Journal of Functional Analysis 255 (2008) 1–12 www.elsevier.com/locate/jfa
Estimates for Hilbertian Koszul homology Xiang Fang 1 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA Received 21 March 2007; accepted 19 December 2007 Available online 14 April 2008 Communicated by G. Pisier
Abstract The objective of this paper is to give new kind of estimates for Hilbertian Koszul homology, inspired by commutative algebra, in multivariable Fredholm theory. © 2008 Elsevier Inc. All rights reserved. Keywords: Multivariable Fredholm theory; Koszul complex; Homology; Lech’s formula
0. Introduction The Fredholm index of a single operator admits a generalization to several variables via Koszul complexes over Hilbert spaces, which is, in general, difficult to calculate. In particular, in sharp contrast with rich results on Noetherian algebraic modules, over Hilbert modules currently there are essentially no systematic estimates for higher Koszul homology groups. In [13–15], we initiated a study of Fredholm theory through the asymptotic behavior of higher powers of a tuple T¯ . See also Eschmeier’s [12]. In this paper, the asymptotic methods lead to estimates for all powers of T¯ . Let T¯ = (T1 , . . . , Tn ) (n ∈ N) be a Fredholm tuple of commuting operators on a Hilbert space H . This means that the homology groups Hi (K(T1 , . . . , Tn )) (i = 0, 1, . . . , n) of the associated Koszul complex K(T1 , . . . , Tn ) of Ti over the Hilbert space H are all finite-dimensional. Let k = (k1 , . . . , kn ) ∈ Nn be a multi-index, and T¯ k = (T1k1 , . . . , Tnkn ). If T¯ is Fredholm, then so E-mail address:
[email protected]. 1 Partially supported by National Science Foundation Grant DMS 0400509.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2007.12.016
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is T¯ k . For convenience, let hi (k1 , . . . , kn ) = dim(Hi (K(T1k1 , . . . , Tnkn ))). The main result of this paper is Theorem 1. For any Fredholm tuple (T1 , . . . , Tn ), there exist e0 , e1 , . . . , en ∈ Z, and a constant C > 0 such that for all i = 0, 1, . . . , n, and k1 , k2 , . . . , kn ∈ N, k1 k2 · · · kn · ei hi (k1 , . . . , kn ) k1 k2 · · · kn
C ei + . min ki
A few remarks follow: • Considering the multi-index k is indeed useful, say, in [14], where n = 2, and supk hi (1, k) < ∞ implies ei = 0. h (T k ,...,T k )
• Clearly, our result implies that ei = limk→∞ i 1 k n n (see Corollary 2.3 in [12]) and that index index(T¯ ) = ni=0 (−1)i ei by the multiplicity formula index(T1k1 , . . . , Tnkn ) = k1 · · · kn index(T1 , . . . , Tn ). • When H is replaced by a finitely generated module over a Noetherian ring, the corresponding function hi is dominated by a polynomial of ki with degree n − i, hence ei = 0 except for exists. possibly e0 [26]. It is not clear whether limk→∞ hi (k,...,k) k n−i Two main ingredients in the proof of Theorem 1. Many arguments in this paper refine those of Eschmeier’s [12] in order to obtain quantitative results. The first set of techniques is sheaf theoretic. First touched upon by Markoe [22], sheaf theory for operators was systematically investigated later [25], and the primary reference is the monograph [11]. The second set is commutative algebra in nature, and is more closely related to our previous work. In particular, we own a deep intellectual debt to C. Lech [14,18,19], from which we borrow many ideas. Both sets of techniques are well known, and in fact easy, to experts in algebra and analysis, respectively. What we do here is to bring them together to yield estimates which appear of value in operator theory. 1. Background Definitions. In order to study the spectral theory of a tuple of commuting operators, instead of a single operator, J.L. Taylor, in 1970, introduced a seminal approach via Koszul complexes over Banach spaces [28,29]. For a commuting tuple T¯ = (T1 , . . . , Tn ) on a Banach space H , its Koszul complex K(T1 , . . . , Tn ) is K(T¯ ):
0→H ⊗
n
Cn → H ⊗
n−1
Cn → · · · → H ⊗
0
Cn → 0.
n Here n Cn is the kth exterior power of Cn . Let {e1 , . . . , en } be an orthonormal basis nfor C , and let ci be the creation operator associated with ei , that is, ci (ξ ) = ei ∧ ξ for ξ ∈ C . Then the boundary operator is B = T1 ⊗ c1∗ + · · · + Tn ⊗ cn∗ . The tuple (T1 , . . . , Tn ) is called invertible if the complex K(T¯ ) is exact.
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Subsequently, a multivariable Fredholm theory is formulated: a tuple T¯ of commuting operators is Fredholm if K(T¯ ) has a finite-dimensional homology group at each stage, that is, dimC Hi K(T¯ ) < ∞ for all i = 0, 1, . . . , n [1,2,7,8,11,30]. We also write Hi (T¯ ) instead of Hi (K(T¯ )) for convenience. The n + 1 homology groups of K(T¯ ) are the multivariable analogs of the kernel ker(T ) and cokernel H /T H of a single operator T ∈ B(H ). When (T1 , . . . , Tn ) is Fredholm, define the multivariable Fredholm index by index(T1 , . . . , Tn ) =
n (−1)i dimC Hi K(T¯ ) , i=0
the Euler characteristic of K(T¯ ). The multivariable index index(T¯ ) is connected with a variety of problems in both classical analysis and algebraic topology [3,11,20,21]. Currently, however, there is essentially no effective computational tools, especially for higher homology groups Hi (·), that is, for those groups with i > 0. Most known examples are, or are reduced to, acyclic tuples: Hi (·) = 0 except for i = 0, hence index(·) = dim(H0 (·)). Consequently, there is a current need to get a better grasp on those higher homology groups. Motivation. Our approach to Hi (·) originates from an effort to generalize the following simple arguments from [14] to several variables: for a single Fredholm operator T acting on a separable Hilbert space H , by the definition of Fredholm index, and the multiplicity formula,
index(T ) = =
index(T k ) k dim(ker(T k )) dim(H /T k H ) − k k dim(ker(T k )) dim(H /T k H ) − lim . k→∞ k→∞ k k
= lim
Here both limits exist, and are in fact integers. This leads to links to commutative algebra through the Hilbert function k → dim(H /T k H ), [10], and a celebrated result of J.-P. Serre, relating the Euler characteristics of Koszul complexes to Samuel multiplicities [27]. 2. Correction modules C(M, L; J ) This section is purely commutative algebra. We introduce a notion of correction modules, which, simple as it is, seems not discussed explicitly in literature. For operator theorists wanting more algebraic references, see standard texts [4,10] for Samuel multiplicity, and see [14,18,19], and [26] for Lech’s formulas.
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Definition 2. Let R be a ring, J ⊂ R be an ideal, and M ⊂ L be a submodule of an R-module L. Define the correction module of M in L with respect to an ideal J to be C(M, L; J ) =
M ∩ JL . JM
Remark. When R and L are Noetherian, the Artin–Rees lemma is useful for the study of the asymptotic behavior of C(M, L; J k ) when k → ∞. Lemma 3. Let R be a local Noetherian ring, I = (x1 , . . . , xn ) ⊂ R be its maximal ideal, and Ik = (x1k1 , . . . , xnkn ) for any k ∈ Nn . If L is a finitely generated R-module, and M ⊂ L is a submodule, then there exists a constant C such that for all k ∈ Nn , length C(M, L; Ik ) k1 k2 · · · kn ·
C . min kj
Proof. Let N = L/M be the quotient module. For any ideal J ⊂ R, applying the functor (·) ⊗R R/J , which is only right-exact, to a short exact sequence of R-modules 0 → M → L → N → 0,
(2.1)
→ M/J M → L/J L → N/J N → 0.
(2.2)
we get a right-exact sequence
By the definition of correction module, it follows 0 → C(M, L; J ) → M/J M → L/J L → N/J N → 0.
(2.3)
Consider J = Ik , and by the algebraic Lech’s formula (see Lemma 4), there exists a constant CE for the modules E = M, L, or N , such that CE . k1 · · · kn · e(E) length(E/Ik E) k1 · · · kn e(E) + min kj
(2.4)
Here length(E/I t E) t→∞ tn
e(E) = n! lim
is the Samuel multiplicity of E with respect to I . By the additivity of Samuel multiplicity over short exact sequence (2.1), we have e(L) = e(M) + e(N ). Now the proof is completed by observing
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length C(M, L; Ik ) = length(M/Ik M) + length(N/Ik N ) − length(L/Ik L) k1 k2 · · · kn ·
C M + CN . min kj
2
Remarks. (1) We derive the name of C(M, L; J ) from (2.3). (2) For the proof of Theorem 1, the only case we need is R = O0 , the local ring of germs of holomorphic functions around the origin in Cn . For readers’ convenience we record the following. Lemma 4 (Lech’s inequality). Let J = (x1 , . . . , xn ) be an ideal of a local ring R, generated by xi , and let M be a Noetherian R-module such that length(M/J M) < ∞, then there exists a constant C such that k1 C kn , k1 · · · kn e(M, J ) length M/ x1 , . . . , xn M k1 · · · kn e(M, J ) + minj kj here e(M, J ) is the Samuel multiplicity of M with respect to J . The original proof of Lech is contained in the proof of Theorem 2 in [18], which in fact only covers the case M = R. The (Hilbert) module case is treated in [14]. Both proofs can be easily adopted to prove Lemma 4. 3. Difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) as correction modules This section is still purely algebraic. Let R be any commutative ring, and L• :
· · · → Lp → Lp−1 → · · ·
be a complex of R-modules, with Hp (L• ) denoting the homology group at the pth stage, p ∈ Z. For any ideal J ⊂ R, we represent the difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) as correction modules in this section. This is also considered in [12]. Here we refine the arguments in [12] and obtain more quantitative results. Since the difference between Hp (L• /J L• ) and Hp (L• )/J Hp (L• ) is often encountered, say in base change theorems, in algebraic geometry, our results here may be of interests to algebraists. Recall that for any R-module M, there exists a natural morphism (say, by [5]) Hp (L• ) ⊗R M → Hp (L• ⊗R M). Here we will consider M = J , and R/J . Let Zp ⊂ Lp be the set of closed elements, that is, Zp = ker(Lp → Lp−1 ), and let Bp ⊂ Lp be the set of boundary elements, that is, Bp = Image(Lp+1 → Lp ). Note that Hp (L• ) = Zp /Bp . Lemma 5. For the natural morphism j : Hp (L• )/J Hp (L• ) → Hp (L• /J L• ), the cokernel is isomorphic to coker(j ) ∼ = C(Bp−1 , Lp−1 ; J ).
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The kernel ker(j ) is resolved by an exact sequence of correction modules 0 → C(Bp , Zp ; J ) → C(Bp , Lp ; J ) → C(Zp , Lp ; J ) → ker(j ) → 0. Proof. The standard strategy in algebra is to analyze the natural morphism j : Hp (L• )/J Hp (L• ) → Hp (L• /J L• ) by embedding it into a commutative diagram. To resolve Hp (L• /J L• ), we consider the long exact sequence associated with 0 → J L• → L• → L• /J L• → 0, and get the first row of the diagram (3.1). To resolve Hp (L• )/J Hp (L• ) we consider the straightforward short exact sequence which leads to the second row of the diagram (3.1). Together with the natural morphisms j1 , j2 = id, and j , we obtain a commutative diagram Hp (J L• ) j1
0
d1
Hp (L• )
d2
Hp (L• /J L• )
j2
J Hp (L• )
δ
Hp−1 (J L• )
d3
Hp−1 (L• ) (3.1)
j
Hp (L• )
Hp (L• )/J Hp (L• )
0.
The cokernel part is easier. By the second commutative square in the diagram (3.1), and the exactness of the first row in (3.1), coker(j ) = coker(d2 ) ∼ = Image(δ) = ker(d3 ). Let Z∗ (J L• ) (respectively B∗ (J L• )) denote the closed (respectively boundary) elements of the complex J L• . Then ker(d3 ) =
Zp−1 (J L• ) ∩ Bp−1 J Lp−1 ∩ Zp−1 ∩ Bp−1 J Lp−1 ∩ Bp−1 = = . Bp−1 (J L• ) J Bp−1 J Bp−1
Now consider ker(j ). By the second commutative square, and exactness of both rows in (3.1), ker(j ) =
Image(d1 ) ker(d2 ) = . J Hp (L• ) J Hp (L• )
Note that Image(d1 ) =
J Lp ∩ Zp + Bp Bp
and J Hp (L• ) =
J Zp + Bp . Bp
Hence we can resolve ker(j ) by C(Zp , Lp ; J ) 0→
J Lp ∩ Zp J Lp ∩ Zp + Bp (J Lp ∩ Zp ) ∩ (J Zp + Bp ) → → → 0. J Zp J Zp J Zp + Bp
(3.2)
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Observe that (J Lp ∩ Zp ) ∩ (J Zp + Bp ) = (J Lp ∩ Zp ) ∩ Bp + J Zp = J Lp ∩ Bp + J Zp . Here the first equality is because if x ∈ J Zp , y ∈ Bp such that x + y ∈ J Lp ∩ Zp , then y ∈ J Lp ∩ Zp since x ∈ J Zp ⊂ J Lp ∩ Zp . Hence the left-hand side of (3.2) is isomorphic to J Lp ∩ Bp J Lp ∩ Bp J Lp ∩ Bp + J Zp ∼ = . = J Zp (J Lp ∩ Bp ) ∩ J Zp J Zp ∩ Bp But the last one is resolved by correction modules C(Bp , Zp ; J ) and C(Bp , Lp ; J ) 0→
J Zp ∩ Bp J Lp ∩ Bp J Lp ∩ Bp → → → 0. J Bp J Bp J Zp ∩ Bp
Now patching (3.2) and (3.3) completes the proof of the lemma.
(3.3)
2
4. Parametrized Koszul complexes In this section sheaf theory comes into the play. For more background interested readers should see [11], especially those arguments related to Lemma 2.1.5, Proposition 9.4.5, and Theorem 10.3.13. Here our approach is slightly more algebraic. It allows conceptual proofs and leads to conjectures for further development. We start with a connection between a Hilbert module H over a ring R, associated with an operator tuple T¯ = (T1 , . . . , Tn ), and its sheaf model [11,25], h = O(H )/(z − T¯ )O(H ), as well as its stalk at the origin h0 = O0 (H )/(z − T¯ )O0 (H ). Here R is any of the following three rings C[z1 , . . . , zn ],
O(Cn ),
and O(U ),
with U being a Stein neighborhood of the Taylor spectrum σ (T¯ ). In any case, and even for O0 , let I = (z1 , . . . , zn ) be the maximal ideal at the origin. We usually assume that dim(H /I H ) < ∞ and 0 ∈ σ (T¯ ). In an effort to relate H to h0 , Douglas and Yan showed in [9] that the Hilbert function of H , with respect to I , is greater than or equal to the Hilbert function of h0 . In [15] we showed that the inequality between the two Hilbert functions is in fact an equality. This plays a key role in the proof of the semi-continuity of Samuel multiplicity over Hilbert modules. The result from [15] can be reformulated as that the completions of H and h0 in the so-called I -adic topology [27] are isomorphic, Hˆ ∼ = hˆ 0 ,
(4.1)
which is better suited for generalization. In particular, an easy consequence of (4.1) is H /I H ∼ = h0 /I h0 .
(4.2)
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Next we aim at the homological generalizations of (4.1) and (4.2). First, rewrite the completion Hˆ as an inverse limit H /I k H. Hˆ = lim ←− k→∞
Let Ik = (z1k , . . . , znk ) ⊂ R. Then, by basic facts on inverse limits H /Ik H. Hˆ = lim ←− k→∞
Observe that H /Ik H can be written as the 0th homology group of the Koszul complex K(T1k , . . . , Tnk ; H ) of (T1k , . . . , Tnk ) on H . On the other hand, the sheaf model h can be written as the 0th homology group H0 (z − T¯ , O(H )) of the Koszul complex of z − T¯ = (z1 − T1 , . . . , zn − Tn ) on O(H ). To generalize Eq. (4.1), observe that, for each i = 0, 1, . . . , n, we can form an inverse system of the Koszul homology groups, [6,16],
k Hi T1 , . . . , Tnk ; H , k = 1, 2, . . . . Definition 6. For each i = 0, 1, . . . , n, we define Hˆ i = lim Hi T1k , . . . , Tnk ; H . ←− k→∞
For the sheaf side, as generalization of the sheaf model h = h(0) , we call h(i) = Hi z − T¯ , O(H ) , i = 0, 1, . . . , n, the homological sheaf models of H . The modules Hˆ i are reminiscent of Grothendieck’s local cohomology modules in algebraic geometry [16]. According to Markoe [22], h(i) is in fact a coherent analytic sheaf around the origin for each i when T¯ is Fredholm. The significance of Hˆ i and h(i) is yet to be understood. As a first step, and as a generalization of (4.1), we offer the following conjecture. Let h(i),0 denote the stalk at the origin, and hˆ (i),0 its I -adic completion. Conjecture. For any Fredholm tuple T¯ and each i = 0, 1, . . . , n, we have a natural isomorphism Hˆ i ∼ = hˆ (i),0 of modules over the ring of power series C z1 , . . . , zn . As for Eq. (4.2), observe that h0 /I h0 can be written as the 0th homology group of the Koszul complex K(z − T¯ ; R/I ⊗ H ) of z − T¯ = (z1 − T1 , . . . , zn − Tn ) on O0 (H )/I O0 (H ) = R/I ⊗C H . Note that O0 /I O0 are isomorphic to R/I , as Artinian rings, for any of R = C[z1 , . . . , zn ], O(Cn ), and O(U ). For each of these three rings, we generalize (4.2) to Lemma 7. Let f = (f1 , . . . , fn ) be a regular sequence in R, and (f ) be the ideal generated by fj . Then Hi f1 (T¯ ), . . . , fn (T¯ ); H ∼ = Hi z − T¯ , R/(f ) ⊗C H .
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Remarks. (1) Here f being regular means that the Koszul complex of f on R yields a free resolution of R/(f ) [10]. In particular, length(R/(f )) < ∞. (2) Under the condition f −1 (0) = 0, Lemma 7 is already covered in [12] which, in turn, is modeled after the proof of Theorem 10.3.13 in [11]. Modulo technical matters, what is new here is just the way it is presented. (3) Our proof is essentially only a series observations in homological algebra, which can establish the result for a larger category, and motivates a further conjecture—see the remark at the end of the paper. Proof. When i = 0, both sides are directly verified to be H /
fj (T¯ )H . In fact one has
H0 z − T¯ , R/(f ) ⊗C H ∼ = R/(f ) ⊗R H. The natural map from the left to the right is the class of x ⊗ y ∈ R/(f ) ⊗C H being sent to the class of x ⊗ y ∈ (R/(f )) ⊗ H . It is clearly surjective with kernel being the submodule generated by rx ⊗C y − x ⊗C ry, which is the same as that generated by (zi − Ti )(x ⊗C y) = zi x ⊗C y − x ⊗C Ti y [10]. For general i, since f is regular, the Koszul homology is also given by the derived functors TorR i (·, ·), Hi f1 (T¯ ), . . . , fn (T¯ ); H = TorR i R/(f ), H . Let Rw denote the ring R with variables written in w = (w1 , . . . , wn ), and consider H as a module over Rw . Then Hi (z − T¯ , R/(f ) ⊗C H ), viewed as a module over Rz ⊗ Rw , is naturally a module over Rw . Hence, the functors Fi : M → Hi (z − T¯ , R/(f ) ⊗C M) can be regarded as over the category of Rw (= R)-modules. To show the sequence of functors Fi : M → Hi (z − T¯ , R/(f ) ⊗C M) coincide with the derived functors M → TorR i (R/(f ), M) for any R-module M, we only need to show that F = (Fi ) is a universal δ-functor—here we use the machinery in homological algebra as explained in Section 3.1, [17]. Being a δ-functor is clear by definition. To show it is universal, it suffices to show that Fi is coeffaceable for i > 0. This can be verified by (1) the category of all R-modules have enough projectives, and (2) Fi (R) = 0 when i > 0. The first is algebraic folklore, and the second follows easily from the definition of regular sequence. Next we give more details for Hi z − w, O(U )/(f ) ⊗C O(U ) = 0 (i > 0) for readers’ convenience [10]. Because (z − w) forms a regular sequence in O(U ) ⊗C O(U ), the Koszul homology can be calculated via O(U )⊗C O(U )
Tori
O(U ) ⊗C O(U )/(z − w), O(U )/(f ) ⊗C O(U ) .
Since (f ) is regular by assumption, the Koszul complex K(f, O(U )) provides a free resolution of O(U )/(f ), hence K(f, O(U )) ⊗C O(U ) a free resolution of O(U )/(f ) ⊗C O(U ). Hence the above Tori can be calculated through the complex
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K f, O(U ) ⊗C O(U ) ⊗O(U )⊗C O(U ) O(U ) ⊗C O(U )/(z − w) ∼ = K f, O(U ) ⊗C O(U )/(z − w). The last term, regarded as a complex of O(U )-modules in the variable z, is isomorphic to K(f, O(U )), which is acyclic, hence Hi (· · ·) = 0. 2 ∼ O0 (H )/J O0 (H ). If L• = K(z − T¯ , Let J = (f ) = (z1k1 , . . . , znkn ), then R/(f ) ⊗C H = ¯ O0 (H )) denotes the Koszul complex of z − T on O0 (H ), then, by Lemma 7, Hi T1k1 , . . . , Tnkn ∼ = Hi (L• /J L• ). Since O0 (H ) in L• is an infinitely generated O0 -module when dim(H ) = ∞, a standard strategy for parametrized complexes is to find a complex of finitely generated O0 -modules with isomorphic homology groups, which will allow us to apply results from Section 3. Lemma 8. If T¯ is Fredholm, then there exists a complex E• of finitely generated O0 -modules: · · · → Ei → Ei−1 → · · · , such that for J = 0, or any k = (k1 , . . . , kn ) ∈ Nn and J = (z1k1 , . . . , znkn ), Hi (L• /J L• ) ∼ = Hi (E• /J E• ),
i ∈ Z.
Proof. This is essentially due to [11] and [12]. The construction of E• is detailed in [11]. The isomorphism between homology groups is verified in [12]. 2 Proof of Theorem 1. Since the components Ei in E• are finitely generated O0 -modules, so are the homology groups Hi (E• ). By Lemma 4, the function φ(k) = dim Hi (E• )/J Hi (E• ) satisfies, for some constant C, φ(k) k1 · · · kn ei +
C , min kj
here ei = ei (T¯ ) is the Samuel multiplicity of Hi (E• ) = Hi (L• ) with respect to I . Now, the estimates on correction modules, together with the representation of the difference between Hi (E• )/J Hi (E• ) and Hi (E• /J E• ) as correction modules, completes the proof of the upper bound in Theorem 1. The lower bound is much easier, and is in fact part of Theorem 2.4 in [12]. Our treatment here is just slightly different. For fixed k, we claim that hi (k1 , . . . , kn ) = ei · k1 · · · kn when the tuple is T¯ − λ, where λ is in a small neighborhood of the origin except for a possibly thin subvariety. Because hi (k1 , . . . , kn ) is upper semi-continuous around the origin as a function of λ in T¯ − λ, we get the lower bound. Since the singularity set of the coherent sheaf Hi (L• ) is thin, we can choose small λ such that, with respect to the tuple T¯ − λ, Hi (L• ) is free, and, for any primary ideal J , the following are naturally isomorphic: Hi (L• /J L• ) ∼ = Hi (L• )/J Hi (L• ) (see Grauert’s comparison theorem [5,17]).
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Here the latter has dimension ei · dim(O0 /J ) since Hi (L• ) is free. Choosing J = (z1k1 , . . . , znkn ) gives us the claim. 2 Because Hi (z − T¯ , O(H )) is coherent around the origin when T¯ is Fredholm [22], and ei is the Samuel multiplicity of the stalk of Hi (z − T¯ , O(H )) at the origin, a straightforward consequence is, by the invariance of Samuel multiplicity of stalks of a coherent analytic sheaf, that is Hi (z − T¯ , O(H )) in our case, the local constancy of ei (T¯ − λ). Corollary 9. If T¯ is Fredholm, then the function λ ∈ Cn → ei (T¯ − λ) is locally constant in a neighborhood of the origin. In other words, let Ω be a connected component of the Fredholm domain Cn \ σe (T¯ ), then ei (T¯ − λ) is a constant for λ ∈ Ω. Motivated by the base change formula for Fredholm index index(f1 (T¯ ), . . . , fn (T¯ )) [24], it is natural to ask whether similar formulas hold for Hi (f1 (T¯ ), . . . , fn (T¯ )) and ei (f1 (T¯ ), . . . , fn (T¯ )). In general, Hi (·) is too unstable to enjoy a nice base change formula. For ei , however, we observe that the proof of the base change formula in Theorem 10.3.16 in [11] goes, roughly, as follows. The key in reduction is that index(T¯ − λ) is locally constant in λ. For a neighborhood U ⊃ σ (T¯ ) of the Taylor spectrum σ (T ), and a map F = (f1 , . . . , fn ) : U → Cn with F (0) = 0, we can consider index(F (T¯ ) − λ) such that the fibre (F − λ)−1 (0) is simple, that is, a collection of k distinct points {p1 , . . . , pk }, here k being the mapping degree of f at 0. Then over each simple point pi , the contribution to index can be counted directly, hence leading to the base change formula. Now, based on Corollary 9, the whole proof in [11] carries over for ei (·). Corollary 10. Let T¯ be a Fredholm tuple, and F ∈ O(U )n be an n-tuple of analytic functions defined on an open neighborhood U of the Taylor spectrum σ (T¯ ). Assume that F (0) = 0 and 0∈ / σe (F (T¯ )), and let mz (F ) denote the multiplicity of F at z. Then, for each i = 0, 1, . . . , n,
ei F (T¯ ) =
mz (F )ei (T¯ − z).
z∈F −1 (0)∩σ (T¯ )
We end the paper with a remark when f = (f1 , . . . , fn ) in R = C[z1 , . . . , zn ], O(Cn ), or O(U ), is not necessarily a regular sequence. If we rewrite R/(f ) as the 0th Koszul homology of f on R, then Lemma 7 becomes Hi f1 (T¯ ), . . . , fn (T¯ ); H ∼ = Hi z − T¯ , H0 (f, R) ⊗C H . Hence it motivates Conjecture. For general f , there exists a spectral sequence, with E 2 page 2 ∼ Epq = Hp z − T¯ , Hq (f, R) ⊗C H , convergent to Hp+q (f1 (T¯ ), . . . , fn (T¯ ); H ).
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X. Fang / Journal of Functional Analysis 255 (2008) 1–12
Adopting this viewpoint, Lemma 7, that is when f is regular, actually follows immediately from Grothendieck’s spectral sequence of composition functors [23,31]. For the general case, we will address the conjecture by constructing spectral sequences directly from double complexes in a coming work. Acknowledgment The author thanks J. Eschmeier for sending several of his manuscripts and papers, and for communications which prompt this work. References [1] C.-G. Ambrozie, F.-H. Vasilescu, Banach Space Complexes, Math. Appl., vol. 334, Kluwer, Dordrecht, 1995. [2] W. Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189 (1) (2002) 53–79. [3] M. Atiyah, A survey of K-theory, in: K-Theory and Operator Algebras, Proc. Conf., Univ. Georgia, Athens, GA, 1975, in: Lecture Notes in Math., vol. 575, Springer-Verlag, Berlin, 1977, pp. 1–9. [4] S. Balcerzyk, T. Jozefiak, Commutative Rings. Dimension, Multiplicity and Homological Methods, Prentice–Hall, Englewood Cliffs, NJ, 1989. [5] C. Banica, O. Stanasila, Algebraic Methods in the Global Theory of Complex Spaces, Wiley, 1976 (translated from Rumanian). [6] M.P. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math., vol. 60, Cambridge Univ. Press, Cambridge, 1998. [7] R.E. Curto, Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1) (1981) 129–159. [8] R.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, Harlow, 1988, pp. 25–90. [9] R. Douglas, K. Yan, Hilbert–Samuel polynomials for Hilbert modules, Indiana Univ. Math. J. 42 (1993) 811–820. [10] D. Eisenbud, Commutative Algebra. With a View toward Algebraic Geometry, Grad. Texts in Math., vol. 150, Springer-Verlag, New York, 1995. [11] J. Eschmeier, M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monogr. (N.S.), vol. 10, Oxford Univ. Press, New York, 1996. [12] J. Eschmeier, Samuel multiplicity and Fredholm theory, preprint. [13] X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math. 186 (2) (2004) 411–437. [14] X. Fang, The Fredholm index of a pair of commuting operators, Geom. Funct. Anal. 16 (2) (2006) 367–402. [15] X. Fang, The Fredholm index of a pair of commuting operators, II, preprint. [16] R. Hartshorne, Local Cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, Berlin, New York, 1967. [17] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer-Verlag, New York, 1977. [18] C. Lech, On the associativity formula for multiplicities, Ark. Mat. 3 (1957) 301–314. [19] C. Lech, Notes on multiplicities of ideals, Ark. Mat. 4 (1959) 63–86. [20] R.N. Levy, The Riemann–Roch theorem for complex spaces, Acta Math. 158 (3–4) (1987) 149–188. [21] R.N. Levy, Algebraic and topological K-functors of commuting n-tuple of operators, J. Operator Theory 21 (2) (1989) 219–253. [22] A. Markoe, Analytic families of differential complexes, J. Funct. Anal. 9 (1972) 181–188. [23] J. McCleary, A User’s Guide to Spectral Sequences, second ed., Cambridge Stud. Adv. Math., vol. 58, 2000. [24] M. Putinar, Base change and the Fredholm index, Integral Equations Operator Theory 8 (5) (1985) 674–692. [25] M. Putinar, Spectral theory and sheaf theory, II, Math. Z. 192 (3) (1986) 473–490. [26] P. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Math., vol. 133, Cambridge Univ. Press, Cambridge, 1998. [27] J.-P. Serre, Local Algebra, Springer Monogr. Math., Springer-Verlag, Berlin, 2000. [28] J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970) 172–191. [29] J.L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970) 1–38. [30] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, translated from the Romanian, Math. Appl. (East European Ser.), vol. 1, D. Reidel Publ. Co., Dordrecht, 1982 [Editura Academiei Republicii Socialiste Romania, Bucharest, 1982]. [31] C.A. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., vol. 38, Cambridge Univ. Press, Cambridge, 1994.
Journal of Functional Analysis 255 (2008) 13–24 www.elsevier.com/locate/jfa
Spectral properties of the canonical solution operator to ∂¯ Friedrich Haslinger 1 , Bernhard Lamel ∗,2 Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Österreich Received 25 July 2007; accepted 17 March 2008 Available online 28 April 2008 Communicated by Paul Malliavin
Abstract We study boundedness, compactness, and Schatten-class membership of the canonical solution operator ¯ restricted to (0, 1)-forms with holomorphic coefficients, on L2 (dμ) where μ is a measure with the to ∂, property that the monomials form an orthogonal family in L2 (dμ). The characterizations are formulated in terms of moment properties of μ. Our results generalize the results of the first author to several variables, contain some known results for several variables, and also cover new ground. © 2008 Elsevier Inc. All rights reserved. Keywords: dbar-Equation; Solution operator; Compactness
1. Introduction and statement of results In this paper, we study spectral properties of the canonical solution operator to ∂¯ acting on spaces of (0, 1)-forms with holomorphic coefficients in L2 (dμ) for measures μ with the property that the monomials zα , α ∈ Nn , are orthogonal in L2 (dμ). This situation covers a number of basic examples:
* Corresponding author.
E-mail addresses:
[email protected] (F. Haslinger),
[email protected] (B. Lamel). 1 Supported by the FWF, Projekt P19147. 2 Supported by the FWF, Projekt P17111.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.013
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F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
• Lebesgue measure on bounded domains in Cn which are invariant under the torus action (θ1 , . . . , θn )(z1 , . . . , zn ) → eiθ1 z1 , . . . , eiθn zn (i.e. Reinhardt domains). • Weighted L2 spaces with radial-symmetric weights (e.g., generalized Fock spaces). • Weighted L2 spaces with decoupled radial weights, that is, dμ = e
j
ϕj (|zj |2 )
dV ,
where ϕj : R → R is a weight function. Sufficient conditions for the weight in order for the Fock space to be infinite-dimensional are known from the work of Shigekawa [12]. Some of these examples have been studied previously; our approach has the advantage of unifying these previous result as well as of being applicable in new situations as well. Our main focus in this paper is the case n > 1; indeed, we generalize results of the first author (see [4–6]) to this setting. The behaviour of the canonical solution operator S is interesting from many points of view. ¯ First, there is a close connection between properties of S and properties of the ∂-Neumann op∗ ¯ erator N ; indeed, S = ∂ N . In particular, noncompactness of S prohibits compactness of N . As is well known, S behaves quite nicely on spaces of (0, 1)-forms with holomorphic coefficients, and we shall exploit this connection. On the other hand, for convex domains, a result of Fu and Straube [2] shows that compactness of S on forms with holomorphic coefficients is also sufficient for compactness on all of L2 . There is also an intriguing connection between the canonical solution operator S and the theory of magnetic Schrödinger operators (see [3] and [7]); this connection has been exploited in the recent paper of the first author and Helffer [8] in order to study compactness of S on general (not rotation-invariant) weighted L2 -spaces on Cn . Let us introduce the notation used in this paper. We denote by A2 (dμ) = zα : α ∈ Nn , the closure of the monomials in L2 (dμ), and write mα = cα−1
=
α 2 z dμ.
We will give necessary and sufficient conditions in terms of these multimoments of the measure ¯ when restricted to (0, 1)-forms with coefficients in μ for the canonical solution operator to ∂, 2 A (dμ) to be bounded, compact, and to belong to the Schatten class Sp . This is accomplished by presenting a complete diagonalization of the solution operator by orthonormal bases with corresponding estimates. In the case of radial-symmetric measures our results specializes to the results of [10] applied to this specific case; we are also able to characterize membership in Sp for all positive p in some cases (a question left open in [10]).
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
15
As usual, for a given function space F , F(0,1) denotes the space of (0, 1)-forms with coefficients in F , that is, expressions of the form n
fj d z¯ j ,
fj ∈ F.
j =0
The ∂¯ operator is densely defined operator ¯ = ∂f
n ∂f d z¯ j . ∂ z¯ j j =1
The canonical solution operator S assigns to each ω ∈ L2(0,1) (dμ) the solution to the ∂¯ equation which is orthogonal to A2 (dμ); this solution need not exist, but if the ∂¯ equation for ω can be solved, then Sω is defined, and is given by the unique f ∈ L2 (dμ) which satisfies ¯ =ω ∂f
in the sense of distributions and f ⊥ A2 (dμ).
Our main interest in this paper is the spectral behaviour of the map S restricted to A2(0,1) (dμ). We first give a criterion for S to be a bounded operator. We will frequently encounter multiindices γ which might have one (but not more than one) entry equal to −1: in that case, we define cγ = 0. We will denote the set of these multi-indices by Γ . We let ej = (0, . . . , 1, . . . , 0) be the multi-index with a 1 in the j th spot and 0, elsewhere. Theorem 1. S : A2(0,1) (dμ) → L2 (dμ) is bounded if and only if there exists a constant C such that cγ +ep cγ +2ep
cγ
−
for all multi-indices γ ∈ Γ . We have a similar criterion for compactness. Theorem 2. S : A2(0,1) (dμ) → L2 (dμ) is compact if and only if
lim γ
cγ +ep cγ +2ep
−
cγ cγ +ep
=0
(1)
for all p = 1, . . . , n. In particular, the only if implication of Theorem 2 implies several known noncompactness statements for S, e.g. of Knirsch and Schneider [9], Schneider [11], as well as the noncompactness of S on the polydisc. The main interest in these noncompactness statements is that if S fails ¯ to be compact, so does the ∂-Neumann operator N . The multimoments also lend themselves to characterizing the finer spectral property of being in the Schatten class Sp . Let us recall that an operator T : H1 → H2 belongs to the Schatten class Sp if the self-adjoint operator T ∗ T has a sequence of eigenvalues belonging to p .
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F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
Theorem 3. Let p > 0. Then S : A2(0,1) (dμ) → A2 (dμ) is in the Schatten-p-class Sp if and only if cγ +ej γ ∈Γ
−
cγ +2ej
j
p/2
cγ
< ∞.
cγ +ej
(2)
The condition above is substantially easier to check if p = 2 (we will show that the sum is actually a telescoping sum then), i.e. for the case of the Hilbert–Schmidt class; we state this as a theorem. Theorem 4. The canonical solution operator S is in the Hilbert–Schmidt class if and only if
lim
k→∞
cγ < ∞. cγ +ep
γ ∈Nn , |γ |=k 1pn
(3)
1.1. Application in the case of decoupled weights Let us apply Theorem 1 to the case of decoupled weights, or more generally, of product measures dμ = dμ1 × · · · × dμn , where each dμj is a (circle-invariant) measure on C. Note that for such measures, there is definitely no compactness by Theorem 2. If we denote by
j
ck =
|z|2j dμk
−1 ,
C
we have that c(γ1 ,...,γn ) =
n
γ
ck k .
k=1
We thus obtain the following corollary. Corollary 5. For a product measure dμ = dμ1 × · · · × dμn as above, the canonical solution operator S : A2(0,1) (dμ) → L2 (dμ) is bounded if and only if there exists a constant C such that j +1
ck
j +2
ck
j
−
ck
j +1
ck
for all j ∈ N and for all k = 1, . . . , n. Equivalently, S is bounded if and only if the canonical solution operator Sj : A2 (dμj ) → L2 (dμj ) is bounded for every j = 1, . . . , n.
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
17
1.2. Application in the case of rotation-invariant measures In the case of a rotation-invariant measure μ, we write md = |z|2d dμ; Cn
a computation (see [10, Lemma 2.1]) implies that cγ =
(n + |γ | − 1)! 1 . (n − 1)!γ ! m|γ |
(4)
In order to express the conditions of our theorems, we compute (setting d = |γ | + 1) cγ +ep p
cγ +2ep
−
d+2n−1 md+1
cγ cγ +ep
=
d+n md 1 md+1 d+n md ,
−
md md−1 ,
γp = −1 for all p, else.
(5)
Note that the Cauchy–Schwarz inequality implies that for large enough d, the first case in (5) always dominates the second case; using this observation and some trivial inequalities, we get the following corollaries, which should be compared to the results of the first author in the onedimensional case [6] and the results of Lovera and Youssfi [10]. Corollary 6. Let μ be a rotation-invariant measure on Cn . Then the canonical solution operator to ∂¯ is bounded on A2(0,1) (dμ) if and only if
sup d∈N
(2n + d − 1)md+1 md − (n + d)md md−1
< ∞.
(6)
Corollary 7. Let μ be a rotation-invariant measure on Cn . Then the canonical solution operator to ∂¯ is compact on A2(0,1) (dμ) if and only if
lim
d→∞
(2n + d − 1)md+1 md − (n + d)md md−1
= 0.
(7)
Corollary 8. Let μ be a rotation-invariant measure on Cn . Then the canonical solution operator to ∂¯ is a Hilbert–Schmidt operator on A2(0,1) (dμ) if and only if
lim
d→∞
n + d − 1 md+1 < ∞. n−1 md
(8)
Corollary 9. Let μ be a rotation-invariant measure on Cn , p > 0. Then the canonical solution operator to ∂¯ is in the Schatten class Sp , as an operator from A2(0,1) (dμ) to L2 (dμ) if and only if
∞
n + d − 2 (2n + d − 1)md+1 d=1
n−1
(n + d)md
md − md−1
p/2 < ∞.
(9)
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F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
In particular, Corollary 9 improves [10, Theorem C] in the sense that it also covers the case 0 < p < 2. We would like to note that our techniques can be adapted to the setting of [10] by considering the canonical solution operator on a Hilbert space H of holomorphic functions endowed with a norm which is comparable to the L2 -norm on each subspace generated by monomials of a fixed degree d, if in addition to the requirements in [10] we also assume that the monomials belong to H; this introduces the additional weights found by [10] in the formulas, as the reader can check. In our setting, the formulas are somewhat “cleaner” by working with A2 (dμ) (in particular, Corollary 8 only holds in this setting). 2. Monomial bases and diagonalization In what follows, we will denote by uα =
√ α cα z
the orthonormal basis of monomials for the space A2 (dμ), and by Uα,j = uα d z¯ j the corre¯ sponding basis of A2(0,1) (dμ). We first note that it is always possible to solve the ∂-equation for the elements of this basis; indeed, ∂¯ z¯ j uα = Uα,j . The canonical solution operator is also easily determined for forms with monomial coefficients: Lemma 10. The canonical solution Szα d z¯ j for monomial forms is given by Szα d z¯ j = z¯ j zα −
cα−ej cα
zα−ej ,
α ∈ Nn .
(10)
Proof. We have ¯zj zα , zβ = zα , zβ+ej ; so this expression is nonzero only if β = α − ej (in particular, if this implies (10) for multi-indices α with αj = 0; recall our convention that cγ = 0 if one of the entries of γ is negative). Thus Szα d z¯ j = z¯ j zα + czα−ej , and c is computed by −1 0 = z¯ j zα + czα−ej , zα−ej = cα−1 + ccα−e , j which gives c = −cα−ej /cα .
2
We are going to introduce an orthogonal decomposition A2(0,1) (dμ) =
Eγ
γ ∈Γ
of A2(0,1) (dμ) into at most n-dimensional subspaces Eγ indexed by multi-indices γ ∈ Γ (we will describe the index set below), and a corresponding sequence of mutually orthogonal finitedimensional subspaces Fγ ⊂ L2 (dμ) which diagonalizes S (by this we mean that SEγ = Fγ ). To motivate the definition of Eγ , note that
Sz d z¯ k , Sz d z¯ = α
β
0,
cα 1 cα cα+e
−
cα−ek cα+e −ek
,
β = α + e − ek , β = α + e − ek ,
(11)
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
19
so that Szα d z¯ k , Szβ d z¯ = 0 if and only if there exists a multi-index γ such that α = γ + ek and β = γ + e . We thus define Eγ = span{Uγ +ej ,j : 1 j n} = span zγ +ej d z¯ j : 1 j n , and likewise Fγ = SEγ . Recall that Γ is defined to be the set of all multi-indices whose entries are greater or equal to −1 and at most one negative entry. Note that Eγ is 1-dimensional if exactly one entry in γ equals −1, and n-dimensional otherwise. We have already observed that Fγ are mutually orthogonal subspaces of L2 (dμ). Whenever we use multi-indices γ and integers p ∈ {1, . . . , n} as indices, we use the convention that the p run over all p such that γ + ep 0; that is, for a fixed multi-index γ ∈ Γ , either the indices are either all p ∈ {1, . . . , n} or there is exactly one p such that γp = −1, in which case the index is exactly this one p. We next observe that we can find an orthonormal basis of Eγ and an orthonormal basis of Fγ such that in these bases Sγ = S|Eγ : Eγ → Fγ acts diagonally. First note that it is enough to do this if dim Eγ = n (since an operator between one-dimensional spaces is automatically diagonal). Fixing γ , the functions Uj := Uγ +ej ,j are an orthonormal basis of Eγ . The operator Sγ is clearly nonsingular on this space, so the functions SUj = Ψj constitute a basis of Fγ . For a basis B of j j vectors v j = (v1 , . . . , vn ), j = 1, . . . , n, of Cn we consider the new basis Vk =
n
j
v k Uj ;
j =1
since the basis given by the Uj is orthonormal, the basis given by the Vk is also orthonormal provided that the vectors vk = (vk1 , . . . , vkn ) constitute an orthonormal basis for Cn with the standard Hermitian product. Let us write Φk = SVk =
j
vk SUj .
j
The inner product Φp , Φq is then given by ⎛
Φ1 , Φ1 · · · ⎜ .. ⎝ .
Φn , Φ1 · · · ⎛ v1 · · · 1 ⎜ . = ⎝ .. vn1 · · ·
⎞
Φ1 , Φn ⎟ .. ⎠ .
Φn , Φn n v1 ⎞ ⎛ Ψ1 , Ψ1 .. .. ⎟ ⎜ . . ⎠⎝ n
Ψn , Ψ1 vn
j k j,k vp v¯ q SUj , SUk .
···
We therefore have
⎞ ⎛ v¯ 1
Ψ1 , Ψn 1 ⎟⎜ . .. ⎠ ⎝ .. .
· · · Ψn , Ψn
v¯1n
···
v¯n1 ⎞ .. ⎟ . . ⎠ v¯nn
(12)
Since the matrix ( Ψj , Ψk )j,k is Hermitian, we can unitarily diagonalize it; that is, we can choose an orthonormal basis B of Cn such that with this choice of B the vectors ϕγ ,k = Vk = j j vk Uγ +ej ,j of Eγ are orthonormal, and their images Φk = SVk are orthogonal in Fγ . Therefore, Φk / Φk is an orthonormal basis of Fγ such that Sγ : Eγ → Fγ is diagonal when expressed in terms of the bases {V1 , . . . , Vn } ⊂ Eγ and {Φ1 / Φ1 , . . . , Φn / Φn } ⊂ Fγ , with entries Φk .
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F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
Furthermore, the Φk are exactly the square roots of the eigenvalues of the matrix ( Ψp , Ψq ) which by (11) is given by
Ψp , Ψq = SUγ +ep ,p , SUγ +eq ,q = cγ +ep cγ +eq Szγ +ep d z¯ p , Szγ +eq d z¯ q
cγ +ep cγ 1 − = cγ +ep cγ +eq cγ +ep cγ +ep +eq cγ +eq =
cγ +ep cγ +eq − cγ cγ +ep +eq . √ cγ +ep +eq cγ +ep cγ +eq
(13)
Summarizing, we have the following proposition. Proposition 11. With μ as above, the canonical solution operator S : A2(0,1) (dμ) → L2(0,1) (dμ) admits a diagonalization by orthonormal bases. In fact, we have a decomposition A2(0,1) = γ Eγ into mutually orthogonal finite-dimensional subspaces Eγ , indexed by the multi-indices γ with at most one negative entry (equal to −1), which are of dimension 1 or n, and orthonormal bases ϕγ ,j of Eγ , such that Sϕγ ,j is a set of mutually orthogonal vectors in L2 (dμ). For fixed γ , the norms Sϕγ ,j are the square roots of the eigenvalues of the matrix Cγ = (Cγ ,p,q )p,q given by Cγ ,p,q =
cγ +ep cγ +eq − cγ cγ +ep +eq . √ cγ +ep +eq cγ +ep cγ +eq
(14)
In particular, we have that n
Sϕγ ,j = trace(Cγ ,p,q )p,q = 2
j =1
n
cγ +ep p=1
cγ +2ep
−
cγ cγ +ep
.
(15)
3. Boundedness: Proof of Theorem 1 In order to prove Theorem 1, we are using Proposition 11. We have seen that we have an orthonormal basis ϕγ ,j , γ ∈ Γ , j ∈ {1, . . . , n}, such that the images Sϕγ ,j are mutually orthogonal. Thus, S is bounded if and only if there exists a constant C such that
Sϕγ ,j 2 C for all γ ∈ Γ and j ∈ {1, . . . , dim Eγ }. If dim Eγ = 1, then γ has exactly one entry (say the j th √ one) equal to −1; in that case, let us write ϕγ = Uγ +ej d z¯ j . We have Sϕγ = cγ +ej z¯ j zγ +ej , and so
Sϕγ 2 =
cγ +ej cγ +2ej
.
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
21
On the other hand, if dim Eγ = n, we argue as follows: writing Sϕγ ,j 2 = λ2γ ,j with λγ ,j > 0, from (15) we find that n
λ2γ ,j
j =1
=
n
cγ +ej j =1
−
cγ +2ej
cγ cγ +ej
.
The last two equations complete the proof of Theorem 1. 4. Compactness In order to prove Theorem 2, we use the following elementary lemma (which is for example contained in [1]). Lemma 12. Let H1 and H2 be Hilbert spaces, and assume that S : H1 → H2 is a bounded linear operator. Then S is compact if and only if for every ε > 0 there exists a compact operator Tε : H1 → H2 such that the following inequality holds:
Sv 2H2 Tε v 2H2 + ε v 2H1 .
(16)
Proof of Theorem 2. We first show that (1) implies compactness. We will use the notation which was already used in the proof of Theorem 1; that is, we write Sϕγ ,j 2 = λ2γ ,j . Let ε > 0. There / Aε , exists a finite set Aε of multi-indices γ ∈ Γ such that for all γ ∈ n j =1
λ2γ ,j =
n
cγ +ej
cγ +2ej
j =1
−
cγ
cγ +ej
< ε.
Hence, if we consider the finite-dimensional (and thus, compact) operator Tε defined by aγ ,j ϕγ ,j = Tε aγ ,j Sϕγ ,j , γ ∈Aε
for any v =
aγ ,j ϕγ ,j ∈ A2(0,1) (dμ) we obtain 2
Sv 2 = Tε v 2 + S a ϕ γ ,j γ ,j γ ∈A / ε
= Tε v 2 +
|aγ ,j |2 Sϕγ ,j 2
γ ∈A / ε
= Tε v 2 +
|aγ ,j |2 λ2γ ,j
γ ∈A / ε
Tε v 2 + ε
|aγ ,j |2
γ ∈A / ε
Tε v + ε v 2 . 2
Hence, (16) holds and we have proved the first implication in Theorem 2.
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F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
We now turn to the other direction. Assume that (1) is not satisfied. Then there exists a K > 0 and an infinite family A of multi-indices γ such that for all γ ∈ A, n
λ2γ ,j =
j =1
n
cγ +ej j =1
cγ +2ej
−
cγ cγ +ej
> nK.
In particular, for each γ ∈ A, there exists a jγ such that λ2γ ,jγ > K. Thus, we have an infinite orthonormal family {ϕγ ,jγ : γ ∈ A} of vectors such that their images Sϕγ ,jγ are orthogonal and √ have norm bounded from below by K, which contradicts compactness. 2 5. Membership in the Schatten classes Sp and in the Hilbert–Schmidt class We keep the notation introduced in the previous sections. We will also need to introduce the usual grading on the index set Γ , that is, we write Γk = γ ∈ Γ : |γ | = k ,
k −1.
(17)
In order to study the membership in the Schatten class, we need the following elementary lemma. Lemma 13. Assume that p(x) and q(x) are continuous, real-valued functions on RN which are homogeneous of degree 1 (i.e. p(tx) = tp(x) and q(tx) = tq(x) for t ∈ R), and q(x) = 0 as well as p(x) = 0 implies x = 0. Then there exists a constant C such that 1 q(x) p(x) C q(x). C
(18)
Proof. Note that the set Bq = {x: q(x) = 1} is compact: it is closed since q is continuous, and since |q| is bounded from below on S N by some m > 0, it is necessarily contained in the closed ball of radius 1/m. Now, the function |p| is bounded on the compact set Bq ; say, by 1/C from below and C from above. Thus for all x ∈ RN ,
x 1 C, p C q(x) which proves (18).
2
Proof of Theorem 3. Note that S is in the Schatten class Sp if and only if
p
λγ ,j < ∞.
γ ∈Γ,j
We rewrite this sum as γ ∈Γ
j
p λγ ,j
=: M ∈ R ∪ {∞}.
(19)
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
23
Lemma 13 implies that there exists a constant C such that for every γ ∈ Γ , p/2
1 2 p/2 p λγ ,j λγ ,j C λ2γ ,j . C j
j
j
Hence, M < ∞ if and only if γ
p/2 < ∞,
λ2γ ,j
j
which after applying (15) becomes the condition (2) claimed in Theorem 3.
2
Proof of Theorem 4. S is in the Hilbert–Schmidt class if and only if
λ2γ ,j < ∞.
(20)
γ ∈Γ,j
We will prove that k
λ2γ ,j =
=−1 γ ∈Γ ,j
α∈Nn , |α|=k+1 1pn
cα , cα+ep
(21)
which immediately implies Theorem 4. The proof is by induction over k. For k = −1, the lefthand side of (21) is n j =1
λ2−ej ,j =
n
zj 2 c0 =
j =1
n c0 , cep j =1
which is equal to the right-hand side. Now assume that the (21) holds for k = K − 1; we will show that this implies it holds for k = K. We write K
λ2γ ,j =
=−1 γ ∈Γ ,j
cα
α∈Nn , |α|=K−1 1pn
=
α∈Nn , |α|=K 1pn
This finishes the proof of Theorem 4.
2
cα+ep cα
cα+ep
.
+
cγ +ej γ ∈ΓK ,j
cγ +2ej
−
cγ cγ +ej
24
F. Haslinger, B. Lamel / Journal of Functional Analysis 255 (2008) 13–24
References [1] J.P. D’Angelo, Inequalities from Complex Analysis, Carus Math. Monogr., vol. 28, Math. Assoc. America, Washington, DC, 2002. [2] S. Fu, E.J. Straube, Compactness of the ∂-Neumann problem on convex domains, J. Funct. Anal. 159 (2) (1998) 629–641. [3] S. Fu, E.J. Straube, Semi-classical analysis of Schrödinger operators and compactness in the ∂-Neumann problem, J. Math. Anal. Appl. 271 (1) (2002) 267–282. [4] F. Haslinger, The canonical solution operator to ∂ restricted to Bergman spaces, Proc. Amer. Math. Soc. 129 (11) (2001) 3321–3329 (electronic). [5] F. Haslinger, Compactness of the canonical solution operator to ∂ restricted to Bergman spaces, in: Functionalanalytic and Complex Methods, Their Interactions, and Applications to Partial Differential Equations, Graz, 2001, World Sci. Publ., River Edge, NJ, 2001, pp. 394–400. [6] F. Haslinger, The canonical solution operator to ∂ restricted to spaces of entire functions, Ann. Fac. Sci. Toulouse Math. (6) 11 (1) (2002) 57–70. [7] F. Haslinger, Magnetic Schrödinger operators and the ∂-equation, J. Math. Kyoto 46 (2) (2006) 249–257. [8] F. Haslinger, B. Helffer, Compactness of the solution operator to ∂ in weighted L2 -spaces, J. Funct. Anal. 243 (2007) 679–697. [9] W. Knirsch, G. Schneider, Continuity and Schatten–von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces, J. Math. Anal. Appl. 320 (1) (2006) 403–414. [10] S. Lovera, E.H. Youssfi, Spectral properties of the ∂-canonical solution operator, J. Funct. Anal. 208 (2) (2004) 360–376. [11] G. Schneider, Non-compactness of the solution operator to ∂ on the Fock-space in several dimensions, Math. Nachr. 278 (3) (2005) 312–317. [12] I. Shigekawa, Spectral properties of Schrödinger operators with magnetic fields for a spin 12 particle, J. Funct. Anal. 101 (2) (1991) 255–285.
Journal of Functional Analysis 255 (2008) 25–45 www.elsevier.com/locate/jfa
Boundedness and compactness of Hankel operators on the sphere ✩ Jingbo Xia Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA Received 14 August 2007; accepted 23 March 2008 Available online 28 April 2008 Communicated by N. Kalton
Abstract We consider Hankel operators on the Hardy space of the unit sphere in Cn . We show that a large amount of information about the function f − Pf can be recovered from the Hankel operator Hf . For example, if Hf is compact, then the function f − Pf is necessarily in VMO. © 2008 Elsevier Inc. All rights reserved. Keywords: Hankel operator; Mean oscillation
1. Introduction Let S denote the unit sphere {z ∈ Cn : |z| = 1} in Cn . Let σ be the positive, regular Borel measure on S which is invariant under the orthogonal group O(2n), i.e., the group of isometries on Cn ∼ = R2n which fix 0. Furthermore we normalize σ such that σ (S) = 1. The Cauchy projection P is defined by the integral formula (Pf )(w) =
✩
f (ζ ) n dσ (ζ ), 1 − w, ζ
|w| < 1.
This work was supported by National Science Foundation grant DMS-0456448. E-mail address:
[email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.017
26
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
See [7, p. 39]. Recall that P is the orthogonal projection from L2 (S, dσ ) onto the Hardy space H 2 (S). For each z ∈ Cn with |z| < 1, we write kz (w) =
(1 − |z|2 )n/2 , (1 − w, z)n
|w| 1.
It is well known that the formula 1/2 d(ζ, ξ ) = 1 − ζ, ξ ,
ζ, ξ ∈ S,
(1.1)
defines metric a on S [7, p. 66]. Throughout the paper, we denote 1/2 B(ζ, r) = x ∈ S: 1 − x, ζ < r for ζ ∈ S and r > 0. There is a constant A0 ∈ (2−n , ∞) such that 2−n r 2n σ B(ζ, r) A0 r 2n
(1.2)
√ for all ζ ∈ S and 0 < r 2 [7, Proposition 5.1.4]. A function f ∈ L1 (S, dσ ) is said to have bounded mean oscillation if 1 f BMO = sup |f − fB(ζ,r) | dσ < ∞, ζ ∈S σ (B(ζ, r)) r>0
B(ζ,r)
where fB = B f dσ/σ (B), the average of f over B. A function f ∈ L1 (S, dσ ) is said to have vanishing mean oscillation if 1 lim sup |f − fB(ζ,r) | dσ = 0. δ↓0 ζ ∈S σ (B(ζ, r)) 0
B(ζ,r)
We denote the collection of functions of bounded mean oscillation on S by BMO. Similarly, let VMO be the collection of functions of vanishing mean oscillation on S. A fundamental result due to Coifman, Rochberg and Weiss asserts that [P , Mf ] is bounded for f ∈ BMO with [P , Mf ] Cf BMO , and [P , Mf ] is compact if f ∈ VMO [3]. Also see [2]. Recently, the author showed that the same boundedness and compactness results for commutators still hold if P is replaced by a more general class of singular integral operators of the Calderón–Zygmund type on S [10]. Given any f ∈ BMO, the Hankel operator Hf : H 2 (S) → L2 (S, dσ ) is defined by the formula Hf = (1 − P )Mf |H 2 (S). That is, Hf h = (1 − P )(f h), h ∈ H 2 (S). We will also identify Hf with the operator (1 − P )Mf P on the Hilbert space L2 (S, dσ ). The commutator result mentioned above implies that Hf is bounded for every f ∈ BMO, and Hf is compact if f ∈ VMO. Note that the operator Hf defined here is different from the operator Kb defined on p. 628 in [3]. Nowadays, in fact, the operator Kb considered in [3] is usually referred to as the “small”
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
27
Hankel operator, whereas the Hf defined above is regarded as the “big” Hankel operator. In this paper, the term “Hankel operator” refers to Hf . From the view point of operator theory, Hf seems to be more natural an operator to study. There are two kinds of problems in the theory of Hankel operators, namely “two-sided” problems and “one-sided” problems [12]. A “two-sided” problem concerns Hf and Hf¯ simultaneously. By virtue of the relation [P , Mf ] = Hf∗¯ − Hf , “two-sided” problems are equivalent to the study of the commutator [P , Mf ]. Therefore there is a large body of literature on “two-sided" problems. By contrast, a “one-sided” problem is the study of Hf alone. The conventional view is that a “one-sided” problem is more difficult than the corresponding “two-sided” problem. In the case n = 1, i.e., on the unit circle, because of the fact f − Pf ∈ H 2 ,
(1.3)
every “one-sided” problem is actually a “two-sided” problem. But when n 2, (1.3) no longer holds, and a difference between “two-sided” problems and “one-sided” problems appears. The main difficulty in “one-sided” problems is the fact that the subspace L2 (S, dσ ) H 2 (S) + H 2 (S)
(1.4)
is huge and intractable when n 2. A good example of a “one-sided” result is the following theorem due to Zheng. Theorem 1.1. (See [11, Theorem 5].) Let f ∈ BMO. Then the Hankel operator Hf is compact if and only if lim Hf kz = 0.
|z|↑1
(1.5)
Although Theorem 1.1 is the best existing result on the compactness of Hf , it still leaves something to be desired. One cannot help but ask, what exactly does (1.5) tell us about f ? Note that the involvement of f in (1.5) is through the operator Hf and, therefore, indirect. A more desirable characterization of the compactness of Hf would be in terms of a more direct condition, such as the membership of f in some easily-defined function class. The purpose of this paper is to report a previously unnoticed fact, namely that the Hankel operator Hf actually tells us a great deal about the commutator [P , Mf −Pf ]. That is, in many situations, a “one-sided” problem actually has a “two-sided” solution! In other words, notwithstanding the size of (1.4), the theory of Hankel operators in the case n 2 resembles the case n = 1 in more ways than we previously realized. What initially led to this investigation was the consideration of the subset A = f ∈ L∞ (S, dσ ): Hf is compact of L∞ (S, dσ ). As Davie and Jewell observed, A is in fact a Banach subalgebra of L∞ (S, dσ ) [4, p. 365]. In the case n = 1, it is well known that A = H ∞ + C [5,12], which is unquestionably a direct condition for compactness. But when n 2, A is known to be strictly larger than
28
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
H ∞ (S) + C(S) [4]. So here at least, there is a genuine difference between the case n = 1 and the case n 2. But wait, for difference is not the whole story. Even for A, there is similarity between the case n = 1 and the case n 2. Let us also consider the subset A1 = f ∈ L∞ (S, dσ ): f − Pf ∈ VMO of L∞ (S, dσ ). If f ∈ A1 , then by the result of Coifman, Rochberg and Weiss we mentioned earlier the commutator [P , Mf −Pf ] is compact. Since Hf = Hf −Pf , it follows that A1 ⊂ A. One might say that A1 is the obvious part of A. Our first result is the reverse inclusion, i.e., A consists of nothing but its obvious part. Theorem 1.2. A ⊂ A1 . This result can be further refined. For each f ∈ L1 (S, dσ ) and each ζ ∈ S, denote 1 sup |f − fB(ξ,r) | dσ, LMO(f )(ζ ) = lim δ↓0 B(ξ,r)⊂B(ζ,δ) σ (B(ξ, r)) B(ξ,r)
which is called the local mean oscillation of f at ζ [6, Definition 5.1]. Theorem 1.3. If f is a function in BMO and ζ is a point in S such that lim Hf kz = 0,
(1.6)
lim Hf kz = 0,
(1.7)
z→ζ |z|<1
then LMO(f − Pf )(ζ ) = 0. Corollary 1.4. If f ∈ BMO and if |z|↑1
then f − Pf ∈ VMO. One might say that Corollary 1.4 “explains” why Theorem 1.1 holds: if f belongs to BMO and satisfies (1.5), then f − Pf ∈ VMO, which implies the compactness of [P , Mf −Pf ], which in turn implies the compactness of Hf −Pf = Hf . Corollary 1.5. Suppose that f ∈ BMO and that f ⊥ H 2 (S) + H 2 (S). Then Hf is compact if and only if Hf¯ is compact. This is obviously reminiscent of a well-known result due to Berger and Coburn about Hankel operators on the Segal–Bargmann space [1, Theorem B].
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
29
Theorem 1.6. There exists a constant 0 < C < ∞ which depends only on the complex dimension n such that f − Pf BMO C sup Hf kz |z|<1
for every f ∈ BMO. Combining this with the result of Coifman, Rochberg and Weiss [3], we have [P , Mf −Pf ] C1 Hf ,
(1.8)
f ∈ BMO. Corollary 1.7. There exists a constant 0 < C < ∞ which depends only on the complex dimension n such that for f ∈ BMO satisfying the condition f ⊥ H 2 (S) + H 2 (S), we have C −1 Hf Hf¯ CHf . Suppose that A is a bounded operator on a Hilbert space H. Recall that the essential norm of A is defined by the formula AQ = inf A + K: K is compact on H . Equivalently, AQ = π(A), where π denotes the quotient map from B(H) into the Calkin algebra Q = B(H)/K(H). An analogue of (1.8) holds for essential norms. Theorem 1.8. There exists a constant 0 < C < ∞ which depends only on the complex dimension n such that [P , Mf −Pf ] CHf Q Q for every f ∈ BMO. Note that in all the results above the condition f ∈ BMO was a part of the assumption. But the bound provided by Theorem 1.6 enables us to deal with symbol functions which are not a priori assumed to be in BMO. For ψ ∈ L2 (S, dσ ), we can still define the Hankel operator Hψ on the dense subset H ∞ (S) of H 2 (S). That is, Hψ h = (1 − P )(ψh) for h ∈ H ∞ (S). Theorem 1.9. If ψ ∈ L2 (S, dσ ) and if sup Hψ kz < ∞,
|z|<1
then ψ − P ψ ∈ BMO. Combining Theorem 1.9 and Corollary 1.4, and using the fact that Hψ = Hψ−P ψ , we have the following improvement of Theorem 1.1.
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J. Xia / Journal of Functional Analysis 255 (2008) 25–45
Corollary 1.10. Suppose that ψ ∈ L2 (S, dσ ) and that lim Hψ kz = 0.
|z|↑1
Then ψ − P ψ ∈ VMO. Consequently Hψ naturally extends to a compact operator that maps H 2 (S) into L2 (S, dσ ) H 2 (S). The rest of the paper consists of the proofs of the above results. More specifically, in Section 2 we establish an inequality involving mean oscillation and P , which is the key to the proofs of these results. The proofs of Theorems and Corollaries 1.2–1.8 are presented in Section 3. The proof of Theorem 1.9 uses a smoothing argument, which is given in Section 4. 2. An estimate of mean oscillation It was shown in [3] that P maps L∞ (S, dσ ) into BMO. In fact, something slightly stronger is also true. Proposition 2.1. If f ∈ BMO, then Pf ∈ BMO. As it turns out, the key to the proofs of the results in Section 1 is the following refinement of Proposition 2.1. Proposition 2.2. There exists a constant 0 < C2.2 < ∞ which depends only on the complex dimension n such that for all f ∈ L2 (S, dσ ) and B = B(ζ, r), where ζ ∈ S and r > 0, we have
1 σ (B)
Pf − (Pf )B 2 dσ
1/2 C2.2
B
1 σ (B1 )
+ C2.2
∞ k=2
1/2
|f − fB1 | dσ 2
B1
2−k σ (Bk )
|f − fBk | dσ, Bk
where Bk = B(ζ, 2k r) for every k 1. Proof. Given f ∈ L2 (S, dσ ) and B = B(ζ, r), we may assume (Pf − (Pf )B )χB = 0, for otherwise there is nothing to prove. Define g=
1 Pf − (Pf )B χB , (Pf − (Pf )B )χB
which is, of course, a unit vector in L2 (S, dσ ). Write 1 for the constant function of value 1 on S. Then obviously 1, g = 0. Thus
1 σ (B)
B
Pf − (Pf )B 2 dσ
1/2 =
Pf, g Pf − (Pf )B , g = 1/2 . σ 1/2 (B) σ (B)
(2.1)
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
31
To estimate Pf, g, note that P 1 = 1, which leads to 1, P g = 1, g = 0. Hence Pf, g = f, P g = f − fB1 , P g ∞ = (f − fB1 )P g dσ +
(f − fB1 )P g dσ.
(2.2)
k=2B \B k k−1
B1
Next we estimate the terms in (2.2), using the properties of g and P . For the first term in (2.2), we have
|f − fB1 ||P g| dσ (f − fB1 )χB1 P g (f − fB1 )χB1 .
B1
By (1.2), σ (B1 ) 23n A0 σ (B). Thus if we set C1 = (23n A0 )1/2 , then B1
1/2 |f − fB1 ||P g| dσ (f − fB1 )χB1 = σ (B1 )
1/2 C1 σ (B)
1 σ (B1 )
1 σ (B1 )
1/2
|f − fB1 | dσ 2
B1
1/2
|f − fB1 |2 dσ
.
(2.3)
B1
To estimate the other terms in (2.2), we first need to show that there is a constant C2 which depends only on n such that 1 |1 − y, ζ |1/2 1 − (1 − x, y)n (1 − x, ζ )n C2 |1 − x, ζ |n+(1/2)
(2.4)
if y ∈ B and x ∈ S \ B1 . Given y ∈ B and x ∈ S \ B1 , let us first estimate |x, y − ζ |. We write x = x, ζ ζ + x ⊥ and y = y, ζ ζ + y ⊥ , where x ⊥ , ζ = 0 = y ⊥ , ζ . Thus x, y − ζ = x, ζ (ζ, y − 1) + x ⊥ , y ⊥ . Therefore x, y − ζ 1 − y, ζ + x ⊥ y ⊥ = 1 − y, ζ + 1 − x, ζ 2 1/2 1 − y, ζ 2 1/2 1/2 1/2 1 − y, ζ + 21 − x, ζ 1 − y, ζ . Since d(x, ζ ) 2r whereas d(y, ζ ) < r, the above implies x, y − ζ 31 − x, ζ 1/2 1 − y, ζ 1/2 .
(2.5)
Also, since d(x, ζ ) 2r > 2d(y, ζ ), we have (1/2)d(x, ζ ) − d(y, ζ ) > 0. Thus d(x, y) d(x, ζ ) − d(y, ζ ) > (1/2)d(x, ζ ). Combining this with (2.5), we find that |x, y − ζ | 1 1 |1 − y, ζ |1/2 = − 12 . 1 − x, y 1 − x, ζ |1 − x, y||1 − x, ζ | |1 − x, ζ |3/2
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J. Xia / Journal of Functional Analysis 255 (2008) 25–45
By simple algebra and another application of the fact d(x, y) > (1/2)d(x, ζ ), we have 4n−1 1 1 |1 − y, ζ |1/2 − (1 − x, y)n (1 − x, ζ )n n · |1 − x, ζ |n−1 · 12 |1 − x, ζ |3/2 , proving (2.4). Applying (2.4) and (1.2), if y ∈ B and x ∈ Bk \ Bk−1 , k 2, then C2 r 1 1 22n+1 C2 1 C3 . − · k 2n k (1 − x, y)n (1 − x, ζ )n (2k−1 r)2n+1 = 2k 2 σ (Bk ) (2 r) By the definition of g, we have g = 0 on S \ B and g dσ = 0. B
Also, by the Cauchy–Schwarz inequality, |g| dσ σ 1/2 (B)g = σ 1/2 (B). B
For x ∈ S \ B1 we have (P g)(x) = B
g(y) dσ (y) = (1 − x, y)n
B
1 1 − g(y) dσ (y). (1 − x, y)n (1 − x, ζ )n
Therefore (P g)(x)
C3 2k σ (Bk )
|g| dσ
C3 σ 1/2 (B) 2k σ (Bk )
if x ∈ Bk \ Bk−1 , k 2.
(2.6)
B
Integrating the above over Bk \ Bk−1 , we see that
C3 σ 1/2 (B) C3 σ (Bk \ Bk−1 ) k σ 1/2 (B) 2k σ (Bk ) 2
|P g| dσ Bk \Bk−1
if k 2.
(2.7)
Applying (2.6) and (2.7), for each k 2 we have
|f − fB1 ||P g| dσ Bk \Bk−1
|f − fBk ||P g| dσ + |fBk − fB1 |
Bk \Bk−1
σ 1/2 (B)
C3 2k σ (Bk )
Bk \Bk−1
|f − fBk | dσ + Bk \Bk−1
|P g| dσ
C3 1/2 σ (B)|fBk − fB1 |. 2k
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
33
But |fBk − fB1 |
k
|fBj
j =2
k σ (Bj ) 1 − fBj −1 | |f − fBj | dσ. σ (Bj −1 ) σ (Bj ) j =2
Bj
Applying (1.2) again, we see that if we set C4 = (1 + 23n A0 )C3 , then
1 C4 1/2 σ (B) k 2 σ (Bj ) k
|f − fB1 ||P g| dσ
j =2
Bk \Bk−1
|f − fBj | dσ. Bj
Therefore ∞
|f − fB1 ||P g| dσ C4 σ 1/2 (B)
k=2 B \B k k−1
k ∞ 1 1 |f − fBj | dσ 2k σ (Bj ) k=2
j =2
Bj
∞ ∞ 1 1 1/2 |f − fBj | dσ = C4 σ (B) 2k σ (Bj ) j =2
= 2C4 σ 1/2 (B)
k=j
Bj
∞ 2−j |f − fBj | dσ. σ (Bj ) j =2
Bj
Combining this with (2.2) and (2.3), we find that
Pf, g C2.2 σ
1/2
(B)
1 σ (B1 )
1/2
|f − fB1 | dσ 2
B1
∞ 2−j + |f − fBj | dσ , σ (Bj ) j =2
where C2.2 = max{C1 , 2C4 }. Recalling (2.1), this completes the proof.
Bj
2
Remark 1. The above proof is inspired by ideas from harmonic analysis, particularly ideas from [8, Chapters III and IV]. Remark 2. In essence Proposition 2.2 tells us that, with regard to mean oscillation, the Cauchy projection P is very well behaved. 3. Proofs of 1.2–1.8 For each z ∈ Cn with 0 < |z| < 1, define the Möbius transform ϕz (w) =
1 w, z w, z 2 1/2 z − 1 − |z| z , w − z− 1 − w, z |z|2 |z|2
|w| 1.
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J. Xia / Journal of Functional Analysis 255 (2008) 25–45
Then ϕz is an involution, i.e., ϕz ◦ ϕz = id [7, Theorem 2.2.2]. Recall that the formula (Uz g)(ξ ) = g ϕz (ξ ) kz (ξ ),
ξ ∈ S and g ∈ L2 (S, dσ ),
defines a unitary operator with the property [Uz , P ] = 0 [9, Section 6]. We have g ◦ ϕz − g ◦ ϕz , 1 = Uz g ◦ ϕz − gkz , kz = g − gkz , kz kz for g ∈ L2 (S, dσ ). By a standard argument, if f ∈ BMO, then f BMO is comparable to sup|z|<1 (f − f kz , kz )kz . Therefore there exist constants 0 < α < β < ∞ such that αf BMO sup f ◦ ϕz − f ◦ ϕz , 1 βf BMO
(3.1)
0<|z|<1
for all f ∈ BMO. By [7, Theorem 2.2.5], if 0 < |a| < 1 and 0 < |z| < 1, then f ◦ ϕa ◦ ϕz − f ◦ ϕa ◦ ϕz , 1 sup f ◦ ϕλ − f ◦ ϕλ , 1 .
(3.2)
0<|λ|<1
Combining (3.1) and (3.2), we see that αf ◦ ϕa BMO βf BMO
(3.3)
for all f ∈ BMO and a ∈ Cn with 0 < |a| < 1. Lemma 3.1. Given any f ∈ BMO and z ∈ Cn with 0 < |z| < 1, there exist functions hz and vz satisfying the following four conditions: (a) (b) (c) (d)
hz ∈ H 2 (S). hz + vz = f − Pf . vz kz = Hf kz . vz BMO C3.1 f BMO , where the constant C3.1 depends only on n.
Proof. Given f ∈ BMO and 0 < |z| < 1, set hz = P (f ◦ ϕz ) ◦ ϕz − Pf
and vz = f − P (f ◦ ϕz ) ◦ ϕz .
Then (a) and (b) are obvious. Using the identities ϕz ◦ ϕz = id and [Uz , P ] = 0, we have Hf kz = Hf ◦ϕz ◦ϕz kz = Uz Hf ◦ϕz 1 = Uz f ◦ ϕz − P (f ◦ ϕz ) = vz kz , proving (c). To verify (d), note that Proposition 2.2 provides a constant C such that P ηBMO CηBMO for every η ∈ BMO. Combining this with (3.3), we have vz BMO f BMO + P (f ◦ ϕz ) ◦ ϕz BMO f BMO + (β/α) P (f ◦ ϕz ) BMO f BMO + (β/α)Cf ◦ ϕz BMO f BMO + (β/α)2 Cf BMO . Thus C3.1 = 1 + (β/α)2 C will do for (d).
2
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
35
Proof of Theorem 1.3. Let f ∈ BMO and ζ ∈ S, and suppose that (1.6) holds. Let > 0 be given. Pick an integer K > 2 such that ∞
C2.2 C3.1
2−k f BMO ,
(3.4)
k=K+1
where C2.2 and C3.1 are the constants in Proposition 2.2 and Lemma 3.1 respectively. Let ξ ∈ S and 0 < r < 2−K−1 . Given such a pair of ξ and r, denote B = B(ξ, r).
(3.5)
Also, given such a pair of ξ and r, define ρ = 2K r and 1/2 ξ. z = 1 − ρ2
(3.6)
Applying Lemma 3.1 to f and this particular z, we obtain hz and vz satisfying (a)–(d). By (a) and (b), we have hz + P vz = P (hz + vz ) = P (f − Pf ) = 0. That is, hz = −P vz . Now apply Proposition 2.2 to vz and the B given by (3.5). We have
1 σ (B)
hz − (hz )B 2 dσ
B
1 = σ (B)
1/2
P vz − (P vz )B 2 dσ
1/2
B
C2.2
K
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2 + C2.2
∞
2−k vz BMO ,
k=K+1
Bk
where Bk = B(ξ, 2k r), k 1. If g ∈ L2 (S, dσ ) and σ (E) > 0, then 1 1 2 g(x) − g(y)2 dσ (x) dσ (y). |g − gE | dσ = 2 σ (E) 2σ (E) E
E E
Therefore K
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2
Bk
K
1/2 1 σ (BK ) vz − (vz )B 2 dσ . K σ (B1 ) σ (BK ) BK
There is a constant C(n, K) which depends only on n and K such that K
σ (BK ) C(n, K). σ (B)
(3.7)
36
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
By (d), vz BMO C3.1 f BMO . Substituting these inequalities in (3.7) and recalling (3.4), we find that
1 σ (B)
hz − (hz )B 2 dσ
1/2
C2.2 C(n, K)
1 σ (BK )
B
vz − (vz )B 2 dσ K
1/2 + . (3.8)
BK
We next consider the kernel function kz . For y ∈ BK = B(ξ, 2K r) = B(ξ, ρ), 1 − y, z = 1 − 1 − ρ 2 1/2 y, ξ 1 − 1 − ρ 2 1/2 + 1 − y, ξ 2ρ 2 . Note that 0 < ρ < 1/2. Combining the above inequality with (1.2), we find that kz (y)2 =
(ρ 2 )n (ρ 2 )n 1 1 = n 2n n 2n 2 2n 8 σ (BK ) |1 − y, z| (2ρ ) 4 ρ
if y ∈ BK . Thus
1 σ (BK )
vz − (vz )B 2 dσ K
1/2
1 σ (BK )
BK
1/2
|vz |2 dσ
8n/2 vz kz .
BK
Combining this with (3.8) and with the fact that vz kz = Hf kz , we now have
1 σ (B)
hz − (hz )B 2 dσ
1/2 C2.2 C(n, K)8n/2 Hf kz + .
B
Also,
1 σ (B)
vz − (vz )B 2 dσ
1/2
1 σ (B)
B
1/2
|vz |2 dσ
C(n, K) σ (BK )
B
1/2
|vz |2 dσ BK
Since hz + vz = f − Pf , the above inequalities yield the estimate
1 σ (B)
f − Pf − (f − Pf )B 2 dσ
1/2 (1 + C2.2 )C(n, K)8n/2 Hf kz +
B
for the B and z given by (3.5) and (3.6) respectively. To complete the proof, we use (1.6). There is a δ1 > 0 such that (1 + C2.2 )C(n, K)8n/2 Hf kz
if |z − ζ | δ1 .
By (3.6) and the fact that ρ = 2K r, there is a 0 < δ2 < 2−K−1 such that |z − ζ | δ1
if |ξ − ζ | δ2 and 0 < r δ2 .
.
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
37
Finally, there is a δ > 0 such that |ξ − ζ | δ2
and r δ2
if B(ξ, r) ⊂ B(ζ, δ).
Combining the above, for this δ we have
1 σ (B)
f − Pf − (f − Pf )B 2 dσ
1/2 2
B
if B = B(ξ, r) ⊂ B(ζ, δ). Since > 0 is arbitrary, this means LMO(f − Pf )(ζ ) = 0.
2
Proof of Corollary 1.4. By Theorem 1.3, if (1.7) holds, then LMO(f − Pf )(ζ ) = 0 for every ζ ∈ S. Since S is compact, this implies f − Pf ∈ VMO. 2 Proof of Theorem 1.2. It is well known that kz converges to 0 weakly as |z| ↑ 1. Therefore if f ∈ A, then f satisfies (1.7). By Corollary 1.4, this means f ∈ A1 . 2 Proof of Corollary 1.5. Suppose that f ∈ BMO and that f ⊥ H 2 (S)+H 2 (S). If Hf is compact, then (1.7) holds. By Corollary 1.4, f = f − Pf ∈ VMO. Thus [P , Mf ] is compact, which implies Hf¯ is compact. Then apply the same argument to f¯. 2 Proof of Theorem 1.6. We first pick an integer L > 2 such that C2.2 C3.1
∞ k=L+1
1 2−k , 4
(3.9)
where C2.2 and C3.1 are the constants in Proposition 2.2 and Lemma 3.1 respectively. Let f ∈ BMO be given and write u = f − Pf. Proposition 2.2 tells us that uBMO < ∞. Thus there exist ξ ∈ S and r > 0 such that 1 1 |u − uB(ξ,r) | dσ uBMO . σ (B(ξ, r)) 2 B(ξ,r)
Write B = B(ξ, r) as in the proof of Theorem 1.3. Also, let ρ = 2L r. Now the proof divides into two cases. (1) Suppose that ρ < 1/2. In this case we define 1/2 z = 1 − ρ2 ξ. Applying Lemma 3.1 to u and z, we obtain hz and vz such that
(3.10)
38
(i) (ii) (iii) (iv)
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
hz ∈ H 2 (S); hz + vz = u − P u = u; vz kz = Hu kz ; vz BMO C3.1 uBMO .
Again, hz = −P vz because of (i) and (ii). Applying Proposition 2.2 to vz and B, we have
1 σ (B)
hz − (hz )B 2 dσ
B
1 = σ (B)
1/2
P vz − (P vz )B 2 dσ
1/2
B L
C2.2
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2 + C2.2
∞
2−k vz BMO ,
k=L+1
Bk
where Bk = B(ξ, 2k r), k 1. Again, L
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2 L
Bk
1/2 1 σ (BL ) vz − (vz )B 2 dσ . L σ (B1 ) σ (BL ) BL
Combining the above with (iv) and with (3.9), we see that 1 σ (B)
hz − (hz )B dσ C2.2 C(n, L)
1 σ (BL )
B
vz − (vz )B 2 dσ L
1/2
1 + uBMO , 4
BL
where C(n, L) depends only on n and L. Just as in the proof of Theorem 1.3, we have
1 σ (BL )
vz − (vz )B 2 dσ L
1/2
1 σ (BL )
BL
1/2
|vz | dσ 2
BL
8
n/2
vz kz = 8n/2 Hu kz .
Just as in the proof of Theorem 1.3, since u = hz + vz , from the above we deduce 1 σ (B)
1 |u − uB | dσ (1 + C2.2 )C(n, L)8n/2 Hu kz + uBMO . 4
B
Recalling (3.10) and using the fact that Hu = Hf , we now have 1 1 uBMO (1 + C2.2 )C(n, L)8n/2 Hf kz + uBMO . 2 4
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
39
Cancelling out (1/4)uBMO form both sides, we obtain 1 uBMO (1 + C2.2 )C(n, L)8n/2 Hf kz 4 in the case ρ < 1/2. (Note that this last step required the fact uBMO < ∞. This is the reason why Theorem 1.9 requires a separate proof.) (2) Suppose that ρ 1/2. Then r 2−L−1 . Clearly, in this case there is a constant C (n, L) which depends only on n and L such that
1 σ (B(ξ, r))
|u − uB(ξ,r) | dσ C (n, L)u.
B(ξ,r)
But u = f − Pf = Hf 1 and 1 = k0 . Thus, by (3.10), we have 1 uBMO C (n, L)Hf k0 2 in the case ρ 1/2. This completes the proof.
2
Proof of Corollary 1.7. Obviously, Hf¯ [Mf , P ] for all f ∈ BMO. If f ⊥ H 2 (S), then [P , Mf ] = [P , Mf −Pf ]. Also, f ⊥ H 2 (S) + H 2 (S) if and only if f¯ ⊥ H 2 (S) + H 2 (S). Combining (1.8) with these trivial facts, the conclusion follows. 2 Proof of Theorem 1.8. For each g ∈ BMO, define gLMO = lim sup δ↓0
ζ ∈S 0
1 σ (B(ζ, r))
|g − gB(ζ,r) | dσ. B(ζ,r)
Note that if G ∈ VMO, then g + GLMO = gLMO . According to [9, Proposition 4.1], there exists a C1 which depends only on n such that for every g ∈ BMO, there is a p ∈ C(S) such that g + pBMO C1 gLMO = C1 g + pLMO . Now let f ∈ BMO be given. Then there is a q ∈ C(S) such that f − Pf + qBMO C1 f − Pf + qLMO . Define b = f − Pf + q. Then we have bBMO C1 bLMO
(3.11)
and, because P maps C(S) into VMO [3, p. 629], P b ∈ VMO.
(3.12)
40
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
Since [P , Mq ] and Hq are compact and since Hf = Hf −Pf , we have [P , Mb ] = [P , Mf −Pf ] Q Q and Hf Q = Hb Q . Thus, to complete the proof, it suffices to show that there is a C which depends only on n such that the inequality [P , Mb ] CHb Q (3.13) Q holds for every b ∈ BMO satisfying (3.11) and (3.12). By Proposition 2.2, there is a C2 such that g − P gBMO C2 gBMO
(3.14)
for every g ∈ BMO. Note that (3.12) implies b − P bLMO = bLMO . Hence for b ∈ BMO satisfying (3.11) and (3.12) we also have b − P bBMO C2 C1 b − P bLMO .
(3.15)
Next we show that there is a 0 < C3 < ∞ such that b − P bLMO C3 lim sup Hb kw t↑1 t|w|<1
(3.16)
for every b ∈ BMO satisfying (3.11) and (3.12). This is similar to the case (1) in the proof of Theorem 1.6. We pick an integer F > 2 such that C2.2 C3.1 C1 C2
∞ k=F +1
1 2−k , 4
(3.17)
where C2.2 , C3.1 , C1 and C2 are the constants in Proposition 2.2, Lemma 3.1, (3.11) and (3.14) respectively. Write u = b − P b. Then (3.15) becomes (3.18) uBMO C2 C1 uLMO . √ √ Let t ∈ ( 3/2, 1) be given and write s = 1 − t 2 . Then 0 < s < 1/2. Since uBMO < ∞, there exist 0 < r < 2−F s and ξ ∈ S such that 1 1 |u − uB(ξ,r) | dσ uLMO . (3.19) σ (B(ξ, r)) 2 B(ξ,r)
Write B = B(ξ, r)
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
41
as before. Note that 0 < 2F r < s < 1/2. Denote 2 1/2 ξ. z = 1 − 2F r Applying Lemma 3.1 to u and z, we obtain hz and vz such that (i) (ii) (iii) (iv)
hz ∈ H 2 (S); hz + vz = u − P u = u; vz kz = Hu kz ; vz BMO C3.1 uBMO .
Again, hz = −P vz because of (i) and (ii). Applying Proposition 2.2 to vz and B, we have
1 σ (B)
hz − (hz )B 2 dσ
B
1 = σ (B)
1/2
P vz − (P vz )B 2 dσ
1/2
B
C2.2
F
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2
∞
+ C2.2
2−k vz BMO ,
(3.20)
k=F +1
Bk
where Bk = B(ξ, 2k r), k 1. Again, F
k=1
1 σ (Bk )
vz − (vz )B 2 dσ k
1/2
C(n, F )
1 σ (BF )
Bk
vz − (vz )B 2 dσ F
1/2 ,
BF
where C(n, F ) depends only on n and F . Just as in the proof of Theorem 1.3, we have
1 σ (BF )
vz − (vz )B 2 dσ F
1/2
1 σ (BF )
BF
1/2
|vz |2 dσ BF
8
n/2
vz kz = 8n/2 Hu kz .
By (iv) and (3.18), we have vz BMO C3.1 uBMO C3.1 C1 C2 uLMO . Thus, by (3.17), the second term on the right-hand side of (3.20) is not more than (1/4)uLMO . Consequently
1 σ (B)
B
hz − (hz )B 2 dσ
1/2
1 C2.2 C(n, F )8n/2 Hu kz + uLMO . 4
42
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
Similar to the situation in the proofs of Theorems 1.3 and 1.6, recalling (3.19) and the fact that u = hz + vz , we now have 1 1 uLMO 2 σ (B)
1 |u − uB | dσ (1 + C2.2 )C(n, F )8n/2 Hu kz + uLMO . 4
B
Let C3 = 4(1 + C2.2 )C(n, F )8n/2 . Then the above implies uLMO C3 Hu kz . Since 0 < r < 2−F s, we have |z| = (1 − (2F r)2 )1/2 > t. Thus we conclude that uLMO C3 sup Hu kw t|w|<1
√ for every t ∈ ( 3/2, 1). Since HP b is compact, this proves (3.16). To prove (3.13), observe that lim sup Hb kw Hb Q . t↑1 t|w|<1
(3.21)
Indeed because kw → 0 weakly as |w| ↑ 1, for any compact operator K we have lim sup Hb kw = lim sup (Hb + K)kw Hb + K. t↑1 t|w|<1
t↑1 t|w|<1
Since this holds for any compact operator K, (3.21) follows. By (3.12) and (3.11), we have [P , Mb ] [P , Mb ] C4 bBMO C4 C1 bLMO = C4 C1 b − P bLMO . Q Combining this with (3.16) and (3.21), (3.13) follows.
2
4. Smoothing As in [7], let U = U(n) denote the collection of unitary transformations on Cn . For each U ∈ U , define the operator WU : L2 (S, dσ ) → L2 (S, dσ ) by the formula (WU g)(ζ ) = g(U ζ ), g ∈ L2 (S, dσ ). By the invariance of σ , WU is a unitary operator on L2 (S, dσ ). Obviously, [P , WU ] = 0 for every U ∈ U . With the usual multiplication and topology, U is a compact group. Following [7], we write dU for the Haar measure on U . It is easy to see that for each g ∈ L2 (S, dσ ), the map U → WU g is continuous with respect to the norm topology of L2 (S, dσ ). Let Φ be a continuous function on U . Then for each g ∈ L2 (S, dσ ) we can define the integral YΦ g =
Φ(U )WU g dU U
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
43
in the sense that YΦ g, f =
Φ(U )WU g, f dU U
for every f ∈ L2 (S, dσ ). Lemma 4.1. If Φ ∈ C(U), then YΦ g∞ < ∞ for every g ∈ L2 (S, dσ ). Proof. Recall that the equality
f (U ζ ) dU =
f dσ
U
holds for all f ∈ C(S) and ζ ∈ S [7, Proposition 1.4.7]. Thus for q, p ∈ C(S) we have
YΦ q, p = Φ(U ) Φ(U )q(U ζ ) dU p(ζ ) dσ (ζ ) q(U ζ )p(ζ ) dσ (ζ ) dU = U
Φ∞
U
q(U ζ ) dU p(ζ ) dσ (ζ ) = Φ∞ |q| dσ |p| dσ.
U
Since YΦ is obviously a bounded operator on L2 (S, dσ ) and since C(S) is dense in L2 (S, dσ ), the above implies YΦ g, f Φ∞ |g| dσ |f | dσ for all g, f ∈ L2 (S, dσ ). This obviously means YΦ g∞ < ∞.
2
Proof of Theorem 1.9. Let η : [0, ∞) → [0, 1] be a continuous function satisfying the conditions that η = 1 on [0, 1] and that η = 0 on [2, ∞). For each j ∈ N, define Φj (U ) =
η(j 1 − U ) , η(j 1 − V ) dV U
U ∈ U.
Then the following properties are obvious: (1) (2) (3) (4)
Φj ∈ C(U). Φj 0 on U . Φj (U ) = 0 if 1 − U 2/j . U Φj (U ) dU = 1.
Now let ψ be given as in the statement of the theorem and denote R = sup Hψ kz . |z|<1
44
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
Furthermore, for each j ∈ N denote ψj = YΦj ψ. By Lemma 4.1, ψj ∞ < ∞. Thus we can apply Theorem 1.6 to obtain ψj − P ψj BMO C sup Hψj kz , |z|<1
(4.1)
where C depends only on the complex dimension n. We claim that sup Hψj kz R
(4.2)
|z|<1
for every j ∈ N. To prove (4.2), we first note that for all U ∈ U and z ∈ Cn with |z| < 1, we have WU Hψ WU ∗ kz = HWU ψ kz and WU ∗ kz = kU z . Thus for all j ∈ N, |z| < 1 and f ∈ L2 (S, dσ ) H 2 (S) we have Hψj kz , f = ψj kz , f = ψj , k¯z f = YΦj ψ, k¯z f = =
Φj (U )kz WU ψ, f dU =
U
Φj (U )WU ψ, k¯z f dU
U
Φj (U )HWU ψ kz , f dU U
=
Φj (U )WU Hψ kU z , f dU. U
By properties (2) and (4) we now have Hψ kz , f j
Φj (U )Hψ kU z f dU Rf U
for all j ∈ N, |z| < 1 and f ∈ L2 (S, dσ ) H 2 (S). This proves (4.2). Now consider an arbitrary B = B(ζ, r), where ζ ∈ S and r > 0. By (4.1) and (4.2), 1 σ (B)
ψj − P ψj − (ψj − P ψj )B dσ CR
(4.3)
B
for every j ∈ N. Clearly, the proof will be complete if we can show limj →∞ ψj − ψ = 0, for this convergence and (4.3) together will give us 1 σ (B)
B
ψ − P ψ − (ψ − P ψ)B dσ CR.
J. Xia / Journal of Functional Analysis 255 (2008) 25–45
45
Thus the proof is now reduced to that of the strong operator convergence lim YΦj = 1
(4.4)
j →∞
on the Hilbert space L2 (S, dσ ). But this is a routine. It is easy to see that if q ∈ C(S), then (YΦj q)(ζ ) = Φj (U )q(U ζ ) dU, ζ ∈ S. U
Applying properties (1)–(4), we have lim YΦj q − q∞ = 0,
j →∞
q ∈ C(S).
(4.5)
Also, by (2) and (4), the norm of the operator YΦj on the Hilbert space L2 (S, dσ ) satisfies the estimate YΦj 1. Obviously, (4.4) follows from (4.5) and this norm bound. 2 References [1] C. Berger, L. Coburn, Toeplitz operators on the Segal–Bargmann space, Trans. Amer. Math. Soc. 301 (1987) 813– 829. [2] S.-Y.A. Chang, A generalized area integral estimate and applications, Studia Math. 69 (1980/1981) 109–121. [3] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976) 611–635. [4] A. Davie, N. Jewell, Toeplitz operators in several complex variables, J. Funct. Anal. 26 (1977) 356–368. [5] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [6] P. Muhly, J. Xia, Calderón–Zygmund operators, local mean oscillation and certain automorphisms of the Toeplitz algebra, Amer. J. Math. 117 (1995) 1157–1201. [7] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980. [8] E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. [9] J. Xia, Bounded functions of vanishing mean oscillation on compact metric spaces, J. Funct. Anal. 209 (2004) 444–467. [10] J. Xia, Singular integral operators and essential commutativity on the sphere, 2006, preprint. [11] D. Zheng, Toeplitz operators and Hankel operators on the Hardy space of the unit sphere, J. Funct. Anal. 149 (1997) 1–24. [12] K. Zhu, Operator Theory in Function Spaces, second ed., Math. Surveys Monogr., vol. 138, Amer. Math. Soc., Providence, RI, 2007.
Journal of Functional Analysis 255 (2008) 46–89 www.elsevier.com/locate/jfa
E0-dilation of strongly commuting CP0 -semigroups Orr Moshe Shalit 1 Department of Mathematics, Technion, Haifa 32000, Israel Received 19 September 2007; accepted 2 April 2008 Available online 9 May 2008 Communicated by D. Voiculescu
Abstract We prove that every strongly commuting pair of CP0 -semigroups has a minimal E0 -dilation. This is achieved in two major steps, interesting in themselves: (1) we show that a strongly commuting pair of CP0 semigroups can be represented via a two parameter product system representation; (2) we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP0 -semigroups on B(H ), H finite-dimensional, has an E0 -dilation. © 2008 Elsevier Inc. All rights reserved. Keywords: Dilation; E0 -semigroups; CP0 -semigroups; Two-parameter semigroups; Strong commutativity; Product system; Representation; Completely contractive covariant representation
1. Introduction Let M be a von Neumann algebra acting on a separable Hilbert space H . A CP0 -semigroup on M is a family Θ = {Θt }t0 of normal, unital, completely positive maps on M satisfying the semigroup property Θs+t (a) = Θs Θt (a) ,
s, t 0, a ∈ M,
E-mail address:
[email protected]. 1 The author is supported by the Jacobs School of Graduate Studies and the Department of Mathematics at the
Technion—I.I.T, and by the Gutwirth Fellowship. This research is part of the author’s PhD thesis, done under the supervision of Professor Baruch Solel. Fax: +972 4 8293388. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.003
O.M. Shalit / Journal of Functional Analysis 255 (2008) 46–89
47
and the continuity condition lim Θt (a)h, g = Θt0 (a)h, g ,
t→t0
a ∈ M, h, g ∈ H.
A CP0 -semigroup is called an E0 -semigroup if each of its elements is a ∗-endomorphism. In the past two decades, E0 -semigroups have been extensively studied (for a thorough introduction, including many references and “historical” remarks, see [1]). Although every E0 -semigroup is a CP0 -semigroup, non-multiplicative CP0 -semigroups are known to be quite different from E0 semigroups. However, it has been proved that, in some sense, every CP0 -semigroup is “part” of a bigger E0 -semigroup. To be more precise, we say that a quadruple (K, u, R, α) is an E0 -dilation of Θ if K is a Hilbert space, u : H → K is an isometry, R is a von Neumann algebra satisfying u∗ Ru = M, and α is an E0 -semigroup such that Θt (u∗ bu) = u∗ αt (b)u,
b ∈ R,
for all t 0. It has been proved by several authors, using several different techniques, that every CP0 -semigroup has an E0 -dilation (Bhat, Skeide [2], SeLegue [7], Muhly, Solel [5] and Arveson [1]. We note that most of the authors have this result also for not necessarily unital semigroups). This is the precise sense in which we mean that every CP0 -semigroup is a “part” of an E0 -semigroup. If S is a topological semigroup, one can define the notions of CP0 and E0 -semigroups over S. It is then natural to ask whether every CP0 -semigroup Θ = {Θs }s∈S over S has an E0 -dilation. In this paper we make a first attempt to prove the existence of a minimal E0 -dilation for a CP0 semigroup over R2+ := [0, ∞) × [0, ∞). Let us now describe what we actually achieve. If {Rt }t0 and {St }t0 are two CP0 -semigroups that commute (that is, for all t, s 0, Rs St = St Rs ) then we can define a two parameter CP0 -semigroup P(s,t) = Rs St . And if we begin with a CP0 -semigroup {P(t,s) }(t,s)∈R2 , then we can define a commuting pair of semigroups by Rt = + P(t,0) and St = P(0,t) (there are some non-trivial continuity issues to take care of. This will be done below, in Lemma 6.2). The main result of this paper is the following theorem. Theorem. Let {Rt }t0 and {St }t0 be two strongly commuting CP0 -semigroups on a von Neumann algebra M ⊆ B(H ), where H is a separable Hilbert space. Then the two-parameter CP0 -semigroup P defined by P(s,t) := Rs St has a minimal E0 -dilation. The condition of strong commutativity that appears in the above theorem is a technical one, and it is not yet completely understood (see Definition 4.1 below). However, there are many pairs of strongly commuting CP0 -semigroups, and in Appendix A we give some sufficient, and in some cases even necessary, conditions for strong commutativity. These give rise to many examples of two-parameter semigroups for which the above theorem applies. In particular, by Proposition A.1 below, if H is finite-dimensional then every pair of commuting CP maps on B(H ) commute strongly, so every pair of commuting CP0 -semigroups on B(H ) has a minimal E0 -dilation (Corollary 6.7).
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Let us now give an overview of the paper, which should also give some idea of how the above theorem is proved. In what follows, we shall use the notation of the theorem stated above. After reviewing some preliminary notions and setting the notation in Section 2, we explain in Section 3 the approach of Muhly and Solel to dilation theory (as it appeared in [5]). This is the approach that we will be using. In Section 4, after introducing the notion of strong commutativity and proving a few related results, we construct a (discrete) product system of M -correspondences X over R2+ , together with a fully coisometric, completely contractive covariant representation (σ, T ) of X on H , such that for all a ∈ M, (s, t) ∈ R2+ , ∗ P(s,t) (a) = T˜(s,t) (IX(s,t) ⊗ a)T˜(s,t) .
It is in the construction of the product system X that strong commutativity plays its role. In Section 5, we prove that every fully coisometric, completely contractive covariant representation of a product system over Rk+ (and over some more general semigroups, as well) can be dilated to an isometric and fully coisometric covariant representation. We do this using the method of “representing product system representations as contractive semigroups on a Hilbert space,” which we have introduced in [8]. In Section 6 we show that the isometric dilation (ρ, V ) of the product system representation (σ, T ) obtained in Section 5 gives rise to our sought after E0 -dilation in the following way (up to a few simplifications that we must make here). Let K be the Hilbert space on which V represents X, put R = ρ(M ) , and let u be the isometric inclusion H → K. The E0 -dilation we are looking for is (K, u, R, α), where the semigroup α = {αs }s∈R2 is defined by +
αs (b) = V˜s (I ⊗ b)V˜s∗ ,
s ∈ R2+ , b ∈ R.
At the end of Section 6 we show that the dilation that we constructed is minimal, and we show that if M = B(H ) then R = B(K). In Section 7 we close this paper by considering the problem of finding an E0 -dilation to a CP0 -semigroup over N × R+ where strong commutativity does not occur. 2. Preliminaries 2.1. C ∗ /W ∗ -correspondences, their products and their representations Definition 2.1. Let A be a C ∗ -algebra. A Hilbert C ∗ -correspondences over A is a (right) Hilbert A-module E which carries an adjointable, left action of A. Definition 2.2. Let M be a W ∗ -algebra. A Hilbert W ∗ -correspondence over M is a self-adjoint Hilbert C ∗ -correspondence E over M, such that the map M → L(E) which gives rise to the left action is normal. The following notion of representation of a C ∗ -correspondence was studied extensively in [4], and turned out to be a very useful tool.
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Definition 2.3. Let E be a C ∗ -correspondence over A, and let H be a Hilbert space. A pair (σ, T ) is called a completely contractive covariant representation of E on H (or, for brevity, a c.c. representation) if (1) T : E → B(H ) is a completely contractive linear map; (2) σ : A → B(H ) is a nondegenerate ∗-homomorphism; and (3) T (xa) = T (x)σ (a) and T (a · x) = σ (a)T (x) for all x ∈ E and all a ∈ A. If A is a W ∗ -algebra and E is W ∗ -correspondence then we also require that σ be normal. Given a C ∗ -correspondence E and a c.c. representation (σ, T ) of E on H , one can form the Hilbert space E ⊗σ H , which is defined as the Hausdorff completion of the algebraic tensor product with respect to the inner product x ⊗ h, y ⊗ g = h, σ x, y g . One then defines T˜ : E ⊗σ H → H by T˜ (x ⊗ h) = T (x)h. As in the theory of contractions on a Hilbert space, there are certain particularly nice representations which deserve to be singled out. Definition 2.4. A c.c. representation (σ, T ) is called isometric if for all x, y ∈ E, T (x)∗ T (y) = σ x, y . (This is the case if and only if T˜ is an isometry.) It is called fully coisometric if T˜ is a coisometry. Given two Hilbert C ∗ -correspondences E and F over A, the balanced (or inner) tensor product E ⊗A F is a Hilbert C ∗ -correspondence over A defined to be the Hausdorff completion of the algebraic tensor product with respect to the inner product x ⊗ y, w ⊗ z = y, x, w · z , x, w ∈ E, y, z ∈ F. The left and right actions are defined as a · (x ⊗ y) = (a · x) ⊗ y and (x ⊗ y)a = x ⊗ (ya), respectively, for all a ∈ A, x ∈ E, y ∈ F . We shall usually omit the subscript A, writing just E ⊗ F . When working in the context of W ∗ -correspondences, that is, if E and F are W *correspondences and A is a W ∗ -algebra, then E ⊗A F is understood do be the self-dual extension of the above construction. Suppose S is an abelian cancellative semigroup with identity 0 and p : X → S is a family of W ∗ -correspondences over A. Write X(s) for the correspondence p −1 (s) for s ∈ S. We say that X is a (discrete) product system over S if X is a semigroup, p is a semigroup homomorphism and, for each s, t ∈ S \ {0}, the map X(s) × X(t) (x, y) → xy ∈ X(s + t) extends to an isomorphism Us,t of correspondences from X(s)⊗A X(t) onto X(s +t). The associativity of the multiplication means that, for every s, t, r ∈ S, Us+t,r (Us,t ⊗ IX(r) ) = Us,t+r (IX(s) ⊗ Ut,r ).
(1)
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We also require that X(0) = A and that the multiplications X(0) × X(s) → X(s) and X(s) × X(0) → X(s) are given by the left and right actions of A and X(s). Definition 2.5. Let H be a Hilbert space, A be a W ∗ -algebra and let X be a product system of Hilbert A-correspondences over the semigroup S. Assume that T : X → B(H ), and write Ts for the restriction of T to X(s), s ∈ S, and σ for T0 . T (or (σ, T )) is said to be a completely contractive covariant representation of X if (1) for each s ∈ S, (σ, Ts ) is a c.c. representation of X(s); and (2) T (xy) = T (x)T (y) for all x, y ∈ X. T is said to be an isometric (fully coisometric) representation if it is an isometric (fully coisometric) representation on every fiber X(s). Since we shall not be concerned with any other kind of representation, we shall call a completely contractive covariant representation of a product system simply a representation. 2.2. CP-semigroups and E-dilations Let S be a unital subsemigroup of Rk+ , and let M be a von Neumann algebra acting on a Hilbert space H . A CP map is a completely positive, contractive and normal map on M. A CPsemigroup over S is a family {Θs }s∈S of CP maps on M such that: (1) for all s, t ∈ S Θs ◦ Θt = Θs+t ; (2) Θ0 = idM ; (3) for all h, g ∈ H and all a ∈ M, the function S s → Θs (a)h, g is continuous. A CP-semigroup is called an E-semigroup if it consists of ∗-endomorphisms. A CP (E)semigroup is called a CP0 (E0 )-semigroup if all its elements are unital. Definition 2.6. Let M be a von Neumann algebra of operators acting on a Hilbert space H , and let Θ = {Θs }s∈S be a CP-semigroup over the semigroup S. An E-dilation of Θ is a quadruple (K, u, R, α), where K is a Hilbert space, u : H → K is an isometry, R is a von Neumann algebra satisfying u∗ Ru = M, and α is an E-semigroup over S such that Θs (u∗ au) = u∗ αs (a)u,
a ∈ R,
for all s ∈ S. If (K, u, R, α) is a dilation of Θ, then (M, Θ) is called a compression of (K, u, R, α).
(2)
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Let us review some basic facts regarding E-dilations. Most of the content of the following paragraphs is spelled out in [1, Chapter 8], for the case where S = R+ . Note that by putting a = uxu∗ in (2), for any x ∈ M, one has Θs (x) = u∗ αs (uxu∗ )u,
x ∈ M.
(3)
If one identifies M with uMu∗ , H with uH , and p with uu∗ , one may give the following equivalent definition, which we shall use interchangeably with Definition 2.6: a triple (p, R, α) is called a dilation of Θ if R is a von Neumann algebra containing M, α is an E-semigroup on R and p is a projection in R such that M = pRp and Θs (pap) = pαs (a)p holds for all s ∈ S, a ∈ R. With this change of notation, we have pαs (a)p = Θs (pap) = Θs p 2 ap 2 = pαs (pap)p, so, taking a = 1 − p, 0 = pαs p(1 − p)p p = pαs (1 − p)p. This means that for all s ∈ S, αs (1 − p) 1 − p. A projection with this property is called coinvariant (note that if α is an E0 -semigroup then p is a coinvariant projection if and only if it is increasing, i.e., αs (p) p for all s ∈ S). Equivalently, uu∗ αs (1) = uu∗ αs (uu∗ ),
s ∈ S.
(4)
One can also show that (3) and (4) together imply (2), and this leads to another equivalent definition of E-dilation of a CP-semigroup. Let Θ = {Θs }s∈S be a CP-semigroup on a von Neumann algebra M, and let (K, u, R, α) be an E-dilation of Θ. Assume that q ∈ R is a projection satisfying uu∗ q. Assume furthermore that q is coinvariant. Then one can show that the maps βs : a → qαs (a)q are the elements of a CP-semigroup on qRq. If the maps {βs } happen to be multiplicative on qRq, then we say that q is multiplicative. In this case, (qK, u, qRq, β) is an E-dilation of Θ, which is in some sense “smaller” than (K, u, R, α). On the other hand, consider the von Neumann algebra ˜ = W∗ R
αs (uMu ) . ∗
s∈S
˜ we This algebra is clearly invariant under α, and it contains uMu∗ . Thus, restricting α to R, obtain a “smaller” dilation. This discussion leads to the following definition.
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Definition 2.7. Let (K, u, R, α) be an E-dilation of the CP-semigroup Θ. (K, u, R, α) is said to be a minimal dilation if there is no multiplicative, coinvariant projection 1 = q ∈ R such that uu∗ q, and if R = W∗
αs (uMu∗ ) .
(5)
s∈S
In [1] Arveson defines a minimal dilation slightly differently: Definition 2.8. Let (K, u, R, α) be an E-dilation of the CP-semigroup Θ. (K, u, R, α) is said to a minimal dilation if the central support of uu∗ in R is 1, and if (5) holds. The two definitions have been shown to be equivalent in the case where Θ is a CP0 -semigroup over R+ [1, Section 8.9]. We now treat the general case. Proposition 2.9. Definition 2.7 holds if 2.8 does. Proof. Assume that Definition 2.7 is violated. If (5) is violated, then Definition 2.8 is, too. So assume that (5) holds, and that there is a multiplicative, coinvariant projection 1 = q ∈ R such that uu∗ q. Denote A = {αs (a): a ∈ uMu∗ , s ∈ S}. By a trivial generalization of [1, Proposition 8.9.4], q commutes with αs (qRq) for all s ∈ S, so q commutes with A, thus q commutes with W ∗ ( s∈S αs (uMu∗ )). In other words, q is central in R. 2 Whether or not the two definitions are equivalent remains an interesting open question. ∗ To prove that they ∗are, it would be enough to show that the central support of p = uu in ∗ W ( s∈S αs (uMu )) is a coinvariant projection, because the central support is clearly a multiplicative projection. This has been done by Arveson in [1, Proposition 8.3.4], for the case of a CP0 -semigroup over S = R+ . Arveson’s proof makes use of the order structure of R+ and cannot be extended to the case R2+ with which we are concerned in this paper. 3. Overview of the Muhly–Solel approach to dilation In this section we describe the approach of Muhly and Solel to dilation of CP-semigroups on von Neumann algebras. This approach was used by Muhly and Solel to dilate CP-semigroups over N and R+ [5], and later by Solel for semigroups over N2 [10]. Our program is to adapt this approach for semigroups over S = R2+ . 3.1. The basic strategy Let Θ be a CP-semigroup over the semigroup S, usually acting on a von Neumann algebra M of operators in B(H ). The dilation is carried out in two main steps. In the first step, a (discrete) product system of M -correspondences X over S is constructed, together with a c.c. representation (σ, T ) of X on H , such that for all a ∈ M, s ∈ S, Θs (a) = T˜s (IX(s) ⊗ a)T˜s∗ ,
(6)
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where Ts is the restriction of T to X(s). In [5, Proposition 2.21], it is proven that for any c.c. representation (σ, T ) of a W ∗ -correspondence E over a W ∗ -algebra N , the mapping ∗ a → T˜s (IX(s) ⊗ a)T˜s is a normal, completely positive map on σ (N ) (for all s). It is also shown that if T is isometric then this map is multiplicative. Having this in mind, one sees that a natural way to continue the process of dilation will be to “dilate” (σ, T ) to an isometric c.c. representation. Definition 3.1. Let A be a C ∗ -algebra, X be a product system of A-correspondences over the semigroup S, and (σ, T ) a c.c. representation of X on a Hilbert space H . An isometric dilation of (σ, T ) is an isometric representation (ρ, V ) of X on a Hilbert space K ⊇ H , such that (i) H reduces ρ and ρ(a)|H = PH ρ(a)|H = σ (a), for all a ∈ A; (ii) for all s ∈ S, x ∈ Xs , one has PH Vs (x)|KH = 0; (iii) for all s ∈ S, x ∈ Xs , one has PH Vs (x)|H = Ts (x). Such a dilation is called minimal in case the smallest subspace of K containing H and invariant under every Vs (x), x ∈ X, s ∈ S, is all of K. It will be convenient at times to regard an isometric dilation as a quadruple (K, u, V , ρ), where (ρ, V ) are as above and u : H → K is an isometry. Constructing a minimal isometric dilation (K, u, V , ρ) of the representation (σ, T ) appearing in Eq. (6) constitutes the second step of the dilation process. Then one has to show that if R = ρ(M ) , and α is defined by αs (a) := V˜s (IX(s) ⊗ a)V˜s∗ ,
a ∈ R,
then the quadruple (K, u, R, α) is an E-dilation for (Θ, M). In [4,5,10], it is proved that any c.c. representation of a product system over N, R+ or N2 (the latter two, X is assumed to be a product system of W ∗ -correspondence, and σ is assumed to be normal), has a minimal isometric dilation. Moreover, it is shown that if X is a product system of W ∗ -correspondences and σ is assumed to be normal then ρ is also normal. When the product system is over N or R+ , the minimal isometric dilation is also unique. From these results, the authors deduce the existence of an E-dilation of a CP-semigroup Θ acting on a von Neumann algebra M. When Θ is a CPsemigroup over S = R+ and H is separable, then α is shown to be an E-semigroup that is a minimal dilation. 3.2. Description of the construction of the product system and representation for one parameter semigroups In this subsection we give a detailed description of Muhly and Solel’s construction of the product system and c.c. representation associated with a one-parameter CP-semigroup [5]. We shall use this construction in Section 4. We note that the original construction in [5] was carried out for CP0 -semigroups, but it works just as well for CP-semigroups, and that no use is made of the continuity with respect to t.
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Let Θ = {Θt }t0 be a CP-semigroup acting on a von Neumann algebra M of operators in B(H ). Let B(t) denote the collection of partitions of the closed unit interval [0, t], ordered by refinement. For p ∈ B(t), we define a Hilbert space Hp,t by Hp,t := M ⊗Θt1 M ⊗Θt2 −t1 M ⊗ · · · ⊗Θt−tn−1 H, where p = {0 = t0 < t1 < t2 < · · · < tn = t}, and the right-hand side of the above equation is the Hausdorff completion of the algebraic tensor product M ⊗ M ⊗ · · · ⊗ H with respect to the inner product T1 ⊗ · · · ⊗ Tn ⊗ h, S1 ⊗ · · · ⊗ Sn ⊗ k
∗ · · · Θt1 T1∗ S1 · · · Sn−1 Sn k . = h, Θt−tn−1 Tn∗ Θtn−1 −tn−2 Tn−1 Hp,t is a left M-module via the action S · (T1 ⊗ · · · ⊗ Tn ⊗ h) = ST1 ⊗ · · · ⊗ Tn ⊗ h. We now define the intertwining spaces
LM (H, Hp,t ) = X ∈ B(H, Hp,t ): ∀S ∈ M.XS = S · X . The inner product X1 , X2 := X1∗ X2 , for Xi ∈ LM (H, Hp,t ), together with the right and left actions (XR)h := X(Rh), and (RX)h := (I ⊗ · · · ⊗ I ⊗ R)Xh, for R ∈ M , X ∈ LM (H, Hp,t ), make LM (H, Hp,t ) into a W ∗ -correspondence over M . The Hilbert spaces Hp,t and W ∗ -correspondences LM (H, Hp,t ) form inductive systems as follows. Let p, p ∈ B(t), p p . In the particular case where p = {0 = t0 < · · · < tk < tk+1 < · · · < tn = t} and p = {0 = t0 < · · · < tk < τ < tk+1 < · · · < tn = t}, we can define a Hilbert space isometry v0 : Hp,t → Hp ,t by v0 (T1 ⊗ · · · ⊗ Tk+1 ⊗ Tk+2 ⊗ · · · ⊗ Tn ⊗ h) = T1 ⊗ · · · ⊗ Tk+1 ⊗ I ⊗ Tk+2 ⊗ · · · ⊗ Tn ⊗ h. This map gives rise to an isometry of W ∗ -correspondences v : LM (H, Hp,t ) → LM (H, Hp ,t ) by v(X) = v0 ◦ X. Now, if p p are any partitions in B(t), then we can define v0,p,p : Hp,t → Hp ,t and vp,p : LM (H, Hp,t ) → LM (H, Hp ,t ) by composing a finite number of maps such as v0 and v constructed in the previous paragraph, and we get legitimate arrow maps. Now one can form two different direct limits: Ht := lim −→(Hp,t , v0,p,p )
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and E(t) := lim −→ LM (H, Hp,t ), vp,p . The inductive limit also supplies us with embeddings of the blocks v0,p,∞ : Hp,t → Ht and vp,∞ : LM (H, Hp,t ) → E(t). One can also define interwining spaces LM (H, Ht ), each of which has the structure of an M -correspondence, and these spaces are isomorphic as W ∗ correspondences to the spaces E(t). {E(t)}t0 is the product system of M -correspondences that we are looking for. We have yet to describe the c.c. representation (σ, T ) that will “represent” Θ as in Eq. (6) (with X(s) replaced by E(s)). The sought after representation is the so called “identity representation,” which we now describe. First, we set σ = T0 = idM . Next, let t > 0. For p = {0 = t0 < · · · < tn = t}, the formula ιp (h) = I ⊗ · · · ⊗ I ⊗ h defines an isometry ιp : H → Hp,t , with adjoint given by the formula ι∗p (X1 ⊗ · · · ⊗ Xn ⊗ h) = Θt−tn−1 Θtn−1 −tn−2 · · · Θt1 (X1 )X2 · · · Xn−1 Xn h. For p a refinement of p, one computes ι∗p = ι∗p ◦ v0,p,p . This induces a unique map ι∗t : Ht → H that satisfies ι∗t ◦ v0,p,∞ = ι∗p . The c.c. representation Tt on E(t) is given by Tt (X) = ι∗t ◦ X, where we have identified E(t) with LM (H, Ht ). 4. Representing strongly commuting CP0 -semigroups In this section and in the next two we prove our main result: every pair of strongly commuting CP0 -semigroups has an E0 -dilation. As we mentioned in the previous section, our program is to prove this result using the Muhly–Solel approach, which consists of two main steps. In this section we concentrate on the first step: the representation of a pair of strongly commuting CPsemigroups using a product system representation via a formula such as Eq. (6) above. This will be done in Section 4.3, whereas Sections 4.1 and 4.2 will be devoted to the notion of strong commutativity and its implications. Throughout this and the two following sections, M will be a von Neumann algebra acting on a Hilbert space H . There is a natural correspondence between two parameter semigroups of maps and pairs of commuting one parameter semigroups. Indeed, if {Rt }t0 and {St }t0 are two semigroups that commute (that is, for all t, s 0, Rs St = St Rs ) then we can define a two parameter semigroup P(s,t) = Rs St . And if we begin with a semigroup {P(t,s) }(t,s)∈R2 , + then we can define a commuting pair of semigroups by Rt = P(t,0) and St = P(0,t) . It is not trivial that P is continuous (in the relevant sense) if and only if R and S are—it follows from the fact that (s, X) → Rs (X) is jointly continuous in the weak topology (we shall make this argument precise in Lemma 6.2). From now on we fix the notation in the preceding paragraph, and we shall use either {P(t,s) }(t,s)∈R2 or the pair {Rt }t0 and {St }t0 to denote a fixed two+ parameter CP-semigroup. Note also that if {αt }t0 and {βt }t0 are commuting E-dilations of {Rt }t0 and {St }t0 acting on the same von Neumann algebra, then {αt βs }t,s0 is an E-dilation of {P(t,s) }(t,s)∈R2 , and vice versa. +
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4.1. Strongly commuting CP maps Let Θ and Φ be CP maps on M. We define the Hilbert space M ⊗Φ M ⊗Θ H to be the Hausdorff completion of the algebraic tensor product M ⊗alg M ⊗alg H with respect to the inner product a ⊗ b ⊗ h, c ⊗ d ⊗ k = h, Θ b∗ Φ(a ∗ c)d k . Definition 4.1. Let Θ and Φ be CP maps on M. We say that they commute strongly if there is a unitary u : M ⊗Φ M ⊗Θ H → M ⊗Θ M ⊗Φ H such that: (i) u(a ⊗Φ I ⊗Θ h) = a ⊗Θ I ⊗Φ h for all a ∈ M and h ∈ H . (ii) u(ca ⊗Φ b ⊗Θ h) = (c ⊗ IM ⊗ IH )u(a ⊗Φ b ⊗Θ h) for a, b, c ∈ M and h ∈ H . (iii) u(a ⊗Φ b ⊗Θ dh) = (IM ⊗ IM ⊗ d)u(a ⊗Φ b ⊗Θ h) for a, b ∈ M, d ∈ M and h ∈ H . The notion of strong commutation was introduced by Solel in [10]. Note that if two CP maps commute strongly, then they commute. The converse is false (for concrete examples see Sections 7 and A.5). In Appendix A we shall give many examples of strongly commuting pairs of CP maps, and for some von Neumann algebras we shall give a complete characterization of strong commutativity. For the time being let us just state the fact that if H is a finite-dimensional Hilbert space, then any two commuting CP maps on B(H ) strongly commute (see Section A.3). The “true” significance of strong commutation comes from a bijection between pairs of strongly commuting CP maps and product systems over N2 with c.c. representations ([10, Propositions 5.6 and 5.7], and the discussion between them). It is this bijection that enables one to characterize all pairs of strongly commuting CP maps on B(H ) [10, Proposition 5.8]. In the next section we will work with the spaces M ⊗P1 M · · · M ⊗Pn H , where P1 , . . . , Pn are CP maps. These spaces are defined in a way analogous to the way that the spaces M ⊗Θ M ⊗Φ H were defined in the beginning of this section. The following results are important for dealing with such spaces. Lemma 4.2. Assume that Pn−1 and Pn commute strongly. Then there exists a unitary v : M ⊗P1 M ⊗P2 · · · ⊗Pn−1 M ⊗Pn H → M ⊗P1 M ⊗P2 · · · ⊗Pn M ⊗Pn−1 H such that (1) v(I ⊗P1 · · · ⊗Pn−1 I ⊗Pn h) = I ⊗P1 · · · ⊗Pn I ⊗Pn−1 h, for all h ∈ H , (2) for all X ∈ M, v ◦ (X ⊗ I ⊗ · · · ⊗ I ⊗ I ) = (X ⊗ I ⊗ · · · ⊗ I ⊗ I ) ◦ v, (3) for all X ∈ M , v ◦ (I ⊗ I ⊗ · · · ⊗ I ⊗ X) = (I ⊗ I ⊗ · · · ⊗ I ⊗ X) ◦ v.
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Proof. Let u : M ⊗Pn−1 M ⊗Pn H → M ⊗Pn M ⊗Pn−1 H be the unitary that makes Pn−1 and Pn commute strongly. Define v = IE ⊗ u, where E denotes the W ∗ -correspondence (over M) M ⊗P1 M ⊗P2 · · · ⊗Pn−3 M equipped with the inner product ∗ · · · P1 a1∗ b1 · · · bn−3 . a1 ⊗ · · · ⊗ an−3 , b1 ⊗ · · · ⊗ bn−3 = Pn−3 an−3 The fact that v commutes with M ⊗ I ⊗ · · · ⊗ I and I ⊗ I · · · I ⊗ M and satisfies the three conditions listed above are clear from the definition and from the properties of u. The fact that u is surjective implies that v is, too. It is left to show that v is an isometry (and this will also show that it is well defined). Let ai ⊗Pn−2 bi ⊗Pn−1 ci ⊗Pn hi be an element of E ⊗Pn−2 M ⊗Pn−1 M ⊗Pn H .
2
v
a ⊗ b ⊗ c ⊗ h i P i P i P i n n−2 n−1
= ai ⊗Pn−2 u(bi ⊗Pn−1 ci ⊗Pn hi ), aj ⊗Pn−2 u(bj ⊗Pn−1 cj ⊗Pn hj ) =
u(bi ⊗Pn−1 ci ⊗Pn hi ), Pn−2 ai , aj u(bj ⊗Pn−1 cj ⊗Pn hj ) = (∗)
i,j
=
u(bi ⊗Pn−1 ci ⊗Pn hi ), u Pn−2 ai , aj bj ⊗Pn−1 cj ⊗Pn hj = (∗∗)
i,j
=
bi ⊗Pn−1 ci ⊗Pn hi , Pn−2 ai , aj bj ⊗Pn−1 cj ⊗Pn hj
i,j
2
a ⊗ b ⊗ c ⊗ h = i Pn−2 i Pn−1 i Pn i
the equality marked by (∗) follows from the fact that u interwines the actions of M on M ⊗Pn−1 M ⊗Pn H and M ⊗Pn M ⊗Pn−1 H , and the one marked by (∗∗) is true because u is unitary. 2 Lemma 4.3. Assume that P and Q are strongly commuting CP maps on M. Then there exists an isomorphism v = vP ,Q of M-correspondences v : M ⊗P M ⊗Q M → M ⊗Q M ⊗P M such that v(I ⊗P I ⊗Q I ) = I ⊗Q I ⊗P I. Proof. For any two CP maps Θ, Φ let WΘ,Φ be the Hilbert space isomorphism WΘ,Φ : M ⊗Θ M ⊗Φ M ⊗I H → M ⊗Θ M ⊗Φ H
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∗ given by WΘ,Φ (a ⊗Θ b ⊗Φ c ⊗I h) = a ⊗Θ b ⊗Φ ch. By a straightforward computation WΘ,Φ ∗ (a ⊗ b ⊗ h) = a ⊗ b ⊗ I ⊗ h, and by even shorter computations is given by WΘ,Φ Θ Φ Θ Φ I ∗ ∗ W and WΘ,Φ WΘ,Φ WΘ,Φ Θ,Φ are identity maps. For all a, b, c, x ∈ M and all y ∈ M we have
WΘ,Φ (xa ⊗Θ b ⊗Φ c ⊗I yh) = xa ⊗Θ b ⊗Φ cyh = xa ⊗Θ b ⊗Φ ych = (x ⊗ I ⊗ y)WΘ,Φ (a ⊗Θ b ⊗Φ c ⊗I h). From this, it also follows that ∗ ∗ WΘ,Φ (x ⊗ I ⊗ y) = (x ⊗ I ⊗ I ⊗ y)WΘ,Φ
(x ∈ M, y ∈ M ).
We now define a map T : M ⊗P M ⊗Q M ⊗I H → M ⊗Q M ⊗P M ⊗I H by ∗ T = WQ,P ◦ u ◦ WP ,Q ,
where u is the map that makes P and Q commute strongly. As a product of such maps, T is a unitary interwining the left actions of M and M . The v that we are looking for is a map v : M ⊗P M ⊗Q M → M ⊗Q M ⊗P M that satisfies T = v ⊗ IH . We will find this v using a standard technique exploiting the self duality of M ⊗Q M ⊗P M. For any x ∈ M ⊗Q M ⊗P M we define a map Lx : H → M ⊗Q M ⊗P M ⊗I H by Lx (h) = x ⊗ h (h ∈ H ). The adjoint is given on simple tensors by L∗x (y ⊗ h) = x, y h. Now, if there is a v such that T = v ⊗ IH , then for all z ∈ M ⊗P M ⊗Q M and x ∈ M ⊗Q M ⊗P M we must have x, v(z) h = L∗x v(z) ⊗ h = L∗x T (z ⊗ h).
This leads us to define, fixing z ∈ M ⊗P M ⊗Q M, a mapping ϕ from M ⊗Q M ⊗P M into M: ϕ(x)h := L∗x T (z ⊗ h). We now prove that x → ϕ(x)∗ is a bounded, M-module mapping into M. into M. For all x ∈ M ⊗Q M ⊗P M, ϕ(x) is linear. L∗x T (z ⊗ h) L∗x T zh, so ϕ(x) ∈ B(H ). So ϕ(x)∗ exists and is also a bounded, linear operator on H . Now take d ∈ M . Then ϕ(x)dh = L∗x T (z ⊗ dh) = L∗x T (I ⊗ d)(z ⊗ h) = L∗x (I ⊗ d)T (z ⊗ h) = dϕ(x)h (L∗x interwines M from its very definition) whence ϕ(x) ∈ M = M. Thus, ϕ(x)∗ ∈ M. M-module mapping. This is because for all x, y ∈ M ⊗Q M ⊗P M and all a ∈ M Lx+y = Lx + Ly and Lax = aLx (and also Lxa = Lx a).
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Bounded mapping. From the inequalities L∗x T (z ⊗ h) L∗x T zh and L∗x x it follows that ϕ(x)∗ = ϕ(x) zx. It now follows from the self-duality of M ⊗Q M ⊗P M that for all z ∈ M ⊗P M ⊗Q M there exists a v(z) ∈ M ⊗Q M ⊗P M such that x, v(z) h = L∗x T (z ⊗ h)
(7)
for all x ∈ M ⊗Q M ⊗P M, h ∈ H . It is easy to see from (7) that v(z) is a right M-module mapping. (7) can be re-written as L∗x v(z) ⊗ h = L∗x T (z ⊗ h), ∗ and, since this⊥ holds for all x, this means that (v(z) ⊗ h) = T (z ⊗ h) (because x Ker(Lx ) = ( x Im(Lx )) = {0}), or, in other words, v ⊗ I = T . This last equality implies that v is unitary, and that it has all the properties required. For example, if a, b, c, X ∈ M and h ∈ H , then v(Xa ⊗ b ⊗ c) ⊗ h = T (Xa ⊗ b ⊗ c ⊗ h) = (X ⊗ I ⊗ I ⊗ I )T (a ⊗ b ⊗ c ⊗ h) = (X ⊗ I ⊗ I ⊗ I ) v(a ⊗ b ⊗ c) ⊗ h = (X ⊗ I ⊗ I )v(a ⊗ b ⊗ c) ⊗ h. Putting v1 = v(Xa ⊗ b ⊗ c) and v2 = (X ⊗ I ⊗ I )(v(a ⊗ b ⊗ c) we have that for all h ∈ H
2 0 = v1 ⊗ h − v2 ⊗ h2 = (v1 − v2 ) ⊗ h = h, v1 − v2 , v1 − v2 h , which implies that v1 − v2 , v1 − v2 = 0, or v(Xa ⊗ b ⊗ c) = (X ⊗ I ⊗ I )(v(a ⊗ b ⊗ c).
2
Remark 4.4. The converse of Lemma 4.3 is also true: if there is an isometry of Mcorrespondences v : M ⊗P M ⊗Q M → M ⊗Q M ⊗P M such that v(I ⊗ I ⊗ I ) = I ⊗ I ⊗ I then P and Q strongly commute. Indeed, to obtain u : M ⊗P M ⊗Q H → M ⊗Q M ⊗P H with the desired properties, we simply reverse the construction above. That is, we define T = v ⊗ I , and u = WQ,P ◦ T ◦ WP∗ ,Q . Lemma 4.5. Assume that Pj and Pj +1 commute strongly, for some j n − 2. Then there exists a unitary u : M ⊗P1 · · · ⊗Pj M ⊗Pj +1 · · · M ⊗Pn H → M ⊗P1 · · · ⊗Pj +1 M ⊗Pj · · · M ⊗Pn H such that (1) u(I ⊗P1 · · · I ⊗Pj I ⊗Pj +1 I · · · I ⊗Pn h) = I ⊗P1 · · · I ⊗Pj +1 I ⊗Pj I · · · I ⊗Pn h,
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(2) for all X ∈ M, u ◦ (X ⊗ I · · · I ⊗ I ) = (X ⊗ I · · · I ⊗ I ) ◦ u, (3) for all X ∈ M , u ◦ (I ⊗ I · · · I ⊗ X) = (I ⊗ I · · · I ⊗ X) ◦ u. Proof. Let v : M ⊗Pj M ⊗Pj +1 M → M ⊗Pj +1 M ⊗Pj M be the unitary that is described in Lemma 4.3. Introduce the notation E = M ⊗P1 · · · ⊗Pj −2 M (understood to be C if j = 1 and M if j = 2) and F = M ⊗Pj +3 · · · M ⊗Pn H (understood to be H if j = n − 2). Define u : E ⊗Pj −1 M ⊗Pj M ⊗Pj +1 M ⊗Pj +2 F → E ⊗Pj −1 M ⊗Pj +1 M ⊗Pj M ⊗Pj +2 F by u := IE ⊗ v ⊗ IF . u is a well-defined, unitary mapping, possessing the properties asserted.
2
Putting together Lemmas 4.2, 4.3 and 4.5, we obtain the following Proposition 4.6. Let R1 , R2 , . . . , Rm , and S1 , S2 , . . . , Sn be CP maps such that for all 1 i m, 1 j n, Ri commutes strongly with Sj . Then there exists a unitary v : M ⊗R1 · · · ⊗Rm M ⊗S1 · · · ⊗Sn H → M ⊗S1 · · · ⊗Sn M ⊗R1 · · · ⊗Rm H such that (1) v(I ⊗R1 I · · · I ⊗Sn h) = I ⊗S1 I · · · I ⊗Rm h, for all h ∈ H ; (2) for all X ∈ M, v ◦ (X ⊗ I · · · I ⊗ I ) = (X ⊗ I · · · I ⊗ I ) ◦ v; (3) for all X ∈ M , v ◦ (I ⊗ I · · · I ⊗ X) = (I ⊗ I · · · I ⊗ X) ◦ v. The existence of v as above is clear: simply apply the isomorphisms from the previous lemmas one by one. One might think that applying these isomorphisms in different orders might lead to different v’s. In the next subsection we will see, however, that the order of application does not influence the total outcome (see Proposition 4.8).
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4.2. Strongly commuting CP-semigroups Definition 4.7. Two semigroups of CP maps {Rt }t0 and {St }t0 are said to commute strongly if for all (s, t) ∈ R2+ the CP maps Rs and St commute strongly. In Appendix A we have collected a few examples of strongly commuting CP-semigroups, and we give some necessary and sufficient conditions for strong commutativity in special cases. From this point on R and S will denote two strongly commuting CP-semigroups. Proposition 4.8. If the CP-semigroups {Rt }t0 and {St }t0 commute strongly, then, for all (s, t), (s , t ) ∈ R2+ , the associated maps vRs ,St : M ⊗Rs M ⊗St M → M ⊗St M ⊗Rs M, and vRs ,St : M ⊗Rs M ⊗St M → M ⊗St M ⊗Rs M (see Lemma 4.3) satisfy the following identity: (I ⊗ I ⊗ vRs ,St )(vRs ,St ⊗ I ⊗ I ) = (vRs ,St ⊗ I ⊗ I )(I ⊗ I ⊗ vRs ,St ).
(8)
Proof. Let a, b, c, d, e ∈ M. Assume that vRs ,St (a ⊗Rs b ⊗St c) = m i=1 Ai ⊗St Bi ⊗Rs Ci , and n that vRs ,St (I ⊗Rs d ⊗St e) = j =1 γi ⊗St δj ⊗Rs j . Operating on a ⊗Rs b ⊗St c ⊗Rs d ⊗St e with the operator on the left-hand side of Eq. (8), we obtain (I ⊗ I ⊗ vRs ,St )(vRs ,St ⊗ I ⊗ I )(a ⊗ b ⊗ c ⊗ d ⊗ e) = (I ⊗ I ⊗ vRs ,St )
m
Ai ⊗ Bi ⊗ Ci ⊗ d ⊗ e = (∗)
i=1
=
m
Ai ⊗ Bi ⊗ Ci · vRs ,St (I ⊗ d ⊗ e)
i=1
=
m n
Ai ⊗ Bi ⊗ Ci γj ⊗ δj ⊗ j ,
i=1 j =1
where the equality marked by (∗) is justified because vRs ,St is a left M-module map. Operating on a ⊗Rs b ⊗St c ⊗Rs d ⊗St e with the operator on the right-hand side of Eq. (8), we obtain (vRs ,St ⊗ I ⊗ I )(I ⊗ I ⊗ vRs ,St )(a ⊗ b ⊗ c ⊗ d ⊗ e) = (∗) = (vRs ,St ⊗ I ⊗ I ) a ⊗ b ⊗ c · vRs ,St (I ⊗ d ⊗ e) =
n (vRs ,St ⊗ I ⊗ I )(a ⊗ b ⊗ cγj ⊗ δj ⊗ j ) j =1
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=
n
vRs ,St (a ⊗ b ⊗ cγj ) ⊗ δj ⊗ j = (∗∗)
j =1
=
n
vRs ,St (a ⊗ b ⊗ c) · γj ⊗ δj ⊗ j
j =1
=
m n
Ai ⊗ Bi ⊗ Ci γj ⊗ δj ⊗ j ,
i=1 j =1
where the equality marked by (∗) is justified because vRs ,St is a left M-module map, and the one marked by (∗∗) is OK because vRs ,St is a right M-module map. So Eq. (8) holds for all s, s , t, t , and this proof is complete. 2 4.3. Representing a pair of strongly commuting CP0 -semigroups via the identity representation—the strongly commuting case Recall the notation that we fixed in this section: M is a von Neumann algebra acting on H , {Rt }t0 and {St }t0 are two strongly commuting CP-semigroups on M, and P(s,t) := Rs St . By the discussion in Section A.5, two CP maps that commute strongly may do so in more than one way. Once and for all we fix for every s, t ∈ R+ a unitary that makes Rs and St commute strongly and we also fix the corresponding associated map vRs ,St . We wish to stress the fact that we have just made infinitely many arbitrary choices that (we believe) affect the structure of all the constructions to follow. Let {E(t)}t0 , {F (t)}t0 denote the product systems (of W ∗ -correspondences over M ) associated with {Rt }t0 and {St }t0 , respectively, and let T E , T F be the corresponding identity E and θ F the unitaries representations (as described in Section 3.2). For s, t 0, we denote by θs,t s,t E : E(s) ⊗M E(t) → E(s + t), θs,t
and F θs,t : F (s) ⊗M F (t) → F (s + t).
Proposition 4.9. For all s, t 0 there is an isomorphism of W ∗ -correspondences ϕs,t : E(s) ⊗M F (t) → F (t) ⊗M E(s).
(9)
The isomorphisms {ϕs,t }s,t0 , together with the identity representations T E , T F , satisfy the “commutation” relation: (10) T˜sE IE(s) ⊗ T˜tF = T˜tF IF (t) ⊗ T˜sE ◦ (ϕs,t ⊗ IH ), t, s 0. Proof. We shall adopt the notation used in Section 3.2 (with a few changes), and follow the proof of [10, Proposition 5.6]. Fix s, t 0. Let p = {0 = s0 < s1 < · · · < sm = s} be a partition of [0, s]. We define HpR = M ⊗Rs1 M ⊗Rs2 −s1 · · · M ⊗Rsm −sm−1 H
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and we define (for a partition q) HqS in a similar manner. If q = {0 = t0 < t1 < · · · < tn = t}, we also define R,S = M ⊗Rs1 · · · ⊗Rsm −sm−1 M ⊗S1 · · · ⊗Stn −tn−1 H. Hp,q S,R S,R,S S,R,S,R Hq,p is defined similarly. We can go on to define Hq,p,p , Hq,p,q ,p , etc. Recall that E(s) is the direct limit of the directed system (LM (H, HpR ), vp,p ). Similarly, we shall write (LM (H, HqS ), uq,q ) for the directed system that has F (t) as its limit. We write vp,∞ , uq,∞ for the limit isometric embeddings. We proceed to construct an isomorphism
ϕs,t : E(s) ⊗ F (t) → F (t) ⊗ E(s) that has the desired property. Let p = {0 = s0 < s1 < · · · < sm = s} and q = {0 = t0 < t1 < · · · < tn = t} be partitions of [0, s] and [0, t], respectively. Denote by Γp,q The map from S,R ) given by X ⊗ Y → (I ⊗ I · · · I ⊗ X)Y . LM (H, HpR ) ⊗ LM (H, HqS ) into LM (H, Hq,p As explained in [5, Lemma 3.2], Γp,q is an isomorphism. We define Γq,p to be the correR,S S,R R,S ). Let u : Hq,p → Hp,q sponding map from LM (H, HqS ) ⊗ LM (H, HpR ) into LM (H, Hp,q S,R R,S be the isomorphism from Proposition 4.6, and define Ψ : LM (H, Hq,p ) → LM (H, Hp,q ) by Ψ (Z) = u ◦ Z. The argument from [10, Proposition 5.6] can be repeated here to show that Ψ is an isomorphism of W ∗ -correspondences. Define tp,q : LM (H, HpR ) ⊗ LM (H, HqS ) → LM (H, HqS ) ⊗ LM (H, HpR ) by −1 tp,q = Γq,p ◦ Ψ ◦ Γp,q .
Define maps W1 : H → HpR and W2 : H → HqS by W1 h = I ⊗R1 · · · I ⊗Rsm −sm−1 h and W2 h = S,R R,S I ⊗S1 · · · I ⊗Stn −tn−1 h. Also, let U1 : HpR → Hq,p and U2 : HqS → Hp,q be the maps U1 ξ = I ⊗S1 I · · · I ⊗Stn −tn−1 ξ and U2 η = I ⊗R1 I · · · I ⊗Rsm −sm−1 η. Just as in [10], we have that
W1∗ U1∗ = W2∗ U2∗ u,
(11)
and that, for X ∈ LM (H, HpR ), we have U1∗ (I ⊗ · · · I ⊗ X) = XW2∗ . Now, for X ∈ LM (H, HpR ) and Y ∈ LM (H, HqS ), U1∗ Γp,q (X ⊗ Y ) = U1∗ (I ⊗ I · · · I ⊗ X)Y = XW2∗ Y.
(12)
If we define (TpR , id)2 to be the identity representation of LM (H, HpR ), and (TqS , id) to be the identity representation of LM (H, HqS ), (see the closing paragraph in Section 3.2), then (12) implies that, for h ∈ H , (13) W1∗ U1∗ Γp,q (X ⊗ Y ) h = TpR (X)TqS (Y )h = T˜pR I ⊗ T˜qS (X ⊗ Y ⊗ h). On the other hand, using (11) and an analog of (13), 2 Watch out—we have here a little problem with notation—this resembles T E , T F that we defined above. t t
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W1∗ U1∗ Γp,q (X ⊗ Y ) h = W1∗ U1∗ Ψ −1 Γq,p ◦ tp,q (X ⊗ Y ) h = W1∗ U1∗ u∗ Γq,p ◦ tp,q (X ⊗ Y ) h = W2∗ U2∗ Γq,p ◦ tp,q (X ⊗ Y ) h = T˜qS I ⊗ T˜pR tp,q (X ⊗ Y ) ⊗ h . Let us summarize what we have accumulated up to this point. For fixed s, t 0, and any two partitions p, q of [0, s] and [0, t], respectively, we have a Hilbert space isomorphism tp,q : LM H, HpR ⊗ LM H, HqS → LM H, HqS ⊗ LM H, HpR satisfying T˜pR I ⊗ T˜qS = T˜qS I ⊗ T˜pR (tp,q ⊗ IH ).
(14)
These maps induce an isomorphism tp,∞ : LM (H, HpR ) ⊗ F (t) → F (t) ⊗ LM (H, HpR ) that satisfies tp,∞ (I ⊗ uq,∞ ) = (uq,∞ ⊗ I )tp,q .
(15)
Plugging (15) in (14) we obtain T˜pR I ⊗ T˜qS = T˜qS u∗q,∞ ⊗ T˜pR (tp,∞ ⊗ IH )(I ⊗ uq,∞ ⊗ IH ). The discussion before Theorem 3.9 in [5] imply that T˜tF (uq,∞ ⊗ I ) = T˜qS , or, letting pq denote the projection in F (t) onto uq,∞ (LM (H, HqS )), T˜tF (pq ⊗ I ) = T˜qS u∗q,∞ ⊗ IH . The last two equations sum up to T˜pR I ⊗ T˜tF (I ⊗ pq ⊗ IH ) = T˜tF pq ⊗ T˜pR (tp,∞ ⊗ IH )(I ⊗ pq ⊗ IH ), which implies, in the limit, T˜pR I ⊗ T˜tF = T˜tF IF (t) ⊗ T˜pR (tp,∞ ⊗ IH ). Repeating this “limiting process” in the argument p, we obtain a map t∞,∞ : E(s) ⊗ F (t) → F (t) ⊗ E(s), which we re-label as ϕs,t , that satisfies (10). The above procedure can be done for all s, t 0, giving isomorphisms {ϕs,t } satisfying the commutation relation (10). 2 Our aim now is to construct a product system X over R2+ and a c.c. representation T of X that will lead to a representation of {P(s,t) }(s,t)∈R2 as in Eq. (6). Proposition 4.9 is a key ingredient + in the proof that the representation that we define below gives rise to such a representation. But before going into that we need to carefully construct the product system X.
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We define X(s, t) := E(s) ⊗ F (t), and θ(s,t),(s ,t ) : X(s, t) ⊗ X(s , t ) → X(s + s , t + t ), by E −1 F θ(s,t),(s ,t ) = θs,s ⊗ θt,t ◦ I ⊗ ϕs ,t ⊗ I . To show that {X(s, t)}t,s0 is a product system, we shall need to show that “the θ ’s make the tensor product into an associative multiplication,” or simply: θ(s,t),(s +s ,t +t ) ◦ (I ⊗ θ(s ,t ),(s ,t ) ) = θ(s+s ,t+t ),(s ,t ) ◦ (θ(s,t),(s ,t ) ⊗ I ),
(16)
for s, s , s , t, t , t 0. Proposition 4.10. X = {X(s, t)}t,s0 is a product system. That is, Eq. (16) holds. Proof. The proof is nothing but a straightforward and tedious computation, using Proposition 4.8. Let s, s , s , t, t , t 0, and let p, p , p , q, q , q be partitions of the corresponding intervals. It is enough to show that the maps on both sides of Eq. (16) give the same result when applied to an element of the form ζ = X ⊗ Y ⊗ X ⊗ Y ⊗ X ⊗ Y , where X ∈ LM (H, HpR ), Y ∈ LM (H, HqS ), etc. Let us operate first on ζ with the right-hand side of (16). Now, E F θ(s,t),(s ,t ) (X ⊗ Y ⊗ X ⊗ Y ) = θp,p X ⊗ tp−1 ⊗ θq,q ,q (Y ⊗ X ) ⊗ Y , E is the restriction of θ E to L (H, H R ) ⊗ L (H, H R ), θ F is defined similarly, where θp,p M M p s,s p q,q and tp ,q is the map defined in Proposition 4.9. Looking at the definition of tp ,q , we see that R,S S,R −1 tp−1 ,q (Y ⊗ X ) = Γp ,q (Up ↔q ◦ (I ⊗ Y )X ). Here Up ↔q denotes the unitary Hp ,q → Hq,p given by Proposition 4.6. Assume that
Up ↔q ◦ (I ⊗ Y )X =
(I ⊗ xi )yi .
i
Then Γp−1 xi ⊗ yi , ,q Up ↔q ◦ (I ⊗ Y )X = i
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therefore, θ(s,t),(s ,t ) (X ⊗ Y ⊗ X ⊗ Y ) =
(I ⊗ X)xi ⊗ (I ⊗ yi )Y . i
So, θ(s+s ,t+t ),(s ,t ) ◦ (θ(s,t),(s ,t ) ⊗ I )ζ = θpE ∨p+s ,p ⊗ θqF ∨q+t ,q i
⊗ Y . × (I ⊗ X)xi ⊗ Γp−1 ,q ∨q+t Up ↔q ∨q+t ◦ I ⊗ (I ⊗ yi )Y X Repeated application of Proposition 4.8 shows that, and this is a crucial point, Up ↔q ∨q+t = (I ⊗ Up ↔q )(Up ↔q ⊗ I ). Thus Up ↔q ∨q+t ◦ I ⊗ (I ⊗ yi )Y X = (I ⊗ Up ↔q )(I ⊗ I ⊗ yi ) Up ↔q (I ⊗ Y )X . Write Up ↔q ◦ (I ⊗ Y )X as
j (I
⊗ aj )bj . Then we have
Up ↔q ∨q+t ◦ I ⊗ (I ⊗ yi )Y X = (I ⊗ Up ↔q )(I ⊗ I ⊗ yi )(I ⊗ aj )bj j
I ⊗ Up ↔q ◦ (I ⊗ yi )aj bj . = j
We now write Up ↔q ◦ (I ⊗ yi )aj as
k (I
⊗ Ai,j,k )Bi,j,k . With this notation, we get
θ(s+s ,t+t ),(s ,t ) ◦ (θ(s,t),(s ,t ) ⊗ I )ζ = (I ⊗ I ⊗ X)(I ⊗ xi )Ai,j,k ⊗ (I ⊗ I ⊗ Bi,j,k )(I ⊗ bj )Y . i,j,k
Now let us operate first on ζ with the left-hand side of (16), repeating all the steps that we have made above: θ(s ,t ),(s ,t ) (X ⊗ Y ⊗ X ⊗ Y ) = θpE ,p ⊗ θqF ,q X ⊗ tp−1 ,q (Y ⊗ X ) ⊗ Y (I ⊗ X )aj ⊗ (I ⊗ bj )Y , = j
thus, θ(s,t),(s +s ,t +t ) ◦ (I ⊗ θ(s ,t ),(s ,t ) )ζ E F = θp,p ∨p +s ⊗ θq,q ∨q +t j
× X ⊗ Γp−1 . ∨p +s ,q Up ∨p +s ↔q ◦ I ⊗ (I ⊗ Y )X aj ⊗ (I ⊗ bj )Y
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As above, we factor Up ∨p +s ↔q as (Up ↔q ⊗ I )(I ⊗ Up ↔q ), to obtain Up ∨p +s ↔q ◦ I ⊗ (I ⊗ Y )X aj = (Up ↔q ⊗ I ) ◦ I ⊗ (I ⊗ xi )yi aj i
=
(I ⊗ I ⊗ xi )(Up ↔q ⊗ I ) ◦ (I ⊗ yi )aj
i
=
(I ⊗ I ⊗ xi )(I ⊗ Ai,j,k )Bi,j,k .
i,k
So we get θ(s,t),(s +s ,t +t ) ◦ (I ⊗ θ(s ,t ),(s ,t ) )ζ = (I ⊗ I ⊗ X)(I ⊗ xi )Ai,j,k ⊗ (I ⊗ I ⊗ Bi,j,k )(I ⊗ bj )Y , i,j,k
and this is exactly the same expression as we obtained for θ(s+s ,t+t ),(s ,t ) (θ(s,t),(s ,t ) ⊗ I )ζ .
2
Theorem 4.11. There exists a two-parameter product system of M -correspondences X, and a completely contractive, covariant representation T of X into B(H ), such that for all (s, t) ∈ R2+ and all a ∈ M, the following identity holds: ∗ T˜(s,t) (IX(s,t) ⊗ a)T˜(s,t) = P(s,t) (a).
(17)
Furthermore, if P is unital, then T is fully coisometric. Proof. As above, define X(s, t) := E(s) ⊗ F (t). By Proposition 4.10, X is a product system. For s, t 0, ξ ∈ E(s) and η ∈ F (t), we define a representation T of X by T(s,t) (ξ ⊗ η) := TsE (ξ )TtF (η). It is clear that for fixed s, t 0, T(s,t) , together with σ = idM , extends to a covariant representation of X(s, t) on H . In addition, (18) T˜(s,t) = T˜sE IE(s) ⊗ T˜tF , so T˜(s,t) 1. By [4, Lemma 3.5], T(s,t) is completely contractive. Also, if P is unital, so are R and S, thus T E and T F are fully coisometric, whence T is fully coisometric. We turn to show that for x1 ∈ X(s1 , t1 ), x2 ∈ X(s2 , t2 ), T(s1 +s2 ,t1 +t2 ) (x1 ⊗ x2 ) = T(s1 ,t1 ) (x1 )T(s2 ,t2 ) (x2 ). Let ξi ∈ E(si ), ηi ∈ F (ti ), i = 1, 2. Put Φ = IE(s1 ) ⊗ ϕs2 ,t1 ⊗ IF (t2 ) . Treating the maps θsE1 ,s2 , θtF1 ,t2 as identity maps, we have that Φ : X(s1 + s2 , t1 + t2 ) → X(s1 , t1 ) ⊗ X(s2 , t2 ). We need to show that T(s1 +s2 ,t1 +t2 ) Φ −1 (ξ1 ⊗ η1 ⊗ ξ2 ⊗ η2 ) = T(s1 ,t1 ) (ξ1 ⊗ η1 )T(s2 ,t2 ) (ξ2 ⊗ η2 ).
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But for this it suffices to show that −1 T(s,t) ϕs,t (η ⊗ ξ ) = T(0,t) (η)T(s,0) (ξ ),
ξ ∈ E(s), η ∈ F (t).
Let h ∈ H . Now, on the one hand, recalling (10), we have −1 T˜(s,0) (IE(s) ⊗ T˜(0,t) ) ϕs,t (η ⊗ ξ ) ⊗ h = T˜(0,t) (IF (t) ⊗ T˜(s,0) )(η ⊗ ξ ⊗ h) = T(0,t) (η)T(s,0) (ξ )h. On the other hand, writing
−1 ξi ⊗ ηi for ϕs,t (η ⊗ ξ ), we have
−1 T˜(s,0) (IE(s) ⊗ T˜(0,t) ) ϕs,t (η ⊗ ξ ) ⊗ h = T˜(s,0) ξi ⊗ T(0,t) (ηi )h = T(s,0) (ξi )T(0,t) (ηi )h = T(s,t) ξi ⊗ ηi h −1 = T(s,t) ϕs,t (η ⊗ ξ ) h −1 so we conclude that T(0,t) (ξ )T(s,0) (η) = T(s,t) (ϕs,t (η ⊗ ξ )), as required. Finally, using [5, Theorem 3.9], we easily compute for a ∈ M:
∗ ∗ ∗ T˜(s,0) = T˜(s,0) (IE(s) ⊗ T˜(0,t) )(IE(s) ⊗ IF (t) ⊗ a) IE(s) ⊗ T˜(0,t) T˜(s,t) (IX(s,t) ⊗ a)T˜(s,t) ∗ = T˜(s,0) IE(s) ⊗ St (a) T˜(s,0) = Rs St (a) = P(s,t) (a). This concludes the proof.
2
5. Isometric dilation of a fully coisometric product system representation In the previous section, given a von Neumann algebra M ⊆ B(H ) and two strongly commuting CP0 -semigroups on M, we constructed a product system X of M -correspondences over R2+ and a product system representation (σ, T ) of X on H such that for all (s, t) ∈ R2+ and all a ∈ M, ∗ = P(s,t) (a). T˜(s,t) (IX(s,t) ⊗ a)T˜(s,t)
In other words, we have completed the first step in our program for dilation. In this section we shall carry out the second step: we shall construct a fully coisometric, isometric dilation (ρ, V ) of (σ, T ) on some Hilbert space K ⊇ H . In the next section we will show that the family of maps given by ∗ α(s,t) (b) := V˜(s,t) (IX(s,t) ⊗ b)V˜(s,t)
for all b ∈ R := ρ(M ) is the E0 -dilation that we are looking for. In fact, we are going to prove a little more than we need: we shall prove that every fully coisometric representation of a product system over a (certain kind of) subsemigroup of Rk+ has an
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isometric dilation (see Section 5.2). This result will be proved by “representing the representation as a contractive semigroup on a Hilbert space” (see Section 5.1), a method that we introduced in [8]. Since in this paper we shall be ultimately interested in applying this result for the product system and representation given in Theorem 4.11, we will not make the construction or statement in the most general possible way, in hope of making the presentation as smooth as possible. For example, one does not have to assume that neither the product system nor the representation is unital, but we shall make these assumptions, as they hold for the output of Theorem 4.11. Also, the reader will note that our construction makes sense for more general semigroups than those we shall consider. 5.1. Representing product system representations as contractive semigroups on a Hilbert space Let S be a subsemigroup of Rk+ (k can be taken to be some infinite cardinal number, but we shall assume k ∈ N to keep things simple). Let A be a unital C ∗ -algebra, and let X be a discrete product system of unital C ∗ -correspondences over S (an A-correspondence is said to be unital if the left action of A is unital. Note that if A is unital, then the right action of A on every Acorrespondence is unital). Let (σ, T ) be a completely contractive covariant representation of X on the Hilbert space H , and assume that σ is unital. Our unital assumptions imply that A ⊗ H = X(0) ⊗ H ∼ = H via the identification a ⊗ h ↔ σ (a)h. This identification will be made repeatedly below. Define H0 to be the space of all finitely supported functions f on S such that for all s ∈ S, f (s) ∈ X(s) ⊗σ H . We equip H0 with the inner product δs · ξ, δt · η = δs,t ξ, η , for all s, t ∈ S, ξ ∈ X(s) ⊗ H, η ∈ X(t) ⊗ H (where the δ’s on the left-hand side are Dirac deltas, the δ on the right-hand side is Kronecker’s delta). Let H be the completion of H0 with respect to this inner product. Note that H∼ =
X(s) ⊗ H,
s∈S
but defining it as we did has a small notational advantage. We define a family Tˆ = {Tˆs }s∈S of operators on H0 as follows. First, we define Tˆ0 to be the identity. Now assume that s > 0. If t ∈ S and t s, then we define Tˆs (δt · ξ ) = 0 for all ξ ∈ X(t) ⊗σ H . And we define Tˆs δt · (xt−s ⊗ xs ⊗ h) = δt−s · xt−s ⊗ T˜s (xs ⊗ h)
(19)
if t s > 0. In [8] we showed that Tˆ = {Tˆs }s∈S extends to a well-defined semigroup of contractions on H. Note that the adjoint of Tˆ is given by Tˆs (δt · xt ⊗ h) = δt+s · xt ⊗ T˜s∗ h, thus, if T is a fully coisometric representation, then Tˆ is a semigroup of coisometries. We summarize the construction in the following proposition.
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Proposition 5.1. Let A, X, S and (σ, T ) be as above, and let H=
X(s) ⊗σ H.
s∈S
There exists a contractive semigroup Tˆ = {Tˆs }s∈S on H such that for all s ∈ S, x ∈ X(s) and h ∈ H, Tˆs (δs · x ⊗ h) = Ts (x)h. If T is a fully coisometric representation, then Tˆ is a semigroup of coisometries. 5.2. Isometric dilation of a fully coisometric representation For any r = (r1 , . . . , rk ) ∈ Rk , we denote r+ := (max{r1 , 0}, . . . , max{rk , 0}) and r− := r+ − r. Throughout this section, S will be a subsemigroup of Rk+ such that for all s ∈ S − S, both s+ and s− are in S. The semigroup that we are most interested in, namely Rk+ , satisfies this condition. In Section 7 we shall need the following theorem for S = N × R+ , and for possible future applications we may need the following theorem for S = Nk , which also satisfy this condition. Theorem 5.2. Let S be as above, let X = {X(s)}s∈S be a product system of unital Acorrespondences over S, and let (σ, T ) be a fully coisometric representation of X on H , with σ unital. Then there exist a Hilbert space K ⊇ H and a minimal, fully coisometric and isometric representation (ρ, V ) of X on K, with ρ unital, such that: (1) PH commutes with ρ(A), and ρ(a)PH = σ (a)PH , for all a ∈ A; (2) PH Vs (x)|H = Ts (x) for all s ∈ S, x ∈ X(s); (3) PH Vs (x)|KH = 0 for all s ∈ S, x ∈ X(s). If σ is nondegenerate and X is essential (that is, AX(s) is dense in X(s) for all s ∈ S) then ρ is also nondegenerate. If A is a W ∗ -algebra, X is a product system of W ∗ -correspondences and (σ, T ) is a representation of W ∗ -correspondences, then (ρ, V ) is also a representation of W ∗ -correspondences. Proof. The proof is very similar to the proof of Proposition 3.2 in [8], so we will not go into all the details whenever they were taken care of in that paper. However, we note that there are some essential differences between the situation at hand and the one treated in [8]. Let H = s∈S X(s) ⊗σ H , and let Tˆ be the semigroup of coisometries constructed in the discussion preceding Proposition 5.1. Since Tˆ is a semigroup of coisometries, there exists a minimal, regular unitary dilation W = {Ws }s∈S of the semigroup {Tˆs∗ }s∈S on a Hilbert space K ⊇ H (this should be well-known folklore, related to the theory of unitary dilations as described in [11]; see [9] for details). We denote Vˆs = Ws∗ . We have for all s ∈ S − S PH Vˆs+ Vˆs∗− PH = Tˆs+ Tˆs∗− .
(20)
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Since the semigroup Vˆ consists of commuting unitaries, and since commuting unitaries doubly commute, we also have PH Vˆs∗− Vˆs+ PH = Tˆs+ Tˆs∗− .
(21)
This triviality turns out to be crucial: it will allow us to compute the inner products in K. Introduce the Hilbert space K, K=
Vˆs δs · (x ⊗ h) : s ∈ S, x ∈ X(s), h ∈ H .
We consider H as embedded in K (or in H or in K) by the identification h ↔ δ0 · (1 ⊗ h). (This is where we use the fact that σ is unital.) We turn to the definition of the representation V of X on K. First, note that σ (a)h is identified with δ0 · 1 ⊗σ σ (a)h = δ0 · a ⊗σ h. Next, we define a left action of A on H by a · (δs · x ⊗ h) = δs · ax ⊗ h, for all a ∈ A, s ∈ S, x ∈ X(s) and h ∈ H . As we have explained in [8], this gives rise to a well defined a ∗-representation that commutes with Tˆ : a Tˆs (δt xt−s ⊗ xs ⊗ h) = δt−s axt−s ⊗ Ts (xs )h = Tˆs (δt axt−s ⊗ xs ⊗ h). Taking adjoints shows that this left action commutes Tˆs∗ (s ∈ S), as well. We shall now define a representation (ρ, V ) of X on K. First, we define ρ by the rule ρ(a)Vˆs (δs · xs ⊗ h) = Vˆs (δs · axs ⊗ h).
(22)
Using (21), one shows that ρ(a) extends to a bounded map on K. It then follows by direct computation that ρ is a ∗-representation. When (σ, T ) is a representation of W ∗ -correspondences, we also have to show that ρ is a normal representation. Let {aγ } ⊆ ball1 (A) be a net converging in the weak operator topology to a ∈ ball1 (A). It is known (for an outline of a proof, see [6]) that the mapping taking b ∈ A to b ⊗ IH ∈ B(X(s) ⊗σ H ) is continuous in the (σ -)weak topologies. Thus, for all s ∈ S, x ∈ X(s) and h ∈ H , aγ x ⊗ h −→ ax ⊗ h in the weak topology of X(s) ⊗σ H . It follows that δs · aγ x ⊗ h −→ δs · ax ⊗ h in the weak topology of K, so Vˆs δs · aγ x ⊗ h −→ Vˆs δs · ax ⊗ h
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weakly. This implies that ρ(aγ ) → ρ(a) in the weak operator topology of B(K), so ρ is normal. Note that H reduces ρ(A), and that ρ(a)|H = σ (a)|H (under the appropriate identifications). Indeed, putting t = 0 in Eq. (22) gives ρ(a)(δ0 · 1 ⊗ h) = δ0 · a ⊗ h = δ0 · 1 ⊗ σ (a)h. The assertions regarding the unitality and nondegeneracy of ρ are clear from the definitions. We have completed the construction of ρ, and we proceed to define the representation V of X on K. For s > 0, we define Vs by the rule Vs (xs )Vˆt (δt · xt ⊗ h) = Vˆs+t (δs+t · xs ⊗ xt ⊗ h).
(23)
One has to use (21) to show that Vs (xs ) can be extended to a well-defined operator on K, but once that is done, it is easy to see that for all s ∈ S, (ρ, Vs ) is a covariant representation of X(s) on K. We now show that it is isometric. This computation is included so the reader has an opportunity to appreciate the role played by Eq. (21). Let s, t, u ∈ S, x, y ∈ X(s), xt ∈ X(t), xu ∈ X(u) and h, g ∈ H . Then Vs (x)∗ Vs (y)Vˆt δt · xt ⊗ h, Vˆu δu · xu ⊗ g = Vˆt+s δt+s · y ⊗ xt ⊗ h, Vˆu+s δu+s · x ⊗ xu ⊗ g
∗ δ · y ⊗ xt ⊗ h, δu+s · x ⊗ xu ⊗ g = (∗) Vˆ = Vˆ(t−u) − (t−u)+ t+s ∗ δ · y ⊗ xt ⊗ h, δu+s · x ⊗ xu ⊗ g = Tˆ(t−u)+ Tˆ(t−u) − t+s ∗ . . . (xt ⊗ h), δu+s · x ⊗ xu ⊗ g = δu+s · y ⊗ (I ⊗ T˜(t−u)+ ) I ⊗ T˜(t−u) − ∗ . . . (xt ⊗ h), δu · y, x xu ⊗ g = δu · (I ⊗ T˜(t−u)+ ) I ⊗ T˜(t−u) − ∗ δ · xt ⊗ h, δu · y, x xu ⊗ g = Tˆ(t−u)+ Tˆ(t−u) − t ∗ δ · x, y xt ⊗ h, δu · y, x xu ⊗ g = (∗) = Tˆ(t−u)+ Tˆ(t−u) − t ∗ δ · x, y xt ⊗ h, δu · xu ⊗ g Vˆ = Vˆ(t−u) − (t−u)+ t = Vˆt δt · x, y xt ⊗ h, Vˆu δu · xu ⊗ g = ρ x, y Vˆt δt · xt ⊗ h, Vˆu δu · xu ⊗ g . (The equations marked by (∗) are where we use (21).) This shows that Vs (x)∗ Vs (y) = ρ( x, y ), so (ρ, V ) is indeed an isometric representation. To see that it is fully coisometric, is enough to show that for all s ∈ S, V˜s is onto. It is clear that Im(V˜s ) =
Vˆt+s (δt+s · xs ⊗ xt ⊗ h): t ∈ S, xs ∈ X(s), xt ∈ X(t), h ∈ H .
But if t ∈ S, xt ∈ X(t) and h ∈ H , then
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Vˆt (δt · xt ⊗ h) = Vˆt Vˆs Vˆs∗ (δt · xt ⊗ h) = (∗) = Vˆt Vˆs Tˆs∗ (δt · xt ⊗ h) = Vˆt+s δt+s · xt ⊗ T˜s∗ h ∈ Im(T˜s ), where (∗) is justified because Vˆs∗ is an extension of Tˆs∗ (as is any unitary dilation of an isometry). This shows that V˜s is onto, so it is a unitary, hence V is fully coisometric. We have yet to show that V is a representation of product systems (that is, that the semigroup property holds) and that it is in fact a dilation of T . Let h ∈ H , s, t, u ∈ S, and let xs , xt , xu be in X(s), X(t), X(u), respectively. Then Vs+t (xs ⊗ xt )Vˆu (δu · xu ⊗ h) = Vˆs+t+u (δs+t+u · xs ⊗ xt ⊗ xu ⊗ h) = Vs (xs )Vˆt+u (δt+u · xt ⊗ xu ⊗ h) = Vs (xs )Vt (xt )Vˆu (δu · xu ⊗ h), so the semigroup property holds. Finally, let s ∈ S, x ∈ X(s) and h = δ0 ·1⊗h ∈ H . We compute: PH Vs (x)|H h = PH Vs (x)δ0 · 1 ⊗ h = PH Vˆs (δs · x ⊗ h) = PH PH Vˆs |H (δs · x ⊗ h) = PH Tˆs (δs · x ⊗ h) = PH δ0 · 1 ⊗ Ts (x)h = Ts (x)h. We remark that V is already a minimal isometric dilation of T , because K= =
Vˆs δs · (x ⊗ h) : s ∈ S, x ∈ X(s), h ∈ H
Vs (x) δ0 · (1 ⊗ h) : s ∈ S, x ∈ X(s), h ∈ H .
Item (3) in the statement of the theorem follows as in [8, Proposition 3.2].
2
6. E0 -dilation of a strongly commuting pair of CP0 -maps In this section we prove the main result of this paper: every pair of strongly commuting CP0 semigroups has a minimal E0 -dilation. In the last two sections we worked out the two main steps in the Muhly–Solel approach to dilation. In this section we will put together these two steps and take care of the remaining technicalities. It is convenient to begin by proving a few technical lemmas. We then turn to prove the existence of the dilation, and we close this section with a discussion of minimality issues.
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6.1. CP-semigroups and some of their continuity properties Lemma 6.1. Let N be a von Neumann algebra, let S be an abelian, cancellative semigroup with unit 0, and let X be a product system of N -correspondences over S. Let W be completely contractive covariant representation of X on a Hilbert space G, such that W0 is unital. Then the family of maps Θs : a → W˜ s (IX(s) ⊗ a)W˜ s∗ ,
a ∈ W0 (N ) ,
is a semigroup of CP maps (indexed by S) on W0 (N ) . Moreover, if W is an isometric (a fully coisometric) representation, then Θs is a ∗-endomorphism (a unital map) for all s ∈ S. Proof. By Proposition 2.21 in [5], {Θs }s∈S is a family of contractive, normal, completely positive maps on W0 (N) . Moreover, these maps are unital if W is a fully coisometric representation, and they are ∗-endomorphisms if W is an isometric representation. All that remains is to check that Θ = {Θs }s∈S satisfies the semigroup condition Θs ◦ Θt = Θs+t . Fix a ∈ W0 (N ) . For all s, t ∈ S, Θs Θt (a) = W˜ s IX(s) ⊗ W˜ t (IX(t) ⊗ a)W˜ t∗ W˜ s∗ = W˜ s (IX(s) ⊗ W˜ t )(IX(s) ⊗ IX(t) ⊗ a) IX(s) ⊗ W˜ t∗ W˜ s∗ ∗ −1 = W˜ s+t (Us,t ⊗ IG )(IX(s) ⊗ IX(t) ⊗ a) Us,t ⊗ IG W˜ s+t ∗ = W˜ s+t (IX(s·t) ⊗ a)W˜ s+t
= Θs+t (a). Using the fact that W0 is unital, we have Θ0 (a)h = W˜ 0 (IN ⊗ a)W˜ 0∗ h = W˜ 0 (IN ⊗ a)(I ⊗ h) = W0 (IN )ah = ah, thus Θ0 (a) = a for all a ∈ N .
2
Lemma 6.2. Let {Rt }t0 and {St }t0 be two CP-semigroups on M ⊆ B(H ), where H is a separable Hilbert space. Then the two-parameter CP-semigroup P defined by P(s,t) := Rs St is a CP-semigroup, that is, for all a ∈ M, the map R2+ (s, t) → P(s,t) (a) is weakly continuous. Moreover, P is jointly continuous on R2+ × M, endowed with the standard × weak-operator topology.
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Proof. Let (sn , tn ) → (s, t) ∈ R2+ , and let an → a ∈ M. By [5, Proposition 4.1], CP-semigroups are jointly continuous in the standard × weak-operator topology, so Stn (an ) → St (a) in the weak operator topology. By the same proposition used once more, P(sn ,tn ) (an ) = Rsn Stn (an ) → Rs St (a) = P(s,t) (a) where convergence is in the weak operator topology.
2
The above lemma show that, given two CP0 -semigroups {Rt }t0 and {St }t0 , we can form a two-parameter CP0 -semigroup {P(s,t) } = {Rs St }s,t0 which satisfies the natural continuity conditions. For the theorem below, we will need P to satisfy a stronger type of continuity. This is the subject of the next two lemmas. Lemma 6.3. Let S be a topological semigroup with unit 0, and let {Ws }s∈S be a semigroup over S of CP maps on a von Neumann algebra R ⊆ B(H ). Let A ⊆ R be a sub-C ∗ -algebra of R such that for all a ∈ A, WOT −−→ a Ws (a) −
as s → 0. Then for all a ∈ A, SOT −→ Wt (a) Wt+s (a) −
as s → 0. Proof. One can repeat, almost word for word, the proof of the first half of Proposition 4.1 in [5], which addresses the case S = R+ . 2 Lemma 6.4. Let Θ = {Θt }t0 be a CP-semigroup on M ⊆ B(H ), where H is a separable Hilbert space. Then Θ is jointly strongly continuous, that is, for all h ∈ H , the map (t, a) → Θt (a)h is continuous in the standard × strong-operator topology. Proof. First, assume that Θ is an E-semigroup. Let (tn , an ) → (t, a) in the standard × strongoperator topology in R+ × M, and h ∈ H .
Θt (an )h − Θt (a)h 2 = Θt (an )h 2 − 2 Re Θt (an )h, Θt (a)h + Θt (a)h 2 , n n n since Θ is continuous in the standard × weak-operator topology, it is enough to show that Θtn (an )h2 → Θt (a)h2 . But
Θt (an )h 2 = Θt a ∗ an h, h → Θt (a ∗ a)h, h = Θt (a)h 2 , n n n because an∗ an → a ∗ a in the weak-operator topology, and Θ is jointly continuous with respect to this topology. Thus Θ is also jointly continuous with respect to the strong-operator topology.
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Now let Θ be an arbitrary CP-semigroup, and let (K, u, R, α) be an E-dilation of Θ. Then for all a ∈ M, t ∈ R+ , Θt (a) = u∗ αt (uau∗ )u, whence Θ inherits the required type of joint continuity from α.
2
From the above lemma one immediately obtains: Proposition 6.5. Let {Rt }t0 and {St }t0 be two CP-semigroups on M ⊆ B(H ), where H is a separable Hilbert space. Then the two-parameter CP-semigroup P defined by P(s,t) := Rs St is strongly continuous, that is, for all a ∈ M, the map R2+ (s, t) → P(s,t) (a) is strongly continuous. Moreover, P is jointly continuous on R2+ × M, endowed with the standard × strongoperator topology. 6.2. The existence of an E0 -dilation We have now gathered enough tools to prove our main result. Theorem 6.6. Let {Rt }t0 and {St }t0 be two strongly commuting CP0 -semigroups on a von Neumann algebra M ⊆ B(H ), where H is a separable Hilbert space. Then the two parameter CP0 -semigroup P defined by P(s,t) := Rs St has a minimal E0 -dilation (K, u, R, α). Moreover, K is separable. Proof. We split the proof into the following steps: (1) (2) (3) (4) (5)
Existence of a ∗-endomorphic dilation (K, u, R, α) for (M, P ). Minimality of the dilation. Continuity of α on M. Separability of K. Continuity of α.
Step 1: Existence of a ∗-endomorphic dilation. Let X and T be the product system (of M -correspondences) and the fully coisometric product system representation given by Theorem 4.11. By Theorem 5.2, there is a covariant isometric and fully coisometric representation ˜ = ρ(M ) , and let u be the iso(ρ, V ) of X on some Hilbert space K ⊇ H , with ρ unital. Put R ˜ We define a semigroup metric inclusion H → K. Note that, since uH reduces ρ, p := uu∗ ∈ R. α˜ = {α˜ s }s∈R2 by +
α˜ s (b) = V˜s (I ⊗ b)V˜s∗ ,
˜ s ∈ R2+ , b ∈ R.
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˜ The (first part By Lemma 6.1 above, α˜ is a semigroup of unital, normal ∗-endomorphisms of R. of the) proof of Theorem 2.24 in [5] works in this situation as well, and shows that ˜ M = u∗ Ru
(24)
Ps (u∗ bu) = u∗ α˜ s (b)u
(25)
and that
˜ s ∈ R2+ . Note that we cannot use that theorem directly, because for fixed s ∈ S, for all b ∈ R, X(s) is not necessarily the identity representation of Ps . For the sake of completeness, we repeat the argument (with some changes). By Theorem 5.2, for all a ∈ M , u∗ ρ(a)u = σ (a), and by definition, σ (a) = a, thus ˜ = u∗ ρ(M ) u = u∗ ρ(M )u = (M ) = M, u∗ Ru where the ⊆ part of the second equality follows from the fact that uH reduces ρ(M ). This ˜ ⊆ R. ˜ To obtain (25), we establishes (24), which allows us to make the identification M = p Rp 2 ˜ fix s ∈ R+ and b ∈ R, and we compute Ps (u∗ bu) = T˜s (I ⊗ u∗ bu)T˜s∗ = (∗) = u∗ V˜s (I ⊗ u)(I ⊗ u∗ bu)(I ⊗ u∗ )V˜s∗ u = (∗∗) = u∗ V˜s (I ⊗ b)V˜s∗ u = u∗ α˜ s (b)u. The equalities marked by (∗) and (∗∗) are justified by items (2) and (3) of Theorem 5.2, respectively. Eq. (25) implies that p is a coinvariant projection. Since α˜ is unital, we have α˜ t (p) p for all t ∈ R2+ , that is, p is an increasing projection. Even though we started out with a minimal isometric representation V of T , we cannot show that α˜ is a minimal dilation of P . We define ∗ R=W α˜ t (M) . (26) t∈R2+
This von Neumann algebra is invariant under α, ˜ and we denote α = α| ˜ R . Now it is immediate that (p, R, α) is a ∗-endomorphic dilation of (M, P ). Indeed, for all b ∈ R and all t ∈ R2+ , pαt (b)p = p α˜ t (b)p = Pt (pbp), ˜ α) because (p, R, ˜ is a dilation of (M, P ). It is also clear that M = pRp. The only issue left to handle is the continuity of α. We now define two one-parameter semigroups on R: β = {βt }t0 and γ = {γt }t0 by βt = α(t,0) and γt = α(0,t) . Clearly, β and γ are semigroups of normal, unital ∗-endomorphisms of R. If we show that K is separable, then by Lemma 6.2, once we show that β and γ are E0 -semigroups—that is, possess the required weak continuity—then we have shown that α is an E0 -semigroup. The rest of the proof is dedicated
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to showing that β and γ are E0 -semigroups and that K is separable. But before we do that, we must show that the dilation is minimal, and, in fact, a bit more. Step 2: Minimality of the dilation. What we really need to prove is that K=
α(sm ,tn ) (M)α(sm ,tn−1 ) (M) · · · α(sm ,t1 ) (M)α(sm ,0) (M)α(sm−1 ,0) (M) · · · α(s1 ,0) (M)H (27)
where in the right-hand side of the above expression we run over all strictly positive pairs (s, t) ∈ R2+ and all partitions {0 = s0 < · · · < sm = s} and {0 = t0 < · · · < tn = t} of [0, s] and [0, t]. We shall also need an analog of (27) with the roles of the first and second “time variables” of α replaced, but since the proof is very similar we shall not prove it. Recall that K=
V(s,t) X(s, t) H : (s, t) ∈ R2+ .
Thus, it suffices to show that for a fixed (s, t) ∈ R2+ , α(sm ,tn ) (M) · · · α(sm ,t1 ) (M)α(sm ,0) (M)α(sm−1 ,0) (M) · · · α(s1 ,0) (M)H V(s,t) X(s, t) H = (28) where in the right-hand side of the above expression we run over all partitions {0 = s0 < · · · < sm = s} and {0 = t0 < · · · < tn = t} of [0, s] and [0, t]. To show that we can consider only s and t strictly positive, we note that if u, v ∈ R2+ , then Vu X(u) H = V˜u (IX(u) ⊗ V˜v ) IX(u) ⊗ V˜v∗ X(u) ⊗ H = V˜u (IX(u) ⊗ V˜v ) IX(u) ⊗ T˜v∗ X(u) ⊗ H = V˜u+v X(u) ⊗ T˜v∗ H ⊆ Vu+v X(u + v) H. We now turn to establish (28). Recall the notation and constructions of Sections 3.2 and 4.3: X(s, t) := E(s) ⊗ F (t), and T(s,t) (ξ ⊗ η) := TsE (ξ )TtF (η), where (E, T E ) and (F, T F ) are the product systems and representations representing R and S via Muhly and Solel’s construction as described in 3.2. By [5, Lemma 4.3(2)], for all r > 0,
∗ (IE(r) ⊗ a) T˜rE h: a ∈ M, h ∈ H = Er ⊗M H,
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where Er = LM (H, HpR ) with the partition p = {0 = r0 < r1 = r}. Similarly,
∗ (IF (r) ⊗ a) T˜rF h: a ∈ M, h ∈ H = Fr ⊗M H. Fix s, t > 0. Under the obvious identifications, if we go over all the partitions {0 = s0 < · · · < sm = s} and {0 = t0 < · · · < tn = t} of [0, s] and [0, t], the collection of correspondences Es1 ⊗ Es2 −s1 ⊗ · · · ⊗ Esm −sm−1 ⊗ Ft1 ⊗ · · · ⊗ Ftn −tn−1 is dense in X(s, t). Using [5, Lemma 4.3(2)] repeatedly, we obtain α(sm ,tn ) (M) · · · α(sm ,t1 ) (M)α(sm ,0) (M)α(sm−1 ,0) (M) · · · α(s1 ,0) (M)H = α(sm ,tn ) (M) · · · V˜(s1 ,0) (I(s1 ,0) ⊗ M)V˜(s∗1 ,0) H ∗ = α(sm ,tn ) (M) · · · V˜(s1 ,0) (I(s1 ,0) ⊗ M) T˜sE1 H = α(sm ,tn ) (M) · · · V˜(s2 ,0) (I(s2 ,0) ⊗ M)V˜(s∗2 ,0) V˜(s1 ,0) (Es1 ⊗ H ). But V˜(s∗2 ,0) V˜(s1 ,0) = I(s1 ,0) ⊗ V˜(s∗2 −s1 ,0) V˜(s∗1 ,0) V˜(s1 ,0) = I(s1 ,0) ⊗ V˜(s∗2 −s1 ,0) , so we get α(sm ,tn ) (M) · · · α(sm ,t1 ) (M)α(sm ,0) (M)α(sm−1 ,0) (M) · · · α(s1 ,0) (M)H = α(sm ,tn ) (M) · · · V˜(s2 ,0) (I(s2 ,0) ⊗ M) I(s1 ,0) ⊗ V˜(s∗2 −s1 ,0) (Es1 ⊗ H ) = α(sm ,tn ) (M) · · · V˜(s2 ,0) (Es1 ⊗ Es2 −s1 ⊗ H ). Continuing this way, we see that α(sm ,tn ) (M) · · · α(sm ,t1 ) (M)α(sm ,0) (M)α(sm−1 ,0) (M) · · · α(s1 ,0) (M)H = V(s,t) (Es1 ⊗ Es2 −s1 ⊗ · · · ⊗ Esm −sm−1 ⊗ Ft1 ⊗ · · · ⊗ Ftn −tn−1 )H. Since this computation works for any partition of [0, s] and [0, t], we have (28). This, in turn, implies (27), which is what we have been after. Now it is a simple matter to show that (p, R, α) is a minimal dilation of (M, P ). First, note that by (27) K = [RpK]. In light of (26), Definitions 2.7 and 2.8 and Proposition 2.9, we have to show that the central support of p in R is IK . But this follows by a standard (and short) argument, which we omit.
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Step 3: Continuity of β and γ on M. We shall now show that function R+ t → βt (a) is strongly continuous from the right for each a ∈ A := C ∗ ( t∈R2 αt (M)). Of course, the same is + true for γ as well. Let k1 = i αsi (mi )hi and k2 = j αtj (nj )gj be in K, where si = (si1 , si2 ), tj = (tj1 , tj2 ) ∈ R2+ , mi , nj ∈ R and hi , gj ∈ H . By (27), we may consider only si1 , tj1 > 0. Take a ∈ M and t > 0. For the following computations, we may assume that k1 and k2 are given by finite sums, and we take t < min{tj1 , si1 }i,j . We will abuse notation a bit by denoting (t, 0) by t. Now compute:
βt (a)k1 , k2
=
αt (a)αsi (mi )hi , αtj (nj )gj
i,j
=
αtj n∗j αt (a)αsi (mi )hi , gj
i,j
αt αtj −t n∗j aαsi −t (mi ) hi , gj = i,j
Pt pαtj −t n∗j papαsi −t (mi )p hi , gj = i,j
Pt Ptj −t pn∗j p aPsi −t (pmi p) hi , gj = i,j
t→0
−−−→
Ptj pn∗j p aPsi (pmi p)hi , gj
i,j
=
aαsi (mi )hi , αtj (nj )gj
i,j
= ak1 , k2 , where we have made use of the joint strong continuity of P (Proposition 6.5). This implies that for all a ∈ M, αt (a) → a weakly as t → 0. It follows from Lemma 6.3 that β is strongly right continuous on t∈R2 αt (M), whence it is also strongly right continuous on + A := C ∗ t∈R2 αt (M) . +
Step 4: Separability of K. As we have already noted in Step 2, from (27) it follows that K=
αu1 (a1 ) · · · αuk (ak )h: ui ∈ R2+ , ai ∈ M, h ∈ H .
We define K0 =
γt1 βs1 (a1 ) · · · γtk βsk (ak ) h: si , ti ∈ Q+ , ai ∈ M, h ∈ H ,
and K1 =
γt1 βs1 (a1 ) · · · γtk βsk (ak ) h: si ∈ R+ , ti ∈ Q+ , ai ∈ M, h ∈ H .
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K0 is clearly separable. Because of the normality of γ , the strong right continuity of β on M and the fact that multiplication is strongly continuous on bounded subsets of R, we can assert that K0 = K1 , thus K1 is separable. Now from the strong right continuity of γ on A and the continuity of multiplication, we see that K = K1 , whence it is separable. Step 5: Continuity of α. Recall that all that we have left to show is that β and γ possess the desired weak continuity. We shall concentrate on β. A short summary of the situation: we have a semigroup β of normal, unital ∗-endomorphisms defined on a von Neumann algebra R (which acts on a separable Hilbert space K), and there is a weakly dense C ∗ -algebra A ⊆ R such that for all a ∈ A, k ∈ K, the function R+ τ → βτ (a)k ∈ K is right continuous. From this, we want to conclude that for all b ∈ R, and all k1 , k2 ∈ K, the map τ → βτ (b)k1 , k2 is continuous. This problem was already handled by Arveson in [1] and by Muhly and Solel in [5]. For completeness, we give some shortened variant of their arguments. For every b ∈ R, there is a sequence {an } in A weakly converging to b. Thus, for every b ∈ R and every k1 , k2 , ∈ K, the function τ → βτ (b)k1 , k2 is the pointwise limit of the sequence of right continuous functions τ → βτ (an )k1 , k2 , so it is measurable. It now follows from [1, Proposition 2.3.1] (which, in turn, follows from well-known results in the theory of operator semigroups) that β is an E0 -semigroup. 2 By Proposition A.1, if H is a finite-dimensional Hilbert space, then every pair of commuting CP-semigroups on B(H ) commutes strongly. Denote by Mn (C) the algebra of n × n complex matrices. We have the following corollary. Corollary 6.7. Every two parameter CP0 -semigroup on Mn (C) has an E0 -dilation. Loosely speaking, the whole point of dilation theory is to present a certain object as part of a simpler, better understood object. Theorem 6.6 tells us that under the strong-commutativity assumption, a two-parameter CP0 -semigroup can be dilated to a two parameter E0 -semigroup. Certainly, E0 -semigroups are a very special case of CP0 -semigroups, so we have indeed made the situation simpler. But did we really? Perhaps P (the CP0 -semigroup) was acting on a very simple kind of von Neumann algebra, but now α (the dilation) is acting on a very complicated one? Actually, we did not say much about the structure of R (the dilating algebra). In this context, we have the following partial, but quite satisfying, result. Proposition 6.8. If M = B(H ), then R = B(K). Proof. Let q ∈ B(K) be a projection in R . In particular, pq = qp = pqp, so qp is a projection B(H ) which commutes with B(H ), thus qp is either 0 or IH . If it is 0 then for all ti ∈ R2+ , mi ∈ M, h ∈ H , qαt1 (m1 ) · · · αtk (mk )h = αt1 (m1 ) · · · αtk (mk )qph = 0, so qK = 0 and q = 0.
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If qp = IH then for all 0 < ti ∈ R2+ , mi ∈ M, h ∈ H , qαt1 (m1 ) · · · αtk (mk )h = αt1 (m1 ) · · · αtk (mk )qph = αt1 (m1 ) · · · αtk (mk )h, so qK = K and q = IK . We see that the only projections in R are 0 and IK , so R = C · IK , hence R = R = B(K). 2 7. Example: E0 -dilation of a CP0 -semigroup over N × R 2+ —without strong commutation In the previous section we proved the main result of this paper, Theorem 6.6, which says that every pair of strongly commuting CP0 -semigroups has an E0 -dilation. In fact, the only place where strong commutativity was used was in showing that the CP0 -semigroup at hand could be represented by a product system representation as in the following equation Θs (a) = T˜s (IX(s) ⊗ a)T˜s∗ .
(29)
Furthermore, in light of our dilation result from Section 5.2, Theorem 5.2, we see that given a subsemigroup S ⊆ Rk such that for all s ∈ S, s− , s+ ∈ S, and a CP0 -semigroup Θ = {Θs }s∈S acting on a von Neumann algebra M ⊆ B(H ), (H separable), an E0 -dilation of Θ can be constructed if we are able to find a product system of M -correspondences X over S and a fully coisometric product system representation T of X on H fulfilling (29). In this section we use this observation to dilate a CP0 -semigroup over N × R+ which does not satisfy strong commutation. Let H = C ⊕ L2 (0, ∞). Denote by U the left-shift semigroup on L2 (0, ∞) given by (Ut f )(s) = f (t + s). Let St = 1 ⊕ Ut , and define a CP0 -semigroup Φ on B(H ) by Φt (a) = St aSt∗ . Next, define k = 1 ⊕ 0 ∈ H , and define the CP map Θ by Θ(a) = ak, k IH ,
a ∈ B(H ).
Peeking into [10, Example 5.5] one sees that for all t ∈ R+ , Θ and Φt commute but not strongly. However, we shall show that the CP0 -semigroup Ψ = {Ψn,t }(n,t)∈N×R+ defined by Ψn,t = Θ n ◦ Φt has an E0 -dilation. In light of the opening remarks of this section, all we have to do is construct an appropriate product system representation. 2 Let {ei }∞ i=1 be an orthonormal basis for L (0, ∞), and set e0 = k. Define Ei,0 to be the infinite square matrix indexed by I = {0, 1, 2, . . .} having 1 in the ith row 0th column, and zeros elsewhere. Abusing notation slightly, we let Ei,0 denote also the operator that this matrix represents ∗ with respect to the basis E = {e0 }∞ i=0 , namely, the rank one operator ei ⊗ e0 . We note that Θ(a) =
i∈I
∗ Ei,0 aEi,0 .
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If (ai,j (t))i,j ∈I is the matrix representing St with respect to E , then we have S t ej =
ai,j (t)ei ,
i∈I
thus St Ej,0 =
ai,j (t)Ei,0 =
i∈I
ai,j (t)Ei,0 St .
i∈I
The matrix function a(t) is a semigroup of coisometric matrices, so there is a semigroup of unitary matrices {u(t)}t0 indexed by I ∪ I , where I is another copy of I , such that the I –I block in u(t) is equal to a(t), and the I –I block in u(t) is 0 (u is simply the matrix representation of the minimal isometric dilation of the semigroup S, which is unitary, because a(t) is coisometric). We now define a family {Ti }i∈I ∪I of operators on H by Ti = Ei,0 when i ∈ I and Ti = 0 when i ∈ I . Because of the block structure that u(t) possesses, we have for all t 0 S t Tj = ui,j (t)Ti St . (30) i∈I ∪I
We shall now construct a product system of Hilbert spaces over N × R+ . Let E = 2 (I ∪ I ), and put E(n) = E ⊗n . We fix an orthonormal basis F = {fi }i∈I ∪I in E. Also, let F be the trivial product system, that is, the product system with F (t) = C for all t ∈ R+ and the obvious multiplication. For all n ∈ N and all t ∈ R+ , we define X(n, t) = E(n) ⊗ F (t). To make X = {X(n, t)}(n,t)∈N×R+ into a product system, we must define unitaries U(m,s)(n,t) : X(m, s) ⊗ X(n, t) → X(m + n, s + t) that are associative in the sense of Eq. (1). This is where u comes in. If λ ∈ F (s), μ ∈ F (t), we define uk,j (t)fi ⊗ fk ⊗ λμ, U(1,s)(1,t) (fi ⊗ λ) ⊗ (fj ⊗ μ) = k∈I ∪I
and we continue this map to all of X. Let k, m, n ∈ N, and s, t, u ∈ R+ . We have to show that U(k,s)(m+n,t+u) (I ⊗ U(m,t)(n,u) ) = U(k+m,s+t)(n,u) (U(k,s)(m,t) ⊗ I ). We shall operate with both sides on a typical element of the form fi1 ⊗ · · · ⊗ fik ⊗ λ ⊗ fj1 ⊗ · · · ⊗ fjm ⊗ μ ⊗ fl1 ⊗ · · · ⊗ fln ⊗ ν, where λ ∈ F (s), μ ∈ F (t) and ν ∈ F (u). Operating first with (I ⊗ U(m,t)(n,u) ) we get l1 ,...,ln
ul1 ,l1 (t) · · · uln ,ln (t)fi1 ⊗ · · · ⊗ fik ⊗ λ ⊗ fj1 ⊗ · · · ⊗ fjm ⊗ fl1 ⊗ · · · ⊗ fln ⊗ μν,
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and following with an application of U(k,s)(m+n,t+u) we get l1 ,...,ln
ul1 ,l1 (t) · · · uln ,ln (t)
j1 ,...,jm
uj1 ,j1 (s) · · · ujm ,jm (s)
l1 ,...,ln
ul1 ,l1 (s) · · · uln ,ln (s)
fi1 ⊗ · · · ⊗ fik ⊗ fj1 ⊗ · · · ⊗ fjm ⊗ fl1 ⊗ · · · ⊗ fln ⊗ λμν which is j1 ,...,jm
uj1 ,j1 (s) · · · ujm ,jm (s)
l1 ,...,ln
ul1 ,l1 (s + t) · · · uln ,ln (s + t)
fi1 ⊗ · · · ⊗ fik ⊗ fj ⊗ · · · ⊗ fjm ⊗ fl ⊗ · · · ⊗ fln ⊗ λμν 1
1
because u is a semigroup. On the other hand, applying first (U(k,s)(m,t) ⊗ I ) we get j1 ,...,jm
uj ,j1 (s) · · · ujm ,jm (s)fi1 ⊗ · · · ⊗ fik ⊗ fj ⊗ · · · ⊗ fjm ⊗ λμ ⊗ fl1 ⊗ · · · ⊗ fln ⊗ ν, 1
1
which becomes, after operating with U(k+m,s+t)(n,u) , j1 ,...,jm
uj ,j1 (s) · · · ujm ,jm (s) 1
l1 ,...ln
ul ,l1 (s + t) · · · uln ,ln (s + t) 1
fi1 ⊗ · · · ⊗ fik ⊗ fj1 ⊗ · · · ⊗ fjm ⊗ fl1 ⊗ · · · ⊗ fln ⊗ λμν which is the same as above. We now proceed to construct a product system representation that will give rise to Ψ . We define T(n,t) (ei1 ⊗ · · · ⊗ ein ⊗ 1) = Ti1 · · · Tin St . The relation (30) is precisely what makes T into a product system representation (it is completely contractive because (Ti )i∈I ∪I is a row contraction). The last thing to check is that for all a ∈ B(H ), ∗ = Ψ(n,t) (a). T˜(n,t) (IX(n,t) ⊗ a)T˜(n,t)
But, after some identifications, S˜t = St , and T˜ is just the row contraction (Ti )i∈I ∪I , so we are done. We note that in this example too many “miracles” have happened, and we do not yet understand how what we have done here can be generalized to other CP0 -semigroups over N × R+ . Appendix A. Examples of strongly commuting semigroups In this appendix we give some examples of strongly commuting CP-semigroups. In special cases we are able to state a necessary and sufficient condition for strong commutativity.
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A.1. Endomorphisms, automorphisms, and composition with automorphisms By [10, Lemma 5.4], there are plenty of examples of CP maps Θ, Φ that commute strongly: (1) If Θ and Φ are endomorphisms that commute then they commute strongly. (2) If Θ and Φ commute and either one of them is an automorphism then they commute strongly. (3) If α is a normal automorphism that commutes with Θ, and Φ = Θ ◦ α, then Θ and Φ commute strongly. We note that item (2) does not remain true if automorphism is replaced by endomorphism. Here is an example: take M = B(2 (N)), and identify every operator with its matrix representation with respect to the standard basis. Let Θ be the map that takes a matrix to its diagonal, and let Φ be given by conjugation with the right shift. Θ is a (unital) CP map, Φ is a (non-unital) ∗-endomorphism, these two maps commute, but not strongly. Because two CP-semigroups Θ and Φ commute strongly if and only if for all s, t ∈ R+ , Θs and Φt commute strongly, it is immediate that: (1) If Θ and Φ are commuting E-semigroups then they commute strongly. (2) If Θ and Φ commute and either one of them is an automorphism semigroup then they commute strongly. (3) If α is a normal automorphism semigroup that commutes with Θ, and Φt = Θt ◦ αt , then Θ and Φ commute strongly. At a first glance, item (1) might not seem very interesting in the context of dilating CPsemigroups to endomorphism semigroups. However, we find this item very interesting, because one expects a good dilation theorem not to complicate the situation in any sense. For example, in Theorem 6.6, in order to prove the existence of an E-dilation we have to assume that the CPsemigroups {Rt }t0 and {St }t0 are unital, but the E-dilation that we construct is also unital. Another example, again from Theorem 6.6: if the CP-semigroups act on a type I factor, then so does the minimal E0 -dilation that we construct. The importance of item (1) is that it ensures that if {αt }t0 and {βt }t0 are an E-dilation of {Rt }t0 and {St }t0 , then α and β commute strongly. A.2. Semigroups on B(H ) It is a well-known fact that if Θ and Φ are CP-semigroups, then for each t there are two m(t) n(t) (2 -independent) row contractions {Ti,t }i=1 and {St,j }j =1 (m(t), n(t) may be equal to ∞) such that for all a ∈ B(H ) Θt (a) =
∗ Tt,i aTt,i ,
(A.1)
∗ St,j aSt,j .
(A.2)
i
and Φt (a) =
j
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We shall call such semigroups conjugation semigroups, as they are given by conjugating an element with a row contraction. It now follows from [10, Proposition 5.8], that Θ and Φ commute strongly if and only if for all (s, t) ∈ R2+ there is an m(t)n(t) × m(t)n(t) unitary matrix u(s, t) = u(s, t)(k,l) (i,j ) (i,j ),(k,l) such that for all i, j , Tt,i Ss,j =
(k,l)
u(s, t)(i,j ) Ss,l Tt,k .
(A.3)
(k,l)
As a simple example, if Φ and Ψ are given by (A.1) and (A.2), and St,j commutes with Ts,i for all s, t, i, j , then Φ and Ψ strongly commute. A.3. Semigroups on B(H ), H —finite-dimensional If H is a finite-dimensional then any two commuting CP-semigroups on B(H ) commute strongly. This follows immediately from the following proposition. Proposition A.1. Let Φ and Ψ be two commuting CP maps on B(H ), with H a finitedimensional Hilbert space. Then Φ and Ψ strongly commute. Proof. Assume that Φ is given by Φ(a) =
m
Si aSi∗
i=1
and that Ψ is given by Ψ (a) =
n
Tj aTj∗ ,
j =1
where {S1 , . . . , Sm } and {T1 , . . . , Tn } are row contractions and m, n ∈ N. Because Φ and Ψ commute, we have that mn
Si Tj aTj∗ Si∗
=
i,j =1
mn
Tj Si aSi∗ Tj∗
i,j =1
for all a ∈ B(H ). By the lemma [3, p. 153] this implies that there exists an mn × mn unitary matrix u such that (k,l) u(i,j ) Tl Sk , S i Tj = (k,l)
and this means precisely that Φ and Ψ strongly commute.
2
O.M. Shalit / Journal of Functional Analysis 255 (2008) 46–89
87
We note here that the lemma cited above is stated in [3] for unital CP maps, but the proof works for the non-unital case as well. The reason that the assertion of the proposition fails for B(H ) with H infinite-dimensional is that in that case we may have mn = ∞, and the lemma is only true for a CP maps given by finite sums. A.4. Conjugation semigroups on general von Neumann algebras Let M be a von Neumann algebra acting on a Hilbert space H . We now show that if Θ and Φ are CP-semigroups on a von Neumann algebra M given as in (A.1) and (A.2), where Tt,i , St,j are all in M, then a sufficient condition for them to commute strongly with each other is that there exists a unitary as in (A.3). To this end, it is enough to show that if Θ and Φ are CP maps given by Θ(a) =
m
Ti aTi∗ ,
i=1
and Φ(a) =
n
Sj aSj∗ ,
j =1
where Ti , Sj are all in M, then a sufficient condition for strong commutation is the existence of a unitary matrix (k,l) u = u(i,j ) (i,j ),(k,l) such that for all i, j , Ti S j =
u(k,l) (i,j ) Sl Tk .
(k,l)
Indeed, by [10, Proposition 5.6], it is enough to show that there are two M correspondences E and F , together with an M -correspondence isomorphism t : E ⊗M F → F ⊗M E and two c.c. representations (σ, T ) and (σ, S) of E and F , respectively, on H , such that: (1) (2) (3)
for all a ∈ M, T˜ (IE ⊗ a)T˜ ∗ = Θ(a), ˜ F ⊗ a)S˜ ∗ = Φ(a), for all a ∈ M, S(I ˜ ˜ ˜ T (IE ⊗ S) = S(IF ⊗ T˜ ) ◦ (t ⊗ IH ).
We construct these correspondences as follows. Let E=
m i=1
M
and F =
n j =1
M ,
88
O.M. Shalit / Journal of Functional Analysis 255 (2008) 46–89
n with the natural inner product and the natural actions of M . If we denote by {ei }m i=1 and {fj }j =1 the natural “bases” of these spaces, then we can define
t (ei ⊗ fj ) =
u(k,l) (i,j ) fl ⊗ ek .
(k,l)
m n We define σ to be the identity representation. Now E ⊗σ H ∼ = i=1 H , and F ⊗σ H ∼ = j =1 H , and on these spaces we define T˜ and S˜ to be the row contractions given by (T1 , . . . , Tm ) and (S1 , . . . , Sn ). Some straightforward calculations shows that items (1)–(3) are fulfilled. A.5. Semigroups on Cn or ∞ We close this paper with a more down-to-earth example of a strongly commuting pair of CP0 semigroups. Let M = Cn or ∞ (N), considered as the algebra of diagonal matrices acting on the Hilbert space H = Cn or 2 (N). In this context, a unital CP map is just a stochastic matrix, that is, a matrix P such that pij 0 for all i, j and such that for all i,
pij = 1.
j
Indeed, it is straightforward to check that such a matrix gives rise to a normal, unital, completely positive map. On the other hand, for all i, the composition of a normal, unital, completely positive map with the normal state projecting onto the ith element must be a normal state, so it has to be given by a nonnegative element in 1 with norm 1. Given two such matrices P and Q, we ask when do they strongly commute. To answer this question, we first find orthonormal bases for M ⊗P M ⊗Q H and M ⊗Q M ⊗P H . If {ei } is the vector with 1 in the ith place and 0’s elsewhere, it is easy to see that the set {ei ⊗P ej ⊗Q ek }i,j,k spans M ⊗P M ⊗Q H , and {ei ⊗Q ej ⊗P ek }i,j,k spans M ⊗Q M ⊗P H . We compute ei ⊗P ej ⊗Q ek , em ⊗P ep ⊗Q eq = ek , Q ej∗ P ei∗ em ep eq = δi,m δj,p δk,q qkj pj i . Thus,
(qkj pj i )−1/2 · ei ⊗P ej ⊗Q ek : i, j, k such that qkj pj i = 0 is an orthonormal basis for M ⊗P M ⊗Q H , and similarly for M ⊗Q M ⊗P H . If u : M ⊗P M ⊗Q H → M ⊗Q M ⊗P H is a unitary that makes P and Q commute strongly, then for all i, k we must have u(ei ⊗P a ⊗Q ek ) = (ei ⊗ 1 ⊗ ek )u(ei ⊗P a ⊗Q ek ) = ei ⊗Q b ⊗P ek , thus for all i, j , the spaces Vi,j := {ei ⊗P a ⊗Q ek : a ∈ M} and Wi,j := {ei ⊗Q a ⊗P ek : a ∈ M} bust be isomorphic. Thus, a necessary condition for strong commutativity is that for all i, k, {j : qkj pj i = 0} = {j : pkj qj i = 0},
(A.4)
O.M. Shalit / Journal of Functional Analysis 255 (2008) 46–89
89
where | · | denotes cardinality. This condition is also sufficient, because we may define a unitary between each pair Vi,j and Wi,j , sending ei ⊗P 1 ⊗Q ek to ei ⊗Q 1 ⊗P ek and doing whatever on the complement. By the way, this example shows that when two CP maps commute strongly, there may be a great many unitaries that “implement” the strong commutation. One can impose certain block structures on P and Q that will guarantee that (A.4) is satisfied. Since we are in particularly interested in semigroups, we shall be content with the following observation. Let P and Q be two commuting, irreducible, stochastic matrices. Then Pt := e−t etP and Qt := e−t etQ are two commuting, stochastic semigroups with strictly positive elements, and thus they commute strongly. For example, let 1 P= 3
1 1 1 1 1 1 , 1 1 1
1/2 0 1/2 Q = 1/4 1/2 1/4 . 1/4 1/2 1/4
One may check that P and Q commute, but do not satisfy (A.4), hence they do not commute strongly. So we see that strong commutativity may fail even in the simplest cases. However, P and Q are both irreducible, thus the semigroups they generate do commute strongly. References [1] W.B. Arveson, Non-Commutative Dynamics and E-semigroups, Springer Monogr. Math., Springer, Berlin, 2003. [2] B.V.R. Bhat, M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000) 519–575. [3] R. Gohm, Noncommutative Stationary Processes, Lecture Notes in Math., vol. 1839, Springer, Berlin, 2004. [4] P. Muhly, B. Solel, Tensor algebras over C ∗ -correspondences: Representations, dilations, and C ∗ -envelopes, J. Funct. Anal. 158 (1998) 389–457. [5] P. Muhly, B. Solel, Quantum Markov processes (correspondences and dilations), Internat. J. Math. 13 (2002) 863– 906. [6] P. Muhly, B. Solel, Hardy algebras, W ∗ -correspondences and interpolation theory, Math. Ann. 330 (2) (2004) 353– 415. [7] D. SeLegue, Minimal dilations of CP maps and C ∗ -extension of the Szeg˝o limit theorem, PhD dissertation, University of California, Berkeley, CA, 1997. [8] O. Shalit, Representing a product system representation as a contractive semigroup and applications to regular isometric dilations, preprint, arXiv: 0706.3178v1 [math.OA], 2007. [9] O. Shalit, Dilation theorems for contractive semigroups, available online on http://tx.technion.ac.il/~orrms. [10] B. Solel, Representations of product systems over semigroups and dilations of commuting CP maps, J. Funct. Anal. 235 (2006) 593–618. [11] B. Sz.-Nagy, C. Foia¸s, Harmonic Analysis of Operators in Hilbert Space, North-Holland, Amsterdam, 1970.
Journal of Functional Analysis 255 (2008) 90–119 www.elsevier.com/locate/jfa
On a class of weighted anisotropic Sobolev inequalities Stathis Filippas a,d , Luisa Moschini b , Achilles Tertikas c,d,∗ a Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece b Dipartimento di Metodi e Modelli Matematici, University of Rome “La Sapienza”, 00185 Rome, Italy c Department of Mathematics, University of Crete, 71409 Heraklion, Greece d Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
Received 19 September 2007; accepted 3 March 2008 Available online 12 May 2008 Communicated by H. Brezis
Abstract In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary. © 2008 Elsevier Inc. All rights reserved. Keywords: Weighted Sobolev inequalities; Anisotropic Sobolev inequalities; Grushin operators; Distance function
1. Introduction and main results In this article, motivated by the work of Caffarelli and Cordoba [6] in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities, that is Sobolev type inequalities where different derivatives have different weight functions.
* Corresponding author at: Department of Mathematics, University of Crete, Knossos avenue, 71409 Heraklion, Greece. E-mail addresses:
[email protected] (S. Filippas),
[email protected] (L. Moschini),
[email protected] (A. Tertikas).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.014
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
91
Phase transitions or interfaces appear in physical problems when two different states coexist and there is a balance between two opposite tendencies: a diffusive effect that tends to mix the materials and a mechanism that drives them into their pure state, which is typically given by a nonnegative potential F (x, u), denoting the energy density of the configuration u. For example it is known that minimizers of the functionals 2 J (u) := |∇u|2 + F (x, u) dx, Ω
for 0 < < 1, F (x, u) = (1 − u2 )δ+ , and Ω ⊂ RN open and bounded, develop free boundaries if 0 < δ < 2, while generate exponential convergence to the states ±1 if δ = 2, that is in the case connected to the Ginzburg–Landau equation, see [6]. The main results of [6] are concerned with the study of regularity properties of interfaces. Their results are closely related to a conjecture of De Giorgi according to which bounded solutions of the Ginzburg–Landau scalar equation on the whole space RN that are monotone in one direction, are one-dimensional (see [15]); in particular they concern the question of De Giorgi under the additional assumption that the level sets are the graphs of an equi-Lipschitz family of functions (see [18] for the case N = 2, see also [2] for the general case). In establishing these results a central role is played by various anisotropic Sobolev type inequalities, see [6, Propositions 4–5]. Moreover, the weighted anisotropic Sobolev inequalities we are dealing with, are also intimately connected to Sobolev inequalities for Grushin type operators. Unweighted local version of this type of inequalities have been studied in [12,13], as well as in [14] where Muckenhoupt weights were considered. As a further motivation to the present study, we mention that Sobolev inequalities, are used in the proof of Liouville type theorems for the corresponding linear elliptic operators in divergence form. For other type of anisotropic Sobolev type inequalities we refer to [1,3,19]. To state our results let us first introduce some notation. We define the infinite cylinder H1 as well as the finite cylinder C1 by H1 := (x , λ) ∈ RN −1 × R: |x | < 1 , C1 := (x , λ) ∈ RN −1 × R: |x | < 1, |λ| < 1 . We will prove weighted Sobolev inequalities on the finite cylinder C1 , the weight being a power of the distance function to the top or the bottom of the cylinder {λ = ±1}. Our first result is the following Theorem 1.1. Let N 2, α > −1 and σ ∈ (−2α, 2). Then, for any Q with 2 Q Qcr (N, α, σ ) :=
2(N + N+
2α+σ 2−σ ) , 2α+σ 2−σ − 2
(1.1)
there exists a positive constant C = C(Q, N, α, σ ), such that for any function f ∈ C0∞ (H1 ) there holds
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S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
Q α 1 − |λ| f (x , λ) dx dλ
C1
C
2
Q
α σ 1 − |λ| |∇x f |2 + 1 − |λ| |∂λ f |2 dx dλ.
(1.2)
C1
In the limit case where σ = 2, estimate (1.2) holds for Q = 2 and any f ∈ C0∞ (H1 ) but fails for Q > 2 and f ∈ C0∞ (C1 ). By taking σ = 2α, 0 < α < 1, our result contains [6, Proposition 4]. When σ 2 we can still have similar inequalities for α < −1. More precisely when σ = 2 we have Theorem 1.2. Let N 2 and α < −1. For any Q with 2 Q N2N −2 , in case N 3, or Q 2 in case N = 2, there exists a positive constant C = C(N, α, Q), such that for any function f ∈ C0∞ (C1 ) there holds
Q α 1 − |λ| f (x , λ) dx dλ
C1
C
2
Q
α 2 1 − |λ| |∇x f |2 + 1 − |λ| |∂λ f |2 dx dλ.
(1.3)
C1
When σ > 2 we obtain the same inequality but this time for exponents Q that satisfy Q Qcr as defined in (1.1). Thus, we have Theorem 1.3. Let N 2, α < −1 and σ ∈ (2, −2α). Then, for any Q with Qcr Q if N = 2 or Qcr Q N2N −2 if N 3, there exists a positive constant C = C(N, Q, α, σ ), such that for any function f ∈ C0∞ (C1 ) there holds
Q α 1 − |λ| f (x , λ) dx dλ
C1
C
2
Q
α σ 1 − |λ| |∇x f |2 + 1 − |λ| |∂λ f |2 dx dλ.
(1.4)
C1
When α > −1 then (1 − |λ|)α is an L1 (−1, 1) function and using Hölder’s inequality one can obtain the inequality for any Q with 2 Q Qcr once it is true for Qcr . However this is not the case when α < −1. We note that for Q = 2 inequality (1.4) is still valid as one can see using Poincaré inequality in the x -variables. The validity or not of (1.4) for 2 < Q < Qcr remains an open question. Finally, as σ > 2 approaches 2, Qcr approaches 2 and therefore the Q-interval of validity of (1.4) approaches the interval [2, N2N −2 ] in complete agreement with the result of Theorem 1.2.
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
93
A central role in the proof of the previous results is played by various weighted isotropic N −1 × R: x > 0}, which are Sobolev inequalities in the upper half-space RN N + := {(x , xN ) ∈ R of independent interest. We present such a result: Theorem 1.4. Let either N = 2,
2 , Q
2 Q,
and B = A −
2N , N −2
and B = A − 1 +
(1.5)
or else, N 3,
2Q
Q−2 N. 2Q
(1.6)
If BQ + 2A = 0, or if A = B = 0 then (i) There exists a positive constant C = C(A, Q, N ), such that for any function f ∈ C0∞ (RN +) there holds
Q BQ xN f (x , xN ) dx dxN
2
Q
C
RN +
2A |∇x f |2 + |∂xN f |2 dx dxN . (1.7) xN
RN +
(ii) If moreover BQ + 2A > 0, or if A = B = 0 inequality (1.7) still holds even if f ∈ C0∞ (RN ). The exponent Q = Q(A, B, N ) given by conditions (1.6) and (1.5) is the best possible, as one can easily see arguing by scaling x = Ry , xN = RyN . In case N 3, part (i) of Theorem 1.4 is due to Maz’ya, see [17, Section 2.1.6]. Here we will provide a simpler proof along the lines of [9–11]. A particular case of (1.7) has been obtained in [7] under an additional assumption on f , by different methods. We next present a direct consequence of Theorem 1.1. Corollary 1.5. For N 2, m > −1 and ∈ (0, 12 ) we set C1, := (x , λ) ∈ RN −1 × R: |x | < 1, |λ| < 1 − 1+m . Let α > −1 and β > 0 satisfy −2α(1 + m) < βm < 2(1 + m), and 2 P Pcr (N, m, α, β) :=
2α(1+m)+βm 2(1+m)−βm ) . 2α(1+m)+βm 2(1+m)−βm − 2
2(N + N+
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Then, there exists a positive constant C = C(N, P , m, α, β) independent of , such that for any function f ∈ C0∞ (C1, ) there holds
P α 1 − |λ| f (x , λ) dx dλ
2
P
C
C1,
α (1 − |λ|)β 2 1 − |λ| |∂ f | |∇x f |2 + dx dλ. λ β
C1,
The above corollary is in the same spirit as the results in [6]. Indeed, when α = 1 and β = 2, Corollary 1.5 entails the weighted Sobolev inequality of [6, Proposition 5] providing a precise range for the Sobolev exponent. Analogous results can be easily obtained in case α < −1, by using Theorems 1.2 and 1.3. We next consider the more general case of weighted anisotropic inequalities where the distance is taken from a higher codimension boundary. More precisely, for x ∈ RN we write x = (x , λ), with x ∈ RN −k and λ ∈ Rk , with 1 < k < N . Let Ω ⊂ Rk be a smooth bounded domain and B1 = {x : |x | < 1} be the unit ball in RN −k . We also set d = d(λ) = dist(λ, ∂Ω). In this case our main result reads Theorem 1.6. Let N 3, 1 < k < N , α > −1 and σ ∈ (−2α, 2) with 2α + σ k 0. Then, for any Q, 2 Q Qkcr :=
2(N + N+
2α+σ k 2−σ ) , 2α+σ k 2−σ − 2
there exists a positive constant C = C(Q, N, α, σ, k), such that for any function f ∈ C0∞ (B1 × Ω) there holds
2 d α |f |Q dx
Q
C
B1 ×Ω
d α |∇x f |2 + d σ |∇λ f |2 dx.
(1.8)
B1 ×Ω
The limit case k = N , corresponds to the following isotropic weighted inequality
2 d |f | dλ α
Ω
Q
Q
C
d α+σ |∇λ f |2 dλ, Ω
which is true when α + σ < 1 but not when α + σ 1; see remark after the proof of Theorem 1.6 for details. To prove the above theorem an important role is played by the following weighted anisotropic Sobolev inequality in the upper half-space RN + . To state the result we first introduce some nota , λ) = (x , x , y), with x ∈ RN −k , x ∈ [0, ∞), , 1 < k < N , we write x = (x tion. For x ∈ RN N N + k−1 and y ∈ R . We also write dx for dx dλ = dx dxN dy. Theorem 1.7. Let γ ∈ R, and either N = 2,
Q 2,
and B = A − 1 +
Q − 2 2 + γ (k − 1) , 2Q
(1.9)
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
95
or else, N 3,
2Q
2N , N −2
and B = A − 1 +
Q − 2 N + γ (k − 1) . 2Q
(1.10)
If BQ + 2A = 0 then (i) There exists a positive constant C = C(A, Q, N, k, γ ), such that for any function f ∈ C0∞ (RN + ) there holds
Q BQ xN f (x) dx
2
Q
C
RN +
2γ 2A |∇x ,xN f |2 + xN |∇y f |2 dx. xN
(1.11)
RN +
(ii) If moreover BQ + 2A > 0, inequality (1.11) still holds even if f ∈ C0∞ (RN ). We note that the exponent Q = Q(A, B, N, γ , k) given by (1.10) is the best possible as one can easily check using the natural scaling x = Rz , xN = RzN and y = R γ +1 w. Inequality (1.11) is a weighted Sobolev inequality for Grushin type operators Lγ := x ,xN + 2γ γ xN y having associated gradient ∇γ := (∇x , ∂xN , xN ∇y ), so that 2γ
|∇γ g|2 = |∇x ,xN g|2 + xN |∇y g|2 . When γ ∈ N then Lγ :=
∂2 ∂x12
2γ ∂ 2 ∂x22
+ x1
belongs to the class of differential operators considered
by [4]; in particular, it is hypoelliptic and satisfies a Harnack inequality since the Lie algebra γ generated by the vector fields ∂x∂ 1 and x1 ∂x∂ 2 has rank two at any point of the plane. When γ > 0 and 2A > −1, local versions of inequality (1.11) were considered in [14]. Our method has the advantage of allowing a bigger range of values for the parameters A and γ . On the other hand the results of [14] cover a bigger variety of weights. We finally note that weighted Sobolev type inequalities of the kind we present in this work play an important role in establishing Harnack inequalities and heat kernel estimates in [11] in the isotropic case, whereas in the non isotropic case, weighted Sobolev inequalities are crucial in establishing Liouville type theorems, see [5]. This paper is organized as follows. In Sections 2, 3 and 4 we consider the case of codimension k = 1. In particular, in Section 2 we study the case σ < 2, in Section 3 the critical case σ = 2, whereas in Section 4 the supercritical case σ > 2. Finally the last Section 5 is devoted to the study of the higher codimension case and in particular we give the proofs of Theorems 1.6 and 1.7. 2. Codimension 1 degeneracy; the case σ < 2 In this section we will give the proofs of Theorems 1.1, 1.4 and Corollary 1.5. We first give the proof Theorem 1.4. Proof of Theorem 1.4. Let us first give the proof of part (ii). For any u ∈ C0∞ (RN ) it is well known that SN u
N
L N−1
∇u L1 ,
(2.1)
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S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119 1
1
where SN := Nπ 2 [Γ (1 + N2 )]− 2 (see, e.g., [17, p. 189]). We apply (2.1) to the function u := a v, for v ∈ C ∞ (RN ) and a > 0. Thus, we have xN 0
SN x a v
N
N
L N−1
a−1 a |∇v|xN + axN |v| dx dxN .
RN +
To estimate the last term of the right-hand side, we integrate by parts,
a−1 xN |v| dx dxN
a
a ∇xN |v| dx dxN
=
RN +
=−
RN +
a xN ∇|v| dx dxN .
(2.2)
RN +
From this we get a
a−1 xN |v| dx dxN
a |∇v|xN dx dxN .
RN +
(2.3)
RN +
Consequently,
a
x v
N
−1 2SN
N
L N−1
a |∇v|xN dx dxN .
(2.4)
RN +
For any 1 p NN−1 and any two functions w and v, the following interpolation inequality can be easily seen to be true:
b
w v
Lp
C1 w a v
N L N−1
+ C2 w a−1 v L1 ,
for b = a − 1 +
p−1 N, p
(2.5)
with two positive constants C1 , C2 independent of w and v. From (2.4) and (2.5) for w := xN we obtain the following:
bp xN |v|p dx dxN
RN +
1
p
C1
a |∇v|xN dx dxN + C2
RN +
a−1 xN |v| dx dxN .
(2.6)
RN +
Using now (2.3) we arrive at the following Lp –L1 weighted estimate RN +
bp xN |v|p dx dxN
1
p
C1
a |∇v|xN dx dxN .
RN +
To pass to the corresponding LQ –L2 estimate we apply (2.7) to v := |f |s , s > 0, to obtain
(2.7)
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
bp xN |f |ps
RN +
97
1
p
dx dxN
a f s−1 |∇f |xN dx dxN
C RN +
bp
=C
a− bp 2
dx dxN
xN2 |f |s−1 |∇f |xN
RN +
xN |f |2s−2 dx dxN bp
C
1 2
RN +
Choosing s =
2 2−p
2a−bp
|∇f |2 xN
dx dxN
1 2
.
RN +
so that 2s − 2 = ps we get
xN |f |ps dx dxN bp
2 −1 p
2a−bp
C
RN +
xN
|∇f |2 dx dxN .
(2.8)
RN +
To arrive at (1.7) we take BQ = bp, Q = ps and 2a − bp = 2A. For this choice of the parameters we arrive at
BQ xN |f |Q dx dxN
2
Q
C
RN +
2A xN |∇f |2 dx dxN
RN +
Q−2 2 with 2 Q N2N −2 and B = A − 1 + 2Q N , in case N 3, or Q 2 and B = A − Q in case N = 2. Since 2a = 2A + BQ, the condition a > 0 is equivalent to BQ + 2A > 0. This completes the proof of part (ii) of Theorem 1.4. Concerning part (i), we note that for v ∈ C0∞ (RN + ), and a ∈ R, it follows from (2.2) that
|a| RN +
a−1 xN |v| dx dxN
a |∇v|xN dx dxN .
(2.9)
RN +
Consequently, estimate (2.4) remains true for any a ∈ R. Estimate (2.6) is still true, and using (2.9) we arrive at (2.7). The use of (2.9) however imposes the condition that a = 0. The rest of the argument remains the same. The condition a = 0 is equivalent to BQ + 2A = 0. We finally note that, when A = B = 0 then (1.7) is the standard Sobolev inequality. 2 As a consequence of Theorem 1.4 we have the following inequality in a strip: Proposition 2.1. Let H1 = {(x , xN ) ∈ RN −1 × R: |x | < 1},
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S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
N = 2,
2 , Q
2 Q,
and B = A −
2N , N −2
and B = A − 1 +
or N 3,
2Q
Q−2 N. 2Q
If BQ + 2A = 0, or if A = B = 0 then, (i) There exists a positive constant C = C(A, Q, N ), such that for any function f ∈ C0∞ (H1 ∩ RN + ) there holds
Q BQ xN f (x , xN ) dx dxN
2
Q
H1 ∩{0<xN <1}
C
2A |∇x f |2 + |∂xN f |2 dx dxN . xN
(2.10)
H1 ∩{0<xN <1}
(ii) If moreover BQ + 2A > 0, inequality (2.10) still holds even if f ∈ C0∞ (H1 ). In the case where 2A = BQ ∈ (0, ∞) and under the more restrictive assumption that f ∈ C0∞ (H1 ∩ {0 < xN < 1}), the result of part (ii) has been established in [7] by different methods (see also [8]). Proof of Proposition 2.1. We prove part (ii), the other case being quite similar. To do this we will use part (ii) of Theorem 1.4. We also need to remove the zero boundary conditions on the hyperplane xN = 1. To this end, let f ∈ C0∞ (H1 ) and we denote by ξ(xN ) a C 1 function such that ξ(xN ) = 1 if xN 12 and ξ(xN ) = 0 if xN 1. We then have
BQ xN |f |Q dx dxN
LHS := C
2
Q
{0<xN <1}
BQ xN |f ξ |Q dx dxN
{0<xN <1}
2
Q
+
Q BQ xN f (1 − ξ ) dx dxN
2
Q
{ 12 <xN <1}
=: I1 + I2 .
(2.11)
Applying Theorem 1.4, part (ii) to the function f ξ , we obtain I1 C
2 2 2A ∇x (f ξ ) + ∂xN (f ξ ) dx dxN xN
{0<xN <1}
C {0<xN <1}
2A |∇x f |2 + |∂xN f |2 + f 2 dx dxN . xN
(2.12)
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99
BQ
2A are uniformly bounded both from above Concerning I2 we note that the weights xN and xN 1 and below for xN ∈ [ 2 , 1], and therefore, applying the standard Sobolev inequality to the function f (1 − ξ ) which is zero for |x | = 1 as well as for xN = 12 we get
2A |∇x f |2 + |∂xN f |2 + f 2 dx dxN . xN
I2 C { 12 <xN <1}
Combining this with (2.11) and (2.12) we get 2A |∇x f |2 + |∂xN f |2 + f 2 dx dxN . xN LHS C
(2.13)
{0<xN <1}
To continue, let B1 := {x ∈ RN −1 : |x | < 1}. For any fixed xN ∈ [0, 1], we have by the Poincaré inequality f 2 (x , xN ) dx C |∇x f |2 dx , B1
B1
whence 1 2
f (x
2A , xN ) dx xN dxN
1 C
0 B1
2A |∇x f |2 dx xN dxN .
0 B1
From this and (2.13) the result follows.
2
We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. It is enough to prove (1.2) in the upper half-cylinder; that is, if f ∈ C0∞ (H1 ) then we will show that
Q (1 − λ)α f (x , λ) dx dλ
{0<λ<1}
C
2
Q
(1 − λ)α |∇x f |2 + (1 − λ)σ |∂λ f |2 dx dλ.
(2.14)
{0<λ<1}
We first consider the case σ < 2. We change variables by x = x , s = (1 − λ) 2 ϕ(x , s) := f (x , 1 − s 2−σ ), it follows that inequality (2.14) is equivalent to
s
{0<s<1}
σ +2α 2−σ
ϕ(x , s)Q dx ds
2
Q
C {0<s<1}
s
σ +2α 2−σ
2−σ 2
thus setting
|∇x ϕ|2 + |∂s ϕ|2 dx ds; (2.15)
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in fact we easily compute that ds =
σ −2 − σ2 2 (1 − λ)
dλ, ∂λ =
ds dλ ∂s
=
σ −2 |∇x f | + (1 − λ) |∂λ f | = |∇x ϕ| + 2 2
2
σ
σ −2 − σ2 ∂s 2 (1 − λ)
and
2 |∂s ϕ|2 .
2
When σ ∈ (−2α, 2) we now use Proposition 2.1, part (ii). Suppose first that N 3. For Q−2 σ +2α σ +2α 2N A = 2(2−σ ) and B = 2(2−σ ) − 1 + 2Q N , with 2 Q N −2 we have that the right-hand side of (2.15) dominates
Q s BQ ϕ(x , s) dx ds
2
Q
.
{0<s<1}
To deduce (2.15) we need
σ +2α 2−σ
σ +2α BQ = ( 2(2−σ ) −1+
Q2
Q−2 2Q N )Q,
which is equivalent to
N+ N+
2α+σ 2−σ . 2α+σ 2−σ − 2
(2.16)
On the other hand, the restriction 2A + BQ > 0 is easily seen to be equivalent to Q>2
N− N+
2α+σ 2−σ 2α+σ 2−σ − 2
¯ =: Q.
(2.17)
We note that Qcr as given by (1.1) satisfies both (2.16) and (2.17) and therefore (1.2) has been proved for Q = Qcr . The full range of Q follows by using Hölder’s inequality in the left-hand side of (1.2). The case N = 2 is treated quite similarly. Thus (1.2) has been proved for any f ∈ C0∞ (H1 ). In the special case σ = 2 and Q = 2 we note that (1.2) is still valid. To see this we change 1 variables by x = x and t = (1 − λ)α+1 thus setting g(x , t) := f (x , 1 − t α+1 ). It follows that inequality (2.14) is equivalent to
Q g(x , t) dx dt
t∈(0,1)
2
Q
C
|∇x g|2 + t 2 |∂t g|2 dx dt,
(2.18)
t∈(0,1)
in fact we easily compute that dt = −(α + 1)(1 − λ)α dλ, ∂λ =
dt dλ ∂t
= −(α + 1)(1 − λ)α ∂t and
|∇x f |2 + (1 − λ)σ |∂λ f |2 = |∇x g|2 + (α + 1)2 t 2 |∂t g|2 . Inequality (2.18) with Q = 2 holds true, as one can easily see using Poincaré inequality for the slices t = constant. It remains to show that (1.2) fails in the case σ = 2, α > −1 and Q > 2 even thought we take f ∈ C0∞ (C1 ). To this end, let us make use of the following different change of variables x = x and λ = tanh xN . Then λ ∈ (−1, 1) goes to xN ∈ (−∞, ∞) and (1 − |λ|) ∼ (1 − λ2 ) = (cosh xN )−2 ∼ e−2|xN | and dλ ∼ (cosh xN )−2 dxN ∼ e−2|xN | dxN . We define
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
101
g(x , xN ) := f (x , tan hxN ). Then it follows from that for any function g ∈ C0∞ (H1 ) the following inequality should be true if (1.2) holds true:
Q e−2(α+1)|xN | g(x , xN ) dx dxN
H1
2
Q
e−2(α+1)|xN | |∇x g|2 + |∂xN g|2 dx dxN .
C
(2.19)
H1
For g ∈ C0∞ (H1 ∩ {xN > 0}) we set gτ (x , xN ) := g(x , xN − τ ), τ > 0. Clearly, gτ ∈ C0∞ (H1 ∩ {xN > 0}) and applying (2.19) to the family gτ we get e
Q g(x , xN ) dx dxN
−2(α+1)xN
H1
Ce
−2τ (α+1)( Q−2 Q )
2
Q
e−2(α+1)xN |∇x g|2 + |∂xN g|2 dx dxN ,
H1
for any τ > 0. Taking the limit τ → +∞ we reach a contradiction for Q > 2, α > −1. This completes the proof of Theorem 1.1. 2 Remark. In case σ = 2α and α ∈ (0, 1), estimate (1.2) is an improvement of Caffarelli and Cordoba [6, Proposition 4]. Indeed, our Sobolev exponent Qcr is strictly bigger than the one 2N coming from the arguments of [6]—which is less than . Moreover, we only assume that 4α N + α+1 −2
f ∈ C0∞ (H1 ) instead of f ∈ C0∞ (C1 ).
Remark. In case σ = −α and α > 0 inequality (1.2) is a Sobolev inequality for a Grushin type α operator corresponding to the vector fields ((1 − |λ|) 2 ∇x , ∂λ ); we refer to [13] where local versions of similar inequalities have been considered. Remark. We note that in the case σ = −2α, estimate (2.15) corresponds to the standard Sobolev inequality in a strip, and the result follows from Proposition 2.1, part (i); thus (1.2) still holds true for any f ∈ C0∞ (C1 ) if σ = −2α. We next show how Corollary 1.5 follows from Theorem 1.1. Proof of Corollary 1.5. It is a consequence of Theorem 1.1. Indeed, for (x , λ) ∈ C1, we have 1
βm (1−|λ|)β > (1 − |λ|) 1+m , β > 0. The result β βm 1+m there; in particular Pcr (N, m, α, β) =
1 − |λ| > 1+m , that is, −1 > (1 − |λ|)− 1+m , and so then follows from Theorem 1.1 by choosing σ := βm ). 2 Qcr (N, α, 1+m
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3. The critical case σ = 2 As we have seen in Theorem 1.1 inequality (1.2) fails for σ = 2, α > −1 and Q > 2. To obtain Sobolev type inequalities in this case, we need to use different weights in the two sides of the inequality. More precisely we have the following Theorem 3.1. Let N 2, and α > −1. For any Q with 2 Q N2N −2 , in case N 3, or Q 2 (Q−2)(α+1) there exists a positive constant C = C(N, α, Q, θ ), in case N = 2, and for any θ > 2 such that for any function f ∈ C0∞ (C1 ) there holds
α+θ f (x , λ)Q dx dλ 1 − |λ|
C1
C
2
Q
α 2 1 − |λ| |∇x f |2 + 1 − |λ| |∂λ f |2 dx dλ.
(3.1)
C1
Proof. It is enough to prove (3.1) in the upper half cylinder; that is, if f ∈ C0∞ (C1 ) then we will show that
Q (1 − λ)α+θ f (x , λ) dx dλ
{0<λ<1}
2
Q
(1 − λ)α |∇x f |2 + (1 − λ)2 |∂λ f |2 dx dλ.
C
(3.2)
{0<λ<1}
We change variables by x = x , s = − K1 ln(1 − λ), for an arbitrary K > 0, thus setting ϕ(x , s) = f (x , 1 − e−Ks ), and arguing as in the proof of Theorem 1.1, we see that inequality (3.2) follows as soon as we prove the following inequality e
−sK(α+θ+1)
Q ϕ(x , s) dx ds
{s>0}
2
Q
C
|∇ϕ|2 e−sK(α+1) dx ds.
(3.3)
{s>0}
In fact we easily compute that dλ = Ke−Ks ds = K(1 − λ) ds, ∂λ = |∇x f |2 + (1 − λ)2 |∂λ f |2 = |∇x ϕ|2 +
ds dλ ∂s
=
1 −1 K (1 − λ) ∂s
1 |∂s ϕ|2 ∼ |∇ϕ|2 . K2
We note that ϕ ∈ C0∞ (H1 ). To continue, we will make use of Proposition 3.3, see below. For K(α + 1) and 2 K(α + 1) Q − 2 1 (N + K(α + 1))(Q − 2) B= + N= K(α + 1) + 2 2Q Q 2 A=
and
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103
we have
Q e−sK(α+1) e−θKs ϕ(x , s) dx ds
RN +
C
e−sK(α+1) |∇ϕ|2 dx ds,
2
Q
∀ϕ ∈ C0∞ (H1 ),
(3.4)
RN + N + α + 1) Q−2 where θ := ( K 2 . Note that θ = 0 if Q = 2 as suggested by Theorem 1.1. Due to the arbitrariness of K this means that we may take any value θ > (α+1)(Q−2) . The restriction 2 Q+2 Q−2 2A + BQ = 0 is easily seen to be equivalent to 2 (K(α + 1) + Q+2 N ) = 0, which is trivially satisfied. The case N = 2 is treated quite similarly. 2
According to Theorem 3.1 one cannot match the weights in the weighted anisotropic Sobolev inequality (3.1) when α > −1 and Q > 2. However, in the case α < −1 we can match the weights, thus proving Theorem 1.2. Proof of Theorem 1.2. The case Q = 2 is a simple consequence of Poincaré inequality. We therefore consider the case Q > 2. Using the same change of variables as in the proof of Theorem 3.1 the sought for inequality is equivalent to the following inequality e
−sK(α+1)
RN +
C
Q ϕ(x , s) dx ds
2
e−sK(α+1) |∇ϕ|2 dx ds,
Q
∀ϕ ∈ C0∞ (H1 ).
(3.5)
e2As |∇ϕ|2 dx ds,
(3.6)
RN +
We will use Proposition 3.5. Thus, we have
eBQs |ϕ|Q dx ds
2
RN +
for B = − K(α+1) and A − 1 = B − Q
Q
C RN +
Q−2 2Q N
= − K(α+1) − Q
Q−2 2Q N . To deduce (3.5) from (3.6) Q−2 Q (N − K(α + 1)). This last in-
we need 2A −K(α + 1) which is equivalent to 2 equality is always satisfied by taking K large enough. On the other hand, BQ + 2(A − 1) = Q−2 −K(α + 1) Q+2 Q − Q N = 0, for K large. The case N = 2 is treated similarly. 2 It remains to give the proof of the auxiliary results we used above. We first have
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Theorem 3.2. Let either N = 2,
2 Q,
and B = A + 1 −
2 , Q
or else, N 3,
2Q
2N , N −2
and B = A +
Q−2 N. 2Q
Then, if BQ+2A = 0, there exists a positive constant C = C(A, Q, N ) such that for any function f ∈ C0∞ (RN + ) there holds e
−BQxN
2
Q
|f | dx dxN Q
e−2AxN |∇f |2 dx dxN .
C
RN +
(3.7)
RN +
Proof. We apply the Gagliardo–Nirenberg–Sobolev inequality (2.1) to the function u := e−axN v, for any v ∈ C0∞ (RN + ) and a = 0, to get
SN e−axN v
N
L N−1
|∇v|e−axN + |a|e−axN |v| dx dxN .
RN +
To estimate the last term of the right-hand side, we integrate by parts, −axN −axN a e |v| dx dxN = − ∇e |v| dx dxN = e−axN ∇|v| dx dxN RN +
RN +
RN +
whence, |a|
e−axN |v| dx dxN
RN +
e−axN |∇v| dx dxN .
(3.8)
RN +
Consequently,
−ax
e N v
N L N−1
C
e−axN |∇v| dx dxN .
(3.9)
RN +
We note that this is true even if a = 0. Using the interpolation inequality (2.5) with w := e−xN , as well as (3.8) and (3.9) we arrive at the following Lp –L1 estimate (e−(a−1)xN e−axN ) RN +
e−bpxN |v|p dx dxN
1
p
C RN +
e−(a−1)xN |∇v| dx dxN ,
(3.10)
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
with 1 p NN−1 , b = a − 1 + need the following inequality e
−(a−1)xN
p−1 p N
105
and a = 1. Indeed in order to reach inequality (3.10) we
e−(a−1)xN |∇v| dx dxN
|v| dx dxN C
RN +
RN +
which follows from inequality (3.8) if a = 1. We next apply (3.10) to v := |f |s , s > 0, to obtain
e−bpxN |f |ps dx dxN
RN +
C
1
p
f s−1 |∇f |e−(a−1)xN dx dxN
RN +
=C
e−
bpxN 2
f s−1 |∇f |e−(a−1)xN +
bpxN 2
dx dxN
RN +
C
e−bpxN f 2s−2 dx dxN
1 2
RN +
Choosing s =
2 2−p ,
e
|∇f |2 e−2(a−1)xN +bpxN dx dxN
1 2
.
RN +
so that 2s − 2 = ps we get
−bpxN
|f |
ps
2 −1 p
dx dxN
C
RN +
e(−2(a−1)+bp)xN |∇f |2 dx dxN .
(3.11)
RN +
To conclude the proof of the lemma we take BQ = bp, Q = ps, and A = a − 1 − condition a = 1 is equivalent to BQ + 2A = 0. 2
bp 2 .
The
As a consequence of the previous theorem, we have the following result which is the analogue of Proposition 2.1. That is, in some cases we can remove the zero boundary condition at xN = 0. Proposition 3.3. Let either N = 2,
2 Q,
and B = A + 1 −
2 , Q
or else, N 3,
2Q
2N , N −2
and B = A +
Q−2 N. 2Q
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Then, if BQ + 2A = 0, there exists a positive constant C = C(A, Q, N ) such that for any function f ∈ C0∞ (H1 ) there holds
e−BQxN |f |Q dx dxN
2
Q
e−2AxN |∇f |2 dx dxN .
C
RN +
(3.12)
RN +
Proof. To deduce (3.12) from (3.7) we will work as in the proof of Proposition 2.1 in order to remove the zero boundary condition on the hyperplane xN = 0. Let ξ(xN ) be a C 1 function such that ξ(xN ) = 1 if xN 2 and ξ(xN ) = 0 if xN ∈ [0, 1], then for any f ∈ C0∞ (H1 ) we have LHS := C
e
−BQxN
2
|f | dx dxN Q
Q
RN +
e
−BQxN
2
|f ξ | dx dxN Q
Q
+
RN +
e
−BQxN
Q f (1 − ξ ) dx dxN
2
Q
RN +
=: I1 + I2 .
(3.13)
Applying (3.7) to the function f ξ , we obtain I1 C
e−2AxN |∇f |2 + f 2 dx dxN .
RN +
On the other hand, since the weights e−BQxN and e−2AxN are uniformly bounded both from above and below in the interval [0, 2], we may apply the standard Sobolev inequality to the function f (1 − ξ ) which is zero when |x | = 1 as well as when xN = 2 to get I2 C
∇ f (1 − ξ ) 2 dx dxN C
RN−1 ×[0,2]
e−2AxN |∇f |2 + f 2 dx dxN .
RN +
Combining the above estimates we have LHS C
e−2AxN |∇f |2 + f 2 dx dxN .
(3.14)
RN +
To conclude we use the Poincaré inequality on the set B1 = {x ∈ RN −1 : |x | < 1}. For any fixed xN B1
f 2 (x , xN ) dx C
B1
|∇x f |2 dx ,
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
107
whence e
−2AxN
∞
f dx dxN = 2
e
RN +
−2AxN
C
∞
f dx dxN C 2
B1
0
e−2AxN
|∇x f |2 dx dxN ,
B1
0
e−2AxN |∇f |2 dx dxN .
RN +
2
From this and (3.14) the result follows.
We next present a new Sobolev inequality which also involves exponential weights. We used this estimate in the proof of Theorem 1.2. Theorem 3.4. Let either N = 2,
2 , Q
2 Q,
and B = A −
2N , N −2
and B = A − 1 +
or else, N 3,
2Q
Q−2 N. 2Q
Then, if BQ+2A = 2, there exists a positive constant C = C(A, Q, N ) such that for any function f ∈ C0∞ (RN + ) there holds
eBQxN |f |Q dx dxN
2
Q
C
RN +
e2AxN |∇f |2 dx dxN .
(3.15)
RN +
Proof. Working as in the proof of Theorem 3.2 we obtain (3.8) and (3.9) that is, |a|
e
axN
|v| dx dxN
RN +
eaxN |∇v| dx dxN ,
(3.16)
RN +
and
ax
e N v
N L N−1
C RN +
which are valid for any a in R.
eaxN |∇v| dx dxN ;
(3.17)
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We next use the interpolation inequality (2.5) with w := exN , as well as (3.16) and (3.17) to arrive at the following Lp –L1 estimate (eaxN e(a−1)xN ) e
bpxN
1
p
|v| dx dxN p
C
RN +
with 1 p ing estimate
N N −1 ,
eaxN |∇v| dx dxN ,
(3.18)
RN +
b = a − 1 + p−1 p N and a = 1. To reach inequality (3.18) we used the follow
e(a−1)xN |v| dx dxN C
RN +
e(a−1)xN |∇v| dx dxN ,
RN +
which is a consequence of (3.16) if a = 1. We next apply (3.18) to v := |f |s , s > 0, to obtain e
bpxN
RN +
C
|f |
ps
1
p
dx dxN
f s−1 |∇f |eaxN dx dxN
RN +
=C
e
bpxN 2
f s−1 |∇f |eaxN −
bpxN 2
dx dxN
RN +
C
ebpxN f 2s−2 dx dxN
1 2
RN +
Choosing s =
2 2−p ,
|∇f |2 e2axN −bpxN dx dxN
1 2
.
RN +
so that 2s − 2 = ps, we get
e
bpxN
|f |
ps
2 −1 p
dx dxN
C
RN +
e(2a−bp)xN |∇f |2 dx dxN .
(3.19)
RN +
To conclude the proof of the lemma we take BQ = bp, Q = ps, and A = a − a = 1 is equivalent to BQ + 2A = 2. 2 We finally have Proposition 3.5. Let either N = 2,
2 Q,
and B = A −
2 , Q
bp 2 .
The condition
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
109
or else, N 3,
2Q
2N , N −2
and B = A − 1 +
Q−2 N. 2Q
Then, if BQ + 2A = 2, there exists a positive constant C = C(A, Q, N ) such that for any function f ∈ C0∞ (H1 ) there holds 2 Q BQxN Q e |f | dx dxN C e2AxN |∇f |2 dx dxN . (3.20) RN +
RN +
Proof. We need to remove the zero boundary condition of f , on the hyperplane xN = 0. As usual, let ξ(xN ) be a C 1 function such that ξ(xN ) = 1 if xN 2 and ξ(xN ) = 0 if xN ∈ [0, 1], then for any f ∈ C0∞ (H1 ) we have f = f ξ + f (1 − ξ ). To conclude the proof we argue as in the proof of Proposition 3.3. We omit further details. 4. The supercritical case σ > 2 In this section we will give the proof of Theorem 1.3. It is a direct consequence of a more general result. We recall that Qcr = Qcr (N, α, σ ) := We also set ¯ Q(N, α, σ ) := and
2(N + N+
2α+σ 2−σ ) . 2α+σ 2−σ − 2
2(N − N+
2α+σ 2−σ ) , 2α+σ 2−σ − 2
2α + σ 2α + σ 2−σ Q N+ −2 − N + 2 2 2−σ 2−σ 2α + σ 2−σ N+ − 2 (Q − Qcr ). = 4 2−σ
θcr :=
(4.1)
We then have Theorem 4.1. Let N 2, α < −1 and σ ∈ (2, −2α). Then, for any θ θcr and any Q = Q¯ with 2 Q N2N −2 , in case N 3, or Q 2 in case N = 2, there exists a positive constant C = C(Q, N, α, σ, θ ), such that for any function f ∈ C0∞ (C1 ) there holds 2 Q α+θ f (x , λ)Q dx dλ 1 − |λ| C1
C
α σ 1 − |λ| |∇x f |2 + 1 − |λ| |∂λ f |2 dx dλ.
C1
To prove the above result we will use the following consequence of Theorem 1.4.
(4.2)
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Proposition 4.2. Let H1 = {(x , xN ) ∈ RN −1 × R: |x | < 1}, N 3,
2Q
2N , N −2
and B = A − 1 +
2 Q,
and B = A −
Q−2 N, 2Q
or N = 2,
2 . Q
If BQ + 2A = 0, or if A = B = 0, then there exists a positive constant C = C(A, Q, N ), such that for any function f ∈ C0∞ (H1 ) there holds
Q BQ xN f (x , xN ) dx dxN
2
Q
2A |∇x f |2 + |∂xN f |2 dx dxN . xN
C
{xN >1}
(4.3)
{xN >1}
Proof. The proof is quite similar to the proof of Proposition 2.1 we therefore sketch it. We use a C 1 cutoff function ξ(xN ) such that ξ(xN ) = 1 in xN 2 and ξ(xN ) = 0 if 0 xN 1. Hence we write f = f ξ + f (1 − ξ ). Now f (1 − ξ ) satisfies the standard Sobolev inequality in 1 xN 2, while f ξ satisfies the assumptions of Theorem 1.4, part (i). Putting things together and using Poincaré inequality in the x -variables we conclude the proof. We omit further details. 2 Proof of Theorems 4.1 and 1.3. We first prove Theorem 4.1. As usual, it is enough to prove (4.2) in the upper half-cylinder. That is, if f ∈ C0∞ (C1 ) then we need to show that
α+θ
(1 − λ) {0<λ<1}
Q f (x , λ) dx dλ
2
Q
(1 − λ)α |∇x f |2 + (1 − λ)σ |∂λ f |2 dx dλ.
C
(4.4)
{0<λ<1}
As in the proof of Theorem 1.1, we change variables by x = x , s = (1 − λ) 2 ϕ(x , s) := f (x , 1 − s 2−σ ), it follows that inequality (4.4) is equivalent to s
σ +2α+2θ 2−σ
ϕ(x , s)Q dx ds
2
Q
C
{s>1}
s
σ +2α 2−σ
2−σ 2
|∇x ϕ|2 + |∂s ϕ|2 dx ds,
B=
−1+
(4.5)
{s>1}
for ϕ ∈ C0∞ (H1 ). We now use Proposition 4.2. Suppose first that N 3. For A = σ +2α 2(2−σ )
thus setting
Q−2 2Q N , with
2Q s {s>1}
2N N −2
BQ
σ +2α 2(2−σ )
and
we have that the right-hand side of (4.5) dominates
Q ϕ(x , s) dx ds
2
Q
.
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
111
Q−2 σ +2α To deduce (4.5) we need σ +2α+2θ BQ = ( 2(2−σ 2−σ ) − 1 + 2Q N )Q, which is satisfied by any θ θcr as defined in (4.1). Let us finally observe that BQ + 2A = 0 corresponds to the assumption θ = −σ − 2α that is ¯ Q = Q. The case N = 2 is treated quite similarly. To prove Theorem 1.3 we note that for Q Qcr (N, α, σ ) we have that θcr 0 and therefore we can take θ = 0. 2
5. The case of codimension k degeneracy 1 < k < N In this section we will prove Theorems 1.6 and 1.7. Proof of Theorem 1.7. We will divide the proof into three steps. Step 1. (The critical L1 weighted anisotropic inequality.) Suppose that either β > 0 and u ∈ C0∞ (RN ) or else β ∈ R and u ∈ C0∞ (RN + ). Then, for a constant C depending only on N there holds:
βN+γ (k−1) N−1
xN
|u|
N N−1
N−1 N
dx
C(N)
RN +
γ β xN |∇x ,xN u| + xN |∇y u| dx.
(5.1)
RN +
The proof follows closely the standard proof of the L1 Gagliardo–Nirenberg–Sobolev inequality. Suppose that β = 0 and u ∈ C0∞ (RN + ). Let us write x = (x1 , . . . , xN −k ) and y = (y1 , . . . , yk−1 ). We then have that for i = 1, . . . , N − k, ∞ u(x) = −
uxi (x1 , . . . , ti , . . . , xN −k , xN , y) dti .
xi
From which it follows easily that β xN u(x)
β
xN |uxi | dti .
(5.2)
R
We similarly have that γ +β
xN
u(x)
γ +β
xN
|uyi | dsi ,
(5.3)
R
where integration is performed in the yi -variable, i = 1, . . . , k − 1. A similar argument shows that β xN u(x)
∞ β ξ |uxN | + |β|ξ β−1 |u| dξ, 0
112
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
from which it follows easily that β xN u(x) 2
∞ ξ β |uxN | dξ
(5.4)
0
which is true also if β = 0. Multiplying (5.2)–(5.4) and raising to the power βN+γ (k−1) N−1
xN
2
1 N −1
we get
N u(x) N−1
N −k
β xN |uxi | dti
∞
i=1 R
ξ |uxN | dξ β
k−1
γ +β xN |uyj | dsj
1 N−1
(5.5)
.
j =1 R
0
We next integrate with respect to x1 and apply Hölder’s inequality in the right-hand side, then we integrate with respect to the x2 variable and so on until we integrate with respect to all variables. This way we reach the following estimate
βN+γ (k−1) N−1
xN
N u(x) N−1 dx
RN +
2
N −k i=1
β xN |uxi | dx
RN +
β xN |uxN | dx
k−1 j =1
RN +
γ +β xN |uyj | dx
1 N−1
.
(5.6)
RN +
To continue we use in the right-hand side of (5.6) the well-known inequality N i=1
N N 1 ai N ai , N
ai 0.
i=1
We then conclude that
βN+γ (k−1) N−1
xN
N N−1 N γ β u(x) N−1 dx C(N) xN |∇x ,xN u| + xN |∇y u| dx ,
RN +
(5.7)
RN +
which is the sought for estimate (5.1). Step 2. (The Lp –L1 estimate.) For 1 p with weight
N N −1
we will use the interpolation inequality (2.5)
N+γ (k−1) N
w = xN
,
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
113
and βN + γ (k − 1) , N + γ (k − 1)
a :=
b=a−1+
p−1 N. p
For these choices we have that
b
w u
C1
Lp
βN+γ (k−1) N−1
xN
N u(x) N−1 dx
N−1 N
+ C2
RN +
β−1
xN |u| dx.
(5.8)
RN +
We will also make use of the estimate β−1 β β |β| xN |u| dx xN |uxN | dx xN |∇x ,xN u| dx RN +
RN +
(5.9)
RN +
which follows easily using an integration by parts if β = 0. From (5.7)–(5.9) and using the specific values of the weight and the parameters we get
b˜
x u
Lp
N
γ β xN |∇x ,xN u| + xN |∇y u| dx,
C
(5.10)
RN +
with γ ∈ R, β = 0 and p − 1 N + γ (k − 1) . b˜ = β − 1 + p
(5.11)
Step 3. (The LQ –L2 estimate.) Here we will apply estimate (5.10) to the function u(x) = |f (x)|s with s > 0. After some elementary calculations and use of Hölder’s inequality we find that
˜ bp xN |f |sp dx
1
p
RN +
C
1
˜ bp
xN |f |2s−2 dx
xN
RN +
We now choose s = choices we get that RN +
˜ 2β−bp
2
2γ |∇x ,xN f |2 + xN |∇y f |2 dx
1 2
.
(5.12)
RN + 2 2−p
˜ and 2A = 2β − bp. ˜ For this (so that sp = 2s − 2), Q = sp, BQ = bp
Q BQ xN f (x) dx
2
Q
C RN +
2γ 2A |∇x ,xN f |2 + xN |∇y f |2 dx xN
114
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
Q−2 with γ ∈ R, 2 Q N2N −2 if N 3 or for Q 2 if N = 2, and B = A − 1 + 2Q (N + γ (k − 1)). The condition β = 0 is equivalent to 2A + BQ = 0. The case where f ∈ C0∞ (RN ) and 2A + BQ > 0, or equivalently β > 0 is practically the same; we just note that (5.8) remains true for β > 0. 2
We are now ready to give the proof of Theorem 1.6. Proof of Theorem 1.6. We will use a2 (finite) partition of unity for Ω which we denote by ϕi , i = 0, . . . , m, such that 1 = m i=0 ϕi . We denote by Ωi the support of each function ϕi . We assume Ω0 Ω and therefore c d(λ, ∂Ω) c−1 for λ ∈ Ω0 . For i 1, in each Ωi we will use i ), i ∈ {1, . . . , m} with y i ∈ Δ := {y i : |y i | β for j = 1, . . . , k − 1} for local coordinates (y i , xN i j some positive constant β < 1. Each point λ ∈ Ω i ∩ ∂Ω is described by λ = (y i , ai (y i )), where the functions ai satisfy a Lipschitz condition on Δi with a constant A > 0 that is i ai y − ai zi Ay i − zi i ): y i ∈ Δ , a (y i ) − β < x i < a (y i ) + β} for y i , zi ∈ Δi . We next define Bˆ i by Bˆ i := {(y i , xN i i i N i i < a (y i )} and Γ = B i i i ˆ so that Bˆ i ∩ Ω = {(y , xN ): y ∈ Δi , ai (y ) − β < xN i i i ∩ ∂Ω = i i i i i ˆ {(y , xN ): y ∈ Δi , xN = ai (y )}. We note that Ωi ⊂ Bi ∩ Ω. Next we observe that for any i ) d(λ) (a (y i ) − x i ) (see, e.g., [16, Coroly ∈ Bˆ i ∩ Ω we have that (1 + A)−1 (ai (y i ) − xN i N i = 0}. From now on lary 4.8]). By straightening the boundary Γi we may suppose that Γi ⊂ {xN we omit the subscript i for convenience. As a first step we will prove that for u ∈ C0∞ (B1 × H1+ ), where B1 := {|x | < 1} and H1+ := {|y | < 1} × {0 < xN < 1} there holds
α xN |u|Q dx dxN dy
B1 ×H1+
2
Q
α σ σ |∇x u|2 + xN xN |∇y u|2 + xN |∂xN u|2 dx dxN dy,
C
(5.13)
B1 ×H1+ 2−σ
where x ∈ RN −k , and λ = (y, xN ) with y ∈ Rk−1 and xN ∈ R. We change variables by t = xN 2 thus obtaining
t
{0
2α+σ 2−σ
C
|u| dx dt dy Q
t
2α+σ 2−σ
2
Q
2σ |∇x ,t u|2 + t 2−σ |∇y u|2 dx dt dy.
(5.14)
{0
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
Next, for f (x) =
m
i=0 ϕi (λ)f (x),
we write
2
Q
d |f | dx α
115
Q
C
m
2 d |f ϕi | dx α
Q
Q
(5.15)
i=0 B ×Ω 1
B1 ×Ω
Using (5.13) for i = 1, . . . , m and the standard Sobolev inequality for i = 0, in the right-hand side of (5.15), after some calculations, we end up with
2
Q
d |f | dx α
Q
d |∇x f |2 + d σ |∇λ f |2 dx +
C
B1 ×Ω
d α+σ f 2 dx.
α
B1 ×Ω
(5.16)
B1 ×Ω
To estimate the last term, we first use Proposition 5.1 to obtain
d
α+σ
f dx C 2
B1 ×Ω
d
α+σ
|∇λ f | dx + 2
B1 ×Ω
d f dx , α
2
(5.17)
B1 ×Ω
and then Poincaré inequality in the x -variables,
d α f 2 dx B1 ×Ω
d α |∇x f |2 dx.
B1 ×Ω
Hence, we end up with
d α+σ f 2 dx C B1 ×Ω
d α |∇x f |2 + d σ |∇λ f |2 dx.
B1 ×Ω
Combining this with (5.16) we conclude the result.
2
We next prove the proposition we used in the proof of Theorem 1.6. Proposition 5.1. Let α + σ < 1. Then there exists a constant C = C(α, σ, Ω) > 0 such that d
α+σ −2 2
f dλ C
Ω
d α+σ |∇λ f |2 dλ,
∀f ∈ C0∞ (Ω),
(5.18)
Ω
the previous inequality fails when α + σ 1. Let α + σ = 1. Then there exists a constant C = C(α, Ω) > 0 such that Ω
X 2 (d) 2 f dλ C d
d|∇λ f |2 dλ +
Ω
d α f 2 dλ ,
Ω
where X(d) := (1 − ln(d/D))−1 and D := supλ∈Ω d(λ).
∀f ∈ C0∞ (Ω),
(5.19)
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S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
Finally, if α + σ > 1, there exists a constant C = C(α, σ, Ω) > 0 such that
d α+σ −2 f 2 dλ C
Ω
d α+σ |∇λ f |2 dλ +
Ω
d α f 2 dλ ,
∀f ∈ C0∞ (Ω).
(5.20)
Ω
Proof of Proposition 5.1. Step 1. (An auxiliary estimate.) Let Ωδ := {λ ∈ Ω: d(λ) δ}. We will establish the following estimate: Given any > 0 there exists a δ0 > 0 such that for any 0 < δ δ0 and any u ∈ C0∞ (Ω) |∇λ u|2 dλ Ωδ
1 4
u2 X 2 (d)u2 1 1 − X(δ) − dλ + dλ + u2 dS. 4 2δ d2 d2
Ωδ
Ωδ
(5.21)
∂Ωδ
To prove this our starting point is the obvious relation ∇d X∇d 2 0 ∇λ u − − u dλ. 2d 2d Ωδ
Expanding the square, integrating by parts and using the fact that |d d| can be made arbitrarily small in Ωδ , for δ sufficiently small, the result follows. Step 2. (Proof of (5.18).) We change variables by u := d leads to the following identity
d
α+σ
|∇λ f | dλ = 2
Ωδ
α+σ 2
f . A straightforward calculation
2 α+σ u α+σ 1− |∇λ u| dλ − dλ 2 2 d2 2
Ωδ
+
α+σ 2
d 2 α+σ u dλ − d 2δ
Ωδ
Ωδ
u2 dS.
(5.22)
∂Ωδ
From (5.21), (5.22) and using the fact that | d| < C in Ωδ , we easily get that there exist positive constants c such that for δ sufficiently small
d α+σ |∇λ f |2 dλ c Ωδ
d α+σ −2 f 2 dλ +
c
f 2 dS.
δ 1−(α+σ )
Ωδ
∂Ωδ
On the other hand, away from the boundary we have that
f2 C Ω\Ωδ
Ω\Ωδ
|∇λ f |2 + C ∂Ωδ
f 2 dS,
(5.23)
S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
from which one can easily deduce α+σ −2 2 d f Cδ Ω\Ωδ
d
α+σ
|∇λ f | + 2
Cδ
f 2 dS.
δ 1−(σ +α)
Ω\Ωδ
117
(5.24)
∂Ωδ
Combining (5.23) and (5.24) the result follows. We note that when α + σ 1, the constants can be approximated by C0∞ (Ω) functions in the norm given by v H 1 (d α+σ ) := Ω d α+σ (|∇v|2 + v 2 ) dλ; see [11, Theorem 2.11]. In particular, one can put a constant function in (5.18) to obtain an obvious contradiction. Step 3. (Proof of (5.19) and (5.20).) We first give the proof of (5.19). For g ∈ C0∞ (Ωδ ) and 1
u = d 2 g we get from (5.21) and (5.22) that for any > 0 there exists a δ0 > 0 such that for any 0 < δ δ0 ,
d|∇λ g| dλ 2
1 − 2 4
Ωδ
X 2 (d)g 2 dλ. d
(5.25)
Ωδ
To establish the result we argue as follows. Let ξ(s) be a C 1 function such that 0 ξ 1, ξ(s) = 0 if s 2, ξ(s) = 1 if 0 s 1, and let us define ϕ(λ) = ξ( dδ ). Whence for f ∈ C0∞ (Ω) we have that f = ϕf + (1 − ϕ)f . Using (5.25) for ϕf and the fact that on the support of (1 − ϕ)f we have
X 2 (d)f 2 dλ c1 d
Ω
X 2 (d) d
δ −1 and d α min{δ α , D α }, we arrive at
d ϕ 2 |∇λ f |2 + |∇λ ϕ|2 f 2 dλ + c2
Ω
C
d|∇λ f |2 dλ +
Ω
d α f 2 dλ .
(1 − ϕ)2 f 2 dλ Ω
(5.26)
Ω
To prove (5.20) we work similarly. We just note that the analogue of (5.25) is
d
α+σ
|∇λ g| dλ 2
1 − (α + σ ) 2
Ωδ
We omit further details.
2
d α+σ −2 g 2 dλ,
g ∈ C0∞ (Ωδ ).
Ωδ
2
Remark. The limit case k = N of Theorem 1.6, corresponds to the following isotropic weighted inequality:
2 d |f | dλ α
Ω
Q
Q
C
d α+σ |∇λ f |2 dλ,
(5.27)
Ω
where Ω is a smooth bounded domain in RN and f ∈ C0∞ (Ω). Using similar arguments one can show that the above inequality is true if α + σ < 1, provided that α > −1, 2α + N σ 0
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S. Filippas et al. / Journal of Functional Analysis 255 (2008) 90–119
2(N +α) and 2 Q QN cr = N +α+σ −2 . On the other hand inequality (5.27) fails if α + σ 1, by the argument of Proposition 5.1. We finally have the following analogue of Corollary 1.5.
Corollary 5.2. For N 3, 1 < k < N , m > −1 and ∈ (0, 12 ) we set C1, := (x , λ) ∈ RN −k × Rk : |x | < 1, |λ| < 1 − 1+m . Let α > −1 and β > 0 satisfy −2α(1 + m) < βm < 2(1 + m)
and 2α(1 + m) + βkm 0.
Then, for any P with 2 P Pcr (N, m, α, β, k) :=
2α(1+m)+βkm 2(1+m)−βm ) , 2α(1+m)+βkm 2(1+m)−βm − 2
2(N + N+
there exists a positive constant C = C(N, P , m, α, β, k) independent of , such that for any function f ∈ C0∞ (C1, ) there holds
P α 1 − |λ| f (x , λ) dx dλ
C1,
2 P
C
α (1 − |λ|)β 2 |∇x f |2 + dx dλ. 1 − |λ| |∇ f | λ β
C1, 1
Proof. It follows from Theorem 1.6. We have that 1 − |λ| > 1+m , that is −1 > (1 − |λ|)− 1+m βm β and consequently (1−|λ|) > (1 − |λ|) 1+m , β > 0. The result then follows from Theorem 1.6 by β choosing σ :=
βm 1+m
βm there; in particular Pcr (N, m, α, β, k) = Qcr (N, α, 1+m , k).
2
Acknowledgments The authors are grateful to Prof. E. Lanconelli for interesting conversations and suggestions which improved the presentation. The authors would like to thank E. Cinti for making her thesis available to them after a first draft of the present paper has been written. L.M. would like to thank Prof. X. Cabré, who has drawn her attention to the problem treated in Sections 2–3, and for several discussions. L.M. acknowledges the support of University of Crete and FORTH during her visit to Greece. A.T. acknowledges the support of Universities of Rome I and Bologna as well as the GNAMPA project “Liouville theorems in Riemannian and sub-Riemannian settings” during his visits in Italy. References [1] M.S. Baouendi, Sur une classe d’opérateurs elliptiques dégénérés, Bull Soc. Math. France 95 (1967) 45–87. [2] M.T. Barlow, R.F. Bass, C. Gui, The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math. 8 (2000) 1007–1038. [3] W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. 129 (2001) 1233–1246. [4] J.M. Bony, Principe de maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969) 277–304.
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[5] X. Cabré, L. Moschini, Liouville type theorems for anisotropic degenerate elliptic problems, in preparation. [6] L. Caffarelli, A. Cordoba, Phase transitions: Uniform regularity of the intermediate layers, J. Reine Angew. Math. 593 (2006) 209–235. [7] E. Cinti, Tesi di laurea, Universitá degli Studi di Bologna, 2006. [8] E. Cinti, in preparation. [9] S. Filippas, V.G. Maz’ya, A. Tertikas, A sharp Hardy Sobolev inequality, C. R. Math. Acad. Sci. Paris 339 (2004) 483–486. [10] S. Filippas, V.G. Maz’ya, A. Tertikas, Critical Hardy–Sobolev inequalities, J. Math. Pures Appl. (9) 87 (1) (2007) 37–56. [11] S. Filippas, L. Moschini, A. Tertikas, Sharp two-sided heat kernel estimates for critical Schrödinger operator on bounded domains, Comm. Math. Phys. 273 (2007) 237–281. [12] B. Franchi, E. Lanconelli, Une métrique associée á une classe d’opérateurs elliptiques dégénérés, in: Proceedings of the Meeting, Linear Partial and Pseudo-Differential Operators, Torino, 1982, Rend. Sem. Mat. Univ. Politec., Torino (1984) 105–114, 1983, special issue. [13] B. Franchi, E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984) 1237–1264. [14] B. Franchi, C.E. Gutiérrez, R.L. Wheeden, Weighted Sobolev–Poincaré inequalities for Grushin-type operators, Comm. Partial Differential Equations 19 (1994) 523–604. [15] E. de Giorgi, Convergence problems for functionals and operators, in: Proc. Internat. Meeting on Recent Methods in nonlinear Analysis, Rome 1978, Pitagora, Bologna, 1979. [16] A. Kufner, Weighted Sobolev Spaces, Teubner-Texte Math., vol. 31, Teubner, Leipzig, 1981. [17] V.G. Maz’ya, Sobolev Spaces, Springer, Berlin, 1985. [18] L. Modica, S. Mortola, Some entire solutions in the plane of nonlinear Poisson equations, Boll. Unione Mat. Ital. Sez. B 17 (1980) 614–622. [19] R. Monti, Sobolev inequalities for weighted gradients, Comm. Partial Differential Equations 31 (2006) 1479–1504.
Journal of Functional Analysis 255 (2008) 120–132 www.elsevier.com/locate/jfa
A dynamical systems approach to the Kadison–Singer problem ✩ Vern I. Paulsen Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA Received 25 September 2007; accepted 7 April 2008 Available online 12 May 2008 Communicated by N. Kalton
Abstract In these notes we develop a link between the Kadison–Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends uniquely to the von Neumann algebra of the group. We prove that if any state arising in the Kadison–Singer problem has a unique extension, then the injective envelope of a C*-crossed product algebra associated with the state necessarily contains the full von Neumann algebra of the group. We prove that this latter property holds for states arising from rare ultrafilters and δ-stable ultrafilters, independent, of the group action and also for states corresponding to non-recurrent points in the corona of the group. © 2008 Elsevier Inc. All rights reserved. Keywords: Kadison–Singer problem; Dynamical system; Ultrafilter; Non-recurrent point
1. Introduction Let H be a separable Hilbert space, and let D ⊂ B(H) be a discrete MASA. The Kadison– Singer problem [10] asks whether or not every pure state on D has a unique extension to a state on B(H). Without loss of generality, one can assume that the Hilbert space is 2 (N), where N ✩ This research was supported in part by NSF grant DMS-0600191. Portions of this research were conducted at the American Institute of Mathematics. E-mail address:
[email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.006
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denotes the natural numbers, with the canonical orthonormal basis, {en }n∈N and that the MASA is the subalgebra of operators that are diagonal with respect to this basis. However, since any two discrete MASA’s on any two separable infinite-dimensional Hilbert spaces are conjugate, one may equally well assume that the Hilbert space is 2 (G), where G is a countable, discrete group, with canonical orthonormal basis {eg }g∈G and that the MASA is the set of operators that are diagonal with respect to this basis. Thus, D = {Mf : f ∈ ∞ (G)}, where Mf denotes the operator of multiplication by the function f . We let 1 denote the identity of G and let Ug = λ(g) denote the unitary operators given by the left regular representation, so that Ug eh = egh . The reason that we prefer this slight change of perspective, is that we are interested in incorporating properties of the group action into results on the Kadison–Singer problem. Indeed, identifying the MASA, D ≡ ∞ (G) ≡ C(βG), where βG denotes the Stone–Cech compactification of G, then pure states on D correspond to the homomorphisms induced by evaluations at points in βG. Moreover, we have that for each g ∈ G, the map g1 → gg1 extends uniquely to a homeomorphism, hg : βG → βG and this family of homeomorphisms satisfy, hg1 ◦ hg2 = hg1 g2 , that is, they induce an action of G on the space βG and we set hg (ω) = g · ω. In this paper we study the extent to which uniqueness or non-uniqueness of extensions of the pure state induced by a point ω ∈ βG is related to the dynamical properties of the point. In particular, we will be interested in the orbit G · ω = {g · ω: g ∈ G} of the point. We will use the fact that the map g → g · ω is one-to-one, which is a consequence of a theorem of Veech [16]. To see why this is so, note that this map being one-to-one is equivalent to requiring that the stabilizer subgroup, Gω = {g ∈ G: g · ω = ω} consist of the identity. When this is the case, we shall say that ω has trivial stabilizer. If every point in βG has trivial stabilizer, then G acts freely (and continuously) on βG. Conversely, if there exists a compact, Hausdorff space X equipped with a continuous G-action that is free, then choosing any x ∈ X, the map g → g · x extends to a continuous G-equivariant map h : βG → X and for any point ω ∈ βG, the stabilizer of ω is contained in the stabilizer of h(ω) and hence is trivial. Thus, every point in βG has trivial stabilizer if and only if G can act freely and continuously on some compact Hausdorff space. Veech’s theorem [16] shows that, in fact, every locally compact group acts freely on a compact, Hausdorff space. Thus, every point in βG has trivial stabilizer. We begin by examining properties of any state extension of the pure state given by evaluation at a point ω. To this end, given Mf ∈ D, we let f ∈ C(βG) denote the corresponding continuous function on βG and let sω : D → C denote the pure state given by evaluation at ω, that is, sω (f ) = f (ω). Let s : B(2 (G)) → C be any state extension of sω and let π : B(2 (G)) → B(Hs ) and v1 ∈ Hs be the GNS representation of s, so that s(X) = π(X)v1 , v1 . We set vg = π(Ug )v1 , g ∈ G, let Ls ⊆ Hs denote the closed linear span of the vg ’s and let φs : B(2 (G)) → B(Ls ) denote the completely positive map given by φs (X) = PLs π(X)|Ls . Note that for any Mf ∈ D we have that π(Mf )vg , vg = π Ug−1 Mf Ug v1 , v1 = f (g · ω).
Hence, these vectors are reducing for π(D) and orthonormal. Hence, they are an orthonormal basis for Ls and the map W eg = vg is a Hilbert space isomorphism between Ls and 2 (G). Setting ψs (X) = W ∗ φs (X)W we obtain a completely positive map on B(2 (G)).
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Also, we have that ψs (X)eh , eg = π(X)vh , vg = π(Ug −1 XUh )v1 , v1 = s(Ug −1 XUh ). This shows that the correspondence s → ψs is one-to-one. In particular, we have that ψs (Ug ) = Ug and that for any Mf ∈ D, ψs (Mf ) = πω (Mf ), where πω : D → D is the *-homomorphism given by πω (Mf )(g) = f (g · ω). From these two equations, we see that the restriction of ψs to the C*-algebra generated by D and the set {Ug : g ∈ G}, is a *-homomorphism, which we will denote by πω × λ, satisfying, πω × λ
n
Mfn Ugn = ψs Mfn Ugn = πω (Mfn )Ugn , n
n
for every finite, or norm convergent sum. This algebra is, in fact, *-isomorphic to the reduced crossed-product C*-algebra, D ×r G as defined in say [14], although we do not need that fact here, we shall adopt that notation for the algebra. By Choi’s theory of multiplicative domains [7] of completely positive maps, we have that for any A1 , A2 ∈ D ×r G and X ∈ B(2 (G)), we have that ψs (A1 XA2 ) = πω × λ(A1 )ψs (X)πω × λ(A2 ). We now characterize the range of this correspondence. Theorem 1. Let ω ∈ βG. If ψ : B(2 (G)) → B(2 (G)) is any completely positive extension of πω × λ and we set s(X) = ψ(X)e1 , e1 , then s is a state extension of sω and ψ = ψs . Consequently, the map, s → ψs is a one-to-one, onto affine map between the convex set of state extensions of sω and the convex set of completely positive extensions of πω × λ. Proof. It is clear that s is a state extension of sω . Now given any X ∈ B(2 (G)) and g, h ∈ G, we have that ψ(X)eh , eg = ψ(Ug −1 XUh )e1 , e1 = s(Ug −1 XUh ) = ψs (X)eh , eg and, thus, ψ(X) = ψs (X). Hence, the map ψ → s defines an inverse to the map s → ψs , so that these correspondences are one-to-one and onto. Finally, it is clear that both of these correspondences preserve convex combinations. 2 Corollary 2. Let ω ∈ βG. Then the following are equivalent: • sω : D → C has a unique extension to a state on B(2 (G)), • πω × λ : D ×r G → B(2 (G)) has a unique extension to a completely positive map on B(2 (G)). There is, of course, always one distinguished state extension of sω . If we let E0 : B(2 (Z)) → D be the canonical projection onto the diagonal, then the regular extension of sω is given by X → sω E0 (X) .
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Corresponding to this regular extension is a unique completely positive map, ψω : B(2 (Z)) → B(2 (Z)) which we also call the regular completely positive extension. We wish to describe this map in some detail. Every X ∈ B(2 (G)) has a formal series, X ∼ g∈G Mfg Ug where Mfg = E0 (XUg−1 ). To compute the (gi , gj )th entry of ψω (X), we note that XUgj e1 , e1 = sω E0 Ug−1 XUgj . ψω (X)gi ,gj = ψω (X)egj , egi = ψω Ug−1 i i But, we have that XUgj ∼ Ug−1 i
Ug−1 Mfg Ugi (Ug −1 ggj ), i i
g∈G
and hence, XUgj = Ug−1 Mf E0 Ug−1 i i
gi gj−1
Ug i .
Thus, ψω (X)gi ,gj = fgi g −1 (gi · ω). j
These observations lead to the following result. Theorem 3. Let ω ∈ βG and let ψω be the regular completely positive extension corresponding to sω . If X ∼ g∈G Mfg Ug , then ψω (X) ∼ g∈G πω (Mfg )Ug . Moreover, sω has a unique extension to a state on B(2 (G)) if and only if ψω is the unique completely positive map on B(2 (G)) extending πω × λ : D ×r G → B(2 (G)). Note that VN(G) is always contained in the range of ψω , since X ∈ VN(G) implies that there are scalars, ag ∈ C such that X ∼ g ag Ug and hence, ψω (X) ∼ g πω (ag I )Ug = g ag Ug . It may seem paradoxical to attempt to make progress on the Kadison–Singer problem by replacing statements about uniqueness of the extension of a state to statements about uniqueness of the extension of a completely positive map, but something is gained by making the domain and range of the map the same space. We make this precise in the following results. Definition 4. Let ω ∈ βG, then we define the uniqueness set for ω to be the set,
U(ω) = X ∈ B 2 (G) : s(X) = sω E0 (X) ∀s , where s denotes an arbitrary state extension of sω . We also define the uniqueness set for πω to be the set,
U(πω × λ) = X ∈ B 2 (G) : ψ(X) = ψω (X) ∀ψ , where ψ denotes an arbitrary completely positive extension of πω × λ. Proposition 5. Let ω ∈ βG, then U(πω × λ) =
g,h∈G
Ug U(ω)Uh .
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Proof. Since every completely positive extension of πω × λ is of the form ψs for some state extension of sω , we have that X ∈ U(πω ), if and only if ψs (X) = ψω (X) for every extension s. But this is equivalent to s(Ug −1 XUh ) = ψs (X)eh , eg = ψω (X)eh , eg = s(E0 (Ug −1 XUh )) for every g, h ∈ G, which is equivalent to Ug −1 XUh ∈ U(ω) for every g, h ∈ G, and the result follows. 2 Of course, if sω has a unique extension, then the sets above are all equal to B(2 (G)), but if some sω fails to have a unique extension, then it should be easier to show that U(πω × λ) = B(2 (G)), than to show that U(ω) = B(2 (G)). We now turn our attention to some results that relate uniqueness of extension to injective envelopes. Definition 6. We let Aω = πω × λ(D ×r G) and we denote the von Neumann algebra generated by {Ug : g ∈ G} by VN(G). Recall that a map φ is called a B-bimodule map for an algebra B if φ(b1 xb2 ) = b1 φ(x)b2 , for every b1 , b2 ∈ B. Proposition 7. Let ω ∈ βG. If sω has a unique extension, then every completely positive map, φ : B(2 (G)) → B(2 (G)) that fixes Aω elementwise, also fixes the range of ψω elementwise and is a VN(G)-bimodule map. Proof. If φ does not fix the range, then φ ◦ ψω would be another completely positive map extending πω . Thus, φ must fix the range of ψω elementwise. But VN(G) is a subset of the range of ψω and so must be fixed. By Choi’s [7] theory of multiplicative domains this implies that φ is a VN(G)-bimodule map. 2 Remark 8. For many countable groups G, even when G = Z, there exist completely positive maps, φ : B(2 (G)) → B(2 (G)) that fix C ∗ (G) elementwise and hence are C ∗ (G)-bimodule maps, but whose range does not contain VN(G) and that are not VN(G)-bimodule maps. Examples of such completely positive maps are constructed in [6]. However, any completely positive map φ : B(2 (G)) → B(2 (G)) that fixes D ×r G elementwise is necessarily the identity map on all of B(2 (G)), since D ×r G contains the compact operators. Thus, since C ∗ (G) ⊆ Aω ⊆ D ×r G, whether or not sω has a unique extension should be related to how small πω (D) can be made. This last result can be interpreted in terms of injective envelopes. Recall, that given a unital C*-subalgebra A ⊆ B(H) and a completely positive idempotent map φ : B(H) → B(H) that fixes A elementwise and is minimal among all such maps, then the range of φ, R(φ) is completely isometrically isomorphic to the injective envelope of A, I (A). Such maps are called minimal A-projections. Thus, in particular, the ranges of any two minimal A-projections are completely isometrically isomorphic via a map that fixes A elementwise. See [11] for further details. For this reason the collection of subspaces that are ranges of minimal A-projections are called the copies of the injective envelope of A. If we let F(A) denote the set of elements of B(H) that are elementwise fixed by every completely positive map that fixes A, then it is clear that F(A) is contained in every copy of I (A). In [12] it is shown that F(A) is, in fact, the intersection of all copies of I (A), but that is not necessary for the following result.
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Corollary 9. Let ω ∈ βG. If sω has a unique extension, then VN(G) ⊆ R(ψω ) ⊆ F(Aω ). Thus, if the Kadison–Singer problem has an affirmative answer, then necessarily VN(G) ⊆ I (Aω ), and this inclusion is as a subalgebra for every ω. However, we know that, generally, VN(G) is not a subalgebra of I (C ∗ (G)). Problem 10. Let Dg ∈ πω (D) be chosen such that Y ∼ g∈G Dg Ug is a bounded operator. Is Y necessarily in the range of ψω ? Can conditions on the orbit of ω be given that guarantee that this is the case? 2. Dynamics and algebra In this section we begin to look at how dynamical properties of points and their behavior with respect to a natural semigroup structure on βG can be related to uniqueness of extension. For more on this structure see [9], but we recall a few basic definitions. Given ω ∈ βG we let ρω : βG → βG be the unique continuous function satisfying, ρω (g) = g · ω, for all g ∈ G so that ρω (βG) is the closure of the orbit of ω. Since ρω ◦ hg (g1 ) = ρω (gg1 ) = gg1 · ω = hg ◦ ρω (g1 ), we have that ρω ◦ hg = hg ◦ ρω , i.e., the map ρω is equivariant for the action of G on βG. This map also defines a semigroup structure on βG by setting, α · ω ≡ ρω (α). We caution that in spite of the notation, this operation is not abelian even when the underlying group is abelian (except for finite groups). However, it is associative and continuous in the left variable and so it gives βG the structure of a compact right topological semigroup. We refer the reader to [9] for these and other basic facts about this algebraic structure on βG. One fact that we shall use is that the corona, G∗ = βG \ G is a two-sided ideal in βG. We now wish to relate dynamical properties of a point ω, to the structure of Aω and to the semigroup properties of ω. Proposition 11. Let ω, α ∈ βG. Then ρω ◦ ρα = ρα·ω , πα ◦ πω = πα·ω and ψα ◦ ψω = ψα·ω . Proof. We have that ρω ◦ ρα (g) = ρω (hg (α)) = hg ◦ ρω (α) = hg (α · ω) = ρα·ω (g), and hence, ρω ◦ ρα = ρα·ω . The second equality comes from the fact that after identifying D = C(βG), then πω (f ) = f ◦ ρω , and hence, πα ◦ πω (f ) = f ◦ ρω ◦ ρα = f ◦ ρα·ω = πα·ω (f ). The proof of the third identity is similar. 2 Definition 12. A point ω ∈ βG is called idempotent if ω · ω = ω. Non-zero idempotent points are known to exist [9], since the corona is a compact right continuous semigroup. As we shall see below they play a special role in the Kadison–Singer problem. Proposition 13. Let ω ∈ βG. Then the following are equivalent: • • • • •
ω is idempotent, ρω ◦ ρω = ρω , πω : D → D is idempotent, πω × λ : D ×r G → D ×r G is idempotent, ψω is an idempotent completely positive map.
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Theorem 14. Let ω ∈ βG be an idempotent. If sω has a unique state extension, then R(ψω ) is completely isometrically isomorphic to the injective envelope of Aω . Moreover, R(ψω ) is the unique copy of the injective envelope inside B(2 (G)) and ψω is the unique projection onto it. Thus, the identity map on Aω extends uniquely to an embedding of I (Aω ) into B(2 (G)). Proof. We have seen earlier that for any ω ∈ βG the range of ψω is contained in any copy of the injective envelope. Thus, when ω is idempotent, since ψω is already a completely positive projection, its range must be a minimal completely positive projection and its range must be the unique copy of the injective envelope. 2 Conjecture 15. We conjecture that if ω is idempotent, then sω possesses non-unique extensions. That is, the Kadison–Singer conjecture is false and idempotent points provide counterexamples. In fact, we believe that idempotent points fail to satisfy the condition of Corollary 9, VN(G) ⊆ F(Aω ). The following result lends some credence to the above conjecture, at least for minimal idempotents. An idempotent ω is minimal if it is minimal in any of several different orders ([9, Definition 1.37] and [4]) and this is shown to be equivalent to the left ideal generated by ω, i.e., {α · ω: α ∈ βG} being a minimal left ideal [9, Theorem 2.9]. Moreover, by [9, Theorem 19.23c], a minimal idempotent is uniformly recurrent (see also [5] where this is proved for the case of N). Recall that ω uniformly recurrent means [9, Definition 19.1] that for every neighborhood U of ω we have that the set S = {g ∈ G: g ·ω ∈ U } is syndetic, i.e., there exists a finite set g1 , . . . , gm ∈ G such that g1 · S ∪ g2 · S ∪ · · · ∪ gm · S = G. Theorem 16. Let G be a countable, abelian, discrete group and let X ∈ VN(G). If there exists a minimal idempotent ω such that X ∈ U(ω), then X ∈ U(α) for every α ∈ βG. Thus, if there exists any state sα which fails to have unique extension for some X ∈ VN(G), then every minimal idempotent fails to have unique extension for that X. Proof. Since U(ω) is an operator system, X ∈ U(ω) if and only if Re(X) = (X + X ∗ )/2 and Im(X) = (X − X ∗ )/(2i) are both in U(ω). Thus, it will be sufficient to assume that X = X ∗ . Moreover, since I ∈ U(ω), it is sufficient to assume also that E(X) = 0. Now by Anderson’s paving results [1] and [2] (see also [13, Theorem 2.7]), X ∈ U(ω) if and only if for each > 0, there exists A in the ultrafilter corresponding to ω such that − PA PA XPA + PA where PA ∈ D is the diagonal projection PA = diag(ag ) with
1, g ∈ A, ag = 0, g ∈ / A. Let U ⊆ βG be the clopen neighborhood of ω satisfying A = U ∩ G, so that under the identification of D with C(βG) the projection PA is identified with the characteristic function of U , χU . We have that S = {g ∈ G: g · ω ∈ U } is syndetic, so let g1 , . . . , gm be as in the definition of syndetic. Note that ψω (PA ) = πω (PA ) = πω (χU ) = diag(bg ) where
1, g · ω ∈ U, bg = 0, g · ω ∈ / U,
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that is ψω (PA ) = PS . Also, since X ∈ VN(G) we have that ψω (PA XPA ) = πω (PA )ψω (X)πω (PA ) = PS XPS . Thus, applying ψω to the above inequality, we have that − PS PS XPS + PS . Next notice that conjugating the first inequality by λ(g), we have that λ(g)PA λ(g −1 ) = PgA and λ(g)PA XPA λ g −1 = λ(g)PA λ g −1 λ(g)Xλ g −1 λ(g)PA λ g −1 = PgA XPgA . Thus, − PgA PgA XPgA + PgA . Applying the map ψω to this inequality and observing that ψω (PgA ) = PgS , we have that − Pgi S Pgi S XPgi S + Pgi S , for i = 1, . . . , m. Since, G = g1 S ∪ · · · ∪ gm S, and > 0 was arbitrary, this shows that X is pavable in Anderson’s sense and so, applying Anderson’s [3] paving results, we have that every pure state on D has a unique extension to X. 2 Conjecture 17. The same result holds for non-abelian groups. Conjecture 18. Assuming that a minimal idempotent has a unique extension, should imply that every point has a unique extension. That is, if we fix any minimal idempotent ω, then the Kadison– Singer conjecture is true if and only if sω has a unique state extension. If ω is any point in the corona, then ψω annihilates any operator all of whose “diagonals” are c0 . Thus, when ω is idempotent, R(ψω ) can contain no operators with any c0 “diagonals” and so in particular, no compact operator. In this sense, πω (D) is a “small” subset of ∞ (G), which is one of the reasons that we conjecture that these points are potential counterexamples. Thus, in this case R(ψω ) ∩ K = (0), where K denotes the compacts. Since any copy of the injective envelope of Aω is contained in R(ψω ) it also follows that for ω idempotent, I (Aω ) ∩ K = (0) and hence cannot be all of B(2 (G)). We now consider the opposite case, points for which πω (D) is a “large” subset of ∞ (D). We believe that these points are good candidates for having unique extensions. The following result characterizes the points for which K ∩ R(ψω ) = (0), at least for many groups and, in particular, for these points, we shall see that the injective envelope of Aω is all of B(2 (G)). Recall that given a dynamical system, a point ω is non-recurrent if there is an open neighborhood U of the point such that g · ω ∈ / U for all g = 1. Also, given a semigroup, an element ω is right cancellative, if α · ω = β · ω implies that α = β. Portions of the following result can also be deduced from [9, Theorem 8.11]. Theorem 19. Let ω ∈ βG and assume that G is torsion free. Then the following are equivalent: (i) (ii) (iii) (iv)
ω is non-recurrent, K ∩ Aω = (0), K ⊆ Aω , πω : D → D is onto,
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(v) Aω = D ×r G, (vi) ρω is one-to-one, (vii) ω is right cancellative. Proof. (i) implies (iii). If ω is non-recurrent there is a clopen neighborhood U of ω containing no other g · ω, g = 1. Hence, πω (χU ) is the rank one projection onto the span of e1 . = πω (Ugi χU Ug−1 ) ∈ Aω Since Ug ∈ Aω , we have the matrix units, Egi ,gj = Ugi πω (χU )Ug−1 j j for all gi , gj ∈ G and hence K ⊆ Aω (ii) implies (i). Suppose Aω contains a nonzero compact K = πω (Mfg )Ug (formal sum). Then all πω (Mfg ) are compact. So there is a non-zero πω (Mf ) ∈ K ∩ Aω . Applying a sequence pn of polynomials to this, tending to the characteristic functions of some eigenspace you find a finite rank “diagonal” projection Q = χS in Aω with S a finite set. Choose such a Q with S = {g1 , . . . , gn } of minimal non-zero cardinality. Conjugating Q by Ug , we obtain another such projection in Aω corresponding to the set g · S and so we may assume that 1 ∈ S. Also, the product of two such projections is also in Aω and is the finite rank projection corresponding to S ∩ gS. But since S is of minimal non-zero cardinality, either S ∩ g · S is empty or S ∩ g · S = S. Hence, for each g ∈ S, g −1 · S = S, and so S is a finite subgroup, but then any non-zero element of S would be a torsion element. Thus, S = {1}, and we have that E1,1 ∈ Aω . Hence, there exists f ∈ C(βG), such that E1,1 = πω (Mf ). Now f (ω) = 1 and f (g · ω) = 0 for all g = 1. Thus, U = {α: |f (α)| > 1/2} is an open set containing ω but no other g · ω and so ω is non-recurrent. Clearly, (iii) implies (ii) and so (i), (ii), and (iii) are equivalent. Also, it is clear that (v) implies (iv) implies (ii). Moreover, since πω is given by composition with ρω , we have that (iv) and (vi) are equivalent. Since, α · ω = ρω (α), it is also clear that ω is right cancellative is equivalent to ρω being one-to-one. Thus, (vi) and (vii) are equivalent. (i) implies (iv). Since ω is non-recurrent there is a neighborhood U1 of ω that contains no other point on its orbit. Hence, g · U1 is a neighborhood of g · ω containing no other point on the orbit. Thus, non-recurrent is equivalent to the set {g · ω} being discrete. Now using the fact that βG is a compact, Hausdorff space and hence normal, and that the set of points is countable, one can choose neighborhoods, Vg of g · ω such that Vg ∩ Vh is empty for g = h, i.e., the points are what is called strongly discrete. To recall the construction, first enumerate the points, ωi = gi · ω, then using normality, choose for each i disjoint open sets Ui , Vi such that ωi ∈ Ui and the closure of {ωj : j = i} is contained in Vi . Then set W1 = U1 , Wi = Ui ∩ V1 ∩ · · · ∩ Vi−1 , i 2. Thus, using the fact that βG is extremally disconnected, we may choose disjoint clopen sets Ug , g ∈ G, with g · ω ∈ Ug . Thus, if we let Ag = Ug ∩ G, then πω (χAg ) is the rank one ∞ ∞ projection onto the span of eg . Given any f1 ∈ (G) define f2 ∈ (G), by f2 (h) = f1 (g) if / g∈G Ag set f2 (h) = λ where λ ∈ C is any number. Then and only if h ∈ Ag , and when h ∈ f2 ∈ ∞ (G) and if fˆ2 denotes the continuous extension of f2 to βG, then fˆ2 (g · ω) = f1 (g). Thus, πω (f2 ) = f1 and so (iv) holds. (iv) implies (v). Given any finite sum, B = nj=1 Mfj Ugj , we may pick continuous n functions hj , such that πω ( j =1 Mhj Ugj ) = B. Thus, the range of the *-homomorphism, πω : D ×r G → D ×r G is dense in D ×r G and hence is onto. 2 Corollary 20. Let G be a countable torsion free group. If ω1 , ω2 ∈ βG are non-recurrent, then ω1 · ω2 is non-recurrent.
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Corollary 21. Let G be a countable torsion free group and let ω ∈ βG. Then I (Aω ) = B(2 (G)) if and only if ω is non-recurrent. Proof. If ω is non-recurrent, then Aω contains all the compacts and hence any completely positive map that fixes Aω fixes all the compacts and hence is the identity map. To see this note that if it fixes every diagonal matrix unit, then it is a Schur product map, which fixes every matrix unit and so is the identity map. Conversely, if ω is not non-recurrent, then Aω contains no non-zero compacts and hence the quotient map to the Calkin algebra is an isometry on Aω . But the injective envelope is an essential extension of Aω and, hence, any embedding of I (Aω ) into B(2 (G)), composed with the quotient map must also be an isometry on I (Aω )and hence, I (Aω ) = B(2 (G)). 2 The above result shows that non-recurrent points all satisfy the condition of Corollary 9 that is necessary for sω to have a unique extension. This lends credence to the following conjectures. Conjecture 22. We conjecture that if ω ∈ βG is non-recurrent, then the state sω extends uniquely to B(2 (G)). Conjecture 23. Let G be a countable torsion free group. If ω ∈ βG is non-recurrent, then we conjecture that R(ψω ) = B(2 (G)). Assuming the last conjecture, one can prove that if ω is non-recurrent and sα has a unique extension, then sα·ω has a unique extension. So these conjectures might shed some light on the algebraic properties of points with unique extensions. 3. Dynamical properties of ultrafilters We now examine the dynamical properties of various classes of ultrafilters that have been studied in relation to the Kadison–Singer problem. An ultrafilter β on a countable set N is called selective [9] if given any partition of N = Pi into subsets, either Pi is in the ultrafilter for some i or there exists a set B in the ultrafilter, such that for every i, B ∩ Pi has cardinality at most 1. An ultrafilter is called rare [8] if for each partition N = Pi into finite subsets, there exists a set B in the ultrafilter such that for every i, B ∩ Pi has cardinality at most 1. An ultrafilter is called δ-stable [8] if for each partition N = Pi , into sets of arbitrary sizes, then either one of the Pi ’s is in the ultrafilter or there exists a set B in the ultrafilter such that for every i, B ∩ Pi is finite. Note that an ultrafilter is selective if and only if it is rare and δ-stable. Finally, a point in a topological space is called a P-point [17] if every Gδ that contains the point contains an open neighborhood of the point. Note that if X is a compact, Hausdorff space, x ∈ X is a P-point and f ∈ C(X), then {y ∈ X: f (y) = f (x)} is a Gδ set and hence contains an open neighborhood of x! Thus, this definition of P-point is antithetical to the concept of p-point that appears in function theory. Choquet [8, Proposition 1] proves that for any discrete set N , ω is a δ-stable if and only if ω is a P-point in the corona βN \ N . For this reason the term δ-stable has fallen into disuse and such ultrafilters are, generally, called P-points without reference to the topological space.
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Finally, we recall a stronger notion than non-recurrent. A point ω in a dynamical system is wandering, if there exists a neighborhood, U of ω such that g · U ∩ h · U is empty for any g = h in the group. In an earlier version of this paper, we proved that rare ultrafilters are wandering points in N and Z and conjectured that they were wandering in every group. Since then, Ken Davidson has verified this conjecture. We present our proof of the case of N and Davidson’s proof for general groups below. Proposition 24. Let ω ∈ βN be rare, then ω is wandering in the corona N∗ . Proof. Let ω be a rare ultrafilter on N. Consider the following partition of N into finite sets: N = {1} ∪ {2} ∪ {3, 4} ∪ {5, 6} ∪ {7, 8, 9} ∪ {10, 11, 12} ∪ {13, 14, 15, 16} . . . . Let σ = {1} ∪ {3, 4} ∪ {7, 8, 9} ∪ {13, 14, 15, 16} . . . (half the sets in the partition) and let σj denote the j th subset in this list. By general properties of ultrafilters, either σ or its complement is in ω. We assume, without loss of generality, that σ ∈ ω. Since ω is rare, there exists B ∈ ω such that B ∩ σj has at most one element for all j . Define γ = σ ∩ B ∈ ω, then |γ ∩ (γ + n)| 2n ∀n ∈ N which can be seen by looking at how the σj ’s in σ are spread out. Thus, if U denotes the clopen set in βN corresponding to γ , then (n + U ) ∩ (m + U ) ⊂ N and is finite. Thus, the relatively open set U ∩ N∗ is wandering in N∗ . 2 We now present Davidson’s proof for general groups. Theorem 25. Let G be a countable, discrete group and let ω ∈ βG be a rare ultrafilter, then ω is wandering in the corona G∗ , and, hence Aω = D ×r G. Proof. Let e denote the ∞identity of G and choose finite subsets, {e} = G0 ⊆ G1 ⊆ · · · with Gn = G−1 and G = n n=0 Gn . Set Pn = Gn · · · G1 · G0 and let An = Pn \Pn−1 , n 1, and A0 = {e}. We have that each An is finite, An ∩ Am is empty for n = m, and G = ∞ n=0 An . If g ∈ Gk and n > k, we claim that gAn ⊆ An−1 ∪ An ∪ An+1 = Pn+1 /Pn−2 . To see this note that gPn ⊆ Pn+1 . If p ∈ Pn and gp ∈ Pn−2 , then p = g −1 (gp) ∈ Pn−1 and, hence p ∈ / An . Thus, if p ∈ An , then gp ∈ / Pn−2 , and the claim follows. Now since ω is rare, we maypick a set U in the ultrafilter ω, such that U ∩ An has at most one element for all n. Let E = An , n even and O = An , n odd. Then E ∩ O is empty and E ∪ O = G. Hence either E is in the ultrafilter, or O is in the ultrafilter.
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If E is in the ultrafilter ω, let V = U ∩ E which is in the ultrafilter ω. If g ∈ Gk , g = e we claim that the cardinality of gV ∩ V is at most k/2. Assuming the claim, we see that the clopen neighborhood V of ω corresponding to V , has the property that, W = V ∩ G∗ is an open neighborhood of ω with gW ∩ W empty for all g = e. To see the claim, write V = {a2m : a2m ∈ A2m , m 0}, so that x ∈ gV ∩ V if and only if x = ga2m = a2j . But if 2m > k, then ga2m ∈ A2m−1 ∪ A2m ∪ A2m+1 , and so j = m, forcing g = e. Thus, ga2m = a2j has no solutions for 2m > k when g = e. The proof for the case that O is in the ultrafilter ω, is identical. 2 Corollary 26. Let G be a countable discrete group. Assuming the continuum hypothesis, G∗ contains a dense set of wandering points. Proof. By [8], if we assume the continuum hypothesis, then the rare ultrafilters are dense in G∗ . 2 We do not know if it is necessary to assume the continuum hypothesis to conclude that the wandering points are dense in G∗ . Problem 27. Let ω be an ultrafilter on N, then for every countable, discrete group G and every labeling, G = {gn : n ∈ N}, we have that ω determines a point in G∗ . The above result shows that when ω is rare then the point obtained in this fashion is wandering for the action of G on G∗ . Conversely, if ω is an ultrafilter on N with this property, then must ω be rare? By the result of Reid [15], we know that the state given by any rare ultrafilter has a unique extension to B(2 (G)). This suggests the following conjecture: Conjecture 28. If ω ∈ G∗ is wandering for the G action on G∗ , then the state corresponding to evaluation at ω extends uniquely to a state on B(2 (G)). Proposition 29. Let G be a countable, discrete group. Then δ-stable ultrafilters are non-recurrent in βG. Proof. Let ω be a δ-stable ultrafilter, we have that {g · ω: g ∈ G} is a distinct set of points. For each g ∈ G that is not equal to the identity, the complement of {g · ω} is an open neighborhood of ω. The intersection of these sets is a Gδ containing ω and hence, applying the equivalence of δ-stable to P-point, contains an open neighborhood of ω. No point on the orbit of ω returns to this open neighborhood. 2 Combining the above result with Theorem 19, we see that δ-stable ultrafilters satisfy the condition of Corollary 9 that is necessary for sω to have a unique extension.
