue it further here. We can rewrite (5.1) in the more convenient form MBpσ (Bn )→Bpσ (Bn ;2 ) ⊂ H ∞ Bn ; 2 ∩ Xpσ Bn ; 2 , where Xpσ Bn ; 2 = f ∈ Bpσ Bn ; 2 : μf is a Carleson measure for Bpσ (Bn ) is normed by p f Xσ (B ;2 ) p n
=
sup ϕ∈Bpσ (Bn )
Bn
|ϕ(z)|p dμf (z) p
ϕB σ (Bn ) p
.
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Conjecture 1. Given g ∈ MBpσ (Bn )→Bpσ (Bn ;2 ) satisfying Mg Bpσ (Bn )→Bpσ (Bn ;2 ) 1, ∞ gj (z)2 δ 2 > 0,
z ∈ Bn ,
j =1
and h ∈ MBpσ (Bn ) , there is a vector-valued function f ∈ Xpσ (Bn ; 2 ) such that f Xpσ (Bn ;2 ) Cn,σ,p,δ hMB σ (Bn ) , p
N
fj (z)gj (z) = h(z),
z ∈ Bn .
j =1
While we are unable to settle this conjecture here, we can prove the weaker conjecture obtained by relaxing the Carleson measure condition slightly in the definition of the space Xpσ . We say that a positive measure μ is a weak Carleson measure for Bpσ (Bn ) if sup
ζ ∈Bn
Sζ
dμ(z)
< ∞.
(1 − |ζ |2 )pσ
Note that a Carleson measure μ for Bpσ (Bn ) is automatically a weak Carleson measure for Bpσ (Bn ). This can be seen by testing the embedding for μ over reproducing kernels. Let W Xpσ Bn ; 2 = f ∈ Bpσ Bn ; 2 : μf is a weak Carleson measure for Bpσ (Bn ) be normed by p f W Xσ (B ;2 ) p n
= sup
ζ ∈Bn
Sζ
dμf (z)
(1 − |ζ |2 )pσ
.
Theorem 5. Given g ∈ MBpσ (Bn )→Bpσ (Bn ;2 ) satisfying Mg Bpσ (Bn )→Bpσ (Bn ;2 ) 1, ∞ gj (z)2 δ 2 > 0,
z ∈ Bn ,
j =1
and h ∈ MBpσ (Bn ) , there is a vector-valued function f ∈ W Xpσ (Bn ; 2 ) such that f W Xpσ (Bn ;2 ) Cn,σ,p,δ hMB σ (Bn ) , p
N j =1
fj (z)gj (z) = h(z),
z ∈ Bn .
S¸ . Costea et al. / Journal of Functional Analysis 258 (2010) 3818–3840
3839
In order to prove Theorem 5 we introduce, in analogy with CM(Bn ), a Banach space WCMσp (Bn ) of measurable functions h on the ball Bn such that |h(z)|p dλn (z) is a weak Carleson measure for Bpσ (Bn ); p Sζ |h(z)| dλn (z) σ <∞ . WCMp (Bn ) = h: sup (1 − |ζ |2 )pσ ζ ∈Bn Note that σ W Xpσ Bn ; 2 = f ∈ H Bn ; 2 : 1 − |z|2 Y m f (z) ∈ WCMσp (Bn ) . Using the argument above we will see below that Theorem 1 follows from: Lemma 6. Let a, b, c ∈ R, 1 < p < ∞ and σ 0. Then the operator Ta,b,c defined by Ta,b,c h(z) = Bn
√ (1 − |z|2 )a (1 − |w|2 )b ( (w, z))c h(w) dV (w) |1 − wz|n+1+a+b+c
is bounded on WCMσp (Bn ) if c > −2n and
− pa < −n < p(b + 1).
In Lemma 5 above we proved that Lemma 6 holds in the special case σ = n2 and p = 2 by exploiting the characterization of Carleson measures for H 2 (Bn ) as being precisely the weak Carleson measures. Thus the proof of Lemma 5 above also applies to prove Lemma 6. The straightforward verification is left to the reader. We will also need the following slight generalization of Proposition 3 in [5]. Proposition 2. Suppose that 1 < p < ∞, 0 σ < ∞, M 1, m > 2( pn − σ ) and α = (α0 , . . . , αM ) ∈ ZM+1 with |α| = m. For g1 , . . . , gM ∈ MBpσ (Bn )→Bpσ (Bn ;2 ) and h ∈ Bpσ (Bn ) we + have
pσ α p α Y 1 g1 (z) . . . Y M gM (z)p Y α0 h (z)p dλn (z) 1 − |z|2
Bn
Cn,M,σ,p
M j =1
p Mgj B σ (B )→B σ (B ;2 ) n n p p
p
hB σ (Bn ) . p
Proposition 3 in [5] proves the case g1 = · · · = gM and the proof given there carries over immediately to the case of different gj here. Now we combine Lemma 6 and Proposition 2 to obtain Theorem 5. Note that (3.3) again shows that boundedness of Ta,b,c on the vector-valued version CMσp (Bn ; 2 ) is equivalent to boundedness on CMσp (Bn ).
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Proof of Theorem 5. We must establish the following two inequalities:
m0
μ
1 − |z|2 σ +m0 ∂ F (z)
∂z
WCMσp (Bn ;2 )
σ μ+1 Cn,δ 1 − |z|2 X mμ Ωμ h (z) WCMσ (B p
n ;
2)
,
(5.2)
and
μ+1
1 − |z|2 σ X mμ Ω
μ h (z)
WCMσp (Bn ;2 )
m +μ
Cn,δ Mg B σμ(B p
σ 2 n )→Bp (Bn ; )
hMB σ (Bn ) . p
(5.3)
Just as for (4.5) in the previous section, inequality (5.2) here follows verbatim the argument in [5] but with the boundedness of Ta,b,c on WCMσp (Bn ) used in place of boundedness on Lp (λn ). To establish (5.3), and even the stronger inequality with the larger CMσp (Bn ; 2 ) norm on the left side, it suffices to show p μ+1 ϕ(z)p 1 − |z|2 σ X mμ Ω μ h (z) dλn (z) Bn (m +μ)p p p ϕB σ (Bn ) . σ 2 hM σ Bp (Bn ) p n )→Bp (Bn ; ) p
Cn,δ Mg B σμ(B
But this follows using Proposition 2 together with the argument used to prove (4.6) at the end of the previous section. 2 References [1] M. Andersson, H. Carlsson, Estimates of the solutions of the H p and BMOA corona problem, Math. Ann. 316 (2000) 83–102. [2] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures and interpolating sequences for Besov spaces on complex balls, Mem. Amer. Math. Soc. 859 (2006), 163 pp. [3] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962) 547– 559. [4] P. Charpentier, Solutions minimales de l’é quation ∂u = f dans la boule et le polydisque, Ann. Inst. Fourier (Grenoble) 30 (1980) 121–153. [5] S. ¸ Costea, E.T. Sawyer, B.D. Wick, The corona theorem for the Drury-Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in Cn , arXiv:0811.0627v7, 8 April 2009, 82 pp. [6] J. Garnett, Bounded Analytic Functions, Pure Appl. Math., vol. 96, Academic Press, 1981. [7] K.C. Lin, The H p corona theorem for the polydisc, Trans. Amer. Math. Soc. 341 (1994) 371–375. [8] N.K. Nikol’ski˘ı, Operators, Functions, and Systems: an Easy Reading, vol. 1: Hardy, Hankel, and Toeplitz, translated from the French by Andreas Hartmann, Math. Surveys Monogr., vol. 92, Amer. Math. Soc., Providence, RI, 2002. [9] J.M. Ortega, J. Fabrega, Pointwise multipliers and decomposition theorems in analytic Besov spaces, Math. Z. 235 (2000) 53–81. [10] E. Sawyer, Function Theory: Interpolation and Corona Problems, Fields Inst. Monogr., vol. 25, Amer. Math. Soc., Providence, RI, 2009. [11] N.Th. Varopoulos, BMO functions and the ∂ equation, Pacific J. Math. 71 (1977) 221–273. [12] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, 2004.
Journal of Functional Analysis 258 (2010) 3841–3854 www.elsevier.com/locate/jfa
Non-vanishing functions and Toeplitz operators on tube-type domains Adel B. Badi The 7th of October University, Faculty of Science, Mathematics Department, PO Box 2478, Misurata, Libya Received 20 October 2009; accepted 16 November 2009 Available online 1 December 2009 Communicated by Alain Connes
Abstract We prove an index theorem for Toeplitz operators on irreducible tube-type domains and we extend our results to Toeplitz operators with matrix symbols. In order to prove our index theorem, we proved a result asserting that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domain, not necessarily irreducible, is equal to a unimodular function defined as the product of powers of generic norms times an exponential function. © 2009 Elsevier Inc. All rights reserved. Keywords: Tube-type domains; Toeplitz operators; Generalized Fredholm index; Topological index; Indicial triple
1. Introduction The well-known I. Gohberg and M. Krein index theorem for Toeplitz operators on the unit circle is a relation between the (analytic) Fredholm index of a Toeplitz operator and the winding number of its symbol (the topological index). The theory of Toeplitz operators have been extended and studied on many generalized Hardy spaces. However, in most cases there is no result similar to Gohberg–Krein index theorem. G.J. Murphy has introduced an elegant index theory for generalized Toeplitz operators in [6] in attempt to generalize Gohberg–Krein theorem. Murphy’s theory is based on C∗ -algebras where he used orthogonal projections to define Toeplitz operators and traces to define the topological and analytical indices. He showed that the classical Gohberg–Krein index theorem for Toeplitz E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.015
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A.B. Badi / Journal of Functional Analysis 258 (2010) 3841–3854
operators on the unit circle can be derived from his results and he proved an index theorem for Toeplitz operators on the group of unitary 2 × 2 matrices. Moreover, he showed that the Gohberg–Krein theorem for Toeplitz operators with matrix symbols on the unit circle follows from his results too. He has proved an index theorem for Toeplitz operators with matrix symbols on the group of 2 × 2 unitary matrices. In [1], we used Murphy’s results to prove an index theorem for Toeplitz operators on the quarter-plane and we extended our results to the related Toeplitz operators with matrix symbols. In this paper we prove an index theorem for Toeplitz operators on irreducible tube-type bounded symmetric domains. Our results can be viewed as an extension of Murphy’s result for Toeplitz operators on the compact group of unitary 2 × 2 matrices. In fact, this group is the Shilov boundary of an irreducible tube-type bounded symmetric domain. We prove our index theorem using H. Upmeier’s theory for Toeplitz operators on bounded symmetric domains; see [9]. All of the examples in [6] are of Toeplitz operators on connected compact groups, hence, the calculation of the topological index is done using the result of E. van Kampen in [10] which is an extension of a result of H. Bohr in [2]. In this paper the symbol algebra is defined on connected compact spaces which are not necessarily groups. Thus, we need a result on these spaces similar to van Kampen’s result. This result asserts that a non-vanishing function on the Shilov boundary of a tube-type bounded symmetric domains is equal to a unimodular function defined as the product of powers of generic norms times an exponential function. Recall that a complex-valued function is called unimodular if its absolute value is 1. This result is valid for all tube-type domains, the reducible and irreducible ones. We prove this result in its full generality in spite of the fact that our index theorem is proved only for Toeplitz operators on irreducible tube-type domains. In Section 2, we introduce tube-type domains and some Jordan triple basic concepts like tripotents and generic norms which are necessary in our work. Bounded symmetric domains have very rich geometric and Jordan algebraic theories, however, only those properties needed in our work will be given. The structure of the Shilov boundary of a tube-type domain is studied and some of its important properties are given. We also give the topological properties of a compact subset of the Shilov boundary called the reduced Shilov domain. Section 3 is dedicated to the proof of our result on non-vanishing functions on Shilov boundaries of tube-type domains. We study Hardy spaces on irreducible tube-type domains in Section 4 where we depended on the results obtained by Upmeier. Finally, we prove our index theorem for Toeplitz operators on irreducible tube-type domains in Section 5 and we extend it to the related Toeplitz operators with matrix symbols. 2. Jordan triples, tripotents and generic norms Let Z be a finite-dimensional vector space on the field C of complex numbers provided with the unique norm topology. Polynomials can be defined on Z without reference to a certain basis and we can define holomorphic functions using polynomials [9, Section 1.1]. Let D be a domain in Z, that is, a connected non-empty open subset of Z. A mapping g : D → D is said to be biholomorphic if it is bijective and g, g −1 are holomorphic. If D is bounded and for every z ∈ D there exists a biholomorphic mapping gz with z as an isolated fixed point of gz and gz ◦ gz is the identity map on D, then D is called a bounded symmetric domain. Any bounded symmetric domain is biholomorphically equivalent to a unique (up to a linear isomorphism) circled bounded symmetric domain [3, Theorem 1.6]. Recall that the set D is
A.B. Badi / Journal of Functional Analysis 258 (2010) 3841–3854
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circled if it is invariant under the map Z → Z, z → λz, λ ∈ T and 0 ∈ D, where T is the unit circle in C. We will assume that all bounded symmetric domains are circled. A Jordan triple product on Z is a mapping Z 3 → Z, (x, y, z) → {xyz} which is linear in the first variable, conjugate-linear in the second variable, symmetric in the first and the third variables and xy{zuv} + z{yxu}v = {xyz}uv + zu{xyv} , for all x, y, z, u, v ∈ Z. If {zzz} = tz for z ∈ Z implies t > 0 or z = 0, then we call Z a positive hermitian Jordan triple system, PJT for short. Let Z be a PJT and define the mapping L(x, y) : Z → Z, z → {xyz} for any x, y ∈ Z. Then the sesquilinear form x | y = trace L(x, y) ,
x, y ∈ Z,
(2.1)
is a positive-definite inner product on Z. If we denote by 1 the identity map on Z, then the set z ∈ Z 1 − L(z, z) is positive definite
(2.2)
is a bounded symmetric domain. Conversely, for every bounded symmetric domain D ⊆ Z, there exists a Jordan triple product on Z such that Z is a PJT and D is equal to the set defined in (2.2). Moreover, there is a unique Jordan triple product such that D is the open unit ball with respect to the so-called spectral norm on Z. Henceforth, we will always relate this unique Jordan triple product on Z to any given bounded symmetric domain D ⊆ Z. Let x be an element in the bounded symmetric domain D ⊆ Z. If the conjugate-linear mapping Qx : Z → Z, z → {xzx} is invertible, then we call x an invertible element. The domain D is said to be of tube-type if Z contains an invertible element. A bounded symmetric domain is said to be irreducible if it is not a direct product of two bounded symmetric domains. Let Di be a bounded symmetric domain with the related PJT Zi , where i = 1, 2. By [3, 4.11], if D = D1 × D2 , then Z = Z1 ⊕ Z2 and Z1 , Z2 are Jordan triple ideals in Z. Recall that a subspace W of Z is called a Jordan triple ideal if {xyz}, {yxz} ∈ W , for all x ∈ W, y, z ∈ Z. The domain D is of tube-type if and only if each of D1 , D2 is of tube-type. Any bounded symmetric domain D ⊆ Z is the direct product of a finite number of irreducible bounded symmetric domains and Z is the direct sum of their related PJTs. We will be interested only in tube-type domains. We give a list of all irreducible tube-type domains. Any tube-type domain is the direct product of a finite number of the following domains. In the following n will denote a positive integer and z is the transpose of the square matrix z ∈ Mn (C). Z = Mn (C), D = {z ∈ Z | z < 1}. Z = {z ∈ M2n (C) | z = −z}, D = {z ∈ Z | z < 1}. Z = {z ∈ Mn (C) | z = z}, D = {z ∈ Z | z < 1}. Z = Cn , n 3, D = {z ∈ Z | z · z ¯ + (z · z¯ )2 − |z · z|2 < 2}. Here z¯ is complex conjugate of z and z · w = nj=1 zj wj . VI: The exceptional tube-type domain in Z = C27 .
In : II2n : IIIn : IVn :
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The Jordan triple product related to the PJT structure on Z in the first three cases is {xyz} = ¯ − (xy ∗ z + zy ∗ x)/2, where y ∗ is the adjoint of the matrix y. In the fourth case, {xyz} = (x · y)z (x · z)y¯ + (z · y)x. ¯ The bounded symmetric domains In,m , n = m, II2n+1 and V are not of tube-type; see [3, Section 4.13]. A tripotent is an element e ∈ Z such that {eee} = e. We say that the two tripotents e1 , e2 ∈ Z are orthogonal if {e1 e1 e2 } = 0. A non-zero tripotent is called primitive if it is not the sum of nonzero orthogonal tripotents. If e ∈ Z is a primitive tripotent and Z is the direct sum of the Jordan triple ideals Z1 , Z2 , then either e ∈ Z1 or e ∈ Z2 and the primitive tripotents in Z1 and Z2 are primitive in Z too. Any non-zero tripotent is the sum of a finite number of pairwise orthogonal primitive tripotents and any primitive tripotent is contained in a maximal set of pairwise orthogonal primitive tripotents. A maximal set of pairwise orthogonal primitive tripotents is called a frame. All frames in Z have the same number of elements and this number is called the rank of Z. The rank of a bounded symmetric domain is the rank of the related PJT. A maximal tripotent is the sum of the elements of some frame. The rank in the cases In , II2n and IIIn is n, the tripotents are exactly the partial isometries and the maximal tripotents are exactly the unitaries. In IVn the rank is 2 and in VI the rank is 3. A generic norm on Z is a homogeneous polynomial N : Z → C with degree equal to the rank of Z such that z ∈ Z is invertible if and only if N (z) = 0 and N (e) = 1, for some tripotent e ∈ Z. Thus, if r is the rank of Z, then N (λz) = λr N (z), for all λ ∈ C, z ∈ Z. The PJT related to an irreducible bounded symmetric domain has a generic norm if and only if the domain is of tubetype. Moreover, any two generic norms differ by a complex factor of absolute value 1. In the case of In and IIIn the determinant polynomial is a generic norm, and in II2n the Pfaffian polynomial is a generic norm. In IVn , the polynomial z → z · z is a generic norm. Let D be a tube-type domain in Z. We can view the polynomial algebra P(Z) as a subset of C(D). The Shilov boundary S of D is the Shilov boundary of D relative to the function algebra P(Z). By [3, Theorem 6.5], S is the set of all maximal tripotents of Z. Thus, if λ ∈ T and e ∈ S, then λe ∈ S. The following proposition describes the structure of the Shilov boundary of a reducible domain. In fact, this elementary result is valid for all bounded symmetric domains. We couldn’t find it in the literature, hence, we state it and we prove it here. Proposition 2.1. If a tube-type domain is the direct product of irreducible domains, then its Shilov boundary is the direct product of the Shilov boundaries of the irreducible domains. Proof. It is sufficient to prove that if the tube-type domain D is the direct product of the domains D1 , D2 , the Shilov boundary of D equals the product of the Shilov boundaries of D1 , D2 . Let Z, Z1 , Z2 be the spaces with Jordan triple systems related to D, D1 , D2 , respectively. Then Z = Z1 ⊕ Z2 . Let S, S1 , S2 be the Shilov boundaries of D, D1 , D2 , respectively and let r, r1 , r2 be the ranks of the domains D, D1 , D2 , respectively. Then r = r1 + r2 . If e ∈ S, then e is a maximal tripotent of Z. Thus, e = e1 + · · · + er , where e1 , . . . , er are primitive orthogonal tripotents of Z. Each of e1 , . . . , er belongs to either Z1 or Z2 . Hence, exactly r1 elements of e1 , . . . , er belong to Z1 and their sum is a maximal tripotent in Z1 . The other r2 elements of e1 , . . . , er are in Z2 and their sum is a maximal tripotent in Z2 . Thus, e ∈ S1 × S2 . Conversely, if e ∈ S1 , d ∈ S2 , then e = e1 + · · · + er1 and d = d1 + · · · + dr2 , where e1 , . . . , er1 are primitive orthogonal tripotents of Z1 and d1 , . . . , dr2 are primitive orthogonal tripotents of Z2 . Thus, e1 , . . . , er1 , d1 , . . . , dr2 is a frame in Z and e1 + · · · + er1 + d1 + · · · + dr2 is a maximal tripotent in Z. Hence, e + d in S. 2
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Let D be an irreducible tube-type domain with Shilov boundary S and let N be a fixed generic norm on Z. Then |N (z)| = 1 for all z ∈ S. The reduced Shilov boundary of the irreducible domain D related to N is defined as the set SN = {z ∈ S | N (z) = 1}. In the following proposition we give the properties of the reduced Shilov boundary we need in proving the main theorem in the following section. Proposition 2.2. Let D ⊆ Z be an irreducible tube-type domain, let N be a generic norm on Z and let SN be the related reduced Shilov boundary. (1) SN is linearly isomorphic to the reduced Shilov boundary of D related to any other generic norm on Z. (2) SN is compact, connected and simply connected. (3) SN is path connected and locally path connected. Proof. (1) Suppose that N1 is a generic norm on Z and SN1 is the related reduced Shilov boundary. Let r be the rank of Z and let λ ∈ T such that N = λN1 . Then the linear isomorphism z → κz maps SN onto SN1 , where λ = κ r . (2) It is obvious that SN is compact. By [9, p. 63], SN is a connected and simply connected manifold. (3) Follows from the fact that SN is a connected manifold. 2 3. Non-vanishing functions on the Shilov boundary of a tube-type domain Let n be a fixed integer and let D = D1 × · · · × Dn , where D1 , . . . , Dn are irreducible tubetype domains. Suppose that S1 , . . . , Sn are the Shilov boundaries of the domains D1 , . . . , Dn , respectively. By Proposition 2.1, S = S1 × · · · × Sn is the Shilov boundary of D. For each 1 j n, let Zj be the PJT associated to Dj and let rj be its rank. Let Nj be a fixed generic norm Shilov boundary. on Zj and let SNj be the related reduced Define the product space Y = nj=1 (R × SNj ). To simplify notation, the point ((t1 , u1 ), . . . , (tn , un )) in Y will be denoted by (tj , uj )nj=1 . The mapping Ψ : Y → S,
n (tj , uj )nj=1 → eitj uj j =1 = eit1 u1 , . . . , eitn un
is continuous and surjective. If Ψ ((tj , uj )nj=1 ) = Ψ ((tj , u j )nj=1 ), then we write (tj , uj )nj=1 ∼ (tj , u j )nj=1 . This is equivalent to the condition tj = tj + mj αj , u j = e−mj αj uj , where mj ∈ Z and αj = 2π/rj for 1 j n. Note that ∼ is an equivalence relation on Y . Let Q be the quotient space of Y by this equivalence relation. Note that Q is a compact space since it is the image of the compact subspace nj=1 ([0, αj ] × SNj ) ⊆ Y under the quotient mapping q : Y → Q. Define the mapping Ψˆ : Q → S by
n (tj , uj )nj=1 → Ψ (tj , uj )nj=1 = eitj uj j =1 , where [(tj , uj )nj=1 ] is the equivalence class of (tj , uj )nj=1 . The mapping Ψˆ is a well-defined bijection and Ψ = Ψˆ ◦ q. By the definition of the quotient topology, Ψˆ is continuous. Therefore, Ψˆ is a homeomorphism. In addition to the properties of the space Y given above, we need the following lemma.
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Lemma 3.1. Let ν : Y → T be a continuous mapping. Then there exists a continuous mapping ψ : Y → R such that ν = eiψ . Proof. By Proposition 2.2, the space Y is simply connected, path connected and locally path connected. Hence, the fundamental group of Y is the trivial group. Consider the covering space R over the space T relative to the mapping p : R → T, t → eit . By applying the general lifting lemma [4, Chapter 8, Lemma 14.2], there exists a continuous function ψ : Y → R such that ν = p ◦ ψ. Hence, ν = eiψ . 2 If ψ0 , ψ1 : Y → R are continuous functions such that eiψ0 = eiψ1 , then there is a unique integer k such that ψ0 = ψ1 + 2πk. This remark will be used frequently in the proof of the following lemma. Lemma 3.2. If ϕ : S → T is continuous, then ϕ = θ eiψ for some continuous function ψ : S → R, where θ : S → C is the unimodular function defined by the unique integers k1 , . . . , kn as n k θ (uj )nj=1 = Nj j (uj ), j =1
(uj )nj=1 ∈ S.
Proof. We denote the Kronecker delta function by δj l . Consider the continuous mapping ν0 : Y → T, where ν0 = ϕ ◦ Ψ . By Lemma 3.1, there exists a continuous function ψ0 : Y → R such that ν0 = eiψ0 . If l is an integer such that 1 l n, then n n exp iψ0 tj + δj l αj , e−iδj l αj uj j =1 = ν0 tj + δj l αj , e−iδj l αj uj j =1 n = ϕ eitj uj j =1 = ν0 (tj , uj )nj=1 = exp iψ0 (tj , uj )nj=1 , for all (tj , uj )nj=1 ∈ Y . Hence, there exists an integer kl such that n ψ0 tj + δj l αj , e−iδj l αj uj j =1 = ψ0 (tj , uj )nj=1 + 2πkl .
(3.1)
Using the n integers k1 , . . . , kn satisfying Eq. (3.1), we define the unimodular function k θ : S → C such that θ ((uj )nj=1 ) = nj=1 Nj j (uj ) for (uj )nj=1 ∈ S. Define the function ϕ1 : S → T such that ϕ1 = θ −1 ϕ and let ν1 : Y → T be the function ν1 = ϕ1 ◦ Ψ . By Lemma 3.1, there exists a continuous function ψ1 : Y → R such that ν1 = eiψ1 . If (tj , uj )nj=1 ∈ Y , then n exp iψ1 (tj , uj )nj=1 = ν1 (tj , uj )nj=1 = ϕ1 eitj uj j =1 n −1 it n ϕ e j uj j =1 = θ eitj uj j =1 n = N1−k1 eit1 u1 · · · Nn−kn eitn un ϕ eitj uj j =1 n = e−i j =1 rj kj tj ν0 (tj , uj )nj=1
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n n = exp iψ0 (tj , uj )j =1 − i rj kj tj . j =1
Hence, there is an integer k such that n ψ1 (tj , uj )nj=1 = ψ0 (tj , uj )nj=1 − rj kj tj + 2πk,
(3.2)
j =1
for all (tj , uj )nj=1 ∈ Y . Let l be an integer such that 1 l n. Then, by Eq. (3.2), we get n n ψ1 tj + δj l αj , e−iδj l αj uj j =1 = ψ0 tj + δj l αj , e−iδj l αj uj j =1 −
n
rj kj tj − rl kl αl + 2πk.
j =1
Now we use Eq. (3.1) and the equality αl = 2π/rl to get n n ψ1 tj + δj l αj , e−iδj l αj uj j =1 = ψ0 (tj , uj )nj=1 + 2πkl − rj kj tj − rl kl αl + 2πk j =1
n = ψ0 (tj , uj )nj=1 − rj kj tj + 2πk
= ψ1 (tj , uj )nj=1 .
j =1
Note that we obtained the last equality from Eq. (3.2). By induction, it follows that the relation n ψ1 tj + mj αj , e−imj αj uj j =1 = ψ1 (tj , uj )nj=1
(3.3)
holds for all (tj , uj )nj=1 ∈ Y and all m1 , . . . , mn ∈ Z. Let Q be the quotient space defined above. Define the function ψˆ 1 : Q → R,
(tj , uj )nj=1 → ψ1 (tj , uj )nj=1 .
By Eq. (3.3) and by the remarks on the definition of the equivalence relation ∼ on Y , the function ψˆ 1 is well defined and ψ1 = ψˆ 1 ◦ q, where q is the quotient canonical mapping. Moreover, it follows that ψˆ 1 is continuous. Let Ψˆ : Q → S be the homeomorphism defined above and define the continuous function ψ : S → R, where ψ = ψˆ 1 ◦ Ψˆ −1 . Let (uj )nj=1 ∈ S. For all j = 1, . . . , n, let sj be a real number such that 0 sj < αj and Nj (uj ) = eirj sj . Hence, exp iψ (uj )nj=1 = exp iψˆ 1 ◦ Ψˆ −1 (uj )nj=1 n = exp iψ1 sj , e−isj uj j =1
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n = ν1 sj , e−isj uj j =1 n = ϕ1 Ψ sj , e−isj uj j =1 = ϕ1 (uj )nj=1 . Thus, ϕ1 = eiψ . Since ϕ1 = θ −1 ϕ, it follows that ϕ = θ eiψ . This proves the existence part of this lemma. To prove the uniqueness part, it is sufficient to show that if k1 , . . . , kn are integers and ψ˜ : S → R is a continuous function, then the equality n k Nj j (uj ) = 1 for all (uj )nj=1 ∈ S, exp iψ˜ (uj )nj=1 j =1
implies k1 = · · · = kn = 0. If (tj , uj )nj=1 ∈ Y , then
n n it n it n kj it 1 = exp iψ˜ e j uj j =1 Nj e j uj = exp iψ˜ e j uj j =1 + i kj rj tj . j =1
j =1
Hence, there exists an integer k such that n n kj rj tj = 2πk. ψ˜ eitj uj j =1 + j =1
Choose a fixed element uj ∈ SNj for each 1 j n. Let m1 , . . . , mn ∈ {0, 1}. By the previous equation we get n n kj rj · 0 = 2πk ψ˜ (uj )nj=1 = ψ˜ ei·0 uj j =1 + j =1
n n = ψ˜ e2iπmj uj j =1 + 2π kj rj mj j =1
n = ψ˜ (uj )nj=1 + 2π kj rj mj . j =1
Thus,
n
j =1 kj rj mj
= 0. It follows easily that k1 = · · · = kn = 0.
2
The next theorem is the main result in this section, it follows almost directly from Lemma 3.2. Theorem 3.3. For every continuous non-vanishing function ϕ on S there is a continuous function ψ on S such that ϕ = θ eψ where θ : S → C is the unimodular function defined as k θ ((uj )nj=1 ) = nj=1 Nj j (uj ) for the unique integers k1 , . . . , kn .
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Proof. Let ϕr : S → R and ϕu : S → T be the continuous functions defined by the relations ϕr = |ϕ|, ϕu = ϕ/|ϕ|, respectively. Since ϕr is positive, it is easy to show that there is a continuous function ψr : S → R such that ϕr = eψr . Lemma 3.2 asserts that there exist unique integers k1 , . . . , kn such that ϕu = θ eiψu where θ : S → C is the unimodular function defined k by θ ((uj )nj=1 ) = nj=1 Nj j (uj ) and ψu : S → R is a continuous function. Define the continuous function ψ : S → C, where ψ = ψr + iψu . Hence, ϕ = θ eψ . Since ϕu = ϕ/|ϕ|, the integers k1 , . . . , kn are unique. 2 Remark 3.4. The integers k1 , . . . , kn in Theorem 3.3 are independent of the choice of the generic norms N1 , . . . , Nn . 4. Harmonic analysis on tube-type domains Throughout this section, we consider a fixed irreducible tube-type domain D ⊆ Z with its related PJT structure and let S be its Shilov boundary. We choose a fixed generic norm N on Z. There is a unique K-invariant regular Borel probability measure μ on S, where K is the group of all invertible linear mappings on Z such that g(D) = D; see [9, p. 125]. Let L2 (S) be the space of square-integrable functions on S with respect to μ. The Hardy space H 2 (S) is the L2 closure of the algebra of all polynomials P(Z) in L2 (S). We denote the orthogonal projection of the space L2 (S) onto H 2 (S) by PD . Define the set VD = {v ∈ Z | N (v) = 0}. Note that the set VD is independent of the choice of the generic norm N . For any u ∈ Z, we define the homogeneous polynomial u of degree 1 on Z by u (z) = z | u, where · | · is the inner product defined in Eq. (2.1). Let H(Z) be the linear subspace of P(Z) spanned by the polynomials kv , where k is a non-negative integer and v ∈ VD . This space is called the space of harmonic polynomials [9, p. 121]. Let SN be the reduced Shilov boundary of D related to N . There exists a unique probability Borel measure on SN which is invariant under the commutator subgroup of K. We define L2 (SN ) as the L2 space relative to this measure. If N1 is another generic norm on Z and SN1 its related reduced Shilov boundary, then, by Proposition 2.2, there is a linear isomorphism SN1 → SN . This isomorphism preserves the invariant measures and it induces a unitary L2 (SN ) → L2 (SN1 ). Proposition 4.1. There is a collection {Hα }α∈I of finite-dimensional subspace of L2 (S) for some indexing set I such that: (1) (2) (3) (4)
For each α ∈ I , the elements of Hα are harmonic polynomials restricted to S. The spaceL2 (SN ) is unitarily equivalent to the orthogonal sum α∈I Hα . L2 (S) = α∈I, l∈Z N l Hα . H 2 (S) = α∈I, l0 N l Hα .
Proof. By [9, Theorem 2.8.68], the space L2 (SN ) can be identified with the orthogonal sum α∈I Hα , for some indexing set I . For each α ∈ I , Hα is a finite-dimensional Hilbert subspace of L2 (S); see [9, pp. 118 and 133]. Moreover, by [9, Proposition 4.11.51], the elements of Hα are harmonic polynomials restricted to S. Thus, we have proved (1) and (2). Statement (3) follows from Theorem 2.8.38 and Eq. (2.8.34) in [9]. The equality in (4) is proved in Theorem 2.8.47 in [9]. 2
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For every α ∈ I , let the positive integer dα be the dimension of Hα and let the set of harmonic (α) (α) (α) polynomials {p1 , . . . , pdα } be an orthonormal basis of Hα . Thus, {N l pk | l ∈ Z, 1 k dα , (α)
α ∈ I } is an orthonormal basis for L2 (S) and {N l pk | l 0, 1 k dα , α ∈ I } is an orthonormal basis for H 2 (S). Therefore, (α) PD N l pk =
(α)
N l pk , 0,
if l 0, if l < 0. (α)
Now we define the unitary mapping U : L2 (S) → L2 (T) ⊗ L2 (SN ) such that U (N l pk ) = (α) zl ⊗ pk for α ∈ I , 1 k dα , l ∈ Z, where L2 (T) is the space of square-integrable functions on T relative to the normalized arc length measure and z is the inclusion map z : T → C, λ → λ. Define the ∗-isomorphism π : B(L2 (S)) → B(L2 (T) ⊗ L2 (SN )) by setting π(a) = U aU ∗ for a ∈ B(L2 (S)). Proposition 4.2. If u ∈ Z, then there exists a unique pair of operators Au , Bu in B(L2 (SN )) such that u p = N −1 Au (p) + Bu (p), for all p ∈ H(Z). Proof. The result follows from Eqs. (4.11.50) and (4.11.58) in [9].
2
The mapping which takes a continuous function on S to its related multiplication operator on L2 (S) is a ∗-isomorphism. Thus, we can consider C(S) as a unital C∗ -subalgebra of B(L2 (S)). Let H 2 (T) be the Hardy space related to L2 (T) and let PT be the orthogonal projection of L2 (T) onto the subspace H 2 (T). Proposition 4.3. Let u ∈ Z and let Au , Bu be the unique pair of operators related to u as in Proposition 4.2. Then we have (1) π(N ) = z ⊗ 1. (2) π(PD ) = PT ⊗ 1. (3) π( u ) = z ⊗ A∗u + 1 ⊗ Bu∗ . (α)
Proof. Let zl ⊗ pk be an arbitrary element in the orthonormal basis of L2 (T) ⊗ L2 (SN ) defined above. Then (α) (α) (α) π(N) zl ⊗ pk = U N U ∗ zl ⊗ pk = U N N l pk (α) (α) = U N l+1 pk = zl+1 ⊗ pk (α) = (z ⊗ 1) zl ⊗ pk . Thus, π(N ) = z ⊗ 1. This proves (1). To show that (2) holds, note first that π(PD )(zl ⊗ pk(α) ) = U (PD (N l pk(α) )). Thus, if l 0, then
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(α) (α) (α) π(PD ) zl ⊗ pk = U N l pk = zl ⊗ pk (α) (α) = PT zl ⊗ pk = (PT ⊗ 1) zl ⊗ pk . (α)
(α)
Similarly, if l < 0, then π(PD )(zl ⊗ pk ) = 0 = (PT ⊗ 1)(zl ⊗ pk ). This proves (2). Now we prove (3). By Proposition 4.2 above, (α) (α) (α) π( u ) zl ⊗ pk = U u N l pk = U N l u pk (α) (α) = U N l N −1 Au pk + Bu pk (α) (α) = U N l−1 Au pk + N l Bu pk (α) (α) = zl−1 ⊗ Au pk + zl ⊗ Bu pk (α) = z−1 ⊗ Au + 1 ⊗ Bu zl ⊗ pk . Thus, π( u ) = z−1 ⊗ Au + 1 ⊗ Bu . Therefore, π( u ) = z ⊗ A∗u + 1 ⊗ Bu∗ .
2
Now we define CD as the unital C∗ -subalgebra of B(L2 (SN )) generated by the operators Au , Bv for all u, v ∈ Z. We need the following property of the C∗ -algebra CD which follows easily from [9, Proposition 4.11.140]. Proposition 4.4. There exists a character τ on the unital C∗ -algebra CD . Proposition 4.5. The ∗-isomorphism π maps C(S) into C(T) ⊗ CD . Proof. Let C0 be the unital ∗-subalgebra of C(S) generated by all of the functions u , where u ∈ Z. Let u1 , u2 ∈ S such that u1 = u2 . If we set v = u1 − u2 , then v (u1 ) = v (u2 ). Thus, C0 separates the points of S and by the Stone–Weierstrass theorem, C0 is dense in C(S). By Proposition 4.3, π( u ) = z ⊗ A∗u + 1 ⊗ Bu∗ ∈ C(T) ⊗ CD , for all u ∈ Z. Therefore, π(C0 ) ⊆ C(T) ⊗ CD and the result follows by density. 2 5. Toeplitz operators and their index theorem We keep the notation of the previous section. If ϕ ∈ C(S), then we define the Toeplitz operator Wϕ with symbol ϕ as Wϕ (g) = PD (ϕg)
g ∈ H 2 (S) .
We define the Toeplitz algebra TD related to the tube-type domain D as the C∗ -algebra generated by all these Toeplitz operators. In this section we prove an index theorem for these operators using the general index theory introduced by Murphy in [6]. Let B be a unital C∗ -algebra. An indicial triple for B is an ordered triple Ω = (L, F, tr) such that • L is a unital C∗ -algebra and B is a unital C∗ -subalgebra of L, • F is a self-adjoint unitary in L,
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• tr is a lower semicontinuous trace on L, • the set {ϕ ∈ B | [F, ϕ] = F ϕ − ϕF ∈ Mtr } is dense in B, where Mtr is the definition ideal of the trace tr. We will denote the closure of Mtr by Ktr . The commutator [F, ϕ] will always be denoted by dϕ. The unital dense ∗-subalgebra {ϕ ∈ B | dϕ ∈ Mtr } will be denoted by BΩ . Let LT be the C∗ -subalgebra of B(L2 (T)) generated by C(T) and the projection PT . Then the triple ΩT = (LT , FT , trT ) is an indicial triple for C(T), where FT = 2PT − 1 and trT is the restriction of the canonical trace on B(L2 (T)) to LT ; see Example 2.7 in [6]. By Proposition 4.5, π(C(S)) ⊆ LT ⊗ CD . We denote the unital C∗ -algebra LT ⊗ CD by LD . From now on, we will ignore the ∗-isomorphism π and we will consider that C(S) is a unital C∗ -subalgebra of LD . By Proposition 4.3, PD ∈ LD . Define the lower semicontinuous trT ⊗ τ on LD by the relation (trT ⊗ τ )(a) = trT (ι ⊗ τ )(a) for a ∈ L+ D, where τ is the character on the C∗ -algebra CD in Proposition 4.4. Let a ∈ MtrT . Then, by [7, + Proposition A1(a)], |a| ∈ M+ trT . Thus, |a ⊗ b| = |a| ⊗ |b| ∈ MtrT ⊗τ for all b ∈ CD . Again by the same proposition we get a ⊗ b ∈ MtrT ⊗τ . By density it follows that KtrT ⊗ CD ⊆ KtrT ⊗τ . Now we define the lower semicontinuous trace trD on LD by the equality trD (a) =
(trT ⊗ τ )(a), +∞,
a ∈ (KtrT ⊗ CD )+ ,
+ a ∈ L+ D \(KtrT ⊗ CD ) .
Moreover, KtrD = KtrT ⊗ CD . The following proposition shows that the trace trD defines an indicial triple for C(S). Proposition 5.1. Let FD = 2PD − 1. Then the triple ΩD = (LD , FD , trD ) is an indicial triple for C(S). Proof. Let BΩD be the unital ∗-subalgebra of all elements ϕ ∈ C(S) such that dϕ ∈ MtrD . By Example 2.7 in [6], d u = [FT , z] ⊗ A∗u ∈ MtrT ⊗ CD for all u ∈ Z. Thus, d u ∈ MtrD . This implies that u ∈ BΩD for all u ∈ Z. Hence, BΩD contains the unital dense ∗-subalgebra C0 defined in the proof of Proposition 4.5. This implies that BΩD is dense in C(S). Therefore, ΩD is an indicial triple for C(S). 2 A topological index on a unital C∗ -algebra B is a locally constant homomorphism from Inv(B), the group of invertible elements in B, to the additive group R. By [6, Theorem 2.6], if Ω = (L, F, tr) is an indicial triple for B, then there is a unique topological index ω on B such that ω(ϕ) = 12 tr(ϕ −1 dϕ), for all ϕ ∈ Inv(BΩ ). Let ωD be the unique topological index on C(S) related to the indicial triple ΩD . Proposition 5.2. Let ϕ be a non-vanishing function on S. Then ωD (ϕ) = k, where k is the unique integer such that ϕ = N k eψ for some ψ ∈ C(S).
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Proof. First we calculate ωD (N ), 1 1 trD N −1 dN = trD z−1 ⊗ 1 (dz ⊗ 1) 2 2 1 1 = (trT ⊗ τ ) z−1 dz ⊗ 1 = trT z−1 dz τ (1) = ωT (z), 2 2
ωD (N ) =
where ωT is the unique topological index on C(T) related to the indicial triple ΩT . Hence, by Example 2.7 in [6], ωD (N ) = 1. By Theorem 3.3, there exists a unique integer k such that ϕ = N k eψ , where ψ ∈ C(S). By elementary properties of topological indices, we have ωD (ϕ) = ωD N k + ωD eψ = k · 1 + 0 = k. Thus, ωD (ϕ) = k.
2
It is obvious that the value of the topological index ωD is independent of the choice of the character τ . Moreover, by Remark 3.4, it is independent of the choice of the generic norm N . Suppose that Ω = (L, F, tr) is an indicial triple for B. Let ϕ ∈ B. The element Tϕ = P ϕP is called a Toeplitz element associated to Ω, where P = (F + 1)/2. Let A be the C∗ -subalgebra generated by the Toeplitz elements associated to Ω. This unital algebra is called the Toeplitz algebra associated to Ω. An element a ∈ A is called an Ω-Fredholm element if it is invertible modulo KTr , where the trace Tr is the restriction of tr to A. If a ∈ A is an Ω-Fredholm element, then there exists b ∈ A such that P −ab, P −ba ∈ MTr and the Ω-Fredholm index of a is defined as index(a) = Tr(ab − ba). The Ω-Fredholm index is a well-defined function. This function is real-valued and it has algebraic properties similar to the classical Fredholm index; see [5,6]. Now by Proposition 5.2 and [6, Theorem 3.1], we have the following immediate result. Theorem 5.3. Let ϕ be a non-vanishing function on S. Then Tϕ is an ΩD -Fredholm element and indexD (Tϕ ) = −k, where k is the unique integer such that ϕ = N k eψ for some ψ ∈ C(S) and indexD is the ΩD -Fredholm index. For any C∗ -algebra A, we denote the closed ideal generated by all of the commutators in A by Com(A) and we will call it the closed commutator ideal of A. We denote the ideal of compact operators on H 2 (T) by K(H 2 (T)). Let AD be the Toeplitz algebra related to the indicial triple ΩD . The algebra AD is ∗isomorphic to the Toeplitz algebra TD of Toeplitz operators on the tube-type domain D defined in the beginning of this section and the ∗-isomorphism TD → AD is defined by a → aPD . By [9, Theorem 4.11.76], Com(TD ) = K(H 2 (T)) ⊗ CD . The latter ∗-isomorphism maps this ideal onto Com(AD ) = PT K(L2 (T))PT ⊗ CD . By Lemma 3.1 in [1], K(L2 (T)) = KtrT . Hence, Com(AD ) = PT KtrT PT ⊗ CD . Theorem 5.4. If ϕ ∈ C(S), then the Toeplitz element Tϕ is ΩD -Fredholm if and only if ϕ is invertible. Proof. If a ∈ KTrD , then a ∈ KtrD = KtrT ⊗ CD . By density, we can choose a sequence (an )n1 in the algebraic tensor product KtrT CD such that an → a. Thus, PD an PD ∈ PT KtrT PT CD . Therefore, a = PD aPD ∈ PT KtrT PT ⊗ CD , since PD an PD → PD aPD . Thus, KTrD ⊆ Com(AD ).
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By [8, Theorem 3.11], there exists a ∗-homomorphism : AD → C(S) such that (Tϕ ) = ϕ, for all ϕ ∈ C(S), and ker() = Com(AD ). Now if Tϕ is an ΩD -Fredholm element, then there is an element S ∈ AD such that both of the elements PD − STϕ and PD − Tϕ S belong to the ideal KTrD and it follows immediately that (S)ϕ = ϕ(S) = 1. Hence, ϕ is an invertible function. The opposite direction follows from Theorem 5.3. 2 Corollary 5.5. If ϕ be a non-vanishing function on S, then indexΩ (Tϕ ) = 0 if and only if ϕ = eψ , for some ψ ∈ C(S). Now we extend our results to Toeplitz operators with M × M matrix symbols defined on the Shilov boundary of an irreducible tube-type domain. These operators act on the Hilbert 2 (S) = H 2 (S) ⊕ · · · ⊕ H 2 (S). The space L2 (S) is defined similarly and we denote the space HM M M times
2 (S) by P . Let C be the space of continuous M × M orthogonal projection of L2M (S) onto HM M M matrix-valued functions on S. Then CM = MM (C(S)). The Toeplitz operator with symbol Φ ∈ 2 (S). These Toeplitz operators can be obtained CM is defined by as TΦ (f) = PM (Φf) for all f ∈ HM LD ⊗ MM (C), from the indicial triple ΩM = (LM , FM , trM ) for CM , where LM = MM (LD ) = FM = FD ⊗ 1M = 2PM − 1 and the trace trM is defined by trM ([aj k ]) = M j =1 trD (ajj ), ; see [6, Theorem 2.9]. The following is the index theorem for Toeplitz operators [aj k ] ∈ L+ M with M × M matrix symbols on an irreducible tube-type domain. The proof of the following theorem is similar to that of Theorem 3.7 in [1], hence, we will not write it.
Theorem 5.6. Let Φ ∈ CM . Then TΦ is an ΩM -Fredholm operator if and only if det Φ is nonvanishing on S and in this case we have indexM (TΦ ) = −k, where k is the unique integer such that det Φ = N k eψ , for some continuous function ψ on S and indexM is the ΩM -index mapping. References [1] A.B. Badi, An index theorem for Toeplitz operators on the quarter-plane, Proc. Amer. Math. Soc. 137 (2009) 3779– 3786. [2] H. Bohr, Uber die Argumentvariation einer Fastperiodischen Funktion, Danske Vidensk. Selskab. X (10) (1930). [3] O. Loos, Bounded Symmetric Domains and Jordan Pairs, Math. Lectures, University of California at Irvine, Irvine, 1977. [4] J.R. Munkres, Topology, a First Course, Prentice Hall, Inc., 1975. [5] G.J. Murphy, Fredholm index theory and the trace, Proc. R. Ir. Acad. Sect. A 94 (1994) 161–166. [6] G.J. Murphy, Topological and analytical indices in C∗ -algebras, J. Funct. Anal. 234 (2006) 261–276. [7] J. Phillips, I. Raeburn, An index theorem for Toeplitz operators with noncommutative symbol space, J. Funct. Anal. 120 (1994) 239–263. [8] H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983) 221–237. [9] H. Upmeier, Toeplitz Operators and Index Theory in Several Complex Variables, Birkhäuser, 1996. [10] E. van Kampen, On almost periodic functions of constant absolute value, J. London Math. Soc. 12 (1937) 3–6.
Journal of Functional Analysis 258 (2010) 3855–3878 www.elsevier.com/locate/jfa
Quasi-hyperbolic semigroups ✩ Charles J.K. Batty a,∗ , Yuri Tomilov b,c a St. John’s College, Oxford OX1 3JP, England, United Kingdom b Faculty of Mathematics and Informatics, Nicholas Copernicus University of Toru´n, 87-100 Toru´n, Poland c Institute of Mathematics, Polish Academy of Sciences, Sniadeckich str. 8, 00-956 Warsaw, Poland
Received 26 October 2009; accepted 7 January 2010 Available online 18 January 2010 Communicated by N. Kalton
Abstract We introduce the notion of quasi-hyperbolic operators and C0 -semigroups. Examples include the pushforward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C0 -semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property. © 2010 Elsevier Inc. All rights reserved. Keywords: Semigroup; Operator; Lower bound; Expansive; Quasi-hyperbolic; Approximate point spectrum; Push-forward; Anosov
1. Introduction The notion of hyperbolicity of operators and operator semigroups is central in differentiable dynamics and the theory of differential equations. For a general view on the subject one may consult [4] and [14]. Recall that a bounded linear operator T on a Banach space X is said to be hyperbolic if there is a splitting X = Xs ⊕ Xu where Xs and Xu are closed T -invariant subspaces of X, T Xu is ✩
This work was completed with the support of the Marie Curie Transfer of Knowledge program, project “TODEQ”. The second author was partially supported by the MNiSzW grant N201384834. * Corresponding author. E-mail addresses:
[email protected] (C.J.K. Batty),
[email protected] (Y. Tomilov). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.005
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invertible, and (T Xs )n 12 and (T Xu )−n 12 for some n ∈ N. Then Xs and Xu are the stable and unstable parts of X for T . It is a standard result that T is hyperbolic if and only if the spectrum σ (T ) of T does not meet the unit circle T. Hyperbolicity for a C0 -semigroup (T (t))t0 on X can be defined in a similar way and it is equivalent to hyperbolicity of T (1); see Section 3 below. If A is the generator of a C0 -semigroup (T (t))t0 on X then for a variety of Banach spaces F of X-valued functions on R the existence and uniqueness of mild solutions to x(t) ˙ = Ax(t) + f (t)
(t ∈ R)
(1.1)
in F for every f ∈ F is equivalent to (T (t))t0 being hyperbolic, see for example [4, Chapters 4–5] and [26, Chapter 8]. Similar results concerning discrete versions of (1.1) can be found in [4, Chapters 4–5] and [32]. The interplay between solvability of (1.1) and hyperbolicity of the associated semigroups has motivated intensive studies of hyperbolic semigroups and exponentially dichotomic evolution families and their relationship to the properties of solutions to (1.1) and fine spectral structure of d the associated differential operator − dt + A. Recent accounts of this area include, in particular, [4, Chapters 4–5], [2,3,19,22–24,37,38]. Moreover, hyperbolic operator semigroups arise very naturally in dynamical systems theory, in particular in the study of Anosov flows and diffeomorphisms. Let M be a compact smooth Riemannian manifold, and let TM be its tangent bundle. Denote by C(TM) the real Banach space of continuous sections of the tangent bundle TM with the supremum norm. With any diffeomorphism ϕ : M → M one can associate a linear operator Eϕ on C(TM), the so-called push-forward operator, defined by (θ ∈ TM), (Eϕ f )(θ ) = Dϕ ϕ −1 θ f ϕ −1 θ where Dϕ stands for the differential of ϕ. Recall that ϕ is said to be Anosov if the tangent bundle TM splits as a direct sum of Dϕ-invariant sub-bundles TM = TMs ⊕ TMu such that Dϕ contracts TMs exponentially for positive times and contracts TMu exponentially for negative times. Anosov diffeomorphisms constitute one of the basic classes of mappings in differentiable dynamics possessing numerous specific features such as structural stability [14]. Consider the complexification of C(TM) and the corresponding extension of Eϕ to the complexification, for which we shall use the same notation. Mather’s famous characterisation of Anosov diffeomorphisms obtained in [29, Theorem] says that ϕ is Anosov if and only if 1∈ / σ (Eϕ ). As noted in [29, Lemma] (see also [4, Lemma 6.28]), for aperiodic ϕ the spectrum of Eϕ is invariant with respect to rotations. Thus, since Anosov diffeomorphisms are aperiodic, Mather’s result can be restated as in [29, Theorem]: ϕ is Anosov if and only if Eϕ is hyperbolic. Answering a question of Hirsch concerning invariant hyperbolic submanifolds, Mañé introduced the notion of a quasi-Anosov diffeomorphism, that is a diffeomorphism ϕ : M → M such that for every θ ∈ M and every non-zero x from the tangent space Tθ M of M the set {Dϕ n (θ )x: n ∈ Z} is unbounded. It is easy to show that every Anosov diffeomorphism is quasi-Anosov but the converse does not hold in general [11]. Mañé proved in [28, Theorem A] (see also [8, Theorem]) that ϕ is quasi-Anosov if and only if the approximate spectrum σap (Eϕ ) of Eϕ does not contain 1. Since quasi-Anosov diffeomorphisms are aperiodic, by the rotational symmetry of σap (Eϕ ) for such ϕ (see the proof of [29, Lemma] and [4, Lemma 6.28]), we infer
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as above that ϕ is quasi-Anosov if and only if σap (Eϕ ) is disjoint from the unit circle T. Similar results hold in the context of smooth flows on Riemannian manifolds where the notion of a quasi-Anosov flow and the associated Mather (evolution) semigroup can be defined and studied similarly to the discrete setting. We refer to [5,6,17,41] for early developments of the operator approach to the study of dynamics of flows, and to the more recent book [4, Chapters 6, 7] for more general abstract results. The Mather–Mañé spectral theory can be extended to the broader framework of Banach space valued cocycles over flows. Let Θ be a locally compact metric space and {ϕ t }t∈R be a continuous flow on Θ. If {Φ t }t0 is a strongly continuous, exponentially bounded, L(X)-valued cocycle over {ϕ t }t∈R then one can define the corresponding Mather (or evolution) C0 -semigroup (Eϕt )t0 on C0 (Θ, X) by t (θ ∈ Θ). Eϕ f (θ ) = Φ t ϕ −t θ f ϕ −t θ
(1.2)
Note that in general (Eϕt )t0 does not extend to a group. Hyperbolicity of (Eϕt )t0 is equivalent to invertibility of the generator of (Eϕt )t0 and is also equivalent to exponential dichotomy of {Φ t }t∈R —see [4, Chapters 6 and 7] for the details of these notions and results. In particular, by [4, Theorem 6.30] if {ϕ t }t∈R is aperiodic and Gϕ is the generator of (Eϕt )t0 then σap Eϕt \ {0} = etσap (Gϕ )
(t > 0).
(1.3)
An analogous spectral mapping theorem is true for Mather semigroups (E t )t0 defined on appropriate Lp -spaces (see [4, Theorem 6.37]). The spectral theory of Mather semigroups including the relation (1.3) turned out to be crucial for the study of dynamics of various partial differential equations such as Euler’s equations—see [4, Chapter 8] and the more recent papers [24,39,40]. While hyperbolic Mather semigroups are well understood and the hyperbolic dynamics of Anosov diffeomorphisms and flows translate into hyperbolic dynamics of associated discrete or continuous operator semigroups on a Banach space of continuous sections, the situation with quasi-Anosov diffeomorphisms and flows is not so explicit at least from an operator-theoretic point of view. Thus we arrive at a need to generalise the notion of hyperbolicity for linear operators. In fact Eisenberg and Hedlund [9,15] introduced a relevant class of operators, without indicating any applications. Let T be an invertible bounded linear operator on a Banach space X. According to [9] the operator T is expansive if for each x ∈ X there exists n ∈ Z (depending on x) such that T n x 2x; and T is uniformly expansive if there exists n ∈ N (independent of x) such that max(T n x, T −n x) 2x for all x ∈ X. Uniform expansiveness of T was characterised in [15, Theorem 1] (see also [9, Theorem 2]) by the condition σap (T ) ∩ T = ∅.
(1.4)
In the case of the push-forward operator T = Eϕ , (1.4) clearly coincides with Mañé’s spectral characterisation of quasi-Anosov diffeomorphisms ϕ. We extend this notion to bounded linear operators T on X which are not necessarily invertible and we introduce a more suggestive terminology as follows. Definition 1.1. We shall say that T is quasi-hyperbolic if there exists n ∈ N (independent of x) such that max(T 2n x, x) 2T n x for all x ∈ X.
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If T is quasi-hyperbolic, then there exist a > 0 and c > 0 such that max(T 2n x, x) cean T n x for all x ∈ X and all n ∈ N. Moreover, for each x ∈ X, the sequence (T n x)n∈N either diverges exponentially or it decays exponentially. It is clear that any hyperbolic operator is quasi-hyperbolic. Moreover, an invertible operator is uniformly expansive if and only if it is quasi-hyperbolic. The push-forward operator associated with a quasi-Anosov diffeomorphism is a quasi-hyperbolic operator. Quasi-hyperbolic operators appear also in the theory of hyperbolic partial differential equations (see Example 2.3). Many weighted shifts are quasi-hyperbolic (see Example 2.6). While quasi-hyperbolic operators have arisen naturally in applications, no systematic treatment of quasi-hyperbolicity has been made so far. The aim of the paper is to fill the gap in the literature and to pursue the study of quasi-hyperbolicity mainly in the context of C0 -semigroups on Banach spaces. Our approach is based on a deep extension theorem of Read and Müller for operators, a new notion of a lower Fourier multiplier and the techniques of evolution semigroups. The paper can be considered as an extension of [21,20,18] where hyperbolic C0 -semigroups on Banach spaces were characterised in terms of resolvents of their generators. The paper is organised as follows. Section 2 is devoted to the study of quasi-hyperbolicity for discrete semigroups, i.e. iterates of a single operator. Quasi-hyperbolic operators are identified there as restrictions of hyperbolic operators to subspaces. This is a striking analogue of a part of [28, Theorem A] identifying quasi-Anosov diffeomorphisms with restrictions of diffeomorphisms to invariant submanifolds with hyperbolic structure. In Section 3 we characterise quasi-hyperbolic C0 -semigroups in terms of their generators and lower Fourier multipliers. In particular we prove in Corollary 3.10 that quasi-hyperbolicity of a C0 -semigroup (T (t))t0 on a Hilbert space X with generator A is equivalent to the lower bound (A − is)x cx s ∈ R, x ∈ D(A) , (1.5) for some c > 0, where D(A) is the domain of A. However the simple condition (1.5) fails to characterise hyperbolicity of semigroups on general Banach spaces as our Example 3.3 shows, and the lower Fourier multiplier property is difficult to verify in Banach spaces. Thus in Section 4 we study what kind of asymptotic properties of (T (t))t0 can be deduced from (1.5) if X is merely a Banach space. We show that (1.5) ensures that non-zero complete trajectories of (T (t))t0 grow faster than polynomially in a pointwise sense (Theorem 4.1) and they grow exponentially in an integral norm (Theorem 4.5). It remains an open question whether they grow exponentially in a pointwise sense. 2. Quasi-hyperbolic operators In order to give a spectral characterisation of quasi-hyperbolic operators, we recall an extension theorem. The appropriate construction was first carried out by Read in the context of Banach algebras [34]. Later Read adapted it to operators [35], while Müller independently exhibited a neat way to deduce the operator case from the algebra case (see [31, Theorem 9.22]). Theorem 2.1 (Read). Let T be a bounded linear operator on X. There is a Banach space Y and a bounded operator S on Y such that X is isometrically embedded in Y , SX = T , S = T and σ (S) = σap (T ). In the following result the equivalence of (i) and (iii) for invertible T was proved by Hedlund [15]. Read’s Theorem allows us to introduce the equivalent condition (iv) which makes the prop-
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erties of quasi-hyperbolic operators rather transparent. It also provides an alternative proof of the non-trivial part of Hedlund’s result. Theorem 2.2. Let T be a bounded linear operator on X. The following are equivalent: (i) (ii) (iii) (iv)
T is quasi-hyperbolic. There exists c > 0 such that (T − λ)x cx for all x ∈ X and λ ∈ T. σap (T ) ∩ T = ∅. There is a Banach space Y and a hyperbolic operator S on Y such that X is isometrically embedded in Y and SX = T .
In condition (iv) one may arrange that S = T and σ (S) = σap (T ). Proof. (i) ⇒ (ii). Suppose that max(T 2n x, x) 2T n x for all x ∈ X, and (ii) is false. We aim to obtain a contradiction. There are sequences (xk )k1 in X and (λk )k1 in T such that xk = 1 for all k and T xk − λk xk → 0. Passing to a subsequence, we may assume that λk → λ ∈ T. Then T 2n xk − λ2n xk → 0 and T n xk − λn xk → 0, so T 2n xk → 1 and T n xk → 1. But max T 2n xk , xk 2T n xk . Letting k → ∞ gives a contradiction. It is immediate from Read’s Theorem that (iii) implies (iv), while (ii) ⇒ (iii) is trivial and (ii) ⇒ (i) is elementary. 2 Example 2.3. The following hyperbolic equation with periodically moving boundary has been studied in [7]. Let sin(πt) Ω := (x, t): 0 < x < 1 + ⊆ R2 . 2π The boundary value problem utt − uxx = 0 in Ω,
(2.1)
u = 0 on ∂Ω,
(2.2)
is well posed for data u(·, 0), ut (·, 0) in the energy space H = H01 (0, 1) × L2 (0, 1), so that the monodromy operator U (2, 0) : H → H, U (2, 0) u(·, 0), ut (·, 0) = u(·, 2), ut (·, 2) , is well defined. Moreover, U (2, 0) is invertible in H, and by [7, p. 304 and Theorem 3.4] √ 1 σ U (2, 0) = λ ∈ C: √ |λ| 3 , 3
σap U (2, 0) ∩ T = ∅.
(2.3)
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By Theorem 2.2, U (2, 0) is quasi-hyperbolic and it follows that there exist a > 0 and c > 0 such that for every non-zero solution u of (2.1) one has either u(t)H ceat
for all t > 0 or u(t)H cea|t|
for all t < 0.
Thus the energy of the system governed by (2.1), (2.2) grows exponentially in either positive or negative time for any choice of non-zero initial values. In fact, as noted in [7, p. 304], lim
t→±∞
log u(t)H ln 3 = . |t| 8
General statements describing fine structure of the spectrum of monodromy operators for (2.1), (2.2) on a large class of domains Ω can be found in [7, Theorem 3.4, Theorem 2.4] and in [27, Theorem 1]. Other examples of quasi-hyperbolic operators on Hilbert space that are not hyperbolic were given in [9, Examples 4, 5] (see Example 2.6). We shall show that those examples are particular cases of a large class of quasi-hyperbolic operators. Example 2.4. Let w : Z → R be a (strictly) positive bounded sequence. Consider the weighted right shift operator Swr on lp (Z) (1 p < ∞), defined by Swr (xn )n∈Z = w(n)xn−1 n∈Z .
(2.4)
Let un =
w(0)w(1) · · · w(n − 1) 1 (w(n) · · · w(−1))−1
(n > 0), (n = 0), (n < 0).
Define
1/n i + Swr = lim inf un+k /uk , n→∞ k>0
r + Swr = lim
sup un+k /uk
n→∞ k>0
1/n
,
1/n i − Swr = lim inf uk /uk−n , n→∞ k<0
1/n r − Swr = lim sup uk /uk−n . n→∞ k<0
Then the spectral radius of Swr is r Swr = max r − Swr , r + Swr , and the inner spectral radius is i Swr := min |λ|: λ ∈ σ Swr = min i + Swr , i − Swr . If Swr is invertible then i(Swr ) = r((Swr )−1 )−1 .
(2.5)
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The following result was proved by Ridge [36, Theorem 3] for the shifts defined on l2 (Z). Its extension to lp (Z)-spaces with p ∈ [1, +∞) presents no difficulties (see [30] and [25, Section I.6]). Proposition 2.5 (Ridge). Let Swr be the shift operator defined in (2.4). Then the following hold. (a) σ (Swr ) = {λ ∈ C: i(Swr ) |λ| r(Swr )}. (b) If r − (Swr ) < i + (Swr ) then σap (Sw ) = λ ∈ C: i − Swr |λ| r − Swr ∪ λ ∈ C: i + Swr |λ| r + Swr . Otherwise, σap Swr = σ Swr . It follows from Proposition 2.5 and Theorem 2.2 that Swr is quasi-hyperbolic, but not hyperbolic, if and only if r − (Swr ) < 1 < i + (Swr ). In particular, if lim w(n) = w±
n→±∞
and w+ > w− ,
then
σap Swr = λ ∈ C: |λ| = w− or |λ| = w+ , and Swr is quasi-hyperbolic, but not hyperbolic, if and only if w− < 1 < w+ . There is a corresponding result for weighted left shifts Swl on p (Z). Define the invertible ˜ = w(−n). Then isometry J : lp (Z) → lp (Z) by J ((xn )n∈Z ) = ((x−n )n∈Z ), and let w(n) Swl (xn )n∈Z = w(n)xn+1 n∈Z = J −1 Swr˜ J (xn )n∈Z . Thus the approximate point spectrum of Swl coincides with that of Swr˜ which allows a description by means of Proposition 2.5. Examples 2.6. 1. A particular case of the weighted shift operator Swr with √ 2 2 (n 0), w(n) = 1 √ (n < 0), 2 2
was considered in [9, Example 4]. 2. We show here that [9, Example 5] can be put into the framework of weighted shifts discussed above. Let A be the annulus {z ∈ C: 12 < |z| < 2} and X be the Hilbert space of all holomorphic functions belonging to the space L2 (A), where A is equipped with planar Lebesgue measure. Let M : X → X be defined by (Mf )(z) = zf (z). It was shown in [13] that 1 1 σap (M) := λ ∈ C: |λ| = or |λ| = 2 . (2.6) σ (M) := λ ∈ C: |λ| 2 , 2 2 Thus, M is quasi-hyperbolic, by Theorem 2.2 or by a direct argument given in [9].
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If en := zn /zn (n ∈ Z), then {en : n ∈ Z} is an orthonormal basis in L2 (A), and moreover Men = wn en+1 ,
−1 where wn := zn+1 zn (n ∈ Z).
Thus M can be identified with the right shift Swr on l2 (Z). By an easy calculation lim wn = 2,
n→∞
1 lim wn = , n→−∞ 2
so we get (2.6) by Ridge’s results (Proposition 2.5). In the spirit of Theorem 2.2(iii) and with a view towards Section 4, given an operator T on X, we would like to characterise the property that (HE) X can be continuously embedded in a Banach space Y such that T = SX for some quasihyperbolic operator S on Y . It follows from Theorem 2.2 that we can assume that S is hyperbolic, and (HE) can be reformulated as follows: (HE) There is a norm · on X and there are constants c > 0 and C such that, for each x ∈ X, (a) x Cx, (b) T x Cx , (c) (T − λ)x cx for all λ ∈ T. If Y and S exist, then we can take · to be the norm of Y . Conversely, if · exists, then we can take Y to be the completion of (X, · ). We aim now to limit the search to a narrow class of norms. A necessary condition for the existence of such a pair (Y, S) is the following property of uniform exponential growth in a bilateral sense: (UE) There exists α > 1 such that, for each x ∈ X \ {0} there exists cx > 0 such that either (a) T n x cx α n for all n ∈ N, or (b) y cx α n whenever n ∈ N, y ∈ X and T n y = x. This implies each of the following two conditions concerning firstly a notion similar to that of an expansive operator, and secondly the point spectrum: (EX) If x ∈ n∈N Ran T n , then there exists n ∈ N such that either T n x 2x or y 2x for each y ∈ X with T n y = x (so, T is expansive if T is invertible); (PS) σp (T ) ∩ T = ∅, where σp (T ) is the point spectrum of T . There are simple examples of invertible operators on Hilbert space satisfying (EX) but not (PS) (a diagonal operator on 2 (N) as in [9, Example 2]), and vice versa (an unweighted shift on both (EX) and 2 (Z)), and also an example of an invertible operator on Hilbert space satisfying (PS) but not (UE) (the shift Swr on 2 (Z) where w(n) 1 for all n ∈ Z, ∞ w(n) = ∞ and n=0 lim|n|→∞ w(n) = 1).
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Let us assume (for simplicity) that T is surjective. For α > 1 and x ∈ X, define y n qα− (x) = inf : y ∈ X, n ∈ N, T y = x . αn
T n x , n∈N α n
qα+ (x) = inf
Note that (UE) is equivalent to max qα+ (x), qα− (x) > 0 x ∈ X \ {0} . However, the functionals qα+ and qα− may not satisfy the triangle inequality, so it is not clear whether (UE) implies (HE) . Instead we define pα+ (x) = inf pα− (x) = inf
n
qα+ (xr ): n ∈ N, xr ∈ X, x =
n
r=1
r=1
n
n
qα− (xr ): n ∈ N, xr ∈ X, x =
r=1
pα (x) = max pα+ (x), pα− (x) .
xr , xr ,
r=1
It is routine to verify that pα+ , pα− and pα are seminorms on X, and the following inequalities hold pα+ (x) x, pα+ (T x) T pα+ (x), qα+ (T x) αqα+ (x),
pα− (x) x, pα− (T x) T pα− (x),
(2.7)
pα− (x) αpα− (T x).
(2.8)
The assumption that T is surjective ensures that pα+ (T x) αpα+ (x).
(2.9)
Theorem 2.7. Let T be a bounded surjection on X. The following are equivalent: (i) There exists α > 1 such that pα is a norm on X. (ii) There exist a Banach space Y and a quasi-hyperbolic operator S on Y such that X is continuously embedded in Y and T = SX . (iii) There exist a Banach space Z and a hyperbolic operator U on Z such that X is continuously embedded in Z and T = UX . Proof. It is immediate from Theorem 2.2 that (ii) implies (iii). Suppose that (iii) holds. Each x ∈ X can be written uniquely as x = zx+ + zx− where zx+ and zx− are respectively in the unstable and stable parts of Z for U . Hence there are constants C, c > 0 and α > 1, independent of x, such that zx+ Z + zx− Z CxX and
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U n zx+ Z cα n zx+ Z and zZ cα n zx− Z if n 0, z ∈ Z and U n z = zx− . Moreover, z(T n x)+ = U n zx+ . Hence n n T x 1 U n zx+ cα zx+ Z . X Z C C
It follows that q+ (x)
c C zx+ Z .
If y ∈ X and T n y = x, then U n zy− = zx− and
yX
1 cα n zy− Z zx− Z . C C
It follows that q− (x) Cc zx− Z . If x = nr=1 xr , then n
q+ (xr )
r=1
n c c zxr + Z zx+ Z . C C r=1
Hence p+ (x) Cc zx+ Z . Similarly, p− (x) Cc zx− X . If x = 0, then either zx+ = 0 or zx− = 0, and it follows that p+ (x) = 0 or p− (x) = 0. This establishes (i). Now suppose that (i) holds. Let Y be the completion of (X, pα ). It follows from (2.7) that T extends to a bounded linear operator S on Y . Let x ∈ X. For n 0, it follows from (2.9) and (2.8) that 2n S x = pα T 2n x pα+ T 2n x α n pα+ T n x , Y xY = pα (x) pα− (x) α n pα− T n x . Thus, max S 2n x Y , xY α n pα T n x = α n S n x Y . It now follows by density of X in Y that S is quasi-hyperbolic on Y .
2
3. Quasi-hyperbolic semigroups Now we consider the corresponding notions for a C0 -semigroup T := (T (t))t0 on a Banach space X. It is standard terminology that T is hyperbolic if there is a splitting X = Xs ⊕ Xu where Xs and Xu are closed T -invariant subspaces of X, T (t)Xu is invertible for some (or all) t > 0, and T (t)Xs < 1 and (T (t)Xu )−1 < 1 for some t > 0. Definition 3.1. We say that T = (T (t))t0 is quasi-hyperbolic if there exists t > 0 (independent of x) such that max(T (2t)x, x) 2T (t)x for all x ∈ X. It is easy to see that T is quasi-hyperbolic or hyperbolic if and only if T (1) is quasi-hyperbolic or hyperbolic, respectively. If T is quasi-hyperbolic, then, by Theorem 2.2, T (1) can be extended to a hyperbolic operator S(1) on a larger space Y . It can be seen from the construction in [35], or more easily from that in the proof of [31, Theorem 9.22], that each operator T (t) also extends to an operator S(t) on Y with S(t) = T (t) and S(t1 )S(t2 ) = S(t1 + t2 ). Replacing
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Y if necessary by the maximal subspace on which S is strongly continuous, we can arrange that (S(t))t0 is a C0 -semigroup on Y , and then it is hyperbolic. Hence quasi-hyperbolic and hyperbolic C0 -semigroups can be characterised in terms of spectral properties of T (1) as in Section 2. It would be more useful to have results involving spectral properties of the generator A of T , but the failure of the spectral mapping theorem for C0 -semigroups means that this is not straightforward. The natural analogue of Theorem 2.2(ii) for A is the lower bound (1.5). One simple general result is the following. Proposition 3.2. Let A be the generator of a quasi-hyperbolic C0 -semigroup (T (t))t0 on a Banach space X. Then (1.5) holds. Proof. Let t be such that max(T (2t)x, x) 2T (t)x for all x ∈ X. For x ∈ D(A) and s ∈ R, t
e−isτ T (τ )(A − is)x dτ = e−ist T (t)x − x.
(3.1)
0
Hence tMt (A − is)x T (t)x − x, where Mt = sup|τ |t T (τ ). Replacing x by T (t)x in (3.1), tM2t (A − is)x T (2t)x − T (t)x . Let c = (2tM2t )−1 . Then (A − is)x 2c max T (t)x − x, T (2t)x − T (t)x 2cT (t)x 2c x − T (t)x − x 2 2cx − (A − is)x . The converse of Proposition 3.2 does not hold, even for a C0 -group of positive operators on a reflexive Banach lattice. This is shown by the following example of a C0 -group with bad spectral properties, adapted from an example originally due to Wolff [43] (see [12, pp. 102–103]). Example 3.3. Let X be the reflexive Banach space Lp (R, e2x dx) ∩ Lq (R, w(x) dx), where 1 < p < 2 < q < ∞, and w(x) =
eax 1
(x 0), (x > 0),
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for some a > 2q/p. Here, f X =
f (x)p e2x dx
1/p
+
R
f (x)q w(x) dx
1/q .
R
Let (T (t)f )(s) = f (s + t) (s, t ∈ R). Then (T (t))t∈R is a C0 -group on X, T (t) = 1 (t 0), σ (A) ∩ iR = ∅ and sups∈R (is − A)−1 < ∞ [12]. Hence, (LB) holds. However, for t > 0, there exists f ∈ X such that f X = 1 and T (−t)f X < 2f X (because T (t) > 1/2). Moreover T (t)f X < 2f X (because T (t) < 2). So T (t) is not quasi-hyperbolic. Note that if Y = Lp (R, e2x dx) then X is continuously and densely embedded in Y , and the C0 -group (S(t))t∈R of shifts on Y is hyperbolic with trivial unstable part because S(t) = e−2t/p (t ∈ R). Moreover, S(t)X = T (t). Thus the continuous-parameter analogue of (HE) holds in this example. We shall now characterise quasi-hyperbolicity of C0 -semigroups in terms of Fourier multiplier properties of linear pencils induced by semigroup generators. In particular this will establish that the converse of Proposition 3.2 holds on Hilbert spaces (Corollary 3.10). Let A be a densely defined, closed linear operator on a Banach space X. By S(R, X) denote the space of X-valued Schwartz functions. For ϕ ∈ S(R) and x ∈ D(A), write (ϕ ⊗ x)(t) = ϕ(t)x (t ∈ R). Let SA (R, X) := lin{ϕ ⊗ x: x ∈ D(A), ϕ ∈ S(R)} and F be the Fourier transform on S(R, X). Note that SA (R, X) is dense in Lp (R, X) for every p ∈ [1, +∞). Definition 3.4. The linear operator MA−i· : SA (R, X) → S(R, X), (MA−i· f )(s) = (A − is)f (s)
(s ∈ R),
is said to be a lower Lp (R, X)-Fourier multiplier if there exists c > 0 such that −1 F MA−i· F f
Lp
cf Lp
(3.2)
for every f ∈ SA (R, X). Note that −1 F MA−i· F f (s) = Af (s) − f (s).
(3.3)
Remark 3.5. If iR ⊂ ρ(A) and R(λ, A) = (λ − A)−1 denotes the resolvent of A then (3.2) is equivalent to the fact that the operator MR(i·,A) , defined on SA (R, X) by (MR(i·,A) f )(s) = R(is, A)f (s)
(s ∈ R),
is an Lp (R, X)-Fourier multiplier, i.e. that R(i·, A) is bounded from R to L(X) and F −1 MR(i·,A) F extends to a bounded linear operator on Lp (R, X). To see this, it is sufficient to show that (3.2) implies the boundedness of R(i·, A)—then the claim is apparent.
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Let x ∈ D(A) and s ∈ R. Choose ϕ ∈ S(R) with ϕLp = 1, and let ϕn (t) = e−ist n−1/p ϕ(t/n) (n ∈ N). Then (3.3) and (3.2) imply that −ϕ ⊗ x + ϕn ⊗ Ax = F −1 (A − i·)F (ϕn ⊗ x) cx. n L L p
p
By passing to the limit in the above inequality one gets (A − is)x cx x ∈ D(A), s ∈ R . Thus, R(i·, A) ∈ L∞ (R, L(X)) and (3.2) implies the claim. It is well known that MR(i·,A) is an Lp (R, X)-Fourier multiplier if and only if (T (t))t0 is hyperbolic, see [21, Theorem 2.7]. Our approach to the study of lower Fourier multipliers will be based on evolution semigroups (see [4]). The same technique was employed for the study of Fourier multiplier properties of resolvents in [20] (see also [21]). Recall that given a strongly continuous evolution family (U (t, s))ts0 on a Banach space X, the associated evolution semigroup (E(t))t0 on Lp (R, X) (1 p < ∞) is defined by (3.4) E(t)f (s) = U (s, s − t)f (s − t) f ∈ Lp (R, X), t 0, s ∈ R . A special case occurs when (U (t, s))ts0 is induced by a C0 -semigroup (T (t))t0 on X with generator A (see [4, Section 2.2]). Then (3.4) takes the form E(t)f (s) = T (t)f (s − t) f ∈ Lp (R, X), t 0, s ∈ R . In this case, (E(t))t0 is the product of two commuting semigroups on Lp (R, X): the multiplication semigroup (M(t))t0 , (M(t)f )(s) = T (t)f (s), and the shift semigroup (S(t))t0 , (S(t)f )(s) = f (s − t), with the generators D, Df = − df dt , and MA , MA f = Af (s), respectively defined on their maximal domains. Thus, by well-known criteria, the generator G of (E(t))t0 can be identified as the closure of the sum D + MA defined on I := D(D) ∩ D(MA ). In other words, I is a core for G. In the next lemma we diagonalise G on an appropriate domain contained in I . Lemma 3.6. Let G be the generator of the evolution semigroup (E(t))t0 on Lp (R, X), associated with an evolution family (U (t, s))ts0 on X, where 1 p < ∞. Then G = F −1 MA−i· FSA .
(3.5)
Proof. Observe that SA (R; X) is dense in D(G) and is (E(t))t0 -invariant. Thus, SA (R, X) is a core for G, i.e. GSA = G. On the other hand, by (3.3) for every f ∈ SA (R, X), Gf = F −1 MA−i· F f.
2
(3.6)
We shall now discuss the relation between the spectral properties of G, E(1) and certain weighted shift operators. Recall that Mξ−1 GMξ = −iξ + G,
Mξ−1 E(1)Mξ = e−iξ E(1)
(ξ ∈ R),
(3.7)
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where (Mξ f )(s) := eiξ s f (s). Hence the spectrum and the approximate point spectrum of G are invariant under vertical translations, and the spectrum and the approximate point spectrum of E(1) are rotationally invariant (see [4, p. 65]). Moreover, the following are equivalent: (G1) σap (G) ∩ iR = ∅, (G2) G is bounded from below, i.e. Gf Lp cf Lp
f ∈ D(G) ,
(3.8)
for some c > 0, (G3) G satisfies (1.5), i.e. (G − is)f
Lp
cf Lp
f ∈ D(G), s ∈ R ,
for some c > 0. r the For a bounded sequence of bounded linear operators (Wn )n∈Z on X, denote by S{W n} associated right shift operator on p (Z, X) defined by
r S{W (xn )n∈Z = (Wn xn−1 )n∈Z n}
(xn )n∈Z ∈ lp (Z, X) .
As in (3.7), r r Mξ−1 S{W Mξ = e−iξ S{W n} n}
(ξ ∈ R),
where Mξ ((xn )n∈Z ) := (einξ xn )n∈Z . Hence, the spectrum and the approximate point spectrum r of S{W are rotationally invariant. n} When the weight {Wn } does not depend on n and Wn = W for all n ∈ Z, the correspondr r . We can relate the approximate point specing shift operator S{W will be denoted by SW n} r trum of SW to the approximate point spectrum of its weight W as in the following lemma. r ) is determined By rotation-invariance and a scaling argument, this lemma shows that σap (SW by σap (W ). Lemma 3.7. Let W ∈ L(X) and p ∈ [1, ∞). Then r ) ∩ T = ∅; (a) σ (W ) ∩ T = ∅ ⇔ σ (SW r ) ∩ T = ∅. (b) σap (W ) ∩ T = ∅ ⇔ σap (SW
Proof. The statement (a) and the implication ⇐ in (b) are proved in [4, Lemma 2.37]. Thus, it remains to show the implication ⇒ in (b). Assume that σap (W ) ∩ T = ∅,
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i.e. W is quasi-hyperbolic. Let V be a hyperbolic extension of W to a Banach space Y ⊃ X provided by Theorem 2.2. Consider the shift operator SVr on lp (Z, X) associated with the operator V . Since σ (V ) ∩ T = ∅, we have σ SVr ∩ T = ∅ by the property (a). Moreover, r SVr x = SW x
x ∈ lp (Z, X) .
r to l (Z, Y ) ⊃ l (Z, X), so Thus, SVr is a hyperbolic extension of SW p p
r σap SW ∩T=∅ by Theorem 2.2.
2
Now we return to our consideration of an evolution family (U (t, s))ts0 , and we recall the relations between the approximate spectra of the functional operator E(1), the difference r operator S{U (n,n−1)} and the differential operator G. Lemma 3.8. Let (E(t))t0 be an evolution semigroup on Lp (R, X) associated with an evolution family (U (t, s))ts0 , and let G be the generator of (E(t))t0 . Then the following properties are equivalent. (i) σap (E(1)) ∩ T = ∅, r (ii) σap (S{U (n,n−1)} ) ∩ T = ∅, (iii) σap (G) ∩ iR = ∅. The equivalence (i) ⇔ (iii) follows from the spectral mapping theorem for approximate point spectrum satisfied by evolution semigroups (see, for example, [4, Lemma 2.41, Theorem 3.13, p. 83] or [33, Proposition 2.2]). For the proof of the equivalence (ii) ⇔ (iii) see [42, Theorem 2], [22, Theorem 1.2 and p. 1004] or [23, Theorem 1.4]. (In [22] and [23] the equivalence was proved in a more general context of closed ranges.) The following theorem and its corollary provide a characterisation of quasi-hyperbolic semigroups on X in terms of lower Fourier multipliers. Their proofs are based on a combination of spectral properties of shifts given in Lemmas 3.7 and 3.8. As one may expect, such a characterisation becomes especially transparent when X is a Hilbert space. Theorem 3.9. Let (T (t))t0 be a C0 -semigroup on a Banach space X with generator A. Then (T (t))t0 is quasi-hyperbolic if and only if MA−i· is a lower Lp (R, X)-Fourier multiplier for all/some p ∈ [1, +∞). Proof. Let p ∈ [1, ∞), (E(t))t0 be the evolution semigroup on Lp (R, X) associated with the semigroup (T (t))t0 , G be the generator of (E(t))t0 , and STr (1) be the (right) shift on lp (Z, X) associated with the operator T (1).
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From Lemma 3.6 one easily gets that MA−i· is a lower Lp (R, X)-Fourier multiplier if and only if (3.8) holds. Now (3.8) is equivalent to (G1), and Lemma 3.8 shows that (G1) is equivalent to σap STr (1) ∩ T = ∅.
(3.9)
In turn, by Lemma 3.7, (3.9) is equivalent to σap T (1) ∩ T = ∅, i.e. T (1) and then (T (t))t0 are quasi-hyperbolic.
2
If X is a Hilbert space, then the Fourier transform is a unitary operator on L2 (R, X), and the following corollary of Theorem 3.9 is almost immediate. Corollary 3.10. Let (T (t))t0 be a C0 -semigroup on a Hilbert space X with generator A. Then (T (t))t0 is quasi-hyperbolic if and only if (1.5) holds. A description of σap (T (t)) (t > 0) can also be given in terms of lower Lp (T)-Fourier multipliers, in a similar way to [4, Section 2.2.1]. By rescaling it suffices to characterise when 1 ∈ σap (T (2π)). Define
n
P (T, X) :=
ξ xk : ξ ∈ T, xk ∈ X, n ∈ N , k
k=−n n
PA (T, X) :=
ξ xk : ξ ∈ T, xk ∈ D(A), n ∈ N . k
k=−n
Definition 3.11. The linear operator MA−i· : PA (T, X) → P (T, X), n n k ξ xk := ξ k (A − ik)xk , MA−i· k=−n
k=−n
is said to be a lower Lp (T, X)-Fourier multiplier, where p ∈ [1, ∞), if there exists c > 0 such that n n ξ k (A − ik)xk c ξ k xk (3.10) k=−n
Lp (T,X)
for all n ∈ N, xk ∈ D(A). The following counterpart of Theorem 3.9 holds.
k=−n
Lp (T,X)
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Theorem 3.12. Let (T (t))t0 be a C0 -semigroup on a Banach space X with generator A. Then 1∈ / σap (T (2π)) if and only if MA−i· is a lower Lp (T, X)-Fourier multiplier for all/some p ∈ [1, +∞). The proof follows the same lines as the proof of Theorem 3.9, so we give only a sketch. Given a C0 -semigroup (T (t))t0 on X define an evolution semigroup (Eper (t))t0 on Lp ([0, 2π], X) by the formula Eper (t)f (s) = T (t)f [s − t](mod 2π)
s ∈ [0, 2π] .
(3.11)
Let Gper be the generator of (Eper (t))t0 . Note that as in the case of evolution semigroups on Lp (R, X) one has 1 ∈ σap Eper (2π)
⇐⇒
0 ∈ σap (Gper )
by the spectral mapping theorem for the approximate point spectrum of evolution semigroups [4, Lemma 2.29]. Since
Eper (2π)f (t) = T (2π)f (t) t ∈ [0, 2π] , 1 ∈ σap Eper (2π) ⇐⇒ 1 ∈ σap T (2π) .
(Unlike the case of multipliers on Lp (R, X) above, there is no need to invoke shift operators on p (Z, X) here.) Finally the property of MA−i· being a lower Lp (T, X)-Fourier multiplier is equivalent to the boundedness from below of Gper since n −1 F Gper F f (ξ ) = (A − ik)xk ξ k
(ξ ∈ T),
k=−n
for f ∈ PA (T, X), f (ξ ) =
n
k=−n xk ξ
k.
Remark 3.13. Theorems 3.9 and 3.12 remain valid if the Lp -spaces are replaced by C0 (R, X) and C(T, X) respectively. The definitions of lower Fourier multipliers on these spaces, and the proofs, remain valid, mutatis mutandis. In the following we describe a class of natural examples of quasi-hyperbolic C0 -semigroups. Example 3.14. Let w : R → R be a (strictly) positive continuous function such that w(s) ceωt w(s − t)
(s ∈ R, t 0),
for some c, ω ∈ R. Define a weighted right shift C0 -semigroup (T (t))t0 on Lp (R) (1 p < ∞), by the formula T (t)f (s) =
w(s) f (s − t). w(s − t)
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Note that (T (t))t0 is an evolution semigroup on Lp (R) associated with the strongly continuous evolution family (Uw (t, s))ts ⊂ L(C), given by Uw (t, s) = w(t)/w(s) (t s). Then by Lemma 3.8 σap T (1) ∩ T = ∅
⇐⇒
σap Swr ∗ ∩ T = ∅,
where Swr ∗ is the right shift operator on p (Z) (1 p < ∞) associated with the weight w ∗ (n) := w(n)/w(n − 1), i.e., Swr ∗ (xn )n∈Z =
w(n) xn−1 w(n − 1)
. n∈Z
Hence quasi-hyperbolicity of T (1) is equivalent to the quasi-hyperbolicity of the operator Swr ∗ , and that is described completely by Proposition 2.5. Thus the class of weighted shift semigroups provides a variety of examples of quasi-hyperbolic semigroups. For instance, for w such that lim
n→±∞
w(n) = w± w(n − 1)
and w+ > w− ,
(3.12)
we have
σap T (1) = σap Swr = λ ∈ C: |λ| = w− or |λ| = w+ . Thus (T (t))t0 is quasi-hyperbolic, but not hyperbolic, if and only if w− < 1 < w+ . Recall also that under the assumptions (3.12),
σ T (1) = λ ∈ C: w− |λ| w+ . 4. Weaker properties Since the lower bounds (1.5) for A − is do not imply quasi-hyperbolicity of T (t) in general, it is of interest to find weaker properties of T (t) which are implied by lower bounds. The spectral mapping theorem for the point spectrum of a semigroup [10, Theorem IV.3.7] shows that (1.5) implies that T (t) satisfies (PS), and we shall see in Corollary 4.2 that it also implies (EX). A reasonable question is whether (1.5) implies the analogue of (HE) that there is a Banach space Y in which X is continuously embedded and a (quasi-)hyperbolic C0 -semigroup (S(t))t0 on Y with T (t) = S(t)X . A positive answer to that question would imply the analogue of (UE) so that, for each non-zero x, either T (t)x increases at an exponential rate in t or inf{y: y ∈ X, T (t)y = x} increases at an exponential rate in t. We are unable to show that is the case, but we shall give two slightly weaker results in Corollary 4.2 and Theorem 4.5. In this section, F denotes the Fourier transform on L1 (R). Theorem 4.1. Let T = (T (t))t0 be a C0 -semigroup with generator A, and assume that the lower bound (1.5) holds for some c > 0. For each n 1 and a > 0, there exists αn (depending on n, a and T ) such that
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a T (t2 )x y cn 1 1 − α + (F f )(τ )T (τ )x dt x + n t2n t1n n! t1 t2
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(4.1)
0
whenever t1 , t2 > a, x, y ∈ X, T (t1 )y = x, f ∈ L1 (R), f L1 = 1 and supp (F f ) ⊆ [0, a]. Proof. Let n 1 and a > 1 be fixed, and let t1 , t2 , x and y be as in the statement of the theorem. Let g(τ ) =
(1 + tτ1 )n (1 −
τ n t2 )
(−t1 τ < 0), (0 τ t2 ).
Let s ∈ R and t2 yr =
g (r) (τ )e−isτ T (τ + t1 )y dτ
(r = 0, 1, . . . , n).
−t1
Then yr ∈ D(A) and (A − is)yr = yr+1 + g (r) (0−) − g (r) (0+) x
(r = 0, 1, . . . , n − 1), (A − is)yn = g (n) (0+)e−ist2 T (t2 )x − g (n) (0−)eist1 y + g (n) (0−) − g (n) (0+) x. By the assumption (1.5), yr+1 cyr − xg (r) (0−) − g (r) (0+) (r = 0, 1, . . . , n − 1), (n) g (0+)e−ist2 T (t2 )x − g (n) (0−)eist1 y cyn − xg (n) (0−) − g (n) (0+). Noting that g (r) (0−) = obtain
n! (n−r)!t1r
and g (r) (0+) =
(−1)r n! (n−r)!t2r ,
and in particular g(0−) = g(0+), we
T (t2 )x y c 1 1 y + − x + n t2n t1n n! t1n t2n c2 n n 1 1 yn−1 − x n−1 + n−1 + n + n n! t1 t2 t1 t2 .. .
n 1 n! 1 cn . y0 − x + n! (n − r)! t1r t2r r=1
Now let f ∈ L1 (R) with f L1 = 1 and supp (F f ) ⊆ [0, a]. Then
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a t2 cn cn g(τ )(F f )(τ )T (τ )x dt = f (s) g(τ )e−isτ T (τ + t1 )y dτ ds n! n! −t1
R
0
n T (t2 )x y 1 n! 1 + n + x + r . t2n t1 (n − r)! t1r t2 r=1
However, for τ > 0, g(τ ) = 1 +
n (−1)r n! r τ . r!(n − r)!t2r r=1
Hence a cn (F f )(τ )T (τ )x dt n! 0
a a n cn n! r g(τ )(F f )(τ )T (τ )x dt + τ (F f )(τ )T (τ )x dτ n! r!(n − r)! r=1
0
T (t2 )x y + n + x t2n t1
n
0
n! a r+1 Ma 1+ (n − r)! r! r=1 1 1 T (t2 )x y + + α x + n t2n t1n t1 t2 where Ma = sup{T (t): 0 t a} and αn is suitably chosen.
1 1 + r t1r t2
2
Corollary 4.2. Let A be the generator of a C0 -semigroup (T (t))t0 on a Banach space X, and assume that (1.5) holds. Let x ∈ t>0 Ran(T (t)) with x = 0. Then at least one of the following holds: (i) For each n ∈ N, T (t)x/t n → ∞ as t → ∞; (ii) For each n ∈ N, inf{y: T (t)y = x} →∞ tn
as t → ∞.
Choose f ∈ L1 (R) with f L1 = 1, supp (F f ) ⊆ [0, a] for some a > 0, and Proof. a (F f )(t)T (t)x dt = 0. This is possible by strong continuity of (T (t))t0 . 0 If neither (i) nor (ii) held, then one could choose values of n ∈ N, t1 > a and t2 > a such that (4.1) failed. 2 Corollary 4.3. Let A be the generator of a C0 -group (T (t))t∈R on a Banach space X, and assume that (1.5) holds. Then (T (t))t0 is expansive.
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Remark 4.4. We can introduce a norm · T ,a on X defined by a xT ,a = sup (F f )(τ )T (τ )x dτ : f ∈ L1 (R), f 1 = 1, supp (F f ) ⊆ [0, a] . 0
The different norms for different values of a are equivalent to each other. Now (4.1) can be written as T (t2 )x y cn 1 1 . + n xT ,a − αn x + t2n t1 n! t1 t2 This suggests that the norm · T ,a is a possible candidate to show that T (t) satisfies (HE) , but it would be necessary to obtain an inequality of this type with this norm on the left-hand side. Our second partial result indicates exponential growth in an integral sense. A very similar result is given in [16, Theorem 2.5]. However the proof there seems to be incomplete (in general, boundedness of the Carleman transform of u near z0 ∈ iR does not imply that z0 is not in the Carleman spectrum of u). So we give a complete proof here. Recall that a function g : R → X is a complete trajectory of T if g(t + s) = T (t)g(s) (t 0, s ∈ R). Any complete trajectory is continuous on R. When T is a C0 -group, the complete trajectories are of the form g(t) = T (t)x (t ∈ R) for some x ∈ X. Theorem 4.5. Let (T (t))t0 be a C0 -semigroup on a Banach space X with generator A, and assume that (1.5) holds. If g : R → X is a non-zero complete trajectory of (T (t))t0 then there exists ε > 0 such that g(·) ∈ / L1 (R, e−ε|t| dt). Proof. Let c > 0 be as in (1.5). Let λ = a + is ∈ C, where |a| c/2. Then (A − λ)x (A − is)x − |a|x c x x ∈ D(A) . 2
(4.2)
Let g be a complete trajectory and suppose that g(·) ∈ L1 (R, e−ε|t| dt) for every ε > 0. Let x = g(0). The Carleman transform gˆ of g is defined for Re λ = 0 by: ∞ 0
g(λ) ˆ =
−
e−λt g(t) dt
∞ λt 0 e g(−t) dt
(Re λ > 0), (Re λ < 0).
Then gˆ is holomorphic, and bounded on {λ: | Re λ| > ε} for each ε > 0. In particular, gˆ is bounded on {λ: | Re λ| c/2}. k Consider λ with Re λ > 0, and let k > 0. Then 0 e−λt g(t) dt ∈ D(A) and k (A − λ)
e
0
−λt
g(t) dt = e−λk g(k) − x.
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Since (A − λ) is closed, letting k → ∞ (through a suitable sequence) shows that g(λ) ˆ ∈ D(A)
and (A − λ)g(λ) ˆ = −x.
(4.3)
A variation of this argument shows that (4.3) also holds for Re λ < 0. It follows from (4.2) that g(λ) ˆ 2x/c for 0 < | Re λ| < c/2. Thus gˆ is bounded on C \ iR. Now suppose that 0 < | Re λ| c/2 and 0 < | Re μ| c/2. Then 2 (A − λ) g(λ) g(λ) ˆ − g(μ) ˆ ˆ − g(μ) ˆ c 2 4 |λ − μ|x. = −x − −x + (μ − λ)g(μ) ˆ c c2 Now by Cauchy’s criterion, gˆ extends continuously to iR, and then gˆ is a bounded entire function. Since g(a) ˆ → 0 as a → ∞, gˆ is identically zero. Then g = 0 by the uniqueness theorem for Laplace transforms. 2 Finally we consider the situation where (1.5) holds but no trajectory grows exponentially in forward time. Recall that the growth bound ω(T ) of (T (t))t0 is the unique quantity in R ∪ {−∞} such that the spectral radius of T (t) is exp(tω(T )) for some/all t > 0. If ω(T ) < 0, then (1.5) holds, and T is exponentially stable and therefore trivially hyperbolic. If ω(T ) = 0, then no trajectory grows exponentially in forward time, and we now show that (1.5) implies that all non-zero trajectories grow exponentially in reverse time, with the same exponent. Example 3.3 is an illustration of this situation. Theorem 4.6. Let (T (t))t0 be a C0 -semigroup with generator A and with growth bound 0, and assume that the lower bound (1.5) holds. Then (i) (T (t))t0 is not quasi-hyperbolic, (ii) A is invertible, (iii) there exist constants ε > 0 and κ > 0 such that y κeεt A−1 x whenever t > 0, x, y ∈ X and T (t)y = x. Proof. Since the growth bound is 0, T contains a point in the boundary of σ (T (1)) and hence in σap (T (1)), so T is not quasi-hyperbolic. Moreover, R(λ, A) exists whenever Re λ > 0 and R(λ, A) is bounded for Re λ a for any a > 0. It follows from (4.2) that R(λ, A) is bounded for 0 < Re λ < a for some a > 0, and then it follows from the Neumann series for the resolvent that R(λ, A) exists and is uniformly bounded for Re λ > −ε for some ε > 0. By a theorem of Weis and Wrobel [1, Theorem 5.1.9], there exists a constant C such that T (t)A−1 Ce−εt (t > 0). Hence y C −1 eεt A−1 x if T (t)y = x. 2 References [1] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001.
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[2] A.G. Baskakov, Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators, Sovrem. Mat. Fundam. Napravl. 9 (2004) 3–151 (in Russian); translation in J. Math. Sci. (N. Y.) 137 (2006) 4885–5036. [3] C.J.K. Batty, S. Srivastava, The non-analytic growth bound of a C0 -semigroup and inhomogeneous Cauchy problems, J. Differential Equations 194 (2003) 300–327. [4] C. Chicone, Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., Providence, RI, 1999. [5] C. Chicone, R.C. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980) 28–39. [6] C. Chicone, R.C. Swanson, Spectral theory for linearisations of dynamical systems, J. Differential Equations 40 (1981) 155–167. [7] J. Cooper, H. Koch, The spectrum of a hyperbolic evolution operator, J. Funct. Anal. 133 (1995) 301–328. [8] G. Couper, Characterization of quasi-Anosov diffeomorphisms, Bull. Austral. Math. Soc. 17 (1977) 321–334. [9] M. Eisenberg, J.H. Hedlund, Expansive automorphisms of Banach spaces, Pacific J. Math. 34 (1970) 647–656. [10] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin, 2000. [11] J. Franks, C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc. 223 (1976) 267–278. [12] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006. [13] P.R. Halmos, G. Lumer, J.J. Schäffer, Square roots of operators, Proc. Amer. Math. Soc. 4 (1953) 142–149. [14] B. Hasselblatt, Hyperbolic dynamical systems, in: Handbook of Dynamical Systems, vol. 1A, North-Holland, Amsterdam, 2002, pp. 239–319. [15] J.H. Hedlund, Expansive automorphisms of Banach spaces. II, Pacific J. Math. 36 (1971) 671–675. [16] S.Z. Huang, Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems, Studia Math. 116 (1995) 23–41. [17] R. Johnson, Analyticity of spectral subbundles, J. Differential Equations 35 (1980) 366–387. [18] M.A. Kaashoek, S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations 112 (1994) 374–406. [19] Yu. Latushkin, A. Pogan, R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory 58 (2007) 387–414. [20] Yu. Latushkin, F. Räbiger, Operator valued Fourier multipliers and stability of strongly continuous semigroups, Integral Equations Operator Theory 51 (2005) 375–394. [21] Yu. Latushkin, R. Shvydkoy, Hyperbolicity of semigroups and Fourier multipliers, in: Systems, Approximation, Singular Integral Operators, and Related Topics, Bordeaux, 2000, in: Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 341–363. [22] Yu. Latushkin, Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois J. Math. 48 (2004) 999–1020. [23] Yu. Latushkin, Yu. Tomilov, Fredholm differential operators with unbounded coefficients, J. Differential Equations 208 (2005) 388–429. [24] Y. Latushkin, M. Vishik, Linear stability in an ideal incompressible fluid, Comm. Math. Phys. 233 (2003) 439–461. [25] K.B. Laursen, M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monogr., vol. 20, Oxford University Press, New York, 2000. [26] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982. [27] O. Lopes, On the structure of the spectrum of a linear time periodic wave equation, J. Anal. Math. 47 (1986) 55–68. [28] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977) 351–370. [29] J. Mather, Characterization of Anosov diffeomorphisms, Indag. Math. 30 (1968) 479–483. [30] T.L. Miller, V.G. Miller, M. Neumann, Local spectral properties of weighted shifts, J. Operator Theory 51 (2004) 71–88. [31] V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Oper. Theory Adv. Appl., vol. 139, Birkhäuser, Basel, 2003. [32] S.Y. Pilyugin, Generalizations of the notion of hyperbolicity, J. Difference Equ. Appl. 12 (2006) 271–282. [33] F. Räbiger, R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum 52 (1996) 225–239. [34] C.J. Read, Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element, Trans. Amer. Math. Soc. 286 (1986) 715–725. [35] C.J. Read, Spectrum reducing extension for one operator on a Banach space, Trans. Amer. Math. Soc. 308 (1988) 413–429.
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[36] W.C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970) 349–356. [37] R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, in: Evolution Equations, Semigroups and Functional Analysis, Milano, 2000, in: Progr. Nonlinear Differential Equations Appl., vol. 50, Birkhäuser, Basel, 2002, pp. 311–338. [38] R. Schnaubelt, Asymptotic behaviour of parabolic nonautonomous evolution equations, in: Functional Analytic Methods for Evolution Equations, in: Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 401–472. [39] R. Shvydkoy, The essential spectrum of advective equations, Comm. Math. Phys. 265 (2006) 507–545. [40] R. Shvydkoy, Cocycles and Mane sequences with an application to ideal fluids, J. Differential Equations 229 (2006) 49–62. [41] R.C. Swanson, The spectrum of vector bundle flows with invariant subbundles, Proc. Amer. Math. Soc. 83 (1981) 141–145. [42] V.M. Tyurin, Invertibility of linear differential operators in some function spaces, Sibirsk. Mat. Zh. 32 (1991) 160– 165 (in Russian); translation in Siberian Math. J. 32 (1991) 485–490. [43] M. Wolff, A remark on the spectral bound of the generator of positive operators with applications to stability theory, in: Functional Analysis and Applications, Proc. Oberwolfach, 1980, Birkhäuser, 1981, pp. 39–50.
Journal of Functional Analysis 258 (2010) 3879–3905 www.elsevier.com/locate/jfa
Particle approximation of the Wasserstein diffusion Sebastian Andres ∗ , Max-K. von Renesse Technische Universität Berlin, Germany Received 26 October 2009; accepted 27 October 2009 Available online 2 December 2009 Communicated by Paul Malliavin
Abstract We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein diffusion (von Renesse and Sturm (2009) [25]), assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya (2003) [19]. © 2009 Elsevier Inc. All rights reserved. Keywords: Measure-valued processes; Hydrodynamic limit; Nonlinear fluctuations; McKean–Vlasov limit
1. Introduction The large scale behaviour of stochastic interacting particle systems is often described by (possibly nonlinear) deterministic evolution equations in the hydrodynamic limit, a fact which can be understood as a dynamic version of the law of large numbers, cf. e.g. [15]. The fluctuations around such deterministic limits usually lead to linear Ornstein–Uhlenbeck type SPDE on the diffusive time scale. In view of this typical two-step scaling hierarchy it is no surprise that only few types of SPDE with nonlinear drift operator admitting a rigorous particle approximation are known (cf. [13] for a survey on lattice models, e.g. [23,17] for exchangeable diffusions and e.g. [5,9] for interactive population models). According to [13] the appearance of a nonlinear SPDE as the scaling limit of some microscopic system is an indication of criticality, i.e. a situation * Corresponding author.
E-mail addresses:
[email protected] (S. Andres),
[email protected] (M.-K. von Renesse). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.029
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when microscopic fluctuations become too large for an averaging principle as in the law of large numbers to become effective. In this work we add one more example to the collection of (in this case conservative) interacting particle systems with a nonlinear stochastic evolution in the hydrodynamic limit. We study a sequence of Langevin-type SDEs with reflection for the positions 0 ≡ xt0 xt1 xt2 · · · xtN xtN +1 ≡ 1 of N moving particles on the unit interval 1 β 1 dt −1 − N +1 xti − xti−1 xti+1 − xti √ + 2 dwti + dlti−1 − dlti , i = 1, . . . , N,
dxti =
(1)
driven by independent real Brownian motions {w i } and local times l i satisfying t dlti
0,
lti
=
1{x i =x i+1 } dlsi . s
s
(2)
0
At first sight Eq. (1) resembles familiar Dyson-type models of interacting Brownian motions with electrostatic interaction, for which the convergence towards (deterministic) McKean– Vlasov equations under various assumptions is known since long, cf. e.g. [22,6]. Except from the fact that (1) models a nearest-neighbour and not a mean-field interaction, the most important difference towards the Dyson model is however, that in the present case for N β − 1 the drift is attractive and not repulsive. One technical consequence is that the system (1) and (2) has to be understood properly because it can no longer be defined in the class of Euclidean semi-martingales (cf. [4] for a rigorous analysis). The second and more dramatic consequence is a clustering of hence strongly correlated particles such that fluctuations are seen on large hydrodynamic scales. More precisely, assuming Markov uniqueness for the corresponding infinite dimensional Kolmogorov operator, cf. Definition 2.2 below, we show that for properly chosen initial condition the empirical probability distribution of the particle system in the high density regime μN t
N 1 = δx i N·t N i=1
converges for N → ∞ to the Wasserstein diffusion (μt ) on the space of probability measures P([0, 1]). This process was introduced in [25] as a conservative model for a diffusing fluid when its heat flow is perturbed by a kinetically uniform random forcing. In particular (μt ) is a solution in the sense of an associated martingale problem to the SPDE dμt = βμt dt + (μt ) dt + div( 2μt dBt ),
(3)
where is the Neumann Laplace operator and dBt is space–time white noise over [0, 1] and, for μ ∈ P([0, 1]), (μ) ∈ D ([0, 1]) is the Schwartz distribution acting on f ∈ C ∞ ([0, 1]) by
(μ), f =
f (I− ) + f (I+ ) f (I+ ) − f (I− ) f (0) + f (1) − − , 2 |I | 2
I ∈gaps(μ)
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where gaps(μ) denotes the set of intervals I = (I− , I+ ) ⊂ [0, 1] of maximal length with μ(I ) = 0 and |I | denotes the length of such an interval. The SPDE (3) has a familiar structure. For instance, the Dawson–Watanabe (‘super-Brownian √ motion’) process solves dμt = βμt dt + 2μt dBt whereas the empirical measure of a countable family of independent Brownian motions satisfies the equation dμt = μt dt + √ div( 2μt dBt ), both again in the weak sense of the associated martingale problems, cf. e.g. [7]. The additional nonlinearity introduced through the operator into (3) is crucial for the construction of (μt ) by Dirichlet form methods because it guarantees the existence of a reversible measure Pβ on P([0, 1]) which plays a central role for the convergence result, too. For β > 0, Pβ can be defined as the law of the random probability measure η ∈ P([0, 1]) defined by 1
f, η =
β f Dt dt,
∀f ∈ C [0, 1] ,
0 β
γ
is the real-valued Dirichlet process over [0, 1] with parameter β and γ where t → Dt = γt·β β denotes the standard Gamma subordinator. It is argued in [25] that Pβ admits the formal Gibbsean representation Pβ (dμ) =
1 −β Ent(μ) 0 e P (dμ) Z
with the Boltzmann entropy Ent(μ) = [0,1] log(dμ/dx) dμ as Hamiltonian and a particular uniform measure P0 on P([0, 1]), which illustrates the non-Gaussian character of Pβ . For instance, Pβ is neither log-concave nor does it project nicely to linear subspaces. However an appropriate version of the Girsanov formula holds true for Pβ , see also [26], which implies the L2 (P([0, 1]), Pβ )-closability of the quadratic form w 2 β ∇ F P (dμ), F ∈ Z, E(F, F ) = μ P ([0,1])
on the class F (μ) = f φ , μ , φ , μ , . . . , φ , μ , 2 k 1 Z = F : P [0, 1] → R f ∈ Cc∞ Rk , {φi }ki=1 ⊂ C ∞ [0, 1] , k ∈ N where w ∇ F = (D|μ F ) (·) 2 μ L ([0,1],μ) and (D|μ F )(x) = ∂t|t=0 F (μ + tδx ). The corresponding closure, still denoted by E, is a local regular Dirichlet form on the compact space (P([0, 1]), τw ) of probabilities equipped with the weak topology. This allows to construct a unique Hunt diffusion process ((Pη )η∈P ([0,1]) , (μt )t0 ) properly associated with E, cf. [11]. Starting (μt ) from equilibrium Pβ yields what shall be called in the sequel exclusively the Wasserstein diffusion. Note that the SPDE (3) may be called singular in several respects. Apart from the curious nonlinear structure of the drift component (μ), the noise is multiplicative non-Lipschitz and de-
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generate elliptic in all states μ which are not fully supported on [0, 1], i.e. the infinite dimensional Kolmogorov operator associated to (3) is not nicely behaved. Our approach for the approximation result is therefore again based on Dirichlet form methods. We use that the convergence of symmetric Markov semigroups is equivalent to an amplified notion of Gamma-convergence [20] of the associated sequence quadratic forms E N . In our situation it suffices to verify that Eqs. (1) and (2) define a sequence of reversible finite dimensional particle systems whose equilibrium distributions converge nicely enough to Pβ . In particular we show that also the logarithmic derivatives converge in an appropriate L2 -sense which implies the Mosco-convergence of the Dirichlet forms. (The pointwise convergence of the same sequence E N to E has been used in a recent work by Döring and Stannat to establish the logarithmic Sobolev inequality for E, cf. [8].) Since the approximating state spaces are finite dimensional we employ [19] for a generalized framework of Mosco-convergence of forms defined on a scale of Hilbert spaces. In case of a fixed state space with varying reference measure the criterion of L2 -convergence of the associated logarithmic derivatives has been studied in e.g. [16]. However, in our case this result does not directly apply because in particular the metric and hence also the divergence operation itself is depending on the parameter N . However, only little effort is needed to see that things match up nicely, cf. Section 4.3. For the sake of a clearer presentation in the proofs we will work with a parametrization of a probability measure on [0, 1] by the generalized right continuous inverse of its distribution function, which however is mathematically inessential. A side result of this parametrization is a diffusive scaling limit result for a (1 + 1)-dimensional gradient interface model with non-convex interaction potential (cf. [12]), see Section 6. 2. Set up and main results Construction of (XtN ). Following [4], for β > 0, the generalized solution to the system (1) and (2) is obtained rigorously by Dirichlet form methods, observing that in the regular case β N +1 the solution XtN = (xt1 , . . . , xtN ) defines a reversible Markov process on ΣN := {x ∈ RN , 0 x 1 x 2 · · · x N 1} ⊂ RN with equilibrium distribution qN dx 1 , . . . , dx N =
(β)
N +1
(( Nβ+1 ))N +1
i=1
i β −1 x − x i−1 N+1 dx 1 . . . dx N .
The generator LN of X N satisfies L f (x) = N
N 1 ∂ β 1 −1 − f (x) N +1 x i − x i−1 x i+1 − x i ∂x i i=1
+ f (x)
for x ∈ Int(ΣN ),
(4)
and E (f, g) = − N
ΣN
f (x)L g(x) qN (dx) = N
∇f (x) · ∇g(x) qN (dx),
ΣN
for all f, g ∈ C ∞ (ΣN ), ∇g · ν = 0 on ∂ΣN , where ν denotes the inner normal field on ∂ΣN .
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For general β > 0, N ∈ N the measure qN ∈ P(ΣN ) is well defined and satisfies the Hamza condition, because it has a strictly positive density with locally integrable inverse, cf. e.g. [1]. This implies that the form E N (f, f ) with domain f ∈ C ∞ (ΣN ) is closable on L2 (ΣN , qN ). The closure of E N , denoted by the same symbol, defines a local regular Dirichlet form on L2 (ΣN , qN ), from which a properly associated qN -symmetric Hunt diffusion ((Px )x∈ΣN , (XtN )t0 ) is obtained. Now we can state the main results of this work as follows. Theorem 2.1. Let E 0 be the limit of any -convergent subsequence of (N · E N )N . Then, there ˜ and exists a Dirichlet form E˜ , not depending on E 0 , which is extending E , i.e. D(E) ⊆ D(E) ˜ E(u) = E(u) for u ∈ D(E), such that ˜ E(u) E 0 (u) E(u)
˜ for all u ∈ D(E).
In particular, every -limit of (N · E N ) coincides with the Wasserstein Dirichlet form E on D(E). For the precise definition of -convergence of Dirichlet forms, see Section 3 below. By a general compactness result (see Theorem 3.3 below) every sequence of Dirichlet forms contains -convergent subsequences. As a corollary to Theorem 2.1 we obtain that the associated sequence of Hunt processes on P([0, 1]) converges weakly, provided E is Markov unique in the following sense. Definition 2.2. The Wasserstein Dirichlet form E is Markov unique if there is no proper extension ˜ where ZN = {F ∈ Z | E˜ of E on L2 (P([0, 1]), Pβ ) with generator L˜ such that ZN ⊂ D(L), F (μ) = f ( φ1 , μ , . . . , φk , μ ), φi (0) = φi (1) = 0, i = 1, . . . , k}. Corollary 2.3. For β > 0, assume that the Wasserstein Dirichlet form E is Markov unique. Let (XtN ) denote the qN -symmetric diffusion on ΣN induced from the Dirichlet form E N , starting 1 N from equilibrium X0N ∼ qN , and let μN t = N i=1 δx i ∈ P([0, 1]), then the sequence of proN·t
cesses (μN . ) converges weakly to (μ.) in CR+ ((P([0, 1]), τw )) for N → ∞. Remark 2.4 (On the Markov uniqueness assumption). Condition 2.2 is a very subtle assumption. By general principles, cf. [2, Theorem 3.4], it is weaker than the essential self-adjointness of the generator of (μt )t0 on ZN = {F ∈ Z | F (μ) = f ( φ1 , μ , . . . , φk , μ ), φi (0) = φi (1) = 0, i = 1, . . . , k} and stronger than the well-posedness, i.e. uniqueness, in the class of Hunt processes on P([0, 1]) of the martingale problem defined by Eq. (3) on the set of test functions ZN . In analytic terms, condition 2.2 is closely related to the Meyers–Serrin (weak = strong) property of the corresponding Sobolev space, cf. Corollary 5.2 and [10,18]. Variants of this assumption appear in several quite similar infinite dimensional contexts as well [14,16] and the verification depends crucially on the integration by parts formula which in the present case of Pβ has a very peculiar structure. None of the standard arguments for Gaussian or even log-concave or regular measures with smooth logarithmic derivative is applicable here. However, in the accompanying paper [4] we have managed to prove Markov uniqueness for E N , using the fact that the reference measure qN admits an extension qˆN lying in the Muckenhoupt class A2 (RN ).
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Remark 2.5. For illustration we present some numerical simulation results, using a regularized version of (1) and (2), by courtesy of Theresa Heeg, Bonn, in the case of N = 4 particles with β = 10, β = 1 and β = 0.3 respectively, at large times.
3. Finite dimensional approximation of Dirichlet forms in Mosco and Gamma sense In this section we recall the concept of Mosco- and -convergence of a sequence of Dirichlet forms in the generalized sense of Kuwae and Shioya, allowing for varying base L2 -spaces, developed in [19]. Our main results will follow by applying these concepts to the sequence of generating Dirichlet forms N · E N of (g·N ) on L2 (ΣN , qN ) and E on L2 (G, Qβ ) introduced in Section 4.1 below. Definition 3.1 (Convergence of Hilbert spaces). A sequence of Hilbert spaces H N converges to a Hilbert space H if there exists a family of linear maps {Φ N : H → H N }N such that limΦ N uH N = uH N
for all u ∈ H.
A sequence (uN )N with uN ∈ HN converges strongly to a vector u ∈ H if there exists a sequence (u˜ N )N ⊂ H tending to u in H such that lim lim supΦ M u˜ N − uM H M = 0, N
M
and (uN ) converges weakly to u if lim uN , vN H N = u, v H , N
for any sequence (vN )N with vN ∈ H N tending strongly to v ∈ H . Moreover, a sequence (BN )N of bounded operators on H N converges strongly (resp. weakly) to an operator B on H if BN uN → Bu strongly (resp. weakly) for any sequence (uN ) tending to u strongly (resp. weakly). Definition 3.2 (-convergence). A sequence (E N )N of quadratic forms E N on H N -converges to a quadratic form E on H if the following two conditions hold:
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(i) If a sequence (uN )N with uN ∈ H N strongly converges to a u ∈ H , then E(u, u) lim inf E N (uN , uN ). N
(ii) For any u ∈ H there exists a sequence (uN )N with uN ∈ H N which converges strongly to u such that E(u, u) = lim E N (uN , uN ). N
The main interest of -convergence relies on the following general compactness theorem (see Theorem 2.3 in [19]). Theorem 3.3. Any sequence (E N )N of quadratic forms E N on H N has a -convergent subsequence whose -limit is a closed quadratic form on H . For the convergence of the corresponding semigroup operators the appropriate notion is Mosco-convergence. Definition 3.4 (Mosco-convergence). A sequence (E N )N of quadratic forms E N on H N converges to a quadratic form E on H in the Mosco sense if the following two conditions hold: (Mosco I) If a sequence (uN )N with uN ∈ H N weakly converges to a u ∈ H , then E(u, u) lim inf E N (uN , uN ). N
(Mosco II) For any u ∈ H there exists a sequence (uN )N with uN ∈ H N which converges strongly to u such that E(u, u) = lim E N (uN , uN ). N
Extending [20] it is shown in [19] that Mosco-convergence of a sequence of Dirichlet forms is equivalent to the strong convergence of the associated resolvents and semigroups. However, we shall prove that the sequence N · E N converges to E in the Mosco sense in a slightly modified fashion, namely the condition (Mosco II) will be replaced by (Mosco II ) There is a core K ⊂ D(E) such that for any u ∈ K there exists a sequence (uN )N with uN ∈ D(E N ) which converges strongly to u such that E(u, u) = limN E N (uN , uN ). Theorem 3.5. Under the assumption that H N → H the conditions (Mosco I) and (Mosco II ) are equivalent to the strong convergence of the associated resolvents. Proof. We proceed as in the proof of Theorem 2.4.1 in [20]. By Theorem 2.4 of [19] strong convergence of resolvents implies Mosco-convergence in the original stronger sense. Hence we need
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to show only that our weakened notion of Mosco-convergence also implies strong convergence of resolvents. Let {RλN , λ > 0} and {Rλ , λ > 0} be the resolvent operators associated with E N and E, respectively. Then, for each λ > 0 we have to prove that for every z ∈ H and every sequence (zN ) tending strongly to z the sequence (uN ) defined by uN := RλN zN ∈ H N converges strongly to u := Rλ z as N → ∞. The vector u is characterized as the unique minimizer of E(v, v) + λ v, v H − 2 z, v H over H and a similar characterization holds for each uN . Since for each N the norm of RλN as an operator on H N is bounded by λ−1 , by Lemma 2.2 in [19] there exists a subsequence of (uN ), still denoted by (uN ), that converges weakly to some u˜ ∈ H . By (Mosco II ) we find for every v ∈ K a sequence (vN ) tending strongly to v such that limN E N (vN , vN ) = E(v, v). Since for every N E N (uN , uN ) + λ uN , uN H N − 2 zN , uN H N E N (vN , vN ) + λ vN , vN H N − 2 zN , vN H N , using the condition (Mosco I) we obtain in the limit N → ∞ ˜ H E(v, v) + λ v, v H − 2 z, v H , E(u, ˜ u) ˜ + λ u, ˜ u ˜ H − 2 z, u which by the definition of the resolvent together with the density of K ⊂ D(E) implies that u˜ = Rλ z = u. This establishes the weak convergence of resolvents. It remains to show strong convergence. Let uN = RλN zN converge weakly to u = Rλ z and choose v ∈ K with the respective strong approximations vN ∈ H N such that E N (vN , vN ) → E(v, v), then the resolvent inequality for RλN yields E N (uN , uN ) + λuN − zN /λ2H N E N (vN , vN ) + λvN − zN /λ2H N . Taking the limit for N → ∞, one obtains lim sup λuN − zN /λ2H N E(v, v) − E(u, u) + λv − z/λ2H . N
Since K is a dense subset we may now let v → u ∈ D(E), which yields lim sup uN − zN /λ2H N u − z/λ2H . N
Due to the weak lower semicontinuity of the norm this yields limN uN − zN /λH N = u − z/λH . Since strong convergence in H is equivalent to weak convergence together with the convergence of the associated norms the claim follows (cf. Lemma 2.3 in [19]). 2 4. Proof of Theorem 2.1 4.1. The G-parameterization We parameterize the space P([0, 1]) in terms of right continuous quantile functions, cf. e.g. [24,25]. The set G = g : [0, 1) → [0, 1] g càdlàg nondecreasing ,
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equipped with the L2 ([0, 1], dx) distance dL2 is a compact subspace of L2 ([0, 1], dx). It is homeomorphic to (P([0, 1]), τw ) by means of the map ρ : G → P [0, 1] ,
g → g∗ (dx),
which takes a function g ∈ G to the image measure of dx under g. The inverse map κ = ρ −1 : P([0, 1]) → G is realized by taking the right continuous quantile function, i.e. gμ (t) := inf s ∈ [0, 1]: μ[0, s] > t . Let now (g· ) = (κ(μ. )) be the G-image of the Wasserstein diffusion under the map κ with invariant initial distribution Qβ , where Qβ denotes the law of the real-valued Dirichlet process with parameter β > 0 as described in the introduction. In [25, Theorem 7.5] it is shown that (g· ) is generated by the Dirichlet form, again denoted by E , which is obtained as the L2 (G, Qβ )-closure of E(u, v) =
∇u|g (·), ∇v|g (·) L2 ([0,1]) Qβ (dg),
u, v ∈ C1 (G),
G
on the class C1 (G) = u : G → R u(g) = U f1 , g L2 , . . . , fm , g L2 , U ∈ Cc1 Rm , 2 {fi }m i=1 ⊂ L [0, 1] , m ∈ N , where ∇u|g is the L2 ([0, 1], dx)-gradient of u at g. We are now going to apply the results of Kuwae and Shioya, summarized in the last section, when H N = L2 (ΣN , qN ), H = L2 (G, Qβ ) and Φ N is defined to be the conditional expectation operator ΦN : H → H N ,
N Φ u (x) := E u|gi/(N+1) = x i , i = 1, . . . , N .
To that purpose, we need to show that this sequence of Hilbert spaces is convergent in the sense of Definition 3.1. Proposition 4.1. H N converges to H along Φ N , for N → ∞. Proof. We have to show that Φ N uH N → uH for each u ∈ H . Let F N be the σ -algebra on G generated by the projection maps {g → g(i/(N + 1)) | i = 1, . . . , N }. By abuse of notation we identify Φ N u ∈ H N with E(u|F N ) of u, considered as an element of L2 (Qβ , F N ) ⊂ H . Since the measure qN coincides with the respective finite dimensional distributions of Qβ on ΣN we have Φ N uH N = Φ N uH . Hence the claim will follow once we show that Φ N u → u in H . For the latter we use the following abstract result, whose proof can be found, e.g. in [3, Lemma 1.3].
