Journal of Functional Analysis 259 (2010) 2193–2214 www.elsevier.com/locate/jfa
Topological centers of module actions induced by unitary representations Pak-Keung Chan Department of Mathematics, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada Received 29 January 2010; accepted 17 June 2010 Available online 13 July 2010 Communicated by N. Kalton
Abstract In this paper, we study the topological centers of bilinear maps induced by unitary representations, giving a characterization when the center is minimal and also conditions which guarantee that the center is maximal. Various examples whose topological centers are maximal, minimal or neither will be given. We will also investigate the topological centers of sub-representations, direct sums and tensor products. © 2010 Elsevier Inc. All rights reserved. Keywords: Arens product; Bounded bilinear map; Banach G-module action; Module action; Unitary representation; Topological center
1. Introduction In 1951, Arens initiated the study of extension of bilinear maps on normed space and introduced the concept of regularity of bilinear maps (see [1] and [2]). The study of Arens regularity of bilinear maps and the topological center problem has attracted some attention. In [16], Ulger showed that the Arens regularity of a bounded bilinear map can be characterized by its weakly compactness or its reflexiveness and simplified proofs of some old results. For more recent results, the reader is referred to [6] and [14]. On the other hand, special attention has been focused on the bilinear maps arisen from Banach algebras. See [13] and [7]. Our purpose in this paper is to study a bounded bilinear map induced by a unitary representation π of a locally compact group G and the topological center problem related to it. This E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.006
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paper is organized as follows. In Section 2, we introduce some notations in abstract harmonic analysis, defining the bounded bilinear map induced by a unitary representation π , giving some preliminary results. In Section 3, we study cases under which the topological center is maximal, minimal or neither. A characterization of the maximality of the topological center will be demonstrated and various examples will be given. In Section 4, we investigate the topological centers of bilinear maps induced by direct sum, tensor product, sub-representation of given representations. 2. Notations and some preliminaries Unless otherwise specified, G always denotes a locally compact group equipped with a fixed left Haar measure dx. Let x ∈ G and f be a complex-valued function defined on G. We define the left translation of f by x as lx f (y) = (x f )(y) = f (xy) (y ∈ G). A function f : G → C is left uniformly continuous if f is bounded, continuous and the map G → CB(G), x → lx f is continuous with respect to the · ∞ -norm topology. We denote the space of all left uniformly continuous functions by LUC(G) and remark that LUC(G) is a two-sided translation invariant, unital C∗ -subalgebra of CB(G), the C∗ -algebra of all bounded, complex-valued continuous functions defined on G. As well known, the dual space of LUC(G), denoted by LUC(G)∗ , can be made into a Banach algebra as follows. For m, n ∈ LUC(G)∗ , f ∈ LUC(G), x ∈ G, we define ml f : G → C by ml f (x) = m, lx f . It is easy to check that ml f ∈ LUC(G). Define mn ∈ LUC(G)∗ by mn, f = m, nl f , then LUC(G)∗ becomes a Banach algebra. The reader is referred to Lau [11] for more details. Let M(G) be the Banach algebra of all the complex Radon measures on G. Lau showed that M(G) can be embedded into LUC(G)∗ as a closed subalgebra (see [11]). At later time, Ghahramani, Lau and Losert improved that result and proved the following lemma in [8]. Lemma 2.1. The map θ : M(G) → LUC(G)∗ defined by θ (μ), f = f (x) dμ(x), (μ ∈ M(G), f ∈ LUC(G)) is an isometric algebra homomorphism. Moreover, we have: (a) LUC(G)∗ = M(G) ⊕1 C0 (G)⊥ , and (b) C0 (G)⊥ is a closed two-sided ideal of LUC(G)∗ . Let X be a Banach space and let G be a locally compact group. We say that X is a Banach G-module if G acts on X as bounded invertible operators with norm less than or equal to one such that the action is continuous with respect to the norm topology. More precisely, it means that there exists a map X × G → X with the following properties: • For each ξ ∈ X, x, y ∈ G, we have ξ · e = ξ and (ξ · x) · y = ξ · (xy). • For each x ∈ G, the map ξ → ξ · x is a bounded, invertible linear operator on X with norm less than or equal to one. • For each ξ ∈ X, the map x → ξ · x is continuous with respect to the norm topology. We refer the reader [12] for details. Let π : G → B(H) be a unitary representation of a locally compact group G. Bekka and Xu defined a unital C∗ -subalgebra of B(H) and a bilinear map as follows. See [3] and [17]. Let x ∈ G and T ∈ B(H). Define a map B(H) × G → B(H) by T · x = π(x −1 )T π(x), then B(H) becomes a G-module. Define UCB(π) = {T ∈ B(H) | the map G → B(H), x → T · x
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is continuous in norm topology}, then UCB(π) is a unital C∗ -subalgebra of B(H). When the G-module action is restricted on UCB(π), UCB(π) becomes a Banach G-module. Lemma 2.2. The mapping UCB(π)∗ × UCB(π) → LUC(G), (M, T ) → MT , defined by MT (x) = M, T · x is bilinear and MT ∞ MT . Proof. Let x ∈ G, then |MT (x)| = |M, T · x | MT · x MT , hence MT is a bounded function on G. Let (xα ) be a net in G such that xα → x ∈ G, then |MT (xα )−MT (x)| MT · xα − T · x → 0. Therefore, MT is a continuous function. Let xα , x, y ∈ G such that xα → x, then x (MT )(y) − x (MT )(y) = MT (xα y) − MT (xy) α = M(T · xα )(y) − M(T · x)(y) = δy M, T · xα − T · x δy MT · xα − T · x MT · xα − T · x → 0 uniformly about y, where δy M ∈ UCB(π)∗ is defined by δy M, T = M, T · y . Therefore MT ∈ LUC(G). The checking of the bilinearity of the mapping (M, T ) → MT is left to the reader. 2 Next, we define a map LUC(G)∗ × UCB(π)∗ → UCB(π)∗ by (m, M) → mM, where mM, T = m, MT , T ∈ UCB(π). It is routine to check that the map is a bounded bilinear map with mM mM. Proposition 2.3. With the mapping (m, M) → mM defined above, UCB(π)∗ becomes a left Banach LUC(G)∗ -module with mM mM and δe M = M. Proof. Let T ∈ UCB(π), x, y ∈ G, then x (MT )(y) = (MT )(xy) = M, T
· xy
= M, (T · x) · y = M(T · x)(y). Therefore x (MT ) = M(T · x). Next, nl (MT )(x) = n, x (MT ) = n, M(T · x) = nM, T · x = (nM)(T )(x) and hence nl (MT ) = (nM)T . Therefore
(mn)M, T = mn, MT = m, nl (MT ) = m, (nM)T = m(nM), T ,
so (mn)M = m(nM). Also
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δe M, T = δe , MT = (MT )(e) = M, T · e = M, T , hence δe M = M.
2
Remark 2.4. The above Banach module structure of UCB(π)∗ is suggested by Mr. M.Y.-H. Cheng. Remark 2.5. The bilinear map (m, M) → mM constructed in above coincides with Arens’ construction when G is discrete. We recall Arens’ construction. Let X, Y , Z be Banach spaces and let θ : X × Y → Z be a bounded bilinear map. Define θ ∗ : Z ∗ × X → Y ∗ , the adjoint of θ , by θ ∗ (z , x), y = z , θ (x, y) (x ∈ X, y ∈ Y, z ∈ Z ∗ ). The above process can be repeated and we define θ ∗∗ = (θ ∗ )∗ : Y ∗∗ × Z ∗ → X ∗ and θ ∗∗∗ = (θ ∗∗ )∗ : X ∗∗ × Y ∗∗ → Z ∗∗ . If G is a discrete group, both UCB(π) and LUC(G) have preduals. Namely, UCB(π) = B(H) = L1 (H)∗ , where L1 (H), equipped with the trace-class norm, is the Banach space of all trace-class operators on the Hilbert space H. (see [15, Chapter II, Section 1]) and LUC(G) = l∞ (G) = l1 (G)∗ . Define a G-module action on the space L1 (H) by G × L1 (H) → L1 (H), (x, L) → x · L = π(x)Lπ(x −1 ). By integration, we obtain a Banach l1 (G)-module L1 (H), namely l1 (G) × L1 (H) → L1 (H), f (x)x · L. (f, L) → f · L = f (x)x · L dx = x∈G
Define a bounded bilinear map θ : l1 (G) × L1 (H) → L1 (H) by m(f, L) = f · L. Let L ∈ L1 (H), T ∈ B(H), M ∈ B(H)∗ , x ∈ G, then ∗ θ (T , δx ), L = T , θ (δx , L) = T , x · L = tr T π(x)Lπ x −1 = tr π x −1 T π(x)L = tr (T · x)L = T · x, L . Therefore θ ∗ (T , δx ) = T ·x and θ ∗∗ (M, T ), δx = M, θ ∗ (T , δx ) = M, T ·x = MT (x), hence θ ∗∗ (M, T ) = MT . Finally, θ ∗∗∗ (m, M), T = m, θ ∗∗ (M, T ) = m, MT = mM, T . Therefore θ ∗∗∗ (m, M) = mM. 3. Topological centers of the left LUC(G)∗ -module action It is obvious that for each fixed M ∈ UCB(π)∗ , the map LUC(G)∗ → UCB(G)∗ , m → mM is weak∗ –weak∗ continuous. However, it is false that for each m ∈ LUC(G)∗ , the map UCB(π)∗ → UCB(π)∗ , M → mM is weak∗ –weak∗ continuous. A natural question arises: For what m is the mapping M → mM weak∗ –weak∗ continuous? Therefore it makes sense to define
Z(π) = m ∈ LUC(G)∗ the map UCB(π)∗ → UCB(π)∗ , M → mM is weak∗ –weak∗ continuous , the topological center of the module action induced by π . Note that Z(π) contains M(G) as we shall prove in Lemma 3.2. Before proving this result, we would first state a proposition which characterizes Z(π).
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Proposition 3.1. Let π : G → B(H) be a unitary representation of a locally compact group G and let UCB(π) and Z(π) be defined as above. For each m ∈ LUC(G)∗ , the following are equivalent: (1) For each T ∈ UCB(π), the map T m : UCB(π)∗ → C defined by M, T m = mM, T (M ∈ UCB(π)∗ ) lies in UCB(π). (2) m ∈ Z(π). (3) The map UCB(π)∗ → UCB(π)∗ , M → mM is weak∗ –weak∗ continuous on all bounded parts of UCB(π)∗ . Proof. The equivalence of (1) and (2) are clear and the implication of (2) ⇒ (3) is trivial. Now we prove that (3) ⇒ (1). Suppose that the mapping M → mM is weak∗ –weak∗ continuous on all bounded parts of UCB(π)∗ . Let T ∈ UCB(π) be fixed, then the linear functional T m ∈ UCB(π)∗∗ is σ (UCB(π)∗ , UCB(π)) continuous on any bounded part of UCB(π)∗ . By [5, V.5.6], T m is a σ (UCB(π)∗ , UCB(π)) continuous linear functional on UCB(π)∗ and hence T m ∈ UCB(π) by [4, p. 125, Theorem 1.3]. 2 Lemma 3.2. Let π : G → B(H) be a unitary representation of a locally compact group G. Then M(G) ⊆ Z(π), where M(G) is regarded as a subspace of LUC(G)∗ as in Lemma 2.1. Proof. Define a map UCB(π) × LUC(G)∗ → UCB(π)∗∗ by (T , m) → T m, where M, T m = mM, T , M ∈ UCB(π)∗ . The map is clearly bilinear and T m T m. Let m ∈ M(G), T ∈ UCB(π) and > 0. First, we assume that the support of m, denoted by K = supp(m) is compact. Choose a finite partition {Ei | i = 1, 2, . . . , n} of K consisting of Borel sets Ei such that T · x − T · y < /|m|(K) whenever x, y ∈ Ei . This is possible by the uniform continuity of the map x → T · x on the compact set K. For each i, fix xi ∈ Ei . Let M ∈ UCB(π)∗ , then
n n m(Ei )T · xi = m, MT − m(Ei )M, T · xi M, T m − i=1
i=1
n m(Ei )M, T · xi = MT (x) dm(x) − i=1
K
n = M, T · x − T · xi dm(x) i=1 E
n i=1
i
M
|m|(Ei ) = M. |m|(K)
Therefore T m − ni=1 m(Ei )T · xi . Since T m can be approximated by a sequence of elements in UCB(π) with respect to the norm topology, T m ∈ UCB(π). If the support of the measure m is not compact, we can choose a sequence of measures (μn )n with compact supports such that m − μn → 0. Then T m − T μn T m − μn → 0, so T m ∈ UCB(π). By Proposition 3.1, m ∈ Z(π). 2
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Proposition 3.3. Let π : G → B(H) be a unitary representation of a locally compact group G and let UCB(π) and Z(π) be defined as above. Then the topological center Z(π) is a Banach subalgebra of LUC(G)∗ containing M(G). Proof. By Proposition 3.1 (1) and the previous lemma, Z(π) is a subalgebra of LUC(G)∗ containing M(G). Let (mk ) be a sequence in Z(π) such that mk → m ∈ LUC(G)∗ with respect to the norm topology. Let (Mα ) be a bounded net in UCB(π)∗ such that Mα → M ∈ UCB(π)∗ with respect to the weak∗ -topology. Let T ∈ UCB(π) and let > 0. Choose K > 0 such that Mα K for all α. Fix k such that mk − m < /(K(T + 1)), then mMα , T − mM, T mk (Mα − M), T + (m − mk )Mα , T + (mk − m)M, T mk (Mα − M), T + 2. Consequently lim supα |mMα , T − mM, T | 2 and hence mMα , T → mM, T , i.e. m ∈ Z(π) by Proposition 3.1. Therefore Z(π) is closed. 2 4. Minimality of the topological center In this section, we state a theorem which characterizes the minimality of the topological center (i.e. Z(π) = M(G)) in terms of a factorization property. It follows immediately that the topological center of the module action induced by the left regular representation of any locally compact group is always minimal. Lastly, we give an example that the topological center of a countable direct sum of finite dimensional representations is minimal. Before stating the main theorem, we need few lemmas. The first lemma may be well known, and it is similar to [10, Theorem 15.9]. However, we include a proof for completeness. Lemma 4.1. Let G be a locally compact group and let K1 , K2 be two disjoint compact subsets of G, then there exists a compact, symmetric neighborhood U of e such that K1 U and K2 U are disjoint. Proof. First we claim that for any y ∈ K2 , there exist open neighborhoods Uy , Vy of e such that K1 Uy ∩ yVy = ∅. Let y ∈ K2 be given. For each x ∈ K1 , there exist open neighborhoods Sx and Tx of e such that xSx Sx ∩ yTx = ∅. Note that {xSx | x ∈ K1 } coversK1 , so we may select a finite subcover {xi Sxi | i = 1, 2, . . . , n}. Define Uy = ni=1 Sxi , Vy = ni=1 Txi , then Uy and Vy are open neighborhoods of e. We show that K1 Uy ∩ yVy = ∅. Let x ∈ K1 , then x ∈ xi Sxi for some i. Consequently xUy ∩ yVy ⊆ xi Sxi Sxi ∩ yTxi = ∅ and hence K1 Uy ∩ yVy = ∅. By the above claim, for each y ∈ K2 , we may choose open neighborhoods Uy and Vy for e such that K1 Uy ∩ yVy Vy = ∅. Note that {yVy | y ∈ K2 } covers K2 , so we may select am finite U , V = subcover {yi Vyi | i = 1, 2, . . . , m}. Define open neighborhoods U = m y i i=1 i=1 Vyi of e. Let y ∈ K2 , then y ∈ yi Vyi for some i. Therefore K1 U ∩ yV ⊆ K1 Uyi ∩ yi Vyi Vyi = ∅ and hence K1 U ∩ K2 V = ∅. We finish the proof by choosing a compact, symmetric neighborhood of e contained in U ∩ V . 2 Lemma 4.2. Let G be a locally compact, non-compact group and let K1 and K2 be two disjoint compact subsets of G. Then there exists a compact, symmetric neighborhood U of e and a sequence (xn ) in G such that:
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(1) K1 U xi ∩ K1 U xj = ∅ whenever i = j , (2) K2 U xi ∩ K2 U xj = ∅ whenever i = j , (3) K1 U xi ∩ K2 U xj = ∅ for any i, j . Moreover the set {xn | n ∈ N} is closed but not compact. Proof. Without loss of generality, we assume that K1 and K2 are non-empty. By Lemma 4.1, we choose a compact, symmetric neighborhood U of e such that K1 U ∩ K2 U = ∅ and construct a sequence (xn ) inductively. Let x1 = e. Suppose that x1 , x2 , . . . , xn have been chosen such that: K1 U xi ∩ K1 U xj = ∅ whenever i = j , and K2 U xi ∩ K2 U xj = ∅ whenever i = j , and K1 U xi ∩ K2 U xj = ∅ for any i, j ∈ {1, 2, . . . n}. We assert that there exists y ∈ G such that: K1 U xi ∩ K1 Uy = ∅ for i = 1, 2, . . . , n, and K2 U xi ∩ K2 Uy = ∅ for i = 1, 2, . . . , n, and K1 U xi ∩ K2 Uy = ∅ for i = 1, 2, . . . , n, and K1 Uy ∩ K2 U xi = ∅ for i = 1, 2, . . . , n. Suppose the contrary that the assertion is false, then for any y ∈ G, we have y∈
n −1 −1 U K1 K1 U xi ∪ U −1 K2−1 K2 U xi ∪ U −1 K2−1 K1 U xi ∪ U −1 K1−1 K2 U xi i=1
and hence G=
n −1 −1 U K1 K1 U xi ∪ U −1 K2−1 K2 U xi ∪ U −1 K2−1 K1 U xi ∪ U −1 K1−1 K2 U xi , i=1
which is a contradiction since the set on the right is compact. Choose xn+1 = y, where y ∈ G is any element which satisfies the above condition. By induction, we obtain a sequence (xn ) in G. Clearly (1) and (2) are satisfied by our construction. For (3), if i = j , K1 U xi ∩ K2 U xi = (K1 U ∩ K2 U )xi = ∅. If i = j , K1 U xi ∩ K2 U xj = ∅ by our construction. We show that {xn | n ∈ N} is a closed, non-compact subset of G. Let (xα )α be a net in {xn | n ∈ N} which converges to x ∈ G. We assert that there exists α0 such that xα = xα0 whenever α α0 . In that case, xα → xα0 ∈ {xn | n ∈ N}. Suppose the contrary. Choose an open neighborhood V of e such that V V −1 ⊆ U . Choose α0 such that xα ∈ V x whenever α α0 . By assumption, there exist ∈ V xx −1 V −1 ⊆ U α1 , α2 α0 such that xα1 = xα2 . Note that xα1 ∈ V x and xα2 ∈ V x, so xα1 xα−1 2 and hence xα1 ∈ U xα2 , which is a contradiction. Therefore {xn | n ∈ N} is closed. Suppose the contrary that {xn | n ∈ N} is compact. For the net (xn )n , it has a subnet (xnα )α which converges to a point, say xk in {xn | n ∈ N}. Choose α0 such that xnα ∈ U xk whenever α α0 . Choose α1 such that nα k + 1 whenever α α1 . Choose α2 such that α2 α1 and α2 α0 , then xnα2 ∈ U xk . Clearly xnα2 ∈ U xnα2 . However nα2 k + 1 implies that xnα2 = xk , contradicting to U xk ∩ U xnα2 = ∅. 2
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Lemma 4.3. Let G be a locally compact, non-compact group and let K1 , K2 be two disjoint, compact subsets of G. Let U , (xn ) be the compact, symmetric neighborhood of e and the sequence in G respectively as in the previous lemma. Then there exists ∞ f ∈ LUC(G)\C0 (G), 0 f 1, ∞ K x and f vanishes outside such that f = 1 on n=1 1 n n=1 K1 U xn . In particular, f = 0 on ∞ K U x . n n=1 2 Proof. By Urysohn lemma, we choose g ∈ C0 (G), 0 g 1,such that g = 1 on K1 and −1 g vanishes outside K1 U . Define f : G → [0, +∞] by f (x) = ∞ n=1 g(xxn ). Note that for −1 each x ∈ G, there exists at most one n such that g(xxn ) = 0. For, if m = n but g(xxn−1 ) = 0 −1 ) = 0, then xx −1 ∈ K U and xx −1 ∈ K U , hence x ∈ K U x ∩ K U x which and g(xxm 1 1 1 n 1 m n m is impossible. It is immediate that 0 f 1 and f is a Borel function. Let (tα )α be a net in G such that tα → t ∈ G. Let s ∈ G be arbitrary. Let > 0. Choose α0 such that tα g − t g < whenever α α0 . Let α α0 . Note that there are at most two integers n such that the term |g(tα sxn−1 ) − g(tsxn−1 )| is non-zero. Moreover, for such non-zero terms, we have −1 g(tsxn−1 )| tα g − t g < . Therefore |tα f (s) − t f (s)| = |f (tα s) − f (ts)| |g(t ∞α sxn ) − −1 |g(t sx ) − g(tsxn−1 )| < 2, so f ∈ LUC(G). By the construction, it is clear that f = 1 n=1 ∞ α n U xn )c . In particular, f ∈ / C0 (G). Since ∞ on n=1 K1 xn and f = 0 on ( ∞ n=1 K1 n=1 K1 U xn ∞ and n=1 K2 U xn are disjoint, f = 0 on ∞ K U x . 2 2 n n=1 Lemma 4.4. Let G be a locally compact group. Given μ ∈ M(G) and f ∈ LUC(G), we define f · μ(x) = f (yx) dμ(y), then f · μ ∈ LUC(G). Moreover, f · μ∞ f ∞ μ. Proof. Let M = μ + 1. Let > 0, then there exists an open neighborhood U of e such that |f (x) − f (y)| < /M whenever xy −1 ∈ U . First, we assume that the support of μ, denoted by K, is compact. For each y ∈ K, there exists a symmetric open neighborhood Vy of e such that Vy Vy Vy ⊆ y −1 Uy. Note that {yVy | y ∈ K} is an open covering of K, so we may choose a finite subcover {yi Vyi | i = 1, 2, . . . , n}. Define V = ni=1 Vyi . Let x1 , x2 ∈ G such that x1 x2−1 ∈ V . Let y ∈ K be arbitrary, then y ∈ yi Vyi for some i. Therefore (yx1 )(yx2 )−1 = yx1 x2−1 y −1 ∈ yi Vyi Vyi (Vyi )−1 yi−1 = yi Vyi Vyi Vyi yi−1 ⊆ U and hence |f (yx1 ) − f (yx2 )| < /M. Consequently f · μ(x1 ) − f · μ(x2 ) f (yx1 ) − f (yx2 ) d|μ|(y) μ/M < . K
This proves that f · μ ∈ LUC(G). It is clear that f · μ∞ f ∞ μ for general μ ∈ M(G), f ∈ LUC(G). Lastly, if the support of μ is not compact, we may, by inner regularity of μ, choose a sequence (μn ) in M(G), with supp(μn ) compact and μn − μ → 0, then f · μ − f · μn f ∞ μn − μ → 0. As f · μn ∈ LUC(G) and LUC(G) is a closed subspace of L∞ (G), f · μ ∈ LUC(G). 2 Remark 4.5. A similar technique was used by Granirer and Lau. We refer the reader to [9, Lemma 4]. Now, we are ready to state the main theorem in this paper which characterizes the minimality of the topological center Z(π) in terms of a factorization property. The forward implication of this theorem is inspired by [6, Theorem 3.1].
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Theorem 4.6. Let G be a locally compact, non-compact group and let π : G → B(H) be a continuous unitary representation. Let F = {MT | M ∈ UCB(π)∗ , T ∈ UCB(π)}, then the following are equivalent: (a) The linear span of F is norm dense in LUC(G). (b) Z(π) = M(G). Proof. We prove that (a) ⇒ (b). Suppose that the condition in (a) holds. Let Z be the topological center of LUC(G)∗ , i.e. Z is the subset of LUC(G)∗ which consists of all m such that the map LUC(G)∗ → LUC(G)∗ , n → mn is weak∗ –weak∗ continuous. Recall that Z = M(G) by [11] and we already know that M(G) ⊆ Z(π), so we will finish the proof once we show that Z(π) ⊆ Z. Let m ∈ Z(π). To prove that m ∈ Z, it suffices that the map LUC(G)∗ → LUC(G)∗ , n → mn is weak∗ –weak∗ continuous on all bounded parts of LUC(G)∗ (see [11]). Let (nα ) be a bounded net in LUC(G)∗ such that nα → n ∈ LUC(G)∗ with respect to the weak∗ -topology. Let f ∈ LUC(G). First, we assume that f ∈ span F . Write f=
k
Mi Ti ,
i=1
where Mi ∈ UCB(π)∗ and Ti ∈ UCB(π), then mnα , f =
k mnα , Mi Ti i=1
=
k
k m, (nα )l (Mi Ti ) = m, (nα Mi )Ti
i=1
=
i=1
k
m(nα Mi ), Ti →
i=1
=
k
m(nMi ), Ti
i=1
k
m, (nMi )Ti =
i=1
k
m, nl (Mi Ti )
i=1
= mn, f . Then we drop the assumption that f ∈ span F . Let > 0. Choose K > 0 such that nα K and m K. Choose f0 ∈ span F such that f − f0 ∞ < /(K 2 ). Since mnα , f0 → mn, f0 , there exists α0 such that |mnα , f0 − mn, f0 | < whenever α α0 . For any α α0 , mnα , f − mn, f mnα , f − mnα , f0 + mnα , f0 − mn, f0 + mn, f0 − mn, f 3. Therefore m ∈ Z.
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We prove the direction (b) ⇒ (a) by contradiction. Suppose the contrary that the closed linear span of F = LUC(G). Pick m ∈ LUC(G)∗ such that m = 0 but m vanishes on the closed linear span of F . Note that for any M ∈ UCB(π)∗ , mM = 0. In particular m ∈ Z(π). By Lemma 2.1, we may write m = m1 + m2 where m1 ∈ M(G) and m2 ∈ C0 (G)⊥ . If m2 = 0, we obtain a contradiction immediately since m2 = m − m1 ∈ Z(π). Suppose that m2 = 0. We denote m = m1 = μ ∈ M(G). Next we shall produce another m ∈ Z(π) with m = 0 and m ∈ C0 (G)⊥ , then arrive at a contradiction. Note that for M ∈ UCB(π)∗ , T ∈ UCB(π), x, y ∈ G, we have (δx M)T (y) = δx M, T · y = δx , M(T · y) = M(T · y) (x) = M, (T · y) · x = M, T · (yx) = (MT )x (y) and hence (MT )x = (δx M)T . Therefore
MT (yx) dμ(y) = μ, (δx M)T = 0,
for each x ∈ G. First we consider the case that μ is a signed-measure. By Jordon decomposition theorem, we have μ = μ+ − μ− . Furthermore, we assume that supp(μ+ ) ⊆ K1 and supp(μ− ) ⊆ K2 , where K1 and K2 are two disjoint compact subsets of G. Choose a compact, symmetric neighborhood U of e, a sequence (xn ) in G, f ∈ LUC(G)\C0 (G) as in Lemmas 4.2 and 4.3. By Lemma 4.4, f · μ ∈ LUC(G). We assert that f · μ ∈ / C0 (G). Note that the constant function 1 ∈ F (for, let T = idH and choose M ∈ UCB(π)∗ such that M, T = 1), so μ(G) = μ, 1 = 0. Therefore − μ+ (G) = μ− (G) = 0. We prove that f · μ+ = μ+ (G) ∞ on {xn | n ∈ N} and f · μ = 0 on {xn | n ∈ N}. Let y ∈ K1 and x ∈ {xn | n ∈ N}, then yx ∈ n=1 K1 xn , so f (yx) = 1. Consequently, +
f · μ (x) =
f (yx) dμ+ (y) = μ+ (K1 ) = μ+ (G)
K1
and hence f · μ+ = μ+ (G) on {xn | n ∈ N}. If y ∈ K2 , x ∈ {xn| n ∈ N}, then yx ∈ ∞ n=1 K2 xn ⊆ ∞ − (x) = − (y) = 0. Therefore K U x , so f (yx) = 0 and consequently f · μ f (yx) dμ 2 n n=1 K2 f · μ = f · μ+ − f · μ− = μ+ (G) on the non-compact set {xn | n ∈ N}. In particular f · μ ∈ / C0 (G). Note that in our case, f ∞ = 1. Then we consider the case that μ is a general signed-measure. Denote λ = μ > 0, then μ+ (G) = μ− (G) = λ/2. By inner regularity of μ+ , μ− , we may choose positive measures − + − + + μ+ 0 , μ0 with supp(μ0 ) ⊆ K1 , supp(μ0 ) ⊆ K2 , K1 , K2 being compact and 0 μ0 μ , + − − + − + − 0 μ− 0 μ , μ − μ0 < λ/100, μ − μ0 < λ/100. Since μ and μ are mutually singular, K1 , K2 can be chosen such that K1 ∩ K2 = ∅. By the previous argument, there exists − + f ∈ LUC(G) with f ∞ = 1 such that f · (μ+ 0 − μ0 ) = μ0 (G) > λ/2 − λ/100 = 49λ/100 on a + − non-compact set {xn | n ∈ N}. Set μ0 = μ0 − μ0 , then μ − μ0 < λ/50, so f · μ − f · μ0 ∞ f ∞ μ − μ0 < λ/50. Therefore |f · μ(x)| > 49λ/100 − λ/50 for any x ∈ {xn | n ∈ N} hence f ·μ∈ / C0 (G). If μ ∈ M(G) is a complex measure, we may write μ = μ1 + iμ2 for some finite signed-measures μ1 , μ2 . Note that at least one of μ1 , μ2 is non-zero. Choose f ∈ LUC(G) as
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before, according to the non-zero measure μi , then f · μ = (f · μ1 ) + i(f · μ2 ). Note that both / C0 (G). f · μ1 , f · μ2 are real-valued and at least one of them is not in C0 (G), so f · μ ∈ We conclude that there exists f0 ∈ LUC(G) such that f0 · μ ∈ LUC(G)\C0 (G). By Hahn– Banach theorem, there exists m ∈ LUC(G)∗ such that m (f0 · μ) = 0 while m = 0 on C0 (G). Define m ∈ LUC(G)∗ by m , f = m , f · μ . Clearly if f ∈ C0 (G), then f · μ ∈ C0 (G), / M(G). If so m ∈ C0 (G)⊥ . m (f0 ) = m , f0 · μ = 0, so m = 0 and in particular m ∈ ∗ , T ∈ UCB(π), then m , MT = m , (MT ) · μ . However (MT ) · μ(x) = M ∈ UCB(π) MT (yx) dμ(y) = 0, so m , MT = 0. Consequently, m M = 0 for all M ∈ UCB(π)∗ . In particular m ∈ Z(π)\M(G). 2 We notice that in our main theorem, if the closed linear span of F is LUC(G), the m ∈ Z(π)\M(G) constructed has the property that mM = 0 for any M ∈ UCB(π)∗ . It is interesting to ask: Is it possible to find such m other than that form? In the following, we give a sufficient condition which guarantees the existence of m ∈ Z(π)\M(G) with the property mM = M for all M ∈ UCB(π)∗ . ˇ We recall some facts about Stone–Cech compactification. Let Ω be a Tychonoff space (i.e. T1 ˇ and completely regular). The Stone–Cech compactification βΩ of Ω is defined as the Gelfand spectrum of CB(Ω), the commutative, unital C∗ -algebra of all bounded, continuous, complexvalued functions defined on Ω. Note that βΩ has the following properties: (1) βΩ is compact. (2) The identity map ι : Ω → βΩ is a topological embedding, i.e. ι(Ω) is dense in βΩ and the map ι : Ω → ι(Ω) is a homeomorphism. The reader is referred to [4, pp. 137–138] for more detail. We remark that a locally compact Hausdorff space is a Tychonoff space. We identify Ω with βΩ and simply write ω for ι(ω). The following lemma is probably well known. However, we cannot find a proof from standard textbooks, so we include a proof here for completeness. Lemma 4.7. Let Ω be a locally compact Hausdorff space. Let f ∈ CB(Ω), then f ∈ C0 (Ω) if and only if f(ω) = 0 for any ω ∈ βΩ\Ω. (f denotes the Gelfand transform of f .) Proof. Let f ∈ C0 (Ω). We prove by contradiction. Suppose that there exists ω0 ∈ βΩ\Ω such that f(ω0 ) = 0. Since Ω is dense in βΩ, we may choose a net (ωα )α is Ω such that ωα → ω0 . Fix 0 > 0 such that |f(ω0 )| > 0 . By passing to a subnet, we may assume, without lose of generality, that |f(ωα )| > 0 for all α. Let K = {ω ∈ Ω | |f(ω)| 0 } which is compact. Choose a subnet (ωα ) of (ωα ) such that (ωα ) converges to some ω0 ∈ K. However, ωα → ω0 and ω0 = ω0 , which is a contradiction. Conversely, let f ∈ CB(Ω) such that f(ω) = 0 for each ω ∈ βΩ\Ω. Let > 0 and define K = {ω ∈ Ω | |f(ω)| }. Let (ωα )α be a net in K. By regarding (ωα )α as a net in βΩ and by the compactness of βΩ, there exists a subnet (ωα ) of (ωα ), and ω0 ∈ βΩ such that ωα → ω0 . Observe that |f(ω0 )| = limα |f(ωα )| , so ω0 ∈ K. Therefore K is compact and hence f ∈ C0 (Ω). 2 Proposition 4.8. Let π : G → B(H) be a unitary representation of a locally compact group. Let N = {x ∈ G | T · x = T for any T ∈ UCB(π)}, the kernel of the G-module action induced by π , which is a closed normal subgroup of G. If N is non-compact, there exists m ∈ Z(π)\M(G) such that mM = M for any M ∈ UCB(π)∗ . In particular, M(G) is strictly contained in Z(G).
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Proof. Regard N as a locally compact Hausdorff topological space. As N is non-compact, we can select ω0 ∈ βN\N and define a character m on CB(N )∗ by m, f = f(ω0 ), where we iden∗ tify the two C -algebras CB(N ), C(βN ) via the Gelfand transform f → f. By the previous lemma, m, f = 0, for any f ∈ C0 (N ). Define m ∈ LUC(G)∗ by m, f = m, f |N . Let f ∈ Cc (G), then clearly f |N ∈ Cc (N ). Therefore m, f = m, f |N = 0, i.e. m ∈ C0 (G)⊥ . Denote the identity functions on G and on N by 1G and 1N respectively, then m, 1G = m, 1N = 1, hence m = 0. Let M ∈ UCB(π)∗ , T ∈ UCB(π) and x ∈ N , then MT (x) = M, T · x = M, T . Therefore MT |N = M, T 1N and hence mM, T = M, T m, 1N = M, T , i.e. mM = M. 2 We also remark the following observation. Proposition 4.9. Using the above notation, if G is non-compact and the kernel N of the Gmodule action is non-trivial, i.e. N = {e}, then the factorization property in the main theorem fails to hold, hence M(G) is properly contained in Z(π). Proof. Suppose that there exists x ∈ N with x = e. Clearly for each M ∈ UCB(π)∗ and T ∈ UCB(π), MT (x) = MT (e). Consequently, f (x) = f (e) for each f in the closed linear span of F . However, by Urysohn lemma, there exists a continuous function g with compact support such that g(x) = g(e) and hence g ∈ LUC(G) but g does not lie in the closed linear span of F . 2 Now we apply our main theorem to some examples. Corollary 4.10. Let λ be the left regular representation of a locally compact group G. Then {MT | M ∈ UCB(λ)∗ , T ∈ UCB(λ)} = LUC(G), hence Z(λ) = M(G). Proof. Let f ∈ LUC(G) be given. Define Tf : L2 (G) → L2 (G) be the multiplication operator induced by f , i.e. Tf (g) = gf (g ∈ L2 (G)). We recall that the map f → Tf is an isometric embedding of LUC(G) into B(H). Note that for any x ∈ G, Tf · x = Tx f . Therefore if (xα ) is a net in G converging to x ∈ G, we have Tf · xα − Tf · x = xα f − x f → 0, hence Tf ∈ UCB(λ). For y ∈ G, we let δy ∈ LUC(G)∗ be the evaluation at y. We regard LUC(G) as a subspace of UCB(λ) and let My ∈ UCB(λ)∗ be any Hahn–Banach extension of δy . If x ∈ G, then (My Tf )(x) = My , Tf · x = My , Tx f = δy , x f = f (xy) = fy (x). Therefore My Tf = fy . In particular, Me Tf = f . It follows that, by the main theorem, Z(λ) = M(G) if G is non-compact. If G is compact, we always have Z(λ) = M(G) since LUC(G)∗ = C(G)∗ = M(G). 2 Example 4.11. Let G = Z be the discrete group of integers. For each n ∈ N, let qn : Z → Zn be the canonical quotient map and let λn : Zn → B(l2 (Zn )) be the left regular representation. Let λn = λn ◦ qn and define π = ∞ λ n=1 n . Then Z(π) = M(G). Proof. We identify l2 (Zn ) ∼ = Cn and let ek (k = 1, 2, . . . , n), be the canonical orthonormal base ∞ (n) of l2 (Zn ). Let H = n=1 l2 (Zn ), then {ek | n ∈ N and k = 1, 2, . . . , n} is an orthonormal base (n) (n) of H. For each x ∈ G, π(x)ek = e[k+x]n , where [k + x]n is the unique integer in {1, 2, . . . , n} such that [k + x]n ≡ k + x (mod n). First, we claim that for any subset A ⊆ N ∪ {0} ⊆ G, (n)
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χA ∈ {MT | M ∈ UCB(π)∗ , T ∈ UCB(π)}. Let A ⊆ N ∪ {0} be given. Define T ∈ B(H) by setting (n) (n) T ek = ek , if n is even, k ∈ {1, 2, . . . , n/2} and k − 1 ∈ A, 0, otherwise. For each n ∈ N, we define Mn = e1 ⊗ e1 ∈ L1 (H) ⊆ B(H)∗ . (If ξ1 , ξ2 ∈ H, we define a rank-one operator ξ1 ⊗ ξ2 on H by ξ1 ⊗ ξ2 (η) = η, ξ2 ξ1 .) Choose a subnet (Mnα )α of (Mn )n such that Mnα → M ∈ B(H)∗ with respect to the weak∗ -topology. Let x ∈ G. Fix n0 ∈ N such that n0 > |x|. Note that (2n)
(2n)
Mn T (x) = tr(Mn T · x) = tr Mn π x −1 T π(x) = tr π(x)Mn π x −1 T (2n) (2n) (2n) (2n) = tr e[1+x]2n ⊗ e[1+x]2n T = T e[1+x]2n e[1+x]2n . Consider two cases. Case I: Suppose that x ∈ A. Let n n0 , then 1 1 + x 2n, so [1 + x]2n = 1 + x. As (2n) (2n) [1 + x]2n − 1 = x ∈ A, T e[1+x] = e[1+x] . Therefore 2n 2n MT (x) = limMnα , T · x = lim Mn , T · x = 1. α
n→∞
Case II: Suppose that x ∈ / A. Let n n0 . If x 0, we still have 1 1 + x 2n, so [1 + (2n) x]2n − 1 = x ∈ / A and consequently T e[1+x] = 0. If x < 0, we have −n < 1 − n < 1 + x 0, 2n (2n)
so [1 + x]2n = 2n + (1 + x) ∈ {n + 1, n + 2, . . . , 2n} and consequently T e[1+x]2n = 0. Therefore MT (x) = lim Mn T (x) = 0. n→∞
This shows that χA = MT . Then we prove that for any A ⊆ {n ∈ Z | n < 0} ⊆ G, χA ∈ {MT | M ∈ UCB(π)∗ , T ∈ UCB(π)}. Let such set A be given. Define T ∈ B(H) by setting: (n) n T ek = ek , 0,
if n is even, k ∈ { n2 + 1, n2 + 2, . . . , n} and k − 1 − n ∈ A, otherwise.
For each n ∈ N, define Mn = e1 ⊗ e1 ∈ L1 (H) ⊆ B(H)∗ . Choose a subnet (Mnα )α of (Mn )n such that Mnα → M ∈ B(H)∗ with respect to the weak∗ -topology. Let x ∈ G and fix n0 ∈ N such that n0 > |x|. We consider two cases. (2n) (2n) Case I: Suppose that x ∈ A. Let n > n0 , then Mn T (x) = T e[1+x] | e[1+x] . As [1 + x]2n = 2n 2n (2n)
(2n)
(2n)
(1 + x) + 2n ∈ {n + 1, n + 2, . . . , 2n}, [1 + x]2n − 1 − 2n = x ∈ A. Consequently, T e[1+x]2n = (2n)
e[1+x]2n so Mn T (x) = 1. Therefore MT (x) = lim Mnα T (x) = lim Mn T (x) = 1. α
n
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Case II: Suppose that x ∈ / A. Let n > n0 . If x < 0, we have [1 + x]2n − 1 − n = x ∈ / A, (2n) so T e[1+x]2n = 0. If x 0, we have [1 + x]2n = 1 + x ∈ / {n + 1, n + 2, . . . , 2n}. Therefore (2n) T (e[1+x] ) = 0 and hence Mn T (x) = 0. Consequently 2n
MT (x) = lim Mn T (x) = 0, n→∞
so χA = MT . It is now clear that for any B ⊆ G, χB lies in the linear span of {MT | M ∈ UCB(π)∗ , T ∈ UCB(π)}. Therefore the linear span of {MT | M ∈ UCB(π)∗ , T ∈ UCB(π)} contains all the simple functions and consequently it is dense in l∞ (G) = LUC(G). By our characterization theorem, Z(π) = M(G). 2 Remark 4.12. In this example, each representation λn is finite dimensional and hence Z(λn ) = λ LUC(G)∗ . However, Z( ∞ n=1 n ) is minimal, i.e. equals to M(G). 5. Maximality of the topological center In this section, we give two sufficient conditions, each of which will guarantee that the topological center is maximal, i.e. Z(π) = LUC(G)∗ . An example whose topological center is maximal is also demonstrated. Proposition 5.1. Let π : G → B(H) be a unitary representation of a locally compact group G. If at least one of the following conditions is satisfied, the topological center Z(π) is maximal, i.e. Z(π) = LUC(G)∗ . (1) dim(π) < ∞. (2) For each > 0, we define N (π, ) = N = {x ∈ G | T · x − T < T for any T ∈ UCB(π)}. Suppose that for each > 0, there exist x1 , x2 , . . . , xm ∈ G satisfying that: for each x ∈ G, there exist i ∈ {1, 2, . . . , m} and y ∈ N such that x = xi y. Proof. Suppose that the first condition holds. Let (Mα )α be a net in UCB(π)∗ such that Mα → M ∈ UCB(π)∗ with respect to the weak∗ -topology. Since dim(π) < ∞, UCB(π)∗ is a finite dimensional vector space and all locally convex topologies coincide. Therefore for any m ∈ LUC(G)∗ and T ∈ UCB(π), |mMα , T − mM, T | = |m, (Mα − M)T | mMα − MT → 0, hence m ∈ Z(π). Suppose that the second condition holds. We assert that for any bounded net (Mα )α in UCB(π)∗ , M ∈ UCB(π)∗ , T ∈ UCB(π), if Mα → M with respect to the weak∗ -topology, then Mα T − MT ∞ → 0. Without loss of generality, we assume that Mα 1, M 1, T 1. Let > 0 be arbitrary. Choose x1 , x2 , . . . , xm ∈ G as in the assumption, then we obtain a partition {A1 , A2 , . . . , Am } of G with the property that for any x ∈ Ai , there exists y ∈ N such that x = xi y. Let x ∈ Ai and write x = xi y for some y ∈ N , then T · x − T · xi = (T · xi ) · y − T · xi < T · xi . Therefore |MT (x) − MT (xi )| = |M, T ·x −T ·xi | MT ·x −T ·xi < . Similarly, we have |Mα T (x)−Mα T (xi )| < . Define λi = MT (xi ) and λαi = Mα T (xi ). Since Mα T → MT pointwisely, we may choose α0 such α i ∈ {1, 2 . . . , m} and α α0 , i.e. |λ that |Mα T (xi ) − MT (xi )| < whenever i − λi | < . By the m m above discussion, it is clear that MT − i=1 λi χAi ∞ and Mα T − i=1 λαi χAi ∞ . Therefore
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m m m Mα T − MT ∞ Mα T − λαi χAi + λαi χAi − λi χAi i=1 i=1 i=1 ∞ ∞ m + λi χAi − MT i=1
∞
3 whenever α α0 and hence Mα T − MT ∞ → 0. Let m ∈ LUC(G)∗ . In order to show m ∈ Z(π), by Proposition 3.1 it suffices that the map M → m · M is weak∗ –weak∗ continuous on all the bounded part of UCB(π)∗ . Let (Mα )α be a bounded net in UCB(π) and suppose that Mα → M with respect to the weak∗ -topology. Let T ∈ UCB(π), then |mMα , T − mM, T | mMα T − MT → 0. Therefore m ∈ Z(π). 2 Remark 5.2. Let N = {x ∈ G | T · x = T for any T ∈ UCB(π)}. If |G/N| < ∞, the second condition will be satisfied and hence the topological center Z(π) is maximal. For, suppose that |G/N| = m. We pick an element xi from each N -coset, then for each > 0, x1 , x2 , . . . , xm and N clearly satisfy the second condition since N ⊆ N . Identify the quotient group R/Z with [0, 1) in a canonical way. Let α ∈ [0, 1)\Q. It is an easy exercise to check that the subgroup of R/Z generated by α, namely {nα | n ∈ Z} is dense in [0, 1). Lemma 5.3. Identify the quotient group R/Z with [0, 1). Given sufficiently small > 0, α ∈ [0, 1)\Q (where α is regarded as an element in R/Z), we define Z0 = {n ∈ Z | nα ∈ (0, )}, then (1) Z0 is an infinite set, and (2) there exists k ∈ N such that |m − n| k whenever m, n ∈ Z0 are two successive elements in Z0 . Proof. Since α is irrational, the map n → nα ∈ R/Z is injective. Since {nα | n ∈ Z} is dense in [0, 1), there exist infinitely many n ∈ Z such that nα ∈ (0, ). This proves the first part. Next, we choose n0 ∈ Z such that 0 < n0 α < /2. Denote β = n0 α and define Z1 = {n ∈ Z | nβ ∈ (0, )}. Let n1 be the smallest positive integer such that n1 β < 1 < (n1 + 1)β. Let m, n be two successive elements in Z1 with m < n. If mβ ∈ (0, − β), we clearly have mβ + β ∈ (0, ) so n = m + 1. If mβ ∈ [ − β, ), then mβ + n1 β mβ + n1 β − 1 > ( − β) − β > 0. Note that mβ + n1 β mβ + (n1 β − 1) < mβ < . Therefore (m + n1 )β ∈ (0, ) and hence n m + n1 . Lastly, if n ∈ Z1 , then nβ ∈ (0, ), nn0 α ∈ (0, ), so nn0 ∈ Z0 . Consequently n0 Z1 ⊆ Z0 . Take k = n0 n1 , then |m − n| k for any two successive elements m, n ∈ n0 Z1 . Since n0 Z1 is an infinite subset of Z0 and every two successive elements in n0 Z1 satisfy the required property, it follows that the elements of Z0 satisfy the property as well. 2 Example 5.4. Let Z be the usual discrete group of integers. Choose θ ∈ [0, 2π) such that θ/(2π) is irrational. Let cos θ − sin θ A= . sin θ cos θ
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Let H = C2 and let π : Z → B(H) M2 (C) be defined by π(n) = An . Let π= ∞ n=1 π be the ∞ countable direct sum of π . Let H = n=1 H. Although N π ) = ∞, we π = Nπ = {0} and dim( still have Z( π ) = l ∞ (G)∗ . Proof. Let > 0. Choose δ > 0 such that cos φ − sin φ − I < /2, sin φ cos φ whenever φ ∈ (−δ, δ) + 2πZ. Therefore ∞ ∞ cos φ − sin φ cos φ − sin φ − I = −I sin φ cos φ sin φ cos φ n=1 n=1 cos φ − sin φ − I = < /2, sin φ cos φ Define N = {n ∈ Z | where I is the identity operator on H. π (−n)T π (n) − T T , for any T ∈ B(H)}. Set Z0 = {n ∈ Z | nθ/(2π) ∈ (0, δ/(2π))+Z}. If n ∈ Z0 , then for any T ∈ B(H), π (n) − T π (−n)T π (n) − T π (n) + T π (−n)T π (n) − T π (n) − IT π (−n) − IT + T , cos nθ − sin nθ cos(−nθ) − sin(−nθ) by observing that π (n) = , π (−n) = sin(−nθ) cos(−nθ) with nθ ∈ (0, δ) + sin nθ cos nθ 2πZ and −nθ ∈ (−δ, 0) + 2πZ. Therefore Z0 ⊆ N . By the previous lemma, Z0 is an infinite set such that the distance between two successive elements in bounded. As N is a superset of Z0 , N has the same property. Now it clear that condition (2) in Proposition 5.1 is fulfilled, so Z( π ) = l ∞ (Z)∗ . 2 6. An example that M(G) Z(π) LUC(G)∗ In this section, we give an example that the topological center Z(π) is neither minimal nor maximal. Example 6.1. Let G = Z × Z and let q : G → Z be the canonical quotient map defined by q(i, j ) = j . Let λ : Z → B(l2 (Z)) be the left regular representation and let π = λ ◦ q, then M(G) Z(π) LUC(G)∗ . Proof. Since the kernel of the G-action induced by π is non-trivial, M(G) is properly contained in Z(π). Let H = l2 (Z) and let {ek | k ∈ Z} be the canonical base of H. We show that Z(π) is not maximal. For each i ∈ N, let ni = δ(0,i) ∈ l1 (G) and let n ∈ l∞ (G)∗ be any weak∗ -cluster point / Z(π). For each j ∈ N, define Mj = e−j ⊗ e−j ∈ L1 (H) → of the net (ni ). We assert that n ∈ ∗ B(H) . Let T0 ∈ B(H) be defined by e if k 1, T0 (ek ) = k 0 if k 0.
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Let T ∈ B(H). We observe that ni Mj , T = ni , Mj T = (Mj T ) (0, i) = Mj , π (0, −i) T π (0, i) = tr Mj π (0, −i) T π (0, i) = tr π (0, i) Mj π (0, −i) T = tr(ei−j ⊗ ei−j T ) = ei−j ⊗ ei−j , T , and hence ni Mj = ei−j ⊗ ei−j . Let M be an arbitrary weak∗ -cluster point of the net (Mj ), then we have nM, T0 = lim ni M, T0 i→∞
= lim lim ni Mj , T0 = lim lim ei−j ⊗ ei−j , T0 i→∞ j →∞
i→∞ j →∞
= lim lim T0 ei−j | ei−j = 0. i→∞ j →∞
On the other hand lim nMj , T0 = lim lim ni Mj , T0
j →∞
j →∞ i→∞
= lim lim ei−j | ei−j = 1. j →∞ i→∞
Let Mjα be a subnet of (Mj ) such that limα Mjα = M with respect to the weak∗ -topology. Now it is clear that nMjα nM with respect to the weak∗ -topology and hence n ∈ / Z(π). 2 7. Direct sums and tensor products of unitary representations and their topological centers In this section, we investigate the relations between the topological centers of sub-representations, finite direct sums, tensor products with that of the underlying representations. We prove that if π1 is a sub-representation of π2 , we always have Z(π2 ) ⊆ Z(π 1 ). We also show that for an arbitrary unitary representation π , and n ∈ N, the finite direct sum ni=1 π = nπ and the original representation π have the same topological centers. Lastly, we give a condition which guarantees that Z(π1 ⊗ π2 ) = M(G). Lemma 7.1. Let (π1 , H1 ), (π2 , H2 ) be unitary representations of G. Suppose that π1 is a subrepresentation of π2 . Let P : H2 → H1 be the canonical projection. For each T ∈ B(H2 ), we define T ∈ B(H1 ) by T = P ◦ T |H1 . If T ∈ UCB(π2 ), then T ∈ UCB(π1 ). Moreover, the map T → T is surjective. Proof. Let x ∈ G. Notice that T · x = π1 x −1 T π1 (x) = π1 x −1 P (T |H1 )π1 (x) = P π2 x −1 T π2 (x)|H1 = P [T · x]|H1 .
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Let (xα ) be a net in G such that xα → x ∈ G, then T · x α − T · x
B(H1 )
P T · xα − T · xB(H2 ) → 0,
i.e. T ∈ UCB(π1 ). Given T0 ∈ UCB(π1 ), we define T = T0 ◦ P ∈ B(H2 ). It is clear that T ∈ UCB(π2 ) with T = T0 , so the map T → T is surjective. 2 ∈ UCB(π2 )∗ by Lemma 7.2. Using the above notation, if M ∈ UCB(π1 )∗ , we define M M, T = M, T . Then MT = MT as an element in LUC(G) for any T ∈ UCB(π2 ). Proof. Let x ∈ G, then (x) = M, T · x = M, P ◦ (T · x)|H = M, T · x = MT (x). MT 1
2
We now state a proposition relating the topological centers of a sub-representation and the original representation. Proposition 7.3. Let (π1 , H1 ), (π2 , H2 ) be unitary representations of a locally compact group G. If π1 is a sub-representation of π2 , then Z(π2 ) ⊆ Z(π1 ). Proof. Let m ∈ Z(π2 ). Let (Mα ) be a net in UCB(π1 )∗ such that Mα → M ∈ UCB(π1 )∗ with re spect to the σ (UCB(π1 )∗ , UCB(π1 ))-topology. Let T ∈ UCB(π2 ), then M α , T = Mα , T → ∗ M, T = M, T , hence Mα → M with respect to the σ (UCB(π2 ) , UCB(π2 ))-topology. Let T0 ∈ UCB(π1 ). Choose T ∈ UCB(π2 ) such that T = T0 , then mMα , T0 = m, Mα T0 = m, Mα T T = m, M α T = mM α , T → mM, = m, MT = m, MT0 = m, MT = mM, T0 . Hence mMα → mM with respect to the σ (UCB(π1 )∗ , UCB(π1 ))-topology, i.e. m ∈ Z(π1 ).
2
Next, we consider direct sum of unitary representations. Let π be a unitary representation of G and let π = α π be the direct sum of α (a cardinal) copies of π . It is interesting to ask: How are Z(π) and Z(π ) related? Since π is a sub-representation of π , we have Z(π ) ⊆ Z(π) by the previous proposition. In fact, if α is finite, we can say more. Before stating and proving the proposition, we first introduce some notations. Let H be the underlying Hilbert space for π and let H = n H, the direct sum of n copies of H. In order to avoid confusion, we let H1 = H2 = · · · = Hn = H and write H = ni=1 Hi . For each i ∈ {1, 2, . . . , n}, we let Pi : H → Hi be the canonical projection and let Ii : Hi → H be the canonical injection. Given T ∈ B(H ), we associate n2 operators on H (here we identify H1 H2 · · · Hn H) as follows.
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For i, j ∈ {1, 2, . . . , n}, we define Tij : Hj → Hi by Tij = Pi ◦ T ◦ Ij . We call {Tij | 1 i n, 1 j n} the components of T . Conversely, given n2 bounded linear operators Tij ∈ B(H), we can associate T ∈ B(H ) by T=
Ii ◦ Tij ◦ Pj .
1in, 1j n
We remark that the above processes of decomposition and composition are converse to each other. More precisely, given T ∈ B(H ), we first decompose it and obtain Tij ∈ B(H), then use these n2 operators Tij to construct T ∈ B(H ). It can be verified that T = T. On the other hand, given n2 operators Tij on H, we compose them and obtain T ∈ B(H ). It can be shown that Tij = Tij . If x ∈ G and T ∈ B(H ) or T ∈ B(H), we denote π (x −1 )T π (x) and π(x −1 )T π(x) by the same symbol T · x. We then prove few lemmas. Lemma 7.4. Using the above notations and let T ∈ B(H ), Tij ∈ B(H) the components of T , then for each x ∈ G, T · x has components Tij · x, i.e. (T · x)ij = Tij · x. Proof. Note that T · x = π x −1 T π (x) = π x −1 ◦ Ii ◦ Tij ◦ Pj ◦ π (x). i,j
But for each i, j , π x −1 ◦ Ii ◦ Tij ◦ Pj ◦ π (x) = Ii ◦ π x −1 ◦ Tij ◦ π(x) ◦ Pj = Ii ◦ (Tij · x) ◦ Pj . Therefore T · x =
i,j Ii
◦ (Tij · x) ◦ Pj and hence (T · x)ij = Tij · x.
2
Lemma 7.5. Let T ∈ B(H ) with components Tij ∈ B(H), then T ∈ UCB(π ) if and only if for each i, j , Tij ∈ UCB(π). Proof. Suppose that T ∈ UCB(π ). Let x, y ∈ G. By the previous lemma, for any i, j , Tij · x − Tij · y = (T · x)ij − (T · y)ij = Pi ◦ (T · x − T · y) ◦ Ij Pi T · x − T · yIj → 0 as x → y. Therefore Tij ∈ UCB(π). Conversely, suppose that for each i, j , Tij ∈ UCB(π). Let x ∈ G, then (Ii ◦ Tij ◦ Pj ) · x = π x −1 ◦ (Ii ◦ Tij ◦ Pj ) ◦ π (x) = Ii ◦ π x −1 ◦ Tij ◦ π(x) ◦ Pj = Ii ◦ (Tij · x) ◦ Pj , so for any x, y ∈ G,
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T · x − T · y = (Ii ◦ Tij ◦ Pj ) · x − (Ii ◦ Tij ◦ Pj ) · y Ii Tij · x − Tij · yPj → 0 as x → y. Therefore T ∈ UCB(π ).
2
Then we consider decomposition of elements in B(H )∗ . We continue to use the notations defined in above. Given M ∈ B(H )∗ , we associate n2 elements in B(H)∗ as follows. For each i, j ∈ {1, 2, . . . , n}, we define Mij ∈ B(H)∗ by Mij , T = M, Ii ◦ T ◦ Pj . We call Mij the components of M. We remark that if M ∈ UCB(π )∗ , then Mij ∈ UCB(π)∗ . We observe that if M ∈ B(H )∗ and T ∈ B(H ) with components Mij and Tij respectively, then M, T = . If M ∈ UCB(π )∗ and T ∈ UCB(π ) with components Mij and Tij respectively, ij Mij , Tij then MT = ij Mij Tij . We also need a lemma which deals with weak∗ -convergence. Lemma 7.6. Let (M α )α be a net in UCB(π )∗ , M ∈ UCB(π )∗ . Let Mijα , Mij be the components of M α and M respectively, then the following are equivalent: (a) M α → M with respect to σ (UCB(π )∗ , UCB(π ))-topology, (b) for each i, j , Mijα → Mij with respect to σ (UCB(π)∗ , UCB(π))-topology. Proof. Let (M α )α be a net in UCB(π )∗ such that M α → M with respect to the weak∗ -topology. Let i, j ∈ {1, 2, . . . , n} and let T ∈ UCB(π), then Mijα , T = M α , Ii ◦ T ◦ Pj
→ M, Ii ◦ T ◦ Pj = Mij , T . Therefore Mijα → Mij with respect to the weak∗ -topology. Conversely, let (M α ) be a net in UCB(π )∗ , M ∈ UCB(π )∗ such that for each i, j , Mijα → Mij with respect to the weak∗ topology. Let T ∈ UCB(π ), then M α , T = ij Mijα , Tij → ij Mij , Tij = M, T . Therefore M α → M with respect to the weak∗ -topology. 2 Now we are able to state and prove the following proposition. Proposition 7.7. Let π be a unitary representation of a locally compact group G and let π = π be the direct sum of n copies of π (n ∈ N), then Z(π) = Z(π ). n Proof. Since π is a sub-representation of π , we have Z(π ) ⊆ Z(π). Therefore, it suffices to show the reversed inclusion. Let m ∈ Z(π). Let (M α ) be a net in UCB(π )∗ such that ∗ ∗ -topology. Let T ∈ UCB(π ), then mM α , T = respect M α → M ∈ UCB(π ) with to the weak α α α m, M T = m, ij Mij Tij = ij mMij , Tij . By the previous lemma, Mijα → Mij with respect to the weak∗ -topology for each i, j . Since m ∈ Z(π), mMijα → mMij with respect α to the weak∗ -topology. Therefore, ij mMij , Tij → ij mMij , Tij = ij m, Mij Tij = m, MT = mM, T , hence m ∈ Z(π ). 2 In the following, we consider the tensor product of two unitary representations. Let π1 : G → B(H1 ), π2 : G → B(H2 ) be unitary representations of a locally compact group G. We denote
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the inner tensor product of π1 and π2 by π1 ⊗ π2 , i.e. π1 ⊗ π2 : G → B(H1 ⊗ H2 ) defined by π1 ⊗ π2 (x) = π1 (x) ⊗ π2 (x), x ∈ G. Lemma 7.8. Using the above notations, if T1 ∈ UCB(π1 ) and T2 ∈ UCB(π2 ), then T2 ⊗ T2 ∈ UCB(π1 ⊗ π2 ). Proof. Let T1 ∈ UCB(π1 ) and T2 ∈ UCB(π2 ) and let x ∈ G. Note that (T1 ⊗ T2 ) · x = (π1 (x −1 ) ⊗ π2 (x −1 ))(T1 ⊗ T2 )(π1 (x) ⊗ π2 (x)) = (π1 (x −1 )T1 π1 (x)) ⊗ (π2 (x −1 )T2 π2 (x)) = (T1 · x) ⊗ (T2 · x). Therefore, if (xα ) is a net in G which converges to x ∈ G, then (T1 ⊗ T2 ) · xα − (T1 ⊗ T2 ) · x = (T1 · xα ) ⊗ (T2 · xα ) − (T1 · x) ⊗ (T2 · x) (T1 · xα ) ⊗ (T2 · xα ) − (T1 · xα ) ⊗ (T2 · x) + (T1 · xα ) ⊗ (T2 · x) − (T1 · x) ⊗ (T2 · x) T1 · xα T2 · xα − T2 · x + T2 · xT1 · xα − T1 · x T1 T2 · xα − T2 · x + T2 T1 · xα − T1 · x → 0. 2 Proposition 7.9. Using the above notations, if there exists i ∈ {1, 2} such that span{MT | M ∈ UCB(πi )∗ , T ∈ UCB(πi )} is norm dense in LUC(G), then Z(π1 ⊗ π2 ) = M(G). Proof. Let F = {MT | M ∈ UCB(πi )∗ , T ∈ UCB(πi )} and F = {M T | M ∈ UCB(π1 ⊗ π2 )∗ , T ∈ UCB(π1 ⊗ π2 )}. We shall show that F ⊆ F . We simplify our notation and assume that i = 1. The case that i = 2 can be proved in exactly the same way. Let M1 ∈ UCB(π1 )∗ and T1 ∈ UCB(π1 ). Choose M2 ∈ UCB(π2 ), such that M2 , I2 = 1, where I2 is the identity operator on the Hilbert space H2 . Note that the map UCB(π1 ) × CI2 → C, (T , λI2 ) → M1 , T M2 , λI2 = λM1 , T is bounded bilinear, so it induces a bounded linear functional M : UCB(π1 ) ⊗ CI2 → C. We extend M (still denoted by M ) and obtain a bounded linear functional on UCB(π1 ⊗ π2 ) by Hahn–Banach theorem. Define T = T1 ⊗ I2 , then T ∈ UCB(π1 ⊗ π2 ) by the previous lemma. If x ∈ G, then M T (x) = M , T · x = M , (T1 · x) ⊗ I2 = M1 , T1 · x = M1 T1 (x). Therefore the linear span of F is norm dense in LUC(G). If G is non-compact, Z(π1 ⊗ π2 ) = M(G) by Theorem 4.6. If G is compact, it is automatic that Z(π1 ⊗ π2 ) = M(G). 2 Acknowledgments This paper constitutes a part of the authors PhD thesis, prepared at the University of Alberta under the supervision of Professor Anthony Lau. The author would like to express his deep gratitude to Professor Anthony Lau for his constant encouragements and guidance and to Mr. Michael Yin-Hei Cheng for his valuable discussion. References [1] [2] [3] [4] [5] [6] [7]
R. Arens, Operators induced in function classes, Monatsh. Math. 55 (1951) 1–19. R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951) 839–848. M.E.B. Bekka, Amenable unitary representations of locally compact groups, Invent. Math. 100 (1990) 383–401. J.B. Conway, A Course in Functional Analysis, second ed., Springer, 1990. N. Dunford, J.T. Schwarz, Linear Operators, Part 1, Interscience Publ. Inc., New York, 1964. M. Eshaghi Gordji, M. Filali, Arens regularity of module actions, Studia Math. 181 (3) (2007) 237–254. M. Filali, A.I. Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras, Gen. Topol. Algebra (Tartu) (2001) 95–124. [8] F. Ghahramani, A.T.M. Lau, V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1) (1990) 273–283. [9] E. Granirer, A.T.M. Lau, Invariant means on locally compact groups, Illinois J. Math. 15 (1971) 249–257.