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References [1] J. Anderson, Extensions, restrictions and representations of states on C*-algebras, Trans. Amer. Math. Soc. 249 (1979) 303–329. [2] J. Anderson, Extreme points in sets of linear maps on B(H ), J. Funct. Anal. 31 (1979) 195–217. [3] J. Anderson, A conjecture concerning pure states of B(H ) and related theorems, in: Proceedings, 5th International Conference Operator Algebras, Timisoara and Herculane, Romania, Pitman, New York/London, 1984. [4] V. Bergelson, Minimal Idempotents and Ergodic Ramsey Theory, in: Topics in Dynamics and Ergodic Theory, in: London Math. Soc. Lecture Note Ser., vol. 310, Cambridge Univ. Press, Cambridge, 2003, pp. 8–39. [5] A. Blass, Ultrafilters: Where Topological Dynamics = Algebra = Combinatorics, preprint. [6] P. Casazza, D. Edidin, D. Kalra, V. Paulsen, Projections and the Kadison–Singer Problem, Matrices and Operators, in press. [7] M.D. Choi, A Schwarz inequality for positive linear maps on C*-algebras, Illinois J. Math. 18 (1974) 565–574. [8] G. Choquet, Deux classes remarquable d’ultrafilters sur N , Bull. Sci. Math. (2) 92 (1968) 143–153. [9] N. Hindman, D. Strauss, Algebra in the Stone–Cech Compactification, de Gruyter Exp. Math., vol. 27, de Gruyter, New York, 1998. [10] R.V. Kadison, I. Singer, Extensions of pure states, Amer. J. Math. 81 (1959) 547–564. [11] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge Univ. Press, Cambridge, 2002. [12] V.I. Paulsen, Injective envelopes and the weak expectation property, preprint. [13] V.I. Paulsen, M. Raghupathi, Some new equivalences of Anderson’s paving conjectures, preprint. [14] G. Pedersen, C*-algebras and Their Automorphism Groups, London Math. Soc. Monogr., vol. 14, Academic Press, London, 1979. [15] G.A. Reid, On the Calkin representations, Proc. London Math. Soc. (3) 23 (1971) 547–564. [16] W.A. Veech, Topological dynamics, Bull. Amer. Math. Soc. 83 (1977) 775–830. [17] R.C. Walker, The Stone–Cech Compactification, Ergebn. Math., vol. 83, Springer-Verlag, Berlin, 1974.
Journal of Functional Analysis 255 (2008) 133–141 www.elsevier.com/locate/jfa
Absolute continuity of Wasserstein geodesics in the Heisenberg group A. Figalli a,∗ , N. Juillet b,c,1 a Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, CNRS UMR 6621 Parc Valrose,
06108 Nice Cedex 02, France b Institut Fourier BP 74, UMR 5582, Université Grenoble 1, 38402 Saint-Martin-d’Hères Cedex, France c Institut für Angewandte Mathematik, Universität Bonn, Poppelsdorfer Allee 82, 53115 Bonn, Germany
Received 5 October 2007; accepted 11 March 2008 Available online 25 April 2008 Communicated by C. Villani
Abstract In this paper we answer to a question raised by Ambrosio and Rigot [L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261–301] proving that any interior point of a Wasserstein geodesic in the Heisenberg group is absolutely continuous if one of the endpoints is. Since our proof relies on the validity of the so-called Measure Contraction Property and on the fact that the optimal transport map exists and the Wasserstein geodesic is unique, the absolute continuity of Wasserstein geodesic also holds for Alexandrov spaces with curvature bounded from below. © 2008 Elsevier Inc. All rights reserved. Keywords: Optimal transport; Wasserstein geodesic; Absolute continuity; Heisenberg group; Alexandrov spaces
1. Introduction The optimal transportation problem is nowadays a very active research domain. After having being intensively studied in a Euclidean and a Riemannian setting by many authors, it has been recently investigated also in a sub-Riemannian framework. In particular, optimal transporta* Corresponding author.
E-mail addresses:
[email protected] (A. Figalli),
[email protected],
[email protected] (N. Juillet). 1 The author is partially supported by the ANR project “Cannon” (ANR-06-BLAN-0366). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.006
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tion in the Heisenberg group Hn has been first studied by Ambrosio and Rigot [1], where it is proved that the Monge problem can been solved, and a Brenier–McCann representation holds (see Proposition 1.1). The books by Villani [12,13] provide an excellent presentation of optimal mass transportation, while two general references about the Heisenberg group are the books by Montgomery [8] and the one by Capogna, Danielli, Pauls and Tyson [4]. The reader is referred to these books for a detailed presentation on these two active mathematical domains. The aim of this paper is to study the absolute continuity of Wasserstein geodesics, and answer to an open problem proposed by Ambrosio and Rigot [1, Section 7(c)]. Before stating our result in Theorem 1.2, we briefly introduce the concepts appearing in this paper. Let n be a non-negative integer. The Heisenberg group Hn can be written in the form R2n+1 Cn × R, and an element of Hn is written as (z; t) = (z1 , . . . , zn ; t). The group structure of Hn is given by n (z1 , . . . , zn ; t) · z1 , . . . , zn ; t = z1 + z1 , . . . , zn + zn ; t + t + 2 zk zk , k=1
where (z) denotes the imaginary part of a complex number. With this structure, Hn is a Lie group (with neutral element 0H = (0Cn ; 0)). As basis for the associated Lie algebra of left invariant vector fields we take as usual X1 , . . . , Xn , Y1 , . . . , Yn , T , where Xk = ∂xk + 2yk ∂t
for k = 1, . . . , n,
Yk = ∂yk − 2xk ∂t
for k = 1, . . . , n,
T = ∂t , with xk , yk ∈ R, xk + iyk = zk . The horizontal distribution (X1 , . . . , Xn , Y1 , . . . , Yn ) allows to define a sub-Riemannian distance, called Carnot–Carathéodory distance, that we denote by dC . This distance is defined as 1 n 2 dC (x, y) := inf
a (s) + b2 (s) ds k
γ
0
k
k=1
where the infimum is taken among all absolutely continuous curves γ from x to y such
that γ˙ (s) = nk=1 [ak (s)Xk (γ (s)) + bk (s)Yk (γ (s))] for a.e. s. We recall that the Carnot– Carathéodory distance restricts to Euclidean lines l of R2n+1 as follows. If for each point p ∈ l the direction of the line l at p is spanned by the horizontal distribution, then the restriction of dC to l equals up to a constant the Euclidean distance dEuc . If it is not, then there is a constant C and a real function F (s) = Cs 1/2 + o(s 1/2 ) as s ↓ 0 such that dC (·, p)|l = F (dEuc (·, p)). In √ 1/2 particular, the restriction of dC on lines directed by T is π dEuc . Inspired by the exponential map in Riemannian geometry, Ambrosio and Rigot introduced in [1] a special exponential map expH , which differs from the isomorphism between the Lie algebra and the Lie group: the numbers A + iB ∈ Cn and w ∈ [−π/2, π/2] parameterize the geodesics starting from 0H which can be written as s → expH (s(A + iB), sw).
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The Monge–Kantorovich problem (with a quadratic cost) is the following: given μ0 and μ1 two probability measures on a complete and separable metric space (X, d), minimize d(p, q)2 dπ(p, q) inf π
X×X
among all couplings π of μ0 and μ1 (that is, among all probability measures π on X × X whose is a minimum) marginals are μ0 and μ1 ). The square root of the above infimum (which indeed gives rise to a distance on the so-called Wasserstein space W2 (X) = {μ | X d 2 (x0 , x) dμ(x) < ∞ for some x0 ∈ X}. It turns out that if (X, d) is geodesic space, W2 (X) is also geodesic space. In this paper we will investigate the absolute continuity of measures staying in a geodesic path from an absolutely continuous measure to an other measure of Hn . Proposition 1.1 proved by Ambrosio and Rigot provides a nice representation of such geodesics using the notion of approximate differential, see [2, Definition 5.5.1]. We recall that f : R2n+1 → R has an approximate differential at x ∈ R2n+1 if there exists a function h : R2n+1 → R differentiable at x such that the set {f = h} has density 1 at x with respect to the Lebesgue measure. In this case the approximate derivatives of f at x are defined as ˜ (x) + iYf ˜ (x), Tf ˜ (x) := Xh(x) + iYh(x), Th(x) Xf = X1 h(x) + iY1 h(x), . . . , Xn h(x) + iYn h(x), Th(x) . It is not difficult to show that this definition makes sense. Proposition 1.1. (See [1, Theorem 5.1 and Remark 5.9].) Let μ0 and μ1 be two Borel probability measures on Hn . Assume that μ0 is absolutely continuous with respect to L2n+1 and that 2 dC (0H , x) dμ0 (x) + dC (0H , y)2 dμ1 (y) < +∞. Hn
Hn
Then there exists a unique optimal transport plan from μ0 to μ1 . Moreover, there exists a function ϕ which is approximately differentiable μ0 -a.e. such that the optimal transport plan is concentrated on the graph of ˜ ˜ ˜ − iYϕ(x), −Tϕ(x) . T (x) := x · expH −Xϕ(x) As a consequence of this theorem, it is observed in [1, Section 7(c)] that the family of measures ˜ ˜ ˜ − is Yϕ(x), −s Tϕ(x) , μs := Ts # μ with Ts (x) := x · expH −s Xϕ(x) with s ∈ [0, 1] is a constant-speed geodesic in W2 (Hn ) between μ0 and μ1 . Moreover, since ϕ is approximately differentiable μ0 -a.e., a simple variant of the proof of [1, Lemma 4.7] shows that for μ0 -a.e. x there exists a unique minimizing geodesic between x and T (x). In particular this implies that the geodesic in W2 (Hn ) between μ0 and μ1 is unique. In [1, Section 7(c)] the following open problem is raised: are all measures μs absolutely continuous for s ∈ [0, 1)?
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This question is motivated by the fact that the above property holds in the Euclidean and the Riemannian setting (see [13, Chapter 8]). The aim of this paper is to give a positive answer to the above question. Since the Heisenberg group is non-branching, by [13, Theorem 7.29] we know that for any time s ∈ [0, 1) the map Ts is μ0 -essentially injective (i.e. its restriction to a set with full μ0 measure is injective), and there exists an inverse transport map Ss uniquely defined up to μs negligible sets such that Ss ◦ Ts = Id μ0 -a.e. (and so Ss # μs = μ0 ). Our main result is the following. Theorem 1.2. Let (μs )s∈[0,1] be a geodesic of the Wasserstein space W2 (Hn ) and assume that μ0 has density ρ with respect to L2n+1 . Then for any s ∈ [0, 1) the measure μs is absolutely continuous with respect to the Lebesgue measure L2n+1 , and its density is bounded by 1 ρ ◦ Ts−1 |Ts (A) , (1 − s)2n+3
(1)
where Ts is the (μ0 -almost uniquely defined) optimal transport map from μ0 to μs , and A is any set of full μ0 -measure on which Ts is injective. We remark that the usual way to prove the absolute continuity of the intermediate measures is to use the Monge–Mather shortening principle (see [13, Chapter 8]). In Section 2 we will see that this approach cannot work for the Heisenberg group. We will also give an example of an optimal transport (μt )t∈[0,1] such that the measure at time 1/2 is concentrated on a set of Hausdorff dimension 1, while the sets of dimension 1 are negligible for μ0 and μ1 . These “bad” results show that strange phenomena can occur in the Heisenberg case, and this made less clear the answer to the absolute continuity question. However, in Section 3 we will see that the absolutely continuity is a consequence of the following two properties: the so-called MCP (Measure Contraction Property), which is indeed true in the Heisenberg group [5], and the fact that the optimal transport map exists and the Wasserstein geodesic is unique. Thanks to this fact, we observe that the same proof of the absolute continuity can be done in Alexandrov spaces with a lower curvature bound. Indeed, in this case the existence of an optimal transport map and the uniqueness of the Wasserstein geodesic were proved by Bertrand [3] under the assumption that μ0 is compactly supported and absolutely continuous with respect to the Hausdorff measure. Moreover, the MCP property holds, see [9, Lemma 2.3 and Proposition 2.8]. Therefore we obtain the following result (see also Remark 2.1). Theorem 1.3. Let (X, d) be an n-dimensional, complete Alexandrov space with curvature K. Let μ0 and μ1 be two compactly supported probability measures, with μ0 absolutely continuous with respect to the n-dimensional Hausdorff measure Hdn . Denote by μs the unique Wasserstein geodesic between μ0 and μ1 . Then, for any s ∈ [0, 1), the measure μs is absolutely continuous with respect to Hdn , and its density is bounded by d(x,T −1 (x)) n−1 sK(n−1) ( √s ) 1 s n−1 ρ ◦ Ts−1 (x)|Ts (A) . −1 d(x,T (x)) 1−s s s √ ) K(n−1) ((1 − s) s n−1
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Here Ts is the (μ0 -almost uniquely defined) optimal transport map from μ0 to μs , A is any set of full μ0 -measure on which Ts is injective, and the function sR (t) is given by √ ⎧ 1 √ sin( Rt) ⎪ ⎨ R sR (t) := t ⎪ ⎩ √ 1 sinh(√−Rt) −R
if R > 0, if R = 0, if R < 0.
2. Failure of the Monge–Mather shortening principle A good presentation of the Monge–Mather shortening principle can be found in [13, Chapter 8]. We give here a simplified picture of it in the particular case of geodesic spaces. Let (X, d) be a geodesic space, and denote by Hd the Hausdorff measure (here, we do not care about the dimension of the Hausdorff measure). The idea of the shortening lemma is the following. Fix a Borel set K, and take 4 points a, b, p, q ∈ K. Suppose that we want to transport a and b on p and q (this is an informal way to say that we want to transport the measure 12 (δa + δb ) onto 12 (δp + δq )), and assume that for the quadratic cost it is optimal to send a on p and b on q, that is d 2 (a, p) + d 2 (b, q) d 2 (a, q) + d 2 (b, p). Consider now two constant-speed geodesics α, β : [0, 1] → X from a to p and from b to q, respectively, and suppose that we can prove the following estimate: there is a constant C(K, s) (depending only on K and on the time s ∈ [0, 1]) such that C(K, s)d α(s), β(s) d(a, b). Then, given any Wasserstein geodesic (μs )s∈[0,1] such that μ0 (K) = μ1 (K) = 1, if μ0 is absolutely continuous with respect to Hd one can easily prove that also μs is absolutely continuous with respect to Hd . The Heisenberg group (Hn , dC ) with the Lebesgue measure can be put in the above framework. 2.1. Horizontal right translations as optimal transport The Lebesgue measure L2n+1 is the Haar measure of the Heisenberg group because the left translations of Hn are affine transformation with determinant 1. The (2n + 2)-dimensional Hausdorff measure is also a Haar measure because dC is left-invariant and 2n + 2 is the correct dimension. Then by uniqueness, both measures are equal up to a constant. We recall that right translations by an horizontal vector provide an optimal transport in the Heisenberg group. This can be proved projecting everything on Cn and comparing any transport with the optimal Euclidean transport (which indeed is a translation), see also [1, Example 5.7]. Let μ0 be the restriction of L2n+1 to (0, 1)2n+1 , and consider the horizontal vector u = (1, 0, . . . , 0; 0). With the notation of the introduction, Ts is given for any s ∈ [0, 1] by the map a → a · (s, 0, . . . , 0; 0). More precisely, writing a as (x + iy, z2 , . . . , zn ; t), we have Ts (a) = (x + s) + iy, z2 , . . . , zn ; t + 2sy .
(2)
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We observe that Ts is affine on R2n+1 with Jacobian determinant 1, so the measure μs = Ts # μ0 is absolutely continuous. However, as we will show, the shortening principle does not hold. Fix a ∈ (0, 1)2n+1 , and let aε := a + ε(i, 0, . . . , 0; −2x − 4s) = x + i(y + ε), z2 , . . . , zn ; t − 2εx − 4εs with ε small enough so that aε ∈ (0, 1)2n+1 . Then, using (2) twice, Ts (aε ) = aε · (s, . . . , 0; 0) = (x + s) + i(y + ε), z2 , . . . , zn ; (t − 2εx − 4εs) + 2s(y + ε) = (x + s) + i(y + ε), z2 , . . . , zn ; (t + 2sy) − 2ε(x + s) = Ts (a) · vε , where vε is the horizontal vector (iε, 0, . . . , 0; 0). Therefore dC (a, aε ) = dC 0H , a −1 · aε = dC 0H , (iε, 0, . . . , 0; −4εs) ∼ 2 π|ε|s as ε → 0, while dC Ts (a), Ts (aε ) = dC (0, vε ) = |ε|. Thus we see that the shortening principle cannot hold. Moreover from this example one can also see that there is no hope to find a decomposition of (0, 1)2n+1 into a family of countable Borel sets such that on each set the shortening principle holds, possibly with a different constant (if such weaker condition holds, one can still prove quite easily the absolute continuity of the interpolation). 2.2. An instructive optimal transport We consider the following transportation problem: the two measures μ0 and μ1 are concentrated on the vertical line L := (z; t) ∈ Hn z = 0Cn ) , with μ0 concentrated on the negative part L− = L ∩ {t 0} and μ1 on the positive one L+ = L ∩ {t 0}. We remark that the restriction of the quadratic cost dC2 on L is linear in the real coordinate, that is dC2 (0Cn ; t), (0Cn ; t ) = π|t − t |. We can then reduce the transportation problem to a L1 -Monge–Kantorovich problem on the real line R. This situation is quite particular because all couplings of μ0 and μ1 are optimal (see [12, Chapter 2]). Let us investigate a concrete example: identifying L = {0Cn } × R with R, let μ0 and μ1 be L1 [−1, 0] and L1 [0, 1], respectively. A (optimal) coupling is given by (Id, T )# μ0 , where the transport map is T : (0Cn ; t) → (0Cn ; −t).
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There is a multiple choice of geodesics between (0Cn ; t) and T (0Cn ; t) (actually the cutlocus of a point p ∈ L is exactly L \ {p}). To construct a Wasserstein geodesic, we select the (unique) geodesic between (0Cn ; t) and (0Cn ; −t) whose √ midpoint is on the horizontal half-line {(r, 0, . . . , 0; 0) | r ∈ [0, +∞)}. This midpoint is exactly ( 2|t|/π, 0, . . . , 0; 0). Using these geodesics, we define a Wasserstein geodesic (μs )s∈[0,1] between μ0 and μ1 which satisfies the following property: although μ0 and μ1 are absolutely continuous with respect to the 2-dimensional Hausdorff measure (induced by the distance dC ), the intermediate measure μ1/2 is concentrated on the horizontal line {(r, 0, . . . , 0; 0) | r ∈ R} whose dimension is 1. This observation could suggest that one can find a measure μ0 absolutely continuous with respect to the Lebesgue measure such that μ1/2 is not absolutely continuous because concentrated on a set of lower dimension. As announced in the introduction, we will prove in Section 3 that this cannot happen. Remark 2.1. As explained in the book by Villani [13, Notes on Chapter 8], it can be proved that the shortening lemma holds for non-negatively curved Alexandrov spaces (this follows from an estimate found by the first author, see [13, Eq. (8.45)]). It is not known if the property is also true for Alexandrov spaces with curvature bounded from below, see [13, Open Problem 8.21]. 3. Proof of Theorem 1.2 The starting point for the proof of the theorem is an estimate of the second author on the size of a set when contracted along geodesics to a point [5]. Given x, y ∈ Hn and s ∈ (0, 1), let us denote by Ms (x, y) the set of points m such that dC (x, m) = sdC (x, y),
dC (m, y) = (1 − s)dC (x, y).
For E ⊂ Hn , we denote by Ms (E, y) the set Ms (E, y) :=
Ms (x, y).
x∈E
We remark that, for fixed y, for L2n+1 -a.e. x the set Ms (x, y) is a single point and the curve s → Ms (x, y) is the unique constant-speed geodesic between x and y. Proposition 3.1. (See [5, Section 2].) Let y ∈ Hn and E a measurable set. Then Ms (E, y) is measurable and for any s ∈ [0, 1], L2n+1 Ms (E, y) (1 − s)2n+3 L2n+1 (E). Remark 3.2. This estimate, in a more elaborate form, is known as MCP(0, 2n + 3). On Riemannian manifolds this property is shown to be equivalent to a Ricci curvature bound, and it can be regarded as a generalized notion of a lower Ricci curvature bound for metric measure spaces [9,11]. This notion is however different from the Curvature–Dimension condition CD(K, N ) introduced by Lott–Villani [6,7] and Sturm [10,11], and is weaker if the metric space is nonbranching. In particular CD(K, N ) does not hold in Hn for any curvature K and any dimension N (see [5]).
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The idea of the proof is now the following. First we approximate the target measure μ1 by a sequence of discrete measures, and using Proposition 3.1 we prove the absolute continuity of the interpolation in the case of a discrete target measure. Then we pass to the limit, and we finally get the upper bound
on the density of the interpolation. Let μk1 = k1 ki=1 δyi be a sequence weakly converging to μ1 , and denote by T k the optimal transport map between μ0 = ρL2n+1 and μk1 . As in the introduction, (μks )s∈[0,1] denotes the unique Wasserstein geodesic between μ0 and μk1 , and Tsk is the transport map from μ0 to μks . We remark that, if we prove the estimate in (1) with a certain set A of full μ0 -measure, then the bound will obviously be true also for any set containing A. Thus, up to a replacement of A with A ∩ {ρ > 0}, we can assume that A ⊂ {ρ > 0}, so that μ0 and L2n+1 are equivalent on A. For each i = 1, . . . , k, let Ai ⊂ A be theset of points x ∈ A such that T k (x) = yi . The sets Ai n are mutually disjoint and μ0 H \ ki=1 Ai = 0. Let us fix i. Since T k (Ai ) = yi , the curve s → Tsk (x) is the unique geodesic from x to yi for 2n+1 L -a.e. x ∈ Ai . Therefore there exists Bi ⊂ Ai such that L2n+1 (Ai \ Bi ) = 0 and s → Tsk (x) is the unique geodesic from x to yi for all x ∈ Bi . Consider now E ⊂ Bi . By the uniqueness of the geodesics from E to yi we have Ms (E, yi ) = Tsk (E). We can therefore apply Proposition 3.1 to obtain that, for any E ⊂ Bi L2n+1 Tsk (E) (1 − s)2n+3 L2n+1 (E). Since L2n+1 (Ai \ Bi ) = 0, the above estimate is still true if E ⊂ Ai . Recalling now that the sets Ai are disjoint and Tsk is essentially injective, we easily obtain ∀E ⊂ A,
L2n+1 Tsk (E) (1 − s)2n+3 L2n+1 (E).
Indeed it suffices to take E ⊂ A, split it as Ei = E ∩ Ai , write the estimate for Ei and add all the estimates for i = 1, . . . , k. The above property can also be stated by saying that, for any F ⊂ Tsk (A), −1 (F ) ∩ A , L2n+1 (F ) (1 − s)2n+3 L2n+1 Tsk or equivalently A
g Tsk (x) dL2n+1 (x)
1 (1 − s)2n+3
g(y) dL2n+1 (y)
(3)
Hn
for all g ∈ Cc (Hn ), with g 0. Since the Wasserstein geodesic between μ0 and μ1 is unique, by the stability of the optimal transport we have that, for any fixed s, the sequence μks weakly converges to μs , and the optimal transport maps Tsk from μ0 to μks converge in μ0 -measure to Ts from μ0 to μs (see [13, Chapter 7 and Corollary 5.21]).
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Thus, up to a subsequence, we can assume that Tsk → Ts μ0 -a.e., which in particular implies that Tsk → Ts for L2n+1 -a.e. x ∈ A. We can therefore pass to the limit in (3), obtaining 1 g Ts (x) dL2n+1 (x) g(y) dL2n+1 (y) (4) (1 − s)2n+3 Hn
A
for all g ∈ Cc (Hn ), g 0. Moreover, arguing by approximation and using the monotone convergence theorem, we obtain that (4) holds for any measurable function g 0 (in this case, both sides of the equation can be infinite). From this fact we can directly conclude that Ts sends a set with positive Lebesgue measure into a set with positive Lebesgue measure, which implies that μs is absolutely continuous. In order to prove the bound on the density of μs , we consider in (4) g(y) := χTs (A) (y)h(y)ρ ◦ Ts−1 (y), with h 0. In this way we get
h(y) dμs (y) =
h Ts (x) dμ0 (x)
A
Ts (A)
=
h Ts (x) ρ(x) dL2n+1 (x)
A
1 (1 − s)2n+3
h(y)ρ ◦ Ts−1 (y) dL2n+1 (y).
Hn
From the arbitrariness of h and the fact that μs is concentrated on Ts (A) the bound follows. References [1] L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2) (2004) 261– 301. [2] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Wasserstein Space of Probability Measures, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2005. [3] J. Bertrand, Optimal transport on Alexandrov spaces, preprint. [4] L. Capogna, D. Danielli, S.D. Pauls, J.T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progr. Math., Birkhäuser, Basel, 2007. [5] N. Juillet, Ricci curvature bounds and geometric inequalities in the Heisenberg group, preprint SFB611, Bonn. [6] J. Lott, C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (1) (2007) 311–333. [7] J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math., in press. [8] R. Montgomery, A Tour of SubRiemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., vol. 91, Amer. Math. Soc, Providence, RI, 2002. [9] S.-I. Otha, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007) 805– 828. [10] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (1) (2006) 65–131. [11] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (1) (2006) 133–177. [12] C. Villani, Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, Amer. Math. Soc., Providence, RI, 2003. [13] C. Villani, Optimal transport, old and new, notes from the Saint-Flour 2005 Summer School, available online at www.umpa.ens-lyon.fr/~cvillani.
Journal of Functional Analysis 255 (2008) 142–183 www.elsevier.com/locate/jfa
Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property Junsheng Fang ∗,1 , Don Hadwin, Eric Nordgren, Junhao Shen 2 Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA Received 9 October 2007; accepted 8 April 2008 Available online 12 May 2008 Communicated by Alain Connes
Abstract In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II1 factors and Mn (C)) and symmetric gauge norms on L∞ [0, 1] and Cn . As the first application, we obtain that the class of unitarily invariant norms on a type II1 factor coincides with the class of symmetric gauge norms on L∞ [0, 1] and von Neumann’s classical result [J. von Neumann, Some matrixinequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300] on unitarily invariant norms on Mn (C). As the second application, Ky Fan’s dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp -theory (e.g., non-commutative Hölder’s inequality, duality and reflexivity of non-commutative Lp -spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M2 (C)) and some extreme points of N(Mn (C)) (n 3). For a type II1 factor M, we prove that if t (0 t 1) is a rational number then the Ky Fan tth norm is an extreme point of N(M). © 2008 Elsevier Inc. All rights reserved.
* Corresponding author.
E-mail addresses:
[email protected] (J. Fang),
[email protected] (D. Hadwin),
[email protected] (E. Nordgren),
[email protected] (J. Shen). 1 Research supported in part by a University of New Hampshire Dissertation Fellowship. 2 Research supported in part by an NSF grant. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.008
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143
Keywords: Finite von Neumann algebras; The weak Dixmier property; Tracial gauge norms; s-Numbers; Ky Fan norms
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 2.1. Nonincreasing rearrangements of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 2.2. Invertible measure-preserving transformations on [0, 1] . . . . . . . . . . . . . . . . . . . . . . . 150 2.3. s-Numbers of operators in type II1 factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.4. s-Numbers of operators in finite von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . 152 3. Tracial gauge semi-norms on finite von Neumann algebras satisfying the weak Dixmier property 153 3.1. Gauge semi-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.2. Tracial gauge semi-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.3. Symmetric gauge semi-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.4. Unitarily invariant semi-norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.5. Weak Dixmier property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.6. A comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4. Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5. Ky Fan norms on finite von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6. Dual norms of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1. Dual norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2. Dual norms of Ky Fan norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.3. Proof of Theorem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7. Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8. Proof of Theorems D and E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9. Proof of Theorem F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10. Extreme points of normalized unitarily invariant norms on finite factors . . . . . . . . . . . . . . . . . 171 10.1. N(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.2. Ne (Mn (C)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.3. Ne (M2 (C)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 10.4. Proof of Theorem K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 11. Proof of Theorem G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 12. Completion of type II1 factors with respect to unitarily invariant norms . . . . . . . . . . . . . . . . . 178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 12.1. Embedding of M|||·||| into M and L1 (M, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 12.2. M 12.3. Elements in M|||·||| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 12.4. A generalization of Hölder’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13. Proof of Theorems H and I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
1. Introduction The unitarily invariant norms were introduced by von Neumann [21] for the purpose of metrizing matrix spaces. Von Neumann, together with his associates, established that the class of unitarily invariant norms of n × n complex matrices coincides with the class of symmetric gauge functions of their s-numbers. These norms have now been variously generalized and
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utilized in several contexts. For example, Schatten [17,18] defined norms on two-sided ideals of completely continuous operators on an arbitrary Hilbert space; Ky Fan [13] studied Ky Fan norms and obtained his dominance theorem. The unitarily invariant norms play a crucial role in the study of function spaces and group representations (see e.g. [12]) and in obtaining certain bounds of importance in quantum field theory (see [20]). For historical perspectives and surveys of unitarily invariant norms, see Schatten [17,18], Hewitt and Ross [9], Gohberg and Krein [7] and Simon [20]. The theory of non-commutative Lp -spaces has been developed under the name “noncommutative integration" beginning with pioneer work of Segal, Dixmier, and Kunze. Since then the theory has been extensively studied, extended and applied by Nelson, Haagerup, Fack, Kosaki, Junge, Xu, and many others. The recent survey by Pisier and Xu [15] presents a rather complete picture on non-commutative integration and contains a lot of references. This theory is still a very active subject of investigation. Some tools in the study of the usual commutative Lp -spaces still work in the non-commutative setting. However, most of the time, new techniques must be invented. To illustrate the difficulties one may encounter in studying the noncommutative Lp -spaces, we mention here one basic well-known fact. Let H be a complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. The basic fact states that the usual triangle inequality for the absolute values of complex numbers is no longer valid for the absolute values of operators, namely, in general, we do not have |S +T | |S|+|T | for S, T ∈ B(H), where |S| = (S ∗ S)1/2 is the absolute value of S. Despite such difficulties, by now the strong parallelism between non-commutative and classical Lebesgue integration is well known. Motivated by von Neumann’s theorem and the analogies between non-commutative and classical Lp -spaces, in this paper, we will systematically study tracial gauge norms on finite von Neumann algebras that satisfy the weak Dixmier property. Before stating the main theorem and its consequences, we explain some of the notation and terminology that will be used throughout the paper. In this paper, a finite von Neumann algebra (M, τ ) means a von Neumann algebra M with a faithful normal tracial state τ . A finite von Neumann algebra (M, τ ) is said to satisfy the weak Dixmier property if for every positive operator T ∈ M, τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable, i.e., τ (S n ) = τ (T n ) for all n = 0, 1, 2, . . .}. Recall that finite factors satisfy the Dixmier property: if T ∈ M, then τ (T ) is in the operator norm closure of the convex hull of {U T U ∗ : U ∈ M is a unitary operator} and hence satisfy the weak Dixmier property. In Section 3.5, we prove that a finite von Neumann algebra (M, τ ) satisfies the weak Dixmier property if and only if either (M, τ ) can be identified as a von Neumann subalgebra of (Mn (C), τn ) that contains all diagonal matrices, where τn is the normalized trace on Mn (C), or M is diffuse. Throughout the paper, we will reserve the notation · for the operator norm on von Neumann algebras. A tracial gauge norm ||| · ||| on a finite von Neumann algebra (M, τ ) is a norm on M satisfying |||T ||| = ||||T |||| for all T ∈ M (gauge invariant) and |||S||| = |||T ||| if S and T are two equi-measurable positive operators in M (tracial). For a finite von Neumann algebra (M, τ ), let Aut(M, τ ) be the set of ∗-automorphisms on M that preserve the trace. A symmetric gauge norm ||| · ||| on a finite von Neumann algebra (M, τ ) is a gauge norm on M satisfying |||θ (T )||| = |||T ||| for all positive operators T ∈ M and θ ∈ Aut(M, τ ). A unitarily invariant norm ||| · ||| on a finite von Neumann algebra (M, τ ) is a norm on M satisfying |||U T W ||| = |||T ||| 1 for all T ∈ M and unitary operators U, W in M. On (L∞ [0, 1], 0 dx) and (Cn , τ ), where n τ ((x1 , . . . , xn )) = x1 +···+x , a norm is a tracial gauge norm if and only if it is a symmetric gauge n norm. A norm on a finite factor is a tracial gauge norm if and only if it is a unitarily invariant
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norm. A normalized norm is one that assigns the value 1 to the identity operator (which is also denoted by 1). In [5], Fack and Kosaki defined μs (T ), the generalized s-numbers of an operator T in a finite von Neumann algebra (M, τ ) by μs (T ) = inf T E: E is a projection in M with τ (1 − E) s ,
0 s 1.
For 0 < t 1, the Ky Fan tth norm, |||T |||(t) , on a finite von Neumann algebra (M, τ ) is defined by |||T |||(t) =
1 t
t μs (T ) ds. 0
Then ||| · |||(t) is a tracial gauge norm on (M, τ ). Note that |||T |||(1) = τ (|T |) = T 1 is the trace norm. Let n ∈ N, a1 a2 · · · an an+1 = 0 and f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + · · · + n n n 1 an χ[ n−1 ,1] (x). For T ∈ M, define |||T |||f = 0 f (s)μs (T ) ds. Then n
|||T |||f =
n k(ak − ak+1 ) k=1
n
|||T |||( k ) . n
Therefore, |||T |||f is a tracial gauge norm on (M, τ ). Note that if f (x) is the constant 1 function on [0, 1], then |||T |||f = |||T |||(1) = T 1 = τ (|T |). Let F = {f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + · · · + an χ[ n−1 ,1] (x): a1 a2 · · · an 0, a1 +···+an n
n
n n
n
1, n = 1, 2, . . .}. In Section 7, we prove the following representation theorem, which is the main result of this paper. Theorem A. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property. If ||| · ||| is a normalized tracial gauge norm on M, then there is a subset F of F containing the constant 1 function on [0, 1] such that for every T ∈ M, |||T ||| = sup |||T |||f : f ∈ F , where |||T |||f is defined as above. To prove Theorem A, we firstly prove the following technical theorem in Section 4. Theorem B. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Then M1,|||·||| = {T ∈ M: |||T ||| 1} is closed in the weak operator topology. The Russo–Dye theorem [16] and the Kadison–Peterson theorem [10] on convex hulls of unitary operators in von Neumann algebras and the idea of Dixmier’s averaging theorem [2] play fundamental roles in the proof of Theorem B. An important consequence of Theorem B is the
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following corollary which enables us to apply the powerful techniques of normal conditional expectations from finite von Neumann algebras to abelian von Neumann subalgebras. Corollary 1. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If A is a separable abelian von Neumann subalgebra of M and EA is the normal conditional expectation from M onto A preserving τ , then |||EA (T )||| |||T ||| for all T ∈ M. The notion of dual norms plays a key role in the proof of Theorem A. Let ||| · ||| be a norm on a finite von Neumann algebra (M, τ ). Then the dual norm ||| · |||# is defined by |||T |||# = sup τ (T X): X ∈ M, |||X||| 1 ,
T ∈ M.
In Section 5, we study the dual norms systematically. By applying Corollary 1 and careful analysis, we prove the following theorem. Theorem C. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and ||| · ||| be a tracial gauge norm on M. Then ||| · |||# is also a tracial gauge norm on M and ||| · |||## = ||| · |||. Combining Corollary 1, Theorem C and the following theorem on non-increasing rearrangements of functions (see [8, 10.13] for instance), we prove Theorem A in Section 7. Hardy–Littlewood–Pólya. Let f (x), g(x) be non-negative Lebesgue measurable functions on [0, 1] and let f ∗ (x), g ∗ (x) be the non-increasing rearrangements of f (x), g(x), respectively, 1 1 then 0 f (x)g(x) dx 0 f ∗ (x)g ∗ (x) dx. Now we state some important consequences of Theorem A. Since there is a natural one-to-one correspondence between Ky Fan tth norms on finite von Neumann algebras (satisfying the weak 1 Dixmier property) and Ky Fan tth norms on (L∞ [0, 1], 0 dx) or (Cn , τ ), the first application of Theorem A is the following. Theorem D. Let (M, τ ) be a diffuse finite von Neumann algebra (or a von Neumann subalgebra of Mn (C), τ = τn |M , such that M contains all diagonal matrices). Then there is a one-toone correspondence between tracial gauge norms on (M, τ ) and symmetric gauge norms on 1 n , respectively). Namely: (L∞ [0, 1], 0 dx) (or (Cn , τ ), τ ((x1 , . . . , xn )) = x1 +···+x n 1 1. If ||| · ||| is a tracial gauge norm on (M, τ ) and θ is an embedding from (L∞ [0, 1], 0 dx) into (M, τ ) (or x1 ⊕ · · · ⊕ xn is the diagonal matrix with diagonal elements x1 , . . . , xn , respectively), then |||f (x)||| = |||θ (f (x))||| defines a symmetric gauge norm on (L∞ [0, 1], 1 n 0 dx) (or |||(x1 , . . . , xn )||| = |||x1 ⊕ · · · ⊕ xn ||| defines a symmetric gauge norm on (C , τ ), respectively). 1 2. If ||| · ||| is a symmetric gauge norm on (L∞ [0, 1], 0 dx) (or (Cn , τ ) respectively), then |||T ||| = |||μs (T )||| (or |||T ||| = |||(s1 (T ), . . . , sn (T ))||| , respectively) defines a tracial gauge norm on (M, τ ).
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As consequences of Theorem D, we have the following corollary and von Neumann’s theorem. Corollary 2. There is a one-to-one correspondence between unitarily invariant norms on a 1 type II1 factor (M, τ ) and symmetric gauge norms on (L∞ [0, 1], 0 dx). Namely: 1 1. If |||·||| is a unitarily invariant norm on M and θ is an embedding from (L∞ [0, 1], 0 dx) into 1 (M, τ ), then |||f (x)||| = |||θ (f (x))||| defines a symmetric gauge norm on (L∞ [0, 1], 0 dx). 1 2. If ||| · ||| is a symmetric gauge norm on (L∞ [0, 1], 0 dx), then |||T ||| = |||μs (T )||| defines a unitarily invariant norm on M. Von Neumann. There is a one-to-one correspondence between unitarily invariant norms on n Mn (C) and symmetric gauge norms on (Cn , τ ), τ ((x1 , . . . , xn )) = x1 +···+x . Namely: n 1. If ||| · ||| is a unitarily invariant norm on Mn (C), then |||(x1 , . . . , xn )||| = |||x1 ⊕ · · · ⊕ xn ||| defines a symmetric gauge norm on (Cn , τ ). 2. If ||| · ||| is a symmetric gauge norm on (Cn , τ ), then |||T ||| = |||(s1 (T ), . . . , sn (T ))||| defines a unitarily invariant norm on Mn (C). Theorem D establishes the one-to-one correspondence between tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property and symmetric gauge norms on abelian von Neumann algebras. The following theorem further establishes the one-to-one correspondence between the dual norms on finite von Neumann algebras satisfying the weak Dixmier property and the dual norms on abelian von Neumann algebras, which plays a key role in the studying of duality and reflexivity of the completion of type II1 factors with respect to unitarily invariant norms. Theorem E. Let (M, τ ) be a diffuse finite von Neumann algebra (or a von Neumann subalgebra of Mn (C), τ = τn |M , such that M contains all diagonal matrices). If ||| · ||| is a tracial gauge 1 norm on (M, τ ) corresponding to the symmetric gauge norm ||| · |||1 on (L∞ [0, 1], 0 dx) (or (Cn , τ ), respectively) as in Theorem D, then ||| · |||# on M is the tracial gauge norm correspond1 ing to the symmetric gauge norm ||| · |||#1 on (L∞ [0, 1], 0 dx) (or (Cn , τ ), respectively) as in Theorem D. The second consequence of Theorem A is the following theorem. Theorem F. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and S, T ∈ M. If |||S|||(t) |||T |||(t) for all Ky Fan t-th norms, 0 t 1, then |||S||| |||T ||| for all tracial gauge norms ||| · ||| on M. As a corollary, we obtain the following Ky Fan’s dominance theorem. (See [13].) If S, T ∈ Mn (C) and |||S|||(k/n) |||T |||(k/n) , i.e., k k i=1 si (S) i=1 si (T ) for 1 k n, then |||S||| |||T ||| for all unitarily invariant norms ||| · ||| on Mn (C).
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A unitarily invariant norm ||| · ||| on a type II1 factor M is called singular if limτ (E)→0+ |||E||| > 0 and continuous if limτ (E)→0+ |||E||| = 0. The following theorem is proved in Section 11. Theorem G. Let ||| · ||| be a unitarily invariant norm on M and let T be the topology induced by ||| · ||| on M1,· = {T ∈ M: T 1}. If ||| · ||| is singular, then T is the operator norm topology on M1,· . If ||| · ||| is continuous, then T is the measure topology (in the sense of Nelson [14]) on M1,· . Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. We denote by be the completion of M with respect M|||·||| the completion of M with respect to ||| · |||. Let M to the measure topology in the sense of Nelson [14]. In Section 12, we prove that there is an that extends the identity map from M onto M. An element injective map from M|||·||| into M in M can be identified with a closed, densely defined operator affiliated with M (see [14]). So generally speaking, an element in M|||·||| should be treated as an unbounded operator. We will consider the following two questions in Section 13: Question 1. Under what conditions is M|||·|||# the dual space of M|||·||| in the following sense: for every φ ∈ M|||·||| # , there is a unique X ∈ M|||·|||# such that φ(T ) = τ (T X),
∀T ∈ M|||·||| ,
and φ = |||T |||? Question 2. Under what conditions is M|||·||| a reflexive Banach space? 1 Let ||| · |||1 be the symmetric gauge norm on (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the same questions can be asked about L∞ [0, 1]|||·|||1 , the completion of L∞ [0, 1] with respect to ||| · |||1 . As further consequences of Theorem A, we prove the following theorems that answer the Questions 1 and 2, respectively. Theorem H. Let M be a type II1 factor, ||| · ||| be a unitarily invariant norm on M and ||| · |||# be the dual unitarily invariant norm on M. Let ||| · |||1 be the symmetric gauge norm on 1 (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the following conditions are equivalent: 1. 2. 3. 4.
M|||·|||# is the dual space of M|||·||| in the sense of Question 1; L∞ [0, 1]|||·|||# is the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1; 1 ||| · ||| is a continuous norm on M; ||| · |||1 is a continuous norm on L∞ [0, 1].
Theorem I. Let M be a type II1 factor, ||| · ||| be a unitarily invariant norm on M and let ||| · |||# be the dual unitarily invariant norm on M. Let ||| · |||1 be the symmetric gauge norm on 1 (L∞ [0, 1], 0 dx) corresponding to ||| · ||| on M as in Corollary 2. Then the following conditions are equivalent:
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M|||·||| is a reflexive space; L∞ [0, 1]|||·|||1 is a reflexive space; both ||| · ||| and ||| · |||# are continuous norms on M; both ||| · |||1 and ||| · |||#1 are continuous norms on L∞ [0, 1].
A key step to proving Theorem H is based on the following fact: if ||| · ||| is a continuous unitarily invariant norm on M and φ ∈ M|||·||| # , then the restriction of φ to M is an ultraweakly continuous linear functional, i.e., φ is in the predual space of M. A significant advantage of our approach is that we develop a relatively complete theory of unitarily invariant norms on type II1 factors before handling unbounded operators. Indeed, unbounded operators are slightly involved only in the last two sections (Sections 12 and 13). Compared with the classical methods (e.g., [19]), which have to do a lot of subtle analysis on unbounded operators, our methods are much simpler. Let M be a finite factor. Recall that a norm ||| · ||| on M is called a normalized norm if |||1||| = 1. Let N(M) be the set of normalized unitarily invariant norms on M. Then N(M) is a convex compact set in the pointwise weak topology. Let Ne (M) be the set of extreme points of N(M). By the Krein–Milman theorem, N(M) is the closure of the convex hull of Ne (M) in the pointwise weak topology. So it is an interesting question of characterizing the set Ne (M). In Section 10, we prove the following theorems. Theorem J. Ne (M2 (C)) = {max{tT , T 1 }: 1/2 t 1}, where T 1 = τ2 (|T |). Theorem K. If M is a type II1 factor and t is a rational number such that 0 t 1, then the Ky Fan tth norm is an extreme point of N(M). This paper is almost self-contained and we do not assume any backgrounds on noncommutative Lp -theory. 2. Preliminaries 2.1. Nonincreasing rearrangements of functions Throughout this paper, we denote by m the Lebesgue measure on [0, 1]. In the following, a measurable function and a measurable set mean a Lebesgue measurable function and a Lebesgue measurable set, respectively. For two measurable sets A and B, A = B means m((A \ B) ∪ (B \ A)) = 0. Let f (x) be a real measurable function on [0, 1]. The non-increasing rearrangement function, f ∗ (x), of f (x) is defined by ∗
f (x) =
sup{y: m({f > y}) > x}, 0 x < 1; ess inf f, x = 1.
(2.1)
We summarize some useful properties of f ∗ (x) in the following proposition. Proposition 2.1. Let f (x), g(x) be real measurable functions on [0, 1]. Then we have the following:
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1. f ∗ (x) is a non-increasing, right-continuous function on [0, 1] such that f ∗ (0) = ess sup f ; 1 1 2. if f (x) and g(x) are bounded functions and 0 f n (x) dx = 0 g n (x) dx for all n = 0, 1, 2, . . . , then f ∗ (x) = g ∗ (x); 1 1 3. f (x) and f ∗ (x) are equi-measurable and 0 f (x) dx = 0 f ∗ (x) dx when either integral is well defined. 2.2. Invertible measure-preserving transformations on [0, 1] Let G = {φ: φ(x) is an invertible measure-preserving transformation on [0, 1]}. It is well known that G acts on [0, 1] ergodically (see [6, pp. 3, 4], for instance), i.e., for a measurable subset A of [0, 1], φ(A) = A for all φ ∈ G implies that m(A) = 0 or m(A) = 1. Lemma 2.2. Let A, B be two measurable subsets of [0, 1] such that m(A) = m(B). Then there is a φ ∈ G such that φ(A) = B. Proof. We can assume that m(A) = m(B) > 0. Since G acts ergodically on [0, 1], there is a φ ∈ G such that m(φ(A)∩B) > 0. Let B1 = φ(A)∩B and A1 = φ −1 (B1 ). Then m(A1 ) = m(B1 ) and φ(A1 ) = B1 . By Zorn’s lemma and maximality arguments, we prove the lemma. 2 Corollary 2.3. Let A1 , . . . , An and B1 , . . . , Bn be disjoint measurable subsets of [0, 1] such that m(Ak ) = m(Bk ) for 1 k n. Then there is a φ ∈ G such that φ(Ak ) = Bk for 1 k n. Proof. We can assume that A1 ∪ · · · ∪ An = B1 ∪ · · · ∪ Bn = [0, 1]. By Lemma 2.2, there is a φk ∈ G such that φk (Ak ) = Bk , 1 k n. Define φ(x) = φk (x) for x ∈ Ak . Then φ ∈ G and φ(Ak ) = Bk for 1 k n. 2 1 For f (x) ∈ L∞ [0, 1], define τ (f ) = 0 f (x) dx. The following theorem is a version of the Dixmier’s averaging theorem (see [3] or [11]) and it has a similar proof. Theorem 2.4. Let f (x) ∈ L∞ [0, 1] be a real function. Then τ (f ) is in the L∞ -norm closure of the convex hull of {f · φ(x): φ ∈ G}. We end this subsection with the following proposition. Proposition 2.5. If φ(x) is an invertible measure-preserving transformation on [0, 1], then θ (f ) = f ◦ φ is a ∗-automorphism of L∞ [0, 1] preserving τ . Conversely, if θ is a ∗-automorphism of L∞ [0, 1] preserving τ , then there is an invertible measure-preserving transformation on [0, 1] such that θ (f ) = f ◦ φ for all f (x) ∈ L∞ [0, 1]. Proof. The first part of the proposition is easy to see. Suppose θ is a ∗-automorphism of L∞ [0, 1]. Let φ(x) = θ (f )(x), where f (x) ≡ x. Then it is easy to see the second part of the proposition. 2
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2.3. s-Numbers of operators in type II1 factors In [5], Fack and Kosaki give a rather complete exposition of generalized s-numbers of τ -measurable operators affiliated with semi-finite von Neumann algebras. For the sake of reader’s convenience and our purpose, we provide sufficient details on s-numbers of bounded operators in finite von Neumann algebras in the following. We will define s-numbers of bounded operators in finite von Neumann algebras from the point of view of non-increasing rearrangement of functions. The following lemma is well known. The proof is an easy exercise. Lemma 2.6. Let (A, τ ) be a separable (i.e., with separable predual ) diffuse abelian von Neumann algebra with a faithful normal trace τ on A. Then there is a ∗-isomorphism α from (A, τ ) 1 1 onto (L∞ [0, 1], 0 dx) such that τ = 0 dx ◦ α. Let M be a type II1 factor and let τ be the unique trace on M. For T ∈ M, there is a separable diffuse abelian von Neumann subalgebra A of M containing |T |. By Lemma 2.6, there is a ∗-iso1 1 morphism α from (A, τ ) onto (L∞ ([0, 1], 0 dx) such that τ = 0 dx ◦ α. Let f (x) = α(|T |) and f ∗ (x) be the non-increasing rearrangement of f (x) (see (2.1)). Then the s-numbers of T , μs (T ), are defined as μs (T ) = f ∗ (s),
0 s 1.
Lemma 2.7. μs (T ) does not depend on A and α. Proof. Let A1 be another separable diffuse abelian von Neumann subalgebra of M containing 1 1 |T | and let β be a ∗-isomorphism from (A1 , τ ) onto (L∞ [0, 1], 0 dx) such that τ = 0 dx · β. 1 n 1 Let g(x) = β(|T |). For every number n = 0, 1, 2, . . . , 0 f (x) dx = τ (|T |n ) = 0 g n (x) dx. ∗ Since both f (x) and g(x) are bounded positive functions, by 2 of Proposition 2.1, f (x) = g ∗ (x) for all x ∈ [0, 1]. 2 Corollary 2.8. For T ∈ M and p 0, τ (|T |p ) =
1 0
μs (T )p ds.
The following lemma says that the above definition of s-numbers coincides with the definition of s-numbers given by Fack and Kosaki. Recall that P(M) is the set of projections in M. Lemma 2.9. For 0 s 1, μs (T ) = inf T E: E ∈ P(M), τ E ⊥ = s . Proof. By the polar decomposition and the definition of μs (T ), we may assume that T is positive. Let A be a separable diffuse abelian von Neumann subalgebra of M containing T and 1 1 let α be a ∗-isomorphism from (A, τ ) onto (L∞ [0, 1], 0 dx) such that τ = 0 dx · α. Let f (x) = α(T ) and let f ∗ (x) be the non-increasing rearrangement of f (x). Then μs (T ) = f ∗ (s). By the definition of f ∗ ,
1 m f ∗ > μs (T ) = lim m f ∗ > μs (T ) + s n→∞ n
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∗ 1 ∗ m f μs (T ) lim m f > μs (T ) − s. n→∞ n Since f ∗ and f are equi-measurable, m({f > μs (T )}) s and m({f μs (T )}) s. Therefore, there is a measurable subset A of [0, 1], {f > μs (T )} ⊂ [0, 1] \ A ⊂ {f μs (T )}, such that m([0, 1] \ A) = s and f (x)χA (x)∞ = μs (T ) and f (x)χB (x)∞ μs (T ) for all B ⊂ [0, 1] \ A such that m(B) > 0. Let F = α −1 (χA ). Then τ (F ⊥ ) = s, T F = α −1 (f χA )∞ = μs (T ) and T F μs (T ) for all non-zero subprojections F of F ⊥ . This proves that μs (T ) inf{T E: E ∈ P(M), τ (E ⊥ ) = s}. Similarly, for every > 0, there is a projection F ∈ M such that τ (F⊥ ) = s + , T F = μs+ (T ) and T F μs+ (T ) for all non-zero subprojections F of F⊥ . Suppose E ∈ M is a projection such that τ (E ⊥ ) = s. Then τ (E ∧ F⊥ ) = τ (E) + τ (F⊥ ) − τ (E ∨ F⊥ ) = 1 + − τ (E ∨ F ⊥ ) > 0. Hence, T E T (E ∧ F⊥ ) μs+ (T ). This proves that inf{T E: E ∈ P(M), τ (E ⊥ ) = s} μs+ (T ). Since μs (T ) is rightcontinuous, μs (T ) inf{T E: E ∈ P(M), τ (E ⊥ ) = s}. 2 Corollary 2.10. Let S, T ∈ M. Then μs (ST ) Sμs (T ) for s ∈ [0, 1]. We refer to [4,5] for other interesting properties of s-numbers of operators in type II1 factors. 2.4. s-Numbers of operators in finite von Neumann algebras Throughout this paper, a finite von Neumann algebra (M, τ ) means a finite von Neumann algebra M with a faithful normal tracial state τ . An embedding of a finite von Neumann algebra (M, τ ) into another finite von Neumann algebra (M1 , τ1 ) means a ∗-isomorphism α from M to M1 such that τ = τ1 ◦ α. Let (L(F2 ), τ ) be the free group factor with the faithful normal trace τ . Then the reduced free product von Neumann algebra M1 = (M, τ ) ∗ (L(F2 ), τ ) is a type II1 factor with a (unique) faithful normal trace τ1 such that the restriction of τ1 to M is τ . So every finite von Neumann algebra can be embedded into a type II1 factor. Definition 2.11. Let (M, τ ) be a finite von Neumann algebra and T ∈ M. If α is an embedding of (M, τ ) into a type II1 factor (M1 , τ1 ), then the s-numbers of T are defined as μs (T ) = μs α(T ) . Similar to the proof of Lemma 2.7, we can see that μs (T ) is well defined, i.e., does not depend on the choice of α and M1 . Let T ∈ (Mn (C), τn ), where τn is the normalized trace on Mn (C). Then |T | is unitarily equivalent to a diagonal matrix with diagonal elements s1 (T ) · · · sn (T ) 0. In the classical matrices theory [1,7], s1 (T ), . . . , sn (T ) are also called s-numbers of T . It is easy to see that the relation between μs (T ) and s1 (T ), . . . , sn (T ) is the following μs (T ) = s1 (T )χ[0,1/n) (s) + s2 (T )χ[1/n,2/n) (s) + · · · + sn (T )χ[n−1/n,1] (s).
(2.2)
Since no confusions will arise, we will use both s-numbers for matrices in Mn (C). We refer to [1,7] for other interesting properties of s-numbers of matrices. We end this section by the following definition.
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Definition 2.12. Positive operators S and T in a finite von Neumann algebra (M, τ ) are equimeasurable if μs (S) = μs (T ) for 0 s 1. By 2 of Proposition 2.1 and Corollary 2.8, positive operators S and T in a finite von Neumann algebra (M, τ ) are equi-measurable if and only if τ (S n ) = τ (T n ) for all n = 0, 1, 2, . . . . 3. Tracial gauge semi-norms on finite von Neumann algebras satisfying the weak Dixmier property 3.1. Gauge semi-norms Definition 3.1. Let (M, τ ) be a finite von Neumann algebra. A semi-norm ||| · ||| on M is called gauge invariant if for every T ∈ M, |||T ||| = |T |. Lemma 3.2. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. Then the following conditions are equivalent: 1. ||| · ||| is gauge invariant; 2. ||| · ||| is left unitarily invariant, i.e., for every unitary operator U ∈ M and operator T ∈ M, |||U T ||| = |||T |||; 3. for operators A, T ∈ M, |||AT ||| A · |||T |||. Proof. “3 ⇒ 2” and “2 ⇒ 1” are easy to see. We only prove “1 ⇒ 3.” We need to prove that if A < 1, then |||AT ||| |||T |||. Since A < 1, there are unitary operators U1 , . . . , Uk such k kT that A = U1 +···+U (see [10,16]). Since |U1 T | = · · · = |Uk T | = |T |, |||AT ||| = ||| U1 T +···+U ||| k k |||U1 T |||+···+|||Uk T ||| |||T |||. 2 k Corollary 3.3. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge invariant semi-norm on M such that |||T V ||| = |||T ||| for every unitary operator V ∈ M and operator T ∈ M. If 0 S T , then |||S||| |||T |||. Proof. Since 0 S T , there is an operator A ∈ M such that S = AT A∗ and A 1. Similar to the proof of Lemma 3.2, |||S||| = |||AT A∗ ||| A · |||T ||| · A∗ |||T |||. 2 Definition 3.4. A normalized semi-norm on a finite von Neumann algebra (M, τ ) is a semi-norm ||| · ||| such that |||1||| = 1. By Lemma 3.2, we have the following corollary. Corollary 3.5. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a normalized gauge semi-norm on M. Then for every T ∈ M, |||T ||| T . A simple operator in a finite von Neumann algebra (M, τ ) is an operator T = a1 E1 + · · · + an En , where E1 , . . . , En are projections in M such that E1 + · · · + En = 1.
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Corollary 3.6. Let (M, τ ) be a finite von Neumann algebra, ||| · |||1 and ||| · |||2 be two gauge invariant semi-norms on M. Then ||| · |||1 = ||| · |||2 on M if |||T |||1 = |||T |||2 for all positive simple operators T ∈ M. Proof. Without loss of generality, assume |||1|||1 = |||1|||2 = 1. Let T ∈ M be a positive operator. By the spectral decomposition theorem, there is a sequence of positive simple operators Tn ∈ M such that limn→∞ T − Tn = 0. By Corollary 3.5, limn→∞ |||T − Tn |||1 = limn→∞ |||T − Tn |||2 = 0. By the assumption of the corollary, |||Tn |||1 = |||Tn |||2 . Hence, |||T |||1 = |||T |||2 . Since both ||| · |||1 and ||| · |||2 are gauge invariant, ||| · |||1 = ||| · |||2 . 2 3.2. Tracial gauge semi-norms Definition 3.7. Let (M, τ ) be a finite von Neumann algebra. A semi-norm ||| · ||| on M is called tracial if |||S||| = |||T ||| for every two equi-measurable positive operators S, T in M. A semi-norm ||| · ||| on M is called a tracial gauge semi-norm if it is both tracial and gauge invariant. Since for a positive operator T in a finite von Neumann algebra (M, τ ), T = 1 limn→∞ (τ (T n )) n , the operator norm · is a tracial gauge norm on (M, τ ). Another less obvious example of a tracial gauge norm on (M, τ ) is the non-commutative L1 -norm: T 1 = 1 τ (|T |) = 0 μs (T ) ds. The less obvious part is to show that · 1 satisfies the triangle inequality. The following lemma overcomes this difficulty. Lemma 3.8. A1 = sup{|τ (U A)|: U ∈ U(M)}, where U(M) is the set of unitary operators in M. Proof. By the polar decomposition theorem, there is a unitary operator V ∈ M such that A = V |A|. By the Schwarz inequality, |τ (U A)| = |τ (U V |A|)| = |τ (U V |A|1/2 |A|1/2 )| τ (|A|)1/2 · τ (|A|)1/2 = τ (|A|). Hence A1 sup{|τ (U A)|: U ∈ U(M)}. Let U = V ∗ , we obtain A1 sup{|τ (U A)|: U ∈ U(M)}. 2 Corollary 3.9. A + B1 A1 + B1 . Lemma 3.10. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge invariant semi-norm on M. Then ||| · ||| is tracial if |||S||| = |||T ||| for every two equi-measurable positive simple operators S, T in M. Proof. We can assume that |||1||| = 1. Let A, B be two equi-measurable positive operators in M. By the spectral decomposition theorem, there are two sequences of positive simple operators An , Bn in M such that An and Bn are equi-measurable and limn→∞ A − An = limn→∞ B − Bn = 0. By Corollary 3.5, limn→∞ |||A − An ||| = limn→∞ |||B − Bn ||| = 0. By the assumption of the lemma, |||An ||| = |||Bn |||. Hence, |||A||| = |||B|||. 2 3.3. Symmetric gauge semi-norms Definition 3.11. Let (M, τ ) be a finite von Neumann algebra and let Aut(M, τ ) be the set of ∗-automorphisms on M preserving τ . A semi-norm ||| · ||| on M is called symmetric if θ (T ) = |||T |||, ∀T ∈ M, θ ∈ Aut(M, τ ).
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A semi-norm ||| · ||| on M is called a symmetric gauge semi-norm if it is both symmetric and gauge invariant. n Example 3.12. Let M = Cn and τ (T ) = x1 +···+x , where T = (x1 , . . . , xn ) ∈ Cn . Then n Aut(M, τ ) is the set of permutations on {1, . . . , n}. So a semi-norm ||| · ||| on M is a symmetric gauge semi-norm if and only if for every (x1 , . . . , xn ) ∈ Cn and a permutation π on {1, . . . , n},
(x1 , . . . , xn ) = |x1 |, . . . , |xn | , and (x1 , . . . , xn ) = (xπ(1) , . . . , xπ(n) ). Lemma 3.13. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. If ||| · ||| is tracial gauge invariant, then ||| · ||| is symmetric gauge invariant. Proof. Let θ ∈ Aut(M, τ ) and T ∈ M. We need to prove that |||θ (T )||| = |||T |||. Since |θ (T )| = θ (|T |) and ||| · ||| is gauge invariant, we can assume that T is positive. Since θ ∈ Aut(M, τ ), T and θ (T ) are equi-measurable. Hence, |||T ||| = |||θ (T )|||. 2 Example 3.14. Let M = C ⊕ M2 (C) and τ (a ⊕ B) = a2 + τ2 (B) 2 , where τ2 is the normalized trace on M2 (C). Define |||a ⊕ B||| = |a|/2 + τ2 (|B|). Then ||| · ||| is a symmetric gauge norm but not a tracial gauge norm. Note that 1 ⊕ 0 and 0 ⊕ 1 are equi-measurable, but 1/2 = |||1 ⊕ 0||| = |||0 ⊕ 1||| = 1. Aut(M, τ ) acts on M ergodically if θ (T ) = T for all θ ∈ Aut(M, τ ) implies T = λ1. Lemma 3.15. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a semi-norm on M. If Aut(M, τ ) acts on M ergodically, then the following are equivalent: 1. ||| · ||| is a tracial gauge semi-norm; 2. ||| · ||| is a symmetric gauge semi-norm. Proof. “1 ⇒ 2” by Lemma 3.13. We need to prove “2 ⇒ 1.” By Corollary 3.6, we need to prove |||S||| = |||T ||| for two equi-measurable simple operators S, T in M. Similar to the proof of Corollary 2.3, there is a θ ∈ Aut(M, τ ) such that S = θ (T ). Hence |||S||| = |||T |||. 2 1 Corollary 3.16. A semi-norm on (L∞ [0, 1], 0 dx) or (Cn , τ ) is a tracial gauge norm if and n only if it is a symmetric gauge norm, where τ ((x1 , . . . , xn )) = x1 +···+x . n 3.4. Unitarily invariant semi-norms Definition 3.17. Let (M, τ ) be a von Neumann algebra. A semi-norm ||| · ||| on M is unitarily invariant if |||U T V ||| = |||T ||| for all T ∈ M and unitary operators U, V ∈ M.
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Proposition 3.18. Let ||| · ||| be a semi-norm on M. Then the following statements are equivalent: 1. ||| · ||| is unitarily invariant; 2. ||| · ||| is gauge invariant and unitarily conjugate invariant, i.e., |||U T U ∗ ||| = |||T ||| for all T ∈ M and unitary operators U ∈ M; 3. ||| · ||| is left-unitarily invariant and |||T ||| = |||T ∗ ||| for every T ∈ M; 4. for all operators T , A, B ∈ M, |||AT B||| A · |||T ||| · B. Proof. “1 ⇒ 4” is similar to the proof of Lemma 3.2. “4 ⇒ 3,” “3 ⇒ 2,” and “2 ⇒ 1” are routine. 2 For a unitary operator U ∈ M, let θ (T ) = U T U ∗ . Then θ ∈ Aut(M, τ ). By Proposition 3.18, we have the following. Corollary 3.19. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a symmetric, gauge invariant semi-norm on M. Then ||| · ||| is a unitarily invariant semi-norm on M. n . Define |||(x1 , . . . , xn )||| = Example 3.20. Let M = Cn , n 2 and τ ((x1 , . . . , xn )) = x1 +···+x n |x1 |. Then ||| · ||| is a unitarily invariant semi-norm but not a symmetric gauge semi-norm on M.
Lemma 3.21. Let (M, τ ) be a finite factor and ||| · ||| be a semi-norm on M. Then the following conditions are equivalent: 1. ||| · ||| is a tracial gauge semi-norm; 2. ||| · ||| is a symmetric gauge semi-norm; 3. ||| · ||| is a unitarily invariant semi-norm. Proof. “1 ⇒ 2” by Lemma 3.13 and “2 ⇒ 3” by Corollary 3.19. We need to prove “3 ⇒ 1.” By Corollary 3.6, we need to prove |||S||| = |||T ||| for two equi-measurable positive simple operators S, T ∈ M. Suppose S = a1 E1 + · · · + an En and T = a1 F1 + · · · + an Fn , where E1 + · · · + En = 1 and F1 + · · · + Fn = 1 and τ (Ek ) = τ (Fk ) for 1 k n. Since M is a factor, there is a unitary operator U ∈ M such that Ek = U Fk U ∗ for 1 k n. Therefore, S = U T U ∗ and |||S||| = |||T |||. 2 3.5. Weak Dixmier property Definition 3.22. A finite von Neumann algebra (M, τ ) satisfies the weak Dixmier property if for every positive operator T ∈ M, τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}. A finite factor (M, τ ) satisfies the Dixmier property (see [2,11]): for every operator T ∈ M, τ (T ) is in the operator norm closure of the convex hull of {U T U ∗ : U ∈ U(M)}. Hence finite factors satisfy the weak Dixmier property. In the following, we will characterize finite von Neumann algebras satisfying the weak Dixmier property. There is a central projection P in a finite von Neumann algebra (M, τ ) such that P M is of type I and (1 − P )M is of type II. A type II von Neumann algebra is diffuse, i.e, there are no non-zero minimal projections in the von Neumann algebra. Furthermore, there are central
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projections P1 , . . . , Pn , . . . in M, such that P1 + · · · + Pn + · · · = P and Pn M = An ⊗ Mn (C), An is abelian. We can decompose An into an atomic part Aan and a diffuse part Acn , i.e., there is a projection Qn in An , Aan = Qn An , such that Qn = En1 + En2 + · · · , where Enk is a minimal projection in Aan and τ (Enk ) > 0, and Acn = (1 − Qn )An is diffuse. Let Ma = ⊕ Aan ⊗ Mn (C) and Mc = ⊕ Acn ⊗ Mn (C) ⊕ (1 − P )M. Then M = Ma ⊕ Mc . We call Ma the atomic part of M and Mc the diffuse part of M. A finite von Neumann algebra (M, τ ) is atomic if M = Ma and is diffuse if M = Mc . Lemma 3.23. Let (M, τ ) be a finite-dimensional von Neumann algebra such that for every two non-zero minimal projections E, F ∈ M, τ (E) = τ (F ). Then (M, τ ) satisfies the weak Dixmier property. Proof. Since M is finite-dimensional, M ∼ = Mk1 (C) ⊕ · · · ⊕ Mkr (C). Since τ (E) = τ (F ) for every two non-zero minimal projections E, F ∈ M, (M, τ ) can be embedded into (Mn (C), τn ), where n = k1 + · · · + kr . So we can assume that (M, τ ) is a von Neumann subalgebra of (Mn (C), τn ) such that M contains all diagonal matrices a1 E1 + · · · + an En . Now for every positive operator T ∈ M, there is a unitary operator U ∈ M such that U T U ∗ = a1 E1 + · · · + an En , (a E +···+aπ(n) En ) n . Then τ (T ) = π π(1) 1 n! . 2 a1 , . . . , an 0 and τ (T ) = a1 +···+a n Lemma 3.24. Let (M, τ ) be a diffuse finite von Neumann algebra. Then (M, τ ) satisfies the weak Dixmier property. Proof. Let A be a separable diffuse abelian von Neumann subalgebra of M. By Lemma 2.6, 1 1 there is a ∗-isomorphism α from (A, τ ) onto (L∞ [0, 1], 0 dx) such that 0 dx · α = τ . For a positive operator T ∈ M, there is an operator S ∈ A such that α(S) = μs (T ). Hence 1 τ (T ) = τ (S) = 0 μs (T ) ds. By Theorem 2.4, for any > 0, there are S1 , . . . , Sn in A such that n < . Hence (M, τ ) satisfies the weak S, S1 , . . . , Sn are equi-measurable and τ (S) − S1 +···+S n Dixmier property. 2 Lemma 3.25. Let (M, τ ) be an atomic finite von Neumann algebra with two minimal projections E and F in M such that τ (E) = τ (F ). Then (M, τ ) does not satisfy the weak Dixmier property. Proof. Since (M, τ ) is an atomic finite von Neumann algebra, M ∼ = Mk1 (C) ⊕ Mk2 (C) ⊕ · · · . Let Eij be minimal projections in Mki such that Eij = 1. Without loss of generality, assume that τ (E11 ) > τ (E21 ) τ (E31 ) · · · . Let ⎛ ⎜ T =⎝
⎞
1 ..
⎟ ⎠ ⊕ A,
. 1
k1
where ⎛ ⎜ A=⎝
1 2
⎞ ..
⎛
⎟ ⎜ ⎠⊕⎝
. ( 12 )k2
( 12 )k2 +1
⎞ ..
⎟ ⎠ ⊕ ···.
. ( 12 )k2 +k3
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If T1 ∈ M and T are equi-measurable, then ⎛ 1 ⎜ .. T1 = ⎝ .
⎞ ⎟ ⎠ ⊕ A1 , 1
k1
where A and A1 are equi-measurable. Hence, if τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}, then τ (T ) = 1. This is a contradiction. 2 Let (M, τ ) be a finite von Neumann algebra and let E ∈ M be a non-zero projection. The induced finite von Neumann algebra (ME , τE ) is the von Neumann algebra ME = EME with E) a faithful normal trace τE (ET E) = τ (ET τ (E) . The proof of the following lemma is similar to the proof of Lemma 3.25. Lemma 3.26. Let (M, τ ) be a finite von Neumann algebra such that Ma = 0 and Mc = 0. Then M does not satisfy the weak Dixmier property. Proof. Let P be the central projection such that Ma = P M and Mc = (1 − P )M. Let A be a separable diffuse abelian von Neumann subalgebra of (Mc , τ1−P ). By Lemma 2.6, there is a positive operator A in Mc such that μs (A) = 1−s 2 with respect to (Mc , τ1−P ). Consider T = P + A(1 − P ). Then 1, 0 s < τ (P ); μs (T ) = 1−s 1 2τ (1−P ) 2 , τ (P ) s 1 with respect to (M, τ ). If T1 ∈ M and T are equi-measurable, then T1 = P + A1 such that A1 and A are equi-measurable. Hence, if τ (T ) is in the operator norm closure of the convex hull of {S ∈ M: S and T are equi-measurable}, then τ (T ) = 1. This is a contradiction. 2 Summarizing Lemmas 3.23–3.26, we can characterize finite von Neumann algebras satisfying the weak Dixmier property as the following theorem. Theorem 3.27. Let (M, τ ) be a finite von Neumann algebra. Then M satisfies the weak Dixmier property if and only if M satisfies one of the following conditions: 1. M is finite-dimensional (hence atomic) and for every two non-zero minimal projections E, F ∈ M, τ (E) = τ (F ), or equivalently, (M, τ ) can be identified as a von Neumann subalgebra of (Mn (C), τn ) that contains all diagonal matrices; 2. M is diffuse. Corollary 3.28. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and E ∈ M be a non-zero projection. Then (ME , τE ) also satisfies the weak Dixmier property. The following example shows that we cannot replace the weak Dixmier property by the following condition: τ (T ) is in the operator norm closure of the convex hull of {θ (T ): θ ∈ Aut(M, τ )}.
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Example 3.29. (C ⊕ M2 (C), τ ), τ (a ⊕ B) = 13 a + 23 τ2 (B), satisfies the weak Dixmier property. On the other hand, let T = 1 ⊕ 2 ∈ C ⊕ M2 (C). Then for every θ ∈ Aut(M, τ ), θ (T ) = T . Hence, τ (T ) is not in the operator norm closure of the convex hull of {θ (T ): θ ∈ Aut(M, τ )}. 3.6. A comparison theorem The following theorem is the main result of this section. Theorem 3.30. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property. If ||| · ||| is a normalized tracial gauge semi-norm on M, then for all T ∈ M, T 1 |||T ||| T . In particular, every tracial gauge semi-norm on M is a norm. Proof. By Corollary 3.5, |||T ||| T for every T ∈ M. To prove T 1 |||T |||, we can assume T 0. Let > 0. Since (M, τ ) satisfies the weak Dixmier property, there are S1 , . . . , Sk in k M such that T , S1 , . . . , Sk are equi-measurable and τ (T ) − S1 +···+S < . By Corollary 3.5, k S1 +···+Sk S1 +···+Sk k |||τ (T ) − ||| τ (T ) − < . Hence T = |τ (T )| ||| S1 +···+S ||| + 1 k k k |||S1 |||+···+|||Sk ||| + = |||T ||| + . 2 k By Theorem 3.30 and Lemma 3.21, we have the following corollary. Corollary 3.31. Let (M, τ ) be a finite factor and let ||| · ||| be a normalized unitarily invariant norm on M. Then T 1 |||T ||| T ,
∀T ∈ M.
In particular, every unitarily invariant semi-norm on a finite factor is a norm. By Theorem 3.30 and Lemma 3.15, we have the following corollary. Corollary 3.32. Let ||| · ||| be a normalized symmetric gauge semi-norm on (L∞ [0, 1], n (or (Cn , τ ), where τ ((x1 , . . . , xn )) = x1 +···+x ). Then n T 1 |||T ||| T ,
1 0
dx)
∀T ∈ L∞ [0, 1] or Cn .
In particular, every symmetric gauge semi-norm on (L∞ [0, 1], is a norm.
1 0
dx) (or (Cn , τ ), respectively)
4. Proof of Theorem B To prove Theorem B, we need the following lemmas. Lemma 4.1. Let E1 , . . . , En be projections in M such that E1 + · · · + En = 1 and T ∈ M. Then S = E1 T E1 + · · · + En T En is in the convex hull of {U T U ∗ : U ∈ U(M)}.
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Proof. Let T = (Tij ) be the matrix with respect to the decomposition 1 = E1 + · · · + En . Let U = −E1 + E2 + · · · + En . Then simple computation shows that ⎛
T11 ⎜ 0 1 ⎜ UT U∗ + T = ⎜ . ⎝ .. 2 0
0 T22 .. .
··· ··· .. .
⎞ 0 T2n ⎟ ⎟ .. ⎟ = E1 T E1 + (1 − E1 )T (1 − E1 ). . ⎠
Tn2
···
Tnn
By induction, S = E1 T E1 + · · · + En T En is in the convex hull of {U T U ∗ : U ∈ U(M)}.
2
Corollary 4.2. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a unitarily invariant norm on M. Let E1 , . . . , En be projections in M such that E1 + · · · + En = 1 and T ∈ M and S = E1 T E1 + · · · + En T En . Then |||S||| |||T |||. Recall that for a (non-zero) finite projection E in M, τE (ET E) = on ME = EME.
τ (ET E) τ (E)
is the induced trace
Lemma 4.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and ||| · ||| be a tracial gauge norm on M. Suppose T , E1 , . . . , En ∈ M, T 0, E1 + · · · + En = 1. Then |||T ||| |||τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En |||. Proof. We may assume that |||1||| = 1. Since M satisfies the weak Dixmier property, by Corollary 3.28, (MEi , τEi ) also satisfies the weak Dixmier property, 1 i n. Let > 0. There are operators Si1 , . . . , Sik in MEi such that Ei T Ei , Si1 , . . . , Sik are equi-measurable and Si1 + · · · + Sik − τ (E T E )E Ei i i i < . k Let Sj = S1j E1 + · · · + Snj En , 1 j k. Then T , S1 , . . . , Sn are equi-measurable and S1 + · · · + Sk − τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En < . k By Corollary 3.5, |||
S1 + · · · + Sk − τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En ||| < . k
Hence, |||τE1 (E1 T E1 )E1 + · · ·+ τEn (En T En )En ||| |||T ||| + . Since > 0 is arbitrary, we obtain the lemma. 2 Corollary 4.4. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If A is a finite-dimensional abelian von Neumann subalgebra of M and EA is the normal conditional expectation from M onto A preserving τ , then for every T ∈ M, |||EA (T )||| |||T |||.
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Proof. Let A = {E1 , . . . , En } such that E1 + · · · + En = 1. Then for every T ∈ M, EA (T ) = τE1 (E1 T E1 )E1 + · · · + τEn (En T En )En . By Corollary 4.2 and Lemma 4.3, |||EA (T )||| |||T |||.
2
Proof of Theorem B. By Lemma 3.13 and Corollary 3.19, ||| · ||| is unitarily invariant. Suppose Tα is a net in M1,|||·||| such that limα Tα = T in the weak operator topology. Let T = U |T | be the polar decomposition of T . Then limα U ∗ Tα = |T | in the weak operator topology. Since ||| · ||| is unitarily invariant, |||U Tα ||| 1 and ||||T |||| = |||T |||. So we may assume that T 0 and Tα = Tα∗ . By the spectral decomposition theorem and Corollary 3.5, to prove |||T ||| 1, we need to prove |||S||| 1 for every positive simple operator S such that S T . Let S = a1 E1 + · · · + an En and > 0. Since limα Tα = T S, limα Ei Tα Ei = Ei T Ei ai Ei for 1 i n. Hence, limα τEi (Ei (Tα + )Ei ) ai + > ai . So there is a β such that τE1 (E1 (Tβ + )E1 )E1 + · · · + τEn (En (Tβ + )En )En S. By Lemma 4.3 and Corollary 3.3, 1 + |||Tβ + ||| |||τ (E1 (Tβ + )E1 )E1 + · · · + τ (En (Tβ + )En )En ||| |||S|||. Since > 0 is arbitrary, |||S||| 1. 2 Proof of Corollary 1. Since A is a separable abelian von Neumann algebra, there is a sequence of finite-dimensionalabelian von Neumann subalgebras An such that A1 ⊂ A2 ⊂ · · · ⊂ A and A is the closure of n An in the strong operator topology. Let EAn be the normal conditional expectation from M onto An preserving τ . Then for every T ∈ M, EA (T ) = limn→∞ EAn (T ) in the strong operator topology. By Theorem B and Corollary 4.4, |||EA (T )||| |||T |||. 2 In the following we give some other useful corollaries of Theorem B. Corollary 4.5. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose 0 T1 T2 · · · T in M such that limn→∞ Tn = T in the weak operator topology. Then limn→∞ |||Tn ||| = |||T |||. Proof. By Corollary 3.3, |||T1 ||| |||T2 ||| · · · |||T |||. Hence, limn→∞ |||Tn ||| |||T |||. By Theorem B, limn→∞ |||Tn ||| |||T |||. 2 Corollary 4.6. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and ||| · |||1 and ||| · |||2 be two tracial gauge norms on M. Then ||| · |||1 = ||| · |||2 on M if |||T |||1 = |||T |||2 for every operator T = a1 E1 + · · · + an En in M such that a1 , . . . , an 0 and τ (E1 ) = · · · = τ (En ) = n1 , n = 1, 2, . . . . Proof. We need only to prove |||T |||1 = |||T |||2 for every positive operator T in M. By Theorem 3.27, M is either a finite-dimensional von Neumann algebra such that τ (E) = τ (F ) for arbitrary two non-zero minimal projections in M or M is diffuse. If M is a finite-dimensional von Neumann algebra such that τ (E) = τ (F ) for arbitrary two non-zero minimal projections in M, then the corollary is obvious. If M is diffuse, by the spectral decomposition theorem, there is a sequence of operators Tn ∈ M satisfying the following conditions: 1. 0 T1 T2 · · · T ,
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2. Tn = an1 En1 + · · · + ann Enn , an1 , . . . , ann 0 and τ (En1 ) = · · · = τ (Enn ) = n1 , 3. limn→∞ Tn = T in the weak operator topology. By the assumption of the corollary, |||Tn |||1 = |||Tn |||2 . By Corollary 4.5, |||T |||1 = |||T |||2 .
2
Corollary 4.7. Let M be a type II1 factor and ||| · |||1 and ||| · |||2 be two unitarily invariant norms on M. Then ||| · |||1 = ||| · |||2 on M if ||| · |||1 = ||| · |||2 on all type In subfactors of M, n = 1, 2, . . . . 5. Ky Fan norms on finite von Neumann algebras Let (M, τ ) be a finite von Neumann algebra and 0 t 1. For T ∈ M, define the Ky Fan tth norm by
|||T |||(t)
T , 1 t t 0 μs (T ) ds,
t = 0; 0 < t 1.
Let M1 = (M, τ ) ∗ (LF2 , τ ) be the reduced free product von Neumann algebra of M and the free group factor LF2 . Then M1 is a type II1 factor with a faithful normal trace τ1 such that the restriction of τ1 to M is τ . Recall that U(M1 ) is the set of unitary operators in M1 and P(M1 ) is the set of projections in M1 . Lemma 5.1. For 0 < t 1, t|||T |||(t) = sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t}. Proof. We may assume that T is a positive operator. Let A be a separable diffuse abelian von Neumann subalgebra of M1 containing T and let α be a ∗-isomorphism from (A, τ1 ) onto 1 1 (L∞ [0, 1], 0 dx) such that τ1 = 0 dx · α. Let f (x) = α(T ) and let f ∗ (x) be the non-increasing rearrangement of f (x). Then μs (T ) = f ∗ (s). By the definition of f ∗ (see (2.1)),
1 t m f ∗ > f ∗ (t) = lim m f ∗ > f ∗ (t) + n→∞ n and
∗ 1 ∗ ∗ ∗ m f f (t) lim m f > f (t) − t. n→∞ n Since f ∗ and f are equi-measurable, m({f > f ∗ (t)}) t and m({f f ∗ (t)}) t. Therefore, there is a measurable subset A of [0, 1], {f > f ∗ (t)} ⊂ A ⊂ {f f ∗ (t)}, such that m(A) = t. t Since f (x) and f ∗ (x) are equimeasurable, A f (s) ds = 0 f ∗ (s) ds. Let E = α −1 (χA ). t ∗ Then τ1 (E ) = t and τ1 (T E ) = A f (s) ds = 0 f (s) ds = t|||T |||(t) . Hence, t|||T |||(t) sup{|τ1 (U T E)|: U ∈ U(M), E ∈ P(M1 ), τ1 (E) = t}. We need to prove that if E is a projection in M1 , τ1 (E) = t, and U ∈ U(M1 ), then t|||T |||(t) |τ1 (U T E)|. By the Schwarz inequality, |τ1 (U T E)| = τ1 (EU T 1/2 T 1/2 E) τ1 (U ∗ EU T )1/2 × 1 τ1 (ET )1/2 . By Corollary 2.8, τ1 (ET ) = 0 μs (ET ) ds. By Corollary 2.10, μs (ET ) t min{μs (T ), μs (E)T }. Note that μs (E) = 0 for s τ1 (E) = t. Hence, τ1 (ET ) 0 μs (T ) ds = t|||T |||t . Similarly, τ1 (U ∗ EU T ) t|||T |||t . So |τ1 (U T E)| t|||T |||t . This proves that t|||T |||(t) sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t}. 2
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Theorem 5.2. For 0 t 1, ||| · |||(t) is a normalized tracial gauge norm on (M, τ ). Proof. We only prove the triangle inequality, since the other parts are obvious. We may assume that 0 < t 1. Let S, T ∈ M. By Lemma 5.1, t|||S + T |||(t) = sup{|τ1 (U (S + T )E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} sup{|τ1 (U SE)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} + sup{|τ1 (U T E)|: U ∈ U(M1 ), E ∈ P(M1 ), τ1 (E) = t} = t|||S|||(t) + t|||T |||(t) . 2 Proposition 5.3. Let (M, τ ) be a finite von Neumann algebra and T ∈ (M, τ ). Then |||T |||(t) is a non-increasing continuous function on [0, 1]. Proof. Let 0 < t1 < t2 1. 1 |||T |||(t1 ) − |||T |||(t2 ) = t1 =
1 t1
t1 0
1 μs (T ) ds − t2
t1 0
μs (T ) ds −
t2 μs (T ) ds 0
t2
1 t2 −t1
t1
μs (T ) ds
t2 (t2 − t1 )
0.
Since μs (T ) is right-continuous, |||T |||(t) is a non-increasing continuous function on [0, 1].
2
Example 5.4. The Ky Fan nk th norm of a matrix T ∈ (Mn (C), τn ) is |||T ||| k = n
s1 (T ) + · · · + sk (T ) , k
1 k n.
6. Dual norms of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property 6.1. Dual norms Let ||| · ||| be a norm on a finite von Neumann algebra (M, τ ). For T ∈ M, define |||T |||#M = sup τ (T X): X ∈ M, |||X||| 1 . When no confusion arises, we simply write ||| · |||# instead of ||| · |||#M . Lemma 6.1. ||| · |||# is a norm on M. Proof. If T = 0, |||T |||# τ (T T ∗ )/|||T ∗ ||| > 0. It is easy to see that |||λT |||# = |λ| · |||T |||# and |||T1 + T2 |||# |||T1 |||# + |||T2 |||# . 2 Definition 6.2. ||| · |||# is called the dual norm of ||| · ||| on M with respect to τ . The next lemma follows directly from the definition of dual norm.
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Lemma 6.3. Let ||| · ||| be a norm on a finite von Neumann algebra (M, τ ) and let ||| · |||# be the dual norm on M. Then for S, T ∈ M, |τ (ST )| |||S||| · |||T |||# . The following corollary is a generalization of Hölder’s inequality for bounded operators in finite von Neumann algebras. Corollary 6.4. Let (M, τ ) be a finite von Neumann algebra and let ||| · ||| be a gauge norm on M. Then for S, T ∈ M, ST 1 |||S||| · |||T |||# . Proof. By Lemma 3.8, ST 1 = sup{|τ (U ST )|: U ∈ U(M)}. By Lemmas 6.3 and 3.2, |τ (U ST )| |||U S||| · |||T |||# = |||S||| · |||T |||# . 2 Proposition 6.5. If ||| · ||| is a unitarily invariant norm on a finite von Neumann algebra (M, τ ), then ||| · |||# is also a unitarily invariant norm on M. Proof. Let U be a unitary operator. Then |||U T |||# = sup{|τ (U T X)|: X ∈ M, |||X||| 1} = sup{|τ (T XU )|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T ||| and |||T U |||# = sup{|τ (T U X)|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T |||. 2 Proposition 6.6. If ||| · ||| is a symmetric gauge norm on a finite von Neumann algebra (M, τ ), then ||| · |||# is also a symmetric gauge norm on (M, τ ). Proof. Let θ ∈ Aut(M, τ ). Then |||θ (T )|||# = sup{|τ (θ (T )X)|: X ∈ M, |||X||| 1} = sup{|τ (θ (T θ −1 (X)))|: X ∈ M, |||X||| 1} = sup{|τ (T θ −1 (X))|: X ∈ M, |||X||| 1} = sup{|τ (T X)|: X ∈ M, |||X||| 1} = |||T |||. 2 Lemma 6.7. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If T ∈ M is a positive operator, then |||T |||# = sup τ (T X): X ∈ M, X 0, XT = T X, |||X||| 1 . Proof. Let A be a separable abelian von Neumann subalgebra of M containing T and let EA be the normal conditional expectation from M onto A preserving τ . For every Y ∈ M such that |||Y ||| 1, let X = EA (Y ). By Corollary 1, ||||X|||| = |||X||| |||Y ||| 1. Furthermore, |τ (T Y )| = |τ (EA (T Y ))| = |τ (T EA (Y ))| = |τ (T X)| τ (T |X|). Hence, |||T |||# = sup τ (T X): X ∈ M, X 0, XT = T X, |||X||| 1 .
2
Lemma 6.8. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose T = a1 E1 + · · · + an En is a positive simple operator in M. Then |||T ||| = sup τ (T X): X = b1 E1 + · · · + bn En 0 and |||X|||# 1 n # = sup ak bk τ (Ek ): X = b1 E1 + · · · + bn En 0 and |||X||| 1 . k=1
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Proof. By Lemma 6.7, |||T |||# = sup{|τ (T X)|: X ∈ M, X 0, XT = T X, |||X||| 1}. Let A = {E1 , . . . , En } and let EA be the normal conditional expectation from M onto A preserving τ . Then S = EA (X) = τE1 (E1 XE1 )E1 + · · · + τEn (En XEn )En is a positive operator, τ (T X) = τ (EA (T X)) = τ (T EA (X)) = τ (T S), and |||S||| |||X||| by Corollary 4.4. Combining the definition of dual norm, this proves the lemma. 2 Corollary 6.9. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Suppose S, T are equi-measurable, positive simple operators in M. Then |||S|||# = |||T |||# . Theorem 6.10. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. Then ||| · |||# is also a tracial gauge norm on M. Furthermore, if |||1||| = 1, then |||1|||# = 1. Proof. By Lemma 3.13, ||| · ||| is a symmetric gauge norm on M. By Proposition 6.6, Corollary 6.9 and Lemma 3.10, ||| · |||# is a tracial gauge norm on M. Note that |||1||| = 1, hence, |||1|||# τ (1 · 1) = 1. On the other hand, by Theorem 3.30, |||1|||# = sup{|τ (X)|: X ∈ M, |||X||| 1} sup{|||X|||: X ∈ M, |||X||| 1} 1. 2 Corollary 6.11. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on M. If N is a von Neumann subalgebra of M satisfying the weak Dixmier property, then ||| · |||#N is the restriction of ||| · |||#M to N . Proof. Let ||| · |||1 = ||| · |||#N and let ||| · |||2 be the restriction of ||| · |||#M to N . By Theorem 6.10, both ||| · |||1 and ||| · |||2 are tracial gauge norms on N . By Lemma 3.6, to prove ||| · |||1 = ||| · |||2 , we need to prove |||T |||1 = |||T |||2 for every positive simple operator T ∈ N . Let A be a finitedimensional abelian von Neumann subalgebra of N containing T . By Lemma 6.8, |||T |||#M = |||T |||#N = |||T |||#A . So |||T |||1 = |||T |||2 . 2 6.2. Dual norms of Ky Fan norms For (x1 , . . . , xn ) ∈ Cn , τ (x) =
x1 +···+xn n
defines a trace on Cn . For 1 k n, the Ky Fan x ∗ +···+x ∗
(Cn , τ ) is |||(x1 , . . . , xn )|||( k ) = 1 k k , where (x1∗ , . . . , xn∗ ) is the decreasing rearn rangement of (|x1 |, . . . , |xn |). Let = {(x1 , . . . , xn ) ∈ Cn : x1 x2 xk = xk+1 = · · · = xn 0, x1 +···+xk 1} and E be the set of extreme points of . k The proof of the following lemma is an easy exercise. k n th norm on
k k Lemma 6.12. E consists of k + 1 points: (k, 0, . . .), ( k2 , k2 , 0, . . .), . . . , ( k−1 , . . . , k−1 , 0, . . .), (1, 1, . . . , 1) and (0, 0, . . . , 0).
The following lemma is well known. For a proof we refer to [8, 10.2]. Lemma 6.13. Let s1 s2 · · · sn 0 and t1 , . . . , tn 0. If t1∗ t2∗ · · · tn∗ is the decreasing rearrangement of t1 , . . . , tn , then s1 t1∗ + · · · + sn tn∗ s1 t1 + · · · + sn tn .
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Lemma 6.14. For T ∈ (Mn (C), τn ),
k |||T |||#( k ) = max T , T 1 . n n Proof. Let |||T |||1 = |||T |||# k and |||T |||2 = max{ nk T , T 1 }. Then both ||| · |||1 and ||| · |||2 are (n)
unitarily invariant norms on Mn (C). To prove ||| · |||1 = ||| · |||2 , we need only to prove |||T |||1 = |||T |||2 for every positive matrix T in Mn (C). We can assume that ⎞ ⎛ s1 ⎟ ⎜ .. T =⎝ ⎠, . sn where s1 , . . . , sn are s-numbers of T such that s1 s2 · · · sn . By Lemmas 6.8 and 6.13,
n
n i=1 si ti i=1 si ti : (t1 , . . . , tn ) ∈ = sup : (t1 , . . . , tn ) ∈ E . |||T |||1 = sup n n Note that T = s1 s2 · · · sn 0. By Lemma 6.12 and simple computations, |||T |||1 = max{ nk T , T 1 } = |||T |||2 . 2 The next lemma simply follows from the definition of dual norms. Lemma 6.15. Let (M, τ ) be a finite von Neumann algebra and ||| · |||, ||| · |||1 , ||| · |||2 be norms on M such that |||T |||1 |||T ||| |||T |||2 ,
∀T ∈ M.
|||T |||#2 |||T |||# |||T |||#1 ,
∀T ∈ M.
Then
Corollary 6.16. Let (M, τ ) be a finite von Neumann algebra and ||| · |||1 , ||| · |||2 be equivalent norms on M. Then ||| · |||#1 and ||| · |||#2 are equivalent norms on M. Theorem 6.17. Let M be a type II1 factor and 0 t 1. Then |||T |||#(t) = max{tT , T 1 },
∀T ∈ M.
Proof. Firstly, we assume t = nk is a rational number. Let Nr be a type Irn subfactor of M. Then the restriction of ||| · |||(t) to Nr is ||| · |||( rk ) . By Lemma 6.14 and Corollary 6.11, |||T |||#(t) = rn
max{tT , T 1 } for T ∈ Nr . By Corollary 4.7, |||T |||#(t) = max{tT , T 1 } for all T ∈ M. Now assume t is an irrational number. Let t1 , t2 be two rational numbers such that t1 < t < t2 . By Lemma 6.15, for every T ∈ M, max t2 T , T 1 |||T |||#(t) max t1 T , T 1 . Since t1 t t2 are arbitrary, |||T |||#(t) = max{tT , T 1 }.
2
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6.3. Proof of Theorem C Lemma 6.18. Let n ∈ N and τ be an arbitrary faithful state on Cn . If ||| · ||| is a norm on (Cn , τ ) and ||| · |||# is the dual norm with respect to τ , then ||| · |||## = ||| · |||. Proof. By Lemma 6.3, |||T |||## = sup{|τ (T X)|: X ∈ Cn , |||X|||# 1} |||T |||. We need to prove |||T ||| |||T |||## . By the Hahn–Banach theorem, there is a continuous linear functional φ on Cn with respect to the topology induced by ||| · ||| on Cn such that |||T ||| = φ(T ) and φ = 1. Since all norms on Cn induce the same topology, there is an element Y ∈ Cn such that φ(S) = τ (SY ) for all S ∈ Cn . By the definition of dual norm, |||Y |||# = φ = 1. By Lemma 6.3, |||T ||| = φ(T ) = τ (T Y ) |||T |||## . 2 Proof of Theorem C. By Theorem 6.10, both ||| · |||## and ||| · ||| are tracial gauge norms on M. By Corollary 3.6, to prove ||| · |||## = ||| · |||, we need to prove that |||T ||| = |||T |||## for every positive simple operator T ∈ M. Let A be the abelian von Neumann subalgebra generated by T . ## By Corollary 6.11 and Lemma 6.18, |||T |||## M = |||T |||A = |||T |||. 2 7. Proof of Theorem A Let (M, τ ) be a finite von Neumann algebra. Lemma 7.1. Let n ∈ N, a1 a2 · · · an an+1 = 0 and f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + n n n · · · + an χ[ n−1 ,1] (x). For T ∈ M, define n
1 |||T |||f =
(7.1)
f (s)μs (T ) ds. 0
Then |||T |||f =
n k(ak − ak+1 )
n
k=1
Proof. Since t|||T |||(t) =
t 0
1
=
2
n f (s)μs (T ) dt = a1
0 n k=1
(7.2)
n
μs (T ) ds, summation by parts shows that
1 |||T |||f =
|||T ||| k .
n μs (T ) ds + a2
0
k(ak − ak+1 ) |||T ||| k . n n
1 μs (T ) ds + · · · + an
1 n
μs (T ) ds n−1 n
2
Corollary 7.2. The norm ||| · |||f defined as above is a tracial gauge norm on M and |||1|||f = 1 a1 +···+an . 0 f (x) dx = n Lemma 7.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let {||| · |||α } be a set of tracial gauge norms on (M, τ ) such that |||1|||α 1 for all α. For
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every T ∈ M, define |||T ||| = sup |||T |||α . α
Then ||| · |||
α ||| · |||α
is also a tracial gauge norm on (M, τ ).
Proof. By Corollary 3.5, |||T ||| T is well defined. It is easy to check that ||| · ||| is a tracial gauge norm on (M, τ ). 2 Proof of Theorem A. Let
F = μs (X): X ∈ M, |||X|||# 1, X = b1 F1 + · · · + bk Fk 0, 1 where F1 + · · · + Fk = 1 and τ (F1 ) = · · · = τ (Fk ) = , k = 1, 2, . . . . k 1 For every positive operator X ∈ M such that |||X|||# 1, 0 μs (X) ds = τ (X) = X1 |||X|||# 1 by Theorem 3.30. Hence F ⊂ F and μs (1) = χ[0,1] (s) ∈ F by Theorem 6.10. For T ∈ M, define |||T ||| = sup |||T |||f : f ∈ F . By Corollary 7.2, |||·||| is a tracial gauge norm on M. To prove that |||·||| = |||·|||, by Corollary 4.6, we need prove that |||T ||| = |||T ||| for every positive operator T ∈ M such that T = a1 E1 + · · · + an En and τ (E1 ) = · · · = τ (En ) = n1 . By Lemma 6.8 and Theorem C, n 1 # |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0 and |||X||| 1 . n k=1
Note that if X = b1 E1 + · · · + bn En is a positive simple operator in M and |||X|||# 1, 1 then μs (X) ∈ F and |||T |||μs (X) = 0 μs (X)μs (T ) ds = n1 nk=1 ak∗ bk∗ , where {ak∗ } and {bk∗ } are non-increasing rearrangements of {ak } and {bk }, respectively. By Lemma 6.13, |||T ||| sup{|||T |||f : f ∈ F } = |||T ||| . We need to prove |||T ||| |||T ||| . Let X = b1 F1 + · · · + bk Fk be a positive operator in M such that F1 + · · · + Fk = 1, τ (F1 ) = · · · = τ (Fk ) = 1k and |||X|||# 1. We need only prove that |||T ||| |||T |||μs (X) . Since (M, τ ) satisfies the weak Dixmier property, by Theorem 3.27, (M, τ ) is either a von Neumann subalgebra of (Mn (C), τn ) that contains all diagonal matrices or M is a diffuse von Neumann algebra. In either case, we may assume that T = a˜ 1 E˜ 1 + · · · + a˜ r E˜ r and X = b˜1 F˜1 +· · ·+ b˜r F˜r , where E˜ 1 +· · ·+ E˜ r = F˜1 +· · ·+ F˜r = 1 and τ (E˜ i ) = τ (F˜i ) = 1r for 1 i r, a˜ 1 · · · a˜ r 0 and b˜1 · · · b˜r 0. Let Y = b˜1 E˜ 1 +· · ·+ b˜r E˜ r . Then X and Y are two equimeasurable operators in M and μs (X) = μs (Y ). By Theorem 6.10, |||Y |||# 1. By Lemma 6.3, 1 ˜ |||T ||| τ (T Y ) = a˜ i bi = r r
i=1
1
1 μs (Y )μs (T ) ds =
0
μs (X)μs (T ) ds = |||T |||μs (X) . 0
2
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Combining Theorem A and Lemma 3.21, we obtain the following corollary. Corollary 7.4. Let (M, τ ) be a finite factor and let ||| · ||| be a normalized unitarily invariant norm on M. Then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ M, |||T ||| = sup{|||T |||f : f ∈ F }. Combining Theorem A and Lemma 3.15 we obtain the following corollary. 1 Corollary 7.5. Let ||| · ||| be a normalized symmetric gauge norm on (L∞ [0, 1], 0 dx). Then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ L∞ [0, 1], |||T ||| = sup{|||T |||f : f ∈ F }. 8. Proof of Theorems D and E 1 Lemma 8.1. Let θ1 , θ2 be two embeddings from (L∞ [0, 1], 0 dx) into a finite von Neumann algebra (M, τ ). If ||| · ||| is a tracial gauge norm on M, then |||θ1 (f )||| = |||θ2 (f )||| for every f ∈ L∞ [0, 1]. Proof. If f ∈ L∞ [0, 1] is a positive function, then θ1 (f ) and θ2 (f ) are equi-measurable operators in M. Hence |||θ1 (f )||| = |||θ2 (f )|||. 2 Proof of Theorem D. We prove Theorem D for diffuse finite von Neumann algebras. The proof of the atomic case is similar. We may assume that the norms on M or L∞ [0, 1] are normalized. By the definition of Ky Fan norms, there is a one-to-one correspondence between Ky Fan tth 1 norms on (M, τ ) and Ky Fan tth norms on (L∞ [0, 1], 0 dx) as in Theorem D. By Lemma 7.1, Theorems 3.27 and A, there is a one-to-one correspondence between normalized tracial norms 1 on (M, τ ) and normalized symmetric gauge norms on (L∞ [0, 1], 0 dx) as in Theorem D. 2 Example 8.2. For 1 p ∞, the Lp -norm on L∞ [0, 1] defined by f (x) = p
1 ( 0 |f (x)|p dx)1/p , ess sup |f |,
1 p < ∞; p=∞
1 is a normalized symmetric gauge norm on (L∞ [0, 1], 0 dx). By Corollaries 2 and 2.8, the induced norm 1 (τ (|T |p ))1/p = ( 0 |μs (T )|p ds)1/p , 1 p < ∞; T p = T , p=∞ is a normalized unitarily invariant norm on a type II1 factor M. The norms { · p : 1 p ∞} are called Lp -norms on M. Corollary 8.3. Let (M, τ ) be a finite von Neumann algebra satisfying the weak Dixmier property and let ||| · ||| be a tracial gauge norm on (M, τ ). If (M, τ ) can be embedded into a finite factor (M1 , τ1 ), then there is a unitarily invariant norm ||| · |||1 on (M1 , τ1 ) such that ||| · ||| is the restriction of ||| · |||1 to (M, τ ).
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The following example shows that without the weak Dixmier property, Corollary 8.3 may fail. Example 8.4. On (C2 , τ ), τ ((x, y)) = 13 x + 23 y, let |||(x, y)||| = 23 |x| + 13 |y|. It is easy to see that ||| · ||| is a tracial gauge norm on (C2 , τ ). Let M1 be the reduced free product of (C2 , τ ) with the free group factor L(F2 ). Then M1 is a type II1 factor with a faithful normal trace τ1 such that the restriction of τ1 to C2 is τ . Suppose ||| · |||1 is a unitarily invariant norm on M1 such that the restriction of ||| · |||1 to C2 is ||| · |||. Let E = (1, 0) and F = (0, 1) in C2 . Then τ1 (E) = τ (E) < τ (F ) = τ1 (F ). So there is a unitary operator U in M1 such that U EU ∗ F . By Corollary 3.3, 23 = |||E||| = |||E|||1 = |||U EU ∗ |||1 |||F |||1 = |||F ||| = 13 . This is a contradiction. Proof of Theorem E. Let ||| · |||2 be the tracial gauge norm on M corresponding to the symmetric 1 gauge norm ||| · |||#1 on (L∞ [0, 1], 0 dx) as in Theorem D. By Lemma 4.6, to prove ||| · |||2 = ||| · |||# on M, we need to prove |||T |||2 = |||T |||# for every positive simple operator T = a1 E1 + · · · + an En in M such that τ (E1 ) = · · · = τ (En ) = n1 . We may assume that a1 · · · an 0. Then μs (T ) = a1 χ[0, 1 ) (s) + · · · + an χ[ n−1 ,1] (s). By Lemma 6.8, n
n
n 1 # |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0 and |||X||| 1 . n #
k=1
By Lemma 6.13,
n 1 |||T ||| = sup ak bk : X = b1 E1 + · · · + bn En 0, b1 · · · bn 0, |||X||| 1 . n #
k=1
By Theorem D and Lemma 6.8, n # 1 |||T |||2 = μs (T ) = sup ak bk : g(s) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s) 0, n n n k=1 g(s) 1 . By Lemma 6.13, n # 1 |||T |||2 = μs (T ) = sup ak bk : g(s) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s) 0, n n n k=1 b1 · · · bn 0, g(s) 1 . Note that if b1 · · · bn 0, then μs (b1 E1 + · · · + bn En ) = b1 χ[0, 1 ) (s) + · · · + bn χ[ n−1 ,1] (s). n n Since ||| · ||| is the tracial gauge norm on (M, τ ) corresponding to the symmetric gauge norm ||| · |||1
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on (L∞ [0, 1],
1 0
171
dx) as in Theorem D, |||b1 E1 + · · · + bn En ||| 1 if and only if |||b1 χ[0, 1 ) (s) +
· · · + bn χ[ n−1 ,1] (s)|||1 1. Therefore, |||T |||2 = |||T |||# . n
2
n
p Example 8.5. If p = 1, let q = ∞. If 1 < p < ∞, let q = p−1 . Then the Lq -norm on L∞ [0, 1] is p ∞ the dual norm of the L -norm on L [0, 1]. By Theorem E, the Lq -norm on a type II1 factor M is the dual norm of the Lp -norm on M.
9. Proof of Theorem F Proof of Theorem F. Let ||| · ||| be a tracial gauge norm on M. By Lemma 7.1, |||S|||f |||T |||f for every f ∈ F . By Theorem A, |||S||| |||T |||. 2 Corollary 9.1. Let M be a type II1 factor and S, T ∈ M. If |||S|||(t) |||T |||(t) for all Ky Fan tth norms, 0 t 1, then |||S||| |||T ||| for all unitarily invariant norms ||| · ||| on M. By Corollary 9.1, we obtain Ky Fan’s dominance theorem [13]. Ky Fan’s dominance theorem. If S, T ∈ Mn (C) and |||S|||(k/n) |||T |||(k/n) , i.e., ki=1 si (S) k i=1 si (T ) for 1 k n, then |||S||| |||T ||| for all unitarily invariant norms ||| · ||| on Mn (C). 10. Extreme points of normalized unitarily invariant norms on finite factors In this section, we assume that M is a finite factor with the unique tracial state τ . 10.1. N(M) Let N(M) be the set of normalized unitarily invariant norms on M. It is easy to see that N(M) is a convex set. Let F(M) be the set of complex functions defined on M. Then F(M) is a locally convex space such that a neighborhood of f ∈ F(M) is N (f, T1 , . . . , Tn , ) = g ∈ F(M): g(Ti ) − f (Ti ) < . In this topology, fα → f means limα fα (T ) = f (T ) for every T ∈ M. We call this topology the pointwise weak topology. Lemma 10.1. N(M) ⊆ F(M) is a compact convex subset in the pointwise weak topology. Proof. It is clear that N(M) is a convex subset of F(M). Suppose ||| · |||α ∈ F(M) and f (T ) = limα |||T |||α for every T ∈ M. It is easy to check that f (T ) defines a unitarily invariant semi-norm on M such that f (1) = 1. By Corollary 3.31, f (T ) is a norm and f ∈ N(M). 2 Let Ne (M) be the subset of extreme points of N(M). By the Krein–Milman theorem, the closure of the convex hull of Ne (M) is N(M) in the pointwise weak topology. It is an interesting question of characterizing Ne (M). In the following, we will provide some results on Ne (M).
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10.2. Ne (Mn (C)) For n 2, let 1 ⊕ s2 ⊕ · · · ⊕ sn be the matrix ⎛ 1 ⎜ s2 ⎜ ⎜ .. ⎝ .
⎞ ⎟ ⎟ ⎟ ∈ Mn (C). ⎠ sn
Let ||| · ||| be a normalized unitarily invariant norm on Mn (C). For 0 sn · · · s2 1, define f (s2 , . . . , sn ) = f|||·||| (s2 , . . . , sn ) = |||1 ⊕ s2 ⊕ · · · ⊕ sn |||.
(10.1)
In the following, let Ωn−1 = {(s2 , . . . , sn ): 0 sn · · · s2 1}. By [7, Lemma 3.2] and Corollary 3.31, we have the following lemma. Lemma 10.2. Let f (s2 , . . . , sn ) be a function defined on Ωn−1 . In order that f (s2 , . . . , sn ) = f|||·||| (s2 , . . . , sn ) for some ||| · ||| ∈ N(Mn (C)), it is necessary and sufficient that the following conditions are satisfied: 1. f (s2 , . . . , sn ) > 0 for all (s2 , . . . , sn ) ∈ Ωn−1 and f (1, . . . , 1) = 1; 2. f (s2 , . . . , sn ) is a convex function on Ωn−1 ; 3. for 0 sn sn−1 · · · s1 , 0 tn tn−1 · · · t1 , if ki=1 si ki=1 ti for 1 k n, then s1 · f ( ss21 , . . . , ssn1 ) t1 · f ( tt21 , . . . , ttn1 ). If f (s2 , . . . , sn ) satisfies the above conditions, then f satisfies 1 + s2 + · · · + sn f (s2 , . . . , sn ) 1 n for all (s2 , . . . , sn ) ∈ Ωn−1 . Let ||| · |||1 , ||| · |||2 ∈ N(Mn (C)). If |||S|||1 = |||S|||2 for all S = 1 ⊕ s2 ⊕ · · · ⊕ sn , (s2 , . . . , sn ) ∈ Ωn−1 , then |||T |||1 = |||T |||2 for every matrix T ∈ Mn (C). This implies the following lemma. Lemma 10.3. Let |||·|||1 , |||·|||2 ∈ N(Mn (C)). Then |||·|||1 = |||·|||2 if and only if f|||·|||1 (s2 , . . . , sn ) = f|||·|||2 (s2 , . . . , sn ) for all (s2 , . . . , sn ) ∈ Ωn−1 . Let 1 m n. Suppose ||| · ||| is a normalized unitarily invariant norm on Mm (C) and g(s2 , . . . , sm ) = g|||·||| (s2 , . . . , sm ) is the function on Ωm−1 induced by ||| · ||| (see (10.1)). Define f (s2 , . . . , sn ) on Ωn−1 by f (s2 , . . . , sn ) = g(s2 , . . . , sm ),
(s2 , . . . , sn ) ∈ Ωn−1 .
It is easy to check that f (s2 , . . . , sn ) is a function on Ωn−1 satisfying Lemma 10.2. By Lemmas 10.2 and 10.3, there is a unique normalized unitarily invariant norm ||| · |||1 ∈ N(Mn (C)) such that f (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) = g(s2 , . . . , sm ) for all (s1 , . . . , sn ) ∈ Ωn−1 . (This fact can also be obtained by Corollary 7.4 and Lemma 10.3.) ||| · |||1 is called the induced norm of ||| · |||.
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Conversely, suppose ||| · |||1 is a normalized unitarily invariant norm on Mn (C) and f (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) is the function on Ωn−1 induced by ||| · |||1 . If f (s2 , . . . , sn ) = g(s2 , . . . , sm ) for all (s2 , . . . , sn ) ∈ Ωn−1 , then g(s2 , . . . , sm ) satisfies Lemma 10.2. Hence, there is a unique normalized unitarily invariant norm ||| · ||| on Mm (C) such that g(s2 , . . . , sm ) = g|||·||| (s2 , . . . , sm ) for all (s2 , . . . , sm ) ∈ Ωm−1 . ||| · ||| is called the reduced norm of ||| · |||1 . Proposition 10.4. For 1 k n, the Ky Fan extreme point of N(Mn (C)).
k n th
norm (see Example 5.4) on Mn (C) is an
Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(Mn (C)) satisfy ||| · |||( k ) = α||| · |||1 + n (1 − α)||| · |||2 . Let f (s2 , . . . , sn ) = f|||·||| k (s2 , . . . , sn ), f1 (s2 , . . . , sn ) = f|||·|||1 (s2 , . . . , sn ) and (n)
f2 (s2 , . . . , sn ) = f|||·|||2 (s2 , . . . , sn ) for (s2 , . . . , sn ) ∈ Ωn−1 . Then f (s2 , . . . , sn−1 ) = αf1 (s2 , . . . , sn−1 ) + (1 − α)f2 (s2 , . . . , sn−1 ). ∂f k , ∂s∂fk+1 = · · · = ∂s = 0. Since f1 (s2 , . . . , sn ), f2 (s2 , . . . , sn ) Since f (s2 , . . . , sn ) = 1+s2 +···+s k n are convex functions on Ωn−1 ,
∂fi ∂sj
0 for i = 1, 2 and k +1 j n. Since f = αf1 +(1−α)f2 ,
∂fi ∂sj
= 0 for i = 1, 2 and k + 1 j n. This implies that fi (s2 , . . . , sn ) = gi (s2 , . . . , sk ) for all (s2 , . . . , sn ) ∈ Ωn−1 and i = 1, 2. By the discussions above the proposition, there are normalized unitarily invariant norms ||| · |||1 , ||| · |||2 on Mk (C) such that gi (s2 , . . . , sk ) = (gi )|||·|||i (s2 , . . . , sk ) for all (s2 , . . . , sk ) ∈ Ωk−1 and i = 1, 2. By Lemma 10.2, gi (s2 , . . . , sk )
1 + s2 + · · · + sk k
for all (s2 , . . . , sk ) ∈ Ωk−1 and i = 1, 2. Since f = αf1 + (1 − α)f2 , 1 + s2 + · · · + sk = αg1 (s2 , . . . , sk ) + (1 − α)g2 (s2 , . . . , sk ). k This implies that g1 (s2 , . . . , sk ) = g2 (s2 , . . . , sk ) =
1+s2 +···+sk . k
So f = f1 = f2 .
2
The proof of the following proposition is similar to that of Proposition 10.4. Proposition 10.5. Let 1 m n and ||| · ||| be a normalized unitarily invariant norm on Mm (C). If ||| · ||| is an extreme point of N(Mm (C)), then the induced norm ||| · |||1 on Mn (C) is also an extreme point of N(Mn (C)). Question. For n 3, find all extreme points of N(Mn (C)). 10.3. Ne (M2 (C)) In this subsection, we will prove Theorem J. We need the following auxiliary results. The following lemma is a corollary of Lemma 10.2 in the case n = 2.
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Lemma 10.6. Let f (s) be a function on [0, 1]. If there is a normalized unitarily invariant norm ||| · ||| on M2 (C) such that f (s) = f|||·||| (s) = |||1 ⊕ s|||, then f (s) is an increasing convex function on [0, 1] satisfying 1+s f (s) 1, 2
∀s ∈ [0, 1].
Corollary 10.7. For 0 a b 1, we have 1 0 f (a−) f (a+) f (b−) f (b+) f (1−) . 2 Proof. Since f (s) is an increasing convex function, 0 f (a−) f (a+) f (b−) f (b+) f (1−). By Lemma 10.6, f (1) − f (1 − h) 1 − (2 − h)/2 1 lim = . h→0+ h→0+ h h 2
f (1−) = lim For
1 2
2
t 1, define ||| · |||t = max{tT , T 1 }.
Lemma 10.8. For 1/2 t 1, ||| · |||t is an extreme point of N(M2 (C)). Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(M2 (C)) such that ||| · |||t = α||| · |||1 + (1 − α)||| · |||2 . Let f (s) = f|||·|||t (s), f1 (s) = f|||·|||1 (s) and f2 (s) = f|||·|||2 (s). Then f (s) = αf1 (s) + (1 − α)f2 (s). Note that t 0 s 2t−1 2 ; f (s) = s+1 2t−1 2 2 s 1. 1 2t−1 Hence, f (s) = 0 if 0 s < 2t−1 2 and f (s) = 2 if 2 < s 1. By Corollary 10.7, f1 (s) = 2t−1 1 2t−1 f2 (s) = 0 if 0 s < 2 and f1 (s) = f2 (s) = 2 if 2 < s 1. Since f (s), f1 (s), f2 (s) are convex functions and hence continuous and f (1) = f1 (1) = f2 (1) = 1, f (s) = f1 (s) = f2 (s) for all 0 s 1. This implies that ||| · |||t = ||| · |||1 = ||| · |||2 . 2
Lemma 10.9. The mapping: t → ||| · |||t is continuous with respect to the usual topology on [1/2, 1] and the pointwise weak topology on N(M2 (C)). In particular, {||| · |||t : 1/2 t 1} is compact in the pointwise weak topology. Proof. For every 0 s 1, |||1 ⊕ s|||t = max{t, 1+s 2 } is a continuous function on [0, 1]. Hence, the mapping: t → ||| · |||t is continuous with respect to the usual topology on [1/2, 1] and the pointwise weak topology on N(M2 (C)). 2 Lemma 10.10. The set
1
S = ||| · |||: ||| · ||| =
||| · |||t dμ(t), μ is a regular Borel probability measure on [1/2, 1]
1/2
is a convex compact subset of N(M2 (C)) in the pointwise weak topology.
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Proof. Suppose {||| · |||α } is a net in S such that ||| · |||α → ||| · ||| ∈ N(M2 (C)) in the pointwise weak topology. Let μα be the regular Borel probability measure on [1/2, 1] corresponding to ||| · |||α . Then there is a subnet of μα that converges weakly to a regular Borel probability measure μ on [1/2, 1], i.e., for every continuous function φ(t) on [1/2, 1], 1
1 φ(t) dμαβ (t) =
lim α
1/2
φ(t) dμ(t).
1/2
In particular, for every T ∈ M2 (C), we have 1 |||T ||| = lim |||T |||αβ = lim αβ
αβ
1 |||T |||t dμαβ (t) =
1/2
Hence ||| · ||| ∈ S.
|||T |||t dμ(t).
1/2
2
Lemma 10.11. Let f (s) be a convex, increasing function on [0, 1] such that 1+s f (s) 1, 2
∀s ∈ [0, 1].
Then there is an element ||| · ||| ∈ S such that f (s) = |||1 ⊕ s|||. Proof. We can approximate f uniformly by piecewise linear functions satisfying the conditions of the lemma. By Lemma 10.10, we may assume that f (s) is a piecewise linear function. Furthermore, we may assume that 0 = a0 < a1 < a2 < · · · < an = 1 and f (s) is linear on [ai , ai+1 ] for 0 i n − 1. Let f (s) = αi /2 on [ai , ai+1 ]. By Corollary 10.7, 0 = α0 α1 · · · αn−1 1. Let g(s) = (1 − αn−1 )1 ⊕ s + (αn−1 − αn−2 )|||1 ⊕ s|||αn−1 + · · · + (α1 − α0 )|||1 ⊕ s|||α1 + α0 1 ⊕ s1 . Then g(1) = f (1) = 1 and g (s) = αi /2 on [ai , ai+1 ]. So g (s) = f (s) except s = αi for 1 i n. Hence f (s) = g(s) for all 0 s 1. 2 Proof of Theorem J. By Lemma 10.8, {||| · |||t : 1/2 t 1} are extreme points of N(M). By Lemmas 10.10, 10.11 and 10.3, the closure of the convex hull of {||| · |||t : 1/2 t 1} in the pointwise weak topology is N(M2 (C)). By Lemmas 10.8, 10.9 and by [11, Theorem 1.4.5], Ne (M2 (C)) = {||| · |||t : 1/2 t 1}. 2 Corollary 10.12. Let f (s) be a function on [0, 1]. Then the following conditions are equivalent: 1. f (s) = f|||·||| (s) = |||1 ⊕ s||| for some normalized unitarily invariant norm ||| · ||| on M2 (C); 2. f (s) is an increasing convex function on [0, 1] such that 1+s 2 f (s) 1 for all s ∈ [0, 1]; 3. f (s) is an increasing convex function on [0, 1] such that f (1) = 1 and f (1−) 12 . In the following, we will show how to write the Lp -norms on M2 (C) in terms of extreme points of N(M2 (C)). Recall that for 1 p < ∞, the Lp -norm of 1 ⊕ s is
1 ⊕ sp =
1 + sp 2
1/p .
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1/p , 0 s 1. Then f (1) = 1 and Let fp (s) = f·p (s) = ( 1+s p 2 )
fp (s) =
s p−1 1 + s p 1/p−1 , 2 2
fp (0) = 0, fp (1) = 12 . Lemma 10.13. For 1 < p < ∞ and 0 s 1, 1
|||1 ⊕ s|||t 4fp (2t − 1) dt.
fp (s) = 1/2
Proof. 1
|||1 ⊕ s|||t 4fp (2t − 1) dt
1/2
1 =
|||1 ⊕ s||| x+1 2fp (x) dx 2
0
s =
1 + s 2fp (x) dx + 2
1
1 + x 2fp (x) dx 2
s
0
1
= (1 + s)f (s) − (1 + s)f (0) + 2f (1) − (1 + s)f (s) −
fp (x) dx
s
= 1 − fp (1) + fp (s) = fp (s).
2
Corollary 10.14. For 1 < p < ∞ and T ∈ M2 (C), 1 T p =
|||T |||t 4fp (2t − 1) dt.
1/2
10.4. Proof of Theorem K Lemma 10.15. Let M be a type II1 factor and let ||| · ||| be a normalized unitarily invariant norm on M. Suppose N1 ⊂ N2 ⊂ · · · is a sequence of type Inr subfactors of M such that Nr ∼ = Mnr (C) and limr→∞ nr = ∞. If the restriction of ||| · ||| to Nr is an extreme point of N(Nr ) for all r = 1, 2, . . . , then ||| · ||| is an extreme point of N(M). Proof. Suppose 0 < α < 1 and ||| · |||1 , ||| · |||2 ∈ N(M) such that ||| · ||| = α||| · |||1 + (1 − α)||| · |||2 on M. Then for every r = 1, 2, . . . , ||| · ||| = α||| · |||1 + (1 − α)||| · |||2 on Nr . By the assumption of
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the lemma, ||| · ||| = ||| · |||1 = ||| · |||2 on Nr . By Corollary 4.7, ||| · ||| = ||| · |||1 = ||| · |||2 on M. So ||| · ||| is an extreme point of N(M). 2 Proof of Theorem K. By the assumption of the theorem, t = nk is a rational number. Then we can construct a sequence of type Irn subfactor Mrn of M such that Mn ⊆ M2n ⊆ · · · . Then the restriction of || · |||(t) on Mrn is || · |||( rk ) . By Proposition 10.4, the restriction of || · |||(t) on Mrn is rn an extreme point of N(Mrn (C)). By Lemma 10.15, || · |||(t) is an extreme point of N(M). 2 Remark 10.16. Here we point out other interesting examples of extreme points of N(M). For 0 t 1, recall that || · |||(t) is the tth Ky Fan norm on M. For any non-negative function c(t) on [0, 1] such that c(t)∞ = 1 and T ∈ M, define T |||[c(t)] = c(t)T |||(t) ∞ . Then it is easy to see that || · |||[c(t)] is a normalized unitarily invariant norm on M. It can be proved that if c(t) is a simple function or if tc(t) is a simple function, then|| · |||[c(t)] is an extreme point of N(M). 11. Proof of Theorem G In this section, we assume that M is a type II1 factor with the unique tracial state τ and ||| · ||| is a unitarily invariant norm on M. For two projections E, F in M, τ (E) τ (F ) if and only if there is a unitary operator U ∈ M such that U EU ∗ F . By Corollary 3.3, if τ (E) τ (F ), |||E||| |||F |||. So we can define r ||| · ||| =
lim
τ (E)→0+
|||E|||.
Definition 11.1. A unitarily invariant norm ||| · ||| on M is singular if r(||| · |||) > 0 and continuous if r(||| · |||) = 0. Example 11.2. The operator norm is singular since r( · ) = limτ (E)→0+ E = 1. If 0 < t 1, the Ky Fan tth norm ||| · |||(t) is continuous since r(||| · |||(1) ) = r( · 1 ) = limτ (E)→0+ τ (E) = 0 and r(||| · |||(t) ) 1t · r(||| · |||(1) ) = 0. If 1 p < ∞, it is easy to see that the Lp -norm on M is also continuous. Lemma 11.3. If ||| · ||| is singular, then ||| · ||| is equivalent to the operator norm · . Indeed, for every T ∈ M, we have r ||| · ||| T |||T ||| |||1||| · T . Proof. By Lemma 3.2, |||T ||| |||1||| · T . We need to prove r(||| · |||)T |||T |||. We may assume that T > 0. For any > 0, let E = χ[T −,T ] (T ) > 0. Then T (T − )E. By Corollary 3.3 and Lemma 3.2, |||T ||| |||(T − )E||| (T − ) · |||E||| (T − )r(||| · |||). Since > 0 is arbitrary, r(||| · |||)T |||T |||. 2
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Recall that a neighborhood N (, δ) of 0 ∈ M in the measure topology (see [14]) is N(, δ) = T ∈ M, there is a projection E ∈ M such that τ (E) < δ and T E ⊥ < Proof of Theorem G. By Lemma 11.3, if ||| · ||| is singular, then T is the operator topology on M1,· . Suppose ||| · ||| is continuous. For , δ > 0 and T ∈ M such that T 1 and |||T ||| < δ, by Corollary 3.31, τ (χ[,1] (|T |) T1 |||T ||| < δ and T · χ[0,) (|T |) < . This implies that {T ∈ M1,· : |||T ||| < δ} ⊆ N (, δ). Conversely, let ω > 0. Since r(||| · |||) = 0, there is an , 0 < < ω/2, such that if τ (E) < then |||E||| < ω/2. For every T ∈ N (, ω/2) and T 1, choose E ∈ M such that τ (E) < and T E ⊥ < ω/2. By Proposition 3.18 and Corollary 3.5, |||T ||| |||T E||| + |||T E ⊥ ||| < T · |||E||| + T E ⊥ < ω/2 + ω/2 = ω. Hence {T ∈ N(, ω/2): T 1} ⊆ {T ∈ M: |||T ||| < ω}. 2 Corollary 11.4. Topologies induced by the Lp -norms, 1 p < ∞, on the unit ball of a type II1 factor are the same. 12. Completion of type II1 factors with respect to unitarily invariant norms In this section, we assume that M is a type II1 factor with the unique tracial state τ and ||| · ||| is a unitarily invariant norm on M. The completion of M with respect to ||| · ||| is denoted by M|||·||| . We will use the traditional notation Lp (M, τ ) to denote the completion of M with respect to be the completion the Lp -norm defined as in Example 11.2. Note that L∞ (M, τ ) = M. Let M of M in the measure topology in the sense of [14]. 12.1. Embedding of M|||·||| into M Lemma 12.1. Let ||| · ||| be a continuous unitarily invariant norm on M and T ∈ M. For every > 0, there is a δ > 0 such that if τ (E) < δ, then |||T E||| < . Proof. Since ||| · ||| is continuous, limτ (E)→0 |||E||| = 0. Hence, for every > 0, there is a δ > 0 such that if τ (E) < δ, then |||E||| < 1+T . By Proposition 3.18, |||T E||| T · |||E||| < . 2 Lemma 12.2. Let ||| · ||| be a continuous unitarily invariant norm on M and let {Tn } in M be a Cauchy sequence with respect to ||| · |||. For every > 0, there is a δ > 0 such that if τ (E) < δ, then |||Tn E||| < for all n. Proof. Since {Tn } is a Cauchy sequence with respect to ||| · |||, there is an N such that for all n N , |||Tn − TN ||| < /2. By Lemma 12.1, there is a δ1 such that if τ (E) < δ1 then |||TN E||| < /2. By Proposition 3.18, for n N , |||Tn E||| |||(Tn − TN )E||| + |||TN E||| < |||(Tn − TN )||| · E + /2 < . A simple argument shows that we can choose 0 < δ < δ1 such that if τ (E) < δ then |||Tn E||| < for all n. 2 The following proposition generalizes Theorem 5 of [14]. Proposition 12.3. Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. that extends the identity map from M to M. There is an injective map from M|||·||| to M
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Proof. If ||| · ||| is singular, by Lemma 11.3, M|||·||| = M. So we will assume that ||| · ||| is continuous. If {Tn } in M is a Cauchy sequence with respect to ||| · |||, then {Tn } is a Cauchy sequence |) in the L1 -norm by Corollary 3.31. For every δ > 0 and T ∈ M, τ (χ(δ,∞) (|T |) τ (|T δ . Hence, 1 if {Tn } is a Cauchy sequence in M in the L -norm, then {Tn } is a Cauchy sequence in the that extends the identity map measure topology. So there is a natural map Φ from M|||·||| to M from M to M. To prove that Φ is injective, we need to prove that if {Tn } in M is a Cauchy sequence with respect to ||| · ||| and Tn → 0 in the measure topology, then limn→∞ |||Tn ||| = 0. Let > 0. By Lemma 12.2, there is a δ > 0 such that if τ (E) < δ then |||Tn E||| < /2 for all n. Since Tn → 0 in the measure topology, there are N and δ1 , 0 < δ1 < δ, such that for all n N , there is a projection En such that τ (En ) < δ1 and Tn En⊥ < /2. By Corollary 3.31, |||Tn ||| |||Tn En⊥ ||| + |||Tn En ||| < Tn En⊥ + /2 < . This proves that limn→∞ |||Tn ||| = 0 and that extends the identity map from M to M. 2 hence Φ is an injective map from M|||·||| to M By the proof of Proposition 12.3, we have the following. Corollary 12.4. There is an injective map from M|||·||| to L1 (M, τ ) that extends the identity map from M to M. The following corollary is very By Proposition 12.3, we will consider M|||·||| as a subset of M. useful. Corollary 12.5. Let M be a type II1 factor and let ||| · ||| be a unitarily invariant norm on M. If {Tn } ⊂ M is a Cauchy sequence with respect to ||| · ||| and limn→∞ Tn = T in the measure topology, then T ∈ M|||·||| and limn→∞ Tn = T in the topology induced by ||| · |||. satisfying the following conditions: Corollary 12.6. M|||·||| is a linear subspace of M 1. if T ∈ M|||·||| , then T ∗ ∈ M|||·||| ; 2. T ∈ M|||·||| if and only if |T | ∈ M|||·||| ; 3. if T ∈ M|||·||| and A, B ∈ M, then AT B ∈ M|||·||| and |||AT B||| A · |||T ||| · B. In particular, ||| · ||| can be extended to a unitarily invariant norm, also denoted by ||| · |||, on M|||·||| . and L1 (M, τ ) 12.2. M The following theorem is due to Nelson [14]. is a ∗-algebra and T ∈ M if and only if T is a closed, Theorem 12.7. (See Nelson [14].) M is a positive operator, then densely defined operator affiliated with M. Furthermore, if T ∈ M limn→∞ χ[0,n] (T ) = T in the measure topology. as in [5]. In the following, we define s-numbers for unbounded operators in M and 0 s 1, define the sth numbers of T by Definition 12.8. For T ∈ M μs (T ) = inf T E: E ∈ M is a projection such that τ E ⊥ = s .
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such Theorem 12.9. (See Fack and Kosaki [5].) Let T and Tn be a sequence of operators in M that limn→∞ Tn = T in the measure topology. Then for almost all s ∈ [0, 1], limn→∞ μs (Tn ) = μs (T ). Let {Tn } be a sequence of operators in M such that T = limn→∞ Tn in the L1 -norm. By Lemma 3.8, {τ (Tn )} is a Cauchy sequence in C. Define τ (T ) = limn→∞ τ (Tn ). It is obvious that τ (T ) does not depend on the sequence {Tn }. In this way, τ is extended to a linear functional on L1 (M, τ ). Lemma 12.10. Let ||| · ||| be a normalized unitarily invariant norm on a type II1 factor M. If T ∈ M|||·||| and X ∈ M, then T X ∈ L1 (M, τ ). Proof. By the proof of Proposition 12.3, limn→∞ Tn = T in the measure topology. Hence limn→∞ Tn X = T X in the measure topology (see [14, Theorem 1]). By Corollary 6.4, Tn X − Tm X1 |||Tn − Tm ||| · |||X|||# . So {Tn X} is a Cauchy sequence in the L1 -norm. By Corollary 12.5, T X ∈ L1 (M, τ ) and limn→∞ Tn X = T X in the L1 -norm. 2 12.3. Elements in M|||·||| Lemma 12.11. For all T ∈ M|||·||| , |||T ||| = sup τ (T X): X ∈ M, |||X|||# 1 . Proof. Let {Tn } be a sequence of operators in M such that limn→∞ Tn = T with respect to ||| · |||. By Corollary 6.4, if X ∈ M and |||X|||# 1, then |τ (T X)| = limn→∞ |τ (Tn X)| limn→∞ |||Tn ||| = |||T |||. Therefore, |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. We need to prove that |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. Let > 0. Since limn→∞ Tn = T with respect to ||| · |||, there is an N such that |||T − TN ||| < /3. For TN , there is an X ∈ M, |||X|||# 1, such that |||TN ||| |τ (TN X)| + /3. By the proof of Lemma 12.10 and Corollary 6.4, τ (T X) − τ (TN X) = lim τ (Tn X) − τ (TN X) n→∞
lim |||Tn − TN ||| · |||X|||# |||T − TN ||| < /3. n→∞
So |τ (T X)| |τ (TN X)| − |τ ((TN − T )X)| |||TN ||| − /3 − /3 |||T ||| − . Therefore, |||T ||| sup{|τ (T X)|: X ∈ M, |||X|||# 1}. 2 The following theorem generalizes Theorem A. Its proof is based on Lemma 12.11 and is similar to the proof of Theorem A. So we omit the proof. Theorem 12.12. If ||| · ||| is a unitarily invariant norm on a type II1 factor M, then there is a subset F of F containing the constant 1 function on [0, 1] such that for all T ∈ M|||·||| , |||T ||| = sup |||T |||f : f ∈ F ,
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where |||T |||f is defined in Lemma 7.1 by Eq. (3) or by Eq. (4) and F = {f (x) = a1 χ[0, 1 ) (x) + a2 χ[ 1 , 2 ) (x) + · · · + an χ[ n−1 ,1] (x): a1 a2 · · · an 0, n n
n
a1 +···+an n
1, n = 1, 2, . . .}.
n
Combining Theorems 12.12 and 12.9, we have the following corollary. Corollary 12.13. Let ||| · ||| be a unitarily invariant norm on a type II1 factor M and let ||| · ||| 1 be the corresponding symmetric gauge norm on (L∞ [0, 1], 0 dx) as in Corollary 2. If T ∈ M, then T ∈ M|||·||| if and only if μs (T ) ∈ L∞ [0, ∞)|||·||| . In this case, |||T ||| = |||μs (T )||| . and 1 p ∞. Then T ∈ Lp (M, τ ) if and only if μs (T ) ∈ Example 12.14. Let T ∈ M 1 ∞ p L ([0, 1]). In this case, T p = ( 0 μs (T )p ds)1/p = ( 0 λp dμ|T | (λ))1/p . 12.4. A generalization of Hölder’s inequality Lemma 12.15. Let ||| · ||| be a unitarily invariant norm on a finite factor M and let T ∈ M|||·||| be a positive operator. Then limn→∞ χ[0,n] (T ) = T with respect to ||| · |||. Proof. If ||| · ||| is singular, then T ∈ M by Lemma 11.3 and the lemma is obvious. We may assume that ||| · ||| is continuous. Let Tn = χ[0,n] (T )) and > 0. By Lemma 12.2, there is a δ > 0 such that if τ (E) < δ then |||T E||| < . There is an N such that μs ([N, ∞)) < δ. So for m > n N , |||Tm − Tn ||| = |||T · χ(m,n] (T )||| < . This implies that {Tn } is a Cauchy sequence of M with respect to ||| · |||. Since limn→∞ Tn = T in the measure topology, by Corollary 12.5, limn→∞ Tn = T in the topology induced by ||| · |||. 2 The following theorem is a generalization of Hölder’s inequality. Theorem 12.16. Let ||| · ||| be a normalized unitarily invariant norm on a finite factor M. If T ∈ M|||·||| and S ∈ M|||·|||# , then T S ∈ L1 (M, τ ) and T S1 |||T ||| · |||S|||# . Proof. By the polar decomposition and Corollary 12.6, we may assume that S and T are positive operators. Let Tn = χ[0,n] (T ) and Sn = χ[0,n] (S). By Lemma 12.15, limn→∞ |||T − Tn ||| = limn→∞ |||S − Sn |||# = 0. Let K be a positive number such that |||Tn ||| K and |||Sn |||# K for all n and > 0. Then there is an N such that for all m > n N , |||Tm − Tn ||| < /(2K) and |||Sm − Sn |||# < /(2K). By Corollary 6.4, Tm Sm − Tn Sn 1 (Tm − Tn )Sm 1 + Tn (Sm − Sn )1 |||Tm − Tn ||| · |||Sm |||# + |||Tn ||| · |||Sm − Sn |||# < . This implies that {Tn Sn } is a Cauchy sequence in M with respect to · 1 . Since limn→∞ Tn Sn = T S in the measure topology, by Proposition 12.3, limn→∞ Tn Sn = T S in · 1 . By Corollary 6.4, Tn Sn 1 |||Tn ||| · |||Sn |||# for every n. Hence, T S1 |||T ||| · |||S|||# . 2 Combining Example 8.5 and Theorem 12.16, we obtain the non-commutative Hölder’s inequality. Corollary 12.17. Let M be a finite factor with the faithful normal tracial state τ . If T ∈ Lp (M, τ ) and S ∈ Lq (M, τ ), then T S ∈ L1 (M, τ ) and T S1 T p · Sq , where 1 p, q ∞ and
1 p
+
1 q
= 1.
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13. Proof of Theorems H and I In this section, we assume that M is a type II1 factor with the unique tracial state τ , ||| · ||| is a unitarily invariant norm on M and ||| · |||# is the dual unitarily invariant norm on M (see 1 Definition 6.2). Let ||| · |||1 be the corresponding symmetric gauge norm on (L∞ [0, 1], 0 dx) as 1 in Theorem D and ||| · |||#1 be the dual norm on (L∞ [0, 1], 0 dx). Lemma 13.1. If M|||·|||# is the dual space of M|||·||| in the sense of Question 1, then L∞ [0, 1]|||·|||# 1
is the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1.
Proof. By Corollary 2 and Lemma 2.6, there is a separable diffuse abelian von Neumann sub1 algebra A of M and a ∗-isomorphism α from A onto L∞ [0, 1] such that τ = 0 dx ◦ α and |||α(T )|||1 = |||T ||| for each T ∈ A. By Theorem E, |||α(T )|||#1 = |||T |||# for each T ∈ A. So we need only prove that A|||·|||# is the dual space of A|||·||| in the sense of Question 1. Let φ ∈ A|||·||| # . By the Hahn–Banach extension theorem, φ can be extended to a bounded linear functional ψ on M|||·||| such that ψ = φ. By the assumption of the lemma, there is an operator X ∈ M#|||·||| such that ψ(S) = τ (SX) for all S ∈ M|||·||| and ψ = |||X|||# . Let X = U |X| be the polar decomposition of X and Xn = U · χ[0,n] (|X|). By Lemma 12.15, limn→∞ Xn = X with respect to the norm ||| · |||# . Let Yn = EA (Xn ) for n = 1, 2, . . . . By Corollary 1, {Yn } is a Cauchy sequence in A with respect to the norm ||| · |||# and |||Yn |||# |||Xn |||# . Let Y = limn→∞ Yn with respect to the norm ||| · |||# . Then Y ∈ A#|||·||| and |||Y |||# |||X|||# = ψ = φ. For T ∈ A|||·||| , φ(T ) = ψ(T ) = τ (T X) = limn→∞ τ (T Xn ) = limn→∞ τ (EA (T Xn )) = limn→∞ τ (T Yn ) = τ (T Y ). By Lemma 12.11, φ = |||Y |||# . 2 Recall that ||| · ||| is a singular norm on M if limτ (E)→0+ |||E||| > 0 and is a continuous norm on M if limτ (E)→0+ |||E||| = 0 (see Section 11). Corollary 13.2. If ||| · ||| is a singular unitarily invariant norm on M, then M|||·|||# is not the dual space of M|||·||| in the sense of Question 1. Proof. Since ||| · ||| is a singular norm on M, by Lemma 11.3, ||| · ||| is equivalent to the operator norm on M and M|||·||| = M. By Corollary 6.16 and Theorem 6.17, ||| · |||# is equivalent to the L1 -norm on M. So ||| · |||1 is equivalent to the L∞ -norm on L∞ [0, 1] and ||| · |||#1 is equivalent to the L1 -norm on L∞ [0, 1] by Theorem E. Note that L∞ [0, 1]|||·|||1 = L∞ [0, 1] is not separable with respect to ||| · |||1 but L∞ [0, 1]|||·|||# is separable with respect to ||| · |||#1 . So L∞ [0, 1]|||·|||# is not 1
1
the dual space of L∞ [0, 1]|||·|||1 in the sense of Question 1. By Lemma 13.1, M|||·|||# is not the dual space of M|||·||| in the sense of Question 1. 2 Lemma 13.3. If ||| · ||| is a continuous unitarily invariant norm on M, then M|||·|||# is the dual space of M|||·||| in the sense of Question 1. Proof. We may assume that |||1||| = 1. By Theorem 12.16, M|||·|||# is a subspace of the dual space of M|||·||| in the sense of Question 1. Let φ be a linear functional in the dual space of M|||·||| . Then for every T ∈ M|||·||| , |φ(T )| φ · |||T |||. By Corollary 3.31, for every T ∈ M,
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|φ(T )| φ · T . So φ is a bounded linear functional on M. Since ||| · ||| is a continuous norm on M, limτ (E)→0 |||E||| = 0. Hence, limτ (E)→0 φ(E) = 0. This implies that φ is an ultraweakly continuous linear functional on M and hence in the predual space of M. So there is an operator X ∈ L1 (M, τ ) such that for all T ∈ M, φ(T ) = τ (T X). By Lemma 12.11, |||X|||# = φ < ∞. This implies that X ∈ M|||·|||# . So φ(T ) = τ (T X) for all T ∈ M|||·||| and φ = |||X|||# . This proves the lemma. 2 Proof of Theorems H and I. Combining Lemmas 13.1, 13.3 and Theorem A gives the proof of Theorems H and I. 2 Example 13.4. If 1 p < ∞ and p1 + L1 (M, τ ) is not the dual space of M.
1 q
= 1, then Lq (M, τ ) is the dual space of Lp (M, τ ).
Example 13.5. For 1 < p < ∞, Lp (M, τ ) is a reflexive space. L1 (M, τ ) and M are not reflexive spaces. By Theorem 6.17, for 0 t 1, M|||·|||(t) is not a reflexive space. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. J. Dixmier, Les anneaux d’operateurs de classe finie, An. Sci. École. Norm. Sup. Paris 66 (1949) 209–261. J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981. T. Fack, Sur la notion de valeur caractéristique, J. Operator Theory 7 (2) (1982) 307–333. T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123 (2) (1986) 269–300. N.A. Friedman, Introduction to Ergodic Theory, Van Nostrand/Reinhold, 1970. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI, 1969. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge Math. Library, Cambridge Univ. Press, Cambridge, 2001. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 2, Springer-Verlag, Berlin, 1970. R.V. Kadison, G.K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (2) (1985) 249–266. R. Kadison, J. Ringrose, Fundamentals of the Theory of Operator Algebras, vols. 1, 2, Academic Press, New York, 1986. R.A. Kunze, Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958) 519–540. K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766. E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974) 103–116. G. Pisier, Q. Xu, Non-commutative Lp -spaces, in: Handbook of the Geometry of Banach Spaces, vol. 2, NorthHolland, Amsterdam, 2003, pp. 1459–1517. B. Russo, H.A. Dye, A note on unitary operators in C ∗ -algebras, Duke Math. J. 33 (1966) 413–416. R. Schatten, A Theory of Cross-spaces, Ann. Math. Studies, vol. 26, Princeton Univ. Press, Princeton, NJ, 1950. R. Schatten, Norm Ideals of Completely Continuous Operators, Springer-Verlag, Berlin, 1960. I.E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953) 401–457. B. Simon, Trace Ideals and Their Applications, second ed., Amer. Math. Soc., Providence, RI, 2005. J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300.
Journal of Functional Analysis 255 (2008) 184–199 www.elsevier.com/locate/jfa
Dual pair correspondences for non-linear covers of orthogonal groups Hung Yean Loke a,∗ , Gordan Savin b a Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore b Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
Received 1 November 2007; accepted 9 April 2008 Available online 12 May 2008 Communicated by P. Delorme
Abstract In this paper we study compact dual pair correspondences arising from smallest representations of nonlinear covers of odd orthogonal groups. We identify representations appearing in these correspondences with subquotients of cohomologically induced representations. © 2008 Elsevier Inc. All rights reserved. Keywords: Lie groups; Representations; Dual pairs
1. Introduction Let p be an odd positive integer and let q be an even positive integer. Let SO0 (p, q) be the identity component of the Lie group SO(p, q) and let G be the central extension of SO0 (p, q) with a maximal compact subgroup Spin(p) × SO(q) if p < q, K0 = SO(p) × Spin(q) if q < p. The group G is not a linear group. In [9], we investigated the smallest representations of G that do not factor through the linear quotient SO0 (p, q). (Such representations are called genuine.) We described the corresponding Harish-Chandra modules: one such module V if p < q and two * Corresponding author.
E-mail addresses:
[email protected] (H.Y. Loke),
[email protected] (G. Savin). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.009
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modules V + and V − if p > q. These representations are interesting for a variety of reasons. For example, if G is split then V (in the case p + 1 = q) or V + and V − (in the case p − 1 = q) lift to a trivial representation (of an appropriate algebraic group) via the local Shimura correspondence [1]. Let g be the complexified Lie algebra of G. (Lie algebras in this paper are complex unless specified otherwise.) Let W be the Harish-Chandra module of one of the smallest representations above. We showed in [9] that W is a (g, K)-module where K ⊇ K 0 is obtained by replacing the SO-factor of K 0 by the corresponding full orthogonal group. This extension is important for investigation of dual pair correspondences arising from W . More precisely, let K2 = O(s). Consider a standard embedding of K2 into the O-factor of K. Note that, by Witt’s lemma, this embedding is unique up to a conjugation. Let g1 be the centralizer of K2 in g. Then g1 =
so(p, r), r = q − s, so(r, q), r = p − s,
if p < q, if p > q.
Let G1 be a connected subgroup of G corresponding to the Lie algebra g1 and let K10 = G1 ∩ K 0 . Then W , when restricted to g1 × K2 , decomposes discretely W=
Θ(τ ) ⊗ τ
τ
where the sum is taken over all irreducible finite-dimensional representations of K2 , and Θ(τ ) is naturally a (g1 , K10 )-module. In [9], we obtained some partial results about Θ(τ ), such as irreducibility of Θ(τ ), which were necessary to established a correspondence of infinitesimal characters. Our objective in this paper is to give a more thorough investigation of the correspondence. Let q m = p−1 2 and m = 2 . Consider a θ -stable maximal parabolic subgroup q1 = l1 + n1 in g1 whose Levi component corresponds to a subgroup L1 =
U(m) × SO0 (1, r) SO(r, 0) × U(m )
if p < q, if p > q
U(m) ⊆ Spin(p) is a two-fold cover of U(m), which is given as a pull-back of in G1 . Here U(m) ⊆ SO(p). We identify Θ(τ ) with subquotients of modules which are cohomologically induced from irreducible representations of L1 which are trivial on the SO-factor and genuine on the U-factor. In particular this implies that these cohomologically induced subquotients are unitarizable and we have a detailed information about their K1 -types, since the types of Θ(τ ) could be computed by the usual branching rules of orthogonal groups. One can consider representations cohomologically induced from representations of L1 which are trivial on the SO-factor and not genuine on the U-factor. It is interesting to note that these representations (of the linear quotient of G1 ) appear as double lifts from compact orthogonal groups in the Howe correspondence [8] and [10]. In Section 6 we highlight a special case. Assume that r > q is an odd integer. Knapp [5] introduced a family πs of (so(r, q), SO(r) × Spin(q))-modules, s = 0, 1, 2 . . . . The module πs is a Harish-Chandra module of a genuine representation of G1 if and only if s is even. If s is even then p = r + s is odd. We show that πs is isomorphic to our Θ(0) where 0 denotes the trivial representation of O(s). These results, therefore, complement the results of Paul and Trapa [11].
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It is shown there that πs for s odd appear as double lifts of trivial representations of compact groups in the Howe correspondence [8,10]. The study of our compact dual pairs unfortunately requires use of disconnected groups for technical reasons. In order to avoid the complications of treating covers of disconnected Lie groups, we will work exclusively with Harish-Chandra modules in this paper. The main results and proofs for V and V ± are similar but each requires slightly different set of notations. Hence we will divide the paper into two parts. The first part consists of Sections 2–4 where we concentrate on one family of dual pairs for the smallest representation V . The main purpose is to explain the main ideas quickly and clearly without being buried by the notations. In Section 5, we will state but without proofs the corresponding results for V ± . 2. The smallest representation In Sections 2–4, we will assume that p < q. Let V be the Harish-Chandra module of the smallest representation of G as in [9]. The module V is unitarizable and it extends to an irreducible (g, K)-module for K = Spin(p) × O(q). We need some notation in order to describe the K-types of V . 2.1. Notation The following convention will be used throughout the paper. Given a multiple of numbers λ = (λ1 , . . . , λr , 0, . . . , 0) then, by adding or removing 0’s at the tail, λ can be considered an s-tuple for every s r. Let 1k := (1, . . . , 1) and 0k := (0, . . . , 0) where there are k copies of 1’s and 0’s respectively. We set εi = (0, . . . , 0, 1, 0, . . . , 0) where 1 appears at the ith position. Given β = (β1 , . . . , βr ) and γ = (γ1 , . . . , γs ), we will denote (β1 , . . . , βr , γ1 , . . . , γs ) by (β, γ ) if there is no fear of confusion. Let Λ(n) denote the set of highest weights λ = (λ1 , . . . , λ[n/2] ) of so(n). For e = 0, 12 , let Λ(n, e) denote the subset of Λ(n) consisting of λ = (λ1 , . . . , λ[n/2] ) where λi ∈ Z + e. Hence Λ(n) = Λ(n, 0) ∪ Λ(n, 12 ). Let τnλ denote the finite-dimensional irreducible representations of so(n) with the highest weight λ. If λ is in Λ(n, 0) then τnλ is an irreducible representation of the compact group SO(n). Otherwise it is an irreducible representation of Spin(n) which does not descend to SO(n). The trivial representation may be denoted by CSO(n) . Let n−4 ρn = ( n−2 2 , 2 , . . .) ∈ Λ(n) denote the half sum of positive roots of so(n). Next we discuss irreducible representations of O(n). Let Λ(O(n)) denote the subset of elements in Zn such of the form (λ1 , . . . , λk , 0n−k )
or
(λ1 , . . . , λk , 1n−2k , 0k )
(1)
where λi are positive integers, and k n2 . Irreducible representations of O(n) are parameterized by Λ(O(n)) (see [2] and [4]). We will call an element λ of Λ(O(n)) a highest weight of O(n). λ Let τO(n) denote the corresponding irreducible finite-dimensional representation of O(n). The trivial representation of O(n) is sometimes denoted by CO(n) . λ λ Finally we recall a branching rule. Suppose n > s, then τO(n) contains τO(s) if and only if λi λi λi+n−s for all 1 i s.
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With this notation in hand, we can now describe the K-types of V . Recall that m = restriction of V to K = Spin(p) × O(q) is
V=
λ+ q−p 2 1m
τp
p−1 2 .
The
λ ⊗ τO(q) .
λ∈Λ(p,0) λ Here λ in τO(q) is considered as an element of Λ(O(q)) by adding 0’s at the tail. In particular, q−p
the minimal K-type of V is τp 2 μp,q
1m
⊗ CO(q) . The infinitesimal character of V is
q −1 p−1 1 3 , , ,..., . = 1, 2, . . . , 2 2 2 2
We now consider the restriction of V to g1 × K2 , where g1 = so(p, r) and K2 = O(s) for some integers r and s such that r + s = q. We obtain a direct sum V=
λ Θ(λ ) ⊗ τO(s) .
(2)
λ ∈Λ(O(s))
Note that every Θ(λ ) is a (g1 , K1 )-module, where K1 = Spin(p) × O(r). Since V is admissible with respect to Spin(p) ⊆ K1 , it follows that each Θ(λ ) is an admissible (g1 , K1 )-module. 2.2. The K1 -types of Θ(τ ) Let λ be in Λ(O(s)). Write λ = (λ1 , . . . , λt , 0, . . . , 0). We will now describe the K1 -types of λ . Θ(λ ) = Θ τO(s) λ+ q−p 2 1m
Let δ1 be a K1 -type of Θ(λ ). Obviously, δ1 must be isomorphic to τp
λ+ q−p 2 1m
some λ = (λ1 , . . . , λm ) in Λ(p, 0), and it has to lie in the K-type δ = τp Furthermore, the multiplicity of δ1 in Θ(λ ) is given by
μ
⊗ τO(r) for
λ ⊗ τO(q) of V .
μ λ λ dimC HomK1 ×K2 δ1 ⊗ τKλ 2 , δ = dimC HomO(r)×O(s) τO(r) ⊗ τO(s) . , τO(q)
(3)
By the branching rule stated after (1), the right-hand side is nonzero only if λi λi for all i m, and λi = 0 for all i > m. In particular Θ(λ ) is nonzero if and only if the number of nonzero integers in λ is not greater than (p − 1)/2, that is, t m. (If that is the case then λ can be viewed as a highest weight for so(p).) Moreover, the branching rule implies that λ + q−p 2 1m
W (λ ) = τp
⊗ CO(r)
appears in Θ(λ ) with multiplicity one and it is the (unique) minimal K1 -type of Θ(λ ). Let K10 = Spin(p) × SO(r) be the identity component of K1 . We can view Θ(λ ) as a (g1 , K10 )-module. The minimal K1 -type restricts irreducibly to K10 , and it is not hard to see that it becomes the unique minimal K10 -type of Θ(λ ).
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We will now state [9, Theorem 9.1]. The use of disconnected K2 is crucial here. (Note that we have just proved the second part.) Theorem 2.1. Recall that g1 = so(p, r), K2 = O(s) and K10 = Spin(p) × SO(r). Let Θ(τ ) be the lift of an irreducible representation τ of K2 . Then (i) The (g1 , K10 )-module Θ(τ ) is either zero or irreducible. (ii) Suppose τ and τ are non-isomorphic irreducible representations of K2 , and suppose Θ(τ ) and Θ(τ ) are nonzero. Then the minimal K10 -types of Θ(τ ) and Θ(τ ) are non-isomorphic. In particular, Θ(τ ) and Θ(τ ) are non-isomorphic (g1 , K10 )-modules. 3. Cohomological induction The purpose of this section is to introduce cohomological induction and realize V in terms of the cohomological induction. 3.1. Notation We recall some basic definitions and notation from [6] and [14]. We use a subscript 0 to denote a real Lie algebra. Those without are complex Lie algebras. Consider a connected Lie group G. Let K 0 be a maximal compact subgroup. Let g0 and k0 be the Lie algebras of G and K 0 , respectively. Let θ be the Cartan involution of g0 fixing k0 . Let q = l + n be a θ -stable parabolic subalgebra of g. Let q denote its opposite parabolic subalgebra. Let L denote the corresponding 0 connected Lie subgroup top of G with Lie algebra l0 . If Z is an irreducible (l, L ∩ K )-module, then
n and we put Z = Z ⊗ g
indq¯ Z = U(g) ⊗q¯ Z. We will write ind Z if it is clear what g and q are. If Z has infinitesimal character λZ , then ind Z has infinitesimal character λZ + ρ(n). Let Li (Z) = Πi ind Z g,K 0
where Πi = (Πg,L∩K 0 )i is the ith derived functor of the Bernstein functor. If Z = Cλ is the onedimensional character of (l, L ∩ K 0 ), then we denote Aq (λ) = Ls0 (Cλ ) and it has infinitesimal character λ + ρ(g). Given a (g, K 0 )-module W , we set W h to be the subspace of K 0 -finite vectors in the conjugate linear dual vector space of W . Let s0 := dim(n ∩ k) and let Γ s0 be the s0 th derived functor of the Zuckerman functor of taking K 0 -finite vectors. By [6, Eq. (6.25)], Ls0 (Z)h = Γ s0 ((ind Z )h ). By [14, Theorem 6.3.5], there is a non-degenerate sesquilinear pairing between Γ s0 ((ind Z )h ) and Γ s0 (ind Z ). Hence if Γ s0 (ind Z ) is K 0 -admissible then Ls0 (Z) = Γ s0 ind Z . In this paper, we find it more convenient to work with Γ s0 (ind Z ) and ignore Ls0 (Z) completely. However we will state all final results in Ls0 (Z) because it is a more widely accepted definition.
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3.2. A positive root system We now specialize to g = so(p, q) and K 0 = Spin(p) × SO(q). Recall that m = p−1 2 and = q2 . Let g0 and k0 be the real Lie algebras of G and K 0 , respectively. Choose a compact Cartan subalgebra h0 ⊆ k0 of g0 and positive root system Φ + with respect to h0 such that the simple roots εi −εi+1 for 1 i m−1 belong to so(p), and εi −εi+1 for m+1 i m+m −1 belong to so(q). The non-compact simple roots are εm − εm+1 and εm+m . √ Let λ0 = (1m , 0m ) ∈ −1h∗0 . Let q = l + n be the maximal parabolic subalgebra in g where l is spanned by roots perpendicular to λ0 . Then q is θ -stable. The Levi factor l corresponds to the subgroup m
L= U(m) × SO0 (1, q) in G. Here U(m) ⊆ Spin(p) is a two-fold cover of U(m) ⊆ SO(p). We note that the weights of finite-dimensional representations of U(m) which do not descend to U(m) can be identified with m-tuples of half-integers. The one-dimensional representation with the weight ( 12 , . . . , 12 ) is 1/2 denoted by detu(m) . Under the adjoint action of L, the radical n decomposes as n = Cm ⊗ C1+q ⊕
2 Cm
representation of U(m) and C1+q is the standard representation of where Cm is the standard 0 SO (1, q). The summand 2 (Cm ) is spanned by long roots εi + εj for 1 i < j m. These long roots and short roots εi for 1 i m are precisely all compact roots contained in n. It follows that s0 = dim(n ∩ k) =
m(m + 1) p 2 − 1 = , 2 8
and this number is independent of q. A maximal compact subgroup of L is L∩K 0 = U(m)×SO(q). However, since our considerations involve a disconnected group, we also need to consider a slightly larger group U(m) × O(q). We view C1+q , in the decomposition of n above, as a natural (so(1, q), O(q))-module. Then, as (l, U(m) × O(q))-modules,
top
n∼ = det u(m) ⊗ det m O(q) . q+m
The action of so(1, q) is, of course, trivial. Recall that if Z is an (l, U(m) × O(q))-module then, using the cohomological induction, Z gives rise to a (g, K 0 )-module Γ s0 (ind Z ). There are two important observations to be made here. First, since ind Z is already SO(q)-finite, the functor Γ s0 is simply the s0 th derived functor of the Zuckerman functor of taking Spin(p)-finite vectors. Using the definition and the treatment of Γ s0 in [14, Chapter 6], Γ s0 (ind Z ) can be computed by considering ind Z as an (so(p), U(m))-module. Furthermore since ind Z is an O(q)-module, and the action of so(p) commutes with the action of O(q), Γ s0 (ind Z ) is naturally an O(q)module. In other words, Γ s0 (ind Z ) extends to a (g, K)-module.
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Let Z0 be a one-dimensional (l, U(m) × O(q))-module such that the action of so(1, q) ⊆ l is trivial and, as U(m) × O(q)-modules, − p+q
Z0 ∼ = det u(m)2 ⊗ det m O(q) .
(4)
We set M0 = Γ s0 (ind Z0 ). One easily checks that the infinitesimal character of M0 is μp,q , the infinitesimal character of V . Lemma 3.1. The (g, K)-module M0 is Spin(p)-admissible so M0 = Ls0 (Z0 ). It contains the Kq−p
1m
⊗ CO(q) with multiplicity one. The K-type W0 is also the minimal K 0 -type type W0 = τp 2 of M0 . In particular, M0 is nonzero. We will derive this lemma as a corollary of the proof of Theorem 4.2 in Section 4. Alternatively the lemma also follows from the Blattner formula (see [6, Theorem 5.64]). Since the K-type W0 appears in Ls0 (Z0 ) with multiplicity one, we define Ls0 (Z0 ) to be the unique irreducible (g, K)-subquotient of Ls0 (Z0 ) containing W0 . Proposition 3.2. The irreducible (g, K)-modules V and Ls0 (Z0 ) are isomorphic. Proof. Both representations have the same infinitesimal character μp,q and the minimal K 0 type W0 . We showed in [9] that V is the unique irreducible (g, K 0 )-module with infinitesimal q−p
1m
character μp,q and minimal K 0 -type τp 2 ⊗ CSO(q) . Hence the two modules are isomorphic (g, K 0 )-modules. There are two ways to extend V from a (g, K 0 )-module to a (g, K)-module. One differs from the other by the determinant character of O(q). Hence V and Ls0 (Z0 ) are the same because they have the same minimal K-type W0 . 2 4. Identifying Θ(λ ) Let r and s be two integers such that r + s = q. Choose a standard embedding of O(s) into O(q), the second factor of K. Let g1 ∼ = so(p, r) be the centralizer of O(s) in g. Note that g1 is θ -invariant. In this section we consider the restriction of V to (g1 , K10 ) × K2 where K10 = Spin(p) × SO(r) and K2 = O(s). Suppose λ = (λ1 , . . . , λs ) is in Λ(O(s)) such that Θ(λ ) in (2) is nonzero. Then by (3), λi = 0 if i > m = p−1 2 . In particular, λ can be considered in element in Λ(p, 0) by adding or removing some 0’s at the tail. The irreducible (g1 , K10 )-module Θ(λ ) has a unique minimal K10 -type λ + q−p 2 1m
W (λ ) = τp
⊗ CSO(r) .
Using the θ -stable parabolic q = l + n in g, we define q1 = q ∩ g1 . Write q1 = l1 + n1 . Then l1 corresponds to a subgroup L1 = U(m) × SO0 (1, r)
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in G1 . For every λ such that Θ(λ ) = 0 (or equivalently λi = 0 for i > m) let Z(λ ) be an irreducible L1 -module such that the action of SO0 (1, r) is trivial and
q−p−2r 1m 2
λ+ Z(λ ) ∼ = τu(m)
(5)
as U(m)-modules. Set M(λ ) := Γ s0 (ind Z(λ ) ). We have explained in the previous section that we may take Γ s0 to be the s0 th derived functor of the Zuckerman functor of taking Spin(p)-finite vectors. Lemma 4.1. The (g1 , K10 )-module M(λ ) is Spin(p)-admissible so M(λ ) = Ls0 (Z(λ )). Any of its Spin(p)-type is isomorphic to λ + q−p 2 1m +κ
τp
where κ is an m-tuple of non-negative integers. The module M(λ ) contains the K10 -type W (λ ) with multiplicity one and it is the minimal K10 -type. We will prove Lemma 4.1 together with Theorem 4.2 below. One could also verify this lemma directly using the Blattner’s formula. Let Ls0 (Z(λ )) denote the unique irreducible subquotient of M(λ ) = Ls0 (Z(λ )) containing the minimal K10 -type W (λ ). We can now state the main result of this section. Theorem 4.2. The irreducible (g1 , K10 )-modules Θ(λ ) and Ls0 (Z(λ )) are isomorphic. In particular Ls0 (Z(λ )) is unitarizable and it has K10 -types given by the branching (3). Remarks. It is interesting to note that Ls0 (Z(λ )) is not always in the good or weakly good range (see [6, Definition 0.49]). Hence it may be reducible. It is of separate interest that the image of the bottom layer map induces an unitarizable subquotient. The infinitesimal character of L(Z(λ )) is
q − p − 2r λ + 1m , 0[ r+1 ] + ρp+r . 2 2
Hence Theorem 4.2 gives an alternative proof of the correspondence of infinitesimal characters of so(p, r) and so(s) [9, Theorem 1.2]. The rest of this section contains the proofs of Lemmas 3.1, 4.1 and Theorem 4.2. It is inspired by the work of [3], [15] and [16]. Recall that n1 ⊆ n. We have a decomposition n = n1 + n2 such that n2 = Cm ⊗ Cs is a tensor product of standard representations of U(m) and O(s), while the group SO0 (1, r) acts trivially on it. We extend n2 to a representation of U(m) × U(s). It is well known that (see [2] and [4]) Symn n2 =
μ
μ
μ
τU(m) ⊗ τU(s)
(6)
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where the sum is taken over all partitions μ of n of length not longer than min(m, s). (So every such partition can be viewed as a highest weight for both U(m) and U(s).) We further restrict the μ summand τU(s) to O(s) μ
τU(s) =
λ τO(s) .
(7)
λ ↑μ
λ The notation λ ↑ μ simply means that τO(s) is a subrepresentation of τU(s) , and the sum is taken μ μ with multiplicities. Note that τO(s) appears in the restriction from τU(s) with multiplicity one. Using this notation, we get
Symn n2 =
μ
μ λ ↑μ
μ
λ τU(m) ⊗ τO(s)
(8)
as a sum of irreducible representations of U(m) × O(s).
We now recall the definition of Z0 from (4). One easily sees that the restriction of Z0 to L1 × O(s) is given by q−1
2 Z0 = det u(m) ⊗ Cso(1,r) ⊗ CO(s) .
Let symm : Sym(g) → U(g) denote the symmetrization map (see [14, §0.4.2 ]). By the Poincaré– Birkhoff–Witt theorem,
ind Z0 = U(n1 ) ⊗ symm Sym(n2 ) ⊗ Z0
(9)
as L1 × O(s)-modules. Let Sn (n2 ) = ni=0 Symi (n2 ). We define Fn to be the (g1 , L1 ∩ K10 )
submodule of ind Z0 generated by 1 ⊗ symm(Sn (n2 )) ⊗ Z0 . Hence {Fn : n = 0, 1, 2, . . .} forms
an exhaustive increasing filtration of g1 × O(s)-submodules of ind Z0 . We will now state a special case of a known fact which is used in proof of the Blattner formula in [6]. Lemma 4.3. For every positive integer n, we have an isomorphism of g1 × O(s)-modules Fn /Fn−1 =
μ λ ↑μ
q−1 )1m g μ+( λ indq¯ 11 τu(m) 2 ⊗ Cso(1,r) ⊗ τO(s)
λ where μ is any partition of n of length not more than min(m, s) and τO(s) is counted with multiμ plicity with which it appears in the restriction of τU(s) .
We shall use the filtration Fn to compute Γ s0 (ind Z0 ). Case 1. We first consider the filtration Fn in the case r = 0 and s = q. In particular, g1 = so(p). Put q−1 )1m g μ+( . V (μ) = indq¯ 11 τu(m) 2
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μ+ q−p 1m
The infinitesimal character of V (μ) is the same as the infinitesimal of τp 2 . In particular, these infinitesimal characters are pairwise different for different partitions μ. It follows that the filtration Fn splits:
λ V (μ) ⊗ τO(q) . (10) ind Z0 = μ λ ↑μ
Here the first sum is taken over all partitions μ of length no more than m = μ counted with multiplicity with which it appears in τU(q) .
p−1 2 ,
λ and τO(q) is
Lemma 4.4. Let μ be a partition of length not more than m. Then V (μ) is an irreducible so(p)module. Proof. Since V (μ) is u(m)-finite generalized Verma module, any proper submodule of V (μ) must be a quotient of some V (μ ) where μ = μ . Note that the lowest u(m)-type τ of V (μ ) is a nonzero u(m)-type of V (μ). Let hm denote the maximal Cartan subalgebra of u(m). We claim that the highest weight of τ is of the form μ + ( q−1 2 )1m + κ where κ is sum of roots of n1 restricted to hm . Indeed by (9), μ+( q−1 2 )1m
V (μ) = Sym(n1 ) ⊗ τu(m)
as a u(m)-module. By [13, Proposition 3.2.12], an irreducible u(m)-module (in particular τ ) on the right-hand side of the above equation has highest weight μ + ( q−1 2 )1m + κ where κ is a hm -weight of Sym(n1 ). This proves our claim. The roots of n1 are of the form εi or εi + εj so κ is a m-tuple of non-negative integers. Since V (μ ) is proper, κ is nonzero. The infinitesimal characters of V (μ) and V (μ ) correspond to the q−p weights μ + q−p 2 1m + ρp and μ + 2 1m + κ + ρp , respectively, under the Harish-Chandra homomorphism. These two weights correspond to partitions of different lengths because the entries of μ, κ, ρp are non-negative, q > p and κ is nonzero. Hence V (μ) and V (μ ) do not have the same infinitesimal character, and V (μ ) cannot map to V (μ). The lemma is proved. 2 We recall the previous section that the Zuckerman functor Γ j is computed in the category of (so(p), U(m))-modules. If we apply Γ j to both sides of (10) then
λ Γ j ind Z0 = Γ j V (μ) ⊗ τO(q) . (11) μ λ ↑μ
Since s0 = dim(n1 ∩ k1 ) =
m(m+1) 2
=
p 2 −1 8 , by the Borel–Weil–Bott–Kostant theorem, μ+ q−p 1m τp 2 . The reader may recognize that we have
Γ j (V (μ)) = 0 if j = s0 and Γ s0 (V (μ)) = essentially followed the proof of the Blattner formula in [6] to compute K-types of Ls0 (Z0 ). Now we have the following conclusions:
μ+ q−p 2 1m
(A) Spin(p)-type of Γ s0 (ind Z0 ) is of the form τp
μ
with multiplicity given by dim τU(q) .
Therefore Γ s0 (ind Z0 ) is admissible with respect to Spin(p). This also follows from a very
general criterion in [7]. We now have Γ s0 (ind Z0 ) = Ls0 (Z0 ).
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(B) A K-type of Ls0 (Z0 ) is of the form τp
λ , where τ λ ⊗ τO(q) O(q) appears in the restriction q−p
μ
1m
⊗ CO(q) which occurs with from τU(q) . In particular, the minimal K-type is W0 = τp 2 multiplicity one. It is also the image of the bottom layer map. With this, we have proven Lemma 3.1. Case 2. Now we return to the general r for g1 = so(p, r). Consider the filtration Fn in this situation. We recall (5) and we abbreviate q−1 )1m g g μ+( L(μ) = indq¯ 11 Z(μ) = indq¯ 11 τu(m) 2 ⊗ Cso(1,r) . Then, Fn /Fn−1 is a direct sum of L(μ) where μ is a partition of n of length not more than min(m, s). By (10) and Lemma 4.4, L(μ) is a direct sum of various V (μ ), and L(μ) is a U(m))(so(p), U(m))-submodule of (10). Since Γ s0 is computed in the category of (so(p),
modules, Γ s0 (L(μ)) is a Spin(p)-submodule of Γ s0 (ind Z0 ). By Conclusion (A) in Case 1,
Γ s0 (ind Z0 ) is Spin(p)-admissible so Γ s0 (L(μ)) is Spin(p)-admissible and Γ s0 (L(μ)) = Ls0 (Z(μ)). This proves the first assertion of Lemma 4.1. In order to understand Spin(p)-types of Γ s0 (L(μ)), we must describe μ such that V (μ ) ⊆ L(μ). Lemma 4.5. If V (μ ) ⊆ L(μ), then μ = μ + κ where κ is an m-tuple of non-negative integers. Proof. The proof is similar to part of the proof of Lemma 4.4. Let hm denote the Cartan subalgebra of u(m). Let τ be the lowest u(m)-type of V (μ ). It has highest hm -weight μ + ( q−1 2 )1m . μ+( q−1 2 )1m
As a u(m)-module L(μ) = Sym(n2 ) ⊗ τu(m)
. Since τ is a u(m)-type in L(μ), by [13,
Proposition 3.2.12], the highest hm -weight of τ is of the form μ + ( q−1 2 )1m + κ where κ is a hm -weight of Sym(n1 ), i.e. sum of roots of n1 . Since the roots of n1 when restricted to hm are of the form εi or εi + εj , κ is an m-tuple of non-negative integers. 2 In addition, L(μ) contains a unique copy of V (μ) and SO(r) acts trivially on it. By the above μ+ q−p 2 1m +κ
lemma, the Spin(p)-types of Γ s0 (L(μ)) are τp integers, and the Lemma 4.1.
K10 -type
μ+ q−p 1m W (μ) = τp 2
, here κ is an m-tuple of non-negative
⊗ CSO(r) occurs with multiplicity one. This proves
Lemma 4.6. Let Fn be the filtration as in Lemma 4.3. Then Γ j (Fn ) = 0 if j = s0 . Furthermore we have an exact sequence 0 → Γ s0 (Fn−1 ) → Γ s0 (Fn ) → Γ s0 (Fn /Fn−1 ) → 0. Proof. Let F¯ n := Fn /Fn−1 . Then Fn and F¯ n are direct sums of V (μ)’s in (10). As explained in U(m))-modules. Hence the previous section, we may compute Γ j F¯ n in the category of (so(p), Γ j (Fn ) and Γ j (F¯ n ) are direct sums of Γ j (V (μ)) and we have shown that these are zeros if j = s0 . Finally we apply the functor Γ to the exact sequence 0 → Fn−1 → Fn → F¯ n → 0 to get the long exact sequence. The exact sequence in the lemma follows immediately.
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Lemma 4.7. In the category of (g1 , K10 )-modules, Γ s0 (Fn ) is an exhaustive increasing filtration of Ls0 (Z0 ) and Γ s0 (Fn )/Γ s0 (Fn−1 ) = Γ s0 (Fn /Fn−1 ) =
λ Γ s0 L(μ) ⊗ τO(s) .
μ λ ↑μ
λ is Here the first sum is taken over all partitions of length no more than min(m, s) and τO(s) μ counted with multiplicity with which it appears in τU(s) .
Proof. This follows from Lemmas 4.6 and 4.3.
2
We are finally ready to prove Theorem 4.2, that is, compute Θ(λ ) where λ is in Λ(O(s)) of length not more than min(m, s). We define S(λ ) as the set of all partitions μ of length not more μ λ . Since V is an irreducible subquotient of L (Z ), than min(m, s) such that τU(s) contains τO(s) s0 0 it follows that Θ(λ ) is an irreducible subquotient of Ls0 (Z0 ), considered as (g1 , K10 )-module. It follows that Θ(λ ) is an irreducible subquotient of Γ s0 (L(μ)) for some μ in S(λ ). We now need the following lemma. λ + q−p 2 1m
Lemma 4.8. Let μ be in S(λ ). The K10 -type W (λ ) = τp if and only if μ = λ .
⊗ CSO(r) occurs in Γ s0 (L(μ))
Proof. We check Spin(p)-types. If W (λ ) is contained in Γ s0 (L(μ)) for some then, as we have μ λ , this just seen, λ = μ + (κ1 , . . . , κm ) where κi 0. On the other hand, since τU(s) contains τO(s) is possible only if μ = λ as desired. 2 Since Θ(λ ) contains W (λ ) the lemma implies that Θ(λ ) is an irreducible subquotient of s0 (Z(λ )). This proves Theorem 4.2.
Γ s0 (L(λ )) = L
5. The smallest representation V + In this section, we will extend Theorems 2.1 and 4.2 to representations V + and V − . Since the proofs are almost identical to those in the previous sections, we will only state the main results. q Let g = so(p, q) and K = O(p) × Spin(q). Recall that m = p−1 2 and m = 2 . Let g0 and k0 be the real Lie algebras of G and K 0 , respectively. Choose a compact Cartan subalgebra h0 ⊆ k0 of g0 and positive root system Φ + such that the simple roots εi − εi+1 for 1 i m − 1 belong to so(q) and, εi − εi+1 for m + 1 i m + m − 1 and εm +m belong to so(p). The non-compact simple root is εm − εm +1 . We refer√ to the notation on cohomological induction introduced in Section 3. We set λ0 = (1m , 0m ) ∈ −1h∗0 and we let q = l + n be the corresponding parabolic subalgebra. The algebra l corresponds to the subgroup L = SO(p) × U(m ) in G. We have s0 =
m (m −1) 2
=
q(q−2) . 8
Let Z0 be a one-dimensional O(p) × U(m )-module
−( p+q )
2 Z0 = det m O(p) ⊗ det u(m ) .
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We consider the (g, K)-module Ls0 (Z0 ). It is equal to Aq (λ) where λ = − p+q 2 λ0 . The following is essentially a result of [5] and [12]. The only difference is that we consider K and not K 0 . See Section 6 for more details. Theorem 5.1. Recall that p > q and K = O(p) × Spin(q). p−q
1
(i) The minimal K-type of Ls0 (Z0 ) is W0 = CO(p) ⊗ τq 2 m and it occurs in Ls0 (Z0 ) with multiplicity 1. (ii) Let V + = Ls0 (Z0 ) denote the irreducible subquotient of Ls0 (Z0 ) generated by W0 . Then V + is an unitarizable (g, K)-module.
Remark. As in the case of V in Section 4, we work with Γ s0 (ind Z0 ) instead of Ls0 (Z0 ). Part of
the proof involves establishing the fact that Γ s0 (ind Z0 ) is K 0 -admissible so that Γ s0 (ind Z0 ) = Ls0 (Z0 ). The same applies to Ls0 (Z(λ )) in Theorem 5.3 below. The restriction of V + to K = O(p) × Spin(q) is
V+ =
λ+ p−q 2 1m
λ τO(p) ⊗ τq
.
λ∈Λ(q+1,0) p−2 p−4 1 + remains irreIts infinitesimal character is ( q2 , q−2 2 , . . . , 1, 2 , 2 , . . . , 2 ). The module V ducible as a (g, K 0 )-module. In [9] we call V + a smallest representation of the non-linear cover of SO(p, q), and there is also an outline of a construction of V + using Gelfand–Zetlin bases.
Remark. We note that by an outer automorphism action of the pair (so(p, q), K) on V + , we get another smallest representation V − . All the results in this paper on V + would immediately give corresponding results for V − via this outer automorphism. Therefore we will only work with V + . Choose a standard embedding of K2 = O(s) into O(p), the first factor of K. Let g1 ∼ = so(r, q) be the centralizer of O(s) in g. Note that g1 is θ -invariant. In this section we consider the restriction of V + to K2 × (g1 , K1 ) where K1 = O(r) × Spin(q). V+ =
λ τO(s) ⊗ Θ(λ ).
λ ∈Λ(O(s))
Since O(s) is compact, the right-hand side is a direct sum. Furthermore V + is admissible with respect to Spin(q), so Θ(λ ) is an admissible (g1 , K1 )-module. The K1 -types of Θ(λ ) can be computed using branching rules similar to (3). More precisely, μ
λ+ q−p 2 1m
suppose δ1 = τO(r) ⊗ τq λ+ q−p 2 1m
λ ⊗ τq τO(p)
is a K1 -type of Θ(λ ). Then δ1 has to lie in the K-type δ =
of V + . The multiplicity of δ in Θ(λ ) is given by
μ λ λ λ dimC HomK1 ×O(s) δ1 ⊗ τO(s) . , δ = dimC HomO(r)×O(s) τO(r) ⊗ τO(s) , τO(p)
(12)
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By the right-hand side of (12), Θ(λ ) is nonzero if and only if nonzero entries of λ is not greater than q2 . The minimal K10 -type of Θ(λ ) is λ + p−q 2 1m
W (λ ) = CSO(r) ⊗ τq
.
(13)
We compare the next theorem with Theorem 2.1. Theorem 5.2. Recall that g1 = so(r, q), K2 = O(s) and K10 = SO(r) × Spin(q). Let Θ(τ ) be the lift of an irreducible representation τ of K2 . Then: (i) The (g1 , K10 )-module Θ(λ ) is either zero or irreducible. (ii) Suppose Θ(λ ) and Θ(η ) are nonzero. Then Θ(λ ) and Θ(η ) are isomorphic (g1 , K10 )modules if and only if λ = η . Part (i) follows the same argument as that of [9, Theorem 9.1]. We will omit the proof. Part (ii) is a consequence of (13) because if λ = η , then Θ(λ ) and Θ(η ) have distinct minimal K10 types. 5.1. Cohomological induction We would like to identify Θ(λ ) as a subquotient of a cohomological induced module. Suppose Θ(λ ) is nonzero. Then the number of nonzero entries in λ is not greater than m . Let q1 = q ∩ g1 be a theta-stable parabolic subalgebra of g1 . Its Levi subalgebra l1 corresponds to a subgroup U(m ) L1 = SO(r) × in G1 . Let Z(λ ) be an irreducible L1 -module which is trivial on SO(r) and such that the restriction to U(m ) is λ + p−q−2r 1m 2
Z(λ ) ∼ = τu(m )
.
We consider the cohomologically induced representation Ls0 (Z(λ )). Its minimal K10 -type is W (λ ) in (13) and it occurs with multiplicity one. Let Ls0 (Z(λ )) denote the unique irreducible (g1 , K10 )-subquotient of Ls0 (Z(λ )) containing W (λ ). The next theorem is proved in the same way as Theorem 4.2. Theorem 5.3. The irreducible (g1 , K10 )-modules Θ(λ ) and Ls0 (Z(λ )) are isomorphic. In particular, Ls0 (Z(λ )) is nonzero and unitarizable. 6. On results of Knapp and Trapa The aim of this section is to relate our results to some results of Knapp and Trapa. Assume that r is an integer and r q. For every non-negative integer s, Knapp [5] defined
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an (so(r, q), K10 )-module πs as a certain (naturally unitarizable) subquotient of Aq (λ) where q = l + n, l = u(m ) + so(r) and λ=
s −r −q 1m , 0[ 2r ] . 2
The module πs contains the minimal K10 -type of Aq (λ). Trapa showed in [12] that πs is irreducible. We now focus our attention to non-negative integral values of s so that Aq (λ) is a faithful representation of K10 . This implies that s−r−q ∈ Z + 12 , that is, r + s is odd. 2 + Consider W = V and the dual pair (g1 , K1 ) × O(s) where g1 = so(r, q), K1 = O(r) × Spin(q) and p = r + s. Let Θ(0) denote the theta lift of the trivial representation of O(s). Then Θ(0) is an (so(r, q), K10 )-module. The next theorem follows from Theorem 5.3. Theorem 6.1. Let r and s be two positive integers such that r q and p = r + s is odd. Then the (so(r, q), K10 )-module Θ(0) is isomorphic to πs . We note that Knapp computed K10 -types of πs . His computation shows that K10 -types of πs coincide with K10 -types of Θ(0). Hence this paper gives an independent proof of the fact that πs is irreducible (see [12]). An interesting way to formulate the above result for odd r is as follows. Let π0 , π2 , . . . be ∼ Θ(0) where Θ(0) is the theta lift of the Knapp’s family for so(p, q) , where p > q. Then π2a = appear in trivial representation of O(2a). Again, we note that Paul and Trapa studied how π2a+1 the Howe correspondence [11]. Acknowledgments The first author would like to thank the hospitality of the Mathematics Department at University of Utah while the initial idea for this paper was first conceived. Likewise, the second author would like to thank the hospitality of the Mathematics Department at National University of Singapore during 2007 when this paper was completed. We would also like to thank Peter Trapa for some very insightful discussions, and the referee for pointing out a few gaps in a previous version. The first author was supported by an NUS grant R-146-000-085-112. The second author was supported by an NSF grant DMS 0551846. References [1] J. Adams, D. Barbasch, A. Paul, P. Trapa, D. Vogan, Shimura correspondences for split real groups, J. Amer. Math. Soc. 20 (2007) 701–751. [2] Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups, Cambridge Univ. Press, Cambridge, 1998, third corrected printing, 2003. [3] Benedict H. Gross, Nolan R. Wallach, On quaternionic discrete series representations, and their continuations, J. Reine Angew. Math. 481 (1996) 73–123. [4] Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in: The Schur Lectures, 1992, Tel Aviv, in: Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. [5] Anthony Knapp, Nilpotent orbits and some unitary representations of indefinite orthogonal groups, J. Funct. Anal. 209 (2004) 36–100. [6] Anthony Knapp, David Vogan, Cohomological Induction and Unitary Representations, Princeton Univ. Press, Princeton, NJ, 1995.
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[7] T. Kobayashi, Discrete decomposability of the restriction of Aq (λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (2) (1998) 229–256. [8] H.Y. Loke, Howe quotients of unitary characters and unitary lowest weight modules, Represent. Theory 10 (2006) 21–47. [9] H.Y. Loke, G. Savin, The smallest representations of non-linear covers of odd orthogonal groups, Amer. J. Math., in press. [10] K. Nishiyama, C.B. Zhu, Theta lifting of unitary lowest weight modules and their associated cycles, Duke Math. J. 125 (3) (2004) 415–465. [11] Annegret Paul, Peter E. Trapa, Some small unipotent representations of indefinite orthogonal groups and the theta correspondence, in: Univ. Aarhus Publ. Ser., vol. 48, Aarhus Univ., Aarhus, 2007, pp. 103–125. [12] Peter E. Trapa, Some small unipotent representations of indefinite orthogonal groups, J. Funct. Anal. 213 (2004) 290–320. [13] David Vogan, Representations of Real Reductive Lie Groups, Birkhäuser Boston, Boston, MA, 1981. [14] Nolan R. Wallach, Real Reductive Groups I, Academic Press, New York, 1988. [15] Nolan R. Wallach, Transfer of unitary representations between real forms, in: Representation Theory and Analysis on Homogeneous Spaces, New Brunswick, NJ, 1993, in: Contemp. Math., vol. 177, Amer. Math. Soc., Providence, RI, 1994, pp. 181–216. [16] Nolan R. Wallach, C.-B. Zhu, Transfer of unitary representations, in: Special Issue in Memory of A. Borel, Asian J. Math. 8 (4) (2004) 861–880.
Journal of Functional Analysis 255 (2008) 200–227 www.elsevier.com/locate/jfa
On non-equilibrium stochastic dynamics for interacting particle systems in continuum Yuri Kondratiev a,b , Oleksandr Kutoviy a,∗ , Robert Minlos c a Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33615 Bielefeld, Germany b Research Center BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany c IITP, Russian Academy of Science, Moscow, Russia
Received 26 November 2007; accepted 11 December 2007 Available online 1 February 2008 Communicated by Paul Malliavin
Abstract We propose a general scheme for construction of Markov stochastic dynamics on configuration spaces in continuum. An application to the Glauber-type dynamics with competitions is considered. © 2007 Elsevier Inc. All rights reserved. Keywords: Configuration space; Glauber dynamics; Non-equilibrium Markov process
1. Introduction Interacting particle systems (IPS) is a large and growing area of probability theory and infinitedimensional analysis which is devoted to the study of certain models that arise in statistical physics, biology, economics, etc. Most of the results in the theory of IPS are related to the study of the so-called lattice systems and their Markov stochastic evolutions. In such systems the spatial structure of the considered model is presented by a lattice (or an infinite graph). Considered processes are usually specified by transition rates and associated Markov generators. The existence problem for the corresponding Markov process on the lattice configuration space can be solved positively under quite general assumptions about the transition rates, see e.g. [22]. * Corresponding author.
E-mail addresses:
[email protected] (Y. Kondratiev),
[email protected] (O. Kutoviy),
[email protected] (R. Minlos). 0022-1236/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2007.12.006
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Comparing with the lattice case, the situation with Markov stochastic dynamics for IPS in continuum is essentially different. In particular, it is true for an important class of birth-and-death processes in continuum (or so-called spatial birth-and-death processes). To this class belongs the Glauber type dynamics in continuum which are under active consideration, see [3,29]. Another class of interesting stochastic processes is formed by the Kawasaki type dynamics in continuum [19] and gradient diffusions [1,17]. Most of the results we have up to now for these processes are related to the equilibrium case (via the Dirichlet forms approach) [13,18] or to the processes in bounded domains (see, e.g., [8,24]). The situation with the non-equilibrium case is much pure. In particular, non-equilibrium spatial birth-and-death processes were constructed recently by Garcia and Kurtz for a special class of transition rates using techniques of stochastic differential equations [6] and a graphical construction was applied in [5]. Note that in both mentioned papers the death rate was considered to be a constant and the latter plays an essential technical role. A continuous version of the lattice contact model was analyzed in [14]. In contrast to the lattice case, constructions of the stochastic dynamics in continuum show essential difference between the Markov processes and Markov functions concepts. The latter notion (due to E. Dynkin) concerns the case of the processes with given initial distributions contrary to the more usual initial points framework. This weaker notion of the Markov function is not so essential in the lattice models because corresponding Markov processes can be constructed (typically) under very general assumptions. A principal role of dynamics with given classes of initial distributions was clarified at first for (deterministic) Hamiltonian dynamics in continuum, see e.g. [4]. In the present paper the role of the Markov functions approach is clarified for an infinite particle stochastic dynamics in continuum. This approach is based on the study of the corresponding (dual) Kolmogorov equation on measures. Such equation can be transported to an equation for corresponding correlation functions. Typically, this correlation functions equation does not admit a direct perturbation theory approach. In fact, the main technical observation made in the paper is related to the consideration of its dual time evolution on the so-called quasi-observables. This approach appeared for the first time in the literature on stochastic IPS in continuum in our paper [16] in the particular case of a Glauber-type dynamics. The idea to move the dynamics to a proper quasi-observables space (as well as the notion of quasi-observables itself) follows naturally from the concepts of harmonic analysis on configuration spaces, see e.g. [11]. We apply a perturbation technique to this dynamics in proper weighted L1 -spaces of functions on finite configurations and produce time evolutions of correlation functions as a dual object. The choice of corresponding weights gives precise description of the class of admissible initial distributions for our processes. One should emphasize also another principal moment of the paper. Namely, even if we have constructed a time evolution of the correlation functions, we need to show that they correspond to a time evolution of measures. In fact, this point is hidden in several works in statistical physics concerning BBGKY-hierarchy, etc. A rigorous mathematical analysis of this problem is based on a proper concept of positive definiteness of the correlation functions which was developed in [2,11]. The power of the described general scheme we illustrate by the application to a particular model of Glauber-type stochastic dynamics in continuum. In this process the birth of points is independent and uniformly distributed in space. Without a death part, the density of the system will grow to the infinity with the time. To prevent such unbounded growth we can introduce a self-regulation in the model. The latter can be done in several ways and one of them is to introduce the competition between points via a proper death rate. This competition (via density
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dependent mortality in terminology of the spatial ecology), we choose in such a way that a Gibbs measure on the configuration space becomes a symmetrizing measure for the considered generator. Note that such type of stochastic dynamics with competition may be also realized as a proper framework for individual based models of complex socio-economic systems. Considered Glauber dynamics has unbounded death rate. Therefore, all known results in this field cannot be applied for the construction of the corresponding Markov process. In the present work, we have constructed a family of Markov functions for the Glauber dynamics with a competition corresponding to a class of initial distributions explicitly defined in the paper and depending on the interaction potential. This place needs an additional explanation: for the continuous IPS it would be too much to expect the existence of the stochastic dynamics for arbitrary initial distribution. The latter is wrong even for the systems without interactions, see e.g. [18]. The class of admissible initial distributions shows “how far” from the a priori reversible state an initial distribution can be chosen to be able to prevent an explosion in the dynamics. Note that the right scale of the deviations from the equilibrium state is an important technical problem for several models of continuous infinite particle dynamics (stochastic or deterministic ones). 2. Foundations We consider the Euclidean 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 configuration 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, Γ0(n) is equivalent to the symmetrization of n n Rd = (x1 , . . . , xn ) ∈ Rd xk = xl if k = l , d )n /S , where S is the permutation group of {1, . . . , n}. Hence, one can introduce i.e. to the (R n n the corresponding topology and Borel σ -algebra, which we denote by O(Γ0(n) ) and B(Γ0(n) ), respectively. The space of finite configurations Γ0 :=
(n)
Γ0
n∈N0
is equipped with the topology O(Γ0 ) of disjoint union. Let B(Γ0 ) denotes the corresponding Borel σ -algebra. A set B ∈ B(Γ0 ) is called bounded if there exists Λ ∈ Bb (Rd ) and N ∈ N such that (n) B⊂ N n=0 ΓΛ . The configuration space Γ := γ ⊂ Rd |γ ∩ Λ| < ∞, for all Λ ∈ Bb Rd
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is equipped with the vague topology O(Γ ). It is a Polish space (see e.g. [12]). 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 ) with respect to this 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; Λ (Γ ), Λ ∈ B (Rd )—the set of function from B (Γ ), whose support is a subset of Λ; • Bbs 0 b bs 0 Λ (Γ )—the set of continuous functions from B • CBΛ 0 bs bs (Γ0 ). On Γ we consider the set of cylinder functions FL0 (Γ ), i.e. the set of all measurable functions G ∈ L0 (Γ ) which are measurable with respect to BΛ (Γ ) for some Λ ∈ Bb (Rd ). These functions are characterized by the following relation: F (γ ) = F ΓΛ (γΛ ). Those cylinder functions which are measurable with respect to BΛ (Γ ) for fixed Λ ∈ Bb (Rd ) we will denote by FL0 (Γ, BΛ (Γ )). Next we would like to describe some facts from the harmonic analysis on the configuration space based on [11]. 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. [20,21]. The summation in the latter expression is taken over all finite subconfigurations of γ , which is denoted by symbol ξ γ . K-transform is linear, positivity preserving, and invertible, with K −1 F (η) :=
(−1)|η\ξ | F (ξ ), ξ ⊂η
F ∈ FL0 (Γ ), η ∈ Γ0 .
(1)
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It is easy to see that for any Λ ∈ Bb (Rd ) and arbitrary F ∈ FL0 (Γ, BΛ (Γ )), K −1 F (η) = 1ΓΛ (η)K −1 F (η),
∀η ∈ Γ0 .
(2)
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., [11]. One can introduce a convolution : L0 (Γ0 ) × L0 (Γ0 ) → L0 (Γ0 ), G1 (ξ1 ∪ ξ2 )G2 (ξ2 ∪ ξ3 ), (G1 , G2 ) → (G1 G2 )(η) :=
(3)
(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 [2]. 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 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 ). A measure ρ ∈ Mlf (Γ0 ) is called positive definite if
(G G)(η)ρ(dη) 0,
∀G ∈ Bbs (Γ0 ),
Γ0
where G is a 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
Γ0
G(η)(K ∗ μ)(dη).
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The measure ρμ := K ∗ μ is called the correlation measure of μ. As shown in [11] for μ ∈ M1fm (Γ ) and any G ∈ L1 (Γ0 , ρμ ) the series
KG(γ ) :=
G(η),
(4)
ηγ
is μ-a.s. absolutely convergent. Furthermore, KG ∈ L1 (Γ, μ) and
G(η)ρμ (dη) =
Γ0
(KG)(γ )μ(dγ ).
(5)
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 A measure μ ∈ M1fm (Γ ) is called locally absolutely continuous with respect to πzσ iff μΛ := −1 Λ = π ◦ p −1 for all Λ ∈ B (Rd ). In this case, μ ◦ pΛ is absolutely continuous with respect to πzσ zσ b Λ ∗ ρμ := K μ is absolutely continuous with respect to λzσ . Let kμ : Γ0 → R+ be the corresponding Radon–Nikodym derivative, i.e. kμ (η) :=
dρμ (η), dλzσ
η ∈ Γ0 .
Remark 2.1. The functions
kμ(n) (x1 , . . . , xn ) :=
kμ(n) : (Rd )n → R+ , kμ ({x1 , . . . , xn }), 0,
d )n , if (x1 , . . . , xn ) ∈ (R otherwise
(6)
are well-known correlation functions in statistical physics, see e.g. [27, 28]. Next, we recall the theorem about characterization of correlation measures (or correlation functions).
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Theorem 2.1. (Cf. [2,11].) Let ρ ∈ Mlf (Γ0 ) be given. Assume that ρ is positive definite, normalized and that for each bounded open Λ ⊂ Rd , for every C > 0 there exists DΛ,C > 0 such that (n) ρ ΓΛ DΛ,C C n ,
n ∈ N0 .
Then, there exists a unique measure μ ∈ M1fm (Γ ) with ρ = K μ. Remark 2.2. A sufficient condition for the bound in the theorem has the following form: for each bounded open Λ ⊂ Rd there exist εΛ > 0 and CΛ > 0 such that (n) ρ ΓΛ (n!)−εΛ (CΛ )n .
(7)
For the 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 )∈Pn (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 Pn (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. [15,23]) and it will be crucial for calculations in many places proposed in the next sections. 3. General approach to the construction of non-equilibrium dynamics for interacting particle systems (IPS) In this section we investigate the existence problem for non-equilibrium Markov processes of IPS in continuum. The mechanism of an evolution of IPS on Γ , which we would like to study, is formally described by the heuristically given generator L, defined on some proper domain of functions on Γ . The problem of construction of the corresponding process in Γ , in mathematically rigorous sense, is related to the problem of construction of a semigroup associated with L on a functional space over Γ . The latter problem in its turn concerns the possibility to find a solution to the Kolmogorov equation, which corresponds to the generator of this semigroup. Formally (only in the sense of action of operator), it has the form dFt = LFt , dt Ft |t=0 = F0 .
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In the most interesting cases the construction of the corresponding semigroup on functional spaces on Γ seems to be a very difficult question. This difficulty is mostly related to the complex structure of infinite-dimensional space Γ . In this section we propose an alternative way for the construction of the corresponding dynamic which uses deeply the harmonic analysis technique described in the previous section. Let := K −1 LK L be the formal K-transform image of L or symbol of the operator L. This object we consider as the starting point on the way to the mathematically rigorous description of the model. Let : Γ0 → R+ be an arbitrary and fixed positive function, such that
(η) C |η| , for some C > 0. We consider : D(L) ⊂ L( ) → L( ) L in the Banach space L = L( ) := L1 (Γ0 , dλ1 ), where λ1 is the Lebesgue–Poisson measure with parameters z = 1 and σ is the Lebesgue measure on Rd . One should emphasize, that the Banach space L has a Fock space structure: ∞
(n) L1 Γ0 , (n) σ (n) ,
n=0 (n)
where (n) is the nth component of the function on Γ0 . plays the crucial role in our technique. The following condition on the operator L D(L)) is a generator of a C0 -semigroup in L( ), which will be denoted Assumption 3.1. (L, t , t 0. by U t , t 0, gives the solution Gt = U t G0 to the following evolutional Remark 3.1. The semigroup U equation for the operator L in the Banach space L( ): dGt = LGt , dt Gt |t=0 = G0 .
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t , t 0, gives the possibilThe functional evolution on L(ρ) constructed via the semigroup U ity to construct the corresponding evolution of locally finite measures on Γ0 . In order to do this we consider the dual space K( ) := k : Γ0 → R k · −1 ∈ L∞ (Γ0 , λ1 ) to the Banach space L( ). The duality is given by the following expression
G, k := G · k dλ1 , G ∈ L( ).
(8)
Γ0
It is clear that K( ) is the Banach space with the norm k := k −1 L∞ (Γ
0 ,λ1 )
.
Note also, that k · −1 ∈ L∞ (Γ0 , λ1 ) means that the function k satisfies the bound k(η) const (η), λ1 -a.e. t , t 0, is constructed in the following way: The evolution on K( ), which corresponds to U t G, k. G, kt := U We denote t k := kt . U t , t 0, is a semigroup on the Banach space K( ). But it is not necessarily Remark 3.2. U a C0 -semigroup. The continuity in 0 of a L∞ -semigroups implies the boundness of the corresponding generators, which is not necessarily the case in our situation. Let k ∈ K( ) be a correlation function of some measure μ ∈ M1 (Γ ), where M1 (Γ ) denotes the class of all probability measures on Γ . Let t k, kt := U
t 0,
be the corresponding evolution of the function k in time. In order to say that there exists the corresponding evolution of probability measures on Γ we assume Assumption 3.2. For any t 0, kt ∈ K(ρ) is a positive definite, normalized function. We set M1ρ (Γ ) := μ ∈ M1 (Γ ) kμ const · ρ, λ1 -a.e. . Under Assumption 3.2, due to Theorem 2.1 about the characterization of correlation measures, one can easily construct a time evolution of the measures on M1ρ :
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209
k → μ, t k → μt , kt = U
t 0,
Ut μ := μt ∈ M1ρ . Remark 3.3. It is not difficult to see that Ut , t 0, is a semigroup. But, of course, not necessarily a C0 -semigroup. Remark 3.4. Suppose that the operator L is a generator of a semigroup on some functional space on Γ . Suppose also that it is possible to define the adjoint operator L to the operator L on M1 (Γ ). Then, the constructed above semigroup Ut , t 0, determines the solution μt := Ut μ0 to the dual Kolmogorov equation for the operator L : ∂μt = L μ t , ∂t μt |t=0 = μ0 ∈ M1ρ . Theorem 3.1. Suppose that Assumptions 3.1, 3.2 are satisfied. Then, for any μ ∈ M1ρ , there μ exists a Markov process (Xt )t0 on the configuration space Γ with the initial distribution μ associated with the generator L. Proof. Let n ∈ N, A1 , . . . , An ∈ B(Γ ) and the moments of time 0 t1 · · · tn be arbitrary and fixed. Then there exists a process, defined on some probability space (Ω, F, P ), the finitedimensional distribution of which is given by the following formula: μ μ P Xt1 ∈ A1 , . . . , Xtn ∈ An =
1An Utn −tn−1 . . . Ut2 −t1 1A1 Ut1 μ (dγ ),
Γ
where for A ∈ B(Γ ) and t 0 the measure 1A Ut μ on Γ is defined by
1A Ut μ(S) :=
1A (γ )Ut μ(dγ ),
S ∈ B(Γ ).
S
Moreover, 1A Ut μ ∈ M1ρ since the indicator function of each A ∈ B(Γ ) is bounded by 1. Eventually, we have constructed the non-equilibrium Markov process. 2 4. Application to the Glauber dynamics with competition The approach proposed in the previous section was successfully applied to a special class of Glauber dynamics on Γ with the birth rate equal to a constant (see [16]). Below we study the model with the death rate equal to some unbounded function, that makes it impossible to apply the approaches developed by [5] and [6]. In applications this death rate may be considered as reflection of the competition between the particles of the system.
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4.1. Potential and Gibbs measures on configuration spaces A pair potential is a Borel, even function φ : Rd → R ∪ {+∞}. Below we list some standard conditions on φ, known from statistical physics: (S) (Stability) There exists B > 0 such that, for any η ∈ Γ0
E(η) :=
φ(x − y) −B|η|.
{x,y}⊂η
Notice that the stability condition implies that the potential φ is semi-bounded from below. (I) (Integrability) For any β > 0,
1 − exp −βφ(x) dx < ∞.
C(β) := Rd
(SI) (Strong integrability) For any β > 0,
Cst (β) :=
1 − exp βφ(x) dx < ∞.
Rd
(P) (Positivity) φ(x) 0 for all x ∈ Rd . 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.
The energy of the configuration η ∈ Γ0 , or the Hamiltonian E φ : Γ0 → R ∪ {+∞}, which corresponds to the potential φ, is defined by E φ (η) =
φ(x − y),
η ∈ Γ0 , |η| 2.
{x,y}⊂η φ
The Hamiltonian EΛ : ΓΛ → R for Λ ∈ Bb (Rd ), which corresponds to the potential φ, is defined by φ
EΛ (η) =
φ(x − y),
η ∈ ΓΛ , |η| 2.
{x,y}⊂η φ
For fixed φ we will write for short E = E φ and EΛ = EΛ . For given γ¯ ∈ Γ we define the interaction energy between η ∈ ΓΛ and γ¯Λc = γ¯ ∩ Λc , Λc = d R \ Λ: WΛ (η|γ¯ ) =
x∈η, y∈γ¯Λc
φ(x − y).
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The interaction energy is said to be well defined if for any Λ ∈ Bb (Rd ), η ∈ ΓΛ and γ¯ ∈ Γ it is finite or +∞. For β > 0 we define EΛ (η|γ¯ ) = EΛ (η) + WΛ (η|γ¯ ) and
ZΛ (γ¯ ) :=
exp −βEΛ (η|γ¯ ) λz (dη)
ΓΛ
the so-called partition function. Let Λ ∈ Bb (Rd ), β > 0, be arbitrary, and let γ¯ ∈ Γ . The finite volume Gibbs measure on the space ΓΛ with the boundary configuration γ¯ is defined by PΛ,γ¯ (dη) =
exp {−βEΛ (η|γ¯ )} λz (dη). ZΛ (γ¯ )
Let {πΛ } denote the specification associated with z and the Hamiltonian E (see [25]) which is defined by
πΛ,γ¯ (A) = PΛ,γ¯ (dη) A
where A = {η ∈ ΓΛ : η ∪ (γ¯Λc ) ∈ A}, A ∈ B(Γ ) and γ¯ ∈ Γ . A probability measure μ on Γ is called a Gibbs measure for E and z if μ πΛ,γ¯ (A) = μ(A) for every A ∈ B(Γ ) and every Λ ∈ Bb (Rd ). This relation is the well-known (DLR)-equation (Dobrushin–Lanford–Ruelle equation), see [7] for more details. 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 a fixed potential φ we will write G(z, β) instead of G(φ, z, β). 4.2. Glauber type dynamics. Generator and the corresponding symbol on the space of finite configurations According to the general scheme the mechanism of an evolution of configurations in Γ should be specified by some formally given generator. The action of such generator in the case of Glauber type dynamics has the following form:
− d(x, γ \ x)Dx F (γ ) + b(x, γ )Dx+ F (γ ) dx, (LF )(γ ) := (Lb,d )F (γ ) = x∈γ
Rd
where Dx− F (γ ) = F (γ \ x) − F (γ ) and Dx+ F (γ ) = F (γ ∪ x) − F (γ ).
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It is known that the Gibbs measure μ ∈ G(z, β) is reversible with respect to the Markov process associated with L (i.e. the operator L is symmetrical in L2 (Γ, μ)) if and only if the following condition on coefficients b and d (birth and death rates) is fulfilled: b(x, γ ) = ze−βE(x,γ ) d(x, γ ).
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In the sequel we will be interested only in the models with birth and death rates of the form • Glauber dynamics (G+ ): b(x, γ ) = ze−βE(x,γ ) ,
d(x, γ ) = 1.
Such model was investigated by many authors, see e.g. [13,15,16]. As it was mentioned before, in the first paper the authors used the particular realization of the general approach proposed in the present work. Under conditions (I) and (P), the non-equilibrium Glauber type dynamics on the configuration space Γ was constructed. In the present paper we consider another example of the Glauber type dynamics • Glauber dynamics (G− ): b(x, γ ) = z,
d(x, γ ) = eβE(x,γ ) .
The generator which corresponds to (G− ) we denote by the same symbol L. Remark 4.1. In the case, when E(x, γ ) is given via a potential with a positive part, the death rate of the operator L will be unbounded. In the considered model, the death rate reflects a competition between points in the configuration. In the spatial ecology models such a case is related to a density dependent mortality notion. For the technical reasons we will be also interested in the model with the birth and death rates localized in some volume Λ ∈ Bb (Rd ): bΛ (x, γ ) = z1Λ (x),
dΛ (x, γ ) = 1Λ (x)eβE(x,γΛ ) .
The corresponding operator we denote by LΛ . 4.3. Symbol of the operator L Let us consider the operator L on functions FL0 (Γ, BΛ (Γ )). 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 about Markov pregenerator. Proposition 4.1. The image of L under the K-transform (or symbol of L) on functions G ∈ Bbs (Γ0 ) is given by
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(LG)(η) := K −1 LKG (η) = −G(η) eβφ(x−y) x∈η y∈η\x
−
G(ξ )
ξ ⊂η, ξ =η
eβφ(x−y)
x∈ξ y∈ξ \x
eβφ(x−y) − 1 + G(η ∪ x)z dx.
y∈η\ξ
Rd
we have Proof. According to the definition of the operator L (LG)(η) = K −1
eβE(x,·\x) Dx− KG(·) + z
x∈·
=
(−1)|η\ξ |
ξ ⊂η
+z
eβE(x,ξ \x) Dx− KG(ξ )
|η\ξ |
(−1)
ξ ⊂η
Dx+ KG(·) dx (η)
Rd
x∈ξ
Dx+ KG(ξ ) dx.
(10)
Rd
At the beginning we transform the first expression in the sum (10) I1 G(η) :=
(−1)|η\ξ |
ξ ⊂η
=
eβE(x,ξ \x) Dx− KG(ξ )
x∈ξ
(−1)|η\ξ |
ξ ⊂η
eβE(x,ξ \x)
G(ρ) −
ρ⊂ξ \x
x∈ξ
G(ρ)
ρ⊂ξ
=− (−1)|η\ξ | eβE(x,ξ \x) G(ρ ∪ x). ξ ⊂η
ρ⊂ξ \x
x∈ξ
Using the definitions of the K-transform and its inverse mapping we obtain I1 G(η) = −
(−1)|η\ξ | eβE(x,ξ \x) K G(· ∪ x) (ξ \ x) ξ ⊂η
=−
x∈ξ
(−1)|η\(ξ ∪x)| eβE(x,ξ ) K G(· ∪ x) (ξ )
x∈η ξ ⊂η\x
=−
|(η\x)\ξ |
(−1)
βφ(x−y) e K − 1 (ξ )K G(· ∪ x) (ξ )
x∈η ξ ⊂η\x
y∈·\x
βφ(x−y) −1 K =− e K − 1 K G(· ∪ x) (η \ x). x∈η
y∈·\x
For any measurable function f on Rd we denote eλ (f, η) :=
x∈η
f (x),
η ∈ Γ0 .
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A direct application of the definition of the convolution yields eλ eβφ(x−·) − 1 G(· ∪ x) (η \ x)
I1 G(η) = −
x∈η
=−
eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x)
x∈η (ξ1 ,ξ2 ,ξ3 )∈P3 (η\x)
=−
eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x).
x∈η ξ1 ⊂η\x (ξ2 ,ξ3 )∈P2 (η\(ξ1 ∪x))
Picking out from the last expression the item which corresponds to ξ1 = ∅ we get I1 G(η) = −
G(η)eλ eβφ(x−·) − 1 (ξ2 )
x∈η (ξ2 ,ξ3 )∈P2 (η\x)
−
x∈η
ξ1 ⊂η\x, ξ1 =∅
eλ eβφ(x−·) − 1 (ξ1 ∪ ξ2 )G(ξ2 ∪ ξ3 ∪ x). (11)
(ξ2 ,ξ3 )∈P2 (η\(ξ1 ∪x))
Changing the summation in (11) we have I1 G(η) = −G(η) −
eλ eβφ(x−·) − 1 (ξ )
x∈η ξ ⊂η\x
x∈η
eλ eβφ(x−·) − 1 (η \ x) \ (ξ2 ∪ ξ3 ) ∪ ξ2 G(ξ2 ∪ ξ3 ∪ x).
(ξ2 ,ξ3 )∈P2 (η\x), ξ1 ∪ξ2 =η\x
Using the fact that Keλ eβφ(x−·) − 1 (η) = eλ eβφ(x−·) (η),
η ∈ Γ0 ,
we obtain I1 G(η) = − G(η)
eβφ(x−y)
x∈η y∈η\x
−
eλ eβφ(x−·) − 1 (η \ x) \ ξ ∪ ρ G(ξ ∪ x).
x∈η ξ ⊂η\x, ξ =η\x ρ⊂ξ
Finally, changing the summation in the last term and using (12) we get I1 G(η) = −G(η)
x∈η y∈η\x
−
ξ ⊂η, ξ =η
G(ξ )
eβφ(x−y) x∈ξ y∈ξ \x
Now, we transform the second item of (10):
eβφ(x−y)
eβφ(x−y) − 1 . y∈η\ξ
(12)
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I2 G(η) := z
|η\ξ |
(−1)
ξ ⊂η
=z
Dx+ KG(ξ ) dx
Rd
(−1)|η\ξ |
ξ ⊂η
Rd
215
G(ρ) −
ρ⊂ξ ∪x
G(ρ) dx.
ρ⊂ξ
A direct use of the definitions of the K-transform and its inverse yields
I2 G(η) = z (−1)|η\ξ | G(ρ ∪ x) dx ξ ⊂η
Rd
ρ⊂ξ
|η\ξ | =z (−1) K G(· ∪ x) (ξ ) dx = z G(η ∪ x) dx. Rd
ξ ⊂η
2
Rd
4.4. Verification of Assumption 3.1 In the following subsections, the Lebesgue–Poisson measure λ1 , defined for the general approach, will be denoted for simplicity by λ. We assume also that the potential φ satisfies conditions (S) and (SI). in the Banach space For arbitrary and fixed C > 0, we consider the operator L (13) LC := L1 Γ0 , C |η| λ(dη) . s
Symbol · stands for the norm of the space (13) and symbol → denotes the strong convergence of operators in LC . Remark 4.2. According to the general scheme (η) := C |η| , η ∈ Γ0 . For any Λ ∈ Bb (Rd ) we set LΛ C := {G ∈ LC | G Γ0 \ΓΛ = 0}.
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Λ It is not difficult to show that LΛ C is a closed linear subset in (LC , · ). Therefore, (LC , · ) is a subspace of (LC , · ). For any ω > 0 we introduce a set H(ω, 0) of all densely defined closed operators T on LC , the resolvent set ρ(T ) of which contains the following sector π π + ω := ζ ∈ C |arg ζ | < + ω , ω > 0, Sect 2 2
and for any ε > 0 (T − ζ 1)−1 Mε , |ζ |
|arg ζ |
π + ω − ε, 2
where Mε does not depend on ζ . Let H(ω, θ ), θ ∈ R denotes the set of all operators of the form T = T0 + θ with T0 ∈ H(ω, 0).
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Remark 4.3. It is well known (see e.g., [10]), that any T ∈ H(ω, θ ) is a generator of a holomorphic semigroup U (t) in the sector |arg t| < ω. The function U (t) is not necessarily uniformly bounded, but it is quasi-bounded, i.e. U (t) consteθt in any sector of the form |arg t| ω − ε. Proposition 4.2. For any C > 0, the operator (L0 G)(η) = (L0,β G)(η) := −G(η) D(L0 ) = G ∈ LC
eβφ(x−y) ,
x∈η y∈η\x
βφ(x−y) e G(η) ∈ LC x∈η y∈η\x
is a generator of a contraction semigroup on LC . Moreover, L0 ∈ H(ω, 0) for all ω ∈ (0, π2 ). Proof. It is not difficult to show that the operator L0 is densely defined and closed. Let 0 < ω < π 2 be arbitrary and fixed. Since the potential V satisfies (S), for all η ∈ Γ0 , |η| > 1, we have
1
eβφ(x−y) |η|e |η|
2β
{x,y}∈η φ(x−y)
|η|e−2Bβ .
(15)
x∈η y∈η\x
This inequality implies that for all ζ ∈ Sect( π2 + ω) βφ(x−y) > 0, e + ζ
η ∈ Γ0 .
x∈η y∈η\x
Therefore, for any ζ ∈ Sect( π2 + ω) the inverse operator (L0 − ζ 1)−1 , the action of which is given by (L0 − ζ 1)−1 G (η) = −
x∈η
1
y∈η\x
eβφ(x−y) + ζ
G(η),
(16)
is well defined on the whole space LC . Moreover, it is a bounded operator in this space and (L0 − ζ 1)−1
1 |ζ | , M |ζ | ,
if Re ζ 0, if Re ζ < 0,
(17)
where the constant M does not depend on ζ . Indeed, the case Re ζ 0 is a direct consequence of (16) and the inequality x∈η y∈η\x
eβφ(x−y) + Re ζ |η|e−2Bβ + Re ζ Re ζ 0.
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We prove now the bound (17) for the case Re ζ < 0. Using (16), we have 1 (L0 − ζ 1)−1 G = G(·) | βφ(x−y) + ζ | e x∈· y∈·\x |ζ | 1 = G(·) . |ζ | | x∈· y∈·\x eβφ(x−y) + ζ | Since ζ ∈ Sect( π2 + ω), π + ω = |ζ | cos ω. | Im ζ | |ζ |sin 2 Hence, |
x∈η
|ζ | 1 |ζ | =: M βφ(x−y) + ζ | | Im ζ | cos ω y∈η\x e
and (17) is fulfilled. The rest statement of the lemma follows now directly from the theorem of Hille–Yosida (see e.g., [10]). 2 An additional parameter of the model. Since the intensity z of the Lebesgue–Poisson measure in the definition of the Banach space LC is equal to 1, let z, which was involved in the structure of the birth rate of L, now plays the role of an additional parameter := z. We set now (L1 G)(η) := (L1,β G)(η) eβφ(x−y) − 1 , G(ξ ) eβφ(x−y) =− ξ ⊂η, ξ =η
x∈ξ y∈ξ \x
y∈η\ξ
D(L1 ) := D(L0 ) and
(L2, G)(η) =
G(η ∪ x) dx, Rd
D(L2 ) := D(L0 ). The well-definiteness of these operators will be clear from the lemma below. We will sometimes to emphasize the dependence on and β. use the notation L ,β instead of L Lemma 4.1. For any δ > 0 there exist 0 > 0 and β0 > 0 such that for all 0 and β β0 (L1,β + L2, )G aL0 G + bG, G ∈ D(L0 ), (18) with a = a(, β) < δ, b = b(, β) < δ.
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Proof. It is not difficult to show that βE(x,ξ \x) |η| βφ(x−y) G(ξ ) 1−e C λ(dη) e
L1 G
Γ0 ξ ⊂η, ξ =η
x∈ξ
y∈η\ξ
βE(x,ξ \x) |η| βφ(x−y) G(ξ ) 1−e C λ(dη) e = Γ0 ξ ⊂η
−
x∈ξ
y∈η\ξ
βE(x,η\x) |η| G(η) e C λ(dη).
(19)
x∈η
Γ0
The application of the Minlos lemma to (19) gives us
L1 G
βE(x,η\x) |η| G(η) e C λ(dη) x∈η
Γ0
= eCst (β)C − 1 L0 G.
1 − eβφ(y) C |ξ | λ(dξ ) − L0 G
Γ0 y∈ξ
We estimate the norm L2 G using the Minlos lemma and the bound (15). Namely,
G(η ∪ x) dx C |η| λ(dη)
L2 G Γ0 Rd
|η|G(η)C |η|−1 λ(dη)
Γ0
e2Bβ
Γ0
eβE(x,η\x) G(η)C |η|−1 λ(dη) = e2Bβ C −1 L0 G.
x∈η
Therefore, (L1 + L2 )G eCst (β)C + e2Bβ C −1 − 1 L0 G. And hence the assertion of the lemma is fulfilled with the coefficients a := eCst (β)C + e2Bβ C −1 − 1,
b := 0
which can be taken less then δ for the appropriate choice of and β.
2
Theorem 4.1. For any C > 0, and for all , β > 0 which satisfy 2eCst (β)C + 2e2Bβ C −1 < 3 the operator L ,β is a generator of a holomorphic semigroup in LC .
(20)
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219
Proof. It follows from the theorem about perturbation of holomorphic semigroup (see e.g., [10]). For the reader’s convenience, below we give its formulation: For any T ∈ H(ω, θ ) and for any ε > 0 there exist positive constants , δ such that if the operator A satisfies Au aT u + bu,
u ∈ D(T ) ⊂ D(A),
with a < δ, b < δ, then T + A ∈ H(ω − ε, ). In particular, if θ = 0 and b = 0, then T + A ∈ H(ω − ε, 0). 2 Remark 4.4. Applying the proof of the theorem about perturbation of the generator of a holomorphic semigroup (see, e.g. [10]) to our case and taking into account the fact that L0 ∈ H(ω, 0), for any ω ∈ (0, π2 ), one can conclude that δ in this theorem can be chosen to be 12 . For our further purposes we have to show that the holomorphic semigroup constructed in Theorem 4.1 can be approximated by semigroups localized in bounded volumes. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Then all results proved in this subsection hold true for the operator L Λ G(η) := −
eβφ(x−y) G(η)
x∈ηΛ y∈ηΛ \x
−
G(ξ )
ξ ⊂η, ξ =η
eβφ(x−y)1Λ (y) − 1 + G(η ∪ x) dx
eβφ(x−y)
x∈ξΛ y∈ξΛ \x
y∈η\ξ
Λ
acting in the functional space LΛ C with the domain D(L Λ ) := G ∈ LC
eβφ(x−y) G(η) ∈ LΛ C .
x∈ηΛ y∈ηΛ \x
Namely, the main result can be formulated as follows. Theorem 4.2. For any Λ ∈ Bb (Rd ), and any triple of constants C, > 0, and β > 0 which satisfy 2eCst (β)C + 2e2Bβ C −1 < 3 Λ the operator L Λ is a generator of a holomorphic semigroup in LC .
Remark 4.5. The arguments, analogous to those which were proposed in the proof of Lemma 4.1, imply that (18) holds for the operators L 0,Λ G(η) := −
eβφ(x−y) G(η),
x∈ηΛ y∈ηΛ \x
L 1,Λ G(η) := −
ξ ⊂η, ξ =η
G(ξ )
x∈ξΛ y∈ξΛ \x
eβφ(x−y)
eβφ(x−y)1Λ (y) − 1 , y∈η\ξ
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and
(L2,,Λ G)(η) =
G(η ∪ x) dx, Λ
with βE(x,ηΛ \x) Λ D(L0,Λ ) = D(L1,Λ ) = D(L2,Λ ) := G ∈ LC e G(η) ∈ LC . x∈ηΛ
Moreover, the bound (18) in this case will be uniform with respect to Λ ∈ Bb (Rd ), i.e. the coefficients a > 0 and b > 0 in (18) can be chosen independent of Λ. Fix any triple of positive constants C, and β which satisfies (20) and any Λ ∈ Bb (Rd ). Λ Remark 4.6. Let U t (C, , β) be a holomorphic semigroup generated by the operator (LΛ , D(LΛ )) Λ Λ on LC . Then Ut (C, , β)PΛ , t 0, where PΛ G(η) := 1ΓΛ (η)G(η),
G ∈ LC ,
is a semigroup on LC generated by the operator L Λ PΛ with the domain D(LΛ PΛ ) := G ∈ LC
e
βφ(x−y)
1ΓΛ (η)G(η) ∈ LC .
x∈ηΛ y∈ηΛ \x
Remark 4.7. The theorem about perturbation of the generator of a holomorphic semigroup, mentioned before in this subsection (see also [10]), implies that for any Λ ∈ Bb (Rd ) and ε > 0 there exists > 0 and a constant M > 0 which does not depend on Λ such that for any ζ from the half-plane Re ζ > the following bound holds: −1 (L Λ PΛ − ζ )
Mε , |ζ − |
arg (ζ − ) π + ω − ε. 2
be a sequence of bounded Borel sets such that Λn ⊂ Λn+1 , for all n ∈ N, and Let {Λn }n1 d . Below, we formulate the following approximation theorem. Λ = R n1 n Λn t (C, , β) and {U Theorem 4.3. Let U t (C, , β), n 1} be holomorphic semigroups generated Λn and {L by L Λn , , n 1} in the spaces LC and LC , respectively. Then, s t (C, , β), UtΛn (C, , β)PΛn → U
uniformly on any finite interval of t 0.
n → ∞,
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Proof. Using the approximation theorem for quasi-bounded semigroups (see e.g. [10]), it is enough to show that −1 s (L → (L − ζ )−1 Λn , PΛn − ζ )
for some ζ ∈ C such that Re ζ > θ . Let ζ ∈ C, Re ζ > θ be arbitrary and fixed. For any G ∈ LC it holds −1 (L − ζ )−1 G Λn , PΛn − ζ ) G − (L −1 − ζ )−1 G. = (L Λn , PΛn − ζ ) [L −L Λn , PΛn ](L ) = D(L0 ) For any G ∈ D(L − L [L Λn , PΛn ]G(η) =− eβφ(x−y) 1 − 1ΓΛn (η) G(η) x∈η y∈η\x
−
G(ξ )
ξ ⊂η, ξ =η
+
eβφ(x−y) − 1 1 − 1ΓΛn (ξ )1ΓΛn (η \ ξ )
eβφ(x−y)
x∈ξ y∈ξ \x
y∈η\ξ
1 − 1Λn (η ∪ x) G(η ∪ x) dx +
G(η ∪ x) dx,
Λcn
Λn
where Λcn = Rd \ Λn . Using the simple inequality 1 − 1Γ (ξ )1Γ (η) 1 − 1Γ (ξ ) + 1 − 1Γ (η), Λn Λn Λn Λn
ξ, η ∈ Γ0 ,
and the estimates analogous to those which were proposed in Lemma 4.1 we obtain [L − L Λn , PΛn ]G(η)
C (β)C βφ(x−y) 2Bβ −1 st e + e C 1 − 1ΓΛn (·) e G(·)
−1
x∈· y∈·\x
|G(·)
| ·Λcn + e C
βφ(x−y) 1 − 1Γ (η)K(η)C |η| λ(dη), + e G(·) Λn 2Bβ
x∈· y∈·\x
ΓΛn
where K(η) :=
1 − eβφ(x) ,
η ∈ Γ0 .
x∈η
All of terms in the right-hand side of the last inequality tends to zero, when n → ∞.
(21)
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Using Remark 4.7 and (21) we easily conclude that the difference in (21) also tends to zero when n → ∞. 2 4.5. Verification of Assumption 3.2 t (C, , β) be a Fix any triple of positive constants C, and β which satisfies (20). Let U and let holomorphic semigroup generated by L ,β KC := k : Γ0 → R k(·)C −|·| ∈ L∞ (Γ0 , λ) be the dual space to the space LC with respect to the following duality:
G, k := Gk dλ.
(22)
(23)
Γ0
It is also called the space of “so-called correlation functions.” Analogously to the general scheme, KC is a Banach space. Note, also that k(·)C −|·| ∈ L∞ (Γ0 , λ) means that the function k satisfies the following bound k(η) const C |η| , a.a. η ∈ Γ0 with respect to λ, (24) which is known as the classical Ruelle bound, see e.g. [27]. (C, , β) be a semigroup on KC determined by According to the general scheme, let U t t (C, , β) via the duality (23). U Next, we solve the following problem. Suppose that k0 ∈ KC is a correlation function, i.e., there exists a probability measure μ0 ∈ M1fm (Γ ), locally absolutely continuous with respect to the Poisson measure, whose correlation function is exactly k0 . We would like to investigate now (C, , β) preserves the property whether the evolution of k0 in time given by the semigroup U t (C, , β)k0 , at any moment of time t > 0, is a correlation described above. Namely, whether U t function or not? In order to answer this question, one can apply, for example, the theorem about characterization of correlation functions, proposed in [11]. The conditions of this theorem, which must be checked for our particular model are the following: (C, , β)k0 0, for any t 0: G G, U t
∀G ∈ Bbs (Γ0 ).
Further explanations will be devoted to the verification of the latter condition. Let μ ∈ G(β, z) and {πΛ,∅ }Λ∈Bb (Rd ) denotes the specification with empty boundary conditions corresponding to the Gibbs measure μ. We define
E(F, G) := Dx+ F (γ )Dx+ G(γ ) πΛ (dγ , ∅), F, G ∈ KCBΛ bs (Γ0 ), Γ
x∈γ
Λ where KCBΛ bs (Γ0 ) is K-image of CBbs (Γ0 ). Now we would like to list some facts the proofs of which are completely analogous to those proposed in [13,17].
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2 d Lemma 4.2. The set KCBΛ bs (Γ0 ) is dense in L (Γ, πΛ,∅ ) for any Λ ∈ Bb (R ).
Lemma 4.3. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Then (E, KCBΛ bs (Γ0 )) is a well-defined bilinear form on L2 (Γ, πΛ,∅ ). Lemma 4.4. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Suppose that conditions (S) and (SI) are satΛ isfied. Then (LΛ , KCBΛ bs (Γ0 )) is an operator associated with the bilinear form (E, KCBbs (Γ0 )) 2 in L (Γ, πΛ,∅ ), i.e.
E(F, G) = LΛ F (γ )G(γ ) πΛ,∅ (dγ ), F, G ∈ KCBΛ bs (Γ0 ). Γ
Lemma 4.5. Let Λ ∈ Bb (Rd ) be arbitrary and fixed. Suppose that conditions (S) and (SI) are satisfied and μ ∈ G(z, β). Then there exists a self-adjoint positive Friedrichs’ extension Λ 2 (L Λ , D(L Λ )) of the operator (LΛ , KCBbs (Γ0 )) in L (Γ, πΛ,∅ ). Moreover, (L Λ , D(L Λ )) is a 2 generator of a contraction semigroup which preserves 1 in L (Γ, πΛ,∅ ), associated with some Markov process. Remark 4.8. It is well known (see e.g. [26]) that under the condition of Lemma 4.5 the semi1 d group generated by (L Λ , D(L Λ )) can be extended to L (Γ, πΛ,∅ ). For any Λ ∈ Bb (R ), the extension of this semigroup in L1 (Γ, πΛ,∅ ) we will denote by (UtΛ )t0 . For the generator of this semigroup we will use the notation (L Λ , D1 (L Λ )), where D1 (L Λ ) ⊃ D(L Λ ) is the domain of 1 L Λ in L (Γ, πΛ,∅ ). Now, we introduce one of the crucial lemmas about the evolution of the “so-called correlation functions.” Lemma 4.6. Let the positive constants C, and β which satisfy (20) be arbitrary and fixed. The (C, , β) on KC preserves positive semi-definiteness, i.e. for any t 0 semigroup U t (C, , β)k 0, ∀G ∈ Bbs (Γ0 ), G G, U t iff G G, k 0,
(25)
for any G ∈ Bbs (Γ0 ). Remark 4.9. Let MC stands for the set of all probability measures on Γ , locally absolutely continuous with respect to the Poisson measure, with locally finite moments, whose correlation functions satisfy the bound (24). As it was pointed out at the beginning of this section, the condition (25) on the function k ∈ KC insures the existence of a unique measure μ ∈ MC whose correlation function is k, see [11]. Proof of Lemma 4.6. Under the assumptions of the lemma we have to show that for any t 0 t (C, , β)(G G), k 0, ∀G ∈ Bbs (Γ0 ). U (26)
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But G G ∈ Bbs (Γ0 ) for any G ∈ Bbs (Γ0 ). Therefore, due to Theorem 4.3 it is enough to show that for any t 0 and any G ∈ Bbs (Γ0 ) there exists Λ ∈ Bb (Rd ) such that for all Λ ∈ Bb (Rd ), Λ ⊃ Λ UtΛ (C, , β)PΛ (G G), k 0.
(27)
Let Λ ∈ Bb (Rd ) be arbitrary and fixed. We set −1 Λ UtΛ := K U t (C, , β)K ,
t 0.
(UtΛ )t0 is a semigroup on −1 Λ L1 := KLΛ C , · 1 := K · L C
which is a Banach space. Moreover, it is not difficult to show that the generator of this semigroup coincides with (LΛ , KD(L Λ )). 1 Proposition 4.3. For any F ∈ LΛ 1 ⊂ L (Γ, πΛ,∅ ),
UtΛ F = UtΛ F,
t 0, in L1 (Γ, πΛ,∅ ),
where (UtΛ )t0 is defined in Remark 4.8. Λ Λ Proof. The fact that (LΛ , KD(L Λ )) is the generator of (Ut )t0 in (L1 , · 1 ) implies the following (see e.g. [9])
−n Λ U F − 1 − t L Λ F → 0, t n 1
n → ∞, for all F ∈ LΛ 1.
Since · 1 · L1 (Γ,πΛ,∅ ) , the latter fact gives −n Λ U F − 1 − t L Λ F → 0, t 1 n L (Γ,πΛ,∅ )
n → ∞, for all F ∈ LΛ 1.
(28)
Λ Analogously, the fact that (L Λ , D1 (L Λ )) is the generator of (Ut )t0 gives us
−n Λ Ut F − 1 − t L F → 0, Λ 1 n L (Γ,πΛ,∅ )
n → ∞, for all F ∈ LΛ 1.
(29)
As was shown before, there exists > 0 such that for any real ζ > −1 −1 −1 −1 (L Λ − ζ 1) F − (LΛ − ζ 1) F = (L Λ − ζ 1) [LΛ − L Λ ](LΛ − ζ 1) F.
The function Fζ := (LΛ − ζ 1)−1 F ∈ KD(L Λ ). Hence, [LΛ − L Λ ]Fζ = 0. The latter fact means that
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−n Λ Ut F − 1 − t L F Λ 1 n L (Γ,πΛ,∅ ) −n Λ t = F → 0, Ut F − 1 − n LΛ 1 L (Γ,πΛ,∅ )
n → ∞, for all F ∈ LΛ 1.
The convergences (28) and (29) imply the assertion of the proposition.
225
(30)
2
Corollary 4.1. Lemma 4.5 implies that for any moment of time t 0 UtΛ F 0,
for all non-negative F ∈ KBbs (Γ0 ).
(31)
Let t 0 and G ∈ Bbs (Γ0 ) be arbitrary and fixed. Suppose that N ∈ N and Λ ∈ Bb (Rd ) are such that G G Γ
N
(n) n=0 ΓΛ
0\
= 0.
d 2 2 Then, K(G G) = |KG|2 ∈ LΛ 1 for all Λ ∈ Bb (R ), Λ ⊃ Λ . Moreover, PΛ |KG| = |KG| . d Hence, the left-hand side of (27) for any Λ ∈ Bb (R ), Λ ⊃ Λ , is equal to the following expression:
UtΛ (C, , β)PΛ (G G), k =
Λ KU t (G G)(γ )μ (dγ )
Γ
UtΛ K(G G)(γ )μ (dγ ) =
= Γ
UtΛ |KG|2 (γ )μΛ (dγ ),
ΓΛ
where μΛ is a projection of μ on ΓΛ (see Remark 4.9). Let us mention that the measure μ is locally absolutely continuous with respect to the Poisson measure π . Therefore, UtΛ (C, , β)PΛ (G G), k =
UtΛ |KG|2 (γ )
ΓΛ
dμΛ (γ )πΛ (dη). dπΛ
Corollary 4.1 implies that there exists a set S ⊂ Γ, πΛ,∅ (S) = 0, such that for all γ ∈ Γ \ S: UtΛ |KG|2 (γ ) 0. But πΛ,∅ is absolutely continuous with respect to πΛ . Furthermore, the corresponding Radon– Nikodym derivative is positive almost surely with respect to πΛ . Hence, πΛ (SΛ ) = 0, where SΛ is a projection of the set S to ΓΛ , and UtΛ (C, , β)PΛ (G G), k =
UtΛ |KG|2 (γ )
ΓΛ \SΛ
The latter proves the assertion of Lemma 4.6.
2
dμΛ (γ )πΛ (dη) 0. dπΛ
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The result obtained in Lemma 4.6 and the characterization of correlation functions in [11] imply the following corollary. Corollary 4.2. Let positive constants C, and β which satisfy (20) be arbitrary and fixed. Let k ∈ KC be such that G G, k 0, for any G ∈ Bbs (Γ0 ). Then for any t 0 there exists a (C, , β)k. unique measure μt ∈ MC whose correlation function is U t In Corollary 4.2, we denote the evolution of the measure μ in time by Ut (C, , β)μ := μt . According to the general scheme (Ut (C, , β))t0 is a semigroup on MC,β . This leads us directly to the construction of a non-equilibrium Markov process (or rather a Markov function) on Γ . Theorem 4.4. Suppose that conditions (S) and (SI) are satisfied. For any triple of positive conμ stants C, and β which satisfies (20) and any μ ∈ MC there exists a Markov process Xt ∈ Γ with initial distribution μ associated with the generator L . Proof. The proof is a direct consequence of Theorem 3.1.
2
Acknowledgments The financial support of the DFG through the SFB 701 (Bielefeld University) and German– Ukrainian Project 436 UKR 113/80 is gratefully acknowledged. R. Minlos gratefully acknowledges the financial support of RFFI No. 06-01-00449 and CRDF research funds N RM1-2085. References [1] S. Albeverio, Yu. Kondratiev, M. Röckner, Analysis and geometry on configuration spaces, J. Funct. Anal. 154 (1998) 444–500. [2] 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. [3] 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. [4] R.L. Dobrushin, Ya.G. Sinai, Yu.M. Suhov, Dynamical system of the statistical mechanics, in: Sovrem. Mat. Fundam. Napravl., vol. 2, VINITI, 1985, pp. 235–284. [5] R. Fernandez, P. Ferrari, G. Guerberoff, Spatial birth-and-death processes in random environment, Math. Phys. Electron. J. 11 (2005) 3–52. [6] N. Garcia, T. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 281–303. [7] H.O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, 1982. [8] R.A. Holley, D.W. Stroock, Nearest neighbor birth and death processes on the real line, Acta Math. 140 (1978) 103–154. [9] K. Ito, F. Kappel, Evolution Equations and Approximations, World Scientific, 2002. [10] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. [11] 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. [12] Yu.G. Kondratiev, O.V. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (7) (2006) 774–783. [13] Yu.G. Kondratiev, E. Lytvynov, Glauber dynamics of continuous particle systems, Ann. Inst. H. Poincaré Probab. Statist. 41 (4) (2005) 685–702. [14] Yu.G. Kondratiev, A. Skorokhod, On contact process in continuum, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2) (2006) 187–198.
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[15] 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. [16] Yu.G. Kondratiev, O.V. Kutoviy, E. Zhizhina, Nonequilibrium Glauber-type dynamics in continuum, J. Math. Phys. 47 (11) (2006) 17 pp. [17] Yu.G. Kondratiev, E. Lytvynov, M. Röckner, Infinite interacting diffusion particles. I. Equilibrium process and its scaling limit, Forum Math. 18 (1) (2006) 9–43. [18] Yu.G. Kondratiev, E. Lytvynov, M. Röckner, Equilibrium Kawasaki dynamics of continuous particle systems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2) (2007) 185–209. [19] Yu. G. Kondratiev, T. Kuna, M.J. Oliveira, J.L. da Silva, L. Streit, Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems SFB-701, preprint, University of Bielefeld, Bielefeld, Germany, 2007. [20] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I, Arch. Ration. Mech. Anal. 59 (1975) 219–239. [21] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. II, Arch. Ration. Mech. Anal. 59 (1975) 241–256. [22] T.M. Liggett, Interacting Particle Systems, Springer-Verlag, 1985. [23] R.A. Minlos, Lectures on statistical physics, Uspekhi Mat. Nauk 23 (1) (1968) 133–190 (in Russian). [24] C. Preston, Spatial birth-and-death processes, in: Proceedings of the 40th Session of the International Statistical Institute, Warsaw, 1975, vol. 2, Bull. Inst. Internat. Statist. 46 (1975) 371–391. [25] C. Preston, Random Fields, Lecture Notes in Math., vol. 534, Springer-Verlag, 1976. [26] M. Reed, B. Simon, Methods of Modern Mathematical Physics, 2. Fourier Analysis, Self-Adjointness, Academic Press, 1972. [27] D. Ruelle, Statistical Mechanics, Benjamin, 1969. [28] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970) 127–159. [29] L. Wu, Estimate of spectral gap for continuous gas, Ann. Inst. H. Poincaré Probab. Statist. 40 (4) (2004) 387–409.
Journal of Functional Analysis 255 (2008) 228–254 www.elsevier.com/locate/jfa
Matrix Riemann–Hilbert problems and factorization on Riemann surfaces M.C. Câmara a,1 , A.F. dos Santos a,∗,1 , Pedro F. dos Santos b,2 a Centro de Análise Funcional e Aplicações, Departamento de Matemática, Instituto Superior Técnico, Portugal b Centro de Análise, Geometria e Sistemas Dinâmicos, Departamento de Matemática,
Instituto Superior Técnico, Portugal Received 7 December 2007; accepted 10 January 2008 Available online 29 February 2008 Communicated by Paul Malliavin
Abstract The Wiener–Hopf factorization of 2 × 2 matrix functions and its close relation to scalar Riemann–Hilbert problems on Riemann surfaces is investigated. A family of function classes denoted C(Q1 , Q2 ) is defined. To each class C(Q1 , Q2 ) a Riemann surface Σ is associated, so that the factorization of the elements of C(Q1 , Q2 ) is reduced to solving a scalar Riemann–Hilbert problem on Σ. For the solution of this problem, a notion of Σ-factorization is introduced and a factorization theorem is presented. An example of the factorization of a function belonging to the group of exponentials of rational functions is studied. This example may be seen as typical of applications of the results of this paper to finite-dimensional integrable systems. © 2008 Elsevier Inc. All rights reserved. Keywords: Riemann–Hilbert problem; Factorization; Riemann surfaces; Integrable systems
* Corresponding author.
E-mail addresses:
[email protected] (M.C. Câmara),
[email protected] (A.F. dos Santos),
[email protected] (P.F. dos Santos). 1 Partially supported by Fundação para a Ciência e a Tecnologia through Program POCI 2010/FEDER and FCT Project PTDC/MAT/81385/2006. 2 Partially supported by Fundação para a Ciência e a Tecnologia through Program POCI 2010/FEDER. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.01.008
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229
1. Introduction The central object of this paper is the interplay between the problem of Wiener–Hopf factorization of 2 × 2 matrix functions and scalar Riemann–Hilbert problems on Riemann surfaces. The first part of the paper deals with the definition of certain classes of matrix functions that play a fundamental role in the study of the factorization of 2 × 2 matrix functions and the second part concentrates on the associated Riemann–Hilbert problems on Riemann surfaces. Before we give an overview of the main results of the paper we recall the definition of Wiener– Hopf factorization of a 2×2 Hölder continuous matrix function G on a piecewise smooth contour ΓC which divides C into disjoint regions ΩC+ , ΩC− such that C = ΩC+ ∪ ΓC ∪ ΩC− (Definition 3.1 below). This is a representation of the form G = G− DG+ ,
(1.1)
where G± and their inverses are analytic in ΩC± and continuous in ΩC± and D = diag(zk1 , zk2 ) with ki ∈ Z and k1 k2 . In the paper we define a family of classes C(Q1 , Q2 ) of all the G ∈ Cμ (ΓC ) satisfying GT Q1 G = hQ2 ,
(1.2)
where the upperscript T denotes transposition and h is an invertible scalar function on ΓC . Denoting by Cμ± (ΓC ) the spaces of functions of Cμ (ΓC ) that have analytic extensions into ΩC± , we assume in (1.2) that Q1 ∈ Cμ− (ΓC ) + R and Q2 ∈ Cμ+ (ΓC ) + R, where R denotes the space of rational functions with poles off ΓC . The first important result of the paper is that all functions that have a factorization of the form (1.1) belong to some class C(Q1 , Q2 ) with the additional condition that det Q1 = det Q2 (Theorems 2.4 and 2.7). This fact means that we may associate with each class C(Q1 , Q2 ) a Riemann surface (which may be the Riemann sphere or even trivial) defined by an algebraic curve of the form μ2 = det Q1 = det Q2 . This, in turn, provides us with a tool that enables us to study the solvability of the factorization problem for important classes of matrix functions and obtain formulas for the factors when the factorization exists. In the paper a general representation for the elements of each class C(Q1 , Q2 ) is derived (Theorem 2.12) and this leads to a technique to obtain a related scalar Riemann–Hilbert problem on the above-mentioned Riemann surface. In Theorem 2.20 we state an important result that gives an alternative characterization of the classes C(Q1 , Q2 ) in terms of an equivalence of multiplication operators on [Cμ (ΓC )]2 and scalar multiplication operators on the preimage of the contour ΓC under the standard projections from the Riemann surface to the complex plane. The results of Section 2 give, for the first time, a general framework for the study of the factorizations of 2 × 2 matrix functions, framework that goes significantly beyond the dispersive results that can be found in the specialized literature. In Section 3 we introduce the concept of Σ -factorization relative to a contour Γ on a Riemann surface Σ (Definition 3.1), which we recall here. A function f ∈ Cμ (Γ ), invertible on the contour Γ is said to have a Σ -factorization if it can be represented in the form f = f− rf+
(1.3)
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with r a rational function on Σ and f+±1 ∈ Cμ+ (Γ ), f−±1 ∈ Cμ− (Γ ) where Cμ± (Γ ) are the subspaces of Cμ (Γ ) of functions that have analytic extensions into the regions that are preimages of Ω ± , continuous in Ω ± . If r is a constant the factorization is called special. It is shown in Theorem 3.4 that this case occurs if and only if we have 1 2πi
(log f )ω ∈ Zg + BZg , Γ
where ω is the vector of normalized holomorphic differentials and Zg + BZg is the lattice of periods of ω. Definition (1.3) is crucial in the development of a technique for solving scalar Riemann– Hilbert problems on a Riemann surface in a rigorous and elegant way. This is done through a factorization theorem (Theorem 3.4) which in general terms states that if f ∈ Cμ (Γ ) is such that log f ∈ Cμ (Γ ) then f possesses a Σ-factorization with factors given by f+ = exp PΓ+ log f h+ , f− = exp PΓ− log f − log h h− ,
(1.4)
where h is a function depending on log f (see Theorem 3.4) whose factorization h = h− rh+ is given in Proposition 3.10. In (1.4) PΓ± are the complementary projections defined by the meromorphic analog of the Cauchy kernel (see Appendix A). The above result has the convenient feature of avoiding the use of the so called discontinuous analog of the Cauchy kernel for Riemann surfaces [17]. The method applies to surfaces with genus greater than 1 although the calculations become considerably more difficult in the general case. Section 4 deals with an example that belongs to the group of exponentials of rational matrix functions which, besides illustrating the techniques developed in Sections 2 and 3, has the additional interest of being typical of problems appearing in the study of finite-dimensional integrable systems [4,13]. To end this introduction we make some brief remarks on references related to the problems that are dealt with in this paper. The study of the class C(Q1 , Q2 ) was initiated in [3], with emphasis on the case Q1 = Q2 . However the theory expounded in the present paper is much more general and the connection with Riemann–Hilbert problems on Riemann surfaces was not touched in [3]. A reduction of the factorization problem of 2 × 2 Daniele–Khrapkov matrix functions to a scalar Riemann–Hilbert problem on a Riemann surface was studied for the first time in [10]. However the treatment followed in [10] is unnecessarily complicated and appears incomplete from the point of view of the relation between the dimension of the spaces of solutions of the two problems. Also the solvability of the resulting Riemann–Hilbert problem on the associated Riemann surface is not studied in that paper. General references on Riemann–Hilbert problems on Riemann surfaces are [9,14,16,17]. In [14] the solvability conditions are presented but no usable formulas are given. [17] is a useful general reference on the Riemann–Hilbert problem on Riemann surfaces, including the question of the analogs of the Cauchy kernel. The paper is almost entirely written in a classical complex analysis perspective.
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The notions of Σ-factorization and the factorization Theorem 3.4 are presented here for the first time and cannot be found in any of the above references. 2. Matrix factorization and the class C(Q1 , Q2 ) In what follows we restrict the study of matrix factorization to 2 × 2 matrix-functions because of its interest in applications and its relation to factorization on a Riemann surface to be developed later. However many results of this section can be generalized to n × n matrix functions at the expense of greater computational complexity. Before we define the class C(Q1 , Q2 ) we introduce some terminology. By H + (∂D) and − H (∂D) we denote, respectively, the Hardy spaces of bounded analytic functions in D (the unit disc in C) and C \ D. These spaces will be identified with closed subspaces of L∞ (∂D). By R(∂D) we denote the space of rational functions on C with poles off ∂D. If A is an algebra, we denote by G(A) the group of invertible elements in the algebra A. Definition 2.1. Let G ∈ G([L∞ (∂D)]2×2 ). G is said to possess a bounded Wiener–Hopf factorization if it can be represented in the form G = G− DG+ ,
(2.1)
where G± ∈ G([H ± (∂D)]2×2 ) and D = diag(zk1 , zk2 ) with ki ∈ Z and k1 k2 . The factorization (2.1) is said to be canonical if k1 = k2 = 0. Remark 2.2. In the above definition we used the expression “Wiener–Hopf factorization” to denote the factorization (2.1). This is the standard designation in the area of singular operator theory where most of the results concerning this concept can be found. However, in other areas of mathematics, in particular in integrable systems, the designation Riemann–Hilbert factorization is commonly used [7]. The expression Birkhoff factorization is also adopted in some areas where the above notion appears. It is, for example, the case of the classification of holomorphic vector bundles over the Riemann sphere [8, Chapter 2]. The concept of factorization presented in Definition 2.1 is a particular case of the concept of generalized factorization, but is sufficient and appropriate for our purpose (for the theory of generalized factorization, see e.g. [5] and [2]). Definition 2.3. Let Q1 , Q2 be symmetric matrix functions such that 2×2 Q1 ∈ G H − (∂D) + R(∂D) ,
2×2 Q2 ∈ G H + (∂D) + R(∂D) .
We denote by C(Q1 , Q2 ) the set of all matrix functions G ∈ G([L∞ (R)]2×2 ) satisfying the relation GT Q1 G = hQ2 ,
(2.2)
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where the upperscript T stands for matrix transposition, and det Q1 h = det G , det Q2 where the square-root is assumed to have positive real part. In the case Q1 = Q2 = Q we have Q ∈ R2×2 and the class C(Q1 , Q2 ) is denoted by C(Q). In the following theorem we show that every factorizable 2 × 2 matrix function belongs to some class in the family of classes C(Q1 , Q2 ). Theorem 2.4. If G ∈ G([L∞ (∂D)]2×2 ) admits a bounded factorization then there exist Q1 , Q2 such that G ∈ C(Q1 , Q2 ). Proof. Let G = G− DG+ be a factorization of G. Take −1 Q1 = GT− G−1 − ,
Q2 = GT+ D 2 G+ .
Obviously Q1 , Q2 are symmetric and Q1 ∈ G[H − (∂D) + R(∂D)]2×2 , Q2 ∈ G[H + (∂D) + R(∂D)]2×2 . A straightforward calculation shows that GT Q1 G = Q2 , i.e., h = 1 in this case. 2 We will assume from now on that Q1 , Q2 are of the form [qij ], where qij ∈ Cμ± (∂D) + R(∂D) and either q11 = 0 or q11 ∈ G(Cμ± (∂D) + R(∂D)), taking the upperscripts + and − as corresponding to Q2 and Q1 , respectively. We also assume that det Q1 and det Q2 admit a square-root in Cμ (∂D). The set of such pairs (Q1 , Q2 ) will be denoted by Q. Moreover we focus our attention in matrix functions G ∈ (Cμ (∂D))2×2 . Indeed, several classes that are relevant from the point of view of applications are of this type and belong to some class C(Q1 , Q2 ) for an appropriate (Q1 , Q2 ) ∈ Q [4,12,13]. In this case, since it is clear that Q1 , Q2 in Theorem 2.4 are not unique, we are able to make a specially interesting choice for Q1 , Q2 . To show this we give next some auxiliary results. Lemma 2.5. Let Q ∈ [Cμ (∂D)]2×2 be a symmetric matrix function of the form q1 q2 . Q= q2 q3 Then (1) If q1 is invertible, then 1 Q = q1 S T J S, 2
(2.3)
where
0 1 J= , 1 0 with 2 = − det Q.
S=
1 (q2 + )q1−1
1 (q2 + )q1−1
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(2) If q1 = 0, 1 Q = S T J S, 2
(2.4)
where S=
0 2q2
1 . q3
Proof. A straightforward calculation shows that (2.3) and (2.4) hold.
2
Corollary 2.6. Let (Q1 , Q2 ) ∈ Q. Then C(Q1 , Q2 ) = ∅ and there is G0 ∈ C(Q1 , Q2 ) ∩ [Cμ (∂D)]2×2 admitting a bounded factorization and such that the function h in (2.2) admits a bounded factorization. Proof. Taking into account our previous assumptions on Q1 , Q2 , we can assume without loss of generality that the element in the first row and first column of both matrices is either 0 or 1. Let 1 Q1 = S1T J S1 , 2
1 Q2 = S2T J S2 , 2
according to Lemma 2.5, and let G0 = S1−1 S2 . We have G0 ∈ C(Q1 , Q2 ) ∩ [Cμ (∂D)]2×2 and det G0 ∈ GCμ (∂D), so that G0 admits a bounded factorization. Moreover, G0 satisfies (2.2) with h = 1. 2
1 , Q
2 ) ∈ Q such that Theorem 2.7. Let (Q1 , Q2 ) ∈ Q. Then there is a pair (Q
1 = det Q
2 = p, det Q where p is a monic polynomial admitting, at most, simple zeros and
1 , Q
2 ). C(Q1 , Q2 ) = C(Q Proof. Let G0 ∈ C(Q1 , Q2 ) admit a bounded factorization and satisfy GT0 Q1 G0 = hQ2 , with h = h− zμ h+ ,
μ ∈ Z, h± ∈ GH ± (∂D),
and let moreover g = det G0 = g− zk g+ , Define
k ∈ Z, g± ∈ GH ± (∂D).
(2.5)
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1 = g− h−1 Q − Q1 , −1 μ−k
2 = h+ g+ z Q2 . Q
(2.6)
2 −2
1 = g− det Q h− det Q1 .
(2.7)
Then
From (2.5) we have (det G0 )2 det Q1 = h2 det Q2 . Substituting this result in (2.7), and using the expression for the factorization of det G0 (det G0 = g− zk g+ ), we get −2 2(μ−k) 2 −2 −2 2
1 = g−
2 . det Q h− g h det Q2 = h2+ g+ z = det Q
(2.8)
Now, since
1 ∈ H − (∂D) + R(∂D) 2×2 , Q
2 ∈ H + (∂D) + R(∂D) 2×2 Q
we have, from the equality (2.8),
1 , det Q
2 ∈ R(∂D). det Q
1 , Q
2 are obtained from Q1 , Q2 through multiplication by scalar functions Noting that Q (cf. (2.6)) it follows that
1 , Q
2 ). C(Q1 , Q2 ) = C(Q
(2.9)
1 , Q
2 ) = C(r Q
1 , r Q
2 ) for any r ∈ G(R(∂D)), we see that (2.9) holds with Q
1 , Q
2 Since C(Q
1 , det Q
2 are monic polynomials admitting at most simple zeros, as we set to such that det Q prove. 2 Theorem 2.7 means that we can associate with each class C(Q1 , Q2 ) a certain polynomial function (which may be a constant). We shall see now that this enables us to associate in a unique way an algebraic curve to each class C(Q1 , Q2 ). We start by considering the case where (Q1 , Q2 ) ∈ Q and Q1 = Q2 = Q, in which case we will say that Q ∈ Q. We will also use the notation
J = αI α ∈ GCμ (∂D) . It is clear that J ⊂ C(Q) for any Q ∈ Q. Theorem 2.8. Let Q, Q∗ ∈ Q. If C(Q) ∩ C(Q∗ ) = J then Q = βQ∗ for some β ∈ R(∂D) and C(Q) = C(Q∗ ).
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Proof. Let Q be given by (2.3) or (2.4), according to Lemma 2.5, and assume that there is G∈ / J , G ∈ C(Q) ∩ C(Q∗ ). Define
= SGS −1 . G Since G ∈ C(Q) and noting that S T J S = q1 Q, we have
= q1 S −1 T GT QGS −1 = hJ,
T J G G
Hence, from the proof of Theorem 2.12 below, it follows that G
is a where h = det G = det G. diagonal matrix,
= diag(a, d) with a = d. G
∈ C(J ∗ ) with J ∗ = (S −1 )T Q∗ S −1 . It folOn the other hand, since G ∈ C(Q∗ ), we have G ∗ ∗ ∗ lows that J = β J with β ∈ G(Cμ (∂D)), which is equivalent to Q∗ = βQ with β ∈ R(∂D) and, consequently C(Q) = C(Q∗ ). 2 Corollary 2.9. If C(Q) = C(Q∗ ), for Q, Q∗ ∈ Q, then Q∗ = βQ with β ∈ R. Proof. We have J Q ∈ C(Q) = C(Q∗ ), for J =
0 1 −1 0
and J Q ∈ / J , so that the result follows from Theorem 2.8.
2
Now we consider the classes C(Q1 , Q2 ) in general. Theorem 2.10. Let (Q1 , Q2 ) ∈ Q. If C(Q1 , Q2 ) = C(Q∗1 , Q∗2 ), then Qi = βi Q∗i with βi ∈ GCμ (∂D) for i = 1, 2. Proof. We recall from [3, Theorem 3.5] that C(Q1 , Q2 ) = C(Q1 )H = C Q∗1 H = C Q∗1 , Q∗2 , for some H ∈ C(Q1 , Q2 ) = C(Q∗1 , Q∗2 ). Thus C(Q1 ) = C(Q∗1 ) and it will follow from Corollary 2.9 that Q1 = β1 Q∗1 . Analogously, C(Q1 , Q2 ) = H C(Q2 ) = C Q∗2 H and Q2 = β2 Q∗2 .
2
As an immediate consequence of Theorems 2.10 and 2.7, we have the following result.
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Theorem 2.11. For every (Q1 , Q2 ) ∈ Q, there is a unique monic polynomial p admitting, at most, simple zeros such that
2
1 = det Q p = det Q
1 , Q
2 ) satisfying the conditions of Theorem 2.7. for any pair (Q Next we give an important result on the a representation of any G belonging to C(Q1 , Q2 ). Theorem 2.12. Let G ∈ C(Q1 , Q2 ) with det Q1 = det Q2 . Then G = S1−1 DS2 , where S1 , S2 correspond to the matrix S of the representation of Q1 , Q2 (cf. Lemma 2.5) and D is diagonal or anti-diagonal. Proof. For G ∈ C(Q1 , Q2 ) we have, for some h, GT Q1 G = hQ2 .
(2.10)
Since S1 and S2 are invertible we may write G = S1 DS2−1 , for some matrix D. Substituting this expression in (2.10) gives T −1 T T S2 D S1 Q1 S1 DS2−1 = hQ2 , from which it follows that D T S1T Q1 S1 D = hS2T QS2 and, in view of Lemma 2.5, DT J D =
hJ,
(2.11)
for same scalar function
h. It is worth noting that (2.11) defines a group as can be easily checked. To obtain all the elements of this group we write D in the general form a b D= . c d Substitution of this matrix in (2.11) leads to the equations ac = 0,
bd = 0
which can only have non-trivial solutions if a = d = 0 or
b = c = 0.
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In this first case D anti-diagonal and in the second it is diagonal (the group defined by (2.11) has two connected components). The other two equations that come from (2.11) simply give
h = bc = − det D,
h = ad = det D,
2
respectively for the first and second cases.
Corollary 2.13. With the same assumptions of Theorem 2.12 the Riemann–Hilbert problem Gφ + = φ − ,
2 φ ± ∈ H ± (∂D) ,
(2.12)
is equivalent to D S2 φ + = S1 φ − ,
2 φ ± ∈ H ± (∂D) .
(2.13)
Remark 2.14. It is clear that, from the point of view of solving the Riemann–Hilbert (2.13), the case where D is anti-diagonal is entirely analogous to the case where D is diagonal. Thus, unless otherwise stated, we shall assume that D is diagonal. Before we leave the results of Theorems 2.12 and 2.4 it is useful to write Eq. (2.13) in system form: d1 q21 φ1+ + q22 φ2+ + φ2+ = q11 φ1− + q12 φ2− + φ2− , d2 q21 φ1+ + q22 φ2+ − φ2+ = q11 φ1− + q12 φ2− − φ2− ,
(2.14)
where the first subscript in the q corresponds to Q1 or Q2 . It will be shown later that the system (2.14) is equivalent to a scalar Riemann–Hilbert problem on the hyperelliptic Riemann surface defined by the equation μ2 = det Q1 = det Q2 = 2
(2.15)
assuming that det Qi is not a constant. This fact yields a powerful tool for solving Eqs. (2.14). In what follows we shall denote by Σ the Riemann surface obtained by the compactification of the above algebraic curve which henceforth we write in the form μ2 = p(λ), where p(λ) is assumed to be a polynomial of degree 2(g + 1) (g 0) with simple roots. Thus, Σ is a hyperelliptic Riemann surface of genus g. It is convenient to view it as a branched cover of C via the meromorphic function λ : Σ → C induced by (λ, μ) → λ. The meromorphic function induced by (λ, μ) → μ will be denoted by μ. We shall assume that p(λ) has an even number 2(g + 1) (with g −1) of zeros inside D and no zeros on ∂D. This implies √ that there is a continuous branch of log p on ∂D. We shall denote by : ∂D → C the branch of p(λ) for which Re > 0. Note also that the contour Γ = λ−1 (∂D) consists of two disjoint closed paths which divide Σ into two disjoint regions.
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Definition 2.15. We denote by Γ1 the component of Γ = λ−1 (∂D) where Re μ > 0. The other component is denoted by Γ2 . The contour Γ = Γ1 Γ2 divides Σ into two disjoint regions defined by Ω + = λ−1 (D) and Ω − = Σ \ Ω + . In the process of reducing our Riemann–Hilbert problem on ∂D to a scalar Riemann–Hilbert on a contour in Σ we shall obtain another characterization of the class C(Q1 , Q2 ). Firstly we define a transformation TΣ from the space of Hölder continuous functions on ∂D to the space of Hölder continuous functions on the contour Γ in Σ . Definition 2.16. Let TΣ : [Cμ (∂D)]2 → Cμ (Γ ) be the linear transformation defined by TΣ (φ1 , φ2 )|Γi = φi . The following proposition gives the main properties of TΣ . Proposition 2.17. Let TΣ be as in Definition 2.16. Then the following assertions hold: (1) TΣ maps (φ1 + φ2 , φ1 − φ2 ) into φ1 + μφ2 (here φ1 , φ2 ∈ Cμ (∂D) and, on Γ , we use φi to abbreviate λ∗ φi = φi ◦ λ); (2) TΣ is invertible with inverse given by 2 TΣ−1 : Cμ (Γ ) → Cμ (∂D) ,
TΣ−1 Ψ = (ψ|Γ1 , ψ|Γ2 ).
Proof. Follows straightforwardly from Definition 2.16.
2
For the next result we recall Eqs. (2.14) which we write again in the form (2.13) (renumbered (2.16) for convenience), D S2 φ + = S1 φ + ,
(2.16)
where Si =
1 1
qi − qi +
(i = 1, 2).
(2.17)
Before we derive a Riemann–Hilbert problem on the Riemann surface defined by (2.15) we need some definitions and notation concerning function spaces on Σ . Definition 2.18. By Cμ+ (Γ ) we will denote the subspace of Cμ (Γ ) whose elements are boundary values of analytic functions in Ω + . Analogously for Cμ− (Γ ). We are now in a position to state the result of the following proposition. Proposition 2.19. Eq. (2.13) is equivalent to the scalar Riemann–Hilbert problem on Γ , d φ1+ + q2 φ2+ + μφ2+ = φ1− + q1 φ2− + μφ2− , where d = TΣ (d1 , d2 ) and φ1± , φ2± ∈ Cμ± (Γ ).
(2.18)
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Proof. Applying TΣ to both sides of (2.16) and using Definition 2.16 gives (i) for the right-hand side, TΣ S1 φ − = TΣ φ1− + q1 φ2− − φ2− , φ1− + q1 φ2− + φ2− = φ1− + q1 φ2− − μφ2− ,
(2.19)
where φi− ∈ Cμ− (Γ ); (ii) for the left-hand side: TΣ DS2 φ = dTΣ S2 φ = d φ1+ + q2 φ2+ + μφ2+ ,
(2.20)
where we have used the fact that TΣ Dψ = dTΣ ψ . 2
From (2.20) and (2.19) we obtain (2.18) as desired.
We are now in a position to state an alternative characterization of the classes C(Q1 , Q2 ) in terms of multiplication operators on [Cμ (∂D)]2 and Cμ (Γ ). In the following we denote by mG the operator of multiplication by G on [Cμ (∂D)]2 and md the operator of multiplication by d on Cμ (Γ ). Theorem 2.20. The matrix valued function G in [G(L∞ (∂D))]2×2 belongs to C(Q1 , Q2 ) where det Q1 = det Q2 = p(λ) ∈ C[λ], if and only if there exists an operator md : Cμ (Γ ) → Cμ (Γ ) such that md = TΣ mS1 mG mS −1 TΣ−1
(2.21)
md = τ ∗ TΣ mS1 mG mS −1 TΣ−1 ,
(2.22)
2
or 2
where Σ is the Riemann surface defined by the equation μ2 = p(λ), S1 , S2 are the matrices given in Eq. (2.17), and τ ∗ : Cμ (Γ ) → Cμ (Γ ) denotes the composition with the hyperelliptic involution τ : Σ → Σ . Proof. Suppose G ∈ C(Q1 , Q2 ). Then, by Theorem 2.12, the matrix D = S1 GS2−1 is either diagonal or anti-diagonal. A direct calculation gives TΣ mD TΣ = md , if D = d01 d02 , and TΣ mD TΣ = τ ∗ md , if D = d02 d01 . Hence either (2.21) or (2.22) holds. Conversely, if (2.21) or (2.22) hold, then the matrix D = S1 GS2−1 is either diagonal or antidiagonal and it follows from Theorem 2.12 that G ∈ C(Q1 , Q2 ). 2 3. Σ-factorization In this section we define a factorization for scalar functions belonging to Cμ (Γ ) where Γ is a contour in a hyperelliptic Riemann surface Σ . This factorization allows us to study scalar
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Riemann–Hilbert problems such as those arising from vector Riemann–Hilbert problems in C as shown in Section 2. Recall from Section 2 that Σ is the Riemann surface associated to the equation μ2 = p(λ), where p(λ) is a polynomial of degree 2g + 2 without multiple roots. Hence Σ is a hyperelliptic Riemann surface of genus g and is obtained from the plane algebraic curve
Σ0 = (λ, μ) ∈ C2 μ2 = p(λ) by adding two points “at infinity,” ∞1 , ∞2 , such that ς = λ−1 is a local parameter at these points. There are two natural meromorphic functions on Σ : those induced by the projections (λ, μ) → λ and (λ, μ) → μ. They will be denoted respectively by λ and μ. The field of rational functions on Σ will be denoted by R(Σ). As usual, it is convenient to view Σ as a 2-branched cover of P(C2 ) under the map λ : Σ → P(C2 ) = C ∪ {∞}. The two points in λ−1 (0) will be denoted by 01 and 02 . Definition 3.1. Let Γ be a contour in Σ and let f ∈ GCμ (Γ ), 0 < μ < 1; f is said to possess a Σ-factorization relative to Γ if it has a representation of the form f = f− rf+ ,
(3.1)
where (f+ )±1 ∈ Cμ+ (Γ ), (f− )±1 ∈ Cμ− (Γ ) (see Definition 2.18) and r ∈ R(Σ). If r is constant (3.1) is called a special Σ -factorization. It is easily seen that, if f = f− f+ and f = f − f + are two special Σ -factorizations for f , then f + = cf+ ,
f − = cf− ,
where c is a constant. Indeed we have f + f+−1 = f− f−−1 = c. Keeping in mind the application to vector valued Riemann–Hilbert problems in C, we consider only the case where Γ is the (oriented) boundary of a region Ω + ⊂ Σ defined in Definition 3.2 below. For a general reference on Riemann surfaces see, for example, [11] or [15]. Definition 3.2. We will denote by Ω + the inverse image under λ of the unit disk D ⊂ C. We assume that D contains 2(g + 1) zeros of p(λ) so that, if g = −1, Ω + is a union of two disjoint disks; if g 0, Ω + is a Riemann surface of genus g with two closed disks removed. We will also consider the open set Ω − = Σ \Ω + . Note that Ω − is a Riemann surface of genus g = g −g −1 with two disks removed. Assumption 3.3. Henceforth we assume Γ = ∂Ω + with the orientation induced from Ω + .
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Next we show that every function f ∈ GCμ (Γ ) such that log f ∈ Cμ (Γ ) has a Σ-factorization. Theorem 3.4. Let f ∈ Cμ (Γ ) be such that log f ∈ Cμ (Γ ) and let h be the function given by μ μ h = exp α1 + · · · + αg g λ λ with αk =
1 4πi
log f
λk−1 dλ μ
(k = 1, . . . , g).
Γ
Then f has a Σ-factorization f = f− rf+
(3.2)
with f+ = exp PΓ+ log f h+ , g μ − αk k h− , f− = exp PΓ log f − λ
(3.3) (3.4)
k=1
where PΓ± are the bounded projections on Cμ (Γ ) defined in Appendix A, and h+ , h− , r are the factors of the Σ -factorization h = h+ rh− given in Proposition 3.10 below. Proof. Putting φ = log f and denoting by PΓ± the bounded projections on Cμ (Γ ) defined in Appendix A, we have
Γ φ + φ = PΓ+ φ + PΓ− φ = PΓ+ φ + P
g
αk
k=1
μ , λk
(3.5)
Γ ∈ Cμ− (Γ ) (cf. proof of Proposition A.1) and where PΓ+ φ ∈ Cμ+ (Γ ), P αk =
1 4πi
ξ k−1 φ(ξ, τ ) dξ. τ
Γ
The result now follows from (3.5) and Proposition 3.10 below.
2
Theorem 3.4 reduces the problem of computing a Σ -factorization g of f to the computation of a Σ-factorization for the exponential h = exp() where = k=1 αk μ/λk . To obtain this factorization we will need some information about the periods of the differential d. We start by fixing bases for the first homology group H1 (Σ; Z) and the space of holomorphic differentials Ω 1 (Σ).
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Let {a1 , . . . , ag , b1 , . . . , bg } be a canonical basis of H1 (Σ; Z) (this means that ai · aj = bi · bj = 0 and ai · bj = δij ) whose elements do not pass through 0j , ∞j . We denote by ζ = (ζ1 , . . . , ζg ) the vector of differentials given by ζk =
λk−1 dλ. μ
It is well known that {ζ1 , . . . , ζg } is a basis of holomorphic differentials [15]. However, it is usually more convenient to use a basis ω = (ω1 , . . . , ωg ) which is dual to the homology basis {aj , bj }, i.e., it satisfies ai ωj = δij . The bases ζ , ω are related by ω = Cζ , where C = (cij ) is the inverse of the matrix A = (aij ) of a-periods of ζ : aij = ζj . ai
We will also need the matrix of b-periods of ω, denoted by B = (bij ): bij = ωj . bi
Notation 3.5. For convenience we introduce the following notation concerning integrals and residues of vectors of differentials. (1) Given a differential ϕ and a vector of 1-cycles c = (c1 . . . , cg ) we denote by c ϕ the vector of c-periods of ϕ: c ϕ = ( c1 ϕ, . . . , cg ϕ). (2) Given a vector of meromorphic differentials ϕ = (ϕ1 , . . . , ϕg ) we denote by Resp (ϕ) the vector of residues (Resp (ϕ1 ), . . . , Resp (ϕg )). (3) Given a vector ϕ = (ϕ1 , . . . , ϕg ) and a 1-cycle c we denote by c ϕ the vector of differentials of periods ( c ϕ1 , . . . , c ϕg ). g Lemma 3.6. Given α = (α1 , . . . , αg ) ∈ Cg consider the function = k=1 αk λμk . Then d = γ∞ − γ0 where γ∞ and γ0 are differentials of the second kind satisfying: (i) γ∞ is holomorphic in Σ \ {∞1 , ∞2 }; in Σ \ {01 , 02 }; (ii) γ0 is holomorphic (iii) aj γ∞ = aj γ0 = 0; = (b1 , . . . , bg ) and C is the matrix defined above (ω = (iv) b γ∞ = b γ0 = 4πiCα, where b Cζ ). Proof. The meromorphic differential d is holomorphic in Σ \ {0j , ∞j | j = 1, 2} and all its residues are zero. That is, d is a differential of the second kind with singularities at 0j , ∞j . It follows from the properties of the differentials of the second kind [15, Chapter 8] that there exists a meromorphic differential γ∞ with the same principal part as d at ∞1 , ∞2 and holomorphic elsewhere. Setting γ0 := γ∞ − d, all the required properties in the statement are satisfied except possibly (iii) and (iv). Changing γ0 , γ∞ by adding an appropriate linear combination of the differentials ωj gives (iii).
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It remains to prove (iv). Let P be a standard polygon for Σ and let P + , denote the preimage of Ω + under the identification map P → Σ . Since P is simply connected there is a holomorphic function f = (f1 , . . . , fg ) : P → Cg such that df = ω. Applying the bilinear relations and the properties of γ0 , γ∞ , we obtain γ0 = 2πi
Resp (γ0 f) = 2πi
p∈P
b
=−
Resp (γ0 f) =
p∈P +
ω =
∂P+
γ0 f
∂P+
(d)f =
∂P+
ω.
∂Ω +
Now, for each j , we have ωj = ∂Ω +
αr
r
= 4πi
∂Ω +
μ ω = αr cj s j λr r,s
λs−r−1 dλ ∂Ω +
cj r αr = 4πi(Cα)j .
r
Hence the second equality in (iv) holds. Since γ∞ − γ0 is exact, the first equality also holds.
2
Definition 3.7. Fix a point p0 ∈ Σ \ {∞j , 0j }. Define p A(p) =
ω ∈ Cg . p0
Of course, the value of this integral depends on the choice of a path between p0 and p. Therefore the expressions involving A(p) are, in general, multivalued. Given a constant v ∈ Cg and p ∈ Σ, we define F (p | v) = θ A(p) − v − K, B ∈ C, where θ is the Riemann theta function, θ (z, B) = n∈Zg exp(2πi( 12 nT Bn + nT z)), B is the matrix of b-periods of ω and K is the vector of Riemann constants [6]. Remark 3.8. The function F (p | v) is multivalued for its definition involves A(p). The effect of changing the integration path between p0 and p is determined by the quasi-periodicity properties of the theta function θ (z, B): 1 t T θ (z + n + Bm, B) = exp −2πi m Bm + m z θ (z, B). 2 We can now obtain the Σ -factorization for the function h that is referred to in Theorem 3.4. The method used to obtain this factorization is closely related to the construction of a Baker– Akhiezer function in [6]. In order to state the result we need one more definition.
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− + − Definition 3.9. Let D+ = p+ 1 + · · · + pg and D− = p1 + · · · + pg be nonspecial divisors + − + − (see [11]) such that pi ∈ Ω and pi ∈ Ω .
Proposition 3.10. Let h : Σ → C be given by μ μ h = exp α1 + · · · + αg g λ λ
(3.6)
and let u = 2Cα (cf. Lemma 3.6). Then for all N ∈ N large enough, h has a Σ-factorization h = h+ rh−
(3.7)
with p h+ (p) = exp
γ∞
p0
F (p | A(D− ) + Nu )N , F (p | A(D− ))N
(3.8)
p F (p | A(D+ ) − Nu )N h− (p) = exp − γ0 , F (p | A(D+ ))N
(3.9)
p0
where the same path from p0 to p is used to evaluate the integrals of γ0 , γ∞ and A(p). The function r is given by F (p | A(D+ ) − Nu ) F (p | A(D− ) + r = h(p0 ) F (p | A(D+ )) F (p | A(D− ))
u −N N)
.
(3.10)
In particular r is a rational functional. − + − Proof. Since D+ = p+ 1 + · · · + pg and D− = p1 + · · · + pg are nonspecial divisors such that + − + − pi ∈ Ω and pi ∈ Ω , it follows [6, Chapter II] that the functions
F p A(D+ ) ,
F p A(D− )
− + − have exactly g zeros at the points p+ 1 , . . . , pg and p1 , . . . , pg , respectively. It also follows that we can choose N ∈ N large enough so that the functions F (p | A(D+ ) + Nu ), F (p | A(D− ) − Nu ) − + − are not identically zero. Their zeros are points q+ 1 , . . . , qg and q1 , . . . , qg such that
u + A q+ 1 + · · · + qg = A(D+ ) + N ,
u − A q− 1 + · · · + qg = A(D− ) − N .
− + − We assume N is large so that q+ i ∈ Ω and qi ∈ Ω . Consider the functions h+ and h− defined in (3.8) and (3.9). To show that h+ is independent of the path of integration we consider the effect of adding to it a cycle homologous to i (ni ai +
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mi bi ). Set n = (n1 , . . . , ng ) and m = (m1 , . . . , mg ). By Lemma 3.6(iv) the first factor in the formula for h+ transforms as follows: p exp
γ∞
p γ∞ , exp 2πi u m exp →
T
p0
p0
while the second factor is multiplied by exp 2πiN[− 12 mT Bm − mT (A(p) + A(D− ) +
u N
− K)]
exp 2πiN [− 12 mT Bm − mT (A(p) + A(D− ) − K)]
= exp 2πi −uT m ,
hence the value of h+ remains unchanged. The verification that h− is well defined is similar. It is clear from the properties of γ∞ , γ0 and the preceding remarks that h±1 + is holomorphic ±1 + − in Ω and h− is holomorphic in Ω . Since d = γ∞ − γ0 (see Lemma 3.6) and h = exp it follows that F (p | A(D+ ) − Nu ) F (p | A(D− ) + h = h+ h− h(p0 ) F (p | A(D+ )) F (p | A(D− ))
u −N N)
.
2
Corollary 3.11. Let h and u be as in Proposition 3.10. If u ∈ Zg + BZg then h has a special Σ -factorization. Proof. Let u = n1 + Bm1 ∈ Zg + BZg . Then taking N = 1 in (3.8) and (3.9) we get well-defined elements of GCμ± (Γ ) given (up to multiplicative constants) by the following expressions: p h+ (p) = exp
γ∞ exp −mT1 A(p) ,
p0
p h− (p) = exp − γ0 exp mT1 A(p) . p0
Since h/(h+ h− ) = h(p0 ) we conclude that h = h+ h− h(p0 ) is a special Σ -factorization.
2
Theorem 3.12. Let f ∈ Cμ (Γ ) be such that log f ∈ Cμ (Γ ). Then f has a special Σ -factoriza 1 g g tion with respect to Γ if and only if 2πi Γ (log f )ω ∈ Z + BZ . Proof. From Theorem 3.4 it follows that f has a special Σ -factorization iff the same is true for function h = exp(α1 μ/λ + · · · + αg μ/λg ) where αk =
1 4πi
(log f )ζk Γ
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and ζk = (λk−1 /μ) dλ. Denoting by ζ the vector of holomorphic differentials (ζ1 , . . . , ζg ) we have 1 1 C (log f )ζ = 2Cα. (log f )ω = 2πi 2πi Γ
Γ
By Corollary 3.11, u = 2Cα ∈ Zg + BZg is a sufficient condition for h to have a special Σfactorization. Hence it remains to show that this condition is necessary. Let P be a standard polygon for Σ and let P + , P − , respectively, denote the preimage of Ω + and Ω − under the identification map P → Σ . Recall from Definition 3.2 that D contains 2(g + 1) branch points (with g −1). If g > 0, P + is simply connected. If D contains no branch points (g = −1) then P + is a disjoint union of two simply connected sets. In either case we can define in P + a continuous branch of log f+ , which is holomorphic in the interior of P + . Hence we get 1 2πi
(log f+ )ω = 0.
(3.11)
∂P+
Proceeding similarly for P − , we obtain a continuous branch of log f− : P − → C such that 1 2πi
(log f− )ω = 0.
(3.12)
∂P−
Denoting by Γ ± the preimage of Γ in P ± (with the orientation induced from ∂P ± ) we obtain
(log f )ω =
(log f )ω =
Γ+
Γ
(log f+ )ω −
Γ+
(log f− )ω
Γ−
for f = f+ f− on Γ and Γ + , Γ − have opposite orientations. Now, if p1 , p2 ∈ P ± are two points with the same image under the map P → Σ then log f± (p1 ) = log f± (p2 ) + 2πi n± for some n± ∈ Z. From this and Eqs. (3.11) and (3.12) we conclude that 1 2πi
(log f ) ≡ Γ
1 2πi
(log f+ )ω −
∂P+
(log f− )ω
mod Zg + BZg
∂P−
≡ 0 mod Z + BZ . g
g
2
Next we illustrate how the factorization Theorem 3.4 can be used to solve a Riemann–Hilbert problem on the Riemann surface Σ. We restrict our study to a homogeneous problem f ψ+ = ψ− ,
(3.13)
where f ∈ Cμ (Γ ) and ψ+ , ψ− are assumed to be holomorphic respectively in Ω + , Ω − and continuous in Ω + , Ω − .
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247
Let f = f+ rf− be a Σ -factorization of f then f ψ+ = ψ−
⇔
rf+ ψ+ = f−−1 ψ− .
Hence, if (ψ+ , ψ− ) is a solution of the Riemann–Hilbert problem (3.13), there exists R ∈ R(Σ) such that rf+ ψ+ in Ω + , R= f−−1 ψ− in Ω − . Let D = div(r|Ω + ) (the divisor of r|Ω + .) Then we have R ∈ L(−D) where we set
L(−D) := g ∈ R(Σ) div(R) − D 0 , following standard notation [11]. Conversely, if g ∈ L(−D) then the pair (ψ+ , ψ− ) = (gr −1 f+−1 , gf− ) is a solution problem (3.13). This proves: Proposition 3.13. Let D = div(r|Ω + ), then the space of solutions of (3.13) is isomorphic to L(−D). The computation of the dimension of the space L(−D) is a classical problem whose answer is given by the Riemann–Roch theorem [11,15]. In the special case where log f ∈ Cμ (Γ ) we obtain the following result. Proposition 3.14. Under the condition log f ∈ Cμ (Γ ), the Riemann–Hilbert problem (3.13) has a non-trivial solution if and only if f has a special Σ -factorization. In this case the dimension of the space of solutions is 1. Proof. If f has a special Σ -factorization it is clear that the space of solutions of problem (3.13) has dimension one. Conversely, assume the space of solutions of (3.13) has dimension one. We start by computing the degree of the divisor D. Clearly deg D = IndΓ (r). Since f±±1 is holomorphic in Ω ± it follows that IndΓ (f± ) = 0 and so IndΓ (r) = IndΓ (f ). The condition log f ∈ Cμ (Γ ) gives IndΓ (f ) = 0, hence deg D = 0. Since deg D = 0 we have dim L(−D) 1 and the equality occurs iff there is a rational function r− ∈ R(Σ) such that div(r− ) = D [15]. In this case, we set r+ = r/r− and r has a special Σ -factorization r = r+ r− . Therefore f = (f+ r+ )(r− f− ) is a special factorization for f . 2 For the example discussed in the next Section it will be convenient to have to the following generalization of Proposition 3.14. Proposition 3.15. Let D be a divisor on Σ. Then the space of solutions (ψ+ , ψ− ) of (3.13) satisfying div(ψ+ ) + div(ψ− ) D is isomorphic to the space of rational functions L(−D − D), where D = div(r|Ω + ).
(3.14)
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Proof. Let f = f+ rf− be a Σ -factorization of f . As in the proof of Proposition 3.14 we obtain a rational function rf+ ψ+ in Ω + , R= f−−1 ψ− in Ω − . From (3.14) it follows that div(R) − D D , i.e., R ∈ L(−D − D). Conversely, if R ∈ L(−D − D) then the pair (Rr −1 f+−1 , Rf− ) is a solution of (3.13) satisfying (3.14). 2 4. Example In this section we illustrate the results of the previous sections by solving a Riemann–Hilbert problem corresponding to a 2 × 2 matrix symbol that belongs to a family of exponentials of rational matrices. Symbols of this form appear for example in the study of finite-dimensional integrable systems (cf. [4,13]). Specifically, let G = exp(tL),
(4.1)
where L is a rational 2 × 2 matrix function and t ∈ R. The symbol (4.1) belongs to the class
with Q
= J L, where C(Q) J =
0 1 . −1 0
Indeed we have
= exp tLT J L exp(tL) = exp tLT J exp(tL)L. GT QG
(4.2)
But for any 2 × 2 matrix A we have AT J A = (det A)J
which, introduced in (4.2), gives
= (det G)J L = (det G)Q,
GT QG
i.e., G ∈ C(Q). For our example we take
v L(λ) = −ku
u , −v
(4.3)
where u and v are Laurent polynomials in λ given by u = aλ − xλ−1 , with a, x, k positive real constants.
v = xλ−1 ,
λ ∈ ∂D,
(4.4)
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The Riemann–Hilbert problem we have to solve is Gφ + = φ − ,
(4.5)
where G is given by (4.1). Recall from Section 2 that the Riemann surface associated with problem (4.5) is given by the equation
μ2 = − det Q.
by Q = λQ
we obtain C(Q) = C(Q)
with det Q a polynomial in λ. Hence the Replacing Q associated Riemann surface Σ has the form considered in Section 3, i.e., it is the compactification of the algebraic curve μ2 = p(λ),
(4.6)
where p is the polynomial p(λ) = −ka 2 λ4 + 2akxλ2 + x 2 (1 − k). The zeros of p(λ) are given by 1
1 x ± xk − 2 x 1 ± k− 2 . λ = = 2a 2a
2
For k > 1 all zeros of p(λ) are real and symmetric in pairs. We shall consider three distinct cases: (i) x < (ii)
a , 1+k −1/2
a 1+k −1/2
(iii) x >
which leads to all zeros of p(λ) inside D;
<x<
a 1−k −1/2
a , 1−k −1/2
corresponding to two zeros inside D and two zeros outside D;
corresponding to all zeros outside D.
Cases (i) and (iii) are analogous from the point of view of the topology of the problem and thus we shall consider only cases (ii) and (iii). Since G ∈ C(Q), the scalar Riemann–Hilbert problem on Σ corresponding to the problem (4.5) can be obtained simply by diagonalizing G, which is equivalent to diagonalizing L. We get for L L = SD0 S −1 ,
(4.7)
where D0 = diag( μλ , − μλ ) and S=
−1
1
v−μ/λ u
− v+μ/λ u
.
(4.8)
Then G = exp(tL) = SDS −1 ,
(4.9)
μ μ D = diag exp , exp − . λ λ
(4.10)
where
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Using (4.9) in (4.5) we get our Riemann–Hilbert problem in the form DS −1 φ + = S −1 φ −
(4.11)
or, in the form of two scalar equations
μ + + φ + uφ2 = v + d1 v + λ 1 μ + φ1 + uφ2+ = v − d2 v − λ
μ − φ + uφ2− , λ 1 μ − φ + uφ2− , λ 1
(4.12)
where d1 = exp(μ/λ) and d2 = exp(−μ/λ). As shown in Proposition 2.19, problem (4.12) is equivalent to the following scalar Riemann–Hilbert problem on the Riemann surface Σ defined by the algebraic curve (4.6), d
v+
μ + μ − φ1 + uφ2+ = v + φ + uφ2− λ λ 1
(4.13)
which can be simplified to give x+μ − x +μ + + d φ + φ2 = φ + φ2− , p2 1 p2 1
(4.14)
where d = di on Γi (i = 1, 2) with Γi the preimages of ∂D in Σ and p2 (λ) = aλ2 − x. Note that in the present example Σ is an elliptic Riemann surface as p(λ) in relation (4.6) is a forth degree polynomial with distinct zeros. Before we apply the results of Section 3 to solving Eq. (4.14) it is useful to make the following observations. Remark 4.1. (1) Cases (ii) and (iii) defined after Eq. (4.6) differ in the following aspects: in case (iii) the region Ω + = λ−1 (D) is a union of two disjoint simply-connected regions; in case (ii) both regions Ω + , Ω − are connected but not simply-connected. (2) The function whose Σ-factorization we have to obtain is, in the present example, simply the function h(λ, μ) = exp(α1 μ/λ) that appears in Proposition 3.10 with g = 1 and α1 = t. We continue our study of Eq. (4.14) precisely by looking at the factorization of d(λ, μ) = exp(tμ/λ). As noted in Remark 4.1 this corresponds to d = h with g = 1 and α1 = t in Proposition 3.10. We assume that the constant t in the expression of G (see (4.1)) is sufficiently small for the factor r in h = h− rh+ to be given by the simplest non-trivial form (this is a consequence of the fact noted in the proof of Proposition 3.10 that the vector u of b-periods of the differential d log h tends to zero with t): r(p) =
F (p | A(D+ ) − u)F (p | A(D− ) + u) , F (p | A(D+ ))F (p | A(D− ))
(4.15)
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251
where D± = p± are effective divisors of degree 1 with support in Ω ± . We may now write Eq. (4.14) in the form rh+
x+μ + x +μ − − = R, φ1 + φ2+ = h−1 φ + φ − 2 p2 p2 1
(4.16)
where R is a rational function on Σ since the left-hand side and the right-hand side are meromorphic functions, respectively, in Ω + and Ω − . We are now in a position to characterize completely the rational function R, i.e., to give its divisor of zeros and poles. Since Σ is a Riemann surface of genus 1, the Abel–Jacobi map A : Σ → J (Σ) = C/(Z + BZ) (see Definition 3.7) is an isomorphism. Therefore, R and r will be considered as doubly periodic meromorphic functions on C under the transformation u = A(p). We have the following conditions on R: (i) a pole at the point of Ω + where r has a pole (denoted by u0 ); (ii) two poles that come from the factor (x + μ)/p2 (denoted u1 , u2 ): noting that x 2 − μ2 = kp2 , it follows that x + μ has two zeros on Σ corresponding to the values of λ where p2 (λ) = 0; since p2 has four zeros in Σ, two of which are not compensated by zeros of x + μ and thus lead to two poles of (x + μ)/p2 ; (iii) a zero coming from the zero of r in Ω + (denoted v0 ); (iv) a zero that we impose at an arbitrary point of Ω + (denoted v1 ); (v) a zero v2 that comes from Abel’s theorem: v0 + v1 + v2 ≡ u0 + u1 + u2
mod Z + BZ.
Hence in terms of Jacobi theta functions R is given by the formula R(u) = kR
θ (u − v0 )θ (u − v1 )θ (u − v2 ) , θ (u − u0 )θ (u − u1 )θ (u − u2 )
(4.17)
where kR is a constant, and θ (u) denotes the theta function θ (u, B) recalled in Definition 3.7 (for a reference to the Jacobi theta functions, see e.g. [1]). A solution to the Riemann–Hilbert problem (4.14) is obtained from (4.16) x +μ + φ + φ2+ = r −1 h−1 + R, p2 1 x +μ − φ + φ2− = h− R p2 1
(4.18) (4.19)
from which φ1± and φ2± can be obtained by separating the left-hand side into its invariant and anti-invariant components with respect to the involution on Σ. To end our analysis we state the following proposition. Proposition 4.2. The space of solutions of the Riemann–Hilbert problem (4.14) has dimension 2. Two linearly independent solutions are given by formulas (4.18) and (4.19) for two distinct values of v1 .
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Proof. We have only to prove that the dimension of the space of solutions is two. This is a consequence of the fact that, in (4.17), the dimension of the space of rational functions with three fixed poles (u0 , u1 , u2 ) and a fixed zero (v0 ) is two, in view of the Riemann–Roch theorem. 2 Appendix A In this appendix we define two complementary projections PΓ± on Cμ (Γ ) and give their fundamental properties. For f ∈ Cμ (Γ ), let PΓ± f (λ, μ) = ±
1 4πi
Γ
τ + μ f (ξ, τ ) dξ, τ ξ −τ
(λ, μ) ∈ Ω ± ,
(A.1)
± the projection defined on Cμ (∂D) by Denoting by P∂D
1 ± = (I + S∂D ), P∂D 2
(A.2)
where I is the identity operator and S∂D is the singular integral operator with Cauchy kernel [2] on Cμ (∂D), and defining fE =
f +f∗ , 2
fO =
f −f∗ , 2μ
(A.3)
where f ∗ is the image of f ∈ Cμ (Γ ) under the usual involution, f ∗ (λ, μ) = f (λ, −μ). It is easy to see that fE and fO can be identified with functions in Cμ (∂D) and ± ± PΓ± (f ) = P∂D (fE ) + μP∂D (fO ).
(A.4)
+ − (ϕ) and P∂D (ϕ), for ϕ ∈ Cμ (∂D), with their analytic extensions to D In (A.4), we identify P∂D and C \ D, respectively. Thus, taking (A.4) into account and, on the other hand, identifying PΓ± (f ) with the corresponding boundary-value functions in Γ , we see that the following proposition holds (in this proposition Cμ+ (Γ ) (Cμ− (Γ )) denotes the space of functions in Cμ (Γ ) that have analytic extensions into Ω + (respectively Ω − )).
Proposition A.1. We have: (i) PΓ± (f ) : Cμ (Γ ) → Cμ (Γ ), with PΓ± (f ) defined by (A.1) for f ∈ Cμ (Γ ), are complementary projections, i.e., Cμ (Γ ) = PΓ+ (Cμ (Γ )) ⊕ PΓ− (Cμ (Γ )); (ii) Im PΓ+ = Cμ+ (Γ ); (iii) Im PΓ− = Cμ− (Γ ) iff g = 0, and Im PΓ− = Cμ− (Γ ) ⊕ span{ λμj , j = 1, . . . , g}, if g 1. ± [2]. As Proof. (i) and (ii) follow immediately from (A.4) and the well-known properties of P∂D − to Im PΓ , we see that its elements are functions in Cμ (Γ ) possessing an analytic extension to Ω − \ {∞1 , ∞2 } and a pole of order no greater than g at ∞1 , ∞2 , due to the second term on the right-hand side of (A.4).
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If g = 0, we conclude that Im PΓ− ⊂ Cμ− (Γ ) and, conversely, for every f ∈ Cμ− (Γ ) it follows from (A.4) that PΓ+ (f ) = 0, so that Im PΓ− = Cμ− (Γ ). If g 1, noting that g 1 1 ξ 1 ξ g−1 ξ = − − 2 − ··· − g + ξ −λ λ λ λ λ ξ −λ we get, for (λ, μ) ∈ Ω − , μ − μP∂D (fO )(λ) = μ PΓ− (f ) O (λ, μ) = − 4πi
Γ
f (ξ, τ ) dξ τ ξ −λ
μ μ μ μ = α1 + α2 2 + · · · + αg g − g Ig (λ, μ), λ λ λ λ
(A.5)
where αk =
1 4πi
ξ k−1 f (ξ, τ ) dξ, τ
k = 1, . . . , g,
(A.6)
Γ
1 Ig (λ, μ) = 4πi
Γ
dξ ξg − g f (ξ, τ ) = −P∂D ξ f O (λ). τ ξ −λ
(A.7)
If follows from (A.7) that λIg (λ, μ) is bounded as (λ, μ) tends to ∞1 or ∞2 and therefore, from (A.4) and (A.5), we conclude that, for any f ∈ Cμ (Γ ),
Γ (f ) + α1 PΓ− (f ) = P
μ μ + · · · + αg g , λ λ
(A.8)
where
Γ (f ) = P − (fE ) − μ Ig ∈ Cμ− (Γ ). P ∂D λg
(A.9)
Thus Im PΓ− ⊂ Cμ− (Γ ) ⊕ span{ λμj , j = 1, . . . , g}. Conversely, if f ∈ Cμ− (Γ ) ⊕ span{ λμj , j = 1, . . . , g}, then fE , fO can be identified with functions in Cμ− (∂D) and it follows from (A.4) that PΓ+ f = 0, so that the second equality in (iii) holds. 2 References [1] N.I. Akhiezer, Elements of the Theory of Elliptic Functions, Transl. Math. Monogr., vol. 79, Amer. Math. Soc., Providence, RI, 1990, translated from the second Russian edition by H.H. McFaden. [2] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, second ed., Springer Monogr. Math., Springer, Berlin, 2006, prepared jointly with A. Karlovich. [3] M.C. Câmara, A.F. dos Santos, M.P. Carpentier, Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems, Proc. Roy. Soc. Edinburgh Sect. A 132 (1) (2002) 45–74. [4] M.C. Câmara, A.F. dos Santos, P.F. dos Santos, Lax equations factorization and Riemann–Hilbert problems, Port. Math. 64 (2007) 509–533. [5] K.F. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory Adv. Appl., vol. 3, Birkhäuser, Basel, 1981.
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[6] B.A. Dubrovin, Theta-functions and nonlinear equations, Russian Math. Surveys 36 (1982) 11–92, with an appendix by I.M. Krichever. [7] M.A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, London Math. Soc. Stud. Texts, vol. 38, Cambridge Univ. Press, Cambridge, 1997. [8] N.J. Hitchin, G.B. Segal, R.S. Ward, Integrable Systems, Oxf. Grad. Texts in Math., vol. 4, Clarendon/Oxford Univ. Press, New York, 1999. [9] W. Koppelman, Singular integral equations, boundary value problems and the Riemann–Roch theorem, J. Math. Mech. 10 (1961) 247–277. [10] E. Meister, F. Penzel, On the reduction of the factorization of matrix functions of Daniele–Khrapkov type to a scalar boundary value problem on a Riemann surface, Complex Variables Theory Appl. 18 (1–2) (1992) 63–71. [11] R. Miranda, Algebraic Curves and Riemann Surfaces, Grad. Stud. Math., vol. 5, Amer. Math. Soc., Providence, RI, 1995. [12] S. Prössdorf, F.-O. Speck, A factorisation procedure for two by two matrix functions on the circle with two rationally independent entries, Proc. Roy. Soc. Edinburgh Sect. A 115 (1–2) (1990) 119–138. [13] A.G. Reyman, A. Semenov-Tian-Shansky, Group-theoretic methods in the theory of finite-dimensional integrable systems, in: S. Novikov, V. Arnold (Eds.), Dynamical Systems, in: Encycl. Math. Sci., vol. VII, Springer, Berlin, 1994. [14] Yu.L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Math. Appl. (Soviet Ser.), vol. 16, Reidel, Dordrecht, 1988. [15] G. Springer, Introduction to Riemann Surfaces, Addison–Wesley, Reading, MA, 1957. [16] È.I. Zveroviˇc, The Behnke–Stein kernel and the solution in closed form of the Riemann boundary value problem on the torus, Soviet Math. Dokl. 10 (1969) 1064–1068. [17] È.I. Zveroviˇc, Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces, Russian Math. Surveys 26 (1) (1971) 117–192.
Journal of Functional Analysis 255 (2008) 255–281 www.elsevier.com/locate/jfa
Properties of the density for a three-dimensional stochastic wave equation Marta Sanz-Solé 1 Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain Received 20 February 2008; accepted 2 April 2008 Available online 7 May 2008 Communicated by Paul Malliavin
Abstract We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x (y) be the density of the law of the solution u(t, x) of such an equation at points (t, x) ∈ ]0, T ] × R3 . We prove that the mapping (t, x) → pt,x (y) owns the same regularity as the sample paths of the process {u(t, x), (t, x) ∈ ]0, T ] × R3 } established in [R.C. Dalang, M. Sanz-Solé, Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it. © 2008 Elsevier Inc. All rights reserved. Keywords: Stochastic wave equation; Correlated noise; Sample path regularity; Malliavin calculus; Probability law
1. Introduction In this paper we consider the stochastic wave equation in space dimension d = 3,
E-mail address:
[email protected]. URL: http://www.mat.ub.es/~sanz. 1 Supported by the grant MTM 2006-01351 from the Dirección General de Investigación, Ministerio de Educación y Ciencia, Spain. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.004
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∂2 − u(t, x) = σ u(t, x) F˙ (t, x) + b u(t, x) , 2 ∂t ∂ u(0, x) = v˜0 (x), ∂t
u(0, x) = v0 (x),
(1)
where t ∈ ]0, T ] for a fixed T > 0, x ∈ R3 , and denotes the Laplacian on R3 . The functions σ and b are Lipschitz continuous, and the process F˙ is the formal derivative of a Gaussian random field, white in time and correlated in space defined as follows. Let f (x) = ϕ(x)kβ (x),
(2)
where ϕ is a smooth positive function and kβ denotes the Riesz kernel kβ (x) = |x|−β , β ∈ ]0, 2[. We shall assume that f defines a tempered measure and R3
μ(dξ ) < ∞, 1 + |ξ |2
(3)
where μ = F −1 (f ) and F denotes the Fourier transform operator. This condition is satisfied for instance for densities of the form (2) with ϕ(x) = exp(−σ 2 |x|2 /2) and β ∈ ]0, 2[ (see [5]). Let D(R4 ) be the space of Schwartz test functions (see [16]). Then, on some probability space, there exists a Gaussian process F = (F (ϕ), ϕ ∈ D(R4 )) with mean zero and covariance functional defined by E F (ϕ)F (ψ) =
R+
ds
˜ dx f (x) ϕ(s) ∗ ψ(s) (x),
(4)
R3
˜ x) = ψ(s, −x). where ψ(s, Riesz kernels are a class of singular correlation functions which have already appeared in several papers related with the stochastic heat and wave equations, for instance in [2,3,7–9]. 2 We recall that the fundamental solution G(t) associated to the wave operator L = ∂t∂ 2 − in 1 dimension three is given by G(t) = 4πt σt , t > 0, where σt denotes the uniform surface measure on the sphere of radius t ∈ [0, T ], hence with total mass 4πt 2 . The properties of G(t) together with the particular form of the covariance of the noise play a crucial role in giving a rigorous formulation to the initial value problem (1). Here, we shall follow the same formulation as in [5] which for the purpose of existence and uniqueness of solution of (1) introduces a localization of the SPDE by means of a set related with the past light cone, as follows. Let D be a bounded domain in R3 . Set KaD (t) = y ∈ R3 : d(y, D) a(T − t) ,
t ∈ [0, T ],
(5)
where a 1 and d denotes the Euclidean distance. Then, a solution to the SPDE (1) in D is an adapted, mean-square continuous stochastic process (u(t)1KaD (t) , t ∈ [0, T ]) with values in L2 (R3 ), satisfying
M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281
257
d G(t) ∗ v0 + G(t) ∗ v˜0 (·) u(t, ·)1KaD (t) (·) = 1KaD (t) (·) dt t + 1KaD (t) (·)
G(t − s, · − y)σ u(s, y) 1KaD (s) (y)M(ds, dy)
0 R3
t + 1KaD (t) (·)
ds G(t − s) ∗ b u(s, ·) 1KaD (s) (·) ,
(6)
0
almost surely, for any t ∈ [0, T ], where we consider the stochastic integral defined in [4] and M denotes the martingale measure derived from F (see [3]). The following result is a quotation of [5, Theorem 4.11] and will be invoked repeatedly in this paper. Theorem 1.1. Assume that: (a) the covariance density is of the form (2) with the covariance factor ϕ bounded and positive, ϕ ∈ C 1 (Rd ) and ∇ϕ ∈ Cbδ (Rd ), for some δ ∈ ]0, 1]; (b) the initial values v0 , v˜0 are such that v0 ∈ C 2 (R3 ), and v0 and v˜0 are Hölder continuous with orders γ1 , γ2 ∈ ]0, 1], respectively; (c) the coefficients σ and b are Lipschitz. 1+δ Then for any q ∈ [2, ∞[ and α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, there is C > 0 such that for (t, x), (t¯, y) ∈ [0, T ] × D, q αq E u(t, x) − u(t¯, y) C |t − t¯| + |x − y| . (7)
In particular, almost surely, the stochastic process (u(t, x), (t, x) ∈ [0, T ] × D) solution of (6) has α-Hölder continuous sample paths, jointly in (t, x), and q sup (8) E u(t, x) < ∞ (t,x)∈[0,T ]×D
for any q ∈ [1, ∞[. In this paper we are interested in studying the properties of the density of the solution of (6) as a function of (t, x) ∈ ]0, T ] × D, where D is a bounded subset of R3 . We shall denote this density by pt,x (y). We shall prove that (t, x) → pt,x (y) is jointly Hölder continuous, uniformly in y on compact sets. This question is trivial in the very particular case where the initial conditions v0 , v˜0 and the coefficient b vanish, and the coefficient σ is a constant function. In fact, with these assumptions and σ = 1 the solution to Eq. (6) is a Gaussian process, centered, stationary in the space variable, and with σt2
2 := E u(t, x) =
t ds 0
μ(dξ ) R3
sin2 (s|ξ |) . |ξ |2
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From the expression pt,x (y) =
√1 2π σt
exp(− |y|2 ) it is not difficult to prove that 2
2σt
pt,x (y) − pt¯,x (y) C(t0 , t1 , D)|t − t¯|, for 0 < t0 t < t¯ t1 , D ⊂ R3 . However, in the general situation that we are considering in this article, the problem becomes much more involved. Suppose that v0 , v˜0 are null functions, assume also that the covariance of the process F is given by (4) with dx f (x) replaced by Γ (dx), where Γ is a non-negative, tempered, non-negative definite measure. Set μ = F −1 (Γ ). We introduce an assumption on μ, denoted by (Hη ), saying
μ(dξ ) < ∞, for some value of η ∈ ]0, 1]. Assume that the coefficients σ and b are of that R3 (1+|ξ |2 )η class C 1 with bounded derivatives and that (Hη ) holds for some η ∈ (0, 1). Then, the existence of the density pt,x at any fixed point (t, x) ∈ ]0, T ] × D has been established in [13]. Moreover, assuming that σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and that (Hη ) holds for some η ∈ (0, 12 ), it is proved in [14] that y → pt,x (y) is a C ∞ function. We refer the reader to [15] for results on applications of Malliavin calculus to the analysis of probability laws of SPDEs. In [12], it is shown that the extension of Walsh’s integral introduced in [2] does not require for the integrands any stationary property in the spatial variable. As a consequence of this fact, the results of [2,13,14] and [15] concerning the stochastic wave equation can be formulated with non-null deterministic initial conditions. In addition, the solution of the equation in this setting coincides with the solution to (6). Furthermore, in the particular case of absolutely continuous covariance measures Γ (dx) = f (x) dx satisfying (3) the existence and smoothness of the density pt,x are proved in [12] under the weaker assumption (H1 ). Hence, on the basis of the above mentioned references and remarks, we can write the next statement, which together with Theorem 1.1 are the starting point of our work.