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Lemma 4.2. Let (Ω, D, μ) be a measure space and (Fn )n∈N a sequence of σ -subalgebras of D. Then E(f |Fn ) → f for all f ∈ Lp , p ∈ [1, ∞) if and only if for all A ∈ D there is a sequence An ∈ Fn such that μ(An A) → 0 for n → ∞. In order to apply this lemma to the given case (G, B(G), Qβ ), where B(G) denotes the Borel σ -algebra on G, let FQβ ⊂ B(G) denote the collection of all Borel sets F ⊂ G which can be approximated by elements FN ∈ F N with respect to Qβ in the sense above. Note that FQβ is again a σ -algebra, cf. the appendix in [3]. Let M denote the system of finitely based open cylinder sets in G of the form M = {g ∈ G | gti ∈ Oi , i = 1, . . . , L} where ti ∈ [0, 1] and Oi ⊂ [0, 1] open. From the almost sure right continuity of g and the fact that g. is continuous at t1 , . . . , tL for Qβ -almost all g it follows that MN := {g ∈ G | g(ti ·(N +1)/(N+1)) ∈ Oi , i = 1, . . . , L} ∈ F N is an approximation of M in the sense above. Since M generates B(G) we obtain B(G) ⊂ FQβ such that the assertion holds, due to Lemma 4.2. 2 Remark 4.3. It is much simpler to prove Proposition 4.1 for a dyadic subsequence N = 2m − 1, m ∈ N when the sequence Φ N uH N is nondecreasing and bounded, because Φ N is a projec tion operator in H with increasing range im(Φ N ) as N grows. Hence, Φ N uH N is Cauchy and thus N Φ u − Φ M u2 = Φ N u2 − Φ M u2 → 0 for M , N → ∞, H H H
i.e. the sequence Φ N u converges to some v ∈ H . Since obviously Φ N u → u weakly in H it follows that u = v such that the claim is obtained from |Φ N uH − uH | Φ N u − uH . 4.2. The upper bound In this subsection we shall prove that the Wasserstein Dirichlet form E is an upper bound for any -limit of (N · E N ) as stated in Theorem 2.1. To that aim we essentially need to show that the condition (Mosco II ) holds for (N · E N ) and E. To simplify notation for f ∈ L2 ([0, 1], dx) denote the functional g → f, g L2 ([0,1]) on G by lf . We introduce the set K of polynomials defined by n ki lfi (g), ki ∈ N, fi ∈ C [0, 1] . K = u ∈ C(G) u(g) =
i=1
Lemma 4.4. The linear span of K is a core of E. Proof. Recall that by [25, Theorem 7.5] the set C1 (G) = u : G → R u(g) = U f1 , g L2 , . . . , fm , g L2 , U ∈ Cc1 Rm , 2 {fi }m i=1 ⊂ L [0, 1] , m ∈ N , is a core for the Dirichlet form E in the G-parametrization. The boundedness of G ⊂ L2 ([0, 1], dx) implies that U is evaluated on a compact subset of Rm only, where U can be
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approximated by polynomials in the C 1 -norm. From this, the chain rule for the L2 -gradient operator ∇ and Lebesgue’s dominated convergence theorem in L2 (G, Qβ ) it follows that the linear span of polynomials of the form u(g) = ni=1 lfkii (g) with ki ∈ N, fi ∈ C([0, 1]), ki ∈ N, is also a core of E . 2 Lemma 4.5. For a polynomial u ∈ K with u(g) = H N , then uN → u strongly.
n
ki i=1 lfi (g)
let uN :=
n
i=1 (Φ
N (l
ki fi ))
∈
Proof. Let u˜ N := ni=1 (Φ N (lfi ))ki ∈ H be the respective product of conditional expectations, where as above Φ N also denotes the projection operator on H = L2 (G, Qβ ). Note that by Jensen’s inequality for any measurable functional u : G → R, |Φ N (u)|(g) Φ N (|u|)(g) for Qβ -almost all g ∈ G, such that in particular Φ N (lfi )L∞ (G ,Qβ ) lfi L∞ (G ,Qβ ) f C([0,1]) . Hence each of the factors Φ N (lfi ) ∈ H is uniformly bounded and converges strongly to lfi in L2 (G, Qβ ), such that the convergence also holds true in any Lp (G, Qβ ) with p > 0. This implies u˜ N → u in H . Furthermore, n n M k k lim limΦ u˜ N − uM H M = lim limΦ M Φ N (lfi ) i − Φ M (lfi ) i N M N M i=1 i=1 H n n k k Φ N (lfi ) i − = lim lfii = 0. 2 N i=1
i=1
H
For the proof of (Mosco II ) we will also need that the conditional expectation of the random variable g w.r.t. Qβ given finitely many intermediate points {g(ti ) = x i } yields the linear interpolation. Lemma 4.6. For X ∈ ΣN define gX ∈ G by gX (t) = x + (N + 1) · t − i x i+1 − x i i
i i +1 , if t ∈ , i = 0, . . . , N, N +1 N +1
then E g|F N (X) = gX . Proof. This quite classical claim follows essentially from the bridge representation of the Dirichlet process, i.e. Qβ is the law of (γ (β · t))t∈[0,1] on G conditioned on γ (β) = 1 where γ is the standard Gamma subordinator, cf. e.g. [25]. Together with the elementary property that EQβ (g(t)) = t for t ∈ [0, 1] the claim follows from the homogeneity of γ together with simple scaling and iterated use of the Markov property. 2 Proposition 4.7. For all u ∈ K there is a sequence uN ∈ D(E N ) converging strongly to u in H and N · E N (uN , uN ) → E(u, u). In particular, condition (Mosco II ) is satisfied. From Proposition 4.7 one can immediately derive the upper bound in Theorem 2.1. To see this fix a subsequence, still denoted by N , such that (N · E N ) has a -limit E 0 . Then, we need
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to show that E 0 E on D(E) (note that by definition E = ∞ on the complement of D(E)). Fix now any u ∈ K. Using (Mosco II ) we find a sequence uN ∈ H N converging strongly to u in H such that E(u) = lim N · E N (uN ) E 0 (u), N
where the last inequality follows from the first property of -convergence. Hence, E 0 (u) E(u) for all u ∈ K and, since K is a core of D(E), by approximation we get that this also holds for all u ∈ D(E). Proof of Proposition 4.7. For u ∈ K let uN := ni=1 (Φ N (lfi ))ki ∈ H N as above then the strong convergence of uN to u is assured by Lemma 4.5. From Lemma 4.6 we obtain that Φ N (lf )(X) =
f, gX . In particular i ∇Φ N (lf )(X) =
1 · ηN ∗ f (ti ), N +1
(5)
where ti := i/(N + 1), i = 1, . . . , N + 1 and ηN denotes the convolution kernel t → ηN (t) = (N + 1) · (1 − min(1, |(N + 1) · t|)). For the convergence of N · EN (uN , uN ) to E(u, u) note first that f ∗ ηN → f in C([0, 1]) as N → ∞. Hence, using (5) we also get 2 N · ∇Φ N (lf )(X) =
N N 2 N η ∗ f (ti ) → f, f L2 ([0,1]) . 2 (N + 1) i=1
Similarly, for arbitrary f1 , f2 ∈ C([0, 1]) N · ∇Φ N (lf1 ), ∇Φ N (lf2 ) RN → f1 , f2 L2 ([0,1]) . Consider now u ∈ K with u(g) = ∇
L2
n
ki i=1 lfi (g).
The chain rule for the L2 -gradient operator ∇ =
yields n
∇u, ∇u L2 (0,1) =
i=j
j,s=1
and analogously for uN =
∇uN , ∇uN RN
n
i=1 (Φ
N (l
k lfii
ki fi ))
r=s
lfkrr
k −1 ks −1 lfs fj , fs L2 (0,1) ,
kj ks lfjj
with ∇ = ∇ R
N
n N N ki kr Φ (lfi ) Φ (lfr ) = j,s=1
i=j
(6)
r=s
k −1 N k −1 × kj ks Φ N (lfj ) j Φ (lfs ) s ∇Φ N (lfj ), ∇Φ N (lfs ) RN .
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Since N · E (uN , uN ) = N ·
∇uN , ∇uN RN dqN
N
ΣN
=N ·
∇uN , ∇uN g(t1 ), . . . , g(tN ) Qβ (dg)
G
and for Qβ -a.e. g Φ N (lf ) g(t1 ), . . . , g(tN ) → lf (g),
as N → ∞,
if f ∈ C([0, 1]) the first assertion of the proposition holds by (6) and dominated convergence. The second assertion follows now from linearity and polarisation together with Lemma 4.4. 2 For later use we make an observation which follows easily from the proof of the last proposition. Lemma 4.8. For u and uN as in the proof of Proposition 4.7 and for Qβ -a.e. g we have (N + 1)ιN ∇uN g(t1 ), . . . , g(tN ) − ∇u|g
→ 0,
L2 (0,1)
as N → ∞,
with ιN : RN → D([0, 1), R) defined as above and tl := l/(N + 1), l = 0, . . . , N + 1. Proof. By the definitions we have for every x ∈ ΣN (N + 1)ιN ∇uN (x) n N N N ki kj −1 l kj Φ (lfj )(x) Φ (lfi )(x) (N + 1) ∇ Φ N (lfj )(x) 1[tl ,tl+1 ] = j =1
=
i=j
n j =1
l=1
N k k −1 Φ (lfi )(x) i kj Φ N (lfj )(x) j
i=j
N
N η ∗ fj (tl )1[tl ,tl+1 ] ,
l=1
where we have used again (5). Furthermore, for every j , 2 1 N N η ∗ fj (tl )1[tl ,tl+1 ] (t) − fj (t) dt 0
l=1 N N 2 η ∗ fj (tl ) − fj (t) dt = tl+1
l=1 tl N N N 2 2 η ∗ fj (tl ) − fj (tl ) dt + 2 fj (tl ) − fj (t) dt, 2 tl+1
l=1 tl
tl+1
l=1 tl
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where the first term tends to zero as N → ∞ since ηN ∗ fj → fj in the sup-norm and the second term tends to zero by the uniform continuity of fj . From this we can directly deduce the claim because Φ N (lf )(g(t1 ), . . . , g(tN )) → lf (g) as N → ∞ for Qβ -a.e. g. 2 4.3. The lower bound For the lower bound, which will be based on the (Mosco I) condition, we exploit that the respective integration by parts formulas of E N and E converge. In case of a fixed state space a similar approach is discussed abstractly in [16]. However, here also the state spaces of the processes change which requires some extra care for the varying metric structures in the Dirichlet forms. Let T N := {f : ΣN → RN } be equipped with the norm f 2T N :=
1 N
f (x)2 N qN (dx), R
ΣN
then the corresponding integration by parts formula for qN on ΣN reads
∇u, ξ T N = −
1
u, divqN ξ H N . N
(7)
To state the corresponding formula for E we introduce the Hilbert space of vector fields on G by T = L2 G × [0, 1], Qβ ⊗ dx , and the subset Θ ⊂ T Θ = span ζ ∈ T ζ (g, t) = w(g) · ϕ g(t) , w ∈ K, ϕ ∈ C ∞ [0, 1] : ϕ(0) = ϕ(1) = 0 . Lemma 4.9. Θ is dense in T . Proof. Let us first remove the condition φ(0) = φ(1), i.e. let us show that the T -closure of Θ coincides with that of Θ¯ = span{ζ ∈ T | ζ (g, t) = w(g)·ϕ(g(t)), w ∈ K, ϕ ∈ C ∞ ([0, 1])}. Since supg∈G w(g) < ∞ for any w ∈ K, it suffices to show that any ζ ∈ Θ¯ of the form ζ (g, t) = ϕ(g(t)) can be approximated in T by functions ζk (g, t) = ϕk (g(t)) with ϕk ∈ C ∞ ([0, 1]) and ϕk (0) = ϕk (1) = 0. Choose a sequence of functions ϕk ∈ C0∞ ([0, 1]) such that supk ϕk C([0,1]) < ∞ and ϕk (s) → ϕ(s) for all s ∈ ]0, 1[. Now for Qβ -almost all g it holds that {s ∈ [0, 1] | g(s) = 0} = {0} and {s ∈ [0, 1] | g(s) = 1} = {1}, such that φk (g(s)) → φ(g(s)) for Qβ ⊗ dx-almost all (g, s). The uniform boundedness of the sequence of functions ζk : G × [0, 1] → R together with dominated convergence w.r.t. the measure Qβ ⊗ dx the convergence is established. In order to complete the proof of the lemma note that Qβ -almost every g ∈ G is a strictly increasing function on [0, 1]. This implies that the set Θ¯ is separating the points of a full measure subset of G × [0, 1]. Hence the assertion follows from the following abstract lemma. 2
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Lemma 4.10. Let μ be a probability measure on a Polish space (X, d) and let A be a subalgebra of C(X) containing the constants. Assume that A is μ-almost everywhere separating on X, i.e. ˜ = 1 and for all x, y ∈ X˜ there is an a ∈ A such there exists a measurable subset X˜ with μ(X) that a(x) = a(y). Then A is dense in any Lp (X, μ) for p ∈ [1, ∞). Proof. We may assume w.l.o.g. that A is stable w.r.t. the operation of taking the pointwise inf and sup. Let u ∈ Lp (X), then we may also assume w.l.o.g. that u is continuous and bounded on X. By the regularity of μ we can approximate X˜ from inside by compact subsets Km such that μ(Km ) 1 − m1 . On each Km the theorem of Stone–Weierstrass tells that there is some am ∈ A such that u|Km − am |Km C(Km ) m1 , and by truncation am C(X) uC(X) . In particular, for > 0, μ(|am − u| > ) μ(X \ Km ) m1 , if m 1/, i.e. am converges to u on X in μprobability. Hence some subsequence am converges to u pointwise μ-a.s. on X, and hence the claim follows from the uniform boundedness of the am by dominated convergence. 2 The L2 -derivative operator ∇ defines a map ∇ : C1 (G) → T which by [25, Proposition 7.3], cf. [26], satisfies the integration by parts formula
∇u, ζ T = − u, divQβ ζ H ,
u ∈ C1 (G), ζ ∈ Θ,
where, for ζ (g, t) = w(g) · ϕ(g(t)), divQβ ζ (g) = w(g) · Vϕβ (g) + ∇w(g)(.), ϕ g(.) L2 (dx) with 1 Vϕβ (g) := Vϕ0 (g) + β
ϕ (0) + ϕ (1) ϕ g(x) dx − 2
0
and Vϕ0 (g) :=
ϕ (g(a+)) + ϕ (g(a−)) a∈Jg
2
−
δ(ϕ ◦ g) (a) . δg
Here Jg ⊂ [0, 1] denotes the set of jump locations of g and ϕ(g(a+)) − ϕ(g(a−)) δ(ϕ ◦ g) (a) := . δg g(a+) − g(a−) By formula (8) one can extend ∇ to a closed operator on D(E) such that E(u, u) = ∇u2T .
(8)
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4.3.1. Definition of E˜ Lemma 4.11. The functional ˜ u)1/2 := sup E(u,
ζ ∈Θ
− u, divQβ ζ H ζ T
˜ ˜ = u ∈ L2 G, Qβ E(u) <∞ on D(E)
(9)
˜ and E(u) ˜ is a Dirichlet form on L2 (G, Qβ ) extending E, i.e. D(E) ⊂ D(E) = E(u) for u ∈ D(E). Proof. First note that it makes no difference to (9) if Θ is replaced by the larger set Θ˜ = {ξ | ξ(g, s) = w(g)ϕ(g(s)), w ∈ C1 (G), ϕ ∈ C ∞ ([0, 1]), φ(0) = φ(1) = 0}. The lower semicontinuity of E˜ in L2 (G, Qβ ) is trivial as well as the fact that E˜ is an extension E, due to (8). To prove that E˜ is Markovian we use the stronger quasi-invariance of Qβ under certain transformations of G, cf. [25, Theorem 4.3]. Let h ∈ G be a C 2 -diffeomorphisms of [0, 1] and let τh−1 : G → G, τh (g) = h−1 ◦ g, then dτh−1 ∗ Qβ β (g) = Xh (g)Yh0 (g), dQβ
(10)
where Xh : G → R,
Xh (g) = e
ln h (g(s)) ds
and Yh0 (g) :=
h (g(a+)) · h (g(a−))
a∈Jg
δ(h◦g) δg (a)
1 . h (g(0)) · h (g(1−)) φ
Given ζ = w(·)φ(·) ∈ Θ we may apply formula (10) in the case when h = ht , where (0, ) × [0, 1] → [0, 1], (t, x) → ht (x) is the flow of smooth diffeomorphisms of [0, 1] induced from the ODE h˙ t (x) = φ(ht (x)) with initial condition h0 (x) = x. Together with the fact that the approxiβ mations of the logarithmic derivative of Xht Yh0t w.r.t. the variable t stay uniformly bounded (cf. [25, Section 5]) this yields lim
t→0
G
u(ht (g)) − u(g) · w(g) Qβ (dg) = − t
u divQβ ζ (g) Qβ (dg) G
˜ More precisely, u ∈ L2 (G, Qβ ) belongs to D(E) ˜ if and only if for any u ∈ D(E). Dφ u(g) := w-lim t→0
u(ht (g)) − u(g) t
exists in L2 G, Qβ
for all φ ∈ C ∞ ([0, 1]) with φ(0) = φ(1) = 0 and ∃Du ∈ T
s.t. Du, ζ T = − u, divQβ ζ L2 (G ,Qβ ) = Dφ u, w L2 (G ,Qβ )
˜ ∀ζ = w(·)φ(·) ∈ Θ.
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˜ u) = Du2 . Now for u ∈ D(E) ˜ and κ ∈ C 1 (R) by Taylor’s formula Moreover, E(u, T κ(u(ht (g))) − κ(u(g)) = κ (θ ) · (u(gt ) − u(g)) for some θ → u(g) for t → 0 such that in case of κ ∞ 1 Dφ κ ◦ u(g) = w-lim t→0
κ ◦ u(ht (g)) − κ ◦ u(g) = κ ◦ u(g) · Dφ u(g) t
in L2 G, Qβ .
Choose some sequence uk ∈ C1 (G) such that uk (g) → u(g) for Qβ -almost all g ∈ G. Then ζ˜k = κ ◦ uk (·) · w(·)φ(·) ∈ Θ. Since Dφ u and w both belong to L2 (G, Qβ ), by dominated convergence
κ ◦ u · Dφ u, w L2 (G ,Qβ ) = lim κ ◦ uk · Dφ u, w L2 (G ,Qβ ) k
= lim Du, ζ˜k T E˜ 1/2 (u, u) lim ζ˜k T k
k
E˜ 1/2 (u, u)ζ T . ˜ and E(κ ˜ ◦ u) E(u). ˜ Hence κ ◦ u in D(E) Applied to a uniformly bounded family κ : R → R which converges pointwise to κ(s) = min(max(s, 0), 1) the lower semicontinuity of E˜ yields the claim. 2 4.3.2. Convergence of integration by part formulas The convergence of (7) to (8) is established by the following lemma whose prove is given below. Lemma 4.12. For ζ ∈ Θ there exists a sequence of vector fields ζN : ΣN → RN such that divqN ζN ∈ H N converges strongly to divQβ ζ in H and such that ζN T N → ζ T for N → ∞. Proof. By linearity it suffices to consider the case ζ (g, t) = w(g) · ϕ(g(t)) with w(g) = n ki i=1 lfi (g). Choose 1 i ζN x , . . . , x N := wN x 1 , . . . , x N · ϕ x i with wN :=
n
i=1 (Φ
N (l
ki fi )) .
Then β
divqN ζN = wN · VN,ϕ + ∇wN , ϕ RN , with ϕ x 1 , . . . , x N := ϕ x 1 , . . . , ϕ x N and β VN,ϕ x 1 , . . . , x N :=
N N ϕ(x i+1 ) − ϕ(x i ) i β −1 + ϕ x . N +1 x i+1 − x i i=0
i=1
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We recall that for all bounded measurable u : [0, 1]N → R
u x 1 , . . . , x N qN (dx) =
u g(t1 ), . . . , g(tN ) Qβ (dg),
G
ΣN
with ti = i/(N + 1), i = 0, . . . , N + 1. Using this we get immediately ζN 2T N =
1 N
N ΣN i=1
→
2 2 wN (x)ϕ x i qN (dx) =
w (g) G
G
1 2
N 1 2 2 g(t1 ), . . . , g(tN ) wN ϕ g(ti ) Qβ (dg) N i=1
2 ϕ g(s) ds Qβ (dg) = ζ 2T .
0
To prove strong convergence of divqN ζN to divQβ ζ , by definition we have to show that there exists a sequence (dN ζ )N ⊂ H tending to divQβ ζ in H such that 2 lim lim supΦ M (dN ζ ) − divqM ζM H M = 0. N
M
The choice dN ζ (g) := divqN ζN g(t1 ), . . . , g(tN ) makes this convergence trivial, once we have proven that in fact (dN ζ )N converges to divQβ ζ in H . This is carried out in the following two lemmas. 2 Lemma 4.13. For Qβ -a.s. g we have β VN,ϕ g(t1 ), . . . , g(tN ) → Vϕβ (g),
as N → ∞,
and we have also convergence in Lp (G, Qβ ), p > 1. β
Proof. We rewrite VN,ϕ (g(t1 ), . . . , g(tN )) as N ϕ(g(ti+1 )) − ϕ(g(ti )) β (ti+1 − ti ) VN,ϕ g(t1 ), . . . , g(tN ) = β g(ti+1 ) − g(ti ) i=0
N −1 ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ(g(t1 )) − ϕ(g(t0 )) ϕ g(ti ) − + − g(t1 ) − g(t0 ) g(ti+1 ) − g(ti ) i=1
ϕ(g(tN +1 )) − ϕ(g(tN )) . + ϕ g(tN ) − g(tN +1 ) − g(tN )
(11)
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Note that all terms are uniformly bounded in g with a bound depending on the supremum norm β of ϕ and ϕ , respectively. Since the same holds for Vϕ (g) (cf. Section 5 in [25]), it is sufficient to show convergence Qβ -a.s. By the support properties of Qβ g is continuous at tN +1 = 1, so that the last line in (11) tends to zero. Using Taylor’s formula we obtain that the first term in (11) is equal to N
β
i=0
N 1 ϕ g(ti ) (ti+1 − ti ) + ϕ (γi ) g(ti+1 ) − g(ti ) (ti+1 − ti ), 2 i=0
1 for some γi ∈ [g(ti ), g(ti+1 )]. Obviously, the first term tends to β 0 ϕ (g(s)) ds and the second one to zero as N → ∞. Thus, it remains to show that the second line in (11) converges to ϕ (g(a+)) + ϕ (g(a−)) 2
a∈Jg
δ(ϕ ◦ g) ϕ (0) + ϕ (1) − (a) − . δg 2
(12)
Note that by the right-continuity of g the first term in the second line in (11) tends to −ϕ (0). Let now a2 , . . . , al−1 denote the l − 2 largest jumps of g on ]0, 1[. For N very large (compared with l) we may assume that a2 , . . . , al−2 ∈ ] N 2+1 , 1 − N 2+1 [. Put a1 := N 1+1 , al := 1 − N 1+1 . For j = 1, . . . , l let kj denote the index i ∈ {1, . . . , N}, for which aj ∈ [ti , ti+1 [. In particular, k1 = 1 and kl = N . Then i∈{k2 ,...,kl−1 }
l−1 ϕ(g(ti+1 )) − ϕ(g(ti )) δ(ϕ ◦ g) −→ (aj ) ϕ g(ti ) − ϕ g(aj −) − N →∞ g(ti+1 ) − g(ti ) δg j =2
−→
l→∞
a∈Jg
δ(ϕ ◦ g) (a). (13) ϕ g(a−) − δg
Provided l and N are chosen so large that g(ti+1 ) − g(ti ) C l for all i ∈ {0, . . . , N} \ {k1 , . . . , kl }, where C = sups |ϕ (s)|/6, again by Taylor’s formula we get for every j ∈ {1, . . . , l − 1} kj +1 −1
i=kj +1
ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ g(ti ) − g(ti+1 ) − g(ti ) kj +1 −1
=−
1 1 2 ϕ g(ti ) g(ti+1 ) − g(ti ) + ϕ (γi ) g(ti+1 ) − g(ti ) 2 6
i=kj +1
1 −→ − N →∞ 2
a j +1 −
g(aj +1 −)
aj +
g(aj +)
1 ϕ g(s) dg(s) + O l −2 = − 2
ϕ (s) ds + O l −2 .
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Summation over j leads to +1 −1 l−1 kj
j =1 i=kj +1
ϕ(g(ti+1 )) − ϕ(g(ti )) ϕ g(ti ) − g(ti+1 ) − g(ti )
1 −→ − N →∞ 2 l−1
g(aj +1 −)
ϕ (s) ds + O l −1
j =1 g(a +) j
1 =− 2
1
1 ϕ (s) ds + 2
0
−→ −
l→∞
l−1
g(a j +)
ϕ (s) ds + O l −1
j =2 g(a −) j
1 1 ϕ (1) − ϕ (0) + ϕ g(a+) − ϕ g(a−) . 2 2 a∈Jg
Combining this with (13) yields that the second line of (11) converges in fact to (12), which completes the proof. 2 Since wN (g(t1 ), . . . , g(tN )) converges to w in Lp (G, Qβ ), p > 0 (cf. proof of Lemma 4.5 above), the last lemma ensures that the first term of dN ζ converges to the first term of divQβ ζ in H , while the following lemma deals with the second term. Lemma 4.14. For Qβ -a.s. g we have ∇wN g(t1 ), . . . , g(tN ) , ϕ g(t1 ), . . . , g(tN ) RN → ∇w|g , ϕ g(.) L2 (0,1) ,
as N → ∞,
and we have also convergence in H . Proof. As in the proof of the last lemma it is enough to prove convergence Qβ -a.s. Note that
∇wN ( g ) L2 (0,1) , g ), ϕ( g ) RN = (N + 1) ιN ∇wN ( g ) , ιN ϕ(
writing g := (g(t1 ), . . . , g(tN )) and using the extension of ιN on RN . By triangle and Cauchy– Schwarz inequality we obtain (N + 1)ιN ∇wN ( g ) L2 (0,1) − ∇w|g , ϕ g(.) L2 (0,1) g ) , ιN ϕ( (N + 1)ιN ∇wN ( g ) L2 (0,1) + ∇w|g , ιN ϕ( g ) − ϕ g(.) L2 (0,1) g ) − ∇w|g , ιN ϕ( (N + 1)ιN ∇wN ( g ) L2 (0,1) g ) − ∇w|g L2 (0,1) ιN ϕ( g ) − ϕ g(.) L2 (0,1) , + ∇w|g L2 (0,1) ιN ϕ( which tends to zero by Lemma 4.8 and by the definition of ιN .
2
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4.3.3. Condition (Mosco I) and the lower bound Proposition 4.15. Let uN ∈ D(E N ) converge weakly to u ∈ H , then ˜ u) lim inf N · E N (uN , uN ). E(u, N →∞
Proof. Let u ∈ H and uN ∈ H N converge weakly to u. Let ζ ∈ Θ and ζ N be as in Lemma 4.12, then − u, divQβ ζ H = − lim uN , divqN ζN H N = lim N · ∇uN , ζN T N 1/2 lim inf N · ∇uN T N · ζN TN = lim inf N · E N (uN , uN ) · ζ T , such that
1/2 − u, divQβ ζ H ˜ u) 1/2 = sup E(u, lim inf N · E N (uN , uN ) . ζ T ζ ∈Θ
2
Finally we prove the lower bound in Theorem 2.1. Let as before N · E N be -convergent to E 0 and let u ∈ H be arbitrary. Then, using the second property of -convergence we find a sequence uN ∈ H N tending strongly to u such that E 0 (u) = limN N · E N (uN ). In particular, uN converges weakly to u and Proposition 4.15 implies that ˜ E(u) lim inf N · E N (uN ) = lim N · E N (uN ) = E 0 (u), N
N
and the claim follows. 5. Weak convergence of processes This section is devoted to the proof of Corollary 2.3. As usual we show compactness of the laws of (μN . ) and in a second step the uniqueness of the limit. 5.1. Tightness Proposition 5.1. The sequence (μN . ) is tight in CR+ ((P([0, 1]), τw )). Proof. According to Theorem 3.7.1 in [7] it is sufficient to show that the sequence ( f, μN . )N ∈N is tight, where f is taken from a dense subset in F ⊂ C([0, 1]). Choose F := {f ∈ C 3 ([0, 1]) | N N f (0) = f (1) = 0}, then f, μN t = F (XN ·t ) with F N (x) =
N 1 i f x . N i=1
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The condition f (0) = f (1) = 0 implies F N ∈ D(LN ) and N +1 N +1 N β f (x i ) − f (x i−1 ) f (x i ) − f (x i−1 ) i − + f x , N +1 x i − x i−1 x i − x i−1
N · LN F N (x) =
i=1
i=1
i=1
which can be written as N · LN F N (x) =
N +1 β f (x i ) − f (x i−1 ) N +1 x i − x i−1 i=1
+
N i=1
i f (x i ) − f (x i−1 ) f (x N +1 ) − f (x N ) f x − − . x i − x i−1 x N +1 − x N
Finally, this can be estimated as follows: N · LN F N (x) f (β + 1) + f =: C β, f 3 C ([0,1]) . ∞ ∞
(14)
This implies a uniform in N Lipschitz bound for the BV part in the Doob–Meyer decompoN ). The process X N has continuous sample paths with square field operator sition of F N (XN. N N (F, F ) = L (F 2 ) − 2F · LN F = |∇F |2 . Hence the quadratic variation of the martingale part N ) satisfies of F N (XN ·
F
N
N XN ·
t
N − F N XN · s =N ·
t
∇F N 2 X N dr = 1 r N
s
t N 2 i f xr dr s
2 (t − s)f ∞ .
i=1
(15)
Since N N 1 β f Di/(N+1) → f Dsβ ds F X0 = N 1
N
i=1
Qβ -a.s.,
0
N ) the law of F N (X0N ) is convergent and by stationarity we conclude that also the law of F N (XN ·t is convergent for every t. Using now Aldous’ tightness criterion the assertion follows once we have shown that
N N N XN ·τN → 0, E F N X N ·(τN +δN ) − F
as N → ∞,
(16)
for any given δN ↓ 0 and any given sequence of bounded stopping times (τN ). Now, the Doob– Meyer decomposition reads as N N N F XN XN ·τN = MτNN +δN − MτNN + ·(τN +δN ) − F
τN+δN
N
τN
N N · LN F X N ·s ds,
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N )] . Using (14) and (15) where M N is a martingale with quadratic variation [M N ]t = [F N (XN · t we get
N N N E F N X N XN ·τN E MτNN +δN − MτNN + C1 δN ·(τN +δN ) − F 2 1/2 E MτNN +δN − MτNN + C 1 δN 1/2 = C2 E M N τ +δ − M N τ + C 1 δN N N N 1/2 C3 δN + δN , for some positive constants Ci , which implies (16).
2
The argument above shows the balance of first and second order parts of N · LN as N tends to infinity. Alternatively one could use the symmetry of (X·N ) and apply the Lyons–Zheng decomposition for the same result. 5.2. Identification of the limit Throughout this section we will assume that Markov-uniqueness holds for the Wasserstein Dirichlet form E. An immediate consequence is that E coincides with the Dirichlet form E˜ defined in the previous section. ˜ Corollary 5.2 (Meyers–Serrin property). Assume E is Markov-unique, then E = E. Proof. The assumption means that E has no proper extension in the class of Dirichlet forms on L2 (G, Qβ ). By Lemma 4.11 we obtain E˜ = E which is the claim. 2 In order to identify the limit of the sequence (μN . ) we will work again with G-parametrization introduced in Section 4.1. For technical reasons we introduce the following modification of (μN . ) which is better behaved in terms of the map κ. Lemma 5.3. For N ∈ N define the Markov process νtN :=
N 1 μN δ0 ∈ P [0, 1] , t + N +1 N +1
then (ν.N ) is convergent on CR+ ((P([0, 1]), τw )) if and only if (μN . ) is. In this case both limits coincide. Proof. Due to Theorem 3.7.1 in [7] it suffices to consider the sequences of real-valued process N
f, μN . and f, ν. for N → ∞, where f ∈ C([0, 1]) is arbitrary. From 2 1 f, μN f ∞ t − f, δ0 N + 1 N +1 t0
N = sup sup f, μN t − f, νt t0
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N it follows that dC ( f, μN . , f, ν. ) → 0 almost surely, where dC is any metric on C([0, ∞), R) inducing the topology of locally uniform convergence. Hence for a bounded and uniformly dC continuous functional F : C([0, ∞), R) → R
E F f, μN − E F f, ν.N → 0 .
for N → ∞.
Since weak convergence on metric spaces is characterized by expectations of uniformly continuous bounded functions (cf. e.g. [21, Theorem 6.1]) this proves the claim. 2
by
Let (g·N ) := (κ(ν·N )) be the process (ν·N ) in the G-parameterization. It can also be obtained N gtN = ι XN ·t
with the imbedding ι = ιN ι : ΣN → G,
ι(x) =
N
x i · 1[ti ,ti+1 ) ,
i=0
with ti := i/(N + 1), i = 0, . . . , N + 1. The convergence of (μN · ) to (μ· ) in CR+ (P([0, 1]), τw ) is thus equivalent to the convergence of (g·N ) to (g· ) in CR+ (G, dL2 ). By Proposition 5.1 and Lemma 5.3 (g·N )N is a tight sequence of processes on G. The following statement identifies (g· ) as the unique weak limit. Proposition 5.4. For any f ∈ C(G l ) and 0 t1 < · · · < tl , N →∞ −→ E f (gt1 , . . . , gtl ) . E f gtN1 , . . . , gtNl Proposition 5.4 will essentially be implied by the following statement, which follows immediately by combining the Markov-uniqueness of E and Corollary 5.2 with Propositions 4.15 and 4.7. Theorem 5.5. (N · E N , H N ) converges to (E, H ) along Φ N in Mosco sense. By the abstract results in [19] Mosco-convergence is equivalent to the strong convergence of the associated semigroup operator, from which we will now derive the convergence of the finite dimensional distributions stated in Proposition 5.4. Lemma 5.6. For u ∈ C(G) let uN ∈ H N be defined by uN (x) := u(ιx), then uN → u strongly. Moreover, for any sequence fN ∈ H N with fN → f ∈ H strongly, uN · fN → u · f strongly. Proof. Let u˜ N ∈ H be defined by u˜ N (g) := u(g N ), where g N := i/(N + 1), then u˜ N → u in H strongly. Moreover,
N
i=1 g(ti )1[ti ,ti+1 ) , ti
lim limΦ M u˜ N − uM H M = lim limΦ M u˜ N − u˜ M H = lim u˜ N − uH = 0, N
M
N
M
N
:=
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where as above we have identified Φ M with the corresponding projection operator in L2 (G, Qβ ). For the proof of the second statement, let H f˜N → f in H such that limN lim supM Φ M f˜N − fM H M = 0. From the uniform boundedness of u˜ N it follows that also u˜ N · f˜N → u · f in H . In order to show H M uM · fM → u · f write M Φ (u˜ N · f˜N ) − uM · fM M Φ M (u˜ N · f˜N ) − uM · Φ M (f˜M ) M H H M ˜ + uM · fM − uM · Φ (fM ) H M . Identifying the map Φ M with the associated conditional expectation operator, considered as an orthogonal projection in H , the claim follows from M Φ (u˜ N · f˜N ) − uM · Φ M (f˜M ) M = Φ M (u˜ N · f˜N ) − u˜ M · Φ M (f˜M ) H H M = Φ (u˜ N · f˜N ) − Φ M (u˜ M · f˜M )H u˜ N · f˜N − u˜ M · f˜M H and uM · fM − uM · Φ M (f˜M ) M u∞ fM − Φ M (f˜N ) M H H M + u∞ Φ (f˜N ) − Φ M (f˜M )H M = u∞ fM − Φ M (f˜N )H M + u∞ Φ M (f˜N ) − Φ M (f˜M )H u∞ fM − Φ M (f˜N ) M H
+ u∞ f˜N − f˜M H such that in fact limN lim supM Φ M (u˜ N · f˜N ) − uM · fM H M = 0.
2
Proof of Proposition 5.4. It suffices to prove the claim for functions f ∈ C(G l ) of the form f (g1 , . . . , gl ) = f1 (g1 ) · f2 (g2 ) · · · · · fl (gl ) with fi ∈ C(G). Let PtN : H N → H N be the semiN )] = P N f, g group on H N induced by g N via Eg·qN [f (XN H N . From Theorem 5.5 and the t ·t N abstract results in [19] it follows that Pt converges to Pt strongly, i.e. for any sequence uN ∈ H N converging to some u ∈ H strongly, the sequence PtN uN also strongly converges to Pt u. Let fiN := fi ◦ ιN , then inductive application of Lemma 5.6 yields N N PtN fl · PtN fl−1 · PtN . . . f2N · PtN fN ... l −tl−1 l−1 −tl−2 l−2 −tl−3 1 1 −→ Ptl −tl−1 fl · Ptl−1 −tl−2 (fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 ) . . .
N →∞
strongly,
which in particular implies the convergence of inner products. Hence, using the Markov property of g N and g we may conclude that
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lim E f1 gtN1 . . . fl gtNl N
= lim E f1N XtN1 . . . flN XtNl N
N N N N N N N = lim 1, PtN −tl−1 fl · Ptl−1 −tl−2 fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 . . . H N l N = 1, Ptl −tl−1 fl · Ptl−1 −tl−2 (fl−1 · Ptl−2 −tl−3 . . . f2 · Pt1 f1 ) . . . H = E f1 (gt1 ) . . . fl (gtl ) . 2 6. A non-convex (1 + 1)-dimensional ∇φ-interface model We conclude with a remark on a link to stochastic interface models, cf. [12]. Consider an interface on the one-dimensional lattice N := {1, .√ . . , N}, whose location at time t is represented by the height variables φt = {φt (x), x ∈ N } ∈ N · ΣN with dynamics determined by√the generator L˜ N defined below and with the boundary conditions φt (0) = 0 and φ(N + 1) = N at ∂N := {0, N + 1}. L˜ N f (φ) :=
β −1 N +1
x∈N
1 1 ∂ − f (φ) + f (φ) φ(x) − φ(x − 1) φ(x + 1) − φ(x) ∂φ(x)
√ √ for φ ∈ Int( N √ · ΣN ) and with φ(0) := 0 and φ(N + 1) := N . L˜ N corresponds to LN as an operator on C 2 ( N · ΣN ) with Neumann boundary conditions. Note that this system involves a non-convex interaction potential function V on (0, ∞) given by V (r) = (1 − Nβ+1 ) log(r) and the Hamiltonian HN (φ) :=
N V φ(x + 1) − φ(x) ,
φ(0) := 0,
φ(N + 1) :=
√
N.
x=0
Then, the natural stationary distribution of the interface is the Gibbs measure μN conditioned on √ N · ΣN : μN (dφ) :=
1 exp −HN (φ) 1{(φ(1),...,φ(N ))∈√N ·ΣN } dφ(x), ZN x∈N
where ZN is √a normalization constant. Note that μN is the corresponding measure of qN on the state space N · ΣN . Suppose now that (φt )t0 is the stationary process generated by L˜ N . Then the space–time scaled process 1 Φ˜ tN (x) := √ φN 2 t (x), N
x = 0, . . . , N + 1,
taking values in ΣN , is associated with the Dirichlet form N · E N . Introducing the G-valued fluctuation field Φ˜ tN (x)1[x/(N+1),(x+1)/(N +1)) (ϑ), ΦtN (ϑ) := ιN Φ˜ tN (ϑ) = x∈N
ϑ ∈ [0, 1),
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by our main result we have weak convergence for the law of the equilibrium fluctuation field Φ N to the law of the nonlinear diffusion process κ(μ. ) on G, which is the G-parametrization of the Wasserstein diffusion. References [1] S. Albeverio, M. Röckner, Classical Dirichlet forms on topological vector spaces—closability and a Cameron– Martin formula, J. Funct. Anal. 88 (2) (1990) 395–436. [2] S. Albeverio, M. Röckner, Dirichlet form methods for uniqueness of martingale problems and applications, in: Stochastic Analysis, Ithaca, NY, 1993, Amer. Math. Soc., Providence, RI, 1995, pp. 513–528. [3] A. Alonso, F. Brambila-Paz, Lp -continuity of conditional expectations, J. Math. Anal. Appl. 221 (1) (1998) 161– 176. [4] S. Andres, M.-K. von Renesse, Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition, 2009, submitted for publication. [5] D. Blount, Diffusion limits for a nonlinear density dependent space–time population model, Ann. Probab. 24 (2) (1996) 639–659. [6] E. Cépa, D. Lépingle, Diffusing particles with electrostatic repulsion, Probab. Theory Related Fields 107 (4) (1997) 429–449. [7] D.A. Dawson, Measure-valued Markov processes, in: École d’Été de Probabilités de Saint-Flour XXI—1991, in: Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1–260. [8] M. Döring, W. Stannat, The logarithmic Sobolev inequality for the Wasserstein diffusion, Probab. Theory Related Fields 145 (1–2) (2009) 189–209. [9] R. Durrett, L. Mytnik, E. Perkins, Competing super-Brownian motions as limits of interacting particle systems, Electron. J. Probab. 10 (35) (2005) 1147–1220. [10] A. Eberle, Uniqueness and Non-uniqueness of Semigroups Generated by Singular Diffusion Operators, SpringerVerlag, Berlin, 1999. ¯ [11] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co., Berlin, 1994. [12] T. Funaki, Stochastic interface models, in: Lectures on Probability Theory and Statistics, in: Lecture Notes in Math., vol. 1869, Springer, Berlin, 2005, pp. 103–274. [13] G. Giacomin, J.L. Lebowitz, E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, in: Stochastic Partial Differential Equations: Six Perspectives, Amer. Math. Soc., Providence, RI, 1999, pp. 107–152. [14] M. Grothaus, Y.G. Kondratiev, M. Röckner, N/V -limit for stochastic dynamics in continuous particle systems, Probab. Theory Related Fields 137 (1–2) (2007) 121–160. [15] C. Kipnis, C. Landim, Scaling Limits of Interacting Particle Systems, Springer-Verlag, Berlin, 1999. [16] A.V. Kolesnikov, Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures, J. Funct. Anal. 230 (2) (2006) 382–418. [17] P. Kotelenez, A class of quasilinear stochastic partial differential equations of McKean–Vlasov type with mass conservation, Probab. Theory Related Fields 102 (2) (1995) 159–188. [18] A.M. Kulik, Markov uniqueness and Rademacher theorem for smooth measures on infinite-dimensional space under successful filtration condition, Ukraïn. Mat. Zh. 57 (2) (2005) 170–186. [19] K. Kuwae, T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11 (4) (2003) 599–673. [20] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (2) (1994) 368–421. [21] K.R. Parthasarathy, Probability Measures on Metric Spaces, Probab. Math. Statist., vol. 3, Academic Press Inc., New York, 1967. [22] A.-S. Sznitman, Topics in propagation of chaos, in: École d’Été de Probabilités de Saint-Flour XIX—1989, in: Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 165–251. [23] J. Vaillancourt, On the existence of random McKean–Vlasov limits for triangular arrays of exchangeable diffusions, Stoch. Anal. Appl. 6 (4) (1988) 431–446. [24] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2003. [25] M.-K. von Renesse, K.-T. Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37 (3) (2009) 1114– 1191. [26] M.-K. von Renesse, M. Yor, L. Zambotti, Quasi-invariance properties of a class of subordinators, Stochastic Process. Appl. 118 (11) (2008) 2038–2057.
Journal of Functional Analysis 258 (2010) 3906–3921 www.elsevier.com/locate/jfa
On the existence of an extremal function in critical Sobolev trace embedding theorem ✩ A.I. Nazarov ∗ , A.B. Reznikov Saint-Petersburg State University, Universitetskii pr. 28, St. Petersburg, Russian Federation Received 28 October 2009; accepted 26 February 2010 Available online 16 March 2010 Communicated by H. Brezis
Abstract Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev–Poincaré inequality are established. It is shown that some of these conditions are sharp. © 2010 Elsevier Inc. All rights reserved. Keywords: Critical trace Sobolev inequality; Critical trace Sobolev–Poincaré inequality; Sharp constants
1. Introduction Let n 2, and let Ω be a domain in Rn with strictly Lipschitz boundary. For 1 < p < n denote by p = (n−1)p n−p the trace Sobolev exponent for p, that is the critical exponent for the trace embedding Wp1 (Ω) → Lq (∂Ω). Since the embedding operator Wp1 (Ω) → Lp (∂Ω) is non-compact, the problem of attainability of the norm of this operator (i.e. the problem of existence of an extremal function in the trace embedding theorem) is non-trivial. Corresponding problem for conventional embedding np is the Sobolev conjugate of p) was treated in many papers, Wp1 (Ω) → Lp∗ (Ω) (here p ∗ = n−p see, e.g., the recent survey [9] and further references therein. ✩
This paper was partially supported by grant NSh.227.2008.1. The first author was also supported by RFBR grant 08-01-00748. * Corresponding author. E-mail address:
[email protected] (A.I. Nazarov). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.018
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
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The problem for the trace embedding is considerably less investigated. Let us consider the inequality K(n, p) ≡
∇vp,Rn+
inf
v∈C˙∞ (Rn+ )\{0}
v(·, 0)p ,Rn−1
> 0,
(1)
where C˙∞ (Rn+ ) is the set of functions on Rn+ with bounded support. Obviously, the functional in (1) is invariant with respect to translations and dilations of v. It is well known that the infimum in (1) is attained on some function with an unbounded support. Escobar [4] conjectured that the minimizer in (1) is − n−p wε (x) = x − x ε p−1 ,
(2)
with x ε = (0, . . . , 0, −ε), and proved it for p = 2. Later his conjecture was proved in full generality in the remarkable paper [8]. The result of [8] implies K(n, p) =
n−p p−1
1 p
1 (n−1)p ωn−2 n−1 n−1 ·B , . 2 2 2(p − 1)
In this paper we consider the critical trace embedding in bounded domains, i.e. the inequality λ1 (n, p, Ω) =
inf
v∈Wp1 (Ω)\{0}
vWp1 (Ω) vp ,∂Ω
p
p
>0
(I) p
(the norm of the numerator is defined as vW 1 (Ω) = ∇vp,Ω + vp,Ω ). p
Our first result reads as follows. Theorem 1. Let n 2, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω) > 0 and for 1 < p < n+1 2 + β, the infimum in (I) is attained. Remark 1. By standard argument it follows that under suitable normalization the extremal function in (I), if it exists, is a positive solution to the non-linear Neumann problem − p u + up−1 = 0 in Ω;
|∇u|p−2
∂u = up −1 ∂n
on ∂Ω
(here p u = div(|∇u|p−2 ∇u)). Our main tool is the concentration-compactness principle of Lions ([6]; see also [7]). It is used in various forms; for the problem (I) it can be reformulated as follows. Proposition 1. Let Ω be a bounded domain in Rn with ∂Ω ∈ C 1 . Let the infimum in (I) satisfy the inequality λ1 (n, p, Ω) < K(n, p). Then the infimum is attained.
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Proof. Let us consider a minimizing sequence {vk }, normalized in Lp (∂Ω). Without loss of generality one can assume that vk v in Wp1 (Ω). By the Lions theorem ([6, Part 2]; see also [7, Lemma 2.2]), |vk |p |v|p + |∇vk |p μ |∇v|p + cj δ x − x j , Cj δ x − x j j
j
(the convergence is in the sense of measures on ∂Ω and on Ω, respectively), where {x j } is at most countable set of distinct points in ∂Ω while cj and Cj are positive constants. Since {vk } is a minimizing sequence, by verbal repetition of the proof of Theorem 2.2 [7] we obtain the alternative — either v is a minimizer of the corresponding problem and the set {x j } is empty, or v = 0 and the set {x j } contains a single point x 0 . In the second case c0 = 1 and p C0 = λ1 (n, p, Ω). Note that until now all the arguments do not use the smoothness of ∂Ω. Let the second case occur. Then, similarly to Corollary 2.1 [7], multiplying vk by a cutoff function we can assume that supports of vk are sufficiently small. Due to the assumption ∂Ω ∈ C 1 , the neighborhood of x 0 in the large scale looks like a half-space. Hence λ1 (n, p, Ω) K(n, p). This contradiction proves the desired statement. 2 Thus, to prove the attainability of the infimum in (I) it is sufficient to present a function such that the quotient in (I) is less than K(n, p). Following [2], see also [7], we succeed, constructing a function with a small support, simulating the behavior of wε (x). Remark 2. The attainability of the infimum in (I) was proved in [5] under some additional assumption on Ω. This assumption means that the quotient in (I) taken on constant function is less than K(n, p). In particular, this is the case for “small” domains. Namely, for a domain with a smooth boundary define Ω as a “dilation” of Ω with the coefficient . Then for any 1 < p < n there exists ∗ > 0 such that the infimum in (I) is attained on Ω for < ∗ (note that, in contrast with (1), the quotient in (I) is not homogeneous with respect to dilations). Further, we slightly strengthen the statement of [5] and show that the infimum in (I) is attained for “small” domains without assumption of smoothness. Theorem 2. Let n 2. Suppose Ω is a bounded domain in Rn with strictly Lipschitz boundary. Then for any 1 < p < n there exists ∗ > 0 such that for < ∗ , the infimum in (I) is attained on Ω. On the other hand, for “large” domains the assumption of smoothness in Theorem 1 is essential. Theorem 3. Let n 2. Suppose Ω is a polyhedron in Rn , and 1 < p < n. Then there exists
∗ > 0 such that for > ∗ , the infimum in (I) is not attained on Ω. Finally, we consider the trace Sobolev–Poincaré inequality λ2 (n, p, Ω) = (here we use the notation v = |∂Ω|−1
inf
v∈Wp1 (Ω)\{c}
∂Ω
v dΣ).
∇vp,Ω >0 v − vp ,∂Ω
(II)
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
3909
Theorem 4. Let n 3, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω) > 0 and for 1 < p < n+1 2 + β, the infimum in (II) is attained. The structure of our paper is the following. In Section 2 we establish required integral estimates and prove Theorem 1. Theorems 2 and 3 are proved in Section 3; inequality (II) is considered in Section 4. In Section 5 we generalize Theorems 1 and 4 to the case of non-euclidean norm of gradient. Let us recall some notation. A point in Rn is denoted by x = (x , xn ) where x ∈ Rn−1 ; r stands for |x |. Rn+ = {x ∈ Rn : xn > 0}. Qρ = {x: r < ρ, 0 < xn < ρ} is a cylinder in Rn , Bρ (x 0 ) = {x: |x − x 0 | < ρ}. n/2 p ωn−1 = 2π is the area of the unit sphere in Rn . p = p−1 is the Hölder conjugate exponent Γ ( n2 ) to p. B is the Euler beta-function. We denote by oρ (1) a quantity which tends to zero as ρ → 0. We use letter C to denote various positive constants. To indicate that C depends on some parameters, we write C(. . .). 2. Proof of Theorem 1 Consider the least ball containing Ω. Let x 0 ∈ ∂Ω be a point of contact of Ω with this ball. Then all the principal curvatures of ∂Ω at x 0 are positive, and therefore H (x 0 ) > 0. We introduce a local coordinate system such that x 0 is the origin, x lies in the tangent plane and the axis xn is directed into Ω. Since ∂Ω ∈ C 2 , in some neighborhood of the origin ∂Ω is the graph of a function xn = F (x ), and F (x ) = (Ax , x ) + o(r 2 ) as r → 0, where A is a positive definite matrix. For sufficiently small ε > 0 and ρ > 0 we introduce the function u(x) = u(r, xn ) = ϕ(r, xn )wε (r, xn ),
(3)
where wε is defined in (2), while ϕ is a smooth cut-off function such that ϕ = 1 in Q ρ ; 2
and |∇ϕ|
ϕ = 0 in Rn \ Qρ ,
C ρ.
2.1. Estimate of ∇up,Ω for 1 < p
n+1 2
Let us apply the estimate ∇(f g)p |f ∇g|p + C |∇f g|p + |∇f g| · |f ∇g|p−1 to ∇u. Since ∇ϕ does not vanish only in Qρ \ Q ρ , one obtains 2
Ω
|∇ϕ · wε |p + |∇ϕ · wε | · |ϕ∇wε |p−1 dx
(4)
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A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
ρ/2 ρ
ρ ρ
C
ρ −p r n−2 dxn dr
+ 0 ρ/2
p(n−p)
(r 2 + xn2 ) 2(p−1)
ρ/2 0
p−1 ρ − n−p Cρ p−1 . · 1+ 2 2 r + xn
(5)
Furthermore, one has |ϕ∇wε |p |∇wε |p =
n−p p−1
p
x − x ε −(n−1)p .
The inequalities (4) and (5) imply
|∇u| dx p
Qρ ∩Ω
Ω
n−p p−1
p
n−p n−p x − x ε −(n−1)p dxn dx + Cρ − p−1 = I1 − I2 + Cρ − p−1 , (6)
where
ρ
I1 = |x |<ρ 0
n−p p−1
F (x )
I2 =
p
n−p p−1
|x |<ρ
0
p ρ ρ
x − x ε −(n−1)p dxn dx ,
p
x − x ε −(n−1)p dxn dx .
It is easy to see that I1 = ωn−2
n−p p−1
0 0
r n−2 dxn dr (r 2 + (xn + ε)2 )
p(n−1) 2(p−1)
E1 ε
− n−p p−1
,
(7)
where E1 = ωn−2
n−p p−1
p ∞ ∞
t n−2 dt ds p(n−1)
2 2 2(p−1) 0 0 (t + (s + 1) ) ωn−2 n − p p−1 n−1 n−1 , . = B 2 p−1 2 2(p − 1)
(8)
Further, since sup|x |<ρ |F (x )| → 0 as ρ → 0, we have I2 = =
n−p p−1 n−p p−1
p |x |<ρ
p ρ 0 S1
F (x ) dx (|x |2 + ε 2 )
p(n−1) 2(p−1)
· 1 + oρ (1)
(Aω, ω) + oρ (1) dω ·
r n dr (r 2 + ε 2 )
p(n−1) 2(p−1)
· 1 + oρ (1)
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
=
ωn−2 n − p n p−1
Thus, for p <
n+1 2
3911
ρ
p ε
1− n−p p−1
ε 0
· H x 0 + oρ (1) .
t n dt (t 2 + 1)
p(n−1) 2(p−1)
the following estimate holds: I2 E 2 ε
1− n−p p−1
1− n−p · H x 0 + oρ (1) − C, ρ p−1 .
(9)
Here ∞ t n dt ωn−2 n − p p ωn−2 n − p p n+1 n−1 E2 = , − 1 . (10) = ·B p(n−1) n p−1 2n p − 1 2 2(p − 1) (t 2 + 1) 2(p−1) 0
For p =
n+1 2
one gets the estimate I2 C ln(ρ/ε) − 1 .
(11)
2.2. Estimates of up ,∂Ω for 1 < p < n We have
p
u ∂Ω
dΣ
1 + |∇F (x )|2 dx p(n−1)
|x |< ρ2
(|x |2 + (ε + F (x ))2 ) 2(p−1)
|x |< ρ2
dx (|x |2 + (ε + F (x ))2 )
p(n−1) 2(p−1)
= |x |< ρ2
dx p(n−1)
(|x |2 + ε 2 ) 2(p−1)
εF (x ) F 2 (x ) × 1 − (n − 1)p 2 − · O (1) =: J1 − J2 − J3 . ρ |x | + ε 2 |x |2 + ε 2
(12)
It is easy to see that ρ
2 J1 = ωn−2 0
r n−2 dr (r 2 + ε 2 )
p(n−1) 2(p−1)
D1 ε
n−1 − p−1
− Cρ
n−1 − p−1
,
(13)
where ∞ D1 = ωn−2
p(n−1)
0
Further, we obtain
t n−2 dt (t 2 + 1) 2(p−1)
n−1 n−1 ωn−2 ·B , . = 2 2 2(p − 1)
(14)
3912
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921 ρ
J2 = (n − 1)p
2
(Aω, ω) + oρ (1) dω ·
0 S1
εr n dr p(n−1)
(r 2 + ε 2 ) 2(p−1)
+1
ρ
n−1 ωn−2 p 1− p−1 = ε n
2ε
t n dt (t 2 + 1)
0
D2 ε
n−1 1− p−1
p(n−1) 2(p−1) +1
· H x 0 + oρ (1)
· H x 0 + oρ (1) ,
(15)
where ωn−2 p D2 = n
∞
t n dt p(n−1)
0
(t 2 + 1) 2(p−1)
+1
=
ωn−2 p n+1 n−1 , . · B 2n 2 2(p − 1)
(16)
Finally, since |F (x )| Cr 2 , we have ρ
2 |J3 | Cρ 0
r n+1 dr (r 2 + ε 2 )
p(n−1) 2(p−1)
Cρε
n−1 1− p−1
(17)
.
2.3. Estimates of up,Ω We have
ρ
ρ ρ up dx C
Ω
p(n−p)
0 0
r n−2 dxn dr (r 2 + (xn + ε)2 ) 2(p−1)
⎧ p 2 −n ⎪ ⎨ Cε p−1 , C(1 + ln(ρ/ε)), ⎪ ⎩ C,
Cε
p 2 −n p−1
ε ∞
t n−2 dt ds p(n−p)
0 0
(t 2 + (s + 1)2 ) 2(p−1)
√ if 1 < p√< n; if p = √n; if p > n.
2.4. Completion of the proof The estimates (12), (13), (15) and (17) yield
up dΣ D1 ε
n−1 − p−1
1− n−1 − D2 H (x0 ) + oρ (1) ε p−1 − C(ρ),
∂Ω
where D1 , D2 are defined in (14), (16). By making use of (6), (7), (9), (11) and (18) we obtain
(18)
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921 p
uW 1 (Ω) E1 ε
3913
− n−p p−1
p
⎧ n−p p 2 −n ⎪ 0 ) + o (1))ε 1− p−1 + Cε p−1 + C(ρ), ⎪ −E (H (x 2 ρ ⎪ ⎪ n−p ⎨ 0 ) + o (1))ε 1− p−1 + C ln ε −1 + C(ρ), −E (H (x 2 ρ + ⎪ 1− n−p ⎪ −E2 (H (x 0 ) + oρ (1))ε p−1 + C(ρ), ⎪ ⎪ ⎩ −C ln ε −1 + C(ρ), where E1 , E2 are defined in (8) and (10). p 2 −n Since 1 − n−p p−1 < p−1 for p > 1, for 1 < p < p
uW 1 (Ω) p
p
up ,∂Ω
E1 p/p
D1
n+1 2
√ if 1 < p < n; √ if p = n; √ if n < p < n+1 2 ; n+1 if p = 2 ,
we get
p D2 E 2 0 H x + o · 1+ − (1) ε + o(ε) . ρ p D1 E 1
(19)
n+1 2
Direct computation shows that for 1 < p <
p D2 E 2 2(n − p)(p − 1) < 0. − =− p D1 E 1 n(n − 2p + 1)
(20)
Hence, for ε and ρ sufficiently small, one has uWp1 (Ω) up ,∂Ω
1/p
<
E1
1/p
.
(21)
D1
For p = n+1 2 the relation (21) also holds true for ε and ρ sufficiently small. By continuity, for some β > 0 and p < n+1 2 +β 1/p
λ1 (n, p, Ω) <
E1
1/p
D1
= K(n, p).
The application of Proposition 1 completes the proof. Remark 3. One can see from the proof that the requirement ∂Ω ∈ C 2 can be relaxed. For example, we can assume that ∂Ω ∈ C 1 , and the normal map of ∂Ω is absolutely continuous. Remark 4. Theorem 1 is also valid for an arbitrary manifold Ω with a smooth boundary, if ∂Ω contains a point with the positive mean curvature (with respect to the inner normal). The proof is carried out in the same way. 3. On “small” and “large” domains Proof of Theorem 2. Assume that the infimum in (I) is not attained on Ω. Then the arguments of Proposition 1 show that a minimizing sequence vk , normalized in Lp (∂Ω), satisfies
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A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
|vk |p δ x − x 0 ,
p |∇vk |p λ1 (n, p, Ω) · δ x − x 0
(the convergence is in the sense of measures on ∂Ω and on Ω, respectively). Multiplying vk by a cut-off function we can assume that supports of vk are arbitrarily small. p This implies that λ1 (n, p, Ω) does not depend on . p Since ∂Ω is strictly Lipschitz, λ1 (n, p, Ω) cannot be zero. However, the quotient (I) on constants tends to zero as → 0. This contradiction proves the theorem. 2 Proof of Theorem 3. The proof is similar to [2, Theorem 7.2], see also [3, Theorem 2]. Suppose there exists an unbounded sequence of ’s such that the infimum in (I) is attained on Ω. Then, by a standard argument, the corresponding minimizers u are positive in Ω. Consider the functions v (x) = C( )u ( x) in Ω, with C( ) given by the normalization condition v p ,∂Ω = 1. Then the weak Euler equation for the v reads as follows:
|∇v |p−2 ∇v , ∇h dx + p Ω
v p−1 h dx = λp Ω
v p
−1
h dΣ,
(22)
∂Ω
where λ = λ1 (n, p, Ω). We claim that λ are bounded. Indeed, let us consider a point xˆ ∈ ∂Ω. It is easy to see that ˆ is a conic surface. Thus, for functions v supported for sufficiently small ρ the set ∂Ω ∩ Bρ (x) ˆ the quotient (I) in Ω does not depend on . We denote by γ (x) ˆ the infimum of the in Bρ (x), quotient (I) over the set of such functions v and arrive at ˆ λ γ := inf γ (x).
(23)
x∈∂Ω ˆ
Put h = v in (22). Then p
p
∇v p,Ω + p v p,Ω = λp .
(24)
Without loss of generality, v v in Wp1 (Ω). From (24) and (23) we conclude that v = 0. Arguing as in the proof of Theorem 2.2 [7] one deduces that there exists a point x 0 ∈ ∂Ω such that |v |p δ(x − x 0 ) in the sense of measures on ∂Ω. Therefore, for a given ρ > 0, v p ,∂Ω\B ρ (x 0 ) → 0.
(25)
2
Remark 5. Really, the more strong property supΩ\Bρ (x 0 ) v → 0 can be proved, but we do not need it. By η we denote a smooth cut-off function such that η = 0 in B ρ x 0 ; η = 1 in Rn \ Bρ x 0 . 2
We substitute h = ηp v into Eq. (22): ηp |∇v |p dx + p ηp−1 |∇v |p−2 ∇v , ∇η dx + p ηp v p dx λp ηp v p dΣ. Ω
Ω
Ω
∂Ω
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
3915
Recalling (23), we arrive at p η∇v p,Ω
+
p
p ηv p,Ω
γ
p−1
ηp v p dΣ + Cv p,Ω · η∇v p,Ω .
p
(26)
∂Ω
The Hölder inequality and (25) give us p p −p p p p η v dΣ = ηp v p dΣ ηv p ,∂Ω · v p ,∂Ω\B ρ (x 0 ) = ηv p ,∂Ω · o (1). ∂Ω
2
∂Ω\B ρ (x 0 ) 2
We apply the trace embedding theorem to the function ηv : p p p p p ηv p ,∂Ω C ∇(ηv )p,Ω + ηv p,Ω C η∇v p,Ω + v p,Ω .
(27)
Substituting these relations to (26), we obtain
η∇v p,Ω v p,Ω
p
η∇v p,Ω C v p,Ω
p−1
η∇v p,Ω + o (1) · 1 + v p,Ω
p .
Therefore, η∇v p,Ω C. v p,Ω
(28)
Now we consider a smooth cut-off function η1 such that η1 = 1 in Bρ x 0 ;
in Rn \ B2ρ x 0 .
η1 = 0
The normalization of v and the inequality (23) yield p
p
v p ,∂Ω∩B
ρ (x
0)
v p ,∂Ω∩B
ρ (x
1 γ p (x 0 )
p
0)
η1 v p ,∂Ω
p
· η1 v W 1 (Ω) p
p 1 p · ∇(η1 v )p,Ω + η1 v p,Ω . p γ
By (4), p
v p ,∂Ω∩B
ρ (x
0)
1 p p p−1 · ∇v p,Ω + C v p,Ω + v p,Ω · ∇v p,Ω\B (x 0 ) . ρ γp
p
Substituting ∇v p,Ω from (24) and recalling (23), we get
p p p p p−1 · v p,Ω 1 − v p ,∂Ω∩B (x 0 ) + C v p,Ω + v p,Ω · ∇v p,Ω\B (x 0 ) . p ρ ρ γ Thus,
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A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
v p ,∂Ω\B (x 0 ) ∇v p,Ω\Bρ (x 0 ) p−1
p ρ . +C 1+ p γp v p,Ω v p,Ω p
(29)
Using normalization of v , (27) and (28), we obtain p
v p ,∂Ω\B
ρ (x
p
0)
p
v p,Ω
v p ,∂Ω\B
ρ (x
0)
p
v p,Ω
p
ηv p ,∂Ω p
v p,Ω
η∇v p,Ω p C, C 1+ v p,Ω
while the second term in (29) is bounded by (28): ∇v p,Ω\Bρ (x 0 ) v p,Ω
η∇v p,Ω C. v p,Ω
Therefore, the right-hand side in (29) is bounded. This gives a contradiction for large enough. 2 4. Proof of Theorem 4 The following statement is proved in the same way as Proposition 1. Proposition 2. Let Ω be as in Proposition 1. Let the infimum in (II) satisfy the inequality λ2 (n, p, Ω) < K(n, p). Then the infimum is attained. To construct a function with zero trace mean-value having the quotient (II) less then K(n, p) we modify the function (3). Following [2, Theorem 5.1], for large p we proceed by subtracting a suitable function with a small support while for small p we will subtract a constant. We need the following estimate for the function u defined in (3):
u dΣ ∂Ω
1 + |∇F (x )|2 dx n−p
|x |<ρ
(|x |2 + (F (x ) + ε)2 ) 2(p−1)
C
dx n−p
|x |<ρ
C |x |<ρ
(|x |2 + (F (x ) + ε)2 ) 2(p−1) ⎧ n−1− n−p ⎪ p−1 , ⎨ Cε dx C(1 + log(ρ/ε)), n−p ⎩ (|x |2 + ε 2 ) 2(p−1) ⎪ C,
if p < if p = if p >
2n−1 n ; 2n−1 n ; 2n−1 n .
(30)
1. First, consider the case p > 2n−1 n . Let u1 be a smooth non-negative function such that u1 |∂Ω ≡ 0 and supp u1 ∩ supp u = ∅. We define the function with zero trace mean-value u dΣ u(x) ˆ = u(x) − u1 (x) ∂Ω . ∂Ω u1 dΣ The estimates of Section 2.4 and (30) yield for p <
n+1 2
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921 p
∇ u ˆ p,Ω p
u ˆ p ,∂Ω
p
∇up,Ω + C(ρ) p
up ,∂Ω
E1 p/p
D1
3917
p D2 E 2 0 H x + oρ (1) ε + o(ε) . · 1+ − p D1 E 1
By (20), for ε and ρ sufficiently small, one has 1/p
E1 ∇ u ˆ p,Ω < 1/p . u ˆ p ,∂Ω D1
(31)
For p = n+1 2 this relation also holds true for ε and ρ sufficiently small. By continuity, for some β > 0 and p < n+1 2 +β 1/p
λ2 (n, p, Ω) < 2. Now we consider the case 1 < p γ ∈ (0, γ ∗ ), where γ ∗ = n+1−2p n−p ∈ (0, 1). The estimates of Section 2.2 yield
up dΣ D1 ε
E1
1/p
D1
2n−1 n .
= K(n, p).
(32)
Define u by (3) and put ρ = ε γ with some
− n−p −γ n−1 − D2 R x 0 + oε (1) ε p−1 − Cε p−1 .
n−1 − p−1
∂Ω
Immediate computation shows that γ∗
n−1 n−p < . p−1 p−1
Therefore, for γ ∈ (0, γ ∗ )
up dΣ D1 ε
n−1 − p−1
− n−p − D2 R x 0 + oε (1) ε p−1 .
∂Ω
Furthermore, the estimates of Section 2.1 imply |∇u|p dx E1 ε
− n−p p−1
1− n−p −γ n−p − E2 R x 0 + oε (1) ε p−1 + Cε p−1 .
Direct calculations result in γ∗
n−p n−p = − 1. p−1 p−1
Hence, for γ ∈ (0, γ ∗ ) |∇u|p dx E1 ε
− n−p p−1
1− n−p − E2 R x 0 + oε (1) ε p−1 .
(33)
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A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
Now we define the function with zero trace mean-value 1 u dΣ. u(x) ˆ = u(x) − |∂Ω| ∂Ω
Using (30) and the Minkowski inequality we conclude that
p
u ˆ p ,∂Ω
|u| ˆ supp u∩∂Ω 1 p
D1 ε
n−p − p(p−1)
1 p
dΣ
1−
1
up ,∂Ω
| supp u ∩ ∂Ω| p − |∂Ω|
u dΣ ∂Ω
n−p 1 D2 0 γ n−1 + p(p−1) p R x + o (1) ε − Cε Φ(ε) , ε p D1
(34)
where Φ(ε) =
ε
n−1− n−p p−1
,
1 + ln ε γ −1
if p =
Immediate calculations show that for n 3 and 1 < p γ
2n−1 n ; 2n−1 n .
if 1 < p <
2n−1 n ,
the inequality
n−p n−1 n−p +n−1− >1 + p p(p − 1) p−1
holds true for γ = γ ∗ and, therefore, for some γ ∈ (0, γ ∗ ). Choose such a γ , then (34) implies p u ˆ p ,∂Ω
1 p
D1 ε
− n−p p−1
p D2 0 1− R x + oε (1) ε . p D1
(35)
The relations (33), (35) and (20) imply (31) for ε small enough. Thus, the inequality (32) is valid in this case. The application of Proposition 2 completes the proof. 5. Some generalizations We can consider the quotients (I)–(II) with a more general definition of ∇vp,Ω in the numerator. Namely, let us consider an arbitrary norm M(x) in Rn . Assume, for simplicity, that the function M is strictly convex in non-radial directions, and M ∈ C 1 (Rn \ {0}). Denote by Mo (x) =
x, ξ M(ξ ) ξ =0
sup x, ξ = sup
M(ξ )1
the dual norm. In particular, if
M(x) = |x|q ≡
n k=1
then Mo (x) = |x|q .
1 q
|xk |q
,
1 < q < ∞,
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
3919
n Let ωn−1,M stand for the area of a unit sphere SM 1 = {x ∈ R : M(x) = 1}. In particular, for 1 n n M(x) = |x|q we have ωn−1,M = ωn−1,q = q[2Γ ( q + 1)] /Γ ( q ). We introduce an equivalent seminorm ∇vp,Ω,M in Wp1 (Ω) by
p
∇vp,Ω,M =
Mp (∇v) dx Ω
and replace ∇vp,Ω in the numerators of (I) and (II) by ∇vp,Ω,M . Denote by (IM ) (respectively, (IIM )) these quotients and by λk (n, p, Ω, M), k = 1, 2, the corresponding infima. It is proved in [8] that K(n, p, M) ≡
inf
v∈C˙∞ (Rn+ )\{0}
∇vp,Rn+ ,M v(·, 0)p ,Rn−1
=
∇wε,M p,Rn+ ,M wε,M (·, 0)p ,Rn−1
,
where − n−p p−1 , wε,M (x) = Mo x − x ε
(36)
and x ε = (0, . . . , 0, −ε). Verbatim repetition of the proof of Proposition 1 gives us Proposition 3. Let Ω be as in Proposition 1. Let the infimum in (IM ) satisfy the inequality λ1 (n, p, Ω, M) < K(n, p, M). Then the infimum is attained. Remark 6. Under suitable normalization the extremal function in (IM ), if it exists, is a positive solution to the Neumann-type problem − p,M u + up−1 = 0 in Ω;
n, ∇M(ξ ) ξ =∇u = up −1
on ∂Ω.
x∈∂Ω
Here
p,M u = div Mp−1 (ξ )∇M(ξ )ξ =∇u is the generalized p-Laplacian, generated by M. In particular, for M(x) = |x|q
p,M u = p,q u ≡
n p−q |∇u|q |uxj |q−2 uxj x . j =1
j
Notice that p,2 = p is the conventional p-Laplacian, while p,p is the so-called pseudo-pLaplacian (see, e.g., [1]). Theorem 5. Let n 2, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω, M) > 0 and for 1 < p < n+1 2 + β, the infimum in (IM ) is attained.
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A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
Proof. We introduce a local coordinate system just as in the proof of Theorem 1 and consider the function u(x) = ϕ(x)wε,M (x), where wε,M is defined in (36), while ϕ is a smooth cut-off function equal to 1 for Mo (x) < ρ/2 and to 0 for Mo (x) > ρ. As in Section 2.1, we obtain for p < n+1 2
p ( n−p p−1 ) dx
M (∇u) dx p
[Mo (x − x ε )]p (n−1)
Qρ ∩Ω
Ω
n−p p−1
p−1
E1M ε
− n−p p−1
−
n−p p−1
p
E2M ε
n−1 − p−1 −1
+ C(ρ).
Here E1M
∞ = Rn−1 0
E2M
= Rn−1
n−p p−1 dxn dx [Mo (x , xn + 1)]p (n−1)
dy = [Mo (y , 1)]p (n−1)
=
0 SM 1
Rn−1
(Ax , x ) dx (n−1) · 1 + oρ (1) = p [Mo (x , 1)]
∞ (Aω, ω) dω
SM 1
and f (t) = Mo (ω, t). Further, ∗ up dΣ
Mo (x ) ρ2
∂Ω
∞
r
− n−p p−1
− n−1 −1
r p−1 dr dω ; f p (n−1) (r −1 )
dr
f p (n−1) (r −1 )
· 1 + oρ (1) ,
0
dx [Mo (x , ε + F (x ))]p (n−1) ρ
=ε
n−1 − p−1
ε 0 SM 1
− n−1 −1
r p−1 dr dω . f p (n−1) (r −1 + (εr)−1 F (εrω))
By the Taylor formula, 1 F (εrω) 1 f (r −1 ) F (εrω) + = · 1−α · oρ (1) , f α (r −1 + (εr)−1 F (εrω)) f α (r −1 ) f (r −1 ) εr εrf (r −1 ) and hence
∗
up dΣ E1M ε
n−1 − p−1
− D2M ε
n−1 1− p−1
− C(ρ),
∂Ω
where D2M
= p (n − 1) SM 1
∞ (Aω, ω) dω 0
n−1
− r p−1 f (r −1 ) dr (n−1)+1 −1 · 1 + oρ (1) . p f (r )
A.I. Nazarov, A.B. Reznikov / Journal of Functional Analysis 258 (2010) 3906–3921
3921
Finally, the estimate (18) remains valid. Substituting these estimates into (IM ), similarly to (19), we arrive at p
p
∇up,Ω,M + up,Ω p
up ,∂Ω
p D2M n − p E2M ε + o(ε) . K p (n, p, M) · 1 + − p E1M p − 1 E1M
Integrating by parts, we obtain D2M =
n−2p+1 M p−1 E2
p λ1 (n, p, Ω, M) K p (n, p, M) ·
· (1 + oρ (1)). Therefore,
n − p E2M 1 + oρ (1) ε + o(ε) 1−2 n − 1 E1M
< K p (n, p, M). If p =
n+1 2
then
Ω
(37)
p−1 E M ε −1 − C ln(ε) + C(ρ). This gives (37) for Mp (∇u) dx ( n−p 1 p−1 )
p = n+1 2 and, by continuity, for some β > 0 and p < completes the proof. 2
n+1 2
+ β. The application of Proposition 3
In a similar way one can prove analog of Theorem 4. Theorem 6. Let n 3, and let Ω be a bounded domain in Rn with ∂Ω ∈ C 2 . Then for some β(Ω, M) > 0 and for 1 < p < n+1 2 + β, the infimum in (IIM ) is attained. References [1] M. Belloni, B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞, ESAIM Control Optim. Calc. Var. 10 (2004) 28–52. [2] A.V. Demyanov, A.I. Nazarov, On the existence of an extremal function in Sobolev embedding theorems with critical exponents, Algebra i Analiz 17 (5) (2005) 105–140 (in Russian). English transl. in: St. Petersburg Math. J. 17 (2006) 108–142. [3] O. Druet, Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds, J. Funct. Anal. 159 (1998) 217–242. [4] J.F. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J. 37 (1988) 687–698. [5] J. Fernandez Bonder, J.D. Rossi, On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. London Math. Soc. 37 (1) (2005) 119–125. [6] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Parts 1 and 2, Rev. Mat. Iberoamericana 1 (1985) 45–121 (1), 145–201 (2). [7] P.-L. Lions, F. Pacella, M. Tricarico, Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J. 37 (1988) 301–324. [8] B. Nazaret, Best constant in Sobolev trace inequalities on the half-space, Nonlinear Anal. 65 (10) (2006) 1977–1985. [9] A.I. Nazarov, Dirichlet and Neumann problems to critical Emden–Fowler type equations, J. Global Optim. 40 (2008) 289–303.
Journal of Functional Analysis 258 (2010) 3922–3953 www.elsevier.com/locate/jfa
Construction of strong solutions of SDE’s via Malliavin calculus Thilo Meyer-Brandis a , Frank Proske b,∗ a Department of Mathematics, LMU, Theresienstr. 39, D-80333 Munich, Germany b CMA, Department of Mathematics, University of Oslo, PO Box 1053 Blindern, N-316 Oslo, Norway
Received 30 October 2009; accepted 9 November 2009 Available online 16 December 2009 Communicated by Paul Malliavin
Abstract In this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients. We illustrate this approach by studying stochastic differential equations driven by the Wiener process. Using Malliavin calculus we derive the result of A.K. Zvonkin (1974) [31] for bounded and measurable drift coefficients as a special case of our analysis of SDE’s. Moreover, our approach yields the important insight that the solutions obtained by Zvonkin are even Malliavin differentiable. The latter indicates that the “nature” of strong solutions of SDE’s is tightly linked to the property of Malliavin differentiability. We also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions. © 2009 Elsevier Inc. All rights reserved. Keywords: Malliavin calculus; Strong solutions of SDE’s
1. Introduction Consider the stochastic differential equation (SDE) given by dXt = b(t, Xt ) dt + σ (t, Xt ) dBt ,
0 t T,
X0 = x ∈ Rd ,
* Corresponding author.
E-mail addresses:
[email protected] (T. Meyer-Brandis),
[email protected] (F. Proske). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.010
(1)
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3923
where Bt is a d-dimensional Brownian motion with respect to a filtration Ft , generated by Bt on a probability space (Ω, F , π). Further the drift coefficient b : [0, T ] × Rd → Rd and dispersion coefficient σ : [0, T ] × Rd → Rd×d are assumed to be Borel measurable functions. It is well known that if b(t, x) + σ (t, x) C 1 + x ,
x ∈ Rd , 0 t T ,
(2)
and b(t, x) − b(t, y) + σ (t, x) − σ (t, y) D x − y ,
x, y ∈ Rd , 0 t T ,
(3)
for constants C, D 0, then there exists a unique global strong solution Xt of (1), that is, a continuous Ft -adapted process Xt solving (1). Moreover we have that
T
Xt2 dt < ∞.
E 0
Stochastic differential equations were first studied by K. Itô (see [10]). Such types of equations can be e.g. used to model the motion of diffusing particles in a liquid. Many other important applications pertain to physics, biology, social sciences and mathematical finance. From a theoretical point of view these processes play a central role in the theory of Markov processes. The importance of SDE’s is emphasized by the fact that a very broad class of continuous Markov processes can be reduced to diffusion or quasidiffusion processes through certain transformations. See e.g. [27]. Stochastic differential equations are natural generalizations of ordinary differential equations, that is, of equations of the type dXt = b(t, Xt ), dt
X0 = x.
(4)
However, there is a peculiar difference between these two equations if the diffusion coefficient of the SDE is non-degenerate: It is known that a solution to (4) may not be unique or even not exist, if b is non-Lipschitzian. On the other hand, if the right-hand side of (4) is superposed by a (small) noise, that is, t Xt = x +
b(s, Xs ) ds + εBt , 0
then the regularization effect of the Brownian motion Bt enforces the existence of a unique global strong solution for all ε > 0, when b is e.g. bounded and measurable. This remarkable fact was first observed by Zvonkin in his celebrated paper [31]. An extension of this important result to the multidimensional case was given by Veretennikov [28]. The authors first employ estimates of solutions of parabolic partial differential equations to construct weak solutions. Then a pathwise uniqueness argument is applied to ensure a unique strong solution.
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
We mention that the latter results, which can be considered a milestone of the theory of SDE’s, have been generalized by Gyöngy and Martínez [8] and Krylov and Röckner [14], who impose deterministic integrability conditions on the drift coefficient to guarantee existence and uniqueness of strong solutions. As in [31] and [28], the authors first derive weak solutions. Here the approach of Gyöngy and Martínez rests on the Skorohod embedding, whereas the method of Krylov and Röckner is based on an argument of Portenko [23]. Finally, the authors resort to pathwise uniqueness and obtain strong solutions. Another technique which relies on an Euler scheme of approximation and pathwise uniqueness can be found in Gyöngy and Krylov [7]. See also [6], where a modified version of Gronwall’s lemma is invoked. We repeat that the common idea in all the references above is the construction of weak solutions together with the subsequent use of the celebrated pathwise uniqueness argument obtained by Yamada and Watanabe [30] which can be formulated as weak existence + pathwise uniqueness
⇒
strong uniqueness.
(5)
In this paper we devise a new method to study strong solutions of SDEs of type (1) with irregular coefficients. Our method does not rely on the pathwise uniqueness argument but gives a direct construction of strong solutions. Further, to conclude strong uniqueness, we derive the following result that in some sense is diametrically opposed to (5): strong existence + uniqueness in law
⇒
strong uniqueness.
Our technique is mainly based on Malliavin calculus. More precisely, we use a compactness criterion based on Malliavin calculus combined with “local time variational calculus” and an approximation argument to show that a certain generalized process in the Hida distribution space satisfies the SDE. As our main result we derive stochastic (and deterministic) integrability conditions on the coefficients to guarantee the existence of strong solutions. Moreover, our method yields the insight that these strong solutions are Malliavin differentiable. As a special case of our technique we obtain the result of Zvonkin [31] for SDE’s with merely bounded and measurable drift coefficients together with the additional and rather surprising information that these solutions are Malliavin differentiable. The latter sheds a new light on solutions of SDE’s and raises the question to which extent the “nature” of strong solutions is tied to the property of Malliavin differentiability. We emphasize that, although in this paper we focus on SDE’s driven by Brownian motion, our technique exhibits the capacity to cover stochastic equations for a broader class of driving noises. In particular, it can be applied to SDE’s driven by Lévy processes, fractional Brownian motion or more general by fractional Lévy processes. Other applications are evolution equations on Hilbert spaces, SPDE’s and anticipative SDE’s. In particular, the applicability of our technique to infinite dimensional equations on Hilbert spaces seems very interesting since very little is known in this case. Such equations cannot be captured by the framework of the above authors. The latter is due to the fact that the authors’ techniques utilize specific estimates, which work in the Euclidean space, but fail in infinite dimensional spaces. See e.g. [8, Lemma 3.1], where an estimate of Krylov [13] for semimartingales is used. Our paper is inspired by ideas initiated in [19] and [25]. In [25] the author constructs strong solutions of SDE’s with functional discontinuous coefficients by using certain estimates for the n-th homogeneous chaos of chaos expansions of solutions. The paper [19] exploits a comparison theorem to derive solutions of SDE’s.
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3925
The paper is organized as follows: In Section 2 we give the framework of our paper. Here we review basic concepts of Malliavin calculus and Gaussian white noise theory. Then in Section 3 we illustrate our method by studying the SDE (1). Our main results are Theorems 4, 5 and 22. Section 4 concludes with a discussion of our method. 2. Framework In this section we recall some facts from Gaussian white noise analysis and Malliavin calculus, which we aim at employing in Section 3 to construct strong solutions of SDE’s. See [9,22,15] for more information on white noise theory. As for Malliavin calculus the reader is referred to [21,17,18,4]. 2.1. Basic elements of Gaussian white noise theory As mentioned in the Introduction we want to use generalized stochastic processes on a certain stochastic distribution space to analyze strong solutions of SDE’s. In the sequel we give the construction of the stochastic distribution space, which goes back to T. Hida (see [9]). From now on we fix a time horizon 0 < T < ∞. Consider a (positive) self-adjoint operator A on L2 ([0, T ]) with Spec(A) > 1. Let us require that A−r is of Hilbert–Schmidt type for some r > 0. Denote by {ej }j 0 a complete orthonormal basis of L2 ([0, T ]) in Dom(A) and let λj > 0, j 0 be the eigenvalues of A such that 1 < λ0 λ1 · · · → ∞. Let us assume that each basis element ej is a continuous function on [0, T ]. Further let Oλ , λ ∈ Γ be an open covering of [0, T ] such that −α(λ)
sup λj
j 0
sup ej (t) < ∞ t∈Oλ
for α(λ) 0. In what follows let S([0, T ]) denote the standard countably Hilbertian space constructed from (L2 ([0, T ]), A). See [22]. Then S([0, T ]) is a nuclear subspace of L2 ([0, T ]). We denote by S ([0, T ]) the corresponding conuclear space, that is, the topological dual of S([0, T ]). Then the Bochner–Minlos theorem provides the existence of a unique probability measure π on B(S ([0, T ])) (Borel σ -algebra of S ([0, T ])) such that eiω,φ π(dω) = e
− 12 φ2 2
L ([0,T ])
S ([0,T ])
holds for all φ ∈ S([0, T ]), where ω, φ is the action of ω ∈ S ([0, T ]) on φ ∈ S([0, T ]). Set Ωi = S [0, T ] ,
Fi = B S [0, T ] ,
for i = 1, . . . , d. Then the product measure
μi = π,
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
×μ d
μ=
(6)
i
i=1
on the measurable space (Ω, F ) :=
d
i=1
Ωi ,
d
Fi
(7)
i=1
is referred to as d-dimensional white noise probability measure. Consider the Doléans–Dade exponential
1 2 e(φ, ˜ ω) = exp ω, φ − φL2 ([0,T ];Rd ) , 2 d (1) (d) d for d ω = (ω1 , . . . , ωd ) ∈ (S ([0, T ])) and φ = (φ , . . . , φ ) ∈ (S([0, T ])) , where ω, φ := i=1 ωi , φi . n be the n-th completed symmetric tensor product of In the following let ((S([0, T ]))d )⊗ ˜ ω) is holomorphic in φ around zero. Hence there (S([0, T ]))d with itself. One verifies that e(φ, n ) such that exist generalized Hermite polynomials Hn (ω) ∈ (((S([0, T ]))d )⊗
e(φ, ˜ ω) =
1 Hn (ω), φ ⊗n n!
(8)
n0
for φ in a certain neighbourhood of zero in (S([0, T ]))d . It can be shown that d ⊗ n Hn (ω), φ (n) : φ (n) ∈ S [0, T ] , n ∈ N0
(9)
is a total set of L2 (μ). Further one finds that the orthogonality relation
Hn (ω), φ (n) Hm (ω), ψ (m) μ(dω) = δn,m n! φ (n) , ψ (n) L2 ([0,T ]n ;(Rd )⊗n )
(10)
S ([0,T ]) n m is valid for all n, m ∈ N0 , φ (n) ∈ ((S([0, T ]))d )⊗ , ψ (m) ∈ ((S([0, T ]))d )⊗ where
δn,m
1 if n = m, 0 else.
Define Lˆ 2 ([0, T ]n ; (Rd )⊗n ) as the space of square integrable symmetric functions f (x1 , . . . , xn ) with values in (Rd )⊗n . Then the orthogonality relation (10) implies that the mappings φ (n) → Hn (ω), φ (n) n to L2 (μ) possess unique continuous extensions from (S([0, T ])d )⊗
⊗n In : Lˆ 2 [0, T ]n ; Rd → L2 (μ)
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3927
for all n ∈ N. We remark that In (φ (n) ) can be viewed as an n-fold iterated Itô integral of φ (n) ∈ Lˆ 2 ([0, T ]n ; (Rd )⊗n ) with respect to a d-dimensional Wiener process (1) (d) Bt = Bt , . . . , Bt
(11)
(Ω, F , μ).
(12)
on the white noise space
It turns out that square integrable functionals of Bt admit a Wiener–Itô chaos representation which can be regarded as an infinite dimensional Taylor expansion, that is, L2 (μ) =
⊗n . In Lˆ 2 [0, T ]n ; Rd
(13)
n0
We construct the Hida stochastic test function and distribution space by using the Wiener–Itô chaos decomposition (13). For this purpose let Ad := (A, . . . , A),
(14)
where A was the operator introduced in the beginning of the section. We define the Hida stochastic test function space (S) via a second quantization argument, that is, we introduce (S) as the space of all f = n0 Hn (·), φ (n) ∈ L2 (μ) such that f 20,p :=
⊗n p 2 n! Ad φ (n) 2
L ([0,T ]n ;(Rd )⊗n )
<∞
(15)
n0
for all p 0. It turns out that the space (S) is a nuclear Fréchet algebra with respect to multiplication of functions and its topology is given by the seminorms · 0,p , p 0. Further one observes that e(φ, ˜ ω) ∈ (S)
(16)
for all φ ∈ (S([0, T ]))d . In the sequel we refer to the topological dual of (S) as Hida stochastic distribution space (S)∗ . Thus we have constructed the Gel’fand triple (S) → L2 (μ) → (S)∗ . The Hida distribution space (S)∗ exhibits the crucial property that it contains the white noise of the coordinates of the d-dimensional Wiener process Bt , that is, the time derivatives (i)
Wt belong to (S)∗ .
:=
d (i) B , dt t
i = 1, . . . , d,
(17)
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We shall also recall the definition of the S-transform which is an important tool to characterize elements of the Hida test function and distribution space. See [24]. The S-transform of a Φ ∈ (S)∗ , denoted by S(Φ), is defined by the dual pairing S(Φ)(φ) = Φ, e(φ, ˜ ω)
(18)
for φ ∈ (SC ([0, T ]))d . Here SC ([0, T ]) is the complexification of S([0, T ]). We mention that the S-transform is a monomorphism from (S)∗ to C. In particular, if S(Φ) = S(Ψ )
for Φ, Ψ ∈ (S)∗ ,
then Φ = Ψ. One checks that (i) S Wt (φ) = φ (i) (t),
i = 1, . . . , d,
(19)
for φ = (φ (1) , . . . , φ (d) ) ∈ (SC ([0, T ]))d . Finally, we need the important concept of the Wick or Wick–Grassmann product, which we want to use in Section 3 to represent solutions of SDE’s. The Wick product can be regarded as a tensor algebra multiplication on the Fock space and can be defined as follows: The Wick product of two distributions Φ, Ψ ∈ (S)∗ , denoted by Φ Ψ , is the unique element in (S)∗ such that S(Φ Ψ )(φ) = S(Φ)(φ)S(Ψ )(φ)
(20)
for all φ ∈ (SC ([0, T ]))d . As an example we find that ψ (m) Hn (ω), φ (n) Hm (ω), ψ (m) = Hn+m (ω), φ (n) ⊗
(21)
n m for φ (n) ∈ ((S([0, T ]))d )⊗ and ψ (m) ∈ ((S([0, T ]))d )⊗ . The latter in connection with (8) shows that
e(φ, ˜ ω) = exp ω, φ
(22)
for φ ∈ (S([0, T ]))d . Here the Wick exponential exp (X) of an X ∈ (S)∗ is defined as exp (X) =
1 X n , n!
n0
where X n = X · · · X, if the sum on the right-hand side converges in (S)∗ .
(23)
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3929
2.2. Some definitions and notions from Malliavin calculus In this section we briefly elaborate a framework for Malliavin calculus. Without loss of generality we consider the case d = 1. Let F ∈ L2 (μ). Then it follows from (13) that F=
Hn (·), φ (n)
(24)
n0
for unique φ (n) ∈ Lˆ 2 ([0, T ]n ). Assume that
2 nn!φ (n) L2 ([0,T ]n ) < ∞.
(25)
n1
Then the Malliavin derivative Dt of F in the direction Bt is defined by Dt F =
n Hn−1 (·), φ (n) (·, t) .
(26)
n1
We introduce the stochastic Sobolev space D1,2 as the space of all F ∈ L2 (μ) such that (25) is fulfilled. The Malliavin derivative D· is a linear operator from D1,2 to L2 (λ × μ), where λ denotes the Lebesgue measure. We mention that D1,2 is a Hilbert space with the norm · 1,2 given by F 21,2 := F 2L2 (μ) + D· F 2L2 ([0,T ]×Ω,λ×μ) .
(27)
We obtain the following chain of continuous inclusions: (S) → D1,2 → L2 (μ) → D−1,2 → (S)∗ ,
(28)
where D−1,2 is the dual of D1,2 . 3. Main results In this section we want to introduce a new technique to analyze strong solutions of stochastic differential equations. We will focus on the analysis of strong solutions of equations driven by Brownian motion with merely measurable drift b : [0, T ] × Rd → Rd , that is, dXt = b(t, Xt ) dt + dBt ,
0 t T,
X0 = x ∈ Rd ,
(29)
where Bt is a d-dimensional Brownian motion with respect to the stochastic basis (Ω, F , μ), {Ft }0tT
(30)
for a μ-augmented filtration {Ft }0tT generated by Bt . At the end of the section we show how equations with more general diffusion coefficients can be reduced to equations of type (29).
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While in this paper we are focusing on SDE’s driven by Brownian motion, we want to point out that our approach relies on a general principle that exhibits potential to study strong solutions of equations driven by a broader class of processes including the infinite dimensional case. Our study of strong solutions is inspired by the following observation, which was made in [16]. See also [20] for an analogue result on equations driven by Lévy processes including the infinite dimensional case. Proposition 1. Suppose that the drift coefficient b : [0, T ] × Rd → Rd in (29) is bounded and Lipschitz continuous. Then there exists a unique strong solution Xt of (29), which allows for the explicit representation ˜ ET (b) ϕ t, Xt(i) (ω) = Eμ˜ ϕ t, B˜ t(i) (ω)
(31)
(i)
for all ϕ : [0, T ] × R → R such that ϕ(t, Bt ) ∈ L2 (μ) for all 0 t T , i = 1, . . . , d. The object E (b) is given by ˜ := exp ET (b)(ω, ω)
d T
(j ) (j ) Ws (ω) + b(j ) s, B˜ s (ω) ˜ d B˜ s (ω) ˜
j =1 0
1 − 2
T
(j ) 2 (j ) ˜ Ws (ω) + b s, Bs (ω) ˜ ds .
(32)
0
˜ F˜ , μ), Here (Ω, ˜ (B˜ t )t0 is a copy of the quadruple (Ω, F , μ), (Bt )t0 in (30). Further Eμ˜ denotes a Pettis integral of random elements Φ : Ω˜ → (S)∗ with respect to the measure μ. ˜ The (j ) Wick product in the Wick exponential of (32) is taken with respect to μ and Wt is the white T (j ) (j ) noise of Bt in the Hida space (S)∗ (see (17)). The stochastic integrals 0 φ(t, ω) ˜ d B˜ s (ω) ˜ in (32) are defined for predictable integrands φ with values in the conuclear space (S)∗ . See [11] for definitions. The other integral type in (32) is to be understood in the sense of Pettis. n = T be a sequence of partitions of the interval [0, T ] Remark 2. Let 0 = t1n < t2n < · · · < tm n n | → 0. Then the stochastic integral of the white noise W (j ) can be apn −1 n with maxm |t − t i i+1 i=1 proximated as follows:
T
Ws (ω) d B˜ s (ω) ˜ = lim (j )
(j )
n→∞
0
mn (j ) (j ) (j ) B˜ n (ω) ˜ − B˜ n (ω) ˜ W n (ω) i=1
ti+1
ti
ti
in L2 (λ × μ; ˜ (S)∗ ). For more information about stochastic integration on conuclear spaces the reader may consult [11]. In what follows let us denote the expression on the right-hand side of (31) for ϕ(t, x) = x by Yti,b , that is, (i) Yti,b := Eμ˜ B˜ t ET (b)
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3931
for i = 1, . . . , d. Further set Ytb = Yt1,b , . . . , Ytd,b .
(33)
In particular, under the assumptions of Proposition 1, the solution is explicitly given as Xt = Ytb . Further, for notational convenience, we will also from now on assume that T = 1. The appearance of the representation formula (31) naturally raises the following questions: (i) Proposition 1 provokes to conjecture that Ytb in (33) also represents strong solutions of (29) with discontinuous drift coefficients b. So, if b is non-Lipschitzian, will it be possible to interpret Ytb under some conditions as a generalized stochastic process in the Hida distribution space which could be verified as a strong solution of (29)? (ii) The result also suggests that one could formally apply the Malliavin derivative Dt (see Section 2.2) to the generalized process Ytb without affecting the coefficient b. The latter gives rise to the following questions: Are strong solutions of (29) for a larger class of non-Lipschitzian drift coefficients Malliavin differentiable? Or is even Malliavin differentiability a property which is intrinsically linked to the “nature” of strong solutions? It turns out that the above questions can be positively affirmed as our main result shows (see Theorem 4). Before we state this theorem we need to introduce some notation and definitions. Assume that (1) (d) Bt = (Bt , . . . , Bt ), 0 t 1, is a Brownian motion starting in x ∈ Rd , whose components (i) Bt are defined on (Ωi , Fi , μi ), i = 1, . . . , d (see (7)). In the sequel we denote by (1) (d) := B1−t , Bˆ t := Bˆ t , . . . , Bˆ t the time-reversed Brownian motion. Note that Bˆ t
(i)
(i) (i) (i) Bˆ t = B1 + W˜ t −
0 t 1,
(34)
satisfies for each i = 1, . . . , d the equation
t ˆ (i) Bs ds, 1−s
0 t 1, a.e.,
(35)
0
ˆ (i)
where W˜ t , 0 t 1, are independent μi -Brownian motions with respect to the filtrations FtB (i) generated by Bˆ t , i = 1, . . . , d. See [5]. (i)
Definition 3. Let L(t) ∈ Rd×d , 0 t 1, be a continuous t matrix function. We shall say that L belongs to the class L ⊂ Rd×d iff L(t) commutes with s L(u) du for all 0 s t 1. We are coming to our main result: Theorem 4. Suppose that Bt = (Bt(1) , . . . , Bt(d) ), 0 t 1, is a Brownian motion starting in x ∈ Rd . Let b : [0, 1] × Rd → Rd be a Borel measurable function. Set b0 = b and let bn : [0, 1] × Rd → Rd , n 1, be a sequence of continuous functions with compact support such that bn (t, ·)
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is continuously differentiable, 0 t 1. Further assume that bn (·, X· ) ∈ L for all n 1, where (n) Xt is the solution to (29) associated with the drift bn and the derivative with respect to the space variable. Denoting by (·)1i,j d , Rd×d -matrices require that the following conditions are satisfied: (n)
(i)
1
sup E exp 512 n0
bn (s, Bs )2 ds
< ∞.
(36)
0
(ii) t (j ) (i) bn (s, Bs ) dBs sup sup E sup exp −λ n1 0tt 1 0λ1
t
1i,j d
36 < ∞.
(37)
(iii) t (j ) (i) sup sup E sup exp −λ bn (1 − s, Bˆ s ) d W˜ s n1 0tt 1 0λ1
t
1i,j d
36 < ∞.
(38)
(iv) t (i) Bˆ s (j ) ˆ ds bn (1 − s, Bs ) sup sup E sup exp λ 1−s n1 0tt 1 0λ1
t
1i,j d
36 < ∞.
(39)
(v) 1 sup E n1
(j ) bn (s, Bs )2+ε ds
12/(2+ε) <∞
(40)
0
for all j = 1, . . . , d for some ε > 2. (vi) 1 24/(2+ε) ˆ s(i) 1+ε/2 (j ) B bn (1 − s, Bˆ s ) sup E ds <∞ 1−s
n1
0
for all j = 1, . . . , d for some ε > 2.
(41)
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3933
(vii) Rn −→ 0,
(42)
n→∞
where the factor Rn is defined as 1/2 Rn = E[Jn ] with 1 d (j ) 2 Jn = bn (s, Bs ) − b(j ) (s, Bs ) ds 2 j =1
0
1 +
(j ) bn (s, Bs ) 2 − b(j ) (s, Bs ) 2 ds
2 .
0
Then there exists a global strong solution Xt to dXt = b(t, Xt ) dt + dBt ,
X0 = x ∈ Rd ,
which is explicitly represented by (33). Moreover, (i)
Xt ∈ D1,2 for all i = 1, . . . , d, 0 t 1. If in addition Xt is unique in law then strong uniqueness holds. As a consequence of Theorem 4 we now obtain the following result in the 1-dimensional case. Theorem 5. Let b : [0, 1] × R → R be bounded and Borel measurable. Then there exists a unique Malliavin differentiable solution to (29) which has the explicit form (33). Remark 6. In the 1-dimensional case the existence and uniqueness of strong solutions to (29) for bounded and measurable drift coefficients were first obtained by Zvonkin in his celebrated paper [31]. The extension to the multidimensional case was given by [28]. We point out that our solution technique grants the important additional insight that such solutions are Malliavin differentiable. We defer the proofs of the above two theorems to a later time point. First, in order to facilitate the following reading, we give an overview of the structure of the proof of Theorem 4. The proof rests on the following three pillars: (α) Compactness criterion for subsets of L2 (μ) via Malliavin calculus (see [3]). (β) Local time–space calculus (see [5]). (γ ) The white noise representation (31).
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Our programme to prove Theorem 4 essentially consists of two steps: Step 1. We consider a drift coefficient b : [0, 1] × Rd → Rd and an approximating sequence bn : [0, 1] × Rd → Rd , n 1, of functions in the sense of Theorem 4. Then we show that for (n) each 0 t 1 the sequence Xt = Ytbn , n 1, is relatively compact in L2 (μ; Rd ) by employing 2 a compactness criterion for L (μ; Rd ) (see [3]). In applying this criterion we resort to a “local time variational calculus” argument which is based on a local time–space integration concept as developed in [5]. Step 2. We show that Ytb is a well-defined object in the Hida distribution space for each 0 t 1, and prove by means of the S-transform (18) that Ytbn −→ Ytb n→∞
in L2 (μ; Rd ) for 0 t 1. Using a certain transformation property for Ytb we can finally verify Ytb as a solution to (29). In order to accomplish the first step of our programme we need some auxiliary results. The first result which is due to [3, Theorem 1] provides a compactness criterion for subsets of L2 (μ; Rd ). Theorem 7. Let {(Ω, A, P ); H } be a Gaussian probability space, that is, (Ω, A, P ) is a probability space and H a separable closed subspace of Gaussian random variables of L2 (Ω), which generate the σ -field A. Denote by D the derivative operator acting on elementary smooth random variables in the sense that n D f (h1 , . . . , hn ) = ∂i f (h1 , . . . , hn )hi ,
hi ∈ H, f ∈ Cb∞ Rn .
i=1
Further let D1,2 be the closure of the family of elementary smooth random variables with respect to the norm F 1,2 := F L2 (Ω) + DF L2 (Ω;H ) . Assume that C is a self-adjoint compact operator on H with dense image. Then for any c > 0 the set G = G ∈ D1,2 : GL2 (Ω) + DGL2 (Ω;H ) c is relatively compact in L2 (Ω). We also need the following technical result, which can be found in [3].
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3935
Lemma 8. Let vs , s 0, be the Haar basis of L2 ([0, 1]). For any 0 < α < 1/2 define the operator Aα on L2 ([0, 1]) by Aα vs = 2kα vs
if s = 2k + j
for k 0, 0 j 2k and Aα 1 = 1. Then for all β with α < β < (1/2), there exists a constant c1 such that 1 1
Aα f c1 f L2 ([0,1]) +
0 0
|f (t) − f (t )|2 dt dt |t − t |1+2β
1/2 .
In the sequel we will make use of the concept of stochastic integration over the plane with respect to Brownian local time. See [5]. Consider elementary functions f : [0, 1] × R → R given by
f (s, x) =
fij χ(sj ,sj +1 ] (s).χ(xi ,xi+1 ] (x),
(43)
(sj ,xi )∈
where (xi )1in , (fij )1in,1j m are finite sequences of real numbers, (sj )1j m a partition of [0, 1] and = {(sj , xi ), 1 i n, 1 j m}. Denote by {L(t, x)}0t1,x∈R the local time of a 1-dimensional Brownian motion B. Then integration of elementary functions with respect to L can be defined by 1 f (s, x) L(ds, dx) 0 R
=
fij L(sj +1 , xi+1 ) − L(sj , xi+1 ) − L(sj +1 , xi ) + L(sj , xi ) .
(44)
(sj ,xi )∈
The latter integral actually extends to integrands of the Banach space (H, · ) of measurable functions f such that 1 f = 2 0 R
1 + 0 R
1/2
2 2 ds dx x f (s, x) exp − √ 2s 2πs
2 ds dx xf (s, x) exp − x < ∞. √ 2s s 2πs
The proof of the following theorem is given in [5, Theorem 3.1, Corollary 3.2].
(45)
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Theorem 9. Let f ∈ H. (i) Then t f (s, x) L(ds, dx),
0 t 1,
0 R
exists and t f (s, x) L(ds, dx) f E 0 R
for 0 t 1. (ii) If f is such that f (t, ·) is locally square integrable and f (t, ·) continuous in t as a map from [0, 1] to L2loc (R), then t
f (s, x) L(ds, dx) = − f (·, B), B t ,
0 t 1,
0 R
where [·,·] stands for the quadratic covariation of processes. (iii) In particular, if f (t, x) is differentiable in x, then t
t f (s, x) L(ds, dx) = −
0 R
f (s, B) ds,
0 t 1,
0
where f (s, x) denotes the derivative in x. The proof of Theorem 4 relies on the following lemma. Lemma 10. Retain the conditions of Theorem 4. Then the sequence Xt(n) = Ytbn , n 1 (see (33)) is relatively compact in L2 (μ; Rd ), 0 t 1. (n)
Proof. We want to prove the relative compactness of the sequence of SDE solutions Xt n 1, by applying Theorem 7. Because of Lemma 8 it is sufficient to show that (n) 2 (n) 2 E X1 + E Aα D· X1 L2 ([0,1];Rd ) M
= Ytbn ,
(46)
for all n 1, where 0 M < ∞. (n) Without loss of generality we study the sequence X1 , n 1. Then by the conditions of (n) Theorem 4 the solutions Xt , n 1, are Malliavin differentiable (see [21,4,29]). Moreover the
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3937
chain rule with respect to the Malliavin derivative Dt implies that
(n) Dt X 1
1 =
bn s, Xs(n) Ds Xs(n) ds + Id ,
0 t 1, n 1.
(47)
t
It follows from our assumptions that the solution to the linear equation (47) for each n 1 is given by
(n) Dt X 1
1 = exp
bn
(n) s, Xs ds ∈ Rd×d ,
0 t 1.
(48)
t
See e.g. [1]. Fix 0 t t 1. Then we find that (n) (n) 2 E Dt X1 − Dt X1 1 1 2 (n) (n) = E exp bn s, Xs ds − exp bn s, Xs ds . t
t
So it follows from Girsanov’s theorem, the properties of supermartingales and (36) that (n) (n) 2 E D t X 1 − D t X 1 1 1 2 1/2 C · E exp bn (s, Bs ) ds − exp bn (s, Bs ) ds t
t
for a constant C which is independent of n. The latter in connection with the properties of evolution operators for linear systems of ODE’s, the mean value theorem and Hölder’s inequality give (n) (n) 2 E Dt X1 − Dt X1 4 1/2 1 4 t const.E exp bn (s, Bs ) ds · exp bn (s, Bs ) ds − Id t
t
4 1 4 const.E exp bn (s, Bs ) ds bn (s, Bs ) ds t
t
t
t 4 1/2 · sup exp λ bn (s, Bs ) ds 0λ1 t
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1 12 1/6 const.E exp bn (s, Bs ) ds t
t 12 1/6 t 12 1/6 · E bn (s, Bs ) ds E sup exp λ bn (s, Bs ) ds . 0λ1 t
(49)
t
In the next step of our proof we want to invoke a “local time variational calculus” argument based on Theorem 9 to get rid of the derivatives bn in (49). In order to simplify notation let us assume that the initial value x of the SDE (29 ) is zero. In the sequel we introduce the notation Li (ds, dx) (i) to indicate local time–space integration in the sense of Theorem 9 with respect to Bt (i.e. the i-th component of Bt ) on (Ωi , μi ), i = 1, . . . , d (see (7)). So using Theorem 9, point (iii), we infer from (49) that (n) (n) 2 E Dt X1 − Dt X1 1 (j ) const.E exp − bn (s, x) Li (ds, dx) t R
t (j ) ·E − bn (s, x) Li (ds, dx) t
1i,j d
R
1i,j d
12 1/6
12 1/6
(j ) · E sup exp λ − bn (s, x) Li (ds, dx) 0λ1
t
t
1i,j d
R
12 1/6
= const.I1 · I2 · I3 ,
(50)
where 1 (j ) I1 := E exp − bn (s, x) Li (ds, dx)
1i,j d
t R
t (j ) I2 := E − bn (s, x) Li (ds, dx) t
1i,j d
R
12 1/6 ,
12 1/6
and t (j ) bn (s, x) Li (ds, dx) I3 := E sup exp λ − 0λ1
t
R
1i,j d
12 1/6 .
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3939
Now we want to use the following decomposition of local time–space integrals (see the proof of Theorem 3.1 in [5]): t
t
fi s, Bs(i) dBs(i) +
fi (s, x) Li (ds, dx) = 0
0
1
fi 1 − s, Bˆ s(i) d W˜ s(i)
1−t
1 − 1−t
Bˆ s(i) fi 1 − s, Bˆ s(i) ds, 1−s
(51)
0 t 1, a.e. for fi ∈ H, i = 1, . . . , d (see Theorem 9). Here Bˆ t is the i-th component of the (i) time-reversed Brownian motion and W˜ t is a Brownian motion on (Ωi , μi ) with respect to the ˆ (i) filtration FtB , i = 1, . . . , d (see (35)). Let us apply (51) to establish some upper bounds for the factors I1 , I2 , I3 . (1) Estimate for I1 : (51) and Hölder’s inequality entail that (i)
1 (j ) (i) I1 = E exp − bn (s, Bs ) dBs t
· exp
1i,j d
1−t
(j ) bn (1 − s, Bˆ s ) d W˜ s(i)
−
1i,j d
0
1−t ˆ s(i) B (j ) ds · exp bn (1 − s, Bˆ s ) 1−s
1i,j d
0
1 (j ) E exp − bn (s, Bs ) dBs(i) t
1i,j d
1−t (j ) (i) · E exp − bn (1 − s, Bˆ s ) d W˜ s 0
12 1/6
36 1/18
1i,j d
(i) Bˆ s (j ) ˆ ds · E exp bn (1 − s, Bs ) 1−s
36 1/18
1−t
1i,j d
0
36 1/18 .
(2) Estimate for I3 : Similarly to (1) we find t (j ) (i) I3 E sup exp −λ bn (s, Bs ) dBs 0λ1
t
1i,j d
36 1/18
(52)
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1−t (j ) (i) ˆ ˜ · E sup exp −λ bn (1 − s, Bs ) d Ws 0λ1
1−t
1i,j d
1−t ˆ s(i) B (j ) ds · E sup exp λ bn (1 − s, Bˆ s ) 1−s 0λ1
36 1/18
1−t
1i,j d
36 1/18 .
(53)
(3) Estimate for I2 : The decomposition (51), Burkholder’s and Hölder’s inequality imply that
I2 const.
d i,j =1
12 1/6 t (j ) E bn (s, x) Li (ds, dx) R
t
12 1/6 12 1/6 1−t t d (j ) (j ) const. +E bn (1 − s, Bˆ s ) d W˜ s(i) E bn (s, Bs ) dBs(i) i,j =1
1−t
t
12 1/6 1−t ˆ s(i) B (j ) ds +E bn (1 − s, Bˆ s ) 1−s 1−t
const.
d i,j =1
6 1/6 1−t 6 1/6 t (j ) (j ) 2 2 bn (s, Bs ) ds bn (1 − s, Bˆ s ) ds +E E 1−t
t
12 1/6 1−t ˆ s(i) B (j ) ds +E bn (1 − s, Bˆ s ) 1−s 1−t
t 12/(2+ε) 1/6 d ε/(2+ε) (j ) 2+ε t − t bn (s, Bs ) E ds const. i,j =1
t
1−t 12/(2+ε) 1/6 ε/(2+ε) (j ) 2+ε bn (1 − s, Bˆ s ) + t − t E ds 1−t
2ε/(2+ε) + t − t E
1−t 24/(2+ε) 1/6 ˆ (i) 1+ε/2 (j ) bn (1 − s, Bˆ s ) Bs ds 1−s 1−t
for > 0. Hence by the estimates in (1), (2), (3) we conclude from (50) that ε/(2+ε) (n) (n) 2 E Dt X1 − Dt X1 C t − t for all 0 t t 1, n 1, where C 0 is a universal constant. Then using Theorem 7 and Lemma 8 completes the proof. 2
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3941
We are coming to Step 2 of our programme. Under the conditions of Theorem 4 we want to gradually prove the following: • Ytb in (33) is a well-defined object in the Hida distribution space (S)∗ , 0 t 1 (Lemma 11). • Ytb ∈ L2 (μ), 0 t 1 (Lemma 13). (n) • We invoke Theorem 9 and show that Xt = Ytbn converges to Ytb in L2 (μ; Rd ) for a subsequence, 0 t 1. • We apply a transformation property for Ytb (Lemma 15) and identify Ytb as a solution to (29). The first lemma gives a criterion under which the process Ytb belongs to the Hida distribution space. Lemma 11. Suppose that
1
Eμ exp 36
b(s, Bs )2 ds
< ∞,
(54)
0
where the drift b : [0, 1] × Rd → Rd is measurable. Then the coordinates of the process Ytb , defined in (33), that is, Yti,b = Eμ˜ B˜ t(i) ET (b) ,
(55)
are elements of the Hida distribution space. Proof. Without loss of generality we consider the case d = 1. Set Φ(ω, ˜ ω) = ϕ B˜ t (ω) ˜ ET (b)(ω, ω) ˜ and 1 1 φ 2 1 E Mt = exp b(t, B˜ t ) + φ(t) d B˜ t − b(t, B˜ t ) + φ(t) dt 2 0
0
for φ ∈ SC ([0, 1]), where s Msφ (ω) ˜ =
b t, B˜ t (ω) ˜ + φ(t) d B˜ t (ω). ˜
0
Using the Kubo–Yokoi delta function (see [15, Theorem 13.4]), the characterization theorem of Hida distributions and the concept of G-entire functions (see [24]) we find that Φ is a well-defined map from Ω˜ to (S)∗ (up to equivalence). Further it follows from our assumption, Hölder’s inequality and the supermartingale property of Doléans–Dade exponentials that
3942
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
φ ˜ ·) (φ) = Eμ˜ ϕ(B˜ t )E Mt Eμ˜ S Φ(ω, 1 1 1 2 2 K · Eμ˜ E 2 b(t, B˜ t ) + Re φ(t) d B˜ t exp a φ(t) dt 0
0
K exp a|φ|20 , for φ ∈ SC ([0, 1]), where a, K 0 are constants and |φ|20 := rem 13.4] yields the result. 2
1 0
|φ(t)|2 dt. Then [15, Theo-
The next auxiliary result gives a justification for the factor Rn in (42) of Theorem 4. Lemma 12. Let bn : [0, 1] × Rd → Rd be a sequence of Borel measurable functions with b0 = b such that the integrability condition (36) is valid. Then 1 2 i,bn i,b S Yt − Yt (φ) const · Rn · exp 34 φ(s) ds 0
for all φ ∈ (SC ([0, 1]))d , i = 1, . . . , d, with the factor Rn as in (42). Proof. For i = 1, . . . , d we obtain by Proposition 1 and (19) that i,bn S Yt − Yti,b (φ) Eμ˜
d 1 (i) (j ) (j ) B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1
1 − 2
1
0
2 b(j ) (s, B˜ s ) + φ (j ) (s) ds
0
d 1 (j ) (j ) · exp bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s j =1 0
1 + 2
1
(j ) b(j ) (s, B˜ s )2 − bn (s, B˜ s )2 ds
0
1 +
φ
(j )
(j ) (j ) (s) b (s, B˜ s ) − bn (s, B˜ s ) ds − 1 .
0
Since exp(z) − 1 |z| exp |z|
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3943
it follows from Hölder’s inequality that i,bn 1 S Yt − Yti,b (φ) Eμ˜ |Qn |2 2 · Eμ˜
d 1 (i) (j ) (j ) B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1
1 − 2
1
(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds
0
2
1 2 exp 2|Qn | ,
0
where d (j ) (j ) 1 (j ) (j ) Qn = bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s + b (s, B˜ s )2 − bn (s, B˜ s )2 ds 2 1
1
j =1 0
1 +
0
(j ) φ (j ) (s) b(j ) (s, B˜ s ) − bn (s, B˜ s ) ds.
0
We find that 1 d 1 2 2 (j ) (j ) φ(s) ds · Eμ˜ bn (s, B˜ s ) − b(j ) (s, B˜ s ) d B˜ s Eμ˜ |Qn |2 9d 2 exp j =1
0
1 +
(j ) (j ) b (s, B˜ s )2 − bn (s, B˜ s )2 ds
0
2
1 +
0
(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds
0
1 = 3 exp
φ(s)2 ds Eμ˜ [Jn ],
0
where 2 1 1 d (j ) (j ) 2 (j ) (j ) 2 2 b (s, B˜ s ) − bn (s, B˜ s ) ds . Jn = bn (s, B˜ s ) − b (s, B˜ s ) ds + 2 j =1
0
0
Further we get that d 1 (i) (j ) B˜ t exp Eμ˜ b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1
1 − 2
1 0
0
(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds
2
exp 2|Qn |
3944
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
Eμ˜
d 1 (i) (j ) (j ) B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1
1 − 2
1
b
(j )
0
(s, B˜ s ) + φ
(j )
2 (s) ds
4 1 2
0
1 1 1 · √ Eμ˜ exp{−8 Re Qn } 2 + Eμ˜ exp{8 Re Qn } 2 2 1 1 + Eμ˜ exp{−8 Im Qn } 2 + Eμ˜ exp{8 Im Qn } 2 . By Hölder’s inequality again and the supermartingale property of Doléans–Dade exponentials we get the estimate Eμ˜ exp{−8 Re Qn } d 1 1 (j ) (j ) 2 (j ) (j ) Eμ˜ exp b (s, B˜ s ) − bn (s, B˜ s ) ds − 8 b (s, B˜ s )2 − bn (s, B˜ s )2 ds 128 j =1
1 +8
0
0
2 Re φ (j ) (s) ds +
0
1
(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds
1 2
0
1 2 Ln exp 4 φ(s) ds , 0
where
Ln = Eμ˜ exp
d j =1
1 +8
1 128
(j ) 2 (j ) b (s, B˜ s ) − bn (s, B˜ s ) ds
0
(j ) b (s, B˜ s )2 − bn(j ) (s, B˜ s )2 ds
1 2
.
0
Similarly we conclude that 1 2 1 2 φ(s) ds . Eμ˜ exp{8 Re Qn } Ln exp 4 0 1
1
Also Eμ˜ [exp{−8 Im Qn }] 2 and Eμ˜ [exp{8 Im Qn }] 2 have the same upper bound as in the previous inequality.
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3945
Finally we see that d 1 (i) (j ) (j ) Eμ˜ B˜ t exp b (s, B˜ s ) + φ (j ) (s) d B˜ s Re j =1
1 − 2
1
0
(j ) 2 b (s, B˜ s ) + φ (j ) (s) ds
4 1 2
0
1 1 2 2 (i) 8 1 18 4 ˜ ˜ b(s, Bs ) ds φ(s) ds . Eμ˜ Bt Eμˆ exp 512 exp 64 0
0
Altogether we get 1 2 i,bn i,b S Yt − Yt (φ) const · Rn · exp 34 φ(s) ds 0
2
with Rn as in (42).
The next lemma, which also is of independent interest, shows that under some conditions Ytb is square integrable. Lemma 13. Let bn : [0, 1] × Rd → Rd with b0 = b be a sequence of Borel measurable functions such that the conditions (36) and (42) are fulfilled. Further impose on each bn , n 1, Lipschitz continuity and a linear growth condition. Then the process Ytb given by (55) is square integrable for all t. Proof. Because of our assumptions on bn , n 1, we conclude from Lemma 11 that Ytbn are square integrable unique solutions to (29). Further, Hölder’s inequality and the supermartingale property of Doléans–Dade exponentials imply that i,bn 2 Yt 2
L (μ)
= Eμ˜
(i) 2 E B˜ t
1
bn (s, B˜ s ) d Bˆ s
0
1 1 4 2 M < ∞. exp 6 bn (s, B˜ s ) ds
const · sup Eμ˜ n1
(56)
0
Thus the sequence Ytbn is relatively compact in L2 (μ; Rd ) in the weak sense. From this we see that there exists a subsequence of Ytbn which converges to an element Zt ∈ L2 (μ; Rd ) weakly. Without loss of generality we assume that Ytbn → Zt
weakly for n → ∞.
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
In particular, since 1 E
φ(s) dBs ∈ Lp (μ),
p > 0,
0
one finds that Eμ Yti,bn E
1
φ(s) dBs
(i) → Eμ Zt E
1
φ(s) dBs
0
for n → ∞.
0
On the other hand the estimate as given in Lemma 12 yields Eμ Yti,bn E
1
φ(s) dBs
=
(i) Eμ˜ B˜ t E
1
0
bn (s, B˜ s ) + φ(s) d B˜ s
0
→ Eμ˜
B˜ t(i) E
1
= S Ytb (φ),
b(s, B˜ s ) + φ(s) d B˜ s
0
d φ ∈ SC [0, 1] .
Hence S Yti,b (φ) = S Zt(i) (φ),
d φ ∈ SC [0, 1] .
Since the S-transform is a monomorphism we get that Yt = Zt ∈ L2 (μ; Rd ).
2
We shall introduce a class M of approximating functions. Definition 14. We denote by M the class of Borel measurable functions b : [0, 1] × Rd → Rd for which there exists a sequence of approximating functions bn : [0, 1] × Rd → Rd in the sense of Rn → 0 in (42) such that the conditions of Theorem 4 hold. We are ready to state the following “transformation property” for Ytb . Lemma 15. Assume that b : [0, 1] × Rd → Rd belongs to the class M. Then ϕ (i) t, Ytb = Eμ˜ ϕ (i) (t, B˜ t )ET (b) a.e. for all 0 t 1, i = 1, . . . , d and ϕ = (ϕ (1) , . . . , ϕ (d) ) such that ϕ(Bt ) ∈ L2 (μ; Rd ). Proof. See [25, Lemma 16] or [19] for a proof.
2
Using the above auxiliary results we can finally give the proof of Theorem 4.
(57)
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3947
Proof of Theorem 4. We aim at employing the transformation property (57) of Lemma 15 to verify that Ytb is a unique strong solution of the SDE (29). To shorten notation we set t d t (j ) (j ) j =1 0 ϕ (s, ω) dBs and x = 0. 0 ϕ(s, ω) dBs := We first remark that Y·b has a continuous modification. The latter can be checked as follows: Since each Ytbn is a strong solution of the SDE (29) with respect to the drift bn we obtain from Girsanov’s theorem and (36) that 1 i,bn (i) i,bn 2 (i) 2 ˜ ˜ ˜ ˜ = Eμ˜ Bt − Bu E bn (s, Bs ) d Bs Eμ Yt − Yu 0
const · |t − u| for all 0 u, t 1, n 1, i = 1, . . . , d. By Lemma 10 we know that (bn )
Yt
(b)
→ Yt
in L2 μ; Rd
for a subsequence, 0 t 1. So we get that 2 Eμ Yti,b − Yui,b const · |t − u|
(58)
for all 0 u, t 1, i = 1, . . . , d. Then Kolmogorov’s lemma provides a continuous modification of Ytb . Since B˜ t is a weak solution of (29) for the drift b(s, x) + φ(s) with respect to the measure 1 dμ∗ = E( 0 (b(s, B˜ s ) + φ(s)) d B˜ s ) dμ we obtain that 1 i,b (i) ˜ ˜ ˜ S Yt (φ) = Eμ˜ Bt E b(s, Bs ) + φ(s) d Bs (i) = Eμ∗ B˜ t 1 = Eμ∗
0
(i) b (s, B˜ s ) + φ (i) (s) ds
0
t =
Eμ˜
b (s, B˜ s )E
1
(i)
0
b(u, B˜ u ) + φ(u) d B˜ u
ds + S Bt(i) (φ).
0
Hence the transformation property (57) applied to b gives S Yti,b (φ) = S
t
(i) i,b b u, Yu du (φ) + S Bt (φ). (i)
0
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
Then the injectivity of S implies that t Ytb
=
b s, Yub ds + Bt .
0
The Malliavin differentiability of Ytb follows from the fact that sup Yti,bn 1,2 M < ∞
n1
for all i = 1, . . . , d and 0 t 1. See e.g. [21]. Finally, if the solution Xt = Ytb is unique in law then our conditions allow the application of Girsanov’s theorem to any other strong solution. Then the proof of Proposition 1 (see e.g. [25, Proposition 1]) shows that any other solution necessarily takes the form Ytb . 2 As a consequence of Theorem 4 we obtain the following result. Corollary 16. Replace in Theorem 4 the conditions (36)–(39) by
1
sup E exp 2592d n0
5
bn (s, Bs )2 ds
<∞
(59)
0
and 1 (i) (j ) B s ds <∞ sup E exp 36 bn (s, Bs ) s
n1
(60)
0
for all 1 i, j d and retain all the other assumptions of Theorem 4. Then there exists a Malliavin differentiable strong solution to (29) which has the explicit representation (33). Moreover, if the solution is unique in law, then strong uniqueness holds. Proof. The proof directly follows from Girsanov’s theorem and Hölder’s inequality.
2
Remark 17. (i) A sufficient condition for uniqueness in law in Corollary 16 is e.g. the assumption that 1
b(t, Yt )2 dt < ∞ a.e.
0
for all weak solutions Yt to (29). See [12, Proposition 5.3.10]. (ii) It is worth mentioning that the Malliavin differentiability of the solution in Corollary 16 implies its quasi-continuity on the sample space. See [2].
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3949
(iii) We shall mention that for d = 2 the commutativity requirement bn (·, X· ) ∈ L (see Definition 3) is fulfilled if e.g. (n)
⎧ ∂ (2) ∂ (1) ⎪ ⎪ b (t, x) = −αn βn b (t, x), ⎨ ∂x1 n ∂x2 n ∂ (2) ∂ (1) ∂ (1) ⎪ ⎪ ⎩ bn (t, x) = bn (t, x) − (αn + βn ) b (t, x) ∂x2 ∂x1 ∂x2 n
(61)
holds for all n 1, 0 t 1, x ∈ R2 , where (αn ) and (βn ) are sequences of real numbers such that αn = βn , n 1. Now let us turn to the proof of Theorem 5. Proof of Theorem 5. Note that in the 1-dimensional case, the commutativity requirement (n) bn (·, X· ) ∈ L is always fulfilled. Then the proof follows from Corollary 16 and Lemma 18 (see below). 2 Lemma 18. 1 Bt dt <∞ E exp k t
0
for all k 0. Proof. See [26].
2
Remark 19. Let f : R → R be a bounded Borel measurable function. Then, if one combines Lemma 18 for d = 2 with Corollary 16 and the relation (61) one obtains that drift coefficients b : [0, 1] × R2 → R2 of the form
f (x1 + x2 ) b(t, x) = f (x1 + x2 ) admit unique Malliavin differentiable solutions Ytb to (29). We remark that many other examples of this type – also for general dimensions – can be constructed. In the next result, integrability conditions imposed on the drift coefficient are of nonexponential type. Corollary 20. Let us assume that the sequence of functions bn , n 0, with b0 = b satisfies all / [0, 1]. Further let S, R ∈ conditions of Theorem 4 aside from (36)–(41). Set bn (t, x) = 0 if t ∈ [0, ∞) and S < R. Suppose that
lim sup
sup
r0 n0 (s,x)∈[S,S+R]×Rd
s+r 2+ε bn (t, x + Bt−s ) E dt = 0 s
(62)
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
and
lim sup
sup
r0 n1 (s,x)∈[S,S+R]×R
s+r (j ) (j ) (x + Bt−s ) 1+ε E dt = 0 bn (t, x + Bt−s ) t d
(63)
s
for all j = 1, . . . , d and some ε > 2. Then uniqueness in law entails the existence of a unique Malliavin differentiable solution to (29). Proof. The proof follows from an argument of N.I. Portenko which exploits the Markovianity of the Wiener process [23]. 2 Remark 21. Using the m-dimensional Gaussian kernel p(t; x, y) the conditions (62) and (63) can be recast in the deterministic form s+r lim sup
sup
r0 n0 (s,y)∈[S,S+R]×Rd
bn (t, x)2+ε p(t − s; x, y) dx dt = 0
(64)
s Rd
and
lim sup
sup
r0 n1 (s,y)∈[S,S+R]×Rd
1+ε s+r (j ) bn (t, x) xj p(t − s; x, y) dx dt = 0 t
(65)
s Rd
for all j = 1, . . . , d and some ε > 2. Finally, we give an extension of Theorem 4 to a class of non-degenerate d-dimensional Itôdiffusions. Theorem 22. Consider the time-homogeneous Rd -valued SDE dXt = b(Xt ) dt + σ (Xt ) dBt ,
X0 = x ∈ Rd ,
0 t T,
(66)
where the coefficients b : Rd → Rd and σ : Rd → Rd × Rd are Borel measurable. Require that there exists a bijection Λ : Rd → Rd , which is twice continuously differentiable. Let Λx : Rd → L(Rd , Rd ) and Λxx : Rd → L(Rd × Rd , Rd ) be the corresponding derivatives of Λ and assume that Λx (y)σ (y) = idRd
for y a.e.
as well as Λ−1 is Lipschitz continuous.