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[10] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 1, second ed., Springer, 1979. [11] A.T.M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986) 273–283. [12] A.T.M. Lau, A. Paterson, Group amenability properties for von Neumann algebras, Indiana Univ. Math. J. 55 (4) (2006) 1363–1388. [13] A.T.M. Lau, A. Ulger, Topological centers of certain dual algebras, Transl. Amer. Math. Soc. 348 (3) (1996) 1191– 1212. [14] S. Mohammadzadeh, H.R.E. Vishki, Arens regularity of module actions and the second adjoint of a derivation, Bull. Austral. Math. Soc. 77 (2008) 465–476. [15] M. Takesaki, Theory of Operator Algebras I, Springer, 1979. [16] A. Ulger, Weakly compact bilinear forms and Arens regularity, Proc. Amer. Math. Soc. 101 (4) (1987) 697–704. [17] Q. Xu, Representations and compactifications of locally compact groups, Math. Jpn. 52 (2) (2000) 323–329.
Journal of Functional Analysis 259 (2010) 2215–2237 www.elsevier.com/locate/jfa
New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equations Afif Ben Amar a , Soufiene Chouayekh b , Aref Jeribi b,∗ a Département de Mathématiques, Faculté des Sciences de Gafsa, Université de Gafsa, Cité Zarrouk,
2112, Gafsa, Tunisia b Département de Mathématiques, Université de Sfax, Faculté des Sciences de Sfax, Route de Soukra Km 3.5, B.P. 1171,
3000, Sfax, Tunisia Received 30 January 2010; accepted 23 June 2010 Available online 10 July 2010 Communicated by K. Ball
Abstract We introduce a class of Banach algebras satisfying certain sequential condition (P) and we prove fixed point theorems for the sum and the product of nonlinear weakly sequentially continuous operators. Later on, we give some examples of applications of these types of results to the existence of solutions of nonlinear integral equations in Banach algebras. © 2010 Elsevier Inc. All rights reserved. Keywords: Banach algebra; Fixed point theorems; Sequentially weakly continuous; Integral equations; Differential equations
1. Introduction The study of functional integral equations and differential equations is the main object of research on nonlinear functional analysis. These equations occur in physical, biological and economic problems. * Corresponding author.
E-mail addresses:
[email protected] (A. Ben Amar),
[email protected] (S. Chouayekh),
[email protected] (A. Jeribi). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.016
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Some of these equations can be formulated into nonlinear operators equations: x = AxBx + Cx
(1.1)
in suitable Banach algebras. In recent years, many authors have focused on the resolution of Eq. (1.1) and obtained a lot of valuable results (see for example [3–6,9,11–14,16] and the references therein). These studies were mainly based on the convexity of the bounded domain, the celebrate Schauder fixed point theorem [16] and properties of operators A, B and C (cf. completely continuous, k-set contractive, condensing and the potential tool of the axiomatic measures of non-compactness). Because the weak topology is the practice setting and natural to investigate the problems of existence of solutions of different types of nonlinear integral equations and nonlinear differential equations in Banach algebras, it turns out that the results mentioned above cannot be easily applied. One of difficulties arising when treating such situations, is that a bounded linear functional ϕ acting on a Banach algebra does not necessarily satisfy the following inequality: ϕ(x.y) cϕ(x)ϕ(y), with c 0 and x, y ∈ X. In the present paper, for this interest, we introduce a class of Banach algebras satisfying certain sequential conditions called here the condition (P) (see Definition 3.1). The main goal of this paper is to prove some new fixed point theorems in a nonempty closed convex subset of any Banach algebras or Banach algebras satisfying the condition (P) under weak topology setting. Our main conditions are formulated in term of weak sequential continuity to the three nonlinear operators A, B and C involved in Eq. (1.1). Besides, no weak continuity conditions are required for this work. Our main results are applied to investigate the existence of solutions for the two following nonlinear functional integral equations in Banach algebra C(J, X)
σ (t) q(t) + p t, s, x(s), x(λs) ds .u ,
x(t) = a(t) + (T1 x)(t)
0 < λ < 1,
0
and x(t) = a(t)x(t) + (T2 x)(t)
σ (t) q(t) + p t, s, x(s), x(λs) ds .u ,
0 < λ < 1,
0
where J is the interval [0, 1] and X is any Banach algebra. The functions a, q, σ are continuous on J ; T1 , T2 , p(.,.,.,.) are nonlinear functions and u is a nonvanishing vector. The organization of this paper is as follows: in the next section, we give some definitions that will be needed in the sequel. Section 3 is devoted to the existence results for Eq. (1.1). So, we present some new fixed point theorems in Banach algebras. The main results of this section are Theorem 3.3 and Corollary 3.1. We end this section by discussing briefly the existence of positive solutions. In Section 4, we will apply the results obtained in precedent section to investigate the two FIE (4.1) and (4.2). The main result of this section is Theorems 4.1 and 4.2. Finally, we close this section by giving an application to resolve a particular example of functional differential equations.
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2. Preliminaries Definition 2.1. An algebra E is a vector space endowed with an internal composition law noted by (.) i.e.,
(.) : E × E → E, (x, y) → x.y
which is associative and bilinear. A normed algebra is an algebra endowed with a norm satisfying the following property for all x, y ∈ E;
x.y xy.
A complete normed algebra is called a Banach algebra. Definition 2.2. Let X be a Banach space. An operator A : X → X is said to be weakly compact if A(B) is relatively weakly compact for every bounded subset B ⊂ X. Definition 2.3. Let X be a Banach space. An operator A : X → X is said to be sequentially weakly continuous on X if for every sequence {xn } with xn x, we have Axn Ax; here denotes weak convergence. Definition 2.4. Let X be a Banach space. An operator A : X → X is said to be strongly continuous on X, if for every sequence {xn } with xn x, we have Axn → Ax; here → denotes convergence in X. Definition 2.5. Let X be a Banach space. A mapping G : X → X is called D-Lipschitzian if there exists a continuous and nondecreasing function φG : R+ → R+ such that Gx − Gy φG x − y for all x, y ∈ X, with φG (0) = 0. Sometimes we call the function φG a D-function of G on X. If φG (r) = kr for some k > 0, then G is called a Lipschitzian function on X with the Lipschitz constant k. Further if k < 1, then G is called a contraction on X with the contraction k. Remark 2.1. Every Lipschitzian mapping is D-Lipschitzian, but the converse may not be true. If φG is not necessarily nondecreasing and satisfies φG (r) < r, for r > 0, the mapping G is called a nonlinear contraction with a contraction function φG . The following fixed points results stated in [2] will be used throughout the next section. The proof follows from O. Arino. S. Gautier and J.P. Penot theorem [1] Theorem 2.1. (See [2, Theorem 2.5].) Let X be a Banach space, S be a nonempty closed convex subset of X and N : S → S be a sequentially weakly continuous map. If N (S) is relatively weakly compact, then N has a fixed point in S.
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3. Fixed point theorems Now, we are ready to state our first fixed point theorems in Banach algebras to provide the existence results of Eq. (1.1). First we have Theorem 3.1. Let E be a Banach algebra and S be a nonempty closed convex subset of E. Let A, C : E → E and B : S → E be three operators such that −1
(i) ( I −C A ) (ii)
exists on B(S).
−1 ( I −C A ) B is sequentially weakly continuous. −1 ( I −C A ) B(S) is relatively weakly compact.
(iii) (iv) x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S. Then Eq. (1.1) has at least one solution in S.
Remark 3.1. Note that the assumption (iv) of above theorem is introduced by Burton [8] instead of assuming that AxBy + Cx ∈ S, for all x, y ∈ S. Proof of Theorem 3.1. From assumption (i), it follows that for each y in S, there is a unique xy ∈ E such that I −C xy = By (3.1) A or, equivalently Axy By + Cxy = xy .
(3.2)
Since the hypothesis (iv) holds, then xy ∈ S. Therefore, we can define ⎧ ⎨ N : S → S, I − C −1 ⎩y → Ny = By. A By using the hypotheses (ii), (iii) and Theorem 2.1, we conclude that N has a fixed point y in S. Hence, y verifies Eq. (1.1) i.e., AyBy + Cy = y.
2
We also have Proposition 3.1. Let E be a Banach algebra and S be a nonempty closed convex subset of E. Let A, C : E → E and B : S → E be three operators such that (i) A and C are D-Lipschitzians with the D-functions φA and φC respectively, (ii) A is regular on E, i.e., A maps E into the set of all invertible elements of E, (iii) B is a bounded function with bound M. Then ( I −C A )
−1
exists on B(S) as soon as MφA (r) + φC (r) < r, for r > 0.
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Proof. Let y be fixed in S and define the mapping
ϕy : E → E, x → ϕy (x) = AxBy + Cx.
Let x1 , x2 ∈ E, the use of the assumption (i) leads to ϕy (x1 ) − ϕy (x2 ) Ax1 By − Ax2 By + Cx1 − Cx2 Ax1 − Ax2 By + Cx1 − Cx2 MφA x1 − x2 + φC x1 − x2 . Now, an application of a fixed point theorem of Boyd and Wong [7] yields that there is a unique element xy ∈ E such that ϕy (xy ) = xy . Hence, xy verifies Eq. (3.2) and so, by virtue of the hypothesis (ii), xy verifies Eq. (3.1). Therefore, the mapping ( I −C A ) is deduced. 2
−1
is well defined on B(S) and ( I −C A )
−1
By = xy and the desired result
In what follows, we will combine Theorem 3.1 and Proposition 3.1 to obtain the following fixed point theorems in Banach algebras. Theorem 3.2. Let E be a Banach algebra and S be a nonempty closed convex subset of E. Let A, C : E → E and B : S → E be three operators such that (i) (ii) (iii) (iv) (v) (vi)
A and C are D-Lipschitzians with the D-functions φA and φC respectively, A is regular on E , B is strongly continuous, B(S) is bounded with bound M, −1 ( I −C is weakly compact on B(S), A ) x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S.
Then Eq. (1.1) has at least one solution in S as soon as MφA (r) + φC (r) < r, for all r > 0. Remark 3.2. Here, we suppose B is strongly continuous and B(S) is bounded but not totally bounded. Thus, Theorem 1.1 in [12] follows as a consequence of Theorem 3.2. −1
Proof of Theorem 3.2. From Proposition 3.1, it follows that ( I −C A ) of assumption (vi), we obtain
I −C A
−1
B(S) ⊂ S.
exists on B(S). By virtue
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Moreover, the use of hypotheses (iv) and (v) leads that ( I −C A )
−1
B(S) is relatively weakly com-
−1 pact. Now, we show that ( I −C A ) B is sequentially weakly continuous. To see this, any sequence in S such that un u in S. By virtue of assumption (iii), we have
let {un } be
Bun → Bu. −1
Since ( I −C A )
is a continuous mapping on B(S) (see [14, Theorem 2.4]), we deduce that
I −C A
−1
Bun →
I −C A
−1 Bu.
−1
This shows that ( I −C A ) B is sequentially weakly continuous. Finally, an application of Theorem 3.1 yields that Eq. (1.1) has a solution in S. 2 Theorem 3.3. Let S be a nonempty closed convex subset of a Banach algebra E. Let A, C : E → E and B : S → E be three operators such that (i) (ii) (iii) (iv) (v)
A and C are D-Lipschitzians with the D-functions φA and φC respectively, B is sequentially weakly continuous and B(S) is relatively weakly compact, A is regular on E, −1 ( I −C is sequentially weakly continuous on B(S), A ) x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S.
Then Eq. (1.1) has at least one solution in S as soon as MφA (r) + φC (r) < r, for all r > 0. Remark 3.3. Recently, some fixed point theorems involving three operators in Banach algebras were established for completely continuous maps. Because every totally bounded subset of X is relatively weakly compact, Theorem 2.1 in [12] follows as a sequence of Theorem 3.3. Further, in general, the continuity condition is not easy to verify. In Theorem 3.3, the continuity is not required. Proof of Theorem 3.3. Similarly to the proof of preceding Theorem 3.2, we obtain ( I −C A ) exists on B(S) and
−1
Since ( I −C A )
I −C A
−1
−1
B(S) ⊂ S.
and B are sequentially weakly continuous, so, by composition we have
−1 ( I −C A ) B
−1
is sequentially weakly continuous. Finally, we claim that ( I −C A ) weakly compact. To see this, let {un } be any sequence in S and let vn =
I −C A
−1 Bun .
B(S) is relatively
A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
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Since B(S) is relatively weakly compact, there is a renamed subsequence {Bun } weakly converging to an element w. This fact, together with hypothesis (iv) gives that vn = We infer that ( I −C A )
−1
I −C A
−1
Bun
I −C A
−1 w.
B(S) is sequentially relatively weakly compact. An application of the −1
Eberlein–Šmulian theorem [10] yields that ( I −C A ) B(S) is relatively weakly compact, which leads our claim. The result is concluded immediately from Theorem 3.1. 2 Because the product of two sequentially weakly continuous functions is not necessarily sequentially weakly continuous, we will introduce: Definition 3.1. We will say that the Banach algebra E satisfies condition (P) if
(P)
For any sequences {xn } and {yn } in E such that xn x and yn y, then xn yn xy; here denotes weak convergence.
Note that, every finite dimensional Banach algebra satisfies condition (P). Even, if X satisfies condition (P) then C(K, X) is also Banach algebra satisfying condition (P), where K is a compact Hausdorff space. The proof is based on Dobrokov’s theorem: Theorem 3.4. (See [15, Dobrakov, p. 36].) Let K be a compact Hausdorff space and X be a Banach space. Let (fn )n be a bounded sequence in C(K, X), and f ∈ C(K, X). Then (fn )n is weakly convergent to f if and only if (fn (t))n is weakly convergent to f (t) for each t ∈ K. Theorem 3.5. Let E be a Banach algebra satisfying condition (P). Let S be a nonempty closed convex subset of E. Let A, C : E → E and B : S → E be three operators such that (i) (ii) (iii) (iv) (v) (vi)
A and C are D-Lipschitzians with the D-functions φA and φC respectively, A is regular on E , A, B and C are sequentially weakly continuous on S, B(S) is bounded with bound M, −1 ( I −C is weakly compact on B(S), A ) x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S.
Then Eq. (1.1) has at least one solution in S as soon as MφA (r) + φC (r) < r, for all r > 0. Proof. Similarly to the proof of Theorem 3.2, we obtain ( I −C A )
I −C A
−1
B(S) ⊂ S
−1
exists on B(S),
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and ( I −C A )
−1
B(S) is relatively weakly compact. In view of Theorem 3.1, it suffices to establish
−1 ( I −C A ) B
that is sequentially weakly continuous. To see this, let {un } be a weakly convergent sequence of S to a point u in S. Now, define the sequence {vn } of the subset S by vn = −1
Since ( I −C A )
I −C A
−1 Bun .
B(S) is relatively weakly compact, so, there is a renamed subsequence such that vn =
I −C A
−1 Bun v.
But, on the other hand, the subsequence {vn } verifies vn − Cvn = Avn Bun . Therefore, from assumption (iii) and in view of condition (P), we deduce that v verifies the following equation v − Cv = AvBu, or, equivalently v=
I −C A
−1 Bu.
Next we claim that the whole sequence {un } verifies
I −C A
−1
Bun = vn v.
Indeed, suppose that this is not the case, so, there is V w a weakly neighborhood of v satisfying for all n ∈ N, there exists an N n such that vN ∈ / V w . Hence, there is a renamed subsequence {vn } verifying the property for all n ∈ N,
vn ∈ / V w.
(3.3)
However for all n ∈ N,
vn ∈
I −C A
Again, there is a renamed subsequence such that vn v .
−1 B(S).
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According to the preceding, we have v =
I −C A
−1 Bu,
and, consequently v = v, which is a contradiction with the property (3.3). This yields that ( I −C A ) weakly continuous. 2
−1
B is sequentially
An interesting corollary of Theorem 3.5 is Corollary 3.1. Let E be a Banach algebra satisfying condition (P) and let S be a nonempty closed convex subset of E. Let A, C : E → E and B : S → E be three operators such that (i) (ii) (iii) (vi) (v)
A and C are D-Lipschitzians with the D-functions φA and φC respectively, A is regular on E , A, B and C are sequentially weakly continuous on S, A(S), B(S) and C(S) are relatively weakly compacts, x = AxBy + Cx ⇒ x ∈ S, for all y ∈ S.
Then Eq. (1.1) has at least one solution in S as soon as MφA (r) + φC (r) < r, for all r > 0. −1
Proof. In view of Theorem 3.5, it is enough to prove that ( I −C A ) compact. To do this, let {un } be any sequence in S and let vn =
I −C A
B(S) is relatively weakly
−1 Bun .
(3.4)
Since B(S) is relatively weakly compact, there is a renamed subsequence {Bun } weakly converging to an element w. On the other hand, by Eq. (3.4), we obtain vn = Avn Bun + Cvn .
(3.5)
Since {vn } is a sequence in S, so, by assumption (iv), there is a renamed subsequence such that Avn x and Cvn y. Hence, in view of condition (P) and the last equation (3.5), we obtain vn xw + y. −1
This shows that ( I −C A )
B(S) is sequentially relatively weakly compact. An application result of −1
the Eberlein–Šmulian theorem [10] yields that ( I −C A )
B(S) is relatively weakly compact.
2
Now, we shall discuss briefly on the existence of positive solutions. Let E1 and E2 be two Banach algebras, with positive closed cones E1+ and E2+ , respectively. An operator G from E1 into E2 is said to be positive if it carries the positive cone E1+ into E2+ (i.e., G(E1+ ) ⊂ E2+ ).
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Theorem 3.6. Let E be a Banach algebra satisfying condition (P) and S be a nonempty closed convex subset of E such that S + = S ∩ E + = ∅. Let A, C : E → E and B : S → E be three operators such that (i) (ii) (iii) (iv) (v)
A and C are D-Lipschitzians with the D-functions φA and φC respectively, A is regular on E, A, B and C are sequentially weakly continuous on S + , A(S + ), B(S + ) and C(S + ) are relatively weakly compacts, x = AxBy + Cx ⇒ x ∈ S + , for all y ∈ S + .
Then Eq. (1.1) has at least one solution in S + as soon as M + φA (r) + φC (r) < r, for all r > 0, where M + = B(S + ). Proof. Obviously S + = S ∩ E + is a closed convex subset of E. From Proposition 3.1, it follows −1 that ( I −C exists on B(S + ). By virtue of assumption (v), we have A )
I −C A
−1
B(S + ) ⊂ S + .
Then we can define the mapping: ⎧ + + ⎨ N : S → S , I − C −1 ⎩y → Ny = By. A Now, an application of Corollary 3.1 yields that N has a fixed point in S + . As a result, by the definition of N , Eq. (1.1) has a solution in S + . 2 4. Functional integral equations In this section we illustrate the applicability of our Corollary 3.1 and Theorem 3.3 by considering the following examples of nonlinear functional integral equations. 4.1. Example Let (X, .) be a Banach algebra satisfying condition (P). Let J = [0, 1] the closed and bounded interval in R, the set of all real numbers. Let E = C(J, X) the Banach algebra of all continuous functions from [0, 1] to X, endowed with the sup-norm ∞ , defined by f ∞ = sup{f (t); t ∈ [0, 1]}, for each f ∈ C(J, X). We consider the nonlinear functional integral equation (in short, FIE): x(t) = a(t) + (T1 x)(t)
σ (t) q(t) + p t, s, x(s), x(λs) ds .u ,
0 < λ < 1,
(4.1)
0
for all t ∈ J , where u = 0 is a fixed vector of X and the functions a, q, σ, p, T1 are given, while x = x(t) is an unknown function.
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We shall obtain the solution of FIE (4.1) under some suitable conditions. Suppose that the functions involved in Eq. (4.1) verify the following conditions: a : J → X is a continuous function. σ : J → J is a continuous and nondecreasing function. q : J → R is a continuous function. The operator T1 : C(J, X) → C(J, X) is such that (a) T1 is Lipschitzian with a Lipschitzian constant α, (b) T1 is regular on C(J, X), (c) T1 is sequentially weakly continuous on C(J, X), (d) T1 is weakly compact. (H5 ) The function p : J × J × X × X → R is continuous such that for arbitrary fixed s ∈ J and x, y ∈ X, the partial function t → p(t, s, x, y) is continuous uniformly for (s, x, y) ∈ J × X × X. (H6 ) There exists r0 > 0 such that (a) |p(t, s, x, y)| r0 −q∞ for each t, s ∈ J ; x, y ∈ X such that x r0 and y r0 , 1 ∞ (b) T1 x∞ (1 − a r0 ) u for each x ∈ C(J, X), (c) αr0 u < 1.
(H1 ) (H2 ) (H3 ) (H4 )
Theorem 4.1. Under assumptions (H1 )–(H6 ), Eq. (4.1) has at least one solution x = x(t) which belongs to the space C(J, X). Remark 4.1. When X is infinite dimensional, the subset Ar0 = {x ∈ X; x r0 } is not compact. So, the restriction p on J × J × Ar0 × Ar0 is not uniformly continuous. Thus, we note that the operator B in Eq. (4.1) is not necessarily continuous on S. Proof of Theorem 4.1. First, we begin by showing that C(J, X) verifies condition (P). To see this, let {xn }, {yn } any sequences in C(J, X) such that xn x and yn y. So, for each t ∈ J , we have xn (t) x(t) and yn (t) y(t) (cf. Theorem 3.4). Since X verify condition (P), then xn (t)yn (t) x(t)y(t), because (xn yn )n is a bounded sequence [18], this, further, implies that xn yn xy
(cf. Theorem 3.4),
which shows that the space C(J, X) verifies condition (P). Let us define the subset S of C(J, X) by S := y ∈ C(J, X), y∞ r0 = Br0 . Obviously S is nonempty, convex and closed. Let us consider three operators A, B and C defined on C(J, X) by (Ax)(t) = (T1 x)(t),
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σ (t) (Bx)(t) = q(t) + p t, s, x(s), x(λs) ds .u,
0 < λ < 1,
0
(Cx)(t) = a(t). We shall prove that the operators A, B and C satisfy all the conditions of Corollary 3.1. (i) From assumption (H4 )(a), it follows that A is Lipschitzian with a Lipschitzian constant α. Clearly C is Lipschitzian with a Lipschitzian constant 0. (ii) From assumption (H4 )(b), it follows that A is regular on C(J, X). (iii) Since C is constant, so, C is sequentially weakly continuous on S. From assumption (H4 )(c), A is sequentially weakly continuous on S. Now, we show that B is sequentially weakly continuous on S. Firstly, we verify that if x ∈ S, then Bx ∈ C(J, X). Let {tn } be any sequence in J converging to a point t in J . Then σ(tn ) σ (t) (Bx)(tn ) − (Bx)(t) = p tn , s, x(s), x(λs) ds − p t, s, x(s), x(λs) ds .u 0
0
σ(tn ) p tn , s, x(s), x(λs) − p t, s, x(s), x(λs) ds u 0
σ(tn ) p t, s, x(s), x(λs) ds u + σ (t)
1
p tn , s, x(s), x(λs) − p(t, s, x(s), x(λs) ds u
0
+ r0 − q∞ σ (tn ) − σ (t)u. Since tn → t, so, (tn , s, x(s), x(λs)) → (t, s, x(s), x(λs)), for all s ∈ J . Taking into account the hypothesis (H5 ), we obtain p tn , s, x(s), x(λs) → p t, s, x(s), x(λs)
in R.
Moreover, the use of assumption (H6 ) leads to p tn , s, x(s), x(λs) − p t, s, x(s), x(λs) 2 r0 − q∞ for all t, s ∈ J , λ ∈ (0, 1). Consider
ϕ : J → R, s → ϕ(s) = 2 r0 − q∞ .
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Clearly ϕ ∈ L1 (J ). Therefore, from the dominated convergence theorem and assumption (H2 ), we obtain (Bx)(tn ) → (Bx)(t)
in X.
It follows that Bx ∈ C(J, X). Next, we prove B is sequentially weakly continuous on S. Let {xn } be any sequence in S weakly converging to a point x in S. So, from assumptions (H5 )–(H6 ) and the dominated convergence theorem, we get 1 lim
n→∞
p t, s, xn (s), xn (λs) ds =
0
1
p t, s, x(s), x(λs) ,
0
which implies lim
n→∞
1
q(t) +
1 p t, s, xn (s), xn (λs) ds .u = q(t) + p t, s, x(s), x(λs) ds .u.
0
0
Hence, (Bxn )(t) → (Bx)(t)
in X.
Since (Bxn )n is bounded by r0 u, then Bxn Bx
(cf. Theorem 3.4).
We conclude that B is sequentially weakly continuous on S. (iv) We will prove that A(S), B(S) and C(S) are relatively weakly compact. Since S is bounded by r0 and taking into account the hypothesis (H4 )(d), it follows that A(S) is relatively weakly compact. Now, we show B(S) is relatively weakly compact. (Step I) By definition, B(S) := B(x), x∞ r0 . For all t ∈ J , we have B(S)(t) = (Bx)(t), x∞ r0 . We claim that B(S)(t) is sequentially weakly relatively compact in X. To see this, let {xn } be any 1 sequence in S, we have (Bxn )(t) = rn (t).u, where rn (t) = q(t) + 0 p(t, s, xn (s), xn (λs)) ds. Since |rn (t)| r0 and (rn (t)) is a real sequence, so, there is a renamed subsequence such that rn (t) → r(t)
in R,
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which implies rn (t).u → r(t).u
in X,
and, consequently (Bxn )(t) → q(t) + r(t) .u
in X.
We conclude that B(S)(t) is sequentially relatively compact in X, then B(S)(t) is sequentially relatively weakly compact in X. (Step II) We prove that B(S) is weakly equicontinuous on J . If we take > 0; x ∈ S; x ∗ ∈ X ∗ ; t, t ∈ J such that t t and t − t . Then σ (t) σ(t ) ∗ ∗ x (Bx)(t) − (Bx) t = p t, s, x(s), x(λs) ds − p t , s, x(s), x(λs) ds x (u) 0
0
σ (t) p t, s, x(s), x(λs) − p t , s, x(s), x(λs) ds x ∗ (u) 0
σ(t ) + p t , s, x(s), x(λs) ds x ∗ (u) σ (t)
w(p, ) + r0 − q∞ w(σ, ) x ∗ (u), where w(p, ) = sup p(t, s, x, y) − p t , s, x, y : t, t , s ∈ J ; t − t ; x, y ∈ Br0 , w(σ, ) = sup σ (t) − σ t : t, t ∈ J ; t − t . Taking into account the hypothesis (H5 ) and in view of the uniform continuity of the function σ on the set J , it follows that w(p, ) → 0 and w(σ, ) → 0 as → 0. An application of the Arzelà–Ascoli theorem [17], we conclude that B(S) is sequentially weakly relatively compact in X. Again an application result of Eberlein–Šmulian theorem [10] yields that B(S) is relatively weakly compact. As C(S) = {a}, hence C(S) is relatively weakly compact. (v) Finally, it remains to prove the hypothesis (v) of Corollary 3.1. To see this, let x ∈ C(J, X) and y ∈ S such that x = AxBy + Cx, or, equivalently for all t ∈ J , x(t) = a(t) + (T1 x)(t)(By)(t).
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But, for all t ∈ J , we have x(t) x(t) − a(t) + a(t). Then x(t) (T1 x)(t)r0 u + a∞ a∞ r0 + a∞ 1− r0 = r0 . From the last inequality and taking the supremum over t, we obtain x∞ r0 , and, consequently x ∈ S. We conclude that the operators A, B and C satisfy all the requirements of Corollary 3.1. Thus, an application of it yields that the FIE (4.1) has a solution in the space C(J, X). 2 4.2. Example To illustrate Theorem 3.3, We consider the nonlinear functional integral equation (in short, FIE) in C(J, X). x(t) = a(t)x(t) + (T2 x)(t)
σ (t) q(t) + p t, s, x(s), x(λs) ds .u ,
0 < λ < 1, (4.2)
0
for all t ∈ J , where u = 0 is a fixed vector of X and the functions a, q, σ , p, T2 are given, while x in C(J, X) is an unknown function. We shall obtain the solution of (FIE (4.2)) under some suitable conditions on the functions involved in (4.2). Suppose that the functions a, q, σ , p and the operator T2 verify the following conditions: (H1 ) (H2 ) (H3 ) (H4 )
a : J → X is a continuous function with a∞ < 1. σ : J → J is a continuous and nondecreasing function. q : J → R is a continuous function. The operator T2 : C(J, X) → C(J, X) is such that (a) T2 is Lipschitzian with a Lipschitzian constant α, (b) T2 is regular on C(J, X), −1 (c) ( TI2 ) is well defined on C(J, X), −1
(d) ( TI2 ) is sequentially weakly continuous on C(J, X). (H5 ) The function p : J × J × X × X → R is continuous such that for arbitrary fixed s ∈ J and x, y ∈ X, the partial function t → p(t, s, x, y) is continuous uniformly for (s, x, y) ∈ J × X × X. (H6 ) There exists r0 > 0 such that
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(a) |p(t, s, x, y)| r0 −q∞ for each t, s ∈ J ; x, y ∈ X such that x r0 and y r0 , 1 ∞ (b) T2 x∞ (1 − a r0 ) u for each x ∈ C(J, X), (c) αr0 u < 1. Theorem 4.2. Under assumptions (H1 )–(H6 ), Eq. (4.2) has at least one solution x = x(t) which belongs to the space C(J, X). Remark 4.2. Note that the operator C in Eq. (4.2) does not satisfy condition (iv) of Corollary 3.1. In fact, if we take X = R and a ≡ 12 , then (Cx)(t) = 12 x(t). Thus 1 C(S) = x x∞ r0 , 2
= B r0 . 2
Because C(J, R) is with infinite dimensional, C(S) is not relatively compact. Furthermore, R is with finite dimensional, so, C(S) is not relatively weakly compact [17]. Proof of Theorem 4.2. Let us consider three operators A, B and C defined on C(J, X) by (Ax)(t) = (T2 x)(t), σ (t)
p t, s, x(s), x(λs) ds .u,
(Bx)(t) = q(t) +
0 < λ < 1,
0
(Cx)(t) = a(t)x(t). We shall prove that the operators A, B and C satisfy all the conditions of Theorem 3.3. (i) From assumption (H4 )(a), A is Lipschitzian with a Lipschitzian constant α. Next, we show that C is Lipschitzian on C(J, X). To see this, fix arbitrarily x, y ∈ C(J, X). Then, if we take an arbitrary t ∈ J , we get (Cx)(t) − (Cy)(t) = a(t)x(t) − a(t)y(t) a∞ x(t) − y(t). From the last inequality and taking the supremum over t, we obtain Cx − Cy∞ a∞ x − y∞ . This proves that C is Lipschitzian with a Lipschitzian constant a∞ . (ii) Arguing as in the proof of Theorem 4.1, we obtain B is sequentially weakly continuous on S and B(S) is relatively weakly compact. (iii) From assumption (H4 )(b), A is regular on C(J, X). −1 (iv) We show that ( I −C is sequentially weakly continuous on B(S). To see this, let x, y ∈ A ) C(J, X) such that I −C (x) = y, A
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or, equivalently (1 − a)x = y. T2 x Since a∞ < 1, so, (1 − a)−1 exists on C(J, X), then
I (x) = (1 − a)−1 y. T2
This implies, from assumption (H3 )(c), that x=
I T2
−1
(1 − a)−1 y .
Thus
I −C A
−1
(x) =
I T2
−1
(1 − a)−1 x
for all x ∈ C(J, X). Now, let {xn } be a weakly convergent sequence of B(S) to a point x in B(S), then (1 − a)−1 xn (1 − a)−1 x, and so, it follows from assumption (H4 )(d) that
I T2
−1
(1 − a)−1 xn
I T2
−1
(1 − a)−1 x ,
we conclude that
I −C A
−1
(xn )
I −C A
−1 (x).
(v) Finally, a similar reasoning as, in the last point of Theorem 4.1, proves that the condition (v) of Theorem 3.3 is fulfilled. We conclude that the operators A, B and C satisfy all the requirements of Theorem 3.3. 2 Corollary 4.1. Let (X, ) be a Banach algebra satisfying condition (P), with positive closed cone X + . Suppose that the assumption (H1 )–(H6 ) hold. Also, assume that −1
u belongs to X + , a(J ) ⊂ X + , q(J ) ⊂ R+ , p(J × J × X + × X + ) ⊂ R+ and ( TI2 ) positive operator from the cone positive C(J, X + ) of C(J, X) into itself. Then Eq. (4.2) has at least one positive solution x in the cone C(J, X + ).
is a
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Proof. Let S + := x ∈ S, x(t) ∈ X + for all t ∈ J . Obviously S + is nonempty, closed and convex. Similarly to the proof of Theorem 4.2, we show that A and C are Lipschitzians with a Lipschitzian constant α and a∞ respectively. A is regular on C(J, X). A, B and C are sequentially weakly continuous on S + . Because S + is a subset of S, so, we have A(S + ), B(S + ) and C(S + ) are relatively weakly compacts. (v) Finally, we shall show that the hypothesis (v) of Theorem 3.6 is satisfied. In fact, fix an arbitrarily x ∈ C(J, X) and y ∈ S + such that
(i) (ii) (iii) (iv)
x = AxBy + Cx. Arguing as in the proof of Theorem 4.2, we get x ∈ S. Moreover, the last equation leads that for all t ∈ J,
x(t) = a(t)x(t) + (T2 x)(t)(By)(t),
thus, x(t)(1 − a(t)) = (By)(t). (T2 x)(t)
for all t ∈ J,
Since for all t ∈ J , a(t) < 1, so, (1 − a(t))−1 exists in X, and +∞ −1 1 − a(t) = a n (t). n=0
Since a(t) belongs to the closed positive cone X + , then (1 − a(t))−1 is positive. Also, we verify that for all t ∈ J , (By)(t) is a positive. Therefore, the map ψ defined on J by −1 ψ(t) = 1 − a(t)
σ (t) p t, s, x(s), x(λs) ds .u q(t) +
0
belongs to the positive cone C(J, X + ) of C(J, X). Then B maps C(J, X + ) into itself. Seeing that
σ (t) −1 I p t, s, x(s), x(λs) ds .u = ψ(t), q(t) + x (t) = 1 − a(t) T2 0
A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
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then x=
I T2
Thus, x ∈ C(J, X + ) and, consequently x ∈ S + .
−1 (ψ). 2
Next, we provide an example of the operator T2 presented in Theorem 4.2. R). Let E = C(J, R) = C(J ) denotes the Banach algebra of Example of the operator T2 in C(J,R all continuous real-valued functions on J with norm x∞ = supt∈J |x(t)|. Clearly C(J ) satisfies condition (P). Let b : J → R is continuous and nonnegative, and define ⎧ ⎨ T2 : C(J ) → C(J ), 1 . ⎩ x → T2 x = 1 + b|x| We obtain the following functional integral equation: σ (t) 1 x(t) = a(t)x(t) + p t, s, x(s), x(λs) ds , q(t) + 1 + b(t)|x(t)|
0 < λ < 1. (4.3)
0
We will prove all the conditions (a)–(d) of (H4 ) in Theorem 4.2 to Eq. (4.3): (a) Fix x, y ∈ C(J ). Then, for all t ∈ J , we have 1 1 (T2 x)(t) − (T2 y)(t) = 1 + b(t)|x(t)| − 1 + b(t)|y(t)| b(t)y(t)| − |x(t) (1 + b(t)|x(t)|)(1 + b(t)|y(t)|) b∞ x(t) − y(t).
=
Taking the supremum over t, we obtain T2 x − T2 y∞ b∞ x − y∞ . Which shows that T2 is Lipschitzian with a Lipschitzian constant b∞ . (b) Clearly T2 is regular on C(J ). −1 (c) We show that ( TI2 ) exists on C(J ). To see this, let x, y ∈ C(J ) such that
I x = y, T2
or, equivalently x 1 + b|x| = y,
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A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
which implies |x| 1 + b|x| = |y|, hence √ 2 b|x| + |x| = |y|. For each t0 ∈ J such that b(t0 ) = 0, we have x = y. Then for each t ∈ J such that b(t) > 0, we obtain 2 1 1 + y(t), b(t) x(t) + √ = 4b(t) 2 b(t) which further implies
1 + b(t)x(t) = − √ 2 b(t)
1 + y(t), 4b(t)
hence 1 b(t)x(t) = − + 2
1 + y(t)b(t), 4
and, consequently x(t) =
y(t) = 1 + b(t)|x(t)|
1 2
+
y(t) 1 4
.
+ b(t)|y(t)|
We remark that the equality is also verified for each t such that b(t) = 0. Consider F the function defined by the expression F : C(J ) → C(J ), x → F (x) = 1 x1 2+
4 +b|x|
.
It is easily to verify for all x ∈ C(J ) I I ◦ F (x) = F ◦ (x) = x. T2 T2 We conclude that
I T2
−1
(x) =
1 2
+
x 1 4
. + b|x| −1
(d) It is an easy exercise to show that T2 and ( TI2 ) on B(S).
is sequentially weakly continuous
A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
Remark 4.3. One can check easily that ( TI2 ) C(J, R+ ) of C(J, R) into itself.
−1
2235
is a positive operator from the positive cone
4.3. Applications to functional differential equations In this section, we prove the existence theorems for the nonlinear functional differential equation in Banach algebra by the applications of the abstract results of the previous section under generalized Lipschitz conditions. We consider the following nonlinear functional differential equation (in short, FDE) in C(J )
x T2 x
t
− q1 (t) =
∂p t, s, x(s), x(λs) ds + p t, t, x(t), x(λt) , ∂t
0
t ∈ J, 0 < λ < 1,
(4.4)
satisfying the initial condition x(0) = ζ ∈ R
(4.5)
where the functions q1 , p and the operator T2 are given with q1 (0) = 0, while x = x(t) is an unknown function. By a solution of the FDE (4.4)–(4.5), we mean an absolutely continuous function x : J → R that satisfies the two equations (4.4)–(4.5) on J . The existence result for the FDE (4.4)–(4.5) is: Theorem 4.3. We consider the following assumptions: (H1 ) The q1 : J → R is a continuous function. (H2 ) The operator T2 : C(J ) → C(J ) is such that (a) T2 is Lipschitzian with a Lipschitzian constant α, (b) T2 is regular on C(J ), −1 (c) ( TI2 ) is well defined on C(J ), −1
(d) ( TI2 ) is sequentially weakly continuous on C(J ), (e) For all x ∈ C(J ), we have (T2 x)∞ 1. (H3 ) The function p : J × J × X × X → R is continuous such that for arbitrary fixed s ∈ J and x, y ∈ R; the partial function t → p(t, s, x, y) is C 1 on J . (H4 ) There exists r0 > 0 such that (a) For all t, s ∈ J ; y, z ∈ [−r0 , r0 ] and x ∈ C(J ), we have p(t, s, y, z) r0 − q1 ∞ − (b) αr0 < 1. Then the FDE (4.4)–(4.5) has at least one solution in C(J ).
|ζ | . |(T2 x)(0)|
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A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
Proof. Note that the FDE (4.4)–(4.5) is equivalent to the integral functional equation:
ζ + x(t) = (T2 x)(t) q1 (t) + (T2 x)(0)
t
p t, s, x(s), x(λs) ds ,
t ∈ J, 0 < λ < 1.
(4.6)
0
Eq. (4.6) represents a particular case of Eq. (4.2) with for all t ∈ J ; σ (t) = t, a(t) = 0, u = 1 ζ . Therefore, we have for all t ∈ J ; (Ax)(t) = (T2 x)(t), (Bx)(t) = and q(t) = q1 (t) + (T2 x)(0) t q(t) + 0 p(t, s, x(s), x(λs)) ds and C(x)(t) = 0. Now, we shall prove that the operators A, B and C satisfy all the conditions of Theorem 3.3. Similarly to the proof of preceding Theorem 4.2, we obtain (i) A and C are Lipschitzians with a Lipschitzian constant α and 0 respectively. (ii) B is sequentially weakly continuous on S and B(S) is relatively weakly compact where S = Br0 := {x ∈ C(J ), x∞ r0 }. (iii) A is regular on C(J ). −1 −1 (iv) ( I −C = ( TI2 ) is sequentially weakly continuous on B(S). A ) It, thus, remains to prove (v) of Theorem 3.3. First, we show that M = B(S) r0 . To see this, fix an arbitrarily x ∈ S. Then, for t ∈ J , we get (Bx)(t) q1 (t) +
|ζ | + |(T2 x)(0)|
q1 (t) +
|ζ | + |(T2 x)(0)|
t
p t, s, x(s), x(λs) ds
0
1
p t, s, x(s), x(λs) ds
0
|ζ | |ζ | + r0 − q1 ∞ − |(T2 x)(0)| |(T2 x)(0)|
q1 ∞ + = r0 . Taking the supremum over t, we obtain
Bx∞ r0 . Thus M r0 , and, consequently αM + β = αM αr0 < 1. Next, fix an arbitrarily x ∈ C(J ) and y ∈ S such that x = AxBy + Cx,
A. Ben Amar et al. / Journal of Functional Analysis 259 (2010) 2215–2237
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or, equivalently for all t ∈ J,
x(t) = (T2 x)(t)(By)(t),
then x(t) T2 x∞ By∞ , and thus, in view of assumption (H2 )(e), yields that x(t) By∞ . Since y ∈ S, this further implies x(t) r0 , taking the supremum over t, we obtain x∞ r0 . As a result, x is in S. This proves (v). Now, applying Theorem 3.3, we see that Eq. (4.6) has at least one solution in C(J ). 2 References [1] O. Arino, S. Gautier, J.P. Penot, A fixed oint theorem for sequentially continuous mapping with application to ordinary differential equations, Funkcial. Ekvac. 27 (3) (1984) 273–279. [2] A. Ben Amar, A. Jeribi, M. Mnif, Some fixed point theorems and application to biological model, Numer. Funct. Anal. Optim. 29 (1–2) (2008) 1–23. [3] J. Banas, L. Lecko, Fixed points of the product of operators in Banach algebras, Panamer. Math. J. 12 (2) (2002) 101–109. [4] J. Banas, K. Sadarangani, Solutions of some functional-integral equations in Banach algebras, Math. Comput. Modelling 38 (3–4) (2003) 245–250. [5] J. Banas, B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 332 (2) (2007) 1371–1379. [6] J. Banas, L. Olszowy, On a class of measures of non-compactness in Banach algebras and their application to nonlinear integral equations, Z. Anal. Anwend. 28 (4) (2009) 475–498. [7] D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. [8] T. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1) (1998) 85–88. [9] J. Caballero, B. Lopez, K. Sadarangani, Existence of non-decreasing and continuous solutions of an integral equations with linear modification of the argument, Acta Math. Sin. (Engl. Ser.) 23 (9) (2007) 1719–1728. [10] J.B. Conway, A Course in Functional Analysis, Springer-Verlag, Berlin, 1990. [11] B.C. Dhage, On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 22 (5) (1988) 603–611. [12] B.C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J. 44 (1) (2004) 145–155. [13] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (3) (2005) 273– 280. [14] B.C. Dhage, On some nonlinear alternatives of Leray–Schauder type and functional integral equations, Arch. Math. (Brno) 42 (1) (2006) 11–23. [15] I. Dobrakov, On representation of linear operators on C0 (T , X), Czechoslovak Math. J. 21 (96) (1971) 13–30. [16] D.R. Smart, Fixed Points Theorems, Cambridge Univ. Press, 1980. [17] Io.I. Vrabie, Ai.I. Cuza, O. Mayer, C0 -Semigroups and Applications, Elsevier, New York, 2003. [18] C. Wagschal, Topologie et Analyse Fonctionnelle, Hermann, Paris, 2003.
Journal of Functional Analysis 259 (2010) 2238–2252 www.elsevier.com/locate/jfa
Turbulence and Araki–Woods factors Román Sasyk a,∗ , Asger Törnquist b a Instituto Argentino de Matemáticas – CONICET, Saavedra 15, Piso 3 (1083), Buenos Aires, Argentina b Kurt Gödel Research Center, University of Vienna, Währinger Strasse 25, 1090 Vienna, Austria
Received 8 February 2010; accepted 23 June 2010 Available online 17 July 2010 Communicated by S. Vaes
Abstract Using Baire category techniques we prove that Araki–Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures. © 2010 Elsevier Inc. All rights reserved. Keywords: Von Neumann algebras; Classification of factors; Descriptive set theory; Borel reducibility; Turbulence
1. Introduction The present paper continues a line of research into the structure of the isomorphism relation for separable von Neumann algebras using techniques from descriptive set theory, which was initiated in [21] and [20]. The central notion from descriptive set theory relevant to this paper is that of Borel reducibility. Recall that if E and F are equivalence relations on Polish spaces X and Y , respectively, we say that E is Borel reducible to F if there is a Borel function f : X → Y such that ∀x, x ∈ X xEx
⇐⇒
f (x)Ff x ,
* Corresponding author.
E-mail addresses:
[email protected] (R. Sasyk),
[email protected] (A. Törnquist). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.018
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and if this is the case we write E B F . Borel reducibility is a notion of relative complexity of equivalence relations and the isomorphism problems they pose, and the statement E B F is interpreted as saying that the points of X are classifiable up to E-equivalence by a Borel assignment of complete invariants that are F -equivalence classes. The requirement that f be Borel is a natural restriction to ensure that the invariants are assigned in a reasonably definable way. Without a definability condition on the function f reducibility would amount only to a consideration of the cardinality of the quotient spaces X/E and Y/F . In [21] it was shown that the isomorphism relation in all the natural classes of separable von Neumann factors, II1 , II∞ , IIIλ , (0 λ 1) do not admit a classification by countable structures. That is, if L is a countable language and Mod(L) is the natural Polish space of countable L-structures (see [10, §2.3]), then there is no Borel reduction of the isomorphism relation of von Neumann factors of any fixed type to the isomorphism relation Mod(L) in Mod(L). This in particular implies that there is no Borel assignment of countable groups, graphs, fields or orderings as complete invariants for the isomorphism problem for factors. Recently, Kerr, Li and Pichot in [16] obtained several non-classification results along the same lines exhibited here but for the automorphism groups of finite factors. For instance they showed that the conjugacy relation for the trace-preserving free weakly mixing actions of discrete groups on a II1 factor is not classifiable by countable structures. These types of results are much stronger than the classical smooth/non-smooth dichotomy, since they give specific information about the complexity of the kind of invariant that can be used in a complete classification. They are also stronger than the traditional smooth/non-smooth dichotomy for equivalence relation, since, for instance, isomorphism of countable groups is not smooth, yet in many cases countable groups are reasonable invariants. The earliest non-smoothness result for the isomorphism relation of von Neumann algebras is Woods’ Theorem [24], which asserts that the isomorphisms relation for ITPFI2 factors is not smooth. Recall that a von Neumann algebra M is called an Araki–Woods factor or an ITPFI factor (short for infinite tensor product of factors of type I) if it is of the form M=
∞
Mnk (C), φk
k=1
where Mnk (C) denotes the algebra of nk × nk matrices and the φk are faithful normal states. In the case when nk = 2 for all k, the factor M is called ITPFI2 . In this paper we will show: Theorem 1.1. The isomorphism relation for ITPFI2 factors is not classifiable by countable structures. This solves a problem posed in [20], and provides a strengthening and a new proof of Woods’ Theorem. It also provides a new and more direct proof that the isomorphism relation for injective type III0 factors is not classifiable by countable structures, a result proven in [21] using Krieger’s Theorem regarding the duality between flows and injective factors, [18]. Results of Krieger [17] and Connes and Woods [6] show that not all injective factors are ITPFI factors. Thus a natural question to ask was whether ITPFI factors are “simpler” objects to classify from the point of view of Borel reducibility. The results of this article show that even for this elementary class of von Neumann factors, the classification problem is too complicated to be distinguished by countable structures invariants, which might be surprising given the simplicity of their construction.
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The ITPFI factors constructed in the proof of Theorem 1.1 correspond to group-measure space factors constructed from the measure-class preserving odometer actions of Z on {0, 1}N , when {0, 1}N is equipped with a finite product measure. Therefore we obtain the following interesting corollary: Theorem 1.2. The odometer actions of Z on {0, 1}N preserving the measure class of a finite non-atomic ergodic product measure are not classifiable, up to orbit equivalence, by countable structures. This stands in contrast to Dye’s Theorem for probability measure preserving actions, and may be compared with the theorem of Ioana, Kechris, Tsankov and Epstein in [12] on the nonclassifiability up to orbit equivalence of probability measure preserving ergodic actions of a countable non-amenable group. 2. A turbulence lemma In this section we establish a general lemma which shows that a wide class of natural actions are turbulent, in the sense of [10]. Recall that if G is a Polish group acting continuously on a Polish space X, then the action is said to be turbulent if the following holds1 : For all x, y ∈ X, all open U ⊆ X with x ∈ U and all open V ⊆ G containing the identity, there is y0 ∈ U in the G-orbit of y, such that for all neighbourhoods U0 of y0 there is a finite sequence xi ∈ U (0 i n) with x0 = x and a sequence gi ∈ V (0 i < n), such that xi+1 = gi · xi and xn ∈ U0 . Recall moreover that a Fréchet space is a completely metrizable locally convex vector space (over R or C). Lemma 2.1. Let F be a separable Fréchet space and let G ⊆ F be a dense subgroup of the additive group (F, +). Suppose (G, +) has a Polish group topology such that the inclusion map i : G → F is continuous, and satisfies (∗) for all g ∈ G and open V ⊆ G with 0 ∈ V there is n ∈ N such that n1 g ∈ V (e.g. when G itself is a Fréchet space). Then either G = F or the action of G on F by addition, g · x = g + x, is turbulent and has meagre dense classes. In particular, if (G, · G ) and (F, · F ) are separable Banach spaces such that G is a dense subspace of F and the inclusion map i : G → F is bounded, then either G = F or the action of (G, +) on F by addition is turbulent and has meagre dense classes. 1 Strictly speaking, Hjorth required a turbulent action to have dense, meagre orbits, but we keep those requirements separate.
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Proof. Note that G is an analytic subset of F since i : G → F is continuous, and so it has the Baire Property in F (see [13, 8.21 and 21.6]). So if G = F then G must be meagre in F , since otherwise by Pettis’ Theorem [13, 9.9] G must contain a neighbourhood of the identity, and so G = F . Since G is dense in F it follows that all the G-orbits are meagre and dense. So it suffices to show that the action of G is turbulent. For this, let x ∈ F and let U ⊆ F be a convex open neighbourhood of F . Let y ∈ U , let U0 be an open neighbourhood of y such that y ∈ U0 ⊆ U , and let V ⊆ G be a neighbourhood of 0 in G. Since G is dense in F we may find g ∈ G such that x + g ∈ U0 . By assumption there is n ∈ N such that gn ∈ V , and since U is convex we have 2 n−1 1 g, x + g ∈ U, x, x + g, x + g, . . . , x + n n n which shows that G acts turbulently.
2
Remark 2.2. A version of Lemma 2.1 was already noted by Kechris in [14]. Many turbulence results found in the literature are special instances of the above lemma. For instance, let (c0 , · ∞ ) denote the real Banach space of real valued sequences that converge to zero, equipped with the sup-norm. The elementary example [10, 3.23] that c0 acts turbulently on RN by addition fits into this framework. Moreover, condition (∗) may be replaced by the weaker condition (∗∗) for all g ∈ G and W, V ⊆ G open neighbourhoods such that g ∈ W and 0 ∈ V , there is z ∈ V and n ∈ N such that nz ∈ W . in which case [10, Proposition 3.25] also follows from the above. The results in [22] also fall into this category. Indeed, let X be a locally compact not compact Polish space. The space C(X, R) = {f : X → R continuous} is a separable Fréchet space with the topology given by uniform convergence in compact sets. C0 (X, R) = {f ∈ C(X, R): limx→∞ f (x) = 0} is a dense subspace of C(X, R) and it is Polish in the topology given by uniform convergence. It follows from the previous lemma that the natural action of C0 (X, R) on C(X, R) is turbulent, has meagre classes and every class is dense. The exponential map then gives [22, Theorem 1.1]. In a similar fashion one could also recover [22, Theorem 1.2]. S ∞ -ergodicity and turbulence. Let S∞ denote the group of all permutations of N. The importance of the notion of turbulence comes from its relation to the notion of S∞ -ergodicity. Recall that an equivalence relation E on a Polish space X is said to be generically S∞ -ergodic if whenever S∞ acts continuously on a Polish space Y giving rise to the orbit equivalence relation ESY∞ , and f : X → Y is a Baire measurable function such that ∀x, x xEx
⇒
f (x)ESY∞ f x
(i.e. f is a homomorphism of equivalence relations), then there is a single S∞ -orbit [y]E Y such S∞ that x ∈ X: f (x) ∈ [y]E Y S∞
is comeagre. Note that if E ⊆ E , where E is also an equivalence relation, and E is generically S∞ -ergodic, then so is E .
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The fundamental theorem in Hjorth’s theory of turbulence is that if a Polish group G acts continuously and turbulently on a Polish space X, then the associated orbit equivalence relation X is generically S -ergodic, see [10, Theorem 3.18]. Since the isomorphism relation Mod(L) EG ∞ in the Polish space Mod(L) of countable L-structures is induced by a continuous S∞ -action, X to Mod(L) if the G-orbits are turbulence provides an obstruction to Borel reducibility of EG meagre. The isomorphism relations for countable groups, graphs, fields, orderings, etc., are special instances of Mod(L) for appropriate choices of the language L, and so turbulence can be used to prove the impossibility of obtaining a complete classification by a reasonable (i.e. Borel or Baire measurable) assignment of such countable objects as invariants. 3. Proof of the main theorem We will now define a family (Mx )x∈c0 of ITPFI factors parameterized by elements of c0 . The family is chosen with great care so that it will be possible to prove for each x ∈ c0 , the set y ∈ c0 : T (My ) = T (Mx ) is meagre, and so in particular, isomorphism of the Mx is meagre in the parameter x ∈ c0 . The motivation behind the definition can be traced back to the results in [9]. For j ∈ N define Nj = 2j ! , and for each x ∈ c0 let ljx = ln(2)j !ex(j )/j ! . Let φjx be the state on M2 (C) given by φjx (a) =
1 1 + e−lj
x
1 0 x Tr a · 0 e−lj
.
Then we define Mx to be the ITPFI2 factor Mx =
∞ ⊗Nj M2 (C), φjx . j =1
In other words, Mx is the ITPFI2 factor with eigenvalue list (λxn , 1 − λxn )n∈N where λxn is given by λxn =
1 1 + e−lj
x
x j −1 j whenever i=1 Ni < n i=1 Ni for some j ∈ N. Since ljx → ∞ and j Nj e−lj = ∞, all the factors Mx are of type III, [3, III.4.6.6].
Theorem 1.1 will be proved by showing that the family of factors (Mx )x∈c0 is not classifiable up to isomorphism by countable structures. An outline of the proof is as follows: First we will show that the equivalence relation x ∼iso x
⇐⇒
Mx is isomorphic to Mx
R. Sasyk, A. Törnquist / Journal of Functional Analysis 259 (2010) 2238–2252
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has meagre classes, thus showing that the family (Mx ) contains uncountably many nonisomorphic factors. Then we will show that there is a subgroup G ⊆ c0 of the additive group (c0 , +) that satisfies the hypothesis of Lemma 2.1, and with the additional property that Mg+x Mx for all g ∈ G and x ∈ c0 , where denotes isomorphism of von Neumann algebras. From this fact it will be easy to deduce that the equivalence relation ∼iso is not classifiable by countable structures. Finally we will show that the map x → Mx is Borel (in a precise way) and thus provides a Borel reduction of ∼iso to . The main tool used to distinguish uncountably many non-isomorphic elements of the family (Mx )x∈c0 is Connes’ invariant T (M). Recall that if M is a von Neumann algebra with a faithful semifinite normal weight ϕ, the Tomita–Takesaki theory associates to it a one parameter group ϕ of automorphisms of M, the so-called modular automorphism group. If σt denotes the modular automorphism group of (M, ϕ), the T -set of M is the additive subgroup of R defined by ϕ T (M) = t ∈ R: σt is an inner automorphism . ϕ
Even though σt depends on ϕ, Connes’ non-commutative Radon–Nikodym Theorem guarantees that T (M) is independent of the choice of the faithful semifinite normal weight ϕ. The T -set is arguably the most important invariant employed to distinguish injective type III0 factors and it can be found already in Araki and Woods’s seminal article [1]. A thorough treatment of these important concepts that are at the heart of the structural theory of factors of type III can be found in [5, 5.3–5.5], [3, III.3, III.4] and [23]. For the purpose of this article we will only need the following lemma. Lemma 3.1. (See [4, Corollaire 1.3.9].) If M is an ITPFI2 factor with eigenvalue list (λn , 1 − λn ) then the T -set is given by the formula ∞
1+it 1+it <∞ . 1 − λn + (1 − λn ) T (M) = t ∈ R:
n=1
The following slightly abusive notation is convenient in this paper: For a real s ∈ R, write s (mod 2π) for the unique element of {s + 2πp: p ∈ Z} ∩ (−π, π]. For x ∈ c0 and t ∈ R define δjx (t) = tljx
(mod 2π).
When the value of t is clear from the context we will usually write δjx for δjx (t). The next lemma is stated only for the family (Mx )x∈c0 , but is a special case of a well-known consequence of Lemma 3.1 which has been observed in many places in the literature (see e.g. [9] and [1]). We include its proof for the sake of completeness.
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Lemma 3.2. For each t ∈ R, t ∈ T (Mx ) iff ∞
2 x Nj e−lj δjx (t) < ∞.
j =1
Proof. By Lemma 3.1, t ∈ T (Mx )
∞
⇐⇒
Nj
1 −
1+it
1
+
1 + e−lj
x
1+it
<∞ −ljx
e−lj
x
1+e
∞
1 −ljx −iljx t <∞ 1 + e Nj 1 − e x 1 + e−lj j =1
∞
1 −ljx −iδjx (t) <∞ Nj 1 − e x 1+e 1 + e−lj j =1 j =1
⇐⇒
⇐⇒
∞
⇐⇒
x Nj e−lj 1 − cos δjx (t) < ∞
j =1 ∞
⇐⇒
Nj e−lj
x
x 2 4 δj (t) + O δjx (t) <∞
j =1 ∞
⇐⇒
2 x Nj e−lj δjx (t) < ∞.
2
j =1
Remark 3.3. The previous lemma sheds light on the motivation behind the definition of the x x(j )/j ! ) family (Mx )x∈c0 . Indeed, since Nj e−lj = 2j !(1−e goes to 1 when j → ∞, then to control ∞ −ljx x 2 the sum j =1 Nj e (δj (t)) , and thus the T -set, it will be enough to control the size of δjx (t). This fact is what we will exploit in the next two lemmas. Lemma 3.4. For each x ∈ c0 , T (Mx ) = {0}. Proof. Define ∞
t=
1 a(j ) ln 2 j !ex(j )/j ! j =1
where a(j ) ∈ (0, 3π] is defined recursively by letting a(1) = 1 and in general for j > 1, a(j ) = −
j −1
k=1
Then 0 < t < ∞ and we have
a(k) j !ex(j )/j ! x(k)/k! k!e
(mod 2π) + 2π.
R. Sasyk, A. Törnquist / Journal of Functional Analysis 259 (2010) 2238–2252
ljx t =
∞
j !ex(j )/j !
a(k) k!ex(k)/k!
j !ex(j )/j !
∞
a(k) a(k) + a(j ) + j !ex(j )/j ! x(k)/k! k!ex(k)/k! k!e
k=1
=
j −1
2245
k=j +1
k=1
and so ∞
δjx (t) =
j !ex(j )/j !
k=j +1
a(k) k!ex(k)/k!
(mod 2π).
If j is large enough so that for all k j we have 1/2 ex(k)/k! 2 then ∞
0
j !ex(j )/j !
k=j +1
∞
a(k) j! 12π k! k!ex(k)/k! k=j +1
12π
∞
k=1
=
1 (j + 1)k
12π . j
Hence for j sufficiently large it holds that δjx (t) =
∞
j !ex(j )/j !
k=j +1
and so by (†) and Lemma 3.2 we have t ∈ T (Mx ).
a(k) 1 ∼ k!ex(k)/k! j 2
Lemma 3.5. For each t ∈ R \ {0} the set {x ∈ c0 : t ∈ / T (Mx )} is a dense Gδ subset of c0 . Proof. Since T (Mx ) is a subgroup of (R, +), we may assume that t > 0. For each K ∈ N let AK = x ∈ c0 : (∃L ∈ N)
L
2 x Nj e−lj δjx
>K .
j =1
The set AK is open since for each j ∈ N the function x → (δjx )2 is continuous. By Lemma 3.2 we have / T (Mx ) = AK x ∈ c0 : t ∈ K∈N
so it suffices to show that j > j0 ,
K∈N AK
is dense. Let y ∈ c0 and ε > 0. Pick j0 ∈ N such that for all y(j ) < ε 2
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and 1 ε √ < . t ln(2) j0 2 a(j ) √ For j j0 define x(j ) = y(j ). For j > j0 define x(j ) = j ! ln(1 + t ln(2)j ), where a(j ) is ! j defined according to the following rule: ⎧ ⎨ 0 if |t ln(2)j ! (mod 2π)| √1 , 2 j a(j ) = ⎩ 1 if |t ln(2)j ! (mod 2π)| < √1 . 2 j
It is clear that if j > j0 then 0 x(j ) j !
a(j ) 1 ε √ √ < , 2 t ln(2)j ! j t ln(2) j
so x ∈ c0 and x − y ∞ < ε. On the other hand we have that
a(j ) a(j ) x x(j )/j ! = t ln(2)j ! + √ . = t ln(2)j ! 1 + tlj = t ln(2)j !e √ t ln(2)j ! j j By the choice of a(j ) we have |δjx | = |tljx (mod 2π)|
1 √ . 2 j
It follows that
∞
2 x 1 2 x(j )/j ! ) Nj e−lj δjx 2j !(1−e = ∞, √ 2 j j =1 j =j +1
∞
0
which shows that x ∈
K∈N AK .
2
Recall that the equivalence relation ∼iso in c0 is defined by x ∼iso x ⇐⇒ Mx Mx . For x ∈ c0 let [x]∼iso = {y ∈ c0 : y ∼iso x}. Lemma 3.6. For each x ∈ c0 , [x]∼iso is meagre. Proof. By Lemma 3.4 there exists t0 ∈ T (Mx ) \ {0}. Then [x]∼iso ⊆ y ∈ c0 : t0 ∈ T (My ) and so [x]∼iso is meagre by Lemma 3.5.
2
Remark 3.7. It will be shown below that, in a precise way, x → Mx is Borel. By (the proof of) [8, Theorem 2.2] the isomorphism relation is analytic (see also [21, Corollary 15].) It follows that ∼iso is analytic, and so by the Kuratowski–Ulam Theorem [13, 8.41] we get from Lemma 3.6 that ∼iso is meagre as a subset of c0 × c0 . We will need the following fact, a proof of which may be found in [1, Lemma 2.13], see also [4, Lemme 1.3.8].
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Proposition 3.8. If M1 and M2 are ITPFI2 factors with eigenvalue lists (λn,1 , 1 − λn,1 )n∈N and (λn,2 , 1 − λn,2 )n∈N , respectively, and ∞
1 1 2 1 1 2 (λn,1 ) 2 − (λn,2 ) 2 + (1 − λn,1 ) 2 − (1 − λn,2 ) 2 < ∞, n=1
then M1 and M2 are unitarily isomorphic (and so they are isomorphic). Remark 3.9. Denote by c00 ⊆ c0 the set of all eventually zero sequences. Then c00 acts continuously on c0 by addition. By Proposition 3.8 it follows easily that if g ∈ c00 then Mg+x Mx for all x ∈ c0 . Thus the action of c00 on c0 preserves ∼iso . Since ∼iso is meagre in c0 × c0 and clearly c00 -orbits are dense, we can now apply [2, Theorem 3.4.5], by which it follows that E0 B ∼iso . Here E0 denotes the equivalence relation in {0, 1}N defined by xE0 y
⇐⇒
(∃N ) (∀n N ) x(n) = y(n).
Below we will show that the assignment x → Mx is Borel, and so it follows that E0 is Borel reducible to isomorphism of ITPFI2 factors. Since E0 is not smooth this provides a new proof of the following: Theorem. (See Woods [24].) E0 is Borel reducible to isomorphism of ITPFI2 factors. In particular the isomorphism relation for ITPFI2 factors is not smooth. Arguably the proof exhibited here is simpler than the argument given in [24], partly because we avoid to construct an explicit Borel reduction from E0 to isomorphism of ITPFI2 factors that made Woods’ original proof quite involved. Observe that since c00 doesn’t admit a Polish group structure, Hjorth’s theory of turbulence does not apply to its actions. In what follows we will overcome this difficulty by defining a group G that can play the role of c00 , but which is also Polish. Specifically, consider the set G = a ∈ c0 :
∞
j!