Theorem 1.2. Assume assumptions (a) and (b) of Theorem 1.1. Suppose also that σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and inf{|σ (z)|, z ∈ R} σ0 > 0. Then, for any fixed (t, x) ∈ ]0, T ] × D, the law of the real valued random variable u(t, x) solution to (6) has a density pt,x ∈ C ∞ . The main purpose of this paper is to prove that with the assumptions of this theorem, for any y ∈ R, the mapping (t, x) ∈ ]0, T ] × R3 → pt,x (y) 1+δ is jointly α-Hölder continuous with α ∈ ]0, inf(γ1 , γ2 , 2−β 2 , 2 )[ (see Theorem 2.1 in Section 1). For stochastic differential equations and some finite-dimensional stochastic evolution systems with an underlying semigroup structure one can find results of this type for instance in [6]. For SPDEs the problem has not been yet very much explored. To the best of our knowledge, this issue has only been studied for the stochastic heat equation in spatial dimension d = 1 in [10] and for the wave equation with d = 2 in [9] (see [1] and [7] for the existence and regularity of the density for these two types of SPDEs). It is worthy noticing that in these two references, the Hölder degree regularity of pt,x (y) in (t, x) is better than for the sample paths of the solution process u(t, x), while in the equation under consideration we obtain the same order. As it will
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become clear from the proof, the reason is the rather degenerate character of the fundamental solution of the wave equation in dimension three. The method of our proof is based on the integration by parts formula of Malliavin calculus, as in the above mentioned references. We next give the main ideas and steps of the proof. Fix y ∈ R and let (gn,y , n 1) be a sequence of smooth functions converging pointwise to the Dirac delta function δ{y} . Fix t, t¯ ∈ ]0, T ], x, x¯ ∈ D and assume that we can prove sup E gn,y u(t, x) − gn,y u(t¯, x) ¯ C |t − t¯|β1 + |x − x| ¯ β2 ,
(9)
n∈N,y∈K
for some β1 , β2 > 0, where K ⊂ R. Then, since pt,x (y) = E[δ{y} (u(t, x))] (see [17, Theorem 1.12] for a rigorous meaning of this identity), by passing to the limit as n → ∞, we will have joint Hölder continuity of the mapping (t, x) ∈ ]0, T ] × D → pt,x (y) ∈ R with degree β1 in t and β2 in x, uniformly in y ∈ K. An estimate like (9) is obtained by the following procedure. For simplicity we write g instead ¯ around u(t, x) up to a certain order r0 of gn,y . We first consider a Taylor expansion of g(u(t¯, x)) chosen in such a way to obtain optimal values of β1 and β2 . Then for any r r0 , we estimate terms of the type (r) r , E g u(t, x) u(t, x) − u(t¯, x) ¯ and the term corresponding to the rest in the Taylor expansion, whose structure is similar. For this we use the version of the integration by parts formula for one-dimensional random variables given in [17, Lemma 2, p. 54] (see also [11, Eqs. (2.29)–(2.31)]) which we now quote as a lemma. Lemma 1.3. On an abstract Wiener space (Ω, H, P ), we consider two real-valued random varip ∞ r ables ξ and Z such that ξ ∈ D∞ , Dξ −1 p2 L (Ω), Z ∈ D . Let g be a function in C , H ∈ for some r 1. Denote by g˜ the antiderivative of g. Then, the following formula holds: ˜ )Hr+1 (Z, ξ ) , E g (r) (ξ )Z = E g(ξ
(10)
where Hr , r 1, is defined recursively by ZDξ , H1 (Z, ξ ) = δ Dξ 2H ZDξ , Hr+1 (Z, ξ ) = δ Hr (Z, ξ ) Dξ 2H
r 1.
In this lemma, δ stands for the adjoint operator of the Malliavin derivative, also termed divergence operator or Skorohod integral and we have used the notations of [11] and [15], as we shall do throughout the paper when referring to notions and results of Malliavin calculus. The abstract Wiener space that we shall consider here is the one associated with the Gaussian process F restricted to the time interval [0, T ], as is described in [15, Section 6.1]. For the sake of completeness and its further use, we recall that H := HT , HT = L2 ([0, T ]; H) and that H is the completion of the inner product space consisting of test functions endowed with the inner product
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ϕ, ψ H =
˜ dx f (x)(ϕ ∗ ψ)(x).
R3
Assume that the function g˜ in Lemma 1.3 is bounded. From (10) it clearly follows that (r) E g (ξ )Z g ˜ ∞ Hr+1 (Z, ξ )L1 (Ω) . (11) Furthermore, as a consequence of the continuity property of the Skorohod integral and the assumptions on ξ , for any r 1, k 1 and p ∈ (1, ∞), Hr (Z, ξ ) C Z k+r,4r p k,p (see [10, Corollary 4.1]). Consequently, under the previous assumptions from (11) we obtain (r) E g (ξ )Z C g ˜ ∞ Z r+1,4r+1 . (12) Let us recall that for a natural number k and a real number p ∈ [1, ∞[, Z k,p = Z Lp (Ω) +
k r D Z r=1
Lp (Ω;HT⊗r )
.
We shall apply (12) mainly to ξ := u(t, x) and Z := (u(t, x) − u(t¯, x)) ¯ r , for natural values of r. Under the hypotheses of Theorem 2.1 the assumptions of Lemma 1.3 are satisfied (see [14] and [15, Chapters 7 and 8]). Thus we face the problem of giving upper bounds for (u(t, x) − u(t¯, x)) ¯ r r+1,4r+1 . Malliavin derivatives of the solution of (1) satisfy evolution equations (see [15, Theorem 7.1] k u(t, x), (t, x) ∈ [0, T ] × D) is a and [14]). Indeed, for x ∈ D, and a natural number k 1, (D·,∗ ⊗k HT -valued process satisfying Dτ,∗ u(t, x) = 0 for τ > t, and for τ t it is the solution of the evolution equation t k k u(t, x) = Zτ,∗ (t, x) + Dτ,∗
G(t − s, x − y) 0 R3
k × Γ k σ, u(s, y) + σ u(s, y) Dτ,∗ u(s, y) M(ds, dy) +
t ds 0
k × Γ k b, u(s, y) + b u(t − s, x − y) Dτ,∗ u(t − s, x − y) .
G(s, dy) R3
(13)
In this equation, (Z k (t, x), (t, x) ∈ [0, T ] × D) is a HT⊗k -valued stochastic process and for a given function g ∈ C k and a random variable X ∈ Dk,2 , Γ k (g, X) = D k (g(X)) − g (X)D k X. The solution of (13) satisfies u(s, y) < +∞ sup (14) k,p (t,x)∈[0,T ]×D
for any p ∈ [1, ∞[ (see [15, Theorem 7.1]).
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In the next section, we shall make use of the explicit form of (13) for k = 1. In this case 1 Zτ,∗ (t, x) = G(t − τ, x − ∗)σ u(τ, ∗)
(15)
and Γ 1 (g, X) = 0. With some effort, using the tools on stochastic integration of Hilbert-valued processes developed in [15] it can be proved that the conclusions of Theorem 1.1 also apply to the HT⊗k valued stochastic process solution to (13). More precisely, for any k 1, q ∈ [2, ∞[ and 1+δ ¯ α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, there is C > 0 such that for (t, x), (t , y) ∈ [0, T ] × D, u(t, x) − u(t¯, y) C |t − t¯| + |x − y| α . k,q
(16)
Hence, with the Hölder continuity property on u(t, x) and its Malliavin derivatives we may be able to prove (9) for specific values of β1 , β2 . We shall fix what is the top order r0 in the Taylor expansion of g(u(t¯.x)). ¯ Clearly, the lower exponents βi should come from the first order term. However, in the examples studied so far, terms of first and second order give the same exponent. For Eq. (6) the situation is different. Already at the first order level of the expansion, we shall see that the contribution of the pathwise integral involving the coefficient b is of the same order than the Hölder continuity exponent given in Theorem 1.1. Clearly, the second order term would provide twice the Hölder continuity degree. Therefore, a Taylor expansion of first order gives the best possible result. However, to conclude whether the regularity of the density pt,x in (t, x) is the same as that of the sample paths of u(t, x), we have to check that the contribution to the first order term in the Taylor expansion of the stochastic integral is not worse than that of the pathwise integral. This explains the strategy of the proof of the main result in the next section. 2. Main result Throughout this section D denotes a fixed bounded domain of R3 and C will be any positive finite constant. We assume that (3) holds. Our purpose is to prove the following theorem. Theorem 2.1. Assume that: (a) the covariance density is of the form (2) and the covariance density factor ϕ is bounded and positive, ϕ ∈ C 1 (Rd ) and ∇ϕ ∈ Cbδ (Rd ) for some δ ∈ ]0, 1]; (b) the initial values v0 , v˜0 are such that v0 ∈ C 2 (R3 ), and v0 and v˜0 are Hölder continuous with orders γ1 , γ2 ∈ ]0, 1], respectively; (c) the coefficients σ and b are C ∞ functions with bounded derivatives of order greater or equal to one, and there exists σ0 > 0 such that inf{|σ (z)|, z ∈ R} σ0 . Then the mapping (t, x) ∈ ]0, T ] × D → pt,x (y) 1+δ is α-Hölder continuous jointly in (t, x) with α ∈ ]0, inf(γ1 , γ2 , 2−β 2 , 2 )[, uniformly in y ∈ R.
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Proof. Fix y ∈ R and let (gn,y , n 1) be a sequence of regular functions converging pointwise to δ{y} as n → ∞; for example, a sequence of Gaussian kernels with mean y and variances converging to zero. We may assume that the corresponding antiderivatives g˜ n,y are uniformly bounded by 1. To simplify the notation, we shall write g instead of gn,y . Step 1 (Time increments). For (t, x) ∈ [0, T ] × D we consider the Taylor expansion E g u(t + h, x) − g u(t, x) = E g u(t, x) u(t + h, x) − u(t, x) 2 ˜ x, h) u(t + h, x) − u(t, x) , + E g u(t,
(17)
where h > 0 and u(t, ˜ x, h) denotes a random variable lying on the segment determined by u(t + h, x) and u(t, x). First order term. Set T1 (t, x, h) = E g u(t, x) u(t + h, x) − u(t, x) . We aim to prove that sup
(t,x)∈[0,T ]×D
T1 (t, x, h) Chα ,
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. By using Eq. (6), we write T1 (t, x, h) 3i=1 T1,i (t, x, h), with
d G(t + h) ∗ v0 + G(t + h) ∗ v˜0 (x) T1,1 (t, x, h) = E g u(t, x) dt d − G(t) ∗ v0 + G(t) ∗ v˜0 (x) , dt t+h T1,2 (t, x, h) = E g u(t, x) ds G(t + h − s, dz)b u(s, x − z) 0
t −
ds 0
R3
G(t − s, dz)b u(s, x − z)
R3
,
t+h T1,3 (t, x, h) = E g u(t, x) G(t + h − s, x − z)σ u(s, x − z) M(ds, dz) 0 R3
t − 0 R3
G(t − s, x − z)σ u(s, x − z) M(ds, dz) .
In fact, by our choice of (t, x) all the indicator functions in (6) take the value 1.
(18)
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We shall apply repeatedly the inequality (12) with r = 1, ξ := u(t, x) and different choices of Z. To begin with, we take Z :=
d d G(t + h) ∗ v0 + G(t + h) ∗ v˜0 (x) − G(t) ∗ v0 + G(t) ∗ v˜0 (x). dt dt
Since Z is deterministic, Z k,p = |Z|, for any k and p. Then, applying (12) and [5, Lemma 4.9] yields sup
(t,x)∈]0,T ]×D
T1,1 (t, x, h) Chγ1 ∧γ2 .
(19)
We next study the term T1,2 (t, x, h). Let t+h T1,2,1 (t, x, h) = E g u(t, x) ds G(t + h − s, dz)b u(s, x − z) . t
We apply (12) to Z :=
t+h t
ds
R3
R3
G(t + h − s, dz)b(u(s, x − z)) and consider the measure on
given by dsG(t + h − s, dz) with total mass [t, t applying Minkowski’s inequality, we obtain + h] × R3
h2 2
and an arbitrary p ∈ [1, ∞[. By
t+h ds G(t + h − s, dz)b u(s, x − z) t
R3
2,p
t+h ds G(t + h − s, dz)b u(s, x − z) 2,p . t
R3
By the chain rule of Malliavin calculus, b u(s, y)
2,p
2 C 1 + u(s, y)2,p + u(s, y)2,2p .
Consequently, sup (s,y)∈[0,T ]×D2T
b u(s, y)
2,p
<∞
(20)
where for a bounded set D ⊂ R3 and a 0, we denote by Da = {z ∈ R3 ; d(z, D) a} and we have applied (14). Thus, we have proved t+h ds G(t + h − s, dz)b u(s, x − z) sup (t,x)∈]0,T ]×D t
R3
Ch2 , 2,p
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for any p ∈ [1, ∞[, and consequently, sup
(t,x)∈]0,T ]×D
T1,2,1 (t, x, h) Ch2 .
(21)
Set t T1,2,2 (t, x, h) = E g u(t, x) ds b u(s, x − z) G(t + h − s, dz) − G(t − s, dz) , R3
0
that according to (12) we can bound as follows, t T1,2,2 (t, x, h) C ds b u(s, x − z) G(t + h − s, dz) − G(t − s, dz) 0
.
2,42
R3
We have
t
b u(s, x − z) G(t + h − s, dz)
ds R3
0
t =
ds(t + h − s) 0
B1 (0)
t
G(1, dz)b u s, x − (t + h − s)z ,
b u(s, x − z) G(t − s, dz)
ds R3
0
t ds(t − s)
= 0
G(1, dz)b u s, x − (t − s)z ,
B1 (0)
z as can be easily checked by applying the change of variables z → t+h−s and z → tively. Then, by the triangular inequality we obtain for any p ∈ [1, ∞[
t ds b u(s, x − z) G(t + h − s, dz) − G(t − s, dz) 0
R3
t Ch ds 0
G(1, dz)b u s, x − (t + h − s)z
B1 (0)
t + C ds 0
(22)
B1 (0)
z t−s ,
respec-
2,p
2,p
G(1, dz) b u s, x − (t + h − s)z − b u s, x − (t − s)z
. (23) 2,p
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For the study of the first term in the right-hand side of (23), we apply Minkowski’s inequality and then (20). This yields t ds 0
G(1, dz)b u s, x − (t + h − s)z
< C.
2,p
B1 (0)
The Lipschitz property of b and (7), (16), (14) yield b u s, x − (t + h − s)z − b u s, x − (t − s)z
2,p
Chα ,
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. Consequently, after having applied Minkowski’s inequality we see that the second term of the right-hand side of (23) is bounded by Chα , uniformly in (t, x) ∈ [0, T ] × D. Thus, we have proved
sup
(t,x)∈[0,T ]×D
T1,2,2 (t, x, h) Chα ,
and along with (21) we obtain sup
(t,x)∈[0,T ]×D
T1,2 (t, x, h) Chα ,
(24)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. Let us remark that in [10] and [9] the contribution of the term analogous to T1,2 (t, x, h) is a power of h of higher order than the Hölder degree of the solution. In fact, for the heat equation and the wave equation in spatial dimension two, by integration of the increments G(t + h − s, dz) − G(t − s, dz) we get powers of h. For the wave equation in dimension three, such an approach is not possible. Instead, “increments” of G(t − s, dz) are transfered to increments of b(u(s, x − z)) (this is the role played by the change of variables that we have performed to obtain (22)) and after this, we can conclude by applying the Lipschitz property of b and the Hölder continuity of the sample paths. The inequality (24) tell us that we are not going to improve the Hölder degree of the mapping t ∈ ]0, T ] → pt,x (y) in more than the given α. But it might happen that the contribution of T1,3 (t, x, h) makes the overall estimation worse. We next carry out a careful analysis of this term and prove that its contribution in terms of powers of h is the same as T1,2 (t, x, h). We write T1,3 (t, x, h) = T1,3,1 (t, x, h) + T1,3,2 (t, x, h) with
T1,3,1 (t, x, h) t+h = E g u(t, x) σ u(s, y) G(t + h − s, x − y)M(ds, dy) , t R3
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T1,3,2 (t, x, h) t = E g u(t, x) σ u(s, y) G(t + h − s, x − y) − G(t − s, x − y) M(ds, dy) . 0 R3
The term T1,3,1 (t, x, h) vanishes, since the random variable g (u(t, x)) is adapted to the natural filtration generated by the martingale measure M. In contrast with T1,2 , for the analysis of T1,3,2 (t, x, h) we do not start by applying (12), which actually would lead to worse results (see Remark 2.2); instead, we apply [15, Proposition 3.9]. Since the mathematical expectation of a Skorohod integral is zero, we obtain T1,3,2 (t, x, h) = E g u(t, x) D·,∗ u(t, x), σ u(·, ∗) × G(t + h − ·, x − ∗) − G(t − ·, x − ∗) 1]0,t] (·) H T = E g u(t, x) D·,∗ u(t, x), σ u(·, ∗) . × G(t + h − ·, x − ∗) − G(t − ·, x − ∗) 1]0,t] (·) HT
For any (t˜, x) ˜ ∈ [0, T ] × R3 , we define h ˜ B·,∗ (t , x) ˜ = σ u(·, ∗) G(t˜ + h − ·, x˜ − ∗) − G(t˜ − ·, x˜ − ∗) . With this notation, and by applying (12) to T1,3,2 (t, x, h) we see that h (t, x) H 3,43 . T1,3,2 (t, x, h) C D·,∗ u(t, x), B·,∗
(25)
T
We shall consider norms · k,p for arbitrary k 1 and p ∈ [1, ∞[, instead of · 3,43 . By virtue of (13) and (15), we write D·,∗ u(t, x), B h (t˜, x) ˜ H k,p ·,∗ T h C B (t, x; t˜, x) ˜ + B h (t, x; t˜, x) ˜ 1
2
k,p
k,p
+ B3h (t, x; t˜, x) ˜ k,p ,
where h B1h (t, x; t˜, x) ˜ = G(t − ·, x − ∗)σ u(·, ∗) , B·,∗ (t˜, x) ˜ H , B2h (t, x; t˜, x) ˜ =
t
T
h ˜ G(t − s, x − y)σ u(s, y) D·,∗ u(s, y)M(ds, dy), B·,∗ (t , x) ˜
0 R3
˜ = B3h (t, x; t˜, x)
t
ds
0
R3
, HT
h ˜ G(s, dy)b u(t − s, x − y) D·,∗ u(t − s, x − y), B·,∗ (t , x) ˜
. HT
y Consider the change of variables (s, y) → (t − s, t−s ); Fubini’s theorem along with Minkowski’s inequality yield
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t h B (t, x; t˜, x) ˜ k,p = ds(t − s) G(1, dy)b u s, x − (t − s)y 3 R3
0
h × D·,∗ u s, x − (t − s)y , B·,∗ (t˜, x) ˜ H T
k,p
C b ∞
ds 0
t C
t
h G(1, dy) D·,∗ u s, x − (t − s)y , B·,∗ (t˜, x) ˜ H k,p T
R3
h ˜ ds sup D·,∗ u(s, y), B·,∗ (t , x) ˜ H k,p .
0
T
y∈KaD (s)
The last inequality is obtained as follows. By definition of the sets KaD (t), it is obvious that for any (t, x) ∈ [0, T ] × D, x belongs to KaD (t). Since the support of the measure G(s, dy) is the boundary of the ball centered at zero and with radius s, the y-variable in the above integrals belongs to KaD (s). Hence, by Gronwall’s lemma h ˜ sup D·,∗ u(t, x), B·,∗ (t , x) ˜ H k,p T
x∈KaD (t)
C sup
h B (t, x; t˜, x) ˜
x∈KaD (t)
1
k,p
+ B2h (t, x; t˜, x) ˜ k,p .
(26)
Our aim is to prove that sup
(t,x)∈[0,T ]×D
α D·,∗ u(t, x), B h (t, x) ·,∗ H k,p Ch , T
(27)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, for k = 3. This is done recursively on k = 0, 1, 2, 3, where by convention · 0,p = · p . To illustrate the method and simplify the presentation, we shall consider in (26) the norm · 1,p instead of · 3,p . That is, we shall deal only with derivatives up to the first order. Thus let as first prove (27) for k = 0, that means for the Lp (Ω)-norm. For this, we start by studying the Lp (Ω)-norm of B2h (t, x; t˜, x). ˜ h (t˜, x) ˜ HT . From Burkholder’s To shorten the notation, set D(s, y; t˜, x) ˜ = D·,∗ u(s, y), B·,∗ inequality it follows that
h p B (t, x; t˜, x) ˜ p 2 p t ˜ dy) = E G(t − s, x − y)σ u(s, y) D(s, y; t˜, x)M(ds, 0 R3
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t CE
ds 0
dy R3
¯ d y¯ G(t − s, x − y)σ u(s, y) f (y − y)
R3
p 2 × G(t − s, x − y)σ ¯ u(s, y) ¯ D(s, y; t˜, x)D(s, ˜ y; ¯ t˜, x) ˜
t C
ds sup y∈KaD (s)
0
p h ˜ E D·,∗ u(s, y), B·,∗ (t , x) ˜ H . T
(28)
Therefore, (26) with k = 0 and Gronwall’s lemma yields h ˜ sup D·,∗ u(t, x), B·,∗ (t , x) ˜ H p C sup B1h (t, x; t˜, x) ˜ p T
x∈KaD (t)
x∈KaD (t)
with a constant C independent of t˜ and x. ˜ Hence, we can fix t˜ = t and x˜ = x in the preceding inequality and obtain sup
h sup D·,∗ u(t, x), B·,∗ (t, x) H p C sup T
t∈[0,T ] x∈KaD (t)
sup B1h (t, x)p ,
t∈[0,T ] x∈KaD (t)
(29)
where B1h (t, x) stands for B1h (t, x; t, x). By the very definition of the inner product in HT we have t h B (t, x)p = E dr dξ dη G(t − r, x − ξ )σ u(r, ξ ) f (ξ − η) 1 p R3
0
R3
p × G(t + h − r, x − η) − G(t − r, x − η) σ u(r, η) . We consider each one of the terms in the difference of the right-hand side of this inequality and apply respectively the change of variables t −r (ξ, η) → x − ξ, (x − η) , t +h−r
(ξ, η) → (x − ξ, x − η).
With this, the increments in time of the measure G are transfered to increments of σ and f . More precisely, we obtain h B (t, x)p C T1,3,2,1 (t, x, h) + T1,3,2,2 (t, x, h) 1
with
p
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t T1,3,2,1 (t, x, h) = E dr G(t − r, dξ )G(t − r, dη)σ u(r, x − ξ ) R3 R3
0
t +h−r t +h−r f η−ξ t −r t −r p t +h−r η − σ u(r, x − η) , × σ u r, x − t −r ×
t G(t − r, dξ )G(t − r, dη)σ u(r, x − ξ ) σ u(r, x − η) T1,3,2,2 (t, x, h) = E dr R3 R3
0
p t +h−r t +h−r f η − ξ − f (η − ξ ) . × t −r t −r
For the analysis of T1,3,2,1 (t, x, h) we consider the measure with support on [0, T ] × Bt−r (0) × Bt−r (0) defined by t + h − r t +h−r f η − ξ . ν(dr; dξ, dη) := dr G(t − r, dξ )G(t − r, dη) t −r t −r
(30)
Following the steps of the proof of [5, Lemma 6.3] we obtain t
ν(dr; dξ, dη) < ∞.
sup 0tt+hT
0 Bt−r (0) Bt−r (0)
Then, we can write t T1,3,2,1 (t, x, h) = E
ν(dr; dη, dξ )σ u(r, x − ξ )
0 Bt−r (0) Bt−r (0)
p t +h−r η − σ u(r, x − η) × σ u r, x − t −r and apply Hölder’s inequality with respect to the measure ν(dr; dη, dξ ). This yields t
T1,3,2,1 (t, x, h) C
ν(dr; dη, dξ ) 0 Bt−r (0) Bt−r (0)
p t +h−r η − σ u(r, x − η) . × E σ u(r, x − ξ ) σ u r, x − t −r
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We now apply Schwarz’ inequality to the factor containing the expectation. Since the coefficient σ is a Lipschitz function, by (7) and (8) we obtain 1 2p 2 t +h−r E σ u r, x − η − σ u(r, x − η) t −r |η|=t−r αp h η Chαp , C sup |η|=t−r t − r
T1,3,2,1 (t, x, h) C sup
(31)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [ and a constant C not depending on (t, x) ∈ [0, T ] × D for an arbitrary D. To study T1,3,2,2 (t, x, h) we consider the measure on [0, t] × Bt−r (0) × Bt−r (0) given by
μh (dr; dξ, dη) = dr G(t − r, dξ )G(t − r, dη) t + h − r t +h−r f η − ξ − f (η − ξ ) . × t −r t −r We also consider two additional measures with the same support as μh (dr; dξ, dη) obtained by applying the triangular inequality to the expression t + h − r t +h−r . f η − ξ − f (η − ξ ) t −r t −r They are defined by h f (η − ξ ), t −r t +h−r μh2 (dr; dξ, dη) = dr G(t − r, ξ )G(t − r, dη) t −r t + h − r η − ξ − f (η − ξ ) . × f t −r μh1 (dr; dξ, dη) = dr G(t − r, ξ )G(t − r, dη)
(32)
With these new ingredients, T1,3,2,2 (t, x, h) C(T1,3,2,2,1 + T1,3,2,2,2 ), where t T1,3,2,2,1 = E
μh1 (dr; dξ, dη)σ
p u(r, x − ξ ) σ u(r, x − η) ,
0 Bt−r (0) Bt−r (0)
t T1,3,2,2,2 = E
p μh2 (dr; dξ, dη)σ u(r, x − ξ ) σ u(r, x − η) .
0 Bt−r (0) Bt−r (0)
t
We next check that 0 Bt−r (0) Bt−r (0) μh1 (dr; dξ, dη) < Ch. Indeed, owing to (2) and by the change of variable r → t − r, we have
M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281
t
271
μh1 (dr; dξ, dη)
0 Bt−r (0) Bt−r (0)
t Ch 0
t = Ch
dr r
G(r, dξ )G(r, dη)kβ (ξ − η) Bt−r (0) Bt−r (0)
dr r
dξ R3
0
|FG(r)(ξ )|2 Ch, |ξ |3−β
uniformly in t ∈ [0, T ], where in the last inequality we have applied [5, Lemma 2.3] with b = 1. Consequently, Hölder’s inequality, the linear growth of the coefficient σ and the property (8) yield T1,3,2,2,1 Chp ,
(33)
uniformly in (t, x) ∈ [0, T ] × D. The next step consists of proving that μh2 (dr; dξ, dη) defines a finite measure as well, and in giving an estimate of its total mass in terms of powers of h. For this, we consider the inequality t +h−r f η − ξ − f (η − ξ ) t −r t +h−r ϕ η − ξ − ϕ(η − ξ ) kβ (η − ξ ) t −r t +h−r t +h−r η − ξ k β η − ξ − kβ (η − ξ ) , + ϕ t −r t −r
(34)
which is a consequence of (2) and the triangular inequality. The properties of ϕ together with [5, Lemma 2.3] yield
t
G(t − r, dξ )G(t − r, dη)
dr 0
Bt−r (0) Bt−r (0)
t +h−r t −r
t +h−r η − ξ − ϕ(η − ξ ) kβ (η − ξ ) × ϕ t −r t
Ch 0
Ch, uniformly in t ∈ [0, T ].
dr t −r
G(t − r, dξ )G(t − r, dη)kβ (η − ξ )
Bt−r (0) Bt−r (0)
(35)
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Let us now consider the contribution to μh2 (dr; dξ, dη) of the second term of the right-hand side of (34). Let α˜ ∈ ]0, 1[, β ∈ ]0, 2[ with α˜ + β ∈ ]0, 2[. By applying [5, Lemma 2.6(a)] with η , we obtain b := α, ˜ a := 3 − (α˜ + β), c := h, u := η − ξ , x := t−r t
G(t − r, dξ )G(t − r, dη)
dr 0
Bt−r (0) Bt−r (0)
t + h − r t + h − r ϕ η − ξ t −r t −r
t +h−r × kβ η − ξ − kβ (η − ξ ) t −r t
ϕ ∞
t +h−r t −r
dr 0
ϕ ∞ hα˜ × R3
t dr 0
Bt−r (0) Bt−r (0)
t +h−r t −r
h η G(t − r, dξ )G(t − r, dη)Dkβ η − ξ, t −r
G(t − r, dξ )G(t − r, dη) Bt−r (0) Bt−r (0)
η , Dk w, dw kα+β (η − ξ − hw) ˜ 3−α˜ t −r
(36)
where we have set Dg(x, y) := g(x + y) − g(x) for a function g : R3 → R. Our next purpose is to prove that the last integral in the above expression is bounded, uniformly in t, t + h ∈ [0, T ]. For this, as in [5, Lemma 6.4] we split the integral on the w-variable in the last expression into the sum of two integrals: on a finite ball containing the origin and on the complementary of this set. In this way we obtain as an upper bound of t
t +h−r t −r
dr 0
× R3
G(t − r, dξ )G(t − r, dη) Bt−r (0) Bt−r (0)
η , Dk w, dw kα+β (η − ξ − hw) ˜ 3−α˜ t −r
the sum of the three terms: t t +h−r (1) I1 = dr t −r 0
G(t − r, dξ )G(t − r, dη)
Bt−r (0) Bt−r (0)
dw kα+β (η − ξ − hw)k3−α˜ w + ˜
× B2 (0) (2)
t
I1 =
dr 0
t +h−r t −r
× B2 (0)
η , t −r
G(t − r, dξ )G(t − r, dη)
Bt−r (0) Bt−r (0)
dw kα+β (η − ξ − hw)k3−α˜ (w), ˜
M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281
(3) I1
t
t +h−r dr t −r
= 0
× B2 (0)c
G(t − r, dξ )G(t − r, dη) Bt−r (0) Bt−r (0)
η . Dk w, dw kα+β (η − ξ − hw) ˜ 3−α˜ t −r
Consider the change of variable w → w + By Fubini’s theorem we obtain (1) I1
t
dw k3−α˜ (w)
B3 (0)
× kα+β ˜
dr 0
dw k3−α˜ (w)
and then η →
t+h−r t−r η
(1)
that we apply to I1 .
G(t − r, dξ )G(t − r, dη)
Bt−r (0) Bt−r (0)
G(t − r, dξ )G(t − r + h, dη)kα+β (η − ξ − hw). ˜
dr 0
B3 (0)
η t−r
t
=
t +h−r t −r
t +h−r η − ξ − hw t −r
273
R3 R3
The properties of the Fourier transform and the expression of this operator applied to Riesz kernels yield, after regularization of G, G(t − r, dξ )G(t − r + h, dη)kα+β (η − ξ − hw) ˜ R3 R3
FG(t − r)(ξ )FG(t − r + h)(ξ )k3−(α+β) (. − hw)(ξ ). ˜
= R3
Hence by applying Schwarz’s inequality, the last integral is bounded by FG(t)(ξ ) 2 k3−(α+β) sup dξ, ˜ t∈[0,T ] R3
which is known to be finite whenever α˜ + β ∈ [0, 2] (see for instance [5, Eq. (2.5)]). Since k3−α˜ (w) is integrable in a neighbourhood of the origin for any α˜ > 0, we finally ob(1) tain I1 , is bounded uniformly in t, h ∈ [0, T ]. (2) Similar but simpler arguments show that the same property hold for I1 . η For any λ ∈ [0, 1] set ψ(λ) = k3−α˜ (w + λ t−r ). It is easy to check that |ψ (λ)| Ck4−α˜ (w + η λ t−r ). Moreover, for |w| 2, and |η| = t − r, w + λ η |w| − λ η |w| − 1 |w| , t −r t − r 2 by the triangular inequality.
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Thus, (3) I1
t C
dr 0
t +h−r t −r
G(t − r, dξ )G(t − r, dη) Bt−r (0) Bt−r (0)
1
×
kα+β (η − ξ − hw) dw ˜
(B2
(0))c
t 2 dr FG(r)(ξ ) k3−(α+β) dw k4−α˜ (w) dξ . ˜ r
C (B2
0
η dλ k4−α˜ w + λ t −r
(0))c
0
(37)
R3
For α˜ ∈ ]0, 1[ the integral (B2 (0))c dw k4−α˜ (w) is finite. Moreover, if α˜ + β ∈ ]0, 2[ the last integral in (37) is also finite, owing to [5, Lemma 2.3] applied to the value b = 1. This lead us to conclude that I1(3) is bounded uniformly in t, h ∈ [0, T ]. Summarizing, as a consequence of (35), (36) and the preceding discussion, we have proved that t
μh2 (dr; dξ, dη) Chα˜ ,
sup t∈[0,t]
(38)
0 Bt−r (0) Bt−r (0)
with α˜ ∈ ]0, 2−β 2 [. We can now apply Hölder’s inequality with respect to the measure μh2 (dr; dξ, dη). By virtue of (38), the linear growth of σ and (8) we obtain sup
(t,x)∈[0,T ]×D
T1,3,2,2,2 Chαp ,
with α ∈ ]0, 2−β 2 [. Finally, the estimates (31), (33) and (39) imply that h B (t, x) Chα , sup 1 p (t,x)∈[0,T ]×D
(39)
(40)
and a fortiori sup
(t,x)∈[0,T ]×D
α D·,∗ u(t, x), B h (t, x) ·,∗ H p Ch , T
(41)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. This finishes the analysis of the · p contribution to the left-hand side of (27). h (t, x) We next consider the Lp (Ω, HT )-norm of D D·,∗ u(t, x), B·,∗ HT . As in the previous h h ˜ with arbitrary t˜ ∈ [0, T ], x˜ ∈ R3 . By virtue of (26) step, we shall replace B·,∗ (t, x) by B·,∗ (t˜, x) and (41) it suffices to study the Lp (Ω, HT )-norm of DBih (t, x; t˜, x) ˜ for i = 1, 2. We start with ˜ the analysis of B2h (t, x; t˜, x).
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By applying the differential rules of Malliavin calculus we have D.∗ B2h (t, x; t˜, x) ˜ ˜ = G(t − ·, x − ∗)σ u(.∗) D(s, y; t˜, x) t +
G(t − s, x − y) σ u(s, y) D.∗ u(s, y)D(s, y; t˜, x) ˜
0 R3
+ σ u(s, y) D.∗ D(s, y; t˜, x) ˜ M(ds, dy). Applying Hölder’s inequality and using that σ is bounded, we obtain, as in (28), p E G(t − ·, x − ∗)σ u(.∗) D(s, y; t˜, x) ˜ H
T
t C
p ˜ . E D(s, y; t˜, x)
ds sup
(42)
y∈KaD (s)
0
For fixed t˜, x˜ we consider the HT -valued process defined by K(s, y; t˜, x) ˜ = σ u(s, y) D.∗ u(s, y)D(s, y; t˜, x) ˜ + σ u(s, y) D.∗ D(s, y; t˜, x), ˜
(43)
(s, y) ∈ [0, T ] × R3 , for which we have p 1 2p 1 p 2 E Du(s, y)2p 2 ˜ H σ ∞ E D(s, y; t˜, x) E K(s, y; t˜, x) ˜ HT T p p ˜ . + σ ∞ E DD(s, y; t˜, x)
(44)
HT
We can apply the Lp -estimates for stochastic integrals with respect to the Gaussian process M of Hilbert-valued integrands (see [15, Eq. (6.8) of Theorem 6.1] and [12, p. 289]) yielding t p G(t − s, x − y)K(s, y; t˜, x)M(ds, ˜ dy) p
L (Ω,HT )
0 R3
t C
ds sup y∈KaD (s)
0
p 2p 1 2 + DD(s, y; t˜, x) ˜ Lp (Ω;H ) . E D(s, y; t˜, x) ˜
(45)
T
By taking t˜ = t and x˜ = x and considering the inequalities (42), (45), we obtain DB h (t, x)p p 2 L (Ω;H
T)
t C
ds sup 0
y∈KaD (s)
p 2p 1 h h (t, x) H + E D·,∗ u(s, y), B·,∗ (t, x) H 2 E D·,∗ u(s, y), B·,∗ T
p h + D D·,∗ u(s, y), B·,∗ (t, x) H Lp (Ω;H ) . T
T
T
(46)
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Then, (41) and Gronwall’s lemma yield p h sup D D·,∗ u(t, x), B·,∗ (t, x) H Lp (Ω;H
sup
t∈[0,T ] x∈KaD (t)
C
sup
T)
T
p sup DB1h (t, x)Lp (Ω;H
T
t∈[0,T ] x∈KaD (t)
αp , + h )
(47)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The last step of the proof consist of checking that for an arbitrary bounded set D ⊂ R3 , DB h (t, x) p sup Chα , (48) 1 L (Ω;H ) (t,x)∈[0,T ]×D
T
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The proof of this fact can be done following the same lines as for (40). We apply the results on the densities ν(dr; dξ, dη), μh1 (dr; dξ, dη), μh2 (dr; dξ, dη), defined in (30), (32) respectively, proved so far. Instead of the process {σ (u(s, y)), (s, y) ∈ [0, T ] × R3 } and the Lp (Ω)-norm, we shall deal here with the HT -valued process {D(σ (u(s, y))), (s, y) ∈ [0, T ] × R3 } and the Lp (Ω; HT )-norm. In addition to (7), we should also apply (16) and (14). We leave the details to the reader. Together with (19) and (24) this proves (18) and concludes the proof of the first step of the proof.
Remark 2.2. Applying first (12) and then estimates for the · 2,p -norm of the stochastic integral leads to t σ u(s, y) G(t + h − s, x − y) − G(t − s.x − y) M(ds, dy) T1,3,2 C 0 R3
2,43
α
Ch 2 . Thus, we loose accuracy. This may be a justification for a pretty tricky approach in the preceding proof. The rest in the time expansion. The second and last term in (17) to be examined is 2 ˜ x, h) u(t + h, x) − u(t, x) . R(t, x, h) = E g u(t, We shall apply (12) to the random variables ξ := u(t, ˜ x, h) and Z := (u(t + h, x) − u(t, x))2 . For this, we have to make sure that the assumptions of Lemma 1.3 are satisfied. For Z := (u(t + h, x) − u(t, x))2 , and the two choices of ξ —u(t, x) and u(t + h, x)—this has been proved in [14]. Then it suffices to remark that the norm · HT as well as · −1 HT define convex functions and use the definition of u(t, ˜ x, h) to conclude. Consequently, 2 R(t, x, h) C u(t + h, x) − u(t, x)3,43 .
M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281
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Owing to (7) and (16) we conclude that sup
(t,x)∈[0,T ]×D
R(t, x, h) Ch2α ,
(49)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The estimates (18) and (49) show that
sup
E g u(t + h, x) − g u(t, x) Chα ,
(t,x)∈[0,T ]×D
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. Therefore the mapping t ∈ ]0, T [ → pt,x (y) is Hölder con1+δ 3 tinuous of degree α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, uniformly in y ∈ R varying on bounded sets.
Step 2 (Space increments). Fix t ∈ ]0, T ] and consider the Taylor expansion ¯ − u(t, x) E g u(t, x) ¯ − g u(t, x) = E g u(t, x) u(t, x) 2 + E g u(t, ˆ x, x) ¯ u(t, x) ¯ − u(t, x) ,
(50)
where x, x¯ ∈ D and u(t, ˆ x, x) ¯ denotes a random variable lying on the segment determined by u(t, x) ¯ and u(t, x). First order term. Our aim is to prove that ¯ α, ¯ − u(t, x) C|x − x| sup E g u(t, x) u(t, x)
t∈[0,T ]
(51)
1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. As we did for the time increments, we consider Eq. (6) and write 3 E g u(t, x) u(t, x) ¯ − u(t, x) = Si (t, x, x), ¯ i=1
with ¯ S1 (t, x, x) d d ¯ − G(t) ∗ v0 + G(t) ∗ v˜0 (x) G(t) ∗ v0 + G(t) ∗ v˜0 (x) , = E g u(t, x) dt dt S2 (t, x, x) ¯ t = E g u(t, x) ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) , 0
R3
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S3 (t, x, x) ¯ t G(t − s, x¯ − z) − G(t − s, x − z) σ u(s, y) M(ds, dz) . = E g u(t, x) 0 R3
Let us consider S1 (t, x, x). ¯ As for the term T1,1 (t, x, h), we first apply the inequality (12) and notice that d d ¯ − G(t) ∗ v0 + G(t) ∗ v˜0 (x) G(t) ∗ v0 + G(t) ∗ v˜0 (x) Z(t; x, x) ¯ := dt dt is deterministic. Thus, it suffices to estimate the absolute value of the random variable Z(t; x, x) ¯ defined before. For this, we apply [5, Lemmas 4.2 and 4.4] which tell us that the fractional Sobolev norm of any integration degree p 2 and differential order ρ < γ1 ∧ γ2 is bounded. Hence, since p is arbitrary, by the Sobolev embedding theorem we have that ¯ ρ, ¯ C|x − x| ¯ sup C Z(t; x, x) (52) sup S1 (t, x, x) t∈[0,T ]
t∈[0,T ]
with ρ < γ1 ∧ γ2 . ¯ By virtue of (12), it suffices to We continue the proof with the study of the term S2 (t, x, x). find an upper bound of t ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) R3
0
2,42
in terms of a power of |x − x|. ¯ The measure on [0, t] × R3 defined by ds G(t − s, dz) is finite. Hence, we can apply Minkowski’s inequality and obtain for any p ∈ [1, ∞[ t ds G(t − s, dz) b u(s, x¯ − z) − b u(s, x − z) R3
0
t
ds 0
2,p
G(t − s, dz)b u(s, x¯ − z) − b u(s, x − z) 2,p
R3
C|x − x| ¯ α, 1+δ with α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [. The last inequality is obtained by using that b and its derivatives are Lipschitz continuous and bounded functions, and by applying (7) and (16). Hence,
¯ C|x − x| ¯ α, sup S2 (t, x, x)
t∈[0,T ]
for any α ∈ ]0, γ1 ∧ γ2 ∧
2−β 2
∧
1+δ 2 [.
(53)
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279
To analyze S3 (t, x, x) ¯ we proceed in a similar manner as for T1,3 (t, x, h) by applying first [15, Proposition 3.9] and then (12). We obtain ¯ = E g u(t, x) D·,∗ u(t, x), σ u(·, ∗) S3 (t, x, x) × G(t − ·, x¯ − ∗) − G(t − ·, x − ∗) 1]0,t] (·) H T C D·,∗ u(t, x), σ u(·, ∗) G(t − ·, x¯ − ∗) − G(t − ·, x − ∗) 1]0,t] (·) H 3,43 . T
Notice that the last expression has a similar structure than the right-hand side of (25) where h (t, x) := G(t + h − ·, x − ∗) − G(t − ·, x − ∗) is replaced by G(t − ·, x¯ − ∗) − G(t − ·, x − ∗). B.∗ Hence, we can proceed as in the analysis of the time increments to see that it suffices to deduce an estimate for G(t − ·, x − ∗)σ u(·, ∗) , σ u(·, ∗) G(t − ·, x¯ − ∗) − G(t − ·, x − ∗)
HT 3,p ,
for any p ∈ [1, ∞[. To pursue the proof, we split the argument of the above expression into two terms
t S3,1 (t, x, x) ¯ =
dr
dη G(t − r, x − ξ )
dξ
0
R3
R3
t
× σ u(r, ξ ) f (ξ − η)σ u(r, η) G(t − r, x¯ − η),
S3,2 (t, x, x) ¯ =
dr 0
dη G(t − r, x − ξ )
dξ R3
R3
× σ u(r, ξ ) f (ξ − η)σ u(r, η) G(t − r, x − η),
and we apply the change of variables (ξ → x − ξ, η → x¯ − η), (ξ → x − ξ, η → x − η), respectively. We obtain S3,1 (t, x, x) ¯ − S3,2 (t, x, x) ¯
t =
dr
G(t − r, dξ )G(t − r, dη)f (ξ − η)σ u(r, x − ξ )
R3 R3
0
× σ u(r, x¯ − η) − σ u(r, x − η)
t +
dr 0
G(t − r, dξ )G(t − r, dη) f x − x¯ − (ξ − η) − f (ξ − η)
R3 R3
× σ u(r, x − ξ ) σ u(r, x¯ − η) .
(54)
By Minkowski’s inequality the · k,p -norm of the first term in the right-hand side of (54) is bounded by
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M. Sanz-Solé / Journal of Functional Analysis 255 (2008) 255–281
t
G(t − r, dξ )G(t − r, dη)f (ξ − η)
dr R3
0
R3
× σ u(r, x − ξ ) σ u(r, x¯ − η) − σ u(r, x − η) k,p
C|x − x| ¯ α,
(55)
where the very last upper bound follows from (7), (16) and (14). For the second term of the right-hand side of (54) we apply [5, Lemma 6.1] which implies
t sup
t∈[0,T ]
dr 0
¯ α˜ , G(r, dξ )G(r, dη) f x − x¯ − (ξ − η) − f (ξ − η) C|x − x|
R3 R3
with α˜ ∈ ]0, (2 − β) ∧ 1[. From this and the properties (8), (14), we obtain the upper bound C|x − x| ¯ α˜ . Hence we conclude ¯ C|x − x| ¯ α, sup S3 (t, x, x)
t∈[0,T ]
for any α ∈ ]0, γ1 ∧ γ2 ∧
2−β 2
∧
1+δ 2 [.
(56)
With (52)–(56), we have proved (51).
The rest term in the space expansion. The contribution of the second order term in (50) comes from the estimate 2 E g u(t, ˆ x, x) ¯ u(t, x) ¯ − u(t, x) C g ∞ h2α , which is a consequence of (7). Hence, we have proved that for any fixed y ∈ R3 the mapping x ∈ D → pt,x (y) is Hölder 1+δ continuous of degree α ∈ ]0, γ1 ∧ γ2 ∧ 2−β 2 ∧ 2 [, uniformly in t ∈ ]0, T ]. The proof of the theorem is now complete.
2
Acknowledgments This paper has been written when the author was visiting the Institute Mittag-Leffler at Djursholm (Sweden) during a semester devoted to SPDEs. She would like to express her gratitude for the inspiring environment, the very kind hospitality and the financial support provided by this institution. References [1] V. Bally, E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998) 27–64. [2] R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s, Electron. J. Probab. 4 (1999). [3] R.C. Dalang, N.E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1) (1998) 187–212.
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[4] R.C. Dalang, C. Mueller, Some non-linear SPDE’s that are second order in time, Electron. J. Probab. 8 (1) (2003) 1–21. [5] R.C. Dalang, M. Sanz-Solé, Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press. [6] S. Kusuoka, D. Stroock, Applications of Malliavin calculus, Part II, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 32 (1985) 1–76. [7] A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab. 27 (1999) 803–844. [8] A. Millet, M. Sanz-Solé, Approximation and support theorem for a wave equation in two space dimensions, Bernoulli 6 (5) (2000) 887–915. [9] A. Millet, P.-L. Morien, On a stochastic wave equation in two space dimensions: Regularity of the solution and its density, Stochastic Process. Appl. 86 (1) (2000) 141–162. [10] P.-L. Morien, The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation, Bernoulli 5 (2) (1999) 275–298. [11] D. Nualart, The Malliavin Calculus and Related Topics, second ed., Probab. Appl. (N. Y.), Springer-Verlag, Berlin, 2006. [12] D. Nualart, L. Quer-Sardanyons, Existence and smoothness of the density for spatially homogeneous SPDEs, Potential Anal. 27 (2007) 281–299. [13] L. Quer-Sardanyons, M. Sanz-Solé, Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal. 206 (1) (2004) 1–32. [14] L. Quer-Sardanyons, M. Sanz-Solé, A stochastic wave equation in dimension 3: Smoothness of the law, Bernoulli 10 (1) (2004) 165–186. [15] M. Sanz-Solé, Malliavin Calculus with Applications to Stochastic Partial Differential Equations, Fund. Sci. Math., EPFL Press, 2005, distributed by CCR Press. [16] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. [17] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./SpringerVerlag, Bombay, 1984.
Journal of Functional Analysis 255 (2008) 283–312 www.elsevier.com/locate/jfa
Gaussian bounds for degenerate parabolic equations D. Cruz-Uribe a,∗ , Cristian Rios b a Department of Mathematics, Trinity College, Hartford, CT 06106-3100, USA b University of Calgary, Calgary, AB T2N1N4, Canada
Received 5 June 2007; accepted 29 January 2008 Available online 15 May 2008 Communicated by Daniel W. Stroock
Abstract Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A2 or QC class. We show that there is a heat kernel Wt (x, y) associated to the parabolic equation wut = div A∇u, and Wt satisfies classic Gaussian bounds: 2 Wt (x, y) C1 exp −C2 |x − y| . t t n/2 We then use this bound to derive a number of other properties of the kernel. © 2008 Published by Elsevier Inc. Keywords: Kernel; Gaussian bounds; Degenerate elliptic; Degenerate parabolic
1. Introduction 1.1. Overview of the problem In this paper we study the degenerate parabolic equation div A∇u = w
∂u . ∂t
* Corresponding author.
E-mail addresses:
[email protected] (D. Cruz-Uribe),
[email protected] (C. Rios). 0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.01.017
(1.1)
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where A = (Aij (x))ni,j =1 is a matrix of complex-valued, measurable functions satisfying the degenerate ellipticity condition ⎧ ⎪ ⎪ ⎨
λw(x)|ξ |2 ReAξ, ξ = Re
⎪ ⎪ ⎩
Aξ, η Λw(x)|ξ ||η|,
n
Aij (x)ξj ξ¯i ,
i,j =1
(1.2)
for some λ, Λ, 0 < λ Λ < ∞, and all ξ, η ∈ Cn . The weight w that controls the degeneracy is a non-negative, locally integrable function which we assume is either in the Muckenhoupt class A2 or in the class of QC weights, which arise in the study of quasi-conformal mappings. For ease of reference we make the following definition. Definition 1.1. Let En (w, λ, Λ) denote the class of n×n matrices of complex-valued, measurable functions satisfying the degenerate ellipticity condition (1.2). Remark 1.2. If A ∈ En (w, λ, Λ), then so is its adjoint, A∗ . Our goal is to show that if A is real symmetric, then there exists a heat kernel associated to this equation: an L∞ function Wt (x, y) such that given a function f ∈ Cc∞ ,
u(x, t) =
Wt (x, y)f (y) dy
(1.3)
Rn
is a solution of (1.1) satisfying the initial condition u(x, 0) = f (x). Further, we will show that the heat kernel satisfies Gaussian bounds: 2 Wt (x, y) C1 exp −C2 |x − y| . t t n/2
(1.4)
A central feature of our results is that the weight does not appear in (1.3) or in (1.4)—the kernel is defined with respect to Lebesgue measure. When w ≡ 1 (that is, A is uniformly elliptic), these results are well known. For the existence of the heat kernel, see Friedman [18]; Gaussian bounds are due to Aronson [1]. Solutions of Eq. (1.1) with A real symmetric were first treated by Chiarenza and Frasca [8]. Chiarenza and Serapioni [11], showed that if n 3 and w ∈ A2 , then solutions of (1.1) satisfied a Harnack inequality on standard parabolic cylinders. Chiarenza and Franciosi [7] proved the same result for n 2 and w a QC weight. (See Proposition 3.8 below.) More recently, Ishige [24] has proved a Harnack inequality and continuity of the solution for a more general parabolic equation with lower order terms, and for A any real matrix that satisfies (1.2) with w ∈ A2 . When A is a complex matrix, much less is known even when A is uniformly elliptic. Auscher [2] showed that Gaussian bounds hold for L∞ perturbations of real symmetric matrices. Auscher, McIntosh and Tchamitchian [5] showed that Gaussian bounds hold if A satisfies (1.2) and is Hölder continuous, and when n 2 smoothness is not necessary. Ouhabaz [27] has shown that if A is a Lipschitz perturbation of a real, uniformly elliptic matrix, then (1.4) holds. (Note that neither of the last two results assumes that A is symmetric.) On the other hand,
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Auscher, Coulhon and Tchamitchian [4], building on an example in [26], showed that if n 5, then Gaussian bounds for the heat kernel need not hold in general. Given a degenerate elliptic matrix A, it might seem more natural to consider the usual parabolic equation ∂u . ∂t
div A∇u =
(1.5)
When w ∈ A2 this equation has been studied by several authors; their results indicate that, surprisingly, (1.1) is the “right” generalization to consider. Chiarenza and Serapioni [9,10] showed that the assumption w ∈ A2 did not imply local regularity of the solution. Local regularity required higher integrability of the weight w −1 ; e.g., the stronger hypothesis w ∈ A1+2/n , n 3. Further, they showed that in this case solutions to (1.5) satisfied a Harnack inequality on the weighted parabolic cylinders Qx,t (r) = B2r (x) × (t − hx (r), t), where
−n/2
hx (r) =
w(y)
2/n dy
.
Br (x)
Gutiérrez and Nelson [22] showed that with the same assumption the heat kernel associated with Eq. (1.5), W˜ t (x, y), satisfied a weighted Gaussian estimate: there exist C1 , C2 , α > 0 such that α W˜ t (x, y) C1 exp −C2 hx (|x − y|) . (1.6) n t h−1 x (t) A sharper but more complicated version of this inequality was later proved by Gutiérrez and Wheeden [23]. 1.2. Statement of the main results Hereafter, let w be either an A2 weight or a QC weight. If w ∈ QC, then we will also assume n 2; otherwise we can take n 1. Define the differential operator Lw = −w −1 div A∇, and let e−t Lw be the semigroup generated by Lw . Given any f in the domain of e−t Lw , u(x, t) = e−t Lw f (x) is a solution of the initial value problem ⎧ ⎨ ∂u = −Lw u, ∂t ⎩ u(x, 0) = f (x).
(1.7)
Our main result is the following. Theorem 1.3. If A ∈ En (w, λ, Λ) is real symmetric, then there exists a heat kernel Wt (x, y) associated to the operator e−t Lw such that for all f ∈ Cc∞ , e−t Lw f (x) =
Wt (x, y)f (y) dy. Rn
(1.8)
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Furthermore, for all t > 0 and x, y ∈ Rn , the kernel Wt satisfies the Gaussian bounds 2 Wt (x, y) C1 exp −C2 |x − y| , t t n/2
(1.9)
and the Hölder continuity estimates μ |h| |x − y|2 Wt (x + h, y) − Wt (x, y) C1 , exp −C 2 t t n/2 t 1/2 + |x − y| μ |h| |x − y|2 Wt (x, y + h) − Wt (x, y) C1 , exp −C2 t t n/2 t 1/2 + |x − y|
(1.10) (1.11)
where h ∈ Rn is such that 2|h| t 1/2 + |x − y|. The constants C1 , C2 and μ depend only on n, w, λ, and Λ. Remark 1.4. Perhaps the most outstanding aspect of our results is that they appear to be independent of the weight. Thus the kernel Wt (x, y) is integrated with respect to Lebesgue measure and the upper bounds (1.9) on Wt are the classic Gaussian bounds for the heat kernel [1]. This is in sharp contrast to the weighted inequalities (1.6) obtained in [22,23] for a related class of degenerate parabolic equations. The classical nature of our estimates will be very useful for the weighted Kato square root problem, among other applications. Remark 1.5. We believe that Theorem 1.3 should hold for all real matrices (i.e., not necessarily symmetric) that satisfy (1.2). Towards proving this, we remark that the approach taken by Ouhabaz [27] for uniformly elliptic matrices can be adapted to the case of degenerate matrices. However, this technique relies on the Sobolev embedding theorem, and in the weighted case this result is only true on bounded domains (see [17]). Central to the proof of Theorem 1.3 is an estimate which is of independent interest. When w ≡ 1 this inequality appears in Auscher et al. [6], without proof; they note that in the special case of the Laplace–Beltrami operator the result is due to Gaffney [19] and that it can be proved by modifying an argument due to Davies [14]. Theorem 1.6. Given A ∈ En (w, λ, Λ), let E and F be two closed subsets of Rn and set d = dist(E, F ). Then there exist constants c, C > 0 such that given any function f ∈ L2 (w) with supp(f ) ⊂ E, −t L e w f
L2 (w,F )
Ce−cd
2 /t
f L2 (w,E) .
The constants depend only n, λ, and Λ. As a consequence of the Gaussian bounds we have the conservation property. Theorem 1.7. Given a matrix A ∈ En (w, λ, Λ), suppose that the heat kernel of the associated semigroup e−t Lw satisfies (1.4). Then e−t Lw 1 = 1, where 1 denotes the function on Rn which equals the constant 1.
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As another consequence we get bounds for the derivative of the semigroup. Theorem 1.8. With the same hypotheses as Theorem 1.7, the semigroup Vt , t 0, defined by ∂ −t Lw f (x) = Lw e−t Lw f (x), e Vt f (x) = − ∂t
is given by a heat kernel Vt (x, y): for all f ∈ Cc∞ ,
Vt f (x) =
Vt (x, y)f (y) dy. Rn
Furthermore, for all t > 0 and x, y ∈ Rn , Vt satisfies the Gaussian bounds Vt (x, y)
C1 t n/2+1
|x − y|2 exp −C2 , t
and the Hölder continuity estimates
|x − y|2 , exp −C2 t t n/2+1 μ |h| |x − y|2 Vt (x, y + h) − Vt (x, y) C1 , exp −C2 t t n/2+1 t 1/2 + |x − y|
Vt (x + h, y) − Vt (x, y)
C1
|h| t 1/2 + |x − y|
μ
where h ∈ Rn is such that 2|h| t 1/2 + |x − y|. Finally, Vt has zero integral: for all x ∈ Rn ,
Vt 1 =
Vt (x, y) dy = 0. Rn
Remark 1.9. In both Theorems 1.7 and 1.8 we do not assume that A is real symmetric; we only assume that the associated heat kernel satisfies Gaussian bounds. 1.3. Organization The remainder of this paper is organized as follows. In Sections 2 and 3 we gather some basic result about weighted Sobolev spaces and semigroups. In Section 4 we prove the Gaffney-type estimate in Theorem 1.6. In Section 5 we prove Theorems 1.3 and 1.8. In Section 6 we prove Theorem 1.7. Throughout this paper, all notation will be standard or defined as needed. Λ and λ will always denote the ellipticity constants in (1.2). Unless otherwise specified, C, c, etc., will denote an arbitrary constant which may depend on the dimension n, λ and Λ, and the A2 or QC constant associated with a weight w. Sometimes for clarity we will specify the dependence of the constant by writing C(n), C(n, w), etc. Given an angle θ , define the sector Σ(θ ) = {z ∈ C: z = 0, |arg(z)| < θ }. Finally, given a bounded operator T on L2 (w, Ω), let
T B(L2 (w,Ω)) denote its operator norm.
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2. Weighted Sobolev spaces The basic theory of weighted Sobolev spaces was developed by Fabes, Kenig and Serapioni [17] and we will follow their development. (See also Turesson [29].) One key difference, however, is that they only dealt with real-valued functions; complex-valued functions introduce a number of minor technical complications. 2.1. Weights By a weight we mean a non-negative, locally integrable function. Given a weight w and a measurable set E, we let
w(E) =
w(x) dx. E
We are concerned with two weight classes: the Muckenhoupt class A2 and the quasiconformal weights QC. The first class is straightforward to define; we give a slightly more general definition which we will need below. Given p, 1 < p < ∞, and a weight w, we say that w ∈ Ap if there exists a constant [w]Ap (referred to as the Ap constant of w) such that p−1
1 1 1−p sup w(x) dx w(x) dx = [w]Ap < ∞, |Q| |Q| Q Q
Q
where the supremum is taken over all cubes Q in Rn . Note that if w ∈ Ap , then it follows by a change of variables that for all a ∈ R, b ∈ Rn , wab (x) = w(ax + b) ∈ Ap , and [wab ]Ap = [w]Ap . Let A∞ denote the union of the Ap classes. The properties of these weights can be found in [16, 20,21]. To define the class QC, fix n 2 and let f : Rn → Rn be a bijection whose components fi have distributional derivatives in LnLoc . Let f (x) denote the Jacobian of f and let |f | denote its determinant. Then f is quasi-conformal if there exists a constant k such that
n ∂fi (x)2 ∂xj
1/2
1/n k f (x) .
i,j =1
Given such an f , the function w = |f |1−2/n is called a QC weight. We will denote the best constant k associated with f , and so with w, by [w]QC . As before we have that [wab ]QC = [w]QC . David and Semmes [12] showed that if w ∈ QC, then w ∈ A∞ , but QC weights have much more structure than an arbitrary A∞ weight. The classes QC and A2 are different. Thus, a QC weight need not be in A2 : for example, w(x) = |x|α(n−2) ∈ QC for all α > −1, but w is not in A2 if α n/(n − 2). Conversely, |x|β , −n < β −(n − 2), is in A2 but not in QC. (See [17, p. 106].)
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2.2. Weighted Sobolev spaces Given an open set Ω in Rn , and a weight w in either A2 or QC, let Lp (w, Ω), 1 p < ∞, be the Banach function space of complex-valued functions with norm
f
Lp (w,Ω)
=
f (x)p w(x) dx
1/p .
Rn
L2 (w, Ω) is a Hilbert space with inner product
f, gw = f (x)g(x)w(x) dx. Ω
When p = ∞, L∞ (w, Ω) consists of functions essentially bounded with respect to the measure w(x) dx; since w ∈ A∞ , L∞ (w, Ω) = L∞ (Ω). Define the Sobolev space H01 (w, Ω) to be the closure of Cc∞ (Ω) with respect to the norm
f H 1 (w,Ω) = f L2 (w,Ω) + ∇f L2 (w,Ω) . 0
(If Ω = Rn , then we write simply L2 (w) and H01 (w).) That this closure is well defined is a consequence of the following lemma [17, pp. 90, 106]. Lemma 2.1. Given a weight w in A2 or QC, suppose {gk } ⊂ Cc∞ (Ω) is such that
gk L2 (w,Ω) → 0. If g is such that ∇gk − g L2 (w,Ω) → 0, then g = 0. Thus, if f ∈ H01 (w, Ω), there exists a vector-valued function in L2 (w, Ω), denoted ∇f , and there exists {gk } ⊂ Cc∞ (Ω) such that in L2 (w, Ω), {gk } converges to f and {∇gk } converges to ∇f . If w ∈ A2 , w −1 is locally integrable, so by Hölder’s inequality, f, ∇f ∈ L1Loc (Ω). Hence, we can integrate by parts: for all φ ∈ Cc∞ (Ω),
∇f (x) φ(x) dx = − f (x) ∇φ(x) dx. Ω
Ω
However, if w ∈ QC, then f and ∇f need not be locally integrable, so this formula does not make sense. This fact complicates the proof of some of the differentiation formulas given below. To establish additional details about the structure of H01 (w, Ω), note first that if f = u + iv ∈ 1 H0 (w, Ω), then by passing to the real and imaginary parts of an approximating sequence of Cc∞ functions, we have that u, v ∈ H01 (w, Ω). Lemma 2.2. Let θ : R2 → C have continuous partial derivatives and be such that |∇θ | M. If f = u + iv ∈ H01 (w, Ω), then the function θ (f ) = θ (u, v) ∈ H01 (w, Ω) and almost everywhere in Ω, ∂u ∂v ∂θ (f ) ∂θ ∂θ (u, v) (u, v) = + , ∂xi ∂s ∂xi ∂t ∂xi
1 i n.
(2.1)
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Proof. Let {φk } ⊂ Cc∞ (Ω) be such that φk → f in H01 (w, Ω). By passing to a subsequence, we may assume that it converges pointwise almost everywhere as well. For each k 1, θ (φk ) is continuously differentiable and has compact support in Ω, and so is in H01 (w, Ω). Let uk = Re(φk ), vk = Im(φk ). By the chain rule, for 1 i n, ∂θ (φk ) ∂θ ∂uk ∂θ ∂vk = + . (uk , vk ) (uk , vk ) ∂xi ∂s ∂xi ∂t ∂xi By assumption, θ is Lipschitz, so |θ (φk )(x) − θ (f )(x)| M|φk (x) − f (x)|. Hence,
θ (φk )(x) − θ (f )(x)2 w(x) dx
Ω
1/2
M
φk (x) − f (x)2 w(x) dx
1/2 .
Ω
Since φk → f in L2 (w, Ω), we have that θ (φk ) → θ (f ) in L2 (w, Ω). Since u, v ∈ L2 (w, Ω) and ∇θ ∈ L∞ , the right-hand side of (2.1) is in L2 (w, Ω), 1 i n. Temporarily denote it by Fi . Then ∂θ (φk ) ∂θ ∂u ∂θ ∂u ∂uk ∂θ ∂uk ∂θ ∂x − Fi ∂s (uk , vk ) ∂x − ∂s (u, v) ∂x + ∂t (uk , vk ) ∂x − ∂t (u, v) ∂x i i i i i = Ak + Bk . To complete the proof it will suffice to show that Ak , Bk → 0 in L2 (w, Ω). For in that case, since H01 (w, Ω) is complete we have that θ (f ) ∈ H01 (w, Ω) and (2.1) holds. We will show Ak converges to 0; the proof for Bk is identical. We have that ∂θ ∂u ∂uk ∂θ ∂u ∂θ ∂u ∂θ A (uk , vk ) (uk , vk ) (uk , vk ) (u, v) + − − ∂s ∂xi ∂s ∂xi ∂s ∂xi ∂s ∂xi ∂u ∂uk ∂u ∂θ ∂θ . + (uk , vk ) − (u, v) M − ∂x ∂x ∂s ∂s ∂x i
i
i
Since u ∈ H01 (w, Ω), the first term tends to 0 in L2 (w, Ω). The second term is dominated point∂u |. Further, since φk → f pointwise and θ has continuous partial derivatives, the wise by 2M| ∂x i second term tends to 0 pointwise. Therefore, by the dominated convergence theorem it also tends to 0 in L2 (w, Ω), and we are done. 2 Lemma 2.3. Let f = u + iv ∈ H01 (w, Ω). Then |f | ∈ H01 (w, Ω) and for 1 i n and almost every x ∈ Ω, ∂u ∂v u(x) ∂x (x) + v(x) ∂x (x) ∂|f | i i χ{f (x) =0} . (x) = ∂xi |f (x)|
(2.2)
√ Proof. For each > 0, define the function θ (s, t) = s 2 + t 2 + 2 − . Then θ has continuous, bounded derivatives, and so by Lemma 2.2, θ (f ) ∈ H01 (w, Ω) and for 1 i n, ∂u ∂v ∂θ (f ) u(x) ∂xi (x) + v(x) ∂xi (x) = ∂xi u(x)2 + v(x)2 + 2
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almost everywhere. Since θ (f )(x) |f (x)| and θ (f ) → |f | pointwise, by the dominated convergence theorem it converges in L2 (w, Ω). ∂u ∂u Denote the right-hand side of (2.2) by Fi . Then |Fi | | ∂x (x)| + | ∂x (x)| ∈ L2 (w, Ω). Furi i
(f ) (f ) ther, we have that | ∂θ∂x | |Fi | and | ∂θ∂x | converges pointwise to Fi . So again by dominated i i 2 convergence it converges in L (w, Ω). This yields the desired result. 2
Corollary 2.4. If f ∈ H01 (w, Ω) is real-valued, then f + = max(f, 0) and f − = min(f, 0) are both in H01 (w, Ω), and for 1 i n, ∂f + ∂f = χ{f >0} , ∂xi ∂xi
∂f − ∂f = χ{f <0} . ∂xi ∂xi
Proof. It suffices to note that f + = (|f | + f )/2 and f − = (|f | − f )/2 and then apply Lemma 2.3. 2 3. Semigroup theory In this section we state some basic properties of semigroup theory and use these to construct a weak solution to (1.1). We follow the approach given by Ouhabaz [27] and we refer the reader to this work and to Kato [25] for further information. Throughout this section A ∈ En (w, λ, Λ) and w is an A2 or QC weight. 3.1. Sesquilinear forms and the associated operators Definition 3.1. For all f, g ∈ H01 (w, Ω) define the sesquilinear form a(f, g) by
a(f, g) =
A∇f (x) · ∇g(x) dx.
(3.1)
Ω
Since Cc∞ (Ω) is dense in H01 (w, Ω), the sesquilinear form a is densely defined on L2 (w, Ω). We will sometimes denote H01 (w, Ω), the domain of a, by D(a). Proposition 3.2. The sesquilinear form a has the following properties: (1) a is accretive: for all f ∈ D(a), Re a(f, f ) 0. (2) a is continuous: |a(f, g)| M f a g a , where the norm is defined by f a = Re a(f, f ) + f 2L2 (w,Ω) . (3) a is closed: (D(a), · a ) is a complete space. (4) a is sectorial: there exists ω ∈ (0, π2 ) such that for all f ∈ D(a), Im a(f, f ) tan(ω)Re a(f, f ).
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Proof. Properties (1), (2) and (4) are immediate consequences of the ellipticity condition (1.2): for all f, g ∈ D(a),
2 Re a(f, f ) λ ∇f (x) w(x) dx 0, Ω
and a(f, g) Λ
∇f (x)∇g(x)w(x) dx
Ω
Λ
∇f (x)2 w(x) dx
Ω
Λ Re λ
1/2
∇g(x)2 w(x) dx
Ω
1/2
A∇f (x) · ∇f (x) dx
1/2
A∇g(x) · ∇g(x) dx
Re
Ω
1/2
Ω
Λ
f a g a . λ
Further,
Im a(f, f ) Λ
∇f (x)2 w(x) dx
Ω
Λ λ
Re A∇f (x) · ∇f (x) dx =
Λ Re a(f, f ). λ
Ω
This yields (4) with ω = arctan(Λ/λ). The proof of (3) follows from the fact that H01 (w, Ω) is a complete space with respect to the norm · H 1 (w,Ω) , and from the fact that · a ≈ · H 1 (w,Ω) , since by the ellipticity condi0
tion (1.2), Re a(f, f ) ≈ ∇f 2L2 (w,Ω) .
0
2
The properties in Proposition 3.2 allow us to use the abstract theory of sesquilinear forms. This immediately yields that there exists a densely defined operator Lw on L2 (w, Ω) such that for all f ∈ D(Lw ), g ∈ D(a), a(f, g) = Lw f, gw . Furthermore, D(Lw ) is dense in D(a) with respect to the norm · a . (See [27, Proposition 1.22].) Note that if A and f, g are smooth functions, then by integration by parts we have that Lw = −w −1 div A∇. Define the adjoint form a∗ by ∗
a (f, g) = a(g, f ) =
A∗ ∇f (x) · ∇g(x) dx.
Ω
Since A∗ ∈ En (w, λ, Λ), everything we have said above holds for a∗ . In particular, we have the densely defined operator L∗w which is the adjoint of Lw .
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The classes En (w, λ, Λ) are preserved under (small) rotations. More precisely, we have the following lemma. Lemma 3.3. Let A ∈ En (w, λ, Λ), and let ω = arctan(Λ/λ), τ = arctan √
λ . Λ2 −λ2
If z ∈ Σ(τ ),
then zA ∈ En (w, λz , Λz ), where λz = λ|z|[cos(arg(z)) − sin(arg(z))/ tan(τ )] and Λz = Λ|z|. Proof. First note that λz > 0: cos arg(z) − sin arg(z) / tan(τ ) = cos arg(z) 1 − tan arg(z) / tan(τ ) > cos arg(z) 1 − tan(τ )/ tan(τ ) = 0. Since A ∈ En (w, λ, Λ), for all ξ ∈ Cn , 2 2 2 ReAξ, ξ + ImAξ, ξ = Aξ, ξ Λ2 w(x)2 |ξ |4 , 2 ReAξ, ξ λ2 w(x)2 |ξ |4 . Together, these imply that ImAξ, ξ Λ2 − λ2 w(x)|ξ |2 , and so RezAξ, ξ = |z| cos(arg z) ReAξ, ξ − sin(arg z) ImAξ, ξ |z| cos(arg z)λw(x)|ξ |2 − sin(arg z) Λ2 − λ2 w(x)|ξ |2 Λ2 2 −1 = λ|z|w(x)|ξ | cos(arg z) − sin(arg z) λ2 λ|z|w(x)|ξ |2 cos arg(z) − sin arg(z) / tan(τ ) . On the other hand it is immediate that for all ξ , η, |zAξ, η| Λ|z|w(x)|ξ ||η|.
2
3.2. Resolvents and semigroups Though the operator Lw is not bounded, there are two closely related bounded operators associated to it: the resolvent (λI + Lw )−1 and the semigroup e−t Lw . The operator Lw is the infinitesimal generator of a strongly continuous, contraction semigroup e−t Lw . The range of e−t Lw is D(Lw ). Further, e−t Lw is a holomorphic semigroup in the sector Σ(π/2 − ω), where ω = arctan(Λ/λ). (See [27, Theorem 1.52].) Proposition 3.4. If the matrix A is real, then the semigroup e−t Lw is real and positive: if f ∈ L2 (w, Ω) is real-valued, then so is e−t Lw f ; if f is non-negative, then so is e−t Lw f .
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Proof. By [27, Theorem 2.5], e−t Lw is real if given any f = u + iv, a(u, v) ∈ R. This is immediate from the definition of a given that A is real. By [27, Theorem 2.6], e−t Lw is positive if, in addition, u+ , u− ∈ H01 (w, Ω), and a(u+ , u− ) 0. The first condition is just Corollary 2.4. The second also follows from Corollary 2.4: ∇u+ and ∇u− have disjoint supports, so a(u+ , u− ) = 0. 2 For every t > 0, tI + Lw is an invertible operator from D(Lw ) into L2 (w, Ω). Its inverse, the resolvent of Lw , (tI + Lw )−1 , is a bounded operator on L2 (w, Ω) satisfying t (tI + Lw )−1 2 B(L (w,Ω)) 1.
(3.2)
An important consequence of the fact that e−t Lw is a holomorphic semigroup is the fact that the resolvent is bounded on a sector of the complex plane larger than the half-plane {Re z > 0}. (See [27, Theorem 1.45].) Proposition 3.5. Let ω = arctan(Λ/λ). Then for every θ ∈ (π/2, π − ω) there exists Mθ > 0 such that sup z(zI + Lw )−1 B(L2 (w,Ω)) Mθ .
(3.3)
z∈Σ(θ)
The next proposition shows that e−t Lw and (zI + Lw )−1 can be characterized in terms of one another. (See [25, pp. 482–489].) Proposition 3.6. Let ω = arctan(Λ/λ). Then for f ∈ L2 (w, Ω) and z ∈ Σ(π/2 − ω), e
−zLw
1 f= 2πi
ezζ (ζ I + Lw )−1 f dζ,
(3.4)
Γ
where the path Γ is the union of the rays γ ± = {z ∈ C: z = re±iθ , r R > 0} and the arc γ0 = {z ∈ C: z = Reiψ , |ψ| θ }, going around the origin counter-clockwise, and π − ω > θ > π/2 + |arg(z)|. Further, if Re(z) > 0, −1
(zI + Lw )
∞ =
e−zt e−t Lw dt.
(3.5)
0
More generally, if z ∈ Σ(π − ω), then this identity remains true if we instead integrate over the ray t = reiθ , r > 0, where θ is such that e−t Lw is bounded and Re(zt) > 0. Inequality (3.5) is referred to as the Laplace identity. Another consequence of the fact that e−t Lw is a holomorphic semigroup is that the operator
∂ −t Lw ∂ −t Lw e e f = −Lw e−t Lw f f= ∂t ∂t
(3.6)
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295
is again a bounded operator on L2 (w, Ω). More precisely, there exists C > 0 such that for all t > 0, Lw e−t Lw
B(L2 (w,Ω))
C . t
(See Yosida [30, p. 239].) ∗ Finally, we note that the operators e−¯zLw and (¯zI + L∗w )−1 are the adjoints of e−zLw and (zI + Lw )−1 , respectively. 3.3. Solutions of the parabolic equation Here we define a weak solution to the parabolic equation (1.7) and show that the semigroup e−t Lw yields a solution. Our definition is the one given by Chiarenza and Frasca [8]. Fix T > 0; we say that u ∈ L2 ([0, T ], H01 (w, Ω)) if for any t ∈ [0, T ] the function u(·, t) ∈ H01 (w, Ω) and
T
u(·, t) 2
H01 (w,Ω)
dt < ∞.
0
We define u ∈ L2 ([0, T ], L2 (w, Ω)) in the same way, but with the H01 norm replaced by the L2 norm. We say that a function v ∈ W0 ([0, T ]) if v ∈ L2 ([0, T ], H01 (w, Ω)), vt ∈ L2 ([0, T ], L2 (w, Ω)), and v(x, 0) = v(x, T ) = 0, x ∈ Ω. A function u is a weak solution of (1.1) if u ∈ L2 ([0, T ], H01 (w, Ω)) and for all v ∈ W0 ([0, T ]),
T
A∇u(x) · ∇v(x) − w(x)u(x)vt (x) dx dt = 0.
(3.7)
0 Ω
Proposition 3.7. Given f ∈ L2 (w, Ω), define the function u(x, t) = e−t Lw f (x), t > 0. Then u ∈ L2 ([0, T ], L2 (w, Ω)) and (3.7) holds. Since u(x, 0) = f (x), u is a solution of (1.7). Proof. We first show that u ∈ L2 ([0, T ], L2 (w, Ω)).
T
T
u(·, t) 2
dt C H 1 (w,Ω) 0
u(x, t)2 w(x) dx dt + C
0 Ω
0
T
∇u(x, t)2 w(x) dx dt.
0 Ω
We estimate each integral separately. The first is straightforward: since u = e−t Lw f and the semigroup is a contraction on L2 (w, Ω),
T
0 Ω
u(x, t)2 w(x) dx dt
T
f (x)2 w(x) dx dt T f 2 2
L (w,Ω)
0 Ω
< ∞.
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To estimate the second integral, note that u(·, t) = e−t Lw f (·) ∈ D(Lw ), and so a(u, u) = Lw u, uw . Therefore, by the ellipticity conditions (1.2), again using the fact that e−t Lw is a contraction,
T
∇u(x, t)2 w(x) dx dt 1 λ
0 Ω
T
Re a u(·, t), u(·, t) dt
0
=
1 λ
1 = λ
T
Lw u(x, t)u(x, t)w(x) dx dt
Re 0
Ω
T
Re
Lw e−t Lw f (x)e−t Lw f (x)w(x) dx dt
Ω
0
T
2 −1 ∂ −t Lw Re e f (x) w(x) dx dt 2λ ∂t
Ω 0 1 Re f (x)2 − e−T Lw f (x)2 w(x) dx = 2λ
=
Ω
C f 2L2 (w) < ∞. To show that u satisfies (3.7), first note that ut = −Lw e−t Lw f = −Lw u. Now fix v ∈ W0 ([0, T ]). Then, since v(x, 0) = v(x, T ) = 0, by Fubini’s theorem and integration by parts in t,
T
w(x)u(x, t)vt (x, t) dx dt = 0 Ω
T u(x, t)v(x, t)|T0
−
Ω
ut (x, t)v(x, t) dt w(x) dx 0
T =
Lw u(x, t)v(x, t) dt w(x) dx Ω 0
T
a u(·, t), v(·, t) dt
= 0
T
=
A∇u(x, t) · ∇v(x, t) dx dt. 0 Ω
Eq. (3.7) now follows immediately.
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297
Finally, we give a precise statement of the Harnack inequality. Given a pair (x0 , t0 ), x0 ∈ Ω, t > 0, define the parabolic cylinders Qρ (x0 , t0 ) = (x, t): |t − t0 | < ρ 2 , |x − x0 | < 2ρ , 3 2 ρ + 2 Qρ (x0 , t0 ) = (x, t): ρ < t − t0 < ρ , |x − x0 | < , 4 2 3 2 1 2 ρ − . Qρ (x0 , t0 ) = (x, t): − ρ < t − t0 < − ρ , |x − x0 | < 4 4 2 Proposition 3.8. Let A ∈ En (w, λ, Λ) be real symmetric with w ∈ A2 (n 1) or w ∈ QC (n 2). Then there exists γ = γ (n, λ, Λ, w) > 0 such that if u(x, t) is a non-negative solution of (1.1) in Qρ (x0 , t0 ), ρ > 0, then sup
Q− ρ (x0 ,t0 )
u(x, t) γ
inf
Q+ ρ (x0 ,t0 )
u(x, t).
Proof. When w ∈ QC and n 2 this result is due to Chiarenza and Franciosi [7]. When w ∈ A2 and n 3, it is due to Chiarenza and Serapioni [11]; the cases n = 1 and 2 for w ∈ A2 follow from the higher-dimensional case via an argument shown to the second author by M. Safonov. Here we sketch the details for n = 2; the case n = 1 is treated in essentially the same way. Let u(x, y, t) be a non-negative solution of (1.1) in Qρ (x0 , y0 , t0 ), ρ > 0, i.e. div A∇u(x, y, t) = ∂t u(x, y, t), ˜ where A ∈ E2 (w, λ, Λ). For ρ z 3ρ define v(x, y, z, t) = zu(x, y, t) and let A(x, y, z) be the 3 × 3 matrix ˜ A(x, y, z) =
A(x, y) 0
0 w(x, y)
.
Then a short calculation shows that v is a solution of the three-dimensional parabolic equation ˜ is ˜ (x,y,z) v = vt in the set Qρ (x0 , y0 , 2ρ, t0 ) ⊂ Qρ (x0 , y0 , t0 ) × (ρ, 3ρ). The matrix A div(x,y,z) A∇ 3 ˜ ˜ λ, Λ), where w(x, ˜ y, z) = w(x, y) ∈ A2 (R ). Further, v is positive real symmetric and A ∈ E3 (w, in Qρ (x0 , y0 , 2ρ, t0 ). Therefore, by the Harnack inequality in dimension n = 3, sup
Q− ρ (x0 ,y0 ,2ρ,t0 )
v(x, y, z, t) γ
inf
Q+ ρ (x0 ,y0 ,2ρ,t0 )
v(x, y, z, t),
and so sup
Q− ρ (x0 ,y0 ,t0 )
u(x, y, t) 3γ
inf
Q+ ρ (x0 ,y0 ,t0 )
u(x, y, t).
2
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4. Proof of Theorem 1.6 The proof of Theorem 1.6 is based on the following lemma. It is a generalization to degenerate elliptic operators and complex time of a result due to Auscher et al. [6, Lemma 2.1], and our proof is based on theirs. Throughout this section we assume that w ∈ A2 or QC and A ∈ En (w, λ, Λ). Lemma 4.1. Let E and F be two closed sets in Rn and let d = dist(E, F ). Let τ = arctan √
λ Λ2 −λ2
and fix ν, 0 < ν < π/2 + τ , and z ∈ Σ(ν). Then there exist positive constants C and c depending on n, Λ, λ, ν such that for all f ∈ L2 (w) with support in E,
√ z(zI + Lw )−1 f (x)2 w(x) dx Ce−cd |z| f (x)2 w(x) dx. (4.1) F
E
Proof. We consider first the case where ν < π/2. Without loss of generality we may also assume ν > π/4, so tan(ν) > 1. Since z(zI + Lw )−1 = (I + z−1 Lw )−1 and |z| > 0, if we make the change of variables z → z−1 , then to establish (4.1) it will suffice to prove
d (I + zLw )−1 f (x)2 w(x) dx Ce−c √|z| f (x)2 w(x) dx. (4.2) F
E
Define uz = (I + zLw )−1 f ; then f = uz + zLw uz . By the definition of Lw , if v ∈ H01 (w), then Lw uz , vw = a(uz , v), so
uz (x)v(x)w(x) dx + z A∇uz (x) · ∇v(x) dx = f (x)v(x)w(x) dx. Rn
Rn
Rn
Let v = η2 ut , where η ∈ C0∞ is a non-negative function with supp(η) ∈ Rn \ E that will be fixed below. Since f and η have disjoint supports, the right-hand side is zero. Hence, if we rearrange terms we get that
z 2 u (x) η(x)2 w(x) dx + z
Rn
= −2z
A∇uz (x) · ∇uz (x)η(x)2 dx
Rn
η(x)uz (x)A∇uz (x) · ∇η(x) dx.
(4.3)
Rn
Now take the absolute value of both sides of (4.3). Since A ∈ En (w, λ, Λ), we can apply (1.2) and Young’s inequality to get for any > 0 that
2z η(x)uz (x)A∇uz (x) · ∇η(x) dx Rn
2|z| Rn
η(x)uz (x)A∇uz (x) · ∇η dx
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2|z|Λ
η(x)uz (x)∇uz (x)∇η(x)w(x) dx
Rn
|z|Λ
299
z 2 u (x) ∇η(x)2 w(x) dx + |z|Λ
2 η(x)2 ∇uz (x) w(x) dx.
(4.4)
Rn
To estimate the absolute value of the left-hand side of (4.3), let
R=
z 2 u (x) η(x)2 w(x) dx,
n s + it = z,
S + iT =
Rn
A∇uz (x) · ∇uz (x)η(x)2 dx. R
Clearly, R 0, and since z ∈ Σ(ν), s > 0 and |t| tan(ν)s. Again by (1.2),
η(x) Re A∇uz (x) · ∇uz (x) dx λ
S=
2
Rn
2 η(x)2 ∇uz (x) w(x) dx 0,
Rn
and
|T |
η(x)2 Im A∇uz (x) · ∇uz (x) dx Λ
Rn
2 Λ η(x)2 ∇uz (x) w(x) dx S. λ
Rn
Let γ = λ(tan(ν)Λ)−1 < 1. Then if we combine these inequalities we get that the absolute value of the left-hand side of (4.3) is equal to
1/2 (R + sS − tT )2 + (sT + tS)2 1/2 R 2 + s 2 S 2 + t 2 T 2 − 2R|tT | + 2RsS + t 2 S 2 1/2 R 2 + t 2 T 2 − 2(1 − γ )R|tT | + |z|2 S 2 1/2 γ R 2 + γ t 2 T 2 + |z|2 S 2 √ γ |z| R + S. 2 2
If we now combine this estimate with (4.4) we get that √
2 γ z 2 |z|λ 2 u (x) η(x) w(x) dx + η(x)2 ∇uz (x) w(x) dx 2 2 Rn
|z|Λ
Rn
z 2 u (x) ∇η(x)2 w(x) dx + |z|Λ
Rn
2 η(x)2 ∇uz (x) w(x) dx.
(4.5)
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λ Let = 2Λ ; then the second term on each side of the inequality cancel. Now let η = eαθ − 1, where supp(θ ) ⊂ Rn \ E, 0 θ 1, θ = 1 on F , ∇θ ∞ Kd , and
1/2 √ λ γ d α= √ . 2 2 16Λ K |z| Then (4.5) yields √
√
2 2 γ 2αθ(x) γ e − 2eαθ(x) + 1 uz (x) w(x) dx e2αθ(x) uz (x) w(x) dx. 2 8 Rn
Rn
If we discard the integral of |uz |2 w on the left-hand side and rearrange terms, we get that
2 2 8 e2αθ(x) uz (x) w(x) dx eαθ(x) uz (x) w(x) dx. 3 Rn
Rn
Since θ 1 and θ = 1 on F , this yields
z 2 u (x) w(x) dx 8 e−α uz (x)2 w(x) dx. 3 Rn
F
Since uz = (I + zLw )−1 f and by Proposition 3.5 the resolvent is bounded on L2 (w) with a constant that depends on ν, inequality (4.2) follows immediately. Now suppose that ν ∈ (π/2, π/2 + τ ). In this case, we can find ν < π/2 and τ < τ such that z = z ζ , where |ζ | = 1, | arg(ζ )| < τ , and z ∈ Σ(ν ). Then we can rewrite the left-hand side of (4.2) as
(I + zLw )−1 f (x)2 w(x) dx = I + z L −1 f (x)2 w(x) dx, w F
F
where L w = ζ Lw is the densely defined operator associated to the sesquilinear form generated by the matrix ζ A. By Lemma 3.3, there exist 0 < λζ < Λζ such that ζ A ∈ En (w, λζ , Λζ ). Therefore, L w and its resolvent have all the properties of Lw that we used in the above argument, so we can repeat it to get the desired inequality. 2 Remark 4.2. Since A∗ ∈ En (w, λ, Λ), Lemma 4.1 holds for L∗w with the same constants. We will now prove Theorem 1.6. Fix t > 0; without loss of generality we may assume that d 2 t. By Proposition 3.6,
1 −t Lw f= etζ (ζ I + Lw )−1 f dζ, e 2πi Γ
where Γ is the union of the rays γ ± = {z ∈ C: z = re±iν , r R > 0} and the arc γ0 = {z ∈ C: z = Reiψ , |ψ| ν}, going around the origin counter-clockwise, and ν ∈ (π/2, π/2 + τ ),
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301
where τ is defined as in Lemma 4.1. We will fix the value of R below. Then by Minkowski’s inequality and Lemma 4.1,
−t L e w f (x)2 w(x) dx
F
=
1 4π 2
2
etζ (ζ I + Lw )−1 f (x) dζ w(x) dx Γ
F
1/2 2 tζ
e 1 ζ (ζ I + Lw )−1 f (x)2 w(x) dx d|ζ | ζ 4π 2 Γ
C 4π 2
F
f (x)2 w(x) dx
tζ 2 e −c d √|ζ | e 2 d|ζ | . ζ
(4.6)
Γ
E
Let β = cos(ν) < 0 and parametrize the last integral using the definition of the path Γ to get
−tζ
∞ tβr
ν √ √ e −c d √|ζ | e −c d2 r −tR cos θ −c d2 R e 2 d|ζ | 2 dr + e e dθ e ζ r Γ
Now let
−ν
R
2 −c d √R e 2 R
∞
R √ d
2 −c e 2 tR|β|
d
etβr dr + 2πetR e−c 2
R tβR
e
d
+ 2πetR e−c 2
√
√
R
R
.
√ R = δd/t, where δ = c/4. Then, since d 2 t,
−tζ e −c d √|ζ | cδ d 2 cδ d 2 d2 2 2 d2 2 d2 ˜ −c˜ t . d|ζ | 2 eβδ t e− 2 t + 2πeδ t e− 2 t Ce ζ e 2 δ |β|
Γ
The desired inequality follows immediately if we substitute this estimate into (4.6). 5. Proof of Theorems 1.3 and 1.8 The bulk of this section is devoted to the proof of Theorem 1.3; at the end we discuss how to derive Theorem 1.8 from it. The heart of the proof of Theorem 1.3 is to show that perturbations of e−t Lw are bounded from L2 (w) to L∞ . More precisely, given any real-valued φ ∈ Cc∞ , there exist constants α and C such that for all f ∈ L2 (w), −φ −t L φ e e w e f
n
L∞
Ct − 4 eαtρ f L2 (w) , 2
(5.1)
where ρ = ∇φ L∞ . Given inequality (5.1), we get the existence of the heat kernel Wt (x, y) via functional analysis; then by an argument due to Davies (cf. [13]) we get the desired Gaussian
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bounds. Hölder continuity estimates then follow from the Harnack inequality and a classical argument. 5.1. The L2 (w), L∞ estimates To prove (5.1) it will suffice to prove it for non-negative functions f and for x = 0 and t = 1: −φ −L φ e e w e f (0) Ceαρ 2 f
(5.2)
L2 (w) .
To see that it suffices to consider non-negative functions, we use the fact that e−t Lw is linear and that given f = u + iv, we can decompose f as u+ − u− + iv + − iv − , where g + = max(g, 0) and g − = − min(g, 0), g L2 (w) g − L2 (w) + g + L2 (w) 2 g L2 (w) . The second reduction follows from homogeneity and the fact that if w is in A2 or QC, then so is wab = w(a · + b) for all a ∈ R, b ∈ Rn . To show that we can take t = 1, suppose that (5.1) holds when√t = 1 for all operators Lw . Define the functions u(x, t) = e−φ e−t Lw eφ f (x) t t ts). Then a straightforward computation shows that v = e−φ e−s Lt eφ f t , and v(y, s) = u( ty, √ √ where f t (y) = f ( ty), φ t (y) = φ( ty), and Lt is the operator induced by the sesquilinear form
at (f, g) = At ∇f (x) · ∇g(x) dx, Rn
√ √ where At (y) = A( ty). The matrix At satisfies (1.2) with w replaced by w t (y) = w( ty). Since w ∈ A2 /QC, w t ∈ A2 /QC with the same constant. Therefore, if (5.1) holds for s = 1 for the operator Lt , then, with ρt = ∇φ t L∞ = t 1/2 ρ, u(·, t) ∞ = v(·, 1) ∞ Ceαρt2 v(·, 0) 2 t L L L (w ) = Ce
αtρ 2
√ 2 √ f ( ty) w( ty) dy
1 2
n
= Ct − 4 eαtρ f L2 (w) . 2
Rn
To show that it suffices to take x = 0, we can repeat the above argument, replacing f by f 0 (x) = f (x + x0 ), w by w 0 (x) = w(x + x0 ), etc., for some fixed x0 ∈ Rn . We will now prove (5.2). Fix f ∈ L2 (w). Let Q0 ⊂ Rn be the cube (with sides parallel to the coordinate axes) centered at the origin with (Q0 ) = 9, and let f 0 = f χQ0 . For each integer k 1, let Qk = 3k Q0 and define f k = f χQk \Qk−1 . Decompose Qk \Qk−1 into 3n − 1 disjoint cubes Qk,j , 1 j 3n − 1, of side length 3k+1 . For k 1, let f k,j = f χQk,j . Then we have that ∞ 3 −1 −L φ k,j −φ −L φ e w e f (0) + e−φ(0) e−Lw eφ f 0 (0). e e w e f (0) e−φ(0) n
(5.3)
k=1 j =1
We first estimate the sum. Let uk,j (x, t) = e−t Lw eφ f k,j (x); then by Proposition 3.4 uk,j is k,j k,j k k,j is a a non-negative solution of Lw u = ut in Rn+1 + . Define v (y, s) = u (3 y, s). Then v n+1 solution to Lk v = vs in R+ , where Lk is the operator induced by the sesquilinear form defined
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303
as above with the matrix Ak (y) = A(3k y). The matrix Ak satisfies (1.2) with w replaced by w k (y) = w(3k y), which is again an A2 /QC weight. Therefore, by Proposition 3.8 applied to the parabolic cylinder Q1 (0, 13 8 ), sup
13 Q− 1 (0, 8 )
v k,j (y, s) γ
inf
13 Q+ 1 (0, 8 )
v k,j (y, s).
13 k,j k,j (y, s). Let B = {x: In particular, since (0, 1) ∈ Q− 13 v r 1 (0, 8 ), |v (0, 1)| γ infQ+ 1 (0, 8 ) |x| < r}. Then
k,j v (0, 1) C(n)γ w k (B 1 )− 12
218
k,j v (y, s)2 w k (y) dy ds
2
19 8
1 2
.
|y|< 12
By a change of variables this becomes k,j u (0, 1) C(n)γ w(B 3k )− 12
218
k,j u (x, t)2 w(x) dx dt
1 2
.
2 19 8
k
|x|< 32
Since eφ f k,j (x) is supported in Qk,j and dist(Qk,j , {|x| < orem 1.6 we have
3k k 2 }) dist(Qk , Qk−2 ) = 3 ,
by The-
e−φ(0) e−Lw eφ f k,j (0) = e−φ(0) uk,j (0, 1) − 12 −c32k
C(n)γ w(B 3k )
218
e
e
2 19 8
2(φ(x)−φ(0)) k,j
f
2 (x) w(x) dx dt
1 2
Rn
1 2k C(n)γ w(B 3k )− 2 e−c3 eφ(·)−φ(0) L∞ (Qk,j ) f k,j L2 (w) 2
1
C(n)γ w(B 3k )− 2 e−c3 e3 2
2k
k+1
√
n 2 ∇φ L∞
k,j f
L2 (w)
.
We estimate the last term on the right-hand side in (5.3) in a similar fashion, except that instead of Theorem 1.6 we use the fact that e−Lw is bounded on L2 (w) to obtain √ n e−φ(0) e−Lw eφ f 0 (0) Ce9 2 ∇φ L∞ f 0 L2 (w) .
Now substitute both these estimates in (5.3) and apply Hölder’s inequality twice to get
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e−φ(0) e−Lw eφ f (0) ∞ 3 −1 n
C(n)γ
1
√
n 2 ∇φ L∞
w(B 3k )− 2 e−c3 e3 2k
k+1
k,j f
2
k=1 j =1
√
L2 (w)
+ Ce9
n 2 ∇φ L∞
0 f
L2 (w)
∞ √ 1/2 1 2k k+1 n ∇φ ∞ k L f 2 C(n)γ 3n − 1 w(B 3k )− 2 e−c3 e3 L2 (w) 2
k=1 √
+ Ce9
n 2 ∇φ L∞
C(n, γ )
∞
C(n, γ )
L2 (w)
1
w(B 3k )− 2 e−c3 e3
∞
2k
k+2
√
n 2 ∇φ L∞
k f
L2 (w)
2
k=0
0 f
1/2 −1
w(B 3k )
k=0
2
∞
2 √ exp −2c32k + 3k+2 n ∇φ L∞ f k L2 (w)
1/2 .
k=0
Since w ∈ A∞ , it satisfies a reverse doubling condition: there exists β > 1 such that βw(Br ) w(B3r ). Thus, ∞
w(B 3k )−1 2
k=0
∞
β −k w(B 1 )−1 < C(w) < ∞. 2
k=0
Therefore, we have shown that e−φ(0) e−Lw eφ f (0)
1/2 ∞ 81n 2 2 f k 2
∇φ 2L∞ C(n, γ , w) exp = C(n, γ , w)eαρ f L2 (w) , L (w) 8c k=0
where α =
81n 8c
and ρ = ∇φ L∞ . This proves (5.2) and so (5.1).
5.2. Gaussian bounds To find the heat kernel and show that it satisfies Gaussian bounds, first note that by duality, (5.1) implies that −φ −t L φ e e w e f
n
L2 (w)
Ct − 4 eαtρ f L1 . 2
Since e−φ e−t Lw eφ is a semigroup, we can combine this inequality with (5.1) to get −φ −t L φ e e w e f
Ct −n/2 eαtρ f L1 . 2
L∞
(5.4)
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By a classical result of Dunford and Pettis [15] (see also [3, p. 42]), this implies that for each φ φ ∈ Cc∞ , there exists a kernel Wt (x, y) such that for all f ∈ L1 , e
−φ −t Lw φ
e
e f (x) =
φ
Wt (x, y)f (y) dy Rn
and φ Wt (x, y) Ct −n/2 eαtρ 2 . In particular, let φ ≡ 0 to get the kernel Wt (x, y) of e−t Lw ; then it is immediate that Wt (x, y) = φ eφ(x)−φ(y) Wt (x, y). Hence, Wt (x, y) Ceαtρ 2 eφ(x)−φ(y) t −n/2 .
(5.5)
Inequality (5.5) is true for every φ ∈ Cc∞ with ∇φ L∞ = ρ, ρ > 0. Therefore, by an approximation argument we may take φ to be a Lipschitz function that satisfies φ(x) − φ(y) = −ρ|x − y|. Then (5.5) becomes Wt (x, y) Ct −n/2 exp αtρ 2 − ρ|x − y| . If we optimize the value of ρ, we get ρ =
|x−y| 2αt ,
and thus
2 Wt (x, y) Ct −n/2 exp − |x − y| . 4αt This is the desired inequality. 5.3. Hölder continuity The proof of inequalities (1.10) and (1.11) follow by standard arguments from the classical theory of elliptic and parabolic operators. Therefore, here we will only briefly sketch the proof. First, since the heat kernel of L∗w is Wt (y, x), (1.11) follows from (1.10) by duality. Given that Wt (x, y) satisfies Gaussian bounds, it is well known (see [3, p. 30]) that to prove (1.10) it suffices to prove that there exist constants C and ν > 0 such that for all t > 0 and all x, y, h ∈ Rn , ν Wt (x + h, y) − Wt (x, y) C|h| . t n/2+ν/2
By a classical result (again see [3, p. 42]), this is equivalent to proving that e−t Lw maps L1 into C|h|ν the space of Hölder continuous functions C ν , with norm t n/2+ν/2 .
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To prove that e−t Lw : L1 → C ν , we first use the Harnack inequality, Lemma 3.8, and an argument due to Trudinger [28] to show that if f ∈ L1 and u(x, t) √ = e−t Lw f (x), then there exist and 0 < ρ < ρ t/2, ν > 0 and C > 0 such that given (x, t) ∈ Rn+1 0 + osc u C
Qρ (x,t)
ρ ρ0
ν osc u,
Qρ0 (x,t)
where oscR u = supR u − infR u. Further, by (5.4) (with φ ≡ 0) we have that osc u 2 u L∞ (Qρ0 (x,t)) Ct −n/2 f L1 .
Qρ0 (x,t)
The desired norm inequality follows by combining these estimates. 5.4. Proof of Theorem 1.8 This result follows from the fact that since e−t Lw is an holomorphic semigroup the Gaussian bounds in Theorem 1.3 can be extended to complex time. More precisely, we have the following. Theorem 5.1. Let A ∈ En (w, λ, Λ) be real symmetric, and let ω = arctan(Λ/λ). Then for all ν, 0 < ν < π/2 − ω, if z ∈ Σ(ν), there exists a heat kernel Wz (x, y) associated to the operator e−zLw . Furthermore, for all x, y ∈ Rn , the kernel Wz satisfies 2 Wz (x, y) C1 exp −C2 |x − y| , |z| |z|n/2 and μ |h| |x − y|2 Wz (x + h, y) − Wz (x, y) C1 , exp −C2 |z| |z|n/2 |z|1/2 + |x − y| μ |h| |x − y|2 Wz (x, y + h) − Wz (x, y) C1 , exp −C2 |z| |z|n/2 |z|1/2 + |x − y| where h ∈ Rn is such that 2|h| |z|1/2 + |x − y|. The constants C1 , C2 and μ depend only on n, ν, λ, and Λ. The proofs of Theorem 1.8 and the Gaussian bounds in Theorem 5.1 are identical to the proofs in the unweighted case as given in Auscher and Tchamitchian [3, p. 48]. We refer the reader there for complete details. The proof that Vt 1 = 0 then follows at once from the conservation property, Theorem 1.7. 6. Proof of Theorem 1.7 Our proof is adapted from the one for uniformly elliptic operators given by Auscher, McIntosh and Tchamitchian [5, Lemma 5.8]. The weighted case differs in many technical details, so we present the complete proof. Hereafter, let ω = arctan(Λ/λ).
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307
It is straightforward to show that since the kernel of e−zLw , z ∈ Σ(π/2−ω), satisfies Gaussian bounds, e−zLw : Lp → Lp , 1 p ∞, with a bound that depends only on arg(z). By the Laplace identity we get the same Lp estimates for (zI + Lw )−1 . The same proof shows that the weighted versions of these inequalities are true if 2 p < ∞; in the case 1 p < 2 we need to assume w ∈ Ap . This need not be the case if w ∈ A2 or QC. However, the following weaker result suffices for our purposes. Lemma 6.1. Let w ∈ A2 or QC, and let z ∈ Σ(π − ω). Then (zI + Lw )−1 is a densely defined operator from L1 (w) into itself with domain containing Cc∞ . In fact, if φ ∈ Cc∞ , then (zI + Lw )−1 φ(x) Ce−c|x| ,
(6.1)
where the constants depend on n, Λ, λ, arg(z) and φ. Proof. If inequality (6.1) holds, then it follows immediately that e−zLw φ ∈ L1 (w): since w ∈ A∞ , there exists D > 1 such that for every r > 0, w(B2r (0)) Dw(Br (0)) [21, p. 695]. Hence, ∞ k (zI + Lw )−1 φ 1 Cw B1 (0) + Ce−c2 w B2k+1 (0) \ B2k (0) L (w) k=0
Cw B1 (0)
∞
D k e−c2 < ∞. k
k=0
To prove (6.1), fix φ ∈ Cc∞ and suppose that supp(φ) ⊂ BR (0), R > 1. By Proposition 3.6, −1
(zI + Lw )
∞ φ(x) =
e−zνt e−νt Lw φ(x) dt,
0
where we take ν ∈ Σ(π/2 − ω) such that |ν| = 1 and Re(zν) > 0. Since e−νt Lw φ ∈ L∞ is uniformly bounded, assume without loss of generality that |x| > 2R, so if x ∈ supp(φ), |x − y| > |x|/2. Therefore, by Theorem 5.1, (zI + Lw )−1 φ(x)
∞ C
e
∞ e
− Re(zν)t
n R
0
∞ C φ ∞ R n 0
−n/2
|νt| Rn
0
C
− Re(zν)t
|x − y|2 φ(y) dy dt exp −c |νt|
c |x|2 φ(y) dy dt t −n/2 exp − 4 t
c |x|2 e− Re(zν)t t −n/2 exp − dt 4 t
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C φ ∞ R
+1 |x|2 2j
c |x|2 dt e− Re(zν)t t −n/2 exp − 4 t
∞
n
j =−∞ ∞
C φ ∞ R n
2j |x|2
2j |x|2 e−2
j =−∞ ∞
C φ ∞ |x|
2
j =−∞
j |x|2 Re(zν)
j 2 −n/2 c 2 |x| exp − 2−j 8
n c −j j 2 −1 j . exp −2 |x| Re(zν) − 2 − log(2) 8 2
Since |x| > 2R 2, there exists J > 0 such that 2J |x| < 2J +1 . We estimate the sum in the last term by splitting it into three pieces depending on the size of j : ∞ j =−∞
c n −1 j exp −2j |x|2 Re(zν) − 2−j − log(2) 8 2
=
∞ j =0
∞
−1
+
j =−J
+
−J −1
j =−∞
c n −1 j exp −2j |x|2 Re(zν) − 2−j − log(2) 8 2
exp −2j |x|2 Re(zν)
j =0
−1 c −j n + exp −|x| Re(zν) −1 j exp − 2 − log(2) 8 2 j =−J
∞ c c n c −1 j exp 2J − 2j + log(2) + exp − |x| 8 4 8 2 j =J +1
Ce−c|x| . Combining these two estimates we get that (zI + Lw )−1 φ(x) C φ ∞ |x|2 e−c|x| C φ ∞ e−c|x| . This completes the proof.
2
Proof of Theorem 1.7. Since the semigroup and the resolvent are bounded on L∞ , by Proposition 3.6 we have that
1 e−t Lw 1 = e−tζ (ζ I + Lw )−1 1 dζ, (6.2) 2πi Γ
where in the definition of Γ we take R = 1/t. Suppose for the moment that we could prove for all ζ ∈ Γ that (ζ I + Lw )−1 1 = ζ −1 .
(6.3)
D. Cruz-Uribe, C. Rios / Journal of Functional Analysis 255 (2008) 283–312
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Then (6.2) becomes 1 2πi
e t Lw 1 =
ζ −1 e−tζ dζ = 1,
Γ
where the last equality follows from a standard contour integral argument. To prove (6.3) it suffices to show that for all φ ∈ Cc∞ ,
−1 −1 (ζ I + Lw ) 1 φ(x)w(x) dx = ζ φ(x)w(x) dx. Rn
By duality,
Rn
(ζ I + Lw )−1 1 φ(x)w(x) dx =
Rn
−1 ζ¯ I + L∗w φ(x)w(x) dx.
Rn
Let h(x) = (ζ¯ I + L∗w )−1 φ(x); then by Lemma 6.1 we have that h ∈ L1 (w). Furthermore, L∗w h = −ζ¯ h + φ, so L∗w h ∈ L1 (w). Therefore,
(ζ I + Lw )−1 1 φ(x)w(x) dx = ζ −1 φ(x)w(x) dx − ζ −1 L∗w h(x)w(x) dx. Rn
Rn
Rn
To complete the proof we will show that the last integral equals zero. Let supp(φ) ⊂ BR (0), and for all r > 2R, let χr ∈ Cc∞ be such that χr ≡ 1 on Br (0), supp(χr ) ⊂ B2r (0), and |∇χr | cr −1 . Then
n ∗ ∗ Lw h(x)w(x) dx = lim Lw h(x)χr (x)w(x) dx r→∞ Rn R
= lim A∗ ∇h(x) · ∇χr (x) dx r→∞
Rn
Λ lim
r→∞ Rn
lim Cr r→∞
∇h(x)∇χr (x)w(x) dx
−1
w B2r (0)
∇h(x)2 w(x) dx
1/2 .
supp(∇χr )
To bound the last integral, we first make a preliminary estimate. let ψ ∈ Cc∞ be such that supp(ψ) ∩ supp(φ) = ∅. Then for all δ > 0,
∇h(x)2 ψ(x)2 w(x) dx λ−1 Re A∗ ∇h(x) · ∇h(x)ψ(x)2 dx Rn
Rn
λ−1 A∗ ∇h(x) · ∇h(x)ψ(x)2 dx Rn
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λ
∗ 2 Lw h(x) h(x)ψ(x) w(x) dx
−1
Rn
+ 2λ
∗ ψ(x)h(x)A ∇h(x) · ψ(x) dx
−1
Rn
λ−1 −ζ¯ h(x) + φ(x) h(x)ψ(x)2 w(x) dx Rn
+ δΛλ
−1
∇h(x)2 ψ(x)2 w(x) dx
Rn
+ δ −1 Λλ−1
h(x)2 ∇ψ(x)2 w(x) dx.
Rn
If we fix δ such that δΛλ−1 = 1/2, then we can rearrange terms to get
∇h(x)2 ψ(x)2 w(x) dx C|ζ |
Rn
h(x)2 ∇ψ(x)2 + ψ(x)2 w(x) dx.
(6.4)
Rn
Now for each r > 2R, fix ψr ∈ Cc∞ such that supp(ψ) ⊂ B3r (0) \ Br/2 (0), ψ ≡ 1 on supp(∇χr ) and |∇ψr | C. Then by (6.4) and Lemma 6.1 we have that
lim Cr
−1
r→∞
w B2r (0)
∇h(x)2 w(x) dx
1/2
supp(∇χr )
= lim Cr −1 w B2r (0)
∇h(x)2 ψr (x)2 w(x) dx
r→∞
1/2
supp(∇χr )
lim Cr
−1
r→∞
w B2r (0)
h(x)2 w(x) dx
1/2
B3r (0)\Br/2 (0)
3/2 lim Cr −1 w B3r (0) exp(−cr) r→∞
= 0. The last equality holds since w ∈ A∞ : as in the proof of Lemma 6.1, there exists D > 1 such that for all r > 1, if 3N < r 3N +1 , then 3/2 −cr 3/2 −c3N 3 e D 2 (N +1) w B1 (0) e C. w B3r (0) This completes the proof.
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Acknowledgments The authors would like to thank Steven Hofmann for suggesting this project to us and for his help and guidance in completing it. Both authors gratefully acknowledge the support of the Stewart-Dorwart faculty development fund at Trinity College. References [1] D.G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967) 890–896. [2] Pascal Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc. (2) 54 (2) (1996) 284–296. [3] Pascal Auscher, Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii + 172. [4] Pascal Auscher, Thierry Coulhon, Philippe Tchamitchian, Absence de principe du maximum pour certaines équations paraboliques complexes, Colloq. Math. 71 (1) (1996) 87–95. [5] Pascal Auscher, Alan McIntosh, Philippe Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1) (1998) 22–73. [6] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn , Ann. of Math. (2) 156 (2) (2002) 633–654. [7] Filippo Chiarenza, Michelangelo Franciosi, Quasiconformal mappings and degenerate elliptic and parabolic equations, Matematiche (Catania) 42 (1–2) (1987) 163–170. [8] Filippo Chiarenza, Michele Frasca, Boundedness for the solutions of a degenerate parabolic equation, Appl. Anal. 17 (4) (1984) 243–261. [9] Filippo Chiarenza, Raul P. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl. (4) 137 (1984) 139–162. [10] Filippo M. Chiarenza, Raul P. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (8) (1984) 719–749. [11] Filippo Chiarenza, Raul P. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova 73 (1985) 179–190. [12] Guy David, Stephen Semmes, Strong A∞ weights, Sobolev inequalities and quasiconformal mappings, in: Analysis and Partial Differential Equations, in: Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 101–111. [13] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92, Cambridge Univ. Press, Cambridge, 1990. [14] E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992) 99–119, Festschrift on the occasion of the 70th birthday of Shmuel Agmon. [15] Nelson Dunford, B.J. Pettis, Linear operations on summable functions, Trans. Amer. Math. Soc. 47 (1940) 323–392. [16] Javier Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, RI, 2001. [17] Eugene B. Fabes, Carlos E. Kenig, Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1) (1982) 77–116. [18] Avner Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs, NJ, 1964. [19] Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 1–11. [20] José García-Cuerva, José L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, 1985. [21] Loukas Grafakos, Classical and Modern Fourier Analysis, Pearson/Prentice Hall, Upper Saddle River, NJ, 2004. [22] Cristian E. Gutiérrez, Gail S. Nelson, Bounds for the fundamental solution of degenerate parabolic equations, Comm. Partial Differential Equations 13 (5) (1988) 635–649. [23] Cristian E. Gutiérrez, Richard L. Wheeden, Bounds for the fundamental solution of degenerate parabolic equations, Comm. Partial Differential Equations 17 (7–8) (1992) 1287–1307. [24] Kazuhiro Ishige, On the behavior of the solutions of degenerate parabolic equations, Nagoya Math. J. 155 (1999) 1–26. [25] Tosio Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss., vol. 132, Springer-Verlag, New York, 1966.
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Journal of Functional Analysis 255 (2008) 313–373 www.elsevier.com/locate/jfa
Stochastic scalar conservation laws Jin Feng ∗,1 , David Nualart 2 Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States Received 11 February 2008; accepted 17 February 2008 Available online 9 April 2008 Communicated by Paul Malliavin
Abstract We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito’s calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L1 contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only. Published by Elsevier Inc. Keywords: Stochastic analysis; Scalar conservation law; Stochastic compensated compactness
1. Introduction We are interested in the well-posedness (existence and uniqueness) for first order nonlinear stochastic partial differential equation (SPDE) of the following type ∂t u(t, x) + divx F u(t, x) =
z∈Z
* Corresponding author.
E-mail address:
[email protected] (J. Feng). 1 Supported in part by US ARO W911NF-08-1-0064. 2 Supported in part by US NSF DMS-0604207.
0022-1236/$ – see front matter Published by Elsevier Inc. doi:10.1016/j.jfa.2008.02.004
σ x, u(t, x); z ∂t W (t, dz).
(1)
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In the above, x ∈ R d , t 0, u(t, x) is a random scalar-valued function, F = (F1 , . . . , Fd ) : R → R d is a vector field (the flux). Regarding the random term on the right-hand side of the equation, Z is a metric space, and W (t, dz) is a space–time Gaussian white noise martingale random measure with respect to a filtration {Ft } (e.g. Walsh [18], Kurtz and Protter [13]) with E W (t, A) ∩ W (t, B) = μ(A ∩ B)t (2) for measurable A, B ⊂ Z, where μ is a (deterministic) σ -finite Borel measure on the metric space Z. In addition, σ : R d × R × Z → R. In the case of σ = 0, (1) reduces to a deterministic partial differential equation known as the scalar conservation law ∂t u(t, x) + divx F u(t, x) = 0, (3) which has been extensively studied in nonlinear partial differential equation theory literature (e.g. Dafermos [3]). A well-known difficulty for (3) is that solutions cannot be interpreted in classical sense: non-differentiability in x for u(t, x) develops in finite time, even if u(0, x) is chosen to be smooth [3, Theorem 5.1.1]. On the other hand, because of the nonlinearity in F , (Schwartz) distributional weak solution will generally not be unique (e.g. Section 4.2 of [3]). Kruzkov [9,10] introduced a method for selecting a weak solution motivated by physical consideration (the entropic solution). Well-posedness of (3) in the entropic solution sense can be proved for u(t) ∈ L1 ∩ L∞ , t 0. There are also other methods of selecting weak solutions. Most of these different approaches can be shown to be equivalent, at least in one space dimension d = 1. It is worth mentioning that, from a physical point of view, vector-valued u version of (3) is ultimately more interesting. However, little is known about well-posedness in that case. A detailed exposition about deterministic conservation law, for scalar- as well as vector-valued u, is given by Dafermos [3]. See also Chen [2] for a survey. Chapter 11 of Evans [7] contains a brief but informative introduction to the scalar case. The goal of this article is to introduce a proper generalization of entropic solution to the stochastic case (1) (Definition 2.5). Such notion will enable us to prove uniqueness of solution under mild assumptions on F and σ (Theorem 3.5). We will also give existence result for slightly more restrictive situations in one space dimension in Section 4. The following example gives us a feel on the scope of application that model (1) covers. Example 1.1. Let Z = {1, 2, . . . , m} and μ be a counting measure on Z, (1) reduces to m σk x, u(t, x) ∂t Wk (t), ∂t u(t, x) + divx F u(t, x) =
(4)
k=1
where W1 , . . . , Wm are independent standard Brownian motions. In particular, taking d = 1 and F (u) = |u|2 /2, the equation reduces to the stochastic Burgers’ equation ∂t u(t, x) + u(t, x)ux (t, x) =
m
σk x, u(t, x) ∂t Wk (t).
k=1
The W (t, dz) term can be extended to general semi-martingale random measure, and the theory developed here is expected to hold as well. We do not pursue this direction in this article.
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There is a well-known connection between Hamilton–Jacobi and conservation law equations. Such connection can be transferred to the stochastic case as well. Let scalar function φ = φ(t, x) : [0, ∞) × R d → R be a solution to (5) ∂t φ(t, x) + F ∇x φ(t, x) = σ x, ∇x φ(t, x); z ∂t W (t, dz). Z
Let vector-valued function u(t, x) = ∇x φ(t, x), then ∇x σ (x, u; z) + ∂u σ (x, u; z)(∇x · u) ∂t W (t, dz). ∂t u + ∇x F (u) = Z
The case of d = 1 and σ = σ (x; z) independent of u gives scalar conservation law as considered in (1). In a series of publications [14], Lions and Souganidis consider equations related to (5): m σk ∇φ(t, x) ◦ dWk (t), ∂t φ + F ∇φ, D 2 φ = k=1
where D 2 is the Hessian operator and ◦ stands for Stratonovich type integral. Stochastic generalizations of viscosity solution are used. 2. Stochastic entropic solution—definition and main result 2.1. Definitions Definition 2.1. (Φ, Ψ ) is called an entropy–entropy flux pair if Φ ∈ C 1 (R) and Ψ = (Ψ1 , . . . , Ψd ) : R d → R d is a vector field satisfying Ψk (r) = Φ (r)(Fk ) (r),
k = 1, . . . , d.
(6)
Remark 2.2. Ψk can be chosen as r Ψk (r) =
Φ (s)(Fk ) (s) ds,
for some fixed v ∈ R.
v
Note that, unlike the usual definition, we do not require Φ to be convex in this definition. A special class of entropy–entropy flux pairs will play a major role in later analysis. We define it next. For each ε > 0, let β = βε ∈ C ∞ (R) be convex, β(r) = 0,
r 0,
β(r) = Cε + r,
Cε > 0, r ε.
K = (Φ, Ψ ) is an entropy–entropy flux pair:
Φ(r) = Φ u (r) = β(u − r), or Φ(r) = Φv (r) = β(r − v), u, v ∈ R .
Throughout this article, we assume the following regularities.
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Condition 2.3. (1) Fk ∈ C 2 (R), and Fk (s) have at most polynomial growth in s, for all k = 1, . . . , d; (2) For each compact subset K ⊂ R d × R d , there exists MK : Z → R and nonnegative, nondecreasing, continuous function ρK : R → R with ρK (0) = 0 such that
σ (y, v; z) − σ (x, u; z) |u − v|1/2 ρK |u − v| + |x − y| MK (z), ∀(x, y) ∈ K, z ∈ Z, where CK ≡
2 MK (z) μ(dz) < ∞.
z∈Z
Example 2.4. Let σk : R d × R → R be Lipschitz for each k = 1, . . . , m and consider (4), then the second part of the above conditions is satisfied. Definition 2.5 (Stochastic entropic solution). Let (Ω, {Ft : t 0} ⊂ F, P ) be a filtered probability space where W (t, ·) is adapted space–time Gaussian white noise martingale random measure satisfying (2). We call an L2 (R d )-valued {Ft } -adapted stochastic process u = u(t) = u(t, x) a stochastic entropic solution of (1), provided (1) for each T > 0, p = 2, 3, 4, . . . , p sup E u(t) p < ∞,
(8)
σ x, u(r, x); z 4 dx μ(dz) dr < ∞.
(9)
0tT
and for each N = 1, 2, . . . fixed,
T
E z∈Z |x|N
0
(2) For each 0 s t, each 0 ϕ ∈ Cc2 (R d ), and each (Φ, Ψ ) ∈ K, Φ u(t, ·) , ϕ − Φ u(s, ·) , ϕ t
Ψ u(r, ·) , ∇x ϕ dr +
s
+ (s,t]×Z
1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2
(s,t]×Z
Φ u(r, ·) · σ ·, u(r, ·); z , ϕ W (dr, dz).
(10)
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Perhaps a more revealing way to re-state integral inequality (10) is to say that u is a (Schwartz distributional) weak solution to ∂t Φ u(t, x) + divx Ψ u(t, x) 1 Φ u(t, x) σ 2 x, u(t, x); z μ(dz) 2 Z
+
∂W (t, dz) . Φ u(t, x) σ x, u(t, x); z ∂t
(11)
Z
When σ = 0, the right-hand side of the above inequality drops to zero. (11) reduces to exactly the defining differential inequality in deterministic entropic solution initially introduced by Kruzkov [9]. Some explanation on the meaning of (10) is necessary: (Φ, Ψ ) ∈ K implies that Φ and Φ are bounded and Ψ has at most polynomial growth. Together with (8) and (9), each term in (10) is well defined. A significant special yet common case satisfying (9) is when σ is uniformly bounded supx,u,z |σ (x, u, z)| < +∞, and μ(Z) < ∞. By an interpolation argument, to verify that (8) holds for p = 2, 3, . . . , it is good enough to show for even positive integer valued cases of p = 2, 4, 6, . . . . Moreover, both imply that (8) holds for all p ∈ [2, ∞). Unlike deterministic scalar conservation law (i.e. the case σ = 0), to prove path-wise uniqueness, we also need to capture more explicitly “noise–noise interaction” between any two possibly different stochastic solutions. We strengthen the definition of solution as follows. Definition 2.6 (Stochastic strong entropic solution). We call an L2 (R d )-valued, {Ft }-adapted process v = v(t) = v(t, x) a stochastic strong entropic solution of (1) if the following holds: (1) it is a stochastic entropic solution (i.e. (8), (9) and (10) hold for u replaced by v); ˜ satisfying (2) for each L2 (R d )-valued, Ft -adapted process u(t) p ˜ p < ∞, T > 0, p = 2, 3, . . . , sup E u(t) 0tT
and for each β ∈ C ∞ (R) of the form (17), 0 ϕ ∈ Cc∞ (R d × R d ), and f (r, z; v, y) =
˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx, β u(r,
x∈R d
there exists a deterministic function {A(s, t): 0 s t} such that
E y (s,t]×Z
f r, z; v(t, y), y W (dr, dz) dy
E (s,t]×Z y
∂ f r, z; u(r, ˜ y), y σ y, v(r, y); z μ(dz) dy dr + A(s, t) ∂v
(12)
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with the following property: for each T > 0, there exist partitions 0 = t1 < t2 < · · · < tm = T satisfying m
lim
maxi |ti+1 −ti |→0+
A(ti , ti+1 ) = 0,
t 0.
i=1
2.2. Main results We list another set of conditions stronger than those in Condition 2.3. Condition 2.7. (1) (2) (3)
d = 1, F ∈ C 2 (R) and the set {r ∈ R: F (r) = 0} is dense in R, ∞ d 2 d there 2exist f ∈ L (R ) ∩ L (R ), deterministic constant C > 0, and M : Z → R such that Z M (z) μ(dz) < ∞,
σ (x, u; z) f (x) 1 + |u| M(z),
(13)
σ (x, u; z) − σ (y, v; z) C |u − v| + |x − y| M(z) + σ (x, u; z) .
(14)
and
The main result of this article is the following. Theorem 2.8. Assume Condition 2.3 holds, and that u0 satisfies p p E u0 p + u0 2 < ∞,
p=1,2,... L
p (R d )-valued
random variable
p = 1, 2, . . . .
(Uniqueness) Suppose that u, v are two stochastic entropic solutions of (1) with the same initial condition u(0) = u0 = v(0), and that one of u, v is a strong stochastic entropic solution. Then almost surely u(t) = v(t) for t 0. (Existence) Assume furthermore that Condition 2.7 holds, then there exists a strong stochastic entropic solution (hence also entropic solution) for (1) with initial value u0 . 2.3. Notations Throughout, we denote the space of smooth, rapidly decreasing functions
S R d = f ∈ C ∞ : sup x m Dxn f (x) < ∞, m, n = 1, 2, . . . .
(15)
x
Let J ∈ Cc∞ (R d ) be the standard mollifier defined by J (x) =
C exp{ |x|21−1 } if |x| < 1, 0 if |x| 1,
(16)
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319
where constant C > 0 is selected so that R d J (z) dz = 1. For each ε > 0, we set Jε (z) = ε −d J (ε −1 |z|). Jε ∈ Cc∞ (R d ) with supp(Jε ) ⊂ [−ε, ε]d . For each f ∈ Lloc (R d ), we define its mollification fε (x) = Jε ∗ f (x) = Jε (x − y)f (y) dy = Jε (y)f (x − y) dy. Rd
Rd
Let A ⊂ R d , then function χA (x) = 1 if x ∈ A and χA (x) = 0 if x ∈ / A. To simplify, with a slight abuse of notation, we denote χ(x) = χ[0,+∞) (x). For a ∈ R, we denote a+ = max{a, 0}. Then |a| = a+ + (−a)+ . We need smooth functions approximating β(r) = r+ ∈ C(R). We consider J in the special case of d = 1 and define r−ε ρε (r) = Jε (s) ds,
r βε (r) =
−∞
ρε (s) ds,
r ∈ R.
(17)
−∞
Then by direct verification, we have the following. Lemma 2.9. The above constructed ρε , βε ∈ C ∞ (R) have the following properties: βε = ρε , βε (r) = Jε (r − ε); ρε is a nondecreasing function and βε (r) = ρε (r) =
0 if r 0, 1 if r 2ε;
(18)
and βε is convex and βε (r) = where Cˆ =
0 if r 0, ε Cˆ + (r − 2ε) if r 2ε,
(19)
1 s −1 ( t=−1 J (t) dt) ds < 2. Furthermore, 0 βε (r) = Jε (r − ε) ε −1 C,
0 r 2ε,
implying 0 rβε (r) 2C,
for 0 r 2ε.
3. Uniqueness 3.1. A doubling lemma Let u be a stochastic entropic solutions and v be a stochastic strong entropic solution. We estimate the evolution of (u(t) − v(t))+ 1 = (u(t) − v(t))+ L1 . First, let β = βε (r) as constructed in Lemma 2.9. We approximate (u(t) − v(t))+ 1 by R d ×R d β(u(t, x)−v(t, y))ϕ(x, y) dx dy (by considering limits βε (r) → r+ and ϕ(x, y) dx dy → δx (dy) dx). Then we develop estimate for time evolution of such approximate. In the deterministic scalar conservation law setting (i.e. σ = 0), Kruzkov [9] appears to be the first who introduced
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such an argument for a uniqueness proof. Our goal in this section is to generalize such arguments properly to a stochastic setting (see Lemma 3.2). Let 0 ϕ ∈ Cc∞ (R d × R d ). For t > s 0,
β u(t, x) − v(t, y) ϕ(x, y) dx dy −
R d ×R d
=
β u(s, x) − v(s, y) ϕ(x, y) dx dy
R d ×R d
β u(t, x) − v(t, y) ϕ(x, y) dx dy −
β u(s, x) − v(t, y) ϕ(x, y) dx dy
β u(s, x) − v(t, y) ϕ(x, y) dx dy −
+
β u(s, x) − v(s, y) ϕ(x, y) dx dy
≡ I1 + I 2 . First, we estimate I2 . We introduce notation α(u, v) = α1 (u, v), . . . , αd (u, v) , where (noting β(r) = 0 for r < 0) ∞ αk (u, v) =
β
(u − w)Fk (w) dw
v
u =
β (u − w)Fk (w) dw.
(20)
v
Lemma 3.1. t
I2
α u(s, x), v(r, y) · ∇y ϕ(x, y) dx dy dr
s R d ×R d
1 + 2
t
β u(s, x) − v(r, y) σ 2 y, v(r, y); z ϕ(x, y) dx dy μ(dz) dr
s Z x,y
β u(s, x) − v(r, y) σ y, v(r, y); z ϕ(x, y) dx dy W (dr, dz).
−
(21)
(s,t]×Z x,y
Proof. Let u ∈ R be fixed. We take Φ(v) = β(u − v), Ψk (v) = apply (10) to v(t, y). Therefore, for each x ∈ R d ,
β u − v(t, y) ϕ(x, y) dy −
y
v
u (−β
)(u − w)F (w) dw, k
β u − v(s, y) ϕ(x, y) dy
y
t
v(r,y)
− s y∈R d
u
d k=1
∂ β (u − w)Fk (w) dw ϕ(x, y) dy dr ∂yk
and
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1 + 2
321
β u − v(r, y) σ 2 y, v(r, y); z ϕ(x, y) dy μ(dz) dr
(s,t]×Z y
+
−β u − v(r, y) σ y, v(r, y); z ϕ(x, y) dy W (dr, dz).
(s,t]×Z y
Taking u = u(s, x) inside the above inequality and integrating x, and applying Fubini’s theorem, we arrive at (21). 2 We estimate I1 next. We introduce notation α(u, ˆ v) = αˆ 1 (u, v), . . . , αˆ d (u, v) , where (noting β (r) = 0 for r 0) u αˆ k (u, v) =
β
(w − v)Fk (w) dw
u =
−∞
β (w − v)Fk (w) dw.
(22)
v
For each v ∈ R d fixed, taking Φ(u) = β(u − v) and Ψ (u) = α(u, ˆ v), we apply (10) for u(t, x), then we take v = v(t, y),
β u(t, x) − v(t, y) ϕ(x, y) dx dy −
R d ×R d
β u(s, x) − v(t, y) ϕ(x, y) dx dy
R d ×R d
t
αˆ u(r, x), v(t, y) · ∇x ϕ(x, y) dx dy dr
s R d ×R d
+
1 2
t
β u(r, x) − v(t, y) σ 2 x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr
s Z R d ×R d
+
β u(r, x) − v(t, y) σ x, u(r, x); z ϕ(x, y) dx W (dr, dz) dy
y∈R d (s,t]×Z x∈R d
≡ I3 + I4 + I5 . To achieve simplicity in exposition, we have slightly abused notation for the term I5 as this should not be understood as an Ito’s integral (the integrand contains anticipative term v(t, y)). The rigorous meaning of it should be understood in the following sense. Let
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f (r, z, v, y) =
β u(r, x) − v σ x, u(r, x); z ϕ(x, y) dx
Rd
and G5 (s, t; v, y) =
(23)
f (r, z, v, y) W (dr, dz).
(s,t]×Z
For each v, y fixed, the above is an Ito’s integral. Then, we define
G5 s, t; v(t, y), y dy.
I5 = I5 (s, t) = y∈R d
A key defining property (12) in stochastic strong entropic solution gives us E[I5 ] E − R d ×R d
β u(r, x) − v(r, y) σ y, v(r, y); z
(s,t]×Z
× σ x, u(r, x); z μ(dz) dr ϕ(x, y) dx dy + A(s, t). Together with the estimate on E[I3 ] and E[I4 ], we have E[I1 ] t
E
αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dr dy
s R d ×R d
1 + E 2
β u(r, x) − v(r, y) σ 2 x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr
(s,t]×Z R d ×R d
−E
β u(¯r , x) − v(¯r , y) σ y, v(¯r , y); z
(s,t]×Z R d ×R d
× σ x, u(¯r , x); z ϕ(x, y) dx dy μ(dz) d r¯
+ A(s, t). Combine this with the estimate on I2 in (21), by arbitrariness of 0 s t, we can arrive at a stochastic version of the doubling of variable estimate first introduced by Kruzkov [9] in deterministic context (see for instance Evans [7, Theorem 3, p. 608]).
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Lemma 3.2. For each t > 0, we have
β u(t, x) − v(t, y) ϕ(x, y) dx dy −
E R d ×R d
β u(0, x) − v(0, y) ϕ(x, y) dx dy
R d ×R d
t
α u(r, x), v(r, y) · ∇y ϕ(x, y) + αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy dr
E 0 R d ×R d
1 + E 2
β u(r, x) − v(r, y)
(0,t]×Z R d ×R d
2 × σ y, v(r, y); z − σ x, u(r, x), z ϕ(x, y) dx dy μ(dz) dr . Proof. We select the sequence of partitions of [0, t] appearing in the defining relation of strong entropic solution (Definition 2.6): 0 = t1 · · · tm = t < ∞. Using the above estimate on I1 and the estimate (21) on I2 (set the s = ti , t = ti+1 there), E
β u(ti+1 , x) − v(ti+1 , y) ϕ(x, y) dx dy − ti+1 ti R d ×R d
β u(ti , x) − v(ti , y) ϕ(x, y) dx dy
α u(r, x), v(r, y) · ∇y ϕ(x, y) + αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy dr
E 1 + E 2
β u(r, x) − v(r, y)
(ti ,ti+1 ]×Z R d ×R d
2 × σ y, v(r, y); z − σ x, u(r, x), z ϕ(x, y) dx dy μ(dz) dr
+ A(ti , ti+1 ). Summing over i,
β u(t, x) − v(t, y) ϕ(x, y) dx dy −
E R d ×R d
t
β u(0, x) − v(0, y) ϕ(x, y) dx dy
R d ×R d
E
α u(r, x), v(r, y) · ∇y ϕ(x, y) + αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy dr
0 R d ×R d
1 + E 2
(0,t]×Z R d ×R d
β u(r, x) − v(r, y)
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2 × σ y, v(r, y); z − σ x, u(r, x); z ϕ(x, y) dx dy μ(dz) dr +
m
A(ti , ti+1 ).
i=1
Taking limm→∞ , we arrive at the desired inequality.
2
3.2. Uniqueness Using Lemma 3.2 as the point of departure, we now let ϕ(x, y) dx dy → δx (dy) dx and β(r) → r+ to arrive at an L1 type estimate for u − v. First, we select test function ϕ in the following manner. Let J be a one-dimensional standard mollifier as defined by (16), and let 0 ψ ∈ Cc∞ (R d ). We choose
d − y x x +y k k −d ∈ Cc∞ R d × R d . ϕδ (x, y) = δ J ψ 2δ 2
(24)
k=1
Then 1 xk − yk x +y −d xj − yj J ∂xj ϕδ (x, y) = δ J ψ 2δ 2δ 2δ 2 k=j
d 1 −d xk − yk x+y , + J δ ∂j ψ 2 2δ 2
(25)
k=1
1 xk − yk x +y −d xj − yj ∂yj ϕδ (x, y) = − J δ J ψ 2δ 2δ 2δ 2 k=j
d 1 −d xk − yk x +y , + J δ ∂j ψ 2 2δ 2
(26)
k=1
and
d − y x x +y k k −d (∂xj + ∂yj )ϕδ (x, y) = δ ∈ Cc∞ R d × R d . J ∂j ψ 2δ 2
(27)
k=1
We let βε be defined according to (17), and take β = βε , ϕ = ϕδ in Lemma 3.2. We note that βε (r) → r+ uniformly in r ∈ R as ε → 0+. Recall the definition of αk , αˆ k in (20) and (22), using Lemma 2.9, for each k = 1, 2, . . . , each u, v ∈ R d fixed and α = (α1 , . . . , αd ),
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
∞ lim αk (u, v) = lim
ε→0+
ε→0+
325
βε (u − w)Fk (w) dw
v
∞ =
χ(u − w)Fk (w) dw = χ(u − v) Fk (u) − Fk (v)
v
= lim αˆ k (u, v). ε→0+
We recall that by earlier convention on notations, χ(r) = χ[0,+∞) (r). The right-hand side of the inequality in Lemma 3.2 consists of two terms. In view of the above limit for limε→0+ αk and limε→0+ αˆ k , the first term is easier to be controlled in the limδ→0+ limε→0+ limit. However, the second term is easier to be controlled in the limε→0+ limδ→0+ limit. Therefore, we need more careful estimates by considering ε → 0+, δ → 0+ at the same time with appropriate speeds. Lemma 3.3. Suppose that ε → 0+, δ → 0+ and εδ −1 → 0+ (e.g. let δ = ε 2/3 ), then
lim sup ε→0+, δ→0+, εδ −1 →0+
t
E
α u(r, x), v(r, y) · ∇y ϕ(x, y)
0 R d ×R d
+ αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy
dr
t d
E χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψ(x) dx dr .
0
Rd
k=1
Proof. We need to estimate the difference between αk (u, v) and χ(u − v)(Fk (u) − Fk (v)) more precisely. When u v, for w v, βε (u − w) = 0, therefore αk (u, v) = 0 = χ(u − v) Fk (u) − Fk (v) . When u > v, then by Lemma 2.9, u αk (u, v) =
βε (u − w)Fk (w) dw
v v∨(u−2ε)
Fk (w) dw +
= v
u
βε (u − w)Fk (w) dw
v∨(u−2ε)
= χ(u − v) Fk (u) − Fk (v) +
u
v∨(u−2ε)
βε (u − w) − 1 Fk (w) dw.
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This, together with the at most polynomial growth assumption on F in Condition 2.3, implies
αk (u, v) − χ(u − v) Fk (u) − Fk (v) 2
u
F (w) dw εCp 1 + |u|p
u−ε
for some p 1, where Cp is independent of ε and u, v and can be chosen to be independent of k = 1, 2, . . . , d as well. Similar conclusion holds for αˆ k (u, v) = χ(u − v) Fk (u) − Fk (v) +
u∧(v+2ε)
β (w − v) − 1 Fk (w) dw.
v
Combine the above estimate with (25), (26) and (27), t
E
α u(r, x), v(r, y) · ∇y ϕ(x, y)
0 R d ×R d
+ αˆ u(r, x), v(r, y) · ∇x ϕ(x, y) dx dy
dr
t
E
d χ u(r, x) − v(r, y) Fk u(r, x) − Fk v(r, y)
R d ×R d
0
k=1
× ∂xk ϕδ (x, y) + ∂yk ϕδ (x, y) dx dy
dr ε + C E δ d
j =1
t
p
p
2 + u(r, x) + v(r, y)
0 R d ×R d
− y x x x + y − y j j k k
× δ −d
J J ψ dx dy dr
2δ 2δ 2 k=j
+ εC
d
t
j =1
× δ
−d
E
p
p
2 + u(r, x) + v(r, y)
0 R d ×R d
d xk − yk x +y dx dy dr . J ∂j ψ 2δ 2
(28)
k=1
This gives the conclusion of the lemma.
2
Next, we estimate the second term on the right-hand side of the inequality in Lemma 3.2.
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327
Lemma 3.4. Suppose that ε → 0+, δ → 0+ and δ 2 ε −1 → 0+ (e.g. let δ = ε 2/3 ), then lim sup
βε u(r, x) − v(r, y)
E
ε→0+, δ→0+, δ 2 ε −1 →0+
(0,t]×Z R d ×R d
2 × σ y, v(r, y); z − σ x, u(r, x), z ϕδ (x, y) dx dy μ(dz) dr 0. Proof. Select compact K = Kψ ⊂ R d × R d to be such that supp(ϕδ ) ⊂ K for all 0 < δ < 1. Under Condition 2.3 on σ (x, u; z), and by the estimate 0 βε ε −1 C, 0 rβε (r) 2Cχ[0,2ε] (r) (e.g. Lemma 2.9) and that βε (r) = 0 for r 2ε or r 0, for (x, y) ∈ K,
2
β (u − v) σ (y, v; z) − σ (x, u, z)
2 2 |u − v| + 2β (u − v)|x − y|2 MK 2β (u − v)|u − v|ρK (z) 2 2 2 4CρK (2ε)MK (z) + 2Cε −1 |x − y|2 MK (z).
We have therefore E
2 β u(r, x) − v(r, y) σ y, v(r, y); z − σ x, u(r, x), z
(0,t]×Z R d ×R d
× ϕδ (x, y) dx dy μ(dz) dr
2 4CρK (2ε)
×
ϕδ (x, y) dx dy
K
2 MK (z) μ(dz)ψ∞ t.
2 MK (z) μ(dz) + 2(Cψ )ε −1 δ 2
Z
(29)
Z
The conclusion follows.
2
Theorem 3.5. Suppose u is a stochastic entropic solution of (1) and v is a stochastic strong entropic solution. Then (1) (L1 contraction) E u(t) − v(t) + 1 E u(0) − v(0) + 1 .
(30)
(2) (Comparison principle) Suppose that v(0, x) u(0, x) a.e. in x holds almost surely, and that E[(u(0, ·) − v(0, ·))+ 1 ] < ∞, then almost surely v(t, x) u(t, x)
a.e. in x.
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Remark 3.6. The following proof can be simplified considerably if we assume sup E u(t) 1 + v(t) 1 < ∞. 0tT
However, we adapted a definition of entropic solution which does not assume the above. We have no effective way of establishing such estimates in the absence of additional structural assumptions on σ . Proof. We define ϕδ according to (24) where δ = ε 2/3 . Taking ε → 0+ limit to the inequality in Lemma 3.2, by Lemmas 3.3 and 3.4, E
u(t, x) − v(t, x) + ψ(x) dx −
Rd
u(0, x) − v(0, x) + ψ(x) dx
Rd
t d
χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψ(x) dx dr . E
0
Rd
(31)
k=1
Let 0 ψN (x) = e−N
−1 |x|
∈ W 1,p (R d ), p = 1, 2, . . . , ∞. Then ∂k ψN (x) = −
1 xk ψN (x), N |x|
x = 0,
and ∂k ψN ∞ N −1 . Noting estimate (8) in the definition of entropic solution, by standard approximation by truncation and mollification arguments, (31) holds with ψ replaced by ψN . By the at most polynomial growth assumption on F in Condition 2.3, there exists p0 1, such that for any integer p > p0 1,
χ(u − v) Fk (u) − Fk (v) χ(u − v)
u
F (r) dr k
v
C1 (u − v)+ + |u|1+p + |v|1+p .
(32)
By (8) for u and v,
t
χ u(r, x) − v(r, x) Fk u(r, x) − Fk v(r, x) ∂k ψN (x) dx dr
E
0
x
C1 N
−1
t E 0
u(r, x) − v(r, x) + ψN (x) dx dr + θN (t),
Rd
where 1+p 1+p θN (t) = C1 N −1 sup E u(r) 1+p + v(r) 1+p . 0rt
(33)
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329
Let wN (t) = E[ R d (u(t, x) − v(t, x))+ ψN (x) dx] and w(t) = E[ R d (u(t, x) − v(t, x))+ dx]. Then wN (t) wN (0) + θN (T ) + C1 N
−1
t wN (r) dr,
0 t T.
0
By (8) for u, v, sup0tT wN (t) < ∞. Therefore by the Gronwall’s inequality, −1 sup wN (t) wN (0) + θN (T ) eC1 N T . 0tT
Send N → ∞. By the monotone convergence theorem, wN (t) → w(t) for every t 0. We arrive at (30). From (30) it follows that, if v(0, x) u(0, x) a.e. in x almost surely, then v(t, x) u(t, x) a.e. in x almost surely. 2 4. A constructive existence theory 4.1. Heuristic outlines We refer to Dafermos [3] (in particular Chapter VI) for background discussions and references on physical motivation of deterministic conservation laws. The stochastic case can be considered similarly. We would like to view (1) as limit of some microscopic stochastic system behaving effectively like ∂t u(t, x) + divx F u(t, x) = σ x, u(t, x); z ∂t W (t, dz) + εxx u, u(0) = u0 , (34) z∈Z
with asymptotically vanishing ε. The εxx term (with ε > 0) has a smoothing effect on solution u. For now, let us pretend that u = uε is a solution of (34) which is sufficiently smooth so that spatial derivatives up to the second order exist in classical sense and are continuous (Lemma 4.10). Let Φ ∈ C 2 (R) and Ψ = (Ψ1 , . . . , Ψd ) be an entropy–entropy flux pair (Definition 2.1). Then by Ito’s formula, at least formally, ∂t Φ u(t, x) + divx Ψ u(t, x) Φ u(t, x) σ x, u(t, x); z ∂t W (t, dz) + εΦ u(t, x) xx u(t, x) = z∈Z
1 + Φ u(t, x) 2
σ 2 x, u(t, x); z μ(dz).
(35)
Z
It is tempting to send ε → 0 and arrive at a limit. This is not correct. With each ε > 0 fixed, we can establish second order derivative information about u by exploiting the smoothing effect of ε. However, we do not know the magnitude of fluctuation for the nonlinear term
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J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
εΦ (u(t, x))xx u(t, x) as ε vanishes (note that u = uε ). This is the exact same difficulty as in deterministic conservation law case (σ = 0) which is handled as follows. First, we observe
2
εΦ u(t, x) xx u(t, x) = εxx Φ u(t, x) − εΦ u(t, x) ∇x u(t, x) .
(36)
We can always view xx Φ(u(t, x)) in (Schwartz) distributional sense, as far as u is locally integrable. We do not have control over |∇x u(t, x)| uniformly over ε > 0, and it is not reasonable to hope so (this quantity can blow up in finite time, even if u(0) ∈ C ∞ ∩ Cb , in the case of σ = 0). However, since Φ is convex (i.e. Φ 0) for ϕ 0, we have a one-sided trivial bound
εΦ (u)|∇x u|2 , ϕ 0.
In summary, we have now for 0 ϕ ∈ Cc2 , Φ u(t, ·) , ϕ − Φ u(s, ·) , ϕ t
Ψ u(r, ·) , ∇x ϕ dr +
= s
1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2
(s,t]×Z
t +ε
2
Φ u(r, ·) , ϕ − Φ u(r, ·) ∇x u(r, ·) , ϕ dr
s
+
Φ u(r, ·) σ ·, u(r, ·); z , ϕ W (dr, dz)
(s,t]×Z
t
Ψ u(r, ·) , ∇x ϕ dr +
s
+ o(ε) +
1 Φ u(r, ·) σ 2 ·, u(r, ·); z , ϕ μ(dz) dr 2
(s,t]×Z
Φ u(r, ·) σ ·, u(r, ·); z , ϕ W (dr, dz).
(37)
(s,t]×Z
Both the left-hand and right-hand sides of the inequality are stable under ε → 0+ limit, provided p we have Lloc type stability of u = uε (which is a lot easier and possible to estimate than ∇uε or uε ). Sending ε → 0, (10) follows. As in the deterministic scalar conservation law case, we face two main issues in order to make the above rigorous. One, we need regularity estimates on the solution u for the approximate equation (34) so that the Ito’s formula can be applied to the transformation of u(t, x) = uε (t, x). Two, we also need to verify relative compactness on u = uε (in some appropriate topology) as ε goes to zero. The first issue is more or less known in various different contexts for slightly different models in stochastic analysis literature. We will adapt existing methods and discuss the issue more carefully (because of the generality here) in the first subsection below. The second issue, however, has never been considered in its current generality (i.e. with the stochastic term). Kim [8] modifies deterministic arguments to construct a very special SPDE which is essentially reformulated as a randomness in coefficients type of conservation law. To handle the general
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
331
situation here, we have to introduce non-trivial new technical ideas. In particular, we derive stochastic versions of the (originally deterministic) compensated compactness results. As in the deterministic case, the compensated compactness argument will eventually restrict our consideration to one space dimension (i.e. d = 1) only. However, most of our estimates are not restricted by dimensionality. In deterministic theory, one can also obtain compactness in C([0, ∞), Lp ) (e.g. p = 1 or p = ∞) by Ascoli–Arzela type argument by estimating corresponding modulus of continuities in time and in space variables (e.g. [3, Section 6.3]). In particular, the spatial modulus of continuity estimate is usually achieved by perturbing conservation law equation through a spatial translation of the solution and then prove a comparison result. Such procedure does not generalize well to the type of SPDEs as in (1). Because of the σ term, spatial translation of solution will generally not be another solution or even an approximate solution in some well controlled sense. Similar observation was also made by Kim [8] in a simpler model context. Theorem 4.1. Suppose that Condition 2.7 holds. Then there exists a stochastic strong entropic solution (hence also a stochastic entropic solution, see Definitions 2.6 and 2.5) u = u(t, x) for (1). We divide proof into several parts below. 4.2. Existence and regularity of approximate equation (34) Throughout this subsection, we assume that F = (F1 , . . . , Fd ) satisfies Fk ∈ C ∞ for each k, (m) and that the mth order derivative of Fk satisfies |Fk (r)| Cm < ∞, m = 0, 1, 2, . . . . We m m also assume that σ (x, u; z), Dx σ (x, u; z), ∂u σ (x, u; z) exist and are continuous and uniformly bounded and Dxm σ (·, u; z) ∈ S(R d ) (see (15)), for all m = 1, 2, . . . . Finally, Z supx,u |σ 2 (x, u; z)| μ(dz) < ∞. 4.2.1. Existence of solution when ε > 0 is held fixed Let the fundamental solution of the heat equation be denoted by G(t, x) = Gε (t, x) =
1 2 e−|x| /(4εt) , d/2 (4πεt)
t > 0.
Let E[u0 22 ] < ∞. First, we define successive approximates to (34): let u0 (t, x) = u0 (x), un (0, x) = u0 (x) and dun (t, x) + ∇ · F un−1 (t, x) dt = εun (t, x) dt + σ x, un−1 (t, x); z W (dt, dz). (38) Z
We consider the mild solution for the above equation given by
t
u (t, x) =
G(t, x − y)u0 (y) dy −
n
y
+ (0,t]×Z y
G(t − s, x − y) 0 y
d
∂yi Fi un−1 (s, y) dy ds
i=1
G(t − s, x − y)σ y, un−1 (s, y); z dy W (ds, dz).
(39)
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The above right-hand side needs some explanation. The first term is always well defined because of E[u0 22 ] < ∞, the second term is well defined provided Dy un−1 ∈ L∞ ([0, T ]; L2 (R d )) for each T > 0; the third term is always well defined because of our earlier assumptions on σ . At this point, it is only clear that u1 is defined. We need regularity information on un−1 to conclude that un is defined for n 2. We claim that all un (t, x) are well defined and un (t, ·) ∈ S(R d ). To verify this claim, we need the following properties. Lemma 4.2. Let h = h(s) = h(s, x) be an adapted process in C([0, T ]; H p (R d )), p = 1, 2, . . . , T > 0 and h(s, ·) ∈ S(R d ). Let V (t, x) = G(t − s, x − y)h(s, y) dy ds. (0,t]×R d
Then V = V (t) = V (t, x) is an adapted process in C([0, T ]; H p (R d )), T > 0 and V (t, ·) ∈ S(R d ) and in particular, G(t − s, x − y)h(s, y) dy ds = − G(t − s, x − y)∂yk h(s, y) dy ds. ∂xk (0,t]×R d
(0,t]×R d
Similarly, if for each z ∈ Z fixed, f (·,·; z) = f (t, y; z) ∈ C([0, T ]; H p (R d )) is an Ft -adapted process in t for p = 1, 2, . . . , T > 0, and T E
m
2 2 f (s, y; z) + Dy f (s, y; z) dy μ(dz) ds < ∞,
0 Rd Z
where m = 1, 2, . . . , and f (s,·; z) ∈ S(R d ), then N(t, x) = G(t − s, x − y)f (s, y; z) dy W (ds, dz),
(40)
(0,t]×Z y
has the following property for each T > 0. Lemma 4.3. N (t) ∈ C([0, T ]; H p (R d )), p = 1, 2, . . . , and in particular, G(t − s, x − y)∂yk f (s, y; z) dy W (ds, dz), ∂xk N (t, x) = −
a.s.
(0,t]×Z y
and N(t, ·) ∈ C ∞ (R d ). Proof. Continuity of N as an L2 -valued process in t can be handled as in Proposition 7.3 of Da Prato and Zabczyk [4]. Regarding ∂xk N , all we need is to show that (Schwartz) distributional derivative of N agrees with the right-hand side. Then, since the right-hand side is continuous in x, the identify is established. For each ϕ ∈ Cc∞ (R d ),
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
E
333
G(t − s) ∗x f (s, x; z)∂xk ϕ(x)
[0,t]×Z x
2
− G(t − s) ∗x (−∂k f )(s, x; z)ϕ(x) dx W (ds, dz)
E
G(t − s) ∗x f (s, x; z)∂k ϕ(x)
= [0,t]×Z
2 − G(t − s) ∗x (−∂k f )(s, x; z)ϕ(x) dx
μ(dz) × ds = 0.
In the above, ∗x means convolution with respect to spatial variable x only. By such representation, it follows that ∂xk N has trajectory in C([0, T ]; L2 (R d )). Replace f by ∂xk f and repeat the above arguments, we conclude that N has trajectory in C([0, T ]; H p (R d )) and N (t, ·) ∈ C p (R d ) for p = 1, 2, . . . . 2 Lemma 4.4. N (t, ·) ∈ S(R d ) almost surely for each t fixed. Proof. From Lemma 4.3, we already know that N (t, ·) ∈ C ∞ (R d ). Therefore, we only need to show
sup |x|m N (t, x) < ∞,
a.s.
x∈R d
On the one hand, by a Sobolev (Morrey’s) inequality (e.g. [7, (23), p. 268]), there exists deterministic constant C > 0 when p d,
sup |x|m N (t, x) C |x|m N (t, ·) W 1,p (R d ) .
x∈R d
On the other hand, direct computation shows that for t > 0, there exist (constant coefficient) mth order polynomials of t, denoted by Ci (t), i = 0, 1, 2, . . . , m, (C0 = 0 is a constant), such that t m ∂xmk G(t, x − y) = C0 xkm + C1 (t)xkm−1 + · · · + Cm (t) G(t, x − y). Therefore, by induction, it is sufficient to prove that for all j = 0, 1, . . . , m,
|t
(0,t]×Z y
=
− s|j ∂xmk G(t
− s, x − y)f (s, y; z) dy W (ds, dz)
|t − s| G(t − s, x j
1,p
Wx
− y)∂ymk f (s, y; z) dy W (ds, dz)
(0,t]×Z y
The above holds because of another Sobolev embedding · W 1,p C · H 1+m ,
p 2, 2m > d,
1,p
Wx
< ∞,
a.s.
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and that because for i = 0, 1, . . . ,
E
x
|t − s| G(t − s, x j
(0,t]×Z y
C
dx
2 n+i G(t − s) ∗x ∂yk f (s, x; z) μ(dz) ds dx
E x
2
− y)∂yn+i f (s, y; z) dy W (ds, dz)
k
(0,t]×Z
n+i ∂ f (s, ·; z) 2 μ(dz) ds < ∞, yk 2
CE [0,t]×Z
where the first inequality follows from Burkholder–Davis–Gundy inequality, and the second follows from Young inequality for convolutions. 2 In view of Lemmas 4.2–4.4, by induction, we conclude the following. Lemma 4.5. For each n = 1, 2, . . . , un (t) ∈ C([0, T ]; H p (R d )), p = 1, 2, . . . , un (t, ·) ∈ S(R d ), t > 0. It is well known that, in the context of stochastic semi-linear equation, under moderate conditions, a mild solution is also a weak solution. The following is a statement of this kind in our present context. Its proof follows, for instance, from a straightforward adaptation of [4, Proposition 6.4]. Lemma 4.6. For each ϕ ∈ Cc∞ (R d ),
t
∇ϕ, F un−1 (s) ds
u (t), ϕ − u (0), ϕ = n
n
0
+
σ x, un−1 (r, x); z ϕ(x) dx W (dr, dz)
[0,t]×Z x
t +ε
ϕ, un−1 (r) dr.
0
We define energy functionals e2m : L2 (R d ) → [0, ∞]: 2 1 e2m (u) = m u 2 , 2
m = 0, 1, 2, . . . .
Lemma 4.7. There exist finite constants Cε,m,T > 0 which is independent of n such that m n E e2k (u0 ) , t T . E e2m u (t) Cε,m,T 1 + k=0
(41)
(42)
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335
Proof. Apply Ito’s formula to e2m , there exists finite constant C˜ ε,m,T > 0, t m n 2 n m n−1 m n n F u (r) , ∇u (r) − ε ∇u (r) 2 dr E e2m u (t) = E e2m u (0) + 1 + E 2
0
m
σ x, un−1 (r, x); z 2 dx μ(dz) dr
x
[0,t]×Z x
E e2m (u0 ) + C˜ ε,m,T
t
m n−1 2 E F u (r) 2
0
+ 1 + E e0 un−1 (r) + · · · + e2m un−1 (r) dr. Denote
m n Mn (t) = 1 + E e2k u (t) ,
m M(0) = 1 + E e2k (u0 ) .
k=0
k=0
Then t Mn (t) cM(0) + c
Mn−1 (s) ds 0
where the constant c > 0 is independent of n. Choose K so large that c follows inductively (in n) that Mn (t) cM(0)eKt .
T 0
e−Kt dt < 1. Then it
2
We now show that un converges in appropriate sense to a limiting process. We adapt a wellknown fixed point argument which can be found in proof of part two of Theorem 7.4 in [4], for instance. Because of the term divx F (u(t, x)), the adaptation requires explanation. Lemma 4.8. There exists a Lp (R d )-valued (p 2), Ft -adapted process u satisfying p lim sup E u(t) − un (t) p = 0, n→∞ 0tT
(43)
and for m = 0, 1, 2, . . . , m E e2k (u0 ) , E e2m u(t) Cε,m,T 1 +
t T.
(44)
k=0
In addition, p sup E u(t) p < ∞.
0tT
(45)
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Moreover, u is a mild solution to (34) in the sense that
t
u(t, x) =
G(t, x − y)u(0, y) dy − y
+
G(t − s, x − y)
d
∂yi Fi u(s, y) dy ds
i=1
0 y
G(t − s, x − y)σ y, u(s, y); z dy W (ds, dz).
(46)
(0,t]×Z y
Proof. First, by direct integration ∂x G(t, ·) = i 1
∂x G(t, x) dx = Ct −1/2 , i
t > 0.
x
We denote n L1 u (t, x) =
t
d
∂i G(t − s, x − y)Fi un (s, y) dy ds.
0 y∈R d i=1
For p 1, t 0, we define a deterministic measure on [0, t] by √ √ m(ds) = mt (ds) = ∂i G(t − s, ·) 1 ds = 2Cd( t − t − s ). Then p E L1 un (t, ·) − L1 um (t, ·) p c1
d i=1
p t n E ∂i G(t − s) ∗ u (s) − um (s) (·) ds p
0
p t d
n
∂i G(t − s) ∗ u (s) − um (s) p ds
E
c2
i=1
0
p t
n
m u (s) − u (s) p m(ds)
c3 E
0
t c4 E 0
c5 t
1/2
t p n m u (s) − u (s) m(ds) = c4 E un (s) − um (s) p m(ds) p p
p sup E un (s) − um (s) p ,
0st
0
(47)
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where the second inequality follows from the Minkowski’s inequality, the third one follows from the Young’s inequality for convolutions, the fourth one follows from Jensen’s inequality. Therefore, p p sup E L1 un (s) − L1 um (s) p ct 1/2 sup E un (s) − um (s) p .
0st
0st
Denote
n L2 u (t, x) =
G(t − s, x − y)σ y, un (s, y); z dy W (ds, dz).
(0,t]×Z y∈R d
By properties of stochastic integral for p 2, p E L2 un (t, ·) − L2 um (t, ·) p
G(t − s, x − y) σ y, un (s, y); z c6 E
x
(0,t]×Z y
p/2 2 m
− σ y, u (s, y); z dy μ(dz) ds dx t c7 E
G(t − s, ·) ∗ un (s) − um (s) (x) 2(p/2) dx ds
0 x
t c8 E
G(t − s, ·) un (s) − um (s) p ds 1 p
0
p c9 t sup E un (s) − um (s) p . 0st
Combine the above, and apply them to (39), there exist ρ ∈ (0, 1) and T0 > 0 which are independent of the initial conditions un (0), n = 1, 2, . . . , such that
n
u − um ≡ sup E un (t) − um (t) p 1/p ρ un−1 − um−1 . p 0tT0
By a fixed point argument and by “pasting” short time existence result to obtain global existence result, we have existence of u satisfying (43). The same type estimates can be used to show (45). Then (44) follows from (42) and Fatou’s lemma, and the mild solution property (46) follows from the fixed point argument applied to (39). 2 Taking limit n → ∞ to the equality in Lemma 4.6 we have
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Lemma 4.9. For each ϕ ∈ Cc∞ (R d ),
u(t), ϕ − u(0), ϕ =
t
∇ϕ, F u(s) ds +
σ x, u(r, x); z ϕ(x) dx W (dr, dz)
[0,t]×Z x
0
t +ε
ϕ, u(r) dr.
0
4.2.2. Regularity of solution when ε > 0 is fixed As in the proof of Proposition 7.3 in [4], we can show that
U (t, x) =
G(t − s, x − y)σ y, u(s, y); z dy W (ds, dz)
(0,t]×Z y
has a continuous modification as L2 (R d )-valued process. Therefore, suppose that E[e2m (u0 )] < ∞, then not only do we have (44), it can also be shown a posterior that u ∈ C([0, ∞); L2 (R d )) for all d = 1, 2, . . . . In the rest of this subsection, we denote u = uε to emphasize its dependence on ε > 0. Lemma 4.10. Suppose that E[e2m (u0 )] < ∞ for 2m [d/2] + 3. Then there exists an Ft adapted process u = u(t) ∈ C([0, ∞); L2 (R d )) satisfying almost surely that (1) e2m (u(t)) < ∞, for all t > 0; (2) ∂ij u = ∂xi ,xj u(t, ·) ∈ C(R d ) for all i, j = 1, . . . , d. Therefore, (34) holds in the classical strong sense. That is, for each x fixed, (34) holds as a finite-dimensional stochastic differential equation. Proof. The conclusions then follow from (44) and from a Sobolev inequality—see Evans [7, Theorem 6, p. 270]. 2 Apply Ito’s formula to (34), we obtain the following. Lemma 4.11. Let Φ ∈ C 2 (R) and convex. If E[e2m (u0 )] < ∞ for 2m > [d/2] + 3, then there exists Ft -adapted solution with properties listed as in Lemma 4.10 such that (37) holds. 4.2.3. Uniform in ε estimate for solutions of (34) We now denote uε , Fε , σε to emphasize their dependence on ε. Throughout this subsection, we assume that those conditions on Fε , σε at the beginning of Section 4.2 still holds. Next, we derive some estimates which are uniform in ε. For such purpose, we require initial conditions satisfy, for some 2m > [d/2] + 3, E e2m uε (0) < ∞,
ε > 0,
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and p p sup E uε (0) p + uε (0) 2 < ∞,
p = 1, 2, . . . .
ε
(48)
Finally, we require that
sup σε (x, u; z) f (x) 1 + |u| M(z),
(49)
ε>0
where
ZM
2 (z) μ(dz) < ∞
and f ∈ L∞ (R d ) ∩ L2 (R d ).
Lemma 4.12. For even positive integers p = 2, 4, 6, . . . , p sup sup E uε (t, ·) p < ∞.
(50)
ε 0tT
p
Proof. By (45), we already know that sup0tT E[uε (t)p ] < ∞ for every p 2 and T 0. u Let Φ(u) = (p)−1 |u|p and Ψ (u) = (Ψ1 (u), . . . , Ψd (u)) be Ψk (u) = 0 Φ (r)(Fε ) k (r) dr. Then for each x ∈ R d fixed, we apply (35), (36), Lemma 4.10 and integration with respect to x to arrive at p p E uε (t) p − E uε (0) p
t p(p − 1)
up−2 (s, x)σε2 ε
E
x, uε (s, x); z dx μ(dz) ds.
Z x
0
Gronwall inequality (noting (49)) then implies p p sup E uε (t) p CT sup E uε (0) p .
2
(51)
ε>0
0tT
In the case of p = 2, since by (37), uε (t) 2 − uε (0) 2 2
t
2
= 0 Z x
σε2 x, uε (s, x); z dx μ(dz) ds − 2ε
+
∇x uε (s, x) 2 dx ds
[0,t]×R d
uε (s, x)σ x, uε (s, x); z dx W (ds, dz).
(0,t]×Z x
This leads to the following estimates:
(52)
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Lemma 4.13. For each p = 1, 2, . . . T p 2 sup E ε ∇uε (t) 2 dt < ∞. ε>0
(53)
0 2p
Proof. Apply Ito’s formula to uε (t)2 using (52), by (50), (48), 2p 2p sup E uε (T ) 2 + uε (0) 2 < ∞. ε>0
Note that, (49) and (50) give T σε2
sup E ε>0
0
x, uε (s, x); z dx μ(dz) ds
p <∞
z x
and by Burkholder–Davis–Gundy inequality and (49),
sup E
ε>0
(0,T ]×Z x
c sup E
ε>0
p
uε (s, x)σ x, uε (s, x); z dx W (ds, dz)
u2ε (s, x)σ 2
p/2
< ∞. x, uε (s, x); z dx μ(dz) ds
[0,T ]×Z
In view of (52), the conclusion follows from the above estimates.
2
More generally, we have the following useful estimate. Lemma 4.14. Let Φ ∈ C 2 (R) with Φ, Φ , Φ having at most polynomial growth. Φ needs not be convex. Then
p T
2
sup E ε Φ uε (t, x) ∇x uε (t, x) dx dt < ∞,
ε>0 0
p = 1, 2, . . . , T > 0.
(54)
Rd
Proof. Let (Φ, Ψ ) be an entropy–entropy flux pair. The equality in (37) holds when u is replaced by uε , ϕ = 1, for general (possibly non-convex) Φ ∈ C 2 . Using (50), the rest of the proof follows that of the previous Lemma 4.13. 2 4.3. Convergence of {uε (t, x): ε > 0} as measure-valued processes We generalize L.C. Young’s relaxed measure approach to treat convergence of nonlinear PDEs in this stochastic setting. We identify uε (t, x) with a random measure-valued function νε (t, x, du) = δuε (t,x) (du).
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With a slight abuse of notation, we also denote νε (t) = νε (t, dx, du) = νε (t, x, du) dx, and view it as a random measure-valued process in the following sense. Let M0 = M(R d × R) be the space of nonnegative Radon measures ν on R d × R with ν(dx, R) = dx. We endow M0 with a topology τ0 so that νn → ν ∈ M0 if and only if f, νn → f, ν for all f ∈ Cb (R d × R) satisfying f (x, u) = 0 when |x| > k for some k > 0. (M0 , τ0 ) is metrizable as follows. We denote Π ν1 ,ν2 = π ∈ M R d × R × R d × R : π dx, du; R d × R = ν1 (dx, du); π R d × R; dy, dv = ν2 (dy, dv) , ν1 , ν2 ∈ M0 ,
(55)
and introduce r(ν1 , ν2 ) =
∞ 1 qk (ν1 , ν2 ) , 2k 1 + qk (ν1 , ν2 ) k=1
where
qk2 (ν1 , ν2 ) = inf
2 2 ν1 ,ν2 . |x − y| + |u − v| ∧ 1 π(dx, du; dy, dv): π ∈ Π
|x|k, |y|k
Note that on each subspace Ak = {(x, u); |x| k, u ∈ R} ⊂ R d × R, ν(Ak ) = Ck is a finite constant which only depends on k; |x − y|2 + |u − v|2 ∧ 1 defines a metric on Ak which gives the same topology as the one induced by usual Euclidean distance. Consequently qk is just a 2-Wasserstein metric on space of measures on Ak with fixed total mass Ck . It induces the usual weak convergence topology on such a sub-space of finite measures. See Ambrosio, Gigli and Savaré [1, Chapter 7] for some properties of such metric. It follows that (M0 , r) is a complete separable metric space. It can be shown that each νε (t) has continuous trajectories in C([0, ∞), M0 ) ⊂ M([0, ∞); M0 ). Here and below, we write M([0, ∞); M0 ) to denote the space of Borel-measurable, M0 -valued processes on [0, ∞) topologized by a metric d ν1 (·), ν2 (·) =
∞
e−t 1 ∧ r ν1 (t), ν2 (t) dt.
(56)
0
(M([0, ∞); M0 ), d) is a complete separable metric space. For properties of such type of space, see Kurtz [12, Section 4]. We have trouble establishing convergence in probability (even along subsequences) of {νε (·): ε > 0} in C([0, ∞); M0 ). We will prove convergence in M([0, ∞); M0 ) instead. By existence of slicing measure (e.g. [6, Theorem 10, p. 14]), for each ν ∈ M0 , there exists a probability measure-valued function ν(x; ·) = ν(x; du) ∈ P(R) such that for each f ∈ Cb (R d × R),
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(1) x → u f (x, u)ν(x; du) is Lebesgue measurable; (2) R d ×R f (x, u)ν(dx, du) = R d R f (x, u)ν(x; du) dx. Therefore, each process ν(t) ∈ M([0, ∞), M0 ) also admits a representation ν(t, dx, du) = ν(t, x; du) dx. From this point on, we assume Condition 2.7 holds for σ and for F , except the condition d = 1 (i.e. d may still be any positive integer). We also assume that p E u0 p < ∞,
p = 1, 2, . . . .
We take σε and Fε to be the following particular approximation of σ and F . Let J ∈ Cc∞ (R) be the one-dimensional mollifier in (16), and let φ ∈ Cc∞ (R) be such that 0 φ 1, φ(r) = 1 for |r| < 1 and φ(r) = 0 for |r| > 2 and |φ (r)| 2. Let Fε = (F1,ε , . . . , Fd,ε ) with Fk,ε (r) = φ ε|r|2 Fk (r) ∗ Jε (r), σε (x, u; z) =
d y v
Jε (xk − yk )Jε (u − v) φ ε |y|2 + |v|2 σ (y, v; z) dy dv,
k=1
where Jε (r) = ε −1 J (r/ε). Then Fε , σε satisfy the conditions required at the beginning of Section 4.2. Under (14),
σε (x, u; z) − σ (x, u, z)
d
Lφ,σ |y| + |v| Jε (yk )Jε (v) dyk dv M(z) + σ (x, u; z)
k=1
y v
εC M(z) + σ (x, u; z) ,
(57)
where Lφ,σ > 0 is a constant. Similarly, we can estimate the error for approximating F by Fε . By part one of Condition 2.3, there exist constants C > 0 and p0 ∈ {1, 2, . . .},
F (r + s) − F (r) |s|C 1 + |r|p0 , k
k
r ∈ R, |s| < 1.
Therefore
F (r) − F (r) εC1 1 + |r|p0 . k,ε k
(58)
We now construct a smooth approximation of u0 . Let uε (0, x) = y∈R d
Jε (x − y) u0 (y)φ ε|y|2 dy ∈ Cc∞ R d .
(59)
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
343
Then for each ε > 0 and m = 1, 2, . . . fixed, 2 1 E e2m uε (0) = E m Jε ∗ u0 φ ε|y|2 2 Cm,ε E u0 22 < ∞, 2 and p p sup E uε (0) p E u0 p < ∞,
p = 1, 2, . . . .
(60)
ε>0
Using the above error estimates, we can derive the following main result of this section. Lemma 4.15. There exists an Ft -adapted process ν0 (·) with trajectory in M([0, ∞); M0 ) such that lim E r νε (t), ν0 (t) = 0,
ε→0+
t 0.
This implies that {νε (·): ε > 0}, as metric space (M([0, ∞); M0 ), d)-valued random variables, converges in probability to ν0 (·). Therefore, for each 0 t1 · · · tm , lim νε (t1 ), . . . , νε (tm ) = ν0 (t1 ), . . . , ν0 (tm ) ,
ε→0+
in probability.
Proof. Let 0 ϕδ ∈ Cc∞ (R d × R d ) be of the form as in (24), and βε be of the form as in Lemma 2.9. We derive an approximate version of the inequality appearing in Lemma 3.2. Notice that xx βε uθ (t, x) − uκ (t, y)
2
= βε uθ (t, x) − uκ (t, y) xx uθ + βε uθ (t, x) − uκ (t, y) ∇x uθ (t, x) . By Ito’s formula,
βε uθ (t, x) − uκ (t, y) ϕδ (x, y) dx dy
R d ×R d
βε uθ (0, x) − uκ (0, y) ϕδ (x, y) dx dy + M(t) + A1 (t) + A2 (t) + A3 (t),
R d ×R d
with non-decreasing processes
1
A
(t) = A1ε,δ,θ,κ (t) =
t
0 R d ×R d
βε uθ (r, x) − uκ (r, y) − divx Fθ uθ (r, x)
+ divy Fκ uκ (r, y) ϕδ (x, y) dx dy
dr,
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2
A
t
1 2
(t) = A2ε,δ,θ,κ (t) =
β uθ (r, x) − uκ (r, y)
ε
σθ x, uθ (r, x); z
z
0 R d ×R d
2 − σκ y, uκ (r, y); z μ(dz)ϕδ (x, y) dx dy dr, t 3
A
(t) = A3ε,δ,θ,κ (t) =
βε uθ (r, x) − uκ (r, y) (θ xx − κyy )ϕδ (x, y) dx dy dr,
0 R d ×R d
and martingale
βε uθ (r, x) − uκ (r, y)
M(t) = Mε,δ,θ,κ (t) = R d ×R d (0,t]×Z
× σθ x, uθ (r, x); z − σκ y, uκ (r, y); z W (dr, dz)ϕδ (x, y) dx dy.
We can invoke the a priori estimates in Lemma 4.12 to estimate the Ak s. First, since ϕδ has compact support in x, y, uniformly in δ > 0, and since 0 βε (r) r,
sup (θ xx − κyy )ϕδ (x, y) C(θ + κ)δ −2 , x,y
by (50), for each t > 0, lim
θ→0+, κ→0+, δ→0+, ε→0+, δ −2 (θ+κ)→0+
E A3 (t) = 0.
Noting 0 β (r) ε −1 C (Lemma 2.9), E A2 (t) I + II + III where 1 I= 2
t E 0
×
β uθ (r, x) − uκ (r, x)
ε
R d ×R d
σθ x, uθ (r, x); z − σκ y, uκ (r, y); z 2 μ(dz)ϕδ (x, y) dx dy dr,
z
II = ε −1 C
t
III = ε
t C
E 0
σθ x, uθ (r, x); z − σ x, uθ (r, x); z 2 μ(dz)ϕδ (x, y) dx dy dr,
R d ×R d z
0 −1
E
R d ×R d z
σ y, uκ (r, y); z − σκ y, uκ (r, y); z 2 μ(dz)ϕδ (x, y) dx dy dr.
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345
By (57), (II + III) = 0.
lim
ε→0+, δ→0+, κ→0+, θ→0+; ε −1 (θ 2 +κ 2 )→0+
By identical arguments as in the proof of Lemma 3.4, we have (29) holds when u(r, x) and v(r, y) are replaced by uθ (r, x) and uκ (r, y), respectively. Therefore lim
ε→0+, δ→0+, θ→0+, κ→0+, δ 2 ε −1 →0+
I = 0.
In summary lim
θ→0+, κ→0+, δ→0+, ε→0+, ε −1 (θ 2 +κ 2 )→0+, ε −1 δ 2 →0+
E A2 (t) = 0.
We now estimate A1 . First, we approximate α = αε , αˆ = αˆ ε in (20) and (22) by αε,θ = (αε,θ;1 , . . .) and αˆ ε,θ = (αˆ ε,θ;1 , . . .) ∞ αε,θ;k (u, v) =
βε (u − w)Fθ,k (w) dw
u =
v
u αˆ ε,θ;k (u, v) =
βε (u − w)Fθ,k (w) dw,
v βε (w − v)Fθ,k (w) dw
−∞
u =
βε (w − v)Fθ,k (w) dw.
v
By (58), there exists p > 1 such that
αε,θ;k (u, v) − αε;k (u, v) Cθ 1 + |u|p + |v|p . Similar estimate holds for αˆ ε,θ;k − αˆ ε;k . Therefore t
A (t) =
E 1
0
αε,θ uθ (r, x), uκ (r, y) ∇y ϕδ (x, y)
R d ×R d
+ αˆ ε,κ uθ (r, x), uκ (r, y) ∇x ϕδ (x, y) dx dy
dr
t
E
0
αε uθ (r, x), uκ (r, y) ∇y ϕδ (x, y)
R d ×R d
+ αˆ ε uθ (r, x), uκ (r, y) ∇x ϕδ (x, y) dx dy
dr
t θ κ +C + E δ δ
s R d ×R d
p
p 1 + uθ (t, x) + vκ (t, y)
− y x x x + y − y j j k k
× δ −d
J J ψ dx dy dr .
2δ 2δ 2 k=j
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We note that the estimates in (28) still hold with u, v replaced by uθ , vκ . We write lim = lim supε→0+,δ→0+,κ→0+,θ→0+, θ+κ →0+, ε →0+ . By (27) and (32), δ
δ
lim A1 (t) t
lim E
0
d χ uθ (r, x) − uκ (r, y) Fj uθ (r, x) − Fj uκ (r, y)
R d ×R d
j =1
x+y
dx dy dr × Jδ (x − y)∂j ψ
2
t
C lim E
x + y
uθ (r, x) − uκ (r, y) Jδ (x − y)∂j ψ dx dy dr , 2
0 R d ×R d
where C is a constant independent of the choice of ψ . By symmetry, we also have similar estimates when the roles of uθ (t, x) and uκ (t, y) are reversed. √ √ Now, we let θ, κ → 0+ and take ε = θ ∨ κ, δ = ε 2/3 . Then (θ + κ)δ −2 → 0+, (θ 2 + κ 2 )ε −1 → 0+, and δ 2 ε −1 → 0+, εδ −1 → 0+. From the construction of βε , there exists a constant C0 > 0 such that
βε (r) + βε (−r) − |r| εC0 . Denote
uθ (t, x) − uκ (t, y) Jδ (x − y)ψ x + y dx dy . 2
mψ (t) = lim E R d ×R d
It follows from the above estimates that
mψ (t) − mψ (s) C
t d
m∂j ψ (r) dr,
0 s t.
s j =1 −1
By simple approximation, the above still holds when ψ = ψN (x) = e−N |x| ∈ W 1,p (R d ), p = 1, 2, . . . , ∞. As in the proof of Theorem 3.5, it follows (by Gronwall inequality) then lim mψN (t) lim mψN (0).
N →∞
N →∞
We now recall the way uθ (0, x) is constructed in (59), by the integrability estimates in (60),
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
lim mψN (0) lim
N →∞
lim E
N →∞ θ→0+
uθ (0, x) − u0 (x) ψN (x) dx
x
uκ (0, y) − u0 (y) ψN (y) dy
+ lim E κ→0+
347
= 0.
y
We now estimate limN mψN (t) using qK (νθ (t), νκ (t)) for K = 1, 2, . . . . Let stochastic measure πt (dx, du; dy, dv) = δuκ (t,y) (dv) Jδ (x − y) dy δuθ (t,x) (du) dx. Then πt ∈ Π νθ (t),νκ (t) (see definition in (55)), for N K,
uθ (t, x) − uκ (t, y) Jδ (x − y)ψN x + y dx dy 2
R d ×R d
e−1
|u − v|πt (dx, du; dy, dv)
u,v∈R;|x|K,|y|K 2 e−1 qK
νθ (t), νκ (t) − e−1
|x − y|2 Jδ (x − y) dx dy.
|x|
This implies that, for each K fixed, 2 νθ (t), νκ (t) lim E qK
θ,κ→0+
e lim
lim
N →∞ κ→0+, θ→0+
uθ (0, x) − uκ (0, y) Jδ (x − y)ψN x + y dx dy 2
E (x,y)∈R d ×R d
lim mψN (0) 0. N →∞
By definition of d in (56) and by dominated convergence theorem lim E d νθ (·), νκ (·) =
∞
θ,κ→0+
e−t
lim E 1 ∧ r νθ (t), νκ (t) dt = 0.
θ,κ→0+
0
There exists a process ν0 (·) ∈ M([0, ∞); M0 ) satisfying the conclusion of the lemma.
2
Remark 4.16. We suspect that limε→0+ νε = ν0 in probability as C([0, ∞); M0 )-valued random variables. However, due to a lack of good control of E[sup0tT |M(t)|] by r(νθ (t), νκ (t)), and due to a lack of uniform estimate on modulus of continuity (in space C([0, ∞); M0 )) for νε , we cannot confirm this.
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4.4. Stochastic compensated compactness and existence of function-valued limit process We generalize the deterministic compensated compactness argument to a stochastic setting, showing that ν0 (t, x, du) as a (random) measure in u has to be point mass. Consequently it can be identified with a random function u0 (t, x). Unlike the rest of the paper, we assume an extra assumption d = 1 throughout this section. Let u ν0 (t, x; du). u(t, ¯ x) = u∈R
The main result of this section is Lemma 4.17. The following holds: F (u)ν0 (t, x, du) dt dx = F u(t, ¯ x) dt dx,
a.s.
u∈R
Furthermore, if the set of r such that F (r) = 0 is dense in r ∈ R, then ν0 (t, dx, du) = δu(t,x) (du) dx, ¯
almost surely.
(61)
We divide the proof into several parts which are proved in subsections that follows. First, it is useful to further relax our view point by viewing νε (dt, dx, du) = νε (t, dx, du) dt (where ε 0) as random measures on [0, ∞) × R d × R. Let M = M([0, ∞) × R d × R) be the space of nonnegative Radon measures ν on [0, ∞) × R d × R satisfying ν(dt, dx, R) = dt × dx. We endow M with a variant of weak topology so that νn → ν if and only if f, νn → f, ν for all f = f (t, x, u) ∈ F ⊂ Cb ([0, ∞) × R d × R). The set F consists of bounded continuous functions f with compact support in t, x uniformly in u. That is, there exists C = Cf such that f (t, x; u) = 0 once t + |x| > C, for all u ∈ R. As in the M0 introduced earlier, there is a metrizable topology on M which coincide with the above notion of sequential convergence and turning (M, τ ) into a Polish space. Therefore, {νε : ε > 0} is a sequence of M-valued random variables. Note that M([0, ∞), M0 ) can be continuously embedded into M. Let (Φ, Ψ ) be a given entropy–entropy flux pair with Φ, Φ , Φ having at most polynomial growth. We define Ito’s integral Mε (t, x) = Φ uε (r, x) σε x, uε (r, x); z W (dr, dz) [0,t]×Z
and let Φε (t, x) = Φ uε (t, x) ,
Ψε (t, x) = Ψ uε (t, x) ,
χε (t, x) = χε,1 + χε,2 ,
ψε (t, x) = ψε,1 + ψε,2 ,
and
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349
where χε,1 (t, x) = ε∂x2 Φ uε (t, x) = ε∂x Φ uε (t, x) ∂x uε (t, x) , χε,2 (t, x) = ∂t Mε (t, x); and ψε,1 (t, x) =
1 2
Φ uε (t, x) σε2 x, uε (t, x); z μ(dz),
Z
2
ψε,2 (t, x) = −εΦ uε (t, x) ∂x uε (t, x) . The meaning of χε,2 is given as follows. Mε (t) = Mε (t, x, ω) is a continuous function in t for each x, ω ∈ Ω fixed. We can take Schwartz distributional derivative ∂t in t of Mε and such derivative is χε,2 . The equality in (37) (Lemma 4.11) is a statement that, ∂t Φε (t, x, ω) + ∂x Ψε (t, x, ω) = χε (t, x, ω) + ψε (t, x, ω)
(62)
holds ω-wise. Note again, ∂t , ∂x above should all be understood in Schwartz distributional sense. Let T > 0 be an arbitrarily given but fixed constant. We denote O = (0, T ) × R. 4.4.1. A priori estimates for several sequences of random fields The main result of this subsection is the following. −1 Lemma 4.18. {∂t Φε + ∂x Ψε : ε > 0} is a sequence of Hloc (O)-valued random variables. As such random variables, the sequence is tight.
Proof. We apply a stochastic generalization of the Murat lemma (see Lemma A.3) to show −1 that, as Hloc (O)-valued random variables, the left-hand side of (62) is tight. This only requires verifying conditions of Lemma A.3. By the integrability conditions on uε in Lemma 4.12, {Φε : ε > 0} and {Ψε : ε > 0} are both p stochastically bounded as Lloc (O)-valued random variables, 2 p < ∞. Therefore, the left−1,p hand side of (62) is a stochastically bounded sequence in Wloc (O). By the moment estimates (50) on uε in Lemma 4.13, {ψε,1 : ε > 0} is stochastically bounded in L2loc (O), hence it is stochastically bounded in as random variable in Mloc (O) (space of Radon signed-measures on each fixed bounded open subset of O) with total variation norm. By (54), {ψε,2 : ε > 0} is also stochastically bounded (in total variation norm), as Mloc (O)-valued random variables. −1 By (53), limε→0+ χε,1 = 0 in probability as sequence of Hloc (O)-valued random vari−1 ables, and is therefore tight. Finally, we claim that the set of Hloc (O)-valued random variables {χε,2 : ε > 0} is tight (which we will prove in Lemma 4.20 below). 2
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We call a deterministic function 0 ϕ(t, x) 1 a cutoff function, if ϕ(t, x) ∈ Cc∞ ([0, ∞) × R d ). We use it to discuss local properties of functions such as local integrability and so on. We localize Mε by t Mεϕ (t, x) =
ϕ(r, x) Mε (dr, x). 0
ϕ
We set Mε (t, x) = 0 for t < 0. In order to estimate some fractional derivatives of Mε locally, it will be convenient for us to introduce the notion of Marchaud fractional derivative for α ∈ (0, 1) (Samko, Kilbas, Marichev [16, Section 5.4]). We define α D± φ (t) =
−α (1 − α)
∞
φ(t ∓ s) − φ(t) ds, s 1+α
0
for those φ where the integrand above is L1 integrable. At least for φ ∈ Cc∞ (R) [16, Section 5.7],
d2 − 2 dt
α/2
φ=
d dt
α α φ = D+ φ.
α f ∈ L (R) for some p > 1 and s −1 = p −1 − α, Provided f ∈ Lsloc (R), D− loc p
∞
∞ α f (t)D+ φ(t) dt
=−
−∞
α φ(t)D− f (t) dt,
φ ∈ Cc∞ (R).
−∞
See [16, Corollary 2 of Theorem 6.2]. Therefore, in such cases, Schwartz distributional derivative αf. ∂ α f = D− ϕ Recall that Mε (t, x) is a continuous (in time) local martingale with each x fixed. Hence it is almost surely Hölder continuous in t (almost everywhere) with exponent 0 < β < 1/2 when x ϕ is fixed (Revuz, Yor [15, Exercise 1.20, p. 187]). Consequently, for 0 < α < β < 1/2, ∂tα Mε = ϕ αM . D− ε Lemma 4.19. Assume that (49) holds. Let ϕ be a cutoff function. Then there exists an 0 < α < 1/2 ϕ −1+α such that, as Hloc (O)-valued random variables, {∂t Mε : ε > 0} are stochastically bounded. That is, for each δ > 0, there exists a constant Cδ > 0 with sup P ∂t Mεϕ −1+α > Cδ < δ. ε>0
Proof. First, the integrability estimates in (50) imply that 2 sup E Mεϕ 2 < ∞. ε>0
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351
Therefore, we only need to prove that, for each δ > 0, there exists a constant Cδ > 0 satisfying sup P ∂tα Mεϕ 2 > Cδ < δ.
(64)
ε>0
We verify this next. Recall that we assume Φ(u) is of at most polynomial growth as u → ∞. Take γ > 6, then for 0 s < t T , there exists p0 > 2 such that γ E Mεϕ (t, ·) − Mεϕ (s, ·) γ C1 |x|
γ /2 t
2
2 2 E (Φ ) uε (r, x) ϕ (r, x) σε x, uε (r, x); z μ(dz) dr
dx
s
Z
t
γ /2
p
0
ϕ 2 (r, x) 1 + uε (r, x) dr dx
C2 E
|x|
T C3 |t − s|
γ /2−1
E
p0 γ /2 1 + uε (r) p γ /2 dr C4 |t − s|γ /2−1 . 0
0
The first inequality above follows from martingale inequalities, the second one follows from (49) and the third one from Jensen’s inequality. The last inequality follows from (50). From the above, T T E 0 0
ϕ
ϕ
Mε (t) − Mε (s)γ |t − s|1/p−1/γ
γ
ds dt C5 < ∞.
By Chebychev’s inequality, T T P 0 0
ϕ
ϕ
Mε (t) − Mε (s)γ |t − s|1/p−1/γ
γ
ds dt > λ C6 λ−1 ,
λ > 0.
(65)
By a normed space version of Garsia’s inequality (e.g. Stroock, Varadhan [17, Exercise 2.4.1, p. 60]), if T T 0 0
ϕ
ϕ
Mε (t) − Mε (s)Lγ |t − s|1/2−1/γ
γ ds dt λ
then ϕ M (t) − M ϕ (s) Cλ |t − s|−3/γ +1/2 , ε ε γ
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for some deterministic constant Cλ . This implies that, for 0 < α < β ≡ 1/2 − 3/γ , α ϕ γ α ∂ M = D M ϕ γ C7 t ε γ −,t ε γ
|Mεϕ (t + s, x) − Mεϕ (t, x)| γ
dx dt ds
s (1+α) t
x
s
γ
∞ ϕ ϕ Mε (t + s) − Mε (t)γ
C8
ds dt C9 Cλ < ∞,
s (1+α) t
ε > 0,
(66)
s=0
α emphasizes that the Marchaud fractional derivative is taken with respect to t. where D−,t Combining of (65) and (66) gives (64). 2
Lemma 4.20. Assume that (49) holds. Then for each δ > 0 and T > 0, there exists a compact set −1 (O) such that K = K(δ, T ) Hloc inf P (χε,2 ∈ K) > 1 − δ.
ε>0
−1+α −1 Proof. The conclusion follows from the compact embedding of Hloc (O) to Hloc (O) and from the results in Lemma 4.19. 2
4.4.2. Identifying ν0 (t, x; du) as a function To simplify notation, for any function f = f (u), we denote f = f (t, x) =
f (u)ν0 (t, x, du)
u∈R
whenever the integral makes sense. In particular, u(t, x) =
u∈R uν0 (t, x; du).
We also write
D
X = Y for random variables X, Y having identical probability law/distribution. Let (Φi , Ψi ), i = 1, 2, be two choices of entropy–entropy flux pairs, where Φi has at most polynomial growth (therefore Ψi will have at most polynomial growth as well). Lemma 4.21. For every deterministic ϕ ∈ Cc∞ (O), D
ϕ, Ψ1 Φ2 − Φ1 Ψ2 = ϕ, Ψ1 · Φ2 − Φ1 · Ψ2 .
(67)
Proof. On the one hand, by Lemma 4.15 and the uniform in ε moment estimates in (50), for each ϕ ∈ Cc∞ ((0, T ) × R d ), the following convergence in probability result holds lim
ε→0+
ϕ(t, x) Ψ1 uε (t, x) Φ2 uε (t, x) − Φ1 uε (t, x) Ψ2 uε (t, x) dx dt
= lim
ε→0+ (t,x)∈[0,T ]×R
ϕ(t, x)
= ϕ, Ψ1 Φ2 − Φ1 Ψ2 .
u∈R
Ψ1 (u)Φ2 (u) − Φ1 (u)Ψ2 (u) νε (t, x, du) dt dx
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
353
On the other hand, we can apply Theorem A.2 by choosing Gε (t, x) = Φ1 uε (t, x) , Ψ1 uε (t, x) ,
Hε (t, x) = −Ψ2 uε (t, x) , Φ2 uε (t, x) .
In view of Lemmas 4.12 and 4.18, we have the following convergence in probability law result: lim
ε→0+
ϕ(t, x) Ψ1 uε (t, x) Φ2 uε (t, x) − Φ1 uε (t, x) Ψ2 uε (t, x) dt dx
D
= ϕ, Ψ1 · Φ2 − Φ1 · Ψ2 . We conclude the proof.
2
We now finish the proof of the main result of this section. ∞ Proof of Lemma 4.17. Let u deterministic 0 ϕ ∈ Cc (O). Take Φ1 (u) = u, Ψ1 (u) = F (u) and Φ2 (u) = F (u), Ψ2 (u) = 0 (F )2 (r) dr. Eq. (67) gives
D ϕ, F 2 − uΨ2 = ϕ, (F )2 − u · Ψ2 .
(68)
On the other hand, for t > 0, x ∈ R, ω ∈ Ω fixed, and u ∈ R, by Schwartz’s inequality, 2 F (u) − F u(t, x) =
u
2
F (v) dv
u − u(t, x) Ψ2 (u) − Ψ2 u(t, x) . (69)
u(t,x) ¯
Integrate the above as a function of u against ν0 (t, x, du), 2 F 2 + F (u) − 2F F (u) uΨ2 − u · Ψ2 .
(70)
Take expectation on both (68) and (70) and combine them, 2 ϕ(t, x)E F − F (u) dt dx 0. By the arbitrariness of ϕ, the following holds almost surely: F dt dx =
F (u)ν0 (t, x, du) dt dx = F u(t, x) dt dx = F (u) dt dx.
R
Therefore
ϕ(t, x)(dt dx) =
2 F (u) − F u(t, x, ω) ν0 (t, x, du; ω)P (dω)
ω∈Ω u∈R
2 ϕ(t, x)(dt dx)E F 2 − F (u) =
ϕ(t, x)(dt dx)E[uΨ2 − u · Ψ2 ]
(71)
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=
ϕ(t, x)(dt dx)
u − u(t, x, ω) Ψ2 (u) − Ψ2 u(t, x; ω) ν0 (t, x, du; ω)P (dω).
ω u
In the above, the second equality follows from (71) and the third one follows from (68). We conclude then, almost surely (69) holds as an equality for all u on the support of random measure ν0 (t, x; du). On the other hand, from the property of Schwartz inequality, this cannot be true ¯ With the condition on F , the support of u has to be on single unless F is constant during u, u. point mass u¯ almost surely. 2 By (50) and Fatou’s lemma
|u| ν0 (t; x, du) dx < ∞,
sup E 0tT
p
p = 2, 4, . . . .
x u
If (61) holds, then
p
u(t, ¯ x) dx < ∞,
sup E 0tT
p = 2, 4, . . . .
(72)
x
4.5. Existence of stochastic entropic solution 4.5.1. Existence of measure-valued solution Let Fε (r) = (Fε,1 , . . . , Fε,d ) be as defined in last section. Let convex Φ ∈ C 2 (R) have at most polynomial growth. Define Ψε = (Ψε,1 , . . . , Ψε,d ) with r
Φ (s)(Fε,k ) (s) ds.
Ψε,k (r) = 0
Then (37) can be written as (Lemma 4.11)
Φϕ, νε (t) − Φϕ, νε (s)
(Ψε · ∇ϕ)νε (r, x, du) dx dr
(s,t]×R d ×R
+
1 2
2 Φ σε ϕ νε (r, x, du) dx dr μ(dz)
Z (s,t]×R d ×R
+ε
(Φϕ)νε (r, x, du) dx dr
(s,t]×R d ×R
+
Φ σε ϕ, νε (r) W (dr, dz),
(s,t]×Z
where shorthand notation Φϕ = Φ(u)ϕ(x) and so on are used. The main result of this section is
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355
Lemma 4.22. Let u(0), F, σ satisfy conditions in Theorem 4.1. Then the Ft -adapted process ν0 (t) has trajectory in C([0, ∞), M0 ), and it satisfies (Ψ · ∇ϕ)ν0 (r, x, du) dx dr Φϕ, ν0 (t) − Φϕ, ν0 (s) (s,t]×R d ×R
+
1 2
2 Φ σ ϕ ν0 (r, x, du) dx dr μ(dz)
Z (s,t]×R d ×R
Φ σ ϕ, ν0 (r) W (dr, dz).
+
(74)
(s,t]×Z
We show the proof in two steps. First, we establish the following. Lemma 4.23. Assume that (61) holds, then lim E r ν0 (t), ν0 (s) = 0, t→s+
s 0.
In particular, this implies that ν0 (·) ∈ C([0, ∞); M0 ). Proof. Let y ∈ R d . Take Φ(u) = |u − uε (s, y)|2 and ϕ(x) = Jδ (x − y) and apply (37) (Lemma 4.11). We notice that u Ψε,k (u) = 2
Fε,k (r)
r − uε (s, y) dr = 2
0
r
Fε,k (r)r dr − 2uε (s, y) Fε,k (u) − Fε,k (0) ,
0
k = 1, 2, . . . , d, have at most polynomial growth, and that
∂x Jδ (x − y) c1 δ −1 ,
x Jδ (x − y) c2 δ −2 . k For ψ 0 and ψ ∈ Cc∞ (R d ), we denote O = int supp(ψ) ,
Oδ = x ∈ R d : dist(x, O) < δ .
Then
uε (t, x) − uε (s, y) 2 Jδ (x − y) dx ψ(y) dy
Fs
E y∈O x∈O δ
uε (s, x) − uε (s, y) 2 Jδ (x − y) dx ψ(y) dy
y∈O x∈O δ
c3 + δ
t
E s
y∈O x∈O δ
2
p1
1 + uε (s, y) + uε (r, x) dx ψ(y) dy Fs dr
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J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
t +
σ x, uε (r, x); z Jδ (x − y) dx ψ(y) dy μ(dz) Fs dr 2
E y∈O x∈O δ z
s
εc4 + 2 δ
t
E
p
1 + uε (r, x) 2 dx ψ(y) dy Fs dr
y∈O x∈O δ
s
for some p1 , p2 2. Let stochastic measure π(dx, du; dy, dv) = δuε (s,y) (dv) Jδ (x − y) dy δuε (t,x) (du) dx. Then direct verification shows that π ∈ Π νε (t),νε (s) (see definition in (55)). Furthermore, for each K > 0 fixed such that {x: |x| < K} ⊂ O, and for all δ > 0,
uε (t, x) − uε (s, y) 2 Jδ (x − y) dx dy
(x,y)∈O ×O δ
=
|u − v|2 π(dx, du; dy, dv)
(x,y)∈O ×O δ
qK νε (t), νε (s) −
|x − y|2 Jδ (x − y) dx dy
O ×O δ
qK νε (t), νε (s) − δ 2 c5 . In view of the convergence in probability result in Lemma 4.15, E qK ν0 (t), ν0 (s)
|u − v|2 Jδ (x − y)ν0 (s; dx, du)ν0 (s; dy, dv) c5 δ 2 + E y x
c6 + δ
t
sup E s
ε>0
+ c7
y∈O x∈O δ
t
2
p1
1 + uε (s, y) + uε (r, x) dx dy dr
sup E s
ε>0
σ x, uε (r, x); z Jδ (x − y) dx dy μ(dz) dr. 2
y∈O x∈O δ z
Noting (50) and (61), lim sup E qK ν0 (t), ν0 (s) c5 δ 2 + E t→s+
y∈O x∈O δ
2
u(s, ¯ x) − u(s, ¯ y) Jδ (x − y) dx dy .
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357
By the arbitrariness of δ > 0 and integrability estimates (72) on u(s) ¯ we arrive at lim E qK ν0 (t), ν0 (s) = 0,
s 0, K = 1, 2, . . . .
t→s+
2
Hence conclude the lemma. Next, we show that Lemma 4.24.
lim E
ε→0+
Φ σε ϕ, νε (r) W (dr, dz) −
(s,t]×Z
2
Φ σ ϕ, ν0 (r) W (dr, dz)
= 0.
(s,t]×Z
Proof.
E
Φ σε ϕ, νε (r) W (dr, dz) −
(s,t]×Z
(s,t]
E
2
Φ σ ϕ, ν0 (r) W (dr, dz)
(s,t]×Z
2 Φ σε ϕ, νε (r) − Φ σ ϕ, ν0 (r) μ(dz) dr.
Z
Therefore, the result follows from Lemma 4.15 and from (50).
2
Assuming (61) holds, then (74) becomes (10). Combined with estimates (72) u(t, ¯ x) is a stochastic entropic solution. 4.6. Existence of stochastic strong entropic solution To be consistent with the notations in the definition of strong entropic solution (as well as the uniqueness proof), we write vε = uε where uε is constructed as in Lemma 4.10, and v = v(t, y) = u(t, y) = uν0 (t, y; dv). We assume all the conditions at the beginning of Section 4.2.3 regarding vε (0), Fε , and σε . We also assume that σ satisfies (13). We assume that (61), a conclusion of Lemma 4.17, holds and consider general dimensions d = 1, 2, . . . . (72) translates into p sup E v(t) p < ∞,
p = 2, 4, . . . .
(75)
0tT
Let u(t) ˜ = u(t, ˜ x) be an arbitrary Ft -adapted Lp (R d )-valued process with p ˜ p < ∞, sup E u(t)
∀T > 0, p = 2, 4, . . . .
0tT
Let β be of the form as in (17). We prove that (12) holds in the following lemma.
(76)
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Lemma 4.25. For each T > 0, there exists a deterministic function {A(s, t): 0 s t < ∞} such that
E
˜ x) − v(t, y) σ x, u(r, ˜ x); z ϕ(x, y) dx dy W (dr, dz) β u(r,
(s,t]×Z y x
E −
R d ×R d
˜ x) − v(r, y) σ y, v(r, y); z β u(r,
Z×(s,t]
× σ x, u(r, ˜ x); z μ(dz) dr ϕ(x, y) dx dy + A(s, t) with the property that, for each sequence of partitions 0 t1 · · · tm T , lim
maxi |ti+1 −ti |→0+
A(ti , ti+1 ) = 0.
i
The proof consists of several involved estimates. We present them in further subsections. 4.6.1. A special martingale N , and its estimates For each α ∈ C 2 such that α, α , α ∈ Cb (R), and each ϕ ∈ Cc∞ (R d × R d ), we denote
N (α, ϕ)(s, t; y, v) =
α u(r, ˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx W (dr, dz), (77)
(s,t]×Z x
where 0 s t, (y, v) ∈ R d × R, and the integral is defined in Ito’s sense. For each s fixed, the above is a martingale in t s. Next, we derive some useful properties and a priori estimates regarding N . We note that N (α, ϕ)(s, t; y, v) = 0 whenever |y| > C for some large C = Cϕ depending only on the support of ϕ. Lemma 4.26. The following identities hold almost surely for each (y, v) ∈ R d × R fixed: ∂v N(α, ϕ)(s, t; y, v) = N (−α , ϕ)(s, t; y, v), ∂yi N(α, ϕ)(s, t; y, v) = N (α, ∂yi ϕ)(s, t; y, v). Proof. The proof of Theorem 7.6 of Kunita [11, p. 180] can be modified to show this.
2
Lemma 4.27. Suppose that α ∈ Cc (R). Then for each T > 0, p > 5, there exist a > 0, C2 > 0, for any δ > 0, E
sup
s,t∈[0,T ], |t−s|<δ
N(α, ϕ)(s, t; ·,·) p < C2 δ a . p
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359
Proof. First, p E N (α, ϕ)(s, t; ·,·) p
p
E
α u(r, ˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx W (dr, dz)
dy dv = v∈R |y|
(s,t]×Z x
p/2
2 t
α u(r, c1 E ˜ x) − v σ x, u(r, ˜ x); z ϕ(x, y) dx
μ(dz) dr dy dv
v y
s
z
x
c2 |t − s|
E v |y|
0 |x|
T c3 |t − s|
(p/2−1)
T c4 |t − s|
p
p ˜ x) dv α∞ 1 + u(r,
E 0 |y|
(p/2−1)
p
p
α u(r, ˜ x) dx dr dy dv ˜ x) − v 1 + u(r,
T
(p/2−1)
1+
E
dx dy dr
|v|u(r,x)|+c ˜ α
p+1
u(r, ˜ x) dx dr c5 |t − s|(p/2−1) .
x
0
In the above, the first inequality follows from martingale inequality, the second one follows from Jensen’s inequality and (13), the third inequality follows from the compact support assumption on α, and the last inequality from (76). The above implies T T E
N(α, ϕ)(s, t; ·, ·)p 1
|t − s| 2
0 0
p
1 − 2p
ds dt c6 < ∞.
(78)
Note N (α, ϕ)(s, t; ·,·) = N (α, ϕ)(0, t; ·,·) − N (α, ϕ)(0, s; ·,·). By a normed linear space version of Garsia inequality [17, Exercise 2.4.1, p. 60], when p > 8,
sup
s,t∈[0,T ];|s−t|<δ
N(α, ϕ)(s, t; ·,·) c7 p
T T
N (α, ϕ; ·, ·)p
0 0
Taking the expectation, the conclusion follows from (78).
|t − s|
1 1 2 − 2p
ds dt δ
2
Lemma 4.28. Suppose that α, α ∈ Cc (R), and p > 8, then E for some a > 0.
sup
0s, tT , |t−s|<δ
N(α, ϕ)(s, t; ·,·) p C2,α,ϕ δ a ∞
p−5 2p
.
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J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
Proof. By a Sobolev inequality (e.g. [7, (23), p. 268]), there exist deterministic constants c1,p , c2,p > 0 N (α, ϕ)(s, t; ·,·) ∞ c1,p N (α, ϕ)(s, t; ·,·) W 1,p (R d ×R) d N (α, ∂yk ϕ)(s, t; ·,·) p . c2,p N (α, ϕ)(s, t; ·,·) p + N (α , ϕ)(s, t; ·, ·) p + k=1
In view of the above, the conclusions follow from Lemma 4.27.
2
4.6.2. Γi,ε and their estimates Let β ∈ C ∞ (R) satisfy the conditions of Lemma 2.9. We will need to estimate the following random variables. Let k = 1, 2, . . . , d be fixed and t Γ1,ε (s, t) = y
s
t
s
y
Γ2,ε (s, t) =
(N β , ϕ) s, r; y, vε (r, y) −∂yk Fε vε (r, y) dr dy, N (β , ϕ) s, r; y, vε (r, y) εyy vε (r, y) dy dr.
Lemma 4.29. Let A(s, t) = A1 (s, t) + A2 (s, t), where
Ak (s, t) = lim inf E Γk,ε (s, t) , ε>0
k = 1, 2.
Then for every sequence of partitions of [0, T ], T > 0, 0 t1 · · · tm T , lim
maxi |ti+1 −ti |→0+
A(ti , ti+1 ) = 0.
i
We divide the proof into several lemmas. First, we estimate Γ1,ε (s, t). Let v Gε (u, v) =
β (u − r)Fε,k (r) dr,
u, v ∈ R.
0
β has compact support, implying that the above integration can be restricted to 0 r |u|. Therefore,
sup Gε (u, v) Cβ 1 + |u|p , ∀u, v ∈ R, ε>0
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
361
for some integer p ∈ {1, 2, . . .}. For each ϕ ∈ Cc∞ (R d × R d ), let
˜ x), v σ x, u(r, ˜ x); z ϕ(x, y) dx W (dr, dz). Gε,k u(r,
Xε (ϕ)(s, t; y, v) = (s,t]×Z x
As in the proof of Lemma 4.26,
∂v Xε (ϕ)(s, t; y, v) =
˜ x), v σ x, u(r, ˜ x); z ϕ(x, y) dx W (dr, dz), ∂v Gε u(r,
(s,t]×Z x
∂yk Xε (ϕ)(s, t; y, v) = Xε (∂yk ϕ)(s, t; y, v). As in the proof of Lemmas 4.27 and 4.28, we have Lemma 4.30. For each p > 2, there exist a constant C = C(β, ϕ, p) independent of ε, and a constant a > 0 such that p E Xε (ϕ)(s, t; ·, ·) ∞ C|t − s|a . By a formal application of integration by parts (note that we are handling integral with anticipating integrand), we have (79). We justify this rigorously next. Lemma 4.31. The following representation holds almost surely: t Γ1,ε (s, t) = s
Xε (∂yk ϕ) s, r; y, vε (r, y) dy dr.
(79)
y
Proof. Let Jδ be a one-dimensional mollifier as defined before (smooth and have compact support). First, through integration by parts, N (β y
, ϕ)(s, r; y, v)Fε (v)Jδ
v − vε (r, y) dv ∂yk vε (r, y) dy
v
=
∂v Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dv ∂yk v(r, y) dy
y v
=
Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dv ∂yk v(r, y) dy
y v
=−
Xε (ϕ)(s, r; y, v)∂yk Jδ v − vε (r, y) dy dv
v y
= v y
∂yk Xε (ϕ)(s, r; y, v)Jδ v − vε (r, y) dy dv
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Xε (∂yk ϕ)(s, r; y, v)Jδ v − vε (r, y) dv dy.
= y v
Sending δ → 0+ and noting the estimates in Lemmas 4.28, 4.30, we arrive at the conclusion.
2
Therefore, by Lemma 4.30,
E Γ1,ε (s, t)
t C1 |r − s|a dy dr C2 |t − s|1+a ≡ A1 (s, t) s |y|cϕ
satisfies that for all partitions of [0, T ], limmaxi |ti+1 −ti |→0+ i |A1 (ti , ti+1 )| = 0. Now we estimate Γ2,ε . First, we note that (36) holds even if Φ is not convex. In particular, β u(¯ ˜ r , x) − vε (t, y) yy vε (r, y)
2
˜ r , x) − vε (r, y) − β u(¯ ˜ r , x) − vε (r, y) ∇y vε (r, y) . = yy β u(¯ Similar to the last lemma, Lemma 4.32. t
N (β , ϕ) s, r; y, vε (r, y) εyy vε (r, y) dy dr
Γ2,ε (s, t) = s |y|
t =ε
N (β , yy ϕ) s, r; y, vε (r, y) dy dr
s |y|
t −
2
N (β , ϕ) s, r; y, vε (r, y) ε ∇y vε (r, y) dy dr.
s |y|
The first terms can be handled using Lemma 4.28. Estimating the second term in the Γ2,ε is more involved. We formulate details in the following Lemma 4.33. There exists a deterministic function A = A(s, t), 0 s t, such that t lim sup E ε→0+
s
2
N (β , ϕ) s, r; y, vε (r, y) ε ∇y vε (r, y) dy dr A(s, t);
y
and for each sequence of partitions 0 t1 · · · tm T , lim
maxi |ti+1 −ti |→0+
i
A(ti , ti+1 ) = 0.
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363
Proof. We define t Λε (t) =
2 ε ∇y vε (r) 2 dr,
Yδ =
0
sup
s,t∈[0,T ]; |t−s|<δ
N (β , ϕ)(s, t; ·,·) , ∞
and let A(s, t) = lim sup E Y|t−s| Λε (t) − Λε (s) . ε→0+
By Lemma 4.13,
p
sup E Λε (T ) < ∞,
p = 1, 2, . . . , T > 0.
(80)
ε>0
Combined with Lemma 4.28, A(s, t) is finite. Let T > 0 and partition 0 = t1 < · · · < tm = T with δm = maxi |ti+1 − ti |. It follows then m i=1
A(ti , ti+1 ) lim sup
m E Yδm Λε (ti+1 ) − Λε (ti ) = lim sup E Yδm Λε (T )
ε→0+ i=1
ε→0+
q 1/p
sup E Λε (T )
E |Yδm |p
1/q
ε>0
a Cδm ,
for some a > 0, p −1 + q −1 = 1 with p > 8. In the above, we invoked (80) and Lemma 4.28 for the last inequality. The conclusion follows. 2 We conclude that Lemma 4.29 holds. 4.6.3. Proof of Lemma 4.25 With the above estimates, we are ready to prove the main result of this section. Proof. Let J, Jδ be mollifiers defined as in (16) in the special case of one dimension. Recall notation (77), we first let
Zε,δ (t) =
N (β , ϕ) s, t; y, v − vε (t, y) Jδ (v) dv dy
|y|
=
N (β , ϕ)(s, t; y, v) Jδ v − vε (t, y) dv dy.
(81)
|y|
Then Zε,δ is a semi-martingale in t s. We recall that N (β , ϕ)(s, t; y, v) = 0 whenever |y| > cϕ for some large c = cϕ depending only on the support of ϕ. Therefore, the integration of y above can be restricted to a bounded set |y| < cϕ .
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The following approximation result holds:
lim E Zε,δ (t) = E lim Zε,δ (t) = E N (β , ϕ) s, t; y, vε (t, y) dy .
δ→0+
δ→0+
y
In the above, we interchanged the order of expectation and limδ→0+ . This follows from dominated convergence theorem, which is justified by observation
Zε,δ (t) C˜ ϕ N (β , ϕ)(s, t; ·,·)
∞
and by estimate in Lemma 4.28 and by assumption in (76). Similarly, lim lim E Zε,δ (t) = E
ε→0+ δ→0+
N (β , ϕ) s, t; y, v(t, y) dy .
y
On the other hand, by Ito’s formula (i.e. d(XY ) = X dY + Y dX + d[X, Y ] for two semimartingales X, Y ) and by integration by parts applied to (81), t Zε,δ (t) =
Jδ v − vε (r, y) dr N (β , ϕ)(s, r; y, v) dy dv
y v r=s
t + y
s
N (β , ϕ)(s, r; y, v)Jδ v − vε (r, y) dv
v
× − divy Fε vε (r, y) dr + εyy vε (r, y) dr +
+ y (s,t]×Z
σε y, vε (r, y); z W (dr, dz) dy
z
1 N (β , ϕ)(s, r; y, v) Jδ v − vε (r, y) dv 2
v
× σε2 y, vε (r, y); z μ(dz) dr dy ˜ x) − v Jδ v − vε (r, y) dv σ x, u(r, β u(r, ˜ x); z ϕ(x, y) − x y (s,t]×Z
v
× σε y, vε (r, y); z μ(dz) dr dx dy. Therefore, E Zε,δ (t) = Iε,δ + II ε,δ + III ε,δ + IV ε,δ , where
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
t Iε,δ = E y
II ε,δ = E 1 III ε,δ = E 2
N (β , ϕ)(s, r; y, ·) ∗ Jδ (·) vε (r, y) − divy Fε vε (r, y) dr dy ,
s
t y
365
N (β , ϕ)(s, r; y, ·) ∗ Jδ (·) vε (r, y) εyy vε (r, y) dr dy ,
s
2 N (β , ϕ)(s, r; y, ·) ∗ Jδ (·) vε (r, y) σε y, vε (r, y); z μ(dz) dr dy ,
y (s,t]×Z
IV ε,δ = −E x y (s,t]×Z
˜ x) − v Jδ v − vε (r, y) dv β u(r,
v
× σ x, u(r, ˜ x); z σε y, vε (r, y); z μ(dz) ϕ(x, y) dx dy dr . Since β (·)∞ < ∞, by dominated convergence theorem,
˜ x) − vε (r, y) σ x, u(r, β u(r, ˜ x); z
lim IV ε,δ = −E
δ→0+
x y (s,t]×Z
× σε y, vε (r, y); z μ(dz) ϕ(x, y) dx dy dr . Furthermore, noting (57), by the estimates (50), (76), a uniform (in ε) integrability arguments gives
˜ x) − v(r, y) σ x, u(r, ˜ x); z β u(r,
lim lim IV ε,δ = −E
ε→0+ δ→0+
x y (s,t]×Z
× σ y, v(r, y); z μ(dz) ϕ(x, y) dx dy dr . Regarding III ε,δ , by (49) and (50), we have t lim inf lim inf III ε,δ CE ε→0+ δ→0+
N (β , ϕ)(s, r; ·,·)
∞
1+
2
v(r, y) dy dr ≡ A3 (s, t).
|y|cϕ
s
In view of Lemma 4.28 and (75), for all partitions of [0, T ], lim
maxi |ti+1 −ti |→0+
m i=1
A3 (ti , ti+1 ) = 0.
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We now estimate Iε,δ . By Lemma 4.28 and the dominated convergence theorem, t lim Iε,δ = E
δ→0+
y
N (β , ϕ) s, r; y, vε (r, y) − divy Fε vε (r, y) dr dy .
s
By Lemma 4.29, lim inf lim Iε,δ A1 (s, t) ε→0+ δ→0+
with A1 satisfying the requirement of the lemma. Similarly, we can handle the II ε,δ term. Take A(s, t) = A1 (s, t) + A2 (s, t) + A3 (s, t), then the lemma follows.
2
Acknowledgments We thank Yaozhong Hu for useful discussions, and Elton Hsu for a careful reading and suggestions. Appendix A. A probabilistic generalization of Div–curl lemma In weak convergence/Young measure approach to construction of deterministic nonlinear PDEs (e.g. [3,6]), it is important to extract information on limit of some nonlinear functionals. The well-known div–curl lemma is such a useful device. Below, we generalize it to a stochastic setting. Throughout this appendix, we assume bounded open set O ⊂ R m has a smooth C ∞ boundary ∂O. For F = (F1 , . . . , Fm ) ∈ L2 (O; R m ), we identify curlF = ∇ × F with an m × m-matrixvalued function with the (i, j )th component defined by (∇ × F )ij = ∂xj Fi − ∂xi Fj ∈ H −1 (O). D
Let Xε , X be Polish space S-valued random variables. By limε→0+ Xε = X, we mean Xε converges in probability law/distribution to X. By tightness of {Xε : ε > 0}, we mean the family of probability distributions (on S) of the random variables is tight. ¯ be a sequence of H p (O; R m ) × H q (O; R m )Lemma A.1. Let {(Fε , Gε ): ε > 0} and (F¯ , G) valued random variables, where p = −q ∈ {0, ±1, ±2, . . .}. Suppose the following conditions hold. (1) {Fε : ε > 0} is stochastically bounded in H p (O; R m ). That is, for each δ > 0, there exists a deterministic constant Cδ ∈ (0, ∞) such that sup P Fε H p > Cδ < δ. ε>0
J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
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(2) Let (·,·)H p denote the inner product in H p (O; R m ). For each finite collection of deterministic φ1 , . . . , φk ∈ H p (O; R m ), k = 1, 2, . . . , sequence of R k × H q (O; R m )-valued random variables converges: D ¯ . lim (φ1 , Fε )H p , . . . , (φk , Fε )H p , Gε = (φ1 , F¯ )H p , . . . , (φk , F¯ )H p , G
ε→0+
(A.1)
Let ·,· be the continuous bilinear pairing between H p (O; R m ) and H q (O; R m ), p = −q. Then D ¯ lim Fε , Gε = F¯ , G,
ε→0+
where the above is joint convergence with that in (A.1). That is, D ¯ F¯ , G ¯ . lim (φ1 , Fε )H p , . . . , (φk , Fε )H p ; Gε ; Fε , Gε = (φ1 , F¯ )H p , . . . , (φk , F¯ )H p ; G;
ε→0+
Proof. By Skorokhod representation theorem (e.g. [5, Theorem 3.18]), without loss of generality, we assume that all random variables are defined on the same probability space and all convergence in probability distribution is convergence almost surely. D ¯ First, for each h, δ > 0, by condition (1) and the assumption that limε→0+ Gε = G,
¯ > h lim P Fε H p Gε − G ¯ Hq > h lim P Fε , Gε − G
ε→0+
ε→0+
¯ H q > hC −1 + P Fε H p > Cδ < δ. lim P Gε − G δ ε→0+
D ¯ ¯ ¯ = F , G. Therefore, to conclude the lemma, it is sufficient to prove that limε→0+ Fε , G Let {f1 , . . . , fk , . . .} and {g1 , . . . , gk , . . .} be a dual system of (deterministic) complete orthonormal bases for H p (O; R m ) and H q (O; R m ), respectively. That is,
f¯, g ¯ =
∞ (f¯, fk )H p (g, ¯ gk )H q ,
∀f¯ ∈ H p , g¯ ∈ H q .
k=1
For every h, δ > 0, by condition (1),
2 ∞
¯ lim sup P
(Fε , fk )H p (G, gk )H q > h
N →∞ ε>0 k=N +1 ∞ 2 2 ¯ gk )H q > h lim sup P Fε H p (G, N →∞ ε>0
lim
N →∞
P
k=N +1 ∞
¯ gk )2H q > hC −2 (G, δ
k=N +1
+ sup P Fε H p > Cδ δ. ε>0
By the above uniform in ε estimate, and by condition (2),
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J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
¯ = lim lim lim Fε , G
ε→∞ N →∞
ε→0+
D
= lim
lim
N →∞ ε→0+
D
= lim
N →∞
N
∞
¯ gk )H q − (Fε , fk )H p (G,
∞
¯ gk )H q (Fε , fk )H p (G,
k=N +1
k=1 N ¯ gk )H q (Fε , fk )H p (G, k=1
¯ gk )H q = F¯ , G. ¯ (F¯ , fk )H p (G,
2
k=1
¯ H¯ ) be a sequence of L2 (O; R m ) × Theorem A.2 (Div–curl). Let (Gε , Hε ), ε > 0, (G, L2 (O; R m )-valued random variables. Suppose the following holds: (1) {Gε : ε > 0} and {Hε : ε > 0} are both stochastically bounded as L2 (O; R m )-valued random variables. That is, for each δ > 0, there exists a deterministic constant Cδ ∈ (0, ∞) such that sup P Gε L2 + Hε L2 > Cδ < δ. ε>0
(2) For each finite collection of deterministic φ1 , . . . , φk ∈ L2 (O; R m ), lim φ1 , Gε , . . . , φk , Gε ; φ1 , Hε , . . . , φk , Hε
ε→0+
D ¯ . . . , φk , G; ¯ φ1 , H¯ , . . . , φk , H¯ . = φ1 , G, (3) Both {∇ · Gε : ε > 0} and {∇ × Hε : ε > 0} are tight as sequences of H −1 (O)-valued random variables. Then for each finite collection of deterministic ϕ1 , . . . , ϕk ∈ Cc∞ (O), D ¯ · H¯ , . . . , ϕk , G ¯ · H¯ , lim ϕ1 , Gε · Hε , . . . , ϕk , Gε · Hε = ϕ1 , G
ε→0+
where the convergence is joint convergence in probability law/distribution with that in condition (2). Proof. Let H 2 (O, R m )-valued random variables hε be defined as (weak) solution to −hε = Hε ,
x ∈ O,
hε = 0,
x ∈ ∂O.
(A.2)
Condition (1) of the theorem implies that {hε : ε > 0} is stochastically bounded as H 2 (O; R m )valued random variables. Since any bounded set in H 2 (O; R m ) is a compact set in L2 (O; R m ), {hε : ε > 0} is a tight sequence as L2 (O; R m )-valued random variable. Selecting subsequence if necessary, there exists a L2 (O; R m )-valued random variable h0 such that D
lim hε = h0 .
ε→0+
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369
Indeed, h0 has to be H 2 (O; R m )-valued. To see this, we invoke the Skorohod representation and assume limε→0+ hε − h0 L2 = 0 almost surely. Then by lower semicontinuity of · H 2 , h0 (·, ω)
H2
lim inf hε (·, ω) H 2 . ε→0+
Consequently for each δ > 0, there exists Cδ > 0, P h0 H 2 > Cδ P lim inf hε H 2 > Cδ = P ε→0+
! " hε H 2 > Cδ κ>0 0<ε<κ
" hε H 2 > Cδ lim sup P hκ H 2 > Cδ < δ. = lim P κ→0+
κ→0+
0<ε<κ
Let fε = −∇ · hε ,
Nε = Hε − ∇fε .
Then {fε : ε > 0} is stochastically bounded as H 1 (O)-valued random variables; and is a tight sequence as L2 (O)-valued random variables. Selecting a subsequence if necessary, we have f0 , a L2 (O; R m )-valued random variable, such that, as L2 (O; R m )-valued random variables D
lim fε = f0 .
ε→0+
Furthermore, by H 1 (O) stochastic boundedness of {fε : ε > 0}, f0 is also a H 1 (O)-valued random variable. Since for each 1 i m, Nεi
= Hεi
− ∂xi fε =
m
−∂x2j hiε + ∂xi ∂xj hjε
j =1
=
m
m ∂xj ∂xi hjε − ∂xj hiε = ∂xj (∇ × hε )ij ,
j =1
(A.3)
j =1
and since by condition (3) of the theorem, {∇ × hε : ε > 0} is tight as H 1 (O; R m )-valued random variables, it follows that {Nε : ε > 0} is tight as L2 (O; R m )-valued random variables. Selecting subsequence if necessary, there exists L2 (O, R m )-valued random variable N0 such that, as L2 (O; R m )-valued random variables, D
lim Nε = N0 .
ε→0+
To summarize, we can find subsequence so that D
lim (hε , fε , Nε ) = (h0 , f0 , N0 ).
ε→0+
(A.4)
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Note that such convergence holds in the sense that the triplet (hε , fε , Nε ) are viewed as metric space E-valued random variables, where E = L2 O; R m × L2 (O) × L2 O; R m with corresponding norm topology. Note that (−) : L2 (O; R m ) → H −2 (O; R m ) is a continuous map, by continuous mapping theorem and (A.2), joint with the convergence in (A.4), as a sequence of H −2 (O; R m )-valued random variables, Hε converges in law/probability distribution to H −2 (O; R m )-valued H0 D
lim Hε = H0 ,
ε→0+
where H0 is defined through −h0 = H0 , x ∈ O (note also that h0 = 0, x ∈ ∂O). Since h0 is H 2 (O; R m )-valued, we conclude that H0 is indeed L2 (O; R m )-valued. By condition (2) of the D theorem, H¯ and H0 has to be of the same probability law/distribution H¯ = H0 . By Skorohod representation, we may assume without lose of generality that all random variables live in the same reference probability space and selecting subsequence if necessary, convergences are convergences in almost sure sense. Then f0 = −∇ · h0 ,
N0 = H0 − ∇f0 .
Finally, for each ϕ ∈ Cc∞ (O),
Gε · Hε ϕ dx =
Gε · (Nε + ∇fε )ϕ dx = Gε , Nε ϕ − Gε , (∇ϕ)fε − ∇ · Gε , fε ϕ,
where ·,· is the continuous bilinear pairing between H −p (O) and H0 (O), p = 0, 1. Assuming ϕ is deterministic, by Lemma A.1, p
D ¯ ¯ f0 ϕ ¯ (∇ϕ)f0 − ∇ · G, lim ϕ, Gε · Hε = G, N0 ϕ − G, ¯ · (N0 + ∇f0 )ϕ dx = ϕ, G ¯ · H¯ . = G 2
ε→∞
The following lemma offers a practical way of verifying condition (3) of Theorem A.2, by exploring structural information in (stochastic) scalar conservation law equations. Lemma A.3 (Murat’s lemma). Suppose that: (1) {φε : ε > 0} is a stochastically bounded sequence in W −1,p (O) for some p > 2. That is, for each δ > 0, there exists a Cδ ∈ (0, ∞) such that sup P φε W −1,p > Cδ < δ. ε>0
(2) φε = χε + ψε . (3) {χε : ε > 0} is tight as H −1 (O)-valued random variables.
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371
(4) {ψε : ε > 0}, as signed Radon measure M(O)-valued random variables, is stochastically bounded in total variation norm · M(O) . That is, for each δ > 0, there exists Cδ ∈ (0, ∞) such that sup P ψε M(O) > Cδ < δ. ε>0
Then the sequence of random fields {φε : ε > 0} is tight as a sequence of H −1 (O)-valued random variables. Proof. First of all, by conditions of the lemma, for each δ > 0, we can find constants C1,δ , C2,δ > 0 and a deterministic compact set K1,δ H −1 (O) such that inf P (Ωε,δ ) > 1 − δ, ε
where Ωε,δ = ω ∈ Ω: φε (·, ω) W −1,p C1,δ ∩ ω: χε (·, ω) ∈ K1,δ ∩ ω: ψε (·, ω) M(O) C2,δ . We repeat proof for the deterministic version of such lemma (e.g. [3, Lemma 15.2.1] or [6, Corollary 1]) in this stochastic context to ensure that choice of certain compact sets does not depend on ω. First, we recall that the embedding from M(O) → W −1,q (O) is compact for m any q ∈ (1, m−1 ) (e.g. Evans [6, Theorem 6]). Therefore there exists a deterministic compact K2,δ = K2,δ (C2,δ ) W −1,q (O) such that ω: ψε M(O) C2,δ ⊂ ω: ψε (·, ω) ∈ K2,δ . We now define new random fields gε , hε ∈ W01,2 (O) as the (unique) weak solutions of −gε = χε ,
gε = 0
on ∂O,
and −hε = ψε ,
hε = 0
on ∂O,
and denote fε = gε + hε . Then by elliptic theory, there exist deterministic compact set K3,δ = 1,q K3,δ (K1,δ ) W01,2 (O) and deterministic compact set K4,δ = K4,δ (K2,δ ) W0 (O) such that ω: χε (·, ω) ∈ K1,δ ⊂ ω: gε (·, ω) ∈ K3,δ ,
ω: ψε (·, ω) ∈ K2,δ ⊂ ω: hε (·, ω) ∈ K4,δ .
Consequently, there exist deterministic compact sets 1,q
K5,δ W0 (O),
−1,q
K6,δ = K6,δ (K5,δ ) W0
(O),
such that {ω: χε ∈ K1,δ } ∩ ω: ψε M(O) C2,δ ⊂ {ω: fε ∈ K5,δ } ⊂ {ω: φε ∈ K6,δ },
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J. Feng, D. Nualart / Journal of Functional Analysis 255 (2008) 313–373
where the last inclusion follows from −fε = φε ,
x ∈ O,
−1,q
−1,p
By interpolation between W0 (O) and W0 K7,δ = K7,δ (K6,δ , C1,δ ) H −1 (O) such that
fε (x) = 0,
x ∈ ∂O.
(O), there exists a deterministic compact set
Ωε,δ = ω: φε W −1,p C1,δ ∩ {ω: χε ∈ K1,δ } ∩ ω: ψε M(O) C2,δ ⊂ ω: φε W −1,p C1,δ ∩ {ω: φε ∈ K6,δ } ⊂ {ω: φε ∈ K7,δ }. Consequently, inf P (φε ∈ K7,δ ) > 1 − δ.
ε>0
Conclusion of the lemma follows.
2
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Journal of Functional Analysis 255 (2008) 374–448 www.elsevier.com/locate/jfa
Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems Pascal Auscher a , Andreas Axelsson b,∗ , Steve Hofmann c a Université de Paris-Sud, UMR du CNRS 8628, 91405 Orsay Cedex, France b Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden c Mathematics Department, University of Missouri, Columbia, MO 65211, USA
Received 6 February 2008; accepted 14 February 2008 Available online 18 April 2008 Communicated by C. Kenig
Abstract We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L2 for small complex L∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices. © 2008 Elsevier Inc. All rights reserved. Keywords: Neumann problem; Dirichlet problem; Elliptic equation; Non-symmetric coefficients; Dirac operator; Functional calculus; Quadratic estimates; Perturbation theory; Carleson measure
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 1.1. Operators and vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
* Corresponding author.
E-mail addresses:
[email protected] (P. Auscher),
[email protected] (A. Axelsson),
[email protected] (S. Hofmann). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.02.007
P. Auscher et al. / Journal of Functional Analysis 255 (2008) 374–448
1.2. Embedding in a Dirac equation . . . . . . . . . . . 1.3. Outline of the paper . . . . . . . . . . . . . . . . . . 2. Operator theory and algebra . . . . . . . . . . . . . . . . . . 2.1. The basic operators . . . . . . . . . . . . . . . . . . . 2.2. Hodge decompositions and resolvent estimates 2.3. Quadratic estimates: generalities . . . . . . . . . . 2.4. Decoupling of the Dirac equation . . . . . . . . . 2.5. Operator equations and estimates for solutions 3. Invertibility of unperturbed operators . . . . . . . . . . . 3.1. Block coefficients . . . . . . . . . . . . . . . . . . . . 3.2. Constant coefficients . . . . . . . . . . . . . . . . . . 3.3. Real symmetric coefficients . . . . . . . . . . . . . 4. Quadratic estimates for perturbed operators . . . . . . . 4.1. Perturbation of block coefficients . . . . . . . . . 4.2. Perturbation of vector coefficients . . . . . . . . . 4.3. Proof of main theorems . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper we prove that the Neumann, Dirichlet and regularity problems are well posed in L2 (Rn ) for divergence form second order elliptic equations divt,x A(x)∇t,x U (t, x) = 0
(1.1)
n on the half-space Rn+1 + := {(t, x) ∈ R × R ; t > 0}, n 1, when A is a small complex L∞ perturbation of either a block matrix, a constant matrix or a real symmetric matrix. Furthermore, the matrix A = (aij (x))ni,j =0 ∈ L∞ (Rn ; L(Cn+1 )) is assumed to be t-independent with complex coefficients and accretive, with quantitative bounds A∞ and κA , where κA > 0 is the largest constant such that
Re A(x)v, v κA |v|2 ,
for all v ∈ Cn+1 , x ∈ Rn .
We shall approach Eq. (1.1) from a first order point of view, rewriting it as the first order system
divt,x A(x)F (t, x) = 0, curlt,x F (t, x) = 0,
(1.2)
where F (t, x) = ∇t,x U (t, x). Recall that a vector field F = F0 e0 + F1 e1 + · · · + Fn en can be written in this way as a gradient if and only if curlt,x F = 0, by which we understand that ∂j Fi = ∂i Fj , for all i, j = 0, . . . , n. We write {e0 , e1 , . . . , en } for the standard basis for Rn+1 with e0 upward pointing into Rn+1 + , and write t = x0 for the vertical coordinate. For the vertical derivative, we write ∂0 = ∂t . Denote also by F := F1 e1 + · · · + Fn en , the tangential part of F , and write curlx F = 0 if ∂j Fi = ∂i Fj , for all i, j = 1, . . . , n. In the formulation of the boundary value problems below, we assume that A = A(x) is a given coefficient matrix with properties as above. Furthermore, by saying that Ft (x) = F (t, x)
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satisfies (1.2) we shall mean that Ft ∈ C 1 (R+ ; L2 (Rn ; Cn+1 )) and that for each fixed t > 0, we have divx (AF ) = −(A(x)∂0 F )0 , ∇x F0 = ∂0 F and curlx F = 0, where the derivatives on the left-hand sides are taken in the sense of distributions. If this holds for F , then in particular we 1 (Rn+1 ), and we see that U satisfies (1.1) in the sense that can write F = ∇t,x U , with U ∈ W2,loc +
A(x)∇t,x U (t, x), ∇t,x ϕ(t, x) dt dx = 0,
. for all ϕ ∈ C0∞ Rn+1 +
Rn+1 +
Neumann problem (Neu-A). Given a function φ(x) ∈ L2 (Rn ; C), find a vector field Ft (x) = such that Ft ∈ C 1 (R+ ; L2 (Rn ; Cn+1 )) and F satisfies (1.2) for t > 0, and furF (t, x) in Rn+1 + thermore limt→∞ Ft = 0 and limt→0 Ft = f in L2 norm, where the conormal part of f satisfies the boundary condition e0 · (Af ) =
n
A0j fj = φ,
j =0
For U , this means that the conormal derivative
on Rn = ∂Rn+1 + .
∂U ∂νA (0, x) = φ(x)
in L2 (Rn ).
Regularity problem (Reg-A). Given a function ψ : Rn → C with tangential gradient ∇x ψ ∈ such that Ft ∈ C 1 (R+ ; L2 (Rn ; Cn+1 )) L2 (Rn ; Cn ), find a vector field Ft (x) = F (t, x) in Rn+1 + and F satisfies (1.2) for t > 0, and furthermore limt→∞ Ft = 0 and limt→0 Ft = f in L2 norm, where the tangential part of f satisfies the boundary condition f = f1 e1 + · · · + fn en = ∇x ψ,
on Rn = ∂Rn+1 + .
For U , this means that ∇x U (x, 0) = ∇x ψ(x), i.e. U (x, 0) = ψ(x) in W˙ 21 (Rn ). Dirichlet problem (Dir-A). Given a function u(x) ∈ L2 (Rn ; C), find a function Ut (x) = U (t, x) 2 n 1 n n+1 )) and ∇ U satisin Rn+1 t,x t + such that Ut ∈ C (R+ ; L2 (R ; C)), ∇t,x Ut ∈ C (R+ ; L2 (R ; C fies (1.2) for t > 0, and furthermore limt→∞ Ut = 0, limt→∞ ∇t,x Ut = 0 and limt→0 Ut = u in L2 norm. We shall also use first order methods based on (1.2) to solve the Dirichlet problem. However, here we use a different relation between (1.1) and (1.2) which we now describe. Assume F (t, x) = nk=0 Fi (t, x)ei is a vector field satisfying (1.2). Applying ∂t to the first equation and using that curlt,x F = 0 yields 0 = ∂t divt,x A(x)F = divt,x A(x)(∂t F ) = divt,x A(x)(∇t,x F0 ), since the coefficients are assumed to be t-independent. Thus the normal component U := F0 satisfies (1.1). Note that when A = I , the functions F1 , . . . , Fn are conjugates to U in the sense of Stein and Weiss [25]. From this we see that solvability of (Dir-A) is a direct consequence of solvability of the following auxiliary Neumann problem. Neumann problem (Neu⊥ -A). Given a function φ(x) ∈ L2 (Rn ; C), find a vector field Ft (x) = such that Ft ∈ C 1 (R+ ; L2 (Rn ; Cn+1 )) and F satisfies (1.2) for t > 0, and F (t, x) in Rn+1 +
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furthermore limt→∞ Ft = 0 and limt→0 Ft = f in L2 norm, where the normal part of f is e0 · f = f0 = φ. The main result of this paper is the following L∞ perturbation result for the boundary value problems. Theorem 1.1. Let A0 (x) = ((a0 )ij (x))ni,j =0 be a t-independent, complex, accretive coefficient matrix function. Furthermore assume that A0 has one of the following extra properties. (b) A0 is a block matrix, i.e. (a0 )0i (x) = (a0 )i0 (x) = 0 for all 1 i n and all x ∈ Rn . (c) A0 is a constant coefficient matrix, i.e. A0 (x) = A0 (y) for all x, y ∈ Rn . (s) A0 is a real symmetric matrix, i.e. (a0 )ij (x) = (a0 )j i (x) ∈ R for all 0 i, j n and all x ∈ Rn . Then there exists ε > 0 depending only on A0 ∞ , the accretivity constant κA0 and the dimension n, such that if A ∈ L∞ (Rn ; L(Cn+1 )) is t-independent and satisfies A − A0 ∞ < ε, then Neumann and Regularity problems (Neu-A), (Neu⊥ -A) and (Reg-A) above have a unique solution F (t, x) with the required properties for every boundary function g(x), being φ(x) and ∇x ψ(x), respectively. Furthermore Dirichlet problem (Dir-A) above has a unique solution U (t, x) with the required properties for every boundary function u(x). The solutions depend continuouslyon the data with the following equivalences of norms. If we ∞ define the triple bar norm |||Gt |||2 := 0 Gt 22 t −1 dt and the non-tangential maximal function ∗ (F )(x) := sup N t>0
−
−
F (s, y) 2 ds dy
1/2 ,
|s−t|
where −E := |E|−1 E and c0 ∈ (0, 1), c1 > 0 are constants, then for Neumann and Regularity problems we have
∗ (F ) , g2 ≈ f 2 ≈ sup Ft 2 ≈ |||t∂t Ft ||| ≈ N 2 t>0
and for Dirichlet problem we have
∗ (U ) . u2 ≈ sup Ut 2 ≈ |||t∂t Ut ||| ≈ |||t∇x Ut ||| ≈ N 2 t>0
Moreover, the solution operators SA , being SA (g) = F or SA (u) = U , respectively, depend Lipschitz continuously on A, i.e. there exists C < ∞ such that SA2 − SA1 L2 (Rn )→X CA2 − A1 L∞ (Rn ) when Ai − A0 ∞ < ε, i = 1, 2, where F X or U X denotes any of the norms above. Throughout this paper, we use the notation X ≈ Y and X Y to mean that there exists a constant C > 0 so that X/C Y CX and X CY , respectively. The value of C varies from one usage to the next, but then is always fixed.
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Let us now review the history of works on these boundary value problems, starting with the case of matrices of the form A˜ which we now describe. By standard arguments, Theorem 1.1 also shows well-posedness of the corresponding boundary value problems on the region Ω above a Lipschitz graph Σ = {(t, x); t = g(x)}, where g : Rn → R is a Lipschitz function. Indeed, if (t, x) := U (t + g(x), x) satisfies the function U (t, x) satisfies divt,x A(x)∇t,x U = 0 in Ω then U n+1 ˜ divt,x A(x)∇t,x U = 0 in R+ , where 1 1 −(∇x g(x))t ˜ A(x) A(x) := 0 I −∇x g(x)
0 . I
Thus Theorem 1.1 gives conditions on A for which Neumann problem (conormal derivative ˜ t,x U = (e0 − ∇x g) · A∇t,x U given), Regularity problem (tangential gradient ∇x U = e0 · A∇ = U given) are well posed. Note that A is real (∂t U )∇x g +∇x U given), and Dirichlet problem (U symmetric if and only if A˜ is, but that A˜ being constant or of block form does not imply the same for A. For the Laplace equation A = I in Ω, solvability of (Neu-I˜) and (Reg-I˜) was first proved by Jerison and Kenig [16], and solvability of (Dir-I˜) was first proved by Dahlberg [12]. Later Verchota [26] showed that these boundary value problems are solvable with the layer potential integral equation method. For general real symmetric matrices A, not being of the “Jacobian type” I˜ above, the wellposedness of (Dir-A) was first proved by Jerison and Kenig [17], and (Neu-A) and (Reg-A) by Kenig and Pipher [20]. These results make use of the Rellich estimate technique. For Neumann and Regularity problems, this integration by parts technique yields an equivalence
∂U
≈ ∇x U L2 (Rn ) , (1.3)
∂ν A L2 (Rn ) which is seen to be equivalent with the first estimate f ≈ g in the theorem above, and shows that the boundary trace f splits into two parts of comparable size. Turning to the unperturbed case where A = A0 satisfies (b), then (1.3) is still valid, but the proof is far deeper than Rellich estimates. In fact, it is equivalent with the Kato square root estimate proved by Auscher, Hofmann, Lacey, Mc Intosh and Tchamitchian in [4]. (For the nondivergence form case a00 = 1, see [10].) For details concerning this equivalence between the Kato problem and the boundedness and invertibility of the Dirichlet-to-Neumann map ∇x U → ∂U ∂ν we refer to Kenig [19, Remark 2.5.6], where also many further references in the field can be found. We now consider what is previously known in the case when A does not satisfy (b), (c) or (s). Here (Dir-A) has been showed to be well posed by Fabes, Jerison and Kenig [14] for small perturbations of (c), using the method of multilinear expansions. More recently, the boundary value problems have been studied in the Lp setting and for real but non-symmetric matrices in the plane, i.e. n = 1. Here Kenig, Koch, Pipher and Toro [22] have obtained solvability of the Dirichlet problem for sufficiently large p, and Kenig and Rule [21] have shown solvability of the Neumann and regularity problems for sufficiently small dual exponent p . In the perturbed case A ≈ A0 when A0 satisfies (c) or (s), the well-posedness of (Neu-A), (Reg-A) and (Dir-A) is also proved in [2] by Alfonseca, Auscher, Axelsson, Hofmann and Kim. With the further assumption of pointwise resolvent kernel bounds, perturbation of case (b) is also implicit in [2]. It is worth comparing the present methods to those of [2]. In [2], due to the presence of kernel bounds, the solvability of the boundary value problems is meant in the sense of non-tangential maximal estimates at the boundary and this follows from the use of layer
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potentials. The first main result in [2] in the unperturbed case (s), is the proof via singular integral operator theory of boundedness and invertibility of layer potentials. The second main result in [2] is the stability of the simultaneous occurrence of both boundedness and invertibility, which hold in the unperturbed cases (c), (s) and (b). Solvability then follows. Here, we setup a different resolution algorithm (forcing us to introduce some substantial material), which consists in solving the first order system (1.2) instead of (1.1), also by a boundary operator method, but acting on the gradient of solutions involving a generalised Cauchy operator EA , the goal being to establish boundedness of EA and invertibility of related operators EA ± NA . Boundedness of EA = sgn(TA ) is obtained via quadratic estimates of an underlaying first order differential operator TA , and the deep fact is here that those quadratic estimates alone are stable under perturbations. Stability of invertibility is then easy. The perturbation argument requires sophisticated harmonic analysis techniques inspired by the strategy of [2]. In particular, the latter uses extensively the technology of the solution of the Kato problem for second order operators in [4], whereas we utilise here the work of Axelsson, Keith and Mc Intosh [10], which adapts and extends this technology to first order operators of Dirac type. Indeed, we note that our Dirac type operators TA are of the form ΠA of [10] in the case of block matrices (b) of Theorem 1.1. But TA has a more complicated structure when A is not a block matrix and we understand how to prove boundedness of EA at the moment only in the cases specified by Theorem 1.1. We also note that the present paper, like [10], makes no use of kernel bounds and only needs L2 off-diagonal bounds for the operators, which always holds. The boundary operator method for first order Dirac type operators, used here to solve second order boundary value problems, was developed in the thesis of Axelsson [6], which has been published as the four papers [5,7–9]. It covers operators on Lipschitz domains as described above and in Example 1.5. The result in [10] pursued the program initiated by Auscher, Mc Intosh and Nahmod in [3], consisting of connecting the Kato problem and the functional calculus of first order differential operators of Dirac type. As said, it thus applies to the boundary value problems for operators of case (b). What is new here is the setup for full matrices encompassing the above. We prove also a sort of meta-theorem (see Theorem 1.3) which roughly says that the set of matrices for which the needed quadratic estimates on TA hold, is open. We also show that non-tangential maximal estimates hold for our solutions. By uniqueness in the class of solutions of (1.1) with non-tangential maximal estimates, this implies that our solutions are the same as those in [2] for perturbations of the real symmetric and constant cases. The non-tangential maximal estimate here also yields an indirect proof of non-tangential limits of solutions of (1.2) which hold for the solutions of (1.1) in [2]. We do not know how to prove this fact directly in the framework of this article. Note also that we prove here that the nontangential maximal functions have comparable L2 -norms for different values of the parameters c0 and c1 , and that the slightly different non-tangential maximal function used in [2] therefore has comparable norm. Before turning to the method of proof for Theorem 1.1, we would like to stress the importance of the final result that the solution operators g → F and u → U depend Lipschitz continuously on L∞ changes of the matrix A around A0 . This is an important motivation for considering complex A, as the authors do not know any proof of this perturbation result which does not make use of boundedness of the operators in a complex neighbourhood of A0 . We also remark that we in fact prove that A → SA is holomorphic, from which we deduce Lipschitz continuity as a corollary.
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In this paper, we shall mainly focus on the boundary value problems (Neu-A) and (Reg-A). The reason is that, assuming the Cauchy operator EA is bounded, we prove in Section 2.5 that well-posedness follows as (Reg-A∗ )
⇐⇒
Neu⊥ -A
⇒
(Dir-A).
That (Reg-A∗ ) implies (Dir-A) has been proved by Kenig and Pipher [20, Theorem 5.4] in the case of real matrices A. 1.1. Operators and vector fields We now explain the basic ideas of the method we use for the proof of Theorem 1.1. The appropriate Hilbert space on the boundary Rn is Hˆ 1 := f ∈ L2 Rn ; Cn+1 ; curlx (f ) = 0 . The condition on f means that its tangential part is curl-free. Indeed, the trace f (x) of a vector field F (t, x) solving (1.2) belongs to Hˆ 1 due to the second equation in (1.2). The basic picture, building on ideas from [7], is that the Hilbert space splits into two different pairs of complementary subspaces as + ˆ1 − ˆ1 Hˆ 1 = EA H ⊕ EA H = NA+ Hˆ 1 ⊕ NA− Hˆ 1 .
(1.4)
± ˆ1 H , consisting of L2 boundary We first discuss the splitting into the Hardy type subspaces EA n+1 traces of vector fields F ± solving (1.2) in R± , respectively. Our main work in this paper is ± for certain A. These projections can to establish boundedness of the projection operators EA ± 1 be written EA = 2 (I ± EA ), where EA for simple A is a singular integral operator of Cauchy type. However, in the general case EA may fail to be a singular integral operator. To handle the ± we make use of functional calculus of closed Hilbert space operators, and show projections EA ± that EA = χ± (TA ) are the spectral projections of an underlaying bisectorial operator TA in Hˆ 1 . The functions χ± (z) are the characteristic functions for the right and left complex half-planes. To find TA , assume F (t, x) satisfies (1.2) in Rn+1 + and solve for the vertical derivative
−1 ∂t F0 = −a00
n
a0i ∂i F0 + ∂i (AF )i ,
i=1
∂t Fi = ∂i F0 ,
i = 1, . . . , n.
The right-hand side defines an operator −TA in Hˆ 1 which on F (t, x), for fixed t > 0, satisfies ∂t F + TA F = 0. Concretely, if we identify f = f0 e0 + f with (f0 , f )t , where f is a tangential curl-free vector field, then −1 A00 ((A0 , ∇x ) + divx A0 ) A−1 f0 00 divx A TA f = , (1.5) −∇x 0 f
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where
ˆ 1 ; ∇x f0 ∈ L2 , divx (Af ) ∈ L2 D(TA ) = f = (f0 , f )t ∈ H
A=
and
A0 . A
A00 A0
If F (t, x) is a vector field in Rn+1 satisfying (1.2), then using this operator TA , we can repro+ duce F provided we know the full trace f = F |Rn , through a Cauchy type reproducing formula F (t, x) = (e−t|TA | f )(x). However, in (Neu-A) and (Reg-A) only “half” of the trace f is known since the boundary conditions for f are e0 · Af = φ and e0 ∧ f = e0 ∧ ∇ψ , respectively. We now turn to the second splitting in (1.4), which is used to split the boundary trace f into the regularity and Neumann data. We define the A-tangential and normal subspaces of Hˆ 1 to be the null spaces of these two operators: NA+ Hˆ 1 := f ∈ Hˆ 1 ; e0 · Af = 0 , NA− Hˆ 1 := f ∈ Hˆ 1 ; e0 ∧ f = 0 . In contrast with the Hardy subspaces, it is straightforward to show that we have a topological splitting Hˆ 1 = NA+ Hˆ 1 ⊕ NA− Hˆ 1 , and therefore that the corresponding pair of projections NA± are bounded. We can now reformulate (Neu-A) and (Reg-A) as follows. Neumann problem (Neu-A) being well posed means that the restricted projection + ˆ1 H → NA− Hˆ 1 NA− : EA
is an isomorphism, since NA− Hˆ 1 is a complement of the null space of e0 · A(·). Similarly Regularity problem (Reg-A) being well posed means that the restricted projection + ˆ1 H → NA+ Hˆ 1 NA+ : EA
is an isomorphism, since NA+ Hˆ 1 is a complement of the null space of e0 ∧ (·). Note that what is important here is which subspace NA± projects along, not what subspace they project onto. + − − EA = sgn(TA ) and NA := We shall also find it convenient to use the operators EA := EA + − ± 2 2 NA − NA . These are reflection operators, i.e. EA = I and NA = I , and we have EA = 12 (I ± EA ) ± 1 and NA = 2 (I ± NA ). Example 1.2. Let n = 1 and A = I . Then the space Hˆ 1 is simply L2 (R; C2 ) and the fundamental operator TA becomes T := TI =
0 d − dx
d dx
0
≈
0 −iξ
iξ , 0
if ≈ denotes conjugation with Fourier transform. Furthermore E := EI = sgn(T ) =
0 −iH
iH , 0
N := NI =
−1 0 , 0 1
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where Hf (x) :=
i p.v. π
f (y) dy. x −y
Note that the operator E is contained in the Borel functional calculus of the self-adjoint operator T , and it follows that E = 1. On the other hand the operator N is outside the Borel functional calculus of T . Indeed, the operators b(T ) in the Borel functional calculus of T all commute, but we have the anticommutation relation EN + N E = 0.
(1.6)
A consequence of this equation is that the boundary value problems (Neu-I ) and (Reg-I ) are well posed, or equivalently that N − : E + L2 → N − L2 and N + : E + L2 → N + L2 are isomorphisms. Consider for example (Reg-I ) and assume we want to solve N + f = g for f ∈ E + L2 . Applying 4E + to the equation gives 4E + g = (I + E)(I + N )f = f + Ef + Nf + ENf = f + f + Nf − Nf = 2f, so f = 2E + g and it follows that N + : E + L2 → N + L2 is an isomorphism. Having solved for f ∈ E + L2 , we can find the solution F (t, x) = Ct+ f (x) in R2+ by using the Cauchy extension Ct+ := e−t|T | E + for t > 0. As a convolution operator, the Fourier multiplier Ct+ has the expression
tu0 (y) − (x − y)u1 (y) 1 + Ct (u0 e0 + u1 e1 ) = dy e0 2π t 2 + (x − y)2
+
R
1 2π
R
(x − y)u0 (y) + tu1 (y) dy e1 , t 2 + (x − y)2
and in particular F (t, x) = Ct+ f (x) = 2Ct+ (g1 e1 ) =
1 π
R
(−(x − y)e0 + te1 )g1 (y) dy. t 2 + (x − y)2
For a more general A, even if A is real symmetric, the operator TA is not self-adjoint and ± is a highly non-trivial problem. Also Eq. (1.6) fails for general A, proving boundedness of EA and therefore such explicit formulae for the solution F (t, x) as in Example 1.2 are not available. However, to show well-posedness of (Neu-A) and (Reg-A) it suffices to show that I ± 12 (EA NA + NA EA ) = 12 (EA ± NA )2 are invertible, as explained in [7]. To summarise, in order to solve (Neu-A) and (Reg-A) we need that: ± + ˆ1 H ⊕ are bounded, so that we have a topological splitting Hˆ 1 = EA (i) the Hardy projections EA − ˆ1 EA H , and + ˆ1 + ˆ1 H → NA+ Hˆ 1 and NA− : EA H → NA− Hˆ 1 are isomor(ii) the restricted projections NA+ : EA phisms.
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To prove Theorem 1.1, we shall prove the following. (i ) That TA satisfies quadratic estimates for all A such that A − A0 ∞ < ε. From this it will ± ± follow that EA are bounded for all such A and that A → EA is continuous (in fact holomorphic). (ii ) That (ii) holds for the unperturbed A0 . From (i ) and since clearly NA± are bounded and depend continuously on A, it then follows from a continuity argument that (ii) holds for all A in a neighbourhood of A0 . ± We emphasise that boundedness of the Hardy projections EA alone does not show that (Neu-A) and (Reg-A) are well posed. In our framework, the boundary value problems being well posed + ˆ1 H , which exists as a closed subspace when the Hardy projections means that the Hardy space EA are bounded, is transversal to the A-tangential and normal subspaces NA± Hˆ 1 . A concrete example showing that (Neu-A) and (Reg-A) may fail to be well posed, even though EA is bounded, is furnished by the matrix
k sgn(x) , 1
1 A(x) = −k sgn(x)
with parameter k ∈ R. The corresponding elliptic equation (1.1) in R2+ was studied by Kenig, Koch, Pipher and Toro in [22], where they showed that (Dir-A) fails to be well posed for certain values of k. Moreover, that (Neu-A) and (Reg-A) also fails for some k, is shown by Kenig and Rule [21]. On the other hand, EA = 1 for all k ∈ R since according to (1.5) TA =
d − k(sgn(x) dx
d dx
sgn(x))
d − dx
d dx
0
± are orthogonal projections. is self-adjoint, and therefore EA
1.2. Embedding in a Dirac equation Unfortunately there is a technical problem in applying harmonic analysis to the operator TA in order to prove (i ): the space Hˆ 1 is defined through the non-local condition curlx (f ) = 0. This prevents us from using multiplication operators, for example when localising with a cutoff f →ηf , as these does not preserve Hˆ 1 . To avoid this problem we embed Hˆ 1 ⊂ H := L2 (Rn ; C Rn+1 ), where
CR
n+1
=
0
⊕
1
⊕
2
⊕··· ⊕
n+1
is the full complex exterior algebra of Rn+1 , which in particular contains the vectors 1 = Cn+1 and the scalars 0 = C along with all k-vectors k . (We identify k-vectors with the dual k-forms in Euclidean space.) In this way we obtain closure, i.e. all operators, including multiplication operators preserve H. Furthermore we embed Eq. (1.2) in a Dirac type equation ∗ −1 dt,x dt,x + B(x) B(x) F (t, x) = 0,
(1.7)
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∗ are the ex ∈ L∞ (Rn ; L( C Rn+1 )) are accretive and B|1 = A. Here dt,x and dt,x where B, B terior and interior derivative operators, as defined in Section 2. In particular, if F (t, x) : Rn+1 + → 1 is a vector field satisfying (1.2) then ∗ dt,x (BF ) = − divt,x (AF ) = 0,
dt,x F = curlt,x F = 0. In the same way as in Thus (1.7) follows from (1.2) for any choice of the auxiliary function B. Section 1.1 we shall solve for the vertical derivative in (1.7) and obtain an operator TB acting in H, such that TA = TB |Hˆ 1 . For applications to the Neumann and regularity problems it suffices to consider B = I ⊕ A ⊕ I ⊕ · · · ⊕ I . The stability result for the operator TA we prove in this paper is the following. Theorem 1.3. Let A0 be a t-independent, complex, accretive coefficient matrix function such that TB0 has quadratic estimates in H, where B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I . Then there exists ε0 > 0 depending only on A0 ∞ , κA0 and n, such that if A ∈ L∞ (Rn ; L(Cn+1 )) is t-independent and satisfies A − A0 ∞ < ε0 , then TB has quadratic estimates in H, where B = I ⊕ A ⊕ I ⊕ · · · ⊕ I . In particular TA has quadratic estimates in Hˆ 1 . Thus Hˆ 1 splits into Hardy subspaces, the spectral subspaces of TA , i.e. each f ∈ Hˆ 1 can be ± f (x) satisfy (1.2) uniquely written f = f + + f − , where f ± = F ± |Rn and F ± (t, x) = e∓t|TA | EA n+1 in R± . Moreover, we have equivalence of the norms f 2 ≈ f + 2 + f − 2 ± ± ∗ (F ± )2 . and f ± 2 ≈ supt>0 F±t 2 ≈ |||t∂t F±t ||| ≈ N
With a more general choice B = B 0 ⊕ B 1 ⊕ B 2 ⊕ · · · ⊕ B n+1 , where B k ∈ L∞ (Rn ; L( k )), we also obtain new perturbation results concerning boundary value problems and more generally transmission problems for k-vector fields. For k-vector fields we consider the function space k Hˆ B := f ∈ L2 Rn ; L k ; dx (f ) = 0 = dx∗ B k f ⊥ ⊂ H, where f and f⊥ denote the tangential and normal parts of f . This is the appropriate function space since traces of k-vector fields F (t, x) in Rn+1 ± satisfying
∗ B k (x)F (t, x) = 0, dt,x dt,x F (t, x) = 0,
(1.8)
k . Note that for 1-vectors, i.e. vectors, the condition d ∗ ((B 1 f ) ) = 0 is void and belong to Hˆ B ⊥ x 1 =H ˆ 1 . For k-vector fields, we consider the following. Hˆ B
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Transmission problem (Tr-B k α ± ). Let B k = B k (x) ∈ L∞ (Rn ; L( k )) be accretive and let k , find k-vector fields F + (x) = α ± ∈ C be given jump parameters. Given a k-vector field g ∈ Hˆ B t n+1 − n+1 ± + − F (t, x) in R+ and Ft (x) = F (t, x) in R− such that Ft ∈ C 1 (R+ ; L2 (Rn ; k )) and F ± satisfies (1.8) for ±t > 0, and furthermore limt→±∞ Ft± = 0 and limt→0± Ft± = f ± in L2 norm, where the traces f ± satisfy the jump conditions
e0 ∧ α − f + − α + f − = e0 ∧ g, k + + e0 B α f − α − f − = e0 B k g .
The second main result of this paper is the following L∞ perturbation result of Transmission problem (Tr-B k α ± ). Theorem 1.4. Let B0k = B0k (x) ∈ L∞ (Rn ; L( k )) be t-independent, accretive and possibly complex, and assume that B0k is a block matrix, i.e. k B0 st = 0,
whenever 0 ∈ (s \ t) ∪ (t \ s)
and s, t ⊂ {0, 1, . . . , n} has lengths |s| = |t| = k. Then there exist ε > 0 and C < ∞ depending only on B0k ∞ , the accretivity constant κB k and dimension n, such that if B k = B k (x) ∈ 0 L∞ (Rn ; L( k )) is t-independent and satisfies
k
B − B k
0 L∞ (Rn )
2 < min ε, C α + /α − + 1 ,
(1.9)
then Transmission problem (Tr-B k α ± ) above is well posed in the sense that for every boundary k , there exist unique k-vector fields F ± (t, x) with properties as in (Tr-B k α ± ). function g ∈ Hˆ B ± The solution Ft depends continuously on g with equivalences of norms g2 ≈ f + + f − 2 ≈ f + 2 + f − 2 and
±
f ≈ sup F ± ≈ t∂t F ± . ±t 2 ±t 2 t>0
This perturbation theorem for transmission problems for k-vectors has two important corollaries. On one hand it specialises when k = 1 to a generalisation of Theorem 1.1(b), giving perturbation results for transmission problems across Rn for the divergence form equation (1.1). The details of this Neumann-regularity transmission problem is stated as (Tr-Aα ± ) in Section 4.3. On the other hand it specialises when either α + = 0 or α − = 0 to a generalisation of Theorem 1.1(b), giving perturbation results for boundary-value problems for k-vectors. Our result for these boundary-value problems (Nor-B k ) and (Tan-B k ) is given as Corollary 4.17 in Section 4.3. ∗ B appear naturally when = B, operators of the form dt,x + B −1 dt,x Example 1.5. In the case B ∗ pulling back the unperturbed Hodge–Dirac operator d + d with a change of variables. As above, consider the region Ωabove a Lipschitz graph Σ = {(t, x); t = g(x)}. We define the pullback of the field F : Ω → Rn+1 to be the field
(ρ ∗ F )(t, x) := ρ T (x)F ρ(t, x)
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in Rn+1 + , where ρ(t, x) = (t + g(x), x) is the parametrisation of Ω, having differential ρ(x)|1 =
1 ∇g(x) 0 I
acting on vector fields and extended naturally to Rn+1 , and ρ T (x) denotes the transposed matrix. From the well-known fact that dt,x commutes with ρ ∗ , we get the intertwining relation ∗ ∗ F , G−1 (ρ ∗ F ) = ρ ∗ dt,x + dt,x dt,x + Gdt,x
(1.10)
where G(x) = (gij (x)) = ρ T (x)ρ(x) is the metric for the parametrisation, being real symmetric. ∗ )F = 0 in Ω, gives us an operator Solving for the vertical derivative in the equation (dt,x + dt,x n+1 DΣ in L2 (Σ; R ), which is similar to the operator (e0 − ∇g(x))−1 (dx + dx∗ ) in H. From (1.10) it follows that TG−1 ρ(x)T f (x) = ρ(x)T (DΣ f )(x). It is known that the operator DΣ satisfies quadratic estimates, and therefore so does TG−1 . For references and further discussion of DΣ , see [10, Consequence 3.6]. From this we get the bounded Clifford–Cauchy singular integral operator EG−1 ρ(x)T f (x) 2 (g(x)e0 + x) − (g(y)e0 + y) e0 − ∇g(y) f (y) dy, = ρ(x)T p.v. 2 2 (n+1)/2 σn (|y − x| + (g(y) − g(x)) ) Rn
where σn is the area of the unit n-sphere in Rn+1 and EG−1 = sgn(TG−1 ). For this reason, we shall refer to EB = sgn(TB ) as generalised Cauchy integral operators and EB± = χ± (TB ) as gen = B. eralised Hardy projection operators, also when B = G−1 and B 1.3. Outline of the paper In Section 2 we explain how we use the exterior algebra C Rn+1 and the exterior and interior derivative operators d and d ∗ . In Section 2.1 we introduce the Dirac type operator TB which extends TA to the full exterior algebra as well as projection operators NB± which extend the Atangential and normal projections NA± from above. Section 2.2 is concerned with the spectral properties of TB , where we prove that TB is a bisectorial operator and that the resolvents (λI − TB )−1 has L2 off-diagonal estimates. Section 2.3 surveys the theory of functional calculus of bisectorial operators like TB . In Section 2.5, Lemmas 2.49 and 2.55 characterise the classes of solutions Ft and Ut , respectively, to the boundary value problems, and are used in particular to prove uniqueness. In Section 3 we prove (i ) quadratic estimates and (ii ) invertibility in the unperturbed case B = B0 . The most involved case is when B0 is real symmetric. In order to prove the quadratic k and H ˇB estimates we use the results from Section 2.4 that TB0 leaves certain subspaces Hˆ B 0 0 invariant and that therefore it suffices to establish quadratic estimates in each subspace separately.
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Section 4 contains the core harmonic analysis results of the paper, where we establish quadratic estimates for TB when B ≈ B0 through a perturbation argument based on Eqs. (4.3)– (4.8). Section 4.1 treats the case when B0 is a block matrix and makes use of techniques from the solution of the Kato square root problem [4]. In Lemma 4.10 we construct a new set of test functions fQw which also can be used in the Carleson measure estimate in [10] to simplify the proof there. Section 4.2 treats matrices of the form B = I ⊕ A ⊕ I ⊕ · · · ⊕ I . The key technique is Lemma 4.14 where we compare B0 with a corresponding block matrix Bˆ 0 for which the results from Section 4.1 applies. The reason why this approximation B0 ≈ Bˆ 0 works is that the normal vector component F 1,0 has additional regularity by Lemma 2.12 when F satisfy the Dirac type equation (1.7). The paper ends with Section 4.3, where we bring the results together and prove Theorems 1.4, 1.1 and 1.3. 2. Operator theory and algebra In this section we develop the operator theoretic framework we use to prove the perturbation theorems stated in the introduction. In particular we introduce our basic operator TB , along with perturbations of the normal and tangential projection operators N − and N + . These all act in the n+1 n n+1 we write Hilbert space H := L2 (R ; C Rn+1 ) on the boundary Rn = ∂Rn+1 + = ∂R− . In R the standard ON basis as {e0 , e1 , . . . , en }, where e0 denote the vertical direction and e1 , . . . , en span the horizontal hyperplane Rn . We write the corresponding coordinates as x0 , x1 , . . . , xn and we also use the notation t = x0 . The corresponding partial derivatives we write as ∂i = ∂x∂ i and ∂t = ∂0 = ∂t∂ . Our functions f ∈ H take values in the full complex exterior algebra over Rn+1
=
CR
n+1
=
0
⊕
1
⊕··· ⊕
n+1
.
This is a 2n+1 -dimensional linear space with n + 2 pairwise orthogonal subspaces k of di mensions n+1 es := es1 ∧ . . . ∧ esk if s = {s1 , . . . , sk } ⊂ {0, 1, . . . , n} and k . With the notation s1 < s2 < · · · < sk , the space k of homogeneous k-vectors is the linear span of {es ; |s| = k}. In particular, identifying e∅ with 1 and the singleton set {j } with j , we have 0 = C and 1 = Cn+1 . A general element in is called a multivector and is a direct sum of k-vectors of different degrees k. Definition 2.1. Introduce the sesqui-linear scalar product (f, g) =
s
f s es ,
t
gt e t
=
fs g s ,
s
on and the bilinear scalar product f · g = s fs gs . Define the counting function σ (s, t) := #{(si , tj ); si > tj }, where s = {si }, t = {tj } ⊂ {0, 1, . . . , n}. (i) The exterior product f
∧g
is the complex bilinear product for which
es ∧ et = (−1)σ (s,t) es∪t if s ∩ t = ∅ and zero otherwise.
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(ii) The (left) interior product f g is the complex bilinear product for which (es et , eu ) = (et , es ∧ eu ) for all s, t, u ⊂ {0, 1, . . . , n}. Explicitly we have es et = (−1)σ (s,t\s) et\s if s ⊂ t
and zero otherwise.
Example 2.2.The most common situation is when forming a product between a vector a = n 1 and a k-vector j =0 aj ej ∈
f=
fs1 ,...,sk es1 ∧ . . . ∧ esk .
0s1 <···<sk n
In this case the interior product a f is a (k − 1)-vector, whereas the exterior product a ∧ f yields k-vectors as a (k + 1)-vector. Note also that we embed all different spaces k of homogeneous pairwise orthogonal subspaces in the 2n+1 -dimensional linear space . Thus we may add the two products to obtain (the Clifford product) a f +a∧f ∈
k−1
k+1
⊕
⊂
.
In the special case when f = b is also a vector, i.e. k = 1, we have ab= a∧b=
n
aj bj ∈
0
,
j =0
(ai bj − aj bi )ei ∧ ej ∈
2
,
0i<j n
where we write b{j } = bj . We see that a b coincide with the bilinear scalar product a · b. Furthermore, in three dimensions n+1 = 3, the exterior product a ∧ b ∈ 2 can be identified with 1 the vector product a × b ∈ by using the Hodge star identifications e{1,2} ≈ e0 , −e{0,2} ≈ e1 and e{0,1} ≈ e2 . The following anticommutativity, associativity and derivation properties of these products summarise the fundamental algebra we shall need in this paper. Lemma 2.3. If a, b ∈
1
are vectors and f , g, h ∈ a ∧ b = −b ∧ a,
f
∧ (g ∧ h) = (f ∧ g) ∧ h,
, then
a ∧ a = 0, f (g h) = (g ∧ f ) h,
a (b ∧ f ) = (a · b)f − b ∧ (a f ). We shall also frequently use that if a ∈
1
is a real vector, then (a f, g) = (f, a ∧ g).
Proof. That a ∧ b = −b ∧ a and a ∧ a = 0 is readily seen from Example 2.2. These and the associativity f ∧ (g ∧ h) = (f ∧ g) ∧ h are well-known properties of the exterior product. To see how f (g h) = (g ∧ f ) h follows, note first that by linearity it suffices to consider the case
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when f , g and h all are real, i.e. have real coefficients in the standard basis {es }. Under this assumption we pair with an arbitrary w ∈ and use that left interior and exterior multiplication are adjoint operations. We get f (g h), w = (g h, f ∧ w) = h, g ∧ (f ∧ w) = h, (g ∧ f ) ∧ w = (g ∧ f ) h, w . For the derivation identity, by linearity it suffices to prove ei (ej
∧ es ) + ej ∧ (ei
es ) =
es , i = j, 0, i =
j,
for all s ⊂ {0, 1, . . . , n}. This is straightforward to verify from Definition 2.1. Definition 2.4. For a multivector f ∈ e0 ∧ f + e0 f = (μ + μ∗ )f . We call
, we write μf := e0
∧
2
f , μ∗ f := e0 f and mf :=
f⊥ = N − f := μμ∗ f = mμ∗ f = μmf, f = N + f := μ∗ μf = mμf = μ∗ mf, the normal and tangential parts of f , respectively. Concretely, if f =
{s1 ,...,sk } fs1 ,...,sk es1 ∧ . . . ∧ esk ,
f⊥ =
then its normal part is
fs1 ,...,sk es1 ∧ . . . ∧ esk
{s1 ,...,sk }0
and its tangential part is f =
fs1 ,...,sk es1 ∧ . . . ∧ esk .
{s1 ,...,sk } /0
/ s. Note that boththe subspace of norIn particular es is normal if 0 ∈ s and tangential if 0 ∈ mal multivectors N − and the subspace of tangential multivectors N + have dimension 2n , although the subspaces N ± k have different dimensions in general. Throughout this paper, upper case letters denote -valued functions F (t, x) = Ft (x) in the n+1 domain Rn+1 or R whereas lower case letters will denote -valued functions f on the + − boundary Rn . We use the sesqui-linear scalar product (f, g) := Rn (f (x), g(x)) dx on the Hilbert space H. Definition 2.5. Using the nabla symbols ∇x = ∇ = nj=1 ej ∂j in Rn and ∇t,x = nj=0 ej ∂j in Rn+1 , we define the operators of exterior and interior derivation as
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dx f = df = ∇ ∧ f :=
n
ej
dx∗ f = d ∗ f = −∇ f := −
∧ ∂j f,
j =1
dt,x F = ∇t,x ∧ F :=
n
n
ej ∂j f,
j =1
ej
∗ dt,x F
∧ ∂j F,
= −∇t,x F := −
j =0
n
ej ∂j F.
j =0
We shall also find it convenient to use the operators d := imd and d ∗ = −id ∗ m = imd ∗ . Remark 2.6. Clearly, in the splitting = 0 ⊕ 1 ⊕ · · · ⊕ n+1 , the operators d and d ∗ map d : L2 (Rn ; k ) → L2 (Rn ; k+1 ) and d ∗ : L2 (Rn ; k ) → L2 (Rn ; k−1 ) as unbounded operators. Moreover, if we further decompose the space of homogeneous k-vectors into its normal and tangential subspaces as
where
k
⊥
=
:= N −
n n n+1 0 1 1 2 2 ⊕ ⊥⊕ ⊕ ⊥ ⊕ ⊕ ··· ⊕ ⊥⊕ ⊕ ⊥ , k
and
k
:= N +
k
, then we see that the operators d and d ∗ map
d : L2 Rn ; k⊥ → L2 Rn ; k , d ∗ : L2 Rn ; k → L2 Rn ; k⊥ ,
d : L2 Rn ; k → L2 Rn ; k+2 , ⊥ d ∗ : L2 Rn ; k⊥ → L2 Rn ; k−2 .
∗ )2 = d 2 = (d ∗ )2 = μ2 = Lemma 2.7. We have, on appropriate domains, (dt,x )2 = (dt,x ∗ 2 (μ ) = 0 and the anti-commutation relations
∗ {m, dt,x } = ∂t = m, −dt,x , {d, d ∗ } = − = −
n
∂k2 ,
m2 = {μ, μ∗ } = I, {d, μ} = {d, μ∗ } = 0,
1
where {A, B} = AB + BA denotes the anticommutator. ∗ }. For Proof. The proofs are straightforward, using Lemma 2.3. Let us prove that ∂t = {m, −dt,x a function F (t, x) we have
∇x,t (mF ) =
n
ei (m∂i F ) =
0
n
ei (e0 ∂i F ) +
n
0
ei (e0 ∧ ∂i F ).
0
Using the anticommutativity and associative properties, the first sum is n n (e0 ∧ ei ) ∂i F = − (ei ∧ e0 ) ∂i F = −e0 0
0
n 0
ei ∂i F ,
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whereas using the derivation property, the second sum is n
ei (e0 ∧ ∂i F ) = ∂0 F − e0 ∧
0
n
ei ∂i F .
0
Adding up we obtain ∇x,t (mF ) = ∂t F − m(∇x,t F ).
2
2.1. The basic operators Definition 2.8. Throughout this paper we denote by B ∈ L∞ (Rn ; L( )) a bounded, accretive and complex matrix function acting on f ∈ H as f (x) → B(x)f (x), with quantitative bounds B∞ and κB > 0, where κB is the largest constant such that Re B(x)w, w κB |w|2 ,
for all w ∈
, x ∈ Rn .
0 1 2 n+1 , where B k ∈ We shall also k assume that B is of the form B = B ⊕ B ⊕ B ⊕ · · · ⊕ B n L∞ (R ; L( )), so that B preserve the space of k-vectors. For the matrix part B 1 acting on := mBm. It is not true in vectors, we use the alternative notation A = B 1 . We also define B general that B preserve k-vectors.
Remark 2.9. It would be more optimal to replace the quantitative bounds B∞ and κB > 0 with KB and kB , where KB and kB are the optimal constants such that kB f 2 (Bf, f ) KB f 2 ,
for all f ∈ H.
We consider F (t, x) satisfying the Dirac type equation
∗ B F (t, x) = 0 mdt,x + B −1 mdt,x
(2.1)
∗ −1 dt,x B F (t, x) = 0. dt,x + B
(2.2)
or equivalently
In order to solve for the vertical derivative ∂t F , we note that ∗ dt,x F = d ∗ F − μ∗ ∂t F.
dt,x F = dF + μ∂t F, Inserted in (2.1) this yields ∂t F +
1 N + −B −1 N − B
md + B −1 md ∗ B F = 0.
Definition 2.10. Write MB := N + − B −1 N − B and define the unbounded operator TB := MB−1 md + B −1 md ∗ B = −iMB−1 d + B −1 d ∗ B = in H with domain D(TB ) := D(d) ∩ B −1 D(d ∗ ).
1 −1 μ∗ B μ−B
−1 d ∗ B d +B
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namely B = B is not the best, A rather surprising fact is that the most obvious choice for B, = mBm. For example, this is the only choice for which a Rellich type formula, as but rather B in Proposition 3.8, holds on all H, in the case of Hermitean coefficients B. Note that TB is closely related to operators of the form ΠB = Γ + B −1 Γ ∗ B, where Γ denotes a first order, homogeneous partial differential operator with constant coefficients such that Γ 2 = 0, which were studied in [10]. Unfortunately, the factor MB−1 does not commute with ΠB for general B. However, it has other useful commutation properties. Lemma 2.11. The operator MB is an isomorphism and B −1 N − B MB−1 = MB−1 N − , −1 + −1 B N B MB = MB−1 N + ,
N − MB−1 = MB−1 B −1 N − B , N + MB−1 = MB−1 B −1 N + B .
Proof. Note that MB = B −1 (BN + − N − B), where the last factor is the diagonal matrix (−B⊥⊥ ) ⊕ B in the splitting H = N − H ⊕ N + H, if B⊥⊥ and B denote the diagonal blocks of B, and thus BN + − N − B commutes with N ± . Furthermore, the diagonal blocks are accretive, so BN + − N − B and thus MB is invertible. This proves the two equations to the right. To obtain the equations to the left, replace B by B −1 and note that −N − + B −1 N + B = N + − B −1 N − B.
2
Let us comment on the terminology “Dirac type equation” for (2.1). Normally this denotes a ∗ , whose square acts componentwise as the Laplace first-order differential operator, like dt,x + dt,x operator. In our situation, the following holds. ∗ Bd (mF ) = 0. In particular it follows that Lemma 2.12. If F (t, x) satisfies (2.1), then dt,x t,x
divt,x A∇t,x F 1,0 = 0 if F = F 0 + (F 1,0 e0 + F 1, ) + F 2 + · · · + F n+1 , i.e. F 1,0 is the normal component of the vector part of F . ∗ = −∂ − d ∗ m from Proof. We use the anticommutation relations mdt,x = ∂t − dt,x m and mdt,x t t,x Lemma 2.7, which shows that (2.1) is equivalent with
∗ dt,x m + B −1 dt,x mB F (t, x) = 0, ∗ B to this equation shows that d ∗ Bd (mF ) = 0, and since ∂t − B −1 ∂t B = 0. Applying dt,x t,x t,x 0 ∗ Bd evaluating the scalar part shows that divt,x A∇t,x F 1,0 = 0 since dt,x t,x preserves kvectors. 2
The following notions are central in our operator theoretic framework. Definition 2.13. Let H be a Hilbert space.
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(i) A (topological) splitting of H is a decomposition H = H1 ⊕ H2 into closed subspaces H1 and H2 . In particular, we have f1 + f2 ≈ f1 + f2 if fi ∈ Hi . (ii) Two bounded operators R + and R − in H are called complementary projections if R + + R − = I , (R ± )2 = R ± and R ± R ∓ = 0. (iii) A bounded operator R in H is called a reflection operator if R 2 = I . We note the following connection between these concepts. Lemma 2.14. There is a one-to-one correspondence R = R + − R − ←→ R ± = 12 (I ± R) ←→ H = R + H ⊕ R − H between reflection operators in H, complementary projections in H and topological splittings of H. We write R ± H = R(R ± ) for the range of the projection R ± . In Definition 2.4 we introduced the complementary projections N ± associated with the splitting of H into the subspaces of tangential and normal multivector fields. The corresponding reflection operator is N := N + − N − = μ∗ μ − μμ∗ = (μ∗ − μ)m = m(μ − μ∗ ). We also introduce B-perturbed versions of the tangential and normal subspaces N − H and N + H as B −1 N + H := B −1 f ; f ∈ N + H , B −1 N − H := B −1 f ; f ∈ N − H . In Definition 2.10 we encountered one of the complementary projections B −1 N + B and B −1 N − B associated with the splitting H = B −1 N + H ⊕ B −1 N − H. However, more important will be the following complementary projections. Definition 2.15. Let Nˆ B+ and Nˆ B− be the complementary projections associated with the splitting H = B −1 N + H ⊕ N − H. We sometimes use the shorter notation NB+ := Nˆ B+ and NB− := Nˆ B− . Also let Nˇ B+ and Nˇ B− be the complementary projections associated with the splitting H = N + H ⊕ B −1 N − H. Let NB = Nˆ B := Nˆ B+ − Nˆ B− and Nˇ B := Nˇ B+ − Nˇ B− be the associated reflection operators.
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With the notation μ∗B := B −1 μ∗ B, these operators are −1 −1 Nˆ B+ = μ∗B μ + μ∗B = μ + μ∗B μ, −1 −1 ∗ = μ + μ∗B μB , Nˆ B− = μ μ + μ∗B −1 −1 μ − μ∗B , = μ + μ∗B Nˆ B = μ∗B − μ μ + μ∗B and similarly for Nˇ B± and Nˇ B . We shall prove in Section 2.2 that all these operators are bounded. It is mainly the operators Nˆ B+ , Nˆ B− and Nˆ B that we shall use. Definition 2.16. Let f, g B := ((BN + − N − B)f, g), for f, g ∈ H. As BN + − N − B = BMB is invertible, ·,· B is a duality, i.e. there exists C < ∞ such that f, g B Cf g, f C sup f, g B /g, g C sup f, g B /f . g =0
f =0
We write T for the adjoint of an operator T with respect to this duality, i.e. if Tf, g B = f, T g B for all f, g ∈ H. Proposition 2.17. We have adjoint operators (Nˆ B ) = Nˇ B ∗ , (Nˇ B ) = Nˆ B ∗ , N = N and (TB ) = −TB ∗ . Proof. To prove that (Nˆ B ) = Nˇ B ∗ , recall that Nˆ B is the reflection operator for the splitting H = B −1 N + H ⊕ N − H and Nˇ B ∗ is the reflection operator for the splitting H = N + H ⊕ (B ∗ )−1 N − H. Thus we need to prove that B −1 N + f1 − N − f2 , N + g1 + (B ∗ )−1 N − g2 B = B −1 N + f1 + N − f2 , N + g1 − (B ∗ )−1 N − g2 B ,
for all fi , gi , i.e. that B −1 N + f1 , (B ∗ )−1 N − g2 B = 0 = N − f2 , N + g1 B . Use Lemma 2.11 to obtain
B −1 N + f1 , (B ∗ )−1 N − g2
B
BN + − N − B B −1 N + f1 , (B ∗ )−1 N − g2 = N − MB B −1 N + f1 , g2 = MB B −1 N − N + f1 , g2 = 0.
=
A similar calculation shows that N − f2 , N + g1 B = 0. The proof of (Nˇ B ) = Nˆ B ∗ is similar. For the unperturbed reflection operator N , the duality N = N follows directly from the fact the BN + − N − B is diagonal in the splitting H = N + H ⊕ N − H. To prove (TB ) = −TB ∗ , we calculate BN + − N − B MB−1 md + B −1 md ∗ B f, g = (Bmd + md ∗ B)f, g = f, (B ∗ dm + d ∗ mB ∗ )g
TB f, g B =
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−1 = − f, N + B ∗ − B ∗ N − N + B ∗ − B ∗ N − (B ∗ md + md ∗ B ∗ )g −1 = − f, N + − (B ∗ )−1 N − B ∗ md + (B ∗ )−1 md ∗ B ∗ g B = f, −TB ∗ g B , where we have used that (B ∗ )−1 N + B ∗ − N − = N + − (B ∗ )−1 N − B ∗ .
2
Remark 2.18. Our main use of these dualities is for proving surjectivity. Recall the following standard technique for proving invertibility of a bounded operator T : H1 → H2 . Assume we have at our disposal two pairs of dual spaces H1 , K1 1 and H2 , K2 2 and that the adjoint operator is T : K2 → K1 . If we can prove a priori estimates Tf f ,
for all f ∈ H1 ,
then T is injective and has closed range. If furthermore T is injective, in particular if it satisfies a priori estimates, then T is surjective and therefore an isomorphism. Remark 2.19. We shall also need to restrict dualities to subspaces. Let H, K be a duality, i.e. let ·,· : H × K → C satisfy the estimates in Definition 2.16. If H1 ⊂ H is a subspace, then a subspace K1 ⊂ K is such that H1 , K1 satisfies estimates as in Definition 2.16 if and only if K1 is a complementary subspace to the annihilator {g ∈ K; f, g = 0, for all f ∈ H1 }, in the sense of Definition 2.13(i). In particular, if R ± are complementary projections in H, the adjoint operators (R ± ) are also complementary projections and the duality H, H restricts to a duality R + H, (R + ) H , since the annihilator of R + H is N((R + ) ), which is complementary to (R + ) H. 2.2. Hodge decompositions and resolvent estimates In this section, we estimate the spectrum of the operator TB . For this we make use of Hodge type decompositions of H as explained below. Definition 2.20. By a nilpotent operator Γ in a Hilbert space H, we mean a closed, densely defined operator such that R(Γ ) ⊂ N(Γ ). In particular Γ 2 f = 0 if f ∈ D(Γ ). We say that a nilpotent operator is exact if R(Γ ) = N(Γ ). If Γ is another nilpotent operator, then we say that Γ and Γ are transversal if there is a constant c = c(Γ, Γ ) < 1 such that (f, g) cf g,
f ∈ R(Γ ), g ∈ R(Γ ),
or equivalently if f + g ≈ f + g for all f ∈ R(Γ ) and g ∈ R(Γ ). Note that any nilpotent operator Γ is transversal to its adjoint Γ ∗ with c = 0, since R(Γ ) ⊂ N(Γ ) = R(Γ ∗ )⊥ . Below we collect the properties of Hodge type splittings which we need in this paper. This generalises results from [10, Proposition 2.2] and [9, Proposition 3.11].
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Lemma 2.21. Let Γ and Γ be two nilpotent operators in a Hilbert space H which are exact and transversal with constant c(Γ, Γ ), and assume also that the adjoints Γ ∗ and Γ ∗ are transversal. Let B be a bounded, accretive multiplication operator and assume that max c(Γ, Γ ), c(Γ ∗ , Γ ∗ ) < κB /B∞ . Define operators Π := Γ + Γ , Γ B := B −1 Γ B and ΠB := Γ + Γ B with domains D(Π) := ), D(Γ B ) := B −1 D(Γ ) and D(ΠB ) := D(Γ ) ∩ D(Γ B ), respectively. D(Γ ) ∩ D(Γ (i) We have a topological splitting H = N(Γ ) ⊕ N(Γ B ), so that f1 + f2 ≈ f1 + f2 , f1 ∈ N(Γ ), f2 ∈ N(Γ B ). The operator ΠB is a closed operator with dense domain and range. Furthermore Γ B is an exact nilpotent operator. The complementary Hodge type projections associated with the splitting are P1B := Γ ΠB−1 = ΠB−1 Γ B = Γ ΠB−2 Γ B , P2B := Γ B ΠB−1 = ΠB−1 Γ = Γ B ΠB−2 Γ, where we identify a bounded, densely defined operator with its bounded extension to H. In the special case when B = I we write P1 and P2 for these projections. (ii) Let B1 and B2 be two bounded, accretive operators in H, with κBi /Bi ∞ > max(c(Γ, Γ ), c(Γ ∗ , Γ ∗ )), i = 1, 2. Then there exists C < ∞, depending only on Bi ∞ , κBi , c(Γ, Γ ) and c(Γ ∗ , Γ ∗ ), such that
Γ + B −1 Γ B2 f C −1 λf , 1
for all f ∈ D(Γ ) ∩ B2−1 D(Γ ),
if Πf λf for all f ∈ D(Π). Proof. Since Γ B is conjugated to Γ , it is an exact nilpotent operator. To prove (i), it suffices to prove the estimate f + g f + g,
f ∈ N(Γ ), g ∈ N(Γ B ).
(2.3)
Indeed, this shows that N(Γ )⊕ N(Γ B ) ⊂ H. Furthermore, replacing (Γ, Γ , B) with (Γ ∗ , Γ ∗ , B ∗ ) shows that N(Γ ∗ ) ⊕ N(ΓB∗∗ ) ⊂ H. Conjugating with B ∗ then shows that N((Γ B )∗ ) ⊕ N(Γ ∗ ) ⊂ H. Therefore a duality argument proves the splitting H = N(Γ ) ⊕ N(Γ B ). From this it follows that
B ) ∩ N(Γ ) ⊕ D(Γ ) ∩ N(Γ B ) D(ΠB ) = D(Γ
and R(ΠB ) = R(Γ ) ⊕ R(Γ B ) are dense and that ΠB is closed, as well as the boundedness of the associated projections. To prove (2.3), we use that Bg ∈ N(Γ ) and thus |(f, Bg)| c(Γ, Γ )f Bg, and estimate:
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f 2 κB−1 Re(Bf, f ) κB−1 Re B(f + g), f − (Bg, f ) κB−1 B∞ f + gf + κB−1 c(Γ, Γ )Bgf κB−1 B∞ f + gf + κB−1 c(Γ, Γ )B∞ f + g + f f κB−1 B∞ 1 + c(Γ, Γ ) f + gf + κB−1 B∞ c(Γ, Γ )f 2 . Solving for f 2 , this shows that f f + g provided κB−1 B∞ c(Γ, Γ ) < 1, which proves (2.3). To prove (ii) we factorise Γ + B1−1 Γ B2 = P1 + B1−1 P2 Π P2 + P1 B2 . Here (P1 + B1−1 P2 )−1 = P1B1 + B1 P2B1 and (P2 + P1 B2 )−1 = P1B2 B2−1 + P2B2 . Indeed, we have 1 P + B1−1 P2 P1B1 + B1 P2B1 = P1 P1B1 + P1 B1 P2B1 + B1−1 P2 P1B1 + B1−1 P2 B1 P2B1 = P1 P1B1 + B1−1 P2 B1 P2B1 = P1 + P2 P1B1 + B1−1 P1 + P2 B1 P2B1 = P1B1 + P2B1 = I, 2 P + P1 B2 P1B2 B2−1 + P2B2 = P2 P1B2 B2−1 + P2 P2B2 + P1 B2 P1B2 B2−1 + P1 B2 P2B2 = P2 P2B2 + P1 B2 P1B2 B2−1 = P2 P1B2 + P2B2 + P1 B2 P1B2 + P2B2 B2−1 = P2 + P1 = I. A similar calculation shows that P1B1 + B1 P2B1 and P1B2 B2−1 + P2B2 also are left inverses. Thus P1 + B1−1 P2 and P2 + P1 B2 are isomorphisms and (ii) follows. 2 We can now prove the following perturbed normal and tangential splittings of H. H = Nˆ B+ H ⊕ Nˆ B− H = B −1 N + H ⊕ N − H, H = Nˇ B+ H ⊕ Nˇ B− H = N + H ⊕ B −1 N − H. To see this, let first Γ = μ and Γ = Γ ∗ = μ∗ in Lemma 2.21(i). It follows that N(Γ ) = N − H, N(ΓB∗ ) = B −1 N + H, P1B = Nˆ B− and P2B = Nˆ B+ . On the other hand, choosing Γ = μ∗ and Γ = Γ ∗ = μ, we see that N(Γ ) = N + H, N(ΓB∗ ) = B −1 N − H, P1B = Nˇ B+ and P2B = Nˇ B− . Lemma 2.21(i) thus shows that all the oblique normal and tangential projections Nˆ B± and Nˇ B± from Definition 2.15 are bounded operators on H, i.e. we have the stated splittings. Next we apply Lemma 2.21(ii) to prove resolvent bounds for the operator TB from Definition 2.10. Define closed and open sectors and double sectors in the complex plane by Sω+ := z ∈ C; |arg z| ω ∪ {0}, o Sν+ := z ∈ C; z = 0, |arg z| < ν ,
Sω := Sω+ ∪ (−Sω+ ), o o . Sνo := Sν+ ∪ −Sν+
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Proposition 2.22. The operator TB is a closed operator in H with dense domain and range, for any accretive, complex matrix function B ∈ L∞ (Rn ; L( )). Furthermore, TB is a bisectorial operator with σ (TB ) ⊂ Sω , where ω := arccos κB / 2B∞ ∈ [π/3, π/2) and if ω < ν < π/2, then there exists C < ∞ depending only on ν, κB and B∞ such that
(λ − TB )−1 C/|λ|, for all λ ∈ / Sν . Proof. It is a consequence of Lemma 2.21(i) that the operator ΠB = d + B −1 d ∗ B is a closed operator with dense domain and range. Since TB = −iMB−1 ΠB with MB an isomorphism, it follows that TB also is closed with dense domain and range. o . We first prove that u To prove the resolvent estimate, write λ = 1/(iτ ) where τ ∈ Sπ/2−ν o −1 μ∗ B) Cf if (I − iτ TB )u = f , uniformly for τ ∈ Sπ/2−ν . Multiply the equation with i(μ − B to obtain −1 μ∗ B f, −1 Γ B u = i μ − B Γ +B where Γ := iμ + τ d and Γ := −iμ∗ + τ d ∗ are nilpotent by Lemma 2.7. It suffices to prove −1 Γ B)u. u (Γ + B (i) By orthogonality we have
(Γ + Γ ∗ )u 2 = Γ u2 + Γ ∗ u2 = μu2 + |τ |2 du2 + 2 Re(iμu, τ du) + μ∗ u2 + |τ |2 d ∗ u2 + 2 Re(−iμ∗ u, τ d ∗ u), where Re(iμu, τ du) + Re(−iμ∗ u, τ d ∗ u) = Re(iτ d ∗ μu, u) + Re(u, iτ μd ∗ u) = Re iτ {d ∗ , μ}u, u = 0, by Lemma 2.7. Thus (Γ + Γ ∗ )u2 = (μ + μ∗ )u2 + |τ |2 (d + d ∗ )u2 mu2 = u2 . In particular Γ is exact. (ii) Next we prove that Γ and Γ are transversal, with a bound c < 1 uniformly for all τ ∈ o Sπ/2−ν . By exactness, it suffices to bound (f, g) for f = Γ u ∈ R(Γ ) and Γ g = 0. Furthermore, using the orthogonal Hodge splitting H = N(Γ ) ⊕ N(Γ ∗ ), we may assume that Γ ∗ u = 0. We get (f, g) = (Γ u, g) = (u, Γ ∗ g) = u, (−iμ∗ + τ d ∗ )g = u, −iμ∗ g + i(τ /τ )μ∗ g = 2(τ − τ )/(2iτ )(u, μ∗ g), and thus |(f, g)| 2|sin(arg τ )|ug cf g by (i), where c < κB /B∞ since π/2−ν < arcsin(κB /(2B∞ )). A similar argument shows that Γ ∗ and Γ ∗ are transversal with the same o . constant c < κB /B∞ uniformly for τ ∈ Sπ/2−ν (iii) To apply Lemma 2.21(ii) it now suffices to prove that (Γ + Γ )u C −1 u uniformly o . From Lemma 2.21(i) with B = I we have (Γ + Γ )u ≈ Γ u + Γ u. for all τ ∈ Sπ/2−ν
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Using the Hodge splitting H = N(Γ ) ⊕ N(Γ ) it suffices to prove Γ u u for u ∈ N(Γ ), and Γ v v for v ∈ N(Γ ). To prove for example the first estimate, write N(Γ ) u = u1 + u2 ∈ N(Γ ) ⊕ N(Γ ∗ ). Then Γ u2 = Γ u2 2 u2 2 = u − u1 2
u2 + u1 2 − 2cuu1 1 − c2 u2 ,
where we have used (i) in the second step and (ii) in the fourth step. This proves that u Cf if (I − iτ TB )u = f . Since (I − iτ TB ) = I − iτ TB ∗ by Proposition 2.17, a duality argument shows that I − iτ TB is onto, and the proof is complete. 2 From the uniform boundedness of the resolvents RtB := (I + itTB )−1 for t ∈ R and the boundedness of the Hodge projections P1B := dΠB−1 and P2B := d ∗B ΠB−1 , where ΠB = d + d ∗B , d = imd B + RB ) and d ∗B = B −1 d ∗ B, we can now deduce boundedness of operators related to PtB = 12 (R−t t 1 B B and QB t = 2i (R−t − Rt ). Corollary 2.23. The following families of operators are all uniformly bounded for t > 0. RtB := (I + itTB )−1 , −1 PtB := I + t 2 TB2 , 2 2 −1 , QB t := tTB I + t TB B t 2 TB2 PtB = tTB QB t = I − Pt , 1 B tdQB t = iPB MB I − Pt , 2 B td ∗B QB t = iPB MB I − Pt , −1 B 2 QB t MB td = i I − Pt PB , −1 ∗ B 1 QB t MB td B = i I − Pt PB ,
tdPtB = iP1B MB QB t , td ∗B PtB = iP2B MB QB t , 2 PtB MB−1 td = iQB t PB , 1 PtB MB−1 td ∗B = iQB t PB , tdPtB MB−1 td = P1B MB PtB − I P2B , tdPtB MB−1 td ∗B = P1B MB PtB − I P1B , td ∗B PtB MB−1 td = P2B MB PtB − I P2B , td ∗B PtB MB−1 td ∗B = P2B MB PtB − I P1B .
These families of operators are not only bounded, but have L2 off-diagonal bounds in the following sense. Definition 2.24. Let (Ut )t>0 be a family of operators on H, and let M 0. We say that (Ut )t>0 has L2 off-diagonal bounds (with exponent M) if there exists CM < ∞ such that −M Ut f L2 (E) CM dist(E, F )/t f whenever E, F ⊂ Rn and supp f ⊂ F . Here x := 1 + |x|, and dist(E, F ) := inf{|x − y|: x ∈ E, y ∈ F }. We write Ut off,M for the smallest constant CM . The exact value of M is normally not important and we write Ut off , where it is understood that M is chosen sufficiently large but fixed. Proposition 2.25. All the operator families from Corollary 2.23 has L2 off-diagonal bounds for all exponents M 0.