T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
3951
Suppose that the function b∗ : Rd → Rd given by b∗ (x) := Λx Λ−1 (x) b Λ−1 (x) d d −1 −1 −1 1 σ Λ (x) [ei ], σ Λ (x) [ei ] + Λxx Λ (x) 2 i=1
i=1
satisfies the conditions of Theorem 4, where ei , i = 1, . . . , d, is a basis of Rd . Then there exists a Malliavin differentiable solution Xt to (66). Proof. Itô’s lemma applied to (66) implies that dYt = Λx Λ−1 (Yt ) b Λ−1 (Yt ) d d −1 −1 −1 1 σ Λ (Yt ) [ei ], σ Λ (Yt ) [ei ] dt + dBt , + Λxx Λ (Yt ) 2 i=1
Y0 = Λ(x),
i=1
0 t T,
where Yt = Λ(Xt ). Because of Theorem 4 there exists a Malliavin differentiable solution Yt to the last equation. Thus Xt = Λ−1 (Yt ) solves (66). Finally, since Λ−1 is Lipschitz continuous, Xt is Malliavin differentiable. See e.g. [21, Proposition 1.2.3]. 2 4. Conclusion In the previous sections we have presented a new method for the construction of strong solutions of SDE’s with discontinuous coefficients. This approach is more direct and more “natural” compared to other techniques in the literature in so far as it evades a pathwise uniqueness argument. Actually, using Malliavin calculus and “local time variational calculus” we directly verify a generalized process to be a strong solution. Our approach covers the result of Zvonkin [31] as a special case. Furthermore, our method gives the additional valuable insight that the solutions derived are Malliavin differentiable. The latter suggests that the nature of solutions of SDE’s is closely linked to the property of Malliavin differentiability. Finally, we mention that our technique relies on a general principle which seems flexible enough to study strong solutions of a broader class of stochastic equations, for example: 1. Infinite dimensional Brownian motion with drift: Q
dXt = b(t, Xt ) dt + dBt , Q
X0 = x ∈ H,
(67)
where Bt is a Q-cylindrical Brownian motion on a Hilbert space H and Q a positive symmetric trace class operator. This necessitates an infinite dimensional version of Theorem 7. Furthermore, it is possible to study Eq. (67) with an additional densely defined operator in the drift term.
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T. Meyer-Brandis, F. Proske / Journal of Functional Analysis 258 (2010) 3922–3953
2. Infinite dimensional jump SDE’s: dXt = γ (Xt− ) dLt ,
L0 = x ∈ H,
(68)
where Lt is an H -valued additive process. 3. Certain types of anticipative SDE’s, that is e.g. Brownian motion with non-adapted drift. 4. Similar equations for fractional Brownian motion or fractional Lévy processes. 5. Certain types of forward–backward SDE’s driven by Brownian motion. References [1] H. Amann, Gewöhnliche Differentialgleichungen, de Gruyter, 1983. [2] N. Bouleau, F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de Gruyter, 1991. [3] G. Da Prato, P. Malliavin, D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris, Sér. I 315 (1992) 1287–1291. [4] G. Di Nunno, B. Øksendal, F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Universitext, Springer, 2009. [5] N. Eisenbaum, Integration with respect to local time, Potential Anal. 13 (2000) 303–328. [6] S. Fang, T. Zhang, A class of stochastic differential equations with non-Lipschitzian coefficients: pathwise uniqueness and no explosion, C. R. Acad. Sci. Paris, Sér. I 337 (2003). [7] I. Gyöngy, N.V. Krylov, Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Related Fields 105 (1996) 143–158. [8] I. Gyöngy, T. Martinez, On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J. 51 (4) (2001) 763–783. [9] T. Hida, H.-H. Kuo, J. Potthoff, J. Streit, White Noise. An Infinite Dimensional Approach, Kluwer, 1993. [10] K. Itô, On a stochastic integral equation, Proc. Japan Acad. 22 (1946) 32–35. [11] G. Kallianpur, J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, IMS Lecture Notes Monogr. Ser., 1995. [12] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1988. [13] N.V. Krylov, Estimates of the maximum of the solution of a parabolic equation and estimates of the distribution of a semimartingale, Mat. Sb. (N.S.) 130 (172) (1986) 207–221 (in Russian). [14] N.V. Krylov, M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2) (2005) 154–196. [15] H.H. Kuo, White Noise Distribution Theory, Probab. Stoch. Ser., CRC Press, Boca Raton, FL, 1996. [16] A. Lanconelli, F. Proske, On explicit strong solutions of Itô-SDE’s and the Donsker delta function of a diffusion, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (3) (2004). [17] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in: Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto, 1976, Wiley, 1978, pp. 195–263. [18] P. Malliavin, Stochastic Analysis, Grundlehren Math. Wiss., vol. 313, Springer-Verlag, Berlin, 1997. [19] T. Meyer-Brandis, F. Proske, On the existence and explicit representability of strong solutions of Lévy noise driven SDE’s, Commun. Math. Sci. 4 (1) (2006). [20] T. Meyer-Brandis, F. Proske, Explicit representation of strong solutions of SDE’s driven by infinite dimensional Lévy processes, J. Theoret. Probab. (2009), in press, online: doi:10.1007/s10959-009-0226-6. [21] D. Nualart, The Malliavin Calculus and Related Topics, Springer, 1995. [22] N. Obata, White Noise Caculus and Fock Space, Lecture Notes in Math., vol. 1577, Springer-Verlag, Berlin, 1994. [23] N.I. Portenko, Generalized Diffusion Processes, Transl. Math. Monogr., vol. 83, Amer. Math. Soc., Providence, RI, 1990. [24] J. Potthoff, L. Streit, A characterization of Hida distributions, J. Funct. Anal. 101 (1991) 212–229. [25] F. Proske, Stochastic differential equations – some new ideas, Stochastics 79 (2007) 563–600. [26] P. Salminen, M. Yor, Properties of perpetual integral functionals of Brownian motion with drift, Ann. Inst. H. Poincare Probab. Statist. 41 (3) (2005) 335–347. [27] V. Skorohod, On the local structure of continuous Markov processes, Teor. Veroyatn. Primen. 11 (1966) 381–423; English transl. in Theory Probab. Appl. 11 (1966). [28] A.Y. Veretennikov, On the strong solutions of stochastic differential equations, Theory Probab. Appl. 24 (1979) 354–366.
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[29] S. Watanabe, Malliavin’s calculus in terms of generalized Wiener functionals, in: G. Kallianpur (Ed.), Theory and Application of Random fields, Springer, 1983. [30] T. Yamada, S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971) 155–167. [31] A.K. Zvonkin, A transformation of the state space of a diffusion process that removes the drift, Math. USSR (Sbornik) 22 (1974) 129–149.
Journal of Functional Analysis 258 (2010) 3955–3976 www.elsevier.com/locate/jfa
Flat orbits and kernels of irreducible representations of the group algebra of a completely solvable Lie group Mounir Elloumi a,∗ , Hanen Koubaa b , Jean Ludwig a a Laboratoire LMAM, UMR CNRS 7122, Département de Mathématiques, Université Paul Verlaine-Metz, Ile du Saulcy,
F-57045 Metz cedex 1, France b Département de Mathématiques, Faculté des Sciences de Sfax, Tunisia
Received 6 April 2009; accepted 5 March 2010 Available online 23 March 2010 Communicated by P. Delorme
Abstract We show that the kernel of an irreducible unitary representation π of the group algebra L1 (G) of a completely solvable Lie group G is given by the functions, whose abelian Fourier transform vanish on the Kirillov orbit Oπ of π if and only if this orbit Oπ is flat. This is a generalization of a result obtained before for nilpotent Lie groups. © 2010 Elsevier Inc. All rights reserved. Keywords: Completely solvable Lie groups; Flat orbits; Group algebras; Kernels of induced representations
1. Introduction Let G = exp(g) be a connected simply connected exponential Lie group with Lie algebra g. ˆ of G, the set of equivalence classes of irreducible unitary representations The unitary dual G of G, has a nice geometric parametrization via the Kirillov orbit method. It is well known that there is a one to one correspondence π → Oπ between the equivalence classes of irreducible representations π of G and the co-adjoint orbits Oπ in g∗ , the dual vector space of g. Furthermore, every unitary representation π of C ∗ (G), the C ∗ -algebra of G, is uniquely determined by * Corresponding author.
E-mail addresses:
[email protected] (M. Elloumi),
[email protected] (H. Koubaa),
[email protected] (J. Ludwig). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.004
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its kernel ker(π). We can therefore expect a description of the kernel in C ∗ (G) of an irreducible unitary representation in terms of the corresponding co-adjoint orbit. If now G = exp(g) is a connected simply connected nilpotent Lie group, then the mapping L1 (G) → L1 (g), f → f ◦ exp is an isometry and J. Ludwig in [5] has shown that the kernel ker(π) in the group algebra L1 (G) is given by the subspace ker(π) := {f ∈ L1 (G); f ◦ exp = 0 on Oπ } if and only if the orbit Oπ of π is flat, i.e. an affine linear subset of g∗ . Nilpotent Lie groups are ∗-regular, i.e. the canonical mapping from the primitive ideal space Prim(C ∗ (G)) to the ∗-primitive ideal space Prim∗ (L1 (G)) : I → I ∩ L1 (G) is a homeomorphism. In particular the kernel kerC ∗ (G) (π) of π in the C ∗ -algebra of G is given by the closure kerL1 (G) (π) in C ∗ (G) of the kernel kerL1 (G) (π) in L1 (G). This shows that in general a “nice” description of the kernel ker(π) in C ∗ (G) in terms of its Kirillov orbit is not available, if the orbit is not flat (see [6]). In the exponential case, the group G is no longer ∗-regular in general (see [2]), but it may ˆ (see [8]). In this paper, we extend the still be that kerC ∗ (G) (π) = kerL1 (G) (π) for every π ∈ G result obtained for nilpotent Lie groups in [5] to the completely solvable ones and show the ˆ such that the co-adjoint orbit Oπ of π is closed. Then Oπ is flat if and only following. Let π ∈ G if ker(π) = {f ∈ L1 (G): [(f ◦ exp).jg ]ˆ(Oπ ) = {0}}, where exp is the exponential mapping of G and jg dx denotes the pull-back of the Haar measure of the group G to the Lie algebra g via the exponential mapping. The paper contains three sections. In the first, we give the necessary definitions and properties of completely solvable Lie groups and of induced representations. In the second section we present several characterizations of flat co-adjoint orbits of completely solvable Lie groups. In the last section, we determine the kernels in the group algebra of the irreducible representations associated to flat orbits. 2. Preliminaries 2.1. Some notations and basic facts A connected, simply connected solvable Lie group with Lie algebra g is called exponential if the exponential mapping exp : g → G is a C ∞ diffeomorphism. In this case we denote by log the inverse mapping of exp. It is well known that G is exponential if and only if for every X ∈ g, the spectrum of the endomorphism ad(X) : gC → gC , ad(X)U := [X, U ], does not contain any number of the form λi with λ ∈ R∗ . If in particular the spectrum of ad(X) is real for every X ∈ g, then we say that G is completely solvable. In this case, there exists a Jordan–Hölder sequence g = g1 ⊃ g2 ⊃ · · · ⊃ gn ⊃ gn+1 = {0} of ideals in g, such that the dimension of gj /gj +1 = 1 for every j = 1, . . . , n. Choosing for every j an element Zj ∈ gj \ gj +1 , we obtain a Jordan– Hölder basis Z = {Z1 , . . . , Zn } of g and for every X ∈ g, we have a real number ρj (X), such that [X, Zj ] = ρj (X)Zj modulo gj +1 , j = 1, . . . , n. The linear functionals ρj : g → R are called the roots of g. If g is nilpotent then of course all the roots are 0. Since for exponential solvable groups G the exponential mapping is a diffeomorphism, we can transfer the multiplication in G to the vector space g and we obtain the so-called Campbell– Baker–Hausdorff multiplication ·g on g: X ·g Y = log(exp X · exp Y ) 1 1 1 X, [X, Y ] + Y, [Y, X] + · · · , = X + Y + [X, Y ] + 2 12 12
X, Y ∈ g.
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Let dg denote a left Haar measure on G. The pull-back exp∗ (dg) of the measure dg is the measure jg (X) dX on g, where jg (X) is the Jacobian of the left translation by X on g: 1 − e−adg (X) jg (X) = det ad (X) g
(see [9]). The group G acts on g by the adjoint representation AdG , i.e., AdG (g)(X) = Ad(g)(X) = ead(log(g)) X,
g ∈ G, X ∈ g,
and on g∗ by the co-adjoint representation Ad∗G , i.e.,
Ad∗G (g)l, X := l, AdG g −1 (X) =: g.l, X ,
g ∈ G, l ∈ g∗ , X ∈ g.
We denote by g∗ /G the space of the co-adjoint G-orbits O(l) = {g.l: g ∈ G}, l ∈ g∗ . Let g(l) = {X ∈ g: l, [X, g] = {0}} be the stabilizer of l ∈ g∗ in g. It is also the Lie algebra of G(l) = {g ∈ G: g.l = l}. A co-adjoint orbit O(l) of l ∈ g∗ is said to be saturated with respect to an ideal g0 of g, if O(l) = O(l) + g⊥ 0 . In this case we have that g(l) ⊂ g0 . So we can say, in the case where g0 is of codimension 1 in g, that
dim O(l0 ) = dim O(l) − 2,
l0 = l|g0 , O(l0 ) = exp (g0 ) · l0 .
Let dg be a left Haar measure on G and let ΔG be the modular function of G, which is defined by the formula
ξ gx −1 dg = ΔG (x)
G
ξ(g) dg, G
for all x ∈ G and for every ξ belonging to the space Cc (G) of continuous functions on G with compact support. We have thus that
−1
ΔG (x) = det Ad(x) = exp − tr ad(log x)
(x ∈ G).
Let H be a closed connected subgroup of G with corresponding Lie algebra h. We denote by ΔH,G the real character of H defined by ΔH,G (h) =
ΔH (h) ΔG (h)
(h ∈ H ).
Hence, we have
ΔH,G (h) = exp tr adg/h (log h)
(h ∈ H ).
It is well known that if H is a normal subgroup of G, then ΔH,G (h) = 1 for all h ∈ H .
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2.2. Induced representation We consider the space E(G, H ) = ξ : G → C, continuous with compact support modulo H ,
such that ξ(gh) = ΔH,G (h)ξ(h), g ∈ G, h ∈ H . The group G acts on E(G, H ) by left translation and there exists a unique (up to a positive multiple) positive G-invariant linear functional on this space, which is denoted by νG/H . Therefore, we can write it in the form of an integral νG/H (ξ ) =
ξ(g) dνG/H (g).
G/H
We remark that if ΔH,G = 1, then νG/H is simply a G-invariant measure on the homogeneous space G/H and the space E(G, H ) coincides with Cc (G/H ). We can write then the Haar integral on G as a double integral over H and the quotient space G/H :
f (g) dg = G
G/H
f (gh)ΔH,G (h)−1 dh dνG/H (g),
f ∈ Cc (G).
(1)
H
For details see [1]. Let ρ be a unitary representation of H on the Hilbert space Hρ and let Cc (G/H, ρ) be the space of continuous functions ξ : G → Hρ , which are compactly supported modulo H satisfying
1 ξ(gh) = ΔH,G (h) 2 ρ h−1 ξ(g),
h ∈ H, g ∈ G.
We define an L2 -norm on Cc (G/H, ρ) as follows ξ 22
=
ξ(g)2 dνG/H (g). H ρ
G/H
The induced representation indG H ρ is just the left regular representation of G on the completion L2 (G/H, ρ) of Cc (G/H, ρ) with respect to the norm .2 defined above. 2.3. The kernel of induced representations ˆ i.e., the space of equivalence classes [π] of all irreducible unitary The unitary dual G, representations π of G has been described via the Kirillov–Bernat–Vergne orbit method (see [4]). Every unitary irreducible representation of G is equivalent to an induced representation ∗ πl,pl = indG P χl for some l ∈ g and a Pukanszky polarization pl at l, where χl denotes the unitary −il(X) , X ∈ pl of the closed connected subgroup Pl := exp(pl ). A pocharacter χl (exp X) := e larization at l ∈ g∗ is by definition a subalgebra pl of g, such that l, [pl , pl ] = {0} and such that dim(pl ) = 12 (dim(g) + dim(g(l))). We say that pl or Pl satisfy Pukanszky’s condition if
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Ad∗ (Pl )l = l + p⊥ l . The representations πl,pl and πl ,pl are equivalent if and only if l and l are in the same G-orbit O and so the mapping
ˆ Θ : g∗ /G → G,
O(l) → [πO(l) ] := indG Pl χl
is a bijection and even a homeomorphism (see [4]). We need the following lemma (see [5] and [2] for the description of the kernels of such induced representations). Lemma 2.1. Let H be a closed subgroup of G. Let ρ be a unitary representation of H on the 1 Hilbert space Hρ and let π = indG H ρ. Then ker(π) is the set of all functions f ∈ L (G) such that there exists a set N of measure 0 in G so that for every x ∈ G \ N , we have a set Nx of measure 0 in G, such that for all y ∈ / Nx the linear operator fρ (x, y) defined on Hρ by fρ (x, y) :=
1 ΔH,G (h)− 2 f xhy −1 ρ(h) dh
H
exists and is 0. Proof. Let f ∈ L1 (G). Let first ρ0 be the left regular representation of G on L2 (G/H, 1). Choose a non-negative continuous function ξ ∈ L2 (G/H, 1), which vanishes nowhere on G. Then there exists a set N of measure 0 in G, such that
|f | ∗ ξ(x) = ρ0 |f | ξ(x) < ∞,
x∈ / N.
Hence for x ∈ / N, ∞ > |f | ∗ ξ(x) = =
f (y)ξ y −1 x dy =
G
ΔG y −1 f xy −1 ξ(y) dy
G
ΔH,G h−1 ΔG h−1 y −1 f xh−1 y −1 ξ(yh) dh dy
G/H H
=
−1/2 ΔH,G (h)ΔG h−1 ΔG y −1 f xh−1 y −1 ξ(y) dh dy
G/H H
=
1/2 ΔH,G (h)ΔG (h)ΔG y −1 f xhy −1 ΔH h−1 ξ(y) dh dy
G/H H
= G/H
− 12
ΔH,G (h)
−1
−1 f xhy dh ξ(y) dy. ΔG y
H
Therefore, by the theorem of Fubini, there exists for every x ∈ G\N a set Mx ⊂ G of measure 0, such that
1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 dh < ∞ H
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1 for every y ∈ / Mx and such that the function y → H ΔH,G (h)− 2 ΔG (y −1 )|f (xhy −1 )| dh is integrable. Whence for x ∈ / N and η ∈ Cc (G/H, ρ),
π(f )η (x) = f (y)η y −1 x dy = ΔG y −1 f xy −1 η(y) dy G
=
G
−1/2 ΔH,G (h)ΔG h−1 ΔG y −1 f xh−1 y −1 ρ h−1 η(y) dh dy
G/H H
= G/H
1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 ρ(h) dh η(y) dy.
(2)
H
We deduce from (2) that f ∈ ker(indG H ρ) if and only if for every x ∈ G \ N , there exists a set Nx ⊃ Mx of measure 0 in G such that the linear operator
1 ΔH,G (h)− 2 ΔG y −1 f xhy −1 ρ(h) dh H
is 0 for every y ∈ / Nx .
2
3. Flat orbits In this section we characterize the flat orbits of a completely solvable Lie group of endomorphisms of a finite dimensional real vector space V. Let D = exp(D) be an exponential Lie group of linear endomophisms of V . We assume that D is completely solvable. This means that the eigenvalues of every D ∈ D, considered as an endomorphism of the complexification VC of V , are real numbers. We denote by D the associative hull in the endomorphism ring of the vector space V generated by D. Then the group D is contained in the algebra RIV + D . Note that D is linearly generated by the set {D j : D ∈ D, j ∈ N}. For l ∈ V ∗ , we define:
ND (l) = x ∈ V : l, D(x) = 0, ∀D ∈ D ,
D(l) = D ∈ D: D t (l) = 0 ,
AD (l) = x ∈ V : l, T (x) = 0, ∀T ∈ D . Here D t denotes the transpose of D: D t l, X := l, D(X) , X ∈ V , l ∈ V ∗ . It follows from the definitions, that AD (l) ⊂ ND (l) and that
AD (l) = X ∈ ND (l): T (X) ∈ ND (l), ∀T ∈ D . Definition 3.1. We say that an orbit O(l) = Dt l ⊂ V ∗ of the exponential completely solvable group D is flat, if the subspace ND (l) of V (and hence ND (q) of every element q ∈ O(l)) is D-invariant.
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Theorem 3.2. Let D = exp(D) be an exponential completely solvable Lie group of endomorphisms of the real finite dimensional vector space V . Let l ∈ V ∗ and O = O(l) = Dt l be the D-orbit of l. The following statements are equivalent: 1) O is flat, i.e. ND (l) is D-invariant ⇔ ND (l) = AD (l). 2) Dt · l|ND (l) = l|ND (l) . 3) There exists an analytic function P : R → R; P (ξ ) = 1 + a2 ξ 2 + a3 ξ 3 + · · · for small ξ , with a2 = 0, such that for every q in the orbit O(l), P (D t )q ∈ O(l) for D ∈ D small enough. Proof. 1) ⇒ 2) Let X ∈ ND (l) = AD (l). Since D j (X) ∈ ND (l) for every j ∈ N∗ , it follows that l, D j (X) = 0,
j ∈ N∗ , X ∈ ND (l), D ∈ D,
and so exp(D t )l, X = l, X . 2) ⇒ 1) Let X ∈ ND (l). For all D ∈ D, s ∈ R, we have then that
l, X = exp sD t (l), X (sD t )k (l), X = k! k0
s 2 t 2 D + · · · (l), X = IV + sD t + 2! s 2 t 2 D (l), X + · · · . = l, X + s D t (l), X + 2! It follows that, s 2 t 2 D (l), X + · · · = 0 s D t (l), X + 2! and therefore for all j 1, (D t )j (l), X = 0. Hence, l, T (X) = 0 for all T ∈ D and thus ND (l) ⊂ AD (l), which completes the proof in this case. 3) ⇒ 1) We proceed by induction on d = dim(V ) + dim(D). The result is obviously true if d = 1. Let d 2. We take V0 = ker(l) ∩ AD (l). We have to treat the following cases: Case 1. V0 = {0}. Let p : V −→ V˜ = V /V0 be the canonical projection and j the transposed map of p. Take ˜ by ˜ = l. We define the Lie algebra D ˜l ∈ V˜ ∗ such that j (l)
D˜ p(x) = p D(x) ,
D ∈ D and x ∈ V .
˜ = P (D t )(q), q ∈ O and D small enough, the induction hypothesis applied to As j (P (D˜ t )(q)) ˜ ˜ = A ˜ (l). ˜ Hence ND (l) = p −1 (N ˜ (l)) ˜ is D-invariant. ˜ V and D implies that ND˜ (l) D D
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Case 2. V0 = {0}. This implies that dim(AD (l)) = 0 or 1. Subcase 2.1. AD (l) = RZ, for some Z ∈ V \ {0}, D(Z) = {0} and l(Z) = 0. In this case, there exists a non-zero vector Y ∈ V and two linear functionals α, β = 0 from D to R such that D(Y ) = α(D)Y + β(D)Z,
∀D ∈ D,
since D is completely solvable. It follows that α is a homomorphism of the Lie algebra D. Therefore, we can suppose that α and β are linearly independent, if α = 0. In the case where α and β are linearly dependent and α = 0 there exists c ∈ R such that β = cα. Thus we have D(Y + cZ) = α(D)(Y + cZ),
∀D ∈ D,
and we are in the same situation as subcase 2.2 which is detailed later. Without loss of generality, we can assume that l(Y ) = 0 and l(Z) = 1. Let D0 be the kernel of β. Then D0 is a subalgebra of G. Let D0 := exp(D0 ). Then D0 is a closed connected subgroup of D. Assume first that α = 0. The D0 -orbit O0 of l is given by:
O0 = q ∈ O(l): q(Y ) = 0 . ˙ = 1, α(D) ˙ = 0 and D = D0 ⊕RD. ˙ Thus D(Y ˙ )=Z In fact, there exists D˙ ∈ D\D0 such that β(D) ˙ ) = 0, for all D ∈ D. So we have and D D(Y ˙ D = D0 exp(RD). Let q ∈ O(l) such that q(Y ) = 0. There exists D0 ∈ D0 , and s ∈ R such that q = exp(D0t ) × exp(s D˙ t )(l). Since q(Y ) = 0, we have
0 = exp D0t exp s D˙ t (l), Y ˙ exp(D0 )(Y ) = l, exp(s D)
˙ eα(D0 ) Y = l, exp(s D) = l, eα(D0 ) (Y + sZ) = seα(D0 ) . This implies that, s = 0 and so q ∈ (D0 )l. Thus {q ∈ O(l): q(Y ) = 0} ⊂ O0 . On the other hand, we evidently have (Dt0 )l(Y ) = 0. As P (Dt0 )(q), Y = {0} for every q ∈ O0 , it follows for D ∈ D0 , q ∈ O0 , that P (D t )q ∈ O0 whenever P (D t )q ∈ O. We can apply the induction hypothesis to D0 and O0 . Hence RY ⊕ ND (l) = ND0 (l) = AD0 (l). We show now that ND (l) is D-invariant. Let v ∈ ND (l). We have l, D 2 (v) = 0, for all D ∈ D. In fact, for all D0 ∈ D0 and s ∈ R small enough,
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P s(D˙ + D0 )t l, Y = l, Y + a2 s 2 (D˙ + D0 )2 (Y ) + o s 3
˙ 0 + D0 D˙ (Y ) + o s 3 = l, Y + a2 s 2 D˙ 2 + D02 + DD
˙ 0 (Y ) + o s 3 = a2 s 2 l, DD
= a2 α(D0 )s 2 + o s 3 =: Q(s). On the other hand, we have
exp −Q(s)D˙ t P s(D˙ + D0 )t l, Y = P s(D˙ + D0 )t l, Y − Q(s)Z
= P s(D˙ + D0 )t l, Y − Q(s) = 0.
It follows that for s ∈ R small enough,
exp −Q(s)D˙ t P s(D˙ + D0 )t l ∈ D0 . Since v ∈ ND (l) ⊂ ND0 (l) = AD0 (l), we have
l, v = exp −Q(s)D˙ t P s(D˙ + D0 )t l, v
˙ = P s(D˙ + D0 )t l, v − Q(s)D(v) + o s3
˙ = l, v − Q(s)D(v) + a2 s 2 (D˙ + D0 )2 (v) + o s 3
= l, v + a2 s 2 (D˙ + D0 )2 (v) + o s 3 . This implies that a2 l, (D˙ + D0 )2 (v) = 0. As a2 = 0, we have l, (D˙ + D0 )2 (v) = 0. Hence
l, D 2 (v) = 0 for all D ∈ D. Now, for D1 , D2 ∈ D and v ∈ ND (l),
0 = l, (D1 + D2 )2 (v) = l, D12 + D22 + 2D1 D2 + [D1 , D2 ] (v) = 2 l, D1 D2 (v) (since [D1 , D2 ] ∈ D). This shows that D(v) ⊂ ND (l) and so ND (l) is D-invariant. The subcase α = 0 is similar. Subcase 2.2. AD (l) = {0}. Then there exists a non-zero Y ∈ V and non-zero homomorphism α on D such that D(Y ) = α(D)Y,
∀D ∈ D.
Without loss of generality, we can assume that l(Y ) = 1. Let D0 be the kernel of α and D0 := ˙ = 1 and D = D0 ⊕ RD. ˙ It is easy to see that exp(D0 ). There exists D˙ ∈ D\D0 such that α(D)
O0 := Dt0 l = q ∈ O: q(Y ) = 1 . On the other hand, for all s ∈ R small enough and D0 ∈ D0
P s(D˙ + D0 )t (l), Y = l, Y + a2 s 2 (D˙ + D0 )2 (Y ) + a3 s 3 (D˙ + D0 )3 (Y ) + · · · = 1 + a2 s 2 + a3 s 3 + · · · = 1 + Q(s) > 0.
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˙ (s(D˙ + D0 )t )(l) ∈ O0 for s small enough Then, for q(s) = ln(1 + Q(s)) we get exp(−q(s)D)P in R. In addition, by the same reasoning as above, using the induction hypothesis, we see that ND0 (l) is D0 -invariant. Let v ∈ ND (l), we compute
l, v = exp −q(s)D˙ P s(D˙ + D0 )t (l), v
˙ = P s(D˙ + D0 )t (l), v − a2 s 2 D(v) + o s3
˙ = l, v − a2 s 2 D(v) + a2 s 2 (D˙ + D0 )2 (v) + o s 3
= l, v + a2 s 2 (D˙ + D0 )2 (v) + o s 3 . It follows that
a2 s 2 l, (D˙ + D0 )2 (v) + θ s 3 = 0,
for all s ∈ R.
Hence, we have l, D 2 (v) = 0 for all D ∈ D and so l, D1 D2 (v) = 0 for all D1 , D2 ∈ D i.e. ND (l) is D-invariant. 1) ⇒ 3) Since now ND (l) is D-invariant, the D-orbit O of l is contained in l + ND (l)⊥ . The dimension of this orbit O is equal to the dimension of V /ND (l) because the dimension of O is equal to the dimension of D modulo the stabilizer D(l) := {D ∈ D, D t (l) = 0} of l and since the bilinear map
D/D(l) × V /ND (l) : D + D(l), v + ND (l) → l, D(v) establishes a duality between the two quotient spaces. Hence O is an open subset of l + ND (l)⊥ . We take the function P (ξ ) := 1 + ξ 2 , ξ ∈ R. Then the mapping
D × l + ND (l)⊥ → l + ND (l)⊥ ;
(D, q) → P (D)q
is continuous and so for every q ∈ O we can find a small neighbourhood U of 0 in D, such that P (D)q ∈ O for every D ∈ U . 2 Corollary 3.3. Let D be an exponential completely solvable Lie group of endomorphisms of the real finite dimensional vector space V . Let l ∈ V ∗ and let O(l) be the D-orbit of l in V ∗ . If O(l) is closed, then the following statements are equivalent: 1) ND (l) is D-invariant: ND (l) = AD (l). 2) Dt · l|ND (l) = l|ND (l) . 3) a) O(l) is affine linear. b) O(l) = l + AD (l)⊥ . 4) There exists an analytic function P : R → R; P (ξ ) = 1 + a2 ξ 2 + a3 ξ 3 + · · · for small ξ , with a2 = 0, such that for every q in the orbit O(l), P (D t )q ∈ O(l) for D ∈ D small enough. Proof. It suffices to proof the implications 1) ⇒ 3)a) and 3)a) ⇒ 3)b). 1) ⇒ 3)a) For X ∈ ND (l) and D ∈ D, we have
1 exp D t (l), X = l, X + l, D(X) + l, D 2 (X) + · · · = l, X . 2!
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Hence O(l) ⊂ l + ND (l)⊥ . On the other hand, reasoning as in the proof of the preceding theorem, we see that O(l) is open in l + ND (l)⊥ . Since by hypothesis it is also closed, it follows that O = l + ND (l)⊥ . 3)a) ⇒ 3)b) We evidently have O(l) ⊂ l + AD (l)⊥ . On the other hand, let W be a subspace of V such that O(l) = l + W ⊥ . For all D ∈ D and s ∈ R, we have
1 t
exp sD (l) − l ∈ O(l) − l ⊂ W ⊥ . s Hence for X ∈ W ,
1 t
exp sD (l) − l , X = 0 s
and so
s t 2 s 2 t 3 D t (l), X + D l, X + D l, X + · · · = 0 2! 3!
and therefore for all j 1, D ∈ D, X ∈ W , D j (X) ∈ ker(l), j 1, i.e. W ⊂ AD (l). Whence, W = AD (l). 2 Corollary 3.4. Let G = exp(g) be a completely solvable Lie group and let l ∈ g∗ . If the G-orbit O(l) of l is closed, then the following statements are equivalent: 1) g(l) is an ideal in g. 2) Ad∗ (G)l|g(l) = l|g(l) . 3) O(l) = l + g(l)⊥ . 4. Representations associated to flat orbits Let l ∈ g∗ and pl be a polarization for l satisfying the Pukanszky condition. Let Pl = exp(pl ) ˆ be the representation indG χl , where χl is the unitary character of Pl defined by and πl ∈ G Pl χl (x) := e−i l,log(x) , x ∈ Pl . Let as in Section 2.1 J = (gi )ni=1 be a Jordan–Hölder sequence and Z = {Z1 , . . . , Zn } be a Jordan–Hölder basis of g adapted to J . We denote by I pl the index set I pl := {i ∈ {1, . . . , n}; pl ∩ gi = pl ∩ gi+1 }. Then for i ∈ I pl , we can take the vector Zi in pl . Let also I g/pl be the index set {1, . . . , n} \ I pl = {i ∈ {1, . . . , n}; gi ∩ pl = gi+1 ∩ pl }. We consider the function ψpl defined on G by ψpl (x) =
i∈I g/pl
. ρi (log(x)) −
ρi (log(x)) e
ρi (log(x)) 2
−e
2
The function ψpl is bounded and Ad(G)-invariant. For p ∈ Pl we have the following identity:
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ΔPl ,G (p)
−1 2
jpl (log p) = ψpl (p). jg (log p)
(3)
Indeed, ΔPl ,G (p)
−1 2
ρi (log(p)) jpl (log p) −ρi (log(p))/2 e = 1 − e−ρi (log(p)) jg (log p) g/p g/p i∈I l i∈I l ρi (log(p)) = ρi (log(p)) ρi (log(p)) − 2 2 g/p e − e i∈I l = ψpl (p).
Let I (l, pl ) be the closed subspace of L1 (G), given by f (uxv)ψpl (x)e−i l,log x dx = 0 . I (l, pl ) = f ∈ L1 (G): ∀u, v ∈ G, G
Then I (l, pl ) is in fact a twosided ideal of the algebra L1 (G), since for every f ∈ I (l, pl ) the left and right translates of f are all contained in I (l, pl ). Proposition 4.1. I (l, pl ) is contained in ker(πl ). Proof. Let g ∈ I (l, pl ) and α ∈ Cc (G) and let f := g ∗ α. Then f ∈ I (l, pl ) too and the function p → f (uxpv) is contained in L1 (Pl ) for every x, u, v ∈ G since
f (uxpv) dp =
Pl
g(y)α y −1 uxpv dp dy
G Pl
ΔG y −1 g uxy −1 α(ypv) dp dy
= G Pl
g ux(yq)−1
−1
=
ΔG (yq)
G/Pl Pl
l
Pl
ΔG (yq)−1 ΔPl ,G (q)−1 g ux(yq)−1
= G/Pl Pl
The function y → Pl
Pl
α(yqpv) dpΔP ,G (q)−1 dq dμG/P (y)
l
α(ypv) dp dq dμG/P (y). l
Pl
|α(ypv)| dp =: α˜ v (y) is uniformly bounded in y and so
f (uxpv) dp =
ΔG (yq)−1 ΔPl ,G (q)−1 g ux(yq)−1 α˜ v (y) dq dμG/Pl (y)
G/Pl Pl
= G
g(uxy)α˜ v y −1 dy < ∞.
M. Elloumi et al. / Journal of Functional Analysis 258 (2010) 3955–3976
Now, for all u, v, x ∈ G and all p ∈ Pl we have that 0=
f upxp −1 v ψpl (x)e−i l,log x dx
G
=
f upxp −1 v ψpl pxp −1 e−i l,log x dx
G
= ΔG (p)
f (uxv)ψpl (x)e−i Ad
∗ (p)l,log x
dx
G
= ΔG (p)
∗ f u exp(Y )v ψpl (exp Y )e−i Ad (p)l,Y jg (Y ) dY.
g ⊥ As Ad∗ (Pl )l = l + p⊥ l , we get for u, v, x ∈ G, q ∈ pl , that:
0=
f u exp(Y )v ψpl exp(Y ) e−i l+q,Y jg (Y ) dY
g
=
e
−i q+l,Y
f u exp(Y + U )v ψpl exp(Y + U ) e−i l,U jg (Y + U ) dU d Y˙ .
pl
g/pl
Hence, for every Y ∈ g, u, v ∈ G, 0=
f u exp(Y + U )v ψpl exp(Y + U ) e−i l,U jg (Y + U ) dU.
pl
Therefore, for Y = 0, u, v ∈ G, 0=
f u exp(U )v ψpl exp(U ) e−i l,U jg (U ) dU.
pl
Hence, by (3), for all u, v ∈ G, 0=
−1
f u exp(U )v ΔPl ,G exp(U ) 2 jpl (U )e−i l,U dU
pl
=
f (upv)ΔPl ,G (p)
−1 2
e−i l,log(p) dp.
Pl
Thus, by Lemma 2.1, f ∈ ker(πl ) and finally g ∈ ker(πl ).
2
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Theorem 4.2. Let G = exp(g) be a completely solvable Lie group. Let l ∈ g∗ such that the Gorbit O(l) is closed. If O(l) is affine linear then
ker(πO(l) ) = f ∈ L1 (G): (f ◦ exp)jg ˆ O(l) = {0} . ⊥ Proof. Let τl := indG G(l) χl , where G(l) := exp(g(l)). As O(l) is affine linear, O(l) = l + g(l) and g(l) is an ideal of g (by Corollary 3.4). Hence G(l) is a closed connected normal subgroup of G. Furthermore, we have
ker(τl ) =
ker(πq ) = ker(πl )
q∈l+g(l)⊥
(see [4]). We show that ker(τl ) = {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(O(l)) = {0}}. Let f ∈ Cc (G) ∗ ker(τl ) ∗ Cc (G). Then, by Lemma 2.1 we get
f exp(s)h χl (h) dh =
G(l)
f ◦ exp s + ϕs (h) χl (h)jg(l) (h) dh = 0,
g(l)
for all s ∈ g, where ϕs (h) = s ·g h − s 1 1 1 s, [s, h] + h, [h, s] + · · · = h + [s, h] + 2 12 12
for small s ∈ g, h ∈ g(l).
We see that the mapping ϕs : g(l) → g(l) is a diffeomorphism, whose inverse ψs is given by: ψs (h) = (−s) ·g (h + s),
h ∈ g(l) (s ∈ g).
On the other hand, for all f ∈ L1 (G) we have
f ◦ exp(s + h)jg (s + h) dh ds
g/g(l) g(l)
f (g) dg =
= G
=
f (sh) dh
G/G(l) G(l)
f ◦ exp s + ϕs (h) jg(l) (h)jg/g(l) (s) dh ds
g/g(l) g(l)
= g/g(l) g(l)
This proves that
f ◦ exp(s + h)jg(l) ψs (h) jg/g(l) (s)Jac(ψs )(h) dh ds.
(4)
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Jac(ψs )(h) =
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jg (s + h) . jg(l) (ψs (h))jg/g(l) (s)
We deduce from Eq. (4) that
f ◦ exp(s + h)jg (s + h)e−il(h) dh = 0,
s ∈ g,
g(l)
since l, ψs (h) = l, h for all h ∈ g(l), because g(l) is an ideal of g. Therefore f ◦ exp(Y )jg (Y )e−i l+q,Y dY = 0, q ∈ g(l)⊥ . g
As Cc (G) ∗ ker(τl ) ∗ Cc (G) is dense in ker(τl ), it follows that ker(τl ) ⊂ {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(l + g(l)⊥ ) = 0}. Let now f ∈ L1 (G) such that [(f ◦ exp)jg ]ˆ(l + g(l)⊥ ) = 0, then by the same computation as above, we can show that G(l) f (sh)χl (tht −1 ) dh = 0 for all s, t ∈ G. That means by Lemma 2.1 that f ∈ ker(τl ) and thus
ker(τl ) = f ∈ L1 (G): (f ◦ exp)jg ˆ O(l) = 0 .
2
We show now the converse direction. We take an exponential solvable Lie group G = exp(g) and an exponential completely solvable Lie group D = exp(D) of automorphisms of g containing the group Ad(G). We also suppose that there is an analytic mapping P : D × g → g such that for small (D, X) P (D, X) = X + aD 2 (X) +
ki+nj 2
a k1 ...kr D k1 adn1 (X) . . . D kr adnr (X)D kr+1 (X), (n1 ...nr )
with a = 0. We write P (D) : g → g for the mapping: P (D)(X) = P (D, X) (D ∈ D) and we suppose that P (D) is a diffeomorphism of g for every D ∈ D. Define for D ∈ D the linear bijection Pˇ (D) of L1 (g) defined by
Pˇ (D)f (X) := f P (D)X JP (D) (X),
X ∈ g,
where JP (D) (X) denotes the Jacobian of P (D) at X ∈ g. Definition 4.3. For an ideal I in the algebra L1 (g), let h(I ) be the set of characters ∗ −i q,Y h(I ) := q ∈ g , 0 = χq (f ) = f (Y )e dY, f ∈ I . g
Then h(I ) is a closed (possibly empty) subset of g∗ . Lemma 4.4. Let g, D and P as above. Let l ∈ g∗ and let I be a closed ideal in L1 (g), so that h(I ) is the closure O(l) of the D-orbit O(l) of l. If I is invariant under the maps Pˇ (D), D ∈ D, then ND (l) is D-invariant.
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Proof. We proceed by induction on the number d := dim(D) + dim(g). If d = 1, the result is obviously true. Suppose now that d 2. Let g0 = ker(l) ∩ AD (l). Then g0 is D-invariant. We first assume that g0 = {0}. Let p be the canonical projection of g onto g˜ = g/g0 and let j be ˜ = l. We define a Lie algebra of exponential the transpose of p. Let l˜ ∈ g˜ ∗ be defined by j (l) ˜ derivations on g˜ in the following way: for all D ∈ D, let D˜ be defined by D(p(x)) = p(D(x)), ˜ l) ˜ ˜ and ad g˜ ⊂ D. x ∈ g. So we have evidently O(l) = j ((exp D) Let I˜ = π(I ), where π : L1 (g) → L1 (˜g) is the canonical surjection: π(f )(x) ˜ :=
f (x + z) dz,
f ∈ L(g), x˜ = x + g0 .
g0
˜ l. ˜ Thus I˜ is a closed ideal in L1 (˜g) (see [7], p. 177) and the hull of I˜ is h(I˜) = (exp D) ˜ ˜ ˜ Define the maps P (D) (D ∈ D) by: ˜ x) P (D)( ˜ = P (D)x + g0 ,
x˜ = x + g0 ,
˜2
= x˜ + a D (x) ˜ + ···,
for small x˜ ∈ g˜ .
Then
˜ p(x) = p P (D)(x) , P (D)
for all x ∈ g.
˜ ˜ and x˜ = p(x) It is easy to see that I˜ is Pˇ (D)-invariant; Indeed, for all f˜ = π(f ) ∈ I˜, D˜ ∈ D
˜ f˜(x) Pˇ (D) ˜ = f˜ p P (D)(x) JP (D) ˜ p(x)
= f P (D)(x + h) JP (D) (x + h) dh g0
= g0
JP (D) (x + Q(D, x)−1 (h)) f P (D)(x) + h dh, JQ(D,x)
where Q(D, x) is a diffeomorphism of g0 given by Q(D, x)(h) = P (D)(x + h) − P (D)(x)
(h ∈ g0 , x ∈ g, D ∈ D).
On the other hand, for all ϕ ∈ L1 (g), we have
ϕ(x) dx =
g
g/g0 g0
= g/g0 g0
and
ϕ P (D)(x + h) JP (D) (x + h) dh d x˜
JP (D) (x + Q(D, x)−1 (h)) ϕ P (D)(x) + h dh d x, ˜ JQ(D,x)(h)
M. Elloumi et al. / Journal of Functional Analysis 258 (2010) 3955–3976
ϕ(x) dx = g
ϕ(x + h) dh d x˜ =
g/g0 g0
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ϕ P (D)(x) + h dh JP (D) ˜ d x. ˜ ˜ (x)
g/g0 g0
This implies that JP (D) ˜ = ˜ (x)
JP (D) (x + Q(D, x)−1 (h)) , JQ(D,x)(h)
x ∈ g, h ∈ g0 .
˜ f˜(x) We obtain thus that Pˇ (D) ˜ = π(Pˇ (D)f (x)). ˜ = A ˜ (l) ˜ and thus ND (l) is D-invariant. By the induction hypothesis we get ND˜ (l) D Suppose now that g0 = {0}. Then dim(AD (l)) = 0 or 1. Case 1. AD (l) = RZ, for some non-zero Z in g. Since AD (l) is D-invariant and l, Z = 0, it follows that D(Z) = 0 for all D ∈ D. In this case, there exists a non-zero Y ∈ g and two linear functionals α, β on D such that D(Y ) = α(D)Y + β(D)Z,
∀D ∈ D
since D is completely solvable with β = 0. If α = 0, then we can assume that α and β are linearly independent. The case where α and β are linearly dependent and α = 0 is tackled similarly as in the Case 2 which is treated later. The linear functional α is a homomorphism of D. Without loss of generality, we can assume that l(Y ) = 0 and l(Z) = 1. We can find an element D˙ ∈ ker(α), ˙ = 1. such that β(D) We apply now the technique of restriction of an ideal to an invariant subgroup developed in [3]. Let g1 be any D-invariant subspace of g containing AD (l), such that dim(g/g1 ) = 1. l = 0 and let χt Such a g1 always exists since D is completely solvable. Let l0 ∈ g⊥ 1 with 0 be the character of g defined by: χt (v) = ei tl0 ,v , v ∈ g. The ideal J = t∈R χt I is Pˇ (D)invariant and h(J ) = h(I ) + Rl0 . Define for x ∈ g and a function f : g → C the function x f by 1 x f (y) := f (x + y), y ∈ g. Let J be the set of all functions f ∈ J , so that x f|g1 ∈ L (g1 ) for all 1 x ∈ g and so that the maps x → x f|g1 from g1 to L (g1 ) are continuous. J is dense in J and Pˇ (D)-invariant, since I is a Pˇ (D)-invariant ideal of L1 (g). We define the ideal I1 of L1 (g1 ) as the closure in L1 (g1 ) of the functions f|g1 f ∈ J . Let D1 be the restriction of the D on g1 and let l1 = l|g1 . The hull of I1 is exactly the restriction of the hull of J on g1 . Thus h(I1 ) = (exp D1 t )l1 . We still have ad g1 ⊂ D1 . The ideal I1 is Pˇ (D1 )-invariant. Indeed, since P (D) maps g1 into g1 , we have that JP (D) (h) is a constant times JP (D1 ) (h), h ∈ g1 . The induction hypothesis applied to I1 , l1 , D1 implies that ND (l) ⊃ ND (l) ∩ g1 = ND1 (l1 ) = AD1 (l1 ) = AD (l). Hence we obtain that either AD (l) = ND (l) (if ND (l) ⊂ g1 ) or dim(AD (l)) + 1 = dim(ND (l)) 2. Let now D0 be the kernel of β. Then D0 is a subalgebra of D. Let D0 = exp(D0 ). We have that ND0 (l) = RY ⊕ ND (l) and the closure of the D0 -orbit of l is given by: t
D0 l = q ∈ Dt l: q(Y ) = 0 . We look at the ideal I1 defined to be the closure in L1 (g) of the sum of the ideal I and the kernel KY of the surjective homomorphism
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πY : L (g) → L (g/RY ), 1
1
π(f )(x + RY ) :=
f (x + yY ) dy. R
It is easy to check that KY is Pˇ (D)-invariant for every D ∈ D0 , since P (D)(RY ) ⊂ RY , D ∈ D0 . The hull of I1 is the subset h(I ) ∩ {q ∈ g∗ ; q, Y = 0} = Dt0 l =: O0 (l). By the induction hypothesis for I1 , D0 and g, we get ND0 (l) = AD0 (l) and so ND0 (l) is D0 -invariant. This implies that dim(ND0 (l)) 3 since dim(ND0 (l)) = dim(ND (l))+1. If dim(ND0 (l)) = 2 then ND0 (l) = AD0 (l) = RY +RZ. Whence ND (l) = AD (l). We prove now that the case dim(ND0 (l)) = 3 cannot happen. Suppose otherwise. If we have a D-invariant subspace g1 of co-dimension 1 containing ND0 (l) ⊃ ND (l), then we have seen that, AD (l) = ND (l) and so ND0 (l) is of dimension 2. Hence no D-invariant subspace can contain ND0 (l). In other terms, either g = AD0 (l) or the smallest D-invariant subspace of g containing AD0 (l) equals g. We show that in the two cases g is abelian. If g = AD0 (l) = RY0 + RY + RZ, then for a D˙ ∈ D ˙ = 1 and α(D) ˙ = 0, we have that with β(D) t
˙ ˙ 0 = l, exp s D(g), exp s D(g) = exp s D˙ l, [g, g] for all s ∈ R. It follows that if we write [Y0 , Y ] = cY for some c, then ˙ = c l, Y + sZ = cs ˙ 0 , Y ] = l, c exp s DY 0 = l, exp s D[Y and so [Y0 , Y ] = 0 and g is abelian. In the second case take again D˙ ∈ D \ D0 and let ˙ 0, Y1 = DY
...,
Yk = D˙ k Y0
(k = 2, 3, . . . , n),
where n is the largest integer such that the set {Y1 , . . . , Yn } is linearly independent modulo ˙ span{Y, Z}. The subspace h of g, spanned by Y0 , Y1 , . . . , Yn , Y and Z is by definition D-invariant. It is also D0 -invariant. Indeed AD0 (l) is D0 -invariant, it follows that D0 (Y0 ) ⊂ AD0 (l) ⊂ h. If the functional α = 0, then D0 is an ideal in D and so we see that inductively on k = 1, . . . , for D ∈ D0 ,
˙ k−1 ) = D(Y ˙ k−1 ) + [D, D](Y ˙ k−1 ) ∈ h. D(Yk ) = D D(Y If α = 0, we can take D˙ in ker(α) ∩ [D, D]. In particular D˙ is a nilpotent endomorphism. Take now D00 = ker(α) ∩ ker(β), which is an ideal of D contained in D0 . The subspace h is therefore D00 -invariant by the argument above. There exists an element D˙ 0 ∈ D0 , such that α(D˙ 0 ) = 1. ˙ = −D˙ modulo D00 , we have that Again, by induction on k, as [D˙ 0 , D] ˙ k−1 + D˙ D˙ 0 Yk−1 ∈ h, D˙ 0 Yk = [D˙ 0 , D]Y k = 1, 2, . . . , n. Thus h is D-invariant and so h = g.
M. Elloumi et al. / Journal of Functional Analysis 258 (2010) 3955–3976
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We show first now that Y is central in g; we have that, since D˙ is a derivation of g and since Y ∈ AD0 (l),
˙ 0 ), Y = l, D˙ [Y0 , Y ] − Y0 , D(Y ˙ ) = α ad(Y0 ) . 0 = l, [Y1 , Y ] = l, D(Y =0
It follows that [Y0 , Y ] = 0 and so by induction on k, ˙ k−1 ), Y [Yk , Y ] = D(Y
˙ ) = D˙ [Yk−1 , Y ] − Yk−1 , D(Y = 0 − [Yk−1 , Z] = 0. Hence Y is central in g. We prove now that [Y0 , g] = 0. We remark that for all j 1, ad(Yj ) is nilpotent, since for these j ’s, ad(Yj ) ∈ [D, D]. This implies that [Y0 , Yj ] = aj Y ∈ RY for some aj ∈ R, because Y0 ∈ AD0 (l) and so [Yj , Y0 ] ∈ RY . On the other hand, by induction on k, we can check that [Yk , Y ] =
k j (−1)j Ck D˙ k−j [Y0 , Y+j ],
∀k, = 1, 2, . . . , n.
j =0
Using the formula 0 = [Yk , Yk ] =
k j (−1)j Ck D˙ k−j [Y0 , Yk+j ] j =0
= (−1)k [Y0 , Y2k ] − (−1)k−1 D˙ [Y0 , Y2k−1 ] = (−1)k a2k Y − (−1)k−1 a2k−1 Z,
k = 1, . . . , n,
we deduce that for any k = 1, 2, . . . , n, ak = 0, and hence ad(Y0 ) = 0. Whence Y0 is contained ˙ 0 ) and inductively all the Yk ’s, k = 2, . . . , n. Finally g in the center of g and so is then Y1 = D(Y is abelian. Then the polynomial maps P (D), D ∈ D, are reduced to the linear maps given by P (D)(x) = x + aD 2 (x) +
bk D 2+k (x)
k0
for some bk ∈ R and for D ∈ D. As I is invariant under these linear maps, the hull h(I ) of I is invariant under the corresponding linear maps P (D)t , which have the form P (D)t = 1 + aD 2 +
k0
bk D 2+k ,
D ∈ D small.
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Since the orbit O(l) is open in its closure (see [1]), we have that for every q ∈ O(l), P (D)t (q) ∈ O(l) for D small enough. Applying now Theorem 3.2 we have that ND (l) = AD (l), but this contradicts the assumption that dim(ND0 (l)) = 3. Case 2. dim(AD (l)) = 0. In this case, there exists a non-zero Y ∈ g and a homomorphism α = 0 on D such that D(Y ) = α(D)Y,
∀D ∈ D.
Without loss of generality, we can assume that l(Y ) = 1. Let D0 be the kernel of α and suppose that ad(g) ⊂ D0 . There exists X ∈ g so that [X, Y ] = Y . Then, the D-orbit is saturated with respect to the D-invariant subspace g1 = {U ∈ g: [U, Y ] = 0}. Let D1 be the restriction of D on g1 and
I1 = h ∗ f|g1 : f ∈ I, h ∈ Cc (G) .1 . Then the ideal I1 is Pˇ (D1 )-invariant and h(I1 ) = h(I )|g1 . Furthermore h(I1 ) = (exp Dt1 )l|g1 and by the induction hypothesis for I1 , D1 , g1 , we have that ND (l) = ND1 (l|g1 ) = AD1 (l|g1 ) = AD (l) = {0}. Assume now that α(ad g) = 0. In this case Y is central in g and we have for D0 := ker(α)
exp Dt0 l = q ∈ O(l): q(Y ) = 1 . Let g1 be a D-invariant subspace of g of co-dimension 1. We define J=
χq I
and I1 = h ∗ f|g1 : f ∈ J, h ∈ Cc (G) .1 .
q∈g⊥ 1 t Then I1 is Pˇ (D|g1 )-invariant, h(J ) = h(I ) + g⊥ 1 and h(I1 ) = D l1 . Hence, by the induction hypothesis applied to I1 , D1 , g1 we obtain:
ND (l) ∩ g1 = ND|g1 (l|g1 ) = AD|g1 (l|g1 ) = {0}. Let now y := RY and let K := {f ∈ L1 (g):
y f (u + y) dy = 0, u almost everywhere}. subspace l + y⊥ and K is Pˇ (D0 )-invariant for
Then the hull of the ideal K is the affine D0 ∈ D0 small enough, since we can write for u ∈ g and y ∈ y:
every
P (D)(u + y) = uD + P (D)y = uD + q(D)y for some uD ∈ g depending only on u and D and some real number q(D). Hence the closure J0 of the ideal I + K is also Pˇ (D0 )-invariant and its hull is equal to the closure of the D0 -orbit of l. Applying the induction hypothesis to D0 and J0 , we see that ND (l) + RY = ND0 (l) = AD0 (l). We have seen above that for any D-invariant co-one dimensional subspace g1 of g the dimension of
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ND (l) ∩ g1 = 1. Hence the dimension of ND0 (l) is less or equal to 2. If this dimension is one, then ND (l) = {0}. If ND (l) is contained in a proper D-invariant subspace, then we have also finished by the argument above. It remains the case where ND0 (l) is of dimension 2 and contained in no Dinvariant proper subspace. We can write ND0 (l) = RU + RY , where l(U ) = 0. Since U ∈ ND0 (l) and ND0 (l) is D0 -invariant, it follows that RU must be itself D0 -invariant. Hence there exists a character γ of g, such that [T , U ] = γ (T )U , T ∈ g. Hence ND0 (l) is contained in the nilradical of g. But then g itself is nilpotent, since the smallest D-invariant subspace containing U and Y is equal to g and the elements of D are derivations of g. Then necessarily α = 0 and so ND0 (l) is contained in the center of g and finally g itself is abelian. Since the hull of I is the closure of a D-orbit, which is P (D)t -invariant for small D in D, we can now apply as before Theorem 3.2 and we have that ND (l) is D-invariant. 2 Theorem 4.5. Let G = exp(g) be a completely solvable Lie group and let l ∈ g∗ . Suppose that ˆ be associated to O(l). The following the coadjoint orbit O(l) of l is closed in g∗ . Let πl ∈ G statements are equivalent: 1) ker(πl ) = {f ∈ L1 (G): [(f ◦ exp)jg ]ˆ(O(l)) = 0}, 2) The orbit O(l) is affine linear. Proof. 1 ⇒ 2) It is clear that Il = {(f ◦ exp)jg : f ∈ ker(πl )} is invariant under the linear maps Pˇ (ad(X)), X ∈ g, defined by:
1 P ad(X) (Y ) = X ·g Y ·g X − 2X = Y + ad(X)2 Y + · · · higher brackets in X, Y ∈ g, 6 since ker(πl ) is translation-invariant by elements of G and Il ⊂ L1 (g) is translation-invariant by elements of g. Furthermore we have that
jg (Y ) ΔG (exp X) dY = f (Y ) dY, X ∈ g, f ∈ L1 (g). f P ad(X)Y jg (Y + 2X) g
g
Finally the hull of h(Il ) is the orbit O(l), because Il = {h ∈ L1 (g), hˆ = 0 on O(l)}. Indeed, we have that L1 (G) = {f : G → C; jg (f ◦ exp) ∈ L1 (g)} and L1 (g) = {h : g → C; (jg−1 h) ◦ log ∈ L1 (G)} and therefore by our assumption 1)
Il = (f ◦ exp)jg : f ∈ ker(πl )
= (f ◦ exp)jg : (f ◦ exp)jg ˆ = 0 on O(l)
= h ∈ L1 (g): hˆ = 0 on O(l) . Then by Lemma 4.4, O(l) is an affine linear orbit (we take D = ad g). 2) ⇒ 1) Theorem 4.2. 2 References [1] M.P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, M. Vergne, Représentations des groupes de Lie résolubles, Monogr. Soc. Math. Fr., vol. 4, Dunod, Paris, 1972.
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[2] J. Boidol, ∗-Regularity of exponential Lie groups, Invent. Math. 56 (1980) 231–238. [3] W. Hauenschild, J. Ludwig, The injection and the projection theorem for spectral sets, Monatsh. Math. 92 (1981) 167–177. [4] H. Leptin, J. Ludwig, Unitary Representation Theory of Exponential Lie Groups, de Gruyter Exp. Math., vol. 18, 1994. [5] J. Ludwig, Good ideals in the group algebra of a nilpotent Lie group, Math. Z. 161 (1978) 195–210. [6] J. Ludwig, On the Hilbert–Schmidt semi-norms of L1 of a nilpotent Lie group, Math. Ann. 273 (1986) 383–395. [7] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Clarendon Press, Oxford, 1968. [8] O. Ungermann, The Jacobson topology of Prim∗ L1 (G) for exponential Lie groups, Thesis, 2007. [9] N.R. Wallach, Harmonic Analysis on Homogeneous Spaces, Pure Appl. Math., vol. 19, Marcel Dekker, Inc., New York, 1973.
Journal of Functional Analysis 258 (2010) 3977–3987 www.elsevier.com/locate/jfa
The Heyde theorem for locally compact Abelian groups G.M. Feldman Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47, Lenin ave., Kharkov, 61103, Ukraine Received 8 April 2009; accepted 10 March 2010 Available online 20 March 2010 Communicated by S. Vaes
Abstract We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with nonvanishing characteristic functions. If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then each μj = γj ∗ ρj , where γj are Gaussian measures, and ρj are distributions supported in G. © 2010 Elsevier Inc. All rights reserved. Keywords: Gaussian measure; Locally compact Abelian group
1. Introduction By the well-known Skitovich–Darmois theorem a Gaussian measure on the real line can be characterized by the independence of two linear forms of independent random variables. A similar result was obtained by C.C. Heyde, where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. He proved the following theorem ([9], see also [10, §13.4]). E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.005
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Theorem A. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables, let αj , βj be nonzero constants such that βi αi−1 ± βj αj−1 = 0 for all i = j . If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then all ξj are Gaussian. In the last years much attention has been devoted to generalizing of the classical characterization theorems into various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, symmetric spaces (see e.g. [7,11], and [5] where one can find additional references). The present article continues these researches. Let X be a locally compact second countable Abelian group and let Y = X ∗ be its character group. Denote by (x, y) the value of a character y ∈ Y at an element x ∈ X. Let Aut(X) be the set of all topological automorphisms of X. Denote by T the circle group (the one-dimensional torus) T = {z ∈ C: |z| = 1}. Let M 1 (X) be the convolution semigroup of probability distributions on X. For μ ∈ M 1 (X) denote by μ(y) ˆ =
(x, y) dμ(x),
y∈Y
X
its characteristic function (the Fourier transform). If μ = μ1 ∗ μ2 , μj ∈ M 1 (X), then μj are called factors of μ. A probability measure μ ∈ M 1 (X) is called Gaussian (in the sense of Parthasarathy) [12, Ch. IV, §6] if its characteristic function can be represented in the form μ(y) ˆ = (x, y) exp −ϕ(y) ,
y ∈ Y,
where x ∈ X and ϕ is a continuous nonnegative function satisfying the equation ϕ(u + v) + ϕ(u − v) = 2 ϕ(u) + ϕ(v) ,
u, v ∈ Y.
(1)
Taking into account that in the article we will deal only with Gaussian measures in the sense of Parthasarathy we will name them Gaussian. Denote by Γ (X) the set of Gaussian measures on the group X. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Consider linear forms L1 = α1 ξ1 + · · · + αn ξn and L2 = β1 ξ1 + · · · + βn ξn . It was proved in [4] that the symmetry of the conditional distribution of L2 given L1 implies that all μj ∈ Γ (X) if and only if X contains no elements of order 2. If a group X contains elements of order 2, then the following natural problem arises: Problem 1. Let X be a locally compact second countable Abelian group, and assume that X contains elements of order 2. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ±βj αj−1 ∈ Aut(X) for all i = j . What distributions μj are characterized by the symmetry of the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn ? This problem was solved in [4] for the case when X is the two-dimensional torus T2 . Namely, the following theorem holds: Let ξ1 , ξ2 be independent random variables with values in T2
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and distributions μj with non-vanishing characteristic functions. Consider linear forms L1 = α1 ξ1 +α2 ξ2 and L2 = β1 ξ1 +β2 ξ2 , where αj , βj ∈ Aut(T2 ) such that β1 α1−1 ±β2 α2−1 ∈ Aut(T2 ). If the conditional distribution of L2 given L1 is symmetric, then μj = γj ∗ρj , where γj ∈ Γ (T2 ), ρj ∈ M 1 (G), j = 1, 2, and G is the subgroup of T2 generated by all elements of order 2. The aim of this article is to solve Problem 1 for groups X containing no subgroup topologically isomorphic to T. It turned out that the answer is the same as in case when X is the two-dimensional torus T2 . We will prove the following theorem. Theorem 1. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T. Let G be the subgroup of X generated by all elements of order 2. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric, then each μj = γj ∗ ρj , where γj ∈ Γ (X), ρj ∈ M 1 (G). The proof of Theorem 1 can be reduced to solving of a functional equation in the class of continuous positive definite functions on a locally compact Abelian group. To prove Theorem 1 we shall use some results of the structure theory for locally compact Abelian groups and the duality theory (see [8]). If L is a subgroup of Y , then denote by A(X, L) = {x ∈ X: (x, y) = 1 for all y ∈ L} its annihilator. Denote by cX the connected component of zero of X and by bX the subgroup of X consisting of all compact elements of X. For each natural number n let fn : X → X be a homomorphism fn (x) = nx, and X (n) = Im fn , X(n) = Ker fn . For μ ∈ M 1 (X) we define the distribution μ¯ ∈ M 1 (X) by the formula μ(E) ¯ = ˆ¯ μ(−E) for all Borel sets E ⊂ X. Observe that μ(y) = μ(y). ˆ We denote by σ (μ) the support of μ ∈ M 1 (X). It should be noted that if μ(y) ˆ = 1 for all L, then σ (μ) ⊂ A(X, L). If α ∈ Aut(X), then the adjoint automorphism α˜ ∈ Aut(Y ) is defined by the formula (x, αy) ˜ = (αx, y) for all x ∈ X, y ∈ Y . A subgroup G of X is called characteristic if α(G) = G for any α ∈ Aut(X). Let ψ : Y → C be an arbitrary function, and let h ∈ Y . We denote by h the finite difference operator h ψ(y) = ψ(y + h) − ψ(y),
y ∈ Y.
A continuous function ψ on Y is called a polynomial if for some natural m the equality m+1 ψ(y) = 0 h is fulfilled for all y, h ∈ Y . The minimal m for which this equality holds is called the degree of ψ. Observe that a nonzero function satisfying (1) is a polynomial of degree 2. 2. Proof of lemmas To prove Theorem 1 we need some lemmas. Lemma 1. (See [4].) Let Y be a locally compact Abelian group, ψj , j = 1, 2, . . . , n, n 2, be functions on Y satisfying the equation
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u, v ∈ Y,
(2)
j =1
where δˆj ∈ Aut(Y ) such that δˆi ± δˆj ∈ Aut(Y ) for i = j . Then each function ψj satisfies the equation ψj (y) = 0, 2k 2n−2 h
(3)
where k, h and y are arbitrary elements of Y . Lemma 2. Let n be a natural number and X be a locally compact Abelian group satisfying the where X is a subgroup of X such that X (n) = X. condition X (n) = cX . Then X = cX × X, Proof. Assume first that X is a compact group. Then Y = X ∗ is a discrete group. Set L = X/cX . It follows from the condition of the lemma that L(n) = L. This implies that (L∗ )(n) = L∗ . Taking into account that L∗ ∼ = A(Y, cX ) = bY , we see that bY is a bounded subgroup of Y . Since bY is a pure subgroup of Y , by the Kulikov theorem (see [6, §27]) bY is a direct factor of Y , i.e. Y = D × bY ,
(4)
∼ Y/bY ∼ where D = = (cX )∗ . Taking into account that cX = A(X, bY ), it follows from (4) that where X ∼ X = cX × X, = (bY )∗ . Thus, the statement of the lemma is proved for a compact group X. Let X be an arbitrary locally compact Abelian group. By the structure theorem X ∼ = Rm × G, where m 0 and G contains a compact open subgroup. Without restricting the generality, we may assume that X = Rm × G. This implies that cX = Rm × cG . Let K be a compact open sub group of G. It is obvious that cK = cG and K (n) = cK . As has been shown above K = cK × K, (n) = K. Let π be a continuous homomorphism π : K → cK , such that the restriction of where K π to cK is the identity automorphism. Since the group cK is divisible, the homomorphism π can be extended to a homomorphism πˆ : G → cK = cG (see [8, §A.7]). Taking into account that the homomorphism πˆ is continuous on an open subgroup K, we conclude that πˆ is continuous on G, [8, §6.22]. Hence, X = Rm × cG × G = cX × X. Lemma 2 is proved. 2 and then G = cG × G Lemma 3. (See [4].) Let X be a locally compact second countable Abelian group. Let ξ1 , . . . , ξn , n 2, be independent random variables with values in X and distributions μj . Assume that δj ∈ Aut(X). The conditional distribution of L2 = δ1 ξ1 + · · · + δn ξn given L1 = ξ1 + · · · + ξn is symmetric if and only if the characteristic functions of the distributions μj satisfy the equation n j =1
μˆ j (u + δ˜j v) =
n
μˆ j (u − δ˜j v),
u, v ∈ Y.
(5)
j =1
Lemma 4. Let X be a locally compact second countable Abelian group. Let αj , βj ∈ Aut(X), j = 1, 2, . . . , n, n 2, such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. Assume that the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given
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L1 = α1 ξ1 + · · · + αn ξn is symmetric. Then the random variables ξj can be substituted by their shifts ξj in such a way that the conditional distribution of L 2 = β1 ξ1 + · · · + βn ξn given L 1 = α1 ξ1 + · · · + αn ξn is symmetric and all supports σ (μ j ) ⊂ {x ∈ X: 2x ∈ cX }. Proof. Passing to the random variables ξ˜j = αj ξj we can assume without loss of generality that L1 = ξ1 + · · · + ξn and L2 = δ1 ξ1 + · · · + δn ξn , where δj ∈ Aut(X) such that δi ± δj ∈ Aut(X) for i = j . By Lemma 3, the characteristic functions μˆ j (y) satisfy Eq. (5). It is clear that the characteristic functions of the distributions μ¯ j satisfy Eq. (5) too. Hence, the characteristic functions of the distributions νj = μj ∗ μ¯ j also satisfy Eq. (5). Observe that νˆ j (y) = |μˆ j (y)|2 > 0, and set ψj (y) = − ln νˆ j (y). It follows from (5) that the functions ψj satisfy Eq. (2) and hence by Lemma 1, the functions ψj satisfy Eq. (3). This implies that the function ψj is a polynomial on the subgroup Y (2) . Particularly, the function ψj is a polynomial on the subgroup bY (2) . It is well known (see [2]) that any polynomial on the subgroup consisting of all compact elements of a group is a constant. So, we have ψj (y) = ψj (0) = 0 for y ∈ bY (2) . Hence, νˆ j (y) = 1 for all y ∈ bY (2) and for this reason σ (νj ) ⊂ K = A(X, bY (2) ). Since cX = A(X, bY ), we have K = {x ∈ X: (x, 2y) = 1 for all y ∈ bY } = {x ∈ X: (2x, y) = 1 for all y ∈ bY } = {x ∈ X: 2x ∈ cX }. Since νˆ j (y) = 1 for y ∈ bY (2) , the restrictions of the characteristic functions μˆ j (y) to the subgroup bY (2) are some characters of the subgroup bY (2) . We can extend these characters to some characters of the group Y . By the duality theorem, there exist elements xj ∈ X such
that μˆ j (y) = (xj , y) for y ∈ bY (2) . Set x1 = −δ1−1 nj=2 δj xj and xj = xj , j = 2, . . . , n.
n Since j =1 δj xj = 0, the functions fj (y) = (xj , y) satisfy Eq. (15) on the group Y . Put λˆ j (y) = μˆ j (y)(xj , y), j = 1, 2, . . . , n. It is clear that the characteristic functions λˆ j (y) possess the following properties: (i) (ii) (iii)
λˆ j (y) satisfy Eq. (5) on the group Y ; λˆ j (y) = 1 for y ∈ bY (2) and hence, σ (λj ) ⊂ K, j = 2, . . . , n;
λˆ 1 (y) = (x , y), where x = x1 + δ −1 nj=2 δj xj , for y ∈ b (2) . 1
1
1
Y
Inasmuch as bY (2) is a characteristic subgroup of the group Y , it follows from (i) and (ii) that the restriction of Eq. (5) to the subgroup bY (2) is of the form λˆ 1 (u + δ˜1 v) = λˆ 1 (u − δ˜1 v),
u, v ∈ bY (2) .
Putting here u = δ˜1 v we conclude that λˆ 1 (y) = 1 for y ∈ bY (4) . This implies that σ (λ1 ) ⊂ F = A(X, bY (4) ) = {x ∈ X: 4x ∈ cX }. Since K ⊂ F and taking into account that F is a characteristic subgroup of the group X, we can assume from the beginning that the group X satisfies the where X (4) = X. It follows from this that Y = condition X (4) = cX . By Lemma 2, X = cX × X, ∗ ∗ ∼ ∼ D ×bY , where D = cX , bY = X . Represent the element x1 in the form x1 = c + x, ˜ where c ∈ cX , This implies that (x , y) = (x, ˜ y) for y ∈ b . Set g (y) = ( x, ˜ y), g (y) = 1, y ∈ Y , x˜ ∈ X. Y 1 j 1 j = 2, . . . , n, and μˆ j (y) = λˆ j (y)gj (y), j = 1, 2, . . . , n. It follows from (iii) that μˆ 1 (y) = 1 for y ∈ bY (2) . Moreover μˆ j (y) = 1 for y ∈ bY (2) , j = 2, . . . , n, and hence, σ (μ j ) ⊂ A(X, bY (2) ) = K, j = 1, 2, . . . , n. Since μˆ j (y) = 1 for y ∈ bY (2) , j = 1, 2, . . . , n, the characteristic functions μˆ j (y) are invariant with respect to the subgroup bY (2) and hence, they satisfy Eq. (5) on the subgroup bY . Taking into account that the characteristic functions μˆ j (y) and λˆ j (y) satisfy Eq. (5) on the sub-
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group bY we conclude that the functions gj (y) also satisfy Eq. (5) on the subgroup bY . It follows from this that (x, ˜ u + δ˜1 v) = (x, ˜ u − δ˜1 v),
u, v ∈ bY ,
and hence, 2x˜ = 0. This implies that the characteristic functions μˆ j (y) satisfy Eq. (5) on the group Y . By Lemma 3, if independent random variables ξj take values in X and have distributions μ j , then the conditional distribution of L 2 = β1 ξ1 +· · ·+βn ξn given L 1 = α1 ξ1 +· · ·+αn ξn is symmetric. Lemma 4 is proved. 2 Remark 1. Taking into account that K = {x ∈ X: 2x ∈ cX } is a characteristic subgroup of X, we draw the following conclusion from Lemma 4. Let ξj , j = 1, 2, . . . , n, n 2, be independent random variables with values in a locally compact second countable Abelian group X and distributions μj with non-vanishing characteristic functions. Let αj , βj ∈ Aut(X) such that βi αi−1 ± βj αj−1 ∈ Aut(X) for all i = j . Assume that the conditional distribution of L2 = β1 ξ1 + · · · + βn ξn given L1 = α1 ξ1 + · · · + αn ξn is symmetric. Studying the possible distributions μj we can suppose, without restricting of generality, that X (2) = cX . Lemma 5. (See [2].) If a locally compact second countable Abelian group X contains no subgroup topologically isomorphic to T and the characteristic function of a distribution μ ∈ M 1 (X) is of the form μ(y) ˆ = exp −ψ(y) , where ψ is a polynomial, then μ ∈ Γ (X). Lemma 6. Let Y be a locally compact Abelian group, let ψ be a continuous function on Y satisfying the equation 2k 2h ψ(y) = 0,
h, k, y ∈ Y,
(6)
and ψ(−y) = ψ(y), ψ(0) = 0. Then the function ψ can be represented in the form ψ(y) = ϕ(y) + rα ,
y ∈ yα + Y (2) ,
where ϕ is a continuous function on Y satisfying Eq. (1), and Y = decomposition of the group Y with respect to the subgroup Y (2) .
(7)
α (yα
+ Y (2) ), y0 = 0, is a
Proof. It should be noted that any function ϕ on the subgroup Y (2) satisfying Eq. (1) can be extended to the function ϕ˜ on the group Y by the formula 1 ϕ(y) ˜ = ϕ(2y), 4
y ∈ Y.
(8)
The function ϕ˜ also satisfies Eq. (1). Analogously, any additive function l, i.e. any function satisfying the Cauchy equation
G.M. Feldman / Journal of Functional Analysis 258 (2010) 3977–3987
l(u + v) = l(u) + l(v)
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(9)
on the subgroup Y (2) can be extended to an additive function l˜ on the group Y by the formula ˜ = 1 l(2y), l(y) 2
y ∈ Y.
(10)
Observe that any polynomial f on Y of degree 2 can be represented as a sum f (y) = ϕ(y) + l(y) + c, where ϕ is a continuous function satisfying Eq. (1), l is a continuous function satisfying Eq. (9), and c is a constant. Note that there exists one-to-one correspondence between functions ϕ satisfying (1) and biadditive functions Φ(u, v). Namely, Φ(u, v) = 12 [ϕ(u + v) − ϕ(u) − ϕ(v)], ϕ(y) = Φ(y, y). Consider an arbitrary coset yα + Y (2) of the group Y with respect to the subgroup Y (2) . The function ψ(yα + y), as a function of y satisfies Eq. (6). For this reason its restriction to the subgroup Y (2) is a polynomial of degree 2. Hence, we have the following representation ψ(yα + y) = ϕα (y) + lα (y) + cα ,
y ∈ Y (2) ,
(11)
where the function ϕα satisfies Eq. (1), the function lα satisfies Eq. (9), and cα is a constant. Extend functions ϕα and lα from the subgroup Y (2) to the functions ϕ˜ α and l˜α on the group Y by formulas (8) and (10) respectively. Passing from the function ϕ˜α to the corresponding biadditive function Φα we find from (11)
ψ(−yα − y) = ψ yα + (−2yα − y) = ϕ˜ α (−2yα − y) + l˜α (−2yα − y) + cα = 4Φα (yα , yα ) + 4Φα (yα , y) + Φα (y, y) − 2l˜α (yα ) − l˜α (y) + cα ,
y ∈ Y (2) .
(12)
Since ψ(yα + y) = Φα (y, y) + l˜α (y) + cα ,
y ∈ Y (2) ,
(13)
and ψ(−y) = ψ(y), we conclude from equalities (12) and (13) that 2Φα (yα , y) = l˜α (y),
y ∈ Y (2) .
(14)
It follows from (13) and (14) that ψ(yα + y) = Φα (yα + y, yα + y) − 2Φα (yα , y) − Φα (yα , yα ) + l˜α (y) + cα = ϕ˜ α (yα + y) + rα , where rα is a constant.
y ∈ Y (2) ,
(15)
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Putting k = h in (6) we obtain the equation ψ(y + 4h) − 2ψ(y + 3h) + 2ψ(y + h) − ψ(y) = 0,
y, h ∈ Y.
(16)
Substituting y = 0, h = yα + 2u, u ∈ Y in (16) we arrive at ψ(4yα + 8u) − 2ψ(3yα + 6u) + 2ψ(yα + 2u) = 0,
u ∈ Y.
Taking into account (15) we infer that ϕ˜ 0 (4yα + 8u) − 2ϕ˜ α (3yα + 6u) + 2ϕ˜ α (yα + 2u) = 0,
u ∈ Y.
Passing here from the functions ϕ˜0 and ϕ˜α to the corresponding biadditive functions Φ0 and Φα we obtain
4 Φ0 (u, u) − Φα (u, u) + 4 Φ0 (yα , u) − Φα (yα , u)
+ Φ0 (yα , yα ) − Φα (yα , yα ) = 0, u ∈ Y. This implies that ϕ˜ α (y) ≡ ϕ˜ 0 (y), and representation (7) follows from (15). Lemma 6 is proved. 2 Denote by Rℵ0 the space of all sequences of real numbers equipped with the product topology. We note that the topological group Rℵ0 is not locally compact. Denote by Rℵ0 ∗ the space of all finitary sequences of real numbers with the topology of strictly inductive limit of spaces Rn . The topological group Rℵ0 ∗ is not locally compact either. Let t = (t1 , . . . , tn , . . .) ∈ Rℵ0 and s = (s1 , . . . , sn , 0, . . .) ∈ Rℵ0 ∗ . Set t, s = ∞ j =1 tj sj , and put (t, s) = exp{i t, s }. ℵ 0 Let μ be a distribution on the group R . We define the characteristic function of μ by the formula μ(s) ˆ = Rℵ0 (t, s) dμ(t), s ∈ Rℵ0 ∗ . Let A = (αij )∞ i,j =1 be a symmetric positive semidefi
ℵ0 ∗ . We can nite matrix, i.e. the square form As, s = ∞ α s s i,j =1 ij i j is nonnegative for all s ∈ R define now a Gaussian measure on the group Rℵ0 (see, e.g. in [3, §5]). A probability measure μ on the group Rℵ0 is called Gaussian if its characteristic function is represented in the form μ(s) ˆ = (t, s) exp − As, s ,
s ∈ Rℵ0 ∗ ,
where t ∈ Rℵ0 , and A = (αij )∞ i,j =1 is a symmetric positive semidefinite matrix. Denote by ℵ 0 Γ (R ) the set of Gaussian measures on the group Rℵ0 . Lemma 7. (See [1].) Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T, let E be either Rm or Rℵ0 , let γ be a symmetric Gaussian measure on X. Then there exists a continuous monomorphism π : E → X, such that γ = π(N ), N ∈ Γ (E). Lemma 8. (See [4].) Assume that a group X is of the form X = E × G, where either E = Rm or E = Rℵ0 and G is a locally compact second countable Abelian group such that all nonzero elements of G have order 2. If τ ∈ M 1 (X), τ = λ ∗ ω, where λ ∈ Γ (E), ω ∈ M 1 (G), and the
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characteristic function of ω does not vanish, then each factor τ1 of τ can be represented in the form τ1 = λ1 ∗ ω1 , where λ1 ∈ Γ (E) and ω1 ∈ M 1 (G). Remark 2. Denote by Z(2) the two-element cyclic group. It is relevant to remark that if G is a locally compact Abelian group and all nonzero elements of G have order 2, then by the structure theorem for such groups [8, §25.29] we have
N∗
M × Z(2) , G∼ = Z(2) where M and N are cardinal numbers, the group (Z(2))M is a direct product of the group Z(2) ∗ considering in the product topology, and the group (Z(2))N is a weak direct product of the group Z(2) considering in the discrete topology. 3. Proof of Theorem 1 Proof of Theorem 1. Passing to the random variables ξj = αj ξj we can assume without loss of generality that L1 = ξ1 + · · · + ξn and L2 = δ1 ξ1 + · · · + δn ξn , where δj ∈ Aut(X) such that δi ± δj ∈ Aut(X) for i = j . Put νj = μj ∗ μ¯ j and ψj (y) = − ln νˆ j (y). By Lemma 3, the symmetry of the conditional distribution of L2 given L1 implies that the characteristic functions μˆ j (y) satisfy Eq. (5). Hence, the characteristic functions νˆ j (y) also satisfy Eq. (5). It follows from this that the functions ψj satisfy Eq. (2). By Lemma 1, equality (3) holds true. It follows from this that the restriction of the function ψj to the subgroup Y (2) is a polynomial. We will prove at first that any distribution νj is represented as a convolution of a Gaussian measure and a distribution supported in G. By Lemma 4 and Remark 1, we can suppose that the group X possesses the property: X (2) = cX , i.e. the mapping π : X → cX , π(x) = 2x is an epimorphism, Ker π = G and cX ∼ = X/G. It follows from this that (cX )∗ ∼ = A(Y, G) = Y (2) . We can assume that cX = {0}, for otherwise the statement of Theorem 1 follows directly from Lemma 4. Taking into account that cX contains no subgroup topologically isomorphic to T and (cX )∗ ∼ = Y (2) we can apply Lemma 5 to the restriction of the function exp{−ψj (y)} to Y (2) . We obtain that this restriction is the characteristic function of a Gaussian measure on X/G ∼ = cX . This implies that the restriction of the function ψj to Y (2) satisfies Eq. (1). If ψj (y) = 0 for y ∈ Y (2) , then σ (νj ) ⊂ A(X, Y (2) ) = G, and the statement of Theorem 1 for the distribution μj follows from the fact that μj is a factor of νj . Therefore we can assume that ψj (y) ≡ 0 on Y (2) . Thus, the restriction of the function ψj to Y (2) is a polynomial of degree 2. Hence, for any fixed k ∈ Y the restriction of the function 2k ψj (y) to Y (2) is a polynomial of degree 1. On the other hand (3) implies that the function 2k ψj (y) is a polynomial on Y . It follows from this that 2k ψj (y) is a polynomial of degree 1 on Y . So, the function ψj satisfies Eq. (6). Applying Lemma 6 we obtain representation (7) for the function ψj , where the function ϕ and rα depend on ψj . Let γ be a Gaussian measure on the group X with the characteristic function γˆ (y) = exp −ϕ(y) ,
y ∈ Y.
(17)
Consider on the group Y the function g(y) = exp{−rα },
y ∈ yα + Y (2) .
(18)
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It follows from g(y) = νˆ j (y)/γˆ (y) that the function g is continuous. Observe also that g is invariant with respect to the subgroup Y (2) . We will verify that g is a positive definite function. For this purpose we take arbitrary elements t1 , . . . , tn ∈ Y , such that tj ∈ yαj + Y (2) and verify that for arbitrary z1 , . . . , zn ∈ C the inequality n
g(ti − tj )zi z¯ j 0
(19)
i,j =1
holds true. Consider a subgroup H in Y generated by cosets yαj + Y (2) , j = 1, . . . , n. Note ∼ (A(X, Y (2) ))∗ = G∗ . We conclude from this that H consists of a finite number of that Y/Y (2) = cosets. Set K = H ∗ , L = A(X, H ). Then K ∼ = X/L, L ⊂ G. Consider the restriction g|H of the function g to H . This restriction is invariant with respect to the subgroup Y (2) and hence, it can be considered as a function on the factor-group H /Y (2) . Note that the factor-group H /Y (2) is finite and all nonzero elements of it have order 2. This implies that any real valued function on H /Y (2) is the characteristic function of a signed measure. From what has been said it follows that g|H is the characteristic function of a signed measure β on a finite subgroup F = A(K, Y (2) ). It follows from (7), (17) and (18) that the restriction of the function νˆ j (y) to H is the characteristic function of a convolution of a Gaussian measure α on K and a signed measure β on F . We will verify that β is a probability distribution and hence, (19) will be proved. We remind that we assume that X (2) = cX . This property implies that cX ∼ = X/G and hence, cX ∼ = (X/L)/(G/L). Taking into account that cX contains no subgroup topologically isomorphic to T it follows from this that the group K also contains no subgroup topologically isomorphic to T. By Lemma 7, there exists a continuous monomorphism p : E → K, where either E = Rm or E = Rℵ0 such that α = p(N), N ∈ Γ (E). Thus, the Gaussian measure α is concentrated on a Borel subgroup B = p(E) of K. Note that all nonzero elements of subgroup F have order 2 and hence, B ∩ F = {0}. Since α ∗ β is a probability distribution, we conclude that β is a probability distribution too. Thus, g is a positive definite function. By the Bochner theorem there exists a distribution ρ ∈ M 1 (X) such that ρ(y) ˆ = g(y). Since g(y) = 1 for all y ∈ Y (2) , we see 1 (2) that σ (ρ) ⊂ A(X, Y ) = G, i.e. ρ ∈ M (G). We inferred that νj = γ ∗ ρ, where γ ∈ Γ (X), ρ ∈ M 1 (G). Prove now that the distribution μj has the required representation. By Lemma 7, there exists a continuous monomorphism q : E → X, where either E = Rm or E = Rℵ0 , such that γ = q(N ), N ∈ Γ (E). Since q(E) ∩ G = {0}, the monomorphism q can be extended to a continuous monomorphism q˜ : E × G → X by the formula q(t, ˜ ζ ) = q(t) + ζ , t ∈ E, ζ ∈ G. Then νj = q(N ˜ ∗ ρ) and hence, the distribution νj is concentrated on the Borel subgroup q(E) + G of X. The distribution μj is a factor of νj . Substituting, if it is necessary, the distribution μj by its shift, we can assume that μj is also concentrated on the Borel subgroup q(E)+G. For this reason μj = q(M ˜ j ), where Mj ∈ M 1 (E × G) and Mj is a factor of N ∗ ρ. By Lemma 8, Mj = Nj ∗ ρj , where Nj ∈ Γ (E), ρj ∈ M 1 (G). Hence, μj = q(M ˜ j ) = q(N ˜ j ∗ ρj ) = q(N ˜ j ) ∗ q(ρ ˜ j ) = γj ∗ ρ j , where γj = q(N ˜ j ). Since γj ∈ Γ (X), Theorem 1 is proved. 2 We note that the following statement results from Lemma 6 and the proof of Theorem 1.
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Proposition 1. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to T, G be the subgroup of X generated by all elements of ¯ and the characteristic function of the distribution ν be order 2. Let μ ∈ M 1 (X), ν = μ ∗ μ, of the form νˆ (y) = exp −ψ(y) , where the function ψ satisfies the equation 2k m h ψ(y) = 0,
k, h, y ∈ Y.
Then μ = γ ∗ ρ, where γ ∈ Γ (X), and ρ ∈ M 1 (G). Proposition 1 can be regarded as one more group analogue of the well-known Marcinkiewicz theorem (compare with Lemma 5). Acknowledgment I thank the referee for carefully reading the manuscript and useful remarks. References [1] G.M. Feldman, Gaussian distributions on locally compact Abelian groups, Theory Probab. Appl. 23 (1978) 529– 542. [2] G.M. Feldman, Marcinkiewicz and Lukacs theorems on Abelian groups, Theory Probab. Appl. 34 (1989) 290–297. [3] G.M. Feldman, Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, Amer. Math. Soc. Transl. Math. Monogr., vol. 116, Amer. Math. Soc., Providence, RI, 1993. [4] G.M. Feldman, On a characterization theorem for locally compact Abelian groups, Probab. Theory Related Fields 133 (2005) 345–357. [5] G.M. Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts Math., vol. 5, European Math. Soc., Zurich, 2008. [6] L. Fuchs, Infinite Abelian Groups. 1, Academic Press, New York/London, 1970. [7] P. Graczyk, J.-J. Loeb, A Bernstein property of measures on groups and symmetric spaces, Probab. Math. Statist. 20 (1) (2000) 141–149. [8] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis. 1, Springer-Verlag, Berlin/Gottingen/Heildelberg, 1963. [9] C.C. Heyde, Characterization of the normal low by the symmetry of a certain conditional distribution, Sankhya Ser. A 31 (1969) 115–118. [10] A.M. Kagan, Yu.V. Linnik, C.R. Rao, Characterization Problems of Mathematical Statistics, Wiley, New York, 1973. [11] D. Neuenschwander, R. Schott, The Bernstein and Skitovich–Darmois characterization theorems for Gaussian distributions on groups, symmetric spaces, and quantum groups, Expo. Math. 15 (4) (1997) 289–314. [12] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York/London, 1967.
Journal of Functional Analysis 258 (2010) 3988–4009 www.elsevier.com/locate/jfa
Homological properties of modules over semigroup algebras Paul Ramsden Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom Received 22 April 2009; accepted 10 March 2010 Available online 23 March 2010 Communicated by S. Vaes
Abstract Let S be a semigroup. In this paper we investigate the injectivity of 1 (S) as a Banach right module over 1 (S). For weakly cancellative S this is the same as studying the flatness of the predual left module c0 (S). For such semigroups S, we also investigate the projectivity of c0 (S). We prove that for many semigroups S for which the Banach algebra 1 (S) is non-amenable, the 1 (S)-module 1 (S) is not injective. The main result about the projectivity of c0 (S) states that for a weakly cancellative inverse semigroup S, c0 (S) is projective if and only if S is finite. © 2010 Elsevier Inc. All rights reserved. Keywords: Banach algebra; Homology; Cohomology; Module; Projective; Injective; Flat; Amenable; Semigroup
1. Introduction Let S be a semigroup, and let 1 (S) be its associated Banach convolution algebra. In this paper we study certain homological properties of modules over 1 (S). The aim is to characterize homological properties of the Banach algebra 1 (S) (and its modules) in terms of the underlying semigroup S. Homological properties of Banach algebras associated with groups and semigroups have been studied by many authors. Some recent papers are [1,6–8]. The notions of projectivity, injectivity, and flatness are fundamental in homology theory. The theory of these concepts in the category of Banach modules is expounded by A.Ya. Helemskii in [10]. In [4], H.G. Dales and M.E. Polyakov undertook a study of these properties for variE-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.007
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ous canonical modules associated with the group algebra L1 (G). For example, they proved the following theorem. Here we regard C0 (G) as a submodule of the dual module L∞ (G) = L1 (G) . Theorem 1.1. (See [4, Theorems 3.1, 4.7 and 4.9].) Let G be a locally compact group. Then: (i) L1 (G) is an injective right L1 (G)-module if and only if G is discrete and amenable. (ii) C0 (G) is a flat left L1 (G)-module if and only if G is amenable. (iii) C0 (G) is a projective left L1 (G)-module if and only if G is compact. In this paper we shall investigate the same questions for the Banach algebra 1 (S), where S is a (discrete) semigroup. We cannot hope for such simple characterizations for a general semigroup. This is partly due to the complexities of describing the amenability of the Banach algebra 1 (S) in terms of the semigroup S; see [3, Theorem 10.12]. 1.1. Overview of contents • Section 2 contains all the background material and definitions about Banach modules and semigroups that we shall need. • In Section 3 we prove some general results about injective Banach modules. • In Section 4 we show how amenability of the underlying semigroup enters the picture. Here we show that the injectivity of certain modules over 1 (S) implies that S is amenable. This follows from a general result which also applies to modules over L1 (G) and M(G) for a locally compact group G. In Theorem 4.10, under the additional assumption that S is cancellative, we give a characterization of the injectivity of 1 (S) (S must be an amenable group). At the end of this section we give an example (Example 4.12) of a finite semigroup S such that 1 (S) is a right injective module, but 1 (S) is not amenable. • In Section 5 we investigate the flatness of the left 1 (S)-predual module c0 (S) for a weakly right cancellative semigroup S. Our main theorem (Theorem 5.5) gives a necessary combinatorial condition on the set of idempotents. This condition is not satisfied by the bicyclic semigroup or (N, max). • In Section 6 we move on to investigating the semigroups S such that c0 (S) has the stronger property of being projective. Here we prove, in Theorem 6.5 that, if S is a weakly cancellative semigroup such that c0 (S) is projective, then S must be finite. An immediate corollary of this result (Theorem 6.6) gives a characterization for the class of inverse semigroups (c0 (S) is projective if and only if S is finite). 2. Preliminaries For n ∈ N, we set Nn = {1, 2, . . . , n}. The indicator function of a subset T of a set S is denoted by χT . Let f : S → E be a function from a set S to a vector space E. For T ⊂ S we define χT f : S → E by (χT f )(s) = f (s) (s ∈ T ) and (χT f )(s) = 0 (s ∈ S \ T ). Let E be a Banach space. We denote the dual space by E ; the action of λ ∈ E on an element x ∈ E is written as x, λ. For a subspace F of a Banach space E we set F 0 = λ ∈ E : x, λ = 0 (x ∈ F ) . Then F 0 is a closed subspace of E . We set F 00 = (F 0 )0 ⊂ E .
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, and B(E, F ) denotes the space of all bounded The projective tensor product is denoted by ⊗ linear operators between Banach spaces E and F . 2.1. Banach modules Throughout this section A is a Banach algebra. We denote by A-mod, and by mod-A the categories of Banach left A-modules, and of Banach right A-modules, respectively. We shall give definitions only for one category of module; similar definitions apply for modules in other categories. Let E ∈ A-mod. Then the dual space E has a natural right A-module structure given by x, λ · a = a · x, λ
a ∈ A, x ∈ E, λ ∈ E .
This module is called the dual module of E. Let S ⊂ A be a subset, and let E ∈ A-mod. We set S · E = {a · x: a ∈ S, x ∈ E} and SE = lin S · E, the linear span of S · E. If S is a left ideal of A then SE is a submodule of E. The essential part of E is the closed submodule AE, and E is essential if AE = E. Now we describe the ‘dual’ concept. Let F ∈ mod-A. For a subset S ⊂ A we set F ⊥S = x ∈ F : x · S = {0} . If S is a left ideal of A then F ⊥S is a closed submodule of F . We set F ⊥ = F ⊥A , which is the annihilator submodule of F . The A-module F is faithful if F ⊥ = {0}, and F is an annihilator module if F ⊥ = F . Now suppose that F = E for some E ∈ A-mod. Then F ⊥S = (SE)0 , and so we have (SE) = F /(SE)0 = F /F ⊥S . Hence, if SE = E, then F ⊥S = {0}. In particular the dual of an essential module is faithful. Similarly, for E ∈ A-mod we set S⊥
E = x ∈ E: S · x = {0} .
For E, F ∈ A-mod we denote by A B(E, F ) the subspace of B(E, F ) consisting of bounded left A-module morphisms. Similarly, BA (E, F ) denotes the space of bounded right A-module morphisms when E, F ∈ mod-A. The unitization of A is denoted by A . If E ∈ A-mod, then we consider E ∈ A -mod in the obvious way. E ∈ A-mod with module operation specified by Let E ∈ A-mod. Then A ⊗ a · (b ⊗ x) = ab ⊗ x
a ∈ A, b ∈ A , x ∈ E .
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E → E by We define the canonical morphism π : A ⊗ a ∈ A , x ∈ E .
π(a ⊗ x) = a · x
Let E ∈ mod-A. Then B(A , E) ∈ mod-A with the module operation a ∈ A, b ∈ A , T ∈ B A , E .
(T · a)(b) = T (ab)
We define the canonical embedding Π : E → B(A , E) by the formula a ∈ A , x ∈ E .
Π(x)(a) = x · a 2.2. Banach homology
We now recall the definitions and basic relationships from Banach homology that this paper is concerned with. For full details see [10] and [11]. We eschew the original homological definitions and give the standard characterizations which have proved most useful for checking projectivity and injectivity in specific cases. Because of the duality involved it is convenient to study injective right modules and projective and flat left modules. Proposition 2.1. Let A be a Banach algebra. (i) Let F ∈ mod-A. Then F is injective if and only if there exists ρ ∈ BA (B(A , F ), F ) with ρ ◦ Π = IF . E) with (ii) Let E ∈ A-mod. Then E is projective if and only if there exists ρ ∈ A B(E, A ⊗ π ◦ ρ = IE . E) ) such that (iii) Let E ∈ A-mod. Then E is flat if and only if there exists ρ ∈ A B(E, (A ⊗ the following diagram commutes ρ
E ιE
E) (A ⊗ π
E . (iv) If either E is essential or F is faithful, then we can replace A by A in the above characterizations. Part (i) is proved in [10, III.1.31] and the case where E is faithful is [4, Proposition 1.7]; part (ii) is [10, IV.1.1, IV.1.2]; part (iii) is similar to [10, Exercise VII2.8], see also [15, Lemma 4.3.22]. The following facts are elementary: a module E is flat if and only if the dual module E is injective; every projective module is flat. The classes of amenable and contractible Banach algebras are particularly important and well studied (see [10, Chapter IV], [12] or [2, §2.8]). The following proposition gives one nice property of these Banach algebras.
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Proposition 2.2. Let A be a Banach algebra, and let E ∈ A-mod or E ∈ mod-A. (i) If A is contractible, then E is projective. (ii) If A is amenable, then E is flat. The following is a long standing open problem. Question 2.3. Let A be a Banach algebra such that every E ∈ A-mod and every E ∈ mod-A is flat [ projective]. Is A amenable [contractible]? The answer is known to be positive for many classes of Banach algebras, in particular for the class of group algebras. Part of the motivation for this work was to answer this question in the class of semigroup algebras. Although we come short of a complete answer, our results strongly suggest that the answer is ‘yes’ in this class. 2.3. Semigroups and semigroup algebras Let S be a semigroup. The set of idempotents in S is denoted by E(S). A semigroup S is a semilattice if S is commutative and E(S) = S. The canonical partial order on E(S) is given by pq
⇐⇒
p = pq = qp
p, q ∈ E(S) .
Let S be a semigroup. The semigroup algebra 1 (S) is the completion in the 1 -norm of the algebra CS. It is the Banach algebra generated by the semigroup. The convolution product
on 1 (S) is uniquely defined by requiring that δs δt = δst (s, t ∈ S). We identify 1 (S) with 1 (S ), where S is the semigroup formed by adjoining an identity to S. These Banach algebras have been studied by many authors. A recent exposition is the memoir [3]. 2.3.1. Cancellativity We shall use the following notation introduced by Grønbæk in [9]. For s, t ∈ S we define the sets −1 = {u ∈ S: ut = s} st
and
−1 t s = {u ∈ S: tu = s}.
Let S be a semigroup. An element t ∈ S is left cancellable if u = v whenever tu = tv. Equivalently we require that |[t −1 s]| 1 (s ∈ S). Right cancellable elements are defined similarly. The semigroup S is cancellative if each element is both left and right cancellative. The semigroup S is weakly left (respectively, right) cancellative if [t −1 s] (respectively, [st −1 ]) is finite for each s, t ∈ S, and weakly cancellative if it is both weakly left cancellative and weakly right cancellative. Lemma 2.4. Let S be a semigroup such that the Banach algebra 1 (S) has a left identity. Suppose that S contains a right cancellable element. Then: (i) S has a left identity eS ; (ii) for each right cancellable element t we have teS = t.
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Proof. (i) Let eA = s∈S es δs be a left identity for 1 (S), and let t be a right cancellable element, so that |[tt −1 ]| 1. We have δt =
s∈S
Hence
s∈[tt −1 ] es
es δst =
r∈S
e s δr .
s∈[rt −1 ]
= 1 and in particular the set [tt −1 ] is non-empty, say [tt −1 ] = {u}. Then δu δt = δt = eA δt .
Since t is right cancellable, it follows that δu = eA , and so u is a left identity for S. (ii) Let t be a right cancellable element and eS a left identity for S. Then t 2 = t (eS t) = (teS )t, which implies that t = teS . 2 2.3.2. Module actions Let S be a semigroup, and let E ∈ 1 (S)-mod or E ∈ mod-1 (S). We shall use the following more compact notation for the module actions t · x = δt · x,
x · t = x · δt
(x ∈ E, t ∈ S).
The standard actions of 1 (S) on 1 (S) are given by (t · a)(s) =
a(u),
(a · t)(s) =
u∈[t −1 s]
a(u)
s, t ∈ S, a ∈ 1 (S) ,
u∈[st −1 ]
where we define the sum over an empty set to be zero. The dual actions of 1 (S) on ∞ (S) are given by (t · λ)(s) = λ(st),
(λ · t)(s) = λ(ts)
s, t ∈ S, λ ∈ ∞ (S) .
For s ∈ S, the indicator function of the set {s} will be denoted by δs when considered as an element of 1 (S) and by λs when considered as an element of ∞ (S). This notation implies that the left module actions satisfy t · δs = δts
and t · λs = χ[st −1 ]
(t ∈ S).
Proposition 2.5. (See [3, Theorem 4.6].) Let S be a semigroup. Then c0 (S) is a left [right] 1 (S)submodule of ∞ (S) if and only if S is weakly right [left] cancellative. 3. Some general results on injective modules In this section we prove some basic intrinsic properties of injective modules. We begin with a generalization of [4, Proposition 1.8].
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Proposition 3.1. Let A be a Banach algebra, and let E ∈ mod-A be injective. Suppose that there exist Q ∈ B(A, E) and a subset S ⊂ A such that Q(ba) = Q(b) · a
(b ∈ S, a ∈ A).
Then there is an element x ∈ E with Q(b) = x · b
(b ∈ S).
Proof. There exists a morphism ρ ∈ BA (B(A , E), E) such that ρ ◦ Π = IE . Extend Q to A by setting Q(e ) = 0. Then Q ∈ B(A , E) and Q · b = Π Q(b)
(b ∈ S).
Set x = ρ(Q) ∈ E. Then we have Q(b) = (ρ ◦ Π) Q(b) = ρ(Q · b) = x · b as required.
(b ∈ S),
2
Corollary 3.2. Let A be a Banach algebra, let I be a complemented ideal in A, and let E ∈ mod-A be injective. Then the map j : E → BA (I, E) given by j (x) : a → x · a : I → E is a Banach A-module epimorphism with kernel E ⊥I . Proof. Take T ∈ BA (I, E), and set Q = T ◦ P , where P : A → I is a projection. Then Q ∈ B(A , E) and satisfies Q(ba) = Q(b) · a (b ∈ I, a ∈ A). By Proposition 3.1, there exists x ∈ E such that T (b) = Q(b) = x · b (b ∈ I ), i.e., T = j (x). The rest is clear. 2 Corollary 3.3. Let A be a Banach algebra. (i) Let A be a subalgebra of a Banach algebra B. Suppose that B is injective in mod-A. Then there exists p ∈ B with pa = a (a ∈ A). (ii) Let I be a closed right ideal in A. Suppose that A/I is injective in mod-A. Then I has a left modular identity. (iii) Let I be a closed, complemented right ideal in A. Suppose that I is injective in mod-A. Then there exists p ∈ I with pa = a (a ∈ I ). (iv) Let I be a closed, complemented right ideal in A. Suppose that I is injective in mod-A. Then I has a bounded left approximate identity. Proof. These all follow from Proposition 3.1 by choosing specific maps Q. (i) Take Q : A → B to be the inclusion map. (ii) Take Q : A → A/I to be the quotient map. (iii) Take Q : A → I to be a projection.
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(iv) Take the map Q : A → I to be a projection onto I followed by the natural embedding into I . Then there exists Φ ∈ I with Φ · a = a (a ∈ I ). The result now follows by a standard argument involving the weak-∗ topology on I , see [2, Proposition 2.9.14(iii)]. 2 Proposition 3.4. Let A be a Banach algebra, and let E ∈ mod-A be injective. Suppose that there exists a0 ∈ A \ {0} with a0 A = 0. Then E ⊥ = E · a0 . In particular, the subspace E · a0 is closed. Proof. The inclusion E · a0 ⊂ E ⊥ is clear. Let x ∈ E ⊥ . Take μ ∈ (A ) with a0 , μ = 1, and let T ∈ B(A , E) be the rank-1 operator given by T (a) = a, μx
a ∈ A .
Let PA , PCe ∈ B(A ) be projections on to the subspaces A and Ce , respectively. Then we have (Πx) ◦ PCe = (T · a0 ) ◦ PCe . Clearly, for any S ∈ B(A , E), we have ρ(S) = ρ(S ◦ PA ) + ρ(S ◦ PCe ). Combining this with the fact that ρ((Πx) ◦ PA ) = ρ((T · a0 ) ◦ PA ) = 0, we have x = ρ (Πx) ◦ PCe = ρ (T · a0 ) ◦ PCe = ρ(T · a0 ) = ρ(T ) · a0 . Therefore E ⊥ = E · a0 .
2
Example 3.5. (See [16, Example 4.4].) Let X be a Banach space with dim X 2, and take ϕ ∈ X \ {0}. Define a product on X by ab = ϕ(a)b
(a, b ∈ X).
With this product X is a Banach algebra which, following [16], we denote by Aϕ (X). By [16], Aϕ (X) is a biprojective Banach algebra. Since Aϕ (X) does not have a right identity, Aϕ (X) is not injective in Aϕ (X)-mod. The Banach algebra Aϕ (X) is faithful in Aϕ (X)-mod, but not faithful in mod-Aϕ (X). Proposition 3.6. The Banach algebra Aϕ (X) is injective in mod-Aϕ (X) if and only if dim X = 2. Proof. Set A = Aϕ (X). Suppose that A is injective in mod-A. Choose a0 ∈ ker ϕ \ {0}. By Proposition 3.4 we have ker ϕ = A⊥ = Aa0 = Ca0 . By the rank-nullity theorem, dim X = 2.
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Conversely, suppose that dim X = 2. Let {a0 , a1 } be a basis of X with a0 , ϕ = 0 and a1 , ϕ = 1. Take λ ∈ X with a0 , λ = 1 and a1 , λ = 0. We define a map ρ : B(A , A) → A by
ρ(T ) = T a0 , λa1 − T a1 , λa0 + T e , λ a0
T ∈ B A , A .
This is a right A-module morphism with ρ ◦ Π = IA . The easiest way to see this is to check that ρ(T · a0 ) = ρ(T ) · a0 , ρ(T · a1 ) = ρ(T ) · a1 (T ∈ B(A , A)), and ρ(Πa0 ) = a0 , ρ(Πa1 ) = a1 . Therefore A is injective in mod-A. 2 4. Amenability and injectivity In this section we show how the injectivity of certain modules over a semigroup algebra implies that the underlying semigroup must be amenable. This is a generalization of the argument in [4, §4]. In contrast to the group case, for a general semigroup, amenability is only part of the story. 4.1. General results Definition 4.1. Let A be a Banach algebra, and let E ∈ mod-A. Then an element λ ∈ E \ {0} is an augmentation-invariant functional if there exists a character ϕ on A, with a · λ = ϕ(a)λ for each a ∈ A. The triple (E, λ, ϕ) is an augmentation-invariant Banach right A-module. Example 4.2. (i) Let A be a Banach algebra with a character ϕ, and let I be a closed left ideal of A with I ⊂ ker ϕ. Then (I, ϕ|I, ϕ) is an augmentation-invariant right A-module. (ii) If E ∈ mod-A is augmentation-invariant, then so is E . Lemma 4.3. Let A be a Banach algebra, let I be a complemented left ideal of A, and let (E, λ, ϕ) be an augmentation-invariant Banach right A-module with I ⊂ ker ϕ. Suppose that E is injective in mod-A. Then there exists Λ ∈ B(I, E) such that: (i) a · Λ = ϕ(a)Λ for each a ∈ A; (ii) Π(x), Λ = x, λ for each x ∈ E. Proof. Since E is injective there is a right A-module morphism ρ : B(A , E) → E with ρ ◦ Π = IE . Set Λ0 = ρ (λ) ∈ B(A , E) . Since ρ is a left A-module morphism, we have a · Λ0 = ϕ(a)Λ0 (a ∈ A). Take T ∈ B(A , E) such that T |I = 0. Pick a ∈ I with ϕ(a) = 1. Then T · a = 0, and 0 = T · a, Λ0 = T , Λ0 . Now take T ∈ B(I, E). We can extend T to T ∈ B(A , E) by setting T = T ◦ P , where P : A → I is a projection. Set T , Λ := T, Λ0
T ∈ B(I, E) .
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Since (T · a − T · a)|I = 0 (a ∈ A), it follows that a · Λ = ϕ(a)Λ (a ∈ A). Similarly, since (Π(x) − Π(x))|I = 0 (x ∈ E), it follows that Π(x), Λ = x, λ. 2 F ) → E = F . If F is a left In the following theorem we set π = Π : B(I, E) = (I ⊗ π |I ⊗ F ⊂ F , and we can replace π by π . A-submodule of F , then Theorem 4.4. Let A be a Banach algebra, let I be a complemented left ideal of A, and let (E, λ, ϕ) be an augmentation-invariant Banach right A-module with I ⊂ ker ϕ. Suppose that E is the dual of a Banach space F , and that E is injective in mod-A. Then there exists a bounded F such that: net (vα ) ⊂ I ⊗ (i) limα a · vα − ϕ(a)vα π = 0 for each a ∈ A; π (vα ) = x, λ for each x ∈ E. (ii) limα x, F , and let σ = σ (X, X ) be the weak topology on X. Proof. Set X = I ⊗ First, a net (uα ) is indexed by the family of all finite subsets of B(I, E), with the ordering specified by inclusion. For each such α = {T1 , . . . , Tk }, choose uα ∈ X such that Ti , uα = Ti , Λ (i = 1, . . . , k), where Λ ∈ X was specified in Lemma 4.3. For each a ∈ A and T ∈ B(I, E), we have T , a · uα = T · a, uα = T · a, Λ = ϕ(a)T , Λ = ϕ(a)T , uα , for each sufficiently large α, and so limα (a · uα − ϕ(a)uα ) = 0 in (X, σ ). Also for each x ∈ E, we have x, π (uα ) = Π(x), uα = Π(x), Λ = x, λ,
for each sufficiently large α, and so limα Π(x), uα − x, λ = 0. Let {a1 , . . . , ak } and {x1 , . . . , x } be finite subsets of A and E, respectively, and let ε > 0. Let C = C {x1 , . . . , x }, ε = z ∈ X: Π(xi ), z − xi , λ < ε (i = 1, . . . , ) , and consider the Banach space Y = X1 ⊕ · · · ⊕ Xk , where each Xi = X (i = 1, . . . , k) and we are taking the 1 -sum. Also consider the linear operator W : z → a1 · z − ϕ(a1 )z, . . . , ak · z − ϕ(ak )z ,
X → Y.
The set C is convex in X, and so W (C) is convex in Y . We have shown that 0 belongs to the σ (Y, Y )-closure of W (C) in Y . By Mazur’s theorem, it follows that 0 belongs to the · -closure of W (C) in Y . The existence of the required net (vα ) follows. 2 4.2. The Banach algebras L1 (G) and M(G) Let G be a locally compact group, and let M(G) denote the measure algebra on G. There is always one character on M(G); this is the augmentation character ϕG , defined by ϕG (μ) = μ(G)
μ ∈ M(G) .
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The restriction of ϕG to L1 (G) regarded as a closed ideal in M(G) has the form ϕG (f ) =
f (s) dm(s)
f ∈ L1 (G) .
G
Let G be a locally compact group. We set P (G) = f ∈ L1 (G): f 0, f = 1 . We shall use the following characterization of amenability, known as Reiter’s condition. Proposition 4.5. (See [13, Proposition (0.8)].) Let G be a locally compact group. Then G is amenable if and only if there is a net (fα ) ⊂ P (G) such that lim t · fα − fα = 0 for each t ∈ G. Theorem 4.6. Let G be a locally compact group, and let (E, λ, ϕG ) be an augmentationinvariant Banach right M(G)-module. Suppose that E is a dual space, and that E is injective in mod-M(G). Then G is amenable. F Proof. Let E have a Banach space predual F . Set A = M(G) and I = L1 (G). Let (vα ) ⊂ I ⊗ be the net given by Theorem 4.4. Take x ∈ E[1] with x, λ = 1. We have π (vα )F x, π (vα ) 1/2 vα π for large enough α. Hence by passing to a subnet we may suppose that, for each α, vα π 1/2. F = L1 (G, F ) to define a net (kα ) in I by We use the identification I ⊗ kα (s) =
vα (s)F vα π
(s ∈ G).
Then kα 0, and kα 1 =
kα (s) dm(s) =
G
G
vα (s)F dm(s) = 1. vα π
Now take t ∈ G. We have 1 t · kα − kα 1 vα π
−1 vα t s − vα (s) dm(s) 2t · vα − vα π . F
G
Hence limα t · kα − kα 1 = 0. Therefore by Proposition 4.5, G is amenable.
2
The same result holds under the hypothesis that E is injective in mod-L1 (G). This combines with Johnson’s theorem and Proposition 2.2 to give the following result, which was proved under additional hypothesis in [4, Theorem 4.6].
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Theorem 4.7. Let G be a locally compact group, and let (E, λ, ϕG ) be an augmentationinvariant Banach right L1 (G)-module. Suppose that E is a dual space. Then E is injective in mod-L1 (G) if and only if G is amenable. These theorems apply to the L1 (G) or M(G)-modules; L1 (G), M(G) and their second duals. Some further results about modules over M(G) are contained in [14]. 4.3. The Banach algebra 1 (S) Theorem 4.8. Let S be a semigroup such that 1 (S) is injective in mod-1 (S). Then S is a left-amenable semigroup and 1 (S) has a left identity. Proof. That S is left-amenable follows in the same way as the proof of Theorem 4.6. That 1 (S) has a left identity is Corollary 3.3(i). 2 Let A be a Banach algebra with a left identity eA . Then eA is an identity for A if and only if {a ∈ A: aA = 0} = {0}, i.e., A is a faithful right A-module. Lemma 4.9. Let S be a semigroup such that 1 (S) is injective in mod-1 (S). Let t ∈ S be a left cancellable element. Then there exists at ∈ 1 (S) with a t δt e A = e A for each left identity eA ∈ 1 (S). Proof. Set A = 1 (S). There is a map T : tS → S : ts → s which extends to a bounded linear operator T : 1 (tS) → A. Set U = T ◦ P , where P : A → 1 (tS) is a projection. Then U satisfies U (b a) = U (b) a
b ∈ 1 (tS), a ∈ A .
By Proposition 3.1, there exists at ∈ A with U (b) = at b (b ∈ 1 (tS)). In particular, eA = U (δt eA ) = at δt eA . 2 Theorem 4.10. Let S be a cancellative semigroup. Then 1 (S) is injective in mod-1 (S) if and only if S is an amenable group. Proof. Sufficiency is obvious; we shall prove necessity. Suppose that 1 (S) is an injective right module. By Theorem 4.8, the Banach algebra 1 (S) has a left identity. By Lemma 2.4, S has an identity eS . By Lemma 4.9, for each s ∈ S, there exists as ∈ 1 (S) with as δs = δeS . It follows that there is an element s −1 ∈ S with s −1 s = eS . From the equation ss −1 s = eS s and right cancellativity we have ss −1 = eS . Therefore S is a group. It is amenable by Theorem 4.8. 2
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Corollary 4.11. The Banach algebra 1 (N) is not injective in mod-1 (N), equivalently c0 (N) is not flat in 1 (N)-mod. Example 4.12. Let S be the right-zero semigroup. The product is given by st = t
(s, t ∈ S).
The Banach algebra 1 (S) belongs to the class of Banach algebras of the form Aϕ (X) defined in Example 3.5. By Proposition 3.6, 1 (S) is right injective if and only if |S| = 2. Since in this case 1 (S) is finite dimensional, this is equivalent to the predual module c0 (S) = ∞ (S) being projective in 1 (S)-mod. Let S2 be the right-zero semigroup on 2 elements. We note that 1 (S2 ) is not amenable. Indeed S2 is left-amenable but not right-amenable; further, S2 has a left identity, but 1 (S2 ) does not have a right identity. Hence the plausible conjecture that 1 (S) is injective in mod-1 (S) only if 1 (S) is amenable is false. 5. Flatness of the predual module c0 (S) Let S be a weakly right cancellative semigroup. Then, by Proposition 2.5 c0 (S) ∈ 1 (S)-mod, and we can identify the dual right 1 (S)-module c0 (S) with 1 (S). Hence for weakly right cancellative semigroups, injectivity of 1 (S) in mod-1 (S) is the same as flatness of c0 (S) in 1 (S)-mod. In this section we shall study this problem for the class of weakly right cancellative semigroups. 5.1. A necessary combinatorial condition For the next two lemmas, we suppose that S is a semigroup, and that E is a Banach space. E. For an element t ∈ S, and F ∈ 1 (S)-mod, we set For a subset T ⊂ S, we set XT = 1 (T ) ⊗ t⊥F = {δt }⊥F . Lemma 5.1. For each t ∈ S, we have t⊥
XS
0
= XS · t.
Proof. Let t ∈ S. The inclusion XS · t ⊂ (t⊥XS )0 is clear. Take λ ∈ (t⊥XS )0 and u, v ∈ S with tu = tv. Then for each x ∈ E we have δu ⊗ x − δv ⊗ x ∈ t⊥XS . Hence, under the identification XS = ∞ (S, E ), we have
0 = δu ⊗ x − δv ⊗ x, λ = x, λ(u) − λ(v)
(x ∈ E).
It follows that λ(u) = λ(v). Hence, for each s ∈ S, λ is constant on the set [t −1 s]. Therefore we can define
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ϕ(s) = Then ϕ ∈ XS and λ = ϕ · t.
if there exists u ∈ [t −1 s], if [t −1 s] = ∅
λ(u), 0,
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(s ∈ S).
2
For a subset T ⊂ S and an element t ∈ S, we define the following subset of T :
G(T , t) =
−1 t s ∩ T = {u ∈ T : ∃v ∈ T , v = u, tu = tv}.
{s∈S: |[t −1 s]∩T |2}
The complement of G(T , t) in T is perhaps simpler to describe: we have T \ G(T , t) = s ∈ T : t −1 (ts) ∩ T = {s} . For example, if t is a left cancellable element, then G(T , t) = ∅. For a subset T ⊂ S, we identify XT with the closed, complemented subspace of XS consisting of functions on S whose support is contained in T . We can then identify XT with XT00 in XS . Lemma 5.2. For each t ∈ S and T ⊂ S, we have: (i) t · XS ⊂ (XtS ) ; (ii) t⊥ (XT ) ⊂ XG(T ,t) . Proof. (i) Let t ∈ S and ϕ ∈ (XtS )0 . Then ϕ · t = 0, and so for each Λ ∈ XS , we have ϕ, t · Λ = ϕ · t, Λ = 0. 00 = X . Therefore t · Λ ∈ XtS tS (ii) Let t ∈ S and T ⊂ S. Take z = s∈S δs ⊗ xs ∈ t⊥XT . The equation t · z = 0 gives
xu = 0
(s ∈ S).
u∈[t −1 s]∩T
Take u ∈ supp z. Then u ∈ [t −1 (tu)] ∩ T . Hence |[t −1 (tu)] ∩ T | 2, and z ∈ XG(T ,t) . We have proved that t⊥XT ⊂ XG(T ,t) . Now by Lemma 5.1 we have (XG(T ,t) )0 ⊂
t⊥
XT
0
Hence t⊥ (XT ) = (XT · t)0 ⊂ (XG(T ,t) )00 = XG(T ,t) .
= XT · t. 2
Theorem 5.3. Let S be a weakly right cancellative semigroup such that, for each N ∈ N, there exist elements (s1 , r1 , t1 ), . . . , (sN , rN , tN ) in S × S × S with the following properties:
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(i) sn ∈ Srn \ Stn rn (n ∈ NN ); (ii) the sets [s1 r1−1 ], . . . , [sN rN−1 ] are pairwise disjoint; (iii) the sets G(r1 S , t1 ), . . . , G(rN S , tN ) are pairwise disjoint. Then c0 (S) is not flat in 1 (S)-mod. Proof. We set E = c0 (S), and for a subset T ⊂ S , we set AT = 1 (T ). Assume towards a contradiction that E is flat in 1 (S)-mod. Then there exists a left 1 (S) E) with π ◦ ρ = iE . Fix N ∈ N, and let (s1 , r1 , t1 ), . . . , module morphism ρ : E → (AS ⊗ (sN , rN , tN ) ∈ S × S × S be the elements given by the hypothesis. For each n ∈ NN , set xn = rn · λsn = χ[sn r −1 ] . n
E) . Since sn ∈ By Lemma 5.2(i), ρ(xn ) ∈ (Arn S ⊗ / Stn rn , we have tn · ρ(xn ) = ρ(tn · xn ) = ρ(χ[sn (tn rn )−1 ] ) = 0. E) . Hence by Lemma 5.2(ii), ρ(xn ) ∈ (AG(rn S ,tn ) ⊗ N E) . By (ii), N Set Φ = ρ( n=1 xn ) ∈ (AS ⊗ n=1 xn ∞ = 1. Hence there is a net (zα ) E)[ρ] such that limα zα = Φ in the weak-∗ topology. For each n ∈ NN , let in (AS ⊗ E → AG(rn S ,tn ) ⊗ E be a projection. By (iii), we have Pn (ρ(xm )) = 0 (n = m). Pn : AS ⊗ Hence limα Pn (zα ) = Pn (Φ) = ρ(xn ) in the weak-∗ topology. For each i ∈ NN , by (i) we can pick un ∈ [sn rn−1 ]. For large enough α, we have Π(δu ), Pn (zα ) − Π(δu ), ρ(xn ) < 1/2. n n Now for each n ∈ NN , we have
Π(δun ), ρ(xn ) = δun , π ◦ ρ(xn ) = xn , δun = 1.
Hence, for large enough α, we have Π(δu ), Pn (zα ) > 1/2, n and so Pn (zα )π 1/2. But now using (iii), for sufficiently large α,
ρ zα π
N
Pn (zα ) N/2. π n=1
This holds for each N ∈ N, the required contradiction.
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5.2. A special case Here we describe a special case of the condition in Theorem 5.3, which is easier to apply to certain examples. The key is the following description of the sets G(qS, p). Lemma 5.4. Let S be a semigroup, and let p, q ∈ E(S) with p < q. Then pS ⊂ qS and G(qS, p) = (qS \ pS) ∪ p(qS \ pS) is a disjoint union of sets. Proof. It is clear that pS ⊂ qS and (qS \ pS) ∩ p(qS \ pS) = ∅. Let π1 : S × S → S be the projection onto the first coordinate. Consider the sets G = (u, v) ∈ qS × qS: u = v, pu = pv and F = (u, pu): u ∈ qS \ pS ∪ (pu, u): u ∈ qS \ pS . We make the identifications π1 (G) = G(qS, p)
and π1 (F ) = (qS \ pS) ∪ p(qS \ pS).
Since F ⊂ G we have π1 (F ) ⊂ π1 (G). Now take u ∈ π1 (G) with u = π1 ((u, v)) for some (u, v) ∈ G. If u ∈ / pS, then (u, pu) ∈ F . If u ∈ pS, then v ∈ qS \ pS and (u, v) = (pv, v) ∈ F . In either case u ∈ π1 (F ). 2 Theorem 5.5. Let S be a weakly right cancellative semigroup. Suppose that there exists an infinite chain of idempotents r1 > s1 > t1 > · · · > rn > sn > tn > rn+1 > sn+1 > tn+1 > · · · such that for each n ∈ N, tn (rn S \ tn S) ∩ rn+1 S = ∅. Then c0 (S) is not flat in 1 (S)-mod. Proof. We shall apply Theorem 5.3; we verify clauses (i)–(iii). Clearly sn ∈ Srn \ Stn = Srn \ Stn rn (n ∈ N), so that clause (i) holds. −1 ]. Then Take n < m. Assume towards a contradiction that there exists u ∈ [sn rn−1 ] ∩ [sm rm urn = sn and urm = sm . Multiplying the first of these equations on the right by rm gives urm = rm . Hence rm = sm , which is a contradiction. Therefore clause (ii) holds. For each l m we have tk S ⊃ rm S. Hence we have (rn S \ tn S) ∩ G(rm S, tm ) ⊂ (S \ tn S) ∩ rm S = ∅.
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Using Lemma 5.4, we have G(rn S, tn ) ∩ G(rm S, tm ) = tn (rn S \ tn S) ∩ G(rm S, tm ) ⊂ tn (rn S \ tn S) ∩ rm S ⊂ tn (rn S \ tn S) ∩ rn+1 S = ∅ (by hypothesis). Therefore condition (iii) of Theorem 5.3 also holds, and hence c0 (S) is not flat in 1 (S)-mod.
2
Example 5.6. We give some examples of semigroups which satisfy the hypothesis of Theorem 5.5. (i) Let N∨ = (N, max). The canonical partial order on N∨ is the reverse of the natural order on N. We set rn = 3n − 2,
sn = 3n − 1,
tn = 3n (n ∈ N).
For any n ∈ S, we have nN∨ = [n, ∞). Hence for each n ∈ N we have tn (rn N∨ \tn N∨ ) = 3n[3n− 2, 3n − 1] = {3n}, which is disjoint from the set rn+1 N∨ = [3n + 1, ∞). Therefore c0 (N∨ ) is not flat in 1 (N∨ )-mod. (ii) Let B be the bicyclic semigroup. Then B = N0 × N0 with the multiplication (m, n)(p, q) = m − n + max{n, p}, q − p + max{n, p} (m, n), (p, q) ∈ B . We set rn = (3n − 2, 3n − 2),
sn = (3n − 1, 3n − 1),
tn = (3n, 3n) (n ∈ N).
For any (m, n) ∈ B, we have (m, n)B = [m, ∞) × N0 . Hence for each n ∈ N we have tn (rn B \ tn B) = (3n, 3n) [1, 3n − 1] × N0 = {3n} × N0 , which is disjoint from the set rn+1 B = [3n + 1, ∞) × N0 . Therefore c0 (B) is not flat in 1 (B)-mod. The next example is ‘at the opposite extreme’ to those above. This semigroup does not satisfy the hypothesis of Theorem 5.3 (and hence also of Theorem 5.5). Example 5.7. Let X be a set, let PX = P(X) be the power set of X, and let FX be the set of all finite subsets of X. Then PX is a semilattice with the multiplication st = s ∪ t
(s, t ∈ PX ),
and FX is a subsemilattice of PX called the free semilattice over X. The empty set is the identity of PX . For s, t ∈ PX , we have −1 −1 ∅ if t ⊂ s, st = t s = {u: ut = s} = {s \ u: u ⊂ t} if t ⊂ s.
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For t ⊂ s ∈ FX we have |[t −1 s]| = 2|t| , and hence FX is weakly cancellative. For each t ∈ PX , we have [tt −1 ] = {u: u ⊂ t}. The canonical partial order on PX is given by st
⇐⇒
t ⊂s
(s, t ∈ PX ).