2 a(j ) < ∞ . 2
j =1
The set G becomes a separable real Hilbert space when equipped with the inner product given j ! a(j )b(j ). Since c ⊂ G, it follows that G is dense in c . Moreover, since by a, b = ∞ 2 00 0 j =1 G = c0 , by Lemma 2.1 the action of G on c0 by addition is turbulent, has meagre classes and all the classes are dense. The following lemma shows that the G-action on c0 preserves the ∼iso classes: Lemma 3.10. If a ∈ G, then Mx is unitarily equivalent to Ma+x . In particular, x ∼iso (a + x).
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Proof. By Proposition 3.8 it is enough to check that the sum ∞
j =1
Nj
1
1 1+e
2
−ljx
−
1 2
1 1+e
2
−lja+x
+
e−lj
x
1+e
−ljx
1 2
−
a+x
e−lj 1+e
1 2 2
−lja+x −s
1
1
e is finite. Since the derivatives of the functions f (s) = (1 + e−s )− 2 and h(s) = ( 1+e −s ) 2 are bounded by 1 whenever s > 0, the previous sum is bounded by ∞
∞
2 2 2Nj ljx − lja+x = 2 ln2 (2) 2j ! (j !)2 e2x(j )/j ! 1 − ea(j )/j !
j =1
j =1
∞
2 3 2j ! (j !)2 a(j )/j ! + O a(j )/j !
j =1
< K˜
∞
2j ! a(j )2
j =1
˜ and this is finite whenever a ∈ G. for appropriate constants K and K,
2
Theorem 3.11. The equivalence relation ∼iso is generically S∞ -ergodic, and ∼iso is not classifiable by countable structures. c
Proof. Let G be as above, and let EG0 denote the orbit equivalence relation induced by the action c of G on c0 . Then by Lemma 3.10 we have EG0 ⊆∼iso . Since G acts turbulently, it follows by c0 [10, Theorem 3.18] that EG is generically S∞ -ergodic, and so as noted in the discussion of S∞ -ergodicity in §2, ∼iso is generically S∞ -ergodic. Suppose, seeking a contradiction, that S∞ acts continuously on the Polish space Y and ∼iso B ESY∞ . If f : c0 → Y were a Borel reduction witnessing this then f would map a comeagre set in c0 to the same S∞ -class. But this would contradict that all ∼iso classes are meagre by Lemma 3.6, and so f can’t be a reduction. Hence ∼iso is not classifiable by countable structures. 2 Remark 3.12. It follows from Theorem 3.11 that the set x ∈ c0 : T (Mx ) is uncountable is comeagre in c0 , since otherwise the assignment x → T (Mx ) would give an ∼iso -invariant assignment of countable subsets of R on a comeagre set, and so by [10, Lemma 3.14] the function x → T (Mx ) would be constant on a comeagre set. But this contradicts Lemma 3.4 and 3.5. It follows from the above and Lemma 3.4 that {x ∈ c0 : Mx is of type III0 }
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is comeagre, as it should be, since ITPFI factors of type IIIλ , 0 < λ 1, are classified by a single real number λ. It actually follows from [9, Proposition 1.3] that for all x ∈ c0 , T (Mx ) is uncountable, thus Mx is of type III0 , but using an entirely different line of argument. Recall from [21] and [20] that if H is a separable complex Hilbert space, then vN(H) denotes the standard Borel space of von Neumann algebras acting on H, equipped with the Effros Borel structure originally introduced in [7] and [8]. Let vN(H) denote the isomorphism relation in vN(H). Theorem 3.13. The isomorphism relation for ITPFI2 factors is not classifiable by countable structures. Proof. It suffices to show that there is a Borel function f : c0 → vN( 2 (N)) such that for all 2 x ∈ c0 we have f (x) Mx , since then by Theorem 3.11 it follows that ∼iso B vN( (N)) . That such a function f exists follows from the next three lemmas. 2 Lemma 3.14. Suppose X is a standard Borel space and (Hx : x ∈ X) is a family of infinite dimensional separable Hilbert spaces, and that (enx )n∈N is an orthonormal basis of Hx for each x ∈ X. Suppose further that Y is a standard Borel space and (Tyx : x ∈ X, y ∈ Y ) is a family of operators such that Tyx ∈ B(Hx ) for all y ∈ Y , x ∈ X and that the functions x X × Y → C : (x, y) → Tyx enx , em are Borel for all n, m. Then there is a Borel function θ : X × Y → B( 2 (N)) and a family (ϕx : x ∈ X) such that (1) ϕx ∈ B(Hx , 2 (N)) satisfies ϕx (enx ) = en , where (en )n∈N is the standard basis for 2 (N). (2) For all x ∈ X, y ∈ Y and ξ ∈ Hx we have θ (x, y)(ϕx (ξ )) = ϕx (Tyx (ξ )). Moreover, if Mx is the von Neumann algebra generated by the family (Tyx : y ∈ Y ), and there are Borel functions ψn : X → Y such that (Tψxn (x) : n ∈ N) generates Mx for each x ∈ X, then there is a Borel function θˆ : X → vN( 2 (N)) such that θˆ (x) Mx for all x ∈ X. Proof. The family (ϕx : x ∈ X) is uniquely defined by (1), and θ is uniquely defined by θ (x, y) = T
⇐⇒
x , (∀n, m) T en , em 2 (N) = Tyx enx , em H x
which also gives a Borel definition of the graph of θ , so θ is Borel by [13, 14.12] since B( 2 (N)) is a standard Borel space when given the Borel structure generated by the weak topology. If we let fn : X → vN 2 (N) : x → θ x, ψn (x) .
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Then the “moreover” part follows from [7, Theorem 2] since Mx is isomorphic to fn (x): n ∈ N ∈ vN 2 (N) .
2
Lemma 3.15. There is a Borel function f : (0, 1)N → vN( 2 (N)) such that for all x ∈ (0, 1)N , f (x) is isomorphic to Nx =
∞ M2 (C), x(n), 1 − x(n) , n=1
the ITPFI factor with eigenvalue list (x(n), 1 − x(n))n∈N . Proof. Let M2 (C) act on itself by multiplication. Then let η : (0, 1)N → (M2 (C)4 )N be a Borel function such that
√ x(n) 0 √ η(x)(n)1 = 0 1 − x(n) and {η(x)(n)i : i ∈ {1, 2, 3, 4}} is an orthonormal basis for M2 (C). For each i ∈ {1, 2, 3, 4}N such that i(k) = 1 eventually, let eix = η(x)(1)i(1) ⊗ η(x)(2)i(2) ⊗ · · · ⊗ η(x)(n)i(n) ⊗ · · · . Then (ex : i(k) = 1 eventually) is an orthonormal basis for the Hilbert space i
Hx =
∞ M2 (C), η(x)(n)1 . n=1
Let M2 (C)
2
1 1+e
: j ∈N −l x j
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Remarks 3.17. 1. Consider Cantor space {0, 1}N and the odometer action of Z on X = {0, 1}N (i.e. “adding one with carry”). Let z ∈ (0, 1)N , and let μz be the product measure μz =
∞ z(n)δ0 + 1 − z(n) δ1 , n=1
where δ0 and δ1 denote the Dirac measures on {0, 1} concentrating on 0 and 1, respectively. The measure class of μz is preserved by the odometer action, and if μz is ergodic for the odometer we let Nz = L∞ (X, μz ) Z be the Krieger factor obtained from the group-measure space construction. Then ∞ M2 (C), z(n), 1 − z(n) , Nz = L∞ X, μz Z
n=1
see [3, III.3.2.18]. By Krieger’s celebrated Theorem ([18, 8.4], see also [3, III.3.2.19]) the group measure space factors Nz and Nz are isomorphic precisely when the corresponding measure class-preserving odometer actions are orbit equivalent. If we now, for each x ∈ c0 , let zx (n) = λxn , where (λxn , 1 − λxn )n∈N is the eigenvalue list of the factor Mx , then since all the factors Mx are type III, the measure μzx is non-atomic and ergodic for the odometer. Thus we obtain the following consequence of Theorem 3.13: Theorem 3.18. The odometer actions of Z on {0, 1}N that preserve the measure class of some ergodic non-atomic μz as above, are not classifiable up to orbit equivalence by countable structures. 2. The observation made in [21, Corollary 8] is equally pertinent to the main result of this paper: Since the proof relies only on Baire category techniques, Theorem 3.13 shows that it is not possible to construct in Zermelo–Fraenkel set theory without the Axiom of Choice a function that completely classifies ITPFI2 factors up to isomorphism by assigning countable structures type invariants. 3. In [21] it was shown that isomorphism of separable factors is Borel reducible to an equivalence relation arising from a Borel action of the unitary group of 2 (N) on a standard Borel space. This in particular then applies to ITPFI factors. Since by Theorem 1.1 we have E0
⇐⇒
(∃n) (∀m > n) x(n) = y(n).
This remark also applies to II1 , II∞ and IIIλ , 0 λ 1, using the results of [21]. 4. In [20] we asked (Problem 4) if all possible Kσ subgroups of R appear as the T -set of some ITPFI factor. Stefaan Vaes has kindly pointed out to us that this is already known not to be the case: This follows from the results of [11, §2], see also [19, §2]. Acknowledgments Research for this paper was mainly carried out at the Hausdorff Institute for Mathematics in Bonn during the ‘Rigidity’ programme in the fall semester of 2009. We wish to thank the
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Hausdorff Institute for kind hospitality and support. A. Törnquist was also supported in part through Austrian Science Foundation FWF grant No. P19375-N18 and a Marie Curie grant No. 249167 from the European Union. References [1] H. Araki, J. Woods, A classification of factors, Publ. RIMS, Kyoto Univ. Ser. A 3 (1968) 51–130. [2] H. Becker, A. Kechris, The Descriptive Set Theory of Polish Group Actions, London Math. Soc. Lecture Note Ser., vol. 232, Cambridge University Press, 1996. [3] B. Blackadar, Operator Algebras: Theory of C ∗ -Algebras and Von Neumann Algebras, Encyclopaedia Math. Sci., vol. 122, Springer-Verlag, Berlin, 2006. [4] A. Connes, Une classification des facteurs de type III, Ann. Sci. École Norm. Sup. (4) 6 (2) (1973) 133–252. [5] A. Connes, Noncommutative Geometry, Academic Press, 1994. [6] A. Connes, J. Woods, A construction of approximately finite-dimensional non-ITPFI factors, Canad. Math. Bull. 23 (1980) 227–230. [7] E.G. Effros, The Borel space of von Neumann algebras on a separable Hilbert space, Pacific J. Math. 15 (4) (1965) 1153–1164. [8] E.G. Effros, Global structure in von Neumann algebras, Trans. Amer. Math. Soc. 121 (1966) 434–454. [9] T. Giordano, G. Skandalis, Krieger factors isomorphic to their tensor square and pure point spectrum flows, J. Funct. Anal. 64 (1985) 209–226. [10] G. Hjorth, Classification and Orbit Equivalence Relations, Math. Surveys Monogr., vol. 75, American Mathematical Society, 2000. [11] B. Host, J.-F. Méla, F. Parreau, Non singular transformations and spectral analysis of measures, Bull. Soc. Math. France 119 (1) (1991) 33–90. [12] A. Ioana, A. Kechris, T. Tsankov, Subequivalence relations and positive definite functions, Groups Geom. Dyn. 3 (4) (2009) 579–625. [13] A. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math., vol. 156, Springer-Verlag, 1995. [14] A.S. Kechris, Actions of polish groups and classification problems, in: Analysis and Logic, in: London Math. Soc. Lecture Note Ser., vol. 262, Cambridge University Press, 2002, pp. 115–187. [15] A.S. Kechris, A. Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1) (1997) 215–242. [16] D. Kerr, H. Li, M. Pichot, Turbulence, representations, and trace-preserving actions, Proc. Lond. Math. Soc. 100 (2010) 459–484. [17] W. Krieger, On the infinite product construction of non-singular transformations of a measure space, Invent. Math. 15 (1972) 144–163. [18] W. Krieger, On ergodic flows and the isomorphism of factors, Math. Ann. 223 (1976) 19–70. [19] S. Popa, S. Vaes, Actions of F∞ whose II1 factors and orbit equivalence relations have prescribed fundamental group, J. Amer. Math. Soc. 23 (2010) 383–403. [20] R. Sasyk, A. Tornquist, Borel reducibility and classification of von Neumann algebras, Bull. Symbolic Logic 15 (2) (2009) 169–183. [21] R. Sasyk, A. Tornquist, The classification problem for von Neumann factors, J. Funct. Anal. 256 (2009) 2710–2724. [22] N. Sofronidis, Turbulence phenomena in real analysis, Arch. Math. Logic 44 (2005) 801–815. [23] M. Takesaki, Theory of Operator Algebras II, Encyclopaedia Math. Sci., vol. 125, Springer-Verlag, 2003. [24] E.J. Woods, The classification of factors is not smooth, Canad. J. Math. 25 (1973) 96–102.
Journal of Functional Analysis 259 (2010) 2253–2295 www.elsevier.com/locate/jfa
Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities ✩ Riccardo Molle a,∗ , Donato Passaseo b a Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy b Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy
Received 16 February 2010; accepted 18 May 2010 Available online 1 June 2010 Communicated by J. Coron
Abstract In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.). © 2010 Elsevier Inc. All rights reserved. Keywords: Jumping nonlinearities; Multiplicity of solutions; Variational methods
1. Introduction Let us consider the following problem u + g(u) = ξ u=0
in Ω, on ∂Ω,
(1.1)
✩ Supported by the Italian national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”. * Corresponding author. E-mail address:
[email protected] (R. Molle).
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.010
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where Ω is a bounded connected domain of Rn , ξ ∈ L2 (Ω) and g : R → R is a continuous function such that lim
s→−∞
g(s) =α s
and
lim
s→+∞
g(s) =β s
(1.2)
with α and β in R. We assume, for example, α β. We denote by λ1 < λ2 λ3 · · · the eigenvalues of the Laplace operator − in H01 (Ω). If there exists some eigenvalue λi such that λi ∈ ]α, β[, we say that g is a jumping nonlinearity and that λi is a jumped eigenvalue. There exists a very extensive literature on the elliptic problems where the nonlinear terms interfere with the spectrum of the linear operator and, in particular, on the elliptic equations with jumping nonlinearities (see [1–8,10–33,38–41] etc.). Here we recall only the following results. If g ∈ C 1 (R) and g (s) = λi ∀s ∈ R, ∀i ∈ N, then one can apply, for example, a well-known result of Caccioppoli (see [9]) which guarantees that there exists a unique solution u for every ξ ∈ L2 (Ω). The first result concerning the case of jumping nonlinearities is due to Ambrosetti and Prodi. In [4] they consider the problem
u + g(u) = ξ0 + te1 u=0
in Ω, on ∂Ω
(1.3)
where g ∈ C 2 (R), ξ0 ∈ L2 (Ω), t ∈ R and e1 is the positive eigenfunction, normalized in L2 (Ω), corresponding to the first eigenvalue λ1 . Under the assumption that g (s) > 0 ∀s ∈ R and 0 < lim g (s) < λ1 < lim g (s) < λ2 , s→−∞
s→+∞
(1.4)
they prove that there exists a smooth function t¯ : L2 (Ω) → R such that problem (1.3) has exactly two solutions if t > t¯(ξ0 ), exactly one solution if t = t¯(ξ0 ) and no solution if t < t¯(ξ0 ). After the result of Ambrosetti and Prodi, several papers have been devoted to describe the right-hand side members ξ for which there exist solutions and to estimate the number of solutions, under suitable assumptions on α and β with respect to the eigenvalues of − in H01 (Ω). In recent years there has been a new interest also in superlinear Ambrosetti–Prodi type problems, i.e. problems where it is allowed to be β = +∞ and g has subcritical or critical growth at +∞ (see [19,23,43] etc.). If no eigenvalue belongs to the interval [α, β], then a well-known theorem of Rabinowitz (see [37]) applies and guarantees that problem (1.1) has at least one solution for every ξ ∈ L2 (Ω). If α < λ1 < β, there exists a function t¯ : L2 (Ω) → R such that problem (1.3) has at least two solutions if t > t¯(ξ0 ), at least one solution if t = t¯(ξ0 ) and no solution if t < t¯(ξ0 ). If α < λ1 < λ2 < β, there exist at least four solutions for t > 0 large enough. If α > λ1 (i.e. the jumping involves higher eigenvalues) then there exist solutions also for t < 0 large enough (see [3,4,7, 28–32,41] etc.). If we look for solutions of problem (1.3) with t (positive or negative) large enough, the following two problems (we denote by P + (e1 ) and P − (e1 )) arise in a natural way and play an important role, namely P ± (e1 )
u − αu− + βu+ = ±e1
in Ω,
u=0
on ∂Ω
(1.5)
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2255
where u+ = max{u, 0} and u− = u+ − u (in fact, if in problem (1.3) we replace u by |t|u and let t → ±∞, we obtain P ± (e1 ) as limit problem). In the early 1980s, Lazer and McKenna raised the following conjecture: there exist at least 2i solutions of problem (1.3) for t > 0 large enough when α < λ1 < λi < β. This conjecture has been proved only in the case n = 1 (see [12,30,39]) while it does not hold in the case n > 1. In fact, in [14] Dancer considered the following problem
u − αu− + βu+ = ξ0 + te1
in Ω,
u=0
on ∂Ω
(1.6)
and showed that for every i 2 there exist a smooth bounded domain Ωi in Rn , with n > 1, and a function ξ0 ∈ L2 (Ωi ) such that problem (1.6), with Ω = Ωi , has only four solutions for t > 0 large enough even if α < λ1 (Ωi ) < λi (Ωi ) < β. In [34] and [35] we prove that, for any fixed domain Ω, the number of solutions may be arbitrarily large, provided the number of jumped eigenvalues is large enough. In fact, in [34] we fix α < λ1 and show that, for all k ∈ N and ξ0 ∈ L2 (Ω), problem (1.6) has, for β and t positive and large enough, a solution uk,β,t such that, as t → +∞, 1t uk,β,t tends to a solution uk,β of problem P + (e1 ), which presents k peaks near the maximum points of e1 and, as β → +∞, e1 (this result might be seen as a weaker version of the Lazer– converges to the solution α−λ 1 McKenna conjecture, which is not in contradiction with the counter-example of Dancer [14]). In [35] we consider the case α > λ1 and, for β > 0 large enough, we construct analogous k1 as β → +∞ and, peaks solutions uk,β of problem P − (e1 ), which converge to the solution λ1e−α
1 for t < 0 large enough, give rise to solutions uk,β,t of problem (1.3), such that |t| uk,β,t → uk,β as t → −∞. In the present paper we consider the case α = λ1 . In this case, the methods used in [34] e1 1 or λ1e−α in the case α = λ1 ) might and [35] do not work, no solution there exists which (as α−λ 1 be the limit of the k-peaks solutions, so more refined arguments are necessary. However, also in this case we can construct k-peaks solutions and describe their behavior. The solutions uk,β we obtain in this case tend to −∞, as β → +∞, a.e. in Ω and the top of the peaks tends to +∞ (while in [34] and [35] the height of the peaks remains bounded). In order to distinguish the effect of the assumption α = λ1 and the role of the eigenfunction e1 , in this paper we consider the right-hand side members ξ of the form ξ = ξ0 + tp where p is any function in L2 (Ω) such that p > 0 a.e. in Ω. Notice that, if we replace e1 by such a p in the case α < λ1 , for example, then the method used in [34] yields solutions uk,β which tend to the solution ( + α)−1 p as β → +∞ and present k peaks localized near the minimum points of ( + α)−1 p (which are the maximum points of e1 if p = e1 ). On the contrary, due to the fact that α = λ1 , the solutions uk,β we obtain in this paper for the problem
u − λ1 u− + βu+ = p u=0
in Ω, on ∂Ω
(1.7)
have k peaks which are always localized near the maximum points of e1 for any p ∈ L2 (Ω) such that p > 0 a.e. in Ω.
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The method we use in this paper is completely variational. The solutions of problem (1.7) are obtained as critical points of the functional fβ : H01 (Ω) → R defined by fβ (u) =
1 2
2 2 |Du|2 − λ1 u− − β u+ dx +
Ω
∀u ∈ H01 (Ω).
pu dx
(1.8)
Ω
The main result of this paper is presented in the following theorem. Theorem 1.1. Let Ω be a bounded domain of Rn , with n 3, and p a function in L2 (Ω) such that p > 0 a.e. in Ω. Then, for every positive integer k there exists β¯k > 0 such that, for all β > β¯k , problem (1.7) has a solution uk,β satisfying the following properties: β
β
I) there exist a positive constant r¯ and k points x1 , . . . , xk in Ω, with β r¯ dist xi , ∂Ω > √ β
β 2¯r β and xi − xj > √ β
for i = 1, . . . , k
such that, for all β > β¯k , uk,β (x) 0 ∀x ∈ Ω \ √ β B(xi , r¯ / β) for i = 1, . . . , k; β β II) the points x1 , . . . , xk satisfy, in addition, β lim e1 xi = max e1
β→+∞
Ω
lim
β→+∞
k
√ β i=1 B(xi , r¯ / β)
for i = 1, . . . , k
for i = j,
(1.9)
and u+ k,β ≡ 0 in
and
β β β xi − xj = ∞ for i = j ;
(1.10)
III) moreover, we have lim
β→+∞
sup
√ β B(xi ,¯r / β)
uk,β = +∞
(1.11)
and, for every δ ∈ ]0, maxΩ e1 [, sup
lim
√ β→+∞ k β Ωδ \ i=1 B(xi ,¯r / β)
uk,β = −∞
(1.12)
δ = {x ∈ Ω: e1 (x) > δ}; where Ω IV) as β → +∞, uk,β L2 (Ω) → ∞ and uk,β −1 u → −e1 in H01 (Ω); L2 (Ω) k,β √ β V) for all i ∈ {1, . . . , k}, the function Uk,β,x β , defined in β(Ω − xi ) by i
x β + x Uk,β,x β (x) = uk,β −1 u √ k,β i , L2 (Ω) i β
(1.13)
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
2257
converges as β → +∞ to the nonconstant radial solution U of the problem
U + U + = 0 in Rn , lim U (x) = − max e1 |x|→∞
(1.14)
Ω
and the convergence is uniform on the compact subsets of Rn . Notice that only for n 3 problem (1.14) has a nontrivial solution. In the cases n = 1 and n = 2, a different approach is necessary in order to construct k-peaks solutions (see [36]). A result analogous to Theorem 1.1, concerning problem (1.6) with α = λ1 and p in place of e1 , is presented in Theorem 4.1. For this problem we prove that, for β and t positive and large enough, there exists a solution uk,β,t such that, as t → +∞, 1t uk,β,t tends to the solution uk,β obtained in Theorem 1.1 for problem (1.7). Moreover, we show that, in addition to the solutions given by Theorems 1.1 and 4.1, there exist also solutions of different type, corresponding to lower critical values of the related functional, which (unlike the solutions uk,β and uk,β,t ) tend to be localized near the boundary of the domain Ω as β and t tend to +∞. The paper is organized as follows. In Section 2 we outline the mini–max scheme we use to construct k-peaks solutions and prove some preliminary results. In Section 3 we describe the behavior, as β → +∞, of the functions obtained by the mini–max method and prove that, actually, they are solutions for β > 0 large enough. In Section 4 we prove Theorems 1.1 and 4.1. Finally, in Section 5 we prove the existence of solutions corresponding to lower critical values. 2. Notations and preliminary results Let us denote by r¯1 the radius of the ball in Rn for which the first eigenvalue is 1, i.e. the Dirichlet problem
u + u = 0 in B(0, r¯1 ), u=0 on ∂B(0, r¯1 )
has a positive solution. Then, for β > 0, set rβ = the set
3¯r1 √ β
(2.1)
and, for every positive integer k, consider
Ωβk = (x1 , . . . , xk ) ∈ Ω k : |xi − xj | 2rβ if i = j, dist(xi , ∂Ω) rβ for i = 1, . . . , k . (2.2) Notice that, for every fixed k, we have Ωβk = ∅ for β large enough. In this paper we always assume that k and β are chosen in such a way that Ωβk = ∅. For every (x1 , . . . , xk ) ∈ Ωβk , let us consider the pairwise disjoint balls B(x1 , rβ ), . . . , B(xk , rβ ), included in Ω. We say that a function u ∈ H01 (Ω) is a k-peaks function with respect to these balls if u+ = ki=1 u+ i where, 1 (Ω) such that u+ ≡ 0 and u+ (x) = 0 is a nonnegative function in H for every i ∈ {1, . . . , k}, u+ i i i 0 β ∀x ∈ Ω \ B(xi , rβ ). Then, let us denote by Sx1 ,...,xk the subset of H01 (Ω) consisting of all the k-peaks functions u, with respect to the balls B(x1 , rβ ), . . . , B(xk , rβ ), such that
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+ 2 Du dx − β i
Ω
+ 2 ui dx +
Ω
pu+ i dx = 0
(2.3)
Ω
and
+ 2 ui (x) (x − xi ) dx = 0
(2.4)
Ω 2 for every i ∈ {1, . . . , k} (notice that (2.3) means that fβ (u)[u+ i ] = 0 and (2.4) that the L -mass + of ui has center in xi ).
Lemma 2.1. Let us fix a positive integer k and, for β > 0 (large enough) such that Ωβk = ∅, set − 2 − 2 β k Du u dx: u ∈ Sx1 ,...,xk , (x1 , . . . , xk ) ∈ Ωβ , dx = 1 . cβ = inf Ω
(2.5)
Ω
Then, we have cβ > λ1 and limβ→+∞ cβ = λ1 . Proof. It is clear that cβ λ1 . In order to prove that cβ > λ1 , we argue by contradiction and assume that there exist a sequence (x1,j , . . . , xk,j )j ∈N in Ωβk and a sequence of functions (uj )j ∈N in H01 (Ω) such that
− 2 uj dx = 1 ∀j ∈ N
uj ∈ Sxβ1,j ,...,xk,j ,
(2.6)
Ω
and lim
− 2 Du dx = λ1 . j
j →∞
(2.7)
Ω 1 It follows that the sequence (u− j )j converges in H0 (Ω) to the eigenfunction e1 . Notice that, since p > 0 in Ω, from (2.3) we obtain
Ω
+ 2 Du dx β j,i
+ 2 ui,j dx.
(2.8)
Ω
−1 + 1 It follows that, for i = 1, . . . , k, the sequence ( u+ j,i L2 (Ω) uj,i )j is bounded in H0 (Ω) and, up
to a subsequence, it converges weakly in H01 (Ω), in L2 (Ω) and almost everywhere in Ω, to a 2 nonnegative function u¯ + ¯+ i such that Ω (u i ) dx = 1. On the other hand, for i = 1, . . . , k, we have Ω
− u+ j,i (x)uj (x) dx = 0 ∀j ∈ N.
(2.9)
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Therefore, as j → ∞, we obtain
u¯ + i (x)e1 (x) dx = 0 for i = 1, . . . , k
(2.10)
Ω
¯+ which is a contradiction because e1 (x) > 0 ∀x ∈ Ω and u¯ + i 0, u i ≡ 0 for i = 1, . . . , k. Thus, we can conclude that cβ > λ1 . In order to prove that limβ→+∞ cβ = λ1 , it suffices to fix k pairwise distinct points x1 , . . . , xk in Ω and observe that, by standard techniques, one can construct a nonnegative function uβ ∈ H01 (Ω \ ki=1 B(xi , rβ )) which, as β → +∞, converges in H01 (Ω) to the eigenfunction e1 . 2 Proposition 2.2. Let us fix k ∈ N, choose β > 0 (large enough) such that Ωβk = ∅ and consider a point (x1 , . . . , xk ) ∈ Ωβk . β
β
Then Sx1 ,...,xk = ∅ and the minimum of the functional fβ on the set Sx1 ,...,xk is achieved. β
Proof. In order to verify that Sx1 ,...,xk = ∅, notice that, because of the choice of r¯1 , since √ βrβ > r¯1 , for i = 1, . . . , k there exist nonnegative functions vi ∈ H01 (Ω), vi ≡ 0, such that vi = 0
in Ω \ B(xi , rβ ),
|Dvi |2 dx < β
Ω
vi2 dx
and
Ω
2 vi (x) (x − xi ) dx = 0.
(2.11)
Ω
Therefore, there exists (a unique) ti > 0 such that fβ (ti vi ) = max{fβ (tvi ): t > 0} and the func β tion ki=1 ti vi belongs to Sx1 ,...,xk . β Notice that infS β fβ > −∞. In fact, for every u ∈ Sx1 ,...,xk , we have fβ (u) = fβ (−u− ) + x1 ,...,xk β fβ (u+ ), where fβ (u+ ) = ki=1 fβ (u+ i ). Since u ∈ Sx1 ,...,xk , from property (2.3) we infer that + + fβ (ui ) = max{fβ (tui ): t > 0} > 0 for i = 1, . . . , k. Moreover, taking into account Lemma 2.1, we obtain 1 − 2 − 2 Du dx − 1 λ1 fβ −u− = u dx − pu− dx 2 2 Ω
1 (cβ − λ1 ) 2
Ω
Ω
− 2 u dx −
Ω
pu− dx
Ω
2 1 (cβ − λ1 )u− L2 (Ω) − p L2 (Ω) u− L2 (Ω) . 2
(2.12)
It follows that inf fβ inf fβ −u− : u ∈ Sxβ1 ,...,xk > −∞.
β Sx1 ,...,xk
(2.13)
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R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295 β
Now, let us consider a minimizing sequence (uj )j for fβ on Sx1 ,...,xk . Taking into account (2.12), we infer that lim supu− j L2 (Ω) < +∞.
(2.14)
j →∞
Let us prove that + 0 < lim infu+ j,i L2 (Ω) lim sup uj,i L2 (Ω) < +∞ j →∞
j →∞
for i = 1, . . . , k.
(2.15)
+ −1 + Since u+ j,i ≡ 0, we can set vj,i = uj,i L2 (Ω) uj,i . From property (2.3), taking into account that p > 0 in Ω, we obtain Ω |Dvj,i |2 dx < β. So, the sequence (vj,i )j is bounded in H01 (Ω) and, up to a subsequence, it converges as j → ∞ to a function vi ∈ H01 (Ω) in L2 (Ω), weakly in H01 (Ω) and a.e. in Ω (therefore vi L2 (Ω) = 1 and vi 0 in Ω). In order to prove that
lim infu+ j,i L2 (Ω) > 0 j →∞
for i = 1, . . . , k,
(2.16)
we argue by contradiction and assume that (up to a subsequence) limj →∞ u+ j,i L2 (Ω) = 0 for some i ∈ {1, . . . , k}. From (2.3) we obtain
+ u
j,i L2 (Ω)
|Dvj,i |2 dx − β u+ j,i L2 (Ω) +
Ω
pvj,i dx = 0
(2.17)
Ω
which, as j → ∞, gives Ω pvi dx = 0, that is a contradiction because p > 0, vi 0 in Ω and vi ≡ 0. In analogous way, in order to prove that lim supu+ j,i L2 (Ω) < +∞ j →∞
for i = 1, . . . , k,
(2.18)
assume that (up to a subsequence) limj →∞ u+ j,i L2 (Ω) = +∞ for some i ∈ {1, . . . , k}. From (2.3) we get
−1 |Dvj,i | dx − β + u+ j,i L2 (Ω)
pvj,i dx = 0,
2
Ω
(2.19)
Ω
which, as j → ∞, yields |Dvj,i |2 dx = β.
lim
j →∞ Ω
As a consequence, we obtain limj →∞ fβ (u+ j,i ) = +∞ because
(2.20)
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
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+ fβ u+ j,i = max fβ tuj,i : t > 0 2 t |Dvj,i |2 dx − β + t pvj,i dx: t > 0 = max 2
=
1 β− 2
Ω
|Dvj,i |2 dx
Ω
−1
pvj,i dx
Ω
2 (2.21)
.
Ω
On the other hand, we have − + − β fβ (uj ) > fβ u+ j,i + fβ −uj fβ uj,i + inf fβ −u : u ∈ Sx1 ,...,xk .
(2.22)
Taking into account (2.13), it follows that limj →∞ fβ (uj ) = +∞ which is a contradiction because (uj )j is a minimizing sequence. + Thus, in particular, we have proved that the sequences (u− j )j and (uj,i )j , for i = 1, . . . , k, are bounded in L2 (Ω) and, as a consequence, also in H01 (Ω). It follows that (up to a subsequence) they converge in L2 (Ω), weakly in H01 (Ω) and almost everywhere in Ω. Let us denote by u β the limit function of the sequence (uj )j . If we prove that u ∈ Sx1 ,...,xk , we can conclude that fβ (u) = minS β fβ . x1 ,...,xk
In order to prove this fact, we have only to verify that lim
j →∞
+ 2 Du dx = j,i
Ω
+ 2 Du dx
for i = 1, . . . , k
i
(2.23)
Ω
because, in this case, we can say that the function u satisfies property (2.3) (the other properties follow easily from the convergence in L2 (Ω) and a.e. in Ω of the sequence (uj )j ). Arguing by contradiction, assume that (2.23) does not hold, that is lim
j →∞
+ 2 Du dx > j,i
Ω
+ 2 Du dx
(2.24)
i
Ω
for some i ∈ {1, . . . , k}. It follows that fβ (u)[u+ i ] < 0 (because uj ∈ Sx1 ,...,xk ∀j ∈ N). As a ¯ + consequence, there exists t¯i ∈ ]0, 1[ such that fβ (t¯i u+ i )[ti ui ] = 0. Nowobserve that, for every nonnegative function w ∈ H01 (Ω), fβ (w)[w] = 0 implies fβ (w) = 12 Ω pw dx, as one can easily verify. Therefore, since p > 0 in Ω, we have β
1 ¯ fβ t¯i u+ i = ti 2
Ω
pu+ i dx
1 < 2
Ω
pu+ i dx
1 = lim j →∞ 2
+ pu+ j,i dx = lim fβ uj,i j →∞
(2.25)
Ω
+ + 2 + 2 (while we have fβ (u+ i ) = limj →∞ fβ (uj,i ) if limj →∞ Ω |Duj,i | dx = Ω |Dui | dx). Thus, we can say that, if (2.24) occurs for some i ∈ {1, . . . , k}, then there exists a function β ¯ u¯ ∈ Sx1 ,...,xk (of the form u¯ = −u− + ki=1 t¯i u+ i with ti ∈ ]0, 1] for i = 1, . . . , k) such that
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fβ (u) ¯ < lim fβ (uj ) = inf fβ . j →∞
Clearly, this gives a contradiction.
β
Sx1 ,...,xk
2
Taking into account Proposition 2.2, we can introduce the function μβ : Ωβk → R defined by μβ (x1 , . . . , xk ) = min fβ
∀(x1 , . . . , xk ) ∈ Ωβk .
β Sx1 ,...,xk
(2.26)
Proposition 2.3. Assume k ∈ N and β > 0 (large enough) such that Ωβk = ∅. Then (a) for all x¯ ∈ Ωβk and for every sequence (xj )j in Ωβk such that limj →∞ xj = x, ¯ we have lim sup μβ (xj ) μβ (x); ¯ j →∞
β
(2.27)
β
(b) there exists (x1 , . . . , xk ) ∈ Ωβk such that β β μβ x1 , . . . , xk = max μβ . Ωβk
(2.28)
Proof. If x¯ = (x¯1 , . . . , x¯k ) and xj = (xj,1 , . . . , xj,k ), we have limj →∞ xj,i = x¯i for i = 1, . . . , k. β By Proposition 2.2, there exists u¯ ∈ Sx¯1 ,...,x¯k such that fβ (u) ¯ = μβ (x¯1 , . . . , x¯k ). For all j ∈ N, k β ¯+ ¯+ let us consider the function uj ∈ Sxj,1 ,...,xj,k such that uj = −u− j =1 u j + j,i , where u j,i (x) = + − u¯ i (x + x¯i − xj,i ) and −uj is the minimizing function for the minimum min fβ (v): v ∈ H01 (Ω), v 0 in Ω, v u¯ + dx = 0 for i = 1, . . . , k . j,i
(2.29)
Ω
Notice that the sequence (uj )j is well defined because there exists a unique minimizing function ¯ − in H01 (Ω). In fact, let us consider the convex cone for (2.29). Moreover, as j → ∞, u− j →u Sj = v ∈ H01 (Ω): v 0 in Ω, v u¯ + dx = 0 for i = 1, . . . , k . j,i
(2.30)
Ω
β For every v ∈ Sj , the function v + ki=1 u¯ + j,i belongs to Sxj,1 ,...,xj,k . Therefore, taking into account Lemma 2.1, we obtain 1 1 |Dv|2 dx − λ1 v 2 dx + pv dx fβ (v) = 2 2 Ω
1 λ1 1− 2 cβ
Ω
|Dv|2 dx + Ω
Ω
pv dx Ω
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
λ1 1 1 v 2H 1 (Ω) − p L2 (Ω) v H 1 (Ω) 1− 0 2 cβ λ1 0
∀v ∈ Sj , ∀j ∈ N.
2263
(2.31)
It follows that the sublevels of fβ on Sj are bounded in H01 (Ω). Moreover, the functional fβ is strictly convex on the convex cone Sj . In fact, for all v1 , v2 in Sj , a direct computation of the second derivative of fβ gives
D(v1 − v2 )2 dx − λ1
Ω
(v1 − v2 ) dx (cβ − λ1 )
(v1 − v2 )2 dx
2
Ω
(2.32)
Ω
β where the inequality holds because of Lemma 2.1, since −|v1 − v2 | + ki=1 u¯ + j,i ∈ Sxj,1 ,...,xj,k . Thus, by standard arguments, we can say that, indeed, there exists a unique minimizing function − for fβ on Sj . We denote this function by −u− ¯ − in H01 (Ω) j for all j ∈ N and show that uj → u − as j → ∞. In fact, since fβ (−u− j ) fβ (0) = 0, from (2.31) we infer that the sequence (uj )j is bounded in H01 (Ω), so (up to a subsequence) it converges to a function u˜ in L2 (Ω), weakly in H01 (Ω) and a.e. in Ω. One can easily verify that −u˜ ∈ S∞ , where S∞ = v ∈ H01 (Ω): v 0 in Ω, v u¯ + dx = 0 ,
(2.33)
Ω
so that the function −u˜ + u¯ + belongs to Sx¯1 ,...,x¯k . Notice that also −u¯ − belongs to S∞ and it minimizes fβ on S∞ . 1 Now, for all j ∈ N, denote by −u¯ − ¯ − on the convex cone Sj j the H0 (Ω)-projection of −u (which converges to −u¯ − in H01 (Ω) as j → ∞). Then, we have β
− ¯j fβ −u− j fβ −u
∀j ∈ N and
− lim fβ −u¯ − ¯ . j = fβ −u
j →∞
(2.34)
It follows that − − ¯ . fβ (−u) ˜ lim inf fβ −u− j lim sup fβ −uj fβ −u j →∞
j →∞
(2.35)
Using again Lemma 2.1, one can verify that −u¯ − is the unique minimizing function for fβ on S∞ . Therefore, we infer that u˜ = u¯ − and limj →∞ fβ (−u− ¯ − ) which, in particular, j ) = fβ (−u ¯ − in H01 (Ω) as j → ∞. implies that u− j →u As a consequence, we obtain also that uj → u¯ and fβ (uj ) → fβ (u) ¯ as j → ∞. On the other hand, we have μβ (xj,1 , . . . , xj,k ) fβ (uj )
∀j ∈ N
because uj ∈ Sxj,1 ,...,xj,k and μβ (xj,1 , . . . , xj,k ) = minS β
fβ .
β
xj,1 ,...,xj,k
(2.36)
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Therefore, we have ¯ = μβ (x¯1 , . . . , x¯k ) lim sup μβ (xj,1 , . . . , xj,k ) fβ (u)
(2.37)
j →∞
which proves assertion (a). The assertion (b) follows easily from (a), by standard arguments, because Ωβk is a compact set. 2 Let us report here a result on the capacity, we shall use later. For every bounded domain A in Rn , with n 3, the capacity of A is defined by cap(A) = min
|Du|2 dx: u ∈ D1,2 Rn , u 1 a.e. in A .
(2.38)
Rn
It is well known that there exists a unique minimizing function uA , which is harmonic in Rn \ A. Lemma 2.4. Let A1 , A2 , . . . , As , with s > 1, be s pairwise disjoint bounded domains in Rn with n 3. Then, we have cap
s
Ai <
i=1
Proof. Let us set A =
s
i=1 Ai
s
cap(Ai ).
i=1
and
uˆ A (x) = max uAi (x): i = 1, . . . , s
∀x ∈ Rn .
(2.39)
Then, we have cap(A) =
|DuA |2 dx <
Rn
|D uˆ A |2 dx <
s
|DuAi |2 dx =
i=1 Rn
Rn
s
cap(Ai ).
2
i=1
3. Asymptotic behavior as β → +∞ Let us fix a positive integer k and let β > 0 (large enough) such that Ωβk = ∅. Taking into β
β
account Propositions 2.2 and 2.3, there exist (x1 , . . . , xk ) ∈ Ωβk and uβ ∈ S β
β
β β β x1 ,...,xk
such that
fβ (uβ ) = μβ (x1 , . . . , xk ) = maxΩ k μβ . Our aim is to describe the asymptotic behavior of uβ as β β → +∞ and to prove that uβ is a solution of problem (1.7) for β large enough. Lemma 3.1. Fix k ∈ N and let β > 0 (large enough) such that Ωβk = ∅. Consider a point β
β
(x1 , . . . , xk ) ∈ Ωβk and a function uβ ∈ S
β β β x1 ,...,xk
β
β
such that fβ (uβ ) = μβ (x1 , . . . , xk ) (see Propo-
− −1 − 1 sition 2.2). Then, limβ→+∞ u− β L2 (Ω) = ∞ and uβ L2 (Ω) uβ → e1 in H0 (Ω) as β → +∞.
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
2265
Proof. First, notice that the function −u− β minimizes the functional fβ on the convex cone + 1 Cβ = v ∈ H0 (Ω): v 0 in Ω, vuβ dx = 0 .
(3.1)
Ω
Since rβ → 0 as β → +∞, one can construct by standard methods a function wβ ∈ Cβ such that wβ → −e1 in H01 (Ω) as β → +∞. Notice that the function twβ ∈ Cβ ∀t 0 and t2 2
fβ (twβ ) =
|Dwβ |2 dx − λ1 Ω
t2 2
wβ2 dx + t Ω
pwβ dx.
(3.2)
Ω
In particular, for t = tβ = (λ1 Ω wβ2 dx − Ω |Dwβ |2 dx)−1 Ω pwβ dx > 0, we obtain limβ→+∞ fβ (tβ wβ ) = −∞. As a consequence, since fβ (−u− β ) fβ (tβ wβ ), we have also lim fβ −u− β = −∞.
β→+∞
(3.3)
It follows that lim
β→+∞
u− β e1 dx = +∞.
(3.4)
Ω
− In fact, set u1,β = ( Ω u− β e1 dx)e1 and u2,β = uβ − u1,β . Then, we have 1 fβ −u− β = 2
− 2 Du dx − λ1 β 2
Ω
1 = 2
Ω
λ1 |Du2,β |2 dx − 2
Ω
− 2 uβ dx −
Ω
pu− β dx
(u2,β )2 dx − Ω
1 (λ2 − λ1 ) u2,β 2L2 (Ω) − p L2 (Ω) 2
pu− β dx
Ω
u− β e1 dx − p L2 (Ω) u2,β L2 (Ω) .
(3.5)
Ω
Thus, taking into account (3.3), we obtain (3.4). Moreover, we say that lim
β→+∞ Ω
In fact, from (3.5) we obtain
u− β e1 dx
−1
u2,β H 1 (Ω) = 0. 0
(3.6)
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−2 λ1 1 − 2 u2,β H 1 (Ω) 1− 0 uβ e1 dx 2 λ2 0 Ω
u− β e1 dx
− p L2 (Ω)
−1
Ω
1 − √ p L2 (Ω) u2,β H 1 (Ω) 0 λ2
u− β e1 dx
−2 (3.7)
Ω
which, taking into account (3.4), easily implies −(3.6). Notice that, since u− = u + u = ( 1,β 2,β β Ω uβ e1 dx)e1 + u2,β , we have
− 2 u 2
β L (Ω)
=
u− β e1 dx
2 + u2,β 2L2 (Ω) ,
(3.8)
Ω
which, taking into account (3.4), implies limβ→+∞ u− β L2 (Ω) = +∞. Moreover, from (3.4) and (3.6), we also infer that − −1 lim uβ L2 (Ω) u− (3.9) β e1 dx = 1. β→+∞
Ω −1 − 1 In order to prove that u− β L2 (Ω) uβ → e1 in H0 (Ω) as β → +∞, by a direct computation we obtain
− − u − u 2 e1 2 1 = β β L (Ω) H (Ω)
0
− u− β e1 dx − uβ L2 (Ω)
2 λ1 + u2,β 2H 1 (Ω) .
(3.10)
0
Ω
It follows that − −1 u 2
− β L (Ω) uβ
2 − e1 H 1 (Ω)
−1 = u− β L2 (Ω)
0
2 −2 2 u− e dx − 1 λ1 + u− β 1 β L2 (Ω) u2,β H 1 (Ω)
(3.11)
0
Ω −1 − which, taking into account (3.9) and (3.6), implies limβ→+∞ u− β L2 (Ω) uβ − e1 H 1 (Ω) = 0. 2 0
β
β
Proposition 3.2. Fix k ∈ N and let β > 0 (large enough) such that Ωβk = ∅. Let (x1 , . . . , xk ) ∈ Ωβk and uβ ∈ S
β β β x1 ,...,xk
β
β
satisfy fβ (uβ ) = μβ (x1 , . . . , xk ) = maxΩ k μβ (see Proposition 2.3). Then β
(a) lim β
β→+∞
2−n 2
2 2 −1 fβ (uβ ) = − 2k cap(¯r1 ) max e1 pe1 dx , Ω
(3.12)
Ω
where ( for short) cap(¯r1 ) denotes the capacity of the ball of radius r¯1 in Rn (see (2.38));
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
2267
(b) β lim e1 xi = max e1
β→+∞
for i = 1, . . . , k,
Ω
(3.13)
β β β xi − xj = ∞ for i = j.
(3.14)
2 2 −1 fβ (uβ ) − 2k cap(¯r1 ) max e1 pe1 dx .
(3.15)
lim
β→+∞
Proof. First, let us prove that lim inf β
β→+∞
2−n 2
Ω
Ω β
β
For β > 0 (large enough) choose (x¯1 , . . . , x¯k ) ∈ Ωβk such that β lim e1 x¯i = max e1
β→+∞
for i = 1, . . . , k
Ω
(3.16)
and lim
β→+∞
β β β x¯i − x¯j = ∞ for i = j.
β
(3.17)
β
β
Consider a function u¯ β ∈ S β ¯ β ) = μβ (x¯1 , . . . , x¯k ). We have fβ (u¯ β ) = β such that fβ (u x¯1 ,...,x¯k k ¯+ ¯+ ¯+ fβ (−u¯ − i=1 fβ (u β)+ β,i ), where fβ (u β,i ) = max{fβ (t u β,i ): t > 0} > 0 for i = 1, . . . , k and −1 ¯− ¯− fβ (−u¯ − β ) = min{fβ (−t v¯ β ): t > 0} < 0, where v¯ β = u β L2 (Ω) u β. By a direct computation, we obtain
1 fβ −u¯ − β =− 2
|D v¯β |2 dx − λ1
Ω
v¯β2 dx Ω
−1
2 p v¯β dx
(3.18)
Ω
where, by Lemma 3.1, limβ→+∞ Ω p v¯β dx = Ω pe1 dx and limβ→+∞ [ Ω |D v¯β |2 dx − 2 λ1 Ω v¯β dx] = 0. Moreover, if we set w¯ β = −v¯β + e1 , we have
v¯β2
|D v¯β | dx − λ1 2
Ω
dx =
|D w¯ β | dx − λ1
w¯ β2 dx
2
Ω
Ω
w¯ β2 dx = β
2−n 2
(3.19)
Ω
and, after rescaling,
|D w¯ β |2 dx − λ1 Ω
where W β (x) = w¯ β ( √xβ ) ∀x ∈
Ω
√ βΩ.
λ1 |DW β |2 − W 2β dx β √ βΩ
(3.20)
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It follows that β
2−n 2
fβ −u¯ − β
−1
2 1 λ1 =− |DW β |2 − W 2β dx p v¯β dx . 2 √ β
fβ (u¯ β ) β
2−n 2
(3.21)
Ω
βΩ
√ β Now, for i = 1, . . . , k, consider the functions W β,x¯ β defined in β(Ω − x¯i ) by W β,x¯ β (x) = i i √ β W β (x + β x¯i ). Taking into account (3.16) and (3.17), one can prove by standard arguments that (up to a subsequence) W β,x¯ β converges as β → +∞ to a function W ∞,i ∈ D1,2 (Rn ), W ∞,i i
maxΩ e1 , W ∞,i is harmonic where W ∞,i < maxΩ e1 and
k λ1 |DW β |2 − W 2β dx |DW ∞,i |2 dx. β→+∞ β √ lim inf
(3.22)
i=1Rn
βΩ
2 2 Notice that Ω |D u¯ + ¯+ ¯ β 0 has β,i | dx β Ω (u β,i ) dx (namely, the subset of B(x¯ i , rβ ) where u the first eigenvalue which is not greater than β). Therefore, the set where W ∞,i = maxΩ e1 is a subset of B(0, 3¯r1 ) which has first eigenvalue not greater than 1. Thus, since the ball of radius r¯1 has the smallest capacity among the domains whose first eigenvalue is less than or equal to 1, we infer that 2 |DW ∞,i |2 dx max e1 cap(¯r1 ) for i = 1, . . . , k. (3.23) Ω
Rn
It follows that 2−n 2
lim inf β
β→+∞
2 2 −1 fβ (u¯ β ) − 2k cap(¯r1 ) max e1 pe1 dx , Ω
(3.24)
Ω β
β
which implies (3.15) because fβ (uβ ) = maxΩ k μβ μβ (x¯1 , . . . , x¯k ) = fβ (u¯ β ). β Now, let us prove that lim sup β
2−n 2
2 2 −1 fβ (uβ ) − 2k cap(¯r1 ) max e1 pe1 dx .
Ω
β→+∞
(3.25)
Ω
For every ρ ∈ ]¯r1 , 3¯r1 [, consider a positive function θρ ∈ H01 (B(0, ρ)) such that
θρ2 (x)x dx = 0 and
B(0,ρ)
Then, denote by u˜ β the function in S
|Dθρ |2 dx <
B(0,ρ) β β β x1 ,...,xk
such that
B(0,ρ)
θρ2 dx.
(3.26)
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
u˜ + β,i (x) = ti θρ
2269
β ρ ∀x ∈ B xi , √ , β
β β x − xi
(3.27)
where ti is the positive number such that fβ (u˜ + ˜+ ˜− β satisfies β,i )[u β,i ] = 0 (for i = 1, . . . , k) and u β ρ k − u˜ β (x) = 0 ∀x ∈ i=1 B(xi , √β ),
k − β ρ 1 fβ −u˜ β = min fβ (v): v ∈ H0 (Ω), v 0 in Ω, v = 0 in B xi , √ β
(3.28)
i=1
(this minimum is achieved by a unique function, as one can prove arguing as in the proof of Proposition 2.2). Notice that limβ→+∞ fβ (u˜ + β,i ) = 0 for i = 1, . . . , k. In fact + fβ u˜ + ˜ β,i : t 0 β,i = max fβ t u 2 + 2 + 2 t + D u˜ β,i − β u˜ β,i dx + t p u˜ β,i dx: t 0 = max 2 Ω
=β
2−n 2
2 t max 2
2−n 2
2 t max 2
Ω
t |Dθρ |2 − θρ2 dx + β
B(0,ρ)
β
x β p √ + xi θρ (x) dx: t 0 β
B(0,ρ)
n |Dθρ |2 − θρ2 dx + tβ 4 −1 p L2 (Ω) θρ L2 (B(0,ρ)) : t 0 ,
B(0,ρ)
(3.29) where the last term tends to zero as β → +∞ (as one can easily verify). Arguing as in the proof of Lemma 3.1, one can verify that the minimality property of u˜ − β
−1 1 implies that limβ→+∞ u˜ − ˜− ˜− β L2 (Ω) = ∞, v˜ β := u β L2 (Ω) u β → e1 in H0 (Ω) as β → +∞ and
1 fβ −u˜ − β =− 2
−1
v˜β2
|D v˜β | dx − λ1 2
Ω
dx
Ω
2 p v˜β dx
(3.30)
.
Ω
Thus, taking also into account that fβ (uβ ) = μβ (x1 , . . . , xk ) fβ (u˜ β ) = fβ (−u˜ − β) + k + ˜ β,i ), it follows that i=1 fβ (u β
lim sup β β→+∞
2−n 2
fβ (uβ ) lim sup β
2−n 2
fβ (u˜ β ) = lim sup β
β→+∞
=−
n−2 1 lim sup β 2 2 β→+∞
β→+∞
2−n 2
β
fβ −u˜ − β
|D v˜β |2 − λ1 v˜β2 dx
−1
Ω
If we set w˜ β = −v˜β + e1 and Wβ (x) = v˜β ( √xβ ) ∀x ∈
Ω
√ βΩ, we have
2 pe1 dx
.
(3.31)
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|D v˜β |2 − λ1 v˜β2 dx =
Ω
|D w˜ β |2 − λ1 w˜ β2 dx
Ω
=β
2−n 2
|D Wβ |2 − √
λ1 2 Wβ dx. β
(3.32)
βΩ β
Notice that there exist x1 , . . . , xk in Ω such that, up to a subsequence, xi → xi as β → +∞, for i = 1, . . . , k. Moreover, there exist h pairwise disjoint subsets P1 , . . . , Ph of {1, 2, . . . , k} √ β β (h k) such that hj=1 Pj = {1, 2, . . . , k} and, as β → +∞, β(xi − xj ) converges if i and j √ β β both belong to the same subset while limβ→+∞ β|xi − xj | = ∞ if i and j belong to different subsets (it is clear that xi = xj if i and j belong to the same subset). √ β Now, let us choose i1 ∈ P1 , i2 ∈ P2 , . . . , ih ∈ Ph and set Wβ,j (x) = Wβ (x + βxij ) ∀x ∈ √ β β(Ω − xij ), for j = 1, . . . , h. Then, standard arguments allow us to say that (up to a subsequence) the functions Wβ,1 , . . . , Wβ,h converge as β → +∞ to functions W∞,1 , . . . , W∞,h in D1,2 (Rn ) and |D Wβ |2 −
lim
β→+∞ √
h λ1 2 |D W∞,j |2 dx. Wβ dx = β
(3.33)
j =1Rn
βΩ
Moreover, if Pj (for j = 1, . . . , h) consists of kj elements, there exist kj pairwise disjoint balls kj with radius ρ in Rn , B(c1 , ρ), . . . , B(ckj , ρ), such that W∞,j ≡ mj in i=1 B(ci , ρ), W∞,i is kj n harmonic in R \ i=1 B(ci , ρ) and
|D W∞,j | dx 2
= m2j cap
kj
(3.34)
B(ci , ρ)
i=1
Rn
(see (2.38)), where mj = e1 (xi ) for any i ∈ Pj (it is clear that different choices of i in Pj give the same constant mj ). It follows that lim sup β
2−n 2
fβ (uβ )
β→+∞
kj "−1 ! h
2 1 2 mj cap B(ci , ρ) pe1 dx − 2 j =1
i=1
∀ρ ∈ ]¯r1 , 3¯r1 [.
(3.35)
Ω
kj B(ci , r¯1 )) kj cap(¯r1 ) (see Thus, if we let ρ → r¯1 and take into account that cap( i=1 Lemma 2.4) and that mj maxΩ e1 , we obtain
lim sup β β→+∞
2−n 2
"−1 ! h
2 1 2 fβ (uβ ) − mj kj cap(¯r1 ) pe1 dx 2 j =1
Ω
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
2271
2 2 −1 − 2k cap(¯r1 ) max e1 pe1 dx ,
(3.36)
Ω
Ω
that is (3.25), which completes the proof of assertion (a). kj For the proof of assertion (b), it suffices to observe that cap( i=1 B(ci , r¯1 )) < kj cap(¯r1 ) if kj > 1 (see Lemma 2.4). In fact, if kj > 1 for some j ∈ {1, 2, . . . , h} or if e1 (xi ) < maxΩ e1 for some i ∈ {1, . . . , k}, we infer from (3.35) that lim sup β
2−n 2
2 2 −1 fβ (uβ ) < − 2k cap(¯r1 ) max e1 pe1 dx ,
(3.37)
Ω
β→+∞
Ω
which is in contradiction with (3.15). In conclusion, we can say that e1 (xi ) = maxΩ e1 for i = 1, . . . , k (that is (3.13)) and, in addition, that h = k and kj = 1 for j = 1, . . . , h (that is (3.14)). 2 β
β
Proposition 3.3. Fix k ∈ N and let β > 0 (large enough) such that Ωβk = ∅. Let (x1 , . . . , xk ) ∈ Ωβk and uβ ∈ S
β β β x1 ,...,xk
β
β
such that fβ (uβ ) = μβ (x1 , . . . , xk ) = maxΩ k μβ (see Proposition 2.3). For β √ β i = 1, . . . , k, let Uβ,x β be the function defined in β(Ω − xi ) by i
Uβ,x β (x) = uβ −1 u L2 (Ω) β i
x β √ + xi β
∀x ∈
β β Ω − xi .
(3.38)
Then, we have u → −e1 in H01 (Ω) as β → +∞; (a) limβ→+∞ uβ L2 (Ω) = ∞ and uβ −1 L2 (Ω) β (b) as β → +∞, Uβ,x β converges to the smooth radial function U such that U (x) = 0 ∀x ∈ i ∂B(0, r¯1 ), lim|x|→∞ U (x) = − maxΩ e1 , U (x) + U (x) = 0 ∀x ∈ B(0, r¯1 ), U (x) = 0 ∀x ∈ Rn \ B(0, r¯1 ); moreover, the convergence is uniform on the compact subsets of Rn . + 2 2 Proof. Since we have uβ 2L2 (Ω) = u− β L2 (Ω) + uβ L2 (Ω) , the assertion (a) will follow from Lemma 3.1 provided we prove that
−1 + lim u− β L2 (Ω) uβ L2 (Ω) = 0
β→+∞
(3.39)
k −1 + + (that is limβ→+∞ u− j =1 Uβ,j where, for j = β L2 (Ω) uβ L2 (Ω) = 1). Notice that Uβ,x β = i √ β β + is a nonnegative function in H01 (B( β(xj − xi ), 3¯r1 )). In particular, for j = i, 1, . . . , k, Uβ,j + ∈ H01 (B(0, 3¯r1 )) and (since uβ ∈ S Uβ,i
B(0,3¯r1 )
β β β) x1 ,...,xk
DU + 2 dx < β,i
B(0,3¯r1 )
+ 2 Uβ,i dx.
(3.40)
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Therefore, there exists a function Ei ∈ H01 (B(0, 3¯r1 )) such that (up to a subsequence) + −1 + L2 (Ω) Uβ,i → Ei , as β → +∞, in L2 (B(0, 3¯r1 )), weakly in H01 (B(0, 3¯r1 )) and a.e. in Uβ,i B(0, 3¯r1 ). It follows that, for i = 1, . . . , k, Ei2 dx = 1, |DEi |2 dx 1 (3.41) B(0,3¯r1 )
and (since uβ ∈ S
B(0,3¯r1 )
β β β) x1 ,...,xk
Ei2 (x)x dx = 0.
(3.42)
B(0,3¯r1 )
√ β β −1 − √x Now, set vβ = u− β L2 (Ω) uβ , wβ = −vβ + e1 and Wβ,xiβ (x) = wβ ( β + xi ) ∀x ∈ β(Ω − xi ). Taking into account the assertion (b) of Proposition 3.2 and arguing as in the proof of that proposition, one can prove that (up to a subsequence) Wβ,x β converges as β → +∞ to a function i
Wi ∈ D1,2 (Rn ) and that ! lim inf β
β→+∞
2−n 2
fβ (uβ ) − 2
k
"−1 |DWi | dx 2
2 pe1 dx
i=1Rn
(3.43)
.
Ω
Moreover, the convergence is uniform on the compact subsets of Rn , Wi maxΩ e1 in Rn and Wi = maxΩ e1 on supp Ei (the support of Ei ). It follows that
2 |DWi |2 dx max e1 cap(supp Ei )
(3.44)
Ω
Rn
which, combined with (3.43), implies ! lim inf β
β→+∞
2−n 2
fβ (uβ ) − 2 max e1 Ω
k 2
"−1 cap(supp Ei )
2 pe1 dx
i=1
.
(3.45)
Ω
Notice that, as a consequence of (3.41), supp Ei is a subset of B(0, 3¯r1 ) such that λ1 (supp Ei ) 1, where |Du|2 dx: u ∈ H01 B(0, 3¯r1 ) , u2 dx = 1, λ1 (supp Ei ) = inf B(0,3¯r1 )
B(0,3¯r1 )
u(x) = 0 ∀x ∈ B(0, 3¯r1 ) such that Ei (x) = 0
(3.46)
(λ1 (supp Ei ) is the first eigenvalue of − in supp Ei ). Therefore, (3.45) is not in contradiction with (3.25) only if supp Ei is a ball of radius r¯1 for every i = 1, . . . , k. In fact, only the balls
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
2273
of radius r¯1 have the smallest capacity among the subsets of Rn whose first eigenvalue is less than or equal to 1. Since these balls have the first eigenvalue equal to 1, (3.41) implies that Ei is an eigenfunction related to the first eigenvalue of the ball supp Ei . This eigenfunction has radial symmetry with respect to the center of this ball, so (3.42) implies that supp Ei = B(0, r¯1 ) for i = 1, . . . , k. Moreover, we infer from (3.43) that 2 |DWi |2 dx = max e1 cap(¯r1 ) (3.47) Ω
Rn
so, for every i = 1, . . . , k, Wi is the continuous function W such that W = maxΩ e1 in B(0, r¯1 ), W = 0 in Rn \ B(0, r¯1 ) and lim|x|→∞ W (x) = 0. −1 + Let us prove that u− β L2 (Ω) uβ L2 (Ω) U β converges as β → +∞ to the positive eigenβ,xi
function E of − in H01 (B(0, r¯1 )) such that the function E + W is a smooth function. Arguing by contradiction, assume that it converges to tE with t = 1. Consider a radial function √ β ψ ∈ C01 (B(0, 3¯r1 )) such that ψ(x) = 1 if |x| = r¯1 and set ψβ (x) = ψ( β(x − xi )). Then, a direct computation gives n−2 − −1 2 lim β uβ L2 (Ω) fβ (uβ )[ψβ ] = (t − 1) (DU · ν) dx = 0 (3.48) β→+∞
∂B(0,¯r1 )
(where ν denotes the outward normal on ∂B(0, r¯1 )). Therefore, taking into account that uβ is the unique maximum point for fβ in the set {uβ + su+ β,i : s −1}, one can prove by standard β
methods that there exists a continuous map η : [−1, 1] → H01 (B(xi , rβ )) such that η(s) = 0 for |s| 12 and, for |s| 12 , + β uβ + su+ ≡ 0 in B xi , rβ , (3.49) fβ uβ + su+ β,i + η(s) < fβ (uβ ), β,i + η(s) + 2 β uβ + su+ x − xi dx = 0. (3.50) β,i + η(s) (x) β
B(xi ,rβ ) + + Notice that fβ (uβ + su+ β,i + η(s))[(uβ + suβ,i + η(s)) ] depends continuously on s, it is positive for s = − 12 and negative for s = 12 . Therefore, there exists s˜ ∈ ]− 12 , 12 [ such that β fβ (uβ + s˜ u+ s ))[(uβ + s˜ u+ s ))+ ] = 0. It follows that uβ + s˜ u+ s) ∈ S β β, β,i + η(˜ β,i + η(˜ β,i + η(˜ x1 ,...,xk
which is a contradiction because fβ (uβ ) = minS β
β β x1 ,...,xk
fβ and fβ (uβ + s˜ u+ s )) < fβ (uβ ). β,i + η(˜
Thus, we can say that t = 1. √ −1 √x It remains to prove (3.39). If we set Uβ (x) = u− β L2 (Ω) uβ ( β ) ∀x ∈ βΩ, we obtain
lim
β→+∞ √
βΩ
+ 2 Uβ dx = k
E 2 dx < +∞
(3.51)
B(0,¯r1 )
which obviously implies (3.39). Now, using (3.39), one can easily verify all the assertions and complete the proof.