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Proof. First consider the resolvents RtB = (I + itTB )−1 . As we already have proved uniform bounds for RtB , it suffices to prove
(I + itTB )−1 f L
2 (E)
M CM |t|/dist(E, F ) f
for |t| dist(E, F ). We prove this by induction on M as in [10, Proposition 5.2]. Let η : Rn → := {x ∈ Rn ; dist(x, E) dist(x, F )} [0, 1] be a bump function such that η|E = 1, supp η ⊂ E and ∇η∞ 1/ dist(E, F ) ≈ 1/ dist(E, F ). Since the commutator is ηI, RtB = tRtB MB−1 [ηI, d] + B −1 [ηI, d ∗ ]B RtB , where [ηI, d], [ηI, d ∗ ] ∇η∞ , we get
B
R f ηR B f = ηI, R B f |t|∇η∞ R B f t
L2 (E)
t
t
t
L2 (E)
,
where we used that ηf = 0. By induction, this proves the off-diagonal bounds for RtB . From this, B off-diagonal bounds for PtB , QB t and I − Pt also follows immediately. B Next we consider tdPt and use Lemma 2.21(i) to obtain
tdP B f t
L2 (E)
ηtdPtB f [ηI, td]PtB f + tdηPtB f
|t|∇η∞ PtB f L (E) + tTB ηPtB f 2
|t|∇η∞ PtB f L (E) + [η, tTB ]PtB f + ηtTB PtB f 2
B
. |t|∇η∞ PtB f L (E) + Qt f L (E) 2
2
(2.4)
This and the corresponding calculation with d replaced by d ∗B proves the off-diagonal bounds for tdPtB and td ∗B PtB . From this the result for PtB MB−1 td and PtB MB−1 td ∗B follows immediately with a duality argument. Indeed, (MB−1 d) = MB−1∗ d ∗B ∗ and (MB−1 d ∗B ) = MB−1∗ d is proved similarly to TB = −TB ∗ in Proposition 2.17. ∗ B B −1 B −1 ∗ The proof for tdQB t , td B Qt , Qt MB td and Qt MB td B is similar, replacing Pt and Qt with Qt and I − Pt . Finally, the last four estimates follows from a computation like (2.4), −1 for example replacing Pt and Qt with PtB MB−1 td and QB t MB td proves the estimate for tdPtB MB−1 td. 2 We finish this section with a lemma to be used in Section 4. This lemma is proved with an argument similar to that in [15, Lemma 2.3]. For completeness, we include a short proof. Lemma 2.26. Assume that (Ut )t>0 and (Vt )t>0 both have L2 off-diagonal bounds with exponent M. Then (Ut Vt )t>0 has L2 off-diagonal bounds with exponent M and Ut Vt off,M 2M+1 Ut off,M Vt off,M . Proof. By Definition 2.24 we need to prove that −M Ut Vt f L2 (E) dist(E, F )/t f
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whenever supp f ⊂ F . To this end let G := {x ∈ Rn ; dist(x, E) ρ/2}, where ρ := dist(E, F ). We get
Ut Vt f L2 (E) Ut (χG Vt f ) L
2 (E)
+ Ut (χRn \G Vt f ) L
2 (E)
Ut C0Ut Vt f L2 (G) + CM ρ/2t −M Vt f L2 (Rn \G) Vt Ut Ut Vt C0Ut CM ρ/2t −M f + CM ρ/2t −M C0Vt f 2M+1 CM CM ρ/t −M f .
2
2.3. Quadratic estimates: generalities In Proposition 2.22 we proved the spectral estimate σ (TB ) ⊂ Sω for some angle ω < π/2 with bounds on the resolvent outside Sω . In this section we survey some general facts about the functional calculus of the operator TB . For a further background and discussion of these matters we refer to [1,10]. Definition 2.27. For ω < ν < π/2, we define the following classes of holomorphic functions f ∈ H (Sνo ) on the open double sector Sνo : Ψ Sνo := ψ ∈ H Sνo ; ψ(z) C min |z|s , |z|−s , z ∈ Sνo , for some s > 0, C < ∞ , H∞ Sνo := b ∈ H Sνo ; b(z) C, z ∈ Sνo , for some C < ∞ , F Sνo := w ∈ H Sνo ; w(z) C max |z|s , |z|−s , z ∈ Sνo , for some s < ∞, C < ∞ . Thus Ψ (Sνo ) ⊂ H∞ (Sνo ) ⊂ F (Sνo ) ⊂ H (Sνo ). For ψ ∈ Ψ (Sνo ), we define a bounded operator ψ(TB ) through the Dunford functional calculus ψ(TB ) :=
1 2πi
ψ(λ)(λI − TB )−1 dλ,
(2.5)
γ
where γ is the unbounded contour {±re±iθ ; r > 0}, ω < θ < ν, parametrised counterclockwise around Sω . The decay estimate on ψ and the resolvent bounds of Proposition 2.22 guarantee that ψ(TB ) < ∞. For general w ∈ F (Sνo ) we define −k k w(TB ) := QB q w (TB ), where k is an integer larger than s if |w(z)| C max(|z|s , |z|−s ), and q(z) := z(1 + z2 )−1 and QB := q(TB ). This yields a closed, densely defined operator w(TB ) in H. Furthermore, we have λ1 w1 (TB ) + λ2 w2 (TB ) = (λ1 w1 + λ2 w2 )(TB ), w1 (TB )w2 (TB ) = (w1 w2 )(TB ),
(2.6)
for all w1 and w2 ∈ F (Sνo ). Here T = S means that the graph G(T ) is dense in the graph G(S).
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The functional calculus w → w(TB ) have the following convergence properties as proved in [1]. Lemma 2.28. If bk ∈ H∞ (Sνo ) is a sequence uniformly bounded on Sνo which converges to b uniformly on compact subsets, and if bk (TB ) are uniformly bounded operators, then bk (TB )f → b(TB )f,
for all f ∈ H,
and b(TB ) lim supk bk (TB ). Definition 2.29. The following operators in the functional calculus are of special importance to us. (1) qt (z) = q(tz) := tz(1 + t 2 z2 )−1 ∈ Ψ (Sνo ), which give the operator QB t . (2) |z|s := (z2 )s/2 ∈ F (Sνo ), which give the operator |TB |s . Note that |z| does not denote absolute value here, but z → |z| is holomorphic on Sνo . (3) e−t|z| ∈ H∞ (Sνo ), which give the operator e−t|TB | . (4) The characteristic functions χ ± (z) =
1 if ±Re z > 0, 0 if ±Re z < 0
which give the generalised Hardy projections EB± := χ ± (TB ). (5) The signum function sgn(z) = χ + (z) − χ − (z) which give the generalised Cauchy integral EB := sgn(TB ). The main work in this paper is to prove the boundedness the projections EB± . As in Lemma 2.14, if these are bounded then they correspond to a splitting H = EB+ H ⊕ EB− H of H into the Hardy subspaces EB± H associated with Eq. (2.1). That the projections are bounded is also equivalent with having a bounded reflection operator EB . Definition 2.30. For a function F (t, x) defined in Rn+1 ± we write ∞ 1/2
2 dt
F (±t, x) |||F |||± := t 0
and for short |||F |||+ =: |||F |||. When F (t, x) = (Θt f )(x) for some family of operators (Θt )t>0 , we use the notation |||Θt |||op := sup |||Θt f |||. f =1
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Our main goal in this paper will be to prove quadratic estimates of the form ∞
B 2 dt
Q f ≈ f 2 , t t
(2.7)
0
for certain coefficients B. We recall the following two basic results concerning quadratic estimates which are proved by Schur estimates. For details we refer to [1]. o and S o , and define ψ (z) := Proposition 2.31. Let ψ ∈ Ψ (Sνo ) be non-vanishing on both Sν+ t ν− ψ(tz). Then there exists 0 < C < ∞ such that C −1 QB f ψt (TB )f C QB f . t
t
Proposition 2.32. If TB satisfies quadratic estimates (2.7), then TB has bounded H∞ (Sνo ) functional calculus, i.e.
b(TB ) b∞ , for all b ∈ H∞ S o . ν Thus H∞ (Sνo ) b → b(TB ) ∈ L(H) is a continuous homomorphism. Before proving quadratic estimates for TB for certain B in Sections 3 and 4, we introduce a dense subspace on which the operator b(TB ) is defined for any b ∈ H∞ (Sνo ). Definition 2.33. Let VB be the dense linear subspace VB := D |TB |s ∩ R |TB |s ⊂ H. s>0
We see that D(|TB |s )∩ R(|TB |s ) increases when s decreases. The density of D(|TB |)∩ R(|TB |), and therefore of VB , follows from the fact that β 2
B 2 dt B = Pα − PβB f → f, Qt f t
α
as (α, β) → (0, ∞), for all f ∈ H. Moreover, if b ∈ H∞ (Sνo ) and f ∈ VB then b(TB )f ∈ VB ⊂ H. To see this, write b(TB )f = (bψ)(TB ) ψ(TB )−1 f , where ψ(z)−1 := (1 + |z|s )/|z|s/2 if f ∈ D(|TB |s ) ∩ R(|TB |s ). Then ψ(TB )−1 f ∈ H and (bψ)(TB ) is bounded since bψ ∈ Ψ (Sνo ). Furthermore, if s < s/2 then |TB |s (ψ(TB )−1 f ) ∈ H and |TB |−s (ψ(TB )−1 f ) ∈ H, so b(TB )f ∈ D(|TB |s ) ∩ R(|TB |s ). Lemma 2.34. We have an algebraic splitting VB = EB+ VB + EB− VB ,
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EB+ VB ∩ EB− VB = {0}, and f ∈ EB± VB is in one-to-one correspondence with F (t, x) = Ft (x) := (e∓t|TB | f )(x) in Rn+1 ± , and lim Ft = f,
t→0±
Ft ∈ D(TB ) = D(d) ∩ D(d ∗ B),
lim Ft = 0,
t→±∞
∂t Ft = −TB Ft ∈ L2 Rn ,
±t > 0.
Proof. That each f ∈ VB can be uniquely written f = f + + f − , where f ± ∈ EB± VB , follows from (2.6). To verify the properties of Ft , it suffices to consider the case f ∈ EB+ VB as the case f ∈ EB− VB is similar. Since ze−t|z| ∈ Ψ (Sν0 ) it follows that Ft ∈ D(TB ). Moreover, since 1 −(t+h)|z| − e−t|z| ) → −|z|e−t|z| uniformly on Sν0 , it follows from Lemma 2.28 that ∂t Ft = h (e −|TB |Ft and since Ft ∈ EB+ VB we have |TB |Ft = TB Ft . To prove the limits, assume that f ∈ D(|TB |s ) ∩ R(|TB |s ) for some 0 < s < 1. Writing f = |TB |−s u, we see that Ft − f = t s ψ(tTB )u,
where ψ(z) = e−|z| − 1 /|z|s .
Similarly with f = |TB |s v, we get Ft = t −s ψ(tTB )v,
where ψ(z) = |z|s e−|z| .
Since in both cases ψ(tTB ) are uniformly bounded in t, using a direct norm estimate in (2.5), it follows that both limits are 0 as t → 0 and t → ∞, respectively. In particular, f = limt→0 Ft is uniquely determined by F . 2 We now further discuss the quadratic estimates (2.7). First note the following consequence of the duality TB = −TB ∗ from Proposition 2.17. Again we refer to [1] for further details. ∗
B Lemma 2.35. If |||QB t f ||| f for all f ∈ H, then |||Qt f ||| f for all f ∈ H.
In Section 3.3 we shall use the following Hardy space reduction of the quadratic estimate. This is a technique due to Coifman, Jones and Semmes [11], and adapted to the setting of functional calculus by Mc Intosh and Qian [23, Theorem 5.2]. Proposition 2.36. Assume that we have reverse quadratic estimates in EB+∗ VB ∗ and EB−∗ VB ∗ , i.e. g |||t∂t Gt |||± ,
g ∈ EB±∗ VB ∗ ,
where Gt = e∓t|TB ∗ | g. Then B Q f f , t
f ∈ H.
In particular, if we have reverse quadratic estimates in both Hardy spaces for both operators TB B∗ o and TB ∗ , then |||QB t f ||| ≈ f ≈ |||Qt f |||, f ∈ H, and thus TB and TB ∗ have bounded H∞ (Sν ) functional calculus for all ω < ν < π/2.
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Proof. Assume first that g ∈ EB+∗ VB ∗ ,
g |||t∂t Gt |||,
and write ψt (z) := tze−t|z| so that t∂t Gt = −ψt (TB ∗ )g by Lemma 2.34. Let f ∈ H and define qt∓ (z) := χ ∓ qt (z) so that qt∓ (TB ) = EB∓ QB t . To prove that − q (TB )f f , t it suffices to bound auxiliary functions
β α
f ∈ H,
qt− (TB )f 2 dtt uniformly for α > 0 and β < ∞. To this end, define the
β ht :=
−
q (TB )f 2 dt t t
−1/2
+ ∗ −1 N B − B ∗ N − qt− (TB )f,
α
β g := −
qt+ (TB ∗ )ht
dt , t
α
so that
β α
ht 2 dtt C and g ∈ EB+∗ VB ∗ , and calculate
β α
−
q (TB )f 2 dt t t
1/2
β
dt = f, g B B t α f g f ψs (TB ∗ )g
=
qt− (TB )f, ht
∞ β f 0
ψs q + (TB ∗ ) ht dt t t
ds s
1/2
α
β f
2
ht
2 dt
t
1/2 f .
α
In the second equality we have used that qt− (TB ) = qt− (−TB ∗ ) = −qt+ (TB ∗ ), in the second estimate we used the hypothesis and the second last estimate is a Schur estimate. We here use that (ψs qt+ )(TB ∗ ) η(t/s), where η(x) := min(x s , x −s ) for some s > 0. With a similar argument |||qt+ (TB )f ||| f , f ∈ H, follows from the reverse quadratic estimate for g ∈ EB−∗ VB ∗ . If both reverse estimates holds for B ∗ , then B − Q f q (TB )f + q + (TB )f f , t t t
f ∈ H,
and if the same holds for B and B ∗ interchanged, then Lemma 2.35 proves that |||QB t f ||| ≈ f , f ∈ H, and by Proposition 2.32 this proves that TB has bounded H∞ (Sνo ) functional calculus, and similarly for TB ∗ . 2
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We end this section with a discussion of the holomorphic perturbation theory for the functional calculus of TB . Definition 2.37. Let z → Uz ∈ L(X , Y) be an operator-valued function defined on an open subset D ⊂ C of the complex plane. We say that Uz is holomorphic if for all z ∈ D there exists an operator Uz ∈ L(X , Y) such that
1
(Uz+w − Uz ) − U → 0, z
w X →Y
w → 0.
Lemma 2.38. Let z → Uz ∈ L(X , Y) be an operator-valued function defined on an open subset D ⊂ C of the complex plane. Then the following are equivalent. (i) z → Uz is holomorphic. and g ∈ Y ∗ , (ii) The scalar function h(z) = (Uz f, g) is a holomorphic function for all f ∈ X ∗ ∗ ⊂ X and Y ⊂ Y are dense, and Uz is locally bounded. where X In particular, if z → Uzk are holomorphic on D for k = 1, 2, . . . and k Uz f, g = hk (z) → h(z) = (Uz f, g),
, g ∈ Y ∗ , for all z ∈ D, f ∈ X
and supz∈K,k1 Uzk < ∞ for each compact subset K ⊂ D, then z → Uz is holomorphic on D. Proof. For the equivalence between (i) and (ii), see Kato [18, Theorem III 3.12]. To prove the convergence result, it suffices to show that h(z) is holomorphic on D. That this is true follows from an application of the dominated convergence theorem in the Cauchy integral formula for hk (z). 2 Below, we shall assume that z → Bz is a given holomorphic matrix-valued function defined on an open subset D ⊂ C such that Bz is a multiplication operator as in Definition 2.8 for each z ∈ D, and that ωD := supz∈D arccos(κBz /(2Bz ∞ )) < π/2. Let ωD < ν < π/2. 0 Lemma 2.39. For τ ∈ Sπ/2−ν , the operator-valued function D z → (I − iτ TBz )−1 is holomorphic.
Proof. Similar to the proof of Proposition 2.22 we have z Γ + Γ Bz )−1 i(B z μ − μ∗ Bz ), (I − iτ TBz )−1 = (B z μ − μ∗ Bz where Γ = iμ + τ d and Γ = −iμ + τ d ∗ . It is clear that the multiplication operator B z Γ + Γ Bz )−1 is holomorphic. depends holomorphically on z, so it suffices to show that z → (B Let z ∈ D, write B := Bz and Bw := Bz+w and calculate + Γ B)−1 w Γ + Γ Bw )−1 − (BΓ (B w Γ + Γ Bw )−1 (B w − B) Γ (BΓ + Γ B)−1 = −(B w Γ + Γ Bw )−1 Γ (Bw − B)(BΓ + Γ B)−1 . − (B
(2.8)
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The first and last factors in both terms on the right-hand side are uniformly bounded by w Γ + Γ Bw )−1 . Furthermore, multiplying Lemma 2.21(i) so we deduce continuity of w → (B Eq. (2.8) from the right with Γ , the first term on the right vanishes as + Γ B)−1 Γ 2 = 0, + Γ B)−1 Γ = (BΓ Γ (BΓ w Γ + Γ Bw )−1 Γ . Thus, dividing Eq. (2.8) by w and letting and we deduce continuity of w → (B w → 0 we see that the limit exists and equals + Γ B)−1 B + Γ B)−1 − (BΓ + Γ B)−1 Γ B (BΓ + Γ B)−1 . Γ (BΓ −(BΓ
2
Lemma 2.40. If ψ ∈ Ψ (Sνo ), then D z → ψ(TBz ) is holomorphic. Proof. Let γ be the unbounded contour {±re±iθ ; r > 0}, ωD < θ < ν, parametrised counterclockwise around SωD . By inspection of the proof of Lemma 2.39 we have
1 −1 −1 −1 (λ − T sup |λ| − ∂ ) − (λ − T ) (λ − T ) Bz+w Bz z Bz
→ 0,
w λ∈γ as w → 0. Thus 1 1 ψ(TBz+w ) − ψ(TBz ) → w 2πi
ψ(λ)∂z (λ − TBz )−1 dλ,
γ
since
γ
|ψ(λ)|| dλ λ | < ∞.
2 B
Lemma 2.41. Assume that TBz satisfy quadratic estimates |||Qt z f ||| ≈ f , f ∈ H, locally uniformly for z ∈ D. If b ∈ H∞ (Sνo ), ψ ∈ Ψ (Sνo ) and 0 < α < β < ∞, then the following operators depend holomorphically on z ∈ D. (i) u(x) → (b(TBz )u)(x) : H → H; (ii) u(x) → v(t, x) = (b(tTBz )u)(x) : H → L2 (Rn × (α, β); ); dtdx (iii) u(x) → v(t, x) = (ψt (TBz )u)(x) : H → L2 (Rn+1 + , t ; ). Proof. (i) Take a uniformly bounded sequence ψk (z) ∈ Ψ (Sνo ) which converges to b(z) ∈ H∞ (Sνo ) uniformly on compact subsets of Sνo . Lemma 2.38 then applies with Uzk = ψk (TBz ) and Uz = b(TBz ), using Lemmas 2.40 and 2.28. β (ii) It suffices by Lemma 2.38 to show that h(z) = α ht (z) dt is holomorphic, where ht (z) = (b(tTBz )f, Gt ), for all f (x) ∈ H and G(t, x) ∈ C0∞ (Rn × (α, β); ). That ht (z) is holomorphic for each t is clear from (i), and for h(z) this follows from an application of the Fubini theorem to the Cauchy integral formula for ht (z). (iii) Consider the truncations Uzk : u(x) → v(t, x) = χk (t)(ψt (TBz )u)(x), where χk denotes the characteristic function of the interval (1/k, k). It is clear from (ii) that Uzk is holomorphic, and letting k → ∞ we deduce from Lemma 2.38 that ψt (TBz ) is holomorphic. 2
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Proposition 2.42. Assume that TB satisfies quadratic estimates |||QB t f ||| ≈ f , f ∈ H, locally uniformly for all B such that B − B0 ∞ < ε, and let ψ ∈ Ψ (Sνo ). Then we have the Lipschitz estimates
b(TB ) − b(TB ) Cb∞ B2 − B1 ∞ , b ∈ H∞ S o , ν 2 1 ψt (TB ) − ψt (TB ) CB2 − B1 ∞ , 2 1 when Bi − B0 < ε/2, i = 1, 2. Proof. Let B(z) := B1 + z(B2 − B1 )/B2 − B1 ∞ , so that B(z) is holomorphic in a neighbourhood of the interval [0, B2 − B1 ∞ ]. In this neighbourhood, we have bounds b(TB(z) ) b∞ by Proposition 2.32 and holomorphic dependence on z by Lemma 2.41(i). Schwarz’ d lemma now applies and proves that dz b(TB(z) ) b∞ for all z ∈ [0, B2 − B1 ∞ ]. This shows that
b(TB ) − b(TB ) 2 1
B 2 −B1
d
b(TB(t) ) dt Cb∞ B2 − B1 ∞ .
dt
0
The proof of the Lipschitz estimate for ψt (TB ) similarly follows from Lemma 2.41(iii).
2
2.4. Decoupling of the Dirac equation In Section 2.1 we introduced the Dirac type equation ∗ mdt,x + B −1 mdt,x B F =0
(2.9)
satisfied by functions F (t, x) : Rn+1 . Of particular interest is when both terms vanish, i.e. ± → when dt,x F = 0, dF = −μ∂t F, or equivalently when (2.10) ∗ dt,x (BF ) = 0, d ∗ (BF ) = μ∗ B∂t F. as in Lemma 2.34, where f, TB f ∈ Consider a solution F (t, x) = e∓t|TB | f to (2.9) in Rn+1 ± EB± VB . Using ∂t F = −TB F and Lemma 2.11, we get (mdt,x F )|t=0 = m(d − μTB )f = md − N + MB−1 md + B −1 md ∗ B f = MB−1 MB − B −1 N + B md − B −1 N + md ∗ B f = MB−1 −N − md − B −1 N + md ∗ B f = MB−1 dμ + B −1 d ∗ μ∗ B f. ∗ (BF ) at t = 0 if and only if dμf = 0 = d ∗ μ∗ Bf . Thus we see that dt,x F = 0 = dt,x We may also rewrite Eq. (2.9) as
∗ dt,x m + B −1 dt,x mB F (t, x) = 0,
(2.11)
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∗ = −∂ − d ∗ m from Lemma 2.7. Consider the case when by using mdt,x = ∂t − dt,x m and mdt,x t t,x both terms vanish, i.e. when
dt,x (mF ) = 0, ∗ (mBF ) = 0, dt,x
or equivalently when
dF = μ∗ ∂t F, d ∗ (BF ) = −μB∂t F.
(2.12)
For F (t, x) = e∓t|TB | f solving (2.9) we also have dt,x (mF )|t=0 = ∂t F |t=0 − mdt,x F |t=0 = MB−1 dm + B −1 d ∗ mB f − MB−1 dμ + B −1 d ∗ μ∗ B f = MB−1 dμ∗ + B −1 d ∗ μB f. ∗ (mBF ) at t = 0 if and only if dμ∗ f = 0 = d ∗ μBf . Thus we see that dt,x (mF ) = 0 = dt,x
Definition 2.43. Introduce the closed, densely defined operators TˆB := −MB−1 dμ∗ + B −1 d ∗ μB ,
TˇB := −MB−1 dμ + B −1 d ∗ μ∗ B ,
with domains
D(TˆB ) := f ∈ H; μ∗ f ∈ D(d), μBf ∈ D(d ∗ ) , D(TˇB ) := f ∈ H; μf ∈ D(d), μ∗ Bf ∈ D(d ∗ ) ,
respectively, and define the closed subspaces Hˆ B := f ∈ H; d(μf ) = 0 = d ∗ (μ∗ Bf ) , Hˇ B := f ∈ H; d(μ∗ f ) = 0 = d ∗ (μBf ) . Proposition 2.44. We have a topological splitting of H into closed subspaces H = Hˆ B ⊕ Hˇ B . Furthermore we have TB = TˆB + TˇB with D(TB ) = D(TˆB ) ∩ D(TˇB ), and ˆ B = N(TˇB ), R(TˆB ) = H
ˇ B = N(TˆB ). R(TˇB ) = H
Thus, if we identify TˆB with its restriction to Hˆ B and TˇB with its restriction to Hˇ B , then these are bisectorial operators with spectral and resolvent estimates as in Proposition 2.22. Proof. Clearly D(TˆB ) ∩ D(TˇB ) = D(d) ∩ B −1 D(d ∗ ) = D(TB ) and TˆB + TˇB = TB . Furthermore Lemma 2.21(i) shows that N(TˇB ) = Hˆ B and N(TˆB ) = Hˇ B . (1) To show R(TˆB ) ⊂ Hˆ B , let f ∈ D(TˆB ) and use Lemma 2.11 to get
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dμ(TˆB f ) = −dμMB−1 dμ∗ + B −1 d ∗ μB f = −dmN + MB−1 dμ∗ + B −1 d ∗ μB f = −dmMB−1 B −1 N + B dμ∗ + B −1 d ∗ μB f = −dmMB−1 B −1 N + B − N − dμ∗ f = −dmdμ∗ f = md 2 μ∗ f = 0, and similarly d ∗ μ∗ B(TˆB f ) = −d ∗ mB B −1 N − B MB−1 dμ∗ + B −1 d ∗ μB f = −d ∗ mBMB−1 N − dμ∗ + B −1 d ∗ μB f = −d ∗ mBMB−1 N − − B −1 N + B B −1 d ∗ μBf = d ∗ md ∗ μBf = −m(d ∗ )2 μBf = 0. A similar calculation shows that R(TˇB ) ⊂ Hˇ B . (2) Next we show that for all f ∈ D(TB ), we have TB f ≈ TˆB f + TˇB f . Using that MB and B are isomorphisms, Lemma 2.21(i) with Γ = d and Γ = d ∗ and orthogonality μg2 + μ∗ g2 = g2 , we obtain TˆB f + TˇB f ≈ μ∗ df + μd ∗ Bf + μdf + μ∗ d ∗ Bf ≈ df + d ∗ Bf ≈ TB f . (3) Clearly Hˆ B ∩ Hˇ B = {0} and R(TB ) ⊂ R(TˆB ) + R(TˇB ). Taking closures, using that TB has dense range in H and using (2) yields H = R(TˆB ) ⊕ R(TˇB ). Thus from (1) it follows that R(TˆB ) = Hˆ B and R(TˇB ) = Hˇ B and that H = Hˆ B ⊕ Hˇ B .
2
It follows that TB is diagonal in the splitting H = Hˆ B ⊕ Hˇ B . We shall now further decompose the subspace Hˆ B into the subspaces of homogeneous k-vector field k , which also are preserved by TB . The same decomposition can be made for Hˇ B , but this is not useful since TB does not preserve these subspaces. k := Hk ∩ H ˆB. Definition 2.45. Let Hk := L2 (Rn ; L( k )) and Hˆ B 0 ,H ˆ 1 , Hˆ 2 , . . . , Hˆ n+1 and Hˇ B (but not Lemma 2.46. The operator TB preserve all subspaces Hˆ B B B B k the subspaces H ) and we have a splitting n+1 0 1 2 Hˆ B = Hˆ B ⊕ Hˆ B ⊕ Hˆ B ⊕ · · · ⊕ Hˆ B .
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Furthermore, for the operators Nˆ B , Nˇ B and N , we have mapping properties k k → Hˆ B , Nˆ B : Hˆ B
Nˇ B : Hˇ B → Hˇ B
and N : Hˆ 1 → Hˆ 1 .
k ) = Tˆ (H k ˆ k . To Proof. As dμ∗ , d ∗ μ, B and N ± all preserve Hk it follows that TB (Hˆ B B ˆ B) ⊂ H B show the splitting of Hˆ B it suffices to prove n+1 0 1 2 Hˆ B ⊂ Hˆ B ⊕ Hˆ B ⊕ Hˆ B ⊕ · · · ⊕ Hˆ B .
To this end, let f ∈ Hˆ B and write fk for the
k
part of f . Since dμf = 0 = d ∗ μ∗ Bf we get
0 = (dμf )k+2 = d (μf )k+1 = dμ(fk ), 0 = (d ∗ μ∗ Bf )k−2 = d ∗ (μ∗ Bf )k−1 = d ∗ μ∗ (Bf )k = d ∗ μ∗ B(fk ), k and f ∈ H ˆk for all k. Thus fk ∈ Hˆ B k B. To prove the mapping properties for Nˆ B , note that if f = f1 + f2 in the splitting H = B −1 N + H ⊕ N − H, then f ∈ Hˆ B if and only if μf2 ∈ D(d) and μBf1 ∈ D(d ∗ ), according to Definition 2.43. Clearly f1 − f2 = Nˆ B (f ) ∈ Hˆ B if f ∈ Hˆ B . Since Nˆ B preserves Hk , the desired mapping property follows. The proofs for Nˇ B and N are similar. 2
Lemma 2.47. With ·,· B denoting the duality from Definition 2.16, we have dual operators (TˆB ) = −TˆB ∗ ,
(TˇB ) = −TˇB ∗ ,
k ,H ˆ k ∗ B for all k. In the case k = 1, and restricted dualities Hˆ B , Hˆ B ∗ B , Hˇ B , Hˇ B ∗ B and Hˆ B B 1 1 we shall write the duality as Hˆ , Hˆ A .
Proof. The proofs of TˆB = −TˆB ∗ and TˇB = −TˇB ∗ are similar to that of TB = −TB ∗ in Proposition 2.17. From this we get that the annihilator of Hˆ B = R(TˆB ) is N(TˆB ∗ ) = Hˇ B ∗ which is a complement of Hˆ B ∗ . Thus we see from Remark 2.19 that Hˆ B , Hˆ B ∗ B is a duality. The proof of the duality Hˇ B , Hˇ B ∗ B is similar. Also, since BN + − N − B in the definition of ·,· B preserves k ,H ˆ k ∗ B for all k. 2 Hk we also have dualities Hˆ B B Remark 2.48. • The subspace of vector fields with a curl-free tangential part 1 Hˆ B = f ∈ L2 Rn ; 1 ; d(e0 ∧ f ) = 0 is independent of B and coincides with the space Hˆ 1 from Section 1.1. Furthermore, the operator TA there coincides with TB |Hˆ 1 = TˆB |Hˆ 1 . • The dense subspace VB ⊂ H splits VB = Vˆ B ⊕ Vˇ B with Proposition 2.44, where Vˆ B := VB ∩ Hˆ B ⊂ Hˆ B and Vˇ B := VB ∩ Hˇ B ⊂ Hˇ B are dense subspaces, and we can further decompose k. Vˆ B into homogeneous k-vector fields, Vˆ B = Vˆ B0 ⊕ Vˆ B1 ⊕ · · · ⊕ Vˆ Bn+1 , where Vˆ Bk := VB ∩ Hˆ B
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k,H ˆ B , Hˇ B splits algebraically into • Furthermore, all these dense subspaces Vˆ Bk , Vˆ B , Vˇ B of Hˆ B + ˆk k ˆ Hardy spaces similar to Lemma 2.34, e.g. VB = EB VB + EB− Vˆ Bk . Here f ∈ EB± Vˆ B if and only if F (t, x) satisfies (2.10), and f ∈ EB± Vˇ B if and only if F (t, x) satisfies (2.12).
2.5. Operator equations and estimates for solutions Our objective in this section is to set up our boundary operator method for solving the boundary value problems (Neu-A), (Reg-A), (Neu⊥ -A) and (Dir-A) as well as the transmission problem (Tr-B k α ± ). Under the assumption that TB has quadratic estimates, which is made throughout this section, we first show that solutions F (t, x) are determined by their traces f through the reproducing Cauchy type formula Ft = e−t|TB | f . Recall from Lemma 2.46 that both k . We shall write E operators EB and NB preserve all subspaces Hˆ B ˆ k = sgn(TB |H ˆk ) B k = EB |H B
and NB k = NB |Hˆ k for the restrictions. In particular, we write A = B 1 when k = 1.
B
B
Lemma 2.49. Assume that TB satisfies quadratic estimates, let f ∈ Hk and let (0, ∞) t → Ft (x) = F (t, x) ∈ Hk be a family of functions. Then the following are equivalent. k and F = e−t|TB | f . (i) f ∈ EB+ Hˆ B t (ii) Ft ∈ C 1 (R+ ; Hk ) and satisfies the equations ∗ dt,x B(x)F (t, x) = 0, dt,x F (t, x) = 0,
and have L2 limits limt→0+ Ft = f and limt→∞ Ft = 0. In fact, such Ft belong to C j (R+ ; Hk ) for all j 1 and are in one-to-one correspondence with k , and we have equivalences of norms the trace f ∈ EB+ Hˆ B f ≈ sup Ft ≈ |||t∂t Ft |||. t>0 k is also valid for F The corresponding reproducing formula Ft = et|TB | f , f = F |Rn ∈ EB− Hˆ B solving the equations in Rn+1 − , and the corresponding estimates hold.
Proof. (i) implies (ii). As in the proof of Lemma 2.34, from Lemma 2.28 it follows that j limt→0 Ft = f and limt→∞ Ft = 0, and also that ∂t Ft = (−|TA |)j e−t|TA | f . Therefore Ft ∈ k . As j k C (R+ ; H ) for all j . For j = 1, we get 0 = (∂t + |TB |)Ft = (∂t + TB )Ft , since Ft ∈ EB+ Hˆ B ∗ explained in Section 2.4 this is equivalent with the two equations dt,x F = 0 and dt,x (BF ) = 0. (ii) implies (i). The two equations can be written (−μ∗ ∂t + d ∗ )BF = 0 and (μ∂t + d)F = 0. Applying μ∗ and μ, respectively, to these equations and using nilpotence, we obtain μ∗ d ∗ BF = 0 k , and since H ˆ k is closed we also have f ∈ Hˆ k . and μdF = 0. Therefore Ft ∈ Hˆ B B B k and similarly f = f + + f − . Next we write Ft = Ft+ + Ft− , where Ft± := EB± Ft ∈ EB± Hˆ B We rewrite the equations satisfied by Ft as ∂t Ft + TB Ft = 0. Applying the projections EB± to this equation yields ∂t Ft+ + |TB |Ft+ = 0,
∂t Ft− − |TB |Ft− = 0,
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since TB Ft± = ±|TB |Ft± . Fix t > 0. Then it follows that e(t−s)|TB | Fs− is constant for s ∈ (t, ∞) and that e(s−t)|TB | Fs+ is constant for s ∈ (0, t). Taking limits, this shows that Ft− = lim e(t−s)|TB | Fs− = lim e(t−s)|TB | Fs− = 0 and s→t +
s→∞
Ft+ = lim e(s−t)|TB | Fs+ = lim e(s−t)|TB | Fs+ = e−t|TB | f + . s→t −
s→0
k and F = e−t|TB | f . Therefore f = f + ∈ EB+ Hˆ B t To prove the norm estimates we note that f = limt→0 Ft . By Proposition 2.32, e−t|TB | are uniformly bounded and thus supt>0 Ft f . Furthermore, using Proposition 2.31 with ψ(z) = ze−|z| shows that |||t∂t Ft ||| ≈ f . 2
Remark 2.50. In proving that (ii) implies (i), it suffices to assume that Ft grows at most polynomially when t → ∞. Indeed, from the equation Ft− = e−(s−t)|TB | Fs− for s > t, it then follows that
k −
T F = (s − t)−k (s − t)TB k e−(s−t)|TB | F − C(s − t)−k Fs → 0, s B t when s → ∞. Since TB is injective, this shows that Ft− = 0 and therefore that Ft = e−t|TB | f ∈ k as before. EB+ Hˆ B We now proceed by showing how, given data g in (Tr-B k α ± ), we can solve for the trace f = F |Rn by using the boundary operators EB and NB . Lemma 2.51. Assume that TB satisfies quadratic estimates, so that the Hardy projections EB± are bounded by Proposition 2.32, let α ± ∈ C be given jump parameters and define the associated spectral point λ := (α + + α − )/(α + − α − ). Then k k → Hˆ B λ − EB k NB k : Hˆ B
is an isomorphism if and only if the transmission problem (Tr-B k α ± ) is well posed. Proof. If we identify the k-vector fields F ± (t, x) in Transmission problem (Tr-B k α ± ) with the boundary traces f ± and write f = f + +f − using Lemma 2.49, then we see that the transmission problem is equivalent with the system of equations
NB+k α − EB+k − α + EB−k f = NB+k g, NB−k α + EB+k − α − EB−k f = NB−k g.
Using EB±k = 12 (I ± EB k ) and adding up the equations, we see that the system is equivalent with the equation (λ − EB k NB k )f = This proves the lemma.
2
α+
2 E k g. − α− B
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Next we consider k = 1 and the boundary value problems (Neu-A), (Reg-A) and (Neu⊥ -A). By Lemma 2.49, we have the following. + ˆ1 H → NA− Hˆ 1 is an (1) (Neu-A) is well posed if and only if the restricted projection NA− : EA isomorphism. + ˆ1 H → NA+ Hˆ 1 is an (2) (Reg-A) is well posed if and only if the restricted projection NA+ : EA + ˆ1 H → N + Hˆ 1 is an isomorphism. Note isomorphism, or equivalently if and only if N + : EA + + − 1 that both NA and N project along N H . + ˆ1 H → N − Hˆ 1 is an (3) (Neu⊥ -A) is well posed if and only if the restricted projection N − : EA isomorphism.
Proposition 2.52. Assume that TA satisfies quadratic estimates. Then (Reg-A) is well posed if and only if (Neu⊥ -A∗ ) is well posed. + ˆ1 H → N + Hˆ 1 is an isomorphism, then so is Proof. We need to show that if N + : EA + ˆ1 − − 1 N : EA∗ H → N Hˆ . The proof uses two facts. First that we have adjoint operators (EA ) = −EA∗ and N = N according to Proposition 2.17 and Lemma 2.47. As in Remark 2.19, this + ˆ1 − ˆ1 + ˆ1 + ˆ1 shows that we have dual spaces EA H , EA ∗ H A and N H , N H A , and we see that
N + f, g
A
− = f, g A = f, EA ∗ g A,
+ ˆ1 + ˆ1 for all f ∈ EA H and g ∈ N + Hˆ 1 . Therefore, the restricted projections N + : EA H → N + Hˆ 1 − − + 1 1 and EA∗ : N Hˆ → EA∗ Hˆ are adjoint. Secondly, if R1± and R2± are two pairs of complementary projections in a Hilbert space H, as in Definition 2.13, then R1− : R2+ H → R1− H has a priori estimates, as in Remark 2.18, if and only if R2− : R1+ H → R2− H has a priori estimates. Indeed, both statements are seen to be equivalent with that the subspaces R1+ H and R2+ H are transversal, i.e. that the estimate f1 + f2 ≈ f1 + f2 holds for all f1 ∈ R1+ H and f2 ∈ R2+ H. To see this, assume that R1− f2 f2 holds for all f2 ∈ R2+ H. Then f2 R1− (f1 + f2 ) f1 + f2 for all fi ∈ Ri+ H, which proves transversality. Conversely, assume that R1+ H and R2+ H are transversal. Then f2 − R1− f2 = R1+ f2 =: f1 ∈ R1+ H for all f2 ∈ R2+ H. Therefore R1− f2 = f2 − f1 ≈ f2 + f1 f2 . The same argument can be used to show that transversality also holds if and only if R2− f1 f1 holds for all f1 ∈ R1+ H. + ˆ1 To prove the proposition, assume that N + : EA H → N + Hˆ 1 is an isomorphism. It follows − − that the adjoint operator EA∗ : N + Hˆ 1 → EA∗ Hˆ 1 also is an isomorphism. Using the second fact above twice, shows that − − ˆ1 H EA : N − Hˆ 1 → EA
+ ˆ1 − ˆ1 and N − : EA ∗H → N H
have a priori estimates. As these are adjoint operators as well, both must in fact be isomorphisms. In particular we have shown that (Neu⊥ -A∗ ) is well posed. The proof of the converse implication is similar. 2
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+ ˆ H, which When perturbing A, it is preferable not to have operators defined on spaces like EA varies with A. We have the following.
Lemma 2.53. Assume that TA satisfies quadratic estimates. (1) If I − EA NA : Hˆ 1 → Hˆ 1 is an isomorphism, then Neumann problem (Neu-A) is well posed. (2) If I + EA NA : Hˆ 1 → Hˆ 1 is an isomorphism, or if I + EA N : Hˆ 1 → Hˆ 1 is an isomorphism, then Regularity problem (Reg-A) is well posed. (3) If I − EA N : Hˆ 1 → Hˆ 1 is an isomorphism, then Neumann problem (Neu⊥ -A) is well posed. Proof. Assume for example that I + EA NA is an isomorphism. We need to prove that + ˆ1 + ˆ1 H → NA+ Hˆ 1 is an isomorphism. Note that if NA+ f = g where f ∈ EA H , then NA+ : EA 1 1 1 g = NA+ f = (I + NA )f = (EA + NA )f = EA (I + EA NA )f. 2 2 2 − f ), If g ∈ NA+ Hˆ 1 , let f := 2(EA + NA )−1 g. Then it follows that 0 = NA− g = 12 (EA + NA )(EA − − − 1 since NA (EA + NA ) = 2 (EA + NA − I − NA EA ) = (EA + NA )EA , so EA f = 0 and therefore + ˆ1 f ∈ EA H . A similar calculation proves well-posedness of the other boundary value problems. 2
Remark 2.54. More generally, letting k = 1 and (α + , α − ) = (1, 0), i.e. λ = 1, in Lemma 2.51, we see that I − EA NA is an isomorphism if and only if the restricted projections − ˆ1 H → NA+ Hˆ 1 NA+ : EA
+ ˆ1 H → NA− Hˆ 1 and NA− : EA
are isomorphisms. Similarly, if (α + , α − ) = (0, 1), i.e. λ = −1 in Lemma 2.51, we see that I + + ˆ1 H → NA+ Hˆ 1 and EA NA is an isomorphism if and only if the restricted projections NA+ : EA − ˆ1 H → NA− Hˆ 1 are isomorphisms. NA− : EA We next turn to the Dirichlet problem (Dir-A), where we aim to prove an analogue of Lemma 2.49 which characterises the solution Ut as a Poisson integral of the boundary trace u. As discussed in the introduction, we shall use (Neu⊥ -A) to construct the solution Ut . Given Dirichlet data u ∈ L2 (Rn ; C), we form ue0 ∈ N − Hˆ 1 . It then follows from Lemma 2.12 that the vector field Ft solving (Neu⊥ -A) with data ue0 , has a normal component U := F0 which satisfies the second order equation (1.1). We now define the Poisson integral of u to be Pt (u) := (Ft , e0 ),
when Ft = e−t|TA | f and (f, e0 ) = u.
Lemma 2.55. Assume that TA satisfies quadratic estimates and that Neumann problem (Neu⊥ A) is well posed. Let u ∈ L2 (Rn ; C) and let (0, ∞) t → Ut (x) = U (t, x) ∈ L2 (Rn ; C) be a family of functions. Then the following are equivalent. (i) Ut = Pt u for all t > 0. (ii) Ut ∈ C 2 (R+ ; L2 (Rn ; C)), ∇t,x Ut ∈ C 1 (R+ ; L2 (Rn ; Cn+1 )) and U satisfies the equation
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divt,x A(x)∇t,x U (t, x) = 0, and we have L2 limits limt→0+ Ut = u, limt→∞ Ut = 0 and limt→∞ ∇t,x Ut = 0. If this holds, then Ut , ∇t,x Ut ∈ C j (R+ ; L2 (Rn )) for all j 1. Furthermore, Ut is in one-to-one correspondence with the trace u, and we have equivalences of norms u ≈ sup Ut ≈ |||t∂t Ut |||. t>0
Proof. (i) implies (ii). Assume that Ut = (Ft , e0 ), where Ft = e−t|TA | f and (f, e0 ) = u. As in the proof of Lemma 2.34, from Lemma 2.28 it follows that Ft → f , and therefore Ut → u, and j also that ∂t Ft = (−|TA |)j e−t|TA | f . Therefore Ft ∈ C j (R+ ; L2 (Rn )) for all j , and so does Ut . For j = 1, we see that F satisfies the Dirac type equation since |TA |Ft = TA Ft , and Lemma 2.12 thus shows that Ut satisfies the second order equation. Furthermore, we note from the expression (1.5) for TA , that ∇x Ut = (∂t Ft ) . Thus ∇x Ut ∈ C j (R+ ; L2 (Rn ; C)) for all j , and yet another application of Lemma 2.28 shows that Ut = o(1) and ∇t,x Ut = o(1/t) when t → ∞. (ii) implies (i). Assume Ut has the stated properties and boundary trace u. Consider the family of vector fields Gt := ∇t,x Ut . Since these satisfy Lemma 2.49(ii) for t s > 0 with boundary trace Gs , we obtain that Gs+t = e−t|TA | Gs for all s, t > 0. For the normal components, this means that ∂0 Us+t = Pt (∂0 Us ), or equivalently that ∂s (Us+t − Pt (Us )) = 0. Since lims→∞ Us = 0, we must have Us+t = Pt (Us ) for all s, t > 0. Letting s → 0, we conclude that Ut = Pt (u). The equivalence of norms u ≈ supt>0 Ut follows from the uniform boundedness of the operators Pt . For the equivalence u ≈ |||t∂t Ut ||| we use that (Neu⊥ -A) is well posed and the corresponding square function estimate for Ft from Lemma 2.49 to get u ≈ f ≈ |||t∂t Ft ||| ≈ |||t∂t Ut |||, since for all t > 0 we have ∂t Ut = N − (∂t Ft ) ≈ ∂t Ft . 2 ∗ (F ) ≈ f in TheoWe end this section with the proof of the non-tangential estimate N rem 1.1. Proposition 2.56. Assume that TA satisfies quadratic estimates. Let Ft = e−t|TA | f , where f ∈ + ˆ1 ∗ (F ), where the non-tangential maximal function is H . Then f ≈ N EA ∗ (F )(x) := sup N t>0
1/2 2 − − F (s, y) ds dy , D(t,x)
and D(t, x) := {(s, y) ∈ Rn+1 + ; |s − t| < c0 t, |y − x| < c1 t}, for given constants c0 ∈ (0, 1) and c1 > 0. The proof uses the following lemma. 1, Lemma 2.57. Let f ∈ Hˆ 1 and define Ht = (1 + itTB )−1 f ∈ Hˆ 1 . Write Ht = Ht1,0 e0 + Ht and 1,0 1, f = f e0 + f .
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(i) The normal component Ht1,0 satisfies the second order divergence form equation a [ 1 it div ] ⊥⊥ a⊥
a⊥ a
1 a⊥⊥ f 1,0 1,0 Ht = [ 1 it div ] , it∇ −a f 1, 1,
where we identify normal vectors ue0 with scalars u, and the tangential component Ht satisfies 1,
Ht
= f 1, + it∇Ht1,0 .
(ii) There exist p < 2 and q > 2 such that for any fixed r0 < ∞ we have
1,0 q 1/q − Ht +
B(x,r0 t)
1, p 1/p 1/p − Ht M |f |p (x), B(x,r0 t)
for all x ∈ Rn and t > 0. Here M(f )(x) := supr>0 −B(x,r) |f (y)| dy denotes the Hardy– Littlewood maximal function. Proof. (i) Multiplying the equation (1 + itTB )Ht = f with MB we get m(μ + itd) + B −1 m(−μ∗ + itd ∗ ) B Ht = MB f. Similar to the proof of Lemma 2.12 we can now use the anticommutation relations from Lemma 2.7 to rewrite this equation as
−(μ + itd)m + B −1 −(−μ∗ + itd ∗ )m B Ht = MB f,
since {m, μ + itd} = I , {m, −μ∗ + itd ∗ } = −I and I − B −1 I B = 0. Then apply (−μ∗ + itd ∗ )B to obtain (μ∗ − itd ∗ )B(μ + itd)(mHt ) = (−μ∗ + itd ∗ ) BN + − N − B f, since −μ∗ + itd ∗ is nilpotent. Evaluating the scalar part of this equation, we get the desired identity. 1, To find the identity for Ht , we use the expression for TA = TˆB |Hˆ 1 from Definition 2.43. Multiplying the equation (I + it TˆA )Ht = f by AN + − N − A yields 1,
a Ht
− a00 Ht1,0 e0 − it A∇Ht1,0 + d ∗ μAHt = a f 1, − a00 ft1,0 e0 .
Evaluating the tangential part of this equation gives the desired identity. (ii) By rescaling, we see from (i) that it suffices to show that
B(x,r0 )
u(y) q dy
1/q
+ B(x,r0 )
∇u(y) p dy
1/p
1/p M |g|p (x),
(2.13)
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for all x ∈ Rn and all u and g satisfying an equation a⊥ a a⊥⊥ g 1,0 1 u = [ ] , [ 1 i div ] ⊥⊥ 1 i div g 1, i∇ −a a⊥ a where A is a matrix with same norm and accretivity constant as for A. Indeed, by rescaling we see that u(x) = Ht1,0 (tx), g(x) = f (tx) and A (x) = A(tx) satisfies this hypothesis. To prove (2.13), we use that the maps g → u and g → ∇u have Lp (Rn ) → Lq (Rn ) and Lp (Rn ) → Lp (Rn ) off-diagonal bounds, respectively, with exponent M for any M > 0, i.e. there exists CM < ∞ such that −M gp uLq (E) + ∇uLp (E) CM dist(E, F )
(2.14)
whenever E, F ⊂ Rn and supp g ⊂ F . To see this, let L := [ 1 i div ]A [ 1 i∇ ]t . Note that L : W21 (Rn ) → W2−1 (Rn ) is an isomorphism and that L : Wp1 (Rn ) → Wp−1 (Rn ) is bounded. Then by the stability result of Šne˘ıberg [24], it follows that there exists ε > 0 such that L : Wp1 (Rn ) → Wp−1 (Rn ) is an isomorphism when |p − 2| < ε. We then fix p0 ∈ (2 − ε, 2) and use Sobolev’s embedding theorem to see that uq0 + ∇up0 uWp1 LuW −1 gp0 . p0
0
By choosing p0 close to 2, we may assume that q0 > 2. Thus we have bounded maps g → u : Lp0 (F ) → Lq0 (E) and g → ∇u : Lp0 (F ) → Lp0 (E), with norms C. Also, by Proposition 2.25, the norms of g → u : L2 (F ) → L2 (E) and g → ∇u : L2 (F ) → L2 (E) are bounded by dist(E, F ) −M0 . Interpolation now proves (2.14) for some p0 < p < 2, 2 < q < q0 and M = M0 (1 − p0 /p)/(1 − p0 /2). Finally we show how (2.14) implies (2.13). Let E = F0 := B(x, r0 ) and for k 1 let Fk := B(x, 2k r0 ) \ B(x, 2k−1 r0 ). This gives
B(x,r0 )
u(y) q dy
1/q
∇u(y) p dy
+
1/p
∞ k=0
B(x,r0 )
∞ k=0
2
−Mk
Fk
g(y) p dy
2(n/p−M)k
1/p p − g(y) dy
B(x,2k r0 )
1/p M |g|p (x) , provided we chose M > n/p.
2
∗ (F ) f , we calculate Proof of Proposition 2.56. To prove that N 2
N ∗ (F ) 2 sup − − F (s, y) ds dy dx t>0
= sup t>0
Rn |y−x|
−
|s−t|
Fs 2 ds sup F(1+c0 )t 2 ≈ f 2 . t>0
1/p
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We have here used that F(1+c0 )t = e−((1+c0 )t−s)|TA | Fs , which by Proposition 2.32 shows that F(1+c0 )t Fs when |s − t| < c0 t. ∗ (F ) f , we note that since curlt,x F = 0, we can write F = ∇t,x U for To prove N some scalar potential U , and we see that U solves the second order equation (1.1). With notation x) = {(s, y); |s − t| < c˜0 t, |y − x| < c˜1 t}, for constants c0 < c˜0 < 1 and c1 < c˜1 < ∞, and D(t, U := − −D(t,x) U (s, y) ds dy, we have
1/2 2 2 1/2 − − t∇U (s, y) ds dy − − U (s, y) − U ds dy D(t,x)
D(t,x)
1/p p − − t∇U (s, y) ds dy , D(t,x)
with 2(n + 1)/(n + 3) < p < 2. The first estimate uses Caccioppoli’s inequality and the second estimate uses Poincaré’s inequality. Thus it suffices to bound the L2 norm of
1/p p . sup − − F (s, y) ds dy t>0
D(t,x)
To this end, we write Fs = Hs + ψs (TA )f , where Hs := (I + isTA )−1 f and ψ(z) := e−|z| − (1 + iz)−1 . Using the quadratic estimates, the second term has estimates sup
2/p p − − ψs (TA )f (y) ds dy dx
t>0
Rn
D(t,x)
sup
Rn
t>0
|s−t|
∞
ψs (TA )f (y) 2 dy ds dx t n+1
ψs (TA )f (y) 2 s −(n+1) dy dx ds
0 Rn |y−x|
∞
2 ds
f 2 . ψs (TA )f s 0
For the first term, we use Lemma 2.57(ii) and obtain Rn
2/p p sup − − Hs (y) ds dy dx t>0
D(t,x)
sup Rn
sup
t>0 |s−t|
−
B(x,c˜1 s/(1−c˜0 ))
Hs (y) p dy
2/p dx
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2/p 2/p M |f |p 2/p |f |p 2/p = f 22 , using the boundedness of the Hardy–Littlewood maximal function on L2/p (Rn ). This completes the proof. 2 3. Invertibility of unperturbed operators In this section we prove Theorems 1.4 and 1.1 for the unperturbed problem, i.e. for B0k and A0 , respectively. We do this by verifying the hypothesis in Lemmas 2.51 and 2.53, i.e. we prove that TB k satisfies quadratic estimates and λ − EB k NB k is an isomorphism, and that TA0 satisfies 0 0 0 quadratic estimates and I ± EA0 NA0 are isomorphisms, respectively. 3.1. Block coefficients In this section we assume that B = B0 ∈ L∞ (Rn ; L( )) have properties as in Definition 2.8 with the extra property that it is a block matrix, i.e. 0 B⊥⊥ B= 0 B in the splitting H = N − H ⊕ N + H. Note that B being of this form is equivalent with the commutation relations N ± B = BN ± . Lemma 3.1. Let B be a block matrix as above. Then TB = Γ + B −1 Γ ∗ B, where Γ = Nmd = −iN d is a nilpotent first order, homogeneous partial differential operator with constant coefficients. Proof. Since N ± B = BN ± , it follows that MB = N + − B −1 N − B = N + − N − = N and TB = N md + B −1 md ∗ B = Γ + B −1 Γ ∗ B, since N 2 = I and N B −1 = B −1 N . The operator Γ is nilpotent since Γ 2 = N mdN md = NmN dmd = −NmN md 2 = 0. 2 Remark 3.2. Note that if ΠB = Γ + ΓB∗ , where ΓB∗ = B −1 Γ ∗ B and Γ is nilpotent, is an operator of the form considered in [10], then ΠB intertwines Γ and ΓB∗ in the sense that ΠB Γ u = ΓB∗ ΠB u for all u ∈ D(ΓB∗ ΠB ) and ΠB ΓB∗ u = Γ ΠB u for all u ∈ D(Γ ΠB ). Thus ΠB2 commutes with both Γ and ΓB∗ on appropriate domains. In particular, if PtB = (1 + t 2 ΠB2 )−1 and QB t = tΠB (1 + B Γ u for all u ∈ D(Γ ) and Γ ∗ P B u = u = Q t 2 ΠB2 )−1 , then we find that Γ PtB u = PtB Γ u, ΓB∗ QB t t B t ∗ B ∗ PtB ΓB∗ u, Γ QB t u = Qt ΓB u for all u ∈ D(ΓB ). Theorem 3.3. Let B be a block matrix as above. Then (i) TB satisfies quadratic estimates, and
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(ii) EB NB + NB EB = 0 and NB = N . In particular λ − EB NB is an isomorphism whenever λ∈ / {i, −i} with (λ − EB NB )−1 =
λ2
1 (λ − NB EB ). +1
Proof. For operators of the form Γ + B −1 Γ ∗ B, quadratic estimates were proved in [10], with essentially the same methods as we use here in Section 4.1. To prove (ii), note that since B is a block matrix, it follows that B −1 N ± H = N ± H. Thus the projections NB± associated with the splitting H = N − H ⊕ B −1 N + H are N ± and the associated reflection operator is NB = N . To prove invertibility of λ − EB NB , we note that TB NB = Γ + B −1 Γ ∗ B N = −N Γ + B −1 Γ ∗ B = −NB TB , since N commutes with d, d ∗ and B, and anticommutes with m. Thus EB NB = sgn(TB )NB = NB sgn(NB TB NB ) = NB sgn(−TB ) = −NB EB , since sgn(z) is odd. Using this anticommutation formula we obtain (λ − NB EB )(λ − EB NB ) = λ2 + 1 − λ(EB NB + NB EB ) = λ2 + 1, and similarly (λ − EB NB )(λ − NB EB ) = λ2 + 1, from which the stated formula for the inverse follows. 2 3.2. Constant coefficients We here collect results in the case when B(x) = B ∈ L( ) is a constant accretive matrix. In this case we make use of the Fourier transform 1 Fu(ξ ) = u(ξ ˆ ) := u(x)e−i(x,ξ ) dx, 2πi Rn
acting componentwise. If we let μξ f (ξ ) := ξ
∧ f (ξ ),
μ∗ξ f (ξ ) := ξ f (ξ ),
then TB , conjugated with F , is the multiplication operator Mξ f (ξ ) := MB−1 imμξ − iB −1 mμ∗ξ B f (ξ ), Lemma 3.4. For all t ∈ R and ξ ∈ Rn we have (it + Mξ )−1 ≈ t 2 + |ξ |2 −1/2 .
ξ ∈ Rn .
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:= mBm, Proof. Let u = (it + Mξ )f . It suffices to prove that u2 (t 2 + |ξ |2 )f 2 . With B it follows from the definition of Mξ that −1 Γ ∗ B f, mMB u = Γ + B where Γ = i(tμ + μξ ). Using Lemma 2.7 we get
(Γ + Γ ∗ )g 2 = Γ g2 + Γ ∗ g2 = (Γ ∗ Γ + Γ ∗ Γ )g, g = t 2 + |ξ |2 g2 . Therefore our estimate follows from Lemma 2.21(ii).
2
Proposition 3.5. If B(x) = B ∈ L( ) is a constant, accretive matrix, then TB satisfies quadratic estimates. Proof. Using the lemma, we obtain the estimate tMξ t|ξ | −1 (i + Mtξ )−1 . 1 + t 2 M 2 t|Mξ | (i − Mtξ ) 1 + t 2 |ξ |2 ξ Thus using Plancherel’s formula we obtain
2
2 ∞ ∞
tTB
tMξ
dt
dt
≈ u u ˆ
1 + t 2T 2 t
1 + t 2M 2 t 0
B
ξ
0
∞
Rn
0
t|ξ | 1 + t 2 |ξ |2
2
2 dt u(ξ ˆ ) dξ ≈ u2 , t
where the last step follows from a change of variables s = t|ξ |.
2
Next we prove that Neumann and Regularity problems are well posed in the case of a complex, constant, accretive matrix A=
a00 a0
a0 . a
For this it suffices to consider B = I ⊕ A ⊕ I ⊕ I ⊕ · · · ⊕ I and the action of TA = TB |Hˆ 1 on the invariant subspace Hˆ 1 . Recall that f ∈ Hˆ 1 means that f is a vector field f : Rn → 1 = Cn+1 such that df = 0. On the Fourier transform side f ∈ Hˆ 1 is seen to correspond to a vector field fˆ : Rn → 1 such that ξ ∧ fˆ = 0, i.e. fˆ is such that its tangential part f is a radial vector field. Thus the space F Hˆ 1 = fˆ ∈ L2 Rn ; Cn+1 ; ξ
∧ fˆ
=0
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can be identified with L2 (Rn ; C2 ) if we use {e0 , ξ/|ξ |} as basis for the two-dimensional space Hˆ ξ1 to which fˆ(ξ ) belongs. Furthermore, the operator TA is conjugated to the multiplication operator Mξ = MA−1 iμ∗ μξ − iA−1 μμ∗ξ A : F Hˆ 1 → F Hˆ 1 , under the Fourier transform, where MA := N + − A−1 N − A. We see from the expression (1.5) for TA that, at a fixed point ξ ∈ S n−1 , the matrix for Mξ in the basis {e0 , ξ } is
+ a0 , ξ ) −i
i a00 (a0
i a00 (a ξ, ξ ) . 0
Theorem 3.6. Let A(x) = A ∈ L( ) be a constant, complex, accretive matrix. Then I ± EA NA : Hˆ 1 → Hˆ 1 are isomorphisms. In particular, Neumann and Regularity problems (Neu-A) and (Reg-A) are well posed. Proof. From Remark 2.54 we see that it suffices to prove that all four restricted projections ± ˆ1 H → NA± Hˆ 1 NA± : EA
are isomorphisms, or equivalently that the constant multiplication operators NA± are isomorphisms on F(Hˆ 1 ). For the projections χ± (Mξ ) conjugated to the Hardy projection operators ± EA , we observe that Mtξ = tMξ and hence χ± (tMξ ) = χ± (Mξ ) for all t > 0. Thus it suffices to verify that NA± : χ± (Mξ )Hˆ ξ1 → NA± Hˆ ξ1 are isomorphisms for each ξ ∈ S n−1 . For such fixed ξ , using the basis {e0 , ξ } from above, NA− Hˆ ξ1 = {fˆ ∈ Hˆ ξ1 ; e0 ∧ fˆ = 0} is spanned by [ 1 0 ]t and NA+ Hˆ ξ1 = {fˆ ∈ Hˆ ξ1 ; (Afˆ, e0 ) = 0} is spanned by [ (a0 , ξ ) −a00 ]t . Indeed (A(ze0 + wξ ), e0 ) = za00 + w(a0 , ξ ). If we call eξ+ and eξ− the two eigenvectors of Mξ , it follows that χ± (Mξ )Hˆ ξ1 are spanned by these, as these subspaces are one-dimensional. It suffices to show that [ 1 0 ]t and [ (a0 , ξ ) −a00 ]t are not eigenvectors of Mξ . We have 1 1 (a + a0 , ξ ) = i a00 0 , 0 −1 1 (a0 + a0 , ξ )(a0 , ξ ) − (a ξ, ξ ) (a0 , ξ ) a 00 . =i Mξ −a00 −(a0 , ξ ) Mξ
Clearly the normal vector is not an eigenvector. To prove that the second is not an eigenvector, note that the cross product is −(a0 , ξ )2 + (a0 + a0 , ξ )(a0 , ξ ) − a00 (a ξ, ξ ) = (a0 , ξ )(a0 , ξ ) − a00 (a ξ, ξ ).
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The right-hand side is non-zero since
a00 a0
a0 a
z z a00 , = wξ wξ (a0 , ξ )
is a non-degenerate quadratic form as A is accretive.
(a0 , ξ ) (a ξ, ξ )
z z , w w
2
Remark 3.7. We note that the method above also can be used to show that I ± EA N : Hˆ 1 → Hˆ 1 , with the unperturbed operator N , are isomorphisms when A is constant. Here we also need to observe that the tangential vector [ 0 1 ]t is not an eigenvector to Mξ . 3.3. Real symmetric coefficients In this section, we assume that B ∗ = B. We first prove a Rellich type estimate. Proposition 3.8. Assume that B ∗ = B and that f ∈ EB+ VB or f ∈ EB− VB . Then (Bf, f ) = 2 Re e0 (Bf ), e0 f = 2 Re e0 ∧ (Bf ), e0 ∧ f . In particular f ≈ Nˆ B− f ≈ Nˇ B− f ≈ Nˇ B+ f ≈ Nˆ B+ f . Proof. It suffices to consider f ∈ EB+ VB as the case f ∈ EB− VB is treated similarly. We use Lemma 2.34 and write Ft := e−t|TB | f . Hence (0, ∞) t → Ft ∈ H is differentiable, limt→0 Ft = f and limt→∞ Ft = 0. Furthermore Ft ∈ D(dx ) and BFt ∈ D(dx∗ ) for all t ∈ (0, ∞). We note the formulae ∗ ∗ mGt + mdt,x Gt = −∂t Gt , dt,x ∗ BFt . Bmdt,x Ft = −mdt,x
The first identity, which we apply with Gt = BFt , follows from Lemma 2.7, whereas the second is equivalent to ∂t Ft + TB Ft = 0, and follows from Lemma 2.34. We get ∞ ∞ (Bf, f ) = − (∂t BFt , Ft ) + (BFt , ∂t Ft ) = −2 Re (∂t BFt , Ft ) 0
0
∞ ∞ ∗ ∗ ∗ dt,x mBFt + mdt,x BFt , Ft = 2 Re dt,x mBFt − Bmdt,x Ft , Ft = 2 Re 0
0
∞
∗ dt,x mBFt , Ft − (mBFt , dt,x Ft )
= 2 Re 0
∞ ∗ (d − μ∗ ∂t )mBFt , Ft − mBFt , (d + μ∂t )Ft = 2 Re 0
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∞ = −2 Re (∂t mBFt , μFt ) + (mBFt , ∂t μFt ) 0
= 2 Re(mBf, μf ) = 2 Re e0 ∧ (Bf ), e0 ∧ f . Note that all integrals are convergent since Ft min(1, t −s ) and since ∂t Ft = −TB Ft with TB Ft min(t s−1 , t −1 ) if f ∈ D(|TB |s ) ∩ R(|TB |−s ). This follows as in the proof of Lemma 2.34. Furthermore, we note that (e0 ∧ Bf, e0 ∧ f ) = Bf, e0 (e0 ∧ f ) = Bf, f − e0 ∧ (e0 f ) = (Bf, f ) − e0 (Bf ), e0 f . Together with the calculation above this proves that (Bf, f ) = 2 Re(e0 (Bf ), e0 f ). To prove that f ≈ Nˆ B+ f ≈ Nˆ B− f , it suffices to show that f Nˆ B± f since Nˆ B± are bounded. From the Rellich type identities above we have f 2 e0 (Bf )e0 f Nˆ B− f f which proves f Nˆ B− f , and using the other identity we obtain f 2 e0 ∧ (Bf )e0 ∧ f f Nˆ B+ f . The proof of f ≈ Nˇ B+ f ≈ Nˇ B− f is similar. 2 We note that Proposition 3.8 also proves that the norms of the two components of f ∈ EB± VB in the unperturbed splitting H = N − H ⊕ N + H are comparable. Corollary 3.9. Assume that B ∗ = B and that f ∈ EB+ VB or f ∈ EB− VB , and decompose B in the splitting H = N − H ⊕ N + H as
B⊥⊥ B= B⊥
B⊥ . B
If f = N + f and f⊥ = N − f , then (B⊥⊥ f⊥ , f⊥ ) = (B f , f ). In particular f ≈ N + f ≈ N − f . Proof. We obtain from Proposition 3.8 that Re(B⊥⊥ f⊥ +B⊥ f , f⊥ ) = Re(B⊥ f⊥ +B f , f ). ∗ =B . 2 The corollary now follows since B⊥⊥ , B κB and B⊥ ⊥ Next we turn to quadratic estimates for the operator TB . As before we assume that B is as in Definition 2.8 and that B = B ∗ . For the rest of this section we shall also assume that j
j
B⊥ = B⊥ = 0,
for j 2.
(3.1)
Note that in particular this is true if B = I ⊕ A ⊕ I ⊕ · · · ⊕ I , where A ∈ L∞ (Rn ; L( 1 )). To prove quadratic estimates, we shall use Proposition 2.36 where we verify the hypothesis k and H ˇ B , which is possible by Lemma 2.46. separately on the subspaces Hˆ B
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Lemma 3.10. Assume that B ∗ = B and (3.1) holds. Then f |||t∂t Ft |||± ,
for all f ∈ EB± Vˇ B ,
where Ft = e∓t|TB | f . Proof. As the two estimates are similar, we only show the estimate for f ∈ EB+ Vˇ B . Note that F (t, x) satisfies (2.12), i.e.
dF = e0 ∂t F, d ∗ (BF ) = −e0 ∧ (B∂t F ).
Splitting both sides of both equations into normal and tangential parts, with notation as in Corollary 3.9, we obtain dF = m∂t F⊥ ,
(3.2)
dF⊥ = 0,
(3.3)
d ∗ (B⊥⊥ F⊥ + B⊥ F ) = −m(B⊥ ∂t F⊥ + B ∂t F ), d ∗ (B⊥ F⊥ + B F ) = 0.
(3.4) (3.5)
We have here used that e0 f = mf if f is normal, and e0 ∧ f = mf if f is tangential. A key observation is that the first term on the right-hand side in (3.4) vanishes since B⊥ F⊥ = 0. To see this, note that n+1 n+1 1 2 F⊥1 + B⊥ F⊥2 + · · · + B⊥ F⊥ . B⊥ F⊥ = B⊥ 2 = · · · = B n+1 = 0. Furthermore, writing F 1 = F e with the function By hypothesis (3.1), B⊥ 0 0 ⊥ ⊥ F0 being scalar, we get from (3.3) that 0 = d(F0 e0 ) = (∇F0 ) ∧ e0 , so F0 is constant and therefore vanishes, and thus so does F⊥1 . Eq. (3.4) reduces to
B ∂t F = −md ∗ (B⊥⊥ F⊥ + B⊥ F ).
(3.6)
We calculate ∞ ∞ ∗ f (B f , f ) = −2 Re (B ∂t F , F ) dt = 2 Re d (B⊥⊥ F⊥ + B⊥ F ), mF dt 2
0
0
∞
∗ d (B⊥⊥ ∂t F⊥ + B⊥ ∂t F ), mF + d ∗ (B⊥⊥ F⊥ + B⊥ F ), m∂t F t dt
= −2 Re 0
∞
(B⊥⊥ ∂t F⊥ + B⊥ ∂t F , ∂t F⊥ ) + (mB ∂t F , m∂t F ) t dt
= 2 Re 0
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∞ = 2 Re (B∂t F, ∂t F )t dt ≈ |||t∂t F |||2 . 0
Here in the third step we use (3.6), in the fourth step we integrate by parts, in the fifth step we use duality and that dmF = −mdF = −∂t F⊥ by (3.2) for the first term and again (3.6) for the second term. Finally in step 6 we use that m is an isometry and that B⊥ F⊥ = 0. 2 Next we turn to the subspace Hˆ 1 . Lemma 3.11. Let A ∈ L∞ (Rn ; L( 1 )) be real symmetric and A κ > 0, and let B = I ⊕ A ⊕ I ⊕ · · · ⊕ I . Then f |||t∂t F |||± ,
for all f ∈ EB± Vˆ B1 ,
where F = e∓t|TB | f . Proof. Recall that if f ∈ EB± Vˆ B1 and F = F0 e0 + F = e∓t|TB | f , then Lemma 2.12 shows that F0 satisfies the equation divt,x A(x)∇t,x F0 (t, x) = 0. Therefore, by the square function estimate of Dahlberg, Jerison and Kenig [13] and the estimates of harmonic measure of Jerison and Kenig [17], we have estimates f0 |||t∇t,x F0 |||± . Hence applying the Rellich estimates in Proposition 3.8, we obtain f ≈ f0 |||t∇t,x F0 |||± |||t∂t F |||± , since ∂i F0 = ∂0 Fi , as dt,x F = 0.
2
We are now in position to prove the main result of this section. Theorem 3.12. Let A ∈ L∞ (Rn ; L( 1 )) be real symmetric and A κ > 0, and let B = I ⊕ A ⊕ I ⊕ · · · ⊕ I . Then TB has quadratic estimates, so that in particular EB = sgn(TB ) : H → H is bounded. Furthermore we have isomorphisms I ± EB NB : H → H. Proof. To prove that TB has quadratic estimates it suffices by Proposition 2.36 to prove f |||t∂t F |||± ,
for all f ∈ EB± VB .
To this end, we split f = f0 + f1 + · · · + fn+1 + fˇ
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j where fj ∈ Vˆ B and fˇ ∈ Vˇ B . Lemma 2.46 shows that e∓t|TB | preserves these subspaces so that similarly
F = F0 + F1 + · · · + Fn+1 + Fˇ , j where (Fj )t ∈ Vˆ B and Fˇt ∈ Vˇ B . It thus suffices to prove that fj |||t∂t Fj |||± , j = 0, 1, . . . , n + 1, and fˇ |||t∂t Fˇ |||± . Lemma 3.10 shows that fˇ |||t∂t Fˇ |||± and Lemma 3.11 shows that f1 |||t∂t F1 |||± . Furthermore, for j = 1 we observe that TB = TˆB = TI on the subspace j Vˆ B , where I denotes the identity matrix. Thus Fj = e∓t|TI | fj and it follows from Proposition 3.5 that fj |||t∂t Fj |||± . Applying Propositions 2.36 and 2.32 now shows that EB = sgn(TB ) : H → H is a bounded operator. To show that I ± EB Nˆ B : H → H is invertible, note that the adjoint with respect to the duality ·,· B from Definition 2.16 is
I ∓ Nˇ B EB = Nˇ B (I ∓ EB Nˇ B )Nˇ B , according to Proposition 2.17. Having established the boundedness of EB , the Rellich estimates clearly extends to a priori estimates for all eight restricted projections Nˆ B± : EB± H → Nˆ B± H,
Nˇ B± : EB± H → Nˇ B± H.
As in Remark 2.54 this translates to a priori estimates for I ± EB Nˆ B and their adjoints, which proves that they are isomorphisms as in Remark 2.18. 2 Remark 3.13. Using instead Corollary 3.9, we also prove as in Theorem 3.12 that I ± EB N : H → H, where N is the unperturbed operator, are isomorphisms. 4. Quadratic estimates for perturbed operators In Section 3 we proved that TB0 satisfies quadratic estimates B0 Qt f ≈ f ,
f ∈ H,
(4.1)
for certain unperturbed coefficients B0 : (b) Block coefficients
(B0 )⊥⊥ B0 = 0
0 . (B0 )
(c) Constant coefficients B0 (x) = B0 , x ∈ Rn , of the form B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I , i.e. B0 only acts non-trivially on the vector part. (s) Real symmetric coefficients of the form B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I . Note that for (c), we did prove quadratic estimates for general constant coefficients B0 in Proposition 3.5. However, since we only prove invertibility I ± EB0 NB0 on the subspace Hˆ 1
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in Theorem 3.6, we shall only prove perturbation results for constant coefficients of the form B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I . In this section we let B0 ∈ L∞ (Rn ; L( )) be a fixed accretive coefficient matrix with properties (b), (c) or (s). Constants C in estimates or implicit in notation and ≈ in this section will be allowed to depend only on B0 ∞ , κB0 and dimension n. Note that this is indeed the case for the constants implicit in (4.1). We now consider a small perturbation B ∈ L∞ (Rn ; L( )) of B0 : B − B0 L∞ (Rn ;L()) ε0 . Throughout this section, we assume in particular that ε0 is chosen small enough so that B has properties as in Definition 2.8 with B∞ 2B0 ∞ and κB 12 κB0 . The goal is to show that B Q f Cf , t
whenever B − B0 ∞ ε, f ∈ H,
(4.2)
where C = C(B0 ∞ , κB0 , n) and ε = ε(B0 ∞ , κB0 , n) ε0 . Since the properties of B0 are stable under taking adjoints, we see that the quadratic estimate (2.7) follows from (4.2) and Lemma 2.35. In order to prove (4.2) we make use of the following identity, where we note that each of the six terms consists of three factors, the first being one of the operators from Corollary 2.23, the second being a multiplication operator E with norm E∞ ε0 and the last factor being one of the operators from Corollary 2.23 but with B replaced by B0 : −1 B B0 B MB − MB−1 MB0 Qt 0 QB t − Qt = Pt 0 B − iPtB MB−1 B −1 − B0−1 B0 td ∗B0 Pt 0 B − i PtB MB−1 td ∗B B −1 (B − B0 ) Pt 0 −1 B − QB MB − MB−1 MB0 tTB0 Qt 0 t 0 −1 −1 B + iQB − B0−1 B0 td ∗B0 Qt 0 t MB B B0 −1 ∗ −1 + i QB t MB td B B (B − B0 ) Qt .
(4.3) (4.4) (4.5) (4.6) (4.7) (4.8)
Recall that TB = −iMB−1 d + B −1 d ∗ B ,
where d = imd, MB = N + − B −1 N − B,
2 2 −1 and similarly for B . The identity (4.3)–(4.8) is PtB = (1 + t 2 TB2 )−1 and QB 0 t = tTB (1 + t TB ) established by using
−1 −1 −1 −1 (Y − X) − Y (Y − X)X 1 + X 2 − X 1 + X2 = 1 + Y2 Y 1 + Y2 with Y = tTB and X = tTB0 , and then inserting Y − X = TB − TB0 MB0 tTB0 − i MB−1 B −1 − B0−1 B0 td ∗B0 = MB−1 − MB−1 0 − i MB−1 td ∗B B −1 (B − B0 ) .
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t )t>0 with L2 Lemma 4.1. Assume that for all E ∈ L∞ (Rn ; L( )) and all operator families (Q t off C as in Definition 2.24, we have the two estimates off-diagonal bounds Q 0 Q t E tdQB t
t |||op + 1 E∞ , C |||Q 0 Q t |||op + 1 E∞ . t E td ∗B QB C |||Q t op 0 op
(4.9) (4.10)
Then there exists ε > 0 such that (4.2) holds. Proof. First consider the terms (4.3), (4.4) and (4.8). Using the uniform boundedness of PtB and −1 ∗ QB t MB td B from Corollary 2.23, we deduce that (4.3)
B B − B0 ∞ Qt 0 op , (4.4) B − B0 ∞ td ∗ PtB0 , B0 op op B0 (4.8) B − B0 ∞ Qt . op op op
Observe that td ∗B0 Pt 0 = iP2B0 MB0 Qt 0 as in Corollary 2.23, where the Hodge projection P2B0 and MB0 are bounded. Thus we see from (4.1) that B
B
(4.3)
op
+ (4.4) op + (4.8) op B − B0 ∞ .
1t := χ(t)QB To handle the terms (4.5)–(4.7) we introduce the truncated operator families Q t and −1 ∗ 2 B −1 Qt := χ(t)Pt MB td B , where χ(t) denotes the characteristic function of the interval [τ , τ ] 1 2 for some large τ . Note that PtB MB−1 td ∗B = iQB t PB as in Corollary 2.23, and therefore |||Qt |||op = 1 1 1 t |||op . To use the hypothesis on the terms (4.5)–(4.7) we note that the last factors t P |||op |||Q |||Q B are B0
Pt
B0
tTB0 Qt −itd ∗B0 Qt
B0
∗ B0 B tdQt 0 + iMB−1 td B0 Qt , = I + iMB−1 0 0 B B tdQt 0 − iMB−1 td ∗B0 Qt 0 , = −iMB−1 0 0 B = −i td ∗B0 Qt 0 ,
respectively. Thus we get from (4.3)–(4.8), after multiplication with χ(t), that 0 χ(t) QB − QB t t
op
C χ(t)QB t op + 1 B − B0 ∞ ,
and thus if B − B0 ∞ ε := 1/(2C) that χQB t
B
op
CB − B0 ∞ + |||Qt 0 |||op C , 1 − CB − B0 ∞
B since |||χ(t)QB t |||op < ∞. Since this estimate is independent of τ , it follows that |||Qt f ||| f . 2
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Before turning to the proofs of (4.9) and (4.10), we summarise fundamental techniques from harmonic analysis that we shall need. We use the following dyadic decomposition of Rn . Let = ∞ j n n j −1 < t 2j . For a dyadic cube Q ∈ , 2j j =−∞ 2j where t := {2 (k +(0, 1] ): k ∈ Z } if 2 n+1 j nj j denote by l(Q) = 2 its sidelength, by |Q| = 2 its volume and by RQ := Q × (0, 2 ] ⊂ R+ the associated Carleson box. Let the dyadic averaging operator At : H → H be given by 1 At u(x) := uQ := − u(y) dy = u(y) dy |Q| Q
Q
for every x ∈ Rn and t > 0, where Q ∈ t is the unique dyadic cube that contains x. We now survey known results for a family of operators Θt : H → H, t > 0. For the proofs we refer to [4] and [10]. Definition 4.2. By the principal part of (Θt )t>0 we mean the multiplication operators γt defined by γt (x)w := (Θt w)(x) for every w ∈ . We view w on the right-hand side of the above equation as the constant function defined on Rn by w(x) := w. We identify γt (x) with the (possibly unbounded) multiplication operator γt : f (x) → γt (x)f (x). Lemma 4.3. Assume that Θt has L2 off-diagonal bounds with exponent M > n. Then Θt extends to a bounded operator L∞ → Lloc 2 . In particular we have well-defined functions γt ∈ n ; L( )) with bounds (R Lloc 2 2 − γt (y) dy Θt 2off Q
for all Q ∈ t . Moreover γt At Θt off uniformly for all t > 0. We have the following principal part approximation Θt ≈ γt . Lemma 4.4. Assume that Θt has L2 off-diagonal bounds with exponent M > 3n and let Ft : Rn → be a family of functions. Then (Θt − γt At )Ft Θt off |||t∇Ft |||, where ∇f = ∇ ⊗ f = nj=1 ei ⊗ (∂j f ) denotes the full differential of f . Moreover, if Pt is a standard Fourier mollifier (we shall use Pt = (1+t 2 Π 2 )−1 where Π = Γ +Γ ∗ and Γ is an exact nilpotent, homogeneous first order partial differential operator with constant coefficients as in [10]) and Ft = Pt f for some f ∈ H, then |||γt At (Pt − I )f ||| Θt off f and |||t∇Pt |||op C. Thus (Θt Pt − γt At )f Θt off f .
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Definition 4.5. A function γ (t, x) : Rn+1 is called a Carleson function if there exists C < + → ∞ such that γ (t, x) 2 dx dt C 2 |Q| t RQ
for all cubes Q ⊂ Rn . Here RQ := Q × (0, l(Q)] is the Carleson box over Q. We define the Carleson norm γt C to be the smallest constant C. We use Carleson’s lemma in the following form. Lemma 4.6. Let γt (x) = γ (t, x) : Rn+1 + → n+1 R+ → be a family of functions. Then
be a Carleson function and let Ft (x) = F (t, x) :
|||γt Ft ||| γt C N∗ (Ft ) , where N∗ (Ft )(x) := sup|y−x|
0 −B(x,r) |f (y)| dy denotes the Hardy–Littlewood maximal function, and thus |||γt At f ||| γt C f . Lemma 4.7. Assume that Θt has L2 off-diagonal bounds with exponent M > n. Then Θt f C |||Θt |||op + Θt off f ∞ , for every f ∈ L∞ (Rn ;
). In particular, choosing f = w = constant we obtain γt C |||Θt |||op + Θt off .
4.1. Perturbation of block coefficients In this section we assume that B0 is a block matrix, i.e. we assume that B0 =
(B0 )⊥⊥ 0
0 , (B0 )
in the splitting H = N − H ⊕ N + H. Our goal is to prove Theorem 1.4 by verifying the hypothesis of Lemma 4.1. We recall from Lemma 3.1 that TB0 = ΠB0 = Γ + B0−1 Γ ∗ B0 ,
where Γ := −iN d,
is an operator of the form treated in [10]. Thus with a slight change of notation for E, we need to prove the following.
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t )t>0 is an operator family with L2 off-diagonal Theorem 4.8. If B0 is a block matrix and if (Q t off C as in Definition 2.24, then with notation as above bounds Q Q t |||op + 1 E∞ , t E t 2 Γ ΓB∗ PtB0 C |||Q op 0 Q t E t 2 Γ ∗ Γ PtB0 C |||Q t |||op + 1 E∞ , B0
B0
where Pt
op
(4.11) (4.12)
= (1 + t 2 ΠB2 0 )−1 .
We note that if (4.11) holds with Γ replaced by Γ ∗ and with B0 replaced by B0−1 , then also (4.12) holds. This follows from the conjugation formula t E t 2 ΓB∗ Γ PtB0 Q 0 t B −1 B0 EB −1 t 2 Γ ∗ Γ = B −1 B0 Q 0
0
0
B0−1
2 −1 1 + t 2 Γ ∗ + B0 Γ B0−1 B0 .
Thus it suffices to prove (4.11), as long as we only use properties of (Γ, B0 ) shared with (Γ ∗ , B0−1 ). To this end, we let Θt be the operator t E t 2 Γ ΓB∗ PtB0 , Θt := Q 0 and denote by γt (x) its principal part as in Definition 4.2. We note that we have a Hodge type splitting H = N(Γ ) ⊕ N(ΓB∗0 ) by Lemma 2.21(i), and since Θt |N(ΓB∗ ) = 0 it suffices to bound 0 |||Θt f ||| for f ∈ N(Γ ). We do this by writing Θt f = Θt (I − Pt )f + (Θt Pt − γt At )f + γt At f,
(4.13)
where Π := Γ + Γ ∗ is the corresponding unperturbed operator and Pt := (1 + t 2 Π 2 )−1 and Qt := tΠ(1 + t 2 Π 2 )−1 . Lemma 4.9. We have, for all f ∈ N(Γ ), the estimate Θt (I − Pt )f C |||Q t |||op + 1 E∞ f . Proof. If f ∈ N(Γ ), then (I − Pt )f = tΓ Qt f ∈ N(Γ ), which shows that t 2 Γ ΓB∗0 Pt 0 (I − B
B
Pt )f = (I − Pt 0 )(I − Pt )f . To prove the estimate, we write −1 0 t E I − PtB0 tΓ Qt f = Q t Ef − Q t EPt f − Q t EQB Θt (I − Pt )f = Q t ΠB0 Γ Qt f, B B t Ef ||| |||Q t |||E∞ f . For the second term, where we recall that Qt 0 = tΠB0 Pt 0 . Clearly |||Q we write Θt := Qt E with principal part γt as in Definition 4.2 and estimate
t Pt − t Pt f ||| (Θ γt At f ||| |||Θ γt At )f + |||
t |||op + 1 E∞ f . E∞ f + γt C f E∞ f + |||Q
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t off E∞ in Lemma 4.4 and for the In the second step, we used for the first term that Θ second term we used Lemma 4.6. In the last step, we used Lemma 4.7 on the last term and that t |||op E∞ . Finally we note that t |||op |||Q |||Θ −1 0 Q t EQB t ΠB0 Γ Qt f E∞ |||Qt f ||| E∞ f , Γ is bounded by Lemma 2.21(i). since the Hodge projection ΠB−1 0
2
To estimate the last term in (4.13) we shall apply a local T (b) theorem as in [4,10]. We here give an alternative construction of test functions to those used in [10], more in the spirit of the original proof of the Kato square root problem [4]. Lemma 4.10. Let Γ be a nilpotent operator in H, which is a homogeneous, first order par tial differential operator with constant coefficients. Denote by ⊂ the image of the Γ linear functions u : Rn → under Γ , where we identify with the constant functions Rn → . Then for each w ∈ Γ with |w| = 1, each cube Q ⊂ Rn and each ε > 0, there exists a test w ∈ H such that f w ∈ R(Γ ), f w |Q|1/2 , function fQ,ε Q,ε Q,ε
∗ w
Γ f B0 Q,ε
1 |Q|1/2 εl(Q)
w and − fQ,ε − w ε 1/2 . Q
Proof. Let u(x) be a linear function such that w = Γ u and sup3Q |u(x)| l(Q), and define wQ := Γ (ηQ u), where ηQ is a smooth cutoff such that ηQ |2Q = 1, supp(ηQ ) ⊂ 3Q and ∇ηQ ∞ 1/ l(Q). It follows that wQ ∈ R(Γ ),
wQ |2Q = w,
supp wQ ⊂ 3Q
and wQ ∞ C.
B
w := P 0 w , where we write l = l(Q). Using Corollary 2.23, Next we define the test function fQ,ε Q εl
w |Q|1/2 and Γ ∗ f w 1 |Q|1/2 and since Γ commutes with P 0 , it follows that fQ,ε εl B0 Q,ε εl(Q) w it follows that fQ,ε ∈ R(Γ ). To verify the accretivity property, we make use of [10, Lemma 5.6] which shows that B
− f w − w = − I − P B0 wQ = − εlΓ QB0 wQ Q,ε εl εl Q
Q
Q
B0 2 1/4 2 1/4 B0 1/2 ε − Qεl wQ − εlΓ Qεl wQ ε 1/2 , Q
where we used that wQ ∈ R(Γ ) in the second step.
Q
2
Proof of Theorem 4.8. We have seen that it suffices to prove (4.11), and to bound each term in (4.13) for f ∈ N(Γ ). The first term is estimated by Lemma 4.9 and the second by Lemma 4.4.
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t |||op + 1)E∞ f , it suffices by To prove that the last term has estimate |||γt At f ||| (|||Q Lemma 4.6 to prove that t |||op + 1 E∞ . γt C |||Q To this end, we apply the local T (b) argument and stopping time argument in [10, Section 5.3]. n; Note that R(Γ ) ⊂ L (R operator, it follows that Γ ) and thus, since Γ is 2 an exact nilpotent N(Γ ) ⊂ L2 (Rn ; Γ ). Furthermore At f ∈ L2 (Rn ; Γ ) if f ∈ L2 (Rn ; Γ ). Thus it suffices to bound the Carleson norm of γt (x) seen as a linear operator γt (x) : Γ → . The conical de composition ν∈V Kν of the space of matrices performed in [10, Section 5.3], here decomposes the space L( Γ ; ) and for the fixed unit matrix ν ∈ L( Γ ; ) we choose w ∈ Γ and wˆ ∈ such that |w| ˆ = |w| = 1 and ν ∗ (w) ˆ = w. With the stopping time argument in [10, Section 5.3], using the new test functions fQw from Lemma 4.10, we obtain γt 2C Q,w∈
1 |Q| , |w|=1
sup Γ
γt (x)At f w (x) 2 dx dt , Q t
RQ
w for a small enough but fixed ε. To estimate the right-hand side we use (4.13) where fQw = fQ,ε with f replaced with the test function fQw , estimate the first two terms with Lemma 4.9 (which works since fQw ∈ R(Γ )) and Lemma 4.4, and obtain
γt At f w 2 dx dt Q t
RQ
γt At f w − Θt f w 2 dx dt + Q Q t
RQ
t |||op + 1 2 E2∞ |Q| + |||Q
Θt f w 2 dx dt Q t
RQ
Θt f w 2 dx dt . Q t
RQ
Using that ΓB∗0 fQw
1 1/2 l(Q) |Q|
we then get
Θt f w = Q t E tΓ PtB0 t ΓB∗ fQw E∞ t |Q|1/2 . Q 0 l(Q) This yields
Θt f w 2 dx dt |Q|E2 ∞ Q t
RQ
l(Q)
t l(Q)
2
dt |Q|E2∞ , t
0
which proves that RQ
γt At f w 2 dx dt |||Q t |||op + 1 2 E2∞ |Q|. Q t
2
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Remark 4.11. (i) Note that using the new test function from Lemma 4.10 simplifies the estimate of the term (γt At − Θt )fQw in the above proof as compared with the proof of [10, Proposition 5.9]. The useful new property of the test functions from Lemma 4.10 is that they belong to R(Γ ). (ii) In the proof of Theorem 4.8, we only estimate the Carleson norm of the restriction of the matrix γt to the subspace Γ as this suffices since we want to bound the quadratic norm to prove (4.11) and (4.12), we use Γ being of γt At f , and At f is always Γ -valued. However, either N md or N md ∗ . In these two cases, the space Γ is either the orthogonal complement of span{1, e0 } or span{e0,1,...,n , e1,...,n }, respectively. Note also that block matrices preserve these ⊥ spaces ⊥ Γ . It is seen that in these two cases γt = 0 on Γ , so actually we do get an estimate of the Carleson norm of the whole matrix γt . 4.2. Perturbation of vector coefficients In this section we assume that the unperturbed coefficients B0 are of the form B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I, i.e. B0 only acts non-trivially on the vector part, and that A0 is a matrix such that TB0 has quadratic estimates. Note that this hypothesis is true if A0 is either real symmetric, constant or of block form, by Theorem 3.12, Proposition 3.5 and Theorem 3.3, respectively. Our goal is to prove Theorem 1.1, by verifying the hypothesis of Lemma 4.1, as well as proving Theorem 1.3. We start by reformulating Lemma 4.1 in terms of e−t|TB0 | , acting only on functions f in one of B the two Hardy spaces EB±0 H, instead of Pt 0 . t off C, then for all f ∈ E + H we have estimates Theorem 4.12. If B0 is as above and if Q B0 Q t |||op + 1 E∞ f , t E(Ft − f ) C |||Q t t |||op + 1 E∞ f , Qt Ed Fs ds C |||Q
(4.14) (4.15)
0
where Ft := e−t|TB0 | f is the extension of f as in Lemma 2.49. The corresponding estimates for f ∈ EB−0 H also hold. Proof that Theorem 4.12 implies (4.9) and (4.10). We first note that it suffices to prove (4.9) and (4.10) for all f ∈ EB+0 H and all f ∈ EB−0 H since we have a Hardy space splitting H = EB+0 H ⊕ EB−0 H. We only consider f ∈ EB+0 H since the proof for f ∈ EB−0 H is similar. Now let f ∈ EB+0 H and use Proposition 2.31, which shows that if ψ ∈ Ψ (Sνo ), then B
|||ψt (TB0 )|||op |||Qt 0 |||op C. For the estimate (4.9), we write −1 0 t EtdQB t E dT −1 ψt (TB0 ), Q I − e−t|TB0 | + Q t = Qt E dTB0 B0 where ψ(z) = e−|z| − (1 + z2 )−1 . Note for the first term that TB−1 (I − e−t|TB0 | )f = TB−1 (f − 0 0 t −1 t −1 Ft ) = −TB0 0 ∂s Fs ds = 0 Fs ds. Therefore Theorem 4.12, the boundedness of dTB0 and Proposition 2.31 give the estimate (4.9).
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For the estimate (4.10), note that it suffices to estimate B0 0 t EtTB0 QB Q t = Qt E I − Pt since iMB0 TB0 = d + d ∗B0 . But this follows immediately from (4.14) since t E(f − Ft ) + Q t E I − PtB0 f = Q t Eψt (TB0 )f, Q with the same ψ as above.
2
t E(I −PtB0 )f directly. Remark 4.13. In the case when B0 is a constant matrix, we can estimate Q We prove that Q t |||op + 1 E∞ f t E I − PtB0 f |||Q t |||op E∞ . For the second term we write Θt := Q t EPtB0 . t E|||op |||Q as follows. Clearly |||Q Inserting a standard Fourier mollifier Pt , we write Θt = Θt (I − Pt ) + Θt Pt . Here |||Θt (I − B Pt )f ||| |||Pt 0 (I − Pt )f ||| f is easily verified using the Fourier transform. On the other hand, |||Θt Pt f ||| (Θt Pt − γt At )f + |||γt At f ||| f + γt C f , using Lemmas 4.4 and 4.6. However, since B0 is constant, we have that TB0 w = 0 if w is a constant function, and therefore t E PtB0 w = Q t Ew. γt w = Q t |||op + 1)E∞ . Therefore Lemma 4.7 shows that γt C (|||Q We now set up some notation for the proof of Theorem 4.12. We decompose the function Ft := e−t|TB0 | f , where f ∈ EB+0 H, as 1, + Ft2 + · · · + Ftn+1 , Ft = Ft0 + Ft1,⊥ + Ft
(4.16)
and similarly for f = limt→0+ Ft . It is important to note the special property that the normal component of the vector part Ft1,⊥ = Ft1,0 e0 has by Lemma 2.12: it satisfies the divergence form second order equation divt,x A0 (x)∇t,x F 1,0 = 0. Furthermore, we decompose the matrix A0 as A0 =
a⊥⊥ a⊥
a⊥ , a
in the splitting H = N − H ⊕ N + H. We view the components a⊥⊥ , a⊥ , a⊥ and a as operators, and write a⊥⊥ (f 1,0 e0 ) = (a00 f 1,0 )e0 , a⊥ f 1, = (a0 · f 1, )e0 and a⊥ (f 1,0 e0 ) = f 1,0 a0 , where a00 is a scalar and a0 and a0 are vectors.
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We introduce an auxiliary block matrix Bˆ 0 = I ⊕ Aˆ 0 ⊕ I ⊕ · · · ⊕ I,
a ˆ A0 = ⊥⊥ 0
0 . a
Lemma 4.14. Let Ft := e−t|TB0 | f , where f ∈ EB+0 H, so that (∂t + TB0 )Ft = 0. Then −1 −1 1, (∂t + TBˆ 0 )Ft = −i a00 a⊥ dFt1,⊥ − id ∗ˆ a a⊥ Ft1,⊥ = ∂t Ft1,⊥ + id ∗ˆ Ft . B0
B0
Proof. To prove the first identity, note that (∂t + TBˆ 0 )Ft = (TBˆ 0 − TB0 )Ft −1 = −i Bˆ 0 N + − N − Bˆ 0 (Bˆ 0 d + d ∗ Bˆ 0 )Ft −1 + i B0 N + − N − B0 (B0 d + d ∗ B0 )Ft −1 (B0 − Bˆ 0 )d + d ∗ (B0 − Bˆ 0 ) Ft = i Bˆ 0 N + − N − Bˆ 0 since Bˆ 0 N + − N − Bˆ 0 = B0 N + − N − B0 . The vector part of this matrix is more 0 a⊥ ⊕ 0 ⊕ · · · ⊕ 0, B0 − Bˆ 0 = 0 ⊕ a⊥ 0
−a⊥⊥ 0
0 a .
Further-
which shows that (B0 − Bˆ 0 )dFt = a⊥ dFt1,⊥
and d ∗ (B0 − Bˆ 0 )Ft = d ∗ a⊥ Ft1,⊥ ,
using the mapping properties of d and d ∗ from Remark 2.6. Thus −1 a⊥ dFt1,⊥ + d ∗ a⊥ Ft1,⊥ (∂t + TBˆ 0 )Ft = −ia⊥⊥ −1 −1 a⊥ dFt1,⊥ − id ∗Bˆ a a⊥ Ft1,⊥ , = −i a00 0
since Bˆ 0 N + − N − Bˆ 0 = −a⊥⊥ on normal vector fields. To prove the second identity, we multiply the Dirac equation (∂t + TB0 )Ft = 0 by B0 N + − − N B0 to obtain B0 N + − N − B0 ∂t Ft − i(B0 d + d ∗ B0 )Ft = 0. The normal component of the vector part on the left-hand side is 1, = 0. −a⊥⊥ ∂t Ft1,⊥ − i a⊥ dFt1,⊥ + d ∗ a⊥ Ft1,⊥ + a Ft Here we have used the expression for the vector part of B0 N + − N − B0 above for the first term. For the other terms we recall from Remark 2.6 that the vector part of dFt is dFt1,⊥ which is
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tangential, and that the normal vector part of d ∗ B0 Ft is d ∗ (B0 Ft )1, . Therefore, after multiplying −1 , we obtain the equation with a00 −1 −1 1, a⊥ dFt1,⊥ − id ∗ˆ a a⊥ Ft1,⊥ = ∂t Ft1,⊥ + id ∗ˆ Ft , −i a00 B0
which proves the lemma.
B0
2
As in Lemma 3.1 we note that since Bˆ 0 is a block matrix TBˆ 0 = ΠBˆ 0 = Γ + Bˆ 0−1 Γ ∗ Bˆ 0 ,
where Γ := −iN d.
To prove Theorem 4.12 we shall need the following corollary to Theorem 4.8. ˆ Corollary 4.15. Let B0 be the block matrix defined above, assume that Qt off C and let v ∈ L∞ (Rn ; ) be a function with norm v∞ C. Then for all f ∈ EB+0 H we have the estimates Q t |||op + 1 E∞ f , t Et 2 Γ Γ ∗ PtBˆ 0 Ft1,0 v |||Q ˆ
(4.17)
Q t EPtBˆ 0 Ft1,0 v |||Q t |||op + 1 E∞ f ,
(4.18)
B0
where Ft := e−t|TB0 | f and Ft1,0 = (Ft , e0 ). The corresponding estimates for f ∈ EB−0 H also hold. We defer the proof until the end of this section, and turn to a lemma in preparation for the proof of Theorem 4.12. Lemma 4.16. If f ∈ EB+0 H and Ft := e−t|TB0 | f , then t s ds (s∂s Fs ) |||t∂t Ft ||| f , t s
(4.19)
0
|||tdFt ||| |||t∂t Ft ||| f .
(4.20)
The corresponding estimates for f ∈ EB−0 H also hold. Proof. The proof of (4.19) uses Schur estimates. Applying Cauchy–Schwarz inequality, we estimate the square of the left-hand side by
2 t 2 ∞ ∞ t
t s s ds s dt ds
dt 2 ds (s∂s Fs ) s∂s Fs
t s t t s t s t 0
0
0
0
∞ ∞ = 0
where ψ(z) := ze−|z| .
s
0
2 s dt ds = ψt (TB0 )f f 2 , s∂s Fs 2 t t s
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To prove (4.20), we note that iMB0 TB0 = d + d ∗B0 and write P1B0 , P2B0 for the two Hodge projections corresponding to the splitting H = N(d) ⊕ N(d ∗B0 ). Note that these projections are bounded by Lemma 2.21(i). We get tdFt = iP1B0 MB0 (tTB0 Ft ) = iP1B0 MB0 ψt (TB0 )f, from which (4.20) follows.
2 Bˆ 0
Proof that Corollary 4.15 implies the estimate (4.14). By inserting I = Pt write the left-hand side in (4.14) as
Bˆ
+ (I − Pt 0 ) we
t EPtBˆ 0 (Ft − f ) + Q t E I − PtBˆ 0 Ft − Q t E I − PtBˆ 0 f =: X1 + X2 − X3 . Q Bˆ 0
t |||op + 1)E∞ f since I − Pt For X3 we get from Theorem 4.8 the estimate |||X3 ||| (|||Q
Bˆ t 2 Γ Γ ∗ˆ Pt 0 B0
Bˆ + t 2 Γ ∗ˆ Γ Pt 0 . B0
=
For the term X2 we use the first identity in Lemma 4.14 to obtain
t EQtBˆ 0 tT ˆ Ft X2 = Q B0 t EQtBˆ 0 (t∂t Ft ) − i Q t EQtBˆ 0 a −1 a⊥ tdFt1,⊥ = −Q 00 t E QtBˆ 0 td ∗ Ft1,0 a −1 a⊥ e0 =: −X4 − iX5 − iX6 . − iQ ˆ B0
We have the estimate |||X4 ||| E∞ |||t∂t Ft ||| E∞ f . For X5 , we see from Remark 2.6 that dFt1,⊥ is the vector part of dFt . Thus |||X5 ||| E∞ |||tdFt ||| E∞ f by (4.20). Bˆ
Bˆ
Bˆ
To handle the term X6 we note that Qt 0 td ∗ˆ = −Qt 0 tΓ ∗ˆ N = −t 2 Γ Γ ∗ˆ Pt 0 N using ReB0
B0
B0
−1 a⊥ e0 , the estimate |||X6 ||| mark 3.2. Thus we obtain from Corollary 4.15, with v = a (|||Qt |||op + 1)E∞ f . It remains to estimate the term X1 . To handle this, we separate the normal vector part as Bˆ 0
t EPt X1 = Q
t EPtBˆ 0 f 1,0 e0 + Q t EPtBˆ 0 (Gt − g) =: X7 − X8 + X9 , Ft1,0 e0 − Q
where Gt := Ft − Ft1,⊥ and g = f − f 1,⊥ . From Corollary 4.15, with v = e0 , we get the es t |||op + 1)E∞ f . For the term X8 , we write PtBˆ 0 = I − t 2 Γ Γ ∗ PtBˆ 0 − timate |||X7 ||| (|||Q ˆ B0
Bˆ t 2 Γ ∗ˆ Γ Pt 0 . B0
t |||op + 1)E∞ f . For From Theorem 4.8 we obtain the estimate |||X8 ||| (|||Q the term X9 , we integrate by parts to obtain t EPtBˆ 0 (t∂t Gt ) − Q t EPtBˆ 0 X9 = Q
t s∂s2 Gs 0
ds =: X10 − X11 .
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We have the estimate |||X10 ||| E∞ |||t∂t Ft ||| E∞ f . For the term X11 we apply ∂t to the last expression for (∂t + TBˆ 0 )Ft in Lemma 4.14 and get ∂s2 Gs + ∂s TBˆ 0 Fs = id ∗Bˆ ∂s Fs1, .
(4.21)
0
Thus X11
t t ds Bˆ 0 Bˆ 0 ∗ s s ds 1, t E Pt tT ˆ t E Pt td (s∂s Fs ) s∂s Fs = −Q + iQ B0 Bˆ 0 t s t s 0
0
=: −X12 + iX13 . Bˆ
Bˆ 0
Both |||X12 ||| and |||X13 ||| can now be estimated with E∞ f by (4.19) since Pt 0 tTBˆ 0 = Qt Bˆ
and Pt 0 td ∗ˆ are uniformly bounded by Corollary 2.23. This proves the estimate (4.14). B0
2
Proof that Corollary 4.15 implies the estimate (4.15). We write the left-hand side in (4.15), using integration by parts, as t Ed Q
t
t EtdFt − Q t Ed Fs ds = Q
0
t s∂s Fs ds =: X1 − X2 . 0 Bˆ 0
t E(tdFt )||| E∞ f by (4.20). For X2 , we write I = Pt For X1 we have |||Q and get t E tdPtBˆ 0 X2 = Q
t
s ds Bˆ 0 (s∂s Fs ) + Q t EtdQt t s
0
Bˆ
+ (I − Pt 0 )
t s∂s TBˆ 0 Fs ds =: X3 + X4 . 0
Bˆ
Using that tdPt 0 C by Corollary 2.23, and (4.19) shows that |||X3 ||| E∞ f . To handle X4 , we use the identity (4.21), which gives t EtdQtBˆ 0 X4 = Q
t t 2 ∗ 1, − s∂s Gs ds + i sd ˆ ∂s Fs ds =: −X5 + iX6 . B0
0 Bˆ
0 Bˆ
Bˆ
Bˆ
For X6 we note that dQt 0 d ∗ˆ = N dQt 0 N d ∗ˆ = −Γ Qt 0 Γ ∗ˆ = −Γ 2 Qt 0 = 0 using ReB0 B0 B0 mark 3.2, and thus X6 = 0. To handle X5 , we rewrite this with an integration by parts as t E tdQtBˆ 0 (t∂t Gt ) − Q t EtdQtBˆ 0 Gt + Q t E tdQtBˆ 0 g =: X6 − X7 + X8 , X5 = Q Bˆ
where g = f − f 1,⊥ . Using Lemma 4.16 and that tdQt 0 C by Corollary 2.23, we get Bˆ
Bˆ
|||X6 ||| E∞ f . For X8 , we note that tdQt 0 = iN t 2 Γ Γ ∗ˆ Pt 0 . Thus we can apply TheoB0 t |||op + 1)E∞ f . rem 4.8 to obtain |||X8 ||| (|||Q
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We now write Gt = Ft − Ft1,0 e0 and get Bˆ 0
Bˆ
Ft1,0 e0 =: X9 − X10 .
t EtdPt 0 (tT ˆ Ft ) − Q t EtdQt X7 = Q B0 Bˆ
Bˆ
Again noting that tdQt 0 = iN t 2 Γ Γ ∗ˆ Pt 0 we obtain from Corollary 4.15, with v = e0 , the B0 t |||op + 1)E∞ f . The term X9 remains, on which we use the first estimate |||X10 ||| (|||Q identity in Lemma 4.14 to obtain t EtdPtBˆ 0 (t∂t Ft ) − i Q t E tdPtBˆ 0 a −1 a⊥ tdFt1,⊥ X9 = −Q 00 t E tdPtBˆ 0 td ∗ Ft1,0 a −1 a⊥ e0 =: −X11 − iX12 − iX13 . − iQ ˆ B0
Bˆ
Using that tdPt 0 C by Corollary 2.23, shows that |||X11 ||| E∞ f . For X12 we see from Remark 2.6 that dFt1,⊥ is the vector part of dFt . Thus |||X12 ||| E∞ |||tdFt ||| E∞ f Bˆ
Bˆ
Bˆ
by (4.20). To handle the final term X13 we note that dPt 0 d ∗ˆ = dN Pt 0 N d ∗ˆ = −Γ Pt 0 Γ ∗ˆ = B0
Bˆ −Γ Γ ∗ˆ Pt 0 B0
B0
using Remark 3.2. Thus we obtain from Corollary 4.15, with v = t |||op + 1)E∞ f . This proves the estimate (4.15). 2 estimate |||X13 ||| (|||Q
B0 −1 a a⊥ e0 , the
Proof of Corollary 4.15. Let f ∈ EB+0 H and consider the functions Ft = e−t|TB0 | f and Gt :=
Pt 0 f . We note that Ft − Gt = ψt (TB0 )f , where ψ(z) = e−|z| − (1 + z2 )−1 ∈ Ψ (Sνo ). Thus t |||op + 1)E∞ f , i = 1, 2, for the two it suffices to prove the estimate |||Θti (Gt1,0 )||| (|||Q families of operators B
t Et 2 Γ Γ ∗ PtBˆ 0 Mv , Θt1 := Q ˆ B0
Bˆ
t EPt 0 Mv , Θt2 := Q
(4.22) (4.23)
where Mv denotes the multiplication operator Mv (f ) := vf . To this end, we write Θti Gt1,0 = Θti − γti At Gt1,0 + γti At Gt1,0 , where γti (x) is the principal part of the operator family Θti as in Definition 4.2. Using the principal part approximation Lemma 4.4 and Carleson’s lemma 4.6 we obtain the estimate i 1,0 i 1,0 i Θ Gt = Θ t∇ Gt + γ N∗ At Gt1,0 . t t off t C By Proposition 2.25 and Lemma 2.26, we have Θti off E∞ . Furthermore, by Lemma 4.7 and Theorem 4.8 we have
i
γ Θ i + Θ i |||Q t |||op + 1 E∞ . t C t op t off t E(I − t 2 Γ Γ ∗ PtBˆ 0 − t 2 Γ ∗ Γ PtBˆ 0 )Mv . For Θt2 we have used that Θt2 = Q ˆ ˆ B0
B0
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443
To bound |||t∇(Gt1,0 )|||, we note that the vector part of dGt is dGt1,⊥ = imd e0 Gt1,0 = −id Gt1,0 = −i∇ Gt1,0 . Thus, similar to the proof of (4.20), it follows that 1,0 t∇ Gt |||tdGt ||| tTB PtB0 f f . 0 Finally, to bound N∗ (At Gt1,0 ) we write Gt = 12 (Ht + H−t ), where Ht := (1 + itTB0 )−1 f . We now observe that the divergence form equation and estimate for Ht1,0 in Lemma 2.57 in fact holds for all f ∈ H, by inspection of the proof. We get 1,0 1,0 N∗ At Ht (x) = sup − Ht (z) dz |y−x|
sup t>0
Q(y,t)
1/q q 1/p − Ht1,0 (z) dz M |f |p (x) ,
B(x,r0 t)
where Q(y, t) denotes the dyadic cube Q ∈ t which contains y. Using the boundedness of the Hardy–Littlewood maximal function on L2/p (Rn ), we obtain
N∗ At Ht1,0 M |f |p 1/p = M |f |p 1/p |f |p 1/p = f 2 . 2/p 2/p 2 2 1,0 ) f , and thus N∗ (At Gt1,0 ) f . We have A similar argument shows that N∗ (At H−t t |||op + 1)E∞ f , and therefore Corollary 4.15. 2 proved that |||Θti (Gt1,0 )||| (|||Q
4.3. Proof of main theorems We are now in position to prove Theorems 1.1, 1.3 and 1.4 stated in the introduction. Proof of Theorem 1.4. Given a perturbation B k ∈ L∞ (Rn ; L( k )) of the unperturbed coeffi cients B0k ∈ L∞ (Rn ; L( k )), we introduce B := I ⊕ · · · ⊕ B k ⊕ · · · ⊕ I ∈ L∞ (Rn ; L( )) and B0 := I ⊕ · · · ⊕ B0k ⊕ · · · ⊕ I ∈ L∞ (Rn ; L( )) acting in all H. By Theorem 3.3(i), TB0 satisfies quadratic estimates and by Lemma 4.1 and Theorem 4.8 there exists ε > 0 such that we have quadratic estimates B Q f ≈ f , whenever B − B0 ∞ ε, f ∈ H. t Therefore, by Propositions 2.32 and 2.42 we have when B − B0 ∞ ε/2 well-defined and bounded operators EB = sgn(TB ) which depend Lipschitz continuously on B, i.e. EB2 − EB1 CB2 − B1 ∞ ,
when Bi − B0 < ε/2, i = 1, 2.
To prove that (Tr-B k α ± ) is well posed, note that by Lemma 2.51 it suffices to show that λ − EB NB is invertible since then in particular λ − EB k NB k = (λ − EB NB )|Hˆ k is invertible. Here the B
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spectral parameter is λ = (α + 1)/(α − 1) and α := α + /α − . By Theorem 3.3(ii), the unperturbed operator λ − EB0 NB0 is invertible when λ2 + 1 = 0. For the perturbed operator we write
1 (λ − NB0 EB0 )(EB0 NB0 − EB NB ) . λ − EB NB = (λ − EB0 NB0 ) I + 2 λ +1 Here EB0 NB0 − EB NB B − B0 ∞ since we clearly have Lipschitz continuity NB2 − NB1 B2 − B1 ∞ . It follows that λ − EB NB is invertible when B − B0 ∞ C|λ2 + 1|. Under the assumption B − B0 ∞ < ε/2 with ε small we can replace λ2 + 1 = 2(α 2 + 1)/ (α − 1)2 with α 2 + 1. This proves that for each boundary function g there exists a unique solution f = f + + f − satisfying the jump conditions in (Tr-B k α ± ) and f + + f − ≈ f ≈ g. The norm estimates for F ± in Theorem 1.4 now follows from Lemma 2.49. A formula for the solution is −1 F ± (t, x) = 2e∓t|TB | EB± α + + α − EB − α + − α − NB g(x). This completes the proof of Theorem 1.4.
2
Before turning to the proofs of Theorems 1.1 and 1.3, we note some corollaries of Theorem 1.4. First, if we let k = 1 in Theorem 1.4, then it proves that the following Neumannregularity transmission problem is well posed for small L∞ perturbations A of a block matrix A0 . Transmission problem (Tr-Aα ± ). Let α ± ∈ C be given jump parameters. Given scalar functions ψ, φ : Rn → C with ∇x ψ ∈ L2 (Rn ; Cn ) and φ ∈ L2 (Rn ; C), find gradient vector fields ± 1 n n+1 )) and F ± satisfies (1.2) F ± (t, x) = ∇t,x U ± (t, x) in Rn+1 ± such that Ft ∈ C (R± ; L2 (R ; C ± ± for ±t > 0, and furthermore limt→±∞ Ft = 0 and limt→0± Ft = f ± in L2 norm, where the traces f ± satisfy the jump conditions ⎧ − + + − ⎨ α ∇x U (0, x) − α ∇x U (0, x) = ∇x ψ(x), + − ∂U ∂U ⎩ α+ (0, x) − α − (0, x) = φ(x), ∂νA ∂νA where ∇x U ± (0, x) = f± (x) and
∂U ± ∂νA
= (Af ± , e0 ) denotes the conormal derivative.
Secondly, Theorem 1.4 give perturbation results for the following boundary value problems for k-vector fields. k , find a k-vector field F (t, x) in Normal BVP (Nor-B k ). Given a k-vector field g ∈ Hˆ B k n+1 1 n R+ such that Ft ∈ C (R+ ; L2 (R ; )) and F satisfies (1.8) for t > 0, and furthermore limt→∞ Ft = 0 and limt→0 Ft = f in L2 norm, where f satisfies e0 B k f = e0 B k g on Rn = ∂Rn+1 + . k , find a k-vector field F (t, x) in Tangential BVP (Tan-B k ). Given a k-vector field g ∈ Hˆ B k n+1 R+ such that Ft ∈ C 1 (R+ ; L2 (Rn ; )) and F satisfies (1.8) for t > 0, and furthermore limt→∞ Ft = 0 and limt→0 Ft = f in L2 norm, where f satisfies
e0 ∧ f = e0 ∧ g
on Rn = ∂Rn+1 + .
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Corollary 4.17. Let B0k = B0k (x) ∈ L∞ (Rn ; L( k )) be accretive and assume that B0k is a block matrix. Then there exists ε > 0 depending only on the constants B0k ∞ and κB k and dimen0 sion n, such that if B k ∈ L∞ (Rn ; L( k )) satisfies B k − B0k ∞ < ε, then Normal and Tangential boundary value problems (Nor-B k ) and (Tan-B k ) above are well posed. − + k ± Proof. (i) For (Nor-B k ) in Rn+1 + , we let α = 0 and α = 1 in (Tr-B α ). Then we obtain two decoupled jump conditions
−e0 ∧ f − = e0 ∧ g, e0 B k f + = e0 B k g .
Discarding the solution F − , we obtain a unique solution F = F + to (Nor-B k ). − + k ± (ii) For (Tan-B k ) in Rn+1 + , we let α = 1 and α = 0 in (Tr-B α ). Then we obtain two decoupled jump conditions
e0 ∧ f + = e0 ∧ g, −e0 B k f − = e0 B k g .
Discarding the solution F − , we obtain a unique solution F = F + to (Tan-B k ).
2
Note that well-posedness of (Neu-A) and (Reg-A) for Theorem 1.1(b) is the special case k = 1 of Corollary 4.17, as well as the special cases (α + , α − ) = (1, 0) and (0, 1), respectively, of (Tr-Aα ± ). Proof of Theorem 1.3. Let A0 be such that TB0 has quadratic estimates in H, where B0 = I ⊕ A0 ⊕ I ⊕ · · · ⊕ I . Thus by Lemma 4.1 and Theorem 4.12 there exists ε > 0 such that we have quadratic estimates B Q f ≈ f , t
whenever B − B0 ∞ < ε, f ∈ H.
Therefore, by Proposition 2.32, we have when B − B0 ∞ < ε well defined and bounded operators EB = sgn(TB ). With Lemma 2.46, these restricts to bounded operators EA in Hˆ 1 . In particu± f and f ≈ f + + f − . lar f ∈ Hˆ 1 can be decomposed as f = f + + f − , where f ± := EA Moreover, Lemma 2.49 and Proposition 2.56 proves the stated norm equivalences for F ± . 2 Proof of Theorem 1.1. Given a perturbation A ∈ L∞ (Rn ; L( 1 )) of the unperturbed coeffi cients A0 ∈ L∞ (Rn ; L( 1 )), which we assume are either of block form, real symmetric or conn ; L( )) and B := I ⊕ A ⊕ I ⊕ · · · ⊕ I ∈ (R stant, we introduce B := I ⊕ A ⊕ I ⊕ · · · ⊕ I ∈ L ∞ 0 0 L∞ (Rn ; L( )) acting in all H. That TB0 satisfies quadratic estimates follows from Theorem 3.3(i), Theorem 3.12 and Proposition 3.5, respectively. Theorem 1.3 now shows that we have quadratic estimates for TB when B − B0 ∞ < ε. In case A0 is a block matrix, we note that this result follows already from Theorem 4.8. By Propositions 2.32 and 2.42 we have when B − B0 ∞ < ε/2 well-defined and bounded operators EB = sgn(TB ) which depend Lipschitz continuously on B, so that EB2 − EB1 CB2 − B1 ∞ ,
when Bi − B0 < ε/2, i = 1, 2.
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To prove that (Neu-A) and (Reg-A) are well posed, note that by Lemma 2.53 it suffices to show that I ± EA NA : Hˆ 1 → Hˆ 1 are invertible, where EA = EB |Hˆ 1 and NA = NB |Hˆ 1 . By Theorems 3.3(ii), 3.12 and 3.6, respectively, the unperturbed operators I ± EA0 NA0 : Hˆ 1 → Hˆ 1 are invertible. For the perturbed operator we write I ± EA NA = (I ± EA0 NA0 ) I ± (I ± EA0 NA0 )−1 (EA NA − EA0 NA0 ) . Here EA NA − EA0 NA0 A − A0 ∞ since we clearly have Lipschitz continuity NA2 − NA1 A2 − A1 ∞ . It follows that I ± EA NA are invertible when A − A0 ∞ < ε . The well-posedness of (Neu⊥ -A) is a consequence of Proposition 2.52, since our hypothesis is stable when taking adjoints A → A∗ . Alternatively, we can replace NA0 with N above, proving that I ± EA N is an isomorphism, using Theorem 3.3(ii), Remarks 3.13 and 3.7. This proves that for (Neu-A), (Reg-A) and (Neu⊥ -A) and each boundary function g (being φ and ∇x ψ, + ˆ1 respectively), there exists a unique solution f = EA H satisfying the boundary condition and f ≈ g. Lemma 2.49 and Proposition 2.56 proves that the stated norms of f and F (t, x) := (e−t|TA | f )(x) are equivalent. The well-posedness of (Dir-A), as well as the first three norm estimates, follows from Lemma 2.55 and (Neu⊥ -A). To show that |||t∇x Ut ||| ≈ |||t∂t Ut |||, we consider the gradient vector field Gt = ∇t,x Ut as in the proof of the lemma. From (Reg-A) and (Neu⊥ -A), it follows that for all t > 0, we have ∂t Ut = N − Gt ≈ N + Gt = ∇x Ut , from which the square ∗ (U ), we consider the vector function estimate for ∇x Ut follows. To show that u ≈ N field Ft of conjugate functions from the proof of the lemma. Proposition 2.56 shows that ∗ (F ) N ∗ (U ). Moreover, the proof of the reverse estimate u N ∗ (U ) u ≈ f ≈ N is similar to the proof of f N∗ (F ) in Proposition 2.56, using the uniform boundedness of Pt . Finally we note that the solution operators for (Neu-A), (Neu⊥ -A), (Reg-A) and (Dir-A) are −1 φe0 , Ft = 2e−t|TA | (EA − NA )−1 a00 Ft = 2e−t|TA | (EA + N )−1 (∇x ψ),
Ft = 2e−t|TA | (EA − N )−1 (φe0 ), Ut = 2 e−t|TA | (EA − N )−1 (ue0 ), e0 .
The Lipschitz continuity of u → U is a consequence of the corresponding result for (Neu⊥ -A). For the norms supt>0 Ft and |||t∂t Ft |||, Lipschitz continuity for the solution operators follows ∗ (F )2 . from Proposition 2.42. It remains to show Lipschitz continuity for the norm F X = N To this end, we consider + f (x) → FAz = e−t|TAz | EA f (x) : Hˆ 1 → X z and the truncations f (x) → FAk z (t, x) = χk (t)FAz (t, x), where χk denotes the characteristic function for (1/k, k) as in the proof of Lemma 2.41(iii). We claim that it suffices to show that, for each fixed k, the operator f (x) → FAk z (t, x) : Hˆ 1 → X depends holomorphically on z. Indeed, using Schwarz’ lemma as in Proposition 2.42, we obtain the Lipschitz estimate
k
F − F k CA2 − A1 ∞ f 2 , A2 A1 X
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447
uniformly for all k, since F k X F X Cf by Proposition 2.56. Furthermore, by the monotone convergence theorem we have
k
F − F k ! FA − FA X , 2 1 A2 A1 X
k → ∞,
so the desired Lipschitz continuity follows after taking limits. To prove that f (x) → FAk z (t, x) is holomorphic, we note that
k 2
F X
sup Rn
t>0
k
|s−t|
Rn
1/k |y−x|
k F (s, y) 2 ds dy dx s n+1
k F (s, y) 2 ds dy s n+1
k dx ≈
k 2
F ds, s
2
1/k
for fixed k, where in the last step we use that 1/k s −1 k. Since f (x) → F k (t, x) : Hˆ 1 → L2 (Rn × (1/k, k)) is holomorphic by Lemma 2.41(ii) and the embedding L2 (Rn × (1/k, k)) → X is continuous and independent of z, it follows that f (x) → F k (t, x) : Hˆ 1 → X is holomorphic for each fixed k. This completes the proof of Theorem 1.1. 2 Acknowledgments This work was mainly conducted at the University of Orsay while Axelsson was post-doctoral fellow supported by the French Ministry of Teaching and Research (Ministère de l’Enseignement et de la Recherche) and then by the European IHP network “Harmonic Analysis and Related Problems” (contract HPRN-CT-2001-00273-HARP). Axelsson gratefully acknowledges the University of Orsay for providing facilities to work. Hofmann also thanks the University of Orsay for partial support of a one week visit that jump started this project. References [1] D. Albrecht, X. Duong, A. Mc Intosh, Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry, Part III, Canberra, 1995, in: Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., Canberra, 1996, pp. 77–136. [2] M. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann, S. Kim, Analyticity of layer potentials and L2 solvability of boundary value problems for divergence form elliptic equations with complex L∞ coefficients, preprint. [3] P. Auscher, A. Mc Intosh, A. Nahmod, The square root problem of Kato in one dimension, and first order elliptic systems, Indiana Univ. Math. J. 46 (3) (1997) 659–695. [4] P. Auscher, S. Hofmann, M. Lacey, A. Mc Intosh, P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn , Ann. of Math. (2) 156 (2) (2002) 633–654. [5] A. Axelsson, Oblique and normal transmission problems for Dirac operators with strongly Lipschitz interfaces, Comm. Partial Differential Equations 28 (11–12) (2003) 1911–1941. [6] A. Axelsson, Transmission problems for Dirac’s and Maxwell’s equations with Lipschitz interfaces, PhD thesis, The Australian National University, 2003. [7] A. Axelsson, Transmission problems and boundary operator algebras, Integral Equations Operator Theory 50 (2) (2004) 147–164. [8] A. Axelsson, Transmission problems for Maxwell’s equations with weakly Lipschitz interfaces, Math. Methods Appl. Sci. 29 (6) (2006) 665–714.
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Journal of Functional Analysis 255 (2008) 449–470 www.elsevier.com/locate/jfa
Polyhedral norms on non-separable Banach spaces V.P. Fonf a,1 , A.J. Pallares b,2 , R.J. Smith c,∗ , S. Troyanski b,2,3 a Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel b Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain c Department of Pure Mathematics, University of Cambridge, United Kingdom
Received 8 January 2008; accepted 2 March 2008 Available online 28 April 2008 Communicated by N. Kalton
Abstract We prove the existence of equivalent polyhedral norms on a number of classes of non-separable spaces, the majority of which being of the form C(K). In particular, we obtain a complete characterization of those trees T , such that C0 (T ) admits an equivalent polyhedral norm. © 2008 Elsevier Inc. All rights reserved. Keywords: Polyhedral norm; Scattered compact; Tree; Orlicz space
1. Introduction A Banach space X is called polyhedral if the unit ball of each of its finite-dimensional subspaces is a polytope [17]. The simplest example of an infinite-dimensional polyhedral space is c0 in the natural norm. No infinite-dimensional dual space is polyhedral [19], or even isomorphic to a polyhedral space [7]. Clearly, polyhedrality is an isometric property, i.e. it can be gained or
* Corresponding author.
E-mail addresses: [email protected] (V.P. Fonf), [email protected] (A.J. Pallares), [email protected] (R.J. Smith), [email protected] (S. Troyanski). 1 Supported by the Spanish government, grant MEC SAB 2005-016, and by the Israel Science Foundation, grant 139/03. 2 Supported by MCYT MTM 2005-08379 and Fundación Séneca 00690/PI/04 CARM. 3 Supported by the Institute of Mathematics and Informatics Bulgarian Academy of Sciences and grant MM-1401/2004 of Bulgarian NSF. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.001
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lost by the introduction of an equivalent norm. In the study of polyhedral spaces the notion of a boundary plays a very important role. Definition 1. Let X be a Banach space. A subset B ⊂ SX∗ is called a boundary of X if for any x ∈ X there is f ∈ B with f (x) = x. Clearly, for an infinite-dimensional Banach space a boundary must be infinite, i.e. at least countable, and it turns out that this is the case in polyhedral spaces (see Theorem 2 below). By the Krein–Milman theorem, the set ext BX∗ is a boundary. In general, a boundary need not contain all extreme points, but it must contain the w∗ -exposed points. Recall that a functional f0 ∈ SX∗ is a w∗ -exposed point of BX∗ if there is a x0 ∈ SX such that f0 (x0 ) = 1 > f (x0 ) for each f ∈ BX∗ , f = f0 , and that f0 is called w∗ -strongly exposed if lim fn − f0 = 0 whenever fn ∈ BX∗ and lim fn (x0 ) = 1. Thus, each boundary B contains the set B0 = w∗ -str exp BX∗ of all the w∗ -strongly exposed points of BX∗ . We summarize some basic properties of polyhedral spaces in the following theorem. Theorem 2. (See [5,7].) (1) Let X be a polyhedral Banach space with the density character w. Then the set B0 = w∗ -str exp BX∗ is a boundary of X with |B0 | = w. B0 is a minimal boundary of X, i.e. it is contained in any other boundary. In particular if X is separable then B0 is countable. (2) If a Banach space X has a countable boundary then X ∗ is separable and X admits an equivalent polyhedral norm, the boundary of which has (∗) (see Definition 4 below). (3) Every polyhedral space is a c0 -saturated, Asplund space. For a simpler proof of the first part of Theorem 2 see [10]. A nice alternative proof of the first part of Theorem 2 is given in [22]. Note that Theorem 2 gives a characterization of the polyhedral spaces in the separable case. Separable polyhedral spaces have many interesting properties, as well as some other characterizations (see [2,5,9–12,18,22]). Unfortunately, not much is known about non-separable polyhedral spaces. Besides general properties of polyhedral spaces described in Theorem 2, we recall that the space C([1, α]) of all continuous functions on the segment of ordinals [1, α], where α is arbitrary, admits an equivalent polyhedral norm [6]. Also, M. Jimenez-Sevilla informed us that, under CH, the Kunen compact K provides an example of a non-separable Asplund space C(K) that admits no polyhedral renorming. Indeed, in her paper with J.P. Moreno [16, Proposition 4.3] they proved that for any equivalent norm on C(K), the set of w∗ -denting points of the dual unit ball is countable. Thus, by Theorem 2, part (1), C(K) admits no polyhedral norms. The purpose of this paper is to identify some new classes of non-separable spaces which admit a polyhedral norm. Most of these spaces take the form C(K), where K is compact. From Theorem 2, part (3), it follows that K must be scattered. In Section 2, we study Talagrand operators, one of our main tools in polyhedral renorming, and prove that C(K) admits such a renorming if K is a finite product of compact ordinal segments or σ -discrete spaces. An important subclass of scattered compact sets is the class of trees. Due to the fundamental paper [15], the renorming theory of spaces C0 (T ), where T is a tree, is very rich, and we use some of its results in our paper. In Section 3 we prove one of our main results, Theorem 10, which states that the space C0 (T ) has a polyhedral renorming if and only if it admits a Talagrand operator. Surprisingly enough, according to [15], this happens if and only if C0 (T ) admits a Fréchet differentiable norm, and
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if and only if C0 (T ) admits a LUR norm. By using Theorem 10, we obtain ZFC examples of scattered compacts K with the property that C(K) does not admit a polyhedral renorming. In Section 4 we establish a sufficient condition for a Banach space with uncountable unconditional basis to have a polyhedral renorming. Then we apply this result to non-separable Orlicz spaces. In the remainder of this introduction, we prove two results used throughout the paper, and end with a discussion of a couple of open problems. Let us first note that the canonical supremum norm in C(K) is not polyhedral if K is infinite. This follows from two well-known facts: C(K) contains an isometric copy of (c, · ∞ ) and (c, · ∞ ) is not polyhedral. The following necessity condition is a generalization of the latter fact (take Tn : (c, · ∞ ) → (c, · ∞ ), n = 1, 2, . . . , defined by (Tn y)i = yi if i n, and zero, otherwise). Lemma 3. Let Y be a polyhedral Banach space. Suppose that there exists a family L of linear maps T : Y → Y such that y = sup T y: T ∈ L for all y ∈ Y. (1.1) Then Y ∗ is the norm closed linear span of T ∈L T ∗ Y ∗ . Proof. From (1.1), it follows that T 1 for any T ∈ L. Put B = w∗ -cl T ∗ (BY ∗ ). T ∈L
By using (1.1) and the fact that B is w∗ -compact it is easy to see that B contains a boundary. -str exp BY ∗ ⊂ B. By the definition of w∗ -strongly exposed points, Now by Theorem 2, part 1, w∗ ∗ we have w -str exp BY ∗ ⊂ cl T ∈L T ∗ (BY ∗ ). Since BY ∗ = cl co w∗ -str exp BY ∗ by [10,22], the conclusion follows. 2 Next, we present our main tool used in constructing polyhedral renormings. Definition 4. We say that a set B ⊂ BX∗ has property (∗) if, given any w∗ -limit point f0 of B (i.e. any w∗ -neighborhood of f0 contains infinitely many points of B), we have f0 (x) < 1 whenever x ∈ SX . If a Banach space in some norm has a boundary with (∗), we simply say that the norm itself has (∗). It is not difficult to see that if B has (∗) and is 1-norming then B contains a boundary. The next proposition demonstrates the relevance of property (∗). We give a proof for the sake of completeness. Proposition 5. (See [12].) Assume that a Banach space X admits a 1-norming subset B ⊂ BX∗ with (∗). Then X is a polyhedral space. Proof. Let E ⊂ X be finite-dimensional. Evidently x = supf ∈B f (x). Assume, for a contradiction, that whenever F ⊂ B is finite, there is x ∈ E such that maxf ∈F f (x) < x. In this ∞ way we obtain a sequence {fi }∞ i=1 ⊂ B of distinct points and {xi }i=1 ⊂ SE such that fi (xi ) → 1. By compactness and by considering subsequences if necessary, we can find x ∈ SE such that xi − x → 0 and a w∗ -accumulation point f ∈ BX∗ of {fi }∞ i=1 . It is clear that f (x) = 1, contradicting property (∗) of B. 2
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To finish, we present two open problems. It was proved in [4,5] that a separable Banach space is isomorphically polyhedral if and only if it has a boundary with (∗), with respect to some equivalent norm. Given that all the polyhedral norms constructed in this paper have boundaries with (∗), it is natural to ask whether this equivalence holds for non-separable Banach spaces also. We mention another property, related to polyhedrality, which has attracted attention in, for example, the theory of smooth renormings. A norm · on X is said to depend locally on finitely many coordinates if, given any non-zero x ∈ X, there exists an open set U containing x, a function Φ and finitely many functionals f1 , . . . , fn ∈ X ∗ such that y = Φ(f1 (x), . . . , fn (x)) whenever y ∈ U . By modifying the proof of Proposition 5, it is clear that if · has (∗) then it depends locally on finitely many coordinates (with Φ = max). In [9], it was shown that a separable space X is isomorphically polyhedral if and only if it has an equivalent norm that depends locally on finitely many coordinates. Again, we can ask whether this result holds in the non-separable case. 2. Scattered compact spaces By Theorem 2, part (3), if, for a compact space K, the space C(K) admits an equivalent polyhedral norm, then C(K) is an Asplund space. It is well known that C(K) is Asplund if and only if K is scattered, so when investigating the problem of polyhedral renormings of C(K) spaces, we only need to consider scattered K. In this section, we develop some general techniques and apply them to two important classes of scattered, compact spaces. First, we show that if K is a compact ordinal segment then C(K) admits a polyhedral renorming, thus providing another proof of the result in [6] stated in the introduction. Second, we show that if K is a σ -discrete space then the same conclusion holds. We go on to prove that the same is also true if K is a finite product of spaces of either class. While we do encounter tree spaces in this section, a fuller treatment is deferred until the following section. We begin stating a natural generalization of Haydon’s definition of Talagrand operators [14], which have been used to considerable effect in the theory of smooth renormings on C(K) spaces. According to [14], given the space C0 (L), with L locally compact, a linear bounded operator T : C0 (L) → c0 (L × M) is a Talagrand operator if, for any x ∈ C0 (L), there is a pair (t, m) ∈ L × M with x(t) = x∞ and (T x)(t, m) = 0. Definition 6. Let X be a Banach space and M be a non-empty set. A linear, bounded operator T : X → c0 (SX∗ × M) is called a Talagrand operator if, for any x ∈ X, there is a pair (f, m) ∈ SX∗ × M with f (x) = x and (T x)(f, m) = 0. Clearly, if an operator satisfies Haydon’s definition then it is also a Talagrand operator in our sense. Some of the general theory of these operators is developed in [21]. The canonical example of a Talagrand operator T is defined on C([0, α]), where [0, α] is a compact ordinal segment. We set (see [14]) x(ξ ) − x(ξ + 1), ξ < α, (T x)(ξ ) = x(α), ξ = α. Note that a similar construction was used in [6]. Here, the set M is a singleton so we can safely ignore it. If x ∈ C([0, α]) is non-zero then (T x)(ξ ) = 0, where ξ is maximal, subject to the condition that x∞ = |x(ξ )|. To see that T maps into c0 ([0, α]), observe that by the
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Stone–Weierstrass theorem, C([0, α]) is the closed linear span of the set of indicator functions {1[0,ξ ] : ξ α}, so it is enough to check that T maps the set of these indicator functions into c0 ([0, α]). This is clear however, as we can see by inspection that the support of T 1[0,ξ ] is the singleton {ξ }. Proposition 7. Assume that a Banach space X admits a Talagrand operator. Then, for any ε > 0, X admits an ε-equivalent polyhedral norm with (∗). Proof. We shall assume that T = ε. For any f ∈ SX∗ set Af = {T ∗ δ(f,m) : m ∈ M} where δ(f,m) is the evaluation functional at (f, m). Then put A = B=
f ∈SX∗
Af and
f ± Af .
f ∈SX∗
Since T acts into c0 (SX∗ × M), it is easy to see that the only w∗ -limit point of the set A is the origin. Introduce in X the following norm |||x||| = sup g(x) , x ∈ X. g∈B
From the definition of a Talagrand operator, it follows that for any non-zero x ∈ X, we have |||x||| > x.
(2.2)
On the other hand, we have |||x||| (1 + ε)x for any x ∈ X. Therefore the norm ||| · ||| is ε-equivalent to the original one. We need to check that B has (∗). Let g be a w∗ -limit point of B. Since the only w∗ -limit point of A is the origin, it follows that g ∈ BX∗ . Assume that there is x ∈ X, |||x||| = 1, with g(x) = 1. We have x g(x) = 1 = |||x||| contradicting (2.2). Thus B has (∗) as required.
2
Corollary 8. (See [6].) If ε > 0 and α is any ordinal then C[0, α] admits an ε-equivalent polyhedral norm with (∗). Recall that a tree (T , ) is a partially ordered set, such that given any t ∈ T , the set of predecessors {s ∈ T : s t} is well ordered. Further definitions relating to trees will be given in the next section. There, we shall see that trees give rise to scattered, locally compact spaces. Example 9. We say that a tree T is special if it is a countable union of antichains; that is to say, T = ∞ i=0 Ai , where distinct elements of any given Ai are incomparable. It is evident that any tree of countable height is special.
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For any special tree T , there is a Talagrand operator T : C0 (T ) → c0 (T ). To construct T , let T = ∞ i=0 Ai , where the sets Ai are pairwise disjoint antichains. Define T : C0 (T ) → l∞ (T ) by (T x)(t) = 2−i x(t),
whenever t ∈ Ai .
If x ∈ C0 (T ) is non-zero then given any t ∈ T such that |x(t)| = x, we have (T x)(t) = 0. As with the ordinal example, we use the Stone–Weierstrass theorem to show that T maps into c0 (T ). Indeed, C0 (T ) is the closed linear span of the set of indicator functions {1(0,t] : t ∈ T }, where (0, t] denotes the set of all predecessors of t in T . Thus, to prove that T acts into c0 (T ), it is sufficient to check that T 1(0,t] ∈ c0 (T ) for every t. However, this is clear because if ε > 0 and 2−n < ε then, by the antichain property, there are at most n elements s ∈ (0, t] satisfying T 1(0,t] (s) ε. Subtler examples of Talagrand operators on C0 (T ) can be found in [15]. A complete characterization of polyhedral renormings on tree spaces is given in the next theorem. Theorem 10. If T is a tree then the following are equivalent. (1) C0 (T ) admits a polyhedral renorming; (2) for any ε > 0, C0 (T ) admits an ε-equivalent polyhedral renorming with (∗); (3) C0 (T ) admits a Talagrand operator. This characterization turns out to be exactly the same as that of equivalent Fréchet norms on C0 (T ) although there is no indication that the same is true of C(K), for general compact K. In fact, we do not know any examples of polyhedral Banach spaces which lack Fréchet renormings. After Proposition 7, to complete the proof of this theorem, it is enough to prove that if C0 (T ) admits no Talagrand operators, then C0 (T ) has no polyhedral renormings. We postpone the proof of this fact to the next section (see Theorem 14). Now we turn to the class of σ -discrete compact spaces. A compact set K is σ -discrete if it can be written as a countable union of sets {Di }∞ i=1 , each of which is discrete in its relative topology. For example, if K is compact and the derived set K (ω1 ) of order ω1 is empty, then K (β) is empty for some β < ω1 and so K = α<β (K (α) \ K (α+1) ) is σ -discrete. The idea behind the proof of the following theorem is based on a result in [13]. Theorem 11. Let K be a σ -discrete compact set. Then for every ε > 0, C(K) admits an εequivalent polyhedral norm with (∗). Proof. Let K = and
∞
i=1 Di , where each Di
is discrete, and let ε > 0. Define It = {i ∈ N: t ∈ cl Di }
ψ(t) = 1 + ε
2−i .
i∈It
We specify a norm · on C(K) by setting x = sup ψ(t)x(t) . t∈K
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It is clear that · ∞ · (1 + ε) · ∞ and the set B = {±ψ(t)δt : t ∈ K} is 1-norming for (C(K), · ). Now we show that B has (∗). Let f be a w∗ -limit point of B. Then f = αδt for some t ∈ K and α ∈ R. We claim |α| < ψ(t), thus giving f < 1. We begin by picking n ∈ N such that t ∈ Dn . As Dn is discrete, we can find an open set V satisfying (cl Dn ) ∩ V = {t}. If s ∈ V and s = t then n ∈ It \ Is . Let us define J = It ∪ {i ∈ N: i > n + 1} and consider the open set U =K \
{cl Di : i ∈ N \ J }.
By the definition of J , it is clear that t ∈ U and Is ⊂ J whenever s ∈ U . Moreover, the way we choose V gives that n ∈ / Is for each s ∈ (U ∩ V ) \ {t} and we have: Is \ It ⊂ J \ It ⊂ {i ∈ N: i > n + 1}. Thus, for such an s we obtain the following inequality ψ(t) − ψ(s) = ε
i∈It \Is
2−i − ε
i∈Is \It
2−i ε2−n − ε
2−i = ε2−n−1 > 0.
i>n+1
Now we can find a net {sλ } ⊂ (U ∩ V ) \ {t} such that lim sλ = t and lim ψ(sλ ) = |α|. From this, it is evident that ψ(t) − |α| ε2−n−1 > 0. 2 We remark that Proposition 7 and Theorem 11 apply to incomparable classes of spaces. The Ciesielski–Pol compact space K, introduced in [1], has the property that there is no injective linear map T : C(K) → c0 (Γ ), for any set Γ . Moreover, K (3) is empty. As Talagrand operators are evidently injective, this means that C(K) cannot admit such an operator. On the other hand, the compact [0, ω1 ] is not σ -discrete. Our last results of this section concern renorming injective tensor products. For convenience, we review some basic facts about these products. Given Banach spaces X and Y , the injective product X ⊗ε Y is the completion of the algebraic product X ⊗ Y with respect to the norm
f (xi )g(yi ): f ∈ BX∗ , g ∈ BY ∗ . xi ⊗ yi = sup If K is compact then C(K) ⊗ε Y identifies with the space C(K; Y ) of continuous Y -valued functions on K. In particular, given compact spaces K1 and K2 , we have C(K1 ) ⊗ε C(K2 ) ≡ C(K1 ; C(K2 )) ≡ C(K1 × K2 ). If f ∈ X ∗ and IY is the identity operator on Y then we define f Y = f ⊗ IY on X ⊗ Y by Y f ( xi ⊗ yi ) = f (xi )yi . We have f Y = f and extend f Y to the completion. Similarly we define g X for g ∈ Y ∗ . Note that g ◦ f Y = f ⊗ g = f ◦ g X whenever f ∈ X ∗ and g ∈ Y ∗ , and u = supf ∈A f Y (u) = sup(f,g)∈A×B (f ⊗ g)(u) = supg∈B g X (u) for all u ∈ X ⊗ε Y and all 1-norming subsets A ⊂ BX∗ , B ⊂ BY ∗ . We shall denote the strong operator topology by SOT. Observe that the map f → f Y is w∗ to-SOT continuous on bounded subsets of X ∗ . Indeed, if {fλ } is a bounded net converging to f in the w∗ -topology then, by inspection, we have fλY ( xi ⊗ yi ) → f Y ( xi ⊗ yi ) in norm; the
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boundedness of the net allows us to extend this convergence to points in the completion. As an immediate corollary, we have u = f Y (u) = g X (u) for some f ∈ SX∗ and g ∈ SY ∗ . Amongst other things, the next result is a generalization of Proposition 5. Theorem 12. Suppose that X admits a boundary with (∗) and Y admits a polyhedral renorming. Then X ⊗ε Y admits a polyhedral renorming. Moreover, if both X and Y admit boundaries with (∗) then so does X ⊗ε Y . Proof. Let · denote the norms on X, Y and X ⊗ε Y , where · is polyhedral on X and Y . Assume that B is a boundary of X with (∗). Since B is 1-norming, u = sup f Y (u) f ∈B
for u ∈ X ⊗ε Y . We claim that the natural tensor norm is polyhedral. First of all, we show that if u = 1 then there is an open neighborhood U of u and a finite set F ⊂ B, such that v = maxf ∈B {f Y (v)} whenever v ∈ U . Indeed, if the negation holds then we can find a sequence {ui }∞ i=1 converging to u ∈ SX⊗ε Y in norm, together with a sequence of distinct Y Y elements {fi }∞ i=1 ⊂ B such that ui − fi (ui ) → 0. It follows that fi (u) → 1. If f is a w∗ -accumulation point of {fi } then, by the remarks above, f Y is a SOT-accumulation point of {fiY }, whence f Y (u) = 1. If we take g ∈ SY ∗ such that g(f Y (u)) = 1, we have 1 = f (g X (u)) g X (u) u = 1, meaning B does not have (∗). Therefore, locally about points on the sphere, · depends on only finitely many elements of B. Let E ⊂ X ⊗ε Y be a finite-dimensional subspace. Given u ∈ SE , we can take U and F as above. The sum H = f ∈F f Y (E) is a finite-dimensional subspace of Y and thus there exists a finite set G ⊂ SY ∗ such that y = maxg∈G {g(y)} whenever y ∈ H . Consequently, v =
max
(f,g)∈F ×G
(f ⊗ g)(v)
whenever v ∈ U ∩ E. By a simple compactness argument applied to SE , it follows that BE is a polytope. We move on to the case where Y also has a boundary B with (∗). This time, the natural tensor norm can be written as u =
sup
(f,g)∈B×B
(f ⊗ g)(u) .
We prove that D = {f ⊗ g: (f, g) ∈ B × B }, which is evidently a 1-norming subset of BX⊗ε Y , has (∗). To this end, suppose that h is a w∗ -accumulation point of D and, for a contradiction, let us assume that h(u) = 1 for some u ∈ SX⊗ε Y . For n ∈ N, consider the set In = f ∈ B: (f ⊗ g)(u) − 1 < n−1 for some g ∈ B . There are 2 cases: (a) In is infinite for every n or (b) there is an n0 such that In0 is finite. ∞ In case (a), we can extract sequences {fi }∞ i=1 ⊂ B, {gi }i=1 ⊂ B such that (fi ⊗ gi )(u) → 1 ∗ and fi = fj whenever i = j . If g is a w -accumulation point of the sequence {gi }, then g X is a SOT-accumulation point of {giX } and thus, by extracting a subsequence if necessary,
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we can assume that (fi ⊗ g)(u) → 1. Finally, if f is a w∗ -accumulation point of {fi } then f (g X (u)) = (f ⊗ g)(u) = 1, which contradicts the fact that B has (∗), as the fi are distinct and 1 g X (u) u = 1. If (b) holds instead, it must be that Jn is infinite for all n, where Jn = g ∈ B : (f ⊗ g)(u) − 1 < n−1 for some f ∈ B . Indeed, otherwise there exists n0 such that In0 and Jn0 are both finite, whence there is a neighborhood of h that only meets finitely many elements of D. Hence we can simply exchange the roles of fi and gi in the argument above to contradict the fact that B has (∗). 2 The following is an immediate corollary of Proposition 7, Theorems 11 and 12. Corollary 13. Let K1 , . . . , Kn be compact spaces.
If, for every ε > 0 and i n, C(Ki ) admits an ε-equivalent norm with (∗), then so does C( ni=1 Ki ). In particular, if Ki is either an ordinal
segment or σ -discrete, then for every ε > 0, C( ni=1 Ki ) admits an ε-equivalent norm with (∗). As shown in [21], not every finite, Cartesian product of ordinals K admits a Talagrand operator on C(K), so we cannot make a direct application of Proposition 7 in this context. For such finite products of ordinals K, it is also the case that C(K) admits both LUR [20] and Fréchet renormings [14]. 3. The proof of Theorem 10 The aim of this section is to complete the proof of Theorem 10. As mentioned before, we need to prove the following result. Theorem 14. Suppose that T is a tree such that C0 (T ) admits no Talagrand operators, then C0 (T ) has no polyhedral renormings. First, we introduce some definitions and theory concerning trees, including Haydon’s characterization of trees T such that C0 (T ) admits a Talagrand operator [15, Theorem 8.1] (see Theorem 22 below). After that we show in Lemma 23 that if C0 (T ) does not admit such an operator, then for any equivalent norm · on C0 (T ) there exists a subspace Y of (C0 (T ), · ), and a sequence of linear operators {Tn : Y → Y : n ∈ N} such that y = sup{Tn y: n ∈ N} for all y ∈ Y , and Y ∗ is not the norm closed linear span of n∈N Tn∗ Y ∗ . Thus, by Lemma 3, Y and (C0 (T ), · ) are not polyhedral. Let T be a tree. For convenience, we fix an element 0, not in T , such that 0 ≺ t for all t ∈ T . Given s ∈ T ∪ {0} and t ∈ T with s ≺ t, we set (s, t] = {ξ : s ≺ ξ t} and [t, ∞) = {u ∈ T : u t}. The meanings of (s, t), [s, t) and (t, ∞) should be clear. The interval topology on T takes as a basis all sets of the form (s, t] as above. This topology is locally compact because each basis element (s, t] is compact, and it is scattered since any minimal element of a given non-empty subset A ⊂ T is isolated in A. In order for the topology to be Hausdorff also, we require that every non-empty, totally ordered subset of T has at most one minimal upper bound. Henceforth we assume that all our trees have this property. Note that the Hausdorff property also allows us to define a meet ∧ on pairs of elements of T that have a common predecessor;
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if (0, s] ∩ (0, t] is non-empty then we set s ∧ t = sup(0, s] ∩ (0, t]. For the rest of this section, we fix an equivalent norm · on C0 (T ), satisfying · · ∞ M · . Central to the proof we seek is the notion of increasing functions on trees. The map ρ : T → R is increasing if ρ(s) ρ(t) whenever s t. Definition 15. Given a function f ∈ C0 (T ), a ∈ R and t, u ∈ T with t u, we define μ(f, a, t, u) = inf f + a1(t,u] + ϕ: supp ϕ ⊂ (u, ∞) . In this way, we extend slightly the definition of Haydon’s so-called μ-functions (see [15, Section 3] and elsewhere throughout the paper). In Haydon’s definitions, f is restricted to a certain subset of C0 (T ) which depends on t, whereas here, f is arbitrary and thus there is a need to specify t explicitly. For convenience, let us also define μ(f, t) = μ(f, a, t, t) = inf f + ϕ: supp ϕ ∈ (t, ∞) . We see immediately that for fixed f and a, the functions u → μ(f, a, t, u) and t → μ(f 1T \(t,∞) , t) are increasing on the domains [t, ∞) and T , respectively. By elementary reasoning, it is apparent that the functions f → μ(f, a, t, u)
and a → μ(f, a, t, u)
are 1-Lipschitz, which is a fact to be exploited in several approximation arguments later on. In this section, we pay attention to the first type of increasing function above. In particular, we describe two situations in which the infimum in the definition of μ(f, a, t, u) is attained. The following material is a mild generalization of [15, Lemma 3.1], [15, Proposition 3.4] and associated remarks. Proofs are provided for convenience. If t ∈ T then we write t + for the set of immediate successors of t. Definition 16. Let ρ : T → R be an increasing function. Then t ∈ T is a bad point for ρ if there + is a sequence of distinct points {ui }∞ i=1 ⊂ t such that lim ρ(ui ) = ρ(t). Observe that if u t is a bad point for μ(f, a, t, ·) then μ(f, a, t, u) = f + a1(t,u] . ∞ + Indeed, we take a sequence of distinct points {vi }∞ i=1 ⊂ u and functions {ϕi }i=1 , where ϕi is supported on (vi , ∞), such that
f + a1(t,vi ] + ϕi μ(f, a, t, vi ) + 2−i . Since (f + a1(t,vi ] + ϕi ) converges pointwise to f + a1(t,u] , we have weak convergence because T is scattered, and therefore f + a1(t,u] μ(f, a, t, u) as required. We move on to the second example of infimum attainment.
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Definition 17. A subset E ⊂ T is said to be ever-branching if, given t ∈ E, there exist incomparable elements u, v ∈ E such that t ≺ u, v. The simplest type of ever-branching subset is a dyadic tree of height ω. The significance of ever-branching subsets for the μ-functions is explained by the following result, which is a cosmetic generalization of [15, Proposition 3.4]. Lemma 18 (Haydon). Let E ⊂ T be an ever-branching subset and suppose that t ∈ T and u ∈ E, with t u. Then there exists a function ψ ∈ C0 (T ), ψ∞ = 1, supported on (t, u] ∪ (u, ∞) and satisfying: (1) ψ(s) = 1 for s ∈ (t, u]; (2) if f ∈ C0 (T ) and μ(f, a, t, ·) is constant on E, then μ(f, a, t, u) = f + aψ. Proof. Let t, u nand E be as above. Since E is ever-branching, we can choose elements uσ ∈ E, σ∈ ∞ n=0 {0, 1} , such that u∅ = u and for each σ , uσ ≺ uσ 0 , uσ 1 and uσ 0 and uσ 1 are incomparable. In this way we construct a dyadic tree inside E of height ω. Define ψ = 1(t,u] +
∞
2−n−1
n=0
(1(uσ ,uσ 0 ] + 1(uσ ,uσ 1 ] ).
σ ∈{0,1}n
Clearly supp ψ ⊂ (t, u] ∪ (u, ∞), ψ∞ = 1 and property (1) above is seen to be satisfied. Now let μ(f, a, t, ·) be constant on E. For every σ , we have some ϕσ supported on (uσ , ∞), such that f + a1(t,uσ ] + ϕσ − 2−n μ(f, a, t, uσ ) = μ(f, a, t, u). Define yn = 2−n
σ ∈{0,1}n (a1(t,uσ ]
2−n
+ ϕσ ). Since
(f + a1(t,uσ ] + ϕσ ) = f + yn
σ ∈{0,1}n
we obtain f + yn 2−n
μ(f, a, t, u) + 2−n = μ(f, a, t, u) + 2−n . σ ∈{0,1}n
Since μ(·, a, t, u) is Lipschitz, all that is necessary to complete the proof is to show that lim aψ − yn = 0. Observe that the support of aψ − yn is contained in the disjoint union of (uσ , ∞) for σ ∈ {0, 1}n . Now ψ(uσ ,∞) ∞ 2−n for σ ∈ {0, 1}n and ϕσ μ(f, a, t, u) + 2−n + f + |a|. Therefore aψ − yn ∞ 2−n a + 2−n M(μ(f, a, t, u) + 2−n + f + |a|) → 0 as required. At this point, it is convenient to give a definition and make a couple of remarks.
2
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Definition 19. Given an increasing function ρ : T → R, we say that t is a fan point of ρ if there exists an ever-branching subset E containing t, such that ρ(E) = {ρ(t)}. We will call such ψ of Lemma 18 fan functions. Remark 20. Note that infimum attainment in the definition of the μ-functions will be satisfied, regardless of the choice of dyadic tree in Lemma 18. Remark 21. Suppose that ρ = i∈I μi is a valid sum of μ-functions on T (where each μ-function is declared to vanish outside its domain). Let t, u, E and ψ be as in Lemma 18. Suppose further that ρ is constant on E. Then every μi whose domain includes t is constant on E, and the infimum in the definition of μi is attained using ψ , independently of i. Likewise, if t is a bad point for ρ then it is also a bad point for μi whenever t is in the domain of μi , thus we have infimum attainment for all such i. Now we can state the result we referred to at the beginning of the section. Theorem 22. (See [15, Theorem 8.1].) The space C0 (T ) admits a Talagrand operator if and only if there exists an increasing function ρ : T → R that has no bad points or fan points. If we suppose that T is a tree, such that C0 (T ) admits no Talagrand operators, then every increasing function ρ : T → R has either a bad point or a fan point. Thus, to obtain a proof of Theorem 14, because of Lemma 3, it is enough to prove the following lemma. Lemma 23. Suppose that T is a tree, such that every increasing, real-valued function defined on it has either a bad point or a fan point, and that · is an equivalent norm on C0 (T ). Then there exists a separable subspace Y of (C0 (T ), · ) and a sequence of linear operators L = = sup{Tn y: n ∈ N} for all y ∈ Y and Y ∗ is not the {Tn : n ∈ N} acting on Y , such that∗ y ∗ norm closed linear span of n∈N Tn Y . As a consequence of Lemma 3, Y and (C0 (T ), · ) are not polyhedral. Note that if T is a tree, such that every increasing function ρ : T → R has either a bad point or a fan point, then one of the following is true: either every increasing function has a bad point, or every such function has a fan point. Indeed, otherwise we could find increasing functions ρ1 and ρ2 with no bad points and no fan points respectively. Summing them would produce an increasing function with neither. So it is possible to split the proof into two cases, the first in which every increasing function has a bad point, and the second in which every such function has a fan point. This we do, due to the unpalatable number of technicalities that arise when trying to tackle both cases simultaneously. It so happens that the machinery required for the bad point case is largely a simplification of that needed for the more complicated fan point case. Thus, to avoid an overly long treatment, we proceed with the fan point case first, and sketch the bad point case afterwards, taking into account all the important differences. Our present undertaking builds upon a construction that features in [15, Theorem 8.1]. We require some more notation regarding fan points and fan functions. Let us suppose, for the mo-
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ment, that we have constructed increasing, real-valued functions ρ1 , . . . , ρn on T . Let Fn be the set of fan points of ρn and, following Haydon [15, Theorem 8.1], define ρn (t) if t is a minimal element of Fn , εn,t = ρn (t) − sup{ρn (s): s ∈ (0, t) ∩ Fn } otherwise. We let Γn be the set of t ∈ Fn such that εn,t > 0. Observe the next important fact: if t ∈ Fn \ Γn then (3.3) ρn (t) = sup ρn (s): s ∈ (0, t) ∩ Γn . Now we depart from Haydon. Given t ∈ Fn , let u ∈ [t, ∞) be minimal, subject to the requirement that there exist distinct v0 , v1 ∈ u+ , and ever-branching subsets Eθ ∈ [vθ , ∞) such that ρ(Eθ ) = {ρn (vθ )}, for θ ∈ {0, 1}. Notice that u is unique, and so we can define πn (t) to be this u. In addition, we set πn,θ (t) = vθ for θ ∈ {0, 1}, although these vθ need not be unique. Now fix fan functions ψn,t,θ , furnished by Lemma 18, with πn,θ (t) in place of u. In particular, ψn,t,θ is supported on (t, πn,θ (t)] ∪ (πn,θ (t), ∞) and ψn,t,θ (s) = 1 whenever s ∈ (t, πn,θ (t)]. We permit a mild abuse of notation while defining πn,t and ψn,t,u by πn,1 (t) if πn,0 (t) u, ψn,t,1 if πn,0 (t) u, πn,t (u) = and ψn,t,u = πn,0 (t) otherwise, ψn,t,0 otherwise. Next, we define some subspaces of C(T ). Let t ∈ T and let Cn,t be the closed linear span of {1(0,t] } ∪
1(s,t] : s ∈ (0, t) ∩ Γk ∪ ψk,s,t : s ∈ (0, t) ∩ Γk .
k
Since s∈(0,t)∩Γn εn,s ρn (t), we see that (0, t) ∩ Γn is countable, hence each Cn,t is separable. Let {ql }∞ l=1 be an enumeration of the non-zero rationals and, for each t ∈ Fn , fix a dense sequence (fn,t,k )∞ k=1 in Cn,t and define the function σn,t : T → R by σn,t (u) =
∞
k,l,m=1 2
−k−l−m μ(fn,t,k +ql ψn,t,u ,qm ,t,u) fn,t,k +|ql |+|qm |
if u ∈ (t, ∞) \ (t, πn (t)],
0 otherwise.
Because μ(f + aψn,t,u , b, t, u) f + aψn,t,u + b1(t,u] f + |a| + |b| we have σn,t (u) 1 for all u. Also, σn,t is increasing, for if u ∈ (t, ∞) \ (t, πn (t)] and u v then v ∈ (t, ∞) \ (t, πn (t)] and thus ψn,t,u = ψn,t,v . Now we simply use the fact that μ(f, a, t, · ) is increasing. Proof of Lemma 23. As previously mentioned, we proceed with the fan point case first. Actually we construct an isomorphic embedding T : c ⊕ c0 → C0 (T ). After this construction we consider Y = (T (c ⊕ c0 ), · ) and define a sequence of maps Tn over the isometric copy X = (c ⊕ c0 , ||| · |||) of Y , where |||(x, y)||| = T (x, y). We begin by defining a sequence of increasing functions {ρn }∞ n=1 . We would like this sequence to satisfy the following properties:
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(1) ρn (t) ρn+1 (t) for all t and ρn (u) − ρn (t) ρn+1 (u) − ρn+1 (t) whenever t u; (2) ρn (t) ∈ (0, 2 − 2−n ); (3) if k < n, t ∈ Γk , t u πk (t) and v ∈ u+ \ (t, πk (t)] then ρn (v) > ρn (u). We begin by setting ρ1 (t) = μ(1(0,t] , t). We have that assume that ρn has been constructed. We define ρn+1 = ρn + 2−n−2
1 M
ρ1 (t) 1, so (2) is clear. Now
εn,t σn,t
t∈Γn
and note that this sum is well defined, for ρn+1 (u) = ρn (u) + 2−n−2
εn,t σn,t (u)
t∈(0,u)∩Γn
ρn (u) + 2
−n−2
εn,t
t∈(0,u)∩Γn
1 + 2−n−2 ρn (u) 1 + 2−n−2 2 − 2−n < 2 − 2−n−1 . So we have (2). Since the second summand in the definition of ρn+1 is an increasing, nonnegative function, we have (1). Therefore, for (3) to be fulfilled, we simply need to check that ρn+1 (v) > ρn+1 (u) whenever t ∈ Γn , t u πn (t) and v ∈ u+ \ (t, πn (t)]. This is indeed so, for σn,t (v) > 0 = σn,t (u). This completes the construction of (ρn ). From (1) and (2) we can define ρ(t) = supn ρn (t). Since ρ is increasing, it has a fan point w by hypothesis. Note that, again from (1), ρn (u) − ρn (t) ρ(u) − ρ(t) for all n and whenever t u, thus w ∈ Fn for all n. By (3), w ∈ / Γn for all n. In fact, if t ∈ Γn and t u πn (t), then u∈ / Fn+1 . Indeed, if u v and ρn+1 (v) = ρn+1 (u) then, necessarily, v πn (t) by (3). So the set of v ∈ [u, ∞) such that ρn+1 (v) = ρn+1 (u) is contained in [u, πn (t)], which is totally ordered and certainly not ever-branching. If we define the set Γ = n Γn , it follows that Γ ∩ (0, w) countable, because, as we have seen, s∈(0,w)∩Γn εn,s ρ(w) for each n. Let v = sup Γ ∩ (0, w) w. Our intention is to construct ∞ strictly increasing sequences {ni }∞ i=1 ⊂ N and {ti }i=1 ⊂ T such that ti ∈ Γni ∩ (0, v), πni (ti ) ∧ w ≺ ti+1 and ti → v. Firstly, we show that if t ∈ Γn ∩ (0, w) then there exists u ∈ Γn+1 ∩ πn (t) ∧ w, w .
(3.4)
We can see that w πn (t). Indeed, w ∈ Fn+1 , so it cannot be that w πn (t) by the corollary of (3) presented above. Let t = πn (t) ∧ w ∈ [t, w); also by (3), if t is the unique element of (t )+ ∩ (0, w] then t ∈ / (t, πn (t)] by maximality of t and so ρn+1 (t ) < ρn+1 (t ) ρn+1 (w). Therefore, by (3.3), there exists u ∈ Γn+1 ∩ (t , w) as required. In particular, Γ ∩ (0, w) has no greatest element; thus, as Γ ∩ (0, w) is also countable, we can fix a strictly increasing sequence {si }∞ / Γ1 , we can i=1 ⊂ Γ ∩ (0, v), such that si → v. Since w ∈ find t1 ∈ Γn1 ∩ (0, v) by (3.3), where n1 = 1. Assume that we have constructed t1 ≺ t2 ≺ · · · ≺ ti ≺ v and corresponding n1 < n2 < · · · < ni , such that tj ∈ Γnj ∩ (0, w) for j i and sj ≺ tj +1 for j < i. If t = max{si , ti } ∈ Γn then, by repeating (3.4) enough times, we can find ni+1 > ni , n and ti+1 ∈ Γni+1 ∩ (πni (ti ) ∧ w, w).
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Since ti+1 ∈ (ti , ∞) \ (ti , πni (ti )] and ti+1 ≺ w, we have πni ,ti (ti+1 ) = πni ,ti (w) and we define ui to be this element of the tree. By construction, we have ensured that ui ti+1 . It follows that the sets Hi = (ti , ui ] ∪ (ui , ∞), i 1, are pairwise disjoint. If we define ψi = ψni ,ti ,ti+1 = ψni ,ti ,w then the support of ψi lies entirely in Hi , and moreover ψi (ui ) = 1. Since w is a fan point of ρ, we can fix a fan function ψ, supported on (w, ∞). Now let us define a linear map T : c ⊕ c0 → C0 (T ) by T (x, y) =
∞ ∞ (xi − xi−1 )(1(ti−1 ,w] + ψ) + yi ψ i , i=1
i=1
∞ where x = {xi }∞ i=1 ∈ c, y = {yi }i=1 ∈ c0 , x0 = 0 and t0 = 0. The second sum in the definition of T is well defined because the supports of ψi are pairwise disjoint and y ∈ c0 . The support of T (x, y) is included in the disjoint union of (0, t1 ], (ti , ti+1 ] ∪ Hi , i 1 and [w, ∞). Hence, we see that
T (x, y) = max |x1 |, sup xi 1(t ,t ] + yi ψi ∞ , | lim xi | i i+1 ∞ i1
sup |xi | + |yi | i1
2 (x, y) ∞ . On the other hand, T (x, y) T (x, y)(tn ) = (xi − xi−1 )1(t ,w] (tn ) = |xn | i−1 ∞ in
and T (x, y)
∞
T (x, y)(un ) = yn ψn (un ) = |yn |
thus T (x, y)∞ (x, y)∞ . So we have that T is an isomorphic embedding. If |||(x, y)||| = T (x, y) then ||| · ||| is equivalent to · ∞ on c ⊕ c0 . Put X = (c ⊕ c0 , ||| · |||) and Y = (T (c ⊕ c0 ), · ). Now define maps Pn , Qn : c ⊕ c0 → c0 by Pn (x, y)i =
xi 0
if i n, otherwise,
yi xn 0
if i < n, if i = n, otherwise,
and Qn (x, y)i =
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and set Tn = Pn ⊕ Qn : X → X. It is obvious that ker Tn = (x, y) ∈ X: xi = 0 for i n and yi = 0 for i < n and if ξ ∈ X ∗ is defined by ξ(x, y) = lim xi then ξ ∈ X ∗ \ cl n (ker Tn )⊥ . Thus the conclusion of Lemma 3 is not satisfied. It remains to show that condition (1.1) of this lemma can be met, and then we will have that X, and thus Y and (C0 (T ), · ), are not polyhedral. Let (x, y) ∈ c ⊕ c0 and set g = T (x, y) and gi = T Ti (x, y). Moreover, set h = g1T \(w,∞) and hi = g1T \(ti ,∞) . For clarity, we note the identities g=
∞ ∞ (xi − xi−1 )1(ti−1 ,w] + yi ψi + (lim xi )ψ = h + (lim xi )ψ = h + h(w)ψ i=1
i=1
and gi =
i j =1
xj 1(tj −1 ,tj ] +
i−1
yj ψj + xi ψi = hi + xi ψi = hi + hi (ti )ψi .
j =1
Condition (1.1) of Lemma 3 will be met if we can show that g = supi gi . This will follow from the following claims: (a) gi = μ(hi , ti ) for all i and g = μ(h, w); (b) μ(h, w) = supi μ(hi , ti ). We prove claim (a) first. Let Ci be the linear span of {1(tj ,ti ] : 0 j < i} ∪ {ψj : 1 j < i}. Evidently, {Ci }∞ i=1 forms an increasing sequence of subspaces and hi ∈ Ci . Note also that since tj ∈ Γnj and nj < ni for j < i, we have Ci ⊂ Cni ,ti . Take i 1. If ti+1 ∈ Fni+1 then ti+1 ∈ Fni +1 also, so by construction of ρni +1 we have that ti+1 is a fan point for σni ,ti . Since we have ensured that ti+1 ∈ (ti , ∞) \ (ti , πni (ti )], it follows that ti+1 is a fan point for μ(fni ,ti ,k + ql ψi , qm , ti , ·) for every k, l and m. Therefore μ(fni ,ti ,k + ql ψi , qm , ti , ti+1 ) = fni ,ti ,k + ql ψi + qm (1(ti ,ti+1 ] + ψi+1 ) for every k, l and m, by Remark 21. By uniform approximation, it follows that μ(f + aψi , b, ti , ti+1 ) = f + aψi + b(1(ti ,ti+1 ] + ψi+1 ) for every f ∈ Cni ,ti and a, b ∈ R; equivalently, μ(f + aψi + b1(ti ,ti+1 ] , ti+1 ) = μ(f + aψi + b1(ti ,ti+1 ] , b, ti+1 , ti+1 ) = (f + aψi + b1(ti ,ti+1 ] ) + bψi+1 .
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The space of functions f + aψi + b1(ti ,ti+1 ] of the above form includes Ci+1 . Thus we have μ(hi+1 , ti+1 ) = hi+1 + hi+1 (ti+1 )ψi+1 = gi+1 . It remains to show that μ(h1 , t1 ) = g1 , but this is straightforward, because C1 is the linear span of {1(0,t1 ] } and t1 is a fan point for ρ1 : t → μ(1(0,t] , t), so by homogeneity μ(b1(0,t1 ] , t1 ) = b(1(0,t1 ] + ψ1 ) . Now we show that g = μ(h, w). By repeating the above argument with w and ψ in place of ti+1 and ψi+1 , respectively, we obtain μ(f + aψi + b1(ti ,w] , w) = μ(f + aψi + b1(ti ,w] , b, w, w) = f + aψi + b(1(ti ,w] + ψ) for every i. Set ki = hi + hi (ti )1(ti ,w] =
i i−1 (xj − xj −1 )1(tj −1 ,w] + yj ψ j . j =1
j =1
From above, we have μ(ki , w) = ki + hi (ti )ψ and since ki − h∞ = ki + hi (ti )ψ − g ∞ → 0 we have μ(h, w) = g by another uniform approximation argument. This completes the proof of claim (a). Now to prove claim (b). First, note that by the remarks about μ-functions made after Definition 15, the sequence (μ(hi , ti )) is increasing and bounded above by μ(h, w). Because w ∈ Fn \ Γn for all n, we have ρn (w) = sup ρn (t): t ∈ (0, w) ∩ Γn for all n by (3.3). Notice that, if ρ = i∈I σi is a valid sum of increasing functions and ρ(u) = sup{ρ(t): t ∈ A}, where A ⊂ (0, u], then σi (u) = sup{σi (t): t ∈ A} for all i ∈ I . Therefore, σni ,ti (w) = σni ,ti (t): t ∈ (0, w) ∩ Γni for all i. By a further uniform approximation argument, we have μ(f + aψi , b, ti , w) = sup μ(f + aψi , b, ti , t): t ∈ πni (ti ) ∧ w, w ∩ Γni
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for all f ∈ Cni ,ti , a, b ∈ R and i 1. Take ε > 0 and let i be such that, for j i, we have kj − ki ∞ ε. In particular h − ki ∞ ε, so by the 1-Lipschitz property of μ-functions we have μ(h, w) − μ(ki , w) ε. (3.5) From above μ(ki , w) = μ hi , hi (ti ), ti , w = sup μ hi , hi (ti ), ti , t : t ∈ πni (ti ) ∧ w, w ∩ Γni thus there exists j i such that μ(ki , w) − μ hi , hi (ti ), ti , tj ε
(3.6)
because μ(hi , hi (ti ), ti , ·) is increasing and tj → v. Now μ hi , hi (ti ), ti , tj = μ hi + hi (ti )1(ti ,tj ] , tj so since hj − (hi + hi (ti )1(ti ,tj ] )∞ = kj − ki ∞ ε, again by the 1-Lipschitz property, we have therefore μ(hj , tj ) − μ hi , hi (ti ), ti , tj hj − hi + hi (ti )1(t ,t ] ε (3.7) i j and putting inequalities (3.5), (3.6) and (3.7) together gives μ(h, w) − μ(hj , tj ) 3ε which proves claim (b) and completes the proof of the fan point case. It remains to sketch a proof of the bad point case. It is a simplification of the material above because fan functions cease to matter. In fact, the construction of ρ largely becomes that which is presented in [15, Theorem 8.1]. As above, let us assume the existence of increasing functions ρ1 , . . . , ρn , and let Bn denote the set of bad points of ρn . Define εn,t and Γn , with Bn in place of Fn . All π -functions and ψ -functions should be ignored. Thus Cn,t becomes the closed linear span of {1(0,t] } ∪ 1(s,t] : s ∈ (0, t) ∩ Γk k
and σn,t reduces to σn,t (u) =
k,l 2
−k−l−2 μ(fn,t,k ,ql ,t,u) fn,t,k +|ql |
if u ∈ (t, ∞),
0 otherwise.
The definition of the sequence {ρn }∞ n=1 remains the same, as do conditions (1) and (2). However condition (3) must be replaced by (3 ) if k < n and t ∈ Γk then infu∈t + ρn (u) − ρn (t) > 0.
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Arguing as above, for (3 ) to be fulfilled, we just need infu∈t + ρn+1 (u) − ρn+1 (t) > 0 whenever t ∈ Γn . This is indeed so, because, for f ∈ Cn,t , a ∈ R and t u μ(f, a, t, u) M −1 f + a1(t,u] ∞ (2M)−1 f ∞ + |a| because all elements of this version of Cn,t have support disjoint from a1(t,u] . Therefore, for t ≺ u, σn,t (u) 1/2M > 0 = σn,t (t) and so ρn+1 (u) − ρn+1 (t) 2n−3 εn,t /M. By hypothesis, let w be a bad point of ρ, defined as before. Then w ∈ Bn for all n and, by (3 ), w∈ / Γn for any n. Setting v = sup Γ ∩ (0, w) as above, we construct strictly increasing sequences ∞ {ni }∞ i=1 ⊂ N and {ti }i=1 ⊂ T such that ti ∈ Γni ∩ (0, v) and ti → v. We can simplify (3.4) to (3.8) thus: if t ∈ Γn ∩ (0, w) then there exists u ∈ Γn+1 ∩ (t, w).
(3.8)
Indeed, by (3 ), ρn+1 (t) < ρn+1 (t ) ρn+1 (w), where t is the unique element of (0, w] ∩ t + . By (3.3), we must have some u ∈ Γn+1 ∩ (t, w). Once the sequences have been constructed, we move straight to the definition of T : (c, · ∞ ) → (C0 (T ), · ∞ ). Let T (x) =
∞
(xi − xi−1 )1(ti−1 ,w] .
i=1
In this case, T is an isometry. Consider X = c with the norm |||x||| = T (x), Y = (T (c), · ), and define the maps Tn : X → X by Tn (x)i =
xi 0
if i n, otherwise.
It is clear that the conclusion of Lemma 3 is not satisfied. To show that condition (1.1) of this lemma holds and conclude that X and thus (C0 (T ), · ) is not polyhedral, we show that claims (a ) gi = μ(gi , ti ) for all i and g = μ(g, w); (b ) μ(g, w) = supi μ(gi , ti ) hold, where g and gi are as above. We set all fan functions ψ and ψi to zero, so that h = g and hi = gi , and replace all mention of ‘fan point’ with ‘bad point.’ We leave it to the reader to verify that the above argument holds with these changes in place. 2 4. Non-separable spaces with unconditional basis In 1994, D. Leung [18] proved that a Banach space X with a shrinking basis {en }∞ n=1 is isomorphically polyhedral if and only if, there exists an equivalent norm ||| · ||| on X which is monotone with respect to {en }∞ and, for every x = a e ∈ X, we may find m ∈ N such that n n n=1 m an en . |||x||| = n=1
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Actually, the sufficient condition of the former result goes back to [3,8]. We extend the above result to Banach spaces with an uncountable unconditional basis. In this case we need essential modifications. Theorem 24. Let X be a Banach space with a monotone unconditional basis {ei }i∈I , i.e. Pσ = 1, for any σ ⊂ I, |σ | < ∞, and Pσ x = i∈σ ei∗ (x)ei , where {ei∗ }i∈I is the biorthogonal system for {ei }i∈I . Assume that for any x ∈ X there is σ ⊂ I , |σ | < ∞, with x = Pσ x. Then X is isomorphic to a polyhedral space. Proof. Fix a decreasing sequence {εk }∞ k=1 of positive numbers with lim εk = 0,
0 < εk < 1/4 and εk+1 < εk2 /4.
k
(4.9)
From (4.9) we easily get (1 + 4εk )(1 − εk )2 > 1 + εk2 > 1 + 4εk+1 .
(4.10)
Let σ ⊂ I , |σ | = n < ∞, Lσ = [ei∗ ]i∈σ . Let Dσ be a symmetric finite εn -net in SLσ . Put D= (1 + 4ε|σ | )Dσ , V ∗ = w∗ -cl co D. σ ⊂I, |σ |<∞
Introduce in X a new norm as follows: |||x||| = max∗ f (x) f ∈V
for x ∈ X. It is not difficult to see that the norm ||| · ||| is equivalent to the original one, and that V ∗ is the dual ball in the norm ||| · |||. Claim 1. (a) |||x||| > x, for any x ∈ X, x = 0. (b) (1 + εn2 )BLσ ⊂ (1 − εn ) co((1 + 4εn )Dσ ). Proof. Let x ∈ X, x = 0, and σ ⊂ I , |σ | = n < ∞, be such that x = Pσ x, and f ∈ SX∗ be such that f (Pσ x) = Pσ x. If g = Pσ∗ f then g(x) = f (Pσ x) = Pσ x = x. In particular, g ∈ SLσ , and hence there is h ∈ Dσ with g − h < εn . We have (by using (4.10)) (1 − εn )|||x||| (1 − εn )
max
f ∈(1+4εn )Dσ
f (x) (1 − εn )(1 + 4εn )h(x)
(1 − εn )(1 + 4εn ) g(x) − g − h × x (1 + 4εn )(1 − εn )2 x > 1 + εn2 x, which finishes the proof of (a). To prove (b) we use the following part of the inequality above (1 − εn ) max f (x) 1 + εn2 x, f ∈(1+4εn )Dσ
and the separation theorem.
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469
Claim 2. Let σ and ν be two finite subsets of I with n = |σ | < |ν| = m. Then Pσ∗ (1 + 4εm )Dν ⊂ (1 − εn ) co (1 + 4εn )Dσ . Proof. By using Pσ∗ = 1, εm εn+1 , (4.10) and Claim 1(b), we get Pσ∗ (1 + 4εm )Dν ⊂ (1 + 4εn+1 )BLσ ⊂ 1 + εn2 BLσ ⊂ (1 − εn ) co (1 + 4εn )Dσ , finishing the proof.
2
Claim 3. The set D has property (∗) in the norm ||| · |||. Proof. Let f0 be a w∗ -limit point of the set D, i.e. any w∗ -neighborhood of f0 contains infinitely many points of D. We will prove that either |||f0 ||| < 1 or, if |||f0 ||| = 1, there is not x ∈ X, |||x||| = 1, with f0 (x) = 1. Put σ0 = i ∈ I : f0 (ei ) = 0 ,
p0 = |σ0 |,
and consider two cases. Case 1. There is an integer M and a w∗ -neighborhood W0 of f0 such that for any w∗ neighborhood W of f0 with W ⊂ W0 , and for any gσ ∈ (1 + 4ε|σ | )Dσ with gσ ∈ W , we have |σ | M. In this case we can assume without loss of generality that there is an integer p M and that f0 is a w∗ -limit point of a net {gσ }, with |σ | = p for any σ . It is not difficult to see that p0 p. Moreover since f0 is a w∗ -limit point of D and each Dσ is finite, it easily follows that p0 < p. Now from Claim 2 we get |||f0 ||| < 1. Case 2. For any integer M and for any w∗ -neighborhood W0 of f0 there is a w∗ -neighborhood W of f0 with W ⊂ W0 , such that there is gσ ∈ (1 + 4ε|σ | )Dσ with gσ ∈ W , with |σ | > M. In this case f0 ∈ BX∗ . Assume that |||f0 ||| = 1 and there is x ∈ X, |||x||| = 1, with f0 (x) = 1. Since f0 ∈ BX∗ it follows that x 1. However from Claim 1 we get |||x||| > x, for any x ∈ X, x = 0, which is a contradiction. 2 The theorem now follows from Claim 3 and Proposition 7.
2
Recall that a non-degenerate Orlicz function M is a non-decreasing convex function defined on t 0, with M(0) = 0, M(t) > 0 for all t > 0, and limt→+∞ M(t) = +∞. For a set Γ , the Orlicz space M (Γ ) consists of all real functions x defined on Γ such that γ ∈Γ M(|x(γ )|/ρ) < +∞ for some ρ > 0, equipped with the norm x = inf ρ > 0: M x(γ ) /ρ 1 . γ ∈Γ
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Let {eγ }γ ∈Γ be the family of the functions eγ (β) = δγ ,β , whose sole non-zero value is 1 at β = γ , and hM (Γ ) be the closed subspace of M (Γ ) generated by this family. Clearly {eγ }γ ∈Γ is a monotone, symmetric basis of the Banach space hM (Γ ). Corollary 25. Let M be a non-degenerate Orlicz function such that there exists a finite number K satisfying lim
t→0
M(Kt) = +∞. M(t)
Then the spaces hM (Γ ) admits a polyhedral renorming, for any Γ . Proof. The construction of the desired norm runs along the lines of the construction in the proof of Theorem 4 from [18]. To prove that this norm is polyhedral we use Theorem 24. 2 References [1] K. Ciesielski, R. Pol, A weakly Lindelöf function space C(K) without any continuous injection into c0 (Γ ), Bull. Polish Acad. Sci. Math. 32 (1984) 681–688. [2] R. Deville, V.P. Fonf, P. Hajek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998) 139–154. [3] V.P. Fonf, Classes of Banach spaces connected with massiveness of the set of extreme points of the dual ball, PhD thesis, Sverdlovsk, 1979. [4] V.P. Fonf, One property of Lindenstrauss–Phelps spaces, Funct. Anal. Appl. 13 (1979) 66–67 (transl. from Russian). [5] V.P. Fonf, Some properties of polyhedral Banach spaces, Funct. Anal. Appl. 14 (1980) 89–90 (transl. from Russian). [6] V.P. Fonf, A property of spaces of continuous functions on segments of ordinals, Sibirsk. Mat. Zh. 21 (3) (1980) 230–232; English transl. in: Siberian Math. J. 6 (1980). [7] V.P. Fonf, Polyhedral Banach spaces, Math. Notes Acad. Sci. USSR 30 (1981) 809–813 (transl. from Russian). [8] V.P. Fonf, Weakly extremal properties of Banach spaces, Mat. Zametki 45 (6) (1989) 83–92; English transl. in: Math. Notes Acad. Sci. USSR 45 (5–6) (1989) 488–494. [9] V.P. Fonf, Three characterizations of polyhedral Banach spaces, Ukrainian Math. J. 42 (9) (1990) 1145–1148 (transl. from Russian). [10] V.P. Fonf, On the boundary of a polyhedral Banach space, Extracta Math. 15 (2000) 145–154. [11] V.P. Fonf, L. Vesely, Infinite-dimensional polyhedrality, Canad. J. Math. 56 (2004) 472–494. [12] V.P. Fonf, J. Lindenstrauss, R.R. Phelps, Infinite-dimensional convexity, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 1, Elsevier, 2001, pp. 599–670. [13] P. Hájek, R. Haydon, Smooth norms and approximation in Banach spaces of the type C(K), Quart. J. Math. 58 (2007) 221–228. [14] R. Haydon, Smooth functions and partitions of unity on certain Banach spaces, Quart. J. Math. 47 (1996) 455–468. [15] R. Haydon, Trees in renorming theory, Proc. London Math. Soc. 78 (1999) 541–584. [16] M. Jimenez-Sevilla, J.P. Moreno, Renorming Banach spaces with the Mazur intersection property, J. Funct. Anal. 144 (1997) 486–504. [17] V. Klee, Polyhedral sections of convex bodies, Acta Math. 103 (1960) 243–267. [18] D.H. Leung, Some isomorphically polyhedral Orlicz sequence spaces, Israel J. Math. 87 (1994) 117–128. [19] J. Lindenstrauss, Notes on Klee’s paper “Polyhedral sections of convex bodies”, Israel J. Math. 4 (1964) 235–242. [20] N.K. Ribarska, V.D. Babev, A stability property for locally uniformly rotund renorming, J. Math. Anal. Appl., in press. [21] R.J. Smith, Bounded linear Talagrand operators on ordinal spaces, Quart. J. Math. 56 (2005) 383–395. [22] L. Vesely, Boundary of polyhedral space—an alternative proof, Extracta Math. 15 (2000) 213–217.
Journal of Functional Analysis 255 (2008) 471–493 www.elsevier.com/locate/jfa
Extreme amenability of L0 , a Ramsey theorem, and Lévy groups ✩ Ilijas Farah a,b , Sławomir Solecki c,∗ a Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 b Matematicki Institut, Kneza Mihaila 35, Belgrade, Serbia c Department of Mathematics, 1409 W. Green St., University of Illinois, Urbana, IL 61801, USA
Received 19 February 2008; accepted 31 March 2008 Available online 12 May 2008 Communicated by J. Bourgain
Abstract We show that L0 (φ, H ) is extremely amenable for any diffused submeasure φ and any solvable compact group H . This extends results of Herer–Christensen, and of Glasner and Furstenberg–Weiss. Proofs of these earlier results used spectral theory or concentration of measure. Our argument is based on a new Ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods. Using this work, we give an example of a group which is extremely amenable and contains an increasing sequence of compact subgroups with dense union, but which does not contain a Lévy sequence of compact subgroups with dense union. This answers a question of Pestov. We also show that many Lévy groups have non-Lévy sequences, answering another question of Pestov. © 2008 Elsevier Inc. All rights reserved. Keywords: Extremely amenable groups; L0 ; Submeasures; Ramsey theory; Borsuk–Ulam theorem
1. Introduction A topological group is called extremely amenable if each of its continuous actions on a compact space has a fixed point. These types of groups and their connections with concentration of measure and with Ramsey theory have received considerable attention. For recent treatments of ✩
Research supported by NSERC (I. Farah) and NSF grant DMS-0400931 (S. Solecki).
* Corresponding author.
E-mail addresses: [email protected] (I. Farah), [email protected] (S. Solecki). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.016
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extreme amenability, see [17] and [8]. The earliest examples of such groups were of the form L0 (φ, H ) for certain locally compact second countable groups H and certain submeasures φ, see [5,7]. While among very disconnected groups—those with a basis at the identity consisting of open subgroups—extreme amenability is well understood through its connection with structural Ramsey theory [8], this is not the case for many connected groups, such as L0 (φ, H ). The present paper contributes to the clarification of this situation. Now we explain the notions of submeasure and of L0 (φ, H ). Let B be an algebra of subsets of a set X. A function φ mapping B to R is called a submeasure if φ(∅) = 0 and, for U, V ∈ B, φ(U ) φ(V ), if U ⊆ V , and φ(U ∪ V ) φ(U ) + φ(V ). A submeasure μ is called a measure if μ(U ∪ V ) = μ(U ) + μ(V ) for U, V ∈ B that are disjoint. Note that we do not require countable additivity. We say that a submeasure φ has no measure below if there is no non-zero measure μ such that μ φ. The generic submeasure turns out to have no measure below [7]. A submeasure φ is called diffused if for any > 0 there is a covering of X by sets from B of φ-submeasure less than . Let φ be a submeasure, and let H be a topological group. By H -valued step functions we understand functions on X with values in H , with finite ranges, and with preimages of points in B. We make two step functions f, g equivalent if φ({x ∈ X: f (x) = g(x)}) = 0. Define S(φ, H ) to be the space of all equivalence classes of H -valued step function with the topology of convergence in the submeasure φ. If H is locally compact, second countable (lcsc), we view S(φ, H ) as equipped with the right invariant metric dφ (f, g) = inf > 0: φ x ∈ X: ρ f (x), g(x) > < ,
(1.1)
where ρ is a right invariant metric on H . (Such a metric ρ exists and is automatically complete as H is lcsc.) Of course, the value of dφ (f, g) depends only on the equivalence classes of f and g in S(φ, H ). We define L0 (φ, H ) as the completion of S(φ, H ) with respect to the metric dφ . As usual, if X = 2N and B is the algebra of all closed and open subsets of X, then elements of L0 (φ, H ) can be identified with equivalence classes of certain Borel functions from 2N to H . It consists of all equivalence classes of Borel functions if and only if φ is exhaustive, i.e., if limn φ(An ) = 0 for every disjoint family An ∈ B. All measures are exhaustive. In an answer to an old question of Maharam, an exhaustive submeasure with no measure below was constructed recently by Talagrand [20]. To fix attention, we first consider the case of the circle group H = T. We come back to the issue of more general groups H below. 1. The following observation goes back to Nikodým [13, pp. 139–141]: φ is not diffused if and only if L0 (φ, T) has a nontrivial continuous homomorphism to T. (In [13] this is proved only for φ a measure and with R in place of T, but the method adapts with minor changes to the present situation.) In particular, if φ is not diffused, then L0 (φ, T) is not extremely amenable. 2. It follows from the results of Herer and Christensen [7] that if φ is a submeasure with no measure below, then L0 (φ, T) is extremely amenable. Note that such a φ is necessarily diffused. On the opposite end of the spectrum, it was proved by Glasner [5], and independently Furstenberg and Weiss, that if μ is a diffused measure, then L0 (μ, T) is also extremely amenable.
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In light of 1 and 2 above, one is compelled to inquire whether L0 (φ, T) is extremely amenable if φ is only assumed to be diffused. We settle this question in the affirmative in the following result, whose sharper version is proved as Theorem 3.1. Theorem 1.1. Let φ be a diffused submeasure, and let H be a solvable compact second countable group. Then L0 (φ, H ) is extremely amenable. It is worth pointing out that Herer and Christensen’s method [7] is based on the spectral theorem while that of Glasner, Furstenberg, and Weiss [5] is based on concentration of measure. Our argument, presented in Sections 2 and 3, is different from both. It is combinatorial and uses a new Ramsey-type result for groups Z/p with p prime. This is Theorem 2.1, whose motivation and proof stem from some applications of algebraic topology to combinatorics. (For background see [2] and [11].) This seems to be the first result relating generalizations of the Borsuk–Ulam theorem to extreme amenability. We now consider the following broader problem. So far there have been two main methods for proving extreme amenability: structural Ramsey theorems [12] and concentration of measure (Lévy property) [9]. The appropriate class of groups for application of structural Ramsey theory are automorphism groups of countable models or, in other words, closed subgroups of the group S∞ of all permutations of N, see [8,14]. Such a group is extremely amenable if an appropriate structural Ramsey theorem holds. Furthermore, by [8], within this class of groups the two phenomena, extreme amenability of a group and existence of an appropriate structural Ramsey theorem, are essentially equivalent. The appropriate class of groups for application of concentration of measure are groups that contain increasing sequences of compact subgroups with dense union, see [5,6]: given such a group if a sequence of compact subgroups with dense union exhibits an appropriate concentration of measure, the group is extremely amenable. A question of Pestov, motivated by an older problem of Furstenberg, asks if this implication is an equivalence within the class of groups having increasing sequences of compact subgroups with dense unions (much like structural Ramsey theory/extreme amenability equivalence within closed subgroups of S∞ ). See [17, §7.2] for a discussion of this problem. We give an example of a group, again of the form L0 (φ, H ) for a certain submeasure φ and a compact group H , that answers this question in the negative. We will now be more precise. The concentration of measure phenomenon is used in proving extreme amenability through the following notion (see e.g., [5,6,16]). Let G be a topological group. An increasing sequence (Kn ) of compact subgroups of G, equipped with their Haar probability measures μn , is a Lévy sequence if for every open V 1 and every sequence An ⊆ Kn of measurable sets such that infn μn (An ) > 0 we have that limn μn (V An ) = 1. A group is a Lévy group if it has a Lévy sequence of compact subgroups whose union is dense in G. Lévy groups were introduced by Gromov and Milman in [6], where it was shown that every Lévy group is extremely amenable. This theorem was applied in a number of situations to obtain various extreme amenability results. This prompted the question, which originated with Furstenberg in the late 1970s and was later reconsidered by Pestov, asking whether each group that is extremely amenable and has an increasing sequence of compact subgroups whose union is dense is a Lévy group [17, p. 155]. For more background see [17, §7.2]. We consider Lévy groups in Section 4. Theorem 4.2 contains a slightly more precise formulation of the following result showing that Lévy sequences cannot be used to prove extreme amenbility of L0 (φ, H ) if H is compact and not connected.
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Theorem 1.2. For every compact second countable group H that is not connected there exists a diffused submeasure ψ on the algebra of all closed and open subsets of 2N such that L0 (ψ, H ) is not Lévy. The Pestov question is now answered as follows. Let H = Z/2, and let ψ be the submeasure from Theorem 1.2. Then L0 (ψ, Z/2) is extremely amenable by Theorem 1.1 as ψ is diffused. Each element of L0 (ψ, Z/2) has order 2, so there are increasing sequences (Kn ) of compact, in fact finite, subgroups of L0 (ψ, Z/2) with n Kn dense. Finally, L0 (ψ, Z/2) is not a Lévy group by Theorem 1.2. One ingredient of the proof of Theorem 1.2 is Lemma 4.1, in which we show that in a Lévy group each increasing sequence of compact subgroups with dense union retains a certain amount of concentration. Developing this theme, in Lemma 4.4, we give a simple method of constructing non-Lévy increasing sequences of compact subgroups with dense unions and apply it in Propositions 4.5 and 4.6 producing non-Lévy sequences in Lévy groups. This answers a question of Pestov [17, p. 155]. In a different direction, we generalize the concentration of measure method from [5] and [16] to prove Proposition 4.7. On the one hand, this generalizes the Glasner–Furstenberg– Weiss–Pestov result that L0 (μ, H ) is Lévy for a compact group H and a diffused measure μ. On the other hand, it gives examples of submeasures φ with no measure below, as in the Herer– Christensen result discussed earlier, such that L0 (φ, H ) is Lévy. Such examples were not known according to the earlier version of [17]. Additional notation and conventions. For a set x, we write |x| for the cardinality of x. The set of natural numbers, denoted by N, includes 0. A natural number n ∈ N will be sometimes identified with the set of all smaller natural numbers, so n = {0, . . . , n − 1}. Given two sets x and y, by x y we denote the set of all functions from y to x. 2. A Ramsey-type result Our proof of extreme amenability will involve a Ramsey result for groups Z/p with p a prime, Theorem 2.1. In fact, we will only need a consequence of this result which is stated as Corollary 2.2 at the end of this section. Let A be a set. Let n l be positive natural numbers. We denote by An:l the set of all partial functions from n to A whose domain has at least n − l elements. If A = Z/p, h ∈ (Z/p)n:l , and r ∈ Z/p, we write r + h for the element of (Z/p)n:l whose domain is equal to that of h and that is such that (r + h)(i) = r + h(i) for any i in the domain of h. A subset L of (Z/p)n:l is called full if there exist h ∈ (Z/p)n and a ⊆ n such that |a| n − l and for any r ∈ Z/p, (r + h) ar ∈ L for some ar ⊆ n with a ⊆ ar .
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Theorem 2.1. Let p1 , . . . , pk be prime numbers. Let d ∈ N, d > 0. Then ∃l1 ∀n1 l1 ∃l2 ∀n2 l2 . . . ∃lk ∀nk lk
for any d-coloring of
k (Z/pi )ni :li i=1
n1 :l1
there exist full sets L1 ⊆ (Z/p1 )
nk :lk
, . . . , Lk ⊆ (Z/pk )
such that L1 × · · · × Lk is monochromatic. It may be instructive for the reader to phrase and prove the above theorem for k = 1, p1 = 2, and d = 2. The proof in this special case is rather simple but, unlike in other Ramsey type results, the 2-color version is not representative of the general case. Needless to say it is the general case that is used in applications. See also the remarks following the proof of Corollary 2.2 below. Let us first review terminology and results from algebraic topology that will be used in the proof. More details can be found in [11, p. 138]. We recall the basic notions around abstract complexes. Let X be a finite set. By a complex with vertex set X, we understand a closed under subsets family F of subsets of X. We call elements of F the simplices of the complex. Given two complexes F1 on X and F2 on Y , a function f : X → Y is called simplicial if {f (x): x ∈ F } ∈ F2 for any F ∈ F1 . For a group H , we say that a complex F with vertex set X is an H -complex if there is an action of H on X such that {hx: x ∈ F } ∈ F for any F ∈ F and h ∈ H . Given two H -complexes F1 on X and F2 on Y , we say that a simplicial map f : X → Y is equivariant if f (hx) = hf (x) for any x ∈ X and h ∈ H . If such an equivariant simplicial function exists, we write H
F1 −→ F2 . For a complex F , define the complex sd(F) by declaring its set of vertices to be F and by letting sd(F) = {C ⊆ F: C = ∅ and ∀F1 , F2 ∈ C, F1 ⊆ F2 or F2 ⊆ F1 }. We say that sd(F) is obtained from F by the barycentric subdivision. By sdk , with k ∈ N, we will indicate the k-fold iteration of the barycentric subdivision operation. If F is an H -complex, then sd(F) with the natural induced action of H is an H -complex as well. If K is a simplicial complex, the polyhedron of its geometric representation is denoted by K . If K is an H -complex, the action of H on K naturally induces an action of H on
K . A simplicial map f between simplicial complexes K and L induces a continuous function from K to L which sends geometric simplexes to simplexes. Moreover, if K and L are H -complexes and f is equivariant, then the induced map is also equivariant. The following complex is a basic object. Let n be a natural number. Let (Z/p)∗ n be the complex whose vertices are ordered pairs (i, r) with i < n and r ∈ Z/p. A set of vertices is a simplex of (Z/p)∗ n if it forms a partial function from n to Z/p. The complex (Z/p)∗ n is a Z/p-complex when considered with the following action on the vertices s, (i, r) → (i, s + r), where s ∈ Z/p and (i, r) is a vertex of (Z/p)∗ n . We will use the following theorem which is an immediate consequence of Proposition 6.2.4 and Theorem 6.3.3 in [11].
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Let k ∈ N. If n d · (p − 1) + 1, then for any continuous function from sdk ((Z/p)∗ n ) to Rd there exists an orbit of the Z/p action on sdk ((Z/p)∗ n ) that is mapped to a single point in Rd . Proof of Theorem 2.1. We start with a claim. Claim. Let p be a prime number. For any d ∈ N, d > 0, there exists l ∈ N such that for any n l any d-coloring of (Zp )n:l has a monochromatic full set. Proof of Claim. We consider (Z/p)n:l to be the set of vertices of the simplicial complex n:l for which for any h , h ∈ C either (Z/p)n:l 1 2 c whose simplices are precisely those C ⊆ (Z/p) h1 ⊆ h2 or h2 ⊆ h1 . Consider it with the action of Z/p given by Z/p × (Z/p)n:l (r, h) → r + h ∈ (Z/p)n:l . Now we relate the complex (Z/p)n:l c to the complexes obtained by iterated application of the barycentric subdivision to (Z/p)∗ (l+1) . An immediate checking gives . (2.1) sd (Z/p)∗ (l+1) = (Z/p)l+1:l c We further show that for any n > l Z/p → (Z/p)n+1:l . sd (Z/p)n:l c c
(2.2)
To see this, let h1 ⊆ h2 ⊆ · · · ⊆ hk be a simplex in (Z/p)n:l c , that is, v = {h1 , . . . , hk } is a vertex ). If the domain of h has > n − l elements, then hk is a vertex in (Z/p)n+1:l , and in sd((Z/p)n:l k c c in this case map v to hk . If the domain of hk has precisely n − l points, in which case hk = h1 and v = {h1 }, map v to hk ∪ {(n, hk (m))}, where m is the minimal element of the domain of hk . Note . One easily checks that this defines an equivariant that hk ∪ {(n, hk (m))} a vertex in (Z/p)n+1:l c simplicial map witnessing (2.2). Combining (2.1) and (2.2), we see that for n > l Z/p sdn−l (Z/p)∗ (l+1) → (Z/p)n:l c .
(2.3)
Let now c : (Z/p)n:l → d be a d-coloring. Let ej , j = 0, . . . , d − 1, be the standard basic vectors in Rd . Let Δ be the simplex whose vertex set is {ej : j = 0, . . . , d − 1} (and whose simplices are all non-empty subsets of the vertex set). Consider the simplicial map f from (Z/p)n:l c to Δ induced by the function (Z/p)n:l h → ec(h) . By composing the function given by (2.3) with f we get a simplicial map sdn−l (Z/p)∗ (l+1) → Δ which induces a continuous map from sdn−l ((Z/p)∗ (l+1) ) to the geometric simplex Δ ⊆ Rd spanned by the vectors e0 , . . . , ed−1 . Now if we take l + 1 d · (p − 1) + 1, by the theorem stated as the final remark preceding this proof, there exists a point in sdn−l ((Z/p)∗ (l+1) ) whose
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whole Z/p orbit is mapped by the above function to a singe point in Δ . Since the function in (2.3) preserves the action, we obtain a point x0 ∈ (Z/p)n:l c such that F (x0 ) = F (r + x0 )
for each r ∈ Z/p,
(2.4)
where F : (Z/p)n:l c → Δ is the continuous induced by f . Since F (x0 ) is a point in Δ , at least one of its coordinates, say i0 , is not equal to 0. Point x0 lies in the geometric realization of a maximal simplex of (Z/p)n:l c , that is, of a simplex of the form hl ⊆ hl−1 ⊆ · · · ⊆ h0 where hi ∈ (Z/p)n:l has n − i elements in its domain. Fix r ∈ Z/p. Point r + x0 lies in the geometric realization of the maximal simplex r + hl ⊆ r + hl−1 ⊆ · · · ⊆ r + h0 . By (2.4), the i0 th coordinate of F (r + x0 ) is non-zero. Thus, there is 0 jr l such that the point of (Z/p)n:l c corresponding to r + hjr is mapped by F to a point with non-zero i0 th coordinate. By the definition of F , this implies that ec(r+hjr ) has non-zero i0 th coordinate, which means that c(r + hjr ) = i0 . Since {r + hjr : r ∈ Z/p} is a full set, Claim is established.
2
We now prove the theorem by induction on k. For k = 1 it is simply Claim. Assume our theorem holds for k. In order to prove it for k + 1, we pick li for i k given l1 , n1 , . . . , li−1 , ni−1 using our inductive assumption. Here is how to pick lk+1 given l1 , n1 , . . . , lk , nk . Apply Claim with the number of colors equal to k d = d · (Z/pi )ni :li i=1
to obtain l. Let lk+1 be equal to this l. Now given a d-coloring c of the product consider the following d -coloring c of (Z/pk+1 )nk+1 :lk+1 :
k+1
ni :li i=1 (Z/pi )
k ni :li c (h) = c(h1 , . . . , hk , h): (h1 , . . . , hk ) ∈ (Z/pi ) .
i=1
Let Lk+1 ⊆ (Z/pk+1 )nk+1 :lk+1 be a c -monochromatic full set. For an h ∈ Lk+1 let c (h1 , . . . , hk ) = c(h1 , . . . , hk , h) be a d-coloring of ki=1 (Z/pi )ni :li . Note that since Lk+1 is c -monochromatic, the definition of c does not depend on the choice of h ∈ Lk+1 . Find full sets L1 ⊆ (Z/p1 )n1 :l1 , . . . , Lk ⊆ (Z/pk )nk :lk
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such that L1 × · · · × Lk is c -monochromatic. Then it is clear that L1 × · · · × Lk × Lk+1 is c-monochromatic. 2 J. Matoušek has pointed out to us that an isomorphic version of the simplicial complex (Z/p)n:l c used in the proof above occurs in [10] (it is denoted there by K), where the lower bound for its Z/p-index is derived by Sarkaria’s trick. In the sequel, we will need only the following consequence of Theorem 2.1. In its statement, for X ⊆ ki=1 (Z/pi )ni we write X − X = {h1 − h2 : h1 , h2 ∈ X}. Corollary 2.2. Let p1 , . . . , pk be prime numbers. Let d ∈ N, d > 0. Then ∃l1 ∀n1 l1 ∃l2 ∀n2 l2 . . . ∃lk ∀nk lk for any d-coloring of ki=1 (Z/pi )ni there exist v1 ⊆ n1 , . . . , vk ⊆ nk and a color X such that |v1 | = l1 , . . . , |vk | = lk and for every f ∈ ki=1 Z/pi there is h ∈ X − X satisfying h(j ) = f (i) for all i k and all j ∈ ni \ vi . Proof. We claim that, given p1 , . . . , pk and d, any sequence l1 , n1 , . . . , lk , nk obtained by using Theorem 2.1 is as required. Given a coloring c of ki=1 (Z/pi )ni define a coloring c of k ni :li by letting c (h) = c(h), ˇ where hˇ ⊇ h is an extension defined by h(j ˇ ) = 1 for i=1 (Z/pi ) j∈ / dom(h). An application of Theorem 2.1 to c completes the proof. 2 Note that Theorem 2.1 becomes false even when k = 1 if Z/p is replaced by Z. In fact, for every l 1, for n = 2l + 1, we have a partition Zn = X1 ∪ X2 such that for every h ∈ Zn and color X ∈ {X1 , X2 } there is a ∈ Z such that |{j : f (j ) = a + h(j )}| l + 1 for every f ∈ X. To see it, let f ∈ X1 if |{j : f (j ) > 0}| l + 1 and f ∈ X2 , otherwise. Since for every h ∈ Z2l+1 we can find a and a such that a + h(j ) > 0 for all j and a + h(j ) < 0 for all j , the conclusion follows. On the other hand, it is not difficult to see that Corollary 2.2 with k = 1 and d = 2 and with Z/p replaced by Z (or any other group) remains true already with l = 1. However, the argument does not seem to generalize to the higher values of d. 3. Extreme amenability of L0 In this section, we will prove the following theorem of which Theorem 1.1 is an immediate consequence. Theorem 3.1. Let φ be a diffused submeasure. Let H be a second countable group such that the closure of each finitely generated subgroup of H is compact and solvable. Then L0 (φ, H ) is extremely amenable. We will be using the following old characterization of extreme amenability due to Pestov [15] (see also [17, Theorem 3.4.9]): G is extremely amenable if and only if SS −1 is dense for every S ⊆ G such that F S = G for some finite F . In the following lemma, we register some well-known preservation properties of extremely amenable groups.
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Lemma 3.2. Let G be a topological group. (i) Let H be a closed normal subgroup of G. If both H and G/H are extremely amenable, then so is G. (ii) If G is the union of a family of extremely amenable subgroups which is directed under inclusion, then G is extremely amenable. (iii) If G contains a dense extremely amenable subgroup, then G is extremely amenable. (iv) Let (Hn ) be a sequence of topological groups, and let πn : Hn+1 → Hn be surjective continuous homomorphisms such that G = lim ←−(Hn , πn ). If each Hn is extremely amenable, then so is G. (v) The product of an arbitrary family of extremely amenable groups is extremely amenable. (vi) Continuous homomorphic images of extremely amenable groups are extremely amenable. Proof. These are all well known. Points (iii) and (vi) are immediate from the definition of extreme amenability, as are (i) and (ii) if one only realizes that the set of fixed points of a continuous action of a topological group on a compact space is compact. Point (v) is a consequence of (i), (ii) and (iii). Point (iv) follows from Pestov’s characterization of extreme amenability stated in the beginning of the present section. Indeed, let G = F S for some finite F ⊆ G. Let πn∞ : G → Hn be the projection. It follows from our assumptions that each πn∞ is surjective. Thus, since each Hn is extremely amenable, πn∞ (S)πn∞ (S)−1 is dense in Hn for each n. This immediately implies that SS −1 is dense in G since the topology on G is generated by sets of the form (πn∞ )−1 (U ) for U ⊆ Hn open in Hn . 2 Now we have a couple of lemmas giving an analysis of groups S(φ, H ). Lemma 3.3. Let H be a compact, second countable, Abelian group. Then H = lim ←−(Hn , πn ), where each Hn is a compact, second countable, Abelian group in which torsion elements are dense, and πn : Hn+1 → Hn is a continuous surjective homomorphism. Proof. Let T = R/Z. Since each compact second countable Abelian group is isomorphic to a closed subgroup of (T)N , it is the inverse limit, with continuous surjective homomorphisms as bonding maps, of closed subgroups of (T)n with n ∈ N. The lemma follows, therefore, from the fact that the torsion elements are dense in any closed subgroup H of (T)n with n ∈ N. This can be seen as follows. There is a surjective homomorphism from the dual group of (T)n , which is Zn , onto the dual group of H . Thus, this last group is finitely generated and, therefore, it is the product of a finite Abelian group and Zk , for some k ∈ N. Now H being the dual of such a group is isomorphic to H × (T)k with H finite. 2 Lemma 3.4. Let φ be a submeasure. (i) Let (Hn ) be a sequence of topological groups, and let πn : Hn+1 → Hn be continuous surjective homomorphisms. Then πn∗ : S(φ, Hn+1 ) → S(φ, Hn ) given by πn∗ (f ) = πn ◦ f is a surjective continuous homomorphism and ∗ S φ, lim ←−(Hn , πn ) = lim ←− S(φ, Hn ), πn .
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(ii) If H is a dense subgroup of a topological group H , then S(φ, H ) is a dense subgroup of S(φ, H ). (iii) If H is the union of a directed by inclusion family of subgroups H, then S(φ, H ) = {S(φ, H ): H ∈ H}. (iv) Let H be a closed normal subgroup of a topological group H . Then S(φ, H /H ) = S(φ, H )/S(φ, H ). ∗ Proof. (i) Define r : lim ←−(S(φ, Hn ), πn ) → S(φ, lim ←−(Hn , πn )) by letting
r (fn ) (x) = fn (x) . We leave it to the reader to check that r is a bijective homomorphism which is a homeomorphism. (ii) is immediate as is (iii) which, however, depends on the fact that each function in S(φ, H ) attains finitely many values. (iv) Define r : S(φ, H ) → S(φ, H /H ) be r(f )(x) = f (x)/H . One checks that r is a continuous surjective homomorphism whose kernel is S(φ, H ). Thus, it induces a bijective continuous homomorphism from S(φ, H )/S(φ, H ) onto S(φ, H /H ). We leave it to the reader to check that this homomorphism is open. 2 Extreme amenability will be proved using the lemma below. Lemma 3.5. Assume G is a topological group such that for every partition G = X1 ∪ · · · ∪ Xd , every finite F ⊆ G and every open neighborhood W of the identity there is t d such that F ⊆ W Xt (Xt )−1 . Then G is extremely amenable. Proof. We will use Pestov’s characterization of extreme amenability stated at the beginning of this section. Let S ⊆ G be such that G = td gt S. Put Xt = gt S. By our assumption, for every W and finite F there is t = t (W, F ) d such that F ⊆ W Xt (Xt )−1 . We claim there is t = t (W ) d such that G = W Xt (Xt )−1 . Otherwise, for every t d there is ht ∈ G \ W Xt (Xt )−1 . Then F = {h1 , . . . , hd } is not included in W Xt (Xt )−1 for any t d, contradicting our assumption. A similar argument shows that there is t0 d such that G = U Xt0 (Xt0 )−1 for every open is dense in G, which neighborhood U of the identity. Therefore, Xt0 (Xt0 )−1 = gt0 SS −1 gt−1 0 makes SS −1 dense, and the extreme amenability follows. 2 A condition easily equivalent to the one in Lemma 3.5 was considered by Christensen already in [3, p. 104]. In fact, the condition from the lemma is equivalent to extreme amenability as proved in [17, Theorem 3.4.9]. Proof of Theorem 3.1. By Lemmas 3.2(iii), 3.4(iii), and 3.2(ii), it suffices to prove the theorem for S(φ, H ), where H second countable, compact and solvable. Now, by Lemmas 3.4(iv) and 3.2(i), it is enough to prove it for H second countable compact and Abelian. By Lemmas 3.3, 3.4(i), and 3.2(iv), we can assume that H is second countable, compact, Abelian, and that the torsion elements are dense in H . Therefore, by applying Lemma 3.4(ii) and (iii) and Lemma 3.2(ii) and (iii), we reduce proving the theorem to the case when H is finite. One more application of Lemmas 3.4(iv) and 3.2(i) shows that we can restrict ourselves to H = Z/p for a prime number p. We consider this case in what follows.
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By Lemma 3.5 it suffices to prove that S(φ, Z/p) has the following property: Let X1 ∪ · · · ∪ Xd = S(φ, Z/p), let F ⊆ S(φ, Z/p) be finite, and let W ⊆ S(φ, Z/p) be an open neighborhood of 0. Then there exists t0 d with F ⊆ W + (Xt0 − Xt0 ). Our intention is to use Corollary 2.2. We can assume that F consists of all elements of S(φ, Z/p) which are constant on sets in a fixed partition b1 , . . . , bk of X into sets from B. This is because each finite subset of S(φ, Z/p) is contained in a set of this form. Since φ is diffused, we can also assume that for some > 0, W = g ∈ S(φ, Z/p): φ x ∈ X: g(x) = 0 < . At this point we are given , the number k of elements in the partition of X, and the number d of elements in the partition of S(φ, Z/p). With these k and d, use Corollary 2.2 to find l1 . Partition b1 into finitely many sets in B of φ-submeasure < δ1 such that 0 < δ1 · l1 < /k. (We use diffusion of φ here.) Let n1 be the number of elements in this partition of b1 . We can assume that n1 l1 . Now given l1 , n1 use Corollary 2.2 to find l2 . Repeat what was done above with b2 replacing b1 . That is, partition b2 into finitely many sets in B of φ-submeasure < δ2 such that 0 < δ2 · l2 < /k. Let n2 , which we can assume to be l2 , be the number of elements in this partition of b2 . Continue in this fashion using Corollary 2.2 until all sets bi are taken care of. For each 1 i k, let a0i , a1i , . . . , ani i −1 be the partition of bi constructed above. Now we describe a coloring of ki=1 (Z/p)ni into d colors. Given c¯ = (c1 , . . . , ck ) ∈
k (Z/p)ni , i=1
consider gc¯ ∈ S(φ, Z/p) defined by gc¯ aji = ci (j ). The color assigned to c¯ = (c1 , . . . , ck ) is equal to the subscript of the element of the partition X0 , . . . , Xd−1 of S(φ, Z/p) to which gc¯ belongs. By our choice of the numbers li and ni there are h1 ∈ (Z/p)n1 , . . . , hk ∈ (Z/p)nk , t0 < d, and v1 ⊆ n1 , . . . , vk ⊆ nk such that |v1 | = l1 , . . . , |vk | = lk , and for every d¯ = (d1 , . . . , dk ) ∈ (Z/p)k there are f¯ and f¯ , both colored by t0 , such that di = f¯(j ) − f¯ (j ) for all i and all j ∈ ni \ vi . Every g ∈ F is constant on each bi , hence by the last observation there are gf¯ and gf¯ satisfying
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k i aj : j ∈ v i φ x: g(x) = (gf¯ − gf¯ )(x) φ i=1
k
φ aji < k · /k = . i=1 j ∈vi
It follows from it that g ∈ W + (gf¯ − gf¯ ). Since g ∈ F was arbitrary and since gf¯ , gf¯ ∈ Xt0 , we see that F ⊆ W + Xt0 − Xt0 , and we are done. 2 4. Extreme amenability and Lévy groups In this section, we study connections between extreme amenability and being Lévy and between being Lévy and having non-Lévy sequences of compact subgroups. This is done principally for groups of the form L0 (φ, H ), but also for the unitary group of the separable Hilbert space and the group of measure preserving transformations. We start with a lemma proving a property of general Lévy groups. This property may be of some independent interest and we make some additional comments on it in Section 5. It says that in a Lévy group all increasing sequences of compact subgroups with dense unions exhibit some degree of concentration as in (4.1). This degree of concentration has to be carefully calibrated since, as shown by Propositions 4.5 and 4.6, in general Lévy groups arbitrary increasing sequences of compact subgroups with dense unions are not Lévy, that is, full concentration fails on them. The immediate usefulness of the lemma below is in proving that a given group is not Lévy. The lemma allows us to replace the necessity of showing that no increasing sequence of compact subgroups with dense union is Lévy by constructing one increasing sequence of compact subgroups with dense union on which concentration fails in a uniform way, that is, on which condition (4.1) is violated. Lemma 4.1. Let a topological group G be Lévy. Then for any increasing sequence (Kn ) of compact subgroups of G with dense union, an open neighborhood V of 1, a compact set F ⊆ G, and Borel sets An ⊆ Kn , we have lim inf νn (V An ) − νn (FAn ) 0, n
(4.1)
where νn is the probability Haar measure on Kn , considered as a measure on G with support Kn . Proof. Assume the conclusion fails and pick (Kn ), V , F , and An witnessing the failure of (4.1). By going to a subsequence of (Kn ), we can suppose that there are δ1 , δ2 so that νn (FAn ) > δ2 > δ1 > νn (V An ).
(4.2)
Let (Hm ) be an arbitrary increasing sequence of compact subgroups of G with dense union. We will show that the sequence is not Lévy, thereby proving that G is not a Lévy group. Let μm
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be the probability Haar measure on Hm . Let W 1 be an open neighborhood of 1 with W 2 ⊆ V . Using compactness of F , find m0 such that F ⊆ Hm0 W and h1 , . . . , hp ∈ Hm so that F⊆
p
(4.3)
hi W.
i=1
We claim that there are constants β1 , β2 > 0 such that for m m0 we can find Borel C ⊆ Hm with μm (C) >
β2 δ2 p
and μm (Hm \ W C) > β1 (1 − δ1 ),
(4.4)
thereby proving that (Hm ) is not a Lévy sequence. Let W = the interior of
f Wf −1 .
(4.5)
f ∈F
The open set W is a neighborhood of 1 since F is compact. Let m m0 be fixed. Fix n large enough so that Hm ⊆ W Kn .
(4.6)
Claim. Let B ⊆ Kn be a measurable set and let α, β 0, α + β = 1. Then νn g ∈ Kn : μm W Bg −1 βνn (B) ανn (B). Proof. Consider the compact group Hm × Kn with the product measure μm × νn . By (4.6) all vertical sections of the set {(h, g) ∈ Hm × Kn : hg ∈ W } are non-empty. By the translation invariance of νn each vertical section of X = (h, g) ∈ Hm × Kn : hg ∈ W B is of νn -measure at least νn (B). By Fubini’s theorem, {g ∈ Kn : the horizontal section of X at g has μm -measure βνn (B)} has νn -measure at least ανn (B). But the horizontal section of X at g is Hm ∩ W Bg −1 . 2 Using (4.2), we can fix 0 < α1 , α2 < 1 such that α1 (1 − δ1 ) + α2 δ2 > 1.
(4.7)
Set β1 = 1 − α1 and β2 = 1 − α2 . We show that for this choice of β1 and β2 (4.4) holds. Applying the claim to the set Kn \ V An with α1 and β1 and using (4.2), we get a set of g ∈ Kn of νn -measure α1 νn (Kn \ V An ) > α1 (1 − δ1 ) such that μm W (Kn \ V An )g −1 β1 νn (Kn \ V An ) > β1 (1 − δ1 ).
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For each such g we have μm Hm \ W V An g −1 > β1 (1 − δ1 ).
(4.8)
Now apply the claim to the set FAn with α2 and β2 in combination with (4.2) to obtain a set of g ∈ Kn of νn -measure α2 νn (FAn ) > α2 δ2 such that μm W FAn g −1 > β2 νn (FAn ) > β2 δ2 .
(4.9)
Therefore, using (4.5) and (4.3), we obtain W FAn = W FAn ⊆ F W An ⊆ h1 W 2 An ∪ · · · ∪ hp W 2 An ⊆ h1 V An ∪ · · · ∪ hp V An . In view of this, (4.9) implies μm h1 V An g −1 ∪ · · · ∪ hp V An g −1 > β2 δ2 . Since μm is invariant under translations by h1 , . . . , hp as they are elements of Hm , we see that for a set of g ∈ Kn of νn -measure > α2 δ2 we have β2 δ2 . μm V An g −1 > p
(4.10)
Taking into account (4.7), we get g0 ∈ Kn for which both (4.8) and (4.10) hold. Thus, (4.4) is fulfilled by C = V An g0−1 ∩ Hm . 2 Theorem 4.2 below is a more precise version of Theorem 1.2. We need some definitions. Let Mi , i ∈ N, be a sequence of non-zero natural numbers. Put Σ = ∞ i=0 j
σ ∈I
The submeasure ψ0 coincides with the first element of the sequence of the so-called auxiliary measures in the definition of the one-dimensional Hausdorff measure on i Mi taken with the metric d(x, y) = 2−min{i: xi =yi } [18, p. 50]. Theorem 4.2. For every compact group H that is not connected there is a sequence of natural numbers Mi , i ∈ N, such that the submeasure ψ0 on the algebra of all clopen subsets of i Mi has the following properties: (i) it is diffused; (ii) L0 (ψ0 , H ) is not a Lévy group.
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In order to state the next lemma, we need some definitions. Let M and d be natural numbers. For g : M → d and j0 < d, let μ(g) = j0 if l < M: g(l) = j0 = max l < M: g(l) = j j
(4.11)
and the value of min{l < M: g(l) = j } for j = j0 is the smallest of all such numbers evaluated for j realizing the maximum on the right-hand side of (4.11). Assume non-zero natural numbers M0 , . . . , Mk and d are given. Let Xk = xσ : σ ∈ M0 × · · · × Mi : ∀σ xσ ⊆ M|σ | and |xσ | < 2|σ | . i
Let f ∈ d M0 ×···×Mk , and let x¯ ∈ Xk . For i k define fx¯i : M0 × · · · × Mi → d as follows. Put fx¯k = f . Assume fx¯i+1 is defined. Let σ ∈ M0 × · · · × Mi . Put fx¯i (σ ) = μ(g), where g : Mi+1 → d is given by i+1 g(l) = fx¯ (σ l), if l ∈ Mi+1 \ xσ ; 0, if l ∈ xσ . This defines fx¯i for all i k and all x¯ ∈ Xk . Lemma 4.3. Let > 0 and d ∈ N be given. There exists a sequence of natural numbers M0 , M1 , . . . such that for each k f ∈ d M0 ×···×Mk : ∃x¯ ∈ Xk f 0 (∅) = 0 < 1 + · d M0 ···Mk . x¯ d Proof. First we specify how fast the sequence Mi , i 0, grows. We will use a probability argument involving the claim below, which is a consequence of the central limit theorem. Claim. Let d 1 be a natural number. Let m ∈ N and Δ > 0 be given. There exist M ∈ N and δ > 0 such that if Y0 , . . . , YM−1 are independent random variables taking values 0, . . . , d − 1 so that for any i < M and k < d Pr(Yi = k) − 1 < δ, (4.12) d then 1 Pr {i < M: Yi = 0} + 2m > max {i < M: Yi = k} < + Δ. d 0k
(4.13)
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Proof of Claim. Let X0 , X1 , . . . be a sequence of independent random variables with Pr(Xi = k) =
1 d
(4.14)
d for k = 0, . . . , d −1. For each i define a random variable Zi with values in R by letting Zi = eXi , d where e0 , . . . , ed−1 is the standard orthonormal basis in R . Put SM = i<M Zi . Note that
Pr {i < M: Xi = 0} + 2m > max {i < M: Xi = k} 0k
SM SM 2m = Pr · e0 + > max · ek , M M 0k
(4.15)
√ where · is the ordinary inner product. For a fixed m, 2m/M goes to 0 faster than 1/ M and the distribution of each Zi is invariant under the orthonormal transformations of Rd induced by any permutation of the vectors ei ; thus, by the central limit theorem [1, Theorem 29.5], for large M not depending on the particular sequence X0 , X1 , . . . , the second term in (4.15) is < d1 + Δ2 . Therefore, 1 Δ Pr {i < M: Xi = 0} + 2m > max {i < M: Xi = k} < + . d 2 0k
(4.16)
Δ Fix M as above. Let δ = 2M . Let Y0 , . . . , YM−1 be independent random variables fulfilling (4.12). It is easy to see that given X0 , . . . , Xi−1 each fulfilling (4.14) and such that X0 , . . . , Xi−1 , Yi , . . . , YM are independent, there is Xi fulfilling (4.14) such that X1 , . . . , Xi−1 , Xi , Yi+1 , . . . , YM−1 are independent and Pr(|Xi − Yi | = 0) < δ. Note now that the probability that one of the events |{i < M: Xi = k}| = |{i < M: Yi = k}| happens with 0 k < d is < δ · M < Δ/2. Thus, it follows from (4.16) that (4.13) holds. 2
Now we are ready to finish proving Lemma 4.3. Let > 0 be given. Let δ−1 = . If δi > 0 is defined, using Claim with m = 2i and Δ = δi , find Mi+1 ∈ N and δi+1 > 0. This describes a sequence (Mi )∞ (δi )∞ i=0 along with an auxiliary sequence of positive real numbers i=−1 . Fix k. We show by backward induction on i0 k that for any σ0 ∈ i
1 + δi0 −1 d M0 ···Mk . d
For i0 = 0 this gives the conclusion of the lemma. The estimate clearly holds for i0 = k. Assume this is true for all σ0 l with l ∈ Mi0 . Define Yl , l < Mi0 , to be independent random variables such that Yl = 0 with probability 1 d M0 ···Mk
f ∈ d M0 ×···×Mk : ∃y¯ ∈ Xk f i0 +1 (σ l) = 0 , 0 y¯
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which by inductive assumption is < d1 + δi0 . Obviously, it is d1 . Further, let Pr(Yl = i) = Pr(Yl = i ) for 1 i, i < d. It follows that for all i < d 1 1 − δi0 Pr(Yl = i) < + δi0 . d d
(4.17)
Fix x¯ ∈ Xk for a moment. Note now that from the definition of f i0 in the following list each condition implies the next one: i
fx¯ 0 (σ0 ) = 0, l: f i0 +1 (σ l) = 0 + 2i0 > max l: f i0 +1 (σ l) = i − 2i0 , 0 0 x¯ x¯
(4.18b)
l: ∃y¯ ∈ Xk f i0 +1 (σ l) = 0 + 2 · 2i0 0 y¯ i +1 i +1 > max l: ∀y¯ ∈ Xk fy¯ 0 (σ0 l) > 0 and fx¯ 0 (σ0 l) = i .
(4.18c)
i>0
0
(4.18a)
The left-hand side of (4.18c) does not depend on x. ¯ We analyze now the right-hand side i +1 M ×···×M k 0 of (4.18c). The set of all f ∈ d with ∀y¯ ∈ Xk fy¯ 0 (σ0 l) > 0 does not depend on i +1
x. ¯ Furthermore, in this set, for a given x, ¯ the fraction of functions f with fx¯ 0 (σ0 l) = i is the same for all 0 < i < d, and this fraction does not depend on x. ¯ Thus, the implications of (4.18) give the first inequality in the formula below; the second inequality follows from (4.17) and from the choice of δi0 and Mi0 . 1
f ∈ d M0 ×···×Mk : ∃x¯ ∈ Xk f i0 (σ0 ) = 0 x¯
d M0 ···Mk Pr {l < Mi0 : Yl = 0} + 2 · 2i0 > max{l < Mi0 : Yl = i} i
1 < + δi0 −1 . d Thus, we obtain our inductive conclusion.
2
Proof of Theorem 4.2. By our assumption H is a compact group that is not connected, hence it contains a proper open subgroup V . The index of V in H , d = [H : V ], is finite. Pick (Mi ) so that the conclusion of Lemma 4.3 holds with d and some 0 < 0 < 1/2. The submeasure ψ0 is as in the paragraph preceding the statement of Theorem 4.2. Point (i) is immediate since for each n ∈ N, i Mi is covered by {[σ ]: |σ | = n}, and ψ0 ([σ ]) 2−n if |σ | = n. We prove (ii). We aim to apply Lemma 4.1. Let Kn consist of all elements of S(ψ0 , H ) that are constant on all [σ ] for σ ∈ Σ with |σ | = n + 1. Clearly Kn is a compact group, Kn ⊆ Kn+1 , and n Kn = S(ψ0 , H ) is dense in L0 (ψ0 , H ).
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Let F be a finite subset of H intersecting each left coset of V . Let F be the finite set of constant functions in S(ψ0 , H ) with values in F . Let V ⊆ L0 (ψ0 , H ) be an open set whose intersection with S(ψ0 , H ) gives the relatively open in S(ψ0 , H ) set
f ∈ S(ψ0 , H ): ψ0 x: f (x) ∈ / V < 1/2 .
Using Lemma 4.3, we will find Borel sets An ⊆ Kn such that F An = Kn
and νn (V An )
1 + 0 , d
(4.19)
where νn is the Haar probability measure on Kn . This clearly implies that (4.1) fails since 0 < 1 − d1 , and therefore L0 (ψ0 , H ) is not Lévy. We identify H /V with d. Since each f ∈ Kn is constant on each [σ ] for σ ∈ Σ with |σ | = n + 1, each such f induces a function fˆ : M0 × · · · × Mn → d by fˆ(σ ) = f (x)/V for any x ∈ [σ ], σ ∈ M0 × · · · × Mn . Recall the notation from Lemma 4.3. Let 0¯ be the element of Xn with all its entries equal to the empty set. The set An = f ∈ Kn : fˆ0¯0 (∅) = 0 is clearly Borel and F An = Kn . It remains to check the second part of (4.19). Let h ∈ V An ∩Kn . Pick f ∈ An such that h ∈ V f . We have ψ0
ˆ ) < 1. [σ ]: σ ∈ Σ, |σ | = n + 1, and fˆ(σ ) = h(σ 2
Let I ⊆ Σ be such that
τ ∈I
ˆ ) ⊆ [σ ]: σ ∈ Σ, |σ | = n + 1 and fˆ(σ ) = h(σ [τ ]: τ ∈ I
and
1 2−|τ | < . 2
(4.20)
Note that the condition in Lemma 4.3 guarantees that Mk 2k for each k. With this in mind, we see that the second part of (4.20) gives that if |σ | = n+1, then [σ ] = {[τ ]: τ ∈ I, σ ⊆ τ, σ = τ } allowing us to remove from I any τ with |τ | > n + 1. Thus, we assume that |τ | n + 1 for each τ ∈ I . For σ ∈ i
i
× ··· ×
hˆ 0x¯ (∅) fˆ0¯0 (∅) = 0. Thus, V An ∩ Kn ⊆ h ∈ Kn : ∃x¯ ∈ Xn hˆ x0¯ (∅) = 0 .
(4.21)
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Since the map Kn h → hˆ ∈ d M0 ×···×Mn is such that for any D ⊆ d M0 ×···×Mn we have νn {h ∈ Kn : hˆ ∈ D} =
|D| , d M0 ···Mn
it follows from (4.21) that νn (V An ) which is <
1 d
1 d M0 ···Mn
+ 0 by Lemma 4.3.
g ∈ d M0 ×···×Mn : ∃x¯ ∈ Xn g 0 (∅) = 0 , x¯
2
Propositions 4.5, 4.6 and 4.7 below complement Theorem 4.2. A non-Lévy sequence for G is an increasing sequence of compact subgroups with dense union that is not a Lévy sequence. Below we prove two propositions on the existence of non-Lévy sequences. In the first one, we show that groups L0 (φ, H ) with H compact disconnected and φ an arbitrary diffused submeasure contain non-Lévy sequences despite the fact that some of them are Lévy groups as proved in [16] or in Lemma 4.7. We start with a lemma on constructing non-Lévy sequences. For > 0 and a subset A of a group G with a right invariant metric d, we write (A) = {g ∈ G: d(g, h) < for some h ∈ A}. Lemma 4.4. Assume a metric group G with a right invariant metric d is the increasing union of compact subgroups Kn , n ∈ N. Assume there are ε > 0 and compact groups Ln , n ∈ N, such that Kn < Ln , supn [Ln : Kn ] < ∞, and Ln (Kn ) . Then G can be written as an increasing union of a non-Lévy sequence. Proof. Our assumption supn [Ln : Kn ] < ∞ guarantees that for each n there is n with Ln ⊆ Kn . By going to a subsequence, we can ensure two things, first that Ln ⊆ Kn+1 , and so Ln ⊆ Ln+1 , and second that for all n, [Ln : Kn ] = m for some fixed m ∈ N. If νn is the Haar measure on Ln , we have νn (Kn ) = 1/m. Further, since d is right invariant, if Ln (Kn ) , then a left coset of Kn in Ln is disjoint from (Kn ) . Thus, νn ((Kn ) ) 1 − (1/m). It follows that (Ln ) is an increasing sequence of compact subgroups whose union is G and which is not Lévy. 2 Proposition 4.5. The group L0 (φ, H ) for a diffused submeasure φ and a compact disconnected Abelian group H contains a non-Lévy sequence. Proof. Fix a diffused φ on an algebra B and a compact disconnected group H with an invariant metric dH . There is α > 0 such that for every finite subalgebra A of B there is b ∈ B satisfying φ(aΔb) > α for all a ∈ A. Otherwise, the group S(φ, Z/2) is totally bounded with respect to dφ , hence its completion L0 (φ, Z/2) is compact, contradicting its extreme amenability. Since H is disconnected, it has a closed normal subgroup G of finite index m. By the compactness of H there is 0 < ε < α such that dH (g, h) > ε for all g, h belonging to different cosets of G. Let Bn be an increasing sequence of finite algebras with union B and let Kn be the subgroup of all Bn -measurable maps. By Lemma 4.4 applied to S(φ, H ), it will suffice to find ε > 0 such that for every n there is a compact group Ln ⊇ Kn with [Ln : Kn ] = m and (Kn )ε ∩ Ln = Kn . For each n find bn such that φ(bn Δa) > ε for all a ∈ Bn . Let Bn be the subalgebra generated by Bn and bn . Then with π : H → H /G being the natural projection, Ln = {f ∈ S(φ, H ): f is
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Bn -measurable and f = f1 + f2 with f1 ∈ Kn and π ◦ f2 measurable with respect to the subalgebra generated by bn } is a compact subgroup of S(φ, H ) including Kn . For h ∈ H let h˜ ∈ L0 (φ, H ) be equal to h on bn and equal to 1 on bn . Then h˜ ∈ Ln and for F ⊆ H such that F G = H we have F˜ Kn = Ln with F˜ = {f˜: f ∈ F }. Therefore [Ln : Kn ] = m. If f and g in Ln belong to different cosets of Kn , then {x: dH (f (x), g(x)) > ε} is of the form aΔbn or aΔbn for some a ∈ Bn , hence dφ (f, g) > ε. Hence Kn is a subset of Ln of Haar measure 1/m such that (Kn )ε ∩ Ln = Kn . 2 The next proposition gives an application of Lemma 4.4 to groups other than L0 (φ, H ). Both groups in this proposition are Lévy; see [17, §4]. Proposition 4.6. The unitary group U (H) of a separable infinite-dimensional Hilbert space (with the strong topology) and the group Aut(X, μ) of measure preserving transformations of a standard measure space (with the weak topology) have non-Lévy sequences. Proof. Fix an orthonormal basis (en ) such that U (n) is the group of all operators that fix all vectors in the closed linear span of {ei | i > n}. Let gn be an involution defined by the permutation of the basis {ei }∞ / {n, n + 1}. Then i=1 that swaps en+1 with en+2 and sends ei to itself for i ∈ gn ∈ U (n + 2) and it clearly commutes with all elements of U (n). Also, for a, b in U (n) the vectors gn a(en ) = en+1 and b(e n ) = en are orthogonal, and the conclusion follows by Lemma 4.4 applied to the dense subgroup n U (n) with Kn = U (n) and Ln = Kn ∪ gn Kn . For Aut(X, μ) it suffices to consider the case when X = [0, 1] and μ is the restriction of the Lebesgue measure. Let Kn be the finite group of all transformations interchanging the intervals Ij n = [j 2−n , (j + 1)2−n ) for 0 j < 2n by translations. Since the weak topology is the weakest topology making all functions f → μ(A ∩ f [A]), for A ⊆ X measurable, continuous, by Lemma 4.4 applied to n Kn it suffices to find gn ∈ Kn+1 of order 2 which commutes with elements of Kn and is such that for some measurable Jn of measure 1/2 and all a, b in Kn we have a[Jn ] ∩ gn b[Jn ] = ∅. (Then Ln = Kn ∪ gn Kn will do the job.) Let τ ∈ S2n+1 be the permutation swapping 2j with 2j + 1 for all j < 2n . Let gn ∈ Kn+1 be the transformation moving each Ij,n+1 to Iτ (j ),n+1 by translations. Then gn ∈ Kn+1 has order 2 and it commutes with all elements of Kn . If Jn = j <2n I2j,n+1 then μ(Jn ) = 1/2, a[Jn ] = Jn for all a ∈ Kn and gn [Jn ] = [0, 1] \ Jn . 2 In contrast with Theorem 4.2, the next result shows that there is a large class of diffused submeasures φ such that L0 (φ, H ) is Lévy for any compact H . A submeasure φ on an algebra B of subsets of X is called strongly diffused if for any > 0 there exists δ > 0 for which there exist partitions P of X into arbitrarily large number of sets from B such that for any A ⊆ P if |A| < δ · |P|, then φ( A) < . Note that all diffused measures are strongly diffused. Many submeasures with no measures below are also strongly diffused, see the examples on [19, p. 98] with σn Cn for some finite constant C 2. Proposition 4.7. Let φ be a strongly diffused submeasure. (i) If H compact, then L0 (φ, H ) is a Lévy group. (ii) If H is an amenable locally compact second countable group, then L0 (φ, H ) is extremely amenable.
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Proof. (i) Fix a metric dφ on S(φ, H ) as in (1.1). Let P be a partition of X into non-empty sets from B. We will refer to such P simply as a partition. We write H (P) for the subgroup of S(φ, H ) consisting of all the functions constant on each set in P. This group is isomorphic to H k where k is the number of elements in P and by λk we will denote the probability Haar measure on it. Further, by dk we will denote the Hamming metric on H (P) given by 1 dk (f, g) = {a ∈ P: f a = g a}. k We produce a Lévy sequence consisting of groups of the form H (P) where P is a partition. If a partition P refines another partition Q, then H (Q) < H (P). Thus, it will suffice to show that given a partition Q and > 0, there is a partition P refining Q such that for any measurable A ⊆ H (P) with λ(A) > , we have λm ({f ∈ H (P): dφ (f, A) < }) > 1 − . Let k = |Q|. Let δ > 0 be as in the definition of strong diffusion chosen for the given > 0. By concentration of measure (e.g., [9]), find n0 ∈ N so that for any partition P with |P| = m n0 and any A ⊆ H (P) with λm (A) we have δ λm f ∈ H (P): dm (f, A) < > 1 − . k
(4.22)
This is possible since H (P) is isomorphic to H |P | . Find a partition P with at least n0 elements such that for any A ⊆ P with |A| < δ · |P | we have φ( A) < . Let P be the partition induced by the elements of P ∪ Q. Put m0 = |P|. Note that n0 m0 |Q| · |P |. Let A ⊆ H (P) be measurable and such that λm0 (A) > . Since (4.22) holds for m = m0 , it suffices to show that for f ∈ H (P), dm0 (f, A) < δ/k implies dφ (f, A) < . Let g ∈ A be such that dm0 (f, g) < δ/k. Then
{a ∈ P: f a = g a} < δ · m0 δ · |Q| · |P | = δ · |P |. k k It follows that {a ∈P: f a = g a} can be covered by at most δ · |P | elements of the partition P . Thus, φ( {a ∈ P: f a = g a}) < , and therefore dφ (f, A) < . (ii) We obtain this point by combining the argument above with a use of Følner sequences as in [16, Theorem 2.2]. 2 5. Remaining questions Herer and Christensen [7] proved that L0 (φ, R) is extremely amenable if φ has no measure below. Pestov [16, Theorem 2.2] showed that L0 (μ, H ) is extremely amenable if μ is a diffused measure and H is an amenable lcsc group. Neither of these results is implied by our Theorem 3.1. In fact, no group of the form L0 (φ, H ) with H Abelian lcsc containing an element which generates a subgroup whose closure is not compact is covered by Theorem 3.1. Therefore, it is natural to ask for generalizations of this theorem as in questions 1 and 2 below. 1. Is every group of the form L0 (φ, H ), for φ diffused and H compact, extremely amenable? 2. For what diffused φ is L0 (φ, Z), or L0 (φ, R), extremely amenable?
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Question 1 is related to proving an analogue of Theorem 2.1, or at least Corollary 2.2, for an arbitrary finite group in place of Z/p. In fact, it would suffice to prove it for simple finite groups. Similarly for question 2. Hence the next question, the answer to which might involve extending the methods of [4]. 3. Is there a Ramsey-type result along the lines of Theorem 2.1 or Corollary 2.2 for arbitrary finite (or simple finite) groups? Is there a Ramsey-type result analogous to Corollary 2.2 with Z replacing Z/p? A rudimentary argument shows that Corollary 2.2 with k = 1 and d = 2 and Z/p replaced by an arbitrary group remains true already with l = 1. However, the argument does not seem to generalize to the higher values of d. Now, a question related to Theorem 4.2. 4. Let φ be a diffused submeasure, and let H be a compact connected second countable group. Is L0 (φ, H ) Lévy? In Lemma 4.1 we introduced the following property of Polish groups G. (∗) For any increasing sequence (Kn ) of compact subgroups of G with dense union, an open neighborhood V of 1, a compact set F ⊆ G, and Borel sets An ⊆ Kn , we have lim inf νn (V An ) − νn (FAn ) 0, n
where νn is the probability Haar measure on Kn , considered as a measure on G with support Kn . In connection with the lemma, Pestov suggested the following question. 5. (Pestov) Let G be a Polish group containing an increasing sequence of compact groups with dense union. Assume G satisfies (∗) above. Is G Lévy? The following proposition, communicated to us by the referee, shows that (∗) implies extreme amenability. Proposition 5.1. Let G be a Polish group that contains an increasing sequence of compact groups with dense union and satisfies (∗). Then G is extremely amenable. Proof. Fix an increasing sequence (Kn ) of compact subgroups of G with dense union, and let νn be the normalized Haar measure on Kn . Fix A ⊆ G such that FA = G for some finite F , and let V be a symmetric neighborhood of the identity. Let A be the closure of A. (We consider the closure of A only because of questions of measurability.) Then (∗) implies lim sup νn V (A ∩ Kn ) lim sup νn F (A ∩ Kn ) = 1. n
n
We claim V A(A)−1 V is dense in G. Otherwise, there is g ∈ disjoint, and consequently lim supn νn (V (A ∩ Kn )) 12 .
n Kn
for which V A and gV A are
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Since V was arbitrary, A(A)−1 is dense in G and, therefore, so is AA−1 . It follows that G is extremely amenable by Pestov’s characterization stated at the beginning of Section 3. 2 Acknowledgments We thank Jiˇrí Matoušek and Vladimir Pestov for valuable comments. References [1] P. Billingsley, Probability and Measure, third ed., Wiley Series Probab. Math. Stat., Wiley, New York, 1995. [2] A. Björner, Topological methods, in: Handbook of Combinatorics, vol. 2, Elsevier, Amsterdam, 1995, pp. 1819– 1872. [3] J.P.R. Christensen, Topology and Borel Structure, North-Holland Math. Stud., vol. 10, North-Holland, Amsterdam, 1974. [4] M. de Longueville, Notes on the topological Tverberg theorem, Discrete Math. 247 (1–3) (2002) 271–297. [5] E. Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1–3) (1998) 119–125. [6] M. Gromov, V.D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (4) (1983) 843–854. [7] W. Herer, J.P.R. Christensen, On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975) 203–210. [8] A.S. Kechris, V.G. Pestov, S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (1) (2005) 106–189. [9] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr., vol. 89, Amer. Math. Soc., Providence, RI, 2001. [10] J. Matoušek, On the chromatic number of Kneser hypergraphs, Proc. Amer. Math. Soc. 130 (9) (2002) 2509–2514. [11] J. Matoušek, Using the Borsuk–Ulam Theorem, Universitext, Springer-Verlag, Berlin, 2003. [12] J. Nešetˇril, Ramsey theory, in: Handbook of Combinatorics, Elsevier, Amsterdam, 1995, pp. 1331–1403. [13] O. Nikodým, Contribution à la théorie des fonctionnelles linéaires en connexion avec la théorie de la mesure des ensembles abstraits, Mathematica (Cluj) 5 (1931) 130–141. [14] V. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (10) (1998) 4149– 4165. [15] V. Pestov, Some universal constructions in abstract topological dynamics, in: Topological Dynamics and Applications, Minneapolis, MN, 1995, in: Contemp. Math., vol. 215, Amer. Math. Soc., Providence, RI, 1998, pp. 83–99. [16] V. Pestov, Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, Israel J. Math. 127 (2002) 317–357. [17] V. Pestov, Dynamics of Infinite-Dimensional Groups: The Ramsey–Dvoretzky–Milman Phenomenon, Univ. Lecture Series, vol. 40, Amer. Math. Soc., Providence, RI, 2006. [18] C.A. Rogers, Hausdorff Measures, Cambridge Univ. Press, London, 1970. [19] M. Talagrand, A simple example of a pathological submeasure, Math. Ann. 252 (1980) 97–102. [20] M. Talagrand, Maharam’s problem, Ann. of Math. (2), in press.
Journal of Functional Analysis 255 (2008) 494–501 www.elsevier.com/locate/jfa
Isometric embeddings of compact spaces into Banach spaces Y. Dutrieux, G. Lancien ∗ Université de Franche-Comté, Laboratoire de Mathématiques UMR 6623, 16 route de Gray, 25030 Besançon Cedex, France Received 31 January 2008; accepted 2 April 2008 Available online 7 May 2008 Communicated by N. Kalton
Abstract We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y , then any separable Banach space is linearly isometric to a subspace of Y . We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c0 or 1 . © 2008 Elsevier Inc. All rights reserved. Keywords: Isometric embeddings; Compact metric spaces; Geometry of Banach spaces
1. Introduction This paper is motivated by questions about universal Banach spaces. In 1925, P.S. Urysohn [9] was the first to give an example of a separable metric space U such that every separable metric space is isometric to a subset of U (we say that U is isometrically universal). However the foundation of the questions about universal Banach spaces is the theorem of S. Banach and S. Mazur [2] asserting that every separable Banach space is linearly isometric to a subspace of C([0, 1]) and therefore every separable metric space is isometric to a subset of C([0, 1]). It is then natural to wonder what are the Banach spaces that are isometrically universal for smaller classes of Banach * Corresponding author.
E-mail addresses: [email protected] (Y. Dutrieux), [email protected] (G. Lancien). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.002
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spaces or metric spaces. For instance, G. Godefroy and N.J. Kalton proved very recently in [5] that if a separable Banach space contains an isometric copy of every separable strictly convex Banach space, then it contains an isometric copy of every separable Banach space. On the other hand, in another recent work [6], N.J. Kalton and the second named author computed the best constant in Aharoni’s theorem (see [1] for the original paper) and, in particular, showed that every metric space with relatively compact balls embeds almost isometrically into the Banach space c0 . The main result of this paper is that a Banach space containing isometrically every compact metric space must contain a subspace linearly isometric to C([0, 1]). The techniques that we use come from classical results on isometries between Banach spaces. The first of them is of course the well-known result of S. Mazur and S. Ulam [8] who proved that a surjective isometry between two Banach spaces is necessarily affine. In other words, the linear structure of a Banach space is completely determined by its isometric structure. Then, one naturally wonders about what can be said when a Banach space X is isometric to a subset of a Banach space Y . The first fundamental result in this direction is due to T. Figiel who showed in [3] that if j : X → Y is an isometric embedding such that j (0) = 0 and Y is the closed linear span of j (X), then there is a (necessarily unique) linear quotient Q : Y → X of norm one and so that Q ◦ j = IdX . Let us notice now that the existence of the above map Q is clearly equivalent to the following:
∀x1 , . . . , xn ∈ X ∀λ1 , . . . , λn ∈ R
n n λk j (xk ) λk xk . k=1
k=1
More recently, as an application of their work on Lipschitz-free Banach spaces, G. Godefroy and N.J. Kalton [4] used Figiel’s result to prove that if a separable Banach space is isometric to a subset of another Banach space Y , then it is actually linearly isometric to a subspace of Y . Let us mention that this is not true in the non separable case and that counterexamples are given in [4]. In Section 2 we recall the necessary background on Lipschitz-free Banach spaces. We also state the version of [4, Theorem 3.1] that we shall use in the sequel. In Section 3 we prove the main result of the paper. More precisely, we produce a compact subset K0 of C([0, 1]) such that any Banach space containing an isometric copy of K0 must contain a subspace which is linearly isometric to C([0, 1]). We also show how our technique can be combined with the results of G.M. Lövblom in [7] on almost isometries between C(K)-spaces. Finally, let us say that M is an isometrically representing subset of the Banach space X if any Banach space Y containing an isometric copy of M contains a subset which is isometric to X. Notice that if M is an isometrically representing subset of a separable Banach space X, then it follows from the result of Godefroy and Kalton that any Banach space containing an isometric copy of M has a subspace which is linearly isometric to X. In the last section we produce compact isometrically representing subsets for the finite dimensional polyhedral spaces and for 1 . We also show that the unit ball of c0 isometrically represents the whole space. 2. Preliminary results We begin this section with a localized version of [4, Theorem 3.1 and Corollary 3.3]. We use the notation of [4] but recall it for the sake of completeness.
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Let (E, d) be a metric space with a specified point that we denote by 0. For Y a Banach space and f : E → Y we write
f (y) − f (x) f L = sup ; x = y in E . d(x, y) The space Lip0 (E) is the space of all f : E → R such that f (0) = 0 and f L < ∞ equipped with the norm · L . It turns to have a canonical predual F(X) which is the closed linear span in the dual of Lip0 (E) of the evaluation functionals δ(x) defined by δ(x)(f ) = f (x), for all f in Lip0 (E) and x in E. If Y is a Banach space and g : E → Y is a Lipschitz map, then there exists a unique linear operator g : F(E) → Y such that g ◦ δ = g. Moreover g = gL . In particular, when E is a Banach space, applying this to the identity map on E, we see that δ admits a normone linear left inverse β. For more information on the subject, the reader is invited to refer to [4] and its bibliography and in particular to the book of N. Weaver [10]. When F is a subset of E which contains 0, we denote by FE (F ) the closed linear span in F(E) of the evaluation functionals δ(x), x ∈ F . Since, by inf-convolution, any real valued Lipschitz function on F can be extended to the whole space E with the same Lipschitz constant, it is clear that the spaces FE (F ) and F(F ) are canonically isometric. In our first lemma, we rephrase [4, Theorem 3.1] for our particular purpose. Lemma 2.1. Let X be a separable Banach space. Let F be a closed convex subset of X such that 0 ∈ F . We assume that the closed linear span of F is X. Then there exists an isometric linear embedding T : X → FX (F ) such that β ◦ T is the identity map on X. Proof. SinceX is separable, there exists a sequence (xn )n1 in F which is total in X and such that the set { ∞ k=1 tk xk ; 0 tk 1 for all k} is a compact subset of F . We introduce the Hilbert cube Hn = (tk )∞ k=1 ; 0 tk 1 for all k and tn = 0 endowed with the product Lebesgue measure λn . Following the proof of [4, Theorem 3.1], we define φn =
δ xn + tk xk dλn (t). tk x k − δ
Hn
Our choice of (xn ) ensures that φn ∈ FX (F ). As proved in [4, Theorem 3.1], the map xn → φn extends to a norm-one linear operator T from X to FX (F ) such that β ◦ T is the identity map on X. 2 From this, we derive the main statement of this section. Theorem 2.2. Let X and Y be Banach spaces. Assume that X is separable. Let F be a closed convex subset of X, containing 0 and such that the closed linear span of F is X. Let j : F → Y
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be an isometric embedding such that j (0) = 0 and denote Z the closed linear span of j (F ). Finally, assume that n n (2.1) λk j (xk ) λk xk . ∀x1 , . . . , xn ∈ F ∀λ1 , . . . , λn ∈ R k=1
k=1
Then X is linearly isometric to a subspace of Z. Proof. Let T : X → FX (F ) be the operator given by Lemma 2.1 and let j : FX (F ) → Z be the linear operator defined by j ◦ δ = j . It follows from the remark made in the introduction that there is a linear operator Q : Z → X satisfying Q 1 and (Q ◦ j )(x) = x for all x in F . So for any x ∈ F , we have Q ◦ j ◦ δ(x) = Q ◦ j (x) = x = β ◦ δ(x). Hence, by linearity and continuity, Q◦j (μ) = β(μ) for all μ ∈ FX (F ). Since T (X) ⊂ FX (F ), we have Q◦j ◦T (x) = β ◦T (x) = x for any x ∈ F and thus, again by linearity and continuity, for any x ∈ X. Finally, the fact that Q is a contraction implies that j ◦ T : X → Z is a linear isometric embedding. 2 3. Isometric embeddings of spaces of continuous functions We begin this section with the main result of this paper. Theorem 3.1. Let (R, d) be a compact metric space. Then there is a compact subset K of C(R) such that whenever K embeds isometrically into a Banach space Y , then C(R) is linearly isometric to a subspace of Y . Proof. We may assume that the diameter of (R, d) is less than or equal to 1. Then we consider K = {f ∈ C(R), f ∞ 1 and f L 1}. Let Y be a Banach space and assume that j : K → Y is an isometry such that j (0) = 0. We denote F = K/5 and Z the closed linear span in Y of j (F ). In view of Theorem 2.2, it is enough to prove inequality (2.1). Our argument is adapted from the original work of T. Figiel [3]. For s, t ∈ R, we define ϕt (s) = 1 − d(s, t). For t in R, the functions ϕt and −ϕt clearly belong to K and j (ϕt ) − j (−ϕt ) = 2. Thus we can pick yt∗ ∈ Y ∗ such that yt∗ = 1 and yt∗ (j (ϕt ) − j (−ϕt )) = 2. Since j is an isometry and j (0) = 0, we clearly have: ∀t ∈ R ∀λ ∈ [−1, 1]
∗ yt ◦ j (λϕt ) = λ.
(3.2)
The key point of our proof is to show that ∀t ∈ R ∀ϕ ∈ F
∗ yt ◦ j (ϕ) = ϕ(t).
(3.3)
So let us assume that there exist t ∈ R and ϕ ∈ F such that (yt∗ ◦ j )(ϕ) = ϕ(t). We set ψ = ϕ − (yt∗ ◦ j )(ϕ)ϕt . Since ψ∞ < 1/2 and ψ(t) = 0, there exists u ∈ {−2, 2} so that 0 < (ϕt − uψ)(t) < 1. Besides, ψL < 1/2 and the diameter of R is less than or equal to 1, so for any s ∈ R: −1 < 1 − d(s, t) − uψ(s) = (ϕt − uψ)(s) (ϕt − uψ)(t) < 1.
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Hence ψ − ϕt = ϕ − y ∗ ◦ j (ϕ) + 1 ϕt < 1 . t u ∞ u 2 ∞ Note that λ = (yt∗ ◦ j )(ϕ) +
1 u
(3.4)
∈ [−1, 1]. So (3.2) yields:
1 ∗ = yt ◦ j (ϕ) − λ = yt∗ ◦ j (ϕ) − yt∗ ◦ j (λϕt ) ϕ − λϕt ∞ . 2 This is in contradiction with inequality (3.4) and finishes the proof of (3.3). We are now able to prove inequality (2.1) and therefore to conclude our proof. Indeed, we have that for all ψ1 , . . . , ψn ∈ F and all λ1 , . . . , λn ∈ R: n n ∗ λk j (ψk ) supyt λk j (ψk ) t∈R k=1 k=1 n n = sup λk ψk (t) = λk ψ k . t∈R k=1
k=1
2
∞
From the universality of C([0, 1]), we immediately deduce the following. Corollary 3.2. Consider the following compact subset of C([0, 1]): K0 = f ∈ C [0, 1] ; f ∞ 1 and f L 1 . If a Banach space Y contains an isometric copy of K0 , then it contains an isometric copy of any separable metric space and any separable Banach space is linearly isometric to a subspace of Y . It is now natural to ask if a metric space that is isometrically universal for all metric compact spaces is isometrically universal for all separable metric spaces. The next proposition shows that, for elementary reasons, this is not the case. Proposition 3.3. There exists a separable metric space V such that every separable and bounded metric space is isometric to a subset of V but so that R cannot be isometrically embedded into V . Proof. Let B denote the unit ball of C([0, 1]). Notice first that rB contains an isometric copy of all separable metric spaces with diameter less than r. Let V be the disjoint union of the sets Vn = nB, for n ∈ N. We now define a metric d on V as follows. On Vn , d is the natural distance in C([0, 1]). For n = m, f ∈ Vn and g ∈ Vm , we set d(f, g) = f ∞ + 1 + g∞ . It is clear that (V , d) is a separable metric space which is isometrically universal for all separable bounded metric spaces. On the other hand, any connected component of V is bounded. Therefore R does not embed isometrically into V . 2 We shall now combine Theorem 3.1 with a result of G.M. Lövblom [7] on almost isometries between C(K) spaces to obtain
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Corollary 3.4. Let R and S be compact metric spaces. Assume there exists a Lipschitz embedding F of the unit ball of C(R) into C(S) such that ∀f, g ∈ BC(R)
15 f − g F (f ) − F (g) f − g. 16
Then C(R) is linearly isometric to a subspace of C(S). 1 Proof. We may assume that F (0) = 0. Using [7, Theorem 2.1] for the particular value of ε = 16 , 1 we obtain an isometry j : 2 BC(R) → BC(S) . In particular, the set of functions on R such that both the supremum and the Lipschitz norms are less than or equal to 12 isometrically embeds into C(S). Then it follows from our previous proof that C(R) embeds linearly isometrically into C(S). 2
4. Isometrically representing subsets In this section, we address the following problem: given a separable Banach space X, we look for a small subset K of X such that whenever K isometrically embeds into a Banach space Y , then X embeds linearly isometrically into Y . We remind the reader that we call such a set K an isometrically representing subset of X. We shall restrict ourselves to considering K to be a compact subset of X or the unit ball of X. We start with a finite dimensional result. Theorem 4.1. Let X be a finite dimensional polyhedral Banach space. Then the unit ball of X is an isometrically representing subset of X. Proof. Let j : BX → Y be an isometric embedding such that j (0) = 0. Let x1∗ , . . . , xl∗ ∈ SX∗ so that BX = li=1 {x ∈ X; |xi∗ (x)| 1}. After removing some of the xi∗ ’s if necessary, we may and do assume that ∀i ∈ {1, . . . , l} ∃xi ∈ SX
xi∗ (xi ) = 1 and ∀j = i
∗ x (xi ) < 1. j
Then ∃r ∈ 0, 12 ∀x ∈ X ∀i ∈ {1, . . . , l} x − xi 4r
⇒
x = xi∗ (x).
(4.5)
We now imitate the proof of Theorem 3.1 with the xi ’s replacing the functions ϕt . So for 1 i l, we can pick yi∗ ∈ SY ∗ so that for any λ ∈ [−1, 1], (yi∗ ◦ j )(λxi ) = λ. Assume now that x ∈ rBX and (yi∗ ◦ j )(x) = xi∗ (x). Then consider w = x − (yi∗ ◦ j )(x)xi . Since 2w 4r and xi∗ (w) = 0, it follows from (4.5) that there is u ∈ {−2, 2} such that xi∗ (xi − uw) = xi − uw < 1. Following the lines of our previous proof, we then get a contradiction. Thus we have ∀x ∈ rBX ∀i ∈ {1, . . . , l}
∗ yi ◦ j (x) = xi∗ (x).
Therefore, we obtain that for all x1 , . . . , xn ∈ rBX and all λ1 , . . . , λn ∈ R:
(4.6)
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n n ∗ λk j (xk ) sup yi λk j (xk ) 1il k=1 k=1 n n ∗ = sup xi λk xk = λk xk . 1il k=1
k=1
Hence, we are in situation to apply Theorem 2.2 and conclude that X is linearly isometric to a subspace of Z. 2 As a consequence, we obtain a similar result for c0 . Corollary 4.2. The Banach space c0 is isometrically represented by its unit ball. Proof. We shall prove that 1 ∀x1 , . . . , xn ∈ Bc0 ∀λ1 , . . . , λn ∈ R 8
n n λk j (xk ) λk xk . k=1
k=1
(4.7)
∞
We may as well assume that x1 , . . . , xn are finitely supported sequences, so that there exists N ∈ N such that x1 , . . . , xn belong to the linear span of {e1 , . . . , eN }, where (ek )k1 is the canonical basis of c0 . If (ek∗ )k1 denotes the dual basis of (ek )k1 , notice that for any x ∈ c0 satisfying x − ek 12 , we have that x = ek∗ (x). Then the inequality (4.7) follows directly from our previous proof. Again, Theorem 2.2 finishes the argument. 2 In the case of 1 , we need to use a completely different method to obtain the following. Proposition 4.3. The Banach space 1 admits a compact isometrically representing subset. Proof. For A ⊂ N, we define μ(A) = k∈A 2−k = k∈A 2−k ek 1 where (ek ) stands for the canonical basis of 1 . We denote by K the space of all subsets of N endowed with the metric d(A, B) = μ(A \ B) + μ(B \ A). It is clear that K is isometric to a compact subset of 1 . Assume now that j : K → Y is an isometric embedding of K into some Banach space Y . We may assume that j (∅) = 0. For any n ∈ N, we denote yn = 2n j ({n}). Notice that yn = 1. For any α = (αk ) ∈ 1 , we define T α = αk yk ∈ Y . T is clearly a norm-one operator. Moreover, given α ∈ 1 , we set P = {k; αk > 0}, Q = {k; αk < 0} and pick y ∗ ∈ SY ∗ such that y ∗ (j (P ) − j (Q)) = d(P , Q). Since all the triangle inequalities are equalities, weinfer that y ∗ (j ({p})) = 2−p and y ∗ (j ({q})) = −2−q for any p ∈ P , q ∈ Q. Hence y ∗ (T α) = |αk | and T is a linear isometric embedding. 2 Questions. We leave open the following questions: (1) Is a Banach space always isometrically represented by its unit ball? (2) Does every separable Banach space admit a compact isometrically representing subset? (3) If two separable Banach spaces have the same compact subsets up to isometry, do they isometrically embed into each other?
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Acknowledgments We owe the presentation of Figiel’s result through inequality (2.1) to the referee, who also suggested question (3) and mentioned that Figiel’s technique does not work in general. We would like to thank Julien Melleray for having shown to us an example of an isometric embedding of the Euclidean disk into some Banach space for which inequality (2.1) is not satisfied on any smaller disk. References [1] I. Aharoni, Every separable metric space is Lipschitz equivalent to a subset of c0+ , Israel J. Math. 19 (1974) 284–291. [2] S. Banach, Théorie des opérations linéaires, Warsaw, 1932. [3] T. Figiel, On non-linear isometric embeddings of normed linear spaces, Bull. Acad. Polon. Sci. Math. Astro. Phys. 16 (1968) 185–188. [4] G. Godefroy, N.J. Kalton, Lipschitz free Banach spaces, Studia Math. 159 (2003) 121–141. [5] G. Godefroy, N.J. Kalton, Isometric embeddings and universal spaces, Extracta Math. 22 (2) (2007) 179–189. [6] N.J. Kalton, G. Lancien, Best constants for Lipschitz embeddings of metric spaces into c0 , Fund. Math. 199 (2008) 249–272. [7] G.M. Lövblom, Isometries and almost isometries between spaces of continuous functions, Israel J. Math. 56 (2) (1986) 143–159. [8] S. Mazur, S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932) 946–948. [9] P.S. Urysohn, Sur un espace métrique universel, C. R. Acad. Sci. Paris 180 (1925) 803–806. [10] N. Weaver, Lipschitz Algebras, World Scientific, 1999.
Journal of Functional Analysis 255 (2008) 502–531 www.elsevier.com/locate/jfa
Spectral analysis of a two body problem with zero-range perturbation M. Correggi a,∗ , G. Dell’Antonio b , D. Finco b a Scuola Normale Superiore (SNS), P.zza dei Cavalieri 7, 56126 Pisa, Italy b Dipartimento di Matematica, Università Di Roma “La Sapienza”, P.zza A. Moro 5, 00185 Rome, Italy
Received 5 October 2007; accepted 6 April 2008 Available online 19 May 2008 Communicated by J. Coron
Abstract We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one, two and three and we study its spectrum. In fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian. © 2008 Elsevier Inc. All rights reserved. Keywords: Solvable models in quantum mechanics; Zero-range interactions
1. Introduction We consider in Rd , d = 1, 2, 3, a system composed of a test particle and a harmonic oscillator interacting through a zero-range force. The Hamiltonian is formally written as Hαω ≡ H0ω + “αδ( x − y)”,
1 1 ω2 y 2 ωd − . H0ω ≡ − x − y + 2 2 2 2
* Corresponding author.
E-mail addresses: [email protected] (M. Correggi), [email protected] (G. Dell’Antonio), [email protected] (D. Finco). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.005
(1.1)
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The precise meaning of the formal expression “αδ( x − y)” will be given shortly. Hamiltonians with formal zero-range forces have been introduced in physics since the early 1930s (see, e.g., [10,15,16,18]), in particular for the study of scattering of low energy atoms and electrons from a target. The operators often considered in literature are approximations of (1.1), which correspond, very roughly speaking, to interactions with “very massive” nuclei through a potential of “very short range” and the nuclei are supposed so massive that they can be regarded as fixed scattering centers. Here we consider a more general model in which the nuclei of the target are regarded as quantum particles harmonically bound to their equilibrium positions. This kind of model is widely used in physics to reconstruct the structure of the target (e.g., the distribution of the equilibrium positions) from scattering data. In this paper we treat only the case of one harmonic oscillator, in which case α is a real parameter. We will come back in a forthcoming paper to the case in which several oscillators are present. From the point of view of mathematics, zero-range interactions are interesting non-trivial models (see, e.g., [1,3,8]), for which it is possible to find simple explicit solutions to the Schrödinger equation and to compute physically relevant quantities. These models can be indexed by a small number of parameters which codify the “strength” and the position of the interactions. The cases d = 1, 2, 3 will be treated in Sections 2, 3 and 4 respectively, and in each section we shall recall some results about the rigorous definition of zero range interactions in the corresponding dimension, often referring to [7] for proofs and further details. Here we only remark that the definition of zero range interaction is much easier in dimension one when it can be given in terms of boundary conditions at x = y. On the contrary, in dimensions two and three the definition requires a more sophisticated analysis in terms of self-adjoint extensions of the restriction of the operator H0ω to smooth functions which vanish in some neighborhood of the hyperplane x = y. This extension may be obtained at a purely formal level, considering a suitable interaction of range and letting → 0, after having applied a suitable renormalization prescription. An equivalent rigorous definition is obtained through the theory of quadratic forms. This is the definition we shall use in our analysis, indeed we shall show that in dimensions 1, 2 and 3 the operators Hαω are determined by simple quadratic forms, closed and bounded below. It is worth remarking that zero-range interactions, as we have defined them, do not exist when d 4 (the restriction mentioned above defines in this case an operator which is essentially selfadjoint) and that we consider only part of the self-adjoint extensions, i.e., those commonly called “δ”-type. This is by no means the only way to define an interaction supported by a manifold of codimension 1 (in our case the manifold x = y). The extension we choose is known in the physical literature as “single layer potential.” Other extensions are possible, among them the ones corresponding, roughly speaking, to double layer potential (dipole layers) and to various forms of the Robin conditions. A detailed mathematical study of the general case for pseudodifferential operators in a bounded domain with regular boundaries can be found, e.g., in [4,11,17]. We remark that in two and three dimensions the interaction is supported on immersed manifolds of codimensions 2 and 3 respectively. In this case one can use a higher order Poisson kernel or equivalently define the Schrödinger Hamiltonian by boundary triple theory (see, e.g., [5,19]). In this approach the auxiliary Hilbert space is a Sobolev space built over the Laplace–Beltrami of
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the manifold and the auxiliary maps are roughly speaking the trace maps on the regular and the singular part of the Poisson kernel at the manifold. In the two-dimensional and three-dimensional cases this construction leads to all self-adjoint extensions of the Laplacian restricted to function which vanish in a neighborhood of the manifold (one can take in place of the Laplacian any strictly elliptic operator with smooth coefficients with a possible addition of a potential and a regular magnetic field). While the method can be used for any immersed manifold, in our case one can, as we have done, give rather explicit analytic formulae and estimates. Notice that when the codimension is greater than three there is only one such extension, the Laplacian defined on the Sobolev space H2 . In the following sections, we shall prove that the essential spectrum of Hαω is the half-line [0, +∞), for all values of the parameters and d = 1, 2, 3, and the wave operators Ω± (Hαω , H0ω ) exist and are complete. We shall also fully characterize the negative part of the spectrum and give estimates of the number of eigenvalues. We plan to come back to the scattering problem in a forthcoming paper and give a complete description of the multi-channel scattering associated with the pair Hαω , H0ω . 1.1. Notation We introduce in this section some notation and basic facts which will be used in the rest of the paper. Vectors in Rd will be denoted by x , the modulus of x by x and x stands for (1 + x 2 )1/2 . Unless stated otherwise · will denote both the norm of functions in L2 (Rd ) and the norm of bounded endomorphism of L2 (Rd ). Given any function f ∈ L2 (Rd ), its Fourier transform, denoted by fˆ, will be defined by ≡ 1 fˆ(k) d x e−i k·x f ( x ). (1.2) d (2π) 2 d R
We shall denote the Sobolev space of order m by Hm (Rd ), i.e., Hm Rd ≡ f ∈ L2 Rd km fˆ ∈ L2 Rd , f Hm = km fˆ, and the logarithmic Sobolev space by Hlog Rd ≡ f ∈ L2 Rd log 1 + k fˆ ∈ L2 Rd , f Hlog = log 1 + k fˆ. ω = 1 (p 2 + ω2 x 2 ), which will We introduce the Hamiltonian of the harmonic oscillator by Hosc 2 (ω) be used as a reference Hamiltonian in some technical estimates, and denote by Ψn ( x ), n ∈ Nd , its normalized eigenvectors. The integral kernel of the semigroup (Mehler kernel, see [2]) is given by
e−Hosc t ( x ; x ) ≡ ω
ω(x 2 + x 2 ) ωx · x + . exp − d d 2 tanh ωt sinh ωt π 2 (1 − e−2ωt ) 2 e−
ωdt 2
(1.3)
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Let us recall for the reader’s convenience some facts about compact operators. Let H and H be two Hilbert spaces, we shall denote the space of compact operators from H to H by B0 (H , H ); if A ∈ B0 (H , H ), we denote by μn (A) its singular values with decreasing ordering μ0 (A) μ1 (A) · · · 0. For 1 p ∞, we shall denote the Schatten ideals1 by
∞ p Bp (H , H ) = A ∈ B0 (H , H ) μn (A) < +∞ ,
n=1
A p =
∞ p μn (A)
1
p
,
n=0
and, for p = ∞, we simply have B∞ (H , H ) = B(H , H ) and A ∞ = A . Let us recall also that, for A ∈ Bp (H , H ), we have μn (A) = μn (A∗ ), where ∗ denotes the adjoint, and A∗ ∈ Bp (H , H ) (see, e.g., [14]). We shall denote the spectrum of an operator A by σ (A), the pure point spectrum by σpp (A) and the essential spectrum by σess (A). The resolvent of the operator 1 1 ω2 y 2 ωd − H0ω ≡ − x − y + 2 2 2 2 is given by the following integral kernel,2 −1 Gλω ( x , y; x , y ) ≡ H0ω + λ ( x , y; x , y ) =
ωd−1
1
d
2 2 πd
λ
dν 0
ν ω −1 d
d
(1 − ν 2 ) 2 (ln ν1 ) 2
ω ω 1−ν 2 ων 2 2 2 y +y − × exp − ( x − x ) − ( y − y ) , 2 1+ν 1 − ν2 2 ln ν1 (1.4) where λ > 0 and x, y, x , y ∈ Rd . The above expression has been obtained in [7]. Note that by separation of variables the kernel (1.4) can be expressed as well as3 Gλω ( x , y; x , y ) = 2
n∈Nd
Ψn ( y )Ψn ( y )[−x + 2ωn + 2λ]−1 ( x ; x ), (ω)
(ω)
(1.5)
1 The norm A will also be denoted by Tr(|A|), the usual trace class norm. 1 2 In the following we shall often omit the suffix ω and set Gλ ≡ Gλ . Similarly we denote by H and H the operators α 0 1 Hα1 and H01 respectively. 3 For any n ∈ Nd , we set n ≡ di=1 ni .
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which in the one-dimensional case becomes Gλω (x, y; x , y ) =
+∞ (ω) (ω) Ψn (y)Ψn (y ) exp − 2(ωn + λ)|x − x | . √ 2(ωn + λ) n=0
(1.6)
The symbol Π will stand for the collision plane Π ≡ ( x , y) ∈ R2d | x = y ,
(1.7)
x , y) for the potential associated with f ∈ L2 (Rd ), i.e., and Gωλ f ( Gωλ f ( x , y) ≡
d x Gλω ( x , y; x , x )f ( x ).
(1.8)
Rd
If we denote by P : Hm (R2d ) → L2 (Rd ), m > d/2, the restriction to the plane Π , we trivially have G λ = Gλω P ∗ . Any positive constant will be denoted by c, whose value may change from line to line. 2. The one-dimensional case 2.1. Preliminary results The easiest way to give the expression (1.1) a rigorous meaning is to consider the (formal) quadratic form associated with such an operator: for any α ∈ R, at least formally, we have u|Hαω |u = u|H0ω |u + α
2 dx u(x, x) .
R
This formal expression identifies a closed quadratic form bounded below (see [7] for the proof). Definition 2.1 (Quadratic form Fαω ). The quadratic form (Fαω , D(Fαω )) is defined as follows, Fαω [u] ≡ F0ω [u] + αFint [u] 2 1 ∂u ≡ dx dy + 2 ∂x R2
2 1 ∂u 2 ω2 y 2 2 ω 2 |u| − |u| + α dx u(x, x) , (2.1) + 2 ∂y 2 2
D Fαω = u ∈ L2 R2 Fαω [u] < +∞ .
R
(2.2)
The main properties of Fαω are summarized in the following theorem. Theorem 2.2 (Closure of the form Fαω ). The quadratic form (Fαω , D(Fαω )) is closed and bounded below on D Fαω = D F0ω = u ∈ L2 R2 u ∈ H1 R2 , yu ∈ L2 R2 .
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We denote by (Hαω , D(Hαω )) the bounded from below self-adjoint operator on L2 (R2 ) defined by (2.1) and (2.2). Concerning the resolvent of Hαω we have the following result. Theorem 2.3 (Operator Hαω ). The domain and the action of Hαω are the following, D Hαω = u ∈ L2 R2 u = ϕ λ + Gωλ q, ϕ λ ∈ D H0ω , q + αPu = 0 , ω Hα + λ u = H0ω + λ ϕ λ .
(2.3) (2.4)
Moreover there exists λ0 > 0 such that, for λ > λ0 and for any f ∈ L2 (R2 ), one has −1 −1 ω Hα + λ f = H0ω + λ f + Gωλ qf ,
(2.5)
where the charge qf is a solution to the following equation, qf + α Kωλ qf + PGλω f = 0
(2.6)
and Kωλ ≡ PGλω P ∗ has integral kernel Kωλ (x; x ) ≡ Gλω (x, x; x , x ). It is straightforward to see that Hαω is an extension of H˜ 0 defined by D(H˜ 0 ) = u ∈ C0∞ R2 \ Π , 1 1 ω2 y 2 ω − u. H˜ 0 u = − x − y + 2 2 2 2 Then, by definition, Hαω is a perturbation of H0ω supported by the null set Π , i.e., a rigorous counterpart of (1.1). It follows from a general argument (see [1, Lemma C.2]) that Hαω is a local operator, i.e., if u = 0 in an open set Ω, then Hαω u = 0 in Ω. The effect of the interaction is equivalent to the boundary condition q + αPu = 0 satisfied by u ∈ D(Hαω ) (see (2.3)) which fixes a unique self-adjoint extension of H˜ 0 . Such a boundary condition is manifestly local, i.e., the value of q at a given point x ∈ R is proportional to the value of u at the point (x, x). In this sense the constructed Hamiltonian Hαω defines a local zero-range interaction. Note that the quadratic form (2.1) is not the most general zero-range perturbation of F0ω : it is clear, for instance, that, if we take a real function α(x) such that α ∈ L∞ , then Fα [u] ≡
R2
2 1 ∂u dx dy + 2 ∂x
2 1 ∂u 2 ω2 y 2 2 ω 2 |u| − |u| + dx α(x)u(x, x) (2.7) + 2 ∂y 2 2 R
define another zero-range perturbation of F0ω . The boundary condition corresponding to Fα is q(x) + α(x)(Pu)(x) = 0. The perturbation we use is distinguished by its invariance under translations along the coincidence manifold Π . In fact, the quadratic form (2.1) gives the simplest “δ”-like zero-range perturbation of F0ω which correspond to a local boundary condition. Similar remarks hold for the two- and threedimensional cases too.
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2.2. Spectral analysis In this section we shall study the spectrum of (1.1). We analyze first the properties of the operator Kωλ , for fixed ω = 1. Proposition 2.4 (Spectral analysis of K1λ ). The operator K1λ : L2 (R) → L2 (R) is compact, positive definite and self-adjoint. Let μn (λ), n ∈ N, be its eigenvalues arranged in a decreasing order. Then μn (λ) is a decreasing function of λ, for any n ∈ N, and limλ→0 μ0 (λ) = +∞, whereas limλ→0 μn (λ) < +∞, for n > 0. Furthermore the following estimate 3 2 μn (λ) − √ 1 (2.8) 2(n + λ) 5 π holds true for any n ∈ N. Proof. In order to simplify the notation, we shall denote by K λ the operator K1λ , i.e., Kωλ for fixed ω = 1. Using the boundedness criterium for integral operators, see [12], it is straightforward to see that λ K c
1 dν 0
ν λ−1
.
(2.9)
1 − ν 2 + ln ν1
1
Estimate (2.9) implies that K λ cλ− 2 for λ → 0 and for λ → +∞. The operator K λ is manifestly self-adjoint. We introduce the following decomposition:
1
K (x; x ) = λ
dν mλ (ν)kν (x; x ),
(2.10)
0
ν λ−1 mλ (ν) ≡ √ √ , 2π 1 − ν 2 ln ν1
(x − x )2 ν(x − x )2 1 1−ν 2 . x + x 2 − − kν (x; x ) ≡ exp − 21+ν 1 − ν2 2 ln ν1
(2.11)
(2.12)
Since kν is a positive operator-valued function and mλ (ν) is a positive function for ν ∈ (0, 1), the operator K λ is positive and has empty kernel. Let us prove now that K λ is a compact operator. Using lemma in [21, p. 65], it is straightforward to prove that 1−δ dν mλ (ν)kν (x; x ) 0
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is the kernel of a trace class operator, for any 0 < δ < 1. By Halmos criterium, it converges to K λ in the uniform topology, therefore K λ is compact and positive. By (2.10), the inequality K λ1 > K λ2 holds for λ1 < λ2 , therefore the eigenvalues are decreasing functions of λ by Min-Max theorem (see, e.g., [22, Theorem XIII.1]). In order to study the behavior of the eigenvalues of K λ for λ → 0, it suffices to notice that, if 2 f is orthogonal to exp{− y2 }, then limλ→0 f |K λ |f < +∞; the statement follows by Min-Max theorem and estimate (2.9). We prove now the eigenvalue upper bound. Note first that (see, e.g., [25]) − 1 1 1 Hosc + λ − 1/2 2 = √ π
∞
1
e−(λ− 2 )t e−Hosc t dt , √ t 1
0
which together with (1.3) yields 1 − 1 Hosc + λ − 1/2 2 (x; x ) 1 = π
1 0
ν ν λ−1 1 1−ν 2 2 . x + x 2 − dν √ √ exp − (x − x ) 21+ν 1 − ν2 2π 1 − ν 2 ln ν1 1
1 + λ − 1/2)− 2 − In order to apply Schur test (see [12]) to the operator (Hosc
dx
√
2K λ , we estimate
√ 1 − 1 Hosc + λ − 1/2 2 (x; x ) − 2K λ (x; x )
R
ν λ−1 (1 − ν 2 )x 2 dν √ exp − 2(1 + ν 2 ) 1 + ν 2 ln ν1 0
(1 + ν 2 ) ln ν1 (1 − ν)3 x 2 × 1− exp − 1 − ν 2 + (1 + ν 2 ) ln ν1 2(1 + ν 2 )[1 − ν 2 + (1 + ν 2 ) ln ν1 ]
1 =√ π
1
1 √ 2 π
1 0
ν λ−1 (1 − ν)3 , dν 3 ln ν1 (1 + ν 2 ) 2 [1 − ν 2 + (1 + ν 2 ) ln ν1 ]
where we have used the inequality b(a2 − a1 ) a 2 − a1 exp −a1 x 2 − b exp −a2 x 2 exp −a2 x 2 , a1 a1 which holds true for any 0 < a1 < a2 , 0 < b < 1 and ba2 > a1 . The last integral can be easily estimated by
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1 dν 0
ν λ−1 (1 − ν)3
∞
dt 0
3
ln ν1 (1 + ν 2 ) 2 [1 − ν 2 + (1 + ν 2 ) ln ν1 ] (1 − e−t )3 t
3 2
1
3 2
∞
dt t + 0
3
dt t − 2
12 , 5
1
so that, using the kernel symmetry, one has 1 1 √ 6 H + λ − 1/2 − 2 − 2K λ √ . osc 5 π The result is thus a simple consequence of Min-Max theorem.
(2.13)
2
We are now able to study the point spectrum of Hαω . Theorem 2.5 (Negative spectrum of Hαω ). For α 0, Hαω has no negative eigenvalues, while, for α < 0, there is a finite number Nω (α) of negative eigenvalues −E0 (α, ω) −E1 (α, ω) · · · 0 satisfying the scaling property √ En (α, ω) = ωEn (α/ ω, 1).
(2.14)
The corresponding eigenvectors are given by un = GωEn qn , where qn is a solution4 of the homogeneous equation qn + αKωEn qn = 0. Furthermore there exists α0 > 0 such that, for −α0 < α < 0, Nω (α) = 1, whereas, for |α| α0 , Nω (α) > 1. For fixed ω > 0, the ground state energy E0 (α, ω) satisfies the asymptotics E0 (α, ω) ∼
α2 2
(2.15)
as α → 0. Proof. First we derive an integral equation equivalent to the eigenvalue problem. Let u be a solution to Hαω u = −Eu, E > 0. Using (2.4), this proves to be equivalent to ω H0 + E φ λ = (λ − E)Gωλ q. The first resolvent identity yields from (2.5), φ λ = GωE q − Gωλ q, which implies u = GωE q. 4 Such a solution is actually unique once the L2 -norm of q is fixed, as it is by the L2 -normalization of u . n n
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511
On the other hand by using the boundary conditions in (2.3), we arrive at the following homogeneous equation for q and E: q + αKωE q = 0.
(2.16)
By scaling the above equation is equivalent to the following one: α E/ω q˜ + √ K1 q˜ = 0 ω
(2.17)
i.e., q solves (2.16) if and only if √ 1 q(x) ˜ ≡ ω− 4 q(x/ ω) solves (2.17), which implies (2.14). The other properties of the negative eigenvalues follow then from Proposition 2.4: if α 0, E/ω (2.17) has no solution, since K1 is a positive operator; if α < 0, by projecting (2.17) onto the E/ω eigenvectors of K1 , one obtains the algebraic equation 1+
αμn (E/ω) = 0, √ ω
(2.18)
and the eigenvalue equation is equivalent to find some n ∈ N and E > 0 satisfying (2.18). The monotonicity of μn together with their asymptotics as E → 0 (see Proposition 2.4) imply that, for any α < 0, (2.18) has only a finite number of solutions En . More precisely, for |α| < α0 , it has only one solution E0 , since limE→0 μ0 (E/ω) = +∞ and limE→0 μn (E/ω) < +∞, for n > 0. For fixed ω, the estimate (2.13) yields μ0 (E/ω, 1) =
√ ω + O( E) 2E
as E → 0 and Eq. (2.15) easily follows from (2.18).
(2.19)
2
Note that the result contained in theorem above yields also the expected asymptotic behavior as ω → 0: the limiting system is given by two particles freely moving on the line with a mutual zero-range interaction. The spectrum of such an operator is absolutely continuous for any sign of α, because of the translation invariance associated with the motion of the center of mass and it is [0, ∞) for α > 0, [−α 2 /2, ∞) for α < 0. If α < 0, the scaling property (2.14) implies that the eigenvalues accumulate at the bottom of the continuous spectrum as ω → 0 and the corresponding bound states eventually disappear. More interesting is the opposite asymptotics, that is the limit ω → ∞: in this case the strength of the harmonic oscillator becomes so large that, roughly speaking, one of the two particles remains fixed at the origin. More precisely we expect that the reduced dynamics of the other particle is generated by an Hamiltonian formally given by 1 hα = − x + “αδ(x)”. 2
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We define hα as the self-adjoint operator corresponding to the closed and bounded from below quadratic form fα given by fα [u] =
1 2
R
2 du dx + α|u(0)|. dx
It is straightforward to compute the domain and the action of hα (see, e.g., [1]): 1 d2 , hα = − 2 dx 2 D(hα ) = u ∈ H1 (R) ∩ H2 R \ {0} u 0+ − u 0− = 2αu(0) ,
(2.20) (2.21)
and, for α < 0, its spectrum contains only one negative eigenvalue −α 2 /2 with (normalized) eigenvector ξα (x) ≡
|α|e−|α||x| .
For fixed α < 0 and ω sufficiently large (larger than α 2 /2), the operator Hαω has only one negative eigenvalue −E0 (α, ω) with normalized eigenvector uα,ω (x, y). We denote by ρα,ω the reduced density matrix associated with the ground state uα,ω (x, y), i.e., the trace class operator ρα,ω : L2 (R) → L2 (R) with integral kernel ρα,ω (x; x ) ≡
dy u∗α,ω (x, y)uα,ω (x , y).
(2.22)
R
Proposition 2.6 (Ground state asymptotics as ω → ∞). For any fixed α < 0 and for ω → ∞, E0 (α, ω) =
α2 + O ω−1 , 2
(2.23)
and the reduced density matrix ρα,ω converges to the one-dimensional projector onto ξα , i.e., ρα,ω −→ |ξα ξα | ω→∞
(2.24)
in the norm topology of B1 (L2 (R)). 1
Proof. We first notice that the bound K λ cλ− 2 (see (2.9)) together with the eigenvalue equation (2.18) imply the bound E0 (α, ω) cα 2 , so that the ground state energy of Hαω is bounded (from above) uniformly in ω and E0 /ω → 0, as ω → ∞, for fixed α. Therefore the first part of the statement can be proved exactly as the asymptotics (2.15) (see, e.g., (2.19)). We now consider the ground state wave function uα,ω , which can be expressed as uα,ω = E Gω 0 q0 (see Proposition 2.5), where q0 is a solution to the homogeneous equation q0 + E αKω 0 q0 = 0. Note that the L2 -norm of q0 is actually fixed by the normalization of uα,ω . Let (ω) (ω) us decompose q0 as q0 = Q0 Ψ0 + ξ , with Ψ0 |ξ = 0. We are going to prove that
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uα,ω − wα,ω 2 Q0 wα,ω (x, y) ≡ √ 2E0
c q0 2
513
(2.25)
, 3 ω2 (ω) Ψ0 (y) exp − 2E0 |x| .
(2.26)
The proof is done in two steps. We first show that uα,ω − vα,ω 2 1 (ω) vα,ω (x, y) ≡ √ Ψ0 (y) 2E0
c q0 2 3
,
ω2 √
dx e−
2E0 |x−x |
Ψ0 (x )q0 (x ). (ω)
R
Indeed, by using the representation (1.6), we can easily estimate uα,ω − vα,ω 2
∞
√
n=1
∞ 2 (ω) 1 c q0 2 Ψ |q0 | 2 c q0 . n 3 3 3 3 2(ωn + E0 ) 2 ω 2 n=1 n 2 ω2
1
√ 1 On the other hand, setting q˜0 (x) ≡ ω− 4 q0 (x/ ω), one has |Q0 |2 wα,ω 2 = , (2.27) 3 2 (2E ) 0 |Q0 |2 1 (1) (1) vα,ω 2 = + dx dx |x − x |Ψ0 (x)Ψ0 (x )q˜0∗ (x)q˜0 (x ) 3 3 2 2 (2E0 ) 2E0 ω R
|Q0
|2
(2E0 )
3 2
+
c q0
2
3
R
,
ω2
and 2vα,ω |wα,ω =
2|Q0 |2 3
(2E0 ) 2 2|Q0
|2
(2E0 )
3 2
Q∗0
+√ 3 ω(2E0 ) 2 −
c q0
(1) dx |x|Ψ0 (x)q˜0 (x) exp
R
2E0 |x| − ω
2
3
ω2 3
so that vα,ω − wα,ω 2 c q0 2 /ω 2 and (2.25) is proven. E Let us now consider the charge q0 . The asymptotic behavior of the operator Kω 0 , as ω → ∞, is given by the following estimate E K 0 − √ 1 Ψ (ω) Ψ (ω) = O ω− 12 , (2.28) 0 0 ω 2E0 which easily follows from (2.13) and the simple inequality
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ω 1 2H + 2E0 − 1 − 2 − √ 1 Ψ (ω) Ψ (ω) = O ω− 12 . osc 0 0 2E0 Therefore by projecting the homogeneous equation (2.16) onto ξ , we get ξ 2 −
c|α| ξ q0 0 √ ω
because of (2.28), which in turn implies q0 2 = |Q0 |2 + O(ω−1 ) and ξ = O(ω−1 ). In order to derive from (2.25) the L2 -convergence of the ground state to wα,ω , we need then to bound q0 uniformly in ω; this can be done by exploiting the L2 -normalization of uα,ω : (2.27), (2.25) and the estimate for ξ yield |Q0 | (2E0 )
3 4
= wα,ω uα,ω +
c q0 ω
3 4
=1+
c q0 ω
3 4
1+
c|Q0 | ω
3 4
5 + O ω− 4
which together with the reverse inequality imply that |Q0 | = 1 + o(1) and q0 = 1 + o(1). Hence the integral operator with kernel (w) ρα,ω (x; x ) ≡
R
∗ dy wα,ω (x, y)wα,ω (x , y) =
|Q0 |2 exp − 2E0 |x| + |x | 2E0
converges in trace class norm to |ξα ξα |, since it is a projector onto a vector which converges in L2 -norm to ξα (|Q0 | → 1 and E0 → α 2 /2, as ω → ∞). On the other hand estimate (2.25) gives (w) ρα,ω − ρα,ω = o(1) and the convergence of the operators in the norm topology implies weak convergence in B1 (L2 (R)), but, since Tr(ρα,ω ) = Tr(|ξα ξα |) = 1, the convergence is actually in trace class norm. 2 Note that in one dimension point interactions share much of the properties of interactions through potentials, which may give the possibility of proving the above result via methods used in spectral theory, such as a variation, due to Grushin (see, e.g., [24]), of Schur complement method. Now we give some partial results on the positive spectrum of (1.1). Theorem 2.7 (Positive spectrum of Hαω ). The essential spectrum of Hαω is equal to [0, +∞) and the wave operators Ω± (Hαω , H0ω ) exist and are complete. Proof. It is sufficient to prove that (Hαω + λ)−1 − (H0ω + λ)−1 is a trace class operator for some λ > 0 and ω = 1, then the thesis follows from Weyl’s theorem (see [22, Theorem XIII.14]) and Kuroda–Birman theorem (see [21, Theorem XI.9]). Let us introduce the operator Qλ ≡ (I + αK1λ )−1 . For λ > λ0 , Qλ is bounded and positive (see Proposition (2.4)); the resolvent equation (2.5) can be cast in the following form (Hα + λ)−1 − Gλ = G λ Qλ G λ∗ .
(2.29)
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515
It is immediate to notice that the right-hand side of (2.29) is a positive operator; using Cauchy– Schwarz inequality and the boundedness of Qλ , one can prove that G λ Qλ G λ∗ (x, y; x , y ) is a continuous bounded function and that λ λ λ∗ G Q G (x, y; x , y ) c
1
λ
dν 0
ν ω −1
2 (2.30)
.
1 − ν 2 + ln ν1
Then using the lemma in [21, p. 65], together again with Cauchy–Schwarz inequality and the boundedness of Qλ , one has dx dy G λ Qλ G λ∗ (x, y; x, y) R2
c
dx dy
c
1/2
R
R2
2 dy G λ (x, y; y , y )
2 dy G λ∗ (y , y ; x, y)
1/2
R
2 dx dy dy G λ (x, y; y , y ) c.
2
R2
3. The two-dimensional case 3.1. Preliminary results In order to rigorously define the operator (1.1), we use again the theory of quadratic forms. We refer to [7] for proofs and the heuristic derivation of the quadratic form associated with (1.1). Definition 3.1 (Quadratic form Fαω ). The quadratic form (Fαω , D(Fαω )) is defined as follows D Fαω = u ∈ L2 R4 ∃q ∈ D Φαλ,ω , ϕ λ ≡ u − Gωλ q ∈ D F0ω ,
(3.1)
λ Fαω [u] ≡ F λ,ω [u] + Φα,ω [q],
(3.2)
where λ > 0 is a positive parameter and
F λ,ω [u] ≡
d x d y
2 2 1 2 ω2 y 2 λ 2 1 , (3.3) ∇x ϕ λ + ∇y ϕ λ + λϕ λ − λ|u|2 − ω|u|2 + ϕ 2 2 2
R4
λ D Φα,ω = q ∈ L2 R2 Φαλ,ω [q] < +∞ , 2 1 2 λ x ) + x ) − q( x ) , (3.4) Φα,ω [q] ≡ d x α + aωλ (x) q( d x d x Gλω ( x , x; x , x )q( 2 R2
R4
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aωλ (x) ≡
1 1 1 4ν λ/ω−1 (1 − ν) C + dν 1− 4π (1 − ν) (1 + ν 2 ) ln ν1 + 1 − ν 2 0 (1 − ν 2 ) ln ν1 + 2(1 − ν)2 2 × exp − ωx , 2[(1 + ν 2 ) ln ν1 + 1 − ν 2 ] 1 1 1 ∞ e− ν e− ν 1 + dν 2 ln dν . C≡− ν ν ν 0
(3.5) (3.6)
1
Note that the decomposition u = ϕ λ + Gωλ q is well defined and unique (for fixed λ), Gωλ q ∈ for any q ∈ L2 (R2 ) and the quadratic form (3.2) is independent of the parameter λ (see [7]); as a matter of fact λ plays here the role of a free parameter and its value will be chosen later. L2 (R4 )
Theorem 3.2 (Closure of the form Fαω ). The quadratic form (Fαω , D(Fαω )) is closed and bounded below on the domain (3.1) for any ω 0. Let us denote by Γωλ the positive self-adjoint operator on L2 (R2 ) associated with the quadratic λ , i.e. form Φα,ω λ q|Γωλ |q ≡ Φα,ω [q] − α q 2 ,
(3.7)
and by Hαω the Hamiltonian defined by Fαω . Then we have Theorem 3.3 (Operator Hαω ). The domain and the action of Hαω are the following D Hαω = u ∈ L2 R4 u = ϕ λ + Gωλ q, ϕ λ ∈ D H0ω , q ∈ D Γωλ , α + Γωλ q = Pϕ λ , (3.8) ω (3.9) Hα + λ u = H0ω + λ ϕ λ , and the resolvent of Hαω can be represented as
Hαω + λ
−1
f = Gλω f + Gωλ qf ,
(3.10)
where, for any f ∈ L2 (R4 ), qf is a solution to α + Γωλ qf = PGλω f.
(3.11)
As in the one-dimensional case, one can recognize in (3.9) a self-adjoint extension of the symmetric operator H˜ 0 defined by D(H˜ 0 ) = u ∈ C0∞ R4 \ Π , 1 1 ωy 2 ˜ H0 u = − x − y + −ω u 2 2 2 so that Hαω is a singular perturbation of H0ω supported on Π .
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517
Note that the unperturbed Hamiltonian, corresponding to the case of no interaction between the particle and the harmonic oscillator, belongs to the family Hαω and is given by α = +∞. This fact, which could seem surprising, if one considers the formal Hamiltonian (1.1), is due to the renormalization procedure required to give a rigorous meaning to such a formal expression. Therefore we stress that in the two- and three-dimensional cases α does not play the role of coupling constant of the system. Also in the two-dimensional case, the one-parameter family of extensions considered has local boundary conditions (see [7]). 3.2. Spectral analysis In order to study the spectrum of Hαω , we first need to state some spectral properties of the operator Γωλ . λ ) can be characterized Proposition 3.4 (Spectral analysis of Γωλ ). For ω > 0 the domain D(Φα,ω in the following way:
λ = q ∈ L2 R2 q ∈ Hlog R2 , qˆ ∈ Hlog R2 . D Φα,ω
(3.12)
λ is closed and defines a self-adjoint operator Γ λ . For any λ > 0 the specOn this domain Φα,ω ω trum, σ (Γωλ ), is purely discrete, i.e., σ (Γωλ ) = σpp (Γωλ ). Let γn (λ), n ∈ N, be the eigenvalues of Γ1λ arranged in an increasing order (limn→∞ γn (λ) = +∞). For every n ∈ N, γn (λ) is a non-decreasing function of λ. Furthermore limλ→0 γ0 (λ) = −∞ and the other eigenvalues remain bounded below, i.e., for any λ 0, there exists a finite constant c such that γn (λ) −c, for any n ∈ N, n > 0.