Take r, t ∈ PX with r t, i.e., t \r = ∅. Take u ∈ rPX . We can write u = r ∪s where s ∩r = ∅. Set u ∪ (t \ u) if t \ u = ∅, v= u \ (s ∩ t) if s ∩ t = ∅. The condition t \ r = ∅ ensures that one of these cases must occur. Then u = v and t ∪ u = t ∪ v. Hence u ∈ G(rPX , t) and G(rPX , t) = rPX . Hence for r1 t1 and r2 t2 we have G(r1 PX , t1 ) ∩ G(r2 PX , t2 ) = r1 r2 PX . Therefore Theorem 5.5 gives no information about the injectivity of the module 1 (PX ). It is proved in [14] that if X is an infinite set, then 1 (PX ) and 1 (FX ) are not right injective Banach algebras. The proof of this result involves a long technical combinatorial calculation and will be presented elsewhere. 6. Projectivity of the predual module c0 (S) Again for a weakly right cancellative semigroup S we now consider when c0 (S) has the stronger property of being projective in 1 (S)-mod. 6.1. Some technical ‘smallness’ results Lemma 6.1. Let S be an infinite, weakly right cancellative semigroup such that, for every finite set F ⊂ S, there exists r ∈ S with rS ∩ F = ∅. Suppose that c0 (S) is projective in 1 (S)-mod c0 (S). Then for each N ∈ N, there exist elements with splitting morphism ρ : c0 (S) → 1 (S ) ⊗ x1 , . . . , xN in c0 (S) and a partition {F1 , . . . , FN } of S with the properties: (i) N i=1 xi ∞ = 1, (ii) ρ(xi )π 1 for each i ∈ NN , and (iii) χFi ρ(xi ) − ρ(xi )π < 1/3i for each i ∈ NN . Proof. We set E = c0 (S), and for a subset T ⊂ S, we set AT = 1 (T ). Fix N ∈ N. To begin, choose r1 , t1 ∈ S with [t1 r1−1 ] = ∅ and set x1 = r1 · λt1 = χ[t1 r −1 ] . 1
E = 1 (r1 S , E), and 1 = x1 ∞ = π ◦ ρ(x1 )∞ ρ(x1 )π . Take a Then ρ(x1 ) ∈ Ar1 S ⊗ finite set F1 ⊂ r1 S with χF1 ρ(x1 ) − ρ(x1 )π < 1/3. Now suppose that x1 , . . . , xk and {F1 , . . . , Fk } are already constructed. Choose rk+1 ∈ S with rk+1 S ∩ ki=1 Fi = ∅. Set G=
k −1 −1 ti ri and H = (srk+1 )rk+1 . i=1
s∈G
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The set H is finite, so we can choose an element u in the complement. Set tk+1 = urk+1
and xk+1 = rk+1 · λtk+1 = χ[tk+1 r −1 ] . k+1
−1 Since u ∈ [tk+1 rk+1 ] we have ρ(xk+1 )π 1. −1 ] is disjoint from G. Assume towards a contradiction We shall show that the set [tk+1 rk+1 −1 −1 ] ⊂ H. that there exists v ∈ [tk+1 rk+1 ] ∩ G. Then vrk+1 = tk+1 = urk+1 , and so u ∈ [(vrk+1 )rk+1 k+1 −1 This is a contradiction, and therefore [tk+1 rk+1 ] ∩ G = ∅, whence i=1 xi ∞ = 1. We have E. Take a finite set Fk+1 ⊂ rk+1 S with χFk+1 ρ(xk+1 ) − ρ(xk+1 )π < ρ(xk+1 ) ∈ Ark+1 S ⊗ 1/3k+1 . −1 In the final stage we may take FN = S \ N i=1 Fi , so that the sets (Fi ) form a partition of S. 2
Theorem 6.2. Let S be an infinite, weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then there exists a finite set F ⊂ S such that, for each r ∈ S, rS ∩ F = ∅. Proof. We set AS = 1 (S ) and E = c0 (S). Since E is projective in 1 (S)-mod, there exists a left 1 (S)-module morphism ρ : E → E with π ◦ ρ = IE . Assume towards a contradiction that the condition is not satisfied, so AS ⊗ that we can apply Lemma 6.1. Fix N ∈ N, and let x1 , . . . , xN and {F1 , . . . , FN } be the elements corresponding to ρ given by Lemma 6.1. Firstly, for each m ∈ NN we have
1 χF ρ χF ρ(xi ) χS\F ρ(xi ) xi m i m π π 3i π
i=m
and χFm ρ(xm )π 1 −
1 3m .
i=m
i=m
i=m
Hence
N
1 1 1 1 χ − 1 = , ρ x 1 − − 1 − Fm i 3m 3i 1 − 1/3 2 i=1
i=m
π
and so, N N N
N ρ ρ xi = χF1 ρ xi + · · · + χFN ρ xi . 2 i=1
π
i=1
i=1
π
This holds for each N ∈ N, the required contradiction.
π
2
Corollary 6.3. Let S be a weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then the set
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tt −1 t∈S
is finite. Hence the set E(S) is also finite. Proof. Let F be the finite set given by Theorem 6.2. Take t ∈ S and u ∈ [tt −1 ]. Then either t ∈ F or f = ts for some s ∈ S and f ∈ F . In the latter case we have uf = uts = ts = f , and so u ∈ [ff −1 ]. Hence −1 tt −1 ⊂ ff , f ∈F
t∈S
and so the set t∈S [tt −1 ] is finite. For each p ∈ E(S) we have p ∈ [pp −1 ], hence the set E(S) is finite.
2
6.2. Main result for weakly cancellative semigroups We shall adapt the argument that works for groups [4, Theorem 3.1], to show that if S is a weakly cancellative semigroup and c0 (S) is projective, then S must be finite. The following technical looking lemma is a key to adapting the group argument. It is trivial that every element t in an infinite group has the property given in the lemma. Lemma 6.4. Let S be an infinite, weakly right cancellative semigroup. Suppose that c0 (S) is projective in 1 (S)-mod. Then there exists an element t ∈ S such that, for every finite set F ⊂ S, there exists r ∈ S \ F with [tr −1 ] = ∅. Proof. Assume towards a contradiction that the conclusion is false. Then, for every t ∈ S, there exists a finite set F (t) ⊂ S such that [tr −1 ] = ∅ for all r ∈ S \ F (t). Let F be the finite set given by Theorem 6.2. Take s ∈ S \ F . Then su = f for some f ∈ F and u ∈ S. Since s ∈ [f u−1 ] it must be that u ∈ F (f ), and hence S⊂
f u−1 ∪ F.
f ∈F u∈F (f )
But the set on the right-hand side is finite, and so S is finite. This is a contradiction. Therefore the conclusion holds. 2 Theorem 6.5. Let S be a weakly cancellative semigroup such that c0 (S) is projective in 1 (S)-mod. Then S is finite. c0 (S) be a left 1 (S)-module morphism with π ◦ ρ = Ic0 (S) . Proof. Let ρ : c0 (S) → 1 (S ) ⊗ Assume towards a contradiction that S is infinite. Let t ∈ S be the element specified in Lemma 6.4. Fix N ∈ N, and take a finite set F ⊂ S with χF ρ(λt ) − ρ(λt )π < 1/N . We shall construct elements r1 , . . . , rN ∈ S with the following properties: (i) the sets [tr1−1 ], . . . , [trN−1 ] are pairwise disjoint and non-empty, and (ii) the sets r1 F, . . . , rN F are pairwise disjoint.
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To begin choose any r1 ∈ S with [tr1−1 ] = ∅. Now suppose that r1 , . . . , rk are already constructed. Set k (ri f )g −1 , X(k) =
k −1 Y (k) = tri ,
i=1 f,g∈F
Z(k) =
i=1
u−1 t . u∈Y (k)
Since the sets X(k) and Z(k) are finite, we can use Lemma 6.4 to choose an element rk+1 ∈ −1 ] = ∅. We now show that clauses (i) and (ii) are satisfied. S \ X(k) ∪ Z(k) with [trk+1 Take 1 i < j N . Assume that there exists u ∈ [tri−1 ] ∩ [trj−1 ]. Then urj = t, and so
rj ∈ [u−1 t] for u ∈ [tri−1 ] ⊂ Y (j − 1). Hence rj ∈ Z(j − 1), which is a contradiction, giving clause (i). Next assume that there exists v ∈ ri F ∩ rj F so that ri f = rj g for some f, g ∈ F . But then rj ∈ [(ri f )g −1 ] ⊂ X(j − 1), which is a contradiction, and so clause (ii) holds. For each i ∈ NN , since [tri−1 ] = ∅, we have ri · ρ(λt )π 1 and we have the norm estimate ri · χF ρ(λt ) ri · ρ(λt ) − ri · χF ρ(λt ) − ρ(λt ) 1 − 1/N. π π π Now, we have N N N
ρ ρ ri · λt ri · χF ρ(λt ) − ri · ρ(λt ) − χF ρ(λt ) i=1 i=1 i=1 π π π N N
ri · χF ρ(λt ) − = ri · ρ(λt ) − χF ρ(λt ) π i=1
i=1
π
N (1 − 1/N) − N/N = N − 2. This holds for each N ∈ N, the required contradiction. Therefore S is finite.
2
Let S be a finite inverse semigroup. Then 1 (S) is contractible [5, Theorem 8], and so every E ∈ 1 (S)-mod is projective. Hence we have the following. Theorem 6.6. Let S be a weakly cancellative inverse semigroup. Then c0 (S) is projective in 1 (S)-mod if and only if S is finite. Acknowledgments This work forms part of the authors PhD thesis at the University of Leeds, which was supported by an EPSRC grant. The author wishes to thank his supervisor Professor H.G. Dales for his advice. References [1] Y. Choi, Hochschild homology and cohomology of 1 (Zk ), published online October 2008, by Quart. J. Math. (Oxford) (2008). [2] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., vol. 24, Clarendon Press, Oxford, 2000.
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[3] H.G. Dales, A.T.-M. Lau, D. Strauss, Banach algebras on semigroups and on their compactifications, Mem. Amer. Math. Soc. 205 (966) (2010). [4] H.G. Dales, M.E. Polyakov, Homological properties of modules over group algebras, Proc. Lond. Math. Soc. (2) 63 (2004) 390–426. [5] J. Duncan, I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978) 309–321. [6] Brian E. Forrest, Hun Hee Lee, Ebrahim Samei, Projectivity of modules over Fourier algebras, preprint, see arXiv: 0807.4361. [7] M. Ghandehari, H. Hatami, N. Spronk, Amenability constants for semilattice algebras, Semigroup Forum 79 (2) (2009) 279–297. [8] F. Gourdeau, A. Pourabbas, M.C. White, Simplicial cohomology of some semigroup algebras, Canad. Math. Bull. 50 (1) (2007) 50–70. [9] N. Grønbæk, Amenability of weighted discrete convolution algebras on cancellative semigroups, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988) 351–360. [10] A.Ya. Helemskii, The Homology of Banach and Topological Algebras, Kluwer Academic Publishers, Dordrecht, 1986. [11] A.Ya. Helemskii, Banach and Locally Convex Algebras, Clarendon Press, Oxford, 1993. [12] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). [13] A.L.T. Paterson, Amenability, Math. Surveys Monogr., vol. 29, American Mathematical Society, Providence, RI, 1988. [14] P. Ramsden, Homological properties of semigroup algebras, thesis, University of Leeds, 2009. [15] V. Runde, Lectures on Amenability, Lecture Notes in Math., vol. 1774, Springer Verlag, 2002. [16] Yu.V. Selivanov, Coretraction problems and homological properties of Banach algebras, in: A.Ya. Helemskii (Ed.), Topological Homology, Nova Science, Huntington, New York, 2000, pp. 145–200.
Journal of Functional Analysis 258 (2010) 4010–4025 www.elsevier.com/locate/jfa
Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents ✩ David Damanik, Zheng Gan ∗ Department of Mathematics, Rice University, Houston, TX 77005, USA Received 9 June 2009; accepted 4 March 2010 Available online 21 March 2010 Communicated by L. Gross
Abstract We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant. © 2010 Elsevier Inc. All rights reserved. Keywords: Limit-periodic Schrödinger operators; Singular continuous spectrum; Lyapunov exponent
1. Introduction This paper is a part of a sequence of papers devoted to the study of spectral properties of discrete one-dimensional limit-periodic Schrödinger operators. The first paper in this sequence [7] contains results in the regime of zero Lyapunov exponents, while the present paper investigates the regime of positive Lyapunnov exponents. Our general aim is to exhibit as rich a spectral picture as possible within this class of operators. In particular, we want to show that all basic ✩
D.D. & Z.G. were supported in part by NSF grant DMS-0800100.
* Corresponding author.
E-mail addresses:
[email protected] (D. Damanik),
[email protected] (Z. Gan). URLs: http://www.ruf.rice.edu/~dtd3 (D. Damanik), http://math.rice.edu/~zg2 (Z. Gan). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.002
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spectral types are possible and, in addition, in the case of singular continuous spectrum and pure point spectrum, we are interested in examples with positive Lyapunov exponents and examples with zero Lyapunov exponents. From this point of view, the present paper will, to the best of our knowledge for the first time, exhibit limit-periodic Schrödinger operators with purely singular continuous spectrum and positive Lyapunov exponents (whereas [7] had the first examples of limit-periodic Schrödinger operators with purely singular continuous spectrum and zero Lyapunov exponents). Examples with purely absolutely continuous spectrum have been known for a long time, dating back to works of Avron and Simon [2], Chulaevsky [4], and Pastur and Tkachenko [15,16] in the 1980s. These examples (must) have zero Lyapunov exponents. Examples with pure point spectrum (and positive Lyapunov exponents at least at many energies in the spectrum) can be found in Pöschel’s paper [17]; compare also the work of Molchanov and Chulaevsky [13] (who have examples with zero Lyapunov exponents). In the third paper of this sequence we use Pöschel’s general theorem from [17] to construct limit-periodic examples with uniform pure point spectrum within our framework (actually these examples have uniform localization of eigenfunctions); see [8]. Our study is motivated by the recent paper [1], in which Avila disproves a conjecture raised by Simon; see [19, Conjecture 8.7]. That is, he has shown that it is possible to have ergodic potentials with uniformly positive Lyapunov exponents and zero-measure spectrum. The examples constructed by Avila are limit-periodic. In fact, the paper [1] proposes a novel way of looking at limit-periodic potentials. In hindsight, this way is quite natural and provides one with powerful technical tools. Consequently, we feel that a general study of limit-periodic Schrödinger operators may be based on this new approach and we have implemented this in [7,8] and the present paper. We anticipate that further results may be obtained along these lines. It has been understood since the early papers on limit-periodic Schrödinger operators, and more generally almost periodic Schrödinger operators, that these operators belong naturally to the class of ergodic Schrödinger operators, where the potentials are obtained dynamically, that is, by iterating an ergodic map and sampling along the iterates with a real-valued function; see [3,5, 14] for general background. Indeed, taking the closure in ∞ of the set of translates of an almost periodic function on Z (i.e., the hull of the function), one obtains a compact Abelian group with a unique translation invariant probability measure (Haar measure). In particular, the shift on the hull is ergodic with respect to Haar measure and each element of the hull may be obtained by continuous sampling (using the evaluation at the origin, for example). As pointed out by Avila, it is quite natural to take this one step further. That is, once a compact Abelian group and a minimal translation have been fixed, one is certainly not bound to sample along the orbits merely with functions that evaluate a sequence at one point. Rather, every continuous real-valued function on the group is a reasonable sampling function. While this is quite standard in the quasi-periodic case, we are not aware of any systematic use of it in the context of limit-periodic potentials before Avila’s work [1]. The ability to fix the base dynamics and independently vary the sampling functions is very useful in constructing examples of potentials and operators that exhibit a certain desired spectral feature. This has been nicely demonstrated in [1] and is also the guiding principle in our present work. As mentioned above, our main motivation is to find examples of limit-periodic Schrödinger operators with prescribed spectral type. From this point of view, the singular continuity result we prove here is the main result of the paper. However, there was additional motivation to improve the zero measure result of Avila to a zero Hausdorff dimension result. Recent work of Damanik and Gorodetski [9,10] focused on the weakly coupled Fibonacci Hamiltonian. This is an ergodic model that is not (uniformly) almost periodic. Among the results obtained in [9,10], there is a
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theorem that states that the Hausdorff dimension of the spectrum, as a function of the coupling constant, is continuous at zero. That is, as the coupling constant approaches zero, the Hausdorff dimension of the spectrum approaches dimH ([−2, 2]) = 1. When presenting this result, the authors of [9,10] were asked whether this is a universal feature, which holds for all potentials. Thus, our purpose here is to show that there are indeed limit-periodic potentials such that continuity at zero coupling fails in the worst way possible, that is, the Hausdorff dimension of the spectrum is zero for all non-zero values of the coupling constant.1 Let us now describe the models and results in detail. We consider discrete one-dimensional ergodic Schrödinger operators acting in 2 (Z) given by ω Hf,T ψ (n) = ψ(n + 1) + ψ(n − 1) + Vω (n)ψ(n)
(1)
with Vω (n) = f T n (ω) , where ω belongs to a compact space Ω, T : Ω → Ω is a homeomorphism preserving an ergodic Borel probability measure μ and f : Ω → R is a continuous sampling function. It is often benω } eficial to study the operators {Hf,T ω∈Ω as a family, as opposed to a collection of individual ω are always μ-almost surely indepenoperators, since the spectrum and the spectral type of Hf,T dent of ω due to ergodicity. Moreover, if T is in addition minimal (i.e., all T -orbits are dense), ω are independent of ω. then both the spectrum and the absolutely continuous spectrum of Hf,T The Lyapunov exponent is defined as 1 n→∞ n
L(E, T , f ) = lim
(E,T ,f ) logAn (ω) dμ(ω),
(2)
Ω (E,T ,f )
where E ∈ R is called the energy and An
is the n-step transfer matrix of (1) defined as
(E,T ,f ) An (ω) = Sn−1 · · · S0 ,
where Si =
E − f (T i (ω)) 1
−1 . 0
(3)
ω is By the Ishii–Pastur–Kotani theorem, the almost sure absolutely continuous spectrum of Hf,T given by the essential closure of the set of energies where the Lyapunov exponent vanishes. Next we make the spaces and homeomorphisms of especial interest to us explicit.
Definition 1.1. Ω is called a Cantor group if it is an infinite totally disconnected compact Abelian topological group. Definition 1.2. Let Ω be a Cantor group. For ω1 ∈ Ω, let T : Ω → Ω be the translation by ω1 , that is, T (ω) = ω1 · ω. T is called minimal if {T n (ω): n ∈ Z} is dense in Ω for every ω ∈ Ω. 1 Our work was carried out right after the preprint leading to the publication [1] had been released. That version proved zero-measure and did not discuss the Hausdorff dimension issue. After we informed Avila about our results, we learned from him that he had added a remark to the final version of [1] stating that a suitable modification of his proof of zero measure yields zero Hausdorff dimension; see [1, Remark 1.1].
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We will restrict our attention to the case where Ω is a Cantor group and T is a minimal ω have a common spectrum which we will translation. As mentioned above, the operators Hf,T denote by Σ(f ). Here is our main result: Theorem 1.3. Suppose Ω is a Cantor group and T is a minimal translation on Ω. Then there exists a dense set F ⊂ C(Ω, R) such that for every f ∈ F and every λ = 0, the following stateω ments hold true: Σ(λf ) has zero Hausdorff dimension, Hλf,T has purely singular continuous spectrum for every ω ∈ Ω, and E → L(E, T , λf ) is a positive continuous function. The proof of this theorem is based on the constructions in [1]. We make several modifications to these constructions to better control the size of the spectrum and to ensure that the potentials we construct are Gordon potentials. The latter property then implies the absence of point spectrum, which in turn yields singular continuity since the absence of absolutely continuous spectrum already follows from zero measure spectrum. Let us state the Gordon property as a separate result. Definition 1.4. A bounded map V : Z → R is called a Gordon potential if there exist positive integers qi → ∞ such that
max V (n) − V (n ± qi ) i −qi
1nqi
for every i 1. Clearly, if V is a Gordon potential, so is λV for every λ ∈ R. A key part in proving Theorem 1.3 is to establish the following result: Theorem 1.5. Suppose Ω is a Cantor group. Then there exists a dense set F ⊂ C(Ω, R) such that for every f ∈ F , every minimal translation T : Ω → Ω, every ω ∈ Ω, and every λ = 0, λf (T n (ω)) is a Gordon potential. 2. Preliminaries 2.1. Hausdorff measures and dimensions For our relatively restricted purposes, we will simply recall the definition of Hausdorff measures and Hausdorff dimension in this subsection. We refer the reader to [18] for more information. Definition 2.1. Let A ⊆ R be a subset. A countable collection of intervals {bn }∞ n=1 is called a ∞ δ-cover of A if A ⊂ n=1 bn with |bn | < δ for all n’s. (Here, | · | denotes Lebesgue measure, and we will adopt this notation throughout the paper.)
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Definition 2.2. Let α ∈ R. For any subset A ⊆ R, the α-dimensional Hausdorff measure of A is defined as hα (A) = lim inf
∞
δ→0 δ -covers
|bn |α .
(4)
n=1
α The quantity hα (A) is well defined as an element of [0, ∞] since infδ -covers ∞ n=1 |bn | is monotonically increasing as δ decreases to zero and therefore the limit in (4) exists. Restricted to the Borel sets, h1 coincides with Lebesgue measure and h0 is the counting measure. If α < 0, we always have hα (A) = ∞ for any A = ∅, while if α > 1, hα (R) = 0. It is not hard to see that for every A ⊆ R, there is a unique α ∈ [0, 1], called the Hausdorff dimension dimH (A) of A, such that hβ (A) = ∞ for every β < α and hβ (A) = 0 for every β > α. In particular, every A ⊆ R with |A| > 0 must have dimH (A) = 1. 2.2. Minimal translations of Cantor groups and limit-periodic potentials In this subsection we recall how the one-to-one correspondence between hulls of limitperiodic sequences and potential families generated by minimal translations of Cantor groups and continuous sampling functions exhibited by Avila in [1] arises. Definition 2.3. Let S : ∞ (Z) → ∞ (Z) be the shift operator, (SV )(n) = V (n + 1). A two-sided sequence V ∈ ∞ (Z) is called periodic if its S-orbit is finite and it is called limit-periodic if it belongs to the closure of the set of periodic sequences. If V is limit-periodic, the closure of its S-orbit is called the hull and denoted by hullV . The first lemma (see [1, Lemma 2.1]) shows how one can write the elements of the hull of a limit-periodic function in the form Vω (n) = f T n (ω) ,
ω ∈ Ω, n ∈ Z,
(5)
with a minimal translation T of a Cantor group and a sampling function f ∈ C(Ω, R): Lemma 2.4. Suppose V is limit-periodic. Then, Ω := hullV is compact and has a unique topological group structure with identity V such that Z k → S k V ∈ hullV is a homomorphism. Moreover, the group structure is Abelian and there exist arbitrarily small compact open neighborhoods of V in hullV that are finite index subgroups. In particular, Ω = hullV is a Cantor group, T = S|Ω is a minimal translation, and every element of Ω may be written in the form (5) with the continuous function f (ω) = ω(0). The second lemma (see [1, Lemma 2.2]) addresses the converse: Lemma 2.5. Suppose Ω is a Cantor group, T : Ω → Ω is a minimal translation, and f ∈ C(Ω, R). Then, for every ω ∈ Ω, the element Vω of ∞ (Z) defined by (5) is limit-periodic and . we have hullVω = {Vω˜ }ω∈Ω ˜ These two lemmas show that a study of limit-periodic potentials can be carried out by considering potentials of the form (5) with a minimal translation T of a Cantor group Ω and a
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continuous sampling function f . As shown for the first time in the context of limit-periodic potentials by Avila in [1], it is often advantageous to fix Ω and T and to vary f . 2.3. Periodic sampling functions, potentials, and Schrödinger operators In this subsection we discuss the periodic case. For example, which sampling functions f ∈ C(Ω, R) give rise to periodic potentials for some or all (ω, T )? Moreover, what can then be said about the associated Schrödinger operators? Definition 2.6. Suppose Ω is a Cantor group and T : Ω → Ω is a minimal translation. We say that a sampling function f ∈ C(Ω, R) is n-periodic with respect to T if f (T n (ω)) = f (ω) for every ω ∈ Ω. Proposition 2.7. Let f ∈ C(Ω, R). If f (T n0 +m (ω0 )) = f (T m (ω0 )) for some ω0 ∈ Ω, some minimal translation T : Ω → Ω and every m ∈ Z, then for every minimal translation T˜ : Ω → Ω, f is n0 -periodic with respect to T˜ . Proof. Let ϕ : Ω → ∞ (Z), ϕ(ω) = (f (T n (ω)))n∈Z . Since T is minimal, the closure of {T n (ω0 ): n ∈ Z} is Ω. By Lemma 2.5 we have ϕ(Ω) = hull(ϕ(ω0 )). Since f (T n0 +m (ω0 )) = f (T m (ω0 )) for any m ∈ Z, hull(ϕ(ω0 )) is a finite set. Then for any ω ∈ Ω, (f (T n (ω)))n∈Z is some element in hull(ϕ(ω0 )). Since every element in hull(ϕ(ω0 )) is n0 -periodic, (f (T n (ω)))n∈Z is n0 -periodic. This shows that f is n0 -periodic with respect to T . That is, we have f (T n0 +m (ω)) = f (T m (ω)) for every ω ∈ Ω and m ∈ Z. Assume T is the minimal translation by ω1 and let T˜ be another minimal translation by ω2 . n +m By the previous analysis, we have f (ω1 0 · ω) = f (ω1m · ω) for every m ∈ Z and every ω ∈ Ω. q q If ω2 is equal to ω1 for some integer q, obviously we have f (T˜ n0 (ω)) = f ((ω1 )n0 · ω) = f (ω) n for any ω ∈ Ω. If not, since {ω1 : n ∈ Z} is dense in Ω (this follows from the minimality of T ), n we have limk→∞ ω1nk = ω2 , and then f (ω2 0 · ω) = limk→∞ f ((ω1nk )n0 · ω) = f (ω). The result follows. 2 The above proposition tells us that the periodicity of f is independent of T . Thus we may say f is n-periodic without making a minimal translation explicit. Next we recall from [1] how periodic sampling functions in C(Ω, R) can be constructed. Given a Cantor group Ω, a compact subgroup Ω0 with finite index (such subgroups can be found in any neighborhood of the identity element; see above), and f ∈ C(Ω, R), we can define a periodic fΩ0 ∈ C(Ω, R) by fΩ0 (ω) =
f (ω · ω) ˜ dμΩ0 (ω). ˜
Ω0
Here, μΩ0 denotes Haar measure on Ω0 . This shows that the set of periodic sampling functions is dense in C(Ω, R). Moreover, as already noted in[1], there exists a decreasing sequence of Cantor subgroups Ωk with finite index nk such that Ωk = {e}, where e is the identity element of Ω. Let Pk be the set of sampling functions defined on Ω/Ωk , that is, the elements in Pk are nk -periodic potentials. Denote by P the set of all periodic sampling functions. Then, we have Pk ⊂ Pk+1 (which implies nk |nk+1 ) and P = Pk .
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Proposition 2.8. Let f be p-periodic. Then, for every ω ∈ Ω, (E,T ,f ) 1 (ω) logAm m→∞ m (E,T ,f ) 1 = log ρ Ap (e) , p
L(E, T , f ) = lim
(E,T ,f )
(6)
(E,T ,f )
(e)) is the spectral radius of Ap (e). In particular, if restricted to periodic where ρ(Ap sampling functions, the Lyapunov exponent is a continuous function of both the energy E and the sampling function. Proof. If f is p-periodic, as in the proof of Proposition 2.7, for every ω, (f (T n (ω)))n∈Z is some element of the orbit of (f (T n (e)))n∈Z , and so its monodromy matrix (i.e., the transfer matrix over one period) is a cyclic permutation of the monodromy matrix associated with f (T n (e)). (E,T ,f ) (E,T ,f ) (ω) is independent of ω, and since det Ap (ω) = 1, we can conclude that Thus Tr Ap (E,T ,f ) (ω) are independent of ω. So the logarithm of the spectral radius of the eigenvalues of Ap (E,T ,f ) Ap (ω) is independent of ω and (6) follows. The continuity statement follows readily. 2 Lemma 2.9. Let fn ∈ C(Ω, R) be a sequence of periodic sampling functions converging uniformly to f∞ ∈ C(Ω, R). Assume limn→∞ L(E, T , fn ) exists for every E and the convergence is uniform. Then we have that L(E, T , f∞ ) coincides with limn→∞ L(E, T , fn ) everywhere. Proof. Since limn→∞ L(E, T , fn ) exists everywhere, from [1, Lemma 2.5], we have L(E, T , fn ) → L(E, T , f∞ ) in L1loc . So L(E, T , f∞ ) coincides with limn→∞ L(E, T , fn ) almost everywhere. From Proposition 2.8, L(E, T , fn ) is a continuous function, and by uniform convergence, we have that limn→∞ L(E, T , fn ) is also a continuous function. Since L(E, T , f∞ ) is a subharmonic function (cf. [6, Theorem 2.1]), we get that L(E, T , f∞ ) = limn→∞ L(E, T , fn ) for every E. The statement follows. 2 To conclude this subsection on the periodic case, we state two lemmas. The first is well known and the second is [1, Lemma 2.4]. Lemma 2.10. Let f ∈ C(Ω, R) be p-periodic. ω is purely absolutely continuous for every ω ∈ Ω and Σ(f ) is made of (i) The spectrum of Hf,t p bands (compact intervals whose interiors are disjoint). (ii) Σ(f ) = {E ∈ R: L(E, T , f ) = 0}.
Lemma 2.11. Let f ∈ C(Ω, R) be p-periodic. (i) The Lebesgue measure of each band of Σ(f ) is at most 2π p . (ii) Let C 1 be such that for every E ∈ Σ(f ), there exist ω ∈ Ω and k 1 such that (E,T ,f ) Ak (ω) C. Then, |Σ(f )| 4πp C .
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3. Proof of the theorems (E,f )
(E,f,T )
(E,f )
(ω) = An (ω), An = Assume Ω and T are given. For convenience, we write An and L(E, f ) = L(E, T , f ). Since T : Ω → Ω is a minimal translation, the homomorphism Z → Ω, n → T n e is injective with dense image in Ω, and we can write f (n) = f (T n (e)) without any conflicts. We need two more lemmas before proving our theorems. More precisely, we will make further use of the constructions which play central roles in the proof of these two lemmas. (E,f,T ) (e), An
Lemma 3.1. Let B be an open ball in C(Ω, R), let F ⊂ P ∩ B be finite, and let 0 < ε < 1. Then there exists a sequence FK ⊂ P ∩ B such that (i) L(E, λFK ) > 0 whenever ε |λ| ε −1 , E ∈ R. (ii) L(E, λFK ) → L(E, λF ) uniformly on compacts (as functions of (E, λ) ∈ R2 ). This is [1, Lemma 3.1]. As in [1], we use the notation L(E, λF ) =
1 L(E, T , λf ), #F f ∈F
where F is a finite family of sampling functions (with multiplicities!) and λ ∈ R. The proof of this lemma is constructive. We will describe this construction explicitly in the proof of Theorem 1.3 for the reader’s convenience. Lemma 3.2. Suppose B is an open ball in C(Ω, R) and F ⊂ P ∩ B is a finite family of sampling functions. Then for every N 2 and K sufficiently large, there exists FK ⊂ PK ∩ B such that (i) L(E, λFK ) → L(E, λF ) uniformly on compacts (as functions of (E, λ) ∈ R2 ). −N/2 (ii) The diameter of FK is at most nK . This lemma is a variation of [1, Lemma 3.2]. We will prove this lemma using suitable modifications of Avila’s arguments. Some of these modifications, which will later enable us to prove the Gordon property, are not apparent from the statement of the lemma. We will give detailed arguments for the modified parts of the proof and refer the reader to [1] for the parts that are analogous. Proof of Lemma 3.2. Assume that F = {f1 , f2 , . . . , fm } ⊂ C(Ω, R) is a finite family of nk periodic sampling functions with nk 2, and let K > k be large enough. We construct FKt as follows. Let nK = mnk r + d, 0 d mnk − 1. Let Ij = [j nk , (j + 1)nk − 1] ⊂ Z and let 0 = j0 < j1 < · · · < jm−1 < jm = nK /nk be a sequence such that ji+1 − ji = r + 1 when 0 i < d/nk and ji+1 − ji = r when d/nk i m − 1. Define an nK -periodic f as follows. For 0 l nK − 1, let j be such that l ∈ Ij and let i be such that ji−1 j < ji and then let f (l) = fi (l). Next, for any sequence t = (t1 , t2 , . . . , tm ) with ti ∈ {0, 1, . . . , r − 1}, we define an nK -periodic fKt as follows. If j = ji − 1 for some 1 i < m, we let fKt (l) = f (l) + r −N ti , and if j = jm − 2, we let fKt (l) = f (l) + r −N tm . Otherwise we let fKt (l) = f (l). Let FKt be the family consisting of all fKt ’s. The statement (ii) is clear for large K. (Note: in [1], Avila’s
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construction is such that if j = ji − 1 for some 1 i m, then fKt (l) = f (l) + r −20 ti ; otherwise, fKt (l) = f (l).) (E,λf t )
For fixed E and λ, we let AnK K = C (tm ,m) B (m) · · · C (t1 ,1) B (1) , where B (i) = (E,λfi ) ji −ji−1 −1 (E,λf ) (Ank ) , 1 i m − 1 and B (m) = (Ank m )jm −jm−1 −2 , and C (ti ,i) = (E−λr −N t ,λf )
(E,λf )
(E−λr −N t ,λf )
m m i i Ank , 1 i m − 1 and C (tm ,m) = Ank m Ank . When E and λ are (t ,i) i in a compact set, the norm of the C -type matrices is bounded as r grows, while the norm of the B (i) -type matrices may get large. Notice that our perturbation here is r −N t (as opposed to Avila’s r −20 t perturbation in [1, Lemma 3.2 ]), so [1, Claim 3.7] should be replaced by the following version: “Let sj be the most contracted direction of Bˆ (j ) and let uj be the image under Bˆ (j ) of the most expanded direction. Call t j -nice, 1 j d, if the angle between Cˆ (j ) uj and Sj +1 (less than π ) is at least r −3N with the convention that j + 1 = 1 for j = d. Let r be sufficiently large, and let t be j -nice. If z is a non-zero vector making an angle at least r −4N with sj , then z = Cˆ (j ) Bˆ (j ) z makes an angle at least r −4N with Sj +1 and z Bˆ (j ) r −5N z .” The proof of [1, Claim 3.7] can be applied to get the above version of the claim with the corresponding quantitative modification. Moreover, we have also made a little shift in the perturbation,
(E,λf )
(E−λr −N t ,λf )
(E−λr −20 t ,λf )
m m m m , while Avila’s C (tm ,m) = Ank . [1, Claim 3.8] so C (tm ,m) = Ank m Ank still holds, but Avila’s proof of [1, Claim 3.8] cannot be applied directly. To this end we prove the following claim:
Claim 3.3. For any M ∈ SL(2, R), there are m1 , m2 ∈ (0, ∞) with the following property. Suppose A and B are two vectors in R2 , and θ is the angle between A and B with 0 < θ π . Let θ˜ be the angle between MA and MB (again so that 0 < θ˜ π ). Then, m1 θ θ˜ m2 θ. Proof. By the singular value decomposition (see [11, Theorem 2.5.1]), there exist O1 and O2 in SO(2, R) such that M = O1 SO2 , where S is a diagonal matrix. Since O1 and O2 are rotations on R2 , it is sufficient to consider S=
μ1
0
0
μ−1 1
.
Without loss of generality, assume μ1 1. Let A = (a, b)t (t denotes the transpose of vectors) and B = (c, d)t be two normalized vectors, and let θA and θB be the argument of A and the argument of B respectively. Let A˜ = SA = (aμ1 , b/μ1 )t with the argument θA˜ and B˜ = SB = (cμ1 , d/μ1 )t with the argument θB˜ . We adopt the following notation for convenience. Let I , II, III, IV denote one of two vectors in the first quadrant (including {(x, 0): x 0}), the second quadrant (including {(0, y): y > 0}), the third quadrant (including {(x, 0): x < 0}) and the fourth quadrant (including {(0, y): y < 0}), respectively. Then (I, I ) denotes that both two vectors are in the first quadrant, (I, II) denotes that one vector is in the first quadrant while the other is in the second quadrant, and so on. We will need the following observation: 0 < θ1 , θ2 < π/2 and
tan θ1
tan θ2 μ21
⇒
θ1
θ2 . 4μ21
(7)
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Indeed, since tan θ1
1 μ21
tan θ2
1 θ 2μ21 2
and 0 <
θ2 2μ21
4019
< 1, we have
θ2 θ2 5 θ2 θ2 θ2 3 3+O θ1 arctan 2 = − . 2 2 2 2μ1 2μ1 2μ1 2μ1 4μ21 For the proof of Claim 3.3, we consider two cases. Case 1. π/2 θ π . Here A and B cannot be in the same quadrant. Notice that the impact of S on vectors is to move them closer to the x-axis and keep them in the same quadrant. Thus, for the subcases (I, II), (I, III), (II, IV) and (III, IV), we can easily conclude that θ/2 θ˜ 2θ . There are two subcases left, (I, IV) and (II, III). We will discuss (I, IV); the method can be readily adapted to (II, III). For (I, IV), if θA = 0 and θB = 3π/2, then θA˜ and θB˜ are also 0 and 3π/2 respectively, and so θ˜ = θ ; if not, without loss of generality, assume that A is in the first quadrant with π/4 θA < π/2, then tan θA˜ = b 2 = tan 2θA , and by (7), we have aμ1
θ˜ θA˜ (θA θ/4 since θA π/4) and then
θ 16μ21
μ1
θA θ 2 4μ1 16μ21
θ˜ 2θ.
Case 2. 0 < θ < π/2. In this case, (I, III) and (II, IV) are impossible. We will divide the following proof into three parts. (1) We discuss (I, I ) here; the argument may be readily adapted to (II, II), (III, III), and (IV, IV). Without loss of generality, assume θ = θA − θB , then we get tan θ˜ =
μ21 (bc − ad) bd
+ μ41 ac
θ θ˜ . Similarly, we will get 4μ21 4μ21 θ follows.
and by (7), we get so
θ 4μ21
θ˜
tan θ , μ21
θ˜ 4μ21 θ since tan θ˜ μ21 tan θ , and
(2) We discuss (I, IV) here; an adaptation handles (II, III). Without loss of generality, assume θA θ/2. Obviously, we have θ˜ θ . Conversely, we have θ 2 θ˜ (it is essentially the same as (I, IV) in Case 1), and so
θ 16μ21
θ˜ θ follows.
16μ1
(3) We discuss (I, II) here, and the method can be applied to (III, IV). Obviously, we have θ θ˜ . Without loss of generality, assume that A is in the first quadrant and makes an angle hA with the y-axis and that B is in the second quadrant and makes an angle hB with the y-axis. Clearly, θ = hA + hB . Let hA˜ and hB˜ be the angle between the y-axis and A˜ and ˜ respectively. By (7), we conclude that h ˜ 4μ2 hA since the angle between the y-axis and B, 1 A tan hA˜ = μ21 tan hA . Similarly, we get hB˜ 4μ21 hB . So it follows that θ θ˜ = hA˜ + hB˜ 4μ21 (hA + hB ) = 4μ21 θ. Through the above analysis, we see that θ 2 θ˜ 16μ21 θ , concluding the proof of Claim 3.3.
2
16μ1
By this claim, we can modify the last paragraph of the proof of [1, Claim 3.8] as stated below and then our lemma follows.
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“If r sufficiently large, we conclude that for every 0 l r − 2, there exists a rotation Rl,j of angle θj with r −2.5N < θj < r −0.3N such that C (l+1,ij ) uj = Rl,j C (l,ij ) uj . It immediately (t ,i ) follows that there exists at most one choice of 0 tij r − 1 such that C ij j uj has angle at most r −3N with sj +1 , as desired.” We would like to explain how to obtain the statement described in the paragraph above. If r is sufficiently large, it is not hard to conclude that for every 0 l r − 2, there exists a rotation R˜ l,j (E−λr −N (l+1),λfij )
of angle θ˜j with r −2N < θ˜j < r −0.5N such that Ank If id = m, we have (E,λfm )
C (l+1,m) um = Ank
(E,λfm )
= Ank (E,λfm )
Since Ank
(E−λr −N (l+1),λfm )
Ank
uj .
um
(E−λr −N (l),λfm ) nk
R˜ l,m A
(E−λr −N l,λfij )
uj = R˜ l,j Ank
um .
(8)
∈ SL(2, R) is independent of r, we can apply Claim 3.3 to (8) so that we have (E,λfm )
C (l+1,m) um = Ank
(E−λr R˜ l,m Ank
(E,λfm )
= Rl,m Ank = Rl,m C
(l,m)
−N (l),λf ) m
(E−λr −N (l),λfm ) nk
A
um um
um ,
where Rl,m is a rotation of angle θm with r −2.5N < θm < r −0.3N . Then the above paragraph follows. 2 Recall the definition of a Gordon potential given in Definition 1.4. The importance of Gordon potentials lies in the following lemma, which (in a slightly weaker form) first appeared in [12]. Lemma 3.4 (Gordon lemma). Suppose V is a Gordon potential. Then the Schrödinger operator with potential V has no eigenvalues. Now we can give Proof of Theorem 1.3. Given a p0 -periodic f ∈ C(Ω, R) and 0 < ε0 < 1, consider Bε0 (f ) ⊂ C(Ω, R). (We will work within this ball. The denseness of periodic potentials then implies the ε0 . denseness of our constructed limit-periodic potentials.) Let N from Lemma 3.2 be 2. Let ε1 = 10 By Lemma 3.1, there exists a finite family F1 = {f1 , f2 , . . . , fm1 } of p1 -periodic sampling functions such that F1 ⊂ Bε0 (f ) and L(E, λF1 ) > δ1 for some 0 < δ1 < 1 whenever ε1 < |λ| < ε11 and E ∈ R (note that L(E, λfi ) 1 if |E| λfi + 4). Our constructions start with F1 and we will divide them into several steps. Construction 1. First, we will apply Lemma 3.1 to F1 in order to enlarge the range 1 ,δ1 } . Then, there exists a finite family of p˜ 1 -periodic potentials F˜1 = of λ’s. Let ε2 = min{ε 10 ˜ ˜ ˜ {f1 , f2 , . . . , fm˜ 1 } ⊂ Bε0 (f ) such that L(E, λF˜1 ) > δ˜1 for some 0 < δ˜1 < 1 whenever ∀ε2 < |λ| <
1 ε2
and E ∈ R, and
D. Damanik, Z. Gan / Journal of Functional Analysis 258 (2010) 4010–4025
L(E, λF˜1 ) − L(E, λF1 ) < ε2 2 whenever |E| <
1 ε2
and |λ| <
4021
(9)
1 ε2 .
Explicitly, the construction of F˜1 follows from the proof of [1, Claim 3.1]. For very large p˜ 1 > p1 (it must obey p1 |p˜ 1 ), choose N1 (p˜ 1 ) such that if |E| < ε12 , |λ| ε12 , fi ∈ F1 and a 2p1 +1 close to fi then |L(E, λf˜) − L(E, λfi )| < ε22 , since the p˜ 1 -periodic potential f˜ which is N 1 (p˜ 1 ) Lyapunov exponent is continuous for periodic potentials (see Proposition 2.8). For 1 j 2p1 + 1, we define p˜ 1 -periodic potentials f˜(i,j ) by f˜(i,j ) (n) = fi (n), 0 n p˜ 1 − 2 and f˜(i,j ) (p˜ 1 − 1) = fi (p˜ 1 − 1) + N1 (jp˜1 ) . By [1, Claim 3.4], there exists j0 such that the spectrum of f˜(i,j0 ) has exactly p˜ 1 components, that is, all gaps of its spectrum are open. For convenience, we write f˜(i) = f˜(i,j0 ) . So there exists h = h(F1 , p˜ 1 , ε2 ) > 0 such that for any fi ∈ F1 and ε2 |λ| ε12 , Σ(λf˜(i) ) has p˜ 1 components and the Lebesgue measure of the smallest gap is at least h. Choose an integer N2 (p˜ 1 ) with N2 (p˜ 1 ) > ε24π hp˜ 1 . 4πl (i,l) (i) For 0 l N2 (p˜ 1 ), let f˜ = f˜ + . Then F˜1 is just the family obtained by colε2 p˜ 1 N2 (p˜ 1 )
lecting the f˜(i,l) for different fi ∈ F1 and 0 l N2 (p˜ 1 ). Order F˜1 as F˜1 = {f˜1 , f˜2 , . . . , f˜m˜ 1 } such that f˜1 = f˜(1,0) and f˜m˜ 1 = f˜(1,1) . We can also assume that N2 (p˜ 1 ) was chosen large enough, so that we have f˜m˜ 1 − f˜1 = ε2 p˜14π N2 (p˜ 1 ) < 1/3 (this will be used to conclude that our limit-periodic potentials are Gordon potentials). Construction 2. Applying Lemma 3.2 to F˜1 , there exists a finite family of p2 -periodic poten
tm
tials F2 = {f2t1 , f2t2 , . . . , f2 2 } such that F2 ⊂ Bp−2 ⊂ Bε2 ⊂ Bε0 (f ) 2
and L(E, λF2 ) > δ2 for some 0 < δ2 < 1 whenever ε2 < |λ| <
1 ε2
and E ∈ R, and
L(E, λF2 ) − L(E, λF˜1 ) < ε2 2 whenever |E|, |λ| <
1 ε2 .
(10)
From (9) and (10), we have
L(E, λF2 ) − L(E, λF1 ) < ε2
for |E|, |λ| < ε12 . Explicitly, we construct F2 as follows (cf. the proof of Lemma 3.2). Let p2 be large and ˜ 1 p˜ 1 r2 + d, 0 d m ˜ 1 p˜ 1 − 1. Let Ij = [j p˜ 1 , (j + 1)p˜ 1 − 1] ⊂ Z and let 0 = j0 < p2 = m j1 < · · · < jm˜ 1 −1 < jm˜ 1 = pp˜21 be a sequence such that ji+1 − ji = r2 + 1 when 0 i < d/p˜ 1 and ji+1 −ji = r2 when d/p˜ 1 i m ˜ 1 p˜ 1 −1. Define a p2 -periodic potential f2 (l) for 0 l p2 −1 as follows. Let j be such that l ∈ Ij and let i be such that ji−1 j < ji and let f2 (l) = f˜i (l). For any sequence t = (t1 , t2 , . . . , tm˜ 1 ) with ti ∈ {0, 1, . . . , r2 − 1}, let f2t be a p2 -periodic potential
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defined as follows. Let 0 l p2 − 1, and let j be such that l ∈ Ij . If j = ji − 1 for some 1i<m ˜ 1 , we let f2t(l) = f2 (l)+r2−4 ti , and j = jm˜ 1 −2 then f2t(l) = f2 (l)+r2−4 tm˜ 1 . Otherwise t we let f2 (l) = f2 (l). Let p2 be sufficiently large so that p2−2 < 1/3. Moreover, we can estimate the Lebesgue measure of the spectrum. For any E ∈ R and ε2 < |λ| < ε12 , we can find f˜i ∈ F˜1 such that L(E, λf˜i ) > δ˜1 since L(E, λF˜1 ) > δ˜1 . If r2 large enough, (E,λf˜ )
˜
we have A(r2 −2)i p˜1 > eδ1 (r2 −2)p˜1 . Then we have (E,λf2tk ) tk (E,λf˜i ) δ˜1 (r2 −2)p˜ 1 = A A . (r2 −2)p˜ 1 > e (r2 −2)p˜ 1 f2 (ji−1 p˜ 1 ) Since E is arbitrary, we can apply Lemma 2.11 to conclude that the total Lebesgue measure of
˜
1/2
Σ(λf2tk ) is at most 4πp2 e−δ1 (r2 −2)p˜1 < e−p˜1 p2 when r2 sufficiently large. (Here f2tk can be any element from F2 .) Construction 3. Repeating the above procedures. Once we have constructed Fi−1 ⊂ Bp−(i−1) ⊂ i−1
Bεi−1 , by Lemma 3.1, we can get a finite family of p˜ i−1 -periodic potentials F˜i−1 ⊂ Bp−(i−1) satisfying the following. Let εi =
min{εi−1 ,δi−1 } , 10
i−1
and we have
L(E, λF˜i−1 ) > δ˜i−1 for some 0 < δ˜i−1 < 1 whenever ∀εi < |λ| <
1 εi
and E ∈ R, and
L(E, λF˜i−1 ) − L(E, λFi−1 ) < εi 2 whenever |E| < ε1i and |λ| < ε1i . Next, as in Construction 2, we will get a finite family Fi of pi -periodic potentials which satisfies the following (here our perturbation is ri−N i t = ri−2i t, t ∈ {0, 1, 2, . . . , ri − 1}). (i) L(E, λFi ) > δi for some 0 < δi < 1 and all E ∈ R and εi < |λ| < εi−1 . (ii) |L(E, λFi ) − L(E, λFi−1 )| < εi for |E| < ε1i and |λ| < ε1i . (iii) Fi ⊂ Bp−i ⊂ Bεi ⊂ Bεi−1 ⊂ Bε0 (f ), i > 2. (Note: Bε2 may not be in Bε1 .)
i
1/2
(iv) ∀fitk ∈ Fi , |Σ(λfitk )| e−p˜i−1 pi when εi < |λ| < εi−1 (here | · | denotes the Lebesgue measure). (v) pi−i < 13 (i − 1)−p˜i−1 since we can let pi be sufficiently large.
tm
1 −p˜ i−1 . Here N (p˜ (vi) fit1 − fi i = εi p˜i−1 4π 2 i−1 ) appears as in Construction 1, N2 (p˜ i−1 ) < 3 (i − 1) and we can ensure that this inequality holds since p˜ i−1 is fixed while N2 (p˜ i−1 ) can be taken as large as needed.
Then we will get a limit-periodic potential f∞ ∈ Bε0 (f ), whose Lyapunov exponent is a positive continuous function of energy E and the Lebesgue measure of the spectrum is zero (Lemma 2.9 implies that L(E, λfit) → L(E, λf∞ )). Moreover, we have the following two claims. Claim 3.5. f∞ is a Gordon potential.
D. Damanik, Z. Gan / Journal of Functional Analysis 258 (2010) 4010–4025
4023
Proof. Let qi = p˜ i . Obviously, qi → ∞ as i → ∞. For i 1, we have
t1 t1 max f∞ (n) − f∞ (n ± qi ) f∞ (n) − fi+1 (n) + f∞ (n ± qi ) − fi+1 (n ± qi )
1nqi
t1
t1 + fi+1 (n) − fi+1 (n ± qi )
−(i+1)
pi+1
−(i+1)
+ pi+1
+
4π εi+1 p˜ i N2 (p˜ i )
1 1 2 (i)−p˜i + (i)−p˜i 3 3 i −qi .
t1 is an element of Fi+1 .) So f∞ is a Gordon potential. (Here fi+1
2
Claim 3.6. Σ(λf∞ ) has zero Hausdorff dimension for every λ = 0. Proof. Let λ = 0 and 0 < α 1 be given. Without loss of generality, assume λ > 0. Choose i large enough so that εi < λ < 1/εi and 1/i < α. For every fitk ∈ Fi , λf∞ − λfitk < λpi−i
implies2 dist(Σ(λf∞ ), Σ(λfitk )) < λpi−i . Since λfitk is pi -periodic, we have i t Σ λfi k = I˜z(tk ,i) ,
p
z=1
where I˜z(tk ,i) = [az , bz ] is a closed interval. (t ,i) t Let Iz k = [az − λpi−i , bz + λpi−i ] and since dist(Σ(λf∞ ), Σ(λfi k )) λpi−i , we have Σ(λf∞ ) ⊂
pi
Iz(tk ,i) .
z=1 1/2
Moreover, bz − az e−p˜i−1 pi
t
1/2
since |Σ(λfi k )| e−p˜i−1 pi . Then we have
pi −p˜ p1/2 α h Σ(λf∞ ) lim e i−1 i + 2λpi−i α
i→∞
z
1/2 α = lim pi e−p˜i−1 pi + 2λpi−i
i→∞
1/2 1/α −i+1/α α = lim pi e−p˜i−1 pi + 2λpi .
i→∞
2 It is well known that for V , W : Z → R bounded, we have dist(σ ( + V ), σ ( + W )) V − W , where ∞ dist(A, B) denotes the Hausdorff distance of two compact subsets A, B of R.
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D. Damanik, Z. Gan / Journal of Functional Analysis 258 (2010) 4010–4025
Since 1/i < α, we have −i + 1/α < 0, and it follows that 1/2 1/α −i+1/α α lim pi e−pi−1 pi + 2λpi = 0.
i→∞
So we have hα (Σ(λf∞ )) = 0 (note: when i → ∞, λ belongs to (εi , ε1i ) for all i large enough since this interval is expanding). So the Hausdorff dimension of the spectrum is less than α. Since α was arbitrary, the Hausdorff dimension must be zero. 2 This implies all the assertions in Theorem 1.3 except for the absence of eigenvalues for every ω. Given the Gordon lemma (see Lemma 3.4 above), this last statement will follow once Theorem 1.5 is established. 2 Remark 3.7. Since δi δi−1 /10, i 1, it is true that when εi < |λ| < ε1i , L(E, λf∞ ) 89 δi for any E ∈ R. This gives information about the range of the Lyapunov exponent on certain intervals. Clearly, δi → 0 when i → ∞ since the Lyapunov exponent will go to zero when λ goes to zero. Proof of Theorem 1.5. Let ω = e first. Relative to any minimal translation T˜ , the selected f in the proof of Theorem 1.3 is still n0 -periodic by Proposition 2.7, so we can start with the same ball Bε0 (f ) and choose the same periodic potentials in Bε0 (f ). Then we get the same f∞ . For the finite family Fi from Construction 3, though the Lyapunov exponent may change, the following properties hold (note that fit1 does not change). (i) Fi ⊂ Bp−i ⊂ Bεi ⊂ Bε0 (f ). i
(ii) pi−i < 13 (i − 1)−p˜i−1 .
tm
(iii) fit1 − fi i =
4π εi p˜ i−1 N2 (p˜ i−1 )
< 13 (i − 1)−p˜i−1 .
˜ if we repeat Then Claim 3.5 holds true, and so f (T˜ n (e)) is a Gordon potential. For arbitrary ω, the same procedures, (i)–(iii) above still hold as stated (since none of them are related to ω), and Theorem 1.5 follows. 2 References [1] A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Comm. Math. Phys. 288 (2009) 907–918. [2] J. Avron, B. Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981) 101–120. [3] R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990. [4] V. Chulaevsky, Perturbations of a Schrödinger operator with periodic potential, Uspekhi Mat. Nauk 36 (1981) 203– 204. [5] V. Chulaevsky, Almost Periodic Operators and Related Nonlinear Integrable Systems, Manchester University Press, Manchester, 1989. [6] W. Craig, B. Simon, Subharmonicity of the Lyapunov index, Duke Math. J. 50 (1983) 551–560. [7] D. Damanik, Z. Gan, Spectral properties of limit-periodic Schrödinger operators, Discrete Contin. Dyn. Syst. Ser. S, in press. [8] D. Damanik, Z. Gan, Limit-periodic Schrödinger operators with uniformly localized eigenfunctions, preprint, arXiv:1003.1695. [9] D. Damanik, A. Gorodetski, The spectrum of the weakly coupled Fibonacci Hamiltonian, Electron. Res. Announc. Math. Sci. 16 (2009) 23–29.
D. Damanik, Z. Gan / Journal of Functional Analysis 258 (2010) 4010–4025
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[10] D. Damanik, A. Gorodetski, Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian, preprint, arXiv:1001.2552. [11] G. Golub, C. van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, 1989. [12] A. Gordon, On the point spectrum of the one-dimensional Schrödinger operator, Uspekhi Mat. Nauk 31 (1976) 257–258. [13] S. Molchanov, V. Chulaevsky, The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator, Funct. Anal. Appl. 18 (1984) 343–344. [14] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Berlin, 1992. [15] L. Pastur, V. Tkachenko, On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential, Soviet Math. Dokl. 30 (1984) 773–776. [16] L. Pastur, V. Tkachenko, Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials, Trans. Moscow Math. Soc. 51 (1984) 115–166. [17] J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Comm. Math. Phys. 88 (1983) 447–463. [18] C. Rogers, Hausdorff Measure, Cambridge University Press, London, 1970. [19] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1 (2007) 713–772.
Journal of Functional Analysis 258 (2010) 4026–4051 www.elsevier.com/locate/jfa
Symmetrization of Lévy processes and applications Rodrigo Bañuelos a,∗,1 , Pedro J. Méndez-Hernández b,2 a Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States b Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica
Received 29 June 2009; accepted 26 February 2010
Communicated by Daniel W. Stroock
Abstract It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs. © 2010 Elsevier Inc. All rights reserved. Keywords: Symmetrization; Lévy processes
1. Introduction Let D be an open connected set in Rd of finite Lebesgue measure. Henceforth we shall refer to such sets simply as domains. We will denote by D ∗ the open ball in Rd centered at the origin 0 with the same Lebesgue measure as D, and |D| will denote the Lebesgue measure of D. There is a large class of quantities which are related to Brownian motion killed upon leaving D that are maximized, or minimized, by the corresponding quantities for D ∗ . Such results often go by the name of generalized isoperimetric inequalities. They include the celebrated Rayleigh–Faber– Krahn inequality on the first eigenvalue of the Dirichlet Laplacian, inequalities for transition densities (heat kernels), Green functions, and electrostatic capacities (see [4,15–18]). * Corresponding author.
E-mail addresses:
[email protected] (R. Bañuelos),
[email protected] (P.J. Méndez-Hernández). 1 Supported in part by NSF Grant # 0603701-DMS. 2 Supported in part by project No. 821-A7-177 of Centro de Investigación en Matemática Pura y Aplicada (CIMPA).
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.013
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Many of these isoperimetric inequalities can be beautifully formulated in terms of exit times of the Brownian motion Bt from the domain D. For example, if τD is the first exit time of Bt from D, then for all x ∈ D P x {τD > 0} P 0 {τD ∗ > 0},
(1.1)
where 0 is the origin of Rd . Inequality (1.1) contains not only the classical Rayleigh–Faber– Krahn inequality but inequalities for heat kernels and Green functions as well. This inequality is now classical and can be found in many places in the literature. For one of its first occurrences, using the Brascamp–Lieb–Luttinger multiple integrals techniques, please see Aizenman and Simon [1]. Similar inequalities can be obtained by these methods for domains of fixed inradius rather than fixed volume. For more on this, we refer the reader to [5] and [12]. Also, versions of some of these results hold for Brownian motion on spheres and hyperbolic spaces, see [8] and references therein. Once these isoperimetric-type inequalities are formulated in terms of exit times of Brownian motion, it is completely natural to enquire as to their validity for other stochastic processes, and particularly for more general Lévy processes whose generators, as pseudo differential operators, are natural extensions of the Laplacian. Such extensions have been obtained in recent years for the so-called “symmetric stable processes” in Rd and for more general processes obtained from subordination of Brownian motion. We refer the reader to [5,6,12,21]. The purpose of this paper is to show that many of these results continue to hold for very general Lévy processes. At the heart of these extensions are the rearrangement inequalities of Brascamp, Lieb and Luttinger [7]. However, the probabilistic structure of Lévy processes enters in a very crucial way. Of particular importance for our method is the fact, derived from the Lévy– Khintchine formula, that our processes are weak limits of sums of a compound Poisson process and a Gaussian process. We begin with a general description of Lévy processes. A Lévy process Xt in Rd is a stochastic process with independent and stationary increments which is “stochastically” continuous. That is, for all 0 < s < t < ∞, A ⊂ Rd , P x {Xt − Xs ∈ A} = P 0 {Xt−s ∈ A}, for any given sequence of ordered times 0 < t1 < t2 < · · · < tm < ∞, the random variables Xt1 − X0 , Xt2 − Xt1 , . . . , Xtm − Xtm−1 are independent, and for all ε > 0, lim P x |Xt − Xs | > ε = 0.
t→s
The celebrated Lévy–Khintchine formula [20] guarantees the existence of a triple (b, A, ν) such that the characteristic function of the process is given by E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x , where 1 Ψ (ξ ) = −ib, ξ + A · ξ, ξ + 2
Rd
1 + iξ, yIB − eiξ ·y dν(y).
(1.2)
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Here, b ∈ Rd , A is a nonnegative d × d symmetric matrix, IB is the indicator function of the ball B centered at the origin of radius 1, and ν is a measure on Rd such that |y|2 dν(y) < ∞ and ν {0} = 0. (1.3) 2 1 + |y| Rd
The triple (b, A, ν) is called the characteristic of the process and the measure ν is called the Lévy measure of the process. Conversely, given a triple (b, A, ν) with such properties there is Lévy processes corresponding to it. We will use the fact that any Lévy process has a version with paths that are right continuous with left limits, so-called “càdlàg” paths. Next we recall the basic facts on symmetrization needed to state our results; more details on the properties of symmetrization used in this paper can be found in Section 4. Given a positive measurable function f , its symmetric decreasing rearrangement f ∗ is the unique function satisfying f ∗ (x) = f ∗ (y),
if |x| = |y|,
f ∗ (x) f ∗ (y),
if |x| |y|,
lim
|x|→|y|+
∗
f (x) = f ∗ (y),
and m{f > t} = m f ∗ > t ,
(1.4)
for all t 0. Following [14], under the assumption that f vanishes at infinity, an explicit expression for this function is ∗
∞
f (x) =
∗ χ{|f |>t} (x) dt.
0
For symmetrization purposes, in this paper we will only consider Lévy measures ν that are absolutely continuous with respect to the Lebesgue measure m. It may be that some of the results in this paper hold for more general Lévy processes but at this stage we are not able to go beyond the absolute continuity case. Let φ be the density of ν and φ ∗ be its symmetric decreasing rearrangement. Since the function ψ(y) = 1 −
|y|2 1 + |y|2
is positive, decreasing and radially symmetric, that is, ψ ∗ = ψ , it follows that (see Theorem 3.4 in [14]) Rd
|y|2 φ ∗ (y) dy 1 + |y|2
Rd
|y|2 φ(y) dy < ∞. 1 + |y|2
Hence the measure φ ∗ (y) dy satisfies (1.3) and it is also a Lévy measure.
(1.5)
R. Bañuelos, P.J. Méndez-Hernández / Journal of Functional Analysis 258 (2010) 4026–4051
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We denote the d × d identity matrix by Id and the determinant of A by det A. Set A∗ = (det A)1/d Id and define Xt∗ to be the rotationally invariant Lévy process in Rd associated to the triple (0, A∗ , φ ∗ (y) dy). We will often refer to Xt∗ as the symmetrization of Xt . Notice that ∗ ∗ E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x ,
(1.6)
where
1 Ψ (ξ ) = A∗ · ξ, ξ + 2 ∗
1 − eiξ ·y φ ∗ (y) dy
Rd
=
1 ∗ A · ξ, ξ + 2
1 − cos(ξ · y) φ ∗ (y) dy,
Rd
where the last inequality follows from the fact that φ ∗ is symmetric and y → sin(ξ · y) is antisymmetric. The next two theorems are the main results of this paper. Theorem 1.1. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and let Xt∗ be the symmetrization of Xt constructed as above. Let f1 , . . . , fm be nonnegative lower semicontinuous functions. Then for all z ∈ Rd , E
z
m
fi (Xti ) E
i=1
0
m
fi∗
∗ X ti ,
(1.7)
i=1
for all 0 t1 · · · tm . One easily proves that this result is not valid when the functions f1 , . . . , fm are not lower semicontinuous. For example, applying the results with compound Poisson processes, it would follow that for all nonnegative bounded measurable functions f , f (z) f ∗ (0) = f L∞ for all z ∈ Rd . This inequality is true for lower semicontinuous functions but not for measurable functions simply by changing the function on a set of measure zero. However, if we assume further that the distributions of Xt and Xt∗ are absolutely continuous with respect to the Lebesgue measure, we can extend Theorem 1.1 to measurable functions. Theorem 1.2. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and let Xt∗ be the symmetrization of Xt as constructed above. Assume further that for all t > 0 the distributions of Xt and Xt∗ are absolutely continuous with respect to the Lebesgue measure. That is, for all t > 0, P x {Xt ∈ A} =
p(t, x, y) dy A
and
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P Xt∗ ∈ A =
x
p ∗ (t, x, y) dy,
A
for any Borel set A ⊂ Rd . Let f1 , . . . , fm , m 1, be nonnegative measurable functions. Then for all z ∈ Rd , E
z
m
fi (Xti ) E
i=1
0
m
fi∗
∗ X ti ,
i=1
for all 0 t1 · · · tm . Remark 1.3. A sufficient condition for the absolute continuity of the law of a Lévy process is given in [20, page 177]. In our case this is satisfied by both Xt and Xt∗ whenever det(A) > 0 or φ∈ / L1 (Rd ). As we shall see below, Theorem 1.1 together with the now standard Brownian motion argument of Aizenman and Simon [1] implies a generalization of (1.1) to Lévy processes whose Lévy measure is absolutely continuous with respect to the Lebesgue measure. In fact, we will obtain a more general result which applies to Schrödinger perturbations of Lévy semigroups. Let D ⊂ Rd be a domain of finite measure, and consider τDX = inf{t > 0: Xt ∈ / D}, ∗
the first exit time of Xt from D. We also have the corresponding quantity τDX∗ for Xt∗ in D ∗ . As explained in Section 5, the following isoperimetric-type inequality is a consequence of Theorem 1.1. Theorem 1.4. Let D be a domain in Rd of finite measure and f and V be nonnegative continuous functions. Suppose Xt is a Lévy process with Lévy measure absolutely continuous with respect to the Lebesgue measure and Xt∗ is the symmetrization of Xt . Then for all z ∈ Rd and all t > 0, E
z
t X f (Xt ) exp − V (Xs ) ds ; τD > t 0
E
0
t ∗ ∗ ∗ X∗ f Xt exp − V Xs ds ; τD ∗ > t . ∗
(1.8)
0
Our symmetrization results are based on the following now classical rearrangement inequality of Brascamp, Lieb and Luttinger [7]. Theorem 1.5. Let f1 , . . . , fm be nonnegative functions in Rd and denote by f1∗ , . . . , fm∗ their symmetric decreasing rearrangements. Then
R. Bañuelos, P.J. Méndez-Hernández / Journal of Functional Analysis 258 (2010) 4026–4051
··· Rd
m
Rd
Rd
fj
k
j =1
···
bj i xi dx1 · · · dxk
i=1
m
Rd
4031
j =1
fj∗
k
bj i xi dx1 · · · dxk ,
i=1
for all positive integers k, m, and any m × k matrix B = [bj i ]. In Section 5, we also show that Theorem 1.2 implies several isoperimetric inequalities for the transition density of the process Xt . There is a similar result for capacities of Lévy symmetric processes. Let CX (A) be the capacity, associated to Xt , of the set A. If Xt is a symmetric processes, that is, P (Xt ∈ A) = P (Xt ∈ −A), for all t > 0, then T. Watanabe [23] proved that, CX (A) CX∗ A∗ .
(1.9)
This inequality can be obtained, from the methods used in this paper only in the case that Xt is an isotropic unimodal Lévy process, see for example [13]. As explained in [5] and [12], in the case that the process Xt is isotropic unimodal, Theorem 1.1 is an immediate consequence of Theorem 1.5. Recall that Xt is isotropic unimodal if it has transition densities p(t, x, y) of the form p(t, x, y) = qt |x − y| ,
(1.10)
where qt is a function such that qt (r1 ) qt (r2 ), for all r1 r2 and all t > 0. Thus for such Lévy processes (with y fixed)
∗ p(t, ·, y) = p(t, ·, 0),
and Xt = Xt∗ . This class of Lévy processes includes the Brownian motion, rotational invariant symmetric α-stable processes, relativistic stable processes and any other subordinations of the Brownian motion. Notice that in our more general setting, and under the assumption that the distribution of Xt is absolutely continuous relative to the Lebesgue measure, we cannot even ensure that [p(t, ·, y)]∗ is the transition density of a Lévy processes. The rest of the paper is organized as follows. In Section 2 we will prove Theorem 1.1 for Compound Poisson processes. We will consider the case of Gaussian Lévy processes in Section 3. Theorem 1.1 and Theorem 1.2 are proved in Section 4, using a weak approximation of Xt and Xt∗ by Lévy processes of the form Gt + Ct , where Gt is a nondegenerate Gaussian process and Ct is an independent compound process. We will then show some of the applications in Section 5.
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2. Symmetrization of compound Poisson processes In this section we prove a version of the inequality (1.7) for compound Poisson processes. This result, combined with the results in Section 3, will lead to a proof of Theorem 1.1. We start by recalling the structure of compound Poisson processes in terms of random walks. If Ct is a compound Poisson process, starting at x, then its characteristic function is given by E x eiξ ·Ct = eix·ξ −tΨC (ξ ) ,
(2.1)
where ΨC (ξ ) = c
1 − eiξ ·y φ(y) dy,
Rd
and φ is a probability density. We now use the fact that Ct can be written in terms of sums of independent random variables. That is, by Theorem 4.3 [20] there exist a Poisson process Nt with parameter c > 0, and a sequence of i.i.d. random variables {Xn }∞ n=1 such that (1) {Nt }t>0 and {Xn }∞ n=1 are independent, (2) φ(y) is the density of the distribution of Xi , i 1, (3) Ct = SNt + x, where Sn = X1 + · · · + Xn and S0 = 0. Hence if f is a nonnegative Borel function, then E x f (Ct ) = E x f (SNt ) =
∞
P [Nt = n]E f (x + Sn ) .
(2.2)
n=0
Let φ ∗ be the symmetric decreasing rearrangement of φ. Since Rd
φ ∗ (y) dy =
φ(y) dy = 1, Rd
we can consider a new sequence of i.i.d. random variables {Xn∗ }∞ n=1 independent of Nt such that φ ∗ (y) is the density of Xn∗ . Define Sn∗ = X1∗ + · · · + Xn∗ to be the corresponding random walk and Ct∗ the compound Poisson process given by ∗ Ct∗ = SN . t
Notice that the distribution μt of Ct is not absolutely continuous with respect to Lebesgue measure. However, if C0 = x we have the following representation μt = P [Nt = 0]δx +
∞ k=1
P [Nt = k]μk (x),
(2.3)
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4033
with μk the distribution of Sk . That is,
E f (Sk ) =
f (x + y) dμk (y)
x
Rd
=
···
Rd
f
k
xj
j =0
Rd
k
φ(xi ) dx1 · · · dxk .
i=1
Thus if f is a bounded measurable function we have that f ∗ S0∗ = f ∗ (0) = f L∞ and the inequality f (S0 + x) = f (x) f ∗ S0∗ can only be asserted to hold almost everywhere. The next result is a version of inequality (1.7) for random walks where the functions are only assumed to be measurable but the conclusion is only a.e. with respect to the Lebesgue measure. We label it as “Theorem” because it may be of some independent interest. Theorem 2.1. Let f1 , . . . , fm nonnegative functions and k1 · · · km nonnegative integers. Then
m m ∗ ∗ (2.4) fi (x0 + Ski ) E fi Ski , E i=1
i=1
almost everywhere in x0 , with respect to Lebesgue measure. In the case that f1 , . . . , fm are continuous, (2.4) holds pointwise. Proof. Given that X1 , . . . , Xkm are i.i.d we can apply Theorem 1.5 to obtain that E
m
fi (x0 + Ski ) = E
i=1
m
fi (x0 + X1 + · · · + Xki )
i=1
=
··· Rd
Rd
··· Rd
m
fi
fi∗
i=1
m ∗ ∗ =E fi Ski .
i=1
ki
xj
j =0
i=1
m
Rd
ki
km
φ(xi ) dx1 · · · dxm
i=1
xj
j =1
km
φ ∗ (xi ) dx1 · · · dxm
i=1
2
(2.5)
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We can now prove the inequality (1.7) for the compound Poisson process Ct . Let f1 , . . . , fm be nonnegative continuous functions. Since Nt is independent of Sk and Sk∗ , we can combine (2.2) and Theorem 2.1 to obtain
m
m ∞ x fi (SNti ) = P [Nt1 = k1 , . . . , Ntm = km ]E fi (x + Ski ) E k1 k2 ···km
i=1
∞
P [Nt1 = k1 , . . . , Ntm = km ]E
k1 k2 ···km
= E0
m
∗ fi∗ SN t
i
i=1 m
fi∗
∗ Ski
i=1
(2.6)
.
i=1
Thus E
x
m
fi (Cti ) E
0
i=1
m
fi∗
∗ Cti ,
(2.7)
i=1
which is desired result. 3. Symmetrization of Gaussian processes Let Gt be a nondegenerate Gaussian process. Then there exist b ∈ Rd and a strictly positive definite symmetric d × d matrix A such that the density of Gt is given by fA,b (t, x) =
1 exp − (x − tb), A−1 · (x − tb) , √ 2t [2tπ]d/2 det A 1
for all x ∈ Rd and all t > 0. Let us first assume that b = 0. Let u > 0, then −1/2
d d −1/2 · x, A · x < t ln x ∈ R : fA,0 (t, x) > u = x ∈ R : A
1 d (2tπ) u2 det A
A change of variables implies that m x ∈ Rd : fA,0 (t, x) > u =
1 m B(rA,d,u,t ) , 1/2 [det A]
where rA,d,u,t = t ln
1 . (2tπ)d u2 det A
Consider the diagonal matrix 1
A∗ = (det A) d Id .
.
R. Bañuelos, P.J. Méndez-Hernández / Journal of Functional Analysis 258 (2010) 4026–4051
4035
Then m x ∈ Rd : fA,0 (t, x) > u = m x ∈ Rd : fA∗ ,0 (t, x) > u , for all u > 0. Given that fA∗ ,0 (t, x) is rotational invariant and radially decreasing, we conclude that ∗ ∗ fA,b (t, x) = fA,0 (t, x − tb) = fA∗ ,0 (t, x).
(3.1)
If Gt is a degenerate Gaussian process, then t E eiξ ·Gt = exp itb · ξ − i A · ξ, ξ , 2
(3.2)
where A is a positive definite d × d matrix such that det A = 0. Let {v1 , . . . , vd } be the orthonormal eigenvectors of A with eigenvalues λ1 , . . . , λd . We can assume that {λ1 , . . . , λk }, 1 k < d, are the nonzero eigenvalues of A. Let W be the subspace spanned by v1 , . . . , vk . Then Gt can be identified with a nondegenerate Gaussian process in the lower dimension space W and P z [Gt ∈ D] = P z Gt ∈ PW (D) , where PW (D) is the projection of D on the space W . Define A∗ to be the symmetric positive defined matrix with eigenvectors v1 , . . . , vd such that A∗ vi = 0,
k < i d,
and A∗ vi = λvi ,
1 i k,
where λ = (λ1 · · · λk )1/k . The arguments of this section imply that P z [Gt ∈ D] = P z Gt ∈ PW (D) ∗ = P 0 G∗t ∈ DW , ∗ is the ball in W , centered at the origin, with the same k-dimension measure as P (D). where DW W Hence the corresponding symmetrization for this processes should be done in lower dimensions.
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4. Symmetrization of Lévy processes: proof of Theorem 1.1 We will now consider general Lévy processes whose Lévy measures are absolutely continuous with respect to the Lebesgue measure. Our proof uses the fact that the symmetrization of functions preserves order, continuity and almost everywhere convergence. For the convenience of the reader, and for completeness, we briefly describe how to prove these properties of the symmetrization. We recall once again that the symmetric decreasing rearrangement f ∗ of f is the function satisfying f ∗ (x) = f ∗ (y),
if |x| = |y|,
f ∗ (x) f ∗ (y),
if |x| |y|,
lim
∗
|x|→|y|+
f (x) = f ∗ (y),
and m{f > t} = m f ∗ > t ,
(4.1)
m{f > t} = m B 0, r(f, t) .
(4.2)
for all t 0. Define r(f, t) as
Whenever f is a radially symmetric nonincreasing function such that f is right continuous at |x0 |, we have that r f, f (x0 ) = sup r > 0: f (r) > f (x0 ) = |x0 |.
(4.3)
Using this properties of r(f, t), one easily proves the following result, see page 81 of [14]. Lemma 4.1. Let f be a nonnegative function. If f is continuous, then f ∗ is continuous. In addition, if g is a nonnegative function such that g(x) f (x) almost everywhere with respect to the Lebesgue measure, then g ∗ (x) f ∗ (x),
(4.4)
for all x ∈ R. We will also need the following result on almost everywhere convergence and symmetrization. Lemma 4.2. Let {φn }∞ n=1 be a sequence of bounded functions such that lim φn = φ,
n→∞
almost everywhere with respect to the Lebesgue measure. If x0 is a point of continuity of φ ∗ then
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lim φ ∗ (x0 ) = φ ∗ (x0 ). n→∞ n In particular lim φ ∗ n→∞ n
= φ∗,
(4.5)
almost everywhere with respect to the Lebesgue measure. Proof. Assume there exists x0 a continuity point of φ ∗ such that lim φ ∗ (x0 ) = φ ∗ (x0 ). n→∞ n Then there exist > 0 and a subsequence nk such that either φn∗k (x0 ) > φ ∗ (x0 ) + ,
(4.6)
φn∗k (x0 ) < φ ∗ (x0 ) − .
(4.7)
or
Let us assume that (4.6) holds. Since x0 is a continuity point of φ ∗ , there exist 0 < δ < and y0 a continuity point of φ ∗ such that φ ∗ (x0 ) + δ = φ ∗ (y0 ),
and |y0 | < |x0 |.
However, thanks to (4.1) and (4.3), m B 0, |x0 | = lim sup m φn∗k > φn∗k (x0 ) nk →∞
lim sup m φn∗k > φ ∗ (x0 ) + δ nk →∞
= m φ ∗ > φ ∗ (y0 ) = m B 0, |y0 | , which is a contradiction. A similar argument shows that φn∗k (x0 ) < φ ∗ (x0 ) − , yields a contradiction.
2
We recall that under our assumptions E x eiξ ·Xt = e−tΨ (ξ )+iξ ·x , where
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1 Ψ (ξ ) = −ib, ξ + A · ξ, ξ + 2
1 + iξ, yIB − eiξ ·y φ(y) dy,
Rd
B is the unit ball centered at the origin and φ is such that Rd
|y|2 φ(y) dy < ∞. 1 + |y|2
(4.8)
Consider the sequence φn (y) = φ(y)I{t∈R:
1 n
|y| ,
and let φn∗ (y) be its symmetric decreasing rearrangement. Thanks to (4.8), cn =
φn (y) dy < ∞, Rd
and |yi |φn (y) dy < ∞,
1 i d,
B
where again B is the unit ball. Consider Cn,t a compound Poisson process with characteristic function E eiξ ·Cn,t = e−tΨC,n (ξ ) ,
(4.9)
where ΨC,n (ξ ) = cn Rd
φn (y) 1 − eiξ ·y dy. cn
Given that all the eigenvalues of A are nonnegative, if {n }∞ n=1 is a sequence of positive numbers converging to zero, then An = A + n Id is a sequence of nonnegative nonsingular matrices. Let Gn,t be a Gaussian process starting at x, independent of Cn,t , and associated with the ma trix An and the vector bn = b − B yφn (y) dy. Set Xn,t = Cn,t + Gn,t . Since Cn,t and Gn,t are independent, E x eiξ ·Xn,t = e−tΨn (ξ )+iξ ·x , where
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1 Ψn (ξ ) = −ibn , ξ + An · ξ, ξ + 2
4039
1 − eiξ ·y φn (y) dy
Rd
1 = −ib, ξ + An · ξ, ξ + 2
1 + iξ, yIB − eiξ ·y φn (y) dy.
(4.10)
Rd
Let Sn,k = X1n +· · ·+Xkn be the random walk associated to Cn,t . If f1 , . . . , fm are nonnegative continuous functions and t1 · · · tm , then E
x
m
fi (Xn,ti )
i=1
=E
m
x
fi (Cn,ti + Gn,ti )
i=1
∞
=
P [Nt1 = k1 , . . . , Ntm = km ]E
x
k1 k2 ···km
m
fi (Sn,ki + Gn,ti ) .
(4.11)
i=1
Now Theorem 1.5 and equality (3.1) imply that E
x
m
fi Gn,ti +
··· Rd
m
Rd
···
m
m
Rd
= Ex
fi
ki
xj fAn ,bn (t, x0 − x)
j =0
fi∗
ki j =0
i=1
Rd
···
i=1
Rd
=
Xjn
j =1
i=1
=
ki
fi∗
ki
i=1
m
fi∗ G∗n,ti
φ(xj ) dx0 · · · dxkm
j =1
xj fA∗n ,0 (t, x0 ) xj fA∗n ,0 (t, x0 )
j =0
Rd
km
km
φ ∗ (xj ) dx0 · · · dxkm
j =1 km
φ ∗ (xj ) dx0 · · · dxkm
j =1
∗ . + Sn,k i
(4.12)
i=1
This implies that E
x
m i=1
fi (Xn,ti ) E
0
m
fi∗
∗ Xn,t i
.
(4.13)
i=1
Theorem 1.1 will be a consequence of (4.13) and the following result on weak convergence.
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Theorem 4.3. Let f1 , . . . , fk be nonnegative bounded continuous functions, and 0 < t1 < · · · < tm . Then for all x ∈ Rd , lim E
x
n→∞
k
fi (Xn,ti ) = E
x
i=1
k
(4.14)
fi (Xti ) ,
i=1
and lim E
x
n→∞
k
k ∗ x ∗ fi Xn,ti = E f i X ti .
i=1
i=1
Proof. Notice that for all ξ ∈ Rd , lim An · ξ, ξ = A · ξ, ξ .
n→∞
Given that there exists C ∈ R+ such that, 1 + iξ, y − eiξ ·y φn (y) C|ξ |2 |y|2 φ(y) < ∞,
(4.15)
1 − eiξ ·y φn (y) 2φ(y) < ∞,
(4.16)
for all y ∈ B, and
for all y ∈ Rd \ B, it follows from the Dominated Convergence Theorem that lim Ψn (ξ ) = lim
n→∞
n→∞
1 −ib, ξ + An · ξ, ξ + 2
1 = −ib, ξ + A · ξ, ξ + 2
1 + iξ, yIB − eiξ ·y φn (y) dy
Rd
iξ ·y 1 + iξ, yIB − e φ(y) dy .
(4.17)
Rd
We conclude that lim E x eiξ ·Xn,t = E x eiξ ·Xt .
n→∞
On the other hand, using the fact that lim det An = det A,
n→∞
we can easily prove that
lim A∗n · ξ, ξ = A∗ · ξ, ξ .
n→∞
Lemma 4.1 implies that
(4.18)
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φn∗ (x) φ ∗ (x), for all x ∈ R, and all n 1. In addition, Lemma 4.2 gives that lim φ ∗ n→∞ n
= φ∗,
a.e.
Thus the same argument used to prove (4.18) yields ∗ ∗ lim E x eiξ ·Xn,t = E x eiξ ·Xt .
n→∞
(4.19)
Now, if ξ1 , . . . , ξm ∈ Rd , then m
ξj · Xn,tj = (ξ1 + · · · + ξm ) · Xn,t1 +
j =1
m (ξm + · · · + ξj ) · (Xn,tj − Xn,tj −1 ). (4.20) j =2
Since t1 < · · · < tm we have that m
x ξj · Xn,tj E exp i = E x exp i(ξ1 + · · · + ξm ) · Xn,t1 j =1
×
m
E 0 exp i(ξm + · · · + ξj ) · (Xn,tj −tj −1 ) .
(4.21)
j =2
The desired result immediately follows from (4.18), (4.19) and the fact that our characteristic functions are continuous at 0. This last observation follows from the Lévy–Khintchine formula. 2 Combining (4.13) and Theorem 4.3, we obtain E
x
m
fi (Xti ) E
0
i=1
m
fi∗
∗ X ti
(4.22)
i=1
for all nonnegative bounded continuous functions f1 , . . . , fm . Let f1 , . . . , fm be a nonnegative continuous functions. For 1 i m consider the sequence fi,n = max{fi , n}. Then, for all x ∈ Rd and all n 1, 0 fi,n (x) fi,n+1 (x) fi (x). Lemma 4.1 implies that 0
m i=1
∗ fi,n (x)
m i=1
fi∗ (x),
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for all x ∈ Rd . Thus E
x
m
fi,n (Xti ) E
0
i=1
E0
m i=1 m
∗ fi,n
∗ X ti
fi∗ Xt∗i .
i=1
The Monotone Convergence Theorem for the law of Xt imply Theorem 1.1 in the case that all functions are nonnegative and continuous. Finally let f1 , . . . , fm be nonnegative lower semicontinuous functions. Consider fi,n the infimal convolution of fi and the function n|z|2 , that is fi,n (x) = inf f (y) + n|z|2 : y + z = x . One easily proves that {fi,n }∞ n=1 is a sequence of nonnegative continuous functions such that, fi,n (x) fi,n+1 (x) fi (x), and lim fi,n (x) = fi (x),
n→∞
for all x ∈ Rd . Lemma 4.1 imply that ∗ (x) fi∗ (x), fi,n
for all x ∈ Rd , and Theorem 1.1 follows from the Monotone Convergence Theorem.
2
Now we will prove Theorem 1.2. Without lost of generality we can assume that the functions f1 , . . . , fm are finite almost everywhere with respect to the Lebesgue measure. For 1 i m, let us assume that there exists a constant Mi fi (x) Mi , for all x ∈ R. Then there exist sequences of nonnegative continuous functions {φi,n }∞ n=1 such that lim φi,n = fi ,
n→∞
almost everywhere with respect to the Lebesgue measure, and φi,n (x) Mi , for all x ∈ R. By Lemma 4.2 lim φ ∗ n→∞ i,n
= fi∗ ,
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almost everywhere with respect to the Lebesgue measure. Thus the absolute continuity of the laws of Xti and Xt∗i with respect to the Lebesgue measure and the Dominated Convergence Theorem yield (4.22) if the functions f1 , . . . , fm are bounded. Finally, as in the proof of Theorem 1.1, if fi is not bounded, consider the sequence fi,n = max{fi , n}. As before, Lemma 4.1 and the Monotone Convergence Theorem imply Theorem 1.1. 5. Some applications In this section we give several applications of Theorem 1.1, we begin with the proof of Theorem 1.4. Recall that / D} τDX = inf{t > 0: Xt ∈ is the first exit time of Xt from a domain D. Let Dk be a sequence of bounded domains with smooth boundaries such that Dk ⊂ Dk+1 , and ∞ k=1 Dk = D. Since any Lévy process has a version with right continuous paths, we have E
z0
t X f (Xt ) exp − V (Xs ) ds ; τD > t 0
=E
z0
t f (Xt ) exp − V (Xs ) ds ; Xs ∈ D, ∀s ∈ [0, t] 0
m t = lim lim E z0 f (Xt ) exp − V (X it ) ; X it ∈ Dk , i = 1, . . . , m m→∞ k→∞ m m m i=1 m t z0 = lim lim E f (Xt ) exp − V (X it ) IDk (X it ) . m→∞ k→∞ m m m
(5.1)
i=1
Since ∗ exp −sV (x) = exp −sV ∗ (x) , for all s > 0 and all x ∈ Rd , and the functions IDk are lower semicontinuous, Theorem 1.1 implies that m t E z0 f (Xt ) exp − V (X it ) IDk (X it ) m m m i=1 m ∗ ∗ t ∗ ∗ 0 ∗ E f Xt exp − V X it IDk∗ X it . m m m i=1
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Hence we have the following Ez
t f (Xt ) exp − V (Xs ) ds ; τDX > t 0
E
0
t ∗ ∗ ∗ X∗ f Xt exp − V Xs ds ; τD ∗ > t , ∗
(5.2)
0
which is Theorem 1.4. Taking V = 0 and f = 1, gives ∗ P z τDX > t P 0 τDX∗ > t ,
(5.3)
which is a generalization of inequality (1.1). Integrating this inequality with respect to t gives the following result. Corollary 5.1. If ψ is a nonnegative increasing function, then ∗ E z ψ τDX E 0 ψ τDX∗ ,
(5.4)
p ∗ p E z τDX E 0 τDX∗ ,
(5.5)
for all z ∈ D. In particular
for all 0 < p < ∞. Our results imply many isoperimetric inequalities for the potentials and the eigenvalues of Schrödinger operators of the form X HD,V = HDX + V ,
where HDX is the pseudo differential operator associated to Xt with Dirichlet boundary conditions on D. For the convenience of the reader we will give a brief description of the operators and semigroups associated to Lévy processes. For purposes of our formulae below we define the Fourier transform of an L2 (Rd ) function as f (ξ ) =
1 (2π)d/2
e−ix·ξ f (x) dx,
Rd
with 1 f (x) = (2π)d/2
eix·ξ f (ξ ) dξ.
Rd
We define the semigroup associated to the Lévy process Xt by
R. Bañuelos, P.J. Méndez-Hernández / Journal of Functional Analysis 258 (2010) 4026–4051
Tt f (x) = E x f (Xt ) = =
1 E0 (2π)d/2
1 (2π)d/2
ei(Xt +x)·ξ f (ξ ) dξ
4045
Rd
eix·ξ E 0 eiXt ·ξ f (ξ ) dξ
Rd
=
1 (2π)d/2
eix·ξ e−tΨ (ξ ) f (ξ ) dξ.
Rd
This semigroup takes C0 (Rd ) into itself. That is, it is a Feller semigroup. From this we see that, at least formally for f ∈ S(Rd ), the infinitesimal generator is H X f (x) = −
∂Tt f (x) 1 = eix·ξ Ψ (ξ )fˆ(ξ ) dξ. ∂t t=0 (2π)d/2 Rd
Then the Lévy–Khintchine formula implies that the operator associated to Xt is given by H X f (x) =
d
bj ∂j f (x) −
j =1
+
d 1 aj k ∂j ∂k f (x) 2 j,k=1
f (x + y) − f (x) − y · ∇f (x)I{|y|<1} dν(y),
Rd
where aj k are the entries of the matrix A. For instance: (1) If Xt is a standard Brownian motion: 1 H X f = − f. 2 (2) If Xt is a symmetric stable processes of order 0 < α < 2: α/2 1 H Xf = − − f. 2 (3) If Xt is a Poisson process of intensity c: H X f (x) = c f (x + 1) − f (x) . (4) If Xt is a compound Poisson process with measure ν and c = 1: H f (x) = X
f (x + y) − f (x) dν(y).
(5.6)
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In this paper we are interested not on the “free” semigroup for Xt but rather on its “killed” semigroup and its perturbation by the potential V . That is, we want properties of the semigroup TtD,V f (z) = E z
t f (Xt ) exp − V (Xs ) ds ; τDX > t ,
(5.7)
0
defined for t > 0, z ∈ D, and f ∈ L2 (D). Recall our assumption that V is nonnegative and continuous. Thus, t z D,V X Tt f (z) = E f (Xt ) exp − V (Xs ) ds ; τD > t 0
E f (Xt ); τDX > t = TtD |f |(z). z
(5.8)
For the rest of the paper we shall assume that the distributions of Xt and Xt∗ have densities ∗ and p X (t, z, w), respectively, which are continuous in both z and w for all t > 0. X (t, z, w) satisfying The killed semigroup has a heat kernel pD,V p X (t, z, w)
TtD,V f (z) =
X pD,V (t, z, w)f (w) dw.
(5.9)
D
Inequality (5.2) is equivalent to
X f (w)pD,V (t, z, w) dw
∗
X f ∗ (w)pD ∗ ,V ∗ (t, 0, w) dw,
(5.10)
D∗
D
for all z ∈ D and all t > 0, and this in fact holds for all nonnegative Borel functions f by Theorem 1.2. Since f is arbitrary, the continuity assumption of the kernels together with (5.8) gives that for all z, w ∈ D, ∗
∗
X X X (t, z, w) pD pD,V ∗ ,V ∗ (t, 0, 0) pD ∗ (t, 0, 0) < ∞.
(5.11)
If in addition Xt is transient, we can integrate (5.10) in time to obtain the following isoperimetric inequality for the potentials associated to Xt and Xt∗ . Corollary 5.2. Suppose both Xt and Xt∗ are transient and have continuous densities for all t > 0. Then for all z ∈ D,
f (w)GX D,V (z, w) dw
∗
f ∗ (w)GX D ∗ ,V ∗ (0, w) dw,
(5.12)
D∗
D ∗
X ∗ where GX D,V (z, w) and GD ∗ ,V ∗ (0, w) are the Green’s functions corresponding to Xt and Xt , respectively.
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4047
By inequalities (5.10), (5.12), and Proposition 2.1 in [2] (see also page 671 of [8]), we have Corollary 5.3. Suppose both Xt and Xt∗ are symmetric, transient and have continuous densities for all t > 0. Then for all increasing convex functions Φ : R+ → R+ ,
X Φ pD,V (t, z, w) dw
X∗ Φ pD ∗ ,V ∗ (t, w, 0) dw,
(5.13)
∗ Φ GX D ∗ ,V ∗ (w, 0) dw,
(5.14)
D∗
D
and
Φ GX D,V (z, w) dw
D∗
D
for all z ∈ D, t > 0. These corollaries extend several results in C. Bandle [4], see for example page 214. X (t, z, w) can also be represented in terms of the multidimensional distriThe heat kernel pD,V butions. One easily proves, see [12], that X pD,V (t, z, w) = p X (t, z, w)E z
t exp − V (Xs ) ds ; τDX > t Xt = w .
(5.15)
0
If 0 = t0 < t1 < · · · < tm < t, the conditional finite dimensional distribution P z0 {Xt1 ∈ dz1 , . . . , Xtm ∈ dzm | Xt = w} is given by p X (t − tm , zm , w) X p (ti − ti−1 , zi , zi−1 )dz1 · · · dzm . p X (t, z0 , w) m
i=1
Combining (5.15) with the arguments used in (5.1) we have that
X pD,V (t, z, w) = lim m→∞
lim
k→∞ Dk
···
e
− mt
m
i=1 V (X it ) m
m+1 i=1
Dk
p
X
t , zi , zi−1 dz1 · · · dzm , m
(5.16)
where z0 = z and zm+1 = w. The proof of Theorem 1.1 can be adapted to obtain
X pD,V (t, w, w) dw
D
D∗
∗
X pD ∗ ,V ∗ (t, w, w) dw < ∞,
(5.17)
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where the last inequality follows from (5.11) and the fact that |D ∗ | < ∞. That is, the trace X is maximized by the trace of the Schrödinger semiof the Schrödinger semigroup for HD,V ∗ X group HD ∗ ,V ∗ . As explained in [22], the amount of heat contained in the domain D at time t, when D has temperature 1 at t = 0 and the boundary of D is kept at temperature 0 at all times, is given by Qt (D) =
B pD (t, z, w) dz dw, D D
where B is a Brownian motion. Also the torsional rigidity of D is given by
∞ Qt (D)dt = 0
GB D (z, w) dz dw. D D
Using the representation (5.16), we obtain the following results for the heat content and torsional rigidity of Lévy processes. Corollary 5.4. Suppose both Xt and Xt∗ are transient and have continuous densities for all t > 0. Then for all z ∈ D and t > 0,
X pD,V (t, z, w) dz dw
D∗
D D
∗
X pD ∗ ,V ∗ (t, z, w) dz dw,
(5.18)
D∗
and
GX D,V (z, w) dz dw
∗
GX D ∗ ,V ∗ (z, w) dz dw.
(5.19)
D∗ D∗
D D
We recall that the semigroup of the process Xt is self-adjoint in L2 if and only if the process Xt is symmetric. That is, for any Borel set A ⊂ Rd , P 0 {Xt ∈ A} = P 0 {Xt ∈ −A}. In terms of the exponent in the Lévy–Khintchine formula this leads to the representation (see [3]) 1 Ψ (ξ ) = A · ξ, ξ − 2
cos(x · ξ ) − 1 dν(x),
Rd
where A is a symmetric matrix and ν is a symmetric Lévy measure. That is, [ν(A) = ν(−A)] for all Borel sets A. In this case the general theory of Dirichlet forms (see [9]) guarantees that the Markovian semigroup generated by Xt gives rise to the self-adjoint generator H X . Recall that HVX,D is the operator obtained by imposing Dirichlet boundary conditions on D to the Schrödinger operator H X + V . That is, the generator of the killed semigroup {TtD,V }t0 . By (5.11) we have that
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X pD,V (t, w, w) dw
4049
∗
X pD ∗ ,V ∗ (t, 0, 0) dw
D∗
D
∗ X∗ < ∞. = pD ∗ ,V ∗ (t, 0, 0) D
(5.20)
That is, the semigroup of the killed process has finite trace. Whenever D is of finite volume, the operator TtD,V maps L2 (D) into L∞ (D) for every t > 0. This follows from (5.11) and the general theory of heat semigroups as described on page 59 of [9]. In fact, under these assumptions it follows from [9] that there exists an orthonormal basis n ∞ ∞ 2 of eigenfunctions {ϕD,V ,X }n=1 for L (D) and corresponding eigenvalues {λn (D, V , X)}n=1 for the semigroup {TtD,V }t0 satisfying 0 < λ1 (D, V , X) < λ2 (D, V , X) λ3 (D, V , X) · · · with λn (D, V , X) → ∞ as n → ∞. That is, the pair n ϕD,V ,X , λn (D, V , X) satisfies n −λn (D,V ,X)t n ϕD,V ,X (z), TtD ϕD,V ,X (z) = e
z ∈ D, t > 0.
n Notice that λn (D, V , X) is a Dirichlet eigenvalue of H X +V on D with eigenfunction ϕD,V ,X (z). Under such assumptions we have
X (t, z, w) = pD,V
∞
n n e−λn (D,V ,X)t ϕD,V ,X (z)ϕD,V ,X (w).
(5.21)
n=1 X (t, z, w) implies that This eigenfunction expansion for pD,V
t 1 z X −λ1 (D, V , X) = lim log E exp − V (Xs ) ds ; τD > t , t→∞ t
(5.22)
0
for all domains D of finite volume. This gives the following corollary. Corollary 5.5 (Faber–Krahn inequality for Lévy processes). Suppose both Xt and Xt∗ are symmetric, transient and have continuous densities for all t > 0. Then λ1 D ∗ , V ∗ , X ∗ λ1 (D, V .X). More generally, we also have the trace inequality ∞ n=1
valid for all t > 0.
e−tλn (D,X,V )
∞ n=1
e−tλn (D
∗ ,X ∗ ,V ∗ )
(5.23)
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Finally, denote by CX (A) the capacity of the set A for the process Xt . In [23], T. Watanabe proved that CX (A) CX∗ A∗ .
(5.24)
(This question, for Riesz capacities of all orders was raised by P. Mattila in [11].) As explained in [13], this inequality can be obtained from the existing rearrangement inequalities of multiple integrals only in the case that Xt is isotropic unimodal. For general Lévy processes we have the following representation of the capacity due to Port and Stone [19] 1 t→∞ t
lim
P z0 τAXc t dz0 = CX (A).
(5.25)
Since
P z0 τAXc t dz0
= lim lim
k→∞ m→∞
···
1−
m j =1
IAck (zj )
m j =1
p
X
t , zj , zj −1 dz0 · · · dzm , m
(5.26)
where Ak is a decreasing sequence of compact sets such that the interior of Ak contains A for all k and ∞ k=1 Ak = A. We expect that (5.24) can be obtained using a result similar to Theorem 1.5 for more general Lévy processes. However, the corresponding rearrangement inequality for this type of multiple integrals is only known for radially symmetric decreasing functions, see Corollary 2 in [10]. That is, only when Xt is an isotropic unimodal Lévy process. We believe, in the case that the sets Ak are compact, the more general rearrangement inequality for multiple integrals needed for this application should be true but at present we are not able to prove it. Acknowledgments We are very grateful to an anonymous referees for the very careful reading of this paper and the many suggestions which improved the presentation and readability of the paper. References [1] M. Aizenman, B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (2) (1982) 209–273. [2] A. Alvino, G. Trombetti, P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal. 13 (1989) 209–273. [3] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Stud. Adv. Math., vol. 93, Cambridge University Press, Cambridge, 2004. [4] C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math., Pitman, 1980. [5] R. Bañuelos, R. Latała, P.J. Méndez-Hernández, A Brascamp–Lieb–Luttinger-type inequality and applications to symmetric stable processes, Proc. Amer. Math. Soc. 129 (2001) 2997–3008. [6] D. Betsakos, Symmetrization, symmetric stable processes, and Riesz capacities, Trans. Amer. Math. Soc. 356 (2004) 735–755, 3821. [7] H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974) 227–237. [8] A. Burchard, M. Schmuckenschläger, Comparison theorems for exit times, Geom. Funct. Anal. 11 (2001) 651–692.
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[9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [10] R. Friedberg, J.M. Luttinger, Rearrangement inequality for period functions, Arch. Ration. Mech. 61 (1976) 35–44. [11] P. Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990) 185– 198. [12] P.J. Méndez-Hernández, Brascamp–Lieb–Luttinger inequalities for convex domains of finite inradius, Duke Math. J. 113 (2002) 93–131. [13] P.J. Méndez-Hernández, An isoperimetric inequality for Riesz capacities, Rocky Mountain J. Math. 36 (2) (2006) 675–682. [14] E. Lieb, M. Loss, Analysis, Grad. Stud. Math., vol. 14, American Mathematical Society, Providence, RI, 2001. [15] J.M. Luttinger, Generalized isoperimetric inequalities, I, J. Math. Phys. 14 (1973) 586–593. [16] J.M. Luttinger, Generalized isoperimetric inequalities, II, J. Math. Phys. 14 (1973) 1444–1447. [17] J.M. Luttinger, Generalized isoperimetric inequalities, III, J. Math. Phys. 14 (1973) 1448–1450. [18] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951. [19] S.C. Port, C.J. Stone, Infinite divisible processes and their potential theory I, Ann. Inst. Fourier (Grenoble) 21 (1971) 157–275. [20] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [21] B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979. [22] M. van den Berg, J.F. Le Gall, Mean curvature and the heat equation, Math. Z. 215 (1994) 437–464. [23] T. Watanabe, The isoperimetric inequality for isotropic unimodal Lévy processes, Z. Wahrscheinlichkeitstheorie 63 (1983) 487–499.
Journal of Functional Analysis 258 (2010) 4052–4074 www.elsevier.com/locate/jfa
A transference theorem for the Dunkl transform and its applications ✩ Feng Dai a,∗ , Heping Wang b a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada b School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Received 28 August 2009; accepted 10 March 2010 Available online 29 March 2010 Communicated by N. Kalton
Abstract For a family of weight functions invariant under a finite reflection group, we show how weighted Lp multiplier theorems for Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1 . The result is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform and to show the convergence of the Bochner–Riesz means of the Dunkl transform of order above the critical index in weighted Lp spaces. © 2010 Elsevier Inc. All rights reserved. Keywords: Hörmander type multiplier theorem; Dunkl transform; Transference theorem; h-Harmonics
1. Introduction Let R be a reduced root system in Rd normalized so that α, α = 2 for all α ∈ R, where ·,· denotes the standard Euclidean inner product. Given a nonzero vector α ∈ Rd , we denote by σα the reflection with respect to the hyperplane perpendicular to α; that is, σα x = x − 2(x, α/α2 )α for all x ∈ Rd , where · denotes the usual Euclidean norm. Let G denote ✩ The first author was partially supported by the NSERC Canada under grant G121211001. The second author was supported by the National Natural Science Foundation of China (Project No. 10871132) and the Beijing Natural Science Foundation (Project No. 1102011). * Corresponding author. E-mail addresses:
[email protected] (F. Dai),
[email protected] (H. Wang).
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.006
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
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the finite subgroup of the orthogonal group O(d) generated by the reflections σα , α ∈ R. Let κ : R → R+ be a nonnegative multiplicity function on R with the property κ(gα) = κ(α) for all α ∈ R and g ∈ G. Associated with the reflection group G and the function κ is the weight function hκ defined by hκ (x) :=
x, ακ(α) ,
x ∈ Rd ,
(1.1)
α∈R+
where R+ is an arbitrary but fixed positive subsystem of R. The function hκ is a homogeneous function of degree γκ := α∈R+ κ(α), and is invariant under the reflection group G. For convenience, we shall set λκ = d−1 2 + γκ for the rest of the paper. Given 1 p ∞, we denote by Lp (Rd ; h2κ ) the weighted Lebesgue space endowed with the norm f κ,p :=
f (y)p h2 (y) dy
1
κ
p
,
Rd
with the usual change when p = ∞. The Dunkl transform, a generalization of the classical Fourier transform, is defined, for f ∈ L1 (Rd ; h2κ ), by Fκ f (x) = cκ
f (y)Eκ (−ix, y)h2κ (y) dy,
x ∈ Rd ,
(1.2)
Rd
x2 where cκ = ( Rd h2κ (x)e− 2 dx)−1 , and Eκ (ix, y) = Vκ [eix,· ](y) is the weighted analogue of the character eix,y . Here Vκ is the Dunkl intertwining operator, whose precise definition will be given in Section 2. The Dunkl transform plays the same role as the Fourier transform in classical Fourier analysis, and enjoys properties similar to those of the classical Fourier transform (see [11]). Several important results in classical Fourier analysis have been extended to the setting of Dunkl transform by Thangavelu and Yuan Xu [18,17]. The problem, however, turns out to be rather difficult in general. One of the difficulties comes from the fact that the generalized translation operator τy , which plays the role of the usual translation f → f (· − y), is not positive in general (see, for instance, [17, Proposition 3.10]). In fact, even the Lp boundedness of τy is not established in general (see [18,17]). In this paper, we shall first prove a transference theorem (Theorem 3.1) between the Lp multiplier of h-harmonic expansions on the unit sphere and that of the Dunkl transform. This theorem, combined with the corresponding results on h-harmonic expansions on the unit sphere recently established in [2–4], is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform (Theorem 4.1), and to show the convergence of the Bochner–Riesz means in the weighted Lp spaces (Theorem 4.3). The paper is organized as follows. In Section 2, we describe briefly some known results on Dunkl transform and h-harmonic expansions, which will be needed in later sections. The transference theorem, Theorem 3.1, is proved in Section 3. As applications, we prove Theorems 4.1 and 4.3 in the final section, Section 4.
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2. Preliminaries In this section, we shall present some necessary material on the Dunkl transform and the h-harmonic expansions, most of which can be found in [8,11,13,14,17]. 2.1. The Dunkl transform Let R, R+ , G, κ and hκ be as defined in Section 1. Recall that a reduced root system is a finite subset R of Rd \ {0} with the properties σα R = R and R ∩ {tα: t ∈ R} = {±α} for all α ∈ R. The Dunkl operators associated with G and κ are defined by Dκ,i f (x) = ∂i f (x) +
κ(α)
α∈R+
f (x) − f (σ (α)x) α, ei , x, α
1 i d,
(2.1)
where e1 , . . . , ed are the standard unit vectors of Rd . Those operators mutually commute, and map Pdn to Pdn−1 , where Pdn is the space of homogeneous polynomials of degree n in d variables (see [5]). We denote by Π d := Π(Rd ) the C-algebra of polynomial functions on Rd . An important result in Dunkl theory states that there exists a linear operator Vκ : Π d → Π d determined uniquely by
Vκ Pdn ⊂ Pdn ,
Vκ (1) = 1,
and Dκ,i Vκ = Vκ ∂i ,
1 i d.
(2.2)
Such an operator is called the intertwining operator. A very useful explicit formula for Vκ was obtained by C.F. Dunkl [6] in the case of G = Z2 , and was later extended to the more general case of G = Zd2 (d ∈ N) by Xuan Xu [21]. In general, one has the following important result of Rösler [13]: Lemma 2.1. (See [13, Theorem 1.2 and Corollary 5.3].) For every x ∈ Rd , there exists a unique probability measure μκx on the Borel σ -algebra of Rd such that Vκ P (x) =
P (ξ ) dμκx (ξ ),
P ∈ Πd.
(2.3)
Rd
Furthermore, the representing measures μκx are compactly supported in the convex hull C(x) := co{gx: g ∈ G} of the orbit of x under G, and satisfy
μκrx (E) = μκx r −1 E and μκgx (E) = μκx g −1 E
(2.4)
for all r > 0, g ∈ G and each Borel subset E of Rd . In particular, the above lemma shows that the intertwining operator Vκ is positive. We point out that Lemma 2.1 will play a crucial role in the analysis of our paper. By means of (2.3), Vκ can be extended to the space C(Rd ) of continuous functions on Rd . We denote this extension by Vκ again. The Dunkl transform associated with G and κ is defined by (1.2) with
Eκ (−ix, y) := Vκ e−ix,· (y),
x, y ∈ Rd .
(2.5)
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If κ = 0 then Vκ = id and the Dunkl transform coincides with the usual Fourier transform, whereas if d = 1 and G = Z2 then it is closely related to the Hankel transform on the real line. We list some of the known properties of the Dunkl transform in the following lemma. Lemma 2.2. (See [7,11].) (i) If f ∈ L1 (Rd ; h2κ ) then Fκ f ∈ C(Rd ) and limξ →∞ Fκ f (ξ ) = 0. (ii) The Dunkl transform Fκ is an isomorphism of the Schwartz class S(Rd ) onto itself, and Fκ2 f (x) = f (−x). (iii) The Dunkl transform Fκ on S(Rd ) extends uniquely to an isometric isomorphism on L2 (Rd ; h2κ ), i.e., f κ,2 = Fκ f κ,2 . (iv) If f and Fκ f are both in L1 (Rd ; h2κ ) then the following inverse formula holds: f (x) = cκ
Fκ f (y)Eκ (ix, y)h2κ (y) dy,
x ∈ Rd .
Rd
(v) If f, g ∈ L2 (Rd ; h2κ ) then
Fκ f (x)g(x) h2κ (x) dx =
Rd
f (x)Fκ g(x) h2κ (x) dx. Rd
(vi) Given ε > 0, let fε (x) = ε −2−2γκ f (ε −1 x). Then Fκ fε (ξ ) = Fκ f (εξ ). (vii) If f (x) = f0 (x) is radial, then Fκ f (ξ ) = Hλκ − 1 f0 (ξ ) is again a radial function, 2 where Hα denotes the Hankel transform defined by Hα g(s) =
1 Γ (α + 1)
∞ g(r)
Jα (rs) 2α+1 r dr, (rs)α
0
and Jα denotes the Bessel function of the first kind. Statements (i)–(vi) of Lemma 2.2 above were proved by M.F.E. de Jeu [11], and were, in fact, contained in Corollaries 4.7 and 4.22, Theorems 4.26 and 4.20, Lemmas 4.13 and 4.3(3) of [11], respectively. Statement (vii) of Lemma 2.2 was proved by Dunkl [7] and was stated more explicitly in [17, Proposition 2.4]. Given y ∈ Rd , the generalized translation operator f → τy f is defined on L2 (Rd ; h2κ ) by Fκ (τy f )(ξ ) = Eκ (−iξ, y)Fκ f (ξ ),
ξ ∈ Rd .
It is known that τy f (x) = τx f (y) for a.e. x ∈ Rd and a.e. y ∈ Rd . In general, the operator τy is not positive (see, for instance, [17, Proposition 3.10]), and it is still an open problem whether τy f can be extended to a bounded operator on L1 (Rd ; h2κ ). On the other hand, however, it was shown in [17, Theorem 3.7] that the generalized translation operator τy can be extended to all p radial functions in Lp (Rd ; h2κ ), 1 p 2, and τy : Lrad (Rd ; h2κ ) → Lp (Rd ; h2κ ) is a bounded p operator, where Lrad (Rd ; h2κ ) denotes the space of all radial functions in Lp (Rd ; h2κ ).
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The generalized convolution of f, g ∈ L2 (Rd ; h2κ ) is defined by f ∗κ g(x) =
f (y)τx g (y)h2κ (y) dy,
(2.6)
Rd p
where g (y) = g(−y). Since τy is a bounded operator from Lrad (Rd ; h2κ ) to Lp (Rd ; h2κ ) for 1 p p 2, it follows that the definition of f ∗κ g can be extended to all g ∈ Lrad (Rd ; h2κ ) and f ∈ 1 1 p d 2 L (R ; hκ ) with 1 p 2 and p + p = 1. The generalized convolution satisfies the following basic property: Fκ (f ∗κ g)(ξ ) = Fκ f (ξ )Fκ g(ξ ).
(2.7)
More properties on the generalized translation operator and the generalized convolution can be found in [17]. 2.2. h-Harmonic expansions Let Sd−1 = {x ∈ Rd : x = 1} denote the unit sphere of Rd equipped with the usual Lebesgue measure dσ (x). For the weight function hκ given in (1.1), we consider the weighted Lebesgue space Lp (h2κ ; Sd−1 ) of functions on Sd−1 endowed with the finite norm f Lp (h2κ ;Sd−1 ) ≡ f κ,p :=
1/p f (y)p h2 (y) dσ (y) , κ
1 p < ∞,
Sd−1
and for p = ∞ we assume that L∞ is replaced by C(Sd−1 ), the space of continuous functions on Sd−1 with the usual uniform norm f ∞ . We shall use the notation · κ,p to denote the weighted norm for functions defined either on Rd or on Sd−1 whenever it causes no confusion. A homogeneous polynomial is called an h-harmonic if it is orthogonal to all polynomials of lower degree with respect to the inner product of L2 (h2κ ; Sd−1 ). Let Hnd (h2κ ) denote the space of all h-harmonics of degree n, and let projκn : L2 (h2κ ; Sd−1 ) → Hnd (h2κ ) denote the orthogonal projection operator. The projection projκn has an integral representation projκn f (x) :=
f (y)Pnκ (x, y)h2κ (y) dσ (y),
x ∈ Sd−1 ,
(2.8)
Sd−1
where Pnκ (x, y) is the reproducing kernel of Hnd (h2κ ) which can be written in terms of the interwining operator Vκ as (see [20, Theorem 3.2, (3.1)]) Pnκ (x, y) =
n + λ k λ k
Vκ Cn x, · (y), λκ
x, y ∈ Sd−1 ,
(2.9)
λ with λ κ := λκ − 12 = γκ + d−2 2 . Here Cn denotes the standard Gegenbauer polynomial of degree n and index λ as defined in [16]. By means of (2.8) and (2.9), the projection projκn f can be extended to all f ∈ L1 (h2κ ; Sd−1 ).
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
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The following Marcinkiewicz type multiplier theorem was proved recently in [2, Theorem 2.3]: Theorem 2.3. Let {μj }∞ j =0 be a sequence of real numbers that satisfies (i) supj |μj | c < ∞, j +1 (ii) supj 1 2j (r−1) 2l=2j | r μl | c < ∞, with r being the smallest integer
d 2
+ γκ ,
where μl = μl − μl+1 and j +1 μl = j μl − j μl+1 . Then {μj } defines an Lp (h2κ ; Sd−1 ) multiplier for all 1 < p < ∞; that is, ∞ μj projκj f j =0
Ap cf κ,p ,
1 < p < ∞,
κ,p
where Ap is independent of {μj } and f . When κ = 0, the theorem becomes part (1) of [1, Theorem 4.9] on the ordinary spherical harmonic expansions. For δ > −1, the Cesàro (C, δ) means of the h-harmonic expansion are defined by n
−1 Snδ h2κ ; f, x := Aδn Aδn−k projκk f (x),
Aδn−k =
k=0
In the case when G = Zd2 and hκ (x) = recently in [3]:
d
κ(ei ) , i=1 |x, ei |
the following result was proved
Theorem 2.4. Let G = Zd2 and let 1 p ∞ satisfy | p1 − 12 | σκ :=
d−2 2
n−k+δ . n−k
1 2σκ +2
with
+ γκ − min κ(ei ). 1id
Then
supSnδ h2κ ; f κ,p cf κ,p ,
n∈N
for all f ∈ Lp h2κ ; Sd−1
if and only if 1 1 1 δ > δκ (p) := max (2σκ + 1) − − , 0 . p 2 2
(2.10)
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3. A transference theorem The main goal in this section is to establish a transference theorem between the Lp multipliers of h-harmonic expansions on the unit sphere Sd := {x ∈ Rd+1 : x = 1} and those of the Dunkl transform in Rd . Let G, R, hκ be as defined in Section 1. Given g ∈ G, we denote by g the orthogonal transformation on Rd+1 determined uniquely by g x = (gx, xd+1 )
for x = (x, xd+1 ) with x ∈ Rd and xd+1 ∈ R.
Then G := {g : g ∈ G} is a finite reflection group on Rd+1 with a reduced root system R := {(α, 0): α ∈ R}. Let κ denote the nonnegative multiplicity function defined on R with the property κ (α, 0) = κ(α). We denote by Vκ the intertwining operator on C(Rd+1 ) associated with the reflection group G and the multiplicity function κ . Define the weight function hκ (x, xd+1 ) := hκ (x) =
x, ακ(α) ,
x ∈ Rd , xd+1 ∈ R.
α∈R+
Recall that projκn : L2 (Sd ; h2κ ) → Hnd+1 (h2κ ) denotes the orthogonal projection onto the space of h-harmonics. Our main result is the following. Theorem 3.1. Let m : [0, ∞) → R be a continuous and bounded function, and let Uε , ε > 0, be a family of operators on L2 (Sd ; h2κ ) given by
projκn (Uε f ) = m(εn) projκn f,
n = 0, 1, . . . .
(3.1)
Assume that sup Uε f Lp (Sd ;h2 ) Af Lp (Sd ;h2 ) , ε>0
κ
κ
∀f ∈ C Sd ,
(3.2)
for some 1 p ∞. Then the function m( · ) defines an Lp (Rd ; h2κ ) multiplier; that is, Tm f Lp (Rd ;h2κ ) cd,κ Af Lp (Rd ;h2κ ) ,
∀f ∈ S Rd ,
where Tm is an operator initially defined on L2 (Rd ; h2κ ) by
Fκ (Tm f )(ξ ) = m ξ Fκ f (ξ ),
f ∈ L2 Rd ; h2κ , ξ ∈ Rd .
(3.3)
In the case of ordinary spherical harmonics (i.e., κ = 0), Theorem 3.1 is due to Bonami and Clerc [1, Theorem 1.1].
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3.1. Lemmas The proof of Theorem 3.1 relies on several lemmas. Lemma 3.2. If f ∈ Π d+1 then for any x ∈ Rd and xd+1 ∈ R,
Vκ f (x, xd+1 ) = Vκ f (·, xd+1 ) (x) =
f (ξ, xd+1 ) dμκx (ξ ),
(3.4)
Rd
where dμκx is given in (2.3). Proof. Clearly, the second equality in (3.4) follows directly from (2.3). To show the first equalκ f (x, xd+1 ) = Vκ [f (·, xd+1 )](x) for f ∈ C(Rd+1 ) and x ∈ Rd . Since Vκ is a linear ity, we set V operator uniquely determined by (2.2), it suffices to show that the following conditions are satisfied:
κ Pd+1 ⊂ Pd+1 V n n ,
κ (1) = 1, V
κ ∂i , κ = V and Dκ ,i V
1 i d + 1.
Indeed, these conditions can be easily verified using the properties of Vκ in (2.2), and the following identities, which follow directly from (2.1):
Dκ ,i g(x, xd+1 ) = Dκ,i g(·, xd+1 ) (x), Dκ ,d+1 g(x, xd+1 ) = ∂d+1 g(x, xd+1 ),
1 i d,
for g ∈ Π d+1 , x ∈ Rd and xd+1 ∈ R.
2
This completes the proof of Lemma 3.2.
To formulate the next lemma, we define the mapping ψ : Rd → Sd by
ψ(x) := ξ sinx, cosx for x = xξ ∈ Rd and ξ ∈ Sd−1 . Given N 1, we denote by NSd := {x ∈ Rd+1 : x = N } the sphere of radius N in Rd+1 , and define the mapping ψN : Rd → N Sd by ψN (x) := N ψ
x N
x x = N ξ sin , N cos N N
(3.5)
with x = xξ ∈ Rd and ξ ∈ Sd−1 . Lemma 3.3. If f : NSd → R is supported in the set {x ∈ N Sd : arccos(N −1 xd+1 ) 1}, then
f (Nx)h2κ (x) dσ (x) = N −2λκ −1
Sd
where B(0, N) = {y ∈ Rd : y N }.
B(0,N )
sin(x/N ) 2λκ f ψN (x) h2κ (x) dx, x/N
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Proof. First, using the polar coordinate transformation (ξ, θ ) ∈ Sd−1 × [0, π] → x := (ξ sin θ, cos θ ) ∈ Sd , and the fact that dσ (x) = sind−1 θ dθ dσ (ξ ), we obtain π
f (N x)h2κ (x) dσ (x) =
f (N ξ sin θ, N
Sd
0
cos θ )h2κ (ξ
sin θ, cos θ ) dσ (ξ ) (sin θ )d−1 dθ
Sd−1
1 =
sin θ d−1+2γκ d−1 f (N ξ sin θ, N cos θ )h2κ (θ ξ ) dσ (ξ ) θ dθ, θ
0 Sd−1
where the last step uses the identity hκ (y, yd+1 ) = hκ (y), the fact that h2κ is a homogeneous function of degree 2γκ , and the assumption that f is supported in the set {x ∈ N Sd : arccos(N −1 xd+1 ) 1}. Using the usual spherical coordinate transformation in Rd , the last double integral equals Ny siny siny 2λκ 2 , N cosy hκ (y) f dy y y
y1
= N −d−2γκ
xN
= N −2λκ −1
B(0,N )
x x 2 x sin(x/N ) 2λκ sin , N cos hκ (x) f N dx x N N x/N sin(x/N ) 2λκ f (ψN x)h2κ (x) dx, x/N
where the first step uses the homogeneity of the weight hκ and the change of variables y = This proves the desired formula. 2
x N.
Remark 3.1. It is easily seen that the restriction ψN |B(0,N ) of the mapping ψN on B(0, N ) is a bijection from B(0, N ) to {x ∈ N Sd : arccos(N −1 xd+1 ) 1}. Thus, given a function f : B(0, N) → R, there exists a unique function fN supported in {x ∈ N Sd : arccos(N −1 xd+1 ) 1} such that fN (ψN x) = f (x),
∀x ∈ B(0, N ).
(3.6)
On the other hand, using Lemma 3.3, we have Sd
fN (Nx)h2κ (x) dσ (x) = N −2λκ −1
B(0,N )
f (x)h2κ (x)
sin(x/N ) x/N
The formula (3.7) will play an important role in our proof of Theorem 3.1.
2λκ dx.
(3.7)
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4061
We also need a small observation on a formula of Rösler [14] for τy f (x): Lemma 3.4. If f (x) = f0 (x) is a continuous radial function in L2 (Rd ; h2κ ), then for a.e. y ∈ Rd and a.e. x ∈ Rd , y 2 2 . τy f (x) = Vκ f0 x + y − 2yx, · y
(3.8)
Formula (3.8) was first proved in [14] under the assumption that f is a radial Schwartz function. Thangavelu and Yuan Xu [17, Proposition 3.3] later observed that it also holds for radial functions f ∈ L(Rd ; h2κ ) with Fκ f ∈ L(Rd ; h2κ ). Clearly, our assumption in Lemma 3.4 is slightly weaker than that of [17, Proposition 3.3]. Lemma 3.4 can be deduced from the result of Rösler [14], using a density argument. Proof. We first choose a sequence of even, C ∞ functions gj on R satisfying sup gj (t) − f0 (t) 2−j
− 1
2j
|t|2j +1
2
s 2λκ ds
.
0
Let ϕj be an even, C ∞ function on R such that χ[2−j ,2j ] (|t|) ϕj (t) χ[2−j −1 ,2j +1 ] (|t|), and let fj (x) ≡ fj,0 (x) := gj (x)ϕj (x) for x ∈ Rd . Then it’s easily seen that {fj } is a sequence of radial Schwartz functions on Rd satisfying lim
sup
j →∞ 2−j |t|2j
fj,0 (t) − f0 (t) = 0
(3.9)
and lim fj − f κ,2 = 0.
j →∞
(3.10)
Since each fj is a radial Schwartz function, by Lemma 2.1 and the already proven case of Lemma 3.4 (see [14]), we obtain
τy (fj )(x) = fj,0 x2 + y2 − 2yx, ξ dμκy/y (ξ ). (3.11) ξ 1
Next, we fix y ∈ Rd , and set An ≡ An (y) := x ∈ Rd : 2−n x − y x + y 2n for n ∈ N and n n0 (y) := [log y/ log 2] + 1. Since
2
2 x − y x2 + y2 − 2yx, ξ x + y for all ξ 1, it follows by (3.9) that
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lim fj,0
j →∞
x2 + y2 − 2yx, ξ = f0 x2 + y2 − 2yx, ξ
uniformly for x ∈ An (y) and ξ 1. This together with (3.11) and Lemma 2.1 implies lim τy (fj )(x) =
j →∞
f0
x2 + y2 − 2yx, ξ dμκy/y (ξ )
ξ 1
y 2 2 = Vκ f0 x + y − 2yx, · y for every x ∈ An (y) \ {0} and n n0 (y). On the other hand, however, by (3.10), we have lim τy (fj ) − τy f κ,2 = 0
j →∞
for all y ∈ Rd . Thus, y 2 2 τy (f )(x) = Vκ f0 x + y − 2yx, · y for a.e. x ∈ An (y) and all n n0 (y). Finally, observing that the set R \ d
∞
An (y) = x ∈ Rd : x = y
n=n0 (y)
has measure zero in Rd , we deduce the desired conclusion.
2
Remark 3.2. By (2.4) and the supporting condition of the measure dμκx , we observe that Vκ F (rx) =
F (rξ ) dμκx (ξ ),
for all F ∈ C Rd , x ∈ Rd , and r > 0.
(3.12)
Rd
Thus, (3.8) can be rewritten more symmetrically as τy f (x) = Vκ f0 x2 + y2 − 2x, · (y).
(3.13)
Lemma 3.5. Let Φ ∈ L1 (R, |x|2λκ ) be an even, bounded function on R, and let TΦ be an operator L2 (Rd ; h2κ ) → L2 (Rd ; h2κ ) defined by
Fκ (TΦ f )(ξ ) := Fκ f (ξ )Φ ξ , Then TΦ has an integral representation TΦ f (x) = f (y)K(x, y)h2κ (y) dy, Rd
f ∈ L2 Rd ; h2κ .
for f ∈ S Rd and a.e. x ∈ Rd ,
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
4063
where
K(x, y) = c
Φ(s)Vκ 0
(s x2 + y2 − 2x, · ) (y)s 2λκ ds. λκ − 12 2 2 (s x + y − 2x, · )
J
∞
λκ − 12
(3.14)
Furthermore, K(x, y) = K(y, x) for a.e. x ∈ Rd and a.e. y ∈ Rd . Proof. Let g(x) = Hλκ − 1 Φ(x), where x ∈ Rd and Hα denotes the Hankel transform. Since Φ 2
is an even function in L1 (R, |x|2λκ )∩L∞ (R), it follows by the properties of the Hankel transform that g is a continuous radial function in L2 (Rd ; h2κ ) and Fκ g(ξ ) = Φ(ξ ). Thus, using (2.7), we have TΦ f (x) = f ∗κ g(x) =
f (y)τy g(x)h2κ (y) dy Rd
for f ∈ L2 (Rd ; h2κ ). Since g is a continuous radial function in L2 (Rd ; h2κ ), by Lemma 3.4 and Remark 3.2 it follows that
K(x, y) := τy g(x) = Vκ Hλκ − 1 Φ x2 + y2 + 2x, · (y) 2 J ∞ 2 2 λκ − 12 (s x + y − 2x, · ) = c Φ(s)Vκ (y)s 2λκ ds, λκ − 12 2 2 (s x + y − 2x, · ) 0
where the last step uses (2.3), the inequality J 1 (rs) Φ(s) λκ − 2 cΦ(s) λκ − 12 (rs) and Fubini’s theorem. This proves the desired equation (3.14). That K(x, y) = K(y, x) follows from the fact that τx g(y) = τy g(x). 2 Our final lemma is a well-known result for the ultraspherical polynomials: Lemma 3.6. (See [16, (8.1.1), p. 192].) For z ∈ C and μ 0, 1 Γ (μ + 12 ) z −μ+ 2 z μ Jμ− 1 (z). lim k 1−2μ Ck cos = 2 k→∞ k Γ (2μ) 2
This formula holds uniformly in every bounded region of the complex z-plane.
(3.15)
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3.2. Proof of Theorem 3.1 We follow the idea of the proof of Theorem 1.1 of [1]. We first prove the theorem under the additional assumption |m(t)| c1 e−c2 t for all t > 0 and some c1 , c2 > 0. By Lemma 3.5, the operator Tm has an integral representation Tm f (x) =
f (y)K(x, y)h2κ (y) dy, Rd
where K(x, y) is given by (3.14) with Φ = m. Thus, it is sufficient to prove that I := f (y)g(x)K(x, y)h2κ (x)h2κ (y) dx dy cA
(3.16)
Rd Rd
whenever f ∈ Lp (Rd ; h2κ ) and g ∈ Lp (Rd ; h2κ ) both have compact supports and satisfy f Lp (Rd ;h2κ ) = gLp (Rd ;h2 ) = 1. κ To this end, we choose a positive number N to be sufficiently large so that the supports of f and g are both contained in the ball B(0, N ). By Remark 3.1, there exist functions fN and gN both supported in {x ∈ N Sd : arccos(N −1 xd+1 ) 1} and satisfying
fN ψN (x) = f (x),
gN ψN (x) = g(x),
x ∈ Rd ,
(3.17)
where ψN is defined by (3.5). It’s easily seen from (3.7) that 2λκ +1 fN (N·) p d 2 cN − p , L (S ;h )
gN (N ·)
κ
Lp (Sd ;h2κ )
Thus, using (2.8), (2.9), (3.1) and the assumption (3.2) with ε =
IN := N
1 N,
cN
− 2λpκ +1
we obtain
∞
κ −1 2 2 m N n Pn (x, y) fN (Ny)gN (N x)hκ (x)hκ (y) dσ (x) dσ (y)
2λκ +1
Sd Sd
n=0
cA,
.