2
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R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295 β
β
Lemma 3.4. Let k ∈ N and β > 0 such that Ωβk = ∅. Let (x1 , . . . , xk ) ∈ Ωβk and uβ ∈ S such that fβ (uβ ) = minS β
β β x1 ,...,xk
fβ .
β β β x1 ,...,xk
Then, for i = 1, . . . , k and for β large enough, there exists λiβ ∈ Rn such that fβ (uβ )[ψ] =
i β dx u+ β,i (x)ψ(x) λβ · x − xi
β ∀ψ ∈ H01 B xi , rβ .
(3.52)
Ω
Proof. Notice that, since p(x) > 0 ∀x ∈ Ω, from Proposition 3.3 we infer that uβ (x) < 0 ∀x ∈ β Ω \ ki=1 B(xi , 23 rβ ) if β > 0 is large enough. Now, arguing by contradiction, assume that there do not exist multipliers λiβ such that (3.52) holds. Then, taking into account that uβ is the unique maximum point for fβ in the set {uβ + su+ β,i : s −1}, one can prove by standard methods that there exists a continuous β
map η : [−1, 1] → H01 (B(xi , rβ )) satisfying the same properties as the map η used in the β proof of Proposition 3.3. As a consequence, there exists a function u˜ β ∈ S β β such that fβ (u˜ β ) < fβ (uβ ), which is a contradiction because fβ (uβ ) = minS β
β β x1 ,...,xk
fβ .
x1 ,...,xk
2
β
β
Lemma 3.5. Fix k ∈ N and let β > 0 (large enough) such that Ωβk = ∅. Let (x1 , . . . , xk ) and β
β
β β β and vβ x1 ,...,xk β β fβ (vβ ) = μβ (y1 , . . . , yk ).
(y1 , . . . , yk ) in Ωβk , uβ ∈ S
∈S
β β β y1 ,...,yk
β
β
such that f (uβ ) = μβ (x1 , . . . , xk ) =
maxΩ k μβ and β Moreover, assume that
lim
β→+∞
β β β xi − yi = 0 for i = 1, . . . , k.
Let us consider the function Vβ,y β defined in i
√
(3.53)
β
β(Ω − yi ) by
x β Vβ,y β (x) = vβ −1 + y v √ β i . L2 (Ω) i β
(3.54)
Then, also Vβ,y β (as Uβ,x β ) converges to the function U (see Proposition 3.3). If in addition i
i
β
β
we assume that, for some i ∈ {1, . . . , k}, xi = yi for β large enough, then uβ = vβ for β large enough and, if we set
−1
x β β ∀x ∈ β Ω − xi , Zβ,x β (x) = sup |uβ − vβ | (uβ − vβ ) √ + xi i β Ω
(3.55)
there exists a function Zi in Rn such that, up to a subsequence, Zβ,x β converges to Zi as i
β → +∞. Moreover, the convergence is uniform on the compact subsets of Rn , supRn |Zi | 1 and Zi is a weak solution of the equation Z + a(x)Z = 0 in Rn , where a(x) = 1 if x ∈ B(0, r¯1 ) and a(x) = 0 otherwise.
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
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√ β Proof. Notice that, taking into account (3.14), condition (3.53) implies that limβ→+∞ β|yi − β β β yj | = ∞ for i = j . Moreover, since fβ (vβ ) = μβ (y1 , . . . , yk ) maxΩ k μβ = fβ (uβ ), from β Proposition 3.2 we infer that
lim sup β
2−n 2
2 2 −1 fβ (uβ ) − 2k cap(¯r1 ) max e1 pe1 dx . Ω
β→+∞
(3.56)
Ω
Thus, one can easily verify that all the properties of Uβ,x β we used, in the proof of Proposii tion 3.3, in order to prove that Uβ,x β → U as β → +∞, are also verified by Vβ,y β . Therefore, i i we can say that also Vβ,y β → U as β → +∞ and that the convergence is uniform on the compact i
subsets of Rn . β β + It is clear that xi = yi implies u+ β,i ≡ vβ,i . Therefore, for β > 0 large enough, we have uβ ≡ vβ . Taking into account the behavior of the functions Uβ,x β and Vβ,y β , one can prove that, up i i to a subsequence, Zβ,x β converges as β → +∞ to a function Zi , with supRn |Zi | 1, which i
satisfies the equation Z + Z = 0 in B(0, r¯1 ) and Z = 0 in Rn \ B(0, r¯1 ). Now, we prove that the interior and the exterior normal derivatives of Zi on the boundary of B(0, r¯1 ) coincide, so we can say that Zi is a weak solution of the equation Z + a(x)Z = 0 in Rn . In order to prove this fact, set uβ,x β (x) = uβ i
x β √ + xi , β
vβ,x β (x) = vβ i
x β pβ,x β (x) = p √ + xi . i β
x β √ + xi , β (3.57)
After rescaling, Lemma 3.4 implies that there exists λu,β ∈ Rn such that
Duβ,x β Dψ + i
B(0,3¯r1 )
=
u+
β
β,xi
λ1 − 1 u β ψ − u+ β ψ + pβ,x β ψ dx β,xi i β β,xi β
(x)ψ(x)(λu,β · x) dx
∀ψ ∈ H01 B(0, 3¯r1 ) .
(3.58)
B(0,3¯r1 )
Moreover, for β > 0 (large enough) such that that there exists λv,β in Rn such that
√ β β β|xi − yi | < r¯1 (see (3.53)), Lemma 3.4 implies
λ1 1 Dvβ,x β Dψ + v − β ψ − v + β ψ + pβ,x β ψ dx β,xi i i β β,xi β
B(0,2¯r1 )
= B(0,2¯r1 )
β β dx v + β (x)ψ(x) λv,β · x − β yi − xi β,xi
∀ψ ∈ H01 B(0, 2¯r1 ) .
(3.59)
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We can prove that lim λu,β = 0 and
β→+∞
lim λv,β = 0.
(3.60)
β→+∞
Consider for example the first limit (the second one can be proved in a similar way). Arguing by contradiction, assume that (up to a subsequence) lim |λu,β | > 0 and
β→+∞
lim
β→+∞
λu,β = λ¯ |λu,β |
(3.61)
¯ = 1. Then, choose in (3.58) ψ(x) = ψλ¯ (x) = ζ (x)(λ¯ · x) where for a suitable λ¯ ∈ Rn such that |λ| ζ is a function in C01 (B(0, 3¯r1 )) such that ζ (x) = 1 ∀x ∈ B(0, r¯1 ). Thus, taking into account that
DU Dψλ¯ − U + ψλ¯ dx = 0 and
B(0,3¯r1 )
U (x)(λ¯ · x)2 dx > 0,
(3.62)
B(0,¯r1 )
as β → +∞ we obtain from (3.58)
U (x)(λ¯ · x)2 dx
0= B(0,¯r1 )
−1
DU Dψλ¯ − U + ψλ¯ dx = lim |λu,β | > 0, β→+∞
(3.63)
B(0,3¯r1 )
that is a contradiction. Combining (3.58) and (3.59), we obtain λ1 − D(uβ,x β − vβ,x β )Dψ + u β − v − β ψ − u+ β − v + β ψ dx β,xi β,xi β,xi i i β β,xi
B(0,2¯r1 )
+ u
=
β β,xi
(x) − v + β (x) ψ(x)(λu,β · x) dx β,xi
B(0,2¯r1 )
+
v + β (x)ψ(x) (λu,β − λv,β ) · x dx β,xi
B(0,2¯r1 )
β β + λv,β · β yi − xi
v + β (x)ψ(x) dx β,xi
∀ψ ∈ H01 B(0, 2¯r1 ) .
(3.64)
B(0,2¯r1 )
We say that −1
β β vβ L2 (Ω) λv,β · β yi − xi < +∞ lim sup sup |uβ − vβ |
(3.65)
−1 lim sup sup |uβ − vβ | vβ L2 (Ω) |λu,β − λv,β | < +∞.
(3.66)
β→+∞
Ω
and
β→+∞
Ω
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2277
For the proof of (3.65) we argue by contradiction and assume that, up to a subsequence, lim
β→+∞
−1
β β sup |uβ − vβ | = +∞ (or −∞). (3.67) vβ L2 (Ω) λv,β · β yi − xi Ω
Then, choose in (3.64) a function ψ such that ψ(x) > 0 ∀x ∈ B(0, r¯1 ) and let β → +∞. Taking into account (3.60), we obtain
DZi Dψ dx − B(0,2¯r1 )
Zi ψ dx
B(0,¯r1 )
= lim
β→+∞
−1 sup |uβ − vβ | Ω
v + β (x)ψ(x) (λu,β − λv,β ) · x dx β,xi
B(0,2¯r1 )
β β + λv,β · β yi − xi
v + β (x)ψ(x) dx .
β,xi
(3.68)
B(0,2¯r1 )
Notice that
lim vβ −1 L2 (Ω) β→+∞ B(0,2¯r1 )
v + β (x)ψ(x) dx β,xi
=
U (x)ψ(x) dx > 0,
(3.69)
B(0,¯r1 )
that vβ −1 L2 (Ω)
v + β (x)ψ(x) (λu,β − λv,β ) · x dx
β,xi
B(0,2¯r1 )
vβ −1 |λ L2 (Ω) u,β
v + β (x)ψ(x)|x| dx
− λv,β |
β,xi
(3.70)
B(0,2¯r1 )
and that lim vβ −1 L2 (Ω)
v + β (x)ψ(x)|x| dx =
β,xi
β→+∞
B(0,2¯r1 )
U (x)ψ(x)|x| dx < +∞.
(3.71)
B(0,¯r1 )
Therefore, from (3.68) we infer that (3.67) implies lim
β→+∞
−1 sup |uβ − vβ | vβ L2 (Ω) |λu,β − λv,β | = +∞.
(3.72)
Ω
It follows that, up to a subsequence, λu,β = λv,β and there exists λ˜ ∈ Rn , |λ˜ | = 1, such that λu,β − λv,β = λ˜ . β→+∞ |λu,β − λv,β | lim
(3.73)
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˜ r¯1 ) and vanishes Thus, we can choose in (3.64) a function ψ˜ which is strictly positive in B( r¯21 λ, 4 elsewhere. It follows that
˜ U (x)ψ(x) dx > 0 and
B(0,¯r1 )
˜ U (x)ψ(x)( λ˜ · x) dx > 0.
(3.74)
B(0,¯r1 )
Therefore, if the limit in (3.67) is equal to +∞, as β → +∞ we obtain from (3.64)
DZi D ψ˜ dx −
B(0,2¯r1 )
Zi ψ˜ dx
B(0,¯r1 )
= lim
β→+∞
−1 sup |uβ − vβ | vβ L2 (Ω) |λu,β − λv,β | Ω
B(0,¯r1 )
−1
β β sup |uβ − vβ | vβ L2 (Ω) λv,β · β yi − xi
+ lim
˜ U (x)ψ(x)( λ˜ · x) dx
β→+∞
Ω
˜ U (x)ψ(x) dx
B(0,¯r1 )
= +∞,
(3.75)
which is a contradiction. If the limit in (3.67) is equal to −∞,an analogous contradiction can ˜ ˜ ˜ be obtained replacing the function ψ(x) by ψ(−x), because B(0,¯r1 ) U (x)ψ(−x) dx > 0 and ˜ ˜ B(0,¯r1 ) U (x)ψ(−x)(λ · x) dx < 0. Thus, the proof of (3.65) is complete. For the proof of (3.66), we argue again by contradiction and assume that, up to a subsequence, (3.72) and (3.73) hold. Then, choose ψ˜ as before and let β → +∞ in (3.64). Thus, taking into account (3.65), we obtain
DZi D ψ˜ dx −
B(0,2¯r1 )
Zi ψ˜ dx
B(0,¯r1 )
lim
β→+∞
−1 sup |uβ − vβ | vβ L2 (Ω) |λu,β − λv,β | Ω
+ lim inf
β→+∞
˜ U (x)ψ(x)( λ˜ · x) dx
B(0,¯r1 )
−1
β β sup |uβ − vβ | vβ L2 (Ω) λv,β · β yi − xi Ω
U ψ˜ dx
B(0,¯r1 )
= +∞,
(3.76)
which is a contradiction. So (3.66) is proved. Now, notice that, for every smooth function ψ¯ : ∂B(0, r¯1 ) → R, we can find a smooth func¯ tion ψ in H 1 (B(0, 2¯r1 )) such that ψ(x) = ψ(x) ∀x ∈ ∂B(0, r¯1 ) and, in addition, 0
U (x)ψ(x)x dx = 0 B(0,¯r1 )
U (x)ψ(x) dx = 0.
and B(0,¯r1 )
(3.77)
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Taking into account (3.65), (3.66) and (3.77), as β → +∞ we obtain from (3.64)
[DZi Dψ − Zi ψ] dx +
B(0,¯r1 )
DZi Dψ dx = 0,
(3.78)
A(¯r1 ,2¯r1 )
where A(¯r1 , 2¯r1 ) = B(0, 2¯r1 ) \ B(0, r¯1 ). It follows that ψ(DZi · ν) dσ + ψ(DZi · ν) dσ = 0 ∂B(0,¯r1 )
(3.79)
∂A(¯r1 ,2¯r1 )
(where ν denotes the outward normal). Since ψ = ψ¯ on ∂B(0, r¯1 ) and ψ = 0 on ∂B(0, 2¯r1 ), taking into account that ψ¯ is an arbitrary function, we infer that the interior and the exterior normal derivatives of Zi on ∂B(0, r¯1 ) coincide, so Zi is a weak solution of the equation Z + a(x)Z = 0 in Rn . 2 Lemma 3.6. Assume that Z is a weak solution of the equation Z + a(x)Z = 0 in Rn where a(x) = 1 if x ∈ B(0, r¯1 ) and a(x) = 0 otherwise. If in addition we have supRn |Z| < +∞, then there exist c ∈ R and τ ∈ Rn such that Z = cU + (DU · τ ) (see Proposition 3.3). Proof. Let us denote by α0 < α1 α2 · · · the eigenvalues of the Laplace–Beltrami operator on the unit sphere S in Rn and by ϕ0 , ϕ1 , ϕ2 , . . . a base of eigenfunctions normalized and orthogonal in L2 (S). It is well known (see [42]) that α0 = 0, that ϕ0 is a constant function, that α1 = α2 = · · · = αn = n − 1 < αn+1 (namely the second eigenvalue has multiplicity n) and that the coordinate functions ϕj (x) = xj (j = 1, . . . , n), and the linear combinations of these, are the corresponding eigenfunctions. Now, let us consider the functions hj (j = 0, 1, 2, . . .) defined for r 0 by hj (r) =
Z(rx)ϕj (x) dσ.
(3.80)
S
Thus, we have Z(x) =
∞ j =0
hj |x| ϕj
x |x|
∀x ∈ Rn .
(3.81)
Taking into account that Z is a weak solution of the equation Z + a(x)Z = 0 in Rn , a direct computation shows that the functions hj are weak solutions of the equation −
1 r n−1
d 1 n−1 d r hj (r) + 2 αj hj (r) = a(r)hj (r) dr dr r
(3.82)
in ]0, +∞[, where we set a(r) = 1 for r ∈ [0, r¯1 ] and a(r) = 0 for r > r¯1 . We say that, if j > n, then hj (r) = 0 ∀r 0. In fact, first notice that hj (0) = 0 because S ψj (x) dσ = 0 for j 1. Then, arguing by contradiction, assume that hj ≡ 0 in [0, +∞[. We say that, in this case, hj cannot have more than one zero in ]0, +∞[. In fact, if z1 and z2 ,
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z1 < z2 , are two positive zeroes of hj , we have hj (z1 ) = 0 (otherwise hj ≡ 0 because it is solution of a Cauchy problem in z1 for the equation (3.82)). It is clear that we may assume h (z1 ) > 0 (otherwise we replace hj by −hj ). Thus, if we set z˜ 2 = sup r ∈ ]z1 , z2 [: hj (t) > 0 ∀t ∈ ]z1 , r[ ,
(3.83)
we have z˜ 2 ∈ ]z1 , z2 ]. d U (r, 0, . . . , 0) (defined for r 0). Now, let us introduce the positive function h(r) = − dr We say that h is a weak solution of the equation (3.82) for αj = n − 1. In order to prove this ∂U assertion, it suffices to take into account that the function ∂x (for example) is a weak solution of 1 ∂U N the equation Z + a(x)Z = 0 in R and that, for Z = ∂x1 , we have h1 (r) = c1 h(r) ∀r 0, for a suitable constant c1 = 0 (as one can easily verify by a direct computation). Let us consider the functions χj defined by χj (r) = h (r)hj (r) − h(r)hj (r) ∀r 0. A direct computation shows that χj (r) +
h(r)hj (r) n−1 χj (r) = [n − 1 − αj ] r r2
∀r > 0.
(3.84)
Taking into account that αj > n − 1 ∀j > n, that h(r) > 0 ∀r > 0 and that hj (r) > 0 ∀r ∈ ]z1 , z˜ 2 [, it follows that χj (r) +
n−1 χj (r) < 0 r
∀r ∈ ]z1 , z˜ 2 [.
(3.85)
As a consequence, the function Θj defined by Θj (r) = r n−1 χj (r) = r n−1 h (r)hj (r) − h(r)hj (r)
∀r 0
(3.86)
is strictly decreasing on ]z1 , z˜ 2 [. Therefore, since Θj (z1 ) = −z1n−1 h(z1 )hj (z1 ) < 0, we have also
Θj (˜z2 ) < 0 which is impossible because Θj (˜z2 ) = −˜z2n−1 h(˜z2 )hj (˜z2 ) and hj (˜z2 ) 0. Thus, hj has at the most one zero in ]0, +∞[ and, if we set r˜ = sup{r: r > 0, hj (t) = 0 ∀t ∈ ]0, r[}, we may have r˜ ∈ ]0, +∞[ or r˜ = +∞. Moreover, it is clear that we can assume hj (r) > 0 ∀r ∈ ]0, r˜ [ (otherwise we replace hj by −hj ). If r˜ < +∞, we have hj (˜r ) = 0 and hj (˜r ) 0. Moreover, Θj (˜r ) < 0 because limr→0 Θj (r) = 0 and Θj is strictly decreasing in ]0, r˜ [ (as we infer since hj (r) > 0 ∀r ∈ ]0, r˜ [). Therefore, we have Θj (˜r ) = −˜r n−1 h(˜r )hj (˜r ) < 0, which is impossible because hj (˜r ) 0. Thus, we have r˜ = +∞ and Θj (r) Θj (¯r1 ) < 0 ∀r r¯1 . Notice that, for r r¯1 , h(r) = 1 − c¯1 Θj (¯r1 ) > 0, we have hj (r) +
c¯1 r n−1
for a suitable constant c¯1 > 0. Therefore, for c =
n−1 hj (r) c > 0 ∀r r¯1 . r
(3.87)
If we set Hj (r) = r n−1 hj (r), it follows that Hj (r) cr n−1 ∀r r¯1 . Therefore, we have H (r)
h (r)
lim infr→∞ rj n nc , that is lim infr→∞ jr nc > 0, which implies limr→∞ hj (r) = +∞. It is clear that we have obtained a contradiction because supRn |Z| < +∞ implies sup[0,+∞[ hj < +∞. Thus, we have proved that hj (r) = 0 ∀r 0, ∀j > n.
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Notice that h0 (¯r1 ) = 0. In fact, since ϕ0 is constant on S, we have h0 (¯r1 ) = c0 ∂B(0,¯r1 ) Z dσ for a suitable constant c0 . Taking into account that Z + Z = 0 in B(0, r¯1 ), we obtain
(Z + Z)U dx = −
0= B(0,¯r1 )
∂B(0,¯r1 )
=−
Z(DU · ν) dσ +
Z(U + U ) dx
B(0,¯r1 )
Z(DU · ν) dσ
(3.88)
∂B(0,¯r1 ) x ∀x ∈ ∂B(0, r¯1 ). Since (DU · ν) is a nonzero constant function on ∂B(0, r¯1 ), it where ν = |x| follows that ∂B(0,¯r1 ) Z dσ = 0, so h0 (¯r1 ) = 0. Therefore, we have
Z(x) =
n
hj (¯r1 )xj
∀x ∈ ∂B(0, r¯1 )
(3.89)
j =1
and (as one can easily verify) there exists τ ∈ Rn such that Z(x) = DU (x) · τ
∀x ∈ ∂B(0, r¯1 ).
(3.90)
On the other hand, a direct computation shows that the function (DU · τ ) is a weak solution of the equation Z + a(x)Z = 0 in Rn and, in particular, it solves the Dirichlet problem
Z + Z = 0 Z = (DU · τ )
in B(0, r¯1 ), on ∂B(0, r¯1 ).
(3.91)
Notice that this problem in B(0, r¯1 ) has only solutions of the form Z = (DU · τ ) + cU , with c ∈ R, which are weak solutions of the equation Z + a(x)Z = 0 in Rn for all c ∈ R (and belong to D1,2 (Rn ) if and only if c = 0). Since the interior and the exterior normal derivatives of Z on ∂B(0, r¯1 ) coincide, it follows that every solution Z, which is bounded on Rn , has this form for a suitable c ∈ R. 2 β
Proposition 3.7. Fix k ∈ N and let β > 0 (large enough) such that Ωβk = ∅. Let (x1 , β
. . . , xk ) ∈ Ωβk and uβ ∈ S
β β β x1 ,...,xk
β
β
such that fβ (uβ ) = μβ (x1 , . . . , xk ) = maxΩ k μβ . β
Then, there exists β¯k > 0 such that uβ is a solution of problem (1.7) for every β > β¯k . Proof. Taking into account Lemma 3.4, we have to prove that there exists β¯k > 0 such that, for every i = 1, . . . , k, λiβ = 0 ∀β > β¯k . Arguing by contradiction, assume that, for some i ∈ {1, . . . , k}, there exists a sequence (βj )j such that limj →∞ βj = +∞ and λiβj = 0 ∀j ∈ N. Clearly, we can assume that
lim sup j →∞
|λm βj | |λiβj |
1 for m = 1, . . . , k
(3.92)
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(otherwise we replace i by some m = i and the sequence (βj )j by a suitable subsequence). Since |λiβj | = 0, there exists λi ∈ Rn , with |λi | = 1, such that (up to a subsequence) β
β
as j → ∞. Then, choose (y1 j , . . . , yk j ) ∈ Ωβkj and vβj ∈ S β β μβj (y1 j , . . . , yk j ),
in such a way that
β ymj
=
β xmj
for m =
εj > 0 ∀j ∈ N and limj →∞ εj = 0 (this choice of
β β limj →∞ βj |xi j − xmj | = ∞ for m = i). Let us set
βj
j
|λiβ |
→ λi
j
βj , satisfying fβj (vβj ) =
βj
y1 ,...,yk β i and yi j
β β y 1 j , . . . , yk j
λiβ
β
ε = xi j + √j λi , with βj
is indeed possible because
v − uβj −1 u . sj = sup vβj −1 L2 (Ω) βj L2 (Ω) βj
(3.93)
Ω
We say that limj →∞ sj = 0. In fact, arguing as in the proof of Proposition 3.3, one can prove that, as uβj −1 u , also vβj −1 v → −e1 in H01 (Ω) as j → ∞. Moreover, taking into L2 (Ω) βj L2 (Ω) βj account Lemma 3.5, we infer that lim
sup vβj −1 v − uβj −1 u = 0 for m = 1, . . . , k. L2 (Ω) βj L2 (Ω) βj
j →∞ B(x i ,r ) m βj
(3.94)
β Notice that in the domain Ωj = Ω \ km=1 B(xmj , rβj ) (for j large enough so that v − uβj −1 u both uβj and vβj are negative in Ωj ) the function ζj = vβj −1 L2 (Ω) βj L2 (Ω) βj
solves the equation ζj + λ1 ζj = ( vβj −1 − uβj −1 )p with limj →∞ vβj L2 (Ω) = L2 (Ω) L2 (Ω) limj →∞ uβj L2 (Ω) = ∞. It follows that we have also limj →∞ supΩj |ζj | = 0 which, combined with (3.94), yields limj →∞ sj = 0. Now, let us consider the functions (supΩ |vβj − uβj |)−1 (vβj − uβj ) (notice that we have supΩ |vβj − uβj | > 0 and sj > 0 because εj = 0). A direct computation gives −1 (vβj − uβj ) sj vβj L2 (Ω) =
1 −1 vβj −1 2 (Ω) vβj − uβj L2 (Ω) uβj L sj +
1 1 − vβj −1 uβj −1 u 2 u . L2 (Ω) βj L (Ω) L2 (Ω) βj sj
(3.95)
By Lemmas 3.5 and 3.6, for every m ∈ {1, . . . , k} there exist τm ∈ Rn and cm ∈ R such that (up to a subsequence) Z βj converges to cm U + (DU · τm ) as j → ∞ and the convergence is βj ,xm
uniform on the compact subsets of Rn . We say that τm = 0 for some m ∈ {1, . . . , k}. In fact, arguing by contradiction, assume that τm = 0 for every m ∈ {1, . . . , k}. In this case, we must have also cm = 0 for every m ∈ {1, . . . , k}. In fact, set Ωj = {x ∈ Ω: uβj (x) < 0, vβj (x) < 0} and notice that, for j large enough, we have β Ω \ Ωj ⊆ km=1 B(xmj , √r ) for every fixed r > r¯1 . Therefore, since τm = 0 ∀m ∈ {1, . . . , k}, βj
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
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we have limj →∞ (supΩ |vβj − uβj |)−1 sup∂ Ωj |vβj − uβj | = 0. Taking into account that vβj and uβj both satisfy the equation u + λ1 u = p in Ωj and that λ1 (Ωj ) = inf |Du|2 dx: u ∈ H01 (Ωj ), u2 dx = 1 > λ1 , Ωj
(3.96)
Ωj
it follows that limj →∞ (supΩ |vβj − uβj |)−1 supΩj |vβj − uβj | = 0. ¯ ∈ {1, . . . , k}, we have If (arguing by contradiction) cm¯ = 0 for some m
0 < |cm¯ | sup |U | = lim
j →∞
∂B(0,r)
−1 sup |vβj − uβj | Ω
βj
lim
j →∞
|vβj − uβj |
sup βj
∂B(xm¯ , √r
)
−1 sup |vβj − uβj | sup |vβj − uβj | = 0 Ω
(3.97)
Ωj
β
for every r > r¯1 (because ∂B(xm¯ j , √r ) ⊂ Ωj for j large enough). βj
It is clear that (3.97) is impossible, so τm = 0 ∀m ∈ {1, . . . , k} implies cm = 0 ∀m ∈ {1, . . . , k}. On the other hand, if τm = 0 and cm = 0 ∀m ∈ {1, . . . , k}, we obtain lim
j →∞
−1 sup |vβj − uβj | Ω
sup βj B(xm ,rβj )
|vβj − uβj | = 0 ∀m ∈ {1, . . . , k}
(3.98)
which (arguing as before) implies lim
j →∞
−1 sup |vβj − uβj | sup |vβj − uβj | = 0. Ω
(3.99)
Ωj
It follows that limj →∞ (supΩ |vβj − uβj |)−1 supΩ |vβj − uβj | = 0 which obviously is a contradiction. Thus, we can conclude that τm = 0 for some m ∈ {1, . . . , k}. As a first consequence of this fact, we infer that lim sup j →∞
1 1 − vβj −1 u 2 < +∞ L2 (Ω) βj L (Ω) sj
(3.100)
which (since limj →∞ sj = 0) implies u 2 = 1. lim vβj −1 L2 (Ω) βj L (Ω)
j →∞
(3.101)
In order to prove (3.100), we argue by contradiction and assume that, up to a subsequence, 1 1 − vβj −1 2 (Ω) uβj L2 (Ω) = +∞. L j →∞ sj lim
(3.102)
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Taking into account (3.95), from (3.102) it follows that −1 = sup |U | lim sup |vβj − uβj | vβj L2 (Ω) − uβj L2 (Ω)
j →∞ Ω
Rn
(3.103)
and lim Z
j →∞
βj βj ,xm
−1 = sup |U | U Rn
∀m ∈ {1, . . . , k}.
(3.104)
This means that cm = (supRn |U |)−1 and τm = 0 ∀m ∈ {1, . . . , k}. Since (as we proved before) τm = 0 for some m ∈ {1, . . . , k}, we have a contradiction and so we can conclude that (3.100) holds. Notice that −1 0 < lim inf sup |vβj − uβj | sj vβj L2 (Ω) j →∞
Ω
−1 lim sup sup |vβj − uβj | sj vβj L2 (Ω) < +∞. j →∞
(3.105)
Ω
In fact, the second inequality follows directly from (3.95) and (3.100). In order to prove the first inequality, assume that (up to a subsequence) 1 −1 1 − vβj −1 2 (Ω) uβj L2 (Ω) uβj L2 (Ω) sup |uβj | = L L j →∞ sj Ω lim
(3.106)
for a suitable constant L 0. If L = 1, the inequality is a simple consequence of the definition of sj . If L = 1, we proceed as follows. Arguing by contradiction, assume that (up to a subsequence) −1 lim sup |vβj − uβj | sj vβj L2 (Ω) = 0.
j →∞ Ω
(3.107)
Taking into account (3.95), the minimality properties of uβ and Proposition 3.3, it follows that (up to a subsequence) s1j ζj → ce1 in L2 (Ω), as j → ∞, for a suitable constant c = 0. On the other hand, a direct computation gives sj Ω
ζj sj
2
dx + 2 Ω
uβj −1 u L2 (Ω) βj
ζj dx = 0 ∀j ∈ N, sj
(3.108)
which, as j → ∞, yields c Ω e12 dx = 0, that is a contradiction. Thus, the first inequality in (3.105) is proved also in the case L = 1. Now, we prove that τm = 0 for m = i (thus, as a consequence, τi = 0).
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
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By rescaling we obtain, for m = 1, . . . , k,
B(0,3¯r1 )
2 x β vβ+j + xmj x dx βj
= B(0,3¯r1 )
2 x βj
u+ + x x dx m βj βj
+2 B(0,3¯r1 )
+ B(0,3¯r1 )
x x x βj βj βj + +
v − u x dx u+ + x + x + x m m m βj βj βj βj βj βj
2 x x β βj
vβ+j + xmj − u+ + x x dx. m βj βj βj
(3.109)
If m = i, we have (for j large enough)
vβ+j
B(0,3¯r1 )
x β
+ xmj βj
2
x dx = B(0,3¯r1 )
2 x βj + uβj + xm x dx = 0. βj
(3.110)
Thus, taking into account (3.101) and (3.105), since limj →∞ sj = 0, we obtain from (3.109)
U (x) cm U (x) + DU (x) · τm x dx = 0
for m = i
(3.111)
B(0,¯r1 )
which (because of the symmetry of U ) easily implies τm = 0 for m = i. Since τm = 0 for some m ∈ {1, . . . , k}, we must have τi = 0. Moreover, if m = i we have (for j large enough) B(0,3¯r1 )
u+ βj
x β
+ xi j βj
2 x dx = 0
(3.112)
and B(0,3¯r1 )
2 x β vβ+j + xi j x dx = εj λi βj
B(0,3¯r1 )
2 x β vβ+j + xi j dx. βj
(3.113)
Taking into account (3.101), (3.105), (3.109), (3.112) and (3.113), since limj →∞ sj = 0, it follows that (up to a subsequence) εj i λ j →∞ sj
U 2 (x) dx = γ¯i
lim
B(0,¯r1 )
B(0,¯r1 )
U (x) ci U (x) + DU (x) · τi x dx
(3.114)
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for a suitable positive constant γ¯i . Since τi = 0, taking into account the properties of U , it follows that εj ∈ ]0, +∞[ j →∞ sj lim
(3.115)
and τi = −γi λi for a suitable constant γi > 0. Notice that, if the sequence (εj )j is chosen in such a way that in addition it satisfies the condition −1 3 lim εj λiβj βj2 = 0,
(3.116)
j →∞
then from (3.105) and (3.115) we infer that i −1 32 β =0 lim uβj −1 sup |v − u | β β 2 j j λβj j L (Ω)
j →∞
(3.117)
Ω
(this property will be used later). A direct computation shows that, for j large enough, fβj (vβj ) fβj (uβj ) + fβ j (uβj )[vβj − uβj ] 2 1 D(vβj − uβj ) − λ1 (vβj − uβj )2 dx + 2 Ωj
−
k βj 2
(vβj − uβj )2 dx.
(3.118)
m=1 βj B(xm ,rβj )
Now, fix a function π ∈ C01 (B(0, 3¯r1 )) such that π(x) = 1 ∀x ∈ B(0, 2¯r1 ) and, for m = 1, . . . , k,
β β set πjm (x) = π[ βj (x − xmj )] ∀x ∈ B(xmj , rβj ) (πjm (x) = 0 elsewhere). β Then, for j (large enough) such that uβj < 0 in Ω \ km=1 B(xmj , 23 rβj ), we have ! fβ j (uβj )
1−
k
πjm
"
(vβj − uβj ) = 0.
m=1
It follows that fβj (vβj ) − fβj (uβj )
k
fβ j (uβj ) πjm (vβj − uβj )
m=1
1 + 2
Ωj
D(vβ − uβ )2 − λ1 (vβ − uβ )2 dx j j j j
(3.119)
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
−
2287
k βj 2
(vβj − uβj )2 dx,
(3.120)
m=1 βj B(xm ,rβj )
where, for m = 1, . . . , k, fβ j (uβj ) πjm (vβj − uβj ) =
m βj m dx u+ βj ,m (x)πj (x)(vβj − uβj )(x) λβj · x − xm
(3.121)
Ω β
because πjm (vβj − uβj ) ∈ H01 (B(xmj , rβj )) (see Lemma 3.4).
Now, set zj = (supΩ |vβj − uβj |)−1 (vβj − uβj ) and Zj (x) = zj ( √x ) ∀x ∈ βj Ω. βj
Taking into account that (vβj − uβj ) + λ1 (vβj − uβj ) = 0 in Ωj , for j large enough, we infer that there exists t ∈ R such that (up to a subsequence) zj → te1 in H01 (Ω) as j → ∞. Since
|Dzj |2 − λ1 zj2 dx =
Ω
D(zj − te1 )2 − λ1 (zj − te1 )2 dx,
(3.122)
Ω
we have
|Dzj |2 − λ1 zj2 dx =
Ωj
D(zj − te1 )2 − λ1 (zj − te1 )2 dx
Ω
−
k
|Dzj |2 − λ1 zj2 dx.
(3.123)
m=1 βj B(xm ,rβj )
Therefore, after rescaling, we obtain from (3.120) −1 n+1 λi uβ 2 sup |vβ − uβ | β 2 fβ (vβ ) − fβ (uβ ) j L (Ω) j j j j j j βj j Ω
k
U+
m=1B(0,3¯r ) 1
βj (x)π(x)Z
βj βj ,xm
βj ,xm
λm βj · x dx (x) |λiβj |
3 −1 1 sup |vβj − uβj |βj2 λiβj uβj L2 (Ω) 2 Ω
2
2 D Zj (x) − te1 x − λ1 Zj (x) − te1 x dx × βj βj βj √
+
βj Ω
−
k
m=1B(0,3¯r ) 1
k λ1 2 |DZ βj | − Z βj dx − βj ,xm βj βj ,xm
2
m=1B(0,3¯r ) 1
Z
2
βj
βj ,xm
dx .
(3.124)
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Set Zm = cm U + (DU · τm ). Then Z
βj
βj ,xm
2
2 D Zj (x) − te1 x − λ1 Zj (x) − te1 x dx βj βj βj
lim
→ Zm as j → ∞ and we have
j →∞
√
βj Ω
=
k
|D Zm |2 dx < +∞.
(3.125)
m=1Rn
Moreover, for m = 1, . . . , k,
|DZ
lim
βj βj ,xm
j →∞ B(0,3¯r1 )
|2 −
λ1 2 Z βj dx = βj βj ,xm
|D Zm |2 dx < +∞
(3.126)
B(0,3¯r1 )
and
lim
j →∞ B(0,3¯r1 )
Z2
βj βj ,xm
dx =
2 dx < +∞. Zm
(3.127)
B(0,3¯r1 )
Therefore, taking also into account (3.92) and (3.117), since τm = 0 for m = i and τi = −γi λi , it follows from (3.124) that −1 n+1 lim inf λiβj uβj L2 (Ω) sup |vβj − uβj | βj 2 fβj (vβj ) − fβj (uβj ) j →∞
−γi
Ω
U (x) DU (x) · λi x · λi dx > 0.
(3.128)
B(0,¯r1 )
It is clear that we have a contradiction because fβj (vβj ) fβj (uβj ) ∀j ∈ N, since fβj (vβj ) = β
β
μβj (y1 j , . . . , yk j ) and fβj (uβj ) = maxΩ k μβj . βj
Thus, we can conclude that uβ is a solution of problem (1.7) for β > 0 large enough.
2
4. Proof of the main results Proof of Theorem 1.1. Taking into account Propositions 2.2 and 2.3, for every positive inteβ β ger k and for β > 0 (large enough) such that Ωβk = ∅, let us consider a point (x1 , . . . , xk ) ∈ Ωβk and a function uβ ∈ S
β β β x1 ,...,xk
β
β
such that fβ (uβ ) = μβ (x1 , . . . , xk ) = maxΩ k μβ . Then, if we set β
uk,β = uβ , all the assertions in Theorem 1.1 follow directly from the propositions proved in Section 3, as one can easily verify (see, in particular, Propositions 3.2, 3.3 and 3.7). 2 Theorem 4.1. Let Ω be a bounded connected domain of Rn with n 3, k ∈ N and p, ξ0 ∈ L2 (Ω) such that p > 0 in Ω and infΩ (ξ0 /p) > −∞. Then, there exists β¯ (depending on k, p and ξ0 )
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
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such that for all β β¯ there exists t¯(β) (which also depends on k, p and ξ0 ) having the following properties. For all β and t, such that β β¯ and t t¯(β), there exists a k-peaks solution uk,β,t of the problem u − λ1 u− + βu+ = ξ0 + tp in Ω, (4.1) u=0 on ∂Ω such that, as t → +∞, 1t uk,β,t converges in H01 (Ω) to the solution uk,β of problem (1.7), given by Theorem 1.1. For the proof, we have only to describe how the method used to prove Theorem 1.1 can be adapted to prove Theorem 4.1. Let us consider (for t > 0) the equivalent problem
1 u − λ1 u− + βu+ = ξ0 + p t u=0
in Ω,
(4.2)
on ∂Ω.
It is clear that a function u ∈ H01 (Ω) is a solution of problem (4.1) if and only if 1t u solves problem (4.2). The solutions of problem (4.2) are the critical points of the functional fβ,t : H01 (Ω) → R defined by 1 ξ0 u dx ∀u ∈ H01 (Ω). (4.3) fβ,t (u) = fβ (u) + t Ω β,t
Let us consider the set Sx1 ,...,xk consisting of all the k-peaks functions, with respect to the balls (u)[u+ ] = 0 for every i ∈ {1, . . . , k}. NoB(x1 , rβ ), . . . , B(xk , rβ ), such that (2.4) holds and fβ,t i tice that, for t > 0 such that infΩ (ξ0 /p) > −t, we have 1t ξ0 + p > 0 in Ω and, as a consequence, + u+ i (for i = 1, . . . , k) is the unique maximum point for fβ,t in the set {tui : t 0}. Then, arguing as in Section 2, for t > max{− infΩ (ξ0 /p), 0} one can consider the function μβ,t : Ωβk → R defined by μβ,t (x1 , . . . , xk ) = min fβ,t β,t Sx1 ,...,xk
∀(x1 , . . . , xk ) ∈ Ωβk
(4.4) β,t
and (as in the proof of Proposition 2.3) one can prove that, if Ωβk = ∅, there exists (x1 , β,t
β,t
β,t
. . . , xk ) ∈ Ωβk such that μβ,t (x1 , . . . , xk ) = maxΩ k μβ,t . β The following proposition describes what happens as t → +∞ (here ξ0 and p satisfy the same conditions as in Theorem 4.1). Proposition 4.2. For k and β > 0 such that Ωβk = ∅ and for t > max{− infΩ (ξ0 /p), 0}, conβ,t
β,t
sider a point (x1 , . . . , xk ) ∈ Ωβk and a function uβ,t ∈ S β,t β,t μβ,t (x1 , . . . , xk ) = maxΩ k β
μβ,t .
β,t β,t β,t x1 ,...,xk
such that fβ,t (uβ,t ) =
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R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295 β
β
Then, there exist (x1 , . . . , xk ) ∈ Ωβk and a function uβ ∈ S β
β
β β β x1 ,...,xk
such that fβ (uβ ) =
μβ (x1 , . . . , xk ) = maxΩ k μβ , which (up to a subsequence) satisfy the following properties: β
β,t lim x t→+∞ i
β
= xi
for i = 1, . . . , k,
(4.5)
uβ,t → uβ in H01 (Ω) as t → +∞ and limt→+∞ fβ,t (uβ,t ) = fβ (uβ ). β
β
Proof. Since Ωβk is compact, there exists (x1 , . . . , xk ) ∈ Ωβk such that (up to a subseβ
β
quence) (4.5) holds. Arguing by contradiction, assume that μβ (x1 , . . . , xk ) = maxΩ k μβ , that is β
β
β
β
β
β
β
there exists (x˜1 , . . . , x˜k ) ∈ Ωβk such that μβ (x1 , . . . , xk ) < μβ (x˜1 , . . . , x˜k ). Then, by standard arguments, one can show that, as t → +∞, uβ,t → uβ in H01 (Ω) and fβ,t (uβ,t ) → fβ (uβ ), β β β where uβ is a function in S β β such that fβ (uβ ) = μβ (x1 , . . . , xk ). x1 ,...,xk
On the other hand, if u˜ β,t is a function in S
β,t β β x˜1 ,...,x˜k
β
β
such that fβ,t (u˜ β,t ) = μβ,t (x˜1 , . . . , x˜k ), we
have that, as t → +∞, u˜ β,t → u˜ β in H01 (Ω) and fβ,t (u˜ β,t ) → fβ (u˜ β ), where u˜ β is a function β β β ˜ β ) = μβ (x˜1 , . . . , x˜k ). in S β β such that fβ (u x˜1 ,...,x˜k
β,t
β,t
β
β
Thus, we have fβ,t (uβ,t ) < fβ,t (u˜ β,t ), i.e. μβ,t (x1 , . . . , xk ) < μβ,t (x˜1 , . . . , x˜k ) for t > 0 β,t β,t large enough, in contradiction with the fact that μβ,t (x1 , . . . , xk ) = maxΩ k μβ,t . 2 β
Proof of Theorem 4.1 (conclusion). Our aim is to prove that uβ,t is a solution of problem (4.1) for β and t positive and large enough. Notice that, since 1t ξ0 + p > 0 in Ω for t > max{− infΩ (ξ0 /p), 0}, one can argue as in Section 3 (see Proposition 3.3 and Lemma 3.4) β,t β,t and exploit the fact that fβ,t (uβ,t ) = μβ,t (x1 , . . . , xk ) in order to prove that uβ,t (x) < 0 k β,t ∀x ∈ Ω \ i=1 B(xi , 23 rβ ), for β and t positive and large enough, and that, for i = 1, . . . , k, there exists λiβ,t ∈ Rn such that (uβ,t )[ψ] = fβ,t
i β,t dx u+ β,t,i (x)ψ(x) λβ,t · x − xi
β,t ∀ψ ∈ H01 B xi , rβ .
(4.6)
Ω
Then, we have to prove that all the multipliers λiβ,t are zero for β and t positive and large enough. ¯ +∞[ → R+ satisfying the following Indeed, we show that there exist β¯ > 0 and a function t¯ : [β, ¯ ¯ property: for all β and t such that β β and t t (β), all the multipliers λiβ,t are zero for i = 1, . . . , k. In fact, arguing by contradiction, assume that for some i ∈ {1, . . . , k} there exists a sequence (βj )j such that limj →∞ βj = +∞ and sup{t ∈ R: |λiβj ,t | = 0} = +∞ for all j ∈ N. It follows that there exists a sequence (tj )j , with limj →∞ tj = +∞, such that λiβj ,tj = 0. Now, we can proceed as in Section 3 in order to describe the asymptotic behavior of the function uβj ,tj . In particular, arguing as in Proposition 3.7, we can exploit the fact that β ,tj
λiβj ,tj = 0 in order to construct a sequence (y1 j
β ,tj
, . . . , yk j
)j in (Rn )k and a sequence (vβj ,tj )j
R. Molle, D. Passaseo / Journal of Functional Analysis 259 (2010) 2253–2295
in H01 (Ω), such that vβj ,tj ∈ S
βj ,tj βj ,tj
y1
β ,tj
βj ,tj
,...,yk
, fβj ,tj (vβj ,tj ) = μβj ,tj (y1 j
2291 β ,tj
, . . . , yk j
) and
lim infj →∞ aj [fβj ,tj (vβj ,tj ) − fβj ,tj (uβj ,tj )] > 0 for a suitable sequence (aj )j of positive numbers, in contradiction with the fact that fβj ,tj (vβj ,tj ) fβj ,tj (uβj ,tj ) ∀j ∈ N, which holds β ,tj
because fβj ,tj (vβj ,tj ) = μβj ,tj (y1 j
β ,tj
, . . . , yk j
) while fβj ,tj (uβj ,tj ) = maxΩ k μβj ,tj . βj
Thus, if we set uk,β,t = tuβ,t , all the assertions of Theorem 4.1 may be easily verified.
2
Remark 4.3. Under suitable assumptions, the result presented in Theorem 4.1 may be extended in order to cover the case of more general nonlinear terms g(s) which behave as −λ1 s − + βs + when s tends to +∞ and to −∞. 5. Existence of lower energy solutions In this section we describe a simple method to obtain other solutions of problems (1.7) and (4.1), corresponding to lower critical values of the related energy functionals, which present a different asymptotic behavior as β and t tend to +∞. Let us consider the set S β = u ∈ H01 (Ω): u+ ≡ 0, fβ (u) u+ = 0 .
(5.1)
Proposition 5.1. Let Ω be a bounded domain of Rn and p a positive function in L2 (Ω). Then, for all β > λ1 , the set S β is nonempty, the minimum minS β fβ is achieved and every minimizing function u¯ β is a solution of problem (1.7). Proof. It is clear that, since β > λ1 and p > 0 in Ω, τ e1 ∈ S β for a suitable τ > 0. Therefore S β = ∅. Indeed, one can easily verify that there exist also sign changing functions which belong to S β . Moreover, arguing as in the proof of Lemma 2.1, one can prove that − 2 − 2 Du dx: u ∈ S β , inf u dx = 1 > λ1 . Ω
(5.2)
Ω
Let us consider a minimizing sequence (uj )j for fβ on S β . The same arguments used in the proof of Proposition 2.2 show that the sequence (uj )j is bounded in H01 (Ω) (as a consequence of (5.2)) and that, up to a subsequence, it converges to a function u¯ β ∈ S β , such that fβ (u¯ β ) = minS β fβ . In order to prove that u¯ β is a critical point for fβ (since S β is not a smooth manifold) we exploit the fact that fβ (u¯ β + t u¯ + ¯ β ) ∀t −1 such that t = 0. In fact, arguing by conβ ) < fβ (u tradiction as in the proof of Lemma 3.4, if we assume that fβ (u¯ β ) = 0 one can construct by standard techniques a continuous map η : [−1, 1] → H01 (Ω) satisfying similar properties as the map η used in the proof of Lemma 3.4 and Proposition 3.3. The existence of such a map η implies the existence of a function u˜ β ∈ S β such that fβ (u˜ β ) < fβ (u¯ β ), in contradiction with the fact that fβ (u¯ β ) = minS β fβ . Thus, we can conclude that u¯ β is a solution of problem (1.7). 2 An analogous result holds for problem (4.1). In fact, using the notation introduced in Section 4, let us consider the set
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S β,t = u ∈ H01 (Ω): u+ ≡ 0, fβ,t (u) u+ = 0, fβ,t u+ > 0
(5.3)
(here we need to require also fβ,t (u+ ) > 0 because we do not have any information on the sign of ( 1t ξ0 + p)). Proposition 5.2. Let Ω be a bounded domain of Rn , p ∈ L2 (Ω), p > 0 in Ω, ξ0 ∈ L2 (Ω). Then, for all β > λ1 , there exists t¯β 0 such that, if β > λ1 and t > t¯β , we have: (a) S β,t = ∅, (b) the minimum minS β,t fβ,t is achieved, (c) if u¯ β,t is a minimizing function for fβ,t on S β,t , u¯ β,t is a critical point of fβ,t . Moreover, as t → +∞, u¯ β,t converges in H01 (Ω) to a critical point u¯ β of fβ , which minimizes fβ on S β . Proof. For all β > λ1 , let us consider the set 1 2 2 Hβ = u ∈ H0 (Ω): |Du| dx β, u dx = 1, u 0 in Ω Ω
(5.4)
Ω
and notice that the minimum m(ξ ) = min ξ u: u ∈ Hβ
(5.5)
Ω
is achieved for all ξ ∈ L2 (Ω). Moreover, since p > 0 in Ω, we have m(p) > 0. Then, if we set 0) t¯β = max{− m(ξ m(p) , 0}, we obtain
ξ0 1 m + p m(ξ0 ) + m(p) > 0 ∀t > t¯β . t t
(5.6)
(0)[u] > 0 for all u ∈ H . In particular, since e ∈ H , we have Thus, for t > t¯β , we have fβ,t β 1 β fβ,t (0)[e1 ] > 0 for all t > t¯β . It follows that, for all β and t such that β > λ1 and t > t¯β , there exists a unique τ (t) 0 satisfying fβ,t (τ (t)e1 ) = max{fβ,t (τ e1 ): τ 0}. Moreover τ (t) > 0 and fβ,t (τ (t)e1 ) > 0. Thus we have τ (t)e1 ∈ S β,t , which proves assertion (a). For the proof of (b), notice that one can argue as in the proof of Proposition 2.2 in order to show that every minimizing sequence for fβ,t on the set S β,t is bounded in H01 (Ω) and, up to a subsequence, it converges to a function u¯ β,t ∈ S β,t such that fβ,t (u¯ β,t ) = minS β,t fβ,t . In fact, in the proof of Proposition 2.2 the assumption p > 0 in Ω is used only to have fβ (0)[u] > 0 (0)[u] > 0 ∀u ∈ H , which holds ∀u ∈ Hβ (see (5.4)). Analogously, here we need only that fβ,t β 1 for all t > t¯β (even if nothing can be said about the sign of ( t ξ0 + p)). For the proof of assertion (c), one can exploit the fact that u¯ β,t is the unique maximum point for fβ,t in the set {u¯ β,t + t u¯ + β,t : t −1} and, as in the proof of Proposition 5.1, apply the technique already used in the proof of Lemma 3.4 and of Proposition 3.3. Finally, standard methods may be applied in order to study the asymptotic behavior of the solution u¯ β,t as t → +∞. 2
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Remark 5.3. Let us point out that the solutions found in this section differ from the k-peaks solutions obtained before, which (even in the case k = 1) correspond to higher critical values of the functionals fβ and fβ,t . β It is clear that, for all (x1 , . . . , xk ) ∈ Ωβk , Sx1 ,...,xk ⊆ S β so we have minS β fβ minS β fβ , x1 ,...,xk
β,t Sx1 ,...,xk
which implies fβ (u¯ β ) fβ (uk,β ). Moreover, we have also ⊆ which implies an analogous inequality for the solutions obtained for problem (4.1). The asymptotic behavior of the solutions as β and t tend to +∞ shows that, indeed, the strict inequality holds. Let us verify this fact, for example, for the solutions uk,β and u¯ β given by Theorem 1.1 and Proposition 5.1 respectively (similar arguments hold for the solutions of problem (4.1)). In fact, by Proposition 3.2, we have lim β
2−n 2
β→+∞
S β,t ,
2 2 −1 fβ (uk,β ) = − 2k cap(¯r1 ) max e1 pe1 dx Ω
(5.7)
Ω
while, for the solution u¯ β , we can prove that lim β
β→+∞
2−n 2
fβ (u¯ β ) = −∞.
(5.8)
It follows that, for β > 0 large enough, we have fβ (u¯ β ) < fβ (uk,β ). In order to prove (5.8), notice that for all x¯ ∈ Ω we have fβ (u¯ β ) minS β fβ for β > 0 large x¯
enough (because x¯ ∈ Ωβ1 for β large enough). Moreover, arguing as in the proof of Proposition 3.2, one can verify that lim β
β→+∞
2−n 2
2 −1 min fβ = − 2 cap(¯r1 ) e1 (x) ¯
pe1 dx
β
Sx¯
2
∀x¯ ∈ Ω.
(5.9)
Ω
Thus, if we let x¯ approach the boundary of Ω, we obtain (5.8). Actually, the solution u¯ β presents one peak which, as β → +∞, concentrates near a point of ∂Ω (see [36]). The asymptotic behavior of the solutions u¯ β and u¯ β,t , as β and t tend to +∞, suggests the natural problem of the existence of k-peaks solutions (for all k ∈ N) having the peaks localized near the boundary of Ω. Indeed, using a different mini–max method, one can prove that for all k ∈ N there exist k-peaks solutions of this type and that their number may be related to the geometrical properties of ∂Ω (see [36]). References [1] H. Amann, P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 84 (1–2) (1979) 145–151. [2] H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 7 (4) (1980) 539–603. [3] A. Ambrosetti, Elliptic equations with jumping nonlinearities, J. Math. Phys. Sci. 18 (1) (1984) 1–12. [4] A. Ambrosetti, G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4) 93 (1972) 231–246. [5] A. Bahri, Résolution générique d’une équation semi-linéaire, C. R. Acad. Sci. Paris Sér. A–B 291 (4) (1980) A251– A254.
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[6] H. Berestycki, Le nombre de solutions de certains problémes semi-linéaires elliptiques, J. Funct. Anal. 40 (1) (1981) 1–29. [7] M.S. Berger, E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1974/1975) 837–846. [8] B. Breuer, P.J. McKenna, M. Plum, Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof, J. Differential Equations 195 (1) (2003) 243–269. [9] R. Caccioppoli, Un principio di inversione per le corrispondenze funzionali e sue applicazioni alle equazioni alle derivate parziali, Atti Acc. Naz. Lincei 16 (1932) 392–400. [10] N.P. Các, On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue, J. Differential Equations 80 (2) (1989) 379–404. [11] N.P. Các, On a boundary value problem with nonsmooth jumping nonlinearity, J. Differential Equations 93 (2) (1991) 238–259. [12] D.G. Costa, D.G. de Figueiredo, P.N. Srikanth, The exact number of solutions for a class of ordinary differential equations through Morse index computation, J. Differential Equations 96 (1) (1992) 185–199. [13] E.N. Dancer, Multiple solutions of asymptotically homogeneous problems, Ann. Mat. Pura Appl. (4) 152 (1988) 63–78. [14] E.N. Dancer, A counterexample to the Lazer–McKenna conjecture, Nonlinear Anal. 13 (1) (1989) 19–21. [15] E.N. Dancer, On the uniqueness of the positive solution of a singularly perturbed problem, Rocky Mountain J. Math. 25 (3) (1995) 957–975. [16] E.N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (4) (1976/1977) 283–300. [17] E.N. Dancer, Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities, Topol. Methods Nonlinear Anal. 1 (1) (1993) 139–150. [18] E.N. Dancer, S. Yan, On the superlinear Lazer–McKenna conjecture, J. Differential Equations 210 (2) (2005) 317– 351. [19] E.N. Dancer, S. Yan, The Lazer–McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc. (2) 78 (3) (2008) 639–662. [20] D.G. de Figueiredo, On the superlinear Ambrosetti–Prodi problem, Nonlinear Anal. 8 (6) (1984) 655–665. [21] D.G. de Figueiredo, S. Solimini, A variational approach to superlinear elliptic problems, Comm. Partial Differential Equations 9 (7) (1984) 699–717. [22] D.G. de Figueiredo, P.N. Srikanth, S. Santra, Non-radially symmetric solutions for a superlinear Ambrosetti–Prodi type problem in a ball, Commun. Contemp. Math. 7 (6) (2005) 849–866. [23] O. Druet, The critical Lazer–McKenna conjecture in low dimensions, J. Differential Equations 245 (8) (2008) 2199– 2242. [24] S. Fuˇcík, Nonlinear equations with noninvertible linear part, Czechoslovak Math. J. 24 (99) (1974) 467–495. ˇ [25] S. Fuˇcík, Boundary value problems with jumping nonlinearities, Casopis Pˇest. Mat. 101 (1) (1976) 69–87. [26] T. Gallouët, O. Kavian, Résultats d’existence et de non-existence pour certains problèmes demi-linéaires à l’infini, Ann. Fac. Sci. Toulouse Math. (5) 3 (3–4) (1981) 201–246. [27] J.A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J. 20 (1970/1971) 983–996. [28] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (4) (1982) 493– 514. [29] A.C. Lazer, P.J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1) (1981) 282–294. [30] A.C. Lazer, P.J. McKenna, On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 95 (3–4) (1983) 275–283. [31] A.C. Lazer, P.J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues, Comm. Partial Differential Equations 10 (2) (1985) 107–150. [32] A.C. Lazer, P.J. McKenna, Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. II, Comm. Partial Differential Equations 11 (15) (1986) 1653–1676. [33] A. Marino, A.M. Micheletti, A. Pistoia, A nonsymmetric asymptotically linear elliptic problem, Topol. Methods Nonlinear Anal. 4 (2) (1994) 289–339. [34] R. Molle, D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2) (2010) 529–553. [35] R. Molle, D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, in preparation. [36] R. Molle, D. Passaseo, in preparation.
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[37] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986. [38] B. Ruf, On nonlinear elliptic problems with jumping nonlinearities, Ann. Mat. Pura Appl. (4) 128 (1981) 133–151. [39] B. Ruf, Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna, Nonlinear Anal. 9 (12) (1985) 1325–1330. [40] S. Solimini, Existence of a third solution for a class of BVP with jumping nonlinearities, Nonlinear Anal. 7 (8) (1983) 917–927. [41] S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (2) (1985) 143–156. [42] G. Sweers, On the maximum of solutions for a semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 108 (3–4) (1988) 357–370. [43] J. Wei, S. Yan, Lazer–McKenna conjecture: the critical case, J. Funct. Anal. 244 (2) (2007) 639–667.
Journal of Functional Analysis 259 (2010) 2296–2332 www.elsevier.com/locate/jfa
A variational method using fractional order Hilbert spaces for tomographic reconstruction of blurred and noised binary images M. Bergounioux, E. Trélat ∗ Université d’Orléans, UFR Sciences, Math., Labo. MAPMO, UMR 6628, Route de Chartres, BP 6759, 45067 Orléans cedex 2, France Received 20 February 2010; accepted 24 May 2010 Available online 1 June 2010 Communicated by J. Coron
Abstract We provide in this article a refined functional analysis of the Radon operator restricted to axisymmetric functions, and show that it enjoys strong regularity properties in fractional order Hilbert spaces. This study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects, for which we have available one single blurred and noised snapshot. We propose a variational approach to handle this problem, consisting in solving a minimization problem settled in adapted fractional order Hilbert spaces. We show the existence of solutions, and then derive first order necessary conditions for optimality in the form of optimality systems. © 2010 Elsevier Inc. All rights reserved. Keywords: Radon operator; Fractional order Hilbert spaces; Minimization
1. Introduction Our study is motivated by a physical experiment led at the CEA1 that consists in reconstructing a three-dimensional binary axially symmetric object from a single X-ray radiography which * Corresponding author.
E-mail addresses:
[email protected] (M. Bergounioux),
[email protected] (E. Trélat). 1 Commissariat à l’Energie Atomique, Bruyères-le-Châtel, France. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.016
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is moreover blurred and noised. The behavior of some heavy material is studied during an implosion process, and a single radiography is performed during the implosion. At some specific moment, a very brief flash of X-rays is fired from a punctual source through the object and arrives at a detector. Since the object is very dense, X-rays must be of high energy, and many drawbacks appear in practice, causing a high level of blur and noise on the radiograph. We stress on the fact that we have available only one radiography and thus, in turn, classic methods of tomographic reconstruction used in medicine, optics, geophysics, etc, which are requiring the knowledge of many projections of the object (taken from different angles), do not apply to our context. Furthermore, the objects under consideration are composed of one homogeneous medium, and of some holes. In the mathematical modeling of the problem, this feature turns into a binary constraint which is difficult to handle, and only few results exist in that direction. It is assumed that, during the implosion, the shape of the object remains axially symmetric, so that, in theory, a single snapshot is enough to reconstruct the whole object. Moreover, since the source is quite far from the object, it is assumed that X-rays are parallel and orthogonal to the symmetry axis of the object. It follows that the Radon transform has a nice expression, derived hereafter. Recall that the aim of radiography is to measure the attenuation of X-rays through the object. Every point of the radiograph, determined by cartesian coordinates (y, z), corresponds to a measure of this attenuation, and the Radon transform of the object is defined by the projection operator (H0 u)(y, ¯ z) =
u(x, ¯ y, z) dx,
(1)
R
where the function u¯ (with compact support) denotes the density of the object, and x is a coordinate along the rays. Since the objects under consideration are bounded and axially symmetric, we make use of cylindrical coordinates (r, θ, z), where the z-axis corresponds to the symmetry axis. Then, setting u(x, ¯ y, z) = u( x 2 + y 2 , z) and H0 u = H0 u, ¯ we arrive at +∞ r (H0 u)(y, z) = 2 u(r, z) dr, 2 r − y2
(2)
|y|
for all y, z ∈ R. In the sequel we adopt the following notations and conventions. We assume that the set of density functions is the set of bounded variation functions on R+ × R, having a compact support contained in the subset Ω = [0, a) × (−a, a) of R2 , where a > 0 is fixed, and taking their values in the binary set {0, 1}. In particular, the upper bound of the integral in (2) can be set to a. Notice that, for every density function u, the function H0 u is of compact support contained in Ω1 = (−a, a)2 . It has been shown in [1] that H0 extends to a linear continuous operator from L2 (Ω) to 2 L (Ω1 ). However, inverting the operator H0 requires more differentiability, and it turns out that H0−1 cannot be extended to a continuous operator from any space Lp (Ω1 ) to any space Lq (Ω).2 This property illustrates the fact that the problem is ill-posed, and the operator is bad-conditioned. 2 It can however be extended to a continuous linear operator from the Sobolev space W 1,2 (Ω ) to L2 (Ω). 1
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Hence, applying the inverse operator to the radiography causes significant errors and leads to a bad reconstruction of the object. Moreover, as mentioned formerly, due to many drawbacks in the physical experiment, the resulting radiography may be strongly blurred and noised, and actually what we observe on the radiography is vd = BH0 u + τ, that is, the projection of the density of the object, which is moreover blurred and noised. Here, B is a linear operator representing the effect of the blur. Usually, it is assumed in practice that B is the convolution with a positive symmetric kernel K with compact support and such that Kdμ = 1, and that τ is an additive Gaussian white noise of zero mean. In the sequel, we set H = BH0 . To deal with this ill-posed problem, we have proposed in [1] a regularization process based on a variational approach. More specifically, let BV(Ω) denote the space of bounded variation functions, defined as the space of functions u ∈ L1 (Ω) whose distributional gradient Du is a finite vector Radon measure, satisfying u div ϕ dx = −Du, ϕ = − ϕ · d(Du) = − ϕ · σu d|Du|, Ω
Ω
Ω
for every ϕ ∈ Cc1 (Ω, R2 ), where Cc1 (Ω, R2 ) denotes the space of continuously differentiable vector functions of compact support contained in Ω, and where σu : Ω → R2 is a |Du|-measurable function satisfying |σu | = 1 almost everywhere on Ω. The total variation of u ∈ BV(Ω) is then defined as the total variation of the Radon measure Du, that is, by u(x) div ϕ(x) dx ϕ ∈ Cc1 Ω, R2 , ϕL∞ 1 = |Du| = |Du|(Ω). Φ(u) = sup Ω
Ω
Endowed with the norm uBV = uL1 + Φ(u), the space BV(Ω) is a Banach space. Since Ω = [0, a) × (−a, a) is bounded and ∂Ω is Lipschitz, functions of BV(Ω) have a trace of class L1 on the subset Γ = {a} × (−a, a) ∪ [0, a) × {−a} ∪ [0, a) × {a}
(3)
of ∂Ω, and the trace mapping T : BV(Ω) → L1 (Γ ) is linear and bounded (see [12]). The space BV 0 (Ω) is then defined as the kernel of T . It is the space of bounded variation functions on Ω vanishing on Γ , and since T is bounded, it is a Banach space, endowed with the induced norm. Let vd be the projected image (observed data), and let α > 0. Assume that vd ∈ L2 (Ω1 ). Since H = BH0 is a linear continuous operator from L2 (Ω) to L2 (Ω1 ), we have considered in [1] the problem of minimizing the functional 1 u −→ H u − vd 2L2 (Ω ) + αΦ(u) 1 2 over all functions u ∈ BV(Ω) satisfying u(x) ∈ {0, 1} almost everywhere on Ω. Solutions of that minimization problem can then be proposed as a tomographic reconstruction in our problem.