Proof. For the sake of simplicity we fix ω = 1 from the outset and omit the dependence on ω in the notation. The self-adjointness of Γ λ immediately follows from the properties of the quadratic form Φαλ (see [7]). Note that Γ λ can be written in the following way Γ λ = a λ + Γ0λ , where a λ is the multiplication operator for the unbounded function (3.5) and Γ0λ is the selfadjoint operator associated with the positive quadratic form Φ0λ [q] ≡
1 2
2 x ) − q( x ) . d x d x Gλ ( x , x; x , x )q(
(3.13)
R2
Since Φ0λ is positive and a λ (x) is bounded below but not above, Γ λ is an unbounded operator. Notice that a λ (x) is a monotone increasing function of x and a λ (x) c log x for x → ∞; furthermore a λ (x) a λ (0), a λ (0) is a monotone increasing function of λ and a λ (0) c log λ for λ → ∞. Hence the following lower bound Φαλ [q] R2
2 x ) α + a λ (0) q 2 d x α + a λ (x) q(
(3.14)
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proves that for any α ∈ R there exists λ0 such that for λ > λ0 the quadratic form Φαλ is positive; for such λ the operator Γ λ is invertible and its inverse is a bounded operator. We shall prove now that σ (Γ λ ) is purely discrete for any λ. By [22, Theorem XIII.64] it suffices to prove that Dη ≡ q ∈ D Φαλ Φαλ [q] η is a compact subset of L2 (R2 ) for any positive η; this will be proved using the Rellich’s criterion [22, Theorem XIII.65]. The positivity of Γ0λ implies that, if Φαλ [q] η, then
2 x ) η. d x a λ (x)q(
R2
Moreover, applying the Fourier transform, Φαλ can be rewritten in the following equivalent form Φαλ [q] =
2 + 1 ˆ k) d k α + a˜ λ (k) q( 2
R2
2 − q( k )q( ˜ λ (k; ˆ k) ˆ k ) , d k d k G
(3.15)
R4
where 1 4ν λ−1 (1 − ν) 1 1 a˜ (k) ≡ 1− C + dν 4π 1−ν (1 + ν 2 ) ln ν1 + 1 − ν 2 λ
× exp − k ) ≡ 1 ˜ λ (k; G 2π 2
0
(1 − ν 2 ) ln ν1 2[(1 + ν 2 ) ln ν1 + 1 − ν 2 ]
1 dν 0
k
2
,
(3.16)
ν λ−1 (1 − ν 2 ) ln ν1 + 2(1 − ν)2
[(1 + ν 2 ) ln ν1 + 1 − ν 2 ](k 2 + k 2 ) × exp − 2[(1 − ν 2 ) ln ν1 + 2(1 − ν)2 ]
[1 − ν 2 + 2ν ln ν1 ]k · k . − (1 − ν 2 ) ln ν1 + 2(1 − ν)2
(3.17)
In order to prove (3.15), it is convenient to introduce a regularized quadratic form Φαλ,δ obtained by restricting the integration domain in ν to the set [0, 1 − δ], for some 0 < δ < 1. It is straightforward to notice that Φαλ,δ is a bounded form and that for every q ∈ L2 (R2 ), λ,δ Φα [q] is a monotone function of δ; therefore for q ∈ D(Φαλ ) we have lim Φαλ,δ [q] = Φαλ [q].
δ→0
(3.18)
On the other hand, due to the regularization, with straightforward calculations one can prove that
M. Correggi et al. / Journal of Functional Analysis 255 (2008) 502–531
Φαλ,δ [q] ≡
519
1−δ dν 1 α+ x , x; x , x )q ∗ ( x )q( x ) C+ q 2 − d x d x Gλ,δ ( 4π 1−ν R4
0
= α+
1 C+ 4π
1−δ 0
dν 1−ν
q ˆ 2−
k )qˆ ∗ (k) q( ˜ λ,δ (k; d k d k G ˆ k ),
(3.19)
R4
which can be rewritten in the following way: Φαλ,δ [q] =
2 + 1 ˆ k) d k α + a˜ λ,δ (k) q( 2
R2
2 − q( k )q( ˜ λ,δ (k; ˆ k) ˆ k ) , (3.20) d k d k G
R4
˜ λ,δ are the regularization of (3.16) and (3.17). Notice that (3.19) shows how where a˜ λ,δ and G λ Φα can be obtained by a renormalization of the formal quantity q|G λ |q. Due to the monotonicity in δ, we can take the limit δ → 0 of (3.20) and, by (3.18), we obtain (3.15). It is immediate to notice that (3.15) has the same structure as (3.4) and in particular, if Φαλ [q] η, then
2 η. ˆ k) d k a˜ λ (k)q(
R2
The function a˜ λ (k) has the same properties of a λ (x), namely it is a monotone function of k, aˆ λ (k) c log k for k → ∞ and aˆ λ (0) c log λ for λ → 0. Hence Rellich’s criterion guarantees that Dη is a compact subset of L2 (R2 ) and therefore Γ λ has only pure point spectrum. Notice that also the following bound holds, Φ0λ [q] c q 2Hlog (R2 ) .
(3.21)
Indeed using the following inequality
1
x , x; x , x ) c G ( λ
dν 0
ν λ−1 (1 − ν 2 ) ln ν1
1 ν 2 2 exp − ( x − x ) − ( x − x ) 1 − ν2 2 ln ν1
in (3.13) and taking the Fourier transform, we have Φ0λ [q] c R2
d k
1 dν 0
ν λ−1 2ν ln ν1 + 1 − ν 2
× 1 − exp − and, since
2(1 − ν 2 ) ln ν1 4(2ν ln ν1 + 1 − ν 2 )
k2
2 q(k) ˆ ,
(3.22)
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1 0
dν 0
1 − exp −
ν λ−1 2ν ln ν1 + 1 − ν 2
2(1 − ν 2 ) ln ν1 4(2ν ln ν1 + 1 − ν 2 )
k
2
c log 1 + k , (3.21) is proven. Therefore, taking into account the behavior of a λ , (3.21) implies that there exists λ0 > 0 such that for λ > λ0 we have Γ λ c logx + logp + log λ .
(3.23)
We can prove also a similar lower bound for Γ λ . Putting together the lower bound (3.14) and a corresponding lower bound for (3.15), we obtain: q|Γ λ |q
1 2
2 1 x ) + d x a λ (x)q( 2
R2
a λ (0) 2
+
aˆ λ (0) 2
2 ˆ k) d k aˆ λ (k)q(
R2
q 2 ,
(3.24)
and in particular (3.12) holds true, due to (3.23) and (3.24). The monotonicity in λ of the eigenvalues γn (λ) follows from the monotonicity of Φαλ [q] with respect to λ. This can be easily seen by observing that the regularized expression (3.19) is a non-decreasing function of λ, as it must be its limit as δ → 0. In order to analyze the asymptotics for λ → 0 of γ0 (λ), we shall show that there exists a function q belonging to D(Γ λ ), such that limλ→0 q|Γ λ |q = −∞. The result is then a consequence (1) of the Min-Max theorem. Indeed taking the ground state of the 2d harmonic oscillator Ψ0 ( x ), one has (1) (1) lim Ψ0 Γ λ Ψ0 = lim λ→0 λ→0
1 8ν λ−1 (1 − ν) 1 1 1− C + dν 4π 1−ν (3 + ν 2 ) ln ν1 + 4(1 − ν) 0
c1 − c2 lim
1 2
dν
λ→0 0
ν λ−1 1 + ln ν1
= −∞.
(3.25)
The boundedness from below of the other eigenvalues can be proved by showing that the quadratic form remains bounded as λ → 0, if q is orthogonal to the above function Ψ0(1) ( x ). Let (1) ⊥ 2 5 λ ⊥ x ) be an L -normalized function in D(Φα ) such that Ψ0 ( x )|q = 0. From the expresq ( sion of the quadratic form (3.4), it is clear that we can restrict the integrations in ν in (3.5) and Gλ to the interval [0, 1 − 1/e], because the remainder is uniformly bounded in λ, i.e., there exists a finite constant c independent of λ such that 5 For instance one can take q ⊥ = Ψ (1) , n = 0 for some i. i n
M. Correggi et al. / Journal of Functional Analysis 255 (2008) 502–531
521
1 e ⊥ 2 1 1 ⊥ λ ⊥ q q Γ q −c + dν 4π 1−ν
0
∗ d x d x Gλ1/e ( x , x; x , x ) q ⊥ ( x ) q ⊥ ( x )
− R4
where we have expanded the second term in the form as in (3.19). The first term on the righthand side of the expression above is again bounded by a finite constant, whereas, as we are going to prove, the only unbounded contribution comes from Gλ1/e , but it contains a projection to the (1)
subspace spanned by Ψ0 :
1
∗ 1 d x d x Gλ1/e ( x , x; x , x ) q ⊥ ( x ) q ⊥ ( x ) 2π 2
R4
≡ q
⊥
e
dν 0
ν λ−1 ⊥ ⊥ q kν q 1 − ν2
K λ q ⊥ ,
where kν is the integral operator whose kernel is the two-dimensional analogous of (2.12). Moreover −1 ⊥ λ ⊥ 1 ⊥ 1 q Hosc + λ − 1 q ⊥ q K q 2π 1
1 + 2π 2
e
dν 0
ν λ−1 ⊥ q kν − k¯ν q ⊥ , 2 1−ν
k¯ν denoting the integral operator with kernel
ν( x − x )2 1 1−ν 2 2 . x + x − k¯ν (x, x ) ≡ exp − 21+ν 1 − ν2 The last term in (3.26) can be estimated as follows: 1
1 2π 2
e
dν 0
ν λ−1 ⊥ q kν − k¯ν q ⊥ 2 1−ν 1
1 4π 2
e
dν 0
1 2 π
1 0
ν λ−1 (1 − ν 2 ) ln
ν λ−1 dν 1 − ν2
1 ν
d x d x | x − x |2 k¯ν ( x ; x )q ⊥ ( x )q ⊥ ( x )
R4
d x d x x 2 k¯ν ( x ; x )q ⊥ ( x )q ⊥ ( x )
R4
−1 q ⊥ 2 1 ⊥ 1 1 q Hosc Hosc + λ − 1 q ⊥ . π π
(3.26)
(3.27)
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1 , Since, for any q ⊥ orthogonal to the ground state of Hosc
1 −1 2 q ⊥ Hosc + λ − 1 q ⊥ q ⊥ ,
we thus obtain
3 q ⊥ 2 q ⊥ K λ q ⊥ 2π
and the boundedness from below of the operator Γ λ on the subspace of functions orthogonal (1) to Ψ0 . 2 The spectral properties of the operator Γωλ allow us to give a complete characterization of the discrete spectrum of Hαω . Theorem 3.5 (Negative spectrum of Hαω ). For any α ∈ R and ω ∈ R+ the discrete spectrum σpp (Hαω ) of Hαω is not empty and it contains a number Nω (α) 1 of negative eigenvalues −E0 (α, ω) −E1 (α, ω) · · · 0, satisfying the scaling En (α, ω) = ωEn (α, 1).
(3.28)
The corresponding eigenvectors are given by un = GωEn qn , where qn is a solution to the homogeneous equation αqn + ΓωEn qn = 0. Moreover there exists α0 ∈ R such that, if α > α0 , Nω (α) = 1 and, for fixed ω and α → −∞, ln Nω (α) c|α|. The ground state energy has the following asymptotic behavior for fixed ω: E0 −cα −1 for α → +∞ and ln E0 c|α| for α → −∞. Proof. Following the proof of Theorem 2.5, we get that uE is an eigenfunction of Hαω relative to the eigenvalue −E, E > 0, only if uE = GωE q
(3.29)
λ ). On the other hand u belongs to the domain of H ω and then it must for some q ∈ D(Φα,ω E α satisfy the boundary condition on Π , which for a function of this form becomes αq + ΓωE q = 0, or E Φα,ω [q] = α q 2 + q|ΓωE |q = 0.
(3.30)
So that there is a one-to-one correspondence between the negative eigenvalues of Hαω and nontrivial solutions to the homogeneous equation above. In other words −E is an eigenvalue of Hα , if and only if 0 is an eigenvalue of α + ΓωE . Note that, by scaling, q solves (3.30), if and only if √ 1 q( ˜ x ) ≡ ω− 2 q( x / ω) is a solution to the homogeneous equation E/ω
α q˜ + Γ1 which implies (3.28).
q˜ = 0,
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The other results are simple consequences of Proposition 3.4. In particular in order to complete the asymptotic analysis for α → +∞ it is sufficient to notice that in fact (3.14) and (3.25) imply that −c1 λ−1 γ0 (λ) −c2 λ−1 as λ → 0, due to the asymptotic behavior of aωλ (0) and aˆ ωλ (0) in such a limit; this is sufficient to conclude that E0 = O(α −1 ) for α → +∞. The previous argument can be repeated for λ → −∞ and gives that γ0 (λ) c ln λ for λ → +∞, which means ln E0 = O(|α|) for α → −∞. In order to conclude the proof, it is sufficient to notice that, for fixed ω = 1, N1 (α) is bounded below by the cardinality of {n ∈ N | γn (0) −α}; therefore any upper bound on γn (0) provides a lower bound on N1 (α). Due to the monotonicity in λ of γn (λ), to (3.23) and to the straight1 + 1), we can use the eigenvalue distribution forward estimate (logx + logp) c log(Hosc of the logarithm of the harmonic oscillator to estimate N1 (α) which gives ln N1 (α) c|α| for α → ∞. 2 We underline that for ω > 0 the interaction is attractive in the sense that there exists at least one bound state irrespective of the sign of α. This fact is essentially due to the renormalization procedure used to rigorously define the quadratic form in (3.2) and to the presence of the harmonic oscillator; this is a common phenomenon in the theory of point interactions (see, e.g., [1] for a similar effect). Note also that the different scaling (3.28) in ω is due to the scaling properties of the Green function (1.4) (more precisely its restriction to the planes Π ), i.e., in d dimensions, √ √ √ λ/ω √ Gλω ( x , y; x , y ) = ωd−1 G1 ( ωx, ωy; ωx , ωy ). The asymptotics for ω → 0 can be easily derived from (3.28): the spacing between different eigenvalues goes to 0 and in the limit they form a continuum, so that no bound state survives in the limit. On the opposite all the eigenvalues corresponding to excited states diverge as ω → ∞. More detailed results on the eigenvalue asymptotics could be obtained by applying usual techniques in semiclassical analysis (see, e.g., [6,9,13]), but such an investigation goes beyond the aim of this paper. Before giving a partial characterization of the positive spectrum, let us prove a technical lemma. Lemma 3.6. Define the operator Tωk ≡ Gωλ∗ (Gλω )k Gωλ : L2 (R2 ) → L2 (R2 ), for any k ∈ N. Then, if λ > k, Tωk ∈ Bp (L2 (R2 ), L2 (R2 )), for any p > 2(k + 1)−1 . Proof. Setting ω = 1 for the sake of clarity and omitting the ω-dependence in the notation, we have the identity d k+1 PGλ P ∗ , (3.31) Tk = − dλ so that, using (1.4), we get the integral kernel of T k , i.e.,
1
T ( x ; x ) = c k
dν 0
ν λ−1 (ln ν1 )k (1 − ν 2 )
( 1 1−ν 2 x − x )2 ν( x − x )2 × exp − . x + x 2 − − 21+ν 1 − ν2 2 ln ν1
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Notice that, using the same argument as in the proof of Proposition 2.4, we can view T k as the integral over the parameter ν of positive operator-valued functions tν , i.e., 1 T =c k
(3.32)
dν mk (ν)tν , 0
mk (ν) =
ν λ−1 (ln ν1 )k , (1 − ν 2 )
(3.33)
( x − x )2 ν( 1 1−ν 2 x − x )2 2 . x +x − x ; x ) = exp − − tν ( 21+ν 1 − ν2 2 ln ν1
(3.34)
Applying Schur test to the operator tν , one has tν ∞ = tν c(1 − ν), whereas a simple calculation yields tν 1 = Tr(tν ) c(1 − ν)−1 . On the other hand Hölder inequality in Schatten ideals (see [23]) gives 1/p
tν p tν 1 tν 1−1/p (1 − ν)1−2/p .
(3.35)
It is then straightforward to check that k T p
1 dν mk (ν) tν p < +∞
(3.36)
0
for any p > 2(k + 1)−1 .
2
Now we present some partial results on the continuous spectrum of (1.1) by means of a characterization of the mapping properties of the resolvent; in the following we shall fix λ > 1, such that (Γωλ + α)−1 exists and is bounded. Therefore Eq. (3.10) can be cast in the following form ω −1 −1 Hα + λ = Gλω − Gωλ Γωλ + α Gωλ∗ .
(3.37)
Theorem 3.7 (Positive spectrum of Hαω ). The essential spectrum of Hαω is equal to [0, +∞) and the wave operators Ω± (Hαω , H0ω ) exist and are complete. Proof. We shall drop the dependence on ω for brevity. It is sufficient to prove that (Hα + λ)−1 − (H0 + λ)−1 is a compact operator and that (Hα + λ)−3 − (H0 + λ)−3 is trace class for some λ > 0, then the thesis follows from Weyl’s theorem (see [22, Theorem XIII.14]) and [21, Corollary 3 of Theorem XI.11]). We first analyze G λ (Γ λ + α)−1 G λ∗ and prove that it is a compact operator. Due to Lemma 3.6 we have G λ ∈ Bp (L2 (R2 ), L2 (R4 )) with p > 4: by taking k = 0, one obtains G λ∗ G λ ∈ Bp (L2 (R2 ), L2 (R2 )) for p > 2, i.e., denoting by gn2 , n ∈ N, its singular values, {gn } ∈ p for p > 4. By a standard argument (see, e.g., [20, the proof of Theorem VI.17]), one can show that {gn } are the singular values of G λ and the result easily follows. This also implies that G λ∗ ∈ Bp (L2 (R4 ), L2 (R2 )), p > 4, and both operators are compact. Moreover (Γ λ + α)−1 is
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a bounded operator and then G λ (Γ λ + α)−1 G λ∗ ∈ Bp (L2 (R4 ), L2 (R4 )) with p > 2, by Hölder inequality, and in particular is a compact operator. In order to prove the existence of wave operators and asymptotic completeness, let us expand the difference of the resolvent to third power: (Hα + λ)−3 − (H0 + λ)−3 2 −1 = G λ Γ λ + α G λ∗ Gλ −1 2 −1 + Gλ G λ Γ λ + α G λ∗ Gλ + Gλ G λ Γ λ + α G λ∗ 2 −1 −1 −1 + G λ Γ λ + α G λ∗ Gλ + G λ Γ λ + α G λ∗ Gλ G λ Γ λ + α G λ∗ 2 3 −1 −1 + Gλ G λ Γ λ + α G λ∗ + G λ Γ λ + α G λ∗ .
(3.38)
All the terms on the right-hand side of (3.38) are trace class operators. Indeed it is sufficient to use Lemma 3.6, with k = 2, and Hölder inequality, as done when studying G λ (Γ λ + α)−1 G λ∗ . As an example let us consider the first term in the above expression: by Lemma 3.6, G λ∗ (Gλ )2 ∈ Bp (L2 (R4 ), L2 (R2 )) for p > 4/5 (by the same argument applied to G λ ) and thus it is a trace class operator. The claim then follows from boundedness of G λ and (Γ λ + α)−1 and Hölder inequality. 2 4. The three-dimensional case 4.1. Preliminary results As in the two-dimensional case, operator (1.1) can be rigorously defined by means of the theory of quadratic forms (see [7]). Definition 4.1 (Quadratic form Fαω ). The quadratic form (Fαω , D(Fαω )) is defined as follows λ λ , ϕ ≡ u − Gωλ q ∈ D F0ω , D Fαω = u ∈ L2 R6 ∃q ∈ D Φα,ω
(4.1)
λ Fαω [u] ≡ F λ,ω [u] + Φα,ω [u],
(4.2)
where λ > 0 is a positive parameter and
F
λ,ω
[u] ≡
d x d y
2 1 ∇x ϕ λ + 2
2 1 ∇y ϕ λ 2
R6
λ 2 3ω 2 ω2 y 2 λ 2 2 , |u| + ϕ + λ ϕ − λ|u| − 2 2
λ λ D Φα,ω = q q ∈ L2 R3 , Φα,ω [q] < +∞ ,
(4.3)
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2 x ) d x α + aωλ (x) q(
λ Φα,ω [q] ≡ R3
1 2
+ √ aωλ (x) ≡
ω 3
2 x ) − q( x ) , d x d x Gλω ( x , x; x , x )q(
R6
1 + 2
1 dν
1 3
3
1−
8ν λ/ω−1 (1 − ν) 2 3
[(1 + ν 2 ) ln ν1 + 1 − ν 2 ] 2 (1 − ν 2 ) ln ν1 + 2(1 − ν)2 2 × exp − ωx . 2[(1 + ν 2 ) ln ν1 + 1 − ν 2 ]
(4π) 2
0
(1 − ν) 2
(4.4)
(4.5)
The well-posedness of the definition above can be shown exactly as in the two-dimensional case. Moreover in the same way one can prove that the form is actually closed and bounded below (see [7] for the proofs). Theorem 4.2 (Closure of the form Fαω ). The quadratic form (Fαω , D(Fαω )) is closed and bounded below on the domain (4.1). Concerning the self-adjoint operators Hαω and Γωλ associated with the quadratic forms Fαω and respectively, i.e.,
λ Φα,ω
λ q|Γωλ |q ≡ Φα,ω [q] − α q 2 ,
(4.6)
we have the following theorem. Theorem 4.3 (Operator Hαω ). The domain and the action of Hαω are the following D Hαω = u ∈ L2 R6 u = ϕ λ + Gωλ q, ϕ λ ∈ D H0ω , q ∈ D Γωλ , α + Γωλ q = Pϕ , ω Hα + λ u = H0ω + λ ϕ λ ,
(4.7) (4.8)
and the resolvent of Hα can be represented as
Hαω + λ
−1
f = Gλω f + Gωλ qf ,
(4.9)
where, for any f ∈ L2 (R6 ), qf is a solution to α + Γωλ qf = PGλω f.
(4.10)
The operators (4.8) give rise to a one-parameter family of self-adjoint operators, which actually coincides with a family of self-adjoint extensions of the three-dimensional analogous of the operator H˜ 0 introduced in the previous section. Note that the free Hamiltonian H0ω belongs to the family and it is given by (4.8) for α = +∞, exactly as in the two-dimensional case.
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4.2. Spectral analysis Most of the results proved in the two-dimensional case apply also to the three-dimensional one and there are only minor differences in the proofs. Hence we shall often omit the details and refer to the two-dimensional case. The spectral properties of Hαω are strictly related to spectral properties of the operator Γωλ , so we shall start by studying the latter. λ ) can be characterized in the Proposition 4.4 (Spectral analysis of Γωλ ). The domain D(Φα,ω following way: λ D Φα,ω = q ∈ L2 R3 q ∈ H1/2 R3 , qˆ ∈ H1/2 R3 . (4.11) λ is closed and defines a self-adjoint operator Γ λ . For any λ > 0 the spectrum On this domain Φα,ω ω λ σ (Γω ) is purely discrete, i.e., σ (Γωλ ) = σpp (Γωλ ). Let γn (λ), n ∈ N, be the eigenvalues of Γ1λ arranged in an increasing order (limn→∞ γn (λ) = +∞). For every n ∈ N, γn (λ) is a non-decreasing function of λ. Furthermore limλ→0 γ0 (λ) = −∞ and the other eigenvalues remain bounded below, i.e., for any λ 0, there exists a finite constant c such that γn (λ) −c, for any n ∈ N, n > 0.
Proof. Let us set again ω = 1 and denote by Γ λ the operator Γ1λ . We can decompose Γ λ = a λ + Γ0λ , where Γ0λ is the self-adjoint operator associated with the positive quadratic form 2 1 x ) − q( x ) . Φ0λ [q] ≡ d x d x Gλ ( x , x; x , x )q( (4.12) 2 R2
Since Φ0 is positive and a λ (x) is an unbounded function, which is however bounded below for any λ > 0, Γ λ is an unbounded operator which is bounded below. Notice that a λ (x) is a monotone increasing function of x and a λ (x) cx for x √ → ∞; furthermore a λ (x) a λ (0), λ λ a (0) is a monotone increasing function of λ and a (0) c λ for λ → ∞. Hence by the lower bound (3.14) we have that for any α ∈ R, there exists λ0 > 0 such that for λ > λ0 the quadratic form Φαλ is positive and bounded below; for such λ the operator Γ λ is invertible. The claim on the spectrum of Γ λ can be proved in the same way as the two-dimensional case, then it is sufficient to prove that Φαλ can be written in the following way: 2 1 2 λ λ − q( k )q( ˜ λ (k; ˆ k) + ˆ k) ˆ k ) , Φα [q] = d k α + a˜ (k) q( d k d k G (4.13) 2 R2
R4
where 1 4ν λ−1 (1 − ν) 1 1 1− a˜ (k) ≡ C + dν 4π 1−ν (1 + ν 2 ) ln ν1 + 1 − ν 2 0 (1 − ν 2 ) ln ν1 2 × exp − k , 2[(1 + ν 2 ) ln ν1 + 1 − ν 2 ] λ
(4.14)
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k )≡ 1 G (k; 2π 2 ˜λ
1 dν 0
ν λ−1 (1 − ν 2 ) ln ν1 + 2(1 − ν)2
[(1 + ν 2 ) ln ν1 + 1 − ν 2 ](k 2 + k 2 ) × exp − 2[(1 − ν 2 ) ln ν1 + 2(1 − ν)2 ]
[1 − ν 2 + 2ν ln ν1 ]k · k . − (1 − ν 2 ) ln ν1 + 2(1 − ν)2
(4.15)
The function a˜ λ (k) has the same asymptotic behavior for k → ∞ as a λ (x), namely aˆ λ (k) ck and by applying Rellich’s criterion, the spectrum of Γ λ is pure point. Notice that the following bound holds: Φ0λ [q] c q 2H1/2 (R3 ) .
(4.16)
Indeed, using the inequality in (3.13),
1
x , x; x , x ) c G ( λ
dν 0
ν λ−1 ((1 − ν 2 ) ln ν1 )3/2
( x − x )2 ν( x − x )2 , exp − − 1 − ν2 2 ln ν1
and taking the Fourier transform, we have
d k
Φ0λ [q] c R
1 dν 0
ν λ−1 (2ν ln ν1 + 1 − ν 2 )3/2
× 1 − exp −
(1 − ν 2 ) ln ν1 2(1 − ν 2 + 2ν ln ν1 )
2 q(k) ˆ . k 2
(4.17)
Since 1 0
dν 0
ν λ−1 (2ν ln ν1 + 1 − ν 2 )3/2
1 − exp −
2(1 − ν 2 ) ln ν1 4(2ν ln ν1 + 1 − ν 2 )
k
2
ck,
(4.16) is proved. Therefore, taking into account the behavior of a λ , (4.16) implies that there exists λ0 > 0 such that Γ λ c x1/2 + p1/2 + λ1/2
(4.18)
holds for λ > λ0 . Also in the three-dimensional case the lower bound (3.24) holds as well, but let us stress that a λ and a˜ λ have a different behavior; in particular in the three-dimensional case (4.11) holds true, due to (4.18).
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Monotonicity of the eigenvalues and unboundedness from below of γ0 (λ), as λ → 0, can be shown exactly as in the two-dimensional case. Note that one has to evaluate the form on the ground state of the three-dimensional harmonic oscillator. Furthermore we have the lower bound, q ⊥ Γ λ q ⊥
1
1
1 + 2
e
1
⊥ 2 q
dν 3 3 (4π) 2 (1 − ν) 2 0 ∗ − d x d x Gλ1/e ( x , x; x , x ) q ⊥ ( x ) q ⊥ ( x ), R6
but, acting as in the proof of (3.26), we get
∗ d x d x Gλ1/e ( x , x; x , x ) q ⊥ ( x ) q ⊥ ( x )
R6
1 (2π)
3 2
2 ⊥ 1 −1 q Hosc + λ − 3/2 q ⊥ + 2q ⊥ ,
so that, if q ⊥ is a normalized function orthogonal to the ground state of the harmonic oscillator, q ⊥ |Γ λ |q ⊥ −c, for some finite constant c. 2 The discrete spectrum of Hαω can now be fully characterized. Theorem 4.5 (Negative spectrum of Hαω ). For any α ∈ R, the discrete spectrum σpp (Hαω ) of Hαω is not empty and it contains a number Nω (α) of negative eigenvalues −E0 (α, ω) −E1 (α, ω) · · · 0 satisfying the scaling √ En (α, ω) = ωEn (α/ ω, 1).
(4.19)
The corresponding eigenvectors are given by un = GωEn qn , where qn is a solution to the homogeneous equation αqn + ΓωEn qn = 0. Moreover there exists α0 ∈ R such that, if α > α0 , Nω (α) = 1 and, for fixed ω and α → −∞, Nω (α) c|α|6 . The ground state energy has the following asymptotic behavior for fixed ω: E0 cα −1 for α → +∞ and E0 cα 2 for α → −∞. Proof. See the proof of Theorem 3.5; notice that in the argument used to estimate the asymptotics of Nω (α), the spectral distribution of the square root of the three-dimensional harmonic oscillator is involved. 2 An interesting consequence of the above theorem is the existence of a bound state for any α and 0 < ω < ∞, in particular even if α > 0 and there is no bound states for the “reduced” system (we shall come back to this question in the concluding comments).
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The asymptotics for ω → 0 and α > 0 is exactly as in the one- and two-dimensional case, whereas the behavior for ω → 0 and α < 0 proves to be much more complicated, due to the ground state asymptotics (see theorem above). If ω → ∞ we expect that the asymptotics depend on a crucial way on the sign of α, since at least one bound state should survive if α < 0, whereas, if α > 0, all bound states should disappear in the limit. Now we shall give a partial characterization of the positive spectrum of (1.1), but we first state a result analogous6 to Lemma 3.6. Lemma 4.6. Define the operator Tωk ≡ Gωλ (Gλω )k Gωλ∗ , for any k ∈ N. Then, if λ > k, Tωk ∈ Bp (L2 (R3 ), L2 (R3 )), for any p > 3(k + 1/2)−1 . Theorem 4.7 (Positive spectrum of Hαω ). The essential spectrum of Hαω is equal to [0, +∞) and the wave operators Ω± (Hαω , H0ω ) exist and are complete. Proof. We shall omit the dependence on ω for brevity. It is sufficient to prove that (Hα + λ)−1 − (H0 + λ)−1 is a compact operator and that [(Hα + λ)−1 ]4 − [(H0 + λ)−1 ]4 is trace class for some λ > 0, then the thesis follows from Weyl’s theorem (see [22, Theorem XIII.14] and [21, Corollary 3 of Theorem XI.11]). We shall fix λ sufficiently large such that (Γ λ + α)−1 exists. Boundedness of (Γ λ + α)−1 , Hölder inequality and the fact that G λ∗ G λ ∈ Bp (L2 (R6 ), L2 (R6 )), p > 6, because of Lemma 4.6, imply compactness of (Hα + λ)−1 − (H0 + λ)−1 , as in the two-dimensional case. Besides one can show that G λ belongs to Bp (L2 (R6 ), L2 (R3 )), p > 12, and G λ∗ ∈ Bp (L2 (R3 ), L2 (R6 )) for the same p. Finally the tedious but straightforward calculation of [(Hα + λ)−1 ]4 − [(H0 + λ)−1 ]4 and the application of Lemma 4.6 with k = 3 to each term of the expansion give the result. 2 5. Conclusions and perspectives We have studied a quantum system composed of a test particle and a harmonic oscillator interacting through a zero-range force. We have given a rigorous meaning to the Hamiltonian Hαω of the system, described the properties of its spectrum and established asymptotic completeness for the scattering operators Ω± (Hαω , H0ω ), where H0ω is the Hamiltonian of the system without the zero-range force. The negative part of the spectrum of Hαω for ω > 0 is discrete and we have given estimates of the number of bound states. There is a peculiar feature of this part of the spectrum: in the threedimensional setting, in the case of a fixed center, i.e., ω = ∞, when the parameter α is negative, there is exactly one bound state, while, in the case α > 0, the spectrum is absolutely continuous. In our case, if α > 0, there is always a bound state and, if α < 0, the number of bound states increases as the strength of the harmonic force goes to zero. We might interpret this feature as due to the fact that bound states of the harmonic oscillator provide a mechanism through which the test particle is bound, even if the interaction due to the zero-range force is “repulsive.” We have privileged explicit expressions because we regard our analysis as preliminary to a detailed treatment of the case in which many oscillators are present. This model is widely used in the physical literature, for instance in kinetic theory, under the name of Rayleigh gas and in that 6 The proof follows exactly the proof of Lemma 3.6 and is omitted for the sake of brevity.
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context one considers as relevant the spectral and scattering properties of a particle interacting with a background of scatterers, in particular detailed estimates on the scattering cross sections. We plan to extend our analysis to this more general setting and obtain rather detailed information through a multichannel scattering approach, the channels being labeled by the bound states of the harmonic oscillators. Acknowledgments The authors are very grateful to A. Teta for many interesting discussions. The authors also thank the anonymous referee for her/his instructive and very helpful comments. References [1] S. Albeverio, F. Gesztesy, R. Hogh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, 1988. [2] J. Bellandi, E.S. Caetano Neto, The Mehler formula and the Green function of multidimensional isotropic harmonic oscillator, J. Phys. A 9 (1976) 683–685. [3] F.A. Berezin, L.D. Faddeev, A remark on Schrödinger equation with a singular potential, Soviet. Math. Dokl. 2 (1961) 372–375. [4] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971) 11–51. [5] J. Brüning, V. Geyler, K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys. 20 (2008) 1–70. [6] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts Monogr. Phys., Springer-Verlag, Berlin, 1987. [7] G. Dell’Antonio, D. Finco, A. Teta, Singularly perturbed Hamiltonians of a quantum Reyleigh gas defined as quadratic forms, Potential Anal. 22 (2005) 229–261. [8] Y.N. Demkov, V.N. Ostrovsky, Zero-range Potentials and Their Applications in Atomic Physics, Plenum, New York, 1988. [9] M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit, London Math. Soc. Lecture Note Ser., vol. 268, Cambridge Univ. Press, Cambridge, 1999. [10] E. Fermi, Sul moto dei neutroni nelle Sostanze Idrogenate, Ricerca Scientifica 7 (1936) 13–52 (in Italian). [11] G. Grubb, Known and unknown results in elliptic boundary value problems, Bull. Amer. Math. Soc. 43 (2006) 227–230. [12] P.R. Halmos, V.S. Sunder, Bounded Integral Operators on L2 Spaces, Springer-Verlag, New York, 1978. [13] B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Math., vol. 1336, Springer-Verlag, Berlin, 1988. [14] T. Kato, Perturbation Theory for Linear Operators, Classics in Math., Springer-Verlag, Berlin, 1995. [15] R. Kronig, W.G. Penney, Quantum mechanics of electrons in crystal lattices, Proc. Roy. Soc. A 130 (1931) 499–513. [16] L.D. Landau, E.M. Lifshitz, Non-Relativistic Quantum Mechanics, Mir, Moscow, 1967. [17] J.L. Lions, E. Magenes, Non-homogeneous Boundary Values Problems and Applications, vol. I, Springer-Verlag, New York, 1972. [18] S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Clarendon, Oxford, 1984. [19] A. Posilicano, Self-adjoint extensions of restrictions, preprint, arXiv: math-ph/0703078v2, 2007. [20] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I: Functional Analysis, Academic Press, San Diego, CA, 1972. [21] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. III: Scattering Theory, Academic Press, San Diego, CA, 1975. [22] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators, Academic Press, San Diego, CA, 1978. [23] B. Simon, Trace Ideals and Their Applications, Math. Surveys Monogr., vol. 120, Amer. Math. Soc., Providence, RI, 2005. [24] J. Sjostrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier 57 (2007) 2095–2141. [25] K. Yosida, Functional Analysis, Classics in Math., Springer-Verlag, Berlin, 1995.
Journal of Functional Analysis 255 (2008) 532–533 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Enhanced negative type for finite metric trees” [J. Funct. Anal. 254 (2008) 2336–2364] Ian Doust a , Anthony Weston b,∗ a School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia b Department of Mathematics and Statistics, Canisius College, Buffalo, New York 14208, United States
Available online 28 April 2008
A major aim of this paper was to provide a quantitative enhancement of the strict 1-negative type inequalities for finite metric trees due to Hjorth et al. [1]. Any such inequalities clearly require a normalization condition, which was unfortunately omitted from the original statement of Theorem 4.16 (and consequently from inequality (1)). The correct statement of this theorem is as follows. Theorem 4.16. Let (T , d) be a finite metric tree. Then for all natural numbers n 2, all finite subsets {x1 , . . . , xn } ⊆ T , and all choices of real numbers η1 , . . . , ηn with η1 + · · · + ηn = 0, we have ΓT · 2 where ΓT = {
n
2 |η |
+
=1
d(xi , xj )ηi ηj 0
1i,j n
−1 −1 e∈E(T ) |e| } .
Proof. Fix a subset {x1 , . . . , xn } ⊆ T (n 2), together with real numbers η1 , . . . , ηn such that η1 + · · · + ηn = 0. Without any loss of generality we may assume that (η1 , . . . , ηn ) = (0, . . . , 0). By relabelling (if necessary) we may assume there exist natural numbers q, t ∈ N such that q + t = n, η1 , . . . , ηq 0, and ηq+1 , . . . , ηn < 0. DOI of original article: 10.1016/j.jfa.2008.01.013. * Corresponding author.
E-mail addresses: [email protected] (I. Doust), [email protected] (A. Weston). 0022-1236/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jfa.2008.03.018
I. Doust, A. Weston / Journal of Functional Analysis 255 (2008) 532–533
As
q
=−
j =1 ηj
n
k=q+1 ηk
α=
533
we may define
q
ηj = −
j =1
n
1 |η | > 0. 2 n
ηk =
k=q+1
=1
For 1 j q, set aj = xj and mj = ηj /α. And for 1 i t, set bi = xn−i+1 and ni = −ηn−i+1 /α. By construction, D = [aj (mj ); bi (ni )]q,t is a normalized (q, t)-simplex. Moreover, arguing as in the proof of Theorem 2.4 with exponent p = 1, we see that 1 · α2
d(xi , xj )ηi ηj
1i,j n
=2
mj1 mj2 d(aji , aj2 ) +
1j1 <j2 q
ni1 ni2 d(bi1 , bi2 ) −
1i1
n
mj ni d(aj , bi ) .
j,i=1
Hence by Theorem 4.12, 1i,j n
d(xi , xj )ηi ηj −2α · 2
e∈E(T )
−1
|e|
−1
ΓT · =− 2
n
2 |η |
.
2
=1
References [1] P. Hjorth, P. Lisonˇek, S. Markvorsen, C. Thomassen, Finite metric spaces of strictly negative type, Linear Algebra Appl. 270 (1998) 255–273.
Journal of Functional Analysis 255 (2008) 534 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “On fully operator Lipschitz functions” [J. Funct. Anal. 253 (2007) 711–728] E. Kissin a,∗ , V.S. Shulman b a Department of Computing, Communications Technology and Mathematics, London Metropolitan University,
166-220 Holloway Road, London N7 8DB, UK b Department of Mathematics, Vologda State Technical University, Vologda, Russia
Available online 19 May 2008
The authors regret the inadvertent omission of the name of M.S. Birman from Ref. [15]. The correct reference is displayed here. [15] M.S. Birman, M.Z. Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003) 131–168.
DOI of original article: 10.1016/j.jfa.2007.08.007. * Corresponding author.
E-mail addresses: [email protected] (E. Kissin), [email protected] (V.S. Shulman). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.010
Journal of Functional Analysis 255 (2008) 535–588 www.elsevier.com/locate/jfa
Gamma limit of the nonself-dual Chern–Simons–Higgs energy Matthias Kurzke a , Daniel Spirn b,∗ a Institut für angewandte Mathematik, Universität Bonn, Germany b School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Received 26 April 2007; accepted 24 April 2008 Available online 2 June 2008 Communicated by H. Brezis
Abstract We consider the Gamma limit of the Abelian Chern–Simons–Higgs energy Gcsh :=
1 2
|∇Aε uε |2 + U
2 μ2 |curl Aε − hex |2 1 + 2 |uε |2 1 − |uε |2 dx 4 |uε |2 ε
on a bounded, simply connected, two-dimensional domain under the ε → 0 limit. As a first step we study the Gamma limit of 1 Ecsh := 2
2 1 |∇uε |2 + 2 |uε |2 1 − |uε |2 dx ε
U
under two different scalings; Ecsh ≈ |log ε| and Ecsh ≈ |log ε|2 . We apply the |log ε|2 -scaling result to the full Chern–Simons–Higgs energy Gcsh , and as a consequence we are able to compute the first critical field H1 = H1 (U, μ) for the nucleation of a vortex. The method entails estimating in certain weak topologies the Jacobian J (uε ) = det(∇uε ) in terms of the Chern–Simons–Higgs energy Ecsh . © 2008 Elsevier Inc. All rights reserved. Keywords: Chern–Simons–Higgs theory; Gamma convergence; Jacobian compactness; Vortices; Critical field
* Corresponding author.
E-mail addresses: [email protected] (M. Kurzke), [email protected] (D. Spirn). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.020
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M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
1. Introduction Abelian Chern–Simons–Higgs (CSH) theory serves as an anyon model [6,13,14,39] and is a classical field theory defined on (2 + 1)-dimensional Minkowski space. Such models have applications to the theory of high temperature superconductivity, quantum Hall effects and carry fractional charge values [6,39]. The model is described on the (2 + 1)-Minkowski space (R1,2 , g) (the metric tensor g = diag[1, −1, −1] is used in the usual way to lower and raise indices) by the following CSH Lagrangian density: Lcsh = Dα uD α u +
1 2 2 μ αβγ ex − 2 |u| 1 − |u|2 , Aα Fβγ − Fβγ 4 ε
where A = −iAα dx α with Aα : R1,2 → R for α = 0, 1, 2 is the gauge potential with covariant derivative DA = d − iA. The corresponding curvature FA = − 12 Fβγ dx β ∧ dx γ with Fβγ = ∂β Aγ − ∂γ Aβ defines the gauge field, and u : R1,2 → C is the Higgs scalar with Dη u = ∂η u − iAη u, η = 0, 1, 2. Furthermore, the antisymmetric Levi-Civita tensor αβγ is fixed by setting ex ) is 0,1,2 = 1 and μ, ε > 0 are the Chern–Simons coupling parameters. Here αβγ Aα (Fβγ − Fβγ ex the Chern–Simons term with applied field tensor F , see (1.3). The associated Euler–Lagrange equations are 1 2 u |u| − 1 3|u|2 − 1 = 0, ε2 μ αβγ ex Fβγ − Fβγ + J α = 0, 4
Dα D α u +
(1.1) (1.2)
where J α = (iu, D α u) is the matter current. Eq. (1.2) is very different from the more conventional Maxwell’s current equation, D β Fαβ + J α = 0, found in the more widely studied Maxwell–Higgs model, which says that the change in the matter current is due to the rate of change of the electromagnetic field. In the Chern–Simons case μ4 αβγ Fβγ + J α = 0 implies the matter current is proportional to the electromagnetic field. This model has been the source of much interest in the physics community; the book of Yang [39] offers an excellent overview of Chern–Simons–Higgs and related theories. To date most rigorous analysis has been restricted to self-duality which occurs when μ = ε and hex = 0, as discovered independently by Hong, Kim, Pac and Jackiw, Weinberg in [13,14]. On the other hand in this paper we consider ε 1 and μ = O(1). Since the α = 0 refers to time coordinates, we replace D0 by ∂Φ = ∂t − iΦ and replace Dα by ∇A = ∇ − iA when α ∈ {1, 2}. Here (Φ, A) is the field potential. The curvature tensor is defined by F=
0 E1 E2
−E1 0 h
−E2 −h 0
,
0 0 0 0 0 hex
0 −hex 0
,
(1.3)
where h = curl A, Eα = ∂t Aα − ∂α Φ are the induced magnetic and electric fields and hex is the applied magnetic field. We write the current J α in a more classical notation by setting J 0 = (iu, ∂Φ u) = q,
J α = (iu, ∇Aα u) = jAα
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
537
for α ∈ {1, 2} which are the charge and supercurrent, respectively. Therefore, the current equation reads μ2 (h − hex ) + q = 0, − μ2 E2 + jA1 = 0, and μ2 E1 + jA2 = 0, and in more classical notation we write the CSH equations as: 2 ∂Φ u = ∇A2 u +
1 2 2 3|u| u 1 − |u| − 1 , ε2
μ q = − (curl A − hex ), 2 μ jA = (E × e3 ). 2
(1.4) (1.5) (1.6)
If we take the curl and div of the equations, then we get the following useful identities μ μ curl E = ∂t q = − ∂t h, 2 2 μ JA (u) = − div E, 4
div jA =
(1.7) (1.8)
where JA (u) = 12 curl jA (u) = J (u) − 12 curl(A|u|2 ) and J (u) =
1 curl j (u) = det ∇u. 2
Well-posedness questions for Eqs. (1.4)–(1.6) were studied in [7,8]. Since u : R2 → C we can easily induce the formation of topological vortices—regions where |u| = 0 and about which the winding number of the phase is nontrivial. Setting u = ρeiϕ ≈ eiϕ over R2 and ϕ = dθ , then JA ≈ 12 curl(∇ϕ − A) = det ∇u − 12 h. Assuming that E → 0 as |x| → +∞, then we can formally integrate (1.8) over R2 and get 2πd = R2 h dx. Furthermore, integrating (1.5) over the plane and assuming that hex = 0 yields the relation d=
1 2π
h dx = − R2
1 μπ
q dx.
(1.9)
R2
As in Ginzburg–Landau theory, we see that the current and the magnetic field are quantized about a topological vortex; however, in CSH theory the magnetic field induces a quantized electric charge, which can have arbitrary values, depending on μ. This quantized electric charge is a fundamental feature of Chern–Simons–Higgs theory. Following the approach of [13] and [14], we will consider static solutions of Chern–Simons– Higgs systems as stationary points of a two-dimensional energy functional. The correct (Hamiltonian) energy density can be calculated by a Legendre transform as in [39, pp. 164–165]; we sketch an approach through just the Euler–Lagrange equations. We focus on the case of bounded domains as discussed in [11] for the self-dual and in [12] for the general, possibly nonselfdual case.
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Setting ∂t u ≡ 0 and ∂t A ≡ 0, Eqs. (1.4)–(1.6) become 1 u 1 − |u|2 3|u|2 − 1 , 2 ε μ μ 2 Φ|u| = (curl A − hex ), jA (u) = ∇Φ × e3 . 2 2
−Φ 2 u = ∇A2 u +
Removing the electric field potential Φ, we are left with an unusual system of coupled elliptic PDE’s: −
1 μ2 | curl A − hex |2 u = ∇A2 u + 2 u 1 − |u|2 3|u|2 − 1 , 4 4 |u| ε 2 μ curl A − hex + jA (u). 0 = − curl 4 |u|2
(1.10) (1.11)
Taking the curl and div of (1.11) yields two more useful equations curl A − hex μ2 , JA (u) = − 4 |u|2 div jA (u) = 0.
(1.12) (1.13)
Eqs. (1.10)–(1.11) can be formally viewed as the Euler–Lagrange equations of the following Chern–Simons–Higgs energy 1 Gcsh (u, A; hex ) = 2
|∇A u|2 + U
2 μ2 | curl A − hex |2 1 + 2 |u|2 1 − |u|2 dx 2 4 |u| ε
(1.14)
for an applied magnetic field, hex , and a bounded, simply connected domain, U ⊂ R2 . The connection appears to be only formal at first glance, as the singularity at |u| = 0 makes it difficult to rigorously justify a naive calculation of the Euler–Lagrange equations. However, it is possible to show by a regularization technique that minimizers of (1.14) satisfying |u| = 1 on ∂U exist for any choice of ε, μ and hex and satisfy the Euler–Lagrange equations (1.10)–(1.11), see [36, 1 Theorem 1.5]. The idea of the regularization is to replace |u|1 2 by |u2 |+δ 2 and to let δ → 0. For appropriate choices of the parameters, such minimizers will have vortices with |u(z)| = 0 at some point z, see [36] or Corollaries 1.4 and 1.5. We note some unusual features of (1.14). Let hex = 0 and suppose u has a topological vortex at 0. Then u must vanish at the origin. But the second term of (1.14) implies that h = curl A must likewise vanish at the origin. On the other hand the quantization relation (1.9) implies there exists a finite mass of magnetic field about this vortex, and consequently the magnetic field concentrates in an annular region about each topological vortex. This is in contrast to Ginzburg–Landau vortices, where the magnetic field concentrates at the site of the vortex. The second term proves to greatly increase the difficulty of analyzing (1.10)–(1.13), including the lack of a maximum principle. Another important feature of (1.14) is that in the ε → 0 limit, |uε | relaxes to S1 ∪ {0}, as opposed to S1 in the Ginzburg–Landau case. This implies that regions where |u| = 0 with trivial winding number about the region are possible and potentially favorable. We show, however, that such regions are small unless uε → 0 on the whole domain.
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539
1.1. Results Up to now, most attention has focussed on the self-dual case where ε = μ. In this case the CSH equations reduce, following Hong, Kim, Pac and Jackiw, Weinberg [13,14], to a system of first order PDE’s. Solutions can be recovered by solving (after a substitution) a Liouville-type elliptic equation, similar to the Jaffe–Taubes approach to solving the self-dual Ginzburg–Landau equations [15]. It is impossible to give an adequate accounting of the extensive results on selfdual solutions to the Chern–Simons–Higgs equations, but we direct the reader to [6,9,11,13,14, 28,37,39] and the references therein. We turn our attention to nonself-dual Chern–Simons–Higgs theory. The only results to our knowledge for small ε and μ = O(1) for the CSH functional are those of Han, Kim [12], who studied among other things sequential minimizers {uε , Aε } of (1.14) with Aε ≡ 0 and Dirichlet boundary condition uε = g on ∂U with |g| = 1. Their proofs are similar in spirit of the methods Bethuel, Brezis, Helein [3] for the simplified Ginzburg–Landau energy (1.24) and rely heavily on the maximum principle for |uε |. The maximum principle fails when gauge field Aε ≡ 0, so another approach is needed. In this paper we make no restrictions on either Aε or the boundary behavior of |uε |; our study yields compactness and Γ -convergence results for two scalings of the energy. In particular, our convergence results are true for non-minimizers and indeed even for sequences of functions that are not solutions of the corresponding equations. Our techniques are related to the approach of Jerrard, Soner [18,19] combined with the Sandier [30] version of the vortex ball construction method of Jerrard [16] and Sandier [30]. Similar to their approach, we first study the simplified functional 1 Ecsh (u) = 2
|∇u|2 +
2 1 2 |u| 1 − |u|2 . 2 ε
(1.15)
U
(In fact our approach is robust enough to deal with more general potentials of the form ε12 W (|u|2 ) with W (s) = s p (1 − s)q for p 0 and q 1.) The results for (1.15) are then used to analyze the case with a magnetic field hex and gauge field A. We have the following results, stated here in the spirit of Γ -convergence; that is, separated into a compactness result combined with a lower bound for the energy and a construction that shows that the lower bound is essentially optimal. Like most results of Γ -convergence type, our theorems imply that minimizers of the approximating energies converge to minimizers of the limit energy, and minimizers of the limit energy can be recovered as limits of sequences of almost minimizers of the approximating energies, see Corollary 1.4. For the sake of unified exposition, we only state the two-dimensional results here, although some of the results are also true in higher-dimensional domains (see Proposition 4.1). Our general assumption is that U ⊂ R2 is a simply connected domain with the CSH extension property (see Definition 4.5) and that {uε } is a sequence of functions in H 1 (U ; C). By Proposition 4.6, every simply connected C 1,α domain has the CSH extension property, but many other domains also fall into this class, see Section 4. Theorem 1.1 (Compactness and Γ -convergence in the |log ε| scaling). Assume Ecsh (uε ) K|log ε| for some constant K > 0. Taking a subsequence, the modulus ρε = |uε | satisfies ρε → ρ strongly in Lp for p < +∞ where ρ is either identically 0 or identically 1. The Jacobians J (uε )
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M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
are precompact in the weak (C 0,β )∗ topology for every β > 0, and every cluster point J satisfies
J = π i di δai for some di ∈ Z. Furthermore, lim inf ε→0
1 Ecsh (uε ) J M . |log ε|
(1.16)
Here J = 0 if ρ = 0.
Conversely, for every measure J of the form J = π i di δai , there exists a sequence {uε } with J (uε ) J and 1 Ecsh (uε ) = J M . ε→0 |log ε| lim
(1.17)
The proof of this theorem is contained in the proofs of Propositions 5.2 and 5.5. The next result concerns the most interesting energy scaling, that where vortex interaction energy and vortex core energy are of the same order, and where we can observe nucleation of vortices in minimizers induced by a magnetic field, see [19,32,34] in the Ginzburg–Landau case. Theorem 1.2 (Compactness and Γ -convergence in the |log ε|2 scaling). Assume Ecsh (uε ) K|log ε|2 , for some constant K. Set vε = |log1 ε| j (uε ) and wε = |log1 ε| J (uε ) = 12 curl vε . Taking a subsequence, the modulus ρε = |uε | satisfies ρε → ρ strongly in Lp for p < +∞ where ρ is either identically 0 or identically 1. The scaled Jacobian, {wε }, is precompact in the weak (C 0,β )∗ topology and {vε } is bounded in Lp for 1 p < 2. Furthermore, if wε w = 12 curl v and vε v, then also |uvεε | v in L2 , and the energy satisfies
lim inf ε→0
1 1 Ecsh (uε ) v2L2 + curl vM . 2 2 |log ε|
(1.18)
If ρ = 0, then v = 0. Conversely, for every v ∈ L2 (U ; R2 ) such that w = 12 curl v is a Radon measure, there exists a sequence {uε } in H 1 (U ; C) with |uε | = 1 on ∂U such that vε = |log1 ε| j (uε ) v in L2 and wε =
1 |log ε| J (uε ) w
in (C 0,β )∗ and the energy satisfies
1 1 Ecsh (uε ) = v2L2 + curl vM . 2 ε→0 |log ε| 2 lim
(1.19)
The proof of this theorem is contained in the proofs of Propositions 5.1 and 5.3. In the energy scaling of Theorem 1.2, we also have a result with an external magnetic field hex and gauge field A. For simplicity, we state the result only in Coulomb gauge, which amounts to considering only pairs (u, A) with ∇ · A = 0 in U and A · ν = 0 on ∂U . These conditions can always be satisfied by an appropriate gauge transformation replacing (u, A) by (ueiχ , A + ∇χ) without changing the energy.
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
541
Theorem 1.3 (Compactness and Γ -convergence with external field). Assume that the external field satisfies hex = H |log ε| for some H > 0, and consider a sequence {uε , Aε } that satisfies the Coulomb gauge condition and Gcsh (uε , Aε ; hex ) K|log ε|2 . Set aε =
1 |log ε| Aε ,
then {aε } is weakly precompact in W 1,p for all p < 2, and for a subsequence
such that aε a there holds
curl aε −H |uε |
curl a − H in L2 .
Additionally, the compactness assertions of the last theorem hold: vε =
1 |log ε| j (uε ) converges w = 12 curl v. Taking a
J (uε ) to v weakly in all Lp with p < 2, |uvεε | v in L2 , and wε = |log ε| subsequence, the modulus ρε = |uε | satisfies ρε → ρ strongly in Lp for p < +∞ where ρ is either identically 0 or identically 1. If ρ = 0, then curl a = H and v = 0. Furthermore, the energy satisfies
lim inf ε→0
1 Gcsh (uε , Aε ; hex ) Gρ (v, a; H ), |log ε|2
(1.20)
with G1 (v, a; H ) =
1 2
|v − a|2 +
μ2 |curl a − H |2 + curl vM 4
(1.21)
U
and G0 (v, a; H ) = 0.
(1.22)
Conversely, for any a ∈ H 1 (U ; R2 ) and v ∈ L2 (U ; R2 ) such that w = 12 curl v is a Radon measure, there exists a sequence {uε } in H 1 (U ; C) with |uε | = 1 on ∂U and a sequence {Aε } ∈ H 1 (U ; C) satisfying the Coulomb gauge conditions such that vε = |log1 ε| j (uε ) v in L2 , wε = 1 |log ε| J (uε ) w
in (C 0,β )∗ , aε =
1 |log ε| Aε
a in H 1 , and such that (1.20) holds with equality
for ρ = 1. For ρ = 0, there exists a sequence (uε , Aε ) with uε → 0 and that Gcsh (uε , Aε ; hex ) → 0.
1 |log ε|
curl Aε → H such
This theorem follows from Propositions 6.1, 6.2, and 6.3. Regarding minimizers under a Dirichlet boundary condition on the modulus (such minimizers exist and satisfy the Euler–Lagrange equations (1.10), (1.11) by the results of [36]), we have the following. Corollary 1.4. Let (uε , Aε ) be a sequence of minimizers of (1.14) under the conditions |uε | = 1 on ∂U and ∇ · Aε = 0 in U , Aε · ν = 0 on ∂U . Then |uε | → 1 in the strong topology of all Lp (U ) with 1 p < ∞. Setting vε = |log1 ε| j (uε ), we have ( for a subsequence) that vε v in Lp for p < 2 and |uvεε | v in L2 while curl vε curl v in (C 0,β )∗ . Also up to extraction of a subsequence,
weakly in
W 1,p
for p < 2 and
curl A−hex |uε ||log ε|
curl a − H in
L2 .
Aε |log ε|
a
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The limit (v, a) is the unique minimizer of the limit functional (1.21) in the space of pairs (v , a ) ∈ L2 (U ; R2 ) × H 1 (U ; R2 ) such that curl v is a Radon measure, ∇ · a = 0 in U and a · ν = 0 on ∂U . The energies satisfy the following limit equality: 1 Gcsh (uε , Aε ; hex ) = G1 (v, a; H ). ε→0 |log ε|2 lim
The proof of this corollary is given in Proposition 6.4 and is mainly an application of the wellknown result that Γ -convergence implies convergence of minimizers and of minimizing energies, see [5] for an introduction. We remark that the corollary is based only on the energy, not on the Euler–Lagrange equations, and remains valid even if we replace sequences of minimizers by sequences of functions whose energies exceed that of a minimizer only by a term that vanishes as ε → 0. As an application, we calculate the critical field hc1 for which vortices appear in nonzero minimizers of Gcsh (uε , Aε ; hex ), for example for minimizers under a boundary condition |uε | = 1. To be precise, we define the critical field as the field for which the rescaled vorticities |log1 ε| J (uε ) converge to a nonzero limit; this implies via the mapping degree that there exist points z where |uε (z)| = 0. Corollary 1.5. As ε → 0, the critical field hc1 is given asymptotically by H1 (μ)|log ε|, where H1 (μ) =
2 μ2 maxU |zμ |
and zμ is the solution of −
μ2 zμ + zμ + 1 = 0 4
with homogeneous Dirichlet boundary conditions. Concerning the dependence on μ, we have that μ2 H1 (μ) → 2 as μ → 0. Furthermore, H1 (μ) is decreasing in μ and converges to a limit H (U ) > 0 as μ → ∞. Finally, when U ≡ BR , a ball of radius R, then H1 (μ, R) =
2I0 ( 2R μ ) μ2 (I0 ( 2R μ ) − 1)
,
where I0 is the modified Bessel function of zeroeth order. The corollary follows from Theorem 1.3 using some analysis of the limit functional, see Proposition 7.1. 1.2. Methodology with a comparison to Ginzburg–Landau We can compare (1.14) with the Ginzburg–Landau energy
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
1 Ggl (u, A; hex ) := 2
|∇A u|2 + |curl A − hex |2 +
2 1 1 − |u|2 dx. 2 2ε
543
(1.23)
U
The asymptotic properties of (1.23) have been a topic of extensive research since the groundbreaking study of the corresponding functional without gauge field 1 Egl (u) := 2
|∇u|2 +
2 1 1 − |u|2 dx 2 2ε
(1.24)
U
in the book of Bethuel, Brezis, Helein [3], and (1.24) is commonly referred to as the BBH energy. Here the authors offer a complete description of the small ε limit of energy minimizers to (1.24) by PDE and comparison methods. Since energy about each vortex core is of size π|log ε|, uniform energy bounds can be found by cutting out the vortices from the domain. Furthermore, the authors expand the energy (1.24) asymptotically to second order, finding, up to boundary effects, a Coulomb potential. There has also been great success in higher dimensions including [1,4,18, 25,29]. In higher dimensions the vorticity concentrates on (n − 2)-dimensional, integer multiplicity, rectifiable currents. In the case of minimizing sequences the current is area minimizing, see [25]. Non-minimizing sequences have also been subject to significant interest, see [4,17,18,21,22, 24,30,31,33] among other places. The Γ -limit result for the BBH energy was proven in Jerrard, Soner [18] in two-dimensional domains. The higher-dimensional Γ -limit was established jointly in Jerrard, Soner [18], who proved the compactness and energy lower bound, and Alberti, Baldo, Orlandi [1], who were able to construct the needed energy upper bound. The calculation of the first critical field H1 for (1.23) can be found in [32,34]. The Γ -convergence for the full Ginzburg– Landau energy was proven in Jerrard, Soner [19], and the authors used this for a new derivation of H1 . We briefly outline of rest of the paper. Our approach to proving the Γ -limit works well under two cases: either |uε | = 1 or |uε | = 0 on ∂U , and Sections 2 and 3 provide basic results under the restrictive and more difficult condition |uε | = 1 on ∂U . In Section 4 we consider the case |uε | = 0 on ∂U , and later on we get rid of any boundary assumption via an extension argument. In Section 2 we provide several basic estimates on the Chern–Simons–Higgs energy. We make use of the Modica–Mortola method to prove strong convergence of |uε | → 1 when |uε | 12 on the boundary. In Section 3 we prove, assuming |uε | = 1 on ∂U , the basic Jacobian estimate φJ (uε ) dx πdφL∞ + Cε γ φ 0,1 C
(1.25)
csh (uε ) ecsh (uε ) where d ≈ π1 e|log ε| dx, γ ∈ (0, 1), and C depends on |log ε| dx. The estimate, in the spirit of [18], follows from integration by parts and the co-area formula,
1 φJ (uε ) dx = 2
∞
1 ∇φ × j (uε ) dl dt ≈ 2
0 ∂Ω(t)
∞ ≈π 0
deg uε , ∂Ω(t) dt,
∞ τ · ∇ϕ dl dt 0 ∂Ω(t)
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where uε = ρeiϕ and Ω(t) = {x ∈ U such that φ(x) > t}. Therefore, controlling φJ (uε ) dx consists of finding estimates on the degree of u on level sets of φ. In order to establish (1.25) we can divide the level sets t ∈ (0, φL∞ ) into high and low degree sets, and, not surprisingly, the lower degree level sets are rather easy to estimate. On the other hand the higher degree level sets are much more difficult to understand. In particular we expect
Dd = t such that deg uε , ∂Ω(t) d + 1 , (uε ) dx where d ≈ π1 U ecsh|log ε| dx, should be of small measure otherwise there should be a violation of our energy bound. In [18,19] this is accomplished via a covering lemma that relies on lower bounds on the Ginzburg–Landau energy. In the CSH case the Jerrard–Soner method fails; however, we provide a new approach to estimating the size of the set |Dd | that is much more topological in nature than previous methods, and we find ηε αε 1 + exp |Dd | 8ε ecsh (uε ) dx ∇φL∞ , π(d + 1) V
where 0 1 − αε 1 and 0 ηε − 1 1, and where V = ˙ Brk with Brk ⊆ spt(φ) and rk = |Dd | 2∇φ ∞ . This bound provides good control on |Dd | and allows us to establish (1.25) for both L
the Ecsh (u) = O(|log ε|) and Ecsh (u) = O(|log ε|2 ) cases. Sections 4, 5 handle the proof of compactness and Γ -convergence of the CSH energy for energy of size O(|log ε|) and O(|log ε|2 ). Our arguments are similar to the approach found in [18,19]. Section 4 establishes the compactness of the Jacobian in a weak Banach space (C 0,β )∗ for energies of size O(|log ε|) and O(|log ε|2 ). Here we can lift the restriction on the domain being two-dimensional, and we can show that the limiting Jacobian is an (n − 2)-dimensional, integer multiplicity rectifiable current, see Propositions 4.1. We make use of estimate (1.25) and methods developed in [18] to establish this result, and rely on the existence of extensions of our sequence with good boundary conditions that is guaranteed by our assumption that the domain is a CSH extension domain as in Definition 4.5. Section 5.1 provides the lim inf condition on the CSH energy, and this lower bound follows almost directly from estimate (1.25). Section 5.2 completes the Γ -limit proof by constructing the upper bounds in both the O(|log ε|) and O(|log ε|2 ) cases. Here we make use of constructions of [20,31]. Section 6 then establishes the Γ -limit in the presence of the magnetic field potential and the external magnetic field. Finally, in Section 7 we study the limiting energy of the full CSH energy functional in the O(|log ε|2 ) case, as computed in Section 6. The critical field calculations are similar in spirit to the critical field calculation for Ginzburg–Landau energy (1.23). 1.3. Discussion Our choice of a purely energy-based method to study the asymptotic behavior of Chern– Simons–Higgs systems provides a good understanding of minimizers under conditions that ensure |uε | → 1 on the whole domain. We have demonstrated that it successfully explains a magnetic field induced formation of vortices in domains with the topological boundary condition |u| = 1, which is related to previous study of solutions on all of R2 with |u| → 1 at infinity. At least in the self-dual case, vortex solutions have also been found with so-called nontopological boundary conditions, |u| → 0 at infinity. Our corresponding limit functional G0 does not yield
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545
any information about these non-minimizing solutions, so other approaches are needed that can extend existence proofs for such nontopological vortices to the nonself-dual case. An obvious question to ask is what happens when μ = μ(ε) varies with ε. Some of the possible limits in this case are studied in [23], especially with regard to μ(ε) → 0 and μ(ε) → +∞. For the energy scaling Ecsh (uε ) ≈ Cε , we expect domain walls between regions where |uε | ≈ 0 and |uε | ≈ 1 to form. The interaction of those with the magnetic field, especially in the case where μ → 0, and a comparison with the phase diagram for type I and type II superconductors is an interesting open problem. Finally, the dynamics of vortices in the full CSH equations (1.10)–(1.13) can also be considered. In this case it is possible to generate more refined Jacobian estimates in terms of the CSH energy that establishes the rate of Γ -convergence, see [21,22]. Such estimates provide sufficient control to establish the dynamics of topological vortices. Rigorous results that establish the vortex motion law are found in [16,24] for the nonlinear wave equation and [10,35] for the Maxwell–Higgs equations. 2. Basic energy bounds Let u = ρeiϕ : U → C and A : U → R2 then we define two CSH energy densities 1 μ2 | curl A − hex |2 1 2 2 2 , |∇A u|2 + 1 − |u| + |u| 2 4 |u|2 ε2 1 1 2 2 2 2 ecsh (u) = |∇u| + 2 |u| 1 − |u| , 2 ε
gcsh (u, A; hex ) =
and set Ecsh (u) =
ecsh (u) dx
and Gcsh (u, A; hex ) =
U
gcsh (u, A; hex ). U
We note gcsh (u, A; hex )
2 1 1 = ecsh |u| ; |∇ρ|2 + 2 ρ 2 1 − ρ 2 2 ε
therefore, Gcsh (u, A; hex ) Ecsh (|u|) provides a simple lower bound that will be exploited throughout. We have the following useful energy bounds. Lemma 2.1. Suppose |U | Ecsh (|u|) and ε 1 then ρH 1 (U ) C Ecsh |u| , jA (u) α Cα Gcsh (u, A; hex ), L (U ) h − hex Lα (U )
Cα Gcsh (u, A; hex ) μ
for all 1 α < 2 and Cα → ∞ as α → 2. If Ecsh (|u|) < |U | then see Remark 2.2.
(2.1) (2.2) (2.3)
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Proof. Let ρ = |u| in the following. 6
3
1. By Young’s inequality ρ 6 + ρ 2 = ρ 6 − 2ρ 4 + ρ 2 + 2ρ 4 ρ 2 (1 − ρ 2 )2 + 2ρ3 + 23 so ρ 2 (1 − ρ 2 )2
+
23 3
ρ6 3
+ ρ2
which implies
|∇ρ|2 + ρ 2 dx
U
U
2 8 |∇ρ|2 + ρ 2 1 − ρ 2 dx + |U | 3
2
8 ecsh |u| dx + |U |. 3
U
By the assumptions on the energy, this shows (2.1). Furthermore, this implies by Sobolev embedding ρLp (U ) Cp ρH 1 (U ) Cp Ecsh |u|
(2.4)
for any p < ∞. 2. Since |∇A u|2 = | jAρ(u) |2 + |∇ρ|2 | jAρ(u) |2 , jA (u) jA (u) jA (u) α ρ = 2α α ρ ρ 2 ρL 2−α L (U ) (U ) L (U ) L (U ) Cα Gcsh (u, A; hex ) Ecsh |u| Cα Gcsh (u, A; hex ) follows from (2.4). 3. Finally, since
μ2 |h−hex |2 U 8 ρ2
dx Gcsh (u, A; hex ) then
h − hex h − hex ρ h − hex Lα (U ) = 2α α ρ ρ 2 ρL 2−α (U ) L (U ) L (U ) Cα Cα Gcsh (u, A; hex ) Gcsh (u, A; hex ) Ecsh |u| μ μ follows from (2.4).
2
Remark 2.2. If Ecsh (|u|) < |U | then replace (2.1) by ρH 1 C, (2.2) by jA (u)Lα √ √ Cα Gcsh , and (2.3) by h − hex Lα Cμα Gcsh . We have an important covering argument for {|u| < 1/4} that exploits the Modica–Mortola trick [26,27], used with great success by Sandier for complex Ginzburg–Landau energies [30].
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Lemma 2.3. Suppose ρ
1 2
on ∂U , then we have {x ∈ U : ρ < 14 } ⊂
547
Brj with
rj CεEcsh |u|
for all ε ε0 small enough. Proof. For any open set A ⊆ U , we define 1 H∞ (A) = inf 2rj : A ⊂ Brj (yj ) 1 (A) H1 (∂A ∩ U ), as noted in Sandier. For any level set ρ −1 (t) with t < 1 , then the then H∞ 2 interior of the level set is completely included in the set U . Note that Cauchy–Schwarz implies
2 1 1 |∇ρ|2 + 2 ρ 2 1 − ρ 2 ρ 1 − ρ 2 |∇ρ|. 2 ε 2ε So
1 ecsh |u| dx ε
U
1 ρ 1 − ρ 2 |∇ρ| dx = ε
U
1 ε
∞
t 1 − t 2 H1 ρ −1 (t) dt
t=0 1 2
t 1 − t 2 H1 ρ −1 (t) dt.
t= 14
From the bound above and the fact that
1 α → H∞ x: ρ(x) α is an increasing function, we have
1
1 ecsh |u| dx ε
U
2
t 1 − t 2 H1 ρ −1 (t) dt
t= 14 1
1 ε
2
1 {x: ρ < t} dt t 1 − t 2 H∞
t= 14 1
1 1 H∞ ε
2 1 x: ρ < t 1 − t 2 dt, 4 t= 14
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where we need the assumption ρ 12 on ∂U between the first and second lines. Noting that 12 81 1 1 2 1 2 1 t (1 − t ) dt = 1024 > 16 . Therefore, H∞ ({x: ρ(x) 4 }) 16ε U ecsh (|u|) dx. t= 4
In particular, this implies control on the rate of convergence of ρ → 1, which will be used in the proof of compactness of the Jacobian. Corollary 2.4. Suppose |u| 12 on ∂U . Then we have bounds on the rate of strong convergence of ρ → 1 when Ecsh (|u|) |U |: 1 − ρ 2
L2
CεEcsh |u| ,
1 − ρLp Cp,γ ε
2 p −γ
(2.5)
1+ 1 Ecsh |u| 2 p
(2.6)
for all 2 < p < +∞ and some small γ > 0. Proof. We decompose U = {ρ 14 } ∪ {ρ 14 } and use our control on the first set.
2 1 − ρ 2 dx =
U
2 1 − ρ 2 dx +
{ρ 14 }
16
2 1 − ρ 2 dx
{ρ 14 }
1 2 2 ρ 1 − ρ dx + ρ 4 2
{ρ 41 }
2 Cε 2 Ecsh |u| + C εEcsh |u| ,
where in the last line we use ( rj )2 rj2 if rj > 0. Finally, we note that the second term dominates for ε 1. To establish the Lp rate recall (2.1), so
∇(1 − ρ) 2 + (1 − ρ)2 1 + ε Ecsh |u| + 4|U | 6Ecsh |u| . 2
(2.7)
U β
1−β
We now interpolate between the norms. Since f Lp f Lq f Lr (2.7) implies β
for
1 p
=
β q
+
1−β r ,
then
1−β
1 − ρLp (U ) 1 − ρL2 (U ) 1 − ρLr (U ) β 1−β Cr 1 − ρ 2 L2 (U ) 1 − ρH 1 (U ) β 1−β 1+β Ecsh |u| 2 Cr ε β Ecsh |u| 2 , Cr εEcsh |u| where p1 = β2 + 1−β r = proves (2.6). 2
β 2
+
γ 2
for some small γ > 0. Hence, we can take β =
2 p
− γ , which
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3. Basic Jacobian bound In this section we show a relationship between the Jacobian J (u) = det ∇u =
1 curl j (u) 2
and the energy density ecsh (u). Let φ ∈ Cc0,1 (U ) be a Lipschitz function vanishing on ∂U . We define Ω(t) = {x ∈ U such that φ(x) > t} then ∂Ω(t) is a level set of φ. Let Reg(φ) :=
t ∈ [0, φL∞ ] such that ∂Ω(t) = φ −1 (t), ∂Ω(t) rectifiable, and H1 (∂Ω(t)) < ∞
.
By the co-area formula | Reg(φ)| = φL∞ and t ∈ Reg(φ) implies ∂Ω(t) is a union of finite Jordan curves, Γi (t). We set, as in [18], 1 , Γ (t) := components of ∂Ω(t) such that min |u| > x∈Γi (t) 2 1 γ (t) := components of ∂Ω(t) such that min |u| . x∈Γi (t) 2 We set d ∈ Z+ and define
D d := t ∈ Reg(φ): deg u; Γ (t) d + 1 or H1 γ (t) ε , A := Reg(φ) \ D d . Furthermore, define
Dγ := t ∈ Reg(φ): H1 γ (t) ε ,
Dd := t ∈ Reg(φ): Γ (t) is nonempty and deg u; Γ (t) d + 1 so that Dγ ∪ Dd = D d and d D |Dγ | + |Dd |. We will choose d in a special way in Section 4. In Section 3.1 we offer a bound on |Dd | in terms of the excess energy. Let us define Eφ (u) =
ecsh (u) dx
spt(φ)
for short. The main results of this section are as follows.
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Proposition 3.1. Suppose U ⊂ R2 and u ∈ H 1 (U ; C) then φJ (u) dx πd + ε 12 φL∞ + ε 13 ∇φL∞ 2E 2 (u) + 3Eφ (u) + |spt(φ)| φ 4 U
+
|Dd | Eφ (u) 4
(3.1)
for any ε 1. We defer the proof of Proposition 3.1. In order to use (3.1), we need to estimate |Dd |. This is controlled by the following result. Proposition 3.2. Suppose |u| = 1 on ∂U and let ε ∈ (0, e−2 ). Define 1 ε αε = ε|log ε|2 ecsh (u) dx, ηε = > 1, 1 − |log2 ε| U
then ηε ecsh (u) dx , |Dd | 8ε ∇φL∞ 1 + exp πd αε
(3.2)
V
where d = d + 1. Here V ≡ ˙ j Brj is the union of disjoint balls Brj with Brj ⊆ spt(φ) ⊆ U and
|Dd | j rj = 2∇φ ∞ . L
Remark 3.3. Estimate (3.2) implies a weaker estimate ηε Eφ (u) |Dd | 8ε ∇φL∞ 1 + exp πd αε
(3.3) E (u)
φ which is sufficient for the E1 (u) ≈ |log ε| case. In this case we choose d + 1 π|log ε| , which 2 provides a good bound on Dd . For the large energy Eφ (u) ≈ |log ε| , we need the refined estimate (3.2).
The proof of Proposition 3.2 is deferred until Section 3.1. Although the proof of Proposition 3.2 employs some ideas of [18], it is fundamentally different. This is because the argument of [18] relies on point-wise lower bounds of ∂Bs egl (u) dH1 > 0 on each radius ∂Bs . In the CSH case the point-wise lower bound of ∂Bs ecsh (u) dH1 is zero for each radius s, even when deg(u, ∂Bs )) = 0. To overcome this difficulty, we use a strongly modified version of the method originating in [30], see also [31]. In fact our method can be used for energies of the form 1 |∇u|2 + 2 W |u|2 dx, ε where W (s) = s p (1 − s)q .
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Proof of Proposition 3.1. The proof follows from a decomposition. In particular φJ (u) dx = U
1 2
j (u) · t dH1 dt + A Γ (t)
1 2
j (u) · t dH1 dt +
1 2
A γ (t)
j (u) · t dH1 dt D d ∂Ωt
= IA,Γ + IA,γ + ID d .
(3.4)
From Lemmas 3.5, 3.6, and 3.8 below we have φJ (u) dx πdφL∞ + 2ε∇φL∞ Eφ (u) + 1 ε 12 ∇φL∞ Eφ (u) + ε 12 φL∞ 2 U
1 1 1 |spt(φ)| + |Dd |Eφ (u). + ε 3 ∇φL∞ 2Eφ2 (u) + Eφ (u) + 2 4 4
The bound follows.
2
In order to finish the proof of Proposition 3.1, we need to establish Lemmas 3.5, 3.6, and 3.8. We start with a basic estimate of Jerrard, Soner [18]. Lemma 3.4. (See Jerrard, Soner [18].) For any set S, S
|S| j (u) · t dH1 dt 2
ecsh (u) dx
(3.5)
f (x) dx.
(3.6)
spt(φ)
∂Ω(t)
and for any non-negative function f , T
f (x) dH dt ∇φL∞ 1
0 ∂Ω(t)
spt(φ)
Proof. For any t ∈ Reg(φ), Stokes’ theorem implies
j (u) · t dH = 2 1
∂Ω(t)
J (u) dx.
Ω(t)
Since |J (u)| = 12 curl j (u) 12 |∇u|2 ecsh (u), then we get the first identity. On the other hand from the co-area formula, T 0 ∂Ω(t)
f dH1 dt =
f |∇φ| dx ∇φL∞
spt(φ)
We bound these three terms via the follow lemmas.
f dx. spt(φ)
2
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Lemma 3.5. We have 1 1 j (u) · t H dt 2πdλ φL∞ + 4ε 2 ∇φL∞ Eφ (u). A Γ (t)
Proof. For t ∈ A, |u| 12 . We set v =
u |u|
and so j (v) =
j (u) . |u|2
Therefore,
j (v) · t dH1 = 2π deg u; Γ (t) .
Γ (t)
Therefore,
j (u) · t dH1 = 2π deg u; Γ (t) +
Γ (t)
j (u)
Γ (t)
|u|2 − 1 · t dH1 . |u|2
This implies (since j (u) |u||∇u|) then |u|||u|2 − 1| 1 dH1 dt j (u) · t dH − 2π deg u; Γ (t) |∇u| |u|2
A Γ (t)
A Γ (t)
4ε
ecsh (u) dH1 A Γ (t)
4ε∇φL∞ Eφ (u). The bound follows from noting |A| φL∞ .
2
Lemma 3.6. Let γ (t) such that H1 (γ (t)) ε with ε 1 then 1 1 1 j (u) · t dH dt ε 2 ∇φL∞ Eφ + ε 2 φL∞ . Reg(φ)\Dγ γ (t)
Proof. 1. We claim that any x 0 satisfies x then by Young’s inequality,
x(x−1)2 b
+ ( b4 + 1) for any b > 0. In particular if x 0
2 3 1 2√ 2√ 1 2√ x 2 x x2 + = x(x − 1) + x+ x(x − 1) + + 3 3 3 3 3 3 3 3 so 23 x
2√ 3 x(x
− 1) + 23 . Applying Cauchy–Schwarz, x
√ b x(x − 1)2 + +1 , x(x − 1) + 1 b 4
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553
and so yields the claim. 2. Therefore, since |j (u)| |u||∇u| then a 1 a 1 |j (u)| |∇u||u| |∇u|2 + |u|2 = |∇u|2 + 2 |u|2 2 2a 2 a 2 2 2 b 1 |u| (1 − |u| ) a |∇u|2 + 1+ . + 2 2 2a 4 a b Set a = ε α and b = ε 2(1−α) then |j (u)| ε α ecsh (|u|) + ε −α so long as ε 1. Since |A| T = φL∞ and H1 (γ (t)) < ε for every t ∈ A then from Lemma 3.4 and the definition of IA,γ in (3.4)
|IA,γ |
ε α ecsh (u) dH1 dt + A γ (t)
ε α ∇φL∞ Eφ (u) + ε −α
ε −α dH1 dt
A γ (t)
H1 γ (t) dt
A
ε α ∇φL∞ Eφ (u) + ε 1−α
dt A
ε ∇φL∞ Eφ (u) + ε α
Setting α =
1 2
finishes the proof.
1−α
φL∞ .
2
In order to bound the D d terms, we prove a lower bound on contours. Lemma 3.7. Suppose H1 (γ (t)) ε and uL∞ (γ (t)) ε α then ecsh (u) dH1
1 1 . 8 ε 1−2α
γ (t)
Proof. The proof is similar to, but weaker than, a result in [18]. Fix a connected component Γi (t) of γ (t) and set ρ := |u| and βi :=
1 |∇ρ|2 dH1 . 2
Γi (t)
By the definition of γ (t) there is a point x ∈ Γi (t) such that ε α x 12 . Let us parametrize Γi (t) by arclength with
Γi (t) = x(s) s ∈ [0, Gi ] , with x = x(0) = x(Gi ). Then since |x(s)| ˙ = 1,
Gi := H1 Γi (t)
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s
ρ x(s) = ρ x(0) +
∇ρ x(r) · x(r) ˙ dr
0
s 2 1 √ 1 3 ∇ρ x(r) dr + sβi + s 2 2 4 0
and ρ(x(s)) ε α − [0, σi ]
√
sβi
εα 2 ,
2α
1 so long as s σi := min{Gi , 16β , ε }. This implies for x ∈ i 4βi
2 1 ε α 2 3 1 2 72 2 1− = ε 2α 4 . ρ x(s) 1 − ρ x(s) 4 4 4 4 4 Therefore,
1 2 σi ε 2α 72 ρ 1 − ρ 2 dH1 βi + 2 4 2 4ε ε 4
ecsh (u) dH1 βi + Γi (t)
Γi (t)
and minimizing over βi we find (for ε 2α 14 ), σi ε 2α−2 72 ε 2α−2 72 ε 2α = βi + min Gi , βi + 4βi 44 44 2α−2 2 2α−1 ε 7 Gi ε 7 min , 44 25 ε 2α 7 Gi 7 = min ,1 . ε 25 ε23 Summing over components, we find ecsh (u) dH1 = γ (t)
ecsh (u) dH1
Γi (t) components of γ (t)Γ (t) i
ε 2α 7 ε 2α 7 ε7 Gi 7 min , 1 min , 1 ε 25 ε 25 ε23 ε23 4 3 2 48 25 ε 2α 72 ε 2α−1 8 = ε 2α−1 8 ε 2α−1 8 , 8 ε 2 2 2 2
which finishes the lower bound.
2
Recall
Dγ := t ∈ Reg(φ): H1 γ (t) ε ,
Dd := t ∈ Reg(φ): Γ (t) is nonempty and deg u; Γ (t) d + 1 .
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Lemma 3.8. 1 1 |spt(φ)| 1 3 ∇φL∞ 4E 2 (u) + Eφ (u) + ε + |Dd |Eφ (u). j (u) · t dH dt φ 2 2 D d γ (t)
Proof. We first consider Dγ bounds. 1. We subdivide Dγ :
Dγ1 := t ∈ Reg(φ): H1 γ (t) ε and u γ (t) L∞ ε α ,
Dγ2 := t ∈ Reg(φ): H1 γ (t) ε and u γ (t) L∞ < ε α , and we consider a bound on Dγ1 first. Since
ecsh (u) dH1 dt
Dγ ∂Ω(t)
|Dα1 | ε 2α−1 dt = 1−2α 8 8ε
Dα1
then 1 D = 8ε 1−2α
ecsh (u) dH1 dt Cε 1−2α ∇φL∞ Eφ (u).
γ
Dγ1 ∂Ω(t)
Therefore, |Dγ1 | 1 j (u) · t dH dt 2 Dγ1 γ (t)
ecsh (u) dx 4ε 1−2α ∇φL∞ Eφ2 (u).
spt(φ)
This implies j (u) · t dH1 dt 4ε 1−2α ∇φL∞ Eφ2 (u). Dγ1 γ (t)
2. Next for Dγ2 we have |j (u)| |u| 2 |∇u||u| 2 |u| |∇u| 2 + 1
Dγ2 γ (t)
2
1
j (u) dH1 dt
|u|
then
|∇u|2 |u| + dH1 dt 2 2
Dγ2 γ (t)
ε α ∇φL∞
1 dx ecsh (u) + 2 spt(φ)
or
|u| 2
(3.7)
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|spt(φ)| . j (u) · t dH1 dt ε α ∇φL∞ Eφ (u) + 2
(3.8)
Dγ2 γ (t)
We choose α =
1 3
and the bound on Dγ follows.
3. Finally for Dd we have |Dd | 1 j (u) · t dH dt 2 Dd Γ (t)
1 ecsh (u) dx |Dd |Eφ (u). 2
spt(φ)
In the following subsection we show that Dd must have small measure.
2
3.1. Control on Dd In order to prove Proposition 3.2 we show that |Dd | is controlled by the energy and the degree d. We first define ωt = {x ∈ U : |u(x)| t} to be the level set of |u|. Here we make use of the boundary condition on |u|. Lemma 3.9. Suppose |u| = 1 on ∂U and set t ∈ (0, 1 −
1 |log ε| ]
then
1 H∞ (ωt ) 2ε|log ε|2 E1 (u)
for ε
(3.9)
1 50 .
Proof. Repeating the argument from Lemma 2.3,
1 ecsh (u) dx ε
U
1
t 1 − t 2 H1 ρ −1 (t) dt
0 1 1− 2|log ε|
1 ε
t 1 − t 2 H1 ρ −1 (t) dt
1− |log1 ε|
1 ({ρ < 1 − H∞
1 | log ε| })
1 1− 2|log ε|
t − t 3 dt
ε 1− |log1 ε|
H1 ({ρ < t }) ∞ ε
1 1− 2|log ε|
t − t 3 dt 1− |log1 ε|
1 1 ({ρ < t }) t 2 t 4 1− 2|log ε| H∞ − = . ε 2 4 1− 1 |log ε|
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557
We evaluate the integral and
1 t 2 t 4 1− 2|log ε| 3 1 7 1 15 1 1 1 − = − + 2 4 1− 1 4 |log ε|2 8 |log ε|3 64 |log ε|4 2 |log ε|2 |log ε|
for |log ε| 72 . Therefore, 1 H∞ (ωt ) 2ε|log ε|2
ecsh (u) dx U
so long as |log ε| 72 .
2
In this part we use the ideas of Sandier in [30] to compute lower bounds on the S1 -valued map. The following covering lemma follows from an iteration process for S1 -valued maps and is a slight modification of the method introduced by Sandier [30]. Lemma 3.10. Suppose ω is a compact subset of V U and let r(ω) = sum of radii of balls Brj (0) covering ω, i.e.
rj (0) be the minimum
1 r(ω) = inf rj (0) such that ω ⊂ Brj (0) = H∞ (ω). Let v : U \ ω → S1 then for each s 0 there exists a collection of balls Ks = {Brk (s) } such that: (1) rk (s) is an increasing function of s for each k. (2) For any subset of balls {Brkj (s) } ⊆ {Brk (s) } in Ks 1 2
|∇v| dx π 2
kj
kj Brkj (s) \ω
kj rkj (s) |dkj | log , r(ω)
where dkj = deg(v, ∂Brkj (s) ). Proof. 1. We start with a family of disjoint balls {Bk (0)} with ω = Bk (0) and set dj = deg(v, ∂Bk ). For a later time t > 0 we define a new family of balls {Bi (t)} with radii ri (t) and degrees di (t). We also define a seed size εi (t) of Bi (t). We set εi (0) = ri (0). Finally we define an expansion function α(t) = log
rj (t) εk (0)
identical for all k. We now grow the balls rk (t). We have two cases:
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(1) If Bi (t) ∩ Bj (t) = ∅ for all i = j then expand. Leave the εi (s) constant as ri (s) increases continuously such that each α(s) = log εrii (s) (s) is the same. This implies α(s) increases continuously. (2) If Bi (t) ∩ Bj (t) = ∅ some i = j then merge. Include both balls in the smallest ball that contains both balls. If the closure hits another ball, continue merging until we have a ball disjoint from all other balls in the family. This ball contains Bi1 (t), . . . , Bik (t) and has radius r(t) ri1 (t) + · · · + rik (t) and degree d(t) = di1 + · · · + dik (t). The only issue is to redefine r(t) for the r(t) defined above. Therefore, we the seed size. In particular we want α(t) = log ε(t) −α(t) take ε(t) = r(t)e . This process can be continued indefinitely, although every ball in the collection may intersect ∂U . 2. We collect some facts about the expansion and merging process. (1) ω ⊂ Bi (t) for all t 0. This is trivial. (2) For any subset {kj } ⊂ {k}:
ri (t) j rkj (t) α(t) = log = log
. εi (t) j εkj (t) This follows from the simple observation that if (3) The upper bound
a c
=
b d
then
εi (t)
a+b c+d
=
a c
= db .
rj (0).
j such that Bj (0)⊂Bi (t)
This fact holds through expansion, so we need only check that it holds through merging. Suppose at time t we merge B i1 (t), . . . , Bik (t) into B with radius r and seed size ε. Then r (t)
ri (t)
log εr = α(t) = log εjj (t) = log k ε k (t) . Therefore, k ik
r i rik (t) = k ε k εik (t) or ε=
r εik (t) εik (t) ik rik (t) ik
since r
ik
ik rik (t),
which concludes the needed facts on the growth process. 3. For any annulus 12 Br \Bε |∇v|2 dx πd 2 log εr π|d| log εr for our S1 -valued function v. Following our growth strategy for the r(t)’s and ε(t)’s we have bounds
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
1 2
|∇v|2 dx π
|dj |log
j
Brj (t)\Bε (t)
=π
j εj (t)
j εj (0) |ω| = r(ω)
rj (t) rj (t) =π |dj | log εj (t) εj (t) j
j
Since
559
j rj (t) . |dj | log
j εj (t)
then we get the required lower bound.
2
Definition 3.11. Recall the different types of domains:
ωt = x ∈ U such that |u| t and
Ω(t) = x ∈ U such that φ(x) < t . Note Γ (t) = ∂Ω. We can now turn to the Proof of Proposition 3.2. We will choose our set V in step 6. We recall an energy bound of [30], see also [31]. 1. Let Θ(t) =
V \ωt
|∇ϕ|2 dx where u = ρeiϕ . Then following [30]
1 ecsh (u) dx = 2
V
∞ ∂ωt ∩V
0
t 2 (1 − t 2 )2 2 dl − t Θ (t) dt |∇ρ| + ε 2 |∇ρ|
by the co-area formula. Cauchy–Schwarz and integration by parts yields 1
ecsh (u) dx V
∂ωt ∩V
0
|∇ρ| t 2 (1 − t 2 )2 + dl + t 2 2ε 2 |∇ρ|
|∇ϕ| dx dt 2
V \ωt
or 1
ecsh (u) dx V
a(t) + 2tb(t) dt 0
with a(t) = ∂ωt ∩V
|∇ρ| t 2 (1 − t 2 )2 + dl, 2 2ε 2 |∇ρ|
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1 b(t) = 2
|∇ϕ|2 dx. V \ωt
Therefore, 1 1− | log ε|
1
2tb(t) dt 0
t 1 |log ε|
|∇ϕ|2 dx dt
V \ωt
t∈
=
for t = 1 −
1 |log ε|
since
V \ωt
inf 1 1 |log ε| ,1− |log ε|
2 1 1− 2 |log ε|
|∇ϕ|2 dx
V \ωt
t dt 1 |log ε|
|∇ϕ|2 dx V \ωt
|∇ϕ|2 dx is a decreasing function in t. This yields a lower bound
1 1− with t = 1 −
1
1−|log ε|
2 |log ε| V
1 ecsh (u) dx 2
|∇ϕ|2 dx
(3.10)
V \ωt
1 |log ε| .
2. To simplify further discussion, we redefine U ≡ {x ∈ U such that φ(x) > 0}. As an artifact of the proof, we use a subdomain U ε ⊂ U which is roughly within a distance ε αε from the boundary of U . We determine the boundary via foliation of the domain by the level sets of our test function φ. We claim there are t ∈ (0, 5εαε ∇φL∞ ) such that ∂Ω(t) ∩ ωt = ∅, and where 1 (ωt ) 4ε αε = 4ε|log ε|2 E1 (u) H∞
is the bound from Lemma 3.9. To prove this let
G = t ∈ 0, 5ε αε ∇φL∞ ∩ Reg(φ) such that ∂Ω(t) ∩ ωt = ∅ . Then G ⊂
φ(Brk (0) ), since ωt ⊂
Brk (0) , a set of disjoint balls, and
|G| 2∇φL∞
rk (0)
4∇φL∞ ε|log ε|2 E1 (u) = 4∇φL∞ ε αε .
(3.11)
Assume that there are no such t ∈ (0, 5ε αε ∇φL∞ ), then |G| = 5ε αε ∇φL∞ which contradicts (3.11). Therefore, there exists a set of t ∈ (0, 5ε αε ∇φL∞ ) ∩ Reg(φ) of positive measure with Γ (t) ∩ Brk (0) = ∅. Choose one such tε .
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561
Note that Γ (tε ) is a finite collection of disjoint, closed Jordan curves, {Γj }, so their interiors are well defined. For each Γj let Ij be its interior, i.e. ∂Ij = Γj . We define U ε = Ij , which satisfies the following properties:
x ∈ U such that φ(x) tε ⊆ U ε , ∂U ∩ ωt = ∅, ε dist ∂U , U ε αε . ε
(3.12) (3.13) (3.14)
The last fact follows from φ ∈ Cc1 (U ); φ vanishes before the boundary. For a y ∈ U and x ∈ U ε the mean value theorem implies |x − y|∇φL∞ φ(x) − φ(y) = φ(x) tε = 2ε αε ∇φL∞ by the definition of tε . We define Ddε = Dd ∩ {t tε } and bound |Ddε | in the following steps. 3. Unleash the expand and merge process of Lemma 3.10 for initial domain ωt , which we can think of as a union of disjoint balls B rj (0) . For each s we have from Lemma 3.10 a set of balls Ks = {Brk (s) } with ωt ⊂ Brk (s) . We define
Ksint = Brk (s) ∈ Ks such that Brk (s) ⊂ U ε , where the balls of Ksint are wholly included in our subdomain U ε . We claim we can continue the expansion process until some time sσ with
rj (sσ ) = σ =
Brj (sσ ) ∈Ksint σ
|Ddε | . 2∇φL∞
(3.15)
Let Υ (s) =
rk (s).
Brk (s) ∈Ksint
From the expansion process defined in Lemma 3.10, each rk (s) with Brk (s) ∈ Ks is continuously increasing until a merging happens. Furthermore, if a merge of Bk1 , . . . , Bkl happens at s, then
the new ball’s radius r(s) j rkj (s). Finally, by definition and the nesting of balls in Ks , balls can only leave Ksint and not enter at a later time. These three facts imply that
Υ (s) is
⎧ ⎪ ⎨ lower semi-continuous, increasing on continuous intervals, ⎪ ⎩ nonincreasing on discontinous points.
(3.16)
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Set σ max = lim sup Υ (s) s>0
to be the largest covering of components of ωt that lie inside U ε and
s = lim sup s such that Υ (s) = 0 s>0 ) to be the last time which contains a vortex ball inside U ε . Note σ max diam(U implies s < ∞. 2 We prove (3.15) by contradiction. Suppose the claim (3.15) is false, then Υ (s) < σ for all s 0. We now prove a contradiction by showing that if Υ (s) < σ for all s ∈ [0, s ], then we can find sδ > s such that Υ (s) > 0 for all s ∈ [0, sδ ], which contradicts the definition of s . Let a and b satisfy 0 a b, then we define
%a = Brl (b) ∈ Kb such that there exists Brk (a) ∈ Kaint with Brk (a) ⊂ Brl (b) K b
and Cba = t ∈ tε , φL∞ ∩ Reg(φ) such that Γ (t) ∩
∅ . Brk (b) =
%a Brk (b) ∈K b
This definition lets us grow balls from a previous time without worrying about balls leaving Kaint . Since the merge and expand process of Lemma 3.10 yields continuously increasing radii of balls except on a countable number of jumps we have two cases. Either we can find an sδ > s with sδ − s 1 and Brl (sδ
rl (sδ ) < σ
(3.17)
%s ) ∈K s δ
or there is a jump at the exit time s . The case where there is a jump at the exit time s is handled in step 4 below. We suppose (3.17) holds. For any 0 a b,
Cba ⊆
φ Brk (b)
%a Brk (b) ∈K b
so by (3.17) s C δ 2∇φL∞ s
rl (sδ ) < 2∇φL∞ σ = Ddε .
%ss Brl (sδ ) ∈K δ
This implies |Ddε | > |Cssδ |, so there exists a t0 ∈ Ddε \ Cssδ . In particular Γ (t0 ) ⊂ U ε \ ωt is a Jordan curve with deg u, Γ (t0 ) d + 1 = 0.
(3.18)
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However, by (3.13) and ωt ∩ U ε ⊂
Brk (sδ ) ,
%ss Brk (sδ ) ∈K δ
we see that |u| t = 1 −
1 |log ε|
> 0, and as a consequence
Uε \
u |u|
is a well-defined vector field in
Brk (sδ ) .
%ss Brk (sδ ) ∈K δ
Since s is the last time that there can exist balls inside U ε and since Brk (s ) ⊂ Brl (sδ ) for some %ssδ then Brl (sδ ) ∩ ∂U ε = ∅ for each Brl (sδ ) ∈ K %ssδ . However, Lemma 3.12 below implies Brl (sδ ) ∈ K
Uε \
Brk (sδ )
%ss Brk (sδ ) ∈K δ
must be a disjoint union of simply connected sets. Therefore, given any Jordan curve γ ⊂ Uε \
Brk (s
Brk (s ) ,
%s ) ∈K s δ
u , γ ) = 0 since each component is simply conthe Brouwer fixed point theorem implies deg( |u| nected. Therefore,
deg u, Γ (t0 ) = 0, %ssδ such that Brl (sδ ) ⊂ Uε , which in turn contradicts (3.18). Therefore, there exists a ball Brl (sδ ) ∈ K and this in turn implies Ksint
= ∅. Hence Υ (sδ ) rl (sδ ) > 0, which contradicts the definition δ of s . This contradiction implies Υ (s) < σ for all s 0 is false; hence, there exists s ∈ [0, s ] such that Υ (s) σ . By the continuity properties (3.16) of Υ (s) and the definitions of Ksint and s , j there exists a set of at least one time(s) {sσ } ∈ [0, s ] such that
Υ sσj =
rk sσj = σ.
int j ∈K j rk (sσ ) sσ
B
j
We choose sσ = minj {sσ } to be the first time s ∈ [0, s ] that achieves (3.15). By (3.16) this sσ is a point of full continuity, a fact which is used in step 5.
4. If in step 3 there is a jump Br (s) ∈K %ss rl (s) at s = s then we restart the entire process by l redefining Uε with a different level set of φ. In particular since (a) there is a set of t’s of positive measure such that ∂Ω(t) ∩ ωt = ∅ with 0 t 5ε αε ∇φL∞ and (b) the number of jumps of
%ss rl (s) is at most countably infinite and independent of φ, then we can choose a new tε Br (s) ∈K l
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so that both ∂Ω(tε ) ∩ ωt = ∅ and (3.17) hold. We then proceed with the rest of the argument in step 3. . Since sσ is a point of full continuity of Υ (s), 5. We now estimate the degree of the balls in Ksint σ there exists a slightly earlier time s− < sσ with sσ − s− 1 such that for all s− s sσ no jumps in Υ (s) occur and the balls in the set Ksint are identical except for their radii. In particular: (3.19)
Υ (s− ) < σ and
deg(u, ∂Br
k (s− )
) =
Brk (s− ) ∈Ksint −
deg(u, ∂Br
k (sσ )
) .
(3.20)
Brk (sσ ) ∈Ksint σ
This follows from (3.16), the definition of sσ , and the conservation of degree in the expand and merge process of Lemma 3.10. Let Ca = t ∈ tε , φL∞ ∩ Reg(φ) such that Γ (t) ∩ Brk (a) = ∅ Brk (a) ∈Kaint
then by (3.19) and the argument in step 3 we have |Cs− | < Ddε , and hence there is t0 ∈ Ddε \ Cs− . So d deg u, Γ (t0 ) =
Brk (s− ) ∈Ω(t0 )∩U ε
deg(u, ∂Br
k (s− )
deg(u, ∂Brk (s− ) )
)
Brk (s− ) ∈Ksint −
by the definition of Ddε . Thus by (3.20) we have
d
deg(u, ∂Br
k (sσ )
Brk (sσ ) ∈Ksint σ
6. Returning to our step 1 estimate, 1 1−
2 |log ε| V
1 ecsh (u) dx 2
|∇ϕ|2 dx V \ωt
1 2
by (3.10)
Br (sσ ) ∈Kint sσ k
|∇ϕ|2 dx Brk (sσ ) \ωt
) .
(3.21)
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
π
rk (sσ ) Brk (sσ ) ∈Ksint σ |dk | log r(ωt ) int
565
by Lemma 3.10
Brk (sσ ) ∈Ksσ
πd log
σ r(ωt )
by (3.15) and (3.21)
so long as V ⊇ Brk (sσ ) ∈ Ksint . We can now define σ
V≡
Brk (sσ )
(3.22)
Brk (sσ ) ∈Ksint σ
with rk (sσ ) = σ . We see V is composed of a union of disjoint balls Brk (σ ) ⊆ spt(φ) such that
rk (sσ ) = σ . 7. Recall d = d + 1 and t = 1 −
1 |log ε|
then by Lemma 3.9
r(ωt ) 4ε|log ε|2 E1 (u) = 4ε αε . Set ηε =
1 1− |log2 ε|
> 1 then by step 6 above
πd log
σ ηε 4ε αε
ecsh (u) dx V
and so ηε σ 4ε αε exp ecsh (u) dx . πd V |D ε |
Since σ = 2∇φd ∞ and |Dd | 5ε αε ∇φL∞ + |Ddε |, the bound follows. This finishes the proof L of Proposition 3.2. 2 We finish this section with a lemma about the connectedness of two-dimensional sets used in step 3 of the proof of Proposition 3.2. Lemma 3.12. Let A ⊂ R2 be a simply connected, open, bounded set, and let {Bk } be a countable collection of balls such that for each k, ∂A ∩ Bk = ∅. Then A \
Bk is a union of disjoint, simply connected sets.
Proof. Since A is simply connected, then A \ Bk = ˙ Cj , a disjoint union of connected open sets. We claim that each Cj is simply connected. Suppose not, then there is a Cj that is not simply connected. Since Cj is open, connected, and non-simply connected, we can find a Jordan curve γ ⊂ Cj which is not contractible to a
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point. Since γ is a Jordan curve, it has a well-defined interior Int(γ ). Furthermore, since γ is not contractible to a point, there exists x0 ∈ Int(γ ) with x0 ∈ / Cj . But γ ⊂ A, since Cj ⊂ A. Since γ is a closed Jordan curve in A, then it is contractible to a / Cj ⊂ A \ Bk . Therefore, x0 ∈ Bn for point. Hence, x0 ∈ A. Finally, we see x0 ∈ A but x0 ∈ some Bn in the collection of balls {Bk }. Since Bn ∩ Cj = ∅ and x0 ∈ Bn then Bn ⊂ Int(γ ) ⊂ A. Thus Bn ∩ ∂A = ∅, which contradicts our hypothesis.
2
4. Compactness of the Jacobian J (uε ) via Ecsh (uε ) bounds Given bounds on J (u) from Propositions 3.1, 3.2, we can establish compactness results of the spirit of those found in [1,18,19,33]. In particular the we show that sequences {J (uε )} are pre0,β compact in the weak norm (Cc )∗ . This section is subdivided into three subsections. In the first subsection we handle sequences with trace |uε | ≡ 1 on ∂U , and in the short second subsection we study sequences with trace |uε | ≡ 1 on ∂U . Finally, we are able to hand general sequences via an extension of uε that has trace either 0 or 1 on the new domain boundary. This reduces to one of the previous two cases. 4.1. Compactness when ρε = 1 on ∂U Here we can lift restriction on uε ∈ H 1 (R2 ; C) and let uε ∈ H 1 (Rm ; C) for m 2. Proposition 4.1. Let U ⊂ Rm , and suppose that uε ∈ H 1 (U ; C) is a collection of smooth functions such that ecsh (uε ) dx = K1U < ∞. sup (4.1) |log ε| ε∈(0,1] U
Suppose U is a simply connected, bounded domain with smooth boundary, and suppose |uε | = 1 on ∂U . Then there exists a subsequence εn → 0 and a Radon measure J such that: (1) J (uε ) converges to a limit J in the (C 0,β )∗ norm for every β > 0; (2) π1 J is (m − 2)-dimensional integer multiplicity rectifiable; and (3) if ν is the weak limit of a subsequence of {νε } ≡ particular, |J |(U ) K1U .
ecsh (uε ) |log ε| ,
then |J | ν and
d|J | dν
1. In
0,β Remark 4.2. When m = 2 then Proposition 4.1 implies J (uε ) π1 J = dj =1 dj δaj in (Cc )∗
for all β ∈ (0, 1] with dj ∈ Z and |J | = π j |dj | K1U . In other words the limiting Jacobian will condense down to a finite number of delta functions with total mass bounded by νε . As a consequence we have Proposition 4.3. Suppose |uε | = 1 and
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
K2U = sup
0<ε<ε0 spt(φ)
Set vε =
j (uε ) |log ε|
and wε =
J (uε ) |log ε|
=
1 2
567
ecsh (uε ) dx < ∞. |log ε|2
(4.2)
curl vε , then wε is precompact in (C 0,β )∗ for any β > 0.
Furthermore, vε Lp C for 1 p <
γ +2 γ +1
and all ε > 0. Finally,
vε |uε |
v weakly in L2 .
The proof of Proposition 4.1 follows from the arguments of [18]. Sketch of the proof of Proposition 4.1. We now sketch the proof of Proposition 4.1, using several technical results of [18]. Since the m 3 case follows from a 2-dimensional slicing argument, we first consider the two-dimensional case uε ∈ H 1 (R2 ; C). 1. Given the assumptions in Proposition 3.2 andsupposing (4.1) holds, then we have the followcsh (uε ) ing Jacobian bound. Fix λ = (1, 2] and dλ = πλ U e|log ε| dx, then φJ (uε ) dx πdλ φL∞ + Cε λ−1 2λ φ 0,1 C
(4.3)
8
for all ε ∈ (0, e− λ−1 ), where C = C(λ, K1U , spt(φ)). To establish (4.3) we set d = dλ in Proposition 3.1, then dλ = dλ + 1 = πλ ηε
2ε − λ where ηε = ity
1 1− |log2 ε|
Eφ (uε ) λ Eφ (uε ) |log ε| + 1 π |log ε| ,
ηε which implies 1 + exp( πd Eφ (uε )) λ
. Therefore, from Proposition 3.2 and Remark 3.3 we have the inequal-
K1U |Ddλ | ηε αε Eφ (uε ) ε |log ε| 1 + exp Eφ (uε ) ∇φL∞ 4 2 πdλ (K1U )2 ηε 4 ε|log ε| 1 + exp Eφ (uε ) ∇φL∞ 2 πdλ Cε
λ−1 2λ
∇φL∞ , ηε
(4.4) λ+1−2ηε
λ−1
so long as there is a constant C ε 1− λ − 2λ |log ε|4 = ε 2λ |log ε|4 . Setting λ − 1 = δ > 0 then δ − |log4 ε| λ + 1 − 2ηε δ so long as ε e−4 . Therefore, for 8
ε e− δ
λ + 1 − 2ηε δ δ , 4λ 2λ 2λ which means ε so
λ+1−2ηε 2λ
δ
b b |log ε|4 ε 4λ |log ε|4 . By a simple calculus calculation ε a |log ε|b ( ae ) ,
δ
ε 4λ |log ε|4
16λ δe
4 ,
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which completes (4.4). The other terms of (3.1) are also controlled by Cε completes (4.3).
λ−1 2λ
for λ ∈ (1, 2]. This
2. We claim that J (uε ) is strongly precompact in (C 0,β )∗ for all β > 0. By (4.3) and Proposition 3.2 of [18] we can decompose J (uε ) = J0ε + J1ε , both signed measures, with J0ε (C 0 )∗ C 0 ∗ 0,β )∗ ; and J1ε (C 1 )∗ Cε α where α = λ−1 2λ . By Lemma 3.4 (C ) is compactly embedded in (C ε 0,β ∗ therefore, {J0 } is precompact in (C ) . Next we show {J1ε } is precompact in (C 0,β )∗ . Since J (uε )(C 0 )∗ = sup | φJ (uε )| J (uε )L1 C∇u2L2 , then J1ε (C 0 )∗ J (uε )(C 0 )∗ + J0ε (C 0 )∗ C|log ε| + C. By Lemma 3.3 of [18] we can interpolate the weak norms ε J
1 (C 0,β )∗
C J1ε (C 0 )∗ J1ε (C 0,1 )∗ Cε α |log ε|,
so {J1ε } is also precompact in (C 0,β )∗ . This finishes the claim. 3. We now establish the limit of these measures {J (uε )}, and the argument in this step is escsh (uε ) sentially the same as in the proof of Theorem 3.1 in [18]. Set νε = eπ|log ε| then νε ν, Radon measure on U . We claim that J is supported only on x0 such that limr→0 νε (Br (x0 ) ∩ U ) π . If there is a subsequence εn with Br (x0 )∩U dνε α < π for all εn sufficiently small then, using Proposition 3.1 with λ =
α+π α
0
yields
φJ dx = lim
n→∞
φJ (uεn ) dx = 0
since dλ = 0 for such φ. Thus, x0 ∈ / spt(J ). Since νε (U ) K1U < ∞, there exists only finitely
many points aj such that limr→0 νε (Br (aj ) ∩ U ) π . Thus, J = π ci δaj for the limiting Jacobian measure, J . We show that cj ’s are integers. Take a1 ⊂ U with dist(a1 , ∂U ) > r1 . We choose Lipschitz test function φ = (r1 − |x − a1 |)+ then dλ = πλ νε (Br1 (aj )) π1 limr→0 νε (Br (aj )). As in [18] we set
An = t ∈ 0, φL∞ ∩ Reg(φ) such that Γ (t) = ∅ and deg uεn , ∂Ω(t) dλ so that An = A ∩ {t: Γ (t) = ∅}, where A is as in Proposition 3.1 and where the n refers to an εn and uεn . From (4.3) we can show that |An | φL∞ − ε α C. If t ∈ An then there is a component Γ (t) nonempty. This implies φ −1 (t) = ∂Br1 −t (a1 ) satisfies min∂Br1 −t (a1 ) |uεn | 12 and |deg(uεn , ∂Br1 −t (a1 ))| dλ π1 limr→0 νε (Br (aj )). We claim that there is an integer dn dλ such that 1 Sndn = r ∈ [0, r1 ] such that min |uεn | , deg uεn , ∂Br (a1 ) = dn ∂Br (ai ) 2
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r1 has measure at least k1 = 3d . Further, we define Σn ⊂ Sndn with |Σn | = k0 . Defining ψn (x) = λ fn (|x − a1 |) where fn (r) = |[r, r1 ] ∩ Σn |. Then for some r, deg(u, ψ −1 (t)) = deg(u, ∂Br (a1 )) = dn for a.e. t ∈ (0, k0 ). It is easy to see from Proposition 3.1 that ψn J (uεn ) dx = πdn k1 + Cε α
and 0 = lim ψn J (uεn ) dx − πc1 ψn (a1 ) = lim ψn J (uεn ) dx − πc1 k1 . n→∞ n→∞ We see that dn = c1 , an integer, with dn dλ . This completes the compactness argument for uε : R2 → C. 4. We now turn to higher-dimensional domains. If u : Rn → C then the supercurrent is the oneform j (u) = (iu, du) =
n (iu, ∂xi u) dx i i=1
whereas the Jacobian J (u) is a two-form J (u) = det(uxi , uxj ) =
1 dj (u). 2
Component-wise we can write J (u) =
J ij (u) dx i ∧ dx j =
i<j
1 ij J (u) dx i ∧ dx j , 2 i,j
where J ij (u) = (i∂xi u, ∂xj u) = det(uxi , uxj ). The Jacobian can be viewed as a co-dimension 2 current acting on (n − 2)-forms via 1 J (u)(φ) = J (u) ∧ φ dx 2 U
for φ ∈ Λn−2 (Rn ). Here Λk (Rn ) is the space of smooth k-forms on Rn and Λk0 (U ) are those forms with compact support in U ⊂ Rn . The proof that π1 J is an integer-multiplicity rectifiable current was shown in the Ginzburg– Landau case by Jerrard, Soner [18] via a two-dimensional slicing argument, see also [1]. Later Sandier, Serfaty in [33] prove that Jacobians with variable metrics have a similar compactness property. In both cases the authors make use of the fact that an (n − 2)-dimensional current is integer-multiplicity rectifiable if and only if almost every projected two-dimensional slice is an
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integer-multiplicity rectifiable 0-current, a fact proved in [20,38] and later [2]. Therefore, we need only identify what happens on each two-dimensional slice of J (uε ), as we have done in steps 1–3. Since the rest of the proof is pretty much identical to arguments in [18,33], we point the reader to [18] for a precise treatment. 2 We now consider the large Ecsh compactness, Proposition 4.3. Proof of Proposition 4.3. Set vε = ε −γ .
j (uε ) |log ε| ,
wε =
J (uε ) |log ε|
=
1 2
curl vε , and
U ecsh (uε ) dx
1. From Lemma 3.2 and the energy bound (4.2) we have |Dd | 8ε ∇φL∞ αε
ηε 1 + exp ecsh (u) dx , πd V
where ηε =
1 1−
,
2 |log ε|
ε αε = ε|log ε|2
ecsh (u) dx K2U ε|log ε|4 , U
V≡
˙ j
Brj ,
rj =
|Dd | . 2∇φL∞
We fix λ ∈ (1, 2] and define & dλ =
λ π
V
' ecsh (uε ) dx . | log ε|
Setting dλ = dλ + 1 then |log ε| 1 λ πdλ
ecsh (uε ) dx; V
which implies ηε
|Ddλ | 16K2U ε|log ε|4 ∇φL∞ ε λ . Returning to the bound in (3.1), we see 2 ηε 1 |Ddλ |Eφ (uε ) 8 K2U ε 1− λ |log ε|6 ∇φL∞ 2 Cε
λ−1 2λ
∇φL∞
gε =
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by following the argument of step 1 of the proof of Proposition 4.1. This implies the bound φwε dx πdλ φL∞ + Cε λ−1 2λ φ 0,1 . C
(4.5)
As in the proof of Proposition 4.1, once we have estimate (4.5), the compactness of wε in (C 0,β )∗ for all β ∈ (0, 1) follows. 2. We claim vε Lp C(K1U ) for all 1 p <
|vε | dx p
|∇uε |2 dx |log ε|2
2+γ 1+γ
for 0 γ < 2. This follows from
p 2
|uε |
1
2p 2−p
2−p 2p
p K2U 2 uε
2p
.
L 2−p
1
From Corollary 2.4 we have 1 − ρε L6 Cε 3 gε3 since |1 − ρε |2 |1 − ρ 2 |, and |∇ρε | |∇uε | 1
then ρε L2 Cgε2 . By Sobolev estimates 1−γ θ − γ 1−θ γ θ 1−θ 1 − ρε Lr 1 − ρε θL6 ∇uε L ε 2 = Cε 6 (2+γ )− 2 q C ε 3 for
1 r
=
θ 6
+ ( q1 − 12 )(1 − θ ) and p < 2. Therefore, letting q get arbitrarily close to 2 implies
2(γ +2) . Returning to the original bound, we get a uniform Lp bound on γ p vε vε so long as 1 p < γγ +2 +1 . Therefore, if vε v in Lloc then |uε | v. Since E1 (uε ) gε ε −2 and |ρε | 12 on the boundary then ρε → 1 strongly in Ls for all 1 s 6 by Corollary 2.4. Therefore, if ρvεε v weakly in L2 then vε = ρvεε ρε converges weakly p to v in Lloc for 1 p0 where p0 = γγ +2 +1 . Finally, if vε v then by strong convergence of ρε → 1, vε vε we can show that ρε v. Since ρε precompact in L2 implies ρvεε v weakly in L2 . 2
θ>
3γ 2+γ
and hence r <
4.2. Compactness when ρε = 0 on ∂U On the other hand if ρ = 0 on the boundary, then j (uε ) → 0 strongly in Lp (U ) for p < 2, which in turn implies that the Jacobian limit is trivial. Proposition 4.4. Suppose Ecsh (uε ) K|log ε| or Ecsh (uε ) K|log ε|2 , and suppose uε = 0 on ∂U then ρε Lp (U ) → 0 in Lp (U ) for all p < +∞, j (uε ) p → 0 in Lq (U ) for all 1 q < 2, L (U ) as ε → 0. Furthermore, J (uε ) converge to zero in (C 0,α )∗ with α ∈ (0, 1).
(4.6) (4.7)
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Proof. From Cauchy–Schwarz
1 ecsh (uε ) dx ε
U
1 |∇uε ||uε | 1 − |uε |2 dx = 2ε
U
∇H (ρε ) dx,
U
where ρε = |uε | and H (ρε ) = ρε2 (1 −
ρε2 2 ). Since H (ρε )|∂U
H (ρε )
W01,1 (U )
= 0 then the above inequality implies
CεEcsh (uε ).
By the Poincaré inequality H (ρε )L1 (U ) CH (ρε )W 1,1 (U ) CεEcsh (uε ). We can now show that ρε → 0 strongly. First note that if ρε2 >
4 3
0
then ρε2 − 1 13 ; therefore, on the set {ρε2 > 43 } we
have ρε2 9ρε2 (1 − ρε2 )2 . On the other hand in the set {ρε2 < 43 } we have 1 − ρε 2L2
ρε2 2
> 13 :
=
ρε2
+
{ρε2 < 43 }
{ρε2 43 }
3
ρε2
ρε2
ρε2 1− +9 2
{ρε2 < 43 }
2 ρε2 1 − ρε2
{ρε2 43 }
C H (ρε )L1 (U ) + Cε 2 Ecsh (ρε ) CεEcsh (ρε ). 1
Hence ρε L2 Cε 3 for ε small enough. By interpolation with the H 1 (U ) bound on ρ implies ρε Lp (U ) Cε γ for any p < +∞ and some γ > 0, which establishes (4.6) Finally, 1 j (uε )Lq (U ) Cρε Lp (U ) Ecsh (uε ) 2 Cε γ Ecsh (uε ) → 0 as ε → 0, and hence (4.7). Therefore, φJ (uε ) dx = U
1 j (uε ) × ∇φ C∇φL∞ (U ) ε γ , 2
(4.8)
U
for some γ > 0. By steps 2 and 3 of the proof of Proposition 4.3 we can use (4.8) to show that J (uε ) converges to zero in (C 0,α )∗ . 2 4.3. Compactness without assumptions on ρε on ∂U We now consider the compactness without assumptions on the boundary behavior. For this, we assume that all functions on U can be extended to a larger domain such that they satisfy one of the previous conditions. Definition 4.5. A simply connected domain U ⊂ R2 is a CSH extension domain if the following holds:
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There exists a domain V ⊃ U and a constant K such that for any sequence uε ∈ H 1 (U ; C) with Ecsh (uε ; U ) Mε 1ε there exists an extension sequence vε ∈ H 1 (V ; C) with vε = uε on U such that Ecsh (vε ; V ) KMε and the boundary values satisfy |vε ||∂V ≡ bε , with constant boundary value bε ∈ {0, 1}. The class of CSH extension domains is quite large. In particular, it contains the class of smooth domains by the following result. Proposition 4.6. Every simply connected domain whose boundary is of class C 1,α is a CSH extension domain. Proof. First, we will assume the domain U to be the unit ball. We will construct a sequence of functions vε on V = B2 (0) that satisfies one of the boundary conditions. 1. Find a good radius. Let ecsh (uε ) 10Mε . Rε = r ∈ 35 , 45 : uε ∈ H 1 (∂Br ) and ∂Br
Then 3
4 5, 5
\ Rε = [ 35 , 45 ]\Rε
1 1 dr 10Mε
ecsh (uε ) dH1 dr
[ 35 , 45 ] ∂Br
1 Ecsh (uε ) . 10Mε 10
1 Therefore, we have that |Rε | 10 > 0. Now we set 1 3 Rε = r ∈ Rε : such that min |uε | reiθ and max |uε | reiθ θ θ 4 4 1
and since uε ∈ H 1 (∂Br ) then uε ∈ C 0, 2 (∂Br ), hence the extrema are achieved. We show that 1 |Rε | 10 (we only need Rε \ Rε = ∅). By Morrey’s inequality uε
C
0, 21
(∂Br )
Cuε H 1 (∂Br ) C Mε
for a universal constant C. Fix r ∈ Rε then by continuity there exist φ1 = φ1 (r) and φ2 = φ2 (r) such that |uε (reiφ1 )| = 14 and |uε (reiφ2 )| = 34 and 14 |uε (reiθ )| 34 for all φ1 θ φ2 (or φ2 θ φ1 —assume the former without loss of generality). We now show that r ∈ Rε implies a large amount of potential energy on ∂Br . In particular ||uε (reiφ2 )| − |uε (reiφ1 )|| |φ2 − φ1 |
1 2
uε
C
0, 21
(∂Br )
C Mε ,
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and since ||uε (reiφ2 )| − |uε (reiφ1 )||
1 2
then
φ2 (r) − φ1 (r)
1 . CMε
This is in fact a long interval, and the potential term will have a large value. Thus for r ∈ Rε and since r ∈ [ 35 , 45 ],
1 ecsh (uε ) dH 2 2ε
1
∂Br
2 ρ 2 1 − ρ 2 dH1
∂Br
r 2 2ε
φ2
2 ρ 2 reiθ 1 − ρ 2 reiθ dθ
φ1
Hence, 1 Cε 2 Mε
∂Br ecsh (uε ) dH
R = ε
2 2 2 3 4 3 1 − |φ2 − φ1 | 5 10ε 2 5
1 . CMε ε 2
1
for r ∈ Rε and C independent of uε and r. Therefore,
1 dr Cε 2 Mε r∈Rε
ecsh (uε ) dH1 dr
r∈Rε ∂Br
Cε 2 Mε Ecsh (uε ) Cε 2 Mε2 1. 2. (Extension by reflection to the inverse of the good radius.) Let rε ∈ Rε \ Rε , i.e. a radius such that uε ∈ H 1 (∂Brε ), Ecsh (uε ; ∂Brε ) 2Mε and |uε | < 34 or |uε | > 14 on ∂Brε . Set Rε = r1ε ∈ [ 54 , 53 ]. We define vε in BRε by reflection, setting vε (reiφ ) = uε ( 1r eiφ ) for r ∈ (1, Rε ). Now |∂r vε |(reiφ ) C|∂r uε |( 1r eiφ ) and |∂θ vε |(reiφ ) |∂θ uε |( 1r eiφ ), and it follows easily that the energy of vε in BRε \ B1 is bounded by a multiple of the energy of uε in B1 \ Brε (in fact, the Dirichlet part of the energy is identical since reflection at the circle is an anticonformal mapping). 3. (Extension by interpolation to the full circle.) In the annulus B2 \ BRε , we define vε by interpolation. There are two cases. (1) If |uε | < 34 on Brε , we interpolate down to 0. 2−r Define vε (reiφ ) = 2−R uε (rε eiφε ). It is clear that |vε | = 0 on ∂B2 . It is not difficult to check ε that Ecsh (vε ; V ) KMε , using the assumption on ∂Br ecsh (uε ) dH1 . ε
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Set ρ0 (φ) = |uε |(rε eiφ ). By assumption, ε −2 ρ02 (1 − ρ02 )2 KMε and ρ0 < 34 , hence (1 − ρ02 )2 c and thus ε −2 ρ02 cKMε . Now |vε | = f (r)ρ0 with 0 f 1 and it follows that ε
−2
∂Br
f 2 ρ02
2 1 − f 2 ρ02 ε −2
ρ02 CKMε
∂Br
for every r ∈ (Rε , 2), so the penalty term is bounded. That the derivative term is bounded correctly is obvious. (2) If |uε | > 14 on Brε , we interpolate up to 1. iφε
s−Rε uε (rε e ) 2−s Define vε (reiφ ) = (|uε (rε eiφ )| 2−R + 2−R ) . We check that the penalty term is ε ε |uε (rε eiφε )| bounded. s−Rε 2−s Setting a = ρ0 2−R + 2−R , we then have (1−a 2 )2 (1−ρ02 )2 and a ρ0 +1 5ρ0 , hence ε ε 2 2 2 2 2 a (1−a ) 25ρ0 (1−ρ0 )2 , and it follows again that the potential term for our interpolation is bounded by a constant times the potential term for uε (rε eiφ ), which is controlled since rε is a good radius. 1+ρ0 For the radial derivatives, we see that |∂r vε | 2−R Cρ0 . However, ∂Br ρ02 CMε by ε ε Sobolev embedding, so this term can be bounded as claimed. The angular derivatives are obviously controlled by those on ∂Brε , finishing the proof.
4. (General domains.) To arrive at the setup used in the proof above, we use Riemann’s mapping theorem to find a conformal mapping of B1 to U . By the Kellogg–Warschawski theorem, this map is of class C 1,α up to the boundary and the derivative does not have a zero on the boundary, hence can be extended as a diffeomorphism to a domain containing B1 . It is straightforward to use this to construct a C 1 diffeomorphism from B2 to a domain V ⊃ U that extends the conformal mapping. If weundo the mapping, we find two domains U ⊂ V and extension vε of uε such that vε |U ≡ uε and V ecsh (vε ) dx KMε for K a fixed constant depending only on the initial domain ,y ) 2 U . This holds since B2 \B1 |∇x vε |2 dx = V \U |Wij ∇j vε |2 ∂(x ∂(x,y) dx K V \U |∇vε | dx by the change of variables, where K depends only on the initial mapping U → B1 . 2 Corollary 4.7. Any rectangular domain A ⊂ R2 is a CSH extension domain. Proof. By reflection, we can extend a function defined on A to a doubly periodic function on R2 . Choose a simply connected C 1,α domain A ⊂ R2 that contains A. By the previous proposition, A is a CSH extension domain, and it follows that A is an extension domain. 2 If U is a extension domain, then we achieve our general compactness result without assumptions on the boundary. Proof of compactness in Theorems 1.1 and 1.2. We now consider our sequence {uε } with Ecsh (uε ) K|log ε| without assumption on the boundary. Sinceour domain is a CSH extension domain, we can find extensions {vε } of {uε } in V ⊃ U such that V ecsh (vε ) C|log ε|. Since we have an infinite sequence, there exists a subsequence vεj with vεj |∂V = 1 or a subsequence with vεj |∂V = 0. Using either Proposition 4.1 in the first case or Proposition 4.4 in the second case shows J (vε ) J in (Cc )∗ (V ), where J ≡ 0 in the second case. 0,β
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Since U is a compact subset of V , then the sequence J (uε ) is precompact in (Cc )∗ (U ) by choosing test functions within the subset U . The case Ecsh (uε ) K|log ε|2 follows in the same manner. 2 0,β
5. Energy bounds and Γ -convergence for Ecsh (uε ) In this section we show the lower and upper bound parts of Theorems 1.1 and 1.2. 5.1. Lower bound for Ecsh (uε ) As in Section 3 we consider only the case ρε = 1 or ρε = 0 on ∂U . The extension of the lower bound to general functions uε follows by the CSH extension domain properties as in Section 4.3. Proposition 5.1. For every sequence {uε } such that sup 0<ε<ε0
U
ecsh (uε ) dx = K1U < ∞ |log ε|
with J (uε ) → J in (C 0,β )∗ , there holds ecsh (uε ) dx J M . lim inf ε→0 |log ε| U
Here J satisfies the structure condition J = π right-hand side is zero.
di δai with di ∈ Z. Note that when ρε → 0, the
Proof. In the case where uε has an extension with boundary value 1 then from (3.1) and our
choice of dλ = πλ
U ecsh (uε )
|log ε|
:
π φJ dx = π lim φJ (uεn ) dx n→∞ U
U
λφL∞ lim inf n→∞
U
ecsh (uεn ) dx |log εn |
(5.1)
for every λ > 1. Letting λ → 1 yields the bound. If uε has an extension with boundary value 0, then the proposition follows from Proposition 4.4. When ρε = 0 on ∂U , Proposition4.4 implies that both ρε and j (uε ) converge strongly to zero csh (uε ) as ε → 0. Trivially we have lim inf U e|log ε| dx J M where J ≡ 0. For general boundary csh (vε ) behavior we build an extension vε on V ⊃ U such that V e|log ε| dx C for some constant C depending on K and U . If our subsequence is converging to zero, then we follow the second case and lower bound follows trivially. On the other hand if our subsequence is converging to 1, we choose our test function in (5.1) with support in U . This restricts our energy to the energy of the initial domain U , and the energy lower bound follows. 2
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577
For Ecsh (uε ) = O(|log ε|2 ) we have the following lower bound. Proposition 5.2. Let the sequence {uε } satisfy sup 0<ε<ε0
Set vε =
j (uε ) |log ε|
J (uε ) |log ε|
and wε =
=
1 2
U
ecsh (uε ) dx = K2U < ∞. |log ε|2
curl vε . Suppose
1 p < 2. Then w = 12 curl v is a measure and lim inf ε→0
vε |uε |
p
v in L2 and vε v in Lloc for
1 ecsh (uε ) dx v2L2 + curl vM . 2 2 |log ε|
(5.2)
Note that when ρ0 → 0, the right-hand side is zero. Proof. Again, this follows from Proposition 4.4 for sequences that can be extended with boundary value 0. For sequences that can be extended with boundary value 1, we use essentially the proof of Theorem 6.1 of [19], but we include it for the sake of completeness. 1. From Proposition 3.2 there is a set V = ˙ Brk with Brk ⊆ spt(φ) where the Jacobian concentrates. We define 1 if x ∈ Brj , χε (x) = 0 otherwise, then for any h we have h |uvεε | χε 2L2 |h|2 χε | |uvεε | |2 . By the dominated convergence theorem, the first integral converges to zero and the second is bounded by the assumptions. Thus |uvεε | χε 0 in L2 and |uvεε | (1 − χε ) v in L2 . Therefore, ε→0
U\
ecsh (uε ) dx lim inf ε→0 |log ε|2
lim inf
U \ Brj
Brj
lim inf ε→0
U \ Brj
lim inf ε→0
U
1 ∇uε 2 dx 2 |log ε| 1 vε 2 dx 2 |uε |
2 1 vε (1 − χ ) ε dx 2 |uε |
1 vL2 . 2 2. We have wε = 12 curl vε w in the sense of distribution with w = 12 curl v. Let ψ ∈ Cc∞ (U ) then
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sup
ψ curl v dx =
ψL∞ 1
U
J (uε ) sup lim ψ dx |log ε| ψL∞ 1 ε→0 U
sup
ψL∞ 1
ecsh (uε ) dx |log ε|2
λψL∞ lim inf ε→0
= λ lim inf ε→0
U∩
U∩
Brj
ecsh (uε ) dx. |log ε|2 Brj
Adding this bound to step 1 yields 1 lim inf λ ε→0
U
ecsh (uε ) 1 1 dx curl vM + vL2 . 2 2 2 |log ε|
We take λ → 1 which finishes (5.2).
2
5.2. Upper bounds for Ecsh (uε ) In this subsection, we establish the upper bounds corresponding to the lower bounds of the last section, thus finishing the proofs of Theorems 1.1 and 1.2 in the case ρε = 1 on ∂U . The constructions we use are heavily based on those of Jerrard, Soner [19]. The upper bound for the case ρε = 0 on ∂U is straightforward, since the Γ -limit is zero. Proposition 5.3. For every v ∈ L2 (U ; R2 ) such that w = exists a sequence {uε } in H 1 (U ; C) such that vε := wε :=
1 |log ε|
1 2
curl v is a Radon measure, there
1 j (uε ) v in L2 U ; R2 , |log ε| ∗ J (uε ) w in C 0,α for every α > 0.
(5.3) (5.4)
Furthermore, the energy of the sequence satisfies 1 lim sup |log ε|2 ε→0
ecsh (uε )
1 v2L2 + curl vM . 2
(5.5)
U
The proof follows from the following proposition by a standard approximation argument, since C ∞ (U ) is “dense in energy” in the limit spaces. Note that by Proposition 4.3 the weak convergence of vε in Lp for 1 p < 2 holds if and only if |uvεε | converges weakly in L2 , and their respective weak limits are the same. Proposition 5.4. Let v ∈ C ∞ (U ; R2 ). Let {dε } be an increasing sequence with dε → ∞ and εdε → 0. Then there exists a sequence of functions {uε } ∈ H 1 (U ; C) with the following properties:
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
|uε | 1 in U , |uε | = 1 on ∂U , j (uε ) vε := v in Lp for all p < 2, dε J (uε ) 1 curl v =: w in W −1,p for all p < 2 wε := dε 2
579
(5.6) (5.7) (5.8)
and the energy satisfies Ecsh (uε )
dε2 v2L2 (U ) + dε |log ε|wL1 (U ) + o dε2 2
(5.9)
as ε → 0. Proof. This follows as in the proof of Proposition 7.1 of [19]. In that proof one sees that |uε | 1 and |uε | = 1 on ∂U . The construction shows that |uε | = ρε satisfies (ρε )p − 1Lq (U ) → 0 for all p, q ∈ [1, ∞). In fact, ρε is a function that is 1 outside Bε (aiε ), i = 1, . . . , Nε , for some points aiε that satisfy −1
−1
|aiε − ajε | cdε 2 and dist(aiε , ∂U ) cdε 2 , and ρε satisfies |ρε | 1 everywhere. Since |uε | 1, we have 2 1 |uε |2 1 − |uε |2 CNε Cdε wM = o dε2 2 ε as in [19]. The statements on the convergence of the other terms in the energy are independent of the difference between our functional and the Ginzburg–Landau functional, we sketch the argument here for the convenience of the reader. The proof in [19] relies on a Hodge-type decomposition of L2 (U ; R2 ) into vector fields that are curls of functions that are zero on the boundary, gradients, and harmonic vector fields. Gradients and harmonic vector fields are easily approximated as in Lemma 7.4 of [19], and the combination is done as in the proof of Proposition 7.1. For the vector fields that are curls with zero boundary conditions, the measure w is first approximated by a sum wε of point masses at appropriately chosen points aiε as above. In the next step, a vector field vˆε is defined as vˆε = −2 curl −1 D wε , where D denotes the Laplacian with zero Dirichlet boundary data. Setting uˆ ε = eiϕε , where ε) ∇ϕε = dε vˆε , the functions uˆ ε then satisfy j (duˆεε ) = vε and J (u dε = wε . Using the cutoff function ρε above, and defining uε = ρε uˆ ε , one obtains a sequence with the desired properties. 2 In the scaling
U ecsh (u) ≈ C|log ε|,
Proposition 5.5. Let J = π 1 |log ε| J (uε ) J and
we have the following upper bound result.
N
i=1 di δai
1 lim ε→0 |log ε|
for ai ∈ U . Then there exists a sequence {uε } with ecsh (uε ) = J M(U ) . U
(5.10)
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Proof. This can be shown by a construction similar to the one above, but simpler (for the case ε → a with |bε − bε | ε|log ε| of |di | = 1, it is already contained in [3]): Choose points bi,j i i,j i,k ε ) and for 1 j < k |di |. Construct uε as a function that satisfies |uε | = 1 in U \ Bε (bi,j ε satisfies deg(uε , ∂Bε (bi,j )) = sgn(di ) for all i, j, k. Similar to the construction in [3], this can be done with an energy bounded by |J ||log ε| + C log|log ε|, or even by |J ||log ε| + C if all di are ±1. 2 6. Extension to the full CSH energy In this section we discuss the generalization of the results on Ecsh (u) to Gcsh (u, A; hex ), for the scaling Gcsh C|log ε|2 and hex = H |log ε|, proving Theorem 1.3. We use 2 1 μ2 | curl A − hex |2 1 + 2 |u|2 1 − |u|2 . Gcsh (u, A; hex ) = |∇A u|2 + 2 4 |u|2 ε The functional has a gauge invariance: it stays invariant under the gauge transformation (u, A) → (ueiχ , A + ∇χ): Gcsh (ueiχ , A + ∇χ; hex ) = Gcsh (u, A; hex ) for any χ ∈ H 1 (U ). The appropriate gauge-invariant quantities are the supercurrent jAε = j (uε ) − |uε |2 Aε and the magnetic field curl Aε . For simplicity, we will fix the gauge invariance by choosing the Coulomb gauge: we can choose A such that ∇ · A = 0 and A · ν = 0. (Otherwise, let ξ be a solution of ξ = −∇ · A, ν · ∇ξ = −A · ν. Then A˜ = A + ∇ξ − U (A + ∇ξ ) satisfies these assumptions.) ˜ with u˜ = ueiξ instead of (u, A). We use (u, ˜ A) Proposition 6.1. Assume Gcsh (uε , Aε ; hex ) K|log ε|2 and |uε | = 1 on the boundary of a domain V containing U . Set aε = |log1 ε| Aε , where Aε is in the Coulomb gauge. Then aε is weakly compact in W 1,p for all p < 2. For a subsequence where aε a, we also have that curl aε −H curl a − H in L2 . In addition, the compactness assertions of Proposition 5.2 hold: |uε | vε = 1 2
1 |log ε| j (uε )
converges to v weakly in all Lp ,
vε |uε |
v in L2 , and wε =
J (uε ) |log ε|
w=
curl v. Furthermore, the energy satisfies 1 Gcsh (uε , Aε ; hex ) ε→0 |log ε|2 1 μ2 2 2 G(v, a; H ) = |curl a − H | + curl vM . |v − a| + 2 4
lim inf
(6.1)
U
Proof. From the energy bound, it follows directly that H=
curl aε −H |uε |, |uε |
curl aε −H |uε |
is bounded in L2 . Since curl aε −
we can use Hölder’s inequality and obtain for p < 2 curl aε − H uε 2 C(p), curl aε − H Lp 2 |u | L 2−p ε
2
L
since |uε | → 1 in L 2−p by (2.6). Using the Coulomb gauge condition and elliptic regularity it follows that aε is bounded in W 1,p .
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581
To see that curl|uaεε −H and curl aε − H converge to the same limit (which is curl a − H ), note | that Hölder’s inequality again implies that curl aε − H curl aε − H − (curl a (1 − |u − H ) = |) ε ε |uε | |uε | Lp Lp curl aε − H 1 − |uε | 2 , |uε | L 2−p L2 which tends to zero by (2.6). Decomposing Gcsh (uε , Aε ; hex ) = Ecsh (uε ) − + we see that all we need to deal with is
j (uε ) · Aε +
1 2
|Aε |2 |uε |2
μ2 |curl Aε − hex |2 , 4 |uε |2
(6.2)
j (u) · A. Adapting again the proof of [19], we have that
2 Aε · j (uε ) 1 |j (uε )| + |uε |2 |Aε |2 4 |uε |2 1 |∇uε |2 + |uε | |uε |2 − 1 + 1 |Aε |2 4 2 1 1 |∇uε |2 + |Aε |2 + 2 |uε |2 1 − |uε |2 + 2ε 2 |Aε |4 4 4ε 1 Ecsh (uε ) + |Aε |2 + 2ε 2 |Aε |4 . 2
By the uniform W 1,p bounds on aε , it follows via (6.2) and Sobolev embedding that 1 Ecsh (uε ) K|log ε|2 + C|log ε|2 + Cε 2 |log ε|4 C|log ε|2 . 2 This shows that the compactness and lower bound results of Proposition 5.2 are applicable for uε , and we obtain that vε is compact as claimed. We now continue as in [19, p. 555] and decompose Gcsh (uε , Aε ; hex ) = G1csh (uε , Aε ) + G2csh (uε , Aε ) + G3csh (uε , Aε ) + G4csh (uε , Aε ), where G1csh (uε , Aε ) = Ecsh (uε ), | curl aε − H |2 |log ε|2 μ2 |log ε|2 G2csh (uε , Aε ) = |aε |2 + , 2 8 |uε |2 U
U
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G3csh (uε , Aε ) =
|log ε|2 2
U
|uε |2 − 1 |aε |2 ,
G4csh (uε , Aε ) = −|log ε|2
aε · v ε . U
For G1csh , we use the lower bound of Proposition 5.2 and obtain lim inf ε→0
1 1 G1csh (uε , Aε ) v2L2 + curl vM . 2 2 |log ε|
By the weak L2 convergence of semicontinuity, we find that lim inf ε→0
curl aε −H |uε |
(6.3)
→ curl a − H , the convergence of aε → a and lower
1 μ2 1 2 curl a − H 2L2 + a2L2 . G (u , A ) ε ε csh 8 2 |log ε|2
(6.4)
For the third term, we need Corollary 2.4 that shows that 1 − |uε |2 2L2 Cε 2 |log ε|4 , hence 3 G (uε , Aε ) csh
U
2 |uε |2 − 1
1/2
1/2 |Aε |
Cε|log ε|4 → 0.
4
(6.5)
U
The convergence 1 G4 (uε , Aε ) → − |log ε|2 csh
a·v U
follows from the weak convergence of vε in Lp with p < 2 and the strong convergence aε → a in Lq for all q < ∞ that is implied by the weak W 1,p convergence and the Rellich–Kondrachov theorem. Summing up the terms, we obtain the lower bound as claimed. 2 Proposition 6.2. Given a ∈ H 1 (U ; R2 ) and v ∈ L2 (U ; R2 ) such that w = 12 curl v is a Radon measure, there exists a sequence {Aε } ∈ H 1 (U ; R2 ) and uε ∈ H 1 (U ; C) such that vε = p vε 1 1 2 1 |log ε| j (uε ) satisfies vε v in Lloc with p < 2, |uε | v in L . Also, |log ε| Aε a in H . Proof. We choose the sequence {uε } constructed in Proposition 5.4. To find Aε , we set Aε = |log ε|aε , where aε is defined as the Coulomb gauge solution of curl aε = H + |uε |(curl a − H ). We thus have curl aε − curl a = (H − curl a) 1 − |uε | , and since |uε | 1, it follows that (curl aε ) is bounded in L2 . The weak limit must be curl a, since
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
|curl aε − curl a|
2 1 − |uε |
1/2
583
1/2 |H − curl a|
2
→ 0.
Elliptic regularity shows that (aε − a) is bounded in H 1 , and we obtain aε a in H 1 . To see the convergence of the energy, we need to check the convergence of the terms G1ε to 4 Gε as above. The convergence of G1ε was dealt with in the proof of Proposition 6.1. We observe that |curl Aε − hex |2 |uε |2 = |log ε|2 | curl a − H |2 , |uε |2 |uε |2 and aε → a strongly in L2 by the compactness of the embedding H 1 ⊂ L2 , so G2ε converges as claimed. The terms G3ε and G4ε converge by the same arguments as used in the lower bound proof. 2 Proposition 6.3. Assume Gcsh (uε , Aε ; hex ) K|log ε|2 and |uε | = 0 on the boundary of a domain V containing U . Set aε = |log1 ε| Aε , where Aε satisfies the Coulomb gauge condition. Then ρε = |uε | → 0 in all Lq with q < ∞. Furthermore, curl aε → H in Lp for all p < 2, and aε converges in W 1,p to the solution of curl a = H , div a = 0 in U , a · ν = 0 on ∂U . For vε = j (uε ), we have vε → 0 in Lp . The energy satisfies the lower bound lim inf Gcsh (uε , Aε ; hex ) 0. Conversely, there exist functions (uε , Aε ) with |log1 ε| curl Aε → H in Lp and |uε | → 0 in L∞ such that |log ε|−2 Gcsh (uε , Aε ; hex ) → 0. Proof. As above, we deduce the weak compactness of curl aε and the boundedness of Ecsh (uε ). By Proposition 4.4, we obtain that ρε → 0 in all Lq and it follows by Hölder’s inequality that curl a − H strongly converges to 0. The lower bound inequality is trivial. The construction of a recovery sequence is simple: Let uε = ε 2 and Aε = a|log ε|, where a is the Coulomb gauge solution of curl a = H . Then 2 1 ε 4 |a|2 + ε 2 1 − ε 4 |log ε|−2 → 0 |log ε|−2 Gcsh (uε , Aε ; hex ) = 2 U
as ε → 0.
2
We now consider the special case of minimizers under the boundary condition |u| = 1. The existence of such minimizers is shown in [36]. Proposition 6.4. Assume that hex = H |log ε| and define the minimal energies gε and g as (6.6) gε := inf Gcsh (u, A): u ∈ H 1 (U ; C), |u| = 1 on ∂U , A ∈ H 1 U ; R2 , 1 g := inf G (v, a; H ): v ∈ L2 U ; R2 , curl v is a Radon measure, a ∈ H 1 U ; R2 . (6.7) Then there exists a unique (v∗ , a∗ ) in the admissible set such that G1 (v∗ , a∗ ) = g, and there exists a sequence (uε , Aε ) ∈ H 1 (U ; C) × H 1 (U ; R2 ) such that |uε | = 1 on ∂U and Gcsh (uε , Aε ) g|log ε|2 + o |log ε|2
(6.8)
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as ε → 0. In particular, minimizers of Gcsh under the condition |u| = 1 satisfy (6.8). For any sequence of (uε , Aε ) that satisfies (6.8), |u| = 1 on ∂U and the Coulomb gauge condition, the compactness statements of Propositions 5.2 and 6.1 hold, and the limit of (vε , aε ) defined as vε = |log1 ε| j (uε ) and aε = |log1 ε| Aε in the appropriate topology is the minimizer (v∗ , a∗ ) of G1 . The energy infima satisfy gε →g |log ε|2 as ε → 0. Proof. The minimizers of G1 are unique (up to the gauge invariance G1 (v, a; H ) = G1 (v + ∇χ, a + ∇χ; H ), but we have excluded this by choosing Coulomb gauge) by the strict convexity of G1 over the admissible set. The existence of an approximating sequence (uε , Aε ) that satisfies (6.8) and |uε | = 1 on ∂U is the content of Proposition 6.2. It follows that minimizers of Gcsh under this boundary condition also satisfy (6.8). By Corollary 2.4, the boundary condition implies that |uε | → 1 in all Lp (U ), hence we can apply the compactness and lower bound results of Theorem 1.3, which shows that (vε , aε ) defined as above converge to some (v0 , a0 ) and G1 (v0 , a0 ; H ) lim inf ε→0
1 Gcsh (uε , Aε ) g. |log ε|2
However, (v∗ , a∗ ) was the unique minimizer of G1 , so v0 = v∗ and a0 = a∗ . As (uε , Aε ) was chosen as (almost) minimizing, the convergence of the infima also follows. 2 7. Application: critical hex In this section we analyze the limit functional 1 μ2 1 curl a − H 2L2 + curl vM . G(v, a; H ) = v − a2L2 + 2 8 2
(7.1)
We look for the critical field H1 such that for H < H1 , minimizers of (7.1) will satisfy curl v = 0, and for H > H1 , curl v = 0. The critical field in the original scaling will then be hext ≈ H1 |log ε|. Similar results for the Ginzburg–Landau functional are due to Sandier, Serfaty [31,32] where also the relation to a free boundary problem is derived; while this can be done here, we omit this for the sake of brevity. We will follow the argument in [19] but keep the dependence on the extra parameter μ. We show Proposition 7.1. The critical field H1 = H1 (μ) can be calculated by H1 (μ) =
2 , where μ2 maxU |zμ |
2
zμ is the solution of − μ4 zμ + zμ + 1 = 0 in U with homogeneous Dirichlet boundary data. The function μ → H1 (μ) has the following properties: (1) H1 (μ) is a decreasing function; (2) μ2 H1 (μ) → 2 as μ → 0; (3) H1 (μ) → H (U ) as μ → ∞, where H (U ) is given by H (U ) = of w = −1 in U , w = 0 on ∂U .
1 2 supU w , and w
is the solution
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
585
We will show this by comparing G with the functional 1 μ2 F (a; H ) = a2L2 + curl a − H 2L2 . 2 8
(7.2)
The minimizer a∗ of F satisfies the Euler–Lagrange equations a∗ +
μ2 curl(curl a∗ − H ) = 0 4
curl a∗ − H = 0 on ∂U .
in U ,
(7.3)
Taking the curl of this equation, we obtain curl a∗ − H −
μ2 (curl a∗ − H ) + H = 0, 4
(7.4)
so setting H zμ = curl a∗ − H we see that zμ is the solution of −
μ2 zμ + zμ + 1 = 0 4
zμ = 0
in U ,
on ∂U .
(7.5)
We now decompose the energy of (7.1) by setting a = a∗ + b. We obtain 1 G(v, a; H ) = 2
|a∗ |2 + |b|2 + 2a∗ · b U
+
μ2 8
−
|curl a∗ − H |2 + | curl b|2 + 2(curl a∗ − H ) curl b U
1 (v · a∗ + v · b) + 2
U
1 |v|2 + curl vM . 2
(7.6)
U
Integrating by parts and using (7.3) and (7.4), we see that
μ2 v · a∗ = − 4
U
v · curl(curl a∗ − H ) U
=−
μ2 4
v · curl zμ =
H U
μ2 H 4
zμ curl v
(7.7)
U
and
μ2 curl(curl a∗ − H ) · b = 0. a∗ + 4
U
Using (7.7) and (7.8) lets us rewrite (7.6) as
(7.8)
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G(v, a; H ) =
μ2 1 1 μ2 | curl b|2 + |v − b|2 + F (a∗ ) + curl vM − H 8 2 2 4
U
F (a∗ ) + curl vM
μ2
1 − 2 4
H max|zμ | . U
curl vzμ U
(7.9)
It follows that the minimizer of G satisfies curl v = 0 if and only if 1 μ2 − H max|zμ | < 0, U 2 4 that is if and only if H < H1 (μ) =
2 . μ2 maxU |zμ |
The dependence of zμ (and hence H1 ) on μ can be calculated explicitly in the case of U = BR (0): the solution of (7.5) is the given by R zμ (r) =
I0 ( 2r μ) I0 ( 2R μ )
− 1,
where I0 is the modified Bessel function of zeroeth order. It follows that R |=1− max|zμ
Since I0 (x) ∼
x √e 2πx
1 I0 ( 2R μ )
.
for x → ∞ and I0 (x) = 1 + 14 x 2 + O(x 4 ) as x → 0, it follows that
R | → 1 as μ → 0 and μ2 max |zR | → R 2 as μ → ∞. Finally we see that max |zμ μ
H1 (μ) =
2I0 ( 2R μ ) μ2 (I0 ( 2R μ ) − 1)
.
The general behavior is similar: Proposition 7.2. The solution zμ of (7.5) has the following properties: (1) −μ2 zμ (x) is monotonically increasing in μ for every x ∈ U . (2) supU |zμ | → 1 as μ → 0. (3)
μ2 4
supU |zμ | → A(U ) as μ → ∞, where A(U ) = supU y and y is the solution of y = −1 in U , y = 0 on ∂U . 2
Proof. We set yμ = − μ4 zμ . Then yμ solves yμ − that wμ =
∂ ∂μ yμ
solves
4 y μ2 μ
+ 1 = 0. Differentiating, we obtain
M. Kurzke, D. Spirn / Journal of Functional Analysis 255 (2008) 535–588
wμ −
4 8 wμ = − 3 yμ 0, 2 μ μ
587
(7.10)
and so by the maximum principle, wμ 0, which proves the first claim. The second claim follows since zμ → −1 in every U U , for the third we observe that yμ as defined above converges to a solution of y = −1, and A(U ) = supU y. 2 Acknowledgments The second author would like to thank Fang-Hua Lin for suggesting the model. The authors would also like to thank Etienne Sandier for his comments and suggestions that helped with the final draft. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. References [1] G. Alberti, S. Baldo, G. Orlandi, Variational convergence for functionals of Ginzburg–Landau type, Indiana Univ. Math. J. 54 (5) (2005) 1411–1472. [2] L. Ambrosio, B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000) 1–80. [3] F. Bethuel, H. Brezis, F. Helein, Ginzburg–Landau Vortices, Birkhäuser Boston, Boston, MA, 1994. [4] F. Bethuel, H. Brezis, G. Orlandi, Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal. 186 (2001) 432–520. [5] A. Braides, Γ -Convergence for Beginners, Oxford Lecture Ser. Math. Appl., vol. 22, Oxford Univ. Press, Oxford, 2002. [6] L. Caffarelli, Y. Yang, Vortex condensation in the Chern–Simons–Higgs model: An existence theorem, Comm. Math. Phys. 168 (1995) 321–336. [7] D. Chae, M. Chae, The global existence in the Cauchy problem of the Maxwell–Chern–Simons–Higgs system, J. Math. Phys. 43 (2002) 5470–5482. [8] D. Chae, K. Choe, Global existence in the Cauchy problem of the relativistic Chern–Simons–Higgs theory, Nonlinearity 15 (2002) 747–758. [9] D. Chae, N. Kim, Topological multivortex solutions of the self-dual Maxwell–Chern–Simons–Higgs system, J. Differential Equations 134 (1997) 154–182. [10] S. Gustafson, I.M. Sigal, Effective dynamics of magnetic vortices, Adv. Math. 199 (2006) 448–498. [11] J. Han, J. Jang, Self-dual Chern–Simons vortices on bounded domains, Lett. Math. Phys. 64 (2003) 45–56. [12] J. Han, N. Kim, Nonself-dual Chern–Simons and Maxwell–Chern–Simons vortices on bounded domains, J. Funct. Anal. 221 (2005) 167–204. [13] J. Hong, Y. Kim, P.Y. Pac, Multivortex solutions of the Abelian Chern–Simons vortices, Phys. Rev. Lett. 64 (1990) 2230–2233. [14] R. Jackiw, W. Weinberg, Self-dual Chern–Simons vortices, Phys. Rev. Lett. 64 (1990) 2234–2237. [15] A. Jaffe, C. Taubes, Vortices and Monopoles. Structure of Static Gauge Theories, Progr. Phys., vol. 2, Birkhäuser Boston, Boston, MA, 1980. [16] R.L. Jerrard, Vortex dynamics for the Ginzburg–Landau wave equation, Calc. Var. Partial Differential Equations 9 (1999) 1–30. [17] R.L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal. 30 (1999) 721–746. [18] R.L. Jerrard, H.M. Soner, The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differential Equations 14 (2002) 151–191. [19] R.L. Jerrard, H.M. Soner, Limiting behavior of the Ginzburg–Landau functional, J. Funct. Anal. 192 (2002) 524– 561. [20] R.L. Jerrard, H.M. Soner, Functions of bounded higher variation, Indiana Univ. Math. J. 51 (2002) 645–677. [21] R.L. Jerrard, D. Spirn, Refined Jacobian estimates and the Ginzburg–Landau energy, Indiana Univ. Math. J. 56 (2007) 135–186. [22] R.L. Jerrard, D. Spirn, Refined Jacobian estimates and the Gross–Pitaevsky equations, Arch. Ration. Mech. Anal., in press. [23] M. Kurzke, D. Spirn, Scaling limits of the Chern–Simons–Higgs energy, Commun. Contemp. Math. 10 (2008) 1–16.
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[24] F.H. Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999) 737–761. [25] F.H. Lin, T. Rivière, Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc. 1 (1999) 237–311. [26] L. Modica, S. Mortola, Il limite nella Γ -convergenza di una famiglia di funczionali elliptici, Boll. Un. Mat. Ital. 14A (1977) 526–529. [27] L. Modica, S. Mortola, Un esempio di Γ -convergenza, Boll. Un. Mat. Ital. 14-B (1977) 185–299. [28] T. Ricciardi, G. Tarantello, Vortices in the Maxwell–Chern–Simons theory, Comm. Pure Appl. Math. 53 (2000) 811–851. [29] T. Rivière, Line vortices in the U (1)-Higgs model, ESAIM Control Optim. Calc. Var. 1 (1996) 77–167. [30] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998) 379–403. [31] E. Sandier, S. Serfaty, On the energy of type-II superconductors in the mixed phase, Rev. Math. Phys. 12 (2000) 1219–1257. [32] E. Sandier, S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. École Norm. Sup. (4) 33 (2000) 561–592. [33] E. Sandier, S. Serfaty, A product-estimate for Ginzburg–Landau and corollaries, J. Funct. Anal. 211 (2004) 219– 244. [34] S. Serfaty, Local minimizers for the Ginzburg–Landau energy near critical magnetic field. I, Commun. Contemp. Math. 1 (1999) 213–254. [35] D. Spirn, Vortex motion laws for dynamic Ginzburg–Landau models in two dimensions, PhD thesis, New York University, New York, 2001. [36] D. Spirn, X. Yan, Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy, Calc. Var. Partial Differential Equations, in press. [37] G. Tarantello, Multiple condensate solutions for the Chern–Simons–Higgs theory, J. Math. Phys. 37 (1996) 3769– 3796. [38] B. White, Rectifiability of flat chains, Ann. of Math. (2) 150 (1999) 165–184. [39] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, New York, 2001.
Journal of Functional Analysis 255 (2008) 589–619 www.elsevier.com/locate/jfa
Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces M.I. Ostrovskii ∗ Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA Received 24 September 2007; accepted 22 April 2008 Available online 27 May 2008 Communicated by K. Ball
Abstract Let BY denote the unit ball of a normed linear space Y . A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y , there exists a linear projection P : Y → X such that P (BY ) ⊂ A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon. © 2008 Elsevier Inc. All rights reserved. Keywords: Banach space; Space tiling zonotope; Sufficient enlargement for a normed linear space; Totally unimodular matrix
1. Introduction
This paper is devoted to a generalization of the main results of [22], where similar results were proved in the dimension two. We refer to [22,23] for more background and motivation.
* Fax: +1 718 990 1650.
E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.012
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M.I. Ostrovskii / Journal of Functional Analysis 255 (2008) 589–619
1.1. Notation and definitions All linear spaces considered in this paper will be over the reals. By a space we mean a normed linear space, unless it is explicitly mentioned otherwise. We denote by BX (SX ) the unit ball (sphere) of a space X. We say that subsets A and B of finite-dimensional linear spaces X and Y , respectively, are linearly equivalent if there exists a linear isomorphism T between the subspace spanned by A in X and the subspace spanned by B in Y such that T (A) = B. By a symmetric set K in a linear space we mean a set such that x ∈ K implies −x ∈ K. Our terminology and notation of Banach space theory follows [12]. By Bpn , 1 p ∞, n ∈ N, we denote the closed unit ball of np . Our terminology and notation of convex geometry follows [27]. We use the term ball for a symmetric, bounded, closed, convex set with interior points in a finite-dimensional linear space. Definition 1. (See [18].) A ball A in a finite-dimensional normed space X is called a sufficient enlargement (SE) for X (or of BX ) if, for an arbitrary isometric embedding of X into a Banach space Y , there exists a projection P : Y → X such that P (BY ) ⊂ A. A sufficient enlargement A for X is called a minimal-volume sufficient enlargement (MVSE) if vol A vol D for each SE D for X. It can be proved, using a standard compactness argument and Lemma 3 below, that minimalvolume sufficient enlargements exist for every finite-dimensional space. Recall that a real matrix A with entries −1, 0, and 1 is called totally unimodular if all minors (that is, determinants of square submatrices) of A are equal to −1, 0, or 1. See [25] and [29, Chapters 19–21] for a survey of results on totally unimodular matrices and their applications. A Minkowski sum of finitely many line segments in a linear space is called a zonotope (see [3, 13,14,27,28] for basic facts on zonotopes). We consider zonotopes that are sums of line segments of the form I (x) = {λx: −1 λ 1}. For a d × m totally unimodular matrix with columns τi (i = 1, . . . , m) and real numbers ai we consider the zonotope Z in Rd given by Z=
m
I (ai τi ).
i=1
The set of all zonotopes that are linearly equivalent to zonotopes obtained in this way over all possible choices of m, of a rank d totally unimodular d × m matrix, and of positive numbers ai (i = 1, . . . , m) will be denoted by T d . Observe that each element of Td is d-dimensional in the sense that it spans a d-dimensional subspace. It is easy to describe all 2 × m totally unimodular matrices and to show that T2 is the union of the set of all symmetric hexagons and the set of all symmetric parallelograms. The class Td of zonotopes has been characterized in several different ways, see [5,6,10,15, 21,31]. We shall use a characterization of Td in terms of lattice tiles. Recall that a compact set K ⊂ Rd is called a lattice tile if there exists a basis {xi }di=1 in Rd such that Rd =
m1 ,...,md ∈Z
d i=1
mi xi + K ,
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and the interiors of the sets ( di=1 mi xi ) + K are disjoint. The set Λ=
d
mi xi : m1 , . . . , md ∈ Z
i=1
is called a lattice. The absolute value of the determinant of the matrix whose columns are the coordinates of {xi }di=1 is called the determinant of Λ and is denoted d(Λ), see [7, §3]. Theorem 1. (See [6,15].) A d-dimensional zonotope is a lattice tile if and only if it is in Td . It is worth mentioning that lattice tiles in Rd do not have to be zonotopes, see [16,17,32], and [33, Chapter 3]. 1.2. Statements of the main results The main result of [21] can be restated in the following way. (A finite-dimensional normed space is called polyhedral if its unit ball is a polytope.) Theorem 2. A ball Z is linearly equivalent to an MVSE for some d-dimensional polyhedral space X if and only if Z ∈ Td . In [22] it was shown that for d = 2 the statement of Theorem 2 is valid without the restriction of polyhedrality of X. The main purpose of the present paper is to prove the same for each d ∈ N. It is clear that it is enough to prove Theorem 3. Each MVSE for a d-dimensional space is in Td . Using Theorem 3 we show that spaces having non-parallelepipedal MVSE cannot be strictly convex or smooth. More precisely, we prove Theorem 4. Let X be a finite-dimensional normed linear space having an MVSE that is not a parallelepiped. Then X contains a two-dimensional subspace whose unit ball is linearly equivalent to the regular hexagon. Remarks. 1. Theorem 4 is a simultaneous generalization of [22, Theorem 4] (which is a special case of Theorem 4 corresponding to the case dim X = 2) and of [19, Theorem 7] (which states that each MVSE for n2 is a cube circumscribed about B2n ). 2. The fact that X contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon does not imply that X has an MVSE that is not a parallelepiped. A simplest example supporting this statement is 3∞ . 2. Proof of Theorem 3 First we show that it is enough to prove the following lemmas. It is worth mentioning that our proof of Theorem 3 goes along the same lines as the proof of its two-dimensional version in [22]. The most difficult part of the proof is a d-dimensional version of the approximation
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lemma [22, Lemma 2, p. 380], it is the contents of Lemma 2 of the present paper. Also, a twodimensional analogue of Lemma 1 is completely trivial. Lemma 1. Let Tn ⊂ Rd , n ∈ N, be such that Tn ∈ Td , and {Tn }∞ n=1 converges with respect to the Hausdorff metric to a d-dimensional set T . Then T ∈ Td . Remark. If a sequence {Tn }∞ n=1 ⊂ Td converges to a lower-dimensional set T , the set T does not have to be in Tdim T . In fact, as it was already mentioned, T2 is the set of all symmetric hexagons and parallelograms. On the other hand, it is easy to find a Hausdorff convergent sequence of elements of T3 whose limit is an octagon. Lemma 2 (Main lemma). For each d ∈ N there exist ψd > 0 and a function td : (0, ψd ) → (1, ∞) satisfying the conditions: (1) limε↓0 td (ε) = 1; (2) If Y is a d-dimensional polyhedral space, B is an MVSE for Y , and A is an SE for Y satisfying vol A (1 + ε)d vol B
(1)
for some 0 < ε < ψd , then A contains a ball A˜ satisfying the conditions: ˜ T ) td (ε) for some T ∈ Td , where by d(A, ˜ T ) we denote the Banach–Mazur dis(a) d(A, tance; (b) A˜ is an SE for Y . Lemma 3. (See [22, Lemma 3].) The set of all sufficient enlargements for a finite-dimensional normed space X is closed with respect to the Hausdorff metric. Proof of Theorem 3. (We assume that Lemmas 1 and 2 have been proved.) Let X be a ddimensional space and let A be an MVSE for X. Let {εn }∞ n=1 be a sequence satisfying ψd > εn > 0 and εn ↓ 0. Let {Yn }∞ be a sequence of polyhedral spaces satisfying n=1 1 BX ⊂ BYn ⊂ BX . 1 + εn
(2)
Then A is an SE for Yn . Let Bn be an MVSE for Yn . Then (1 + εn )Bn is an SE for X. Since A is a minimal-volume SE for X, we have
vol A vol (1 + εn )Bn = (1 + εn )d vol Bn . By Lemma 2 for every n ∈ N there exists an SE A˜ n for Yn satisfying A˜ n ⊂ A and d(A˜ n , Tn ) td (εn ) for some Tn ∈ Td .
(3)
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The condition (2) implies that (1 + εn )A˜ n is an SE for X. The sequence {(1 + εn )A˜ n }∞ n=1 is bounded (all of its terms are contained in (1 + ε1 )A). By the Blaschke selection theorem [27, p. 50] the sequence {(1 + εn )A˜ n }∞ n=1 contains a subsequence convergent with respect to the Hausdorff metric. We denote its limit by D, and assume that the sequence {(1 + εn )A˜ n }∞ n=1 itself converges to D. ˜ Observe that each An contains (1/(1 + ε1 ))BX and is contained in A. By (3) we may assume without loss of generality that Tn are balls in X satisfying 1 BX ⊂ A˜ n ⊂ Tn ⊂ td (εn )A˜ n ⊂ td (εn )A. 1 + ε1
(4)
It is clear that D is the Hausdorff limit of {A˜ n }∞ n=1 . From (4) we get that D is the Hausdorff . By Lemma 1 we get D ∈ T . limit of {Tn }∞ d n=1 By Lemma 3 the set D is an SE for X. Since (1 + εn )A˜ n ⊂ (1 + εn )A, and (1 + εn )A is Hausdorff convergent to A, we have D ⊂ A. On the other hand, A is an MVSE for X, hence D = A and A ∈ Td . 2 Proof of Lemma 1. By Theorem 1 the sets Tn are lattice tiles. Let {Λn }∞ n=1 be lattices corresponding to these lattice tiles. Since volume is continuous with respect to the Hausdorff metric (see [27, p. 55]), the supremum supn vol(Tn ) is finite. Since Tn is a lattice tile with respect to Λn , the determinant of Λn satisfies d(Λn ) = vol(Tn ). (Although I have not found this result in the stated form, it is well known. It can be proved, for example, using the argument from [7, Proof of Theorem 2, pp. 42–43].) Hence supn d(Λn ) < ∞. Since T is d-dimensional, there exists r > 0 such that rB2d ⊂ T . Choosing a smaller r > 0, if necessary, we may assume that rB2d ⊂ Tn for each n. Therefore the lattices {Λn }∞ n=1 satisfy the conditions of the selection theorem of Mahler (see, for example, [7, §17], where the reader can also find the standard definition of convergence for lattices). Hence the sequence {Λn }∞ n=1 contains a subsequence which converges to some lattice Λ. It is easy to verify that T tiles Rd with respect to Λ. On the other hand, the number of possible distinct columns of a totally unimodular matrix with columns from Rd is bounded from above by 3d , because each entry is 0, 1, or −1. (Actually a much better exact bound is known, see [29, p. 299].) Using this we can show that T is a zonotope by a straightforward argument. Also we can use the argument from [27, Theorem 3.5.2] and the observation that a convergent sequence of measures on the sphere of d2 , each of whom has a finite support of cardinality 3d , converges to a measure supported on 3d points. Thus, T is a zonotope and a lattice tile. Applying Theorem 1 again, we get T ∈ Td . 2 3. Proof of the main lemma 3.1. Coordinatization Proof of Lemma 2. In our argument the dimension d is fixed. Many of the parameters considered below depend on d, although we do not reflect this dependence in our notation. m Since Y is polyhedral, we can consider Y as a subspace of m ∞ . Let P : ∞ → Y be a linear m m ). projection satisfying P (B∞ ) ⊂ A (such a projection exists because A is an SE). Let A˜ = P (B∞ ˜ ˜ It is easy to see that A is an SE for Y . It remains to show that A is close to some T ∈ Td with respect to the Banach–Mazur distance.
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We consider the standard inner product on m ∞ . (The unit vector basis is an orthonormal basis with respect to this inner product.) Let {q1 , . . . , qm−d } be an orthonormal basis in ker P . Let {y1 , . . . , yd } be an orthonormal basis in Y . Let q˜1 , . . . , q˜d be such that {q˜1 , . . . , q˜d , q1 , . . . , qm−d } is an orthonormal basis in m ∞. Lemma 4 (Image shape lemma). Let P and q˜1 , . . . , q˜d be as above. Denote by Q˜ = [q˜1 , . . . , q˜d ] the columns of the transpose mathe matrix whose columns are q˜1 , . . . , q˜d . Let z1 , . . . , zm be m m ) is linearly equivalent to the zonotope d ˜ T . Then P (B∞ trix Q i=1 I (zi ) ⊂ R . Proof. It is enough to observe that: m under two linear projections with the same kernel are linearly equivalent. (i) Images of B∞ m Hence, P (B∞ ) is linearly equivalent to the image of the orthogonal projection with the kernel ker P . ˜ T is the matrix of the orthogonal projection with the kernel ker P . 2 (ii) The matrix Q˜ Q
By Lemma 4 we may replace A˜ by Z=
m
(5)
I (zi )
i=1
in the estimate (a) of Lemma 2. Let M = m d . We denote by ui (i = 1, . . . , M) the d × d minors of [y1 , . . . , yd ] (ordered in some way). We denote by wi (i = 1, . . . , M) the d × d minors of [q˜1 , . . . , q˜d ] ordered in the same m = M) their complementary (m − d) × (m − d) way as the ui . We denote by vi (i = 1, . . . , m−d minors of [q1 , . . . , qm−d ]. Using the word complementary we mean that all minors are considered as minors of the matrix [q˜1 , . . . , q˜d , q1 , . . . , qm−d ], see [1, p. 76]. By the Laplacian expansion (see [1, p. 78]) det[y1 , . . . , yd , q1 , . . . , qm−d ] =
M
θi ui vi
i=1
and det[q˜1 , . . . , q˜d , q1 , . . . , qm−d ] =
M
θ i wi v i
(6)
i=1
for proper signs θi . Since the matrix [q˜1 , . . . , q˜d , q1 , . . . , qm−d ] is orthogonal, we have det[q˜1 , . . . , q˜d , q1 , . . . , qm−d ] = ±1.
(7)
We need the following result on compound matrices. (We refer to [1, Chapter V] for necessary definitions and background.) A compound matrix of an orthogonal matrix is orthogonal (see [1, Example 4, p. 94]).
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M This result implies, in particular, that the Euclidean norms of the vectors {wi }M i=1 and {vi }i=1 M in R are equal to 1. From (6) and (7) we get that either (a) wi = θi vi for every i, or (b) wi = −θi vi for every i. Without loss of generality, we assume that wi = θi vi for all i (we replace q1 by −q1 if it is not the case). We compute the volume of A˜ and B with the normalization that comes from the Euclidean structure introduced above. It is well known (see [20, p. 318]) and is easy to verify that with this normalization
2d |vi | vol A˜ = M | i=1 θi ui vi | i=1 M
and vol B =
2d maxi |ui |
for each MVSE B for Y . Remark. After the publication of [20] I learned that the formula for the volume of a zonotope used in [20] can be found in [2, Appendix, Section VI]. Since vol A˜ vol A, the inequality (1) implies that max |ui | i
M i=1
M |vi | (1 + ε)d θi ui vi .
(8)
i=1
By (a) the inequality (8) can be rewritten as max |ui | i
M i=1
M |wi | (1 + ε)d ui wi .
(9)
i=1
We need the following two observations: d (i) 2d M i=1 |wi | is the volume of Z in R . M (ii) The vector {ui }i=1 is what is called the Grassmann coordinates, or the Plücker coordinates of the subspace Y ⊂ Rm , see [9, Chapter VII] and [30, p. 42]. Recall that Y is spanned by the columns of the matrix [y1 , . . . , yd ]. It is easy to see that if we choose another basis in Y , the Grassmann (Plücker) coordinates will be multiplied by a constant. We denote by Zε (ε > 0) the set of all d-dimensional zonotopes in Rd satisfying the condition (9) with an equality. More precisely, we define Zε as the set of those d-dimensional zonotopes Z in Rd for which
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(1) There exists m ∈ N and a rank d matrix Q˜ of size m × d such that Z = m i=1 I (zi ), where d ˜ zi ∈ R , i = 1, . . . , m, are rows of Q. ˜ (2) There exists a rank d matrix m Y of size m × d such that, if we denote the d × d minors of Q ∞ by {wi }i=1 , where M = d , and the d × d minors of Y , ordered in the same way as the wi , by {ui }∞ i=1 , then max |ui | i
M i=1
M |wi | = (1 + ε) ui wi , d
(10)
i=1
and there is no Y for which max |ui | i
M i=1
M |wi | < (1 + ε) ui wi . d
i=1
Remarks. 1. It is clear that the zonotope property of being in Zε is invariant under changes of the system of coordinates. 2. We do not consider the class Z0 because, as it was shown in [21], this class is contained in Td . Many objects introduced below depend on Z and ε, although sometimes we do not reflect this dependence in our notation. Let Z ∈ Zε . We shall change the system of coordinates in Rd twice. First we introduce in Rd a new system of coordinates such that the unit (Euclidean) ball B2d of Rd is the maximal volume ellipsoid in Z. From now on we consider the vectors zi introduced in Lemma 4 as vectors in Rd and not as d-tuples of real numbers. It is easy to see that the support function of Z is given by hZ (x) =
m x, zi . i=1
It is more convenient for us to write this formula in a different way. We consider the set
z1 zm z1 zm ,..., ,− ,...,− . (11)
z1
zm z1
zm
If the vectors in (11) are pairwise distinct, we let μ to be the atomic measure on the unit (Euclidean) sphere S whose atoms are given by μ(zi / zi ) = μ(−zi / zi ) = zi /2. It is easy to see that (12) hZ (x) = x, z dμ(z). S
The defining formula for μ should be adjusted in the natural way if some of the vectors in (11) are equal. Conversely, if μ is a nonnegative measure on S supported on a finite set, then (12) is a support function of some zonotope (see [27, Section 3.5] for more information on this matter).
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Dealing with subsets of S we use the following terminology and notation. Let x0 ∈ S, r > 0. The set Δ(x0 , r) := √ {x ∈ S: x − x0 < r or x + x0 < r}, where · is the 2 -norm, is called a cap. If 0 < r < 2, then Δ(x0 , r) consists of two connected components. In such a case both x0 and −x0 will be considered as centers of Δ(x0 , r). We are going to show that if ε > 0 is small, then the inequality (9) implies that all but a very small part of the measure μ is supported on a union of small caps centered at a set of vectors which are multiples of a set of vectors satisfying the condition: if we write their coordinates with respect to a suitably chosen basis, we get a totally unimodular matrix. Having such a set, it is easy to find T ∈ Td which is close to Z with respect to the Banach–Mazur distance, see Lemma 15. For any two numbers ω, δ > 0 we introduce the set
Ω(ω, δ) := x ∈ S: μ Δ(x, ω) δ (recall that by S we denote the unit sphere of d2 ). In what follows c1 (d), c2 (d), . . . , C1 (d), C2 (d), . . . denote quantities depending on the dimension d only. Since d is fixed throughout our argument, we regard them as constants. First we find conditions on ω and δ under which the set Ω(ω, δ) contains a normalized basis {ei }di=1 whose distance to an orthonormal basis can be estimated in terms of d only. 1 and δ c1 (d)ωd−1 Lemma 5. There exist 0 < c1 (d), C1 (d), C2 (d) < ∞, such that for ω 6d d d there is a normalized basis {ei }i=1 in the space R satisfying the conditions:
(a) μ(Δ(ei , ω)) δ. (b) If {oi }di=1 is an orthonormal basis in Rd , then the operator N : Rd → Rd given by N oi = ei satisfies N C1 (d) and N −1 C2 (d), where the norms are the operator norms of N, N −1 considered as operators from d2 into d2 . Proof. We need an estimate for μ(S). Observe that if K1 and K2 are two symmetric zonotopes and K1 ⊂ K2 , then μ1 (S) μ2 (S) for the corresponding measures μ1 and μ2 (defined as even measures satisfying (12) with Z = K1 and Z = K2 , respectively). To prove this statement we integrate the equality (12) with respect to x over the Haar measure on S. Now we use the assumption that B2d is the maximal volume ellipsoid in Z. Let ni=1 γi xi ⊗ xi be the F. John representation of the identity operator corresponding to Z (see [12, p. 46]). Then Z ⊂ x: x, xi 1 ∀i ∈ {1, . . . , n} . n n d Since n x = i=1 x, xi γi xi for each x ∈ R , we have Z ⊂ i=1 [−γi xi , γi xi ]. Since i=1 γi = d, this implies μ(S) d. Using the well-known computation, which goes back to B. Grünbaum ([8, (5.2), p. 462], √ see, also, [11, pp. 94–95]) one can find estimates for μ(S) from below, which imply μ(S) d. For our purposes the trivial estimate μ(S) 1 is sufficient (this estimate follows immediately from Z ⊃ B2d , because this inclusion implies hZ (x) x ). We denote the normalized Haar measure on S by η. It is well known that there exists c2 (d) > 0 such that
η Δ(x, r) c2 (d)r d−1
∀r ∈ (0, 1) ∀x ∈ S.
(13)
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Using a standard averaging argument and μ(S) 1, we get that there exists e1 ∈ S such that
μ Δ(e1 , ω) c2 (d)ωd−1 . 1 + ω)-neighborhood (in the d2 metric) of the line L1 spanned by e1 . Consider the closed ( 3d Let Δ1 be the intersection of this neighborhood with S. Our purpose is to estimate μ(S \ Δ1 ) from below. Let x ∈ S be orthogonal to e1 . Then 1 1 hZ (x) 1 · μ(S \ Δ1 ) + + ω · d, 3d
where the left-hand side inequality follows from the fact that Z contains B2d . Therefore 1 μ(S \ Δ1 ) 1 − ( 3d + ω)d. We erase all measure μ contained in Δ1 , use a standard averaging argument again, and find a vector e2 such that
1 +ω d . μ Δ(e2 , ω) \ Δ1 c2 (d)ωd−1 1 − 3d 1 Since μ(Δ(e2 , ω) \ Δ1 ) > 0, the vector e2 is not in the 3d -neighborhood of L1 . 1 Let Δ2 be the intersection of S with the closed ( 3d + ω)-neighborhood of L2 = lin{e1 , e2 } (that is, L2 is the linear span of {e1 , e2 }). Let x ∈ S be orthogonal to L2 . Then 1 1 hZ (x) 1 · μ(S \ Δ2 ) + + ω · d, 3d
where the left-hand side inequality follows from the fact that Z contains B2d . Therefore 1 μ(S \ Δ2 ) 1 − ( 3d + ω) d. Using the standard averaging argument in the same way as in the previous step we find a vector e3 such that
1 d−1 1− +ω d . μ Δ(e3 , ω) \ Δ2 c2 (d)ω 3d 1 Since μ(Δ(e3 , ω) \ Δ2 ) > 0, the vector e3 is not in the 3d -neighborhood of L2 . We continue in an obvious way. As a result we construct a normalized basis {e1 , . . . , ed } satisfying the conditions: 1 (i) μ(Δ(ei , ω)) c2 (d)ωd−1 (1 − ( 3d + ω)d). i−1 1 (ii) dist(ei , lin{ej }j =1 ) 3d , i = 2, . . . , d, where dist(·,·) denotes the distance from a vector to a subspace.
If ω <
1 6d ,
the inequality (i) implies
1 μ Δ(ei , ω) c2 (d)ωd−1 , 2
and we get the estimate (a) of Lemma 5 with c1 (d) = c2 (d)/2.
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To estimate N and N −1 , we let {oi }di=1 be the basis obtained from {ei } using the Gram– Schmidt orthonormalization process. Let N : Rd → Rd be defined by N oi = ei . The estimate √
N C1 (d) with C1 (d) = d follows because the vectors {ei }di=1 are normalized and the vectors {oi }di=1 form an orthonormal set. To estimate N −1 we observe that the matrix of N with respect to the basis {oi } is of the form ⎞ ⎛ N11 N12 . . . N1d N22 . . . N2d ⎟ ⎜ 0 , N =⎜ .. .. ⎟ .. ⎝ ... . . . ⎠ 0 and that the inequality (ii) implies Nii ⎛
N11 ⎜ 0 T =⎜ ⎝ ... 0
0 N22 .. .
... ... .. .
0
...
0
1 3d .
...
Ndd
We have
⎛ ⎞ 1 0 ⎜ 0 ⎟ ⎜0 ·⎜ .. ⎟ ⎜ . ⎠ ⎝ ... Ndd 0
N12 N11
...
1 .. . 0
... .. . ...
N1d N11 N2d N22
⎞
⎟ ⎟ ⎟ = D(I + U ), .. ⎟ . ⎠ 1
where I is the identity matrix, ⎛ ⎛
N11 ⎜ 0 D=⎜ ⎝ ... 0
0 N22 .. .
... ... .. .
0
...
⎞
0 0 ⎟ , .. ⎟ . ⎠ Ndd
0
⎜ ⎜0 ⎜ ⎜ and U = ⎜ ... ⎜ ⎜ ⎝0 0
N1,d−1 N11 N2,d−1 N22
N1d N11 N2d N22
...
0
Nd−1,d Nd−1,d−1
...
0
0
N12 N11
...
0 .. .
... .. .
0 0
.. .
.. .
⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
Therefore
N −1 = (I + U )−1 D −1 = I − U + U 2 − · · · + (−1)d−1 U d−1 D −1 ,
(14)
the identity (I + U )−1 = (I − U + U 2 − · · · + (−1)d−1 U d−1 ) follows from the obvious equality 1 imply that columns of U are vectors with Euclidean U d = 0. The definition of U and Nii 3d 3
norm at most 3d, hence U 3d 2 . Therefore the identity (14) implies the following estimate for N −1 : 3d
−1 U d − 1 −1 3d d 2 − 1 N · D · 3d. 3
U − 1 3d 2 − 1 Denoting the right-hand side of this inequality by C2 (d) we get the desired estimate.
2
Remark. We do not need sharp estimates for c1 (d), C1 (d), and C2 (d) because d is fixed in our argument, and the dependence on d of the parameters involved in our estimates is not essential for our proofs.
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We use the following notation: for a set Γ ⊂ S and a real number r > 0 we denote the set {x ∈ S: inf{ x − y : y ∈ Γ } r} by Γr . Lemma 6. Let c2 (d) be the constant from (13), then μ(S \ ((Ω(ω, δ))ω ))
δ . c2 (d)ωd−1
δ Proof. Assume the contrary, that is, μ(S \ ((Ω(ω, δ))ω )) > c (d)ω d−1 . Then, using a standard 2 averaging argument as in Lemma 5, we find a point x such that
μ Δ(x, ω) \ Ω(ω, δ) ω c2 (d)ωd−1 ·
δ = δ. c2 (d)ωd−1
By the definition of Ω(ω, δ) this implies x ∈ Ω(ω, δ). On the other hand, since the set Δ(x, ω) \ ((Ω(ω, δ))ω ) is non-empty, it follows that x ∈ / Ω(ω, δ). We get a contradiction. 2 3.2. Notation and definitions used in the rest of the proof For each Z ∈ Zε we apply Lemma 5 with ω = ω(ε) = ε 4k and δ = δ(ε) = ε 4dk , where 0 < k < 1 is a number satisfying the conditions k<
1 6 + 4d 2
and k <
1 , 2d + 4d 2
(15)
we choose and fix such number k for the rest of the proof. It is clear that there is Ξ0 = 1 and δ(ε) c1 (d)(ω(ε))d−1 are satisfied for Ξ0 (d, k) > 0 such that the conditions ω(ε) 6d all ε ∈ (0, Ξ0 ), where c1 (d) is the constant from Lemma 5. In the rest of the argument we consider ε ∈ (0, Ξ0 ) only. Let {ei }di=1 be one of the bases satisfying the conditions of Lemma 5 with the described choice of ω and δ. Now we change the system of coordinates in Rd ⊃ Z the second time. The new system of coordinates is such that {ei }di=1 is its unit vector basis. We shall modify the objects introduced so far (Ω, μ, etc.) and denote their versions corresponding to the new ˇ μ, system of coordinates by Ω, ˇ etc. All these objects depend on Z, ε, and the choice of {ei }di=1 . We denote by Sˇ the Euclidean unit sphere in the new system of coordinates. We denote by N : S → Sˇ the natural normalization mapping, that is, N(z) = z/ z , where z is the Euclidean norm of z with respect to the new system of coordinates. The estimates for N and N −1 from Lemma 5 imply that the Lipschitz constants of the mapping N and its inverse N−1 : Sˇ → S can be estimated in terms of d only. We introduce a measure μˇ on Sˇ as an atomic measure supported on a finite set and such that μ(N(z)) ˇ = μ(z) z for each z ∈ S, where z is the norm of z in the new system of coordinates. Using the definition of the zonotope Z it is easy to check that the function hˇ Z (x) =
x, zˇ d μ(ˇ ˇ z),
Sˇ
where ·,· is the inner product in the new coordinate system, is the support function of Z in the new system of coordinates. ˇ ˇ Everywhere below we mean We define Ωˇ = Ω(ω, δ) as N(Ω(ω, δ)). It is clear that ei ∈ Ω. coordinates in the new system of coordinates (when we refer to · , Δ, etc.).
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The observation that N and N−1 are Lipschitz, with Lipschitz constants estimated in terms of d only, implies the following statements: • There exist C3 (d), C4 (d) < ∞ such that
ˇ μˇ Sˇ \ Ω(ω, δ) C
3 (d)ω(ε)
C4 (d)
δ ωd−1
(16)
(we use Lemma 6). • There exist c3 (d) > 0 and C5 (d) < ∞ such that
μˇ Δ x, C5 (d)ω c3 (d)δ
ˇ ∀x ∈ Ω(ω, δ)
(17)
ˇ (we use the definitions of Ω(ω, δ) and Ω(ω, δ)). • There exists a constant C6 (d) depending on d only, such that vol(Z) C6 (d).
(18)
ˇ be the transpose of the matrix whose columns are the coordinates of zi in the new Let Q ˇ ordered in the system of coordinates. We denote by wˇ i (i = 1, . . . , M) the d × d minors of Q M . Therefore (10) implies same way as the wi . The vector {wˇ i }M is a scalar multiple of {w } i i=1 i=1 max |ui | i
M i=1
M |wˇ i | = (1 + ε)d ui wˇ i .
(19)
i=1
The volume of Z in the new system of coordinates is 2d
M
ˇ i |. i=1 |w
3.3. Lemma on six large minors To show that if ε > 0 is small, then the inequality (19) implies that all but a very small part of the measure μˇ is supported “around” multiples of vectors represented by a totally unimodular matrix in some basis, we need the following lemma. It shows that the inequality (19) implies that the measure μˇ cannot have non-trivial “masses” near (d + 2)-tuples of vectors satisfying certain condition. Lemma 7. Let χ(ε), σ (ε), and π(ε) be functions satisfying the following conditions: (1) (2) (3) (4)
limε↓0 χ(ε) = limε↓0 σ (ε) = limε↓0 π(ε) = 0; ε = o((χ(ε))2 (σ (ε))d ) as ε ↓ 0; π(ε) = o(χ(ε)) as ε ↓ 0; There is a subset Φ0 ⊂ (0, Ξ0 ) such that the closure of Φ0 contains 0, and for each ε ∈ Φ0 ˇ such that there exist Z ∈ Zε and points x1 , . . . , xd−2 , p1 , p2 , p3 , p4 in the corresponding S,
μˇ Δ z, π(ε) σ (ε)
∀z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 }.
(20)
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Let U0 be the set of pairs (ε, Z) in which ε ∈ Φ0 and Z satisfies the condition from (4). Let Φ1 ⊂ Φ0 be the set of those ε ∈ Φ0 for which there exists (ε, Z) ∈ U0 such that the corresponding points x1 , . . . , xd−2 , p1 , p2 , p3 , p4 satisfy the condition det(Hα,β ) χ(ε)
(21)
for all matrices Hα,β whose columns are the coordinates of {x1 , . . . , xd−2 , pα , pβ }, α, β ∈ {1, 2, 3, 4}, α = β, with respect to an orthonormal basis {ei }di=1 in Rd . Then there exists Ξ1 > 0 such that Φ1 ∩ (0, Ξ1 ) = ∅. Proof. We assume the contrary, that is, we assume that 0 belongs to the closure of Φ1 . For each ε ∈ Φ1 we choose Z ∈ Zε such that (ε, Z) ∈ U0 and the condition (20) is satisfied. We show that for sufficiently small ε > 0 this leads to a contradiction. We consider the following perturbation of the matrix Hα,β : each column vector z in it is replaced by a vector from Δ(z, π(ε)). We denote the obtained perturbation of the matrix Hα,β p by Hα,β . We claim that p det H α,β χ(ε) − d · π(ε).
(22)
To prove this claim we need the following lemma, which we state in a bit more general form than is needed now, because we shall need it later. Lemma 8. Let x1 , . . . , xd , z ∈ d2 be such that max2id xi m and z − x1 l. Then det[z, x2 , . . . , xd ] − det[x1 , x2 , . . . , xd ] l · md−1 . This lemma follows immediately from the volumetric interpretation of determinants. To get the inequality (22) we apply Lemma 8 d timeswith m = 1 and l = π(ε). Since Z ∈ Zε , it can be represented in the form Z = i I (zi ). First we complete our proof in a special case when the following condition is satisfied: (∗) All vectors zi whose normalizations zi / zi belong to the sets Δ(z, π(ε)), z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 }, have the same norm τ and there are equal amounts of such vectors in each of the sets Δ(z, π(ε)), z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 }, we denote the common value of the amounts by F . The inequality (20) implies F · τ σ (ε). We denote by Λ the set of all numbers i ∈ {1, . . . , M} satisfying the condition: the normalizap tions of columns of the minor wˇ i form a matrix of the form Hα,β , for some α, β ∈ {1, 2, 3, 4}. We need an estimate for i∈Λ |wˇ i |. The inequality (22) implies |wˇ i | τ d (χ(ε) − d · π(ε)) for each i ∈ Λ. d−2 ways to choose On the other hand, the cardinality |Λ| of Λ is 6F d . In fact, 4 there are F zi / zi in the sets Δ(xj , π(ε)), j = 1, . . . , d − 2. There are 2 = 6 ways to choose two of the
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sets Δ(pj , π(ε)), j = 1, 2, 3, 4, and there are F 2 ways to choose one vector zi / zi in each of them. Therefore |Λ| = 6F d and
d
|wˇ i | 6F d τ d χ(ε) − d · π(ε) 6 σ (ε) χ(ε) − d · π(ε) .
(23)
i∈Λ
We assume for simplicity that maxi |ui | = 1 (if it is not the case, some of the sums below should be multiplied by maxi |ui |). The ui are defined above the equality (10). Then the condition (19) can be rewritten as M ui wˇ i |wˇ i | + |wˇ i |. (1 + ε)d i∈Λ
i=1
(24)
i ∈Λ /
On the other hand, M d (1 + ε) ui wˇ i (1 + ε) ui wˇ i + (1 + ε)d |wˇ i |. d
From (24) and (25) we get
(1 + ε)d ui wˇ i |wˇ i | − (1 + ε)d − 1 |wˇ i |. i∈Λ
(25)
i ∈Λ /
i∈Λ
i=1
(26)
i ∈Λ /
i∈Λ
As is well known, 2d M ˇ i | is the volume of Z, hence M ˇ i | 2−d C6 (d). Using this i=1 |w i=1 |w observation and the inequalities (23) and (26) we get 1 ((1 + ε)d − 1)C6 (d)2−d u w ˇ − |wˇ i |. i i (1 + ε)d 6(σ (ε))d (χ(ε) − d · π(ε)) i∈Λ
i∈Λ
(We use the fact that χ(ε) − d · π(ε) > 0 if ε > 0 is small enough.) The conditions (2) and (3) imply that there exists ψ > 0 such that
1 ((1 + ε)d − 1)C6 (d)2−d − (1 + ε)d 6(σ (ε))d (χ(ε) − d · π(ε))
> 1 − 0.04 χ(ε) − d · π(ε)
is satisfied if ε ∈ (0, ψ). The right-hand side is chosen in the form needed below. Let ψ > 0 be such that the statement above is true. Then for ε ∈ (0, ψ) we have
ui wˇ i 1 − 0.04 χ(ε) − d · π(ε) |wˇ i |. i∈Λ
(27)
(28)
i∈Λ
Recall that ui are d × d minors of some matrix [y1 , . . . , yd ]. We need the Plücker relations, see [9, p. 312] or [30, p. 42]. The result that we need can be stated in the following way: if γ1 , . . . , γd−2 , κ1 , κ2 , κ3 , κ4 are indices of d + 2 rows of [y1 , . . . , yd ], then t1,2 t3,4 − t1,4 t3,2 + t2,4 t3,1 = 0,
(29)
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where tα,β is the determinant of the d × d matrix whose rows are the rows of [y1 , . . . , yd ] with the indices γ1 , . . . , γd−2 , κα , and κβ . Note that (29) can be verified by a straightforward computation (which is very simple if we make a suitable change of coordinates before the computation). Now we show that (28) cannot be satisfied. Let Ψ be a set consisting of d + 2 vectors zκ1 , zκ2 , zκ3 , zκ4 , zγ1 , . . . , zγd−2 , formed in the following way. We choose vectors (zκi / zκi ) ∈ Δ(pi , π(ε)), i = 1, 2, 3, 4, and choose vectors (zγi / zγi ) ∈ Δ(xi , π(ε)), i = 1, . . . , d − 2. To p each such selection there corresponds a set of 6 minors wˇ i of the form τ d det(Hα,β ), we denote this set of six minors by {wˇ i }i∈M(Ψ ) . One of the immediate consequences of the Plücker relation (29) is that for any such (d + 2)tuple Ψ 1 |ui | √ 2
for some i ∈ M(Ψ ).
(30)
(Here we use the assumption that maxi |ui | = 1.) For each Ψ we choose one such i ∈ M(Ψ ) and denote it by s(Ψ ). The estimate (22) and the condition (∗) imply that
τ d |wˇ i | τ d χ(ε) − d · π(ε)
(31)
for every i ∈ Λ. Hence for every (d + 2)-tuple Ψ of the described type we have √ 2−1 1 |wˇ i | + √ |wˇ s(Ψ ) | |wˇ i | − √ |wˇ s(Ψ ) | 2 2 i∈M(Ψ )\{s(Ψ )} i∈M(Ψ ) √ ( 2 − 1)|wˇ s(Ψ ) | = |wˇ i | 1 − √ 2 i∈M(Ψ ) |wˇ i | i∈M(Ψ ) √ ( 2 − 1)τ d (χ(ε) − d · π(ε)) |wˇ i | 1 − √ 2 · 6τ d i∈M(Ψ )
< |wˇ i | 1 − 0.04 χ(ε) − d · π(ε) .
ui wˇ i i∈M(Ψ )
i∈M(Ψ )
Thus
< u w ˇ |wˇ i | 1 − 0.04 χ(ε) − d · π(ε) . i i i∈M(Ψ )
(32)
i∈M(Ψ )
Recall that F is the number of vectors zi corresponding to each of the sets Δ(z, π(ε)), z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 }. Simple counting shows that for an arbitrary collection {Υi }i∈Λ of numbers we have Ψ i∈M(Ψ )
Υi = F 2
i∈Λ
Υi .
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Using (32) we get that F 2 ui wˇ i = ui wˇ i u w ˇ i i Ψ i∈M(Ψ )
i∈Λ
<
Ψ
i∈M(Ψ )
|wˇ i | 1 − 0.04 χ(ε) − d · π(ε)
Ψ i∈M(Ψ )
= F2
|wˇ i | 1 − 0.04 χ(ε) − d · π(ε) .
i∈Λ
If ε ∈ (0, ψ), we get a contradiction with (28). To see that the general case can be reduced to the case (∗) we need the following observation: ˇ by Let τ1 , τ2 > 0 be such that τ1 + τ2 = 1. We replace the row with the coordinates of zj in Q two rows, one of them is the row of coordinates of τ1 zj and the other is the row of coordinates of τ2 zj . The zonotope generated by the rows of the obtained matrix coincides with Z. In the matrix [y1 , . . . , yd ] we replace the j th row by two copies of it. It is easy to see that if we replace the sequences {ui }M ˇ i }M i=1 and {w i=1 by sequences of d × d minors of these new matrices, the condition (19) is still satisfied. We can repeat this ‘cutting’ of vectors zj into ‘pieces’ with (19) still being valid. Therefore, we may assume the following: among zj corresponding to each of the sets Δ(z, π(ε)), z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 } there exists a subset Φ(z, π(ε)) consisting of vectors having the same length τ , and such that the sum of norms of vectors from Φ(z, π(ε)) is σ (ε) 2 , moreover, we may assume that the numbers of such vectors in the subsets Φ(z, π(ε)) are the same for all z ∈ {x1 , . . . , xd−2 , p1 , p2 , p3 , p4 }. Lemma 7 in this case can be proved using the same argument as before, but with Λ being the set of those minors wˇ i for which rows are from Φ(z, π(ε)). Everything starting with the inequality (23) can be shown in the same way as before; only some constants will be changed (because we need to replace σ (ε) by σ (ε) 2 ). 2 3.4. Searching for a totally unimodular matrix Let ρ(ε) = ε k , ν(ε) = ε 3k . For a vector s we denote its coordinates with respect to {ei }di=1 by {si }di=1 . (Here k and {ei }di=1 are the same as in Section 3.2.) Lemma 9. If k<
1 , 6 + 4d 2
(33)
ˇ then there exists Ξ2 > 0 such that for ε ∈ (0, Ξ2 ), s, t ∈ Ω(ω(ε), δ(ε)), and α, β ∈ {1, . . . , d}, the inequality min |sα |, |sβ |, |tα |, |tβ | ρ(ε),
(34)
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implies det sα sβ
tα < ν(ε). tβ
(35)
Proof. Assume the contrary, that is, there exists a subset Φ2 ⊂ (0, 1), having 0 in its closure ˇ and such that for each ε ∈ Φ2 there exist Z ∈ Zε , s, t ∈ Ω(ω(ε), δ(ε)) and α, β satisfying the condition (34), and such that det sα tα ν(ε). sβ tβ We apply Lemma 7 with {x1 , . . . , xd−2 } = {ei }i=α,β , {p1 , p2 , p3 , p4 } = {eα , eβ , s, t}. Using a straightforward determinant computation we see that the condition (21) is satisfied with χ(ε) = min{1, ρ(ε), ν(ε)} = ε 3k (we consider ε < 1). The inequality (17) implies that the condition (4) of Lemma 7 is satisfied with π(ε) = C5 (d)ω(ε) = C5 (d)ε 4k and σ (ε) = c3 (d)δ(ε) = c3 (d)ε 4dk . It is clear that the conditions (2) and (3) of Lemma 7 are satisfied. To get (2) we use the condition (33). Applying Lemma 7, we get the existence of the desired Ξ2 . 2 ˇ For each vector from Ω(ω(ε), δ(ε)) we define its top set as the set of indices of coordinates whose absolute values ρ(ε). The collection of all possible top sets is a subset of the set of all subsets of {1, . . . , d}, hence its ˇ δ(ε)) in the following cardinality is at most 2d . We create a collection Θ(ω(ε), δ(ε)) ⊂ Ω(ω(ε), ˇ way: for each subset of {1, . . . , d} which is a top set for at least one vector from Ω(ω(ε), δ(ε)), we choose one of such vectors; the set Θ(ω(ε), δ(ε)) is the set of all vectors selected in this way. ˇ In our next lemma we show that each vector from Ω(ω(ε), δ(ε)) can be reasonably well approximated by a vector from Θ(ω(ε), δ(ε)). Therefore (as we shall see later), to prove Lemma 2 it is sufficient to find a “totally unimodular” set approximating Θ(ω(ε), δ(ε)). Lemma 10. Let ρ(ε) and ν(ε) be as above and let k and Ξ2 be numbers satisfying the conditions ˇ of Lemma 9. Let ε ∈ (0, Ξ2 ), Z ∈ Zε , and let s, t ∈ Ω(ω(ε), δ(ε)) be two vectors with the same top set Σ. Then min t + s , t − s Proof. Observe that if ρ(ε) = ε k >
√1 , d
2
ν(ε) + 4dρ(ε)2 . (ρ(ε))2
(36)
the statement of the lemma is trivial. Therefore we may
assume that ρ(ε) In such a case Σ contains at least one element. First we show that the signs of different components of s and t “agree” on Σ in the sense that either they are the same everywhere on Σ, or they are the opposite everywhere on Σ. In fact, assume the contrary, and let α, β ∈ Σ be indices for which the signs “disagree.” Then, as it is easy to check, √1 . d
det sα sβ
2 tα = |sα ||tβ | + |sβ ||tα | 2 ρ(ε) > ν(ε), tβ
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and we get a contradiction. We consider the case when the signs of tα and sα are the same for each α ∈ Σ , the other case can be treated similarly (we can just consider −s instead of s). We may assume without loss of generality that |tα | |sα | for some α ∈ Σ . We show that in this case ν(ε) |sβ | |tβ | 1 − (ρ(ε))2 for all β ∈ Σ . In fact, if |tβ | < (1 − s ν(ε) > det α sβ
ν(ε) )|sβ | (ρ(ε))2
for some β ∈ Σ , then
ν(ε) tα |tα ||sβ | − |sα ||tβ | |sα ||sβ | ν(ε), tβ (ρ(ε))2
a contradiction. We have
t − s 2 = t 2 + s 2 − 2t, s 2 − 2
(1 −
α∈Σ
2
ν(ε) +4 (ρ(ε))2
α ∈Σ /
ρ(ε)2 2
ν(ε) )sα2 + 2 ρ(ε)2 2 (ρ(ε))
ν(ε) + 4dρ(ε)2 . (ρ(ε))2
α ∈Σ /
2
Let Θ(ω(ε), δ(ε)) = {bj }Jj=1 , where J 2d . We may and shall assume that {ei (ε)}di=1 ⊂ Θ(ω(ε), δ(ε)) (see Lemma 5 and Section 3.2). We denote d ·2d by n and introduce d ·n functions: ϕ1 (ε), . . . , ϕd·n (ε), such that ϕ1 (ε) · · · ϕd·n (ε) = ρ(ε) = ε k ,
1 ϕα (ε) = ϕα+1 (ε) d+1 .
(37) (38)
We consider the matrix X whose columns are {bj }Jj=1 . We order the absolute values of entries of this matrix in non-increasing order and denote them by a1 a2 · · · ad·J . Let j0 be the least index for which ϕd·j0 (ε) > aj0 .
(39)
The existence of j0 follows from {ei (ε)}di=1 ⊂ Θ(ω(ε), δ(ε)). The definition of j0 implies that aj ϕd·j (ε) for j < j0 , hence aj ϕd·(j0 −1) (ε) for j j0 − 1. We replace all entries of the matrix X except a1 , . . . , aj0 −1 by zeros and denote the obtained matrix by G = (Gij ), i = 1, . . . , d, j = 1, . . . , J , and its columns by {gj }Jj=1 . It is clear that
gj − bj d · ϕdj0 (ε).
(40)
¯ . . . , d} ¯ ∪ {1, . . . , J }, where we use bars in We form a bipartite graph G on the vertex set {1, ¯1, . . . , d¯ because these vertices are considered as different from the vertices 1, . . . , d, which are in the set {1, . . . , J }. The edges of G are defined in the following way: the vertices i¯ and j are adjacent if and only if Gij = 0. So there is a one-to-one correspondence between edges of G
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and non-zero entries of G. We choose and fix a maximal forest F in G. (We use the standard terminology, see, e.g. [29, p. 11].) For each non-zero entry of G we define its level in the following way: • The level of entries corresponding to edges of F is 1. • For a non-zero entry of G which does not correspond to an edge in F we consider the cycle in G formed by the corresponding edge and edges of F . We define the level of the entry as the half of the length of the cycle (recall that the graph G is bipartite, hence all cycles are even). Observation. One of the classes of the bipartition has d vertices. Hence no cycle can have more than 2d edges, and the level of each vertex is at most d. To each entry Gij of level f we assign a square submatrix G(ij ) of G all other entries in which are of levels at most f − 1. We do this in the following way. To entries corresponding to edges of F we assign the 1 × 1 matrices containing these entries. For an entry Gij which does not correspond to an edge in F we consider the corresponding edge e in G and the cycle C formed by e and edges of F . Then we consider the entries in G corresponding to edges of C and the minimal submatrix in G containing all of these entries. Now we consider all edges in G corresponding to non-zero entries of this submatrix. We choose and fix in this set of edges a minimum-length cycle M containing e. We define G(ij ) as the minimal submatrix of G containing all entries corresponding to edges of M. It is easy to verify that: • G(ij ) is a square submatrix of G. • Non-zero entries of G(ij ) are in one-to-one correspondence with entries of M. • The expansion of the determinant of G(ij ) according to the definition contains exactly two non-zero terms. • All non-zero entries of G(ij ) except Gij have level f − 1. ˜ Lemma 11. Let k < 1/(2d + 4d 2 ). If ε > 0 is small enough, then there exists a d × J matrix G such that: ˜ is also zero. (1) If some entry of G is zero, the corresponding entry of G ˜ (2) The entries of level 1 of G are the same as for G. ˜ are perturbations of entries of G satisfying the following (3) All other non-zero entries of G conditions: ˜ ij | < ϕd·j0 −f +1 (ε). (a) If Gij is of level f , then |Gij − G ˜ ) of (b) For each non-zero entry Gij of level 2 of G the determinant of the submatrix G(ij ˜ G corresponding to G(ij ) is zero. Proof. Let Gij be an entry of level f . Since, as it was observed above, all entries of G(ij ) have level f − 1, we can prove the lemma by induction as follows. ˜ ij = Gij for all Gij of level one. (1) We let G (2) Let f 2.
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Induction hypothesis. We assume that for all entries Gij of levels (Gij ) satisfying 2 ˜ ij satisfying (Gij ) f − 1 we have found perturbations G ˜ ij | ϕd·j0 −(Gij )+1 (ε), |Gij − G ˜ )) = 0. (Note that this assumption is vacuous if f = 2.) such that det(G(ij Inductive step. Let Gij be an entry of level f . If ε > 0 is small enough we can find a number ˜ ij such that |G ˜ ij − Gij | ϕd·j0 −f +1 (ε) and det(G(ij ˜ )) = 0. Observe that by the induction G hypothesis and the observation that all other entries of G(ij ) have levels f − 1, all other ˜ ) have already been defined. entries of G(ij So let Gij be an entry of level f , and G(ij ) be the corresponding square submatrix. Renumbering rows and columns of the matrix G we may assume that the matrix G(ij ) looks like the one sketched below for some h f . ⎛
a1 ⎜ b1 ⎜ . G(ij ) = ⎜ ⎜ .. ⎝0 0
0 a2 .. .
... ... .. .
0 0 .. .
0 0
... ...
ah−1 bh−1
⎞ Gij 0 ⎟ .. ⎟ ⎟ . ⎟. 0 ⎠ ah
Therefore the matrix G (possibly, after renumbering of columns and rows) has the form ⎛
a1 ⎜ b1 ⎜ . ⎜ . ⎜ . ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜∗ ⎜ ⎜∗ ⎜ . ⎜ .. ⎜ ⎝∗ ∗
0 a2 .. .
... ... .. .
0 0 .. .
Gij 0 .. .
0 0 .. .
0 0 .. .
... ... .. .
0 0 .. .
0 0 .. .
1 0 .. .
0 1 .. .
0 0 ∗ ∗ .. .
... ... ... ... .. .
ah−1 bh−1 ∗ ∗ .. .
0 ah ∗ ∗ .. .
0 0 1 0 .. .
0 0 0 1 .. .
... ... ... ... .. .
∗ ∗
... ...
∗ ∗
∗ ∗
0 0
0 0
... ...
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .. .. .. .. . . . . 1 0 0 0 0 1 0 0
⎞ ... ...⎟ ⎟ .. ⎟ .⎟ ⎟ ...⎟ ⎟ ...⎟ ⎟. ...⎟ ⎟ ...⎟ .. ⎟ .⎟ ⎟ ...⎠ ...
(41)
˜ We have assumed that we have already found entries {a˜ n }hn=1 and {b˜n }h−1 n=1 of G which are h−1 h perturbations of {an }n=1 and {bn }n=1 . The entries 1 shown (41) are the only non-zero entries in their columns, therefore the corresponding edges of G should be in F . Let us denote the ˜ ij . The condition (b) of Lemma 11 can be written as perturbation of Gij we are looking for by G h n=1
a˜ n + (−1)h−1
h−1
˜ ij = 0. b˜n · G
(42)
n=1
˜ ij , found as a solution of (42) satisfies |G ˜ ij − Gij | < So it suffices to show that the number G ϕd·j0 −f +1 (ε). To show this we assume the contrary. Since there are finitely many possibilities
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for j0 and f , the converse can be described as existence of j0 and f , such that there is a subset Φ3 ⊂ (0, 1), whose closure contains 0, satisfying the condition: For each ε ∈ Φ3 there is Z ∈ Zε such that after proceeding with all steps of the construction we get: all the conditions above are satisfied, but h h−1 h−1 h−1 ˜ a˜ n + (−1) |b˜n |. bn · Gij > ϕd·j0 −f +1 (ε) n=1
n=1
(43)
n=1
We need to get from here an estimate for |det(G(ij ))| from below. To get it we observe that the inequality (43) is an estimate from below of the determinant of the matrix ⎛ a˜
1
⎜ b˜1 ⎜ . G (ij ) = ⎜ ⎜ .. ⎝ 0 0
0 a˜ 2 .. .
... ... .. .
0 0 .. .
0 0
... ...
a˜ h−1 b˜h−1
Gij ⎞ 0 ⎟ .. ⎟ ⎟ . ⎟. ⎠ 0 a˜ h
To get from here an estimate for det(G(ij )) from below we observe the following: the 2 -norm of each column of Gij is 1, the 2 -distance between a column of Gij and the corresponding column of G (ij ) is at most 2ϕdj0 −f +2 (ε). Hence the 2 -norm of each column of G (ij ) is 1 + 2ϕdj0 −f +2 (ε). Applying Lemma 8 h times we get
det G(ij ) det G (ij ) − h · 2ϕdj
0 −f +2
h−1 (ε) 1 + 2ϕdj0 −f +2 (ε) .
The induction hypothesis implies |b˜i | ϕd(j0 −1) (ε) − ϕdj0 −f +2 (ε), we get
det G(ij ) ϕdj
h−1 (ε) · ϕd(j0 −1) (ε) − ϕdj0 −f +2 (ε)
h−1 − h · 2ϕdj0 −f +2 (ε) 1 + 2ϕdj0 −f +2 (ε) . 0 −f +1
(44)
Let us keep the notation {gj }Jj=1 for columns of the matrix (41). We consider the following six d × d minors of this matrix: the corresponding submatrices contain the columns {g2 , . . . , gh−1 , gh+1 , . . . , gd }, and two out of the four columns {g1 , gh , gd+1 , gd+2 }. Observe that gh+1 = eh+1 , . . . , gd = ed , gd+1 = e1 , gd+2 = e2 . The absolute values of the minors are equal to det G(ij ) ,
h a n , n=2
h−1 b n , n=1
h−1 |a1 | · b n , n=2
h b n , n=2
h−1 b n .
(45)
n=2
The first number in (45) was estimated in (44). All other numbers are at least (ϕd(j0 −1) (ε))h−1 , it is clear that this number exceeds the number from (44).
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We are going to use Lemma 7 with {x1 , . . . , xd−2 } = {N(g2 ), . . . , N(gh−1 ), N(gh+1 ), . . . , N(gd )} and {p1 , p2 , p3 , p4 } = {N(g1 ), N(gh ), N(gd+1 ), N(gd+2 )}. (Recall that N(z) = z/ z .) Our definitions imply that bj = 1 and gj 1, because gj is obtained from bj by replacing some of the coordinates by zeros. Hence the inequality (44) and the remark above on the numbers (45) imply that the condition (21) is satisfied with
h−1 χ(ε) = ϕdj0 −f +1 (ε) · ϕd(j0 −1) (ε) − ϕdj0 −f +2 (ε)
h−1 − h · 2ϕdj0 −f +2 (ε) 1 + 2ϕdj0 −f +2 (ε) .
(46)
ˇ The inequality (40), the inclusion bj ∈ Ω(ω(ε), δ(ε)) and (17) imply that the condition (20) is satisfied with π(ε) = 2d · ϕdj0 (ε) + C5 (d)ω(ε) and σ (ε) = c3 (d)δ(ε). So it remains to show that the condition (38) implies that the conditions (2) and (3) of Lemma 7 are satisfied. By (38), (46), the inequality 2 h f d, and the trivial observation that all functions ϕα (ε) do not exceed 1 for 0 ε 1, we have
d
ϕdj0 −f +1 (ε) = O χ(ε) .
(47)
Now we verify the condition (3) of Lemma 7. The part (b) can be verified as follows. The conditions (37) and (38), together with f 2 and ω(ε) = ε 4k , imply that π(ε) = O(ϕdj0 (ε)) = o((ϕdj0 −f +1 (ε))d ) = o(χ(ε)). To verify the condition (2) of Lemma 7 it suffices to observe that (47) and (37) imply (ρ(ε))d = O(χ(ε)). Hence (2) is satisfied if 2dk + 4d 2 k < 1. This inequality is among the conditions of Lemma 11. Hence we can apply Lemma 7 and get the conclusion of Lemma 11. 2 ˜ be an approximation of G by a matrix satisfying the conditions of Lemma 11. We Now let G use the same maximal forest F in G as above. It is easy to show (and the corresponding result is well known in the theory of matroids, see, for example, [24, Theorem 6.4.7]) that multiplying ˜ by positive numbers we can make entries corresponding to edges of F columns and rows of G to be equal to ±1. Denote the obtained matrix by G. ˜ satisfies the conditions of Lemma 11, then G is a matrix with entries −1, 0, Lemma 12. If G and 1. ij which are not in the set {−1, 0, 1}. Proof. Assume the contrary, that is, there are entries G ij be one of such entries satisfying the additional condition: the level (Gij ) is the minimal Let G ij which are not in {−1, 0, 1}. Denote by G(ij ) the submatrix of G possible among all entries G which corresponds to G(ij ). ) contains two nonThen, by observations preceding Lemma 11, the expansion of det G(ij ij or −G ij . Our assumptions imply that zero terms: one of them is 1 or −1, the other is G ˜ ) = 0, because G ˜ using multipli ) = 0. This contradicts det G(ij is obtained from G det G(ij cations of columns and rows by numbers. 2 should be totally In Lemma 13 we show that for functions ϕα (ε) chosen as above, the matrix G unimodular for sufficiently small ε. In Lemma 15 we show how to estimate the Banach–Mazur is totally unimodular. distance between Z and Td in the case when G
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is totally unimodular. Lemma 13. If ε > 0 is small enough, the matrix G ˜ is a ϕd(j0 −1)+1 (ε)-apProof. The conclusion of Lemma 11 implies that each entry of G proximation of an entry from G. Therefore for small ε the absolute value of each non-zero entry ˜ is at least ϕd(j0 −1) (ε)/2. This implies the following observation. of G ˜ is a product of the corresponding minor of G and a number Observation. Each d × d minor of G d ζ satisfying (ϕd(j0 −1) (ε)/2) ζ 1. ˜ and the corresponding submatrix S in G. If the Proof. Consider a square submatrix S˜ in G corresponding minor is zero, there is nothing to prove. If it is non-zero, we reorder columns and rows of S˜ in such a way that all entries on the diagonal become non-zero, and do the same Let ri , cj > 0 be such that after multiplying rows of S by ri and columns of reordering with S. ˜ Then the resulting matrix by cj we get S. ˜ = det(S) det(S)
ri cj . i
j
On the other hand, ri ci ϕd(j0 −1) (ε)/2, because the diagonal entry of S is ±1, and the absolute value of the diagonal entry of S˜ is ϕd(j0 −1) (ε)/2. The conclusion follows. 2 Lemma 14. Let D be a d ×J matrix with entries −1, 0, and 1, containing a d ×d identity submaxd−2 , p 1 , p 2 , p 3 , p 4 } trix. If D is not totally unimodular, then it contains d + 2 columns { x1 , . . . , 2 , p 3 , p 4 } minors obtained by such that for all six choices of two vectors from the set { p1 , p xd−2 } are non-zero. joining them to { x1 , . . . , Proof. Our argument follows [4, pp. 1068–1069] (see, also, [29, pp. 269–271]), where a similar statement is attributed to R. Gomory. Suppose that D is not totally unimodular, then it has a square submatrix S with |det(S)| 2. Let S be of size h × h. Reordering columns and rows of D (if necessary), we may assume that D is of the form: S 0 Ih ∗ , D= ∗ Id−h 0 ∗ where Ih and Id−h are identity matrices of sizes h × h and (d − h) × (d − h), respectively, 0 denote matrices with zero entries of the corresponding dimensions, and ∗ denote matrices of the corresponding dimensions with unspecified entries. We consider all matrices which can be obtained from D by a sequence of the following operations: • Addition or subtraction a row to or from another row. • Multiplication of a column by −1, provided that after each such operation we get a matrix with entries −1, 0, and 1. which satisfies the Among all matrices obtained from D in such a way we select a matrix D following conditions:
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(1) Has all unit vectors among its columns; (2) Has the maximal possible number ξ of unit vectors among the first d columns. Observe that ξ < d because the operations listed above preserve the absolute value of the determinant and at the beginning the absolute value of the determinant formed by the first d which is not a unit vector. Let columns was 2. Let dr be one of the first d columns of D {i1 , . . . , it } be indices of its non-zero coordinates. Then at least one of the unit vectors ei1 , . . . , eit (the first d columns of D are linearly independent). Assume is not among the first d columns of D adding/subtracting the that ei1 is not among the first d columns of D. We can try to transform D row number i1 to/from rows number i2 , . . . , it (and multiplying the column number r by (−1), if necessary) into a new matrix D˜ which satisfies the following conditions: • Has among the first d columns all the unit vectors it had before; • Has ei1 as its column number r; • Has all the unit vectors among its columns. It is not difficult to verify that the only possible obstacle is that there exists another column dt such that for some s ∈ {2, . . . , t} in D, det Di1 r Di r s
Di1 t = 2, Di s t
(48)
By the maximality assumption, a submatrix satisfying (48) where by Dij we denote entries of D. exists. It is easy to see that letting p 1 , p 2 , p 3 , p 4 = {dr , ds , ei1 , eis },
and { x1 , . . . , xd−2 } = {e1 , . . . , ed } \ {ei1 , eis },
satisfying the required condition. we get a set of columns of D Since the operations listed above preserve the absolute values of d × d minors, the corresponding columns of D form the desired set. 2 Remark. Lemma 14 can also be obtained by combining known characterizations of regular and binary matroids, see [24] (we mean, first of all, Theorems 9.1.5, 6.6.3, Corollary 10.1.4, and Proposition 3.2.6). We continue our proof of Lemma 13. Assume the contrary. Since there are finitely many possible values of j0 , there is j0 and a subset Φ4 ⊂ (0, 1), whose closure contains 0, satisfying the condition: For each ε ∈ Φ4 there is Z ∈ Zε such that following the construction, we get the preselected is not totally unimodular. value of j0 , and the obtained matrix G Since the entries of G are integers, the absolute values of the minors are at least one. We are going to show that the corresponding minors of G are also ‘sufficiently large,’ and get a contradiction using Lemma 7.
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˜ are at least (ϕd(j0 −1) (ε)/2)d . The By the observation above the corresponding minors of G ˜ Euclidean norm of a column in G is at most 1 + dϕd(j0 −1)+1 (ε). Applying Lemma 8 d times we get that the corresponding minor of G is at least
d
d−1 ϕd(j0 −1) (ε)/2 − d 2 ϕd(j0 −1)+1 (ε) · 1 + dϕd(j0 −1)+1 (ε) . We are going to use Lemma 7 for x1 , . . . , xd−2 , p1 , p2 , p3 , p4 defined in the following way. Let xˇ1 , . . . , xˇd−2 , pˇ 1 , pˇ 2 , pˇ 3 , pˇ 4 be the columns of G corresponding to the columns and x1 , . . . , xd−2 , p1 , p2 , p3 , p4 be their normalizations (that xd−2 , p 1 , p 2 , p 3 , p 4 of G, x1 , . . . , is, x1 = xˇ1 / xˇ1 , etc.). Since norms of columns of G are 1, the condition (21) of Lemma 7 is satisfied with
d
d−1 χ(ε) = ϕd(j0 −1) (ε)/2 − d 2 ϕd(j0 −1)+1 (ε) · 1 + dϕd(j0 −1)+1 (ε) . ˇ Now we recall that columns {gj } of G satisfy (40) for some vectors bj ∈ Ω(ω(ε), δ(ε)). Hence the distance from x1 , . . . , xd−2 , p1 , p2 , p3 , p4 to the corresponding vectors bj is 2dϕdj0 (ε). By (17) the condition (20) is satisfied with π(ε) = 2dϕdj0 (ε) + C5 (d)ω(ε) and σ (ε) = c3 (d)δ(ε). The fact that the conditions (2) and (3) of Lemma 7 are satisfied is verified in the same way as at the end of Lemma 11, the only difference is that instead of (47) we have (ϕd(j0 −1) (ε))d = O(χ(ε)). This does not affect the rest of the argument. Therefore, under the same condition on k should be totally unimodular if ε > 0 is small as in Lemma 11 we get, by Lemma 7, that G enough. is totally unimodular, then there exists a zonotope T ∈ Td such that Lemma 15. If G d(Z, T ) td (ε), where td (ε) is a function satisfying limε↓0 td (ε) = 1. ˜ can be obtained from G using multiplications of rows and Proof. Observe that the matrix G columns by positive numbers. Hence, re-scaling the basis {ei }, if necessary, we get: columns ˜ with respect to the re-scaled basis are of the form ai τi , where τi are columns of a totally of G unimodular matrix (see the definition of Td in the introduction). We are going to approximate the measure μˇ by a measure μ supported on vectors which are ˜ Recall that μˇ is supported on a finite subset of S. ˇ normalized columns of G. The approximation is constructed in the following way. We erase the measure μˇ supported ˇ outside (Ω(ω(ε), δ(ε)))C3 (d)ω(ε) . The total mass of the measure erased in this way is small ˇ by (16). As for the measure supported on B := (Ω(ω(ε), δ(ε)))C3 (d)ω(ε) , we approximate each ˜ atom of it by the atom of the same mass supported on the nearest normalized column of G.
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˜ by A(z). If there are several such We denote the nearest to z ∈ supp μˇ normalized column of G columns, we choose one of them. ˇ Now we estimate the distance from a point of (Ω(ω(ε), δ(ε)))C3 (d)ω(ε) to the nearest nor˜ ˇ malized column of G. The distance from this point to Ω(ω(ε), δ(ε)) is C3 (d)ω(ε), the distance ˇ from a point from Ω(ω(ε), δ(ε)) to the point from Θ(ω(ε), δ(ε)) with the same top set (or its opν(ε) 2 posite), by Lemma 10, can be estimated from above by 2 (ρ(ε)) 2 + 4dρ(ε) . The distance from a point in Θ(ω(ε), δ(ε)) to the corresponding column of G is estimated in (40), it is d · ϕdj0 (ε), ˜ is so it is d · ϕ1 (ε), and the distance from a column of G to the corresponding column of G d · ϕd(j0 −1)+1 (ε) d · ϕ1 (ε). Since we have to normalize this vector, the total distance from a ˜ can be estimated from ˇ point of (Ω(ω(ε), δ(ε)))C3 (d)ω(ε) to the nearest normalized column of G above by C3 (d)ω(ε) +
2
ν(ε) + 4dρ(ε)2 + 4d · ϕ1 (ε). (ρ(ε))2
It is clear that this function, let us denote it by ζ (ε), tends to 0 as ε ↓ 0, recall that ρ(ε) = ek , 1 dn−1 . The obtained measure corresponds to a zonotope ν(ε) = ε 3k , ω(ε) = ε 4k , ϕ1 (ε) = ε ( d+1 ) from Td . Let us denote this zonotope by T . Since the dual norms to the gauge functions of Z and T are their support functions, we get the estimate d(T , Z) sup u∈Sˇ
hˇ Z (u) hˇ T (u) · sup . hˇ T (u) u∈Sˇ hˇ Z (u)
So it is enough to show that C1 (d, ε)
hˇ T (u) C2 (d, ε), hˇ Z (u)
(49)
where limε↓0 C1 (d, ε) = limε↓0 C2 (d, ε) = 1. Observe that Lemma 5 implies that there exists a constant 0 < C7 (d) < ∞ such that C7 (d) hˇ Z (u),
ˇ ∀u ∈ S.
(50)
We have hˇ Z (u) =
u, z d μ(z) ˇ
Sˇ
u, z d μ(z) ˇ +
ˇ B S\
C4 (d)
u, z d μ(z) +
Sˇ
δ(ε) ˇ + hˇ T (u) + ζ (ε)μ( ˇ S), ωd−1 (ε)
z∈supp μ∩ ˇ B
ˇ ∀u ∈ S.
u, z − u, A(z) μ(z) ˇ
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In a similar way we get hˇ T (u) =
u, z d μ(z)
u, z − u, A(z) μ(z) ˇ
z∈supp μ∩ ˇ B
B
Sˇ
u, z d μ(z) ˇ +
ˇ hˇ Z (u) + ζ (ε)μ( ˇ S),
∀u ∈ S.
Using (50) we get 1−
δ(ε) C4 (d) ωd−1 (ε)
C7 (d)
It is an estimate of the form (49).
−
ˇ ˇ ζ (ε)μ( ˇ S) hˇ T (u) ζ (ε)μ( ˇ S) . 1+ C7 (d) C7 (d) hˇ Z (u)
2
It is clear that Lemma 15 completes our proof of Lemma 2.
2
4. Proof of Theorem 4 Proof. We start by proving Theorem 4 for polyhedral X. In this case we can consider X as a subspace of m ∞ for some m ∈ N. Since X has an MVSE which is not a parallelepiped, there m exists a linear projection P : m ∞ → X such that P (B∞ ) has the minimal possible volume, but m ) is not a parallelepiped. Let d = dim X, let {q , . . . , q P (B∞ 1 m−d } be an orthonormal basis in ker P and let {q˜1 , . . . , q˜d } be an orthonormal basis in the orthogonal complement of ker P . As m ) is linearly equivalent to the zonotope spanned by rows of it was shown in Lemma 4, P (B∞ ˜ Q = [q˜1 , . . . , q˜d ]. By the assumption this zonotope is not a parallelepiped. It is easy to see that this assumption is equivalent to: there exists a minimal linearly dependent collection of rows of Q˜ containing 3 rows. This condition implies that we can reorder the coordinates in m ∞ and ˜ 1 ˜ from the right by an invertible d × d matrix C1 in such a way that QC multiply the matrix Q has a submatrix of the form ⎞ ⎛ 1 0 ... 0 ⎜ 0 1 ... 0 ⎟ ⎜ . .. . . . ⎟ ⎜ .. . .. ⎟ . ⎟, ⎜ ⎝ 0 0 ... 1 ⎠ a 1 a 2 . . . ad where a1 = 0 and a2 = 0. Let X be a matrix whose columns form a basis of X. The argument of [21] (see the conditions (1)–(3) on p. 96) implies that X can be multiplied from the right by an invertible d × d matrix C2 in such a way that X C2 is of the form ⎛
1 0 .. .
⎜ ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ sign a 1 ⎝ .. .
0 1 .. .
... ... .. .
0 sign a2 .. .
... ... .. .
⎞ 0 0⎟ .. ⎟ .⎟ ⎟ , 1⎟ ⎟ ⎟ ∗⎠ .. .
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where at the top there is an d × d identity matrix, and all minors of the matrix X C2 have absolute values 1. Changing signs of the first two columns, if necessary, we get that the subspace X ⊂ m ∞ is spanned by columns of the matrix ⎞ ⎛ ±1 0 0 ... 0 ⎜ 0 ±1 0 ... 0⎟ ⎟ ⎜ ⎜ 0 0 1 ... 0⎟ ⎜ . .. .. . . .⎟ ⎜ .. . .. ⎟ . . ⎟ ⎜ ⎜ 0 0 0 ... 1⎟ ⎟ ⎜ . (51) ⎜ 1 1 ∗ ... ∗⎟ ⎟ ⎜ ⎟ ⎜ b c1 ∗ ... ∗⎟ ⎜ 1 ⎜ b c2 ∗ ... ∗⎟ ⎟ ⎜ 2 ⎜ . .. .. . . .. ⎟ ⎝ .. . .⎠ . . bm−l−1 cm−l−1 ∗ . . . ∗ The condition on the minors implies that |bi | 1, |ci | 1, and |bi − ci | 1 for each i. Therefore 2 the subspace, spanned in m ∞ by the first two columns of the matrix (51) is isometric to R with the norm
(α, β) = max |α|, |β|, |α + β| . It is easy to see that the unit ball of this space is linearly equivalent to a regular hexagon. Thus, Theorem 4 is proved in the case when X is polyhedral. Proving the result for general, not necessarily polyhedral, space, we shall denote the space by Y . We use Theorem 3. Actually we need only the following corollary of it: Each MVSE is a polyhedron. Therefore we can apply the following result to each MVSE. Lemma 16. (See [22, Lemma 1].) Let Y be a finite-dimensional space and let A be a polyhedral MVSE for Y . Then there exists another norm on Y such that the obtained normed space X satisfies the conditions: (1) X is polyhedral; (2) BX ⊃ BY ; (3) A is an MVSE for X. So we consider the space Y as being embedded into a polyhedral space X with the embedding satisfying the conditions of Lemma 16. By the first part of the proof the space X satisfies the conditions of Theorem 4 and we may assume that X is a subspace m ∞ in the way described in the first part of the proof. So X is spanned by columns—let us denote them by e1 , . . . , ed —of the matrix (51) in m ∞ . It is easy to see that to finish the proof it is enough to show that the vectors e1 , e2 , e1 − e2 are in BY . It turns out each of these points is the center of a facet of a minimum-volume parallelepiped m containing BX . In fact, let {fi }m i=1 be the unit vector basis of ∞ . Let P1 and P2 be the projections
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onto Y with the kernels lin{fd+1 , . . . , fm } and lin{f1 , fd+2 , . . . , fm }, respectively (recall that Y , m) as a linear space, coincides with X). The analysis from [20, pp. 318–319] shows that P1 (B∞ m ) have the minimal possible volume among all linear projections of B m into X. It is and P2 (B∞ ∞ m ) and P (B m ) are parallelepipeds. easy to see that P1 (B∞ 2 ∞ m ), and that e − e is the center of a We show that e1 , e2 are centers of facets of P1 (B∞ 1 2 m m ) coincide with P (f ), . . . , P (f ), facet of P2 (B∞ ). In fact, the centers of facets of P1 (B∞ 1 1 1 d and it is easy to check that P1 (fi ) = ei for i = 1, . . . , d. As for P2 , we observe that e1 − e2 ∈ lin{f1 , f2 , fd+2 , . . . , fm }, and the coefficient near f2 in the expansion of e1 − e2 is ±1. Therefore P2 (f2 ) = ±(e1 − e2 ). Since the projections P1 and P2 satisfy the minimality condition from [21, Lemma 1] (see, m ) and P (B m ) are MVSE for X. Hence, by also [20, pp. 318–319]), the parallelepipeds P1 (B∞ 2 ∞ the conditions of Lemma 16, they are MVSE for Y also. Hence, they are minimum-volume parallelepipeds containing BY . On the other hand, it is known, see [26, Lemma 3.1], that centers of facets of minimal-volume parallelepipeds containing BY should belong to BY , we get e1 , e2 , e1 − e2 ∈ BY . The theorem follows. 2 Acknowledgment I would like to thank Gideon Schechtman for turning my attention to the fact that the class Td was studied in works on lattice tiles. References [1] A.C. Aitken, Determinants and Matrices, reprint of the 4th ed., Greenwood Press, Westport, CT, 1983. [2] W. Blaschke, Kreis und Kugel, Veit, Leipzig, 1916; reprinted by Chelsea, 1949; Russian transl. of the second ed., Nauka, Moscow, 1967. [3] E.D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969) 323–345. [4] P. Camion, Characterization of totally unimodular matrices, Proc. Amer. Math. Soc. 16 (1965) 1068–1073. [5] H.S.M. Coxeter, The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl. (9) 41 (1962) 137–156; reprinted in: H.S.M. Coxeter, Twelve Geometric Essays, Southern Illinois Univ. Press, Carbondale, IL, 1968; Feffer & Simons, London, 1968. [6] R.M. Erdahl, Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra, European J. Combin. 20 (1999) 427– 449. [7] P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers, second ed., North-Holland, Amsterdam, 1987. [8] B. Grünbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451–465. [9] W.V.D. Hodge, D. Pedoe, Methods of Algebraic Geometry, vol. I, Cambridge Univ. Press, Cambridge, 1947. [10] F. Jaeger, On space-tiling zonotopes and regular chain-groups, Ars Combin. B 16 (1983) 257–270. [11] G.J.O. Jameson, Summing and Nuclear Norms in Banach Space Theory, London Math. Soc. Stud. Texts, vol. 8, Cambridge Univ. Press, Cambridge, 1987. [12] W.B. Johnson, J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 1, Elsevier, Amsterdam, 2001, pp. 1–84. [13] H. Martini, Some results and problems around zonotopes, in: K. Böröczky, G. Fejes Tóth (Eds.), Intuitive Geometry, North-Holland, Amsterdam, 1987, pp. 383–418. [14] P. McMullen, On zonotopes, Trans. Amer. Math. Soc. 159 (1971) 91–109. [15] P. McMullen, Space tiling zonotopes, Mathematika 22 (2) (1975) 202–211. [16] P. McMullen, Convex bodies which tile space by translation, Mathematika 27 (1) (1980) 113–121; see also: P. McMullen, Acknowledgement of priority: “Convex bodies which tile space by translation”, Mathematika 28 (2) (1981) 191 (1982). [17] P. McMullen, Convex bodies which tile space, in: The Geometric Vein, Springer, New York, 1981, pp. 123–128. [18] M.I. Ostrovskii, Generalization of projection constants: Sufficient enlargements, Extracta Math. 11 (3) (1996) 466– 474.
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[19] M.I. Ostrovskii, Projections in normed linear spaces and sufficient enlargements, Arch. Math. 71 (4) (1998) 315– 324. [20] M.I. Ostrovskii, Minimal-volume shadows of cubes, J. Funct. Anal. 176 (2) (2000) 317–330. [21] M.I. Ostrovskii, Minimal-volume projections of cubes and totally unimodular matrices, Linear Algebra Appl. 364 (2003) 91–103. [22] M.I. Ostrovskii, Sufficient enlargements of minimal volume for two-dimensional normed spaces, Math. Proc. Cambridge Philos. Soc. 137 (2004) 377–396. [23] M.I. Ostrovskii, Compositions of projections in Banach spaces and relations between approximation properties, Rocky Mountain J. Math. 38 (4) (2008) 1253–1262. [24] J.G. Oxley, Matroid Theory, Oxford Grad. Texts Math., vol. 3, Oxford Univ. Press, Oxford, 1992. [25] M.W. Padberg, Total unimodularity and the Euler-subgraph problem, Oper. Res. Lett. 7 (1988) 173–179. [26] A. Pełczy´nski, S.J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in Rn , Math. Proc. Cambridge Philos. Soc. 109 (1991) 125–148. [27] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1993. [28] R. Schneider, W. Weil, Zonoids and related topics, in: P.M. Gruber, J.M. Wills (Eds.), Convexity and Its Applications, Birkhäuser, Basel, 1983, pp. 296–317. [29] A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1986. [30] I.R. Shafarevich, Basic Algebraic Geometry, vol. I, Springer, Berlin, 1994. [31] G.C. Shephard, Space-filling zonotopes, Mathematika 21 (1974) 261–269. [32] B.A. Venkov, On a class of Euclidean polyhedra, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (2) (1954) 11–31 (in Russian). [33] C. Zong, Strange Phenomena in Convex and Discrete Geometry, Springer, Berlin, 1996.
Journal of Functional Analysis 255 (2008) 620–641 www.elsevier.com/locate/jfa
Stability of closed geodesics on Finsler 2-spheres Yiming Long a,∗,1 , Wei Wang b,2 a Chern Institute of Mathematics, Key Lab of Pure Mathematics and Combinatorics of Ministry of Education,
Nankai University, Tianjin 300071, People’s Republic of China b School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Received 25 September 2007; accepted 1 May 2008 Available online 2 June 2008 Communicated by H. Brezis
Abstract In this paper, we prove that on every Finsler 2-dimensional sphere, either there exist infinitely many prime closed geodesics or there exist at least two irrationally elliptic prime closed geodesics. © 2008 Elsevier Inc. All rights reserved. Keywords: Finsler spheres; Closed geodesics; Index iteration; Mean index identity; Morse inequality; Stability
1. Introduction and main results This paper is devoted to a study on closed geodesics on Finsler 2-spheres. Let us recall firstly the definition of the Finsler metrics. Definition 1.1. (Cf. [28].) Let M be a finite-dimensional manifold. A function F : T M → [0, +∞) is a Finsler metric if it satisfies: (F1) F is C ∞ on T M \ {0}. (F2) F (x, λy) = λF (x, y) for all y ∈ Tx M, x ∈ M, and λ > 0. * Corresponding author.
E-mail addresses: [email protected] (Y. Long), [email protected] (W. Wang). 1 Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, SRFDP, LPMC of
MOE of China, and Nankai University. 2 Partially supported by LMAM in Peking University in China and China Postdoctoral Science Foundation No. 20070420264. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.001
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(F3) For every y ∈ Tx M \ {0}, the quadratic form gx,y (u, v) ≡
1 ∂2 2 F (x, y + su + tv)t=s=0 , 2 ∂s∂t
∀u, v ∈ Tx M,
is positive definite. In this case, (M, F ) is called a Finsler manifold. F is reversible if F (x, −y) = F (x, y) holds for all y ∈ Tx M and x ∈ M. F is Riemannian if F (x, y)2 = 12 G(x)y · y for some symmetric positive definite matrix function G(x) ∈ GL(Tx M) depending on x ∈ M smoothly. A closed curve in a Finsler manifold is a closed geodesic if it is locally the shortest path connecting any two nearby points on this curve (cf. [28]). As usual, on any Finsler 2-sphere S 2 = (S 2 , F ), a closed geodesic c : S 1 = R/Z → S 2 is prime if it is not a multiple covering (i.e., iteration) of any other closed geodesics. Here the mth iteration cm of c is defined by cm (t) = c(mt). The inverse curve c−1 of c is defined by c−1 (t) = c(1 − t) for t ∈ R. We call two prime closed geodesics c and d distinct if there is no θ ∈ (0, 1) such that c(t) = d(t + θ ) for all t ∈ R. We shall omit the word distinct when we talk about more than one prime closed geodesic. For a closed geodesic c on (S 2 , F ), denote by Pc the linearized Poincaré map of c (cf. [31, p. 143]). Then c is called hyperbolic if all the eigenvalues of Pc avoid the unit circle in C, elliptic if all the eigenvalues of Pc are on the unit circle, irrationally elliptic if all the eigenvalues of Pc locate on the unit circle and are of the form eiθ with θ/π being irrational numbers. It was quite surprising when A. Katok [13] in 1973 found some non-symmetric Finsler metrics on S n with only finitely many prime closed geodesics and all closed geodesics are non-degenerate and elliptic. In Katok’s examples the spheres S 2n and S 2n−1 have precisely 2n closed geodesics (cf. also [31]). Based on the results of W. Klingenberg in [14], W. Ballmann, G. Thorbergsson and W. Ziller in [1] and [2] on the existence and stability of closed geodesics under pinching conditions, H.-B. Rademacher obtained existence and stability results on closed geodesics on Finsler spheres in [25,26] and [27] under pinching or bumpy conditions. Note that results of H. Hofer, K. Wysocki and E. Zehnder in [12] imply that there exist either two or infinitely many prime closed geodesics on every Finsler 2-sphere provided all the closed geodesics together with their iterations are non-degenerate and the stable and unstable manifolds of every hyperbolic closed geodesic intersect transversally. In recent [3], V. Bangert and Y. Long proved that on every Finsler 2-sphere, there always exist at least two prime closed geodesics (cf. also [21]). On the other hand, H. Hofer, K. Wysocki and E. Zehnder in [11] proved that there exist either two or infinitely many distinct closed characteristics on every convex compact smooth hypersurface in R4 . Then Y. Long in [19] proved that if there exist precisely two closed characteristics on such a hypersurface in R4 , both of them must be elliptic. Note that recently W. Wang, X. Hu and Y. Long in [29] further proved that both of them must be irrationally elliptic. Motivated by these results, we prove the following theorem in this paper. Theorem 1.2. On every Finsler 2-sphere with only finitely many prime closed geodesics, there exist always at least two irrationally elliptic prime closed geodesics. Remark 1.3. Comparing with the results in [3], Theorem 1.2 further proves the linear stability of at least two closed geodesics. A. Katok’s metric on S 2 possesses precisely two prime closed geodesics c+ and c− whose linearized Poincaré maps are rotation matrices with rotation angles
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for some α ∈ (0, 1) \ Q, i.e., both of them are irrationally elliptic. Thus Theorem 1.2 is sharp. On the other hand, note that if there exist exactly 2 prime closed geodesics on a Finsler (S 2 , F ), they must behave analytically precisely like those in A. Katok’s S 2 (cf. Corollary 5.4 below). Because all the geodesics on the standard Riemannian 2-sphere of constant curvature are closed and possess linearized Poincaré maps I2 , one cannot hope that Theorem 1.2 holds for all Finsler 2-spheres which possess infinitely many prime closed geodesics. 2π 1±α
Our proof of Theorem 1.2 contains mainly four ingredients: Morse theory, the precise index iteration formulae of Y. Long proved in [19], existence theorems of N. Hingston, and Rademacher’s mean index identity. Section 2 contains some preliminary materials used in our proof including basic properties of critical modules of closed geodesics, Hingston’s theorems, Betti numbers and Morse inequalities. In Section 3, we use results from [19] to classify closed geodesics on S 2 into nine classes. Note that an abstract mean index identity for closed geodesics was first established by H.-B. Rademacher in [25]. In Section 4 we derive this identity with precise information on the dependence of coefficients in this identity on closed geodesics. Based on these preparations, Theorem 1.2 is proved in Section 5. In this paper, let N, N0 , Z, Q, R, and C denote the sets of natural integers, non-negative integers, integers, rational numbers, real numbers, and complex numbers, respectively. We denote by [a] = max{k ∈ Z | k a} for any real number a. Let (p, q) denotes the greatest common divisor of p and q ∈ N. We use only singular homology modules with Q-coefficients. For k ∈ N, we denote by Qk the direct sum Q ⊕ · · · ⊕ Q of k copies of Q and Q0 = 0. When S 1 acts on a topological space X, we denote by X the quotient space X/S 1 . 2. Preliminary results on critical modules of closed geodesics In this section, we will review materials related to critical modules of closed geodesics. Details can be found in [26] and [3] as well as [9] and [10]. On a compact Finsler manifold (M, F ), we choose an auxiliary Riemannian metric. This endows the space Λ = ΛM of H 1 -maps γ : S 1 → M with a natural Riemannian Hilbert manifold structure on which the group S 1 = R/Z acts continuously by isometries, cf. [15, Chapters 1 and 2]. This action is defined by translating the parameter, i.e., (s · γ )(t) = γ (t + s) for all γ ∈ Λ and s, t ∈ S 1 . The Finsler metric F defines an energy functional E and a length functional L on Λ by E(γ ) =
1 2
S1
2 F γ (t), γ˙ (t) dt,
L(γ ) =
F γ (t), γ˙ (t) dt.
(2.1)
S1
Both functionals are invariant under the S 1 -action. The critical points of E of positive energies are precisely the nonconstant closed geodesics c : S 1 → M of the Finsler structure. If c ∈ Λ is a nonconstant closed geodesic then c is a regular curve, i.e., c(t) ˙ = 0 for all t ∈ S 1 , and this implies
that the second differential E (c) of E at c exists. As usual we define the index i(c) of c as the maximal dimension of subspaces of Tc Λ on which E
(c) is negative definite, and the nullity ν(c) of c so that ν(c) + 1 is the dimension of the null space of E
(c).
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For m ∈ N we denote the m-fold iteration map φm : Λ → Λ by φm (γ )(t) = γ (mt),
∀γ ∈ Λ, t ∈ S 1 .
(2.2)
We also use the notation φm (γ ) = γ m . For a closed geodesic c, the mean index is defined by m ˆ = lim i(c ) . i(c) m→∞ m
(2.3)
If γ ∈ Λ is not constant then the multiplicity m(γ ) of γ is the order of the isotropy group {s ∈ S 1 | s · γ = γ }. If m(γ ) = 1 then γ is called prime. Hence m(γ ) = m if and only if there exists a prime curve γ˜ ∈ Λ such that γ = γ˜ m . For a closed geodesic c or a real number κ ∈ R, we denote by Λ(c) = {γ ∈ Λ | E(γ ) < E(c)} and Λκ = {d ∈ Λ | E(d) κ}, respectively. If A ⊆ Λ is invariant under some subgroup Γ of S 1 , we denote by A/Γ the quotient space of A with respect to the action of Γ . We abbreviate Λκ = Λκ /S 1 . Using singular homology with rational coefficients we will consider the following critical Q-module of a closed geodesic c ∈ Λ: C ∗ (E, c) = H∗ Λ(c) ∪ S 1 · c /S 1 , Λ(c)/S 1 .
(2.4)
The reason we use Λ/S 1 instead of Λ is that the Morse series of the first is lacunary, i.e., the Betti numbers bi (Λ/S 1 , Λ0 /S 1 ) satisfy b2i = 0 for all i ∈ N0 . In order to relate the critical modules to the index and nullity of c we use the results of D. Gromoll and W. Meyer from [7,8]. Following [26, Section 6.2], we introduce finite-dimensional approximations to Λ. We √ choose an arbitrary energy value a > 0 and k ∈ N such that every F -geodesic of length < 2a/k is minimal. Then Λ(k, a) = γ ∈ Λ E(γ ) < a and γ |[i/k,(i+1)/k] is an F -geodesic for i = 0, . . . , k − 1 is a (k · dim M)-dimensional submanifold of Λ consisting of closed geodesic polygons with k vertices. The set Λ(k, a) is invariant under the subgroup Zk of S 1 . Closed geodesics in Λa− = {γ ∈ Λ | E(γ ) < a} are precisely the critical points of E|Λ(k,a) , and for every closed geodesic c ∈ Λ(k, a) the index of (E|Λ(k,a) )
(c) equals i(c) and the null space of (E|Λ(k,a) )
(c) coincides with the nullspace of E
(c), cf. [26, p. 51], and [3, §3]. We say that a closed geodesic satisfies the isolation condition, if the following holds: (Iso)
For all m ∈ N the orbit S 1 · cm is an isolated critical orbit of E.
Note that if the number of prime closed geodesics on a Finsler manifold is finite, then all the closed geodesics satisfy (Iso). Now we can apply the results of D. Gromoll and W. Meyer [7] to a given closed geodesic c satisfying (Iso). If m = m(c) is the multiplicity of c, we choose a finite-dimensional approximation Λ(k, a) ⊆ Λ containing c such that m divides k. Then the isotropy subgroup Zm ⊆ S 1 of c acts on Λ(k, a) by isometries. Let D be a Zm -invariant local hypersurface transverse to S 1 · c in c ∈ D. According to [7, Lemma 1], for every such D we can find a product neighborhood
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B+ × B− × B0 of 0 ∈ Rdim Λ(k,a)−1 such that dim B− = i(c), dim B0 = ν(c), and a diffeomorphism ψ : B = B+ × B− × B0 → ψ(B+ × B− × B0 ) ⊆ D from B onto an open subset ψ(B) ⊆ D such that ψ(0) = c and ψ is Zm -invariant, and there exists a smooth function f : B0 → R satisfying f (0) = 0 and f
(0) = 0
(2.5)
E ◦ ψ(x+ , x− , x0 ) = |x+ |2 − |x− |2 + f (x0 ),
(2.6)
and
for (x+ , x− , x0 ) ∈ B+ × B− × B0 . As usual, we call N = {ψ(0, 0, x0 ) | x0 ∈ B0 } a local characteristic manifold at c, and U = {ψ(0, x− , 0) | x− ∈ B− } a local negative disk at c. Both N and U are Zm -invariant. It follows from (2.6) that c is an isolated critical point of E|N . We set N − = N ∩ Λ(c), U − = U ∩ Λ(c) = U \ {c} and D − = D ∩ Λ(c). Using (2.6), the fact that c is an isolated critical point of E|N , and the Künneth formula, one concludes H∗ D − ∪ {c}, D − = H∗ U − ∪ {c}, U − ⊗ H∗ N − ∪ {c}, N − ,
(2.7)
− Q, if q = i(c), − Hq U ∪ {c}, U = Hq U, U \ {c} = 0, otherwise,
(2.8)
where
cf. [26, Lemma 6.4 and its proof]. As studied in [26] and [3] for a Zm -space pair (X, A), let H∗ (X, A)±Zm = [ξ ] ∈ H∗ (X, A) T∗ [ξ ] = ±[ξ ] ,
(2.9)
where T is a generator of the Zm -action. The following proposition was proved in [26] and [3]. Proposition 2.1. (Cf. Satz 6.6 and 6.11 of [26] or Proposition 3.12 of [3].) Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). Then we have C q E, cm ≡ Hq Λ cm ∪ S 1 · cm /S 1 , Λ cm /S 1 +Zm = Hi(cm ) Uc−m ∪ cm , Uc−m ⊗ Hq−i(cm ) Nc−m ∪ cm , Nc−m . (i) If ν(cm ) = 0, then Q, m C q E, c = 0,
if i(cm ) − i(c) ∈ 2Z, and q = i(cm ), otherwise.
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(ii) If ν(cm ) > 0, then β(cm )Zm C q E, cm = Hq−i(cm ) Nc−m ∪ cm , Nc−m , m
where β(cm ) = (−1)i(c )−i(c) . (iii) For all q ∈ Z and m ∈ N, +Zm Hq Nc−m ∪ cm , Nc−m = Hq Nc−m ∪ cm /Zm , Nc−m /Zm . As in [3,7] and [22], we introduce the following Definition 2.2. Suppose c is a closed geodesic of multiplicity m(c) = m satisfying (Iso). If N is a local characteristic manifold at c, N − = N ∩ Λ(c) and j ∈ Z, we define kj (c) ≡ dim Hj N − ∪ {c}, N − , ±Zm kj±1 (c) ≡ dim Hj N − ∪ {c}, N − . Clearly the integers kj (c) and kj±1 (c) are equal to 0 when j < 0 or j > ν(c) and can only take values 0 or 1 when j = 0 or j = ν(c). Proposition 2.3. (See Lemma 3.10 of [3], Lemma 2.4 of [22].) Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). (i) For any m ∈ N and j ∈ Z, 0 kj±1 cm kj cm . (ii) For any m ∈ N, k0+1 cm = k0 cm ,
k0−1 cm = 0.
(iii) In particular, if cm is non-degenerate, i.e., ν(cm ) = 0, then k0+1 cm = k0 cm = 1,
k0−1 cm = 0.
Following the ideas of D. Gromoll and W. Meyer on the degenerate part of the critical module in Theorem 3 of [8], we have the following results. Proposition 2.4. (Cf. Theorem 3 of [8], Section 7.1 of [26] and Theorem 3.11 of [3].) Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). Suppose for some integer m = np 2 with n and p ∈ N the nullities satisfy ν cm = ν cn . Then the following holds for the degenerate part of the critical modules of E with coefficient Q:
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(i) For any integer j , Hj Nc−m ∪ cm , Nc−m ∼ = Hj Nc−n ∪ cn , Nc−n , kj c m = kj c n . (ii) For any integer j , ±Zm ±Zn ∼ Hj Nc−m ∪ cm , Nc−m , = Hj Nc−n ∪ cn , Nc−n kj±1 cm = kj±1 cn . Proposition 2.5. (Cf. Satz 6.13 of [26], Lemma 3.10 of [3], and Proposition 2.6 of [22].) Let c be a prime closed geodesic on a Finsler manifold (M, F ) satisfying (Iso). For any m ∈ N, we have (i) if k0 (cm ) = 1, then kj±1 (cm ) = 0 for 1 j ν(cm ),
+1 −1 ±1 m m m m (ii) if kν(c m ) (c ) = 1 or kν(cm ) (c ) = 1, then kj (c ) = 0 for 0 j ν(c ) − 1,
±1 m m (iii) if kj+1 (cm ) 1 or kj−1 (cm ) 1 for some 1 j ν(cm ) − 1, then kν(c m ) (c ) = 0 = k0 (c ), m m (iv) in particular, if ν(c ) 2, then only one of the numbers kj (c ) can be non-zero.
We need also the relative homological structure of the quotient space Λ ≡ ΛS 2 and the version of the Morse inequality, where Λ0 = Λ0 S 2 = {constant point curves 2 2 ∼ in S } = S . S 1 -equivariant
Theorem 2.6. (See H.-B. Rademacher [25, Theorem 2.4].) We have the Poincaré series P ΛS 2 , Λ0 S 2 (t) = t d−1
1 1 − t dn t d(n+1)−2 + =t +2 t 2k+1 , d 2 d(n+1)−2 1−t 1−t 1−t k1
where d = 2 and n = 1. Thus for q ∈ Z, we have ⎧ 1, if q = 1, 2 0 2 2 0 2 ⎨ bq = bq ΛS , Λ S = rank Hq ΛS , Λ S = 2, if q = 2k + 1, k ∈ N, ⎩ 0 otherwise. We need the following version of the Morse inequality. Theorem 2.7. (See Theorem 6.1 of [26].) Suppose that there exist only finitely many prime closed geodesics {cj }1j k on (M, F ), and 0 a < b ∞ are regular values of the energy functional E. For each q ∈ Z define Mq (Λb , Λa ) = 1j k a<E(cm )
bq (Λb , Λa ) = rank Hq (Λb , Λa ). Then q q (−1)q−j Mj Λb , Λa (−1)q−j bj Λb , Λa , j =0
j =0
Mq Λ b , Λ a b q Λ b , Λ a .
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In particular, if C q (E, cjm ) = 0 for 1 j k and a < E(cjm ) < b, then we have Hq Λb , Λa = 0. We will use the following theorem of N. Hingston for the S 2 case. Note that by Theorems 10.1.2 and 10.1.3 of [20], originally proved in [17], the inequalities on indices in [9] and [10] become equalities. Theorem 2.8. (Follows from Proposition 1 of [9] and the main theorem of [10].) Let c be a closed geodesic with length L on a Finsler 2-sphere S 2 = (S 2 , F ) such that as a critical orbit of the energy function E on ΛS 2 , every orbit S 1 · cm is isolated. Suppose one of the following two conditions holds: (i) k0 (c) > 0, i(cm ) = m(i(c) + 1) − 1 and ν(cm ) = ν(c) for all m ∈ N, or (ii) kν(c) (c) > 0, i(cm ) + ν(cm ) = m(i(c) + ν(c) − 1) + 1 and ν(cm ) = ν(c) for all m ∈ N. Then there exist infinitely many prime closed geodesics on (S 2 , F ). Remark 2.9. Note that the condition on k0 (c) or kν(c) (c) means that c is a local minimum or maximum in the local characteristic manifold Nc at c, respectively. 3. Classification of closed geodesics on S 2 and existence theorems Let c be a closed geodesic on a Finsler 2-sphere S 2 = (S 2 , F ). Denote the linearized Poincaré map of c by Pc : R2 → R2 . Note that the index iteration formulae in [19] of 2000 (cf. Chapter 8 of [20]) work for Morse indices of iterated closed geodesics (cf. [17], Chapter 12 of [20], also [1, 2,4]). Since every closed geodesic on a sphere is orientable, by Theorem 1.1 of [16] of C. Liu (cf. also [30]), the Morse index of a closed geodesic c on a 2-dimensional Finsler sphere coincides with the index of a corresponding symplectic path introduced by C. Conley, E. Zehnder, and Y. Long in 1984–1990 (cf. [6,18,23], as well as [20]). Therefore the iteration formulae of Morse indices of c must be one of the following nine cases in Section 8.1 of [20]. Case 1. Pc is conjugate to a matrix
1 b 01
for some b > 0.
In this case, by 1◦ of Theorem 8.1.4 of [20], we have i(c) = 2p − 1 for some p ∈ N, and i cm = 2mp − 1,
ν cm = 1,
∀m 1.
(3.1)
Case 2. Pc = I2 , the 2 × 2 identity matrix. In this case by 2◦ of Theorem 8.1.4 of [20], we have i(c) = 2p − 1 for some p ∈ N, and i cm = 2mp − 1, Case 3. Pc is conjugate to a matrix
1 −b 0 1
ν cm = 2,
for some b > 0.
∀m 1.
(3.2)
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In this case by 3◦ of Theorem 8.1.4 of [20], we have i(c) = 2p for some p ∈ N0 , and i cm = 2mp, Case 4. Pc is conjugate to a matrix
ν cm = 1,
−1 −b 0 −1
∀m 1.
(3.3)
for some b > 0.
In this case by 1◦ of Theorem 8.1.5 of [20], we have i(c) = 2p + 1 for some p ∈ N0 , and 1 + (−1)m i cm = m(2p + 1) − , 2
1 + (−1)m ν cm = , 2
∀m 1.
(3.4)
Case 5. Pc = −I2 . In this case by 2◦ of Theorem 8.1.5 of [20], we have i(c) = 2p + 1 for some p ∈ N0 , and 1 + (−1)m i cm = m(2p + 1) − , ν cm = 1 + (−1)m , 2 b Case 6. Pc is conjugate to a matrix −1 for some b > 0. 0 −1
∀m 1.
(3.5)
In this case by 3◦ of Theorem 8.1.5 of [20], we have i(c) = 2p + 1 for some p ∈ N0 , and i cm = m(2p + 1),
1 + (−1)m ν cm = , 2
∀m 1.
Case 7. Pc is rationally elliptic, i.e., it is conjugate to some rotation matrix R(θ ) = with some θ ∈ (0, π) ∪ (π, 2π) and θ/π ∈ Q.
(3.6) cos θ
− sin θ sin θ cos θ
In this case by Theorem 8.1.7 of [20], we have i(c) = 2p + 1 for some p ∈ N0 , and
i c
m
=
2mp + 2 2mp + 2
mθ 2π
+ 1,
ν(cm ) = 0,
if mθ = 0 mod 2π,
2π
− 1,
ν(cm ) = 2,
if mθ = 0 mod 2π.
mθ
Case 8. Pc is irrationally elliptic, i.e., it is conjugate to some rotation matrix R(θ ) = with some θ ∈ (0, π) ∪ (π, 2π) and θ/π ∈ / Q.
(3.7) cos θ
− sin θ sin θ cos θ
In this case by Theorem 8.1.7 of [20], we have i(c) = 2p + 1 for some p ∈ N0 , and
i c
m
mθ + 1, = 2mp + 2 2π
ν cm = 0,
Case 9. Pc is hyperbolic, i.e., it is conjugate to the matrix
0 0 1/b
b
∀m 1.
(3.8)
for some b > 0 or b < 0.
In this case, by Theorem 8.1.6 of [20], we have i(c) = p for some p ∈ N0 , and i cm = mp,
ν cm = 0,
∀m 1.
(3.9)
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4. A mean index equality on (S 2 , F ) An important abstract version of a mean index identity for closed geodesics on Finsler manifolds with only finitely many prime closed geodesics was established by H.-B. Rademacher in Theorem 7.9 of [26]. In the following Theorem 4.4, we derive precise dependence of coefficients in this mean index identity on closed geodesics. This dependence is crucial in our proof in Section 5 below. Definition 4.1. Let M = (M, F ) be a compact Finsler manifold of dimension n, c be a prime closed geodesic on M. For each m ∈ N, the critical type numbers of cm is defined by the following 2n − 1 tuple of integers via Definition 2.2 β β β K cm ≡ k0 cm , k1 cm , . . . , k2n−2 cm β(cm ) m β(cm ) m β(cm ) c , k1 c , . . . , kν(cm ) cm , 0, . . . , 0 , = k0 m
where β = β(cm ) = (−1)i(c )−i(c) . We call a prime closed geodesic c homologically invisible if K(cm ) = 0 for all m ∈ N, or homologically visible otherwise. Lemma 4.2. Let c be a prime closed geodesic on a Finsler 2-sphere (S 2 , F ). Then there exists a minimal integer N ∈ N such that ν cm+N = ν cm , i cm+N − i cm ∈ 2Z, K cm+N = K cm , ∀m ∈ N.
∀m ∈ N,
In particular according to the classification in Section 3, we have N = 1, N = 2,
if c is of Case 1, 2, 3, or 8, if c is of Case 4, 5, or 6,
N = P,
if c is of Case 7 with
N = 1, N = 2,
if c is of Case 9 with i(c) ∈ 2Z, if c is of Case 9 with i(c) ∈ 2Z + 1.
θ 2π
=
Q P,
(P , Q) = 1,
Proof. According to Section 3, the closed geodesic c belongs to one and only one of the 9 cases there. We apply Proposition 2.4 to each case, respectively. In Case 1, 2, 3, or 8, ν(cm ) = const for all m ∈ N, and i(cm ) − i(c) is always even, so we obtain N = 1 by Proposition 2.4. m In Case 4 or 5, i(cm ) − i(c) is always even, and we have ν(cm ) = 1+(−1) or ν(cm ) = 1 + 2 m (−1) , respectively. So we obtain N = 2. In Case 6, we have even, i cm − i(c) = (m − 1)(2p + 1) ∈ odd, Thus we have N = 2.
ν(cm ) = 0, ν(cm ) = 1,
if m is odd, if m is even.
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θ m In Case 7, denote by 2π =Q P with P , Q ∈ N and (P , Q) = 1. Since i(c ) − i(c) is always even, we obtain N = P . In Case 9, i(cm ) − i(c) = (m − 1)p. If p is odd, we have
even, if m is odd, i cm − i(c) = (m − 1)p ∈ odd, if m is even. Thus we obtain N = 2 in this case. If p is even, i(cm ) − i(c) is always even, so in this case we have N = 1. The proof is complete. 2 Because Morse indices are always non-negative, i.e., i(cm ) 0 for every prime closed ˆ 0. geodesic c and m ∈ N, we always have i(c) Lemma 4.3. Suppose that there are only finitely many prime closed geodesics on (S 2 , F ). If ˆ = 0 holds for a prime closed geodesic c, then this c must belong to Case 3 in Section 3 and i(c) must be homologically invisible. Proof. Denote all the prime closed geodesics on (S 2 , F ) by {cj }1j k . Let c be a prime closed ˆ = 0. Then c must belong to Case 3 or Case 9 by our classifigeodesic on (S 2 , F ) satisfying i(c) cation in Section 3, In either case, we have i cm = 0,
∀m ∈ N.
(4.1)
In Case 3, we have i(cm ) + ν(cm ) = 2mp + 1, m(i(c) + ν(c) − 1) + 1 = m(2p + 1 − 1) + 1 = 2mp + 1, and ν(cm ) = ν(c) = 1 for all m ∈ N. If k1 (c) = 0, we can apply Theorem 2.8 to get infinitely many prime closed geodesics, which contradicts to the assumption. Hence we must have k1 (c) = 0. If k0 (c) = 0, then by Proposition 2.1 this c is homologically invisible as claimed. If c is of Case 3 and k0 (c) = 1 further holds, then by Proposition 2.4 we obtain k0 cm = 1,
∀m ∈ N.
(4.2)
If c belongs to Case 9, each cm is a local minimum point of E by (3.9) and (4.1), which then yields (4.2). Thus we can handle these two cases together, i.e., there is a prime closed geodesic c ≡ ck on (S 2 , F ) satisfying (4.1) and (4.2). ˆ j ) > 0 for Without loss of generality, we suppose among all cj with 1 j k there hold i(c ˆ j ) = 0 for l + 1 j k for some l ∈ {1, . . . , k − 1}. Then by our argument in 1 j l and i(c the previous paragraph, we must have k1 (cj ) = 0,
∀l + 1 j k.
(4.3)
ˆ j ) > 0 and then limm→∞ i(cm ) = ∞ for 1 j l, we can choose an a ∈ R large Since i(c j enough such that i(cjm ) > 1 holds whenever E(cjm ) > a and 1 j l. In particular, by Proposition 2.1 this implies C 1 E, cjm = 0,
for 1 j l,
(4.4)
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whenever E(cjm ) > a. Note that by (4.1), (4.3) and Proposition 2.1, we also obtain (4.4) for all l + 1 j k and m ∈ N. Then by the last conclusion of Theorem 2.7, we obtain H1 (Λ, Λb ) = 0 for all b a. Considering the exact sequence for the triple (Λ, Λb , Λ0 ), we have ∂1∗ i0∗ → H0 Λb , Λ0 −− → H0 Λ, Λ0 = 0, 0 = H1 Λ, Λb −− where the right-hand side module vanishes by Theorem 2.6. Hence we obtain H0 Λb , Λ0 = 0,
∀b a.
Now we choose an m ∈ N great enough such that κ ≡ E(cm ) > a for c = ck . Then because the total number of prime closed geodesics is finite, we can choose two regular values b1 and b2 of E such that a < b1 < κ = E(cm ) < b2 and κ is the only critical value of E in [b1 , b2 ]. Then the following exact sequence for the triple (Λb2 , Λb1 , Λ0 ), 0 = H0 Λb2 , Λ0 → H0 Λb2 , Λb1 → H−1 Λb1 , Λ0 = 0, yields H0 Λb2 , Λb1 = 0.
(4.5)
On the other hand, by an S 1 -equivariant version of Theorem 1.4.2 on p. 35 of [5], we obtain k q C 0 E, cj , H0 Λb2 , Λb1 = j =1 E(cq )=κ j
and the right-hand side of above has at least one term C 0 (E, cm ) = Q. This yields a contradiction to (4.5) and proves the lemma. 2 Theorem 4.4. Suppose that there exist only finitely many prime closed geodesics {cj }1j p on (S 2 , F ). Denote the homologically visible ones by cj with 1 j k for some k ∈ {0, 1, . . . , p}. ˆ j ) > 0 holds for 1 j k by Lemma 4.3. Then the following equality holds Note that i(c
β (−1)i(cj )+l β cjm kl cjm
1j k 1mNj , 0l2
1 = −1, ˆ j) Nj i(c
β
(4.6)
where Nj = N(cj ) ∈ N is the number defined in Lemma 4.2 for cj , kl (cjm ) are the critical type i(cjm )−i(cj )
numbers of cjm , and β(cjm ) = (−1)
.
Proof. Because the proof is similar to that of Theorem 5.4 in [22], we give a sketch here.
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Because dim C q (E, cjm ) can be non-zero only for q = i(cjm ) + l with 0 l 2 by Proposi tion 2.1, the formal Morse series M(t) = q0, m1, 1j k dim C q (E, cjm )t q becomes
M(t) =
m β kl cjm t i(cj )+l =
1j k, 0l2 m1
sNj +m β )+l kl cjm t i(cj ,
1j k, 0l2 1mNj , s0
h where the last equality follows from Lemma 4.2. Denote M(t) by M(t) = ∞ h=0 wh t . Then we have sN +m β # wh = +l =h . (4.7) kl cjm s ∈ N0 i cj j 1j k, 0l2 1mNj
Claim 1. {wh }h0 is bounded. In fact, we have #
sN +m s ∈ N 0 i cj j +l =h sN +m sN +m ˆ j ) 1 = # s ∈ N 0 i cj j + l = h, i cj j − (sNj + m)i(c ˆ j ) 1 # s ∈ N0 h − l − (sNj + m)i(c h − l − 1 − mi(c ˆ j) ˆ j) h − l + 1 − mi(c = # s ∈ N0 s ˆ j) ˆ j) Nj i(c Nj i(c
2 + 2, ˆ Nj i(cj )
ˆ 2 − 1 = 1 (cf. Theorem 1.4 of where the first equality follows from the fact |i(cm ) − mi(c)| [25]). Hence Claim 1 holds. Next we estimate M n (−1). Using (4.7) we obtain M (−1) ≡ n
n
wh (−1)h
h=0
=
sN +m β + l n . (4.8) (−1)i(cj )+l β cjm kl cjm # s ∈ N0 i cj j
1j k, 0l2 1mNj
The equality holds because sNj +m
(−1)i(cj
)+l
m m = (−1)i(cj )+l = (−1)i(cj )+l (−1)i(cj )−i(cj ) = (−1)i(cj )+l β cjm ,
in which the first equality follows from Lemma 4.2.
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Claim 2. There is a real constant C > 0 such that n M (−1) −
i(cj )+l
(−1)
β β cjm kl cjm
1j k, 0l2 1mNj
n C. ˆ j) Nj i(c
(4.9)
In fact, we have #
sN +m s ∈ N 0 i cj j +l n sN +m sN +m ˆ j ) 1 = # s ∈ N 0 i cj j + l n, i cj j − (sNj + m)i(c ˆ j) n − l + 1 # s ∈ N0 0 (sNj + m)i(c ˆ j) n − l + 1 − mi(c = # s ∈ N0 0 s ˆ j) Nj i(c
n−l+1 + 2. ˆ j) Nj i(c
On the other hand, we have #
sN +m s ∈ N 0 i cj j +l n sN +m ˆ j) + 1 n − l # s ∈ N 0 i cj j (sNj + m)i(c ˆ j) n − l − 1 − mi(c # s ∈ N0 0 s ˆ j) Nj i(c
n−l−1 − 2. ˆ j) Nj i(c
Combining these two estimates together, we obtain (4.9). As in p. 198 of [22], note that by Satz 7.8 of [26] we have B(2, 1) = −1, where B(d, m) = m(m+1)d . Therefore using the above two claims and the fact that − 2d(m+1)−4 lim
n→∞
1 n 1 M (−1) = lim P n (−1) = B(2, 1) = −1, n→∞ n n
by the same proof of Theorem 5.4 of [22] we obtain (4.6).
2
Remark 4.5. If c is a hyperbolic prime closed geodesic, then we have
β (−1)i(cj )+l β cjm kl cjm
1mNj , 0l2
γcj 1 = , ˆ j ) i(c ˆ j) Nj i(c
(4.10)
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where γcj =
− 12 , if i(cj ) is odd, 1, if i(cj ) is even.
In particular, (4.10) coincides with the formula obtained in [25]. Lemma 4.6. Suppose that there exists a closed geodesic d on (S 2 , F ) satisfying (Iso) such that in the classification of Section 3, d belongs to Case 1 with k0 (d) = 1, Case 2 with k0 (d)+k2 (d) = 1, or Case 3 with k1 (d) = 1. Then there exist infinitely many prime closed geodesics on (S 2 , F ). Proof. When d belongs to Case 1, we have i(d) = 2p − 1, i(d m ) = 2mp − 1 = m(2p − 1 + 1) − 1 = m(i(d) + 1) − 1, and ν(d m ) = ν(d) = 1 for all m ∈ N. Thus if k0 (d) = 1, we can apply Theorem 2.8 to obtain infinitely many prime closed geodesics. When d belongs to Case 2, we have i(d) = 2p − 1, i(d m ) = 2mp − 1, and ν(d m ) = 2 for all m ∈ N. Thus i(d m ) + ν(d m ) = 2mp − 1 + 2 = 2mp + 1 = m(2p − 1 + 2 − 1) + 1 = m(i(d) + ν(d) − 1) + 1, i(d m ) = 2mp − 1 = m(2p − 1 + 1) − 1 = m(i(d) + 1) − 1, and ν(d m ) = ν(d) = 2 hold for all m ∈ N. Hence d satisfies the relations of indices and nullities in Theorem 2.8. Thus if k0 (d) = 1 or k2 (d) = 1, Theorem 2.8 implies the existence of infinitely many prime closed geodesics. When d belongs to Case 3, we have i(d) = 2p, i(d m ) + ν(d m ) = 2mp + 1 = m(2p + 1 − 1) + 1 = m(i(d) + ν(d) − 1) + 1, and ν(d m ) = ν(d) = 1 for all m ∈ N. If k1 (d) = 0, we can apply Theorem 2.8 to get infinitely many prime closed geodesics. 2 5. Proof of the main Theorem 1.2 Lemma 5.1. Suppose that there are only finitely many prime closed geodesics on (S 2 , F ). If there exists one irrationally elliptic closed geodesic on (S 2 , F ), then there must be anther irrationally elliptic closed geodesic on (S 2 , F ). Proof. Suppose that there exist only finitely many prime closed geodesics {cj }1j p on (S 2 , F ). Denote the homologically visible ones by cj with 1 j k for some k ∈ {1, . . . , p}. By assumption, we can assume c1 is irrationally elliptic. Then by Theorem 4.4, the following identity holds −1 ˆ 1) i(c
+
β (−1)i(cj )+l β cjm kl cjm
2j k 1mNj , 0l2
1 = −1, ˆ j) Nj i(c
(5.1)
β
where Nj = N(cj ) ∈ N is the number defined in Lemma 4.2 for cj , kl (cjm ) ∈ N0 are the i(cjm )−i(cj )
critical type numbers of cjm and β(cjm ) = (−1) . In fact, by Lemma 4.2 we have N1 = N(c1 ) = 1. By (3.8) we have i(c1 ) is odd and c1 is non-degenerate. Hence by Propoβ(c ) sition 2.1 and Definition 2.2 we have kl 1 (c1 ) = kl+1 (c1 ) = δl0 for l ∈ N0 , where we have j j used the fact that β(c1 ) = (−1)i(c1 )−i(c1 ) = 1. Here δi = 1 when i = j and δi = 0 when i = j . Thus the first term in the left-hand side of (5.1) follows from (4.6). By (3.8) we have ˆ j ) for some ˆ 1 ) = 2p + θ ∈ R \ Q for some p ∈ N0 . Then (5.1) implies that at least one i(c i(c π 2 j k must be irrational, which must be irrationally elliptic by the classification of closed geodesics in Section 3. 2
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Lemma 5.2. Suppose that there are only finitely many prime closed geodesics on (S 2 , F ). Let c be a closed geodesic c of Case 3 in Section 3 and be homologically visible. Then homologically c can be viewed as a hyperbolic closed geodesic with an even initial index in Case 9 of Section 3. Let c be a closed geodesic of Case 6 in Section 3, then homologically c can be viewed as a hyperbolic closed geodesic with an odd initial index in Case 9 of Section 3. Proof. In Case 3, since c is homologically visible, i.e., k0 (c) + k1 (c) = 1 must hold. If k1 (c) = 1, then we apply Lemma 4.6 to get infinitely many prime closed geodesics, which contradicts to the finiteness assumption. Hence there must hold k0 (c) = 1.
(5.2)
By (3.3) and Lemma 4.3, we have i(cm ) = mi(c), i(c) ∈ 2N, and β(cm ) = 1 for all m ∈ N. Thus q C q (E, cm ) = δi(cm ) Q by Proposition 2.1 and (5.2). Therefore homologically c can be viewed as a hyperbolic closed geodesic with an even initial index in Case 9 of Section 3 by Proposition 2.1. In Case 6, let d = c2 . Then d belongs to Case 3, and we have kl (d) = kl (c2 ), for l ∈ {0, 1}. If k1 (d) = 1, then we can apply Lemma 4.6 to get infinitely many prime closed geodesics, which contradicts to the finiteness assumption. Hence k1 (d) = 0, and then by Lemma 2.3 we obtain k1−1 c12 k1 c12 = 0
and k0−1 c12 = 0.
(5.3)
Now by (3.6) we have i(c) ∈ 2N − 1, i(cm ) = mi(c) and then β(cm ) = (−1)m−1 for all m ∈ N. Thus for m ∈ N and q ∈ N0 Proposition 2.1 and (5.3) yield C q E, c2m−1 = Q if q = i c2m−1 , C q E, cm = 0 otherwise. Therefore homologically c can be viewed as a hyperbolic closed geodesic with an odd initial index in Case 9 of Section 3 by Proposition 2.1. 2 Lemma 5.3. Suppose a closed geodesic c belongs to Case 4 or 5 in Section 3, then homologically c can be viewed as a rational elliptic closed geodesic with rotation angle π in Case 7 of Section 3. Suppose a closed geodesic c belongs to Case 1 in Section 3, then homologically c can be viewed as a closed geodesic of Case 2. Proof. In Case 4 or 5, we have i(c) = 2p + 1 for some p ∈ N0 . Then by (3.4) and (3.5), the following holds 1 + (−1)m i cm = m(2p + 1) − 2 m 2mp + 2 2 + 1, ν(cm ) = 0, = 2mp + 2 m2 − 1, ν(cm ) = j ,
if m = 0 mod 2, if m = 0 mod 2,
where j = 1 or 2 when c belongs to Case 4 or 5, respectively. When c belongs to Case 4, by β(c2m )
Definition 4.1 of critical type numbers, K(c2m ) = (k0
β(c2m )
(c2m ), k1
(c2m ), 0). Hence c can
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Y. Long, W. Wang / Journal of Functional Analysis 255 (2008) 620–641 β(c2m )
be viewed homologically as one in Case 5 with critical type number k2 (c2m ) = 0. Thus Pc plays the same role as a rotation matrix with rotation angle π . A similar argument yields the latter part of the lemma. 2 In the following for the notation introduced in Section 2 we use specially Mj = Mj (ΛS 2 , Λ0 S 2 ) and bj = bj (ΛS 2 , Λ0 S 2 ) for j = 0, 1, 2, . . . . Proof of Theorem 1.2. By [24], there exists at least a closed geodesic on (S 2 , F ). Suppose that there are only finitely many prime closed geodesics on (S 2 , F ). By Lemma 5.1, in order to prove there exist at least two irrationally elliptic closed geodesics on (S 2 , F ), it suffices to prove that there must be one. We prove this by contradiction. In the following, we suppose that there exist only finitely many prime closed geodesics on (S 2 , F ) and none of them is irrationally elliptic. We denote the prime closed geodesics on (S 2 , F ) by {cj }1j p . We classify these geodesics into 5 classes: (i) c belongs to Case 1 or 2 in Section 3 and c is homologically visible. (ii) c belongs to Case 4, 5 or 7 in Section 3. (iii) c belongs to Case 3 in Section 3 and c is homologically visible, or c belongs to Case 9 in Section 3 with an even initial index. (iv) c belongs to Case 6 in Section 3, or c belongs to Case 9 in Section 3 with an odd initial index. (v) c is homologically invisible. By Lemmas 5.2 and 5.3, we see that closed geodesics in the same class have the same homological property. We assume in the following that {cj }ni−1 <j ni belong to class i, for 1 i 5 with n0 = 0 and n5 = p. For 0 = n0 < j n1 , we have (5.4) i cjm = 2mpj − 1, ∀m ∈ N, for some pj ∈ N by (3.1) and (3.2). For n1 < j n2 , we have ⎧ m(pj Nj +Lj ) + 1, m ⎨ 2 Nj i cj = m(pj Nj +Lj ) ⎩2 − 1, Nj L
ν(cjm ) = 0,
if m = 0 mod Nj ,
ν(cjm ) = 2,
if m = 0 mod Nj ,
(5.5)
θ
for some pj ∈ N0 and Njj = 2πj ∈ (0, 1) with (Lj , Nj ) = 1 by (3.7), (3.4) and (3.5). Here we view a closed geodesic c in Case 4 as one in Case 5 by Lemma 5.3. For n2 < j n3 , we have (5.6) i cjm = 2mpj , ∀m ∈ N, ˆ j ) > 0. for some pj ∈ N by (3.9) and (3.3). Note that by Lemma 4.3 we obtain 2pj = i(c For n3 < j n4 , we have i cjm = mpj , ∀m ∈ N, for some pj ∈ 2N − 1 by (3.9) and (3.6).
(5.7)
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637
β
We have β(cjm ) = 1 for all m ∈ N and 1 j n2 by (5.4) and (5.5). kl (cjm ) = δl0 for l ∈ N0 , 1 m Nj − 1 and n1 < j n2 by Proposition 2.1. Then from Lemma 4.2, Theorem 4.4, Remark 4.5, and (iv) of Proposition 2.5, we obtain β
(−1)i(cj )+lj klj (cj1 )
1j n1
ˆ j) i(c
1
n2 <j n3
ˆ j) i(c
+
N
−(Nj − 1) + (−1)i(cj )+lj klj (cj j )
n1 <j n2
ˆ j) Nj i(c
+
1
n3 <j n4
ˆ j) 2i(c
−
β
= −1,
(5.8)
ˆ j ) = 2pj for 1 j n1 , i(c ˆ j ) = 2pj + j for n1 < j n2 , for some lj ∈ {0, 1, 2}. Plugging i(c Nj ˆ j ) = pj for n3 < j n4 into (5.8) yields ˆ j ) = 2pj for n2 < j n3 and i(c i(c 2L
β
(−1)i(cj )+lj klj (cj1 ) 2pj
1j n1
+
n2 <j n3
+
β
2(pj Nj + Lj )
n1 <j n2
1 − 2pj
n3 <j n4
N
−(Nj − 1) + (−1)i(cj )+lj klj (cj j )
1 = −1. 2pj
(5.9)
N
Let Nj = 1 for 1 j n1 . Let dj = cj j for 1 j n2 . Then dj belongs to Case 1 or Case 2 of Section 3. Now if 1 = k0 (cj j ) = k0+1 (dj ) k0 (dj ) or 1 = k2 (cj j ) = k2+1 (dj ) k2 (dj ) holds for some 1 j n2 , we can apply Lemma 4.6 to obtain infinitely many prime closed geodesics, which contradicts to the finiteness assumption. Hence we have N
β
β
N N k0+1 cj j = k2+1 cj j = 0,
N
for 1 j n2 .
(5.10)
Thus we have lj = 1 for 1 j n2 . Let N kj = k1+1 cj j ,
for 1 j n2 .
(5.11)
Then (5.9) becomes 1j n1
kj + 2pj
n1 <j n2
−(Nj − 1) + kj + 2(pj Nj + Lj )
n2 <j n3
1 − 2pj
n3 <j n4
1 = −1. 2pj
(5.12)
Let T=
1j n1
pj
n1 <j n2
Multiplying 2T to both sides of (5.12) yields:
(pj Nj + Lj )
n2 <j n4
pj .
(5.13)
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1j n1
T kj + pj
n1 <j n2
T (kj − (Nj − 1)) + (pj Nj + Lj )
n2 <j n3
T − pj
n3 <j n4
T = −2T . pj
(5.14)
We have pT i cj j = 2T − 1,
T Nj pj Nj +Lj
i cj
= 2T − 1,
1 j n1 ,
(5.15)
n1 < j n2 ,
(5.16)
pT i cj j = 2T ,
n2 < j n3 ,
(5.17)
p2T i cj j = 2T ,
n3 < j n4 ,
(5.18)
by (5.4)–(5.7). By Proposition 2.1, we have q C q E, cjm = δi(cm )+1 Qkj , j
C q E, cjm = 0,
q ∈ N0 , 1 m
0 q 2T , m
T , pj
T + 1, pj
(5.19) (5.20)
for 1 j n1 . (5.19) holds by (5.4), (5.10) and (5.11). (5.20) holds because i(cjm ) 2T + 1 by (5.4) and (5.15) when m > pTj . We have q C q E, cjm = δi(cm ) Q, j
q ∈ N0 , 1 m
q C q E, cjm = δi(cm )+1 Qkj , j
C q E, cjm = 0,
T Nj , m = 0 mod Nj , pj N j + L j
q ∈ N0 , 1 m
0 q 2T , m
T Nj , m = 0 mod Nj , pj N j + Lj
T Nj + 1, pj N j + Lj
(5.21) (5.22) (5.23)
for n1 < j n2 . Here (5.21) holds because cjm are non-degenerate for m described there. (5.22) holds by (5.5), (5.10) and (5.11). (5.23) holds because i(cjm ) 2T + 1 by (5.5) and (5.16) when TN
j . m > pj Nj +L j We have
q C q E, cjm = δi(cm ) Q, j
C q E, cjm = 0,
q ∈ N0 , 1 m
0 q 2T , m
T , pj
T + 1, pj
(5.24) (5.25)
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for n2 < j n3 . Here (5.24) holds because cjm are non-degenerate for m described there and i(cjm ) − i(cj ) is always even. (5.25) holds because i(cjm ) 2T + 1 by (5.6) and (5.17) when m > pTj . We have q C q E, cj2m−1 = δ
i(cj2m−1 )
C q E, cjm = 0,
Q,
q ∈ N0 , 1 m
0 q 2T , m >
C q E, cj2m = 0,
T , pj
(5.26)
2T , pj
(5.27)
q ∈ N0 , m ∈ N,
(5.28)
for n3 < j n4 . (5.26) holds since cj2m−1 is non-degenerate. (5.27) holds since i(cjm ) > 2T by 2m (5.7) and (5.18) when m > 2T pj . (5.28) holds since i(cj ) − i(cj ) is odd. Because all i(cjm ) are odd for 1 j n2 , M1 + M3 + · · · + M2T −1 consists of the sum of ranks of non-trivial modules produced by cjm in (5.21) and (5.26), hence we have
M1 + M3 + · · · + M2T −1 =
n1 <j n2
T (Nj − 1) + (pj Nj + Lj )
n3 <j n4
T , pj
(5.29)
by (5.5) and (5.7). On the other hand, because all i(cjm )s are even for n2 < j n3 , M0 + M2 + · · · + M2T consists of the sum of ranks of non-trivial modules produced by cjm in (5.19), (5.22) and (5.24), hence we have M0 + M2 + · · · + M2T =
1j n1
T kj + pj
n1 <j n2
T kj + (pj Nj + Lj )
n2 <j n3
T , pj
(5.30)
by (5.4)–(5.6). Therefore together with Theorems 2.6 and 2.7, (5.14), (5.29) and (5.30), we obtain −2T = M2T − M2T −1 + · · · − M3 + M2 − M1 + M0 b2T − b2T −1 + · · · − b3 + b2 − b1 + b0 = −(2T − 1). This contradiction proves the theorem.
2
Corollary 5.4. Suppose that there are precisely two prime closed geodesics c+ and c− on a Finsler sphere (S 2 , F ). Then there exists a number α ∈ (0, 1) \ Q such that the linearized Poincaré maps of c± are rotation matrices R(θ± ) with θ± = 2π/(1 ± α), respectively. Therefore the Morse indices of their iterations and their mean indices coincide with those of the two closed geodesics in the example of A. Katok.
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Proof. By Theorems 1.2 and 4.4, the two closed geodesics c+ and c− must have irrational mean indices which satisfy 1 1 + = 1. ˆi(c+ ) i(c ˆ −)
(5.31)
ˆ − ) > 2. Thus there exists an irrational number α ∈ Without loss of generality, we suppose i(c ˆ − ) = 2/(1 − α). By (5.31) this implies i(c ˆ + ) = 2/(1 + α). Therefore we (0, 1) such that i(c obtain the conclusion on the linearized Poincaré maps, which further imply the properties of Morse indices. 2 Acknowledgments Y. Long would like to express his sincere thanks to Professor Victor Bangert for many helpful discussions with him on closed geodesics. The authors thank the referee for his/her careful reading of the manuscript and valuable comments. References [1] W. Ballmann, G. Thorbergsson, W. Ziller, Closed geodesics on positively curved manifolds, Ann. of Math. 116 (1982) 213–247. [2] W. Ballmann, G. Thorbergsson, W. Ziller, Existence of closed geodesics on positively curved manifolds, J. Differential Geom. 18 (1983) 221–252. [3] V. Bangert, Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, preprint, 2005 arXiv: 0709.1243v1 [math.SG], 9 Sep. 2007. [4] R. Bott, On the iteration of closed geodesics and Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956) 176–206. [5] K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. [6] C. Conley, E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984) 207–253. [7] D. Gromoll, W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969) 361–369. [8] D. Gromoll, W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Differential Geom. 3 (1969) 493– 510. [9] N. Hingston, On the growth of the number of closed geodesics on the two-sphere, Int. Math. Res. Not. 9 (1993) 253–262. [10] N. Hingston, On the length of closed geodesics on a two-sphere, Proc. Amer. Math. Soc. 125 (1997) 3099–3106. [11] H. Hofer, K. Wysocki, E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. 148 (1998) 197–289. [12] H. Hofer, K. Wysocki, E. Zehnder, Finite energy foliations of tight three spheres and Hamiltonian dynamics, Ann. of Math. (2) 157 (2003) 125–255. [13] A.B. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR 37 (1973) (in Russian); Math. USSR-Izv. 7 (1973) 535–571. [14] W. Klingenberg, Closed geodesics, Ann. of Math. 89 (1969) 68–91. [15] W. Klingenberg, Lectures on Closed Geodesics, Springer, Berlin, 1978. [16] C. Liu, The relation of the Morse index of closed geodesics with the Maslov-type index of symplectic paths, Acta Math. Sin. (Engl. Ser.) 21 (2005) 237–248. [17] C. Liu, Y. Long, Iterated index formulae for closed geodesics with applications, Sci. China 45 (2002) 9–28. [18] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China (Scientia Sinica) Ser. A 7 (1990) 673–682 (Chinese ed.); Sci. China (Scientia Sinica) Ser. A 33 (1990) 1409–1419 (English ed.). [19] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math. 154 (2000) 76–131.
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[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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Y. Long, Index Theory for Symplectic Paths with Applications, Progr. Math., vol. 207, Birkhäuser, Basel, 2002. Y. Long, Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc. 8 (2006) 341–353. Y. Long, W. Wang, Multiple closed geodesics on Riemannian 3-spheres, Calc. Var. 30 (2007) 183–214. Y. Long, E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems, in: S. Albeverio, et al. (Eds.), Stoc. Proc. Phys. and Geom., World Sci., 1990, pp. 528–563. L.A. Lyusternik, A.I. Fet, Variational problems on closed manifolds, Dokl. Akad. Nauk SSSR (N.S.) 81 (1951) 17–18 (in Russian). H.-B. Rademacher, On the average indices of closed geodesics, J. Differential Geom. 29 (1989) 65–83. H.-B. Rademacher, Morse Theorie und geschlossene Geodatische, Bonner Math. Schriften, vol. 229, 1992. H.-B. Rademacher, Existence of closed geodesics on positively curved Finsler manifolds, preprint, 2005. Z. Shen, Lectures on Finsler Geometry, World Sci., Singapore, 2001. W. Wang, X. Hu, Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J. 139 (2007) 411–462. B. Wilking, Index parity of closed geodesics and rigidity of Hopf fibritions, Invent. Math. 144 (2001) 281–295. W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1982) 135–157.
Journal of Functional Analysis 255 (2008) 642–656 www.elsevier.com/locate/jfa
Operator norm localization property of relative hyperbolic group and graph of groups ✩ Xiaoman Chen, Xianjin Wang ∗ Department of Mathematics, Fudan University, Shanghai 200433, China Received 3 October 2007; accepted 23 April 2008 Available online 3 June 2008 Communicated by Alain Connes
Abstract In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H1 , . . . , Hn } has operator norm localization property if and only if each Hi , i = 1, 2, . . . , n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP . If GP has operator norm localization property for all vertices P then π has operator norm localization property. © 2008 Elsevier Inc. All rights reserved. Keywords: Operator norm localization property; Coarse invariant; Roe algebras; Finite propagation; Strongly relative hyperbolic group; Graph of groups
0. Introduction Let X be a discrete, bounded geometry metric space. Associated to X there is a C ∗ -algebra Usually it is called Roe algebra. The coarse Baum–Connes conjecture states that the coarse assembly map C ∗ (X).
μ : KX∗ (X) → K∗ C ∗ (X) ✩
The authors are supported in part by NSFC, FANEDD (No. 200416) and the Ministry of Education, PR China.
* Corresponding author.
E-mail addresses: [email protected] (X. Chen), [email protected] (X. Wang). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.022
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is an isomorphism. It plays a very important role in topology and geometry. The coarse Baum– Connes conjecture has been proved in a number of cases. Most notably, Yu [13] has shown that the coarse Baum–Connes conjecture holds in case of X which is uniformly embedded in Hilbert space. Using Gromov’s expander graph structure Higson [7] gave a counterexample to the coarse Baum–Connes conjecture. The relevant construction is the box space X(Γ ) of an infinite group Γ with property T, residually finite and linear type, that is the coarse disjoint union of the quotient groups Γ /Γn . Recently Gong, Wang and Yu [6] have established relations between the Coarse Geometric Novikov Conjecture for the box space X(Γ ) and the Strong Novikov Conjecture for an infinite group Γ with property T, residually finite. Since the Strong Novikov Conjecture holds for many infinite groups with property T, this implies the Coarse Geometric Novikov Conjecture for a large class of sequences of expanders. In Higson’s original construction [8] and in Gong, Wang and Yu’ construction [6] there is an algebraic lifting principle, that is, an operator T ∈ ∗ (X(Γ )) will restrict to an operator on C ∗ (Γ /Γ ) for all but finitely many n, and such an Calg n alg operator can then be lifted to a Γn -invariant element of Roe algebra of Γ . In general such lifting can be extended to the maximal norm closure [6]. Using some kind of localization estimation of operator norm in case of asymptotic finite dimension, Higson proved that the lifting can also be extended to the reduced norm closure. This was important in his original construction of counterexample to the Coarse Baum–Connes Conjecture. The natural question is what kind of coarse geometric conditions will be needed to guarantee the algebraic level lifting to extend to the reduced norm level. Our focus in this paper is on operator norm localization property, which generalizes the local estimation property in case of asymptotic finite dimension metric space. Guoliang Yu introduced this definition to us when he visited Fudan University in 2005. 1. Preliminaries In this section, we introduce the definition and basic properties of operator norm localization property for a discrete metric space. Let X be a proper metric space, i.e., every close ball in the metric space is compact. Definition 1.1. (See Roe [10].) Let X be a discrete metric space, and H be a separable and infinite-dimensional Hilbert space. A bounded operator T : l 2 (X) ⊗ H → l 2 (X) ⊗ H , is said to have propagation at most r if for all ϕ, ψ ∈ l 2 (X) ⊗ H with d(Supp(ϕ), Supp(ψ)) > r such that T ϕ, ψ = 0. Note that if X is discrete, then we can write l 2 (X) ⊗ H =
(δx ⊗ H ), x∈X
where δx is the Dirac function at x. Every bounded operator acting on l 2 (X) ⊗ H has a corresponding matrix representation T = (Tx,y )x,y∈X , where Tx,y : δy ⊗ H → δx ⊗ H is a bounded operator. We call T is locally compact if Tx,y is a compact operator for all x, y in X. For T to have propagation r, it is equivalent to saying that
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the matrix coefficient Tx,y of T vanishes when d(x, y) > r. The space of operators acting on l 2 (X) ⊗ H with propagation at most r will be denoted by Ar (X). Let T denote the operator norm of a bounded linear operator T . Definition 1.2. The collection of all locally compact, finite propagation operators on l 2 ⊗ H is a ∗-subalgebra of B(l 2 (X) ⊗ H ). Its norm-completion, denoted by C ∗ (X), is the Roe algebra of X. Definition 1.3. Let X be a discrete metric space. Let f : N → N be a (non-decreasing) function. We say that X has operator norm localization property relative to f with constant c 1 if, for all k ∈ N, and every T ∈ Ak (X), there exists nonzero ϕ ∈ l 2 (X) ⊗ H satisfying (1) Diam(Supp(ϕ)) f (k), (2) T ϕ cT ϕ. The infimum over all possible c is called the operator localization number of X. Remark 1.4. The operator norm localization property is called strong property L in [3]. A discrete metric space X has bounded geometry, if for every R > 0, there is a uniform bound on the number of elements in the ball of radius R in X. It is not necessary to assume the metric space to be with bounded geometry in the above definition. But our interest is in the bounded geometric case. Remark 1.5. Recall that a Borel measure on a metric space is said to be locally finite if every bounded Borel subset has finite measure. In our joint work with Tessera and Yu [4], the definition is defined on a metric space with a positive Borel measure ν and we have showed that on locally compact metric space the operator norm localization is independent of the choice Borel measure, i.e., a locally compact metric space X has operator norm localization property if (X, ν0 ) has operator norm localization property for some positive locally finite Borel measure ν0 such that there exists r0 > 0 for which every closed ball with radius r0 has positive measure. Therefore in case that X is locally compact, these definitions are the same, since we can take the counting measure on X. Definition 1.6. A Borel map f from a proper metric space X to another metric space Y is called coarse if (1) f is proper, i.e., the inverse image of any bounded set is bounded, (2) for every R > 0, there exists R > 0 such that d(f (x), f (y)) R for all x, y ∈ X satisfying d(x, y) R. Definition 1.7. We say that the proper metric spaces X and Y are coarsely equivalent if there exist r > 0 and coarse maps ϕ : X → Y , ψ : Y → X such that dY (ϕ ◦ ψ(y), y) r and dX (ψ ◦ ϕ(x), x) r.
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In [3], we have proved that the operator norm localization property is coarsely invariant, i.e., if X and Y are coarsely equivalent metric spaces, then X has operator norm localization property with constant c if and only if Y has operator norm localization property with constant c. Definition 1.8. Let Γ be a countable discrete group. A length function on Γ is a non-negative real-valued function l satisfying, for all x an y in Γ (1) l(xy) l(x) + l(y), (2) l(x −1 ) = l(x), (3) l(x) = 0 if and only if x = 1. This defines a metric on Γ by dΓ (f, g) = lΓ (f −1 g). A length function l is proper if for all C > 0 the subset l −1 ([0, C]) is finite which induces a proper metric on Γ . If Γ is a finitely generated group and its generating set S is symmetric, i.e., S = S −1 , then the length lΓ (g) of an element g ∈ Γ is defined to be the length of a shortest word in S representing g. In this case, dΓ is left invariant in the sense that dΓ (hf, hg) = dΓ (f, g). Let Γ be a group acting on a metric space X. For every k 0, the k-stabilizer Wk (x0 ) of a point x0 ∈ X is defined to be the set of all g ∈ Γ with gx0 ∈ B(x0 , k), where B(x0 , k) is the closed ball with center x0 and radius k. The concept of k-stabilizer is introduced by Bell and Dranishnikov in their work on permanence properties of asymptotic dimension [1]. In [3] we have proved the following useful proposition, it will be used in the proof of the main theorem of the paper. Proposition 1.9. (See [3].) Let Γ be a finite generated group acting freely and isometrically on a metric space X (and X without assuming to have bounded geometry). If X has operator norm localization property with constant cX and there exists x0 ∈ X such that for each k > 0, Wk (x0 ) has operator norm localization property with constant cΓ , where cΓ is independent on k. Then Γ has operator norm localization property with constant c = cX cΓ . Remark 1.10. It is easy to see from the proof of Proposition 1.9 in [3] that when Γ has bounded geometry, the assumption that Γ is finite generated in Proposition 1.9 can be replaced by the following condition, the map π : Γ → X, γ → γ x0 , satisfying for every R > 0, there exists R > 0 such that d(π(g1 ), π(g2 )) R for all g1 , g2 ∈ Γ with d(g1 , g2 ) R. 2. Union theorem Let Γ1 and Γ2 be metric spaces. If both Γ1 and Γ2 have operator norm localization property, a nature question is whether Γ1 ∪ Γ2 has operator norm localization property. Its answer is yes and as an application we apply it to show the permanence properties of relative hyperbolic groups in next section. From [11] we know that if a discrete metric space has finite asymptotic dimension then it has operator norm localization property. By the proofs of Proposition 3.7 and Theorem 3.9 in [4], the following is an example of a group with infinite asymptotic dimension and operator norm localization property.
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Example 2.1. Let X = Z × 2Z × 3Z × 4Z × · · · , where Z is the space of all integers. Then X is an abelian group with infinite asymptotic dimension and still satisfies the above geometric property. So the category of operator norm localization property is properly larger than the category of finite asymptotic dimension. Definition 2.2. A family of metric spaces {Γα }α∈J is said to have operator norm localization property uniformly if there exist a common constant c 1 and a common (non-decreasing) function f : N → N such that, for each α ∈ J , Γα has operator norm localization property relative to f with constant c. Theorem 2.3. Let {Γα }α∈J be a family of metric spaces which has operator norm localization property uniformly with constant c. Let X = α∈J Γα . If X has bounded geometry and for all t > 0 there exists Yt ⊂ X such that {Γα \ Yt } is t-disjoint and Yt has operator norm localization property with constant c. Then X has operator norm localization with constant (1 + )c for any
> 0. Proof. Let d be the metric on X and metrize each Γα , α ∈ J , as a subset of X. Take any T ∈ Ar , 1 without loss of generality we assume T = 1, choose δ > 0 such that (1 − δ)2 − (1+ ) 2 > 0, then there is a ξ ∈ l 2 (X) ⊗ H with ξ = 1 such that T ξ 1 − δ. Let t = 10r such that {Γα \ Yt } is t-disjoint. Denote V0 = Yt , V1 = x ∈ X: d(x, Yt ) 3r \ Yt , V2 = x ∈ X: d(x, Yt ) 6r \ x ∈ X: d(x, Yt ) 3r , .. .