(3.18)
where Pnκ (x, y) =
λκ n+λκ λκ Vκ [Cn (x, ·)](y).
HN (x, y) = N
−2λκ −1
Setting
∞
−1 κ x y ,ψ , m N n Pn ψ N N n=0
and invoking (3.17) and Lemma 3.3, we obtain
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
4065
sin(x/N ) 2λκ 2 2 IN = HN (x, y)f (y)g(x)hκ (x)hκ (y) x/N Rd Rd
sin(y/N) × y/N
2λκ
dx dy .
(3.19)
On the other hand, setting −1 N ∞ x y n −2λκ −1 κ 2λκ bN (ρ, x, y) = N Pn ψ ,ψ m t dt χ[ n , n+1 ) (ρ), N N N N N n+1
n=0
n N
we have ∞ HN (x, y) =
bN (ρ, x, y)ρ 2λκ dρ. 0
Hence, by (3.19), ∞ 2λκ IN = bN (ρ, x, y)ρ dρ f (y)g(x)h2κ (x)h2κ (y) Rd Rd
0
sin(x/N) × x/N
2λκ
sin(y/N ) y/N
2λκ
dx dy .
(3.20)
The key ingredient in our proof is to show that limN →∞ IN = cI , where c is a constant depending only on d and κ. In fact, once this is proven, then the desired estimate (3.16) will follow immediately from (3.18). To show limN →∞ IN = cI , we make the following two assertions: Assertion 1. For any N > 0 and x, y ∈ Rd , bN (ρ, x, y) ce−c2 ρ , where c is independent of x, y and N . Assertion 2. For any fixed x, y ∈ Rd and ρ > 0, lim bN (ρ, x, y) = cm(ρ)Vκ
N →∞
where u(x, y, ξ ) =
J
λκ − 12 (ρu(x, y, ·)) 1
(ρu(x, y, ·))λκ − 2
(y),
(3.21)
x2 + y2 − 2x, ξ , and c is a constant depending only on d and κ.
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For the moment, we take the above two assertions for granted, and proceed with the proof of Theorem 3.1. By Assertion 1 and Hölder’s inequality, we can apply the dominated convergence theorem to the integrals in (3.20), and obtain ∞ 2λκ 2 2 lim IN = lim bN (ρ, x, y)ρ dρ f (y)g(x)hκ (x)hκ (y) dx dy , N →∞ N →∞ Rd Rd
0
which, using Assertion 2, equals ∞ J λκ − 12 (ρu(x, y, ·)) 2λκ 2 2 (y)ρ m(ρ)Vκ dρ f (y)g(x)h (x)h (y) dx dy = c κ κ λκ − 12 (ρu(x, y, ·)) Rd Rd
0
2 2 = c K(x, y)f (y)g(x)hκ (x)hκ (y) dx dy = cI, Rd Rd
where the second step uses (3.14). Thus, we have shown the desired relation limN →∞ IN = cI , assuming Assertions 1 and 2. Now we return to the proofs of Assertions 1 and 2. We start with the proof of Assern −c2 Nn tion 1. Assume that Nn ρ < n+1 ce−c2 ρ , and N for some n ∈ Z+ . Then |m( N )| c1 e n+1 N t 2λκ dt cN −1 ρ 2λκ . Hence, n N
−1 N n x y −2λ −1 κ 2λ bN (ρ, x, y) = N κ m Pn ψ ,ψ t κ dt N N N n+1
n N
cN
−2λκ −2λκ −c2 ρ n + λκ
ρ
e
λκ
y Vκ C λκ ψ x , · ψ n N N
c(Nρ)−2λκ e−c2 ρ n2λκ ce−c2 ρ , where we used (2.9) in the second step, and the positivity of Vκ and the estimate |Cnλκ (t)| cn2λκ −1 in the third step. This proves Assertion 1. Next, we show Assertion 2. A straightforward calculation shows that for Nn ρ n+1 N and ρ > 0, n+1 N
−1 t
n N
2λκ
dt
=
N
1 + oρ (1) , 2λ ρ κ
as N → ∞.
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
This implies that for
n N
ρ
n+1 N
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and ρ > 0,
x y n2λκ −2λκ κ bN (ρ, x, y) = m(ρ) ,ψ 1 + oρ (1) n Pn ψ 2λ κ N N (Nρ) x y y y ,· sin , cos + oρ (1), = cm(ρ)n−2λκ +1 Vκ Cnλκ ψ N y N N
where we used the continuity of m in the first step, and the estimate n−2λκ |Pnκ (ψ( Nx ), ψ( Ny ))| c, as well as the fact that limN →∞ ing Lemma 3.2 and (3.12), we obtain bN (ρ, x, y) = cm(ρ)n
−2λκ +1
Cnλκ
y y sin N
= cm(ρ)n
(Nρ)2λκ
= 1 in the last step (see [4]). Thus, us-
x x y 1 sin cos xj ξj + cos x N N N d
j =1
Rd
× dμκ y
n2λκ
(ξ ) + oρ (1)
−2λκ +1
Cnλκ cos θN (x, y, ξ ) dμκy (ξ ) + oρ (1),
(3.22)
ξ y
where θN (x, y, ξ ) ∈ [0, π] satisfies cos θN (x, y, ξ ) =
d 1 y x y x sin + cos cos . xj ξj sin xy N N N N j =1
Since d
1 2 2 xj ξj + Ox,y N −4 cos θN (x, y, ξ ) = 1 − x + y − 2 2 2N j =1
=1−
1 u(x, y, ξ )2 + Ox,y N −4 , 2N 2
it follows that
1 2 −2 θN (x, y, ξ ) = 2 arcsin u(x, y, ξ ) + Ox,y N 2N
1 u(x, y, ξ )2 + Ox,y N −2 + Ox,y N −2 = N ρu(x, y, ξ ) + ox,y,ρ (1) , = n √ where the last step uses the uniform continuity of the function t ∈ [0, M] → t for any M > 0, n = 1. and the relation limN →∞ Nρ
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F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
Thus, by (3.22) and (3.15), we have lim bN (ρ, x, y) = cm(ρ) lim
N →∞
N →∞ ξ y
ρu(x, y, ξ ) + ox,y,ρ (1) n−2λκ +1 Cnλκ cos dμκy (ξ ) n
−λ + 1
ρu(x, y, ξ ) κ 2 Jλκ − 1 ρu(x, y, ξ ) dμκy (ξ )
= cm(ρ)
2
ξ y
−λ + 1
= cm(ρ)Vκ ρu(x, y, ·) κ 2 Jλκ − 1 ρu(x, y, ·) (y), 2
where we used the fact that Cnλκ ∞ cn2λκ −1 , the bounded convergence theorem and (3.15) in the last step. This proves Assertion 2. In summary, we have shown the theorem with the additional assumption |m(t)| c1 e−c2 t . Finally, we prove that the conclusion of Theorem 3.1 remains true without the additional assumption |m(t)| c1 e−c2 t . To this end, let mδ (t) = m(t)e−δt for δ > 0, and define Tmδ : L2 (Rd , h2κ ) → L2 (Rd ; h2κ ) by
f ∈ L2 Rd ; h2κ .
Fκ (Tmδ f )(ξ ) = mδ (ξ )Fκ f (ξ ), It is known (see [8, p. 191]) that given any ε > 0, f → on Lp (Sd ; h2κ ) that satisfies ∞ −nε κ sup e projn f ε>0 n=0
2) Lp (Sd ;h κ
∞
n=0 e
κ −nε proj nf
is a positive operator
f Lp (Sd ;h2 ) . κ
Indeed, this follows from [6, Theorem 4.2] and the fact that Vκ is positive, which was proved in [13]. Thus, applying Theorem 3.1 for the already proven case, we have supTmδ f Lp (Rd ;h2κ ) cAf Lp (Rd ;h2κ ) .
(3.23)
δ>0
On the other hand, from the definition we can decompose the operator Tmδ as Tmδ f = Pδ (Tf ),
(3.24)
where Fκ (Tf )(ξ ) = m(ξ )Fκ f (ξ ) and Fκ (Pδ f )(ξ ) = e−δξ Fκ f (ξ ). The function Pδ f is called the Poisson integral of f , and it can be expressed as a generalized convolution (see [3]) Pδ f (x) := (f ∗κ Pδ )(x)
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
4069
with d
Pδ (x) := 2γκ + 2
Γ (γκ + d+1 ) δ . √ 2 d+1 2 π (δ + x2 )γκ + 2
It was shown in [3, Theorem 6.2] that lim Pδ f (x) = f (x),
δ→0+
a.e. x ∈ Rd
for any f ∈ Lq (Rd ; h2κ ) with 1 q < ∞. Since m is bounded, Tf ∈ L2 (Rd ; h2κ ) for f ∈ L2 (Rd ; h2κ ). Thus, for any f ∈ S, using (3.24), lim Tmδ f (x) = lim Pδ (Tf )(x) = Tf (x),
δ→0+
δ→0+
a.e. x ∈ Rd ,
(3.25)
which combined with (3.23) and the Fatou theorem implies the desired estimate Tf Lp (Rd ;h2κ ) cAf Lp (Rd ;h2κ ) . This completes the proof of the theorem. 4. Applications 4.1. Hörmander’s multiplier theorem and the Littlewood–Paley inequality As a first application of Theorem 3.1, we shall prove the following Hörmander type multiplier theorem: Theorem 4.1. Let m : (0, ∞) → R be a bounded function satisfying m∞ A and Hörmander’s condition 1 R
2R (r) m (t) dt AR −r ,
for all R > 0,
R
where r is the smallest integer λκ + 1. Let Tm be an operator on L2 (Rd ; h2κ ) defined by
Fκ (Tm f )(ξ ) = m ξ Fκ f (ξ ), Then Tm f κ,p Cp Af κ,p for all 1 < p < ∞ and f ∈ S(Rd ).
ξ ∈ Rd .
(4.1)
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Proof. Let μ = m(ε) for ε > 0 and = 0, 1, . . . . Then r μ = ε r
m (εt1 + · · · + εtr + ε) dt1 · · · dtr (r)
[0,1]r
(r) m (t1 + · · · + tr + ε) dt1 · · · dtr ε r−1
[0,ε]r
ε(r+)
(r) m (t) dt.
ε
This implies that for 2j r,
2
j (r−1)
j +1 2
2 r μl 2j (r−1) ε r−1
j +1
l=2j
l=2j
ε(r+)
(r) m (t) dt
ε j +1 +r) ε(2
(r − 1)2
(r) m (t) dt
j (r−1) r−1
ε
2j ε 2j +2 ε
2
j (r−1)
(r − 1)ε
(r) m (t) dt cr A,
r−1 2j ε
where the last step uses (4.1). On the other hand, however, for 2j r, we have
2
j (r−1)
j +1 2
r μl cr max|μj | cr A.
l=2j
j
Thus, using Theorem 2.3, we deduce ∞ κ sup m(εn) projn f ε>0 n=0
Lp (Sd ;h2κ )
The desired conclusion then follows by Theorem 3.1.
cf Lp (Sd ;h2 ) . κ
2
Remark 4.1. Hörmander’s condition is normally stated in the following form
1 R
1 2R 2 (r) 2 m (t) dt AR −r ,
for all R > 0.
(4.2)
R
See, for instance, [10, Theorem 5.2.7]. Clearly, the condition (4.1) in Theorem 4.1 is weaker than (4.2). On the other hand, however, Theorem 4.1 is applicable only to radial multiplier m( · ).
F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
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Corollary 4.2. Let Φ be an even C ∞ -function that is supported in the set {x ∈ R: and satisfies either
Φ 2−j ξ = 1,
9 10
|x|
21 10 }
ξ ∈ R \ {0},
j ∈Z
or
Φ 2−j ξ 2 = 1,
ξ ∈ R \ {0}.
j ∈Z
Let j be an operator defined by
Fκ ( j f )(ξ ) = Φ 2−j ξ Fκ f (ξ ),
ξ ∈ Rd .
Then we have 1 2 2 | j f | f κ,p ∼κ,p j ∈Z
κ,p
holds for all f ∈ Lp (Rd ; h2κ ) and 1 < p < ∞. Proof. Corollary 4.2 follows directly from Theorem 4.1. Since the proof is quite standard (see, for instance, [15]), we omit the details. 2 4.2. The Bochner–Riesz means Given δ > −1, the Bochner–Riesz means of order δ for the Dunkl transform are defined by BRδ
2 hκ ; f (x) = c
yR
y2 1− 2 R
δ Fκ f (y)Eκ (ix, y)h2κ (y) dy,
R > 0.
(4.3)
Convergence of the Bochner–Riesz means in the setting of Dunkl transform was studied recently by Thangavelu and Yuan Xu [17, Theorem 5.5], who proved that if δ > λκ := d−1 2 + γκ and 1 p ∞ then
sup BRδ h2κ ; f κ,p cf κ,p .
(4.4)
R>0
Our next result concerns the critical indices for the validity of (4.4) in the case of G = Zd2 : Theorem 4.3. Suppose that G = Zd2 , f ∈ Lp (Rd ; h2κ ), 1 p ∞, and | p1 − 12 | (4.4) holds if and only if 1 1 1 δ > δκ (p) := max (2λκ + 1) − − , 0 . p 2 2
1 2λκ +2 .
Then
(4.5)
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F. Dai, H. Wang / Journal of Functional Analysis 258 (2010) 4052–4074
It should be pointed out that the result of [17, Theorem 5.5] is applicable to the case of a general finite reflection group G, while Theorem 4.3 above applies to the case of Zd2 only. Proof. We start with the proof of the sufficiency. Assume that κ := (κ1 , . . . , κd ) and hκ (x) := d κj d j =1 |xj | . Let κ = (κ, 0) and hκ (x, xd+1 ) = hκ (x) for x ∈ R and xd+1 ∈ R. Set m(t) = 2 δ (1 − t )+ . By the equivalence of the Riesz and the Cesàro summability methods of order δ 0 (see [9]), we deduce from Theorem 2.4 ∞ m(εn) projκn f sup ε>0 n=0
Lp (Sd ;h2κ )
cf Lp (Sd ;h2
κ
)
whenever | p1 − 12 | 2σ 1 +2 and δ > δκ (p), where σκ = λκ and δκ (p) = δκ (p). Thus, invoking κ Theorem 3.1, we conclude that for δ > δκ (p), δ 2 B h ; f κ
1
κ,p
cf κ,p .
The estimate (4.4) then follows by dilation. This proves the sufficiency. The necessity part of the theorem follows from the corresponding result for the Hankel transform. To see this, let f (x) = f0 (x) be a radial function in Lp (Rd , h2κ ). Using (4.3) and Lemma 2.2(vii), we have (vii), we have
BRδ
2 hκ ; f (x) =
R
r2 δ 1 − 2 Hλκ − 1 f0 (r)r 2λκ Eκ ix, ry h2κ y dσ y dr. 2 R Sd−1
0
However, by [17, Proposition 2.3] applied to n = 0 and g = 1, we have
1
rx −λκ + 2 2 Eκ ix, ry hκ y dσ y = c Jλκ − 1 rx . 2 2
Sd−1
It follows that
BRδ
2 hκ ; f (x) = c
R
r2 1− 2 R
δ
1
rx −λκ + 2 Hλκ − 1 f0 (r) Jλκ − 1 rx r 2λκ dr 2 2 2
0
Rδ f0 x , = cB δ denotes the Bockner–Riesz mean of order δ for the Hankel transform H where B R λκ − 12 . However, δ p 2λ , 0 < δ < λκ , is bounded on L ((0, ∞), t κ ) if and only if it is known (see [19]) that B R
2λκ + 1 2λκ + 1
(4.6)
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4073
Thus, to complete the proof of the necessity part of the theorem, by (4.6), we just need to observe that if f (x) = f0 (x) is a radial function in Lp (Rd ; h2κ ), then f κ,p = cf0 Lp (R;|x|2λκ ) .
2
Remark 4.2. In the unweighted case, for the classical Fourier transform, Theorem 4.3 is well known, and in fact, it follows from the following Tomas–Stein restriction theorem (see, for instance, [10, Section 10.4]): f L2 (Sd−1 ) Cp f Lp (Rd ) ,
1p
2d + 2 , d +3
(4.7)
where f denotes the usual Fourier transform of f . In the weighted case, while estimates similar to (4.7) can be proved for the Dunkl transform Fκ f (see [12, Theorem 4.1]), they do not seem to be enough for the proof of Theorem 4.3. A similar fact was indicated in [3] for the case of the Cesàro means for h-harmonic expansions on the unit sphere, where global estimates for the projection operators have to be replaced with more delicate local estimates, which are significantly more difficult to prove than the corresponding global estimates. Acknowledgments The authors would like to thank the referee for many valuable comments and suggestions, which help to improve the current version of our paper. References [1] A. Bonami, J.-L. Clerc, Sommes de Cesàro et multiplicateurs des développe-ments en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973) 223–263. [2] Feng Dai, Yuan Xu, Maximal function and multiplier theorem for weighted space on the unit sphere, J. Funct. Anal. 249 (2) (2007) 477–504. [3] Feng Dai, Yuan Xu, Boundedness of projection operators and Cesàro means in weighted Lp space on the unit sphere, Trans. Amer. Math. Soc. 361 (6) (2009) 3189–3221. [4] Feng Dai, Yuan Xu, Cesàro means of orthogonal expansions in several variables, Constr. Approx. 29 (1) (2009) 129–155. [5] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) 167–183. [6] C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (6) (1991) 1213–1227. [7] C.F. Dunkl, Hankel transforms associated to finite reflection groups, in: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Tampa, FL, 1991, in: Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123–138. [8] C.F. Dunkl, Yuan Xu, Orthogonal Polynomials of Several Variables, Cambridge Univ. Press, 2001. [9] J.J. Gergen, Summability of double Fourier series, Duke Math. J. 3 (2) (1937) 133–148. [10] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. [11] M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993) 147–162. [12] L.M. Liu, Harmonic analysis associated with finite reflection groups, PhD thesis, 2004, Capital Normal University, Beijing, China. [13] M. Rösler, Positivity of Dunkl intertwining operator, Duke Math. J. 98 (1999) 445–463. [14] M. Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003) 2413–2438. [15] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton University Press, Princeton, NJ, 1970. [16] G. Szegö, Orthogonal Polynomials, 4th edition, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.
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[17] S. Thangavelu, Yuan Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005) 25–55. [18] S. Thangavelu, Yuan Xu, Riesz transform and Riesz potentials for Dunkl transform, J. Comput. Appl. Math. 199 (1) (2007) 181–195. [19] G.V. Welland, Norm convergence of Riesz–Bochner means for radial functions, Canad. J. Math. 27 (1) (1975) 176–185. [20] Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997) 2963–2973. [21] Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997) 175–192.
Journal of Functional Analysis 258 (2010) 4075–4121 www.elsevier.com/locate/jfa
Matricially free random variables Romuald Lenczewski Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland Received 9 September 2009; accepted 11 March 2010 Available online 29 March 2010 Communicated by D. Voiculescu
Abstract We show that the framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a concept which not only leads to a natural generalization of freeness, but also underlies other fundamental types of noncommutative independence, such as monotone independence and boolean independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws. © 2010 Elsevier Inc. All rights reserved. Keywords: Free probability; Matricial freeness; Strong matricial freeness; Matricially free random variable; Matricially free Fock space; Random matrix; Random pseudomatrix
1. Introduction It has been shown by Voiculescu [25] that free random variables arise naturally as limits of random matrices. In particular, if we take symmetric matrices whose entries form a family of independent Gaussian random variables and we let the size of these matrices go to infinity, their joint distribution (with respect to normalized trace composed with classical expectation) converges to the joint distribution of freely independent random variables with the semicircle distribution obtained by Wigner [29] as the limit distribution of one Gaussian random matrix. Therefore, we can study free random variables, using at least two different frameworks: operator algebras and random matrices. However, it is to some extent surprising that a random E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.010
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matrix framework, of quite different nature than that of operator algebras, exists for free random variables, and the connection between these two approaches does not seem to be very transparent. In this connection, our first motivation is to better understand the relation between the operatorial approach to free probability and random matrices. The second motivation comes from the question whether different types of independence, like freeness of Voiculescu, monotone independence of Muraki [19] and boolean independence based on the regular free product of Bo˙zejko [5], can be included in one natural framework. Note in this context that models with more than one state on a given algebra, like conditional freeness of Bo˙zejko and Speicher [6] and freeness with infinitely many states of Cabanal-Duvillard and Ionesco [8,9] extend free probability and include, as shown by Franz [12], certain elements of monotone probability. For instance, this can be done for convolutions, but including monotone independence in the framework of conditional freeness can be done only under additional (rather restrictive) assumptions on the considered algebras. We show in this paper that one can remedy this situation by introducing a concept of ‘independence’ which reminds freeness, but at the same time has some ‘matricial’ features which places it somewhere between freeness and the model of random matrices. A different reason to look for a new concept of independence arises from concrete examples of interpolations between free probability and monotone probability [17,18]. In particular, the continuous (p, q)-Brownian motions with Kesten distributions and related Poisson processes studied in [2,18] lead to the first example of a two-mode interacting Fock space introduced directly and not by means of orthogonal polynomials. This example has certain matricial features which also call for a new model of ‘independence’ that would be related to freeness. The main result of our paper is the construction of a model, called matricial freeness, which is related to the concept of the free product of states introduced and studied by Ching [10] in the context of von Neumann algebras and by Avitzour [3] and Voiculescu [24] in the context of C ∗ -algebras. The underlying concept is that of the matricially free product of an array of Hilbert spaces with distinguished unit vectors (H, ξ ) = ∗M i,j (Hi,j , ξi,j ) which reminds the free product of Hilbert spaces, to which we associate the matricially free product of states. Multiplication of ‘matricially free random variables’ under this product of states reminds the free product but it also satisfies the condition imitating matrix multiplication. Similarities between free probability and ‘matricially free probability’ hold also on other levels, some of which are studied in this paper. Strictly speaking, however, one needs to take a ‘restriction’ of matricial freeness, in which not all products imitating matrix multiplication are included, called strong matricial freeness, to recover freeness and monotone independence without using the asymptotics. In particular, let (Xi,j ) be a finite array of strongly matricially free random variables from some unital algebra A which includes the diagonal. Then (A) the sums corresponding to the rows of a square array, Ai :=
j
are free with respect to ϕ,
Xi,j
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(B) the sums corresponding to the rows of a lower-triangular array, Bi :=
Xi,j
j i
are monotone independent with respect to ϕ, (C) the diagonal variables Xj,j , j ∈ I , are boolean independent with respect to ϕ, where ϕ is a distinguished state on A which lies everywhere on the diagonal of the associated array of states (ϕi,j ). In the case of triangular arrays, we tacitly assume that the index sets involved are linearly ordered. Let us add that upper triangular arrays give anti-monotone random variables. In this paper, of main interest to us are limit theorems, in which we consider a sequence (Xi,j (n))1i,j n of matricially free arrays of variables taken from unital ∗-algebras A(n), respectively, equipped with distinguished states φ(n) and associated states (φj (n))1j n called ‘conditions’. We study sums
S(n) =
n
Xi,j (n)
i,j =1
called random pseudomatrices, and their asymptotic distributions with respect to the states φ(n) and with respect to normalized traces 1 φj (n), n n
ψ(n) =
j =1
respectively. We assume that the distributions of the Xi,j (n) in the states φ(n) and φj (n) depend on n in a suitable way and are block-identical. It turns out that the central limit theorem for the distributions of random pseudomatrices in the states φ(n) may be viewed as an analog of the central limit theorem for free random variables (especially, if we take square arrays). In turn, the ‘tracial’ central limit theorem for random pseudomatrices in the states ψ(n) is related to the limit theorem for random matrices (especially, if we take square arrays with block-symmetric variances). The limit distributions play then the role of multivariate generalizations of the semicircle distributions. Let us point out, however, that when we consider lower-triangular arrays, the central limit laws can be viewed as generalizations of the arcsine law. More importantly, refinements of our limit theorems give a deeper analogy between random pseudomatrices and random matrices. By a refinement we understand the scheme in which pseudomatrices are replaced by smaller sums which play the role of blocks. Namely, in the case of block-symmetric variances (D) the ψ(n)-distributions of symmetric blocks of random pseudomatrices agree asymptotically with those of random symmetric blocks,
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where random symmetric blocks are symmetric blocks of random matrices in the approach of Voiculescu [25] and Dykema [11]. It is worth mentioning that in this scheme the difference between blocks built from matricially free random variables and strongly matricially free random variables disappears as n → ∞ (in both cases blocks are asymptotically matricially free). This and other asymptotic properties of random pseudomatrices and their blocks are studied in a subsequent paper [16]. Among these properties are also asymptotic versions of (A)–(C) which refer to matricial freeness. Thus, the subarrays corresponding to row blocks of pseudomatrices are asymptotically free, asymptotically monotone independent, or asymptotically boolean independent, depending on whether the pseudomatrices are square, block lower-triangular, or block diagonal, respectively. Therefore, matricial freeness can be treated as a notion of independence which underlies the fundamental types of noncommutative independence as well as the asymptotic structure of random matrices. In Section 2, we introduce the concepts of the ‘matricially free product of states’ and the ‘matricially free Fock space’. We obtain from these structures their strong counterparts in Section 3. In Section 4, we introduce the notions of ‘matricial freeness’ and ‘strong matricial freeness’ and discuss the example of the discrete (strongly) matricially free Fock space. In Section 5, of combinatorial nature, we define and study certain real-valued functions on the set of non-crossing partitions, defined in terms of traces of certain matrices. In Section 6, we study the asymptotic behavior of random pseudomatrices and we prove standard and tracial central limit theorems. The limit distributions, which can be interpreted as matricial multivariate generalizations of semicircle laws, are studied in Section 7. Their decompositions in terms of s-free additive convolutions in the case of two-dimensional arrays are proved in Section 8. Two geometric realizations of the limit distributions, in terms of walks on weighted binary trees and in terms of weighted Catalan paths, are given in Section 9. 2. Matricially free products In this section we introduce the notion of the matricially free product of states as well as the corresponding notions of the matricially free product of Hilbert spaces and the matricially free Fock space. When speaking of arrays indexed by two indices, say i, j , we shall assume that (i, j ) ∈ J ⊆ I × I , where {(j, j ): j ∈ I } = ⊂ J and I is an index set. This refers to the situation when we deal with arrays which contain the diagonal. In particular, this includes lower-triangular arrays of the form J := {(i, j ): i j, i, j ∈ I }, in which case we shall tacitly assume that I is linearly ordered. Without loss of generality we can use the square array formulation most of the time and take their subarrays if needed. Of special interest will be the finite-dimensional case when I = [n] := {1, 2, . . . , n}. := (Hi,j ) be an array of complex Hilbert spaces. By the matricially free Definition 2.1. Let H we understand the Hilbert space direct sum Fock space over H = CΩ ⊕ M(H)
∞
m=1 (i1 ,i2 ) =··· =(im ,im ) n1 ,...,nm ∈N
m 1 2 Hi⊗n ⊗ Hi⊗n ⊗ · · · ⊗ Hi⊗n , m ,im 1 ,i2 2 ,i3
where Ω is a unit vector, with the canonical inner product.
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Let us observe that the Hilbert space tensor powers which appear in the above direct sum have the following three properties: (P1) the ‘matricial property’ – the second index of the preceding power agrees with the first index of the following power, (P2) the ‘freeness property’ – the consecutive pairs of indices are different, (P3) the ‘diagonal subordination property’ – the last pair is ‘diagonal’. The last property is needed to ensure existence of limit mixed moments in limit theorems with the square-root normalization. The term ‘subordination’ follows from the ‘subordination’ of the related additive convolution of an array of measures to the diagonal measures. Although we do not study this convolution in this paper, the basic idea can be understood on the example of the binary tree in Section 9, which can be viewed as a comb product of two graphs, of which the first one is labelled by a diagonal variance. is just one Hilbert space H, Of course, if the index set I consists of one element and thus H the corresponding matricially free Fock space reduces to the usual free Fock space F (H). In ∼ is a (usually, proper) subspace of the free Fock space F ( general, however, M(H) i,j Hi,j ) = ∗i,j F (Hi,j ). Related to the ‘matricially free Fock space’ is the ‘matricially free product of Hilbert spaces’. The terminology parallels that introduced in free probability [24,28]. Definition 2.2. Let (Hi,j , ξi,j ) be an array of Hilbert spaces with distinguished unit vectors. By the matricially free product of (Hi,j , ξi,j ) we understand the pair (H, ξ ), where
H = Cξ ⊕
∞
m=1 (i1 ,i2 ) =··· =(im ,im )
Hi01 ,i2 ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im ,
0 =H M with Hi,j i,j Cξi,j and ξ being a unit vector. We denote it (H, ξ ) = ∗i,j (Hi,j , ξi,j ).
Proposition 2.1. It holds that ξ ∼ M(H), = ∗M i,j M(Hi,j ), ξi,j . Proof. We use the definition of the matricially free Fock space, the isomorphism M(Hi,j ) ∼ = F (Hi,j ) for any i, j and regroup terms. 2 For any j , introduce diagonal subspaces of H of the form
H(j, j ) = Cξ ⊕
∞
m=2 (j,i2 ) =··· =(im ,im ) i2 =j
0 Hj,i ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im , 2
and the associated diagonal partial isometries Vj,j : Hj,j ⊗ H(j, j ) → H:
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ξj,j ⊗ ξ → ξ,
ξj,j 0 Hj,j
0 0 ⊗ ξ → Hj,j , Hj,j 0 0 ⊗ Hj,j1 ⊗ · · · ⊗ Hj0m ,jm → Hj,j ⊗ · · · ⊗ Hj0m ,jm , 1 0 0 0 ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm → Hj,j ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm , 1 1
where m > 1. For any i = j we introduce off-diagonal subspaces of H of the form H(i, j ) =
∞
m=1 (j,i2 ) =··· =(im ,im )
0 Hj,i ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im 2
and the associated off-diagonal partial isometries Vi,j : Hi,j ⊗ H(i, j ) → H for i = j : 0 0 ⊗ · · · ⊗ Hj0m ,jm → Hj,j ⊗ · · · ⊗ Hj0m ,jm , ξi,j ⊗ Hj,j 1 1 0 0 0 0 Hi,j ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm → Hi,j ⊗ Hj,j ⊗ · · · ⊗ Hj0m ,jm , 1 1 where m 1. 0 and Each H(i, j ) is spanned by simple tensors which do not begin with vectors from Hi,j for that reason it is suitable for the left free action of the operators creating such vectors. Thus, roughly speaking, both types of partial isometries jointly replace the unitary maps used in free probability. It is the diagonal subordination property which is responsible for distinguishing two types of isometries. Consider an array of C ∗ -algebras (Ai,j ), each with a unit 1i,j and a state ϕi,j , and let (Hi,j , πi,j , ξi,j ) be the associated GNS triples, so that ϕi,j (a) = πi,j (a)ξi,j , ξi,j for any a ∈ Ai,j . For any i, j , let λi,j be the ∗-representation of Ai,j given by ∗ λi,j (a) = Vi,j πi,j (a) ⊗ IH(i,j ) Vi,j
for a ∈ Ai,j ,
where IH(i,j ) denotes the identity on H(i, j ). Note that these representations are, in general, non-unital. In fact, ∗ = ri,j + si,j , λi,j (1i,j ) = Vi,j Vi,j
where ri,j and si,j are canonical projections in B(H) given by ri,j = PH(i,j )
and si,j = PK(i,j ) ,
0 ⊗ H(i, j ). For given i, j , the projections r where K(i, j ) = Hi,j i,j and si,j are orthogonal and their sum is the canonical projection onto the subspace of H onto which λi,j (Ai,j ) acts nontrivially.
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The λi,j ’s remind the representations λi of free probability [24,28], but the corresponding operators λi,j (a) have larger kernels. Using λi,j ’s, we shall define product representations on A :=
Ai,j ,
i,j
the free product without identification of units, equipped with the unit 1A , and products of states which are analogs of the free product representation and the free product of states, respectively. Definition 2.3. The matricially free product representation πM = ∗M i,j πi,j is the unital ∗-homomorphism λ : A → B(H) given by the linear extension of λ(1A ) = 1 and λ(a1 a2 . . . an ) = λi1 ,j1 (a1 )λi2 ,j2 (a2 ) . . . λin ,jn (an ) for any ak ∈ Aik ,jk , k = 1, . . . , n, with (i1 , j1 ) = (i2 , j2 ) = · · · = (in , jn ). The associated state ϕ = ∗M i,j ϕi,j : A → C is given by
ϕ(a) = πM (a)ξ, ξ and will be called the matricially free product of (ϕi,j ). Basic properties of the product state ϕ are collected in the proposition given below. Roughly speaking, they show that this state (on the free product of C ∗ -algebras without identification of units) has similar properties as the free product of states (on the free product of C ∗ -algebras with identification of units) except that the units of these algebras act as units only on ‘matricial’ tensor products and otherwise they act as null projections. For that purpose, it will be useful to introduce sets of indices associated with ‘matricial’ tensor products: Λn =
(i1 , i2 ), (i2 , i3 ), . . . , (in , in+1 ) : (i1 , i2 ) = (i2 , i3 ) = · · · = (in , in+1 )
and their union Λ = ∞ n=1 Λn . Finally, I stands for the unital subalgebra of A generated by the units 1i,j . In analogy to the notion of marginal laws in classical probability, by marginal moments we shall understand moments of the form ϕi,j (a1 . . . ak ), where a1 . . . ak ∈ Ai,j and i, j are arbitrary. Proposition 2.2. Let ϕ be the matricially free product of states (ϕi,j ) and let ak ∈ Aik ,jk , where k ∈ [n] and (i1 , j1 ) = · · · = (in , jn ). 1. If ak ∈ Ker ϕik ,jk for k ∈ [n], then ϕ(a1 a2 . . . an ) = 0. 2. If ar = 1ir ,jr and am ∈ Ker ϕim ,jm for r < m n, then ϕ(a1 . . . an ) =
ϕ(a1 . . . ar−1 ar+1 . . . an ) 0
if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise.
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3. For any a ∈ A, u1 , u2 ∈ I and i, j ∈ I , it holds that ϕ(u1 au2 ) = ϕ(u1 )ϕ(a)ϕ(u2 ) and ϕ(1i,j ) = δi,j . 4. The restriction of ϕ to Aj,j is ϕj,j for any j ∈ I . 5. The mixed moments ϕ(a1 a2 . . . an ) are uniquely expressed in terms of marginal moments. Proof. If ak ∈ Ker ϕik ,jk for k ∈ [n], where (i1 , j1 ) = · · · = (in , jn ), then it follows from the definition of the λi,j that / Λ or in = jn , 1. πM (a1 . . . an )ξ = 0 if ((i1 , j1 ), . . . , (in , jn )) ∈ 2. πM (a1 . . . an )ξ ∈ Hi01 ,j1 ⊗ · · · ⊗ Hi0n ,jn if ((i1 , j1 ), . . . , (in , jn )) ∈ Λ and in = jn . In both cases we obtain a vector orthogonal to ξ on the RHS, which proves (1). Suppose now that the assumptions of (2) hold. If ((ir , jr ), . . . , (in , jn )) ∈ Λ, then λir ,jr (1ir ,jr ) acts as a unit on Hi0r+1 ,jr+1 ⊗ · · · ⊗ Hi0n ,jn by the definition of the representations λi,j . On the other hand, λir ,jr (1ir ,jr ) kills any simple tensor beginning with h ∈ Hi0r+1 ,jr+1 if jr = ir+1 or ((ir+1 , jr+1 ), . . . , (in , jn )) ∈ / Λ since Vir ,jr does, which completes the proof of (2). In turn, (3) follows from the action of the λ(1i,j ) onto ξ . That ϕ agrees with ϕj,j on Aj,j for any j ∈ I follows from the action of the λj,j (a), a ∈ Aj,j , onto ξ , namely πM (a)ξ = (πj,j (a)ξ )0 + ϕj,j (a)ξ , which gives (4). Finally, (5) is a consequence of (1)–(2). 2 In a similar way we can define states associated with other unit vectors from H. We shall 0 , j ∈ I , which are in consider the simplest case of states associated with unit vectors ej ∈ Hj,j the ranges of πj,j (Aj,j ), respectively, namely ϕj : A → C defined by the formulas
ϕj (a) = πM (a)ej , ej , called conditions associated with ϕ, which will be used for computing normalized traces. Most properties of the states ϕj are inherited from ϕ as the proposition given below demonstrates. However, ϕj |I is quite different than ϕ|I due to different normalization conditions. Proposition 2.3. Let ϕj , j ∈ I , be the conditions associated with ϕ and let ak ∈ Aik ,jk , where k ∈ [n] and (j, j ) = (i1 , j1 ) = · · · = (in , jn ) = (j, j ). 1. If ak ∈ Ker ϕik ,jk for k ∈ [n], then ϕj (a1 a2 . . . an ) = 0 for each j . 2. If ar = 1ir ,jr and am ∈ Ker ϕim ,jm for r < m n, then ϕj (a1 . . . an ) =
ϕj (a1 . . . ar−1 ar+1 . . . an ) 0
if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise.
3. For any a ∈ A, u1 , u2 ∈ I and i, j, k ∈ I , it holds that ϕj (u1 au2 ) = ϕj (u1 )ϕj (a)ϕj (u2 )
and ϕj (1i,k ) = δj,k .
4. The restriction of ϕj to Ai,j is ϕi,j for any i = j . 5. The mixed moments ϕj (a1 a2 . . . an ) are uniquely expressed in terms of marginal moments.
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Proof. Properties (1) and (2) follow from (1) and (2) of Proposition 2.2. The normalization in 0 . In this context, notice that the unit (3) follows directly from the action of λi,k (1i,k ) onto Hj,j vectors ej play the same role with respect to the action of the λi,j (a) for any i = j as ξi,j plays with respect to the action of πi,j (a), where a ∈ Ai,j , and thus ϕj (a) = λi,j (a)ej , ej =
πi,j (a)ξi,j , ξi,j = ϕi,j (a), which gives (4) for i = j . Finally, there exists bj ∈ Aj,j ∩ Ker ϕ such that ϕj (w) = ϕ bj∗ wbj for any w ∈ i,j Ai,j , which reduces computations of mixed moments in each state ϕj to computations of mixed moments in the state ϕ. This proves (5). 2 Remark 2.1. The states ϕ and (ϕj ) share together the property of extending the array of states (ϕi,j ). Thus, ϕ extends the diagonal states ϕj,j for all j , but it does not extend the off-diagonal states ϕi,j for i = j since πM (a)ξ = 0 for any a ∈ Ai,j . In turn, ϕj extends the off-diagonal states ϕi,j , where i = j , but it does not extend ϕj,j . This is a natural consequence of differences in the definitions of the diagonal and off-diagonal partial isometries. Finally, let us denote by λ(I) the unital commutative ∗-subalgebra of B(H) generated by the λ(1i,j ), where i, j ∈ I . By abuse of notation, λ(1i,j ) will also be denoted by 1i,j (in general, these projections are not mutually orthogonal). 3. Strongly matricially free products Of special importance is the subspace of the matricially free Fock space, called the ‘strongly matricially free Fock space’, in which the diagonal Hilbert spaces appear only at the end of tensor products. The main reason is that it is related to both free and monotone Fock spaces. We also study the associated product states which can be viewed as direct generalizations of both free and monotone products of states. := (Hi,j ) we understand the Definition 3.1. By the strongly matricially free Fock space over H of the form subspace of M(H) = CΩ ⊕ R(H)
∞
m=1 i1 =··· =im n1 ,...,nm ∈N
m 1 2 Hi⊗n ⊗ Hi⊗n ⊗ · · · ⊗ Hi⊗n , m ,im 1 ,i2 2 ,i3
with the canonical inner product. A justification for the word ‘strong’ is that in this case the words i1 i2 . . . im which label the tensor products in the above definition satisfy i1 = i2 = · · · = im . Example 3.1. The simplest space of this type is associated with a two-dimensional square ar Then ray H. = R(H)
∞ m=0
R(m) (H),
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where the first few summands are of the form = CΩ, R(0) (H) = H1,1 ⊕ H2,2 , R(1) (H) = H⊗2 ⊕ H⊗2 ⊕ (H1,2 ⊗ H2,2 ) ⊕ (H2,1 ⊗ H1,1 ), R(2) (H) 1,1 2,2 ⊗3 = H ⊕ H⊗3 ⊕ H2,1 ⊗ H⊗2 ⊕ H1,2 ⊗ H⊗2 ⊕ H⊗2 ⊗ H1,1 R(3) (H) 1,1 2,2 1,1 2,2 2,1 ⊗2 ⊕ H1,2 ⊗ H2,2 ⊕ (H1,2 ⊗ H2,1 ⊗ H1,1 ) ⊕ (H2,1 ⊗ H1,2 ⊗ H2,2 ), we do not have tensor products like H2,2 ⊗ H2,1 ⊗ H1,1 and H1,1 ⊗ etc. In contrast to M(H), H1,2 ⊗ H2,2 in the summand of the third order. = (Hi,j ), we have inclusions Remark 3.1. For a given array of Hilbert spaces H ⊆ M(H) ⊆F R(H)
Hi,j
i,j
which, in most cases, are proper. Moreover, if we have a square array and Hi,j ∼ = Hi for any i, j ∈ I , where (Hi )i∈I is a family of Hilbert spaces, then there is a natural isomorphism ∼ R(H) =F
Hi
i∈I ⊗n ⊗n ⊗n m ∼ 1 2 since Hi⊗n ⊗ Hi⊗n ⊗ · · · ⊗ Hi⊗n = Hi1 1 ⊗ Hi2 2 ⊗ · · · ⊗ Him m for any i1 , i2 , . . . , im ∈ I , m ,im 1 ,i2 2 ,i3 n1 , . . . , nm , m ∈ N. Similarly, if we have a lower-triangular array and Hi,j ∼ = Hi for any i j , is isomorphic to the monotone Fock space. then R(H)
Moreover, as expected, there is a product of Hilbert spaces related to the strongly matricially free Fock space, and an analog of Proposition 2.1 holds. Definition 3.2. By the strongly matricially free product of (Hi,j , ξi,j ) we understand the pair (G, ξ ), where G is the subspace of H of the form G = Cξ ⊕
∞
m=1 i1 =··· =im
Hi01 ,i2 ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im .
We denote it (G, ξ ) = ∗Si,j (Hi,j , ξi,j ). If we consider a family of unital C ∗ -algebras (Ai )i∈I , each equipped with a family of states (ϕi,j )j ∈I , then we can look at this product space as follows. If (Hi,j , πi,j , ξi,j ) is the GNS triple associated with the pair (Ai , ϕi,j ), then the Hilbert space Hi,j (as well as the corresponding state and representation) taken as the representation space for the algebra Ai at some given tensor site depends on the algebra Aj represented at the following tensor site on the space Hj,k for some k. It is worth noting that in this framework we can assume that for fixed i ∈ I all vectors ξi,j ,
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j ∈ I , are identified since we can take the tensor product of Hilbert spaces j Hi,j and set ξi = j ∈I ξi,j for each i ∈ I . It is not hard to see that in this framework our model is related to freeness with infinitely many states [8,9]. The construction of the product state is similar to that of the matricially free product. The only difference in all definitions is that the sets Λn are replaced by Γn =
(i1 , i2 ), (i2 , i3 ), . . . , (in , in+1 ) : i1 = i2 = · · · = in
and their union Λ by Γ = ∞ n=1 Γn . Note that the conditions which define Γn ’s are stronger than those which define Λn ’s and therefore all objects constructed in the strong case are obtained from the standard ones by a projection-type operation. The partial isometries in the strong case, denoted by Wi,j ’s, remind Vi,j ’s except that they refer to G rather than H. In particular, the diagonal partial isometries Wj,j : Hj,j ⊗ G(j, j ) → G are given by ξj,j ⊗ ξ → ξ
0 0 and Hj,j ⊗ ξ → Hj,j ,
where G(j, j ) = Cξ for any j , whereas the off-diagonal partial isometries Wi,j and the associated subspaces G(i, j ) are similar to those in the standard case. Definition 3.3. Let ρi,j be the ∗-representation of Ai,j on G given by the formula ∗ ρi,j (a) = Wi,j πi,j (a) ⊗ IG (i,j ) Wi,j
where a ∈ Ai,j ,
for any (i, j ) ∈ J . The corresponding strongly matricially free product representation πS = ∗Si,j πi,j and strongly matricially free product of states ∗Si,j ϕi,j are defined in terms of the ρi,j as in the matricially free case. Using appropriate direct sums of these representations, we can reproduce products of C ∗ probability spaces in free probability of Voiculescu and in monotone probability of Muraki. If (Hi , ξi ) is a family of Hilbert spaces with distinguished unit vectors, then in the decomposition theorem given below, ∗i∈I (Hi , ξi ), i∈I (Hi , ξi ) and i∈I (Hi , ξi ), respectively, stand for free, monotone and boolean products of Hilbert spaces. Theorem 3.1 (Decomposition theorem). Let (Ai,j , ϕi,j ) = (Ai , ϕi ) for any i, j and let (πi , Hi , ξi ) be the GNS triple associated with (Ai , ϕi ) for any i ∈ I . 1. If (Ai,j , ϕi,j ) is a square array, then (G, ξ ) ∼ = ∗i∈I (Hi , ξi ) and each λi = j ∈I ρi,j is the canonical ∗-representation of Ai on ∗i∈I (Hi , ξi ). ∼ 2. If (Ai,j , ϕi,j ) is a lower-triangular array, then (G, ξ ) = i∈I (Hi , ξi ) and each τi = ij ρi,j is the canonical ∗-representation of Ai on i∈I (Hi , ξi ). 3. If (Ai,j , ϕi,j ) is a diagonal array, then (G, ξ ) ∼ = i∈I (Hi , ξi ) and each ρi,i is the canonical ∗-representation of Ai on i∈I (Hi , ξi ). Proof. First, let us remark that therepresentations λi and τi are well defined since the corresponding direct sums of operators j ρi,j (a) are convergent in the strong-operator topology on
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B(G) for any a ∈ Ai and i ∈ I . Now, since Hi,j ∼ = Hi for any i, j , we have a natural isomorphism Hi01 ,i2 ⊗ Hi02 ,i3 ⊗ · · · ⊗ Hi0m ,im ∼ = Hi01 ⊗ Hi02 ⊗ · · · ⊗ Hi0m for any i1 = i2 = · · · = im or any i1 > i2 > · · · > im , in the case of square or lower-triangular arrays, respectively, which leads to the corresponding isomorphisms (G, ξ ) ∼ = ∗i∈I (Hi , ξi )
or
(G, ξ ) ∼ = i∈I (Hi , ξi )
(recall our tacit assumption that I is linearly ordered when dealing with triangular arrays). Then partial isometries which lie in the same row of the array (Wi,j ) are orthogonal in the sense that Wi,j Wi,k = δj,k Wi,j for any i, j , k (in the monotone case the array of partial isometries is lower-triangular and we have here i > j > k). Therefore, we have an orthogonal direct sum decomposition
Wi,j = Vi ,
i ∈ I,
j ∈I
of the unitaries Vi used in the definition of the free product representation [21,24], and an analogous decomposition in the monotone case. The direct sum decompositions of λi (a) and τi (a) in terms of ρi,j (a)’s, where a ∈ Ai , follow then immediately from Definition 3.3, which completes the proof of (1) and (2). In particular, it follows that each λi is unital, but τi are, in general, non-unital. Since the case of a diagonal array is rather elementary, the proof is completed. 2 Consequently, if A is the C ∗ -algebra generated by the family {λi (Ai )} of subalgebras of B(G) and ϕ(.) = .ξ, ξ , then (A, ϕ) is the free product of C ∗ -probability spaces. Similar statements hold for the monotone and boolean products of C ∗ -probability spaces, except that the identity I ∈ B(G) has to be added to the generators. In the natural way this leads to properties (A), (B) and (C) of the introduction, of which the first two can be viewed as decompositions of free and monotone independent random variables in terms of strongly matricially free ones. The above theorem allows us to view the strongly matricially free product of states as a deformation of the free product of states, obtained by a natural direct sum decomposition of the free product of Hilbert spaces. The matricially free product of states is then obtained from its strong counterpart by an extension to a slightly larger Hilbert space which includes all products which arise naturally in matrix multiplication. It is clear that basic results on the strongly matricially free product of states are similar to those for the matricially free product and therefore we state them in an abbreviated form without a proof. Proposition 3.1. Let ϕ be the strongly matricially free product of states (ϕi,j ) and let ϕj , j ∈ I , be the associated conditions. Then the statements of Propositions 2.2–2.3 remain true, with Λ replaced by Γ . As we have mentioned earlier, one of the advantages of using the strong structures is that they are straightforward generalizations of those in free probability (in the case of square arrays) and monotone probability (in the case of lower-triangular arrays). However, it is the matricial freeness which gives a connection with random matrices.
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Remark 3.2. Slightly more general is the case when the diagonal and off-diagonal states differ, but the latter stay the same within each row, namely (Ai,i , ϕi,i ) = (Ai , ϕi )
and (Ai,j , ϕi,j ) = (Ai , ψi )
for i = j,
where each Ai is equipped with two states, ϕi and ψi , respectively. Similar reasoning to that in the free case leads then to the conditionally free product of states (for square arrays) and conditionally monotone products of states (for lower-triangular arrays). 4. Matricial freeness Guided by the notion of the matricially free product of states, we shall introduce now the associated concept of independence called ‘matricial freeness’, and closely related to it, ‘strong matricial freeness’. They involve arrays of noncommutative probability spaces and to some extent they remind models with many states [6,8,9], but they cannot be reduced in a natural way to any of these (freeness with infinitely many states has some non-empty intersection with ‘strong matricial freeness’ and conditional freeness is its special case). Moreover, we will study discrete (strong) matricially free Fock spaces. Let A be a unital algebra with an array (Ai,j ) of subalgebras of A. We will assume that each Ai,j has an internal unit 1i,j which may be different from the unit of A, and we assume that the unital subalgebra I generated by all internal units is commutative. Let ϕ be a distinguished state on A and let {ϕj : j ∈ I } be a family of additional states on A, where by a state we understand a normalized linear functional. If A is a unital ∗-algebra, then we assume that Ai,j ’s are ∗subalgebras and all states are positive functionals. Further, Definition 4.1. States ϕj , j ∈ I , will be called conditions associated with ϕ if for any j ∈ I there exist bj , cj ∈ Aj,j ∩ Ker(ϕ) such that ϕj (a) = ϕ(cj abj ) for any a ∈ A. The array of states on A given by ϕj,j = ϕ
and ϕi,j = ϕj
for any i = j
will be said to be defined by ϕ and the associated conditions ϕj . In addition, if A is a ∗-algebra, we assume that cj = bj∗ for any j . Throughout this paper we will assume the following normalization conditions: ϕ(1i,j ) = δi,j
and ϕj (1i,k ) = δj,k
for any i, j , k. Although they are natural in view of the Hilbert space formulation presented before and will be assumed in this paper, we prefer to exclude them from the definition given below. In particular, the normalization of each ϕj implies that if a ∈ Ai,k and j = k, then ϕj (a) = 0. Moreover, it is equivalent to scaling the variables cj , bj according to ϕ(cj bj ) = 1. Definition 4.2. Let (ϕi,j ) be the array defined by ϕ and the associated conditions ϕj . We say that (1i,j ) is a matricially free array of units associated with (Ai,j ) and (ϕi,j ) if
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1. ϕ(u1 au2 ) = ϕ(u1 )ϕ(a)ϕ(u2 ) for any a ∈ A and u1 , u2 ∈ I, 2. for ak ∈ Aik ,jk ∩ Ker ϕik ,jk , where r < k n, ϕ(a1ir ,jr ar+1 . . . an ) =
ϕ(aar+1 . . . an ) 0
if ((ir , jr ), . . . , (in , jn )) ∈ Λ, otherwise,
where a ∈ A is arbitrary and (ir , jr ) = · · · = (in , jn ). The array (1i,j ) is called a strongly matricially free array of units if Λ is replaced by Γ . The above definition enables us to define the concepts of matricial freeness and its strong version called strong matricial freeness. They both bear some resemblance to freeness, but the main difference is that the identified unit in the context of freeness is replaced by the (strongly) matricially free array of units. In fact, as we have already remarked, it is the strong matricial freeness which can be viewed as a direct generalization of freeness. Definition 4.3. We say that (Ai,j ) is matricially free with respect to the array (ϕi,j ) defined by ϕ and the associated conditions ϕj if 1. for any ak ∈ Ker ϕik ,jk ∩ Aik ,jk , where k ∈ [n] and (i1 , j1 ) = · · · = (in , jn ), ϕ(a1 a2 . . . an ) = 0, 2. (1i,j ) is a matricially free array of units associated with (Ai,j ) and (ϕi,j ). In an analogous manner we define strongly matricially free arrays of subalgebras. Definition 4.4. The array of variables (ai,j ) in a unital algebra A will be called (strongly) matricially free with respect to the array (ϕi,j ) of states on A if the array (C[ai,j , 1i,j ]) is (strongly) matricially free with respect to (ϕi,j ) for some array of elements (1i,j ) of A which is a (strongly) matricially free array of units. If A is a unital ∗-algebra, then, in addition, we require that the functionals ϕi,j are positive, the Ai,j are ∗-subalgebras and the 1i,j are projections. Then an array of variables (ai,j ) will be called ∗-(strongly) matricially free if the array of ∗-algebras ∗ , 1 ) is (strongly) matricially free. (C ai,j , ai,j i,j Using the above definitions, we can uniquely express mixed moments under ϕ and thus under ϕj ’s of arbitrary matricially free random variables in terms of marginal moments under ϕi,j ’s. To see how this computation works, it is convenient to assume that neighboring variables come from different algebras and use the recurrence given below. Remark 4.1. Suppose that (Ai,j ) is matricially free with respect to the array (ϕi,j ) defined by ϕ and the associated conditions ϕj . Writing ak = ak0 + ϕik ,jk (ak )1ik ,jk ∈ Aik ,jk for any 1 k n, we obtain the recurrence
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ϕ(a1 . . . an ) =
ϕik ,jk (ak )ϕ a10 . . . 1ik ,jk . . . an0
1kn
+
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ϕik ,jk (ak )ϕil ,jl (al )ϕ a10 . . . 1ik ,jk . . . 1il ,jl . . . an0
1k
+ ··· + ϕi1 ,j1 (a1 ) . . . ϕin ,jn (an )ϕ(1i1 ,j1 . . . 1in ,jn ). If we assume that neighboring variables come from different algebras, we can reduce the moments under ϕ which appear on the RHS to sums of products of marginal moments (repeated application of Definitions 4.1 and 4.2 is needed). Let us remark that the recurrence generalizes that for free random variables [3] except that the units on the RHS are not equal to the unit of A. The above recurrence can also be used even if neighboring variables belong to the same algebra. In that case, with each product a = a1 . . . an we can associate a partition π = {π1 , . . . , πs } of the set {1, . . . , n} in a natural way. Namely, numbers k, r will belong to the same block of π if and only if (ik , jk ) = (ir , jr ). This allows us to derive basic properties of the mixed moments, collected in the lemma given below (compare with the free case [22]). Moreover, observe that in the definition of (strong) matricial freeness, the mixed moments under ϕ and ϕj ’s are uniquely determined by marginal moments under ϕk |Ai,k for i = k and those under ϕ|Ak,k for any k (cf. Proposition 2.3). Therefore, in order to determine ϕ uniquely on i,j Ai,j , it suffices to specify linear functionals ϕi,j on subalgebras Ai,j such that ϕi,j (1i,j ) = 1, where i, j are arbitrary, and set ϕ(1) = 1. Then ϕj ’s are determined uniquely on i,j Ai,j , provided a pair cj , bj ∈ Aj,j ∩ Ker ϕj,j , such that ϕj,j (cj bj ) = 1, is chosen for any j . We assume now that (Ai,j ) is an array of subalgebras of a unital algebra A, equipped with an array of states (ϕi,j ) defined by ϕ and associated conditions ϕj , with respect to which (Ai,j ) is matricially free. Without loss of generality, we assume in the lemma given below that neighbors come from different algebras. Lemma 4.1. Let a = a1 . . . an , where ak ∈ Aik ,jk for 1 k n and (i1 , j1 ) = · · · = (in , jn ), and let π = {π1 , . . . , πs } be the associated partition of the set {1, . . . , n}. 1. 2. 3. 4. 5.
If there exists k such that (ir , jr ) = (ik , jk ) for r = k, and ak ∈ Ker ϕik ,jk , then ϕ(a) = 0. If π is arbitrary, then ϕ(a) is a sum of products of at least s marginal moments. If π is crossing, then ϕ(a) is a sum of products of more than s marginal moments. If ik = jk for any k, then ϕ(a) = ϕ(a1 ) . . . ϕ(an ). If among i1 , j1 , . . . , im , jm there are more than s different indices, then ϕ(a) = 0.
Proof. The proof of (1) is similar to that of the recurrence of Remark 4.1. In fact, it is an easy consequence of the equation ϕ(a) = ϕik ,jk (ak )ϕ(a1 . . . ak−1 1ik ,jk ak+1 . . . an ), which, in turn, follows from ϕ a1 . . . ak−1 ak0 ak+1 . . . an = 0.
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To justify the latter, we assume (without loss of generality) that neighboring variables come from different algebras. Then, decomposing ar = ar0 + ϕir ,jr (ar )1ir ,jr for any r = k, we repeatedly apply Definitions 4.2–4.3 to the above moment until we are left with a sum of products of moments which do not contain units, have neighboring variables in the kernels of the corresponding states and come from different algebras. However, this means that each of these products vanishes since the variable ak0 does not participate in any reductions as it is the only one from Aik ,jk and that is why ak0 appears in one of the factors. This proves (1). Next, (2) follows from a repeated application of the recurrence of Remark 4.1 and Definitions 4.1–4.2. In order to prove (3), it suffices to consider the case when π has no singletons since if π is crossing, the partition obtained from π by removing singletons is still crossing (note that in the case when neighbors come from different algebras, each non-crossing partition has a singleton). Then consider a summand from the RHS of the same recurrence of the form ϕik ,jk (ak ) . . . ϕir ,jr (ar )ϕ(a1 . . . 1ik ,jk . . . 1ir ,jr . . . an ). There are two possibilities. If among (ik , jk ), . . . , (ir , jr ) there are no repetitions, among the remaining pairs there are still s different ones and further reduction of the above summand gives products of more than s marginal moments in view of (2). In turn, if among (ik , jk ), . . . , (ir , jr ) there is a repetition, then there must be at least s −(r −1) different remaining pairs, and therefore, in view of (2), we obtain again a sum of products of more than s marginal moments, which completes the proof of (3). Property (4) follows from the proof of (1). In order to prove (5), it is enough to consider ak ∈ {ak0 , 1ik ,jk } for any k. It is obvious that the number of different indices among i1 , j1 , . . . , in , jn is smaller or equal to 2s, where s is the number of blocks of π . However, in order to obtain a non-vanishing moment, the last variable in the product a1 . . . an which belongs to a given block πr , say ap , must have the second index equal to the first index of one of the variables which follow ap and are associated with blocks different than πr (this is not necessarily ap+1 if some variables which follow ap are units). In particular, in order that ϕ a1 . . . ap ap+1 . . . aq−1 aq0 . . . an0 = 0, where ap+1 , . . . , aq−1 are units, we must have jq−1 = iq and thus, by Definition 4.2, jk = iq for p < k < q − 1. In that case, the moment reduces to ϕ(a1 . . . ap aq0 . . . an0 ), which implies that we must have jp = iq . This shows how the indices associated with different blocks are tied together if they are separated by units. Therefore, the number of different indices among i1 , j1 , . . . , in , jn is reduced to a number smaller or equal to s (strictly smaller if among them there are diagonal variables other than the last one), which completes the proof of (5). 2 Let us note in this context that, in contrast to the free case, property (1) does not entail the factorization of pyramidally ordered products (see the computation of ϕ(ab a) in Example 4.1). In general, strong matricial freeness generalizes freeness, monotone independence as well as boolean independence. This fact is rather natural in view of Theorem 3.1, but we shall state it below on the more abstract level of independence in the category of noncommutative probability spaces rather than on the level of products of states. For that purpose, assume that (Ai,j ) is an array of subalgebras of a unital algebra A which is strongly matricially free with respect to some array (ϕi,j ) defined by a distinguished state ϕ and the associated conditions ϕj .
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Proposition 4.1. Let Ai,j be a replica of Ai,i with ai,j being a replica of ai,i ∈ Ai,i such that ϕi,j -distribution of ai,j agrees with ϕ-distribution of ai,i for any i, j . 1. If (Ai,j ) is a finite square array, then the family {ai := j ai,j , i ∈ I } is free with respect to ϕ. 2. If (Ai,j ) is a finite lower-triangular array, then the family {ai := j ai,j , i ∈ I } is monotone independent with respect to ϕ. 3. If (Ai,j ) is a diagonal array, then the family {ai,i , i ∈ I } is boolean independent with respect to ϕ. 0 and 1i := j 1i,j for any i, where the summation runs over Proof. Let us denote ai0 := j ai,j 0 =a − those j ’s for which (i, j ) ∈ J , with J denoting the index set of the array, and where ai,j i,j ϕ(ai,j )1i,j (in the case of lower-triangular arrays, the summation runs over j i). Moreover, ϕi,j (ai,j ) = ϕ(ai,i ) = ϕ(ai )
for any (i, j ) ∈ J
by the assumption on identical distributions of replicas and by the normalization ϕ(1i,j ) = δi,j . In the case of a square array, we obtain the implication ϕ ai01 ,j1 . . . ai0n ,jn = 0
⇒
ϕ ai01 . . . ai0n = 0
for i1 = · · · = in . Moreover, using the assumption about identical distributions of replicas once again, we can write ai = ai0 + ϕ(ai )1i for any i, and thus it is enough to observe that for any r it holds that ϕ ai1 ,j1 . . . 1ir ai0r+1 ,jr+1 . . . ai0n ,jn = ϕ ai1 ,j1 . . . ai0r+1 ,jr+1 . . . ai0n ,jn for any r n, which can be used repeatedly to conclude that the 1i ’s can be identified with the unit of A in all mixed moments under ϕ which involve the ak ’s. This completes the proof for the free case. In the monotone case we have to slightly modify the approach since we have triangular arrays and units do not become identified. Note that monotone independence of the ai ’s essentially follows from the following two equations: ϕ(w1 1i 1j 1i w2 ) = ϕ(w1 1i w2 )
and ϕ w1 1i aj0 1i w2 = 0
for any i < j , where w1 , w2 are products of ak ’s. These equations are proved by induction with repeated use of Definitions 4.2–4.3. In fact, it is enough to show that such equations hold if 0 ’s and 1 ’s, and that can be reduced to the proof for mixed w2 is an alternating product of ak,l k,l moments of the above type, in which w2 is an alternating product of variables from the kernels, 0 ∈ A ∩ Ker ϕ . To obtain the first equation, it then suffices to observe that in any such say bk,l k,l i,j mixed moment 1i 1j can be identified with 1i since the latter acts as a projection onto products 0 for k i, and therefore 1 maps 1 w onto itself. The same kind w2 which begin with some bk,l j i 2 of argument is used to prove the second equation, which completes the proof of (2). The proof of (3) is omitted since it is straightforward. 2
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We assumed above that sums j ai,j are finite since in the considered category we do not have a topology. However, appropriately defined inductive limits of mixed moments of these sums considered above can be shown to exist for arbitrary arrays since under ϕ all mixed moments of these sums reduce to finite sums of mixed moments of the ai,j ’s. This implies that in the category of C ∗ -probability spaces these sums converge in the strong-operator topology on some Hilbert space (isomorphic to the associated free product of Hilbert spaces). By a straightforward modification of the free case in Proposition 4.1 we obtain conditional freeness. We just need to take the array (ϕi,j ) defined by ϕ and associated conditions ϕj which are assumed to agree with another state ψ on off-diagonal replicas (as in the free case, the conditions remain different as states on A). Then the considered family of sums is conditionally free with respect to (ϕ, ψ). Some simple examples of mixed moments are given below, where we can also see that the factorization of mixed moments of (strongly) matricially free random variables depends not only on the associated partition, but also on the algebras to which variables belong. In particular, in contrast to the free case, a non-crossing partition associated with a given product may give products of more marginal moments than the number of blocks. Example 4.1. Consider a two-dimensional array of strongly matricially free subalgebras of A, with variables and states ϕ ϕ2 A1,1 A1,2 a a and ∈ , b b A2,1 A2,2 ϕ1 ϕ respectively. Using Remark 4.1 and Definitions 4.1–4.2, we obtain ϕ(aba) = ϕ 2 (a)ϕ(b),
ϕ ab a = ϕ a 2 − ϕ 2 (a) ϕ1 b .
Similar computations give higher-order mixed moments ϕ ab ab = ϕ(b)ϕ1 b ϕ a 2 − ϕ 2 (a) , ϕ(abab) = ϕ 2 (a)ϕ 2 (b), ϕ aba b = ϕ(a)ϕ2 a ϕ b2 − ϕ 2 (b) . These are all non-vanishing moments that contribute to ϕ(ABA) and ϕ(ABAB), respectively, where A = a + a and B = b + b . However, the latter agree with the mixed moments of A and B which are conditionally independent with respect to (ϕ, ψ), where ψ agrees with ϕ1 on C[B] and with ϕ2 on C[A]. In turn, ϕ(aBa) and ϕ(aBaB), corresponding to the associated lowertriangular array, agree with the mixed moments of a, B which are monotone independent with respect to ϕ if ϕ agrees with ϕ1 on C[B]. where Definition 4.5. By a discrete matricially free Fock space we understand M = M(H), Hi,j = Cei,j for any (i, j ) ∈ N, and the array (ei,j ) forms an orthonormal basis of some Hilbert space. In an analogous manner we define the discrete strongly matricially free Fock space R = R(H). Both M and R are subspaces of the discrete free Fock space F ( i,j Cei,j ) and thus allow for the canonical action of free creation and annihilation operators. In order to specify the arrays of units, let us distinguish two types of their subspaces.
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In the case of M these are 1. M(i, j ), spanned by simple tensors which begin with ej,k for some k, where (j, k) = (i, j ), and, in addition, by Ω if i = j , 2. K(i, j ), spanned by vectors which begin with ei,j , where i, j are arbitrary, and the direct sum K(i, j ) ⊕ M(i, j ) is the subspace of M onto which the ∗-algebra generated by (ei,j ) restricted to M acts non-trivially, where (ei,j ) denotes the canonical free creation operator associated with vector ei,j . The canonical projections onto such direct sums are natural candidates for the matricially free units 1i,j and therefore we set 1i,j := PK(i,j )⊕M(i,j )
and i,j = (ei,j )1i,j
for any i, j . The adjoint of i,j will be denoted by ∗i,j . Note that the projection 1i,j is an internal unit in the ∗-algebra Ai,j = C i,j , ∗i,j and ∗i,j i,j = 1i,j for any i, j . However, in the algebra C i,j , ∗i,j : i, j ∈ N there are more relations as the proposition given below demonstrates. Proposition 4.2. In the algebra C i,j , ∗i,j : i, j ∈ N the following relations hold: 1. 2. 3. 4.
∗i,j i,j = 1i,j for any i, j , ∗i,j k,l = 0 whenever (i, j ) = (k, l), i,j k,l = 0 and 1i,j k,l = 0 whenever (i, j ) = (k, l) and j = k, 1i,j j,k = j,k for any i, j , k.
Proof. We omit the elementary proof.
2
In the case of R, we proceed in a completely analogous fashion and distinguish the following subspaces: R(i, j ) = M(i, j ) ∩ R and L(i, j ) = K(i, j ) ∩ R which lead to the definitions of the strongly matricially free array of units and of the creation operators, respectively, 1i,j = PL(i,j )⊕R(i,j )
and ki,j = (ei,j )1i,j ,
where, slightly abusing notation, we use the same symbols for the units as before. Finally, we specify the states: ϕ will denote the vacuum state associated with Ω, the ϕj will be the states associated with the ej,j , j ∈ N, and (ϕi,j ) will be the array defined by ϕ and the ϕj . Here, the same notation is used for states on B(M) and B(R). Proposition 4.3. The array of ∗-subalgebras Ai,j = C i,j , ∗i,j of B(M), where i, j ∈ N, is ∗ of B(R), matricially free with respect to (ϕi,j ). The array of ∗-subalgebras Bi,j = C ki,j , ki,j where i, j ∈ N, is strongly matricially free with respect to (ϕi,j ).
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Proof. The proof is similar to that of Voiculescu for the discrete free Fock space given in [27]. We shall look at the case of (Ai,j ) since the case of (Bi,j ) is analogous. Each algebra Ai,j is spanned by operators of the form q
∗p
i,j i,j ,
where p + q > 0,
and the projection 1i,j . However, the corresponding moments vanish: q ∗p ϕi,j i,j i,j = 0 for any i, j since p + q > 0. Moreover, ϕi,j (1i,j ) = 1 for any i, j . Therefore, in order to show that condition (1) of Definition 4.2 holds, it is enough to show that q ∗p q ∗p ϕ i11,j1 i1 ,j11 . . . inn,jn in ,jnn = 0 whenever (i1 , j1 ) = · · · = (in , jn ) and p1 + q1 > 0, . . . , pn + qn > 0. The same argument as in [27] allows us to reduce the proof to the case when q1 = · · · = qn = 0, which implies that p1 > 0, . . . , pn > 0. But then the moment clearly vanishes. This proves condition (1) of Definition 4.2. Condition (2) follows easily from the definition of the projections 1i,j in view of the relations given in Proposition 4.1. This completes the proof. 2 Example 4.2. For i, j ∈ I , let Gi,j ∼ = F (1) be the free group on one generator gi,j with unit i,j . 2 of l 2 (∗ G ) spanned by vectors of the form δ(g), where g is either Consider the subspace lM i,j i,j the unit e of the free product ∗i,j Gi,j , or a product of the form g1 g2 . . . gm , where gk ∈ G0ik ,ik+1 := 2 , which Gik ik+1 \ {ik ,ik+1 } for each k, with (i1 , i2 ) = · · · = (im , im+1 ) and im+1 = im . The space lM is a simple example of the matricially free Fock space, can be viewed as the space of square integrable functions on the ‘matricially free product of groups’ (which is not a group). Let 1i,j 2 spanned by vectors δ(g), where denote the projection from l 2 (∗i,j Gi,j ) onto the subspace of lM 0 0 g begins with an element from Gi,j or Gj,k for some k or, in the case of i = j , also g = e. We can now define λi,j (g) = λi,j (g)1i,j for g ∈ Gi,j , where by λi,j we denote the left regular representation of Gi,j on l 2 (∗i,j Gi,j ). Then the array (Ai,j ), where Ai,j is the ∗-subalgebra of B(l 2 (∗i,j Gi,j )) generated by λi,j (gi,j ) and 1i,j , with the standard involution, is matricially free with respect to the array (ϕi,j ), where the diagonal states coincide with ϕ(.) = .δ(e), δ(e) and the associated conditions are ϕj (.) = .δ(gj,j ), δ(gj,j ), where j ∈ I . In a similar way we proceed in the case of functions on the ‘strongly matricially free product of groups’. For classical results on random walks on free groups, see [13]. Example 4.3. To be more concrete, we consider a two-dimensional square array of free groups on one generator. Then our Hilbert space is 2 lM = span δ(e), δ(g): g ∈ H1 ∪ H2 ,
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where k k l k l m k l m , g1,2 g2,2 , g1,1 g1,2 g2,2 , g1,2 g2,1 g1,1 , . . . : k, l, m ∈ Z0 H1 := g1,1 and k k l k l m k l m H2 := g2,2 , g2,1 g1,1 , g2,2 g2,1 g1,1 , g2,1 g1,2 g2,2 , . . . : k, l, m ∈ Z0 , respectively, with Z0 denoting the set of non-zero integers. The diagonal units 11,1 and 12,2 are projections onto subspaces spanned by δ(e) and δ(g), where g ∈ H1 and g ∈ H2 , respectively. The off-diagonal units 11,2 and 12,1 are defined in a similar way, so that we have decompositions 2 . 11,1 + 11,2 = 12,1 + 12,2 = 1 of the unit 1 on lM Example 4.4. In the case of square integrable functions on the ‘strongly matricially free product of groups’, the Hilbert space is smaller, namely lS2 = span δ(e), δ(g): g ∈ K1 ∪ K2 , where k k l k l m K1 := g1,1 , g1,2 g2,2 , g1,2 g2,1 g1,1 , . . . : k, l, m ∈ Z0 and k k l k l m K2 := g2,2 , g2,1 g1,1 , g2,1 g1,2 g2,2 , . . . : k, l, m ∈ Z0 , which reflects the fact that the diagonal generators act non-trivially only on δ(e). Internal units are reduced accordingly. Note that in this case lS2 ∼ = l 2 (F (2)), which conforms with Theorem 3.1 and the resulting decomposition of the left regular representation. Example 4.5. In the case of an n-dimensional square array of copies of F (1), we form a tree (a subtree of the homogeneous tree H2n2 ) which corresponds to the ‘matricially free product of n2 free groups’. Suppose the root e corresponds to the ‘father’. We distinguish ‘sons’ and −1 , and gi,j ‘daughters’ in each ‘generation’ which correspond to the left action of gj,j or gj,j
−1 or gi,j , respectively, where i = j . The rules of drawing the tree follow from matricial freeness and are the following: each ‘son’ has 1 ‘son’ and 2n − 2 ‘daughters’, whereas each ‘daughter’ has 2 ‘sons’ and 2n − 1 ‘daughters’. Therefore, ‘sons’ and ‘daughters’ correspond to vertices of valencies 2n and 2n + 2, respectively. In Fig. 1 we draw such a tree for n = 2 (black and empty circles are assigned to ‘sons’ and ‘daughters’, respectively). If we make an additional assumption, for instance that ‘daughters’ cannot have ‘sons’ (this fact corresponds to strong matricial freeness, where diagonal generators kill words beginning with the off-diagonal ones), we recover H2n of free probability.
be the discrete strongly matricially free Fock space and let R(H) ∼ Example 4.6. Let R(H) = F ( j Cej ) be the natural isomorphism of Remark 3.1, where {ej : j ∈ N} is an orthonormal basis of some Hilbert space. If (wi,j ) is an infinite matrix with non-negative parameters p and q
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Fig. 1. Matricially free analog of H4 .
above and below the main diagonal, respectively, and 1’s on the diagonal, then it is easy to see that the (p, q)-creation operators studied in [18] can be identified (with the use of this isomorphism) with the strongly convergent sums Ai =
wi,j ki,j ,
j
where i ∈ N and the (p, q)-annihilation operators are their adjoints. A similar approach can be applied to square arrays of arbitrary Hilbert spaces and ∗-representations, which leads to some notion of ‘(p, q)-independence’. Moreover, it can be carried out for more general matrices (wi,j ) within the framework of the strong matricial freeness, which generalizes notions of independence of this type. 5. Traces In this section we introduce some real-valued functions on the set of non-crossing pairpartitions. These functions are obtained by computing traces of a square real-valued matrix. We will assume later that this matrix has non-negative entries which represent variances of probability measures on the real line MR and we will demonstrate that the functions introduced in this section describe the asymptotics of matricially free random variables in central limit theorems. Let N C m denote the set of non-crossing partitions of the set [m], i.e. if π = {π1 , π2 , . . . , πk } ∈ N C m , then there are no numbers i < p < j < q such that i, j ∈ πr and p, q ∈ πs for r = s. The block πr is inner with respect to πs if p < i < q for any i ∈ πr and p, q ∈ πs (then πs is outer with respect to πr ). It is clear that if πr has outer blocks, then there exists a unique block among them, say πs , which is nearest to πr , i.e. if another block, say πt , is outer with respect to πr , then we must have a < p < b for any a, b ∈ πt and p ∈ πs . In that case we shall write πs = o(πr ) and call the pair (πr , o(πr )) the nearest inner–outer pair of blocks. If πi does not have an outer block, it is called a covering block. It is convenient to extend each partition π ∈ N C 2m to the
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partition π obtained from π by adding one block, say π0 = {0, m + 1}, called the imaginary block. If B(π) is the set of blocks of π , we shall denote by Fr (π) the set of all mappings f : B(π) → [r] called colorings of the blocks of π by the set [r] := {1, 2, . . . , r}. Then the pair (π, f ) plays the role of a colored partition. Let N CC m denote the set of non-crossing covered partitions of [m], by which we understand the subset of N C m consisting of those partitions in which 1 and m belong to the same block (if m = 1, we understand that the partition consists of one block). In terms of diagrams, all blocks of π ∈ N CC m , where m > 1, are covered by the block containing 1 and m. We denote by N C 2m and of [m] and non-crossing pair-partitions N CC 2m the sets of non-crossing pair-partitions
∞ covered 2 2 2 N C and N CC = N CC . of [m], respectively, and we set N C 2 = ∞ m m m=1 m=1 It is easy to see that each π ∈ N C m can be decomposed as π = π (1) ∪ π (2) ∪ · · · ∪ π (p) ,
(5.1)
where π (1) , . . . , π (p) are non-crossing covered partitions of subintervals I1 , I2 , . . . , Ip of [m] whose union gives [m]. By a partition of a set I consisting of r elements we understand the corresponding partition of [r]. On the other hand, each π ∈ N CC m can be decomposed as π = π (0) ∪ π (1) ∪ · · · ∪ π (r) ,
(5.2)
where π (0) is the block containing 1 and m and π (1) , π (2) , . . . , π (r) are non-crossing covered partitions of subintervals I1 , I2 , . . . , Ir of [m] \ π0 . Consider now a square real-valued matrix V = (vi,j ) ∈ Mn (R), where n ∈ N. The usual trace and the normalized trace will be denoted Tr(V ) =
n
1 vj,j , n n
vj,j
and tr(V ) =
j =1
j =1
respectively. For each n, we define the ‘diagonalization mapping’ τ : Mn (R) → Dn (R),
τ (V ) = diag vj,1 , . . . , vj,n , j
j
where Dn (R) is the set of square diagonal real-valued matrices of dimension n (by abuse of notation, the same symbol τ is used for all n). In other words, τ computes the sum of all elements of V in each column separately and puts this value on the diagonal. Using the above trace operations, we shall define two real-valued functions on the set N C 2 , denoted v and v0 , associated with given V ∈ Mn (R). Although there is a close similarity between these functions when restricted to N CC 2 (in particular, they are defined as traces of certain matrix-valued quasi-multiplicative functions), note that they are extended to N C 2 in two different ways. Definition 5.1. For a given matrix V ∈ Mn (R), we define a mapping from N C 2 to Dn (R) by assigning to each π ∈ N C 2 the matrix V (π) by the following recursion:
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Fig. 2. Colored partitions with imaginary blocks.
1. if π consists of one block, we set V (π) = τ (V ), 2. if π ∈ N CC 2 consists of more than one block, then V (π) = τ V π (1) . . . V π (r) V
(5.3)
according to the decomposition (5.2), 3. if π ∈ N C 2 , then V (π) = V π (1) V π (2) . . . V π (p)
(5.4)
according to the decomposition (5.1). Let v : N C 2 → R be the function defined by v(π) = tr(V (π)). Example 5.1. For some V ∈ Mn (R), consider three partitions given in Fig. 2. Color each block by a number from the set [n]. Computation of the corresponding values of the function v gives 1 vi,j vj,k , v(π) = tr τ τ (V )V = n i,j,k
1 v(η) = tr τ τ (V )V τ (V ) = vi,j vj,l vk,l , n i,j,k,l
1 v(ζ ) = tr τ τ (V )τ (V )V = vi,k vj,k vk,l . n i,j,k,l
One can see that to each nearest inner–outer pair of blocks (πr , πs ) we assign the matrix element vp,q , where p and q are the colors of πr and πs respectively. Moreover, if a block does not have any outer blocks and is colored by q, then we assign to it the matrix element vq,t , where t is assumed to be the same for all such blocks (one can imagine that we have an additional ‘imaginary block’ colored by t which covers all other blocks). At the end we sum over all colorings. The function v0 is defined on the set N CC 2 in a very similar manner, except that we replace the right multiplier V in (5.3) by its main diagonal V0 := diag(v1,1 , . . . , vn,n ),
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which corresponds to changing only the contribution of the covering block. Then we extend v0 to all of N C 2 by multiplicativity of v0 . Definition 5.2. For a given matrix V0 ∈ Mn (R), we define a mapping from N CC 2 to Dn (R) by assigning to each π ∈ N CC 2 the matrix V0 (π) by the following recursion: 1. if π consists of one block, we set V0 (π) = V0 , 2. if π ∈ N CC 2 consists of more than one block, then V0 (π) = τ V π (1) . . . V π (r) V0
(5.5)
according to the decomposition (5.2). Let v0 : N C 2 → R be the function defined by v0 (π) = Tr(V0 (π)) for π ∈ N CC 2 , extended to N C 2 by multiplicativity v0 (π) = v0 (π (1) ) . . . v0 (π (p) ) according to the decomposition (5.1). Example 5.2. Let us compute the values of v0 corresponding to the partitions in Fig. 2. We have vi,j vj,j , v0 (π) = Tr τ (V )V0 = i,j
v0 (η) = Tr τ (V )V0 Tr(V0 ) = vi,j vj,j vk,k , i,j,k
v0 (ζ ) = Tr τ τ (V )τ (V )V0 = vi,k vj,k vk,k . i,j,k
Note that the main difference (apart from normalization) between v0 (π) and v(π) concerns the blocks which do not have outer blocks. Here, if such a block is colored by q, we assign to it the matrix element vq,q and we do not use ‘imaginary blocks’. Below we shall prove a lemma which gives explicit combinatorial formulas for v(π) and v0 (π). For that purpose, to the blocks B(π, f ) = (π1 , f ), (π2 , f ), . . . , (πk , f ) of each colored non-crossing partition (π, f ), where f ∈ Fr (π), we assign entries of a given real-valued matrix V ∈ Mr (R) according to the definition given below. If the imaginary block is used, it is convenient to assume that it is also colored by a number from the set [r]. Definition 5.3. Let (π, f ) be a colored non-crossing partition with blocks as above, where f ∈ Fr (π) and let V ∈ Mr (R) be given. For any 0 j r we define vj (π, f ) = vj (π1 , f )vj (π2 , f ) . . . vj (πk , f ), where the functions vj : B(π, f ) → R are given by the following rules:
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1. vj (πi , f ) = vp,q if f (πi ) = p and f (o(πi )) = q, where 0 j r, 2. vj (πi , f ) = vp,j if f (πi ) = p and πi does not have outer blocks, where 1 j r, 3. v0 (πi , f ) = vp,p if f (πi ) = p and πi does not have outer blocks. We are ready to prove purely combinatorial formulas for vj (π) for any 0 j n and any π ∈ N C 2 . In particular, if V is a square n-dimensional matrix with all entries equalto 1/n and f runs over Fn (π), then v0 (π) = 1 and vj (π) = 1/n for any π . In that case we get π∈N C 2 v0 (π) = 2m n j =1 π∈N C 22m vj (π) = cm , the m-th Catalan number. Limit theorems studied in Section 6 will give matricial deformations of Catalan numbers induced by the formulas given by the lemma proven below. Lemma 5.1. For any V ∈ Mn (R), where n ∈ N, and π ∈ N C 2 , it holds that v0 (π) =
v0 (π, f )
and v(π) =
f ∈Fn (π)
1 n
vj (π, f ),
f ∈Fn (π) j ∈[n]
where the summation over j corresponds to all colorings of the imaginary block. Proof. We provide an induction proof for the function v (the proof for v0 is similar). The main induction step will be carried out on the level of the (diagonal) matrices V (π). We claim that its diagonal entries are of the form V (π) q,q = vq (π, f ) f ∈Fn (π)
for any π ∈ N CC 2 and q ∈ [n]. In view of (5.4), the required formula for v(π) is then a straightforward consequence of the claim. Of course, if π consists of one block, then V (π) q,q = vj,q j
and thus our assertion easily follows. Assume now that π has k 2 blocks and suppose the assertion holds for non-crossing covered partitions which have less than k blocks. Since π has a decomposition of type (5.2), the assertion holds for π (1) , π (2) , . . . , π (r) in this decomposition. We know that the matrix assigned to π has the form (5.3) and therefore the product of (diagonal) matrices corresponding to these subpartitions has (diagonal) matrix elements of the form V (π) q,q = τ (W V ) q,q = Wj,j vj,q j
where q is the color of the imaginary block of π and Wj,j =
f1 ∈Fn (π (1) )
vj π (1) , f1 . . .
fr ∈Fn (π (r) )
vj π (r) , fr ,
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by the inductive assumption. Now, since the blocks of π (1) , . . . , π (r) are colored independently, we have vj π (1) , f1 . . . vj π (r) , fr vj,q = vq (π, f ) for a uniquely determined coloring f of the blocks of π in which j can be interpreted as the color of π (0) since π (0) covers π (1) , . . . , π (r) and q can be viewed as the color of the imaginary block of π . This and the above formula for Wj,j gives the desired formula V (π)q,q =
vq (π, f ),
f ∈Fn (π)
which proves our claim and thus completes the proof of the theorem.
2
Under suitable assumptions on V , there is a simple connection between functions v and v0 if V0 = aIn /n, where In is a unit matrix and a is a positive number. We shall express this relation in terms of the corresponding formal Laurent series, which turn out to be Cauchy transforms of (compactly supported) probability measures on the real line associated with appropriately constructed random variables. Proposition 5.1. Let V ∈ M n (R) be such that V0 = aIn /n, where a > 0. If G(z) = ∞ ∞ −2k−2 and G (z) = −2k−2 are formal Laurent series, where 0 k=0 a2k z k=0 b2k z a2k =
v(π)
and b2k =
π∈N C 22k
v0 (π),
π∈N C 22k
and both v(π) and v0 (π) are associated with V , then G0 (z) = 1/(z − aG(z)). Proof. Observe that in the case when vj,j = a/n for all j ∈ [n], we have v(π) =
v0 (π ) a
for any π ∈ N C 2m , where the partition π ∈ N CC 2m+2 is obtained from π by adding to π the block that covers all blocks of π , say {0, m + 1}. This leads to
A(z) :=
∞
a2m z2m = 1 +
v(π)z2m
m=1 π∈N C 2 2m
m=0
= 1+
∞
∞ 1 a
m=1 π∈N CC 2 2m+2
v0 (π)z2m =
C(z) − 1 , az2
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where C(z) = we obtain
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∞
m=0 c2m z
2m
B(z) :=
and c2m =
∞
π∈N CC 22m v0 (π). Now, using the multiplicativity of v0 ,
b2m z2m = 1 +
m=0
∞ m C(z) − 1 = m=1
1 , 2 − C(z)
which leads to 1 1 1 A(z) = 2 1 − and G0 (z) = , B(z) z − aG(z) az where 1 1 1 1 G(z) = A and G0 (z) = B , z z z z which completes the proof.
2
6. Random pseudomatrices In this section we will study the asymptotic behavior of random pseudomatrices for two sequences of states: the sequence of distinguished states, with respect to which our variables will be matricially free, and the sequence of traces which are normalized sums of conditions in the definition of matricial freeness. Under suitable assumptions, we will later obtain two types of limit theorems: the ‘standard’ central limit theorem as well as the ‘tracial’ central limit theorem for matricially free random variables, the latter being related to random matrix models. Suppose we have a sequence (A(n)) of unital ∗-algebras, each equipped with a distinguished state φ(n) and associated conditions {φj (n): 1 j n}. For each n, let (Xi,j (n))1i,j n be an array of self-adjoint random variables in A(n). We are going to study the asymptotic behavior of random pseudomatrices S(n) =
n
Xi,j (n),
(6.1)
i,j =1
where we assume that each array (Xi,j (n)) is matricially free with respect to the array (φi,j (n)) defined by φ(n) and φj (n)’s. In the first setting we shall compute the asymptotic moments of random pseudomatrices with respect to φ(n). This corresponds to the ‘standard CLT’ reminding the CLT for free (or, monotone) random variables [24]. In the second setting, we shall compute the asymptotic moments of random pseudomatrices with respect to the convex linear combinations of conditions 1 φj (n) n n
ψ(n) :=
(6.2)
j =1
which play the role of traces in the context of random pseudomatrices, with φj (n) corresponding the classical expectation E composed with the state associated with vector ei of the canonical
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orthonormal basis (ei )1in in Cn . This will lead to the ‘tracial CLT’ for matricially free random variables reminding the limit theorem for random matrices [25]. Thanks to the properties of the matricially free product of states which essentially boil down to properties (P1) and (P3) of Section 2, the normalization of square root type works in both cases since the number of summands in S(n) which give a non-zero contribution to the limits is in both cases of order n. We partition the set [n] := {1, 2, . . . , n} into disjoint non-empty intervals, [n] = N1 ∪ N2 ∪ · · · ∪ Nr , where r ∈ N, such that their relative sizes nj /n → dj as n → ∞, where nj is the cardinality of Nj for any j ∈ [r]. Then the numbers dj form a diagonal matrix D of trace one called the dimension matrix. Our results will involve the following assumptions: (A1) (Xi,j (n)) is matricially free with respect to (φi,j (n)) for any n ∈ N, (A2) the variables have zero expectations, φi,j (n) Xi,j (n) = 0 for all i, j ∈ [n] and n ∈ N, (A3) their moments are uniformly bounded, i.e. ∀m ∃Mm 0 such that φi,j (n) X m (n) Mm i,j nm/2 for all i, j ∈ [n] and n ∈ N, (A4) their variances are block-identical and are of order 1/n, namely 2 up,q φi,j (n) Xi,j (n) = n for any i ∈ Np , j ∈ Nq , where each up,q is a non-negative real number. Of course, √ the uniform boundedness assumption is satisfied if we take variables of type Xi,j (n) = Xi,j / n, where (Xi,j ) is an infinite array of random variables whose distributions in the states φi,j (n), respectively, are identical and do not depend on n. Although it is convenient to think of the Xi,j (n) as if they were of this form, we want to study similar variables, whose variances are of order 1/n and stay the same within blocks whose sizes become infinite as n → ∞. If the tuple ((i1 , j1 ), . . . , (im , jm )) ∈ (I × I )m , where I is an index set, defines a partition π = {π1 , . . . , πk } of the set [m], i.e. (ip , jp ) = (iq , jq ) if and only if there exists r such that p, q ∈ πr , we will write P (i1 , j1 ), . . . , (im , jm ) = π. Of course, if π is a non-crossing pair-partition, then each πr is a two-element set. By a k-block we will understand a block consisting of k elements. In particular, a 1-block will be called a singleton. We will also adopt the convention that if m is odd, then N C 2m = ∅ and the summation
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over π ∈ N C 2m gives zero. This allows us to state results for moments of the S(n) of all orders without distinguishing even and odd moments. Lemma 6.1. Let (A(n), ψ(n)) be a sequence of noncommutative probability spaces, each with an array (φi,j (n)) defined by φ(n) and φj (n)’s, for which (A1)–(A3) hold, where ψ(n) is given by (6.2). Then 1 ψ(n) S m (n) = v(π, n) + O √ , n 2 π∈N C m
where v(π, n) = tr(V (π, n)) is given by Definition 5.1 and corresponds to the variance matrix 2 (n)). V (n) = (vi,j (n)), where vi,j (n) = φ(n)(Xi,j Proof. We have ψ(n) S m (n) =
i1 ,j1 ,...,im ,jm
=
ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n)
π∈Pm
i1 ,j1 ,...,im ,jm P ((i1 ,j1 ),...,(im ,jm ))=π
ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) ,
where Pm denotes the set of all partitions of [m]. Arguments presented below allow us to conclude that for large n only non-crossing pair-partitions give relevant contributions. 1. If π has a singleton associated with some (ik , jk ) = (j, j ), then the moment φj (n) Xi1 ,j1 (n) . . . Xim ,jm (n) vanishes by Lemma 4.1 since in that case the variable Xik ,jk is the only variable from Aik ,jk in the corresponding moment under φ(n), by which we mean φ(n)(bj∗ abj ), where a = Xi1 ,j1 (n) . . . Xim ,jm (n) and bj ∈ C[Xj,j (n), 1j,j ] ∩ Ker φ(n). For that reason, in the remaining cases we can assume that there are no such singletons. 2. If π has exactly one singleton associated with some (ik , jk ) = (j, j ), then the extended partition π associated to bj∗ abj does not have singletons and has at least one 3-block (this is the imaginary block colored with j ). Therefore, π has the same number of blocks as π and s (m + 1)/2. Since with each block we can associate at most one independent index to sum over (by Lemma 4.1), the sum of the above moments over i1 , j√ 1 , . . . , im , jm and j for the given π (with the 1/n normalization coming from ψ(n)) is O(1/ n ) by (A3). 3. If π has no singletons and is not a pair-partition, then the number of blocks of π is either s (if (j, j ) is present among (i1 , j1 ), . . . , (im , jm )) or s + 1 (in the opposite case). In both cases, we have that s (m + 1)/2 and the summation of the considered moments over √ i1 , j1 , . . . , im , jm and j for the given π is O(1/ n ). 4. If π is a crossing pair-partition, then the associated π is a crossing partition for any j . It suffices to consider the case when π is a pair-partition since otherwise it has a 4-block and a similar argument to that in (3) works. In turn, if π is a pair-partition for some j , then by Lemma 4.1 and the mean zero assumption, the corresponding moment under φj (n) vanishes, which implies that the corresponding moment under ψ(n) also vanishes.
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The above arguments imply that only non-crossing pair-partitions contribute to the limit and we can write m 1 ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) + O √ . ψ(n) S (n) = n 2 π∈N C m
i1 ,j1 ,...,im ,jm P ((i1 ,j1 ),...,(im ,jm ))=π
Now, suppose that m is even, π ∈ N C 2m and the sequence of pairs ((i1 , j1 ), . . . , (im , jm )) is compatible with the matricial multiplication, i.e. such that if (ik , jk ) and (ir , jr ) label an inner– outer pair of blocks, then jk = ir . If πj = {r, r + 1} is the last block which has no inner blocks, then we can pull out the variance corresponding to that block, namely: ψ(n) Xi1 ,j1 (n) . . . Xim ,jm (n) = vir ,jr ψ(n) Xi1 ,j1 (n) . . . Xir−1 ,jr−1 (n)Xir+2 ,jr+2 (n) . . . Xim ,jm (n) , where we use the proof of Lemma 4.1. Continuing this procedure with other blocks which have no inner blocks, and summing over indices i1 , j1 , . . . , im , jm , for which it holds that P ((i1 , j1 ), . . . , (im , jm )) = π , we arrive at 1 n
k0 ,k1 ...,km
1 vk1 ,ko(1) (n) . . . vkm ,ko(m) (n) + O √ , n
where o(r) = 0 if πr has no outer blocks (k0 labels the imaginary block) and o(r) = j if the nearest outer block of πr is labelled by kj . Let us observe that we included in the above sum all possible labellings of the blocks of π . This is done for convenience since it enables us to express the final result in terms of v(π). More explicitly, we allow k0 , k1 , . . . , km to assume arbitrary values from the set [m] (in particular, they can all be equal), which produces certain terms which cannot be obtained from the summation over all ((i1 , j1 ), . . . , (im , jm )) which define π . For example, no nearest inner–outer pair of blocks can √ contribute vj,j vj,j , which appears in the above sum. However, all such terms are of order 1/ n due to insufficient number of different summation indices (there are fewer than m/2 independent indices) and therefore they can be included in the sum without changing the asymptotics. Using Lemma 5.1, we obtain our assertion. 2 Lemma 6.2. Let (A(n), φ(n)) be a sequence noncommutative probability spaces, each with an array (φi,j (n)) of states defined by φ(n) and conditions φj (n), for which (A1)–(A3) hold. Then m 1 v0 (π, n) + O √ , φ(n) S (n) = n 2 π∈N C m
where v0 (π, n) = Tr(V0 (π, n)) is given by Definition 5.2 and corresponds to the variance matrix V (n) = (vi,j (n)). Proof. The proof is similar to that of Lemma 6.1 (in fact, it is simpler since we only need to compute the moments under φ(n) and thus the complications involving moments under φj (n)’s do not appear). 2
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In order to ensure existence of the limits of v(π, n) and v0 (π, n) as n → ∞, we need to specify the sequence of matrices (V (n))n∈N more closely. We shall assume that (A4) holds. Assuming that the variance matrices V (n) are of this form, we can now state the standard and tracial central limit theorems, with limit distributions described in terms of traces of Section 5. Theorem 6.1 (Tracial central limit theorem). Under the assumptions of Lemma 6.1, if V (n) is of the block form (A4) for each n ∈ N, then lim ψ(n) S m (n) = b(π),
n→∞
(6.3)
π∈N C 2m
for any m ∈ N, where b(π) = Tr(B(π)D) and B(π) is the diagonal matrix of Definition 5.1 corresponding to π and the matrix B = DU . Proof. Clearly, if m is odd, we get zeros on both sides of the above formula (we use our convention that in this case N C 2m = ∅). The proof for m = 2k, where k ∈ N, is based on Lemmas 5.1 and 6.1. If, in the combinatorial expression for v(π, n), we substitute for the matrix elements of V (n) the assumed block form, then, using the partition of the set of colors [n] = N1 ∪ N2 ∪ · · · ∪ Nr , we can perform summations over the colorings which belong to each interval Nj separately. Thus, the contributions of various [n]-colorings of π to the limit laws reduce to those corresponding to [r]-colorings and are described in terms of numbers uj (πi , f ), where i ∈ [k], j ∈ [r] and f ∈ Fr (π) (the number j is the color of the imaginary block). We have lim v(π, n) = lim
n→∞
1
nf (1) uj (π1 , f ) . . . nf (k) uj (πk , f )
nj nk+1 j ∈[r] f ∈Fr (π) dj df (1) uj (π1 , f ) . . . df (k) uj (πk , f ) = Tr B(π)D , = n→∞
j ∈[r]
f ∈Fr (π)
where π → B(π) is the matrix-valued function which corresponds to the matrix B = DU in accordance with Definition 5.1. In terms of matrix multiplication, the expression on the righthand side is obtained from that of Lemma 5.1 corresponding to matrix U by multiplying U from the left by the dimension matrix D and multiplying the whole product of matrices from the right by D. This proves our assertion. 2 Theorem 6.2 (Central limit theorem). Under the assumptions of Lemma 6.2, if V (n) is of the block form (A4) for each n ∈ N, then b0 (π), lim φ(n) S m (n) =
n→∞
(6.4)
π∈N C 2m
for any m ∈ N, where π → b0 (π) is the real-valued function of Definition 5.2 corresponding to the matrix B = DU . Proof. The proof is similar to that of Theorem 6.1 and is based on Lemmas 5.1 and 6.2. The only difference is that we do not use imaginary blocks to describe the colorings of all blocks of π . Thus, the contributions of various [n]-colorings of π to the limit laws reduce to those
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corresponding to [r]-colorings and are described in terms of numbers u0 (πi , f ), where i ∈ [k] and f ∈ Fr (π). Namely, we have lim v0 (π, n) = lim
n→∞
n→∞
=
1 nk
nf (1) u0 (π1 , f ) . . . nf (k) u0 (πk , f )
f ∈Fr (π)
df (1) u0 (π1 , f ) . . . df (k) u0 (πk , f ) = Tr B0 (π)
f ∈Fr (π)
as n → ∞, where π → B0 (π) is the function defined by Definition 5.2, which proves our assertion. 2 7. Matricial semicircle distributions The results of Section 6 lead to combinatorial formulas for the asymptotic moments in the corresponding central limit theorems. In this section we are going to express the limits in terms of their Cauchy transforms represented in the form of continued fractions. They play the role of the (standard and tracial) ‘matricial semicircle distributions’. For that purpose let us recall definitions of certain convolutions of distributions, or more generally, of probability measures. If Fμ is the reciprocal Cauchy transform of some probability measure μ ∈ MR , then the K-transform of μ is given by Kμ (z) = z − Fμ (z). The boolean additive convolution μ ν can be defined by the equation Kμν (z) = Kμ (z) + Kν (z), where μ, ν ∈ MR and z ∈ C+ , respectively. In fact, this equation shows that the K-transform is the boolean analog of the logarithm of the Fourier transform [23]. We will also need another convolution, which reminds the monotone convolution [20], called the orthogonal additive convolution and defined by the equation Kμν (z) = Kμ Fν (z) , where μ, ν ∈ MR and z ∈ C+ . It was introduced in [14], where we showed that the above formula defines a unique probability measure on the real line. Moreover, if μ and ν are compactly supported, both μ ν and μ ν are compactly supported. Using these convolutions, we will now define certain important continued fractions (or, ‘continued multifractions’) which converge uniformly on the compact subsets of C+ to the Ktransforms of some probability measures μi,j ∈ MR . Lemma 7.1. For given B ∈ Mr (R) with non-negative entries, continued fractions of the form Ki,j (z) =
z−
k
bi,j bk,i bp,k z − p z − ···
,
where i, j ∈ [r], converge uniformly on the compact subsets of C+ to the K-transforms of some μi,j ∈ MR with compact supports.
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Proof. Let us define a sequence of functions which approximate the Ki,j . Namely, set Ki,j (z) = bi,j /z for any i, j ∈ [r], which are the K-transforms of probability measures on R for any i, j (Bernoulli measures if bi,j > 0 and δ0 if bi,j = 0). In order to use an inductive argument, let us establish the recurrence (m) Ki,j (z) =
z−
bi,j
(m−1) (z) k Kk,i
(m−1)
(m−1)
for m 1. If the Kk,i are the K-transforms of some μk,i ∈ MR for any i and k, respec (m−1) tively, then the sums k Kk,i are the K-transforms of their boolean convolution (m−1)
μi
(m−1)
:= μ1,i
(m−1)
μ2,i
(m−1)
· · · μr,i
∈ MR ,
(m)
and next, each Ki,j is the K-transform of the orthogonal convolution (m)
(m−1)
μi,j := κi,j μi
∈ MR ,
where the κi,j ’s are the Bernoulli measures with K-transforms Ki,j (z) = bi,j /z, respectively. It is easy to see that all these measures are compactly supported. Moreover, the properties of the orthogonal additive convolution (Corollary 5.3 in [14]) say that the moments of μ(n) i,j of orders (m)
2m agree with the corresponding moments of μi,j for any n > m and any given i, j . More precisely, in the formal Laurent series expansion (m)
Ki,j (z) =
∞
c−2n−1 z−2n−1 ,
n=0
the coefficients c−1 , . . . , c−2m−1 are uniquely determined by the constants which appear in the (m) continued fraction of the corresponding Cauchy transform Gi,j at depths 2m, and these de(m)
termine the moments of μi,j of orders 2m. The recurrence for the K-transform given above is (m)
(m−1)
agree down to depth m − 1 for such that the corresponding Cauchy transforms Gi,j and Gi,j any given i, j , which proves our assertion. Therefore, we have weak convergence (m)
w − lim μi,j = μi,j m→∞
to some μi,j ∈ MR for any i, j . These measures are also compactly supported since supi,j bi,j is finite. In turn, this implies that the corresponding Cauchy transforms (and thus K-transforms) converge uniformly to the Cauchy transform (K-transforms) of μi,j on compact subsets of C+ . This completes the proof. 2 We are ready to state a theorem, which is another version of the tracial central limit theorem for matricially free random variables.
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Theorem 7.1. Under the assumptions of Theorem 6.1, the ψ(n)-distributions of Sn converge weakly to the distribution given by the convex linear combination μ=
r
dj μj ,
j =1
where μj = μ1,j μ2,j · · · μr,j for each j = 1, . . . , r and μi,j is the distribution defined by Ki,j for any i, j . Proof. By Theorem 6.1, we have combinatorial formulas for the moments Mm of the limit law in the tracial central limit theorem. The associated distribution extends to a unique compactly supported probability measure μ on the real line since its moments are bounded by the moments of the Wigner semicircle distribution σa with variance a = supi,j bi,j . Using the multiplicative formula (5.4) for B(π), we can formally write the Cauchy transform of μ in the form Gμ (z) =
∞
M2k z−2k−1
k=0
=
∞ 1 + Tr B(π)D z−2k−1 z 2 k=1
π∈N C 2k
−1 = Tr z − K(z) D , which can be called the ‘trace formula’ for Gμ , where K(z) =
∞
B(π)z−2k+1
(7.1)
k=1 π∈N CC 2
2k
is a diagonal-matrix-valued formal power series. Moreover, we will show below that each function Kj on its diagonal is, in fact, the K-transform of some μj ∈ MR . Then, the formal power series given by the trace formula is the Cauchy transform of μ as a convex linear combination of Cauchy transforms of probability measures. In fact, using the definition of B(π) and (5.3), we obtain the equation −1 K(z) = τ z − K(z) B ,
(7.2)
where K(z) = diag(K1 (z), . . . , Kr (z)). By analogy with the scalar-valued case, we can find its solution in the form of a continued fraction. Namely, observe that each Kj (z) has the form of a formal Laurent series Kj (z) =
∞ n=0
c−2n−1 z−2n−1
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for some c−1 , c−3 , . . . , and therefore the above vector equation can be solved by successive approximations. Namely, we set μj tobe the (compactly supported) probability measure associated with the K-transform Kj (z) = i Ki,j (z) for each j ∈ [r], where the Ki,j are given by Lemma 7.1 for B = DU . These K-transforms solve (7.2). This, together with the trace formula for Gμ , gives Gμ =
r
dj Gμj ,
j =1
where Gμj (z) = 1/(z − Kj (z)) is the Cauchy transform of μj for j = 1, . . . , r. That completes the proof. 2 Remark 7.1. The Cauchy transform of each μj can be written as a continued fraction of the form Gμj (z) =
z−
1
i
z−
k
bi,j bk,i bp,k z − p z − ···
which converges on the compact subsets of C+ . Next, we state a theorem, which is another version of the standard central limit theorem for matricially free random variables. Theorem 7.2. Under the assumptions of Theorem 6.2, the φ(n)-distributions of Sn converge weakly to the distribution μ0 = μ1,1 μ2,2 · · · μr,r , where μj,j is the distribution defined by Kj,j for each j . Proof. By Theorem 6.2, we have combinatorial expressions for the limit moments Mm . The proof is similar to that of Theorem 7.1 and is based on the trace formula for the K-transform of μ0 −1 Kμ0 (z) = Tr z − K(z) B0 derived from the definition of the function b0 , which leads to the equation for the Cauchy transform Gμ0 (z) = which completes the proof.
2
z−
1 , j Kj,j (z)
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Remark 7.2. The Cauchy transform Gμ0 can be written as a continued fraction of the form Gμ0 (z) =
z−
1
j
z−
bj,j
i
bi,j bk,i z − k z − ···
which gives a matricial extension of the continued fraction of the Wigner semicircle distribution. 8. Decompositions in terms of subordinations In the one-dimensional case, the limit distributions μ0 and μ are related by Proposition 5.1. In particular, if each variance matrix V (n) has identical entries equal to one, both central limit theorems (standard and tracial) give the Wigner semicircle distribution with variance 1 (of course, the standard case also follows from free probability, whereas the tracial case is related to random matrices). In this section we will analyze in more detail the limit distributions for the two-dimensional case, namely when each variance matrix V (n) consists of four blocks. They will be expressed in terms of two-dimensional arrays of distributions. Finding simple analytic formulas for the corresponding four-parameter Cauchy transforms and densities does not seem possible in the general case. However, we shall derive decomposition formulas for those measures in terms of s-free additive convolutions [14], which gives some insight into their structure (see also [21] for recent results on the multivariate case). The s-free additive convolution refers to the subordination property for free additive convolution, discovered by Voiculescu [26] and generalized by Biane [4]. As shown in [14] and [15], there is a notion of independence, called freeness with subordination, or simply s-freeness, associated with the s-free additive convolution and its multiplicative counterpart. Recall that the s-free additive convolution of μ, ν ∈ MR is the unique probability measure μ ν ∈ MR defined by the subordination equation ν μ = ν (μ
ν),
where denotes the monotone additive convolution [20]. Equivalently, the above subordination property can be written in terms of Cauchy transforms or their reciprocals. Using s-free additive convolutions and the boolean convolution, we obtain a decomposition of the free additive convolution of the form μ ν = (μ
ν) (ν
μ),
which allows us to interpret both s-free additive convolutions appearing here as (in general, non-symmetric) halves of μ ν. We find it interesting that the limit distributions in the twodimensional case will turn out to be deformations of the free additive convolution of semicircle laws implemented by this decomposition. In other words, the subordination property and the associated convolutions give a natural framework for studying matricial generalizations of the semicircle law. For simplicity, it will be convenient to use the indices-free notation for the two-dimensional matrix of K-transforms:
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a(z) c(z)
b(z) d(z)
=
K1,1 (z) K2,1 (z)
K1,2 (z) K2,2 (z)
and A=
α γ
β δ
b1,1 = b2,1
√ b 1,2 = B b2,2
where the square root is interpreted entry-wise. Moreover, we will distinguish two laws by special notations: we denote by σα the Wigner semicircle distribution with the Cauchy transform √ z2 − 4α 2 , Gσα (z) = 2α 2 √ √ where the branch of z2 − 4α 2 is chosen so that z2 − 4α 2 > 0 if z ∈ R and z ∈ (2α, ∞), and by κγ we denote the Bernoulli law with the Cauchy transform z−
Gκγ (z) =
1 , z − γ 2 /z
i.e. σγ = 1/2(δ−γ + δγ ). Finally, we will also use the boolean compressions of μ ∈ MR , where t 0, defined by multiplying its K-transform by t, namely we define Tt μ to be the (unique) probability measure on R, for which KTt μ = tKμ . These transformations were introduced and studied in [7] and called ‘t-transformations’ of μ. We allow t = 0, in which case T0 μ = δ0 . In particular, we shall use two-parameter boolean compressions of semicircle distributions, σα,β = Tt σα for t = (β/α)2 , with √ (2α 2 − β 2 )z − β 2 z2 − 4α 2 Gσα,β (z) = (2α 2 − 2β 2 )z2 + 2β 4 being their Cauchy transforms, where the branch of the square root is the same as in the case of Gσα . Theorem 8.1. If α, β, γ , δ = 0, then the diagonal measures μj,j defined by Kj,j , where 1 j 2, have the form μ1,1 = T1/t (σα,β
σδ,γ ),
μ2,2 = T1/s (σδ,γ
σα,β ),
with the off-diagonal measures given by μ1,2 = Tt μ1,1 and μ2,1 = Ts μ2,2 , where t = (β/α)2 and s = (γ /δ)2 ,
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Proof. It is easy to see that the following algebraic relations hold: a(z) =
α2 z − a(z) − c(z)
and d(z) =
δ2 , z − b(z) − d(z)
b(z) =
β2 z − a(z) − c(z)
and c(z) =
γ2 . z − b(z) − d(z)
Thus, b(z) = ta(z) and c(z) = sd(z), which gives μ1,2 = Tt μ1,1 and μ2,1 = Ts μ2,2 . In turn, from the equation for a(z), we get a(z) =
z − c(z) −
(z − c(z))2 − 4α 2 = Kσα z − c(z) 2
and thus μ1,1 = σα μ2,1 . In a similar manner we obtain μ2,2 = σδ μ1,2 . Therefore, we arrive at the equations μ1,1 = σα (Ts σδ Tt μ1,1 ), μ2,2 = σδ (Tt σα Ts μ2,2 ) since Tt (μ ν) = (Tt μ) ν. In order to express the μj,j in terms of s-free additive convolutions, we need to use the properties of the orthogonal convolution. We have shown in [14] that the moment of order k of μ ν depends on the moments of orders k of μ and the moments of orders k − 2 of ν. This leads to the conclusion that for any compactly supported μ, ν ∈ MR and the associated sequence of measures (μ m ν), defined recursively by μ m ν = μ (ν m−1 μ)
with μ 1 ν = μ ν,
we have weak convergence w − limm→∞ μ m ν = μ ν. If we take μ = Tt σα and ν = Ts σδ (these measures are compactly supported), we get the desired formulas. 2 Corollary 8.1. The measures μ0 , μ1 , μ2 can be decomposed as μ0 = T1/t (σα,β
σδ,γ ) T1/s (σδ,γ
μ1 = T1/t (σα,β
σδ,γ ) (σδ,γ
σα,β ),
μ2 = T1/s (σδ,γ
σα,β ) (σα,β
σδ,γ ),
σα,β ),
where the assumptions and notations are the same as in Theorem 8.1. Proof. These decompositions follow immediately from Theorems 7.1, 7.2 and 8.1.
2
Remark 8.1. The formulas for the diagonal measures μj,j in the proof of Theorem 8.1 remind those for two-periodic continued fractions if we take t = s = 1. The latter are of the same form, except that the semicircle distributions are replaced by much simpler Bernoulli laws. Nevertheless, if t = s = 1, the formulas for the μj take a simple form μj = σ α σ δ ,
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where j = 0, 1, 2. By Theorem 7.1 and Corollary 8.1, the same formula holds for μ. Therefore, all measures μ0 , μ1 , μ2 , μ can be viewed as deformations of the free additive convolution of two semicircle distributions, implemented by means of boolean compressions. Let us consider now the situation in which some of the numbers α, β, γ , δ vanish. Suitable formulas can be derived algebraically, as we did in the proof of Theorem 8.1. However, one can also obtain the same results by taking weak limits in the formulas for the measures μi,j , using the fact that all measures involved have compact supports. For that purpose, let us state a few useful facts about weak limits which will be of interest to us. Then, we consider eight cases, to which the remaining cases are similar (for instance, α = β = 0 is similar to γ = δ = 0). Proposition 8.1. Let t = β 2 /α 2 , where α, β > 0, and let μ ∈ MR be compactly supported. 1. If α → 0+ , then (a) w − lim σα = δ0 , (b) w − lim(σα,β ) = κβ , (c) w − lim(T1/t (σα,β μ)) = δ0 . 2. If β → 0+ , then (a) w − lim σα,β = δ0 , (b) w − lim(Tt μ) = δ0 , (c) w − lim(T1/t (σα,β μ)) = σα μ. Proof. If α → 0+ , then Kσα (z) → 0, which proves 1(a). Here, as well as in the remaining cases, convergence is uniform on compact subsets of C+ . Moreover, Kσα,β = β 2 /(z − α 2 Kσα ) → β 2 /z = Kκβ , which gives 1(b). In turn, if β → 0+ , then Kσα,β (z) = β 2 /(z − Kσα (z)) → 0, thus also w − lim σα,β = δ0 , which proves 2(a). If, in addition, t → 0+ , then Tt μ → δ0 for any μ ∈ MR , which proves 2(b). Finally, T1/t (σα,β
μ) = T1/t Tt σα (μ
σα,β ) = σα (μ
σα,β )
and thus the right-hand side tends weakly to δ0 as α → 0+ , which gives 1(c), and tends weakly to σα (μ δ0 ) = σα μ as β → 0+ , which gives 2(c). This holds for any μ ∈ MR , and we also use the right unit property of δ with respect to the s-free additive convolution, namely μ δ0 = μ. 2 Corollary 8.2. If some of the entries of the matrix A vanish, we can distinguish eight different cases, for which the distributions μi,j are given by Table 1. Proof. If ai,j = 0, then μi,j = δ0 , which easily follows from the algebraic equations for the corresponding K-transforms. An alternative proof can be given by taking weak limits of the formulas of Theorem 8.1, as we proceed with the remaining measures. Thus, if δ → 0+ , then μ1,1 = w − lim T1/t (σα,β
σδ,γ ) = T1/t (σα,β
μ1,2 = w − lim(σα,β
σδ,γ ) = σα,β
μ2,1 = w − lim(σδ,γ
σα,β ) = κγ
κγ , σα,β ,
κγ ),
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Table 1 Distributions μi,j in the case A has zero entries. a1,1
a1,2
a2,1
a2,2
μ1,1
α 0 α 0 0 α α 0
β β 0 β 0 0 0 β
γ γ γ 0 γ 0 0 0
0 0 δ δ δ δ 0 0
T1/t (σα,β δ0 σα σδ,γ δ0 δ0 σα σα δ0
κγ )
μ1,2
μ2,1
μ2,2
σα,β κγ κβ κγ δ0 κβ δ0 δ0 δ0 κβ
κγ σα,β κγ κβ σδ,γ δ0 σδ,γ δ0 δ0 δ0
δ0 δ0 σδ σδ κβ σδ σδ δ0 δ0
by 1(b) of Proposition 8.1, which proves the first case in Table 1. The remaining cases are proved in a similar manner. 2 In the case of arbitrary matrix A, finding the four-parameter densities of μ0 and μ is unwieldy. Below we shall just consider two special cases, in which we can find nice formulas for these measures for matrices A of arbitrary dimension. These two cases are of special interest since they are associated with (asymptotic) freeness and (asymptotic) monotone independence. Proposition 8.2. If A is a square r-dimensional matrix with identical positive entries αj in the j -th row, then μj = σα1 σα2 · · · σαr for each j ∈ [r], and the measures μj coincide with μ and μ0 . Proof. Since the columns of A are identical, the functions Ki,j are the same for all j ’s. Denote them Li = Ki,j , where i, j ∈ [r]. Moreover, Li (z) =
bi
z−
for any i ∈ [r]. Therefore,
r
j =1 Lj (z)
r
i=1 Li (z)
and
r i=1
r bi ri=1 Li (z) = z − j =1 Lj (z)
is the K-transform of the measure
σα1 σα2 · · · σαr and since Kμj (z) = ri=1 Ki,j (z) = ri=1 Li (z), the proof for μj is completed. It is then easy to see that we get the same result for μ and μ0 . 2 Proposition 8.3. If A is a lower-triangular r-dimensional matrix with identical positive entries αj in the j -th row below and on the main diagonal, then μj = σαj σαj +1 · · · σαr for each j ∈ [r]. Moreover, μ0 = μ1 and μ is the convex linear combination of the measures μj as in Theorem 7.1.
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Proof. As in the proof of Proposition 8.2, note that the measures μi,j do not depend on j and thus we can set Li = Ki,j for any i j . If i = r, we have Lr (z) =
br , z − Lr (z)
using the continued fraction for the Kr,j of Lemma 7.1. Therefore, μr,j = σαr for any j r. Next, we have Lk (z) =
z−
bk r
i=k Li (z)
,
which leads to Lk (z) = Kσαk z −
r
Li (z) ,
i=k+1
giving the orthogonal decomposition of μk,j , μk,j = σαk (μk+1,j · · · μr,j ) for any j k < r. Now, we claim that μi,j · · · μr,j = σαi σαi+1 · · · σαr for any 1 j i r. Clearly, it holds for i = r and any j r since we have already shown that μr,j = σαr for any j r. Suppose now that this formula holds for i > k and any j i. We will show that it holds for i = k and any j k. Using the orthogonal decomposition of μk,j given above and the inductive assumption, we obtain μk,j = σαk (σαk+1 · · · σαr ). However, for any μ, ν ∈ MR , we have a simple relation (μ ν) ν = μ ν, which gives μk,j · · · μr,j = σαk σαk+1 · · · σαr and the desired expression for μj . In a similar manner we obtain μ0 and μ.
2
Example 8.1. If α = δ = 0 and β = γ = 0, then we can use√Table 1 to obtain μ0 = σα σα , which is the arcsine law with Cauchy transform Gμ0 (z) = 1/ z2 − α 2 whereas μ is the Wigner semicircle distribution σα . In turn, if α = δ = 0 and β = γ = 0, then μ0 = δ0 , whereas μ = σβ .
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9. Weighted binary trees and Catalan paths In this section we show how to express the limit distributions in terms of walks on weighted binary trees, or equivalently, in terms of weighted Catalan paths. The binary tree serves here as an example of the strongly matricially free Fock space. The usual framework which gives a description of distributions in terms of walks on graphs is the following. Let W (n) denote the set of root-to-root walks of length n on a rooted graph (G, e) and let μ be the spectral distribution of (G, e), i.e. the distribution given by the moments of the adjacency matrix A(G) in the state ϕ associated with the vector δe on the space of square integrable functions on the set V (G) of the vertices of G. Then the n-th moment of A(G) in the state ϕ is equal to the cardinality of the set W (n). In particular, it is well known that the moments of the Wigner semicircle distribution of variance 1 can be expressed in terms of walks on the half-line (T1 , e) with the first vertex denoted by e and chosen as the root. For many distributions we have to use a more general framework, in which the moments of these distributions are expressed in terms of root-to-root (random or, more generally, weighted) walks on some rooted graph, except that to each walk w on this graph we have to assign a realvalued weight ξ(w). Then we can write Mμ (n) =
ξ(w)
w∈W (n)
for any n 1, where μ is the considered distribution. In particular, we obtain the moments of σα for any α > 0 by putting ξ(w) = α n , where n = |w| is the length of w. In the cases which are of interest to us, the
weight function ξ is first defined on the set of edges E(G) of G and then is extended to W = n1 W (n) by multiplicativity. Namely, if we are given a mapping ξ : E(G) → R, we set ξ(w) = ξ(E1 )ξ(E2 ) . . . ξ(En ), where w = (E1 , E2 , . . . , En ) and E1 , E2 , . . . , En are the edges of w (we choose to describe walks on graphs as sequences of edges). Such extension, by abuse of notation denoted also by ξ , will be called multiplicative. For instance, it is easy to see that the moments of μ = σα σδ can be expressed in this form. It is enough to take the free product of two half-lines, which is the binary tree (T2 , e) with root e. Let us color this graph in the natural way, namely each edge which belongs to a copy of the first half-line is colored by 1 and each edge which belongs to a copy of the second half-line is colored by 2. Then the above formula holds for the moments of μ if we take ξ(E) = α whenever E is colored by 1 and ξ(E) = δ whenever E is colored by 2. We will demonstrate below that we can express our distributions μ0 , μ1 , μ2 in a similar form (in particular, we can use the binary tree), except that the weight function ξ will depend on all four parameters which appear in the matrix A. Before formulating the theorem, let us introduce special weight functions related to matricial freeness. Definition 9.1. Let Tr be an r-ary rooted tree with root e and let A ∈ Mr (R). The weight function ξ : E(Tr ) → R which assigns the entries of A to the edges of Tr is called matricial if, for any pair of edges E1 , E2 ∈ E(Tr ), incident on the same vertex and such that E1 is the ‘father’ of E2 ,
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Fig. 3. Binary tree with a matricial weight function.
the following implication holds: ξ(E1 ) = ai,j
for some i, j
⇒
ξ(E2 ) = ak,i
for some k.
The unique multiplicative extension of this weight function to the set of all walks on Tr will also be called matricial. We specialize to p = 2 and the binary tree. Note that any matricial weight function ξ on the binary tree is uniquely determined (up to equivalence) by the set of those entries of the matrix A which are assigned to the set {E1 , E2 } of two edges incident on the root of the tree, called the initial weights. In order to establish a connection with our limit distributions, we will assume, as in Section 8, that A is the ‘square root’ of B = DU of Section 6. Then, in particular, the binary tree with the matricial weight function associated with A and the initial weights {α, δ}, shown in Fig. 3, describes μ0 as we show below. Finally, recall after [1,14] that if (G1 , e1 ) and (G2 , e2 ) are two locally finite simple graphs and μ1 and μ2 are the associated spectral distributions, then the s-free product of G1 and G2 (in that order) can be interpreted as this half of the free product G1 ∗ G2 which ‘begins’ (starting from the root) with a copy G1 . Moreover, the associated spectral distribution is given by μ1 μ2 . Let us add that a similar result holds in the multiplicative case: both the s-free multiplicative convolution and the associated s-free loop product of graphs were introduced in [15]. Below we shall use the s-free product of half-lines, which are (left and right) halves of the binary tree. Theorem 9.1. Let ξ0 , ξ1 , ξ2 be the multiplicative matricial weight functions on the set of walks on T2 associated with matrix A and the initial weights {α, δ}, {β, δ} and {α, γ }, respectively. Then Mμj (n) =
ξj (w)
(9.1)
w∈W (n)
for j = 0, 1, 2 and any n ∈ N, where W (n) denotes the set of root-to-root walks on T2 of length n. Proof. We need to translate the result of Corollary 8.1 to the language of graphs. It is well known that the moments of σα can be interpreted in terms of walks on the half-line with the weight α
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assigned to each edge. The boolean compression Tt of √ σα changes only the weight assigned to the edge incident on the root, namely it multiplies it by t = β/α, which can be illustrated as . Now, the s-free additive convolutions of compressed semicircle distributions which appear in the decompositions of Corollary 8.1, namely σα,β
σδ,γ
and σδ,γ
σα,β
are the spectral distributions of the s-free products of these half-lines (taken in two different orders). These turn out to be the spectral distributions of the two halves of the binary tree T2 with the weight function defined by the initial set {β, γ }. In order to obtain μ0 , we still need to apply T1/t and T1/s , respectively, to the left and right halves of the tree, which amounts to changing the initial weights from β and γ , respectively, to α and δ. As a result, we obtain the weight function associated with A and the initial set {α, δ}. This proves the statement concerning μ0 (the corresponding weight function on the binary tree is shown in Fig. 3). The cases of μ1 and μ2 are very similar (at the end, a boolean compression is applied to only one half of the binary tree). 2 Another geometric interpretation of Corollary 8.1 can be given in terms of Catalan paths. In order to define a Catalan path, we begin with a function f : [2n] → [n], such that f (0) = f (2n) and |f (i) − f (i − 1)| = 1 for any 1 i 2n, and then we define a Catalan path as its unique extension f : [0, 2n] → [0, n] (by abuse of notation, denoted by the same symbol) obtained by connecting each (i − 1, f (i − 1)) with (i, f (i)) with a segment, where 1 i 2n. Clearly, each Catalan path consists of segments of two types: ‘rises’ R1 , R2 , . . . , Rn , and ‘falls’ F1 , F2 , . . . , Fn . Moreover, to each ‘rise’ Rj there corresponds the closest ‘fall’ Fτ (j ) lying to the right of Rj and on the same (vertical) level. There is a natural mapping from the set W (2n) of walks of length 2n on the binary tree and the set C(n) of Catalan paths of length 2n. In order to rephrase Theorem 8.1 in terms of Catalan paths, we need to take the sets of weighted Catalan paths, by which we understand pairs (f, ξ ), where f is a Catalan path and ξ is a real-valued weight function defined on the set of segments of f . The multiplicative formula ξ(f ) = ξ(R1 ) . . . ξ(Rn )ξ(F1 ) . . . ξ(Fn ) assigns the corresponding weight to f . Weighted Catalan paths of special type defined below allow us to rephrase Theorem 8.1. Definition 9.2. A weighted Catalan path (f, ξ ) of length 2n is called matricially weighted if ξ assigns entries of A ∈ Mp (R) to the segments of f in such a way that the following implications hold: ξ(R1 ) = ai,j
for some i, j
⇒
ξ(R2 ) = ak,i
for some k,
ξ(F1 ) = ai,j
for some i, j
⇒
ξ(F2 ) = aj,k
for some k,
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Fig. 4. A weighted Catalan path associated with A.
for any two consecutive ‘rises’ R1 , R2 and two consecutive ‘falls’ F1 , F2 , and the same weights are assigned to Ri and Fτ (i) for each i ∈ [n]. We will consider below the set of matricially weighted Catalan paths of length 2n associated with the matrix A. In order to rephrase Theorem 9.1, using weighted Catalan paths, we need to restrict the set of weights which can be assigned to the first segment of each path (by analogy with trees, we call them initial weights). An example of a weighted Catalan path contributing to μ0 is given in Fig. 4. Corollary 9.1. Let C0 (n), C1 (n) and C2 (n) be the sets of matricially weighted Catalan paths associated with A, with the initial weights {α, δ}, {β, δ} and {γ , α}, respectively. Then Mμj (2n) =
ξ(f )
(f,ξ )∈Cj (n)
for j = 0, 1, 2 and any n ∈ N. Proof. This is an easy consequence of Theorem 9.1 since there is a natural bijection between each Cj (n) and the pair (W (2n), ξj ) for any n. 2 Acknowledgments The remarks and suggestions of the anonymous referees were very helpful in the preparation of the revised version of the manuscript. References [1] L. Accardi, R. Lenczewski, R. Sałapata, Decompositions of the free product of graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007) 303–334. [2] M. Anshelevich, Free Meixner states, Comm. Math. Phys. 276 (2007) 863–899. [3] D. Avitzour, Free products of C ∗ -algebras, Trans. Amer. Math. Soc. 271 (1982) 423–465. [4] Ph. Biane, Processes with free increments, Math. Z. 227 (1998) 143–174. [5] M. Bo˙zejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987) 170–186. [6] M. Bo˙zejko, R. Speicher, ψ -independent and symmetrized white noises, in: L. Accardi (Ed.), Quantum Probability and Related Topics VI, World Scientific, Singapore, 1991, pp. 170–186. [7] M. Bo˙zejko, J. Wysocza´nski, Remarks on t -transformations of measures and convolutions, Ann. Inst. H. Poincaré Probab. Statist. 37 (6) (2001) 737–761. [8] T. Cabanal-Duvillard, Probabilités libres et calcul stochastique. Application aux grandes matrices aléatoires, PhD thesis, l’Université Pierre et Marie Curie, 1998. [9] T. Cabanal-Duvillard, V. Ionescu, Un théorème central limite pour des variables aléatoires non-commutatives, C. R. Acad. Sci. Paris, Sér. I 325 (1997) 1117–1120. [10] W.M. Ching, Free products of von Neumann algebras, Trans. Amer. Math. Soc. 178 (1993) 147–163. [11] K. Dykema, On certain free product factors via an extended matrix model, J. Funct. Anal. 112 (1993) 31–60.