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Using a penalization procedure to tackle the nonconvex constraint, we have proposed some numerical methods that however do not provide very satisfactory results, due to the fact that we do not take into account the deep regularity properties of the projection operator. The Radon transform and its regularity properties have been investigated in a large number of works (see e.g. [5,4,6,10,13–19,22–24] and the references therein), where range characterizations of the Radon transform and their potential applications to tomography are described. Regularity properties are in general derived in the spaces Lp ; however, as mentioned above, in our tomography problem the use of Lebesgue spaces does not lead to satisfactory practical results, which incites to derive stronger regularity features, taking into account the specific expression of the Radon transform, so as to propose a minimization problem settled with a more adapted norm. In the present article, we provide a refined functional analysis of the Radon projection operator H0 defined by (2), and show that it enjoys strong regularity properties in fractional order Hilbert spaces (Section 2). In turn, we propose in Section 3 a modified minimization problem settled in adapted fractional order Hilbert spaces. We show the existence of solutions, and, using a penalization procedure to deal with the nonconvex binarity constraint, we derive first order necessary conditions for optimality in the form of optimality systems. Since many properties of fractional order Hilbert spaces are used throughout the article, and that not all of them are so standard, we provide Appendix A, gathering different equivalent definitions and characterizations of those spaces, defined on Rn or on some bounded subset, in particular in terms of Fourier transform and fractional Laplacian. The development of algorithms based on the theoretical results of this article will be the subject of investigation of a next work. 2. Functional analysis of the projection operator 2.1. Preliminaries Recall that the densities of the objects under consideration are represented by bounded variation functions defined on the set Ω = [0, a) × (−a, a), having a compact support contained in Ω, and taking their values in {0, 1}. For every function u ∈ BV(Ω), the projection operator is defined by a (H0 u)(y, z) = 2 |y|
u(r, z)
r r2
− y2
dr,
for |y| < a and |z| < a. Note that (H0 u)(y, z) = (H0 u)(−y, z), for almost all y, z ∈ R. Notice that, for every u ∈ BV(Ω) having a compact support contained in Ω, extending u by 0 outside Ω, the function H0 u has a compact support as well, contained in Ω1 = (−a, a)2 . In this section we investigate the regularity of H0 u. First of all, observe that, for y fixed, the function z → (H0 u)(y, z) is a bounded variation function on (−a, a), and a stronger regularity property cannot be expected for such functions u. However, since the function (y, z) → H0 (y, z) is a kind of convolution of the function u with respect to the variable y, more regularity is expected with respect to this variable. Before stating the main result, we first recall a definition of fractional order Hilbert spaces.
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Let U be an open subset of Rn . For k ∈ N, the Hilbert space H k (U ) is defined as the space of all functions of L2 (U ), whose partial derivatives up to order k, in the sense of distributions, can be identified with functions of L2 (U ). Endowed with the norm f H k (U ) =
β 2 D f p
1/2
L (U )
|β|k
,
H k (U ) is a Hilbert space. For k = 0, there holds H 0 (U ) = L2 (U ). For s ∈ (0, 1), the fractional order Hilbert space H s (U ) is defined as the space of all functions f ∈ L2 (U ) such that U ×U
|f (x) − f (y)|2 dx dy < +∞. |x − y|n+2s
Endowed with the norm
f H s (U ) =
f 2L2 (U )
+ U ×U
|f (x) − f (y)|2 dx dy |x − y|n+2s
1/2 ,
H s (U ) is a Hilbert space. It is possible to define the Hilbert spaces H s (U ) in other equivalent ways. In particular, the relations with the Fourier transform or with the fractional Laplacian operator are surveyed in Appendix A. These characterizations will be used repeatedly throughout the article. 2.2. Functional properties of the projection operator The next theorem is our first main result. Theorem 1. For every u ∈ BV(Ω), the function (z, y) → (H0 u)(y, z) belongs to the Banach space BV(Ω1 ) ∩ L1 (−a, a; H s (−a, a)), for every s ∈ [0, 1). Moreover, for every s ∈ [0, 1), there exists C > 0 such that, for every u ∈ BV(Ω), there holds H0 uBV(Ω1 ) + H0 uL1 (−a,a;H s (−a,a)) CuBV(Ω) ;
(4)
in other words, the operator H0 : BV(Ω) −→ BV(Ω1 ) ∩ L1 −a, a; H s (−a, a) is linear and continuous. For every s ∈ [0, 1), the operator H0 is linear and continuous as well for the following spaces: • H0 : BV 0 (Ω) −→ BV 0 (Ω1 ) ∩ L1 (−a, a; H s (−a, a)); • H0 : L1 (−a, a; BV(0, a)) −→ BV(Ω1 ) ∩ L1 (−a, a; H s (−a, a)); • H0 : L1 (−a, a; BV 0 (0, a)) −→ BV 0 (Ω1 ) ∩ L1 (−a, a; H s (−a, a)).
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Moreover, for s = 1/2, the statements above can be strengthened by replacing H s (−a, a) by the 1/2 Lions–Magenes space3 H00 (−a, a). In the above statement, the Banach space L1 (−a, a; BV(0, a)) is endowed with the norm a
u(·, z)
BV(0,a)
dz.
−a
The Banach space L1 (−a, a; BV 0 (0, a)) is a closed subspace of L1 (−a, a; BV(0, a)) and thus is endowed with the induced norm. Recall that the space BV 0 (Ω) is the space of bounded variation functions of Ω vanishing on the subset Γ defined by (3). The space BV 0 (0, a) is defined similarly as the space of bounded variation functions on [0, a) vanishing at a. The Banach space L1 (−a, a; H s (−a, a)) is endowed with the norm a
v(·, z)
H s (−a,a)
dz.
−a
In the inequality (4), the function H0 u is considered as a function of (z, y) instead of (y, z). The result means in particular that, for almost every z ∈ (−a, a), the function y → (H0 u)(y, z) belongs to H s (−a, a) for every s ∈ [0, 1), and the resulting function of z is of class L1 . Similarly, every u ∈ L1 (−a, a; BV(0, a)) is considered as a function of (z, r) instead of (r, z); this means that, for almost every z ∈ (−a, a), the function r → u(r, z) belongs to BV(0, a), and the resulting function of z is of class L1 on (−a, a). Remark 1. It actually follows from the proof below (see Lemma 3 and Remark 3) that BV(Ω) (resp., BV 0 (Ω)) is continuously embedded in L1 (−a, a; BV(0, a)) (resp., L1 (−a, a; BV 0 (0, a))). Remark 2. Theorem 1 and Remark 1 hold as well for the blurred projection operator H = BH0 = K H0 . Proof of Theorem 1. Let us first prove that H0 is linear and continuous from L1 (Ω) into L1 (Ω1 ). Lemma 1. For every u ∈ L1 (Ω), there holds H0 uL1 (Ω1 ) 2πauL1 (Ω) . Proof of Lemma 1. For every z ∈ (−a, a), one has a
(H0 u)(y, z) dy 2
−a
a a
−a |y|
u(r, z) r dr dy, r 2 − y2
3 The Lions–Magenes space H 1/2 (−a, a) such that ρ −1/2 f ∈ L2 (−a, a), 00 (−a, a) is the subset of functions f ∈ H where the function ρ is defined on (−a, a) by ρ(y) = a − |y|. General definitions and properties of the Lions–Magenes space are recalled in Appendix A.2.2. 1/2
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and, using Fubini’s Theorem and the fact that a
r
−r
√
r r 2 −y 2
(H0 u)(y, z) dy 2πa
−a
a
dy = rπ , one arrives at
u(r, z) dr.
0
2
Integrating with respect to z, the result follows.
We next prove that H0 is linear and continuous from BV(Ω) into BV(Ω1 ). Lemma 2. There exists C0 > 0 such that H0 uBV(Ω1 ) C0 uBV(Ω) , for every u ∈ BV(Ω). Proof of Lemma 2. Using Lemma 1, it suffices to prove the existence of a constant C0 > 0 such that (H0 u)(y, z) div ξ(y, z) dy dz C0 uBV(Ω) ξ L∞ (Ω1 ) , Ω1
for every u ∈ BV(Ω) and every ξ = (ξ1 , ξ2 ) ∈ Cc1 (Ω1 , R2 ). Using Fubini’s Theorem, one has (H0 u)(y, z) div ξ(y, z) dy dz Ω1
a a a u(r, z)
=2 −a −a |y|
a a =2
−r
∂ξ1 ∂ξ2 (y, z) + (y, z) dr dy dz ∂y ∂z
∂ξ1 ∂ξ2 (y, z) + (y, z) dy dr dz ∂y ∂z
r 2 − y2
r u(r, z)
−a 0
r
r r 2 − y2
a a =
u(r, z) div ϕ(r, z) dr dz
−a 0
=
u(r, z) div ϕ(r, z) dr dz Ω
where the function ϕ = (ϕ1 , ϕ2 ) is defined on [0, a] × [−a, a] by r τ
ϕ1 (r, z) = 2 0 −τ
τ τ2
− y2
r
r ϕ2 (r, z) = 2 −r
r2
∂ξ1 (y, z) dy dτ, ∂y
− y2
ξ2 (y, z) dy.
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An easy computation shows that r ϕ1 (r, z) = 2 −r
y r2
− y2
ξ1 (y, z) dy.
The function ϕ is of class C 1 , but is not of compact support contained in Ω. Hence, we must take into account the trace of u on ∂Ω. Recall that, since Ω is bounded and ∂Ω is Lipschitz, functions of BV(Ω) have a trace on ∂Ω of class L1 , and we denote by T∂Ω : BV(Ω) → L1 (∂Ω) the corresponding bounded linear trace mapping (see [12]). Using Green’s formula, one has u(r, z) div ϕ(r, z) dr dz = − ϕ · d(Du) + (ϕ.ν)T∂Ω u dλ, Ω
Ω
∂Ω
where ν denotes the outer unit normal on ∂Ω, and λ denotes the standard one-dimensional Lebesgue measure (note that ∂Ω is made of four segments). The first integral is bounded by ϕ · d(Du) = ϕ · σu d|Du| ϕL∞ |Du|(Ω) ϕL∞ (Ω) uBV(Ω) , Ω
Ω
and the second integral is bounded by (ϕ.ν)T∂Ω u dλ CT ϕL∞ (∂Ω) uBV(Ω) , ∂Ω
where CT > 0 is the norm of the trace operator T∂Ω . Clearly, there exists C1 > 0 such that ϕL∞ (Ω) + ϕL∞ (∂Ω) C1 ξ L∞ (Ω1 ) . The proof follows.
2
Lemma 3. Let a < b and c < d be real numbers, let O = (a, b) × (c, d), and let g ∈ BV(O). For almost every x ∈ (a, b), the marginal function g x : y → g(x, y) is of bounded variation on b (c, d). Moreover, |Dg|(O) a |Dg x |(c, d) dx. Remark 3. It follows from this lemma that BV(O) is continuously embedded in the space L1 (a, b; BV(c, d)). This fact justifies the end of Remark 1. Proof. The proof of this lemma is actually contained in [6] (see also [12, Theorem 2, p. 220]), however since this result is used repeatedly in the proof of the theorem, we provide a proof for the convenience of the reader. First of all, since g ∈ L1 (O), it follows from Fubini’s Theorem that g x ∈ L1 (c, d) for almost every x ∈ (a, b). Recall that W 1,1 (O) is dense in BV(O) in the sense of the intermediate convergence, that is, there exists a sequence of functions gk ∈ W 1,1 (O) such that gk converges to g in L1 (O) and |Dgk |(O) → |Dg|(O) (see e.g. [12]). Note that, since gk ∈ W 1,1 (O), there holds Dgk = ∇gk and |Dgk |(O) = O ∇gk (x, y) dx dy. From this result, we deduce two properties.
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x ) of the sequence of functions g x : First, we infer that there exists a subsequence (gϕ(k) k y → gk (x, y) that converges to g x : y → g(x, y) in L1 (c, d), for almost every x ∈ (a, b) (with ϕ independent on x). Indeed, since gk converges to g in L1 (O), denoting hk (x) = d c |gk (x, y) − g(x, y)| dy, it follows from Fubini’s Theorem that
b
b d hk (x) dx =
a
a
gk (x, y) − g(x, y) dy dx → 0,
c
i.e., hk converges to 0 in L1 (a, b). Therefore, there exists a subsequence of (hk ) converging almost everywhere to 0 on (a, b). In other words, a subsequence of (gkx ) converges to g x in L1 (c, d), for almost every x ∈ (a, b). Second, we infer that lim infDgkx (c, d) < +∞,
k→+∞
for almost every x ∈ (a, b). Indeed, we have |Dgk |(O) → |Dg|(O), and |Dgk |(O) =
∇gk (x, y) dx dy
b d b ∂gk x Dg (c, d) dx. k ∂y (x, y) dy dx = a
O
c
a
Note that the latter equality holds because, since gk ∈ W 1,1 (O), it follows from Fubini’s Theorem that, for almost every x ∈ (a, b), the function gkx belongs to W 1,1 (c, d), and thus in particd k ular its total variation is |Dgkx |(c, d) = c | ∂g ∂y (x, y)| dy. From Fatou’s Lemma, the function x → lim infk→+∞ |Dgkx |(c, d) is measurable on (a, b), and b
lim infDgkx (c, d) dx lim inf
k→+∞
b
k
k→+∞
a
x Dg (c, d) dx |Dg|(O).
(5)
a
It follows that lim infk→+∞ |Dgkx |(c, d) < +∞ for almost every x ∈ (a, b). From these two points, we can achieve the proof of the lemma, as follows. Let ψ ∈ Cc1 ((c, d), R) such that ψL∞ 1. Then, for almost every x ∈ (a, b), d
d
g (y)ψ (y) dy = lim x
k→+∞
c
x gϕ(k) (y)ψ (y) dy lim infDgkx (c, d) < +∞ k→+∞
c
and therefore g x ∈ BV(c, d). Moreover, integrating this inequality on [a, b] and using (5) leads b to a |Dg x |(c, d) dx |Dg|(O). 2
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In the sequel, we denote by Fy v the Fourier transform of an integrable function v : R×R → R with respect to the first variable, that is,
v(y, z)e−2iπyξ dy,
(Fy v)(ξ, z) = R
for all ξ, z ∈ R. Recall, for every u ∈ L1 (Ω), the function H0 u is of compact support contained in Ω1 . In 2 the lemma below, and in the sequel, u˜ (resp. H 0 u) denotes the extension by 0 to R of the 0 the operator defined by H 0 u = H function u (resp. H0 u). Similarly, we denote by H 0 u, for every u ∈ L1 (Ω). Lemma 4. There holds (Fy H 0 u)(ξ, z) = 2π
a r u(r, ˜ z)J0 (2πξ r) dr,
(6)
0
for every u ∈ L1 (Ω), every ξ ∈ R and almost every z ∈ R, where J0 is the Bessel function of the first kind defined by 2 J0 (x) = π
1 0
cos(tx) dt. √ 1 − t2
(7)
0 (with L2 as a pivot space) is given by The adjoint of Fy H 0 )∗ v (r, z) = 2πr (Fy H
v(ξ, z)J0 (2πξ r) dξ,
(8)
R
for every v ∈ L1 (R2 ), every r ∈ [0, a) and almost every z ∈ (−a, a). Proof. Applying Fubini’s Theorem, we compute, for every ξ ∈ R and almost every z ∈ (−a, a), (Fy H 0 u)(ξ, z) =
a
H0 u(y, z)e−2iπyξ dy
−a
a a u(r, z)
=2 −a |y|
a r u(r, z)
=2 0 −r
r r2
− y2 r
r 2 − y2
e−2iπyξ dr dy
e−2iπyξ dy dr
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1
a =2
√
ru(r, z) −1
0
1 1 − t2
e
−2iπrtξ
dt dr
a =2
ru(r, z)ˆν (rξ ) dr 0
where 1 ν(t) = √ 1[−1,1] (t), 1 − t2 and νˆ is the Fourier transform of the function ν. The function νˆ can be computed using the Bessel function of the first kind J0 defined by (7) (see [2]). Since ν is even, its Fourier transform is 1 νˆ (ω) = 2 0
cos(2πωt) dt = πJ0 (2πω), √ 1 − t2
0 , with L2 as a pivot space. and the formula (6) follows. Let us now compute the adjoint of Fy H 1 2 ∞ For every v ∈ L (R ) and every u ∈ L (Ω), we have ∗ (Fy H0 ) v, u = v, Fy H0 u = v(ξ, z)Fy H 0 u(ξ, z) dξ dz R R
a r u(r, ˜ z)v(ξ, z)J0 (2πξ r) dr dξ dz
= 2π R R 0
a = 2π
r u(r, ˜ z) 0 R
0 )∗ v(r, z) = 2πr and hence (Fy H
v(ξ, z)J0 (2πξ r) dξ dz dr R
R v(ξ, z)J0 (2πξ r) dξ .
2
To prove the theorem, we next make use of the asymptotic properties of the Bessel functions J0 and J1 , where the function J1 is defined by x J1 (x) = √ π Γ (3/2)
1
cos(tx) 1 − t 2 dt.
0
Recall that J0 (x) 1, J0 (x) = −J1 (x),
J1 (x) √1 , 2 d xJ1 (x) = xJ0 (x), dx
(9) (10)
M. Bergounioux, E. Trélat / Journal of Functional Analysis 259 (2010) 2296–2332
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for every x ∈ R, and J1 (x) √1 x
(11)
as x → +∞ (see e.g. [2]). Lemma 5. There exists C2 > 0 (only depending on a) such that, for every u ∈ L1 (−a, a; BV(0, a)), there holds (Fy H 0 u)(ξ, z)
C2 uz (a) + uz 1 L (0,a) + |Duz |(0, a) , 2 3/4 (1 + ξ )
(12)
for every ξ ∈ R and almost every z ∈ (−a, a). In the above statement, recall that u ∈ L1 (−a, a; BV(0, a)) is seen as a function of (z, r); in particular, for almost every z ∈ (−a, a), the function r → uz (r) = u(r, z) is of bounded variations on [0, a), and its total variation is denoted |Duz |(0, a). Also, note that uz (a) exists for almost every z ∈ (−a, a). Proof of Lemma 5. Using the formula (6) and the estimate (9), it is first clear that (Fy H 0 u)(ξ, z) 2πauz
L1 (0,a) ,
(13)
for every ξ ∈ R and almost every z ∈ (−a, a). From (10), there holds d 2πξ rJ1 (2πξ r) = (2πξ )2 rJ0 (2πξ r), dr and, using Green’s formula (integration by parts), one gets, for every ξ = 0 and almost every z ∈ (−a, a) (such that uz (a) exists), 1 (Fy H 0 u)(ξ, z) = 2πξ 2
a uz (r)(2πξ )2 rJ0 (2πξ r) dr 0
a 1 = J1 (2πξ a)uz (a) − ξ ξ
rJ1 (2πξ r) d(Duz )
(14)
[0,a]
and hence, using (11), it follows that (Fy H 0 u)(ξ, z)
1 |ξ |3/2
a uz (a) + |Duz |(0, a) 2π
as |ξ | → +∞. The estimate (12) finally follows from (13) and (15).
2
(15)
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We are now in a position to estimate H0 uL1 (−a,a;H s (−a,a)) . Using (12), it first follows that, 4 for almost every z ∈ (−a, a), the H s norm of the function y → (H 0 u)(y, z) is estimated by (H 0 u)(·, z)
H s (R)
=
2 s 1 + ξ 2 (Fy H0 u)(ξ, z) dξ
1/2
R
C2 uz (a) + uz
L1 (0,a) + |Duz |(0, a)
s−3/2 1 + ξ2 dξ
1/2 .
R
The integral R (1 + ξ 2 )s−3/2 dξ converges if and only if 2s − 3 < −1, that is, s < 1. It follows s that, for almost every z ∈ (−a, a), the function y → (H 0 u)(y, z) belongs to H (R), for every s ∈ [0, 1). Now, for almost every z ∈ (−a, a), the function y → (H0 u)(y, z) is the restriction to (−a, a) of the function y → (H 0 u)(y, z) (which is by definition equal to 0 outside (−a, a)). It then follows from the characterization of fractional Hilbert spaces on a subset by the quotient norm (see Appendix A.2.1) that this function belongs to H s (−a, a), for every s ∈ [0, 1), and that, up to some constant, (H0 u)(·, z)H s (−a,a) (H 0 u)(·, z)H s (R) , for almost every z ∈ (−a, a). Hence, for every s ∈ [0, 1), there exists C3 > 0 such that, for every u ∈ L1 (−a, a; BV(0, a)), there holds H0 u(·, z) s C3 uz (a) + uz L1 (0,a) + |Duz |(0, a) , (16) H (−a,a) for almost every z ∈ (−a, a). As a byproduct, note that the function y → H0 u(y, z), defined on (−a, a), can be extended (by 0) to a function of H s (R), for every s ∈ [0, 1), for almost every z ∈ (−a, a). It follows from [25, Lemma 37.1] (see results recalled in Appendix A.2.1) that the function y → ρ(y)−s H0 u(y, z) belongs to L2 (−a, a), where ρ denotes the distance to the boundary of (−a, a), that is, ρ(y) = a − |y| for every y ∈ (−a, a). In turn, for s = 1/2, the function 1/2 y → H0 u(y, z) belongs to the Lions–Magenes space H00 (−a, a) (see Appendix A.2.2), for almost every z ∈ (−a, a). Integrating (16) with respect to z leads to a H0 uL1 (−a,a;H s (−a,a)) C3
u(a, z) dz + u
a L1 (Ω)
−a
+
|Duz |(0, a) dz .
(17)
−a
This inequality implies the remaining items of the theorem. Indeed, let us first consider functions u ∈ BV(Ω). It has already been mentioned that the trace operator is continuous from BV(Ω) into L1 (∂Ω), hence it follows that a
u(a, z) dz C4 uBV(Ω)
−a
4 Here, we use the definition of the H s norm in terms of Fourier transform, recalled in Appendix A.1.1.
(18)
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for some constant C4 > 0. Moreover, from Lemma 3, there holds a |Duz |(0, a) dz |Du|(Ω).
(19)
−a
The estimate (4) follows from (17), (18), (19), and Lemma 2. The other items follow similarly. This ends the proof of the theorem.
2
Theorem 1 states a strong functional property of the projection operator, which is however not very suitable in view of a variational approach. In order to derive necessary conditions for optimality, it would be better to establish functional properties of H0 in some Hilbert spaces. This is the object of the next section. 2.3. Hilbertian functional properties of the projection operator We have already mentioned that we handle functions of bounded variation on Ω that take their values in {0, 1} almost everywhere. Denote by BV(Ω, {0, 1}) the set of such functions. First of all, notice that such functions belong to L1 (−a, a; BV([0, a), {0, 1})), as already mentioned in Remarks 1 and 3; they also share the following property. Lemma 6. For every u ∈ BV(Ω, {0, 1}), the function (z, r) → u(r, z) belongs to the Banach space L1 (−a, a; H s (0, a)), for every s ∈ [0, 1/2). Proof of Lemma 6. Let u ∈ BV(Ω, {0, 1}). As mentioned above, from Lemma 3, the function uz : r → uz (r) = u(r, z) is of bounded variation on [0, a), for almost every z ∈ (−a, a). Since uz takes its values in {0, 1}, its set of discontinuities is finite. It follows that, for almost every z ∈ (−a, a), there exist an integer nz and real numbers (αi )1inz , (βi )1inz satisfying 0 α1 < β1 < α2 < β2 < · · · < αnz < βnz a, such that uz (r) =
nz
1[αi ,βi ] (r),
(20)
i=1
for almost every r ∈ [0, a). Note that the total variation of the function uz is From Lemma 3, there holds
a |Du| 2
Ω
nz dz,
−a
and hence the function z → nz belongs to L1 (−a, a).
[0,a) |Duz | = 2nz .
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The function uz is extended by 0 outside [0, a), into a function u˜ z ∈ L1 (R). Using (20), one easily computes, for almost every z ∈ (−a, a), the Fourier transform of u˜ z as (F u˜ z )(ξ ) =
nz sin(π(βi − αi )ξ )
πξ
i=1
e−iπ(βi +αi )ξ ,
for every ξ ∈ R. In particular, there holds nz (F u˜ z )(ξ ) |βi − αi | a,
(21)
i=1
for every ξ ∈ R, and (F u˜ z )(ξ ) nz , π|ξ |
(22)
for every ξ ∈ R \ {0}. Using the definition of the H s norm in terms of Fourier transform (recalled in Appendix A.1.1), and using (21) and (22), one has the estimate u˜ z 2H s (R) =
2 s 1 + ξ 2 F u˜ z (ξ ) dξ
R
=
2 s 1 + ξ 2 F u˜ z (ξ ) dξ +
|ξ |1
|ξ |1
s n2z 1 + ξ2 dξ π 2ξ 2
+
2 s 1 + ξ 2 F u˜ z (ξ ) dξ
|ξ |1
s 1 + ξ 2 a 2 dξ
|ξ |1
which is convergent if s < 1/2. Hence u˜ z ∈ H s (R), for almost every z ∈ (−a, a) and every s ∈ [0, 1/2). Since uz is the restriction of u˜ z to (0, a), it follows, using the definition of H s (0, a) in terms of quotient norm (see Appendix A.2.1), that uz ∈ H s (0, a), for almost every z ∈ (−a, a) and every s ∈ [0, 1/2). Moreover, there exists a constant C > 0, depending only on s and a, such that uz H s (0,a) Cnz . Since the function z → nz belongs to L1 (−a, a), we infer that the function (z, r) → u(r, z) belongs to L1 (−a, a; H s (0, a)), for s ∈ [0, 1/2). 2 To comply with the variational approach that we propose next, it would be better to deal with Hilbert spaces and, for instance, to replace L1 with L2 in the previous statements. Unfortunately, we have the following negative remark. Remark 4. There exist some functions u ∈ BV(Ω, {0, 1}) such that the function (z, r) → u(r, z) does not belong to L2 (−a, a; BV(0, a)).
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Fig. 1. Fractal object.
Let us provide an example of such a function.5 Consider in the plane, endowed with coordinates (x, y), the unit square [0, 1] × [0, 1]. We append to this square, on its right side, the two smaller squares 1 1 1 3 1, 1 + × 1, and 1, 1 + × ,1 . 4 4 4 4 Then, we apply a similar appending procedure to each of these latter squares, and so forth, iteratively. We obtain a fractal object (see Fig. 1). Then, we claim that the function u defined on [0, 2] × [0, 1] as the characteristic function of this fractal domain is of bounded variation. Indeed, the L1 norm of u is the sum of the areas of all squares, that is +∞
2
k=0
k
1 4k
2 =
+∞ 1 < +∞. 2k k=0
To prove that u ∈ BV([0, 2] × [0, 1], {0, 1}), it suffices to show that the marginal functions ux : y → u(x, y) and uy : x → u(x, y) are of bounded variation (see [12, Theorem 2, p. 220]). This property is obvious, since for any y the marginal functions uy have at most one jump, and for every x the marginal functions ux have a finite number nx of jumps. More precisely, nx =
k−1
2k
if 1 +
0
otherwise.
1 i=1 4i
x 1+
5 This example has been indicated to us by Simon Masnou.
k
1 i=1 4i ,
∀k 1,
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There holds 2 nx dx =
+∞ k 2 k=1
0
4k
=
+∞ 1 < +∞, 2k k=1
that is, the function x → nx belongs to L1 (0, 2), as expected (see Lemma 6 and its proof), however it does not belong to L2 (0, 2) since 2 n2x dx =
+∞ (2k )2 k=1
0
4k
=
+∞
1 = +∞.
k=1
This example shows that the functions considered in our framework, belonging to BV(Ω, {0, 1}), do not necessarily belong to the Banach space L2 (−a, a; BV(0, a)). In what follows, we are however going to work within this latter space. More precisely, since our functions vanish on the set Γ defined by (3), we are going to work within the space L2 (−a, a; BV 0 (0, a)). Indeed, this Banach space is better suited for our tomography problem, because we are able to prove that the projection operator H0 is linear and continuous from L2 (−a, a; BV 0 (0, a)) into the Hilbert space L2 (−a, a; H s (−a, a)), for every s ∈ [0, a) (see Theorem 2 further). The Hilbert space L2 (−a, a; H s (−a, a)) is then far more suitable for numerical purposes than the Banach space L1 (−a, a; H s (−a, a)), and the use of a scalar product makes easier the derivation of an optimality system (first order necessary conditions for optimality). Hence, although the space L2 (−a, a; BV 0 (0, a)) differs from the usual space BV 0 (Ω), it happens to be relevant in our problem. Actually, in practice, the binary functions considered in our imaging process belong to the space L∞ (−a, a; BV 0 (0, a)) (and thus, a fortiori, to L2 (−a, a; BV 0 (0, a))). Indeed, in practice the functions z → nz are uniformly bounded with respect to z ∈ (−a, a). Concretely, this means that the number of jumps of our binary images is uniformly bounded in every direction. The next theorem, which is our second main result, provides nice functional properties of the projection operator H0 in this new framework. We first give some notations. Denote by X = L2 −a, a; BV 0 (0, a) ,
(23)
the set of all functions u ∈ L2 (Ω) such that the function (z, r) → u(r, z) belongs to L2 (−a, a; BV 0 (0, a)). Recall that BV 0 (0, a) is the closed subset of the set of functions f ∈ BV(0, a) vanishing at a; the total variation, which is a semi-norm, is a norm on BV 0 (0, a). It follows that the space X is a closed subspace of the Banach space L2 (−a, a; BV(0, a)), and can be endowed with the norm a uX = −a
2 |Duz |(0, a) dz
1/2 .
(24)
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Recall that, for s 0, H0s (−a, a) is defined as the closure in H s (−a, a) of the set of all smooth functions having a compact support contained in (−a, a). Note that, for s ∈ [0, 1/2], there holds H0s (−a, a) = H s (−a, a) (see Appendix A). For every s ∈ [0, 1), s = 1/2, denote by Ys = L2 −a, a; H0s (−a, a) ,
(25)
the set of all functions v ∈ L2 (Ω1 ) such that the function (z, y) → v(y, z) belongs to L2 (−a, a; H0s (−a, a)). It is a closed subspace of L2 (−a, a; H s (−a, a)), and, endowed with the norm a vYs =
v(·, z) 2
H s (−a,a)
1/2 dz
,
(26)
−a
Ys is a Hilbert space. For s = 1/2, define, similarly, the Hilbert space 1/2 Y1/2 = L2 −a, a; H00 (−a, a) .
(27)
Theorem 2. For every s ∈ [0, 1), the operator H0 is linear and continuous from X into Ys . Remark 5. Theorem 2 holds as well for the blurred projection operator H = BH0 = K H0 . Proof of Theorem 2. Using the reasoning of the proof of Theorem 1, the inequality (16) implies that, for every u ∈ L2 (−a, a; BV 0 (0, a)), H0 u(·, z)
H s (−a,a)
C3 uz L1 (0,a) + |Duz |(0, a) ,
for almost every z ∈ (−a, a). Integrating with respect to z the square of this inequality leads to the result. 2 3. Variational approach for tomographic reconstruction 3.1. Minimization problem in fractional Sobolev spaces Let vd be the projected image (observed data), and let α > 0. As explained in Section 1, we have proposed in [1] an approach for tomographic reconstruction based on the consideration of the minimization problem ⎧ 1 ⎪ ⎪ ⎨ min F (u), with F (u) = H u − vd 2L2 (Ω ) + αΦ(u), 1 2 u ∈ BV(Ω), ⎪ ⎪ ⎩ u(r, z) ∈ {0, 1} a.e. on Ω. Note that the pointwise constraint, u(r, z) ∈ {0, 1} a.e. on Ω, is a very hard constraint. The constraint set is not convex and its interior is empty for most usual topologies. Based on the functional analysis of Section 2, we propose here to define the functional to be minimized in another way, according to the regularity properties proved in Theorems 1 and 2.
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In view of deriving an optimality system and taking benefit of some Hilbertian structure, we use the results of Theorem 2 and Remark 5, according to which the blurred projection operator H is linear and continuous from X into Ys , for every s ∈ [0, 1). Assuming that vd ∈ Ys (observed data), we are led to consider the minimization problem ⎧ 1 ⎪ s s 2 ⎪ ⎪ min F1 (u), with F1 (u) = H u − vd Ys + αΦ(u), ⎪ 2 ⎪ ⎪ s ⎨ u ∈ BV(Ω) ∩ X, P1 ⎪ ⎪ ⎪ uX η, ⎪ ⎪ ⎪ ⎩ u(r, z) ∈ {0, 1} a.e. on Ω, where s ∈ [0, 1) and α > 0 are fixed parameters, and η > 0 is a fixed (large) real number. The constraint uX η happens to be necessary in order to derive an existence theorem. For η large enough, this constraint is however not active6 and does not change anything to the above minimization problem. The parameter α > 0 is the weight of the total variation. This term is a usual regularization term in image processing. In our framework there is however another possibility, namely, we can replace this term with a regularization term involving the norm of X. In that case, we rather consider the minimization problem ⎧ 1 α ⎪ s s 2 2 ⎪ ⎪ min F2 (u), with F2 (u) = H u − vd Ys + uX , 2 2 s ⎨ P2 u ∈ X, ⎪ ⎪ ⎪ ⎩ u(r, z) ∈ {0, 1} a.e. on Ω. Note that, in the latter minimization problem, the use of η > 0 is not needed. Theorem 3. For all s ∈ [0, 1), α > 0 and η > 0, the minimization problems (P1s ) and (P2s ) admit at least one solution. Proof. Let s ∈ [0, 1), α > 0 and η > 0 be fixed. Let (un ) be a minimizing sequence in BV(Ω)∩X of the minimization problem (P1s ), satisfying un X η and un (x) ∈ {0, 1} almost everywhere on Ω. Then, the sequences (H un − vd Ys )n∈N , (Φ(un ))n∈N and (un X )n∈N are bounded, and we deduce three things. First, the sequence (H un )n∈N is bounded in the Hilbert space Ys , and therefore, up to a subsequence it converges weakly to some w ∈ Ys ; in particular, using the continuous embedding of BV(−a, a) in L2 (−a, a), it converges as well to w, up to some subsequence, for the weak topology of L2 (Ω1 ). Second, the sequence (un )n∈N is bounded in BV(Ω) ∩ L∞ (Ω), and hence, up to a subsequence, it converges to some u ∈ BV(Ω) ∩ L∞ (Ω) for the weak-star topology, i.e., up to a subsequence, (un )n∈N converges strongly to some u ∈ L1 (Ω) and (Dun )n∈N converges to Du in the space of Radon measures for the weak-star topology. 6 As explained formerly, the slices of the binary images treated in practice have a finite number of connected components, uniformly with respect to the slice; this means that there exists R > 0 such that uL∞ (−a,a;BV 0 (0,a)) η, for every u in the set under consideration, and this implies the constraint considered here.
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Third, the sequence (un )n∈N is bounded in X (by η). The space X inherits of the weak compactness properties of L2 and BV; hence, the sequence (un )n∈N converges, up to a subsequence, to some u¯ ∈ X for the weak-star topology, and the lower semi-continuity of the norm yields the inequality u ¯ X η. In particular, (un )n∈N converges weakly to u¯ in L2 (Ω), and thus u¯ = u. From Theorem 2, the operator H is linear and continuous from X into Ys . Hence, up to a subsequence, (H un )n∈N converges weakly to H u in Ys . It follows that w = H u. Since Φ is lower semi-continuous with respect to the weak-star topology, there holds Φ(u) lim inf Φ(un ). n→∞
Using the weak convergence, up to a subsequence, of (H un )n∈N to H u in Ys , we infer that
inf F1s = lim
n→+∞
1 H un − vd 2Ys + αΦ(un ) H u − vd 2Ys + αΦ(u) = F1s (u), 2
and hence F1s (u) = inf F1s . Finally, (un ) converges to u in L1 (Ω), and thus, converges almost everywhere (up to a subsequence) to u. Hence, the pointwise constraint u(x) ∈ {0, 1} is satisfied almost everywhere, and therefore u is a solution of (P1s ). For the minimization problem (P2s ), the proof is similar but simpler. 2 3.2. Penalization of the binarity constraint Due to the binarity constraint, u(r, z) ∈ {0, 1} a.e. on Ω, which is not convex and is of empty interior for most usual topologies, the minimization problems (P1s ) and (P2s ) are not directly tractable for numerical purposes. To deal with the binarity constraint, we propose as in [1] a penalization method. Let ε > 0, β 0, and let u¯ si be a solution of (Pis ), for i = 1, 2. Define s Fi,ε (u) = Fis (u) +
2 1 β u − u2 2 2 + u − u¯ si L2 (Ω) , (Ω) L 2ε 2
for i = 1, 2, and consider the penalized minimization problems s P1,ε
⎧ s (u), min F1,ε ⎪ ⎪ ⎨ u ∈ BV(Ω) ∩ X, ⎪ uX η, ⎪ ⎩ uL∞ (Ω) η,
and s P2,ε
⎧ s ⎨ min F2,ε (u), u ∈ X, ⎩ uL∞ (Ω) η.
Actually in what follows η is chosen large enough, and the resulting constraint should not be active. This constraint is however necessary in order to derive existence results, but does not affect the numerical process. In the sequel we do not mention the dependence of the penalized s ), i = 1, 2 with respect to the parameter η that can be chosen as large as desired problems (Pi,ε
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but is fixed. A contrario the penalization parameter ε will tend to 0. The term β2 u − u¯ si 2L2 (Ω) is an additional penalization term permitting to focus, from the theoretical point of view, on a particular solution u¯ si of (Pis ). In practice, the solution u¯ si is of course not known and we choose β = 0. s ) has at least one solution us , for Theorem 4. The penalized minimization problem (Pi,ε i,ε i = 1, 2. s ). First, in Proof. Let (un )n∈N be a minimizing sequence of the minimization problem (P1,ε particular, the sequence (un )n∈N is bounded in BV(Ω) and in L∞ (Ω), and thus, up to a subsequence, converges to some us1,ε ∈ BV(Ω) for the weak-star topology, and hence for the strong L1 (Ω) topology. Since it is moreover bounded (by η) in L∞ (Ω), it follows from the Lebesgue dominated convergence theorem that the convergence holds for the strong topology of Lp (Ω), for every p ∈ [1, +∞). In addition, since every closed ball of L∞ (Ω) is compact for the L∞ weak star topology, we infer that us1,ε η. The rest of the proof is similar to the proof of Theorem 3. s ), the proof is similar but simpler. 2 For the minimization problem (P2,ε
Theorem 5. 1. Every cluster point u∗1 in BV(Ω) ∩ X of the family (us1,ε ) at ε = 0 is a solution of (P1s ), and every cluster point u∗2 in X of the family (us2,ε ) at ε = 0 is a solution of (P2s ). If moreover β > 0 then u∗i = u¯ si , for i = 1, 2. s (u ) = inf F s , and lim s ∗ 2. There holds limε→0 Fi,ε i,ε ε→0 Ω |Du1,ε | = Ω |Du1 |. i s (us ) F s (u s s ¯s ) = Proof. Since u¯ s1 ∈ BV(Ω) is a solution of (P1s ), one has F1,ε 1,ε 1,ε ¯ 1 ) = F1 (u 1 s s inf F1 , for every ε > 0. Therefore, the family (u1,ε ) is bounded in BV(Ω), and us1,ε − (us1,ε )2 L2 (Ω) tends to 0 as ε tends to 0. Let u∗1 be a (strong) cluster point of (us1,ε ) in BV(Ω) ∩ X of the family (us1,ε ) at ε = 0. Using the same reasoning as in the proof of Theorem 4, it follows that u∗1 is a (strong) cluster point of (us1,ε ) in Lp (Ω), for every p ∈ [1, +∞). Then,
∗ ∗ 2 u − u 1
1
L2 (Ω)
2 lim inf us1,ε − us1,ε L2 (Ω) = 0, ε→0
so that u∗1 (1 − u∗1 ) = 0 almost everywhere on Ω, which is the binarity constraint. Since s (us ) inf F s , one gets F s (u∗ ) F s (u∗ ) + βu∗ − u F1s (u1,ε ) + βus1,ε − u¯ s1 22 F1,ε ¯ s1 22 1 1,ε 1 1 1 1 1 s s s ∗ ∗ inf F1 . Therefore u1 is a solution of (P1 ). In addition, if β > 0, then u1 = u¯ 1 . Finally, since s (u∗ ) lim inf F s (us ) inf F s , (since F s (us ) inf F s ,) it follows inf F1s = F1s (u∗ ) F1,ε 1,ε 1,ε 1 1,ε 1,ε 1 s s that limε→0 F1,ε (u1,ε ) = inf F1s . Moreover, writing 2 2 1 lim inf F1s us1,ε + lim sup us1,ε − us1,ε 2 ε→0 ε ε→0
2 2 1 lim F1s us1,ε + us1,ε − us1,ε 2 = F1s u∗1 ε→0 ε s s lim inf F1 u1,ε , ε→0
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it follows that limε→0 1ε us1,ε − (us1,ε )2 22 = 0, and then that limε→0 F1s (us1,ε ) = F1s (u∗1 ). The conclusion follows then from the continuity properties of H . The proof is similar for the family (us2,ε ). 2 3.3. Optimality system of the penalized minimization problems This section is devoted to the derivation of first order necessary conditions for the penalized s ), i = 1, 2. Throughout the section, let s ∈ [0, 1), α > 0, η > 0, ε > 0 minimization problems (Pi,ε and β 0 be fixed. Denote by BV(Ω) ∩ X if i = 1, Vi = X if i = 2, Φ(u) if i = 1, Ji (u) = 1 2 u X if i = 2, 2 and
Bη =
{u ∈ BV(Ω) ∩ X | uX η and uL∞ (Ω) η} if i = 1, {u ∈ X | uL∞ (Ω) η} if i = 2.
For i = 1, 2, for every u ∈ Vi ∩ Bη , define 1 Gs (u) = H u − vd 2Ys , 2 and Gsi,ε (u) = Gs (u) +
2 1 β u − u2 2 2 + u − u¯ si L2 (Ω) . L (Ω) 2ε 2
With these notations, one has Fis (u) = Gs (u) + αJi (u), s Fi,ε (u) = Gsi,ε (u) + αJi (u), s ) is the problem of minimizing the for i = 1, 2, and the penalized minimization problem (Pi,ε s functional Fi,ε over all functions u ∈ Bη ∩ Vi . Denote by usi,ε a solution of the minimization s ), for i = 1, 2. problem (Pi,ε s is not differentiable, due to the term J that involves a total variation. The functional Fi,ε i However the functional Gsi,ε is differentiable, for i = 1, 2, and ∇Gsi,ε (u) = ∇Gs (u) + qε (u) + β u − u¯ si ,
for the pivot space L2 , where qε (u) = for every u ∈ Vi .
(u − u2 )(1 − 2u) , ε
(28)
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In what follows, we first provide an expression of the differential of Gs in terms of Fourier transform (Section 3.3.1), and then using the fractional Laplacian operator or the fractional Dirichlet Laplacian operator (Section 3.3.2). Then, using results of nonsmooth analysis to deal with the nonsmooth character of the functional Ji , we derive first order necessary conditions s ), i = 1, 2 (Secin terms of optimality systems for both penalized minimization problems (Pi,ε tions 3.3.3 and 3.3.4). 3.3.1. Computation of ∇Gs in terms of Fourier transform Recall that Gs (u) = 12 H u − vd 2Ys , with Ys defined by (25) and (27). Using the results recalled in Appendix A (Appendices A.2.1 and A.2.2), for every s ∈ [0, 1), for every u ∈ Ys , the function H u − vd can be extended by 0 to a function of L2 (−a, a; H s (R), and its norm can be computed in terms of Fourier transform, by 1 G (u) = 2
a
s
Fy (H u − vd )(ξ, z)2 1 + ξ 2 s dξ dz
−a R
2 1 = Fy (H u − vd ) L2 (ω ) , s 2 where L2 (ωs ) is the weighted Hilbertian space of all complex valued functions f defined on R × (−a, a) such that f (ξ, z)2 ωs (ξ ) dξ dz < +∞, R×(−a,a)
where s ωs (ξ ) = 1 + ξ 2 .
(29)
∇Gs (u) = (Fy H )∗ ωs (Fy H )(u − wd ),
(30)
Setting wd = H −1 (vd ), we get finally
with L2 as a pivot space. To make this expression more explicit, we next compute the Fourier transform of the blurred projection operator H . Recall that v˜ denotes the extension by 0 to R2 of any function v and that, from Lemma 4, (Fy H 0 u)(ξ, z) = 2π
a ru(r, z)J0 (2πξ r) dr, 0
for every u ∈ L1 (Ω), every ξ ∈ R and almost every z ∈ (−a, a), where J0 is the Bessel function of the first kind, and 0 )∗ v (r, z) = 2πr v(ξ, z)J0 (2πξ r) dξ, (Fy H R
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for every v ∈ L1 (R2 ), every r ∈ [0, a) and almost every z ∈ (−a, a). As explained in the introduction, the radiography is blurred, and the blur is modeled by a linear operator B writing as a convolution with a positive symmetric kernel K (in practice, a Gaussian kernel) with compact support. Lemma 7. The Fourier transform of the blurred projection operator H = BH0 with respect to the first variable is (Fy B H 0 u)(ξ, z) = (Fy K)(ξ, ·) 2 (Fy H 0 u)(ξ, ·)(z),
(31)
for every u ∈ L1 (Ω), every ξ ∈ R and almost every z ∈ (−a, a), where the notation 2 stands for the convolution product with respect to the second variable. Its adjoint (with L2 as a pivot space) is 0 )∗ v (r, z) = (Fy H 0 )∗ (Fy g 2 v)(r, z), (32) (Fy B H for every v ∈ L1 (R2 ), every r ∈ [0, a) and almost every z ∈ (−a, a). Proof. Since B H 0 u = K (H 0 u), one computes −2iπξy u)(ξ, z) = K(y − x, z − s)(H dy dx ds (Fy B H 0 0 u)(x, s)e R3
=
−2iπξ x (H 0 u)(x, s)e
=
dy dx ds
R
R2
K(y − x, z − s)e
−2iπξ(y−x)
−2iπξ x (Fy K)(ξ, z − s)(H dx ds 0 u)(x, s)e
R2
(Fy K)(ξ, z − s)(Fy H0 u)(ξ, s) ds
= R
= (Fy K)(ξ, ·) 2 (Fy H0 u)(ξ, ·)(z), for every u ∈ L1 (Ω), every ξ ∈ R and almost every z ∈ (−a, a). Let us now compute the adjoint. For every v ∈ L1 (R2 ) and every u ∈ L∞ (Ω), we have
0 )∗ v, u = v, Fy B H (Fy B H 0 u = v(ξ, z)(Fy B H 0 u)(ξ, z) dξ dz R2
=
v(ξ, z)(Fy K)(ξ, z − s)(Fy H 0 u)(ξ, s) ds dξ dz
R3
a a v(ξ, z)(Fy K)(ξ, z − s)ru(r, s)J0 (2πξ r) dr ds dξ dz
= 2π R R −a 0
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a a =
2πru(r, s) v(ξ, z)(Fy K)(ξ, z − s)J0 (2πξ r) dξ dz ds dr,
0 −a
R2
and hence, we infer that 0 )∗ v(r, s) = 2πr (Fy B H
J0 (2πξ r) (Fy K)(ξ, z − s)v(ξ, z) dz dξ.
R
R
Since the kernel K of B is symmetric, it follows that 0 )∗ v(r, s) = 2πr (Fy B H
J0 (2πξ r) (Fy K)(ξ, ·) 2 v(ξ, ·) (s) dξ.
R
The formula (32) then follows from (8).
2
3.3.2. Computation of ∇Gs in terms of fractional Laplacian By definition, there holds 1 G (u) = 2
a
s
−a
(H u − vd )(·, z) 2
H0s (−a,a)
dz,
for every u ∈ Ys , and every s ∈ [0, 1), s = 1/2. For s = 1/2, H0s (−a, a) is replaced with 1/2 H00 (−a, a). Using the results of Appendix A.2.3, it follows that, for every f ∈ H0s (−a, a) 1/2 whenever s ∈ [0, 1), s = 1/2, or f ∈ H00 (−a, a) whenever s = 1/2, the norm of f within these spaces is equivalent to 2 1/2 f 2L2 (U ) + (−)s/2 f L2 (Rn ) , where f is extended by 0 outside (−a, a) (notice that (−)s/2 f is not of compact support). Here, (−)α is the fractional Laplacian operator on Rn , defined, using the Fourier transform F f of f , by (−)α f = F −1 (|ξ |2α F f ) (see Appendix A.1.2). It follows easily that ∇Gs (u) = RΩ1 id + (−)s (H u − v˜d ),
(33)
with L2 as a pivot space, where H u − v˜d is the extension of H u − vd by 0 outside (−a, a), and RΩ1 is the restriction to Ω1 . Another possibility is to express the differential of Gs using the fractional Dirichlet Laplacian operator A defined in Appendix A.2.4. We get, similarly, ∇Gs (u) = id + As (H u − vd ), with L2 as a pivot space.
(34)
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s ) 3.3.3. Optimality system of (P1,ε In this case the regularization term is the total variation semi-norm. First of all, recall the next result of [8] that will be used to tackle the nonsmooth character of the total variation, and then to s . derive optimality conditions for the penalized problem P1,ε
Theorem 6. (See [8, Theorem 2.3].) Let A be a Borelian subset of Rn . Let u¯ ∈ K ∩ BV(A) be the solution of ⎧ ⎪ ⎨ min J (u) + α |Du|, ⎪ ⎩
A
u ∈ K ∩ BV(A),
where K is a closed convex subset of Lp (A) and J is continuous and Gâteaux differentiable from Lp (A) to R (1 p < +∞), and either K is bounded or J is coercive. Then, there exists λ¯ ∈ (M(A)n ) (the dual space of Radon measures) such that ¯ − α div λ¯ , u − u¯ 0, J (u)
(35)
for every u ∈ K ∩ BV(A), and
¯ μ − D u λ, ¯ +
|D u| ¯ Ω
|μ|,
(36)
Ω
for every μ ∈ (M(A))n , where D : BV(A) → (M(A))n and div λ¯ , u = −λ¯ , Du,
(37)
for every u ∈ BV(A). This result cannot be applied to the original problem (P1s ) since the set of constraints is not s ). The proof is similar to the one convex, but can be used to handle the penalized problem (P1,ε ε 2 of [8], and yields the existence of λ ∈ (M(Ω) ) such that ∇Gs1,ε uε − α div λε , u − uε 0,
(38)
for every u ∈ Bη , and
λε , μ − Duε +
ε Du
Ω
|μ|, Ω
for every μ ∈ (M(Ω))2 . Considering μ = Dv with v ∈ BV(Ω) in (39) leads to
λε , D v − uε +
Ω
ε Du
|Dv|, Ω
(39)
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for every v ∈ BV(Ω), that is, Φ(v) Φ uε − divλε , v − uε , which is equivalent to με ∈ ∂Φ(uε ), where με = −div λε . We thus get the following result. s ). Then there exist λε ∈ (M(Ω)2 ) , q ε = q ε (us ) ∈ Theorem 7. Let us1,ε be a solution of (P1,ε 1,ε L∞ (Ω) defined by (28), and με = −div λε such that
∇Gs us1,ε + q ε + αμε , u − us1,ε 0,
(40)
με ∈ ∂Φ us1,ε .
(41)
for every u ∈ Bη , and
s ) 3.3.4. Optimality system of (P2,ε In this case,
1 1 J2 (u) = u2X = 2 2
a −a
2 1 |Duz |(0, a) dz = 2
a
2 ϕ(uz ) dz,
−a
for every u ∈ X = L2 (−a, a; BV 0 (0, a)). Here and in the sequel, the notation ϕ(f ) is used to denote the total variation of a function f ∈ BV(0, a). There holds X = L2 (−a, a; (BV 0 (0, a)) ). For every λ ∈ X , viewed as function of z ∈ (−a, a) of class L2 with values in (BV 0 (0, a)) , denote λz = λ(z) ∈ (BV 0 (0, a)) , for almost every z ∈ (−a, a). The duality product between X and X is defined by a λ, v
X ,X
=
λz , vz BV 0 ,BV 0 dz,
−a
for every λ ∈ X and every v ∈ X. Lemma 8. The functional J2 is convex and locally Lipschitzian on X. Proof. The convexity is obvious. To establish the local Lipschitzian property, we use the fact that the total variation ϕ is Lipschitzian and the Cauchy–Schwarz inequality, getting the estimate J2 (v) − J2 (u) 1 2
a
ϕ(uz ) + ϕ(vz ) ϕ(uz ) − ϕ(vz ) dz
−a
uX + vX u − vX 2uX + ρ u − vX , for all u, v ∈ X such that u − vX ρ.
2
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It follows from this lemma that J2 is subdifferentiable and that the classical subdifferential and the generalized Clarke subdifferential of J2 coincide (see [9]). Moreover, the Clarke generalized directional derivative and the classical directional derivative coincide as well. s ) consists in minimizing Gs (u) + α J (u) over all Recall that the penalization problem (P2,ε 2,ε 2 2 functions u ∈ Bη . This problem is equivalent to the minimization problem min Gs2,ε (u) + u∈X
α J2 (u) + χBη (u), 2
where χBη is defined by χBη (u) = 0 whenever u ∈ Bη , and χBη (u) = +∞ else. s ) is A necessary condition for us2,ε to be an optimal solution to (P2,ε 0 ∈ ∂ Gs2,ε + αJ2 + χBη us2,ε , and hence, using the standard rules of the subdifferential calculus, 0 ∈ ∇Gs2,ε us2,ε + α∂J2 us2,ε + ∂χBη us2,ε ,
(42)
since the considered functions are convex and locally Lipschitzian (see [9]). Lemma 9. Let λ ∈ X . Then, for every u ∈ X, λ ∈ ∂J2 (u) if and only if λz ∈ ϕ(uz )∂ϕ(uz ), for almost every z ∈ (−a, a). Proof. The statement of this lemma is natural, and the proof is quite easy, however it is not possible to apply directly results of [9]. We next provide a proof for the convenience of the reader. Since J2 is a proper convex locally Lipschitzian function, ∂J2 (u) is nonempty for every u ∈ X, and ∂J2 (u) = λ ∈ X J2 (u; v) λ, vX ,X ∀v ∈ X , where J2 (u; v) denotes the directional derivative at u in the direction v. In addition, J2 (u; v) = sup λ, vX ,X λ ∈ ∂J2 (u) . We next compute J2 (u; v), for all u, v ∈ X. One has J2 (u + tv) − J2 (u) 1 = t 2 1 = 2
a −a
a −a
ϕ(uz + tvz )2 − ϕ(uz )2 dz t
ϕ(uz + tvz ) − ϕ(uz ) dz. ϕ(uz + tvz ) + ϕ(uz ) t
By definition of the subdifferential, there holds ϕ(uz + tvz ) − ϕ(uz ) μ, vz BV 0 ,BV 0 , t
(43)
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for every μ ∈ ∂ϕ(uz ), every vz ∈ BV 0 (0, a), and almost every z ∈ (−a, a). Since ϕ(uz + tvz ) − ϕ(uz ) |t|ϕ(vz ), we get ϕ(uz + tvz ) − ϕ(uz ) max ϕ(vz ), λz , vz BV ,BV , 0 0 t for every t ∈ (0, 1] and almost every z ∈ (−a, a). Note that μ, vz BV ,BV μBV ϕ(vz ), 0 0 0 and μBV 0 =
sup
μ, vBV 0 ,BV 0 ,
vBV 0 1
for almost every z ∈ (−a, a). Since μ ∈ ∂ϕ(uz ), it follows that μ, vBV 0 ,BV 0 ϕ(uz + v) − ϕ(uz ) ϕ(v), for every v ∈ BV 0 (0, a). Hence, μBV 0 1 and |μ, vz BV 0 ,BV 0 | ϕ(vz ). Therefore, ϕ(uz + tvz ) − ϕ(uz ) ϕ(vz ), t
(44)
for every t ∈ (0, 1] and almost every z ∈ (−a, a), and we infer that ϕ (uz ; vz ) ϕ(vz ).
(45)
Moreover, 0 ϕ(uz + tvz ) + ϕ(uz ) 2ϕ(uz ) + tϕ(vz ), hence, using (44),
ϕ(uz + tvz ) + ϕ(uz ) ϕ(uz + tvz ) − ϕ(uz ) 2ϕ(uz ) + ϕ(vz ) ϕ(vz ). t The function z → (2ϕ(uz ) + ϕ(vz ))ϕ(vz ) is integrable on (−a, a), since the functions z → ϕ(uz )2 , z → ϕ(vz )2 and z → ϕ(uz )ϕ(vz ) are integrable (indeed, u, v ∈ X). Using (45), we infer that the function z → ϕ(uz )ϕ (uz ; vz )
(46)
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is integrable on (−a, a), for every v ∈ X. Therefore, applying the Lebesgue theorem to (43), we get J2 (u + tv) − J2 (u) = lim t→0 t
a
ϕ(uz )ϕ (uz ; vz ) dz.
−a
Finally J2 (u; v) =
a
a
ϕ(uz )ϕ (uz ; vz ) dz
−a
ϕ(uz )λz , vz
−a
BV 0 ,BV 0
dz,
(47)
for every λz ∈ ∂ϕ(uz ), since λz , vz BV 0 ,BV 0 ϕ (uz ; vz ). For every u ∈ X, define E(u) = μ ∈ X μ : z → ϕ(uz )λz with λz ∈ ∂ϕ(uz ) for a.e. z ∈ (−a, a) . We claim that E(u) = ∂J (u). Indeed, let us first prove that E(u) ⊂ ∂J (u). For every μ ∈ E(u), there holds a μ, vX ,X = −a
a μz , vz BV 0 ,BV 0 dz =
ϕ(uz )λz , vz BV 0 ,BV 0 dz,
−a
for every v ∈ X. Since λz ∈ ∂ϕ(uz ), using (47) we get that μ, vX ,X J (u; v), for every v ∈ X. This implies that μ ∈ ∂J (u) (see [9]), and the inclusion follows. The proof of the converse inclusion readily follows the one of [9, p. 77], and is thus skipped. The key point is the measurability of the function z → ϕ(uz )ϕ (uz ; vz ), for every v ∈ X, which follows in particular from (46). We have thus proved that every μ ∈ ∂J (u) is such that μ : z → ϕ(uz )λz , with λz ∈ ∂ϕ(uz ). The lemma follows. 2 s ). We are now in a position to derive the optimality system of (P2,ε s ). Then, there exist με ∈ X , q ε = q ε (us ) ∈ L∞ (Ω) Theorem 8. Let us2,ε be a solution of (P2,ε 2,ε defined by (28), such that ∇Gs us2,ε + q ε + αμε , u − us2,ε X ,X 0, (48)
for every u ∈ Bη , and μεz ∈ ϕ us2,ε z ∂ϕ us2,ε z , for almost every z ∈ (−a, a).
(49)
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Appendix A. Fractional order Hilbert spaces In this appendix we gather different definitions and characterizations of fractional order Hilbert spaces, on Rn and on bounded subsets, not all of them being so standard. The main references are [3,7,11,20,21,25,26]. Let U be an open bounded subset of Rn . For k ∈ N, the Hilbert space H k (U ) is defined as the space of all functions of L2 (U ), whose partial derivatives up to order k, in the sense of distributions, can be identified with functions of L2 (U ). Endowed with the norm f H k (U )
β 2 D f p = |β|k
1/2
L (U )
H k (U ) is a Hilbert space. For k = 0, there holds H 0 (U ) = L2 (U ). For s ∈ (0, 1), the fractional order Hilbert space H s (U ) is defined as the space of all functions f ∈ L2 (U ) such that U ×U
|f (x) − f (y)|2 dx dy < +∞. |x − y|n+2s
Endowed with the norm
1/2 |f (x) − f (y)|2 f H s (U ) = f 2L2 (U ) + dx dy , |x − y|n+2s U ×U
H s (U ) is a Hilbert space. For a positive noninteger real number s > 0, denote by [s] the floor of s, and let α ∈ (0, 1) such that s = [s] + α. The fractional order Hilbert space H s (U ) is defined as the space of all functions f ∈ L2 (U ), whose partial derivatives of order [s], in the sense of distributions, can be identified with functions of H α (U ). Endowed with the norm
f H s (U ) =
f 2H [s] (U )
1/2 |D β f (x) − D β f (y)|2 + dx dy , |x − y|n+2α |β|=[s] U ×U
H s (U ) is a Hilbert space. Let D(U ) denote the space of C ∞ functions on U , having a compact support contained in U . For every s 0, define H0s (U ) as the closure of D(U ) in H s (U ). The space H0s (U ) is a closed subspace of H s (U ) and thus inherits of its Hilbertian structure. There holds H0s (U ) = H s (U ) if and only if 0 s 1/2 (see [21, Theorem 11.1]), or whenever U = Rn . The space H −s (U ) is defined as the dual of H0s (U ). It is possible to define the fractional order Hilbert spaces H s (U ) in other equivalent ways, in particular, using the Fourier transform or using the fractional Laplacian operator. The situation is quite simple for U = Rn but is more intricate for a bounded domain U .
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A.1. Other characterizations for U = Rn A.1.1. Fourier transform Another possible definition of H s (Rn ) goes by using the Fourier transform F , as follows. For every s > 0, define s/2 F f ∈ L2 R n , H s Rn = f ∈ L2 Rn 1 + |ξ |2 endowed with the norm
f H s (Rn ) =
2 s 1 + |ξ |2 F (f )(ξ ) dξ
1/2 .
Rn
Note that f ∈ H s (Rn ) if and only if (id − )s/2 f ∈ L2 (Rn ), where the operator (id − )s/2 is defined by its symbol (1 + |ξ |2 )s/2 , or, in other words, is defined using the Fourier transform by (id − )s/2 f = F −1 ((1 + |ξ |2 )s/2 F f ). Note that, for s < 0, one has s/2 H s Rn = f ∈ S Rn 1 + |ξ |2 F f ∈ L2 R n , where S(Rn ) denotes the Schwartz space of rapidly decreasing C ∞ functions on Rn . A.1.2. Fractional Laplacian operator Define the fractional Laplacian operator (−)α , using the Fourier transform F f of f , by (−)α f = F −1 (|ξ |2α F f ). This definition actually makes sense for α ∈ (−n/2, 1] and f ∈ S(Rn ). Note that (−)α f ∈ / S since |ξ |2α introduces a singularity at the origin in its Fourier α transform; however, (−) f is of class C ∞ (see e.g. [7]). Clearly, (−)1 = −, (−)0 = id, and (−)α1 ◦ (−)α2 = (−)α1 +α2 . Moreover, the operator (−)α is selfadjoint on L2 (Rn ). An easy computation shows that, for every α ∈ (0, 1), there exists a constant Cn,α such that, for every f ∈ S(Rn ), (−)α f (x) coincides with the principal value of the singular integral Cn,α Rn
f (x) − f (y) dy. |x − y|n+2α
Actually, Cn,α is a positive constant such that Rn
1 − cos(ξ.y) |ξ |2α dy = , Cn,α |y|n+2α
for every ξ ∈ Rn . Note that the above singular integral is well defined whenever 0 < α < 1/2, and in that case it is not necessary to consider the principal value; for 1/2 α < 1, the singularity is near x = y.