Vj +1 = x ∈ X: d(x, Yt ) 3(j + 1)r \ x ∈ X: d(x, Yt ) 3j r , .. . Choose n such that
1 n2
< [(1 − δ)2 −
Ui =
1 ]. (1+ )2
Let
Vj : j ≡ i mod (n) ,
i = 0, 1, 2, . . . , n − 1.
Then Ui ∩ Uj = ∅ if i = j , and there is an i0 such that T Ui0 ξ 1/n, where operators T Ui = U (Tx,yi ) ∈ B(l 2 (X) ⊗ H ), i = 1, 2, . . . , n, are defined by Ui Tx,y
Since (1 − δ)2
n
i=1 T
Ui ξ 2 ,
=
Tx,y , 0,
this implies that
if x ∈ Ui , otherwise.
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2 (X\U ) i0 T ξ (1 − δ)2 − T Ui0 ξ
n2 (1 − δ)2 − 1 , n2 where the operator T (X\Ui0 ) is defined similarly. Let
j
Aα = Γα ∩ (V(nj +i0 +1) ∪ · · · ∪ V(n(j +1)+i0 −1) ), A0 = V0 ∪ V1 ∪ · · · ∪ V(i0 −1)
j = 0, 1, 2, . . . ,
and let A = A0 ∪ Ajα : α ∈ J, j ∈ N ∪ {0} . Note that Yt has operator norm localization property with constant c, by assumption it is coarsely equivalent to A0 . Therefore A0 has operator norm localization property with constant c. Each j j Aα with α ∈ J, j ∈ N ∪ {0} is a subset of Γα , hence Aα is being of operator norm localization property with constant c. Note that every two sets in A are 3r-separated and X \ Ui0 is the union of A, which implies that 1
n
(X\U ) i0 T ξ
n2 (1 − δ)2 − 1 n n2 (1 − δ)2 − 1 n
T (X\Ui0 )
sup T A
n2 (1 − δ)2 − 1 A∈A nc sup T A ηA : ηA ∈ l 2 (X) ⊗ H, n2 (1 − δ)2 − 1 A∈A ηA = 1, diam Supp(ηA ) f (r) nc T η: diam Supp(η) f (r) 2 2 n (1 − δ) − 1 (1 + )c T η: diam Supp(η) f (r) .
That completes the proof.
2
From the above theorem we have Corollary 2.4. Let Γ = Γ1 ∪ Γ2 ∪ · · · ∪ Γn be a discrete metric space with bounded geometry. If Γ1 , Γ2 , . . . , Γn have operator norm localization property with constants cΓ1 , cΓ2 , . . . , cΓn , respectively. Then for any > 0, Γ has operator norm localization property with constant (1 + ) max{cΓ1 , cΓ2 , . . . , cΓ2 }.
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Proof. We prove the corollary only when n = 2, n > 2 is similar. Take the family {Γα } consisting of the sets Γ1 and Γ2 . For each t > 0, we put Yt = Γ1 , then {Γ1 , Γ2 } satisfies the conditions of Theorem 2.3. 2 3. Relative hyperbolic groups Let Γ be a finitely generated group which is hyperbolic relative to a finite family of subgroups {H1 , . . . , Hn }. We will prove that Γ has operator norm localization property if and only if each subgroup Hi has operator norm localization property. If A is a symmetric set of finite generators of Γ , we denote by dA the corresponding left invariant metric on Γ . If B is another such set with A ⊂ B, then the identity map p : (Γ, dA ) → (Γ, dB ) is equivariant and dB (p(x), p(y)) dA (x, y). Let S be a finite symmetric set generating Γ . Denote H=
(Hk − e).
k
Let dS and dS∪H be the left invariant metrics on Γ induced by S and S ∪ H , respectively. For n 1, denote B(n) = g ∈ Γ : dS∪H (g, e) n . In this section, we always view B(n) as a subspace of Γ equipped with the metric dS . The following useful recursive decomposition of B(n) is contained in the proof of theorem in [9]: B(1) = S ∪ B(n) =
k
B(n − 1)Hk ∪
k
Hk ,
B(n − 1)Hk =
(3.1)
B(n − 1)x ,
(3.2)
x∈S
(3.3)
gHk ,
g∈R(n−1)
where the final equality represents a partition of B(n − 1)Hk into disjoint cosets according to a fixed set R(n − 1) of coset representative, R(n − 1) ⊂ B(n − 1). Proposition 3.1. (See [9,5].) For every L > 0 there exists K(L) > 0 such that if Y = x ∈ Γ : dS x, B(n − 1) K(L) , then for each k B(n − 1)Hk ⊂ Y ∪
gHk \ Y
g∈R(n−1)
and the subspaces gHk \ Y , g ∈ R(n − 1), are L-separated with distance dS .
(3.4)
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Proposition 3.2. If each Hk has operator norm localization property with constant c then for any > 0, B(n) has operator norm localization property with constant (1 + )c. Proof. Fix > 0. We prove it by induction. It is obvious that B(1) has operator norm localization property with constant (1 + 1 )c for any 1 > 0. Assume that B(n − 1) has operator norm localization property with constant (1 + n−1 )c for any n−1 > 0. Especially we choose n−1 = 4 . Set 2 n
n and δn = 1+
. Since δn = 1+2
n n 1 + δn−1 (1 + n−1 ) = 1 + δn−1 + n−1 + δn−1
n−1 = 1 + 2 n−1 , by the infinite union Theorem 2.3 and decomposition (3.4), we obtain that B(n − 1)Hk has operator norm localization property with constant (1 + 2 n−1 ). Note that B(n) is the finite union of B(n − 1)Hk and B(n − 1)x. Since (1 + δn−1 )(1 + 2 n−1 ) = 1 + δn−1 + 2 n−1 + 2δn−1 n−1 = 1 + 4 n−1 = 1 + , by Corollary 2.4 we get that B(n) has operator norm localization property with constant (1 + )c, that complete the proof. 2 Corollary 3.3. If each Hk has operator norm localization property with constant c, then B(n) has operator norm localization with constant 2c. The following proposition is contained in [5], proved by Osin in [9]. Proposition 3.4. (See [9,5].) The metric space (Γ, dS∪H ) has finite asymptotic dimension. Proposition 3.5. The metric space (Γ, dS∪H ) has operator norm localization property. Proof. Since (Γ, dS∪H ) has finite asymptotic dimension (cf. [3]), it is easy to verify.
2
Theorem 3.6. Let Γ be a finite generated group which is hyperbolic relative to a finite family {H1 , . . . , Hn } of subgroups. Then Γ has operator norm localization property if and only if each subgroup Hi has operator norm localization property. Proof. If Γ has operator norm localization property, so are its subgroups. Therefore Hk , k = 1, . . . , n, have operator norm localization property. For the converse, assume that each subgroup Hk has operator norm localization property. Without loss of generality, we assume their constant to be c. Let X = (Γ, dS ) and Y = (Γ, dS∪H ). We choose x = e to be the unit in Γ and define π :X → Y γ → γ x,
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then Wn (x) = B(n) = π −1 (BY (e, n)) has operator norm localization property with constant 2c by Corollary 3.3. Note that Γ acts freely, isometrically and transitively on the metric space Y , the conclusion follows from Proposition 1.9. 2 4. Graph of groups In [4] we have proved that the operator norm localization property is preserved by free products, amalgamated free products and HNN extensions. In this section we consider the case of the fundamental group of graph of groups and generalize the results we have obtained in [4]. First of all we recall some basic constructions related to graph of groups which are well-known facts. Definition 4.1. (Cf. Serre [12].) A graph Γ consists of a set X = Vert Γ , a set Y = Edge Γ and two maps Y →X×X y → o(y), t (y) and Y →Y y → y¯ which satisfy the following condition: for each y ∈ Y , y¯¯ = y, y¯ = y and o(y) = t (y). ¯ An element p ∈ X is called a vertex of Γ ; an element y ∈ Y is called an (oriented) edge, and y¯ is called ¯ is called the the inverse edge. The vertex o(y) = t (y) ¯ is called the origin of y, and t (y) = o(y) terminus of y. These two vertices are called the extremities of y. It is possible for o(y) = t (y) and in this case y is termed a loop. We say that two vertices are adjacent if they are the extremities of some edge. An orientation of a graph of Γ is a subset Y+ of Y = Edge Γ such that Y is the disjoint union of Y+ and Y + . Definition 4.2. (Cf. Serre [12].) Let G be a finite generated group and let S be its (symmetric) generating set. We let Γ = Γ (G, S) denote the oriented graph having G as its set of vertices, G × S = (Edge Γ )+ as its orientation, with o(g, s) = g
and t (g, s) = gs
for each edge (g, s) ∈ G × S. The Γ (G, S) is called Cayley graph of G. The graph Γ can be regarded as a metric space if we endow it with a combinatorial metric. This means that the length of every edge of Γ is assumed to be equal to 1. Let Y be a non-empty connected graph. Each vertex P ∈ Vert Y and each edge y ∈ Edge Y associate a group GP and a group Gy = Gy¯ , respectively. There are two injective homomorphisms, φy : Gy → Gt (y) and φy¯ : Gy¯ → Go(y) . Define the group F (G, Y ) to be the group generated by the elements of the GP and the elements y ∈ Edge Y subject to the relations: y = y¯
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and yφy (a)y −1 = φy¯ (a) if y ∈ Edge Y and a ∈ Gy . Let c be a path in Y starting at some vertex P0 . Denote y1 , y2 , . . . , yn to be the edges associated to c, where t (yi ) = Pi . Then the length of path c is n, that is L(c) = n. It is obviously that o(c) = P0 , t (c) = Pn and o(yi+1 ) = t (yi ). Definition 4.3. (See [12].) A word of type c in F (G, Y ) is a pair (c, μ), where c is a path as above and μ = (r0 , r1 , . . . , rn ) is a sequence of elements ri ∈ GPi ,
i = 0, 1, . . . , n.
Definition 4.4. (See [12].) Let (c, μ) be a word of type c. Then we define |c, μ| = r0 y1 r1 y2 r2 · · · yn rn
∈ F (G, Y )
and say that |c, μ| is the element or word in F (G, Y ) associated with (c, μ). Notice that when n = 0, |c, μ| = r0 . Serre gives two equivalent definitions of the fundamental group of the graph of groups (G, Y ). In the first definition, let P0 denote a fixed vertex and GP0 the associated group. Define π = π1 (G, Y, P0 ) to be the set of elements of F (G, Y ) associated to a path c in Y with o(c) = t (c) = P0 . Obviously, π ⊂ F (G, Y ) is a subgroup. In the second definition, let T be a maximal subtree of Y , and define π = π1 (G, Y, T ) to be the quotient of F (G, Y ) of by the normal subgroup generated by the elements t ∈ Edge T . If gy denotes the image of y ∈ Edge Y , then the group π is the group generated by the groups GP and elements gy subject to the relations gy¯ = gy−1 , gy φy (a)gy−1 = φy¯ (a) and gt = e, where a ∈ Gy and t ∈ Edge T . So, in particular, φt (a) = φt¯(a) for all t ∈ Edge T . The equivalence of the definitions is proven in [12]. Example 4.5. (1) If Y is the graph with two vertices P , Q and one edge y, then π1 (G, Y, P ) = π1 (G, Y, Q) = GP ∗Gy GQ is the free product of GP and GQ amalgamated over Gy .
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(2) If Y is the graph with one vertex P and one edge y, then φy¯ (Gy ) is a subgroup of GP , and π1 (G, Y, P ) is precisely the HNN extension of GP over the subgroup φy¯ (Gy ) by means of φy φy−1 ¯ . Theorem 4.6. Let (G,Y) be a finite graph of groups with finitely generated vertex groups GP which have operator norm localization property. Then for any vertex P0 the fundamental group π1 (G, Y, P0 ) has operator norm localization property. To prove the theorem we need to construct a metric space X with operator norm localization property and on which π1 (G, Y, P0 ) acts freely and isometrically. Recall that a tree T is a connected non-empty graph without circuits, consists of two sets, a set Vert T of vertices and a set Edge T of edges, together with two endpoint maps Edge T → Vert T associating each edge to its endpoints. Every pair of two vertices is connected by a unique geodesic edge path, that is, a path without backtracking. A tree of spaces X (cf. [5]) consists of a family of metric spaces {Xv , Xe } indexed by the vertices v ∈ Vert T and edges e ∈ Edge T of T together with maps φe : Xe → Xt (e) , φe¯ : Xe → Xt (e) ¯ . The {φe , φe¯ }e∈Edge T are called the structural maps of X , and the metrics on the vertex and edge spaces are integer-valued. The total space X of the tree of spaces X is the metric space defined as follows. The underlying set of X is the disjoin union of the vertex spaces Xv ; the metric on X is the metric envelope d of the partial metric dˆ (cf. [5]) defined by ˆ y) = d(x,
dv (x, y), 1,
if ∃v ∈ Vert T such that x, y ∈ Xv , if ∃e ∈ Edge T and z ∈ Xe such that x = φe (z), y = φe¯ (z)
for all (x, y) in the domain (x, y): x, y ∈ Xv , v ∈ Vert T ∪ φe (z), φe¯ (z) : z ∈ Xe , e ∈ Edge T . We call (x, y) an adjacency in the total space X if there exists an edge e and an element z ∈ Xe , such that φe (z) = x and φe¯ (z) = y or φe (z) = y and φe¯ (z) = x. Observe that d|Xv = dv and d(x, y) = 1 if x and y are adjacent. For convenience, we give the explicit metric on the total space X which was proved in Proposition 5.5 in [5]. For all x ∈ Xv , y ∈ Xw then d(x, y) = 1 if (x, y) is an adjacency. Further, ˆ 0 , x1 ) + d(x ˆ 2 , x3 ) + · · · + d(x ˆ p−1 , xp ) , d(x, y) = dT (v, w) + inf d(x where dT is the distance on the tree T and the infimum is taken over all sequences x0 , . . . , xp such that (1) (2) (3) (4)
p = 2dT (v, w) + 1, x = x0 , y = xp , (x2k−1 , x2k ) is an adjacency for k = 1, . . . , dT (v, w), x2k , x2k+1 ∈ Xvk , for k = 0, . . . , dT (v, w),
where v = v0 , . . . , vdT (v,w) = w are the vertices along the unique geodesic path in T from v to w.
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Now we construct a Bass–Serre tree X˜ associated to group π . Let T be a maximal tree in the graph Y and πP denote the canonical image of GP in π , obtained via conjugation by the path c, where c is the unique path in T from the base point P0 to the vertex P . Similarly, let πy denote the image of φy (Gy ) in πt (y) . Then, we set
Vert X˜ =
π/πP
P ∈Vert Y
and Edge X˜ =
π/πy .
y∈Vert Y
From now on we will implicitly identify the edge groups and vertex groups with their images in the group F (G, Y ). An edge in X˜ connects the vertex xπo(y) to the vertex xyπt (y) for all y ∈ Edge Y and all x ∈ π . Observe that the stabilizer of the vertices are conjugates of the corresponding vertex groups, while the stabilizer of the edge connecting xπo(y) and xyπy is xyπy y −1 x −1 , a conjugate of the image of the edge group. This obviously stabilizes the second vertex, and it stabilizes the first vertex since yπy y −1 = πy¯ ⊂ πo(y) . It is known (see [12]) that the action of left multiplication on X˜ is isometric. Now we will assume that the graph Y is finite and that the groups associated to the edges and vertices are finitely generated with some fixed set of generators chosen for each group. We let S denote the disjoint union of the generating sets for the groups, and require that S = S −1 . Let lGP be the length function on GP associated to the generating set S. We endow each of the groups the word metric given by the length function lGP . We extend this metric to the group F (G, Y ) and hence to the subgroup π1 (G, Y, P0 ) in the nature way, by adjoining to S the collection {y, y −1 : y ∈ Edge Y }. Proof of Theorem 4.6. Let π = π1 (G, Y, P0 ) and assume that {GP }P ∈Vert Y has operator norm localization property uniformly with constant c. Let dS be the left invariant metric on π induced ˜ The metric on by S. Consider the total space YX˜ of the tree of space Xπ associated to the tree X. YX˜ is metric envelope d of the partial metric dˆ defined by ˆ y) = d(x,
d (x, y) 1,
if ∃P ∈ Vert Y , such that x, y ∈ gGP , if x and y are adjacency,
where d is a metric on π with generating set π , i.e., every two elements of π are at distance 1. The left multiplication by elements of π from the left defines an action of π on YX˜ . It is clear that the action is free and isometric. Note that X˜ and YX˜ are coarsely equivalent and ˜ = 1, so asym dim(Y ˜ ) = 1, this implies that Y ˜ has operator norm localization asym dim(X) X X property with constant less than 2. Fixed x0 = e ∈ GP0 for some P0 ∈ Vert Y , it suffices to show that the m-stabilizers Wm (x0 ) in (π, dS ) have operator norm localization property with constant less than 2c. To complete the proof we need the following proposition. 2 Proposition 4.7. (See [2].) Let Y be a non-empty, finite, connected graph, and (G, Y ) the associated graph of finitely generated groups. Let P0 be a fixed vertex of Y , then under the action of
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˜ the m-stabilizer Wm (P0 ) is precisely the set of elements π1 (G, Y, P0 ) on the Bass–Serre tree X, of type c in F (G, Y ) with L(c) R. Note that the m-stabilizer Wm (x0 ) of discrete group (π, dS ) acting on (YX˜ , d) is contained in ˜ d). ˜ the m-stabilizer Wm (P0 ) of (π, dS ) acting on the Bass–Serre tree (X, Consider the subset K ⊂ F (G, Y ) of all words whose associated path c satisfies o(c) = P0 . For k 1, let Hk = g ∈ K and its associated path c with L(c) = k . The following recursive decomposition of Hk is contained in [2]
Hk =
H0 = GP0 , (Hk−1 yGt (y) ∩ Hk )
(4.1) (4.2)
y∈Edge Y
and for each y ∈ Edge Y , we have
(Hk−1 yGt (y) ∩ Hk ) ⊂
(4.3)
xyGt (y) ,
x∈R(k−1)
where R(k − 1) is the subset of Hk−1 with words which do not end with an element of φy¯ (Gy ). Proposition 4.8. (See [2].) Fixed y ∈ Edge Y , for every r > 0, if we set Yr = Hk−1 yx: x ∈ dS x, φy (Gy ) r , then (Hk−1 yGt (y) ∩ Hk ) ⊂ Yr ∪
xyGty \ Yr
x∈R(k−1)
and subspaces xyGt (y) \ Yr , x ∈ R(k − 1), of (Γ, dS ) are r-separated. Proof. The statements are implicit in the proof of [2, Lemma 3].
2
Lemma 4.9. Assume as the above theorem, then for any m ∈ N, Qm has operator norm localization property with constant (1 + )c for any > 0 where Qm = g ∈ K and its associated path c with L(c) m . Proof. Fix > 0. We proceed by induction. The base case is clear since Q0 is precisely GP0 and by assumption GP0 hasoperator norm localization property with constant c. For m > 1, by the above notation, Qm = km Hk , so by Corollary 2.4, it suffices to prove that Hk has operator norm localization with constant less than (1 + 12 )c for all k m. Since Q0 and H0 coincide, we proceed to the induction step on Hk .
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For the induction step, for each k m − 1, we assume Hk has operator norm localization with
constant (1 + k )c for any k > 0. Especially we choose k = (1 + 8 )c. Let δ1 = 8+
, δ2 = 4+
and δ3 = 2+ . For each y ∈ Edge Y and for all r > 0, let Yr = Hm−1 yx: x ∈ dS x, φy (Gy ) r . Note that Yr is coarsely equivalent to Hm−1 yφy (Gy ), and by the relations on F (G, Y ), this set is Hm−1 φy¯ (Gy )y = Hm−1 y. Hence, Yr is coarsely equivalent to Hm−1 , which, by the assumption, has operator norm localization property with constant (1 + 8 )c. By Proposition 4.8 we have (Hm−1 yGt (y) ∩ Hm ) ⊂ Yr ∪
xyGty \ Yr .
x∈R(m−1)
Since multiplication from the left is an isometry, the hypothesis of the theorem implies that each subset xyGt (y) \ Yr , ∀x ∈ R(m − 1), has operator norm localization property with constant c. Note that the collection of subsets xyGt (y) \ Yr : x ∈ R(m − 1) are r-separated and therefore, by the infinite union Theorem 2.3, we get that Hm−1 yGt (y) ∩ Hm has operator norm localization property with constant (1 + δ1 )(1 + 8 )c = (1 + 4 )c. Observe that Hm ⊂ y∈Edge Y Hm−1 yGt (y) , which is a finite union since Y is assumed to be a finite graph. So by Corollary 2.4, Hm has operator norm localization property with constant (1 + δ2 )(1 + 4 )c = (1 + 2 )c. By inductive hypothesis that Hk have operator norm localization property with constant (1 + 8 ) for each k m − 1, hence for all 0 k m, Hk has operator norm localization property with constant less than (1 + 2 ). From Qm = km Hk and Corollary 2.4 we have Qm has operator norm localization with constant (1 + δ3 )(1 + 2 )c = (1 + )c. 2 The end of proof of Theorem 4.6. For any n ∈ N, the n-stabilizer of (π, dS ) acting on (YX˜ , d) is contained in Qn , from Proposition 1.9 we complete the proof of theorem. 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. Bell, A. Dranishnikov, On asymptotic dimension of groups, Algebr. Geom. Topol. 1 (2001) 57–71. G. Bell, A. Dranishnikov, On asymptotic dimension of groups acting on trees, Geom. Dedicata 103 (2004) 89–101. X. Chen, X. Wang, Remarks on the property L, preprint. X. Chen, R. Tessera, X. Wang, G. Yu, Metric sparsification and operator norm localization, preprint. M. Dadarlat, E. Guentner, Constructions preserving Hilbert space uniform embeddability of discrete groups, preprint. G. Gong, Q. Wang, G. Yu, Geometrization of the strong Novikov conjecture for residually finite groups, preprint. N. Higson, Counterexamples to the coarse Baum–Connes conjecture, preprint. N. Higson, V. Lafforgue, G. Skandalis, Counterexamples to the Baum–Connes conjecture, Geom. Funct. Anal. 12 (2) (2002) 330–354. D. Osin, Asymptotic dimension of relatively hyperbolic groups, preprint, GR/0411585, 2004. J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993), x+90 pp.
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[11] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., Providence, RI, 2003. [12] J.-P. Serre, Trees, Springer-Verlag, Berlin, 1980. [13] 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 255 (2008) 657–680 www.elsevier.com/locate/jfa
The Segal–Bargmann transform for Lévy white noise functionals associated with non-integrable Lévy processes Hsin-Hung Shih 1 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan 811 Received 10 October 2007; accepted 23 April 2008 Available online 29 May 2008 Communicated by L. Gross
Abstract By using a method of truncation, we derive the closed form of the Segal–Bargmann transform of Lévy white noise functionals associated with a Lévy process with the Lévy spectrum without the moment condition. Besides, a sufficient and necessary condition to the existence of Lévy stochastic integrals is obtained. © 2008 Elsevier Inc. All rights reserved. Keywords: Lévy process; Lévy white noise measure; Lévy–Itô decomposition theorem; Segal–Bargmann transform
1. Introduction The theory of white noise analysis initiated by T. Hida has been known as a useful tool in many branches of mathematics and physics. The detailed exposition of the theory can be found in [2,3,8]. Recently, the Lévy processes are extensively applied in quantum probability, stochastic analysis, mathematical finance, and stochastic differential equation, etc. (see e.g. [1,14,16–18]). In view of this, it is important to develop the parallel theory of white noise analysis for such processes. The first study was traced back to the work by Y. Ito [6] in 1988, who constructed a Poissonian counterpart of Hida’s theory. For the further development of Lévy white noise anal-
E-mail address: [email protected]. 1 Research supported by the National Science Council of Taiwan.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.013
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ysis, we refer the reader to Lytvynov et al. [15], Øksendal et al. [16], and Lee, Shih [10] as well as the references cited therein. The processes considered in the above literatures were confined to the class of Lévy processes the corresponding Lévy spectrum β (see (2.1)) satisfying the moment condition: +∞ with n β(du) < +∞ for any n ∈ N. Under this condition, the mean and variance exist for |u| −∞ such Lévy processes, and moreover the associated Lévy white noise space can be immediately built on the space S of tempered distributions on R by applying the Minlos theorem (see [9, Lemma 3.2 below]). In Hida’s white noise analysis, he studied generalized white noise functionals by employing the Segal–Bargmann transform as machinery. To extend Hida’s theory to Lévy processes, Lee and Shih [9] derived the closed form of the Segal–Bargmann transform of Lévy white noise functionals, which played an important role in the development of Lévy white noise analysis, especially in studying the renormalization and quantum decomposition of Lévy white noises. See Lee, Shih [10] for the related discussion. However, many techniques used to derive formulae in the above papers were based on the assumption of the moment condition of Lévy spectrums, which are not fulfilled by many other interesting and important cases of Lévy processes, such as Cauchy processes and α-stable processes, 0 < α < 2 (see e.g. [13]). It is natural to ask if the above theory of Lévy white noise analysis can be developed for all Lévy processes without the help of the moment condition. It does not seem to be evident. For example, we cannot show the existence of Lévy white noise measures on S directly by applying the Minlos theorem. In [12] Lee and Shih began to study Lévy white noise analysis associated with the general cases of Lévy processes without the moment condition. In replacing S by the dual space K of the space of C ∞ functions on R with compact support, it was shown that the Lévy white noise measures always exist as Borel measures on K , and then the representation of the associated Lévy process and the Lévy stochastic integrals were successfully done. In Sections 2 and 3 of this paper, we will improve those results (see Propositions 2.3 and 2.4). In particular, the sufficiency and necessity of the existence of Lévy stochastic integrals will be shown in Proposition 3.5. The main aim of this paper is to show that the result concerning the closed form of Segal– Bargmann transforms in [9, Theorem 5.8] actually holds for all Lévy processes. We outline the proof as follows. To begin with, we truncate the Lévy spectrum β on {0} ∪ [−n, −1/n] ∪ [1/n, n], denoted by β(n) , which satisfies the moment condition. One notes that if the support of β is not concentrated on the union of {0} and a compact subset of R \ {0}, every β(n) corresponds to different Lévy white noise measure Λ(b,β(n) ) and Lévy process X(b,β(n) ) (see Section 2). By employing the Segal–Bargmann transform as machinery, we embed {L2 (K , Λ(b,β(n) ) )} in L2 (K , Λ(b,β) ) (see Propositions 4.1 and 4.2). Apply the results in Lee, Shih [9] to the Segal–Bargmann transforms on L2 (K , Λ(b,β(n) ) ) and show that the limit exists as n tends to infinity. Finally, after proving that such a limit is exactly the desired closed form, we arrive at the main result (see Theorem 4.8). 2. Lévy white noise measures Let Ka be the space of all infinitely differentiable functions on R having compact supports in the interval [−a, a], a > 0. Then Ka is a nuclear space with a family {| · |a,n } of countable a dm 2 norms defined by |η|a,n = 0mn −a | dt m η(t)| dt, η ∈ Ka . Let K be the union of the spaces Ka endowed with the inductive limit topology, and K the dual of K with the weak topology. In the sequel, the pair (x, η), x ∈ K , η ∈ K, will always stand for a K –K pairing.
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We denote by M+ (R) the space of all positive finite Borel measures on R; and for any r ∈ M+ (R), by γ = |γ |(R) the total variation of γ . For every pair (b, β) ∈ R × M+ (R), let gβ , hβ be two complex-valued functions on R given by
i ur e − 1 − i ur β0 (du);
gβ (r) =
hβ (r) =
(2.1)
|u|>1
0<|u|1
where β0 (du) = is,
i ur e − 1 β0 (du),
1+u2 β(du), u2
u = 0; and let f(b,β) be the Lévy function generated by (b, β), that
f(b,β) (r) = i br −
β({0}) 2 r + gβ (r) + hβ (r), 2
r ∈ R.
(2.2)
Consider the complex-valued functional C(b,β) on the space K given by +∞ f(b,β) η(t) dt , C(b,β) (η) = exp
η ∈ K.
(2.3)
−∞
Then, by [12, Theorem 2.1], there is a unique probability measure Λ(b,β) on (K , B(K )), B(K ) being the σ -field generalized by all cylinder sets in K , the characteristic functional of which is C(b,β) , that is, C(b,β) (η) =
ei (x,η) Λ(b,β) (dx)
for any η ∈ K.
K
We call Λ(b,β) the Lévy white noise measure corresponding to the pair (b, β). Let Θβ be the class of all real-valued L1 ∩ L2 (R)-function ψ satisfying lim
n→∞ |t|n
hβ rψ(t) dt
exists in C for any r ∈ R.
(2.4)
It is obvious that the condition (2.4) is fulfilled by all L1 ∩L2 (R)-functions with compact support. In fact, the class Lβ = {ψ ∈ L1 ∩ L2 (R); f(b,β) (rψ) ∈ L1 (R), ∀r ∈ R} is contained in Θβ . When β satisfies the absolute moment of order one or β0 is the Lévy measure associated with certain non-trivial α-stable probability measure on R, it can be shown that Lβ = Θβ (see Lee, Shih [12] for the detailed discussion). An interesting question arises: is there β ∈ M+ (R) such that Lβ is a proper subset of Θβ ? The question is still open. In this section, we will establish the following results on the class Θβ : (1) The domain of the functional C(b,β) can be extended to the class Θβ . (2) For any ψ ∈ Θβ , there is associated a random variable Yψ on (K , B(K ), Λ(b,β) ) such that the characteristic function of Yψ is exactly C(b,β) (rψ), r ∈ R. It will be shown that such a random variable is nothing but a Lévy stochastic integral with respect to some Lévy process in Section 3.
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(3) Θβ is a linear space, while it is not clear whether Lβ is a linear space. For analogous results to the above on the class Lβ , we refer the reader to [12]. Lemma 2.1. For any ψ ∈ L1 ∩ L2 (R), +∞
i uψ(t) e − 1 − i uψ(t) β0 (du) dt < +∞.
−∞ 0<|u|1
Moreover, if φn → ψ in L2 (R), then gβ (φn ) → gβ (ψ) in L1 (R). Proof. We note that for any x, y ∈ R, x−y t+y ix i y i s e − e − i (x − y) = e ds dt 0
0
|x−y|
t + |y| dt |y − x|2 + |y||y − x|.
(2.5)
0
Then this lemma immediately follows from (2.1) and (2.5).
2
Theorem 2.2. For any ψ ∈ L1 ∩ L2 (R) with compact support, let {ηn } ⊂ K be convergent to ψ in L1 ∩ L2 (R). Suppose that all of the supports of ηn ’s and ψ are contained in some compact interval D. Then {(·, ηn )} converges in probability to a random variable on (K , B(K ), Λ(b,β) ), denoted by ·, ψ (b,β) , such that
log E(b,β) ei r ·,ψ (b,β) =
+∞ f(b,β) rψ(t) dt
∀r ∈ R,
(2.6)
−∞
where E(b,β) [f ] is the expectation of f with respect to Λ(b,β) and log f (r) is the unique complexvalued continuous determination of the logarithm of a continuous function f on R with f (0) = 1. Proof. Since hβ (rηn ) → hβ (rψ) in measure on (D, m|D ), m being the Lebesgue measure on R, and hβ (rφ) 4β · 1D , ∀r ∈ R, (2.7) for φ = ηn , ψ , we have hβ (rηn ) → hβ (rψ) in L1 (R) for any r ∈ R by the Lebesgue dominated convergence theorem. On the other hand, by Lemma 2.1, gβ (rηn ) → gβ (rψ) in L1 (R). Then f(b,β) (rηn ) → f(b,β) (rψ) in L1 (R); hence +∞
i r(·,η ) n E(b,β) e → exp ∀r ∈ R. f(b,β) rψ(t) dt −∞
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
661
+∞ Applying (2.7) and Lemma 2.1 we see that the map: r ∈ R → −∞ f(b,β) (rψ(t)) dt is con +∞ tinuous with value 0 at r = 0, which implies that exp( −∞ f(b,β) (rψ(t)) dt), r ∈ R, is the characteristic function of a random variable on (K , B(K ), Λ(b,β) ), denoted by ·, ψ (b,β) . On the other hand, by [7, Lemma 2.16, p. 258], there is a constant c > 0 such that K
|(x, ηn − ηm )| Λ(b,β) (dx) c 1 + |(x, ηn − ηm )|
1
1 − C(b,β) r(ηn − ηm ) dr
0
→ 0 as n, m → ∞. Thus, {(·, ηn )} converges in probability to ·, ψ (b,β) . Finally, the formula (2.6) is obtained by a direct application of [18, Lemma 7.6]. We complete the proof. 2 Theorem 2.3. For ψ ∈ Θβ , let ψn = ψ|[−n,n] be the restriction of ψ on [−n, n], n ∈ N. Then ·, ψn (b,β) converges in probability to a random variable on (K , B(K ), Λ(b,β) ) as n → ∞, denoted by ·, ψ (b,β) . Moreover, for any r ∈ R,
log E(b,β) ei r ·,ψ (b,β) = lim
n→∞ |t|n
= i br
f(b,β) rψ(t) dt
+∞ +∞ β({0}) 2 r ψ(t) dt − ψ(t)2 dt 2
−∞
−∞
+∞ + gβ rψ(t) dt + lim −∞
n→∞ |t|n
hβ rψ(t) dt.
Proof. For each n ∈ N, take a sequence {ηn,k } ⊂ K such that suppηn,k ’s are all contained in a compact interval and ηn,k → ψn in L1 ∩ L2 (R) as k → ∞. By Theorem 2.2, (·, ηn,k ) → ·, ψn (b,β) , (·, ηm,k ) → ·, ψm (b,β) , and (·, ηn,k − ηm,k ) → ·, ψn − ψm (b,β) (n > m) in probability with respect to Λ(b,β) as k → ∞. Thus, ·, ψn (b,β) − ·, ψm (b,β) = ·, ψn − ψm (b,β) Λ(b,β) -a.e., and then we have
log E(b,β) ei r( ·,ψn (b,β) − ·,ψm (b,β) ) +∞ = f(b,β) r ψn (t) − ψm (t) dt −∞
C(b, β) |ψn − ψm |L1 (R) |r| + |ψn − ψm |2L2 (R) r 2 +
hβ rψ(t) dt ,
m<|t|n
which converges to zero as n, m → ∞, where C(b, β) is a positive constant depending only on (b, β). By [7, Lemma 2.16, p. 258], there is a constant c > 0 such that
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K
2 | x, ψn (b,β) − x, ψm (b,β) | Λ(b,β) (dx) 1 + | x, ψn (b,β) − x, ψm (b,β) | 1
c
1 − E(b,β) ei r( ·,ψn (b,β) − ·,ψm (b,β) ) dr
0
for any n, m ∈ N. The right-hand side of this inequality approaches to zero as n, m → ∞; hence { ·, ψn (b,β) } converges in probability to a random variable, denoted by ·, ψ (b,β) , with respect to Λ(b,β) . Since (·, ηn,k )’s are all infinitely divisible, so are ·, ψn (b,β) ’s and ·, ψ (b,β) . It implies that their characteristic functions do not vanish in (−∞, +∞). Therefore, we apply [18, Lemma 7.7] to see that log E(b,β) [ei r ·,ψn (b,β) ] → log E(b,β) [ei r ·,ψ (b,β) ], for any r ∈ R. Moreover, by Lemma 2.1 and Theorem 2.2,
log E(b,β) ei r ·,ψn (b,β) =
+∞ f(b,β) rψn (t) dt −∞
→ i br
+∞ +∞ β({0}) 2 r ψ(t) dt − ψ(t)2 dt 2
−∞
−∞
+∞ + gβ rψ(t) dt + lim −∞
n→∞ |t|n
from which the assertion of this theorem immediately follows.
hβ rψ(t) dt,
2
Proposition 2.4. (i) Θβ is a linear space. Moreover, for any α ∈ R and ψ, φ ∈ Θβ , ·, αψ + φ (b,β) = α ·, ψ (b,β) + ·, φ (b,β) ,
Λ(b,β) -a.e.
(ii) If the support of Λ(b,β) is contained in the dual space S of the Schwartz space S on R, then S is contained in Θβ . Proof. The proof of (ii) can be found in [12, Proposition 4.4]. For the proof of (i), we first note that by Theorem 2.2 ·, αψn + φn (b,β) = α ·, ψn (b,β) + ·, φn (b,β) , Λ(b,β) -a.e., for any n ∈ N, where ψn = ψ|[−n,n] and φn = φ|[−n,n] ; and by Theorem 2.3, ·, αψn +φn (b,β) → α ·, ψ (b,β) + ·, φ (b,β) in probability with respect to Λ(b,β) . Then, by applying [18, Lemmas 7.7 and 7.8] we at once see that
log E(b,β) ei ·,αψn +φn (b,β) → log E ei (α ·,ψ (b,β) + ·,φ (b,β) ) as n → ∞. Consequently,
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lim
n→∞ |t|n
hβ αψ(t) + φ(t) dt = log E(b,β) ei (α ·,ψ (b,β) + ·,φ (b,β) ) − i b
663
+∞ αψ(t) + φ(t) dt
−∞
β({0}) + 2
+∞ +∞ 2 (αψ + φ) (t) dt − gβ αψ(t) + φ(t) dt, −∞
−∞
so that αψ + φ ∈ Θβ . Since (αψ + φ)n = αψn + φn , it follows from Theorem 2.3 that ·, αψ + φ (b,β) = α ·, ψ (b,β) + ·, φ (b,β) , We finish the proof.
Λ(b,β) -a.e.
2
Define a stochastic process {X(b,β) (t); t ∈ R} on (K , B(K ), Λ(b,β) ) by X(b,β) (t; x) =
if t 0, x, 1[0,t] (b,β) , − x, 1[t,0] (b,β) , if t < 0.
Then, by Theorem 2.2, X(b,β) is a version of Lévy process, the corresponding Lévy function of which is exactly f(b,β) . By [18, Theorem 11.5], X(b,β) can be assumed to be a Lévy process with X(b,β) (0) = 0. Proposition 2.5. For any (b, β) ∈ R × M+ (R), there is a Λ(b,β) -null set Δ such that for every x ∈ K \ Δ, x is exactly the distributional derivative of the sample function: t ∈ R → X(b,β) (t; x). Proof. First, for each η ∈ K, it follows from the proof of [12, Proposition 3.5(ii)] that (·, η) +∞ coincides with the Riemann–Stieltjes integral −∞ η(t) X(b,β) (dt; ·) Λ(b,β) -a.e. on K . By the separability of K, we can choose a countable dense subset D of K. Then there is a Λ(b,β) null set Δ ⊂ K with Λ(b,β) (Δ) = 0 such that for any x ∈ K \ Δ and ζ ∈ D, (x, ζ ) equals +∞ −∞ ζ (t) X(b,β) (dt; x). Thus, for any η ∈ K, if we take a convergent sequence {ζn } ⊂ D to η, then for any x ∈ K \ Δ, +∞ +∞ ζn (t) X(b,β) (dt; x) = η(t) X(b,β) (dt; x) (x, η) = lim (x, ζn ) = lim n→∞
n→∞ −∞
−∞
+∞ =−
X(b,β) (t; x)η(t) ˙ dt = − X(b,β) (·; x), η˙ .
(2.8)
−∞
The proof is complete.
2
The Skorokhod space. Let D(R) be the class of all càdlàg functions φ defined on R with φ(0) = 0. It is well known that D(R) is a Polish space under the Skorokhod metric ρ, where for any φ, ψ ∈ D(R), the metric ρ(φ, ψ) of φ and ψ is given by
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inf
∞ ϑ(t) − ϑ(s) 1 min 1, sup φ ϑ(t) n ϑ(t) − ψ(t)n (t) , sup ln + t −s 2n
ϑ∈Θ s
t∈R
n=1
Θ being the set of all continuous functions ϑ of R onto itself that are strictly increasing with ϑ(0) = 0; and 1 n (t) =
0 n + 1 − |t|
if |t| n, if |t| n + 1, if n |t| n + 1.
Remark 2.6. It should be emphasized that the above Skorokhod metric ρ is constructed on “D(R)” instead of “D(R+ )” as in the book [7], where the domains of ϑ ∈ Θ and the truncation n are extended in a natural way to the whole R. All the characterizations of the Skorokhod metric on D(R+ ) presented in [7] remain valid for the metric ρ. ˙ Let T be the mapping from the Skorokhod space (D(R), ρ) into the space K by T(φ) = φ, ˙ where φ is the distributional derivative of φ. By [11, Proposition 2.1 and Corollary 2.2], T is injective and continuous for the Skorokhod topology; and for any Borel set B of (D(R), ρ) (i.e., B ∈ B(D(R))), T(B) is a Borel set of K . In particular, T(D(R)) ∈ B(K ). Recall that the σ field B(D(R)) equals the σ -field generated by all maps: φ ∈ D(R) → φ(s) for s ∈ R (see [7]). Combining this with Proposition 2.5 yields the following result. Theorem 2.7. Let X be a real-valued function on R × K given by X(t; x) =
T−1 (x)(t) 0
if x ∈ T(D(R)), if x ∈ K \ T(D(R)),
t ∈ R.
Then, for any t ∈ R, X(t; ·) is B(K )-measurable. Moreover, for any (b, β) ∈ R × M+ (R), X is a càdlàg version of X(b,β) on (K , B(K ), Λ(b,β) ). Henceforth, we will replace X(b,β) by X for any (b, β) ∈ R × M+ (R). Here, one notes that the definition of X is independent of the choice of (b, β). 3. The Lévy–Itô decomposition theorem Fix a (b, β) ∈ R × M+ (R). Let Bb (R2∗ ) be the class of bounded Borel subsets of R2∗ (= R2 \ {(t, 0); t ∈ R}), away from the t-axis; and the measure νβ (dt, du) = β0 (du) dt on B(R2∗ ). For E ∈ Bb (R2∗ ), let N (E; x) = (t, u) ∈ E; jX (t; x) = u ,
x ∈ K ,
(3.1)
where jX (t; x) = X(t; x) − X(t−; x). Then N (E; x) = 0 if x ∈ K \ T(D(R)). Theorem 3.1 (Lévy–Itô decomposition theorem). (See [18].) On the probability space (K , B(K ), Λ(b,β) ), the sample functions of X enjoy the following decomposition: (i) {N (E; x); E ∈ Bb (R2∗ ), x ∈ K } is a Poisson random measure with the intensity measure νβ .
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(ii) For every n ∈ N \ {1}, define ⎧ t β (ds, du) + t ⎨ 0 1 |u|1 u N if t 0, 0 1<|u|n u N (ds, du) n (3.2) Yβ,n (t) = 0 0 ⎩− 1 t |u|1 u Nβ (ds, du) − t 1<|u|n u N (ds, du) if t < 0, n
β (E; x) ≡ N (E; x)−νβ (E); E ∈ Bb (R2∗ ), x ∈ K } is an independent random meawhere {N sure with zero mean, and the above integrals are defined pathwise as Lebesgue–Stieltjes integrals. Then there is Ω(b,β) ∈ B(K ) ∩ T(D(R)) with Λ(b,β) (Ω(b,β) ) = 1 such that for each x ∈ Ω(b,β) , (a) Yβ (t; x) ≡ limn→∞ Yβ,n (t; x) exists uniformly in t on any bounded interval; and (b) B(b,β) (t; x) ≡ X(t; x) − bt − Yβ (t; x) is continuous in t ∈ R and, for s < t,
1 2 E(b,β) ei r(B(b,β) (t)−B(b,β) (s)) = e− 2 β({0})r |t−s|
∀r ∈ R.
(iii) The two systems {N (E); E ∈ Bb (R2∗ )} and {B(b,β) (t); t ∈ R} are independent. Remark 3.2. From Theorems 2.7 and 3.1 it follows that for any x ∈ Ω(b,β) , ˙ x) = b + B˙ (b,β) (·; x) + Y˙β (·; x) x = X(·; = b + B˙ (b,β) (·; x) + lim Y˙β,n (·; x). n→∞
Let λβ be a positive measure on B(R2 ) defined by λβ (dt, du) = (1 + u2 ) β(du) dt. 2 L (K , Λ(b,β) )-valued function M(b,β) on {E ∈ B(R2 ); λβ (E) < +∞} by +∞ β (dt, du). M(b,β) (E) = 1E (t, 0) B(b,β) (dt) + u1E (t, u) N −∞
Define a
(3.3)
R2∗
Then the system {M(b,β) (E; x); E ∈ B(R2 ) with λβ (E) < +∞, x ∈ K } forms an independent random measure with zero mean which means that E(b,β) [M(b,β) (E)] = 0 and M(b,β) (E) and M(b,β) (F ) are independent to each other for E, F ∈ B(R2 ) with E ∩ F = ∅. In addition, E(b,β) [M(b,β) (E)M(b,β) (F )] = λβ (E ∩ F ) for all E, F ∈ B(R2 ). The mth order Lévy–Itô multiple stochastic integral Ib,β;m (gm ) ≡ . . . gm (s1 , . . . , sm ) M(b,β) (ds1 ) . . . M(b,β) (dsm ), R2
R2
2 ((R2 )m , λ⊗m ), the closed with respect to M(b,β) was introduced by K. Itô in [5], where gm ∈ L c β subspace consisting of all symmetric complex-valued functions in L2c ((R2 )m , λ⊗m β ). Here, for a real Banach space V , Vc denotes the complexification of V . Then we have Ib,β;m (gm ) 2 2 gm (s1 , . . . , sm ) 2 λβ (ds1 ) . . . λβ (dsm ). = m! . . . L (K ,Λ ) (b,β)
R2
R2
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2 ((R2 )m , λ⊗m ) is given by In particular, if gm ∈ L c β
gm =
ai 1E i ×···×E i ,
i=1
m
1
ai ’s ∈ C,
(3.4)
Eji ’s ∈ B(R2 ) with λβ (Eji ) < +∞ and Eji < Eji +1 (see [4]), then
Ib,β;m (gm ) =
i=1
ai
m
M(b,β) Eji .
(3.5)
j =1
For the detailed description and related properties of Ib,β;m (gm ), we refer the reader to [5]. The following is the Itô’s chaos decomposition theorem for square integrable Lévy white noise functionals. Theorem 3.3. (See [5].) Let ϕ be given in L2 (K , Λ(b,β) ). Then 2 ((R2 )m , λ⊗m ), m ∈ N ∪ {0}, such (i) there exists a unique series of kernel functions φm ∈ L c β ∞ that ϕ is equal to the orthogonal direct sum m=0 Ib,β;m (φm ). In notation, we write ϕ ∼ (φm )b,β . (ii) L2 (K , Λ(b,β) ) is isomorphic to the√ Boson Fock space Fs (L2c (R2 , λβ )) over L2c (R2 , λβ ) by √ √ carrying ϕ ∼ (φm )b,β into ( 0!φ0 , 1!φ1 , . . . , m!φm , . . .). Remark 3.4. For g ∈ L2c (R2 , λβ ), +∞ +∞ Ib,β;1 (g) = g(t, 0) B(b,β) (dt) + −∞
β (dt, du), ug(t, u) N
−∞ |u|>0
with the characteristic function E(b,β) [ei rIb,β;1 (g) ], r ∈ R, being
β({0}) 2 r exp − 2
+∞ ei urg(t,u) − 1 − i rug(t, u) β0 (du) dt . g(t, 0)2 dt +
−∞
(3.6)
R2∗
If β has finite second moment, then the function (t, u) ∈ R2 → 1[c,d] (t) is in L2c (R2 , λβ ), and we see that Λ(b,β) -a.e. on K
X(d) − X(c) = Ib,β;1 (1[c,d] ) + b + |u|>1
−1 u+u β(du) (d − c).
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For the general case of β ∈ M+ (R), in particular, without the first moment, X(d) − X(c) = lim Ib,β;1 (1[c,d]×({0}∪{u;1/n|u|n}) ) n→∞
! + b+ u + u−1 β(du) (d − c) 1<|u|n
in probability with respect to Λ(b,β) . Here, one notes that limn→∞ 1<|u|n u β(du) may not exist if β has not the first absolute moment. For example, if X is the α-stable process on (K , B(K ), Λ(b,β) ), then the above limit exists if and only if 1 < α 2. Lévy stochastic integrals. For ψ ∈ L1 ∩ L2 (R), the Lévy–Itô integral Ib,β;1 (ψ · 1R×[−1,1] ) always exists because ψ · 1R×[−1,1] (t, u) ≡ ψ(t) · 1[−1,1] (u), (t, u) ∈ R2 , is in L2c (R2 , λβ ) for any β ∈ M+ (R). On the other hand, if ψ has compact support, the Lebesgue–Stieltjes integral +∞ uψ(t)N (dt, du; x) =
jX (t; x) · 1(1,+∞) |jX (t; x)| ψ(t)
t∈supp ψ
−∞ |u|>1
is convergent for any x ∈ K . In fact, the right-hand summation of the above equality is just a finite sum. Then, it is easy to see that "
#
+∞
log E(b,β) exp i r
uψ(t) N(dt, du)
+∞ = hβ rψ(t) dt
−∞ |u|>1
∀r ∈ R.
−∞
For an arbitrarily given ψ ∈ L1 ∩ L2 (R), we define on (K , B(K ), Λ(b,β) ) +∞
uψ(t) N (dt, du) ≡ lim
n→∞ |t|n |u|>1
−∞ |u|>1
uψ(t) N (dt, du)
(3.7)
in probability, provided that such a limit exists. Since the right-hand integrals in (3.7) are in +∞ finitely divisible, so is −∞ |u|>1 uψ(t) N(dt, du). By [18, Lemma 7.7], "
+∞
log E(b,β) exp i r −∞ |u|>1
# uψ(t) N (dt, du)
= lim
n→∞ |t|n
hβ rψ(t) dt
(3.8)
exists. Thus ψ ∈ Θβ , and vice versa. Summing up the above argument we have the following result.
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+∞ Proposition 3.5. For ψ ∈ L1 ∩ L2 (R), −∞ |u|>1 uψ(t) N (dt, du) exists on the probability space (K , B(K ), Λ(b,β) ) if and only if ψ ∈ Θβ . Moreover, Λ(b,β) -a.e. on K , +∞ +∞ ·, ψ (b,β) = b ψ(t) dt + Ib,β;1 (ψ · 1R×[−1,1] ) + −∞
(3.9)
uψ(t) N (dt, du).
−∞ |u|>1
Proof. The formula (3.9) immediately follows from Theorem 2.3, (3.6), and (3.8).
2
For η ∈ K and x ∈ Ω(b,β) , it follows from Remark 3.2 that ˙ x), η = (x, η) = X(·;
+∞ η(t) X(dt; x)
−∞
+∞ +∞ +∞ =b η(t) dt + η(t) B(b,β) (dt; x) + η(t) Yβ (dt; x), −∞
−∞
(3.10)
−∞
where all of the above integrals in (3.10) are understood pathwise in the sense of Riemann– Stieltjes integrals; and for c < d it follows from Theorem 3.1 that Λ(b,β) -a.e. on Ω(b,β) , Yβ (d) − Yβ (c) =
β (dt, du) + u1[c,d]×{u;0<|u|1} (t, u) N
d u N (dt, du). c |u|>1
R2∗
The above argument leads to the following definition concerning Lévy stochastic integrals. Definition 3.6. For any ψ ∈ Θβ , we define Lévy stochastic integrals: +∞ −∞ ψ(t) X(dt) on (K , B(K ), Λ(b,β) ) by +∞ +∞ ψ(t) Yβ (dt) = uψ(t) · 1{u;0<|u|1} (u) Nβ (dt, du) + −∞
+∞ −∞
ψ(t) Yβ (dt) and
uψ(t) N (dt, du);
−∞ |u|>1
R2∗
and +∞ +∞ +∞ +∞ ψ(t) X(dt) = b ψ(t) dt + ψ(t) B(b,β) (dt) + ψ(t) Yβ (dt). −∞
−∞
−∞
−∞
4. The Segal–Bargmann transform To begin with, the following two notations are fixed in this section: 1. Dn ≡ {u ∈ R \ {0}; 1/n |u| n};
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2. For any β ∈ M+ (R), β(n) (E) ≡ β(E ∩ ({0} ∪ Dn )), ∀E ∈ B(R), n ∈ N. For (b, β) ∈ R × M+ (R), let ϕ ∈ L2 (K , Λ(b,β) ) with ϕ ∼ (φm )b,β . Then the Segal– Bargmann (or the S-) transform S(b,β) ϕ of ϕ is a complex-valued analytic functional on L2c (R2 , λβ ) defined by S(b,β) ϕ(g) =
ϕ(x)E(b,β) (g)(x) Λ(b,β) (dx),
g ∈ L2c R2 , λβ ,
K
1 ⊗m ), called the coherent state functionals associated with g where E(b,β) (g) = ∞ m=0 m! Ib,β;m (g corresponding to (b, β). In fact, S(b,β) ϕ(g) =
∞ m=0
...
R2
φm (s1 , . . . , sm )g(s1 ) · · · g(sm ) λβ (ds1 ) . . . λβ (dsm ).
(4.1)
R2
In this section, we will be devoted to derive the closed form of the Segal–Bargmann transform of square-integrable Lévy white noise functionals. Equivalently, we are looking for an explicit form of the coherent state functionals. Let H be a complex Hilbert space and F 1 (H ) be the class of all analytic functions on H with norm ·F 1 (H ) given by f 2F 1 (H ) =
∞ 2 1 m D f (0) L(m) (H ) , (2) m!
m=0
(m)
where D is the Fréchet derivative of f and L(2) (H ) is the space of all m-linear Hilbert–Schmidt operators on H . Then F 1 (H ) is called the Bargmann–Segal–Dwyer space. It is well known that the S(b,β) -transform is a unitary operator from L2 (K , Λ(b,β) ) onto F 1 (L2c (R2 , λβ )) (see [9]). Proposition 4.1. Let H be a complex separable Hilbert space and K the closed subspace of H . on H by For any F ∈ F 1 (K), we define a functional F (h) = F (hK ), F
h ∈ H,
⊥ ⊥ where h = hK + h⊥ K , hK ∈ K and hK ∈ K , the orthogonal complement of K on H . Then 1 F ∈ F (H ) and F F 1 (H ) = F F 1 (K) .
is an analytic function on K. Let {ei }, {fj } be orthonormal Proof. First, it is obvious that F bases of K and K ⊥ , respectively. We note that for any xi ∈ H , i = 1, 2, . . . , m, if there is k ∈ {1, 2, . . . , m} such that xk ∈ K ⊥ , then (0)(x1 , . . . , xm ) = D m F (0) (x1 )K , . . . , (xk−1 )K , 0, (xk+1 )K , . . . , (xm )K = 0. DmF It implies that
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m D F (0) 2 (m)
= L(2) (H ) =
∞
m D F (0)(ei1 , . . . , eim ) 2
i1 ,...,im =0 ∞
m D F (0)(ei , . . . , ei ) 2 m 1
i1 ,...,im =0
2 = D m F (0) L(m) (K) , (2)
F 1 (H ) = F F 1 (K) . whence F
∀m ∈ N ∪ {0},
2
By Proposition 4.1, F 1 (K) is unitarily isometric to a closed subspace of F 1 (H ). Since, for any n ∈ N, L2c (R2 , λβ(n) ) can be regarded as a closed subspace of L2c (R2 , λβ ) by the isometry: h ∈ L2c (R2 , λβ(n) ) → h · 1R×({0}∪Dn ) ∈ L2c (R2 , λβ ), we apply Proposition 4.1 to define a map Φb,β(n) ,β from L2 (K , Λ(b,β(n) ) ) into L2 (K , Λ(b,β) ) by −1 (S(b,β Φb,β(n) ,β (ϕ) = S(b,β) (n) ) ϕ),
ϕ ∈ L2 (K , Λ(b,β(n) ) ).
Therefore Φb,β
(n) ,β
(ϕ) L2 (K ,Λ
(b,β) )
= S(b,β = ϕL2 (K ,Λ(b,β 2 (n) ) ϕF 1 (L2 c (R ,λβ ))
(n) )
).
On the other hand, let Φb,β,β(n) be the map from L2 (K , Λ(b,β) ) into L2 (K , Λ(b,β(n) ) ) by −1 (S(b,β) ϕ|L2c (R2 ,λβ Φb,β,β(n) (ϕ) = S(b,β (n) )
(n)
) ),
ϕ ∈ L2 (K , Λ(b,β) ).
Proposition 4.2. Let (b, β) ∈ R × M+ (R) and n ∈ N. (i) If ϕ ∈ L2 (K , Λ(b,β(n) ) ) with the chaos decomposition ϕ ∼ (φm )b,β(n) , then Φb,β(n) ,β (ϕ) ∼ φm · (1R×({0}∪Dn ) )⊗m b,β . (ii) If ψ ∈ L2 (K , Λ(b,β) ) with the chaos decomposition ψ ∼ (ψm )b,β , then Φb,β,β(n) (ψ) ∼ (ψm )b,β(n) . Proof. It is straightforward by comparing the S-transform of both-side functionals. For any n ∈ N, we define a mapping Jb,β,n from K into K by Jb,β,n (x) =
b + B˙ (b,β) (·; x) + Y˙β,n (·; x) 0
if x ∈ Ω(b,β) , if x ∈ / Ω(b,β) .
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For each η ∈ K and x ∈ K ,
+∞ +∞ +∞ η(t) dt + η(t) B(b,β) (dt; x) + η(t) Yβ,n (dt; x) · 1Ω(b,β) (x), Jb,β,n (x), η = b
−∞
−∞
−∞
hence the function x ∈ K → (Jb,β,n (x), η) is obviously B(K )-measurable, and so J−1 b,β,n ({x ∈ K ; (x, η) a}) lies in B(K ) for any a ∈ R. Then Jb,β,n is B(K )-measurable by applying the fact that B(K ) is a σ -field generated by all cylinder sets.
Proposition 4.3. Let (b, β) ∈ R × M+ (R) and n ∈ N. (i) For any t ∈ R and x ∈ Ω(b,β) , jX t; Jb,β,n (x) = Yβ,n (t; x) − Yβ,n (t−; x) = jX (t; x) · 1Dn jX (t; x) ; Yβ(n) t; Jb,β,n (x) = Yβ,n (t; x); X t; Jb,β,n (x) = bt + B(b,β) (t; x) + Yβ,n (t; x). (ii) For any B ∈ B(K ), Λ(b,β(n) ) (B) = Λ(b,β) (J−1 b,β,n (B)). (iii) For any complex-valued bounded continuous function ϕ on K ,
ϕ(x) Λ(b,β(n) ) (dx) =
lim
n→∞
K
ϕ(x) Λ(b,β) (dx). K
(iv) For any t ∈ R, B(b,β(n) ) (t; Jb,β,n (x)) = B(b,β) (t; x), Λ(b,β) -a.e. x ∈ Ω(b,β) . Proof. Since Jb,β,n (x) = b + T(B(b,β) (·; x) + Yβ,n (·; x)) for any x ∈ Ω(b,β) , the assertion in (i) immediately follows from the definitions of Jb,β,n , Yβ(n) , Yβ,n , and Theorem 2.7, where T is the map defined several lines before Theorem 2.7. Observe that for any η ∈ K, +∞ C(b,β(n) ) (η) = exp f(b,β(n) ) η(t) dt −∞
=
+∞ +∞ exp i b η(t) dt + i η(t) B(b,β) (dt; x) −∞
K
−∞
+∞ +i η(t) Yβ,n (dt; x) Λ(b,β) (dx)
=
−∞
ei (Jb,β,n (x),η) Λ(b,β) (dx) K
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H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
=
ei (x,η) Λ(b,β) J−1 b,β,n (dx) .
K
This implies that Λ(b,β(n) ) and Λ(b,β) ◦ J−1 b,β,n coincide on the σ -field generated by all random variables (·, η), η ∈ K, hence on B(K ) and we obtain the assertion in (ii). The assertion in (iii) is a simple consequence of (ii) and Remark 3.2. Finally, we observe that 1=
exp i X(t; x) − bt − B(b,β(n) ) (t; x) − Yβ(n) (t; x) Λ(b,β(n) ) (dx)
K
(by Theorem 3.1) = exp i X t; Jb,β,n (x) − bt − B(b,β(n) ) t; Jb,β,n (x) − Yβ(n) t; Jb,β,n (x) Λ(b,β) (dx) K
by (ii) = exp i X t; Jb,β,n (x) − bt − B(b,β(n) ) t; Jb,β,n (x) − Yβ,n (t; x) Λ(b,β) (dx). K
by (i) .
Then X t; Jb,β,n (x) = bt + B(b,β(n) ) t; Jb,β,n (x) + Yβ,n (t; x), Combining this with (i), the assertion in (iv) immediately follows.
Λ(b,β) -a.e. x ∈ Ω(b,β) . 2
For any (b, β) ∈ R × M+ (R), by Proposition 4.3, there is Ξ(b,β) ∈ Ω(b,β) ∩ B(K ) with Λ(b,β) (Ξ(b,β) ) = 1 such that for any x ∈ Ξ(b,β) , Jb,β,n (x) ∈ Ω(b,β(n) ) and the formula in Proposition 4.3(iv) holds for any n ∈ N and t ∈ R. Proposition 4.4. Let (b, β) ∈ R × M+ (R) and ϕ ∈ L2 (K , Λ(b,β(n) ) ) for some n ∈ N. Then Φb,β(n) ,β (ϕ) = ϕ ◦ Jb,β,n ,
Λ(b,β) -a.e. on Ξ(b,β) .
2 ((R2 )m , λ⊗m ) of the form Proof. Let Sm , m ∈ N, be the class of all functions gm ∈ L c β(n) gm =
i=1
1E i ×···×E i , ai 1
m
ai ’s ∈ C,
where Eji ∈ B(R2 ) with λβ(n) (Eji ) < +∞, Eji < Eji +1 , and Eji ∩ (R × {0}) is a union of disjoint intervals of the form (p, q] for any i, j . By Proposition 4.3, we see that
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
673
M(b,β(n) ) Eji ; Jb,β,n (x) +∞ +∞ = 1E i (t, 0) B(b,β(n) ) dt; Jb,β,n (x) + j
−∞
β(n) dt, du; Jb,β,n (x) u1E i (t, u) N j
−∞ |u|>0
+∞ +∞ β (dt, du; x) = 1E i (t, 0) B(b,β) (dt; x) + u1E i (t, u) N j
j
−∞
−∞ Dn
= M(b,β) Eji ∩ R × {0} ∪ Dn ; x ,
∀x ∈ Ξ(b,β) ,
(4.2)
and therefore, for gm ∈ Sm as above, m ai M(b,β(n) ) Eji ; Jb,β,n (x) Ib,β(n) ;m (gm ) Jb,β,n (x) = j =1
i=1
=
i=1
ai
m
by (4.2) M(b,β) Eji ∩ R × {0} ∪ Dn ; x
j =1
= Ib,β;m gm · (1R×({0}∪Dn ) )⊗m (x).
(4.3)
2 ((R2 )m , λ⊗m ) (see [4, Theorem Now, assume that ϕ ∼ (φm )b,β(n) . Since Sm , m ∈ N, is dense in L c β(n) 2.1]), given an > 0, we can choose ψm ∈ Sm such that |ψm − φm |2L2 ((R2 )m ,λ⊗m ) c
β(n)
, m! · 2m+4
for each m ∈ N ∪ {0}.
By the Cauchy–Schwarz inequality, (4.3), Propositions 4.3(ii), and 4.2(i), ϕ ◦ Jb,β,n − Φb,β
(n) ,β
2 (ϕ) L2 (K ,Λ
(b,β) )
2 N ⊗m (x) Λ(b,β) (dx) 2 Ib,β;m ψm · (1R×({0}∪Dn ) ) ϕ Jb,β,n (x) − m=0
Ξ(b,β)
N 2 + 2 Ib,β;m ψm · (1R×({0}∪Dn ) )⊗m − Φb,β(n) ,β (ϕ) 2
L (K ,Λ(b,β) )
m=0
2 N =2 Ib,β(n) ;m (ψm ) Jb,β,n (x) Λ(b,β) (dx) ϕ Jb,β,n (x) − m=0
Ξ(b,β)
N 2 + 2 Ib,β;m ψm · (1R×({0}∪Dn ) )⊗m − Φb,β(n) ,β (ϕ) 2 m=0
L (K ,Λ(b,β) )
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2 N = 4 ϕ − Ib,β(n) ;m (ψm ) 2
L (K ,Λ(b,β(n) ) )
m=0
=4
N m=0
m!|φm − ψm |2L2 ((R2 )m ,λ⊗m ) c β (n)
∞
+
m=N +1
∞ +4 m!|φm |2L2 ((R2 )m ,λ⊗m ) → c β(n) 2 2
m!|φm |2L2 ((R2 )m ,λ⊗m ) c β (n)
as N → ∞.
m=N +1
Since > 0 is arbitrarily given, the proposition is proved.
2
Corollary 4.5. Let (b, β) ∈ R × M+ (R) and g ∈ L2c (R2 , λβ(n) ), n ∈ N. Then E(b,β(n) ) (g) ◦ Jb,β,n = E(b,β) (g · 1R×({0}∪Dn ) ),
Λ(b,β) -a.e. on K .
In Lee, Shih [9] the closed form of the Segal–Bargmann transform was derived for those +∞ β ∈ M+ (R) satisfying the moment condition: −∞ |u|n β(du) < +∞ for any n ∈ N. In the rest of this paper, we will show that this result is actually fulfilled by all β ∈ M+ (R), even without the moment condition. In fact, we will show that the limit “limn→∞ E(b,β(n) ) (Jb,β,n (x))” exists for Λ(b,β) -a.e. x ∈ K , and find its explicit form. First, for (b, β) ∈ R × M+ (R), let Dβ = h ∈ L2c (R2 , λβ ); h∗ ∈ L1c (R2∗ , νβ ) ,
h∗ (t, u) = uh(t, u), (t, u) ∈ R2∗ .
Then Dβ is a complex Banach space with the |·|β -norm defined by |h|β =
1/2 h(t, u) 2 λβ (dt, du) uh(t, u) νβ (dt, du), +
R2
h ∈ Dβ ,
R2∗
and Dβ ⊂ Dβ(n) , n ∈ N. For each h ∈ Dβ , by [9, Proposition 2.1] we can take a Δ(b,β) (h) ∈ B(K ) with Λ(b,β) (Δ(b,β) (h)) = 1 such that ∗ h∗ t, jX (t; x) = h (t, u) N (dt, du; x) < +∞, t∈R
x ∈ Δ(b,β) (h).
R2∗
∗ We note that for x ∈ Δ(b,β) (h), $ there are ∗only a finite number of 1 + h (t, jX (t; x)) which are zero; and the infinite product t∈R (1 + h (t, jX (t; x))) is absolutely convergent. Thus, for each h ∈ Dβ , there is associated a functional on K defined by
$ Υ(b,β) (h; x) =
0,
∗ t∈R (1 + h (t, jX (t; x)))
if x ∈ Δ(b,β) (h), otherwise.
(4.4)
In addition, define two complex functions G(b,β) and P(b,β) on Dβ × (K , B(K ), Λ(b,β) ) by
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
β({0}) G(b,β) (h; ·) = exp − 2 P(b,β) (h; x) =
exp(−
675
+∞ +∞ h(t, 0)2 dt + h(t, 0) B(b,β) (dt) −∞
−∞
∗ R2∗ h (t, u) νβ (dt, du))Υ(b,β) (h; x)
0,
if x ∈ Δ(b,β) (h), otherwise.
(4.5)
Proposition 4.6. Let (b, β) ∈ R × M+ (R). Then, for any h ∈ Dβ and n ∈ N, (i) G(b,β(n) ) (h; Jb,β,n (x)) = G(b,β) (h; x), Λ(b,β) -a.e. x ∈ Ξ(b,β) ; (ii) P(b,β(n) ) (h; Jb,β,n (x)) = P(b,β) (h · 1R×Dn ; x), Λ(b,β) -a.e. x ∈ Ξ(b,β) ∩ Δ(b,β) (h). Proof. First, we observe that
β({0}) G(b,β) (h; ·) = exp − 2
+∞ 2 h(t, 0) dt + Ib,β;1 (h · 1R×{0} ) . −∞
Let ϕ = Ib,β(n) ;1 (h · 1R×{0} ). By Proposition 4.2, Φb,β(n) ,β (ϕ) = Ib,β;1 (h · 1R×{0} ). The assertion (i) immediately follows from Proposition 4.4. Next, by Proposition 4.3(ii), we see that for Λ(b,β) a.e. x ∈ Ξ(b,β) ∩ Δ(b,β) (h), Jb,β,n (x) ∈ Δ(b,β(n) ) (h). Then P(b,β(n) ) h; Jb,β,n (x)
∗ h (t, u) νβ(n) (dt, du) Υ(b,β(n) ) h; Jb,β,n (x) = exp − R2∗
= exp − (h · 1R×Dn )∗ (t, u) νβ (dt, du) Υ(b,β) (h · 1R×Dn ; x) R2∗
= P(b,β) (h · 1R×Dn ; x). We obtain the assertion (ii) and the proof is complete. Lemma 4.7. (See [9, Theorem 5.8].) Let (b, β) ∈ R × M+ (R). Assume that β satisfies the moment condition. Then, for any ϕ ∈ L2 (K , Λ(b,β) ) and h ∈ Dβ , S(b,β) ϕ(h) =
ϕ(x)G(b,β) (h; x)P(b,β) (h; x) Λ(b,β) (dx). K
Now, we are ready to prove the main theorem. Theorem 4.8. Let (b, β) ∈ R × M+ (R). Then, for any ϕ ∈ L2 (K , Λ(b,β) ) and h ∈ Dβ , S(b,β) ϕ(h) =
ϕ(x)G(b,β) (h; x)P(b,β) (h; x) Λ(b,β) (dx). K
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H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
Proof. Assume that ϕ is a bounded continuous function on K . Then ϕ ∈ L2 (K , Λ(b,β(n) ) ) for any n ∈ N. Observe that for h ∈ Dβ , S(b,β) ϕ(h) = lim S(b,β) Φb,β(n) ,β (ϕ) (h · 1R×({0}∪Dn ) ) n→∞ ϕ Jb,β,n (x) E(b,β(n) ) (h) Jb,β,n (x) Λ(b,β) (dx), = lim n→∞
(4.6)
K
where the first equality is obtained by (4.1) and Proposition 4.2, and the second equality by Proposition 4.4 and Corollary 4.5. Since β(n) satisfies the moment condition, by Proposition 4.3(ii) and Lemma 4.7 we see that
ϕ Jb,β,n (x) E(b,β(n) ) (h) Jb,β,n (x) Λ(b,β) (dx)
K
=
ϕ(x)E(b,β(n) ) (h)(x) Λ(b,β(n) ) (dx) K
=
ϕ(x)G(b,β(n) ) (h; x)P(b,β(n) ) (h; x) Λ(b,β(n) ) (dx) K
=
ϕ Jb,β,n (x) G(b,β(n) ) h; Jb,β,n (x) P(b,β(n) ) h; Jb,β,n (x) Λ(b,β) (dx)
K
=
ϕ Jb,β,n (x) G(b,β) (h; x)P(b,β) (h · 1R×Dn ; x) Λ(b,β) (dx),
(4.7)
K
where the last equality is obtained by Proposition 4.6. We note that ϕ(Jb,β,n (x)) → ϕ(x) as n → ∞ since Jb,β,n (x) → x in the weak topology of K , x ∈ Ω(b,β) . Now, for x ∈ Δ(b,β) (h) ∩
∞ %
Δ(b,β) (h · 1R×Dn ),
n=1
the Λ(b,β) -measure of which is 1, we have P(b,β) (h; x) − P(b,β) (h · 1R×D ; x) n |h∗ |L1 (R2 ,ν ) Υ(b,β) (h; x) − Υ(b,β) (h · 1R×D ; x) c ∗ β e n + exp
R2∗ \(R×Dn )
!
∗ h (t, u) νβ (dt, du) − 1 Υ(b,β) (h; x) .
(4.8)
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
677
Let Rn (x) denote the term in the brace of (4.8). Then Rn (x) → 0 as n → ∞, and
∗ |h∗ | +1 Rn (x) 2 + e L1c (R2∗ ,νβ ) exp ln 1 + h (t, u) N (dt, du; x) , R2∗
where the right-hand term is in L1 (K , Λ(b,β) ). Thus, by applying the Lebesgue dominated argument to (4.7) we conclude that this theorem holds for any bounded continuous function ϕ in L2 (K , Λ(b,β) ). Since the class of all bounded continuous functions on K is dense in L2 (K , Λ(b,β) ), the theorem follows. 2 The following proposition gives the analyticity of G(b,β) and P(b,β) for any (b, β) ∈ R × M+ (R). Proposition 4.9. Let (b, β) ∈ R × M+ (R). Then, the following two functions: h → G(b,β) (h; ·)
and h → P(b,β) (h; ·)
are entire from the complex Banach space Dβ into L2 (K , Λ(b,β) ). Proof. From (4.5) it is easy to see that for any x ∈ K ,
∗ Υ(b,β) (h; x) exp ln 1 + h (t, u) N (dt, du; x) , R2∗
which implies that P(b,β) (h; ·) 2 L (K ,Λ
(b,β) )
exp 2|h∗ |L1c (R2∗ ,νβ ) + (1/2)|h|2L2 (R2 ,λ ) . c
β
(4.9)
On the other hand, G(b,β) (h; ·)
β({0}) = exp L2 (K ,Λ(b,β) ) 2
+∞ 2 h(t, 0) dt . −∞
Thus, the maps h → G(b,β) (h; ·) and h → P(b,β) (h; ·), h ∈ Dβ , are locally bounded. In fact, it is obvious that h → G(b,β) (h; ·) is analytic. To see the analyticity of the map: h → 2 P(b,β) (h; ·), it is sufficient to show that for any g, h ∈ Dβ and ϕ ∈ L (K , Λ(b,β) ), the function K ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx), z ∈ C, is entire in C. Take a sequence {zn } ⊂ C which converges to z. Let Ξ = Δ(b,β) (g) ∩ Δ(b,β) (h) ∩ Δ(b,β) (zg + h) ∩
∞ %
Δ(b,β) (zn g + h) .
n=1
Then Λ(b,β) (Ξ ) = 1. Fix an x ∈ Ξ . Let {t1x , t2x , . . .} be the enumerable set of all t ∈ R with jX (t; x) = 0. Since, for any n, k ∈ N,
678
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680 ∞ (zn g ∗ + h∗ ) t x , jX (t x ; x) j
j
j =k ∞ ∗ 1 + sup |zn |; n ∈ N |g | + |h∗ | tjx , jX tjx ; x ,
(4.10)
j =k
where the last sum tends to zero as k → ∞, there is k0 ∈ N such that (zn g ∗ + h∗ ) t x , jX t x ; x < 1 j j 2 for all n ∈ N and j k0 + 1. So, for each j k0 + 1, lim log 1 + (zn g ∗ + h∗ ) tjx , jX tjx ; x = lim log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,
n→∞
n→∞
(4.11) where log stands for the principle branch of the logarithm on C \ {w ∈ R; w 0}. By (4.11), using the inequality: log(1 + w) (3/2)|w| for |w| < 1/2, (4.12) and applying the dominated convergence argument, ∞
lim
n→∞
log 1 + (zn g ∗ + h∗ ) tjx , jX tjx ; x
j =k0 +1
=
log 1 + (zn g ∗ + h∗ )(t, u) Cx (dt, du)
{(tjx ,jX (tjx ;x));j k0 +1}
=
∞
log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,
(4.13)
j =k0 +1
where Cx is the counting measure on {(tjx , jX (tjx ; x)); j k0 + 1}. By (4.11) and (4.13), lim Υ(b,β) (zn g ∗ + h∗ ; x)
n→∞
=
k 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x j =1
× exp
∞
log 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x ,
k k0 + 1.
j =k+1
Since, by (4.10) and (4.12), the above sum tends to zero as k → ∞, we obtain that lim Υ(b,β) (zn g ∗ + h∗ ; x) = Υ(b,β) (zg ∗ + h∗ ; x),
n→∞
x ∈ Ξ.
H.-H. Shih / Journal of Functional Analysis 255 (2008) 657–680
679
Therefore, ϕ(x)P(b,β) (zn g + h; x) Λ(b,β) (dx)
lim
n→∞
K
=
ϕ(x) lim P(b,β) (zn g + h; x) Λ(b,β) (dx) n→∞
by (4.9)
Ξ
(zg ∗ + h∗ )(t, u) νβ (dt, du) = exp − × =
R2∗
ϕ(x) lim Υ(b,β) (zn g ∗ + h∗ ; x) Λ(b,β) (dx) n→∞
Ξ
ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx), K
by which we conclude that the map: z ∈ C → K ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx) is continuous. For every triangular path T in the complex plane, by (4.9) and applying the dominated convergence argument we have ϕ(x)P(b,β) (zg + h; x) Λ(b,β) (dx) dz T K
=
∗ ∗ ϕ(x) lim exp − (zg + h )(t, u) νβ (dt, du) r→∞
Ξ
×
T
R2∗
! r 1 + (zg ∗ + h∗ ) tjx , jX tjx ; x dz Λ(b,β) (dx) = 0. j =1
By the Morera’s theorem, the function z ∈ C → P(b,β) (zg + h; ·) is entire. The proof is complete. 2 Acknowledgments The author would like to thank the referee for pointing out several misprints and making useful comments to improve this paper. He would like to express his sincere gratitude to Professor YuhJia Lee for his constant encouragement and many valuable suggestions during the preparation of this paper. References [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Univ. Press, Cambridge, 2004. [2] T. Hida, Analysis of Brownian Functionals, second ed., Carleton Math. Lecture Notes, vol. 13, Carleton Univ. Press, Carleton, 1978.
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[3] T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise: An Infinite Dimensional Calculus, Kluwer Academic Publishers, Dordrecht, 1993. [4] K. Itô, Multiple Wiener integral, Japan J. Math. 22 (1952) 157–169. [5] K. Itô, Spectral type of shift transformations of differential process with stationary increments, Trans. Amer. Math. Soc. 81 (1956) 253–263. [6] Y. Ito, Generalized Poisson functionals, Probab. Theory Related Fields 77 (1988) 1–28. [7] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. [8] H.-H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, FL, 1996. [9] Y.-J. Lee, H.-H. Shih, The Segal–Bargmann transform for Lévy functionals, J. Funct. Anal. 168 (1999) 46–83. [10] Y.-J. Lee, H.-H. Shih, Analysis of generalized Lévy white noise functionals, J. Funct. Anal. 211 (2004) 1–70. [11] Y.-J. Lee, H.-H. Shih, Quantum decomposition of Lévy processes, in: T. Hida (Ed.), Stochastic Analysis: Classical and Quantum, World Scientific, Singapore, 2005, pp. 86–99. [12] Y.-J. Lee, H.-H. Shih, Lévy white noise measures on infinite dimensional spaces: Existence and characterization of the measurable support, J. Funct. Anal. 237 (2006) 617–633. [13] Y.-J. Lee, H.-H. Shih, Analysis of stable white noise functionals, in: A.B. Cruzeiro, H. Ouerdiane, N. Obata (Eds.), Mathematical Analysis of Random Phenomena, World Scientific, Singapore, 2007, pp. 121–140. [14] J.M. Lindsay, Quantum stochastic analysis in quantum independent increment processes, in: M. Schurmann, U. Franz (Eds.), I: From Classical Probability to Quantum Stochastic Calculus, in: Lecture Notes in Math., vol. 1865, Springer, Berlin, 2005, pp. 181–271. [15] E.W. Lytvynov, A.L. Rebenko, G.V. Schepa’nuk, Wick calculus on spaces of generalized functions of compound Poisson white noise, Rep. Math. Phys. 39 (1997) 219–248. [16] G.D. Nunno, B. Øksendal, F. Proske, White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004) 109–148. [17] K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992. [18] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math., Cambridge Univ. Press, Cambridge, 1999.
Journal of Functional Analysis 255 (2008) 681–725 www.elsevier.com/locate/jfa
Inviscid limit for the energy-critical complex Ginzburg–Landau equation Chunyan Huang, Baoxiang Wang ∗ LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China Received 22 October 2007; accepted 25 April 2008 Available online 29 May 2008 Communicated by C. Kenig
Abstract In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg–Landau equation in Rn , n 3. In lower dimension case (3 n 6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in C(0, T , H˙ s (Rn )), T > 0, s = 0, 1, as a by-product, we get the regularity of solutions in H 3 for the nonlinear Schrödinger equation. In higher dimension case (n > 6), we get the similar convergent behavior in C(0, T , L2 (Rn )). In both cases we obtain the optimal convergent rate. © 2008 Elsevier Inc. All rights reserved. Keywords: Complex Ginzburg–Landau equation; Nonlinear Schrödinger equation; Inviscid limit; Energy-critical power
1. Introduction We are interested in the inviscid limit behavior between the energy-critical complex Ginzburg–Landau equation (CGL) ut − (μ + i)u + (a + i)|u|α u = 0,
u(0, x) = u0 (x),
(1.1)
and the nonlinear Schrödinger equation (NLS) vt − iv + i|v|α v = 0,
v(0, x) = v0 (x),
* Corresponding author.
E-mail addresses: [email protected] (C. Huang), [email protected] (B. Wang). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.017
(1.2)
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+ n + where √ u(t, x) and v(t, x) are complex-valued functions of (t, x) ∈ R × R , R = [0, ∞), i = −1, μ > 0, a > 0, ut = ∂u/∂t, vt = ∂v/∂t, = ∂ 2 /∂ 2 x1 + · · · + ∂ 2 /∂ 2 xn , u0 and v0 are known complex-valued functions of x ∈ Rn . In Eqs. (1.1) and (1.2), 0 < α < 4/(n − 2) (0 < α < ∞, if n = 1, 2) is said to be an energy-subcritical power, α = 4/(n − 2) is said to be an energy-critical power. Eq. (1.1) was first discovered by Ginzburg and Landau for a phase transition in superconductivity [8], which was also subsequently derived in many other areas, such as instability waves in hydrodynamics and pattern formation; cf. [11]. When α is an energysubcritical or an energy-critical power, the global well-posedness of (1.1) has been established, see for instance, Ginibre and Velo [7] and references therein. Eq. (1.2) is a basic model equation for diverse physical phenomena, including Bose–Einstein condensates, description of the envelop dynamics of a general dispersive wave in a weakly nonlinear medium (cf. [10,16]). The global well-posedness of solutions of (1.2) has been studied by many authors (cf. [3–5,13]). When α is an energy-subcritical power, the global well-posedness of (1.2) can be found in Kato [12]. When α is an energy-critical power, Bourgain [3] and Grillakis [9] showed the global well-posedness and the existence of scattering operators for Eq. (1.2) with radial and finite energy data in three spatial dimension, and the radial condition was removed by Colliander, Keel, Staffilani, Takaoka and Tao [5]. Tao [17] showed the global well-posedness of Eq. (1.2) with radial and finite energy data in higher spatial dimensions. Recently, the radial condition in higher spatial dimensions was removed by Ryckman and Visan (cf. [15,19]). In this paper, we consider the following question: does the solution of (1.1) tend to the solution of (1.2) as the parameters μ and a tend to zero? This problem has been studied by several authors; cf. Wu [23], Bechouche and Jüngel [1], Wang [20] and Machihara and Nakamura [14]. But as far as the authors can see, in the energy-critical case α = 4/(n − 2), except that Bechouche and Jüngel [1] obtained the convergent behavior in the weak-star topology, the inviscid limit behavior in the strong topology between (1.1) and (1.2) remains unsolved; cf. [20]. When α = 4/(n − 2), (1.1) and (1.2) can be rewritten as 4
ut − (μ + i)u + (a + i)|u| n−2 u = 0, vt − iv + i|v|
4 n−2
v = 0,
u(0, x) = u0 (x),
v(0, x) = v0 (x).
(1.3) (1.4)
We will show that when the spatial dimension n 3, the solution of (1.3) strongly converges to that of (1.4) in C(0, T ; L2 ) for any T > 0. Moreover, when 3 n 6, we can prove the stronger convergent behavior in C(0, T ; H˙ 1 ) for any T > 0. Let Lr (Rn ) be the Lebesgue space, f r := f Lr (Rn ) , f ˙ k := ∇ k f r , H˙ k := H˙ k , H k := L2 ∩ H˙ k . Put Hr
E u(t) =
2
Rn
2 n − 2 2n 1 u(t, x) n−2 dx, ∇u(t, x) + 2 2n
(1.5)
which is said to be the energy for the CGL (1.3) and the NLS (1.4). E(u) is invariant under the scaling u(t, x) → uλ (t, x) = λ Our main results are the following.
n−2 2
u λ2 t, λx ,
λ > 0.
(1.6)
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Theorem 1.1. Let 3 n 6. Let u0 ∈ H 1 , v0 ∈ H 3 , 0 < δ 1. Assume that T > 1 is arbitrary and u0 − v0 H 1 δ,
0 < μT , a δ.
(1.7)
Then the solutions of (1.3) and (1.4) satisfy the following approximate behavior: k ∇ (u − v)
2 n L∞ t Lx ([0,T ]×R )
C ∇ k (u0 − v0 )2 + μT + a ,
k = 0, 1,
(1.8)
where C := C(u0 H 1 , v0 H 3 ) denotes a constant that depends only on u0 H 1 and v0 H 3 . Theorem 1.2. Let n 5. Let u0 ∈ H 1 , v0 ∈ H 1 , 0 < δ 1. Assume that T > 1 is arbitrary and u0 − v0 H 1 δ,
0 < μT , a δ.
(1.9)
Then the solutions of (1.3) and (1.4) satisfy the following approximate behavior: 1/2 +a , u − vL∞ 2 n C u0 − v0 2 + (μT ) t Lx ([0,T ]×R )
(1.10)
where C := C(n, u0 H 1 , v0 H 1 ) denotes a constant that depends only on n, u0 H 1 and v0 H 1 . Theorem 1.3. Let n > 6. u0 ∈ H 1 , 0 < a, μ 1. Then the solutions u of (1.3) have the following upper bound estimate: u
2(n+2) Lt,xn−2 (R+ ×Rn )
C E(u0 ) ,
(1.11)
where C(E(u0 )) denotes a constant that depends only on E(u0 ) and is independent of μ, a. We now first briefly sketch the proofs of Theorems 1.1 and 1.2. On the basis of our earlier work [20], if we get the upper bounds of the solutions u and v of Eqs. (1.3) and (1.4) in the spaces L2(n+2)/(n−2) (R+ × Rn ), then we can prove our Theorems 1.1 and 1.2 by iteration method. Using Bourgain’s, Colliander, Keel, Staffilani, Takaoka and Tao’s, and Ryckman and Visan’s results (cf. [3,5,15,19]), we see that v in the space L2(n+2)/(n−2) (R+ × Rn ) has an upper bound that depends only on v0 H˙ 1 . A natural idea seems to apply a similar way as in [5,15,19] obtaining a uniform upper bound of u in L2(n+2)/(n−2) (R+ × Rn ) for all 0 < μ, a 1. However, in [5,15,19], the proofs of the global well-posedness of (1.4) rely upon the symmetric property of the Schrödinger semi-group eit (i.e., eit has the same dispersive properties in the cases t < 0 and t > 0) and this symmetric property is invalid for the Ginzburg–Landau semi-group e(μ+i)t . Moreover, the proof will become very complicated if we imitate the procedures as in [5,15,19]. When 3 n 6, we will give a simple proof of Theorem 1.1 by using the perturbation analysis method, i.e., treating the NLS as a perturbative equation of the CGL: wt − (μ + i)w + g(v, w) = 0,
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4
4
where w = u − v, g(v, w) = −μv + (a + i)(|w + v| n−2 (w + v) − |v| n−2 v) + a|v| n−2 v. Hence, it suffices to show that w is uniformly bounded in L2(n+2)/(n−2) (R+ × Rn ) for all 0 < μ, a 1 as the initial data w0 is small enough. Following the idea as in [20], we treat μv as a part of the nonlinear terms. We consider the following integral equation: t w=e
(μ+i)t
w0 −
e(μ+i)(t−τ ) g(v, w)(τ ) dτ. 0
In view of the Strichartz type estimates for the Ginzburg–Landau semi-group (which are similar to those for the Schrödinger semi-group), in order to control w in the space L2(n+2)/(n−2) (R+ × Rn ), we need to control v in H 3 . For n = 3, 4, the upper bound of v in H 3 can be easily obtained, since the nonlinearity |v|4/(n−2) v is a C ∞ function (cf. [3,20]). For n = 5, 6, |v|4/(n−2) v is only two-times differentiable, we need to use the “formal time-differentiation method” to overcome the regularity loss of the nonlinearity. After showing the upper bound of v in H 3 , we can perform the perturbation analysis and finally obtain our result, as desired. For higher spatial dimensions n > 6, since |v|4/(n−2) v is not two-times differentiable, the “formal time-differentiation method” seems invalid to get the upper bound of v in H 3 . This is why we assume that n 6 in Theorem 1.1. When n > 6, we shall bound u in L2(n+2)/(n−2) (R+ × Rn ) by applying the ideas as in Bourgain [3], Colliander, Keel, Staffilani, Takaoka and Tao [5], and Visan [19], i.e., the induction method on the energy E. In detail, for any energy E 0 and T > 0, define MT (E) := sup u
2(n+2)
,
Lt,xn−2 ([0,T ]×Rn )
where u ranges over all of the S˙ 1 solutions of (1.3) on [0, T ] × Rn with E(u) E and the parameters 0 μ, a 1. Put M(E) := sup MT (E): T > 0 . Noticing that both E(u) and MT (E) are invariant under the scaling (1.6), we immediately have Proposition 1.4. For any E, T > 0, we have M(E) = MT (E) ≡ M1 (E).
(1.12)
For E < 0, define M(E) = 0. Our goal is to show M(E) < ∞ for any energy E. To be more comprehensive, we write Q = E: M(E) < ∞ . The perturbation analysis implies that if E ∈ Q then E + ε ∈ Q for some 0 < ε 1, and so, Q is open, see Lemma 2.9 below for details. Moreover, Q contains zero and is connected. We use the induction-on-energy argument as in [3,5,19]. Assume for contradiction that M(E) can be infinite, consider the minimal energy Ec = inf E: M(E) = ∞ < ∞,
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then naturally M(E) < ∞ for all E < Ec . Since Q is open and contains zero, then Ec > 0. Using this critical energy Ec , we can construct a near blowup solution whose energy is near to Ec with 2(n+2)
an enormous Lt,xn−2 norm. We will prove that such near blowup solution does not exist, then eventually we get a contradiction. In detail, by the critical property of Ec and Lemma 2.9, we have Lemma 1.5 (Induction hypothesis). Let 0 < μ, a 1 be variable parameters. Let t0 ∈ R+ , suppose u(t0 ) is an H˙ 1 function with E(u(t0 )) Ec − η for some η > 0, then there exists a global solution u to (1.3) on [t0 , +∞) × Rn with initial data u(t0 ) at time t0 satisfying u
2(n+2)
Lt,xn−2 ([t0 ,+∞)×Rn )
M(Ec − η).
(1.13)
We need eight small parameters 1 η0 η 1 η 2 η 3 η 4 η 5 η 6 η 7 > 0 where each ηj is chosen sufficiently small depending on previous η0 , . . . , ηj −1 and the critical energy Ec . Since M(Ec ) is infinite, we can find an S˙ 1 solution u of (1.3) with Ec /2 E(u) Ec such that for some T > 0, μ, a ∈ (0, 1], u
2(n+2) Lt,xn−2 ([0,T ]×Rn )
=
1 , η7
(1.14)
which leads to the following definition. Definition 1.6. A minimal energy blowup solution (MEBS for short) to (1.3) is an S˙ 1 Schwartz solution which satisfies (1.14) and with energy 1 Ec E(u) Ec . 2
(1.15)
Using the energy estimate for the CGL, we claim that for the MEBS, both μ and a must be very small. In fact, we have Proposition 1.7. If u is a MEBS, then we have max(μ, a) η6 . Proof. By the energy estimate for the CGL (see Theorem 2.7 below), we have T μ
u(τ, x)2 dx dτ + a
0 Rn
T
2(n+2) u(τ, x) n−2 dx dτ E(u0 ).
0 Rn
If a η6 , we see that u
2(n+2) Lt,xn−2 ([0,T ]×Rn )
E(u0 ) η6
(n−2)/2(n+2) ,
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which contradicts (1.14). So, it suffices to consider the case μ η6 . Interpolating Lt,xn−2 be2n
2n
n−2 tween L∞ and L2t Lxn−4 , we have t Lx
u
2(n+2) Lt,xn−2 ([0,T ]×Rn )
u
4/(n+2) 2n n−2 L∞ t Lx
u
(n−2)/(n+2) 2n
L2t Lxn−4
E(u0 )4/(n+2) u E(u0 )
4/(n+2)
which also contradicts (1.14).
(n−2)/(n+2) L2t,x
E(u0 ) η6
(n−2)/2(n+2) ,
2
In the following we always assume that 0 μ, a η6 . By choosing a subinterval [0, T ∗ ] ⊂ [0, T ], such that u
2(n+2) Lt,xn−2 ([0,T ∗ ]×Rn )
=
1 . η5
(1.16)
We divide [0, T ∗ ] = I0 ∪ I1
(1.17)
2(n+2)
with each subinterval containing half of the Lt,xn−2 ([0, T ∗ ] × Rn ) norm. We want to show that u
2(n+2) Lt,xn−2 (I0 ×Rn )
C η1 , . . . , η4 , E(u0 ) .
(1.18)
Following [5,19], the proof of (1.18) proceeds in the following four steps. (1) Using the same way as in [5,19], we can show that the MEBS is localized in physical and frequency space for any time t ∈ I0 . (2) We generalize the frequency localized Morawetz estimate (FLME) of the NLS to the CGL and get a uniform version of the FLME which is independent of μ, a. Our 4 idea is to apply the known Morawetz estimate for the NLS [5,19] and treat μu and a|u| n−2 u as the perturbative terms, which can be controlled by the energy estimates for the CGL. (3) We show the frequency-localized L2 mass conservation law to prevent energy evacuation to the high frequency. Comparing with the NLS, the new difficulty lies in the fact that the L2 norm of the solutions of the CGL has some dissipation (see (2.13) for details) and we need to show that this dissipation vanishes when μ, a 0. Another difficulty is that the Ginzburg–Landau semi-group e(μ+i)t is not symmetric with respect to time t, one needs to carefully deal with the backward case t < 0. (4) Following [5] and using the perturbation analysis, we can finish the proof of (1.18).
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2. Notations and known results In this section, we recall some notations, basic facts and the Strichartz estimates for the NLS and the CGL. 2.1. Notations q
Let I ⊂ R be an interval. We use Lt Lrx to denote the space–time norm: uLq Lr (I ×Rn ) := t
x
I
u(t, x)r dx
q/r
1/q (2.1)
dt
Rn
with the usual modification when q or r is equal to infinity. We denote uLq (I,H˙ k ) := r q q ∇ k uLqt Lr (I ×Rn ) . If there is no confusion, we will write Lt∈I Lrx := Lt Lrx (I ×Rn ) and Lrt∈I,x := x Lrt Lrx (I × Rn ). We introduce the Littlewood–Paley projection operators in following way. Let ϕ(ξ ) be a radially symmetric bump function adapted to the ball {ξ ∈ Rn : |ξ | 2} which equals 1 on the unit ball. Let ψ(ξ ) := ϕ(ξ ) − ϕ(2ξ ). For each dyadic number N , we define the Fourier multipliers: PN f := F −1 ϕ(ξ/N )Ff, P>N f := F −1 1 − ϕ(ξ/N ) Ff, PN f := F −1 ψ(ξ/N )Ff. Recall that for any dyadic number N , there hold the telescoping identities PN f =
PM f ;
P>N f =
MN
PM f ;
f=
M>N
PM f,
M
for all Schwartz f , where M ranges over dyadic numbers. Define PM<·N := PN − PM =
PN ,
M
where N , M N are all dyadic numbers. Similarly we define P
(2.2) (2.3) (2.4)
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⎧ u , 2n ⎪ 2 n−2 (I ×Rn ) 2 ⎨ L∞ t Lx ∩Lt Lx uS˙ 0 (I ×Rn ) := 2 ⎪ ⎩ N ∈2Z PN u
n 6, 1/2
2n 2 n−2 (I ×Rn ) 2 L∞ t Lx ∩Lt Lx
,
n > 6,
and for any s > 0, we define the Strichartz norm S˙ s (I × Rn ) by uS˙ s (I ×Rn ) := |∇|s uS˙ 0 (I ×Rn ) . One can easily check that for any admissible pairs (q, r), k ∇ u
q
Lt Lrx (I ×Rn )
uS˙ k (I ×Rn ) ,
k = 0, 1.
2.2. Strichartz inequalities Denote 4 2∗ = 2 + , n
2∗∗ =
2n , n−2
n 3.
Throughout this paper, p will stand for the dual number of p ∈ [1, ∞], i.e. 1/p + 1/p = 1. Let us follow Cazenave and Weissler’s notation (cf. [4]): Definition 2.1. Let n 3. (q, r) is said to be an admissible pair if 2/q = n(1/2 − 1/r), 2 q, r 2∗∗ . We say that (q , r ) is a dual admissible pair if (q, r) is an admissible pair. From the definition above, we can easily find that (∞, 2), (2∗ , 2∗ ), (2, 2∗∗ ) and (q, r) =
2(n + 2) 2n(n + 2) , 2 n−2 n +4
are admissible pairs and (1, 2), ((2∗ ) , (2∗ ) ), (2, (2∗∗ ) ) are dual admissible pairs. We denote S(t) := e
it
=F
−1 −it|ξ |2
e
t F,
A :=
S(t − τ ) · dτ, 0
Sμ (t) := e
(μ+i)t
=F
−1 −(μ+i)t|ξ |2
e
t F,
Aμ :=
Sμ (t − τ ) · dτ. 0
Eqs. (1.3) and (1.4) can be formally rewritten as u(t) = Sμ (t)u0 + Aμ ut − (μ + i)u ,
(2.5)
v(t) = S(t)v0 + A (vt − iv).
(2.6)
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We use X Y to denote the estimate X CY , C is a universal positive constant. It is well known that S(t) satisfies the decay estimate S(t)ϕ |t|−n(1/2−1/p) ϕp , p
1/p + 1/p = 1, 2 p ∞.
(2.7)
Since e−|ξ | is an Lp multiplier, i.e., e−|ξ | ∈ Mp , Mp is Hörmander’s multiplier space, and 2 Mp is isometrically invariant under surjective affine transformation, we have e−|ξ | Mp = 2
2
e−μt|ξ | Mp (cf. [2]). It follows from (2.7) that for any 2 p < ∞ and t > 0, 2
Sμ (t)ϕ C S(t)ϕ Ct −n(1/2−1/p) ϕp , p p
(2.8)
where C is independent of μ and t, ϕ ∈ Lp , 1/p + 1/p = 1. Combining (2.8) with the standard Strichartz estimates for the Schrödinger equation, one can prove the following time–space Lp –Lp estimates. We omit the details here, one can see Wang [20] for nonendpoint case and [21] for endpoint case. Lemma 2.2. Let n 3, k ∈ N ∪ {0}. Let (q, r) be any admissible pair and (q1 , r1 ) be any dual admissible pair. Then we have Sμ (t)ϕ
Lq (0,∞;H˙ rk )
ϕH˙ k ,
Aμ f Lq (0,T ;H˙ k ) f r
(2.9)
q
L 1 (0,T ;H˙ k )
(2.10)
,
r1
for all ϕ ∈ H˙ k , f ∈ Lq1 (0, T ; H˙ rk ), 0 < T ∞. When μ = 0, it reduces to the standard 1 Strichartz estimates for the Schrödinger equation (see [13]).
We emphasize that the estimates (2.9) and (2.10) are independent of μ > 0. Using Lemma 2.2 together with Duhamel’s formula (2.5) and the triangle inequality, we can verify the following multi-linear form of the Strichartz estimates. Corollary 2.3. Let I be a time interval, k = 0, 1, and let u : I × Rn → C be a solution to ut − (μ + i)u =
M
Fm
m=1
for some functions F1 , . . . , FM , where μ 0, then for any admissible pair (q, r),
M k u(t0 ) ˙ k n + ∇ Fm q r C n H (R ) L L (I ×R )
k ∇ u
t
x
m=1
q
r
Lt m Lxm (I ×Rn )
,
(2.11)
, r ) are dual admissible pairs. where t0 = min I , (q1 , r1 ), . . . , (qm m
We need the following improved Strichartz estimate for the nonhomogeneous term, which can be proved by following the method in [6] and the Strichartz estimates for the CGL.
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Proposition 2.4. Let I ⊂ [0, +∞) be a compact time interval. Let (q, r) and (q, ˜ r˜ ) be two Schrödinger-acceptable pairs satisfying the scaling condition q1 + q1˜ = n2 (1 − 1r − 1r˜ ) and either n−2 r n < < , n r˜ n − 2
1 1 + = 1, q q˜
1 1 , r q
and
1 1 r˜ q˜
or n−2 r n . n r˜ n − 2
1 1 + < 1, q q˜ Then e(μ+i)(t−s) F (s) ds s
q
Lt Lrx (I ×Rn )
Here we say (q, r) Schrödinger-acceptable means that (∞, 2).
1 q
F
q˜
Lt Lrx˜ (I ×Rn )
.
< n( 12 − 1r ), 1 q, r ∞, or (q, r) =
Remark 2.5. We point out here that the pairs (q, r) = (2, n2n(n−2) ˜ r˜ ) = (2, n2n(n−2) 2 −3n−2 ), (q, 2 −5n+10 ) satisfying the hypothesis in Proposition 2.4. Remark 2.6. When we prove a property of the CGL, if we only use the (improved) Strichartz estimates and solve the integral equation (2.5) forward, then we can use an analogous way to the NLS, for the CGL satisfies the same (improved) Strichartz estimates as the NLS. This fact is very important for our later discussion. 2.3. Global well-posedness Theorem 2.7 (Global well-posedness for CGL). (Cf. [7,20].) Let μ, a 0, 0 < max(μ, a) < ∞, 2n u0 ∈ H˙ 1 . Then there exists a unique global solution u with ∇u ∈ Ct L2x ∩ L2t Lxn−2 to (1.3) such that E u(t) + μ
t 0
u(τ, x)2 dx dτ + a
Rn
t 0
2(n+2) u(τ, x) n−2 dx dτ E(u0 ).
(2.12)
Rn
2n
Moreover, if u0 ∈ L2 , then u ∈ Ct L2x ∩ L2t Lxn−2 and 1 u(t)2 + μ 2 2
t
∇u(τ )2 dτ + a 2
0
In [7], the upper bound of u
t
2n 1 u(τ ) n−2 u0 22 . 2n dτ = 2 n−2
(2.13)
0 2(n+2)
Lt,xn−2 (R+ ×Rn )
in Theorem 2.7 relies upon max(μ, a) and it
tends to infinity as max(μ, a) → 0 (which can be seen from the proof of Proposition 1.7).
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We also need the following result which can be found in Colliander, Keel, Staffilani, Takaoka and Tao [5] for n = 3, and Ryckman and Visan (cf. [15,19]) for n 4. Theorem 2.8 (Global well-posedness for NLS). Let n 3. If E(v0 ) < ∞, then there exists a 2(n+2)
unique global solution v ∈ Ct H˙ x1 ∩ Lt,xn−2 to (1.4) such that v(t, x) 2(n+2) C E(v0 ) Lt,xn−2 ∩S˙ 1 (R×Rn )
for some constant C(E(v0 )) depending only on the energy. 2.4. Perturbation lemma We need the following perturbation lemma which is useful in the proof of the main Theorem 1.3. Lemma 2.9 (Long-time perturbation theory). Let n > 6. Let I ⊂ [0, +∞) be a compact time interval and let u˜ be an approximate solution to (1.3) on I × Rn in the sense that 4
˜ n−2 u˜ + e, iu˜ t + (1 − μi)u˜ = (1 − ai)|u| for some function e. Assume that u ˜
2(n+2)
Lt,xn−2 (I ×Rn )
M,
u ˜ L∞ n E ˙1 t H (I ×R ) x
˜ 0 ) in the sense that for some constants M, E > 0. Let t0 = min I and let u(t0 ) close to u(t u(t0 ) − u(t ˜ 0 )H˙ 1 (Rn ) E (2.14) for some E > 0. Assume also the smallness conditions
2 ∇PN e(μ+i)(t−t0 ) u(t0 ) − u(t ˜ 0 ) 2(n+2)
2n(n+2) 2 Lt n−2 Lx n +4 (I ×Rn )
N
∇e
2n
L2t Lxn+2 (I ×Rn )
1 2
ε,
ε,
(2.15) (2.16)
for some 0 < ε < ε1 , where ε1 = ε1 (E, E , M) > 0 is a small constant. Then there exists a solution u to (1.3) on I × Rn with initial data u(t0 ) satisfying ∇(u − u) ˜
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I ×Rn )
7 C(E, E , M) ε + ε (n−2)2 ,
7 u − u ˜ S˙ 1 (I ×Rn ) C(E, E , M) E + ε + ε (n−2)2 ,
uS˙ 1 (I ×Rn ) C(E, E , M).
(2.17) (2.18) (2.19)
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Here C(E, E , M) > 0 is a non-decreasing function of E, E , M, and the dimension n. Remark 2.10. Recall that the Ginzburg–Landau semi-group Sμ (t) satisfies the same Strichartz estimates as the corresponding Schrödinger semi-group S(t). Noticing that t0 = min I , we only need to consider the CGL forward in Lemma 2.9, then the proof is parallel to that of the NLS (see Tao and Visan [18]) and we omit the details of the proof. 3. Proof of Theorem 1.1 In this section, we prove the main Theorem 1.1. We will use the perturbation analysis to give a short proof of Theorem 1.1 when 3 n 6. 3.1. Nonlinear estimates The purpose of this subsection is to derive some nonlinear estimates, which will be applied in 4 the next subsection. Write F (z) = |z| n−2 z, let Fz , Fz¯ be the complex derivatives
∂F ∂F 1 ∂F 1 ∂F Fz := −i , Fz¯ := +i , 2 ∂x ∂y 2 ∂x ∂y then we see that Fz (z) =
4 n |z| n−2 , n−2
Fz¯ (z) =
4 z2 2 |z| n−2 2 , n−2 |z|
(3.1)
4
which implies that Fz (z) and Fz¯ (z) are both O(|z| n−2 ). Note that the difference of two nonlinear terms satisfies the integral identity: 1 F (u) − F (v) =
Fz v + θ (u − v) (u − v) + Fz¯ v + θ (u − v) (u − v) dθ.
(3.2)
0
Hence, 4 4 F (u) − F (v) |u − v| |u| n−2 + |v| n−2 .
(3.3)
Observe that F (u) obeys the fractional chain rule: ∇F u(x) = Fz u(x) ∇u(x) + Fz¯ u(x) ∇u(x).
(3.4)
By (3.1), (3.4) and Hölder’s inequality, we can get the following estimate. Proposition 3.1. For k = 0, 1, we have. k ∇ F (u)
4
2n L2t Lxn+2 (I ×Rn )
u n−2 2(n+2) Lt,xn−2 (I ×Rn )
k ∇ u
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
. (I ×Rn )
(3.5)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
693
Similarly, we can invoke Hölder’s inequality and (3.1), (3.3), (3.4) to get: Proposition 3.2. For k = 0, 1, we have k ∇ F (u) F (u) − F (v)
(2∗ ) Lt,x (I ×Rn )
k ∇ u
4
(2∗ ) Lt,x (I ×Rn )
u n−2 2(n+2)
4 u n−2 2(n+2)
∗
L2t,x (I ×Rn )
Lt,xn−2 (I ×Rn ) 4
+ v n−2 2(n+2)
Lt,xn−2 (I ×Rn )
Lt,xn−2 (I ×Rn )
(3.6)
,
u − vL2∗ (I ×Rn ) .
(3.7)
t,x
Now we estimate the difference with one-order derivative. Write w = u − v. When n 6, in view of the chain rule (3.4), we calculate ∇ F (u) − F (v) = Fz (u)∇u + Fz¯ (u)∇ u¯ − Fz (v)∇v − Fz¯ (v)∇ v¯ = Fz (u) − Fz (v) ∇v + Fz¯ (u) − Fz¯ (v) ∇ v¯ + Fz (u)∇(u − v) + Fz¯ (u)∇(u − v) Fz (u) − Fz (v) + Fz¯ (u) − Fz¯ (v) |∇v| + Fz (u) + Fz¯ (u)∇(u − v) 6−n 6−n 4 |∇v||u − v| |u| n−2 + |v| n−2 + ∇(u − v)|u| n−2 6−n 6−n 4 |∇v||w| |v| n−2 + |w| n−2 + |∇w||w + v| n−2 . (3.8) Therefore by Hölder’s inequality and (3.8), we get Proposition 3.3. When n 6, there hold ∇ F (u) − F (v)
(2∗ ) Lt,x
6−n 6−n n−2 ∇vL2∗ v n−2 + w 2(n+2) 2(n+2) w t,x
Lt,xn−2
Lt,xn−2
2(n+2)
Lt,xn−2
4 4 n−2 + ∇wL2∗ v n−2 2(n+2) + w 2(n+2) , t,x
∇ F (u) − F (v)
2n L2t Lxn+2
∇v
Lt,xn−2
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4
+ ∇w
6−n 6−n n−2 v n−2 2(n+2) + w 2(n+2) w
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4
(3.9)
Lt,xn−2
Lt,xn−2
Lt,xn−2
4 4 n−2 v n−2 2(n+2) + w 2(n+2) .
Lt,xn−2
2(n+2)
Lt,xn−2
(3.10)
Lt,xn−2
We need the following estimates. Proposition 3.4. We have F (u) ∇F (u)
L2
Cu
4
L2
6−n n−2 H˙ 1
CuHn−2 ˙ 1 uH˙ 2 ,
u2H˙ 2 Cu
4 n−2 H˙ 1
uH˙ 3 ,
(3.11) n 6.
(3.12)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
Proof. By Hölder’s inequality and Sobolev’s embedding, F (u)
L2
n+2 4 n−2 n+2 n+2 n+2 n−2 Cu n−2 . 2n+4 C ∇uL2 uL2
L
n−2
Similarly when n 6, ∇F (u)
L2
4 4 C |u| n−2 ∇uL2 C |u| n−2 Ln ∇u
C u
6−n 4 H˙ 1
u
n−2 4 H˙ 2
4 n−2
4
2n L n−2
Cu n−24n uL2 L n−2
uH˙ 2 .
(3.13)
When n 6, we need to compute F (u). Noticing that Fzz (z) = 6−n 2n z |z| n−2 |z| , (n−2)2
Fz¯ z¯ =
6−n 3 8−2n z |z| n−2 |z| 3, (n−2)2
6−n 2n |z| n−2 |z| z , Fz¯z (n−2)2
= Fz¯ z =
we find that F (u(x)) obeys the chain rule:
2 2 2 F u(x) = Fzz u(x) ∇u(x) + 2Fz¯z u(x) ∇u(x) + Fz¯ z¯ u(x) ∇u(x) + Fz u(x) u(x) + Fz¯ u(x) u(x). (3.14) Therefore, it satisfies the estimate: 4 F u(x) C |u| 6−n n−2 |∇u|2 + |u| n−2 |u| ,
n 6.
2
(3.15)
3.2. Proof of Theorem 1.1 Suppose that u and v are solutions of (1.3) and (1.4), respectively. Defining w = u − v, then we see that w satisfies:
4 4 4 wt − (μ + i)w − μv + (a + i) |w + v| n−2 (w + v) − |v| n−2 v + a|v| n−2 v = 0, w(0) = w0 .
w can be rewritten as 4 4 4 w = Sμ (t)w0 − Aμ −μv + (a + i) |w + v| n−2 (w + v) − |v| n−2 v + a|v| n−2 v .
(3.16)
If the initial data w0 is small enough, then we can prove the following estimate for the solution of (3.16), whose proof will be given in the end of this section. Proposition 3.5. Let 3 n 6. Suppose that w is a solution to (3.16), T > 0 is arbitrary, w0 H 1 δ 1, u0 ∈ H 1 , v0 ∈ H 3 , 0 < μT , a δ. Then we have w
2(n+2) Lt,xn−2 ([0,T ]×Rn )
C u0 H 1 , v0 H 3 .
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
695
Proposition 3.6. Let 3 n 6. Suppose v is a solution to (1.4) and v0 ∈ H 3 , then for k = 0, 1, 2, 3, v(t)
n ˙k + L∞ t Hx (R ×R )
C.
(3.17)
Proof. If n = 3, 4, (3.17) is shown by Bourgain [3]. If n = 5, 6, since |v|4/(n−2) v is not threetimes differentiable, we need to use the “formal time-differentiation method” to overcome the regularity loss of the nonlinearity. This method has been used by Kato [12], Cazenave and Weissler [4]. The case k = 0, 1 follows immediately from the mass and energy preserving fact u(t)2 = u0 2 and E(v(t)) = E(v0 ) k ∇ v
2 + n L∞ t Lx (R ×R )
C ∇ k v0 2 ,
k = 0, 1.
(3.18)
When k = 2, v can be written as the following integral form,1 t v = S(t)v0 − i
S(t − τ )F (v) dτ. 0
In view of Theorem 2.8, we see that v
2(n+2) Lt,xn−2 (R+ ×Rn )
+ ∇v
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (R+ ×Rn )
C v0 H˙ 1 .
(3.19)
So, for a given small parameter 0 < η 1, we can divide R+ into J disjoint intervals Ij such that R+ = Jj=1 Ij and v
2(n+2)
Lt,xn−2 (Ij ×Rn )
+ ∇v
2(n+2) n−2
Lt
η,
2n(n+2) 2 +4
Lx n
j = 1, . . . , J.
(3.20)
(Ij ×Rn )
By Corollary 2.3 and (3.15), 4 vS˙ 0 (Ij ×Rn ) v(min Ij )2 + |v| n−2 v 6−n + |v| n−2 |∇v|2
(2∗ )
Lt,x (Ij ×Rn )
2n
L2t Lxn+2 (Ij ×Rn )
.
(3.21)
Using Hölder’s inequality and Sobolev’s embedding, one has that 6−n |v| n−2 |∇v|2
2n
L2t Lxn+2 (Ij ×Rn )
1 Strictly speaking, in order to guarantee the following the a priori estimates hold, we should first show the local wellposedness of solutions in H 3 by using the standard contraction mapping argument, namely, one needs to construct a metric space in which every norm of the solution appeared in the following estimates is finite, see Remark 3.7 below.
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725 6−n
v n−2 2(n+2)
∇v
Lt,xn−2 (Ij ×Rn ) 6−n
v n−2 2(n+2)
∇v
Lt,xn−2 (Ij ×Rn )
2(n+2)
Lt,xn−2 (Ij ×Rn )
∇v
2(n+2) n−2
Lt
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
2n(n+2) 2 +4
Lx n
2 ∇ v
(Ij ×Rn )
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(Ij ×Rn )
4
η n−2 vS˙ 0 (Ij ×Rn ) ,
(3.22)
and 4 |v| n−2 v
(2∗ ) Lt,x (Ij ×Rn )
4
v n−2 2(n+2) Lt,xn−2 (Ij ×Rn )
vL2∗ (Ij ×Rn ) t,x
4
η n−2 vS˙ 0 (Ij ×Rn ) .
(3.23)
Hence, in view of (3.20)–(3.23), we have 4 vS˙ 0 (Ij ×Rn ) v(min Ij )2 + η n−2 vS˙ 0 (Ij ×Rn ) .
(3.24)
Taking j = 1 and noticing that v(min I1 ) = v0 , we obtain that vS˙ 0 (I1 ×Rn ) 2v0 H˙ 2 ,
vL∞ 2 n 2v0 H˙ 2 , t Lx (I1 ×R )
(3.25)
which implies that v(min I2 )2 2v0 H˙ 2 . Repeating the procedure above, we have vS˙ 0 (I2 ×Rn ) 22 v0 H˙ 2 ,
2 vL∞ 2 n 2 v0 H˙ 2 . t Lx (I2 ×R )
By iteration, for any j = 1, . . . , J , we see that vS˙ 0 (Ij ×Rn ) 2j v0 H˙ 2 ,
j vL∞ 2 n 2 v0 H˙ 2 . t Lx (Ij ×R )
Hence, vL∞ 2 (R+ ×Rn ) C v0 H˙ 1 v0 H˙ 2 , L t x
(3.26)
which concludes the conclusion in the case k = 2. Finally, we consider the case k = 3 and use the equation 4 ∇vt − i∇v = −i∇ |v| n−2 v = −i∇F (v)
(3.27)
to estimate vH˙ 3 . It suffices to calculate ∇vt 2 and ∇(F (v))2 . It is easy to see that ∇vt satisfies the following integral equation: t ∇vt = iS(t)∇v0 − iS(t)∇F (v0 ) − i 0
S(t − τ )∇∂τ F v(τ ) dτ.
(3.28)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
697
Let Ij (j = 1, . . . , J ) be the decomposition of R+ as in (3.20). We have 4 ∇vt S˙ 0 (Ij ×Rn ) v(minIj )H˙ 3 + ∇F v(min Ij ) 2 + |v| n−2 ∇vt
(2∗ )
Lt,x (Ij ×Rn )
6−n + |v| n−2 ∇vvt
2n
L2t Lxn+2 (Ij ×Rn )
(3.29)
.
Using the same way as in Proposition 3.2, we get 4 |v| n−2 ∇vt
4
(2∗ ) Lt,x (Ij ×Rn )
v n−2 2(n+2) Lt,xn−2 (Ij ×Rn )
∇vt L2∗ (Ij ×Rn ) t,x
4
η n−2 ∇vt S˙ 0 (Ij ×Rn ) .
(3.30)
In view of Hölder’s inequality and Sobolev’s embedding, we have 6−n |v| n−2 ∇vvt
2n
L2t Lxn+2 (Ij ×Rn )
6−n
v n−2 2(n+2)
∇v
Lt,xn−2 (Ij ×Rn ) 6−n
v n−2 2(n+2)
∇v
Lt,xn−2 (Ij ×Rn )
η
4 n−2
2(n+2) n−2
Lt
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
2n(n+2) 2 +4
Lx n
vt (Ij ×Rn )
2(n+2)
Lt,xn−2 (Ij ×Rn )
∇vt (Ij ×Rn )
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(Ij ×Rn )
∇vt S˙ 0 (Ij ×Rn ) .
(3.31)
Inserting the estimates above into (3.29) and using Proposition 3.4, we conclude that ∇vt S˙ 0 (Ij ×Rn ) v(min Ij )H˙ 3 + ∇F v(min Ij ) 2 4/(n−2) . v(min Ij )H˙ 3 1 + v(min Ij )H˙ 1
(3.32)
Taking notice of v(min I1 ) = v0 , we have 4 n−2 1 + v . ∇vt L∞ v 2 n 3 ˙ 0 0 H t Lx (I1 ×R ) H˙ 1
(3.33)
On the other hand, from Proposition 3.4, (3.18) and (3.26), it follows that for any t > 0, ∇F v(t)
L2
6−n 2 v(t)Hn−2 ˙ 1 v(t) H˙ 2 C v0 H˙ 1 v0 H˙ 3 .
(3.34)
Hence, by Eq. (3.27), ∇vL∞ 2 n ∇vt L∞ L2 (I ×Rn ) + ∇F v(t) ∞ 2 Lt Lx (I1 ×Rn ) t Lx (I1 ×R ) t x 1 C v0 H˙ 1 v0 H˙ 3 .
(3.35)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
(3.35) implies that v(min I2 )
H˙ 3
C v0 H˙ 1 v0 H˙ 3 .
(3.36)
Combining (3.32) with (3.34) and (3.36), we have ∇vt S˙ 0 (I2 ×Rn ) C v0 H˙ 1 v0 H˙ 3 . Again, using the same way as in (3.35), we obtain that ∇vL∞ 2 n C v0 H˙ 1 v0 H˙ 3 . t Lx (I2 ×R ) Using standard iteration method, we get for any j = 1, . . . , J , ∇vL∞ 2 (I ×Rn ) C v0 H˙ 1 v0 H˙ 3 , L j t x which implies that ∇vL∞ 2 + n C v0 H˙ 1 v0 H˙ 3 . t Lx (R ×R ) This completes the conclusion.
(3.37)
2
Remark 3.7. In order to guarantee Proposition 3.6 holds, we should first prove the local well posedness in H 3 , which can be shown by using a standard contraction mapping argument (see [4], for instance) and combining the proof of Proposition 3.6. Define t
S(t − τ )F v(τ ) dτ.
T : v → S(t)v0 − i 0
Now let η, T be sufficiently small numbers, M = 4Cv0 H 3 . Define D = v: ∇ j v S˙ 0 ([0,T ]×Rn ) M, j = 0, 1, 2, ∂t vS˙ 1 ([0,T ]×Rn ) M, ∇v
2(n+2) 2n(n+2) 2 ∗ L2t,x ∩Lt n−2 Lx n +4 ([0,T ]×Rn )
η, ∇vL∞ 2 n M t Lx ([0,T ]×R )
equipped with the metric d(u, v) = u − vL∞ L2 ∩L2∗ ([0,T ]×Rn ) . t
x
t,x
Then (D, d) is a complete metric space. We claim that T : D → D . Indeed, for any v ∈ D , by the Strichartz estimate and ∇v
∗
2(n+2)
2n(n+2) 2 +4
n−2 n L2t,x ([0,T ]×Rn )∩Lt∈[0,T ] Lx
Following the proof of Proposition 3.6, one has that
η 1.
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
j ∇ T v ˙ 0
S ([0,T ]×Rn )
∇T v
M,
0 j 2,
2(n+2)
∗
699
2n(n+2) 2 +4
η.
n−2 n L2t,x ([0,T ]×Rn )∩Lt∈[0,T ] Lx
Taking formal time derivative and spatial derivative with respect to T v, we have i∂t ∇T v + ∇T v = ∇F (v). Using the same way as in (3.29)–(3.31), we estimate 4
∂t T vS˙ 1 ([0,T ]×Rn ) Cv0 H˙ 3 + Cη n−2 M. 4
Since Cη n−2 1, we get ∂t T vS˙ 1 ([0,T ]×Rn ) 3M/4. In (3.34), we have verified that ∇F (v)L∞ 2 n M/4, therefore ∇T vL∞ L2 ([0,T ]×Rn ) M. So, T : D → D . For t Lx ([0,T ]×R ) t x any u, v ∈ D , 4 d(T u, T v) u n−2 2(n+2)
4
+ v n−2 2(n+2)
Lt,xn−2 ([0,T ]×Rn )
Lt,xn−2 ([0,T ]×Rn )
u − vL2∗ ([0,T ]×Rn ) . t,x
So T : D → D is strictly contractive, then there exists a unique fixed point v ∈ D . We have shown that there exists a solution v ∈ C([0, T ]; H 3 ) to (1.4). We can repeat the above procedures step by step, then find a T ∗ > 0 such that v ∈ C([0, T ∗ ); H 3 ) is a solution of (1.4). Now we can prove our main result. Proof of Theorem 1.1. First, we consider the convergence in L2 space. In view of Corollary 2.3 and Proposition 3.2, we have 4 wS˙ 0 (I ×Rn ) C w(min I )2 + μCvL1 L2 (I ×Rn ) + aC |v| n−2 v t
4 4 + C |w + v| n−2 (w + v) − |v| n−2 v
(2∗ )
Lt,x (I ×Rn )
x
(2∗ )
Lt,x (I ×Rn )
4 n−2 C w(min I )2 + μC|I |vL∞ 2 n + aCv 2(n+2) t Lx (I ×R )
Lt,xn−2 (I ×Rn )
4 + C w + v n−2 2(n+2)
4
+ v n−2 2(n+2)
Lt,xn−2 (I ×Rn )
Lt,xn−2 (I ×Rn )
wL2∗ (I ×Rn ) . t,x
vL2∗ (I ×Rn ) t,x
(3.38)
By Theorem 2.8 and Proposition 3.5, we can divide [0, T ] into N = N (u0 H 1 , v0 H 3 ) disjoint time intervals {Ik }N k=1 such that 4 C w + v n−2 2n+4
n−2 Lt,x (Ik ×Rn )
4
+ v n−2 2n+4
n−2 Lt,x (Ik ×Rn )
Replace I by Ik in (3.38), and with (3.39), then
1 , 2
1 k N.
(3.39)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
wS˙ 0 (Ik ×Rn ) CwL∞ 2 n + μC|Ik |vL∞ L2 (I ×Rn ) t Lx (Ik−1 ×R ) t x k a 1 + vL2∗ (Ik ×Rn ) + wL2∗ (Ik ×Rn ) . t,x t,x 2 2
(3.40)
Using iteration, we get wS˙ 0 (Ik ×Rn ) 2CwL∞ 2 n + 2Cμ|Ik |vL∞ L2 (I ×Rn ) + av 2∗ Lt,x (Ik ×Rn ) t Lx (Ik−1 ×R ) t x k 2C 2CwL∞ 2 n + 2Cμ|Ik−1 |vL∞ L2 (I n + av 2∗ L (Ik−1 ×Rn ) t Lx (Ik−2 ×R ) t x k−1 ×R ) t,x
+ 2Cμ|Ik |vL∞ 2 n + av 2∗ L (Ik ×Rn ) t Lx (Ik ×R ) t,x
2 · · · (2C) w0 2 + μ(2C)|Ik |vL∞ 2 n + μ(2C) |Ik−1 |vL∞ L2 (I n t Lx (Ik ×R ) t x k−1 ×R ) k
+ · · · + μ(2C)k |I1 |vL∞ 2 n + av 2∗ L (Ik ×Rn ) t Lx (I1 ×R ) t,x
+ 2CavL2∗ (Ik−1 ×Rn ) + · · · + (2C) avL2∗ (I1 ×Rn ) t,x t,x
k (2C) w0 2 + μ |Ij |vL∞ 2 n t Lx ( j k Ij ×R ) k
j k
+ a sup vL2∗ (Ij ×Rn ) 1 + 2C + · · · + (2C)k j k
t,x
∗ |Ij |vL∞ + a sup v (2C)k+1 w0 2 + μ 2 n 2 n L (Ij ×R ) . t Lx ( j k Ij ×R ) j k
j k
t,x
(3.41)
In view of Corollary 2.3 and inequality (3.39), one can easily check that sup vL2∗ (Ij ×Rn ) 2C E(v0 ) v0 2 .
j k
t,x
(3.42)
Summarizing the above discussion, we obtain that N +2 wL∞ w0 2 + μT vL∞ 2 n (2C) 2 n + av0 2 , t Lx ([0,T ]×R ) t Lx ([0,T ]×R )
(3.43)
where N = N(u0 H 1 , v0 H 3 ) is a large constant that depends only onu0 H 1 and v0 H 3 , and is independent of μ and a. Combining (3.43) with (3.26), we get for any T > 0 N +2 w0 2 + μT v0 H 2 + av0 2 wL∞ 2 n (2C) t Lx ([0,T ]×R )
(3.44)
which implies the desired results. The proof of (1.8) for k = 1 proceeds in a similar way as that for k = 0, which is essentially implied by the proof of Proposition 3.5, see below and we omit the details. 2 Proof of Proposition 3.5. Let T > 0 and 0 < η 1 be a sufficiently small parameter. Following the proof of Proposition 3.6, we can divide [0, T ] into J (depending on v0 H 3 ) disjoint intervals {Ij }Jj=1 such that on each interval Ij there holds
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
v
2(n+2)
Lt,xn−2 (Ij ×Rn )
+ ∇v
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
701
η.
(3.45)
(Ij ×Rn )
Choose 0 < δ 1 small enough satisfying (4C)J δ η. Let w0 H 1 δ/2, 0 < μT , a δ/4. By Sobolev’s embedding, Corollary 2.3, Propositions 3.1–3.3 and 3.6, w
+ ∇wL∞ 2 n t Lx (Ij ×R )
2(n+2)
Lt,xn−2 (Ij ×Rn )
∇w
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(Ij ×Rn )
+ ∇wL∞ 2 n t Lx (Ij ×R )
C w(min Ij )H˙ 1 + μC∇vL1 L2 (Ij ×Rn ) + Ca ∇F (v) t
+ C ∇ F (u) − F (v)
x
2n
L2t Lxn+2 (Ij ×Rn )
2n
L2t Lxn+2 (Ij ×Rn )
C w(min Ij )H˙ 1 + μC|Ij |∇vL∞ 2 n t Lx (Ij ×R ) 4
+ Cav n−2 2(n+2)
∇v
Lt,xn−2 (Ij ×Rn )
+ C∇v
2(n+2) 2n(n+2) 2 Lt∈In−2 Lx n +4 j
+ C∇w
2(n+2) n−2
Lt
Lt∈In−2,x
Lt∈In−2,x
j
H˙ 1
v
2(n+2)
Lt∈In−2,x j
j
4 n−2 2(n+2) Lt,xn−2 (Ij ×Rn )
4
+ w n−2 2(n+2) Lt,xn−2 (Ij ×Rn )
+ Cδ
6−n 6−n + Cη η n−2 + ∇w n−2 2(n+2)
2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
+ C∇w
(Ij ×Rn )
6−n 6−n n−2 v n−2 2(n+2) + w 2(n+2) w
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
C w(min Ij )
2n(n+2) 2 +4
Lx n
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
∇w
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(Ij ×Rn )
4
4
η n−2 + C∇w n−2 2(n+2)
4 C w(min Ij )H˙ 1 + Cδ + Cη n−2 ∇w
2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
2(n+2) n−2
Lt 4
2n(n+2) 2 +4
Lx n
(Ij ×Rn )
n+2
+ Cη∇w n−2 2(n+2)
2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
+ C∇w n−2 2(n+2) Lt
n−2
2n(n+2) 2 Lx n +4 (Ij ×Rn )
.
(3.46)
Taking j = 1 and noticing that w(min I1 ) = w0 , by the standard continuity method, we get ∇w
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(I1 ×Rn )
+ ∇wL∞ 2 n 2Cδ, t Lx (I1 ×R )
(3.47)
where the constant C depends only on u0 H 1 and v0 H 3 . By (3.47), ∇w(min I2 )2 2Cδ. Again, it follows from (3.46) that
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∇w
2(n+2) n−2
Lt
2n(n+2) 2 +4
Lx n
(I2 ×Rn )
+ ∇wL∞ 2 n t Lx (I2 ×R ) 4
4
4C 2 δ + Cη n−2 ∇w + C∇w
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )
n+2 n−2 2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )
+ Cη∇w n−2 2(n+2) Lt
n−2
2n(n+2) 2 +4
Lx n
(I2 ×Rn )
(3.48)
.
The continuity method yields that ∇w
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 (I2 ×Rn )
2 + ∇wL∞ 2 n (4C) δ. t Lx (I2 ×R )
Noticing that (4C)J δ η, we can repeat the procedure above step by step and finally get the result, as desired. 2 4. Proof of Theorem 1.3 4.1. Concentration for the MEBS Using the induction hypothesis (Lemma 1.5) and bilinear estimate, we construct a near solution to (1.3), then applying the long-time perturbation lemma, one can prove the following proposition which means that the MEBS cannot be included in this class, that is to say, the MEBS can only concentrate in one particular frequency in the phase space. Since the proof is similar to NLS (1.4) as in [5,19], we omit the details. Proposition 4.1 (Frequency localization of energy at each time). Let u be a minimal energy blowup solution to (1.3), then for any t ∈ I0 , there exists a dyadic frequency N (t) ∈ 2Z such that for every η4 η η0 , we have Pc(η)N (t) u(t)
PC(η)N (t) u(t)
H˙ x1
η,
(4.1)
H˙ x1
η,
(4.2)
and Pc(η)N (t)<·
H˙ x1
∼ 1,
(4.3)
where 0 < c(η) 1 C(η) < ∞. On the other hand, the minimal energy blowup solution is concentrated in physical space. First, we show that: Proposition 4.2 (Potential energy is bounded below). For any minimal energy blowup solution to (1.3), for any t ∈ I0 , there holds u(t) 2n η1 . Lxn−2
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
703
Outline of the proof. Suppose for a contrary that there exists some time t0 ∈ I0 such that u(t0 ) 2n < η1 . Normalize N (t0 ) = 1. Since u is a MEBS, by Lemma 2.9, we may assume that
Lxn−2
(μ+i)(t−t ) 0 e u(t0 )
2(n+2)
Lt,xn−2 ([t0 ,+∞)×Rn )
1.
Define Plo = PC(η0 ) , Pme = 1−Plo −Phi , Strichartz estimate and Proposition 4.1 imply that for any t > t0 , (μ+i)(t−t ) 0 e Plo u(t0 )
2(n+2) Lt,xn−2 ([t0 ,+∞)×Rn )
+ e(μ+i)(t−t0 ) Phi u(t0 )
2(n+2)
Lt,xn−2 ([t0 ,+∞)×Rn )
η0 .
Hence (μ+i)(t−t ) 0 e Pme u(t0 )
2(n+2)
Lt,xn−2 ([t0 ,+∞)×Rn )
∼ 1.
On the other hand, by the Strichartz estimate, (μ+i)(t−t ) 0 e Pme u(t0 )
2(n+2) Lt,x n ([t0 ,+∞)×Rn )
Pme u(t0 )L2 C(η0 ). x
The above two estimates together with Hölder’s inequality implies that (μ+i)(t−t ) 0 e Pme u(t0 )
n L∞ t,x ([t0 ,+∞)×R )
c(η0 ).
Hence, there exist a time t1 > t0 , and a point x1 ∈ Rn such that (μ+i)(t −t ) 1 0 e Pme u(t0 )(x1 ) c(η0 ).
(4.4)
Choosing time t2 which is symmetric to t1 with respect to t0 , i.e., t0 − t2 = t1 − t0 . Define f (t2 ) := Pme δx1 , where δx1 is the Dirac mass at x1 . Define f (t) := e(μ+i)(t−t2 ) f (t2 ) for any t > t2 . Then n n − we can easily check that for any t > t2 , 1 p ∞, there holds f (t)Lpx C(η0 )t − t2 p 2 . Hence c(η0 ) e(μ+i)(t0 −t2 ) Pme u(t0 ), δx1 = u(t0 ), f (t0 ) f (t0 ) 2n u(t0 ) 2n η1 C(η0 )t1 − t0 , Lxn+2
(4.5)
Lxn−2
which means that both t1 and t2 are far from t0 . Hence f
2(n+2) 2n(n+2) 2 Lt n−2 Lx n +4 ([t0 ,∞)×Rn )
n−2 C(η0 )· − t2 − n+2
2(n+2) n−2
Lt n−2
([t0 ,∞)) n−2
C(η0 )|t2 − t0 |− 2(n+2) C(η0 )η12(n+2) .
(4.6)
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Then, following the same step as in Visan [19], we can show that u
2(n+2)
Lt,xn−2 ([t0 ,+∞)×Rn )
C(η0 , η1 ).
Since [t0 , +∞) contains the interval I1 , this contradicts (1.16).
2
We now state an interpolation lemma which is useful in proving the spatial concentration of energy. Lemma 4.3. (See [22, Corollary 4.2].) Suppose that 1 p0 < p < ∞, −∞ < s1 < s < s0 < ∞, 0 < θ < 1 and 1 θ 1−θ = , + p p0 ∞
s = θ s0 + (1 − θ )s1 ,
then we have uH˙ s (Rn ) Cuθ˙ s0
Bp0 ,p0 (Rn )
p
u1−θ ˙ s1
B∞,∞ (Rn )
.
Using the ideas in Bourgain [3] and Lemma 4.3, we can prove the following. Proposition 4.4 (Spatial concentration of energy at each time). For any minimal energy blowup solution u to (1.3) (or only suppose u 2n η1 , uH˙ 1 1), and for any t ∈ I0 , there exist a dyadic number M(t) and a position x0 n/2
η1 M(t)
n−2 2
x
Lxn−2
(t) ∈ Rn
such that
n−2 PM(t) u t, x0 (t) M(t) 2 .
Furthermore, there holds the inverse Sobolev inequality: M(t)
n−2 2 p−n+kp
p PM(t) (−)k/2 u(t, x)Lp (|x−x
0 (t)|C(η1 )/M(t))
np/2
η1
M(t)
n−2 2 p−n+kp
,
k = 0, 1,
(4.7)
for all 1 < p < ∞. In particular, we have
∇PM(t) u(t, x)2 dx ηn . 1
(4.8)
|x−x0 (t)|C(η1 )/M(t)
Proof. We only prove the second inequality in (4.7), by Proposition 4.2 and Lemma 4.3, we have for any t ∈ I0 , η1 u
n−2
2n Lxn−2
2
2
2 B˙ ∞,∞
2 B˙ ∞,∞
uH˙n1 u n − n−2 u n − n−2 . x
(4.9)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
705
It follows that there exist a dyadic number M(t) and x0 (t) ∈ Rn such that n PM(t) u t, x0 (t) M(t) n−2 2 η2.
1
Recall that PN = F −1 ψ(ξ/N )F . Define φ(ξ ) = ψ(ξ/2) + ψ(ξ ) + ψ(2ξ ), then φ is a smooth function with compact support in [1/4, 4]. Thus n
n−2
M(t) 2 η12 PM(t) u(t, x0 ) = F −1 φ ξ/M(t) |ξ |−k ∗ PM(t) (−)k/2 u(t, x0 ) −1 −k n−k k/2 F = M(t) |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx
M(t)
−1 −k k/2 F |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx
n−k
|x−x0 |C(η1 )/M(t)
n−k
+ M(t)
−1 −k k/2 F |ξ | φ M(t)(x0 − x) PM(t) (−) u(t, x) dx
|x−x0 |>C(η1 )/M(t)
−1 −k PM(t) (−)k/2 u p F M(t) |ξ | φ Lp L (|x−x0 |C(η1 )/M(t)) + PM(t) (−)k/2 uL2 F −1 |ξ |−k φ L2 (|x−x |>C(η )/M(t)) . n p −k
0
1
(4.10)
Since φ is a rapidly decreasing function, we can choose C(η1 ) sufficiently large such that the second term in the last inequality is smaller than M(t) PM(t) (−)k/2 u p L (|x−x
n−2 n 2 − p +k
0 |C(η1 )/M(t))
thus the conclusion follows.
n
η12 /2, which implies that
M(t)
n−2 n 2 − p +k
n
η12 /2,
2
Remark 4.5. M(t) in Proposition 4.4 is equivalent to N (t) in Proposition 4.1. More precisely, M(t) ∈ c(η1 )N (t), C(η1 )N (t) . Proof. By Proposition 4.4, for any t ∈ I0 , there exists a dyadic number M(t) such that PM(t) u(t, x)
n/2
2n n−2
η1 .
(4.11)
While Proposition 4.1 tells us that there exists dyadic number N (t) satisfying Pc(η )N (t) u(t) 2n + PC(η )N (t) u(t) 2n η100n . 1 1 1 n−2
(4.12)
n−2
Then M(t) ∈ c(η1 )N (t), C(η1 )N (t) .
(4.13)
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Hence, for the MEBS, we can regard M(t) in Proposition 4.4 as to be identical with N (t) in Proposition 4.1. 2 Corollary 4.6. Let I0 be defined as in (1.17), then −2 |I0 | η5−1 Nmin .
(4.14)
Proof. By Proposition 4.4, we know that 2 Nmin
N (t) C(η1 )
|PN (t) u|
2
2(n+2) n−2
dx.
|x−x0 (t)|C(η1 )/N (t)
Integrating it on I0 , and together with (1.16), we get 2 |I0 |Nmin
C(η1 )
|PN (t) u|
2(n+2) n−2
dx C(η1 )η5−1 ,
I0 |x−x0 (t)|C(η1 )/N (t)
2
which is the result, as desired.
Remark 4.7. If we normalize Nmin = 1, this corollary tells us that the length of I0 on which the MEBS is defined is not too long, in fact it can be controlled by Cη5−1 . 4.2. N(t) takes finitely many values Lemma 4.8. Let u be a MEBS. Let I = [t1 , t2 ] be any subinterval of I0 with n/2
|I |N (t1 )2 η1 /4C. Then we have c(η1 )N (t1 ) N (t) C(η1 )N (t1 ),
∀t ∈ I.
Proof. By Proposition 4.4 we see that for any t = t1 , there exists N (t1 ) such that N (t1 )−
n−2 2
PN (t ) u(t1 ) ∞ ηn/2 . 1 1 L x
(4.15)
On the other hand, for any t2 t1 , we use Bernstein’s and Hölder’s inequalities to estimate N(t1 )−
n−2 2
PN (t ) u(t1 ) − PN (t ) u(t2 ) ∞ 1 1 L
x
PN (t1 ) u(t1 ) − u(t2 ) 2
n − n−2 2 +2
N(t1 )
t2 N(t1 ) t1
PN (t ) ∂τ u(τ ) dτ 1 2
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
t2 N (t1 )
707
4 PN (t ) u(τ ) + PN (t ) |u| n−2 u2 dτ 1 1 2
t1
t2 N(t1 ) |t1 − t2 |∇uL∞ ([t1 ,t2 ],L2 ) + N (t1 ) 2
4 N (t1 )|u| n−2 u
2n n+2
dτ
t1
CN (t1 )2 |t1 − t2 | + CN (t1 )2 |t1 − t2 |u
n+2 n−2
.
2n
n−2 L∞ ([t1 ,t2 ]×Rn ) t Lx
Since the energy is conserved, we obtain N(t1 )−
n−2 2
PN (t ) u(t1 ) − PN (t ) u(t2 ) ∞ ηn/2 /2, 1 1 1 L x
t1 , t2 ∈ I.
(4.16)
(4.15) and (4.16) yield N (t1 )−
n−2 2
PN (t ) u(t2 ) ∞ ηn/2 /2. 1 1 L x
Following the same step as in Proposition 4.4, we can show that for p =
2n n−2 ,
PN (t ) u(t2 , x)p dx ηnp/2 N (t1 ) n−2 2 p−n , 1 1
|x−x0 (t2 )|C(η1 )/N (t1 )
which gives that PN (t ) u(t2 ) 1
n/2
2n n−2
η1 .
(4.17)
On the other hand, Proposition 4.1 tells us that Pc(η
1 )N (t2 )
u(t2 )
2n n−2
+ PC(η1 )N (t2 ) u(t2 )
2n n−2
η1100n .
(4.18)
By (4.17) and (4.18), we get N (t1 ) ∈ c(η1 )N (t2 ), C(η1 )N (t2 ) , which is the result, as desired.
2
Corollary 4.9. Let {Ij }Jj=1 be a pairwise disjoint decomposition of I0 with Ij = [tj , tj +1 ] and n/2
|Ij |N (tj )2 η1 /4C. Then we have N(t) ≡ N (tj ) for any t ∈ Ij , i.e., N (t) is a step function.
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Proof. By Lemma 4.8, we can regard N (t) as N (t) ≡ N (tj ), So, the result follows.
t ∈ [tj , tj +1 ].
2
Corollary 4.10. We have Nmin := min N (t) > 0 t∈I0
and N(t)−1 is a Lebesgue integrable function on I0 . Proof. Since N(t) takes finitely many values N (tj ) (1 j J ) and each N (tj ) is a dyadic number, we have the result. 2 4.3. Energy is almost conserved as μ, a → 0 By Theorem 2.7, we see that the energy has a dissipation for the solutions of the CGL. However, for the MEBS u, we show that the energy E(u(t)) tends to Ec as μ, a → 0. More precisely, we have Lemma 4.11. Let u be a minimal energy blowup solution, then μ
u(τ, x)2 dx dτ + a
I 0 Rn
2(n+2) u(τ, x) n−2 dx dτ η4 .
I 0 Rn
Proof. Let I0 = [0, t0 ]. Since u is a solution of Eq. (1.3), we see from Theorem 2.7 that E u(t0 ) + μ
t0
u(τ, x)2 dx dτ + a
0 Rn
t0
2(n+2) u(τ, x) n−2 dx dτ Ec .
0 Rn
If the conclusion of Lemma 4.11 does not hold, then E u(t0 ) Ec − η4 . Therefore the induction hypothesis Lemma 1.5 tells us that u
2(n+2)
Lt,xn−2 (I1 ×Rn )
M(Ec − η4 ),
which contradicts the definition of minimal energy blowup solution on interval I1 .
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
709
4.4. Morawetz estimate Proposition 4.12. Suppose n > 6. Let u be a minimal energy blowup solution to (1.3) and N∗ < c(η2 )Nmin , Nmin := inft∈I0 N (t), suppose furthermore that 0 < a, μ < η6 , then I 0 Rn Rn
|PN∗ u(t, y)|2 |PN∗ u(t, x)|2 dx dy dt |x − y|3
+ I 0 Rn Rn
2n
|PN∗ u(t, y)|2 |PN∗ u(t, x)| n−2 dx dy dt η1 N∗−3 . |x − y|
(4.19)
To prove this proposition, we first recall a known statement for general Schrödinger equation: iφt + φ = N .
(4.20)
Proposition 4.13. (Interaction Morawetz inequality for general NLS [19].) Let φ be a solution to (4.20), then φ satisfies (n − 1)(n − 3) I 0 Rn Rn
+2
I 0 Rn Rn
|φ(t, y)|2 |φ(t, x)|2 dx dy dt |x − y|3
φ(t, y)2 x − y {N , φ}p (t, x) dx dy dt |x − y|
4φ3L∞ L2 φL∞ ˙1 t Hx t x
+4 I0
Rn
{N , φ}m (t, y)∇φ(t, x)φ(t, x) dx dy dt,
(4.21)
Rn
where we denote the mass bracket {f, g}m := Im(f g) ¯ and the momentum bracket {f, g}p := Re(f ∇ g¯ − g∇ f¯). It is easy to see that the CGL equation (1.3) can be rewritten as iut + u = μiu + 4 4 (1 − ai)|u| n−2 u := |u| n−2 u + N . We can regard N as an additional nonlinear term for the NLS, then we need to compute the new terms {N , u}p and {N , u}m . On the other hand, the first term φL∞ 2 in the right-hand side of (4.21) is not scaling invariant and it can become t Lx very large after scaling, so we should throw away the low frequency part of φ when we apply Proposition 4.13. Let us now take up the main business of this section. Proof of Proposition 4.12. Writing the left-hand side of (4.19) as L(N∗ , u), we claim that we only need to consider the case N∗ = 1, i.e., L(1, u) η1 for any minimal energy blowup solution u to (1.3) on I0 . In fact, if u is a minimal energy blowup solution to (1.3) on I0 , then by the scaling 2(n+2)
n−2
invariance of Lt,xn−2 norm and energy norm, uλ = λ− 2 u( λt2 , xλ ) is a minimal energy blowup solution to (1.3) on λ2 I0 . So there holds L(1, uN∗ ) η1 , that is
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
L 1, uN∗ =
N∗2 I0 Rn Rn
|P1 uN∗ (t, y)|2 |P1 uN∗ (t, x)|2 dx dy dt |x − y|3
+ N∗2 I0 Rn Rn
2n
|P1 uN∗ (t, y)|2 |P1 uN∗ (t, x)| n−2 dx dy dt |x − y|
η1 ,
(4.22)
in which
N∗
P1 u
1− n (t, x) = N∗ 2 (PN∗ u)
t x . , N∗2 N∗
(4.23)
By (4.22), we obtain the result, as desired. So our main task is to show the following L(1, u) = I 0 Rn Rn
|P1 u(t, y)|2 |P1 u(t, x)|2 dx dy dt |x − y|3
+ I 0 Rn Rn
2n
|P1 u(t, y)|2 |P1 u(t, x)| n−2 dx dy dt |x − y|
η1 .
(4.24)
To do this, we define uhi = P1 u, ulo = P<1 u. By assumption 1 = N∗ < c(η2 )Nmin , we have ˜ 2 )N (t), 1 < c(η2 )N(t) for any t ∈ I0 . Choosing c(η2 ) small enough, say, 1 < c(η2 )N (t) < η2 c(η c(η ˜ 2 ) is another small number depends on η2 , then combining Proposition 4.1 with Sobolev embedding, we have uη−1 L∞ n + uη−1 ˙1 t H (I0 ×R ) x
2
2
2n
n−2 L∞ (I0 ×Rn ) t Lx
η2 .
(4.25)
This in particular implies ulo L∞ n + ulo ˙1 t H (I0 ×R ) x
2n
n−2 L∞ (I0 ×Rn ) t Lx
η2 .
(4.26)
(4.25) and Bernstein’s inequality yield ∞ ∞ uhi L∞ 2 n P 1·η−1 uLt L2x (I0 ×Rn ) + P>η−1 uLt L2x (I0 ×Rn ) t Lx (I0 ×R ) 2
2
P1·η−1 uL∞ n + η2 P>η−1 uL∞ H˙ 1 (I ×Rn ) ˙1 0 t H (I0 ×R ) t 2
x
2
η2 .
x
(4.27)
Our goal is to prove (4.24), which implies particularly that I0
Rn
Rn
|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt η1 , |x − y|3
(4.28)
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711
which can be rewritten as |uhi |2
1
− n−3 L2t H˙ x 2 (I0 ×Rn )
η12 .
(4.29)
By the standard continuity argument, it suffices to prove (4.29) under the bootstrap hypothesis: |uhi |2
1
− n−3 L2t H˙ x 2 (I0 ×Rn )
(C0 η1 ) 2 ,
(4.30)
where C0 is a large constant depending on energy but not on any ηi . We claim that (4.30) implies − n−3 |∇| 4 uhi
1
L4t,x (I0 ×Rn )
(C0 η1 ) 4 ,
(4.31)
as can be seen by taking f = uhi in the following lemma. Lemma 4.14. (See [19, Lemma 5.6].) − n−3 |∇| 4 f
L4x
1 n−3 |∇|− 2 |f |2 22 .
To prove Proposition 4.12, we first prove a weaken form. Proposition 4.15. Under the same assumption as Proposition 4.12, we have I 0 Rn Rn
|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt + |x − y|3
η2 + η 2
I 0 Rn Rn
2n
|uhi (t, y)|2 |uhi (t, x)| n−2 dx dy dt |x − y|
4 4 4 uhi (t, x)Phi |u| n−2 u − |uhi | n−2 uhi − |ulo | n−2 ulo (t, x) dx dt
(4.32)
I 0 Rn
+ η2 I0
4 uhi (t, x)Plo |uhi | n−2 uhi (t, x) dx dt
(4.33)
4 uhi (t, x)Phi |ulo | n−2 ulo (t, x) dx dt
(4.34)
4 ∇ulo (t, x)ulo (t, x) n−2 uhi (t, x) dx dt
(4.35)
n+2 ∇ulo (t, x)uhi (t, x) n−2 dx dt
(4.36)
4 ∇Plo |u| n−2 u (t, x)uhi (t, x) dx dt
(4.37)
Rn
+ η2 I 0 Rn
+ η22 I0
Rn
+ η22 I 0 Rn
+ η22 I0
Rn
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n+2
|uhi (t, y)|2 |ulo (t, x)| n−2 |uhi (t, x)| dx dy dt |x − y|
+ I0
Rn
Rn
n+2
|uhi (t, y)|2 |ulo (t, x)||uhi (t, x)| n−2 dx dy dt |x − y|
+ I0
Rn
Rn
I0
(4.39)
4
|uhi (t, y)|2 |Plo (|uhi | n−2 uhi )(t, x)||uhi (t, x)| dx dy dt. |x − y|
+ Rn
(4.38)
Rn
4
(4.40)
4
Proof. We apply Proposition 4.13 with φ = uhi , N = Phi (|u| n−2 u)+μiuhi −aiPhi (|u| n−2 u) := N1 + N2 + N3 , then we obtain (n − 1)(n − 3) I 0 Rn Rn
+2
I 0 Rn Rn
|uhi (t, y)|2 |uhi (t, x)|2 dx dy dt |x − y|3
uhi (t, y)2 x − y {N1 + N2 + N3 , uhi }p dx dy dt |x − y|
4uhi 3L∞ L2 uhi L∞ ˙1 t Hx t x {N1 + N2 + N3 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt. +4 I0
Rn
(4.41)
Rn
By energy conservation and (4.27), 3 4uhi 3L∞ L2 uhi L∞ ˙ 1 η2 . t H t
x
x
Visan [19] has shown that
{N1 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt (4.32) + (4.33) + (4.34),
I 0 Rn Rn
I 0 Rn Rn
uhi (t, y)2 x − y {N1 , uhi }p (t, x) dx dy dt |x − y|
I 0 Rn Rn
2n
|uhi (t, y)|2 |uhi (t, x)| n−2 dx dy dt + (4.35) + · · · + (4.40). |x − y|
So, it suffices to consider the corresponding estimates for N2 and N3 . We first begin with the momentum bracket term:
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
{N2 , uhi }p = μ Re(iuhi ∇uhi − uhi ∇iuhi ) = −μ Im ∇(uhi uhi ) − 2∇uhi uhi ,
713
(4.42)
and 4 4 {N3 , uhi }p = a −i|u| n−2 u, uhi p − a −iPlo |u| n−2 u , uhi p 4 4 = a −i|u| n−2 u, u p − a −i|ulo | n−2 ulo , ulo p 4 4 4 − a −i |u| n−2 u − |ulo | n−2 ulo , ulo p − a −iPlo |u| n−2 u , uhi p 4 4 4 = −a −i |u| n−2 u − |ulo | n−2 ulo , ulo p − a −iPlo |u| n−2 u , uhi p := I1 + I2 ,
(4.43) 4
where in the last equality, we use the fact that {−i|u| n−2 u, u}p = 0. This can be seen in the following: 4 4 4 −i|u| n−2 u, u p = Re −i|u| n−2 u∇ u¯ − u∇ −i|u| n−2 u 4 4 = Im |u| n−2 u∇ u¯ − u∇ |u| n−2 u¯ = 0. Now we handle the mass bracket term, {N2 , uhi }m = μ Im(iuhi uhi ) = μ Re(uhi uhi ),
(4.44)
and 4 {N3 , uhi }m = −a Re Phi |u| n−2 u uhi 4 4 2n = −a Re Phi |u| n−2 u uhi − |uhi | n−2 uhi uhi − a|uhi | n−2 4 4 4 = −a Re Phi |u| n−2 u − |uhi | n−2 uhi − |ulo | n−2 ulo uhi 4 4 2n + a Re Plo |uhi | n−2 uhi uhi − a Re Phi |ulo | n−2 ulo uhi − a|uhi | n−2 . Then (4.42) corresponds to uhi (t, y)2 x − y {N2 , uhi }p (t, x) dx dy dt |x − y| I 0 Rn Rn
= (n − 1)μ I 0 Rn Rn
+ 2μ
I 0 Rn Rn
= II1 + II2 .
uhi (t, y)2
1 Im(uhi uhi ) dx dy dt |x − y|
uhi (t, y)2 x − y Im ∇uhi uhi (t, x) dx dy dt |x − y|
(4.45)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
We first estimate II1 , using the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality, II1 = (n − 1)μ I 0 Rn
1 ∗ |uhi |2 (x) Im(uhi uhi ) dx dt |·|
1 2 μ ∗ |uhi | |·| I0
μ
p
uhi uhi
Lx
|uhi |2 q uhi L
2p 2−p
x
uhi 2 2q uhi Lx
I0
uhi L2x dt
2p 2−p Lx
uhi L2x dt, 1 q
where p is chosen large enough (at least p > n), and 1 uhi L2q uhi θL12 ∇uhi 1−θ , L2 x
dt
Lx
I0
μ
p
Lx
x
uhi
x
=
1 p
2p 2−p Lx
+
(4.46)
n−1 n .
Using interpolation,
2 uhi θL22 ∇uhi 1−θ , L2 x
x
3n , θ1 = 56 , θ2 = 13 . where 0 < θi < 1, i = 1, 2. For convenience, we choose p = 32 n, then q = 3n−1 Bernstein’s inequality tells us that uhi 2 ∇uhi 2 uhi 2 . Replace the above estimates into (4.46), and applying the energy estimate in Theorem 2.7, μu2 2 E(u0 ), we get Lt,x
II1 μ
5
1
1
2
uhi L3 2 ∇uhi L3 2 uhi L3 2 ∇uhi L3 2 uhi L2x dt x
I0 5 3
x
η2 μ
x
x
5
uhi 22 dt η23 .
(4.47)
I0
The second term can be bounded by II2 μ
∇uhi L2x uhi L2x uhi 2L2 dt y
I0
μη22
uhi 22 dt η22 . I0
In the sequel, we deal with the contribution of the momentum bracket (4.43). We denote by = ˙ ˙ ∇(f g) + f ∇g, then that we omit the conjugate symbols in the equality. Writing {f, g}p = 4 4 I1 = −a −i |u| n−2 u − |ulo | n−2 ulo , ulo p 4 4 4 4 = ˙ a∇ i |u| n−2 u − |ulo | n−2 ulo ulo + a i |u| n−2 u − |ulo | n−2 ulo ∇ulo .
(4.48)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
715
Integrating by parts, we find that the first term in (4.48) corresponds to a scalar multiple of
4
I 0 Rn Rn
4
|uhi (t, y)|2 ||u| n−2 u − |ulo | n−2 ulo |(t, x)|ulo (t, x)| dx dy dt (4.38) + (4.39). |x − y|
The second term corresponds to
4 4 uhi (t, y)2 |u| n−2 u − |ulo | n−2 ulo (t, x)∇ulo (t, x) dx dy dt (4.35) + (4.36).
I 0 Rn Rn 4
Now we estimate I2 in (4.43), when the derivative is taken on Plo (|u| n−2 u), we estimate it by 4 a I0 Rn Rn |uhi (t, y)|2 |∇Plo (|u| n−2 u)||uhi (t, x)| dx dy dt (4.37), while when the derivative falls on uhi , it is a bad term, so we integrate by parts, then we obtain a
4 uhi (t, y)2 ∇Plo |u| n−2 u (t, x)uhi (t, x)
I 0 Rn Rn 4
+
|uhi (t, y)|2 |Plo (|u| n−2 u)(t, x)||uhi (t, x)| dx dy dt |x − y|
(4.37) + · · · + (4.40). Now let us deal with the mass bracket as in (4.41). (4.44) corresponds to Cμ
Re uhi uhi (t, y)∇uhi (t, x)uhi (t, x) dx dy dt
I 0 Rn Rn
uhi L∞ 2 uhi L∞ L2 μ t Lx t y
∇uhi L2x uhi L2y dt η22 .
(4.49)
I0
(4.45) corresponds to
{N3 , uhi }m (t, y)∇uhi (t, x)uhi (t, x) dx dy dt
I 0 Rn Rn
(4.32) + (4.33) + (4.34) + a
2n |uhi | n−2 (t, y)∇uhi (t, x)uhi (t, x) dx dy dt
I 0 Rn Rn
(4.32) + (4.33) + (4.34) + a
2n
∇uhi L2x uhi L2x uhi n−22n dt I0
Lyn−2
(4.32) + (4.33) + (4.34) + aη2
2n
uhi n−22n dt, I0
Lyn−2
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
in which
2n
uhi n−22n dt auhi L2
a I0
n uhi x,t (I0 ×R )
Lyn−2
n+2 n−2 2(n+2) Lt,xn−2 (I0 ×Rn )
n−2 a 1/2 |I0 |1/2 η2 a 2(n+2) uhi 1/2 −1/2
η6 η5
1/2
η2 η4
n+2 2(n+2) Lt,xn−2 (I0 ×Rn )
n−2
η2 ,
(4.50)
where in the last second inequality we apply Corollary 4.6 and Lemma 4.11. Up to now, we finish the proof of Proposition 4.15. 2 Now let us develop estimates on the low and high frequency parts to u, which we will use to bound the error terms in Proposition 4.15. Proposition 4.16. The low frequency part of u satisfies 4 2
ulo S˙ 1 (I0 ×Rn ) C1 η2(n−2) .
(4.51)
The high frequency part of u can be split into a good and a bad term, i.e., uhi = g + b, they satisfy the estimates 2
gS˙ 0 (I0 ×Rn ) C1 η2n−2 ,
(4.52)
gS˙ 1 (I0 ×Rn ) C1 , − 2 |∇| n−2 b
2n(n−2) 2 −3n−2
L2t Lxn
(4.53) 1 4
(I0 ×Rn )
C1 η 1 .
(4.54)
Proof. We define g and b to be the unique solutions to the following initial value problems respectively i∂t + (1 − μi) g = G + Phi F (ulo ) + Phi F (ulo + g) − F (g) − F (ulo ) , g(t0 ) = uhi (t0 ) and i∂t + (1 − μi) b = B + Phi F (u) − F (ulo + g) , b(t0 ) = 0 4
where F (z) = (1 − ai)|z| n−2 z, G + B = Phi (F (g)). The aim of defining b(t0 ) here is to make use of the improved Strichartz estimate (Proposition 2.4). Then following the same way as in [19], using bootstrap argument, we can prove our Proposition 4.16. 2 With Proposition 4.16 at hand, we can bound the error terms in the right-hand side of Proposition 4.15, we note here that ulo is a good term, we apply the estimation (4.51) directly, while
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
717
uhi is a bad term, we split it into g and b, making use of (4.52)–(4.54), we can also treat it, since the argument is the same as in [19], we omit it. So, we get that (4.32)–(4.40) can all be bounded by η1 , then Proposition 4.12 is proven. 2 By Proposition 4.16 and scaling, there holds Corollary 4.17. Suppose n > 6, u is a minimal energy blowup solution to (1.3), N∗ < c(η2 )Nmin . Then we can split PN∗ u = g + b satisfying: 2
gS˙ 0 (I0 ×Rn ) η2n−2 N∗−1 ,
(4.55)
gS˙ 1 (I0 ×Rn ) 1,
(4.56)
− 2 |∇| n−2 b
1
−3
η14 N∗ 2 .
(4.57)
1 − 3 n−2 η14 N∗ 2 n ,
(4.58)
2n(n−2) 2 L2t Lxn −3n−2 (I0 ×Rn )
Upon scaling, we can prove b b
2n2 2n (n+1)(n−2) Ltn−2 Lx (I0 ×Rn )
2n(n+2) 2(n+2) (n−2)(n+3) Lt n−2 Lx (I0 ×Rn )
PN∗ u
1 − 3 n−2 η14 N∗ 2 n+2 ,
(4.59)
1
6n L3t Lx3n−4 (I0 ×Rn )
η16 N∗−1 .
(4.60)
4.5. Contradiction argument Proposition 4.18. Let u be a minimal energy blowup solution to (1.3), 0 < a, μ η6 . Then for any t ∈ I0 , we have N (t) C(η4 )Nmin . Proof. N(t) is a step function, therefore there exists tmin ∈ I0 such that N (tmin ) = Nmin . By Bernstein inequality and energy conservation, we have −1 u(tmin )L2 c(η0 )Nmin , x P>C(η )N u(tmin ) 2 c(η0 )N −1 . 0 min min L
Pc(η
0 )Nmin <·
x
(4.61)
Suppose for contradiction that there exists tevac ∈ I0 such that N (tevac ) C(η4 )Nmin , we may assume tmin < tevac (since the case tevac < tmin is more easier). By Proposition 4.1, for any η4 η η0 and for every t ∈ I0 , P
<η4−1 Nmin u(tevac ) H˙ x1
η4 .
(4.62)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
Define ul = P<η10n Nmin u, uh = Pη10n Nmin u, Ph = Pη10n Nmin , then from (4.61), we know 3
3
3
uh (tmin )
L2x
−1 η1 Nmin .
(4.63)
1 −1 η1 Nmin , 2
(4.64)
If we could show uh (tevac )
L2x
noticing that the Bernstein inequality tells us P>C(η )N uh (tevac ) 2 c(η1 )N −1 1 min min L
(4.65)
x
then (4.64) and (4.65) together with the triangle inequality imply PC(η
1 )Nmin
1 −1 uh (tevac )L2 η1 Nmin . x 4
Again by Bernstein, we have PC(η
1 )Nmin
u(tevac )H˙ 1 η310n Nmin PC(η1 )Nmin uh (tevac )L2 c(η1 , η3 ), x
x
which would contradict with (4.62) if we choose η4 small enough. So now we only need to prove (4.64). We use the standard continuity method, suppose that there exists a time t∗ satisfying tmin t∗ tevac and inf
tmin tt∗
1 −1 uh L2x η1 Nmin . 2
(4.66)
We will show that this can be bootstrapped to inf
tmin tt∗
3 −1 uh L2x η1 Nmin . 4
(4.67)
Then {t∗ ∈ [tmin , tevac ]: (4.66) holds} is both open and closed in [tmin , tevac ], then (4.64) holds. Now we show that (4.66) implies (4.67). Define L(t) = Rn |uh (t, x)|2 dx. By (4.63), we have t∗ −2 1 2 −2 L(tmin ) η12 Nmin . If we could show that tmin |∂t L(t)| 100 η1 Nmin , then t L(t) L(tmin ) − tmin
which means that (4.67) holds. Note that
∂t L(t) dt 99 η2 N −2 , 100 1 min
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
∂t L(t) = 2
4 Ph |u| n−2 u , uh m dx − 2a
Rn
=2
719
4 Re Ph |u| n−2 u uh dx − 2μ
|∇uh |2 dx
Rn
4 4 Im Ph |u| n−2 u − |uh | n−2 uh uh dx
− 2a
4 4 Re Ph |u| n−2 u − |uh | n−2 uh uh dx
Rn
− 2a
2n
|uh | n−2 dx − 2μ
Rn
|∇uh |2 dx,
(4.68)
Rn
then we only need to bound the following three terms respectively: t∗ tmin Rn
4 4 1 2 −2 Ph |u| n−2 u − |uh | n−2 uh |uh | dx dt η N , 100 1 min t∗
|uh | n−2 dx dt
1 2 −2 η N , 100 1 min
(4.70)
|∇uh |2 dx dt
1 2 −2 η N . 100 1 min
(4.71)
2n
a tmin Rn
t∗ μ tmin Rn
(4.69)
To estimate (4.69), we need to prove the following Stirchartz estimate for low frequencies. Proposition 4.19. Under the assumptions above, there holds − 3 3 − n+2 n+2 PN uS˙ 1 ([tmin ,tevac ]×Rn ) η4 + max η3 2 N 2 , η3 n−2 N n−2 , for all N η3 Nmin . To prove Proposition 4.19, we now set up an inverse Strichartz estimate for the low frequency part of the solution of the CGL (1.3). That is to say, when restricting to low frequency, we in fact can solve the CGL backward: Lemma 4.20. Suppose u is a minimal energy blowup solution to (1.3), 0 < a < μ η6 . Let [t, tevac ] ⊂ I0 . Then for any admissible pairs (q, r) and (q1 , r1 ), there holds PN u(s) ˙ 1
S ([t,tevac ]×Rn )
for any N η3 Nmin .
4 PN u(tevac )H˙ 1 + ∇PN |u| n−2 u x
q
r
Ls 1 Lx1
,
(4.72)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725 4
Proof of Lemma 4.20. Write F = |u| n−2 u. Applying the Littlewood–Paley projections PN and PM to (1.3), we have ∂t PM PN u − (μ + i)PM PN u + (a + i)PM PN F = 0
(4.73)
holds for any dyadic number M. Then (4.73) can be rewritten as s PM PN u(s) = e
(μ+i)(s−tevac )
PM PN u(tevac ) − (a + i)
e(μ+i)(s−τ ) PM PN F (τ ) dτ,
tevac
where t s tevac . Hence, for any admissible pair (q, r), we have ∇PM PN uLqs Lr e(μ+i)(s−tevac ) ∇PM PN u(tevac )Lq Lr x
s
+
x
e(μ+i)(s−τ ) ∇PM PN F (τ ) dτ
s
tevac
q
.
(4.74)
Ls Lrx
Claim. Let 1 < p < ∞. Suppose that the conditions of Lemma 4.20 are verified. Then 2 2 2 ϕ(ξ/2N)eμ(tevac −s)|ξ | , ϕ(ξ/2N )e−μs|ξ | , ϕ(ξ/2N )eμτ |ξ | ∈ Mp . Claim is a direct consequence of the Mihlin multiplier estimate and Corollary 4.6. Combining this claim with the Strichartz estimate for the NLS, we get (μ+i)(s−t ) μ(s−tevac ) evac ∇P P e ∇PM PN u(tevac )L2 M N u(tevac ) Lqs Lr P2N e x x PM PN u(tevac ) H˙ 1 , x
and s (μ+i)(s−τ ) e ∇PM PN F (τ ) dτ
q
Ls Lrx
tevac
s μs i(s−τ ) −μτ P2N e e P2N e ∇PM PN F (τ ) dτ s i(s−τ ) −μτ e P2N e ∇PM PN F (τ ) dτ tevac
P2N e−μτ ∇PM PN F (τ ) ∇PM PN F (τ )
q
Ls Lrx
tevac
q
r
Lτ 1 Lx1
.
q
q
Ls Lrx
r
Lτ 1 Lx1
(4.75)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
721
That is 4 ∇PM PN uLqs Lr PM PN u(tevac )H˙ 1 + ∇PM PN |u| n−2 u x
q
r
Ls 1 Lx1
x
Squaring and summing this inequality over all M, we obtain (4.72).
.
2
With this lemma together with (4.62), following the same step as in Visan [19], we can prove Proposition 4.19 and then the estimate (4.69). Now let us bound (4.70), by Corollary 4.6, we get 2n
2n
auh n−22n
n−2 Lt,x (I0 ×Rn )
a|I0 |uh n−2
2n n−2 L∞ t Lx
1 2 −2 η N . 100 1 min
(4.76)
Now we estimate (4.71). By Bernstein’s inequality and Lemma 4.11, we have t∗ μ
t∗
|∇uh | dx dt 2
tmin Rn
−2 η3−20n Nmin μ tmin Rn
−2 |uh |2 dx dt η3−20n Nmin η4
1 2 −2 η N , 100 1 min
2
which is (4.71), as desired.
Now we have enough estimates to complete the contradiction argument. Lemma 4.21. For any minimal energy blowup solution u to (1.3), we have
−3 N (t)−1 C(η1 , η2 )Nmin .
(4.77)
I0
Furthermore, by Proposition 4.18, we have −2 |I0 | C(η1 , η2 , η4 )Nmin .
(4.78)
Proof. By (4.60), we have
6n
3n−4 2n
|PN∗ u| 3n−4 dx I0
1
dt η12 N∗−3 ,
∀N∗ < c(η2 )Nmin .
(4.79)
Rn
Let N∗ = c(η2 )Nmin , then
6n
|PN∗ u| 3n−4 dx I0
3n−4 2n
−3 dt C(η1 , η2 )Nmin .
Rn
On the other hand, Bernstein and energy conservation tell us that
(4.80)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
6n 6n P
|x−x(t)|C(η1 )/N (t)
C(η1 )N (t)−n N (t)
3n(n−2) 3n−4
6n c(η2 )P
Lxn−2
2n
c(η2 )N (t)− 3n−4 .
(4.81)
By Proposition 4.4, we have
6n 2n u(t) 3n−4 dx c(η1 )N (t)− 3n−4 .
(4.82)
|x−x(t)|C(η1 )/N (t)
Hence the triangle inequality offers 2n − 3n−4
c(η1 )N (t)
6n PN u(t, x) 3n−4 dx. ∗
(4.83)
|x−x(t)|C(η1 )/N (t)
Integrating over I0 and comparing with (4.80), we get (4.77). Proposition 4.22. Suppose u is a minimal energy blowup solution, then u
2(n+2)
Lt,xn−2 (I0 ×Rn )
C(η0 , η1 , η2 , η4 ).
(4.84)
Proof. Let δ = δ(η0 , Nmax ) > 0 be chosen later. Divide I0 into O(|I0 |/δ) subintervals I1 , . . . , IJ with |Ij | δ. Let tj = min Ij . Since N (tj ) Nmax , by Proposition 4.1 PC(η
0 )Nmax
u(tj )H˙ 1 η0 .
(4.85)
x
Let u(t) ˜ = e(μ+i)(t−tj ) P
H˙ x1
η0 ,
(4.86)
which implies on each time–space interval Ij × Rn ,
1
(μ+i)(t−t ) 2 j e PN ∇ u(tj ) − u(t ˜ j ) 2(n+2)
2
2n(n+2) 2 Lt n−2 Lx n +4 (Ij ×Rn )
N
Bernstein, Sobolev inequalities, together with energy conservation yield u(t) ˜ hence
2(n+2) Lx n−2
n−2 n+2 u(t ˜ j ) C(η0 )Nmax
2n Lxn−2
n−2
n+2 C(η0 )Nmax ,
η0 .
(4.87)
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
u(t) ˜
n−2
2(n+2) Lt,xn−2
723
n−2
n+2 2(n+2) C(η0 )Nmax δ .
(4.88)
Similarly, 4 ∇ u(t) ˜ n−2 u(t) ˜
2n Lxn+2
˜ ∇ u(t)
2n Lxn−2
4 u(t) ˜ n−22n
Lxn−2
4 ˜ L2 u(t) ˜ Hn−2 C(η0 )Nmax ∇ u(t) ˙1 x
x
C(η0 )Nmax , which means that 4 ∇ u(t) ˜ n−2 u(t) ˜
1
2n L2t Lxn+2 (Ij ×Rn )
C(η0 )Nmax δ 2 . 4
˜ we find that With the above preparation, by Lemma 2.9 with e = −(1 − ai)|u| ˜ n−2 u, u
2(n+2)
Lt,xn−2 (Ij ×Rn )
1.
Adding them together, we get u
2(n+2) Lt,xn−2 (I0 ×Rn )
2 C(η0 )Nmax |I0 | C(η0 , . . . , η4 ).
2
This proposition contradict with the definition of minimal energy blowup solution, hence Theorem 1.3 is proven. 5. Proof of Theorem 1.2 Suppose u, v are solutions to (1.3) and (1.4), respectively, let w = u − v, then w satisfies the following equation: 4 4 4 wt − iw + i |u| n−2 u − |v| n−2 v − μu + a|u| n−2 u = 0,
w0 = u0 − v0 .
(5.1)
Applying the Strichartz estimate for NLS on any time–space interval I × Rn , we obtain 4 ∗ wL∞ L2 ∩L2∗ (I ×Rn ) C w(min I )2 + μCuL1 L2 + aCu n−2 2(n+2) uL2 t
x
t
t,x
4 + C u n−2 2(n+2)
Lt,xn−2 (I ×Rn )
x
t,x
Lt,xn−2
4
+ v n−2 2(n+2) Lt,xn−2 (I ×Rn )
wL2∗ . t,x
(5.2)
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C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
Owing to Theorems 1.3 and 2.8, for any T > 0, we can divide[0, T ] into K (depends on E(u0 ) and E(v0 )) mutually disjoint intervals Ik such that [0, T ] = K k=1 Ik satisfying 4 C u n−2 2(n+2)
Lt,xn−2
4
+ v n−2 2(n+2) Lt,xn−2
(Ik ×Rn )
(Ik ×Rn )
1 , 2
∀1 k K.
(5.3)
Then on each Ik × Rn , there holds wL∞ 2 n t Lx (Ik ×R ) 1 1 2 μ 2 u 2Cu − vL∞ 2 (I n ) + 2C μ|Ik | L ×R L2 k−1 t x
n t,x (Ik ×R )
1 1 (2C)k u0 − v0 2 + 2C μ|Ik | 2 μ 2 uL2
t,x (Ik ×R
1 1 + (2C)2 μ|Ik−1 | 2 μ 2 uL2
n t,x (Ik−1 ×R )
+ 2CauL2∗ (Ik ×Rn ) t,x
n)
1 1 + · · · + (2C)k μ|I1 | 2 μ 2 uL2
n t,x (I1 ×R )
+ 2CauL2∗ (Ik ×Rn ) + (2C)2 auL2∗ (Ik−1 ×Rn ) + · · · + (2C)k auL2∗ (I1 ×Rn ) t,x t,x t,x
1 1 μ|Ij | 2 μ 2 uL2 (Ij ×Rn ) (2C)k w0 2 + t,x
j k
+ a sup uL2∗ (Ij ×Rn ) 1 + 2C + · · · + (2C)k t,x
j k
1 (2C)k+1 w0 2 + μ|Ij | 2 E(u0 ) + a sup uL2∗ (Ij ×Rn ) . j k
j k
t,x
(5.4)
In view of Corollary 2.3 and inequality (5.3), one can easily check that sup uL2∗ (Ij ×Rn ) 2C E(u0 ) u0 2 .
j k
t,x
(5.5)
Summarizing the above discussion, we obtain that for any T > 0, K+2 wL∞ w0 2 + (μT )1/2 + au0 2 , 2 n (2C) t Lx ([0,T ]×R )
(5.6)
where K = K(E(u0 ), E(v0 )) is a large constant that is independent of μ and a. This implies the desired results. 2 Acknowledgments The authors are grateful to Professor Carlos E. Kenig for his suggestion on this problem. They also want to express their great thanks to the referee for his/her many helpful suggestions and comments. This work is supported in part by the National Science Foundation of China, grant 10571004 and the 973 Project Foundation of China, grant 2006CB805902.
C. Huang, B. Wang / Journal of Functional Analysis 255 (2008) 681–725
725
References [1] P. Bechouche, A. Jüngel, Inviscid limits of the complex Ginzburg–Landau equation, Comm. Math. Phys. 214 (2000) 201–226. [2] J. Bergh, J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, 1976. [3] J. Bourgain, The Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc., Providence, RI, 1999. [4] T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990) 807–836. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3 , Ann. of Math. (2) 166 (2007) 1–100. [6] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005) 1–24. [7] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation II, Contraction methods, Comm. Math. Phys. 187 (1997) 45–79. [8] V. Ginzburg, L. Landau, On the theory of superconductivity, Zh. Eksperiment. Fiz. 20 (1950) 1064; English transl. in: D. Ter Haar (Ed.), Men of Physics: L.D. Landau, vol. L, Pergamon Press, New York, 1965, pp. 546–568. [9] M. Grillakis, On nonlinear Schrödinger equation, Comm. Partial Differential Equations 25 (2000) 1827–1844. [10] E. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4 (1963) 195–207. [11] M. Gross, P. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (1993) 851–1089. [12] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987) 113–129. [13] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980. [14] S. Machihara, Y. Nakamura, The inviscid limit for the complex Ginzburg–Landau equation, J. Math. Anal. Appl. 281 (2003) 552–564. [15] E. Ryckman, M. Visan, Global well posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R1+4 , Amer. J. Math. 129 (2007) 1–60. [16] C. Sulem, P.L. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, Appl. Math. Sci., vol. 139, Springer-Verlag, Berlin, 1999. [17] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. Math. 11 (2005) 57–80. [18] T. Tao, M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005) 1–28. [19] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007) no, .2, 281-374. [20] B. Wang, The limit behavior of solutions for the complex Ginzburg–Landau equation, Comm. Pure Appl. Math. 55 (2002) 481–508. [21] B. Wang, Large time behavior for critical and subcritical complex Ginzburg–Landau equations in H 1 , Sci. China Ser. A 46 (1) (2003) 64–74. [22] B. Wang, Concentration phenomenon for the L2 super critical nonlinear Schrödinger equation in energy spaces, Commun. Contemp. Math. 8 (2006) 309–330. [23] J.H. Wu, The inviscid limit of the complex Ginzburg–Landau equation, J. Differential Equations 142 (2) (1998) 413–433.
Journal of Functional Analysis 255 (2008) 726–754 www.elsevier.com/locate/jfa
On the equipartition of energy for the critical NLW Luis Vega a,1 , Nicola Visciglia b,∗ a Universidad del Pais Vasco, Apdo. 64, 48080 Bilbao, Spain b Dipartimento di Matematica Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy
Received 31 October 2007; accepted 25 April 2008 Available online 2 June 2008 Communicated by C. Kenig
Abstract We study some qualitative properties of global solutions to the following focusing and defocusing critical NLW: ∗ 2u + λu|u|2 −2 = 0, λ ∈ R, ∂t u(0) = g ∈ L2 Rn u(0) = f ∈ H˙ 1 Rn ,
2n . We will consider the global solutions of the defocusing NLW whose on R × Rn for n 3, where 2∗ ≡ n−2 existence and scattering property is shown in [J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. (7) (1994) 303–309 (electronic); H. Bahouri, J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (6) (1998) 783–789] and [H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175], without any restriction on the initial data (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ). As well as the solutions constructed in [H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (2) (1984) 261– 270] to the focusing NLW for small initial data and to the ones obtained in [C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, preprint], where a sharp condition on the smallness of the initial data is given. We prove that the solution u(t, x) satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see [C. Morawetz, Time decay for the nonlinear Klein–Gordon equation, Proc. Roy. Soc. London Ser. A 306 (1968) 291–296]). As a by-product we deduce that any global solution to critical NLW belonging to a natural functional space satisfies:
* Corresponding author. Fax: +39 0502213224.
E-mail addresses: [email protected] (L. Vega), [email protected] (N. Visciglia). 1 Fax: +34 946012516.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.016
L. Vega, N. Visciglia / Journal of Functional Analysis 255 (2008) 726–754
1 lim R→∞ R
∇x u(t, x)2 dx dt
R |x|
1 = lim R→∞ 2R
727
R |x|
∗ ∇t,x u(t, x)2 + 2λ u(t, x)2 dx dt 2∗
2∗ 2 2λ ∇t,x u(0, x) + ∗ u(0, x) dx. = 2 Rn
© 2008 Elsevier Inc. All rights reserved. Keywords: Critical NLW; Nonlinear scattering; Equipartition of energy; Morawetz estimates; Dispersive estimates
0. Introduction In this paper we study some qualitative properties of solutions to the following family of Cauchy problems: ∗ −2
2u + λu|u|2 u(0) = f ∈ H˙ 1 Rn ,
= 0,
λ ∈ R,
∂t u(0) = g ∈ L2 Rn ,
(t, x) ∈ R × Rn ,
n 3,
(0.1)
2n and 2 ≡ ∂t2 − ni=1 ∂x2i . where 2∗ ≡ n−2 Notice that if λ ≡ 0, then (0.1) reduces to the linear wave equation. Since now on we shall refer to the Cauchy problem (0.1) with λ 0, as to the defocusing critical NLW (similarly the focusing critical NLW will be the Cauchy problem (0.1) with λ < 0). Along this paper we shall work with solutions u(t, x) belonging to the following space: 2(n+1) 1 n 2(n+1) 2 R . (0.2) X ≡ C R; H˙ 1 Rn ∩ C 1 R; L2 Rn ∩ L n−2 R × Rn ∩ Llocn−1 R; B˙ 2(n+1) n−1
We shall also assume that the conservation of the energy is satisfied by the solutions u(t, x), i.e. ∗ ∇t,x u(T , x)2 + 2λ u(T , x)2 dx ≡ const ∀T ∈ R. (0.3) 2∗ Rn
Let us recall that the global well-posedness of the defocusing NLW has been studied in [7] provided that the initial data (f, g) are regular. Actually the global well-posedness of the defocusing Cauchy problem (0.1) has been studied in [16] for initial data (f, g) in the energy space H˙ 1 (Rn ) × L2 (Rn ). In [3] and [2] the same problem has been analysed from the point of view of scattering theory (see also [13]). In particular by combining the results in [3] and [16] it can be shown that for every λ 0 and for every initial data (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ), there exists a unique solution u(t, x) to (0.1) that belongs to the space X introduced in (0.2) and moreover (0.3) is satisfied.
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Concerning the global well-posedness of the focusing NLW, it is well known that for every initial data (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ) small enough, i.e.
|∇x f |2 + |g|2 dx < |λ|
Rn
for a suitable (|λ|) > 0, there exists a unique global solution to (0.1) belonging to the space X above and moreover (0.3) is satisfied. For a proof of this fact see [14]. In the paper [8] a much more precised version of the smallness assumption required on the initial data is given in order to guarantee the global well-posedness of the focusing critical NLW. In order to describe the result in [8] let us introduce the function W (x) ∈ H˙ 1 (Rn ) defined as follows: W (x) ≡
1 (1 +
n−2 |x|2 2 n(n−2) )
.
Then in [8] it is shown that the Cauchy problem (0.1) with λ = −1, has a unique global solution in the space X introduced in (0.2), provided that
n − 2 2∗ n−2 2 2∗ |∇x f | + |g| − dx < |∇x W | − dx |f | |W | n n 2
2
Rn
(0.4)
Rn
and
|∇x f |2 dx <
|∇x W |2 dx.
(0.5)
Moreover in [8] it is proved that blow-up occur provided that f and g satisfy (0.4) and |∇x f |2 dx > |∇x W |2 dx. It is also well known that (0.3) is satisfied by the solutions constructed in [8]. Our aim in this paper is to analyse some qualitative properties of global solutions to (0.1) in both focusing and defocusing case, provided that such a global solutions exist and belong to the space X. We are mainly interested on the asymptotic behaviour for large R > 0, of the following localized energies associated to the solutions u(t, x) of (0.1): 1 R 1 R
|∇t,x u|2 dx dt
R |x|
|∇x u|2 dx dt,
R |x|
and
1 R
2λ ∗ |∇t,x u|2 + ∗ |u|2 dx dt. 2
(0.6)
(0.7)
R |x|
Let us recall that the localized energies (0.6) were first obtained in [9] following the ideas in [1], see also [15] for the non-linear case. In this work we shall describe the asymptotic behaviour of the energies (0.6) and (0.7) as a consequence of a family of energy identities satisfied by the global solutions to (0.1).
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In our opinion those identities have its own interest since they represent a precised version of the classical Morawetz inequalities, first proved in [12]. Since now on we shall denote by X the space defined in (0.2). Next we state the first result of this paper. Theorem 0.1. Let (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ) and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1). Assume moreover that u(t, x) satisfies (0.3). Let ψ : Rn → R be a radially symmetric function such that:
1 + |x|2 x ψ, 2x ψ,
∂ 2ψ ∈ L∞ R n ∂xi ∂xj
∀i, j = 1, . . . , n,
and lim ∂|x| ψ = ψ (∞).
|x|→∞
Then we have the following identity:
1 λ ∗ ∇x uDx2 ψ∇x u − |u|2 2x ψ + |u|2 x ψ dx dt 4 n
R Rn
= ψ (∞) Rn
2λ 2∗ 2 |∇x f | + ∗ |f | + |g| dx. 2 2
(0.8)
Remark 0.1. Let us point out that the hypothesis of Theorem 0.1 are satisfied by the solutions constructed in [16] for defocusing NLW and in [8,14] for the focusing NLW. Remark 0.2. Let us underline that the identity (0.8) represents a precised version of an inequality proved in [12], where (0.8) is stated as an inequality and not as an identity in the special case ψ ≡ |x|. Remark 0.3. The same type of identities as in Theorem 0.1, have been proved in the context of the linear Schrödinger equation in [18] and [20] respectively in the free and in the perturbative case. The L2 -critical NLS has been analysed from the same point of view in [19]. However the result stated for the critical NLW in Theorem 0.1 is much more precise compared with the one in [19] for NLS. One of the main differences between NLS and NLW is that in the former case an explicit representation of the asymptotic behavior of the solutions to the free Schrödinger equation is involved in the argument, while in the case of NLW it is not necessary. In this case one of the fundamental ingredients is the equipartition of energy, see Proposition 2.1 below. Remark 0.4. Another difference between NLW and NLS, is that on the right-hand side in (0.8) we get a quantity that is preserved along the evolution for NLW, while in case of NLS we get the 1 H˙ 2 -norm of the initial data, that is not preserved along the evolution for NLS.
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Due to the freedom allowed to the function ψ in Theorem 0.1, we can deduce the following result. Theorem 0.2. Let (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ), λ ∈ R and u(t, x) ∈ X be as in Theorem 0.1. Then we have: 1 R→∞ R
|∂|x| u|2 dx dt =
lim
R |x|
2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx. 2
(0.9)
Rn
Moreover 1 lim R→∞ R
1 |∇τ u| dx dt = lim R→∞ R
R |x|
∗
|u|2 dx dt = 0,
2
(0.10)
R |x|
where ∂|x| and ∇τ represent the radial derivative and the tangential part of the gradient, respectively. Notice that Theorem 0.2 concerns mainly the concentration of the spatial gradient of the solution. Next we shall present another family of identities that will allow us to study also the behaviour of the energies connected with the time derivative of u(t, x). In order to prove it we shall make use of Levine’s identity given in [10] (see (2.15)). This identity plays also a fundamental role in [8]. Theorem 0.3. Let (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ), λ ∈ R and u(t, x) ∈ X be as in Theorem 0.1. Let ϕ : Rn → R be a function that satisfies the following conditions: x ϕ, x ϕ ∈ L∞ Rn . Then the following identity holds: R
1 2 2 2 2∗ |∂t u| − |∇x u| − λ|u| ϕ + |u| x ϕ dx dt = 0. 2
(0.11)
Rn
In particular we get: 1 R→∞ R
lim
|∇x u|2 − |∂t u|2 dx dt = 0,
(0.12)
R |x|
and 1 R→∞ R
|∇t,x u|2 +
lim
R |x|
2λ 2∗ 2λ 2∗ 2 2 dx dt = 2 |∇ dx. |u| f | + |f | + |g| x 2∗ 2∗ Rn
(0.13)
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Remark 0.5. Notice that (0.12) can be considered as a different version of the classical equipartition of the energy (see [4]), whose classical version can be stated as follows: lim
t→±∞ Rn
∂t u(t, x)2 − ∇x u(t, x)2 dx = 0.
(0.14)
Next we shall fix some notation. Notation. For any 1 p, q ∞ p
Lx
p
q
and Lt Lx
denote the Banach spaces Lp R n
and Lp R; Lq Rn ,
and in the specific case p = q we also use the notation p Lt,x ≡ Lp R; Lp Rn . s (Rn ) the Besov spaces. We shall denote by Lp,q (Rn ) the usual Lorentz spaces and by B˙ p,2 For every 1 p ∞ we shall use the following mixed norm for functions defined on R3 : p
f L∞ Lp r θ
≡ sup
u(rω)p dω
(0.15)
r>0
S2
where
S 2 ≡ ω ∈ R3 |ω| = 1 and dω denotes the volume form on S. We shall denote by H˙ x1 the homogeneous Sobolev space H˙ 1 (Rn ). Given any couple of Banach spaces Y and Z, we shall denote by L(Y, Z) the space of linear and continuous functionals between Y and Z. We shall denote by Ct (Y )
and Ct1 (Y )
respectively the spaces C(R; Y )
and C 1 (R; Y )
where Y is a generic Banach space. p We shall denote by Lt (Y ) the space of Lp functions defined on R and valued in Y . We shall denote by X the functional space introduced in (0.2). Given a space–time dependent function w(t, x) we shall denote by w(t0 ) the trace of w at fixed time t ≡ t0 , in case that it is well-defined.
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We shall denote by . . . dx, . . . dt and . . . dx dt the integral of suitable functions with respect to space, time, and space–time variables, respectively. When it is not better specified we shall denote by ∇v the gradient of any time-dependent function v(t, x) with respect to the space variables. Moreover ∇τ and ∂|x| shall denote respectively the angular gradient and the radial derivative. If ψ ∈ C 2 (Rn ), then D 2 ψ will represent the hessian matrix of ψ . Given a set A ⊂ Rn we denote by χA its characteristic function. The ball of radius R > 0 in Rn shall be denoted as BR . We shall use the function
x ≡ 1 + |x|2 . 1. On the Strichartz estimates for critical NLW Recall that by combining the papers [16] and [3], it follows that the defocusing NLW is globally well-posed in the Banach space introduced in (0.2) and moreover the following properties hold: lim u(t)L2∗ = 0 t→±∞
x
and 2(n+1) 1 n 2 R . u(t, x) ∈ Lt n−1 B˙ 2(n+1) n−1
In the next proposition we gather some known facts that we shall use later on. The main point is that it applies to both focusing and defocusing NLW. Proposition 1.1. Let (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ) and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1). Then we have: ˙1 u(t, x) ∈ L∞ t Hx ; lim u(t)L2∗ = 0;
t→±∞
x
2(n+1) 1 n 2 u(t, x) ∈ Lt n−1 B˙ 2(n+1) R .
(1.1) (1.2) (1.3)
n−1
Proof. For simplicity we shall prove (1.2) only in the case t → ∞ and we shall also show boundedness of u(t) H˙ 1 only for t > 0. The other cases can be treated in a similar way. x
˙1 First step: u(t, x) ∈ L∞ t Hx . Since we are assuming 2(n+1)
u(t, x) ∈ X ⊂ Lt,xn−2 , we can deduce by standard techniques in nonlinear scattering that u(t, x) is asymptotically free. This means that there exists (f + , g + ) ∈ H˙ x1 × L2x such that
L. Vega, N. Visciglia / Journal of Functional Analysis 255 (2008) 726–754
lim u(t) − u+ (t)H˙ 1 + ∂t u(t) − ∂t u+ (t)L2 = 0,
t→∞
x
x
733
(1.4)
where 2u+ = 0, u+ (0) = f + ,
∂t u+ (0) = g + .
The following computation is trivial: sup ∇x u(t)L2 + ∂t u(t)L2 x
t∈R
x
sup∇x u(t) − ∇x u+ (t)L2 + sup∇x u+ (t)L2 t∈R
+ sup∂t u(t) − ∂t u+ (t) t∈R
x
L2x
t∈R
x
+ sup∂t u+ (t)L2 < ∞, t∈R
x
(1.5)
where at the last step we have used (1.4) and the conservation of the energy for solutions to free wave equation. ∗
2 Second step: u(t, x) ∈ L∞ t Lx and proof of (1.2). By combining the previous step with the Sobolev embedding ∗ H˙ x1 ⊂ L2x ,
(1.6)
we deduce that ∗
2 u(t, x) ∈ L∞ t Lx .
On the other hand, by combining again the Sobolev embedding with (1.4) we get: lim u(t)L2∗ lim u(t) − u+ (t)L2∗ + u+ (t)L2∗ = 0,
t→∞
x
t→∞
x
x
(1.7)
where at the last step we have used Proposition A.1 in Appendix A. Third step: proof of (1.3). Once (1.2) has been shown, then the proof of (1.2) follows as in [3]. 2 2. On the asymptotic behaviour of free waves First we present a proposition whose content is well known in the literature. However in Appendix B we shall present a self-contained proof. Proposition 2.1. Let u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) be the unique solution to 2u = 0,
(t, x) ∈ R × Rn , n 3,
u(0) = f ∈ H˙ x1 ,
∂t u(0) = g ∈ L2x .
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Then the following facts occur: ∂t u(t) ± ∂|x| u(t)
= o(1) as t → ±∞; ∇x u(t)2 dx = lim ∂t u(t)2 dx = 1 |∇x f |2 + |g|2 dx; t→±∞ 2 ∂t u(t)2 + ∇x u(t)2 dx = o(1) as t → ±∞;
lim
t→±∞
2|x|<|t|
L2x
∇τ u(t)2 dx = o(1)
as t → ±∞;
(2.1) (2.2) (2.3)
(2.4)
In particular, in the case (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ) we get the following stronger version of (2.1) and (2.3): ∂t u(t) ± ∂|x| u(t)
L2x
=O
1 |t|
as t → ±∞,
(2.5)
and
∂t u(t)2 + ∇x u(t)2 dx = O
1 t2
as t → ±∞.
(2.6)
2|x|<|t|
Proof. See Appendix B.
2
Next we shall study some asymptotic expressions involving solutions to the free wave equation with initial data in the energy space H˙ x1 × L2x . Those expressions will play a fundamental role in the sequel. Lemma 2.1. Assume that u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) solves: 2u = 0, u(0) = f ∈ H˙ x1 ,
∂t u(0) = g ∈ L2x .
(2.7)
Let ψ : Rn → R be a radially symmetric function such that the following limit exists: lim ∂|x| ψ = ψ (∞) ∈ (0, ∞).
|x|→∞
Then lim
t→±∞
1 ∂t u(t)∇x u(t) · ∇x ψ dx = ∓ ψ (∞) 2
|∇x f |2 + |g|2 dx.
(2.8)
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735
Proof. We shall study only the case t → ∞ (in fact the case t → −∞ reduces to the previous one since v(t, x) ≡ u(−t, x) is still a solution to the free wave equation and its behaviour at infinity is related to the behaviour of u(t, x) as t → −∞). Notice that we have
1 ∂t u(t)∇x u(t) · ∇x ψ dx ∇x ψ L∞ x 2
2|x|
∂t u(t)2 + ∇x u(t)2 dx
(2.9)
2|x|
that due to (2.3) implies lim
t→∞ 2|x|
∂t u(t)∇x u(t) · ∇x ψ dx = 0.
Next notice that due to (2.1) we have lim
t→∞ 2|x|>t
∂t u(t)∇x u(t) · ∇x ψ dx = lim
t→∞ 2|x|>t
∂t u(t)∂|x| u(t)∂|x| ψ dx
= − lim
t→∞ 2|x|>t
∂t u(t)2 ∂|x| ψ dx.
(2.10)
On the other hand, we have
∂t u(t)2 dx
inf (∂|x| ψ)
2|x|>t
2|x|>t
∂t u(t)2 ∂|x| ψ dx
2|x|>t
∂t u(t)2 dx.
sup (∂|x| ψ) 2|x|>t
2|x|>t
Then due to the assumption done on ∂|x| ψ implies lim
t→∞ 2|x|>t
∂t u(t)2 ∂|x| ψ dx = ψ (∞) lim
t→∞ 2|x|>t
∂t u(t)2 dx
(2.11)
provided that the last limit exists. By combining (2.9)–(2.11) we deduce that the proof will be concluded provided that we can show lim
t→∞ 2|x|>t
On the other hand, we have
∂t u(t)2 dx = 1 2
|∇x f |2 + |g|2 dx.
(2.12)
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lim
t→∞ 2|x|>t
∂t u(t)2 dx = lim
t→∞
=
1 2
∂t u(t)2 dx − lim
t→∞ 2|x|
∂t u(t)2 dx
|∇x f |2 + |g|2 dx
(2.13)
2
where we have used (2.2) and (2.3).
Lemma 2.2. Assume that u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) solves (2.7). Let φ : Rn → R be a radially symmetric function such that x φ ∈ L∞ x . Then we have lim
t→±∞
∂t u(t)u(t)φ dx = 0.
(2.14)
Proof. As in the proof of Lemma 2.1 it is sufficient to consider the limit for t → ∞. First notice that by combining the decay assumption done on φ with Hardy’s inequality we get ∂t u(t)u(t)φ dx ∂t u(t) 2 u(t)φ 2 C ∂t u 2 2 + ∇x u 2 2 ≡ const Lx Lx Lx Lx
∀t ∈ R.
Due to this fact it is easy to show that by a density argument it is sufficient to prove (2.14) under the assumptions that (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ). Notice that if u(t, x) is a regular solution to (2.7), then we have d2 dt 2
u(t)2 dx = 2
∂t u(t)2 − ∇x u(t)2 dx,
(2.15)
that due to (2.2) implies d2 dt 2
u(t)2 dx = o(1)
as t → ∞.
After integration of this identity we get
u(t)2 dx =
|f | dx + 2t 2
f g dx + o t 2 ,
and hence
Next notice that we have
u(t)2 dx = o t 2
as t → ∞.
(2.16)
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737
∂t u(t)u(t)φ dx = I (t) + II(t), where
I (t) ≡
and II(t) ≡
∂t u(t)u(t)φ dx,
2|x|>t
∂t u(t)u(t)φ dx.
2|x|
Notice that due to the decay assumption done on φ we have I (t)
C t
u(t)∂t u(t) dx C u(t) 2 ∂t u(t) 2 = o(1)∂t u(t) 2 , L L Lx x x t
2|x|>t
where we have used (2.16). In particular we get lim I (t) = 0.
t→∞
On the other hand, due to (2.16) and (2.6) we have II(t) φ L∞
u(t)2 dx
x
1 2
2|x|
∂t u(t)2 dx
1 2
1 , = Co(t)O t
2|x|
and hence lim II(t) = 0.
t→∞
The proof is complete.
2
3. Proof of Theorem 0.1 We shall need the following propositions. Proposition 3.1. Assume that u(t, x) ∈ X is a global solution to (0.1) in dimension n 3 for some λ ∈ R and (f, g) ∈ H˙ x1 × L2x . Assume moreover that u(t, x) satisfies (0.3). Then we have lim
t→±∞
1 ∂t u(t)∇x u(t) · ∇x ψ(x) dx = ∓ ψ (∞) 2
2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx, 2
Rn
where ψ : Rn → R is a radially symmetric function such that the following limit exists lim ∂|x| ψ = ψ (∞) ∈ (0, ∞).
|x|→∞
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Proof. As in the proof of Proposition 1.1 we only treat the case t → ∞. Let u+ (t, x), f + , g + be as in the proof of Proposition 1.1. As a consequence of (1.4) we deduce that ∂t u(t)∇x u(t) · ∇x ψ dx ∂t u+ (t)∇x u+ (t) · ∇x ψ dx = lim
lim
t→∞
t→∞
1 = lim − ψ (∞) t→∞ 2 1 = lim − ψ (∞) t→∞ 2
|∇x f + |2 + |g + |2 dx ∇x u+ (t)2 + ∂t u+ (t)2 dx,
(3.1)
where we have used Lemma 2.1 and the conservation of the energy for the free wave equation. ∗ Next notice that by combining (1.4), (3.1), the Sobolev embedding H˙ x1 ⊂ L2x and the conservation of the energy for solutions to critical NLW we get: lim
t→∞
∂t u(t)∇x u(t) · ∇x ψ dx
∗ ∗ ∇x u+ (t)2 + ∂t u+ (t)2 + 2λ u+ (t)2 − 2λ u+ (t)2 dx 2∗ 2∗ 2∗ 2λ + 2∗ 2 2 2λ 1
∇x u(t) + ∂t u(t) + ∗ u(t) − ∗ u (t) = lim − ψ (∞) dx t→∞ 2 2 2 2∗ 1 2λ λ ∗ = − ψ (∞) |∇x f |2 + |g|2 + ∗ |f |2 dx + ∗ ψ (∞) lim u+ (t) dx t→∞ 2 2 2 1 2λ ∗ |∇x f |2 + |g|2 + ∗ |f |2 dx, = − ψ (∞) 2 2 1 = lim − ψ (∞) t→∞ 2
where we have used Lemma 2.8 and the property lim
t→∞
+ 2∗ u (t) dx = 0.
The proof of (3.2) can be found in Appendix A.
(3.2)
2
Proposition 3.2. Assume that u(t, x) ∈ X is a global solution to (0.1) in dimension n 3 for some λ ∈ R and (f, g) ∈ H˙ x1 × L2x . Let φ : Rn → R be a radially symmetric function such that x φ ∈ L∞ x , then we have lim
t→±∞
∂t u(t)u(t)φ dx = 0.
(3.3)
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Proof. As in Proposition 1.1 it is sufficient to treat the case t → ∞. Let u+ (t, x), f + (x) and g + (x) be as in the proof of Proposition 1.1. Notice that due to the decay assumption done on φ and due to the Hardy inequality we deduce that lim φ u(t) − u+ (t) L2 C lim ∇u(t) − ∇u+ (t)L2 = 0
t→∞
t→∞
x
x
(3.4)
where at the last step we have used (1.4). On the other hand, due again to (1.4) we have: lim ∂t u(t) − ∂t u+ (t)L2 = 0.
t→∞
x
(3.5)
By combining (3.4) and (3.5) we deduce that
∂t u(t)u(t)φ dx = lim
lim
t→∞
t→∞
where at the last step we have used (2.14).
∂t u+ (t)u+ (t)φ dx = 0
2
Proof of Theorem 0.1. Following [15] we multiply Eq. (0.1) by ∇x ψ · ∇x u + 12 x ψu in order to get after integration by parts: T −T Rn
=
1 λ ∗ ∇x uDx2 ψ∇x u − |u|2 2x ψ + |u|2 x ψ dx dt 4 n 1 ∓ ∂t u(±T )∇x u(T ) · ∇x ψ + ∂t u(±T )u(±T )x ψ dx . 2 ± Rn
The proof can be completed by taking the limit as T → ∞ and by using Propositions 3.1 and 3.2. 2 4. Proof of Theorem 0.2 We start this section with some preliminary results that will be useful along the proof of Theorem 0.2. Proposition 4.1. Let (f, g) ∈ H˙ 1 (Rn ) × L2 (Rn ) and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1) for n 3. Then
1 ∗ |u|2 dx dt < ∞, x
(4.1)
and in particular 1 R→∞ R
∗
|u|2 dx dt = 0.
lim
BR
(4.2)
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Proof. Notice that due to the Hölder inequality in Lorentz spaces we get:
1 1 ∗ ∗ 2∗ (1−θ) 2∗ |u| dx
u 2 2∗ n ,2∗ C u 2 θ2n(n+1) ,2
u L2∗ ,2 (Rn ) , x x Ln,∞ L n−1 L n2 −2n−1 (Rn ) x x
where n−1 θ (n2 − 2n − 1) 1 − θ + ∗ = ∗ , 2n(n + 1) 2 2 n i.e. θ=
(n + 1)(n − 2) . n(n − 1)
By combining the previous inequality with the Sobolev embedding: ∗ H˙ x1 ⊂ L2 ,2 Rn and 1
2 B˙ 2(n+1) n−1
,2
2n(n+1) n ,2 R ⊂ L n2 −2n−1 Rn ,
we get
4
2(n+1) 1 ∗ |u|2 dx C u n−1 2(n+1) 1 x L n−1 B˙ 2 t
2(n+1) ,2 (R n−1
n)
u L(n−1)(n−2) < ∞, ∞ H˙ 1 t
x
where at the last step we have used (1.1) and (1.3). The proof of (4.1) is complete. Notice that (4.2) follows by combining (4.1) with the dominated convergence theorem. 2 Proposition 4.2. Let (f, g) ∈ H˙ x1 × L2x and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1) with n 3. Then we have lim
R→∞
∗
|x φR ||u|2 dx dt = 0,
where φ is a radially symmetric function such that |x φ| and φR = Rφ( Rx ).
C x
(4.3)
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Proof. By assumption we have
2∗
C ∗ |u|2 dx dt → 0 as R → ∞, R + |x|
|x φR ||u| dx dt C
where at the last step we have combined the dominated convergence theorem with (4.1).
(4.4) 2
Proposition 4.3. Let (f, g) ∈ H˙ x1 × L2x and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1) for n 3. Then u(t, x) satisfies 1 R→∞ R 3
|u|2 dx dt = 0.
lim
(4.5)
BR
In order to prove Proposition 4.3 we shall need some lemma. Our next result will be particularly useful along the proof of Proposition 4.3 in the case n = 3. Lemma 4.1. Let u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) be the unique solution to (t, x) ∈ R × R3 ,
2u = F ∈ L1t L2x , u(0) = f ∈ H˙ x1 ,
∂t u(0) = g ∈ L2x .
For every 1 p < ∞ there exists a constant C ≡ C(p) > 0 such that the following a priori estimate holds:
u L2 L∞ Lp C f H˙ 1 + g L2x + F L1 L2 . t
r
x
θ
t
x
Proof. In [11] it is proved the following estimate for every 1 p < ∞:
u L2 L∞ Lp C f H˙ 1 + g L2x t
r
x
θ
where u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) is the unique solution to 2u = 0, u(0) = f
∈ H˙ x1 ,
∂t u(0) = g ∈ L2x .
The proof of Lemma 4.1 in the case F (t, x) = 0 follows easily by combining the previous estimate with the Minkowski inequality and the Duhamel formula. 2 Lemma 4.2. Let (f, g) ∈ H˙ x1 × L2x and λ ∈ R be such that there exists a unique global solution u(t, x) ∈ X to (0.1) for n 3. Then we have: u(t, x) and
2n
L2t Lxn−3
< ∞ when n 4,
(4.6)
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u L2 L∞ Lp < ∞ t
r
∀1 p < ∞,
θ
when n = 3.
(4.7)
Proof. We split the proof in two parts. Proof of (4.6). Notice that by combining the Sobolev embedding 2n(n+1) 2 −2n−1
1
2 B˙ 2(n+1) n−1
,2
⊂ Lxn
with (1.3) we get u(t, x)
2(n+1) n−1
Lt
2n(n+1) 2 −2n−1
< ∞.
Lxn
On the other hand, (1.1) implies u(t, x)
∗
2 L∞ t Lx
<∞
and hence by interpolation u(t, x)
2(n+2) n−2
Lt
2n(n+2) (n−2)(n+1)
< ∞.
(4.8)
Lx
Recall that u(t, x) solves: ∗
2u = −λu|u|2 −2 , (t, x) ∈ R × Rn , n 4, u(0) = f ∈ H˙ 1 Rn , ∂t u(0) = g ∈ L2 Rn , and hence by Strichartz estimates (see [6]) we deduce: u(t, x)
2n L2t Lxn−3
n+2 C f H˙ 1 + g L2x + |λ|u n−2 x
2n L2t Lxn+1
.
(4.9)
By combining this estimate with (4.8) we get (4.6). Proof of (4.7). Notice that the proof of (4.6) fails in dimension n = 3 since in this case the endpoint Strichartz estimate (i.e. a version of (4.9) for n = 3) is false. Next we shall overcome this difficulty by using Lemma 4.1. By combining (1.3) (where we choose n = 3) with the Sobolev embedding: 1 2 B˙ 4,2 R3 ⊂ L12 R3 ,
we deduce that u(t, x) ∈ L4t L12 x .
(4.10)
On the other hand, due to (1.1) and due to the Sobolev embedding H˙ 1 (R3 ) ⊂ L6 (R3 ), we get
L. Vega, N. Visciglia / Journal of Functional Analysis 255 (2008) 726–754
u(t, x)
6 L∞ t Lx
743
< ∞.
Hence by interpolation we get
u L5 L10 < ∞. t
(4.11)
x
Next notice that u(t, x) ∈ X solves the following Cauchy problem with forcing term: 2u = −λu5 , u(0) = f ∈ H˙ x1 ,
(t, x) ∈ R × R3 , ∂t u(0) = g ∈ L2x .
By combining this fact with Lemma 4.1 (where we choose F = −λu5 ) and (4.11) we deduce:
u L2 L∞ Lp C f H˙ 1 + g L2x + |λ| u 5L5 L10 < ∞. t
r
x
θ
t
x
2
Proof of Proposition 4.3. We shall prove Proposition 4.3 in dimension n = 3 by using as a basic tool (4.7). It will be clear that the same argument works in dimension n > 3 provided that we use (4.6) instead of (4.7). Since now on we shall assume n = 3. Notice that for every T > 0 we have: ∞R
∞ |u| dx dt = 2
T BR
T
0
u(t, rω)2 dω r 2 dr dt
S2
∞R R
2 T
0
S2
∞
R3
sup T
= R3
u(t, rω)2 dω dr dt
r∈(0,R)
u(t, rω)2 dω dt
S2
2 sup u(t, rω)L2 dt = R 3 u 2L2 ((T ,∞);L∞ L2 ) .
r∈(0,∞)
ω
r
By combining this fact with (4.7) we get the following implication: 1 ∀ > 0 there exists T1 () > 0 s.t. lim sup 3 R→∞ R
∞ |u|2 dx dt . T1 () BR
Of course by a similar argument we can prove that: 1 ∀ > 0 there exists T2 () > 0 s.t. lim sup 3 R→∞ R
−T 2 ()
|u|2 dx dt . −∞ BR
ω
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In particular, if we choose T () = max{T1 (), T2 ()}, then we get: ∀ > 0 there exists T () > 0 s.t. 1 |u|2 dx dt . lim sup 3 R R→∞
(4.12)
R\(−T ();T ()) BR
Hence the proof of Proposition 4.3 (in the case n = 3) will follow from the following fact: 1 ∀T > 0 we have lim sup 3 R R→∞
T |u|2 dx dt = 0.
(4.13)
−T BR
Notice that by using the Hölder inequality we get
u(t)2 dx R 2 u(t)2 6 , Lx
BR
and this implies 1 R3
T
C |u| dx dt R
T
2
−T BR
−T
u(t)2 6 dt 2CT u 2 ∞ 6 . Lx Lt Lx R
By combining this fact with (1.1) and with the Sobolev embedding H˙ x1 ⊂ L6x , we finally get (4.13). 2 Proof of Theorem 0.2. First of all let us recall the following identity 2 ∇x uD ¯ x2 ψ∇x u = ∂|x| ψ|∂|x| u|2 +
∂|x| ψ |∇τ u|2 , |x|
(4.14)
where ψ is a radially symmetric function. By using this identity and by choosing in the identity (0.8) the function ψ ≡ x , then it is easy to deduce that |x|>1
|∇τ u|2 dx dt < ∞. |x|
In particular we deduce lim
R→∞
and
|x|>R
|∇τ u|2 dx = 0 |x|
(4.15)
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1 lim R→∞ R
745
|∇τ u|2 dx dt = 0.
(4.16)
BR
By combining (4.2) with (4.16) we get (0.10). Next we shall prove (0.9). For any k ∈ N we fix a function hk (r) ∈ C0∞ (R; [0, 1]) such that: hk (r) = 1
∀r ∈ R s.t. |r| < 1,
hk (r) = 0
∀r ∈ R s.t. |r| >
hk (r) = hk (−r)
∀r ∈ R.
k+1 , k (4.17)
Let us introduce the functions ψk (r), Hk (r) ∈ C ∞ (R): r ψk (r) =
r (r − s)hk (s) ds
and Hk (r) =
0
(4.18)
hk (s) ds. 0
Notice that
ψk
(r) = hk (r),
ψk (r) = Hk (r)
∞ ∀r ∈ R and
lim ∂r ψk (r) =
hk (s) ds.
r→∞
(4.19)
0
Moreover an elementary computation shows that x ψk
C x
∀x ∈ Rn , n 3,
and 2x ψk (x) =
C |x|3
∀x ∈ Rn s.t. |x| 2 and n 4,
2x ψk (x) = 0 ∀x ∈ R3 s.t. |x| 2,
(4.20) (4.21)
where 2x is the biLaplacian operator. Thus the functions φ ≡ ψk satisfy the assumptions of Proposition 4.2. In the sequel we shall need the rescaled functions x ∀x ∈ Rn , k ∈ N and R > 0, ψk,R (x) ≡ Rψk R
(4.22)
where ψk is defined in (4.18). Notice that by combining the general identity (4.14) with (0.8), where we choose ψ = ψk,R defined in (4.22), and recalling (4.19) we get:
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∂|x| ψk,R 1 λ ∗ |∇τ u|2 − |u|2 2x ψk,R + |u|2 x ψk,R dx dt |x| 4 n ∞ 2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx ∀k ∈ N, R > 0. = (4.23) hk (s) ds 2 2 ∂|x| ψk,R |∂|x| u|2 +
0
Notice also that due to (4.21) we get: 2 ψk,R |u|2 dx dt C |u|2 dx dt x R3 BR
R3
provided that n = 3, and in particular by using (4.5) we get 2 ψk,R |u|2 dx dt = 0. lim x
(4.24)
R→∞
R3
In the case n 4 we use (4.20) in order to deduce:
2 ψk,R |u|2 dx dt C x
Rn
1 R3
|u|2 dx dt +
Rn \BR
BR
|u|2 dx dt . |x|3
(4.25)
On the other hand, an explicit computation shows that if we choose in (0.8) ψ ≡ x , when n 4, then we get: Rn
|u|2 dx dt < ∞ for n 4, |x|3
(4.26)
that in conjunction with the Lebesgue dominated convergence theorem, (4.5) and (4.25) implies 2 ψk,R |u|2 dx dt = 0 for n 4. (4.27) lim x R→∞
Rn
By using (4.24), (4.27), (4.3) and (4.15) we get: lim
∂|x| ψk,R
R→∞
|∇τ u|2 1 2 λ ∗ − x ψk,R |u|2 + x ψk,R |u|2 dx dt = 0 |x| 4 n
(4.28)
for every k ∈ N and for every dimension n 3. We can combine this fact with (4.23) in order to deduce: 2 lim ψk,R |∂|x| u|2 dx dt ∂|x| R→∞
∞
=
hk (s) ds 0
|∇x f |2 +
2λ 2∗ 2 |f | + |g| dx 2∗
∀k ∈ N.
(4.29)
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On the other hand, due to the properties of hk (see (4.17)), we get 1 R
2 ∂|x| ψk,R |∂|x| u|2 dt dx
|∂|x| u| dx dt 2
BR
1 R
1 = R
x |∂|x| u|2 dt dx hk R
|∂|x| u|2 dx dt |x|< k+1 k R
that due to (4.29) implies: 1 lim sup R→∞ R
∞
|∂|x| u| dx dt 2
BR
0
1 k+1 lim inf k R→∞ R
2λ 2∗ 2 |f | + |g| dx 2∗
|∇x f |2 +
hk (s) ds
|∂|x| u|2 dx dt
∀k ∈ N.
(4.30)
BR
Since k ∈ N is arbitrary and since the following identity is trivially satisfied: ∞ hk (s) ds = 1,
lim
k→∞ 0
we can deduce (0.9) by using (4.30). The proof is complete. 2 5. Proof of Theorem 0.3 First step: proof of (0.11). Following [15] we multiply Eq. (0.1) by ϕu and integrating the corresponding identity on the strip (−T , T ) we get: T −T
Rn
1 ∗ |∂t u|2 − |∇x u|2 − λ|u|2 ϕ + |u|2 x ϕ dx dt = ± 2 ±
∂t u(±T )u(±T )ϕ dx. (5.1)
By taking the limit as T → ∞ and by using Proposition 3.2 we get (0.11). Second step: proof of (0.12). For any k ∈ N we fix a function ϕk (r) ∈ C0∞ (R; [0, 1]) such that: ϕk (r) = 1
∀r ∈ R s.t. |r| < 1,
ϕk (r) = 0
∀r ∈ R s.t. |r| >
ϕk (r) = ϕk (−r)
∀r ∈ R.
We also introduce the rescaled functions
k+1 , k (5.2)
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ϕk,R ≡
x 1 ϕk . R R
Notice by combining the cut-off property of the functions ϕk with (4.2) and (4.5), we get: ∗ lim |u|2 ϕk,R dx dt = lim |u|2 x ϕk,R dx dt = 0 ∀k ∈ N, R→∞
R→∞
in any dimension n 3. By using this fact in conjunction with (0.11), where we choose ϕ ≡ ϕk,R , we get: lim (5.3) |∂t u|2 − |∇x u|2 ϕk,R dx dt = 0 ∀k ∈ N. R→∞
Notice that by combining (5.3) with the cut–off properties of ϕk we get:
1 R
BR
∀k ∈ N there exists R(k) > 0 s.t. 1 k+1 1 2 |∂t u| dx dt |∇x u|2 dx dt + k+1 k R( k ) k
∀R > R(k).
BR( k+1 ) k
By combining (5.4) with (0.9) and (0.10) , we get: lim sup R→∞
1 R
|∂t u|2 dx dt
k+1 k
2λ 1 ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx + 2 k
∀k ∈ N,
BR
and in particular 1 lim sup R R→∞
2λ 2∗ 2 2 |∇x f | + ∗ |f | + |g| dx. |∂t u| dx dt 2 2
BR
Similarly one can show that 1 lim inf R→∞ R
|∂t u| dx dt 2
BR
2λ 2∗ 2 |∇x f | + ∗ |f | + |g| dx, 2 2
Rn
and finally we get 1 R→∞ R
|∂t u|2 dx dt =
lim
2λ ∗ |∇x f |2 + ∗ |f |2 + |g|2 dx 2
BR
1 R→∞ R
= lim
|∇x u|2 dx dt BR
where at the last step we have combined (0.9) with (0.10). The proof of (0.12) is complete. Finally notice that by combining (0.9), (0.10) and (0.12), we get (0.13). 2
(5.4)
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Acknowledgment The authors should like to thank the anonymous referee for interesting suggestions that improved the original version of this paper. Appendix A ∗
The aim of this appendix is to show that the L2 -norm of the solution to the following Cauchy problem: 2u = 0, u(0) = f ∈ H˙ x1 ,
∂t u(0) = g ∈ L2x ,
(A.1)
goes to zero as t → ±∞. Notice that this fact represents a slight improvement compared with the usual Strichartz estimate u(t, x)
∗
2 L∞ t Lx
C f H˙ 1 + g L2x . x
On the other hand, in Proposition A.2 we shall show that in general no better result can be expected. In fact we shall show that there cannot exist a priori any rate on the decay of the ∗ L2 -norm of the solution to (A.1). Along this section, when it is not better specified, we shall denote by T (t)(f, g) the solution to the Cauchy problem (A.1) with initial data (f, g) computed at time t, i.e.: T (t) : H˙ x1 × L2x (f, g) → u(t) ∈ H˙ x1 , where u(t, x) solves (A.1). Proposition A.1. Let u(t, x) ∈ Ct (H˙ x1 ) ∩ Ct1 (L2x ) be the unique solution to (A.1), then lim u(t)L2∗ = 0.
t→±∞
x
Proof. We treat for simplicity the case t → ∞ (the case t → −∞ can be treated in a similar way). Notice that due to the Sobolev embedding and the conservation of the energy we have: u(t)2 2∗ S ∇x u(t)2 2 + ∂t u(t)2 2 Lx Lx Lx = S ∇x f 2L2 + g 2L2 ∀(f, g) ∈ H˙ x1 × L2x . x
x
(A.2)
In particular the operators T (t) introduced above, are uniformly bounded for every t > 0 in the ∗ space L(H˙ x1 × L2x , L2x ). On the other hand, we have the following dispersive estimate (see [17]): u(t, x)
L∞ x
t
C
f
n−1 2
m−1 B˙ 1,12
+ g
m+1 B˙ 1,12
(A.3)
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s denotes the standard Besov spaces). Notice also that the Fourier representation of the (here B˙ p,q solution to (A.1) implies
u(t)
L2x
C f L2x + g H˙ −1 . x
(A.4)
In particular by combining (A.3) with (A.4) we deduce u(t)
∗
L2x
n−2 2 C u(t)Ln ∞ u Ln2 n−1 x x t n
∀(f, g) ∈ C0∞ Rn × C0∞ Rn ,
where C ≡ C(f, g) > 0. As a consequence we get lim u(t)L2∗ = 0 ∀(f, g) ∈ C0∞ Rn × C0∞ Rn .
t→∞
(A.5)
It is now easy to remove in (A.5) the regularity assumption (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ) by a classical density argument. 2 Notice that the previous result represents a slight improvement compared with the usual Strichartz estimate: u(t, x)
∗
2 L∞ t Lx
C f H˙ 1 + g L2x . x
On the other hand, next proposition shows that in general no better result can be expected, ∗ since there cannot exist a priori any rate on the decay of the L2 -norm of the solution to (A.1). Proposition A.2. Let γ ∈ C([0, ∞); R) be any function such that lim γ (t) = ∞.
t→∞
Then there exists g ∈ L2x such that u(tn )
∗
L2x
>
1 , γ (tn )
where {tn } is a suitable sequence that goes to +∞ and 2u = 0, ∂t u(0) = g ∈ L2x .
u(0) = 0,
(A.6)
Proof. We claim the following fact: S(t)
L(L2x ,L2x∗ )
∗
0 > 0,
(A.7)
where S(t) : L2x g → u(t) ∈ L2x is the solution operator associated to the Cauchy problem (A.6).
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751
Notice that due to (A.7) we get lim γ (t)S(t)L(L2 ,L2∗ ) = ∞,
t→∞
x
x
and in particular due to the Banach–Steinhaus theorem the operators γ (t)S(t) cannot be pointwisely bounded, or in an equivalent way there exists at least one g ∈ L2x such that sup γ (t)S(t)g L2∗ = ∞.
(A.8)
x
[0,∞)
On the other hand, the function γ (t) S(t)g L2∗ is bounded on bounded sets of [0, ∞), and hence x (A.8) implies that lim sup γ (t)S(t)g L2∗ = ∞ x
t→∞
and it completes the proof. Next we shall prove (A.7). Let us fix h ∈ L2x such that:
h L2x = 1 and S(1)hL2∗ = u(1, x)L2∗ = η0 > 0 x
x
where u(t, x) denotes the unique solution to (A.6) with g = h. n A rescaling argument implies that u (t, x) ≡ 2 −1 u(t, x) solves (A.6) with initial data g ≡ n h ≡ 2 h(x). In particular this implies that: n 1 h = 2 −1 u(1, x) S
and h L2x = 1,
and hence: 1 1 n −1 S u(1, x)L2∗ = u(1)L2∗ 2 2∗ S h 2∗ = 2 x x L(Lx ,Lx ) Lx = S(1)hL2∗ = η0 > 0 ∀ > 0. x
2
The proof of (A.7) is complete. Appendix B
This section is devoted to the proof of Proposition 2.1. Let us underline that its content is well known in the literature, in particular it contains the equipartition of the energy principle first proved in [4] by using Fourier analysis. The aim of this section is to present a proof that involves the conformal energy. Proof of Proposition 2.1. First of all notice that (2.1) implies: lim
t→∞
∂t u(t)2 = lim
t→∞
∂|x| u(t)2 dx.
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By combining this fact with (2.2) and with the following trivial identity: |∇x u|2 = |∂|x| u|2 + |∇τ u|2 , we can deduce (2.4). Hence it is enough to prove (2.1) and (2.2) in order to deduce (2.4). Next notice that by a density argument it is sufficient to prove (2.2), (2.5) and (2.6) under the stronger assumption (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ) in order to deduce (2.1)–(2.3) under the weaker assumption (f, g) ∈ H˙ x1 × L2x . Since now on we shall assume that (f, g) ∈ C0∞ (Rn ) × C0∞ (Rn ). Following [5] we introduce the conformal energy factor 2 t + |x|2 ∂t u + 2ntxj ∂j u . Since 2u = 0 we get for every T > 0 the following identity: 0=
n T 2 t + |x|2 ∂t u + 2txj ∂j u 2u dx dt j =1 0
2 2 n 2 T + |x|2 ∂t u(T ) + ∇x u(T ) + 2nT r∂|x| u(T )∂t u(T ) dx 2
=
−
n 2 |x| |∇x f |2 + |g|2 dx + n(n − 1) 2
T
t |∂t u|2 + |∇x u|2 dx dt,
0
where we have used the Stokes formula. Notice that this identity implies the following inequality: 2 2 T + |x|2 ∂t u(T ) + |∇x u|2 + 4T |x|∂|x| u(T )∂t u(T ) dx |x|2 |∇x f |2 + |g|2 dx.
(B.1)
On the other hand, we have the trivial pointwise inequality |∂|x| u|2 |∇x u|2 that can be combined with (B.1) in order to give: 2 2 T + |x|2 ∂t u(T ) + |∂|x| u|2 + 4T |x|∂|x| u(T )∂t u(T ) dx |x|2 |∇x f |2 + |g|2 dx, (B.2) and
2 2 2 T + |x|2 ∂t u(T ) + ∇x u(T ) − 4T |x|∇x u(T )|∂t u(T )| dx |x|2 |∇x f |2 + |g|2 dx.
(B.3)
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753
Next recall the basic inequality (a + b)2 (c + d)2 + (a − b)2 (c − d)2 4 a 2 + b2 c2 + d 2 + 16abcd,
∀a, b, c, d ∈ R,
whose proof is completely elementary. By combining this inequality with (B.2) and (B.3) we get respectively:
2 2 2 2 T + |x| ∂t u(T ) + ∂|x| u(T ) + T − |x| ∂t u(T ) − ∂|x| u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx,
and
2 2 2 2 T + |x| ∂t u(T ) − ∇x u(T ) + T − |x| ∂t u(T ) + ∇x u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx.
This in turn implies:
∂t u(T ) + ∂|x| u(T )2 dx 4 |x|2 |∇x f |2 + |g|2 dx, 2 T 2 ∂t u(T ) − ∇x u(T ) dx 4 |x|2 |∇x f |2 + |g|2 dx, 2 T
(B.4) (B.5)
and
∂t u(T ) − ∇x u(T )2 + ∂t u(T ) + ∇x u(T )2 dx
2|x|
16 2 T The proof is complete.
|x|2 |∇x f |2 + |g|2 dx.
(B.6)
2
References [1] S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976) 1–38. [2] H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175. [3] H. Bahouri, J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (6) (1998) 783–789. [4] A.R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967) 207–208. [5] G. Dassios, M. Grillakis, Equipartition of energy in scattering theory, SIAM J. Math. Anal. 14 (5) (1983) 915–924. [6] J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1) (1995) 50–68.
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[7] M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (6) (1992) 749–774. [8] C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, preprint. [9] C. Kenig, G. Ponce, L. Vega, On the Zakharov and Zakharov–Schulman systems, J. Funct. Anal. 127 (1) (1995) 204–234. [10] H. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = Au+(u), Trans. Amer. Math. Soc. 192 (1974) 121. [11] S. Machihara, M. Nakamura, K. Nakanishi, T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation, J. Funct. Anal. 219 (1) (2005) 1–20. [12] C. Morawetz, Time decay for the nonlinear Klein–Gordon equation, Proc. Roy. Soc. London Ser. A 306 (1968) 291–296. [13] K. Nakanishi, Scattering theory for the nonlinear Klein–Gordon equation with Sobolev critical power, Int. Math. Res. Not. (1) (1999) 31–60. [14] H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (2) (1984) 261– 270. [15] B. Perthame, L. Vega, Morrey–Campanato estimates for Helmholtz equations, J. Funct. Anal. 164 (2) (1999) 340– 355. [16] J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. (7) (1994) 303–309 (electronic). [17] J. Shatah, M. Struwe, Geometric Wave Equations, Courant Lect. Notes Math., vol. 2, New York Univ., Courant Inst. Math. Sci., New York, 1998. [18] L. Vega, N. Visciglia, On the local smoothing for the Schrödinger equation, Proc. Amer. Math. Soc. 135 (1) (2007) 119–128. [19] L. Vega, N. Visciglia, On the local smoothing for a class of conformally invariant Schrödinger equations, Indiana Univ. Math. J. 56 (5) (2007) 2265–2304. [20] L. Vega, N. Visciglia, Asymptotic lower bounds for a class of Schrödinger equations, Comm. Math. Phys. 279 (2) (2008) 429–453.
Journal of Functional Analysis 255 (2008) 755–767 www.elsevier.com/locate/jfa
Absolutely continuous spectrum of one random elliptic operator O. Safronov Department of Mathematics and Statistics, University of North Carolina at Charlotte, 376 Fretwell Bldg, Charlotte, USA Received 26 December 2007; accepted 24 April 2008 Available online 2 June 2008 Communicated by J. Bourgain
Abstract We consider an elliptic random operator, which is the sum of the differential part and the potential. The potential considered in the paper is the same as the one in the Andersson model, however the differential part of the operator is different from the Laplace operator. We prove that such an operator has absolutely continuous spectrum on all of (0, ∞). © 2008 Elsevier Inc. All rights reserved. Keywords: Absolutely continuous spectrum; Random operators; Trace formulas
1. Formulation of the main result In dimension d 5, we consider the differential operator H0 = − + τ ζ (x)|x|−ε (−θ ),
ε > 0, τ > 0,
(1.1)
where θ is the Laplace–Beltrami operator on the unit sphere S = {x ∈ Rd : |x| = 1} and ζ is the characteristic function of the complement to the unit ball {x ∈ Rd : |x| 1}. Before describing the results we need to give a definition of the operator θ . Let us introduce the polar coordinates E-mail address: [email protected]. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.019
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O. Safronov / Journal of Functional Analysis 255 (2008) 755–767
in Rd and denote them by r = |x| and θ = operator is hidden in the relation =
1 r d−1
x |x| .
One of the definitions of the Laplace–Beltrami
1 ∂ d−1 ∂ r + 2 θ . ∂r ∂r r
The standard argument with separation of variables allows one to define the operator (1.1) as the orthogonal sum of one-dimensional Schrödinger operators, which implies that H0 is essentially self-adjoint on C0∞ (Rd ). The spectrum of this operator has an absolutely continuous component, which coincides with the positive half-line [0, ∞) as a set. We perturb now the operator H0 by a real-valued potential V = Vω that depends on the random parameter ω = {ωn }n∈Zd : ωn χ(x − n). Vω = n∈Zd
The function χ in this formula is the characteristic function of the cube [0, 1)d , and ωn are independent random variables taking their values in the interval [−1, 1]. We will assume that E[ωn ] = 0 for all n. This condition guarantees the presence of oscillations of V . Set H± = H0 ± Vω . Theorem 1.1. Let 0 < ε < 2/(d + 1). Then, for any τ > 0, the operator H± has an absolutely continuous spectrum, whose essential support covers the positive half-line [0, ∞). In other words, the spectral projection EH± (δ) corresponding to a set δ ⊂ R+ of positive Lebesgue measure, is different from zero. Remark. The condition d 5 is related to the necessity to use the estimate (1.13) for the mean value V¯ of the potential over the sphere of radius |x|. Our method requires that V¯ ∈ L(d+1)/2 , while V¯ behaves as |x|−(d−1)/2 at the infinity. Proof of Theorem 1.1. We shall consider only the case when τ = 1, because the only property of τ that matters is that it is positive. The proof of the theorem is based on two sufficiently deep observations. 1. The entropy of the spectral measure of the operator H+ (as well as H− ) can be estimated by the negative eigenvalues of the operators H+ and H− . Let us clarify this statement. Let μ be the spectral measure of H+ , constructed for the element f , which means that (H+ − z)−1 f, f =
∞
−∞
dμ(t) . t −z
Let λj (∓V ) be the negative eigenvalues of the operator H± . Then one can find such an element f of the space L2 (Rd ), that the measure constructed for this element will satisfy the condition b a
λj (V )1/2 + λj (−V )1/2 , log μ (λ) dλ −C 1 + j
j
(1.2)
O. Safronov / Journal of Functional Analysis 255 (2008) 755–767
757
where 0 < a < b < ∞ and the constant C depends only on a and b. The proof of this statement b can be found in [5]. Due to Jensen’s inequality, the integral a log μ (λ) dλ can diverge only to −∞. But, if it converges, then μ (λ) > 0 almost everywhere on [a, b], which leads to a certain conclusion about the absolutely continuous spectrum of H+ . Let us draw attention of the reader to the main difficulty of application of (1.2): it is derived only for compactly supported perturbations and one has to make sure that it survives in the limit, when V is approximated by compactly supported functions. 2. Within the conditions of the theorem,
λj (±V )1/2 < ∞, E
(1.3)
j
which implies λj (±V )1/2 < ∞,
almost surely.
j
Actually, it is much better to take the expectation in both sides of (1.2) and then talk about approximations of V by compactly supported functions, instead of doing it directly. Let us introduce the notation V¯ for the mean value of Vω over the sphere of radius |x| 1 V¯ (x) = |S|
Vω |x|θ dθ.
S
In order to establish (1.3) we will show that Vω = V¯ + div Q where Q is a vector potential having no radial component, i.e. x, Q(x) = 0, ∀x. Besides this, we will show that Q can be chosen in such a way that
E |x|=R
|Q|p dS CE
|Vω |p dS ,
R > 2,
(1.4)
|x|=R
where C depends only on the dimension d and the parameter p 2. Our arguments will be based on the fact that the operator H0 + ζ (x) ±2V ∓ 2V¯ + 4|x|−2+ε Q2 0
(1.5)
is positive, and therefore it does not have negative eigenvalues. The reason why the relation (1.5) holds is that the operator in its left-hand side is representable in the form ∗ − + ζ (x) |x|−ε/2 ∇θ ∓ 2|x|−1+ε/2 Q |x|−ε/2 ∇θ ∓ 2|x|−1+ε/2 Q , ∂ + 1r ∇θ (here er = x/|x|). We will keep the relawhere ∇θ is defined by the relation ∇ = er ∂r tion (1.5) in mind and leave it for the moment. In order to apply (1.5) we have to understand how
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O. Safronov / Journal of Functional Analysis 255 (2008) 755–767
the eigenvalue sums j |λj (±V )|1/2 behave with respect to operation of addition of two potentials. Unfortunately the eigenvalue sums are not additive, but there is something that is similar to additivity. We will show that λj (V1 + V2 )γ λj ε −1 V1 γ + λj (1 − ε0 )−1 V2 γ 0 j
j
(1.6)
j
for any ε0 ∈ (0, 1). In order to do that we need to recall the Birman–Schwinger principle that reduces the study of eigenvalues λj (V ) to the study of the spectrum of a certain compact operator. For any self-adjoint operator T and s > 0 we define n+ (s, T ) = rank ET (s, +∞), where ET (·) denotes the spectral measure of T . Recall the following relation (see [4]) n+ (s + t, T + S) n+ (s, T ) + n+ (t, S).
(1.7)
The next statement is known as the Birman–Schwinger principle. Lemma 1.1. Let V be a real-valued function defined on the space Rd . Let N (λ, V ) be the number of eigenvalues of H0 − V below λ < 0. Then N (λ, V ) = n+ 1, (H0 − λ)−1/2 V (H0 − λ)−1/2 . Combining this lemma with (1.7) we obtain Corollary 1.1. For any ε0 ∈ (0, 1), N (λ, V1 + V2 ) N λ, ε0−1 V1 + N λ, (1 − ε0 )−1 V2 .
(1.8)
∞ Now, since j |λj (V )|γ = 0 γ s γ −1 N (−s, V ) ds, the inequality (1.6) holds for the Lieb– Thirring sums. Without loss of generality we can assume that V and Q are equal to zero for |x| < 1. This can be achieved by the methods of the scattering theory: one can pass from V to ζ V without changing the absolutely continuous spectrum of the differential operator. Due to the representation V = (V − V¯ − 2|x|−2+ε Q2 ) + V¯ + 2|x|−2+ε Q2 , we obtain from (1.6), that λj (V )γ λj 2V − 2V¯ − 4|x|−2+ε Q2 γ + λj 2V¯ + 4|x|−2+ε Q2 γ . (1.9) Since the operator (1.5) is positive, the first sum in the right-hand side of (1.9) equals zero. Thus λj (Vω )γ λj 2V¯ + 4|x|−2+ε Q2 γ .
(1.10)
O. Safronov / Journal of Functional Analysis 255 (2008) 755–767
759
Now formula (1.10) and the classical Lieb–Thirring estimate (see [6] and [7]) lead to the following important intermediate result: −1+ε/2 d+2γ d/2+γ λj (Vω )γ C |x| ¯ Q dx + |V | dx . (1.11) j
Indeed, Theorem 1.2 (Lieb–Thirring). If d 3. Then the negative eigenvalues νj of − − V satisfy the estimate γ +d/2 |νj |γ C V (x) dx, γ 0. j
The Lieb–Thirring estimate is valid for the eigenvalues of a Schrödinger operator. The operator whose eigenvalues we have to estimate has an additional term −|x|− θ . The quadratic form of this operator is positive. Therefore (1.11) follows by monotonicity. What is left to prove at this moment? If we take the expectation in both sides of (1.11), then we shall reduce the problem to the proof of the two relations:
−1+ε/2 d+1 |x| Q dx < ∞ (1.12) E |x|>1
and
(d+1)/2 ¯ E dx < ∞. |V |
(1.13)
The relation (1.12) for ε < 2/(d + 1) immediately follows from (1.4). We shall establish (1.13) in the next section. 2. Proof of (1.4) and (1.13) Let us now prove the necessary estimates (1.4) and (1.13). The proof of these relations is based on the observation that if one has independent random variables τn with zero expectations, then
2
= E τn E (τn )2 . n
n
We shall begin with the following statement. Lemma 2.1. The relation (1.4) holds for even integers p = 2q with q 1. Proof. The mapping V → Q is given by the formula ¯ Q = |x|∇θ −1 θ (V − V ).
(2.1)
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The kernel k(x, y) of this mapping has a singularity of order |x − y|−(d−2) on the diagonal, but this singularity is integrable. Thus, k0 (x, ˆ y) ˆ , |x − y|d−2
k(x, y) =
xˆ =
y x , yˆ = , |x| |y|
where k0 ∈ L∞ , Q(x) =
k(x, y)Vω (y) dS(y).
{y: |y|=|x|}
All the statements about the kernel k will be proved in Appendix A (see Proposition A.1). We represent Q in the form of a sum Q = Q1 + Q2 , where
k0 χ0 (x − y)V (y) dS(y) |x − y|d−2
Q1 = {y: |y|=|x|}
and χ0 is the characteristic function of the unit ball {x: |x| < 1}. We will establish the estimates
E |Qj |2q dS(x) CR d−1 ,
j = 1, 2,
(2.2)
|x|=R
separately. The estimate (2.2) for Q1 is obvious. Indeed,
Q1 (x) C
{y: |y|=|x|}
χ0 (x − y) dS(y). |x − y|d−2
For large values of the radius R = |x| the piece of the sphere intersecting the ball {y: |y − x| < 1} becomes flat. Therefore the latter integral approximately equals the integral over the unit ball in Rd−1 : Q1 (x) C˜
{y∈Rd−1 : |y|<1}
dy C1 . |y|d−2
Let us prove estimate (2.2) for Q2 . Fix x ∈ Rd and denote Δn = ([0, 1)d + n) ∩ {y ∈ Rd : |y| = |x|}. Since E[ωn ] = 0, we obtain that
2q E Q2 (x) C2
m1 +···+mk =2q j
mj 2q! dS(y) , m1 ! · · · mk ! n (1 + |x − y|)d−2 Δn
where all numbers mj are even. Applying the Hölder inequality for Lp -functions, we get
O. Safronov / Journal of Functional Analysis 255 (2008) 755–767
n
Δn
dS(y) (1 + |x − y|)d−2
mj
C3
n
C4
n
dS(y) (1 + |x − y|)2(d−2)
Δn
Δn
761
mj /2
dS(y) (1 + |x − y|)2(d−2)
= C4 {y: |y|=|x|}
dS(y) (1 + |x − y|)2(d−2)
simply because all mj 2 and Δn are uniformly bounded. In oder to estimate the integral {y: |y|=|x|}
dS(y) (1 + |x − y|)2(d−2)
one has to break the domain of integration into two pieces: {y: |x − y| < δ|x|} and {y: |x − y| δ|x|}. If δ > 0 is sufficiently small, then one of these pieces of the sphere will be sufficiently flat, so one can substitute this piece by the plane of dimension d − 1: {y: |y|=|x|, |x−y|<δ|x|}
dS(y) C (1 + |x − y|)2(d−2)
Rd−1
dy . (1 + |y|)2(d−2)
The integral over the second piece can be estimated by homogeneity: {y: |y|=|x|, |x−y|δ|x|}
dS(y) 1 . = O |x|d−3 |x − y|2(d−2)
Consequently,
2q E Q2 (x) C5 . Integrating this inequality with respect to x we obtain (2.2) for j = 2. Thus the statement of the lemma follows from the triangle inequality in the Banach space L2q :
E Q2q (x) dS
(Recall that E[f ] =
Ω
1/2q
f (ω) dω.)
2q E Q1 (x) dS
1/2q
+
2q E Q2 (x) dS
1/2q .
2
Estimate (1.4) is proved only for even integer p. It follows for arbitrary p 2 by interpolation arguments. Indeed, consider the analytic function
f (z) = E
|Q| dS . z
{x: |x|=R}
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Since this function is bounded by the constant Cp R d−1 on each vertical line Re z = 2p, with p ∈ N, it is bounded by CRe(z) R d−1 for Re z 2 according to the following statement. Theorem 2.1. Let f (z) be a bounded analytic function defined on an open domain containing the region a Re z b. Suppose that it is bounded on the two vertical lines by the constants Ca and Cb : f (b + it) Cb ,
f (a + it) Ca ,
∀t ∈ R.
Then f (z) C θ C 1−θ , a
b
for Re z = aθ + b(1 − θ ), θ ∈ (0, 1).
Now we obtain a certain information about the mean values of V over the sphere of radius R. Lemma 2.2. Under conditions of Theorem 1.1
E |V¯ |(d+1)/2 dx < ∞.
(2.3)
−(d−1)q E |V¯ |2q (x) C 1 + |x|
(2.4)
Proof. First we shall prove that
for any positive integer q. Then we will interpolate two such inequalities to obtain (2.4) for q = (d + 1)/4. The mapping V → V¯ is given by the formula V¯ (x) =
c0 |x|d−1
Vω (y) dS(y). {y: |y|=|x|}
Fix x ∈ Rd and denote Δn = [0, 1)d + n ∩ {y ∈ Rd : |y| = |x|} Since E[ωn ] = 0, we obtain that
E |V¯ |2q (x) C5
m1 +···+mk =2q j
mj 2q! dS(y) , m1 ! · · · mk ! n (1 + |x|)d−1 Δn
where all numbers mj are even. It is clear that n
Δn
dS(y) (1 + |x|)d−1
mj
C6
n
1 1 = C7 m (d−1) (d−1)(m j j −1) (1 + |x|) (1 + |x|)
simply because Δn are uniformly bounded. Consequently, (2.4) holds for any positive integer q and therefore for any q 1 (according to Theorem 2.1). Integrating this inequality with respect to x with q = (d + 1)/4, we obtain (2.3). 2
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3. Remarks and references There are some impressive results devoted to the operator −s − + 1 + |x| Vω ,
s > 1/2,
in the mathematical literature. One can study either the discrete or the continuous model of this operator. For the discrete model, the presence of the absolutely continuous spectrum in dimension d = 2 was proved by Bourgain [2,3]. In dimension d = 3, the corresponding result was obtained by Denissov [1]. In spite the fact that the main result of this paper pertains to the theory of random operators, we would like now to formulate a result of the deterministic type. This result will be related to the operator − − εΓ + V ,
(3.1)
where Γ u(x) =
1 |S|
u |x|θ dθ.
S
The potential V in this model is not random: on the contrary, it is a fixed potential. Theorem 3.1. Let d 2 and ε > 0. Assume that V ∈ L∞ , V (rθ ) dθ = 0,
∀r > 1,
(3.2)
S
and
V 2 (x) dx < ∞. (1 + |x|)d−1
(3.3)
Then the absolutely continuous spectrum of the operator (3.1) is essentially supported by [−ε, ∞). Proof. The proof of this theorem relies on the fact that the negative eigenvalues βj (∓V ) of the operator − − εΓ + εI ± V satisfy the condition βj (∓V )1/2 < ∞.
(3.4)
j
The proof of (3.4) is based on the circumstance that the behaviour of the eigenvalues near zero depends only on the structure of the edge of the spectrum of the unperturbed operator. But in the suggested model, this edge has the same structure as the one of the one-dimensional Schrödinger
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operator. Let us introduce the notation P1 for the spectral projection of the operator A = − − εΓ + εI corresponding to the interval [0, ε). Set also P2 = I − P1 . Then V = 2 Re P1 V P2 + P2 V P2 , due to the condition (3.2) on the mean values of V . It was noticed before that βj (V )1/2 = n+ 1, (A + λ)−1/2 V (A + λ)−1/2 dλ √ . 2 λ j ∞
0
Besides the distribution function n+ we shall need distribution functions of singular values of non-selfadjoint operators n(s, T ) = n+ s 2 , T ∗ T , s > 0 (here T is a compact operator). Two of the important properties of this function are s ,T , n(s, T S) n S and n(s1 + s2 , T1 + T2 ) n(s1 , T1 ) + n(s2 , T2 ). Using these properties, we obtain dλ βj (V )1/2 n c1 , (A + λ)−1/2 P1 V √ λ j ∞
0
∞ + 0
dλ n+ c2 , (A + λ)−1/2 P2 V P2 (A + λ)−1/2 √ . 2 λ
Let us remark that the second term equals zero if the norm V L∞ is sufficiently small. In the general case, this term can be well estimated by the integral (3.3). It remains to consider the first term ∞ 0
dλ n c1 , (A + λ)−1/2 P1 V √ λ ∞ = 0
∞ 0
dλ n+ c12 , (A + λ)−1/2 P1 V 2 P1 (A + λ)−1/2 √ λ dλ n+ c12 , (A + λ)−1/2 Γ V 2 Γ (A + λ)−1/2 √ = 2 |Λj |1/2 , λ j
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where Λj are the eigenvalues of the operator −Γ − c1−2 Γ V 2 Γ, which, as a matter of fact, is a one-dimensional Schrödinger operator with the potential αd /r 2 − V 2 . Therefore, according to the Lieb–Thirring bound for a one-dimensional operator (see [6,7]), the sum j |Λj |1/2 can be estimated by the integral (3.3). 2 At the end of this section the author would like to mention one idea, which might be useful for the reader in the study of Anderson’s model. Let us look at the result from a different point of view. Consider an operator which is in a certain sense close to the operator ωn χ(x − n). (3.5) − + Vω , where Vω = n
First, we introduce the class S of perturbations B for which the wave operators exist: B ∈ S if and only if ∃ s-lim exp −it (− + B) exp it (−) . t→±∞
Note, that this class is very rich (meaning “large”) and it can include even differential operators whose coefficients do not decay at infinity. For example, the operator −ζ (x)|x|−s θ ,
s > 1,
belongs to the class S (here ζ is the characteristic function of the exterior of the unit ball), but the coefficients of this operator behave at infinity as |x|2−s . Theorem 3.2. Let ε > 0 and let d 3. Assume that ωn are bounded independent random variables with the property E[ωn ] = 0, for all n. Then for almost every ω, there exists a perturbation B ∈ S such that the operator − + B + (1 + |x|)−ε Vω has a.c. spectrum all over the positive half-line [0, ∞). Proof. Indeed, let Q and V¯ be the same as in the proof of Theorem 1.1. In particular, it means that x, Q(x) = 0, ∀x, div Q = Vω , and V¯ = |S|−1 Vω |x|θ dθ. S
Define −ε −1−ε 2 Q . B = −ζ (x)|x|−(1+ε) θ − ζ (x) 1 + |x| V¯ + 1 + |x| It is easy to check that B ∈ S. On the other hand the operator − + B + ζ (x)(1 + |x|)−ε Vω is positive. So, there is no necessity to estimate eigenvalues of this operator and the trace formula obtained in [5] gives the relation b E
−1−ε 2 1−d log μ (λ) dλ −C 1 + 1 + |x| E Q |x| dx
a
for the spectral measure μ of the operator − + B + ζ (x)(1 + |x|)−ε Vω .
2
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Appendix A 1. Finally consider a technical and less interesting question about the kernel of the operator 2 ∇θ −1 θ on the unit sphere. Let Γ be the orthogonal projection in L (S) onto the subspace of constant functions. 2 Proposition A.1. The kernel of the operator ∇θ −1 θ (I − Γ ) acting in L (S) is a function of the form
k(x, y) =
k0 (x, y) , |x − y|d−2
x, y ∈ S,
where k0 is bounded. Proof. Let us fix the point y. Clearly, the kernel k(x, y) possesses the symmetry with respect to the axis connecting y and −y. Let s be the distance between x and y along the geodesic curve. Then k(x, y) = ρ(s)e(s), where ρ(s) is a certain scalar function and e(x) is the unit vector, tangent to the mentioned geodesic curve at the point x. That means s = 2 arcsin(|x − y|/2). Since div k = δ(x − y) − 1/|S|, we obtain that ρ is a solution of an equation of the form ρ + q(s)ρ = −1/|S|,
s ∈ (0, π),
where the function q(s) = div(e) has two singularities: at the point s = 0 corresponding to y and at the point s = π corresponding to −y. Moreover the character of the singularities at both points y and −y is the same, the only difference is the sign of the leading term: q(s) ∼
d −2 , s
s → 0,
q(s) ∼
d −2 , s−π
s → π.
x−y Indeed, if x ∈ S is close to y, then e is close to the vector |x−y| . Therefore div(e) ∼ function ρ(s) must be smooth at the point s = π , therefore
1 ρ= |S|f (s)
π f (s) ds,
d−2 |x−y| ,
The
where f (s) = exp q(s) ds .
s
Let us clarify the situation with the point s = 0. Since div ρe = −1/|S| everywhere except the point y, we conclude automatically that the function div ρe has to have a singularity at y. The c as s → 0, with some constant c. In other only possible singularity is the one of the type ρ ∼ f (s) words, k(x, y) ∼ as x → y.
2
c |x − y|d−2
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2. We see that an important role in the proof of Theorem 1.1 was played by one of the results of the paper [5]. It would be nice to understand which exactly result of the paper [5] is refered to. Suppose that we have an operator A+ = −
d2 + T (r), dr 2
where T (r) is a matrix-valued function. Certainly this operator has to act in the space of vectorvalued functions, for example in L2 (R+ , Cm ). Assume that T has a finite support. Then there exists a function f ∈ L2 (R+ , Cm ) such that the spectral measure corresponding to f satisfies b
∞ 1/2 λj (A+ ) + T (r)e, e dr , log μ (λ) dλ −C 1 +
a
0
where λj (A+ ) are negative eigenvalues of A+ and e is the vector with coordinates (1, 0, . . . , 0). Moreover, b
λj (A+ )1/2 + λj (A− )1/2 , log μ (λ) dλ −C 1 +
a 2
d where λj (A− ) are negative eigenvalues of A− = − dr 2 − T (r). Here a > 0 and b < ∞.
References [1] S.A. Denisov, Absolutely continuous spectrum of multidimensional Schrödinger operator, Int. Math. Res. Not. 74 (2004) 3963–3982. [2] J. Bourgain, On random Schrödinger operators on Z2 , Discrete Contin. Dyn. Syst. 8 (1) (2002) 1–15. [3] J. Bourgain, Random lattice Schrödinger operators with decaying potential: Some multidimensional phenomena, in: V.D. Milman, G. Schechtman (Eds.), Geometric Aspects of Functional Analysis, Israel Seminar 2001–2002, in: Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 70–98. [4] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. 37 (1951) 760–766. [5] A. Laptev, S. Naboko, O. Safronov, Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials, Comm. Math. Phys. 253 (3) (2005) 611–631. [6] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976) 751–753. See also: The number of bound states of one body Schrödinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 241–252. [7] E.H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in: Studies in Math. Phys., Essays in Honor of Valentine Bargmann, Princeton, NJ, 1976, pp. 269–303.
Journal of Functional Analysis 255 (2008) 768–775 www.elsevier.com/locate/jfa
The fixed point property via dual space properties P.N. Dowling ∗ , B. Randrianantoanina, B. Turett 1 Department of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA Received 3 April 2008; accepted 24 April 2008 Available online 6 June 2008 Communicated by N. Kalton
Abstract A Banach space has the weak fixed point property if its dual space has a weak∗ sequentially compact unit ball and the dual space satisfies the weak∗ uniform Kadec–Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property. © 2008 Elsevier Inc. All rights reserved. Keywords: Fixed point property; E-convexity
Determining conditions on a Banach space X so that every nonexpansive mapping from a nonempty, closed, bounded, convex subset of X into itself has a fixed point has been of considerable interest for many years. A Banach space has the fixed point property if, for each nonempty, closed, bounded, convex subset C of X, every nonexpansive mapping of C into itself has a fixed point. A Banach space is said to have the weak fixed point property if the class of sets C above is restricted to the set of weakly compact convex sets; and a Banach space is said to have the weak∗ fixed point property if X is a dual space and the class of sets C is restricted to the set of weak∗ compact convex subsets of X. A well-known open problem in Banach spaces is whether every reflexive Banach space has the fixed point property for nonexpansive mappings. The question of whether more restrictive * Corresponding author.
E-mail addresses: [email protected] (P.N. Dowling), [email protected] (B. Randrianantoanina), [email protected] (B. Turett). 1 Current address: Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.021
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classes of reflexive spaces, such as the class of superreflexive Banach spaces or Banach spaces isomorphic to the Hilbert space 2 , have the fixed point property has also long been open and has been investigated by many authors [8,14,15,17]. Recently, J. García-Falset, E. Llorens-Fuster, and E.M. Mazcuñan-Navarro [7] proved that uniformly nonsquare Banach spaces, a sub-class of the superreflexive spaces, have the fixed point property. In this article, it is shown that the larger class of E-convex Banach spaces have the fixed point property. The E-convex Banach spaces, introduced by S.V.R. Naidu and K.P.R. Sastry [18], are a class of Banach spaces lying strictly between the uniformly nonsquare Banach spaces and the superreflexive spaces (see also [1]). The second geometric property of Banach spaces that is considered in this article is the weak∗ uniform Kadec–Klee property in a dual Banach space. A dual space X ∗ has the weak∗ uniform Kadec–Klee property if, for every ε > 0, there exists δ > 0 such that, if (xn∗ ) is a sequence in def
the unit ball of X ∗ converging weak∗ to x ∗ and the separation constant sep(xn∗ ) = inf{xn∗ − ∗ : m = n} > ε, then x ∗ < 1 − δ. It is well known [6] that, if X ∗ has weak∗ uniform Kadec– xm Klee property, then X ∗ has the weak∗ fixed point property. If, in addition, the unit ball of X ∗ is weak∗ sequentially compact, more is true: Theorem 3 notes that, if X ∗ has weak∗ uniform Kadec–Klee property and the unit ball of X ∗ is weak∗ sequentially compact, then X has the weak fixed point property. As a consequence of Theorem 3, it is noted that several nonreflexive Banach spaces such as quotients of c0 and C(T )/A0 , the predual of H 1 , have the weak fixed point property. Since the proofs of the main theorems in this paper will require elements of the proof that uniformly nonsquare Banach spaces have the fixed point property, a complete proof of this known result is presented. The proof presented here is a distillation of the original proof and combines elements from [5, Theorem 2.2] and [7, Theorem 3.3]. Recall that a Banach space X is uniformly nonsquare [10] if there exists δ > 0 such that, if x and y are in the unit ball of X, then either (x + y)/2 < 1 − δ or (x − y)/2 < 1 − δ. The general set-up in proving that a Banach space has the weak fixed point property has, by now, become standard fare. If a Banach space X fails to have the weak fixed point property, there exists a nonempty, weakly compact, convex set C in X and a nonexpansive mapping T : C → C without a fixed point. Since C is weakly compact, it is possible by Zorn’s lemma to find a minimal T -invariant, weakly compact, convex subset K of C such that T has no fixed point in K. Since the diameter of K is positive (otherwise K would be a singleton and T would have a fixed point in K), it can be assumed that the diameter of K is 1. It is well known that there exists an approximate fixed point sequence (xn ) for T in K and, without loss of generality, we may assume that (xn ) converges weakly to 0. For details on this general set-up, see [8, Chapter 3]. Theorem 1. (See García-Falset, Llorens-Fuster, and E.M. Mazcuñan-Navarro [7].) Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. Proof. Assume that a Banach space X fails to have the fixed point property. Since uniformly nonsquare spaces are reflexive [10], the fixed point property and the weak fixed point property coincide for X. Therefore there exists a nonexpansive map T : K → K without a fixed point where K is a minimal T -invariant set in X with diameter 1. Let (xn ) be an approximate fixed point sequence for T in K and assume that (xn ) converges weakly to 0. Consider the space ∞ (X)/c0 (X) endowed with the quotient norm given by [wn ] = lim supn wn where [wn ] denotes the equivalence class of (wn ) ∈ ∞ (X). For a bounded set
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C in X, the set [C] in ∞ (X)/c0 (X) is defined by [C] = {[wn ]: wn ∈ C for all n ∈ N}. Using the notation in [5, p. 840], let [W ] = [zn ] ∈ [K]: [zn ] − [xn ] 1/2 and lim sup lim supzm − zn 1/2 . n
m
It is easy to check that [W ] is a closed, bounded, convex, nonempty (since [ 12 xn ] is in [W ]) subset of [K], and is [T ]-invariant where [T ] : [K] → [K] is defined by [T ][zn ] = [T (zn )]. So, by a result of Lin [13], sup[zn ]∈[W ] [zn ] − x = 1 for each x ∈ K. In particular, with x = 0, sup[zn ]∈[W ] [zn ] = 1. Let ε > 0 and choose [zn ] ∈ [W ] with [zn ] > 1 − ε. Let (yj ) = (znj ) be a subsequence of (zn ) such that limyn = [zn ] and (yn ) converges weakly to an element y in K. There is no loss in generality in assuming that yn > 1 − ε for all n ∈ N and in choosing yn∗ ∈ X ∗ so that yn∗ = 1, yn∗ (yn ) = yn , and (yn∗ ) converges weak∗ to y ∗ . (This is possible because the fixed point property is separably-determined [8, p. 35]; so there is no loss in generality in assuming that BX∗ is weak∗ -sequentially compact.) From the definition of [W ] and the weak lower semicontinuity of the norm, it follows that, if n is large enough, yn − y lim infyn − ym < m
Therefore, with un =
2 1+ε (yn
1+ε 2
− y) and u =
1 and y lim infyj − xnj . j 2
2 1+ε y,
2 2 = 2 yn > 2 1 − ε > 2(1 − 2ε) (y y − y) + un + u = 1 + ε n 1+ε 1+ε 1+ε if n ∈ N is large enough. Applying the weak lower semicontinuity of the norm again, it follows that lim inf(un − um ) + u un + u > 2(1 − 2ε) m
if n ∈ N is large enough. So, by taking another subsequence if necessary, we can assume that un + u > 2(1 − 2ε) and (un − um ) + u > 2(1 − 2ε) for all n and all m > n. w∗ ∗ − Furthermore, since ym −→ y ∗ , lim inf(un − um ) − u = lim inf(um + u) − un m m ∗ lim inf ym (um + u) − un m ∗ (un ) = lim inf um + u − ym m
2(1 − 2ε) − y ∗ (un ). w → 0, it follows that lim infm (un − um ) − u > 2(1 − 3ε) if n is large enough. Then, since un − Therefore, for n large enough and m > n also large enough, both (un − um ) + u > 2(1 − 3ε) and (un − um ) − u > 2(1 − 3ε)
P.N. Dowling et al. / Journal of Functional Analysis 255 (2008) 768–775
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hold. Since ε > 0 is arbitrary, un − um < 1, and u < 1, the above inequalities imply that X fails to be uniformly nonsquare, a contradiction which finishes the proof. 2 We want to refine the sequences (xnj ), (yj ), and (yj∗ ) that appear in the proof of Theorem 1. Recall the result of Goebel and Karlovitz [8, p. 124]: If K is a minimal T -invariant, weakly compact, convex set for the nonexpansive map T and (xn ) is an approximate fixed point sequence for T in K, then the sequence (xn − x) converges to the diameter of K for every x in K. Fix ε > 0 and set x˜1 = xn1 , y˜1 = y1 , and y˜1∗ = y1∗ . Then, by the Goebel–Karlovitz Lemma, there exists j1 > 1 such that min{x˜1 − xnj1 , y˜1 − xnj1 } > 1 − ε. Set x˜2 = xnj1 , y˜2 = yj1 , and y˜2∗ = yj∗1 . Another application of the Goebel–Karlovitz Lemma yields j2 > j1 such that mini=1,2 {x˜i − xnj2 , y˜i − xnj2 } > 1 − ε. Set x˜3 = xnj2 , y˜3 = yj2 , and y˜3∗ = yj∗2 . Continuing in this manner, we obtain sequences (x˜n ) and (y˜n ) in K (and BX ) and a sequence (y˜n∗ ) in SX∗ satisfying mini 1 − ε for all n ∈ N and y˜n∗ (y˜n ) = y˜n > 1 − ε for all n ∈ N. In the following, these sequences are renamed by omitting the tildes. The following result is a summary of several easy computations. Lemma 2. Let X be a Banach space whose dual unit ball is weak∗ sequentially compact and assume that X fails the weak fixed point property. Given ε > 0, there exist sequences (yn ) in BX and (yn∗ ) in SX∗ and elements y ∈ BX and y ∗ ∈ BX∗ satisfying: (1) (2) (3) (4) (5) (6) (7)
∗
w w → y and yn∗ − −→ y ∗ ; yn − for every n ∈ N, 1 − ε < yn = yn∗ (yn ) 1; 1+ε ∗ for every n ∈ N, 1−3ε 2 < yn (y) y < 2 ; 1−3ε 1+ε 2 < yn − y < 2 ; 1+ε if n = m, then 1−3ε 2 < yn − ym < 2 ; 1−3ε 1+2ε ∗ if n = m, then 2 < yn (ym ) < 2 ; 1−3ε 1+ε ∗ 2 y (y) 2 .
Proof. Claims (1) and (2), the third inequality in (3), and the second inequalities in (4) and (5) are immediate from the proof of Theorem 1. Then y yn∗ (y) = yn∗ (yn ) − yn∗ (yn − y) > (1 − ε) −
1 + ε 1 − 3ε = 2 2
proving (3). Also yn − y yn∗ (yn − y) = yn − yn∗ (y) > (1 − ε) −
1 + ε 1 − 3ε = 2 2
which finishes the proof of (4). From our refinement of (xn ) and (yn ) done just prior to the lemma and the definition of [W ] in the proof of Theorem 1, if n > m, yn − ym = (yn − xn ) + (xn − ym ) xn − ym − yn − xn
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> (1 − ε) − =
1+ε 2
1 − 3ε 2
showing that (5) holds. The lower inequality in (6) follows from (2) and (5): If n = m, 1 − ε < yn∗ (yn ) = yn∗ (yn − ym ) + yn∗ (ym ) yn − ym + yn∗ (ym ) <
1+ε + yn∗ (ym ). 2
∗ Therefore, if n = m, 1−3ε 2 < yn (ym ). In order to show the upper inequality in (6), we consider subsequences of the sequences (yn ) and (yn∗ ) obtained so far. Note that all of the previous conditions will remain true for subsequences of the current (yn ) and (yn∗ ). Let y˜1 = y1 and y˜1∗ = y1∗ . Since (yn ) converges weakly 1+ε 1−3ε 1+ε ∗ ∗ to y and 1−3ε 2 < y1 (y) < 2 , there exists n1 ∈ N such that, if n n1 , 2 < y1 (yn ) < 2 . 1+ε ∗ Set y˜2 = yn1 and y˜2∗ = yn∗1 . Since (yn ) converges weakly to y and 1−3ε 2 < yn1 (y) < 2 , there 1−3ε 1+ε ∗ ∗ ∗ exists n2 ∈ N such that, if n n2 , 2 < yn1 (yn ) < 2 . Set y˜3 = yn2 and y˜3 = yn2 . Continuing in this manner generates sequences (y˜n ) and (y˜n∗ ) satisfying conditions (1)–(5) and satisfying y˜n∗ (y˜m ) < 1+ε 2 if n < m. Again, we simplify the notation by considering these new sequences but omitting the tildes in the notation. To show the upper inequality in (6) for n > m, first combine (3) with the weak∗ convergence of (yn∗ ) to y ∗ to obtain (7). Then, since (yn ) converges weakly to y, there is no loss in generality 1+2ε ∗ ∗ in assuming that y ∗ (yn ) < 1+2ε 2 for all n ∈ N. In particular, y (y1 ) < 2 . Therefore, since (yn ) converges weak∗ to y ∗ , there exists n1 ∈ N such that, if n n1 , yn∗ (y1 ) < 1+2ε 2 . Setting y˜1 = y1 , ∗ (y˜ ) < 1+2ε and (y ∗ ) y˜1∗ = y1∗ , y˜2 = yn1 , and y˜2∗ = yn∗1 gives y˜2∗ (y˜1 ) < 1+2ε . Then, since y 2 n 2 2 . Set converges weak∗ to y ∗ , there exists n2 ∈ N such that n2 > n1 , and, if n n2 , yn∗ (y˜2 ) < 1+2ε 2 y˜3 = yn2 and y˜3∗ = yn∗2 . Continuing in this manner generates sequences (y˜n ) and (y˜n∗ ) satisfying all of the conditions of the lemma. 2
As a consequence of these computations, we have the following Theorem 3. Let X be a Banach space such that BX∗ is weak∗ sequentially compact. If X ∗ has the weak∗ uniform Kadec–Klee property, then X has the weak fixed point property for nonexpansive mappings. Proof. If X fails to have the weak fixed point property, consider the sequences (yn ) and (yn∗ ) determined in Lemma 2. In particular, note that yn∗ = 1 for all n ∈ N and that (yn∗ ) converges weak∗ to y ∗ . Note also that, if n = m, ∗ ∗ ∗ ∗ yn − ym (yn − ym ) = yn∗ (yn ) − yn∗ (ym ) − ym (yn ) + ym (ym ) > (1 − ε) − = 1 − 4ε.
1 + 2ε 1 + 2ε − + (1 − ε) 2 2
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It follows that ∗ yn − ym 1 − 4ε 1 − 4ε ∗ ∗ yn − ym > =2 > 2 − 10ε. 2 yn∗ − ym yn − ym (1 + ε)/2 1+ε 1 Thus, if ε < 10 , (yn∗ ) is a sequence in the unit sphere of X ∗ , (yn∗ ) converges weak∗ to y ∗ , and ∗ sep{yn } > 1. Therefore, by the weak∗ uniform Kadec–Klee property of X ∗ , there exists δ > 0 such that
y ∗ < 1 − δ.
(∗)
But, by (3) and (7), 1+ε ∗ 1 − 3ε y ∗ (y) yy ∗ < y . 2 2 1 δ Therefore, if ε < min{ 10 , 4}
y ∗ >
1 − 3ε > 1 − 4ε > 1 − δ, 1+ε
a contradiction to (∗). Therefore X has the weak fixed point property for nonexpansive mappings. 2 Of course the theorem implies that c0 with its usual norm has the weak fixed point property which was a result first proven by Maurey [16]. Since H 1 has the weak∗ uniform Kadec–Klee property [2], its predual C(T )/A0 has the weak fixed point property by this theorem. In the same manner, since C1 (H ), the ideal of trace class operators on a Hilbert space H , has the weak∗ uniform Kadec–Klee property [12], its predual C∞ (H ), the ideal of compact operators in B(H ), has the weak fixed point property. Since quotients of Banach spaces with weak∗ sequentially compact dual unit balls have weak∗ sequentially compact dual unit balls [4, p. 227], it is easy to check that the following holds. Corollary 4. Let X be a Banach space such that BX∗ is weak∗ sequentially compact. If X ∗ has the weak∗ uniform Kadec–Klee property and Y is a closed subspace of X, then X/Y has the weak fixed point property for nonexpansive mappings. We note that the corollary implies that the quotients of c0 have the weak fixed point property. This is implicit in the work of Borwein and Sims [3]. The authors had hoped that the sequences identified in Lemma 2 would be useful in establishing connections between superreflexive Banach spaces and the fixed point property for nonexpansive mappings. Consider the sequences generated in Lemma 2 for each ε = k1 , k ∈ N. ∗ ) denote the sequences (y ) and (y ∗ ) constructed in That is, for each k ∈ N, let (yk,n ) and (yk,n n n ∗ Lemma 2 with ε = k1 ; let yk,∞ denote the weak limit of the sequence (yk,n ); and let yk,∞ denote ∗ ∗ the weak limit of the sequence (yk,n ). For a non-trivial ultrafilter U on N, let XU denote the ultrapower of X with respect to U . (For information on ultraproducts in Banach space theory, see [9] or [20].) Define sequences (yn ) in XU , (y∗n ) in (X ∗ )U , and the point y in XU by
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yn = (y1,n , y2,n , y3,n , . . .)U , ∗ ∗ ∗ , y2,n , y3,n , . . . U , and y∗n = y1,n y = (y1,∞ , y2,∞ , y3,∞ , . . .)U . elements The pair of sequences (2(yn − y)) and (y∗n ) forms a biorthogonal system of norm-one ∗ ) and, for each sequence (α ) of nonnegative real numbers, ∞ α y∗ = and (X in X n U n=1 n n ∞ U ∗ ∗ n=1 αn . Moreover, as is clear from the proof of Theorem 3, ym − yn = 2 for all m = n. Initially, the authors felt that, if X was a renorming of 2 , this “positive 1 -type behavior” should not occur in (X ∗ )U since (X ∗ )U would also be a renorming of a Hilbert space. However, as first pointed out to us by Professor V.D. Milman, there do exist renormings of 2 with this behavior. A second example resulted from a discussion with Professors A. Pełczy´nski and M. Wojciechowski. In fact, every infinite-dimensional Banach space can be renormed to exhibit this 1 -type behavior for nonnegative linear combinations. To see this, let (xi , xi∗ ) in X × X ∗ be a biorthogonal system with xi = 1 and xi∗ 2. (Such a biorthogonal system exists by applying a theorem of Ovsepian and Pełczy´nski [4, p. 56] to a separable subspace of X and then extending to functionals on all of X via the Hahn–Banach theorem.) Then |||x||| = max{|x1∗ (x)|, 12 x, supi=j ; i,j 2 (|xi∗ (x)| + |xj∗ (x)|)} defines an equivalent norm on X ∞ with |||x1 + xn ||| = 1 and ||| ∞ n=1 αn (x1 + xn )||| = n=1 αn if αn 0. (For related examples, see Example 3.13 in [18] and Theorem 7 in [11].) Despite the above disappointment, the sequence (y∗n ) in (XU )∗ or the sequences (yn∗ ) in X ∗ for a given ε in Lemma 2 can be used to generalize Theorem 1. A subset A of X is symmetrically ε-separated if the distance between any two distinct points of A ∪ (−A) is at least ε and a Banach space X is O-convex if the unit ball BX contains no symmetrically (2 − ε)-separated subset of cardinality n for some ε > 0 and some n ∈ N [18]. O-convex spaces are superreflexive. Therefore the proof of Theorem 3 combines with property (3) in Lemma 2 to show that, if X fails to have the fixed point property, then, for every ε > 0, there exists a countably infinite set A = {y1∗ , y2∗ , . . .} in the unit sphere of X ∗ such that A ∪ (−A) is (2 − ε)-separated. In particular, this implies: Theorem 5. If X ∗ is O-convex, then the Banach space X has the fixed point property for nonexpansive mappings. Since uniformly nonsquare Banach spaces are O-convex, Theorem 5 is a generalization of Theorem 1. Naidu and Sastry [18] also characterized the dual property to O-convexity. For ε > 0, a convex subset A of BX is an ε-flat if A ∩ (1 − ε)BX = ∅. Note that the convex hulls of the sets {y1∗ , y2∗ , . . .} from Lemma 2 are 3ε-flats. A collection D of ε-flats is jointly complemented if, for each distinct ε-flats A and B in D, the sets A ∩ B and A ∩ (−B) are nonempty. Define E(n, X) = inf{ε: BX contains a jointly complemented collection of ε-flats of cardinality n}. A Banach space X is E-convex if E(n, X) > 0 for some n ∈ N. In [19], S. Saejung noted that E-convex spaces may fail to have normal structure and asked if E convex spaces have the fixed point property. Since a Banach space is E-convex if and only if its dual space is O-convex, Theorem 5 can be restated to give a positive answer to Saejung’s question. Theorem 6. E-convex spaces have the fixed point property for nonexpansive mappings.
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For a detailed analysis of O-convex, E-convex, and related properties in the hierarchy between Hilbert spaces and reflexive spaces, see [1,18,19]. Note added in proof The authors thank Jesús García-Falset for pointing out that X ∗ having the weak∗ uniform Kadec–Klee property implies that R(X) < 2 which, by Theorem 3 in [J. García-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (2) (1997) 532–542], shows that X has the weak fixed point property. Thus Theorem 3 in this article is a special case of Theorem 3 in the article of García-Falset. Acknowledgments The authors thank V.D. Milman, A. Pełczy´nski, N. Randrianantoanina, and M. Wojciechowski for helpful discussions during the preparation of this article. We also thank the referee for several helpful comments. References [1] D. Amir, C. Franchetti, The radius ratio and convexity properties in normed linear spaces, Trans. Amer. Math. Soc. 282 (1984) 275–291. [2] M. Besbes, S.J. Dilworth, P.N. Dowling, C.J. Lennard, New convexity and fixed point properties in Hardy and Lebesgue–Bochner spaces, J. Funct. Anal. 119 (1994) 340–357. [3] Jon M. Borwein, Brailey Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math. 10 (1984) 339–356. [4] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984. [5] T. Domínguez-Benavides, A geometrical coefficient implying the fixed point property and stability results, Houston J. Math. 22 (1996) 835–849. [6] D. van Dulst, B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in: Lecture Notes in Math., vol. 991, 1983, pp. 35–43. [7] J. García-Falset, E. Llorens-Fuster, E.M. Mazcuñan-Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal. 233 (2006) 494–514. [8] Kazimierz Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. [9] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1981) 221–234. [10] R.C. James, Uniformly nonsquare Banach spaces, Ann. of Math. 80 (1964) 542–550. [11] Clifford A. Kottman, Subsets of the unit ball that are separated by more than one, Studia Math. 53 (1975) 15–27. [12] Chris Lennard, C1 is uniformly Kadec–Klee, Proc. Amer. Math. Soc. 109 (1990) 71–77. [13] P.-K. Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985) 69–76. [14] P.-K. Lin, Stability of the fixed point property of Hilbert spaces, Proc. Amer. Math. Soc. 127 (1999) 3573–3581. [15] E. Llorens-Fuster, The fixed point property for renormings of 2 , in: Seminar of Mathematical Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 121–159. [16] B. Maurey, Points fixes des contractions de certains faiblement compacts de L1 , Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, Exp. No. VIII, 1981, 19 pp. [17] E.M. Mazcuñan-Navarro, Stability of the fixed point property in Hilbert spaces, Proc. Amer. Math. Soc. 134 (2005) 129–138. [18] S.V.R. Naidu, K.P.R. Sastry, Convexity conditions in normed linear spaces, J. Reine Angew. Math. 297 (1976) 35–53. [19] Satit Saejung, Convexity conditions and normal structure of Banach spaces, J. Math. Anal. Appl., in press. [20] Brailey Sims, “Ultra”-Techniques in Banach Spaces, Queen’s Papers in Pure and Appl. Math., vol. 60, Queen’s Univ., Kingston, ON, 1982.
Journal of Functional Analysis 255 (2008) 777–818 www.elsevier.com/locate/jfa
Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case Pablo Ramacher 1 Georg-August-Universität Göttingen, Institut für Mathematik, Bunsenstr. 3-5, 37073 Göttingen, Germany Received 4 October 2007; accepted 25 February 2008 Available online 9 April 2008 Communicated by Paul Malliavin
Abstract Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn , and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2 (Rn ) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator res ◦ 2 A0 ◦ ext : C∞ c (X) → L (X) is discrete, and derive asymptotics for the number Nχ (λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ -isotypic component of L2 (X) as λ → ∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L2 (X) is asymptotically proportional to its dimension. © 2008 Elsevier Inc. All rights reserved. Keywords: Pseudodifferential operators; Asymptotic distribution of eigenvalues; Multiplicities of representations of finite groups; Peter–Weyl decomposition
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Weyl calculus for pseudodifferential operators in Rn . . . . . . . . . . . . . . . . Reduced spectral asymptotics and the approximate spectral projection operators Estimates from below for the reduced spectral counting function . . . . . . . . . . .
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E-mail address: [email protected]. 1 The author was supported by the grant RA 1370/1-1 of the German Research Foundation (DFG) during the
preparation of this work. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.02.012
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5. 6. 7.
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Estimates from above for the reduced spectral counting function . . . . . . . . . . . . . . . . . . . . . . 802 The finite group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814
1. Introduction Let G ⊂ O(n) be a compact group of isometries acting on Euclidean space Rn , and let X be a bounded domain in Rn which is transformed into itself under the action of G. Consider the regular representation of G T(g)φ(x) = φ g −1 x in the Hilbert spaces L2 (Rn ) and L2 (X) of square integrable functions by unitary operators. As a consequence of the Peter–Weyl theorem, the representation T decomposes into isotypic components according to Hχ , L2 Rn =
L2 (X) =
ˆ χ∈G
res Hχ ,
ˆ χ∈G
ˆ denotes the set of irreducible characters of G, and res : L2 (Rn ) → L2 (X) is the natural where G 2 n restriction operator. Similarly, ext : C∞ c (X) → L (R ) will denote the natural extension operator. Let A0 be a symmetric, classical pseudodifferential operator in L2 (Rn ) of order 2m with Ginvariant Weyl symbol a and principal symbol a2m , and assume that (A0 u, u) cu2m for some s n c > 0 and all u ∈ C∞ c (X), where · s is a norm in the Sobolev space H (R ). Consider further the Friedrichs extension of the lower semi-bounded operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X),
which is a self-adjoint operator in L2 (X), and denote it by A. Finally, let ∂X be the boundary of X, which is not assumed to be smooth, and assume that for some sufficiently small ρ > 0, vol(∂X)ρ Cρ, where (∂X)ρ = {x ∈ Rn : dist(x, ∂X) < ρ}. Since A commutes with the action of G due to the invariance of a, the eigenspaces of A are unitary G-modules that decompose into irreducible subspaces. In 1972, Arnol’d [1] conjectured that by studying the asymptotic behavior of the spectral counting function Nχ (λ) = dχ
μχ (t),
tλ
where μχ (λ) is the multiplicity of any irreducible representation of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue λ, one should be able to show that the multiplicity of each unitary irreducible representation in the above decomposition of L2 (X) is asymptotically proportional to its dimension as λ → +∞. The aim of this work is to show that this is indeed the case, giving a precise description of the leading term of Nχ (λ), together with an estimate for the remainder.
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The asymptotic distribution of eigenvalues was first studied by Weyl [18] for certain second order differential operators in Rn using variational techniques. Another approach, which also gives an asymptotic description for the eigenfunctions, was introduced by Carleman [4]. His idea was to study the kernel of the resolvent, combined with a Tauberian argument. Minakshisundaram and Pleijel [15] showed that one can study the Laplace transform of the spectral function as well, and extended the results of Weyl to closed manifolds, and Gårding [7] generalized Carleman’s approach to higher order elliptic operators on bounded sets in Rn . Hörmander [11] then extended these results to elliptic differential operators on closed manifolds using the theory of Fourier integral operators. Further developments in this direction were given by Duistermaat and Guillemin, Helffer and Robert, and Ivrii. The first ones to study Weyl asymptotics for elliptic operators on closed Riemannian manifolds in the presence of a compact group of isometries in a systematic way were Donnelly [5] together with Brüning and Heintze [3], giving first-order Weyl asymptotics for the spectral distribution function for each of the isotypic components, together with an estimate for the remainder in some special cases. Later, Guillemin and Uribe [8] described the relation between the spectrum of the considered operators, and the reduction of the corresponding bicharacteristic flow, and Helffer and Robert [9,10] studied the situation in Rn . Our approach is based on the Weyl calculus of pseudodifferential operators developed by Hörmander [12], and the method of approximate spectral projections, first introduced by Tulovsky and Shubin [17]. This method is somehow more closely related to the original work of Weyl, and starts from the observation that the asymptotic distribution function N (λ) for the eigenvalues of an elliptic, self-adjoint operator is given by the trace of the orthogonal projection on the space spanned by the eigenvectors corresponding to eigenvalues λ. By introducing suitable approximations to these spectral projections in terms of pseudodifferential operators, one can then derive asymptotics for N (λ), and also obtain estimates for the remainder term. Nevertheless, due to the presence of the boundary, the original method of Shubin and Tulovsky cannot be applied to our situation, and one is forced to use more elaborate techniques, which were subsequently developed by Feigin [6] and Levendorskii [14]. Recently, Bronstein and Ivrii have obtained even sharp estimates for the remainder term in the case of differential operators on manifolds with boundaries satisfying the conditions specified above [2,13]. This work is structured as follows. Part I provides the foundations of the calculus of approximate spectral projection operators, and addresses the case where G is a finite group of isometries. The case of a compact group of isometries will be the subject of Part II. The main result of Part I is Theorem 8. It states that if G is a finite group, the spectrum of A is discrete, and that, as λ → +∞, the spectral counting function Nχ (λ) is given by
Nχ (λ) = dχ
μχ (t) =
tλ
−1 vol(a2m ((−∞, 1])) n/2m λ + O λ(n−)/2m (2π)n |G|
dχ2
for arbitrary ∈ (0, 12 ), where |G| denotes the cardinality of G, and dχ the dimension of any irreducible representation of G corresponding to the character χ . Consequently, the multiplicity in L2 (X) of any irreducible representation corresponding to the character χ is given asymptotically by
−1 ((−∞,1])) n/2m dχ vol(a2m λ . (2π)n |G|
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2. The Weyl calculus for pseudodifferential operators in Rn We begin by reviewing some basic facts in the Weyl calculus for pseudodifferential operators that will be needed in the sequel. We study then the pullback of symbols, and the composition of pseudodifferential operators with linear transformations. In order to introduce the relevant symbol classes, let g be a slowly varying Riemannian metric in Rl , regarded as a positive definite quadratic form, and assume that m is a positive, g-continuous function on Rl as defined in [12, Definitions 2.1 and 2.2]. Definition 1. The class of symbols S(g, m) is defined as the set of all functions u ∈ C∞ (Rl ) such that, for every integer k 0, k
(k) 1/2 gx (tj ) m(x) < ∞. νk (g, m; u) = sup sup u (x; t1 , . . . , tk ) x∈Rl tj ∈Rl
j =1
Here u(k) stands for the kth differential of u. Note that with the topology defined by the above semi-norms, S(g, m) becomes a Fréchet space. Consider now Rl = Rn ⊕ Rn , regarded as a symplectic space with the symplectic form σ (x, ξ ; y, η) = ξ, y − x, η , where ·,· denotes the usual Euclidean product of two vectors. Thus, σ = dξj ∧ dxj . Assume that g is σ -temperate, and that m is σ, g-temperate (see [12, Definition 4.1]). If a ∈ S(g, m) is interpreted as a Weyl symbol, the corresponding pseudodifferential operator is given by
x +y , ξ u(y) dy dξ, ei(x−y)ξ a Opw (a)u(x) = ¯ 2 where dξ ¯ = (2π)−n dξ . Here and it what follows, it will be understood that each integral is to be performed over Rn , unless otherwise specified. According to [12, Theorem 5.2], Opw (a) defines a continuous linear map from S(Rn ) to S(Rn ), and from S (Rn ) to S (Rn ), and the corresponding class of operators will be denoted by L(g, m). Moreover, one has the following result concerning the L2 -continuity of pseudodifferential operators. Theorem 1. Let g be a σ -temperate metric in Rn ⊕ Rn , g σ the dual metric to g with respect to σ , and g g σ . Let a ∈ S(g, m), and assume that m is σ , g-temperate. Then Opw (a) : L2 (Rn ) → L2 (Rn ) is a continuous operator if, and only if, m is bounded. Proof. See [12, Theorem 5.3].
2
The composition of pseudodifferential operators is described by the main theorem of Weyl calculus. Theorem 2. Let g be a σ -temperate metric in Rn ⊕ Rn , and g g σ . Assume that a1 ∈ S(g, m1 ), a2 ∈ S(g, m2 ), where m1 , m2 are σ, g-temperate functions. Then the composition of Opw (a1 ) with Opw (a2 ) in each of the spaces S(Rn ) or S (Rn ) is a pseudodifferential operator with Weyl symbol σ w (Opw (a1 ) Opw (a2 )) in the class S(g, m1 m2 ). Moreover,
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j 1 w w iσ (Dx , Dξ ; Dy , Dη ) a1 (x, ξ )a2 (y, η)|(y,η)=(x,ξ ) /j ! σ Op (a1 ) Op (a2 ) (x, ξ ) − 2 w
j
∈ S g, hN σ m1 m2
(1)
for every integer N , where Dj = −i∂j , D = (D1 , . . . , Dn ), and h2σ (x, ξ ) = sup y,η
Proof. See [12, Theorems 4.2 and 5.2].
gx,ξ (y, η) σ (y, η) . gx,ξ
2
Note that g can always be written in the form g(y, η) = 2 (yj /λj + ηj2 /μj ), so that
(λj yj2 + μj ηj2 ). Then g σ (x, ξ ) =
hσ (x, ξ ) = max(λj μj )1/2 .
(2)
The following proposition describes the asymptotic expansion of symbols, see [14, Theorem 3.3]. N
Proposition 1. Let aj ∈ S(g, hσ j m) be a sequence of symbols such that 0 = N1 < N2 < · · · → ∞. Then there exists a symbol a ∈ S(g, m) such that: (a) supp a ⊂ j supp aj ; Nl (b) a − l−1 j =1 aj ∈ S(g, hσ m), l > 1. In this case, one writes a ∼
aj .
We will further write S −∞ (g, m) =
∞ S g, hN σm , N =1
and denote the corresponding operator class by L−∞ (g, m). We introduce now certain hypoelliptic symbols which will be needed in the sequel. They were introduced by Levendorskii in [14]. Definition 2. The class of symbols SI(g, m) consists of all a ∈ S(g, m) that can be represented in the form a = a1 + a2 , where cm < |a1 | and a2 ∈ S(g, hσ m) for some constants c, > 0. The corresponding class of operators is denoted by LI(g, m). If instead of cm < |a1 | one has cm < a1 , one writes a ∈ SI + (g, m) and LI + (g, m), respectively. For a proof of the following lemmas, we refer the reader to [14, Lemmas 5.5, 8.1, and 8.2]. Lemma 1. Let a ∈ SI(g, m). Then there exists a symbol b ∈ SI(g, m−1 ) such that Opw (a) Opw (b) − 1 ∈ L−∞ (g, 1),
Opw (b) Opw (a) − 1 ∈ L−∞ (g, 1).
The operator Opw (b) is called a parametrix for Opw (a).
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Lemma 2. If a ∈ SI + (g, m), then there exists a symbol b ∈ S(g, m1/2 ) such that Opw (a) − Opw (b)∗ Opw (b) ∈ L−∞ (g, m), where Opw (b)∗ is the adjoint of Opw (b). Lemma 3. Let > 0, and at ∈ S(g, hσ ), t ∈ R, be a family of symbols depending on a parameter. Furthermore, assume that the corresponding seminorms νk (g, hσ ; at ) are bounded by some constants independent of t, and let c > 0 be arbitrary. Then there exists a subspace L ⊂ L2 (Rn ) of finite codimension such that w Op (at )u
L2
cuL2
for all u ∈ L and all t ∈ R.
Remark 1. Lemma 3 is a consequence of the fact that, for a ∈ S(g, 1), one has the uniform bound w Op (a)
L2
C max νk (g, 1; a), kN
where C > 0 and N ∈ N depend only on the constants characterizing g, but not on a (see the proof of the sufficiency in [12, Theorem 5.3], and [14, Theorem 4.2]). In general, the pullback of symbols under C∞ mappings is described by the following lemma.
Lemma 4. Let g1 , g2 be slowly varying metrics on Rl , respectively Rl , and χ ∈ C∞ (Rl , Rl ). Then χ ∗ S(g2 , 1) ⊂ S(g1 , 1) if, and only if, for every k > 0, k g2χ(x) χ (k) (x; t1 , . . . , tk ) Ck g1x (tj ),
x, t1 , . . . , tk ∈ Rl .
j =1
In particular, if m is g2 -continuous, then χ ∗ m is g1 -continuous and χ ∗ S(g2 , m) ⊂ S(g1 , χ ∗ m). Proof. See [12, Lemma 8.1].
2
In all our applications, we will be dealing mainly with metrics g on R2n of the form δ −ρ 2 gx,ξ (y, η) = 1 + |x|2 + |ξ |2 |y|2 + 1 + |x|2 + |ξ |2 |η| ,
(3)
where 1 ρ > δ 0. The conditions of Theorem 2 are satisfied then, and h2σ (x, ξ ) = (1 + |x|2 + |ξ |2 )δ−ρ by (2). For the rest of this section, assume that g is of the form (3), and put h(x, ξ ) = (1 + |x|2 + |ξ |2 )−1/2 . In this case, the space of symbols S(g, m) can also be characterized as follows.
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Definition 3. Let g be of the form (3), and m a g-continuous function. The class Γρ,δ (m, R2n ), 0 δ < ρ 1, consists of all functions u ∈ C∞ (R2n ) which for all multiindices α, β satisfy the estimates α β ∂ ∂ u(x, ξ ) Cαβ m(x, ξ ) 1 + |x|2 + |ξ |2 (−ρ|α|+δ|β|)/2 . ξ x
l (R2n ) for Γ (h−l , R2n ), where l ∈ R. In particular, we will write Γρ,δ ρ,δ
An easy computation then shows that S(g, m) = Γρ,δ (m, R2n ). For future reference, note β that u ∈ S(g, m) implies ∂ξα ∂x u ∈ S(g, mhρ|α|−δ|β| ). The pullback of symbols for metrics of the form (3) can now be described as follows. Lemma 5. Let δ +ρ 1, and g be a metric of the form (3). Assume that χ(x, ξ ) = (y(x), η(x, ξ )) is a diffeomorphism in R2n such that η is linear in ξ , and the derivatives of y and η are bounded for |ξ | < 1. Furthermore, let 1 gx,ξ (t) gχ(x,ξ ) (t) Cgx,ξ (t), C
1 m(x, ξ ) χ ∗ m(x, ξ ) Cm(x, ξ ), C
where m is a g-continuous function, and C > 0 is a suitable constant. Then χ ∗ S(g, m) ⊂ S(g, χ ∗ m). Proof. Instead of verifying the necessary and sufficient condition in Lemma 4, we will prove the statement directly. Let b ∈ S(g, m), and let s, t, . . . be k vectors in R2n . The kth differential (b ◦ χ)(k) (x, ξ ; s, t, . . .) = t, D s, D . . . (b ◦ χ)(x, ξ ) is given by a sum of terms of the form si tj . . . ∂ α (b ◦ χ)(x, ξ ), where we can assume that all the coefficients si , tj , . . . are different from zero; in particular, (b ◦ χ)(1) (x, ξ ) = b(1) (χ(x, ξ ))χ (1) (x, ξ ), where
χ
(1)
(x, ξ ) =
y (1) (x) A(x, ξ )
0 , B(x)
A being linear in ξ . The derivatives ∂ α (b ◦ χ)(x, ξ ) are sums of expressions of the form β ∂ b χ(x, ξ ) ∂ γ1 χi1 (x, ξ ) . . . ∂ γl χil (x, ξ ), where γ1 + · · · + γl = α and l = |β|. Since additional powers of ξ only appear in companion with additional derivatives of b with respect to η that originate from derivatives of b ◦ χ with respect to x, each of the terms of (b ◦ χ)(k) (x, ξ ; s, t, . . .) can be estimated from above by some constant times an expression of the form si tj . . . ∂ β ∂ β b χ(x, ξ ) P d (x, ξ ), y
η
784
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where P d (x, ξ ) is a homogeneous polynomial in ξ of degree d which is bounded for |ξ | < 1, and d = |β | − N = |β | − k + N ; here N = |α | and N = |α | denote the number of x- and ξ -components in the product si tj , . . . , respectively. Indeed, if we differentiate in ∂ α (b ◦ χ)(x, ξ ) first with respect to ξ we get n
∂ξα (b ◦ χ)(x, ξ ) =
ηj1 ,...,ηj
where ∂ξα = ∂ξi1 . . . ∂ξi that
N
|β |.
|α |
|α |
=1
(∂ηj1 . . . ∂ηj
|α |
∂ηj|α | ∂ηj1 b) χ(x, ξ ) (x) . . . (x), ∂ξi1 ∂ξi|α |
, and differentiating now with respect to x yields the assertion. Note
In order to prove the assertion of the lemma, we have to show that β β
sup sup x,ξ s,t,...
|si tj . . . (∂y ∂η b)(χ(x, ξ ))P d (x, ξ )| 1/2
1/2
gx,ξ (s)gx,ξ (t) . . . m(x, ξ )
< ∞,
(4)
where it suffices to consider only those s, t, . . . whose only non-zero components are si , tj , . . . . Since N d, there are d vectors p, q, . . . among the vectors s, t, . . . contributing with xcomponents to the product si tj pk ql . . . . Furthermore, let w, z, . . . be d vectors such that wn+k = pk , zn+l = ql , . . . , their other components being zero. We then obtain the estimate β β
1/2
|si tj pk ql . . . (∂x ∂ξ b)(χ(x, ξ ))| 1/2
1/2
1/2
1/2
m(χ(x, ξ ))gχ(x,ξ ) (s)gχ(x,ξ ) (t) . . . gχ(x,ξ ) (w)gχ(x,ξ ) (z) . . . d(1−δ−ρ)/2 C 1 + |x|2 + |ξ |2
·
1/2
|P d (ξ )gx,ξ (w)gx,ξ (z) . . . | 1/2
1/2
gx,ξ (p)gx,ξ (q) . . .
for all x, ξ, s, t, . . . . Indeed, −ρ/2 1/2 gx,ξ (w) = |pk | 1 + |x|2 + |ξ |2 ,
δ/2 1/2 gx,ξ (p) = |pk | 1 + |x|2 + |ξ |2 ,
....
On the other hand, besides the d vectors p, q, . . . there are still N − d = k − |β | |β | vectors among the remaining vectors s, t, . . . contributing with x-components to the product si tj . . . . 1/2 Since the corresponding quotients |rl |/gχ(x,ξ ) (r) can be estimated from above by some constant, we can assume that there are precisely |β | of them. Also note that there are exactly d + N = |β | vectors among the vectors s, t, . . . , w, z, . . . contributing with ξ -components to si tj . . . . We can therefore assume that the components of s, t, . . . , w, z, . . . are prescribed by the multiindex β = (β , β ) in such a way that β si tj pk ql . . . ∂xβ ∂ξ b χ(x, ξ ) = b(|β|) χ(x, ξ ); s, t, . . . , w, z, . . . . The desired estimate (4) now follows by using the assumptions that b ∈ S(g, m) and δ + ρ 1. 2
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If a ∈ S(g, m) is regarded as a right, respectively left, symbol, the corresponding pseudodifferential operators are given by Op (a)u(x) = l
ei(x−y)ξ a(x, ξ )u(y) dy dξ, ¯
Op (a)u(x) = r
ei(x−y)ξ a(y, ξ )u(y) dy dξ, ¯
where g is assumed to be of the form (3). By [12, Theorem 4.5], the three sets of operators Opw (a), Opl (a), and Opr (a) coincide. Theorem 2 can then also be formulated in terms of left and right symbols. In what follows, we would like to treat left, right, and Weyl symbols on the same grounding by introducing the notion of the τ -symbol. To do so, we introduce yet another l (R2n ), compare [16, Chapter 4]. class of amplitudes which is closely related to the space Γρ,δ l (R3n ) consists of all functions u ∈ C∞ (R3n ) which for a suitable Definition 4. The class Πρ,δ l ∈ R satisfy the estimates
α β γ ∂ ∂ ∂y u(x, y, ξ ) Cαβγ 1 + |x|2 + |y|2 + |ξ |2 (l−ρ|α|+δ|β+γ |)/2 ξ x (l +ρ|α|+δ|β+γ |)/2 × 1 + |x − y|2 . l (R3n ) and Γ l (R2n ) is described by the following The relationship between the spaces Πρ,δ ρ,δ lemma.