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[12] U. Franz, Multiplicative monotone convolutions, in: M. Bo˙zejko, et al. (Eds.), Quantum Probability, in: Banach Center Publ., vol. 73, 2006, pp. 153–166. [13] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959) 336–354. [14] R. Lenczewski, Decompositions of the free additive convolution, J. Funct. Anal. 246 (2007) 330–365. [15] R. Lenczewski, Operators related to subordination for free multiplicative convolutions, Indiana Univ. Math. J. 57 (2008) 1055–1103. [16] R. Lenczewski, Asymptotic properties of random matrices and pseudomatrices, arXiv:1001.0667 [math.OA], 2010. [17] R. Lenczewski, R. Sałapata, Discrete interpolation between monotone probability and free probability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006) 77–106. [18] R. Lenczewski, R. Sałapata, Noncommutative Brownian motions associated with Kesten distributions and related Poisson processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008) 351–375. [19] N. Muraki, Monotonic independence, monotonic central limit theorem and monotonic law of small numbers, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001) 39–58. [20] N. Muraki, Monotonic convolution and monotonic Levy–Hincin formula, preprint, 2001. [21] A. Nica, Multi-variable subordination distribution for free additive convolution, J. Funct. Anal. 257 (2009) 428–463. [22] R. Speicher, A new example of ‘Independence’ and ‘White Noise’, Probab. Theory Related Fields 84 (1990) 141– 159. [23] R. Speicher, R. Woroudi, Boolean convolution, in: D. Voiculescu (Ed.), Free Probability Theory, in: Fields Inst. Commun., vol. 12, Amer. Math. Soc., 1997, pp. 267–279. [24] D. Voiculescu, Symmetries of some reduced free product C ∗ -algebras, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, in: Lecture Notes in Math., vol. 1132, Springer-Verlag, 1985, pp. 556–588. [25] D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991) 201–220. [26] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, I, Comm. Math. Phys. 155 (1993) 71–92. [27] D. Voiculescu, Lectures on free probability theory, in: Lectures on Probability Theory and Statistics, Saint-Flour, 1998, in: Lecture Notes in Math., vol. 1738, Springer, Berlin, 2000, pp. 279–349. [28] D. Voiculescu, K. Dykema, A. Nica, Free Random Variables, CRM Monogr. Ser., vol. 1, Amer. Math. Soc., Providence, 1992. [29] E. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958) 325–327.
Journal of Functional Analysis 258 (2010) 4122–4153 www.elsevier.com/locate/jfa
Confluent operator algebras and the closability property ✩ H. Bercovici a,∗ , R.G. Douglas b , C. Foias b , C. Pearcy b a Department of Mathematics, Indiana University, Bloomington, IN 47405, United States b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States
Received 18 September 2009; accepted 10 March 2010 Available online 26 March 2010 Communicated by N. Kalton Dedicated to the memory of our good friends and mentors, Paul R. Halmos and Béla Sz˝okefalvi-Nagy
Abstract Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence. © 2010 Elsevier Inc. All rights reserved. Keywords: Confluent algebra; Closability property; Completely nonunitary contraction; Rationally strictly cyclic vector; Quasisimilarity
✩
H.B. and R.G.D. were supported in part by grants from the National Science Foundation.
* Corresponding author.
E-mail addresses:
[email protected] (H. Bercovici),
[email protected] (R.G. Douglas),
[email protected] (C. Pearcy). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.009
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1. Introduction Probably the best known problem in operator theory is the question of whether every bounded linear operator on a complex, separable, infinite dimensional Hilbert space H has a nontrivial invariant subspace. Despite considerable effort by many researchers for more than half a century, the general problem remains open. A generalization, still unresolved, asks whether every transitive algebra of operators must be dense in the weak operator topology. (Recall that an algebra is said to be transitive if there is no common nontrivial invariant subspace for the operators in it.) In the sixties, Arveson approached this problem iteratively, starting from an observation going back essentially to von Neumann. Namely, assume that A is an algebra of operators on a Hilbert space H, and n 1 is an integer. The algebra A is said to be n-transitive if every invariant subspace for A(n) = X (n) = X ⊕ X ⊕ · · · ⊕ X: X ∈ A n times
is invariant for every operator of the form Y (n) where Y is an operator on H. Then A is dense, in the weak operator topology, if and only if it is n-transitive for every n 1. Arveson observed that 2-transitivity is equivalent to the following statement: every closed linear transformation commuting with A is a scalar multiple of the identity operator. For n 3, n-transitivity is implied by a similar statement for densely defined linear transformations commuting with A. Thus, provided that every densely defined linear transformation commuting with A is closable, 2-transitivity implies n-transitivity for all n. As a consequence, Arveson was able to prove that transitive algebras containing certain kinds of subalgebras are indeed dense in the weak operator topology. His results apply to algebras on an L2 -space, containing the algebra L∞ of all bounded measurable multipliers, and to algebras on the Hardy space H 2 (D), containing the algebra H ∞ (D). A few similar results were obtained by others for closely related algebras in the following years; see for instance [19, Chapter 8]. A year ago, Haskell Rosenthal became interested in the question of which algebras of operators on Hilbert space had what he called the closability property which means that every densely defined linear transformation in its commutant is closable. A key step in Arveson’s proofs was to show that the algebras L∞ acting on L2 , and H ∞ (D) acting on H 2 (D), have the closability property. Rosenthal showed that various algebras have the closability property and asked the authors a specific followup question. In finding the answer, the question piqued our interest and resulted in a series of questions related to this topic. Our investigation took us in some unexpected directions, making surprising connections with other topics in operator theory. After some preliminaries in Section 2, we begin in Section 3 by investigating the closability property and determining some algebras which have it, as well as some that do not. In Section 4 we introduce the concept of a rationally strictly cyclic vector, and show that the existence of such a vector for a commutative algebra A implies the closability property. In Section 5 we discuss the invariance of the closability property, and of the existence of rationally strictly cyclic vectors, under an appropriate notion of quasisimilarity. We deduce, for instance, that the commutant of any contraction of class C0 has the closability property. In the course of our study, the importance of something like a functional calculus for quotients became clear. To make this idea precise, in Section 6 we study the related notion of confluence (introduced in Section 4) as it applies to the algebra obtained by applying the H ∞ functional calculus to a completely nonunitary contraction. Confluence implies the existence of a rationally strictly cyclic vector, and therefore the closability
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property as well. Section 7 contains a thorough study of confluence in the context of functional models for contractions. In particular, a characterization is obtained for those contractions which are quasisimilar to the unilateral shift of multiplicity one. This characterization involves the ‘size’ of the analytic functions in the reproducing kernel representative for the operator. The analysis of confluence is somewhat subtle and rests on the harmonic analysis of contractions [23], the theory of the class C0 [4], the theory of dual algebras [5], and results about the class B1 (D) [10]. We thank Haskell Rosenthal for the questions which led to this research. We are also grateful to the referee, who pointed out several errors and numerous misprints in our original manuscript. 2. Preliminaries We will work with operators on Hilbert spaces over the complex numbers C. The algebra of bounded linear operators on a Hilbert space H is denoted L(H). Given T ∈ L(H), PT denotes the smallest unital algebra containing T ; that is, the set of all polynomials in T . The closure of PT in the weak operator topology (also known as WOT) is denoted WT . The norm closure of a subset M ⊂ H is denoted M. The orthogonal projection of H onto a closed linear subspace M ⊂ H is denoted PM . Several special operators play an important role. The space L2 is the space of functions defined on the unit circle T which are square integrable relative to arclength measure. The bilateral shift U ∈ L(L2 ) is the unitary operator defined by (Uf )(ζ ) = ζf (ζ ) for f ∈ L2 and a.e. ζ ∈ T. The Hardy space H 2 ⊂ L2 is the cyclic subspace for U generated by the constant function 1, and S ∈ L(H 2 ) is the unilateral shift of multiplicity 1 defined as S = U | H 2 . More generally, denote by H ∞ = H ∞ (D), the algebra of bounded analytic functions in the unit disk D. For each u ∈ H ∞ one defines an analytic Toeplitz operator Tu ∈ L(H 2 ) as the operator of pointwise multiplication by u. Here one takes advantage of the fact that functions in H ∞ have a.e. defined radial limits on T. Given a subset A ⊂ L(H), A denotes the set of operators commuting with every element of A. The set A is called the commutant of A, and it is a unital algebra, closed in the weak operator topology. An important example is {S} = WS = Tu : u ∈ H ∞ . A function m ∈ H ∞ is inner if |m(ζ )| = 1 for a.e. ζ ∈ T. For every inner function m ∈ H ∞ , the space mH 2 = Tm H 2 is closed and invariant for S. The compression of S to H(m) = H 2 mH 2 is denoted S(m). In other words, S(m) = PH(m) S | H(m) or, equivalently, S(m)∗ = S ∗ | H(m). Another important example of a commutant is
S(m)∗ = WS(m)∗ = Tu∗ H(m): u ∈ H ∞ .
This was proved by Sarason [20]. An operator T ∈ L(H) is a contraction if T 1. A contraction T is completely nonunitary if it has no invariant subspace on which it acts as a unitary operator. For completely nonunitary contractions T , there is a homomorphism u → u(T ) ∈ L(H) which extends the polynomial functional calculus to functions u ∈ H ∞ . This is called the Sz.-Nagy–Foias functional calculus. For instance, u(S) = Tu , and u(S(m)) = PH(m) Tu | H(m).
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A completely nonunitary contraction T ∈ L(H) is of class C0 if u(T ) = 0 for some u ∈ H ∞ \ {0}. The ideal {u ∈ H ∞ : u(T ) = 0} ⊂ H ∞ is principal, and it is generated by an inner function, uniquely determined up to a constant factor of absolute value 1. This function is called the minimal function of T . The most basic example is S(m), whose minimal function is m. We refer the reader to [23] for further background on the analysis of contractions, to [5] for dual algebras, and to [4] for the class C0 . We will refer as needed to these and other original sources for specific results. 3. The closability property Consider a unital subalgebra A of the algebra L(H) of bounded operators on the complex Hilbert space H. The algebra A is not assumed to be norm closed. Definition 3.1. A linear transformation X : D(X) → H is said to commute with A if for every h ∈ D(X) and every T ∈ A we have T h ∈ D(X) and XT h = T Xh. We define now the main concept we study in this paper. Definition 3.2. The algebra A is said to have the closability property if every linear transformation X which commutes with A, and whose domain D(X) is dense in H, is closable. We recall that a linear transformation X is closable if the closure of its graph G(X) = h ⊕ Xh: h ∈ D(X) is again the graph of a linear transformation, usually denoted X and called the closure of X. Equivalently, X is closable if given a sequence hn ∈ D(X) such that limn→∞ hn = 0 and the limit k = limn→∞ Xhn exists, it follows that k = 0. The following observation is a trivial consequence of the fact that a linear transformation commuting with an algebra also commutes with smaller algebras. Lemma 3.3. Assume that A ⊂ B ⊂ L(H) are unital algebras. If A has the closability property then so does B. In particular, if A is commutative and has the closability property, then its commutant A also has the closability property. We start with some examples of algebras which do not have the closability property. The arguments are based on the following simple fact. Lemma 3.4. Let A be a unital subalgebra of L(H). Assume that there exist linear manifolds M, N ⊂ H such that (1) T M ⊂ M and T N ⊂ N for every T ∈ A; (2) M ∩ N = {0} and M + N = H; (3) M ∩ N = {0}. Then A does not have the closability property.
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Proof. Define a linear transformation with domain D(X) = M + N by setting Xh = 0 for h ∈ M and Xh = h for h ∈ N . If X were closable, its closure would satisfy Xh = 0 and Xh = h for any h ∈ M ∩ N , and this is absurd for h = 0. 2 Proposition 3.5. The following algebras do not have the closability property: (1) (2) (3) (4) (5)
The algebra PS generated by the unilateral shift S. The algebra PS(m) , where m is an inner function which is not a finite Blaschke product. The WOT-closed algebra WS ∗ . The WOT-closed algebra WU generated by the bilateral shift U on L2 . Any algebra of the form A ⊗ IK , where A ⊂ L(H) is a unital algebra, and K is an infinite dimensional Hilbert space.
Proof. For the first example, choose an outer function f ∈ H 2 which is not rational, and define M to consist of all polynomials and N = {pf : p a polynomial}. The hypotheses of Lemma 3.4 are verified trivially since both of these spaces are dense in H 2 . Next, assume that m is an inner function but not a finite Blaschke product, and consider a factorization m = m1 m2 such that the inner functions mj are not finite Blaschke products. We can define then subspaces M, N ⊂ H(m) by M = {PH(m) p: p a polynomial} and N = {PH(m) (pm2 ): p a polynomial}. The space M is dense in H(m), so to verify the hypotheses of Lemma 3.4 it suffices to show that M ∩ N = {0}. Consider indeed two polynomials p, q such that PH(m) p = PH(m) (qm2 ). In other words, we have p = qm2 + rm1 m2 for some r ∈ H 2 . If p = 0, this equality implies that the inner factor of p (obviously a finite Blaschke product) is divisible by m2 , contrary to our choice of factors. For example (3), we choose M = {p: p a polynomial} ⊂ H 2 , and we denote by N the linear manifold generated by the functions kλ (z) = (1 − λz)−1 , λ ∈ D \ {0}. These spaces are dense in H 2 , and the identities ∗
p(z) − p(0) S p (z) = , z
S ∗ kλ = λkλ
easily imply that they are invariant under WS ∗ . Finally, a function p in their intersection must be both a polynomial, and a rational function vanishing at ∞, hence p = 0. For example (4), define two sets ω± = {e±it : 0 < t < 3π/2} ⊂ T, denote by χ± their characteristic functions, and set M = χ+ H 2 and N = χ− H 2 . Since M = χ+ L2 and N = χ− L2 , we clearly have M + N = L2 and M ∩ N = χω+ ∩ω− L2 . The fact that M ∩ N = {0} follows easily from the F. and M. Riesz theorem. Finally, assume that K is an infinite dimensional Hilbert space, and let M0 , N0 ⊂ K be two dense linear manifolds such that M0 ∩ N0 = {0}. Then M = H ⊗ M0 and N = H ⊗ N0 will satisfy the hypotheses of Lemma 3.4 for the algebra A ⊗ IK . 2 The first two examples above indicate that an algebra with the closability property must be reasonably large, while the last one shows that it should not have uniform infinite multiplicity. In this paper we will focus on algebras which have multiplicity one. The first example of an algebra with the closability property was of this kind: any maximal abelian selfadjoint subalgebra of L(H) has the closability property, as shown in [3]. This, along with the examples described in Proposition 3.7 (the first of which also appeared in [3], while the second was proved indepen-
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dently in [21]), will be treated in a unified manner in Section 4. The proofs of these particular cases do in fact suggest the more general methods. First, a useful observation about bounded outer functions. This is certainly known, but we could not find a reference. We use the notation u 2 for the norm of a function u ∈ L2 . ∞ with Lemma 3.6. Let v ∈ H ∞ be an outer function. There exist outer functions (wn )∞ n=1 ⊂ H 2 the property that limn→∞ u − wn vu 2 = 0 for every u ∈ L . In particular, if T is a completely nonunitary contraction, the sequence (wn (T )v(T ))∞ n=1 converges to I in the strong operator topology.
Proof. The functions wn will be specified by the requirements that (wn v)(0) > 0, and (wn v)(ζ ) = 1 |v(ζ )|
if |v(ζ )| 1/n, if |v(ζ )| < 1/n
for a.e. ζ ∈ T, so that wn ∞ n. Observe that 1 (wn v)(0) = 2π
logv(ζ ) |dζ | → 1
|v(ζ )|<1/n
as n → ∞, and it follows easily that (wn v)(λ) → 1 uniformly on every compact subset of D. This also implies that wn v → 1 in the weak* topology of H ∞ (given by its duality with L1 /H01 ). Fix now u ∈ L2 of unit norm, and use this weak convergence to deduce that
lim sup u − wn vu 22 n→∞
= lim sup wn vu 22 n→∞
1 − 2 Re lim n→∞ 2π
2π wn v|u|2 |dζ | + 1 0, 0
= lim sup wn vu 22 n→∞ The lemma follows.
− 1 0.
2
Proposition 3.7. The algebras WS and WS(m) have the closability property. Proof. Recall first that every function in H 2 is the quotient of two bounded functions in H ∞ . For instance, given a nonzero function f ∈ H 2 , denote by vf the unique outer function defined by the requirements that vf (0) > 0 and vf (ζ ) = min 1,
1 |f (ζ )|
for almost every ζ ∈ T. The functions vf and uf = f vf belong to H ∞ , and in fact uf (ζ ) = min 1, f (ζ )
a.e. ζ ∈ T.
(3.1)
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Consider first the algebra WS which consists precisely of the analytic Toeplitz operators Tu with u ∈ H ∞ . Let X be a densely defined linear transformation commuting with this algebra, and let f, g ∈ D(X). Observe first that uf = vf f = Tvf f ∈ D(X), and therefore we can write vg uf Xg = Tvg uf Xg = XTvg uf g = X(vg uf g) = X(uf ug ) = XTug uf = Tug Xuf = ug Xuf . Let now gn ∈ D(X) be a sequence converging to zero such that the limit h = limn→∞ Xgn exists. Passing if necessary to a subsequence, we may assume that gn (ζ ) → 0 for almost every ζ ∈ T. By virtue of (3.1) we also have |vgn (ζ )| → 1 and ugn (ζ ) → 0 for a.e. ζ , and therefore ugn Xuf 22
1 = 2π
2π it 2 ug e (Xuf ) eit 2 dt → 0 n 0
as n → ∞ by the dominated convergence theorem. The identity vgn uf Xgn = ugn Xuf proved earlier, along with the fact that |vgn | → 1 a.e., implies that uf h = 0 for every f ∈ D(X). Choosing a function f which is not identically zero, we deduce that h = 0, thus proving that X is closable. Consider now an inner function m, and define a map J : H(m) → H(m) by setting (Jf )(ζ ) = ζf (ζ )m(ζ ),
ζ ∈ T.
(3.2)
This is a conjugate linear surjective isometry on H(m) satisfying the equation J S(m) = S(m)∗ J . More generally, we have the identity J A = A∗ J,
A ∈ WS(m) .
This identity is easily verified when A is a polynomial in S(m), and it extends to arbitrary A using the continuity properties of the functional calculus with H ∞ functions. Let us denote by ξ = 1 − m(0)m ∈ H(m) the projection of 1 onto H(m). This is a separating vector for H(m). That is, the equality Aξ = 0 implies A = 0 for A ∈ H(m). Indeed, if A = u(S(m)), we have Aξ = PH(m) u, and this function is zero if and only if u divides m, in which case u(S(m)) = 0. Consider now a densely defined linear transformation X commuting with WS(m) . We will show that X is closable by proving the identity Xh1 , J h2 = h1 , J Xh2 ,
h1 , h2 ∈ D(X),
which shows that J X ⊂ X ∗ J , and hence X ∗ is densely defined. Indeed, fix h1 , h2 ∈ D(X), and choose an outer function v ∈ H ∞ such that the functions a1 = vh1 ,
a2 = vh2 ,
b1 = vXh1 ,
b2 = vXh2
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are bounded. Set V = v(S(m)), Aj = aj (S(m)), Bj = bj (S(m)) so that V hj = Aj ξ and V Xhj = Bj ξ for j = 1, 2. Observe first that A1 B2 ξ = A1 V Xh2 = XA1 V h2 = XA1 A2 ξ, and a similar calculation shows that A1 B2 ξ = A2 B1 ξ . We conclude that A1 B2 = A2 B1 because ξ is separating. Next we use Lemma 3.6 to find operators Wn ∈ WS(m) such that Wn V converges to I in the strong operator topology. We have then Xh1 , J h2 = lim Wn V Xh1 , J Wn V h2 n→∞
= lim Wn B1 ξ, J Wn A2 ξ n→∞ = lim Wn ξ, B1∗ J Wn A2 ξ n→∞
= lim Wn ξ, J Wn B1 A2 ξ n→∞
= lim Wn ξ, J Wn A1 B2 ξ n→∞ = lim Wn ξ, A∗1 J Wn B2 ξ n→∞
= lim Wn A1 ξ, J Wn B2 ξ = h1 , J Xh2 , n→∞
where we used the identity A1 B2 = A2 B1 and the fact that Wn commutes with Aj and Bj . The theorem is proved. 2 Note incidentally that the example of WS shows that closability of an algebra A is not generally inherited by the algebra {T ∗ : T ∈ A}. We conclude this section with a simple fact which will be used in thestudy of closability for quasisimilar algebras. Let Ai ⊂ L(Hi ), i ∈ I , be algebras. The algebra i∈I Ai ⊂ L( i∈I Hi ) consists of those operators of the form i∈I Ti , where Ti ∈ Ai for each i, and sup{ Ti : i ∈ I } < ∞. Lemma 3.8. A direct sum A = property for every i ∈ I .
i∈I
Ai has the closability property if and only if Ai has this
Proof. Assume first that A has the closability property, and Xi0 is a densely defined linear transformation on Hi0 commuting with Ai0 for some i0 ∈ I . We define a linear transformationX with dense domain D(X) = i∈I Di , where Di0 = D(Xi0 ), Di = Hi for i = i0 , and X( hi ) = ki , where ki0 = Xi0 hi0 and ki = 0 for i = i0 . The linear transformation X commutes with A, hence it is closable. It follows that Xi0 must be closable as well. Conversely, assume that each Ai has the closability property, and let X be a densely defined linear transformation commuting with A. If Pj ∈ A denotes the orthogonal projection onto the j th component of i∈I Hi , we have then Pj X ⊂ XPj , and the linear transformation Xj : Dj = Pj D(X) → Hj defined by Xj = X | Dj commutes with Aj . It follows that each Xj is closable, and then it is easy to verify that X is closable as well. 2
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4. Rationally strictly cyclic vectors and confluence The examples in Proposition 3.7, as well as maximal abelian selfadjoint subalgebras (also known as masas), can actually be treated in a unified manner. For this purpose we need a new concept. Definition 4.1. Let A ⊂ L(H) be a unital algebra. A vector h0 ∈ H is called a rationally strictly cyclic vector for A if for every h ∈ H there exist A, B ∈ A such that Bh = Ah0 and ker B = {0}. Recall that h0 is said to be strictly cyclic for A if Ah0 = H. Thus, a strictly cyclic vector is rationally strictly cyclic, but not conversely. The examples considered in this paper do not have strictly cyclic vectors except in trivial cases. Lemma 4.2. The following algebras have rationally strictly cyclic vectors: (1) WS . (2) WS(m) . (3) Any masa on a separable Hilbert space. More generally, any masa with a cyclic vector. Proof. The vector 1 ∈ H 2 is rationally strictly cyclic for WS , while 1 − m(0)m = PH(m) 1 is rationally strictly cyclic for WS(m) . For (3), we may assume that H = L2 (μ), where μ is a Borel probability measure on some compact topological space, and A = {Mu : u ∈ L∞ (μ)}, where Mu f = uf,
u ∈ L∞ (μ), f ∈ L2 (μ).
Since every function in L2 (μ) is the quotient of two bounded functions, the constant function 1 is rationally strictly cyclic for A. 2 Here are two useful properties of algebras with rationally strictly cyclic vectors. Lemma 4.3. Let A ⊂ L(H) be a unital algebra with a rationally strictly cyclic vector h0 . (1) If T ∈ A \ {0} then T h0 = 0. (2) If A is commutative and D ⊂ H is a dense linear manifold, invariant for A, then {ker T : T ∈ A, T h0 ∈ D} = {0}. Proof. Assume that T ∈ A and T h0 = 0. Given x ∈ H, choose Ax , Bx ∈ A such that Bx x = Ax h0 and ker Bx = {0}. We have then Bx T x = T Bx x = T Ax h0 = Ax T h0 = 0, and therefore T x = 0. This implies that T = 0 since x is arbitrary. Assume now that A is commutative and D ⊂ H is a dense linear manifold, invariant for A. Let h ∈ H be a vector such that Ah = 0 whenever A ∈ A and Ah0 ∈ D. Using the notation above, we have Ax h0 = Bx x ∈ D whenever x ∈ D, and therefore Ax h = 0 for x ∈ D. Thus
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0 = Bh Ax h = Ax Bh h = Ax Ah h0 = Ah Ax h0 = Ah Bx x = Bx Ah x for x ∈ D, which implies Ah x = 0 for such vectors x. From the density of D we deduce that Ah = 0, and thus Bh h = Ah h0 = 0 and h = 0, as desired. 2 We can now prove a generalization of Proposition 3.7. Theorem 4.4. Any unital commutative algebra A ⊂ L(H) which has a rationally strictly cyclic vector has the closability property. Proof. Let h0 ∈ H be a rationally strictly cyclic vector for the algebra A, and let X be a linear transformation with dense domain D(X) commuting with A. Consider a sequence {xn } ⊂ D(X) such that xn → 0 and Xxn → h as n → ∞. By Lemma 4.3(2), it will suffice to show that T h = 0 whenever T ∈ A and T h0 ∈ D(X). Fix such an operator T , and choose operators An , Bn , B, A ∈ A satisfying Bn xn = An h0 , BXT h0 = Ah0 and ker Bn = ker B = {0} for all n 1. Using the commutativity of A, and the fact that X commutes with A we deduce that Bn (BT Xxn − Axn ) = BT XBn xn − ABn xn = BT XAn h0 − AAn h0 = BXAn T h0 − AAn h0 = An (BXT h0 − Ah0 ) = 0. Since Bn is one-to-one, we have BT Xxn = Axn . Letting n → ∞ we obtain BT h = 0 and hence T h = 0, as desired.
2
Observe that if an algebra B ⊂ L(H) contains a unital commutative algebra A with the closability property, then B also has the closability property. Therefore Theorem 4.4 and the result of Arveson mentioned in the introduction have the following consequence. Corollary 4.5. There exists no proper subalgebra of L(H) which is 2-transitive, closed in the strong operator topology and contains a unital commutative subalgebra with a rationally strictly cyclic vector. The calculations in the preceding proof can be used to relate closed, densely defined linear transformations commuting with A with linear transformations of the form B −1 A with A, B ∈ A and ker B = {0}. Note that
G B −1 A = {h ⊕ k ∈ H ⊕ H: Ah = Bk}, and this is generally larger than
G AB −1 = {Bh ⊕ Ah: h ∈ H}.
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Also observe that two linear transformations of this form, say B −1 A, B1−1 A1 , which agree on a dense linear manifold D, must in fact be equal. Indeed, the equality on D implies that BA1 h = B1 Ah for h ∈ D, and therefore B1 A = BA1 . Thus for h ⊕ k ∈ G(B −1 A) we have B(B1 k − A1 h) = B1 (Bk − Ah) = 0, and hence h ⊕ k ∈ G(B1−1 A1 ) because B is injective. Proposition 4.6. Let A be a commutative algebra with a rationally strictly cyclic vector h0 . For every densely defined linear transformation X commuting with A such that h0 ∈ D(X), there exist A, B ∈ A such that ker B = {0} and X ⊂ B −1 A. If X ∈ L(H), we have X = B −1 A. In particular, the commutant A is a commutative algebra. Proof. As in the preceding proof, we choose for each h ∈ H operators Ah , Bh ∈ A such that ker Bh = {0} and Bh h = Ah h0 . Assume now that h0 ∈ D(X) and X commutes with A. We have then for h ∈ D(X), Bh BXh0 Xh = BXh0 XBh h = BXh0 XAh h0 = Ah BXh0 Xh0 = Ah AXh0 h0 = AXh0 Bh h = Bh AXh0 h, −1 from which we conclude that X ⊂ BXh AXh0 because Bh is injective. The remaining assertions 0 follow easily from this. 2
Sometimes an algebra with a rationally strictly cyclic vector has the stronger property defined below. Definition 4.7. Let A ⊂ L(H) be a unital algebra. We will say that A is confluent if for every two vectors h1 , h2 ∈ H \ {0} there exist injective operators B1 , B2 ∈ A such that B1 h1 = B2 h2 . Proposition 4.8. For a commutative unital algebra A ⊂ L(H), the following two assertions are equivalent: (1) A has a rationally strictly cyclic vector and ker B = {0} for every B ∈ A \ {0}; (2) A is confluent. If these equivalent conditions are satisfied, then every nonzero vector is rationally strictly cyclic for A; moreover, every densely defined linear transformation commuting with A is contained in B −1 A for some A, B ∈ A such that ker B = {0}. Proof. Assume first that (1) holds, and h1 , h2 ∈ H \ {0}. With the notation used earlier, we have Ah2 Bh1 h1 = Ah2 Ah1 h0 = Ah1 Bh2 h2 . The operators Ah1 , Ah2 are not zero, and therefore Ah2 Bh1 , Ah1 Bh2 are injective by hypothesis.
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Conversely, assume that A is confluent. Clearly, every nonzero vector is then rationally strictly cyclic. It remains to show that every B ∈ A \ {0} is injective. Assume to the contrary that Bh1 = 0 / ker B. If B1 , B2 are as in Definition 4.7, we obtain for some h1 = 0, and choose h2 ∈ 0 = B1 Bh1 = BB1 h1 = BB2 h2 = B2 Bh2 . This implies Bh2 = 0, contrary to the choice of h2 . The last assertion follows from Proposition 4.6. 2 As an application of the results in this section, we show that some other algebras of Toeplitz operators have the closability property. Consider a bounded, connected open set Ω ⊂ C bounded by n + 1 analytic simple Jordan curves, and fix a point ω0 ∈ Ω. The algebra H ∞ (Ω) consists of the bounded analytic functions on Ω, while Hω20 (Ω) is defined as the space of analytic functions f on Ω with the property that |f |2 has a harmonic majorant in Ω. The norm on Hω20 (Ω) is defined as f 22 = inf u(ω0 ): u a harmonic majorant of |f |2 ,
f ∈ Hω20 (Ω).
Multiplication by a function u ∈ H ∞ (Ω) determines a bounded operator Tu on Hω20 (Ω). Proposition 4.9. The constant function 1 ∈ Hω20 (Ω) is a rationally strictly cyclic vector for the algebra {Tu : u ∈ H ∞ (Ω)}. In particular, this algebra has the closability property. The statement is equivalent to the following result. We refer to [1] and [13] for the function theoretical background. Lemma 4.10. For every function f ∈ Hω20 (Ω) there exist u, v ∈ H ∞ (Ω) such that v ≡ 0 and vf = u. Proof. Denote by π : D → Ω a (universal) covering map such that π(0) = ω0 , and denote by Γ the corresponding group of deck transformations. Thus, Γ consists of those analytic automorphisms ϕ of D with the property that π ◦ ϕ = π . The map f → f ◦ π is an isometry from Hω20 (Ω) onto the space of those functions g ∈ H 2 such that g ◦ ϕ = g for every ϕ ∈ Γ . Fix now f ∈ Hω20 (Ω) \ {0}, and construct an outer function w ∈ H 2 such that |w(ζ )| = min{1, 1/|f ◦ π(ζ )|} for almost every ζ ∈ T. The function w is obviously modulus automorphic in the sense that |w ◦ ϕ| = |w| for every ϕ ∈ Γ . It follows that there is a group homomorphism γ : Γ → T such that w ◦ ϕ = γ (ϕ)w for every ϕ ∈ Γ . Choose a modulus automorphic Blaschke product b ∈ H ∞ such that b ◦ ϕ = γ (ϕ)b for γ ∈ Γ ; see [13, Theorem 5.6.1] for the construction of such products. Then there exist functions u, v ∈ H ∞ (Ω) such that v ◦ π = bw and u ◦ π = bw(f ◦ π). These functions satisfy the requirements of the lemma. 2 5. Quasisimilar algebras We will now study the effect of quasisimilarity on the closability property and the existence of rationally strictly cyclic vectors. Recall that an operator Q ∈ L(H1 , H2 ) is called a quasiaffinity if it is injective and has dense range.
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Definition 5.1. An algebra A1 ⊂ L(H1 ) is a quasiaffine transform of an algebra A2 ⊂ L(H2 ) if there exists a quasiaffinity Q ∈ L(H1 , H2 ) such that for every T2 ∈ A2 , we have QT1 = T2 Q for some T1 ∈ A1 . We write A1 ≺ A2 if A1 is a quasiaffine transform of A2 . The relation A1 ≺ A2 can simply be written as Q−1 A2 Q ⊂ A1 for some quasiaffinity Q. Proposition 5.2. Assume that A1 ⊂ L(H1 ) and A2 ⊂ L(H2 ) are unital algebras such that A1 ≺ A2 . Then (1) If A1 is commutative, then A2 is commutative as well. (2) If A2 has the closability property, then so does A1 . (3) If A2 is confluent, then so is A1 . Proof. Let Q be as in Definition 5.1. Since the map T → Q−1 T Q is obviously an injective algebra homomorphism on A2 , part (1) is immediate. To prove (2), let X be a densely defined linear transformation commuting with A1 . Define the linear transformation Y = QXQ−1 on the dense subspace D(Y ) = QD(X). Since all the operators T2 ∈ A2 have the property that Q−1 T2 Q is in A1 , it follows easily that Y commutes with A2 . Assume now that A2 has the closability property, so that Y is closable. We will verify that X is closable as well. Assume that hn ∈ D(X) are such that hn → 0 and Xhn → k as n → ∞. Obviously then D(Y ) Qhn → 0 and Y Qhn → Qk. We deduce that Qk = 0, and therefore k = 0 since Q is a quasiaffinity. Finally, assume that A2 is confluent and let h1 , h2 ∈ H1 \ {0}. We choose injective C1 , C2 ∈ A2 so that C1 Qh1 = C2 Qh2 , and observe that B1 h1 = B2 h2 , where Bj = Q−1 Cj Q ∈ A1 are injective. 2 Definition 5.3. An algebra A1 ⊂ L(H1 ) is quasisimilar to an algebra A2 ⊂ L(H2 ) if there exist quasiaffinities Q ∈ L(H1 , H2 ) and R ∈ L(H2 , H1 ) such that Q−1 A2 Q ⊂ A1 , R −1 A1 R ⊂ A2 , QR ∈ A2 , and RQ ∈ A1 . We write A1 ∼ A2 if A1 is quasisimilar to A2 . Using the proofs of parts (1) and (2) of the following result, it is easy to see that quasisimilarity is an equivalence relation. Proposition 5.4. Assume that A1 and A2 are commutative quasisimilar algebras, and Q, R satisfy the conditions of Definition 5.3. Then (1) We have Q−1 A2 Q = A1 and R −1 A1 R = A2 . (2) The maps T2 → Q−1 T2 Q and T1 → R −1 T1 R are mutually inverse algebra isomorphisms between A1 and A2 . (3) The commutant A1 is commutative if and only if A2 is commutative. (4) If h1 ∈ H1 is rationally strictly cyclic for A1 then Qh1 is rationally strictly cyclic for A2 . (5) The algebra A1 is confluent if and only if A2 is confluent. (6) The algebra A1 is confluent if and only if A2 is confluent. (7) The algebra A1 has the closability property if and only if A2 does.
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Proof. Define Φ : A2 → A1 and Ψ : A1 → A2 by setting Φ(T2 ) = Q−1 T2 Q and Ψ (T1 ) = R −1 T1 R. We have
Ψ Φ(T2 ) = R −1 Q−1 T2 QR = R −1 Q−1 QRT2 = T2 ,
T2 ∈ A 2 ,
and similarly Φ(Ψ (T1 )) = T1 for T1 ∈ A1 . This proves (2), and (1) follows from (2). Assume now that A1 is commutative and A, B ∈ A2 . We claim that RAQ and RBQ belong to A1 . Indeed,
T1 RAQ = R R −1 T1 R AQ = RA R −1 T1 R Q = RAR −1 T1 (RQ) = RAR −1 (RQ)T1 = RAQT1 for T1 ∈ A1 . We deduce that RAQRBQ = RBQRAQ and hence AQRB = BQRA. Taking A or B to be the identity operator, we deduce that QR commutes with B and A, and therefore QRAB = QRBA, and finally the desired equality AB = BA. To prove (4), assume that h1 is rationally strictly cyclic for A1 . Proposition 4.6 implies the existence of A1 , B1 ∈ A1 such that ker B1 = {0} and RQ = B1−1 A1 . Set A2 = R −1 A1 R, B2 = R −1 B1 R ∈ A2 , and observe that B2 QR = R −1 (B1 RQ)R = R −1 A1 R = A2 , that is QR = B2−1 A2 . Since QR is a quasiaffinity, it follows that ker A2 = {0}. To show that Qh1 is rationally strictly cyclic for A2 , fix a vector h2 ∈ H2 , and choose S1 , T1 ∈ A1 such that ker T1 = {0} and T1 Rh2 = S1 h1 . Set now T2 = R −1 T1 R, S2 = R −1 S1 R ∈ A2 , and note that ker T2 = {0}. We have RQRT2 h2 = RQT1 Rh2 = RQS1 h1 = S1 RQh1 = RS2 Qh1 , so that QRT2 h2 = S2 Qh1 . Applying B2 to both sides we obtain A2 T2 h2 = B2 S2 Qh1 , and strict cyclicity follows because A2 T2 , B2 S2 ∈ A2 and ker(A2 T2 ) = {0}. Assertion (5) follows easily from (4) and Proposition 4.8, or directly from Proposition 5.2(3). Assume now that A1 is confluent, and let h, k ∈ H2 be two nonzero vectors. Then there exist injective operators A1 , B1 ∈ A1 such that A1 Rh = B1 Rk. Thus we have A2 h = B2 k, where A2 = QA1 R and B2 = QB1 R are injective operators in A2 . This proves (6). Finally, assume that A2 has the closability property, and let X be a densely defined linear transformation commuting with A1 . As in the proof of Proposition 5.2(2), to prove (7) it will suffice to show that the linear transformation Y0 = QXQ−1 defined on the dense space D(Y0 ) = QD(X) is closable. To show this, we will define a linear transformation Y ⊃ Y0 which commutes with A2 . Its domain D(Y ) consists of all the finite sums of the form n Tn Qhn , where Tn ∈ A2 and hn ∈ D(X), and Y
n
Tn Qhn =
n
Tn QXhn .
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Toshow that Y is well defined, it will suffice to prove that n Tn Qhn = 0 implies R n Tn QXhn = 0. Indeed, since RTn Q ∈ A1 , we have RTn Qhn ∈ D(X) and
RTn QXhn =
n
XRTn Qhn = XR
n
The fact that Y commutes with every T ∈ A2 is easily verified. If n T Tn Qhn ∈ D(Y ), and YT
n
Tn Qh =
Tn Qhn = 0.
n
T Tn QXhn = T Y
n
The inclusion Y ⊃ Y0 is obvious since A2 is unital.
n Tn Qhn
∈ D(Y ) then clearly
Tn Qhn .
n
2
We will be using the results in this section for the special case of algebras generated by a completely nonunitary contraction T ∈ L(H). For such a contraction we will write H ∞ (T ) = u(T ): u ∈ H ∞ . Parts (1) and (2) of the following lemma are easily verified; in fact Definition 5.3 was formulated so as to make part (2) correct. Lemma 5.5. Let T1 and T2 be two completely nonunitary contractions. (1) (2) (3) (4)
If T1 ≺ T2 then H ∞ (T1 ) ≺ H ∞ (T2 ). If T1 ∼ T2 then H ∞ (T1 ) ∼ H ∞ (T2 ). If H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 is of class C0 , then T2 is also of class C0 . If H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 is not of class C0 , then T1 ∼ ϕ(T2 ) for some conformal automorphism ϕ of D.
Proof. To prove (3), observe that H ∞ (T1 ) ∼ H ∞ (T2 ) implies that H ∞ (T2 ) is isomorphic to H ∞ (T1 ). Assume that T1 is of class C0 . If T1 is a scalar multiple of the identity, then H ∞ (T1 ) = CI , and therefore H ∞ (T2 ) = CI and then T2 must be a scalar multiple of the identity, hence of class C0 . If T1 is not a scalar multiple of the identity, then H ∞ (T1 ) has zero divisors. Indeed, in this case the minimal function m of T1 can be factored into a product m = m1 m2 of two nonconstant inner functions, and then m1 (T1 ) = 0 = m2 (T1 ), while m1 (T1 )m2 (T1 ) = 0. We conclude that H ∞ (T2 ) must also have zero divisors, and this obviously implies that T2 is of class C0 . Finally, assume that H ∞ (T1 ) ∼ H ∞ (T2 ) and T1 (as well as T2 by part (3)) is not of class C0 . Let Q and R be quasiaffinities satisfying the conditions of Definition 5.3 for the algebras A1 = H ∞ (T1 ) and A2 = H ∞ (T2 ). The hypothesis implies that the maps u → u(T1 ) and u → u(T2 ) are algebra isomorphisms from H ∞ to H ∞ (T1 ) and H ∞ (T2 ), respectively. Thus, for every u ∈ H ∞ there exists a unique v ∈ H ∞ satisfying v(T2 ) = R −1 u(T1 )R. The map Φ : u → v is an algebra automorphism of H ∞ . In particular, the function ϕ = Φ(idD ) must have spectrum (in H ∞ ) equal to D, so that ϕ(D) = D. We claim that Φ(u) = u ◦ ϕ for every u ∈ H ∞ . Indeed, given λ ∈ D, we can factor u(z) − u(ϕ(λ)) = (z − ϕ(λ))w for some w ∈ H ∞ , so that Φ(u) − u(ϕ(λ)) = (ϕ − ϕ(λ))Φ(w). The equality (Φ(u))(λ) = u(ϕ(λ)) follows immediately.
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Since Φ is an automorphism, it follows that ϕ is a conformal automorphism of D, and clearly T1 ∼ ϕ(T2 ). 2 Corollary 5.6. Let T be a completely nonunitary contraction. If T ∼ S then H ∞ (T ) is confluent. If T ∼ S(m) then H ∞ (T ) has a rationally strictly cyclic vector. Proof. It suffices to observe that H ∞ (S) = WS , H ∞ (S(m)) = WS(m) , and to apply Proposition 5.4(2), (5) and (4). 2 For operators of class C0 , the converse of the preceding result is also true. The case of confluent algebras of the form H ∞ (T ) will be discussed more thoroughly in the remaining two sections of the paper. Proposition 5.7. Assume that T is a completely nonunitary contraction such that H ∞ (T ) has a rationally strictly cyclic vector. (1) If there exists f ∈ H ∞ \ {0} such that ker f (T ) = {0}, then T is of class C0 and T ∼ S(m), where m is the minimal function of T . (2) If ker f (T ) = {0} for every f ∈ H ∞ \ {0}, then H ∞ (T ) is confluent. Proof. Part (2) follows immediately from Proposition 4.8. To verify (1), assume that f ∈ H ∞ \ {0}, ker f (T ) = {0}, and H ∞ (T ) has a rationally strictly cyclic vector h0 ∈ H. Choose a nonzero vector h1 ∈ ker f (T ), and functions u1 , v1 ∈ H ∞ such that v1 (T ) is injective and v1 (T )h1 = u1 (T )h0 . The function u1 is not zero since v1 (T )h1 = 0. We claim that f (T )u1 (T ) = 0. Indeed, let h be an arbitrary vector in H. Choose u, v ∈ H ∞ such that v(T ) is injective and v(T )h = u(T )h0 . We have then v(T ) f (T )u1 (T )h = f (T )u1 (T ) v(T )h = f (T )u1 (T )u(T )h0 = f (T )u(T )u1 (T )h0 = f (T )u(T )v1 (T )h1 = 0, and therefore f (T )u1 (T )h = 0. Thus T is of class C0 because (f u1 )(T ) = 0 and f u1 ∈ H ∞ \ {0}. Finally, let m be the minimal function of T , denote by M the cyclic space for T generated by h0 , and set N = M⊥ . Let T = PN T | N be the compression of T to N . Clearly we have m(T ) = 0. Now let h ∈ N be a vector, and pick u, v ∈ H ∞ such that v(T ) is injective and v(T )h = u(T )h0 . In particular, we have v(T )h = 0. The injectivity of v(T ) is equivalent to the condition v ∧ m = 1, and this implies that v(T ) is injective as well, so that h = 0 We proved therefore that M = H. In other words, T has a cyclic vector, and thus T ∼ S(m) by the results of [25] (see also [4, Theorem III.2.3]). 2 We conclude this section with a result about arbitrary operators of class C0 . Proposition 5.8. For any operator T of class C0 , the commutant {T } has the closability property. Proof. The operator T is quasisimilar to an operator of the form T = i∈I S(mi ), where each mi is an inner function; see [4, Theorem III.5.1]. By Proposition 5.4(7), it suffices to show that
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{T } has the closability property. Now, {T } ⊃ i∈I {S(mi )} , and Lemma 3.8 shows that it suffices to show that {S(m)} has the closability property for each inner function m. This follows from Proposition 3.7 because {S(m)} = WS(m) . 2 6. Confluent algebras of the form H ∞ (T ) Consider a completely nonunitary contraction T ∈ L(H) such that H ∞ (T ) has a rationally strictly cyclic vector. According to Proposition 5.7, we have T ∼ S(m) if some nonzero operator in H ∞ (T ) has nonzero kernel. Therefore we will restrict ourselves now to operators T such that f (T ) is injective for every nonzero element of H ∞ . In other words, we will assume that H ∞ (T ) is a confluent algebra (cf. Proposition 4.8) and dim H > 1. In this case, the space H can be identified with a space of meromorphic functions. Let us denote by N the Nevanlinna class consisting of those meromorphic functions in D which can be written as u/v, with u, v ∈ H ∞ . Lemma 6.1. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Let h, h0 be two vectors such that h0 = 0, and choose u, v ∈ H ∞ , v = 0, such that v(T )h = u(T )h0 . Then the function u/v ∈ N is uniquely determined by h and h0 . We have u/v = 0 if and only if h = 0. Proof. Choose another pair of functions u1 , v1 ∈ H ∞ , v1 = 0, satisfying v1 (T )h = u1 (T )h0 . We have
v1 (T )u(T ) − v(T )u1 (T ) h0 = v1 (T )v(T ) − v(T )v1 (T ) h = 0, and therefore h0 ∈ ker(v1 u − vu1 )(T ). The hypothesis implies that v1 u = vu1 and hence u/v = u1 /v1 . 2 The function u/v will be denoted h/ h0 . It is clear that the map h → h/ h0 is an injective linear map from H to N , and u(T )h/u(T )h0 = h/ h0 if u ∈ H ∞ \ {0}. We also have h h h1 = · h0 h1 h0 provided that h0 , h1 ∈ H \ {0}. Now let h, h0 ∈ H \ {0}. There exists a unique integer n such that the nonzero function h/ h0 can be written as h u(z) (z) = zn h0 v(z) with u, v ∈ H ∞ and u(0) = 0 = v(0). The number n will be denoted ord0 (h/ h0 ). It will be convenient to write ord0 (h/ h0 ) = ∞ if h = 0. Lemma 6.2. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then 0 inf{ord0 (h/ h0 ): h ∈ H} > −∞ for every h0 ∈ H \ {0}. Proof. Clearly ord0 (h0 / h0 ) = 0. For each integer n, the set Hn = h ∈ H: ord0 (h/ h0 ) −n
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is a linear manifold, and n0 Hn = H. Given integers m, k such that k 1, we denote by Dm,k the set of all vectors h ∈ H for which h/ h0 can be written as h u(z) (z) = z−m h0 v(z) with u ∞ , v ∞ k and |u(0)|, |v(0)| 1. Observe that Dm,k = Hn \ {0}. mn,k1
The proposition will follow if we can show that one of the sets Dm,k has an interior point, and this will follow from the Baire category theorem once we prove that each Dm,k is closed. Assume indeed that (hi )∞ i=0 ⊂ Dm,k is a sequence such that hi → h as i → ∞. For each i write hi ui (z) = z−m h0 vi with ui ∞ , vi ∞ k and |ui (0)|, |vi (0)| 1. By the Vitali–Montel theorem we can assume, after dropping to a subsequence, that there exist functions u, v ∈ H ∞ such that ui (z) → u(z) and vi (z) → v(z) uniformly for z in each compact subset of D. Clearly u ∞ , v ∞ k and |u(0)|, |v(0)| 1. Moreover, we have ui (T )h0 → u(T )h0 and vi (T )hi → v(T )h in the weak topology. (For the second sequence we need to write
vi (T )hi − v(T )h = vi (T )(hi − h) + vi (T ) − v(T ) h, and use the fact that the first term tends to zero in norm, while the second tends to zero weakly by [23, Theorem III.2.1].) The identities T m vi (T )hi = ui (T )h0 for m 0 (resp., vi (T )hi = T −m ui (T )h0 for m < 0) therefore imply T m v(T )h = u(T )h0 (resp., v(T )h = T −m u(T )h0 ) so that h/ h0 = z−m u/v, and thus h ∈ Dm,k , as desired. 2 Lemma 6.3. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then T is injective and T H is a closed subspace of codimension 1. Thus T is a Fredholm operator with index(T ) = −1. Proof. The operator T belongs to a confluent algebra, hence it is injective. Note next that ord0 (T h/ h0 ) = ord0 (h/ h0 ) + 1 and hence inf ord0 (h/ h0 ): h ∈ H + 1 = inf ord0 (h/ h0 ): h ∈ T H . Since these numbers are finite, we cannot have T H = H. To conclude the proof, it will suffice to show that T H has codimension one, since this implies that it is closed as well. Choose h0 ∈ H \ T H, and note that ord0 (h/ h0 ) 0 for every h. Indeed, ord0 (h/ h0 ) = −n < 0 implies an identity of the form T n v(T )h = u(T )h0
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with u(0) = 0. Factoring u(z) − u(0) = zw(z), we obtain h0 =
1 n−1 v(T )h − w(T )h0 ∈ T H, T T u(0)
a contradiction. Thus the function h/ h0 is analytic at 0, and we can therefore define a linear functional Φ : H → C by setting Φh = (h/ h0 )(0). We will show that ker Φ ⊂ T H. Indeed, h ∈ ker Φ implies that v(T )h = T u(T )h0 for some u, v ∈ H ∞ with v(0) = 0. Factoring again v(z) − v(0) = zw(z), we obtain h=
1 T u(T )h0 − w(T )h ∈ T H, v(0)
as claimed. Thus T H has codimension 1, and the lemma is proved.
2
The preceding results allow us to describe completely the spectral picture of T , as well as its commutant. The argument for (3) already appears in [10], and is included for the reader’s convenience. Theorem 6.4. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) We have σ (T ) = D and σe (T ) = T. (2) For each λ ∈ D, λI − T is injective and has closed range of codimension 1. (3) {ker(λI − T ∗ ) : λ ∈ D} = H. More generally, {ker(λI − T ∗ ): λ ∈ S} = H whenever the set S ⊂ D has an accumulation point in D. (4) For every nonzero invariant subspace M of T , there exists an inner function m ∈ H ∞ such that m(T )H = M and the compression TM⊥ of T to M⊥ is quasisimilar to S(m). Conversely, for every inner function m, the minimal function of T(m(T )H)⊥ is m. (5) {T } = H ∞ (T ). (6) The operator T is of class C10 . Thus, the powers T ∗n converge strongly to zero and limn→∞ T n h = 0 for h ∈ H \ {0}. In particular, properties (2) and (3) say that T ∗ belongs to the class B1 (D) defined in [10]. Proof. For λ ∈ D, the operator Tλ = (I − λT )−1 (T − λI ) is also a completely nonunitary contraction, and H ∞ (Tλ ) = H ∞ (T ) is confluent. Thus Lemma 6.3 implies immediately (2). In turn, (1) follows from (2) since T is a contraction. Next we prove (4). Let M = {0} be invariant for T , set N = M⊥ , and choose h0 ∈ M \ {0}. Denote by T = PN T | N the compression of T to N . Given h ∈ N , an equality of the form v(T )h = u(T )h0 implies v(T )h ∈ M, and therefore v(T )h = 0. The fact that h0 is rationally strictly cyclic for H ∞ (T ) implies that T is locally of class C0 , and hence of class C0 by [24] (see also [4, Theorem III.3.1]). Denote by m the minimal function of T . Note that in particular PN m(T ) | N = m(T ) = 0, so that m(T )N ⊂ M.
(6.1)
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We show next that T has a cyclic vector, and hence it is quasisimilar to S(m). Assume to the contrary that T does not have a cyclic vector, and let N1 , N2 be cyclic spaces for T generated by two nonzero vectors h1 , h2 such that T | N1 ∼ S(m) and N1 ∩ N2 = {0} (see [25] or [4, Theorem III.2.13]). There exist nonzero functions u1 , u2 ∈ H ∞ such that u1 (T )h1 = u2 (T )h2 . Dividing these functions by their greatest common inner divisor, we may assume that u1 and u2 do not have any (non constant) common inner factor. We also have u1 (T )h1 = u2 (T )h2 ∈ N1 ∩ N2 , and hence these vectors are equal to zero. We deduce that m divides u1 , and hence m ∧ u2 = 1. This last equality implies that u2 (T ) is a quasiaffinity, hence u2 (T )h2 = 0, a contradiction. Thus T is indeed cyclic. Using (6.1), we observe that m(T )H = m(T )M + m(T )N ⊂ M. Denote now M1 = m(T )H, N1 = M⊥ 1 , and T1 = PN1 T | N1 . Clearly m(T1 ) = 0, and T ∗ = T1∗ | N . It follows that the minimal function of T1 is also m. Applying to T1 the argument showing that T has a cyclic vector yields the same for T1 . Hence M = M1 by the results of [25] (see also [4, Theorem III.2.13]). We start next with a given inner function m, and denote by m1 the minimal function of T(m(T )H)⊥ . The function m1 must divide m, so that m = m1 m2 for some other inner function m2 . With the notation H1 = m1 (T )H = m(T )H, T1 = T | H1 , the algebra H ∞ (T1 ) is confluent, and m2 (T1 )H1 = m2 (T )m1 (T )H = m(T )H = H1 , so m2 (T1 ) has dense range. We claim that m2 (T1 )M = M for every invariant subspace M for T1 . Indeed, from the first part of (4) we know that M = m3 (T1 )H1 for some inner function m3 . Hence m2 (T1 )M = m2 (T1 )m3 (T1 )H1 = m3 (T1 )m2 (T1 )H1 = m3 (T1 )H1 = M, as claimed. Since H ∞ (T1 ) is confluent, we have σ (T1 ) = D by part (1) of the theorem. This implies that T1 belongs to the class A defined in [5]. By the results of [8], there exist vectors x, y ∈ H1 such that
1 u(T1 )x, y = 2π
2π
1 − m2 (0)m2 eit u eit dt 0
for all u ∈ H ∞ . In particular, v(T1 )m2 (T1 )x, y = 0 for v ∈ H ∞ . Set M = {T1n x: n 0}, and observe now that y ⊥ m2 (T1 )M, and therefore y ⊥ M as well. In particular, 1 0 = x, y = 2π
2π 2
1 − m2 (0)m2 eit dt = 1 − m2 (0) , 0
and this implies that m2 is a constant function. We reach the desired conclusion that the minimal function of T(m(T )H)⊥ is m. To prove (3),∗ assume that S ⊂ D has an ∗accumulation point in⊥D, and note that the space N = {ker(λI − T ): λ ∈ S} is invariant for T . Therefore M = N is invariant for T . If M = {0}, we have then m(T )H ⊂ M for some inner function m, and therefore ker m(T )∗ ⊃ N . Given λ ∈ S, choose a nonzero vector fλ ∈ ker(λI − T ∗ ), and observe that 0 = m(T )∗ fλ = m(λ)fλ .
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Thus m(λ) = 0 for λ ∈ S, and we conclude that m = 0, which is impossible. This contradiction implies that M = {0}, thus verifying (6.1). Consider next an operator X ∈ {T } = H ∞ (T ) . By Proposition 4.6, there exist u, v ∈ H ∞ , v = 0, so that v(T )X = u(T ). With fλ as above, we have
∗ v(λ)X ∗ fλ = v(T )X fλ = u(T )∗ fλ = u(λ)f (λ), and thus u(λ) X ∗ fλ ∗ v(λ) = f X . λ We deduce that w = u/v ∈ H ∞ and X = w(T ). The fact that the powers of T ∗ tend strongly to zero follows from (3) because T ∗n fλ = n λ fλ → 0 as n → ∞ for λ ∈ D. It remains to prove that the space M = h ∈ H: lim T n h = 0 n→∞
is equal to {0}. Assume to the contrary that M = {0}, and observe that H ∞ (T | M) is also confluent. In particular, σ (T | M) = D and T | M is of class C00 . According to [7] and [5, Theorem 6.6], T | M belongs to the class Aℵ0 , and, by [5, Corollary 5.5], T has a further invariant subspace N ⊂ M such that N T N has infinite dimension. This space must however have dimension 1 because H ∞ (T | N ) is confluent. This contradiction shows that we must have M = {0}, as claimed. 2 Recall that N+ ⊂ N denotes the collection of functions of the form u/v, where u, v ∈ H ∞ and v is outer. ∞ Corollary 6.5. Let T ∈ L(H) be a completely nonunitary contraction n such that H (T ) is con∗ fluent, and fix a vector h0 ∈ ker T , h0 = 0. Assume that H = {T h0 : n 0}; that is, h0 is cyclic for T . Then h/ h0 ∈ N+ for every h ∈ H.
Proof. We can assume that h = 0. Now choose functions u, u0 ∈ H ∞ \ {0} such that u0 /u = h/ h0 . Thus we have u0 (T )h0 = u(T )h. Consider the factorizations u = mv and u0 = m0 v0 , where m, m0 are inner and v, v0 are outer. By [23, Proposition III.3.1], the operator v0 (T ) is a quasiaffinity, and therefore m0 (T )H =
n0
T n v0 (T )m0 (T )h0 =
T n v(T )m(T )h ⊂ m(T )H.
n0
It follows that (m(T )H)⊥ ⊂ (m0 (T )H)⊥ , and thus m divides m0 by Theorem 6.4(4). It follows that h u0 v0 (m0 /m) = ∈ N+ , = h0 u v as claimed.
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We will denote by A the disk algebra. This consists of those functions in H ∞ which are restrictions of continuous functions on D. If T is a completely nonunitary contraction, we set A(T ) = {u(T ): u ∈ A}. Corollary 6.6. Consider an operator T ∈ L(H), where H is an infinite dimensional Hilbert space. Then (1) The algebra PT is not confluent. (2) If T is a completely nonunitary contraction, then A(T ) is not confluent. Proof. In proving (1), there is no loss of generality in assuming that T < 1 since PT = PαT for any α > 0. Under this assumption, we have PT ⊂ A(T ), so it suffices to prove part (2). Assume therefore that T is a completely nonunitary contraction and A(T ) is confluent. The larger algebra H ∞ (T ) is confluent as well, and Proposition 4.6 implies that for every f ∈ H ∞ , the operator f (T ) ∈ {T } = A(T ) can be written as f (T ) = v(T )−1 u(T ) with u, v ∈ A, v = 0. We have then v(T )f (T ) = u(T ), and thus f = u/v. It is known, however, that there are functions in H ∞ which cannot be represented as quotients of elements of A. An example is provided by any singular inner function f (λ) = e−
ζ +λ T ζ −λ
dμ(ζ )
,
λ ∈ D,
such that the closed support of the singular measure μ is the entire circle T.
2
The assertion in Proposition 4.6, concerning unbounded linear transformations can be improved when H ∞ (T ) is confluent. Proposition 6.7. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Then every closed, densely defined linear transformation commuting with T is of the form v(T )−1 u(T ), where u, v ∈ H ∞ and v is an outer function. Proof. Let X be a closed, densely defined linear transformation commuting with T . Since X is closed, it must also commute with every operator in H ∞ (T ). By Proposition 4.6, there exist u, v ∈ H ∞ such that v ≡ 0 and X ⊂ v(T )−1 u(T ). Let us set
T1 = (T ⊕ T ) | G v(T )−1 u(T ) , and observe that the quasiaffinity Q : h ⊕ k → h from G(v(T )−1 u(T )) to H satisfies QT1 = T Q. Thus H ∞ (T1 ) ≺ H ∞ (T ), and therefore H ∞ (T1 ) is confluent by Proposition 5.2(3). The subspace G(X) is invariant for T1 , so
G(X) = m(T1 )G v(T )−1 u(T ) for some inner function m. To prove the equality X = v(T )−1 u(T ), it suffices to show that m is in fact constant. Indeed, we have
m(T )H = m(T )QG v(T )−1 u(T ) = Qm(T1 )G v(T )−1 u(T ) = QG(X) = D(X) = H,
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and the desired conclusion follows from the second assertion in Theorem 6.4(4). There is no loss of generality in assuming that u and v do not have any nonconstant common inner divisor. We conclude the proof by showing that in this case v must be outer. Let m be an inner divisor of v, and note that for every h ⊕ k ∈ G(X) we have u(T )h = v(T )k ∈ m(T )H, and therefore u(T )D(X) ⊂ m(T )H. Since D(X) is dense in H, we conclude that u(T(m(T )H)⊥ ) = 0, and therefore m divides u. Thus m is constant, and hence v is outer. 2 It follows from the results of [10] that the one-dimensional spaces ker(λI − T )∗ depend analytically on λ and, in fact, there exists an analytic function f : D → H such that ker(λI − T )∗ = Cf (λ) for λ ∈ D. A local version of this result is easily proved. Indeed, set L = (T ∗ T )−1 T ∗ . Given a unit vector f0 ∈ ker T ∗ , the function ∞
−1 λn L∗n f0 f (λ) = I − λL∗ f0 =
(6.2)
n=0
is analytic for |λ| < 1/ L , and obviously T ∗ f (λ) = λf (λ). This calculation is valid for any left inverse of T . The operator L has the advantage that L∗ H = T H, and therefore T n f0 , f0 = f0 , L∗n f0 = 0 for n 1. These relations, along with LT = I , obviously imply
T n f0 , L∗m f0 = δnm ,
n, m 0.
(6.3)
Proposition 6.8. Let T ∈ L(H) be a completely nonunitary contraction such that H ∞ (T ) is confluent. Define L = (T ∗ T )−1 T ∗ and fix a unit vector f0 ∈ ker T ∗ . Then (1) The f0 is cyclic for L∗ . vector n (2) {T H: n 0} = {0}. (3) {L∗n H: n 0} = H [ {T n f0 : n 0}]. Proof. We have seenthat ker(λI − T ∗ ) = Cf (λ) for λ close to zero, where f (λ) is given by (6.2) and belongs to {L∗n f0 : n 0}. Thus (1) follows from Theorem 6.4(3). To prove (2), let f be a nonzero element in the intersection, and set k = inf ord0 (h/f0 ): h ∈ H ,
m = ord0 (f/f0 ) < ∞.
For each integer n we can write f = T n g for some g = 0, and therefore m = n + ord0 (g/f0 ) n + k. This yields a contradiction for large n. The orthogonality relations (6.3) imply the inclusion n0
L∗n H ⊂ H
n0
T n f0 .
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Conversely, consider a vector h ∈ H [ {T n f0 : n 0}]. Given n 1, we have h = L∗n T ∗n h +
n−1
L∗k I − L∗ T ∗ T ∗k h.
k=0
Since I − L∗ T ∗ is the orthogonal projection onto Cf0 , and
T ∗k h, f0 = h, T k f0 = 0,
we deduce that h = L∗n T ∗n h ∈ L∗n H, thus proving the opposite inclusion.
2
7. Confluence and functional models The results in Section 6 show that completely nonunitary contractions T for which H ∞ (T ) is confluent share many of the properties of the unilateral shift S. In this section we will describe some quasiaffine transforms of such operators T . These quasiaffine transforms are in fact functional models associated with purely contractive inner functions of the form Θ=
θ1 , θ2
where θ1 , θ2 ∈ H ∞ . The condition that Θ be inner amounts to the requirement that θ1 (ζ )2 + θ2 (ζ )2 = 1,
a.e. ζ ∈ T,
while pure contractivity means simply that θ1 (0)2 + θ2 (0)2 < 1. We recall the construction of the functional model associated with such a function Θ. The subspace θ1 u ⊕ θ2 u: u ∈ H 2 ⊂ H 2 ⊕ H 2 is obviously invariant for S ⊕ S, and thus the orthogonal complement H(Θ) = H 2 ⊕ H 2 θ1 u ⊕ θ2 u: u ∈ H 2 is invariant for S ∗ ⊕ S ∗ . The operator S(Θ) ∈ L(H(Θ)) is the compression of S ⊕ S to this space or, equivalently, S(Θ)∗ = (S ∗ ⊕ S ∗ ) | H(Θ). Observe that I − S(Θ)∗ S(Θ) has rank one, while I − S(Θ)S(Θ)∗ has rank two. It follows that σ (S(Θ)) = D, and in particular S(Θ) is not of class C0 . Lemma 7.1. Let Θ = θθ21 be a purely contractive inner function. The algebra H ∞ (S(Θ)) is confluent if and only if the functions θ1 and θ2 do not have a nonconstant common inner factor.
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Proof. If either of the functions θj is equal to zero, the other one must be inner. The lemma is easily verified in this case. Indeed, assume that θ1 is inner and θ2 = 0. If θ1 is not constant then ker θ1 (S(Θ)) = {0}, so that H ∞ (S(Θ)) is not confluent. Also, θ1 is a common inner divisor of θ1 and θ2 , so that both conditions in the statement are false. On the other hand, if θ1 is constant then Θ is not pure, so this case does not arise. For the remainder of this proof, we consider the case in which both functions θj are different from zero. Assume first that θj = mϕj , where m is a nonconstant inner function and ϕj ∈ H ∞ for j = 1, 2. The nonzero vector h ∈ H(Θ) defined by h = PH(Θ) (ϕ1 ⊕ ϕ2 ) satisfies m(S(Θ))h = 0, and therefore m(S(Θ)) has nontrivial kernel. Thus H ∞ (S(Θ)) is not confluent. Assume now that θ1 and θ2 do not have a nonconstant common inner factor. We verify first that ker u(S(Θ)) = {0} for u ∈ H ∞ \ {0}. It suffices to consider the case of an inner function u. A vector f1 ⊕ f2 ∈ ker u(S(Θ)) must satisfy uf1 = θ1 g and uf2 = θ2 g for some g ∈ H 2 . The fact that θ1 ∧ θ2 = 1 implies that u divides g, and therefore f1 ⊕ f2 = θ1 (g/u) ⊕ θ2 (g/u) belongs to H(Θ)⊥ ; the equality f1 ⊕ f2 = 0 follows. To conclude the proof, we will show that h = PH(Θ) (1 ⊕ 0) is a rationally strictly cyclic vector for H ∞ (S(Θ)). Indeed, assume that f = f1 ⊕ f2 ∈ H(Θ) \ {0}, and write f1 = a1 /b and f2 = a2 /b, where a1 , a2 , b ∈ H ∞ and b is outer. Define functions u = −bθ2 , v = θ1 a2 − θ2 a1 , and note that
v S(Θ) h − u S(Θ) f = PH(Θ) (v ⊕ 0 − uf1 ⊕ uf2 ) = PH(Θ) (θ1 a2 ⊕ θ2 a2 ) = 0. The lemma follows because u ≡ 0, and hence u(S(Θ)) is injective.
2
Let us remark that the condition θ1 ∧ θ2 = 1 is equivalent to the fact that the function Θ is ∗-outer. In other words, the operators S(Θ) described in the preceding lemma are of class C10 . This is in agreement with Theorem 6.4(6). Proposition 7.2. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) Either S ≺ T or there exists a purely contractive inner function Θ = S(Θ) ≺ T and H ∞ (S(Θ)) is confluent. (2) We have S ≺ T if and only if T has a cyclic vector.
θ1 θ2
such that
Proof. Denote by U+ ∈ L(K+ ) the minimal isometric dilation of T . Thus H ⊂K+ and T PH = PH U+ . Since T ∈ C10 , the operator U+ is a unilateral shift. Let us set M = {T n h1 : n 0}, where h1 ∈ H \ {0}, and let h2 ∈ H M be a cyclic vector for the compression of T to this subspace. Such a vector exists by Theorem 6.4(4). Observe that H = {T n h1 , T n h2 : n 0}. We define now a space E=
U+n h1 , U+n h2 : n 0
and an operator Y ∈ L(E, H) by setting Y = PH | E. The space E is invariant for U+ , Y (U+ | E) = T Y , and Y has dense range. Moreover, the restriction U+ | E is a unilateral shift of multiplicity 1 or 2. Finally, set H = E ker Y , X = Y | H , and denote by T the compression
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of U+ | E to the space H . Then XH = Y E so that X is a quasiaffinity, and XT = T X. Thus we have T ≺ T and hence H ∞ (T ) is confluent by Proposition 5.2(3). We will now prove that at least one of the following alternatives must θ1 hold: either S ≺ T or T is unitarily equivalent to an operator of the form S(Θ), where Θ = θ2 is a purely contractive inner function. For this, we first note that U+ | E is the minimal isometric lifting of T . Therefore T is of class C•0 and its functional model is the compression of the canonical shift on H 2 (DT ∗ ) to H(ΘT ) = H 2 (DT ∗ ) ΘT H 2 (DT ), where ΘT (z) : DT → DT ∗ ,
z ∈ D,
is the characteristic function of T . Therefore ΘT is a purely contractive inner function and according to [23, Chapter VI] dim DT dim DT ∗ = dim E. Thus we must consider the following possibilities: (i) (ii) (iii) (iv) (v)
dim DT dim DT dim DT dim DT dim DT
= dim DT ∗ = 2; = 1, dim DT ∗ = 2; = 0, dim DT ∗ = 2; = dim DT ∗ = 1; and = 0, dim DT ∗ = 1.
In cases (i) and (iv) T is of class C00 , hence of class C0 (see [23, Proposition VI.3.5 and Theo rem VI.5.2]), thus H ∞ (T ) is not θ1 confluent. In case (ii) T is unitarily equivalent to an operator of the form S(Θ), where Θ = θ2 is a purely contractive inner function. In case (iii) T is unitarily equivalent to S ⊕ S and H ∞ (S ⊕ S) is not confluent; to see this consider the vectors 1 ⊕ 0 and 0 ⊕ 1. Finally, in case (v) T is unitarily equivalent to S, and thus S ≺ T . If T has a cyclic vector h1 , we can take h2 = 0, and then U+ | E is a shift of multiplicity 1. In this case, we must have ker Y = {0} so that U+ | E ≺ T . Conversely, S ≺ T implies that T has a cyclic vector since S has one. 2 The argument in the preceding proof appeared earlier in the classification of contractions of class C•0 [26,27], and even earlier in [14] and in the study of the class C0 [22]. When T has a cyclic vector, it is natural to ask under what conditions we actually have T ∼ S. Lemma 7.3. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then the following assertions are equivalent: (1) T ≺ S. (2) T | M ≺ S for some invariant subspace M of T . (3) T | M ≺ S for every nonzero invariant subspace M of T .
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Proof. The implications (3) ⇒ (1) ⇒ (2) are obvious. Next we show that T ≺ T | M for every nonzero invariant subspace M of T . By Theorem 6.4(4), there is an inner function m such that m(T )H = M. Then the operator X : H → M defined by Xh = m(T )h, h ∈ H, is a quasiaffinity and XT = (T | M)X. Using this fact, it is easy to show that (2) ⇒ (1). Indeed, if (2) holds we have T | M ≺ S for some M, and the relations T ≺ T | M ≺ S imply the desired conclusion T ≺ S. Finally, we prove that (1) ⇒ (3). Assume that (1) holds, so that Y T = SY for some quasiaffinity Y . If M is a nonzero invariant subspace for T , the operator Z = Y | M : M → Y M is a quasiaffinity realizing the relation T | M ≺ S | Y M. We conclude that (3) is true since S | Y M is unitarily equivalent to S. 2 We can now state some conditions equivalent to the relation T ∼ S. Theorem 7.4. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent and has a cyclic vector. Let f : D → H be an analytic function such that f (0) = 1 and ker(λI − T ∗ ) = Cf (λ) for every λ ∈ D, and denote H0 = {T n f (0): n 0}. Then the following conditions are equivalent: (1) (2) (3) (4)
T ∼ S. T | H0 ≺ S. There exists an outer function b ∈ H ∞ such that b(h/f (0)) ∈ H 2 for every h ∈ H0 . There exists an outer function b ∈ H ∞ such that b
¯ h, f (λ) ∈ H2 f (0), f (λ¯ )
for every h ∈ H0 . Proof. Since T has a cyclic vector, we have S ≺ T by Proposition 7.2(2). Therefore T ∼ S is equivalent to T ≺ S, and this is equivalent to condition (2) by Lemma 7.3. This establishes the equivalence (1) ⇔ (2). For an arbitrary h ∈ H \ {0}, write the function h/f (0) as a quotient u/v of functions in H ∞ . We have then ¯ = h, v(λ)f (λ) ¯ = v(λ) h, f (λ) ¯ , ¯ = h, v(T )∗ f (λ) v(T )h, f (λ)
and analogously u(T )f (0), f (λ¯ ) = u(λ)f (0), f (λ¯ ). Since v(T )h = u(T )f (0), we conclude that b(λ)
¯ h, f (λ) h (λ) = b(λ) ¯ f (0) f (0), f (λ)
(7.1)
for those λ for which the denominators do not vanish. This proves the equivalence (3) ⇔ (4). ¯ cannot be identically zero since it takes the value 1 Note that the analytic function f (0), f (λ) for λ = 0. It remains to prove the equivalence (2) ⇔ (3), and for this purpose we may as well assume that case. Thus, H = H0 . We apply the construction in the proof of Proposition 7.2 for this particular consider the minimal isometric dilation U+ ∈ L(K+ ) of T , and denote E = {U+n f (0): n 0}.
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Since U+∗ f (0) = T ∗ f (0) = 0, there exists a unitary operator W : H 2 → E such that W 1 = f (0) and W S = (U+ | E)W . One can then construct a quasiaffinity Y : H 2 → H, namely Y = PH W , such that T Y = Y S and Y 1 = f (0). Since an equality of the form v(S)x = u(S)1 for x ∈ H 2 is equivalent to v(T )Y x = u(T )f (0), we deduce that Yx = x, f (0)
x ∈ H 2,
and, conversely, that any vector h ∈ H such that k = h/f (0) ∈ H 2 must belong to Y H 2 ; namely h = Y k. With this preparation, assume that (2) holds, and let X ∈ L(H, H 2 ) be a quasiaffinity such that XT = SX. Then the operator XY is a quasiaffinity in the commutant of S, and therefore XY = b(S) for some outer function b ∈ H ∞ . The equality
X b(T ) − Y X = b(S)X − (XY )X = 0 implies that we also have Y X = b(T ). For any h ∈ H \ {0} we have then b
b(T )h Y Xh h = = = Xh ∈ H 2 , f (0) f (0) Y1
thus proving (3). Conversely, if (3) holds, we can define a linear map X : H → H 2 by setting Xh = b(h/f (0)) for h ∈ H, and this map obviously satisfies XT = SX. Using (7.1) it is easy to verify that X is a closed linear transformation, and hence X is continuous. It is also immediate that XY = b(S) and Y X = b(T ), and this implies that X is a quasiaffinity since b is outer. 2 Corollary 7.5. Assume that T ∈ L(H) is a completely nonunitary contraction such that T ∼ S. ∗ Let f : D → H be an analytic function nsuch that f (0) = 1 and ker(λI − T ) = Cf (λ) for every λ ∈ D, and assume that H = {T f (0): n 0}. Then f (0), f (λ) = 0 for every λ ∈ D. Proof. Let b be an outer function satisfying condition (4) of Theorem 7.4. Assume that ¯ = 0 for some λ ∈ D. Since b(λ) = 0, it follows that h, f (λ) ¯ = 0 for every h ∈ H, f (0), f (λ) and therefore f (λ¯ ) = 0, which is impossible since this vector generates ker(λI − T )∗ . 2 The relation T ≺ S can also be studied in terms of the minimal unitary dilation of T . We will denote by R∗ ∈ L(R∗ ) the ∗-residual part of this minimal unitary dilation; see [23, Section II.3] for the relevant definitions. The facts we require about this operator are as follows: (a) R∗ is a unitary operator with absolutely continuous spectral measure relative to arclength measure on T. (b) There exists an operator Z : H → R∗ (namely, the orthogonal projection onto R∗ ) such that ZT = R∗ Z and Zh = lim T n h. n→∞
In particular, Z is injective if and only if T is of class C1· . (c) The smallest reducing subspace for R∗ containing ZH is R∗ .
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Proposition 7.6. Assume that T is a completely nonunitary contraction such that H ∞ (T ) is confluent. Then (1) The ∗-residual part R∗ of the minimal unitary dilation of T has spectral multiplicity at most 1. (2) We have T ≺ S if and only if R∗ is a bilateral shift of multiplicity 1. (3) We have T ≺ R∗ | ZH, and T ≺ S if and only if ZH = R∗ . (4) T ∗ has a cyclic vector. Proof. Given h1 , h2 ∈ H \ {0}, select u1 , u2 ∈ H ∞ \ {0} such that u1 (T )h1 = u2 (T )h2 . Then we have u1 (R∗ )Zh1 = u2 (R∗ )Zh2 . Since u1 (ζ ) and u2 (ζ ) are different from zero a.e. relative to the spectral measure of R∗ , it follows that the vectors Zh1 and Zh2 generate the same reducing space for R∗ . Therefore R∗ has a ∗-cyclic vector, and this implies (1). Next we prove (3). The fact that T ≺ R∗ | ZH is immediate. If ZH is not reducing, then R∗ | ZH is unitarily equivalent to S and hence T ≺ S. Conversely, if T ≺ S, let W be a quasiaffinity such that W T = SW . For any h ∈ H we have W h = lim S n W h = lim W T n h W Zh , n→∞
n→∞
so there exists an operator X : ZH → H 2 such that X W and XZ = W . Since the range of X contains the range of W , we have X = 0. Pick a vector f ∈ H 2 such that X ∗ f = 0, and observe that lim (R∗ ZH)∗n X ∗ f = lim X ∗ S ∗n f = 0. n→∞
n→∞
Therefore R∗ | ZH is not unitary, and consequently ZH = R∗ . Assume now that T ≺ S. The fact that R∗ is a bilateral shift follows from (3) because the only absolutely continuous unitary operator of multiplicity 1 which has nonreducing invariant subspaces is the bilateral shift. Conversely, if R∗ is a bilateral shift, the results of [17] imply the existence of an invariant subspace M for T such that T | M ≺ S. We deduce that T ≺ S by Lemma 7.3. This proves (2). Finally, (4) also follows from (3) because (R∗ | ZH)∗ has a cyclic vector. 2 Corollary 7.7. Assume that Θ = θθ21 is inner, ∗-outer, and purely contractive. Then S(Θ) ≺ S. More precisely, the operator Q : H(Θ) → H 2 defined by Q(f1 ⊕ f2 ) = θ1 f2 − θ2 f1 , f1 ⊕ f2 ∈ H(Θ), is a quasiaffinity and QS(Θ) = SQ. Proof. We will show that PR∗ H(Θ) = R∗ . To do this, we observe first that the minimal unitary dilation of S(Θ) is the operator U ⊕ U on L2 ⊕ L2 . The space R∗ is the orthogonal complement of the smallest reducing space for U ⊕ U containing {θ1 u ⊕ θ2 u: u ∈ H 2 }. Thus
R∗ = L2 ⊕ L2 θ1 u ⊕ θ2 u: u ∈ L2 , and it follows that PR∗ is the operator of pointwise multiplication by the matrix |θ2 |2 −θ2 θ1 I − ΘΘ ∗ = . −θ2 θ1 |θ1 |2
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Finally, we have PR∗ H(Θ) = PR∗ (H 2 ⊕ H 2 ), and therefore PR∗ H(Θ) is the invariant subspace for U generated by PR∗ (1 ⊕ 0) and PR∗ (0 ⊕ 1). These two vectors are precisely |θ2 |2 ⊕ (−θ2 θ1 ) = (−θ2 u) ⊕ θ1 u, (−θ2 θ1 ) ⊕ |θ1 |2 = (−θ2 v) ⊕ θ1 v, with u = −θ2 and v = θ1 . Since θ1 and θ2 do not have nonconstant common inner divisors, the invariant subspace for S they generate is the entire H 2 . It follows that PR∗ H(Θ) = (−θ2 w) ⊕ θ1 w: w ∈ H 2 , and R∗ | PR∗ H(Θ) is unitarily equivalent to S. The final assertion is verified by noting that (see (b) above)
Z(f1 ⊕ f2 ) = PR∗ (f1 ⊕ f2 ) = −θ2 Q(f1 ⊕ f2 ) ⊕ θ1 Q(f1 ⊕ f2 ) for f1 ⊕ f2 ∈ H(Θ).
2
The preceding result can be extended considerably. As seen in the proof below, the assumption that I − T ∗ T has finite rank can be replaced by the requirement that I − T T ∗ have finite rank. Corollary 7.8. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent and I − T ∗ T has finite rank. Then (1) We have T ≺ S. (2) If in addition T has a cyclic vector, then T ∼ S. Proof. Denote by n the rank of I − T ∗ T , and observe that the characteristic function ΘT is inner, ∗-outer, and coincides with an (n + 1) × n matrix over H ∞ . Indeed, ΘT (0) is a Fredholm operator of index −1. It follows that I − ΘT (ζ )ΘT (ζ )∗ has rank 1 for a.e. ζ ∈ T, and therefore R∗ is a bilateral shift by [23, Section VI.6]. Thus (1) follows from Proposition 7.6(2). Part (2) follows from (1) and Proposition 7.2(2). 2 The result shows that there exist some purely contractive inner functions of the form following Θ = θθ21 with the property that S(Θ) ≺ S. Thus part (1) of Proposition 7.2 could be restated as follows: ∞ there ex(1) If T is a completely nonunitary contraction such θ1 that H (T ) is confluent, then ists a purely contractive inner function Θ = θ2 such that S(Θ) ≺ T , and H ∞ (S(Θ)) is confluent.
Corollary 7.9. Assume that Θ =
θ1 θ2
is purely contractive, inner and ∗-outer.
(1) If f1 ⊕ f2 ∈ H(Θ) is cyclic for S(Θ), then θ1 f2 − θ2 f1 is an outer function. (2) Conversely, if θ1 f2 − θ2 f1 is outer for some f1 , f2 ∈ H 2 , then PH(Θ) (f1 ⊕ f2 ) is cyclic for S(Θ).
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(3) There exists Θ such that S(Θ) does not have a cyclic vector. (4) We have S(Θ) ∼ S if and only if S(Θ) has a cyclic vector. Proof. With the notation of Corollary 7.7, Q(f1 ⊕ f2 ) must be cyclic for S if f1 ⊕ f2 is cyclic for S(Θ). This proves (1). Conversely, assume that u = θ1 f2 − θ2 f1 is outer for some f1 , f2 ∈ H 2 . Upon multiplying f1 , f2 by some outerfunction, we may assume that f1 , f2 ∈ H ∞ . Let g1 ⊕ g2 ∈ H(Θ) be a vector orthogonal to {S(Θ)n PH(Θ) (f1 ⊕ f2 ): n 0}. We have then g1 ⊕ g2 , θ1 p ⊕ θ2 p = g1 ⊕ g2 , f1 p ⊕ f2 p = 0 for every polynomial p. Equivalently, θ1 g1 + θ2 g2 and f1 g1 + f2 g2 belong to L2 H 2 , and therefore the functions ug1 = f2 (θ1 g1 + θ2 g2 ) − θ2 (f1 g1 + f2 g2 ), ug2 = θ1 (f1 g1 + f2 g2 ) − f1 (θ1 g1 + θ2 g2 ) are also in L2 H 2 . Thus gj , up = 0 for all polynomials p, and hence gj = 0, j = 1, 2, because u is outer. Assertion (2) follows. To prove (3), let m1 and m2 be two relatively prime inner functions, and set θ1 = 35 m1 and θ2 = 45 m2 . Nordgren [18] showed that it is possible to choose m1 and m2 so that no function of the form m1 f2 − m2 f1 is outer if f1 , f2 ∈ H 2 . The corresponding operator S(Θ) does not have a cyclic vector. Finally (4) follows from Corollary 7.7 and Proposition 7.2(2). 2 Let us also note a related result which follows easily from [28]. Proposition 7.10. Assume that Θ = θθ21 is inner and ∗-outer. Then the operator S(Θ) is similar to S if and only if there exist f1 , f2 ∈ H ∞ such that θ1 f2 − θ2 f1 = 1. Proof. It was shown in [28] that S(Θ) is similar to an isometry if and only if Θ is left invertible. To conclude, one must observe that the only possible isometry is a unilateral shift of multiplicity 1. 2 The proof of the following proposition follows easily from the above arguments, along with the corresponding properties of S. Proposition 7.11. Let T be a completely nonunitary contraction such that T ∼ S. Then T is of class C10 , both T and T ∗ have cyclic vectors, and the ∗-residual part R∗ of the minimal unitary dilation of T is a bilateral shift of multiplicity 1. The converse of this proposition is not true. Indeed, it was shown in [6] (see also [16]) that there exist operators T of class C10 , with a cyclic vector, such that R∗ is a bilateral shift of multiplicity 1, and σ (T ) ⊃ D. For such operators we will have R∗∗ ≺ T ∗ , so T ∗ also has a cyclic vector, but T ⊀ S. The following partial converse follows from Propositions 7.2(2) and 7.6(2).
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Proposition 7.12. Let T be a completely nonunitary contraction such that H ∞ (T ) is confluent, T has a cyclic vector, and the ∗-residual part R∗ of the minimal unitary dilation of T is a bilateral shift of multiplicity 1. Then T ∼ S. Remark 7.13. For more information about which operators are or can be quasisimilar to the unilateral shift, see [2,9,11,12,15,29] and the references therein. References [1] M.B. Abrahamse, R.G. Douglas, A class of subnormal operators related to multiply connected domains, Adv. Math. 19 (1976) 106–148. [2] J. Agler, J.E. Franks, D.A. Herrero, Spectral pictures of operators quasisimilar to the unilateral shift, J. Reine Angew. Math. 422 (1991) 1–20. [3] W.B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967) 635–647. [4] H. Bercovici, Operator Theory and Arithmetic on H ∞ , Amer. Math. Soc., Providence, RI, 1988. [5] H. Bercovici, C. Foias, C. Pearcy, Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, Amer. Math. Soc., Providence, RI, 1985. [6] H. Bercovici, L. Kérchy, Spectral behaviour of C10 -contractions, preprint. [7] S.W. Brown, Contractions with spectral boundary, Integral Equations Operator Theory 11 (1988) 49–63. [8] S.W. Brown, B. Chevreau, C. Pearcy, On the structure of contraction operators. II, J. Funct. Anal. 76 (1) (1988) 30–55. [9] S. Clary, Quasisimilarity and subnormal operators, PhD thesis, University of Michigan, 1973. [10] M.J. Cowen, R.G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978) 187–261. [11] R.E. Curto, Raúl, L.A. Fialkow, Similarity, quasisimilarity, and operator factorizations, Trans. Amer. Math. Soc. 314 (1989) 225–254. [12] L.A. Fialkow, Quasisimilarity and closures of similarity orbits of operators, J. Operator Theory 14 (1985) 215–238. [13] S.D. Fisher, Function Theory on Planar Domains. A Second Course in Complex Analysis, John Wiley & Sons, New York, 1983. [14] C. Foias, A classification of doubly cyclic operators, in: Hilbert Space Operators, Tihany, Colloquia Math. Soc. J. Bolyai 5 (1970) 155–161. [15] D.A. Herrero, On multicyclic operators, Integral Equations Operator Theory 1 (1978) 57–102. [16] L. Kérchy, On the spectra of contractions belonging to special classes, J. Funct. Anal. 67 (1986) 153–166. [17] L. Kérchy, Shift-type invariant subspaces of contractions, J. Funct. Anal. 246 (2007) 281–301. [18] E.A. Nordgren, The ring N+ is not adequate, Acta Sci. Math. (Szeged) 36 (1974) 203–204. [19] H. Radjavi, P. Rosenthal, Invariant Subspaces, second ed., Dover Publications, Mineola, NY, 2003. [20] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967) 179–203. [21] D. Sarason, Unbounded operators commuting with the commutant of a restricted backward shift, Oper. Matrices 4 (2010) 293–300. [22] B. Sz.-Nagy, C. Foias, Modèle de Jordan pour une classe d’opérateurs de l’espace de Hilbert, Acta Sci. Math. (Szeged) 31 (1970) 91–115. [23] B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. [24] B. Sz.-Nagy, C. Foias, Local characterization of operators of class C0 , J. Funct. Anal. 8 (1971) 76–81. [25] B. Sz.-Nagy, C. Foias, Compléments l’étude des opérateurs de classe C0 . II, Acta Sci. Math. (Szeged) 33 (1972) 113–116. [26] B. Sz.-Nagy, C. Foias, Jordan model for contractions of class C·0 , Acta Sci. Math. (Szeged) 36 (1974) 305–322. [27] B. Sz.-Nagy, C. Foias, Injections of shifts into strict contractions , in: Linear Operators and Approximation. II, Birkhäuser, Basel, 1974, pp. 29–37. [28] B. Sz.-Nagy, C. Foias, On contractions similar to isometries and Toeplitz operators, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976) 553–564. [29] M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Trans. Amer. Math. Soc. 319 (1990) 405–415.
Journal of Functional Analysis 258 (2010) 4154–4182 www.elsevier.com/locate/jfa
Landesman–Lazer type results for second order Hamilton–Jacobi–Bellman equations Patricio Felmer a,∗ , Alexander Quaas b , Boyan Sirakov c,d a Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile b Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla: V-110, Avda. España 1680,
Valparaíso, Chile c UFR SEGMI, Université de Paris 10, 92001 Nanterre Cedex, France d CAMS, EHESS, 54 bd. Raspail, 75006 Paris, France
Received 22 September 2009; accepted 11 March 2010
Communicated by H. Brezis
Abstract We study the boundary-value problem
F (D 2 u, Du, u, x) + λu = f (x, u) in Ω, u=0 on ∂Ω,
where the second order differential operator F is of Hamilton–Jacobi–Bellman type, f is sub-linear in u at infinity and Ω ⊂ RN is a regular bounded domain. We extend the well-known Landesman–Lazer conditions to study various bifurcation phenomena taking place near the two principal eigenvalues associated to the differential operator. We provide conditions under which the solution branches extend globally along the eigenvalue gap. We also present examples illustrating the results and hypotheses. © 2010 Elsevier Inc. All rights reserved. Keywords: Hamilton–Jacobi–Bellman equation; Landesman–Lazer condition; Bifurcation from infinity; Principal eigenvalues
* Corresponding author.
E-mail addresses:
[email protected] (P. Felmer),
[email protected] (A. Quaas),
[email protected] (B. Sirakov). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.012
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1. Introduction We study the boundary-value problem
F (D 2 u, Du, u, x) + λu = f (x, u) u=0
in Ω, on ∂Ω,
(1.1)
where the second order differential operator F is of Hamilton–Jacobi–Bellman (HJB) type, that is, F is a supremum of linear elliptic operators, f is sub-linear in u at infinity, and Ω ⊂ RN is a regular bounded domain. HJB operators have been the object of intensive study during the last thirty years – for a general review of their theory and applications we refer to [25,31,40,14]. Well-known examples include the Fucik operator u + bu+ + au− [26], the Barenblatt operator max{au, bu} 2 [10,30], and the Pucci operator M+ λ,Λ (D u) [35,15]. To introduce the problem we are interested in, let us first recall some classical results in the case when F is the Laplacian and λ ∈ (−∞, λ2 ) (we shall denote with λi the i-th eigenvalue of the Laplacian). If f is independent of u, the solvability of (1.1) is a consequence of the Fredholm alternative, namely, if λ = λ1 , problem (1.1) has a solution for each f , while if λ = λ1 (resonance) it has solutions if and only if f is orthogonal to ϕ1 , the first eigenfunction of the Laplacian. The existence result in the non-resonant case extends to nonlinearities f (x, u) which grow sub-linearly in u at infinity, thanks to Krasnoselski–Leray–Schauder degree and fixed point theory, see [1]. A fundamental result, obtained by Landesman and Lazer [32] (see also [29]), states that in the resonance case λ = λ1 the problem u + λ1 u = f (x, u)
in Ω,
u = 0 on ∂Ω
is solvable provided f is bounded and, setting f ± (x) := lim sup f (x, s), s→±∞
f± (x) := lim inf f (x, s) s→±∞
(1.2)
(this notation will be kept from now on), one of the following conditions is satisfied:
−
f ϕ1 < 0 < Ω
f− ϕ1 > 0 >
f+ ϕ1 , Ω
Ω
f+ ϕ1 .
(1.3)
Ω
This result initiated a huge amount of work on solvability of boundary-value problems in which the elliptic operator is at, or more generally close to, resonance. Various extensions of the results in [32] for resonant problems were obtained in [2,4,12]. Further, Mawhin and Schmitt [34] – see also [19,18] – considered (1.1) with F = for λ close to λ1 , and showed that the first (resp. the second) condition in (1.3) implies that for some δ > 0 problem (1.1) has at least one solution for λ ∈ (λ1 − δ, λ1 ] and at least three solutions for λ ∈ (λ1 , λ1 + δ) (resp. at least one solution for λ ∈ [λ1 , λ1 + δ) and at least three solutions for λ ∈ (λ1 , λ1 + δ)). These results rely on degree theory and, more specifically, on the notion of bifurcation from infinity, studied by Rabinowitz in [37].
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The same results naturally hold if the Laplacian is replaced by any uniformly elliptic operator in divergence form. Further, they do remain true if a general linear operator in non-divergence form L = aij (x)∂x2i xj + bi (x)∂xi + c(x)
(1.4)
is considered, but we have to change ϕ1 in (1.3) by the first eigenfunction of the adjoint operator of L. This fact is probably known to the experts, though we are not aware of a reference. Its proof – which will also easily follow from our arguments below – uses the Donsker–Varadhan [21] characterization of the first eigenvalue of L and the results in [11] which link the positivity of this eigenvalue to the validity of the maximum principle and to the Alexandrov–Bakelman–Pucci inequality (the degree theory argument remains the same as in the divergence case). The interest in this type of problems has remained high in the PDE community over the years. Recently a large number of works have considered the extensions of the above results to quasilinear equations (for instance, replacing the Laplacian by the p-Laplacian), where somewhat different phenomena take place, see [6,20,22,23]. There has also been a considerable interest in refining the Landesman–Lazer hypotheses (1.3) and finding general hypotheses on the nonlinearity which permit to determine on which side of the first eigenvalue the bifurcation from infinity takes place, see [3,5,27]. It is our goal here to study the boundary-value problem (1.1) under Landesman–Lazer conditions on f , when F is a Hamilton–Jacobi–Bellman (HJB) operator, that is, the supremum of linear operators as in (1.4): F [u] := F D 2 u, Du, u, x = sup tr Aα (x)D 2 u + bα (x).Du + cα (x)u , α∈A
(1.5)
where A is an arbitrary index set. The following hypotheses on F will be kept throughout the paper: Aα ∈ C(Ω), bα , cα ∈ L∞ (Ω) for all α ∈ A and, for some constants 0 < λ Λ, we have λI Aα (x) ΛI , for all x ∈ Ω and all α ∈ A. We stress however that all our results are new even for operators with smooth coefficients. Let us now describe the most distinctive features of HJB operators – with respect to the operators considered in the previous papers on Landesman–Lazer type problems – which make our work and results different. The HJB operator F [u] defined in (1.5) is nonlinear, yet positively homogeneous (that is, F [tu] = tF [u] for t 0), thus one may expect it has eigenvalues and eigenfunctions on the cones of positive and negative functions, but they may be different to each other. This fact was established by Lions in 1981, in the case of operators with regular coefficients, see [33]. In that paper he proved F [u] has two real “demi”- or “half”-principal − + − eigenvalues λ+ 1 , λ1 ∈ R (λ1 λ1 ), which correspond to a positive and a negative eigenfunction, respectively, and showed that the positivity of these numbers is a sufficient condition for the solvability of the related Dirichlet problem. Recently in [36] the second and the third author extended these results to arbitrary operators and studied the properties of the eigenvalues and the eigenfunctions, in particular the relation between the positivity of the eigenvalues and the validity of the comparison principle and the Alexandrov–Bakelman–Pucci estimate, thus obtaining extensions to nonlinear operators of the results of Berestycki, Nirenberg, and Varadhan in [11]. − In what follows we always assume that F is indeed nonlinear in the sense that λ+ 1 < λ1 – note + − the results in [36] easily imply that λ1 = λ1 can occur only if all linear operators which appear in (1.5) have the same principal eigenvalues and eigenfunctions.
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In the subsequent works [39,24] we considered the Dirichlet problem (1.1) with f independent of u, and we obtained a number of results on the structure of its solution set, depending − on the position of the parameter λ with respect to the eigenvalues λ+ 1 and λ1 . In particular, we − p proved that for each λ in the closed interval [λ+ 1 , λ1 ] and each h ∈ L , p > N , which is not a + ∗ multiple of the first eigenfunction ϕ1 , there exists a critical number tλ,F (h) such that the equation F [u] + λu = tϕ1+ + h
in Ω,
u=0
on ∂Ω
(1.6)
∗ (h) and has no solutions for t < t ∗ (h). We remark this is in sharp has solutions for t > tλ,F λ,F contrast with the case of linear F , say F = , when (1.6) has a solution if and only if t = ∗ (h) = − 2 tλ, Ω (hϕ1 ) (we shall assume all eigenfunctions are normalized so that their L -norm is one). Much more information on the solutions of (1.6) can be found in [39] and [24]. The value of tλ∗+ ,F (h) in terms of F and h was computed by Armstrong [7], where he obtained an extension 1
to HJB operators of the Donsker–Varadhan minimax formula. We now turn to the statements of our main results. A standing assumption on the function f will be the following (F0) f : Ω × R → R is continuous and sub-linear in u at infinity: lim
|s|→∞
f (x, s) = 0 uniformly in x ∈ Ω. s
Remark 1.1. For continuous f it is known [16,41,42] that all viscosity solutions of (1.1) are actually strong, that is, in W 2,p (Ω), for all p < ∞. Without serious additional complications we could assume that the dependence of f in x is only in Lp , for some p > N . Remark 1.2. Some of the statements below can be divided into subcases by supposing that f is sub-linear in u only as u → ∞ or as u → −∞ (such results for the Laplacian can be found in [19,18]). We have chosen to keep our theorems as simple as possible. Now we introduce the hypotheses which extend the Landesman–Lazer assumptions (1.3) for the Laplacian to the case of general HJB operators. From now on we write the critical t-values ∗ = t ∗ (h) = t ∗ ∗ = t ∗ (h) = t ∗ at resonance as t+ (h) and t− (h), and p > N is a fixed number. + − λ+ ,F λ− ,F We assume there are (F+ ) (F− ) (F+r ) (F−r )
1
1
∗ (c ) < 0, a function c+ ∈ Lp (Ω), such that c+ (x) f+ (x) in Ω and t+ + − p − − ∗ (c− ) > 0, a function c ∈ L (Ω), such that c (x) f (x) in Ω and t− ∗ (c+ ) > 0, a function c+ ∈ Lp (Ω), such that c+ (x) f + (x) in Ω and t+ p ∗ a function c− ∈ L (Ω), such that c− (x) f− (x) in Ω and t− (c− ) < 0.
Remark 1.3. Note that, decomposing h(x) = ( Ω hϕ1+ )ϕ1+ (x) + h⊥ (x), where ϕ1+ is the eigenfunction associated to λ+ 1 , we clearly have tλ∗ (h) = tλ∗ h⊥ −
Ω
+ hϕ1
−
for each λ ∈ λ+ 1 , λ1 .
(1.7)
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So when F = hypotheses (F+ )–(F− ) and (F+r )–(F−r ) reduce to the classical Landesman– Lazer conditions (1.3), since for the Laplacian the critical t-value of a function orthogonal to ϕ1 is always zero, by the Fredholm alternative. We further observe that whenever one of the limits f± , f ± is infinite, the functions c± , c± with the required in (F+ )–(F− ), (F+r )–(F−r ) property always exist, while if any of f± , f ± is in Lp (Ω), we take the corresponding c to be equal to this limit. Note also that the strict inequalities in (F+ )–(F− ) and (F+r )–(F−r ) are important, see Section 7. Throughout the paper we denote by S the set of all pairs (u, λ) ∈ C(Ω) × R which satisfy Eq. (1.1). For any fixed λ we set S(λ) = {u | (u, λ) ∈ S} and if C ⊂ S we denote C(λ) = C ∩ S(λ). − Our first result gives a statement of existence of solutions for λ around λ+ 1 and λ1 , under the above Landesman–Lazer type hypotheses. We recall that for some constant δ0 > 0 (all constants − in the paper will be allowed to depend on N , λ, Λ, γ , diam(Ω)), λ+ 1 , λ1 are the only eigenvalues of F in the interval (−∞, λ− 1 + δ0 ) – see Theorem 1.3 in [36]. Theorem 1.1. Assume (F0) and (F+ ) hold. Then there exist δ > 0 and two disjoint closed connected sets of solutions of (1.1), C1 , C2 ⊂ S such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 ], + (2) C1 (λ) = ∅ and C2 (λ) = ∅ for all λ ∈ (λ+ 1 , λ1 + δ). The set C2 is a branch of solutions “bifurcating from plus infinity to the right of λ+ 1 ”, + that is, C2 ⊂ C(Ω) × (λ1 , ∞) and there is a sequence {(un , λn )} ∈ C2 such that λn → λ+ 1 and un ∞ → ∞. Moreover, for every sequence {(un , λn )} ∈ C2 such that λn → λ+ 1 and un ∞ → ∞, un is positive in Ω, for n large enough. If we assume (F− ) holds, then there is a branch of solutions of (1.1) “bifurcating from minus − infinity to the right of λ− 1 ”, that is, a connected set C3 ⊂ S such that C3 ⊂ C(Ω) × (λ1 , ∞) for − which there is a sequence {(un , λn )} ∈ C3 such that λn → λ1 and un ∞ → ∞. Moreover, for every sequence {(un , λn )} ∈ C3 such that λn → λ− 1 and un ∞ → ∞, un is negative in Ω for n large. Under the sole hypothesis (F+ ) it cannot be guaranteed that the sets of solutions C1 and C2 extend much beyond λ+ 1 . This important fact will be proved in Section 7, where we find δ0 > 0 such that for each δ ∈ (0, δ0 ) we can construct a nonlinearity f (x, u) which satisfies (F0), (F+ ) and (F− ), but for which S(λ+ 1 + δ) is empty. It is clearly important to give hypotheses on f under which we can get a global result, that is, existence of continua of solutions which extend over the gap between the two principal eigenvalues (this gap accounts for the nonlinear nature of the HJB operator!). The next theorems deal with that question, and use the following additional assumptions. (F1) f (x, 0) 0 and f (x, 0) ≡ 0 in Ω. (F2) f (x, ·) is locally Lipschitz, that is, for each R ∈ R there is CR such that |f (x, s1 ) − f (x, s2 )| CR |s1 − s2 | for all s1 , s2 ∈ (−R, R) and x ∈ Ω. A discussion on these hypotheses, together with examples and counterexamples, will be given in Section 7.
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Theorem 1.2. Assume (F0), (F1), (F2), (F+ ) and (F− ) hold. Then there exist a constant δ > 0 and three disjoint closed connected sets of solutions C1 , C2 , C3 ⊂ S, such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 ], − (2) Ci (λ) = ∅, i = 1, 2, for all λ ∈ (λ+ 1 , λ1 ], − − ∅, i = 1, 2, 3, for all λ ∈ (λ1 , λ1 + δ). (3) Ci (λ) = The sets C2 end C3 have the same “bifurcation from infinity” properties as in the previous theorem. While Theorem 1.2 deals with bifurcation branches going to the right of the corresponding eigenvalues, the next theorem takes care of the case where the branches go to the left of the eigenvalues. Theorem 1.3. Assume (F0), (F1), (F2), (F+r ) and (F−r ) hold. Then there exist δ > 0 and disjoint closed connected sets of solutions C1 , C2 ⊂ S such that (1) C1 (λ) = ∅ for all λ ∈ (−∞, λ+ 1 − δ], + (2) C1 (λ) = ∅, C2 (λ) contains at least two elements for all λ ∈ (λ+ 1 − δ, λ1 ), and C2 is a branch + “bifurcating from plus infinity to the left of λ1 ”, − (3) C1 (λ) = ∅ and C2 (λ) = ∅ for all λ ∈ [λ+ 1 , λ1 ), and either: (i) C1 is the branch “bifurcating from minus infinity to the left of λ− 1 ”, (ii) there is a closed connected set of solutions C3 ⊂ S, disjoint of C1 and C2 , “bifurcating from minus infinity to the left of λ− 1 ” such that C3 (λ) has at least two elements for all − − λ ∈ (λ1 − δ, λ1 ), − (4) C2 (λ) = ∅ for all λ ∈ [λ− 1 , λ1 + δ]. In case (ii) in (3), C2 (λ) = ∅ and C3 (λ) = ∅ for all − − λ ∈ [λ1 , λ1 + δ]. Note alternative (3)(ii) in this theorem is somewhat anomalous. While we are able to exclude it in a number of particular cases (in particular for the model nonlinearities which satisfy the hypotheses of the theorem), we do not believe it can be ruled out in general. See Proposition 6.1 in Section 6. Going back to the case when F is linear, a well-known “rule of thumb” states that the number of expected solutions of (1.1) changes by two when the parameter λ crosses the first eigenvalue of F . A heuristic way of interpreting our theorems is that, when F is a supremum of linear operators, crossing a “half”-eigenvalue leads to a change of the number of solutions by one. The following graphs illustrate our theorems.
The paper is organized as follows. The next section contains some definitions, known results, and continuity properties of the critical values t ∗ . In Section 3 we obtain a priori bounds for the
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solutions of (1.1), and construct super-solutions or sub-solutions in the different cases. In Section 4 bifurcation from infinity for HJB operators is established through the classical method of Rabinowitz, while in Section 5 we construct and study a bounded branch of solutions of (1.1). These results are put together in Section 6, where we prove our main theorems. Finally, a discussion on our hypotheses and some examples which highlight their role are given in Section 7. 2. Preliminaries and continuity of t ∗ First, we list the properties shared by HJB operators of our type. The function F : SN × RN × R × Ω → R satisfies (with S, T ∈ SN × RN × R): (H0) F is positively homogeneous of order 1: F (tS, x) = tF (S, x) for t 0. (H1) There exist λ, Λ, γ > 0 such that for S = (M, p, u), T = (N, q, v) M− λ,Λ (M − N ) − γ |p − q| + |u − v| F (S, x) − F (T , x)
M+ λ,Λ (M − N ) + γ |p − q| + |u − v| .
(H2) The function F (M, 0, 0, x) is continuous in SN × Ω. (DF) We have −F (T − S, x) F (S, x) − F (T , x) F (S − T , x) for all S, T . + + In (H1) M− λ,Λ and Mλ,Λ denote the Pucci extremal operators, defined as Mλ,Λ (M) = supA∈A tr(AM), M− λ,Λ (M) = infA∈A tr(AM), where A ⊂ SN denotes the set of matrices whose eigenvalues lie in the interval [λ, Λ], see for instance [15]. Note under (H0) assumption (DF) is equivalent to the convexity of F in S – see Lemma 1.1 in [36]. Hence for each φ, ψ ∈ W 2,p (Ω) we have the inequalities F [φ + ψ] F [φ] + F [ψ] and F [φ − ψ] F [φ] − F [ψ]. We recall the definition of the principal eigenvalues of F from [36]
+ λ+ 1 (F, Ω) = sup λ Ψ (F, Ω, λ) = ∅ ,
− λ− 1 (F, Ω) = sup λ Ψ (F, Ω, λ) = ∅ ,
where Ψ ± (F, Ω, λ) = {ψ ∈ C(Ω) | ±(F [ψ] + λψ) 0, ±ψ > 0 in Ω}. Many properties of the eigenvalues (simplicity, isolation, monotonicity and continuity with respect to the domain, relation with the maximum principle) are established in Theorems 1.1–1.9 of [36]. We shall repeatedly use these results. We shall also often refer to the statements on the solvability of the Dirichlet problem, given in [36] and [24]. We recall the following Alexandrov–Bakelman–Pucci (ABP) and C 1,α estimates, see [28,17, 42]. Theorem 2.1. Suppose F satisfies (H0), (H1), (H2), and u is a solution of F [u] + cu = f (x) in Ω, with u = 0 on ∂Ω. Then there exist α ∈ (0, 1) and C0 > 0 depending on N , λ, Λ, γ , c and Ω such that u ∈ C 1,α (Ω), and u C 1,α (Ω) C0 u L∞ (Ω) + f Lp (Ω) . Moreover, if one chooses c = −γ (so that by (H1) F − γ is proper) then this equation has a unique solution which satisfies u C 1,α (Ω) C0 f L∞ (Ω) . More precisely, any solution of F [u] − γ u f (x) satisfies supΩ u sup∂Ω u + C f LN .
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For readers’ convenience we state a version of Hopf’s lemma (for viscosity solutions it was proved in [9]). Theorem 2.2. Let Ω ⊂ RN be a regular domain and let γ > 0, δ > 0. Assume w ∈ C(Ω) 2 is a viscosity solution of M− λ,Λ (D w) − γ |Dw| − δw 0 in Ω, and w 0 in Ω. Then either w ≡ 0 in Ω or w > 0 in Ω and at any point x0 ∈ ∂Ω at which w(x0 ) = 0 we have 0) lim supt 0 w(x0 +tν)−w(x < 0, where ν is the interior normal to ∂Ω at x0 . t The next theorem is a consequence of the compact embedding C 1,α (Ω) → C 1 (Ω), Theorem 2.1, and the convergence properties of viscosity solutions (see Theorem 3.8 in [17]). Theorem 2.3. Let λn → λ in R and fn → f in Lp (Ω). Suppose F satisfies (H1) and un is a viscosity solution of F [un ] + λn un = fn in Ω, un = 0 on ∂Ω. If {un } is bounded in L∞ (Ω) then a subsequence of {un } converges in C 1 (Ω) to a function u, which solves F [u] + λu = f in Ω, u = 0 on ∂Ω. As a simple consequence of this theorem, the homogeneity of F and the simplicity of the eigenvalues we obtain the following proposition. p Proposition 2.1. Let λn → λ± 1 in R and fn be bounded in L (Ω). Suppose F satisfies (H1) and un is a viscosity solution of F [un ] + λn un = fn in Ω, un = 0 on ∂Ω. If {un } is unbounded in L∞ (Ω) then a subsequence of uunn converges in C 1 (Ω) to ϕ1± . In particular, un is positive (negative) for large n, and for each K > 0 there is N such that |un | Kϕ1+ for n N .
For shortness, from now on the zero boundary condition on ∂Ω will be understood in all differential (in)equalities we write, and · will refer to the L∞ (Ω)-norm. We devote the remainder of this section to the definition and some basic continuity properties of the critical t-values for (1.6). These numbers are crucial in the study of existence of solutions − p at resonance and in the gap between the eigenvalues. For each λ ∈ [λ+ 1 , λ1 ] and each d ∈ L , + which is not a multiple of the first eigenfunction ϕ1 , the number tλ∗ (d) = inf t ∈ R F [u] + λu = sϕ1+ + d has solutions for s t − is well defined and finite. The non-resonant case λ ∈ (λ+ 1 , λ1 ) was considered in [39], while the + − resonant cases λ = λ1 and λ = λ1 were studied in [24]. − In what follows we prove the continuity of tλ∗ : Lp (Ω) → R for any fixed λ ∈ [λ+ 1 , λ1 ]. Ac− tually, in [39] the continuity of this function is proved for all λ ∈ (λ+ 1 , λ1 ), so we only need to + − ∗ ∗ . In doing so, it is take care of the resonant cases λ = λ1 and λ = λ1 , that is, to study t+ and t− ∗ ∗ convenient to use the following equivalent definitions of t+ and t− (see [24])
∗ (d) = inf t ∈ R for each s > t and λn λ+ t+ 1 there exists un
such that F [un ] + λn un = sϕ1+ + d and un is bounded
and
(2.1)
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∗ t− (d) = inf t ∈ R for each s > t and λn λ− 1 there exists un
such that F [un ] + λn un = sϕ1+ + d and un is bounded .
(2.2)
∗ , t ∗ : Lp (Ω) → R are continuous. Proposition 2.2. The functions t+ − ∗ is not continuous, then there is d ∈ Lp (Ω), ε > 0 and a sequence d → d Proof. If we assume t+ n p ∗ (d ) t ∗ (d) + 3ε for all n ∈ N or t ∗ (d ) t ∗ (d) − 3ε for all n ∈ N. in L (Ω) such that either t+ n + + n + ∗ (d ) t ∗ (d) + 3ε for all n ∈ N. Then, for any sequence λ λ+ we First we suppose that t+ n m + 1 find solutions um n of the equation
∗ + m F um n + λm un = t+ (dn ) − 2ε ϕ1 + d
in Ω,
and the sequence { um n } is bounded as m → ∞, for each fixed n – see (2.1). We also consider the solutions wn of F [wn ] − γ wn = dn − d. By Theorem 2.1 we know that wn → 0 in C 1 (Ω). Then by the structural hypotheses on F (recall F [u + v] F [u] + F [v]) we have
m ∗ + F um n + wn + λm un + wn t+ (dn ) − 2ε ϕ1 + dn + (γ + λm )wn ∗ t+ (dn ) − ε ϕ1+ + dn , where the last inequality holds if n is large, independently of m. Fix one such n. On the other hand we can take solutions znm of
∗ F znm + λm znm = t+ (dn ) − ε ϕ1+ + dn
m F um n + wn + λm un + wn .
By (2.1) for any n we have znm → ∞ as m → ∞. By the comparison principle (valid by m m m λm < λ+ 1 and Theorem 1.5 in [36]) we obtain zn un + wn in Ω, hence zn is bounded above + m as m → ∞. Since zn is bounded below, by Theorem 1.7 in [36] and λm λ1 < λ− 1 , we obtain a contradiction. ∗ (d ) t ∗ (d) − 3ε. Let u be a solution of Assume now that t+ n n + ∗ + F [un ] + λ+ 1 un = t+ (d) − 2ε ϕ1 + dn
in Ω,
∗ (d ) < t ∗ (d) − 2ε – Theorem 1.2 in [24]. Let w be the solution of F [w ] + which exists since t+ n n n + cwn = d − dn in Ω, with wn → 0 in C 1 (Ω). Then there exists n0 large enough so that (λ+ 1 + γ )wn0 < εϕ1+ , and consequently un0 + wn0 is a super-solution of
∗ + F [u] + λ+ 1 u = t+ (d) − ε ϕ1 + d
in Ω.
(2.3)
Now, if w is the solution of F [w] − γ w = −d in Ω, by defining vk = kϕ1− − w we obtain + + ∗ + − − F [vk ] + λ+ 1 vk k λ1 − λ1 ϕ1 + d − λ1 + γ w > t+ (d) − ε ϕ1 + d, for k large enough. By taking k large we also have vk < un0 − wn0 in Ω, thus Eq. (2.3) possesses ordered super- and sub-solutions. Consequently it has a solution (by Perron’s method – see for
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∗ (d). This completes the instance Lemma 4.3 in [36]), a contradiction with the definition of t+ ∗. proof of the continuity of the function t+ ∗ . Assuming t ∗ is not conThe rest of the proof is devoted to the analysis of continuity of t− − ∗ (d ) t ∗ (d) + 3ε or tinuous, there is ε > 0 and a sequence dn → d in Lp (Ω) such that either t− n − ∗ (d ) t ∗ (d) − 3ε. In the first case, let us consider a sequence λ λ− , and a solution v of t− n m m − 1 the equation
∗ F [vm ] + λm vm = t− (d) + ε ϕ1+ + d
in Ω.
∗ (d ) t ∗ (d) + We recall vm exists, by the results in [7] and [24]. We have shown in [24] that t− n − ∗ 3ε > t− (d) + ε implies that vm can be chosen to be bounded as m → ∞ (see (2.2)). Let wn be the solution to F [wn ] − γ wn = dn − d in Ω, as above. Then znm0 = vm + wn0 satisfies for some large n0
∗ ∗ F znm0 + λm znm0 t− (d) + ε ϕ1+ + (λm + γ )wn + dn t− (d) + 2ε ϕ1+ + dn , since again wn → 0 in C 1 (Ω). On the other hand, we consider a solution of
∗ + m F um n + λm un = t− (d) + 2ε ϕ1 + dn
in Ω.
∗ (d ) t ∗ (d) + 3ε > t ∗ (d) + 2ε for all n, the sequence um is not bounded (again by (2.2) As t− n − − n − m and [24]) and um n / un ∞ → ϕ1 as m → ∞, for each fixed n. Therefore for large m the function Ψ = um n0 − (vm + wn0 ) < 0 and F [Ψ ] + λm Ψ 0, which is a contradiction with the definition − of λ1 , since λm > λ− 1. ∗ (d ) Let us assume now that for ε > 0 and the sequence dn → d in Lp (Ω) we have t− n − ∗ t− (d) − 3ε, for all n. Let λm λ1 and vm be a solution of the equation F [vm ] + λm vm = ∗ (d) − ε)ϕ + + d in Ω (by (2.2) v is unbounded), and let w be the solution to F [w ] − γ w = (t− m n n n 1 d − dn in Ω. Then, vm / vm ∞ → ϕ1− and wn → 0 in C 1 (Ω). We take a solution um n to
∗ + m F um n + λm un = t− (d) − 2ε ϕ1 + dn
in Ω,
∗ (d ) t ∗ (d) − 3ε < t ∗ (d) − 2ε, for any given n there exists a constant c and note that, since t− n n − − m such that un ∞ cn , for all m. Now, as above, we define Ψ = vm − (um + w ), and see that n n
∗ ∗ F [Ψ ] + λm Ψ t− (d) − ε ϕ1+ + d − t− (d) − 2ε ϕ1+ − dn − F [wn ] − λm wn εϕ1+ − (λm + γ )wn . We choose n large enough to have (λm + γ )wn < εϕ1+ in Ω. Then, keeping n fixed, we can choose m large enough to have Ψ < 0 in Ω, a contradiction with the definition of λ− 1 , since − λm > λ 1 . 2 Finally we prove that the function tλ∗ (d) is also continuous in λ at the end points of the interval when d is kept fixed. This fact will be needed in Section 7.
− [λ+ 1 , λ1 ],
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Proposition 2.3. For every d ∈ Lp (Ω) ∗ lim tλ∗ (d) = t+ (d) and
λ λ+ 1
∗ lim tλ∗ (d) = t− (d).
λλ− 1
∗ ∗ Proof. Let us assume that there are ε > 0 and a sequence λn λ+ 1 such that tλn < t+ − ε (since ∗ d is fixed, we do not write it explicitly). Then by the definition of tλn there is a function un satisfying
∗ F [un ] + λn un = t+ − ε ϕ1+ + d
in Ω.
Since λn λ+ 1 , un cannot be bounded, for otherwise we get a contradiction with the defini∗ by finding a solution with t < t ∗ – from Theorem 2.3. Then by Proposition 2.1 tion of t+ + un / un ∞ → ϕ1+ , un is positive for large n, and ∗ + + ∗ + F [un ] + λ+ 1 un = t+ − ε ϕ1 + d + λ1 − λn un < t+ − ε ϕ1 + d, ∗ , let u be a solution of F [u] + λ+ u = that is, un is a super-solution. On the other hand, for t > t+ 1 + ∗ − ε)ϕ + + d as a right-hand tϕ1 + d, in Ω, then u is a sub-solution for this equation with (t+ 1 side. By taking n large enough, we have un u, so that the equation
∗ + F [u] + λ+ 1 u = t+ − ε ϕ1 + d
in Ω
∗. has a solution, a contradiction with the definition of t+ ∗ ∗ Now we assume that there are ε > 0 and a sequence λn λ+ 1 such that tn = tλn > t+ + 2ε. Let v be a solution to
∗ + F [v] + λ+ 1 v = t+ + ε/2 ϕ1 + d
in Ω,
∗ + ε)ϕ + + d − ε/2ϕ + + (λ − λ+ )v. Since t ∗ + ε < t ∗ − ε/2, by choosing then F [v] + λn v = (t+ n + λn 1 1 1 n large we find F [v] + λn v < (tλ∗n − ε/2)ϕ1+ + d, so that v is a super-solution of
F [u] + λn u = tλ∗n − ε/2 ϕ1+ + d.
(2.4)
− Next we consider a solution uK of F [u] + (λ+ 1 + ν)u = K (where we have set ν = (λ1 − + λ1 )/2 > 0), for each K > 0. Such a solution exists by Theorem 1.9 in [36], and it further satisfies uK < 0 in Ω and uK ∞ → ∞ as K → ∞, so |uK | C(K)ϕ1+ , where C(K) → ∞ as K → ∞. Let w be the (unique) solution of F [w] − γ w = −d in Ω. Since F [uK − w] F [uK ] − F [w], we easily see that the function uK − w is a sub-solution of (2.4) and uK − w < v, for large K. Then Perron’s method leads again to a contradiction with the definition of tλn . This shows tλ∗ is right-continuous at λ+ 1. Now we prove the second statement of Lemma 2.3. Assume there are ε > 0 and a sequence ∗ ∗ λn λ− 1 such that tλn < t− − ε. Let un be a solution to
∗ − ε ϕ1+ + d F [un ] + λn un = t−
in Ω.
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− Since λn → λ− 1 , un cannot be bounded (as before) and then un / un ∞ → ϕ1 . Thus, for large n we have un < 0 and
∗ + − ∗ + F [un ] + λ− 1 un = t− − ε ϕ1 + d + λ1 − λn un < t− − ε ϕ1 + d, so that uk is a super-solution for some large (fixed) k. Consider now a sequence λ˜ n λ− 1 and let vn be the solution to ∗ − ε ϕ1+ + d F [vn ] + λ˜ n vn = t−
in Ω,
whose existence was proved in [7] and [24]. Then vn cannot be bounded, so vn / vn ∞ → ϕ1− , and for large n we have ∗ + − ∗ + ˜ F [vn ] + λ− 1 vn = t− − ε ϕ1 + d + λ1 − λn vn > t− − ε ϕ1 + d, that is, vn is a sub-solution. For the already fixed uk , we can find n sufficiently large so that uk > vn , which implies that the equation ∗ + F [u] + λ− 1 u = t− − ε ϕ1 + d
in Ω
∗. has a solution, a contradiction with the definition of t− ∗ ∗ ∗ Finally, assume that there are ε > 0 and a sequence λn λ− 1 such that tλn > tλn − 2ε > t− + ε. By Theorem 1.4 in [24] we can find a function u which solves the equation F [u] + λ− 1u= + ∗ (t− + ε)ϕ1 + d in Ω. Then
+ ∗ + F [u] + λn u < tλ∗n − ε ϕ1+ + d − εϕ1+ + λn − λ− 1 ϕ1 < tλn − ε ϕ1 + d, so that u is a super-solution of F [u] + λn u = (tλ∗n − ε)ϕ1+ + d, for some large fixed n. As we explained above, since λn < λ− 1 , by Theorem 1.9 in [36] we can construct an arbitrarily negative sub-solution of this problem, hence a solution as well, contradicting the definition of tλ∗n . 2 3. Resonance and a priori bounds In this section we assume that the nonlinearity f (x, s) satisfies the one-sided Landesman– Lazer conditions at resonance, that is, one of (F+ ), (F− ), (F+r ) and (F−r ). Under each of these conditions we analyze the existence of super-solutions, sub-solutions and a priori bounds when − λ is close to the eigenvalues λ+ 1 and λ1 . This information will allow us to obtain existence of solutions by using degree theory and bifurcation arguments. In particular we will get branches bifurcating from infinity which curve right or left depending on the a priori bounds obtained here. We start with the existence of a super-solution and a priori bounds at λ+ 1 , under hypothesis (F+ ). Proposition 3.1. Assume f satisfies (F0) and (F+ ). Then there exists a super-solution z such that + F [z] + λz < f (x, z) in Ω, for all λ ∈ (−∞, λ+ 1 ]. Moreover, for each λ0 < λ1 there exist R > 0 + and a super-solution z0 such that if u is a solution of (1.1) with λ ∈ [λ0 , λ1 ], then u R and u z0 in Ω.
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Proof. We first replace c+ by a more appropriate function: we claim that for each ε > 0 there exist R > 0 and a function d ∈ Lp (Ω) such that d − c+ Lp (Ω) ε
and u Rϕ1+
implies f x, u(x) d(x) in Ω.
In fact, setting σ =
ε , we can find s0 such that f (x, s) c+ (x) − σ in Ω, for all s s0 . Let 2|Ω|1/p + R Ω = {x ∈ Ω | Rϕ1 (x) > s0 } and define the function dR as dR (x) = c+ (x) − σ if x ∈ Ω R , and dR (x) = −M for x ∈ Ω \ Ω R , where M is such that f (x, s) −M, for all s ∈ [0, s0 ]. It is then trivial to check that the claim holds for d = dR , if R is taken such that |Ω \ Ω R | < (ε/2M)p . ∗ (Proposition 2.2) we can fix ε so small that the function Now, by (F+ ) and the continuity of t+
d chosen above satisfies
∗ t+ (d) < 0.
(3.1)
Let zn be a solution to + F [zn ] + λ+ 1 zn = tn ϕ1 + d
in Ω,
∗ (d) < 0, t t ∗ (d), is a sequence such that z can be chosen to be unbounded – where tn → t+ n n + such a choice of tn and zn is possible thanks to Theorem 1.2 in [24]. Then zn / zn → ϕ1+ , which implies that for large n
F [zn ] + λ+ 1 zn < d
and zn Rϕ1+ ,
by (3.1), where R is as in the claim above. Thus zn is a strict super-solution and, since zn is positive, F [zn ] + λzn < f (x, zn ), for all λ ∈ (−∞, λ+ 1 ]. From now on we fix one such n0 and drop the index, calling the super-solution z. Suppose there exists an unbounded sequence un of solutions to F [un ] + λn un = f (x, un )
in Ω,
+ + with λn ∈ [λ0 , λ+ 1 ] and λn → λ. If λ < λ1 then a contradiction follows since λ1 is the first + eigenvalue (divide the equation by un and let n → ∞). If λ = λ1 then un / un → ϕ1+ , so that for n large we have un > z and un Rϕ1+ , consequently f (x, un ) d(x) in Ω. Thus, setting w = un − z we get, by λn λ+ 1 , w > 0, + F [w] + λ+ 1 w F [un ] − F [z] + λ1 (un − z) > f (x, un ) − d 0.