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To avoid the use of principal values, one has the other equivalent expression (obvious to obtain with a change of variable) f (x + y) + f (x − y) − 2f (x) 1 (−)α f (x) = − Cn,α dy. 2 |y|n+2α Rn
Then, for every s ∈ (0, 1) and every f ∈ H s (Rn ), one easily gets f 2H s (Rn )
= f 2L2 (Rn )
+ Rn ×Rn
= f 2L2 (Rn ) +
|f (x) − f (y)|2 dx dy |x − y|n+2s
2 (−)s/2 f 2 2 n , L (R ) Cn,s
and therefore H s (Rn ) can be equivalently defined as the space of all functions of L2 (Rn ) such that the distribution (−)s/2 f can be identified with a function of L2 (Rn ). The relation of the Hilbert spaces H s with the domains of the fractional Laplacian operator is the following. The operator −, defined by Fourier transform, is a selfadjoint positive operator on L2 (Rn ), of domain D(−) = H 2 (Rn ). The fractional operator (−)s has been defined above by Fourier transform, for s > 0. Using the interpolation theory and results from [20,21], one can establish that D (−)s = H 2s Rn , for every s ∈ [0, 1]. A.2. Other characterizations on a bounded domain The situation on a bounded subset U of Rn is more delicate. A.2.1. Quotient norm and extensions The space H s (U ) can be as well defined as the set of restrictions of functions of H s (Rn ) to U , with the quotient norm f H s (U ) = inf f˜H s (Rn ) f˜ ∈ H s Rn , f˜|U = f . If U is bounded with a smooth boundary, then, for every f ∈ L2 (U ) such that U ×U
|f (x) − f (y)|2 dx dy < +∞, |x − y|n+2s
there exists an extension f˜ ∈ L2 (Rn ) of f (defined by symmetry, locally at the boundary of the domain) for which Rn ×Rn
|f˜(x) − f˜(y)|2 dx dy < +∞ |x − y|n+2s
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(see [25, Lemma 36.1]). This extension by symmetry is a specific one. Concerning extensions outside U , note that, for U bounded with smooth boundary, for every s ∈ R (H s with s < 0 is defined further), there exists a continuous linear extension mapping PU ∈ L(H s (U ), H s (Rn )), satisfying PU f|U = f a.e. for every f ∈ H s (U ), and a continuous restriction mapping RU ∈ L(H s (Rn ), H s (U )), such that RU PU = idH s (U ) . Hence, H s (U ) can be as well defined using the equivalent norm
1/2 |PU f (x) − PU f (y)|2 f 2L2 (U ) + dx dy . |x − y|n+2s Rn ×Rn
It would be more interesting (in view of using the fractional Laplacian, see further) to extend f by 0 outside U and to use the above double integral on Rn × Rn . The extension by 0 is however more delicate. Denoting f˜ the extension of f by 0 outside U , the linear mapping f ∈ H s (U ) → f˜ ∈ H s (Rn ) is well defined and continuous if and only if 0 s < 1/2. This means that, for 0 s < 1/2, the space H s (U ) can be as well defined using the equivalent norm
f 2L2 (U )
+ Rn ×Rn
|f (x) − f (y)|2 dx dy |x − y|n+2s
1/2 ,
where f is extended by 0 outside U (but this fact is not true for s 1/2 because of phenomena on the boundary). Actually, the following result holds true [25, Lemma 37.1]. Denoting f˜ the extension of f by 0 outside U , for U bounded with Lipschitz boundary, for 0 < s < 1, f˜ ∈ H s (Rn ) if and only if f ∈ H s (U ) and ρ −s f ∈ L2 (U ), where ρ denotes the distance to the boundary of U . The following extension result holds for the space H0s (U ). Denoting f˜ the extension of f by 0 outside U , for s 0, the linear mapping f ∈ H0s (U ) → f˜ ∈ H s (Rn ) is well defined and continuous if and only if s ∈ / N + 1/2.7 It follows that, for instance, for every s ∈ (0, 1), s = 1/2, the space H0s (U ) can be as well defined using the equivalent norm
f 2L2 (U ) + Rn ×Rn
|f (x) − f (y)|2 dx dy |x − y|n+2s
1/2 ,
where f is extended by 0 outside U (but this fact is not true for s = 1/2). Actually, for s = 1/2, the extension by 0 is linear and continuous for a subspace of H 1/2 (U ), which is next defined as 1/2 H00 (U ). 7 For instance, 1 ∈ H 1/2 (U ) = H 1/2 (U ) and the extension by 0 is piecewise smooth and discontinuous, hence is not 0 in H 1/2 (R). Indeed, although functions of H 1/2 are not continuous in general, piecewise smooth functions that are discontinuous at one point do not belong to H 1/2 . For example, consider f , a Heaviside function that is multiplied by some smooth plateau function; then, f = δ0 + ψ with ψ smooth, hence iξ(F u)(ξ ) = 1 + (F ψ)(ξ ), so that |F f | behaves like 1/|ξ | at infinity, and hence (1 + |ξ |2 )1/4 |F f | ∈ / L2 (R), i.e., f ∈ / H 1/2 (R) (see [25, Chapter 33]).
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A.2.2. The Lions–Magenes space H00 (U ) 1/2 The Lions–Magenes space H00 (U ) is defined as the set of functions f ∈ H 1/2 (U ) such that −1/2 2 ρ f ∈ L (U ), where ρ denotes the distance to the boundary of U (see [21, Theorem 11.7]). 1/2 It is a strict subspace of H 1/2 (U ) = H0 (U ), equipped with the Hilbertian norm
2
1/2 f f 2H 1/2 (U ) + . √ d(·, ∂U ) 2 L (U )
1/2 Equivalently, H00 (U ) is the subspace of functions f ∈ H 1/2 (U ) such that their extension f˜ 1/2 by 0 outside U belongs to H 1/2 (Rn ), and the space H00 (U ) can be endowed with the equivalent norm f˜H 1/2 (Rn ) (see [25, Chapter 33]); for instance this latter norm can be computed by Fourier transform.
A.2.3. Fractional Laplacian operator For U bounded with a smooth boundary, the relation between f H s (U ) and (−)α/2 f 2L2 (Rn ) is more intricate than in the case U = Rn . First, for 0 s < 1/2, every f ∈ H s (U ) can be extended by 0 outside U into a function of H s (Rn ), and hence for such values of s the space H s (U ) can be as well defined using the equivalent norm 2 1/2 f H s (U ) = f 2L2 (U ) + (−)s/2 f L2 (Rn ) , where f is extended by 0 outside U . Note however that, although f has a compact support, the function (−)s/2 f is not of compact support. 1/2 / N + 1/2, and for the space H00 (U ), The same fact holds for the spaces H0s (U ), for s 0, s ∈ s n since functions of these spaces can be extended by 0 to functions of H (R ). For other values of s (and actually, for every s ∈ R), the existence of a continuous linear extension mapping PU , previously mentioned, permits to endow H s (U ), for instance, with the equivalent norm 2 1/2 f H s (U ) = f 2L2 (U ) + (−)s/2 PU f L2 (Rn ) . A.2.4. Relation with the domain of the fractional Dirichlet Laplacian operator Let A denote the opposite of the Dirichlet Laplacian on L2 (U ), of domain D(A) = 1 H0 (U ) ∩ H 2 (U ). It must not be confused with the previous Laplacian operator. The operator A is positive, selfadjoint, and has a discrete spectrum. The domains of its real powers define a scale of Hilbert spaces D(Aα ) (see [11]), which can be characterized as follows. Let (en )n∈N denote an orthonormal basis of eigenvectors of A, and let (λn )n∈N be the associated eigenvalues. Then, α 2 2α 2 λk f, ek < +∞ , D A = f ∈ L (U ) k∈N
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for every α ∈ R. Using the interpolation theory of [21], one can establish that ∀s ∈ [0, 1)
D As =
H02s (U )
if s = 1/4,
1/2 H00 (U )
if s = 1/4,
1/2
where the Lions–Magenes space H00 (U ) has been defined previously (note that, for s = 1, D(A) = H01 (U ) ∩ H 2 (U )). In particular, for every s ∈ [0, 1], s H0 (U ) 2 s 2 f ∈ L (U ) λk f, ek < +∞ = 1/2 H00 (U ) k∈N
if s = 1/2, if s = 1/2.
Note that As f must be distinguished from (−)s f , when both functions can be given a sense. 1/2 For instance, for every f ∈ H0s (U ) with s ∈ (0, 1), s = 1/2 (and f ∈ H00 (U ) for s = 1/2), one has f ∈ D(As/2 ) and thus As/2 f ∈ L2 (U ) by definition, whereas (−)s f ∈ L2 (Rn ) (where f is extended8 by 0 outside U ) is not even of compact support. The space D(As/2 ) is a Hilbert space, when equipped with the graph norm (f 2L2 (U ) + As/2 f 2L2 (U ) )1/2 . It follows that H0s (U ), for s ∈ (0, 1), s = 1/2, can be equiv1/2
alently defined with this graph norm, and similarly the space H00 (U ) can be endowed with the equivalent norm (f 2L2 (U ) + A1/4 f 2L2 (U ) )1/2 . References [1] R. Abraham, M. Bergounioux, E. Trélat, A penalization approach for tomographic reconstruction of binary radially symmetric objects, Appl. Math. Optim. 58 (3) (2008) 345–371. [2] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Tables, Dover, 1970. [3] R.A. Adams, J.F. Fournier, Sobolev Spaces, second ed., Pure Appl. Math., vol. 140, Elsevier Academic Press, Amsterdam, 2003. [4] M. Agranovsky, P. Kuchment, E.T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal. 248 (2) (2007) 344–386. [5] G. Ambartsoumian, P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal. 38 (2) (2006) 681–692. [6] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University Press, 2000. [7] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (7–9) (2007) 1245–1260. [8] E. Casas, K. Kunisch, C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim. 40 (1999) 229–257. [9] F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1983. [10] S. Cuccagna, Sobolev estimates for fractional and singular Radon transforms, J. Funct. Anal. 139 (1) (1996) 94–118. [11] K.K. Engel, R. Nagel, A Short Course on Operator Semigroups, Universitext, Springer, New York, 2006. [12] L.C. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [13] P. Gérard, F. Golse, B. Wennberg, A compactness result for generalized Radon transforms, Math. Res. Lett. 3 (4) (1996) 491–497. [14] M. Greenblatt, A method for proving Lp boundedness of singular Radon transforms in codimension one for 1 < p < ∞, Duke Math. J. 108 (2) (2001) 363–393. 8 Recall that functions of H s (U ) can be extended by 0 to H s (Rn ) for every s 0 such that s ∈ / N + 1/2; for s = 1/2 0 1/2 for instance, functions of H00 (U ) can be extended by 0 to H 1/2 (Rn ).
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[15] M. Greenblatt, Metrics and smoothing of translation-invariant Radon transforms along curves, J. Funct. Anal. 206 (2) (2004) 307–321. [16] S. Helgason, The Radon Transform, Birkhäuser, Basel, 1980. [17] S. Helgason, Ranges of Radon transforms, in: Computed Tomography, Cincinnati, Ohio, 1982, in: Proc. Sympos. Appl. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1982, pp. 63–70. [18] A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space, Math. Z. 184 (2) (1983) 165–192. [19] A. Hertle, On the range of the Radon transform and its dual, Math. Ann. 267 (1) (1984) 91–99. [20] J.-L. Lions, Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs, J. Math. Soc. Japan 14 (2) (1962) 233–241. [21] J.-L. Lions, E. Magenes, Problèmes aux Limites Non-Homogènes et Applications, vol. 1, Travaux Recherches Math., vol. 17, Dunod, Paris, 1968. [22] F. Natterer, Exploiting the ranges of Radon transforms in tomography, in: Numerical Treatment of Inverse Problems in Differential and Integral Equations, Heidelberg, 1982, in: Progr. Sci. Comput., vol. 2, Birkhäuser, Boston, MA, 1983, pp. 290–303. [23] E.T. Quinto, Null spaces and ranges for the classical and spherical Radon transforms, J. Math. Anal. Appl. 90 (2) (1982) 408–420. [24] A. Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (4) (1998) 869–897. [25] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital., vol. 3, Springer, UMI, Berlin, Bologna, 2007. [26] H. Triebel, Theory of Function Spaces, Birkhäuser, 1992.
Journal of Functional Analysis 259 (2010) 2333–2365 www.elsevier.com/locate/jfa
Initial boundary value problem and asymptotic stabilization of the Camassa–Holm equation on an interval Vincent Perrollaz Université Pierre et Marie Curie–Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France Received 4 March 2010; accepted 21 June 2010 Available online 3 July 2010 Communicated by J. Coron
Abstract We investigate the nonhomogeneous initial boundary value problem for the Camassa–Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law. © 2010 Elsevier Inc. All rights reserved. Keywords: Camassa–Holm equation; Initial boundary value problem; PDE control; Asymptotic stabilization
1. Introduction 1.1. Origins of the equation and presentation of the problems This article presents results concerning the initial boundary value problem and the possibility of asymptotic stabilization of the Camassa–Holm equation on a compact interval by means of a stationary feedback law acting on the boundary. The Camassa–Holm equation reads as follows (with κ a real constant): 3 2 3 ∂t v − ∂txx v + 2κ.∂x v + 3v.∂x v = 2∂x v.∂xx v + v.∂xxx v
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.007
for (t, x) ∈ [0, T ] × [0, 1].
(1)
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The Camassa–Holm equation describes one-dimensional surface waves at a free surface of shallow water under the influence of gravity. Here v(t, x) represents the fluid velocity at time t and position x. It is interesting to note that according to [2], it can equally represents the water elevation. Eq. (1) was first introduced by Fokas and Fuchssteiner [17] as a bi-Hamiltonian model, and was derived later as a water wave model by Camassa and Holm [2]. It turns out that this equation was also obtained as a model for propagating waves in cylindrical elastic rods, see Dai [12]. Eq. (1) shares many features with the KdV equation, see [21]. It is bi-Hamiltonian, completely integrable, and admits soliton solutions see [2,7,9,17,23]. However, it can also model breaking waves, in fact in H s (T) (s > 32 ) the solution generally develops singularity in finite time, see [4–6]. The Cauchy problem of (1) has been investigated in great details both on the torus and on the real line, see [1,3,8,13,14,20,24,26]. On the other hand, the study of the initial boundary value problem is much less complete, the homogeneous case was treated in [15] and in a more general setting in [16]. Finally a special case of the inhomogeneous case is considered in [28] (the boundary condition is that there is a constant C such that ∀t 0 we have v(t, x) −→ C). |x|→+∞
The first part of this article will be devoted to the proofs of a local in time existence theorem and of a weak-strong uniqueness result for the initial boundary value problem of (1). To explain our boundary formulation of (1), let us first remark that (1) is equivalent to the system: ∂t y + v.∂x y = −2y.∂x v, (2) 2 v. y − κ = 1 − ∂xx This formulation of (1) and the vorticity formulation of the two-dimensional Euler equation for incompressible perfect fluids (U is the speed and ω its vorticity) share similarities:
∂t ω + (U.∇)ω = 0, div U = 0, curl U = ω.
(3)
In both (2) and (3) there is a coupling between a transport equation and a stationary elliptic one. The initial boundary value problem for the two-dimensional incompressible Euler equation was treated by Yudovitch in [27], where he showed that the problem is well-posed in a classical sense with strong solutions if one prescribes the initial velocity or vorticity, the normal velocity on the boundary and also the vorticity of the fluid on the parts of the boundary where fluid enters. Similarly we will study the initial boundary value problem of (2) with v prescribed on the boundary, and y prescribed at time 0 and on the parts of the boundary where fluid enters. Remark 1. Note that (2) is even more similar to the vorticity formulation of the threedimensional incompressible Euler equation which reads:
∂t ω + (U.∇)ω = (ω.∇)U, div U = 0, curl U = ω
(4)
because here we have a stretching term (ω.∇)U similar to the term −2y ∂x v in (2). Kazhikov has studied the local in time initial boundary value problem in three dimensions see [22]. However
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the Euler equation is much less understood in three dimensions. For example it is still unknown whether a singularity may appear in finite time, see [25]. Furthermore the asymptotic stabilization problem is still open for the three-dimensional incompressible Euler equation which is not the case in two dimensions thanks to the papers of Coron [11] and Glass [18]. In the second part of the article we will investigate Eq. (1) from the perspective of control theory. For a general control system
x˙ = f (x, u), x(t0 ) = x0
(5)
(x being the state of the system and u the so-called control), we can consider two classical problems among others in control theory. 1. First the exact controllability problem which asks, given two states x0 and x1 and a time T to find a certain function u(t) such that the solution to (5) satisfies x(T ) = x1 . 2. If f (0, 0) = 0, the problem of asymptotic stabilization by a stationary feedback law asks to ˙ = f (x(t), u(x(t))), find a function u(x), such that for any state x0 a solution x(t) to x(t) x(t0 ) = x0 is global, satisfies x(t) −→ 0 and also t→+∞
∀R > 0, ∃r > 0 such that x0 r
⇒
∀t ∈ R, x(t) R.
(6)
It may seem that if we have controllability, the asymptotic stabilization property is weaker. Indeed for any initial state x0 , we can find T and u(t) such that the solution to (5) satisfies x(T ) = 0 in this way we stabilize 0 in finite time. However this control suffers from a lack of robustness with respect to perturbation. Indeed with any error on the model, or on the initial state, the state at time T will only be approximately 0. This can be disastrous if x = 0 is unstable for the equation x˙ = f (x, 0). This motivates the problem of asymptotic stabilization by a stationary feedback law which is clearly more robust. In fact in finite dimension, it automatically provides a Lyapunov function. Concerning the Camassa–Holm equation, O. Glass provided in [19] the first results for the controllability and stabilization. More precisely he considered: 3 2 3 v + 2κ.∂x v + 3v.∂x v = 2∂x v.∂xx v + v.∂xxx v + g(t, x)1ω (x) ∂t v − ∂txx
for (t, x) ∈ [0, T ] × T,
(7)
where the control is the function g, and ω is a nonempty open subset of the torus T. He proved that for any time T > 0 we have exact controllability in H s (T) (s > 32 ), and also proposed a stationary feedback law g : H 2 (T) → H −1 (ω) that stabilizes the state v = −κ in H 2 (T). We will consider those problems, but in our case the control will be the boundary values of v and y. Since [0, 1] can be seen as T \ ω the result of Glass on exact controllability by a distributed term on the torus implies a controllability result by boundary terms as soon as the initial boundary value problem makes sense, which will be the case by the end of the first part of this article (we also need enough regularity on the solution). Therefore we will only investigate the asymptotic stabilization by a stationary feedback law acting on the boundary of (1). This time again we will consider the analogy with the asymptotic
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stabilization of the two-dimensional Euler equation of incompressible fluids results by Coron [11] for a simply connected domain and Glass [18] for a general domain. It should be remarked that in three dimensions the problem of asymptotic stabilization is still open. In both cases one of the main difficulty is that the linearized system around the equilibriums (which are (y, v) = (0, −κ) for (2) and (ω, U ) = (0, 0) for (3)) is not stabilizable, so we will use the so-called return method introduced by Coron in [10]. Since the evolution equation of (2) is on y, it will be much easier to work if we consider y and not v to be the state of the system. 1.2. Results We begin with a general remark that will be used many times later. Remark 2. Changing v(t, x) in −v(t, 1 − x) and y(t, x) in −y(t, 1 − x) we change κ into −κ, therefore from now on we will suppose that κ 0 (this choice is more convenient for the stabilization part). Let T be a positive number. In the following we take ΩT = [0, T ] × [0, 1]. Let vl and vr be in C 0 ([0, T ], R) and y0 ∈ L∞ (0, 1). We set Γl = t ∈ [0, T ] vl (t) > 0 and Γr = t ∈ [0, T ] vr (t) < 0 . In the following, we will always suppose that the sets Pl = t ∈ [0, T ] vl (t) = 0 and Pr = t ∈ [0, T ] vr (t) = 0
(8)
have a finite number of connected components. Finally let yl ∈ L∞ (Γl ) and yr ∈ L∞ (Γr ). The functions vl , vr , yl and yr will be the boundary values for the equation and y0 is the initial data. Let now A be the auxiliary function which lifts the boundary values vl and vr and is defined by: 2 A(t, x) = 0, ∀(t, x) ∈ Ω , 1 − ∂xx T A(t, 0) = vl (t), A(t, 1) = vr (t), ∀t ∈ [0, T ].
(9)
Setting v = u + A, we can further rewrite the system (2) as:
2 u(t, x), dx, y(t, x) − κ = 1 − ∂xx u(t, 0) = u(t, 1) = 0, dt a.e.,
(10)
∂t y + (u + A).∂x y = −2y.∂x (u + A), y(0, .) = y0 , y(., 0)|Γl = yl and y(., 1)|Γr = yr .
(11)
The meaning of being a solution to (10)–(11) will be specified later but we will have u ∈ L∞ ((0, T ); Lip([0, 1])) and y ∈ L∞ (ΩT ). In the first part of this article, we will be interested in the initial boundary value problem on the interval for the system (10)–(11). We will first prove a local in time existence theorem:
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Theorem 1. For T˜ > 0, we consider vl , vr ∈ C 0 ([0, T˜ ]) such that the sets Pl and Pr have only a finite number of connected components. Let y0 ∈ L∞ (0, 1), yl ∈ L∞ (Γl ) and yr ∈ L∞ (Γr ). There exist T > 0, and (u, y) a weak solution of the system (10)–(11) with u ∈ L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip([0, T ]; H01 (0, 1)) and y ∈ L∞ (ΩT ). Moreover any such solution 1,p u is in fact in C 0 ([0, T ]; W 2,p (0, 1)) ∩ C 1 ([0, 1]; W0 (0, 1)), ∀p < +∞. Furthermore the existence time of a maximal solution is larger than min(T˜ , T ∗ ), with
T ∗ = max
ln(1 + β/C0 ) , β>0 2(C1 + (2 + sinh(1))(C0 + |κ| + β)) C0 = max y0 L∞ (0,1) , yl L∞ (Γl ) , yr L∞ (Γl ) , C1 =
1 . vr L∞ (0,T ) + vl L∞ (0,T ) . tanh(1)
(12) (13) (14)
In a second step, we will show a weak-strong uniqueness property: Theorem 2. Let (u, y) ∈ L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip([0, T ]; H01 (0, 1)) × L∞ ([0, T ]; Lip([0, 1])) be a weak solution of (10) and (11) then it is unique in L∞ ((0, T ); C 1,1 ([0, 1])) × L∞ (ΩT ). In the second part of the paper, we will be interested in the asymptotic stabilization of the system (1) by a boundary feedback law. Let Al > 2. sinh(1), Ar > Al . cosh(1) + sinh(2), M > 0 and T > 0. Our feedback law for (2) reads: ⎧ ⎨ vl (y) = Al .yC 0 ([0,1]) − κ, 0 y ∈ C [0, 1] → vr (y) = Ar .yC 0 ([0,1]) − κ, ⎩ y˙l (t) + M.yl (t) = 0.
(15)
This allows us to get the following theorem: Theorem 3. For any y0 ∈ C 0 ([0, 1]) there exists (y, v) ∈ C 0 (ΩT ) × C 0 ([0, T ], C 2 ([0, 1])) a weak solution of (2) and (15) satisfying ∀x ∈ [0, 1],
y(0, x) = y0 (x).
(16)
Furthermore any maximal solution of (2), (15) and (16) is global, and if we let
2.c.y0 C 0 ([0,1]) 1 Ar − Al . cosh(1) − sinh(2) and τ = . ln c = min Al − 2. sinh(1), sinh(1) M M then we have: ∀t τ
y(t, .)
C 0 ([0,1])
M 1 . . 2c 1 + M(t − τ )
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2. Initial boundary value problem We first define what we mean by a weak solution to (11). Our test functions will be in the space: Adm(ΩT ) = ψ ∈ C 1 (ΩT ) ψ(t, x) = 0 on [0, T ] \ Γl
× {0} ∪ [0, T ] \ Γr × {0} ∪ {T } × [0, 1] .
(17)
Definition 1. When u ∈ L∞ ((0, T ); Lip([0, 1])), a function y ∈ L∞ (ΩT ) is a weak solution to (11) if ∀ψ ∈ Adm(ΩT ):
y ∂t ψ + (u + A)∂x ψ − ∂x (u + A)ψ dt dx
ΩT
1 =−
T y0 (x)ψ(0, x) dx +
0
ψ(t, 1)vr (t)yr (t) − ψ(t, 0)vl (t)yl (t) dt.
0
Remark 3. It is obvious that C01 (ΩT ) ⊂ Adm(ΩT ) therefore a weak solution to (11) is also a solution to (11) in the distribution sense. And it is then clear that a regular weak solution is a classical solution. 2.1. Strategy In this part we will prove Theorems 1 and 2. Let us first explain the general strategy. We want to solve (10) and (11). Eq. (10) is a linear elliptic equation, and with u fixed (11) is a linear transport equation in y, with boundary data. Even when the flow is regular enough (and it will be in our case) to use the method of characteristics to solve the equation, singularity will generally appear, no matter how smooth the initial and boundary datas are, because of the boundary.
It is therefore useful to deal with weak solution of (11) belonging to L∞ (ΩT ). This is done in Appendix A. Once we know how to deal with each equation separately and have appropriate linear estimates, we use a fixed point strategy. It is interesting to remark that Yudovitch dealt with
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the two-dimensional incompressible Euler equation with nonhomogeneous boundary conditions in a similar way. However with y only essentially bounded, we cannot easily estimate the difference of two couples (u1 , y1 ) and (u2 , y2 ), therefore we will rather use a compactness argument and a Schauder fixed point instead of a Banach fixed point. The auxiliary function A may be less regular in time than u and this is why we will be able to transfer the time regularity of y on u. We will only prove a weak-strong uniqueness property, for the same reason that prevented us from using a Banach fixed point theorem. Therefore in Section 2.2 we will define precisely the fixed point operator F and study some of its properties. In Section 2.3 we will precise the domain on which we will apply Schauder’s fixed point theorem, we will prove the continuity of F in Section 2.4 and also study the additional properties of a fixed point. Finally in Section 2.5 we will prove the weak-strong uniqueness property. 2.2. The operator F The operator F is obtained as follows. Given u in L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip([0, T ]; we will define y to be the solution of (11), and once we have y in L∞ (ΩT ), we introduce u˜ solution of H01 (0, 1))
2 1 − ∂xx u˜ = y − κ.
(18)
Then F is defined as the operator associating u˜ to u. Now let us describe the auxiliary function A once and for all. Proposition 2.1. The function A defined by (9) satisfies: ∀(t, x) ∈ ΩT
1 . sinh(x).vr (t) + sinh(1 − x).vl (t) , sinh(1) 0 A ∈ C [0, T ]; C ∞ [0, 1] , and hence A(t, x) =
AL∞ ((0,T );C 1,1 ([0,1]))
cosh(1) . vr L∞ (0,T ) + vl L∞ (0,T ) . sinh(1)
As in Section A.1, for a function u ∈ L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip([0, T ]; H01 (0, 1)) we consider φ the flow of u + A. For (t, x) ∈ ΩT , φ(., t, x) is defined on a set [e(t, x), h(t, x)], here e(t, x) is basically the entrance time in ΩT of the characteristic curve going through (t, x). Lemma 1. The flow φ satisfies the following properties: 1. φ is C 1 with the following partial derivatives ∂1 φ(s, t, x) = (u + A) s, φ(s, t, x) , s ∂x (u + A) r, φ(r, t, x) dr , ∂2 φ(s, t, x) = −(u + A)(t, x). exp t
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s ∂3 φ(s, t, x) = exp
∂x (u + A) r, φ(r, t, x) dr ,
t
2. ∀j ∈ {1, 2, 3}, ∂j φC 0 (1 + u + AC 0 (ΩT ) )e 3. if e(t, x) > 0 then φ(e(t, x), t, x) ∈ {0, 1}, 4. if h(t, x) < T then φ(h(t, x), t, x) ∈ {0, 1}.
T .∂x (u+A)C 0 (Ω
T)
,
We introduce a partition of ΩT , which allows us to distinguish the different influence zones in ΩT . Definition 2. Let • P = (t, x) ∈ ΩT ∃s ∈ e(t, x), h(t, x) for which φ(s, t, x) = 0 and vl (s) = 0 or φ(s, t, x) = 1 and vr (s) = 0 ∪ φ(s, 0, 0) s h(0, 0) ∪ φ(s, 0, 1) s h(0, 1) , • I = {(t, x) ∈ ΩT \ P | e(t, x) = 0}, • L = {(t, x) ∈ ΩT \ P | e(t, x) > 0 and φ(e(t, x)t, x) = 0}, • R = {(t, x) ∈ ΩT \ P | e(t, x) > 0 and φ(e(t, x)t, x) = 1}. Remark 4. The set P is constituted of the problematic points. Indeed those points belong to the characteristics tangent to the boundary, which are precisely the singular points of e and h. Proposition 2.2. We have the following properties. 1. The sets P , I , L and R constitute a partition of ΩT . 2. The set P is negligible and each spatial section of P is negligible for the 1d lebesgue measure. 3. The function e is C 1 on L ∪ R ∪ I . 4. If (t, x) ∈ L then e(t, x) ∈ Γl and if (t, x) ∈ R then e(t, x) ∈ Γr . 5. All those sets are invariant by the flow φ. 6. If (t, x) ∈ L then ∀x˜ ∈ [0, x], (t, x) ˜ ∈ P ∪ L, if (t, x) ∈ R then ∀x˜ ∈ [x, 1], (t, x) ˜ ∈P ∪R ˜ ∈ P ∪ I. and if (t, x) ∈ I and (t, x + x ) ∈ I then ∀x˜ ∈ [x, x + x ], (t, x) Proof. The points 1, 4, 5, 6 are easy. The second point is true because for any t ∈ [0, T ] the set {(t, x) | x ∈ [0, 1]} ∩ P } is injected in the set of connected components of Pl and Pr , so it is countable and therefore 1d negligible. It implies that P itself is 2d negligible. And the third point is shown in Proposition A.3.
2
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For u ∈ L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip([0, T ]; H01 (0, 1)), we define y ∈ L∞ (ΩT ) by: t • if (t, x) ∈ I , y(t, x) = y0 (φ(0, t, x)). exp(−2 0 ∂x (u + A)(s, φ(s, t, x)) ds), t • if (t, x) ∈ L, y(t, x) = yl (e(t, x)). exp(−2 e(t,x) ∂x (u + A)(s, φ(s, t, x)) ds), t • if (t, x) ∈ R, y(t, x) = yr (e(t, x)). exp(−2 e(t,x) ∂x (u + A)(s, φ(s, t, x)) ds). And we have: 1. the function y is the unique weak solution of (11) in the sense of Definition 1, thanks to Theorem 6 and Proposition A.7 (which can be applied because u ∈ C 0 (ΩT ) and ∂x u ∈ C 0 (ΩT )), 2. since y ∈ L∞ (ΩT ) and satisfies (11), we immediately get y ∈ W 1,∞ (0, T , H −1 (0, 1)), 3. the function y satisfies the estimates: yL∞ (ΩT ) max y0 L∞ , yl L∞ , yr L∞ × exp 2T ∂x uL∞ (ΩT ) + ∂x AL∞ (ΩT ) , ∂t yL∞ ((0,T ),H −1 ) 3. max y0 L∞ (0,1) , yl L∞ (Γl ) , yr L∞ (Γl ) × exp 2T ∂x uL∞ (ΩT ) + ∂x AL∞ (ΩT ) × uL∞ ((0,T );Lip([0,1])) + AL∞ ((0,T );Lip([0,1])) ,
(19)
(20)
4. if (t, x) ∈ I ∪ L ∪ R and if (s, s ) ∈ [e(t, x), h(t, x)]2 , one has the following property: s y s, φ(s, t, x) = y s , φ s , t, x . exp −2 ∂x (u + A) r, φ(r, t, x) dr . s
We can now focus on the elliptic equation (10). Lemma 2. There exists a unique u˜ ∈ L∞ ((0, T ), H01 (0, 1)) such that 2 u(t, ˜ .) y(t, .) − κ = 1 − ∂xx
∀t ∈ (0, T ),
in D (0, 1).
Furthermore u˜ ∈ L∞ ((0, T ); C 1,1 ([0, 1])) ∩ Lip((0, T ), H01 (0, 1)) since y ∈ L∞ (ΩT ) ∩ Lip([0, T ]; H −1 (0, 1)). Moreover we have the bounds u ˜ L∞ ((0,T );C 1,1 ([0,1])) 1 + 2 sinh(1) . |κ| + yL∞ (ΩT ) ,
(21)
∂t u ˜ L∞ ((0,T );H 1 (0,1)) ∂t yL∞ ((0,T ),H −1 (0,1)) .
(22)
0
Proof. In the first point, the constant comes from: x u(t, ˜ x) =
sinh(x) sinh(x − x). ˜ κ − y(t, x) ˜ d x˜ − . sinh(1)
0
The second point is classical
1 0
2
sinh(x). ˜ κ − y(t, x) ˜ d x. ˜
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Finally we can define F by: ∀u ∈ L∞ (0, T ); C 1,1 [0, 1] ∩ Lip [0, T ]; H01 (0, 1) , F (u) = u˜ ∈ L∞ (0, T ); C 1,1 ([0, 1]) ∩ Lip [0, T ]; H01 (0, 1) .
(24)
We now introduce a domain for the operator F . 2.3. The domain Let B0 and B1 be positive numbers, then we set: CB0 ,B1 ,T = u ∈ L∞ (0, T ); C 1,1 ([0, 1]) ∩ Lip [0, T ]; H01 (0, 1) such that both uL∞ ((0,T );C 1,1 ([0,1])) B0 and uLip([0,T ];H 1 (0,1)) B1 . 0
(25)
Obviously CB0 ,B1 ,T is convex. We will endow CB0 ,B1 ,T with the norm .L∞ ((0,T );Lip([0,1])) . Lemma 3. There exist positive numbers B0 , B1 , T , such that F maps CB0 ,B1 ,T into itself. Proof. Let us first introduce the two following constants depending only on the initial and boundary conditions C0 = max y0 L∞ (0,1) , yl L∞ (Γl ) , yr L∞ (Γr ) , C1 =
cosh(1) . vr L∞ (0,T ) + vl L∞ (0,T ) . sinh(1)
Estimates (19), (20), (21) and (22) on y and u˜ now read: yL∞ (ΩT ) C0 . exp 2T ∂x uL∞ (ΩT ) + C1 , u ˜ L∞ ((0,T );C 1,1 ([0,1])) 1 + 2 sinh(1) . |κ| + yL∞ (ΩT ) , ∂t yL∞ ((0,T );H −1 (0,1)) 3.C0 . exp 2T ∂x uL∞ (ΩT ) + C1 . uL∞ ((0,T );Lip([0,1])) + C1 , ˜ L∞ ((0,T );H 1 (0,1)) ∂t yL∞ ((0,T );H −1 (0,1)) . ∂t u 0
Combining those estimates we get: u ˜ L∞ ((0,T );C 1,1 ([0,1])) 1 + 2 sinh(1) . |κ| + C0 . exp 2T ∂x uL∞ (ΩT ) + C1 , ∂t u ˜ L∞ ((0,T );H 1 (0,1)) 3.C0 . exp 2T ∂x uL∞ (ΩT ) + C1 . uL∞ ((0,T );Lip([0,1])) + C1 . 0
Now if u ∈ CB0 ,B1 ,T we have u ˜ L∞ ((0,T );C 1,1 ([0,1])) 1 + 2 sinh(1) . |κ| + C0 . exp 2T (B0 + C1 ) , ∂t u ˜ L∞ ((0,T );H 1 (0,1)) 3.C0 . exp 2T (B0 + C1 ) .(B0 + C1 ). 0
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Finally, to obtain u˜ ∈ CB0 ,B1 ,T it is sufficient that 1 + 2 sinh(1) . |κ| + C0 . exp 2T (B0 + C1 ) B0 and B0 + 3.C0 . exp 2T (B0 + C1 ) .(B0 + C1 ) B1 . Once we have chosen T and B0 , it is easy to choose B1 to satisfy the second inequality. For the first one we just choose B0 sufficiently large and then T close to 0. More precisely: B0 > 1 + 2 sinh(1) . |κ| + C0 , T
0 ln( 1+2 Bsinh(1) − |κ|) − ln(C0 )
2(B0 + C1 )
.
It only to remains to maximize the bound of T to get the minimum existence, and with B0 1+2 sinh(1) = |κ| + C0 + β we get the result announced. 2 Let us now prove the compactness of the domain. Proposition 2.3. CB0 ,B1 ,T is compact with respect to the norm .L∞ ((0,T );Lip([0,1])) . Proof. The fact that CB0 ,B1 ,T is closed in L∞ ((0, T ); Lip([0, 1])) follows from the weak∗ compactness of the domain in L∞ ((0, T ); C 1,1 ([0, 1])) and in Lip([0, T ]; H01 (0, 1)), and a classical use of a limit uniqueness. We now show the relative compactness of CB0 ,B1 ,T in L∞ ((0, T ); Lip([0, 1])). Let (un ) be a 1 sequence of CB0 ,B1 ,T . Since H01 (0, 1) → C 2 ([0, 1]) we can extract by Ascoli’s theorem a subse∞ quence (un ) converging in L (ΩT ). But since we have ∀u ∈ L∞ (0, T ); W 2,∞ (0, 1) ,
2 u ∂x uL∞ (ΩT ) 2. uL∞ (ΩT ) .∂xx , L∞ (Ω )
we can conclude that (un ) actually converges in L∞ ((0, T ); Lip([0, 1])).
T
2
Before applying Schauder’s fixed point theorem, it only remains to prove the continuity of the operator F . 2.4. Continuity of F and properties of the fixed points We begin with a result about the continuity of F . Proposition 2.4. The operator F : CB0 ,B1 ,T → CB0 ,B1 ,T is continuous with respect to .L∞ ((0,T );Lip([0,1])) . Proof. Let us take a sequence (un ) which tends to u with respect to .L∞ ((0,T );Lip([0,1])) . We call u˜ n = F (un ) and u˜ = F (u). Denote by φn the flow of un + A and φ the flow of
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u + A. Thanks to Proposition A.4, we have that φn −→ φ locally in C 1 . Let us show first that n→+∞
yn (t, .) − y(t, .)L1 (0,1) −→ 0 dt a.e. n→0
Let t ∈ [0, T ], having supposed that Pl and Pr have only a finite number of connected components (see (8)), we can assume, reducing t if necessary that vl and vr do not change sign on [0, t]. We will focus on the case where vl 0 and vr 0, the situation:
The characteristics of φn and φ may or may not cross before time t, but we are only interested in their relative positions at time t, which here correspond to φ(t, 0, 0) φn (t, 0, 0) φ(t, 0, 1) φn (t, 0, 1). The other cases are proved in the same way. We first point out that since un ∈ CB0 ,B1 ,T we have a bound for (yn ) in L∞ (ΩT ). Now 1
y(t, x) − yn (t, x) dx =
0
φ(t,0,0)
y(t, x) − yn (t, x) dx +
0
φn (t,0,0)
y(t, x) − yn (t, x) dx
φ(t,0,0) φ(t,0,1)
y(t, x) − yn (t, x) dx +
+ φn (t,0,0)
1 +
φn (t,0,1)
y(t, x) − yn (t, x) dx
φ(t,0,1)
y(t, x) − yn (t, x) dx
φn (t,0,1)
= I1 + I2 + I3 + I4 + I5 . Since φn (t, 0, 0) −→ φ(t, 0, 0) and φn (t, 0, 1) −→ φ(t, 0, 1) and thanks to the uniform n→+∞
n→+∞
bound on yn L∞ (ΩT ) we see that both I2 and I4 tend to 0 when n goes to infinity. For I1 we have:
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φ(t,0,0)
t ∂x (un + A) r, φn (r, t, x) dr yl en (t, x) . exp −2
I1 = 0
en (t,x)
− yl e(t, x) . exp −2
t
∂x (u + A) r, φ(r, t, x) dr dx.
e(t,x)
But thanks to Proposition A.2, if (t, x) ∈ / P (defined by φ) we have en (t, x) −→ e(t, x). This n→+∞
implies that if yl were continuous, since we have a uniform bound on un L∞ ((0,T );Lip([0,1])) the dominated convergence theorem would provide: φ(t,0,0)
y(t, x) − yn (t, x) dx −→ 0.
I1 =
n→+∞
0
The same idea can be applied to I3 and I5 . Hence for yl , yr and y0 continuous we have yn (t, .) − y(t, .)L1 (0,1) −→ 0. n→+∞
But now thanks to inequality (56), we have: y(t, .) 1 y0 L1 (0,1) + yl L1 ((0,t)∩Γl ) + yr L1 ((0,t)∩Γr ) L (0,1) × u + AL∞ (ΩT ) .e3t.∂x (u+A)L∞ (ΩT ) , yn (t, .) 1 y0 L1 (0,1) + yl L1 ((0,t)∩Γl ) + yr L1 ((0,t)∩Γ −r) L (0,1) × un + AL∞ (ΩT ) .e3t.∂x (un +A)L∞ (ΩT ) .
(26)
(27)
So by density of C 0 in L1 , and with the uniform bound on un L∞ ((0,T );Lip([0,1])) , the general case follows, yn (t, .) − y(t, .)
−→ 0.
L1 (0,1) n→+∞
Now only the restriction on t remains, we recall that until now we supposed that vl and vl did not change sign on [0, t]. But if vl and vr do not change sign on [0, t1 ] and then on [t1 , t], we have yn (t1 , .) − y(t1 , .)L1 (0,1) −→ 0. Let us call y˜n the solution of ∂t y˜n + (un + A)∂x y˜n = n→+∞
−2.y˜n .∂x (un + A) on [t1 , t] × [0, 1] with initial value y(t1 , .) and boundary values yl , yr . Due to what precedes we have y˜n (t, .) − y(t, .)L1 (0,1) −→ 0. Now we can conclude that: n→+∞
yn (t, .) − y(t, .) 1 L (0,1) yn (t, .) − y˜n (t, .)L1 (0,1) + y˜n (t, .) − y(t, .)L1 (0,1) yn (t1 , .) − y˜n (t1 , .)L1 (0,1) .un + AL∞ (ΩT ) .e3(t−t1 )∂x (un +A)L∞ (ΩT ) + y˜n (t, .) − y(t, .)L1 (0,1)
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yn (t1 , .) − y(t1 , .)L1 (0,1) .un + AL∞ (ΩT ) .e3(t−t1 )∂x (un +A)L∞ (ΩT ) + y˜n (t, .) − y(t, .)L1 (0,1) −→ 0.
n→+∞
Therefore the convergence in L1 (0, 1) propagates on each interval where vl and vr do not change sign, thanks to the hypothesis on Pr and Pl we have: ∀t ∈ [0, T ]
yn (t, .) − y(t, .)
−→ 0.
L1 (0,1) n→+∞
(28)
Combining this first convergence result with the uniform bound of yn − y in L∞ (ΩT ) and using the dominated convergence theorem in the time variable we obtain: yn → y
in L1 (ΩT ).
In term of u˜ and u˜ n it implies that u˜ n → u˜ in L1 0, T , W 2,1 (0, 1) . But we also have ∀n ∈ N F (un ) ∈ CB0 ,B1 ,T , and we know (see 2.3) that CB0 ,B1 ,T is compact therefore u˜ n → u˜ in CB0 ,B1 ,T (as the unique accumulation point of the sequence). 2 Now we can apply Schauder’s fixed point theorem to F and we get a solution u ∈ L∞ (0, T ); C 1,1 ([0, 1]) ∩ Lip [0, T ]; H01 (0, 1) . The additional regularity properties of any solution u, meaning 1,p ∀p > +∞ u ∈ C 0 [0, T ], W 2,p (0, 1) ∩ C 1 [0, 1], W0 (0, 1) , follow directly from the construction of F and from Proposition A.8. To obtain the minimum existence time announced we just have to realize that the only possible reduction of T occured in Section 2.3. This concludes the proof of Theorem 1. 2.5. Uniqueness To conclude the part about the initial boundary value problem, we prove a weak-strong uniqueness property. Theorem 4. Let (y, u) and (y, ˜ u) ˜ be two solutions of (10) and (11) for the same initial and ˜ boundary data, and such that y˜ ∈ L∞ ((0, T ); Lip([0, 1])). Then y = y˜ and u = u. Proof. Define Y = y˜ − y and U = u˜ − u. Then we have: U ∈ Lip [0, T ]; H01 (0, 1) ,
2 U (t, .) = Y (t, .) dt a.e., 1 − ∂xx
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and Y ∈ L∞ (ΩT ) is the unique weak solution of: ∂t Y + (u + A)∂x Y = −2.Y.∂x (u + A) − ∂x y.U ˜ − 2y.∂ ˜ x U, with Y0 = 0, Yl = 0, Yr = 0. Using Theorem 6 and formula (50) we get with b = −2.∂x (u + A) and f = −U.∂x y˜ − 2y.∂ ˜ xU: For (t, x) ∈ P , t For (t, x) ∈ I,
Y (t, x) =
Y (t, x) = 0, t f r, φ(r, t, x) . exp b r , φ r , t, x dr dr, r
0
t For (t, x) ∈ L,
Y (t, x) =
t
f r, φ(r, t, x) . exp
r
e(t,x)
t For (t, x) ∈ R,
Y (t, x) =
b r , φ r , t, x dr dr,
t
f r, φ(r, t, x) . exp
b r , φ r , t, x dr dr.
r
e(t,x)
˜ ∂x y˜ bounded, we see that for some Now since U (t, .)L∞ (0,1) 5.Y (t, .)L∞ (0,1) and y, C > 0: f (t, .)
L∞ (0,1)
C.Y (t, .)L∞ (0,1) dt a.e.,
and since b is bounded, we get that for some C > 0: Y (t, .)
L∞ (0,1)
t
C .
Y (s, .)
L∞ (0,1)
ds dt a.e.,
0
and we conclude using Gronwall’s lemma.
2
3. Stabilization In this part we prove Theorem 3. Here again we suppose that κ 0. We begin by reformulating (2) and we also give the corresponding statement to Theorem 3 for this new formulation. Rather than (2) we will work on: ⎧ ˇ ˇ ⎪ ⎨ ∂t y + (uˇ + A − κ).∂x y = −2y.∂x (uˇ + A), 2 1 − ∂xx uˇ = y, u(t, ˇ 0) = u(t, ˇ 1) = 0, ⎪ ⎩ 2 ˇ 0) = vl (t) + κ, A(t, ˇ 1) = vr (t) + κ. 1 − ∂xx Aˇ = 0, A(t, This system is equivalent to (2) with the change of unknown v = Aˇ + uˇ − κ.
(29)
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And our stationary feedback law still reads (15). One can check that Theorem 3 can be reformulated in terms of those new unknowns as: Theorem 5. Let Al > 2. sinh(1), Ar > Al . cosh(1) + sinh(2), M > 0, T > 0. For any y0 ∈ C 0 ([0, 1]) there exists y ∈ C 0 (ΩT ) such that if we define uˇ and Aˇ by: 2 ∀(t, x) ∈ ΩT 1 − ∂xx u(t, ˇ x) = y(t, x), u(t, ˇ 0) = u(t, ˇ 1) = 0, 2 ˇ x) = 0, ˇ 0) = Al .y(t, .) 0 1 − ∂xx A(t, A(t, ∀(t, x) ∈ ΩT C ([0,1]) and ˇ 1) = Ar .y(t, .) 0 A(t, C ([0,1]) , then y is the weak solution of ˇ ∂t y + (uˇ + Aˇ − κ).∂x y = −2.y.∂x (uˇ + A).
(30)
This function y also satisfies: ∀t ∈ [0, T ]
∂t y(t, 0) + M.y(t, 0) = 0,
∀x ∈ [0, 1]
y(0, x) = y0 (x).
Besides, if y is a maximal solution of the closed loop system (15), (29) then y is defined on ) and τ = [0, +∞) × [0, 1]. And finally if we let c = min(Al − 2. sinh(1), Ar −Al . cosh(1)−sinh(2) sinh(1) 2.c.y0 C 0 ([0,1]) 1 ), M . ln( M
we have: ∀t τ
y(t, .)
C 0 ([0,1])
1 M . . 2c 1 + M(t − τ )
(31)
We now prove Theorem 5. 3.1. Strategy Let us first describe the main steps of the proof of Theorem 5. In terms of the new unknowns, the equilibrium state that we want to stabilize is y = 0, uˇ = Aˇ = 0. A first natural idea would be to look at the linearized system around the equilibrium state. Its stabilization would provide a local stabilization result on the nonlinear system. But the linearized system reads: ⎧ ⎪ ⎨ ∂ t y − κ.∂x y = 0, 2 u 1 − ∂xx ˇ = y, u(t, ˇ 0) = u(t, ˇ 1) = 0, (32) ⎪ ⎩ 1 − ∂ 2 Aˇ = 0, A(t, ˇ ˇ 0) = v (t) + κ, A(t, 1) = v (t) + κ. l r xx In the case κ = 0, the state y is constant therefore the system is not stabilizable. In this situation we will apply a rough version of the return method that J.-M. Coron introduced in [10]. We will try to use the control in order to put the system in a simpler dynamic where it is easier to stabilize. ˇ When we look at the transport equation we see that the sign of u+ ˇ A−κ controls the geometry ˇ of the characteristics, and the sign of ∂x (uˇ + A) controls the growth of y along the characteristics.
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Therefore we would like our feedback law to provide uˇ + Aˇ 0 (since −κ 0) and ˇ 0. Considering the estimates ((33), (34)) on uˇ we can get from the elliptic equation ∂x (uˇ + A) of (29) we see that with vl (t) = Al .y(t, .)C 0 ([0,1]) − κ, vr (t) = Ar .y(t, .)C 0 ([0,1]) − κ, Aˇ will dominate uˇ and we will have the desired signs. For the existence of a solution we cannot adapt our proof of existence for the initial boundary value problem completely. Our feedback law makes us lose some regularity in time because Aˇ is now an unknown and it has exactly the time regularity of y(t, .)C 0 ([0,1]) . To compensate for this, we will work in the space of continuous functions for y. This is now possible because the flow will always point toward x = 1. Therefore we have to prescribe yl , and we just need to make a continuous transition at (t, x) = (0, 0) and have yl decreasing in time. This is garanteed by ∂t yl (t) + M.yl (t) = 0. In the next part we will prove the existence part of Theorem 5. The asymptotic properties will be proved in the last part. 3.2. Existence of a solution to the closed loop system Once again, we use a fixed point strategy on an operator S we describe now. We begin by defining the domain of the operator. Definition 3. Let X be the space of (g, N ) ∈ C 0 ([0, T ] × [0, 1]) × C 0 ([0, T ]) satisfying: 1. ∀(t, x) ∈ [0, T ] × [0, 1] g(0, x) = y0 (x), g(t, 0) = y0 (0).e−M.t , 2. ∀t ∈ [0, T ] g(t, .)C 0 ([0,1]) N (t), 3. N is nonincreasing and N (0) y0 C 0 ([0,1]) . Proposition 3.1. The domain X is nonempty, convex, bounded and closed with respect to the uniform topology. The proof is elementary and one notices that (y0 (x).e−Mt , y0 C 0 ([0,1]) .e−Mt ) ∈ X. Now for (y, N) ∈ X we define uˇ and Aˇ as the solutions of: 2 ∀(t, x) ∈ ΩT 1 − ∂xx u(t, ˇ x) = y(t, x) and u(t, ˇ 0) = u(t, ˇ 1) = 0, 2 ˇ x) = 0, ˇ 0) = Al N (t) and A(t, ˇ 1) = Ar N (t). ∀(t, x) ∈ ΩT 1 − ∂xx A(t, A(t, One has the following exact formulas: x ∀(t, x) ∈ ΩT
u(t, ˇ x) = −
sinh(x − x).y(t, ˜ x) ˜ d x, ˜ 0
∀(t, x) ∈ ΩT
ˇ x) = N (t) . Ar . sinh(x) + Al . sinh(1 − x) . A(t, sinh(1)
Therefore we have the following inequalities: ˇ x) 2 sinh(1)y(t, .)C 0 ([0,1]) , ∀(t, x) ∈ [0, T ] × [0, 1] u(t, ∂x u(t, ˇ x) 2 cosh(1)y(t, .)C 0 ([0,1]) ,
(33)
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2 ∂ u(t, x) 1 + 2 sinh(1) y(t, .)C 0 ([0,1]) , xx ˇ ∂x A(t, ˇ x) Ar − 2 cosh(1)Al .N (t), sinh(1)
A(t, ˇ x) Al .N (t).
(34) (35)
And in turn those provide: ∀(t, x) ∈ [0, T ] × [0, 1]
ˇ (uˇ + A)(t, x) Al − 2. sinh(1) .y(t, .)C 0 ([0,1]) ,
∀(t, x) ∈ [0, T ] × [0, 1] Ar − 2. cosh(1).Al − sinh(2) ˇ ∂x (uˇ + A)(t, .y(t, .)C 0 ([0,1]) . x) sinh(1)
(36)
(37)
Now if φ is the flow of uˇ + Aˇ − κ, φ is C 1 and since uˇ + Aˇ − κ 0 (thanks to the inequalities above), φ(., t, x) is nondecreasing. This allows us to define the entrance time and then the operator S as follows. Let e(t, x) = min{s ∈ [0, T ] | φ(s, t, x) = 0} with the convention that min ∅ = 0. Now for (t, x) ∈ [0, T ] × [0, 1], S(y, N ) = (y, ˜ N˜ ) with: t ˇ φ(s, t, x)) ds), 1. if x φ(t, 0, 0) y(t, ˜ x) = y0 (φ(0, t, x)). exp(−2 0 ∂x (uˇ + A)(s, t −M.e(t,x) ˇ 2. if x φ(t, 0, 0) y(t, ˜ x) = y0 (0).e . exp(−2. e(t,x) ∂x (uˇ + A)(s, φ(s, t, x)) ds), ˜ 3. N(t) = y(t, ˜ .)C 0 ([0,1]) . From Theorem 6 we know that y˜ is the weak solution of: ˇ ˜ x (uˇ + A), ∂t y˜ + (uˇ + Aˇ − κ)∂x y˜ = −2y∂
y(0, ˜ .) = y0 ,
y(t, ˜ 0) = y0 (0)e−Mt . (38)
Before applying Schauder’s fixed point theorem to S we prove the following statements. Proposition 3.2. 1. The operator S maps X to X. 2. The family S(X) is uniformly bounded and equicontinuous. 3. S is continuous w.r.t. the uniform topology. Proof. 1. It will be useful to distinguish the cases where y0 (0) = 0 (case 1) and y0 (0) = 0 (case 2). First remark that y˜ being continuous, N˜ is continuous. Now in case 1, we have that ∀(t, x) ∈ ΩT , x φ(t, 0, 0) ⇒ y(t, ˜ x) = 0 and both the continuity on {(t, x) ∈ ΩT | x > φ(t, 0, 0)} and the continuity at the interface {(t, x) ∈ ΩT | x = φ(t, 0, 0)} are obvious. In case 2, one must first remark that ∀t ∈ [0, T ], y(t, 0) = 0, so ∀t ∈ [0, T ], 0 < y(t, .)C 0 ([0,1]) N (t). This implies that every characteristic curve points to the right and so e corresponds to Definition A.1. Therefore e is C 1 on {(t, x) ∈ ΩT | x < φ(t, 0, 0)} and continuous at the interface {(t, x) ∈ ΩT | x = φ(t, 0, 0)}, once again we see that y˜ is continuous in ΩT , and so is N˜ . Now it is straightforward from its definition that ∀(t, x) ∈ [0, T ] × [0, 1],
y(0, ˜ x) = y0 (x),
y(t, ˜ 0) = y0 (0).e−M.t .
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ˇ 0 It only remains to see that N˜ = y(t, ˜ .)C 0 ([0,1]) is nonincreasing. Since ∂x (uˇ + A) (see (37)), we see from the definition of y˜ that |y| ˜ does not increase along the characteristics, and since |y(t, ˜ 0)| is also nonincreasing we can conclude. 2. Since X is already bounded and thanks to the first part of the proof, S(X) is bounded. ˜ being implied by the one of the family {y}, The equicontinuity of the family {N} ˜ we will show that we have a common continuity modulus for all {y}. ˜ For now let us focus only on {(t, x) ∈ ΩT | x φ(t, 0, 0)}. On this set y(t, ˜ x) = 0 in case 1. In the second case, we need ˇ the following inequalities valid on ΩT and which follow from the definition of uˇ and A: u ˇ C 0 (ΩT ) 2. sinh(1).y0 C 0 ([0,1]) ,
(39)
∂x u ˇ C 0 (ΩT ) 2. cosh(1).y0 C 0 ([0,1]) , 2 ∂ uˇ 0 xx C (ΩT ) 1 + 2. sinh(1) .y0 C 0 ([0,1]) , 2 ˇ C 0 (Ω ) = ∂xx Aˇ C 0 (Ω ) (Ar + Al )y0 C 0 ([0,1]) , A T
(40) (41)
ˇ C 0 (Ω ) Ar + Al .y0 C 0 ([0,1]) . ∂x A T tanh(1)
(43)
T
(42)
And since φ is the flow of uˇ + Aˇ − κ we also have: ∂1 φC 0 ([0,1]) −κ + 2 sinh(1) + Al + Ar y0 C 0 ([0,1]) , ∂2 φC 0 ([0,1]) −κ + 2 sinh(1) + Al + Ar y0 C 0 ([0,1])
Ar + Al y0 C 0 ([0,1]) , × exp 2.T . cosh(1). 2 + sinh(1)
Ar + Al ∂3 φC 0 ([0,1]) exp 2.T . cosh(1). 2 + y0 C 0 ([0,1]) . sinh(1) Now since we have y(t, ˜ x) = y0 (0).e
−M.e(t,x)
t
. exp −2.
ˇ ∂x (uˇ + A) r, φ(r, t, x) dr ,
e(t,x)
we see that we only need a uniform bound on eC 1 to conclude about the equicontinuity on {(t, x) ∈ ΩT | x φ(t, 0, 0)}. We have 0 e(t, x) T , and thanks to the definition of e, to (39), (42) and y(t, .)C 0 ([0,1]) |y(t, 0)| = |y0 (0)|.e−M.t |y0 (0).e−M.T | we get: ∂t e(t, x)
(κ + (2 sinh(1) + Al + Ar )y0 C 0 ([0,1]) ). exp(2.T . cosh(1).(2 + (Al − 2 sinh(1)).e−M.T .|y0 (0)|
Ar +Al sinh(1) )y0 C 0 ([0,1]) )
.
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In the same way: r +Al exp(2.T . cosh(1).(2 + A sinh(1) )y0 C 0 ([0,1]) ) ∂x e(t, x) . (Al − 2 sinh(1)).e−M.T .|y0 (0)|
In the end, we see that both in case 1 and case 2, the family {y} ˜ is uniformly Lipschitz on {(t, x) ∈ ΩT | x φ(t, 0, 0)}. Now on {(t, x) ∈ ΩT | x φ(t, 0, 0)}, we know t ˇ y(t, ˜ x) = y0 φ(0, t, x) . exp −2. ∂x (uˇ + A) r, φ(r, t, x) dr . 0
Clearly y0 is continuous on [0, 1] therefore it is both bounded and uniformly t continuous, ˇ the family of functions φ is uniformly Lipschitz and the family {exp(−2. 0 ∂x (uˇ + A)× (r, φ(r, t, x)) dr)} is uniformly bounded and equicontinuous. We can conclude that the family {y} ˜ is also equicontinuous on {(t, x) ∈ ΩT | x φ(t, 0, 0)}. Since we have continuity on {(t, x) ∈ ΩT | x = φ(t, 0, 0)}, we can conclude that the family S(X) is uniformly bounded and equicontinuous on ΩT , S(X) is therefore relatively compact in X. 3. It remains to prove that S is continuous w.r.t. to the uniform convergence. Let (yn ) be a sequence in X converging uniformly to y ∈ X. We only have to show that y˜n converges uniformly to y, ˜ since it immediately implies that N˜ n converges uniformly to N˜ . First the uniform convergence of yn and Nn implies the uniform convergence of uˇ n and Aˇ n . Then by Gronwall’s lemma, we also have φn → φ uniformly in C 1 (ΩT ). Using Proposition A.2, we then obtain en → e uniformly in C 0 (ΩT ). Now we decompose ΩT in three parts depending on n. Ln = (t, x) ∈ ΩT Rn = (t, x) ∈ ΩT
x min φn (t, 0, 0), φ(t, 0, 0) , x max φn (t, 0, 0), φ(t, 0, 0) ,
In = ΩT \ (Ln ∪ Rn ). Let us point out first that when n → +∞: lim inf Rn = (t, x) ∈ ΩT x φ(t, 0, 0) , lim inf Ln = (t, x) ∈ ΩT x φ(t, 0, 0) , and lim sup In = (t, x) ∈ ΩT x = φ(t, 0, 0) . • For (t, x) ∈ Ln if y0 (0) = 0 then yn and y˜ are equal to zero otherwise we have the formulas: y(t, ˜ x) = y0 (0).e−M.e(t,x) . exp −2
t
ˇ r, φ(r, t, x) dr , ∂x (uˇ + A)
e(t,x)
y˜n (t, x) = y0 (0).e
−M.en (t,x)
t
. exp −2 en (t,x)
ˇ ∂x (uˇ n + An ) r, φn (r, t, x) dr ,
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and the uniform convergence of y˜n follows from the uniform boundedness and convergence of ∂x uˇ n , ∂x Aˇ n , en and φn . • For (t, x) ∈ Rn the proof is similar. • It remains only to prove the convergence in In . But the width of In tends to zero, and the family {y˜n } is equicontinuous. Therefore the uniform convergence of y˜n in In follows from those in Ln and Rn . 2 Now we can apply Schauder’s fixed point theorem to S and get (y, N ) fixed point of S. It remains to show that it satisfies all of the properties of Theorem 5 except (31) which will be proven in the next subsection. First we have y(t, 0) = y(t, ˜ 0) = y0 (0).e−M.t and it implies ∂t y(t, 0) = −M.y(t, 0). But also N(t) = N˜ (t) = y(t, ˜ .)C 0 ([0,1]) = y(t, .)C 0 ([0,1]) , therefore y(t, .)C 0 ([0,1]) is nonincreasing and, thanks to Theorem 6, y = y˜ is a weak solution of ⎧ 2 u 1 − ∂ ˇ = y, u(t, ˇ 0) = u(t, ˇ 1) = 0, ⎪ xx ⎨ 2 Aˇ = 0, A(t, ˇ 0) = Al .y(t, .) 0 1 − ∂xx C ([0,1]) , ⎪ ⎩ ˇ ∂t y + (uˇ + Aˇ − κ).∂x y = −2y.∂x (uˇ + A).
ˇ 1) = Ar .y(t, .) 0 A(t, C ([0,1]) , (44)
Remark 5. • Since (uˇ + Aˇ − κ)(t, 1) = Ar .y(t, .)C 0 ([0,1]) − κ 0 we had all along Γr = ∅. • Since (uˇ + Aˇ − κ)(t, 0) = Al .y(t, .)C 0 ([0,1]) − κ, we see that a priori, Γl depends on y. But in fact if y0 (0) = 0 then ∀t, y(t, 0) = 0 and Γl = R+. And if y0 (0) = 0 then ∀t, yl (t) = y(t, 0) = 0 and it makes no difference in the weak formulation (53) if we enlarge Γl to R+. Therefore the space of test functions is always: Adm(ΩT ) = φ ∈ C 1 (ΩT ) ∀x ∈ [0, 1] φ(T , x) = 0, ∀t ∈ [0, T ] φ(t, 1) = 0 . • It must be noted that while we required T < ∞, we did not need T to be small. 3.3. Stabilization and global existence To finish the proof of Theorem 5 we have to prove the global existence of a maximal solution and estimate (31). Proof. First we rewrite (36), (37) as: ∀(t, x) ∈ ΩT
ˇ (uˇ + A)(t, x) cy(t, .)C 0 ([0,1]) , ˇ x) cy(t, .)C 0 ([0,1]) . ∂x (uˇ + A)(t,
But y is the solution of the transport equation (30) and it satisfies: t ˇ r, φ(r, t, x) dr . y(t, x) = y s, φ(s, t, x) . exp −2 ∂x (uˇ + A) s
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Combining those facts, we get for t s: t y(t, x) y s, φ(s, t, x) . exp −2 c.y(r, .) 0 C ([0,1]) dr . s
This implies that |y| decreases along the characteristics (strictly for the times where y(t, .) ≡ 0). But we have also imposed y(t, 0) = y(s, 0).e−M(t−s) , therefore |y| also decreases along x = 0. This already shows, thanks to the existence theorem that a maximal solution of the closed loop system is global. To get a more precise statement, we consider all the characteristics between time t and s and we obtain: for 0 s t
y(t, .) 0 C ([0,1]) y(s, .)
max C 0 ([0,1]) . r∈[s,t]
e
−M(r−s)
t
. exp −2c
y(α, .)
C 0 ([0,1]) dα
.
r
t Now we define g(r) = e−M(r−s) . exp(−2c r y(α, .)C 0 ([0,1]) dα), then g (r) = (2cy(r, .)C 0 ([0,1]) − M)g(r) and we know that as long as the quantity y(r, .)C 0 ([0,1]) is not equal to zero, it strictly decreases. So if y0 C 0 ([0,1]) > M 2c , for t small enough y(t, .)C 0 ([0,1]) M and we have: 2c y(t, .) which implies y(τ, .)C 0 ([0,1]) clear when y0 C 0 ([0,1]) M 2c ) y(t, .)
C 0 ([0,1])
C 0 ([0,1])
M 2c .
y0 C 0 ([0,1]) .e−M.t
This provides for τ s t, the inequality (which was
y(s, .)
C 0 ([0,1]) . exp
t
−2c
y(r, .)
C 0 ([0,1]) dr
.
s
And we conclude with a classical comparison principle for ODES.
2
Remark 6. • For κ = 0 the result is easily improved.
2 sinh(1)+Al +Ar we have −κ + uˇ + Aˇ − κ2 . κ.c 2 l +Ar τ − 2 sinh(1)+A − κ2 ⇒ y(t, .)C 0 ([0,1]) |y0 (0)|.e−M(t+ κ ) . κ.c
Indeed if t τ − And therefore t
• In particular if y0 (0) = 0 we see that we stabilize the null state in finite time. • Of course similar results hold for κ 0 thanks to Remark 2.