Lemma 6. Let 0 δ < ρ 1, and p : R2n → Rn be a linear map such that (x, y) → l (R2n ), and define (p(x, y), x − y) is an isomorphism. Let a(w, η) ∈ Γρ,δ b(x, y, ξ ) = a p(x, y), ψ(x, y)ξ , where ψ : Ξ → GL(n, R) is a C∞ mapping on some open subset Ξ ⊂ R2n , having bounded l (Ξ × Rn ). derivatives. If δ + ρ 1, then b ∈ Πρ,δ β γ
Proof. We will proof the assertion by induction on |α + β + γ |. First note that ∂ξα ∂x ∂y b(x, y, ξ ) is given by a sum of terms of the form α β ∂η ∂w a p(x, y), ψ(x, y)ξ P d (x, y, ξ ),
(5)
where P d (x, y, ξ ) is a polynomial in ξ of degree d. Each of these summands can be estimated from above by 2 (l−ρ|α |+δ|β |)/2 d 2 P (ξ ), C 1 + p(x, y) + ψ(x, y)ξ where P d (ξ ) is a polynomial in ξ of degree d with constant coefficients, and C > 0 is a constant. We assert that the inequality −ρ|α | + δ|β | + d −ρ|α| + δ|β + γ |
(6)
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holds for all |α + β + γ | = N , and all occurring combinations of α , β , and d. It is not difficult to verify the assertion for N = 1. Let us now assume that (6) holds for |α + β + γ | = N . Differentiating (5) with respect to ξj yields n ∂ηi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ψ(x, y)ij P d (x, y, ξ ) i=1
+ ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂ξj P d (x, y, ξ ),
and we get the inequalities −ρ |α | + 1 + δ|β | + d −ρ(|α| + 1) + δ|β + γ |, −ρ|α | + δ|β | + d − 1 −ρ(|α| + 1) + δ|β + γ |. Similarly, differentiation with respect to, say xj , gives n ∂wi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ (∂xj pi )(x, y)P d (x, y, ξ ) i=1
+
n ∂ηi ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂xj ψ(x, y)ξ i P d (x, y, ξ ) i=1
+ ∂ηα ∂wβ a p(x, y), ψ(x, y)ξ ∂xj P d (x, y, ξ ), and we arrive at the inequalities −ρ|α | + δ |β | + 1 + d −ρ|α| + δ |β + γ | + 1 ,
−ρ |α | + 1 + δ|β | + d + 1 −ρ|α| + δ|β + γ | − ρ + 1 −ρ|α| + δ |β + γ | + 1 , −ρ|α | + δ|β | + d −ρ|α| + δ |β + γ | + 1 , where, in particular, we made use of the assumption δ + ρ 1. This proves (6) for |α + β + γ | = N + 1. Summing up, we get the estimate α β γ ∂ ∂ ∂y b(x, y, ξ ) ξ x l−ρ|α|+δ|β+γ | C1 1 + p(x, y) + |ξ | l−ρ|α|+δ|β+γ | |l|+ρ|α|+δ|β+γ | 1 + |x − y| C2 1 + p(x, y) + |x − y| + |ξ | , where the latter inequality follows by using the easily verified inequality |s| (1 + |p(x, y)| + |ξ |)s C 1 + |x − y| , (1 + |p(x, y)| + |x − y| + |ξ |)s
s ∈ R,
compare the proof of Proposition 23.3 in [16]. Since |x| + |y| and |p(x, y)| + |x − y| define equivalent metrics, the assertion of the lemma follows. 2
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l (R3n ), where 1 ρ > δ 0. Then the oscillatory integral Proposition 2. Let a(x, y, ξ ) ∈ Πρ,δ
Au(x) =
ei(x−y)ξ a(x, y, ξ )u(y) dy dξ ¯
(7)
defines a continuous linear operator from S(Rn ) to S(Rn ), and from S (Rn ) to S (Rn ). 3n ∞ n Proof. Consider first the case a ∈ C∞ c (R ), and assume that u ∈ C (R ) has bounded derivatives. Then the integration in (7) is carried out over a compact set, and partial integration gives Au(x) = ei(x−y)ξ x − y −M Dξ M Dy N ξ −N a(x, y, ξ )u(y) dy dξ, ¯
where M, N are even non-negative integers, and x stands for (1 + x12 + · · · + xn2 )1/2 . Let now l (R3n ), and assume that M, N are such that l − N (1 − δ) < −n, l + l + 2δN − M < −n. a ∈ Πρ,δ The latter integral then becomes absolutely convergent, defining a continuous function of x, and represents the regularization of the oscillatory integral (7). Increasing M and N we will obtain integrals which are convergent also after differentiation with respect to x. In view of the inequality x k y k x − y k , where k > 0, one finally sees that A defines a continuous map from S(Rn ) to S(Rn ), which, by duality, can be extended to a continuous map from S (Rn ) to S (Rn ). 2 We can now introduce the notion of the τ -symbol. In what follows, m will be a g-continuous function. Corollary 1. Let a ∈ S(g, m) = Γρ,δ (m, R2n ), 0 1 − ρ δ < ρ 1, and τ ∈ R. Then Au(x) =
ei(x−y)ξ a (1 − τ )x + τy, ξ u(y) dy dξ ¯
defines a continuous operator in S(Rn ), respectively S (Rn ). In this case, a is called the τ symbol of A, and the operator A is denoted by Opτ (a). Proof. For simplicity, we restrict ourselves to the case m = h−l . By Lemma 6 we then have l (R3n ), and the assertion follows with the previous propob(x, y, ξ ) = a((1 − τ )x + τy, ξ ) ∈ Πρ,δ sition. The case of a general m is proved in a similar way. 2 Our next aim is to prove the following. Theorem 3. Let 0 1 − ρ δ < ρ 1, τ, τ ∈ R be arbitrary, a(x, ξ ) ∈ S(g, m) = Γρ,δ (m, R2n ), and assume that κ : Rn → Rn is an invertible linear map. Furthermore, assume that m is invariant under κ in the sense that m(κ −1 (x), t κ(ξ )) = m(x, ξ ), and set A = Opτ (a). Then A1 u = A(u ◦ κ) ◦ κ −1 ,
u ∈ S Rn ,
defines a pseudodifferential operator with a uniquely defined τ -symbol σ τ (A1 ) ∈ S(g, m).
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Proof. Let us consider first the case m = h−l . Putting κ1 = κ −1 , one sees that A1 is a Fourier integral operator given by ei(κ1 (x)−y)·ξ a (1 − τ )κ1 (x) + τ y, ξ u κ(y) dy dξ A1 u(x) = ¯ = ei(κ1 (x)−κ1 (y))·ξ a (1 − τ )κ1 (x) + τ κ1 (y), ξ det κ1 (y)u(y) dy dξ, ¯ and performing the change of variables ξ → t κ (ξ ), we get A1 u(x) = ei(x−y)·ξ a1 (x, y, ξ ) u(y)dy dξ, ¯ where we put a1 (x, y, ξ ) = a((1 − τ )κ1 (x) + τ κ1 (y),t κ (ξ ))|det κ1 ||det t κ|. Applying Lemma 6 l (R3n ) for arbitrary a ∈ with p(x, y) = (1 − τ )κ1 (x) + τ κ1 (y), one obtains a1 (x, y, ξ ) ∈ Πρ,δ l (R2n ) = S(g, h−l ). Next, let us introduce the coordinates v = (1 − τ )x + τy, w = x − y, and Γρ,δ expand a1 (x, y, ξ ) = a1 (v + τ w, v − (1 − τ )w, ξ ) into a Taylor series at w = 0, compare [16, pp. 180–182]. This yields
a1 (x, y, ξ ) =
|β+γ |N −1
(−1)|γ | |β| γ τ (1 − τ )|γ | (x − y)β+γ ∂xβ ∂y a1 (v, v, ξ ) + rN (x, y, ξ ), β!γ !
where rN (x, y, ξ ) =
1
cβγ (x − y)
β+γ
|β+γ |=N
γ (1 − t)N −1 ∂xβ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ dt,
0 β γ
cβγ being constants. Since the operator with amplitude (x − y)β+γ (∂x ∂y a1 )(v, v, ξ ) coincides β+γ β γ with the one with amplitude (−1)|β+γ | (∂ξ Dx Dy a1 )(v, v, ξ ), we can write A1 also as A1 = BN + RN , where BN is the operator with τ -symbol bN (x, ξ ) =
|β+γ |N −1
1 |β| β+γ γ τ (1 − τ )|γ | ∂ξ (−Dx )β Dy a1 (x, y, ξ )|y=x , β!γ !
and RN has amplitude rN (x, y, ξ ). Similarly, we can assume that RN is given by a sum of terms having amplitudes of the form 1
β+γ β γ ∂ξ ∂x ∂y a1 v + tτ w, v − t (1 − τ )w, ξ (1 − t)N −1 dt,
0
where |β + γ | = N . In view of the estimate β+γ β γ ∂ ∂ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ ξ
x
l−N (ρ−δ) l +N (ρ+δ) C 1 + |v| + |wt| + |ξ | 1 + |tw| ,
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for some l and |β + γ | = N , one can then show that rN (x, y, ξ ) ∈ Πρ,δ (R3n ), where, by assumption, ρ − δ > 0. Define now A1 as the pseudodifferential operator with τ -symbol l−N (ρ−δ)
a1 (x, ξ ) ∼
∞ bN (x, ξ ) − bN −1 (x, ξ ) .
(8)
N =0
Then A1 − A 1 has kernel and τ -symbol belonging to S(R2n ). Since bN (x, ξ ) ∈ S(g, h−l ) for all N , the assertion of the theorem follows in view of the uniqueness of the τ -symbol, and l (R2n ). Let us consider now the case of a general m. By examining the proof of σ τ (A1 ) ∈ Γρ,δ Lemma 6, we see that a1 (x, y, ξ ) must satisfy an estimate of the form α β γ ∂ ∂ ∂y a1 (x, y, ξ ) C1 m p(x, y), t κ(ξ ) 1 + p(x, y) + |ξ | −ρ|α|+δ|β+γ | ξ x
−ρ|α|+δ|β+γ | C2 m p(x, y), t κ(ξ ) 1 + |x| + |y| + |ξ | ρ|α|+δ|β+γ | × 1 + |x − y| .
Consequently, β+γ β γ ∂ ∂ ∂y a1 v + tτ w, v − t (1 − τ )w, ξ ξ
x
−N (ρ−δ) Cm p v + tτ w, v − t (1 − τ )w , t κ(ξ ) 1 + |v| + |wt| + |ξ | l +N (ρ+δ) × 1 + |tw| , where |β + γ | = N , and we can again define A 1 = Opτ (a1 ) by the asymptotic expansion (8), such that A1 − A 1 has kernel and τ -symbol belonging to S(R2n ). The assertion of the theorem now follows by noting that bN (x, ξ ) ∈ S(g, m) = Γρ,δ (m, R2n ) for all N , due to the invariance of m. In particular, one has the asymptotic expansion r
σ τ (A1 )(x, ξ ) − ∈ S g, hN σm
|β+γ |N −1
1 |β| β+γ γ τ (1 − τ )|γ | ∂ξ (−Dx )β Dy a1 (x, y, ξ )|y=x β!γ ! (9)
for arbitrary integers N , where the first summand is given by a1 (x, x, ξ ) = a(κ −1 (x), t κ(ξ )).
2
Theorem 3 allows us, in particular, to express the τ -symbol of an operator in terms of its τ -symbol. More generally, one has the following. Corollary 2. In the setting of Theorem 3 assume that, in addition, a(κ −1 (x), t κ(ξ )) = a(x, ξ ), and det κ = ±1. Then A1 = A, and the τ -symbol of A = Opτ (a) is given by σ τ (A)(x, ξ ) ∼
1 |γ | β+γ β+γ τ |β| (1 − τ )|γ | (τ − 1)|β| τ ∂ξ Dx a(x, ξ ). β!γ ! β,γ
(10)
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Proof. With a1 (x, y, ξ ) defined as in the proof of Theorem 3, we have a1 (x, y, ξ ) = a((1−τ )x + τ y, ξ ), so that A1 = Opτ (a) = A. The corollary then follows with the asymptotic expansion (9). 2 3. Reduced spectral asymptotics and the approximate spectral projection operators In this section, we shall introduce the method of approximate spectral projection operators, and apply it to the problem considered in the introduction. Thus, let G ⊂ O(n) be a compact group of isometries acting on Euclidean space Rn , and X a bounded domain in Rn which is invariant under G. Consider the regular representation T in the Hilbert spaces L2 (Rn ) and L2 (X) of square integrable functions by unitary operators. Let A0 be a symmetric, classical pseudodifferential operator of order 2m with principal symbol a2m as defined in [16], and regard it as an n operator in L2 (Rn ) with domain C∞ c (R ). Furthermore, assume that A0 is G-invariant, i.e. that it commutes with the operators T (g) for all g ∈ G, and that (A0 u, u) cu2m ,
u ∈ C∞ c (X),
(11)
for some c > 0, where (·,·) denotes the scalar product in L2 (Rn ), and · s is a norm in the Sobolev space H s (Rn ). Consider next the decomposition of L2 (Rn ) and L2 (X) into isotypic components, Hχ , L2 (X) = res Hχ , L2 Rn = ˆ χ∈G
ˆ χ∈G
ˆ is the set of all irreducible characters of G, and res denotes the restriction of functions where G 2 defined on Rn to X. Similarly, ext : C∞ c (X) → L (X) will denote the natural extension operator. The Hχ are closed subspaces, and the corresponding projection operators are given by Pχ = dχ χ(k)T (k) dk, G
where dχ is the dimension of any irreducible representation corresponding to the character χ , and dk denotes Haar measure on G. If G is just finite, dk becomes the counting measure, and one simply has Pχ =
dχ χ(k)T (k). |G| k∈G
Since T (k) is unitary, one computes for u, v ∈ L2 (Rn ) (u, Pχ v) = dχ χ(k) u, T (k)v dk = dχ χ k −1 T k −1 u, v dk = (Pχ u, v), G
G
where we made use of χ(g) = χ(g −1 ). Hence Pχ is self-adjoint. Let now A be the Friedrichs extension of the lower semi-bounded operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X).
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A is a self-adjoint operator in L2 (X), and is itself lower semi-bounded. Its spectrum is real, and consists of the point spectrum and the continuous spectrum. Recall that, in general, a symmetric operator S in a separable Hilbert space is called lower semi-bounded, if there exists a real number c such that for all u ∈ D(S),
(Su, u) cu2
where D(S) denotes the domain of S. Now, if V is a subspace contained in D(S), the quantity N (S, V ) = sup dim L: (S u, u) < 0 ∀0 = u ∈ L , L⊂V
can be used to give a qualitative description of the spectrum of S. More precisely, one has the following classical variational result of Glazman. Lemma 7. Let S be a self-adjoint, lower semi-bounded operator in a separable Hilbert space, and define N (λ, S) to be equal to the number of eigenvalues of S, counting multiplicities, less or equal λ, if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. Then N (λ, S) = N S − λ1, D(S) . Proof. See [14, Lemma A.1].
2
In particular, the lemma above allows one to determine whether S has essential spectrum or not, where the latter is given by the continuous spectrum and the eigenvalues of infinite multiplicity. Let us now return to the situation above. Since A commutes with the action of G on L2 (X), the eigenspaces of A are unitary G-modules that decompose into irreducible subspaces. The purpose of this paper is to investigate the spectral counting function Nχ (λ) = dχ
μχ (t),
tλ
where μχ (λ) is the multiplicity of any irreducible representation of dimension dχ corresponding to the character χ in the eigenspace of A with eigenvalue λ. Nχ (λ) is equal to the number of eigenvalues of A, counting multiplicities, less or equal λ and with eigenfunctions in res Hχ , if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. One has then the following. Lemma 8. Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)). Proof. Let Aχ be the Friedrichs extension of res ◦A0 ◦ ext : C∞ c (X) ∩ Hχ → res Hχ . Then Nχ (λ) = N(λ, Aχ ), and the assertion follows with [14, Lemma A.2]. 2 In order to estimate N (A0 − λ1, Hχ ∩ C∞ c (X)), we will apply the method of approximate spectral projection operators. It was first introduced by Tulovsky and Shubin, and later developed
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and generalized by Feigin and Levendorskii, and we will mainly follow [14] in our construction. Thus, let us consider on R2n the metric gx,ξ (y, η) = |y|2 + h(x, ξ )2 |η|2 ,
−1/2 h(x, ξ ) = 1 + |x|2 + |ξ |2
(12)
which is clearly of the form (3). Our symbol classes will be mainly of the form S(h−2δ g, p) = Γ1−δ,δ (p, R2n ) where p is a σ, h−2δ g-temperate function, and 0 δ < 1/2. In this case, 2δ−1 h2σ (x, ξ ) = 1 + |x|2 + |ξ |2 , by Eq. (2), which amounts to hσ = h1−2δ . Also note that u ∈ S(h−2δ g, p) implies ∂ξα ∂x u ∈ l S(h−2δ g, h(1−δ)|α|−δ|β| p). In particular, S(h−2δ g, h−l ) = Γ1−δ,δ (R2n ), where l ∈ R. The symbols and functions used will also depend on the spectral parameter λ. Nevertheless, their membership to specific symbol classes will be uniform in λ, which means that the values of their seminorms in the corresponding symbol classes will be bounded by some constant independent of λ. Now, if a denotes the left symbol of the classical pseudodifferential operator A0 , clearly a ∈ S(g, h−2m , K × Rn ) for any compact set K ⊂ Rn , so that σ l (A0 − λ1) ∈ S(g, q˜λ2 , K × Rn ) uniformly in λ 1, where β
q˜λ2 (x, ξ ) = h−2m (x, ξ ) + λ
(13)
is a σ, g-temperate function. But for u ∈ C∞ c (X), the quadratic form ((A0 − λ1)u, u) entering in l n the definition of N (A0 − λ1, Hχ ∩ C∞ c (X)) depends only on values of σ (A0 − λ1) on X × R . 2 n l By changing the latter symbol outside X × R we can achieve that σ (A0 − λ1) ∈ S(g, q˜λ ) uniformly in λ 1. In view of Corollary 2 we can therefore assume that A0 − λ1 can be represented as a pseudodifferential operator with Weyl symbol a˜ λ = σ w (A0 − λ1) ∈ S(g, q˜λ2 ). In particular, we may take σ w (A0 ) ∈ S(g, h−2m ). But by Eq. (11) and [14, Lemma 13.1] we even have a2m (x, ξ ) c
for all (x, ξ ) ∈ X × S n−1 and some constant c > 0.
Since a − a2m ∈ S(g, h−2m+1 , K × Rn ), we can therefore assume that A0 ∈ LI + (g, h−2m ), obtaining the following. Lemma 9. Let A0 be a classical pseudodifferential operator satisfying (11). Then A0 and A0 −λ1 can be represented as pseudodifferential operators with Weyl symbols σ w (A0 ) ∈ SI + (g, h−2m ) and a˜ λ ∈ SI + (g, q˜λ2 ), respectively. Note that if σ w (A0 ), and consequently also a˜ λ , are G-invariant in the sense that σ w (A0 ) σg (x, ξ ) = σ w (A0 )(x, ξ ),
a˜ λ σg (x, ξ ) = a˜ λ (x, ξ ),
where σg is the symplectic transformation given by σg (x, ξ ) = (κg (x), t κg (x)−1 (ξ )) = (κg (x), κg (ξ )), and κg (x) = gx denotes the action of g, the operators A0 and A0 − λ1 will commute with the action of G by Corollary 2. We can therefore formulate the assumption about the G-invariance of A0 also in terms of its Weyl symbol, and shall henceforth assume that the Weyl symbol and the principal symbol a2m of A0 are invariant under σg for all g ∈ G. In order
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to define the approximate spectral projection operators, we will introduce now the relevant symbols. Having in mind Lemma 5, let aλ ∈ S(g, 1), and d ∈ S(g, d) be G-invariant symbols which, on Xρ × {ξ : |ξ | > 1}, Xρ = {x: dist(x, X) < ρ}, are given by
λ 1 1− , aλ (x, ξ ) = a2m (x, ξ ) 1 + λ|ξ |−2m d(x, ξ ) = |ξ |−1 , where ρ > 0 is some fixed constant, and in addition assume that d is positive and that d(x, ξ ) → 0 as |x| → ∞. We need to define smooth approximations to the Heaviside function, and to certain characteristic functions on X. Thus, let χ˜ be a smooth function on the real line satisfying 0 χ˜ 1, and 1 for s < 0, χ(s) ˜ = 0 for s > 1. Let C0 > 0 and δ ∈ (1/4, 1/2) be constants, and put ω = 1/2 − δ. We then define the G-invariant function (14) χλ = χ˜ ◦ aλ + 4hδ−ω + 8C0 d h−δ , where 0 < δ − ω < 1/2. 0 (R2n ) uniformly in λ. Lemma 10. χλ ∈ S(h−2δ g, 1) = Γ1−δ,δ
Proof. We first note that h−δ ∈ S g, h−δ ,
aλ + 4hδ−ω + 8C0 d ∈ S(g, 1),
since d ∈ S(g, d) ⊂ S(g, 1), and hδ−ω ∈ S(g, hδ−ω ) ⊂ S(g, 1). Now, each of the derivatives of χλ with respect to x and ξ can be estimated by a sum of derivatives of (aλ + 4hδ−ω + 8C0 d)h−δ . β β But because of ∂ξα ∂x (aλ + 4hδ−ω + 8C0 d) ∈ S(g, h|α| ), ∂ξα ∂x h−δ ∈ S(g, h−δ+|α| ), we obtain α β ∂ ∂ χλ (x, ξ ) Cα,β h(1−δ)|α| = Cα,β 1 + |x|2 + |ξ |2 −(1−δ)|α|/2 , ξ x
0 0 where Cα,β is independent of λ. We therefore obtain χλ ∈ Γ1−δ,0 (R2n ) ⊂ Γ1−δ,δ (R2n ) uniformly in λ, and the assertion follows. 2
Next, let U be a subset in R2n , c > 0, and put U (c, g) = (x, ξ ) ∈ R2n : ∃(y, η) ∈ U : g(x,ξ ) (x − y, ξ − η) < c ; according to Levendorskii [14, Corollary 1.2], there exists a smoothened characteristic function ψc ∈ S(g, 1) belonging to the set U and the parameter c, such that supp ψc ⊂ U (2c, g), and ψc |U (c,g) = 1. Let now Mλ = (x, ξ ) ∈ R2n : aλ < 4hδ−ω + 8C0 d .
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Both Mλ and ∂X × Rn are invariant under σk for all k ∈ G, as well as (∂X × Rn )(c, h−2δ g), and Mλ (c, h−2δ g), due to the invariance of a2m (x, ξ ), and the considered metrics and symbols. Now, let η˜ c , ψλ,c ∈ S(h−2δ g, 1) be smoothened characteristic functions corresponding to the parameter c, and the sets ∂X × Rn and Mλ , respectively. According to Lemma 5, we can assume that they are invariant under σk for all k ∈ G; otherwise consider G η˜ c ◦ σk dk, G ψλ,c ◦ σk dk, respectively. We then define the functions ηλ,−c (x, ξ ) = ηc (x, ξ ) =
0, (1 − η˜ c (x, ξ ))ψλ,1/c (x, ξ ), η˜ c (x, ξ ), 1,
x∈ / X, x ∈ X,
x∈ / X, x ∈ X.
(16) (17)
Only the support of ψλ,c depends on λ, but not its growth properties, so that ηc , ηλ,−c ∈ S(h−2δ g, 1) uniformly in λ. Furthermore, since η˜ 2c = 1 on supp η˜ c , and ψλ,1/c = 1 on supp ψλ,1/2c , one has ηλ,−c = 1 on supp ηλ,−2c , which implies ηλ,−2c ηλ,−c = ηλ,−2c . Similarly, one verifies ηc η2c = ηc . We are now ready to define the approximate spectral projection operators. Definition 5. The approximate spectral projection operators of the first kind are defined by E˜λ = Opw (ηλ,−2 ) Opw (χλ ) Opw (ηλ,−2 ), while the approximate spectral projection operators of the second kind are Eλ = E˜λ2 (3 − 2E˜λ ). Remark 2. E˜λ is a smooth approximation to the spectral projection operator Eλ of A using Weyl calculus, while Eλ is an approximation to Eλ2 (3 − 2Eλ ) = Eλ . Note that, since ηλ,−2 and χλ are G-invariant, Corollary 2 implies that the operators Opw (ηλ,−2 ), Opw (χλ ), and consequently also E˜λ and Eλ , commute with the action T (g) of G. The choice of Eλ was originally due to the fact that its trace class norm can be estimated from above by the operator norm of 3 − 2E˜λ , and the Hilbert–Schmidt norm of E˜λ , which are easier to handle. This construction was first used by Feigin [6]. Both E˜λ and Eλ are integral operators with kernels in S(R2n ). Indeed, the asymptotic expansion (1), together with Proposition 1, imply that the Weyl symbols of E˜λ and Eλ can be written in the form a + r, where a has compact support, and r ∈ S −∞ (h−2δ g, 1), because χλ has compact support in ξ , and ηλ,−2 has x-support in X. Thus, σ w (E˜λ ) and σ w (Eλ ) are rapidly decreasing Schwartz functions, and the same holds for the corresponding τ -symbols. By [12, Lemma 7.2], this also implies that E˜λ and Eλ are of trace class and, in particular, compact operators in L2 (Rn ). In addition, by Theorem 2, and the asymptotic expansion (10), one has σ τ (E˜λ ), σ τ (Eλ ) ∈ S(h−2δ g, 1) uniformly in λ. On the other hand, the functions ηλ,−2 and χλ are realvalued, which by general Weyl calculus implies that Opw (ηλ,−2 ), Opw (χλ ), and consequently also E˜λ , and Eλ , are self-adjoint operators in L2 (Rn ). By construction, Eλ commutes with the projection Pχ , so that Pχ Eλ = Eλ Pχ is a self-adjoint operator of trace class as well. Although the decay properties of σ τ (Eλ ) are independent of λ, its support does depend on λ, which will
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result in estimates for the trace of Pχ Eλ in terms of λ that will be used in order to prove Theorem 8. In particular, the estimate for the remainder term in Theorem 8 is determined by the particular choice of the range (1/4, /1/2) for the parameter δ, which guarantees that 1 − δ > δ. By the general theory of compact, self-adjoint operators, zero is the only accumulation point of the point spectra of E˜λ and Eλ , as well as the only point that could possibly belong to the continuous spectrum. The following proposition and its corollary give uniform bounds for the number of eigenvalues away from zero. They are based on certain L2 -estimates for pseudodifferential operators. Proposition 3. The number of eigenvalues of E˜λ lying outside the interval [− 14 , 54 ] is bounded by some constant independent of λ. Proof. Since χλ , ηλ,−c ∈ S(h−2δ g, 1), Theorem 2 yields σ w (E˜λ ) ∈ S(h−2δ g, 1) uniformly in λ. Furthermore, taking into account the asymptotic expansion (1), we have 2 σ w (E˜λ ) = ηλ,−2 χλ + rλ , 2 where rλ ∈ S(h−2δ g, h1−2δ ). Now, since 0 χλ , ηλ,−2 1, for each > 0 there exists a constant 2 2 c > 0 such that + ηλ,−2 χλ c and (1 + ) − ηλ,−2 χλ c. Consequently, the symbols of 1 + E˜λ and (1+)1− E˜λ admit a representation of the form a1 +a2 , where a1 c, a2 ∈ S(h−2δ g, h1−2δ ); thus
1 + E˜λ ∈ LI + h−2δ g, 1 ,
(1 + )1 − E˜λ ∈ LI + h−2δ g, 1
uniformly in λ. According to Lemma 2, this implies that for each λ there exist two operators T1 , T2 such that 1 + E˜λ T1 and (1 + )1 − E˜λ T2 , and Ti ∈ L−∞ (g, 1) uniformly in λ. Therefore, by Lemma 3, there exist two subspaces Li ⊂ L2 (Rn ) of finite codimension such that Ti uL2 uL2 for u ∈ Li and all λ, which implies, via Cauchy–Schwarz, that −u2L2 (Ti u, u) u2L2 on Li . Putting everything together we arrive at the L2 -estimates (E˜λ u, u) (T1 − 1)u, u −2u2L2 , (E˜λ u, u) (1 + )1 − T2 u, u (1 + 2)u2L2 , where u ∈ L1 ∩ L2 , and taking =
1 8
yields the desired result, since codim L1 ∩ L2 < ∞.
2
Corollary 3. The number of eigenvalues of Eλ lying outside the interval [0, 1] is bounded by some constant independent of λ. Proof. If ν˜ i denote the eigenvalues of E˜λ , then the eigenvalues of Eλ are given by νi = ν˜ i2 (3 − 2˜νi ). 2 Let now NχEλ denote the number of eigenvalues of Eλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ). Since zero is the only accumulation point of the point spectrum of Eλ , NχEλ is clearly finite. The next lemma will show that it
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can be estimated by the trace of the operator Pχ Eλ , and its square, so that it is natural to expect that it will provide a good approximation for Nχ (λ) = tr Pχ Eλ = N (A0 − λ1, Hχ ∩ C∞ c (X)). Lemma 11. There exist constants c1 , c2 > 0 independent of λ such that 2 tr(Pχ Eλ )2 − tr Pχ Eλ − c1 NχEλ 3 tr Pχ Eλ − 2 tr(Pχ Eλ )2 + c2 .
(18)
Proof. Since Eλ ∈ L(h−2δ g, 1), Theorem 1 implies that Eλ is L2 -continuous. Moreover, by Remark 1, there is a constant C independent of λ such that Eλ L2 C; hence all eigenvalues of the operators Eλ are bounded by C. Let now νi,χ denote the eigenvalues of Eλ with eigenfunctions in Hχ . Taking into account Corollary 3 and the previous remark, we obtain the estimate NχEλ
νi,χ 1/2
νi,χ +
(1 − νi,χ ) + c1
1/2νi,χ 1
νi,χ + 2
νi,χ 1/2
νi,χ (1 − νi,χ ) + c1 ,
1/2νi,χ 1
where c1 > 0, like all other constants ci > 0 occurring in this proof, can be chosen independent of λ. Consequently, the right-hand side can be estimated from above by 3 tr Pχ Eλ − 2 tr Pχ Eλ · Pχ Eλ + c2 . In the same way one computes NχEλ =
νi,χ +
νi,χ 1/2
i
νi,χ − 2
(1 − νi,χ )
νi,χ 1/2
i
νi,χ (1 − νi,χ ) − c3
0νi,χ 1/2
νi,χ −
νi,χ − c3
0νi,χ 1/2
i
νi,χ − 2
νi,χ (1 − νi,χ ) − c4 ,
i
where the right-hand side can be estimated from below by 2 tr Pχ Eλ · Pχ Eλ − tr Pχ Eλ − c4 . This completes the proof of (18). 2 As the next section will show, NχEλ will provide us with a lower bound for the spectral counting function Nχ (λ). Nevertheless, in order to obtain an upper bound as well, it will be necessary to introduce new approximations to the spectral projection operators. Namely, let χλ+ = χ˜ aλ+ h−δ ,
aλ+ = aλ − 4hδ−ω − 8C0 d,
where χ˜ is defined as in (14). As in Lemma 10, one verifies that χλ+ ∈ S(h−2δ g, 1) uniformly in λ. Definition 6. The approximate spectral projection operators of the third kind are F˜ λ = Opw η22 χλ+ , while the approximate spectral projection operators of the fourth kind are Fλ = F˜ λ2 (3 − 2F˜ λ ).
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Like the projection operators of the first and second kind, F˜ λ and Fλ are self-adjoint operators in L2 (Rn ) with kernels in S(R2n ), and therefore of trace class. Since Fλ commutes with T (k), Pχ Fλ is a self-adjoint operator of trace class, too. Let MχFλ denote the number of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Since Proposition 3 and Corollary 3 hold for F˜ λ and Fλ as well, we obtain Lemma 12. There exist constants c1 , c2 > 0 independent of λ such that 2 tr(Pχ Fλ )2 − tr Pχ Fλ − c1 MχFλ 3 tr Pχ Fλ − 2 tr(Pχ Fλ )2 + c2 . Proof. The proof is a verbatim repetition of the proof of Lemma 11 with Eλ replaced by Fλ .
(19) 2
4. Estimates from below for the reduced spectral counting function In this section, we shall estimate the spectral counting function Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)) from below by adapting techniques developed in [14] to our situation. The main result will be Theorem 4. Let NχEλ be the number of eigenvalues of Eλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Then there exists a constant C > 0 independent of λ such that Eλ N A0 − λ1, Hχ ∩ C∞ c (X) Nχ − C.
(20)
As a first step towards the proof, let q˜λ be defined as in (13), and qλ ∈ SI(g, q˜λ−1 ) be a Ginvariant symbol which, on X × {ξ : |ξ | > 1} is given by −1/2 , qλ (x, ξ ) = a2m (x, ξ ) 1 + |ξ |−2m λ and consider the G-invariant function π = (hδ−ω + C0 d)−1/2 ∈ SI(g, π), together with the operators Π = Opw (π),
Qλ = Opw (qλ ).
Since π q˜λ−1 is bounded, ΠQλ is a continuous operator in L2 (Rn ). The parametrices of Π and Qλ , which exist according to Lemma 1, will be denoted by RΠ and RQλ . Furthermore, an examination of the proof of Lemma 1 shows that if a ∈ SI(g, m) is G-invariant, then the Weyl symbol b of the parametrix of Opw (a) can be assumed to be G-invariant. Consequently, the parametrices RΠ and RQλ commute with the operators T (k). Lemma 13. Let LEχλ = Span{u ∈ S(Rn ) ∩ Hχ : Eλ u = νu, ν 12 } and L˜ Eχλ = Opl (ηλ,−1 ) · Qλ ΠLEχλ . Then dim L˜ Eχλ dim LEχλ − C = NχEλ − C for some constant C > 0 independent of λ.
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Proof. Let us first note that since ηλ,−1 has support in X × Rn , and Opl (ηλ,−1 ) Qλ Π commutes with Pχ , we have L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ . Next, we will prove that RΠ RQλ Opl (ηλ,−1 )Qλ ΠEλ = Eλ + T ,
(22)
where T ∈ L−∞ (g, 1). Indeed, the Weyl symbol of Opl (ηλ,−1 )Qλ ΠEλ is given by a linear combination of products of derivatives of the Weyl symbols of Qλ , Π , Eλ , and Opl (ηλ,−1 ). By the asymptotic expansion (10), 1 −1 |β| β l σ Op (ηλ,−1 ) (x, ξ ) ∼ ∂ξ Dxβ ηλ,−1 (x, ξ ). β! 2 w
β
Now, Eq. (51) implies that, up to terms of order −∞, the support of σ w (Eλ ) is contained in supp ηλ,−2 , and we shall express this by writing supp∞ σ w (Eλ ) ⊂ supp ηλ,−2 . For the same reason, we must have supp∞ σ w (Opl (ηλ,−1 )Qλ ΠEλ ) ⊂ supp ηλ,−2 . But ηλ,−1 = 1 on supp ηλ,−2 implies that all terms in the expansion of σ w (Opl (ηλ,−1 )) vanish on supp ηλ,−2 , except for the zero order terms. Proposition 1 then yields σ w Opl (ηλ,−1 ) (x, ξ ) = ηλ,−1 (x, ξ ) on supp ηλ,−2 , up to a term of order −∞. On this set, the Weyl symbol of Opl (ηλ,−1 )Qλ ΠEλ therefore reduces to ηλ,−1 = 1 times a linear combination of products of derivatives of the Weyl symbols of Qλ , Π and Eλ supported in supp ηλ,−2 , which corresponds to the Weyl symbol of Qλ ΠEλ , plus an additional term of order −∞. Thus, Opl (ηλ,−1 )Qλ ΠEλ = Qλ ΠEλ + T˜ ,
T˜ ∈ L−∞ g, π q˜λ−1 ,
(23)
and (22) follows by taking into account the definition of the parametrix. Now, Eλ : LEχλ → LEχλ is clearly surjective, and 1 Eλ u u, 2
u ∈ LEχλ ,
implies that Eλ is injective on LEχλ as well. Eq. (22) therefore means that on LEχλ RΠ RQλ Opl (ηλ,−1 )Qλ Π = 1LEλ + T Eλ−1 . χ
(24)
According to Lemma 3, there exists a subspace of finite codimension M such that T u u/8 for all u ∈ M and all λ. This gives −1 T E u 2T u 1 u for all u ∈ LEλ ∩ M. χ λ 4
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Let now v, w ∈ LEχλ ∩ M, and assume that Opl (ηλ,−1 )Qλ Πv = Opl (ηλ,−1 )Qλ Πw. By (24) we deduce w + T Eλ−1 w = v + T Eλ−1 v and consequently (1 + T Eλ−1 )(v − w) = 0. But for u ∈ M ∩ LEχλ one computes
1 + T E −1 u u − T E −1 u 1 − 1 u; λ λ 4 hence 1 + T Eλ−1 is injective, and v = w. Thus we have shown that Opl (ηλ,−1 )Qλ Π : LEχλ ∩ M −→ L˜ Eχλ is injective, and the assertion of the lemma follows with C = codim M < ∞.
(25) 2
Since L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ , the next proposition will provide us with a suitable reference subspace in order to prove Theorem 4. Its dimension will be estimated from below with the help of the preceding lemma. Note that the parametrices of Π and Qλ were needed to show the injectivity of (25). Proposition 4. There exists a subspace L ⊂ L˜ Eχλ such that dim L dim LEχλ − C for some constant C > 0 independent of λ, and (A0 − λ1)u, u L2 < 0 for all 0 = u ∈ L. Eλ ∞ Note that, by construction, L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ , while Lχ ⊂ Cc (X). It is this proposition n that accomplishes the transition from R to X, which, according to (23), is achieved by a perturbation of order −∞.
Proof. Let v ∈ LEχλ and w = Opl (ηλ,−1 )Qλ ΠEλ v ∈ L˜ Eχλ . Eq. (23) implies that w = Qλ ΠEλ v + T˜ v,
T˜ ∈ L−∞ g, π q˜λ−1 .
Consequently, one computes ∗ (A0 − λ1)w, w = Π ∗ Q∗λ A0 − λ1 + 4RQ Opw hδ−ω + C0 d RQλ Qλ ΠEλ v, Eλ v λ ∗ − 4 Π ∗ Q∗λ RQ Opw hδ−ω + C0 d RQλ Qλ ΠEλ v, Eλ v + (T v, v) λ =: (D1 Eλ v, Eλ v) − 4(D2 Eλ v, Eλ v) + (T v, v),
(26)
where T is of order −∞. Now, since Qλ RQλ − 1 ∈ L−∞ (g, 1), we have D2 = 1 + K2 ,
K2 ∈ L(g, h);
indeed, by definition, the Weyl symbol of Π is equal to π = (hδ−ω + C0 d)−1/2 ∈ SI(g, π). Now, according to Lemma 9, A0 − λ1 = Opw (a˜ λ ), where a˜ λ ∈ SI + (g, q˜λ2 ). Thus, D1 = Π ∗ Q∗λ Opw (a˜ λ )Qλ + 4 Opw (hδ−ω + C0 d) Π + K1 ,
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where K1 ∈ L−∞ (g, 1). Furthermore, we can assume that qλ ∈ S(g, q˜λ−1 ) is such that aλ = (a2m − λ)qλ2 ∈ S(g, 1), and using Theorem 2 one computes aλ − σ w Q∗λ Opw (a˜ λ )Qλ = aλ − qλ2 a˜ λ + r = qλ2 (a2m − λ − a˜ λ ) + r ∈ S(g, d),
(28)
where r ∈ S(g, h). But this implies aλ − σ w (Q∗λ Opw (a˜ λ )Qλ ) + 4C0 d cd for some sufficiently large C0 and some c > 0; hence aλ − σ w Q∗λ Opw (a˜ λ )Qλ + 4C0 d ∈ SI + (g, d).
(29)
Using Lemma 2, we conclude from (29) that there exists a T4 ∈ L−∞ (g, d) such that Q∗λ Opw (a˜ λ )Qλ Opw (aλ ) + 4C0 Opw (d) + T4 .
(30)
Together with Eλ v2 14 v2 , Eqs. (26)–(30) therefore yield the estimate (A0 − λ1)w, w = (T v, v) − 4(K2 Eλ v, Eλ v) − 4(Eλ v, Eλ v) + (K1 Eλ v, Eλ v) + Π ∗ Q∗λ Opw (a˜ λ )Qλ + 4C0 Opw (d) + 4 Opw hδ−ω ΠEλ v, Eλ v Π ∗ Opw (aλ ) + 8C0 Opw (d) + 4 Opw hδ−ω ΠEλ v, Eλ v − v2 + (K3 v, v), where K3 ∈ S(h−2δ g, h). We therefore set aλ− := aλ + 8C0 d + 4hδ−ω ∈ S(g, 1), and obtain the estimate (A0 − λ1)w, w Π ∗ Opw aλ− ΠEλ v, Eλ v − v2 + (K3 v, v).
(31)
Thus, it remains to show that Eλ∗ Π ∗ Opw (aλ− )ΠEλ − 1 + K3 is negative definite on some subspace of finite codimension. In order to do so, we will show that Eλ∗ Π ∗ Opw (aλ− )ΠEλ − 1 + K3 −1 + K4 , where K4 ∈ L(h−2δ g, hω ) and ω > 0. As it shall become apparent in the following discussion, the key to this is contained in the fact that, although aλ− ∈ S(g, 1), there exists a K5 ∈ L(h−2δ g, hδ ) such that Opw (χλ aλ− χλ ) K5 ! Now, ΠEλ = Π E˜λ Dλ = Π Opw (ηλ,−2 ) Opw (χλ ) Opw (ηλ,−2 )Dλ = Π Opw (ηλ,−2 ), Opw (χλ ) Opw (ηλ,−2 )Dλ + Opw (χλ )Π Opw (ηλ,−2 ) Opw (ηλ,−2 ))Dλ =: W1 + W2 , where we put Dλ = E˜λ (3 − 2E˜λ ). Since Π and E˜λ are self-adjoint, we obtain Eλ Π Opw aλ− ΠEλ = W2∗ Opw aλ− W2 + R,
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where R = W1∗ Opw (aλ− )W2 + W2∗ Opw (aλ− )W1 + W1∗ Opw (aλ− )W1 is given by a sum of terms which contain either [Π Opw (ηλ,−2 ), Opw (χλ )], or its adjoint [Opw (χλ ), Opw (ηλ,−2 )Π], as factors. Now, the crucial remark is that supp∞ σ w Π Opw (ηλ,−2 ), Opw (χλ ) ⊂ suppdiff χλ ⊂ (x, ξ ): aλ− (x, ξ ) hδ (x, ξ ) , (33) (k)
where suppdiff χλ = {(x, ξ ): ∃k > 0: χλ (x, ξ ) = 0}. To see this, first note that by Theorem 2 and Proposition 1, we have the trivial inclusion supp∞ σ w ([Π Opw (ηλ,−2 ), Opw (χλ )]) ⊂ supp χλ . But since the terms in the asymptotic expansion of the Weyl symbol of [Π Opw (ηλ,−2 ), Opw (χλ )] are of order 1, they vanish unless (x, ξ ) ∈ suppdiff χλ , and one obtains the first inclusion. The second inclusion follows by noting the implications (k)
χλ = 0 ∀k > 0
⇐
χλ = 0
or χλ = 1
⇐
aλ− h−δ 1 or aλ− h−δ 0.
While computing the Weyl symbol of R, we can therefore replace aλ− with bλ−
= aλ− θλ ,
1 − −δ θλ = θ , a h 2 λ
(34)
where θ ∈ C∞ c (R) is a real-valued function taking values between 0 and 1, which is equal 1 on [−1, 1], and which vanishes outside [−2, 2], so that θλ = 1 on {(x, ξ ): |aλ− (x, ξ )| hδ (x, ξ )}. Indeed, this replacement adds at most a term of order −∞ to the Weyl symbol of R. Now, the advantage of performing this replacement resides in the fact that, on supp θλ , one has |aλ− | 4hδ , which together with α β − −|α| −δ−(1−δ)|α|+δ|β| 2 ∂ ∂ a (x, ξ ) C 1 + |x|2 + |ξ |2 2 C 1 + |x|2 + |ξ |2 , ξ x λ
|α| 1,
i.e. νk (h−2δ g, hδ ; aλ− ) < ∞, k 1, yields aλ− ∈ S(h−2δ g, hδ , supp θλ ), in contraposition to aλ− ∈ S(g, 1). Consequently, bλ− ∈ S(h−2δ g, hδ ), and we obtain R ∈ L h−2δ g, hδ π 2 ⊂ L h−2δ g, hω ,
(35)
since W1 , W2 ∈ L(h−2δ g, π), Dλ ∈ L(h−2δ g, 1), and hδ π 2 = hδ (hδ−ω + C0 d)−1 = (h−ω + C0 h−δ d)−1 ∼ hω . Eqs. (31), (32), and (35) therefore yield the estimate (A0 − λ1)w, w W2∗ Opw aλ− W2 v, v − v2 + (K4 v, v),
(36)
where K4 = K3 + R ∈ L(h−2δ g, hω ). To examine W2∗ Opw (aλ− )W2 more closely, let us consider the operator S = Opw (χλ ) Opw aλ− Opw (χλ ) − Opw χλ aλ− χλ . By the usual argument, the asymptotic expansion (1) and Proposition 1 yield supp∞ σ w (S) ⊂ suppdiff χλ . In the computation of the Weyl symbol of S we can therefore again replace aλ−
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with bλ− , getting at most an additional term of order −∞. Since Opw (χλ ) ∈ L(h−2δ g, 1) by Lemma 10, we obtain S ∈ L h−2δ g, hδ .
(37)
Now, by construction, aλ− χλ hδ , since 0 χλ 1 and χλ = 0 for aλ− h−δ > 1, so that one infers α β ∂ ∂ χλ a − χλ (x, ξ ) C 1 + |x|2 + |ξ |2 (−δ−(1−δ)|α|+δ|β|)/2 ξ x
λ
for some constant C > 0. But this implies Opw (χλ aλ− χλ ) ∈ L(h−2δ g, hδ ). Using (36) and (37), we therefore get (A0 − λ1)w, w W3∗ Opw χλ aλ− χλ W3 v, v − v2 + (K5 v, v) = −v2 + (K6 v, v), with W3 = Π Opw (ηλ,−2 ) Opw (ηλ,−2 )Dλ ∈ L(h−2δ g, π), K5 = K4 + W3∗ Opw (χλ ) Opw aλ− Opw (χλ ) − Opw χλ aλ− χλ W3 ∈ L h−2δ g, hω , K6 = K5 + W3∗ Opw χλ aλ− χλ W3 ∈ L h−2δ g, hω . Since hσ = h1−2δ , Lemma 3 implies that the operator −1 + K6 is negative definite on a subspace U ⊂ L2 (Rn ) of finite codimension which does not depend on λ. Putting L := Opl (ηλ,−1 )Qλ ΠEλ (U ∩ LEχλ ∩ M) ⊂ L˜ Eχλ with M as in (25), we finally get (A0 − λ1)w, w < 0 ∀0 = w ∈ L,
(38)
where dim U ∩ LEχλ ∩ M − codimM dim M ∩ Eλ (U ∩ LEχλ ∩ M) dim L, since Eλ is bijective on LEχλ , and dim LEχλ dim U ∩ LEχλ ∩ M + codimU ∩ M. The assertion of the proposition now follows. 2 We can now prove Theorem 4. Proof of Theorem 4. Let L ⊂ L˜ Eχλ ⊂ C∞ c (X) ∩ Hχ be as in the previous proposition. Then (38) ∞ holds, and N (A0 − λ1, Hχ ∩ Cc (X)) dim L. Furthermore, dim L dim LEχλ − C = NχEλ − C, and the assertion of the theorem follows. 2 5. Estimates from above for the reduced spectral counting function In what follows, we shall prove an estimate from above for Nχ (λ) = N (A0 − λ1, Hχ ∩ in terms of the number MχFλ of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . In order to do so, we first prove the following. C∞ c (X))
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Proposition 5. There exists a constant C > 0 independent of λ such that n l w + r ∞ + C. N A0 − λ1, Hχ ∩ C∞ c (X) N Op (η1 )Π Op aλ Π Op (η1 ) + 1, Hχ ∩ Cc R Note that this proposition accomplishes the transition from variational quantities related to Rn to quantities related to the bounded subdomain X. Now, the proof of Proposition 5 relies on the following. ∞ Lemma 14. There exists a subspace L ⊂ C∞ c (X) of finite codimension in Cc (X) such that
(A0 − λ1)u, u Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 RΠ RQλ u, RΠ RQλ u for all 0 = u ∈ L, and all λ. Proof of Proposition 5. Let us assume Lemma 14 for a moment, and introduce the notation Aλ [u] = (A0 − λ1)u, u ,
Bλ [u] =
l Op (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 u, u .
According to that lemma, there exists a subspace L in C∞ c (X) of finite codimension such that Aλ [u] Bλ [RΠ RQλ u],
0 = u ∈ L,
for all λ. Let now m be a positive σ, g-tempered function such that 1/m is bounded. Following [14], we introduce the weight spaces of Sobolev type H(g, m) = span T w: w ∈ L2 Rn , T ∈ L(g, 1/m) ⊂ L2 Rn , and endow them with the strongest topology in which each of the operators T : L2 (Rn ) → H(g, m), T ∈ L(g, 1/m), is continuous. It can then be shown that there exists an operator Λm ∈ L(g, m) such that Λm : H(g, m) → L2 (Rn ) is a topological isomorphism. In particular, H(g, m) becomes a Hilbert space with the norm um = Λm uL2 . Furthermore, we have the continuous embedding S(Rn ) ⊂ H(g, m), and if m1 is a bounded, σ, g-tempered function, and A ∈ L(g, mm1 ), then A : H(g, m) → H(g, m−1 1 ) defines a continuous map. Now, by Theorem 3 and the asymptotic expansion (9), RΠ RQλ ∈ LI(g, π −1 q˜λ ), so that by Lemma 1 the opera−1 tor Λπ RΠ RQλ Λ−1 q˜λ ∈ LI(g, 1) has a parametrix Z ∈ LI(g, 1) satisfying ZΛπ RΠ RQλ Λq˜λ = −∞ 1 + K, where K ∈ L (g, 1). Since by Lemma 3 the kernel of 1 + K must be finite-dimensional, Ker Λπ RΠ RQλ Λ−1 q˜λ < ∞; consequently r = dim Ker RΠ RQλ : H(g, q˜λ ) → H (g, π) < ∞.
(39)
Next, let U ⊂ C∞ c (X) ∩ Hχ be a subspace such that Aλ [u] < 0,
∀0 = u ∈ U.
Then, for all 0 = u ∈ V := U ∩ L ∩ H(g,q˜λ ) (Ker RΠ RQλ : H(g, q˜λ ) → H(g, π)), 0 > Bλ [RΠ RQλ u].
(40)
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Because RΠ RQλ is injective on V , (39) yields the inequality dim U dim V + C dim RΠ RQλ V + C for some constant C > 0 independent of λ. Since RΠ RQλ commutes with the operators T (k) of the representation of G, RΠ RQλ V ⊂ Hχ ∩ H(g, π), and we obtain the estimate dim U sup dim W : Bλ [w] < 0 ∀0 = w ∈ W + C. W ∈H(g,π)∩Hχ
n But C∞ c (R ) ∩ Hχ is dense in H(g, π) ∩ Hχ , and the assertion of the proposition follows.
2
Let us now prove Lemma 14. Proof of Lemma 14. Let u ∈ C∞ c (X). Then r ei(x−y)ξ ηc (y, ξ )u(y) dy dξ Op (ηc )u(x) = ¯ = u(x), since ηc is equal to one on X × Rn . Now, for general B ∈ L(g, p), σ r (Opr (ηc )B) is given by an asymptotic expansion j aj , where the first term is equal to ηc σ r (B). Consequently, σ r Opr (ηc )B = ηc σ r (B) + a − ηc σ r (B) + r, with a as in Proposition 1, and r ∈ S −∞ (h−2δ g, p). But a − ηc σ r (B) = 0 on X × Rn , and we obtain Opr (ηc )Bu = Bu + T u, T ∈ L−∞ h−2δ g, p . (41) Using Lemma 9, and setting u˜ = RΠ RQλ u, one computes (A0 − λ1)u, u = Opw (a˜ λ )Qλ Π u, ˜ Qλ Π u˜ + (T1 u, u) ˜ u˜ = Π ∗ Q∗λ Opw (a˜ λ )Qλ − 4 Opw hδ−ω + C0 d Π u, ˜ u˜ + (T1 u, u) + 4 Π ∗ Opw hδ−ω + C0 d Π u, =: Π ∗ D1 Π Opr (η1 )u, ˜ Opr (η1 )u˜ + 4(D2 u, ˜ u) ˜ + (T2 u, u), where we took (41) into account together with RΠ RQλ − RΠ RQλ ∈ L−∞ (g, q˜λ π −1 ), and Ti ∈ L−∞ . The reason for including Opr (η1 ) will become apparent in the proof of the next theorem. Now, by (28), aλ − σ w (Q∗λ Opw (a˜ λ )Qλ ) ∈ S(g, d), which implies that for sufficiently large C0 D1 − Opw aλ+ = Q∗λ Opw (a˜ λ )Qλ + 4C0 Opw (d) − Opw (aλ ) ∈ LI + (g, d), so that according to Lemma 2, there exists a T3 ∈ L−∞ (g, d) such that D1 − Opw (aλ+ ) T3 . On the other hand, since π 2 = (hδ−ω + C0 d)−1 , D2 − 1 ∈ L(g, h), and we obtain (A0 − λ1)u, u Opl (η1 )Π ∗ Opw aλ+ Π Opr (η1 )u, ˜ u˜ + 2u ˜ 2 + (T4 u, u), (42)
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where T4 ∈ L(g, π −2 q˜λ2 h); hereby we used the fact that Opl (η1 ) is the adjoint of Opr (η1 ), compare [16, p. 26]. Furthermore, since by (9) the Weyl symbol of RΠ RQλ is equal to π −1 q˜λ modulo terms of lower order, (RΠ RQλ )∗ RΠ RQλ + T4 ∈ LI + g, π −2 q˜λ2 . Lemmata 1–3 now allow us to deduce the existence of a subspace L ⊂ C∞ c (X) of finite codimension in L2 (X) such that u ˜ 2 + (T4 u, u) =
(RΠ RQλ )∗ RΠ RQλ + T4 u, u > 0
(43)
for all 0 = u ∈ L, and all λ. Indeed, according to Lemma 2, Λπ 2 [(RΠ RQλ )∗ RΠ RQλ + T4 ]Λ−1 2 ∈ q˜λ
LI + (g, 1) can be written in the form B ∗ B + T5 , where B ∈ LI(g, 1) and T ∈ L−∞ (g, 1). By a reasoning similar to the one that led to (39), one can infer from Lemma 1 that the kernel of B must be finite-dimensional, and together with Lemma 3 conclude that there exists a subspace L˜ ⊂ L2 (Rn ) of finite codimension such that BuL2 cuL2 ,
T5 uL2 <
c2 uL2 , 2
for all u ∈ L˜ and some constant c > 0. Thus, we obtain (43), and together with (42) we get (A0 − λ1)u, u Opl (η1 )Π ∗ Opw aλ+ Π Opr (η1 ) + 1 u, ˜ u˜ for all 0 = u ∈ L. This concludes the proof of the lemma.
2
We are now in position to prove an estimate from above for N (A0 − λ1, Hχ ∩ C∞ c (X)). Theorem 5. Let MχFλ be the number of eigenvalues of Fλ which are 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ . Then there exists a constant C > 0 independent of λ such that Fλ N A0 − λ1, Hχ ∩ C∞ c (X) Mχ + C. Proof. We shall continue with the notation introduced in the proof of Proposition 5. According to that proposition, it suffices to prove a similar estimate for n N Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1, Hχ ∩ C∞ c R from above. For this sake, we will show that there exists a subspace L ⊂ UχFλ = Span u ∈ S(Rn ) ∩ Hχ : Fλ u = νu, ν < 1/2 , whose codimension in UχFλ is finite and uniformly bounded in λ, such that Bλ [u] 0 for all u ∈ L.
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Indeed, let us assume this statement for a moment. Since Fλ is a compact self-adjoint operan tor in L2 (Rn ), there exists an orthonormal basis of eigenfunctions {uj }∞ j =1 in S(R ). But Fλ commutes with the action T (g) of G, so that each of the eigenspaces of Fλ is an invariant subspace, and must therefore decompose into a sum of irreducible G-modules. Consequently, Hχ has an orthonormal basis of eigenfunctions lying in S(Rn ) ∩ Hχ . Hence, Hχ = UχFλ ⊕ VχFλ , where VχFλ = Span{u ∈ S(Rn ) ∩ Hχ : Fλ u = νu, ν 1/2}. Now, if W ⊂ S(Rn ) ∩ Hχ is a subspace with Bλ [u] < 0 for all 0 = u ∈ W, then L ∩ W = {0}, and therefore W ⊂ VχFλ ⊕ U , where U is a finite-dimensional subspace of UχFλ whose dimension is bounded by some constant C > 0 independent of λ. Consequently, dim W dim VχFλ + C. But this implies n N Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 , Hχ ∩ C∞ c R sup dim W : Opl (η1 )Π Opw aλ+ Π Opr (η1 ) + 1 u, u < 0 ∀0 = u ∈ W W ⊂S (R n )∩Hχ
dim VχFλ + C = MχFλ + C, and the assertion of the theorem follows with the previous proposition. Let us now show the existence of the subspace L. Take v ∈ UχFλ ⊂ L2 (Rn ), and put v˜ = (1 − Fλ )v. We then expect ˜ 0. Now, one computes that Bλ [v] 2 1 − Fλ Opl (η1 )Π Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (1 − Fλ )v + (K1 v, v) 2 (Dv, v) + v − Fλ v + (K1 v, v)
˜ = Bλ [v]
1 (Dv, v) + v2 + (K1 v, v), 4
(44)
where we put Fλ = Opw (χλ+ )2 (3 − 2 Opw (χλ+ )), D = 1 − Fλ Opl (η1 )Π Opw aλ+ Π Opr (η1 ) 1 − Fλ , and K1 ∈ L−∞ . Indeed, one has Fλ v 12 v, and Opr (η1 )Fλ −Opr (η1 )Fλ ∈ L−∞ (h−2δ g, 1), since the terms in the asymptotic expansions of the Weyl symbols of Opr (η1 )Fλ and Opr (η1 )Fλ coincide because of η2 = 1 on supp η1 . Next we note that, similarly to (33), (45) supp∞ σ w Fλ , Opl (η1 )Π ⊂ suppdiff χλ+ ⊂ (x, ξ ): aλ+ (x, ξ ) hδ (x, ξ ) , and we set bλ+ = aλ+ θλ ,
θλ = θ
1 + −δ aλ h , 2
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with θ as in (34). An argument similar to that concerning bλ− shows that bλ+ ∈ S(h−2δ g, hδ ). Now, because of bλ+ = aλ+ on supp∞ σ w ([Fλ , Opl (η1 )Π]), we have (Dv, v) =
(1 − Fλ ), Opl (η1 )Π Opw bλ+ Π Opr (η1 ) 1 − Fλ v, v + Opl (η1 )Π 1 − Fλ Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (K2 v, v),
where K2 is of order −∞. Since [(1 − Fλ ), Opl (η1 )Π] Opw (bλ+ )Π Opr (η1 )(1 − Fλ ) ∈ L(h−2δ g, hδ π 2 ) ⊂ L(h−2δ g, hω ), we therefore obtain (Dv, v) = Opl (η1 )Π 1 − Fλ Opw aλ+ Π Opr (η1 ) 1 − Fλ v, v + (K3 v, v), where K3 ∈ L(h−2δ g, hω ). Using a similar argument to commute Π Opr (η1 ) with 1 − Fλ , we finally get (46) (Dv, v) = Opl (η1 )Π 1 − Fλ Opw aλ+ (1 − Fλ )Π Opr (η1 )v, v + (K3 v, v), where K3 ∈ L(h−2δ g, hω ). Now, the asymptotic expansion of the Weyl symbol of the operator (1 − Fλ ) Opw (aλ+ )(1 − Fλ ) gives 2 2 σ w 1 − Fλ Opw aλ+ 1 − Fλ = 1 − χλ+ 3 − 2χλ+ aλ+ + r
(47)
with supp∞ r ⊂ suppdiff χλ+ . While computing r, we can therefore replace aλ+ by bλ+ , so that r ∈ S(h−2δ g, hδ ). As a consequence, (46) and (47) yield 2 (Dv, v) = Opl (η1 )Π Opw 1 − (χλ+ )2 3 − 2χλ+ aλ+ Π Opr (η1 )v, v + (K4 v, v), where K4 ∈ L(h−2δ g, hω ). Hereby we used again the fact that π 2 hδ ∼ hω . Next, one verifies that [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ + C1 hδ ∈ SI + (h−2δ g, [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ + C1 hδ ) for some C1 > 0, since χλ+ = 1 for aλ+ < 0, so that [1 − (χλ+ )2 (3 − 2χλ+ )]2 aλ+ 0. According to Lemma 2, we therefore have 2 2 Opw 1 − χλ+ 3 − 2χλ+ aλ+ K5 ∈ L h−2δ g, hδ , and we arrive at the estimate K6 ∈ L h−2δ g, hω .
(Dv, v) (K6 v, v), Together with (44) we finally obtain the estimate
1 ˜ (v, v) + (K7 v, v), Bλ [v] 4
K7 ∈ L h−2δ g, hω .
Using the already familiar argument of Lemma 3, one infers the existence of a subspace M ⊂ L2 (Rn ) of finite codimension on which 1/4 + K7 is positive definite. Putting L := (1 − Fλ )(UχFλ ∩ M) ⊂ UχFλ we therefore get Bλ [w] 0,
for all w ∈ L.
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Furthermore, since 1 − Fλ is injective on UχFλ , codimU Fλ L = codimU Fλ (M ∩ UχFλ ) codimM, χ
as desired. This completes the proof of the theorem.
χ
2
Remark 3. The leading idea in the proof of the last theorem was that each v ∈ UχFλ has to be, approximately, an eigenvector of the corresponding spectral projection operator of A with eigenvalue zero. For this reason, such a v cannot satisfy (Av, v) < λv2 , nor be an element of W . 6. The finite group case For the rest of Part I, we shall concentrate on the case where G is a finite group, while the compact group case will be treated in Part II. The two preceding sections showed that, in view of Lemmata 11 and 12, the spectral counting function Nχ (λ) = N (A0 − λ1, Hχ ∩ C∞ c (X)) can be estimated from below and from above in terms of the traces of Pχ Eλ and Pχ Fλ , and their squares. We will therefore now proceed to estimate these traces in terms of the reduced Weyl volume. For this sake, we introduce first certain sets associated to the support of the symbols of the approximate spectral projection operators; their significance will become apparent later. Thus, let Wλ = (x, ξ ) ∈ X × Rn : aλ < 0 , Ac,λ = (x, ξ ) ∈ X × Rn : aλ < c hδ−ω + d , Bc,λ = X × Rn − Ac,λ , Dc = ∂X × Rn c, h−2δ g , Fλ = (x, ξ ) ∈ X × Rn : χλ = 0 or ηλ,−2 = 0 or χλ = ηλ,−2 = 1 , RV c,λ = (x, ξ ) ∈ X × Rn : |aλ | < c hδ−ω + d ∪ (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d . Note that Dc = {(x, ξ ) ∈ R2n : dist(x, ∂X) < −2δ
h
√
c(1 + |x|2 + |ξ |2 )−δ/2 }, since for
δ (x, ξ )g(x,ξ ) (x − y, ξ − η) = 1 + |x|2 + |ξ |2
|ξ − η|2 2 + |x − y| < c 1 + |x|2 + |ξ |2
to hold for some (y, η) ∈ ∂X × Rn , it is necessary and sufficient that |x − y|2 (1 + |x|2 + |ξ |2 )δ < c is satisfied for some y ∈ ∂X. Now, recall that |G| = χ∈Gˆ dχ2 . We then have the following. Proposition 6. For sufficiently large c > 0 we have tr Pχ Eλ − Vχ X × Rn , aλ c vol RV c,λ ,
(48)
where dχ2 Vχ X × Rn , aλ = |G|
X×Rn
dχ2 vol Wλ 1(−∞,0] aλ (x, ξ ) dx dξ ¯ = (2π)n |G|
(49)
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is the expected approximation given in terms of the reduced Weyl volume, and 1(−∞,0] denotes the characteristic function of the interval (−∞, 0]. Furthermore, a similar estimate holds for tr Pχ Eλ · Pχ Eλ , too. Proof. The proof will require several steps. Let σ r (Eλ )(x, ξ ) denote the right symbol of Eλ . n Then, for u ∈ C∞ c (R ), dχ −1 χ(h) ei(h x−y)ξ σ r (Eλ )(y, ξ )u(y) dy dξ. Pχ Eλ u(x) = ¯ |G| h∈G
The kernel of Pχ Eλ , which is a rapidly decreasing function, is given by dχ −1 χ(h) ei(h x−y)ξ σ r (Eλ )(y, ξ ) dξ. KPχ Eλ (x, y) = ¯ |G| h∈G
The trace of Pχ Eλ can therefore be computed by tr Pχ Eλ =
KPχ Eλ (x, x) dx
dχ = χ(h) tr Eλ + |G| |G| dχ2
−1 x−x)ξ
ei(h
σ r (Eλ )(x, ξ ) dx dξ, ¯
h=e
where we made use of the relation χ(e) = dχ , and the fact that tr Eλ = σ r (Eλ )(x, ξ ) dx dξ ¯ . As a next step, we will prove that, for all e = h ∈ G, there exists a sufficiently large constant c > 0 such that i(h−1 x−x)ξ r e σ (Eλ )(x, ξ ) dx dξ (50) ¯ c vol(RV c,λ ). As already noticed, the decay properties of σ τ (Eλ )(x, ξ ) ∈ S(h−2δ g, 1) are independent of λ for arbitrary τ ∈ R, while its support does depend on λ. Indeed, by Theorem 2 and Corollary 2, together with the asymptotic expansions (1) and (10) and Proposition 1, 2 2 2 χλ 3 − 2ηλ,−2 χλ + fλ + rλ , σ τ (Eλ ) = ηλ,−2
(51)
where rλ ∈ S −∞ (h−2δ g, 1), and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ; in addition, fλ (x, ξ ) = 0 if (x, ξ ) ∈ Fλ . To see this, note that σ τ (Eλ )(x, ξ ) is given asymptotically as a linear combination of products of derivatives of σ w (Eλ ) at (x, ξ ), which in turn is given asymptotically by a linear combination of terms involving derivatives of ηλ,−2 , χλ . The τ -symbol of Eλ is therefore asymptotically given by aj ∈ S h−2δ g, h(1−2δ)N , aj ∈ S h−2δ g, h(1−2δ)j , σ τ (Eλ ) − 0j
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2 2 fλ = a − (ηλ,−2 χλ )2 (3 − 2ηλ,−2 χλ ) ∈ S(h−2δ g, h1−2δ ) must vanish on Fλ , and we obtain (51). Now, since |rλ (x, ξ )| C (1 + |x|2 + |ξ |2 )−N/2 for some constant C independent of λ and N arbitrarily large, we get the uniform bound rλ (x, ξ ) dx dξ ¯ C; j 1 supp aj ,
note that the x-dependence of h(x, ξ ) is crucial at this point. For this reason, and in order to show (50), where now τ = 1, we can restrict ourselves to the study of 2 2 −1 2 (52) ei(h x−x)ξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) dx dξ, ¯ X×Rn
where we took into account that ηλ,−2 has compact x-support in X. Next, we examine the geometry of the action of G in more detail. Thus, let Σ = {x ∈ Rn : gx = x for some e = g ∈ G} denote the set of not necessarily simultaneous fixed points of G. In other words, Σ=
Σg ,
Σg = x ∈ Rn : gx = x .
e=g∈G
Note that every connected component of Σg is a closed, totally geodesic submanifold. We then have the following. Lemma 15. There exists a constant κ > 0 such that d(gx, x) κd(x, Σg ) for all x ∈ Rn , and arbitrary e = g ∈ G. Proof. Let x ∈ Rn − Σg be an arbitrary point, and p the closest point to x belonging to Σg . Write x = expp t0 X, where expp denotes the exponential mapping of Rn , and (p, X) ∈ Tp Rn ,
|X| = 1.
Then t0 = (x, Σg ). Consider next the direct sum decomposition Tp (Rn ) = U ⊕ V , where U = (p, Y ) ∈ Tp Rn : dgp (Y ) = Y , and V = U ⊥ . Since p is a fixed point of g, we also have the identity g expp Y = expp dgp (Y ), which implies expp tY ∈ Σg
if, and only if,
(p, Y ) ∈ U,
where t ∈ R.
Consequently, U = Tp (Σg ). Now, with expp tY = p + tY , and x, p as above, one computes
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2 2 |gx − x|2 = expp t0 dgp (X) − expp t0 X = p + t0 dgp (X) − p − t0 X 2 = d 2 (x, Σg )dgp (X) − X . Because of (x − p) ⊥ Σg , we must have (p, X) ∈ Tp (Σg )⊥ = V , and therefore dgp (X) − X 2 = 0. The latter expression depends continuously on (p, X) ∈ {(p, Y ) ∈ Tp (Σg )⊥ : |Y | = 1}, and is actually independent of p, so that it can be estimated from below by some positive constant uniformly for all x. The assertion of the proposition now follows. 2 Returning now to our previous computations, we split the integral in (52) in an integral over −δ/2 , D = (x, ξ ) ∈ X × Rn : dist(x, Σ) 1 + |ξ |2 and a second integral over the complement of D in X × Rn . Since supp χλ ⊂ {(x, ξ ): aλ + 4hδ−ω + 8C0 hδ }, the integral over Ω×X D can be estimated by a constant times the volume of the set {(x, ξ ) ∈ Ω×X D: aλ + 4hδ−ω + 8C0 d hδ }, which is contained in the set {(x, ξ ) ∈ X × Rn : dist(x, Σ) < (1 + |ξ |2 )−δ/2 , aλ c(hδ−ω + d)} for some sufficiently large c > 0. By examining the proof of Lemma 18, one sees that the volume of the latter can be estimated from above by vol(Σc2 |ξ |−δ ∩ X) dξ + c3 K|ξ |c1 λ1/2m
for some suitable constants K, ci > 0, and consequently has the same asymptotic behaviour in λ as the volume of RV c,λ . In studying the asymptotic behaviour of the integral (52), we can therefore restrict the domain of integration to D. By the previous lemma, there exists a constant κ > 0 such that −1 h x − x κ 1 + |ξ |2 −δ/2 for all (x, ξ ) ∈ D and e = h ∈ G. 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) + fλ has compact support in ξ , this implies that Since (ηλ,−2 −1
2 ei(h x−x)ξ α 2 2 ∂ξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) −1 2 |h x − x| is integrable on D, as well as rapidly decreasing in ξ . Integrating by parts with respect to ξ we therefore get for (52) the expression D
−1
2 2 ei(h x−x)ξ 2 2 −∂ξ1 − · · · − ∂ξ2n ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ (x, ξ ) dx dξ ¯ ; (53) −1 2 |h x − x|
in particular notice that, by Fubini’s theorem, the boundary contributions vanish. Now, if (x, ξ ) ∈ 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) + fλ is constant, so its derivatives with respect to ξ Fλ , the function (ηλ,−2
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are zeros, and we can restrict the integration in (53) to the set X×Rn Fλ ∩ D, where X×Rn Fλ denotes the complement of Fλ in X × Rn . Lemma 16. For sufficiently large c > 0, the set X×Rn Fλ is contained in RV c,λ . Proof. This assertion is already stated in [14, p. 55]. For the sake of completeness, we give a proof here. Thus, consider / D4 , aλ < −4hδ−ω − 8C0 d , Eλ = (x, ξ ) ∈ X × Rn : (x, ξ ) ∈ and let Mλ be defined as in (15). Since supp η˜ 2 ⊂ D4 , and ψλ,1/2 = 1 on Mλ (1/2, h−2δ g), it is clear that (54) Eλ ⊂ (x, ξ ) ∈ X × Rn : χλ = ηλ,−2 = 1 ⊂ Fλ , and consequently X×Rn Fλ ⊂ X×Rn Eλ . Next, we are going to prove that, for sufficiently large c, / Mλ (1, h−2δ g). Thus, assume (x, ξ ) ∈ Bc,λ ; on X × {ξ : |ξ | > 1} (x, ξ ) ∈ Bc,λ implies (x, ξ ) ∈ we have
|ξ |2m 1 λ 1 2m + . 1− c |ξ | (1 + |x|2 + |ξ |2 )(δ−ω)/2 a2m (x, ξ ) |ξ | + λ Therefore, as c becomes large, |ξ | must become large, too. On the other hand, if (y, η) ∈ Mλ , |η| must be bounded. For large c we therefore have |ξ − η|2 ∼ |ξ |2 , which means that h−2δ (x, ξ )g(x,ξ ) (x − y, ξ − η) ∼ (1 + |x|2 + |ξ |2 )δ → ∞ as c → ∞. Hence, for sufficiently large c, (x, ξ ) ∈ / Mλ (1, h−2δ g). Since supp ψλ,1/2 ⊂ Mλ (1, h−2δ g), we arrive in this case at the inclusions (55) Bc,λ ⊂ (x, ξ ) ∈ X × Rn : ηλ,−2 (x, ξ ) = 0 ⊂ Fλ , and combining (54) and (55) we get X×Rn Fλ ⊂ Ac,λ ∩ X×Rn Eλ ⊂ RV c,λ , as desired.
(56)
2
As a consequence of the foregoing lemma, the integral in (53) is bounded from above by the volume of RV c,λ , times a constant independent of λ, since the integrand is uniformly bounded with respect to λ. Thus, we have shown (50). The assertion of the proposition now follows by observing that tr Eλ − vol Wλ c vol RV c,λ . (57) (2π)n −1 Indeed, similarly to our previous discussion of the integral ei(h x−x)ξ σ r (Eλ )(x, ξ ) dx dξ ¯ , the integral 2 2 2 σ r (Eλ )(x, ξ ) dx dξ ηλ,−2 χλ 3 − 2ηλ,−2 χλ + fλ + rλ (x, ξ ) dx dξ tr Eλ = ¯ ¯ =
P. Ramacher / Journal of Functional Analysis 255 (2008) 777–818
813
can be split into three parts; the contribution coming from rλ (x, ξ ) is bounded by some constant independent of λ, while the contribution coming from fλ can be estimated in terms of the volume of RV c,λ , since supp fλ ⊂ X×Rn Fλ ⊂ RV c,λ , by the previous lemma. Now, 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) must be equal 1 on Wλ ∩ X×Rn RV c,λ , since according to (56) we (ηλ,−2 have X×Rn RV c,λ ⊂ Bc,λ ∪ Eλ , and hence Wλ ∩ X×Rn RV c,λ ⊂ Eλ ⊂ {(x, ξ ) ∈ X × Rn : χλ = 2 2 χλ )2 (3 − 2ηλ,−2 χλ ) vanishes ηλ,−2 = 1}, due to the fact that Wλ ∩ Bc,λ = ∅. Furthermore, (ηλ,−2 −2δ / Mλ (1, h g), by the proof of the preon Bc,λ , since for large c, (x, ξ ) ∈ Bc,λ implies (x, ξ ) ∈ vious lemma. Taking into account that Wλ and RV c,λ are subsets of Ac,λ , we therefore obtain for sufficiently large c 2 2 2 χλ (x, ξ ) dx dξ ηλ,−2 χλ 3 − 2ηλ,−2 ¯ vol(Wλ ∩ Ac,λ RV c,λ ) = + (2π)n
2 2 2 ηλ,−2 χλ 3 − 2ηλ,−2 χλ (x, ξ ) dx dξ. ¯
Ac,λ −(Wλ ∩Ac,λ RV c,λ )
Now, since Ac,λ RV c,λ ⊂ Wλ , one has Ac,λ − Wλ ∩ Ac,λ RV c,λ = RV c,λ . The estimate (57) now follows, and together with (50) we obtain (48). Finally, if in the previous computations Eλ is replaced by Eλ2 , we obtain a similar estimate for the trace of Pχ Eλ · Pχ Eλ = Pχ Eλ2 . This concludes the proof of Proposition 6. 2 As a consequence, we get the following. Theorem 6. Let NχEλ be the number of eigenvalues of Eλ which are 1/2 and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ). Then E N λ − Vχ X × Rn , aλ c vol RV c,λ χ
(58)
for some sufficiently large c > 0. Proof. From the preceding proposition, and the estimate (18), one deduces that for some sufficiently large c > 0 NχEλ 3 tr Pχ Eλ − 2 tr Pχ Eλ · Pχ Eλ + c2 Vχ X × Rn , aλ + c vol RV c,λ , NχEλ 2 tr Pχ Eλ · Pχ Eλ − tr Pχ Eλ − c1 Vχ X × Rn , aλ − c vol RV c,λ , which completes the proof of (58).
2
In analogy to the previous considerations, one proves the following. Theorem 7. For sufficiently large c > 0 one has the estimate F M λ − Vχ X × Rn , aλ c vol RV c,λ , χ where MχFλ is the number of eigenvalues of Fλ , counting multiplicities, greater or equal 1/2, and whose eigenfunctions are contained in the χ -isotypic component Hχ of L2 (Rn ).
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Proof. The proof is similar to the one of Theorem 6, and uses Lemma 12. In particular, as in Eq. (51), one has 2 σ τ (Fλ ) = η22 χλ+ 3 − 2η22 χλ+ + fλ + rλ ,
(59)
where rλ ∈ S −∞ (h−2δ g, 1), and fλ ∈ S(h−2δ g, h1−2δ ), everything uniformly in λ. The asymptotic analysis for tr Pχ Fλ and tr(Pχ Fλ )2 now follows the lines of the proof of Proposition 6. 2 7. Proof of the main result We can now give a description of the asymptotic behavior of the spectral counting function Nχ (λ) as λ → +∞ in the finite group case. Collecting all the results obtained so far one deduces Proposition 7. There exist constants C1 , C2 > 0 which do not depend on λ, such that for all λ N A0 − λ1, Hχ ∩ C∞ (X) − Vχ X × Rn , aλ C1 vol RV C ,λ + C2 . c 1 Proof. By Theorems 4 and 5, there exist constants Ci > 0 independent of λ such that NχEλ −C1 Fλ N (A0 − λ1, Hχ ∩ C∞ c (X)) Mχ + C2 . Theorems 6 and 7 then yield the estimate n −c vol RV c,λ − C1 N A0 − λ1, Hχ ∩ C∞ c (X) − Vχ X × R , aλ c vol RV c,λ + C2 for some sufficiently large c > 0.
2
In order to formulate the main result, we need two last lemmata. Lemma 17. For λ > 0 one has −1 dχ2 vol(a2m ((−∞, 1])) n/2m λ + C, Vχ X × Rn , aλ = n (2π) |G|
where C > 0 is some constant independent of λ. Proof. Let us define bλ (x, ξ ) = aλ (x, λ1/2m ξ ). Then for x ∈ X and |ξ | > λ−1/2m one has
1 1 bλ (x, ξ ) = 1 − . a2m (x, ξ ) 1 + |ξ |−2m Furthermore, by [14, Lemma 13.1], condition (11) implies that a2m (x, ξ ) ι > 0 for all (x, ξ ) ∈ X × S n−1 , so that for x ∈ X and |ξ | > λ−1/2m the condition bλ (x, ξ ) < 0 is equivalent to a2m (x, ξ ) < 1. Now, departing from (49) one computes dχ2 Vχ X × Rn , aλ = |G| =
X Rn
dχ2 (2π)n |G|
1(−∞,0] bλ x, λ−1/2m ξ dx dξ ¯ X Rn
1(−∞,0] bλ (x, ξ ) dx dξ · λn/2m
P. Ramacher / Journal of Functional Analysis 255 (2008) 777–818
=
815
−1 vol(a2m ((−∞, 1])) n/2m + O(1), λ (2π)n |G|
dχ2
and the assertion of the lemma follows.
2
Lemma 18. Assume that for some sufficiently small ρ > 0 there exists a constant C > 0 such that vol(∂X)ρ Cρ. Then, as λ → +∞, vol RV c,λ = O(λ(n−)/2m ), where ∈ (0, 12 ). Proof. According to the definition of RV c,λ at the beginning of Section 6, we have vol RV c,λ vol (x, ξ ) ∈ X × Rn : |aλ | − c hδ−ω + d < 0 + vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d , √ where Dc = {(x, ξ ): dist(x, ∂X) < c(1 + |x|2 + |ξ |2 )−δ/2 }, and 0 < δ − ω < 1/2. In what follows, let us assume that λ 1. It is not difficult to see that, for |ξ | > 1, there exists a constant c1 > 0 independent of λ such that aλ (x, ξ ) − c hδ−ω + d (x, ξ ) < 0
⇒
|ξ | < c1 λ1/2m .
(60)
Indeed, let c1 be such that c12m max(2, 2/ι),
1 c hδ−ω + d (x, ξ ) , 3 x∈X, |ξ |>c1 sup
where ι > 0 is a lower bound for a2m (x, ξ ) on X × S n−1 . Since 1−
λ 1 a2m (x, ξ ) 2
⇐⇒
|ξ |2m
2λ , a2m (x, ξ/|ξ |)
one computes for |ξ | c1 λ1/2m that aλ (x, ξ )
1 1 1 1 1 , 2 1 + λ|ξ |−2m 2 1 + c1−2m 3
while, on the other hand, c(hδ−ω + d)(x, ξ ) 13 , so that aλ (x, ξ ) − c(hδ−ω + d)(x, ξ ) 0. This proves (60). As a consequence, we obtain the estimate vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d vol (x, ξ ) ∈ X × Rn : |ξ | K, dist(x, ∂X) < c2 |ξ |−δ , aλ < c hδ−ω + d + vol (x, ξ ) ∈ X × Rn : |ξ | K vol (∂X)c2 |ξ |−δ ∩ X dξ + c3 , K|ξ |c1 λ1/2m
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P. Ramacher / Journal of Functional Analysis 255 (2008) 777–818
where δ ∈ ( 14 , 12 ), and K 1 is some sufficiently large constant; here and in all what follows, ci > 0 denote suitable, positive constants independent of λ. Now, since vol(∂X)ρ Cρ, for some ρ > 0, vol (x, ξ ) ∈ Dc : x ∈ X, aλ < c hδ−ω + d c2 c4
r n−1−δ dr dS n−1 (η) + c3
S n−1 Krc1 λ1/2m
= c5 λ(n−δ)/2m − K n−δ + c3 . Next, let |ξ | K, and assume that the inequality |aλ (x, ξ )| c(hδ−ω + d)(x, ξ ) is fulfilled. As before, we have |ξ |2m < c12m λ, as well as λ c 1 + λ|ξ |−2m d + hδ−ω (x, ξ ) c6 1 + λ|ξ |−2m |ξ |−(δ−ω) . 1 − a2m (x, ξ )
(61)
Combining (60) and (61), one deduces for sufficiently large K that |ξ |2m −c6 |ξ |2m + λ |ξ |−(δ−ω) +
λ c7 λ. a2m (x, ξ/|ξ |)
Let us now introduce the variable R(x, ξ ) = λ/a2m (x, ξ ) = λ|ξ |−2m /a2m (x, ξ/|ξ |). Performing the corresponding change of variables one computes vol (x, ξ ) ∈ X × Rn : |aλ | − c hδ−ω + d < 0 vol (x, ξ ) ∈ X × Rn : c1 λ1/2m > |ξ | K, 1 − R(x, ξ ) c6 1 + λ|ξ |−2m |ξ |−(δ−ω) + c8 r n−1 dr dS n−1 (η) dx + c8 X S n−1 {rK: |1−R|c9 λ−(δ−ω)/2m }
c10
R X S n−1 {R: |1−R|c9 λ−(δ−ω)/2m }
c11 λ
n 2m
−1
λ Ra2m (x, η)
n 2m
dR dS n−1 (η) dx + c8
n R − 2m −1 dR + c8 = O λ(n−(δ−ω))/2m + c8 .
{R: |1−R|c9 λ−(δ−ω)/2m }
Hereby we made use of the fact that (1 + z)β − (1 − z)β = O(|z|) for arbitrary z ∈ C, |z| < 1, and β ∈ R. 2 We are now in position to prove the main result of Part I, which generalizes [14, Theorem 13.1] to bounded domains with symmetries in the finite group case. Theorem 8. Let G be a finite group of isometries in Euclidean space Rn , and X ⊂ Rn a bounded domain which is invariant under G such that, for some sufficiently small ρ > 0, vol(∂X)ρ Cρ. Let further A0 be a symmetric, classical pseudodifferential operator in L2 (Rn ) of order 2m with G-invariant Weyl symbol σ w (A0 ) ∈ S(g, h−2m ) and principal symbol a2m , and assume that
P. Ramacher / Journal of Functional Analysis 255 (2008) 777–818
817
(A0 u, u) cu2m for some c > 0 and all u ∈ C∞ c (X). Consider further the Friedrichs extension of the operator 2 res ◦A0 ◦ ext : C∞ c (X) −→ L (X),
and denote it by A. Finally, let Nχ (λ) be the number of eigenvalues of A less or equal λ and with eigenfunctions in the χ -isotypic component res Hχ of L2 (X), if (−∞, λ) contains no points of the essential spectrum, and equal to ∞, otherwise. Then, for all ∈ (0, 12 ), Nχ (λ) =
−1 −1 vol(a2m ((−∞, 1])) vol(a2m ((−∞, 1])) n/2m λ + O λ(n−)/2m n (2π) |G| |G|
dχ2
as λ → +∞, where dχ denotes the dimension of any irreducible representation of G corresponding to the character χ . In particular, A has discrete spectrum. Proof. By Lemma 8 and Proposition 7 we have Nχ (λ) − Vχ X × Rn , aλ C1 vol RV C
1 ,λ
+ C2
for some suitable constants C1 , C2 > 0 independent of λ. Lemmata 17 and 18 then imply −1 dχ2 vol(a2m ((−∞, 1])) n/2m λ −O λ(n−)/2m Nχ (λ) − O λ(n−)/2m n (2π) |G|
with arbitrary ∈ (0, 1/2). In particular, Nχ (λ) remains finite for λ < ∞, so that the essential spectrum of A must be empty. The assertion of the theorem now follows. 2 Remark 4. An alternative description of the expected approximation Vχ (X × Rn , aλ ) defined 1 −n/2m dx dS n−1 (η). Then for in (49) can be given as follows. Let γ = n|G| X S n−1 (a2m (x, η)) λ > 0 one has dχ2 γ · λn/2m + O(1). Vχ X × Rn , aλ = (2π)n Indeed, as already noticed, a2m (x, ξ ) ι > 0 for all (x, ξ ) ∈ X × S n−1 , so that aλ is strictly negative on Xρ × {ξ : |ξ | > 1} if, and only if, a2m (x, ξ ) − λ < 0, which in turn is equivalent to −1 1/2m |ξ | < λa2m x, ξ/|ξ | , due to the homogeneity of the principal symbol. For λ > 0 one therefore concludes from this Vχ X × Rn , aλ =
vol (x, ξ ) ∈ X × Rn : |ξ | 1 (2π)n |G| −1 + vol (x, ξ ) ∈ X × Rn : |ξ |2m < λa2m x, ξ/|ξ | dχ2
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= O(1) +
= O(1) +
dχ2 (2π)n |G| dχ2 (2π)n |G|
(λ/a2m (x,η))1/2m
r n−1 dr dS n−1 (η) dx X S n−1
0
1 (λ/a2m (x, η))n/2m dS n−1 (η) dx. n
X S n−1
Acknowledgment The author wishes to thank Professor Mikhail Shubin for introducing him to this subject, and for many helpful discussions and useful remarks. References [1] V. Arnol’d, Frequent representations, Moscow Math. J. 3 (4) (2003) 1209–1221. [2] M. Bronstein, V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients I. Pushing the limits, Comm. Partial Differential Equations 28 (2003) 83–102. [3] J. Brüning, E. Heintze, Representations of compact Lie groups and elliptic operators, Invent. Math. 50 (1979) 169– 203. [4] T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes, in: C.R. Séme Cong. Math. Scand., Stockholm, 1934, Lund, 1935, pp. 34–44. [5] H. Donnelly, G-spaces, the asymptotic splitting of L2 (M) into irreducibles, Math. Ann. 237 (1978) 23–40. [6] V.I. Feigin, Asymptotic distribution of eigenvalues for hypoelliptic systems in Rn , Math. USSR Sb. 28 (4) (1976) 533–552. [7] L. Gårding, On the asymptotic distribution of eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand. 1 (1953) 237–255. [8] V. Guillemin, A. Uribe, Reduction and the trace formula, J. Differential Geom. 32 (2) (1990) 315–347. [9] B. Helffer, D. Robert, Etude du spectre pour un opératour globalement elliptique dont le symbole de Weyl présente des symétries I, Amer. J. Math. 106 (1984) 1199–1236. [10] B. Helffer, D. Robert, Etude du spectre pour un opératour globalement elliptique dont le symbole de Weyl présente des symétries II, Amer. J. Math. 108 (1986) 973–1000. [11] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193–218. [12] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979) 359–443. [13] V. Ivrii, Sharp spectral asymptotics for operators with irregular coefficients II. Domains with boundaries and degenerations, Comm. Partial Differential Equations 28 (2003) 103–128. [14] S.Z. Levendorskii, Asymptotic Distribution of Eigenvalues, Kluwer Academic, Dordrecht, Boston, MA, 1990. [15] S. Minakshisundaram, Å. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949) 242–256. [16] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd ed., Springer-Verlag, Berlin, 2001. [17] V.N. Tulovsky, M.A. Shubin, On the asymptotic distribution of eigenvalues of pseudodifferential operators in Rn , Math. Trans. 92 (4) (1973) 571–588. [18] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912) 441–479.
Journal of Functional Analysis 255 (2008) 819–853 www.elsevier.com/locate/jfa
Atomic representations of rank 2 graph algebras Kenneth R. Davidson a,∗,1 , Stephen C. Power b,2 , Dilian Yang a,3 a Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L-3G1, Canada b Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
Received 30 May 2007; accepted 16 May 2008 Available online 19 June 2008 Communicated by D. Voiculescu
Abstract We provide a detailed analysis of atomic ∗-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal ∗-dilations of defect free representations modelled on finite groups of rank 2. © 2008 Elsevier Inc. All rights reserved. Keywords: Higher rank graph; Atomic ∗-representation; Dilation
1. Introduction Kumjian and Pask [10] introduced higher rank graphs and their associated C*-algebras as a generalization of graph C*-algebras that are related to the generalized Cuntz–Kreiger algebras of Robertson and Steger [16]. The C*-algebras of higher rank graphs have been studied in a variety of papers [7,11,14,15,17]. See also Raeburn’s survey [13]. In [9], Kribs and Power examined the nonself-adjoint operator algebras which are associated with these higher rank graphs. More recently, Power [12] presented a detailed analysis of single * Corresponding author.
E-mail addresses: [email protected] (K.R. Davidson), [email protected] (S.C. Power), [email protected] (D. Yang). 1 Partially supported by an NSERC grant. 2 Partially supported by EPSRC grant EP/E002625/1. 3 Partially supported by the Fields Institute. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.008
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vertex rank 2 case. As this case already contains many new and intriguing algebras, we were motivated to study them more closely. The rank 2 graphs on one vertex form an intriguing family of semigroups with a rich representation theory. There are already many interesting and non-trivial issues. Our purpose in this paper is to completely classify the atomic ∗-representations of these semigroups. These representations combine analysis with some interesting combinatorial considerations. They provide a rich class of representations of the associated C*-algebra which have proven effective in understanding the underlying structure. The algebras are given concretely in terms of a finite set of generators and relations of a special type. Given a permutation θ of m × n, form a unital semigroup F+ θ with generators e1 , . . . , em and f1 , . . . , fn which is free in the ei ’s and free in the fj ’s, and has the commutation relations ei fj = fj ei where θ (i, j ) = (i , j ) for 1 i m and 1 j n. This is a cancellative semigroup with unique factorization [10]. + A ∗-representation of the semigroup F+ θ is a representation π of Fθ as isometries with the property that m
π(ei )π(ei )∗ = I =
i=1
n
π(fj )π(fj )∗ .
j =1
An atomic ∗-representation acts on a Hilbert space with an orthonormal basis which is permuted, up to unimodular scalars, by each of the generators. The C*-algebra C∗ (F+ θ ) is the universal C*algebra generated by these ∗-representations. The motivation for studying these representations comes from the case of the free semigroup ∗ + F+ m generated by e1 , . . . , em with no relations. The C*-algebra C (Fm ) is just the Cuntz algebra Om [1] (see also [2]). Davidson and Pitts [3] classified the atomic ∗-representations of F+ m and showed that the irreducibles fall into two types, known as ring representations and infinite tail representations. They provide an interesting class of C*-algebra representations of Om which are amenable to analysis. Furthermore the atomic ∗-representations feature significantly in the dilation theory of row contractions [5]. The 2-graph situation turns out to be considerably more complicated than the case of the free semigroup. Whereas the irreducible atomic ∗-representations of F+ n are of two types, the irreducible atomic ∗-representations of F+ fall into six types. Nevertheless, we are able to put θ them all into a common framework modelled on abelian groups of rank 2. 2. Background Rank 2 graphs. The semigroup F+ θ is generated by e1 , . . . , em and f1 , . . . , fn . The identity is denoted as ∅. There are no relations among the e’s, so they generate a copy of the free semigroup + on m letters, F+ m ; and there are no relations on the f ’s, so they generate a copy of Fn . There are commutation relations between the e’s and f ’s given by a permutation θ in Sm×n of m × n: ei f j = f j e i
where θ (i, j ) = (i , j ).
A word w ∈ F+ θ has a fixed number of e’s and f ’s regardless of the factorization; and the degree of w is d(w) := (k, l) if there are k e’s and l f ’s. The degree map is a homomorphism of + 2 F+ θ into N0 . The length of w is |w| = k + l. The commutation relations allow any word w ∈ Fθ
K.R. Davidson et al. / Journal of Functional Analysis 255 (2008) 819–853
821
to be written with all e’s first, or with all f ’s first, say w = eu fv = fv eu . Indeed, one can factor w with any prescribed pattern of e’s and f ’s as long as the degree is (k, l). It is straightforward to see that the factorization is uniquely determined by the pattern and that F+ θ has the unique factorization property and cancellation. See also [9,10,12]. Example 2.1. With n = m = 2 we note that the relations e1 f 1 = f 2 e1 ,
e1 f 2 = f 1 e2 ,
e2 f 1 = f 1 e1 ,
e2 f 2 = f 2 e2 ,
arise from the 3-cycle permutation θ = ((1, 1), (1, 2), (2, 1)) in S2×2 . We will refer to F+ θ as the forward 3-cycle semigroup. The reverse 3-cycle semigroup is the one arising from the 3-cycle ((1, 1), (2, 1), (1, 2)). It was shown by Power in [12] that the 24 permutations of S2×2 give rise to 9 isomorphism classes of semigroups F+ θ , where we allow isomorphisms to exchange the ei ’s for fj ’s. In particular, the forward and reverse 3-cycles give non-isomorphic semigroups. Example 2.2. With n = m = 2 the relations e1 f 1 = f 1 e 1 ,
e1 f 2 = f 1 e2 ,
e2 f 1 = f 2 e1 ,
e2 f 2 = f 2 e2 ,
are those arising from the 2-cycle permutation ((1, 2), (2, 1)) and we refer to F+ θ in this case as the flip semigroup because of the commutation rule: ei fj = fi ej for 1 i, j 2. This is an example which has periodicity, a concept which will be explained in more detail later. Representations. We now define two families of representations which will be considered here: the ∗-representations and defect free (partially isometric) representations. Definition 2.3. A partially isometric representation of F+ θ is a semigroup homomorphism σ : → B(H) whose range consists of partial isometries on a Hilbert space H. The representation F+ θ σ is isometric if the range consists of isometries. A representation is atomic if it is partially isometric and there is an orthonormal basis which is permuted, up to scalars, by each partial isometry. That is, σ is atomic if there is a basis {ξk : k 1} so that for each w ∈ F+ θ , σ (w)ξk = αξl for some l and some α ∈ T ∪ {0}. We say that σ is defect free if m i=1
σ (ei )σ (ei )∗ = I =
n
σ (fj )σ (fj )∗ .
j =1
An isometric defect free representation is called a ∗-representation of F+ θ . For a (partially) isometric representation, the defect free condition is equivalent to saying that the ranges of the σ (ei )’s are pairwise orthogonal and sum to the identity, and that the same holds for the ranges of the σ (fj )’s. Equivalently, this says that [σ (e1 ) . . . σ (em )] is a (partial)
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isometry from the direct sum of m copies of H onto H, and the likewise for [σ (f1 ) . . . σ (fn )]. A representation which satisfies the property that these row operators are isometries into H is called row isometric. The left regular representation λ of F+ θ is an example of a representation which is row isometric, but is not defect free. Definition 2.4. The C*-algebra C∗ (F+ θ ) is the universal C*-algebra for ∗-representations. This is the C*-algebra generated by isometries E1 , . . . , Em and F1 , . . . , Fn which are defect free: m n ∗=I = ∗ , and satisfies the commutation relations of F+ , with the universal E E F F i j i=1 j =1 θ i j ∗ + property that every ∗-representation σ of F+ θ extends to a ∗-homomorphism of C (Fθ ) onto C∗ (σ (F+ θ )). In the case of the free semigroup F+ n , Davidson and Pitts [3] classified the atomic row isometric representations. They showed that in the irreducible case there are three possibilities, namely (i) the left regular representation, (ii) a ring representation, determined by a primitive word u in F+ n and a unimodular scalar λ, and (iii) a tail representation, constructed from an aperiodic infinite word in the generators of F+ n . The left regular representation is the only one which is not defect free. The ring and tail representations provide the irreducible atomic ∗-representations. The universal C*-algebra of F+ n is the Cuntz algebra On . A ring representation is determined by a set of k basis vectors which are cyclically permuted, modulo λ, according to the k letters of u. The primitivity of u means that u is not a proper power of a smaller word. On the other hand, a tail representation σ is determined by an infinite word z = z0 z−1 z−2 . . . in the generators of F+ n . There is a subset of basis elements ξ0 , ξ−1 , ξ−2 , . . . for which σ (zk )ξk−1 = ξk for k 0. In both cases the subspace M spanned by the basis vector subset is coinvariant for σ (i.e. σ (w)∗ M ⊂ M for all w ∈ F+ ) and cyclic for σ (i.e. σ (w)M = n w∈F+ n ⊥ H). On the complementary invariant subspace M , σ decomposes as a direct sum of copies of the left regular representation. We shall meet these representations, as restrictions, in the classification of irreducible atomic representations of F+ θ . + Dilations. If σ is a representation of F+ θ on a Hilbert space H, say that a representation π of Fθ on a Hilbert space K ⊃ H is a dilation of σ if
σ (w) = PH π(w)|H
for all w ∈ F+ θ .
The dilation π of σ is a minimal isometric dilation if π is isometric and the smallest invariant subspace containing H is all of K, which means that K = w∈F+ π(w)H. This minimal isometθ ric dilation is called unique if for any two minimal isometric dilations πi on Ki , there is a unitary operator U from K1 to K2 such that U |H is the identity map and π2 = Ad U π1 . It is straightforward to see that an isometric dilation of a defect free representation is still defect free, and hence is a ∗-representation. An important result from our paper [6] is that every defect free representation has a unique minimal ∗-dilation. Both existence and uniqueness are of critical importance. Moreover, if the original representation is atomic, then so is the ∗-dilation. Theorem 2.5. (See [6, Theorems 5.1, 5.5].) Let σ be a defect free (atomic) representation. Then σ has a unique minimal dilation to a (atomic) ∗-representation.
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Conversely, if π is a ∗-representation, one can obtain a defect free representation σ on a subspace H if H is co-invariant, i.e. H⊥ is invariant for π(F+ θ ), by setting σ (w) = PH π(w)|H . By Dilation Theorem 2.5, σ has a unique minimal ∗-dilation. But such a ∗-dilation is evidently the restriction of π to the reducing subspace K = w∈F+ π(w)H. If the subspace H is cyclic, θ i.e. K = K, then this minimal ∗-dilation is π itself. In this case, π is uniquely determined by σ . The significance of this for us is that every atomic ∗-representation has a particularly nice coinvariant subspace on which the compression σ has a very tractable form. It is by determining these smaller defect free representations that we are able to classify the atomic ∗-representations. 3. Examples of atomic representations We begin with a few examples of atomic representations. Example 3.1. Given a permutation θ of m × n, select a cycle of θ , say (i1 , j1 ), (i2 , j2 ), . . . , (ik , jk ) . Form a Hilbert space of dimension k with basis ξs for 1 s k. Define σ (ei ) =
∗ ξs ξs−1
and σ (fj ) =
is =i
ξs−1 ξs∗ .
js =j
That is, σ (fjs ) maps ξs to ξs−1 and σ (eis ) maps ξs−1 back to ξs . Likewise σ (eis+1 ) maps ξs to ξs+1 and σ (fjs+1 ) maps ξs+1 back to ξs . This corresponds to the commutation relation eis fjs = fjs+1 eis+1 indicated by the cycle of θ . i1 i3
i2
ξ2
ξ1 j2
ik
...... j3
ξk jk
j1
So it is not difficult to verify that this defines a defect free atomic representation of F+ θ . By . Dilation theorem 2.5, this can be dilated to a unique ∗-representation of F+ θ One can further adjust this example by introducing scalars. For example, if α, β ∈ T, define σα,β (ei ) = ασ (ei ) and σα,β (fj ) = βσ (fj ). Two such representations will be shown to be unitarily equivalent if and only if α1k = α2k and β1k = β2k . Example 3.2 (Inductive representations). An important family of ∗-representations were introduced in [6]. Recall that the left regular representation λ acts on 2 (F+ θ ), which has orthonormal }, by left multiplication: λ(w)ξ = ξ for all w, v ∈ F+ basis {ξv : v ∈ F+ v wv θ θ . Start with an arbitrary infinite word or tail τ = ei0 fj0 ei1 fj1 . . . . Let Fs = F := F+ θ , for s = act as injective maps by right 0, 1, 2, . . . , viewed as a discrete set on which the generators of F+ θ multiplication, namely,
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ρ(w)g = gw
for all g ∈ F.
Consider ρs = ρ(eis fjs ) as a map from Fs into Fs+1 . Define Fτ to be the injective limit set Fτ = lim(Fs , ρs ); →
and let ιs denote the injections of Fs into Fτ . Thus Fτ may be viewed as the union of F0 , F1 , . . . with respect to these inclusions. The left regular action λ of F+ θ on itself induces corresponding maps on Fs by λs (w)g = wg. Observe that ρs λs = λs+1 ρs . The injective limit of these actions is an action λτ of F+ θ on Fτ . Let + 2 λτ also denote the corresponding representation of Fθ on (Fτ ). The standard basis of 2 (Fτ ) is {ξg : g ∈ Fτ }. A moment’s reflection shows that this provides a defect free, isometric (atomic) representation of F+ θ ; i.e. it is a ∗-representation. We now describe a coinvariant cyclic subspace that contains all of the essential information ∗ about this representation. Let H = λτ (F+ θ ) ξι0 (∅) . This is coinvariant by construction. As it contains ξιs (∅) for all s 1, it is easily seen to be cyclic. Let στ be the compression of λτ to H. ...
i−2,0
•
i−1,0
•
j−2,0
...
i−2,−1
•
i−1,−1
i−2,−2
•
i−1,−2
•
.. .
i0,−1
j0,0
•
j−1,−1
•
j−2,−2
.. .. .. ...
•
j−1,0
j−2,−1
...
i0,0
i0,−2
j0,−1
•
j−1,−2
.. .
j0,−2
.. .
Since λτ is a ∗-representation, for each (s, t) ∈ (−N0 )2 , there is a unique word eu fv of degree (|s|, |t|) such that ξι0 (∅) is in the range of λτ (eu fv ). Set ξs,t = λτ (eu fv )∗ ξι0 (∅) . It is not hard to see that this forms an orthonormal basis for H. For each (s, t) ∈ (−N0 )2 , there are unique integers is,t ∈ {1, . . . , m} and js,t ∈ {1, . . . , n} so that στ (eis,t )ξs−1,t = ξs,t
for s 0 and t 0,
στ (fjs,t )ξs,t−1 = ξs,t
for s 0 and t 0,
στ (ei )ξs,t = 0
if i = is+1,t or s = 0,
στ (fj )ξs,t = 0
if j = js,t+1 or t = 0.
Note that we label the edges leading into each vertex, rather than leading out. This choice reflects the fact that the basis vectors for a general atomic partial isometry representation are each in the range of at most one of the partial isometries π(ei ) and at most one of the π(fj ).
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Consider how the tail τ = ei0 fj0 ei1 fj1 . . . determines these integers. It defines the path down the diagonal; that is, is,s := i|s|
and js−1,s := j|s|
for s 0.
This determines the whole representation uniquely. Indeed, for any vertex ξs,t with s, t 0, take T |s|, |t|, and select a path from (−T , −T ) to (0, 0) that passes through (s, t). The word τT = ei0 fj0 . . . eiT −1 fjT −1 satisfies στ (τT )ξ−T ,−T = ξ0,0 . Factor it as τT = w1 w2 with d(w1 ) = (T − |s|, T − |t|) and d(w2 ) = (|s|, |t|), so that στ (w2 )ξ−T ,−T = ξs,t and στ (w1 )ξs,t = ξ0,0 . Then w1 = eis,t w = fjs,t w . It is evident that each στ (ei ) and στ (fj ) is a partial isometry. Moreover, each basis vector is in the range of a unique στ (ei ) and στ (fj ). So this is a defect free, atomic representation with minimal ∗-dilation λτ . The symmetry group. An important part of the analysis of these atomic representations is the recognition of symmetry. Definition 3.3. The tail τ determines the integer data Σ(τ ) = (is,t , js,t ): s, t 0 . (k)
(k)
Two tails τ1 and τ2 with data Σ(τk ) = {(is,t , js,t ): s, t 0} are said to be tail equivalent if the two sets of integer data eventually coincide; i.e. there is an integer T so that (1) (1) (2) (2) for all s, t T . is,t , js,t = is,t , js,t Say that τ1 and τ2 are (p, q)-shift tail equivalent for some (p, q) ∈ Z2 if there is an integer T so that (2) (2) (1) (1) for all s, t T . is+p,t+q , js+p,t+q = is,t , js,t Then τ1 and τ2 are shift tail equivalent if they are (p, q)-shift tail equivalent for some (p, q) ∈ Z2 . The symmetry group of τ is the subgroup Hτ = (p, q) ∈ Z2 : Σ(τ ) is (p, q)-shift tail equivalent to itself . A sequence τ is called aperiodic if Hτ = {(0, 0)}. The semigroup F+ θ is said to satisfy the aperiodicity condition if there is an aperiodic infinite word. Otherwise we say that F+ θ is periodic. In our classification of atomic ∗-representations, an important step is to define a symmetry group for the more general representations which occur. The ∗-representation will be irreducible precisely when the symmetry group is trivial. This will yield a method to decompose atomic ∗-representations as direct integrals of irreducible atomic ∗-representations. The graph of an atomic representation. We need to develop a bit more notation. Let σ be an atomic representation. Let the corresponding basis be {ξk : k ∈ S}. Write ξ˙k to denote the
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subspace Cξk . Form a graph Gσ with vertices ξ˙k . If σ (ei )ξ˙k = ξ˙l , draw a directed blue edge from ξ˙k to ξ˙l labelled i; and if σ (fj )ξ˙k = ξ˙l , draw a directed red edge from ξ˙k to ξ˙l labelled j . This is the graph of the representation, and it contains all of the information about σ except for the scalars in T. In our analysis of atomic representations, one can easily split a representation into a direct sum of atomic representations which have a connected graph. So we will generally work with representations with connected graph. Lemma 3.4. Let σ be a defect free atomic representation with connected graph Gτ . Let ξ˙1 and ˙ ξ˙2 be two vertices in Gσ . Then there is a vertex η˙ and words w1 , w2 ∈ F+ θ so that ξi = σ (wi )η˙ for i = 1, 2. Proof. The connectedness of the graph means that there is a path from ξ˙1 to ξ˙2 . We will modify this path to first pull back along a path leading into ξ˙1 , and then move forward to ξ˙2 . The original path can be written formally as ak ak−1 . . . a1 where each al has the form ei or fj if it is moving forward or ei∗ or fj∗ if pulling back. After deleting redundancies, we may assume that there are no adjacent terms fj∗ fj or ei∗ ei . At each vertex, there is a unique blue (red) edge leading in; so moving forward along a blue (red) edge and pulling back along the same colour is just one of these redundancies. Thus if there are adjacent terms of the form a ∗ b which do not cancel, then one of a, b is an e and the other an f . For definiteness, suppose that this section of the path moves from η˙ 1 to η˙ 2 along the path fj∗ ei . That means that σ (ei )η˙ 1 = η˙ = σ (fj )η˙ 2 . As σ is defect free, there is a basis vector η˙ 0 and an fj so that σ (fj )η˙ 0 = η˙ 1 . Factor ei fj in the other order as fj ei . Then σ (fj )η˙ 2 = η˙ = σ (ei fj )η˙ 0 = σ (fj ei )η˙ 0 = σ (fj ) σ (ei )η˙ 0 . However there is a unique red edge into η, and thus fj = fj
and σ (ei )η˙ 0 = η˙ 2 .
It is now clear that there is a red–blue diamond in the graph with apex η˙ 0 , and red and blue edges leading to η˙ 1 and η˙ 2 , respectively, and from there, red and blue edges, respectively, leading into η. ˙ So the path fj∗ ei may be replaced by ei fj∗ . Similarly, the path ei∗ fj from η˙ 2 to η˙ 1 may be replaced by fj ei∗ . Repeated use of this procedure replaces any path from ξ˙1 to ξ˙2 by a path of the form w2 w1∗ ; and hence η = σ (w1 )∗ ξ1 is the intermediary vector. 2 4. Classifying atomic representations Consider an atomic ∗-representation π of F+ θ with connected graph Gπ . Observe that if we restrict π to the subalgebra generated by the ei ’s, we obtain an atomic, defect free representation of the free semigroup F+ m . The graph splits into the union of its blue components. By [3], this decomposes the restriction into a direct sum of ring representations and infinite tail representations. Our first result shows that the connections provided by the red edges force a parallel structure in the blue components.
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Lemma 4.1. Let π be an atomic ∗-representation of F+ θ with connected graph Gπ . Let H1 and H2 be two blue components of Gπ and assume that there is a red edge leading from a vertex in H1 to a vertex in H2 . Then every red edge into H2 comes from H1 . The two components are either both of infinite tail type or both are of ring type; and in the ring case the length of the (unique) ring for H2 is an integer multiple t of the ring length in H1 , where 1 t n. Proof. This is an exercise in using the commutation relations. Suppose first that H1 is an infinite tail graph. Fix a vertex ξ˙0 ∈ H1 and a red edge fj0 so that π(fj0 )ξ˙0 = ζ˙0 is a vertex in H2 . There is a unique infinite sequence of vertices ξk for k < 0 and integers ik so that π(eik )ξ˙k−1 = ξ˙k
for k 0.
Now there are unique integers ik and jk so that ...i f jk fj0 ei0 i−1 ...ik+1 = ei0 i−1 k+1
for k < 0.
Let ζ˙k := π(fjk )ξ˙k for k < 0. Then it is evident that π(eik )ζ˙k−1 = ζ˙k
for k 0.
The images of each vertex under the various red edges are all distinct. So in particular, the ζ˙k are all distinct, and so H2 is also an infinite tail graph. Now one similarly can follow each vertex ζ˙k forward under a blue path eu to reach any vertex ζ˙ in H2 . Then ζ˙ = π(eu )ζ˙k = π(eu )π(fjk )ξ˙k = π(fj )π(eu )ξ˙k . Thus the vertex π(eu )ξ˙k in H1 is mapped to ζ˙ by π(fj ). Hence the red edge leading into ζ˙ comes from H1 . The case of a ring representation is similar. Starting with any vertex in H1 which is connected to H2 by a red edge, one can pull back along the blue edges until one is in the ring. So we may suppose that ξ˙0 lies in the ring of H1 and that π(fj0 )ξ˙0 = ζ˙0 in H2 . Let u be the unique minimal word such that π(eu )ξ˙0 = ξ˙0 , and let p = |u|. As in the first paragraph, we continue to pull back from ζ˙0 and from ξ˙0 along the blue edges. Call these edges ζ˙k and ξ˙k , respectively, for k < 0. After p steps, we return to ξ˙0 = ξ˙−p in H1 , and we have obtained p distinct vertices in H2 and reach ζ˙−p . So there is a word u of length p so that π(eu )ζ˙−p = ζ˙0 . The commutation relations yield π(eu )ζ˙−p = ζ˙0 = π(fj0 )π(eu )ξ˙0 = π(eu )π(fj )ξ˙0 . There is a unique blue path of length p leading into ζ˙0 . Therefore u = u and π(fj )ξ˙0 = ζ˙−p . Notice that if j = j0 , then ζ˙−p = ζ˙0 . However if j = j0 , then ζ˙−p is a different vertex in H2 . Repeat the process, pulling back another p blue steps, to reach a vertex ζ˙−2p . Along the way, we obtain vertices which are distinct from the previous ones, and each is the image of some vertex in the ring of H1 . As before, ζ˙−2p is the image of ξ˙0 under some red edge. Eventually this
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process must repeat, because there are only n red edges out of ξ˙0 . That is, there are integers s and t with 1 t n so that ζ˙−(s+t)p = ζ˙−sp . This is a ring in H2 of length tp. The argument that each edge in H2 is in the range of a red edge coming from H1 is identical to the infinite tail case. 2 One might hope that the red edges from H1 to H2 provide a nice bijection, or a t-to-1 map that preserves the graph structure. Even though these edges are determined algebraically by the commutation relations, such a nice pairing does not occur as the following examples demonstrate. Example 4.2. Consider m = n = 3 with θ given by ((1, 2), (2, 1)), or equivalently by the relations e1 f 2 = f 1 e 2 ,
e2 f 1 = f 2 e1 ,
and ei fj = fj ei ,
otherwise.
There is a 1-dimensional defect free representation ρ(e3 ) = ρ(f3 ) = 1 and ρ(ei ) = ρ(fi ) = 0 for i = 1, 2. This has a dilation to a ∗-representation π . Let the initial vector be called ξ0 . Define ζj 0 = π(fj )ξ0 for j = 1, 2; and let ξi = π(ei )ξ0 and ζj i = π(ei )ζj 0 for i = 1, 2 and j = 1, 2. The blue component H0 containing ξ0 has a ring of length one. 3
3
ξ0
1
2 1
3
3
ξ1
2
ζ10 2
1
3
ζ20 2
1
2
ξ2
3
1 2
1
ζ11
ζ12
ζ21
ζ22
It is easy to check that π(e3 )ζj 0 = ζj 0 for j = 1, 2. Hence each is a ring in a separate blue component Hj . But the commutation relations also show that π(fj )ξi = ζij . This means that the vertex ξ˙1 has two red edges leading to the component H1 ; and ξ˙2 has two red edges leading to H2 . Example 4.3. With the same algebra, consider the 1-dimensional representation ρ(e1 ) = ρ(f3 ) = 1 and ρ(ei ) = ρ(fj ) = 0 otherwise. Again this has a dilation to an isometric defect free representation π . Let the initial vector be called ξ . Define ζj = π(fj )ξ for j = 1, 2. A computation with the relations shows that π(e1 )ζ1 = ζ1 . So this is a ring of length one in a component H1 . But π(e2 )ζ1 = ζ2 . So ζ2 is also in H1 ; but even though it is in the range of a red edge from the ring of H0 , it does not lie on the ring of H1 .
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1
3
ξ
1
1
829
2 2
ζ1
ζ2
Example 4.4. Consider m = n = 2 and the reverse 3-cycle semigroup of Example 2.1 given by the 3-cycle ((1, 1), (2, 1), (1, 2)). There is a 1-dimensional defect free representation ρ(e2 ) = ρ(f2 ) = 1 and ρ(e1 ) = ρ(f1 ) = 0. This has a dilation to a ∗-representation π . Let the initial vector be called ξ . Define η = π(f1 )ξ and ζj = π(fj )η for j = 1, 2. Then an exercise with the relations shows that π(e1 )η = η and π(e1 )ζ2 = ζ1 and π(e2 )ζ1 = ζ2 . 2
ξ
2
1 1
η
1
2 2
ζ1
ζ2 1
Thus the initial component H0 has a ring of length one at ξ , as does the component H1 connected to it (at η), but it connects to a component H2 in which ζi are the vertices of a ring of length 2. One can show that there are components with rings of length 2k for all k 0. Splitting into cases. Evidently the reasoning of Lemma 4.1 also applies when we decompose the graph into its red components. It is now possible to split the analysis into several cases. 1. (Ring by ring type.) Both blue and red components are ring representations. In this case, the set of vertices which are in both a red and a blue ring determines a finite-dimensional coinvariant subspace on which the representation is defect free, and there is exactly one red and blue edge beginning at each vertex. Moreover this is a cyclic subspace for the representation because starting at any basis vector, pulling back along the blue and red edges eventually ends in the ring by ring portion. So this finite-dimensional piece determines the full representation. Since each ring is obtained by pulling back from any of the others, it follows from Lemma 4.1 that all of the blue rings have the same length, say k; and likewise all of the red rings have the same length, say l. 2. (Mixing type.) For one colour, the components are ring representations while the for the other colour, the components are infinite tail representations. There are two subcases.
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2a. (Ring by tail type.) The blue components are ring representations, and the red components are infinite tail type. If one begins at any blue ring and pulls back along the red edges, one obtains an infinite sequence of blue ring components. By Lemma 4.1, the size of the ring is decreasing as one pulls back. Hence it is eventually constant. From this point on back, there is a unique red edge from each ring to the corresponding point on the next. Thus one obtains a semi-infinite cylinder of fixed circumference k which is coinvariant and cyclic, and determines the full representation. 2b. (Tail by ring type.) The red components are ring representations, and the blue components are infinite tail type. 3. (Tail by tail type.) Both the red and the blue components are of infinite tail type. This actually has several subtypes. Start with a basis vector ξ˙ in a blue component H0 . Pull back along the red edges to get an infinite sequence of blue components Ht for t 0. The union of these components for t 0 forms a coinvariant cyclic subspace that determines the full representation. This sequence may be eventually periodic, or they may all be distinct. If they are eventually periodic, we may assume that we begin with a component H0 in the periodic sequence. 3a. (Inductive type.) The sequence of components are all distinct. In this case, starting at any basis vector ξ˙0,0 , one may pull back along both blue and red edges to obtain basis vectors ξs,t for (s, t) ∈ (−N0 )2 . The restriction of the representation to this coinvariant subspace is defect free, and determines the whole representation as in Example 3.2. So this is an inductive representation. 3b. The sequence of components repeats after l steps. Thus by Lemma 4.1, there are blue components H0 , . . . , Hl−1 so that every red edge into each Hi comes from Hi−1 (mod l) . This is further refined by comparing the point of return to the starting point ξ˙ . It is not apparent at this point that the only possibilities are the following. 3bi. (Return below.) The return is eventually below the start. In this case, there is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 obtained by pulling back l red steps using the word v0 from ξ˙0 satisfies π(eu0 )ξ˙0 = ζ˙0
and π(fv0 )ζ˙0 = ξ˙0 .
This same type of relationship persists when pulling back along both blue and red edges from ξ˙0 . 3bii. (Return above.) The return is eventually above the start. Here there is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 obtained by pulling back l red steps using the word v0 from ξ˙0 satisfies π(eu0 )ζ˙0 = ξ˙0 = π(fv0 )ζ˙0 . This same type of relationship persists when pulling back along both blue and red edges from ξ˙0 .
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The ring by ring representations are finitely correlated, meaning that there is a finitedimensional coinvariant, cyclic subspace. Equivalently, this means that the ∗-representation is the minimal ∗-dilation of a finite-dimensional defect free representation. Conversely, every finitely correlated atomic ∗-representation with connected graph is of ring by ring type, as the other cases clearly do not have a finite-dimensional non-zero coinvariant and cyclic subspace. In the following sections, each case will be considered in more detail. Eventually a common structure emerges. This will be codified by the general construction given in the next section. Symmetry. In each case, we associate a symmetry group to the picture. In the ring by ring case, it is easiest to describe because there is no equivalence relation. If the minimal coinvariant subspace consists of blue cycles of length k and red cycles of length l, then we associate the representation to a quotient group G of Ck × Cl ; and the symmetry group is a subgroup of G. We show that there is a finite-dimensional representation on Ckl which reflects the full symmetry, and that certain quotients yield a decomposition into irreducible summands. In type 3a, the inductive case, we have already seen how to define a symmetry subgroup of Z2 . Again, an inductive representation will be irreducible precisely when this symmetry group is trivial. In the other cases, the symmetry group is a subgroup of Ck × Z in the type 2a case or of Z2 /Z(a, b) in the 3b cases. In types 2 and 3, the decomposition into irreducibles may require a direct integral rather than a direct sum. 5. A group construction In this section, we will describe a general class of examples, and explain how to decompose them into irreducible representations. Start with an abelian group G with two designated generators g1 and g2 . We consider a defect 2 free atomic representation of F+ θ on (G) which is given by the following data: i : G → {1, . . . , m},
i(g) =: ig ,
j : G → {1, . . . , n},
j (g) =: jg ,
α : G → T,
α(g) =: αg ,
β : G → T,
β(g) =: βg .
We wish to define a representation σ of F+ θ by σ (ei )ξg = δi ig αg ξg+g1 , σ (fj )ξg = δj jg βg ξg+g2 . In order for this to actually be a representation, we require that eig+g2 fjg = fjg+g1 eig
for all g ∈ G
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and αg+g2 βg = βg+g1 αg
for all g ∈ G.
Let Gσ be the graph of this representation, which has vertices ξ˙g for g ∈ G and blue edges labelled ig from ξ˙g to ξ˙g+g1 , and red edges labelled jg from ξ˙g to ξ˙g+g2 . These will be called group construction representations of F+ θ . It is evident that there is a unique blue and red edge leading into each vertex ξ˙g , and so this is a defect free atomic representation. Thus it has a unique minimal ∗-dilation. Decomposing this representation into irreducible summands or a direct integral must simultaneously decompose the ∗-dilation into a direct sum or direct integral of the minimal ∗-dilations of the summands. The symmetry group of σ is defined as H = {h ∈ G: ig+h = ig and jg+h = jg for all g ∈ G}. This will play a central role in this decomposition. We will address the non-trivial issue of how to actually define the functions i and j later when considering the various cases. There are obstructions, and the way by which this difficulty is overcome is not immediately apparent. Example 3.1 is an example of this type in which G is the cyclic group Ck and g1 = −g2 = 1. The case of G = Z2 can be seen in Example 3.2. Here we describe a coinvariant subspace which is identified with (−N0 )2 , but it can be extended to all of Z2 . (We apologize to the reader that the notation is not consistent between Example 3.2 and this section.) The issue of the scalar functions α and β is more elementary. We shall see that it suffices to consider the case in which α and β are constant, and will determine when two are unitarily equivalent. For the moment, we assume that such a representation is given and consider how to analyze it. The group G, being abelian with two generators, is a quotient of Z2 . The subgroups of Z2 are {0}, singly generated Z(a, b), or doubly generated. In the doubly generated case, it is an easy exercise to see that the quotient is finite. In this case, if gi has order pi for i = 1, 2, then G is in fact a quotient of Cp1 × Cp2 . We shall show that the different groups correspond to the different representation types as follows: Type 1. G finite. Type 2. (a) G = Ck × Z; (b) G = Z × Cl . Type 3. (a) G = Z2 ; (bi) G = Z2 /Z(k, l), kl > 0; (bii) G = Z2 /Z(k, l), kl < 0. Scalars. Let us dispense with the scalars first. This is actually quite straightforward in spite of the notation. For each group G, there is a canonical homomorphism κ of Z2 onto G sending the standard generators (1, 0) and (0, 1) to g1 and g2 . Let K be the kernel of κ, so that G Z2 /K.
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Theorem 5.1. Let G = Z2 /K as above. A group construction representation σ on 2 (G) is unitarily equivalent to another of the same type with the same functions i, j for which the scalar functions are constants α0 and β0 . The constants determine a unique character ψ of K. They are unique up to a choice of an extension ϕ of ψ to a character on Z2 ; and they are given by α0 = ϕ(1, 0) and β0 = ϕ(0, 1). The choice of ϕ is unique up to a character χ of G, which changes α0 and β0 to α0 χ(g1 ) and β0 χ(g2 ). Proof. For each (s, t) ∈ Z2 , there is a unique path from ξ˙0 to ξ˙κ(s,t) in Gσ with s blue edges and t red edges, where a negative number indicates traversing the arrow in the backward direction. Corresponding to this, there is a unique partial isometry W in C∗ (σ (F+ θ )) of the form σ (eu )σ (fv ), σ (fv )∗ σ (eu ), σ (eu )∗ σ (fv ) or σ (eu )∗ σ (fv )∗ with |u| = |s| and |v| = |t|, depending on the signs of s and t, so that W ξ˙0 = ξ˙κ(s,t) . The subgroup K corresponds to those paths which return ξ˙0 to itself. Thus for (s, t) ∈ K, W ξ0 = ψ(s, t)ξ0 for a unique scalar ψ(s, t) ∈ T. In any unitarily equivalent representation given in the same form on 2 (G), the vector come from sent to ξ0 must be a vector with exactly the same functions i, j ; and thus must span{ξh : h ∈ H }. Moreover there are constraints on the scalars, namely if U ∗ ξ0 = h∈A ah ξh where A = {h: ah = 0}, then α(h + g) = α(h + g) and β(h + g) = β(h + g) for all h, h ∈ A and all g ∈ G. (For otherwise, the representation would not correspond to a graph.) Hence exactly the same words return each ξh to itself, for h ∈ A. We claim that the function ψ on K is independent of the unitary equivalence. Let ψ be the function obtained in this equivalent representation. The argument of the previous paragraph shows that instead of computing the function ψ using U ∗ ξ0 , we can use ξh for any h ∈ A and obtain the same result. Let W be the partial isometry found in the first paragraph which carries ξ˙0 to ξ˙h . Let (s, t) ∈ K, and let W0 and Wh be the partial isometries of degree (s, t) which take ξ˙0 and ξ˙h to themselves, respectively. Now W ∗ W W0 and W ∗ Wh W are partial isometries of the same combined and absolute degrees which map ξ˙0 to itself. The uniqueness of factorization means that these two words are equal! Therefore
ψ(s, t) = W ∗ W W0 ξ0 , ξ0 = W ∗ Wh W ξ0 , ξ0 = ψ (s, t). So we see that the original choice of scalars forces certain values, namely the function ψ, to be fixed independent of unitary equivalence. Next we show that ψ is a character. Given two words (s1 , t1 ) and (s2 , t2 ) in K, let W1 and W2 be the corresponding partial isometries. Then the partial isometry W corresponding to the sum (s1 + s2 , t1 + t2 ) need not equal W1 W2 . However the unique factorization means that W1 W2 = V ∗ V W for some partial isometry V containing ξ0 in its domain. Thus one computes
ψ(s1 + s2 , t1 + t2 ) = V ∗ V W ξ0 , ξ0 = W1 W2 ξ0 , ξ0 = ψ(s1 , t1 )ψ(s2 , t2 ). So ψ is multiplicative. It is routine to extend ψ to a character ϕ of Z2 . In our context, one can easily do this ‘bare hands.’ But it is a general fact for characters on any subgroup of any abelian group [8, Corollary 24.12]. If ϕ1 and ϕ2 are two characters extending ψ , then ϕ2 ϕ1 is a character which takes the constant value 1 on all of K. Thus it induces a character χ of G. We see that ϕ2 (s, t) = ϕ1 (s, t)χ(sg1 + tg2 ). Conversely, any choice of χ yields an extension.
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The unitary equivalence between σ and the representation σ on the same graph but with constants α0 = ϕ(1, 0) and β0 = ϕ(0, 1) can be accomplished by a diagonal unitary U = diag(γg ). Define γg by selecting a partial isometry W as in the first paragraph so that W ξ˙0 = ξ˙g . Let (a, b) be the degree of the word W . Define γg = ϕ(a, b) W ξ0 , ξg . While the choice of W is not unique, if W is another such partial isometry, W ∗ W ξ˙0 = ξ˙0 corresponds to a word (s, t) in K. Then W has degree (a + s, b + t). Thus
ϕ(a + s, b + t) W ξ0 , ξg = ϕ(a, b)ϕ(s, t) W W ∗ W ξ0 , ξg = ϕ(a, b)ϕ(s, t)ϕ(s, t) W ξ0 , ξg = ϕ(a, b) W ξ0 , ξg . So U is well defined. Now if γg is computed using W , calculate
γg+g1 = ϕ(a + 1, b) σ (eig )W ξ0 , ξg+g1 = ϕ(a + 1, b)αg W ξ0 , ξg . Then we see that U σ (eig )U ∗ ξg = U σ (eig )ϕ(a, b) W ξ0 , ξg ξg = U αg ϕ(a, b) W ξ0 , ξg ξg+g1 = ϕ(a + 1, b)αg W ξ0 , ξg αg ϕ(a, b) W ξ0 , ξg ξg+g1 = ϕ(1, 0)ξg+g1 = α0 ξg+g1 . One deals with σ (fjg )ξg in the same manner. Thus σ is unitarily equivalent to the representation σ with scalars α0 and β0 . It was irrelevant which extension ϕ of ψ was used; so all are unitarily equivalent to each other. 2 Symmetry. The key to the decomposition is to look for symmetry in the graph of σ . Recall that H = {h ∈ G: ig+h = ig and jg+h = jg for all g ∈ G}. It is clear that H is a subgroup of G. Note that we ignore the scalars for this purpose which is justified by Theorem 5.1. Indeed, we shall suppose that the scalars are constants α0 and β0 . For each coset [g] = g + H of G/H , let W[g] = span{ξg+h : h ∈ H }. Pick a representative gk for each coset of G/H , selecting 0 ∈ [0]. Each of the subspaces W[gk ] can be identified with 2 (H ). This identification depends on the choice of representative. Define J[gk ] : 2 (H ) → W[gk ] by J[gk ] ξh = ξgk +h .
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Since ig = ig+h and jg = jg+h , these integers are dependent only on the coset. So we will write i[g] and j[g] . Observe that σ (ei[g] ) maps W[g] to W[g+g1 ] . To understand this map, note that for each gk , there is a unique element hk1 ∈ H so that the representative for [gk + g1 ] is gk + g1 − hk1 . Then it is easy to see that ∗ σ (ei[gk ] )|W[gk ] = α0 J[gk +g1 ] Lhk1 J[g , k]
where Lh is the (left) regular translation by h on 2 (H ). Likewise, there is an hk2 ∈ H so that gk + g2 − hk2 is the chosen representative for [gk + g2 ]. Then ∗ σ (fj[gk ] )|W[gk ] = β0 J[gk +g2 ] Lhk2 J[g . k]
It is now a routine matter to diagonalize σ . The unitary operators Lh for h ∈ H all commute, and so can be simultaneously diagonalized by the Fourier transform which identifies 2 (H ) with L2 (Hˆ ), and carries Lh to the multiplication operator Mh given by Mh f (χ) = χ(h)f (χ). We explain in more detail how this works in the finite case. Here Lh are unitary matrices, and σ decomposes as a finite direct sum of irreducible representations. For each χ ∈ Hˆ , let
ζ0 = |H |−1/2 χ
χ(h)ξh .
h∈H χ
χ
Then a routine calculation shows that Lh ζ0 = χ(h)ζ0 . Consider the subspaces χ χ Mχ = span ζ[g] := J[g] ζ0 : [g] ∈ G/H . The choice of representative for each coset only affects the scalar multiple of the vectors, and the subspace Mχ is independent of this choice. It is easy to see that these are reducing subspaces for 2 σ (F+ θ ). This decomposes σ into a direct sum of representations σχ acting on (G/H ). Indeed, if [gk + g1 ] = [gl ], then we calculate σχ (ei[gk ] )ζ[gk ] = σ (ei[gk ] )J[gk ] |H |−1/2 χ
−1/2
= σ (ei[gk ] )|H | = α0 |H |−1/2
χ(h)ξh
h∈H
χ(h)ξgk +h
h∈H
χ(h)ξgk +g1 +h
h∈H
= α0 |H |−1/2
χ(h)ξgl +hk1 +h
h∈H
= α0 J[gl ] χ(hk1 )|H |−1/2
χ(h + hk1 )ξh+hk1
h∈H χ
χ
= α0 χ(hk1 )ζ[gl ] = α0 χ(hk1 )ζ[gk +g1 ] .
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Similarly, χ
χ
σχ (fj[gk ] )ζ[gk ] = β0 χ(hk2 )ζ[gk +g2 ] . So the representations σχ all act on 2 (G/H ) with the same functions i, j , but with different constants. Since H is finite, so is G; and we may write G = Z2 /K where K is a subgroup of finite index. Then G/H = Z2 /H K =: Z2 /L. We wish to calculate the character ψχ on L which distinguishes σχ . Lemma 5.2. Let σ be a group construction representation on 2 (Z2 /K) with symmetry group ˆ Then the summands σχ are determined L/K, and scalars determined by the character ψ ∈ K. by the set of characters ψχ ∈ Lˆ satisfying ψχ |K = ψ. The enumeration is given by elements which are related by ψχ (l) = ψ0 (l)χ(l + K); and this enumerates all possible of Hˆ = L/K extensions of ψ from K to L. ˙χ ˙χ Proof. For each l ∈ L, there is a unique word wl ∈ F+ θ of degree l so that σχ (wl )ξ0 = ξ0 ; and then χ
χ
σχ (wl )ξ0 = ψχ (l)ξ0 . When k ∈ K, one has σ (wk )ξh = ψ(k)ξh
for all h ∈ H.
Therefore it follows that χ
χ
σχ (wk )ξ0 = ψ(k)ξ0
for all h ∈ H.
So ψχ |K = ψ . In general, we fix an extension ψ0 of ψ to Z2 and use this to calculate ψχ as in Theorem 5.1. For l ∈ L, let hl := l + K ∈ L/K = H . 1 σ (wl ) ψχ (l) = χ(h1 )ξh1 , χ(h2 )ξh2 |H | h1 ∈H
=
1 |H |
h2 ∈H
χ(h1 )χ(h2 )ψ0 (l) ξh1 +hl , ξh2
h1 ∈H h2 ∈H
= χ(hl )ψ0 (l). Hence we obtain the desired relationship between ψ0 and ψχ . Note that the definition of the subspaces Mχ depends on the choice of ϕ, but that the decomposition is unique. All extensions of ψ occur in this manner, so we have enumerated all possibilities. 2 Lastly we explain why these summands are irreducible. Clearly, if H = {0}, the representation is reducible because we have exhibited a non-trivial collection of reducing subspaces. The
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restriction to each of these subspaces yields a representation of G/H on 2 (G/H ). By construction, the symmetry group of each of these representations is H /H = {0}. So irreducibility follows from the following lemma. 2 Lemma 5.3. If a group construction representation of F+ θ on (G) has symmetry group {0}, then it is irreducible.
Proof. The complete lack of symmetry means that for any element g ∈ G \ {0}, there is a word w ∈ F+ θ so that σ (w)ξ0 = 0 and σ (w)ξg = 0. Indeed, if we write G = {g0 = 0, g1 , g2 , . . .}, there are words wk so that σ (wk )ξ0 = 0 and σ (wk )ξgi = 0
for 1 i k.
Thus Pk = σ (wk )∗ σ (wk ) are projections such that Pk ξ0 = ξ0 for all k 1 and Pk ξgi = 0 when k i. Therefore WOT-lim Pk = ξ0 ξ0∗ . ∗ For any gi ∈ G, there are words wi , xi ∈ F+ θ so that σ (wi ) σ (xi )ξ0 = ξgi . Set Vi := ∗ ∗ ∗ ∗ σ (wi ) σ (xi ). Then Vi ξ0 ξ0 Vj = ξgi ξgj . This is a complete set of matrix units for the compact operators in the von Neumann algebra generated by σ (F+ θ ). Therefore σ is irreducible. 2 A similar analysis works in the case of infinite groups. Of course, there are no subspaces corresponding to the representations σχ ; but the procedure is just a measure theoretic version of the same. Putting all of this together, we obtain the following decomposition. 2 Theorem 5.4. Let σ be a group construction representation of F+ θ on G = Z /K. If the symmetry group H is finite, then σ decomposes as a direct sum of irreducible atomic representations σχ on 2 (G/H ), one for each χ ∈ Hˆ . If H is infinite, then σ decomposes as a direct integral over the dual group Hˆ of irreducible atomic representations σχ on 2 (G/H ). The representations σχ all have the same graph, and the scalars are given by all possible extensions of the character ψ on K to L = H K.
Further considerations. Because of Dilation theorem 2.5, we know that an atomic ∗-dilation is uniquely determined by its restriction to any cyclic coinvariant subspace. In the cases examined in this section, one can often select a smaller subspace which will suffice. In the case of a finite group, the subspace has no proper subspace which is cyclic and coinvariant. But when the group is infinite, there are many such subspaces—and none are minimal. For example, in Example 3.2 we saw that the restriction of a representation on Z2 to (−N0 )2 is such a subspace. Indeed, if the subspace is spanned by standard basis vectors, then whenever if contains ξs0 ,t0 , it must contain all ξs,t for s s0 and t t0 . Likewise, if G = Z2 /Z(p, q) with pq 0, then span{ξ[s,t] : s s0 , t t0 } is a proper cyclic coinvariant subspace. In the case pq > 0, it is easy to see that there is no proper cyclic invariant subspace. However we shall also see that Z(p, q) with pq > 0 can never be the full symmetry group of any representation of F+ θ . This discussion suggests that we need to put an equivalence relation on these representations. It is evident that two equivalent representations must have unitarily equivalent ∗-dilations. 2 Definition 5.5. Two group construction representations σ and σ of F+ θ on G = Z /K, with data {α0 , β0 , ig , jg : g ∈ G} and {α0 , β0 , ig , jg : g ∈ G}, respectively, are said to be equivalent if there ˆ so that is an integer T and a character χ ∈ G
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α0 = χ(g1 )α0 , ig = ig
β0 = χ(g2 )β0 ,
and jg = jg
for all g = [s, t], s, t T .
Conversely, a group-like construction which is defined only on a coinvariant cyclic atomic subspace span{ξg : g = [s, t], s, t T } may be extended to the full group, usually in many ways. To see this, take the minimal ∗-dilation π . There are two cases, G = Z2 and G = Z2 /Z(p, q) with pq 0. In the first case, take the basis vector ξT ,T and any infinite word . . . fjk eik . . . fjT eiT . Define ξk,k = π(fjk−1 eik−1 . . . fjT eiT )ξT ,T . The minimal coinvariant subspace Mk generated by ξk,k can be identified with an atomic basis {ξg : g = [s, t], s, t k}. These subspaces are nested, and their union yields a defect free atomic representation on 2 (G) as desired. In the case of G = Z2 /Z(p, q) with pq 0, we may suppose that q = 0. Then using an infinite word . . . eik . . . eiT will work in exactly the same way. The other issue to discuss here is how the decomposition of the ∗-dilation corresponds to the decomposition of the restriction to the coinvariant subspace. This is a direct consequence of the uniqueness of minimal dilations. Indeed in the case of a representation which decomposes into a direct sum of irreducible representations, the direct sum of the ∗-dilations of the summands is clearly a minimal ∗-dilation. Hence it is the unique ∗-dilation. That is, the ∗-dilation of a direct sum is the direct sum of the ∗-dilations of the summands. 2 Theorem 5.6. Let σ be a group construction representation of F+ θ on (G) with symmetry group H . Then the minimal ∗-dilation decomposes as a direct integral of the ∗-dilations of the irreducible integrands in the direct integral decomposition of σ .
Proof. Let π denote the minimal ∗-dilation of σ on K. We first show that there is a spectral measure on K over measurable subsets of Hˆ which is absolutely continuous with respect to Haar measure and extends the spectral measure on 2 (G). Write σ as a direct integral over Hˆ . For each measurable subset A ⊂ Hˆ , let E(A) be the spectral projection onto span{Mχ : χ ∈ A}. The restriction σA of σ to this subspace has a minimal ∗-dilation πA . By uniqueness, π πA ⊕ πAc . This decomposition splits K F (A)K ⊕ F (Ac )K. It is routine to check that F is countably additive, that E(A) = PH F (A) = F (A)PH , and that F is absolutely continuous. The rest follows from standard arguments. Since we do not actually need any explicit formulae for the direct integral decomposition for any of our analyses, we will not subject the reader to the technicalities. 2 The main theorem. The central result of this paper is the following: Theorem 5.7. Every atomic ∗-representation of F+ θ with connected graph is the minimal ∗dilation of a group construction representation. It is irreducible if and only if its symmetry group is trivial. In general, it decomposes as a direct sum or direct integral of irreducible group construction representations. The proof evolves from a case by case analysis of the various types.
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6. Finitely correlated atomic representations In this section, we restrict our attention to atomic ∗-representations of F+ θ which are finitely correlated. Such representations are particularly tractable. As in the case of the free semigroup [5], the whole class of finitely correlated ∗-representations may turn out to be classifiable. This general problem is not considered here. We assume that the graph is connected. By the discussion in Section 4, we obtain a coinvariant cyclic subspace spanned by the ring by ring portion of the graph. We wish to show that it always arises from a group construction. To this end, we need a criterion for when this construction is possible. ˙ Lemma 6.1. Let ρ be an atomic representation of F+ θ . If ξ is a vertex of Gρ and u, v are words ˙ ˙ ˙ such that ρ(eu )ξ = ξ = ρ(fv )ξ , then eu fv = fv eu . Proof. There are words u and v so that eu fv = fv eu . So ρ(fv eu )ξ˙ = ξ˙ = ρ(fv )ξ˙ . Since |v | = |v|, ρ(fv ) would have range orthogonal to ρ(fv ) unless v = v. Therefore they must be equal. Similarly u = u. 2 Lemma 6.2. Suppose that eu0 fv0 = fv0 eu0 where |u0 | = k and |v0 | = l. Then there is an atomic defect free representation σ on Ck × Cl with σ (eu0 )ξ0 = ξ0 = σ (fv0 )ξ0 . Arbitrary constants α, β ∈ T yield a representation σα,β given by σα,β (ei ) = ασ (ei ) and σα,β (fj ) = βσ (fj ). Then σα,β σα ,β if and only if α k = α k and β l = β l . Proof. Write u0 = ik−1,0 . . . i0,0 and v0 = j0,l−1 . . . j0,0 . The commutation relations show that there are unique words ut for 0 t l, so that fv0 eu0 = fj0,l−1 . . . fj0,t eut fj0,t−1 . . . fj0,0 . Write ut = ik−1,t . . . is,t . . . i0,t for 0 t l and note that ul = u0 . Similarly, there are unique words vs so that fv0 eu0 = eik−1,0 . . . eis,0 fvs eis−1,0 . . . ei0,0 . Write vs = js,l−1 . . . js,0 for 0 s k; and again one has vk = v0 . It follows from unique factorization that fv0 eu0 = eik−1,0 . . . eis+1,0 fjs+1,l−1 . . . fjs+1,t eis,t fjs,t−1 . . . fjs,0 eis−1,0 . . . ei0,0 = eik−1,0 . . . eis+1,0 fjs+1,l−1 . . . fjs+1,t+1 eis,t+1 fjs,t . . . fjs,0 eis−1,0 . . . ei0,0 . Now cancellation shows that fjs+1,t eis,t = eis,t+1 fjs,t
for all s ∈ Ck and t ∈ Cl .
These are the relations needed to allow the construction of a homomorphism on 2 (Ck × Cl ) as in Section 5. Namely,
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σα,β (eis,t )ξs,t = αξs+1,t
and σα,β (fjs,t )ξs,t = βξs,t+1 .
One calculates that σα,β (eu0 )ξ0,0 = α k ξ0,0 and σα,β (fv0 )ξ0,0 = β l ξ0,0 . Clearly σα,β is also a defect free atomic representation for any α, β ∈ T. Indeed, scalars can be assigned to each edge arbitrarily; but by Theorem 5.1, there is no loss in making the constants all the same. The characters of Ck × Cl have the form χ(s, t) = ω1s ω2t where ω1k = 1 = ω2l . Thus Theorem 5.1 also shows that σα,β σα ,β if and only if α = χ(g1 )α = ω1 α and β = χ(g2 )β = ω2 β; and this is equivalent to α k = α k and β l = β l . 2 Corollary 6.3. Let H be the symmetry subgroup for the representation σα,β constructed in Lemma 6.2. For any subgroup K H , there is a group construction representation on 2 (G/K) such that σ (eu0 )ξ0 = α k ξ0
and σ (fv0 )ξ0 = β l ξ0 .
This representation decomposes as a direct sum of irreducible atomic defect free representations on 2 (G/H ) indexed by H /K. Proof. It is evident that the induced representation on 2 (G/K) fits the conditions of Section 5. The symmetry group is clearly H /K. Since H /K is finite, Decomposition theorem 5.4 yields a finite direct sum of irreducible representations on 2 (G/H ). 2 We are now ready to establish the following result. Theorem 6.4. Any defect free atomic representation of F+ θ on a finite-dimensional space with connected graph is isometrically isomorphic to a dilation of a group construction representation for a finite group G. Proof. Let σ be a given finitely correlated defect free representation on a finite-dimensional Hilbert space H, and with a connected graph. Pulling back along the blue (respectively red) edges eventually reaches a periodic state in the ring by ring portion of the representation. Fix a standard basis vector ξ0 in the ring by ring. One can find the unique minimal words u0 and v0 so that eu0 ξ˙0 = ξ˙0 and fv0 ξ˙0 = ξ˙0 . Then eu0 fv0 = fv0 eu0 by Lemma 6.1. For each (s, t) ∈ (N0 )2 , there is a unique word ws,t ∈ F+ θ of degree (s, t) such that σ (ws,t )ξ˙0 =: ξ˙s,t = 0. In particular, wk,0 = eu0 and w0,l = fv0 . From the uniqueness of this path, ws+ak,t+bl = ws,t eua0 fvb0 . So ξ˙s+ak,t+bl = ξ˙s,t ; thus the set of vectors ξs,t is periodic, and so may be indexed by an element g of Ck × Cl , say ξg . Let K denote the set K = {g ∈ Ck × Cl : ξ˙g = ξ˙0 }. Clearly K is closed under addition, and hence is a subgroup of Ck × Cl . The cosets each determine a distinct basis vector. So the subspace H0 spanned by all the basis vectors ξs,t in the ring by ring portion of the graph is naturally identified with 2 (G), where G = Ck × Cl /K. Observe that H0 is a cyclic coinvariant subspace for σ . Thus there is a setup exactly as in Section 5 using the group G, with the functions i, j , α and β determined by the action of σ on the basis {ξg : g ∈ G} for H. The compression σ of σ to H0 is therefore unitarily equivalent to this group construction. The representation σ has a unique
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minimal ∗-dilation π , and this evidently coincides with the minimal ∗-dilation of σ . Hence there is a subspace of the ∗-dilation of σ which corresponds to H, exhibiting σ as a dilation of σ . 2 By Theorem 6.4 one can now easily obtain the following. It is sufficient to decompose the graph into connected components. Corollary 6.5. Any finitely correlated atomic ∗-representation of F+ θ is unitarily equivalent to the direct sum of irreducible atomic ∗-representations which dilate group construction representations on finite abelian groups. Example 6.6. Let us take another look at Example 3.1. Let θ be a permutation in Smn , and fix a cycle of θ : (i0 , j0 ), (i1 , j1 ), . . . , (ik−1 , jk−1 ) . Let G = Ck with g1 = 1 and g2 = −1. Set i(g) = ig and j (g) = jg−1 for g ∈ G. Use this to define a representation ρα,β . Then at each vertex ξg , ρα,β (fjg eig )ξg = αρα,β (fjg )ξg+1 = αβξg = βρα,β (eig−1 )ξg−1 = ρα,β (eig−1 fjg−1 )ξg . Thus the commutation relations show that this is a representation. It is not difficult to see that there are no symmetries. So it is irreducible. Let u0 = ik−1 . . . i1 i0 and v0 = j0 j1 . . . jk−1 . These elements satisfy the identities ρα,β (eu0 )ξ0 = α k ξ0 and ρα,β (fv0 )ξ0 = β k ξ0 . It is easy to check that eu0 fv0 = fv0 eu0 . Therefore we may consider the atomic representation σ on Ck × Ck given by Lemma 6.2. It is easy to obtain, for 0 s < k, that us = ik−1−s . . . i1 i0 ik−1 . . . ik−s
and vs = js js+1 . . . jk−1 j0 . . . js−1 .
It follows that the subgroup of symmetries includes H = (0, 0), (1, 1), . . . , (k − 1, k − 1) . However, as this comes from a cycle of θ , a little thought shows that there are no other symmetries. Compute Ck × Ck /H Ck . The characters of Ck are given by χ(1) = ω where ωk = 1. Therefore by Theorem 5.4, we can decompose σ ωk =1 ρω,ω . Long commuting words. We now show that there are many infinitely many finitely correlated representations for any F+ θ by exhibiting arbitrarily long primitive commuting words. Proposition 6.7. For any given F+ θ , there are commuting pairs eu and fv which determine irreducible atomic representations of arbitrarily large dimension.
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Proof. Consider the two sets of words U := {eu : |u| = N !} and V := {fv : |v| = N !}. Since |U| = mN ! and |V| = nN ! , for simplified notation, we write U = {u1 , . . . , umN! } and V = {v1 , . . . , vnN! }. Given ui ∈ U and vj ∈ V, the relation θ uniquely determines ui ∈ U and vj ∈ V such that eui fvj = fvj eui . We obtain a permutation θ of mN ! × nN ! so that θ (ui , vj ) = (ui , vj ). If eu fv = fv eu for all u ∈ U and v ∈ V, then pick any primitive words u ∈ U and v ∈ V. Then eu fv = fv eu leads to an atomic representation on a quotient G of CN ! × CN ! by the symmetry subgroup H . Since u has no symmetries, (s, 0) ∈ / H for any 1 s < |u|. So |G| N !. Otherwise, θ has a cycle, say C, of length t 2. Without loss of generality, we can assume that C = ((u0 , v0 ), . . . , (ut−1 , vt−1 )). As in Example 6.6, we obtain a representation ρ on 2 (CN !t ). Therefore N! does not belong to the symmetry subgroup H . But H = p where p = 0 or p divides N!t. The t pairs (ui , vi ) are all distinct, and so at least one of the words U or V is not N !-periodic. So p does not divide N !. Therefore p N + 1 implying |CN !t /H | N + 1. Consequently there is an irreducible representation on a space of finite dimension greater than N . 2 Example 6.8. Consider the forward 3-cycle algebra of Example 2.1 given by the permutation ((1, 1), (1, 2), (2, 1)) in S2×2 . The relations have the succinct form fj ei = ei+j fi where addition is modulo 2. Observe that fik ei1 ,i2 ,...,ik = eik +i1 ,i1 +i2 ,...,ik−1 +ik fik , fik−1 +ik eik +i1 ,i1 +i2 ,...,ik−1 +ik = eik−1 +2ik +i1 ,ik +2i1 +i2 ,...,ik−2 +2ik−1 +ik fik−1 +ik , .. . f (n−1)i p
k−p
e (n−1)i p
1−p ,...,
= e (n )i1−p ,..., (n )ik−p f (n−1)i . (n−1 k−p p p p p )ik−p
n The binomial sums are over p 0. If we take k = l = 2n − 1, then 2 p−1 ≡ 1 for all p. So in order that the last e term equal the original, it suffices that ls=1 is ≡ 0. The f word is then uniquely determined so that fv eu = eu fv . For example, f1222212 e1121212 = e1121212 f1222212 . It follows that there are primitive commuting pairs of arbitrarily great length. 7. The ring by tail case Now consider the case 2a, the ring by tail type in which the blue components are ring representations and the red components are infinite tail representations. To simplify the presentation, we introduce some notation. For a word u = ik−1 . . . i0 of length k 1, define u(s, 0] := is−1 . . . i0
and u(s) := is−1 . . . i0 ik−1 . . . is
for 0 s < k,
and observe that u(k) = u. Also if v = v−1 v−2 v−3 . . . is an infinite word, for 0 t < t write v(−t, −t ] = v−t−1 . . . v−t
and v(−t, −∞) = v−t−1 v−t−2 v−t−3 . . . .
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Start with a basis vector, ξ0,0 say, in one of the blue rings of minimal size k. One can find a unique word u0 = ik−1,0 . . . i0,0 so that ρ(eu0 )ξ˙0,0 = ξ˙0,0 . Choose a scalar α so that ρ(eu0 )ξ0,0 = α k ξ0,0 . For 1 s k − 1, renormalize the basis vectors so that ξs,0 = α s ρ(eu0 (s,0] )ξ0,0 . Then ρ(eis,0 )ξs,0 = αξs+1,0
for s ∈ Ck .
Pull back along the red edges from ξ0,0 to obtain an infinite red tail v0 := j0,−1 j0,−2 . . . and basis vectors ξ˙0,t for t < 0 satisfying ρ(fj0,t )ξ˙0,t = ξ˙0,t+1
for t < 0.
Normalize the basis vectors so that ρ(fj0,−t )ξ0,−t = ξ0,1−t for t 1. There is a unique word ut = ik−1,t . . . i0,t of length k for each t < 0 so that eu0 fv0 (0,t] = fv0 (0,t] eut . It follows that ρ(eut )ξ˙0,t = ξ˙0,t ; and hence one can deduce that ρ(eut )ξ0,t = α k ξ0,t for t < 0. Define ξs,t = α s ρ(ut (s, 0])ξ0,t . Then ρ(eis,t )ξs,t = αξs+1,t
for s ∈ Ck and t 0.
Similarly one can pull back along the red edges from ξs,0 to obtain an infinite word vs := js,−1 js,−2 . . . for 1 s k − 1. Again using the commutation relations, one obtains that ρ(fjs,t )ξs,t = ξs,t+1 (s)
It is also easy to verify that the cycles ut
satisfy
ρ(eu(s) )ξs,t = α k ξs,t t
for t < 0 and s ∈ Ck .
for t 0 and s ∈ Ck .
It is evident that the vectors {ξs,t : 0 s < k and t 0} span a coinvariant subspace. By the connectedness of the graph, it is also a cyclic subspace. Thus this subspace determines the representation by the uniqueness of the isometric dilation in Theorem 2.5. There are only mk words of length k in m letters. So by the pigeonhole principle, there is some word u which is repeated infinitely often in the sequence {ut : t 0}. Without loss of generality, we may assume that there is a sequence t0 = 0 > t1 > t2 > · · · such that utk = u0 for k 1. Then eu0 fv0 (0,tk ] = fv0 (0,tk ] eu0
for k 1.
Conversely, given a word u0 , α ∈ T, an infinite tail v0 = j0,−1 j0,−2 . . . and a sequence 0 < t1 < t2 < · · · such that eu0 fv0 (0,−tk ] = fv0 (0,−tk ] eu0 for k 1, one can build a representation of type case 2a. We will do this by our group construction by extending the definition of this subspace indexed by Ck × −N0 to the group G = Ck × Z. This may be accomplished in many ways, but a simple way is to make it periodic on Ck × N0 by repeating the segment on Ck × [0, t1 ) using the fact that ut1 = u0 and eu0 fv0 (0,t1 ] = fv0 (0,t1 ] eu0 . The finitely correlated representation of Lemma 6.2 can be unfolded to obtain a representation on Ck × Z. Just cut off the left half
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of the cylinder and glue it to the one we have. Any choice of extension is tail equivalent to any other. Note that this analysis is necessary, and it is not the case that an arbitrary tail eu0 and infinite word fv0 determines a representation. For example, this may force a red edge and blue edge into the same vertex which is not possible from the commutation relations. We have obtained: Theorem 7.1. A ring by tail representation is determined by a word u0 of length k, a scalar α ∈ T, an infinite word v0 := j0,−1 j0,−2 . . . and a sequence 0 > t1 > t2 > · · · such that eu0 fv0 (0,tk ] = fv0 (0,tk ] eu0 for k 1. It corresponds to a group construction on the group Ck × Z. Since a ring by tail representation is given by the group construction on G = Ck × Z, we can use the decomposition results of Section 5. Theorem 7.2. Let π be a (connected) ring by tail representation with symmetry subgroup H Ck × Z. If H Ck × {0}, then π decomposes as a finite direct sum of irreducible ring by tail representations. Otherwise, it decomposes as a direct integral of irreducible ring by ring atomic representations. Proof. In the first case, H = (d, 0) is a subgroup of Ck × {0}. Since H is finite, Theorem 5.4 splits π as a direct sum of finitely many irreducible representations for the quotient group G/H Ck/d × Z. So these are irreducible ring by tail representations. In the second case, H contains an element (a, b) with b = 0. Therefore H contains the element (0, kb). Thus G = Ck × Z/H is a quotient of Ck × Ckb ; and in particular, G is finite. Since H is infinite, Theorem 5.4 yields a direct integral of irreducible representations on 2 (G). Evidently, these are ring by ring type. 2 The case 2b is handled by exchanging the role of the red and blue edges in the case 2a. 8. The tail by tail case In the case 3, both the red and the blue components are infinite tail representations. The most basic is the case 3a, in which the red tail never intersects the original blue component again. Case 3a. As we saw in Example 3.2, the 3a case is an inductive limit of copies of the left regular representation. So in a certain sense, they are the easiest. Given a representation of type 3a, start with any basis vector ξ0,0 . Pull back on both blue and red edges to determine integers is,t and js,t and basis vectors ξ˙s,t for s, t 0 so that π(eis,t )ξ˙s−1,t = ξ˙s,t
and
π(fjs,t )ξ˙s,t−1 = ξ˙s,t
for s, t 0.
The assumption of the case 3a ensures that the ξ˙s,t are all distinct. Thus we have found a cyclic coinvariant subspace M = span{ξs,t : s, t 0} on which we have a representation on (−N0 )2 . The representation is determined by the tail
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τ = e0,0 f−1,0 e−1,−1 f−2,−1 . . . since the other is,t and js,t for s, t 0 are determined by the commutation relations. One could use Theorem 5.1 to make the scalars all equal to 1; but in fact that is not necessary. Instead one selects the appropriate unit vector ξs,t in ξ˙s,t recursively so that π(eis,t )ξs−1,t = ξs,t
and
π(fjs,t )ξs,t−1 = ξs,t
for s, t 0.
Example 3.2 explains how to dilate this to a ∗-representation which is an inductive limit of copies of the left regular representation. By Theorem 2.5, this is the unique minimal ∗-dilation; and hence it is π . While the infinite tail is sufficient to describe the representation, as is done in Example 3.2, it is not unique, and the equivalence relation on tails is not at all transparent. A much more useful collection of data associated to τ is the set Σ(π, ξ0,0 ) = Σ(τ ) = (is,t , js,t ): s, t 0 . In Definition 3.3, we put an equivalence relation of shift tail equivalence on these sets. Definition 8.1. For each inductive ∗-representation π , define Σ(π) to be the equivalence class of Σ(π, ξ ) modulo shift tail equivalence for any standard basis vector ξ ∈ Hπ . That this definition makes sense is part of the following result. Theorem 8.2. If π is an inductive (type 3a) atomic ∗-representation, then Σ(π) is independent of the choice of initial vector. Two inductive ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ). Proof. Start with two standard basis vectors, ξ0,0 and ζ0,0 . The connectedness of the graph means that there is a path from ξ0,0 to ζ0,0 . By Lemma 3.4, there is a path from ξ0,0 to ζ0,0 of the form uv ∗ . Let d(v) = (s1 , t1 ) and d(u) = (s2 , t2 ). Then ξ˙−s1 ,−t1 = ζ˙−s2 ,−t2 . Therefore the data agrees on all basis vectors obtained by pulling back from that common vector. So the two data sets are (s1 − s2 , t1 − t2 )-shift tail equivalent. Clearly, if two inductive ∗-representations σ1 and σ2 have shift tail equivalent data Σ(πi , ξi ), then they are unitarily equivalent. Consider the converse. Suppose that π1 and π2 are unitarily equivalent, say via a unitary U in B(Hπ1 , Hπ2 ). Fix the basis vector ξ0,0 for π1 , and corresponding basis ξs,t for s, t 0 and data Σ(π1 , ξ0,0 ). From the unitary equivalence, ξ0,0 is identified with a vector η0,0 = U ξ ∈ Hπ2 . Write η in the standard basis for π2 , say η = ai ζi ; and choose a standard basis vector ζ = ζi0 for which ai0 = 0. For any s, t 0, there is a word ws,t ∈ F+ θ with d(ws,t ) = (|s|, |t|) so that π1 (ws,t )ξs,t = ξ0,0 . Therefore η0,0 = π2 (ws,t )U ξs,t is in the range of the partial isometry π2 (ws,t ). Since π2 is atomic, the range of π2 (ws,t ) is spanned by standard basis vectors. Consequently ζ is in the range of π2 (ws,t ) for every s, t 0. Restating this another way, it says that Σ(π2 , ζ ) = Σ(π1 , ξ0,0 ). Hence Σ(π1 ) = Σ(π2 ). 2
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Example 8.3. It is tempting to think that an inductive ∗-representation can be determined by the blue and red components containing a standard basis vector. That is, start at a basis vector ξ0,0 . Pull back along the blue edges to obtain the infinite word u = i−1 i−2 . . . determining the blue infinite tail component containing ξ0,0 . Likewise pull back along the red edges to get a red infinite tail v = j−1 j−2 . . . determining the red component containing ξ0,0 . The infinite words u and v may not determine the representation uniquely! Consider the permutation ((1, 1), (1, 3))((1, 2), (2, 1)) in S3×3 with two 2-cycles; and the two infinite words u = e1 e3 e3 e3 . . . and v = f1 f3 f3 f3 . . . . Two inequivalent ∗-representations π1 and π2 will be constructed to produce a coinvariant subspace spanned by vectors ξ−s,−t for s, t 0 with πi (e1 )ξ−1,0 = ξ0,0 ,
πi (e3 )ξ−s−1,0 = ξ−s,0
πi (f1 )ξ0,−1 = ξ0,0 ,
πi (f3 )ξ0,−t−1 = ξ−0,−t
for s 1, for t 1.
So they have the same infinite blue and red components. We define in addition π1 (e3 )ξ−s−1,−t = ξ−s,−t π1 (e2 )ξ−1,−t = ξ0,−t π1 (f3 )ξ−s,−t−1 = ξ−s,−t π1 (f2 )ξ−s,−1 = ξ−s,0
for s 1 and t 0, for t 1, for s 0 and t 1, for s 1,
and π2 (e3 )ξ−s−1,−t = ξ−s,−t π2 (e1 )ξ−1,−t = ξ0,−t π2 (f1 )ξ−s,−t−1 = ξ−s,−t π2 (f3 )ξ−s,−1 = ξ−s,0
for s 1 and t 0, for t 1, for s 0 and t 1, for s 1.
These two ∗-representations are evidently not shift tail equivalent. Thus they are not unitarily equivalent. This means that these representations are not some form of twisted product of an infinite tail representation for Am with an infinite tail representation for An . Case 3b. Now suppose that we have a ∗-representation π of type 3b. Pulling back along the red edges yields a periodic sequence of blue components of infinite tail type, say H0 , . . . , Hl−1 , so that every red edge into each Hi comes from Hi−1 (mod l) . Start with a basis vertex ξ˙ . Let v be the unique word of length l so that fv maps onto ξ˙ . Then there is a vertex ζ˙ ∈ H0 so that π(fv )ζ˙ = ξ˙ . Our first goal is to explain why these two vertices are comparable in H0 if ξ˙ is sufficiently far up the tail.
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Lemma 8.4. Suppose that π is a type 3b atomic ∗-representation in which blue components H0 , . . . , Hl−1 are periodic of period l. There is a vertex ξ˙0 in H0 and a word u0 so that the vertex ζ˙0 on H0 obtained by pulling back l red steps from ξ˙0 via a word v0 satisfies either π(fv0 )ζ˙0 = ξ˙0
and π(eu0 )ξ˙0 = ζ˙0
(3bi)
or π(fv0 )ζ˙0 = ξ˙0 = π(eu0 )ζ˙0 .
(3bii)
This same type of relationship persists for each vertex obtained from pulling back along every path leading into ξ˙0 . Proof. Since H0 is an infinite tail representation of Am , there is a vertex ζ˙0 in H0 and words u1 and u2 so that π(eu1 )ζ˙0 = ξ˙
and π(eu2 )ζ˙0 = ζ˙ ;
and thus π(fv eu2 )ζ˙0 = ξ˙ . Use the commutation relations to write fv eu2 = eu2 fv0 . Set ξ˙0 = π(fv0 )ζ˙0 . Then π(eu1 )ζ˙0 = ξ˙ = π(fv eu2 )ζ˙0 = π(eu )π(fv0 )ζ˙0 = π(eu )ξ˙0 . 2
2
It follows that the vertices ζ˙0 and ξ˙0 are obtained from ξ˙ by pulling back along the blue edges |u1 | and |u2 | = |u2 | steps, respectively. So they are comparable, and the relationship depends on whether |u2 | is less than, equal to or greater than |u1 |. If |u2 | < |u1 |, then uniqueness of the pull back along blue edges means that u1 = u2 u0 . So π(u0 )ζ˙0 = ξ˙0 . Similarly, if |u2 | > |u1 |, then u2 = u1 u0 and π(eu0 )ζ˙0 = ξ˙0 . In the case |u2 | = |u1 |, we have ξ˙0 = ζ˙0 and so ξ˙0 = π(fv0 )ξ˙0 . This is the tail by ring case, which has been excluded. In the first case, suppose that η˙ is any vertex in these l components which is obtained by ˙ pulling back from ξ0 . That is, there is a word w ∈ F+ θ so that π(w)η˙ = ξ0 . There is a basis vector ζ˙ in the same component and a word v with |v | = l so that π(fv )ζ˙ = η. ˙ From the commutation relations, there are words v of length l and w of degree d(w) so that wfv = fv w . Therefore, π(fv )π(w )ζ˙ = ξ˙0 = π(fv0 )ζ˙0 . Uniqueness implies that v = v0 and π(w )ζ˙ = ζ˙0 . Now factor eu0 w = w eu with |u | = |u0 | and d(w ) = d(w ). Then ξ˙0 = π(eu0 w )ζ˙ = π(w )π(eu )ζ˙ . Since w is a word with the same degree, say (s, t), as w and w, we deduce that π(eu )ζ˙ is the unique vertex obtained by pulling back from ξ˙0 by s blue and t red edges, namely η. ˙ That is, π(eu )ζ˙ = η; ˙ and ζ˙ lies above η˙ in its blue component. The other case is handled in a similar manner. 2
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We now proceed as in the case 3a. Start with the basis vertex ξ˙0,0 in H0 provided by Lemma 8.4. Pull back on both blue and red edges to determine integers is,t and js,t for s, t 0 and vertices ξ˙s,t so that π(eis,t )ξ˙s−1,t = ξ˙s,t
and
π(fjs,t )ξ˙s,t−1 = ξ˙s,t
for s, t 0.
The difference in the case 3b is that there is periodicity. In the case 3bi, there are unique words u0 of length k and v0 of length l so that π(fv0 eu0 )ξ˙0 = ˙ξ0 . By Lemma 8.4, for each vertex ξs,t , pulling back k blue edges and l red edges will return to the same vertex. That is, ξ˙s−k,t−l = ξ˙s,t for all s, t 0. So there is Z(k, l) periodicity. This allows us to extend the definitions to all of Z2 using the periodicity; and to then collapse this to a representation on Z2 /Z(k, l). Write ξ[s,t] or ξg for the vector associated to an element g = [s, t] coming from the equivalence class of (s, t). It is now easy to see that the vectors ξg are distinct because one can always choose the representative (s, t) with 0 t < l, determining the component Ht , and within this component, s determines the position on the infinite tail. Indeed, the (k, l)-periodicity allows us to select a distinguished spine because pulling back l steps along the red edges moves us forward k (specific) blue edges. We shall see soon that in this case, there is always additional symmetry. In the case 3bii, there are unique words u0 of length k and v0 of length l so that π(fv0 )ζ˙0 = π(eu0 )ζ˙0 = ξ˙0,0 . Hence ζ˙0 = ξ˙0,−l = ξ˙−k,0 . By Lemma 8.4, for each vertex ξs,t , pulling back k blue edges or pulling l red edges will result in the same vertex. That is, ξ˙s−k,t = ξ˙s,t−l for all s, t 0. So there is Z(k, −l) periodicity. In this case, there is no canonical way to carry forward. However, as in the previous case, we obtain a parameterization of the basis vectors as a semi-infinite subset of Z2 /Z(k, −l); and we will write ξ[s,t] or ξg for the vector associated to an element g = [s, t] coming from the equivalence class of (s, t) when there is a representative with s, t 0. In both cases, the subspace M = span{ξ[s,t] : s, t 0} is a coinvariant cyclic subspace. So it is sufficient to determine shift tail equivalence. As usual, we define the symmetry subgroup Hπ from the shift tail symmetry of the data Σ(π, ξ0,0 ) = {(i[s,t] , j[s,t] ): s, t 0}, but consider it as a subgroup of Z2 /Z(k, ±l) by modding out by the known symmetry. Let Σ(π) denote the shift tail equivalence class of Σ(π, ξ0,0 ). The following result follows in an identical manner to the case 3a, so it will be stated without proof. The converse to the second statement does not follow because there are issues with the scalars that will be dealt with soon. Theorem 8.5. If π is a type 3b atomic ∗-representation, then Σ(π) is independent of the choice of initial vector. If two type 3b ∗-representations π1 and π2 are unitarily equivalent, then Σ(π1 ) = Σ(π2 ). We now consider these two cases in more detail. Case 3bi. Here we have symmetry Z(k, l) with k, l > 0. In this case, there are words u0 = i[−1,0] . . . i[−k,0] and v0 = j[0,l−1] . . . j[0,0] so that π(eu0 fv0 )ξ˙[0,0] = ξ˙[0,0] .
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The basis ξ˙[s,t] satisfies ξ˙[ks,0] = ξ˙[0,−ls] . Since we can pull back along either blue or red edges, we find that these vectors are defined for all s ∈ Z. Indeed, all ξ˙[s,t] are defined for (s, t) ∈ Z2 . This leads to the important observation about the case 3bi that this representation is determined by u0 and v0 . Lemma 8.6. An atomic ∗-representation of type 3bi is determined by words u0 and v0 and a constant β ∈ T satisfying π(eu0 fv0 )ξ0,0 = βξ0,0 . The symmetry subgroup Hπ is always nonzero. Indeed, there is full symmetry (without tail equivalence), namely there is an integer p > 0 so that i[s−p,t] = i[s,t]
and j[s−p,t] = j[s,t]
for all s, t ∈ Z.
Proof. We can find words u1 , v1 , u−1 and v−1 to factor e u 0 f v0 = f v1 e u 1
and fv0 eu0 = eu−1 fv−1 .
Continuing recursively, we obtain words ur in {1, . . . , m}k and vr in {1, . . . , n}l for r ∈ Z so that eur fvr = fvr+1 eur+1
for all r ∈ Z.
This determines the doubly infinite paths τe = . . . u1 u0 u−1 . . .
and τf = . . . v1 v0 v−1 . . . .
The e’s and f ’s move in opposite directions, and the two spines intersect every k steps forward along the blue path for every l steps backward along the red path. The commutation relations allow us to compute the l infinite blue paths and k infinite red paths, completing the picture of this coinvariant subspace. The commutation of words eu of length k with words fv of length l is given by a permutation θ in Smk ×nl determined by θ so that eu fv = fv eu , where θ (u, v) = (u , v ). The pairs (ur , vr ) therefore satisfy θ (ur , vr ) = (ur+1 , vr+1 ). It follows that the pairs (ur , vr ) move repeatedly through a cycle of the permutation θ . Consequently, the sequence (ur , vr ) is periodic of length p, where p is the length of the cycle. It follows that i[s,t] = i[s+pk,t]
and j[s,t] = j[s,t−pl] = j[s+pk,t] .
Therefore Hπ contains [pk, 0] = [0, −pl]; and thus is a non-trivial symmetry subgroup. Moreover the symmetry is global, not just for s, t T . The scalars are determined by Theorem 5.1. The character ψ of Z(k, l) is given by ψ(k, l) = β where π(eu0 fv0 )ξ0,0 = βξ0,0 . One can extend this to a character on Z2 ; and we will do this by setting ϕ(1, 0) = 1 and ϕ(0, 1) = β0 where β0 is an lth root of β. 2 As an immediate consequence of the previous two results, we obtain: Corollary 8.7. Two type 3bi ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ) and β1 = β2 , where β1 and β2 are the scalars of Lemma 8.6.
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Conversely, we obtain Theorem 8.8. Given words u0 in {1, . . . , m}k and v0 in {1, . . . , n}l and scalar β ∈ T, there is an atomic ∗-representation of F+ θ of type 3bi determined by this data. Proof. The proof of Lemma 8.6 explains how this is done. The pair (u0 , v0 ) lies in a cycle of θ (u0 , v0 ), (u1 , v1 ), . . . , (up−1 , vp−1 ) . So eui fvi = fvi+1 eui+1 for i ∈ Cp . As in the construction of Examples 3.1 and 6.6, we obtain that eup−1 ...u1 u0 commutes with fv0 v1 ...vp−1 . Indeed, one has the useful relations eui fvi vi+1 ...vp−1 v0 ...vi−1 = fvi+1 ...vp−1 v0 ...vi eui . This is the algebraic form of the (k, l)-periodicity. By Lemma 6.2, there is a finitely correlated defect free representation on Cpk × Cpl determined by this commuting pair. The relations above ensure (k, l) periodicity; so there is a corresponding construction on Cpk × Cpl / (k, l) . The idea is to ‘unfold’ this to obtain a 3bi ∗-representation. It is probably easier to envisage unfolding the representation on Cpk × Cpl to a representation on Z2 with pkZ × plZ symmetry, and then observing that the (k, l)-periodicity of the original picture becomes Z(k, l) periodicity of the type 3a representation. So one now goes to the quotient Z2 /Z(k, l) to obtain the desired representation of type 3bi. One can deal with scalars as before. 2 Thus we obtain: Corollary 8.9. Every ∗-representation of type 3bi comes from a group construction for a group of the form Z2 /Z(k, l) with kl > 0. It always has a non-trivial symmetry group, and so is never irreducible. It decomposes as a direct integral of irreducible ∗-representations of type 1. Case 3bii. This is the trickiest case. Here we have Z(k, −l) symmetry, where k, l > 0. That is, our basis is ξ[s,t] for s, t 0 where the equivalence class consists of cosets of Z(k, −l). We have a word u0 of length k and v0 of length l so that π(eu0 )ξ˙[−k,0] = π(fv0 )ξ˙[−k,0] = ξ˙[0,0] . Thus there is a scalar β ∈ T so that π(fv0 )ξ[−k,0] = βπ(eu0 )ξ[−k,0] . Pulling back from ξ˙[0,0] along the blue edges yields the infinite tail τe = i[0,0] i[−1,0] i[−2,0] . . . = u0 u1 u2 . . . , where ud = i[dk,0] i[dk−1,0] . . . i[(d−1)k+1,0] for d 0 are the consecutive words of length k. Similarly, pulling back from ξ˙[0,0] along the red edges yields the infinite tail τf = j[0,0] j[0,−1] j[0,−2] . . . = v0 v−1 v−2 . . .
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where vd = i[0,dl] i[0,dl−1] . . . i[0,(d−1)l+1] for d 0 are the consecutive words of length l. Let β0 be chosen so that β0l = β. Then as before, we may normalize the basis vectors so that π(ei[s,t] )ξ[s−1,t] = ξ[s,t]
and π(fj[s,t] )ξ[s,t−1] = β0 ξ[s,t]
for sl + tk 0. Since [dk, 0] = [0, dl], we have π(fvd+1 eud )ξ˙[dk,0] = ξ˙[(d+2)k,0] = π(eud+1 fvd )ξ˙[dk,0] ; and therefore we obtain the commutation relations fvd+1 eud = eud+1 fvd
for d < 0.
This is a rather strong compatibility condition, and suggests why not all pairings are possible. For each d 0, there are at most mk nl possible pairs (ud , vd ). By the Pigeonhole principle, one of these pairs is repeated infinitely often. Without loss of generality, we may suppose that this sequence begins at 0; so that there are integers 0 = d0 > d1 > d2 > · · · so that udr = u0
and vdr = v0
for r 1.
A computation now shows that eu0 u1 ...udr +1 and fv0 v−1 ...vdr +1 commute. Indeed, one readily computes that eui ...uj −1 fvj = fvi eui+1 ...uj
and eui fvi+1 ...vj = fvi ...vj −1 euj .
This encodes the Z(k, −l) symmetry. Repeated application of this yields eu0 u1 ...udr +1 fv0 v−1 ...vdr +1 = eu0 u1 ...udr +2 eudr +1 fvdr fv−1 v−2 ...vdr +1 = eu0 u1 ...udr +2 fvdr +1 eudr fv−1 v−2 ...vdr +1 = fv0 eu1 ...udr +1 eu0 fv−1 v−2 ...vdr +1 = fv0 eu1 ...udr +1 fv0 ...vdr +2 eudr +1 .. . = fv0 v−1 ...vdr +1 eu0 u1 ...udr +1 . Now we can construct a sequence of ring by ring representations on the finite groups Gr = C|dr |k × C|dr |l / (k, −l) by building a ring by ring representation on Gr using these commuting words and the fact that they have the Z(k, −l) symmetry. That is, consider a finitely correlated representation ρr on a basis ζg for g = [s, t] in Gr for dr k s < 0 and 0 t < l by ρr (ei )ζ[s−1,t] = δi is,t ζ[s,t]
and ρr (fj )ζ[s,t−1] = δj js,t β0 ζ[s,t] .
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These observations also provide a way (indeed many ways) to extend the definition of the coinvariant subspace to the full group Z2 /Z(k, l) by making it (d1 k, −d1 l) periodic moving forward. That is, one may define i[s,t] = i[s ,t ]
and j[s,t] = i[s ,t ]
for sk + tl > 0,
where (s , t ) is chosen in (d1 k, 0] × (d1 l, 0] so that s ≡ s (mod d1 k) and t ≡ t (mod d1 l). Lemma 8.10. An atomic ∗-representation of type 3bii is determined by a scalar β ∈ T and two infinite tails τe = u0 u1 u2 . . . and τf = v0 v−1 v−2 . . . , where |ud | = k and |vd | = l, satisfying fvd+1 eud = eud+1 fvd
for d < 0.
Proof. The discussion above shows that β and two infinite words with the desired properties are associated to each ∗-representation of type 3bii. Conversely, suppose that such data is given. Then by Section 6, there are finitely correlated representations ρr as defined above for r 1. The data {(is,t , js,t ): s, t 0} has Z(k, −l) symmetry, and may be extended in a compatible way to all of Z2 /Z(k, −l). Therefore we may construct a representation on Z2 /Z(k, −l) using the group construction of Section 5. 2 Again we consider Σ(π, ξ ) to be defined on the group Z2 /Z(k, −l); and define Σ(π) to be its equivalence class modulo shift tail equivalence. The scalars are determined by Theorem 5.1 by the character ψ on Z(k, −l) given by π(eu0 )π(fv0 )∗ ξ0,0 = ψ(k, −l)ξ0,0 . Combining the previous results yields: Corollary 8.11. Two type 3bii ∗-representations π1 and π2 are unitarily equivalent if and only if Σ(π1 ) = Σ(π2 ) and ψ1 = ψ2 , where ψi is the character on Z(k, −l) determined by πi . Decomposition. It now follows that a type 3 ∗-representation has the form of one of the group constructions. In particular, by Lemma 5.3, a ∗-representation π is irreducible if and only if the symmetry group Hπ is trivial. In general, we obtain a direct integral decomposition into irreducibles. The case 3bi is never irreducible, so these representations decompose as a direct integral of finitely correlated ∗-representations. Theorem 8.12. A (connected atomic) tail by tail ∗-representation π with symmetry group Hπ Gπ decomposes as a direct integral of irreducible atomic ∗-representations dilating a family of representations on 2 (Gπ /Hπ ). Consider the possibilities for the case 3a, the inductive type. The subgroup Hπ = (k, l) for kl > 0 cannot occur, because this would require an irreducible ∗-representation of type 3bi. Corollary 8.13. If π is a tail by tail ∗-representation of inductive type, then the symmetry group is one of the following: (1) Hπ = {(0, 0)} when π is irreducible. (2) Hπ = (k, l) where kl < 0, and π is a direct integral of irreducible 3bii ∗-representations. (3) Hπ = (k, l) where kl = 0, and π is a direct integral of irreducible type 2 ∗-representations.
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(4) Hπ has rank 2, and π is a direct integral of irreducible finitely correlated ∗-representations. It is natural to ask whether F+ θ has irreducible ∗-representations of inductive type. This is equivalent to the existence of an infinite tail without any periodicity. This is exactly the aperiodicity condition introduced by Kumjian and Pask [10]. This property is generic, but there are periodic examples such as the flip semigroup of Example 2.2. In [4], this property is explored in detail. The semigroup F+ θ is either aperiodic, or it has Z(k, −l) periodicity for some kl > 0. This leads to structural differences in C∗ (F+ θ ). It would be interesting to determine whether all aperiodic F+ θ have irreducible ∗-representations of type 3bii. References [1] J. Cuntz, Simple C*-algebras generated by isometries, Comm. Math. Phys. 57 (1977) 173–185. [2] K.R. Davidson, C*-Algebras by Example, Fields Institute Monogr. Ser., vol. 6, Amer. Math. Soc., Providence, RI, 1996. [3] K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semi-group algebras, Proc. London Math. Soc. 78 (1999) 401–430. [4] K.R. Davidson, D. Yang, Periodicity in rank 2 graph algebras, Canad. J. Math., in press. [5] K.R. Davidson, D.W. Kribs, M.E. Shpigel, Isometric dilations of non-commuting finite rank n-tuples, Canad. J. Math. 53 (2001) 506–545. [6] K.R. Davidson, S.C. Power, D. Yang, Dilation theory for rank 2 graph algebras, J. Operator Theory, in press. [7] C. Farthing, P.S. Muhly, T. Yeend, Higher-rank graph C*-algebras: An inverse semigroup and groupoid approach, Semigroup Forum 71 (2005) 159–187. [8] E. Hewitt, K. Ross, Abstract Harmonic Analysis, vol. I, Grundlehren Math. Wiss., vol. 115, Springer, New York, 1963. [9] D.W. Kribs, S.C. Power, The analytic algebras of higher rank graphs, Math. Proc. R. Ir. Acad. 106 (2006) 199–218. [10] A. Kumjian, D. Pask, Higher rank graph C*-algebras, New York J. Math. 6 (2000) 1–20. [11] D. Pask, I. Raeburn, M. Rordam, A. Sims, Rank-two graphs whose C*-algebras are direct limits of circle algebras, J. Funct. Anal. 239 (2006) 137–178. [12] S.C. Power, Classifying higher rank analytic Toeplitz algebras, New York J. Math. 13 (2007) 271–298. [13] I. Raeburn, Graph Algebras, CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., Providence, RI, 2005. [14] I. Raeburn, A. Sims, T. Yeend, Higher-rank graphs and their C*-algebras, Proc. Edinb. Math. Soc. (2) 46 (2003) 99–115. [15] I. Raeburn, A. Sims, T. Yeend, The C*-algebras of finitely aligned higher-rank graphs, J. Funct. Anal. 213 (2004) 206–240. [16] G. Robertson, T. Steger, Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras, J. Reine Angew. Math. 513 (1999) 115–144. [17] A. Sims, Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs, Canad. J. Math. 58 (2006) 1268–1290.
Journal of Functional Analysis 255 (2008) 854–876 www.elsevier.com/locate/jfa
Norm controlled inversions and a corona theorem for H ∞-quotient algebras Pamela Gorkin a , Raymond Mortini b,∗ , Nikolai Nikolski c,1 a Department of Mathematics, Bucknell University, Lewisburg, 17837 PA, USA b Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, F-57045 Metz, France c IMB, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence, France
Received 8 November 2007; accepted 19 May 2008 Available online 13 June 2008 Communicated by N. Kalton
Abstract Let Θ be an inner function on the unit disc D. We give a description of those Θ for which the quotient algebra H ∞ /ΘH ∞ has no corona with respect to the visible part of its spectrum, that is for which ∞ M(H ∞ /ΘH ∞ ) = {z ∈ D: Θ(z) = 0}M(H ) . It happens that this property is equivalent to the norm con∞ ∞ trolled inversion property for H /ΘH , as well as to a kind of weakened Carleson type embedding theorem. The quotient algebra A(D)/ΘH ∞ is also considered. An interpretation of our main results in terms of model operators is given, too. © 2008 Elsevier Inc. All rights reserved. Keywords: H ∞ -quotient algebras; Corona theorems; Inner functions; Weak embedding property; Norm estimates of corona-solutions; Model operators; H ∞ -calculus
1. Introduction The problem we are interested in here is that of efficient inversion, as it is treated in [1,11], and subsequent papers. This problem is described in detail below.
* Corresponding author.
E-mail addresses: [email protected] (P. Gorkin), [email protected] (R. Mortini), [email protected] (N. Nikolski). 1 Partially supported by the Marie Curie Actions European Grant 030042. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.011
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Let A be a commutative unital Banach algebra with unit e and M(A) its maximal ideal space. For a ∈ A, let aˆ denote the Gelfand transform of a. We let ˆ . δ(a) = min a(t) t∈M(A)
Note that δ(a) a ˆ ∞ a. When a = (a1 , . . . , an ) ∈ An we define ˆ , δn (a) = min a(t) t∈M(A)
where |a(t)| ˆ = ( nj=1 |aˆ j (t)|2 )1/2 for t ∈ M(A) and we let a =
n
1/2 aj
2
.
j =1
Let δ be a real number satisfying 0 < δ 1. We are interested in finding, or bounding, the functions c1 (δ) = sup a −1 : a 1, δ(a) δ and
cn (δ) = sup inf b:
n
aj bj = e , a 1, δn (a) δ .
j =1
In fact, often only a part of M(A) is available, say Λ ⊂ M(A). We call this a visible part of M(A). (We will present examples below.) In this case, we would like to bound a −1 in terms of the visible part of the spectral data. Thus, we define the counterpart of δ(a) as ˆ . δ(a, Λ) = inf a(t) t∈Λ
In a similar way, we consider δn (a, Λ). This, in turn, yields modified functions cn (Λ, δ) for n 1 (if it is not readily apparent which algebra we mean, we write cn (A, Λ, δ)). If a is not invertible, we define a −1 = ∞. If δn (a, Λ) δ and a 1, we let b = ∞, if the set of all b = (b1 , . . . , bn ) for which nj=1 aj bj = e is empty. It should be clear that 1 cn (Λ, δ) cn+1 (Λ, δ) and, if 0 < δ δ 1, then cn (Λ, δ) cn (Λ, δ ). This implies the existence of a critical constant, denoted here by δn (A, Λ) (or simply δn (A) if Λ = M(A)) such that cn (Λ, δ) = ∞
for 0 < δ < δn (A, Λ)
and cn (Λ, δ) < ∞ for δn (A, Λ) < δ 1.
If A is a uniform algebra, then δ1 (A) = 0. For elements a with 0 < δn (a, Λ) < δn (A, Λ) we can say that the inversion problem is ill posed in the sense that there is no control of inverses in terms of visible spectral data. For elements with δn (A, Λ) < δn (a, Λ) 1 the problem is said to be well posed; that is, there is such an estimate.
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Let H ∞ be the algebra of bounded holomorphic functions on the unit disc D = {z ∈ C: |z| < 1} endowed with the supremum norm f ∞ = sup|z|<1 |f (z)| and let A(D) = H ∞ ∩ C(D) be the disc algebra. In this paper, given an inner function Θ, we characterize those quotient algebras A = H ∞ /ΘH ∞ or A = A(D)/ΘH ∞ for which the inversion problem is well posed; that is for which δ1 (A, Λ) = 0. Note that we interpret A(D)/ΘH ∞ to mean the canonical image of A(D) in the quotient algebra H ∞ /ΘH ∞ . It is known that A(D)/ΘH ∞ is a closed subalgebra of H ∞ /ΘH ∞ if and only if A(D) + ΘH ∞ is norm closed in H ∞ . It is also an interesting fact [8,17] that the sum C(T) + ΘH ∞ is closed in L∞ (T) if and only if either m(σ (Θ) ∩ T) = 0 or m(σ (Θ) ∩ T) = 1, where m denotes normalized Lebesgue measure on the unit circle T = {z ∈ C: |z| = 1}. Having an inner function Θ and a function f ∈ H ∞ , we define the visible spectrum of an element f + ΘH ∞ in A = H ∞ /ΘH ∞ as the range f (Λ) = f (λ): λ ∈ Λ , where Λ = σ (Θ) ∩ D = {z ∈ D: Θ(z) = 0}, σ (Θ) being the spectrum of Θ (see below). We show (Theorems 3.3 and 3.4) that δ1 (A, Λ) = 0 if and only if δn (A, Λ) = 0 for every n 1, and this happens if and only if Λ is dense in the maximal ideal space M(A). We also characterize the latter property in terms of the canonical factorization Θ = Sμ B, where B(z) =
bλ (z) =
bλj (z),
j 1
|λ| λ − z · λ 1 − λz
is the Blaschke product with the zeros of Θ (repeated according to their multiplicities), and
z+ζ Sμ (z) = exp dμ(ζ ) z−ζ T
stands for the corresponding singular inner function (μ is a measure on the circle T = {z: |z| = 1}, μ 0, singular with respect to Lebesgue measure m). The spectrum of an inner function Θ is defined as C σ (Θ) = supp(μ) ∪ z ∈ D: Θ(z) = 0 . Recall that, following the standard terminology (see [2,7,10]), a function algebra A on a set Λ has no corona if the set Λ is dense in the maximal ideal space M(A). With this notation we show (Theorems 3.3 and 3.4) that the algebra H ∞ /ΘH ∞ , being considered on the set Λ = σ (Θ) ∩ D has no corona if and only if the following weak embedding property (WEP) holds: for every > 0 there exists η > 0 such that
z ∈ D: Θ(z) < η ⊂ z ∈ D: inf bλ (z) < . λ∈Λ
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We call this an embedding property because the latter condition is related to the famous Carleson embedding property (CEP) appearing in the Carleson interpolation theorem. Namely, the Carleson embedding means that (1 − |λj |2 )(1 − |z|2 ) j 1
|1 − λj z|2
C
for every z ∈ D (see [10, p. 151]), whereas for a Blaschke product Θ with zero sequence (λj ) the WEP is equivalent to (1 − |λj |2 )(1 − |z|2 ) j 1
|1 − λj z|2
C
for every z ∈ D \ λ∈Λ {ζ : |bλ (ζ )| < }. It is known that CEP is equivalent to the set Λ being a finite union of Carleson interpolating sequences. Therefore, the condition “Θ = B is a finite product of interpolating Blaschke products,” is sufficient for H ∞ /BH ∞ to have no corona. A surprising example (Example 3.7, invented in a collaboration with S. Treil and V. Vasyunin) shows that there exist Blaschke products satisfying WEP but not CEP, and so there exist algebras H ∞ /BH ∞ without corona (or, equivalently, having the norm controlled inversion property) such that the generating Blaschke product B is not a finite product of interpolating Blaschke products. We mention that, according to Theorems 3.3 and 3.4, the norm controlled inversion property (meaning δ1 (H ∞ /ΘH ∞ ) = 0 and/or the existence of the joint majorant for inverses (solutions of Bezout equations) cn (Λ, δ) < ∞ for all 0 < δ < 1) is equivalent to the simple inverse stability of the restriction algebra H ∞ |Λ: f ∈ H ∞ , δ(f, Λ) = inf f (λ) > 0 λ∈Λ
⇒
f + ΘH ∞ is invertible in H ∞ /ΘH ∞ .
The latter equivalence fails for the quotient disc algebras A(D)/ΘH ∞ . Indeed, Lemma 4.1 shows that every function f ∈ A(D) satisfying δ(f, σ (Θ)) > 0 is invertible in A(D)/ΘH ∞ if and only if m(σ (Θ) ∩ T) = 0, whereas the norm controlled inversions (i.e. δ1 (A(D)/ΘH ∞ ) = 0) hold true if and only if m(σ (Θ) ∩ T) = 0 and the WEP are satisfied (see Theorem 4.2 below). Moreover, in the case where m(σ (Θ) ∩ T) = 0, the maximal ideal space M(A(D)/ΘH ∞ ) can be identified with the spectrum σ (Θ). Therefore, as for some other Banach algebras (see examples in [11,12]), the lack of the norm controlled inversion property for quotient algebras is not necessarily related to the existence of a corona (meaning the existence of a large “invisible spectrum” or difference between the entire maximal ideal space and the “visible part”), but rather a subtle discrepancy between the Gelfand transform norm and the original Banach algebra norm. The geometric meaning of WEP sequences is still, unfortunately, unclear. Nothing similar to the Carleson density characterization for CEP sequences (see [10, p. 153]) is known. It is worth mentioning a couple of known related properties (for details see the end of Section 3 below): a separated WEP sequence is interpolating and a WEP sequence that is a finite union of separated sequences is a CEP sequence. Recall that a sequence, (λj ), is separated if infj =k |bλj (λk )| > 0. Finally, we interpret our results in terms of the spectral properties of the model operators. In fact, the interest in studying just this class of operators was the primary motivation for this paper. Now, let Θ be an inner function and
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KΘ = H 2 ΘH 2 be the model space; that is, the orthogonal complement of the shift-invariant subspace ΘH 2 of the Hardy space 2 k 2 ˆ ˆ H = f= f (k)z : f = f (k) < ∞ 2
k0
k0
on the disc D. The model operator (having Θ as its characteristic function, see [10,13]) is MΘ : KΘ → KΘ , MΘ f = PΘ (zf ),
f ∈ KΘ ,
where PΘ denotes the orthogonal projection on KΘ . It is known that every C00 Hilbert space contraction T having unit defect indices, rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1, is unitarily equivalent to a model operator MΘ . Now we can provide an operator theoretic interpretation of our function theoretic results (see [10,13,18]). (i) The spectrum of MΘ coincides with σ (Θ), and the point spectrum (the set of eigenvalues) is Λ = σ (Θ) ∩ D. (ii) The commutant of MΘ , {MΘ } = {A : KΘ → KΘ , AMΘ = MΘ A} coincides with the set of H ∞ functions of MΘ , and is isometrically isomorphic to the quotient algebra H ∞ /ΘH ∞ . (iii) For f ∈ H ∞ , f (λ) are eigenvalues of A = f (MΘ ), where λ ∈ Λ. (iv) The operator f (MΘ ) is invertible if and only if the class f + ΘH ∞ is invertible in the algebra H ∞ /ΘH ∞ . (v) The inner function Θ is a Blaschke product if and only if the eigen- and associated-vectors of MΘ are complete in KΘ . We are interested in the following questions. Is it possible to guarantee the invertibility of f (MΘ ) knowing that the eigenvalues f (λ), for λ ∈ Λ, are bounded away from zero? Is it possible to bound the norm f (MΘ )−1 in terms of the minimum modulus of the eigenvalues δ(f, Λ) = infλ∈Λ |f (λ)| > 0? Clearly, these questions are equivalent to those answered in Theorems 3.3, 3.4, and 4.2. The paper is organized as follows. Section 2 contains the main notation and preliminaries. Section 3 is devoted to the equivalence of the norm controlled inversions and the WEP, as well as to a discussion of the latter property. Section 4 treats the case of the algebra A(D)/ΘH ∞ . 2. Preliminaries This section contains the notation and definitions we will need. The main contribution of this section is the newly-developed definition of WEP, or the Weak Embedding Property. λ λ−z Let bλ = |λ| be a single Blaschke factor and let N = {0, 1, 2, . . .} denote the natu1−λz ral numbers. Letting κ : λ → kλ be a map from D to N satisfying the Blaschke condition k (1 − |λ|) < ∞ (that is the Blaschke mass of κ is finite) we may define the Blaschke λ λ∈D product B = B(κ, ·) by
P. Gorkin et al. / Journal of Functional Analysis 255 (2008) 854–876
B(κ, z) =
859
bλ (z)kλ .
kλ >0
Note that kλ is just the multiplicity of the zero λ. A singular inner function S is defined by
ξ +z dμ(ξ ) for z ∈ D, Sμ (z) = exp − ξ −z T
where μ is a nonnegative, finite Borel measure on T that is singular with respect to Lebesgue measure m. Every inner function Θ on D can be factored as Θ = eiθ BSμ , where B is a Blaschke product and S is a singular inner function. We let Λ denote the sequence of zeros of Θ inside D. Recall that a Blaschke product B with (simple) zeros (zn ) is said to be an interpolating Blaschke product if H ∞ |Λ = ∞ (Λ). The classical Carleson theorem says that B is an interpolating Blaschke product if and only if B δC (B) := inf (zn ) = inf 1 − |zn |2 B (zn ) > 0. n bzn n The quantity δC (B) is called the Carleson separation constant for B (or for the corresponding zero-sequence Λ). If we write supp(κ) = {λ ∈ D: kλ > 0} for the zeros of the Blaschke factor in D, the spectrum of an inner function Θ is (by definition) σ (Θ) := supp(κ) ∪ supp(μ) (see, for example, [10, p. 62]). If Θ = B is a Blaschke product with simple zeros (that is, kλ 1 on D), then H ∞ /ΘH ∞ is the space of traces of H ∞ on the set Λ; that is, H ∞ /BH ∞ = H ∞ |Λ endowed with the trace norm given by f = inf{g∞ : g ∈ H ∞ , g|Λ = f |Λ}. For Blaschke products with higher multiplicities, the algebra H ∞ /BH ∞ can be similarly interpreted as a space of germs of height kλ on Λ. Let ρ(z, w) = |(z − w)/(1 − zw)| denote the pseudohyperbolic distance between two points z, w ∈ D. For λ ∈ D and > 0, let Dρ (λ, ) denote the pseudohyperbolic disc with center λ and radius . For a function ϕ : D → C and η > 0 we let Ωη (ϕ) = z ∈ D: ϕ(z) < η denote the η-level set of ϕ. The following concept will be the major feature in what follows.
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Definition. Let Θ = BSμ be an inner function and let Λ be the zero sequence of Θ in D. We say that Θ satisfies the Weak Embedding Property (WEP) [for short: Θ ∈ WEP], if for every > 0 there exists η > 0 such that Ωη (Θ) ⊆
Dρ (λ, ).
λ∈Λ
For a WEP inner function we define Dρ (λ, ) , ηΘ () = sup η > 0: Ωη (Θ) ⊆ λ∈Λ
and we call ηΘ the WEP characteristic of Θ. For a Blaschke product B = BΛ associated with a Blaschke sequence Λ, we also write ηB = ηΛ . We note that
Dρ (λ, ) = Ω inf |bλ | . λ∈Λ
λ∈Λ
We also note that every WEP inner function has a nontrivial Blaschke factor and the union in our definition above does not depend on the multiplicities of the zeros of the Blaschke factor. The motivation as to why this property refers to a “weak embedding” is given in the introduction. The WEP inner functions are studied in Sections 3, 4 of this paper. However, we mention here four basic properties of WEP inner functions and WEP characteristics. (P1) If infλ∈Λ |bλ (z)| > 0, then |Θ(z)| ηΘ (). (P2) ηΘ () for every > 0. Proof. Indeed, since bλ is a factor of Θ for every λ ∈ Λ we see that |Θ(z)| |bλ (z)| for every z ∈ D and λ ∈ Λ. Therefore, if ηΘ () > we would be able to find an η and a z such that ηΘ () > |Θ(z)| = η > . Applying the definition of ηΘ () we see that for some λ we would have < Θ(z) bλ (z) < , a contradiction.
2
(P3) Let B be a Blaschke product that is the product of N interpolating Blaschke products Bj , 1 j N . Then B is a WEP Blaschke product and ηB () c · N for all with 0 < < 1 and a convenient constant c > 0. Proof. To see this, we note that it is well known (see, for example [10, p. 218]) that an interpolating Blaschke product Bj associated with the zero set Λj is characterized by the lower estimate
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|Bj (z)| cj infλ∈Λj |bλ (z)| for every z ∈ D, where cj > 0 is a convenient constant. Multiplying these inequalities we obtain a constant c such that B(z) c · inf bλ (z)N λ∈Λ
for every z ∈ D. The result follows.
2
(P4) Let Λ be a sequence in D satisfying CEP (the definition of which appears in the introduction). Then Λ satisfies WEP with a lower bound for ηΛ , as indicated in (P3) above. Proof. Indeed, it is well known (see [2,10], or [13]) that a CEP sequence is a finite union of interpolating sequences. The result follows from (P3) above. 2 It will be convenient to have two notations for the zeros of an inner function when dealing with purely topological properties (see (6) below). Thus, we write Z(Θ) for the zero set {m ∈ M(H ∞ ): m(Θ) = 0} of Θ in the maximal ideal space M(H ∞ ) and ZD (Θ) for the zero set of Θ in the disc. 3. Norm controlled inversions in H ∞ /ΘH ∞ and the WEP The main results of this section are that the norm controlled inversion property for the quotient algebra H ∞ /ΘH ∞ is equivalent to the WEP (Theorems 3.3 and 3.4) and that the WEP is not equivalent to the CEP (Examples 3.7 and 3.8). We also give an upper estimate of the function cn (H ∞ /ΘH ∞ , Λ, δ) in terms of the WEP characteristic ηΘ , where Λ is the zero set of Θ in D. We begin by recalling two known lemmas. The first one is a version of a classical result of Kerr-Lawson [9] and Hoffman [6]. For a proof, see [2, Lemma 1.4, p. 404]. Lemma 3.1 (Hoffman’s lemma). Suppose b is an interpolating Blaschke product with zeros {zn : n √∈ N} and let δ(b) be its Carleson separation constant. If 0 < δ < δ(b) and 0 < < (1 − 1 − δ 2 )/δ, then {z ∈ D: |b(z)| < 2 } is the union of pairwise disjoint domains Vn with zn ∈ Vn and Vn ⊂ {z: ρ(z, zn ) < }. The second lemma is also well known (see [10, p. 218]). Lemma 3.2. Suppose that for z, λ ∈ D we have ρ(z, λ) < δ/3. Then |λ − z| δ/2 min 1 − |λ|2 , 1 − |z|2 . In this section we prove the following theorem. Theorem 3.3. Let Θ be a inner function on D. The following are equivalent. (1) The quotient algebra H ∞ /ΘH ∞ has no corona; that is, Λ = σ (Θ) ∩ D is dense in M(H ∞ /ΘH ∞ ). (2) If f ∈ H ∞ and δ1 (f, Λ) > 0, then f + ΘH ∞ is invertible in H ∞ /ΘH ∞ . (3) δn (H ∞ /ΘH ∞ , Λ) = 0 for every n 1. (4) δ1 (H ∞ /ΘH ∞ , Λ) = 0.
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(5) Θ satisfies the WEP. (6) Θ = BSμ where ZD (B) = Z(B) = Z(Θ). Moreover, if (1)–(6) hold, then 1 ) log( ηΘ (δ/3) √ , cn H ∞ /ΘH ∞ , Λ, δ 18 n + 1 2 [ηΘ (δ/3)]
where ηΘ () is the WEP-characteristic of Θ. There is an alternative way of looking at the theorem above. For instance, statement (2) says that f + ΘH ∞ ∈ H ∞ /ΘH ∞ is invertible if and only if infλ∈Λ |f (λ)| > 0. Statement (3) means that for every f = (f1 , . . . , fn ) ∈ (H ∞ )n satisfying 0 < δn (f, Λ) f 1 there exist solutions h ∈ H ∞ and g = (g1 , . . . , gn ) ∈ (H ∞ )n to the Bezout equation n
fj gj + Θh = 1
j =1
with a norm control in terms of δn (f, Λ) only: g =
n
1/2 gj 2A
cn Λ, δn (f, Λ) .
j =1
The proof of Theorem 3.3 makes use of Carleson’s corona theorem. The aforementioned estimate will follow from the known estimates in the corona theorem (the best known one is given by S. Treil and B. √ Wick in [19]). The extra factor of n + 1 appears because, in place of the norm used in [10,19] or [2], which is n+1 1/2 2 Fj (z) F H ∞ = sup F (z) , where F (z) = , n+1
z∈D
j =1
we work (following the general setting) with the norm F =
n+1 j =1
1/2 Fj 2∞
that arises from the corresponding expression in H ∞ /ΘH ∞ satisfying √ ∞ . F n + 1 F Hn+1 This choice is quite natural, because H ∞ /ΘH ∞ is not, in general, a uniform algebra and therefore · Hn∞ is not defined. Moreover, for other Banach algebras in which Bezout equations are studied (see [12,14]) the use of norms like · is common. We will prove the following equivalent form of Theorem 3.3.
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Theorem 3.4. Let Θ be an inner function on D and let Λ be its zero set in D. The following are equivalent. (1) The quotient algebra H ∞ /ΘH ∞ has no corona; that is, Λ is dense in M(H ∞ /ΘH ∞ ). (2) Given f ∈ H ∞ there exist g, h ∈ H ∞ such that f g + Θh = 1 if and only if δ(f, Λ) := infλ∈Λ |f (λ)| > 0. (3) Given f = (f1 , . . . , fn ) ∈ (H ∞ )n with f :=
n j =1
1/2 fj 2∞
1
and δn (f, Λ) = inf
λ∈Λ
n fj (λ)2
1/2 > 0,
j =1
the Bezout equation n
fj gj + Θh = 1
j =1
has a solution (g, h) := (g1 , . . . , gn , h) ∈ (H ∞ )n+1 such that g :=
n
1/2 gj
2
cn H ∞ /ΘH ∞ , Λ, δn (f, Λ) .
j =1
(4) For f ∈ H ∞ with 0 < δ(f, Λ) f 1, there is a solution to the Bezout equation f g + Θh = 1 with g c1 (δ(f, Λ)). (5) Θ satisfies the WEP. (6) Θ = BSμ where ZD (B) = Z(B) = Z(Θ); in particular Z(Sμ ) ⊆ Z(B). Moreover, if (1)–(6) hold, then, for any δ ∈ ]0, 1[, 1
log( ηΘ (δ/3) ) √ , cn H ∞ /ΘH ∞ , Λ, δ 18 n + 1 [ηΘ (δ/3)]2 where ηΘ () is the WEP-characteristic of Θ. Proof. Since M(H ∞ /ΘH ∞ ) can be identified with Z(Θ) (which is well known), it follows that (1) ⇔ (6). Moreover, the implications (3) ⇒ (4) ⇒ (2) as well as (3) ⇒ (1) ⇒ (2) are obvious. So it remains to prove (2) ⇒ (5) and (5) ⇒ (3). . . . , fn ) ∈ (H ∞ )n and suppose 0 < δ = δn (f, Λ) = First we show that (5) ⇒ (3). Let f = (f1 , infλ∈Λ |f (λ)| f 1, where |f (λ)| = ( nj=1 |fj (λ)|2 )1/2 and f = ( nj=1 fj 2∞ )1/2 . Let z ∈ D satisfy infλ∈Λ |bλ (z)| < δ/3 and let λ be such that |bλ (z)| δ/3. As in [10, p. 218]
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we note that for h ∈ H ∞ if we define the conformal map τ (z) = (w − z)(1 − wz)−1 and write g = h ◦ τ −1 , then h (w) = g (0)τ (w) g∞ τ (w) = h∞ 1 − |w|2 −1 , where the inequality follows from Cauchy’s formula. Therefore, for λ ∈ Λ, f (z) f (λ) − f (λ) − f (z)
2 1/2 δ − |λ − z| · max fj (t) j
δ − |λ − z| ·
j
= δ − |λ − z| ·
t∈[λ,z]
−1 max fj ∞ 1 − t 2
t∈[λ,z]
fj / min 1 − |λ|2 , 1 − |z|2
j
δ − |λ − z|/ min 1 − |λ|2 , 1 − |z|2 . If |bλ (z)| δ/3, by Lemma 3.2 |λ − z| δ/2 · min 1 − |λ|2 , 1 − |z|2 . Consequently, for z satisfying infλ∈Λ |bλ (z)| < δ/3 we have f (z) δ/2. On the other hand, if z satisfies infλ∈Λ |bλ (z)| δ/3, then (see property (P1), Section 2), |Θ(z)| ηΘ (δ/3). Thus by property (P2), Section 2, ηΘ (δ/3) δ/3 < 1/e, we obtain f (z)2 + Θ(z)2 min (δ/2)2 , ηΘ (δ/3)2 = ηΘ (δ/3)2 for every z ∈ D. By Carleson’s corona theorem and the estimates of solutions of corona equations from [19], there exists a solution (g, h) := (g1 , . . . , gn , h) of the Bezout equation n
fj gj + Θh = 1
j =1
such that g
√
n gj (z)2 + h(z)2 n + 1 sup z
√
1/2
j =1
17 1 1 + n+1 log ηΘ (δ/3) ηΘ (δ/3)2 ηΘ (δ/3)
√ 18 1 log n+1 . ηΘ (δ/3) ηΘ (δ/3)2
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This completes the proof of (5) ⇒ (3) and the estimate of inverses claimed in Theorems 3.3 and 3.4. We turn to the final implication showing that (2) ⇒ (5). k First assume that Θ is a Blaschke product; that is, Θ = B = j bλjj , where (λj ) denotes the zero sequence of Θ. Let be a positive number so that < 1. Consider the sets z ∈ D: bλj (z) < , Ω() := Ω inf |bλj | = j
j
and Ω (B) = z: B(z) < . Clearly, Ω() ⊆ Ω (B). Note also that for every r ∈ ]0, 1[ and every > 0, there exists an η > 0 such that Ωη (B) ∩ z ∈ D: |z| r ⊆ Ω(); N kj for example, we may take η = M min|z|r ∞ j =N +1 |bλj (z)|, where M = j =1 kj and N is chosen so large that |λj | > r for every j N . We want to prove that for every > 0 there exists η > 0 such that Ωη (B) ⊆ Ω()
(3.1)
(in other words, {|B| < η} ⊆ j Dρ (λj , )). To this end, suppose to the contrary that there exists > 0 such that for all η > 0 we have Ωη (B) \ Ω() = ∅. Thus, if z1 ∈ Ω1/2 (B) \ Ω(), then |bλj (z1 )| > for every j and |B(z1 )| 1/2. Now let r = 1 − (1 − |z1 |)/2 and choose n2 > 2 so that Ω1/n2 (B) ∩ z: |z| r ⊆ Ω(). Then Ω1/n2 (B) \ Ω() ⊆ z ∈ D: 1 − |z| < 1 − |z1 | /2 . Taking z2 ∈ Ω1/n2 (B)\Ω() we get 1−|z2 | < (1−|z1 |)/2, |bλj (z2 )| > for all j , and |B(z2 )| 1/n2 . Continuing in this way, we obtain sequences (nk ) and (zk ) such that (a) |bλj (zk )| = |bzk (λj )| > for all j and all k; (b) |B(zk )| 1/nk where nk > nk−1 ; (c) 1 − |zk | < (1 − |zk−1 |)/2. Using [10, p. 159] and property (c), we see that (zk ) is an interpolating sequence. Let B1 = ∞ k=1 bzk be the corresponding interpolating Blaschke product. Then [10, p. 218] implies that there exists a constant γ such that B1 (z) γ infbz (z). (3.2) j j
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Now property (a) implies that |B1 (λj )| γ for every j . Property (b) implies that 2 2 2 inf B1 (z) + B(z) : z ∈ D infB(zk ) = 0. k
Therefore, we cannot find H ∞ functions g and h such that B1 g + Bh = 1. This contradicts our hypothesis (2) of Theorem 3.4. Hence our assumption that Ωη (B) \ Ω() = ∅ was wrong. Thus we have established (3.1) and so we proved statement (5) of Theorem 3.4 in the case when Θ is a Blaschke product. Now suppose that Θ = BSμ is an arbitrary inner function satisfying (2). Note that since ΘH ∞ = BSμ H ∞ ⊆ BH ∞ , it follows that for all f if f + ΘH ∞ is invertible in H ∞ /ΘH ∞ , then f + BH ∞ is invertible in H ∞ /BH ∞ . Therefore, the Blaschke factor of Θ also satisfies property (2). We turn now to the singular factor of Θ. First we show that for every α > 0 there exists β > 0 such that {|Sμ | < β} ⊆ {|B| < α}. Suppose this is not the case. Then there exists α0 > 0 such that for every β = 1/m, where m = 1, 2, . . . , there is a point zm ∈ D such that |S(zm )| < 1/m, but |B(zm )| α0 . Without loss of generality, we may assume that (zm ) is an interpolating sequence. Let b be the associated interpolating Blaschke product. Recall that (λk ) denotes the zero sequence of B. Since |bλk (zm )| |B(zm )| α0 for every m and k, we obtain (by Lemma 3.1 or formula (3.2)), that |b(λk )| is bounded away from zero. Thus assertion (2) implies that b + ΘH ∞ is invertible in H ∞ /ΘH ∞ ; that is there exist f and g in H ∞ such that f b + gΘ = 1. But 2 2 2 inf b(z) + Θ(z) : z ∈ D infSμ (zm ) = 0. m
This is a contradiction. Thus, for every α > 0 there exists β > 0 such that Ωβ (Sμ ) ⊆ Ωα (B). Without loss of generality, we may assume that β α. We have already shown that B ∈ WEP, so for > 0 we may choose α > 0 so that Ωα (B) ⊆ Ω (infλ |bλ |). Let β α be as above. We claim that by setting η = β 2 , we obtain Ωη (Θ) ⊆ Ω (infλ |bλ |); that is, we claim that assertion (5) of the theorem holds. In fact, if |Θ(z)| = |B(z)Sμ (z)| < β 2 , then either |B(z)| < β α and hence z ∈ Ω (infλ |bλ |), or |Sμ (z)| < β and then |B(z)| < α, and again z ∈ Ω (infλ |bλ |). 2 We derive two immediate corollaries. First we formalize the “splitting property” of WEP inner functions that appeared at the end of the previous proof. Corollary 3.5. Let Θ = BSμ be an inner function. The following assertions are equivalent: (a) Θ ∈ WEP; (b) B ∈ WEP and for every α > 0 there exists β > 0 such that Ωβ (Sμ ) ⊆ Ωα (B). Proof. By the theorem above, (a) implies assertion (2) of Theorems 3.3, 3.4. Then the second to the last paragraph of the previous proof shows that (b) holds. The last paragraph of the same proof above shows that (b) implies (a). 2
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Corollary 3.6. Let N be an integer and B a product of N interpolating Blaschke products. Then assertions (1)–(6) in Theorem 3.4 hold true for B. Moreover, for every δ ∈ ]0, 1[ and n 1 there exists a constant a > 0 (depending on B, and in particular on N ), such that 1
log( δ/3 ) √ cn H ∞ /BH ∞ , Λ, δ a n + 1 . δ 2N Proof. This follows from Theorem 3.4 and property (3) of Section 2.
2
In Section 2 we discussed Kerr-Lawson’s lemma, which shows that if B is a finite product of interpolating Blaschke products with zero sequence (zn ), then for every ∈ ]0, 1[ there exists η > 0 such that z ∈ D: B(z) < η ⊆ Dρ (zn , ). (3.3) n
Thus, as we noted in Section 2 (property (P3)), every such Blaschke product satisfies the weak embedding property. Kerr-Lawson [9] also proved that if u is an inner function and for some η > 0 and ∈ ]0, 1[ the set {z ∈ D: |u(z)| < η} is contained in the disjoint union n Dρ (zn , ) of pseudohyperbolic discs of fixed radius and centers zn , then u is a finite product of interpolating Blaschke products. In the last paragraph of his paper he asserted that this implies, in particular, that every Blaschke product that is not a finite product of interpolating Blaschke products is arbitrarily small arbitrarily far away from its zeros inside D. The following example shows that this is not the case; indeed the class of WEP Blaschke products is strictly larger than the class of finite products of interpolating Blaschke products. As indicated in the introduction, the main idea for these examples is due to S. Treil and was realized in a correspondence between V. Vasyunin and the third author of this paper. We are indebted to Professors Treil and Vasyunin for permitting us to include the example in this paper. Clearly, WEP, as defined in Section 2, is conformally invariant. In particular, we can replace the disc D by the upper-half plane C+ = {z ∈ C: Im z > 0} without changing the definitions. Since the pseudohyperbolic geometry of C+ is more transparent than that of D, we give our principal example in C+ (Example 3.7 below). Example 3.7. There exists a WEP Blaschke product B that does not satisfy the CEP (that is, B satisfies the WEP, but B is not a finite product of interpolating Blaschke products). Proof. The construction is done using techniques from [10]. Let a > 0 and let Θ(z) = eiaz be an inner function on C+ = {z: Im z > 0}. For t > 0, consider the Frostman shift Ba,t of Θ given by Ba,t (z) =
Θ(z) − e−t . 1 − e−t Θ(z)
Then Ba,t is a Blaschke product with zeros zk,a,t = 2πk/a + it/a, k ∈ Z. It satisfies the Carleson interpolation condition and the Blaschke mass of its zeros is k
t2
at ∼ a. + 4π 2 k 2
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Now take a = 1/n2 and t = n for n = 1, 2, . . . . Define the Blaschke product B by B=
B1/n2 ,n .
n
This will be the Blaschke product we seek. Note that B has the zeros zk,n = 2πkn2 + in3 , for k ∈ Z and n 1. Moreover, we have the following: (1) For every > 0 the level set Ω =
z ∈ C+ : |z − zk,n | < n3 k,n
3 }. The zeros of B outside contains a half-plane; for example Π = {z ∈ C+ : Im z > (100/) (1) Π form a Carleson interpolating sequence, hence B := n100/ B1/n2 ,n is an interpolating Blaschke product. (2) A direct look at
eiz/n2 − e−n 1 − e−n eiz/n2 shows that the subproduct B (2) := n>100/ B1/n2 ,n is bounded away from zero on the strip S = {z ∈ C+ : 0 < Im z (100/)3 }. (3) Since (1) and (2) are valid for every > 0 and B = B (1) B (2) , for sufficiently small η > 0, the part of the level set S ∩ {z ∈ C+ : |B(z)| < η} is included in the union
z ∈ C+ : |z − zk,n | < n3 ,
k∈Z, n<100/
and hence {z ∈ C+ : |B(z)| < η} ⊆ Ω . (4) The three statements above show that for every > 0 there exists η > 0 such that {z ∈ C+ : |B(z)| < η} ⊆ Ω , but B is not a finite product of interpolating Blaschke products (because, for example, the Carleson embedding theorem is not satisfied: max x+iy∈Q y, where the summation runs over all zeros x + iy of B in the square Q = {0 x, y a}, is at least of the order a 4/3 as a → ∞, and not of the order of a as required for Carleson measures). 2 It is of interest to know whether there are inner functions satisfying the WEP that are not Blaschke products. The next example shows that such inner functions exist. Example 3.8. Let B be the Blaschke product of Example 3.7 and let S(z) = eiaz , where a > 0. Then the function Θ = SB is a WEP inner function. Proof. Indeed, keeping the notation of the previous example, we will prove that for every > 0, there exists η > 0 such that Ωη (Θ) ⊂ Ω . Let Π be the half-plane defined in Example 3.7 above. For z ∈ C+ \ Π , we have Θ(z) e−α B(z),
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where α = a(100/)3 . Hence, by Example 3.7, Ω(Θ) ∩ (C+ \ Π ) ⊆ Ωηeα (B) ⊂ Ω for η > 0 sufficiently small. As we saw in Example 3.7 above, Π ⊂ Ω , so the inclusion Ωη (Θ) ⊂ Ω follows. 2 At this point, we mention a few additional properties of WEP Blaschke products and WEP sequences. Properties (P1)–(P4) can be found in Section 2. (P5) A separated WEP sequence is interpolating. Proof. Let Λ = (λj ) be a separated WEP-sequence and let B be the associated Blaschke product. Then, for every ∈ ]0, 1[ there exists η = η() > 0 such that |B| < η ⊆ Dρ (λ, ). λ∈Λ
Since Λ is separated, there exists > 0 such that the discs D = Dρ (λ, ), λ ∈ Λ, are pairwise disjoint. On the boundary, ∂D, of such a disc we have B(z) η( ) = η( ) bλ (z).
By the minimum modulus principle, the function ϕ : z → B(z)/bλ (z), which is holomorphic and |ϕ(z)| η( )/ on this disc. Taking z = λ, we get zero free on D, satisfies the same estimate the Carleson interpolation condition j =k |bλj (λk )| η( )/ > 0 for every λk ∈ Λ. 2 (P6) Let Λ be a finite union of separated sequences and suppose that Λ ∈ WEP. Then Λ ∈ CEP. Proof. Assume that Λ is a union of N separated sequences and let B = λ∈Λ bλ .1 Consider the open set Ω (infλ∈Λ |bλ |) = λ∈Λ Dρ (λ, ) and its open connected components Ω , Ω 2 , . . . . The triangle inequality implies that for > 0 small enough every Ω j contains no more than N points of Λ. Fixing such an > 0, we apply WEP, which gives Ωj Ωη (B) ⊆ Ω inf |bλ | = λ∈Λ
j
for some η > 0. Then every connected component of Ωη (B) is contained in one of Ω j , hence contains no more than N points of Λ. But it is known that a Blaschke product is the product of at most N interpolating Blaschke products if and only if there exists η > 0 such that every connected component of the level set {z: |B(z)| < η} contains at most N zeros of B. This fact is implicitly contained in [11, pp. 229, 230]. It can also be found in [13,15] and [14, vol. 2, p. 189]. Thus Λ ∈ WEP. 2 (P7) Let ζ ∈ ∂D and let 0 < θ < π/2. Let Λ be a sequence in a Stolz angle S(ζ, θ ) (that is the convex hull of the disk |z| sin θ and the point ζ ). Then Λ ∈ WEP if and only Λ ∈ CEP.
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Proof. By definition, we see that a function satisfying CEP also satisfies WEP. To prove the converse, since a separated sequence in a Stolz angle is interpolating, it suffices to show that Λ is a finite union of separated sequences. It is also clear that for the latter property it is sufficient to prove that there exist β and N , 0 < β < 1, such that every disc Dρ (λ, β), λ ∈ Λ, contains no more than N points of Λ (the standard dyadic “white-and-black-boxes” reasoning gives the proof, see for instance [2], or [10, p. 159]). In order to ensure the last property, observe that if θ < θ < π/2, then α := inf bz (λ): z ∈ ∂S(ζ, θ ), λ ∈ Λ > 0. On the other hand, there exists ∈ ]0, 1[, close to 1, such that Dρ (w, ) ∩ ∂S(ζ, θ ) = ∅ for every w ∈ S(ζ, θ ). Hence, for every λ ∈ Λ, there exists z = z(λ) ∈ ∂S(ζ, θ ) such that |bz (λ)| < . Therefore, if β > 0 and γ := + β < 1, then we have Dρ (λ, β) ⊆ Dρ (z(λ), γ ). Now, denoting N(λ) = card Dρ (λ, β) ∩ Λ, we get B z(λ) bμ z(λ) γ N (λ) , μ
where the product is taken over all μ ∈ Dρ (λ, β) ∩ Λ. Since the WEP implies that |B(z(λ))| ηB (α), where B is the Blaschke product associated with Λ (see (P1), Section 2), we get supλ∈Λ N (λ) < ∞. 2 Alternatively, by using maximal ideal space techniques, we can prove the statement in (P7) as follows. Let Θ satisfy the WEP and assume that the zero set Λ of Θ is contained in a Stolz-angle. By Theorem 3.3, (1) ⇔ (6), we have that ZD (Θ) = Z(Θ). By [6, Theorem 6.4] every point m ∈ M(H ∞ ) in the closure of a Stolz-angle belongs to the closure of an interpolating sequence. Thus, by [4], Θ is a finite product of interpolating Blaschke products and so Λ ∈ CEP. 2 (P8) A finite product of WEP Blaschke products has the WEP property. Proof. Let Bj be the corresponding Blaschke products with zero sequences Λj , j = 1, . . . , n. By the WEP for Bj , there exists for every > 0 some δj such that Ωδj (Bj ) ⊆ Ω Now, if B =
n
j =1 Bj ,
Λ=
Ωδ (B) ⊆
n
n
j =1 Λj ,
and δ =
Ωδj (Bj ) ⊆
j =1
Thus B is a WEP-inner function.
n j =1
inf |bλ | .
λ∈Λj
n
j =1 δj ,
Ω
then
inf |bλ | ⊆ Ω inf |bλ | .
λ∈Λj
λ∈Λ
2
Since the definition of a WEP sequence does not involve the multiplicities of the zeros of the associated Blaschke product B, one can ask whether there exist WEP sequences whose multiplicities are unbounded.
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(P9) There exist WEP sequences whose point multiplicities are not bounded. Proof. In fact, using√ the notation of the Example for a slowly growing sequence pn → ∞ 3.7, pn (for example, pn = [ n]), the product B = n1 B1/n 2 ,n satisfies the WEP property. This can be checked in the same manner as Example 3.7. 2 (P10) Let Λ = (λj ) be a WEP sequence with associated Blaschke product B and for > 0 let Λ() its subsequence of -isolated points; that is, Λ() = λj ∈ Λ: inf bλj (λk ) . λk =λj
Then, the multiplicities Nj of points λj in Λ() are uniformly bounded, namely, Nj
log ηΛ (/2) . log(/2)
Proof. To see this, let λj ∈ Λ() and z ∈ D such that |bλj (z)| = /2. Then |bλ (z)| /2 for every λ ∈ Λ. By the WEP-property (property (P1) of Section 2), |B(z)| ηB (/2). On the other hand, (/2)Nj = |bλj (z)|Nj |B(z)|. The result now follows. 2 Finally we remark that by [3], if Θ is WEP-inner function, then for every a ∈ D \ {0} with |a| sufficiently small, the Frostman transform (Θ − a)/(1 − aΘ) is in CEP. In particular, Θ can be uniformly approximated by interpolating Blaschke products. D)/ΘH ∞ 4. The quotient algebra A(D We note that it is still possible to make sense of Theorem 3.3 when we replace Λ = σ (Θ) ∩ D with Λ = σ (Θ). Rather than looking at H ∞ functions, we consider functions from the disc algebra, denoted here by A(D). Recall that A(D)/ΘH ∞ is the canonical image of A(D) in the quotient algebra H ∞ /ΘH ∞ . As mentioned in the introduction, the algebra A(D)/ΘH ∞ is closed in H ∞ /ΘH ∞ if and only if either m(σ (Θ) ∩ T) = 0 or m(σ (Θ) ∩ T) = 1. As we will see later on (Theorem 4.2), the latter case is not of interest for the efficient inversion problem. The former one, to the contrary, is very interesting. In this case, for the problem of norm controlled inversions, the algebra A(D)/ΘH ∞ is even more significant than H ∞ /ΘH ∞ . The reason is that for H ∞ /ΘH ∞ , the norm controlled inversion property (incidentally) coincides with the corona property, and hence the metric problem on the critical constants δn is, in a sense, hidden behind the topological fact that the visible spectrum Λ is dense in M(H ∞ /ΘH ∞ ). For A(D)/ΘH ∞ , these two properties are distinct: the algebra A(D)/ΘH ∞ never has a corona with respect to σ (Θ), but it may or may not have the norm-controlled inversion property. This phenomenon, which does appear in different situations (see [2,14,16]), is a specific internal property of a Banach algebra measuring the discrepancy between the Gelfand transform norm and the original algebra norm. Lemma 4.1. Let Θ be an inner function. The natural restriction embedding f + ΘH ∞ → f |σ (Θ) is a (contractive) homomorphism from the quotient algebra A(D)/ΘH ∞ into C(σ (Θ)). If m(σ (Θ) ∩ T) = 0, the maximal ideal space M(A(D)/ΘH ∞ ) coincides with σ (Θ) with respect to this mapping. Consequently, given fj ∈ A(D) the Bezout equation nj=1 gj fj = 1 is
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solvable in A(D)/ΘH ∞ if and only if nj=1 |fj (λ)|2 > 0 for every λ ∈ σ (Θ) (or if and only if n infλ∈Λ ( j =1 |fj (λ)|2 ) > 0, in the case in which Θ is a Blaschke product B and Λ = B −1 ({0})). Proof. It is known that σ (Θ) = {λ ∈ D: lim infζ →λ |Θ(ζ )| = 0} (see [13, p. 63]). Hence, for every λ ∈ σ (Θ), f ∈ A(D), and g ∈ H ∞ , we have f (λ) = lim f (ζ ) lim supf (ζ ) + Θ(ζ )g(ζ ) f + Θg∞ , ζ →λ
ζ →λ
and therefore, f |σ (Θ)C(σ (Θ)) f H ∞ /ΘH ∞ . Now, assume that m(σ (Θ) ∩ T) = 0. If ϕ is a complex continuous homomorphism on A(D)/ΘH ∞ and λ = ϕ(z + ΘH ∞ ) (where z(ζ ) ≡ ζ ), then λ ∈ σ (Θ). Indeed, λ ∈ D since ϕ 1, and hence ϕ extends naturally to a homomorphism of A(D), f → f (λ). Moreover, there exists an outer function g in A(D) such that the zero set of g is σ (Θ) ∩ T, which gives 0 = ϕ(Θg) = (Θg)(λ); that is, λ ∈ σ (Θ). Since the polynomials are dense in A(D), we get ϕ(f + ΘH ∞ ) = f (λ) for every f ∈ A(D). Therefore, M(A(D)/ΘH ∞ ) ⊆ σ (Θ). Note that A(D) distinguishes the points of σ (Θ). The reverse inclusion is obvious. 2 Theorem 4.2. Let Θ be an inner function on D. The following are equivalent. (1) δn (A(D)/ΘH ∞ , σ (Θ)) = 0 for every n 1. (2) δ1 (A(D)/ΘH ∞ , σ (Θ)) = 0. (3) m(σ (Θ) ∩ T) = 0 and Θ ∈ WEP. Moreover, if (1)–(3) hold, then for every δ ∈ ]0, 1[, 1
log( ηΘ (δ/3) ) √ cn A(D)/ΘH ∞ , Λ, δ 18 n + 1 , [ηΘ (δ/3)]2 where, as usual, Λ = σ (Θ) ∩ D is the zero set of Θ and ηΘ () is the WEP-characteristic of Θ. Before proceeding to the proof, we note that the essential part of Theorem 4.2 will follow from estimates of solutions of the Bezout equation in the disc algebra A(D) as well as a generalization of the Rudin–Carleson theorem on free A(D)-interpolation. By the latter, we mean the following: let E ⊂ D be a closed set. Then A(D)|E = C(E) if and only if m(E ∩ D) = 0 and E ∩ D is an interpolating sequence (see [5]). For our proof we will need the following lemma suggested by Sergei Treil. This lemma can also be found in [16]. Lemma 4.3. Let 0 < δ 1. Then cn (H ∞ , D, δ) = cn (A(D), D, δ). Proof. The inequality cn (H ∞ , D, δ) cn (A(D), D, δ) is well known (see, for example, [10, Appendix 3]). For the reverse inequality, let n 1 and let f = (f1 , . . . , fn ) ∈ (A(D))n satisfy δn (f, D) = ∞ n ∞ solution of the equation inf nλ∈D |f (λ)| > 0. Let φ = (φ1 , . . . , φn ) ∈ (H ) be an H j =1 φj fj = 1.
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Now, given > 0 there exists r with 0 < r < 1 such that n n φj (rz)fj (z) − 1 = φj (rz) fj (z) − fj (rz) < , j =1
j =1
for every z ∈ D. Therefore, the functions φj (rz) , j =1 φj (rz)fj (z)
gj (z) = n
z ∈ D,
give a solution in A(D) satisfying nj=1 gj fj = 1 with norm arbitrarly close to that of the solution given by φj . The inequality follows. 2 Proof of Theorem 4.2. It is clear that (1) ⇒ (2). We show that (2) ⇒ (3). Note first that if F ∈ A(D), then F is continuous at every point of the unit circle. If the inner factor of F is discontinuous at a point λ, its cluster set at the point λ is the closed unit disc [2, p. 80]. Therefore, F must vanish at the point λ. Now suppose f, g ∈ A(D) and f g + Θh = 1. Letting Θh play the role of F above we conclude that f g = 1 on σ (Θ). If m(σ (Θ) ∩ T) > 0, then f g ≡ 1. Since this must hold for every f that is bounded away from zero on σ (Θ) (and, in particular, for those f having a zero outside σ (Θ)) we see that the condition m(σ (Θ) ∩ T) = 0 is necessary for (2). Now let Λ = σ (Θ) ∩ D and let f ∈ H ∞ be such that 0 < δ = δ(f, Λ) = inf f (λ) f ∞ < 1. λ∈Λ
By R. Nevanlinna’s theorem (see [10, p. 204] or [13, vol. 1, p. 234]) there exists an inner function ϕ such that ϕ|Λ = f |Λ. Moreover, the same is true for any Blaschke set Λ ⊃ Λ. By using this fact for (1 + n1 )f instead of f and for Λn = Λ ∪ { 12 , 13 , . . . , n1 } instead of Λ, and uniformly approximating the corresponding inner function ϕ by Blaschke products, we obtain a sequence of Blaschke products Bn such that 1 f (z) − Bn (z) δ/n. sup 1 + n z∈Λn In particular |Bn | (1 + n1 )|f | − |Bn − (1 + n1 )f | δ on Λ. Choosing convenient finite Blaschke subproducts Cn of Bn , we get a sequence (Cn ) of functions in A(D) such that |Cn | δ on Λ and such that (Cn ) converges locally uniformly on D to f . Now Cj ∈ A(D) and 0 < δ = δ(f, Λ) |Cj (λ)| Cj ∞ = 1 for every λ ∈ σ (Θ). By (2), there exist gn ∈ A(D) and hn ∈ H ∞ satisfying Cn gn + Θhn = 1 and gn ∞ c1 A(D)/ΘH ∞ , δ . Using Montel’s theorem we obtain functions g and h in H ∞ with g c1 A(D)/ΘH ∞ , δ and f g + Θh = 1.
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In particular, statement (2) of Theorem 3.4 holds. Thus, it follows from Theorem 3.4 (statement (5)) that Θ has the desired property. The last assertion of the theorem follows from the corresponding statement in Theorem 3.3 and Lemma 4.3. Next we show (3) ⇒ (1). Let Λ = σ (Θ) ∩ D and f = (f1 , . . . , fn ) ∈ (A(D))n be such that 0 < δ = δn (f, Λ) = minf (λ) f 1. λ∈Λ
The proof now proceeds in the same manner as that of Theorem 3.4. This yields the following estimation: f (z)2 + Θ(z)2 ηΘ (δ/3)2 for every z ∈ D. Since f is continuous on D and σ (Θ) ⊆ Λ, we see that |f | ηΘ (δ/3) on σ (Θ). Also, since m(σ (Θ) ∩ D) = 0, there exists a peak function p ∈ A(D) such that p = 1 on σ (Θ) ∩ T and |p(z)| < 1 for z ∈ D \ (σ (Θ) ∩ T) [7, p. 80]. Now given > 0, we may choose n sufficiently large, so that the function φ = 1 − p n satisfies f (z)2 + Θ(z)φ(z)2 (1 − )ηΘ (δ/3)2 for every z ∈ D. Now all the data, fj and Θφ, are in A(D), so we may use Lemma 4.3 to conclude that there exists g1 , . . . , gn+1 ∈ A(D) such that n
fj gj + Θφgn+1 = 1
j =1
and the norm g of g = (g1 , . . . , gn+1 ) is arbitrarily close to the norm of the best possible H ∞ solution. This shows that the lower bound for A(D)-solutions is the same as for the best H ∞ -solutions. 2 5. Open questions We conclude this paper with several open questions. (1) Find a geometric description of WEP sequences, introducing a “weak Carleson density” in place of the classical one that gives a description of the CEP sequences: μ(Qh ) ch, where μ = κ(λ)>0 (1 − |λ|2 )δλ and where Qh is a Carleson square with side h (see [2] or [10] for information on Carleson measures). (2) Describe possible singular factors Sμ of WEP inner functions Θ = Sμ B. We remark that it follows from Example 3.8 that a singular inner function Sμ with finite support, supp(μ), is admissible. (3) Is it true that finite products of interpolating Blaschke products are characterized by property (P3) of Section 2? It is sufficient to prove that if Λ is not in WEP, then c1 (Λ, δ) grows faster than any power of 1/δ as δ → 0.
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(4) Let Θ be a non-WEP inner function, Θ = Sμ BΛ . Is it true that δ1 H ∞ /ΘH ∞ , Λ = 1 or
δ1 A(D)/ΘH ∞ , σ (Θ) = 1?
The latter statement makes sense even when Λ = ∅ (in which case the question is about an estimate of f −1 A(D)/ΘH ∞ for functions f ∈ A(D) satisfying 0 < δ f (z) f ∞ 1 for all z ∈ σ (Θ)); in this case the answer is known to be “yes,” see [11]). (5) Can every inner function be multiplied into the class WEP by a WEP-Blaschke product? This would solve, in particular, the long-standing open problem whether every point outside the Shilov boundary and having a trivial Gleason part lies in the closure of a Blaschke sequence. Acknowledgments The authors are grateful to Sergei Treil and his wife Marina who improvised an informal 48-hour seminar with the third author on the WEP property at their home in Providence (Fall 2006), and to Vassily Vasyunin who made a 1000-mile trip to Providence in order to add his mathematical expertise to these efforts. After several late night attempts to prove or to disprove the conjecture “WEP = CEP,” Sergei arrived on the morning of the second day with glasses of juice and the inspired idea of an example of a sequence that is WEP but not CEP (presented in Example 3.7). References [1] O. El-Fallah, N.K. Nikolski, M. Zarrabi, Estimates for resolvents in Beurling–Sobolev algebras, Algebra i Analiz 10 (6) (1998) 1–92 (in Russian, Russian summary); translated in: St. Petersburg Math. J. 10 (6) (1999) 901–964. [2] J.B. Garnett, Bounded Analytic Functions, Pure Appl. Math., vol. 96, Academic Press, New York, 1981, xvi+467. [3] P. Gorkin, R. Mortini, Two new characterizations of Carleson–Newman Blaschke products, preprint, 2007. [4] C. Guillory, K. Izuchi, D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. R. Ir. Acad. 84 (1984) 1–7. [5] E. Heard, J. Wells, An interpolation problem for subalgebras of H ∞ , Pacific J. Math. 28 (1969) 543–553. [6] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967) 74–111. [7] K. Hoffman, Banach Spaces of Analytic Functions, reprint of the 1962 original. Dover, New York, 1988. [8] K. Izuchi, Y. Izuchi, Algebras of bounded analytic functions containing the disk algebra, Canad. J. Math. 38 (1986) 87–108. [9] A. Kerr-Lawson, Some lemmas on interpolating Blaschke products and a correction, Canad. J. Math. 21 (1969) 531–534. [10] N.K. Nikolski, Treatise on the Shift Operator. Spectral Function Theory, with an appendix by S.V. Hrušˇcev [S.V. Khrushchev] and V.V. Peller; translated from the Russian by J. Peetre, Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin, 1986. [11] N.K. Nikolski, In search of the invisible spectrum, Ann. Inst. Fourier (Grenoble) 49 (6) (1999) 1925–1998. [12] N.K. Nikolski, The problem of efficient inversions and Bezout equations, in: Twentieth Century Harmonic Analysis—A Celebration, Il Ciocco, 2000, in: NATO Sci. Ser. II Math. Phys. Chem., vol. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 235–269. [13] N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, vols. 1, 2. Model Operators and Systems, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, RI, 2002. [14] N.K. Nikolski, Condition numbers of large matrices, and analytic capacities, Algebra i Analiz 17 (4) (2005) 125–180 (in English, English summary).
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[15] N.K. Nikolski, S.V. Khrushchëv, A functional model and some problems of the spectral theory of functions, Tr. Mat. Inst. Steklov. 176 (1987) 97–210, 327 (in Russian); translated in: Proc. Steklov Inst. Math. 3 (1988) 101–214. [16] S. Sidney, Sunwook Hwang, Sequence spaces of continuous functions, Rocky Mountain J. Math. 31 (2001) 641– 659. [17] D. Stegenga, Sums of invariant subspaces, Pacific J. Math. 70 (1977) 567–584. [18] B. Szökefalvi-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, New York, 1970. [19] S. Treil, B. Wick, The matrix-valued H p corona problem in the disk and polydisk, J. Funct. Anal. 226 (2005) 138–172.
Journal of Functional Analysis 255 (2008) 877–890 www.elsevier.com/locate/jfa
Ornstein–Uhlenbeck processes on Lie groups ✩ Fabrice Baudoin a , Martin Hairer b , Josef Teichmann c,∗ a Université Paul Sabatier, Institut des Mathématiques, 118, Rue de Narbonne, Toulouse, Cedex 31062, France b The University of Warwick, Mathematics Department, CV4 7AL Coventry, United Kingdom c Department of Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8–10,
A-1040 Wien, Austria Received 15 November 2007; accepted 12 May 2008 Available online 20 June 2008 Communicated by Daniel W. Stroock
Abstract We consider Ornstein–Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional Lie groups: let L be a hypo-elliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-processes by adding negative horizontal gradient drifts of functions U . In the natural case U (x) = − log p(1, x), where p(1, x) is the density of the law of X starting at identity e at time t = 1 with respect to the right-invariant Haar measure on G, we show the Poincaré inequality by applying the Driver–Melcher inequality for “sum of the squares” operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypo-elliptic diffusion on G. We prove the global strong existence of these OU-type processes on G under an integrability assumption on U . The Poincaré inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypo-elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M. © 2008 Elsevier Inc. All rights reserved. Keywords: Lie group; Hypo-elliptic diffusion; Spectral gap; Simulated annealing
✩ The first and third authors gratefully acknowledge the support from the FWF-grant Y 328 (START prize from the Austrian Science Fund). The second author gratefully acknowledges the support from the EPSRC fellowship EP/D071593/1. All authors are grateful for the warm hospitality at the Mittag-Leffler Institute in Stockholm and at the Hausdorff Institute in Bonn. * Corresponding author. E-mail addresses: [email protected] (F. Baudoin), [email protected] (M. Hairer), [email protected] (J. Teichmann).
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.004
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1. Introduction We consider a left-invariant hypo-elliptic diffusion process X, dXtx =
d
Vi Xtx ◦ dBti ,
X0x = x,
i=1
on a connected Lie group G together with a right-invariant Haar measure μ, which is then also invariant for the diffusion process X. The vector fields V1 , . . . , Vd are assumed to be left-invariant vector fields, and their brackets generate the Lie algebra. In the spirit of sub-Riemannian geometry the hypo-elliptic diffusion process can be used to define a metric (the Carnot–Carathéodory metric d), a geodesic structure and the notion of a gradient (the horizontal gradient gradhor ) on G. The density of the law of Xte with respect to μ is denoted by p(t, .) and smooth by Hörmander’s theorem. Fix t > 0. The natural Ornstein–Uhlenbeck process on G associated to X will be 1 gradhor p(τ, Yt ) dt + Vi (Yt ) ◦ dBti 2 d
dYt =
i=1
out of two reasons: first, the law of Xte is an invariant measure for Y and, second, there is a spectral gap for Y if X satisfies a Driver–Melcher inequality (see Section 4 for precise details). We can easily extend all results to compact homogeneous spaces M. The possible exponential convergence rate is then applied for new simulated annealing algorithms. The interest in those new algorithms lies in the fact that there are less Brownian motions than space dimensions of the optimization problem involved, and that the stochastic differential equations might have a smaller complexity and is therefore easier to evaluate. For this purpose we consider cases where the size of the spectral gap for the previously intro1 for some constant K > 0 (see Section 5 for precise details). This holds duced process Y is 2Kτ true for instance on SU(2) or on the Heisenberg tori. Let U be a smooth potential on M. We regard an equation of the type 1 dZt = − gradhor U (Zt ) dt + ε Vi (Zt ) ◦ dBti 2 d
i=1
as a compact perturbation of the natural OU process in order to get an estimate for its spectral gap, which can be expressed by U (x) + ε 2 log p ε 2 , x0 , x D for all x ∈ M and some constant D. This amounts to a comparison of U with the square of the Carnot–Carathéodory metric d(x0 , x) due to short-time asymptotics of the heat kernel on the compact manifold M. A concatenation of trajectories of such equations under a cooling schedule t → ε(t) = √ c leads then to the desired simulated annealing algorithms. The analytical arguments log(R+t)
are strongly inspired by the seminal works of R. Holley, S. Kusuoka and D. Stroock [7] and [8].
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2. Preparations from functional analysis We consider a finite-dimensional, connected Lie group G with Lie algebra g, its right-invariant Haar measure μ and a family of left-invariant vector fields V1 , . . . , Vd ∈ g. We assume that Hörmander’s condition holds, i.e. the sub-algebra generated by V1 , . . . , Vd coincides with g. We consider furthermore a stochastic basis (Ω, F, P) with a d-dimensional Brownian motion B and the Lie group-valued process dXtx =
d
Vi Xtx ◦ dBti ,
X0x = x ∈ G.
(2.1)
i=1
The generator of this process is denoted by L, we have 1 2 Vi , 2 d
L=
i=1
where we interpret the vector fields as first-order differential operators on C ∞ (G, R). Furthermore, we define a semigroup Pt acting on bounded measurable functions f : G → R by Pt f (x) = E f Xtx . This semigroup can be extended to a strongly continuous semigroup on L2 (G, μ), which we will denote by the same letter Pt . The carré du champ operator Γ is defined for functions f , where it makes sense, by Γ (f, g) = L(f g) − f Lg − gLf.
(2.2)
In our particular case, we obtain immediately Γ (f, f ) =
d
(Vi f )2 .
i=1
Notice that the carré du champ operator does not change if we add a drift to the generator L. Due to the right-invariance of the Haar measure μ and the left-invariance of the vector fields Vi , the operator L is symmetric (reversible) with respect to μ and therefore μ is an infinitesimal invariant measure in the sense that Lf (x) μ(dx) = 0 for all smooth compactly supported test functions f . Furthermore, due to the symmetry of L we have from (2.2) the relation 2 f Lg μ = − Γ (f, g) μ (2.3) for all f ∈ C0∞ (G) and g ∈ C ∞ (G). Let now U : G → R be an arbitrary smooth function such that Z := exp −U (x) μ(dx) < ∞. G
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We consider the modified generator 1 LU := L − Γ (U, ·). 2 Notice that μU = exp(−U )μ is an infinitesimal invariant (finite) measure for LU , since, by (2.3),
1 L f μ = (Lf ) exp(−U ) μ − Γ (U, f ) exp(−U ) μ 2 1 1 Γ f, exp(−U ) μ − Γ (f, U ) exp(−U ) μ = 0, =− 2 2 U
U
and notice that LU is symmetric on L2 (μU ) in the sense that 1 f LU g μU = gLU f μU = − Γ (f, g) μU 2 for all smooth compactly supported test functions f, g : G → R. The last equality is often referred to as integration by parts. By definition and by integration by parts the operator LU has a spectral gap at 0 of size a > 0 in L2 (μU ) if and only if Γ (f, f )(x) μU (dx) 2a f (x)2 μU (dx), G
G
for all compactly supported smooth functions f on G satisfying f (x) μU (dx) = 0. G
If we want to write an inequality for all test functions f , it reads like
Γ (f, f )(x) μ (dx) 2a G
μ (dx) −
U
U
f (x) μ (dx) G
2
U
G
2 (2.4) f (x) μ (dx) U
G
for all test functions f ∈ C0∞ (G). 3. Strong existence of OU-processes with values in Lie groups Let G be a finite-dimensional, connected Lie group. We consider now the special case of the ‘potential’ Wt (x) = − log p(t, x), t > 0, where p(t, x) is the density of the law of Xte with respect to μ. By Hörmander’s theorem [9,10], the function (t, x) → p(t, x) is a positive and smooth function, hence the potential Wt is as in the previous section. We write for short Lt instead of LWt and we call the associated Markov process the natural OU-process on G associated to the diffusion X. We show that we have in fact global strong solutions for the corresponding Stratonovich SDE with values in G. The next proposition is slightly more general.
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Proposition 3.1. Consider a smooth potential U : G → R such that exp −U (x) μ(dx) < ∞. Consider the following Stratonovich SDE with values in G: d y y y Vi Yt ◦ dBti , dYt = V0 Yt dt +
y
Y0 = y ∈ G,
(3.1)
i=1
where V0 f = − 12 Γ (U, f ) for smooth test functions f . Then there is a global strong solution to (3.1) for all initial values y ∈ G. Proof. Since the coefficients defining (3.1) are smooth by assumption, there exists a unique strong solution up to the explosion time
ζy = inf t: lim Yτy = ∞ . τ →t
We then define a semigroup Pt on L2 (μU ) by y (Pt f )(y) = E f Yt 1ζy >t .
(3.2)
It can be shown in exactly the same way as in [4,14] that Pt is a strongly continuous contraction semigroup and that its generator A coincides with LU on the set C0∞ (G) of compactly supported smooth functions. On the other hand, setting D(LU ) = C0∞ (G), one can show as in [4,14] that LU is essentially self-adjoint, so that one must have A = LU = (LU )∗ . In particular, since the constant function 1 belongs to L2 (G, μU ) by the integrability of exp(−U ) and since (LU ψ)(x) μU (dx) = 0 for any test function ψ ∈ C0∞ (G), 1 belongs to the domain of (LU )∗ and therefore also to the domain of A. This then implies that Pt 1 = 1 by the same argument as in [14]. In particular, coming back to the definition (3.2) of Pt , we see that P(ζy = ∞) = 1 for every y, which is precisely the non-explosion result that we were looking for. 2 Remark 3.2. While this argument shows that, given a fixed initial condition y, there exists a y unique global strong solution Yt to (3.1), it does not prevent more subtle kinds of explosions, see for example [6]. By Proposition 3.1 and since p(t, x) is smooth and integrable, it follows immediately that the OU-process exists globally in a strong sense. Corollary 3.3. For any given τ > 0, the process d y y y dYt = V0 Yt dt + Vi Yt ◦ dBti , i=1
with V0 f = − 12 Γ (Wτ , f ) has a global strong solution.
y
Y0 = y ∈ G,
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Remark 3.4. More traditional Lyapunov-function based techniques seem to be highly non-trivial to apply for this situation, due to the lack of information on the behaviour of U (y) at large y. In view of [1,11,12], it is tempting to conjecture that one has the asymptotic lim τ 2 ∂τ log p(τ, y) = d 2 (e, y),
τ →0
(3.3)
and that the limit holds uniformly over compact sets K that do not contain the origin e. (Note that it follows from [1] that this is true provided that K does not intersect the cut-locus.) If it were the case that (3.3) holds, possible space–time scaling properties of p(τ, x) could imply that, for / K. On the every τ > 0, there exists a compact set K such that Lp(τ, x) = ∂τ p(τ, x) > 0 for x ∈ other hand, one has 1 Lp(τ, .) , Lτ Wτ = − Γ (Wτ , Wτ ) + LWτ = − 2 p(τ, .) so that this would imply that Wτ is a Lyapunov function for the corresponding OU-process leading to another proof of the previous corollary. 4. Spectral gaps for natural OU-processes Next we consider the question if Lt admits a spectral gap on L2 (pt μ) for t > 0, which turns out to be a consequence of the Driver–Melcher inequality (see [15]). Theorem 4.1. The following assertions are equivalent: • The operator Lt has a spectral gap of size at > 0 on L2 (pt μ) for all t > 0, and a positive function a : R>0 → R>0 . • The local estimate 2 Pt Γ (f, f ) (g) 2at Pt f 2 (g) − (Pt f )(g) holds true for all test functions f : G → R, for all t > 0 and a positive function a : R>0 → R>0 at one (and therefore all) point g ∈ G. Furthermore, if we know that Γ (Pt f, Pt f )(e) φ(t)Pt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (G), all t 0, and a strictly positive locally integrable function φ : R0 → R>0 , then we can choose at by t at
1 φ(t − s) ds = , 2
0
for t > 0 and the two equivalent assertions hold true.
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Proof. Since μWt is equal to the law of Xte , one has f μWt = Pt f (e) for every f ∈ C0∞ (G). The equivalence of the first two statements then follows from (2.4) and the fact that the translation invariance of (2.1) implies that if the bound holds at some g, it must hold for all g ∈ G. We fix a test function f : G → R as well as t > 0 and consider H (s) = Ps (Pt−s f )2 for 0 s t. The derivative of this function equals H (s) = Ps Γ (Pt−s f, Pt−s f ) and therefore—assuming the third statement—we obtain H (s) φ(t − s)Pt Γ (f, f ) . Whence we can conclude t H (t) − H (0)
φ(t − s) dsPt Γ (f, f ) ,
0
which is the second of the two equivalent assertions for an appropriately chosen a.
2
Remark 4.2. We can replace the Lie group G by a general manifold M, on which we are given a hypo-elliptic, reversible diffusion X with “sum of the squares” generator L. Then the analogous statement holds, in particular local Poincaré inequalities on M for L lead to a spectral gap for the OU-type process Lt with t > 0. Corollary 4.3. Let G be a free, nilpotent Lie group with d generators e1 , . . . , ed of step m 1, and consider 1 2 ei , 2 d
L=
i=1
then the operator Lt has a spectral gap of size at =
1 2Kt
on L2 (pt μ) for some constant K > 0.
Proof. Due to the results of the very interesting PhD thesis [15] (see also [5]), there is a constant K such that the bound Γ (Pt f, Pt f )(e) KPt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (G) and for all times t 0. This shows that at Kt = 12 , due to the assertions of Theorem 4.1. 2
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Corollary 4.4. Let G = SU(2) be the Lie group of unitary matrices on C2 with Lie algebra g = su(2) = i, j, k R with the usual commutation relations, i.e. [i, j ] = 2k and its cyclic variants. Consider L=
1 2 i + j2 , 2
where we understand the elements i, j as left-invariant vector fields on G, then Lt has a spectral 1 gap of size at = 2Kt on L2 (pt μ) for some constant K > 0. Proof. The result follows from [3, Proposition 4.20].
2
4.1. Generalisation to homogeneous spaces Consider now M a compact homogeneous space with respect to the Lie group G, i.e. we have a (right) transitive action πˆ : G × M → M of G on M. We assume that there exists a measure μM on M which is invariant with respect to this action. We also assume that we are given a family V1 , . . . , Vd of left-invariant vector fields on G that generate its entire Lie algebra g as before. These vector fields induce fundamental vector fields ViM on M by means of the action πˆ . By choosing an ‘origin’ o ∈ M, we obtain a surjection π : G → M by π(g) = π(g, ˆ o). The vector fields V1 , . . . , Vd and V1M , . . . , VdM are consequently π -related. Due to the invariance of μM with respect to the action πˆ , the vector fields ViM are anti-symmetric operators on L2 (μM ) and the generator 1 M 2 Vi 2 d
LM =
i=1
is consequently symmetric on L2 (μM ). In particular we have M Vi f ◦ π = Vi (f ◦ π) for i = 1, . . . , d. The local Driver–Melcher inequality on G translates to the same inequality on M by means of PtM (f ) ◦ π = Pt (f ◦ π) for test functions f : M → R, hence we obtain the corresponding Driver–Melcher inequality on M with the same constants, too. 5. A simple result on simulated annealing By comparison with natural OU-processes on the homogeneous space M we can obtain spectral gap results for quite general classes of potentials. For later purposes, namely for applications to simulated annealing algorithms, we shall state a parametrized version of a simple perturbation result.
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Theorem 5.1. Let M be a homogeneous space. Let Vε : M → R be a family of potentials Vε with Vε + log p(ε, ·) Dε for 0 < ε 1, and constants Dε > 0, where p(t, ·) denotes the density of the invariant measure of the process Xt starting at x0 . Assume furthermore that a Poincaré inequality holds for Lε , i.e. aε Pε f 2 (e) Pε Γ (f, f ) (e)
(5.1)
for test functions f ∈ C0∞ (M) with Pε f (e) = 0 and some constant aε > 0 and 0 < ε 1. Then one has exp(−Vε ) ∈ L1 (μ) and the Poincaré inequality
f 2 (x) exp −Vε (x) μ(dx) Cε
Γ (f, f )(x) exp −Vε (x) μ(dx)
(5.2)
holds for all test functions f ∈ C0∞ (M) with f (x) exp(−Vε (x)) μ(dx) = 0 and some constant ε) Cε = exp(2D > 0. In particular, this leads to a spectral gap of size at least aε 1 aε = Cε exp(2Dε ) for LVε . Proof. It follows immediately from the inequality p(ε, x) = exp −Vε (x) exp Vε (x) p(ε, x) exp(−Dε ) exp −Vε (x) that exp(−Vε ) ∈ L1 (μ). Furthermore, exp −Vε (x) =
1 p(ε, x) exp(−Dε )p(ε, x) p(ε, x) exp(Vε (x))
for all x ∈ M by assumption. Hence we deduce (5.2) with Cε =
exp(2Dε ) aε
from (5.1).
2
Remark 5.2. See [2] for results on unbounded perturbations. Throughout the remainder of this section we assume that M is a compact homogeneous space with respect to a connected Lie group G. We consider the same structures as in Section 4.1 on M, but we omit the index M on vector fields and measures in order to improve readability. We shall furthermore impose the following assumption on the spectral gap. Assumption 5.3. There is a constant K > 0 such that Γ (Pt f, Pt f )(e) KPt Γ (f, f ) (e) holds true for all test functions f ∈ C0∞ (M) and 0 t 1.
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We prepare now a quantitative simulated annealing result under the previous Assumption 5.3 on M and follow closely the lines of [7]. Let U : M → R be a smooth potential. The idea is to search for minima of U by sampling the measure 1 U exp − 2 μ. Zε ε Recall that we do always assume that Zε = M exp(− εU2 )μ < ∞. Sampling this measure is performed by looking at the invariant measure of U 1 Lε = L − Γ 2 , · . 2 ε Notice that the previous operator satisfies 1 ε 2 Lε = ε 2 L − Γ (U, .), 2 and a spectral gap for ε 2 Lε is a spectral gap for the diffusion process d y y y dYt = V0 Yt dt + εVi Yt ◦ dBti ,
y
Y0 = y ∈ G,
i=1 y
with V0 f = − 12 Γ (U, f ). Consequently we know—given strong existence—that the law of Yt converges to Z1ε exp(− εU2 )μ and concentrates therefore around the minima of U . In this consideration ε is considered to be fixed. Next we try to obtain a time-dependent version of the previous considerations, leading to a process concentrating precisely at the minima of U . In the following theorem we quantify the speed of convergence towards the invariant measure. We denote by με the probability measure invariant for Lε and we use the notation 2 varε (f ) = f − f ε ε with f ε = M f (g) με (dg) for the variance with respect to this measure. First we estimate the spectral gap along a cooling schedule t → ε(t), then we prove that the measure concentrates around the minima of U even in a time-dependent setting. Theorem 5.4. Let U : M → R be a smooth functions, D a constant, x0 ∈ M a point such that U (x) + ε 2 log p ε 2 , x0 , x D, for all x ∈ M. Then there exist constants R, c > 0 such that for ε(t) = √
c log(R+t)
1 varε(t) (f ) K(R + t) Γ (f, f ) , 2 ε(t) for all test functions f ∈ C0∞ (M) and t 0.
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Proof. We can start to collect results. Combining Assumption 5.3 with Theorem 5.1 applied for Vε = εU2 , we obtain that spectral gap for the operator Lε has size at least 1 2D exp − 2 Kε 2 ε for 0 < ε 1, so that ε 2 Lε has a spectral gap of size at least 2D 1 exp − 2 . K ε We choose c2 = 2D and R sufficiently large so that ε(t) 1 for t 0, and we conclude that 2D K(R + t) K exp ε(t)2 for all t 0, which yields the desired result.
2
We denote by Z the process with cooling schedule t → ε(t) as in the previous theorem, d dZtz = V0 Ztz dt + ε(t)Vi Ztz ◦ dBti , i=1
where the drift vector field is given through V0 f = − 12 Γ (U, f ). Then the previous conclusion leads to the following proposition. Proposition 5.5. Let ft denote the Radon–Nikodym derivative of the law of Ztz with respect to με(t) and let u(t) := ft − 1 2L2 (μ
ε(t) )
denote its distance in L2 (με(t) ) to 1 (which corresponds to varε(t) (ft )), then u (t) −
N 2N 1 u(t) + 2 u(t) + 2 u(t) K(R + t) c (R + t) c (R + t)
for the constants R, c and K from Theorem 5.4, and N = max U − min U . Remark 5.6. We find c2 > 3N K, such that supt0 u(t) is bounded from above by a constant depending on f0 , c, R, N and K. Proof. The proof follows closely the lines of [7]. By assumption we know that varε(t) (ft ) = u(t) = ft 2L2 (μ ) − 1 and hence with the notation β(t) = 1 2 , ε(t)
ε(t)
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u (t) = − Γ (ft , ft ) ε(t) − β (t)
U − U ε(t) ft2 μt
= − Γ (ft , ft ) ε(t) − β (t) U − U ε(t) (ft − 1)2 μt − 2β (t) (U − U ε(t) )(ft − 1) μt
−
2N 1 N u(t) + 2 u(t). u(t) + 2 K(R + t) c (R + t) c (R + t)
Here, we used the Cauchy–Schwarz inequality and the conclusions of the previous Theorem 5.4 in the last line. 2 Assumption 5.7. Let U : M → R be a potential on M such that U (x) − d(x, x0 )2 D1 for some positive constant D1 , a point x0 ∈ M and for all x ∈ M. Here we denote by d(x, x0 ) the Carnot–Carathéodory metric on M, see for instance [11,12] and [16]. Remark 5.8. For non-compact manifolds M the limit lim t log p(t, x0 , x) = −d(x0 , x)2
t→0
is uniform on compact subsets of M, but usually not on the whole of M. An abelian, noncompact example where the limit is globally uniform is M = Rd . On the simplest non-compact and non-abelian example, the Heisenberg group G2d , the limit is not uniform, see recent work of H.-Q. Li [13]. Therefore we consider in our perturbation argument only compact manifolds M. On compact manifolds M we know due to R. Léandre’s beautiful results [11,12] that there is D2 such that d(x0 , x)2 + ε 2 log p ε 2 , x0 , x D2 for x ∈ M. Hence we can conclude by the triangle inequality that the potential U satisfies the assumptions of Theorem 5.4. Theorem 5.9. Assume Assumptions 5.3 and 5.7, and assume that sup ft L2 (με(t) ) < ∞. t0
Define U0 = infx∈M U (x) and, for every δ > 0, denote by Aδ the set Aδ = {x ∈ M | U (x) U0 + δ}. Then we can conclude that P Ztz ∈ Aδ M με(t) (Aδ ) for every t > 0 and every δ 0.
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Proof. It follows from the Cauchy–Schwarz inequality that P Ztz ∈ Aδ = ft με(t) M με(t) (Aδ ), Aδ
as required.
2
Remark 5.10. Since limε→0 με (Aδ ) = 0 for every δ > 0, we obtain that for all continuous bounded test functions f , we have E f Ztz → f (xmin ), provided that there is only one element xmin ∈ M such that U (xmin ) = U0 . Remark 5.11. We can improve the previous result from L2 -estimates to Lq -estimates for ft − 1 Lq (μt ) , for q > 2: this follows from [8, Theorem 2.2] and the fact that the proof of [8, Theorem 2.7] applies due to a valid Sobolev inequality, i.e. for every q > 2 there is a constant C0 > 0 such that 1 2 2 f Lq (μ) C0 Γ (f, f )μ + f L2 (μ) 2 M
holds for all test functions f : M → R. Such a Sobolev inequality can be found for instance in [9, Section 3] or in [16]. Remark 5.12. We can apply hypo-elliptic simulated annealing to potentials on compact nilmanifolds with the respective sub-Riemannian structure due to Corollary 4.3, or we can apply it to potentials on SU(2) due to Corollary 4.4. The implementation of those new algorithms can yield some advantages with respect to elliptic simulated annealing algorithms on Riemannian manifolds as described in [8]. On the one hand the number of Brownian motions involved is smaller, such as the complexity of the SDE d dZtz = V0 Ztz dt + ε(t)Vi Ztz ◦ dBti i=1
as a whole, since less vector fields have to evaluate and the gradient V0 is less complex being a horizontal gradient. The price to pay is a larger constant c > 0 in the rate of convergence. In cases where hypo-elliptic simulated annealing can be directly compared with elliptic simulated annealing on flat space (for instance on 3-torus T3 , where we have the flat Euclidean structure and the sub-Riemmanian structure of the Heisenberg torus with two generators) the elliptic algorithm is superior. This is due to the fact the one can choose the vector fields in the elliptic algorithm constant (on flat T3 ) which reduces the complexity considerably, whereas one has to apply more sophisticated vector fields in the case of the Heisenberg group. For optimization on SU(2), where our theory applies due to Corollary 4.4, we have a visible advantage over the elliptic simulated annealing, since we need one more Brownian motion and one more vector field for the elliptic algorithm, and we cannot simplify the vector fields in the elliptic algorithm.
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Acknowledgment We would like to thank an anonymous referee for helpful comments on a first version of our paper. References [1] G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4) 21 (3) (1988) 307–331. [2] D. Bakry, M. Ledoux, F.-Y. Wang, Perturbations of functional inequalities using growth conditions, J. Math. Pures Appl. (9) 87 (4) (2007) 394–407. [3] F. Baudoin, M. Bonnefont, The subelliptic heat-kernel on SU(2): Representations, asymptotics and gradient bounds, preprint, arXiv: 0802.3320. [4] P.R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Funct. Anal. 12 (1973) 401–414. [5] B.K. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, Stoch. Process. Appl., in press. [6] K.D. Elworthy, Stochastic dynamical systems and their flows, in: Stochastic Analysis, Proc. Internat. Conf., Northwestern Univ., Evanston, IL, 1978, Academic Press, New York, 1978, pp. 79–95. [7] R. Holley, D. Stroock, Simulated annealing via Sobolev inequalities, Comm. Math. Phys. 115 (4) (1988) 553–569. [8] R.A. Holley, S. Kusuoka, D.W. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal. 83 (2) (1989) 333–347. [9] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967) 147–171. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators, Classics in Math., Springer, Berlin, 2007. Reprint of the 1994 edition. [11] R. Léandre, Majoration en temps petit de la densité d’une diffusion dégénérée, Probab. Theory Related Fields 74 (2) (1987) 289–294. [12] R. Léandre, Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal. 74 (2) (1987) 399– 414. [13] H.-Q. Li, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg, C. R. Math. Acad. Sci. Paris. [14] X.-M. Li, Stochastic flows on noncompact manifolds, PhD thesis, 1992. [15] T. Melcher, Hypoelliptic heat kernel inequalities on Lie groups, PhD thesis, University of California at San Diego. [16] A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1) (1984) 143–160.
Journal of Functional Analysis 255 (2008) 891–939 www.elsevier.com/locate/jfa
Free pluriharmonic majorants and commutant lifting ✩ Gelu Popescu Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA Received 13 December 2007; accepted 30 April 2008 Available online 3 June 2008 Communicated by D. Voiculescu
Abstract In this paper we initiate the study of sub-pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball 1/2 <1 , B(H)n 1 := (X1 , . . . , Xn ) ∈ B(H)n : X1 X1∗ + · · · + Xn Xn∗ where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Several classical results from complex analysis have analogues in this noncommutative multivariable setting. We present basic properties for sub-pluriharmonic curves, characterize the class of sub-pluriharmonic curves that admit free pluriharmonic majorants and find, in this case, the least free pluriharmonic majorants. We show that, for any free holomorphic function Θ on [B(H)n ]1 , the map ϕ : [0, 1) → C ∗ (R1 , . . . , Rn ),
ϕ(r) := Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R1 , . . . , Rn on the full Fock space with n generators. We prove that Θ is in the noncommutative Hardy 2 if and only if ϕ has a free pluriharmonic majorant. In this case, we find Herglotz–Riesz and space Hball Poisson type representations for the least pluriharmonic majorant of ϕ. Moreover, we obtain a character2 and provide a parametrization and concrete representations for all free ization of the unit ball of Hball 2 . pluriharmonic majorants of ϕ, when Θ is in the unit ball of Hball In the second part of this paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem which extends, to our noncommutative multivariable setting, several lifting problems including the classical Sz.-Nagy–Foia¸s commutant lifting problem and the extensions obtained by Treil–Volberg, Foia¸s– Frazho–Kaashoek, and Biswas–Foia¸s–Frazho, as well as their multivariable noncommutative versions. We ✩
Research supported in part by an NSF grant. E-mail address: [email protected].
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.023
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solve the GNCL problem and, using the results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem. © 2008 Elsevier Inc. All rights reserved. Keywords: Multivariable operator theory; Noncommutative Hardy space; Fock space; Creation operators; Free holomorphic function; Free pluriharmonic function; Sub-pluriharmonic curves; Commutant lifting
0. Introduction Noncommutative multivariable operator theory has received a lot of attention, in the last two decades, in the attempt of obtaining a free analogue of Sz.-Nagy–Foia¸s theory [62], for row contractions, i.e., n-tuples of bounded linear operators (T1 , . . . , Tn ) on a Hilbert space such that T1 T1∗ + · · · + Tn Tn∗ I. Significant progress has been made regarding noncommutative dilation theory and its applications to interpolation in several variables [1,2,5,8,12,16,19,36–38,41,43,45,47,50–52,54–57], and unitary invariants for n-tuples of operators [3,4,6,38,48,49,51,56]. In [53] and [55], we developed a theory of holomorphic (resp. pluriharmonic) functions in several noncommuting (free) variables and provide a framework for the study of arbitrary ntuples of operators on a Hilbert space. Several classical results from complex analysis have free analogues in this noncommutative multivariable setting. This theory enhances our program to develop a free analogue of Sz.-Nagy–Foia¸s theory. In related areas of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials [25–29], and the work of Muhly and Solel on representations of tensor algebras over C ∗ correspondences (see [31,32]). The present paper is a natural continuation of [53] and [55]. We initiate here the study of subpluriharmonic curves and free pluriharmonic majorants on noncommutative balls. We are led 2 in terms of free pluriharmonic to a characterization of the noncommutative Hardy space Hball 2 . These results are used to majorants, and to a Schur type description of the unit ball of Hball solve a multivariable commutant lifting problem and provide a description of all solutions. To put our work in perspective and present our results, we need to set up some notation. Let F+ n be the unital free semigroup on n generators g1 , . . . , gn and the identity g0 . The length of α ∈ F+ n is defined by |α| := 0 if α = g0 and |α| := k if α = gi1 · · · gik , where i1 , . . . , ik ∈ {1, . . . , n}. If (X1 , . . . , Xn ) ∈ B(H)n , where B(H) is the algebra of all bounded linear operators on an infinitedimensional Hilbert space H, we denote Xα := Xi1 · · · Xik and Xg0 := IH . We recall (see [53, 55]) that a map f : [B(H)n ]1 → B(H) is called free holomorphic function with scalar coefficients if it has a representation
f (X1 , . . . , Xn ) =
∞ k=0 |α|=k
aα X α ,
X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
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where {aα }α∈F+n are complex numbers with lim supk→∞ ( |α|=k |aα |2 )1/2k 1. The function f ∞ if is in the noncommutative Hardy space Hball f ∞ :=
sup
X∈[B(H)n ]1
f (X) < ∞,
2 if f := ( and in Hball |aα |2 )1/2 < ∞. 2 α∈F+ n n We say that h : [B(H) ]1 → B(H) is a self-adjoint free pluriharmonic function on [B(H)n ]1 if h = Re f for some free holomorphic function f . An arbitrary free pluriharmonic function is a linear combination of self-adjoint free pluriharmonic functions. In the particular case when n = 1, a function X → h(X) is free pluriharmonic on [B(H)]1 if and only if the function λ → h(λ) is harmonic on the open unit disc D := {λ ∈ C: |λ| < 1}. If n 2 and h is a free pluriharmonic function on [B(H)n ]1 , then its scalar representation (z1 , . . . , zn ) → h(z1 , . . . , zn ) is a pluriharmonic function on the open unit ball Bn := {z ∈ Cn : z2 < 1}, but the converse is not true. As shown in [53,55], several classical results from complex analysis, regarding holomorphic (resp. harmonic) functions, have free analogues in this noncommutative multivariable setting. Let Hn be an n-dimensional complex Hilbert space with orthonormal basis e1 , e2 , . . . , en , where n ∈ {1, 2, . . .}. We consider the full Fock space of Hn defined by F 2 (Hn ) := C1 ⊕
Hn⊗k ,
k1
where Hn⊗k is the (Hilbert) tensor product of k copies of Hn . Define the left (resp. right) creation operators Si (resp. Ri ), i = 1, . . . , n, acting on F 2 (Hn ) by setting Si ϕ := ei ⊗ ϕ,
ϕ ∈ F 2 (Hn )
(resp. Ri ϕ := ϕ ⊗ ei , ϕ ∈ F 2 (Hn )). The noncommutative disc algebra An (resp. Rn ) is the norm closed algebra generated by the left (resp. right) creation operators and the identity. The noncommutative analytic Toeplitz algebra Fn∞ (resp. R∞ n ) is the weakly closed version of An (resp. Rn ). These algebras were introduced in [40] in connection with a noncommutative version of the classical von Neumann inequality [64] (see [35] for a nice survey). They have been studied in several papers [2,38,39,42–44,46], and recently in [12–15,17,48,49], and [53]. In Section 1, we introduce the class of sub-pluriharmonic curves and present basic properties. We prove that a self-adjoint (i.e. g(r) = g(r)∗ ) map g : [0, 1) → A∗n + An · is a subpluriharmonic curve in the Cuntz–Toeplitz C ∗ -algebra C ∗ (S1 , . . . , Sn ) (see [11]) if and only if g(r) P γr S g(γ )
for 0 r < γ < 1,
where PX [u] is the noncommutative Poisson transform of u at X. We obtain a characterization for the class of all sub-pluriharmonic curves that admit free pluriharmonic majorants, and prove the existence of the least pluriharmonic majorant. More precisely, we show that a self-adjoint mapping g : [0, 1) → A∗n + An · has a pluriharmonic majorant if and only if sup τ g(r) < ∞, 0
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where τ is the linear functional on B(F 2 (Hn )) defined by τ (f ) := f (1), 1 . In this case, there is a least pluriharmonic majorant for g, namely, the map [0, 1) r → u(rS1 , . . . , rSn ) ∈ A∗n + An · , where the free pluriharmonic function u is given by u(X1 , . . . , Xn ) := lim P 1 X g(γ ) γ →1
γ
for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 and the limit is in the norm topology. Let θ be an analytic function on the open disc D. It is well known that the map ϕ : D → R+ defined by ϕ(λ) := |θ (λ)|2 is subharmonic. A classical result on harmonic majorants (see [18, Section 2.6]) states that θ is in the Hardy space H 2 (D) if and only if ϕ has a harmonic majorant. Moreover, the least harmonic majorant of ϕ is given by the formula 1 h(λ) = 2π
2π 0
eit + λ it 2 θ e dt, eit − λ
λ ∈ D.
In Section 2, we obtain free analogues of these results. We show that, for any free holomorphic function Θ on the noncommutative ball [B(H)n ]1 , the mapping ϕ : [0, 1) → C ∗ (R1 , . . . , Rn ),
ϕ(r) = Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
is a sub-pluriharmonic curve in the Cuntz–Toeplitz algebra generated by the right creation operators R1 , . . . , Rn on the full Fock space with n generators. We prove that a free holomorphic func2 if and only if ϕ has a pluriharmonic majorant. tion Θ is in the noncommutative Hardy space Hball In this case, the least pluriharmonic majorant ψ for ϕ is given by ψ(r) := Re W (rR1 , . . . , rRn ), r ∈ [0, 1), where W is the free holomorphic function having the Herglotz–Riesz [30,58] type representation W (X1 , . . . , Xn ) = (μθ ⊗ id)
I+
n i=1
Ri∗
⊗ Xi
I−
n
−1 Ri∗
⊗ Xi
i=1
for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where μθ : R∗n + Rn → C is a positive linear map uniquely determined by Θ. Poisson type representations for the least pluriharmonic majorant are also considered. In Section 3, we provide a parametrization and concrete representations for all pluriharmonic 2 . We show that, majorants of the sub-pluriharmonic curve ϕ, where Θ is in the unit ball of Hball up to a normalization, all pluriharmonic majorants of ϕ have the form Re F , where F is a free holomorphic function given by −1 F (X) = W (X) + (DΓ ⊗ I ) I + G(X) I − G(X) (DΓ ⊗ I ),
X ∈ B(H)n 1 ,
∞ with coefficients in B(D ), where G is in the unit ball of the noncommutative Hardy space Hball Γ Γ is the symbol of Θ, and Re W is the least pluriharmonic majorant of ϕ considered above.
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Moreover, F and G uniquely determine each other. Using this result, we obtain a characterization 2 . More precisely, we show that Θ is a free holomorphic function in the of the unit ball of Hball 2 unit ball of Hball if and only if it has a Schur type representation Θ(X) = L(X) IH −
n
−1 Xi Mi (X)
for X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
i=1
where [ L M1 . . . Mn ]t (t is the transpose) is a free holomorphic function in the unit ball of ∞ with coefficients in Cn . the noncommutative Hardy space Hball We should mention that all the results of this paper are obtained in the more general setting of sub-pluriharmonic curves (resp. free pluriharmonic majorants) with operator-valued coefficients. The celebrated commutant lifting theorem (CLT), due to Sz.-Nagy and Foia¸s [61] for the general case and Sarason [59] for an important special case, provides the natural geometric framework for classical and modern H ∞ interpolation problems. For a detailed analysis of the CLT and its applications to various interpolation and extension problems, we refer the reader to the monographs [21] and [22]. In the present paper, we introduce a generalized noncommutative commutant lifting (GNCL) problem, which extends to a multivariable setting several lifting problems including the classical Sz.-Nagy–Foia¸s commutant lifting [61] and the extensions obtained by Treil–Volberg [63], Foia¸s–Frazho–Kaashoek [23], and Biswas–Foia¸s–Frazho [7], as well as their multivariable noncommutative versions [37,41,45,50,51,54]. While, in the classical case, there is a large literature regarding parametrizations and Schur [60] type representations of the set of all solutions in the CLT (see [20–23], etc.), very little is known in the noncommutative multivariable case (see [51]). In the present paper, we try to fill in this gap. A lifting data set {A, T , V , C, Q} for the GNCL problem is defined as follows. Let T := (T1 , . . . , Tn ), Ti ∈ B(H), be a row contraction and let V := (V1 , . . . , Vn ), Vi ∈ B(K), be the minimal isometric dilation of T on a Hilbert space K ⊃ H, in the sense of [37]. Let Q := (Q1 , . . . , Qn ), Qi ∈ B(Gi , X ), and C := (C1 , . . . , Cn ), Ci ∈ B(Gi , X ), be such that δij Ci∗ Cj n×n Q∗i Qj n×n . Consider a contraction A ∈ B(X , H) such that Ti ACi = AQi ,
i = 1, . . . , n.
We say that B is a contractive interpolant for A with respect to {A, T , V , C, Q} if B ∈ B(X , K) is a contraction satisfying the conditions PH B = A
and Vi BCi = BQi ,
i = 1, . . . , n,
where PH is the orthogonal projection from K onto H. The GNCL problem is to find contractive interpolants B of A with respect to the data set {A, T , V , C, Q}. In Section 4, we solve the GNCL problem and, using our results regarding sub-pluriharmonic curves and free pluriharmonic majorants on noncommutative balls, we provide a Schur type description of all solutions in terms of the elements of the unit ball of an appropriate noncom∞ (see Theorem 4.2). Our results are new, in particular, even in the mutative Hardy space Hball
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multivariable settings considered in [37,41,45,50,51], and [54]. Here we point out some remarkable special cases. If the data set {A, T , V , C, Q} is such that Gi = X , Ci = IX , Qi = Yi ∈ B(X ), for each i = 1, . . . , n, and Y := (Y1 , . . . , Yn ) is a row isometry, we obtain (see Theorem 4.8) a parametrization and a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem [37]. On the other hand, when Gi = X and Ci = IX for i = 1, . . . , n, we obtain a description of all solutions of the multivariable version [50] of Treil–Volberg commutant lifting theorem [63]. The GNCL theorem also implies the multivariable version [50] of the weighted commutant lifting theorem of Biswas–Foia¸s–Frazho [7]. Finally, we remark that, when applied to the interpolation theory setting, our results provide complete descriptions of all solutions to the Nevanlinna–Pick [33,34], Carathéodory–Fejér [9, ∞ 10], and Sarason [59] type interpolation problems for the noncommutative Hardy spaces Hball 2 , as well as consequences to (norm constrained) interpolation on the unit ball of Cn . and Hball These issues will be addressed in a future paper. 1. Sub-pluriharmonic curves in Cuntz–Toeplitz algebras In this section we initiate the study of sub-pluriharmonic curves and present some basic properties. We obtain a characterization for the class of all sub-pluriharmonic curves which admit free pluriharmonic majorants, and find the least pluriharmonic majorants. Some of the results of this section can be extended to the class of C ∗ -subharmonic curves. We need to recall from [55] a few facts concerning the noncommutative Berezin transform associated with a completely bounded linear map μ : B(F 2 (Hn )) → B(E), where E is a separable Hilbert space. This is the map
Bμ : B F 2 (Hn ) × B(H)n 1 → B(E) ⊗min B(H) defined by ∗ Bμ (f, X) := μ BX (f ⊗ IH )BX ,
f ∈ B F 2 (Hn ) , X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
where the operator BX ∈ B(F 2 (Hn )) ⊗ H) is given by
−1 BX := (IF 2 (Hn ) ⊗ ΔX ) I − R1 ⊗ X1∗ − · · · − Rn ⊗ Xn∗ , ΔX := (IH −
n
∗ 1/2 , i=1 Xi Xi )
and
μ := μ ⊗ id : B F 2 (Hn ) ⊗min B(H) → B(E) ⊗min B(H) is the completely bounded linear map uniquely defined by μ(f ⊗ Y ) := μ(f ) ⊗ Y for f ∈ B(F 2 (Hn )) and Y ∈ B(H). An important particular case is the Berezin transform Bτ , where τ is the linear functional on B(F 2 (Hn )) defined by τ (f ) := f (1), 1 . This will be called Poisson transform because it coincides with the noncommutative Poisson transform introduced in [46]. More precisely, we have Bτ (f, X) = PX (f ) := KX∗ (f ⊗ I )KX ,
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where KX = BX |1⊗H : H → F 2 (Hn ) ⊗ H is the noncommutative Poisson kernel. We recall from [46] that the restriction of PX to the C ∗ -algebra C ∗ (S1 , . . . , Sn ), generated by the left creation operators, can be extended to the closed ball [B(H)n ]− 1 by setting − X ∈ B(H)n 1 , f ∈ C ∗ (S1 , . . . , Sn ),
∗ (f ⊗ I )KrX , PX [f ] := lim KrX r→1
where rX := (rX1 , . . . , rXn ), r ∈ (0, 1), and the limit exists in the operator norm topology of B(H). In this case, we have
PX Sα Sβ∗ = Xα Xβ∗
for any α, β ∈ F+ n.
When X := (X1 , . . . , Xn ) is a pure n-tuple, i.e., |α|=k Xα Xα∗ → 0, as k → ∞, in the strong operator topology, then we have PX (f ) = KX∗ (f ⊗ I )KX . The operator-valued Poisson transform at X ∈ [B(H)n ]1 is the map PX : B(E ⊗ F 2 (Hn )) → B(E ⊗ H) defined by
PX [u] := IE ⊗ KX∗ (u ⊗ IH )(IE ⊗ KX ) for any u ∈ B(E ⊗ F 2 (Hn )). We refer to [46,48,49], and [56] for more on noncommutative Poisson transforms on C ∗ -algebras generated by isometries. Let Pn be the set of all polynomials in S1 , . . . , Sn and the identity, and let Pn denote (throughout the paper) the spatial tensor product B(E) ⊗ Pn . A pluriharmonic curve in the (spatial) tensor product B(E) ⊗min C ∗ (S1 , . . . , Sn ) is a map ϕ : [0, γ ) → Pn∗ + Pn · satisfying the Poisson mean value property, i.e., ϕ(r) = P rt S ϕ(t)
for 0 r < t < γ ,
where S := (S1 , . . . , Sn ). According to [55], there exists a one-to-one correspondence u → ϕ between the set of all free pluriharmonic functions on the noncommutative ball of radius γ , [B(H)n ]γ , and the set of all pluriharmonic curves ϕ : [0, γ ) → Pn∗ + Pn · in the tensor product B(E) ⊗min C ∗ (S1 , . . . , Sn ). Moreover, we have u(X) = P 1 X ϕ(r) r
for X ∈ B(H)n r and r ∈ (0, γ ),
and ϕ(r) = u(rS1 , . . . , rSn ) if r ∈ [0, γ ). We also solved in [55] a Dirichlet type extension problem which implies that a free pluriharmonic function u has continuous extension to the closed · such ∗ ball [B(H)n ]− γ if and only if there exists a pluriharmonic curve ϕ : [0, γ ] → Pn + Pn n that u(X) = P 1 X [ϕ(r)] for any X ∈ [B(H) ]r and r ∈ (0, γ ]. We add that u and ϕ uniquely r determine each other and satisfy the equations u(rS1 , . . . , rSn ) = ϕ(r) if r ∈ [0, γ ) and ϕ(γ ) = limr→γ u(rS1 , . . . , rSn ), where the convergence is in the operator norm topology. Now, we can introduce the class of sub-pluriharmonic curves in the tensor algebra B(E) ⊗min C ∗ (S1 , . . . , Sn ). We say that a map ψ : [0, 1) → Pn + Pn ·
with ψ(r) = ψ(r)∗ , r ∈ [0, 1),
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is a sub-pluriharmonic curve provided that for each γ ∈ (0, 1) and each self-adjoint pluriharmonic curve ϕ : [0, γ ] → Pn∗ + Pn · , if ψ(γ ) ϕ(γ ), then ψ(r) ϕ(r)
for any r ∈ [0, γ ].
Our first result is the following characterization of sub-pluriharmonic curves. Theorem 1.1. Let g : [0, 1) → Pn∗ + Pn · be a map with g(r) = g(r)∗ for r ∈ [0, 1). Then g is sub-pluriharmonic if and only if g(r) P γr S g(γ ) for 0 r < γ < 1. The equality holds if g is pluriharmonic. Proof. Assume that g is sub-pluriharmonic and let 0 r < γ < 1. Since g(γ ) ∈ Pn∗ + Pn · , one can use Theorem 4.1 from [55], to deduce that the map u : [B(H)n ]γ → B(E) ⊗min B(H) defined by u(X) = P 1 X g(γ ) γ
for X ∈ B(H)n γ
is free pluriharmonic on [B(H)n ]γ and has a continuous extension to [B(H)n ]− γ . In this case, we have g(γ ) = limt→γ u(tS1 , . . . , tSn ). Moreover, if ϕ : [0, γ ] → Pn∗ + Pn · is the pluriharmonic curve uniquely associated with the free pluriharmonic function u, then ϕ(r) = P γr S g(γ ) for r ∈ [0, γ ]. Since g is a sub-pluriharmonic curve and g(γ ) = ϕ(γ ), we deduce that g(r) ϕ(r) = P γr S g(γ ) for any r ∈ [0, γ ]. Conversely, assume that g has the property that g(r) P γr S g(γ ) for any 0 r < γ < 1.
(1.1)
Let ϕ : [0, γ ] → Pn∗ + Pn · be a pluriharmonic curve such that ϕ(r) = ϕ(r)∗ for r ∈ [0, γ ], and assume that g(γ ) ϕ(γ ). Since g(γ ) and ϕ(γ ) are in B(E) ⊗min C ∗ (S1 , . . . , Sn ) and the noncommutative Poisson transform is a positive map, we deduce that P γr S g(γ ) P γr S ϕ(γ ) ,
0 r < γ.
(1.2)
On the other hand, since ϕ is a pluriharmonic curve on [0, γ ], we have ϕ(r) = P γr S [ϕ(γ )] for 0 r < γ . Hence, using relations (1.1) and (1.2), we deduce that g(r) ϕ(r) for any r ∈ [0, γ ]. The proof is complete. 2 As a consequence of Theorem 1.1, we remark that if u1 , . . . , uk are sub-pluriharmonic curves and λ1 , . . . , λk are positive numbers then λ1 u1 + · · · + λk uk is sub-pluriharmonic. Notice also
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that a self-adjoint function u : [0, 1) → Pn∗ + Pn · is pluriharmonic if and only if both u and −u are sub-pluriharmonic. We recall (see [55, Lemma 2.3]) that if γ1 > 0 and 0 γj 1 for j = 2, . . . , k, then the noncommutative Poisson transform has the property Pγ1 ···γk X = Pγ1 X ◦ Pγ2 S ◦ · · · ◦ Pγk S for any X ∈ [B(H)n ] 1 , where S := (S1 , . . . , Sn ) is the n-tuple of left creation operators on the γ1
Fock space F 2 (Hn ). Moreover, we have Pγ1 ···γk X [g] = (Pγ1 X ◦ Pγ2 S ◦ · · · ◦ Pγk S )[g]
(1.3)
for any g ∈ B(E) ⊗min B(F 2 (Hn )). Corollary 1.2. Let g : [0, 1) → Pn∗ + Pn · be a sub-pluriharmonic curve and let τ be the linear functional on B(F 2 (Hn )) defined by τ (f ) := f (1), 1 . Then (i) P γr S [g(γ1 )] P γr S [g(γ2 )] for 0 < r < γ1 < γ2 < 1; 1
2
τ [g(γ2 )] for 0 < γ1 < γ2 < 1, where τ := τ ⊗ id; (ii) g(0) τ [g(γ1 )] (iii) τ [g(0)] τ [g(γ1 )] for 0 γ1 < 1. Proof. According to Theorem 1.1, we have g(r) P γr S [g(γ2 )]
for 0 r < γ2 < 1,
2
(1.4)
which implies g(γ1 ) P γ1 S g(γ2 ) γ2
for 0 r < γ1 < γ2 < 1.
Hence, using (1.3) and the positivity of the noncommutative Poisson transform, we deduce that P γr
1
S
g(γ1 ) (P γr
1
S
◦ P γ1 S ) g(γ2 ) = P γr S g(γ2 ) γ2
(1.5)
2
for 0 < r < γ1 < γ2 < 1. Passing to the limit in (1.5), as r → 0, and using the continuity of the noncommutative Berezin transform, we deduce that τ g(γ1 ) = P0 g(γ1 ) P0 g(γ2 ) = τ g(γ2 ) . Notice also that relation (1.4) implies g(0) P0 g(γ1 ) = τ g(γ1 ) Part (iii) is now obvious. This completes the proof.
for 0 < γ1 < 1.
2
If g is a sub-pluriharmonic curve on [0, 1) and h is pluriharmonic on the same interval such that g(r) h(r), r ∈ [0, 1), we say that h is a pluriharmonic majorant for g. The next result
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provides a characterization of all sub-pluriharmonic curves which admit free pluriharmonic majorants. In this case, we find the least pluriharmonic majorant. Theorem 1.3. Let g : [0, 1) → Pn∗ + Pn · be a sub-pluriharmonic curve. Then there exists a pluriharmonic majorant of g if and only if τ g(r) < ∞, sup
(1.6)
0
where τ := τ ⊗ id and τ is the linear functional on B(F 2 (Hn )) defined by τ (f ) := f (1), 1 . In this case, there is a least pluriharmonic majorant for g, namely, the map [0, 1) r → u(rS1 , . . . , rSn ) ∈ Pn∗ + Pn · , where the free pluriharmonic function u is given by u(X1 , . . . , Xn ) := lim P 1 X g(γ ) γ →1
γ
for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 and the limit is in the norm topology. Proof. Assume that u is a pluriharmonic majorant for g, i.e., g(γ ) u(γ )
for any γ ∈ [0, 1).
Since τ is a positive map, we deduce that τ [g(γ )] τ [u(γ )], γ ∈ [0, 1). According to Theorem 1.1, we have g(0) τ g(γ )
for γ ∈ (0, 1).
Since u is a pluriharmonic function, we have τ [u(γ )] = u(0). Using these relations, we deduce that g(0) τ g(γ ) u(0)
for γ ∈ (0, 1).
Taking into account that the operators g(0), u(0), and τ [g(γ )], γ ∈ (0, 1), are selfadjoint, one can easily obtain (1.6). Conversely, assume that relation (1.6) holds. Define hγ : [0, γ ) → Pn∗ + Pn · by setting hγ (r) := P γr S g(γ )
for 0 r < γ .
(1.7)
Since g is sub-pluriharmonic, hγ is a pluriharmonic majorant for g on [0, γ ). Notice that if f : [0, 1) → Pn∗ + Pn · is continuous and pluriharmonic on [0, γ ] such that f (r) g(r) for any r ∈ [0, γ ], then f (r) hγ (r) for r ∈ [0, γ ). Indeed, since f (γ ) g(γ ), the Poisson transform is a positive map, and f is pluriharmonic, we have f (r) = P γr S f (γ ) P γr S g(γ ) = hγ (r)
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for r ∈ [0, γ ). This shows that hγ is the least pluriharmonic majorant of g on [0, γ ). Now let 0 < γ < γ < 1. Since hγ is pluriharmonic majorant for g on [0, γ ), it is also a pluriharmonic majorant for g on [0, γ ). Due to our result above, we have hγ (r) hγ (r)
for any r ∈ [0, γ ).
Due to relation (1.7), we have hγ (0) = τ [g(γ )] for 0 < γ < 1 and, therefore, τ g(γ ) < ∞. sup hγ (0) = sup 0<γ <1
0<γ <1
Now, each hγ generates a unique pluriharmonic function on [B(H)n ]γ by setting uγ (X) := P 1 X hγ (t) t
for X ∈ B(H)n t and t ∈ [0, γ ).
(1.8)
If 0 < γ < γ < 1, then for X ∈ B(H)n γ .
uγ (X) uγ (X) Since
sup hγ (0) = sup uγ (0) < ∞, 0<γ <1
0<γ <1
we can use the Harnack type convergence theorem from [55] to deduce the existence of a pluriharmonic function u on [B(H)n ]1 such that its radial function satisfies u(r) := u(rS1 , . . . , rSn ) = limγ →1 uγ (r) for any r ∈ [0, 1), where the convergence is in the operator norm topology. On the other hand, using relation (1.8), the fact that hγ is pluriharmonic on [0, γ ), and relation (1.7), we have uγ (r) = P rt S hγ (t) = hγ (r) = P γr S g(γ )
for 0 r < t < γ .
Since hγ g on [0, γ ) for each γ ∈ (0, 1), we deduce that u g on [0, 1). If f is any pluriharmonic majorant for g on [0, 1), then, as previously shown, we have f hγ on [0, γ ) for each γ ∈ (0, 1). Hence, f u on [0, 1). Therefore, we have shown that u : [B(H)n ]1 → B(E) ⊗min B(H) defined by u(X) = lim P 1 X hγ (t) = lim P 1 X P t S g(γ ) γ →1
t
γ →1
γ
= lim P 1 X g(γ )
γ →1
t
γ
for X ∈ [B(H)n ]t and t ∈ (0, γ ), is a pluriharmonic function on [B(H)n ]1 . Moreover, its radial function u(t) = limr→1 P t S [g(r)], t ∈ [0, 1), is the least pluriharmonic majorant of g. This r completes the proof. 2 Now we can prove the following maximum principle for sub-pluriharmonic curves.
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Theorem 1.4. If g is a sub-pluriharmonic curve on [0, 1) and g(0) g(r) for any r ∈ [0, 1), then g is a constant. Proof. Since τ is a positive map, we have τ [g(r)] τ [g(0)] for r ∈ [0, 1). Hence and using Corollary 1.2, we deduce that g(0) τ g(r) τ g(0)
for r ∈ (0, 1).
(1.9)
τ [g(0)] − g(0). Moreover, ϕ is posOn the other hand, since g(0) is in Pn∗ + Pn · , so is ϕ := itive with τ (ϕ) = 0. Using Theorem 4.1 from [55], the pluriharmonic function h associated with ϕ, i.e., h(X) = PX [ϕ], X ∈ [B(H)n ]1 , is positive and h(0) = 0. Due to the maximum principle for free pluriharmonic functions (see [55, Theorem 2.9]), we deduce that h = 0. Since τ [g(0)]. Due to relation (1.9), ϕ = limr→1 h(rS1 , . . . , rSn ), we also have ϕ = 0, whence g(0) = we have τ g(r) = g(0)
for r ∈ [0, 1).
(1.10)
Hence and using the fact that g(0) g(r), r ∈ [0, 1), we deduce that τ [g(r)] − g(r) 0 for r ∈ (0, 1). A similar argument as before implies g(r) = τ [g(r)] for r ∈ (0, 1). Now taking into τ [g(0)] = g(0) for r ∈ account (1.10), we get g(r) = g(0) for r ∈ [0, 1). Therefore, τ [g(r)] = [0, 1). This completes the proof. 2 A few remarks are necessary. We say that a map ϕ : [0, γ ) → B(E) ⊗min C ∗ (S1 , . . . , Sn ) is a curve if it satisfies the Poisson mean value property, i.e.,
C ∗ -harmonic
ϕ(r) = P rt S ϕ(t)
for 0 r < t < γ .
According to [55], there exists a one-to-one correspondence u → ϕ between the set of all C ∗ harmonic functions on the noncommutative ball [B(H)n ]γ and the set of all C ∗ -harmonic curves ϕ : [0, γ ) → B(E) ⊗min C ∗ (S1 , . . . , Sn ). We say that a map ψ : [0, 1) → B(E) ⊗min C ∗ (S1 , . . . , Sn )
with ψ(r) = ψ(r)∗ , r ∈ [0, 1),
is a C ∗ -subharmonic curve, provided that for each γ ∈ (0, 1) and each C ∗ -harmonic curve ϕ on the closed interval on [0, γ ] such that ϕ(r) = ϕ(r)∗ for r ∈ [0, γ ], if ψ(γ ) ϕ(γ ), then ψ(r) ϕ(r) for any r ∈ [0, γ ]. We remark that Theorem 1.1 and Corollary 1.2 have analogues for C ∗ -subharmonic curves. Since the proofs are similar, we shall omit them. Notice also that any sub-pluriharmonic curve is C ∗ -subharmonic. Finally, we mention that all the results of this section can be written for sub-pluriharmonic curves in the tensor algebra B(E) ⊗min C ∗ (R1 , . . . , Rn ), where R1 , . . . , Rn are the right creation operators on the full Fock space. 2 2. Free pluriharmonic majorants and a characterization of Hball
In this section we show that, for any free holomorphic function Θ on [B(H)n ]1 with coefficients in B(E, Y), the mapping ϕ : [0, 1) → B(E) ⊗min C ∗ (R1 , . . . , Rn ),
ϕ(r) = Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
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2 if and only if ϕ has is a sub-pluriharmonic curve. We prove that Θ is in the Hardy space Hball a pluriharmonic majorant. In this case, we find Herglotz–Riesz and Poisson type representations for the least pluriharmonic majorant of ϕ. First, we introduce some notation. Recall that Hn is an n-dimensional complex Hilbert space with orthonormal basis e1 , e2 , . . . , en , and the full Fock space of Hn is defined by F 2 (Hn ) := C1 ⊕ k1 Hn⊗k . Let F+ n be the unital free semigroup on n generators g1 , . . . , gn , and the identity g0 . We denote eα := ei1 ⊗ · · · ⊗ eik if α = gi1 · · · gik , where i1 , . . . , ik ∈ {1, . . . , n}, and eg0 := 1. Note that {eα }α∈F+n is an orthonormal basis for F 2 (Hn ). We denote by α˜ the reverse ˜ = gik · · · gik if α = gi1 · · · gik ∈ F+ of α ∈ F+ n , i.e., α n. Let Θ : [B(H)n ]1 → B(E, Y) ⊗min B(H) be a free holomorphic function on [B(H)n ]1 with coefficients in B(E, Y) (in this case we denote Θ ∈ Hball (B(E, Y))). Assume that Θ has the representation
Θ(X1 , . . . , Xn ) :=
∞
A(α) ⊗ Xα .
(2.1)
k=0 |α|=k 2 We say that Θ is ∗in the noncommutative Hardy space Hball if there is a constant c > 0 such that α∈F+n A(α) A(α) cIE . When we want to emphasize that the coefficients of Θ are 2 (B(E, Y)). If we set Θ := in B(E, Y), we denote Θ ∈ Hball A∗(α) A(α) 1/2 < ∞, 2 α∈F+ n 2 , · ) is a Banach space. We associate with each Θ ∈ H 2 the operator Γ : E → then (Hball 2 ball 2 Y ⊗ F (Hn ) defined by
Γ x :=
A(α) x ⊗ eα˜ ,
x ∈ E.
(2.2)
α∈F+ n
We call Γ the symbol of Θ. We will see later that Γ x = limr→1 Θ(rR1 , . . . , rRn )(x ⊗ 1), x ∈ E. Conversely, if Γ is an operator given by (2.2), then relation (2.1) defines a free holomorphic 2 . Moreover, one can show that Θ ∈ H 2 and its symbol Γ uniquely determine function Θ in Hball ball each other. 2 (B(E, Y)) and let Γ be its symbol. Lemma 2.1. Let Θ be a free holomorphic function in Hball Then Θ has the state space realization
−1 n ∗ ∗ Θ(X1 , . . . , Xn ) = EY (IY ⊗ PC ) ⊗ IH IY ⊗ IF 2 (Hn )⊗H − Si ⊗ Xi (Γ ⊗ IH ) i=1
for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where EY : Y → Y ⊗ F 2 (Hn ) is defined by setting EY y = y ⊗ 1, and PC denotes the orthogonal projection of F 2 (Hn ) on C. Proof. Assume that Θ has the representation Θ(X1 , . . . , Xn ) := ∞ k=0 |α|=k A(α) ⊗ Xα for 2 some coefficients A(α) ∈ B(E, Y), and let Γ : E → Y ⊗ F (Hn ) be its symbol defined by relation (2.2). Notice that
∗ (IY ⊗ PC ) IY ⊗ Sα∗˜ Γ, A(α) = EY
α ∈ F+ n.
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Therefore, we have Θ(X1 , . . . , Xn ) =
∞ ∗
EY (IY ⊗ PC ) IY ⊗ Sα∗˜ Γ ⊗ Xα k=0 |α|=k
∞
∗ ∗ IY ⊗ Sα˜ ⊗ Xα (Γ ⊗ IH ) = EY (IY ⊗ PC ) ⊗ IH ∗ (IY ⊗ PC ) ⊗ IH = EY
k=0 |α|=k
IY ⊗F 2 (Hn )⊗H −
n
−1 IY ⊗ Si∗ ⊗ Xi
(Γ ⊗ IH )
i=1
for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 . This completes the proof.