Since w > 0, Theorem 1.2 in [36] implies the existence of a constant k > 0 such that w = kϕ1+ , a contradiction with the last strict inequality. Now that we have an a priori bound for the solutions, we may choose an appropriate n0 for the definition of z0 = zn0 , which makes it larger than all solutions. 2 Next we state an analogous proposition on the existence of a sub-solution to our problem at
λ− 1 under hypothesis (F− ).
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Proposition 3.2. Assuming that f satisfies (F0) and (F− ), there exists a strict sub-solution z such that F [z] + λz > f (x, z) in Ω for all λ ∈ (−∞, λ− 1 ]. Moreover, for each δ > 0 there exist R > 0 − and a sub-solution z such that if u solves (1.1) with λ ∈ [λ+ 1 + δ, λ1 ] then u ∞ R and u z in Ω. Proof. By using essentially the same proof as in Proposition 3.1, we can find R > 0 and a func∗ (d) > 0, and u −Rϕ + implies f (x, u(x)) d(x) in Ω. Consider tion d ∈ Lp (Ω) such that t− 1 ∗ a sequence tn t− (d) and solutions zn to + F [zn ] + λ− 1 zn = tn ϕ1 + d
in Ω,
(3.2)
chosen so that zn is unbounded and zn / zn ∞ → ϕ1− – see Theorem 1.4 in [24]. Hence for n large enough F [zn ] + λ− 1 zn > d
and zn −Rϕ1+ .
(3.3)
Thus zn is a strict sub-solution and, since zn is negative for sufficiently large n, F [zn ] + λzn > f (x, zn ), for all λ ∈ (−∞, λ− 1 ]. Fix one such n0 and set z = zn0 . If un is an unbounded sequence of solutions to F [un ] + λn un = f (x, un ), in Ω, with λn ∈ − [λ+ 1 + δ, λ1 ] and λn → λ we obtain a contradiction like in the previous proposition. Namely, if − λ ∈ [λ+ 1 + δ, λ1 ) then the conclusion follows since there are no eigenvalues in this interval. If − λ = λ1 then un / un → ϕ1− , so that for n large un < z and un −Rϕ1+ , hence f (x, un ) d(x), which leads to the contradiction F [z − un ] + λ− 1 (z − un ) 0 and z − un > 0. Then, given the a priori bound, we can choose n0 such that zn0 is smaller than all solutions. 2 The next two propositions are devoted to proving a priori bounds under hypotheses (F+r ) and (F−r ). Proposition 3.3. Under assumptions (F0) and (F+r ) for each δ > 0 the solutions to (1.1) with − λ ∈ [λ+ 1 , λ1 − δ] are a priori bounded. Proof. As in the proof of Proposition 3.1, we may choose R > 0 and a function d so that + + ∗ (d) > 0, that is, ∗ ⊥ dϕ t+ Ω 1 < t+ (d ) (recall (1.7)), and whenever u Rϕ1 then f (x, u) d. + ∗ (d ⊥ ). If the proposition were not true, then there would Let t˜ be fixed such that Ω dϕ1 < t˜ < t+ + be sequences λn λ1 and un of solutions to F [un ] + λn un = f (x, un ), such that un is unbounded. Then un / un → ϕ1+ , in particular, un is positive for large n. Then ⊥ ˜ + F [un ] + λ+ 1 un f (x, un ) d < t ϕ1 + d , ⊥ ˜ + that is, un is a super-solution of F [un ] + λ+ 1 un = t ϕ1 + d . Next, take the solution w of F [w] − ⊥ γ w = −d in Ω, where, as before, γ is the constant from (H1), so that F − γ is proper. For α > 0 we define v = αϕ1− − w, then
+ + − − ⊥ F [v] + λ+ 1 v α λ1 − λ1 ϕ1 − λ1 + γ w + d ,
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exactly like in the proof of Proposition 2.2. If we choose α large enough, we see that v is a sub⊥ ˜ + solution for F [un ] + λ+ 1 un = t ϕ1 + d , and v is smaller than the super-solution we constructed ∗ (d ⊥ ) and before. The existence of a solution to this equation contradicts the definition of t+ ∗ ⊥ t˜ < t+ (d ). 2 Proposition 3.4. Under assumptions (F0) and (F−r ) there exists δ > 0 such that the solutions to − (1.1) with λ ∈ [λ− 1 , λ1 + δ] are a priori bounded. ∗ (d ⊥ ), and Proof. We proceed like in the proof of the previous proposition. Now Ω dϕ1+ > t− + + ∗ ⊥ ˜ ˜ whenever u −Rϕ1 then f (x, u) d. If t is such that Ω dϕ1 > t > t+ (d ), and we assume there are sequences λn λ− 1 and un of solutions to F [un ] + λn un = f (x, un ) in Ω, such that un is unbounded, we get un / un → ϕ1− , consequently ⊥ ˜ + F [un ] + λ− 1 un > t ϕ1 + d . ⊥ ˜ + On the other hand if z solves F [z] + λ− 1 z = t ϕ1 + d in Ω (such z exists by Theorem 1.4 − in [24]), then F [un − z] + λ1 (un − z) > 0, and un − z < 0 in Ω, for large n. Thus, we may apply Theorem 1.4 in [36] to obtain k > 0 so that un − z = kϕ1− , a contradiction with the strict inequality. 2 − 4. Bifurcation from infinity at λ+ 1 and λ1
In this section we prove the existence of unbounded branches of solutions of (1.1), bifurcating − from infinity at the eigenvalues λ+ 1 and λ1 . Then, thanks to the a priori bounds obtained in Section 3, for the two types of Landesman–Lazer conditions (see Propositions 3.1–3.4), we may determine to which side of the eigenvalues these branches curve. We recall that F (M, q, u, x) + cu is decreasing in u for any c −γ , in other words, F + c is a proper operator. Given v ∈ C 1 (Ω) we consider the problem F [u] + cu = (c − λ)v + f (x, v)
in Ω,
u = 0 on ∂Ω,
(4.1)
see Theorem 2.1. We define the operator K : R × C 1 (Ω) → C 1 (Ω) as follows: K(λ, v) is the unique solution u ∈ C 1,α (Ω) of (4.1). The operator K is compact in view of Theorem 2.1 and the compact embedding C 1,α (Ω) → C 1 (Ω). With these definitions, our Eq. (1.1) is transformed into the fixed point problem u = K(λ, u), u ∈ C 1 (Ω), with λ ∈ R as a parameter. We are going to show that the sub-linearity of the function f (x, ·), given by assumption (F0), implies bifurcation at infinity at the eigenvalues of F . The proof follows the standard procedure for the linear case, see for example [38] or [8], so we shall be sketchy, discussing only the main differences. We define
v , G(λ, v) = v 2C 1 K λ, v 2C 1 for v = 0, and G(λ, 0) = 0. Finding u = 0 such that u = K(λ, u) is equivalent to solving the fixed point problem v = G(λ, v), v ∈ C 1 (Ω), for v = u/ u 2C 1 . The important observation is that bifurcation from zero in v is equivalent to bifurcation from infinity for u.
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Let u = G0 (λ, v) be the solution of the problem F [u] + cu = (c − λ)v
in Ω,
u = 0 on ∂Ω,
(4.2)
and set G1 = G − G0 , so that G(λ, v) = G0 (λ, v) + G1 (λ, v). Lemma 4.1. Under the hypothesis (F0) we have lim v C 1 →0
G1 (λ,v) v C 1
= 0.
Proof. Let g = G(λ, v) and g0 = G0 (λ, v). Then we have
1 v F [g] − F [g0 ] + c(g − g0 ) = v C 1 f x, . v C 1 v 2C 1 The right-hand side here goes to zero as v C 1 → 0, by (F0). Then by (DF)
v 1 , F |g − g0 | + c|g − g0 | − v C 1 f x, v C 1 v 2 1 C
so the ABP inequality (Theorem 2.1) implies
v 1 , sup |g − g0 | C v C 1 f x, v C 1 v 2 1 Lp Ω
C
and the result follows.
2
The next proposition deals with the equation v = G0 (λ, v), v ∈ C 1 (Ω) (recall we want to solve v = G0 (λ, v) + G1 (λ, v)), which is equivalent to F D 2 v, Dv, v, x = −λv
in Ω,
v = 0 on ∂Ω.
(4.3)
+ − Proposition 4.1. There exists δ > 0 such that for all r > 0 and all λ ∈ (−∞, λ− 1 + δ) \ {λ1 , λ1 }, the Leray–Schauder degree deg(I − G0 (λ, ·), Br , 0) is well defined. Moreover
⎧ if λ < λ+ 1, ⎨1 deg I − G0 (λ, ·), Br , 0 = 0 < λ < λ− if λ+ 1 1, ⎩ − −1 if λ1 < λ < λ− 1 + δ. Proof. We recall it was proved in [36] that problem (4.3) has only the zero solution in + − (−∞, λ− 1 + δ) \ {λ1 , λ1 }, for certain δ > 0. The compactness of G0 follows from Theorem 2.1, so the degree is well defined in the given ranges for λ. Suppose λ < λ+ 1 and consider the operator I − tG0 (λ, ·) for t ∈ [0, 1]. Since tλ is not an eigenvalue of (4.3), we have for t ∈ [0, 1] deg I − G0 (λ, ·), Br , 0 = deg I − tG0 (λ, ·), Br , 0 = deg(I, Br , 0) = 1.
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− The case λ+ 1 < λ < λ1 was studied in [39]. Consider the problem
F [u] + cu = (c − λ)v − tϕ1+
in Ω,
u = 0 on ∂Ω
(4.4)
0 (λ, v, t). It follows from the results in for t ∈ [0, ∞), whose unique solution is denoted by G [36,39] that for t > 0 the equation F [u] + cu = (c − λ)u − tϕ1+
in Ω,
u=0
on ∂Ω
(4.5)
does not have a solution. On the other hand, since λ is not an eigenvalue, there is R > 0 such that the solutions of (4.5), for t ∈ [0, t], are a priori bounded, consequently 0 (λ, ·, 0), BR , 0 deg I − G0 (λ, ·), Br , 0 = deg I − G 0 (λ, ·, t), BR , 0 = 0. = deg I − G − If λ− 1 < λ < λ1 + δ we proceed as in [24], where the computation of the degree was done by making a homotopy with the Laplacian (see the proof of Lemma 4.2 in that paper). 2
Now we are in position to apply the general theory of bifurcation to v = G(λ, v), see for instance the surveys [38] and [8], and obtain bifurcation branches emanating from (λ+ 1 , 0) and + , 0), exactly like in [13]. In short, from (λ , 0) bifurcates a continuum of solutions of v = (λ− 1 1 G(λ, v), which is either unbounded in λ, or unbounded in u, or connects to (λ, 0), where λ = λ+ 1 − − is an eigenvalue (recall λ+ and λ are the only eigenvalues in (−∞, λ + δ), for some δ > 0). 1 1 1 A similar situation occurs at (λ− 1 , 0). Inverting the variables we obtain bifurcation at infinity for our problem (1.1): Theorem 4.1. Under the hypotheses of Theorem 1.1 there are two connected sets C2 , C3 ⊂ S such that: 1) There is a sequence (λn , un ) with un ∈ C2 (λn ) (un ∈ C3 (λn )), and un ∞ → ∞, λn → λ+ 1 (λ− 1 ). − 2) If (λn , un ) is a sequence such that un ∈ C2 (λn ) (C3 (λn )), un ∞ → ∞ and λn → λ+ 1 (λ1 ), then un is positive (negative) for large n. 3) The branch C2 satisfies one of the following alternatives, for some δ > 0: (i) C2 (λ) = ∅ for − − all λ ∈ (λ+ 1 , λ1 + δ); (ii) there is λ ∈ (−∞, λ1 + δ] such that 0 ∈ C2 (λ); (iii) C2 (λ) = ∅ for + all λ ∈ (−∞, λ1 ); (iv) there is a sequence (λn , un ) such that un ∈ C2 (λn ), un ∞ → ∞, − λn → λ− 1 , and λn λ1 . 4) The branch C3 satisfies one of the following alternatives, for some δ > 0: (i) C3 (λ) = ∅ for − − all λ ∈ (λ− 1 , λ1 + δ); (ii) there is λ ∈ (−∞, λ1 + δ] such that 0 ∈ C3 (λ); (iii) C3 (λ) = ∅ for all λ ∈ (−∞, λ− 1 ); (iv) there is a sequence (λn , un ) such that un ∈ C3 (λn ), un ∞ → ∞, . and λn → λ+ 1 We remark that (F1) excludes alternatives 3)(ii) and 4)(ii) in this theorem.
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5. A bounded branch of solutions In this section we prepare for the proof of our main theorems by establishing the existence of a continuum of solutions of (1.1) which is not empty for all λ ∈ (−∞, λ+ 1 + δ), for some δ > 0. Our first proposition concerns the behavior of solutions of (1.1) when λ → −∞. Proposition 5.1. (1) Assume f satisfies (F0). Then there exists a constant C0 > 0, depending only on F , f , and Ω, such that any solution of (1.1) satisfies u ∞ C0 λ−1 as λ → −∞. (2) If in addition f is Lipschitz at zero, that is, for some ε > 0 and some C > 0 we have |f (x, s1 ) − f (x, s2 )| C|s1 − s2 | for s1 , s2 ∈ (−ε, ε), then (1.1) has at most one solution when λ is sufficiently large and negative. Proof. (1) Let uλ be a sequence of solutions of (1.1), with λ → −∞. We first claim that uλ ∞ is bounded. Suppose this is not so, and say u+ λ ∞ → ∞ (with the usual notation for the positive , on the set Ωλ+ = {uλ > 0} we have the inequality part of u). Then, setting vλ = uλ / u+ λ ∞ F [vλ ] − γ vλ
f (x, uλ ) → 0, u+ λ ∞
as λ → −∞.
The ABP estimate (see Theorem 2.1) then implies supΩ + vλ → 0, which is a contradiction with λ
supΩ + vλ = 1. In an analogous way we conclude that u− λ ∞ is bounded. λ
Hence there exists a constant C such that |f (x, uλ (x))| C in Ω, so λ , F [uλ ] − γ uλ −(λ + γ )uλ − C 0 on the set Ω λ = {uλ > C/(|λ| + γ )}. Applying the maximum principle or the ABP inequality in this where Ω set implies it is empty, which means uλ C/(|λ| + γ ) in Ω. By the same argument we show uλ is bounded below, and (1) follows. (2) From statement (1) we conclude that for λ small, all solutions of (1.1) are in (−ε, ε). If u1 , u2 are two solutions of (1.1) then for |λ| > γ + C we have F [u1 − u2 ] − γ (u1 − u2 ) 0 on {u1 > u2 } which means this set is empty. 2 The next result is stated in the framework of Theorem 1.1 and gives a bounded family of solutions (uλ , λ), for λ ∈ (−∞, λ+ 1 + δ). No assumption of Lipschitz continuity on f is needed. Proposition 5.2. Assume f satisfies (F0) and (F+ ). Then there is a connected subset C1 of S such that C1 (λ) = ∅, for all λ ∈ (−∞, λ+ 1 + δ). Proof. According to Proposition 3.1, given λ0 < λ+ 1 , there is R > 0 so that all solutions of (1.1) ] belong to the ball B . In particular, the equation does not have a solution with λ ∈ [λ0 , λ+ R 1 ] × ∂B . Moreover, there is δ > 0 such that (1.1) does not have a solution in (λ, u) ∈ [λ0 , λ+ R 1 + + [λ1 , λ1 + δ] × ∂BR – otherwise we obtain a contradiction by a simple passage to the limit. Consequently the degree deg(I − K(λ, ·), BR , 0) is well defined for all λ ∈ [λ0 , λ+ 1 + δ] (K(λ, ·) is defined in the previous section). We claim that its value is 1.
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To compute this degree, we fix λ < λ+ 1 and analyze the equation F [u] + λu = sf (x, u)
in Ω,
for s ∈ [0, 1]. Since λ is not an eigenvalue of F in Ω, the solutions of this equation are a priori bounded, uniformly in s ∈ [0, 1], that is, there is R1 R, such that no solution of the equation exists outside of the open ball BR1 . Given v ∈ C 1 (Ω) we denote by Ks (λ, v) the unique solution of the equation F [u] + λu = sf (x, v) in Ω. Then we have deg I − K(λ, ·), BR , 0 = deg I − K1 (λ, ·), BR1 , 0 = deg I − K0 (λ, ·), BR1 , 0 = 1, where the last equality is given by Proposition 4.1. Hence, again by the homotopy invariance of the degree, we have deg(I − K(λ, ·), BR , 0) = 1, for all λ ∈ (λ0 , λ+ 1 + δ). The last fact together with standard degree theory implies that for every λ ∈ [λ0 , λ+ 1 + δ] there is at least one (λ, u), solution of (1.1), and, moreover, there is a connected subset C1 of S such that C1 (λ) = ∅ for all λ in the interval [λ0 , λ+ 1 + δ]. Since λ0 is arbitrary, we can use the same argument for each element of a sequence {λn0 }, with λn0 → −∞. Then, by a limit argument (like the one in the proof of Theorem 1.5.1 in [24]), we find a connected set C1 with the desired properties. 2 Next we study a branch of solutions driven by a family of super- and sub-solutions, assuming that f is locally Lipschitz continuous. In this case the statement of the previous proposition can be made more precise. Specifically, we assume that f satisfies (F2), and there exist u, u ∈ C 1 (Ω), such that u is a super-solution and u is a sub-solution of (1.1), for all λ λ, where λ is fixed. We further assume that u and u are not solutions of (1.1), and u
in Ω,
u=u=0
and
∂u ∂u < ∂ν ∂ν
on ∂Ω.
(5.1)
We define the set ∂u ∂v ∂u < < on ∂Ω , O = v ∈ C 1 (Ω) u < v < u in Ω and ∂ν ∂ν ∂ν
(5.2)
which is open in C 1 (Ω). Since O is bounded in C(Ω), we see that for every λ0 < λ the set of solutions of (1.1) in [λ0 , λ] × O is bounded in C 1 (Ω), that is, all solutions of (1.1) in [λ0 , λ] × O are inside the ball BR , for some R > 0. Lemma 5.1. With the definitions given above, we have deg I − K(λ, ·), O ∩ BR , 0 = 1,
for all λ ∈ [λ0 , λ].
Proof. First we have to prove that the degree is well defined. We just need to show that there are no fixed points of K(λ, ·) on the boundary of O ∩ BR . For this purpose it is enough to prove that, given v ∈ C 1 (Ω) such that u v u in Ω, we have u < K(λ, v) < u in Ω. In what follows we write u = K(λ, v).
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By (F2) we can assume that the negative number c, chosen in Section 4, is such that the function s → f (x, s) + (c − λ)s is decreasing, for s ∈ (−τ, τ ), where τ = max{ u ∞ , u ∞ }. Then F [u] = F [u] − f (x, u) − (c − λ)u + f (x, u) + (c − λ)u F [u] − f (x, v) − (c − λ)v + f (x, u) + (c − λ)u = −cu + f (x, u) + (c − λ)u = c(u − u) + f (x, u) − λu F [u] + c(u − u). By (H1) this implies M+ (D 2 (u − u)) + γ |Du − Du| + (γ − c)(u − u) > 0 in Ω. It follows from ∂u Theorem 2.2 that u < u in Ω and ∂u ∂ν < ∂ν on ∂Ω. The other inequality is obtained similarly. By using its homotopy invariance, the degree we want to compute is equal to the degree at λ0 . But the latter was shown to be one in the proof of Proposition 5.2, which completes the proof of the lemma. 2 Now we can state a proposition on the existence of a branch of solutions for λ ∈ (−∞, λ], whose proof is a direct consequence of Lemma 5.1 and general degree arguments. Proposition 5.3. Assume f satisfies (F0) and (F2). Suppose there are functions u, u ∈ C 1 (Ω) such that u is a super-solution and u is a sub-solution of (1.1) for all λ λ, these functions are not solutions of (1.1) and satisfy (5.1). Then there is a connected subset C1 of S such that C1 (λ) = ∅ for all λ ∈ (−∞, λ) and each u ∈ C1 (λ) is such that u u u. Remark 5.1. In the next section we use this proposition with appropriately chosen sub-solutions and super-solutions. Remark 5.2. The branch C1 is isolated of other branches of solutions by the open set O, since we know there are no solutions on ∂O. 6. Proofs of the main theorems In this section we put together the bifurcation branches emanating from infinity obtained in Theorem 4.1 with the bounded branches constructed in Section 5, and study their properties. Proof of Theorem 1.1. This theorem is a consequence of Proposition 5.2, for the definition of C1 , and of Theorem 4.1, 1)–2), for the definition of C2 and C3 . Both C2 and C3 curve to the right − of λ+ 1 and λ1 , respectively – as a consequence of the a priori bounds obtained in Propositions 3.1 and 3.2. 2 Proof of Theorem 1.2. We first construct the branch C1 , through Proposition 5.3. In view of (F1) we may take as a super-solution the function u ≡ 0. In order to define the corresponding sub-solution we use Proposition 3.2, where a sub-solution is constructed for all λ ∈ (−∞, λ− 1 ]. We can rewrite inequality (3.2) in the following way ˜ + F [zn ] + λ− 1 + δ zn = tn ϕ1 + d,
with t˜n (x) =
δzn (x) + tn . ϕ1+ (x)
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Since zn / zn ∞ → ϕ1− < 0 in C 1 (Ω) we find that for some c > 0 |zn (x)| c, zn ∞ ϕ1+ (x)
∀x ∈ Ω.
Consequently, once n is chosen so that (3.3) holds, we can fix δ > 0 such that t˜n (x) −δc + ∗ (d) > 0, for all x ∈ Ω, which means that z is a sub-solution also for F [u] + (λ− + δ)u = t− n 1 f (x, u), as in the proof of Proposition 3.2. Now we define u = zn , chosen as above, and take λ = λ− 1 + δ in Proposition 5.3. Clearly u and u satisfy also (5.1), so the existence of the branch C1 (with the properties stated in Theorem 1.2) follows from Proposition 5.3. Further, the branches C2 and C3 are given by Theorem 4.1 and both of them curve to the right − of λ+ 1 and λ1 , respectively. Neither C2 or C3 connects to C1 , since C1 is isolated from the exterior of the open set O, see Remark 5.2. Observe that the elements of C2 (resp. C3 ) are outside O for − λ close to λ+ 1 (resp. λ1 ). Therefore the uniqueness statement of Proposition 5.1 excludes the alternatives in Theorem 4.1, 3)(iii) and 4)(iii), since we already know that C1 contains solutions for arbitrary small λ. We already noted cases 3)(ii) and 4)(ii) are excluded by (F1). Finally, case 3)(iv) in Theorem 4.1 is excluded by the a priori bound in Proposition 3.2, so only case 3)(i) remains. 2 Proof of Theorem 1.3. We fix a small number ε > 0 and for each K > 0 consider a solution uK of F [uK ] + (λ− 1 − ε)uK = K in Ω, uK = 0 on ∂Ω, uK < 0 in Ω. We know such a function uK exists, by Theorem 1.9 in [36]. By (F0) we can fix K0 such that K0 > f (x, K0 ) in Ω, hence u = uK0 is a sub-solution of (1.1), for all λ ∈ (−∞, λ− 1 − ε). The super-solution to consider is u ≡ 0, as given by hypothesis (F1). Then Proposition 5.3 yields the existence of a branch C1ε such that C1ε (λ) = ∅ for all λ ∈ (−∞, λ− 1 − ε). Next we pass to the limit as ε → 0, like in the proofs of Proposition 5.2 and Theorem 1.5.1 in [24], and obtain either a connected component of S which bifurcates from infinity to the left − of λ− 1 , or a bounded branch of solutions which “survives” up to λ1 , and hence “continues” in − some small right neighborhood of λ1 , again like in the proof of Proposition 5.2. The first of these alternatives is (3)(i). In case the second alternative is realized there is a connected set of solutions C3 bifurcating from minus infinity towards the left of λ− 1 , as predicted in Theorem 4.1. We claim this branch contains only negative solutions. To prove this, we set , max A = (λ, u) ∈ C3 λ ∈ −∞, λ− u > 0 . 1 Ω
The set A is clearly open in C3 , and A = C3 . Hence if A is not empty, then A is not closed in C3 , by the connectedness of C3 . This means there is a sequence (λn , un ) ∈ A such that λn → λ, un → u, and the limit function u satisfies u 0 in Ω, u vanishes somewhere in Ω, and solves the equation F [u] + (λ − c)u = f (x, u) − cu 0 in Ω, for some large c. Hence by Hopf’s lemma u ≡ 0, a contradiction with (F1). Therefore C3 cannot connect with the branch bifurcating from plus infinity at λ+ 1 . It is not connected to C1 either – by the isolation property of C1 (λ), see Remark 5.2. Further, C3 cannot contain solutions for arbitrarily small λ, since C1 does, and we know solutions are unique for sufficiently small λ. Hence C3 must eventually curve to the right, so extra solutions appear, proving (3)(ii) and (4).
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Finally, a branch C2 bifurcating from plus infinity towards the left of λ+ 1 exists thanks to Theorem 4.1. This branch is kept away from C1 and C3 , as we already saw, and, again by the uniqueness of solutions for sufficiently small λ, C2 has to curve to the right. This completes the proof. 2 The occurrence of alternative (3)(ii) in Theorem 1.3 can be avoided if f satisfies some further hypotheses. Proposition 6.1. Under the hypotheses of Theorem 1.3, if in addition we make one of the following assumptions (1) f (x, s) is concave in s for s < 0, (2) for each a0 > 0 there exists k0 > 0 such that f (x, −kϕ1+ ) < f x, −aϕ1+ , k
for all a ∈ (0, a0 ), k > k0 ,
(6.1)
then alternative (3)(ii) in Theorem 1.3 does not occur. Remark. Note the model example of a sub-linear nonlinearity which satisfies the hypotheses of Theorem 1.3 f (x, s) = −s|s|α−1 + h(x),
α ∈ (0, 1), h 0,
satisfies both hypotheses in the above proposition. Proof of Proposition 6.1. We are going to prove the following stronger claim: under the hypotheses of the proposition, there cannot exist sequences λn , un , vn , such that λn < λn+1 , λn → λ − 1 , un , vn < 0 in Ω, un is bounded, vn → ∞ and un and vn are solutions of (1.1) with λ = λn . Assume this is false and (1) holds. Then (passing to subsequences if necessary) un is convergent in C 1 (Ω), and vn / vn → ϕ1− in C 1 (Ω), so there is n0 such that for all n n0 we have vn < un+1 in Ω. The negative function un+1 is clearly a strict sub-solution of F [u] + λn u = f (x, u), and, since the zero function is a strict super-solution of this equation, it has a negative solution which is above un+1 . We define v n = inf v un+1 < v < 0, v is a super-solution of F [u] + λn u = f (x, u) . Then v n is a solution of F [u]+λn u = f (x, u) such that between un+1 and v n no other solution of this problem exists. Indeed, v n is a super-solution (as an infimum of super-solutions), so between un+1 and v n there is a minimal solution, with which v n has to coincide, by its definition. Note Hopf’s lemma trivially implies that for some ε > 0 we have vn < un+1 − εϕ1+ < v n − 2εϕ1+ . Next, by the convexity of F and the concavity of f we easily check that the function uα = αvn + (1 − α)v n is a super-solution of F [u] + λn u = f (x, u), for each α ∈ [0, 1]. This gives a contradiction with the definition of v n , for α small enough but positive.
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Assume now our claim is false and (2) holds. We again have −C0 ϕ1+ un −c0 ϕ1+ < 0 and vn / vn → ϕ1− , so the numbers εn := sup{ε > 0 | un εvn in Ω} clearly satisfy εn > 0 and εn → 0. Hypothesis (6.1) implies that for sufficiently large n we have εn f (x, vn ) < f (x, un ), that is, F [εn vn ] + λn εn vn < F [un ] + λn un , and Hopf’s lemma yields a contradiction with the definition of εn . 2 7. Discussion and examples The main point of this section is to provide some examples showing that when (F1) or (F2) fails, then the bifurcation diagram for (1.1) may look very differently from what is described in Theorems 1.2–1.3. However, we begin with some general comments on our hypotheses and their use. Hypothesis (F0) is classical sub-linearity for f , which guarantees bifurcation from infinity and also ensures the solutions of (1.1) tend to zero as λ → −∞. Condition (F1) guarantees the existence of a strict super-solution of (1.1) for all λ, while (F2) is used in some comparison statements and to prove uniqueness of solutions of (1.1) for sufficiently negative λ. Further, conditions (F+ )–(F− ) and (F+r )–(F−r ) are the Landesman–Lazer type hypotheses which give a priori bounds when λ stays on one side of the eigenvalues, and thus provide a solution at resonance and determine on which side of each eigenvalue the bifurcation from infinity takes place. The strict inequalities in (F+ )–(F−r ) are important and cannot be relaxed in √ general – for instance the problem F [u] + λ+ 1 u = − max{1 − u, 0} has no solutions (and hence Theorem 1.1 fails), as Theorems 1.6 and 1.4 in [36] show, even though the nonlinearity satisfies the hypotheses of Theorem 1.1, except for the strict inequality in (F+ ). On the other hand, for F = it is known that in the case of equalities in (F+ )–(F−r ) one can give supplementary assumptions on f and the rate of convergence of f to its limits f± , f ± , so that results like Theorem 1.1 still hold, see for instance Remark 21 in [5]. Extensions of these ideas to HJB operators are out of the scope of this work and could be the basis of future research. Now we discuss examples where (F1) or (F2) fails. Example 1. Our first example shows that for all sufficiently small δ > 0 we can construct a nonlinearity f which does not satisfy (F1) and for which the set S(λ+ 1 + δ) is empty. This means that, in the framework of Theorems 1.1–1.2, the branch bifurcating from infinity to the right of + λ+ 1 “turns back” before it reaches λ1 + δ. A similar situation can be described for the branch − bifurcating from minus infinity to the left of λ− 1 that “turns right”, before reaching λ1 − δ. In + − particular there cannot be a continuum of solutions along the gap between λ1 and λ1 . Consider the Dirichlet problem F [u] + λu = tϕ1+ + h
in Ω,
u = 0 on ∂Ω,
(7.1)
∗ at resonance, that is, for λ = λ+ 1 . When t = t+ (h) Eq. (7.1) may or may not have a solution, depending on F and h. An example of such a situation was given in [7] and we recall it here. Take F [u] = max{u, 2u}, and h ∈ C(Ω) such that Ω hϕ1 = 0 and h changes sign on ∂Ω. Here − + − λ+ 1 = λ1 , λ1 = 2λ1 , ϕ1 = −ϕ1 = ϕ1 , where λ1 and ϕ1 are the first eigenvalue and eigenfunction
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∗ =0 of the Laplacian. Then (see Example 4.3 in [7]) under the above hypotheses on h we have t+ + ∗ and problem (7.1) has no solutions if λ = λ1 and t = t+ . By exactly the same reasoning it is ∗ possible to show that problem (7.1) has no solutions if λ = λ− 1 and t = t− . ∗ Lemma 7.1. If Eq. (7.1) with λ = λ+ 1 and t = t+ does not have a solution then there exists δ0 + + ∗ ∗ ∗ such that tλ > t+ provided λ ∈ (λ1 , λ1 + δ0 ). Similarly, if (7.1) with λ = λ− 1 and t = t− does − − ∗ ∗ not have a solution then there exists δ0 such that tλ > t− whenever λ ∈ (λ1 − δ0 , λ1 ).
Before proving the lemma, we use it to construct a nonlinearity with the desired properties. ∗ ∗ ∗ ∗ For λ sufficiently close to λ+ 1 we have t+ < tλ so that we can choose t ∈ (t+ , tλ ). We then define ⎧ + if u −M, ⎪ ⎨ tϕ1 ∗+ h t−t+ +ε f (x, u) = ( (u + M) + t)ϕ1+ + h if −2M u −M, ⎪ ⎩ ∗M (t+ − ε)ϕ1+ + h if u −2M,
(7.2)
where ε and M are some positive constants. We readily see that f satisfies (F0) and (F+ ), the hypotheses of Theorem 1.1, but S(λ) is empty. Indeed, if u ∈ S(λ), then u is a super-solution for (7.1). On the other hand, by Theorem 1.9 in [36], the equation F [u] + λu = K tϕ1+ + h L∞ (Ω) with λ < λ− 1 has a solution uK , for each K > 0. Moreover, for large K, uK is a sub-solution of (7.1) and uK < u. Then by Perron’s method (7.1) has a solution, a contradiction. − ∗ ∗ Similarly, for λ < λ− 1 sufficiently close to λ1 we choose t ∈ (t− , tλ ) and define f (x, u) being + + ∗ equal to tϕ1 + h if u M and to (t− − ε)ϕ1 + h if u 2M. By the same reasoning we find that S(λ) is empty. We summarize: with these choices of λ and f there is a region of non-existence in the gap − between λ+ 1 and λ1 . In other words, the connected sets of solutions of (1.1) C2 (resp. C3 ), predicted in Theorem 1.1, do not extend to the right (resp. to the left) of λ. The first graph at the end of this section is an illustration of this situation. We observe that if we take M sufficiently large then all solutions of (7.1) and (1.1), with f as given in (7.2), coincide. In fact, we can take −M to be a lower bound for all solutions of the inequality F [u] + λu c + h, where c is such that f (x, u) c + h in Ω. Such an M exists by the one-sided ABP inequality given in Theorem 1.7 in [36]. Now we see that (1.1) with this f has a unique solution for λ < λ+ 1 , as an application of Theorem 1.8 in [36], and then the branch of solutions bifurcating from plus infinity must turn left and go towards infinity near the λ-axis, as drawn on the picture. − ∗ Proof of Lemma 7.1. Given λ ∈ (λ+ 1 , λ1 ), let vλ be a solution of
F [u] + λu = tλ∗ ϕ1+ + h in Ω,
(7.3)
whose existence is guaranteed by the results in [39]. We notice that vλ∗ is unbounded as + ∗ ∗ λ λ+ 1 , as otherwise vλ a subsequence of vλ would converge to a solution of (7.1) with λ = λ1 + ∗ ∗ , which is excluded by assumption. That t → t ∗ as λ → λ was proved in Proposiand t = t+ + λ 1 + + ∗ ∗ tion 2.3. Then, by the simplicity of λ+ 1 , we find that vλ / vλ ∞ → ϕ1 as λ → λ1 , in particular, + ∗ vλ becomes positive in Ω, for λ larger than and close enough to λ1 . Suppose for contradiction
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∗ t ∗ , then v ∗ 0 satisfies that t+ λ λ
∗ ∗ ∗ ∗ + ∗ + F vλ∗ + λ+ 1 vλ F vλ + λvλ = tλ ϕ1 + h t+ ϕ1 + h, ∗ so vλ∗ is a super-solution for (7.1) with λ = λ+ 1 and t = t+ . As we already showed above, (7.1) ∗ has a sub-solution below vλ , providing a contradiction. ∗ In the same way, we see that vλ∗ / vλ∗ ∞ → ϕ1− as λ λ− 1 and then vλ becomes negative in Ω, − − ∗ ∗ ∗ for λ < λ1 and close enough to λ1 . Then t− tλ would imply that vλ 0 satisfies
∗ ∗ ∗ ∗ + ∗ + F vλ∗ + λ− 1 vλ F vλ + λvλ = tλ ϕ1 + h t− ϕ1 + h, ∗ so vλ∗ is a super-solution for (7.1) with λ = λ− 1 and t = t− . To construct a sub-solution we − + ∗ consider vε a solutions of F [vε ] + λ1 vε = (t− + ε)ϕ1 + h, with ε > 0. By our assumption, vε / vε ∞ → ϕ1− as ε → 0 (see also Theorem 1.4 in [24]). Hence there exists ε = ε(λ) such that ∗ + vε < vλ∗ and F [vε ] + λ− 1 vε t− ϕ1 + h and then Perron’s method gives a contradiction again. 2
Remark 7.1. The claim gives an idea of the behavior of tλ∗ , with respect to λ, near the extremes of − ∗ the interval [λ+ 1 , λ1 ]. However we do not have any idea about the global behavior of tλ , actually ∗ ∗ we do not even know how t+ and t− compare. For completeness we give a direct proof of the fact that in the above examples condition (F1) is not satisfied by nonlinearities like in (7.2). In this direction we have the following lemma, which is of independent interest. Lemma 7.2. For any h ∈ Lp (Ω), p > N , which is not a multiple of ϕ1+ , ∗ (h) < 0 and t ∗ (h) < 0; (a) if h 0 and h ≡ 0 then t+ − ∗ (h) > 0 and t ∗ (h) > 0; (b) if h 0 and h ≡ 0 then t+ − ∗ (h)ϕ + + h and t ∗ (h)ϕ + + h change sign in Ω. (c) the functions t+ − 1 1 ∗ (h) 0 then, as h 0, by Theorem 1.9 in [36] the problem F [u] + λ+ u = Proof. (a) If t+ 1 ∗ (h)ϕ + + h has a solution. Then by Theorem 1.2 in [24] u + kϕ + is a solution of the same t+ 1 1 problem, for all k > 0. Since u + kϕ1+ is positive for sufficiently large k, by Theorem 1.2 in [36] we get that u is a multiple of ϕ1+ , a contradiction, since h = 0. ∗ (h) 0, by Theorem 1.5 in [24] either there exist sequences ε → 0 and u of solutions If t− n n + ∗ of the problem F [un ] + λ− 1 un = (t− (h) + εn )ϕ1 + h such that un is unbounded and un is negative − + ∗ for large n, or F [u + kϕ1− ] + λ− 1 (u + kϕ1 ) = t− (h)ϕ1 + h for some u and all k > 0. In both cases − we get a negative solution of F [u] + λ1 u 0, which by Theorem 1.4 in [36] is then a multiple of ϕ1− , a contradiction. ∗ (h) 0 then F [u] + λ+ u = t ∗ (h)ϕ + + h has no solution by Theorems 1.6 and 1.4 (b) If t+ + 1 1 ∗ (h)ϕ + + h 0. If t ∗ (h) 0 we again have t ∗ (h)ϕ + + h 0, then F [u] + λ− u = in [36], since t+ − − 1 1 1 ∗ (h)ϕ + + h has no solutions by the anti-maximum principle, see for instance Proposition 4.1 t− 1 − in [24]. Hence by Theorems 1.2 and 1.4 in [24] there exist sequences εn → 0, u+ n and un of ± ± + ± ∗ ± ± ± solutions of F [un ] + λ1 un = (t± (h) + εn )ϕ1 + h such that un / un ∞ → ϕ . Fix w to be the solution of the Dirichlet problem F (w) − γ w = −h in Ω. This problem is uniquely solvable,
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with w < 0 in Ω, since by (H1) the operator F − γ is decreasing in u (see for instance [17] and [36]). Then by the maximum principle and Hopf’s lemma εn ϕ1+ + (λ+ 1 + γ )w < 0 in Ω, + + + + if n is sufficiently large. Hence un + w is positive and F [un + w] + λ1 (un + w) < 0 in Ω, which is a contradiction with Theorem 1.4 in [36]. Similarly, u− n + w is negative and satisfies − − F [u− n + w] + λ1 (un + w) < 0 in Ω, which is a contradiction with Theorem 1.2 in [36]. (c) This is an immediate consequence of (a) and (b). Indeed, if (c) is false we just replace h ∗ (h)ϕ + + h in (a) or (b). 2 by t± 1 ∗ in the preceding lemma also follow from Theorem 1.1 and Remark 7.2. The statements on t+ formula (1.12) in [7].
The following example illustrate the role of hypothesis (F2), which allows the use of the method of sub- and super-solutions, and prevents the branches which bifurcate from infinity to survive for arbitrarily negative λ. Example 2. Consider the function ω(u) =
√u , |u|
ω(0) = 0 and the problem
u + λu = −ω(u)
in Ω.
(7.4)
This problem is variational and its associated functional is J (u) = |∇u|2 − λu2 − |u|3/2 dx, Ω
which is even, bounded below, takes negative values and attains its minimum on H01 (Ω), for each λ < λ1 . The same is valid for J+ (u) = J (u+ ) and J− (u) = J (u− ), whose minima are then a positive and a negative solutions of (7.4). In the context of nonlinear HJB operators we may consider max{u, 2u} + λu = −ω(u),
in Ω,
u = 0 on ∂Ω.
(7.5)
For this problem we have bifurcation from plus infinity to the left of λ1 and from minus infinity to the left of 2λ1 . These branches cannot reach the trivial solution set R × {0}, since bifurcation of positive or negative solutions from the trivial solution does not occur for (7.4). Exactly as in the proof of Theorem 1.3 (see the definition of the set A in the previous section) we can show that they contain only positive or negative solutions. Actually these branches are curves which can never turn, since positive and negative solutions of (7.5) are unique – this can be proved in the same way as Proposition 7.1 below. Example 3. Finally, let us look at an example of a sub-linear nonlinearity f which satisfies (F2) but f (x, 0) ≡ 0. For any HJB operator F satisfying our hypotheses consider F [u] + λu = f˜(u) :=
−u if |u| 1, −ω(u) if |u| 1.
(7.6)
In this situation we have positive (resp. negative) bifurcation from zero at λ = λ+ 1 − 1 (resp. + + − − − 1), more precisely, (λ − 1, kϕ ) and (λ − 1, kϕ ) are solutions for k ∈ [0, 1] (for λ = λ− 1 1 1 1 1
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more general results on bifurcation from zero see [13]). Further, note that there are only positive (resp. negative) solutions on these branches, as well on the branches which bifurcate from plus (resp. minus) infinity, given by Theorem 1.3. This is a simple consequence of the strong maximum principle and the fact that the right-hand side of (7.6) is positive (resp. negative) if u is negative (resp. positive), so if u () 0 and u vanishes at one point then u is identically zero. The bifurcation branches connect, as shown by the following uniqueness result. Proposition 7.1. If u and v are two solutions of (7.6) having the same sign and u > 1 or v > 1 then u ≡ v. If u 1 and v 1 then (by the simplicity of the eigenvalues) λ = λ± 1 −1 and u = v + kϕ1± for some k ∈ [0, 1]. Proof. Say u > 0, v > 0, v > 1. Set τ := sup{μ > 0 | u μv in Ω}. By Hopf’s lemma τ > 0 and we have u τ v. First, suppose τ < 1. By the definition of f˜ in (7.6) and v > 1 we easily see that f˜(u) f˜(τ v) τ f˜(v)
in Ω.
Hence (7.6) and the hypotheses on F imply 2 M− λ,Λ D (u − τ v) − γ D(u − τ v) − (γ + λ)(u − τ v) 0 and u − τ v 0 in Ω, so Hopf’s lemma implies u (τ + ε)v for some ε > 0, a contradiction with the definition of τ . Second, if τ 1 we repeat the above argument with u and v interchanged. This leaves u v and v u as the only case not excluded. 2 The following picture summarizes the above examples.
Acknowledgments P.F. was partially supported by Fondecyt Grant #1070314, FONDAP and BASAL-CMM projects and Ecos-Conicyt project C05E09. A.Q. was partially supported by Fondecyt Grant #1070264 and USM Grant #12.09.17 and Programa Basal, CMM, U. de Chile.
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Journal of Functional Analysis 258 (2010) 4183–4209 www.elsevier.com/locate/jfa
Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes Nicholas Michalowski a , David J. Rule b , Wolfgang Staubach b,∗ a School of Mathematics and the Maxwell Institute of Mathematical Sciences, University of Edinburgh,
James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom b Department of Mathematics and the Maxwell Institute of Mathematical Sciences, Heriot–Watt University, Colin Maclaurin Building, Edinburgh, EH14 4AS, United Kingdom Received 24 October 2009; accepted 12 March 2010
Communicated by I. Rodnianski
Abstract We prove weighted norm inequalities for pseudodifferential operators with amplitudes which are only measurable in the spatial variables. The result is sharp, even for smooth amplitudes. Nevertheless, in the case 1−ρ when the amplitude contains the oscillatory factor ξ → ei|ξ | , the result can be substantially improved. p We extend the L -boundedness of pseudo-pseudodifferential operators to certain weights. End-point results are obtained when the amplitude is either smooth or satisfies a homogeneity condition in the frequency variable. Our weighted norm inequalities also yield the boundedness of commutators of these pseudodifferential operators with functions of bounded mean oscillation. © 2010 Elsevier Inc. All rights reserved. Keywords: Weighted norm inequality; Pseudodifferential operator; Pseudo-pseudodifferential operator; BMO commutator
* Corresponding author.
E-mail addresses:
[email protected] (N. Michalowski),
[email protected] (D.J. Rule),
[email protected] (W. Staubach). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.03.013
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1. Introduction Weighted norm inequalities have a long history in harmonic analysis and partial differential equations. Since the introduction of Ap weights by B. Muckenhoupt [21] (see Definition 2.4), there has been a large amount of activity establishing weighted Lp estimates with weights belonging to Ap . Such estimates have concerned singular integral operators of Calderón–Zygmund type, strongly singular integral operators (or singular integrals of Hirschman–Wainger type), maximal functions, standard pseudodifferential operators, and mildly regular pseudodifferential operators. However, there are still some gaps in the knowledge of weighted norm inequalities for pseudodifferential operators, and the aim of this paper is to fill in some of these. We do this first in the context of symbols introduced by C.E. Kenig and W. Staubach in [18] which are only measurable in the spatial variable. Our methods also extend to amplitudes and we obtain some sharp results in this direction. Most of these weighted boundedness results are new even in the smooth case. However, in the smooth case we can go on to consider end-point cases by using an interpolation technique. The results of this paper extend the applications derived in [18] and as a separate application we prove new boundedness results for the commutators of pseudodifferential operators with functions of bounded mean oscillation (written BMO, see Definition 4.1). Recall that for a function u ∈ C0∞ (Rn ) a pseudodifferential operator is an operator given formally by Ta u(x) :=
1 (2π)n
Rn
a(x, y, ξ )eix−y,ξ u(y) dy dξ,
(1.1)
Rn
whose amplitude (x, y, ξ ) → a(x, y, ξ ) is assumed to satisfy certain growth conditions. The most common class of amplitudes are those introduced by L. Hörmander in [15] and we refer the reader to Definitions 2.1–2.3 below for the specific amplitudes we will consider here and the standard notation we will use. By the weighted boundedness of a pseudodifferential operator Ta or a weighted norm inequality for a pseudodifferential operator we mean the existence of an estimate of the form Ta u Lpw C u Lpw ,
(1.2)
p
for some weight w and 1 p ∞. As usual Lw denotes the weighted Lp space with weight w: u Lpw =
u(x)p w(x) dx
1
p
.
Rn
All our results will concern w ∈ Aq (see Definition 2.4) for some q which may not always be equal to p. In the remainder of this section we summarize the results to follow. Section 2 sets out some definitions, fixes some notation and records some well-known results we will need later. In Section 3 our first main result is Theorem 3.3. It is a pointwise bound of operators in OPL∞ Sρm by a maximal function and corresponds to the Lp -boundedness of OPL∞ Sρm in the region C ∪ D of Fig. 1 obtained in [18]. Such an estimate immediately leads to weighted boundedness results of the form (1.2).
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Fig. 1. Regions of Lp -boundedness.
In Theorem 3.4 we go on to consider symbols of the form (x, ξ ) → ei|ξ | σ (x, ξ ), with σ ∈ L∞ S1m and m < n(ρ − 1)/2. We build on the methods developed by S. Chanillo and A. Torchinsky [7] to prove that operators corresponding to these symbols can be bounded p pointwise by certain maximal functions and consequently are bounded on Lw for w ∈ Ap and 1 < p < ∞. This estimate is obtained when m < n(ρ − 1)/2 and corresponds to the region B ∪ C ∪ D of Fig. 1. In Corollary 3.8 we observe the result can be generalized to amplitudes 1−ρ (x, y, ξ ) → ei|ξ | σ (x, y, ξ ), with σ ∈ L∞ Am 1 . In the case of symbols of the aforementioned form, if we make the additional assumption that the symbol is homogeneous in the ξ variables, we can extend the weighted boundedness result to the end-point value m = n(ρ − 1)/2. This is formulated as Theorem 3.6. These results are particularly interesting in connection to weighted norm inequalities for maximal functions associated to strongly singular integrals. In the case of ρ = 1, Theorem 3.6 yields the weighted version of an Lp -boundedness result due to R. Coifman and Y. Meyer [10]. The weighted boundedness of operators corresponding to symbols can be used to prove pointwise bounds by a maximal function for operators corresponding to amplitudes. This idea is first used to prove Theorem 3.7, where weighted boundedness results are shown for OPL∞ Am ρ in region D of Fig. 1. This is shown to be sharp. The amplitudes, and in particular the rough amplitudes, considered here are interesting because of the connection to the Weyl quantization of rough symbols, and the potential applications of the latter in semiclassical microlocal analysis. The Weyl quantization is when the amplitude takes the form (x, y, ξ ) → a((x + y)/2, ξ ). We can then use a fairly general method of interpolation, based on complex interpolation, to obtain end-point results for smooth amplitudes. The idea is to use properties of Ap -weights to enable us to interpolate between the weighted boundedness for values of m less than the end-point and the unweighted boundedness which is known for values of m greater than the endpoint. An example of this is Theorem 3.10, where we show weighted boundedness for operators n(ρ−1) with 0 < ρ 1 and 0 δ < 1. The interpolation technique can also, for example, in OPAρ,δ be used to prove weighted boundedness results for smooth symbols in the region A ∪ B ∪ C ∪ D of Fig. 1 when δ < ρ (see Theorem 3.11). In Section 4 we use the weighted norm inequalities to prove a variety of boundedness results for the commutators of pseudodifferential operators with BMO functions. When we have the p Lw -boundedness of pseudodifferential operators for all weights w ∈ Ap , we are actually able to show weighted boundedness of k-th commutators as a direct consequence of a general result due to J. Álvarez, R.J. Bagby, D.S. Kurtz and C. Pérez [1], see Section 4 for the definition of these commutators. For these results see Theorem 4.5. However, in Theorem 4.4, we also obtain 1−ρ
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unweighted Lp -boundedness of a commutator knowing only the weighted Lw -boundedness of the original operator for w in a much smaller class of weights. Theorems 4.4 and 4.5 extend the Lp boundedness of BMO commutators with OPS01,0 due to R. Coifman, R. Rochberg and G. Weiss [11], and the Lp boundedness of commutators of bmo functions with rough pseudodifferential operators arising from homogeneous symbols of order zero (the symbol class L∞ Scl0 ), proved by F. Chiarenza, M. Frasca and P. Longo [8]. The space of bmo functions is a localized version of the class BMO. To our knowledge, there are no results in the existing literature concerning the boundedness of BMO commutators of operators with rough amplitudes, even in the case of the Weyl quantization. We would like to thank Andrew Hassell for prompting us to consider the problem of weighted norm inequalities for pseudodifferential operators arising from amplitudes. 2. Definitions, notation and preliminaries First we introduce a standard Littlewood–Paley partition of unity {ϕk }k0 . Let ϕ0 : Rn → R be a smooth radial function which is equal to one on the unit ball centred at the origin and supported on its concentric double. Set ϕ(ξ ) = ϕ0 (ξ ) − ϕ0 (2ξ ) and ϕk (ξ ) = ϕ(2−k ξ ). Then ϕ0 (ξ ) +
∞
ϕk (ξ ) = 1 for all ξ ∈ R,
k=1
and supp(ϕk ) ⊂ {ξ | 2k−1 |ξ | 2k+1 } for k 1. One also has, for all multi-indices α and N 0, α ∂ ϕ0 (ξ ) cα,N ξ −N , ξ 1
where ξ := (1 + |ξ |2 ) 2 , and α ∂ ϕk (ξ ) cα 2−k|α| ξ
for some cα > 0 and all k 1.
(2.1)
We now fix some common notation and terminology. Definition 2.1. A function a : Rn × Rn × Rn → Rn is called an amplitude when it belongs to any one of the following sets. Let m ∈ R, ρ ∈ [0, 1] and δ ∈ [0, 1]. (a) We say a ∈ Am ρ,δ when for each triple of multi-indices α, β and γ there exists a constant Cα,β,γ such that α β γ ∂ ∂ ∂y a(x, y, ξ ) Cα,β,γ ξ m−ρ|α|+δ|β+γ | . ξ x
(b) We say a ∈ L∞ Am ρ when for each multi-index α there exists a constant Cα such that α ∂ a(·,·, ξ ) ξ
L∞ (Rn ×Rn )
Cα ξ m−ρ|α| .
Therefore, here we are only assuming measurability in the (x, y)-variables.
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Given an amplitude a we define the pseudodifferential operator Ta associated to a formally by (1.1), although it is not clear that the integral in (1.1) is well-defined. This can be made rigorous by considering the partial sums of the Littlewood–Paley pieces as follows. We define ak = aϕk . The integral in (1.1) converges absolutely when a is replaced with ak , so in this way we obtain a family of operators {Tak }k . The methods of proof we use will show in each case that the partial sums N
Tak
k=0
converge in operator norm as N → ∞. Consequently, we obtain our desired operator as Ta := limN →∞ N k=0 Tak . An important special case of these operators is when there is no dependency on the y-variable. It is convenient to have a slightly different terminology in this case. Definition 2.2. A function a : Rn × Rn → Rn is called a symbol when it belongs to any one of the following sets. Let m ∈ R, ρ ∈ [0, 1] and δ ∈ [0, 1]. m when for each pair of multi-indices α and β there exists a constant C (a) We say a ∈ Sρ,δ α,β such that
α β ∂ ∂ a(x, ξ ) Cα,β ξ m−ρ|α|+δ|β| . ξ x (b) We say a ∈ L∞ Sρm when for each multi-index α there exists a constant Cα such that α ∂ a(·, ξ ) ξ
L∞
Cα ξ m−ρ|α| .
Therefore, here we are only assuming measurability in the x-variable. m ⊂ Am , L∞ S m ⊂ L∞ Am , S m ⊂ L∞ S m and Am ⊂ L∞ Am . The Obviously, we have Sρ,δ ρ ρ ρ ρ ρ,δ ρ,δ ρ,δ amplitudes of (a) in both Definitions 2.1 and 2.2 were first introduced in [14,15]. If the amplitude is smooth and δ < ρ, then operators arising from amplitudes in Am ρ,δ can be rewritten as a sum m , see, for example, [23]. However, if δ ρ, then the of operators arising from symbols in Sρ,δ boundedness results are known to be different [15,19], as we will see played out in this paper. The symbols of (b) in Definition 2.2 were introduced in [18], and Definition 2.2(b) is the natural generalization of this to amplitudes. They are, for example, much rougher than those considered by S. Nishigaki [22] and K. Yabuta [25] in their investigations of weighted norm inequalities for pseudodifferential operators.
Definition 2.3. Given a class X of symbols or amplitudes, operators which arise from elements in X are denoted by OPX, that is, we say T ∈ OPX when there exists a ∈ X such that T = Ta , defined as in (1.1). When the amplitude of an operator is only measurable in the spatial variables, that is the operator belongs to OPL∞ Sρm or OPL∞ Am ρ , then following [18], we say it is a pseudopseudodifferential operator. It is well known that for standard pseudodifferential operators
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the singular support of the Schwartz kernel of the operator is contained in the diagonal —this is called the pseudo-local property. Singularities in the Schwartz kernel of pseudopseudodifferential operators will in general go beyond the diagonal, so they do not have the pseudo-local property. p Given u ∈ Lloc , the Lp maximal function Mp (u) is defined by
1 Mp (u)(x) = sup B x |B|
u(y)p dy
1
p
(2.2)
B
where the supremum is taken over balls B in Rn containing x. Clearly then, the Hardy– Littlewood maximal function is given by M(u) := M1 (u). An immediate consequence of Hölder’s inequality is that M(u)(x) Mp (u)(x) for p 1. We shall use the notation uB :=
1 |B|
u(y) dy
B
for the average of the function u over B. One can then define the class of Muckenhoupt Ap weights as follows. Definition 2.4. Let w ∈ L1loc be a positive function. One says that w ∈ A1 if there exists a constant C > 0 such that Mw(x) Cw(x),
for almost all x ∈ Rn .
(2.3)
One says that w ∈ Ap for p ∈ (1, ∞) if sup
B balls in Rn
− 1 p−1 wB w p−1 B < ∞.
(2.4)
The Ap constants of a weight w ∈ Ap are defined by [w]A1 :=
sup
wB w −1 L∞ (B)
(2.5)
sup
− 1 p−1 wB w p−1 B .
(2.6)
B balls in Rn
and [w]Ap :=
B balls in Rn
The following results are well known and can be found in, for example, [17,23].
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Theorem 2.5. Suppose p > 1 and w ∈ Ap . There exists an exponent q < p, which depends only on p and [w]Ap , such that w ∈ Aq . There exists ε > 0, which depends only on p and [w]Ap , such that w 1+ε ∈ Ap . q
Theorem 2.6. For 1 < q < ∞, the Hardy–Littlewood maximal operator is bounded on Lw if and p only if w ∈ Aq . Consequently, for 1 p < ∞, Mp is bounded on Lw if and only if w ∈ Aq/p . Theorem 2.7. Suppose that φ : Rn → R is integrable non-increasing and radial. Then, for u ∈ L1 , we have φ(y)u(x − y) dy φ L1 M(u)(x) for all x ∈ Rn . It is useful to record here results of J. Álvarez and J. Hounie [2], see also [15] and [16]. These will be used in Section 3. Theorem 2.8. Let 1 < q < ∞, 0 < ρ 1, 0 δ < 1 and suppose either: 1 1 (a) a ∈ Am ρ,δ and m n(ρ − 1)| q − 2 | + min{0, n(ρ − δ)}; or m and m n(ρ − 1)| 1 − 1 | + min{0, n(ρ − δ)/2}. (b) a ∈ Sρ,δ q 2
Then Ta is bounded on Lq . Part (a) is a consequence of remark (d) on page 11 of [2], together with straightforward adjustments of Theorem 2.2, Lemma 3.1 and Theorem 3.4 therein to the case of amplitudes. Part (b) is explicitly stated in [2] as Theorem 3.4 on page 13. As is common practice, we will denote constants which can be determined by know parameters in a given situation, but whose value is not crucial to the problem at hand, by C. Such parameters in this paper would be, for example, m, ρ, δ, p, n, [w]Ap , and the constants Cα,β,γ , Cα,β and Cα in Definitions 2.1 and 2.2. The value of C may differ from line to line, but in each instance could be estimated if necessary. We sometimes write a b as shorthand for a Cb. 3. Pseudodifferential operators and their weighted Lp boundedness The first fairly simple lemma yields a classical kernel estimate, which in particular implies the rapid decrease off the diagonal of the Schwartz kernel of pseudodifferential operators with certain amplitudes. Lemma 3.1. Let a ∈ L∞ Am ρ with m ∈ R and ρ ∈ [0, 1]. Let ak (x, y, ξ ) = a(x, y, ξ )ϕk (ξ ), for k 0 with ϕk as in the above Littlewood–Paley decomposition. Then, for each l 0, iz,ξ |z| ak (x, y, ξ )e dξ 2k(n+m−ρl) , l
for all x, y, z ∈ Rn .
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Proof. Using the definition of L∞ Am ρ , inequality (2.1) and the Leibniz rule we see that α ∂ ak (x, ξ ) cα 2k(m−ρ|α|) , ξ
for some cα > 0 and k = 1, 2, . . . ,
(3.1)
where we have also used the assumption ρ 1. First suppose l is an integer. For a multi-index α with |α| = l, integration by parts then yields α
iz,ξ α iz,ξ z e dξ (x, y, ξ )e dξ = a (x, y, ξ )∂ a k k ξ = ∂ξα ak (x, y, ξ )eiz,ξ dξ 2k(n+m−ρl) . Summing over all α with |α| = l proves this special case of the lemma. The general result of non-integer values of l follows by interpolation of the inequality for k and k + 1, where k < l < k + 1. 2 The kernel of operators of the form (1.1) is K(x, x − y) = Lemma 3.1 it is easy to conclude K(x, x − y) |x − y|−N
1 (2π)n
a(x, y, ξ )eix−y,ξ dξ . From
for N > 0, |x − y| 1,
(3.2)
provided either ρ > 0 and m ∈ R, or ρ = 0 and m < −n. It was shown in [18] that under certain conditions on the order m pseudo-pseudodifferential operators are bounded on Lp spaces. We record that result here. Theorem 3.2. Fix p ∈ [1, 2] and let a ∈ L∞ Sρm with 0 ρ 1 and m < pn (ρ − 1). Then Ta is a bounded operator on Lq for each q p. Now, our goal is to show that under the same assumptions, OPL∞ Sρm are also bounded on weighted Lp spaces. More precisely, we prove the following theorem. The proof uses a similar method to [18]. Theorem 3.3. Fix p ∈ [1, 2] and let a ∈ L∞ Sρm with 0 ρ 1 and m < pn (ρ − 1). Then there exists a constant C, depending only on n, p, m, ρ and a finite number of the constants Cα in Definition 2.1, such that Ta (u)(x) CMp u(x), q
for all x ∈ Rn . Consequently, Ta is a bounded operator on Lw for each q > p and w ∈ Aq/p . q
Proof. The Lw -boundedness follows immediately from the pointwise estimate, by Theorem 2.6. To prove the pointwise estimate we use the Littlewood–Paley partition of unity introduced in Section 2, we decompose the symbol as a(x, ξ ) = a0 (x, ξ ) +
∞ k=1
ak (x, ξ )
N. Michalowski et al. / Journal of Functional Analysis 258 (2010) 4183–4209
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with ak (x, ξ ) = a(x, ξ )ϕk (ξ ), k 0. First we consider the operator Ta0 . We have 1 Ta0 (u)(x) = (2π)n
a0 (x, ξ )e
ix−y,ξ
u(y) dy dξ =
K0 (x, y)u(x − y) dy,
with K0 (x, y) =
1 (2π)n
a0 (x, ξ )eiy,ξ dξ.
Lemma 3.1 gives us the estimate K0 (x, y) y−M , for each M > n. Theorem 2.7 yields Ta (u)(x) 0
y−M u(x − y) dy Mu(x) Mp u(x),
(3.3)
for all 1 p 2. 1 ak (x, ξ )u(ξ ˆ )eix,ξ dξ for k 1. We note, just as Now let us analyse Tak (u)(x) = (2π) n before, that Tak (u)(x) can be written as Tak (u)(x) =
Kk (x, y)u(x − y) dy
with Kk (x, y) =
1 (2π)n
ak (x, ξ )eiy,ξ dξ = aˇ k (x, y),
where aˇ k here denotes the inverse Fourier transform of ak (x, ξ ) with respect to ξ . One observes that p p Ta (u)(x)p = Kk (x, y)u(x − y) dy = Kk (x, y)σk (y) 1 u(x − y) dy , k σk (y) with weight functions σk (y) which will be chosen momentarily. Therefore, Hölder’s inequality yields Ta (u)(x)p k where
1 p
+
1 p
Kk (x, y)p σk (y)p dy
p p
|u(x − y)|p dy , |σk (y)|p
= 1. Now for an l > pn , we define σk by σk (y) =
2 2
−kρn p
|y| 2−kρ ,
,
−kρ( pn −l)
|y|l ,
|y| > 2−kρ .
(3.4)
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By Hausdorff–Young’s theorem and the estimate (3.1), first for α = 0 and then for |α| = l, we have 2
−kp ρn p
−kp ρn Kk (x, y)p dy 2 p
2
−kp ρn p
ak (x, ξ )p dξ
p p
p 2
pmk
p
dξ
2
kp (m− pn (ρ−1))
,
|ξ |∼2k
and 2
−kρp ( pn −l)
p
−kρp ( pn −l) Kk (x, y) |y|p l dy 2 2
−kρp ( pn −l)
l ∇ ak (x, ξ )p dξ
p p
ξ
p 2
kp(m−ρl)
p
dξ
2
kp (m− pn (ρ−1))
.
|ξ |∼2k
Hence, splitting the integral into |y| 2−kρ and |y| > 2−kρ yields
Kk (x, y)p σk (y)p dy
p p
n kp (m− n (ρ−1)) p p p = 2kp(m− p (ρ−1)) . 2
Furthermore, once again using Theorem 2.7, we have
p |u(x − y)|p dy Mp u(x) p |σk (y)|
with a constant that only depends on the dimension n. Thus (3.4) yields n
Ta u(x)p 2k(m− p (ρ−1)) Mp u(x) p . k
(3.5)
Summing in k using (3.3) and (3.5), we obtain ∞ ∞ p p p n
p k(m− (ρ−1)) Ta u(x) a0 (x, D)u(x) + ak (x, D)u(x) Mp u(x) p 2 1+ . k=1
k=1
We observe that the series above converges if m < pn (ρ − 1). This ends the proof of the theorem. 2 For symbols in L∞ Sρm which contain the oscillatory factor ξ → ei|ξ | , one can improve the p Lw -boundedness result of Theorem 3.3 to all weights w ∈ Ap for m < n2 (ρ − 1). 1−ρ
N. Michalowski et al. / Journal of Functional Analysis 258 (2010) 4183–4209
Theorem 3.4. Let σ ∈ L∞ S1m , with m < n2 (ρ − 1) and set a(x, ξ ) = ei|ξ | ρ < 1. For each 1 < p < ∞, one has the pointwise estimate
1−ρ
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σ (x, ξ ), with 0 <
Ta (u)(x) M(Mu)(x) + Mp u(x),
(3.6)
with a constant depending only on n, p, m, ρ and a finite number of the constants Cα in Definition 2.1. Consequently Ta (u)
q
Lw
u Lqw ,
for all 1 < q < ∞ and w ∈ Aq . Proof. It is clear from Theorems 2.5 and 2.6 that the weighted Lp boundedness will follow from (3.6). To prove this we will use the Littlewood–Paley partition of unity {ϕk }k of Section 2 just as we did above. Observe that we may choose {ϕk }k so that
ϕk (ξ ) = ϕk (ξ ) ϕk−1 (ξ ) + ϕk (ξ ) + ϕk+1 (ξ ) ,
(3.7)
for all ξ ∈ Rn and each k 1. Now, again for k 0, we set ak (x, ξ ) = a(x, ξ )ϕk (ξ ). As in the proof of Theorem 1.2 in [7], we can deal with the piece of the operator containing ϕ0 (ξ ) using a result from [19]. For k 1, we compute Tak (u)(x) =
ak (x, ξ )e
iy·ξ
dξ u(x − y) dy = Kk (x, y)u(x − y) dy.
(3.8)
Using (3.7), we can see Kk (x, y) =
ak (x, ξ )eiy·ξ dξ =
=
a(x, ξ )ϕk (ξ )eiy·ξ dξ
a(x, ξ )ϕk (ξ ) ϕk−1 (ξ ) + ϕk (ξ ) + ϕk+1 (ξ ) eiy·ξ dξ
= Tak (x,·) (ψk )(y), where Tak (x,·) is the operator with the symbol ak (x, ξ ) with x fixed (and hence is a multiplier) and
(ϕk−1 + ϕk + ϕk+1 )ˇ := 2nk ψ 2k · := ψk ,
(3.9)
for some function ψ in the Schwartz class. Now the fact that Tak (x,·) is a multiplier operator, that multipliers are translation invariant, and that Ta∗k (x,·) = Tak (x,·) enables us to write
Kk (x, y)u(x − y) dy =
Tak (x,·) (ψk )(y)u(x − y) dy =
Combining this with (3.8), we have
ψk (y)Tak (x,·) (u)(x − y) dy.
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Ta (u)(x) = ψk (y)Ta (x,·) (u)(x − y) dy ψk (y)Ta (x,·) (u)(x − y) dy. k k k We observe that |ψk | can be majorized by, say ∞
−n αj 2−k dj χQk , j
j =1
where, for each k and j , χQk is the characteristic function of a cube Qkj centred at the origin with j diameter 2−k dj , and j |αj | < ∞. We are also free to assume dj 1 for all j . Therefore, ∞
−n Ta (u)(x) αj 2−k dj k j =1
Ta
k (x,·)
(u)(x − y) dy.
Qkj
In what follows we omit the bar over ak for notational simplicity. Since ak and ak satisfy the same symbol estimates, and since the proof is identical for ak replaced by ak , this shouldn’t cause any confusion for the reader. Now we consider two cases. First, suppose d := 21−k dj < 1/4. For such j and k we dek,j k,j k,j k,j k,j k,j compose u = u1 + u2 + u3 , where ui = uχi and χ1 is the characteristic function of k,j {y | |x − y| 22−k dj }, χ2 is the characteristic function of {y | 22−k dj < |x − y| (21−k dj )ν }, k,j and χ3 is the characteristic function of {y | |x − y| > (21−k dj )ν }. The number ν will be fixed depending only on n, m, ρ and p (as we shall see below) and will be such that ν < ρ. We consider −k −n
−k −n Ta (x,·) uk,j (x − y) dy Tak (x,·) (u)(x − y) dy = 2 dj 2 dj k 1 Qkj
Qkj
−n + 2−k dj
Ta
k,j
u2 (x − y) dy
Ta
k,j
u3 (x − y) dy
k (x,·)
Qkj
−n + 2−k dj
k (x,·)
Qkj
= J1 + J2 + J3 . We can estimate J1 using the L1 boundedness of multipliers of this form: Indeed, since |∂ξα ak (·, ξ )| 2k(m−ρ|α|) , following [18], if Kk (x, y) denotes the kernel of Tak (x,·) (recall again that x is fixed here) then Kk (x, y) dy = Kk (x, y) dy + Kk (x, y) dy := J1,1 + J1,2 . |y|2−kρ
|y|>2−kρ
To estimate J1,1 we use the Cauchy–Schwarz inequality and the kernel estimates prior to the proof of (3.5) for the case p = p = 2. Therefore
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1
2
J1,1
dy
Kk (x, y)2 dy
1 2
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n
2k(m− 2 (ρ−1)) .
|y|2−kρ
To estimate J1,2 we use again the Cauchy–Schwarz inequality and once again the kernel estimates prior to the proof of (3.5) for the case p = p = 2. This yields
−2l
J1,2
|y|
1 2
dy
Kk (x, y)2 |y|2l dy
1 2
n
2k(m− 2 (ρ−1)) .
|y|>2−kρ
Thus, by Young’s inequality, the L1 norm of the multiplier Tak (x,·) has the size 2−εk where ε = n2 (ρ − 1) − m, which in turn yields
−n J1 2−k dj C2
−εk
Ta
k (x,·)
−k −n 2 dj
k,j
u1 (x − y) dy
k,j u (x − y) dy 1
C2−εk Mu(x). To estimate J3 we use Lemma 3.1 with l so large that n2 (1 − ρ) n(1 − ν) + l(ν − ρ) to obtain the estimate Kk (x, z) C2k(n+m−ρl) |z|−l
(3.10)
and so conclude, for any y ∈ Qkj ,
Ta (x,·) uk,j (x − y) = Kk (x, z)uk,j (x − y − z) dz k 3 3
C2k(n+m−ρl)
|z|−l u(x − y − z) dz
|y+z|>(21−k dj )ν
C2k(n+m−ρl)
|z|−l u(x − y − z) dz
|z|>(2ν −1)2−kν djν
(n−l)
C2k(n+m−ρl) 2ν − 1 2−kν djν Mu(x − y)
(n−l) Mu(x − y) C2k(n+m−ρl) 2−kν C2−εk Mu(x − y) since dj 1 and 21−k dj < 1/4, so |z| |y + z| − |y| > (21−k dj )ν − 2−k dj (2ν − 1)2−kν djν . Consequently, J3 C2−εk M(Mu)(x).
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For the case d < 1, it remains only to estimate J2 . We will use the arguments in Theorem 1.2 1−ρ of [7]. We observe that we may write ak (x, ξ ) = 2−εk bk (x, ξ ) where bk (x, ξ ) = ei|ξ | τk (x, ξ ) n
(ρ−1)
and τk ∈ L∞ S12 , with constants Cα independent of k. Suppressing the indices j and k, we γ k,j further partition the function u2 = γ0=2 gγ , with gγ supported in {y | 2γ −1 d < |y − x| 2γ d} and 2γ0 d ∼ d ν , where d = 21−k dj . We obtain, then, the estimate analogous to (3.1) in [7]: J2 C2
−εk
Mu(x) + 2
−εk
γ0
d
−n
γ =2
Sγ (gγ )(y) dy
Qkj
where Sγ is defined as follows. Let δ = n(1 + ρ)( p1 − 12 ) and set
−1/ρ
ξ ηγ (ξ ) = η (1 − ρ)/ 2γ d where η is radial, smooth and such that ⎧ 1/ρ ⎨ 0, if |ξ | < (1/8) , 1/ρ η(ξ ) = 1, if (1/7) < |ξ | < 301/ρ , ⎩ 0, if |ξ | > 401/ρ . Let θ be a smooth cut-off function which is equal to one near infinity and zero near the origin, then define Sγ as Sγ (u)(y) =
ei(y·ξ +|ξ |
1−ρ )
Rn
θ (ξ )τk (x, ξ ) n(1−ρ)/2 |ξ | ηγ (ξ )|ξ |δ+n(ρ−1)/2 u(ξ ˆ ) dξ. |ξ |δ
Following the reasoning of [7], we obtain for fixed x that Sγ (gγ )
Lp
−(δ+ n (ρ−1))/ρ
−(δ+ n (ρ−1))/ρ+n/p 2 2 C 2γ d gγ Lp C 2γ d Mp u(x).
Therefore, observing that −(δ + n2 (ρ − 1))/ρ + n/p = n/(ρp ), we have γ0 γ =2
d
−n
γ0 γ n/(ρp ) Sγ (gγ )(y) dy Cd −n/p Mp u(x) 2 d γ =2
Qkj
ν
= Cd n( ρ −1)/p Mp u(x). So we choose ν so that n(1 − ρν )/p = ε/2, and then ν
−ε/2
2−εk d n( ρ −1)/p = 2−εk/2 dj and
2−εk/2
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J2 C2−εk/2 Mp u(x). k,j
k,j
k,j
Now consider the case d 1/4. We decompose u = u1 + (u − u1 ), where u1 is defined as before. We have −k −n 2 dj
Ta
k (x,·) (u)(x
−n − y) dy = 2−k dj
Qkj
Ta
k (x,·)
Qkj
−n + 2−k dj
Ta
k,j
u1 (x − y) dy
k (x,·)
k,j
u − u1 (x − y) dy
Qkj
= L1 + L2 . The term L1 is identical to J1 , and in fact the same analysis works for d 1/4 as did for d < 1/4. For L2 , by (3.10) with l sufficiently large, we see that, for y ∈ Qkj , k,j
k,j
K = u − u (x − y) (x − y − z) dz (x, z) u − u k k (x,·) 1 1
Ta
C2
k(n+m−ρl)
|z|−l u(x − y − z) dz
|y+z|>22−k dj
C2
k(n+m−ρl)
|y + z|−l u(x − y − z) dz
|y+z|>22−k dj
C2k(n+m−ρl) Mu(x) C2−εk Mu(x), so, L2 C2−εk Mu(x). Collecting all these estimates together, we find ∞ Ta (u)(x) Ta (u)(x) k k=0
M(u)(x) +
αj (J1 + J2 + J3 ) +
d<1/4
Mu(x) +
∞ ∞ k=1 j =1
2
αj (L1 + L2 )
d1/4
αj 2−εk M(Mu)(x) + 2−εk/2 Mp u(x)
M(Mu)(x) + Mp u(x) , which proves (3.6).
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By considering a homogeneous version of the symbols in the previous theorem, we are able to prove the weighted Lp boundedness of the corresponding operators at the end-point m = n(ρ − 1)/2. We now define this class precisely. Definition 3.5. The class L∞ Sclm consists of symbols which are bounded and measurable in the spatial variable and satisfy (1) ∂ξα a(·, ξ ) L∞ cα ξ m−|α| , for each multi-index α. (2) a(x, tξ ) = t m a(x, ξ ), t 1, |ξ | 1. Now we are ready to state and prove our boundedness result for a sub-class of operators in n(ρ−1)/2 . L∞ S ρ Theorem 3.6. Let σ ∈ L∞ Scl and set a(x, ξ ) = ei|ξ | σ (x, ξ ), with 0 < ρ 1. Then for p each 1 < p < ∞ and each w ∈ Ap , the operator Ta is bounded from Lw to itself. 1−ρ
n(ρ−1)/2
Proof. Let us introduce a radial cut-off function θ (ξ ) which is smooth and equal to one for |ξ | 1 and zero for |ξ | 12 . Then once again following the reasoning in the proof of Theorem 1.2 in [7], which in turn relies on [19], yields that the “low frequency” portion of Ta given by 1 Ta1 u(x) = (2π)n
ˆ ) dξ σ (x, ξ )eix,ξ 1 − θ (ξ ) u(ξ
ei|ξ |
1−ρ
p
is bounded on Lw . So, it is enough to only consider the operator Ta2 u(x) =
1 (2π)n
ei|ξ |
1−ρ
σ (x, ξ )eix,ξ θ (ξ )u(ξ ˆ ) dξ.
For this operator we separate the spatial and frequency variables of the symbol a(x, ξ ) using spherical harmonics [9]. Namely let {wk }, k 1 be an orthogonal basis for L2 (Sn−1 ) consisting of eigenfunctions of the Laplacian S on the sphere Sn−1 . We decompose σ (x, ξ )ei|ξ |
1−ρ
= σ0 (x, ξ )e
θ (ξ ) i|ξ |1−ρ
θ (ξ ) +
fk (x)|ξ |
n 2 (ρ−1)
e
i|ξ |1−ρ
θ (ξ )wk
k
with χ(ξ ) ∈ C0∞ , χ(ξ ) = 1 for |ξ |
1 4
and σ0 (x, ξ ) supported in |ξ | 12 . Furthermore fk L∞
CN k−N , for all N > 0. For σ0 (x, ξ ) one has σ0 (x, ξ )ei|ξ | supports of σ0 and θ , therefore we have Ta2 u(x) =
fk (x)
n
=:
k
1−ρ
|ξ | 2 (ρ−1) ei|ξ |
k
ξ 1 − χ(ξ ) , |ξ |
fk (x)Uk (D)u(x).
1−ρ
θ (ξ )wk
θ (ξ ) = 0 because of the disjoint
ξ 1 − χ(ξ ) eix,ξ u(ξ ˆ ) dξ |ξ |
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Now it is clear that Uk (D)u(x) = (Sk (D) ◦ T (D)u)(x) with
Sk (D)u(x) :=
wk
ξ 1 − χ(ξ ) eix,ξ u(ξ ˆ ) dξ |ξ |
and T (D)u(x) :=
n
|ξ | 2 (ρ−1) ei|ξ |
1−ρ
θ (ξ )eix,ξ u(ξ ˆ ) dξ.
0 , a result of N. Miller [20] concerning weighted Lp boundNow since wk ( |ξξ | )(1 − χ(ξ )) ∈ S1,0 0 yields edness of operators with symbols in S1,0
Uk (D)u
p
Lw
T (D)uLp , w
where the constants of the inequality only depend on finite number of derivatives of wk ( |ξξ | )(1 − χ(ξ )), and are of polynomial growth in k. Furthermore, a result of S. Chanillo concerning weighted Lp boundedness of strongly singular integral operators of Hirschman–Wainger type [5], yields T (D)u Lpw u Lpw from which it follows that Uk (D)u Lpw u Lpw with polynomial bounds in k and hence summing in k and using the fact that fk L∞ CN k−N , for all N > 0 yields Ta2 u Lpw u Lpw , which concludes the proof of the theorem.
2
Now we turn to the problem of weighted boundedness of operators arising from amplitudes. To this end we have the following result, which is sharp modulo the end-point m = n(ρ − 1). Theorem 3.7. Suppose 0 ρ 1, m < n(ρ − 1) and a ∈ L∞ Am ρ , then, for each p > 1, we have Ta u(x) Mp u(x), and consequently Ta u Lpw u Lpw . Proof. The weighted norm inequality follows from the pointwise estimate by Theorems 2.5 and 2.6. 1 a(x, y, ξ )eiz,ξ dξ , then we have Let K(x, y, z) := (2π) n Ta u(x) = |x−y|1
K(x, y, x − y)u(y) dy +
K(x, y, x − y)u(y) dy = I + II.
|x−y|>1
However since m < n(ρ − 1), we also know from estimate (3.2) that |K(x, y, x − y)| C|x − y|−N for sufficiently large N and |x − y| 1. So II can be easily majorized by M(u)(x).
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As mentioned earlier we will again use the Littlewood–Paley partition of unity {ϕk }k just as we did in Theorem 3.4. Using that partition of unity and setting Kk (x, y, z) :=
1 (2π)n
ak (x, y, ξ )eiz,ξ dξ
yields I=
∞
Kk (x, y, x − y)u(y) dy =
k=0 |x−y|1
∞
Ik .
k=0
Now once again for k = 0 we observe that |K0 (x, y, x − y)| x − y−N for all N > 0, hence |I0 | Mu(x). If we consider an individual term with k 1, we have |Ik | =
Kk (x, y, x − y)u(y) dy
|x−y|1
=
r Kk (x, y, x − y)b(x − y)
|x−y|1
1 , u(y) dy |b(x − y)|r
where b and r are parameters to be chosen later. Therefore, Hölder’s inequality yields
Kk (x, y, x − y)p b(x − y)rp dy
|Ik |
1
p
|x−y|1
|x−y|1
|u(y)|p dy |b(x − y)|rp
1
p
.
By Theorem 2.7, for r < pn , we have
|x−y|1
|u(y)|p dy |b(x − y)|rp
1
p
Cb−r Mp u(x),
therefore |Ik | C
Kk (x, x − z, z)p |bz|rp dz
1 p
b−r Mp u(x).
(3.11)
Considering the remaining integral, setting σkx (z, ξ ) := ak (x, x − z, ξ ) and using (3.9) we have
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Kk (x, x − z, z) =
ak (x, x − z, ξ )eiz·ξ dξ
=
σkx (z, ξ )eiz·ξ dξ
= =
σkx (z, ξ ) φk−1 (ξ ) + φk (ξ ) + φk+1 (ξ ) eiz·ξ dξ σkx (z, ξ )ψˆ k (ξ )eiz·ξ dξ = Tσkx (ψk )(z).
Therefore, taking b = 2k ,
Kk (x, x − z, z)p |bz|rp dy
1 p
=
Tσ x (ψk )(z)p 2k zrp dz
1
k
p
.
Now we observe that since x is fixed, σkx belongs to the symbol class L∞ Sρm with semi-norms that
are uniform in x. Furthermore, the weight z → |z|rp is in Ap if and only if −n/p < r < n/p.
Now since p > 2, we may apply (3.5) with the p in that estimate taken equal to 1, and use Theorem 2.6 to obtain
Tσ x (ψk )(z)p 2k zrp dz
1 p
k
C2k(m−n(ρ−1)) C2
k(m−n(ρ−1))
M(ψk )(z)p 2k zrp dz ψk (z)p 2k zrp dz
= C2k(m−n(ρ−1)+n/p) .
1 p
1 p
(3.12)
Combining this with (3.11) we obtain |Ik | C2k(m−n(ρ−1)−(r−n/p)) Mp (u)(x). Therefore choosing r such that r − n/p = (m − n(ρ − 1))/2 and summing in k proves the theorem. 2 Theorem 3.7 is sharp in m up to the end-point n(ρ − 1). Indeed, suppose operators in p OPL∞ Am ρ were bounded on L for some p and m > n(ρ − 1). Since taking the adjoint of an operator in OPL∞ Am gives an operator which is also in OPL∞ Am ρ ρ , such operators would also
p be bounded on L . By interpolation they would be bounded on L2 . But this would contradict ∞ m Theorem 3 in [15] as Am ρ,1 ⊂ L Aρ . We also have a generalization of the result in Theorem 3.4 to the settings of amplitudes. Corollary 3.8. Let a(x, y, ξ ) = ei|ξ | σ (x, y, ξ ) with 0 < ρ 1 and assume that σ (x, y, ξ ) ∈ p n L∞ Am 1 with m < 2 (ρ − 1), then Ta is bounded on Lw , for all p ∈ (1, ∞) and w ∈ Ap . 1−ρ
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Proof. In the case ρ = 1, we are exactly in the situation of Theorem 3.7. For the case 0 < ρ < 1 we first observe that in the proof of Theorem 3.4 we have, in fact, proved for the Littlewood– Paley pieces that Ta (u)(x) 2−εk M(Mu)(x) + 2−εk/2 Mp u(x), k so Theorem 2.6 gives us the estimate Ta (u) p 2−εk/2 u p k Lw L w
(3.13)
for 1 < p < ∞ and w ∈ Ap . Now we may repeat the proof of Theorem 3.7, the bound (3.13) enabling us to prove an analogue of (3.12). We leave the details to the interested reader. 2 We now prove an interpolation result that will allow us to extend some of our previous results when we also assume the amplitudes are smooth. Lemma 3.9. Let 0 ρ 1, 0 δ 1, 1 < p < ∞ and m1 < m2 . Suppose that ∞ m1 1 (a) operators in OPAm ρ,δ (or OPL Aρ ) are bounded on Lw for a fixed w ∈ Ap , and ∞ m2 2 p (b) operators in OPAm ρ,δ (or OPL Aρ ) are bounded on L , p
where the bounds depend only on a finite number of Cα,β,γ (or Cα ) in Definition 2.1. Then, for p ∞ m ν each m ∈ (m1 , m2 ), operators in OPAm ρ,δ (or OPL Aρ ) are bounded on Lμ , where μ = w and ν=
m2 − m . m2 − m1
∞ m Proof. For a ∈ Am ρ,δ (or a ∈ L Aρ ) we introduce a family of symbols az (x, y, ξ ) := z ξ a(x, y, ξ ), where z ∈ Ω := {z ∈ C; m1 − m Re z m2 − m}. It is easy to see that, for |α + β| C1 with C1 large enough and z ∈ Ω,
α β γ
∂ ∂ ∂y az (x, y, ξ ) 1 + | Im z| C2 ξ Re z+m−ρ|α|+δ|β+γ | , ξ x for some C2 . (We only require β = γ = 0 if a ∈ L∞ Am ρ .) We introduce the operator m2 −m−z − m2 −m−z
Tz u := w p(m2 −m1 ) Taz w p(m2 −m1 ) u .
First we consider the case of p ∈ [1, 2]. In this case, Ap ⊂ A2 which in turn implies that both 1
−1
p
w p and w p belong to Lloc and therefore for z ∈ Ω, Tz is an analytic family of operators in the sense of Stein and Weiss [24]. Now we claim that for z1 ∈ C with Re z1 = m1 − m, the operator (1 + | Im z1 |)−C2 Taz1 p is bounded on Lw with bounds uniform in z1 . Indeed the amplitude of this operator is 1 −C ∞ m1 2 (1 + | Im z1 |) az1 (x, y, ξ ) which belongs to Am ρ,δ (or L Aρ ) with constants uniform in z1 . Thus, the claim follows from assumption (a).
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Consequently, we have − m2 −m−z1 p
pC
−C2 m2 −m−z1 p w p(m2 −m1 ) Taz1 w p(m2 −m1 ) u Lp Tz1 u Lp = 1 + | Im z1 | 2 1 + | Im z1 |
pC − m2 −m−z1 p 1 + | Im z1 | 2 w p(m2 −m1 ) uLp
w
pC p = 1 + | Im z1 | 2 u Lp , m2 −m−z1
where we have used the fact that |w (m2 −m1 ) u| = w.
m2 −m−z2
Similarly if z2 ∈ C with Re z2 = m2 − m, then |w (m2 −m1 ) u| = 1 and the amplitude of the oper2 ∞ m2 ator (1 + | Im z2 |)−C2 Taz2 belongs to Am ρ,δ (or L Aρ ) with constants uniform in z2 . Assumption (b) therefore implies that
pC p p Tz2 u Lp 1 + | Im z2 | 2 u Lp . Therefore the complex interpolation of operators in [4] implies that for z = 0 we have m2 −m − m2 −m p p p T0 u Lp = w p(m2 −m1 ) Ta w p(m2 −m1 ) u Lp C u Lp . Hence, setting v = w
m −m 2 1)
− p(m2 −m
u this reads Ta (v)p p C u p p , L L μ
μ
where μ = w ν and ν = (m2 − m)/(m2 − m1 ). This ends the proof in the case 1 p 2. p At this point we recall the fact that if a linear operator T is bounded on Lw , then its adjoint T ∗ p is bounded on L 1−p . Therefore, in the case p > 2, we apply the above proof to Ta ∗ , with w
p
p ∈ [1, 2) and v = w 1−p , which yields that Ta ∗ is bounded on Lv ν and since w ∈ Ap , we have p p p v ∈ Ap and so Ta is bounded on Lv (1−p)ν = L (1−p )(1−p)ν = Lμ , which concludes the proof of w the theorem. 2 For smooth amplitudes we can use Lemma 3.9 to show the following end-point versions of Theorems 3.7 and 3.3. n(ρ−1)
Theorem 3.10. If a ∈ Aρ,δ
with 0 < ρ 1, 0 δ < 1, then for all 1 < p < ∞ and all w ∈ Ap , p
n(ρ−1)/2
the corresponding pseudodifferential operator Ta is bounded on Lw . If a ∈ Sρ,δ with 0 < ρ < 1, 0 δ < 1, then for all 2 p < ∞ and all w ∈ Ap/2 , the corresponding pseudodifferential p operator Ta is bounded on Lw . Proof. We begin by proving the first statement for 0 < ρ < 1. By the Extrapolation Theorem of J. Rubio De Francia (see [12]) it is sufficient to show the boundedness of Ta on L2w spaces for each w ∈ A2 , with constants depending only on [w]A2 . Fix m2 such that n(ρ − 1) < m2 < min{0, n(ρ − δ)}. By Theorem 2.5, given w ∈ A2 we can find ε > 0 so that w 1+ε ∈ A2 . For
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this ε take m1 < n(ρ − 1) in such a way that the straight line L that joins points with coordinates (m1 , 1 + ε) and (m2 , 0), passes through the point (n(ρ − 1), 1). Clearly this is possible due to the fact that we can choose the point m1 as close as we like to n(ρ − 1). 1 2 By Theorem 3.7, OPAm ρ,δ are bounded operators on Lw 1+ε for w ∈ A2 and, by Theorem 2.8(a), n(ρ−1)
2 2 are bounded operators on L2w . OPAm ρ,δ are bounded on L . Therefore, by Lemma 3.9, OPAρ,δ 0 In the case a ∈ A1,δ , one just uses the fact that Ta = Ta0 + Ta1 , where Taj , j = 0, 1, are
−j (1−δ)
pseudodifferential operators belonging to OPS1,δ . Now a straightforward modification of the proof of the weighted boundedness of operators in OPS01,0 in [20], yields that Taj are both p bounded on Lw for w ∈ Ap , and therefore the same is true for Ta . The proof of the second statement is similar. By the extrapolation theorem of P. Auscher and J.M. Martell (see [3, Thm. 4.9]) it is sufficient to show the boundedness of Ta on L2w spaces for each w ∈ A1 , with constants depending only on [w]A1 . We now repeat the argument above with n(ρ − 1)/2 instead of n(ρ − 1), n(ρ − δ)/2 instead of n(ρ − δ), and Theorem 3.3 and Theorem 2.8(b) replacing Theorem 3.7 and 2.8(a), respectively. 2 In connection to Theorem 3.10, it should be mentioned that the second part is the extension of the result in [7] to the case of δ ρ. The first part of the theorem extends the weighted Lp n(ρ−1) boundedness of Ta ∈ OPSρ,δ proved in [2] for the range 0 < ρ 12 to the range 0 < ρ 1. However, for symbols we can extend the first part of the theorem to ρ = 0. Indeed, Theorem 3.7 m with m < −n. Furthermore yields the L2w1+ε boundedness of operators with symbols in S0,δ −n m by Theorem 3.2, operators with symbols in S0,δ with m < 2 are bounded on L2 . Hence, an p
n(ρ−1)
for interpolation procedure as above yields the boundedness in Lw of operators in OPSρ,δ 0 ρ < 1, 0 δ < 1. We also mention in passing that the first part of the above theorem yields in particular a sharp weighted boundedness result for the Weyl quantization of symbols. Also using Lemma 3.9, we can extend the range of m at the price of obtaining boundedness for fewer weights. Theorem 3.11. Let 1 < q < ∞, 0 < ρ < 1, 0 δ < 1 and suppose either: 1 1 (a) a ∈ Am ρ,δ and m < n(ρ − 1)| q − 2 | + min{0, n(ρ − δ)}; or m and m < n(ρ − 1)| 1 − 1 | + min{0, n(ρ − δ)/2}. (b) a ∈ Sρ,δ q 2
Then, for all w ∈ Aq , there exists α ∈ (0, 1), depending on m, ρ, δ, q and [w]Ap , such that, for q all ε ∈ [0, α], Ta is bounded on Lwε . Proof. If m < n(ρ −1), then, by Theorem 3.7, there is nothing to prove, so assume m > n(ρ −1). First assuming (a) we fix m2 such that m < m2 < n(ρ − 1)| q1 − 12 | + min{0, n(ρ − δ)} and m1 < p 1 n(ρ − 1). Then since OPAm ρ,δ are a bounded operators on Lw for all w ∈ Ap by Theorem 3.7, and m2 p OPAρ,δ are bounded on L , by Theorem 2.8(a). We see that all the assumptions of Lemma 3.9 are fulfilled and therefore we obtain the desired result. The proof under assumption (b) is the same, except we replace Theorem 2.8(a) by Theorem 2.8(b). 2
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4. Some applications in harmonic analysis In this section we show how our weighted norm inequalities can be used to derive the Lp boundedness of commutators of functions of bounded mean oscillation with a wide range of pseudodifferential operators. We start with the precise definition of a function of bounded mean oscillation. Definition 4.1. A locally integrable function u is of bounded mean oscillation if f BMO := sup B
1 |B|
f (x) − fB dx < ∞,
(4.1)
B
where the supremum is taken over all balls in Rn . We denote the set of such functions by BMO. For u ∈ BMO it is well known that er|u(x)| is locally integrable for r < 1. This is a consequence of the John–Nirenberg theorem, see, for example, [13, p. 524]. Furthermore, for all γ < 21n e , there exits a constant Cn,γ so that for u ∈ BMO and all balls B, 1 |B|
eγ |u(x)−uB |/ u BMO dx Cn,γ .
(4.2)
B
For this see [13, p. 528]. The following abstract lemma will enable us to prove the Lp boundedness of the BMO commutators of pseudodifferential operators. p
Lemma 4.2. For 1 < p < ∞, let T be a linear operator which is bounded on Lwα for all w ∈ Ap for some fixed α ∈ (0, 1]. Then given a function f ∈ BMO, if Φ(z) := ezf (x) T (e−zf (x) u)(x) × v(x) dx is holomorphic in a disc |z| < λ, then the commutator [f, T ] is bounded on Lp . Proof. Without loss of generality we can assume that f BMO = 1. We take u and v in C0∞ with u Lp 1 and v Lp 1, and an application of Hölder’s inequality to the holomorphic function Φ(z) together with the assumption on v yield Φ(z)p
p ep Re zf (x) T e−zf (x) u dx.
Our first goal is to show that the function Φ(z) defined above is bounded on a disc with centre at the origin and sufficiently small radius. At this point we recall a lemma due to Chanillo [6] which states that if f BMO = 1, then for 2 < s < ∞, there is an r0 depending on s such that for all r ∈ [−r0 , r0 ], erf (x) ∈ A 2s . Taking s = 2p in Chanillo’s lemma, we see that there is some r1 depending on p such that for |r| < r1 , erf (x) ∈ Ap . Then we claim that if R := min(λ, αrp1 ) and |z| < R then |Φ(z)| 1. Indeed since R <
αr1 p
αr1 p and p Re z f (x) α
we have | Re z| <
z therefore | p Re α | < r1 . Therefore Chanillo’s
∈ Ap and since ep Re zf (x) = w α , the assumption lemma yields that for |z| < R, w := e p of weighted boundedness of T and the L bound on u, imply that for |z| < R
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Φ(z)p
e
−zf (x) p T e u dx =
p Re zf (x)
p w α e−zf (x) u dx =
p w α T e−zf (x) u dx
w α w −α |u|p dx 1,
and therefore |Φ(z)| 1 for |z| < R. Finally, using the holomorphicity of Φ(z) in the disc |z| < R, Cauchy’s integral formula applied to the circle |z| = R < R, and the estimate |Φ(z)| 1, we conclude that Φ (0) 1 2π
|z|=R
|Φ(ζ )| |dζ | 1. |ζ 2 |
By construction of Φ(z), we actually have that Φ (0) = v(x)[f, T ]u(x) dx and the definition of the Lp norm of the operator [f, T ] together with the assumptions on u and v yield at once that [f, T ] is a bounded operator from Lp to itself for p. 2 The following lemma guarantees the holomorphicity of Φ(z) := v(x) dx, when T is a bounded pseudo-pseudodifferential operator.
ezf (x) T (e−zf (x) u)(x) ×
2 Lemma 4.3. Let Ta ∈ OPL∞ Am ρ,δ be an L bounded operator. Given f ∈ BMO with f BMO = 1 and u and v in C0∞ , there exists λ > 0 such that the function Φ(z) := ezf (x) Ta (e−zf (x) u)(x) × v(x) dx is holomorphic in the disc |z| < λ.
Proof. By the definition of pseudo-pseudodifferential operators above Definition 2.2 as the operator limit of the partial sums of the Littlewood–Paley pieces, N k=0 Tak := TN , it follows that for all g ∈ L2 lim TN g − Ta g L2 = 0.
N →∞
(4.3)
Now recall the fact mentioned in the beginning of this section concerning functions f ∈ BMO, namely the local integrability of er|f (x)| for r < 1. This means er|f | ∈ L2loc if r < 12 , and hence for u ∈ C0∞ and |z| < 12 , one has ue±zf ∈ L2 . Now, if we define ΦN (z) = v(x)ezf (x) TN (e−zf (x) u)(x) dx, then since the integral defining ΦN is absolutely convergent and its integrand is holomorphic in z for |z| < 1, it follows that ΦN is a holomorphic function in |z| < 1. Now we claim that for γ as in (4.2), lim sup ΦN (z) − Φ(z) = 0.
N →∞ |z|< γ
2
Indeed, since
γ 2
< 12 , for |z| < γ2 ,
ΦN (z) − Φ(z) = v(x)ezf (x) (TN − Ta ) e−zf u (x) dx zf −zf 2 ve L2 (TN − Ta ) ue L
N. Michalowski et al. / Journal of Functional Analysis 258 (2010) 4183–4209
v L∞
1 e
2 Re zf (x)
2
dx
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(TN − Ta ) ue−zf 2 . L
supp v
the assumption f BMO = 1 and (4.2), it follows that for any compact set K, Using ±2 Re zf (x) dx C (K), for |z| < γ . Therefore, (4.3) yields e γ K 2 lim sup ΦN (z) − Φ(z) = 0
N →∞ |z|< γ
2
and hence Φ(z) is a holomorphic function in |z| < γ2 .
2
Lemmas 4.2 and 4.3 yield our main result concerning commutators with BMO functions, namely Theorem 4.4. Suppose either: 1 1 (a) a ∈ Am ρ,δ , 0 < ρ 1, 0 δ < 1 and m < n(ρ − 1)| q − 2 | + min{0, n(ρ − δ)}; or m , 0 < ρ 1, 0 δ < 1 and m < n(ρ − 1)| 1 − 1 | + min{0, n(ρ − δ)/2}. (b) a ∈ Sρ,δ q 2
Then, for f ∈ BMO, the commutator [f, Ta ] is bounded on Lq . If (c) a ∈ L∞ Sρm with 0 ρ 1 and m < pn (ρ − 1) with p ∈ [1, 2], then, for f ∈ BMO, the commutator [f, Ta ] is bounded on Lq for all q ∈ (p, ∞). Proof. (a) By Theorem 3.11 part (a), there is an α ∈ (0, 1) for which Ta ∈ OPAm ρ,δ with m < q n(ρ − 1)( q1 − 12 ) + min{0, n(ρ − δ)} is Lwα -bounded. Furthermore if we define the function Φ(z) :=
ezf (x) Ta e−zf (x) u (x)v(x) dx
then Lemma 4.3 yields that Φ(z) is a holomorphic function in a disc around the origin. So an application of Lemma 4.2 with T = Ta yields that the commutator [f, Ta ] is a bounded operator from Lp to itself. (b) We repeat the argument above, but with (a) from Theorem 3.11 replaced by (b). (c) If a ∈ L∞ Sρm with 0 ρ 1, m < pn (ρ − 1), then by Theorem 3.3 we know Ta is q bounded on Lw for w ∈ Aq/p . But for α > 0 sufficiently small, it follows from Definition 2.4 ε that w ∈ Aq/p when w ∈ Aq and 0 ε α. Therefore, from Lemma 4.2, we conclude that [f, a(x, D)]u Lp u Lp . 2 Theorem 4.5. Suppose either: (a) a ∈ L∞ Am ρ with m < n(ρ − 1) and 0 ρ 1; or (b) a(x, y, ξ ) = ei|ξ |
1−ρ
n σ (x, y, ξ ) and σ ∈ L∞ Am 1 with 0 < ρ 1 and m < 2 (ρ − 1); or
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(c) a ∈ Aρ,δ
with 0 δ < 1 and 0 < ρ 1; or
(d) a(x, ξ ) = ei|ξ |
1−ρ
n
σ (x, ξ ) and σ ∈ L∞ Scl2
(ρ−1)
with 0 < ρ 1.
Then, for f ∈ BMO and k a positive integer, the k-th commutator defined by
k
Ta,f,k u(x) := Ta f (x) − f (·) u (x) q
is bounded on Lw for each w ∈ Aq and q ∈ (1, ∞). Proof. The claims follow from Theorem 2.13 in [1], and Theorem 3.7, Corollary 3.8, Theorem 3.10 and Theorem 3.6, respectively. Once again, part (c) of the theorem establishes in particular the boundedness of the commutator of a BMO function and the Weyl quantization of symbols. 2 References [1] Josefina Álvarez, Richard J. Bagby, Douglas S. Kurtz, Carlos Pérez, Weighted estimates for commutators of linear operators (English summary), Studia Math. 104 (2) (1993) 195–209. [2] J. Álvarez, J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators, Ark. Mat. 28 (1) (1990) 1–22. [3] J. Álvarez, M. Milman, Vector valued inequalities for strongly singular Calderón–Zygmund operators, Rev. Mat. Iberoamericana 2 (4) (1986) 405–426. [4] C. Bennett, R.C. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press, 1988. [5] S. Chanillo, Weighted norm inequalities for strongly singular integral operators, Trans. Amer. Math. Soc. 281 (1) (1984) 77–107. [6] S. Chanillo, Remarks on commutators of pseudo-differential operators, in: Multidimensional Complex Analysis and Partial Differential Equations, São Carlos, 1995, in: Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 33–37. [7] S. Chanillo, A. Torchinsky, Sharp function and weighted Lp estimates for a class of pseudodifferential operators, Ark. Mat. 24 (1) (1986) 1–25. [8] F. Chiarenza, M. Frasca, P. Longo, Interior W 2,p estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1) (1991) 149–168. [9] R. Coifman, Y. Meyer, Au dela des operateurs pseudo-différentiels, Asterisque, vol. 57, Societe Mathematique de Paris, France, 1978. [10] R. Coifman, Y. Meyer, Wavelets. Calderón–Zygmund and multilinear operators, translated from the 1990 and 1991 French originals by David Salinger, Cambridge Stud. Adv. Math., vol. 48, Cambridge University Press, Cambridge, 1997. [11] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (3) (1976) 611–635. [12] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, in: Notas de Matemática [Mathematical Notes], 104, in: North-Holland Math. Stud., vol. 116, North-Holland Publishing Co., Amsterdam, 1985. [13] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. [14] L. Hörmander, Pseudo-differential operators and hypoelliptic equations, in: Singular Integrals, Chicago, IL, 1966, in: Proc. Sympos. Pure Math., vol. X, 1967, pp. 138–183. [15] L. Hörmander, On the L2 continuity of pseudo-differential operators, Comm. Pure Appl. Math. 24 (1971) 529–535. [16] J. Hounie, On the L2 continuity of pseudo-differential operators, Comm. Partial Differential Equations 11 (1986) 765–778. [17] J.-L. Journe, Calderón–Zygmund Operators, Pseudodifferential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math., vol. 994, Springer-Verlag, Berlin, 1983. [18] C.E. Kenig, W. Staubach, Ψ -pseudodifferential operators and estimates for maximal oscillatory integrals, Studia Math. 183 (3) (2007) 249–258.
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[19] D.S. Kurtz, R.L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979) 343–362. [20] N. Miller, Weighted Sobolev spaces and pseudodifferential operators with smooth symbols, Trans. Amer. Math. Soc. 269 (1) (1982) 91–109. [21] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (March 1972) 207–226. [22] S.-I. Nishigaki, Weighted norm inequalities for certain pseudodifferential operators, Tokyo J. Math. 7 (1) (1984) 129–140. [23] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. [24] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., vol. 32, Princeton University Press, Princeton, NJ, 1971, x+297 pp. [25] K. Yabuta, Weighted norm inequalities for pseudodifferential operators, Osaka J. Math. 23 (3) (1986) 703–723.
Journal of Functional Analysis 258 (2010) 4210–4228 www.elsevier.com/locate/jfa
Classification of generalized multiresolution analyses ✩ Lawrence W. Baggett a , Veronika Furst b , Kathy D. Merrill c,∗ , Judith A. Packer a a Department of Mathematics, University of Colorado, Boulder, CO 80309, USA b Department of Mathematics, Fort Lewis College, Durango, CO 81301, USA c Department of Mathematics, Colorado College, Colorado Springs, CO 80903, USA
Received 2 November 2009; accepted 1 December 2009 Available online 6 January 2010 Communicated by N. Kalton
Abstract We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H . Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H . An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m, H ). This classification system is applied to MRAs and other classical examples in L2 (Rd ) as well as to previously studied abstract examples. © 2009 Elsevier Inc. All rights reserved. Keywords: Wavelet; Generalized multiresolution analysis; Filter; Ruelle operator
1. Introduction A generalized multiresolution analysis (GMRA) is a Hilbert space structure traditionally associated with classical wavelets, that is, functions whose dilates of translates provide an orthonormal basis for L2 (Rd ). Given a wavelet, the nested sequence of subspaces Vj that result ✩
This research was supported by the National Science Foundation through grant DMS-0701913.
* Corresponding author.
E-mail addresses:
[email protected] (L.W. Baggett),
[email protected] (V. Furst),
[email protected] (K.D. Merrill),
[email protected] (J.A. Packer). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.001
L.W. Baggett et al. / Journal of Functional Analysis 258 (2010) 4210–4228
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from taking only dilation powers less than j are dense and have trivial intersection, with Vj +1 the dilate of Vj , and with V0 invariant under translation. Such a structure is called a GMRA [6], and was developed to understand such wavelets as the famous example given by Journé, whose V0 space does not have an orthonormal basis given by translates of a single function called a scaling function. When V0 has this stronger property, the nested sequence {Vj } is called a multiresolution analysis (MRA) [22,23]. Both MRAs and GMRAs have been extensively exploited to produce and understand wavelets, which in turn have proven useful for applications such as image and signal processing. While wavelets and multiresolution structures were first studied in the Hilbert space L2 (Rd ), analogous definitions make sense in other Hilbert spaces that have appropriate dilation and translation operators. Dutkay and Jorgensen [16] pioneered the study of wavelets in function spaces on fractals, with later work by D’Andrea et al. [14]. Larsen, Raeburn and coworkers then showed that these and other interesting examples can be constructed via direct limits [19,4,5]. Dutkay et al. [9,17] constructed MRAs and super-wavelets in Hilbert spaces formed by direct sums of L2 (Rd ) to orthonormalize examples such as the Cohen wavelet. Tensor products of known examples lead to more exotic specimens (see Section 5). Our purpose in this paper is to construct a set of classifying parameters for GMRAs in order to unify and allow comparison of all these disparate examples. We also provide an explicit construction of a canonical GMRA equivalent to each of them. Accordingly, we will consider GMRA structures in an abstract Hilbert space H, equipped with “translations” given by a unitary representation π of a countable abelian group Γ acting in H, and a “dilation” given by a unitary operator δ. We assume that these operators are related by δ −1 πγ δ = πα(γ )
(1)
for all γ ∈ Γ , where αis an isomorphism of Γ into itself such that the index of α(Γ ) in Γ equals N > 1, and such that α n (Γ ) = {0}. These definitions generalize the classical case of ordinary translation by the integer lattice in L2 (Rd ), given √ by πn f (x) = f (x − n), and dilation by an expansive integer matrix A, given by δf (x) = |det A|f (Ax). The structure of a GMRA, and thus the parameters that uniquely identify it, are revealed via Stone’s Theorem on unitary representations of abelian groups. Using this theorem, we know that the representation π restricted to V0 is completely determined by a measure μ on the dual group Γ and a Borel multiplicity function m : Γ → {0, 1, 2, . . . , ∞}, which essentially describes a unitary equivhow many times each character occurs in the decomposition of π|V0 . There is L2 (σi ), where alence J between the action of π on V0 and multiplication by characters on σi = {ω: m(ω) i}. Because of this, we think of J as a partial alternative Fourier transform. For simplicity, in this paper we will restrict our attention to the commonly studied case where μ is Haar measure, and m is finite a.e. The multiplicity function m is one of the parameters that determine a GMRA. As we will see in Section 4, the other parameter is a “filter” that shows how the operator J interacts with dilation. Classical filters were periodic functions h and g in L2 (Rd ) that described inverse dilates of Fourier transforms of bases of V1 in terms of those of V0 . Starting with an MRA in L2 (Rd ), such functions could be shown to satisfy certain orthogonality relations. Mallat, Meyer and Daubechies [22,23,15] turned this process around by using functions h and g satisfying orthogonality together with additional low-pass and non-vanishing conditions to construct MRAs and wavelets. Lawton [20] and Bratteli and Jorgensen [12] were able to relax the non-vanishing
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condition by allowing Parseval frames in place of orthonormal bases, and Baggett, Courter, Jorgensen, Merrill, Packer [1,3] generalized this work to the GMRA setting by replacing h and g by matrix-valued functions H and G. In [11], Bratteli and Jorgensen related filters h and g to Ruelle operators Sh and Sg , which satisfy relations similar to those of Cuntz operators, and can be used to represent inverse dilations. This work was extended to generalized filters in [3] and later [4]. In the next section, we recall the relationship between abstract GMRAs, multiplicity functions and generalized filters. In particular, we describe conditions on a multiplicity function m and a filter H that guarantee that they will produce a GMRA. It turns out that these conditions are considerably more relaxed in an abstract Hilbert space than in L2 (Rd ). In Section 3 we describe a construction procedure that produces an abstract GMRA from any m and H meeting the required conditions. This construction gives an explicit realization of the abstract direct limit GMRAs built in [4]. While the procedure relies on first choosing a filter G complementary to H , we show in Section 4 that the equivalence between GMRAs does not depend on the choice of G. Thus, the classifying set described there depends only on the pair m and H . In this section, we also give a necessary and sufficient condition on the equivalence class of a filter so that SH is a pure isometry and thus associated with a GMRA, in the case of a finite multiplicity function. We conclude in Section 5 with a variety of examples that illustrate our main theorems, including an example of a GMRA where the translation group is not isomorphic to Zd . 2. GMRAs, multiplicity functions and filters Let H be an abstract, separable Hilbert space, equipped with operators πγ and δ satisfying Eq. (1). Definition 1. A collection {Vj }∞ −∞ of closed subspaces of H is called a generalized multiresolution analysis (GMRA) relative to π and δ if (1) (2) (3) (4)
Vj ⊆ Vj +1 for all j. all j. V j +1 = δ(Vj ) for Vj = {0}, and Vj is dense in H. V0 is invariant under the representation π.
The subspace V0 is called the core subspace of the GMRA {Vj }. For each j , write Wj for the orthogonal comLet {Vj } be a GMRA in a Hilbert space H. plement to Vj in Vj +1 . It follows that H = ∞ j =−∞ Wj . Also, for each j 0, Wj is an invariant subspace for the representation π. We apply Stone’s Theorem on unitary representations of abelian groups to the subrepresentations of π acting in V0 and W0 . Accordingly, there exists a finite, Borel measure μ (unique up to equivalence of measures) on Γ, Borel subsets σ1 ⊇ σ2 ⊇ · · · of Γ(unique up to sets of μ measure 0), and a (not necessarily unique) unitary operator J : V0 → i L2 (σi , μ) satisfying J πγ (f ) (ω) = ω(γ ) J (f ) (ω)
L.W. Baggett et al. / Journal of Functional Analysis 258 (2010) 4210–4228
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for all γ
∈ Γ , all f ∈ V0 , and μ almost all ω ∈ Γ. We write m for the function on Γ given by m(ω) = i χσi (ω), and call it the multiplicity function associated to the representation π|V0 . The GMRA {Vj } is an MRA if and only if m ≡ 1. Analogously, there exists a finite, Borel measure μ, Borel subsets σk , and an operator J : W0 → k L2 ( σk , μ) satisfying J πγ (f ) (ω) = ω(γ ) J (f ) (ω) for all γ ∈ Γ , f ∈ W0 , and μ almost all ω. We write m for the function on Γ given by m (ω) =
χ (ω), and call it the multiplicity function associated to the representation π| . σk W0 k In this paper, we will assume that the measures μ and μ are absolutely continuous with respect to Haar measure, and thus take μ and μ to be the restrictions of Haar measure to the subsets σ1 and σ1 , respectively. We also assume that the multiplicity function m associated to the representation π|V0 is finite almost everywhere. Let α ∗ be the dual endomorphism of Γ onto itself defined by [α ∗ (ω)](γ ) = ω(α(γ )), and note that the kernel of α ∗ contains exactly N elements and that α ∗ is ergodic with respect to the Haar measure μ on Γ. Indeed, suppose that for some γ ∈ Γ and some n > 0, we have γ ◦ (α ∗ )n = γ . Since the characters of Γ separate the points of Γ , it follows that α n γ = γ , and hence γ ∈ α n Γ = {0}. That is, the trivial character is the only γ ∈ Γ that satisfies γ ◦ (α ∗ )n = γ for some n > 0, and it follows that any function f ∈ L2 (Γ) that satisfies f ◦ α ∗ = f must be a constant function. The last statement is equivalent to the definition of the ergodicity of α ∗ . Using α ∗ to relate the representations π|V1 and π|V0 , it is shown in [6] and more generally in [4] that multiplicity functions for a GMRA must satisfy the following consistency equation:
m(ω) + m (ω) =
m(ζ ).
(2)
α ∗ (ζ )=ω
It follows, since the function m is finite a.e., that the sets σi and σk are completely determined by the multiplicity function m. It also follows that a multiplicity function m associated with a GMRA must satisfy the consistency inequality: m(ω)
m(ζ ).
(3)
α ∗ (ζ )=ω
We will see in the next section that the consistency inequality is a sufficient as well as necessary condition for a function m : Γ → {0, 1, 2, . . .} to be a multiplicity function associated to an abstract GMRA. Accordingly, we make the following definition. Definition 2. A multiplicity function is a Borel function m : Γ → {0, 1, 2, . . .} that satisfies the consistency inequality (3). In contrast, Bownik, Rzeszotnik and Speegle [10] and Baggett and Merrill [7] showed that an additional technical condition related to dilates of the translates of the support of m is required for m to be a multiplicity function for a GMRA in L2 (Rd ). We will need the following observation about multiplicity functions.
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Proposition 3. Suppose m : Γ → {0, 1, 2, . . .} satisfies the consistency inequality. If m is not identically 0, then there exists a set F of positive measure in Γ such that
m(ω) <
m(ζ )
α ∗ (ζ )=ω
for all ω ∈ F. That is, the consistency inequality is a strict inequality on a set of positive measure. Proof. Suppose
m(ω) =
m(ζ )
α ∗ (ζ )=ω
for almost all ω ∈ Γ. It follows directly by induction that
m(ω) =
m(ζ )
α ∗n (ζ )=ω
for almost all ω. Let k be a positive integer for which there exists a set E ⊆ Γ of positive measure such that m(ω) k for all ω ∈ E. Choose n such that N n > k. Then, for almost every ω ∈ α ∗ −n (E), we have
m(ωζ ) = m α ∗ n (ω) k,
ζ ∈ker(α ∗n )
implying that there exists some ζ ∈ ker(α ∗ n ), and a subset E ⊆ α ∗ −n (E) of positive measure, such that m(ωζ ) = 0 for all ω ∈ E . Hence, m(ω) = 0 on a set F of positive measure. But, from the equation m α ∗ (ω) = m(ωζ ), α ∗ (ζ )=1
it follows that the sequence {m(α ∗ n (ω))} is nondecreasing. Because α ∗ is ergodic, we must have that the sequence {α ∗ n (ω)} intersects the set F infinitely often for almost all ω. Hence m(ω) = 0 a.e. 2 The other ingredients we will need for our GMRA construction are filters, which are defined in terms of a multiplicity function m as follows: Definition 4. Let m be a multiplicity function, and write σi = {ω: m(ω) i}. Set m (ω) =
m(ζ ) − m(ω),
α ∗ (ζ )=ω
and set σk = {ω: m (ω) k}. Let H = [hi,j ] and G = [gk,j ] be (possibly infinite) matrices of Borel, complex-valued functions on Γ such that for every j , hi,j and gk,j are supported in σj . Suppose further that H and G satisfy the following “filter equations”:
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α ∗ (ζ )=ω
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hi,j (ζ )hi ,j (ζ ) = N δi,i χσi (ω),
(4)
gk,j (ζ )gk ,j (ζ ) = N δk,k χ σk (ω),
(5)
j
α ∗ (ζ )=ω j
and
gk,j (ζ )hi,j (ζ ) = 0.
(6)
α ∗ (ζ )=ω j
Then H is called a filter relative to m and α ∗ , and G is called a complementary filter to H . We note that it will sometimes be useful to consider filters and complementary filters to be matrix valued functions on Γ rather than a matrix of complex valued functions. It is then a consequence of the definition above that the nonzero portion of the matrix H (ω) is contained in the upper left block of dimensions m(α ∗ (ω)) × m(ω), while the nonzero portion of G(ω) is contained in the upper left block of dimensions m (α ∗ (ω)) × m(ω). Given a filter H relative to m and α ∗ , we may define a “Ruelle” operator SH on i L2 (σi ) by SH (f ) (ω) = H t (ω)f α ∗ (ω) . Similarly, a complementary filter G defines a Ruelle operator SG from by
i
L2 ( σi ) to
i
L2 (σi )
SG (f ) (ω) = Gt (ω)f α ∗ (ω) . The filter equations satisfied by H and G translate to the following Cuntz-like conditions for the Ruelle operators (see [3,4]): Lemma 5. If H is a filter relative to m and α ∗ , and G is a complementary filter to H , then the Ruelle operators they define satisfy ∗ S = I , S∗ S = I , (1) SH H G G ∗ (2) SH SG = 0, and ∗ + S S∗ = I , (3) SH SH G G
where I is the identity operator on
i
L2 (σi , μ) and I is the identity operator on
kL
2 ( σk , μ).
Filters, like multiplicity functions, arise naturally out of GMRAs. Let {Vj } be a GMRA (with finite multiplicity function and associated measure absolutely continuous with respect to Haar), μ, { σk }, and J be as in the Stone’s Theorem discussion above. Write Ci for and let μ, {σi }, J , the element of the direct sum space j L2 (σj , μ) whose ith coordinate is χσi and whose other 2 k for the element in coordinates are 0, and C σl , μ) whose kth coordinate σk and l L ( is χ whose other coordinates are 0. Let j hi,j be the element J (δ −1 (J −1 (Ci ))) and j gk,j be the k ))), both in j L2 (σj , μ). It was shown in [2] that the matrix H = [hi,j ] element J (δ −1 (J −1 (C
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is then a filter relative to m and α ∗ , and the matrix G = [gk,j ] is a complementary filter to H . We call these filters constructed from a GMRA, and note that they are not unique, but rather depend on the choice of the maps J and J . The operators J ◦ δ −1 ◦ J −1 and J ◦ δ −1 ◦ J −1 are the corresponding Ruelle operators SH and SG respectively. It follows directly from their definitions that SH and SG are isometries, and the GMRA requirement that Vj = {0} implies that SH = J ◦ δ −1 ◦ J −1 is a pure isometry. Just as with multiplicity functions, this necessary condition on a filter to be associated with a GMRA turns out to be sufficient 2 as well. In Theorem 5 of [4], it is shown that if SH is a pure isometry on a Hilbert space L (σi ), then it is possible to construct a generalized multiresolution analysis via a direct limit process. Our construction in the next section will give a concrete realization under the same hypotheses. Again, as with multiplicity functions, we see that this necessary and sufficient condition on the filter H is much weaker than what is required for a filter to be associated with a GMRA in L2 (Rd ). For example, in that context, the “refinement equation”, (ω) = √ φ
−1 t −1 1 A ω , H At ω φ |det A|
suggests some sort of convergence of the infinite product
∞
j =1
√ 1 H ((At )−j ω), |det A| √
(7) which in
turn requires that the filter H satisfies some low-pass condition of being close to |det A| times a partial identity near the origin [3,4]. Theorems from [2,5] indicate that in the abstract setting, a much weaker condition is sufficient to guarantee that SH is a pure isometry. In the case where the matrix H is 1 × 1, the simple condition that |H (ω)| = 1 on a set of positive measure is sufficient to show that SH is a pure isometry [11,5]. In particular, filters traditionally labeled “high-pass” can be used as H . Proposition 19 in Section 4 of this paper gives a new, more general result of this type. 3. Explicit construction of GMRAs on abstract Hilbert spaces Let m be a multiplicity function on Γ, as in Definition 2 and let H be a filter relative to m and α ∗ . Using Proposition 3, define m (ω) =
m(ζ ) − m(ω),
(8)
α ∗ (ζ )=ω
σk } as in the preceding section. As is shown in [1], given a filter and define the sets {σi } and { H relative to m and α ∗ , there always exists a complementary filter G. For the purposes of this construction, let G be any filter complementary to H . a J map guaranteed by Stone’s Theorem takes the core subspace V0 of any GMRA to Because L2 (σi ), this direct sum of L2 spaces is a natural candidate for the core subspace of an abstract GMRA built out of a multiplicity function and a filter. The group Γ acts on this space in a natural way via multiplication by characters. Similarly, the space L2 ( σk ) is an obvious candidate for −1 the abstract W0 = V1 V0 , and the relationships J ◦ δ ◦ J −1 = SH and J ◦ δ −1 ◦ J −1 = SG suggest that the Ruelle operators SH and SG provide natural abstract inverse dilations on these spaces. Thus the main task remaining in building a GMRA given m and H is to describe positive dilates of W0 ; such subspaces could then be used to fill out the rest of the Hilbert space.
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If the group Γ = Zd , then embedding Γ = Td as [− 12 , 12 ]d in Rd provides us with a sim 2 L ( σk ). Since in this case, α is an ple candidate for the dilate of our constructed W0 = d , we must have α(n) = An, for a matrix A. We define positive dilations isomorphism of Z σk ) → L2 (At σ k ) by Dj : L2 ( D
j
k
−j 1 fk (ω) = f At ω . √ j k |det A| k
For more general Γ , we will use an abstract construction to define the positive dilation D in terms of a cross section for the map α ∗ . Just as in√the case of Γ = Zd , our dilated space will be a direct sum of L2 spaces such that the map f → N f ◦ α ∗ determines an isometry of the dilated space onto the original one. Let c be a regular Borel section for the map α ∗ ; i.e., c is a Borel map from Γ ≡ α ∗ (Γ) ≡ Γ / ker(α ∗ ) into Γ, for which α ∗ (c(ω)) = ω for all ω ∈ Γ. (See, e.g. [21, Lemma 1.1].) Define τ : Γ → ker(α ∗ ) by τ (ω) = c α ∗ (ω) ω−1 . For example, in the simple case associated with dilation by 2 in L2 (R), where Γ = T ≡ [− 12 , 12 ), with α ∗ (ω) = 2ω, we can take c(ω) = ω2 . Thus, here τ (ω) =
0
if ω ∈ [− 14 , 14 ),
1 2
otherwise.
(9)
Now, let ν be a finite Borel measure on Γ. Let E be a Borel subset of Γ, let ζ be an element of the kernel of α ∗ , and set Eζ = ω ∈ E: τ (ω) = ζ . Proposition 6. The set E is the disjoint union
ζ
Eζ , and α ∗ is 1–1 on each Eζ into Γ.
Proof. The first statement is clear. For ω ∈ Eζ we have c α ∗ (ω) = τ (ω)ω = ζ ω, which shows that α ∗ must be 1–1 on Eζ .
2
Now let E1 , E2 , . . . be a (countable) collection of Borel subsets of Γ. For each i let νi be the of νi to the restriction to Ei of the measure ν. Write Ei,ζ for [Ei ]ζ , and let νi,ζ be the restriction = α ∗ (E ), and subset Ei,ζ of Ei . Write K = i L2 (Ei , νi ). For each ζ ∈ ker(α ∗ ), define Ei,ζ i,ζ equal to the measure N α ∗ (ν ) that is defined on E by set νi,ζ ∗ i,ζ i,ζ νi,ζ (F ) = N νi,ζ α ∗ −1 (F ) .
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For example, in the simple setting described by Eq. (9), if we take E1 = [− 38 , 38 ) and E2 = [ 18 , 12 ), then 1 3 1 1 3 1 E1,0 = − , , E1, 1 = − , − ∪ , , 2 4 4 8 4 4 8 1 1 1 1 E2,0 = , , E2, 1 = , , (10) 2 8 4 4 2 so that E1,0
1 1 1 1 1 1 , E1, 1 = − , − ∪ , , = − , 2 2 2 4 4 2 2 1 1 1 E2,0 , E2, , 0 . = , = − 1 4 2 2 2
here is just Haar measure restricted to E . The measure νi,ζ i,ζ Using our newly defined sets and measures, we let
K =
. L2 Ei,ζ , νi,ζ
i,ζ
Proposition 7. For each f ∈ K, set D(f ) equal to the element of K given by 1 D(f ) i,ζ (ω) = √ fi ζ −1 c(ω) . N Then the operator D is an isometry of K onto K . Proof. D(f )2 = ζ
i
=
Ei,ζ
ζ
Ei,ζ
fi ζ −1 c α ∗ (η) 2 dνi,ζ (η) i
=
D(f ) (ω)2 dν (ω) i,ζ i,ζ
2 1 −1 fi ζ c(ω) dνi,ζ (ω) N i
=
ζ E i,ζ
fi ζ −1 τ (η)η 2 dνi,ζ (η) i
ζ E i,ζ
fi (η)2 dνi,ζ (η) = i
ζ E i,ζ
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fi (η)2 dνi (η) = i E i
= f 2 , where the second to last step is justified because, for η ∈ Ei,ζ , we have τ (η) = ζ . Thus, D is an isometry. To see that D is onto K , it suffices to note that the inverse of D is given by √ −1 D (f ) i (ω) = N fi,τ (ω) α ∗ (ω) .
2
We will refer to the space K = D(K) as a dilation by α ∗ of K. Note that this general definition of D is consistent with the definition given at the beginning of this section for the special case of Γ = Zd . For example, in the situation defined by Eq. (10), we can see this equivalence by using integer translation in R to identify L2 ([− 12 , 12 )) ⊕ L2 ([− 12 , − 14 ) ∪ [ 14 , 12 )) with L2 ([− 34 , 34 )) and L2 ([ 14 , 12 )) ⊕ L2 ([− 12 , 0)) with L2 ([ 14 , 1)). We are now ready to construct explicitly a GMRA from the parameters m, H , and G. Theorem 8. Suppose m : Γ → {0, 1, 2, . . .} is a Borel function that satisfies the consistency ∗ inequality, and that H = [h i,j ] is a filter relative to m and α such that the Ruelle operator 2 SH is a pure isometry on be defined from m by the consistency equation i L (σi ). Let m 2 (as in Eq. (8)), and let G be a complementary filter to H . Define V = 0 i L (σi ) and W0 = 2 ∞ σk ). For n 1, inductively set Wn = D(Wn−1 ), and set H = V0 ⊕ n=0 Wn . Define a k L ( representation π of Γ , acting in H, by πγ (f ) (ω) = ω(γ )f (ω). Finally, define an operator T on H by SH (fV0 ) + SG (fW0 ), T (f ) a = D−1 (fWn+1 ),
a = V0 , a = Wn , n 0,
(11)
where we represent an element f of H by {fV0 , fW0 , fW1 , . . .}. Then (1) T is a unitary operator on H. (2) T πγ T −1 = πα(γ ) for all γ ∈ Γ . (3) If Vj is defined to be T −j (V0 ), then the collection {Vj } is a GMRA relative to π and δ, where δ = T −1 . (4) The multiplicity function associated to the core subspace V0 is the given function m, and the given H is a filter constructed from the GMRA {Vj }. Proof. To prove the first claim, note that by Proposition 7, D−1 is an isometry from Wn+1 onto Wn . By Lemma 5, we also have that the definition of T above gives an isometry from V0 ⊕ W0 onto V0 . The second claim follows immediately from the definitions, since the operators SH , SG and D−1 all change the argument of the function from ω to α ∗ (ω).