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Appendix A. Initial boundary value problem for a linear transport equation In this section we will consider the initial boundary value problem for the following linear transport equation: ∂t y + a(t, x).∂x y = b(t, x).y + f (t, x).
(45)
We will look at strong and weak solutions of (45) on ΩT = [0, T ] × [0, 1]. It should be noted that the backward problem is transformed in a standard one by the change of variables: t → T − t. A.1. Properties of the flow Let a ∈ C 0 (ΩT ) be uniformly Lipschitz in the second variable with constant L = aL∞ ((0,T ),Lip([0,1])) . Since we want to use the method of characteristics to solve (45) we need to study the flow of a. Definition 4. For (t, x) ∈ ΩT , let φ(., t, x) be the C 1 maximal solution to: ∂s φ(s, t, x) = a s, φ(s, t, x) , φ(t, t, x) = x,
(46)
which is defined on a certain set [e(t, x), h(t, x)] (which is closed because [0, 1] is compact) and with possibly e(t, x) and/or h(t, x) = t. Remark 7. Obviously e(t, x) > 0 ⇒ φ(e(t, x), t, x) ∈ {0, 1}. Now we take into account the influence of the boundaries by introducing the sets: P = (t, x) ∈ ΩT ∃s ∈ e(t, x), h(t, x) such that φ(s, t, x) ∈ {0, 1} and a s, φ(s, t, x) = 0 ∪ s, φ(s, 0, 0) ∀s ∈ [0, T ] ∪ s, φ(s, 0, 1) ∀s ∈ [0, T ] , I = (t, x) ∈ ΩT \ P e(t, x) = 0 , L = (t, x) ∈ ΩT \ P φ e(t, x), t, x = 0 , R = (t, x) ∈ ΩT \ P φ e(t, x), t, x = 1 , Γl = t ∈ [0, T ] a(t, 0) > 0 , Γr = t ∈ [0, T ] a(t, 1) < 0 . Proposition A.1. The function φ is uniformly Lipschitz on its domain. Proof. This is easily deduced from the standard case by the use of a Lipschitzian extension of a. 2 We can now study the regularity of e. Proposition A.2. Let (t, x) ∈ ΩT \ P , (an ) ∈ C 0 (ΩT ) ∩ L∞ ((0, T ); Lip([0, 1])) a sequence such that an − aC 0 (ΩT ) → 0, an L∞ (0,1;Lip([0,1])) is bounded and (tn ; xn ) ∈ ΩT such that (tn , xn ) → (t, x). Then en (tn , xn ) → e(t, x).
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Proof. Once again we will use a Lipschitzian extension operator Π and we set a˜ n = Π(an ) and a˜ = Π(a). Now let φ˜ n and φ˜ be their respective flows. Using Gronwall’s lemma we have: ˜ L∞ ((0,T );Lip([0,1])) (φ˜ n − φ)(s, ˜ t, x) T .a˜ n − a ˜ C 0 (ΩT ) .eT .a . (47) But we can see that: en (tn , xn ) = min s ∈ [0, tn ] ∀r ∈ [s, tn ], φ˜ n (r, t, x) ∈ [0, 1] . • If (t, x) ∈ I since we have excluded the characteristics coming from (0, 0) and (0, 1) we have that infs∈[0,T ] (d(φ(s, t, x), [0, t] × {0} ∪ [0, t] × {1})) > 0. So we can conclude from (47) that for n large enough φn (., t, x) is defined back to 0 that is en (t, x) = 0. From now on (t, x) ∈ L ∪ R. ˜ t, x) ∈ • Now we can take s strictly lower and close enough to e(t, x), φ(s, / [0, 1], since ˜ t, x), therefore for n large enough (t, x) ∈ / P ⇒ e(t, x) ∈ Γl ∪ Γr . But φ˜ n (s, tn , xn ) → φ(s, φ˜ n (s, tn , xn ) ∈ / [0, 1] and s < tn and we can conclude that lim inf en (tn , xn ) s. But s is arbitrarily close to e(t, x) and we get lim inf en (tn , xn ) e(t, x). • If e(t, x) = t then lim sup en (tn , xn ) lim sup tn = t and en (tn , xn ) → e(t, x). Otherwise since (t, x) ∈ / P then ∀s ∈ ]e(t, x), t[, φ(s, t, x) ∈ ]0, 1[. And now ∀ > 0, ∃α > 0 such that ∀s ∈ [e(t, x) + , t − ] min(φ(s, t, x), 1 − φ(s, t, x)) α. But for n large enough we have: α , 4 φn (s, tn , xn ) − φn (s, t, x) α 4 φn − φC 0 (ΩT )
(the second estimate comes from the uniform bound on an L∞ ((0,1);Lip([0,1])) ). But now, combining those two inequalities we see that for n large and for all s between e(t, x) + and t − we have min(φn (s, tn , xn ), 1 − φn (s, tn , xn )) α2 , this provides lim sup en (tn , xn ) e(t, x) + , and since is arbitrarily small we obtain: lim sup en (tn , xn ) e(t, x).
2
Remark 8. • For an = a it shows that e is continuous outside of P . • If P = ∅, since ΩT is compact the proposition implies that en converges uniformly toward e. Proposition A.3. If we assume that ∂x a ∈ C 0 (ΩT ) then φ is C 1 and e is C 1 on ΩT \ P with: s a(t, x). exp( e(t,x) ∂x a(r, φ(r, t, x))dr) , ∂t e(t, x) = a(e(t, x), φ(e(t, x), t, x)) s exp( e(t,x) ∂x a(r, φ(r, t, x))dr) ∂x e(t, x) = − . (48) a(e(t, x), φ(e(t, x), t, x))
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Proof. The regularity of φ is a classical result. If (t, x) ∈ I , e(t, x) = 0 and it is obvious. For (t, x) ∈ L we have φ(e(t, x), t, x) = 0 and e(t, x) ∈ Γl therefore ∂1 φ(e(t, x), t, x) > 0 and the implicit function theorem let us conclude, we can proceed in the same way for R. The inclusion of the characteristics of (0, 0) and (0, 1) in P is needed here. 2 Proposition A.4. Let (an ) be a sequence of C 0 ([0, T ]; C 1 ([0, 1])) and a ∈ C 0 ([0, T ]; C 1 ([0, 1])) such that an − aL∞ ((0,T );Lip([0,1])) −→ 0. If we call φn the flow of an and φ the flow of a then n→+∞
φn −→ φ locally in C 1 . n→+∞
Proof. Once again using a C 1 extension operator on an and a we deduce the result from the classical standard case, which follows from applications of Gronwall’s lemma. 2 A.2. Strong solutions Here we consider the case of data a ∈ C 0 ([0, T ]; C 1 ([0, 1])), yl ∈ Cc1 (Γl ), yr ∈ Cc1 (Γr ), y0 ∈ b ∈ C 1 (ΩT ) and f ∈ Cc1 (ΩT \ P ). We define the function y in the following way:
Cc1 (0, 1),
for (t, x) ∈ P
for (t, x) ∈ I
y(t, x) = 0, t y(t, x) = y0 φ(0, t, x) . exp b r, φ(r, t, x) dr
(49)
0
t +
t f r, φ(r, t, x) . exp b r , φ r , t, x dr dr, r
0
t b r, φ(r, t, x) dr for (t, x) ∈ L y(t, x) = yl e(t, x) . exp e(t,x)
t +
t f r, φ(r, t, x) . exp b r , φ r , t, x dr dr, (50) r
e(t,x)
for (t, x) ∈ R
t b r, φ(r, t, x) dr y(t, x) = yr e(t, x) . exp e(t,x)
t + e(t,x)
t f r, φ(r, t, x) . exp b r , φ r , t, x dr dr. r
Proposition A.5. We have y ∈ C 1 (ΩT ), supp(y) ⊂ ΩT \ P and y is a strong solution of (45) with the additional conditions that for all x in [0, 1] y(0, x) = y0 (x), for all t in Γl y(t, 0) = yl (t) and for all t in Γr y(t, 1) = yr (t). Besides we have the estimate: T .b 0 C (ΩT ) . (51) yC 0 (ΩT ) max(y0 C 0 (0,1) , yl C 0 (Γl ) , yr C 0 (Γr ) ) + T .f C 0 (ΩT ) .e
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Proof. First, y is equal to 0 in a neighbourhood of P because we chose y0 , yl , yr , f to be null close to P and because of (50). Outside of this neighbourhood, the regularity of y comes from the integral formulas (50) and from the regularity of y0 , yl , yr , f , b, φ and e (proved in Proposition A.3). The fact that y satisfies (45) is a straightforward calculation. 2 Remark 9. We have that: ∀(t, x) ∈ ΩT and ∀s ∈ e(t, x), h(t, x) t y(t, x) = y s, φ(s, t, x) . exp b r, φ(r, t, x) dr s
t +
t f r, φ(r, t, x) . exp b r , φ r , t, x dr dr.
s
(52)
r
A.3. Weak solutions In this section we will consider the case of data a ∈ C 0 ([0, T ]; C 1 ([0, 1])), b, f ∈ L∞ (ΩT ), y0 ∈ L∞ (0, 1), yl ∈ L∞ (Γl ) and yr ∈ L∞ (Γr ). We introduce the space of test functions: Adm(ΩT ) = φ ∈ C 1 (ΩT ) ∀x ∈ [0, 1] φ(T , x) = 0, ∀t ∈ [0, T ] \ Γl φ(t, 0) = 0, ∀t ∈ [0, T ] \ Γr φ(t, 1) = 0 . Proposition A.6. For y ∈ C 1 (ΩT ), y is a strong solution of (45), if and only if it satisfies ∀φ ∈ Adm(ΩT ) y. ∂t φ + a.∂x φ + (b + ∂x a)φ dx dt ΩT
=−
1 f (t, x).φ(t, x) dt dx −
ΩT
T +
φ(0, x).y(0, x) dx 0
a(t, 1).φ(t, 1).y(t, 1) − a(t, 0).φ(t, 0).y(t, 0) dt.
(53)
0
This legitimates the following definition of a weak solution. Definition 5. For a ∈ L∞ (0, T , Lip(0, 1)), b, f ∈ L1 (ΩT ), y0 ∈ L1 (0, 1), yl ∈ L1 (Γl ) and yr ∈ L1 (Γr ), we say that y ∈ L∞ (ΩT ) is a weak solution of (45) if it satisfies (53). Theorem 6. Let a ∈ C 0 ([0, T ]; C 1 ([0, 1])), b, f ∈ L∞ (ΩT ), y0 ∈ L∞ (0, 1), yl ∈ L∞ (Γl ) and yr ∈ L∞ (Γr ). We will also suppose that the sets Pl = t ∈ [0, T ] a(t, 0) = 0 and Pr = t ∈ [0, T ] a(t, 1) = 0
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have at most a countable number of connected components. Then the function y defined by the formula (50), is a weak solution of (45) and satisfies: yL∞ (ΩT ) max y0 L∞ (0,1) , yl L∞ (Γl ) , yr L∞ (Γr ) + T .f L∞ (ΩT ) .eT .bL∞ (ΩT ) .
(54)
Proof. If we let Pt˜ = P ∩ {(t, x) ∈ ΩT | t = t˜}, we can see that each points of a Pt˜ corresponds to at least one connected component of Pl ∪ Pr (since only one characteristic curve goes through the whole connected component) therefore, Pt˜ is at most countable and thus 1d negligible, this implies that P is 2d negligible. Now we have: • • • •
Cc1 (ΩT \ P ) is dense in L1 (ΩT ), Cc1 (0, 1) is dense in L1 (0, 1), Cc1 (Γl ) is dense in L1 (Γl ), Cc1 (Γr ) is dense in L1 (Γr ).
And we can take, thanks to the hypothesis on b, f , y0 , yl and yr : • (bn ) ∈ C 1 (ΩT ) such that bn − bL1 (ΩT ) → 0 and bn L∞ (ΩT ) is bounded, • (fn ) ∈ Cc1 (ΩT \ P ) such that fn − f L1 (ΩT ) → 0 and fn L∞ (ΩT ) is bounded, • (y0,n ) ∈ Cc1 (0, 1) such that y0,n − y0 L1 (0,1) → 0 and y0,n L∞ (0,1) is bounded, • (yl,n ) ∈ Cc1 (Γl ) such that yl,n − yl L1 (Γl ) → 0 and yl,n L∞ (Γl ) is bounded, • (yr,n ) ∈ Cc1 (Γr ) such that yr,n − yl L1 (Γr ) → 0 and yr,n L∞ (Γr ) is bounded. We call (yn ) the sequence of strong solutions to (45). Thanks to (51) we can extract so that: ∃y ∈ L∞ (ΩT ) such that yn converges to y for the weak-∗ topology of L∞ (ΩT ). Now we take the limit in (53) and conclude that y is a weak solution to (45). We can also suppose (we just need to extract again) that we have pointwise convergence almost everywhere of: bn → b,
fn → f,
y0,n → y0 ,
yl,n → yl ,
yr,n → yr .
Thanks to the dominated convergence theorem and to the limit uniqueness, we see that y satisfies (50) and (52) almost everywhere, and this provides (54). 2 A.4. Uniqueness of the weak solution We have proved the existence of a weak solution to (45) and we have the bound (54), therefore the initial boundary value problem will be well posed once we have shown the uniqueness of the weak solution.
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Proposition A.7. Under the hypothesis of Theorem 6, there is only one weak solution to (45). Proof. By linearity we only need to prove the uniqueness for f = 0, y0 = 0, yl = 0, yr = 0. Which is ∀y ∈ L∞ (ΩT ):
∀φ ∈ Adm(ΩT ) y. ∂t φ + a.∂x φ + (b + ∂x a).φ dx dt = 0
⇒
y = 0 a.e.
ΩT
Let y be such as above, we take: • yn ∈ Cc1 (ΩT \ P ) such that yn − yL2 (ΩT ) → 0 and yn L∞ (ΩT ) is bounded, • dn ∈ C 1 (ΩT ) such that dn − (b + ∂x a)L2 (ΩT ) → 0 and dn L∞ (ΩT ) is bounded. We want φn ∈ Adm(ΩT ) to be a strong solution of ∂t φn + a.∂x φn + dn .φn = yn , but the boundary conditions for functions in Adm(ΩT ) makes it a backward problem. Indeed for φn to be a test function we must have ∀x ∈ [0, 1], φn (T , x) = 0, ∀t ∈ [0, T ] \ Γl , φn (t, 0) = 0 and ∀t ∈ [0, T ] \ Γr , φn (t, 1) = 0. As we said previously the change of variables t → T − t transforms a backward problem in a regular forward one, which we can solve thanks to Section A.2. We just need to realize that the change of variables t → T − t sends the old P on the new P , the old [0, T ] \ Γl on the new Γl ∪ Pl and the old [0, T ] \ Γr on the new Γl ∪ Pr . And therefore: ∀n ∈ N,
y. yn + φn (b + ∂x a − dn ) dx dt = 0.
ΩT
Now thanks2 to the hypothesis on yn and dn , and to (51), when n → +∞ we get ΩT |y(t, x)| dx dt = 0. 2 A.5. Additional properties of y Until now weak solutions had only the L∞ regularity but in fact we have more. Lemma 4. If a and ∂x a are continuous and if the sets Pl = {t ∈ [0, T ] | a(t, 0) = 0} and Pr = {t ∈ [0, T ] | a(t, 1) = 0} have a finite number of connected components, and if b and f are in L∞ (ΩT ) then ∀p < +∞ we have yLp (0,1) ∈ C 0 ([0, T ]). Proof. Let t ∈ [0, T ] and 0. Reducing if necessary we can suppose that a(s, 0) and a(s, 1) have a constant sign on [t, t + ]. Hence we will prove the result in the case a(t, 0) 0 and a(t, 1) 0 (the other cases being similar). This implies h(t, 0) t + and e(t + , 1) t:
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Now we have: y(t + , .)p p L
φ(t+,t,0)
1
y(t + , x)p dx +
= (0,1) 0
y(t + , x)p dx
φ(t+,t,0)
since φ(t + , t, 0) −→ 0 and y ∈ L∞ (ΩT ) the first integral tends to 0. Then, if x ∈ →0
[φ(t + , t, 0), 1] we recall that thanks to (52) and after performing the change of variables x˜ = φ(t, t + , x) one has: 1
y(t + , x)p dx =
φ(t,t+,1)
˜ exp y(t, x).
t+ b s, φ(s, t, x) ˜ ds t
0
φ(t+,t,0)
t+ p t+ + f t, φ(r, t, x) ˜ . exp b r , φ r , t, x˜ dr dr t
r
t+ × exp ∂x a s, φ(s, t, x) ˜ ds d x. ˜
(55)
t
And finally since φ(t, t + , 1) −→ 1, f, b, y ∈ L∞ (ΩT ) and ∂x a ∈ C 0 (ΩT ) we get →0+
1
y(t + , x)p dx −→
1
→0+
φ(t+,t,0)
y(t, x)p dx.
0
The other geometries of the characteristics are treated in the same way. And the argument is clearly reversible in time so we also have the case 0. 2 Now we can get some additional regularity for y.
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Proposition A.8. If a and ∂x a are continuous, if the sets Pl = {t ∈ [0, T ] | a(t, 0) = 0} and Pr = {t ∈ [0, T ] | a(t, 1) = 0} have a finite number of connected components, if y0 , yl yr are essentially bounded and if b and f are in L∞ (ΩT ) then ∀p < +∞ we have y ∈ C 0 ([0, T ], Lp (0, 1)). Proof. We take t = 0 and > 0. Reducing if necessary, we can suppose that a(s, 0) and a(s, 1) have a constant sign on [t, t + ]. We will prove the result in the case a(t, 0) 0 and a(t, 1) 0 (the others can be treated in the same way). This implies h(0, 1), h(0, 0) . Let γ > 0, since y0 ∈ L∞ (0, 1) we have a function y˜0 ∈ C 0 ([0, 1]) such that y0 − y˜0 Lp (0,1) γ . We now consider y˜ the weak solution of (45) with boundary value yl and yr and initial value y˜0 . Now by linearity it is clear that y − y˜ is solution to (45) with boundary value 0 and initial value y0 − y˜0 . Therefore the previous lemma asserts that y(t, .) − y(t, ˜ .)Lp (0,1) is continuous and we see that for t sufficiently small y(t, .) − y(t, ˜ .)Lp (0,1) 2.γ . Now since y˜ satisfies (52), since b, f, y˜ ∈ L∞ (ΩT ) and more importantly since y˜0 continuous, we obtain y(, ˜ x) −→ y˜0 (x) for any x in (0, 1), therefore we can conclude that →0+ φ(,0,1) p ˜ x) − y˜0 (x)| dx −→ 0. And finally we conclude that for sufficiently small φ(,0,0) |y(, →0+
y(, ˜ .) − y˜0 (.)Lp (0,1) γ , which implies that for small enough: y(, .) − y0 (.)
Lp (0,1)
We can both translate and reversen the argument in time.
4γ . 2
To finish this part we will prove an inequality about the continuity property of the linear operator providing y in term of f , y0 , yl and yr . Proposition A.9. If a and ∂x a are continuous and if the sets Pl = {t ∈ [0, T ] | a(t, 0) = 0} and Pr = {t ∈ [0, T ] | a(t, 1) = 0} have a finite number of connected components then we have the inequality: ∀t ∈ [0, T ] y(t, .)
L1 (0,1)
f L1 ((0,t)×(0,1)) + y0 L1 (0,1) + yl L1 ((0,t)∩Γl ) + yr L1 ((0,t)∩Γr ) × aL∞ (ΩT ) ).et (∂x aL∞ (ΩT ) +bL∞ (ΩT ) ) .
(56)
Proof. Let us first suppose that a(s, 0), a(s, 1) 0 on [0, T ], this implies h(0, 0) t and e(t, 1) = 0, therefore we can write: y(t, .)
φ(t,0,0) L1 (0,1)
t b r, φ(r, t, x) dr dx yl e(t, x) . exp
0
(57)
e(t,x) φ(t,0,0)
t t f t, φ(r, t, x) . exp b s, φ(s, t, x) ds dr dx (58)
+ 0
e(t,x)
s
V. Perrollaz / Journal of Functional Analysis 259 (2010) 2333–2365
1 +
t b r, φ(r, t, x) dr dx y0 φ(0, t, x) . exp
+ φ(t,0,0)
(59)
0
φ(t,0,0)
1
2363
t t b s, φ(s, t, x) ds dr dx f t, φ(r, t, x) . exp
(60)
s
0
= I1 + I2 + I3 + I4 .
(61)
Now we will treat each Ik separately. In I1 we perform the change of variables: s = e(t, x) (or equivalently x = φ(t, s, 0)) and we get: t I1 =
yl (s).a(s, 0). exp
t
b(r, φ(r, s, 0)) + ∂x a r, φ(r, s, 0) dr ds.
s
0
Therefore we have I1 yl L1 (0,t) .aL∞ (ΩT ) .et.(∂x aL∞ (ΩT ) +bL∞ (ΩT ) ) . For the second integral we have: φ(t,0,0)
t t f t, φ(r, t, x) . exp b s, φ(s, t, x) ds dr dx
I2 = 0
t
(62)
s
e(t,x)
t f t, φ(r, t, x) . exp b s, φ(s, t, x) ds dx dr.
φ(t,0,0)
(63)
s
0 φ(s,0,0)
This time we perform the change of variables: x˜ = φ(r, t, x). And we get:
I2 e
t (∂x aL∞ (ΩT ) +bL∞ (ΩT ) )
t
φ(r,0,0)
f (t, x) ˜ d x˜ dr.
× 0
0
In the same way we obtain:
I3 e
t (bL∞ (ΩT ) +∂x aL∞ (ΩT ) )
φ(0,t,1)
y0 (x) ˜ ˜ d x.
× 0
And finally for I4 we use x˜ = φ(r, t, x) to obtain:
I4 e
t (bL∞ (ΩT ) +∂x aL∞ (ΩT ) )
t
φ(r,t,1)
0 φ(r,0,0)
f (t, x) ˜ d x˜ dr.
(64)
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Combining the inequalities on I1 , I2 , I3 and I4 we get (56). However we supposed that a(s, 0) and a(s, 1) did not change signs between on [0, T ]. Therefore if either a(s, 0) or a(s, 1) change sign at time t1 we only have the desired estimates separately on [t0 , t1 ] and on [t1 , t2 ] where on each interval, a(s, 0) and a(s, 1) do not change sign. More precisely if t ∈ [t1 , t2 ] we have: y(t1 , .)
L1 (0,1)
f L1 ((t0 ,t1 )×(0,1)) + y(t0 , .)L1 (0,1) + yl L1 ((t0 ,t1 )∩Γl ) + yr L1 ((t0 ,t1 )∩Γr )
× aL∞ (ΩT ) ).e(t1 −t0 )(∂x aL∞ (ΩT ) +bL∞ (ΩT ) ) , y(t, .) 1 f L1 ((t1 ,t)×(0,1)) + y(t1 , .)L1 (0,1) + yl L1 ((t1 ,t)∩Γl ) + yr L1 ((t1 ,t)∩Γr ) L (0,1) × aL∞ (ΩT ) ).e(t−t1 )(∂x aL∞ (ΩT ) +bL∞ (ΩT ) ) . And now we can substitute y(t1 , .)L1 (0,1) in the right side of (65) with the right side of (65), which provides (56) on the whole interval [t0 , t2 ]. Finally since we know that a(s, 0) and a(s, 1) change sign only a finite number of time, the previous argument allows us to extend (56) to [0, T ]. 2 Remark 10. The previous estimate and the well posedness in L∞ (ΩT ) of the initial boundary value problem (11) for data y0 , yl , yr and f in L∞ show that the same problem is well posed in C([0, T ]; L1 (0, 1)) with data in L1 . And then since the equation is linear and because we have both the well-posedness in L∞ (ΩT ) with essentially bounded data, and also the well-posedness in C 0 ([0, T ]; L1 (0, 1)) with summable data we can interpolate the two results and get well posedness in C 0 ([0, T ]; Lp (0, 1)) with data in Lp . References [1] A. Bressan, A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Ration. Mech. Anal. 183 (2) (2007) 215–239. [2] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (11) (1993) 1661–1664. [3] G.M. Coclite, H. Holden, K.H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal. 37 (4) (2005) 1044–1069. [4] A. Constantin, J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26 (2) (1998) 303–328. [5] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (2) (1998) 229–243. [6] A. Constantin, J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (1) (2000) 75–91. [7] A. Constantin, V.S. Gerdjikov, R.I. Ivanov, Inverse scattering transform for the Camassa–Holm equation, Inverse Problems 22 (6) (2006) 2197–2207. [8] A. Constantin, L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (1) (2000) 45–61. [9] A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610. [10] J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift, Math. Control Signals Systems 5 (1992) 295–312. [11] J.-M. Coron, On the null asymptotic stabilization of 2-D incompressible Euler equation in a simply connected domain, SIAM J. Control Optim. 37 (6) (1999) 1874–1896. [12] H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mech. 127 (1998) 193–207. [13] R. Danchin, A few remarks on the Camassa–Holm equation, Differential Integral Equations 14 (8) (2001) 953–988.
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[14] R. Danchin, A note on well-posedness for Camassa–Holm equation, J. Differential Equations 192 (2) (2003) 429– 444. [15] J. Escher, Z. Yin, Initial boundary value problems of the Camassa–Holm equation, Comm. Partial Differential Equations 33 (1–3) (2008) 377–395. [16] J. Escher, Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal. 256 (2) (2009) 479–508. [17] A.S. Fokas, B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1) (1981/1982) 47–66. [18] O. Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case, SIAM J. Control Optim. 44 (3) (2005) 1105–1147. [19] O. Glass, Controllability and asymptotic stabilization of the Camassa–Holm equation, J. Differential Equations 245 (6) (2008) 1584–1615. [20] A. Himonas, G. Misiolek, Well posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations 162 (2000) 479–495. [21] R.S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for water waves, J. Fluid Mech. 455 (2002) 63–82. [22] A.V. Kazhikhov, Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid, J. Appl. Math. Mech. 44 (5) (1980) 947–950 (in Russian). [23] J. Lenells, Traveling wave solutions of the Camassa–Holm equation, J. Differential Equations 217 (2) (2005) 393– 430. [24] Y.A. Li, P.J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (1) (2000) 27–63. [25] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math., vol. 27, Cambridge University Press, Cambridge, 2002, ISBN: 0-521-63057-6; 0-521-63948-4. [26] Z. Xin, P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (11) (2000) 1411–1433. [27] V.I. Yudovich, A two dimensional problem of unsteady flow of an ideal incompressible fluid across a given domain, Mat. Sb. 31 (1840) (1964) 562–588 and Mat. Sb. 64 (106) (1964) 277–304. [28] G. Zhang, Z. Qiao, F. Liu, Cusp and smooth solitons of the Camassa–Holm equation under an inhomogeneous boundary condition, Pacific J. Appl. Math. 1 (1) (2008) 105–121.
Journal of Functional Analysis 259 (2010) 2366–2383 www.elsevier.com/locate/jfa
De Branges–Rovnyak spaces and Dirichlet spaces Nicolas Chevrot, Dominique Guillot 1 , Thomas Ransford ∗,2 Département de Mathématiques et de Statistique, Université Laval, 1045, Avenue de la Médecine, Québec (Québec), Canada G1V 0A6 Received 12 March 2010; accepted 8 July 2010 Available online 31 July 2010 Communicated by N. Kalton
Abstract Sarason has shown that the local Dirichlet spaces Dλ may be considered as manifestations of de Branges– Rovnyak spaces H(b), and has used this identification to give a new proof that the spaces Dλ are starshaped. We investigate which other Dirichlet spaces D(μ) arise as de Branges–Rovnyak spaces, and which other de Branges–Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces inside Dλ . © 2010 Elsevier Inc. All rights reserved. Keywords: Hilbert space; de Branges–Rovnyak space; Dirichlet space
1. Introduction The spaces now called de Branges–Rovnyak spaces were introduced by de Branges and Rovnyak in the appendix of [1] and further studied in [2]. Subsequently, thanks in large part to the work of Sarason [6–10], it was realized that these spaces have numerous connections with other topics in complex analysis and operator theory. De Branges–Rovnyak spaces on the unit disk D are a family of subspaces H(b) of the Hardy space H 2 , parametrized by b in the closed unit ball of H ∞ . We shall give the precise definition * Corresponding author.
E-mail addresses:
[email protected] (N. Chevrot),
[email protected] (D. Guillot),
[email protected] (T. Ransford). 1 Supported by scholarships from NSERC (Canada) and FQRNT (Québec). 2 Supported by grants from NSERC, FQRNT and the Canada Research Chairs program. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.004
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of H(b) in Section 2. In general H(b) is not closed in H 2 , but it carries its own norm · b making it a Hilbert space. The general theory of H(b)-spaces subdivides into two cases, according to whether or not b is an extreme point of the unit ball of H ∞ . Perhaps the most important examples of extreme b are inner functions. If b is inner, then it turns out that H(b) = (bH 2 )⊥ , sometimes called the model space associated to b. These spaces have been studied extensively in the literature. In this article we shall concentrate on the case where b is not extreme. An interesting example is√obtained by taking bλ (z) := (1 − τ )λz/(1 − τ λz), where λ ∈ T, the unit circle, and τ = (3 − 5)/2. With this choice, it turns out that H(bλ ) = Dλ , the so-called local Dirichlet space at λ. The space Dλ was studied in detail by Richter and Sundberg in [5], and the identification H(bλ ) = Dλ is due to Sarason [9]. The underlying theme of the present paper is to investigate to what extent this example may be considered typical. Both the local Dirichlet spaces Dλ and the classical Dirichlet space D are instances of a more general family of Dirichlet spaces D(μ), indexed by finite positive measures μ on the unit circle T. Indeed, Dλ = D(δλ ), where δλ is the unit mass at λ, and D = D(m), where m is normalized Lebesgue measure on T. The spaces D(μ) first arose in [4], in connection with the problem of classifying the shift-invariant subspaces of D. For which measures μ does D(μ) arise as a de Branges–Rovnyak space H(b)? In Section 3 we shall show that only such measures are multiples of δλ (λ ∈ T), at the same time recovering Sarason’s identification of the corresponding functions b. The proof of this result is based on a formula for the inner products of monomials in H(b). This is a special case of a formula, established in Section 4, for the H(b)-norm of functions holomorphic on a neighborhood of D. To extend this further and treat general holomorphic functions in D, we are led to consider the problem of approximation of a function f by its expansions fr (z) := f (rz) (r < 1). The spaces Dλ (and more generally D(μ)) enjoy the property of being star-shaped, in the sense that fr always converges to f . Is the same true of de Branges–Rovnyak spaces H(b)? In [9], it is mentioned that a counterexample can be constructed, but, as far as we know, it has never been published. In Section 5 we shall provide two different families of counterexamples, as well as a sufficient condition for H(b) to be star-shaped which covers the case H(bλ ) = Dλ . In Section 6 we prove a transfer principle which represents H(b) inside Dλ . Thus, despite our results to the effect that H(b) is almost never a local Dirichlet space, it can always be represented inside such a space. The paper concludes with some open problems. 2. Background 2.1. De Branges–Rovnyak spaces For χ ∈ L∞ (T), the Toeplitz operator Tχ : H 2 → H 2 is defined by Tχ f := P+ (χf ), where P+ : L2 (T) → H 2 is the canonical projection. Given b in the unit ball of H ∞ , the de Branges– Rovnyak space H(b) is the image of H 2 under the operator (I − Tb Tb )1/2 . We define an inner product on H(b) so as to make (I − Tb Tb )1/2 an isometry from H 2 onto H(b), namely
(I − Tb Tb )1/2 f, (I − Tb Tb )1/2 g b := f, g 2
⊥ f, g ∈ ker(I − Tb Tb )1/2 .
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The norm of f in H(b) is denoted by f b . The space H(b) is a reproducing kernel Hilbert space, with reproducing kernel b kw (z) :=
1 − b(w)b(z) 1 − wz
(z, w ∈ D).
For example, if b ≡ 0, then H(b) = H 2 , and if b is inner, then H(b) = (bH 2 )⊥ , the model subspace of H 2 . The book [7] contains a wealth of information about the spaces H(b). As mentioned in the introduction, the study of de Branges–Rovnyak spaces is governed by a fundamental dichotomy, namely whether or not b is an extreme point of the unit ball of H ∞ (see [7, Chapters IV and V]). For instance, H(b) contains all functions holomorphic in a neighborhood of D if and only if b is non-extreme [7, Theorem V-1]. In what follows, we are only interested in the non-extreme case. According to a well-known theorem [3, p. 138], the function b is non-extreme if and only if log(1 − |b|2 ) ∈ L1 (T). In this case, there is an outer function a ∈ H ∞ for which |a|2 + |b|2 = 1 a.e. on T. Multiplying a by a constant, we may further suppose that a(0) > 0, and a is then uniquely determined. Following Sarason [8], we call (b, a) a pair. Using the pair (b, a), we can express the norm in H(b) in terms of two H 2 -norms. Theorem 2.1. (See [6, Lemma 2, p. 77].) Let (b, a) be a pair. A function f ∈ H 2 belongs to H(b) if and only if Tb f belongs to Ta H 2 . In this case there exists a unique function f + ∈ H 2 such that Tb f = Ta f + , and 2 f 2b = f 22 + f + 2 . Many properties of H(b) can be expressed in terms of the pair (b, a) and more particularly, in terms of the quotient φ := b/a. Notice that φ ∈ N + , the Smirnov space. Conversely, for every function φ ∈ N + , there exists a unique pair (b, a) such that φ = b/a [10, Proposition 3.1]. We next consider Toeplitz operators with unbounded symbols. Given φ holomorphic on D, we define Tφ to be the operator of multiplication by φ on the domain D(Tφ ) := {f ∈ H 2 : φf ∈ H 2 }. The bounded analytic Toeplitz operators are those with a symbol in H ∞ , and the norm of Tφ is then equal to φ∞ . For a general φ, it can be shown that Tφ is densely defined on H 2 if and only if φ ∈ N + [10, Lemma 5.2]. In this case, Tφ has a unique adjoint Tφ∗ , and we henceforth define Tφ := Tφ∗ . The next theorem shows that de Branges–Rovnyak spaces occur naturally as the domain of such adjoint operators. Theorem 2.2. (See [10, Proposition 5.4].) Let (b, a) be a pair and let φ := b/a. Then the domain of Tφ is H(b), and Tφ f = f + (f ∈ H(b)). Consequently, f 2b = f 22 + Tφ f 22
f ∈ H(b) .
(1)
In what follows, we shall sometimes need to exchange the order of Toeplitz operators. According to a classical lemma, if φ, ψ ∈ L∞ (T) and if at least one of them belongs to H ∞ , then Tφ Tψ = Tφψ (see, e.g., [7, p. 9]). As an obvious consequence, if both φ and ψ are in H ∞ , then Tφ and Tψ commute. The next result extends this to the case when one of the symbols belongs to N + .
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Theorem 2.3. (See [10, Proposition 6.5].) Let φ ∈ N + and ψ ∈ H ∞ . Then Tφ Tψ f = Tφψ f = Tψ Tφ f
f ∈ D(Tφ ) .
2.2. Dirichlet spaces For λ ∈ T and f ∈ H 2 , the local Dirichlet integral of f at λ is defined by Dλ (f ) :=
1 2π
f (eit ) − f (λ) 2 dt. eit − λ T
Here f (λ) denotes the value of the radial limit of f at λ, assuming that it exists. If f does not have a radial limit at λ, then we set Dλ (f ) := ∞. The local Dirichlet space at λ is the Hilbert space
Dλ := f ∈ H 2 : f 2λ := f 22 + Dλ (f ) < ∞ . Given a finite positive Borel measure μ on T, we define Dμ (f ) :=
Dλ (f ) dμ(λ)
f ∈ H2 ,
T
and we associate to μ the Hilbert space
D(μ) := f ∈ H 2 : f 2μ := f 22 + Dμ (f ) < ∞ . Note that Dλ is just D(δλ ), where δλ is the Dirac measure at λ. The Dirichlet integral Dμ (f ) can also be expressed as an area integral on the disk. Writing P μ for the Poisson integral of μ, and dA for area Lebesgue measure, we have 1 Dμ (f ) = π
2 f (z) P μ(z) dA(z)
f ∈ H2 .
(2)
D
For a proof of this, see, e.g., [5, Proposition 2.2]. Thus, in particular, if μ is normalized Lebesgue measure on T, then Dμ (f ) is just the usual Dirichlet integral of f and D(μ) is the classical Dirichlet space. For further information on the local Dirichlet integral, we refer to [5]. 3. Coincidence of de Branges–Rovnyak spaces and Dirichlet spaces Our goal in this section is to identify the functions b and the measures μ for which H(b) = D(μ).
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Theorem 3.1. Let b be an element of the unit ball of H ∞ , and let μ be a finite positive Borel measure on T. Then H(b) = D(μ), with equality of norms, if and only if and b(z) =
μ = cδλ
√ τ αλz 1 − τ λz
,
where λ ∈ T, c 0, α ∈ C with |α|2 = c, and τ ∈ (0, 1] with τ + 1/τ = 2 + c. The proof is based on a comparison of inner products of monomials in the two spaces H(b) and D(μ). We begin by computing these inner products in H(b). The first part of the following lemma was already proved in [6, p. 81]. Lemma 3.2. Let (b, a) be a pair and let φ := b/a, say φ(z) = n n 2 z = 1 + |cj |2 b
z
n+k
,z
n
b
j 0 cj z
j.
Then
(n 0),
j =0
=
n
cj +k cj
(n 0, k 1).
j =0
Proof. By (1) and the polarization identity, we have f, g b = f, g 2 + Tφ f, Tφ g 2
f, g ∈ H(b) .
(3)
It therefore suffices to compute Tφ (zn+k ), Tφ (zn ) 2 . For each n 0, we can write φ(z) = n k n+1 ψ (z), where ψ ∈ N + . Thus n n k=0 ck z + z n ck Tzk zn + Tzn+1 ψ zn . Tφ z n = n
k=0
Now Tzk (zn ) = zn−k (0 k n). Also, by Theorem 2.3, we have Tzn+1 ψ zn = Tψ n Tzn+1 zn = Tψ n (0) = 0. n
It follows that Tφ (zn ) =
n
m=0 cn−m z
m.
Hence
n n cn+k−m cn−m = cj +k cj . Tφ zn+k , Tφ zn 2 =
m=0
Together with (3) this gives the result.
j =0
2
The next lemma is the corresponding result for D(μ). We denote by ·,· μ the inner product in D(μ). Also we write μ(k) := T e−ikt dμ(eit ) (k ∈ Z).
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Lemma 3.3. Let μ be a finite positive measure on T. Then n 2 z = 1 + nμ(T) (n 0), μ n+k n μ(−k) (n 0, k 1). z , z μ = n Proof. By (2) and the polarization identity, we have f, g μ = f, g 2 +
1 π
f (z)g (z)P μ(z) dA(z)
f, g ∈ D(μ) .
D
It thus suffices to compute the last integral with f (z) = zn+k and g(z) = zn . With this choice of f, g, we get 1 π
(n + k)z
n+k−1
D
nz
n−1
1 P μ(z) dA(z) = π
1 2π (n + k)nr 2n+k−1 eikt P μ reit dt dr 0 0
1 =
2(n + k)nr 2n+2k−1 μ(−k) dr 0
= n μ(−k). The result follows.
2
Proof of Theorem 3.1. Suppose that H(b) = D(μ), with equality of norms. Notice first that every function holomorphic on a neighborhood of D belongs to D(μ), and therefore also to H(b). By [7, p. 37], this implies that b is not an extreme point in the unit ball of H ∞ . Thus there exists an outer function a such that (b, a) forms a pair, and we may consider φ(z) := b(z)/a(z) = j j 0 cj z , say. The next step is to determine the coefficients cj . Since zn b = zn μ for all n, Lemmas 3.2 and 3.3 give 1+
n
|cj |2 = 1 + nμ(T) (n 0).
j =0
Hence c0 = 0 and |cj |2 = μ(T) for all j 1. Also, since zn+1 , zn b = zn+1 , zn μ for all n, the same lemmas imply that n
cj +1 cj = n μ(−1)
(n 0).
j =0
μ(−1) for all j 1. Putting these facts together, it follows that cj = αλj for all Hence cj +1 cj = j 1, where λ ∈ T and α ∈ C with |α|2 = μ(T).
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Next, we determine μ. Using Lemmas 3.2 and 3.3 once again, we have 1
μ(−k) = zk+1 , z μ = zk+1 , z b = cj +k cj = |α|2 λk = μ(T)λk
(k 1).
j =0
Since μ is a real measure, the same relation holds for all k −1, and clearly it is also true for k = 0. Thus μ has the same Fourier coefficients as the measure cδλ , where c = μ(T), and we conclude that μ = cδλ . It remains to determine b. To do this, we follow the method in [9]. Note first that φ(z) =
cj z j =
j 0
αλj zj =
j 1
αλz 1 − λz
(z ∈ D).
Since φ = b/a and |a|2 + |b|2 = 1 a.e. on T, it follows that |a|2 = 1/(1 + |φ|2 ) a.e. on T. Thus a(z)2 =
|1 − λz|2
a.e. on T.
|1 − λz|2 + |α|2
A simple calculation shows that |1 − λz|2 + |α|2 = τ −1 |1 − τ λz|2 for z ∈ T, where τ ∈ (0, 1] is chosen so that τ + 1/τ = 2 + |α|2 = 2 + c. As a is an outer function, it follows that a(z) =
√ 1 − λz τ 1 − τ λz
(z ∈ D).
Hence, finally, b(z) = a(z)φ(z) =
√ τ αλz 1 − τ λz
(z ∈ D).
This completes the proof of the “only if”. For the “if”, note that with the given choice of b, μ, working back through the calculations above we get zn+k , zn b = zn+k , zn μ for all n, k 0. Since polynomials are dense both in H(b) [7, IV-3, p. 25] and in D(μ) [4, Corollary 3.8], we deduce that H(b) = D(μ), with equality of norms. 2 What if H(b) = D(μ) without equality of norms? Since both H(b) and D(μ) embed boundedly into H 2 , using the closed graph theorem it is easy to see that the norms · b and · μ must be equivalent. Do there exist measures μ, other than point masses, for which D(μ) = H(b) with equivalence of norms? 4. Formulas for the norm in de Branges–Rovnyak spaces Lemma 3.2 provides a formula for the inner product of monomials in H(b), expressed in terms of the coefficients cj of the function φ. Since polynomials are dense in H(b), we might expect there to be an analogous formula for the norms of more general functions. The following theorem, which is implicit in [10], is a first step in this direction.
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Theorem 4.1. Let (b, a) be a pair, and let φ := b/a, say φ(z) = j 0 cj zj . Let f be holo morphic in a neighborhood of D, say f (z) = j 0 f (j )zj . Then the series j 0 f (j + k)cj converges absolutely for each k, and f 2b
2 2
= f (j + k)cj . f (k) +
(4)
k0 j 0
k0
Proof. Suppose first that f is a polynomial, of degree n say. In this case, we argue as in the proof of Lemma 3.2. Writing φ(z) = nj=0 cj zj + zn+1 ψn (z), where ψn ∈ N + , we have Tzn+1 ψ (f ) = Tψ n Tzn+1 (f ) = Tψ n (0) = 0, n
and so Tφ (f ) =
n
cj Tzj (f ) =
j =0
n
cj
j =0
n−j
f (j + k)zk =
n n−k
f (j + k)cj zk .
k=0 j =0
k=0
Using Theorem 2.2, we obtain
f 2b
= f 22
+ Tφ f 22
n−k 2 n n 2
= f (j + k)cj , f (k) + k=0 j =0
k=0
which proves the theorem in this case. For the general case, let us write fn (z) := nj=0 f (j )zj . By what we have already proved, we have
fn 2b
n−k 2 n n 2 f (k) + = f (j + k)cj .
(5)
k=0 j =0
k=0
Fix R > 1 such that f is holomorphic in a neighborhood of D(0, R). Then f (j ) = O(R −j ) as j j → ∞. Since cj = O(R ) for each R ∈ (1, R), it follows that the series j 0 f (j + k)cj converges absolutely for each k. Thus, as n → ∞, the right-hand side of (5) converges to the right-hand side of (4). Also, using Lemma 3.2, we have, for each R ∈ (1, R), 1/2 k k f (k)zk = f (k) 1 + as k → ∞. |cj |2 = O R /R b j =0
Thus the Taylor series of f converges in the norm of H(b). The norm limit agrees with f on the unit disk, because norm convergence implies pointwise convergence. Therefore the left-hand side of (5) converges to the left-hand side of (4) as n → ∞. This completes the proof. 2
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It is instructive to look at what formula (4) gives when H(b) = Dλ . Recall that, in this case, c0 = 0 and cj = λj for all j 1. Thus formula (4) becomes 2 j
Dλ (f ) = f (j + k)λ . k0 j 1
Writing S for the shift operator on H 2 , and S ∗ for its adjoint, namely S ∗ f (z) := (f (z)−f (0))/z, we obtain Dλ (f ) =
S ∗k f (λ)2 .
(6)
k1
This formula is already known. It is implicit in [5], and explicit in [11]. Although we have proved (6) only for functions holomorphic on a neighborhood of D, when suitably interpreted it is actually valid for all functions holomorphic in D, thereby providing a test for membership of Dλ . For the formula to make sense, we interpret S ∗k f (λ) as the radial limit of S ∗k f at λ if this limit exists, and we set |S ∗k f (λ)| := ∞ otherwise. This version of the formula can be deduced from the more restricted version by considering the functions fr (z) := f (rz) and using the fact that Dλ (fr ) → Dλ (f ) as r → 1 (see [5, p. 377]). This naturally raises the question of whether a similar approximation procedure is possible in general H(b)-spaces. This is the subject of the next section. 5. Star-shapedness of de Branges–Rovnyak spaces Throughout this section we assume that b is a non-extreme point of the unit ball of H ∞ , that (b, a) is a pair, and that φ = b/a is the associated function in N + . Given f ∈ H(b) and r ∈ (0, 1), we write fr (z) := f (rz). As fr is holomorphic on a neighbourhood D, we certainly have fr ∈ H(b). By the closed graph theorem, Cr : H(b) → H(b), defined by Cr f := fr , is bounded linear map. We seek to determine whether limr→1 fr − f b = 0 for all f ∈ H(b). A space H(b) for which this holds is called star-shaped. The following proposition provides some criteria for H(b) to be star-shaped. Proposition 5.1. The following are equivalent: (i) limr→1 fr − f b = 0 for all f ∈ H(b); (ii) supr<1 fr b < ∞ for all f ∈ H(b); (iii) supr<1 Cr H(b)→H(b) < ∞. Proof. Obviously (i) implies (ii), and the Banach–Steinhaus theorem shows that (ii) implies (iii). Finally (iii) implies (i), because limr→0 fr − f b = 0 when f is a polynomial, and polynomials are dense in H(b) (see [7, p. 25]). 2 The following weak version of (ii) always holds.
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Theorem 5.2. If f ∈ H(b), then
1 log fr b = o 1−r +
as r → 1.
(7)
Proof. Let g ∈ aH 2 . Then T fr , g 2 = fr , φg 2 = f, φr gr 2 f 2 φr ∞ g2 . φ As a is outer, aH 2 is dense in H 2 , and therefore Tφ fr 2 φr ∞ f 2 . From (1) we get fr b max{φr ∞ , 1}f 2 . Thus, to prove the theorem, it suffices to show that log+ φr ∞ = o(1/(1 − r)) as r → 1− . Let us write φ ∗ for the radial limit function of φ on T. Then, for all z ∈ D, all r ∈ (0, 1) and all K > 1, we have 1 log φ(rz) 2π
+
2π 0
1 − |rz|2 + ∗ it φ e dt log |eit − rz|2
1 log K + 2π
2π 0
∗ it 1 − |rz|2 + |φ (e )| dt log K |eit − rz|2
2 1 log K + 1 − r 2π
2π
log+
|φ ∗ (eit )| dt. K
0
Therefore 2 1 log φr ∞ log K + 1 − r 2π +
2π
log+
|φ ∗ (eit )| dt. K
0
As K is arbitrary, we get log+ φr ∞ = o(1/(1 − r)) as r → 1, as required.
2
We shall see shortly that (7) cannot be improved, in general. However, the first part of the preceding argument can be adapted to provide a simple condition on φ which guarantees that H(b) is star-shaped. Theorem 5.3. If φr /φ is bounded on D, then
Cr H(b)→H(b) max φr /φ∞ , 1 . Consequently, if supr<1 φr /φ∞ < ∞, then H(b) is star-shaped.
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Proof. Let f ∈ H(b) and let g ∈ aH 2 . Then T fr , g 2 = fr , φg 2 = f, φr gr 2 = T f, (φr /φ)gr φ φ 2 Tφ f 2 φr /φ∞ g2 .
(8)
As a is outer, aH 2 is dense in H 2 , and therefore Tφ fr 2 Tφ f 2 φr /φ∞ . From (1) we get fr b max{φr /φ∞ , 1}f b , whence the result. 2 As a special case, we recover a result that we cited in Section 4. This is essentially Sarason’s proof in [9]. Corollary 5.4. The space Dλ is star-shaped and Cr Dλ →Dλ 1 for all r ∈ (0, 1). Proof. We have Dλ = H(b) with φ(z) = λz/(1 − λz). Therefore, for r ∈ (0, 1), we obtain Cr Dλ →Dλ max{2r/(1 + r), 1} = 1. 2 If supr<1 φr /φ∞ < ∞, then necessarily φ(z) = zk φo (z), where φo is outer and 1/φo is bounded (or, equivalently, b(z) = zk bo (z), where bo is outer and 1/bo is bounded). The example in Corollary 5.4 is thus rather typical. Based on this, one might guess that H(b) is star-shaped whenever 1/b is bounded. We shall now prove that this is not the case. Theorem 5.5. Let ρ : (0, 1) → R+ be a function such that ρ(r) = o(1/(1 − r)) as r → 1. Then there exist b (non-extreme in the ball of H ∞ ) and f ∈ H(b) such that lim sup r→1
log fr b = ∞. ρ(r)
The function b may be chosen to be outer with 1/b bounded. Remarks. (i) Taking ρ ≡ 1 in the theorem, we obtain the promised example showing that H(b) need not be star-shaped, even if 1/b is bounded. (ii) The theorem also shows that the estimate (7) cannot be improved. The proof of the theorem is based on two lemmas. Lemma 5.6. Given b, φ as above, 1 + |φ(rw)|2 1 − |w|2 . 2 2 2 w∈D 1 + |φ(w)| 1 − r |w|
Cr 2H(b)→H(b) sup
Proof. Let kw (z) := 1/(1−wz) be the reproducing kernel for H 2 . Then Tφ kw = φ(w)kw . Hence 2 1 + |φ(w)|2 kw 2b = kw 22 + Tφ kw 22 = kw 22 + φ(w) kw 22 = . 1 − |w|2
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Likewise Cr kw 2b = krw 2b = The result follows.
1 + |φ(rw)|2 . 1 − r 2 |w|2
2
Lemma 5.7. Let ρ : (0, 1) → R+ be a function such that ρ(r) = o(1/(1 − r)) as r → 1. Then there exists an outer function φ on D such that |φ| 1 and lim sup r→1
log(|φ(r 2 )|/|φ(r)|) = ∞. ρ(r)
Proof. Fix a positive sequence ( n ) such that the series k k converges and satisfies k>n k = 2 o( n ) as n → ∞. For example, n := e−n will do. Since limr→1 ρ(r)(1 − r) = 0, there exists an increasing sequence rn → 1 such that ρ(rn )(1 − rn )/ n → 0 as n → ∞. Define sn :=
1 − rn 1 + rn
and tn :=
1 − rn2 . 1 + rn2
Let ψ be the outer function on the upper half-plane whose non-tangential limit ψ ∗ on R satisfies logψ ∗ = ( k /tk )1[tk ,2tk ]
a.e. on R.
k1
Note that 1 π
R
log |ψ ∗ (x)| dx k < ∞, 2 1+x k1
so ψ is well defined and |ψ| 1. Define φ on the unit disk by 1−z φ(z) := ψ i |z| < 1 . 1+z Then φ is also an outer function and |φ| 1. We shall show that this function φ satisfies the conclusion of the lemma. For each n 1, we have logφ rn2 /φ(rn ) = logψ(itn ) − logψ(isn ) 1 tn sn = logψ ∗ (x) dx − 2 2 2 2 π tn + x sn + x R
2tk 1 k tn sn = dx. − π tk tn2 + x 2 sn2 + x 2 k1
tk
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Now, if x tn , then tn sn (tn − sn )(x 2 − sn tn ) − = tn2 + x 2 sn2 + x 2 (tn2 + x 2 )(sn2 + x 2 )
tn −1 sn
2
1 rn2 . tn 1 − rn
Therefore, if 1 k n, then k tk
2tk tk
tn rn2 sn dx − . k 1 − rn tn2 + x 2 sn2 + x 2
Also, for every k, we clearly have k tk
2tk tk
2tk tn k sn sn k 2 k dx − − dx − − . tk sn 1 − rn tn2 + x 2 sn2 + x 2 sn2 + x 2 tk
Putting this information together, we deduce that logφ rn2 /φ(rn )
rn2 2 k − k . 1 − rn 1 − rn k>n
kn
Since k>n k = o( n ), it follows that log |φ(rn2 )/φ(rn )| C n /(1 − rn ), where C is a positive constant independent of n. Hence, finally, C n log |φ(rn2 )/φ(rn )| →∞ ρ(rn ) (1 − rn )ρ(rn ) This completes the proof.
as n → ∞.
2
Proof of Theorem 5.5. Let φ be the function given by Lemma 5.7, and let b be the associated element of the unit ball of H ∞ . Note that b is outer and 1/b is bounded. By Lemma 5.6, applied with w = r, we have Cr 2H(b)→H(b)
1 1 + |φ(r 2 )|2 1 |φ(r 2 )|2 2 1 + |φ(r)|2 4 |φ(r)|2
(0 < r < 1).
Therefore, lim sup r→1
log Cr H(b)→H(b) log |φ(r 2 )/φ(r)| lim sup = ∞. ρ(r) ρ(r) r→1
Thus, there exist sequences rn → 1 and An → ∞ such that −A ρ(r ) e n n C r n
H(b)→H(b)
→ ∞.
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By the Banach–Steinhaus theorem, there exists f ∈ H(b) such that lim supe−An ρ(rn ) Crn f b = ∞. n→∞
This gives the desired conclusion.
2
If we multiply b by an inner function u, then a does not change, and the corresponding φ is also multiplied by u. How does multiplication by an inner function affect the star-shapedness of the corresponding de Branges–Rovnyak space? There is one simple case: if H(b) is star-shaped, then so is H(zk b) for every k. Indeed, a calculation like (8) shows that Cr H(zk b)→H(zk b) r k Cr H(b)→H(b)
(0 < r < 1).
For general inner factors, however, the situation is very different. Theorem 5.8. If b∞ = 1, then there is a Blaschke product u such that H(ub) is not starshaped. Remark. In the other case, namely when b∞ < 1, the space H(ub) is star-shaped for every ∗ )1/2 is an invertible operator on H 2 , and the inner function u. Indeed, Tub < 1, so (I − Tub Tub 2 inclusion H(ub) ⊂ H is a surjection. To prove the theorem, we need a further lemma. Lemma 5.9. Let (θn ), (sn ) be sequences in [0, 2π] and (0, 1) respectively. Then there exist a sequence rn ∈ (sn , 1) and Blaschke product u such that u(rn eiθn ) = 0 for all n and infn |u(rn2 eiθn )| > 0. Proof. Let σ denote the pseudo-hyperbolic metric on D, defined by z−w σ (z, w) := 1 − zw
(z, w ∈ D).
For w fixed, we have σ (z, w) → 1 as |z| → 1. Thus, we may inductively choose a sequence rn ∈ (0, 1) so that, if zn := rn eiθn and wn := rn2 eiθn , then σ (zn , wm ) exp −2−n (m = 1, . . . , n − 1), σ (wn , zm ) exp −2−m (m = 1, . . . , n − 1). We may further suppose that rn ∈ (sn , 1) for all n, and that Blaschke product defined by u(z) :=
∞ |zm | zm − z . zm 1 − zm z
m=1
n (1
− rn ) < ∞. Let u be the
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Clearly u(zn ) = 0 for all n. Also, for each n, ∞ u(wn ) = σ (zm , wn ) m=1
= σ (zn , wn )
σ (zm , wn )
rn − rn2 1 − rn3
exp −2−m exp −2−m
1m
m>n
1 rn − rn2 −1 e → e−1 3 1 − rn3
This completes the proof of the lemma.
σ (zm , wn )
m>n
1m
as n → ∞.
2
Proof of Theorem 5.8. Since b∞ = 1, it follows that φ is unbounded (in fact the two conditions are equivalent). Choose (θn ) such that the radial limit φ(eiθn ) exists and satisfies |φ(eiθn )| > n. Then choose (sn ) such that |φ(r 2 eiθn )| > n for all r ∈ (sn , 1). Let (rn ) and u be as given by Lemma 5.9. By Lemma 5.6 (applied with w = rn eiθn ), we have 1 + |(uφ)(rn2 eiθn )|2 1 − rn2 1 + |(uφ)(rn eiθn )|2 1 − rn4 2 2 1 u rn2 eiθn φ rn2 eiθn → ∞ as n → ∞. 2
Crn 2H(ub)→H(ub)
Now apply Proposition 5.1.
2
The counterexamples in this section still leave open the possibility that, given any b and any f ∈ H(b), there exists a sequence rn → 1, depending on b, f , such that f − frn b → 0. Can this be ruled out? 6. A transfer principle The preceding sections demonstrate that the local Dirichlet spaces Dλ are not typical de Branges–Rovnyak spaces. In this section we shall prove a result that points in the other direction, to the effect that de Branges–Rovnyak spaces can always be represented inside local Dirichlet spaces. Theorem 6.1. Let (b, a) be a pair. Set ψ(z) := (1 − z)b(z)/a(z), and define W : H(b) → H 2 by W (f ) := zTψ f
f ∈ H (b) .
Then: (i) W is well defined on H(b); (ii) the kernel of W equals (bi H 2 )⊥ , where bi is the inner factor of b;
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(iii) the image of W is contained in the local Dirichlet space D1 , and f + (z) =
Wf (z) − Wf (1) z−1
f ∈ H(b) .
(9)
A great deal is known about the local Dirichlet spaces Dλ . For example, there is a remarkable formula due to Richter and Sundberg (generalizing an earlier formula of Carleson in the classical Dirichlet space) expressing Dλ (f ) in terms of the factorization f = OSB (outer function, singular inner function, Blaschke product). For more on this see [5, Theorem 3.1]. In principle, at least, Theorem 6.1 allows us to exploit this knowledge to obtain information about general H(b)spaces. In practice, the success of this endeavor depends on being able to identify the operator W , which means understanding the Toeplitz operator Tψ . We shall deduce Theorem 6.1 from an abstract transfer principle. To be able to state this principle, we need an alternative notation for the function f + , one that indicates the dependence on b. Accordingly, we shall write [f ]b := f + . Theorem 6.2. Let (b, a) and (B, A) be pairs. Let B = Bi Bo be the inner-outer factorization of B, and suppose that 1/Bo is bounded. Set ψ := bA/aBo , and define W : H(b) → H 2 by W (f ) := Bi Tψ f
f ∈ H (b) .
Then: (i) W is well defined on H(b); (ii) the kernel of W equals (bi H 2 )⊥ , where bi is the inner factor of b; (iii) the image of W is contained in H(B), and [Wf ]B = [f ]b
f ∈ H(b) .
Proof. (i) Let us begin by noting that ψ ∈ N + , so the Toeplitz operator Tψ is defined at least on polynomials. Also, using Theorem 2.3, we have Tψ = TA/B o Tb/a , so the domain of Tψ includes the domain of Tb/a , which equals H(b). Thus W is well defined on H(b). (ii) Let b = bi bo be the inner-outer factorization of b. By [7, II-6, p. 10], we have H(b) = H(bi ) ⊕ bi H(bo ), where the direct sum is orthogonal with respect to the inner product in H(b). The first summand H(bi ) is just the model space (bi H 2 )⊥ , and we shall now show that it is exactly the kernel of W . Let f ∈ H(b). Then using the fact that outer functions are cyclic in H 2 , we have f ∈ ker W
⇔ ⇔ ⇔ ⇔
Tψ f = 0
Tψ f, aBo h 2 = 0 h ∈ H 2 f, bAh 2 = 0 h ∈ H 2 ⊥ ⊥ f ∈ bAH 2 = bi H 2 .
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(iii) Let f ∈ H(b). Then, in the notation introduced just before the statement of the theorem, we have [f ]b = Tb/a f ∈ H 2 . Therefore, TB (Wf ) = TB (Bi Tψ f ) = TBo Tψ f = TAb/a f = TA [f ]b . By Theorem 2.1, it follows that Wf ∈ H(B) and that [Wf ]B = [f ]b .
2
Proof of Theorem 6.1. Let (B, A) be the pair for which B/A = z/(1 − z). As we have seen in Section 3, H(B) is then just the local Dirichlet space D1 . Thus all of Theorem 6.1 follows immediately from Theorem 6.2, except for the final formula (9). For this, we need to identify [Wf ]B . Given g ∈ D1 , we have [g]B = TB/A g. Hence, for h ∈ AH 2 , B zh g − g(1) ,h . = [g]B , h 2 = g, h = g, A 2 1−z 2 z−1 2
As (g − g(1))/(z − 1) ∈ H 2 and AH 2 is dense in H 2 , it follows that [g]B =
g(z) − g(1) . z−1
Taking g = Wf , we obtain (9). This completes the proof.
2
7. Some open problems (1) Do there exist measures μ on T, other than point masses, such that D(μ) = H(b) for some b? We do not assume equality of norms, though, as observed earlier, the norms must be equivalent. In these circumstances, we can no longer expect b to be determined by μ. For example, if μ = 0, then D(μ) = H 2 , and there are many b for which the inclusion of H(b) in H 2 is surjective—indeed any b ∈ H ∞ with b∞ < 1 will do. (2) A simple weak compactness argument shows that, if f is holomorphic on D and lim infr→1 fr b < ∞, then f ∈ H(b) and f b lim infr→1 fr b . In the other direction, does f ∈ H(b) imply that lim infr→1 fr b < ∞? If so, then do we also have lim infr→1 fr − f b = 0? We have seen that the answer to both questions is ‘no’ if ‘lim inf’ is replaced by ‘lim sup’. (3) Is it possible to characterize those b (or those φ) for which H(b) is star-shaped? As the inequality (8) makes clear, the problem boils down to being able to estimate | f, φr gr 2 | in terms of g2 and f b . (4) Another possible approach to problem (3) is via the H 2 -reproducing kernels kw (z) := 1/(1 − wz). Recall that Tφ kw = φ(w)kw and Cr kw = krw for all w ∈ D and r ∈ (0, 1). This remark was used in Lemma 5.6 to obtain a lower bound for Cr H(b)→H(b) , and hence a necessary condition for H(b) to be star-shaped. Since the family {kw : w ∈ D} spans a dense subspace of H 2 , it could in principle be used to determine Cr H(b)→H(b) exactly. However, this gives rise to a Pick-type problem which we have been unable to solve up to now. (5) Let f ∈ H(b). Although fr → f in H(b), in general, it is always true that f can be approximated by functions holomorphic on a neighborhood of D, indeed even by polynomials. This is proved in [7, IV-3]. However, the proof given there is by duality and is not constructive. Is there a constructive scheme by which f may be approximated in H(b) by functions holomorphic in a neighborhood of D?
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Acknowledgment The authors thank the anonymous referee for several helpful suggestions which greatly improved the paper. References [1] L. de Branges, J. Rovnyak, Canonical models in quantum scattering theory, in: Perturbation Theory and Its Applications in Quantum Mechanics, Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965, Wiley, New York, 1966, pp. 295–392. [2] L. de Branges, J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [3] K. Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1988. [4] S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991) 325– 349. [5] S. Richter, C. Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991) 355–379. [6] D. Sarason, Doubly shift-invariant spaces in H 2 , J. Operator Theory 16 (1986) 75–97. [7] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, John Wiley & Sons Inc., New York, 1994. [8] D. Sarason, Kernels of Toeplitz operators, in: Toeplitz Operators and Related Topics, Santa Cruz, CA, 1992, in: Oper. Theory Adv. Appl., vol. 71, Birkhäuser, Basel, 1994, pp. 153–164. [9] D. Sarason, Local Dirichlet spaces as de Branges–Rovnyak spaces, Proc. Amer. Math. Soc. 125 (1997) 2133–2139. [10] D. Sarason, Unbounded Toeplitz operators, Integral Equations Operator Theory 61 (2008) 281–298. [11] S. Shimorin, Reproducing kernels and extremal functions in Dirichlet-type spaces, J. Math. Sci. (N. Y.) 107 (2001) 4108–4124.