2
Now we introduce a large class of sub-pluriharmonic functions. Theorem 2.2. Let Θ be a free holomorphic function on [B(H)n ]1 with coefficients in B(E, Y). Then the map ϕ(r) := Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
r ∈ [0, 1),
is a sub-pluriharmonic curve in the tensor algebra B(E) ⊗min C ∗ (R1 , . . . , Rn ). 2 and let Γ be its symbol (see Proof. First, assume that Θ is a free holomorphic function in Hball n (2.2)). Define the free holomorphic function on [B(H) ]1 by setting
n
∗ ∗ W (X1 , . . . , Xn ) := Γ ⊗ IH IY ⊗F 2 (Hn )⊗H + IY ⊗ Si ⊗ Xi i=1
IY ⊗F 2 (Hn )⊗H −
n
IY ⊗ Si∗
−1
⊗ Xi
(Γ ⊗ IH ).
(2.3)
i=1
Consider the noncommutative Cauchy kernel Φ(X1 , . . . , Xn ) := IY ⊗F 2 (Hn )⊗H −
n
−1 IY ⊗ Si∗ ⊗ Xi
,
(X1 , . . . , Xn ) ∈ B(H)n 1 .
i=1
Notice that Φ is a free holomorphic function on [B(H)n ]1 and Φ(X1 , . . . , Xn ) = IY ⊗F 2 (Hn )⊗H + Φ(X1 , . . . , Xn ) = IY ⊗F 2 (Hn )⊗H +
n
IY ⊗ Si∗ ⊗ Xi
i=1 n i=1
IY ⊗ Si∗
⊗ Xi Φ(X1 , . . . , Xn ).
(2.4)
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A closer look at the definition of the free holomorphic function W (see (2.3)) reveals that n
∗
∗ W (X1 , . . . , Xn ) = Γ Γ ⊗ IH + 2 Γ ⊗ IH IY ⊗ Si∗ ⊗ Xi Φ(X1 , . . . , Xn )(Γ ⊗ IH ). i=1
(2.5) Now, we use Lemma 2.1 when H := F 2 (Hn ) and Xi := rRi , i = 1, . . . , n, for r ∈ [0, 1). Note that due to the fact that PC = IF 2 (Hn ) − S1 S1∗ − · · · − Sn Sn∗ and using relation (2.4), we deduce that Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) −1 n
∗ ∗ IY ⊗ Si ⊗ rRi = Γ ⊗ IF 2 (Hn ) IY ⊗F 2 (Hn )⊗F 2 (Hn ) − i=1
× (IY ⊗ PC ⊗ IF 2 (Hn ) ) IY ⊗F 2 (Hn )⊗F 2 (Hn ) −
n
−1 IY ⊗ Si∗ ⊗ rRi
i=1
n
∗ ∗ ∗ = Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ IF 2 (Hn ) − Si Si ⊗ IF 2 (Hn ) i=1
× Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) )
∗ = Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn )∗ Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) ) n ∗
∗ ∗ − Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ Si Si ⊗ IF 2 (Hn ) i=1
× Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) ) n ∗
∗ ∗ = Γ ⊗ IF 2 (Hn ) IY ⊗F 2 (Hn )⊗F 2 (Hn ) + IY ⊗ Si ⊗ rRi Φ(rR1 , . . . , rRn ) i=1
× IY ⊗F 2 (Hn )⊗F 2 (Hn ) + Φ(rR1 , . . . , rRn )
n
IY ⊗ Si ⊗ rRi∗
(Γ ⊗ IF 2 (Hn ) )
i=1
n
∗ ∗ ∗ Si Si ⊗ IF 2 (Hn ) − Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ i=1
× Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) )
n ∗
∗
∗ = Γ Γ ⊗ IF 2 (Hn ) + Γ ⊗ IF 2 (Hn ) IY ⊗ Si ⊗ rRi i=1 ∗
× Φ(rR1 , . . . , rRn ) (Γ ⊗ IF 2 (Hn ) )
n ∗
∗ + Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ Si ⊗ rRi (Γ ⊗ IF 2 (Hn ) ) i=1
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
n
∗ ∗ ∗ + Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ Sj ⊗ rRj ×
n
j =1
IY ⊗ Si∗ ⊗ rRi Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) )
i=1
n
∗ ∗ ∗ − Γ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn ) IY ⊗ Sj Sj ⊗ IF 2 (Hn ) j =1
× Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) ). Hence and using relation (2.5), we deduce that Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn )
1 = W (rR1 , . . . , rRn ) + W (rR1 , . . . , rRn )∗ − 1 − r 2 Γ ∗ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn )∗ 2 n ∗ × IY ⊗ Sj Sj ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn )(Γ ⊗ IF 2 (Hn ) ) (2.6) j =1
for any r ∈ [0, 1). Consequently, we have Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn )
1 W (rR1 , . . . , rRn ) + W (rR1 , . . . , rRn )∗ 2
(2.7)
for any r ∈ [0, 1), which proves that W is a free holomorphic function with positive real part. For each s ∈ [0, 1), define the operator Λs : Y ⊗ F 2 (Hn ) → Y ⊗ F 2 (Hn ) by setting Λs
α∈F+ n
hα ⊗ eα := hα ⊗ s |α| eα ,
α∈F+ n
α∈F+ n
hα 2 < ∞.
It is easy to see that Λs is a positive operator such that Λs 1 and lims→1 Λs = I in the strong operator topology. Note also that Λs (IY ⊗ Si ) = s(IY ⊗ Si )Λs ,
i = 1, . . . , n.
(2.8)
Fix s ∈ [0, 1) and set Θs (X1 , . . . , Xn ) := Θ(sX1 , . . . , sXn ) for X1 , . . . , Xn ) ∈ [B(H)n ]1 . It clear that Θs is free holomorphic on an open neighborhood of [B(H)n ]1 and, therefore, continuous on the closed ball [B(H)n ]− 1 . Let Γs be the symbol operator associated with Θs (see relation (2.2)) and let Ws be the operator associated to Γs (by relation (2.5)). Then we have Γs = Λs Γ , where Γ is the symbol associated with Θ. Due to relations (2.5) and (2.8), we deduce that n
∗
∗ 2 ∗ Ws (X1 , . . . , Xn ) = Γ Λs Γ ⊗ IH + 2 Γ Λs ⊗ IH IY ⊗ Si ⊗ Xi i=1
× Φ(X1 , . . . , Xn )(Λs Γ ⊗ IH )
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
907
n
∗ 2
∗ 2 ∗ = Γ Λ s Γ ⊗ IH + 2 Γ Λ s ⊗ IH IY ⊗ Si ⊗ sXi i=1
× Φ(sX1 , . . . , sXn )(Γ ⊗ IH ). Consequently, Ws is a free holomorphic function on an open set containing the closed ball [B(H)n ]− 1 , and SOT- lim Ws = W. s→1
(2.9)
Now, from the first part of the proof (see (2.6)), we have Re Ws (rR1 , . . . , rRn ) − Θs (rR1 , . . . , rRn )∗ Θs (rR1 , . . . , rRn ) n
= 1 − r 2 Γs∗ ⊗ IF 2 (Hn ) Φ(rR1 , . . . , rRn )∗ IY ⊗ r 2 Sj Sj∗ ⊗ IF 2 (Hn ) j =1
× Φ(rR1 , . . . , rRn )(Γs ⊗ IF 2 (Hn ) )
= 1 − r 2 Γ ∗ ⊗ IF 2 (Hn ) Φ(srR1 , . . . , srRn )∗ (Λs ⊗ IF 2 (Hn ) ) n 2 ∗ × IY ⊗ r Sj Sj ⊗ IF 2 (Hn ) (Λs ⊗ IF 2 (Hn ) )Φ(srR1 , . . . , srRn )(Γ ⊗ IF 2 (Hn ) ) j =1
for any r ∈ [0, 1). Hence we deduce that Re Ws (rR1 , . . . , rRn ) Θs (rR1 , . . . , rRn )∗ Θs (rR1 , . . . , rRn )
(2.10)
for any r ∈ [0, 1). Moreover, taking into account the continuity (in the norm operator topology) of the free holomorphic functions Θs and Ws on the closed ball [B(H)n ]− 1 , the continuity of the map [0, 1] r → Φ(srR1 , . . . , srRn ), we deduce that Re Ws (R1 , . . . , Rn ) = Θs (R1 , . . . , Rn )∗ Θs (R1 , . . . , Rn )
(2.11)
for any s ∈ [0, 1). Now, we prove that the map ϕ is a sub-pluriharmonic curve. Fix s ∈ (0, 1) and assume that u is a free holomorphic function on [B(H)n ]s and continuous on the closed ball [B(H)n ]− s such that Θ(sR1 , . . . , sRn )∗ Θ(sR1 , . . . , sRn ) Re u(sR1 , . . . , sRn ).
(2.12)
2 , let W be the free holomorphic function Let t ∈ (s, 1) and set Θ := Θt . Since Θ is in Hball associated with Θ (according to relation (2.3)). Now, we can apply the results above (see (2.11)) to Θ and W and obtain
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
Re Wτ (R1 , . . . , Rn ) = Θτ (R1 , . . . , Rn )∗ Θτ (R1 , . . . , Rn )
(2.13)
for any τ ∈ [0, 1). Using (2.13) and (2.12), we have Re W s (R1 , . . . , Rn ) = Θ s (R1 , . . . , Rn )∗ Θ s (R1 , . . . , Rn ) t
t
t
= Θ(sR1 , . . . , sRn )∗ Θ(sR1 , . . . , sRn ) Re u(sR1 , . . . , sRn ). Employing the noncommutative Poisson transform, we deduce that Re W s (γ R1 , . . . , γ Rn ) = Pγ R Re W s (R1 , . . . , Rn ) t t Pγ R Re u(sR1 , . . . , sRn ) Re u(γ sR1 , . . . , γ sRn ) for any γ ∈ [0, 1). Consequently, applying inequality (2.10) to Θ and W , we have Θ s (γ R1 , . . . , γ Rn )∗ Θ s (γ R1 , . . . , γ Rn ) Re W s (γ R1 , . . . , γ Rn ) Re u(γ sR1 , . . . , γ sRn ) t
t
t
for any γ ∈ [0, 1). Hence, we deduce that Θ(sγ R1 , . . . , sγ Rn )∗ Θ(sγ R1 , . . . , sγ Rn ) Re u(sγ R1 , . . . , sγ Rn ) for any γ ∈ [0, 1). Therefore, Θ(tR1 , . . . , tRn )∗ Θ(tR1 , . . . , tRn ) Re u(tR1 , . . . , tRn ) for any t ∈ [0, s]. This proves that ϕ is a sub-pluriharmonic curve. The proof is complete.
2
Let P (m) , m = 0, 1, . . . , be the set of all polynomials of degree m in e1 , . . . , en , i.e., 2 P (m) := span eα : α ∈ F+ n , |α| m ⊂ F (Hn ), (m)
and define the nilpotent operators Ri
: P (m) → P (m) by
Ri(m) := PP (m) Ri |P (m) ,
i = 1, . . . , n,
where R1 , . . . , Rn are the right creation operators on the Fock space F 2 (Hn ) and PP (m) is the (m) orthogonal projection of F 2 (Hn ) onto P (m) . Notice that Rα = 0 if |α| m + 1. 2 in terms of We can provide now a characterization of the noncommutative Hardy space Hball pluriharmonic majorants.
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
909
Theorem 2.3. Let Θ be a free holomorphic function on [B(H)n ]1 with coefficients in B(E, Y). 2 if and only if the map ϕ defined by Then Θ is in Hball ϕ(r) := Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
r ∈ [0, 1),
has a pluriharmonic majorant. In this case, the least pluriharmonic majorant ψ for ϕ is given by ψ(r) := Re W (rR1 , . . . , rRn ),
r ∈ [0, 1),
(2.14)
where W is the free holomorphic function having the Herglotz–Riesz type representation W (X1 , . . . , Xn ) = (μθ ⊗ id)
I+
n
Ri∗
⊗ Xi
I−
i=1
n
−1 Ri∗
⊗ Xi
(2.15)
i=1
for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where μθ : R∗n + Rn → B(E) is the completely positive linear map uniquely determined by the equation ∗
∗ ∗ μθ R α x, y := lim Θ(rR1 , . . . , rRn ) IY ⊗ S α Θ(rR1 , . . . , rRn )(x ⊗ 1), (y ⊗ 1)
r→1
(2.16)
for α ∈ F+ n and x, y ∈ E. Proof. According to Theorem 2.2, ϕ is a sub-pluriharmonic curve in B(E) ⊗min C ∗ (R1 , . . . , Rn ). Assume that ϕ has a pluriharmonic majorant. Let u be a free pluriharmonic function on [B(H)n ]1 with coefficients in B(E) such that ϕ(r) u(rR1 , . . . , rRn ) for any r ∈ [0, 1). Let Θ have the representation Θ(X1 , . . . , Xn ) :=
∞
A(α) ⊗ Xα ,
k=0 |α|=k
(X1 , . . . , Xn ) ∈ B(H)n 1 ,
(2.17)
and let u have the representation u(X1 , . . . , Xn ) =
∞
∗ B(α) ⊗ Xα∗ +
k=1 |α|=k
∞
B(α) ⊗ Xα ,
k=0 |α|=k
(X1 , . . . , Xn ) ∈ B(H)n 1 .
Notice that B(0) 0 and, for each h ∈ E and r ∈ [0, 1), we have α∈F+ n
2 r 2|α| A(α) h2 = Θ(rR1 , . . . , rRn )(h ⊗ 1) = ϕ(r)(h ⊗ 1), h ⊗ 1 u(rR1 , . . . , rRn )(h ⊗ 1), h ⊗ 1 = u(0)(h ⊗ 1), h ⊗ 1 = B(0) h, h .
Hence, we deduce that there is a constant c > 0 such that 2 . that Θ is in Hball
α∈F+ n
A∗(α) A(α) cIE . This shows
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2 . A closer look at the proof of Theorem 2.2 (see relation Conversely, assume that Θ is in Hball (2.7)) shows that Re W is a pluriharmonic majorant for ϕ, where W is given by relation (2.3). It remains to show that Re W is the least pluriharmonic majorant for ϕ and satisfies relation (2.15). Let G be a free holomorphic function on [B(H)n ]1 with coefficients in B(E) such that Re G 0 and
Θ(tR1 , . . . , tRn )∗ Θ(tR1 , . . . , tRn ) Re G(tR1 , . . . , tRn ) for any t ∈ [0, 1). If s ∈ (0, 1), then we have Θs (R1 , . . . , Rn )∗ Θs (R1 , . . . , Rn ) Re G(sR1 , . . . , sRn ). Using (2.11), we deduce that Re Ws (R1 , . . . , Rn ) Re G(sR1 , . . . , sRn ). Taking the compression to E ⊗ P (m) (see the definition preceding this theorem), we obtain (m)
(m)
Re Ws R1 , . . . , Rn(m) Re G sR1 , . . . , sRn(m) . Now, using relation (2.9) and taking s → 1, we obtain (m)
(m)
Re W R1 , . . . , Rn(m) Re G R1 , . . . , Rn(m) for any m = 0, 1, . . . . According to [55], we deduce that Re W Re G, which shows that the map ψ(r) := Re W (rR1 , . . . , rRn ), r ∈ [0, 1), is the least pluriharmonic majorant for ϕ. Notice that, due to relation (2.5), we have W (X1 , . . . , Xn ) =
∞ k=0 |α|=k
C(α) ⊗ Xα ,
(X1 , . . . , Xn ) ∈ B(H)n 1 ,
where C(0) = Γ ∗ Γ
∗ and C(α) = 2Γ ∗ IY ⊗ S α Γ
if |α| 1.
(2.18)
Using the definition of Γ and the representation (2.17), we deduce that, for any α ∈ F+ n and x, y ∈ E,
∗ ∗ A(β) x ⊗ eβ˜ , IY ⊗ S A(γ ) x ⊗ eγ˜ Γ IY ⊗ S α Γ x, y = α
∗
γ ∈Fn
β∈Fn
=
A(β) x, A(γ ) y eβ˜ , eα˜ γ˜
β,γ ∈F+ n
=
γ ∈F+ n
A(γ α) x, A(γ ) y .
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
911
∗ Therefore Γ ∗ (IY ⊗ S A(γ ) A(γ α) , α ∈ F+ n , where the convergence is in the week γ ∈F+ α )Γ = n operator topology. On the other hand, notice that the limit in (2.16) exists. Indeed, since Θ is in the Hardy space 2 , we have Aβ x2 cx2 for some constant c > 0. This implies Hball β∈F+ n lim
r→1
r |β| A(β) x ⊗ eβ˜ =
β∈Fn
A(β) x ⊗ eβ˜ ,
β∈Fn
whence
∗ lim Θ(rR1 , . . . , rRn )∗ IY ⊗ S α Θ(rR1 , . . . , rRn )(x ⊗ 1), (y ⊗ 1) r→1
|γ | |β| ∗ r A(β) x ⊗ eβ˜ , IY ⊗ S r A(γ ) x ⊗ eγ˜ = α γ ∈Fn
β∈Fn
=
∗
A(β) x ⊗ eβ˜ , IY ⊗ S α
A(γ ) x ⊗ eγ˜
γ ∈Fn
β∈Fn
∗ = Γ ∗ IY ⊗ S α Γ x, y .
∗ ∗ ∗ ∗ + Therefore μθ (R α ) = Γ (IY ⊗ S α )Γ for any α ∈ Fn . Conseα ) = Γ (IY ⊗ S α )Γ and μθ (R quently,
μθ p(R1 , . . . , Rn ) = Γ ∗ IY ⊗ p(S1 , . . . , Sn ) Γ for any polynomial p(R1 , . . . , Rn ) = |α|m (bα Rα∗ + aα Rα ) in R∗n + Rn . This implies that μθ has a unique extension to a completely positive linear map on R∗n + Rn . According to relation (2.18), we have ∗ if |α| 1. and C(α) = 2μθ R α
C(0) = μθ (I )
Hence and due to the fact that μθ is a bounded linear map, we have μθ
I+
n
Ri∗
⊗ Xi
∞ k=1 |α|=k
= μθ (I ) + 2
∞ k=1 |α|=k
=
−1 Ri∗
⊗ Xi
i=1
= μθ I + 2
∞
I−
i=1
n
Rα∗˜
⊗ Xα
μ Rα∗˜ ⊗ Xα
C(α) ⊗ Xα = W (X1 , . . . , Xn ),
k=0 |α|=k
which proves relation (2.15). This completes the proof.
2
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We recall [55] that, in the particular case when f = I , the identity on F 2 (Hn ), and μ is a bounded linear functional on C ∗ (R1 , . . . , Rn ), then the Berezin transform Bμ (I, ·) coincides with the noncommutative Poisson transform Pμ : [B(H)n ]1 → B(H) associated with μ, i.e., (Pμ)(X1 , . . . , Xn ) := (μ ⊗ id) P (R, X) ,
X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
where the Poisson kernel is given by P (R, X) :=
∞
∗ R α ⊗ Xα + I +
k=1 |α=k
∞ k=1 |α=k
∗ R α ⊗ Xα
and the series are convergent in the operator norm topology. In the particular case when n = 1, H = C, X = reiθ ∈ D, and μ is a complex Borel measure on T, the Poisson transform Pμ can be identified with the classical Poisson transform of μ, i.e., 1 2π
π Pr (θ − t) dμ(t), −π
2
1−r where Pr (θ − t) := 1−2r cos(θ−t)+r 2 is the Poisson kernel. Using now Theorems 1.3 and 2.3, we can deduce the following result. 2 and have the representation Corollary 2.4. Let Θ be in Hball
Θ(X1 , . . . , Xn ) :=
∞
A(α) ⊗ Xα ,
k=0 |α|=k
(X1 , . . . , Xn ) ∈ B(H)n 1 .
Under the conditions of Theorem 2.3, the least pluriharmonic majorant Re W satisfies the relations Re W (X1 , . . . , Xn ) = lim P 1 X Θ(tR1 , . . . , tRn )∗ Θ(tR1 , . . . , tRn ) t→1
t
= (Pμθ )(X1 , . . . , Xn ) =
∞
∗ D(α) ⊗ Xα∗ +
k=1 |α|=k
∞
D(α) ⊗ Xα
k=0 |α|=k
for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where Pμθ is the noncommutative Poisson transform of μθ and D(α) =
A(γ ) A(γ α) ,
γ ∈F+ n
where the convergence is in the week operator topology.
α ∈ F+ n,
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2 , geometric structure, and representations 3. The unit ball of Hball 2 and provide a parametrizaIn this section we obtain a characterization of the unit ball of Hball tion and concrete representations of all pluriharmonic majorants for the sub-pluriharmonic curve
ϕ(r) := Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ),
r ∈ [0, 1),
2 . where Θ is in the unit ball of Hball We need a few notations. As in the previous section, we denote by Hball (B(E, Y)) the set of all free holomorphic functions on the noncommutative ball [B(H)n ]1 and coefficients in B(E, Y). ∞ (B(E, Y)) denote the set of all elements F in H Let Hball ball (B(E, Y)) such that
F ∞ := supF (X1 , . . . , Xn ) < ∞, where the supremum is taken over all n-tuples of operators (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where + (B(E)) the set of all free holomorH is an infinite-dimensional Hilbert space. Denote by Hball n phic functions f on the noncommutative ball [B(H) ]1 with coefficients in B(E), where E is a separable Hilbert space, such that Re f 0. Consider the following sets:
+ H+ 1 B(E) := f ∈ Hball B(E) : f (0) = I
∞ H∞ 0 B(E) := g ∈ Hball B(E) : g(0) = 0 .
and
According to [57], the noncommutative Cayley transform is a bijection
∞
C : H+ 1 B(E) → H0 B(E) 1 defined by C[f ] := g, −1 where g ∈ [H∞ 0 (B(E))]1 is uniquely determined by the formal power series (f − 1)(1 + f ) , where f is the power series associated with f . In this case, we have −1 C −1 [G](X) = I + G(X) I − G(X) ,
X ∈ B(H)n 1 .
We recall that if T : H → H is a contraction, then DT := (I − T ∗ T )1/2 and DT := DT H. The first result of this section provides a parametrization for the pluriharmonic majorants of Θ ∗ Θ. 2 (B(E, Y)), and let F Theorem 3.1. Let Θ be a free holomorphic function in the unit ball of Hball be a free holomorphic function in Hball (B(E)) such that F (0) = I . Then
Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) Re F (rR1 , . . . , rRn )
for r ∈ [0, 1),
(3.1)
if and only if there exists G in the unit ball of H∞ 0 (B(DΓ )) such that −1 F (X) = W (X) + (DΓ ⊗ I ) I + G(X) I − G(X) (DΓ ⊗ I ),
X ∈ B(H)n 1 , (3.2)
where Γ is the symbol of Θ and W is defined by relation (2.3). Moreover, F and G uniquely determine each other.
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Proof. Let G be in the unit ball of H∞ 0 (B(DΓ )) and define F by relation (3.2). Due to (2.5), we have W (0) = Γ ∗ Γ ⊗ IH . Since G(0) = I , we deduce that −1 F (0) = W (0) + (DΓ ⊗ IH ) I + G(0) I − G(0) (DΓ ⊗ IH )
= Γ ∗ Γ + DΓ2 ⊗ IH = I. According to Theorem 2.2 (see (2.7)), we have Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) Re W (rR1 , . . . , rRn )
for r ∈ [0, 1).
(3.3)
Due to the properties of the noncommutative Cayley transform, we have Re C −1 [G](rR1 , . . . , rRn ) 0 for any r ∈ [0, 1). Consequently, we have Re F (rR1 , . . . , rRn ) = Re W (rR1 , . . . , rRn ) + (DΓ ⊗ I ) Re C −1 [G](rR1 , . . . , rRn ) (DΓ ⊗ I ) Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) for any r ∈ [0, 1). Therefore, F satisfies inequality (3.1). Conversely, assume that F is a free holomorphic function with F (0) = I and satisfying relation (3.1). According to Theorem 2.3, Re F (rR1 , . . . , rRn ) Re W (rR1 , . . . , rRn )
for any r ∈ [0, 1).
Hence, Ψ := F − W is a free holomorphic function with positive real part and
Ψ (0) = F (0) − W (0) = I − Γ ∗ Γ ⊗ IH = DΓ2 ⊗ IH . We claim that Ψ has a unique factorization of the form Ψ (X) = (DΓ ⊗ IH )Λ(X)(DΓ ⊗ IH ),
X ∈ B(H)n 1 ,
(3.4)
that Λ(0) = I and where Λ is a free holomorphic function with coefficients in B(DΓ ) such Re Λ 0. To this end, assume that Ψ has the representation Ψ (X) = α∈F+n Q(α) ⊗ Xα , Q(α) ∈ B(E), with Q(0) = DΓ2 . For each k 1, consider the subspace M := span{1, eα : α ∈ F+ n and |α| = k}. Notice that, for each r ∈ [0, 1), the positive operator PE ⊗M Ψ (rR1 , . . . , rRn )∗ + Ψ (rR1 , . . . , rRn ) E ⊗M has the operator matrix representation ⎡
2DΓ2 ⎡ |α| ⎢ r Q(α) ⎤ ⎢ M(r) := ⎢ ⎢ ⎥ .. ⎣⎣ ⎦ . |α| = k
[r |α| Q(α) : |α| = k] ⎤ ⎡ 2D 2 · · · 0 ⎤⎥ Γ ⎥ , ⎢ . .. ⎥ ⎥ .. ⎦ ⎦ ⎣ .. . . 0 · · · 2DΓ2
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
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where [r |α| Q(α) : |α| = k] is the row operator with the entries r |α| Q(α) , |α| = k. The notation for the column operator is now obvious. Taking r → 1, we deduce that M := M(1) 0. We recall (see [24]) that a block operator matrix BA∗ CB , where A ∈ B(H), B ∈ B(G, H), and C ∈ B(G), is positive if and only if A and C are positive and there exists a contraction T : CG → AH satisfying B ∗ = C 1/2 T ∗ A1/2 . Applying this result to the operator M, we find a contraction ⎡ Z ⎤ (α) ⎢ .. ⎥ DΓ ⎣ . ⎦ : DΓ → |α|=k |α| = k such that ⎡ Q ⎤ ⎡ 2DΓ (α) ⎢ .. ⎥ ⎢ .. ⎣ . ⎦=⎣ . 0 |α| = k
··· .. . ···
⎤⎡ Z ⎤ 0 (α) .. ⎥ ⎢ .. ⎥ . ⎦ ⎣ . ⎦ DΓ 2DΓ |α| = k
for each k 1. Hence, we have Q(α) = DΓ K(α) DΓ , where K(0) = I and K(α) := 2Z(α) for any ∗ α ∈ F+ n with |α| = k 1. Since |α|=k Z(α) Z(α) I , we deduce that 1 ∗ 2k K(α) K(α) lim 1. k→∞ |α|=k
This implies (see [53]) that Λ(X1 , . . . , Xn ) =
k=0 |α|=k
K(α) ⊗ Xα ,
(X1 , . . . , Xn ) ∈ B(H)n 1 ,
is a free holomorphic function on [B(H)n ]1 . Using the relations above, we deduce the factorization Ψ (X) = (DΓ ⊗ IH )Λ(X)(DΓ ⊗ IH ), X ∈ [B(H)n ]1 . Now, since the operator Λ(rR1 , . . . , rRn ), r ∈ [0, 1), is acting on the Hilbert space DΓ ⊗ F 2 (Hn ) and Ψ (rR1 , . . . , rRn )∗ + Ψ (rR1 , . . . , rRn ) = (DΓ ⊗ IF 2 (Hn ) ) Λ(rR1 , . . . , rRn )∗ + Λ(rR1 , . . . , rRn ) (DΓ ⊗ IF 2 (Hn ) ), we deduce that Λ(rR1 , . . . , rRn )∗ + Λ(rR1 , . . . , rRn ) 0 for any r ∈ [0, 1). Using the noncommutative Poisson transform, we have Re Λ 0, which proves our claim. Due to the properties of the Cayley transform, G := C[Λ] is in the unit ball of H0∞ (B(DΓ )). Finally, since Ψ := F − W and using the factorization (3.4), we deduce (3.2). The fact that F and G uniquely determine each other is now obvious since the Cayley transform is a bijection. This completes the proof. 2 Corollary 3.2. Under the conditions of Theorem 3.1, there is only one F satisfying (3.1) if and only if Γ is an isometry. In this case, F = W .
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∞ (B(E, Y)) can be identified According to [53] and [55], the noncommutative Hardy space Hball ¯ Fn∞ (the weakly closed operator space generated by the spatial to the operator space B(E, Y) ⊗ tensor product), where Fn∞ is the noncommutative analytic Toeplitz algebra (see [39,40,43]). We consider the noncommutative Schur class
∞ B(E, Y) : G∞ 1 , Sball B(E, Y) := G ∈ Hball ¯ Fn∞ . We also use the which can be identified to the unit ball of the operator space B(E, Y) ⊗ notation
0 B(E, Y) := G ∈ Sball B(E, Y) : G(0) = 0 . Sball In what follows we use the notation E (n) for the direct sum of n-copies of the Hilbert space E. 2 . The next results provides a Schur type representation for the unit ball of Hball 2 (B(E, Y)) be such that Θ 1, and let Γ : E → Y ⊗ F 2 (H ) be Theorem 3.3. Let Θ ∈ Hball 2 θ n its symbol. Then there is a one-to-one correspondence JΘ between ⎡ the ⎤ noncommutative Schur
class
(n) Sball (B(DΓθ , DΓθ ))
L M1
and the set of all function matrices ⎣ .. ⎦ in the noncommutative .
Schur class Sball (B(E, Y ⊕ E (n) )) satisfying the equation
−1 n Θ(X) = L(X) IE ⊗H − (IE ⊗ Xi )Mi (X)
Mn
for X := (X1 , . . . , Xn ) ∈ B(H)n 1 . (3.5)
i=1
More precisely, the map
(n)
JΘ : Sball B DΓθ , DΓθ → Sball B E, Y ⊕ E (n) is defined by setting ⎡ ⎤ L ⎤ ϕ1 ⎢M ⎥ 1⎥ ⎢ .. ⎥ ⎢ ⎥ JΘ ⎣ . ⎦ = ⎢ ⎢ .. ⎥ , ⎣ . ⎦ ϕn Mn ⎡
where L ∈ Hball (B(E, Y)) is given by −1 , L(X) = 2Θ(X) F (X) + I
X ∈ B(H)n 1 ,
the free holomorphic function F ∈ Hball (B(E)) is defined by n ∗
∗ F (X) := Γθ ⊗ IH IY ⊗F 2 (Hn )⊗H + IY ⊗ Si ⊗ Xi i=1
(3.6)
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
× IY ⊗F 2 (Hn )⊗H −
−1 IY ⊗ Si∗
⊗ Xi
(Γθ ⊗ IH )
i=1
+ (DΓθ
n
917
−1 n n ⊗ IH ) I + (IE ⊗ Xi )ϕi (X) I − (IE ⊗ Xi )ϕi (X) (DΓθ ⊗ IH ), i=1
i=1
(3.7) and M1 , . . . , Mn ∈ Hball (B(E)) are uniquely determined by the equation −1 (IE ⊗ X1 )M1 + · · · + (IE ⊗ Xn )Mn = F (X) − I F (X) + I , X := (X1 , . . . , Xn ) ∈ B(H) 1 .
(3.8)
In particular, the representation (3.5) is unique if and only if Γθ is an isometry. ϕ .1 Proof. Assume that Θ ∈ Hball (B(E, Y)) and Θ2 1. Consider Φ := .. in the noncomϕn (n)
(n)
mutative Schur class Sball (B(DΓθ , DΓθ )). It is easy to see that Φ ∈ Sball (B(DΓθ , DΓθ )) if and only if the function χ(X) := (IE ⊗ X1 )ϕ1 (X) + · · · + (IE ⊗ Xn )ϕn (X),
X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
0 (B(D , D )). Since is in Sball Γθ Γθ
I+
n i=1
(IE ⊗ Xi )ϕi (X)
−1 n I− (IE ⊗ Xi )ϕi (X) i=1
is the noncommutative Cayley transform of χ , it makes sense to define F by relation (3.7). According to Theorem 3.1, F ∈ Hball (B(E) has the properties F (0) = I , Re F 0, and Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) Re F (rR1 , . . . , rRn )
for r ∈ [0, 1).
(3.9)
Since F ∈ H+ 1 (B(E)), the noncommutative Cayley transform of F , i.e., C[F ] := (F − I )(F + −1 I ) , is in H∞ 0 (B(E)) and C[F ]∞ 1. Consequently, there are some unique M1 , . . . , Mn ∈ Hball (B(E)) such that C[F ](X) = (IE ⊗ X1 )M1 (X) + · · · + (IE ⊗ Xn )Mn (X),
X := (X1 , . . . , Xn ) ∈ B(H) 1 .
Since C[F ]∞ 1, we deduce that ni=1 (IE ⊗rSi )Mi (rS1 , . . . , rSn ) 1 for r ∈ [0, 1). Taking into account that S1 , . . . , Sn are isometries with orthogonal ranges, we obtain the inequality n i=1
r 2 Mi (rS1 , . . . , rSn )∗ Mi (rS1 , . . . , rSn ) I.
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Hence, we deduce that ⎡ ⎡ ⎤ ⎤ M1 (rS1 , . . . , rSn ) M1 ⎢ ⎢ .. ⎥ ⎥ .. ⎣ . ⎦ = lim ⎣ 1, ⎦ . r→1 Mn ∞ Mn (rS1 , . . . , rSn )
M1
.. .
which shows that
is in the noncommutative Schur class Sball (B(E, E (n) )). Now, consider
Mn
the free holomorphic function Ψ (X) := (IE ⊗ X1 )M1 (X) + · · · + (IE ⊗ Xn )Mn (X),
X := (X1 , . . . , Xn ) ∈ B(H) 1 , (3.10)
and notice that I − Ψ = I − C[F ](X) = (F + I ) − (F − I ) (F + I )−1 = 2(F + I )−1 .
(3.11)
Hence, and defining L by relation (3.6), we deduce that L = Θ(I − Ψ ). Consequently, L ∈ Hball (B(E, Y)) and −1 n (IE ⊗ Xi )Mi (X) Θ(X) = L(X) IE ⊗H −
for X := (X1 , . . . , Xn ) ∈ B(H)n 1 .
i=1
⎡
L M1
⎤
Therefore, relation (3.5) holds. Now we show that ⎣ .. ⎦ is in the Schur class Sball (B(E, Y ⊕ . Mn
E (n) )). Since F is the inverse Cayley transform of Ψ , i.e., F = (I + Ψ )(I − Ψ )−1 , for any r ∈ [0, 1), we have Re F (rR1 , . . . , rRn )
−1
1 I − Ψ (rR1 , . . . , rRn )∗ = I + Ψ (rR1 , . . . , rRn )∗ 2
−1 + I + Ψ (rR1 , . . . , rRn ) I − Ψ (rR1 , . . . , rRn )
−1
1 I − Ψ (rR1 , . . . , rRn )∗ I + Ψ (rR1 , . . . , rRn )∗ I − Ψ (rR1 , . . . , rRn ) 2
−1 + I − Ψ (rR1 , . . . , rRn )∗ I + Ψ (rR1 , . . . , rRn ) I − Ψ (rR1 , . . . , rRn )
−1 = I − Ψ (rR1 , . . . , rRn )∗ I − Ψ (rR1 , . . . , rRn )∗ Ψ (rR1 , . . . , rRn )
−1 × I − Ψ (rR1 , . . . , rRn ) . =
Hence, and using relations (3.5), (3.9), and (3.10), we obtain
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
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−1
−1 I − Ψ (rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn ) I − Ψ (rR1 , . . . , rRn ) = Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) Re F (rR1 , . . . , rRn )
−1 I − Ψ (rR1 , . . . , rRn )∗ Ψ (rR1 , . . . , rRn ) = I − Ψ (rR1 , . . . , rRn )∗
−1 × I − Ψ (rR1 , . . . , rRn ) for any r ∈ [0, 1). Consequently, we deduce that Ψ (rR1 , . . . , rRn )∗ Ψ (rR1 , . . . , rRn ) + L(rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn ) I for any r ∈ [0, 1). Due to relation (3.10), we have Ψ (rR1 , . . . , rRn )∗ Ψ (rR1 , . . . , rRn ) =
n
r 2 Mi (rR1 , . . . , rRn )∗ Mi (rR1 , . . . , rRn )
i=1
for r ∈ [0, 1). Combining these relations, we deduce that r2
n Mi (rR1 , . . . , rRn )x 2 + L(rR1 , . . . , rRn )x 2 x2
(3.12)
i=1 ∞ (B(N , M)) can be identified to for any x ∈ E ⊗ F 2 (Hn ) and any r ∈ [0, 1). Remember that Hball ¯ Rn∞ for any Hilbert spaces N and M. In particular, since L and the operator space B(N , M) ⊗ i ∈ B(E) ⊗ ¯ R∞ Mi are bounded free holomorphic functions, according to [55], there exist M n and ∞ ¯ Rn such that L ∈ B(E, Y) ⊗
i = SOT- lim Mi (rR1 , . . . , rRn ) M r→1
Passing to the limit in (3.12), we obtain ⎡ L ⎤
n
= SOT- lim L(rR1 , . . . , rRn ). and L r→1
i=1 Mi x
2 + Lx 2
x2 for any x ∈ E ⊗ F 2 (Hn ). ⎡ L⎤ M
M1 ⎣ .. 1 ⎦ is in ¯ R∞ Therefore, ⎣ .. ⎦ is a contraction in B(E, Y ⊕ E (n) ) ⊗ n , which shows that . . n Mn M ϕ .1 the noncommutative Schur class Sball (B(E, Y ⊕ E (n) )). We remark that .. and F uniquely ϕn M1 .. determine each other by Theorem 3.1. On the other hand, F and . uniquely determine Mn
each other via the noncommutative Cayley transform. Therefore, ⎡ Jθ is⎤a one-to-one mapping. L M
1 Now, let us prove the surjectivity of the correspondence Jθ . Let ⎣ .. ⎦ be in the Schur class .
Mn
Sball (B(E, Y ⊕ E (n) )) such that relation (3.5) holds. Therefore, we have
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L(rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn ) +
n
Mi (rR1 , . . . , rRn )∗ Mi (rR1 , . . . , rRn ) I. (3.13)
i=1
Hence
M1
.. .
is in Sball (B(E, E (n) )), which implies that the free holomorphic function
Mn
Ψ (X) := (IE ⊗ X1 )M1 (X) + · · · + (IE ⊗ Xn )Mn (X),
X := (X1 , . . . , Xn ) ∈ B(H) 1 ,
0 (B(E)). Let F be the inverse Cayley transform of Ψ . Then Re F 0, F (0) = I , and, is in Sball due to relations (3.5) and (3.13), we have
Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn )
−1
−1 = I − Ψ (rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn )∗ L(rR1 , . . . , rRn ) I − Ψ (rR1 , . . . , rRn )
−1 I − Ψ (rR1 , . . . , rRn )∗ Ψ (rR1 , . . . , rRn ) I − Ψ (rR1 , . . . , rRn )∗
−1 × I − Ψ (rR1 , . . . , rRn ) = Re F (rR1 , . . . , rRn ) for any r ∈ [0, 1). The latter equality was proved before, using the fact that F is the inverse Cayley transform of Ψ . In particular, since F (0) = I , we can use the inequality above to deduce that Θ(rR1 , . . . , rRn )(h ⊗ 1)2 F (0)(h ⊗ 1), h ⊗ 1 = h2 for any h ∈ E and r ∈ [0, 1). Hence Θ2 1. Moreover, since Θ(rR1 , . . . , rRn )∗ Θ(rR1 , . . . , rRn ) Re F (rR1 , . . . , rRn ),
r ∈ [0, 1),
we can apply Theorem 3.1 to show that F has the form (3.7) for some ⎤ ϕ1 ⎢ .. ⎥ ⎣ . ⎦ ⎡
(n)
in Sball B DΓθ , DΓθ .
ϕn Now, since F is the inverse Cayley transform of Ψ we have Ψ = (F − I )(F + I )−1 . Notice also that due to relations (3.5) and (3.11), we deduce that −1 , L(X) = 2Θ(X) F (X) + I
X ∈ B(H)n 1 .
ϕ ⎡ L ⎤ M1 .1 Therefore, JΘ .. = ⎣ .. ⎦. Finally, the representation (3.5) is unique if and only if DΓθ = 0, . ϕn Mn
i.e., Γθ is an isometry. The proof is complete.
2
A closer look at the proof of Theorem 3.3 reveals that we have also proved the following result.
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
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Corollary 3.4. Let Θ : [B(H)n ]1 → B(E, Y) ⊗min B(H) be a function. Then Θ is a free holo2 (B(E, Y)) with Θ 1 if and only if it has a representation morphic function in Hball 2
−1 n Θ(X) = L(X) IE ⊗H − (IE ⊗ Xi )Mi (X)
for X := (X1 , . . . , Xn ) ∈ B(H)n 1 ,
i=1
⎡
L M1
⎤
where ⎣ .. ⎦ is in the noncommutative Schur class Sball (B(E, Y ⊕ E (n) )). . Mn
4. All solutions of the generalized noncommutative commutant lifting problem In [23], Foia¸s, Frazho, and Kaashoek obtained a relaxed commutant lifting theorem (RCLT) which extends the celebrated commutant lifting theorem (CLT) due to Sz.-Nagy–Foia¸s [61] for the general case and to Sarason [59] for an important special case. The RCLT leads to solutions of many metric constrained interpolation problems and their H 2 versions (see [22]). We begin by recalling the RCL problem. Let A : H → H and T : H → H be contractions and let R : H0 → H and A : H0 → H be bounded operators satisfying the constraints T AR = AQ
and R ∗ R Q∗ Q.
The RCL problem is to find all contractions B : H → K such that U BR = BQ and PH B = A, where U is the minimal isometric dilation of T on a Hilbert space K ⊃ H and PH is the orthogonal projection of K onto H . We point out a few remarkable particular cases. (i) When H0 = H, R = IH and Q = T is an isometry on H, one can see that the RCLT implies the Sz.-Nagy–Foia¸s commutant lifting theorem. (ii) If R = I , we obtain the Treil–Volberg [63] version of the CLT. (iii) The RCLT implies the weighted CLT of Biswas, Foia¸s, and Frazho [7]. We remark that, when applied to interpolation, the RCLT provides new interpolation problems with variations on the norm constraint. In this section, we introduce a generalized noncommutative commutant lifting (GNCL) problem, which extends to a multivariable setting all the above-mentioned commutant lifting theorems, as well as their multivariable noncommutative versions obtained in [37,41,45,50,51,54]. We solve the GNCL problem and, using the results regarding sub-pluriharmonic functions and free pluriharmonic majorants on noncommutative balls, we provide a complete description of all solutions. In particular, we obtain a concrete Schur type description of all solutions in the noncommutative commutant lifting theorem. An n-tuple T := (T1 , . . . , Tn ) of bounded linear operators acting on a common Hilbert space H is called contractive (or row contraction) if T1 T1∗ + · · · + Tn Tn∗ IH .
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The defect operators associated with T are ΔT ∗ := IH −
n
1/2 Ti Ti∗
∈ B(H)
and ΔT :=
1/2
δij IH − Ti∗ Tj n×n ∈ B H(n) ,
i=1 (n) (n) ∗ ∗ while n the defect spaces of T are D∗ = DT := ΔT H and D = DT := ΔT H , where H := i=1 H denotes the direct sum of n copies of H. We say that an n-tuple V := (V1 , . . . , Vn ) of isometries on a Hilbert space K ⊃ H is a minimal isometric dilation of T if the following properties are satisfied:
(i) V1 V1∗ + · · · + Vn Vn∗ IK ; (ii) Vi∗ |H = Ti∗ , i = 1, . . . , n; (iii) K = α∈F+n Vα H. The isometric dilation theorem for row contractions (see [8,19,37]) asserts that every row contraction T has a minimal isometric dilation V , which is uniquely determined up to an isomorphism. Let Δi : H → D ⊗ F 2 (Hn ) be defined by Δi h := [ΔT (0, . . . , 0, h, 0, . . . , 0) ⊗ 1] ⊕ 0 ⊕ 0 ⊕ · · · . ! "# $ i−1 times
Consider the Hilbert space K := H ⊕ (D ⊗ F 2 (Hn )) and embed H and D in K in the natural way. For each i = 1, . . . , n, define the operator Vi : K → K by
Vi h ⊕ (ξ ⊗ d) := Ti h ⊕ Δi h + (ID ⊗ Si )(ξ ⊗ d)
(4.1)
for any h ∈ H, ξ ∈ F 2 (Hn ), d ∈ D, where S1 , . . . , Sn are the left creation operators on the full Fock space F 2 (Hn ). The n-tuple V := (V1 , . . . , Vn ) is a realization of the minimal isometric dilation of T . Note that & % 0 Ti (4.2) Vi = Δi ID ⊗ Si with respect to the decomposition K = H ⊕ [D ⊗ F 2 (Hn )]. Let us introduce our generalized noncommutative commutant lifting (GNCL) problem. A lifting data set {A, T , V , C, Q} for the GNCL problem is defined as follows. Let T := (T1 , . . . , Tn ), Ti ∈ B(H), be a row contraction and let V := (V1 , . . . , Vn ), Vi ∈ B(K), be the minimal isometric dilation of T on a Hilbert space K ⊃ H given by (4.2). Let Q := (Q1 , . . . , Qn ), Qi ∈ B(Gi , X ), and C := (C1 , . . . , Cn ), Ci ∈ B(Gi , X ), be such that δij Ci∗ Cj n×n Q∗i Qj n×n .
(4.3)
Let A ∈ B(X , H) be a contraction such that Ti ACi = AQi ,
i = 1, . . . , n.
(4.4)
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
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We say that B is a contractive interpolant for A with respect to {A, T , V , C, Q} if B ∈ B(X , K) is a contraction satisfying the conditions PH B = A
and Vi BCi = BQi ,
i = 1, . . . , n,
where PH is the orthogonal projection from K onto H. The GNCL problem is to find contractive interpolants B of A with respect to the data set {A, T , V , C, Q}. Note that B satisfies relation PH B = A if and only if it has a matrix decomposition %
A B= Γ DA
&
: X → H ⊕ D ⊗ F 2 (Hn ) ,
(4.5)
where Γ : DA → D ⊗ F 2 (Hn ) is a contraction. Here DA := (IX − A∗ A)1/2 and DA := DA X . We mention that B and Γ determine each other uniquely. Note that B satisfies the equations Vi BCi = BQi for i = 1, . . . , n, if and only if % &% & & % A 0 A Ti Ci = Qi , i = 1, . . . , n, Γ DA Δi ID ⊗ Si Γ DA which, due to relation (4.4), is equivalent to Δi ACi + (ID ⊗ Si )Γ DA Ci = Γ DA Qi ,
i = 1, . . . , n.
(4.6)
Therefore, the GNCL problem is equivalent to finding contractions Γ : DA → D ⊗ F 2 (Hn ) such that relation (4.6) holds. Using relations (4.3) and (4.4), we deduce that n 2 n 2 n 2 DA Qi yi = Qi yi − AQi yi i=1
i=1
n i=1
=
n i=1
i=1
2 n Ci yi 2 − Ti AQi yi i=1
2 2 n n n 2 2 ACi yi − Ti AQi yi + Ci yi − ACi yi i=1
i=1
i=1
n 2 n = ΔT ACi yi + DA Ci yi 2 i=1
i=1
for any yi ∈ Gi , i = 1, . . . , n. Consider the subspace F ⊂ DA given by ' F :=
n i=1
(− DA Qi yi : yi ∈ Gi , i = 1, . . . , n
.
(4.7)
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939 (n)
Due to the estimations above, we can introduce the operator Ω : F → D ⊕ DA by Ω := where Ω1 and Ω2 are defined as follows: Ω1 : F → D,
Ω1
n
DA Qi yi := ΔT
i=1
Ω2 : F →
n
DA ,
Ω2
i=1
n
n
)
Ω1 Ω2
* ,
and
ACi yi
i=1
DA Qi yi :=
n
i=1
(4.8)
D A Ci y i .
i=1
Consequently, Ω is a contraction. We remark that Ω is an isometry if and only if we have equality in (4.3). Now, notice that n
Δi (ACi yi ) = ΔT
i=1
n
ACi yi ⊗ 1,
yi ∈ Gi .
i=1
It is clear that relation (4.6) is equivalent to ΔT
n
ACi yi ⊗ 1 +
i=1
n n (ID ⊗ Si )Γ DA Ci yi = Γ DA Qi yi , i=1
yi ∈ Gi ,
(4.9)
i=1
which is equivalent to Ω1
n
DA Qi yi ⊗ 1 + [ID ⊗ S1 , . . . , ID ⊗ Sn ]
i=1
=Γ
n
n
Γ Ω2
i=1
n
DA Qi yi
i=1
DA Qi yi .
i=1
Therefore, we have ⎤ Γ P1 ⎢ . ⎥ ED Ω1 + [ID ⊗ S1 , . . . , ID ⊗ Sn ] ⎣ .. ⎦ Ω2 = Γ |F , ⎡
(4.10)
Γ Pn (n)
where ED : D → D ⊗ F 2 (Hn ) is defined by ED y = y ⊗ 1 and Pj : DA → DA is the orthogonal (n) projection of DA onto its j th coordinate. Let B be a fixed solution of the GNCL problem and let Γ : DA → D ⊗ F 2 (Hn ) be the unique contraction determined by B (see (4.5)). Since relation (4.6) holds and S1 , . . . , Sn are isometries with orthogonal ranges, we deduce that 2 n 2 n 2 n DΓ DA Qi yi = DA Qi yi − Γ DA Qi yi i=1
i=1
i=1
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
925
2 n 2 n 2 n = DA Qi yi − Δi ACi yi − (ID ⊗ Si )Γ DA Ci yi i=1
i=1
i=1
2 n n 2 n = DA Qi yi − ΔT ACi yi − Γ DA Ci yi 2 i=1
i=1
i=1
2 n n 2 n 2 = DΓ DA Ci yi + DA Qi yi − ΔT ACi yi i=1
−
i=1
n
i=1
DA Ci yi 2
i=1
=
n i=1
2 n n 2 DΓ DA Ci yi + DA Qi yi − Ω DA Qi yi 2
i=1
n 2 n 2 = DΓ DA Ci yi + DΩ DA Qi yi i=1
n
i=1
i=1
DΓ DA Ci yi 2
i=1
for any yi ∈ Gi , i = 1, . . . , n. Therefore, n n 1/2 DΓ DA Qi yi DΓ DA Ci yi 2 , i=1
yi ∈ Gi ,
i=1
where the equality holds if and only if Ω is an isometry. Consequently, we can define a contrac(n) tion Λ : FΓ := DΓ F → DΓ by setting Λ
n
DΓ DA Qi yi
n (DΓ DA Ci yi ), :=
i=1
yi ∈ Gi .
(4.11)
i=1
Using the definition of Ω2 , we deduce that ΛDΓ x =
n
DΓ Ω2 x,
x ∈ F.
(4.12)
i=1
We remark that Λ is an isometry if and only if Ω is an isometry. Λ (B(D , D (n) )) of all bounded free holoWe introduce the noncommutative Schur class Sball Γ Γ ∞ (B(D , D (n) )) with Φ 1 such that C| morphic functions Φ ∈ Hball Γ ∞ FΓ = Λ. More preΓ cisely, if C has the representation Φ(X1 , . . . , Xn ) = α∈F+n C(α) ⊗ Xα for some coefficients (n)
C(α) ∈ B(DΓ , DΓ ), the latter condition means C(0) |FΓ = Λ and C(α) |FΓ = 0 if |α| 1. Equivalently, Φ(rR1 , . . . , rRn )|FΓ ⊗1 = (Λ ⊗ IF 2 (Hn ) )|FΓ ⊗1 , i.e.,
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
Φ(rR1 , . . . , rRn )(x ⊗ 1) = Λx ⊗ 1
(4.13)
for any x ∈ FΓ and r ∈ [0, 1). Moreover, notice also that the latter condition is equivalent to is the boundary function of Φ. F ⊗F 2 (H ) = Λ ⊗ IF 2 (H ) , where Φ Φ| Γ
n
n
(n)
Λ (D , D ) is nonempty. Indeed, we can take C = ΛP We remark that the set Sball Γ FΓ ⊗ I , Γ where PFΓ is the orthogonal projection of DΓ onto FΓ . ∞ (B(D , D ⊕ D (n) )) is a Schur We say that a bounded free holomorphic function Ψ ∈ Hball A A function associated with the data set {A, T , V , C, Q} if Ψ ∞ 1 such that Ψ |F = Ω. Equivalently,
Ψ (rR1 , . . . , rRn ) 1 and Ψ (rR1 , . . . , rRn )(y ⊗ 1) = Ωy ⊗ 1
(4.14)
Ω (B(D , D ⊕ D (n) )) the set of all Schur functions for any r ∈ [0, 1) and y ∈ F . We denote by Sball A A associated with the data set {A, T , V , C, Q}. Let B be a solution of the GNCL problem with the data set {A, T , V , C, Q} and consider the contraction Γ : DA → D ⊗ F 2 (Hn ) uniquely determined by B (see (4.5)). Let Θ ∈ 2 (B(D , D)) be the free holomorphic function with symbol Γ . Define the map Hball A
(n)
(n)
JΓ : Sball B DΓ , DΓ → Sball B DA , D ⊕ DA by setting ⎡
⎤ L ϕ1 ⎢M ⎥ 1⎥ ⎢ .. ⎥ ⎢ ⎥ JΓ ⎣ . ⎦ := ⎢ ⎢ .. ⎥ , ⎣ . ⎦ ϕn Mn ⎡
⎤
(4.15)
where L ∈ Hball (B(DA , D)) is given by −1 L(Z) := 2Θ(Z) F (Z) + I ,
Z ∈ B(Z)n 1 ,
(4.16)
the free holomorphic function F ∈ Hball (B(DA )) is defined by n ∗
∗ F (Z) := Γ ⊗ IZ ID⊗F 2 (Hn )⊗Z + ID ⊗ Si ⊗ Zi i=1
× ID⊗F 2 (Hn )⊗Z −
n
ID ⊗ Si∗
−1 ⊗ Zi
(Γ ⊗ IZ )
i=1
−1 n n + (DΓ ⊗ IZ ) I + (IDA ⊗ Zi )ϕi (Z) I − (IDA ⊗ Zi )ϕi (Z) (DΓ ⊗ IZ ), i=1
i=1
(4.17) and M1 , . . . , Mn ∈ Hball (B(DA )) are uniquely determined by the equation
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
927
−1 (IDA ⊗ Z1 )M1 (Z) + · · · + (IDA ⊗ Zn )Mn (Z) = F (Z) − I F (Z) + I
(4.18)
for any Z := (Z1 , . . . , Zn ) ∈ [B(Z)n ]1 . Here Z is an infinite-dimensional Hilbert space. In what follows we need the following lemma, which is the core of the main result of this section. 2 Lemma 4.1. Let {A, T , V , C, Q} be a data set for the GNCL problem. Let Γ : D ⎡AL→ ⎤ D ⊗F (Hn ) ϕ M1 .1 be a contraction and let Φ := .. be in Sball (B(DΓ , DΓ(n) )). Define Ψ := ⎣ .. ⎦ by relations . ϕn Mn
(4.16)–(4.18). Then the following statements hold: (i) If Γ satisfies relation (4.6), then the free holomorphic function M :=
M1
.. .
has the property
Mn (n)
Λ (B(D , D )); that M|F = Ω2 if and only if Φ ∈ Sball Γ Γ (n) Ω Λ (B(D , D (n) )). (ii) Ψ is in Sball (B(DA , D ⊕DA )) if and only if Γ satisfies (4.6) and Φ is in Sball Γ Γ 2 (B(D , D)) be the free holomorphic function with symbol Γ . Due to relaProof. Let Θ ∈ Hball A tion (4.17), we have
−1 n
∗ ∗ F (Z) = Γ Γ ⊗ IZ + 2 Γ ⊗ IZ ID⊗F 2 (Hn )⊗Z − ID ⊗ Si ⊗ Zi ∗
×
n
i=1
ID ⊗ Si∗ ⊗ Zi (Γ ⊗ IZ )
i=1
+ DΓ2 ×
−1 n ⊗ IZ + 2(DΓ ⊗ IZ ) I − (IDA ⊗ Zi )ϕi (Z)
n
i=1
(IDA ⊗ Zi )ϕi (Z) (DΓ ⊗ IZ )
i=1
for any Z := (Z1 , . . . , Zn ) ∈ [B(Z)n ]1 . Since Γ ∗ Γ + DΓ2 = I , we deduce that −1 n
∗ ∗ ID ⊗ Si ⊗ Zi F (Z) − I = 2 Γ ⊗ IZ ID⊗F 2 (Hn )⊗Z − ×
n
i=1
ID ⊗ Si∗ ⊗ Zi (Γ ⊗ IH )
i=1
−1 n n (IDA ⊗ Zi )ϕi (Z) (IDA ⊗ Zi )ϕi (Z) + 2(DΓ ⊗ IH ) I − i=1
× (DΓ ⊗ IZ )
i=1
(4.19)
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
for any Z := (Z1 , . . . , Zn ) ∈ [B(Z)n ]1 . Similarly, we obtain −1 n ∗
∗ F (Z) + I = 2 Γ ⊗ IZ ID⊗F 2 (Hn )⊗Z − ID ⊗ Si ⊗ Zi (Γ ⊗ IZ ) i=1
−1 n (IDA ⊗ Zi )ϕi (Z) (DΓ ⊗ IZ ) + 2(DΓ ⊗ IZ ) I −
(4.20)
i=1
for any Z := (Z1 , . . . , Zn ) ∈ [B(Z)n ]1 . Now, that Γ satisfies relation (4.10) (which is equivalent to (4.6)). Let us show that assume M1 . Λ (B(D , D (n) )). Let x ∈ F M := .. has the property that M|F = Ω2 if and only if Φ ∈ Sball Γ Γ Mn
and r ∈ [0, 1). Since S1 , . . . , Sn are isometries with orthogonal ranges, relation (4.10) implies
n
ID ⊗ Si∗ ⊗ rRi (Γ ⊗ IF 2 (Hn ) )(x ⊗ 1)
i=1
⎡
⎤ ⎤ ⎞ Γ P1
⎢ ⎥ ⎢ . ⎥ ⎟ ∗ ⎜ = ⎣ ID ⊗ Si ⎝ED Ω1 x + [ID ⊗ S1 , . . . , ID ⊗ Sn ] ⎣ .. ⎦ Ω2 x ⎠ ⊗ rei ⎦ i=1 Γ Pn ⎛
⎡
n
=
n n (Γ Pi Ω2 x ⊗ rei ) = (Γ ⊗ IF 2 (Hn ) ) (Pi Ω2 x ⊗ rei ). i=1
i=1
Using relation (4.19), we have F (rR1 , . . . , rRn ) − I = Ar + Br , r ∈ [0, 1), where rR := (rR1 , . . . , rRn ), −1 n
∗ ∗ ID ⊗ Si ⊗ rRi Ar := 2 Γ ⊗ IF 2 (Hn ) ID⊗F 2 (Hn )⊗H − ×
n
i=1
ID ⊗ Si∗ ⊗ rRi (Γ ⊗ IF 2 (Hn ) ),
and
i=1
−1 n n (IDA ⊗ rRi )ϕi (rR) (IDA ⊗ rRi )ϕi (rR) Br := 2(DΓ ⊗ IF 2 (Hn ) ) I − i=1
× (DΓ ⊗ IF 2 (Hn ) ). Taking into account the above calculations, we deduce that F (rR1 , . . . , rRn ) − I (x ⊗ 1)
i=1
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
929
−1 n
∗ ∗ = 2 Γ ⊗ IF 2 (Hn ) ID⊗F 2 (Hn )⊗H − ID ⊗ Si ⊗ rRi (Γ ⊗ IF 2 (Hn ) ) i=1
×
n
(Pi Ω2 x ⊗ rei ) + Br (x ⊗ 1)
i=1
for x ∈ F . Hence and due to (4.20), we obtain F (rR1 , . . . , rRn ) − I (x ⊗ 1) n = F (rR1 , . . . , rRn ) + I (Pi Ω2 x ⊗ rei )
i=1
−1 n n (IDA ⊗ rRi )ϕi (rR) (DΓ ⊗ IF 2 (Hn ) ) (Pi Ω2 x ⊗ rei ) − 2(DΓ ⊗ IF 2 (Hn ) ) I − i=1
i=1
+ Br (x ⊗ 1) n (Pi Ω2 x ⊗ rei ) + 2(DΓ ⊗ IF 2 (Hn ) ) = F (rR1 , . . . , rRn ) + I i=1
−1 n (IDA ⊗ rRi )ϕi (rR) χr , × I−
i=1
where χr :=
n n (IDA ⊗ rRi )ϕi (rR)(DΓ ⊗ IF 2 (Hn ) )(x ⊗ 1) − (DΓ ⊗ IF 2 (Hn ) ) (Pi Ω2 x ⊗ rei ). i=1
i=1
Consequently, we have −1 F (rR1 , . . . , rRn ) + I F (rR1 , . . . , rRn ) − I (x ⊗ 1) =
n −1 (Pi Ω2 x ⊗ rei ) + 2 F (rR1 , . . . , rRn ) + I (DΓ ⊗ IF 2 (Hn ) ) i=1
−1 n (IDA ⊗ rRi )ϕi (rR) χr . × I−
(4.21)
i=1 Λ (B(D , D (n) )), then due to relation (4.13), we have If Φ ∈ Sball Γ Γ
Φ(rR1 , . . . , rRn )(y ⊗ 1) = Λy ⊗ 1 for any y ∈ FΓ and r ∈ [0, 1). Using the definition of FΓ and relations (4.11), (4.12), we deduce that, for any x ∈ F ,
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
ϕj (rR1 , . . . , rRn )(DΓ x ⊗ 1) = (Pj ⊗ IF 2 (Hn ) )Φ(rR1 , . . . , rRn )(DΓ x ⊗ 1) = (Pj ⊗ IF 2 (Hn ) )(ΛDΓ x ⊗ 1) n DΓ Ω2 x ⊗ 1 = Pj i=1
= DΓ Pj Ω2 x ⊗ 1 for any x ∈ F . Now, it is clear that χr = 0. Due to relation (4.21), we have n −1 F (rR1 , . . . , rRn ) + I F (rR1 , . . . , rRn ) − I (x ⊗ 1) = (Pi Ω2 x ⊗ rei ). i=1
Hence and using (4.18), we have Mj (rR1 , . . . , rRn )(x ⊗ 1) =
−1 1 IDA ⊗ Rj∗ F (rR1 , . . . , rRn ) + I r × F (rR1 , . . . , rRn ) − I (x ⊗ 1)
= Pj Ω2 x ⊗ 1 for any x ∈ F and j = 1, . . . , n. Consequently, M(rR1 , . . . , rRn )(x ⊗ 1) = Ω2 x ⊗ 1 for any x ∈ F , i.e., M|F = Ω2 . Conversely, if M|F = Ω2 , then, due to relation (4.18), we have
n −1 F (rR1 , . . . , rRn ) + I F (rR1 , . . . , rRn ) − I (x ⊗ 1) = Pj Ω2 x ⊗ rei . i=1
Using relation (4.21), we obtain −1 n −1 2 F (rR1 , . . . , rRn ) + I (DΓ ⊗ IF 2 (Hn ) ) I − (IDA ⊗ rRi )ϕi (rR) χr = 0. i=1
Since χr has the range in DΓ ⊗ F 2 (Hn ), the operator I − ni=1 (IDA ⊗ rRi )ϕi (rR) is invertible on the Hilbert space DΓ ⊗ F 2 (Hn ), and DΓ ⊗ IF 2 (Hn ) is one-to-one on DΓ ⊗ F 2 (Hn ), we deduce that χr = 0. Consequently, we have n
(IDΓ ⊗ rRi ) ϕi (rR1 , . . . , rRn )(DΓ x ⊗ 1) − DΓ Pi Ω2 x ⊗ 1 = 0,
i=1
Since R1 , . . . , Rn are isometries with orthogonal ranges, we deduce that ϕi (rR1 , . . . , rRn )(DΓ x ⊗ 1) = DΓ Pi Ω2 x ⊗ 1,
i = 1, . . . , n,
x ∈ F.
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
931
for any x ∈ F . On the other hand, due to (4.11) and (4.12), we have DΓ Pj Ω2 x ⊗ 1 = (Pj ⊗ IF 2 (Hn ) )(ΛDΓ x ⊗ 1),
i = 1, . . . , n.
Combining these relations, we deduce that Φ(rR1 , . . . , rRn )(y ⊗ 1) = Λy ⊗ 1 for any y ∈ FΓ Λ (B(D , D (n) )), which proves part (i). and r ∈ [0, 1). Therefore, Φ ∈ Sball Γ Γ ϕ .1 Λ (B(D , D (n) )). The result of Assume now that Γ satisfies (4.6) and Φ := .. is in Sball Γ Γ ϕn M1 . part (i) shows that M := .. has the property that M|F = Ω2 . In what follows we will use the Mn ϕ ⎡ L ⎤ M1 .1 fact that JΓ .. = ⎣ .. ⎦ and that relations (4.16)–(4.18) hold. First, notice that (4.18) implies . ϕn Mn
I−
n
−1 (IDA ⊗ rRi )Mi (rR1 , . . . , rRn ) = I − F (rR1 , . . . , rRn ) − I F (rR1 , . . . , rRn ) + I
i=1
−1 = 2 F (rR1 , . . . , rRn ) + I
for any r ∈ [0, 1). Hence and using relation (4.16), we get −1 L(rR1 , . . . , rRn ) = 2Θ(rR1 , . . . , rRn ) F (rR1 , . . . , rRn ) + I n = Θ(rR1 , . . . , rRn ) I − (IDA ⊗ rRi )Mi (rR1 , . . . , rRn ) . (4.22) i=1
Therefore, since M|F = Ω2 , we have L(rR1 , . . . , rRn )(x ⊗ 1) = Θ(rR1 , . . . , rRn )(x ⊗ 1) − Θ(rR1 , . . . , rRn )
n (IDA ⊗ rRi )Mi (rR1 , . . . , rRn )(x ⊗ 1) i=1
n (Pi Ω2 x ⊗ rei ) = Θ(rR1 , . . . , rRn )(x ⊗ 1) − Θ(rR1 , . . . , rRn ) i=1
for any x ∈ F . Since L is a bounded free holomorphic function, L˜ := SOT- limr→1 L(rR) exists ¯ n∞ . Taking r → 1 in the relation above and using and it is in the operator space B(DA , D)⊗R (4.10), we obtain L(rR1 , . . . , rRn )(x ⊗ 1) = Γ x − lim Θ(rR1 , . . . , rRn ) r→1
n (Pi Ω2 x ⊗ rei ) i=1
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G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
= Ω1 x ⊗ 1 +
n (ID ⊗ Si )Γ Pi Ω2 x i=1
− lim Θ(rR1 , . . . , rRn ) r→1
n (Pi Ω2 x ⊗ rei ). i=1
Now, assume that Θ has the representation Θ(Z1 , . . . , Zn ) = ∞ |α|=k A(α) ⊗ Zα on k=0 [B(Z)n ]1 , with α∈Fn+ A∗(α) A(α) I . Then Γ y = α∈Fn+ A(α) y ⊗ eα˜ , y ∈ DA , and n (ID ⊗ Si )Γ Pi Ω2 x = A(α) Pi Ω2 x ⊗ egi α˜ ,
x ∈ F.
α∈Fn+
i=1
On the other hand, we have
n n Θ(rR1 , . . . , rRn ) (Pi Ω2 x ⊗ rei ) = A(α) Pi Ω2 x ⊗ r |α|+1 egi α˜ . i=1 α∈F+ n
i=1
Consequently,
n n lim Θ(rR1 , . . . , rRn ) (Pi Ω2 x ⊗ rei ) = (ID ⊗ Si )Γ Pi Ω2 x.
r→1
i=1
i=1
˜ ⊗ 1) = Ω1 x ⊗ 1 for x ∈ F , which implies L|F = Ω1 . Hence, L(x ⎡ ⎤
L M
1 Ω (B(D , D ⊕ D (n) )) and let M := Conversely, assume that Ψ := ⎣ .. ⎦ is in Sball A A .
we have
(4.23)
Mn
L(rR1 , . . . , rRn )(y ⊗ 1) = Ω1 y ⊗ 1,
y ∈ F,
M(rR1 , . . . , rRn )(y ⊗ 1) = Ω2 y ⊗ 1,
y ∈ F.
M1
.. .
. Then
Mn
and
Due to relation (4.22), we deduce that Ω1 y ⊗ 1 = L(rR1 , . . . , rRn )(y ⊗ 1) n (IDA ⊗ rRi ) = Θ(rR1 , . . . , rRn )(y ⊗ 1) − Θ(rR1 , . . . , rRn ) i=1
× Mi (rR1 , . . . , rRn )(y ⊗ 1) = Θ(rR1 , . . . , rRn )(y ⊗ 1) − Θ(rR1 , . . . , rRn )
n (IDA ⊗ rRi )(Pi Ω2 y ⊗ 1) i=1
G. Popescu / Journal of Functional Analysis 255 (2008) 891–939
933
for any y ∈ F . As before (see (4.23)), taking r → 1, we get Ω1 y ⊗ 1 = Γ y −
n (IDA ⊗ rRi )(Pi Ω2 y ⊗ 1),
y ∈ F,
i=1
which shows that Γ satisfies relation (4.10). Hence, and using part (i), we deduce that Φ ∈ Λ (B(D , D (n) )). The proof is complete. 2 Sball Γ Γ Now we can prove the following generalized noncommutative commutant lifting theorem, which is the main result of this section. Theorem 4.2. Let {A, T , V , C, Q} be a data set. Then any solution of the GNCL problem is given by % B=
A Γ DA
&
: X → H ⊕ D ⊗ F 2 (Hn ) ,
(4.24)
2 (B(D , D)) where Γ : DA → D ⊗ F 2 (Hn ) is the symbol of a free holomorphic function Θ ∈ Hball A given by
−1 n Θ(Z) = L(X) IDA ⊗Z − (IDA ⊗ Zi )Mi (Z)
for Z := (Z1 , . . . , Zn ) ∈ B(Z)n 1 ,
i=1
(4.25) ⎡
L M
⎤
1 Ω (B(D , D ⊕D (n) )). where ⎣ .. ⎦ is an arbitrary element in the noncommutative Schur class Sball A A .
Mn
⎡
L M1
⎤
Ω (B(D , D ⊕ D (n) )) and let Proof. Assume that Ψ := ⎣ .. ⎦ is an arbitrary element in Sball A A . Mn
Θ be given by (4.25). According to Corollary 3.4, Θ is a free holomorphic function in 2 (B(D , D)) and Θ 1. Using Theorem 3.3, we deduce that Ψ = J Φ for a unique Hball A 2 Θ (n) Ω (B(D , D ⊕ D (n) )), we can use Lemma 4.1 to Φ in Sball (B(DΓ , DΓ )). Now, since Ψ ∈ Sball A A deduce that Γ satisfies relation (4.6). Therefore, B is a solution of the GNCL problem. Conversely, assume that B is a solution of the GNCL problem. Then B has a representation (4.24), where Γ : DA → D ⊗ F 2 (Hn ) is a contraction satisfying (4.6).⎡We ⎤recall that Λ (B(D , D (n) )) Sball Γ Γ
is nonempty. Let Φ ∈
Λ (B(D , D (n) )) Sball Γ Γ
L M1
and set Ψ := ⎣ .. ⎦ := JΓ Φ . Mn
(n)
Ω (B(D , D ⊕ D )). (see (4.15)). Applying again Lemma 4.1, we deduce that Ψ is in Sball A A Now, using Theorem 3.3, we deduce that Γ is the symbol of a free holomorphic function 2 (B(D , D)) satisfying (4.25). This completes the proof. 2 Θ ∈ Hball A
To obtain a refinement of Theorem 4.2, we need the following result.
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Lemma 4.3. Let M, L be Hilbert spaces and Φ be a bounded free holomorphic function with coefficients in B(M, L). Let N be a subspace of M and A ∈ B(N , L). Then Φ∞ 1 and Φ|N = A if and only if Φ = (APN ⊗ I ) + (DA∗ ⊗ I )Ψ (PN ⊥ ⊗ I )
(4.26)
∞ (B(N ⊥ , D ∗ )) with Ψ 1, where N ⊥ := M N . Moreover, Φ and Ψ for some Ψ ∈ Hball A ∞ in (4.26) determine each other uniquely.
= Proof. Let Φ C(α) ⊗ Rα be the Fourier representation of Φ. The condition Φ|N = A α∈F+ n is equivalent to C(0) |N = A and C(α) |N = 0 for |α| 1. The latter condition is also equiva N ⊗F 2 (H ) = A ⊗ IF 2 (H ) . With respect to the decomposition M ⊗ F 2 (Hn ) = [N ⊗ lent to Φ| n n : M ⊗ F 2 (Hn ) → L ⊗ F 2 (Hn ) has the matrix repreF 2 (Hn )] ⊕ [N ⊥ ⊗ F 2 (Hn )], the operator Φ sentation Φ = [Φ|N ⊗F 2 (Hn ) Φ|N ⊥ ⊗F 2 (Hn ) ]. Taking into account the structure of row contractions N ⊥ ⊗F 2 (H ) ] is a contraction if and only if (see [24]), [A ⊗ IF 2 (H ) Φ| n
n
N ⊥ ⊗F 2 (H ) = (DA∗ ⊗ IF 2 (H ) )Ψ Φ| n n
(4.27)
is unique. Since Φ : N ⊥ ⊗ F 2 (Hn ) → DA∗ ⊗ F 2 (Hn ). Moreover, Ψ for a unique contraction Ψ is a multi-analytic operator, i.e., Φ(IM ⊗ Si ) = (IL ⊗ Si )Φ, i = 1, . . . , n, so is its restriction N ⊥ ⊗F 2 (H ) , i.e., Φ| n N ⊥ ⊗F 2 (H ) (IN ⊥ ⊗ Si ) = (IL ⊗ Si )Φ| N ⊥ ⊗F 2 (H ) , Φ| n n
i = 1, . . . , n.
Hence, we deduce that (IN ⊥ ⊗ Si ) − (ID ∗ ⊗ Si )Ψ = 0, (DA∗ ⊗ IF 2 (Hn ) ) Ψ A
i = 1, . . . , n.
(IN ⊥ ⊗ Si ) = (ID ∗ ⊗ Since DA∗ ⊗ IF 2 (Hn ) is one-to-one on DA∗ ⊗ F 2 (Hn ), we obtain Ψ A Si )Ψ , i = 1, . . . , n, which proves that Ψ is a multi-analytic operator. According to [53] (see is the boundary function of a unique bounded free holomorphic function Ψ ∈ also [55]), Ψ ⊥ Sball (B(N , DA∗ )). The proof is complete. 2 Using Lemma 4.3, we obtain the following refinement of Theorem 4.2. Remark 4.4. In Theorem 4.2, there is a one-to-one correspondence between the noncommutative Ω (B(D , D ⊕ D (n) )) and the Schur class S Schur class Sball A ball (B(G, DΩ ∗ )), given by the formula A Ψ = (ΩPF ⊗ I ) + (DΩ ∗ ⊗ I )Ψ1 (PG ⊗ I ), where Ω is defined by (4.8), G := DA F , and Ψ1 ∈ Sball (B(G, DΩ ∗ )). Consequently, TheΩ (B(D , D ⊕ D (n) )) can be replaced by orem 4.2 can be restated and the Schur class Sball A A Sball (B(G, DΩ ∗ )). The following result is an addition to Theorem 4.2.
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Theorem 4.5. Let B be a solution of the GNCL problem with the data set {A, T , V , C, Q}, let Γ 2 (B(D , D)) be the free holomorphic function be the contraction determined by B, and Θ ∈ Hball A Λ (B(D , D (n) )) is a with symbol Γ . Then the restriction of the map JΓ (defined by (4.15)) to Sball Γ Γ ⎡ L⎤ M
1 Ω (B(D , D ⊕D (n) )) one-to-one function onto the set of all functions ⎣ .. ⎦ in the Schur class Sball A A .
Mn
and satisfying the equation
−1 n (IDA ⊗ Zi )Mi (Z) Θ(Z) = L(Z) IDA ⊗Z −
for Z := (Z1 , . . . , Zn ) ∈ B(Z)n 1 .
i=1
Proof. Since B is a solution of the GNCL problem, we have Γ satisfies relation (4.6). Then ⎡ L⎤ M
1 Γ = Γθ , where Θ is given as above and Ψ := ⎣ .. ⎦ is in the noncommutative Schur class .
Mn
(n)
(n)
Ω (B(D , D ⊕ D )). Due to Theorem 3.3, there exists a unique Φ ∈ S Sball A ball (B(DΓ , DΓ )) such A (n) Λ that JΓ Φ = Ψ . By Lemma 4.1, we deduce that Φ ∈ Sball (B(DΓ , DΓ )) and the restriction of JΘ Λ (B(D , D (n) )) is a one-to-one function onto the Schur class S Ω (B(D , D ⊕ D (n) )). The to Sball Γ A Γ A ball proof is complete. 2
Using Lemma 4.3, we obtain the following refinement of Theorem 4.5. Remark 4.6. In Theorem 4.5, there is a one-to-one correspondence between the noncommutative Λ (B(D , D (n) )) and the Schur class S Schur class Sball Γ ball (B(GΓ , DΛ∗ )), given by the formula Γ Φ = (ΛPFΓ ⊗ I ) + (DΓ ∗ ⊗ I )Φ1 (PGΓ ⊗ I ), where Λ is defined by (4.11), GΛ := DΓ FΓ , and Φ1 ∈ Sball (B(GΓ , DΛ∗ )). Consequently, Λ (B(D , D (n) )) can be replaced by Theorem 4.5 can be restated and the Schur class Sball Γ Γ Sball (B(GΓ , DΛ∗ )). Now, we consider a few remarkable particular cases. Corollary 4.7. Let {A, T , V , C, Q} be a data set. In the particular case when Gi = X and Ci = IX for i = 1, . . . , n, Theorem 4.2 provides a description of all solutions of the multivariable generalization [50] of Treil–Volberg commutant lifting theorem [63]. Let T := (T1 , . . . , Tn ), Ti ∈ B(H), be a row contraction and let V := (V1 , . . . , Vn ), Vi ∈ B(K), be the minimal isometric dilation of T on a Hilbert space K ⊃ H. Let Y := (Y1 , . . . , Yn ), Yi ∈ B(X ), be a row isometry and let A ∈ B(X , H) be a contraction such that Ti A = AYi , i = 1, . . . , n. The noncommutative commutant lifting (NCL) problem (see [37]) is to find B ∈ B(X , K) such that B 1, PH B = A,
and Vi B = BYi ,
i = 1, . . . , n.
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A parametrization of all solutions in the noncommutative commutant lifting theorem (NCLT) in terms of generalized choice sequences was obtained in [41]. Using the results of this section, we obtain a more refined parametrization and, moreover, a concrete Schur type description of all solutions. Theorem 4.8. Let {A, T , V , C, Q} be a data set in the particular case when, for each i = 1, . . . , n, Gi = X , Ci = IX , Qi = Yi ∈ B(X ), and Y := (Y1 , . . . , Yn ) is a row isometry. Then any solution of the NCL problem is given by %
A B= Γ DA
&
: X → H ⊕ D ⊗ F 2 (Hn ) ,
(4.28)
2 (B(D , D)) where Γ : DA → D ⊗ F 2 (Hn ) is the symbol of a free holomorphic function Θ ∈ Hball A given by
−1 n (IDA ⊗ Zi )Mi (Z) Θ(Z) = L(X) IDA ⊗Z −
for Z := (Z1 , . . . , Zn ) ∈ B(Z)n 1 ,
i=1
(4.29) ⎡
L M1
⎤
Ω (B(D , D ⊕ where Ψ := ⎣ .. ⎦ is an arbitrary element in the noncommutative Schur class Sball A . (n)
Mn
DA )). Moreover, the solution B and the Schur function Ψ uniquely determine each other via the relations (4.28) and (4.29). There is a unique solution to the NCL problem if and only if F = DA (n) or ΩF = D ⊕ DA , where F and Ω are defined by (4.7) and (4.8), respectively. Proof. The first part of the theorem follows from Theorem 4.2. To prove the second part, let B be a solution of the NCL problem, let Γ be the contraction determined by B, and 2 (B(D , D)) be the free holomorphic function with symbol Γ . Due to Theorem 4.5, to Θ ∈ Hball A Λ (B(D , D (n) )) prove that B and Ψ uniquely determine each other, it is enough to show that Sball Γ Γ has just one element. Since (Y1 , . . . , Yn ) is a row isometry, the operator Ω (see (4.8)) is an isometry and, consequently, so is Λ. From the definition of Λ (see (4.11)), we deduce that the (n) (n) range of Ω coincides with DΛ . Therefore, Λ : FΓ → DΛ is a unitary operator. Consequently, DΛ∗ = {0}. According to Remark 4.6, we deduce that Sball (B(GΓ , DΛ∗ )) is a singleton and, Λ (B(D , D (n) )). Therefore, we have proved that any solution B corresponds therefore, so is Sball Γ Γ to a unique Schur function Ψ . Now, due to Remark 4.4, there is a unique solution of the NCL problem if and only if Sball (B(G, DΩ ∗ )) = {0}. The latter equality holds if and only if G = {0} or (n) DΩ ∗ = {0}. Since Ω is an isometry, the condition DΩ ∗ = {0} is equivalent to ΩF = D ⊕ DA . The proof is complete. 2 We remark that one can easily obtain a version of Theorem 4.8 in the more general setting of the NCL problem when the row isometry (Y1 , . . . , Yn ) is replaced by an arbitrary row contraction. Another consequence of Theorem 4.2 is the following multivariable version [50] of the weighted commutant lifting theorem of Biswas–Foia¸s–Frazho [7].
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Corollary 4.9. Let T := (T1 , . . . , Tn ), Ti ∈ B(H), be a row contraction and let V := (V1 , . . . , Vn ), Vi ∈ B(K), be the minimal isometric dilation of T on a Hilbert space K ⊃ H. Let A ∈ B(X , H) be such that A∗ A P , where P ∈ B(X ) be a positive operator. Let Y := (Y1 , . . . , Yn ), Yi ∈ B(X ), be such that [Yi∗ P Yj ]n×n [δij P ]n×n and Ti A = AYi ,
i = 1, . . . , n.
Then there exists B ∈ B(X , K) such that PH B = A, B ∗ B P , and Vi B = BYi ,
i = 1, . . . , n.
: X → H satisfying := P 1/2 X . Since A∗ A P , there exists a unique contraction A Proof. Let X 1/2 . The condition Ti A = AYi , i = 1, . . . , n, implies Ti AC i, i = i = AQ the equation A = AP 1, . . . , n, where Ci = P 1/2 and Qi = P 1/2 Yi : X → X for i = 1, . . . , n. Applying Theorem 4.2, : X → K such that PH B i for i = 1, . . . , n. Setting = A and Vi BC i = BQ we find a contraction B 1/2 , we have B := BP i = BP 1/2 Yi = BYi i = BQ Vi B = Vi BC 1/2 A. Since B is a contraction, we have 1/2 = AP for i = 1, . . . , n. Note also that PH B = PH BP ∗ BP 1/2 P . This completes the proof. 2 B ∗ BP 1/2 B In a future paper, we apply the results of the present paper to the interpolation theory setting to obtain parametrizations and complete descriptions of all solutions to the Nevanlinna–Pick, Carathéodory–Fejér, and Sarason type interpolation problems for the noncommutative Hardy ∞ and H 2 , as well as consequences to (norm constrained) interpolation on the unit spaces Hball ball n ball of C . References [1] J. Agler, J.E. McCarthy, Pick Interpolation and Hilbert Function Spaces, Grad. Stud. Math., vol. 44, Amer. Math. Soc., Providence, RI, 2002, xx+308 pp. [2] A. Arias, G. Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000) 205–234. [3] W.B. Arveson, Subalgebras of C ∗ -algebras III: Multivariable operator theory, Acta Math. 181 (1998) 159–228. [4] W.B. Arveson, The curvature invariant of a Hilbert module over C[z1 , . . . , zn ], J. Reine Angew. Math. 522 (2000) 173–236. [5] J.A. Ball, T.T. Trent, V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernels Hilbert spaces, in: Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume, in: Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. [6] J.A. Ball, V. Vinnikov, Lax–Phillips scattering and conservative linear systems: A Cuntz-algebra multidimensional setting, Mem. Amer. Math. Soc. 837 (2005). [7] A. Biswas, C. Foia¸s, A.E. Frazho, Weighted commutant lifting, Acta Sci. Math. (Szeged) 65 (1999) 657–686. [8] J.W. Bunce, Models for n-tuples of noncommuting operators, J. Funct. Anal. 57 (1984) 21–30. [9] C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen, Math. Ann. 64 (1907) 95–115. [10] C. Carathéodory, L. Fejér, Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard–Landau’chen Satz, Rend. Circ. Mat. Palermo 32 (1911) 218–239. [11] J. Cuntz, Simple C ∗ -algebras generated by isometries, Comm. Math. Phys. 57 (1977) 173–185. [12] K.R. Davidson, D.R. Pitts, Nevanlinna–Pick interpolation for noncommutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998) 321–337.
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Journal of Functional Analysis 255 (2008) 940–993 www.elsevier.com/locate/jfa
Stochastic evolution equations in UMD Banach spaces ✩ J.M.A.M. van Neerven a , M.C. Veraar a,∗ , L. Weis b a Delft Institute of Applied Mathematics, Delft University of Technology,
P.O. Box 5031, 2600 GA Delft, The Netherlands b Mathematisches Institut I, Technische Universität Karlsruhe, D-76128 Karlsruhe, Germany
Received 31 December 2007; accepted 23 March 2008 Available online 28 April 2008 Communicated by N. Kalton
Abstract We discuss existence, uniqueness, and space–time Hölder regularity for solutions of the parabolic stochastic evolution equation
dU (t) = (AU (t) + F (t, U (t))) dt + B(t, U (t)) dWH (t), U (0) = u0 ,
t ∈ [0, T0 ],
where A generates an analytic C0 -semigroup on a UMD Banach space E and WH is a cylindrical Brownian motion with values in a Hilbert space H . We prove that if the mappings F : [0, T ] × E → E and B : [0, T ] × E → L(H, E) satisfy suitable Lipschitz conditions and u0 is F0 -measurable and bounded, then this problem has a unique mild solution, which has trajectories in C λ ([0, T ]; D((−A)θ ))) provided λ 0 and θ 0 satisfy λ + θ < 12 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations. © 2008 Elsevier Inc. All rights reserved.
✩
The first and second named authors are supported by a ‘VIDI subsidie’ (639.032.201) in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organisation for Scientific Research (NWO). The second named author is also supported by the Humboldt Foundation. The third named author is supported by a grant from the Deutsche Forschungsgemeinschaft (We 2847/1-2). * Corresponding author. Current address: Mathematisches Institut I, Technische Universität Karlsruhe, D-76128 Karlsruhe, Germany. E-mail addresses: [email protected] (J.M.A.M. van Neerven), [email protected] (M.C. Veraar), [email protected] (L. Weis). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.03.015
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Keywords: Parabolic stochastic evolution equations; UMD Banach spaces; Stochastic convolutions; γ -Radonifying operators; L2γ -Lipschitz functions
1. Introduction and statement of the results In this paper we prove existence, uniqueness, and space–time regularity results for the abstract semilinear stochastic Cauchy problem (SCP)
dU (t) = AU (t) + F t, U (t) dt + B t, U (t) dWH (t), U (0) = u0 .
t ∈ [0, T0 ],
Here A is the generator of an analytic C0 -semigroup (S(t))t0 on a UMD Banach space E, H is a separable Hilbert space, and for suitable η 0 the functions F : [0, T ] × D((−A)η ) → E and B : [0, T ] × D((−A)η ) → L(H, E) enjoy suitable Lipschitz continuity properties. The driving process WH is an H -cylindrical Brownian motion adapted to a filtration (Ft )t0 . In fact we shall allow considerably less restrictive assumptions on F and B; both functions may be unbounded and may depend on the underlying probability space. A Hilbert space theory for stochastic evolution equations of the above type has been developed since the 1980s by the schools of Da Prato and Zabczyk [8]. Much of this theory has been extended to martingale type 2-spaces [2,3]; see also the earlier work [34]. This class of Banach spaces covers the Lp -spaces in the range 2 p < ∞, which is enough for many practical applications to stochastic partial differential equations. Let us also mention an alternative approach to the Lp -theory of stochastic partial differential equations has been developed by Krylov [22]. Extending earlier work of McConnell [26], the present authors have developed a theory of stochastic integration in UMD spaces [32,33] based on decoupling inequalities for UMD-valued martingale difference sequences due to Garling [14,15]. This work is devoted to the application of this theory to stochastic evolution equations in UMD spaces. In this introduction we will sketch in an informal way the main ideas of our approach. For the simplicity of presentation we shall consider the special case H = R and make the identifications L(R, E) = E and WR = W , where W is a standard Brownian motion. For precise definitions and statements of the results we refer to the main body of the paper. A solution of equation (SCP) is defined as an E-valued adapted process U which satisfies the variation of constants formula t U (t) = S(t)u0 +
S(t − s)F s, U (s) ds +
0
t
S(t − s)B s, U (s) dW (s).
0
The relation of this solution concept with other type of solutions is considered in [43]. The principal difficulty to be overcome for the construction of a solution, is to find an appropriate space of processes which is suitable for applying the Banach fixed point theorem. Any such space V should have the property that U ∈ V implies that the deterministic convolution t t → 0
S(t − s)F s, U (s) ds
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and the stochastic convolution t t →
S(t − s)B s, U (s) dW (s)
0
belong to V again. To indicate why this such a space is difficult to construct we recall a result from [29] which states, loosely speaking, that if E is a Banach space which has the property that f (u) is stochastically integrable for every E-valued stochastically integrable function u and every Lipschitz function f : E → E, then E is isomorphic to a Hilbert space. Our way out of this apparent difficulty is by strengthening the definition of Lipschitz continuity to L2γ Lipschitz continuity, which can be thought of as a Gaussian version of Lipschitz continuity. From the point of view of stochastic PDEs, this strengthening does not restrict the range of applications of our abstract theory. Indeed, we shall prove that under standard measurability and growth assumptions, Nemytskii operators are L2γ -Lipschitz continuous in Lp . Furthermore, in type 2 spaces the notion of L2γ -Lipschitz continuity coincides with the usual notion of Lipschitz continuity. Under the assumption that F is Lipschitz continuous in the second variable and B is L2γ Lipschitz continuous in the second variable, uniformly with respect to bounded time intervals in their first variables, the difficulty described above is essentially reduced to finding a space of processes V having the property that φ ∈ V implies that the pathwise deterministic convolutions t t →
S(t − s)φ(s) ds 0
and the stochastic convolution integral t S(t − s)φ(s) dW (s)
t →
(1.1)
0
define processes which again belong to V . The main tool for obtaining estimates for this stochastic integral is γ -boundedness. This is the Gaussian version of the notion of R-boundedness which in the past years has established itself as a natural generalization to Banach spaces of the notion of uniform boundedness in the Hilbert space context and which played an essential role in much recent progress in the area of parabolic evolution equations. The power of both notions derives from the fact that they connect probability in Banach spaces with harmonic analysis. From the point of view of stochastic integration, the importance of γ -bounded families of operators is explained by the fact that they act as pointwise multipliers in spaces of stochastically integrable processes. This would still not be very useful if it were not the case that one can associate γ -bounded families of operators with an analytic C0 -semigroup (S(t))t0 with generator A. In fact, for all η > 0 and ε > 0, families such as η+ε t (−A)η S(t): t ∈ (0, T0 )
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are γ -bounded. Here, for simplicity, we are assuming that the fractional powers of A exist; in general one has to consider translates of A. This suggests to rewrite the stochastic convolution (1.1) as t t →
(t − s)η+ε (−A)η S(t − s) (t − s)−η−ε (−A)−η φ(s) dW (s).
(1.2)
0
By γ -boundedness we can estimate the Lp -moments of this integral by the Lp -moments of the simpler integral t t →
(t − s)−η−ε (−A)−η φ(s) dW (s).
(1.3)
0 p
Thus we are led to define Vα,∞ ([0, T0 ] × Ω; D((−A)η )) as the space of all continuous adapted processes φ : [0, T0 ] × Ω → D((−A)η ) for which the norm p φVα,∞ ([0,T0 ]×Ω;D ((−A)η ))
p 1
1 p := EφC([0,T0 ];D((−A)η )) p + sup E (t − ·)−α φ(·) γ (L2 (0,t),D((−A)η )) p t∈[0,T0 ]
is finite. Here, γ (L2 (0, t), F ) denotes the Banach space of γ -radonifying operators from L2 (0, t) into the Banach space F ; by the results of [30], a function f : (0, t) → F is stochastically integrable on (0, t) with respect to W if and only if it is the kernel of an integral operator belonging to γ (L2 (0, t), F ). Now we are ready to formulate a special case of one of the main results (see Theorems 6.2, 6.3, 7.3). Theorem 1.1. Let E be a UMD space and let η 0 and p > 2 satisfy η +
1 p
< 12 . Assume that:
(i) A generates an analytic C0 -semigroup on E; (ii) F : [0, T0 ] × D((−A)η ) → E is Lipschitz continuous and of linear growth in the second variable, uniformly on [0, T0 ]; (iii) B : [0, T0 ] × D((−A)η ) → L(H, E) is L2γ -Lipschitz continuous and of linear growth in the second variable, uniformly on [0, T0 ]; (iv) u0 ∈ Lp (Ω, F0 ; D((−A)η )). Then: (1) (Existence and uniqueness) For all α > 0 such that η + p1 < α < 12 the problem (SCP) admits p a unique solution U in Vα,∞ ([0, T0 ] × Ω; D((−A)η )). (2) (Hölder regularity) For all λ 0 and δ η such that λ + δ < 12 the process U − S(·)u0 has a version with paths in C λ ([0, T0 ]; D((−A)δ )).
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For martingale type 2 spaces E, Theorem 1.1 was proved by Brze´zniak [3]; in this setting the L2γ -Lipschitz assumption in (iii) reduces to a standard Lipschitz assumption. As has already been pointed out, the class of martingale type 2 spaces includes the spaces Lp for 2 p < ∞, whereas the UMD spaces include Lp for 1 < p < ∞. The UMD assumption in Theorem 1.1 can actually be weakened so as to include L1 -spaces as well; see Section 9. The assumptions on F and B as well as the integrability assumption on u0 can be substantially weakened; we shall prove versions of Theorem 1.1 assuming that F and B are merely locally Lipschitz continuous and locally L2γ -Lipschitz continuous, respectively, and u0 is F0 -measurable. Let us now briefly discuss the organization of the paper. Preliminary material on γ radonifying operators, stochastic integration in UMD spaces, and γ -boundedness of families of operators, is collected in Section 2. In Sections 3 and 4 we prove estimates for deterministic and stochastic convolutions. After introducing the notion of L2γ -Lipschitz continuity in Section 5 we take up the study of problem (SCP) in Section 6, where we prove Theorem 1.1. The next two sections are concerned with refinements of this theorem. In Section 7 we consider arbitrary F0 -measurable initial values, still assuming that the functions F and B are globally Lipschitz continuous and L2γ -Lipschitz continuous respectively. In Section 8 we consider the locally Lipschitz case and prove existence and uniqueness of solutions up to an explosion time. In Section 9 we discuss how the results of this paper can be extended to a larger class of Banach spaces including the UMD spaces as well as the spaces L1 . The final Section 10 is concerned with applications to stochastic partial differential equations. On bounded smooth domains S ⊆ Rd we consider the parabolic problem ∂u (t, s) = A(s, D)u(t, s) + f t, s, u(t, s) ∂t ∂w (t, s), s ∈ S, t ∈ (0, T ], + g t, s, u(t, s) ∂t Bj (s, D)u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], u(0, s) = u0 (s),
s ∈ S.
Here A is of the form
A(s, D) =
aα (s)D α
|α|2m
with D = −i(∂1 , . . . , ∂d ) and for j = 1, . . . , m, Bj (s, D) =
bjβ (s)D β ,
|β|mj
where 1 mj < 2m is an integer. As a sample existence result, we prove that if f and g satisfy standard measurability assumptions and are locally Lipschitz and of linear growth in the third 2mη,p variable, uniformly with respect to the first and second variables, and if u ∈ H{Bj } (S), then the 2mδ,p
above problem admits a solution with paths in C λ ([0, T ]; H{Bj } (S)) for all δ > that satisfy δ + λ < obtained as well.
1 2
−
d 4m
and 2mδ −
1 p
d 2mp
and λ > 0
= mj , for all j = 1, . . . , m. Uniqueness results are
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All vector spaces in this paper are real. Throughout the paper, H and E denote a separable Hilbert space and a Banach space, respectively. We study the problem (SCP) on a time interval [0, T0 ] which is always considered to be fixed. In many estimates below we are interested on bounds on sub-intervals [0, T ] of [0, T0 ] and it will be important to keep track of the dependence upon T of the constants appearing in these bounds. For this purpose we shall use the convention that the letter C is used for generic constants which are independent of T but which may depend on T0 and all other relevant data in the estimates. The numerical value of C may vary from line to line. We write Q1 A Q2 to express that there exists a constant c, only depending on A, such that Q1 cQ2 . We write Q1 A Q2 to express that Q1 A Q2 and Q2 A Q1 . 2. Preliminaries The purpose of this section is to collect the basic stochastic tools used in this paper. For proofs and further details we refer the reader to our previous papers [30,33], where also references to the literature can be found. Throughout this paper, (Ω, F, P) always denotes a complete probability space with a filtration (Ft )t0 . For a Banach space F and a finite measure space (S, Σ, μ), L0 (S; F ) denotes the vector space of strongly measurable functions φ : S → F , identifying functions which are equal almost everywhere. Endowed with the topology induced by convergence in measure, L0 (S; F ) is a complete metric space. γ -Radonifying operators. A linear operator R : H → E from a separable Hilbert space H into a Banach space E is called γ -radonifying if for some (and then for every) orthonormal basis (hn )n1 of H the Gaussian sum n1 γn Rhn converges in L2 (Ω; E). Here, and in the rest of the paper, (γn )n1 is a Gaussian sequence, i.e., a sequence of independent standard real-valued Gaussian random variables. The space γ (H, E) of all γ -radonifying operators from H to E is a Banach space with respect to the norm
2 1
2
γ Rh . Rγ (H,E) := E
n n
n1
This norm is independent of the orthonormal basis (hn )n1 . Moreover, γ (H, E) is an operator ideal in the sense that if S1 : H → H and S2 : E → E are bounded operators, then R ∈ γ (H, E) implies S2 RS1 ∈ γ (H , E ) and S2 RS1 γ (H ,E ) S2 Rγ (H,E) S1 .
(2.1)
We will be mainly interested in the case where H = L2 (0, T ; H), where H is another separable Hilbert space. The following lemma gives necessary and sufficient conditions for an operator from H to an Lp -space to be γ -radonifying. It unifies various special cases in the literature, cf. [4,42] and the references given therein. In passing we note that by using the techniques of [24] the lemma can be generalized to arbitrary Banach function spaces with finite cotype. Lemma 2.1. Let (S, Σ, μ) be a σ -finite measure space and let 1 p < ∞. For an operator T ∈ L(H, Lp (S)) the following assertions are equivalent:
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(1) T ∈ γ (H, Lp (S)); 2 12 p (2) For some orthonormal basis (hn )∞ n=1 of H the function ( n1 |T hn | ) belongs to L (S); 1 ∞ 2 2 p (3) For all orthonormal bases (hn )∞ n=1 of H the function ( n=1 |T hn | ) belongs to L (S); p (4) There exists a function g ∈ L (S) such that for all h ∈ H we have |T h| hH · g μ-almost everywhere; (5) There exists a function k ∈ Lp (S; H ) such that T h = [k(·), h]H μ-almost everywhere. 2 12 Moreover, in this situation we may take k = ( ∞ n=1 |T hn | ) and have
∞ 1
2
T γ (H,Lp (S)) p
|T hn |2 gLp (S) .
(2.2)
n=1
Proof. By the Kahane–Khintchine inequalities and Fubini’s theorem we have, for all f1 , . . . , fN ∈ Lp (S),
N 1
2
2 |fn |
n=1
Lp (S)
2 1
N 2
= E γn f n
Lp (S)
n=1
p N
= E
γn f n
p
n=1
L (S)
N p 1
p
p E γn f n
Lp (S)
n=1
2 N
p E
γn f n
p
1 p
n=1
1 2
.
L (S)
The equivalences (1) ⇔ (2) ⇔ (3) follow by taking fn := T hn , n = 1, . . . , N . This also gives the first part of (2.2). N 2 12 (2) ⇒ (4). Let g ∈ Lp (S) be defined as g = ( ∞ n=1 |T hn | ) . For h = n=1 an hn we have, for μ-almost all s ∈ S, N N 1 N 1 2 2 2 2 T h(s) = an T hn (s) |an | |T hn (s)| g(s)hH . n=1
n=1
n=1
The case of a general h ∈ H follows by an approximation argument. (4) ⇒ (5). Let H0 be a countable dense set in H which is closed under taking Q-linear combinations. Let N ∈ Σ be a μ-null set such that for all s ∈ N and for all h ∈ H0 , |T h(s)| g(s)f H and h → T h(s) is Q-linear on H0 . By the Riesz representation theorem, applied for each fixed s ∈ N , the mapping h → T h(s) has a unique extension to an element k(s) ∈ H with T h(s) = [h, k(s)]H for all h ∈ H0 . By an approximation argument we obtain that for all h ∈ H we have T h(s) = [h, k(s)]H for μ-almost all s ∈ S. For all s ∈ N ,
k(s) = H
sup
hH 1,h∈H0
h, k(s) =
sup
hH 1,h∈H0
T h(s) g(s).
Putting k(s) = 0 for s ∈ N , we obtain (5) and the last inequality in (2.2).
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(5) ⇒ (3). Let (hn )∞ n=1 be an orthonormal basis for H . Let N ∈ Σ be a μ-null set such that for all s ∈ N and all n 1 we have T hn (s) = [hn , k(s)]. Then for s ∈ N ,
∞ T hn (s)2 n=1
1 2
=
∞ hn , k(s) 2
1 2
= k(s) H .
n=1
This gives (3) and the middle equality of (2.2).
2
Recall that for domains S ⊆ Rd and λ > d2 one has H λ,2 (S) → Cb (S) (cf. [41, Theorem 4.6.1]). Applying Lemma 2.1 with g ≡ C · 1S we obtain the following result. Corollary 2.2. Assume S ⊆ Rd is a bounded domain. If λ > d2 , then for all p ∈ [1, ∞), the embedding I : H λ,2 (S) → Lp (S) is γ -radonifying. From the lemma we obtain an isomorphism of Banach spaces Lp (S; H ) γ H, Lp (S) , which is given by f → (h → [f (·), h]H ). The next result generalizes this observation: Lemma 2.3. (See [33].) Let (S, Σ, μ) be a σ -finite measure space and let p ∈ [1, ∞) be fixed. Then f → (h → f (·)h) defines an isomorphism of Banach spaces Lp S; γ (H, E) γ H, Lp (S; E) . Stochastic integration. In this section we recall some aspects of stochastic integration in UMD Banach spaces. For proofs and more details we refer to our paper [33], whose terminology we follow. A Banach space E is called a UMD space if for some (equivalently, for all) p ∈ (1, ∞) there exists a constant βp,E 1 such that for all Lp -integrable E-valued martingale difference sequences (dj )nj=1 and all {−1, 1}-valued sequence (εj )nj=1 we have
p 1
1 n n
p
p p
E
εj dj
βp,E E
dj
.
j =1
(2.3)
j =1
The class of UMD spaces was introduced in the 1970s by Maurey and Burkholder and has been studied by many authors. For more information and references to the literature we refer the reader to the review articles [5,37]. Examples of UMD spaces are all Hilbert spaces and the spaces Lp (S) for 1 < p < ∞ and σ -finite measure spaces (S, Σ, μ). If E is a UMD space, then Lp (S; E) is a UMD space for 1 < p < ∞. Let H be a separable Hilbert space. An H -cylindrical Brownian motion is family WH = (WH (t))t∈[0,T ] of bounded linear operators from H to L2 (Ω) with the following two properties: (1) WH h = (WH (t)h)t∈[0,T ] is real-valued Brownian motion for each h ∈ H , (2) E(WH (s)g · WH (t)h) = (s ∧ t) [g, h]H for all s, t ∈ [0, T ], g, h ∈ H .
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The stochastic integral of the indicator process 1(a,b]×A ⊗ (h ⊗ x), where 0 a < b < T and the subset A of Ω is Fa -measurable, is defined as T
1(a,b]×A ⊗ (h ⊗ x) dWH := 1A WH (b)h − WH (a)h x.
0
By linearity, this definition extends to adapted step processes Φ : (0, T ) × Ω → L(H, E) whose values are finite rank operators. In order to extend this definition to a more general class of processes we introduce the following terminology. A process Φ : (0, T ) × Ω → L(H, E) is called H -strongly measurable if Φh is strongly measurable for all h ∈ H . Here, (Φh)(t, ω) := Φ(t, ω)h. Such a process is called stochastically integrable with respect to WH if it is adapted and there exists a sequence of adapted step processes Φn : (0, T ) × Ω → L(H, E) with values in the finite rank operators from H to E and a pathwise continuous process ζ : [0, T ] × Ω → E, such that the following two conditions are satisfied: 0 (1) limn→∞ Φ ·n h = Φh in L ((0,0T ) × Ω; E) for all h ∈ H ; (2) limn→∞ 0 Φn dWH = ζ in L (Ω; C([0, T ]; E)).
In this situation, ζ is determined uniquely as an element of L0 (Ω; C([0, T ]; E)) and is called the stochastic integral of Φ with respect to WH , notation: · ζ=
Φ dWH . 0
The process ζ is a continuous local martingale starting at zero. The following result from [32,33] states necessary and sufficient conditions for stochastic integrability. Proposition 2.4. Let E be a UMD space. For an adapted H -strongly measurable process Φ : (0, T ) × Ω → L(H, E) the following assertions are equivalent: (1) the process Φ is stochastically integrable with respect to WH ; (2) for all x ∗ ∈ E ∗ the process Φ ∗ x ∗ belongs to L0 (Ω; L2 (0, T ; H )), and there exists a pathwise continuous process ζ : [0, T ] × Ω → E such that for all x ∗ ∈ E ∗ we have ζ, x ∗ =
·
Φ ∗ x ∗ dWH
in L0 Ω; C [0, T ] ;
0
(3) for all x ∗ ∈ E ∗ the process Φ ∗ x ∗ belongs to L0 (Ω; L2 (0, T ; H )), and there exists an operator-valued random variable R : Ω → γ (L2 (0, T ; H ), E) such that for all f ∈ L2 (0, T ; H ) and x ∗ ∈ E ∗ we have ∗
T
Rf, x = 0
f (t), Φ ∗ (t)x ∗ H dt
in L0 (Ω).
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In this situation we have ζ = (1, ∞),
·
0 Φ dWH
949
in L0 (Ω; C([0, T ]; E)). Furthermore, for all p ∈
t
p
p E sup Φ dWH p,E ERγ (L2 (0,T ;H ),E) .
t∈[0,T ]
0
In the situation of (3) we shall say that R is represented by Φ. Since Φ is uniquely determined almost everywhere on (0, T ) × Ω by R and vise versa (this readily follows from [33, Lemma 2.7 and Remark 2.8]), in what follows we shall frequently identify R and Φ. The next lemma will be useful in Section 7. Lemma 2.5. Let Φ : (0, T ) × Ω → L(H, E) be stochastically integrable with respect to WH . Suppose A ∈ F is a measurable set such that for all x ∗ ∈ E ∗ we have Φ ∗ (t, ω)x ∗ = 0
for almost all (t, ω) ∈ (0, T ) × A. t Then almost surely in A, for all t ∈ [0, T ] we have 0 Φ dWH = 0. Proof. Let x ∗ ∈ E ∗ be arbitrary. By strong measurability it suffices to show that, almost surely in A, for all t ∈ [0, T ] we have t Mt :=
Φ ∗ x ∗ dWH = 0.
0
For the quadratic variation of the continuous local martingale M we have T [M]T =
∗
Φ (s)x ∗ 2 ds = 0 a.s. on A.
0
Therefore, M = 0 a.s. on A. Indeed, let τ := inf t ∈ [0, T ]: [M]t > 0 , where we take τ = T if the infimum is taken over the empty set. Then M τ is a continuous local martingale with quadratic variation [M τ ] = [M]τ = 0. Hence M τ = 0 a.s. This implies the result. 2 R-boundedness and γ -boundedness. Let E1 and E2 be Banach spaces and let (rn )n1 be a Rademacher sequence, i.e., a sequence of independent random variables satisfying P{rn = −1} = P{rn = 1} = 12 . A family T of bounded linear operators from E1 to E2 is called R-bounded if N there exists a constant C 0 such that for all finite sequences (xn )N n=1 in E1 and (Tn )n=1 in T we have
2
2
N
N
2
rn Tn xn C E
rn xn . E
n=1
n=1
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The least admissible constant C is called the R-bound of T , notation R(T ). By the Kahane– Khintchine inequalities the exponent 2 may be replaced by any p ∈ [1, ∞). This only affects the value of the R-bound; we shall use the notation Rp (T ) for the R-bound of T relative to exponent p. Upon replacing the Rademacher sequence by a Gaussian sequence we arrive at the notion of a γ -bounded family of operators, whose γ -bound will be denoted by γ (T ). A standard randomization argument shows that every R-bounded family is γ -bounded, and both notions are equivalent if the range space has finite cotype (the definitions of type and cotype are recalled in the next section). The notion of R-boundedness has played an important role in recent progress in the regularity theory of parabolic evolution equations. Detailed accounts of these developments are presented in [11,23], where more about the history of this concept and further references to the literature can be found. Here we shall need various examples of R-bounded families, which are stated in the form of lemmas. Lemma 2.6. (See [45].) If Φ : (0, T ) → L(E1 , E2 ) is differentiable with integrable derivative, the family TΦ = Φ(t): t ∈ (0, T ) is R-bounded in L(E1 , E2 ), with
R(TΦ ) Φ(0+) +
T
Φ (t) dt.
0
We continue with a lemma which connects the notions of R-boundedness and γ -radonification. Let H be a Hilbert space and E a Banach space. For each h ∈ H we obtain a linear operator Th : E → γ (H, E) by putting Th x := h ⊗ x,
x ∈ E.
Lemma 2.7. (See [17].) If E has finite cotype, the family T = Th : hH 1 is R-bounded in L(E, γ (H, E)). Following [20], a Banach space E is said to have property (Δ) if there exists a constant CΔ
N
such that if (rn )N n=1 and (rn )n=1 are Rademacher sequences on probability spaces (Ω , P ) and (Ω
, P
), respectively, and (xmn )N m,n=1 is a doubly indexed sequence of elements of E, then
N n
2
N N
2
2
EE
rm rn xmn CΔ E E
rm rn xmn .
n=1 m=1
n=1 m=1
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951
Every UMD space has property (Δ) [6] and every Banach space with property (Δ) has finite cotype. Furthermore the spaces L1 (S) with (S, Σ, μ) σ -finite have property (Δ). The space of trace class operators does not have property (Δ) (see [20]). The next lemma is a variation of Bourgain’s vector-valued Stein inequality for UMD spaces [1,6] and was kindly communicated to us by Tuomas Hytönen. Lemma 2.8. Let WH be an H -cylindrical Brownian motion, adapted to a filtration (Ft )t∈[0,T ] , on a probability space (Ω, P ). If E is a Banach space enjoying property (Δ), then for all 1 p < ∞ the family of conditional expectation operators Ep = E(·|Ft ): t ∈ [0, T ] is R-bounded, with R-bound CΔ , on the closed linear subspace Gp (Ω; E) of Lp (Ω; E) spanned T by all random variables of the form 0 Φ dWH with Φ ∈ γ (L2 (0, T ; H ), E). Proof. Let 1 p < ∞ be fixed and choose E1 , . . . , EN ∈ Ep , say En = E(·|Ftn ) with 0 tn T . By relabeling the indices we may assume that t1 · · · tN . We must show that for T all F1 , . . . , FN ∈ Lp (Ω; E) of the form Fn = 0 Φn dWH we have
N
2
N
2
2
E
rn En Fn CΔ E
rn Fn .
n=1
We write En =
n=1
n
where Dj := Ej − Ej −1 with the convention that E0 = 0. The imT portant point to observe is that if Ψj ∈ γ (L2 (0, T ; H ), E) and Gj := 0 Ψj dWH , the random variables Dj Gj are symmetric and independent. Hence, by a standard randomization argument, j =1 Dj ,
N
2
E
rn En Fn
p
n=1
G (Ω;E)
N n
2
=E
rn Dj Fn
p
n=1 j =1
N
2 N
=E
Dj rn Fn
p
G (Ω;E)
j =1
N
2 N
=EE
rj Dj rn Fn
p j =1
n=j
n=j
G (Ω;E)
G (Ω;E)
N
2 N
2
CΔ E E
rj Dj rn Fn
p j =1
n=1
N
2 N
2
= CΔ E
Dj rn Fn
p j =1
n=1
N
2
2
CΔ E
rn Fn
p n=1
G (Ω;E)
G (Ω;E)
.
2
= CΔ E EN
2
rn Fn
p n=1 G (Ω;E)
N
2
G (Ω;E)
The next lemma, obtained in [19] for the case H = R, states that γ -bounded families act boundedly as pointwise multipliers on spaces of γ -radonifying operators. The proof of the general case is entirely similar.
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Lemma 2.9. Let E1 , E2 be Banach spaces and let H be a separable Hilbert space. Let T > 0. Let M : (0, T ) → L(E1 , E2 ) be function with the following properties: (1) for all x ∈ E1 the function M(·)x is strongly measurable in E2 ; (2) the range M = {M(t): t ∈ (0, T )} is γ -bounded in L(E1 , E2 ). Then for all step functions Φ : (0, T ) → L(H, E1 ) with values in the finite rank operators from H to E1 we have MΦγ (L2 (0,T ;H ),E2 ) γ (M)Φγ (L2 (0,T ;H ),E1 ) .
(2.4)
Here, (MΦ)(t) := M(t)Φ(t). As a consequence, the mapping Φ → MΦ has a unique extension to a bounded operator from γ (L2 (0, T ; H ), E1 ) to γ (L2 (0, T ; H ), E2 ) of norm at most γ (M). In [19] it is shown that under slight regularity assumptions on M, the γ -boundedness is also a necessary condition. 3. Deterministic convolutions After these preliminaries we take up our main line of study and begin with some estimates for deterministic convolutions. The main tool will be a multiplier lemma for vector-valued Besov spaces, Lemma 3.1, to which we turn first. Let E be a Banach space, let I = (a, b] with −∞ a < b ∞ be a (possibly unbounded) interval, and let s ∈ (0, 1) and 1 p, q ∞ be fixed. Following [21, Section 3.b], the Besov space s (I ; E) is defined as follows. For h ∈ R and a function f : I → E, we define T (h)f : I → E Bp,q as the translate of f by h, i.e., f (t + h) T (h)f (t) := 0
if t + h ∈ I, otherwise.
Put I [h] := {t ∈ I : t + h ∈ I } and, for f ∈ Lp (I ; E) and t > 0,
p (f, t) := sup T (h)f − f Lp (I [h];E) . |h|t
Now define s s (I ;E) < ∞ , Bp,q (I ; E) := f ∈ Lp (I ; E): f Bp,q where 1 s (I ;E) := f Lp (I ;E) + f Bp,q
0
−s q dt t p (f, t) t
1 q
(3.1)
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s s (I ;E) , B with the obvious modification for q = ∞. Endowed with the norm · Bp,q p,q (I ; E) is a Banach space. The following continuous inclusions hold for all s, s1 , s2 ∈ (0, 1), p, q, q1 , q2 ∈ [1, ∞] with q1 q2 , s2 s1 : s s (I ; E) → Bp,q (I ; E), Bp,q 1 2
s1 s2 Bp,q (I ; E) → Bp,q (I ; E).
If I is bounded, then also Bps 1 ,q (I ; E) → Bps 2 ,q (I ; E) for 1 p2 p1 ∞. The next lemma will play an important role in setting up our basic framework. We remind the reader of the convention, made at the end of Section 1, that constants appearing in estimates may depend upon the number T0 which is kept fixed throughout the paper. Lemma 3.1. Let 1 q < p < ∞, s > 0 and α 0 satisfy s < q1 − p1 and α < q1 − p1 − s, and let s (0, T ; E) the function t → t −α φ(t)1 1 r < ∞. For all T ∈ [0, T0 ] and φ ∈ Bp,r (0,T ) (t) belongs s to Bq,r (0, T0 ; E) and there exists a constant C 0, independent of T ∈ [0, T0 ], such that
t → t −α φ(t)1(0,T ) (t)
s (0,T ;E) Bq,r 0
1 1 q − p −s−α
CT
s (0,T ;E) . φBp,r
Proof. We prove the lemma under the additional assumption that α > 0; the proof simplifies for case α = 0. We shall actually prove the following stronger result
t → t −α φ(t)1(0,T ) (t)
s (R;E) Bq,r
CT
1 1 q − p −s−α
s (0,T ;E) φBp,r
with a constant C independent of T ∈ [0, T0 ]. Fix u ∈ [0, T ] and |h| u. First assume that h 0. Then I [h] = [0, T − h] and, by Hölder’s inequality,
1 q
φ(t + h)1(0,T ) (t + h) − φ(t)1(0,T ) (t) q
dt
(t + h)α R
0
1 T−h
1
q q
φ(t + h) q
φ(t + h) − φ(t) q
dt + dt
(t + h)α
(t + h)α −h
0
T
1
q
φ(t) q
dt +
(t + h)α
T −h
Cu
1 1 q − p −α
φLp (0,T ;E) + CT
1 1 q − p −α
I [h]
φ(t + h) − φ(t) p dt
1
p
.
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Again by Hölder’s inequality,
1
q
φ(t)1(0,T ) (t) φ(t)1(0,T ) (t) q
dt
(t + h)α −
α t R
T
pq (t + h)−α − t −α p−q dt
p−q pq
φLp (0,T ;E)
0
with T
pq (t + h)−α − t −α p−q dt
0
T
− αpq − αpq 1− αpq 1− αpq t p−q − (t + h) p−q dt Ch p−q Cu p−q .
0
Combining these estimates with the triangle inequality we obtain
1
q
φ(t + h)1(0,T ) (t + h) φ(t)1(0,T ) (t) q
dt −
(t + h)α tα R
Cu
1 1 q − p −α
φLp (0,T ;E) + CT
1 1 q − p −α
φ(t + h) − φ(t) p dt
1
p
.
I [h]
A similar estimate holds for h 0. Next we split [0, 1] = [0, T ∧ 1] ∪ [T ∧ 1, 1] and estimate the integral in (3.1). For the first we have T∧1 1
r r
φ(t + h)1 (t + h) (t) du φ(t)1 (0,T ) (0,T )
u−sr sup
−
t →
(t + h)α tα |h|u Lq (R;E) u 0
T∧1 C u−sr T
1 1 q − p −α
|h|u
0 (i)
CT
1 1 q − p −α
1 0
T +C
−sr
u
u
sup φ(· + h) − φ(·)
u−sr
sup φ(· + h) − φ(·)
|h|u
(p−q)r pq −αr
du u
Lp (I [h];E)
+u
r du Lp (I [h];E)
p−q pq −α
r du φLp (0,T ;E) u
1 r
1 r
u
1 r
φLp (0,T ;E)
0 (ii)
CT
1 1 q − p −α
s (0,T ;E) + CT φBp,r
1 1 q − p −s−α
φLp (0,T ;E) .
In (i) we used the triangle inequality in Lr (0, T ∧ 1, du u ) and in (ii) we noted that α <
1 q
− p1 − s.
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Next, T
1
1 q q
φ(t + h)1(0,T ) (t + h) φ(t)1(0,T ) (t) q
φ(t) q
dt
dt − 2
tα
(t + h)α tα R
0
CT
1 1 q − p −α
φLp (0,T ;E) .
Using this we estimate the second part: 1 T ∧1
1
r r
φ(t + h)1 (t + h) (t) du φ(t)1 (0,T ) (0,T )
u−sr sup
−
q (t + h)α tα |h|u L I [h];E u 1
CT
1 1 q − p −α
CT
1 1 q − p −s−α
φLp (0,T ;E)
−sr
u T ∧1
du u
1 r
φLp (0,T ;E) .
Putting everything together and using Hölder’s inequality to estimate the Lq -norm of t −α φ(t) we obtain
t → t −α φ(t)
s (0,T ;E) Bq,r
= t → t −α φ(t)
1 +
−sr
u 0
CT
1 1 q − p −α
Lq (0,T ;E)
1
r
φ(t + h)1(0,T ) (t + h) φ(t)1(0,T ) (t) r du
sup
−
(t + h)α tα |h|u Lq (R;E) u
φLp (0,T ;E) + CT
1 1 q − p −α
s (0,T ;E) + CT φBp,r
1 1 q − p −s−α
φLp (0,T ;E) .
2
A Banach space E has type p, where p ∈ [1, 2], if there exists a constant C 0 such that for all x1 , . . . , xn ∈ E we have
2 1 n n 1 p
2
p E
rj xj
C xj .
j =1
j =1
Here (rj )j 1 is a Rademacher sequence. Similarly E has cotype q, where q ∈ [2, ∞], if there exists a constant C 0 such that for all x1 , . . . , xn ∈ E we have
n j =1
1 q
xj q
2 1 n
2
C E
rj xj
.
j =1
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In these definitions the Rademacher variables may be replaced by Gaussian variables without changing the definitions; for a proof and more information see [13]. Every Banach space has type 1 and cotype ∞, the spaces Lp (S), 1 p < ∞, have type min{p, 2} and cotype max{p, 2}, and Hilbert spaces have type 2 and cotype 2. Every UMD space has nontrivial type, i.e., type p for some p ∈ (1, 2]. In view of the basic role of the space γ (L2 (0, T ; H ), E) in the theory of vector-valued stochastic integration, it is natural to look for conditions on a function Φ : (0, T ) → L(H, E) ensuring that the associated integral operator IΦ : L2 (0, T ; H ) → E, T IΦ f :=
Φ(t)f (t) dt,
f ∈ L2 (0, T ; H ),
0
is well defined and belongs to γ (L2 (0, T ; H ), E). The next proposition, taken from [32], states such a condition for functions Φ belonging to suitable Besov spaces of γ (H, E)-valued functions. Lemma 3.2. If E has type τ ∈ [1, 2), then Φ → IΦ defines a continuous embedding 1
− 12
τ Bτ,τ
0, T0 ; γ (H, E) → γ L2 (0, T0 ; H ), E ,
where the constant of the embedding depends on T0 and the type τ constant of E. 1
−1
τ 2 (0, T0 ; γ (H, E)) → Conversely, if Φ → IΦ defines a continuous embedding Bτ,τ 2 γ (L (0, T0 ; H ), E), then E has type τ (see [18]); we will not need this result.
Lemma 3.3. Let E be a Banach space with type τ ∈ [1, 2). Let α 0 and q > 2 be such that α < 1
−1
τ 2 − q1 . There exists a constant C 0 such that for all T ∈ [0, T0 ] and Φ ∈ Bq,τ (0, T ; γ (H, E)) we have
1 1 − −α sup s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E) CT 2 q Φ τ1 − 1 .
1 2
Bq,τ
t∈(0,T )
2
0,T ;γ (H,E)
Proof. Fix T ∈ [0, T0 ] and t ∈ [0, T ]. Then,
s → (t − s)−α Φ(s)
γ L2 (0,t;H ),E
= s → s −α Φ(t − s) γ L2 (0,t;H ),E
= s → s −α Φ(t − s)1(0,t) (s) γ L2 (0,T
0 ;H ),E
C s → s −α Φ(t − s)1(0,t) (s)
(i)
(ii)
1
Ct 2 CT
s → Φ(t − s)
− q1 −α
1 1 2 − q −α
Φ
1−1 2 (0,T ;γ (H,E)) 0
τ Bτ,τ
1−1 2 (0,t;γ (H,E))
τ Bq,τ
1−1 2 (0,T ;γ (H,E))
τ Bq,τ
In (i) we used Lemma 3.2 and (ii) follows from Lemma 3.1.
2
.
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In the remainder of this section we assume that A is the infinitesimal generator of an analytic C0 -semigroup S = (S(t))t0 on E. We fix an arbitrary number w ∈ R such that the semigroup generated by A − w is uniformly exponentially stable. The fractional powers (w − A)η are then well defined, and for η > 0 we put Eη := D (w − A)η . This is a Banach space with respect to the norm
xEη := x + (w − A)η x . As is well known, up to an equivalent norm this definition is independent of the choice of w. The basic estimate
S(t)
L(E,Eη )
Ct −η ,
t ∈ [0, T0 ],
(3.2)
valid for η > 0 with C depending on η, will be used frequently. The extrapolation spaces E−η are defined, for η > 0, as the completion of E with respect to the norm
xE−η := (w − A)−η x . Up to an equivalent norm, this space is independent of the choice of w. We observe at this point that the spaces Eη and E−η inherit all isomorphic Banach space properties of E, such as (co)type, the UMD property, and property (Δ), via the isomorphisms (w − A)η : Eη E and (w − A)−η : E−η E. The following lemma is well known; a sketch of a proof is included for the convenience of the reader. Lemma 3.4. Let q ∈ [1, ∞) and τ ∈ [1, 2) be given, and let η 0 and θ 0 satisfy η + θ < 3 1 ∞ 2 − τ . There exists a constant C 0 such that for all T ∈ [0, T0 ] and φ ∈ L (0, T ; E−θ ) we 1
−1
τ 2 have S ∗ φ ∈ Bq,τ (0, T ; Eη ) and
S ∗ φ
1
1 1 τ −2 Bq,τ (0,T ;Eη )
CT q φL∞ (0,T ;E−θ ) .
Proof. Without loss of generality we may assume that η, θ > 0. Let ε > 0 be such that η + θ < 3 1 2 − τ − ε. Then S ∗ φ
1
1 1 τ −2 Bq,τ (0,T ;Eη )
CT q S ∗ φ
1
1 − 1 −ε 2 ([0,T ];Eη )
Cτ
CT q φL∞ (0,T ;E−θ ) .
The first estimate is a direct consequence of the definition of the Besov norm, and the second follows from [25, Proposition 4.2.1]. 2
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From the previous two lemmas we deduce the next convolution estimate. Proposition 3.5. Let E be a Banach space with type τ ∈ [1, 2] and let 0 α < 12 . Let η 0 and θ 0 satisfy η + θ < 32 − τ1 . Then there is a constant C 0 such that for all 0 t T T0 and φ ∈ L∞ (0, T ; E),
s → (t − s)−α (S ∗ φ)(s)
γ L2 (0,t),Eη
CT
1 2 −α
φL∞ (0,T ;E−θ ) .
Proof. First assume that 1 τ < 2. It follows from Lemmas 3.3 and 3.4 that for any q > 2 such that α < 12 − q1 ,
s → (t − s)−α S ∗ φ(s)
γ L2 (0,t),Eη
CT
1 1 2 − q −α
CT
1 2 −α
S ∗ φ
1−1 2 (0,T ;E
τ Bq,τ
η)
φL∞ (0,T ;E−θ ) .
For τ = 2 we argue as follows. Since Eη has type 2, we have a continuous embedding L2 (0, t; Eη ) → γ (L2 (0, t), Eη ); see [36]. Therefore, using (3.2),
s → (t − s)−α S ∗ φ(s) 2 C s → (t − s)−α S ∗ φ(s) 2 γ L (0,t),Eη L (0,t;Eη )
C s → (t − s)−α L2 (0,t) S ∗ φL∞ (0,T ;Eη ) CT
1 2 −α
T 1−η−θ φL∞ (0,T ;E−θ ) .
2
The following lemma, due to Da Prato, Kwapie´n and Zabczyk [10, Lemma 2] in the Hilbert space case, gives a Hölder estimate for the convolution 1 Rα φ(t) := (α)
t (t − s)α−1 S(t − s)φ(s) ds. 0
The proof carries over to Banach spaces without change. Lemma 3.6. (See [10].) Let 0 < α 1, 1 < p < ∞, λ 0, η 0, and θ 0 satisfy λ + η + θ < α − p1 . Then there exist a constant C 0 and an ε > 0 such that for all φ ∈ Lp (0, T ; E) and T ∈ [0, T0 ], Rα φC λ ([0,T ];Eη ) CT ε φLp (0,T ;E−θ ) . 4. Stochastic convolutions We now turn to the problem of estimating stochastic convolution integrals. We start with a lemma which, in combination with Lemma 2.9, can be used to estimate stochastic convolutions involving analytic semigroups.
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Lemma 4.1. Let S be an analytic C0 -semigroup on a Banach space E. For all 0 a < 1 and ε > 0 the family a+ε t S(t) ∈ L(E, Ea ): t ∈ [0, T ] is R-bounded in L(E, Ea ), with R-bound of order O(T ε ) as T ↓ 0. Proof. Let N : [0, T ] → L(E, Ea ) be defined as N (t) = t a+ε S(t). Then N is continuously differentiable on (0, T ) and N (t) = (a + ε)t a+ε−1 S(t) + t a+ε AS(t), where A is the generator of S. Hence, by (3.2),
ε−1
N (t)
for t ∈ (0, T ). L(E,E ) Ct a
By Lemma 2.6 the R-bound on [0, T ] can now be bounded from above by T
N (t)
L(E,Ea ) dt
CT ε .
2
0
We continue with an extension of the Da Prato–Kwapie´n–Zabczyk factorization method [10] for Hilbert spaces to UMD spaces. For deterministic Φ, the assumption that E is UMD can be dropped. A related regularity result for arbitrary C0 -semigroups is due to Millet and Smole´nski [27]. It will be convenient to introduce the notation t S Φ(t) :=
S(t − s)Φ(s) dWH (s) 0
for the stochastic convolution with respect to WH of S and Φ, where WH is an H -cylindrical Brownian motion. Proposition 4.2. Let 0 < α < 12 , λ 0, η 0, θ 0, and p > 2 satisfy λ + η + θ < α − p1 . Let A be the generator of an analytic C0 -semigroup S on a UMD space E and let Φ : (0, T ) × Ω → L(H, E−θ ) be H -strongly measurable and adapted. Then there exist ε > 0 and C 0 such that ES
p ΦC λ ([0,T ];E ) η
T C T p
εp
p E s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E
−θ )
dt.
0
Here, and in similar formulations below, it is part of the assumptions that the right-hand side is well defined and finite. In particular it follows from the proposition there exist ε > 0 and C 0 such that
p p C p T εp sup E s → (t − s)−α Φ(s) 2 ES Φ λ C ([0,T ];Eη )
provided the right-hand side is finite.
t∈[0,T ]
γ (L (0,t;H ),E−θ )
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Proof. The idea of the proof is the same as in [10], but there are some technical subtleties which justify us to outline the main steps. Let β ∈ (0, 12 ) be such that λ + η < β − p1 < α − θ − p1 . It follows from Lemmas 2.9 and 4.1 that, for almost all t ∈ [0, T ], almost surely we have
s → (t − s)−β S(t − s)Φ(s)
γ L2 (0,t;H ),E
Ct α−β−θ s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E
−θ )
(4.1)
.
By Proposition A.1, the process ζβ : [0, T ] × Ω → E, 1 ζβ (t) := (1 − β)
t
(t − s)−β S(t − s)Φ(s) dWH (s),
0
is well defined for almost all t ∈ [0, T ] and satisfies
p 1
p
E ζβ (t) p Ct α−β−θ E s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E
1
p
−θ )
.
By Proposition A.1 the process ζβ is strongly measurable. Therefore, by Fubini’s theorem, T ζβ Lp (Ω;Lp (0,T ;E)) CT
α−β−θ
p E s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E
−θ )
dt.
0
By Lemma 3.6, the paths of Rβ ζβ belong to C λ ([0, T ]; Eη ) almost surely, and for some ε > 0 independent of T ∈ [0, T0 ] we have Rβ ζβ Lp (Ω;C λ ([0,T ];Eη ))
CT ε ζβ Lp (Ω;Lp (0,T ;E)) T
p
CT α−β−θ+ε E s → (t − s)−α Φ(s) γ
1
p
L2 (0,t;H ),E
−θ
dt
.
(4.2)
0
The right ideal property (2.1), (4.1), and Proposition 2.4 imply the stochastic integrability of s → S(t − s)Φ(s) for almost all t ∈ [0, T ]. The proof will be finished (with ε = α − β − θ + ε ) by showing that almost surely on (0, T ) × Ω, S Φ = Rβ ζβ . It suffices to check that for almost all t ∈ [0, T ] and x ∗ ∈ E ∗ we have, almost surely,
1 S Φ(t), x = (β) ∗
t 0
(t − s)β−1 S(t − s)ζβ (s), x ∗ ds.
(4.3)
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This follows from a standard argument via the stochastic Fubini theorem, cf. [10], which can be applied here since almost surely we have, writing Φ(r), x ∗ := Φ ∗ (r)x ∗ , t
(t − s)β−1 S(t − s)(s − ·)−β S(s − ·)Φ(·)1[0,s] (·), x ∗
L2 (0,t;H )
ds
0
t =
(s − ·)−β S(s − ·)Φ(·), (t − s)β−1 S ∗ (t − s)x ∗
L2 (0,t;H )
ds
0
t
(s − ·)−β S(s − ·)Φ(·)
γ (L2 (0,t;H ),E)
(t − s)β−1 S ∗ (t − s)x ∗ ds,
0
which is finite for almost all t ∈ [0, T ] by Hölder’s inequality.
2
Remark 4.3. The stochastic integral S Φ in Proposition 4.2 may be defined only for almost all t ∈ [0, T ]. If in addition one assumes that Φ ∈ Lp ((0, T ) × Ω; γ (H, E−θ )), then S Φ(t) is well defined in Eη for all t ∈ [0, T ]. This follows readily from (4.3), [33, Theorem 3.6(2)] and the density of E ∗ in (Eη )∗ . Since we will not need this in the sequel, we leave this to the interested reader. As a consequence we have the following regularity result of stochastic convolutions in spaces with type τ ∈ [1, 2). We will not need this result below, but we find it interesting enough to state it separately. Corollary 4.4. Let E be a UMD space with type τ ∈ [1, 2). Let p > 2, q > 2, λ 0, η 0, θ 0 be such that λ + η + θ < 12 − p1 − q1 . Then there is an δ > 0 such that for all H -strongly strongly measurable and adapted Φ : (0, T ) × Ω → L(H, E−θ ), p
ES ΦC λ ([0,T ];E ) C p T δp EΦ η
p 1−1 2 (0,T ;γ (H,E
τ Bq,τ
(4.4)
. −θ ))
Proof. By assumption we may choose α ∈ (0, 12 ) such that λ + η + θ + p1 < α < 12 − q1 . The result now follows from Proposition 4.2 and Lemma 3.3 (noting that E−θ has type τ ):
p p ES ΦC λ ([0,T ];E ) C p T εp sup E s → (t − s)−α Φ(s) γ (L2 (0,t;H ),E η
t∈[0,T ]
CpT
( 12 − q1 −α+ε)p
EΦ
p 1−1 2 (0,T ;γ (H,E
τ Bq,τ
. −θ ))
2
−θ )
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The main estimate of this section is contained in the next result. Proposition 4.5. Let E be a UMD Banach space. Let η 0, θ 0, α > 0 satisfy 0 η + θ < α < 12 . Let Φ : (0, T ) × Ω → L(H, E−θ ) be adapted and H -strongly measurable. Then for all 1 < p < ∞ and all 0 t T T0 ,
p
p
1 E (t − ·)−α S Φ(·) γ (L2 (0,t;H ),E ) C p T ( 2 −η−θ)p E (t − ·)−α Φ(·) γ (L2 (0,t;H ),E η
−θ )
.
Proof. Fix 0 t T T0 . As in Proposition 4.2 one shows that the finiteness of the righthand side implies that s → S(t − s)Φ(s) is stochastically integrable on [0, t]. We claim that s → S(t − s)Φ(s) takes values in Eη almost surely and is stochastically integrable on [0, t] as an Eη -valued process. Indeed, let ε > 0 be such that β := η + θ + ε < α and put Nβ (t) := t β (μ − A)η+θ S(t). It follows from Lemmas 2.9 and 4.1 that
p
p E S(t − ·)Φ(·) γ (L2 (0,t;H ),E ) CE Nβ (t − ·)(t − ·)−β Φ(·) γ (L2 (0,t;H ),E ) η −θ
p CT εp E (t − ·)−β Φ(·) γ (L2 (0,t;H ),E ) , −θ
and the expression on the right-hand side is finite by the assumption. The stochastic integrability now follows from Proposition 2.4. This proves the claim. Moreover, by Proposition A.1, the stochastic convolution process S Φ is adapted and strongly measurable as an Eη -valued process. Eη ) denote the closed subspaces in Lp (Ω; Eη ) and Let Gp (Ω; Eη ) and Gp (Ω × Ω; T T p H , respec L (Ω × Ω; Eη ) spanned by all elements of the form 0 Ψ dWH and 0 Ψ d W H is an independent copy of WH and Ψ ranges over all adapted elements in tively, where W Lp (Ω; γ (L2 (0, T ; H ), E)). Since Eη is a UMD space, by Proposition 2.4 the operator T Dp 0
H := Ψ dW
T Ψ dWH , 0
Eη ) to Gp (Ω; Eη ). Using the Fubini isomorphism is well defined and bounded from Gp (Ω × Ω; of Lemma 2.3 twice, we estimate
s → (t − s)−α S Φ(s)
Lp (Ω;γ (L2 (0,t),Eη ))
s
−α s → (t − s) S(s − r)Φ(r) dWH (r)
0
γ (L2 (0,t),Gp (Ω;Eη ))
t
−α H (r)
= s → Dp 1(0,s) (r)(t − s) S(s − r)Φ(r) d W
0
γ (L2 (0,t),Gp (Ω;Eη ))
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t
H (r)
s → 1(0,s) (r)(t − s)−α S(s − r)Φ(r) d W
0
s
−α H (r)
s → (t − s) S(s − r)Φ(r) d W
0
963
η )) γ (L2 (0,t),Gp (Ω×Ω;E
. η ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
Rewriting the right-hand side in terms of the function Nβ (t) = t β (μ − A)η+θ S(t) introduced above and using the stochastic Fubini theorem to interchange the Lebesgue integral and the stochastic integral, the right-hand side can be estimated as
s
−α H (r)
s → (t − s) S(s − r)Φ(r) d W
0
η ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
s
−α η+θ H (r)
s → (t − s) (μ − A) S(s − r)Φ(r) d W
0
s
−α −β H (r)
= s → (t − s) (s − r) N (s − r)Φ(r) d W
0
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
s s−r
−α −β
H (r)
= s → (t − s) (s − r) Nβ (w)Φ(r) dw d W
0
0
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
s−w s
−α −β H (r) dw
= s → Nβ (w) (t − s) (s − r) Φ(r) d W
0
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
0
t
= s → Nβ (w)1(0,s) (w)EFs−w
0
s ×
−α
(t − s)
−β
(s − r)
H (r) dw
Φ(r) d W
,
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
0
t ) is the conditional expectation with respect to F t = σ (W H (s)h: where EFt (ξ ) := E(ξ |F 0 s t, h ∈ H ). Next we note that t
N (w) dw T ε . β
0
Applying Lemmas 2.8 and 2.9 pointwise with respect to ω ∈ Ω, we may estimate the right-hand side above by
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t
N (w)
s → 1(w,t) (s) β
0
s × EFs−w
H (r)
(t − s)−α (s − r)−β Φ(r) d W
0
dw −θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
s
H (r)
T ε s → EFs−w (t − s)−α (s − r)−β Φ(r) d W
0
s
−α −β T s → (t − s) (s − r) Φ(r) d WH (r)
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
ε
0
−θ ))) Lp (Ω;γ (L2 (0,t),Lp (Ω;E
ε
−α −β T s → r → (t − s) (s − r) 1(0,s) (r)Φ(r)
.
Lp (Ω;γ (L2 (0,t),γ (L2 (0,t;H ),E−θ )))
Using the isometry γ H1 , γ (H2 , F ) γ H2 , γ (H1 ; F ) , and the Fubini isomorphism, the right-hand side is equivalent to
T ε s → r → (t − s)−α (s − r)−β 1(0,s) (r)Φ(r) Lp (Ω;γ (L2 (0,t),γ (L2 (0,t;H ),E ))) −θ
ε
−α −β
T r → s → (t − s) (s − r) 1(0,s) (r)Φ(r) Lp (Ω;γ (L2 (0,t;H ),γ (L2 (0,t),E ))) . −θ
To proceed further we want to apply, pointwise with respect to Ω, Lemma 2.9 to the multiplier M : (0, t) → L E−θ , γ L2 (0, t), E−θ defined by M(r)x := fr,t ⊗ x,
s ∈ (0, t), x ∈ E−θ ,
where fr,t ∈ L2 (0, t) is the function fr,t (s) := (t − r)α (t − s)−α (s − r)−β 1(r,t) (s). We need to check that the range of M is γ -bounded in L(E−θ , γ (L2 (0, t), E−θ )). For this we invoke Lemma 2.7, keeping in mind that R-bounded families are always γ -bounded and that UMD spaces have finite cotype. To apply the lemma we check that functions fs,t are uniformly bounded in L2 (0, t):
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t
fr,t (s)2 ds = (t − r)2α
t
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(t − s)−2α (s − r)−2β ds
r
0
1 = (t − r)
1−2β
(1 − u)−2α u−2β du
0
1 T 1−2β
(1 − u)−2α u−2β du.
0
It follows from Lemma 2.9 that
s → (t − s)−α (s − ·)−β 1(0,s) (·)Φ(·)
CT
1 2 −β
r → (t − r)−α Φ(r)
Lp (Ω;γ (L2 (0,t;H ),γ (L2 (0,t),E−θ )))
Lp (Ω;γ (L2 (0,t;H ),E−θ ))
1 = CT 2 −η−θ−ε r → (t − r)−α Φ(r) Lp (Ω;γ (L2 (0,t;H ),E Combining all estimates we obtain the result.
−θ ))
.
2
5. L2γ -Lipschitz functions Let (S, Σ) be a countably generated measurable space and let μ be a finite measure on (S, μ). Then L2 (S, μ) is separable and we may define L2γ (S, μ; E) := γ L2 (S, μ); E ∩ L2 (S, μ; E). Here, γ (L2 (S, μ); E) ∩ L2 (S, μ; E) denotes the Banach space of all strongly μ-measurable functions φ : S → E for which φL2γ (S,μ;E) := φγ (L2 (S,μ);E) + φL2 (S,μ;E) is finite. One easily checks that the simple functions are dense in L2γ (S, μ; E). Next let H be a nonzero separable Hilbert space, let E1 and E2 be Banach spaces, and let f : S × E1 → L(H, E2 ) be a function such that for all x ∈ E1 we have f (·, x) ∈ γ (L2 (S, μ; H ), E2 ). For simple functions φ : S → E1 one easily checks that s → f (s, φ(s)) ∈ γ (L2 (S, μ; H ), E2 ). We call f L2γ -Lipschitz function with respect to μ if f is strongly continuous in the second variable and for all simple functions φ1 , φ2 : S → E1 ,
f (·, φ1 ) − f (·, φ2 ) 2 Cφ1 − φ2 L2γ (S,μ;E1 ) . (5.1) γ (L (S,μ;H ),E ) 2
In this case the mapping φ → Sμ,f φ := f (·, φ(·)) extends uniquely to a Lipschitz mapping from γ L2γ (S, μ; E1 ) into γ (L2 (S, μ; H ), E2 ). Its Lipschitz constant will be denoted by Lμ,f . It is evident from the definitions that for simple functions φ : S → E1 , the operator Sf (φ) ∈ γ (L2 (S, μ; H ), E2 ) is represented by the function f (·, φ(·)). The next lemma extends this to arbitrary functions φ ∈ L2γ (S, μ; E1 ).
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Lemma 5.1. If f : S ×E1 → L(H, E2 ) is an L2γ -Lipschitz function, then for all φ ∈ L2γ (S, μ; E1 ) the operator Sμ,f φ ∈ γ (L2 (S, μ; H ), E2 ) is represented by the function f (·, φ(·)). Proof. Let (φn )n1 be a sequence of simple functions such that φ = limn→∞ φn in L2γ (S, μ; E1 ). We may assume that φ = limn→∞ φn μ-almost everywhere. It follows from (5.1) that (f (·, φn (·)))n1 is a Cauchy sequence in γ (L2 (S, μ; H ), E2 ). Let R ∈ γ (L2 (S, μ; H ), E2 ) be its limit. We must show that R is represented by f (·, φ(·)). Let x ∗ ∈ E2∗ be arbitrary. Since R ∗ x ∗ = limn→∞ f ∗ (·, φn (·))x ∗ in L2 (S, μ; H ) we may choose a subsequence (nk )k1 such that R ∗ x ∗ = limk→∞ f ∗ (·, φnk (·))x ∗ μ-almost everywhere. On the other hand since f is strongly continuous in the second variable we have lim f ∗ s, φnk (s) x ∗ = f ∗ s, φ(s) x ∗
k→∞
for μ-almost all s ∈ S.
This proves that for all h ∈ H we have R ∗ x ∗ = f ∗ (·, φ(·))x ∗ μ-almost everywhere and the result follows. 2 Justified by this lemma, in what follows we shall always identify Sμ,f φ with f (·, φ(·)). If f is L2γ -Lipschitz with respect to all finite measures μ on (S, Σ) and γ γ Lf := sup Lμ,f : μ is a finite measure on (S, Σ) is finite, we say that f is a L2γ -Lipschitz function. In type 2 spaces there is the following easy criterium to check whether a function is L2γ -Lipschitz. Lemma 5.2. Let E2 have type 2. Let f : S × E1 → γ (H, E2 ) be such that for all x ∈ E1 , f (·, x) is strongly measurable. If there is a constant C such that
f (s, x)
f (s, x) − f (s, y)
γ (H,E2 )
C 1 + x ,
γ (H,E2 )
Cx − y,
s ∈ S, x ∈ E1 ,
(5.2)
s ∈ S, x, y ∈ E1 ,
(5.3)
γ
then f is a L2γ -Lipschitz function and Lf C2 C, where C2 is the Rademacher type 2 constant of E2 . Moreover, it satisfies the following linear growth condition
f (·, φ)
γ (L2 (S,μ;H ),E2 )
C2 C 1 + φL2 (S,μ;E1 ) .
If f does not depend on S, one can check that (5.1) implies (5.2) and (5.3). Proof. Let φ1 , φ2 ∈ L2 (S, μ; E1 ). Via an approximation argument and (5.3) one easily checks that f (·, φ1 ) and f (·, φ2 ) are strongly measurable. It follows from (5.2) that f (·, φ1 ) and f (·, φ2 ) are in L2 (S, μ; γ (H, E2 )) and from (5.3) we obtain
f (·, φ1 ) − f (·, φ2 )
L2 (S,μ;γ (H,E2 ))
Cφ1 − φ2 L2 (S,μ;E1 ) .
(5.4)
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Recall from [31] that L2 (S, μ; γ (H, E1 )) → γ (L2 (S, μ; H ), E1 ) where the norm of the embedding equals C2 . From this and (5.4) we conclude that
f (·, φ1 ) − f (·, φ2 ) 2 C2 Cφ1 − φ2 L2 (S,μ;E1 ) . γ (L (S,μ;H ),E ) 2
This clearly implies the result. The second statement follows in the same way.
2
A function f : E1 → L(H, E2 ) is said to be L2γ -Lipschitz if the induced function f˜ : S × E1 → L(H, E2 ), defined by f˜(s, x) = f (x), is L2γ -Lipschitz for every finite measure space (S, Σ, μ). Lemma 5.3. For a function f : E1 → L(H, E2 ), the following assertions are equivalent: (1) f is L2γ -Lipschitz; (2) There is a constant C such that for some (and then for every) orthonormal basis (hm )m1 of N H and all finite sequences (xn )N n=1 , (yn )n=1 in E1 we have
N
2
E
γnm f (xn )hm − f (yn )hm
n=1 m1
N
2 N
C E
γn (xn − yn ) + C 2 xn − yn 2 .
2
n=1
n=1
N Proof. (1) ⇒ (2). Let (hm )m1 be an orthonormal basis and let (xn )N n=1 and (yn )n=1 in E1 be arbitrary. Take S = (0, 1) and μ the Lebesgue measure and choose disjoint sets (Sn )N n=1 in N 1 (0, 1) such that μ(Sn ) = N for all n = 1, . . . , N . Now define φ1 := n=1 1Sn ⊗ xn and φ2 := N n=1 1Sn ⊗ yn . Then (2) follows from (5.1). (2) ⇒ (1). Since the distribution of Gaussian vectors is invariant under orthogonal transformations, if (2) holds for one orthonormal basis (hm )m1 , then it holds for every orthonormal basis (hm )n1 . By a well-known argument (cf. [16, Proposition 1]), (2) implies that for all (an )N n=1 in R we have
2
N
an γnm f (xn )hm − f (yn )hm
E
n=1 m1
N
2 N
C E
an γn (xn − yn ) + C 2 an2 xn − yn 2 .
2
n=1
n=1
Now (5.1) follows for simple functions φ, and the general case follows from this by an approximation argument. 2 Clearly, every L2γ -Lipschitz function f : E1 → γ (H, E2 ) is a Lipschitz function. It is a natural question whether Lipschitz functions are automatically L2γ -Lipschitz. Unfortunately, this is not true. It follows from the proof of [29, Theorem 1] that if dim(H ) 1, then every Lipschitz function f : E1 → γ (H, E2 ) is L2γ -Lipschitz if and only if E2 has type 2.
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A Banach space E has property (α) if for all N 1 and all sequences (xmn )N m,n=1 in E we have
2
N
2
N
E
rmn xmn E E
rm rn xmn .
m,n=1
m,n=1
)
Here, (rmn )m,n1 , (rm m1 , and (rn )n1 are Rademacher sequences, the latter two independent of each other. By a randomization argument one can show that the Rademacher random variables can be replaced by Gaussian random variables. It can be shown using the Kahane–Khintchine inequalities that the exponent 2 in the definition can be replaced by any number 1 p < ∞. Property (α) has been introduced by Pisier [35]. Examples of spaces with this property are the Hilbert spaces and the spaces Lp for 1 p < ∞. The next lemma follows directly from the definition of property (α) and Lemma 5.3.
Lemma 5.4. Let E2 be a space with property (α). Then f : E1 → γ (H, E2 ) is L2γ -Lipschitz if N and only if there exists a constant C such that for all finite sequences (xn )N n=1 and (yn )n=1 in E1 we have
2
N
γn f (xn ) − f (yn )
E
n=1
N
2 N
C E
γn (xn − yn ) + C 2 xn − yn 2 .
2
γ (H,E2 )
n=1
n=1
In particular, every f ∈ L(E1 , γ (H, E2 )) is L2γ -Lipschitz. When H is finite-dimensional, this result remains valid even if E2 fails to have property (α). The next example identifies an important class of L2γ -Lipschitz continuous functions. Example 5.5 (Nemytskii maps). Fix p ∈ [1, ∞) and let (S, Σ, μ) be a σ -finite measure space. Let b : R → R be a Lipschitz function; in case μ(S) = ∞ we also assume that b(0) = 0. Define the Nemytskii map B : Lp (S) → Lp (S) by B(x)(s) := b(x(s)). Then B is L2γ -Lipschitz with respect to μ. Indeed, it follows from the Kahane–Khintchine inequalities that
2 1 N 1 N p 2 p
2 2
b xn (s) − b yn (s) E
γn B(xn ) − B(yn )
p dμ(s)
n=1
n=1
S
1 N p p 2 2 xn (s) − yn (s) Lb dμ(s) S
n=1
2 1 N
2
p Lb E
γn (xn − yn )
.
n=1
Now we apply Lemma 5.3.
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6. Stochastic evolution equations I: integrable initial values On the space E we consider the stochastic equation: dU (t) = (AU (t) + F (t, U (t))) dt + B(t, U (t)) dWH (t), (SCP) U (0) = u0 ,
t ∈ [0, T0 ],
where WH is an H -cylindrical Brownian motion. We make the following assumptions on A, F , B, u0 , the numbers η, θF , θB 0: (A1) The operator A is the generator of an analytic C0 -semigroup S on a UMD Banach space E. (A2) The function F : [0, T0 ] × Ω × Eη → E−θF is Lipschitz of linear growth uniformly in [0, T0 ] × Ω, i.e., there are constants LF and CF such that for all t ∈ [0, T0 ], ω ∈ Ω and x, y ∈ Eη ,
F (t, ω, x) − F (t, ω, y)
F (t, ω, x)
E−θF E−θF
LF x − yEη , CF 1 + xEη .
Moreover, for all x ∈ Eη , (t, ω) → F (t, ω, x) is strongly measurable and adapted in E−θF . (A3) The function B : [0, T0 ] × Ω × Eη → L(H, E−θB ) γ
γ
is L2γ -Lipschitz of linear growth uniformly in Ω, i.e., there are constants LB and CB such that for all finite measures μ on ([0, T0 ], B[0,T0 ] ), for all ω ∈ Ω, and all φ1 , φ2 ∈ L2γ ((0, T0 ), μ; Eη ),
B(·, ω, φ1 ) − B(·, ω, φ2 )
γ
γ (L2 ((0,T0 ),μ;H ),E−θB )
LB φ1 − φ2 L2γ ((0,T0 ),μ;Eη ) ,
and
B(·, ω, φ)
γ (L2 ((0,T0 ),μ;H ),E−θB )
γ CB 1 + φL2γ ((0,T0 ),μ;Eη ) .
Moreover, for all x ∈ Eη , (t, ω) → B(t, ω, x) is H -strongly measurable and adapted in E−θB . (A4) The initial value u0 : Ω → Eη is strongly F0 -measurable. We call a process (U (t))t∈[0,T0 ] a mild Eη -solution of (SCP) if (i) U : [0, T0 ] × Ω → Eη is strongly measurable and adapted, (ii) for all t ∈ [0, T0 ], s → S(t − s)F (s, U (s)) is in L0 (Ω; L1 (0, t; E)), (iii) for all t ∈ [0, T0 ], s → S(t − s)B(s, U (s)) H -strongly measurable and adapted and in γ (L2 (0, t; H ), E) almost surely,
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(iv) for all t ∈ [0, T0 ], almost surely U (t) = S(t)u0 + S ∗ F (·, U )(t) + S B(·, U )(t). By (ii) the deterministic convolution is defined pathwise as a Bochner integral, and since E is a UMD space, by (iii) and Proposition 2.4 the stochastic convolutions are well defined. We shall prove an existence and uniqueness result for (SCP) using a fixed point argument in a suitable scale of Banach spaces of E-valued processes introduced next. Fix T ∈ (0, T0 ], p p ∈ [1, ∞), α ∈ (0, 12 ). We define Vα,∞ ([0, T ] × Ω; E) as the space of all continuous adapted processes φ : [0, T ] × Ω → E for which p φVα,∞ ([0,T ]×Ω;E)
p 1
1 p := EφC([0,T ];E) p + sup E s → (t − s)−α φ(s) γ (L2 (0,t),E) p t∈[0,T ]
p
is finite. Similarly we define Vα,p ([0, T ] × Ω; E) as the space of pathwise continuous and adapted processes φ : [0, T ] × Ω → E for which p φVα,p ([0,T ]×Ω;E)
1 p := EφC([0,T ];E) p +
T
p E s → (t − s)−α φ(s) γ (L2 (0,t),E) dt
1
p
0 p
is finite. Identifying processes which are indistinguishable, the above norm on Vα,p ([0, T ] × Ω; p E) and Vα,∞ ([0, T ] × Ω; E) turn these spaces into Banach spaces. The main result of this section, Theorem 6.2 below, establishes existence and uniqueness of a mild solution of (SCP) with initial value u0 ∈ Lp (Ω, F0 ; Eη ) in each of the p p spaces Vα,p ([0, T0 ] × Ω; E) and Vα,∞ ([0, T0 ] × Ω; E). Since we have a continuous emp p bedding Vα,∞ ([0, T0 ] × Ω; E) → Vα,p ([0, T0 ] × Ω; E), the existence result is stronger for p p Vα,∞ ([0, T0 ] × Ω; E) while the uniqueness result is stronger for Vα,p ([0, T0 ] × Ω; E). p For technical reasons, in the next section we will also need the space V˜α,p ([0, T ] × Ω; E) which is obtained by replacing ‘pathwise continuous’ with ‘pathwise bounded and B[0,T ] ⊗ F p measurable’ and C([0, T ]; E) with Bb ([0, T ]; E) in the definition of V˜α,p ([0, T ] × Ω; E). Here Bb ([0, T ]; E) denotes the Banach space of bounded strongly Borel measurable functions on [0, T ] with values in E, endowed with the supremum norm. Consider the fixed point operator LT (φ) = t → S(t)u0 + S ∗ F (·, φ)(t) + S B(·, φ)(t) . In the next proposition we show that LT is well defined on each of the three spaces introduced above and that it is a strict contraction for T small enough. Proposition 6.1. Let E be a UMD space with type τ ∈ [1, 2]. Suppose that (A1)–(A4) are satisfied and assume that 0 η + θF < 32 − τ1 and 0 η + θB < 12 . Let p > 2 and α ∈ (0, 12 ) be such
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that η + θB < α − p1 . If u0 ∈ Lp (Ω; Eη ), then the operator LT is well defined and bounded on each of the spaces p p p V ∈ Vα,∞ [0, T ] × Ω; Eη , Vα,p [0, T ] × Ω; Eη , V˜α,p [0, T ] × Ω; Eη , and there exists a constant CT , with limT ↓0 CT = 0, such that for all φ1 , φ2 ∈ V ,
LT (φ1 ) − LT (φ2 ) CT φ1 − φ2 V . V
(6.1)
Moreover, there is a constant C 0, independent of u0 , such that for all φ ∈ V ,
1
LT (φ) C 1 + Eu0 p p + CT φV . Eη V p
(6.2) p
Proof. We give a detailed proof for the space Vα,∞ ([0, T ] × Ω; Eη ). The proof for Vα,p ([0, T ] × p Ω; Eη ) is entirely similar. For the proof for V˜α,p ([0, T ] × Ω; Eη ) one replaces C([0, T ]; E) by Bb ((0, T ); E). Step 1. Estimating the initial value part. Let ε ∈ (0, 12 ). From Lemmas 2.9 and 4.1 we infer that
s → (t − s)−α S(s)u0
γ (L2 (0,t),E
s → (t − s)−α s −ε u0 2 C γ (L (0,t),Eη ) η)
−α −ε
= C s → (t − s) s u L2 (0,t) 0 Eη Cu0 Eη .
p
For the other part of the Vα,∞ ([0, T ] × Ω; Eη )-norm we note that Su0 C([0,T ];Eη ) Cu0 Eη . It follows that p Su0 Vα,∞ ([0,T ]×Ω;Eη ) Cu0 Lp (Ω;Eη ) .
Step 2. Estimating the deterministic convolution. We proceed in two steps. p (a) For ψ ∈ C([0, T ]; E−θF ) we estimate the Vα,∞ ([0, T ] × Ω; Eη )-norm of S ∗ ψ. By Lemma 3.6 (applied with α = 1 and λ = 0) S ∗ ψ is continuous in Eη . Using (3.2) we estimate: t S ∗ ψC [0,T ];Eη C
(t − s)−η−θF ds ψC([0,T ];E−θF )
0
CT 1−η−θF ψC([0,T ];E−θF ) .
(6.3)
Also, since E has type τ , it follows from Proposition 3.5 that
s → (t − s)−α S ∗ ψ(s)
γ (L2 (0,t),Eη )
T
1 2 −α
ψC([0,T ];E−θF ) .
(6.4)
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Now let Ψ ∈ Lp (Ω; C([0, T ]; E−θF )). By applying (6.3) and (6.4) to the paths Ψ (·, ω) one p obtains that S ∗ Ψ ∈ Vα,∞ ([0, T ] × Ω; Eη ) and 1
min{ 2 −α,1−η−θF } p Ψ Lp (Ω;C([0,T ];E−θF )) . S ∗ Ψ Vα,∞ ([0,T ]×Ω;Eη ) CT
(6.5)
p
(b) Let φ1 , φ2 ∈ Vα,∞ ([0, T ] × Ω; Eη ). Since F is of linear growth, F (·, φ1 ) and F (·, φ2 ) belong to Lp (Ω; C([0, T ]; E−θF )). From (6.5) and the fact that F is Lipschitz continuous in its p Eη -variable we deduce that S ∗ (F (·, φ1 )), S ∗ (F (·, φ2 )) ∈ Vα,∞ ([0, T ] × Ω; Eη ) and
S ∗ F (·, φ1 ) − F (·, φ2 )
p
Vα,∞ ([0,T ]×Ω;Eη )
1 CT min{ 2 −α,1−η−θF } F (·, φ1 ) − F (·, φ2 ) Lp (Ω;C([0,T ];E CT
min{ 12 −α,1−η−θF }
−θF
))
p LF φ1 − φ2 Vα,∞ ([0,T ]×Ω;Eη ) .
(6.6)
Step 3. Estimating the stochastic convolution. Again we proceed in two steps. (a) Let Ψ : [0, T ] × Ω → L(H, E−θB ) be H -strongly measurable and adapted and suppose that
p sup E s → (t − s)−α Ψ (s) γ (L2 (0,t;H ),E
t∈[0,T ]
−θB )
< ∞.
(6.7)
p
We estimate the Vα,∞ ([0, T ] × Ω; Eη )-norm of S Ψ . From Proposition 4.2 we obtain an ε > 0 such that
p
p 1
E S Ψ C([0,T ];E ) p CT ε sup E s → (t − s)−α Ψ (s) γ (L2 (0,t;H ),E η
t∈[0,T ]
1
p
−θB )
.
For the other part of the norm, by Proposition 4.5 we obtain that
p
E s → (t − s)−α S Ψ (s)
γ (L2 (0,t;H ),Eη )
CT
1
p
p
E s → (t − s)−α Ψ (s) γ (L2 (0,t;H ),E
1 2 −η−θB
1
p
−θB )
.
Combining things we conclude that p S Ψ Vα,∞ ([0,T ]×Ω;Eη ) 1
CT min{ 2 −η−θB ,ε}
p
sup E s → (t − s)−α Ψ (s) γ (L2 (0,t;H ),E
t∈[0,T ]
p1 −θB )
(b) For t ∈ [0, T ] let μt,α be the finite measure on ((0, t), B(0,t) ) defined by t μt,α (B) = 0
(t − s)−2α 1B (s) ds.
.
(6.8)
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Notice that for a function φ ∈ C([0, t]; E) we have φ ∈ γ L2 (0, t), μt,α , E
s → (t − s)−α φ(s) ∈ γ L2 (0, t), E .
⇐⇒
Trivially,
1 φL2 ((0,t),μt,α ;E) = (t − ·)−α φ(·) L2 (0,t;E) Ct 2 −α φC([0,T ];E) . p
Now let φ1 , φ2 ∈ Vα,∞ ([0, T ] × Ω; Eη ). Since B is L2γ -Lipschitz and of linear growth and φ1 and φ2 belong to L2γ ((0, t), μt,α ; Eη ) uniformly, B(·, φ1 ) and B(·, φ2 ) satisfy (6.7). Since B(·, φ1 ) and B(·, φ2 ) are H -strongly measurable and adapted, it follows from (6.8) that B(·, φ1 ), p B(·, φ2 ) ∈ Vα,∞ ([0, T ] × Ω; Eη ) and
S B(·, φ1 ) − B(·, φ2 ) p V
α,∞ ([0,T ]×Ω;Eη )
1
T min{ 2 −η−θB ,ε}
p × sup E s → (t − s)−α B s, φ1 (s) − B s, φ2 (s) γ L2 (0,t;H ),E t∈[0,T ]
p E B(·, φ1 ) − B(·, φ2 ) γ (L2 ((0,t),μ
1
= T min{ 2 −η−θB ,ε} sup
t∈[0,T ]
1
LB T min{ 2 −η−θB ,ε} sup γ
t∈[0,T ]
1
LB T min{ 2 −η−θB ,ε} γ
+ T
p 2 −αp
γ
p
1
p
t,α ;H ),E−θB )
p
t,α ;Eη )
p
1 sup E s → (t − s)−α [φ1 − φ2 ] γ (L2 (0,t),E ) p
t∈[0,T ] p
Eφ1 − φ2 C([0,T ];Eη )
η
1 p
1
p LB T min{ 2 −η−θB ,ε} φ1 − φ2 Vα,∞ ([0,T ]×Ω;Eη ) .
γ
1
1
p
Eφ1 − φ2 L2 ((0,t),μ
−θB
(6.9)
Step 4. Collecting the estimates. It follows from the above considerations that LT is well defined p on Vα,∞ ([0, T ] × Ω; Eη ) and there exist constants C 0 and β > 0 such that for all φ1 , φ2 ∈ p Vα,∞ ([0, T ] × Ω; Eη ) we have
LT (φ1 ) − LT (φ2 )
p
Vα,∞ ([0,T ]×Ω;Eη )
p CT β φ1 − φ2 Vα,∞ ([0,T ]×Ω;Eη ) .
(6.10)
The estimate (6.2) follows from (6.10) and
LT (0)
p
Vα,∞ ([0,T ]×Ω;Eη )
p 1 C 1 + Eu0 Eη p .
2
Theorem 6.2 (Existence and uniqueness). Let E be a UMD space with type τ ∈ [1, 2]. Suppose that (A1)–(A4) are satisfied and assume that 0 η + θF < 32 − τ1 and 0 η + θB < 12 . Let p > 2 and α ∈ (0, 12 ) be such that η + θB < α − p1 . If u0 ∈ Lp (Ω, F0 ; Eη ), then there exists a mild
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p
solution U in Vα,∞ ([0, T0 ] × Ω; Eη ) of (SCP). As a mild solution in Vα,p ([0, T ] × Ω; Eη ), this solution U is unique. Moreover, there exists a constant C 0, independent of u0 , such that p p1 p U Vα,∞ . ([0,T0 ]×Ω;Eη ) C 1 + Eu0 Eη
(6.11)
Proof. By Proposition 6.1 we can find T ∈ (0, T0 ], independent of u0 , such that CT < 12 . It follows from (6.1) and the Banach fixed point theorem that LT has a unique fixed point U ∈ p Vα,∞ ([0, T ] × Ω; Eη ). This gives a continuous adapted process U : [0, T ] × Ω → Eη such that almost surely for all t ∈ [0, T ], U (t) = S(t)u0 + S ∗ F (·, U )(t) + S B(·, U )(t).
(6.12)
p
Noting that U = limn→∞ LnT (0) in Vα,∞ ([0, T ] × Ω; Eη ), (6.2) implies the inequality p p1 p p + CT U Vα,∞ U Vα,∞ ([0,T ]×Ω;Eη ) C 1 + Eu0 Eη ([0,T ]×Ω;Eη ) , and then CT <
1 2
implies p p1 p . U Vα,∞ ([0,T ]×Ω;Eη ) C 1 + Eu0 Eη
(6.13)
Via a standard induction argument one may construct a mild solution on each of the intervals [T , 2T ], . . . , [(n − 1)T , nT ], [nT , T0 ] for an appropriate integer n. The induced solution U on [0, T0 ] is the mild solution of (SCP). Moreover, by (6.13) and induction we deduce (6.11). For small T ∈ (0, T0 ], uniqueness on [0, T ] follows from the uniqueness of the fixed point of p LT in Vα,p ([0, T ] × Ω; Eη ). Uniqueness on [0, T0 ] follows by induction. 2 In the next theorem we deduce regularity properties of the solution. They are formulated for U − Su0 ; if u0 is regular enough, regularity of U can be deduced. Theorem 6.3 (Regularity). Let E be a UMD space with type τ ∈ [1, 2] and suppose that (A1)– (A4) are satisfied. Assume that 0 η + θF < 32 − τ1 and 0 η + θB < 12 − p1 with p > 2. Let λ 0 and δ η satisfy λ + δ < min{ 12 − such that for all u0 ∈ Lp (Ω; Eη ),
1 p
− θB , 1 − θF }. Then there exists a constant C 0
p EU − Su0 C λ ([0,T
1
0 ];Eδ )
p
p 1 C 1 + Eu0 Eη p .
(6.14)
Proof. Choose r 1 and 0 < α < 12 such that λ + δ < 1 − 1r − θF , η + θB < α − p1 , and p λ + δ + θB < α − 1 . Let U˜ ∈ Vα,∞ ([0, T0 ] × Ω; Eη ) be the mild solution from Theorem 6.2. It p
follows from Lemma 3.6 (with α = 1) that we may take a version of S ∗ F (·, U˜ ) with
p
E S ∗ F (·, U˜ ) C λ ([0,T
0 ];Eδ )
p
CE F (·, U˜ ) Lr (0,T ;E ) 0 −θF
p
CE F (·, U˜ ) C([0,T ];E ) . 0
−θF
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Similarly, via Proposition 4.2 we may take a version of S ∗ B(·, U˜ ) with
p
E S B(·, U˜ ) C λ ([0,T ];E ) δ 0
p C sup E s → (t − s)−α B ·, U˜ (s) γ (L2 (0,t;H ),E t∈[0,T0 ]
−θB )
.
Define U : [0, T0 ] × Ω → Eη as U (t) = S(t)u0 + S ∗ F (·, U˜ )(t) + S B(·, U˜ )(t), where we take the versions of the convolutions as above. By uniqueness we have almost surely U ≡ U˜ . Arguing as in (6.9) deduce that p
EU − Su0 C λ ([0,T
0 ];Eδ )
p C 1 + U V p
α,∞ ([0,T0 ]×Ω;Eη )
.
2
Now (6.14) follows from (6.11).
7. Stochastic evolution equations II: measurable initial values So far we have solved the problem (SCP) for initial values u0 ∈ Lp (Ω, F0 ; Eη ). In this section we discuss the case of initial values u0 ∈ L0 (Ω, F0 ; Eη ). 0 ([0, T ] × Ω; E) as the linear Fix T ∈ (0, T0 ]. For p ∈ [1, ∞) and α ∈ (0, 12 ) we define Vα,p space of continuous adapted processes φ : [0, T ] × Ω → E such that almost surely, T φC([0,T ];E) +
s → (t − s)−α φ(s) p
γ (L2 (0,t),E)
1
p
dt
< ∞.
0
As usual we identify indistinguishable processes. Theorem 7.1 (Existence and uniqueness). Let E be a UMD space of type τ ∈ [1, 2] and suppose that (A1)–(A4) are satisfied. Assume that 0 η + θF < 32 − τ1 and η + θB < 12 . If α ∈ (0, 12 ) and 0 ([0, T ] × p > 2 are such that η + θB < α − p1 , then there exists a unique mild solution U ∈ Vα,p 0 Ω; Eη ) of (SCP). For the proof we need the following uniqueness result. p
Lemma 7.2. Under the conditions of Theorem 6.2 let U1 and U2 in Vα,∞ ([0, T ] × Ω; Eη ) be the mild solutions of (SCP) with initial values u1 and u2 in Lp (Ω, F0 ; Eη ). Then almost surely on the set {u1 = u2 } we have U1 ≡ U2 . Proof. Let Γ = {u1 = u2 }. First consider small T ∈ (0, T0 ] as in Step 1 in the proof of Theorem 6.2. Since Γ is F0 -measurable we have
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p p U1 1Γ − U2 1Γ Vα,∞ ([0,T ]×Ω;Eη ) = LT (U1 )1Γ − LT (U2 )1Γ Vα,∞ ([0,T ]×Ω;Eη )
= LT (U1 1Γ ) − LT (U2 1Γ ) 1Γ V p ([0,T ]×Ω;E α,∞
η)
1 p U1 1Γ − U2 1Γ Vα,∞ ([0,T ]×Ω;Eη ) , 2 hence almost surely U1 |[0,T ]×Γ ≡ U2 |[0,T ]×Γ . To obtain uniqueness on the interval [0, T0 ] one may proceed as in the proof of Theorem 6.2. 2 Proof of Theorem 7.1. (Existence) Define (un )n1 in Lp (Ω, F0 ; Eη ) as un := 1{u0 Eη n} u0 . p
By Theorem 6.2, for each n 1 there is a unique solution Un ∈ Vα,∞ ([0, T ] × Ω; Eη ) of (SCP) with initial value un . By Lemma 7.2 we may define U : (0, T0 ) × Ω → Eη as U (t) = limn→∞ Un (t) if this limit exists and 0 otherwise. Then, U is strongly measurable and adapted, and almost surely on {u0 Eη n}, for all t ∈ (0, T0 ) we have U (t) = Un (t). Hence, U ∈ 0 ([0, T ] × Ω; E ). It is routine to check that U is a solution of (SCP). Vα,p η (Uniqueness) The argument is more or less standard, but there are some subtleties due to the presence of the radonifying norms. 0 ([0, T ] × Ω; E ) be mild solutions of (SCP). For each n 1 let the stopping Let U, V ∈ Vα,p 0 η U times μU n and νn be defined as
μU n
T0
p
−α
s → (t − s) U (s)1[0,r] (s) γ (L2 (0,t),E ) dt n , = inf r ∈ [0, T0 ]: η
0
νnU = inf r ∈ [0, T ]: U (r)Eη n . This is well defined since T0
p
r → s → (t − s)−α U (s)1[0,r] (s)
γ (L2 (0,t),Eη )
dt
0
is a continuous adapted process by [33, Proposition 2.4] and the dominated convergence theorem. The stopping times μVn and νnV are defined similarly. For each n 1 let U V V τn = μU n ∧ νn ∧ μn ∧ νn , p and let Un = U 1[0,τn ] and Vn = V 1[0,τn ] . Then for all n 1, Un and Vn are in V˜α,p ([0, T0 ] × Ω; Eη ). One easily checks that
τ Un = 1[0,τn ] LT (Un ) n
τ and Vn = 1[0,τn ] LT (Vn ) n ,
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where LT is the map introduced preceding Proposition 6.1 and (LT (Un ))τn (t) := (LT (Un ))(t ∧ τn ). By Proposition 6.1 we can find T ∈ (0, T0 ] such that CT 12 . A routine computation then implies 1 p p Un − Vn V˜α,p ([0,T ]×Ω;Eη ) 2 Un − Vn V˜α,p ([0,T ]×Ω;Eη ) . We obtain that Un = Vn in V˜α,p ([0, T ] × Ω; Eη ), hence P-almost surely, Un ≡ Vn . Letting n tend to infinity, we may conclude that almost surely, U ≡ V on [0, T ]. This gives the uniqueness on the interval [0, T ]. Uniqueness on [0, T0 ] can be obtained by the usual induction argument. 2 p
p Note that in the last paragraph of the proof we needed to work in the space V˜α,p ([0, T ] × Ω; p Eη ) rather than in Vα,p ([0, T ] × Ω; Eη ) because the truncation with the stopping time destroys the pathwise continuity. By applying Theorem 6.3 to the unique solution Un with initial value un := 1{u0 Eη n} u0 , the solution U := limn→∞ Un constructed in Theorem 7.1 enjoys the following regularity property.
Theorem 7.3 (Hölder regularity). Let E be a UMD space and type τ ∈ [1, 2] and suppose that (A1)–(A4) are satisfied. Assume that 0 η + θF < 32 − τ1 and 0 η + θB < 12 . Let λ 0 and δ η satisfy λ + δ < min{ 12 − θB , 1 − θF }. Then the mild solution U of (SCP) has a version such that almost all paths satisfy U − Su0 ∈ C λ ([0, T0 ]; Eδ ). Proof of Theorem 1.1. Part (1) is the special case of Theorem 6.2 corresponding to τ = 1 and θF = θB = 0. For part (2) we apply Theorem 7.3, again with τ = 1 and θF = θB = 0. 2 8. Stochastic evolution equations III: the locally Lipschitz case Consider the following assumptions on F and B. (A2) The function F : [0, T0 ] × Ω × Eη → E−θF is locally Lipschitz, uniformly in [0, T0 ] × Ω, i.e., for all R > 0 there exists a constant LR F such that for all t ∈ [0, T0 ], ω ∈ Ω and xEη , yEη R,
F (t, ω, x) − F (t, ω, y)
E−θF
LR F x − yEη .
Moreover, for all x ∈ Eη , (t, ω) → F (t, ω, x) ∈ E−θF is strongly measurable and adapted, and there exists a constant CF,0 such that for all t ∈ [0, T0 ] and ω ∈ Ω,
F (t, ω, 0)
CF,0 . E −θF
(A3) The function B : [0, T0 ] × Ω × Eη → L(H, E−θB ) is locally L2γ -Lipschitz, uniformly in Ω, i.e., there exists a sequence of L2γ -Lipschitz functions Bn : [0, T0 ] × Ω × Eη → L(H, E−θB ) such that B(·, x) = Bn (·, x) for all xEη < n. Moreover, for all x ∈ Eη , (t, ω) → B(t, ω, x) ∈ E−θB is H -strongly measurable and adapted, and there exists a constant CB,0 such that for all finite measures μ on ([0, T0 ], B[0,T0 ] ) and all ω ∈ Ω,
t → B(t, ω, 0) 2 CB,0 . γ (L ((0,T ),μ;H ),E ) 0
−θB
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One may check that the locally Lipschitz version of Lemma 5.2 holds as well. This gives an easy way to check (A3) for type 2 spaces E. Let be a stopping time with values in [0, T0 ]. For t ∈ [0, T0 ] let Ωt () = ω ∈ Ω: t < (ω) , [0, ) × Ω = (t, ω) ∈ [0, T0 ] × Ω: 0 t < (ω) , [0, ] × Ω = (t, ω) ∈ [0, T0 ] × Ω: 0 t (ω) . A process ζ : [0, ) × Ω → E (or (ζ (t))t∈[0,) ) is called admissible if for all t ∈ [0, T0 ], Ωt () ω → ζ (t, ω) is Ft -measurable and for almost all ω ∈ Ω, [0, (ω)) t → ζ (t, ω) is continuous. Let E be a UMD space. An admissible Eη -valued process (U (t))t∈[0,) is called a local solution of (SCP) if ∈ (0, T0 ] almost surely and there exists an increasing sequence of stopping times (n )n1 with = limn→∞ n such that (i) for all t ∈ [0, T0 ], s → S(t − s)F (·, U (s))1[0,n ] (s) ∈ L0 (Ω; L1 (0, t; Eη )), (ii) for all t ∈ [0, T0 ], s → S(t − s)B(·, U (s))1[0,n ] (s)∈L0 (Ω; γ (L2 (0, t; H ), Eη )), (iii) almost surely for all t ∈ [0, ρn ], U (t) = S(t)u0 + S ∗ F (·, U )(t) + S B(·, U )(t). By (i) the deterministic convolution is defined pathwise as a Bochner integral. Since E is a UMD space, by (ii) and Proposition 2.4 we may define the stochastic convolution as t S B(·, U )(t) =
S(t − s)B s, U (s) 1[0,n ] (s) dWH (s),
t ∈ [0, ρn ].
0
A local solution (U (t))t∈[0,) is called maximal for a certain space V of Eη -valued admissible processes if for any other local solution (U˜ (t))t∈[0,) ˜ and ˜ in V , almost surely we have ˜ U ≡ U |[0,) ˜ . Clearly, a maximal local solution for such a space V is always unique in V . We say that a local solution (U (t))t∈[0,) of (SCP) is a global solution of (SCP) if = T0 almost surely and U has an extension to a solution Uˆ : [0, T0 ] × Ω → Eη of (SCP). In particular, almost surely “no blow” up occurs at t = T0 . We say that is an explosion time if for almost all ω ∈ Ω with (ω) < T0 ,
lim sup U (t, ω) E = ∞. t↑(ω)
η
Notice that if = T0 almost surely, then is always an explosion time in this definition. However, there need not be any “blow up” in this case.
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Let be a stopping time with values in [0, T0 ]. For p ∈ [1, ∞), α ∈ [0, 12 ) and η ∈ [0, 1] we 0,loc define Vα,p ([0, ) × Ω; E) as all E-valued admissible processes (φ(t))t∈[0,) such that there exists an increasing sequence of stopping times (n )n1 with = limn→∞ n and almost surely T φC([0,n ];E) +
s → (t − s)−α φ(s)1[0, ] (s) p dt n γ (0,t;E)
1
p
< ∞.
0
In the case that for almost all ω, n (ω) = T for n large enough, 0,loc 0 Vα,p [0, ) × Ω; E = Vα,p [0, T ] × Ω; E . Theorem 8.1. Let E be a UMD space with type τ ∈ [1, 2] and suppose that (A1), (A2) , (A3) , (A4) are satisfied, and assume that 0 η + θF < 32 − τ1 . (1) For all α ∈ (0, 12 ) and p > 2 such that η + θB < α −
1 p
there exists a unique maximal local
0,loc Vα,p ([0, ) × Ω; Eη )
of (SCP). solution (U (t))[0,) in (2) For all λ > 0 and δ η such that λ + δ < min{ 12 − θB , 1 − θF }, U has a version such that for almost all ω ∈ Ω, λ t → U (t, ω) − S(t)u0 (ω) ∈ Cloc 0, (ω) ; Eδ . If in addition the linear growth conditions of (A2) and (A3) hold, then the above function U is 0 ([0, T ] × Ω; E ) and the following assertions hold: the unique global solution of (SCP) in Vα,p 0 η (3) The solution U satisfies the statements of Theorems 7.1 and 7.3. (4) If α ∈ (0, 12 ) and p > 2 are such that α > η + θB + p1 and u0 ∈ Lp (Ω, F0 ; Eη ), then the p solution U is in Vα,∞ ([0, T0 ] × Ω; Eη ) and (6.11) and the statements of Theorem 6.3 hold. Before we proceed, we prove the following local uniqueness result. Lemma 8.2. Suppose that the conditions of Theorem 8.1 are satisfied and let (U1 (t))t∈[0,1 ) 0,loc 0,loc in Vα,p ([0, 1 ) × Ω; Eη ) and (U2 (t))t∈[0,2 ) in Vα,p ([0, 2 ) × Ω; Eη ) be local solutions of (SCP) with initial values u10 and u20 . Let Γ = {u10 = u20 }. Then almost surely on Γ , U1 |[0,1 ∧2 ) ≡ U2 |[0,1 ∧2 ) . Moreover, if 1 is an explosion time for U1 , then almost surely on Γ , 1 2 . If 1 and 2 are explosion times for U1 and U2 , then almost surely on Γ , 1 = 2 and U1 ≡ U2 . Proof. Let = 1 ∧ 2 . Let (μn )n1 be an increasing sequences of bounded stopping times such p that limn→∞ μn = and for all n 1, U1 1[0,μn ] and U2 1[0,μn ] are in V˜α,p ([0, T0 ] × Ω; Eη ). Let
νn1 = inf t ∈ [0, T0 ]: U1 (t) E n and νn2 = inf t ∈ [0, T0 ]: U2 (t) E n η
η
and let σni = μn ∧ νni and let σn = σn1 ∧ σn2 . On [0, T0 ] × Ω × {x ∈ Eη : xEη n} we may replace F and B by Fn (for a possible definition of Fn , see the proof of Theorem 8.1) and Bn which satisfy (A2) and (A3). As in the proof of Theorem 7.1 it follows that for all 0 < T T0 ,
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σn
U 1[0,σ
− U2σn 1[0,σn ]×Γ V˜ p ([0,T ]×Ω;E ) η α,p
σn
σn
= LT U1 1[0,σn ]×Γ − LT U2 1[0,σn ]×Γ 1[0,σn ]×Γ V˜ p ([0,T ]×Ω;E ) η α,p
σn σn
LT U 1[0,σ ]×Γ − LT U 1[0,σ ]×Γ ˜ p 1
n ]×Γ
1
2
n
CT U σn 1[0,σ 1
n ]×Γ
− U2σn 1[0,σn
n
]×Γ ˜ p
Vα,p ([0,T ]×Ω;Eη )
Vα,p ([0,T ]×Ω;Eη )
,
where CT satisfies limT ↓0 CT = 0. Here 1[0,σn ]×Γ should be interpreted as the process (t, ω) → 1[0,σn (ω)]×Γ (t, ω). For T small enough it follows that U1σn 1[0,σn ]×Γ = U2σn 1[0,σn ]×Γ p in V˜α,p ([0, T ] × Ω; Eη ). By an induction argument this holds on [0, T0 ] as well. By path continuity it follows that almost surely, U1 ≡ U2 on [0, σn ] × Γ . Since = limn→∞ σn we may conclude that almost surely, U1 ≡ U2 on [0, ) × Γ . If 1 is an explosion time, then as in [39, Lemma 5.3] this yields 1 2 on Γ almost surely. Indeed, if for some ω ∈ Γ , 1 (ω) < 2 (ω), then we can find an n such that 1 (ω) < νn2 (ω). We 1 (ω) < (ω). If we combine both assertions we have U1 (t, ω) = U2 (t, ω) for all 0 t νn+1 1 obtain that
1
1
n + 1 = U1 νn+1 (ω), ω E = U2 νn+1 (ω), ω E n. a
a
This is a contradiction. The final assertion is now obvious.
2
Proof of Theorem 8.1. We follow an argument of [3,39]. For n 1 let Γn = {u0 n2 } and un = u0 1Γn . Let (Bn )n1 be the sequence of L2γ -Lipschitz functions from (A3) . Fix an integer n 1. Let Fn : [0, T0 ] × Ω × Eη → E−θF be defined by Fn (·, x) = F (·, x)
for xEη n,
nx ) otherwise. Clearly, Fn and Bn satisfy (A2) and (A3). It follows from and Fn (·, x) = F (·, x E η
p
Theorem 6.2 that there exists a solution Un ∈ Vα,∞ ([0, T0 ] × Ω; Eη ) of (SCP) with u0 , F and B replaced by un , Fn and Bn . In particular, Un has a version with continuous paths. Let n be the stopping time defined by
n (ω) = inf t ∈ [0, T0 ]: Un (t, ω) E n . η
It follows from Lemma 8.2 that for all 1 m n, almost surely, Um ≡ Un on [0, m ∧ n ] × Γm . By path continuity this implies m n . Therefore, we can define = limn→∞ n and on Γn , 0,loc ([0, ) × U (t) = Un (t) for t n . By approximation and Lemma 2.5 it is clear that U ∈ Vα,p Ω; Eη ) is a local solution of (SCP). Moreover, is an explosion time. This proves the existence part of (1). Maximality is a consequence of Lemma 8.2. Therefore, (U (t))t∈[0,) is a maximal local solution. This concludes the proof of (1). We continue with (2). By Corollary 6.3, each Un has the regularity as stated by (2). Therefore, the construction yields the required pathwise regularity properties of U . Turning to (4), let (Un )n1 be as before. As in the proof of Proposition 6.1 one can check that by the linear growth assumption,
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p Un Vα,∞ ([0,T ]×Ω;Eη ) = LT (Un ) V p
α,∞ ([0,T ]×Ω;Eη )
p CT Un Vα,∞ ([0,T ]×Ω;Eη ) + C + Cνn Lp (Ω;Eη ) ,
where the constants do not depend on n and u0 and we have limT ↓0 CT = 0. Since νn Lp (Ω;Eη ) u0 Lp (Ω;Eη ) , it follows that for T small we have p Un Vα,∞ ([0,T ]×Ω;Eη ) C 1 + u0 Lp (Ω;Eη ) , where C is a constant independent of n and u0 . Repeating this inductively, we obtain a constant C p independent of n and u0 such that Un Vα,∞ ([0,T0 ]×Ω;Eη ) C(1 + u0 Lp (Ω;Eη ) ). In particular,
p
p E sup Un (s) E C p 1 + u0 Lp (Ω;Eη ) . η
s∈[0,T0 ]
It follows that
P sup Un (s) E n C p n−p . s∈[0,T0 ]
Since
n1 n
−p
η
< ∞, the Borel–Cantelli lemma implies that
P sup Un (s) E n = 0. k1 nk s∈[0,T0 ]
η
This gives that almost surely, n = T0 for all n large enough, where n is as before. In particular, = T0 and by Fatou’s lemma p p U Vα,∞ ([0,T0 ]×Ω;Eη ) lim inf Un Vα,∞ ([0,T0 ]×Ω;Eη ) C 1 + u0 Lp (Ω;Eη ) . n→∞
Via an approximation argument one can check that U is a global solution. The final statement in (4) can be obtained as in Theorem 6.3. For the proof of (3) one may repeat the construction of Theorem 7.1, using Lemma 8.2 instead of Lemma 7.2. 2 9. Generalizations to one-sided UMD spaces In this section we explain how the theory of the preceding sections can be extended to a class of Banach spaces which contains, besides all UMD spaces, the spaces L1 . A Banach space E is called a UMD+ -space if for some (equivalently, for all) p ∈ (1, ∞) + there exists a constant βp,E 1 such that for all E-valued Lp -martingale difference sequences n (dj )j =1 we have
p 1
1 n n
p
p p
+ rj dj
βp,E dj
, E
E
j =1
j =1
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where (rj )nj=1 is a Rademacher sequence independent of (dj )nj=1 . The space E is called a UMD− space if the reverse inequality holds:
1
p 1 n n
p
p p
− dj
βp,E E
rj dj
. E
j =1
j =1
Both classes of spaces were introduced and studied by Garling [15]. By a standard randomization argument, every UMD space is both UMD+ and UMD− , and conversely a Banach space which is both UMD+ and UMD− is UMD. At present, no examples are known of UMD+ -spaces which are not UMD. For the UMD− property the situation is different: if E is UMD− , then also L1 (S; E) is UMD− . In particular, every L1 -space is UMD− (cf. [28]). Assume that (Ft )t0 is the complete filtration induced by WH . If E is a UMD− -space, condition (3) still gives a sufficient condition for stochastic integrability of Φ, and instead of a norm equivalence one obtains the one-sided estimate
p
T
p E Φ dWH p,E ERγ (L2 (0,T ;H ),E)
0
for all p ∈ (1, ∞), where we use the notations of Proposition 2.4. The condition on the filtration is needed for the approximation argument used in [14]. By using Fubini’s theorem it is obvious that the result also holds if the probability space has the following product structure Ω = Ω1 × Ω2 , F = F ⊗ G, P = P1 ⊗ P2 , and the filtration is of the form (Ft ⊗ G)t0 . Mutatis mutandis, the theory presented in the previous sections extends to UMD− spaces E, with two exceptions: (i) Proposition 4.5 relies, via the use of Lemma 2.8, on the fact that UMD spaces have property (Δ); this property should now be included into the assumptions. (ii) One needs the above assumption on the filtration. We note that it follows from [7] that for E = L1 the assumption on the filtration is not needed. 10. Applications to stochastic PDEs Case of bounded A. We start with the case of a bounded operator A. By putting F˜ := A + F it suffices to consider the case A = 0. Let E be a UMD− space with property (α) (see Section 5). Consider the equation dU (t) = F t, U (t) dt + B U (t) dWE (t), U (0) = u0 ,
t ∈ [0, T ], (10.1)
where WE is an E-valued Brownian motion. With every E-valued Brownian motion WE one can canonically associate an H -cylindrical Brownian motion WH , where H is the so-called reproducing kernel Hilbert space associated with WE (1) (see the proof of Theorem 10.1 below). Using this H -cylindrical Brownian motion WH , the problem (10.1) can be rewritten as a special instance of (SCP).
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We make the following assumptions: (1) F : [0, T ] × Ω × E → E satisfies (A2) with a = θF = 0; (2) B ∈ L(E, L(E)); (3) u0 : Ω → E is F0 -measurable. Theorem 10.1. Under these assumptions, for all α > 0 and p > 2 such that α < 12 − p1 there 0 ([0, T ] × Ω; E). exists a unique strong and mild solution U : [0, T ] × Ω → E of (10.1) in Vα,p Moreover, for all 0 λ < 12 , U has a version with paths in C λ ([0, T ]; E). Proof. Let H be the reproducing kernel Hilbert space associated with WE (1). Then H is a separable Hilbert space which is continuously embedded into E by means of an inclusion operator i : H → E which belongs to γ (H, E). Putting WH (t)i ∗ x ∗ := WE (t), x ∗ (cf. [30, Example 3.2]) we obtain an H -cylindrical Brownian motion. Assumption (A1) is trivially fulfilled, and (A2) and (A4) hold by assumption. Let Bˆ ∈ ˆ L(E, γ (H, E)) be given by B(x)h = B(x)ih. Using Lemma 5.4 one checks that Bˆ satisfies (A3) with a = θB = 0. Therefore, the result follows from Theorems 7.1 and 7.3 (applied to Bˆ and the H -cylindrical Brownian motion WH ). Here we use the extension to UMD− space as explained in Section 9. 2 Elliptic equations on bounded domains. Below we will consider an elliptic equation of order 2m on a domain S ⊆ Rd . We will assume the noise is white in space and time. The regularizing effect of the elliptic operator will be used to be able to consider the white-noise in a suitable way. Space–time white noise equations seem to be studied in the literature in the case m = 1 (cf. [3,9]). Let S ⊆ Rd be a bounded domain with C ∞ boundary. We consider the problem ∂u (t, s) = A(s, D)u(t, s) + f t, s, u(t, s) ∂t ∂w (t, s), s ∈ S, t ∈ (0, T ], + g t, s, u(t, s) ∂t Bj (s, D)u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], u(0, s) = u0 (s),
s ∈ S.
Here A is of the form A(s, D) =
aα (s)D α
|α|2m
where D = −i(∂1 , . . . , ∂d ) and for j = 1, . . . , m, Bj (s, D) =
|β|mj
bjβ (s)D β ,
(10.2)
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where 1 mj < 2m is an integer. We assume that a α ∈ C(S) for all |α| = 2m. For |α| < 2m the coefficients aα are in L∞ (S). For the principal part |α|=2m aα (s)D α of A we assume that there is a κ > 0 such that (−1)m+1
aα (s)ξ α κ|ξ |2m ,
s ∈ S, ξ ∈ Rd .
|α|=2m
For the coefficients of the boundary value operator we assume that for j = 1, . . . , m and |β| mj we have bjβ ∈ C ∞ (S). The boundary operators (Bj )m j =1 define a normal system of Dirichlet type, i.e. 0 mj < m (cf. [40, Section 3.7]). The C ∞ assumption on the boundary of S and on the coefficients bjβ is made for technical reasons. We will need complex interpolation spaces for Sobolev spaces with boundary conditions. It is well known to experts that one can reduce the assumption to S has a C 2m -boundary and bjβ ∈ C 2m−mj (S). However, this seems not to be explicitly contained in the literature. The functions f, g : [0, T ] × Ω × S × R → R are jointly measurable, and adapted in the sense that for each t ∈ [0, T ], f (t, ·) and g(t, ·) are Ft ⊗ BS ⊗ BR -measurable. Finally, w is a space–time white noise (see, e.g., [44]) and u0 : S × Ω → R is an BS ⊗ F0 -measurable initial value condition. We say that u : [0, T ] × Ω × S → R is a solution of (10.2) if the corresponding functional analytic model (SCP) has a mild solution U and u(t, s, ω) = U (t, ω)(s). Consider the following conditions: (C1) The functions f and g are locally Lipschitz in the fourth variable, uniformly on [0, T ] × R Ω × S, i.e., for all R > 0 there exist constants LR f and Lg such that f (t, ω, s, x) − f (t, ω, s, y) LR |x − y|, f g(t, ω, s, x) − g(t, ω, s, y) LR |x − y|, g for all t ∈ [0, T ], ω ∈ Ω, s ∈ S, and |x|, |y| < R. Furthermore, f and g satisfy the boundedness conditions supf (t, ω, s, 0) < ∞,
supg(t, ω, s, 0) < ∞,
where the suprema are taken over t ∈ [0, T ], ω ∈ Ω, and s ∈ S. (C2) The functions f and g are of linear growth in the fourth variable, uniformly in [0, T ] × Ω × S, i.e., there exist constants Cf and Cg such that f (t, ω, s, x) Cf 1 + |x| ,
g(t, ω, s, x) Cg 1 + |x| ,
for all t ∈ [0, T ], ω ∈ Ω, s ∈ S, and x ∈ R. Obviously, if f and g are Lipschitz and f (·, 0) and g(·, 0) are bounded, i.e., if (C1) holds with constants Lf and Lg not depending on R, then (C2) is automatically fulfilled. s The main theorem of this section will be formulated in the terms of the spaces Bp,1,{B (S). j} For their definition and further properties we refer to [41, Section 4.3.3] and references therein. For p ∈ [1, ∞], q ∈ [1, ∞] and s > 0, let
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1 s,p H{Bj } (S) := f ∈ H s,p (S): Bj f = 0 for mj < s − , j = 1, . . . , m , p s s C{Bj } (S) := f ∈ C (S): Bj f = 0 for mj s, j = 1, . . . , m . 2m,p
For p ∈ (1, ∞) let Ap be the realization of A on the space Lp (S) with domain H{Bj } (S). In this way −Ap is the generator of an analytic C0 -semigroup (Sp (t))t0 . Since we may replace A and f in (10.2) by A − w and w + f , we may assume that (Sp (t))t0 is uniformly exponentially stable. From [38, Theorem 4.1] and [41, Theorem 1.15.3] (also see [12]) we deduce that if θ ∈ (0, 1) and p ∈ (1, ∞) are such that 2mθ −
1 = mj , p
for all j = 1, . . . , m,
(10.3)
then p 2m,p 2mθ,p L (S), D(Ap ) θ = Lp (S), H{Bj } (S) θ = H{Bj } (S) isomorphically. Theorem 10.2. Assume that (C1) holds, let d (1) If η ∈ ( 2mp , 12 −
d m
< 2, and let p ∈ (1, ∞) be such that
d 2mp
d < 12 − 4m . 2mη,p
is such that (10.3) holds for the pair (η, p) and if u0 ∈ H{Bj } (S)
d 4m )
almost surely, then for all r > 2 and α ∈ (η +
(2)
d 1 1 4m , 2 − r ) there exists a unique maximal 2mη,p 0,loc solution (u(t))t∈[0,) of (10.2) in Vα,r ([0, ) × Ω; H{Bj } (S)). d d Moreover, if δ > 2mp and λ 0 are such that δ + λ < 12 − 4m and (10.3) holds for the pair d m− ,p λ ([0, τ ); H 2mδ,p (S)) (δ, p), and if u0 ∈ H{Bj }2 (S) almost surely, then u has paths in Cloc {Bj }
almost surely. Furthermore, if condition (C2) holds as well, then: d (3) If η ∈ ( 2mp , 12 −
2mη,p
is such that (10.3) holds for the pair (η, p) and if u0 ∈ H{Bj } (S)
d 4m )
d 1 almost surely, then for all r > 2 and α ∈ (η + 4m , 2 − 1r ) there exists a unique global solution 2mη,p
0 ([0, T ] × Ω; H u of (10.2) in Vα,r {Bj } (S)).
(4) Moreover, if δ >
and λ 0 are such that δ + λ <
d 2mp
m− d2 ,p {Bj }
(δ, p), and if u0 ∈ H almost surely.
1 2
−
d 4m
and (10.3) holds for the pair 2mδ,p
(S) almost surely, then u has paths in C λ ([0, T ]; H{Bj } (S))
Remark 10.3. (i) For p ∈ [2, ∞) the uniqueness result in (1) and (3) can be simplified. In that case one obtains a unique solution in 2mη,p 2mη,p 0 [0, T ] × Ω; H{Bj } (S) . L0 Ω; C [0, T ]; H{Bj } (S) ⊆ Vα,r
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For this case on could also apply martingale type 2 integration theory from [3] to obtain the result. (ii) By the Sobolev embedding theorem one obtains Hölder continuous solutions in time and m− d
space. For instance, assume in (4) that u0 ∈ C{Bj }2 (S) almost surely. It follows from m− d η,p C{Bj }2 (S) → H{Bj } (S) = E, D(−Ap )
η 2m
η−ε → D (−Ap ) 2m
for all p ∈ (1, ∞) and η < m − d2 and ε > 0, that t → S(t)u0 is in C λ ([0, T ]; D((−Ap )δ ) d for all δ, λ > 0 that satisfy δ + λ < 12 − 4m . Since 2m(δ−ε),p D (−Ap )δ → E, D(−Ap ) δ−ε = H{Bj } (S) for all p ∈ (1, ∞) and ε > 0, by Sobolev embedding we obtain that the solution u has paths d in C λ ([0, T ]; CB2mδ (S)) for all δ, λ > 0 that satisfy δ + λ < 12 − 4m . Proof of Theorem 10.2. Let p ∈ (1, ∞) be as in the theorem and take E := Lp (S). For b ∈ (0, 1) let Eb denote the complex interpolation space [E, D(Ap )]b . Note that we use the notation Eb for complex interpolation spaces instead of fractional domain spaces as we did before. This will be more convenient, since we do not assume that Ap has bounded imaginary powers, and therefore we do not know the fractional domain spaces explicitly. Recall (cf. [25]) that Ea → D((−A)b ) and that D((−A)a ) → Eb for all a ∈ (b, 1) for all b ∈ (0, 1). d If b > 2mp , then by [41, Theorem 4.6.1] we have E, D(Ap ) b → C(S). d , 12 − Assume now that η ∈ ( 2mp
d 4m ).
Let F, G : [0, T ] × Ω × Eη → L∞ (S) be defined as
F (t, ω, x) (s) = f t, ω, s, x(s)
and
G(t, ω, x) (s) = g t, ω, s, x(s) .
We show that F and G are well defined and locally Lipschitz. Fix x, y ∈ Eη and let R := max ess supx(s), ess supy(s) < ∞. s∈S
s∈S
From the measurability of x, y and f it is clear that s → (F (t, ω, x))(s) and s → (F (t, ω, y))(s) are measurable. By (C1) it follows that for almost all s ∈ S, for all t ∈ [0, T ] and ω ∈ Ω we have F (t, ω, x) (s) − F (t, ω, y) (s) = f t, ω, s, x(s) − f t, ω, s, y(s) LR f x(s) − y(s) LR f x − yL∞ (S) η LR f x − yEη .
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Also, by the second part of (C1), for almost all s ∈ S, for all t ∈ [0, T ] and ω ∈ Ω we have F (t, ω, 0) (s) = f (t, ω, s, 0) < sup f (t, ω, s, 0) < ∞. t,s,ω
Combing the above results we see that F is well defined and locally Lipschitz. In a similar way one shows that F has linear growth (see (A2)) if (C2) holds. The same arguments work for G. Since L∞ (S) → Lp (S) = E we may consider F as an E-valued mapping. It follows from the Pettis measurability theorem that for all x ∈ Eη , (t, ω) → F (t, ω, x) is strongly measurable in E and adapted. 2 To model the term g(t, x, u(t, s)) ∂w(t,s) ∂t , let H := L (S) and let WH be a cylindrical Brownian motion. Define the multiplication operator function Γ : [0, T ] × Ω × Eη → L(H ) as Γ (t, ω, x)h (s) := G(t, ω, x) (s)h(s),
s ∈ S.
Then Γ is well defined, because for all t ∈ [0, T ], ω ∈ Ω we have G(t, ω, x) ∈ L∞ (S). d Now let θB > θB > 4m be such that θB + η < 12 and (10.3) holds for the pair (θB , 2). Define −θ (−A) B B : [0, T ] × Ω × Eη → γ (H, E) as (−A)−θB B(t, ω, x)h = i(−A)−θB G(t, ω, x)h,
where i : H 2mθB ,2 (S) → Lp (S) is the inclusion operator. This is well defined, because (−A)−θB :
H → H 2mθB ,2 (S) is a bounded operator and therefore by the right-ideal property and Corollary 2.2 it follows that
i(−A)−θB
(−A)−θB
γ (H,E)
L(L2 (S),H 2mθB ,2 (S))
i
γ (H 2mθB ,2 (S),Lp (S))
< ∞.
Moreover, B is locally Lipschitz. Indeed, fix x, y ∈ Eη and let R := max ess supx(s), ess supy(s) < ∞. s∈S
s∈S
It follows from the right-ideal property that
i(−A)θB B(t, ω, x) − B(t, ω, y)
i(−A)−θB
i(−A)θB
γ (H,E)
γ (H,E)
i(−A)θB
γ (H,E)
Γ (t, ω, x) − Γ (t, ω, y)
G(t, ω, x) − G(t, ω, y)
L(H )
L∞ (S)
LR x γ (H,E) g
− yL∞ (S)
a,p i(−A)θB γ (H,E) LR g x − yEη . In a similarly way one shows that B has linear growth. Notice that B is H -strongly measurable and adapted by the Pettis measurability theorem. If p ∈ [2, ∞), then E has type 2 and it follows from Lemma 5.2 that (−A)−θB B is locally 2 Lγ -Lipschitz and B has linear growth in the sense of (A3) if (C2) holds.
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In case p ∈ (1, 2) the above result holds as well. This may be deduced from the previous case. Indeed, for each n define (−A)−θB Bn : [0, T ] × Ω × Eη → γ (H, E) as (−A)−θB Bn (x) = nx ) otherwise. Define (−A)−θB B(x) for all xEη n and (−A)−θB Bn (x) = (−A)−θB Bn ( x −θ ∞ ∞ −θ ∞ B B (−A) Bn : [0, T ] × Ω × L (S) → γ (H, H ) as (−A) Bn (x) = (−A)−θB Bn (x). Replacing E with L2 (S) in the above calculation it follows that Bn∞ is a Lipschitz function uniformly on [0, T ] × Ω. Since H has type 2, (−A)−θB Bn∞ is L2γ -Lipschitz. Fix a finite Borel measure μ on (0, T ) and fix φ1 , φ2 ∈ L2γ ((0, T ), μ; Eη ). Since H → E continuously, it follows that
(−A)−θB Bn (t, ω, φ1 ) − Bn (t, ω, φ2 ) 2 γ (L ((0,T ),μ;H ),E)
∞
−θB ∞
Bn (t, ω, φ1 ) − Bn (t, ω, φ2 ) γ (L2 ((0,T ),μ;H ),H ) C (−A) Cφ1 − φ2 L2 ((0,T ),μ;L∞ (S)) Cφ1 − φ2 L2 ((0,T ),μ;Eη ) , where C also depends on n. In a similarly way one shows that B has linear growth in the sense of (A3) if g has linear growth. 2mβ,p d d , 12 − 4m ] is such that (10.3) holds for the If u0 ∈ H{Bj } (S) almost surely, where β ∈ ( 2mp 2mβ,p
pair (β, p), then ω → u0 (·, ω) ∈ H{Bj } (S) = Eβ is strongly F0 -measurable. This follows from the Pettis measurability theorem. (1) It follows from Theorem 8.1 with η, θB as above and with η + θB < 12 and θF = 0, τ = 0,loc p ∧ 2 that there is a unique maximal local mild solution (U (t))t∈[0,) in Vα,r ([0, ) × Ω; Eη ) 1 1 for all α > 0 and r > 2 satisfying η + θB < α < 2 − r . In particular U has almost all paths in C([0, ), Eη ). Now take u(t, ω, s) := U (t, ω)(s) to finish the proof of (1). d d d (2) Let δ = η > 2mp and λ 0 be such that λ+δ < 12 − 4m . Choose θB > 4m such that λ+δ < 1 2
2mδ,p
λ ([0, (ω)); H − θB . It follows from Theorem 8.1 that almost surely, U − Su0 ∈ Cloc {Bj } (S)).
m− d2 ,p d (S) = E 1 − d ⊆ 4m , p) satisfies (10.3). Since u0 ∈ H{Bj } 2 4m 2mδ,p 1 d λ Eδ almost surely and λ + δ < 2 − 4m we have Su0 ∈ C ([0, T ]; H{Bj } (S)) almost surely. λ ([0, (ω)); H 2mδ,p (S)). In the case ( 1 − d , p) does Therefore, almost all paths of U are in Cloc {Bj } 2 4m 1 d not satisfy (10.3) one can redo above argument with 2 − 4m − for > 0 small. This proves (2).
First consider the case that ( 12 −
(3), (4) This follows from Theorems 8.1 and parts (3), (4) of 8.1.
2
Remark 10.4. The above approach also works for systems of equations. Laplacian in Lp . Let S be an open subset (not necessarily bounded) of Rd . Consider the following perturbed heat equation with Dirichlet boundary values: ∂Wn (t) ∂u (t, s) = u(t, s) + f t, s, u(t, s) + , bn t, s, u(t, s) ∂t ∂t
s ∈ S, t ∈ (0, T ],
n1
u(t, s) = 0,
s ∈ ∂S, t ∈ (0, T ],
u(0, s) = u0 (s),
s ∈ S.
The functions f, bn : [0, T ] × Ω × S × R → R are jointly measurable, and adapted in the sense that for each t ∈ [0, T ], f (t, ·) and bn (t, ·) are Ft ⊗ BS ⊗ BR -measurable. We assume that
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(Wn )n1 is a sequence of independent standard Brownian motions on Ω and u0 : S × Ω → R is a BS ⊗ F0 -measurable initial value condition. We say that u : [0, T ] × Ω × S → R is a solution of (10.4) if the corresponding functional analytic model (SCP) has a mild solution U and u(t, s, ω) = U (t, ω)(s). Let p ∈ [1, ∞) be fixed and let E := Lp (S). It is well known that the Dirichlet Laplacian p generates a uniformly exponentially stable and analytic C0 -semigroup (Sp (t))t0 on Lp (S), and 1,p under a regularity assumption on ∂S one can identify D( p ) as W 2,p (S) ∩ W0 (S). Consider the following p-dependent condition: (C) There exist constants Lf and Lbn such that f (t, ω, s, x) − f (t, ω, s, y) Lf |x − y|, bn (t, ω, s, x) − bn (t, ω, s, y) Lb |x − y|, n for all t ∈ [0, T ], ω ∈ Ω, s ∈ S, and x, y ∈ R. Furthermore, f satisfies the boundedness condition
sup f (t, ω, ·, 0) Lp (S) < ∞, where the supremum is taken over all t ∈ [0, T ] and ω ∈ Ω, and the bn satisfy the following boundedness condition: for all finite measures μ on (0, T ),
T 1
2
2
bn (t, ω, ·, 0) dμ(t)
sup
< ∞,
Lp (S)
0 n1
where the supremum is taken over all ω ∈ Ω. Theorem 10.5. of Rd and let p ∈ [1, ∞). Assume that condition (C) Let 2S be an open subset p holds with n1 Lbn < ∞. If u0 ∈ L (S) almost surely, then for all α > 0 and r > 2 such that 0 ([0, T ] × Ω; Lp (S)). Moreover, α < 12 − 1r , the problem (10.4) has a unique solution U ∈ Vα,r 1 for all λ 0 and δ 0 such that λ + δ < 2 there is a version of U such that almost surely, t → U (t) − Sp (t)u0 belongs to C λ ([0, T ]; [Lp (S), D( p )]δ ). Remark 10.6. Under regularity conditions on ∂S and for p ∈ (1, ∞) one has p 1 L (S), D( p ) δ = x ∈ H 2δ,p (S): x = 0 on ∂S if 2δ − > 0 p provided δ ∈ (0, 1) is such that 2δ −
1 p
= 0.
Proof. We check the conditions of Theorem 7.1 (for p = 1 we use the extensions of our results to UMD− spaces described in Section 9, keeping in mind that L1 -spaces have this property). It was already noted that (A1) is fulfilled. Let E := Lp (S) and define F : E → E as F (t, x)(s) := f (t, s, x(s)). One easily checks that F satisfies (A2) with θF = η = 0.
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Let H := l 2 with standard unit basis (en )n1 , and let B: [0, T ] × Ω × E → L(H, E) be defined as (B(t, ω, x)en )(s) := bn (t, ω, s, x(s)). Then for all finite measures μ on (0, T ) and all φ1 , φ2 ∈ γ (L2 ((0, T ), μ; H ), E),
B(·, φ1 ) − B(·, φ2 )
γ (L2 ((0,T ),μ;H ),E)
T 1
2
2
bn t, ·, φ1 (t)(·) − bn t, ·, φ2 (t)(·) dμ(t)
p
0 n1
E
T 1
2
2
2
Lbn φ1 (t)(·) − φ2 (t)(·) dμ(t)
0 n1
E
p Lφ1 − φ2 γ (L2 ((0,T ),μ),E) , 1 where L = ( n1 L2bn ) 2 . Moreover,
B(·, 0)
T 1
2
2
bn (t, ·, 0) dμ(t)
< ∞. p 2 γ (L ((0,T ),μ;H ),E)
0 n1
From these two estimates one can obtain (A3).
E
2
Acknowledgment We thank Tuomas Hytönen for suggesting an improvement in Proposition 4.5. Appendix A. Measurability of stochastic convolutions In this appendix we study progressive measurability properties of processes of the form t t →
Φ(t, s) dWH (s), 0
where Φ is a two-parameter process with values in L(H, E). Proposition A.1. Let E be a UMD space. Assume that Φ : R+ × R+ × Ω → L(H, E) is H strongly measurable and for each t ∈ R+ , Φ(t, ·) is adapted and has paths in γ (L2 (R+ ; H ), E) almost surely. Then the process ζ : R+ × Ω → E, t ζ (t) =
Φ(t, s) dWH (s), 0
has a version which is adapted and strongly measurable.
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Proof. It suffices to show that ζ has a strongly measurable version ζ˜ , the adaptedness of ζ˜ being clear. Below we use strong measurability for metric spaces as in [42]. Let L0F (Ω; γ (L2 (R+ ; H ), E)) denote the closure of all adapted strongly measurable processes which are almost surely in γ (L2 (R+ ; H ), E). Note that by [33] the stochastic integral mapping extends to L0F (Ω; γ (L2 (R+ ; H ), E)). Let G ⊆ R+ × Ω be the set of all (t, ω) such that Φ(t, ·, ω) ∈ γ (L2 (R+ ; H ), E). Since Φ is H -strongly measurable, we have G ∈ BR+ ⊗ A. Moreover, letting Gt = {ω ∈ Ω: (t, ω) ∈ G} for t ∈ R+ , we have P(Gt ) = 1 and therefore Gt ∈ F0 . Define the H -strongly measurable function Ψ : R+ × R+ × Ω → B(H, E) as Ψ (t, s, ω) := Φ(t, s, ω)1[0,t] (s)1G (t, ω). It follows from [33, Remark 2.8] that the map R+ × Ω (t, ω) → Ψ (t, ·, ω) ∈ γ (L2 (R+ ; H ), E) is strongly measurable. Hence, the map R+ t → Ψ (t, ·) ∈ L0 (Ω; γ (L2 (R+ ; H ), E)) is strongly measurable. Since it takes values in L0F (Ω; γ (L2 (R+ ; H ), E)) it follows from an approximation argument that it is strongly measurable as an L0F (Ω; γ (L2 (R+ ; H ), E))-valued map. Since the elements which are represented by an adapted step process are dense in L0F (Ω; γ (L2 (R+ ; H ), E)), it follows from [42, Proposition 1.9] that we can find a sequence of processes (Ψn )n1 , where each Ψn : R+ → L0F (Ω; γ (L2 (R+ ; H ), E)) is a countably-valued simple function of the form Ψn =
1Bkn Φkn ,
with Bkn ∈ BR+
and Φkn ∈ L0F Ω; γ L2 (R+ ; H ), E ,
k1
such that for all t ∈ R+ we have Ψ (t) − Ψn (t)L0 (Ω;γ (L2 (R+ ;H ),E)) 2−n , where with a slight abuse of notation we write ξ L0 (Ω;F ) := E(ξ F ∧ 1) keeping in mind that this is not a norm. Notice that by the Chebyshev inequality, for a random variable ξ : Ω → F , where F is a normed space, and ε ∈ (0, 1], we have P ξ F > ε = P ξ F ∧ 1 > ε ε −1 ξ L0 (Ω;F ) . It follows from [33, Theorems 5.5 and 5.9] that for all t ∈ R+ , for all n 1 and for all ε, δ ∈ (0, 1],
Cδ 2 1
> ε 2 + n. Ψ (t, s) − Ψ (t, s) dW (s) P
n H
δ2 ε R+
Taking ε ∈ (0, 1] arbitrary and δ = n1 , it follows from the Borel–Cantelli lemma that for all t ∈ R+ ,
= 0. P
Ψ (t, s) − Ψn (t, s) dWH (s) > ε N 1 nN
R+
Since ε ∈ (0, 1], was arbitrary, we may conclude that for all t ∈ R+ , almost surely, ζ (t, ·) = Ψ (t, s) dWH (s) = lim Ψn (t, s) dWH (s). R+
n→∞ R+
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Clearly, Ψn (·, s) dWH (s) = R+
k1
1Bkn (·)
Φkn (s) dWH (s)
R+
has a strongly BR+ ⊗ F∞ -measurable version, say ζn : R+ × Ω → E. Let C ⊆ R+ × Ω be the set of all points (t, ω) such that (ζn (t, ω))n1 converges in E. Then C ∈ BR+ ⊗ F∞ and we may define the process ζ˜ as ζ˜ = limn→∞ ζn 1C . It follows that ζ˜ is strongly BR+ ⊗ F∞ -measurable and for all t ∈ R+ , almost surely, ζ˜ (t, ·) = ζ (t, ·). 2 References [1] J. Bourgain, Vector-valued singular integrals and the H1 -BMO duality, in: Probability Theory and Harmonic Analysis, Cleveland, OH, 1983, in: Monogr. Textbooks Pure Appl. Math., vol. 98, Dekker, New York, 1986, pp. 1–19. [2] Z. Brze´zniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1) (1995) 1–45. [3] Z. Brze´zniak, On stochastic convolution in Banach spaces and applications, Stoch. Stoch. Rep. 61 (3–4) (1997) 245–295. [4] Z. Brze´zniak, J.M.A.M. van Neerven, Space–time regularity for linear stochastic evolution equations driven by spatially homogeneous noise, J. Math. Kyoto Univ. 43 (2) (2003) 261–303. [5] D.L. Burkholder, Martingales and singular integrals in Banach spaces, in: Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 233–269. [6] P. Clément, B. de Pagter, F.A. Sukochev, H. Witvliet, Schauder decompositions and multiplier theorems, Studia Math. 138 (2) (2000) 135–163. [7] S. Cox, M.C. Veraar, Some remarks on tangent martingale difference sequences in L1 -spaces, Electron. Commun. Probab. 12 (2007) 421–433. [8] G. Da Prato, J. Zabczyk, A note on stochastic convolution, Stochastic Anal. Appl. 10 (2) (1992) 143–153. [9] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, in: Encycl. Math. Appl., vol. 44, Cambridge Univ. Press, Cambridge, 1992. [10] G. Da Prato, S. Kwapie´n, J. Zabczyk, Regularity of solutions of linear stochastic equations in Hilbert spaces, Stochastics 23 (1) (1987) 1–23. [11] R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (788) (2003). [12] R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old results of R.T. Seeley, Math. Ann. 328 (4) (2004) 545–583. [13] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge Univ. Press, Cambridge, 1995. [14] D.J.H. Garling, Brownian motion and UMD-spaces, in: Probability and Banach Spaces, Zaragoza, 1985, in: Lecture Notes in Math., vol. 1221, Springer, Berlin, 1986, pp. 36–49. [15] D.J.H. Garling, Random martingale transform inequalities, in: Probability in Banach Spaces VI, Sandbjerg, 1986, in: Progr. Probab., vol. 20, Birkhäuser Boston, Boston, MA, 1990, pp. 101–119. [16] R.C. James, Nonreflexive spaces of type 2, Israel J. Math. 30 (1–2) (1978) 1–13. [17] C. Kaiser, L. Weis, Wavelet transform for functions with values in UMD spaces, Studia Math., in press. [18] N.J. Kalton, J.M.A.M. van Neerven, M.C. Veraar, L. Weis, Embedding vector-valued Besov spaces into spaces of γ -radonifying operators, Math. Nachr. 281 (2008) 238–252. [19] N.J. Kalton, L. Weis, The H ∞ -calculus and square function estimates, in preparation. [20] N.J. Kalton, L. Weis, The H ∞ -calculus and sums of closed operators, Math. Ann. 321 (2) (2001) 319–345. [21] H. König, Eigenvalue Distribution of Compact Operators, Oper. Theory Adv. Appl., vol. 16, Birkhäuser, Basel, 1986. [22] N.V. Krylov, An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, in: Math. Surveys Monogr., vol. 64, Amer. Math. Soc., Providence, RI, 1999, pp. 185–242. [23] P.C. Kunstmann, L. Weis, Maximal Lp -regularity for parabolic equations, Fourier multiplier theorems and H ∞ functional calculus, in: Functional Analytic Methods for Evolution Equations, in: Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311.
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[24] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. II, Ergeb. Math. Grenzgeb., vol. 97, Springer, Berlin, 1979. [25] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., vol. 16, Birkhäuser, Basel, 1995. [26] T.R. McConnell, Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10 (2) (1989) 283–295. [27] A. Millet, W. Smole´nski, On the continuity of Ornstein–Uhlenbeck processes in infinite dimensions, Probab. Theory Related Fields 92 (4) (1992) 529–547. [28] J.M.A.M. van Neerven, M.C. Veraar, On the stochastic Fubini theorem in infinite dimensions, in: Stochastic Partial Differential Equations and Applications VII, Levico Terme, 2004, in: Lecture Notes Pure Appl. Math., vol. 245, CRC Press, 2005, pp. 323–336. [29] J.M.A.M. van Neerven, M.C. Veraar, On the action of Lipschitz functions on vector-valued random sums, Arch. Math. (Basel) 85 (6) (2005) 544–553. [30] J.M.A.M. van Neerven, L. Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2) (2005) 131–170. [31] J.M.A.M. van Neerven, L. Weis, Weak limits and integrals of Gaussian covariances in Banach spaces, Probab. Math. Statist. 25 (1) (2005) 55–74. [32] J.M.A.M. van Neerven, M.C. Veraar, L. Weis, Conditions for stochastic integrability in UMD Banach spaces, in: Banach Spaces and Their Applications in Analysis, in: de Gruyter Proc. Math., de Gruyter, 2007, pp. 127–146. [33] J.M.A.M. van Neerven, M.C. Veraar, L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab. 35 (4) (2007) 1438–1478. [34] A.L. Neidhardt, Stochastic integrals in 2-uniformly smooth Banach spaces, PhD thesis, University of Wisconsin, 1978. [35] G. Pisier, Some results on Banach spaces without local unconditional structure, Compos. Math. 37 (1) (1978) 3–19. [36] J. Rosi´nski, Z. Suchanecki, On the space of vector-valued functions integrable with respect to the white noise, Colloq. Math. 43 (1) (1980) 183–201. [37] J.L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, in: Probability and Banach Spaces, Zaragoza, 1985, in: Lecture Notes in Math., vol. 1221, Springer, Berlin, 1986, pp. 195–222. [38] R. Seeley, Interpolation in Lp with boundary conditions, Studia Math. 44 (1972) 47–60. [39] J. Seidler, Da Prato–Zabczyk’s maximal inequality revisited. I, Math. Bohem. 118 (1) (1993) 67–106. [40] H. Tanabe, Equations of Evolution, Monogr. Stud. Math., vol. 6, Pitman, Boston, MA, 1979. [41] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library, vol. 18, North-Holland, Amsterdam, 1978. [42] N.N. Vakhania, V.I. Tarieladze, S.A. Chobanyan, Probability Distributions on Banach Spaces, Math. Appl., vol. 14, Reidel, Dordrecht, 1987. [43] M.C. Veraar, Stochastic integration in Banach spaces and applications to parabolic evolution equations, PhD thesis, Delft University of Technology, 2006. [44] J.B. Walsh, An introduction to stochastic partial differential equations, in: École d’été de probabilités de Saint-Flour XIV—1984, in: Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. [45] L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp -regularity, Math. Ann. 319 (4) (2001) 735– 758.
Journal of Functional Analysis 255 (2008) 994–1007 www.elsevier.com/locate/jfa
Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w) Oleksandra V. Beznosova Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA Received 18 January 2008; accepted 30 April 2008 Available online 6 June 2008 Communicated by C. Kenig
Abstract The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp (w) if and only if the weight w belongs to the Muckenhoupt class Adp . However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2 (w) case. In this paper we prove that the bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L2 (w) depends linearly on the Ad2 characteristic of the weight w using Bellman function techniques and extrapolate this result to the Lp (w) case. © 2008 Elsevier Inc. All rights reserved. Keywords: Dyadic paraproduct; Weighted Lebesgue space; Bellman functions
1. Introduction Let D be the collection of dyadic intervals on R: D = {I = [k2−j ; (k + 1)2−j ) | k, j ∈ Z}, andlet mI f stand for the average of a locally integrable function f over the interval I : mI f := 1 |I | I f . The dyadic paraproduct is defined as πb f :=
mI f bI hI ,
I ∈D
E-mail address: [email protected]. URL: http://www.math.unm.edu/~beznosik. 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.04.025
O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
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where {hI }I ∈D is the Haar basis normalized in L2 : 1 hI (x) = √ χI + (x) − χI − (x) , |I | where I + and I − are the left and right halves of the dyadic interval I ; bI := b; hI , where ; stands for the dot product in the unweighted L2 , and b is a locally integrable function on R. In order for the dyadic paraproduct to be bounded on Lp we need b to be in BMOd , i.e.: bBMOd
1/2 2 1 b(x) − mI b dx := sup < ∞. I |I | I
We are going to use the fact that the BMOd norm of b can also be written as 1 2 bI . J ∈D |J |
b2BMOd = sup
I ∈D(J )
Paraproducts first appeared in the work of Bony on nonlinear partial differential equations (see [1]). The celebrated T(1) theorem of David and Journé [3] makes paraproducts one of the most important operators in harmonic analysis. The dyadic paraproduct operator is bounded on the weighted Lp (w) if and only if the weight w belongs to the Muckenhoupt class Ad2 (see [5]): w ∈ Ad2 : wAd := sup mI wmI w −1 . 2
I ∈D
The best known bound on the norm of the dyadic paraproduct until recently, was
d
π L2 (w) → L2 (w) C w d 2 b b BMOd . A 2
This statement can be found in [4]. Let us state the main result now. Theorem 1.1 (Main result). The norm of dyadic paraproduct on the weighted Lebesgue space L2 (w) is bounded from above by a constant multiple of the Ad2 characteristic of the weight w times the BMOd norm of b, i.e. for all f ∈ L2 (w) and all g ∈ L2 (w −1 ) πb f ; gL2 CwAd bBMOd f L2 (w) gL2 (w−1 ) . 2
(1.1)
This theorem, together with the sharp version of the Rubio De Francia’s extrapolation theorem from [4], produces Lp bounds of the following type. Theorem 1.2. Let w ∈ Adp and b ∈ BMOd . Then the norm of the dyadic paraproduct operator πb on the weighted Lp (w) space is bounded by πb Lp (w)→Lp (w) C1 (p)wAdp bBMOd
when p 2,
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and by 1
πb Lp (w)→Lp (w) C2 (p)wAp−1 d bBMOd
when p < 2,
p
where C1 (p) and C2 (p) are constants that only depend on p. 2. Proof of the main result Proof. In order to prove Theorem 1.1 it is enough to show that ∀f, g ∈ L2 Σ1 := πb f w −1/2 ; gw 1/2 CwAd bBMOd f 2 g2 . 2
We are going to decompose this sum using the weighted Haar system of functions (see [2]). Let HIw be defined in the following way:
HIw := hI |I | − Aw I χI
and Aw I :=
mI + w − mI − w . 2mI w
Then {w 1/2 HIw } is orthogonal in L2 with norms satisfying the inequality w 1/2 HIw L2 √ |I |mI w. By Bessel’s inequality we have:
∀g ∈ L2
I ∈D
We can break
1
2 1 g; w 1/2 HIw L g2L2 . 2 |I |mI w
(2.1)
into two sums: Σ1 =
mI f w −1/2 bI gw 1/2 ; hI
I ∈D
=
1 mI f w −1/2 bI √ g; w 1/2 HIw |I | I ∈D +
1 mI f w −1/2 bI √ gw 1/2 ; Aw I χI |I | I ∈D
=: Σ2 + Σ3 . We claim that both sums, at most linearly: Σ2 =
2
and
3,
can be bounded with a bound that depends on wAd 2
1 mI f w −1/2 bI √ g; w 1/2 HIw CwAd bBMOd f L2 gL2 2 |I | I ∈D
(2.2)
and Σ3 =
I ∈D
mI f w −1/2 bI Aw |I |mI gw 1/2 CwAd bBMOd f L2 gL2 . I 2
(2.3)
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The bound on 2 is very straight forward. We decompose 2 into the product of two sums using Cauchy–Schwarz: Σ2 =
1 mI f w −1/2 bI √ g; w 1/2 HIw |I | I ∈D
m2I
fw
−1/2
2 bI mI w
1/2
I ∈D
I ∈D
2 1 g; w 1/2 HIw |I |mI w
1/2 .
By (2.1) I ∈D
2 1 g; w 1/2 HIw g2L2 . |I |mI w
So, for (2.2) it is enough to show that I ∈D
m2I f w −1/2 bI2 mI w Cw2Ad b2BMOd f 2L2 .
(2.4)
2
By the weighted Carleson embedding theorem, which can be found, for example, in [6], (2.4) holds if and only if ∀J ∈ D
1 |J |
I ∈D(J )
m2I w −1 mI wbI2 Cw2Ad b2BMOd mJ w −1 . 2
Since ∀I ∈ D, mI wmI w −1 wAd , it is enough to verify that 2
∀J ∈ D
1 |J |
I ∈D(J )
mI w −1 bI2 CwAd b2BMOd mJ w −1 . 2
(2.5)
Inequality (2.5) follows from the fact that b ∈ BMOd and hence the sequence {bI2 }I ∈D is a Carleson sequence with Carleson constant b2 d : BMO
∀J ∈ D
1 |J |
I ∈D(J )
bI2 b2BMOd ,
(2.6)
and the following proposition, which we are going to prove in Section 3. Proposition 2.1. Let w ∈ Ad2 and {λI } be a Carleson sequence of nonnegative numbers, that is, there exists a constant Q > 0 such that ∀J ∈ D
1 λI Q. |J | I ∈D(J )
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Then ∀J ∈ D 1 λI 4QmJ w |J | mI w −1
(2.7)
1 mI wλI 4QwAd mJ w. 2 |J |
(2.8)
I ∈D(J )
and
I ∈D(J )
Estimate (2.8) applied to λI = bI2 and w −1 (w −1 ∈ Ad2 and w −1 Ad = wAd ) provides (2.5), 2 2 so inequality (2.2) for 2 holds. Now we need to prove the inequality (2.3). It is a little bit more involved. We want to show that Σ3 =
bI Aw |I |mI f w −1/2 mI gw 1/2 CwAd bBMOd f 2 g2 . I 2
I ∈D
We are going to use the bilinear embedding theorem from [6] using the observation of J. Wittwer (see [8]), stated in the form similar to the bilinear embedding theorem of S. Petermichl from [7]. Theorem 2.1 (Nazarov, Treil, Volberg). Let v and w be weights. Let {αI } be a sequence of nonnegative numbers such that for all dyadic intervals J ∈ D the following three inequalities hold with some constant Q > 0: 1 αI mI wmI v|I | Q, |J | I ∈D(I )
1 αI mI w|I | QmJ w |J | I ∈D(J )
and 1 αI mI v|I | QmJ v. |J | I ∈D(J )
Then for any two nonnegative functions f, g ∈ L2
αI mI f v 1/2 mI gw 1/2 |I | CQf L2 gL2
I ∈D
holds with some numerical constant C > 0. So, in order to complete the proof it is enough to show that the following three inequalities hold:
O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
999
1 |I |mI wmI w −1 Cb bI Aw I BMOd wAd2 , |J |
∀J ∈ D
(2.9)
I ∈D(J )
∀J ∈ D
1 |I |mI w Cb bI Aw I BMOd wAd2 mJ w, |J |
(2.10)
1 −1 |I |mI w −1 Cb bI Aw I BMOd wAd2 mJ w , |J |
(2.11)
I ∈D(J )
∀J ∈ D
I ∈D(J )
The following proposition helps us handle first inequality (2.9). Proposition 2.2. Let w be a weight from Ad2 , then ∀J ∈ D mI + w − mI − w 2 1 1/4 1/4 1/4 1/4 |I |mI wmI w −1 CmJ wmJ w −1 |J | mI w I ∈D(J )
and therefore, 1 mI + w − mI − w 2 |I |mI wmI w −1 CwAd . 2 |J | mI w I ∈D(J )
The proof of Proposition 2.2 can be found in Section 4. Note that by Cauchy–Schwarz
1 −1 |bI Aw I | |I |mI wmI w |J | I ∈D(J )
2 2 1 w 2 1 2 −1 −1 AI |I |mI wmI w bI mI wmI w . |J | |J | 1
1
I ∈D(J )
I ∈D(J )
It follows from (2.6) that 1 2 1 2 bI mI wmI w −1 wAd bI wAd b2BMOd , 2 |J | 2 |J | I ∈D(J )
I ∈D(J )
and by Proposition 2.2 1 w 2 mI + w − mI − w 2 1 −1 AI |I |mI wmI w = |I |mI wmI w −1 |J | |J | mI w I ∈D(J )
I ∈D(J )
CwAd . 2
To prove inequality (2.10) we need the following result by J.Wittwer (see [8]).
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Lemma 2.2 (J.Wittwer). Let w ∈ Ad2 be a weight, then ∀J ∈ D
mI− w − mI+ w 2 1 |I |mI w CwAd mJ w 2 |J | 2mI w
(2.12)
I ∈D(J )
and this result is sharp. The linear bound (2.10) follows by Cauchy–Schwarz from (2.12) and Proposition 2.2 (inequality (2.8)) applied to λI = bI2 : ∀J ∈ D
1 2 bI mI w CwAd b2BMOd mJ w. 2 |J | I ∈D(J )
And the following proposition, together with (2.5) allows us to establish the inequality (2.11) in a similar way. Proposition 2.3. Let w be a weight in Ad2 , then for all dyadic intervals J , mI + w − mI − w 2 1 |I |mI w −1 CwAd mJ w −1 . 2 |J | mI w I ∈D(J )
The proof of Proposition 2.3 can be found in Section 5. Which completes the proof of the Theorem 1.1. 2 3. Bellman function proof of Proposition 2.1 We are going to show that for any Carleson sequence {λI }I ∈D with constant Q, λI 0 and ∀J ∈ D
1 λI Q, |J | I ∈D(J )
the inequality (2.7) holds for any dyadic interval J , that is, 1 λI 4QmJ w. |J | mI w −1 I ∈D(J )
Note that inequality (2.8) follows from inequality (2.7). Lemma 3.1. Suppose there exists a real-valued function of three variables B(x) = B(u, v, l), whose domain D is given by those x = (u, v, l) ∈ R3 such that u, v 0,
uv 1,
whose range is given by 0 B(x) u,
0 l 1,
O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
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and such that the following convexity property holds: x + + x− = (0, 0, α), 2 B(x+ ) + B(x− ) 1 B(x) − α. 2 4v
∀x, x± ∈ D such that x −
(3.1)
Then Proposition 2.1 holds. Proof. Fix a dyadic interval J . Let xJ = (uJ , vJ , lJ ) where uJ = mJ w, vJ = mJ w −1 and lJ = 1 I ∈D(J ) λI . Clearly for each dyadic interval J , xJ belongs to the domain D. Let x± := |J |Q xJ ± ∈ D. By definition, xJ − where αJ :=
1 |J |Q λJ .
xJ + + xJ − = (0, 0, αJ ), 2
Then, by the convexity condition (3.1) and since 2|J + | = 2|J − | = |J |,
|J |mJ w |J |B(xJ )
1 |J |B(xJ + ) |J |B(xJ − ) + + λJ 2 2 4QmJ w −1
= |J + |B(xJ + ) + |J − |B(xJ − ) +
1 λJ . 4QmJ w −1
We can use the same lower bound estimate for |J + |B(xJ + ) and |J − |B(xJ − ) now. Iterating this procedure and using the assumption that B 0 on D we get mJ w
λI 1 4|J |Q mI w −1 I ∈D(J )
which implies Proposition 2.1.
2
So, Proposition 2.1 will hold if we can show existence of the function B of the Bellman type, satisfying the conditions of Lemma 3.1. Lemma 3.2. The following function B(u, v, l) := u −
1 v(1 + l)
is defined on D, 0 B(x) u for all x = (u, v, l) ∈ D and satisfies the following differential inequalities on D: 1 ∂B (u, v, l) ∂l 4v and
(3.2)
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O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
⎛
⎞ du −(du, dv, dl) d 2 B ⎝ dv ⎠ 0, dl
(3.3)
where d 2 B(u, v, l) denotes the Hessian matrix of the function B evaluated at (u, v, l). Moreover, conditions (3.2) and (3.3) imply the convexity condition (3.1). Proof. Range conditions are obvious. It is nothing but a calculus exercise to check the differential conditions as well: ∂B 1 1 (u, v, l) = 2 ∂l 4v v(1 + l) since 0 l 1 and ⎛ ⎞ 0 du ⎜ 2 ⎝ −(du, dv, dl) d B dv ⎠ = (du, dv, dl) ⎝ 0 dl 0 ⎛
0
0
2 v 3 (1+l) 1 v 2 (1+l)2
1 v 2 (1+l)2 2 v(1+l)3
⎞⎛
⎞ du ⎟⎝ ⎠ ⎠ dv 0. dl
And finally let us see how differential conditions (3.2) and (3.3) imply the convexity condition (3.1): B(x) −
x + + x− x+ + x− B(x+ ) + B(x− ) B(x+ ) + B(x− ) = B(x) − B + B − 2 2 2 2 ∂B = (u, v, l )α − ∂l
1
1 − |t| b
(t) dt,
−1
1−t where b(t) := B(s(t)), s(t) := 1+t 2 s+ + 2 s− , −1 t 1, note that s(t) ∈ D whenever s+ and s− do since D is a convex domain and s(t) is a point on the line between s+ and s− . The first summand in (3.4) appears as an application of the Mean Value Theorem. The second is an exercise in calculus, which we describe now. It is easy to see that
1 − 2
1 −1
b(1) − b(−1) . 1 − |t| b
(t) dt = b(0) − 2
Note also that b(0) = B(x(0)) = B(x), similarly b(−1) = B(x− ) and b(1) = B(x+ ). The differential inequalities trivially imply that −b
(t) 0 and 1 B(x+ ) + B(x− ) ∂B B(x) − = (u, v, l )α − 2 ∂l 2
1 −1
1 1 − |t| b
(t) dt α. 4v
This completes the proofs of both Lemma 3.2 and Proposition 2.1.
2
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1003
4. Bellman function proof of Proposition 2.2 We are going to prove that there is a numerical constant C > 0, such that for all dyadic intervals J ∈ D 1 mI + w − mI − w 2 1/4 1/4 1/4 1/4 |I |mI wmI w −1 CmJ wmJ w −1 |J | mI w
(4.1)
I ∈D(J )
using the Bellman function technique. Lemma 4.1. Suppose there exists a real-valued function of two variables B(x) = B(u, v), whose domain D is given by those x = (u, v) ∈ R2 such that u, v 0 and uv 1,
(4.2)
whose range is given by 0 B(x)
√ 4 uv,
x ∈ D,
and such that the following convexity property holds for all x, x± ∈ D: if x =
x+ + x− 2
then B(x) −
B(x+ ) + B(x− ) v 1/4 C 7/4 (u+ − u− )2 2 u
(4.3)
with a numerical constant C independent of everything. Then Proposition 2.2 will be proved. Proof. Let uI := mI w, vI := mI w −1 , v+ = vI + , v− = vI − and similarly for u± . Then by Hölder’s inequality (u, v) and (u± , v± ) belong to the domain D. Fix J ∈ D, by the convexity and range conditions
4 |J | mJ wmJ w −1 |J |B(uJ , vJ ) |J + |B(u+ , v+ ) + |J − |B(u− , v− ) mJ w −1 1/4
+ |J |C
7/4
mJ w
(mJ + w − mJ − w)2 .
Iterating this process and using the fact that B(u, v) 0 we get 1/4
m w −1 4 |I | I 7/4 (mJ + w − mJ − w)2 , |J | mJ wmJ w −1 C mI w I ∈D(J )
which completes the proof of Lemma 4.1.
2
Now, in order to complete the proof of (4.1) we need to show the existence of the Bellman type function B which satisfies the conditions of Lemma 4.1.
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O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
Lemma 4.2. The following function B(u, v) := is defined on D, 0 B(u, v) inequality in D:
√ 4
√ 4 uv
uv for all (u, v) ∈ D, and satisfies the following differential
−(du, dv) d 2 B
du dv
1 v 1/4 |du|2 . 8 u7/4
(4.4)
Furthermore, this implies the convexity condition (4.3) of Lemma 4.1. Proof. Since u and v are√positive in the domain D, function B = condition 0 B(u, v) 4 uv is trivially satisfied. Let us prove the differential inequality (4.4) now: −(du, dv) d 2 B
du dv
√ 4 uv is well defined on D and
1 −7 −3 −3 du 1 −v 4 u 4 3v 4 u 4 (du, dv) −3 −3 −7 1 16 dv −v 4 u 4 3v 4 u 4 1 −7 du 1 0 v4u 4 = (du, dv) −7 1 8 4 4 dv 0 v u 1 −7 −3 −3 du 1 −v 4 u 4 v4u 4 + (du, dv) −3 −3 −7 1 16 dv −v 4 u 4 v 4 u4
=
1 1 −7 v 4 u 4 |du|2 , 8 as we wanted to show. Now we only need to check the convexity condition (4.3). We fix an interval I and let b(t) := B(ut , vt ), 1 1 1 1 vt := (t + 1)v+ + (1 − t)v− , ut := (t + 1)u+ + (1 − t)u− , 2 2 2 2 −1 t 1. In order to prove inequality (4.3) it is enough to establish b(0) −
v 1/4 b(1) + b(−1) C 7/4 |du|2 . 2 u
It is easy to see that 1 1 b(0) − b(−1) + b(1) = − 2 2
1 −1
1 + |t| b
(t) dt.
O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
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Note that −b
(t)
1 1/4 −7/4 v ut (u1 − u−1 )2 32 t
(4.5)
and that ∀t ∈ [−1/2; 1/2]ut = u0 + 12 t (u1 − u−1 ), since domain D is convex ut ∈ D, and |u1 − u−1 | |u1 | + |u−1 |,
|t| 1/2, u1 , u−1 0,
1 1 −u0 = − (u1 + u−1 ) t (u1 − u−1 ) (u1 + u−1 ) = u0 , 2 2 so ut 32 u0 and similarly vt 12 v0 for t ∈ [−1/2; 1/2]. Together with (4.5) it makes 1/4 −7/4
−b
(t) Cv0 u0
(u1 − u−1 )2 .
So, B(u, v) −
v 1/4 1 1 B(u+ , v+ ) − B(u− , v− ) = b(0) − b(1) + b(−1) C 7/4 |du|2 2 2 u
with numerical constant C independent of everything. Which completes the proof of Lemma 4.2 and Proposition 2.2. 2 5. Proof of the Proposition 2.3 First note that since for every dyadic interval I we have mI wmI w −1 wAd , it is enough 2 to show that ∀J ∈ D
1 (mI + w − mI − w)2 |I | CmJ w −1 |J | m3I w
(5.1)
I ∈D(J )
for some numerical constant C for any weight w, such that expressions mI w and mI w −1 are meaningful for all dyadic intervals I (for example, both functions w and w −1 are locally integrable). Lemma 5.1. Suppose there exists a real-valued function of two variables B(x) = B(u, v), whose domain D is given by those x = (u, v) ∈ R2 such that u, v 0,
(5.2)
uv 1,
(5.3)
whose range is given by 0 B(x) v
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O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
and such that the following convexity property holds for all x, x± ∈ D: if x =
x+ + x− 2
B(x+ ) + B(x− ) 1 C 3 (u+ − u− )2 2 u
then B(x) −
(5.4)
with some numerical constant C. Then Proposition 2.3 will be proved (inequality (5.1) holds for all dyadic intervals J ). Proof. Let uI := mI w, vI := mI w −1 , v+ = vI + , v− = vI − and similarly for u± . Then by Hölder’s inequality (u, v) and (u± , v± ) belong to the D. Fix J ∈ D, by the convexity property and range conditions |J |mJ w −1 |J |B(uJ , vJ ) |J + |B(u+ , v+ ) + |J − |B(u− , v− ) + C|J |
1 m3J w
2 mJ + w − mJ − w .
Iterating this process and using positivity of function B, we get |J |mJ w −1 C
|I |
I ∈D(J )
which completes the proof of Lemma 5.1.
2 1 mI + w − mI − w , 3 mI w
2
To prove inequality (5.1) and Proposition 2.3 we need to show the existence of the function B of the Bellman type satisfying conditions of Lemma 5.1. Lemma 5.2. The following function B(u, v) = v −
1 u
defined on domain D, 0 B(u, v) v for all (u, v) ∈ D and satisfies the following differential inequality in D: du 2 2 −(du, dv) d B 3 |du|2 . u dv Moreover, it implies the convexity condition (5.4) with some numerical constant C independent of everything. Proof. First note that since uv 1 and u and v are both positive in the domain D, B is well defined and 0 B(u, v) =
1 uv − 1 =v− v u u
du = 2u−3 |du|2 . on D and −(du, dv) d 2 B dv Convexity condition (5.4) follows from this in practically the same way as in Proposition 2.2. 2
O.V. Beznosova / Journal of Functional Analysis 255 (2008) 994–1007
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Acknowledgments Special thank to my graduate adviser María Cristina Pereyra. References [1] J. Bony, Calcul symbolique et propagation des singularites pour les equations aux derivees non-lineaires, Ann. Sci. École Norm. Sup. (4) 14 (1981) 209–246. [2] R. Coifman, P. Jones, S. Semmes, Two elementary proofs of the L2 boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. (3) 2 (1989) 553–564. [3] J.-L. Journé, G. David, A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. (2) 20 (1984) 371–397. [4] O. Dragicevic, L. Grafakos, M.C. Pereyra, S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005) 73–91. [5] N.H. Katz, M.C. Pereyra, Haar multipliers, paraproducts and weighted inequalities, in: Analysis of Divergence Control and Management of Divergent Processes, Birkhäuser, 1998, pp. 145–170. [6] F. Nazarov, S. Treil, A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999) 909–928. [7] S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of classical Ap characteristic, Amer. J. Math. 129 (2007) 1355–1375. [8] J. Wittwer, A sharp estimate on the norm of the martingale transform, Math. Res. Lett. 7 (2000) 1–12.
Journal of Functional Analysis 255 (2008) 1008–1023 www.elsevier.com/locate/jfa
A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow Shilong Kuang, Qi S. Zhang ∗ Department of Mathematics, University of California, Riverside, CA 92521, USA Received 25 January 2008; accepted 15 May 2008 Available online 20 June 2008 Communicated by L. Gross
Abstract We establish a point-wise gradient estimate for all positive solutions of the conjugate heat equation. This contrasts to Perelman’s point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman’s estimate, the most general form of our gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, we also prove a localized gradient estimate similar to the Li–Yau estimate for the linear Schrödinger heat equation. The main difference with the linear case is that no assumptions on the derivatives of the potential (scalar curvature) are needed. A classical Harnack inequality follows. © 2008 Elsevier Inc. All rights reserved. Keywords: Conjugate heat equation; Ricci flow; Gradient estimates
1. Introduction In the paper [11], Perelman discovers a monotonicity formula for the W entropy of positive solutions of the conjugate heat equation.
u − Ru + ∂t u = 0, ∂t g = −2Ric.
* Corresponding author. Fax: 011 1 951 827 7314.
E-mail addresses: [email protected] (S. Kuang), [email protected] (Q.S. Zhang). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.014
(1.1)
S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
1009
Here u = u(x, t) with x ∈ M, a compact manifold and t ∈ (0, T ), T > 0; is the Laplace– Beltrami operator under the metric g and Ric is the Ricci curvature tensor. Here and though out it is assumed the metric g is smooth in the region M × (0, T ) with T > 0, unless stated otherwise. Moreover, he shows that this formula implies a point-wise gradient estimate for the fundamental solution of the conjugate heat equation [11, Corollary 9.3]. Namely, let u be the fundamental solution of (1.1) in M × (0, T ) and f be the function such that u = (4πτ )−n/2 e−f with τ = T − t. Then τ 2f − |∇f |2 + R + f − n u 0 in M × (0, T ). This formula can be regarded as a generalization of the Li–Yau–Hamilton gradient estimate for the heat equation. By now it is clear that the importance of Perelman’s monotonicity formula and gradient estimate can hardly be overstated. See for example [1,3,7] and [9]. In particular it can be regarded as a monotonicity formula for the best constant in Gross’ log Sobolev inequality [4]. However, there is one place where some improvement is still desirable, namely the gradient estimate does not apply to all positive solutions to the conjugate heat equation, see [6], e.g. Also for instance, for the Ricci flat manifold S 1 × S 1 . The constant 1 is a solution to the conjugate heat equation. Clearly it does not satisfy Perelman’s gradient estimate stated above. Whether a Perelman type gradient estimate exists for all positive solutions of the conjugate heat equation is a question circulating for a few years. The main goal of this paper is to establish a gradient estimate that works for all positive solutions of the conjugate heat equation. Like Perelman’s estimate for the fundamental solution, the most general form of the new gradient estimate does not require any curvature assumption. Moreover, assuming only lower bound on the Ricci curvature, it also has a local version which appears similar to the Li–Yau estimate for the linear heat equation. An immediate consequence of the gradient estimate is a classical Harnack inequality for positive solutions of the conjugate heat equation. It is well known that a Harnack inequality is very desirable for elliptic and parabolic equations. The rest of the paper is organized as follows. The results concerning the conjugate heat equation under Ricci flow is given in Sections 2, 3. As another application, in Section 2, we prove an uniqueness result for the conjugate heat equation which generalizes a known result [6]. The main part of the paper was originally posted on arXiv on November 2006 (arXiv:math.DG/0611298). 2. New gradient estimate and Harnack inequality for positive solutions to the conjugate heat equation The main result of this section is Theorem 2.1. Suppose g(t) evolve by the Ricci flow, that is, ∂g ∂t = −2Ric on a closed manifold M for t ∈ [0, T ), and u : M × [0, T ) → (0, ∞) be a positive C 2,1 solution to the conjugate heat −f equation 2∗ u = −u − ut + Ru = 0. Let u = e n and τ = T − t. Then: (4πτ ) 2
(i) if the scalar curvature R 0, then for all t ∈ (0, T ) and all points, 2f − |∇f |2 + R
2n ; τ
(2.1)
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(ii) without assuming the non-negativity of R, then for t ∈ [ T2 , T ) and all points, 2f − |∇f |2 + R
3n . τ
(2.2)
Remark 2.1. Since f = − ln u − n2 ln(4πτ ), if we replace f by u accordingly, then we get 2n |∇u|2 uτ −2 −R , 2 u τ u
if R 0;
|∇u|2 3n uτ −2 −R , u τ u2
if R changes sign and t T /2.
(2.3)
It is similar to the Li–Yau gradient estimate for the heat equation on manifolds with nonnegative Ricci curvature, i.e. |∇u|2 ut n − 2 u 2t u for positive solutions of u − ∂t u = 0. Remark 2.2. Some related gradient estimates with various dependence on the Ricci and other curvatures can be found in [5] and [10]. Proof of Theorem 2.1. By a standard approximation argument as in [3, vol. 2] e.g., we can assume without loss of generality that g = g(t) is smooth in the closed time interval [0, T ] and that u is strictly positive everywhere. Indeed, by Theorem A.23 in [3, vol. 2] (due to W.X. Shi), the curvature tensor is uniformly bounded in the time interval [0, T − δ] with the bound depending only on the initial data and δ, a positive number. Moreover the lower bound of the scalar curvature is nondecreasing since the scalar curvature R satisfies (cf. [2, p. 209]) 2 R − ∂t R + R 2 0. n Therefore, we can just work on the interval [0, T − δ] first. In the proof, it will be clear that all constants are independent of the curvature tensor. They only depend on the lower bound of the scalar curvature, which is improving. Hence we can take δ to zero to get the desired result on [0, T ). (i) By standard computation (one can consult various sources for more details ([2], e.g.)),
∂ ∂f + (f ) = + 2 Ric, Hess(f ) + (f ) ∂t ∂t
n 2 + 2 Ric, Hess(f ) + (f ) = −f + |∇f | − R + 2τ
= 2 Ric, Hess(f ) + |∇f |2 − R .
Also using the evolution equation of g,
(2.4)
S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
∂ ∂f + |∇f |2 = 2Ric(∇f, ∇f ) + 2 ∇f, ∇ + |∇f |2 ∂t ∂t
= 2Ric(∇f, ∇f ) + 2 ∇f, ∇ −f + |∇f |2 − R + |∇f |2 .
1011
(2.5)
Notice also
∂ + R = 2R + 2|Ric|2 . ∂t
(2.6)
Combining these three expressions, we deduce
∂ + 2f − |∇f |2 + R ∂t
= 4 Ric, Hess(f ) + |∇f |2 − 2Ric(∇f, ∇f )
− 2 ∇f, ∇ −f + |∇f |2 − R + 2|Ric|2 .
(2.7)
Denote q(x, t) = 2f − |∇f |2 + R. By Bochner’s identity, |∇f |2 = 2|fij |2 + 2∇f ∇(f ) + 2Rij fi fj , the above equation becomes
∂ + q = 4Rij fij + 2|fij |2 + 2∇f ∇(f ) + 2Rij fi fj − 2Rij fi fj ∂t − 2∇f ∇ −f + |∇f |2 − R + 2Rij2 = 4Rij fij + 2|fij |2 + 2Rij2 + 2∇f ∇ 2f − |∇f |2 + R = 2|Rij + fij |2 + 2∇f ∇q,
that is,
∂ 2 + q − 2∇f ∇q = 2|Rij + fij |2 (R + f )2 . ∂t n
(The referee kindly pointed out that Eq. (2.8) was also shown in [3].) Since q = 2f − |∇f |2 + R = 2(f + R) − |∇f |2 − R, and hence R + f =
1 q + |∇f |2 + R , 2
(2.8)
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S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
we have
2 ∂ 1 + q − 2∇f ∇q q + |∇f |2 + R . ∂t 2n
(2.9)
By direct computation, we also have, for any > 0
2 ∂ 2n 2n 2n 1 . + − 2∇f ∇ = ∂t T −t + T −t + 2n T − t +
(2.10)
Combine the above two expressions, we get
2n 2n ∂ + q − − 2∇f ∇ q − ∂t T −t + T −t + 2n 2n 1 2 2 q+ + |∇f | + R q − + |∇f | + R . 2n T −t + T −t +
(2.11)
We deal with the above inequality in two cases: Case 1. At a point (x, t), q +
2n T −t+
q−
+ |∇f |2 + R 0. Then also 2n + |∇f |2 + R 0 T −t +
thus,
∂ + ∂t
q−
2n T −t +
− 2∇f ∇ q −
2n T −t +
0.
(2.12)
2n Case 2. At a point (x, t), q + T −t+ + |∇f |2 + R > 0. Then the inequality (2.11) can be transformed to ∂ 2n 2n + q − − 2∇f ∇ q − ∂t T −t + T −t + 2n 1 2n + |∇f |2 + R q − − q+ 2n T −t + T −t + 1 2n 2 2 |∇f | + R q + + |∇f | + R 0. (2.13) 2n T −t +
Let us define a potential term by
V = V (x, t) =
if q +
0, 1 2n (q
+
2n T −t+
+ |∇f |2 + R),
if q +
2n T −t+ 2n T −t+
+ |∇f |2 + R 0 at (x, t), + |∇f |2 + R 0 at (x, t). (2.14)
We know V is continuous. Further, by the above two cases, we conclude
S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
∂ + ∂t
q−
2n T −t +
− 2∇f ∇ q −
2n T −t +
−V q −
2n T −t +
1013
0. (2.15)
Since we assumed that the Ricci flow is smooth in [0, T ] and that u(x, t) is a positive C 2,1 solution to the conjugate heat equation, we have q = 2f − |∇f |2 + R =
|∇u|2 2u +R − u u2
is bounded for t ∈ [0, T ]. If we choose sufficiently small, then q(x, T ) 2n . Thus by the 2n . Letting → 0, we have for all maximum principle ([2], e.g.), for all t ∈ [0, T ], q(x, t) T −t+ t ∈ [0, T ], q(x, t)
2n . T −t
(2.16)
Recall q = 2f − |∇f |2 + R, τ = T − t. Then we have 2n . τ
(2.17)
|∇u|2 2uτ 2n −R . − u τ u2
(2.18)
2f − |∇f |2 + R Further, f = − ln u − n2 (4πτ ). Then the above yields
Proof of (ii). Next we prove the gradient estimate without the non-negativity assumption for the scalar curvature R. Let c 2n be a constant to be determined later; denote B = |∇f |2 + R. Similarly to the inequality (2.11), we also have,
q−
∂ + ∂t
c T −t +
− 2∇f ∇ q −
c T −t +
1 c (q + B)2 − 2n (T − t + )2 1 c2 c2 2cn 2 (q + B) − = + − 2n (T − t + )2 (T − t + )2 (T − t + )2 c c c(c − 2n) 1 . q− +B q + +B + = 2n T −t + T −t + (T − t + )2
We deal with the previous inequality at a given point (x, t) in three cases.
(2.19)
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Case 1. B 0, and q +
c T −t+
+ B 0. Then also q−
c +B 0 T −t +
thus,
∂ + ∂t
Case 2. B 0, and q +
q−
c T −t+
− 2∇f ∇ q −
c T −t +
0.
(2.20)
+ B > 0. Then the inequality (2.19) can be changed to
∂ + ∂t
c T −t +
c q− T −t +
− 2∇f ∇ q −
c T −t + 1 c c − q+ +B q − 2n T −t + T −t + c 1 + B 0. B q+ 2n T −t +
(2.21)
Case 3. B 0. Then the inequality (2.19) can be changed to ∂ c c + q − − 2∇f ∇ q − ∂t T −t + T −t + 1 c c q+ +B q − 2n T −t + T −t + 1 c 1 2Bc c(c − 2n) + B q− . + + 2n T −t + 2n T − t + (T − t + )2
(2.22)
To continue, we need the following estimate of the scalar curvature R under the Ricci flow: R−
n 2(t + )
(2.23)
for some > 0 depending on the initial value of R (it comes from the weak minimum principle 2 2 for a differential inequality ∂R ∂t R + n R ([2], e.g.)). Thus B = |∇f |2 + R R − for t
T 2
because t
T 2
n n − 2(t + ) 2(T − t + )
⇒t T −t ⇒t + T −t + ⇒
1 t+
1 T −t+
1 2Bc c(c − 2n) + 2n T − t + (T − t + )2 n 2c c(c − 2n) 1 − + 2n 2(T − t + ) T − t + (T − t + )2 c(c − 3n) 1 = . 2n (T − t + )2
(2.24) and
(2.25)
S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
1015
Therefore
∂ + ∂t
− 2∇f ∇ q −
c T −t + c c 1 q+ + 2B q − − 2n T −t + T −t +
q−
c T −t +
c(c − 3n) . 2n(T − t + )2
(2.26)
Taking c = 3n, we have,
q−
∂ + ∂t
2n T −t +
− 2∇f ∇ q −
2n T −t +
2n −V q − T −t +
0 (2.27)
where V = V (x, t) is a bounded function defined by
V=
⎧ ⎪ ⎨ 0, ⎪ ⎩
1 2n (q 1 2n (q
if B 0, q + + +
2n T −t+ 2n T −t+
2n T −t+ 2n T −t+
+ B),
if B 0, q +
+ 2B),
if B < 0 at (x, t).
+ B 0 at (x, t), + B > 0 at (x, t),
(2.28)
Following the similar argument for the inequality (2.1), by the maximum principle again, we have, after letting → 0, 2f − |∇f |2 + R
3n τ
and
|∇u|2 2uτ 3n −R , − 2 u τ u
t T /2.
2
(2.29)
An immediate consequence of the above theorem is: Corollary 2.1 (Harnack inequality). Given a smooth Ricci flow on a closed manifold M, let u : M × [0, T ) → (0, ∞) be a positive C 2,1 solution to the conjugate heat equation. (a) Suppose the scalar curvature R 0 for t ∈ [0, T ). Then for any two points (x, t1 ), (y, t2 ) in M × (0, T ) such that t1 < t2 , it holds 1 n [4|γ (s)|2 + (τ1 − τ2 )2 R] ds τ1 . u(y, t2 ) u(x, t1 ) exp 0 τ2 2(τ1 − τ2 )
(2.30)
Here τi = T − ti , i = 1, 2, and γ (s) : [0, 1] → M is a smooth curve from x to y. (b) Without assuming the non-negativity of the scalar curvature R, for t2 > t1 T /2, it holds 1 3n/2 [4|γ (s)|2 + (τ1 − τ2 )2 R] ds τ1 . u(y, t2 ) u(x, t1 ) exp 0 τ2 2(τ1 − τ2 )
(2.31)
Here R = R(γ (s), T − τ ) with τ = τ2 + (1 − s)(τ1 − τ2 ) and |γ (s)|2 = gT −τ (γ (s), γ (s)).
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Proof. We will only prove (a) since the proof of (b) is similar. Denote τ (s) := τ2 + (1 − s)(τ1 − τ2 ), 0 τ2 < τ1 T , define (s) := ln u γ (s), T − τ (s) where (0) = ln u(x, t1 ), (1) = ln u(y, t2 ). By direct computation, ∇u ∂γ uτ (τ1 − τ2 ) ∂ (s) us = = − ∂s u u ∂s u √ 2γ (s) uτ ∇u = (τ1 − τ2 ) √ · − u 2u τ1 − τ2 2|γ (s)|2 |∇u|2 uτ + − (τ1 − τ2 ) u (τ1 − τ2 )2 2u2 2 2 τ1 − τ2 |∇u| 2uτ 2|γ (s)| + . − = (τ1 − τ2 ) 2 u u2
(2.32)
By our gradient estimate, if R 0, then |∇u|2 2uτ 2n R+ − 2 u τ u where τ = τ2 + (1 − s)(τ1 − τ2 ). Therefore ∂ (s) 2|γ (s)|2 τ1 − τ2 2n + R+ . ∂s (τ1 − τ2 ) 2 τ
(2.33)
Integrating with respect to s on [0, 1], we have (1) − (0)
2
1
1
0
0
|γ (s)|2 ds (τ1 − τ2 ) + (τ1 − τ2 ) 2
R ds
+ n ln
τ1 . τ2
(2.34)
Recall (0) = ln u(x, t1 ), (1) = ln u(y, t2 ). Then u(y, t2 ) ln u(x, t1 )
1 0
n [4|γ (s)|2 + (τ1 − τ2 )2 R] ds τ1 + ln . 2(τ1 − τ2 ) τ2
(2.35)
Therefore, given any two points (x, t1 ), (y, t2 ) in the space–time, we have 1 n [4|γ (s)|2 + (τ1 − τ2 )2 R] ds τ1 u(y, t2 ) u(x, t1 ) exp 0 . τ2 2(τ1 − τ2 )
2
(2.36)
We end the section by presenting an application of the above Harnack inequality on the uniqueness of solutions to the conjugate heat equation. We thank Professor L.F. Tam for very helpful explanations. In the interesting paper [6], Hamilton and Sesum proved the following results.
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Theorem. Suppose M is a compact Kähler manifold with positive first Chern class. Then there exists only one positive solution to the conjugate heat equation satisfying Perelman’s differential Harnack inequality if the metric evolves according to Kähler Ricci flow and has a type I singularity. Moreover, in case the dimension is 2, there is only on positive solution if only the Kähler Ricci flow has a type I singularity. Here we prove that in all dimensions the uniqueness of positive solutions holds if only the Kähler Ricci flow has a type I singularity, i.e., Theorem 2.2. Let M be a compact Kähler manifold with positive first Chern class. If g(t) is the unnormalized Kähler Ricci flow such that |Ric(g(t)| TC−t (T being the singular time) for a uniform constant C, which holds if the flow has type I singularity, then the conjugate heat equation has only one positive solution up to a constant. Proof. We will prove that all positive solutions to the conjugate heat equation satisfy an elliptic type Harnack inequality if |Ric(g(t)| TC−t , i.e., Let u be a positive solution to the conjugate heat equation. There is a uniform constant C0 such that max u C0
M×[0,T )
inf
M×[0,T )
(2.37)
u.
This inequality was proven in [6] under the additional assumption that the solution satisfies Perelman’s differential Harnack inequality. Once the Harnack inequality is established, the rest of the proof follows from the blow up method in [6]. In order to prove (2.37), we observe from Corollary 1 and the condition on the scalar curvature that 1 (τ1 − τ2 )
R γ (s), T − τ2 + (1 − s)(τ1 − τ2 ) ds
0
1 C(τ1 − τ2 ) 0
τ1 1 ds = C ln , τ2 + (1 − s)(τ1 − τ2 ) τ2
and u(y, t2 ) cu(x, t1 )e
2
1 0
|γ (s)|2 ds/(τ1 −τ2 )
T − t1 T − t2
b
for some b > 0. Note that Theorem 2.1, part (ii) and Corollary 2.1, part (b) is valid for all time here due to the lower bound on the scalar curvature coming out of type I singularity assumption. I.e. we can assume that R − 2(Tn−t) after suitable normalization such as |Ric| 1/(2(T − t)). The only modification in the proof is that (2.24) is valid for all time.
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As in [6], using Perelman’s estimate that diam(M, g(t)) 1
2 γ (s)
0
T − t1 ds g(T −τ ) T − t2
1
√ T − t, one has
2 γ (s)
g(t1 )
ds
0
where one also uses the normalized assumption that Ric(g(t)) −g(t)/(T − t). Using reparametrization, the above shows 1 0
2 2 T − t1 γ (s) ds T − t1 . g(T −τ ) T − t2
Therefore u(y, t2 ) cu(x, t1 )e
(T −t1 )2 2 )(τ1 −τ2 )
2 (T −t
T − t1 T − t2
b .
Fixing an arbitrary t1 , we pick t2 = (T + t1 )/2. Then, we conclude u(y, t2 ) Cu(x, t1 ).
(2.38)
Given t > 0, let U (t) = supx∈M u(x, t). Then by the assumption on the Ricci curvature, it holds U (t) −
C U (t). T −t
Hence U (t2 ) c1 U (t1 ). This and (2.38) show that U (t1 ) C inf u(x, t1 ). x∈M
Thus (2.37) is proven. The rest of the proof is identical to that of [6].
2
3. Localized version of the gradient estimate in Section 2 In this section we prove a localized version of the previous gradient estimate. Here we apply Li–Yau’s idea of using certain cut-off functions to the new equations derived in the last section. However the computation is more complicated for two reasons. One is that the metric is also evolving. The another is that the equations coming from the last section have a more complex structure.
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Theorem 3.1. Let M be a compact Riemannian manifold equipped with a family of Rieman∂g nian metrics evolving under Ricci flow, that is, ∂tij = −2Rij . Given x0 ∈ M and r0 > 0, let u be a smooth positive solution to the conjugate heat equation 2∗ u = −u − ut + Ru = 0 in the cube Qr0 ,T := {(x, t) | d(x, x0 , t) r0 , 0 < t T } and τ = T − t. Suppose Ric −K throughout Qr0 ,T for some positive constant K. Then for (x, t) in the half-cube Q r0 , T := {(x, t) | d(x, x0 , t)
r0 2,
0
T 2 },
2
2
we have |∇u|2 2uτ c c − R cK + + 2 − 2 u T u r0
(3.1)
where c > 0 is a constant depending only on the dimension n. Proof. As before let f be a function defined by u = q = 2f − |∇f |2 + R =
e−f n (4πτ ) 2
and
|∇u|2 2uτ − R. − u u2
From inequality (2.9) in the last section, we have q − qτ − 2∇f ∇q
2 1 q + |∇f |2 + R . 2n
(3.2)
For the fixed point x0 in M, let ϕ(x, t) be a smooth cut-off function (mollifier) with support in the cube Qr0 ,T := (x, t) x ∈ M, d(x, x0 , t) r0 , 0 < t T
(3.3)
possessing the following properties: (1) ϕ = ϕ(d(x, x0 , t), t) ≡ ψ(r(x, t))η(t), r(x, t) = d(x, x0 , t); ∂ψ ∂r 0, r T 0 (2) ϕ(x, t) ≡ 1 in Q r0 , T := {(x, t) | d(x, x0 , t) 2 , 0 < t 2 }; 2
(3) | ∂ψr ψa | ∂t η (4) | √ η|
∂η ∂t
0, τ = T − t;
2
c(n,a) ∂rr ψ c(n,a) r0 , | ψ a | r 2 , for some 0 c , for some c depending on n. T
c(n, a), 0 < a < 1;
Now we focus on the product (ϕq)(x, t). Since ϕ has support in Qr0 ,T , we can assume ϕq reaches its maximum at some point (y, s) ∈ Qr0 ,T . If q(y, s) = 2f (y, s)−|∇f (y, s)|2 +R(y, s) is negative, then the theorem is trivially true. Thus we can assume q(y, s) 0. By direct computation, (ϕq) − (ϕq)τ − 2∇f ∇(ϕq) − 2
∇ϕ ∇(ϕq) ϕ
= ϕ(q − qτ − 2∇f ∇q) + (ϕ)q − 2
|∇ϕ|2 q − qϕτ − 2q∇f ∇ϕ ϕ
2 ϕ |∇ϕ|2 q + |∇f |2 + R + (ϕ)q − 2 q − qϕτ − 2q∇f ∇ϕ. 2n ϕ
(3.4)
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At the point (y, s) where the maximum value for ϕq is attained, there hold: (1) (ϕq)(y, s) 0, (2) ∇(ϕq)(y, s) = 0, (3) (ϕq)τ (y, s) 0. The last inequality comes from the fact that (ϕq)|t=T = 0 since ϕ|t=T = 0, ϕq can only take its maximum for t ∈ [0, T ). We have also borrowed the idea of Calabi as used in [8] to circumvent the possibility that (y, s) is in the cut locus of g(s). Thus at the point (y, s), inequality (3.4) becomes 2 |∇ϕ|2 ϕ q + |∇f |2 + R (y, s) ϕτ q + 2q∇f ∇ϕ + 2 q − (ϕ)q 2n ϕ = (I ) + (II) + (III) + (IV).
(3.5)
We estimate each term on the right-hand side as follows: (I ) = ψτ η(τ )q + ψητ q = ψr rτ η(τ )q + ψητ q. From the lower bound assumption on the Ricci curvature Ric −K, we have (see [2], e.g.) ∂r ∂r = − −Kr. ∂τ ∂t
(3.6)
By construction of ψ , we have ψr 0. Therefore, ψr
∂r |ψr | c Kr|ψr | = Kr √ ψ Kr ψ cK ψ. ∂τ r0 ψ
(3.7)
This shows c√ |ητ | √ √ √ (I ) cK ψηq + ψητ q cK ϕq + ψ √ ηq cK ϕq + ψ ηq. η T Recall that ϕ = ψη, for a parameter to be chosen later, we have c√ c c √ (I ) cK ϕq + ϕq K 2 + 2 + 2ϕq 2 , T T |∇ϕ| √ ϕq (II) 2|∇f ||∇ϕ|q 2|∇f | √ ϕ
(3.8)
1 |∇ϕ|2 4|∇f |2 + ϕq 2 ϕ |∇ϕ|2 1 √ = 4 ϕ|∇f |2 3 + ϕq 2 ϕ2 ϕ|∇f |4 +
1 c + ϕq 2 , 3 r02
(3.9)
S. Kuang, Q.S. Zhang / Journal of Functional Analysis 255 (2008) 1008–1023
∇ϕ 2 √ |∇ϕ|2 √ 1 c (III) = 2 3/2 ϕq 2 3/4 ϕq ϕq 2 + , r02 ϕ ϕ ∂r ϕ √ (IV) = −(ϕ)q = − ∂rr ϕ + (n − 1) + ∂r ϕ∂r log g q r |∂r ϕ| ∂r ϕ |∂rr ϕ| √ q+ √ ϕq + (n − 1) K ϕq √ ϕ r ϕ K |∂r ϕ| 2 ∂r ϕ 1 ∂rr ϕ 2 2 q + + ϕq + (n − 1) + ϕq 2 √ √ ϕ r ϕ 2ϕq 2 +
1 r
2 r0 .
(3.10)
c cK ∂r ϕ q. + 2 + (n − 1) 4 r r0 r0
Notice that ϕ ≡ 1 in Q r0 , T . Thus ∂r ϕ = 0 for 0 r 2
1021
2
r0 2,
and we can just focus on r
r0 2,
Then (IV) can be estimated as follows: (IV) 2ϕq 2 +
c cK 2∂r ϕ + 2 + (n − 1) q r0 r04 r0
c cK 4(n − 1)2 ∂r ϕ 2 2 2ϕq + 4 + 2 + ϕq + √ϕ r0 r0 r02 2
3ϕq 2 +
c cK + 2. 4 r0 r0
Combining (I)–(IV), we have 2 ϕ c c cK c q + |∇f |2 + R (y, s) 7ϕq 2 + cK 2 + 2 + 2 + 2 + 4 + ϕ|∇f |4 2n T r0 r0 r0 c c 7ϕq 2 + cK 2 + 2 + 4 + ϕ|∇f |4 . (3.11) T r0 Notice we assumed q(y, s) 0, otherwise the theorem is trivially true, since q 0 is better than what we want to proof. 2 2 q + |∇f |2 + R (y, s) = q + |∇f |2 + R + − R − (y, s) 2 2 1 q + |∇f |2 + R + (y, s) − R − (y, s) 2 2 2 1 q + |∇f |2 (y, s) − R − (y, s) 2 2 1 q 2 + |∇f |4 (y, s) − sup R − 2 Qr ,T
0
1 2 q + |∇f |4 (y, s) − n2 K 2 . 2
(3.12)
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Here we have used the inequalities 2(a − b)2 a 2 − 2b2 , (a + b)2 a 2 + b2 for a, b 0 and the lower bound assumption for the Ricci curvature Ric −K ⇒ R −nK ⇒ R − nK since R = −R − if R < 0. Substituting into (3.11) and reorganizing, we have
1 1 c c 2 ϕ|∇f |4 + cK 2 + 2 + 4 . − 7 ϕq (y, s) − 4n 4n T r0
Take such that 7
1 4n .
(3.13)
Then the above inequality becomes ϕq 2 (y, s) cK 2 +
c c + . T 2 r04
(3.14)
By using inequality a12 + a22 + · · · + an2 (a1 + a2 + · · · + an )2 , c 2 c (ϕq)2 (y, s) ϕq 2 (y, s) cK + + 2 . T r0
(3.15)
If (x, t) ∈ Q r0 , T , then ϕ(x, t) ≡ 1. Thus for any (x, t) ∈ Q r0 , T , 2
2
2
2
q(x, t) = ϕ(x, t)q(x, t) max (ϕq)(x, t) max (ϕq)(x, t) = (ϕq)(y, s) Q r0 , T 2
cK +
2
Qr0 ,T
c c + 2. T r0
(3.16)
Therefore we just proved that in Q r0 , T , 2
2
q(x, t) cK +
c c + 2. T r0
(3.17)
If we bring back u, recall q = 2f − |∇f |2 + R, f = − ln u − n2 ln(4πτ ), then we have |∇u|2 2uτ c c − R cK + + 2 . − u T u2 r0
2
(3.18)
Acknowledgments We would like to thank Professor Bennet Chow and Professor Lei Ni for their support; also thank the referee whose suggestions improved this paper. References [1] Huai-Dong Cao, Xi-Ping Zhu, A complete proof of Poincare and geometrization conjectures—Application of the Hamilton–Perelman theory of the Ricci flow, Asian J. Math. 10 (2) (2006) 165–492. [2] Bennett Chow, Dan Knopf, The Ricci Flow: An Introduction, Math. Surveys Monogr., vol. 110, Amer. Math. Soc., Providence, RI, 2004.
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[3] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni, The Ricci Flow: Techniques and Applications, vols. I, II, Amer. Math. Soc., Providence, RI, 2007. [4] Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (4) (1975) 1061–1083. [5] C. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002) 425– 436. [6] Richard Hamilton, Natasa Sesum, Properties of the solutions of the conjugate heat equations, http://arXiv.org/ math.DG/0601415v1. [7] Bruce Kleiner, John Lott, Notes on Perelman’s papers, http://arXiv.org/math.DG/0605667v1, May 25, 2006. [8] Peter Li, S.T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153–201. [9] John W. Morgan, Gang Tian, Ricci flow and the Poincare conjecture, Clay Math. Monogr., vol. 3, Amer. Math. Soc., Providence, RI, 2007; available at: http://arXiv.org/math.DG/0607607v1, 25 July, 2006. [10] Lei Ni, A matrix Li–Yau–Hamilton inequality for Kaehler–Ricci flow, J. Differential Geom. 75 (2007) 303–358. [11] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/math.DG/ 0211159v1, 11 November, 2002.
Journal of Functional Analysis 255 (2008) 1024–1038 www.elsevier.com/locate/jfa
Differential Harnack estimates for backward heat equations with potentials under the Ricci flow Xiaodong Cao 1 Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA Received 25 February 2008; accepted 15 May 2008 Available online 20 June 2008 Communicated by L. Gross
Abstract In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman’s Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow. © 2008 Elsevier Inc. All rights reserved. Keywords: Differential Harnack estimate; Backward heat equation; Ricci flow
1. Introduction In [16], P. Li and S.-T. Yau proved a differential Harnack inequality by developing a grading estimate for positive solutions of the heat equation (with fixed metric). More precisely, they proved that, for any positive solution f of the heat equation ∂f = f ∂t on Riemannian manifolds with nonnegative Ricci curvature, then E-mail address: [email protected]. 1 Research partially supported by the Jeffrey Sean Lehman Fund from Cornell University.
0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.05.009
X. Cao / Journal of Functional Analysis 255 (2008) 1024–1038
1025
∂ n n ln f − |∇ ln f |2 + = ln f + 0. ∂t 2t 2t The idea was brought to study general geometric evolution equations by R. Hamilton. The differential Harnack estimate has since become an important technique in the studies of geometric evolution equations. For the Ricci flow, R. Hamilton [12] proved a Harnack estimate on Riemannian manifolds with weakly positive curvature operator. The Harnack quantities are also know as curvatures of a degenerate metric in space–time thanks to the work of B. Chow and S.-C. Chu [7]. In dimension two, R. Hamilton [11] proved a Harnack estimate for the scalar curvature when it is positive, the Harnack estimate in the general case was proved by B. Chow in [5]. B. Chow and R. Hamilton generalized their results for the heat equation and for the Ricci flow on surfaces in [8]. For other geometric flows, R. Hamilton proved a Harnack estimate for the mean curvature flow in [14]. B. Chow proved Harnack estimates for Gaussian curvature flow in [4] and for Yamabe flow in [6]. H.-D. Cao [1] proved a Harnack estimate and L. Ni [18] proved a matrix Harnack estimate (of the forward conjugate heat equation) for the Kähler–Ricci flow. For the heat equation, besides the classical result of P. Li and S.-T. Yau, R. Hamilton proved a matrix Harnack estimate for the heat equation in [13]. C. Guenther [10] studied the fundamental solution and Harnack inequality of time-dependent heat equation. In [2], H.-D. Cao and L. Ni proved a matrix Harnack estimate for the heat equation on Kähler manifolds. In an earlier paper [3], R. Hamilton and the author proved several Harnack estimates for positive solutions of the heat-type equation with potential when the metric is evolving by the Ricci flow (a more detailed discussion about the literature of Harnack estimates can also be found in that paper). In [19], G. Perelman proved a Harnack estimate for the fundamental solution of the conjugate heat equation under the Ricci flow. Namely, let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow on a closed manifold, f be the positive fundamental solution to the conjugate heat equation ∂ f = −f + Rf, ∂t τ = T − t, and u = −ln f − n2 ln(4πτ ). Then for t ∈ [0, T ), G. Perelman proved that 2u − |∇u|2 + R +
u n − 0 τ τ
(see [17] or [9, Chapter 16] for a detailed proof). In the present paper, we will first derive a general evolution equation for possible Harnack quantities, we will then prove Harnack estimates for all positive solutions of the backward heattype equation with potentials when the metric is evolving under the Ricci flow. Suppose (M, g(t)), t ∈ [0, T ], is a solution to the Ricci flow on a closed manifold. Let f be a positive solution of the backward heat equation with potential 2R, i.e., ∂gij = −2Rij , ∂t
(1.1)
∂f = −g(t) f + 2Rf. ∂t
(1.2)
Our first main theorem is the following.
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Theorem 1.1. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow on a closed manifold. Let f be a positive solution to the backward heat-type Eq. (1.2), u = −ln f , τ = T − t and H = 2u − |∇u|2 + 2R −
2n . τ
Then for all time t ∈ [0, T ), H 0. If we further assume that our solution to the Ricci flow is of type I, i.e., |Rm|
d0 T −t
for some constant d0 , here T is the blow-up time, then we shall prove the following theorem. Theorem 1.2. Let (M, g(t)), t ∈ [0, T ), be a type I solution to the Ricci flow on a closed manifold. Let f be a positive solution to the backward heat Eq. (1.2), u = −ln f , τ = T − t and n H = 2u − |∇u|2 + 2R − d , τ here d = d(d0 , n) 2 is some constant such that H (τ ) < 0 for small τ . Then for all time t ∈ [0, T ), H 0. We will consider the conjugate heat equation under the Ricci flow. In this case, we also assume that our initial metric g(0) has nonnegative scalar curvature, it is well known that this property is preserved by the Ricci flow. We shall prove Theorem 1.3. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow (1.1) on a closed manifold, and suppose that g(t) has nonnegative scalar curvature. Let f be a positive solution to the conjugate heat equation ∂f = −g(t) f + Rf, ∂t u = −ln f , τ = T − t and n H = 2u − |∇u|2 + R − 2 . τ Then for all time t ∈ [0, T ), H 0.
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Remark 1.1. S. Kuang and Q. Zhang [15] have also proved a similar estimate as in Theorem 1.3 (see Remark 3.1). Our proof follows from a direct calculation of more general evolution equation in Lemma 2.1. The rest of this paper is organized as follows. In Section 2, we will first derive a general evolution equation of Harnack quantity H, for backward heat-type equation with potentials, then we prove Theorems 1.1 and 1.2. We will also prove an integral version of the Harnack inequality (Theorem 2.3). In Section 3, we will prove Theorem 1.3, then we will derive a general evolution formula for a Harnack quantity similar to Perelman’s, as a consequence, we will prove Perelman’s Harnack estimate. In Section 4, we will define two entropy functionals and prove that they are monotone. In Section 5, we will prove a gradient estimate for the backward heat equation (without the potential term). This is also a consequence of the general evolution formula of our Harnack quantity. 2. General evolution equation and proof of Theorem 1.1 Let us first consider positive solutions of general evolution equations ∂ f = −f − cRf ∂t for all constant c which we will fix later. Let f = e−u , then ln f = −u. We have ∂ ∂ ln f = − u, ∂t ∂t and ∇ ln f = −∇u,
ln f = −u.
Hence u satisfies the following equation, ∂ u = −u + |∇u|2 + cR. ∂t
(2.1)
∂f = f + cRf, ∂τ
(2.2)
∂u = u − |∇u|2 − cR. ∂τ
(2.3)
Let τ = T − t, then we have
and u satisfies
We can now define a general Harnack quantity and derive its evolution equation.
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Lemma 2.1. Let (M, g(t)) be a solution to the Ricci flow, and u satisfies (2.3). Let H = αu − β|∇u|2 + aR + b
n u +d , τ τ
where α, β, a, b and d are constants that we will pick later. Then H satisfies the following evolution equation, 2 α ∂H λ 2α − 2β λ = H − 2∇H · ∇u − (2α − 2β)∇∇u + Rij − gij − H ∂τ 2α − 2β 2τ α τ − 2(α − 2β)Rij ui uj + 2(a + βc)∇R · ∇u + (2α − 2β)
nλ2 4τ 2
|∇u|2 α2 2α − 2β λβ − (αc + 2a)R + − 2a |Rc|2 + b− α τ 2α − 2β u n R 2α − 2β 2α − 2β 2α − 2β λ−1 b 2 + λ−1 d 2 + aλ − αλ − bc , + α α α τ τ τ where λ is also a constant that we will pick later. Proof. The proof follows from a direct computation. We first calculate the first two terms in H , ∂(u) = (u) − |∇u|2 − cR − 2Rij uij , ∂τ and ∂(|∇u|2 ) = 2∇u · ∇u − 2Rc(∇u, ∇u) − 2∇u · ∇ |∇u|2 − 2c∇u · ∇R ∂τ = |∇u|2 − 2|∇∇u|2 − 2∇u · ∇ |∇u|2 − 2c∇u · ∇R − 4Rij ui uj , here we used |∇u|2 = 2∇u · ∇u + 2|∇∇u|2 , and ∇u = ∇u + Rc(∇u, ·). Using the evolution equation of R, ∂R = −R − 2|Rc|2 , ∂τ and (2.3), we have
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∂ H = H − α |∇u|2 − αcR − 2αRij uij + 2β|∇∇u|2 + 2β∇u · ∇ |∇u|2 ∂τ + 2βc∇u · ∇R + 4βRij ui uj − 2aR − 2a|Rc|2 − b
cR u |∇u|2 n −b −b 2 −d 2 τ τ τ τ
= H − 2∇H · ∇u − 2(α − 2β)Rij ui uj − (2α − 2β)|∇∇u|2 + 2(a + βc)∇R · ∇u |∇u|2 u n cR − (αc + 2a)R − 2αRij uij − 2a|Rc|2 − b 2 − d 2 − b τ τ τ τ 2 α λ Rij − gij = H − 2∇H · ∇u − 2(α − 2β)Rij ui uj − (2α − 2β)∇∇u + 2α − 2β 2τ +b
αλ nλ2 λ R − (2α − 2β) u − 2 τ τ 4τ |∇u|2 α2 u n cR +b − (αc + 2a)R + − 2a |Rc|2 − b 2 − d 2 − b τ 2α − 2β τ τ τ 2 α λ Rij − gij = H − 2∇H · ∇u − 2(α − 2β)Rij ui uj − (2α − 2β)∇∇u + 2α − 2β 2τ + 2(a + βc)∇R · ∇u + (2α − 2β)
nλ2 2α − 2β λ H + 2(a + βc)∇R · ∇u + (2α − 2β) 2 α τ 4τ 2 |∇u| α2 2α − 2β λβ − (αc + 2a)R + − 2a |Rc|2 + b− α τ 2α − 2β 2α − 2β 2α − 2β 2α − 2β R u n + λ−1 b 2 + λ−1 d 2 + aλ − αλ − bc . α α α τ τ τ −
2 In the above lemma, let us take α = 2, β = 1, a = 2, c = −2, λ = 2, b = 0, d = −2, as a consequence of the above lemma, we have Corollary 2.2. Let (M, g(t)) be a solution to the Ricci flow, f be a positive solution of the following backward heat equation ∂ f = −f + 2Rf, ∂t let u = −ln f , τ = T − t and n H = 2u − |∇u|2 + 2R − 2 . τ Then we have ∂ 2 2 1 2 2 2 H = H − 2∇H · ∇u − H − |∇u| − 2|Rc| − 2∇i ∇j u + Rij − gij . ∂τ τ τ τ
(2.4)
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Now we can finish the proof of Theorems 1.1 and 1.2. Proof of Theorems 1.1 and 1.2. To prove Theorem 1.1, it is easy to see that for τ small enough then H (τ ) < 0. It follows from (2.4) and maximum principle that H 0 for all time τ . From the above proof, we can easily see that if the solution to the Ricci flow is of type I and d 2 is large enough, such that H (τ ) < 0 for τ small, then Theorem 1.2 is true for all time t
n 0 τ
along a space–time path and get a classical Harnack inequality. Theorem 2.3. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow on a closed manifold. Let f be a positive solution to the equation ∂ f = −f + 2Rf. ∂t Assume that (x1 , t1 ) and (x2 , t2 ), 0 t1 < t2 < T , are two points in M × [0, T ). Let t2 Γ = inf γ
1 2 |γ˙ | + R dt, 2
t1
where γ is any space–time path joining (x1 , t1 ) and (x2 , t2 ). Then we have T − t1 n f (x2 , t2 ) f (x1 , t1 ) expΓ . T − t2 Proof. Since H 0, τ = T − t and u = −ln f satisfies ∂u = u − |∇u|2 + 2R, ∂τ we have 2
∂u 2n + |∇u|2 − 2R − 0. ∂τ τ
If we pick a space–time path γ (x, t) joining (x1 , τ1 ) and (x2 , τ2 ) with τ1 > τ2 > 0. Along γ , we have
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du ∂u = + ∇u · γ˙ dτ ∂τ n 1 − |∇u|2 + R + + ∇u · γ˙ 2 τ n 1 |γ˙ |2 + 2R + . 2 τ Hence 1 u(x1 , τ1 ) − u(x2 , τ2 ) inf 2 γ
τ1 2 τ1 . |γ˙ | + 2R dτ + n ln τ2
τ2
Or we can write this as 1 u(x1 , t1 ) − u(x2 , t2 ) inf 2 γ
t2 t1
If we denote Γ = infγ
t2 t1
2 T − t1 . |γ˙ | + 2R dt + n ln T − t2
( 12 |γ˙ |2 + R) dt, then we have T − t1 n expΓ , f (x2 , t2 ) f (x1 , t1 ) T − t2
this finishes the proof.
2
3. On the conjugate heat equation In this section, we consider positive solutions of general evolution equations ∂ f = −f − cRf ∂t on [0, T ], for the special case c = −1, which is the case of conjugate heat equation. In Lemma 2.1, let us take α = 2, β = 1, a = 1, c = −1, λ = 2, b = 0, d = −2, and we arrive at Corollary 3.1. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow, f be a positive solution of the conjugate heat equation ∂ f = −f + Rf, ∂t let u = −ln f , τ = T − t and n H = 2u − |∇u|2 + R − 2 . τ
(3.1)
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Then we have 2 ∂ 1 2 2 R H = H − 2∇H · ∇u − 2uij + Rij − gij − H − |∇u|2 − 2 . ∂τ τ τ τ τ
(3.2)
Now we can finish the proof of Theorem 1.3. Proof of Theorem 1.3. It is easy to see that for τ small enough then H (τ ) < 0. It follows from (3.2) and maximum principle that H 0 for all time τ .
2
As in Section 2, we can see that if the solution to the Ricci flow is of type I, i.e., |Rm|
d0 , T −t
here T is the blow-up time, then the Harnack estimate is true for all time t < T . Theorem 3.2. Let (M, g(t)), t ∈ [0, T ), be a type I solution to the Ricci flow on a closed manifold with nonnegative scalar curvature. Let f be a positive solution to the conjugate heat Eq. (3.1), u = −ln f , τ = T − t and n H = 2u − |∇u|2 + R − d , τ here d = d(d0 , n) 2 is some constant such that H (τ ) < 0 for small τ . Then for all time t ∈ [0, T ), H 0. We can also derive a classical Harnack inequality by integrating along a space–time path. Theorem 3.3. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow on a closed manifold with nonnegative scalar curvature. Let f be a positive solution to the conjugate heat equation ∂ f = −f + Rf. ∂t Assume that (x1 , t1 ) and (x2 , t2 ), 0 t1 < t2 < T , are two points in M × [0, T ). Let t2 Γ = inf γ
2 |γ˙ | + R dt,
t1
where γ is any space–time path joining (x1 , t1 ) and (x2 , t2 ). Then we have T − t1 n expΓ /2 . f (x2 , t2 ) f (x1 , t1 ) T − t2
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In the rest of this section, we will derive Perelman’s Harnack inequality for the conjugate heat Eq. (3.1). First let f = (4πτ )−n/2 e−v , then ln f = − n2 ln(4πτ ) − v. We have ∂ n ∂ ln f = − v + , ∂t ∂t 2τ and ∇ ln f = −∇v,
ln f = −v.
Hence v satisfies the following equation, ∂ n v = −v + |∇v|2 + cR + . ∂t 2τ
(3.3)
Or if we write the equation in backward time τ , we have ∂f = f + cRf, ∂τ
(3.4)
n ∂v = v − |∇v|2 − cR − . ∂τ 2τ
(3.5)
and v satisfies
We can now define a general Harnack quantity and derive its evolution equation. Lemma 3.4. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow, and v satisfies (3.5). Let P = αv − β|∇v|2 + aR + b
n v +d , τ τ
where α, β, a, b and d are constants that we will pick later. Then P satisfies the following evolution equation, 2 α ∂P λ 2α − 2β λ = P − 2∇P · ∇v − (2α − 2β)∇∇v + Rij − gij − P ∂τ 2α − 2β 2τ α τ − 2(α − 2β)Rij vi vj + 2(a + βc)∇R · ∇v + (2α − 2β)
nλ2 4τ 2
|∇v|2 α2 2α − 2β λβ − (αc + 2a)R + − 2a |Rc|2 + b− α τ 2α − 2β v n 2α − 2β 2α − 2β bn λ−1 b 2 + λ−1 d 2 − 2 + α α τ τ 2τ R 2α − 2β aλ − αλ − bc , + α τ
here λ is also a constant that we will pick later.
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Proof. The proof again follows from the same direct computation as in the proof of Lemma 2.1. bn 2 Notice that the only extra term − 2τ 2 comes from the evolution of v. In the above lemma, let us take α = 2, β = 1, a = 1, c = −1, λ = 1, b = 1, d = −1, as a consequence of Lemma 3.4, we have Perelman’s Harnack inequality. Theorem 3.5 (Perelman). Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow, f be the positive fundamental solution of the conjugate heat equation ∂ f = −f + Rf, ∂t let v = −ln f − n2 ln(4πτ ), τ = T − t and P = 2v − |∇v|2 + R +
v n − . τ τ
Then we have 2 ∂ 1 1 P = P − 2∇P · ∇v − 2vij + Rij − gij − P . ∂τ 2τ τ
(3.6)
Moreover, P 0 on [0, T ). In the same spirit of searching Perelman’s Harnack inequality, we shall also take α = 2, β = 1, a = 1, c = −1, λ = 2, b = 0, d = −2 in the above Lemma 3.4. We have a Harnack inequality for all positive solutions of the conjugate heat Eq. (3.1). Theorem 3.6. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow, suppose that g(t) has nonnegative scalar curvature. Let f be a positive solution of the conjugate heat equation ∂ f = −f + Rf, ∂t let v = −ln f − n2 ln(4πτ ), τ = T − t and P = 2v − |∇v|2 + R −
2n . τ
Then we have ∂ |∇v|2 R 1 2 2 P = P − 2∇P · ∇v − 2vij + Rij − gij − P − 2 −2 . ∂τ τ τ τ τ Moreover, for all time t ∈ [0, T ), P 0.
(3.7)
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Proof. It is easy to see that for τ small enough then P (τ ) < 0. It follows from (3.7) and maximum principle that P 0 for all time τ , hence for all t.
2
Remark 3.1. Theorem 3.6 can be deduced from Theorem 1.3 directly since v = u − n2 ln(4πτ ) and b = 0. But as we can see in the direct calculation, it will also lead to Perelman’s Harnack inequality (by choosing different coefficients). S. Kuang and Q. Zhang proved this estimate in [15]. Remark 3.2. We can prove similar estimate for P if we have a type I solution to the Ricci flow. We can also prove a classical Harnack inequality by integrating along a space–time path. 4. Entropy formulas and monotonicities In this section, we will define two entropies which are similar to Perelman’s entropy functionals as in [19], and we will show that both of them are monotone under the Ricci flow. Let (M, g(t)) be a solution to the Ricci flow on a close manifold, we shall also assume that g(t) has nonnegative scalar curvature, we first prove Theorem 4.1. Assume that (M, g(t)), t ∈ [0, T ], is a solution to the Ricci flow on a Riemannian manifold with nonnegative scalar curvature. Let f be a positive solution of ∂ f = −f + 2Rf, ∂t u = −ln f and τ = T − t. Define H = 2u − |∇u|2 + 2R − 2
n τ
and F=
τ 2 H e−u dμ,
M
then ∀t ∈ [0, T ), we have F 0 and d F 0. dt Proof. The fact that F 0 follows directly from H 0. We calculate its time derivative, using (2.4) in Lemma 2.1 and ∂t∂ dμ = −R dμ, we have
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dF d F =− =− dt dτ
M
=−
2τ H e−u + τ 2 e−u
∂ ∂ H + τ 2 H e−u + Rτ 2 H e−u dμ ∂t ∂τ
1 2 τ 2 H e−u − 2τ 2 e−u uij + Rij − gij − 2τ e−u |∇u|2 t
M 2 −u
−τ e
2 R + 2|Rc| dμ 0.
2
We now define an entropy associate with the conjugate heat equation. Theorem 4.2. Assume that (M, g(t)), t ∈ [0, T ], is a solution to the Ricci flow on a closed Riemannian manifold with nonnegative scalar curvature. Let f be a positive solution of ∂ f = −f + Rf, ∂t u = −ln f and τ = T − t. Let H = 2u − |∇u|2 + R − 2
n τ
and W=
τ 2 H e−u dμ,
M
then ∀t ∈ [0, T ), we have W 0 and d W 0. dt Proof. The fact that W 0 follows directly from H 0. To calculate its time derivative, using (3.2) and ∂t∂ dμ = −R dμ, we have dW d W =− =− dt dτ
∂ −u −u 2 −u ∂ 2 2 −u 2τ H e + τ e dμ H + τ H e + Rτ H e ∂τ ∂τ M
=−
1 2 τ 2 H e−u − 2τ 2 e−u uij + Rij − gij − 2τ e−u |∇u|2 t
M
− 2τ Re
−u
dμ 0.
2
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Remark 4.1. As noted in [19], if we consider the system ⎧ ∂ ⎪ ⎨ gij = −2(Rij + ∇i ∇j u), ∂t ⎪ ⎩ ∂ u = −u − R, ∂t then the measure dm = e−u dμ is fixed. This system differs from the original system of the conjugate heat equation under the Ricci flow ⎧ ∂ ⎪ ⎨ gij = −2Rij , ∂t ⎪ ⎩ ∂ u = −u + |∇u|2 − R ∂t by a diffeomorphism. 5. Gradient estimate for the backward heat equation In this section, we consider a gradient estimate for positive solutions f to the backward heat equation ∂ f = −f. ∂t
(5.1)
Since the equation is linear, without loss of generality, we may assume that 0 < f < 1. Let f = e−u , then u satisfies ∂ u = −u + |∇u|2 , ∂t
(5.2)
and u > 0. In the proof of Lemma 2.1, let take α = 0, β = −1, a = c = 0, b = −1 and d = 0, then u H = |∇u|2 − , τ and we have 1 ∂ H = H − 2∇H · ∇u − 4Rc(∇u, ∇u) − H − 2|∇∇u|2 . ∂τ τ
(5.3)
This leads to the following theorem. Theorem 5.1. Let (M, g(t)), t ∈ [0, T ], be a solution to the Ricci flow on a closed manifold with nonnegative curvature operator. Let f (< 1) be a positive solution to the backward heat Eq. (5.1), u = −ln f , τ = T − t and u H = |∇u|2 − . τ
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Then for all time t ∈ [0, T ), H 0, i.e., |∇f |2
f 2 ln(1/f ) . T −t
Proof. Notice that as τ small enough, H < 0, now the proof follows from (5.3) and the maximum principle. 2 Acknowledgments The author would like to thank Laurent Saloff-Coste for general discussion on Harnack inequalities. He would also like to thank Leonard Gross and Richard Hamilton for helpful suggestions. References [1] Huai-Dong Cao, On Harnack’s inequalities for the Kähler–Ricci flow, Invent. Math. 109 (2) (1992) 247–263. [2] Huai-Dong Cao, Lei Ni, Matrix Li–Yau–Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann. 331 (4) (2005) 795–807. [3] Xiaodong Cao, Richard Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, preprint, 2007. [4] Bennett Chow, On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (4) (1991) 469–483. [5] Bennett Chow, The Ricci flow on the 2-sphere, J. Differential Geom. 33 (2) (1991) 325–334. [6] Bennett Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (8) (1992) 1003–1014. [7] Bennett Chow, Sun-Chin Chu, A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Lett. 2 (6) (1995) 701–718. [8] Bennett Chow, Richard S. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math. 129 (2) (1997) 213–238. [9] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, Lei Ni, The Ricci Flow: Techniques and Applications. Part II, Math. Surveys Monogr., vol. 144, Amer. Math. Soc., Providence, RI, 2007; Analytic aspects. [10] Christine M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (3) (2002) 425–436. [11] Richard S. Hamilton, The Ricci flow on surfaces, in: Mathematics and General Relativity, Santa Cruz, CA, 1986, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. [12] Richard S. Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1) (1993) 225–243. [13] Richard S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1) (1993) 113–126. [14] Richard S. Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1) (1995) 215–226. [15] Shilong Kuang, Qi Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, preprint, 2007. [16] Peter Li, Shing Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (3–4) (1986) 153– 201. [17] Lei Ni, A note on Perelman’s LYH-type inequality, Comm. Anal. Geom. 14 (5) (2006) 883–905. [18] Lei Ni, A matrix Li–Yau–Hamilton estimate for Kähler–Ricci flow, J. Differential Geom. 75 (2) (2007) 303–358. [19] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159, 2002.