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Next, we show that the collection {Vj } is a GMRA. The fact that Vj ⊆ Vj +1 follows from T (V0 ) ⊂ V0 , which is immediate from the definition of T . That Vj +1 = δ(Vj ) follows immediately from the definition of δ = T −1 . The trivial intersection property follows from our assumption that SH is a pure isometry, and the dense union from the fact noted in the previous paragraph that T −1 V0 = V0 ⊕ W0 and T −1 Wn = Wn+1 . As a component of the direct sum space, V0 is clearly invariant under the multiplication operators ω(γ ) that define the representation π . The given function m is clearly the multiplicity function of that representation. To establish that H is a corresponding filter, we note that we can take J to be the identity for this V0 , and calculate J δ −1 J −1 (Ci )(ω) = T (Ci )(ω) = SH (Ci )(ω) = hi,j (ω)χσi α ∗ (ω) j
=
hi,j (ω),
j
where the last equality follows from the fact that by the filter equation, hi,j is supported on α ∗ −1 (σi ). 2 Remark 9. We will denote the GMRA {Vj } constructed above by {Vjm,H,G } and refer to it as the canonical GMRA having these parameters. In Section 5 we will construct canonical GMRAs related to classical examples, as well as new ones. First, we establish in the next section conditions under which two GMRAs are the same. While our construction procedure requires the choice of a complementary filter G, we will see that the equivalence classes depend only on the two parameters m and H . 4. A classifying set for GMRAs Let {Vj } be a GMRA in a Hilbert space H, relative to a representation π of Γ and a unitary operator δ, and let {Vj } be a GMRA in a Hilbert space H , relative to a representation π of Γ and a unitary operator δ . Definition 10. We say that the GMRAs {Vj } and {Vj } are equivalent if there exists a unitary operator U : H → H that satisfies: (1) U (Vj ) = Vj for all j. (2) U ◦ πγ = πγ ◦ U for all γ ∈ Γ. (3) U ◦ δ = δ ◦ U. For classical examples in L2 (Rd ), the Fourier transform F gives an equivalence between any j }. Further, if an operator U gives an equivalence between {Vj } and {V }, GMRA {Vj } and {V j = F ◦ U ◦ F −1 is two GMRAs for dilation by A and translation by Zd in L2 (Rd ), then U multiplication by a function u with absolute value 1, and such that u(A∗j ω) = u(ω) for all
L.W. Baggett et al. / Journal of Functional Analysis 258 (2010) 4210–4228
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integers j [8]. Thus equivalence between GMRAs for the same dilation in L2 (Rd ) generalizes the notion of different MSF wavelets attached to the same wavelet set. Recall that we consider only GMRAs with a finite multiplicity function m and with the associated measure μ absolutely continuous with respect to Haar measure. Our first aim is to prove that every such GMRA is equivalent to one of the canonical GMRAs constructed in the preceding section. We will then describe the equivalence relation among these GMRAs in terms of the parameters m, H and G. We will need the following lemma. Lemma 11. The GMRAs {Vj } and {Vj } are equivalent if and only if there exists a unitary operator P mapping V0 onto V0 that satisfies: (1) P ◦ πγ = πγ ◦ P for all γ ∈ Γ. (2) P ◦ δ −1 = δ −1 ◦ P .
Proof. We first assume that the conditions above are satisfied and show that {Vj } and {Vj } are equivalent. For each n 0, define an operator Qn : Wn → Wn by Qn = δ
n+1
◦ P ◦ δ −(n+1) .
∞ Now, define U = P ⊕ ∞ n=0 Qn on H = V0 ⊕ n=0 Wn . One checks directly that U satisfies the required conditions. For the converse, assume that {Vj } and {Vj } are equivalent, with U : H → H implementing the equivalence. Define P = U |V0 . By the definition of equivalence, P maps V0 to V0 , and conditions (1) and (2) follow. 2 Theorem 12. Let {Vj } be a GMRA. Let m be its (finite) associated multiplicity function, and let H = [hi,j ] be a filter constructed from the GMRA using the map J . Let G be a complementary filter to H . Then the GMRA {Vj } is equivalent to the canonical GMRA {Vjm,H,G }. Proof. Define P : V0 → L2 (σi ) by P = J . Condition (1) of Lemma 11 follows immediately. The fact that J ◦ δ −1 ◦ J −1 = SH proves the second condition of that lemma. 2
Theorem 13. The canonical GMRAs {Vjm,H,G } and {Vjm ,H ,G } are equivalent if and only if m = m , and there exists a matrix-valued function A on Γ such that (1) A(ω) = A10(ω) 00 , where A1 (ω) is a unitary matrix of dimension m(ω). (2) H (ω)At (ω) = At (α ∗ (ω))H (ω). Proof. Suppose first that m = m and that there exists a matrix-valued function A satisfying the conditions. Let τr be the subset of Γ on which m(ω) = m (ω) = r. Then both subspaces V0m,H,G
and V0m ,H
,G
are equal to i
L2 (σi ) ≡
r
L2 τ r , C r .
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Define P : V0m,H,G → V0m ,H
,G
by [P (f )](ω) = A(ω)f (ω). It follows directly that P satisfies
the conditions of Lemma 11, and hence {Vjm,H,G } and {Vjm ,H ,G } are equivalent. Conversely, suppose an operator P exists and satisfies the conditions of Lemma 11. The first condition on P implies that the two representations of Γ on V0m,H,G and V0m ,H ,G are unitarily equivalent, whence m must equal m , and V0m,H,G = V0m ,H ,G = i L2 (σi ) = r L2 (τr , Cr ). It is known (e.g. [8]) that any unitary operator P on the direct sum of vector-valued L2 spaces that commutes with all the multiplication operators γ (ω), is itself a multiplication operator of the form P (f ) (ω) = A(ω)f (ω), where A(ω) = A10(ω) 00 , and A1 (ω) is a unitary matrix whose dimension is r = m(ω) for ω ∈ τr . The second condition of Lemma 11 then implies that A satisfies condition (2) of the theorem. 2 Corollary 14. Let m be a multiplicity function and let H be a filter relative to m and α ∗ for which SH is a pure isometry. If G and G are any two complementary filters to H , then the GMRAs {Vjm,H,G } and {Vjm,H,G } are equivalent. The preceding theorem introduces a notion of equivalence among filters that we will use to build a set of classifying parameters for the equivalence classes of GMRAs. In the following definition, we use our knowledge of the form of A to rewrite the equivalence using the conjugate transpose A∗ . ∗ Definition 15. Let m be a multiplicity function. Filters H and H relative to m and α are called A1 (ω) 0 equivalent if there exists a matrix-valued function A on Γ , with A(ω) = , where A1 (ω) 0 0 is a unitary matrix of dimension m(ω), and such that
H (ω) = A α ∗ (ω) H (ω)A∗ (ω) for almost all ω ∈ Γ. Remark 16. If H and H are two filters constructed from the same GMRA using different Stone’s Theorem operators J and J , then H and H are equivalent according to this definition. Here the matrix-valued function A comes from the multiplication operator J J −1 . Lemma 17. Let H be a filter relative to m and α ∗ , and let A be a matrix-valued function of the form described in the preceding theorem. Define the matrix-valued function H by H (ω) = A α ∗ (ω) H (ω)A∗ (ω). Then H is a filter relative to m and α ∗ , i.e., H satisfies the filter equation. Proof. We note that if we write H1 (ω) for the upper left m(α ∗ (ω)) × m(ω) block of H (ω), and let Λ(ω) be N times the m(ω) × m(ω) identity, then the filter equation (4) can be rewritten as
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H1 (ζ )H1∗ (ζ ) = Λ(ω).
4223
(12)
α ∗ (ζ )=ω
We must show that if H satisfies Eq. (12), then so does H . We have α ∗ (ζ )=ω
∗
H1 (ζ )H 1 (ζ ) =
A1 (ω)H1 (ζ )A∗1 (ζ )A1 (ζ )H1∗ (ζ )A∗1 (ω)
α ∗ (ζ )=ω
= A1 (ω)Λ(ω)A∗1 (ω) = Λ(ω).
2
Let H be a filter relative to m and α ∗ . In [2] it was shown that the operator SH fails to be a pure isometry if and only if it has an eigenvector, i.e., if and only if there exists an element F ∈ L2 (σi ) and a complex number λ for which H t (ω)F (α ∗ (ω)) = λF (ω), where |λ| = 1 = F (ω) for almost all ω. Motivated by this result, we make the following definition: Definition 18. A filter H is called an eigenfilter if there exists a constant λ with |λ| = 1 such that for almost all ω, H1,1 (ω) = λ and H1,j (ω) = 0 for j > 1. Using this definition, we have the following restatement of the result from [2]: Proposition 19. SH fails to be a pure isometry if and only if H is equivalent to an eigenfilter. Proof. If there exists a matrix-valued function A such that H (ω)A(ω) = A α ∗ (ω) H (ω), where H (ω) is an eigenfilter, then, computing the first rows of both sides, we see that the first row of A is the desired eigenvector F . Conversely, if SH has an eigenvector F , build a unitary-valued matrix A(ω) having F (ω) as its first row. Set H (ω) = A(α ∗ (ω))H (ω)A∗ (ω). By the previous lemma, H is a filter relative = λ. Because H is a filter, it follows that the to m and α ∗ . Moreover, one can see that H1,1 elements H1,j (ω) are all 0 for j > 1. Hence, H has the desired form. 2 Now, let S be the set of all pairs (m, H ), where m is a multiplicity function and H is a filter relative to m and α ∗ . Let S0 be the subset of S comprising those pairs (m, H ) for which H is equivalent to an eigenfilter, and let S1 = S \ S0 . Finally let E = S1 / ≡ be the set of equivalence classes of S1 with respect to the equivalence relation (m1 , H1 ) ≡ (m2 , H2 ) if m1 = m2 and H1 is equivalent to H2 . Theorem 20. The set E is a classifying set for the equivalence classes of GMRAs (with finite multiplicity functions and associated measures absolutely continuous with respect to Haar measure), in the sense that there is a 1–1 correspondence between E and the classes of GMRAs, and this correspondence can be described explicitly.
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Proof. Given an element s ∈ E, let (m, H ) be a representative of the equivalence class s. Let G be a filter complementary to H , and define κ(s) to be the equivalence class of the GMRA {V m,H,G }. By Theorem 13, the map κ is both well defined and one-to-one, and by Theorem 12, it is onto. 2 5. Examples We will now use the technique outlined in Section 3 to construct examples of canonical GMRAs, and apply the ideas of Section 4 to discuss their equivalence. We work first in the classical setting of MRAs (so m ≡ 1) with single wavelets (so m ≡ 1) for dilation by 2 in L2 (R). Since m is determined for MRAs, their equivalence depends only on the filter H . Example 21. Any MRA for dilation by 2 in L2 (R) with m = m ≡ 1 has canonical Hilbert space L (T) ⊕ L (T) ⊕ 2
2
∞
∞ m,H,G j m,H,G m,H,G L 2 T = V0 ⊕ W0 ⊕ Wj 2
j =1
(13)
j =1
with πn ( fl ) = en · fl , where en (x) = e2πinx , and δ
−1
fV0 ⊕ fW0 ⊕
∞
fWj
√ (ω) = h(ω)fV0 (2ω) + g(ω)fW0 (2ω) ⊕ 2fW1 (2ω)
j =1
⊕
∞ √ 2fWj (2ω) .
(14)
j =2
Equivalence for two different MRAs with single wavelets is equivalent to the existence of a period 1 function a such that |a(ω)| = 1 and h (ω) = a(2ω)h(ω)a(ω), where h and h are filters constructed from the two MRAs. Thus, in particular, equivalence requires that |h| = |h |. However, this is not sufficient, as we will see below. Determining which filters give equivalent MRAs requires determining exactly which functions on the 1-torus are coboundaries where cohomological equivalence is given by Definition 15. √ √ 0 = L2 ([− 1 , 1 ]), we have h = 2χ 1 1 and g = 2χ 1 1 For the Shannon MRA, with V [− , ] ±[ , ] 2 2 4 4
4 2
in the above formula. By mapping Wjm,H,G → L2 (±2j [ 12 , 1]), we can map this canonical GMRA to the Fourier transform of the Shannon GMRA. For the Haar MRA, with V0 spanned by translates of χ[0,1] , we have h = √1 (1 + e−1 ), g = 2
− 1) in the above formula. Here there is no obvious mapping between the canonical GMRA and either the original or its Fourier transform. However, we know all three are equivalent by Theorem 12. In either the Shannon or Haar examples, we can switch the roles of h and g to get a new MRA that cannot be realized in L2 (R) (since iterating the refinement equation (7) leads to a scaling function that must be identically 0). The canonical Hilbert space will still be given by Eq. (13), and the operators πn will be as above. However, in the dilation formula (14), we will now have h given by the old g, and g by the old h. Proposition 19 shows that we still have Sh a pure isometry, m,h,g m,h,g so that the canonical construction does produce a GMRA. Looking at the V−j and W−j that √1 (e−1 2
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result in the case of the reversed Shannon GMRA shows how this example differs from Shannon MRA itself: 1 1 1 1 3 1 2 2 − , → L ± , → L ± , → · · · , δ =L 2 2 4 2 8 2 1 1 1 1 1 3 m,h,g δ −1 : W0 → L2 − , → L2 ± , → · · · . = L2 − , 2 2 4 4 4 8 −1
m,h,g : V0
2
Since the absolute values of the filters in the three examples discussed here are all different on sets of positive measure, the three are seen to be inequivalent MRAs. To see that for MRAs with wavelets, the filters having equal absolute value almost everywhere is not sufficient for equivalence, consider the MRA built from h = −h, where h is the filter for the Haar example. A simple Fourier analysis argument shows that there is no solution to h (ω) = a(2ω)h(ω)a(ω), so this MRA must be inequivalent to the Haar MRA. We note that it has the same canonical Hilbert space as Haar, and the same subspaces Vj , but its dilation on V0 is the negative of the Haar dilation. This negative sign causes problems in the iteration of the refinement equation, so this example cannot be realized in L2 (R). A fourth example in this setting begins with the Cohen filters h = √1 (1 + e−3 ) and g = 2 √1 (1 − e−3 ). The infinite product construction which follows from the refinement equation 2 in L2 (R) yields the functions φ = 13 χ[0,3) and ψ = 13 (χ[0, 3 ) − χ[ 3 ,3) ), which fail to be an or2
2
thonormal scaling function and orthonormal wavelet, respectively, since neither has orthonormal translates. However, it can be shown that the negative dilate space (for dilation by 2) of the Cohen Parseval wavelet coincides with that of the Haar orthonormal wavelet. Hence, the Cohen GMRA equals the Haar MRA. We may apply Theorem 8 to the Cohen filters and multiplicity functions m ≡ 1, m ≡ 1. The canonical Hilbert space will be that given by Eq. (13), on which the integers act by multiplication by exponentials. We see that the spaces Vj for the canonical Cohen GMRA are the same as those for the canonical Haar MRA whenever j 0. However, since √1 (1 + e−3 ) and √1 (1 + e−1 ) 2 2 have different moduli, the two filters must be inequivalent. Therefore the two canonical GMRAs must be inequivalent. Lastly, we remark that while the Cohen wavelet ψ = 13 (χ[0, 3 ) − χ[ 3 ,3) ) is only a Parseval 2
2
wavelet in L2 (R), the element (0, χ[− 1 , 1 ) , 0, 0, 0, . . .) is an orthonormal wavelet for the canon2 2 ical Hilbert space (13), with respect to πn and δ defined by Eq. (14) using the Cohen filters. Dutkay et al. [9,17] also produced an orthonormal wavelet from the Cohen filter, using a “superwavelet” construction. The associated GMRA in L2 (R) ⊕ L2 (R) ⊕ L2 (R) can be seen to be equivalent to our canonical Cohen GMRA by defining the map P in Lemma 11 in the natural way to take the nth translate of the Dutkay scaling function to e2πinx in the canonical V0 . Of course, this example cannot be realized in L2 (R). Next, we consider two non-MRA examples for dilation by 2 in L2 (R): the Journé GMRA, and the example for the Journé multiplicity function with low-pass filter of rank a = 2 described in [4, Example 13]. As is noted there, a GMRA cannot be constructed for this second example using the infinite product construction. However, by Proposition 19, the construction of this paper can be carried out to give such a GMRA.
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Example 22. Let m be the multiplicity function corresponding to the Journé wavelet: ⎧ ⎪ ⎨2 m(x) = 1 ⎪ ⎩ 0
if x ∈ [− 17 , 17 ), if x ∈ ±[ 17 , 27 ) ∪ ±[ 37 , 12 ), otherwise,
so σ1 = [− 12 , − 37 ] ∪ [− 27 , 27 ] ∪ [ 37 , 12 ] and σ2 = [− 17 , 17 ]. Since we know the Journé GMRA has an associated single orthonormal wavelet, m ≡ 1. Filters that give rise to the Journé wavelet via the infinite product construction are described in [13,1,3]. In particular, we may take √ 2χ[− 2 ,− 1 ]∪[− 1 , 1 )∪[ 1 , 2 ] 7 7 4 7 √ 7 4 H= 2χ[− 1 ,− 3 ]∪[ 3 , 1 ] 2
7
0
0
7 2
and G=
√
2χ[− 1 ,− 1 ]∪[ 1 , 1 ] 4
7
7 4
√
2χ[− 1 , 1 ] . 7 7
Here V0m,H,G = L2 (σ1 ) ⊕ L2 (σ2 ), and Wjm,H,G = L2 (2j T), j 0. This canonical GMRA can be mapped to the usual Journé GMRA by integrally translating σ1 and σ2 to the scaling set to form V0 , and T ≡ [− 12 , 12 ] to the wavelet set to form W0 . In [4], an alternative filter H for the same multiplicity function, but which satisfies the lowpass condition of rank a = 2 is constructed: h 1,1 =
√
h 1,2 = h 2,1 = 0,
2χ[− 2 ,− 1 )∪[− 1 , 1 )∪[ 1 , 2 ) , 7
4
7 7
4 7
and h 2,2 =
√ 2χ[− 1 , 1 ) . 14 14
By partitioning R/Z as described in [13,1], we can build the following complementary filter G : g1,1 =
√ 2χ[− 1 ,− 3 )∪[− 1 ,− 1 )∪[ 1 , 1 )∪[ 3 , 1 ) , 2
7
4
7
7 4
7 2
g1,2 =
√ 2χ[− 1 ,− 1 )∪[ 1 , 1 ) . 7
14
14 7
The spaces V0m,H ,G and Wjm,H ,G for j 0 are the same as those for the canonical GMRA corresponding to the standard Journé filters. However, the different filters in the rank 2 exam ple will change the dilation, and thus change the spaces Vjm,H ,G and Wjm,H ,G for j < 0. m,H For example, we have V−1
,G
1 1 = L2 ([− 17 , 17 ] ∪ ±[ 14 , 27 ]) ⊕ L2 ([− 14 , 14 ]), while the standard
m,H,G m,H ,G = L2 ([− 17 , 17 ] ∪ ±[ 14 , 27 ] ∪ ±[− 37 , 12 ]) ⊕ 0. The fact that all the V−j allow nonzero V−1 second components with support overlapping that of the first component suggests the impossibility of mapping the rank 2 example into L2 (R) as we mapped the standard example. Indeed, since iterating the refinement equation would lead to a scaling function with a degenerate multiplicity function [4], the rank 2 example cannot be realized in L2 (R). Thus, these two examples must not be equivalent.
For our next example, we consider dilation by 3, both in L2 (R) and in the Dutkay/Jorgensen enlarged Cantor fractal space [16].
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Example 23. The MRA Haar 2-wavelet for dilation by 3 in L2 (R) has canonical Hilbert space L (T) ⊕ L2 (T) ⊕ L2 (T) ⊕
2
∞
j 2 j L 3 T ⊕L 3 T . 2
j =1
−j The canonical δ −1 = Sh ⊕ (Sg1 ⊕ Sg2 ) ⊕ ( ∞ j =1 D ), where 1 h = √ (1 + e1 + e2 ), 3
1 g1 = √ (e1 − e2 ) 2
1 and g2 = √ (−2 + e1 + e2 ), 6
√ j and D−j (f1 ⊕ f2 )(ω) = 3 (f1 ⊕ f2 )(3j ω). The Cantor set MRA has the same canonical GMRA except with 1 h = √ (1 + e2 ), 2
g1 = e1
1 and g2 = √ (1 − e2 ). 2
These two examples must be inequivalent since √ their h’s have different absolute values. The latter cannot be realized in L2 (R), since h(0) = 2, so that the iterated refinement equation (7) would again force the scaling function to be identically 0. Our final example uses a group Γ different from Zd . ∞ Example 24. Let Γj = ∞ i=j [Z2 ]i = i=j {1, −1}i , embedded as a subgroup of D = Γ−∞ = ∞ j −1 ∞ i=−∞ [Z2 ]i by Γj = i=j [Z2 ]i . Let α be defined on Γ0 by α(γ )n = γn−1 for i=−∞ {1}i ⊕ n > 0 and α(γ )0 = 1. Let H = l 2 (D), and let π be the restriction to Γ0 of the regular representation of D. Define S on D by [S(d)]n = dn−1 , and note that S(γ ) ≡ α(γ ) for γ ∈ Γ0 . Define δ f (S(d)), and note that δ −1 πγ δ = πα(γ ) . on H = l 2 (D) by [δ(f )](d) = We have Γj +1 ⊆ Γj , and ∞ j =−∞ Γj = {eD }, where eD = (. . . , 1, 1, 1, . . .) denotes the 2 multiplicative identity element of D = Γ−∞ , so that l 2 (Γj +1 ) ⊆ l 2 (Γj ) and ∞ j =−∞ l (Γj ) = l 2 ({eD }). If we let Vj = l 2 (Γ−j ), then {Vj } is almost a GMRA. It fails only because constant multiples of the function χ{eD } belong to Vj . We will make it into a GMRA by tensoring it with the dilation by 2 Haar GMRA. It is known (as in [18]) that the tensor product of two GMRAs gives a GMRA. By tensoring our almost GMRA with an actual one, we will preserve all the properties of the almost GMRA, and eliminate the non-trivial intersection. 2 √ Accordingly, let Γ = Z act in H = L (R) by πn f (x) = f (x − n), and let δ f (x) = 2f (2x). We have α acting on Γ by α (n) = 2n. Write {Vj } for the usual Haar GMRA that results from taking V0 to be the closed linear span of translates of χ[0,1] . Set H = H ⊗ H , equipped with the representation π × π of Γ = Γ0 × Γ and the operator δ ⊗ δ . We let ∗ α = α × α and note that α ∗ acts on Γ = ∞ i=0 [Z2 ]i × T by α ((ω0 , ω1 , ω2 , . . .) × x) = 1 1 (ω1 , ω2 , . . .) × 2x, where we parameterize T by [− 2 , 2 ). We have N = 4, and ker(α ∗ ) =
1 ({−1, 1} × ∞ i=1 [{1}]i ) × {0, 2 }. To build the dilation described in Section 3, we can take the ≡ 3, so cross section c((ω0 , ω 1 , ω2 , . . .) × x) = (1, ω0 , ω1 , . . .) × x2 . We have m ≡ 1 and m σ1 = σ2 = σ3 = ∞ σ1 = i=0 [Z2 ]i × T.
∞ √ We define our filter H = h1 ⊗ h2 , where h1 is the filter on i=0 [Z2 ]i given by h1 = 2χ{1}0 × ∞ , and h2 is the low-pass filter for the Haar GMRA described in Example 21, i=1 [Z2 ]i
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√
√1 (1+e−1 ). For our filter complementary to H , we define g1 = 2χ{−1} × ∞ [Z ] 0 i=1 2 i 2 and let g2 be the high-pass filter for the Haar GMRA, g2 = √1 (e−1 − 1). We then take our com2 plementary filter G to be the matrix whose rows are h1 ⊗ g2 , g1 ⊗ h2 , and g1 ⊗ g2 . − χ{−1}0 × ∞ and For an alternative GMRA, we can replace h1 by h 1 = χ{1}0 × ∞ i=1 [Z2 ]i i=1 [Z2 ]i
∞ + χ . These could be viewed as more fractal-like when g1 by g1 = −χ{1}0 × ∞ {−1}0 × i=1 [Z2 ]i i=1 [Z2 ]i
that is, h2 =
combined with the h2 and g2 in the standard tensor product construction. References [1] L.W. Baggett, J.E. Courter, K.D. Merrill, The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ), Appl. Comput. Harmon. Anal. 13 (2002) 201–223. [2] L.W. Baggett, V. Furst, K.D. Merrill, J.A. Packer, Generalized filters, the low-pass condition, and connections to multiresolution analysis, J. Funct. Anal. 257 (2009) 2760–2779. [3] L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005), #083502, 28 pp. [4] L.W. Baggett, N.S. Larsen, K.D. Merrill, J.A. Packer, I. Raeburn, Generalized multiresolution analyses with given multiplicity functions, J. Fourier Anal. Appl. 15 (2009) 616–633. [5] L.W. Baggett, N.S. Larsen, J.A. Packer, I. Raeburn, A. Ramsay, Direct limits, multiresolution analyses, and wavelets, J. Funct. Anal. 258 (8) (2010) 2714–2738. [6] L.W. Baggett, H.A. Medina, K.D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn , J. Fourier Anal. Appl. 5 (1999) 563–573. [7] L.W. Baggett, K.D. Merrill, Abstract harmonic analysis and wavelets in Rn , in: The Functional and Harmonic Analysis of Wavelets and Frames, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, 1999, pp. 17–27. [8] R. Beals, Operators in function spaces which commute with multiplications, Duke Math. J. 35 (1968) 353–362. [9] S. Bildea, D. Dutkay, G. Picioroaga, MRA super-wavelets, New York J. Math. 11 (2005) 1–19. [10] M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001) 71–92. [11] O. Bratteli, P.E.T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution analyses of scale N , Integral Equations Operator Theory 28 (1997) 382–443. [12] O. Bratteli, P. Jorgensen, Wavelets Through a Looking Glass, Birkhäuser, Boston/Basel/Berlin, 2002. [13] J. Courter, Construction of dilation-d wavelets, in: The Functional and Harmonic Analysis of Wavelets and Frames, in: Contemp. Math., vol. 247, Amer. Math. Soc., Providence, 1999, pp. 183–205. [14] J. D’Andrea, K.D. Merrill, J.A. Packer, Fractal wavelets of Dutkay–Jorgensen type for the Sierpinski gasket space, in: Frames and Operator Theory in Analysis and Signal Processing, in: Contemp. Math., vol. 451, Amer. Math. Soc., Providence, 2008, pp. 69–88. [15] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Lecture Notes, vol. 61, SIAM, 1992. [16] D. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, Rev. Math. Iberoamericana 22 (2006) 131–180. [17] D. Dutkay, P.E.T. Jorgensen, Fourier series on fractals: a parallel with wavelet theory, in: Radon Transforms, Geometry and Wavelets, in: Contemp. Math., vol. 464, Amer. Math. Soc., Providence, 2008, pp. 75–101 (English summary). [18] S. Jaffard, Y. Meyer, Les ondelettes, in: Harmonic Analysis and Partial Differential Equations, El Escorial, 1987, in: Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 182–192. [19] N.S. Larsen, I. Raeburn, From filters to wavelets via direct limits, in: Operator Theory, Operator Algebras and Applications, in: Contemp. Math., vol. 414, Amer. Math. Soc., Providence, 2006, pp. 35–40. [20] W. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990) 1898–1901. [21] G.W. Mackey, Induced representations of locally compact groups I, Ann. of Math. 55 (1952) 101–139. [22] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. 315 (1989) 69–87. [23] Y. Meyer, Wavelets and Operators, Cambridge Stud. Adv. Math., vol. 37, Cambridge University Press, Cambridge, England, 1992.
Journal of Functional Analysis 258 (2010) 4229–4250 www.elsevier.com/locate/jfa
Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces Guozheng Cheng, Kunyu Guo ∗ , Kai Wang School of Mathematical Sciences, Fudan University, Shanghai, 200433, China Received 8 November 2009; accepted 24 January 2010 Available online 2 February 2010 Communicated by N. Kalton
Abstract In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H(k) with complete Nevanlinna–Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H(k) ⊗ Cm have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems. © 2010 Elsevier Inc. All rights reserved. Keywords: Transitive algebra property; Reductive algebra property; Fiber dimension; Complete NP kernel
1. Introduction Throughout this paper, let H denote a complex separable Hilbert space, and B(H ) the algebra of all bounded linear operators on H . For a set A of operators in B(H ), we write Lat A for all invariant subspaces of A. Let N be all of the positive integers. For N ∈ N, let IN be the identity matrix acting on CN . Set A ⊗ IN = {A ⊗ IN : A ∈ A}, a set of diagonal operators acting on H ⊗ CN . A unital subalgebra A of B(H ) is called transitive if it is closed in weak operator topology and Lat A = {{0}, H }. * Corresponding author.
E-mail addresses:
[email protected] (G. Cheng),
[email protected] (K. Guo),
[email protected] (K. Wang). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.021
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The well-known transitive algebra problem is: if A is a transitive algebra on a Hilbert space H , is A equal to B(H )? This problem was firstly explicitly stated by Arveson [4], and it remains open up to now. The basic results in [4] led to research on the transitive algebra problem by a number of authors [1,3,7,16,20,23]. Recently, some interesting progress on the connection with other topics in operator theory has been made in [6]. A unital subalgebra A of B(H ) is called reductive if it is closed in weak operator topology and its invariant subspaces are reducing, that is, if M ∈ Lat A, then M⊥ ∈ Lat A. The reductive algebra problem, as stated in [22], is the question: if A is a reductive algebra, must A be self-adjoint? In short, the reductive algebra problem asks if a reductive algebra is a von Neumann algebra. It is a long-standing problem in operator theory. As observed in [23], an affirmative answer to the reductive algebra problem would imply an affirmative answer to the transitive algebra problem. The reductive algebra problem is also an unsolved problem, but a few partial results have been obtained [2,3,21,25,23,22]. In this paper, we study both the transitive algebra and reductive algebra problems on reproducing analytic Hilbert spaces. Let Ω be a domain (an open and connected subset) in Cn , and suppose H is a Hilbert space consisting of some analytic functions on Ω. We call H a reproducing analytic Hilbert space provided that the evaluation functional Eλ : f → f (λ) is continuous for each λ ∈ Ω. It follows that there exists a unique function kλ ∈ H such that f (λ) = f, kλ for each f ∈ H. We call the function kλ (z) = k(z, λ), defined on Ω × Ω, the reproducing kernel of H. Let H(k) denote a reproducing analytic Hilbert space on a domain Ω ⊆ Cn with a reproducing kernel k. An analytic function ϕ on Ω is a multiplier of H(k) if ϕf ∈ H(k) for every f ∈ H(k). We shall write M(k) for the algebra of all multipliers on H(k). The closed graph theorem implies that each ϕ ∈ M(k) induces a bounded multiplication operator Mϕ : f → ϕf on H(k). Throughout the paper, Mϕ ⊗ IN acting on H(k) ⊗ CN is written as Mϕ for simplicity. Write W (k) for the weakly closed algebra {Mϕ : ϕ ∈ M(k)}. A subspace M of H(k) ⊗ CN is called multiplier invariant if it is invariant for W (k). Write WM (k) for the restriction of W (k) to M. To continue we need the following definition. Definition 1.1. Let H(k) be a reproducing analytic Hilbert space on a domain Ω ⊆ Cn . If M is a linear manifold of H(k) ⊗ CN , the fiber dimension f .d. M of M, is defined by f .d. M = sup dim f (z): f ∈ M . z∈Ω
It is straightforward to check that f .d. M = f .d. M, where M is the closure of M in H(k) ⊗ CN . Using the same method as in the proof of Theorem 4.3 [18], one can prove that for almost every z ∈ Ω, f .d. M = dim f (z): f ∈ M . Fiber dimension is a useful invariant in both function theory and operator theory, and paid more and more attention by many authors [11,13,14,12,17,18]. For any element f in H⊗ CN , we shall write it as f = (f1 , . . . , fN ) with respect to the N decomposition H ⊗ CN = N i=1 H ⊗ ei for a given orthonormal basis e1 , . . . , eN of C .
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Definition 1.2. A linear manifold M of H ⊗ CN (N 2) is called a graph if there exist linear transformations T1 , . . . , TN −1 on a linear manifold D of H distinct from {0}, such that M = (x, T1 x, . . . , TN −1 x): x ∈ D . When M is closed, it is called a graph subspace. For a subalgebra A of B(H ), when a graph subspace M of H ⊗ CN is invariant for A ⊗ IN , then M is called an invariant graph subspace for A ⊗ IN and each Ti is called a graph transformation for A. Moreover, the corresponding graph transformations satisfy Ti A = ATi on D for each A ∈ A. If A is a transitive algebra, then it is easy to see that the domain D is dense in H since the closure of D is an invariant subspace for A. Our first result is the following, whose proof is given in Section 2. Theorem A. Let H(k) be a reproducing analytic Hilbert space and M be a subspace of H(k) ⊗ Cm . If A is a transitive algebra on M, then A = B(M) if and only if for each positive integer N 2, and each invariant graph subspace N for A ⊗ IN , f .d. M = f .d. N . Let Ω be a domain containing the origin in Cn . A function k : Ω × Ω → C is called a complete Nevanlinna–Pick kernel [18] (a complete NP kernel for short) if k0 ≡ 1 and 1 − 1/kλ (z) is positive definite on Ω × Ω. For example, let Bn be the open unit ball in Cn , and k(z, w) =
1 , 1 − z, w
z, w ∈ Bn ,
it is easy to check that k is a complete NP kernel. The analytic Hilbert space defined by this kernel is usually called Drury–Arveson space [8,5]. When n = 1, it coincides with the Hardy space on the unit disk D. While the Dirichlet kernel k : D × D → C, k(z, w) =
∞ (zw)n n=0
n+1
is also a complete NP kernel [26]. A subset A of B(H ) is said to have the transitive algebra property if the only transitive algebra containing A is B(H ). An operator T ∈ B(H ) is said to have the transitive algebra property if the singleton {T } has the transitive algebra property. Based on Theorem A, in Section 3 we will prove that W (k) acting on H(k) ⊗ Cm has the transitive algebra property when k is a complete NP kernel, and more generally, we prove that WM (k) acting on a multiplier invariant subspace M of H(k) ⊗ Cm also has the transitive algebra property. Theorem B. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. (1) If A is a transitive algebra on H(k) ⊗ Cm which contains W (k) then A = B H(k) ⊗ Cm .
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(2) If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a transitive algebra on M which contains WM (k), then A = B(M). Compared with transitive algebra property, we say that a subset A of B(H ) has the reductive algebra property if any reductive algebra containing A is self-adjoint. Theorem 8.7 [23] implies that if A has reductive algebra property, then it also has transitive algebra property. Based on Theorem B, in Section 4 we get the following stronger result. Theorem C. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. (1) If A is a reductive algebra on H(k) ⊗ Cm which contains W (k) then A is self-adjoint. (2) If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a reductive algebra on M which contains WM (k), then A is self-adjoint. 2. Transitive algebra problem via fiber dimension This section is mainly devoted to prove Theorem A. Some additional terminology is needed. Definition 2.1. A linear transformation T is said to have a compression spectrum if there exists λ ∈ C such that the range of T − λ is not dense in H , that is, if D is the domain of T , then closure of {(T − λ)x: x ∈ D} is not H . Now we present the Arveson’s lemma, which can be inferred from [4] and plays an important role in the proof of Theorem A. Lemma 2.2. (See [23, Lemma 8.15].) If A is a transitive algebra such that every graph transformation for A has a compression spectrum, then we have A = B(H ). Based on the above lemma, we turn to the proof of Theorem A. Theorem A. Let H(k) be a reproducing analytic Hilbert space and M be a subspace of H(k) ⊗ Cm . If A is a transitive algebra on M, then A = B(M) if and only if for each positive integer N 2, and each invariant graph subspace N for A ⊗ IN , f .d. M = f .d. N . Proof. Let N = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ M be an invariant graph subspace for A ⊗ IN . If A = B(M), then D = M since D is invariant for B(M). Define T : M → M ⊗ CN −1 by Tf = (T1 f, . . . , TN −1 f ),
f ∈ M,
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then the graph of T is N . The closed graph theorem implies that T is bounded, which implies that each Ti is bounded. Applying the fact Ti A = ATi for each A ∈ A = B(M) shows that each Ti is a multiple of the identity operator. In other words, N = (f, λ1 f, . . . , λN −1 f ): f ∈ M for some λ1 , . . . , λN −1 ∈ C. By a simple verification, one has f .d. M = f .d. N . Conversely, we assume that f .d. N = f .d. M for each invariant graph subspace N , and we write d = f .d. M. Since D is dense in M, we have f .d. D = f .d. N = d. This means for almost every z ∈ Ω, dim f (z): f ∈ D = dim f (z), (T1 f )(z), . . . , (TN −1 f )(z) : f ∈ D = d.
(2.1)
Now fix such a z0 . Since dim{f (z0 ): f ∈ D} = d, then there exist f1 , . . . , fd ∈ D, such that these vectors f1 (z0 ), . . . , fd (z0 ) are linearly independent in Cm and for each f ∈ D, f (z0 ) =
d
λj (z0 , f )fj (z0 )
(2.2)
j =1
for some λ1 (z0 , f ), . . . , λd (z0 , f ) ∈ C. The d vectors f1 (z0 ), (T1 f1 )(z0 ), . . . , (TN −1 f1 )(z0 ) , .. . fd (z0 ), (T1 fd )(z0 ), . . . , (TN −1 fd )(z0 ) are linearly independent in CmN . By (2.1), the above d vectors form a basis for the space f (z0 ), (T1 f )(z0 ), . . . , (TN −1 f )(z0 ) : f ∈ D . Combining this with the formula (2.2), one has (Ti f )(z0 ) =
d j =1
λj (z0 , f )(Ti fj )(z0 ),
1 i N − 1.
(2.3)
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Set E = {f (z0 ): f ∈ D} and Fi = {(Ti f )(z0 ): f ∈ D}, then both E and Fi are subspaces in Cm . Furthermore, since D is dense in M, we have E = {f (z0 ): f ∈ M} and hence Fi ⊆ E. For 1 i N − 1, define Ai : E → E by Ai f (z0 ) = (Ti f )(z0 ),
f ∈ D.
(2.4)
Using formulas (2.2) and (2.3), each Ai is a well-defined linear transformation on E. Next we will show that Ti has a compression spectrum. Since each Ai is a linear transformation on the finite dimensional vector space E, we can take an eigenvalue μi of Ai , then μi is a compression spectrum of Ti . Otherwise, {Ti f − μi f : f ∈ D} is dense in M and hence {(Ti f − μi f )(z0 ): f ∈ D} is dense in E. From the fact that E is finite dimensional, we see that
(Ti f − μi f )(z0 ): f ∈ D = E.
By (2.4) the range of Ai − μi I is E, contradicting that μi is an eigenvalue of Ai . It follows that every graph transformation for A has a compression spectrum, hence by Lemma 2.2, A = B(M), as desired.
2
The following corollary is a consequence of Theorem A. Corollary 2.3. Let H(k) be a reproducing analytic Hilbert space. If A is a transitive algebra on H(k) ⊗ Cm , then A = B(H(k) ⊗ Cm ) if and only if for each positive integer N 2, and each invariant graph subspace M for A ⊗ IN , f .d. M = m. In the remainder of this section, we establish two properties of the fiber dimension. The following proposition shows that the fiber dimension can reflect the structure of the linear manifolds. Proposition 2.4. Let H(k) be a reproducing analytic Hilbert space over Ω. If a linear manifold M of H(k) ⊗ CN has f .d. M = 1, then M is a graph. Proof. Without loss of generality, take f = (f1 , . . . , fN ) ∈ M with f1 = 0. Let Z(f1 ) be the zeros of f1 in Ω. Since f .d. M = 1, then for each g = (g1 , . . . , gN ) ∈ M and z ∈ Ω \ Z(f1 ), there exists λ(z, g) ∈ C such that g(z) = λ(z, g)f (z). In this case, λ(z, g) = z ∈ Ω,
g1 (z) f1 (z) .
Since the measure of Z(f1 ) is zero, one has that for almost every
gi (z) = λ(z, g)fi (z) =
fi (z) g1 (z), f1 (z)
i = 2, . . . , N.
(2.5)
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Let D be the set of the first entries of all elements in M. Consequently, by (2.5) for 2 i N , define Ti−1 : D → H(k) by Ti−1 g1 = gi . Then each Ti−1 is a well-defined linear transform. Thus M can be written as M = (g1 , T1 g1 , . . . , TN −1 g1 ): g1 ∈ D . This finishes the proof.
2
We end this section with the following proposition, which may be useful to calculate fiber dimension. Proposition 2.5. Let H(k) be a reproducing analytic Hilbert space. If M1 , M2 are multiplier invariant subspaces of H(k) ⊗ CN with M2 ⊆ M1 and dim M1 /M2 < ∞, then f .d. M1 = f .d. M2 . Proof. It is clear that f .d. M2 f .d. M1 , so we need only to prove that f .d. M1 f .d. M2 . Recall that in this paper we assume that M(k) is nontrivial, so we can take a nonconstant multiplier ϕ and define Sϕ on R = M1 M2 by Sϕ f = PR (ϕf ),
f ∈ R,
here PR is the projection from M1 onto R. Since M2 is a multiplier invariant subspace, it is easy to check that for each positive integer m, (Sϕ )m = Sϕ m , and hence the condition dim R = dim M1 /M2 < ∞ implies there exists a nonzero polynomial in ϕ, denoted by ψ, such that Sψ = 0, which means that ψR ⊆ M2 , and hence ψM1 ⊆ M2 . We have f .d.(ψM1 ) f .d. M2 . From the definition of fiber dimension, we see f .d.(ψM1 ) = f .d. M1 . Combining this fact and (2.6), one has f .d. M1 f .d. M2 , as desired.
2
(2.6)
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3. Transitive algebras on complete NP kernel spaces The aim of this section is to prove Theorem B, the transitive algebra property of both W (k) and WM (k) when k is a complete NP kernel. From Theorem A we see that the transitive algebra problem on H(k) ⊗ Cm can be reduced to the calculation of fiber dimension. So we firstly investigate further properties of the fiber dimension. In this paper, for a subspace M ⊆ H(k) ⊗ CN , let PM denote the projection onto M. Now we present the “occupy” invariant, which first was introduced by Fang [13] for the multiplier invariant subspaces of vector valued Drury–Arveson space. Definition 3.1 is a natural extension of Fang’s definition. Definition 3.1. Let H(k) be a reproducing analytic Hilbert space. For a linear manifold M ⊆ H(k) ⊗ CN , define the “occupy” invariant of M, denoted by lM , to be the maximal dimension of a subspace E of CN with the following property: there exist an orthonormal basis e1 , . . . , el (l = lM ) of E and h1 , . . . , hl ∈ M such that PH(k)⊗E hi ( = 0) ∈ H(k) ⊗ ei ,
i = 1, . . . , l.
When E has the above property we say that M occupies H(k) ⊗ E in H(k) ⊗ CN . There are many important properties of the “occupy” invariant, which were originally established by Fang for the vector valued Drury–Arveson space [13, Lemmas 22 and 23]. The conclusion of Lemma 22 [13] was obtained in a more general setting by Gleason, Richter and Sundberg (Lemma 3.2 [17]). Proposition 3.2. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ CN and it occupies H(k) ⊗ E for some E ⊆ CN , then for any nonzero vector e ∈ E, there exists an element h ∈ M such that PH(k)⊗E h( = 0) ∈ H(k) ⊗ e. Proposition 3.3. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. For any multiplier invariant subspace M of H(k) ⊗ CN , we have lM = f .d. M. One can prove the above two propositions by the same methods as in [13], which we omit here. We point out that Fang’s proof depends on the fact that any multiplier invariant subspace of vector valued Drury–Arveson space is generated by elements having multiplier entries. For complete NP kernel spaces, this is true. Lemma 3.4. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ CN , then the subset {(f1 , . . . , fN ): fi ∈ M(k)} ∩ M is dense in M. We say that f = (f1 , . . . , fN ) ∈ H(k) ⊗ CN has multiplier entries if each fi ∈ M(k) with respect to the decomposition H(k) ⊗ CN =
N i=1
H(k) ⊗ ei
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for some orthonormal basis e1 , . . . , eN of CN . It is easy to see that the definition is independent of the choice of the orthonormal basis of CN . Proof. Recall that functions in H(k) are defined on a domain Ω ⊆ Cn . Theorem 0.7 [19] implies that there exist an auxiliary Hilbert space F and an inner multiplier Φ : Ω → L(F , CN ) (all bounded linear operators from F to CN ) such that ∗ PM = MΦ MΦ ,
M = Φ H(k) ⊗ F ,
here (MΦ f )(λ) = Φ(λ)f (λ) for f ∈ H(k) ⊗ F , λ ∈ Ω. For ϕ ∈ M(k), x ∈ F , it is easy to check that MΦ (ϕ ⊗ x) has multiplier entries. Using the same argument as in the proof of Lemma 2.2(a) [18], one has that kλ ∈ M(k),
λ ∈ Ω.
Therefore, for e ∈ CN and λ ∈ Ω, ∗ M Φ MΦ (kλ ⊗ e) = MΦ kλ ⊗ Φ(λ)∗ e has multiplier entries. Then the conclusion follows from the fact that {kλ : λ ∈ Ω} is dense in H(k). 2 Remark 3.5. By the above lemma and the same argument in [13], we can get a stronger version of Proposition 3.2. That is, under the condition of Proposition 3.2, for any nonzero vector e ∈ E, there exists an element f ∈ M such that PH(k)⊗E f ( = 0) ∈ H(k) ⊗ e, here f has multiplier entries. Indeed, from Proposition 3.2, there exists h ∈ M such that PH(k)⊗E h( = 0) ∈ H(k) ⊗ e. By Lemma 3.4, the multiplier invariant subspace generated by h contains a nonzero function f which has multiplier entries. This implies that PH(k)⊗E f ( = 0) ∈ H(k) ⊗ e, as desired. In fact, this result has been obtained in [17]. Let H(k) be a reproducing analytic Hilbert space. A linear manifold M of H(k)⊗CN (N 2) is called an m-graph (m N ) if there exists an orthonormal basis e1 , . . . , eN of CN such that with respect to this basis, M has the form M = (f1 , . . . , fm , T1 f, . . . , TN −m f ): f = (f1 , . . . , fm ) ∈ D where D is the linear manifold of the first m entries of elements in M with f .d. D = m, and each Ti is a linear transform from D to H(k). M is called an m-graph subspace if it is closed. We need the following extension of Proposition 2.4 for further discussion, which is of independent interest. Firstly, we point out that the sufficiency of the following theorem is also observed by Gleason, Richter and Sundberg in Lemma 3.3 [17]. Theorem 3.6. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. For a multiplier invariant subspace M ⊆ H(k) ⊗ CN , then M is an m-graph subspace if and only if f .d. M = m.
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Proof. Suppose f .d. M = m, then the “occupy” invariant lM = m by Proposition 3.3. This means that there exists a subspace E ⊆ CN with an orthonormal basis e1 , . . . , em , and h1 , . . . , hm ∈ M such that PH(k)⊗E hi ( = 0) ∈ H(k) ⊗ ei ,
i = 1, . . . , m.
N N Extend {ei }m i=1 to an orthonormal basis {ei }i=1 of C . With respect to this basis, for any f ∈ H(k) ⊗ CN , we write f = (f1 , . . . , fN ). In particular, h1 , . . . , hm can be written as
h1 = h11 , 0, . . . , 0, h1m+1 , . . . , h1N , .. . m m hm = 0, . . . , 0, hm m , hm+1 , . . . , hN , where hii = 0 since PH(k)⊗E hi = 0 for 1 i m. i 1 Let Z(hii ) be the zeros of hii in Ω. Then for z ∈ Ω \ m i=1 Z(hi ), the m vectors h (z), . . . , m N h (z) are linearly independent in C . From the equality f .d. M = m, these vectors consist of a basis of {f (z): f ∈ M}. Then for each f = (f1 , . . . , fN ) ∈ M, there exist λ1 (z, f ), . . . , λm (z, f ) ∈ C such that f (z) = λ1 (z, f )h1 (z) + · · · + λm (z, f )hm (z). Obviously, λi (z, f ) =
fi (z) , hii (z)
i = 1, . . . , m.
It follows that fj (z) =
h1j (z) h11 (z)
f1 (z) + · · · +
hm j (z) hm m (z)
fm (z),
j = m + 1, . . . , N.
Note that the above identities hold for almost every z ∈ Ω since the measure of zero. For (f1 , . . . , fN ) ∈ M, define πm (f1 , . . . , fm , fm+1 , . . . , fN ) = (f1 , . . . , fm ),
(3.1) m
i i=1 Z(hi )
is
(3.2)
and let D = πm (M), the set of first m entries of elements in M. Consequently, by (3.1), for m + 1 j N , define Tj −m : D → H(k) by Tj −m (f1 , . . . , fm ) = fj . Then each Tj −m is a well-defined linear transform on D. Since for almost every z ∈ Ω, the m vectors πm h1 (z), . . . , πm hm (z)
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are linearly independent, and hence f .d. D = m. The above reasoning shows that M is an mgraph subspace. Conversely, suppose M = (f1 , . . . , fm , T1 f, . . . , TN −m f ): f = (f1 , . . . , fm ) ∈ D is an m-graph subspace with respect to an orthonormal basis e1 , . . . , eN of CN . By Proposition 3.3 it is enough to show that the “occupy” invariant lM = m. From the definition of m-graph subspace, f .d. D = m, and hence lM = f .d. M f .d. D m. It remains to show that lM m. We prove that by contradiction. Suppose M occupies H(k) ⊗ F in H(k) ⊗ CN with dim F = lM > m, then there exists an orthonormal basis e1 , . . . , el (l = lM ) of F and h1 , . . . , hl ∈ M such that PH(k)⊗F hi ( = 0) ∈ H(k) ⊗ ei ,
i = 1, . . . , l,
(3.3)
here by Remark 3.5 we can assume that each hi has multiplier entries. Write hi = hi1 , . . . , him , T1 hi , . . . , TN −m hi , i = 1, . . . , l,
where hi = (hi1 , . . . , him ) ∈ D. Let d = f .d.( li=1 {hi }), then d m and there exists z ∈ Ω such
that dim( li=1 {hi (z)}) = d. Without loss of generality, we assume that h1 (z), . . . , hd (z) are lin
early independent in Cm , which implies that f .d.( di=1 {hi }) = d. Furthermore, we may assume that the determinant of the d × d matrix Θ = (hij )di,j =1 is not zero at z. Note that the determinant det(Θ) is a nonzero multiplier on H(k). Using Cramer’s rule shows that there exist not all zero multipliers g1 , . . . , gl satisfying the following system of equations ⎧ d+1 d l 1 ⎪ ⎨ g1 h1 + · · · + gd h1 + gd+1 h1 + · · · + gl h1 = 0, .. . ⎪ ⎩ + · · · + gl hld = 0. g1 h1d + · · · + gd hdd + gd+1 hd+1 d Hence g1 h1 + · · · + gl hl = ( 0, . . . , 0, r1 , . . . , rm−d ). d
Moreover, since f .d.
d i=1
d hi , g1 h1 + · · · + gl hl = f .d. hi = d, i=1
we have that for almost every z ∈ Ω, the rank of the (d + 1) × m matrix ⎞ h11 (z) · · · h1d (z) h1d+1 (z) · · · h1m (z) ⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ⎟ ⎠ ⎝ d h1 (z) · · · hdd (z) hdd+1 (z) · · · hdm (z) · · · rm−d (z) 0 ··· 0 r1 (z) ⎛
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does not exceed d. This implies that each ri = 0 since det(Θ) is a nonzero analytic function, and hence g1 h1 + · · · + gl hl = 0.
(3.4)
Combining (3.4) and the fact that for 1 i N − m, Ti Mg = Mg Ti on D for each g ∈ M(k), we have g1 h1 + · · · + gl hl = 0. Therefore, by (3.3) and the fact that e1 , . . . , el are orthogonal in F , for each 1 i l, 0 = PH(k)⊗ei g1 h1 + · · · + gl hl = PH(k)⊗ei PH(k)⊗F g1 h1 + · · · + gl hl = gi PH(k)⊗ei hi , and hence each gi = 0. This leads to a contradiction since g1 , . . . , gl are not all zero multipliers. Therefore, we have lM m. Based on the above discussion, it follows that f .d. M = lM = m. 2 The following corollary is a direct consequence of the above theorem and is used in the proof of Theorem B. Corollary 3.7. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If a graph subspace M = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ H(k) ⊗ Cm is multiplier invariant, and f .d. D = m, then f .d. M = m. Theorem B(1). Let H(k) be a reproducing analytic Hilbert space with complete NP kernel k. If A is a transitive algebra on H(k) ⊗ Cm which contains W (k), then A = B H(k) ⊗ Cm . Proof. We will combine Corollaries 2.3 and 3.7 to complete the proof. Let M = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ H(k) ⊗ Cm be an invariant graph subspace for A ⊗ IN . It is easy to see that D is dense in H(k) ⊗ Cm since it is invariant for the transitive algebra A. Therefore, f .d. D = f .d. H(k) ⊗ Cm = m. Since A contains W (k), M is multiplier invariant and hence by Corollary 3.7, f .d. M = m.
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The required identity A = B H(k) ⊗ Cm follows from Corollary 2.3.
2
Remark 3.8. If H(k) is a reproducing analytic Hilbert space on the open unit ball Bn with a complete NP kernel k and the coordinate functions z1 , . . . , zn are multipliers of H(k). Suppose A is a transitive algebra on H(k) ⊗ Cm which contains {Mzi : 1 i n}, then Lemma 4.1 [18] implies that A contains W (k). Therefore, by Theorem B(1), we have that A = B(H(k) ⊗ Cm ). As pointed in the Introduction, both the Hardy kernel and Dirichlet kernel are complete NP kernels. Therefore, as an immediate consequence, both the unilateral shift and the Dirichlet shift of finite multiplicity have the transitive algebra property. These results were obtained by Arveson [4], Radjavi, Rosenthal [23] and Richter [24]. Now we finish the proof of Theorem B(2). Theorem B(2). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a transitive algebra on M which contains WM (k), then A = B(M). Proof. By Theorem A, it is sufficient to prove that for each positive integer N 2, and each invariant graph subspace N = (f, T1 f, . . . , TN −1 f ): f ∈ D ⊆ M for A ⊗ IN , f .d. M = f .d. N . Assume f .d. M = d(d m), then by Theorem 3.6, M is a d-graph subspace, that is, there exists a linear manifold D1 with f .d. D1 = d such that M = (g1 , . . . , gd , S1 g, . . . , Sm−d g): g = (g1 , . . . , gd ) ∈ D1 with respect to an orthonormal basis e1 , . . . , em of Cm , where each Sj is a linear transform from D1 to H(k). We turn to the invariant graph subspace N . Then by representation of M, the linear manifold D can be written as D = g1 , . . . , gd , S1 g , . . . , Sm−d g : g = g1 , . . . , gd ∈ D1 with D1 ⊆ D1 . Moreover, since D is dense in M, one has that D1 is dense in D1 , and hence f .d. D1 = d. Because both S1 , . . . , Sm−d and T1 , . . . , TN −1 are all linear transforms, there exist linear transforms T1 , . . . , TmN −d from D1 to H(k) such that N=
g1 , . . . , gd , T1 g , . . . , TmN −d g : g = g1 , . . . , gd ∈ D1 .
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Therefore, N is a d-graph subspace. Since A contains WM (k), N is multiplier invariant. Hence by Theorem 3.6, f .d. M = d = f .d. N , as required.
2
4. Reductive algebras on complete NP kernel spaces In this section we mainly consider the reductive algebra property of analytic multiplication operators on complete NP kernel spaces. Firstly, fix some notation. For a subset A of B(H ), let A∗ = {A∗ : A ∈ A}, and A = {B ∈ B(H ): BA = AB, A ∈ A}, the commutant of A. Proposition 4.1. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. Suppose A is a reductive algebra on H(k) which contains W (k), then A = B(H(k)). In particular, A is self-adjoint. Proof. Suppose M is an invariant subspace for A, then it is reducing because A is a reductive algebra. Hence one has PM A = APM
for A ∈ A.
(4.1)
As observed in the proof of Lemma 3.4, M(k) is dense in H(k). So by a routine verification, one has W (k) = W (k).
(4.2)
By (4.2) and the hypothesis that A contains W (k), the PM in (4.1) must be equal to Mg for some multiplier g in M(k). PM = Mg forces g = 0 or 1, which implies M = {0} or H(k). In other words, A has no nontrivial invariant subspace, hence A is a transitive algebra. The required identity A = B H(k) follows from Theorem B(1).
2
The above proposition is a special case of Theorem C(1). In general, in the case of the vector valued H(k) ⊗ Cm (m > 1), a reductive algebra which contains W (k) is properly contained in B(H(k) ⊗ Cm ). An example is the diagonal von Neumann algebra generated by W (k). To continue we need the following lemma. Lemma 4.2. (See [23, Theorem 9.11].) If A is a reductive algebra on a Hilbert space H , and if there exists a collection of invariant subspaces {Mi }ni=1 of A such that H = ni=1 Mi and A|Mi = B(Mi ) for each i, then A is self-adjoint. Now we complete the proof of Theorem C(1).
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Theorem C(1). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If A is a reductive algebra on H(k) ⊗ Cm which contains W (k), then A is self-adjoint. Proof. Suppose A is a reductive algebra on H(k) ⊗ Cm which contains W (k). Firstly we have the following claim. Claim. The von Neumann algebra V = {A, A∗ } is finite dimensional. To see this, for each projection P ∈ V, one has that P M ϕ = Mϕ P ,
ϕ ∈ M(k).
(4.3)
Write PM = (Aij )m×m for some Aij ∈ B(H(k)). Then by (4.3), Aij Mϕ = Mϕ Aij ,
ϕ ∈ M(k).
Consequently, Aij = Mϕij for some ϕij ∈ M(k) by (4.2). Since P is a projection, this implies that Mϕij = Mϕ∗j i . Noticing k0 = 1, and hence for any λ ∈ Ω, Mϕij 1, kλ = Mϕ∗j i 1, kλ . A direct calculus shows that ϕij (λ) = ϕj i (0), which implies that each ϕij is a constant. Combining this observation and the fact that any von Neumann algebra is generated by projections in it, we have that V is finite dimensional, as desired. Based on the Claim, there exist finitely many mutually orthogonal minimal projections P1 , . . . , Ps ∈ V such that P1 + · · · + Ps = I, here I is the identity operator on H(k) ⊗ Cm . Let Mi be the range of minimal projection Pi for i = 1, . . . , s. Then each Mi is a minimal reducing subspace of A, thus the operator algebra A|Mi acting on Mi is transitive. Therefore, by Theorem B(2), A|Mi = B(Mi ), and the desired conclusion follows from Lemma 4.2.
2
Let H(k) be a reproducing analytic Hilbert space on the open unit ball Bn with a complete NP kernel k such that the coordinate functions z1 , . . . , zn are multipliers. If A is a reductive algebra on H(k) ⊗ Cm which contains {Mzi : 1 i n}, as pointed in Remark 3.8, then A contains W (k) and hence A is self-adjoint by Theorem C(1). As an immediate consequence, the unilateral shift of finite multiplicity has the reductive algebra property. This result was obtained by Nordgren, Rosenthal [21] and Ansari [2] respectively. In what follows we will consider the reductive algebra problem on a multiplier invariant subspace M of H(k) ⊗ Cm . We need to develop some new techniques. In the remainder of this section, let e1 , . . . , em be any given orthonormal basis of Cm . For a subset E of a Hilbert space H ,
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then the span of E, denoted by E, is the intersection of all subspaces containing E. For any subspace M ⊆ H(k) ⊗ Cm , set M(λ) = {f (λ): f ∈ M}, λ ∈ Ω. We begin with the following lemmas. Lemma 4.3. Let H(k) be a reproducing analytic Hilbert space over a domain Ω ⊆ Cn . For any subspace M ⊆ H(k) ⊗ Cm , one has that m PM (kλ ⊗ ei )(λ) . M(λ) = i=1
Proof. It is clear that M(λ) ⊇ m i=1 {PM (kλ ⊗ ei )(λ)}, thus we need only to prove the opposite inclusion. To this end we write f ∈ H(k) ⊗ Cm as f = (f1 , . . . , fm ) with respect to the basis e1 , . . . , em , then f (λ) = f, kλ ⊗ e1 , . . . , f, kλ ⊗ em . Therefore, for any f ∈ M, one also has f (λ) = f, PM (kλ ⊗ e1 ) , . . . , f, PM (kλ ⊗ em ) .
(4.4)
Moreover, for each f ∈ M, it can be expressed as f =f +
m
ci PM (kλ ⊗ ei )
i=1
with ci ∈ C and f ⊥PM (kλ ⊗ ei ), i = 1, . . . , m. By (4.4), f (λ) = 0, and hence f (λ) =
m
ci PM (kλ ⊗ ei )(λ),
i=1
which means that f (λ) ∈
m
i=1 {PM (kλ
⊗ ei )(λ)}, as desired.
2
Lemma 4.4. Let H(k) be a reproducing analytic Hilbert space and M be any subspace of H(k) ⊗ Cm . For each subset {ei1 , . . . , eid } ⊆ {e1 , . . . , em } and λ ∈ Ω, d vectors PM (kλ ⊗ ei1 ), . . . , PM (kλ ⊗ eid ) are linearly independent in M if and only if the d vectors PM (kλ ⊗ ei1 )(λ), . . . , PM (kλ ⊗ eid )(λ) are linearly independent in Cm .
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d Proof. In fact, if j =1 cj PM (kλ ⊗ eij ) = 0 for some c1 , . . . , cd ∈ C, then obviously d d j =1 cj PM (kλ ⊗ eij )(λ) = 0. Conversely, we assume that j =1 cj PM (kλ ⊗ eij )(λ) = 0, then for any 1 i m,
d
cj PM (kλ ⊗ eij ), kλ ⊗ ei = 0,
j =1
and hence ! d !2 d d ! ! ! ! c P (k ⊗ e ) = c P (k ⊗ e ), cs PM (kλ ⊗ eij ) ! j M λ ij ! j M λ ij ! ! j =1
=
j =1
d j =1
s=1
cj PM (kλ ⊗ eij ),
d
cs (kλ ⊗ ej )
s=1
= 0, which implies that holds. 2
d
j =1 cj PM (kλ
⊗ eij ) = 0. Based on the above discussion, the lemma
Next we present a key proposition, which is of independent interest. For λ ∈ Ω, let Mλ (k) = {ϕ ∈ M(k): ϕ(λ) = 0}. If M is a multiplier invariant subspace of H(k) ⊗ Cm , let Mλ = Mλ (k)M. It is well known that dim(M Mλ ) is an important invariant in operator theory [13, 10,9,15,17]. Proposition 4.5. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then for every λ ∈ Ω with f .d. M = dim M(λ), m PM (kλ ⊗ ei ) = M Mλ . i=1
In fact, the above result has been observed by Gleason, Richter and Sundberg [17]. Now we present a different proof here. Proof. Firstly, it is easy to verify that "
# ker PM Mϕ∗ #M = M Mλ .
ϕ∈Mλ (k)
From the above identity, it is easy to check that the inclusion “⊆” holds in Proposition 4.5. Next we show the opposite inclusion. The following proof is essentially the same as that of the necessity of Theorem 3.6.
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Assume f .d. M = d (d m). By Lemma 4.3 and the property of fiber dimension, for almost every λ ∈ Ω, d = f .d. M = dim M(λ) = dim
m
PM (kλ ⊗ ei )(λ) .
i=1
Fix such a λ. Write f ∈ H(k) ⊗ Cm as f = (f1 , . . . , fm ) with respect to the basis e1 , . . . , em , and write i PM (kλ ⊗ ei ) = g1i , . . . , gm , i = 1, . . . , m. (4.5) From the proof of Lemma 3.4, we see that each gji ∈ M(k). Without loss of generality, we assume that the vectors PM (kλ ⊗ e1 )(λ), . . . , PM (kλ ⊗ ed )(λ) are linearly independent in Cm . Furthermore, we assume the determinant of the d × d matrix Θ = (gij )di,j =1 , denoted by det(Θ), is not zero at λ. Note that det(Θ) is a nonzero multiplier. Now for any g = (g1 , . . . , gm ) ∈ M Mλ , using Cramer’s rule shows that there exist h1 , . . . , hm ∈ H(k) satisfying the following system of equations ⎧ d+1 d m 1 ⎪ ⎨ h1 g1 + · · · + hd g1 + hd+1 g1 + · · · + hm g1 = det(Θ)g1 , .. . ⎪ ⎩ h1 gd1 + · · · + hd gdd + hd+1 gdd+1 + · · · + hm gdm = det(Θ)gd . Hence by (4.5) h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) − det(Θ)g = ( 0, . . . , 0, r1 , . . . , rm−d ). d
Set h = h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) − det(Θ)g. Since g ∈ M, then by Lemmas 4.3 and 4.4 d d PM (kλ ⊗ ei ), h = f .d. PM (kλ ⊗ ei ) = d. f .d. i=1
i=1
Thus for almost every z ∈ Ω, the rank of the (d + 1) × m matrix 1 (z) · · · g 1 (z) ⎞ g11 (z) · · · gd1 (z) gd+1 m ⎜· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ⎟ ⎝ d d (z) · · · d (z) ⎠ gm g1 (z) · · · gdd (z) gd+1 · · · rm−d (z) 0 ··· 0 r1 (z)
⎛
does not exceed d. This implies that each ri = 0 since det(Θ) is a nonzero multiplier function, and hence h1 PM (kλ ⊗ e1 ) + · · · + hm PM (kλ ⊗ em ) = det(Θ)g.
(4.6)
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Write hi PM (kλ ⊗ ei ) = hi − hi (λ) PM (kλ ⊗ ei ) + hi (λ)PM (kλ ⊗ ei ), det(Θ)g = det(Θ) − det(Θ)(λ) g + det(Θ)(λ)g. It is easy to see that (det(Θ) − det(Θ)(λ))g ∈ Mλ . Since M(k) is dense in H(k) and each PM (kλ ⊗ ei ) has multiplier entries, it is easy to check that (hi − hi (λ))PM (kλ ⊗ ei ) ∈ Mλ for i = 1, . . . , m. Therefore, we apply the projection PMMλ to two sides of (4.6) and have m
hi (λ)PM (kλ ⊗ ei ) = det(Θ)(λ)g,
i=1
which means that g ∈
m
i=1 {PM (kλ
⊗ ei )} since det(Θ)(λ) = 0. This completes the proof.
2
The above proposition means that for every λ ∈ Ω with f .d. M = dim M(λ), dim
m PM (kλ ⊗ ei ) = dim(M Mλ ). i=1
Combining this observation as well as Lemmas 4.3 and 4.4, we immediately have the following corollary. Corollary 4.6. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then for every λ ∈ Ω with f .d. M = dim M(λ), dim(M Mλ ) = f .d. M. Corollary 4.7. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M1 and M2 are mutually orthogonal invariant subspaces of H(k) ⊗ Cm for W (k), then f .d.(M1 ⊕ M2 ) = f .d. M1 + f .d. M2 . Proof. Let M = M1 ⊕ M2 . By Lemma 4.3 and the definition of fiber dimension, we can take λ ∈ Ω such that for j = 1, 2, the following hold: m f .d. M = dim M(λ) = dim PM (kλ ⊗ ei )(λ) i=1
and f .d. Mj = dim Mj (λ) = dim
m PMj (kλ ⊗ ei )(λ) . i=1
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Thus combining Lemma 4.4 and Proposition 4.5, one has f .d. M = dim
m PM (kλ ⊗ ei ) = dim(M Mλ )
(4.7)
i=1
and f .d. Mj = dim
m PMj (kλ ⊗ ei ) = dim(Mj Mj λ ).
(4.8)
i=1
By the fact that both M1 and M2 are invariant subspaces for W (k), it follows that for each ϕ ∈ M(k), M ϕM = (M1 ϕM1 ) ⊕ (M2 ϕM2 ) and hence M Mλ = (M1 M1λ ) ⊕ (M2 M2λ ). Combining this fact with (4.7) and (4.8), we have f .d. M = f .d. M1 + f .d. M2 .
2
In the proof of Theorem C(1), we use the fact that the commutant of the von Neumann algebra generated by W (k) is finite dimensional. In what follows we generalize this fact to multiplier invariant subspaces, which plays an important role in the proof of Theorem C(2). Recall that for a multiplier invariant subspace M of H(k) ⊗ Cm , WM (k) = {Mϕ |M : ϕ ∈ M(k)}. Proposition 4.8. Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , then the von Neumann algebra W = WM (k), WM (k)∗ is finite dimensional. Proof. By Lemma 4.3, we can take λ ∈ Ω such that f .d. M = dim M(λ) = dim
m PM (kλ ⊗ ei )(λ) .
(4.9)
i=1
Let E = m i=1 {PM (kλ ⊗ ei )}, then it is a finite dimensional subspace of M which is reducing for W by Proposition 4.5. Now we define a C ∗ -homomorphism τ : W → W|E
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by τ (A) = A|E . Now we show that it is injective. Let A ∈ ker τ , equivalently, E ⊆ ker A. Since A commutes with both Mϕ |M and PM Mϕ∗ |M , M ker A and ker A are invariant for WM (k), hence they are also invariant for W (k). Therefore, M has a decomposition as M = (M ker A) ⊕ ker A, then by Corollary 4.7, f .d. M = f .d.(M ker A) + f .d.(ker A).
(4.10)
m f .d.(ker A) f .d. E dim PM (kλ ⊗ ei )(λ) = f .d. M,
(4.11)
Meanwhile,
i=1
where the last equality follows from (4.9). Combining (4.10) and (4.11), one has that f .d.(M ker A) = 0, and hence M ker A = {0}. It follows that ker A = M, that is A = 0, as desired. The above argument shows that τ is a C ∗ -isomorphism, and hence W is finite dimensional. 2 Now we give the proof of Theorem C(2). Theorem C(2). Let H(k) be a reproducing analytic Hilbert space with a complete NP kernel k. If M is a multiplier invariant subspace of H(k) ⊗ Cm , and A is a reductive algebra on M which contains WM (k), then A is self-adjoint. Proof. Let V = {A, A∗ } . Since A contains WM (k), then the von Neumann algebra V ⊆ W, which is given in Proposition 4.8, and hence V is finite dimensional by Proposition 4.8. Based on this fact, the conclusion follows from the same argument as in the proof of Theorem C(1). 2 Acknowledgments The authors thank the referee for helpful suggestions which make this paper more readable. This work is partially supported by NKBRPC (2006CB805905) and NSFC (10525106), and Foundation of Fudan University (EYH1411039) and Laboratory of Mathematics for Nonlinear Science, Fudan University. References [1] M. Ansari, Transitive algebra containing triangular operator matrics, J. Operator Theory 14 (1985) 173–180. [2] M. Ansari, Reductive algebras containing a direct sum of the unilateral shift and a certain other operators are self adjoint, Proc. Amer. Math. Soc. 93 (1985) 284–286. [3] M. Ansari, S. Richter, A strong transitive algebra property of operators, J. Operator Theory 22 (1989) 267–276. [4] W. Arveson, A density theorem for operator algebras, Duke Math. J. 24 (1967) 635–647.
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[5] W. Arveson, Subalgebras of C ∗ algebras III: Multivariable operator theory, Acta Math. 181 (1998) 159–228. [6] H. Bercovici, R. Douglas, C. Foias, C. Pearcy, Confluent operators algebras and the closability property, arXiv:0908. 0729v2. [7] R. Douglas, C. Pearcy, Hyperinvariant subspaces and transitive algebras, Michigan Math. J. 19 (1972) 1–12. [8] S. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978) 300–304. [9] X. Fang, Hilbert polynomials and Arveson’s curvature invariant, J. Funct. Anal. 198 (2003) 445–464. [10] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2004) 411–437. [11] X. Fang, Invariant subspaces of the Dirichlet space and commutative algebra, J. Reine Angew. Math. 569 (2004) 189–211. [12] X. Fang, The Fredholm index of quotient Hilbert modules, Math. Res. Lett. 12 (2005) 911–920. [13] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16 (2006) 367–402. [14] X. Fang, Additive invariant on the Hardy space over the polydisc, J. Funct. Anal. 253 (2007) 359–372. [15] X. Fang, The Fredholm index of a pair of commuting operators. II, J. Funct. Anal. 256 (2009) 1669–1692. [16] J. Fang, D. Hadwin, M. Ravichandran, On transitive algebras containing a standard finite von Neumann subalgebra, J. Funct. Anal. 252 (2007) 581–602. [17] J. Gleason, S. Richter, C. Sundberg, On the index of invariant subspaces in spaces of analytic function of several complex variables, J. Reine Angew. Math. 587 (2005) 49–76. [18] D. Greene, S. Richter, C. Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna Pick kernels, J. Funct. Anal. 194 (2002) 311–331. [19] S. McCullough, T. Trent, Invariant subspaces and Nevanlinna–Pick kernel kernels, J. Funct. Anal. 178 (2000) 226– 249. [20] E. Nordgren, H. Radjavi, P. Rosenthal, On density of transitive algebras, Acta Sci. Math. (Szeged) 30 (1969) 175– 179. [21] E. Nordgren, P. Rosenthal, Algebras containing unilateral shifts or finite rank operators, Duke Math. J. 40 (1973) 419–424. [22] H. Radjavi, P. Rosenthal, A sufficient condition that an operator algebra be self-adjoint, Canad. J. Math. 23 (1971) 588–597. [23] H. Radjavi, P. Rosenthal, Invariant Subspaces, Springer-Verlag, Berlin, 1973. [24] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 286 (1988) 205–220. [25] P. Rosenthal, On reductive algebras containing compact operators, Proc. Amer. Math. Soc. 47 (1975) 338–340. [26] H. Shapiro, A. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962) 217–229.
Journal of Functional Analysis 258 (2010) 4251–4278 www.elsevier.com/locate/jfa
On the global existence and wave-breaking criteria for the two-component Camassa–Holm system Guilong Gui a,b , Yue Liu c,∗ a Department of Mathematics, Jiangsu University, Zhenjiang 212013, China b Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China c Department of Mathematics, University of Texas, Arlington, TX 76019, United States
Received 9 November 2009; accepted 5 February 2010 Available online 24 February 2010 Communicated by H. Brezis
Abstract Considered herein is a two-component Camassa–Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space H s , s > 32 by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space H s . © 2010 Elsevier Inc. All rights reserved. Keywords: Global solutions; Transport equation theory; Two-component Camassa–Holm system; Wave breaking
1. Introduction We consider here the coupled two-component Camassa–Holm shallow water system [12,23, 30,31], namely, ⎧ ⎨ mt + umx + 2ux m − Aux + ρρx = 0, t > 0, x ∈ R, m = u − uxx , t > 0, x ∈ R, ⎩ ρt + (uρ)x = 0, t > 0, x ∈ R, * Corresponding author.
E-mail addresses:
[email protected] (G. Gui),
[email protected] (Y. Liu). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.02.008
(1.1)
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where the variable u(t, x) represents the horizontal velocity of the fluid, and ρ(t, x) is related to the free surface elevation from equilibrium (or scalar density) with the boundary assumptions, u → 0 and ρ → 1 as |x| → ∞. The parameter A > 0 characterizes a linear underlying shear flow so that (1.1) models wave-current interactions (see the discussions in [15,25,26] and see also [4,24]). All of those are measured in dimensionless units. Recently, Ivanov [23] gave a rigorous justification of the derivation of the system (1.1) which is a valid approximation to the governing equations for water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case arises for example in situations with underlying shear flow [24]. Set g(x) = 12 e−|x| , x ∈ R. Then (I − ∂x2 )−1 f = g ∗ f for f ∈ L2 (R), where ∗ denotes the spatial convolution. Let η := ρ − 1, (1.1) can be rewritten as a quasi-linear nonlocal evolution system of the type ⎧ ⎨ u + uu = −∂ g ∗ u2 + 1 u2 − Au + 1 η2 + η , t x x 2 x 2 ⎩ ηt + uηx + ηux + ux = 0,
t > 0, x ∈ R,
(1.2)
t > 0, x ∈ R,
or equivalently, ⎧ ut + uux + ∂x P = 0, ⎪ ⎪ ⎨ 1 1 −∂x2 P + P = u2 + u2x − Au + η2 + η, ⎪ 2 2 ⎪ ⎩ ηt + uηx + ηux + ux = 0,
t > 0, x ∈ R, t > 0, x ∈ R, t > 0, x ∈ R.
For A = ρ = 0 in (1.1), one obtains the classical Camassa–Holm model [5], whose relevance for water waves was established in [10,27]. The system (1.1) is formally integrable [19,23,31] as it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter ζ , that is, 1 A 2 2 + Ψ, Ψxx = −ζ ρ + ζ m − 2 4 1 1 − u Ψx + ux Ψ Ψt = 2ζ 2 and has a bi-Hamiltonian structure corresponding to the Hamiltonian H1 =
1 2
mu + (ρ − 1)2 dx
R
with m = u − uxx and the Hamiltonian H2 =
1 2
u(ρ − 1)2 + 2u(ρ − 1) + u3 + uu2x − Au2 dx.
R
There are two Casimirs, i.e. ρ → 1 as |x| → ∞.
ρ − 1 and
m with boundary conditions are taken as u → 0 and
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The system (1.1) without vorticity, i.e. A = 0, was also rigorously justified by Constantin and Ivanov [12] to approximate the governing equations for shallow water waves. M. Chen, S. Liu and Y. Zhang [8] established a reciprocal transformation between the two-component Camassa– Holm system and the first negative flow of the AKNS hierarchy. More recently, Holm, Nraigh and Tronci [22] proposed a modified two-component Camassa–Holm system which possesses singular solutions in component ρ. Mathematical properties of (1.1) with A = 0 have been also studied further in many works. For example, Escher, Lechtenfeld and Yin [18] investigated local wellposedness for the two-component Camassa–Holm system with initial data (u0 , ρ0 ) ∈ H s × H s−1 with s 2 and derived some precise blow-up scenarios for strong solutions to the system. Constantin and Ivanov [12] provided some conditions of wave breaking and small global solutions. Gui and Liu [21] recently obtained results of local well-posedness in the Besov spaces (especially in the Sobolev space H s × H s−1 with s > 32 ) and wave breaking for certain initial profiles. More recently, Guan and Yin [20] studied global existence and blow-up phenomena for the system (1.2) with initial data (u0 , ρ0 − 1) ∈ H s × H s−1 with s 52 . It is known that different from the Korteweg–de Vries (KdV) equation, the Camassa–Holm (CH) equation has a remarkable property, that is, the presence of breaking waves [5,11], which means, the solution remains bounded while its slope becomes unbounded in finite time [9,11]. After wave breaking the solutions of the CH equation can be continued uniquely as either global conservative [2] or global dissipative solutions [3]. The goal of the present paper is to investigate whether or not the two-component Camassa–Holm system has the similar wave-breaking phenomena as the classical Camassa–Holm equation in a lower Sobolev space H s × H s−1 for s > 32 . In other words, whether or not both of two components u and ρ of the solution remain bounded while their slopes become unbounded in finite time. As we know, a crucial ingredient to obtain wave breaking in finite time or global solution for the CH equation is the following invariant property [9].
m t, q(t, x) qx2 (t, x) = m0 (x),
(t, x) ∈ [0, T ) × R,
where m(t, x) = u(t, x) − uxx (t, x) and the function q ∈ C 1 is an increasing diffeomorphism of R and satisfies the following differential equation, ⎧ ⎨ ∂q = u(t, q), 0 < t < T , ∂t ⎩ q(0, x) = x, x ∈ R. This is related to the geodesic equation which is on the diffeomorphism group of the circle [14] or on the Bott–Virasoro group [13,28,29]. Without such a nice invariant property of the CH equation, the issue of whether or not particular initial data of the two-component Camassa–Holm system generate a global solution or wave breaking is more subtle. Our work is motivated in the study of nonlinear models, especially of the transport equation, that is, ∂t f + v∂x f = g, (t, x) ∈ R+ × R, f |t=0 = f0 . It is well known that most of estimates are available when v has enough regularity. Roughly speaking, the regularity of the initial data is expected to be preserved as soon as v belongs to L1loc (R+ ; Lip).
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We give the following remark to explain the meaning of the lowest Sobolev space corresponding to the system (1.1) or (1.2). Remark 1.1. We say H s × H s−1 with s > 32 is the lowest Sobolev space for the two-component Camassa–Holm system based on the following facts. (a) For every Sobolev index s 32 , the Sobolev space H s cannot be embedded in the Lipschitz space (Lip), which is the lowest condition for preserving the regularity of the strong solution to the two-component Camassa–Holm system according to the localized analysis in the transport equation theory. (b) Without the effect of linear dispersion, i.e. A = 0, the system (1.1) has peakon solitons of the form, u(x, t) = ce−|x−ct| , c = 0 with ρ ≡ 0 as the solution of the Camassa–Holm equation. It is noted that the peakon soliton ce−|x−ct| is the weak solution in the Sobolev space H s only for s < 32 . (c) Following the proof of Proposition 4 in [17], one can see that (1.1) is not locally well posed 3 2 in B2,∞ in the following sense. 3
2 ) and ρ ≡ 0 to (1.1) such that for any positive There exists a global solution u ∈ L∞ (R+ ; B2,∞ 3
2 ) and ρ ≡ 0 with T > 0 and > 0 there exists a solution v ∈ L∞ (0, T ; B2,∞
u(0) − v(0)
3
2 B2,∞
Therefore, the exponent s =
3 2
and u − v
3
2 ) L∞ (0,T ;B2,∞
1.
s for r ∈ [1, ∞]. is critical in the range of Besov spaces B2,r
Inspired by [12], we use the properties of invariance of the component ρ associated to a transport equation with more delicate localization analysis in the transport equation theory to derive a new wave-breaking criterion for solutions for the system (1.1) in the lowest Sobolev spaces H s × H s−1 with s > 32 . In this case, due to the Hamiltonian H1 , the horizontal velocity component u is uniformly bounded by the Sobolev imbedding of H 1 into L∞ . It is shown that the slope of u is bounded below, then the slope of the component ρ cannot break in finite time. This implies that the wave breaking of the solution is determined only by the slope of the component u of the solution definitely. Note in [12,18] that the wave breaking in finite time is determined by either the slope of the first component u or the slope of the second component ρ in the Sobolev space H s × H s−1 with s 52 . It is, however, established in Theorem 4.1 and Theorem 4.2 that the wave breaking in finite time only depends on the slope of the first component u in the Sobolev space H s × H s−1 with s > 32 . In other words, the wave breaking in the first component u must occur before that in the second component ρ in finite time. On the other hand, we find a sufficient condition for global solutions which determined only by a nonzero initial profile of the free surface component ρ of the system in H s × H s−1 with s > 32 . This can be done because the slope of the component u can be controlled by the component ρ in finite time provided the sign of ρ does not change. These of improved results of global solutions and wave breaking indeed reveal more important features of wave propagation to the system (1.1). Our main results of the present paper are Theorem 4.1 (Wave-breaking criterion), Theorem 4.2 (Precise wave-breaking criterion) and Theorem 5.1 (Global solution). The remainder of the paper is organized as follows. In Section 2, we recall some basic facts on the Littlewood–Paley theory, which the localization technique is constantly used in the whole
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paper. Section 3 is devoted to the transport equation theory, where Theorem 3.2 is specially interesting to the system (1.2). Using the transport equation theory in the Besov spaces, two wave-breaking criteria to solutions in the lowest Sobolev space H s × H s−1 with s > 32 are demonstrated in Section 4. Finally, a result of global existence of solution in the lowest Sobolev space H s × H s−1 with s > 32 is obtained in the last section, Section 5. Notation. Let A, B be two operators, we denote [A; B] = AB − BA, the commutator between A and B; a b means that there is a uniform constant C that may be different on different lines, such that a Cb. We denote (cj )j ∈N (or (cj (t))j ∈N ) to be a sequence in 2 with norm 1. All of different positive constants might be denoted by the uniform constant C which may depend only on initial data. 2. Littlewood–Paley analysis For convenience of the reader, we shall recall some basic facts on the Littlewood–Paley theory, one may check [1,6,7,16,32] for more details. def
Proposition 2.1 (Littlewood–Paley decomposition). (See [6].) Let B = {ξ ∈ Rd , |ξ | 43 } and def
C = {ξ ∈ Rd , that
3 4
|ξ | 83 }. There exist two radial functions χ ∈ Cc∞ (B) and ϕ ∈ Cc∞ (C) such χ(ξ ) +
∀ξ ∈ Rd ,
q0
q − q 2 q 1
ϕ 2−q ξ = 1,
⇒ ⇒
Supp ϕ 2−q · ∩ Supp ϕ 2−q · = ∅,
Supp χ(·) ∩ Supp ϕ 2−q · = ∅,
and
2 1 χ(ξ )2 + ϕ 2−q ξ 1, 3
∀ξ ∈ Rd .
q0
def def Let h = F −1 ϕ and h˜ = F −1 χ . Then the dyadic operators q and Sq can be defined as follows
def −q
qd h 2q y f (x − y) dy, for q 0, q f = ϕ 2 D f = 2 Rd
def Sq f = χ 2−q D f =
q f = 2qd
−1kq−1 def
−1 f = S0 f
h˜ 2q y f (x − y) dy,
Rd def
and q f = 0 for q −2.
(2.1)
Lemma 2.1 (Bernstein’s inequality). (See [6].) Let B be a ball with center 0 in Rd and C a ring with center 0 in Rd . A constant C exists so that, for any positive real number λ, any nonnegative
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integer k, any smooth homogeneous function σ of degree m, and any couple of real numbers (a, b) with b a 1, there hold Supp uˆ ⊂ λB Supp uˆ ⊂ λC
⇒
⇒
C
Supp uˆ ⊂ λC
1 1 sup ∂ α u Lb C k+1 λk+d( a − b ) uLa ,
|α|=k
λ uLa sup ∂ α u La C 1+k λk uLa ,
−1−k k
⇒
σ (D)u
Lb
|α|=k
1
1
Cσ,m λm+d( a − b ) uLa ,
(2.2)
for any function u ∈ La . Definition 2.1 (Besov spaces). (See [6].) Let s ∈ R, 1 p, r ∞. The inhomogeneous Besov s (Rd ) (B s for short) is defined by space Bp,r p,r d def
s s <∞ , Bp,r R = f ∈ S Rd ; f Bp,r where s f Bp,r
1 ( q∈Z 2qsr q f rLp ) r , = supq∈Z 2qs q f Lp ,
def
for r < ∞, for r = ∞.
def s s∈R Bp,r .
∞ = If s = ∞, Bp,r
Remark 2.1. s (i) f ∈ Bp,r if and only if there exist a constant C and a sequence (cq )q∈N∪{−1} in r with norm 1 satisfying
q f Lp Ccq 2−qs .
(2.3)
s coincides with the Sobolev space H s . (ii) For s ∈ R, p = r = 2, the Besov space Bp,r
Proposition 2.2 (Gagliardo–Nirenberg inequality). For s > 12 , the following statement holds:
log e + f H s , f L∞ C 1 + f B∞,∞ 0 where the constant C = C(s) is independent of f . The proof of this proposition is trivial, which can be found in [6], and we omit it. The following proposition is devoted to dealing with the pseudo-differential operator ∂x (1 − ∂x2 )−1 (or ∂x g∗). Proposition 2.3. (See [6].) Let m ∈ R and f be an S m -multiplier (that is, f : Rd → R is smooth and satisfies that for all multi-index α, there exists a constant Cα such that ∀ξ ∈ Rd , |∂ α f (ξ )| Cα (1 + |ξ |)m−|α| ). Then for all s ∈ R and 1 p, r ∞, the operator f (D) is cons to B s−m . tinuous from Bp,r p,r
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In this paper, we are going to use Bony’s decomposition which consists of writing uv = Tu v + Tv u + R(u, v),
(2.4)
where
Tu v =
Sq−1 u q v
and R(u, v) =
q u q v =
|q−q |1
q−1
q v, q u
q−1
q := q−1 + q + q+1 . where Proposition 2.4 (1-D Moser-type estimates). The following estimates hold. (i) For s 0,
f gH s (R) C f H s (R) gL∞ (R) + f L∞ (R) gH s (R) .
(2.5)
f ∂x gH s (R) C f H s+1 (R) gL∞ (R) + f L∞ (R) ∂x gH s (R) .
(2.6)
(ii) For s > 0,
(iii) For s1 12 , s2 >
1 2
and s1 + s2 > 0, f gH s1 (R) Cf H s1 (R) gH s2 (R) ,
(2.7)
where C’s are constants independent of f and g. Proof. The proof of this lemma is rather classical, and similar estimates can be found in [6]. (2.5) is a standard Moser-type estimate, and (2.7) was used in [16] and [21]. For completeness, we present the detailed proof of (2.6) here. Thanks to Bony’s decomposition (2.4), we decompose f ∂x g as follows: f ∂x g = Tf ∂x g + T∂x g f + R(f, ∂x g). Thanks to Bernstein’s inequalities (2.2), we have
q (Tf ∂x g)
L2
q (Sq −1 f q ∂x g) |q−q |5
Cf L∞
|q−q |5
and
L2
Sq −1 f L∞ q ∂x gL2
|q−q |5
cq 2−sq ∂x gH s Ccq 2−sq f L∞ ∂x gH s
(2.8)
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q (T∂ g f ) 2 x L
q (Sq −1 ∂x g q f ) |q−q |5
Cf H s+1 gL∞
L2
Sq −1 ∂x gL∞ q f L2
|q−q |5
cq 2−(s+1)q 2q
|q−q |5
Ccq 2−sq f H s+1 gL∞ .
(2.9)
While for s > 0, using Bernstein’s inequalities (2.2) again and Young’s inequality, we get
q R(f, ∂x g)
L2
q ( q f q ∂x g) q q−5
C
L2
q ∂x gL∞ q f L2
q q−5
q gL∞ q f L2
q =−1, q q−5
+C
q gL∞ q ∂x f L2
q 0, q q−5
Ccq 2−sq f H s+1 gL∞ , which, together with (2.8), (2.9) and (2.3), completes the proof of (2.6).
2
3. Transport equation theory To study the well-posedness problem of the system (1.2), we need the following theorem on the transport equation (especially taking the space dimension d = 1), which has been used in [21]. Theorem 3.1. (See [16].) Suppose that s > − d2 . Let v be a vector field such that ∇v belongs d
to L1 ([0, T ]; H s−1 ) if s > 1 + d2 or to L1 ([0, T ]; H 2 ∩ L∞ ) otherwise. Suppose also that f0 ∈ H s , F ∈ L1 ([0, T ]; H s ) and that f ∈ L∞ ([0, T ]; H s ) ∩ C([0, T ]; S ) solves the d-dimensional linear transport equations (T )
∂t f + v · ∇f = F, f |t=0 = f0 .
Then f ∈ C([0, T ]; H s ). More precisely, there exists a constant C depending only on s, p and d, and such that the following statements hold: (1) If s = 1 + d2 , t f
Hs
f0
Hs
+ 0
or hence,
F (τ )
t Hs
dτ + C 0
V (τ ) f (τ ) H s dτ,
(3.1)
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
t
f H s eCV (t) f0 H s +
e−CV (τ ) F (τ )
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Hs
dτ
(3.2)
0
t 0
∇v(τ )
H
d 2
∩L∞
dτ if s < 1 +
d 2
and V (t) =
t
∇v(τ )H s−1 dτ else. t (2) If f = v, then for all s > 0, the estimates (3.1) and (3.2) hold with V (t) = 0 ∂x u(τ )L∞ dτ . with V (t) =
0
The following theorem (Theorem 3.2) is crucial to prove wave-breaking criterion (Theorem 4.1 in Section 4). Compared with Theorem 3.1, the following theorem is also specially interesting to the regularity propagation of the solution to the second equation of the twocomponent Camassa–Holm system (1.2) (where ρ − 1 = η = u), since only one derivative of u is involved in V (t) in (3.3) below. It is noted that the estimate (3.3) is quite different from (3.1) in Theorem 3.1, because there is (1 + 12 )-order derivative of u involved. This then makes the problem more difficult to deal with. The proof actually needs more delicate localization analysis in details. Theorem 3.2. Let 0 < σ < 1. Suppose that f0 ∈ H σ , g ∈ L1 ([0, T ]; H σ ), v, ∂x v ∈ L1 ([0, T ]; L∞ ) and that f ∈ L∞ ([0, T ]; H σ ) ∩ C([0, T ]; S ) solves the 1-dimensional linear transport equation (T )
∂t f + v∂x f = g, f |t=0 = f0 .
Then f ∈ C([0, T ]; H σ ). More precisely, there exists a constant C depending only on σ and such that the following statement holds:
f (t) σ f0 H σ + C H
t
g(τ )
t Hσ
dτ + C
0
f (τ )
Hσ
V (τ ) dτ
(3.3)
0
or hence,
f (t)
Hσ
e
CV (t)
t
f0 H σ + C
g(τ )
Hσ
dτ
0
with V (t) =
t
0 (v(τ )L
∞
+ ∂x v(τ )L∞ ) dτ .
Proof. The proof of this theorem is motivated by the one of Theorem 3.1 (see [16]). Applying the localization operator q to the transport equation (T ), we transform the transport equation (T ) along the flow of v, in the following equation (Tq ) on q f , which is a transport equation along the flow of Sq v (Tq )
∂t q f + Sq v∂x q f = q g − Rq , q f |t=0 = q f0 ,
where Rq = Rq (v, f ) := q (v∂x f ) − v q ∂x f + (v − Sq v) q ∂x f .
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To deal with Rq , we need to use the following lemma, which we admit for the time being. Lemma 3.1. For all 0 < σ < 1,
Rq (t)
L2
cq (t)2−qσ f (t) H σ vL∞ + ∂x vL∞
with cq (t) ∈ l 2 and cq (t)l 2 ≡ 1. With Lemma 3.1 in hand, we can continue the proof of Theorem 3.2. Taking the inner product between the first equation of (Tq ) and q f in L2 , we have 1 d 1 q f 2L2 = − 2 dt 2
Sq v∂x | q f |2 + ( q g| q f )L2 − Rq (v, f )| q f L2
R
∂x Sq vL∞ q f 2L2 + Rq L2 + q gL2 q f L2 , which implies
1 d q f 2L2 ∂x vL∞ q f L2 + Rq L2 + q gL2 q f L2 2 dt
Ccq (t)2−qσ f (t) σ vL∞ + ∂x vL∞ + g(t) H
Hσ
q f L2 .
Therefore, one has
q f (t)
t L2
q f0 L2 + C
cq (τ )2−qσ f (τ ) H σ vL∞ + ∂x vL∞ + g(τ ) H σ dτ.
0
(3.4) Multiplying (3.4) by 2qσ , then taking the l 2 norm over q and applying Minkowski’s inequality, we reach
f (t)
t Hσ
f0 H σ + C
g(τ )
t Hσ
dτ + C
0
This ends the proof of Theorem 3.2.
f (r)
Hσ
vL∞ + ∂x vL∞ dτ.
0
2
We now are in a position to prove Lemma 3.1. Proof of Lemma 3.1. Firstly, using Bony’s decomposition, we decompose the term Rq as follows
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
4261
Rq = q (v∂x f ) − v q ∂x f + (v − Sq v) q ∂x f = [ q ; Tv ]∂x f + q T∂x f v − T q ∂x f v + q R(v, ∂x f ) − R(v, q ∂x f ) + (v − Sq v) q ∂x f :=
6
Rqi .
(3.5)
i=1
For Rq1 =
|q−q |5 [ q ; Sq −1 v]∂x q f ,
[ q ; Sq −1 v]∂x q f = 2q
thanks to (2.1), one has
h 2q (x − y) Sq −1 v(y) − Sq −1 v(x) ∂x q f (y) dy.
R
Hence, (2.2) applied ensures
1
R (t) q
L2
|q −q|5
∂x Sq −1 vL∞ 2−q ∂x q f L2 cq (t)2−qσ f (t) H σ ∂x vL∞ . (3.6)
For Rq2 = q T∂x f v = q
2
R (t) q
L2
q 0, |q−q |5 Sq −1 ∂x f q v,
Sq −1 ∂x f L2 q vL∞
q 0, |q −q|5
(2.2) applied again implies
k ∂x f L2 2−q q ∂x vL∞
q 0, |q −q|5 kq −2
2k k f L2 2−q q ∂x vL∞ ,
q 0, |q −q|5 kq −2
which yields that
2
R (t) q
L2
2k ck 2−kσ f H σ 2−q q ∂x vL∞
q 0, |q −q|5 kq −2
2(k−q )(1−σ ) ck f H σ 2−qσ ∂x vL∞
q 0, |q −q|5 kq −2
cq (t)2−qσ f H σ ∂x vL∞ ,
(3.7)
where we used the assumption σ < 1. Similarly, for Rq3 = −T q ∂x f v = − q 0, q q−5 Sq −1 q ∂x f q v, we have
3
R (t) q
L2
Sq −1 q ∂x f L2 q vL∞
q 0, q q−5
q f L2 2q−q q ∂x vL∞
q 0, q q−5
q 0, q q−5
2q−q cq 2−qσ f H σ ∂x vL∞ cq (t)2−qσ f H σ ∂x vL∞ .
(3.8)
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Since Rq4 = q R(v, ∂x f ) = q
4
R (t) q
L2
q q−5 q v q ∂x f ,
we get from (2.2) that
q ∂x f L2 q vL∞
q q−5
=
q =−1, q q−5
q ∂x f L2 + q vL∞
q ∂x f L2 q vL∞
q 0, q q−5
q f L2 + vL∞
q =−1, q q−5
q f L2 , q ∂x vL∞
q 0, q q−5
which gives rise to
4
R (t) q
L2
cq 2(q−q )σ vL∞ 2−qσ f H σ
q =−1, q q−5
+
cq 2(q−q )σ ∂x vL∞ 2−qσ f H σ
q 0, q q−5
cq (t)2−qσ f H σ vL∞ + ∂x vL∞ ,
(3.9)
where we used the assumption σ > 0. q q ∂x f , we have While for Rq5 = R(v, q ∂x f ) = |q−q |5 q v
5
R (t) q
L2
q q ∂x f L2 q vL∞
|q−q |5
=
q q ∂x f L2 q vL∞
q =−1, |q−q |5
+
q q ∂x f L2 , q vL∞
q 0, |q−q |5
from which and (2.2), we get
5
R (t) q
L2
2−q σ cq 2(q−q )σ vL∞ 2−qσ f H σ
q =−1, |q−q |5
+
cq 2(q−q )σ ∂x vL∞ 2−qσ f H σ
q 0, |q−q |5
cq (t)2−qσ f H σ vL∞ + ∂x vL∞ .
(3.10)
Finally, for Rq6 = (v − Sq v) q ∂x f ,
6
R (t) 2 q vL∞ q ∂x f L2 q L q q
=
q =−1, q q
q vL∞ q ∂x f L2 +
q 0, q q
q vL∞ q ∂x f L2 ,
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4263
from which and (2.2), we reach
6
R (t) q
L2
vL∞ q f L2 +
q =−1, q q
2q−q q ∂x vL∞ q f L2
q 0, q q
cq (t)2−qσ f H σ vL∞ + ∂x vL∞ ,
which together with (3.5)–(3.10) completes the proof of Lemma 3.1.
2
4. Wave-breaking criteria Let us first state the following local well-posedness result of (1.2), which was obtained in [21] (up to a slight modification). Lemma 4.1. Suppose that u0 = (u0 , η0 ) ∈ H s × H s−1 , s > 32 . Then there exist T = T (u0 H s ×H s−1 ) > 0 and a unique solution u = (u, η) ∈ C([0, T ); H s × H s−1 ) ∩ C 1 ([0, T ); H s−1 × H s−2 ) of (1.2) with u(0) = u0 . Moreover, the solution u depends continuously on the initial value u0 and the maximal time of existence T > 0 is independent of s. In addition, the Hamiltonian
1 2 u + u2x + η2 dx (4.1) H = H (u, η) = 2 R
is independent of the existence time T . With Lemma 4.1 in hand, we establish the associated Lagrangian scale of (1.2) the initialvalue problem ⎧ ⎨ ∂q = u(t, q), 0 < t < T , (4.2) ∂t ⎩ q(0, x) = x, x ∈ R, where u ∈ C([0, T ), H s ) is the first component of the solution (u, η) of (1.2) with initial data (u0 , ρ0 − 1) ∈ H s × H s−1 with s > 32 , and T > 0 being the maximal time of existence. A direct calculation also yields qtx (t, x) = ux (t, q(t, x))qx (t, x). Hence for t > 0, x ∈ R, we have qx (t, x) = e
t 0
ux (τ,q(τ,x)) dτ
> 0,
which implies that q(t, ·) : R → R is a diffeomorphism of the line for every t ∈ [0, T ). This is inferred that the L∞ norm of any function u(t, ·) ∈ L∞ (R), t ∈ [0, T ) is preserved under the family of diffeomorphisms q(t, ·) with t ∈ [0, T ), that is,
u(t, ·)
L∞ (R)
= u t, q(t, ·) L∞ (R) ,
Similarly, one gets
sup u(t, x) = sup u t, q(t, x) . x∈R
x∈R
t ∈ [0, T ).
(4.3)
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The following wave-breaking criterion shows that the wave breaking only depends on the slope of u but not the slope of ρ. This improves the wave-breaking criterion in [21] and [20], where the slopes of both components u and ρ must be considered. The proof of the following result strongly depends on Theorem 3.2 on the localization analysis for the transport equation. Theorem 4.1. Let u0 = (u0 , η0 ) ∈ H s × H s−1 be as in Lemma 4.1 with s > 32 and u = (u, η) being the corresponding solution to (1.2). Assume Tu∗0 > 0 is the maximal time of existence. Then T
Tu0 < ∞
⇒
u0
∂x u(τ )
L∞
dτ = ∞.
0
Proof. We shall prove this theorem by an inductive argument with respect to the index s. To this end, let us first give a control on η(t)L∞ . In fact, applying the maximal principle to the transport equation about ρ, ρt + uρx + ρux = 0, we have
ρ(t)
t
L∞
ρ0 L∞ + C
∂x uL∞ ρL∞ dτ. 0
A simple application of Gronwall’s inequality implies
ρ(t)
L∞
ρ0 L∞ eC
t 0
∂x uL∞ dτ
,
which gives rise to
t
η(t) ∞ ρ(t) ∞ + 1 1 + 1 + η0 L∞ eC 0 ∂x uL∞ dτ . L L
(4.4)
Now let us concentrate our attentions to the proof of Theorem 4.1. This can be achieved as follows. Step 1. For s ∈ ( 32 , 2), applying Theorem 3.2 to the transport equation with respect to η, ηt + uηx + ηux + ux = 0,
(4.5)
we have (for every 1 < s < 2, indeed)
η(t)
t H s−1
η0 H s−1 + C
t η∂x u + ∂x uH s−1 dτ + C
0
Thanks to the Moser-type estimate (2.5), one has
0
ηH s−1 uL∞ + ∂x uL∞ dτ.
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
η∂x u + ∂x uH s−1 ∂x uH s−1 + C ∂x uH s−1 ηL∞ + ηH s−1 ∂x uL∞ .
4265
(4.6)
Therefore, we have t
η(t)
H s−1
η0 H s−1 + C
∂x u(τ )
H s−1
1 + η(τ ) L∞ dτ
0
t +C
η(τ )
H s−1
u(τ )
L∞
+ ∂x u(τ ) L∞ dτ.
(4.7)
0
On the other hand, Theorem 3.1 applied to the equation about u, 1 2 1 2 2 ut + uux + ∂x g ∗ u + ux + η + η − Au = 0, 2 2 implies (for every s > 1, indeed)
u(t) s u0 H s + C H
t
∂x g ∗ u2 + 1 u2 + 1 η2 + η − Au (τ ) dτ x
s 2 2 H 0
t +C
u(τ )
Hs
∂x u(τ )
L∞
dτ.
0
Thanks to the Moser-type estimate (2.5) and Proposition 2.3, one has
∂x g ∗ u2 + 1 u2 + 1 η2 + η − Au C u2 + 1 u2 + 1 η2 + η − Au x
s
s−1 2 x 2 2 2 H H C uH s−1 uL∞ + ∂x uH s−1 ∂x uL∞
+ ηH s−1 ηL∞ + uH s−1 + ηH s−1 . From this, we reach
u(t)
Hs
t u0 H s + C
u(τ )
Hs
u(τ ) ∞ + ∂x u(τ ) ∞ + 1 dτ L L
0
t +C
η(τ )
0
which together with (4.7) ensures that
H s−1
η(τ )
L∞
+ 1 dτ,
(4.8)
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u(t)
Hs
+ η(t)
t H s−1
u0 H s + η0 H s−1 + C
η(τ )
H s−1
+ u(τ ) H s
0
× u(τ ) L∞ + ∂x u(τ ) L∞ + η(τ ) L∞ + 1 dτ.
(4.9)
Thanks to the Gronwall’s inequality again, one can see
u(t)
Hs
t + η(t) H s−1 u0 H s + η0 H s−1 eC 0 (uL∞ +∂x uL∞ +ηL∞ +1) dτ . (4.10)
Using the Sobolev embedding theorem H s → L∞ (for s > 12 ), we get from (4.1) that
u(t)
L∞
C u0 H 1 + η0 L2 ,
(4.11)
which together with (4.4) and (4.10) implies that
u(t)
Hs
t
+ η(t) H s−1 u0 H s + η0 H s−1 eC1 (t+1) exp{ 0 C∂x u(τ )L∞ dτ } ,
where C1 = C1 (u0 H 1 , η0 L2 , η0 L∞ ). Therefore, if the maximal existence time Tu0 < ∞ satisfies obtain from (4.12) that
Tu0 0
(4.12)
∂x u(τ )L∞ dτ < ∞, we
lim sup u(t) H s + η(t) H s−1 < ∞
(4.13)
t→Tu
0
contradicts the assumption on the maximal existence time Tu0 < ∞. This completes the proof of Theorem 4.1 for s ∈ ( 32 , 2). Step 2. For s ∈ [2, 52 ), applying Theorem 3.1 to the transport equation (4.5), we have
η(t)
t H s−1
η0 H s−1 + C
(η∂x u + ∂x u)(τ )
t H s−1
dτ + C
0
ηH s−1 ∂x u
1
dτ.
1
dτ,
L∞ ∩H 2
0
(4.6) applied implies that t
η(t)
H s−1
η0 H s−1 + C
∂x uH s−1 1 + ηL∞ dτ + C
0
t ηH s−1 ∂x u
L∞ ∩H 2
0
which together with (4.8) yields
u(t)
Hs
+ η(t) H s−1 u0 H s + η0 H s−1 t +C 0
η(τ )
H s−1
+ u(τ ) H s u
H
3 + 2 0
+ η(τ ) L∞ + 1 dτ
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1
with 0 < 0 < 12 , where we used the fact H 2 +0 → L∞ ∩ H 2 . Gronwall’s inequality applied gives that
u(t)
Hs
+ η(t)
u0 H s
H s−1
C + η0 H s−1 e
t
0 (u 3 +0 +η(τ )L∞ +1) dτ H2
.
(4.14)
Therefore, thanks to the uniqueness of solution in Lemma 4.1, (4.1) and (4.13), we get that: if the Tu maximal existence time Tu0 < ∞ satisfies 0 0 ∂x u(τ )L∞ dτ < ∞, then (4.14) implies that
lim sup u(t) H s + η(t) H s−1 < ∞ t→Tu
0
contradicts the assumption on the maximal existence time Tu0 < ∞. This completes the proof of Theorem 4.1 for 2 s < 52 . Step 3. For 2 < s < 3, by differentiating once (4.5) with respect to x, we have ∂t ηx + u∂x (ηx ) + 2ux ηx + ηuxx + uxx = 0.
(4.15)
Theorem 3.2 applied to (4.15) implies that
ηx (t)
t η0x H s−2 + C
H s−2
(2ηx ux + η∂x ux + ∂xx u)(τ )
H s−2
dτ
0
t +C
ηx (τ )
H s−2
u(τ )
L∞
+ ∂x u(τ ) L∞ dτ
0
t η0x H s−2 + C
ηH s−1 + uH s uL∞ + ∂x uL∞ + ηL∞ + 1 dτ,
0
(4.16) where we used the following Moser-type estimates (from (2.6)):
ηx ux H s−2 C ∂x uH s−1 ηL∞ + ∂x ηH s−2 ux L∞ and
η∂x ux H s−2 C ηH s−1 ∂x uL∞ + uxx H s−2 ηL∞ . (4.16), together with (4.8) and (4.7) (where s − 1 is replaced by s − 2), implies that
η(t)
H
s−1 + u(t)
t Hs
η0 H s−1 + u0 H s + C
× u(τ )
η(τ )
0
∞ + ∂x u(τ )
L
L∞
H s−1
+ u(τ ) H s
+ η(τ ) L∞ + 1 dτ.
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Gronwall’s inequality applied again gives (4.10). Hence, using arguments as in Step 1, it completes the proof of Theorem 4.1 for 2 < s < 3. Step 4. For s = k ∈ N, k 3, by differentiating (4.5) k − 2 times with respect to x, we have
∂t ∂xk−2 η + u∂x ∂xk−2 η +
C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−2 u + ∂xk−1 u = 0.
1 +2 =k−3, 1 ,2 0
(4.17) Applying Theorem 3.1 to the transport equation (4.17), we have
k−2
∂ η(t) x
H1
t
∂ k−2 η0 x
H1
+C
k−2
∂ η(τ ) x
H1
∂x u(τ )
1
L∞ ∩H 2
dτ
0
t
+C
0
C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x
1 +2 =k−3, 1 ,2 0
k−2
k−1 ∂x u + ∂x u (τ )
dτ. H1
Since H 1 is an algebra, we have
k−2
η∂x ∂ u 1 Cη 1 ∂ k−1 u 1 Cη 1 uH s H H x x H H and
C1 ,2 ∂x1 +1 u∂x2 +1 η
1 H 1 +2 =k−3, 1 ,2 0
C
1 +2 =k−3, 1 ,2 0
C1 ,2 ∂x1 +1 u H 1 ∂x2 +1 η H 1 CuH s−1 ηH s−1 .
Hence,
k−2
∂ η(t) x
H1
t
∂ k−2 η0 x
H1
+C
ηH s−1 + uH s uH s−1 + ηH 1 + 1 dτ. (4.18)
0
(4.18), together with (4.8) and (4.7) (where s − 1 is replaced by 1), implies that
η(t)
H s−1
+ u(t) H s t
η0 H s−1 + u0 H s + C
η(τ )
0
Gronwall’s inequality applied yields that
H s−1
+ u(τ ) H s u(τ ) H s−1 + η(τ ) H 1 + 1 dτ.
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
u(t)
Hs
4269
t + η(t) H s−1 u0 H s + η0 H s−1 eC 0 (uH s−1 +ηH 1 +1) dτ .
Therefore, if the maximal existence time Tu0 < ∞ satisfies the uniqueness of solution in Lemma 4.1, we get that
u(t)
H s−1
Tu0 0
(4.19)
∂x u(τ )L∞ dτ < ∞, thanks to
+ η(t) H 1
is uniformly bounded by the induction assumption, which together with (4.19) implies
lim sup u(t) H s + η(t) H s−1 < ∞. t→Tu
0
This leads to a contradiction. Step 5. For k < s < k + 1 with k ∈ N, k 3, by differentiating (4.5) k − 1 times with respect to x, we have
∂t ∂xk−1 η + u∂x ∂xk−1 η +
C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−1 u + ∂xk u = 0.
1 +2 =k−2, 1 ,2 0
Theorem 3.2 applied again implies that
k−1
∂ η(t) x
H s−k
t
∂ k−1 η0 x
H s−k
+C
k−1
∂ η(τ ) x
H s−k
u(τ )
L∞
+ ∂x u(τ ) L∞ dτ
0
t
+C 0
1 +2 =k−2, 1 ,2 0
C1 ,2 ∂x1 +1 u∂x2 +1 η + η∂x ∂xk−1 u + ∂xk u (τ )
dτ. H s−k
Using the Moser-type estimate (2.6) and the Sobolev embedding inequality, we have for ∀0 < 0 < 12
k−1
η∂x ∂ u x
H s−k
C ηL∞ ∂xk u H s−k + ηH s−k+1 ∂xk−1 u L∞
C ηL∞ uH s + ηH s−k+1 u k− 1 +0 H
and
1 +2 =k−2, 1 ,2 0
C1 ,2 ∂x1 +1 u∂x2 +1 η
C
1 +2 =k−2, 1 ,2 0
C u
H
k− 21 +0
2
H s−k
C1 ,2 ∂x1 +1 u L∞ ∂x2 +1 η H s−k + ∂x2 η L∞ ∂x1 +1 u H s−k+1
ηH s−1 + η
H
k− 23 +0
uH s .
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Hence,
k−1
∂ η(t) x
H s−k
t
∂ k−1 η0 x
H s−k
× u
+C
η(τ )
0 H
k− 21 +0
+ η
H
k− 23 +0
H s−1
+ u(τ ) H s
+ 1 dτ.
(4.20)
(4.20), together with (4.8) and (4.7) (where s − 1 is replaced by s − k), implies that
η(t) s−1 + u(t) s η0 s−1 + u0 H s + C H H H × u
t
η(τ )
0 H
k− 21 +0
+ η
H
k− 23 +0
H s−1
+ u(τ ) H s
+ 1 dτ.
Applying Gronwall’s inequality then gives that
u(t)
Hs
+ η(t)
H s−1
C u0 H s + η0 H s−1 e
t
0 (u k− 1 +0 +η k− 3 +0 +1) dτ 2 2 H H
In consequence, if the maximal existence time Tu0 < ∞ satisfies thanks to the uniqueness of solution in Lemma 4.1, then we get that
u(t)
H
k− 21 +0
+ η(t)
H
Tu0 0
.
∂x u(τ )L∞ dτ < ∞,
k− 23 +0
is uniformly bounded by the induction assumption, which implies
lim sup u(t) H s + η(t) H s−1 < ∞, t→Tu
0
which leads to a contradiction. Therefore, from Step 1 to Step 5, we complete the proof of Theorem 4.1.
2
Theorem 4.2. Let (u0 , η0 ) be as in Lemma 4.1 with s > 32 and u = (u, η) being the corresponding solution to (1.2). Then the corresponding solution blows up in finite time if and only if lim inf ux (t, x) = −∞.
t→Tu x∈R 0
(4.21)
Lemma 4.2. Let (u, η) (with η := ρ − 1) be the solution of (1.2) with initial value (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R), s > 32 , and T the maximal existence time. If there is M1 0 such that inf
(t,x)∈[0,T )×R
then
ux (t, x) −M1 ,
(4.22)
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
ρ(t, ·)
L∞
ρ0 L∞ eM1 t ,
4271
(4.23)
sup ux (t, x) u0,x L∞ + C1 + ρ0 L∞ eM1 t
(4.24)
x∈R
hold for t ∈ [0, T ), with C1 =
1 + A2 2
1 2
(u0 , ρ0 − 1)
H 1 ×L2
,
(4.25)
and C a positive constant depending only on A, M1 and the norm (u0 , ρ0 − 1)H s ×H s−1 . Proof. By Lemma 4.1 and a simple density argument, it is needed only to show the desired results are valid when s 3. So in the sequel of this section s = 3 is taken for simplicity of notation. Differentiating both sides of the first equation of (1.2) with respect to x and using the identity −∂x2 g ∗ f = f − g ∗ f lead to 1 1 1 1 utx + uuxx + u2x = A∂x g ∗ ux + u2 + ρ 2 − g ∗ u2 + u2x + ρ 2 . 2 2 2 2
(4.26)
Given x ∈ R, let
M(t) = ux t, q(t, x) ,
γ (t) = ρ t, q(t, x) ,
(4.27)
t ∈ [0, T ), with q(t, x) determined in (4.2). Using these notations, Eq. (4.26) and the second one of (1.2) can be rewritten, respectively, as
1 1 M (t) = − M 2 + γ 2 + f t, q(t, x) , 2 2
γ (t) = −γ M,
(4.28)
for t ∈ [0, T ), where the notation denotes the derivative with respect to t and f represents the function 1 1 f = A∂x g ∗ ux + u2 − g ∗ u2 + u2x + ρ 2 . 2 2
(4.29)
Hence, we have 1 1 1 f = Agx ∗ ux + u2 − g ∗ u2 + u2x − g ∗ 1 − g ∗ (ρ − 1) − g ∗ (ρ − 1)2 2 2 2 1 1 |A||gx ∗ ux | + u2 − + g ∗ (ρ − 1), 2 2 with the help of g ∗ (u2 + 12 u2x ) 12 u2 (cf. [9]). Applying Young’s inequality and g(x) = 12 e−|x| leads to
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1 1 1 |A||gx ∗ ux |(t, x) |A|gx L2 ux L2 |A|ux L2 + A2 ux 2L2 , 2 4 4 1 1 g ∗ (ρ − 1)(t, x) g 2 ρ − 1 2 ρ − 1 2 + 1 ρ − 12 2 , L L L L 2 4 4
(4.30) (4.31)
for (t, x) ∈ [0, T ) × R. On the other hand, the continuous embedding of H 1 (R) into L∞ (R) gives (cf. [11], for example, for the best embedding constant) 2u (t, x) 2
2
u R
+ u2x
2
u
+ u2x
+ (ρ − 1) =
R
u20 + u20x + (ρ0 − 1)2 ,
2
(4.32)
R
for all (t, x) ∈ [0, T ) × R, where we used the fact that H (u, ρ − 1) is the conservation law of the system (1.2) in the last identity. Combining (4.30), (4.31), and (4.32) together gives
2
2
1 1 1 1 + A2 (u, ρ − 1) H 1 ×L2 = 1 + A2 (u0 , ρ0 − 1) H 1 ×L2 = C12 , (4.33) 4 4 2 2 where the conservation law H (u, η) = R u + u2x + η2 of (1.2) was used again in the second identity and C1 was introduced in (4.25). Similarly, we have f
1 1 1 −f |A||gx ∗ ux | + g ∗ u2 + u2x + + g ∗ (ρ − 1) + g ∗ (ρ − 1)2 2 2 2
1 1 1 2 1 2 1 1 1 1 u + ux + A2 ux 2L2 + + + + ρ − 12L2 + ρ − 12L2 , 4 4 2 2 L1 2 4 4 4 where we used the estimate g ∗ (u2 + 12 u2x )(t, x) gL∞ u2 + 12 u2x L1 12 u2H 1 . Therefore, we get −f 1 +
2
1 + A2
(u, ρ − 1) 2 1 2 1 + 1 + A (u0 , ρ0 − 1) 2 1 2 1 + C 2 . 1 H H ×L ×L 2 2
(4.34)
In view of the definition of M(t) in (4.27), the assumption (4.22) is now expressed as, for each x ∈ R, M(t) −M1 ,
for t ∈ [0, T ).
In view of this condition, it then follows from the second equation of (4.28) that, for each x ∈ R, t
ρ t, q(t, x) = γ (t) = γ (0)e 0 −M(τ ) dτ ρ0 L∞ eM1 t ,
(4.35)
for t ∈ [0, T ). Hence combining this with (4.3) leads to (4.23). Given any x ∈ R, let us define P (t) = M(t) − u0,x L∞ − C1 − ρ0 L∞ eM1 t , with M(t) = ux (t, q(t, x)) and C1 in (4.25). Observe that P (t) is a C 1 -differentiable function in [0, T ) and satisfies
G. Gui, Y. Liu / Journal of Functional Analysis 258 (2010) 4251–4278
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P (0) = M(0) − u0,x L∞ − C1 − ρ0 L∞ u0,x (x) − u0,x L∞ 0. We now claim P (t) 0,
for all t ∈ [0, T ).
(4.36)
Assume the contrary that there is t0 ∈ [0, T ) such that P (t0 ) > 0. Let t1 = max t < t0 ; P (t) = 0 . Then P (t1 ) = 0 and P (t1 ) 0, or equivalently, M(t1 ) = u0,x L∞ + C1 + ρ0 L∞ eM1 t1 ,
(4.37)
M (t1 ) M1 ρ0 L∞ eM1 t1 > 0.
(4.38)
and
From (4.32), (4.35), (4.37), and the first equation of (4.28), it follows that
1 1 M (t1 ) = − M 2 (t1 ) + γ 2 (t1 ) + f t1 , q(t1 , x) 2 2
2 1 1 1 − u0,x L∞ + C1 + ρ0 L∞ eM1 t1 + ρ0 2L∞ e2M1 t1 + C12 2 2 2 0, a contradiction to (4.38), so the claim (4.36) is valid. Therefore, the arbitrarily chosen of x and (4.3) imply (4.24). 2 Proof of Theorem 4.2. Assume (4.21) is not valid. Then there is some positive number M1 > 0 such that ux (t, x) −M1 holds for (t, x) ∈ [0, T ) × R. It now follows from (4.24) in Lemma 4.2 that ux (t, x) CeM1 t , with C a positive constant depending only on A, M1 and the norm (u0 , ρ0 − 1)H s ×H s−1 . Theorem 4.1 applied implies that the maximal existence time Tu0 = ∞, which contradicts the assumption on the maximal existence time Tu0 < ∞. Conversely, the Sobolev embedding theorem H s (R) → L∞ (R) (with s > 12 ) implies that if (4.21) holds, the corresponding solution blows up in finite time, which completes the proof of Theorem 4.2. 2
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Theorem 4.3. Assume that the initial value (u0 , η0 ) ∈ H s × H s−1 with s > 32 . Let Tu0 > 0 be the maximal time of existence for the corresponding solution (u, η) to the system (1.2). Then we have T
Tu0 < ∞
⇒
u0
∂x u(τ )
0 B∞,∞
+ ρ(τ ) − 1 B 0
∞,∞
dτ = ∞.
0
Proof. We only need to prove this theorem for the case 32 < s < 2. For s 2, the induction argument as in the proof of Theorem 4.1 will complete the proof of Theorem 4.3. Thanks to Proposition 2.2, we have for s > 32 ,
log e + ∂x uH s−1 ∂x uL∞ C 1 + ∂x uB∞,∞ 0
(4.39)
ηL∞ C 1 + ηB∞,∞ log e + ηH s−1 . 0
(4.40)
and
Plugging (4.39) and (4.40) into (4.10), and using the fact (4.11), we get
u(t)
Hs
+ η(t) H s−1
Ct+C u0 H s + η0 H s−1 e
t
+η(τ )B 0 ) log(e+u(τ )H s +η(τ )H s−1 ) dτ 0 0 (∂x u(τ )B∞,∞ ∞,∞
Therefore,
log e + u(t) H s + η(t) H s−1
log e + u0 H s + η0 H s−1 + Ct t +C
∂x u(τ )
0 B∞,∞
+ η(τ ) B 0
∞,∞
log e + u(τ ) H s + η(τ ) H s−1 dτ.
0
Applying Gronwall’s inequality yields
log e + u(t) H s + η(t) H s−1 e
C
t
+η(τ )B 0 ) dτ 0 0 (∂x u(τ )B∞,∞ ∞,∞
Hence, the proof of Theorem 4.3 is complete.
log(e + u0 H s + η0 H s−1 ) + Ct . 2
.
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5. Global existence In view of the criterion for wave breaking (Theorem 4.1), a sufficient condition of global solutions can be obtained in the following. Theorem 5.1 (Global solution). Let (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R) with s > 32 , and T > 0 being the maximal time of existence of the solution (u, ρ) to the system (1.1) with initial data (u0 , ρ0 ). If inf ρ0 (x) > 0,
(5.1)
x∈R
then T = +∞, and the solution (u, ρ) is global. Remark 5.1. Theorem 5.1 improves the result of the global solutions in [20], where the special case s = 2 is required. To prove Theorem 5.1, we need the following lemma. Lemma 5.1. Assume (u, ρ) is the local solution of (1.1) with the initial value (u0 , ρ0 − 1) ∈ H s (R) × H s−1 (R), s > 32 , and T the maximal existence time. If infx∈R ρ0 (x) > 0, then
ux t, q(t, x) , ρ t, q(t, x)
1 C5 eC4 t |ρ0 (x)|
(5.2)
hold for all t ∈ [0, T ), with (see (4.25) for C1 ) C4 =
2
3 3 1 + C12 = + 1 + A2 (u0 , ρ0 − 1) H 1 ×L2 , 2 2 2
C5 = 1 + u0,x 2L∞ + ρ0 2L∞ , and positive constant C depending only on A and the norm (u0 , ρ0 − 1)H s ×H s−1 . Proof. In view of the proof of Lemma 4.2, by Lemma 4.1 and a simple density argument, it suffices to show that the desired results are valid when s 3. So in the sequel of this section s = 3 is taken for simplicity of notation. Observe that the system (1.2) leads to the following ordinary differential equations (see Lemma 4.2 for derivation) for a fixed x ∈ R,
1 1 M (t) = − M 2 + γ 2 + f t, q(t, x) , 2 2
γ (t) = −γ M,
(5.3)
for t ∈ [0, T ) with notation M(t) = ux (t, q(t, x)), γ (t) = ρ(t, q(t, x)) defined in (4.27) and f in (4.29). The second equation of (5.3) implies that γ (t) and γ (0) are of the same sign. For every x ∈ R satisfying γ (0) = ρ0 (x) > 0, define the Lyapunov function (cf. [12]), w(t) := γ (0)γ (t) +
γ (0) 1 + M 2 (t) , γ (t)
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which is a positive function of t ∈ [0, T ). By (5.3), it yields
2 γ (0) γ 1 + M 2 + γ (0)MM
2 γ γ
1 2 = γ (0)M f t, q(t, x) + γ 2
1 γ (0) 2 1+M C4 w(t), f t, q(t, x) + γ 2
w (t) = γ (0)γ −
in [0, T ), where |f | 1 + C12 is derived from (4.33) and (4.34). The preceding differential inequality gives w(t) w(0)eC4 t = C5 eC4 t ,
t ∈ [0, T )
(5.4)
with the help of w(0) = ρ02 (x) + 1 + u20,x (x) 1 + u0,x 2L∞ + ρ0 2L∞ = C5 . Recalling that γ (t) and γ (0) are of the same sign, the definition of w implies γ (0)γ (t) w(t) and |γ (0)||M(t)| w(t). By (5.4),
ux t, q(t, x) = M(t)
ρ t, q(t, x) = γ (t)
1 1 w(t) C5 eC4 t , |γ (0)| |ρ0 (x)| 1 1 w(t) C5 eC4 t |γ (0)| |ρ0 (x)|
are valid for t ∈ [0, T ). Thus the conclusions of (5.2) are obtained.
2
Proof of Theorem 5.1. Assume the contrary that T < ∞ and the solution blows up in finite time. It then transpires from Theorem 4.1 that T
ux (t, x)
L∞
dt = ∞.
(5.5)
0
Note that infx∈R ρ0 (x) > 0. By (5.2) in Lemma 5.1, we have ux (t, x)
1 1 CeCt CeCT < ∞ |ρ0 (x)| infx∈R ρ0 (x)
for all (t, x) ∈ [0, T ) × R, a contrary to (5.5). So T = +∞, and the solution (u, ρ) is global.
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