Journal of Functional Analysis 259 (2010) 2384–2403 www.elsevier.com/locate/jfa
Automorphisms of the bipartite graph planar algebra R.D. Burstein ∗ Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN 37240, United States Received 13 March 2010; accepted 18 May 2010 Available online 19 June 2010 Communicated by D. Voiculescu
Abstract For any abstract subfactor planar algebra P , there exists a finite index extremal subfactor M0 ⊂ M1 with P as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and obtain subfactor planar subalgebras by taking fixed points under groups of automorphisms. This construction provides both new examples of subfactors and new descriptions of the planar algebras of previously known examples. © 2010 Elsevier Inc. All rights reserved. Keywords: Subfactor; Planar algebra
1. Introduction Planar algebras were introduced by Jones in [13]. They are a powerful tool for studying subfactors, providing a graphical calculus on the standard invariant of a finite index extremal subfactor of type II 1 [13,22]. The planar operad is the set of planar tangles with zero or more internal disks and a checkerboard shading, with a distinguished region of the boundary and each internal disk, all taken up to isotopy. The operation is gluing: a tangle may be pasted into an internal disk of another tangle (matching up the distinguished boundary regions) if the number of strands and the shading are compatible. A planar algebra is a graded vector space P = (Vn± ), n 0, along with an associative action of the planar operad. If M0 ⊂ M1 is a finite index extremal II 1 subfactor with Jones tower M0 ⊂ M1 ⊂ M2 ⊂ . . . , then we may take Vn+ = M0 ∩ Mn , Vn− = M1 ∩ Mn+1 . There is an operad action defined in [13] * Fax: +1 615 343 0215.
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.009
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making this into a planar algebra. Conversely, Jones describes a certain list of additional properties which make P into a Popa system, implying that there exists a subfactor M0 ⊂ M1 of which P is the standard invariant [22]. A planar algebra with this list of properties is called a subfactor planar algebra (SPA). Other methods of constructing a subfactor from an SPA have since been obtained [11,15,16], which use more diagrammatic notation than Popa’s original method. Constructing an SPA abstractly therefore implies the existence of a corresponding subfactor. This method may be used to find new subfactors (as in [6] or [1]), or provide new proofs of the existence of subfactors with specified properties. Abstract constructions of SPAs of previously known subfactors can provide new insight into the structure of their standard invariants (e.g. [3, 4,19]). A planar algebra may be constructed from any finite bipartite graph [14], and some infinite graphs as well [11]. These bipartite graph planar algebras (BGPAs) are almost never of subfactor type, because their vector spaces are too large. However, they possess several of the necessary properties required for SPAs, which are inherited by planar subalgebras. We may therefore try to find SPAs by looking at small planar subalgebras of BGPAs. It is generally difficult to show that a graded subspace of a BGPA is closed under the action of the planar operad, although progress has been made in the single generator case (e.g. [7,8,19,1]). In this paper, we describe a method for doing so by taking fixed points of a BGPA under a group of automorphisms. In Section 2 we describe BGPAs, with particular attention to infinite graphs. We introduce a slightly different notation from [11] and [14]; this simplifies the computations of Section 3, where we compute the automorphism group of an arbitrary BGPA PΓ . In Section 4 we find conditions for a planar subalgebra of a BGPA to be an SPA. We conclude in Section 5 by presenting some examples of SPAs obtained by this planar fixed point construction. The set of these subfactors is quite large, and we will do no more than scratch the surface. We will provide planar fixed point descriptions of several previously known classes of subfactors obtained from groups, as well as new group-like examples having both finite and infinite depth. We will also describe some exotic infinite depth √ examples, including at least two distinct extremal non-irreducible subfactors of index (1 + 2)2 , the smallest possible index for such a subfactor (see [20]). We also obtain an uncountably infinite family of non-extremal subfactors, all having the same principal graphs but with pairwise distinct indices. 2. The bipartite graph planar algebra The bipartite graph planar algebra (BGPA) is described by Jones in [14]. The data required are a finite bipartite graph Γ and a function μ from the vertices of the graph to the positive real numbers R+ . Given such data, there is a planar algebra PΓ = (Vn± ), n 0, where each Vn± has a basis labelled by loops of length 2n in the graph, starting at even or odd vertices depending on sign. We will consider BGPAs on infinite graphs as well. In this case, the vector spaces will be infinite dimensional, so there are some new topological considerations. To simplify later computations, we will describes the vector spaces of the planar algebra as operators on a certain Hilbert space. In our diagrams, we will use the convention of [15] that a thick line represents as many parallel strands as necessary. Also we will omit shading in a diagram when both shadings can occur. When not otherwise specified, the distinguished boundary region of a tangle is on the left.
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The discussion that follows is taken from [14], adjusted where necessary to allow for infinite graphs. Note that the values μ(v) below are the squares of the corresponding components of the spin vector in [14]; this will simplify later computations. Let Γ be a locally finite bipartite graph. We associate to Γ a function μ from the vertices of Γ to the positive real numbers obeying the following local boundedness condition: there is some M > 0 such that for any two adjacent vertices v and w, we have μ(v)/μ(w) < M. Let ln± be the set of loops of length 2n on Γ , starting at an even (+) or odd (−) vertex. Then the vector space Vn± of the bipartite graph planar algebra PΓ is the set of bounded functions from ln± to C. The boundary type of a tangle is the ordered pair (n, ±), where 2n strands intersect the tangle boundary transversely and the distinguished region of the tangle is shaded (+) or unshaded (−). The same definition is used for the boundary type of each internal disk of a tangle. Let T be a planar tangle with k internal disks. Let the boundary type of T be (n, ±), and let the internal disks of T have boundary types respectively (n1 , ±), (n2 , ±), . . . , (nk , ±). Then to describe the planar operad, for each set of inputs (x1 , . . . , xk ) (with xk ∈ Vn±k ) we must assign the output Z(T ) ∈ Vn± . This assignment must agree with gluing, be multilinear in the inputs, and be isotopy invariant. We define Z(T ) following [14]. A state of the tangle is a function σ which maps the strands of T to edges of Γ , and the regions of T to vertices of Γ . Shaded regions are mapped to positive vertices, and unshaded regions to negative ones. A state must obey a compatibility condition: if a strand S is adjacent to a region R, then σ (R) must be one of the endpoints of σ (S). A state σ is compatible with a given loop if reading the output of σ counterclockwise around the boundary of T , starting from the distinguished region, produces that loop. Each strand in a tangle is a smooth curve in the plane, which may be thought of as a smooth function v(t) = (x(t), y(t)) from the closed unit interval to R 2 (with v (t) > 0 for all t). A singularity of T is a local maximum or local minimum of a strand, i.e. a point (x, y) = (x(t), y(t)) on the strand with y (t) = 0. Fix a state σ ; let v be the vertex associated to the concave side of the singularity by σ , and w the vertex associated to the convex side. Then the value of the singularity √ is μ(v)/μ(w). A state σ associates a loop Li to each internal disk Di of T , obtained by reading the output of σ counterclockwise starting from the distinguished region of the disk. The value of the state on Di is then the value of the ith input xi at this loop Li . Now we can define the value of Z(T ) on a specified loop L ∈ ln± . This is
σ compatible with L
s ∈ singularities of T
σ (s)
σ (Di )
Di ∈ internal disks
where σ is evaluated on disks and internal singularities as above. By local finiteness of Γ , only finitely many states are compatible with L. Specifically, let m be the largest number of strands that must be crossed to get from the distinguished boundary region of T to any other region (boundary or internal). Then if v is the first vertex of L, every compatible state must assign all regions of T to vertices within a distance of m from v on the graph. A state is determined by its value on strands. Since Γ is locally finite, there is some emax such that each vertex of Γ contacts at most emax edges. Then each strand must be chosen from at m possibilities, and if T contains a distinct strands, each loop is compatible with at most most emax am emax states.
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The local boundedness condition on μ (μ(v)/μ(w) √ < M < ∞ for all adjacent vertices v and w) means that each singularity has value at most M, for any state. Let T have b singularities. Elements of Vn±i are bounded; let N be the largest bound of any input xi , 1 i k. Putting this together, we find that the tangle output Z(T ) may be evaluated on each loop L as am M b/2 N k , so the output is a bounded function a finite sum. Moreover, this sum is bounded by emax ± on ln and this evaluation rule produces an element of Vn± . The proofs of [14] that this map is multilinear, isotopy invariant and respects gluing may be used without alteration, since local finiteness of Γ implies that all necessary sums are finite. Therefore the above definition of Z(T ) produces a planar algebra. There is a natural antilinear involution, which we refer to as ∗ , of each vector space Vn± (see [14]). For each loop L ∈ ln± , the reversed loop L consists of the same list of vertices and edges, taken in the opposite order. Then the involution is defined by A∗ (L) = A(L ), for all A ∈ Vn± and L ∈ ln± . Let ρT be the map corresponding to some tangle T with k internal disks, i.e. ρT (x1 ⊗ x2 ⊗ · · · ⊗ xk ) is equal to Z(T ) when the inputs are (x1 , x2 , . . . , xk ). Then as in [14] we have ρT (x1∗ ⊗ x2∗ ⊗ · · · ⊗ xk∗ ) = ρT (x1 ⊗ x2 ⊗ · · · ⊗ xk )∗ , where the tangle T is the mirror image of T . From [13], this means that the BGPA PΓ is a planar ∗-algebra, with involution as above. For ease of later computation, it will be convenient to describe Vn± as a set of bounded linear operators on a certain Hilbert space. Let Hn± be the Hilbert space with basis {xp } labelled by the paths of length n on Γ whose initial vertex is even (+) or odd (−). These vector spaces are all infinite dimensional when Γ is infinite. We also define the total path Hilbert space H as the direct sum of the Hn± ’s: H = n,± Hn± . Each element A of Vn± naturally defines a linear map on Hn± . A loop L ∈ ln± may be described as a pair of paths (π, ) of length n whose endpoints are the same. Then π and correspond to basis vectors xπ , x in Hn± , and we take A(x ), xπ = A(L). If L ∈ ln± is equal to the pair of paths (π, ), then the reversed loop L is equal to (, π). It follows from the above definition of the involution that A∗ (x ), xπ = A(xπ ), x . In other words, the involution acts on Vn± as the adjoint operation for B(Hn± ). Every path p has a starting vertex s(p) and a terminal vertex t (p). For each vertex v of Γ we may define a projection sv ∈ B(Hn± ) which fixes the closed linear span of {xp |s(p) = v}, and likewise tv which fixes the closed span of {xp |t (p) = v}. The sv ’s and tw ’s form an abelian algebra; in fact the projections {sv tw |v, w ∈ vertices of Γ } are a partition of unity. Since π and above always have the same endpoints, we have each A ∈ Vn± commuting with sv and tv for all vertices v. This means that A ∈ Vn± may be thought of as a (potentially infinite) sum of operators Avw each acting on the subspace sv tw (Hn± ). Each sv tw is of finite rank, and this rank is universally bounded by local finiteness of Γ , so these subspaces have bounded dimension. Since the components {Avw } of A are bounded by definition, it follows that A is bounded in norm as a linear operator on H . We then have Vn± ⊂ B(Hn )± , and in fact Vn± ⊂ {sv , tw |v, w ∈ vertices of Γ } . Each sv tw B(Hn± ) is a finite dimensional matrix algebra, with matrix units given by partial isometries from xp to xq where p and q are paths from of length n from v to w. All such partial isometries are contained in Vn± , so sv tw B(Hn± ) ⊂ Vn± for all v, w. {sv , tw } consists of bounded formal sums of elements of sv tw B(Hn± ), so from the definition of Vn± it follows that Vn± = {sv , tw } as operators on Hn± . This is a von Neumann algebra by the bicommutant theorem.
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We will freely refer to scalar elements of the Vn± ’s; these are just the scalars as operators on the appropriate Hilbert space. To assist with computations using the above notation, we now define a concatenation operation c on the Hn± ’s: Definition 2.1. Let p and q be two paths on the graph Γ . Then c(xp , xq ) is zero if t (p) = s(q), and otherwise is xr where r is the path obtained by first following p and then q. This operation extends linearly and continuously to maps Hn± ⊗ Hm± → Hk± , where k = n + m and the signs are chosen appropriately. From the definition of the bases of the Vn± ’s, every concatenation map is surjective. Furthermore, these maps are associative: c(c(x, y), z) = c(x, c(y, z)). This means we may freely apply c to multiple inputs via the inductive definition c(x1 , x2 , . . . , xn ) = c(c(x1 , x2 , . . . , xn−1 ), xn ). We also define an antilinear path reversal operator rev: Definition 2.2. Let p be a path on the graph Γ . Let q be the reverse path of p, i.e. the same edges and vertices taken in the opposite order. Then rev(xp ) = xq . This operation extends antilinearly and continuously to maps from Hn± to Hn± or Hn∓ , depending on the value of n. Both c and rev are bounded, so they extend to the total Hilbert space H . The map rev is an antilinear involution. Later, we will be interested in demonstrating that certain maps on the Vn± ’s commute with the action of the planar operad. To do this it suffices to show that such maps commute with particular tangles that generate the planar operad (see e.g. [4]). We now describe the action of one set of such generating tangles in terms of the above notation. All of these actions may be readily verified directly from the operad definition. The multiplication tangle:
This tangle corresponds to operator multiplication. The output is AB. The left embedding tangle l(A):
∓ For A ∈ Vn± , we have l(A) ∈ Vn+1 defined by l(A)(c(v, x)) = c(v, A(x)), where x ∈ Hn± and ∓ v ∈ H1 . This operation is bounded in norm by emax .
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The right embedding tangle r(A):
± For A ∈ Vn± , we have r(A) ∈ Vn+1 defined by r(A)(c(x, v)) = c(A(x), v), where x ∈ Hn± and + − v ∈ H1 or H1 depending on the value of n. This operation is norm bounded by emax as well. Both the left and right embedding operators are strongly continuous; this follows directly from the definition.
and Temperley–Lieb generators (e+ and e− ): We now explicitly describe the action of these two projections on the path basis. These calculations will be useful in the next section. Let a and b be paths of length 1 on Γ which start at positive vertices, c and d paths of length 1 which start at negative vertices, with xa , xb , xc , xd the corresponding basis vectors in V1+ and V1− . Then we should have e+ (c(xa , xc )), c(xb , xd ) = 0 unless a is the reverse of c and b is the reverse√of d, with additionally s(a) = s(b). If these conditions hold, then the inner product (a))μ(t (c)) . should be μ(tμ(s(a)) In other words, for each vertex v ∈ P0+ , let μ t (e) c xe , rev(xe )
yv =
e|s(e)=v
where the sum is taken over all paths of length 1 on Γ . Then
+
e (yv ) =
e1 ,e2 |s(e1,2 )=v
μ(te1 )μ(te2 ) c xe2 , rev(xe2 ) μ(v)
e1 |s(e1 )=v μ(t (e1 ))
=
μ(v)
μ t (e) c xe , rev(xe ) e|s(e)=v
=
e|s(e)=v μ(t (e))
μ(v)
yv
For each positive vertex v, yv is an eigenvector for e+ with eigenvalue δv =
μ t (e) /μ(v)
e|s(e)=v
and e+ is zero off the closed linear span of these yv ’s. We also have an e− element. This is the same diagram as above but with reversed shading, and is defined by reversing all signs in the above definition. yv is defined as above, but now for v being a negative vertex. Then e− has each
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such yv as an eigenvector with eigenvalue e|s(e)=v μ(t (e))/μ(v), and is zero off the closed linear span of these yv ’s. ∓ and We can now compute left and right capping operators, respectively LC(A) : Vn± → Vn−1 ± ± RC(A) : Vn → Vn−1 .
RC(A) =
LC(A) =
These operators may be described in terms of embedding and the Temperley–Lieb generators: r n−1 e± l(A)r n−1 e± = r n−2 (T L)l 2 LC(A) l n−1 e± r(A)l n−1 e± = l n−2 (T L)r 2 RC(A) where the signs of the Temperley–Lieb generators are determined by n and the parity of A. This uniquely defines the capping operation: from the definition of left and right embedding r n−2 (e± )l 2 (LC(A)) and l n−2 (e± )l 2 (RC(A)) are each zero only if LC(A), RC(A) respectively are zero. Note that a graded bounded linear map on the Vn± ’s which fixes the Temperley–Lieb algebra and commutes with multiplication and embedding also commutes with capping; for such a map ω we have l n−2 e± r 2 RC ω(A) = ω l n−2 (T l)r 2 RC(A) = l n−2 e± r 2 ω RC(A) by the above definition, and the same holds for LC. We now describe the interaction of the involution on Vn± (defined above) with these tangles. Since conjugation by the involution corresponds to tangle reflection (see [13]), it follows that (AB)∗ = B ∗ A∗ , e± are self-adjoint, and the involution commutes with left and right embedding. This is consistent with the above description of this involution as the adjoint operation on each B(Hn± ). One important property of the BGPA in [14] was the existence of a positive definite V0± valued sesquilinear form on Vn± , namely x, y = RC n (y ∗ x) in the above notation. We would like this form to be positive definite here as well. Let p, q, r, s be path basis elements in Vn± , and A, B rank one partial isometries from (respectively) p to q and r to s. Then it follows directly from the operad definition that A, B is a positive scalar multiple of a rank one projection in V0+ if p = q and r = s, and is zero otherwise. It follows that the form is positive definite on bounded formal sums of such elements, which constitute all of Vn± . In order for a BGPA to be useful in a subfactor context, we should have both shaded and unshaded circles being equal to some scalar δ. This condition on a planar algebra is called modulus δ [13]. By capping off the single vertical strand l(1), we see this is true when (e+ )2 = δe+ , (e− )2 = δe− . This occurs when δv =
e|s(e)=v
μ t (e) /μ(v)
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is independent of v, or
μ t (e) = δμ(v)
e|s(e)=v
for all v. Another way of describing this situation is that the (potentially unbounded) vector with components μ(v) is an eigenvector for the adjacency matrix of the graph Γ , with eigenvalue δ, or that μ is a positive dimension function on the graph. Under these circumstances we will say that the BGPA has modulus δ. √ For modulus δ BGPAs, μ is necessarily locally bounded and Γ is locally finite, with M δ and emax δ 2 . Note that when Γ is finite, δ must be the unique Perron–Frobenius eigenvalue and μ is uniquely determined up to normalization. If Γ is infinite, there may be many functions μ which produce modulus δ BGPAs, with δ potentially depending on the choice of μ. We will not generally require our BGPAs to be modulus δ, as this assumption is not necessary for the results of the next section. 3. Automorphisms of BGPAs An automorphism of a planar algebra is a graded linear map on the Vn± ’s which commutes with the entire planar operad. If the planar algebra has an involution, we will require the map to commute with the involution as well. In this section we describe the automorphism group of an arbitrary BGPA. These automorphisms of planar algebras may be viewed as a generalization of the automorphisms of the standard invariant of a subfactor (see [18]; cf. [23,12] for specific examples). In the BGPA case, these automorphisms are similar to those of a Jones tower of finite dimensional C ∗ -algebras, computed in [10]. Lemma 3.1. Let Γ be a bipartite graph, with path Hilbert space H and function μ as in Section 2. Let U be a unitary operator on H which respects the grading and commutes with the concatenation operator c. Then the action of U on H0± is described by a graph automorphism. Moreover, Ad U leaves the BGPA PΓ invariant. Proof. Let x = xv be a standard basis element of H0+ , corresponding to a vertex v. We have c(x, x) = x, so c(U (x), U (x)) = U (x) as well by the properties of U . The only elements of H0+ which have this property are of the form w∈S xw , where S is some subset of the even vertices of Γ . Unitarity of U implies that in fact U (xv ) = xw for some w. So U acts by permutation on the even vertices of Γ , and likewise on the odd vertices by an identical argument. Let this permutation be σ . As in Section 2, let sv be the projection onto the basis elements corresponding to paths starting at v, and tv the projection onto paths terminating at v. The dimension of tv sw (H1+ ) is the number of edges between v and w, or zero if they are not adjacent. Let n(v, w) be this number of edges. For any path p from v to w (with corresponding basis element xp ), we have c(xv , xp , xw ) = xp , implying that c(xσ (v) , U (xp ), xσ (w) ) = U (xp ). In other words, U tv sw (H1+ ) ⊂ tσ (v) sσ (w) (H1+ ). Moreover U ∗ also commutes with concatenation and respects grading, so applying the above argument to U ∗ gives us equality of the above subspaces. This means that n(σ (v), σ (w)) = n(v, w), and σ is a graph automorphism.
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The above also implies that Ad U leaves the algebra generated by the s’s and t’s invariant, since U sv tw U ∗ = sσ (v) tσ (w) . Therefore Ad U leaves the commutant of this algebra invariant as well. This commutant is precisely Vn± , implying that Ad U acts on the planar algebra as desired. 2 We recall that the path reversal operator rev defined in Section 2 is an involution sending H1+ to H1− . From the definition, this operator obeys rev(x), rev(y) = y, x where ·,· is the inner product on H1+ or H1− . Theorem 3.1. Let Γ be a locally finite bipartite graph, and μ a locally bounded function from the vertices of Γ to R+ . Let H be the path Hilbert space on Γ as above. Let U be a unitary element of B(H ) which preserves the grading of H and commutes with the concatenation operator. Assume further that the restriction of U to H1± commutes with the path reversal operator, and that the vertex permutation induced by U scales the measure on H0 . Then Ad U commutes with the entire planar operad (and adjoint) on Vn± . Proof. We know from Lemma 3.1 that Ad U acts on the planar algebra. Recall l is the left embedding operator. Let x be in Vn± , v ∈ H1± and w ∈ Hn∓ . Then since U commutes with concatenation, U ∗ l(x)U c(v, w) = U ∗ l(x)c U (v), U (w) = U ∗ c U (v), xU (w) = c v, U ∗ xU (w) while l U ∗ xU c(v, w) = c v, U ∗ xU (w) which is the same. So Ad U commutes with left embedding, and with right embedding as well by a similar argument. Since U is unitary, Ad U commutes with operator multiplication and taking adjoint. It remains only to show that Ad U commutes with the Temperley–Lieb diagrams, i.e. with the elements of the operad representing tangles with no internal disks. This algebra is generated by the Temperley–Lieb generators e+ and e− along with multiplication and embedding, so we need to show that Ad U fixes these generators, i.e. that U commutes with them. Let v and w be two adjacent vertices of Γ . Let H1vw be the subspace of H1 spanned by paths from v to w, and H2vw the subspace of H2 spanned by paths from v to w and back to v. Let {ai } be any orthonormal basis for H1vw . Take yvw = i c(ai , rev(ai )), using the reversal operator defined above. Because of the interaction of rev with inner product described above, the inner product of yvw with any element of the form c(v, rev(v)) is equal to the squared norm of v regardless of which basis is chosen. Such elements span H2vw , so yvw is independent of the choice of specific basis. This is true if v is odd or even. Let σ be the permutation action of U on vertices of Γ . As in Lemma 3.1, we have U (H1vw ) = σ (v)σ (w) H1 , and so U maps an orthonormal basis for the first vector space to one for the second. This means that U (yvw ) = yσ (v)σ (w) .
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We recall from Section 2 that the Temperley–Lieb generators in B(H2± ) each leave the closed linear span of certain vectors {xv } invariant, where v is taken from the set of positive or negative vertices depending on the sign of the generator, and are zero off the span of these vectors. We defined μ t (e) c xe , rev(xe ) yv = e|s(e)=v
for each vertex v but we can also write yv =
μ(w)yvw
w|n(v,w)=0
using the notation above. From Lemma 3.1 σ is a graph automorphism of Γ . By assumption, σ scales the function μ by a fixed constant λ. Therefore U
μ(w)yvw = λ−1/2 μ σ (w) yσ (v)σ (w)
Since σ is a graph automorphism, it maps the set of vertices adjacent to v to the set of vertices adjacent to σ (v), and we have as well
U
w|n(w,v)=0
μ(w)yvw =
λ−1/2 μ(w)yσ (v)w
w|n(w,σ (v))=0
So U maps one standard basis vector of the subspace acted on by either Temperley–Lieb generator to λ−1/2 times another such vector. We have e± (yv ) = δv yv , where δv = e|s(e)=v μ(t (e))/μ(v) and the sign of e is chosen to match the parity of the vertex v. From the properties of σ , δσ (v) = =
μ t (e) /μ σ (v) = μ σ t (e) /μ σ (v)
e|s(e)=σ (v)
e|s(e)=v
e|s(e)=v
e|s(e)=v
λμ t (e) /λμ σ (v) = μ t (e) /μ(v) = δv
So U leaves each eigenspace of both Temperley–Lieb generators invariant. This means that U commutes with the Temperley–Lieb generators. So U commutes with a set of generating tangles for the planar operad, and hence with the entire operad. 2 Now we describe two classes of linear maps on a BGPA which meet all the above conditions. Definition 3.1. Let Γ be a bipartite graph, with function μ as in Section 2. Let κ be an automorphism of Γ , i.e. a permutation of the vertices of Γ which preserves numbers of edges connecting pairs of vertices and sends even vertices to even vertices. Assume also that κ preserves or scales μ. Label each n-fold multiple edge by {1, . . . , n}. We may then extend κ to the edges of Γ by asserting that it preserves this numbering. Then κ gives rise to a permutation of
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the paths on Γ , and therefore a map U on the path Hilbert space H . Then Ad U is the graph automorphism operator associated with κ. Lemma 3.2. Graph automorphism operators are automorphisms of the BGPA. Proof. Let Ad U be a graph automorphism operator as above. It follows directly from the definition that Ad U commutes with path reversal and concatenation, and agrees with the grading. We have also assumed that the underlying graph automorphism is trace scaling. So all of the conditions of Theorem 3.1 are satisfied, and Ad U is a planar algebra automorphism. 2 Definition 3.2. Let O be an element of V1+ which is unitary as an operator acting on H1+ . Let O = rev ◦ O ◦ rev, acting on H1− . For 1 i n, let pi be a path basis element of H1+ (for i even) or H1− (for i odd). Then for n > 0, let U c(p1 , p2 , . . . , pn ) = c O(p1 ), O (p2 ), O(p3 ) . . . and this map U extends to a unique bounded linear operator on Hn+ . Define U similarly on Hn− for (n < 0) as the extension of the map U c(p1 , p2 , . . . , pn ) = c O (p1 ), O(p2 ), O (p3 ) . . . where the pi ’s are again in H1+ or H1− as appropriate. Let U act trivially on H0± . Then Ad U is the multiplication operator associated with O. If O acts non-trivially on only one of the subspaces sv tw (H1+ ), while leaving all others fixed, then we will call it a basic multiplication operator associated to the vertex pair {v, w}. Every multiplication operator is a product of basic multiplication operators, and basic multiplication operators associated to different vertex pairs commute with each other. A multiplication operator is scalar if the restriction to B(sv tw (H1+ )) for each v, w is a scalar multiple of the identity. The scalar multiplication operators are the center of the multiplication operator group. If the graph has no multiple edges then every multiplication operator is scalar. Lemma 3.3. Multiplication operators are automorphisms of the BGPA. Proof. Let Ad U be a multiplication operator as above. From the definition, it commutes with concatenation and path reversal and respects grading. Since O comes from an element of V1+ , it commutes with sv and tv for vertices v. Therefore the associated graph automorphism σ is trivial, and preserves the trace. So all the conditions of Theorem 3.1 are satisfied, and Ad U is a planar algebra isomorphism. 2 Direct computation shows that conjugating a basic multiplication operator by a graph automorphism operator produces a basic multiplication operator associated to different vertex pair. So the group of automorphisms generated by these two types of operators has a crossed product structure: the subgroup of multiplication operators is normal. Now we will show that the two types of operators described above in fact generate the entire automorphism group of a BGPA.
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Lemma 3.4. Any automorphism of a BGPA is strongly continuous. Proof. Let α be an automorphism of a BGPA. From the definition in Section 3.2, we have Vn± ⊂ B(Hn± ), where Vn± is a (potentially infinite) direct sum of type I factors. Every automorphism of such a von Neumann algebra can be written Ad U , where U is a unitary element of B(Hn± ). Since α|Vn± commutes with multiplication and involution, this restriction of α is a von Neumann algebra isomorphism, and may be written Ad Un± for unitary U ∈ B(Hn± ). Summing all the Un± s provides a unitary operator on the total Hilbert space H whose adjoint action on the graded vector space Vn± agrees with α. Multiplication is strongly continuous. 2 This means that two automorphisms of a BGPA are equal if and only if they agree on loops, since the loops span a strongly dense set in each Vn± . Lemma 3.5. Let α be an automorphism of a BGPA. Then there is a graph automorphism operator β such that α and β agree on V0± . Proof. Let pv , qw be the atomic projections in V0+ , V0− associated with even and odd vertices v and w. Since α is a BGPA automorphism, it must send pv to some other atomic projection pσ (v) in V0+ , and likewise for qw . Therefore α induces a permutation σ on the vertices of Γ . To see that this permutation is a graph automorphism, note that l(pv )r(qw ) is a minimal central projection of V1+ (or zero), and the dimension of l(pv )r(qw )V1+ is the square of the number of edges between v and w. This dimension is preserved by α, i.e. it agrees with the dimension of l(pσ (v) )r(qσ (w) )V1+ . Therefore (v, w) and (σ (v), σ (w)) have the same number of edges between them, i.e. n(v, w) = n(σ (v), σ (w)). Next we must show that σ preserves or scales the trace. For this we note that pv with a circle around it evaluates to μ(v) qw μ(w) w|n(v,w)=0
Since α commutes with the tangle, we must have μ(v)/μ(w) = μ σ (v) /μ σ (w) for all adjacent v, w, implying the desired result. From the above description of graph automorphism operators, there is a such an operator β whose induced permutation action on the vertices of Γ is the same as α’s. These two operators then agree on V0± . 2 Lemma 3.6. Let α be an automorphism of a BGPA which acts trivially on V0± . Then there is a multiplication operator β such that β and α agree on V1± . Proof. Since α acts trivially on V0± , it fixes all elements of the form r(pv )l(qw ) in V1+ . These elements are the center of V1+ taken as a von Neumann algebra, so α acts as an inner automorphism on V1+ . There is a multiplication operator β whose action on V1+ is any desired inner automorphism. Then α and β agree on V1+ . Both of these automorphisms commute with the half rotation
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which is a bijective map from V1+ to V1− , so they agree on V1− as well.
2
Lemma 3.7. Let α be an automorphism of a BGPA which acts trivially on V1± . Then α is a scalar multiplication operator. Proof. From the proof of Lemma 3.4, we can write the action of α on each Vn± as Ad Un± , where Un± is a unitary in B(Hn± ). First note that for any path p on the graph Γ , with corresponding basis vector xp ∈ Hn± , there is a rank one projection onto xp contained in Vn± . This projection may be written as the product r n (pe1 )r n−1 l(pe2 ) . . . rl n−1 (pen−1 )l n (pen ) where ei is the ith edge of p and pei is the rank one projection in V1+ or V1− onto the vector xei corresponding to the path ei . Since α acts trivially on V1± , it fixes this projection, and Un± must therefore map each xp to a scalar multiple of itself. Now let bl ∈ Vn± be a rank one partial isometry corresponding to a loop l ∈ ln± . We have bl = xp bl xq for certain rank one projections xp , xq as above, implying that α(bl ) = α(xp bl xq ) = xp α(bl )xq The only way this can be true is if α sends bl to a scalar multiple of itself as well. In other words, every loop is an eigenvector of α, and α induces a map ρ from the set of loops to the complex scalars of modulus 1. Since α commutes with the half rotation, ρ(l) is independent of the base point of l. Because α fixes V1± , ρ(l) = 1 for any loop l of length 2. Finally, since α commutes with this diagram
if l3 is the concatenation of the loops l1 and l2 then we have ρ(l3 ) = ρ(l1 )ρ(l2 ). Putting this together we find that ρ is necessarily a 1-dimensional representation of the fundamental group of Γ . But any such representation may be obtained from a scalar multiplication operator. We may always find a set of free generators {l1 , l2 , . . .} for the fundamental group with the property that each generator li contains some edge ei which does not appear in any other loop. Then a basic scalar multiplication operator with value λ associated to the endpoints of some ek corresponds to the representation of π1 (Γ ) sending lk to λ (or possible λ, depending on the direction of lk ) and all other generators to 1. All other representations of π1 may be obtained similarly from basic scalar multiplication operators associated to various ei ’s.
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This implies that there is a scalar multiplication operator which agrees with α on loops. Since the loops span a strongly dense subset of each Vn± , and automorphisms of a BGPA are strongly continuous by Lemma 3.4, it follows that α itself is a scalar multiplication operator. 2 Since scalar multiplication operators themselves act trivially on V1± , this lemma in fact shows that the scalar multiplication operators of a BGPA are isomorphic to the 1-dimensional representations of the fundamental group of the graph. Theorem 3.2. Let PΓ be a BGPA, with multiplication operators E and graph automorphism operators A. Let α be an automorphism of PΓ . Then α = ae for some a ∈ A, e ∈ E. Proof. This follows from the proceeding lemmas. There is β1 ∈ A such that β1−1 α acts trivially on V0± . There is β2 ∈ E such that β2−1 β1−1 α acts trivially on V1± . So β2−1 β1−1 α is a scalar multiplication operator β3 , and α = β1 β2 β3 with β1 ∈ A and β2 β3 ∈ E. 2 Since conjugation by graph automorphisms leaves the multiplication operator group invariant, we may in fact write Aut PΓ = E A with notation as above. 4. Planar fixed point subfactors A subfactor planar algebra is a planar algebra with the following additional properties [13]: • dim V0± = 1. • dim Vn± < ∞ ∀n. • Spherical: Since V0+ and V0− are one-dimensional, we may identify these with C, sending the empty diagram to 1 (in both shadings). A planar algebra is spherical if for any A ∈ V1+ , the left and right caps LC(A) and RC(A) are the same complex scalar. • Involution: There is an antilinear isometry on each Vn± which interacts with tangles as reflection. • Positive definiteness: Involution gives us a scalar sesquilinear form, namely x, y = RC n (y ∗ x) where the multiplication and right capping tangles are as in Section 2. This form should be positive definite. The standard invariant of any finite index extremal II 1 subfactor may be described as a subfactor planar algebra [13,22]. We take Vn+ = M0 ∩ Mn , Vn− = M1 ∩ Mn+1 , and the operad definition is given in [13]. Conversely, if P is a subfactor planar algebra, then there exists a finite index extremal II 1 subfactor such that the standard invariant of this subfactor is P ([22], cf. [13,11,16,15]). Any subfactor planar algebra is modulus δ, where δ is the square root of the Jones index of the corresponding subfactor (see [13]). We will now show that any sufficiently small planar subalgebra of a modulus δ BGPA is of subfactor type. Lemma 4.1. Let PΓ be a bipartite graph planar algebra with spin vector μ. Let x be in V1+ ; let xl represent the element of V0− obtained by capping off to the left, and xr ∈ V0+ from capping off to the right. Suppose both xl and xr are scalars. Then there is some constant α, independent of x, such that xl = αxr .
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Proof. From [14], there is a partition function defined on V0+ and V0− as the linear extension of xv → μ2 (v), and this function has the same value . on xl and xr xl = λ1 v xv , xr = λ2 v xv . If xl and xr are scalars, this means The partition function on x1 is λ1 v μ2 (xv ), and on xr is λ2 v μ2 (xv ). Since these are the
μ2 (x )
same we must have λ1 = λ2 v μ2 (xv ) . In other words α = v v that xl and xr are scalars. 2
2 μ (x ) v 2 v v μ (xv )
and xl = αxr for all x such
Proposition 4.1. Let PΓ be a finite modulus δ bipartite graph planar algebra. Let X be a planar ∗-subalgebra of PΓ such that X ∩ V0+ = X ∩ V0− = C. Then X is a subfactor planar algebra. Proof. The BGPA PΓ has an involution, giving rise to a positive definite V0+ -valued sesquilinear form (see Section 2). The involution and form are inherited by X, and the restriction of the form to X is scalar valued since X ∩ V0± = C. Since Γ is finite, each Vn± is finite dimensional, and so this is true of their intersections with X as well. To show that X is of subfactor type, it remains only to demonstrate sphericality. Let x ∈ X be an element of V1+ . Capping off to left or right produces scalars, since X ∩ V0± is scalar. Therefore the conditions of the lemma above are satisfied. Because μ has modulus δ, shaded and unshaded circles represent the same scalar. These diagrams are the left and right caps of a single vertical strand, so the constant α in the lemma is equal to 1. It follows that xl and xr are equal as scalars, and X is spherical. 2 Using a result of Burns, we can show that a small subalgebra of an infinite BGPA also corresponds to a subfactor. Burns described a class of rigid planar C ∗ -algebras in [9], generalizing Jones’ definition of an SPA. Every rigid planar C ∗ -algebra is the standard invariant of a finite index II 1 subfactor, but this subfactor need not be extremal. From [9], a planar algebra having all the characteristics of an SPA except sphericality is a rigid planar C ∗ -algebra. Theorem 4.1. Let PΓ be a locally finite bipartite graph planar algebra. Let X ⊂ PΓ be a planar ∗-subalgebra with X ∩ V0± = C. Then X is a rigid planar C ∗ -algebra. Proof. Let pv ∈ V0+ be a minimal projection corresponding to some even vertex v, and q = r n (pv ). Take x ∈ X ∩ Vn+ , and let RC n be the diagram consisting of capping off all strands to the right (this is the form from Section 2). Then RC n (x) is a scalar from the properties of X, and RC n (qx) = qRC n (x) by isotopy invariance. q commutes with x and x ∗ with respect to the usual multiplication tangle. If qx = 0, then qx ∗ x = 0, giving RC n (qx ∗ x) = 0 = qRC n (x ∗ x). Since RC n (x ∗ x) is a scalar, this means that px = 0 implies RC n (x ∗ x) = 0. But RC n gives a positive definite form on Vn+ , so x itself is zero in this case. This means that the map x → qx is injective on X ∩ Vn+ . But qVn+ has basis labelled by loops of length 2n which start and end at v. By local finiteness of Γ , this set is finite. Therefore pVn+ is finite dimensional, and X ∩ Vn+ is as well. The same argument shows that X ∩ Vn− is finite dimensional. As in the previous proposition, X inherits an involution and positive definite sesquilinear form from PΓ , so it is a rigid planar C ∗ -algebra. 2 Putting this together, let PΓ be a modulus δ BGPA coming from an infinite graph Γ , and X ⊂ PΓ a planar subalgebra with dim(X ∩ V0± ) = 1. Then X is the planar algebra of a subfactor.
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The subfactor is extremal (and X is an SPA) if and only if X is spherical. This is automatic when X ∩ V1+ is one-dimensional, but is not true in general. 5. Examples 5.1. Introduction We can construct a wide range of subfactor planar algebras using this fixed point technique. In order for the fixed points PΓG to be the planar algebra of a subfactor, we need it to have 1dimensional intersection with V0+ and V0− , which is equivalent to having the graph automorphism part of G act transitively on both positive and negative vertices. We should also check sphericality for non-irreducible infinite graph examples, to see if the subfactor is extremal. Some SPAs thus obtained are the standard invariants of previously known subfactors, while others seem to be previously unclassified. A few such examples are described below. Definition 5.1. If PΓ is a BGPA, and G ⊂ Aut PΓ with PΓG an SPA, then the corresponding subfactor is a planar fixed point subfactor. If such a group of automorphisms of PΓ exists, then there must exist integers p and q such that every even vertex of Γ has degree p and every odd vertex has degree q. When Γ is finite, any planar fixed point subfactor obtained from Γ has index pq. The same is true for arbitrary Γ when the planar fixed point subfactor is irreducible. In general, however, the index can take on other values, and need not be an integer. 5.2. Group-subgroup subfactors A specific example of the planar fixed point construction is found in [12]. In this paper, Gupta starts with the BGPA on the graph with n odd vertices all connected to one even vertex. Then G is some group acting by permutation on the set of odd vertices, and H is the subgroup which fixes some specified vertex. Gupta shows that the fixed points of the BGPA by this action constitute a subfactor planar algebra, and that this is in fact the standard invariant of the group-subgroup subfactor (see e.g. [17]) corresponding to the inclusion H ⊂ G, namely M G ⊂ M H for some outer action of some finite group G on a II 1 factor M. 5.3. Wassermann subfactors Here we let Γ be the graph with two vertices connected by an n-fold multiple edge. Let G be any compact subgroup of the unitaries Mn (C). Then G may be embedded in the multiplication operators on this graph. The fixed points of this G-action are of subfactor type. They are identical to the standard invariant of the Wasserman subfactor
1 ⊗ Mn ⊗ Mn ⊗ · · ·
st G
st G ⊂ Mn ⊗ Mn ⊗ Mn ⊗ · · ·
where G acts pointwise on the tensor products (see [24]).
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5.4. Diagonal subfactors Let G be a finitely generated group of outer automorphisms of a II 1 factor M. Let G have distinct generators {g1 , . . . , gn }, and take Γ to be the graph which has one odd and one even vertex for each element of G. Two vertices vx+ and vy− are connected by a single edge if y = xh, for h ∈ {1, g1 , . . . , gn }. Take μ to be the function which is 1 at every vertex of Γ ; it has modulus n + 1. Then G acts on the graph by left translation: αx (vg± ) = vxg . Associating each such graph automorphism with the corresponding graph automorphism operator on PΓ gives an action of G on PΓ . This action is transitive on even and odd vertices. When the graph is infinite, sphericality may be directly verified: V1+ has dimension n + 1, and each minimal projection has left and right trace 1/(n + 1). Therefore PΓG is of subfactor type. Since vertices are labelled by group elements, loops of length 2n in this graph may be written as a list of vertices −1 =x x − xa1 − xa1 a2−1 xa1 a2−1 a3 . . . − xa1 a2−1 . . . a2n
where each ai comes from the set {1, g1 , . . . , gn }. So we may think of this loop as a starting point x along with a list of generators (a1 , a2 , . . .) whose alternating product is the identity in G. The group action moves the base point (via left translation) while keeping the generator list invariant. It follows that these generator lists label a basis for the intersection of PΓG with Vn± . This basis is precisely that described in [3] for the planar algebra of the diagonal subfactor (see [21,2]). It may be verified that the operations of left and right embedding, involution, and multiplication on PΓG agree with the planar algebra of [3], and the Temperley–Lieb algebra embeds in the same way, so these planar algebras are isomorphic. It follows that the planar algebra constructed in this way is that of a diagonal subfactor without cocycle. 5.5. Bisch–Haagerup subfactors Let G be a group of outer automorphisms of a II 1 factor M, generated by finite subgroups H and K. For simplicity we require H ∩ K = {1}. Let Γ be the graph which has one even vertex for each H right coset in G, and one odd vertex for each K right coset. Then the edges of Γ are labelled by group elements. The endpoints of value H on each even each edge eg are the even vertex vgH and the odd vertex vgK . Let μ have √ vertex and value K on each odd vertex. The BGPA PΓ then has modulus |H ||K|. We may write a loop of length 2n as a list of edges: ex1 − ex2 . . . − ex2n − e1 Assume first that the loop begins at a positive vertex. For the path to be connected, ex1 and ex2 must share a vertex, so x1 and x2 are in the same K-coset and x2 = x1 k1 . Likewise x2 and x3 are in the same H -coset. So we can view this loop as a starting point x along with a list of alternating elements of k and h, with the restriction that k1 h1 k2 h2 . . . kn hn is equal to the identity in G. Again G acts on the graph by left translation, and this gives a G-action on PΓ . This action shifts the base point of loops while leaving the list of ki ’s and hi ’s the same. So we may identify a basis for PΓG ∩ Vn+ , namely n-tuples of elements of K and H obeying k1 h1 k2 h2 . . . kn hn = 1;
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the basis for Vn− is obtained by reversing the roles of H and K. This labelling set is the same as that of the planar algebra of [4] (cf. [5]); again it may be shown that the two planar algebras are isomorphic. It follows that this fixed point planar algebra is the standard invariant of the Bisch–Haagerup subfactor M H ⊂ M K. 5.6. The cube graph Let Γ be the graph of a cube, μ the function sending each vertex of Γ to 1, and PΓ the corresponding BGPA. We obtain several subfactor planar algebras by taking fixed points under various group actions. We note that the automorphism group of the bipartite graph is S4 , since each such automorphism may be described uniquely as a certain permutation of the even vertices. The only modulus δ spin vector assigns weight 3 to every vertex; δ = 3, so every subfactor planar subalgebra produces an index 9 subfactor. We can write down a biprojection which is invariant under every automorphism, so there is always an index 3 intermediate subfactor. We mention some of the possibilities described above. If G is the subgroup generated by (12)(34) and (13)(24), then PΓG is the planar algebra of the diagonal subfactor; the group generators are each order 2, and the group is Z22 . If G is A4 , then PΓG is the Bisch–Haagerup subfactor M Z3 ⊂ M Z3 , where the two order-3 automorphisms of M together generate the group A4 . If we take G = S4 , then PΓG is some other subfactor planar algebra. Its principal graph may be directly computed from the group action:
The dual principal graph is the same. It does not appear to be of any previously categorized type, although we have tentatively identified it as a composition of two group-subgroup subfactors. Finally, we may take G = Aut PΓ . Since the multiplication operator group of PΓ is (S 1 )5 , this group is infinite, and PΓG is infinite depth. 5.7. The degree (3, 2) tree graph Let Γ be the graph which branches twice √ at each even vertex and three times at each odd vertex. The operator norm of this graph is 4 2. There are no multiplication operators on PΓ , so all automorphisms will come from graph automorphisms. First define √μ by μ(v) = 2 when v is even and μ(v) = 3 when v is odd. The BGPA then has modulus 6. Every automorphism, of Γ here gives rise to an automorphism of PΓ . Taking G = Aut(PΓ ), we obtain a subfactor planar algebra PΓG . This subfactor is irreducible, hence automatically spherical, and is non-amenable; we conjecture it is obtained as a composition of two group-subgroup subfactors, where the groups involved are variations of the Grigorchuk lamplighter group. We may also obtain a transitive subgroup as follows: color the edges of the graph with three colors so that no vertex contacts two edges of the same color, and then consider all automorphisms which leave the coloring invariant or permute the colors. This group is Z3 ∗ Z2 , and the
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resulting subfactor is the index 6 Bisch–Haagerup subfactor described in [5] corresponding to this group. Now we describe a non-irreducible example. We color the edges of the graph red and blue so that each even vertex contacts a red and blue edge, and each odd vertex contacts two reds and a blue. We now consider G to be the group of color preserving automorphisms. It may be seen that this group is transitive on odd and even vertices, but the fixed points have 2-dimensional intersection with V1+ space, corresponding to the 2 G-orbits of edges. With the above function μ, sphericality fails: the ‘blue edge’ element has left and right traces of 1/3 and 1/2. To find a subfactor planar subalgebra of this BGPA, we will need to define μ differently. We take μ(v0 ) = 1 for some arbitrary base vertex v0 , and then define μ elsewhere so that for adjacent vertices x even and y odd, we have μ(y) = μ(x)√ if the x − y edge is red, but μ(y) = 21/4 μ(x) if the edge is blue. This trace has modulus 1 + 2. It may be seen that any color preserving automorphism of Γ will multiply μ by a constant factor, hence providing an automorphism of PΓ . With this choice of μ and G the √ color preserving graph automorphisms, the left and right traces of the ‘blue’ element are both 2 − 1, and sphericality holds. So from Section 5.4, PΓG is a subfactor planar algebra. √ √ The resulting subfactor is infinite depth non-amenable. Its index is (1 + 2)2 = 3 + 2 2, and it does not have any intermediate subfactors. From [20], this is the minimum index for an extremal non-irreducible subfactor. The above construction is a new way of getting such a subfactor. Any group which is transitive on even and odd vertices but the same two orbits of edges as above will also produce a subfactor with this list of properties, so we actually have many such examples. For example, we might partition the red edges into ‘red’ and ‘white’ so that each odd vertex contacts one vertex of each color; G might then be the group of graph automorphisms which either preserve color or swap the colors red and white. √ For any δ > 1+ 2, there exists some function μ of modulus δ which is preserved or scaled by any element of G. Construction PΓ and taking fixed points then gives a non-extremal subfactor, as in [9]. So we obtain a continuous family of subfactors with the same principal graphs, but different indices. √ √ Other non-irreducible subfactors with index (1 + 3)2 = 4 + 2 3 may be constructed similarly from a graph which branches twice at each even vertex and four times at each odd vertex. This is the third on the list from [20]. Acknowledgment I would like to thank Professor Dietmar Bisch for his useful suggestions and corrections. References [1] Stephen Bigelow, Scott Morrison, Emily Peters, Noah Snyder, Constructing the extended Haagerup planar algebra, arXiv:0909.4099 [math.OA], 2009, 45 pp. [2] Dietmar Bisch, Entropy of groups and subfactors, J. Funct. Anal. 103 (1) (1992) 190–208. [3] Dietmar Bisch, Das Paramita, Shamindra K. Ghosh, The Planar Algebra of Diagonal Subfactors, Clay Math. Proc., vol. 10, 2008, 25 pp. [4] Dietmar Bisch, Das Paramita, Shamindra K. Ghosh, The planar algebra of group-type subfactors, J. Funct. Anal. 257 (1) (2009) 20–46. [5] Dietmar Bisch, Uffe Haagerup, Composition of subfactors: new examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. (4) 29 (3) (1996) 329–383.
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[6] Dietmar Bisch, Vaughan F.R. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1) (1997) 89–157. [7] Dietmar Bisch, Vaughan F.R. Jones, Singly generated planar algebras of small dimension, Duke Math. J. 101 (1) (2000) 41–75. [8] Dietmar Bisch, Vaughan F.R. Jones, Singly generated planar algebras of small dimension, II, Adv. Math. 175 (2) (2003) 297–318. [9] Michael Burns, Subfactors, planar algebras and rotations, PhD dissertation, University of California, Berkeley, Department of Mathematics, 2003. [10] Richard D. Burstein, Group-type subfactors and Hadamard matrices, Trans. Amer. Math. Soc., in press. [11] Alice Guionnet, Vaughan F.R. Jones, Dimitri Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, arXiv:0712.2904, 2008, 31 pp. [12] Ved P. Gupta, Planar algebra of the subgroup-subfactor, Proc. Indian Acad. Sci. Math. Sci. 118 (4) (2008) 583–612. [13] Vaughan F.R. Jones, Planar algebras, I, arXiv:math/9909027 [math.OA], 1999, 122 pp., New Zealand J. Math., in press. [14] Vaughan F.R. Jones, The planar algebra of a bipartite graph, in: Knots in Hellas ’98, Delphi, World Sci. Publ., River Edge, NJ, pp. 94–117. [15] Vaughan F.R. Jones, Dimitri Shlyakhtenko, Kevin Walker, An orthogonal approach to the subfactor of a planar algebra, arXiv:0806.4146 [math.OA], 2008, 12 pp. [16] Vijay Kodiyalam, V.S. Sunder, From subfactor planar algebras to subfactors, Internat. J. Math. 20 (10) (2009) 1207–1231. [17] Hideki Kosaki, Shigeru Yamagami, Irreducible bimodules associated with crossed product algebras, Internat. J. Math. 3 (5) (1992) 661–676. [18] Phan H. Loi, On automorphisms of subfactors, J. Funct. Anal. 141 (2) (1996) 275–293. [19] Emily Peters, A planar algebra construction of the Haagerup subfactor, arXiv:0808.0764 [math.OA], 2008, 57 pp., Internat. J. Math., doi:10.1142/S0129167X10006380, in press. [20] Mihai Pimsner, Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986) 57–106. [21] Sorin Popa, Sousfacteurs, actions des groupes et cohomologie, C. R. Acad. Sci. Paris Sér. I Math. 309 (12) (1989) 771–776. [22] Sorin Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (3) (1995) 427–445. [23] Anna L. Svendsen, Automorphisms of subfactors from commuting squares, Trans. Amer. Math. Soc. 356 (6) (2004) 2515–2543 (electronic). [24] Antony Wassermann, Coactions and Yang–Baxter equations for ergodic actions and subfactors, in: Operator Algebras and Applications, vol. 2, Cambridge Univ. Press, Cambridge, 1988, pp. 203–236.
Journal of Functional Analysis 259 (2010) 2404–2423 www.elsevier.com/locate/jfa
On the classical limit of Bohmian mechanics for Hagedorn wave packets Detlef Dürr, Sarah Römer ∗ Mathematisches Institut der LMU, Theresienstr. 39, 80333 München, Germany Received 26 March 2010; accepted 21 July 2010 Available online 31 July 2010 Communicated by C. Villani
Abstract We consider the classical limit of quantum mechanics in terms of Bohmian trajectories. For wave packets as defined by Hagedorn we show that the Bohmian trajectories converge to Newtonian trajectories in probability. © 2010 Elsevier Inc. All rights reserved. Keywords: Quantum mechanics; Bohmian mechanics; Classical limit; Semiclassical wave packets
1. Introduction There are many ways to formulate the classical limit of quantum mechanics. The strongest assertion would be about “quantum particle trajectories” becoming Newtonian. Particle trajectories, however, are not ontological elements of orthodox quantum theory and thus the “classical limit” must be defined in some operational way. In contrast, Bohmian mechanics, which for all practical purposes is equivalent to quantum mechanics, is a quantum theory of point particles moving, so the study of the classical limit becomes a straightforward task [1,5]: Under which circumstances are the Bohmian trajectories of particles approximately Newtonian trajectories? Here “approximately” can be understood in various manners. The technically simplest but also weakest is that at every time t the Bohmian particle’s position is close to the center of a “classi* Corresponding author.
E-mail addresses:
[email protected] (D. Dürr),
[email protected] (S. Römer). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.011
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cally moving” very narrow wave packet ψ. This essentially amounts to showing that |ψ(t)|2 is more or less transported along a Newtonian flow (see [11] for a recent work on this). The strongest and clearly most direct assertion would be that almost every Bohmian trajectory converges to a Newtonian trajectory in the uniform topology. We shall prove here a slightly weaker statement, namely that the uniform closeness holds in probability. We shall establish this result for a particular class of wave packets which were defined by Hagedorn in [9] and which move along classical paths. To formulate the precise result let us recall that in Bohmian mechanics the state of a particle is described by a wave function ψ(y, s), where y ∈ R3 , s ∈ R, and by its position Y ∈ R3 . The wave function evolves according to Schrödinger’s equation (h¯ = m = 1) 1 ∂ i ψ(y, s) = H ψ(y, s) := H0 + V (y) ψ(y, s) := − y + V (y) ψ(y, s) ∂s 2
(1)
with the potential1 V . The wave function governs the motion of the particle by ∇y ψ(Y (y 0 , s), s) d ψ Y (y 0 , s) = v Y (y 0 , s), s := Im , ds ψ(Y (y 0 , s), s)
Y (y 0 , 0) = y 0 .
(2)
For a wave function ψ the position Y is a random variable the distribution of which is given by the equivariant probability measure Pψ with density |ψ(y)|2 (Born’s statistical rule; see [6,5] for a precise assertion). This means that at any time t the particle will typically be somewhere in the “main” support of |ψ(y, t)|. Thus for a narrow wave packet which, according to Ehrenfest’s theorem, moves – at least for some time – along a classical trajectory, at every instance of time t the position of the particle will typically be close to a classical position. To be sure: this does not imply that a typical Bohmian trajectory stays close to the classical trajectory for the whole duration of a given time interval, since it may every now and then make a large excursion. We shall consider a sufficiently smooth potential and a special class of initial wave functions where the potential V varies on a much larger scale than the wave functions, see, e.g., [1] for a physical discussions of the scales. More precisely, we choose V ε (y) := V (εy) for some small parameter ε, thus defining a microscopic (y, s) and a macroscopic scale (x, t) := (εy, εs). As initial wave functions we take the semiclassical wave packets Φkε (a(0), η(0), ·) defined by Hagedorn in [8,9]. They are non-isotropic 3-dimensional generalized Hermite polynomials of order k := |k| multiplied by a Gaussian wave packet centered around the classical phase space point (a(0), η(0)). On the macroscopic scale, i.e. on the scale of variation of the potential, their √ standard deviation is of order ε both in position and momentum, that is they vary on an intermediate scale. This is the best order of ε allowed, since by Heisenberg’s uncertainty relation σy σp ∼ 1 on the microscopic scale, so on the macroscopic scale σx σp = εσy σp must be of order ε. 1 1 More rigorously: H is a self-adjoint extension of H | ∞ 3 C0 (Ω) = − 2 + V (with V : Ω ⊆ R → R) on the Hilbert 2 3 space L (R ) with domain D(H ).
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In the following, we change to macroscopic coordinates (x, t) = (εy, εs). With := x , 3 ∇ := ∇x and ψ ε (x, t) := ε − 2 ψ( xε , εt ) Schrödinger’s equation then reads iε
2 ε ∂ ε ψ (x, t) = H ε ψ ε (x, t) = − + V (x) ψ ε (x, t). ∂t 2
(3)
√ In this setting Hagedorn [8,9] proved: With an error of order ε in L2 -norm the solution ψkε (x, t) of (3) with initial data ψkε (x, 0) = Φkε (a(0), η(0), x) is given by Φkε (a(t), η(t), x), where (a(t), η(t)) is the corresponding classical phase space trajectory, that is the solution of the Newtonian law of motion with initial data (a(0), η(0)). Now consider the Bohmian trajectories on the macroscopic scale, i.e. solutions of the differential equation ε ∇ψ ε (Xε (x 0 , t), t) d ε ψ , X (x 0 , t) = v X (x 0 , t), t = ε Im dt ψ ε (X ε (x 0 , t), t)
Xε (x 0 , 0) = x 0 .
(4)
Our main result is their convergence in probability: For all T > 0 and γ > 0 there exists some R < ∞ such that √
ε Pψk (·,0) x 0 ∈ R3 max Xε (x 0 , t) − a(t) R ε > 1 − γ t∈[0,T ]
for all ε small enough. It is clearly desirable to have an analogous result for the velocities, so that convergence of “phase space” trajectories is achieved. However, the control of velocities introduces further technicalities. We shall shortly discuss and present some results on the convergence of velocities in Section 4. Next (Section 2) we give the mathematical setup: We briefly introduce the dynamics we want to compare and Hagedorn’s result that we shall refine for our needs. Section 3 describes our result on the classical limit. Proofs are in Section 5. 2. Mathematical framework Definition 1. The potential V ∈ C ∞ (R3 , R) is in GV if for all multi-indices α ∈ N3 max Dα V ∞ CV
|α|4
(5)
for some CV < ∞ and if multiplication by V maps the Schwartz space S(R3 ) into itself, i.e. if α Vf ∈ S(R3 ) for all f ∈ S(R3 ). Here Dα denotes the (weak) derivative ∂xα11 ∂xα22 ∂x33 . The requirement that V maps S into itself is needed to get Pψ -almost sure global existence of Bohmian mechanics [3,13] for initial wave functions ψ ∈ S. The quantum dynamics is given by Bohmian mechanics, i.e. by (3) and (4). Equivariance of the measure Pψ means that if Xε (·, 0) is |ψ(·, 0)|2 -distributed then Xε (·, t) is |ψ(·, t)|2 distributed [6]. By U ε (t) we denote the unitary propagator generated by H ε : d ε i U (t) = − H ε. dt ε t=0
(6)
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The classical dynamics is given by Newtonian mechanics, so the classical state of a particle at the macroscopic time t is given by its classical position and velocity at that time, which we denote by (a(t), η(t)). For any given initial value (a(0), η(0)) it is the unique global solution of Hamilton’s equations of motion: ˙ = η(t), a(t) ˙ = −∇V a(t) . η(t)
(7)
We introduce now the class of Hagedorn’s wave functions for which we shall establish the classicality of Bohmian trajectories. Hagedorn’s wave packets are the eigenfunctions of some generalized 3-dimensional harmonic oscillator. As such they are generalized Hermite functions, i.e. products of generalized Hermite polynomials and the Gaussian ground state ϕ0 . More precisely, for every ε > 0, every phase space point (a, η) and every pair of admissible matrices (A, B) ∈ C3×3 Hagedorn constructed an orthonormal basis of L2 (R3 ) consisting of semiclassical wave packets 1 ϕk (A, B, ε, a, η, x) := √ A∗ (A, B, ε, a, η)k ϕ0 (A, B, ε, a, η, x), k!
k ∈ N3 ,
(8)
with the ground state ϕ0 (A, B, ε, a, η, x)
i 3 1 1 := (πε)− 4 det(A)− 2 exp − (x − a), BA−1 (x − a) + η, (x − a) 2ε ε
(9)
and the formal vector of raising operators 1 A∗ (A, B, ε, a, η) := √ B ∗ (x − a) − iA∗ (p − η) . 2ε Here p = −i∇y = −iε∇, ·,· is the canonical scalar product on Cn and (A, B) are admissible if AT B − B T A = 0 and A∗ B + B ∗ A = 2.
(10)
In particular, (10) implies that A is invertible, Re(BA−1 ) = (AA∗ )−1 and thus that for some < ∞ constants 0 < C C 3
ε − 4 Ce−C
x−a 2 ε
x−a 2 3 −C ε . ϕ0 (A, B, ε, a, η, x) ε − 4 Ce
Moreover, for any multi-index α ∈ N3
x −a √ ε
α ϕk (A, B, ε, a, η, x) =
|k−k ||α| |k−k |+|α| even
α Ckk (A)ϕk (A, B, ε, a, η, x)
(11)
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and
p−η √ ε
α
ϕk (A, B, ε, a, η, x) =
|k−k ||α| |k−k |+|α| even
α Ckk (B)ϕk (A, B, ε, a, η, x)
(12)
α depends continuously on A resp. B. This in turn entails, denoting by ·,· also the where Ckk scalar product on L2 (R3 ),
ϕk (A, B, ε, a, η, x), xϕk (A, B, ε, a, η, x) = a(t),
ϕk (A, B, ε, a, η, x), pϕk (A, B, ε, a, η, x) = η(t)
(13)
and, for any multi-index α ∈ N3 , |α| (x − a)α ϕk (A, B, ε, a, η, x) C α (A)ε 2 , k 2 |α| (p − η)α ϕk (A, B, ε, a, η, x) C α (B)ε 2 . k 2
(14)
The ϕk s and their gradients scale in ε as follows: There is a constant C < ∞, depending on k, A and B such that a(t) ϕk (A, B, ε, a, η, x) = ε − 34 ϕk A, B, 1, 0, 0, x − √ ε k 1 |x−a(t)| 3 |x − a(t)| − C( √ε )2 ε− 4 C 1 + e 2 (15) √ ε and −η ∇ − i η(t) ϕk (A, B, ε, a, η, x) = ε − 12 p√ ϕk (A, B, ε, a, η, x) ε ε |x − a(t)| k+1 − 12 C( |x−a(t)| √ )2 − 54 ε ε C 1+ e √ ε
(16)
for all x ∈ R3 . Hagedorn’s wave packets yield approximate solutions to Schrödinger’s equation (3): Let a(t), η(t) be a solution of (7) and A(t), B(t) a solution of ˙ = iB(t), A(t) ˙ = iV (2) a(t) A(t), B(t) with initial data A(0), B(0) fulfilling (10). Call i Φkε (x, t) := e ε S(t) ϕk A(t), B(t), ε, a(t), η(t), x the semiclassically time evolved wave packet and
(17)
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ψkε (x, t) := U ε (t)Φkε (x, 0) = U ε (t)ϕk A(0), B(0), ε, a(0), η(0), x
(18)
t the Schrödinger evolved wave packet, where S(t) = 0 [ 12 η2 (s) − V (a(s))] ds is the usual classical action. Then for every T > 0 there is some C < ∞ (depending on T , k, a(t), η(t), A(t)) such that ε √ ψ (x, t) − Φ ε (x, t) < C ε k k 2
(19)
for all t ∈ [0, T ] [9, Theorem 3.5]. Moreover, the semiclassical evolution of the packet is of Schrödinger type: Define the truncated, time dependent quadratic Hamiltonian 2 ε a(t) := − ε + V0,2 x, a(t) ε (t) := H H 2
(20)
with V0,2 the quadratic approximation of V at a(t), 2 α 1 α V0,2 x, a(t) := D V a(t) x − a(t) , α! |α|=0
ε (t, s) be the unitary propagator generated by H ε , i.e. with and let U d ε i ε (s). U (t, s) = − H dt ε t=s
(21)
ε (t, s)Φ ε (x, s) Φkε (x, t) = U k
(22)
Then
for any t, s ∈ R [9, Theorem 3.4]. 3. Bohmian trajectories of Hagedorn wave packets 2
Theorem. Let H ε = − ε2 + V (x), D(H ε ) ⊂ L2 (R3 ) with V ∈ GV . For k ∈ N3 let ψkε (x, t) be given by (18), Xε (x 0 , t) by (4) and a(t) by (7). Then: ε
(i) For all ε > 0 the Bohmian trajectories Xε (x 0 , t) exist globally in time for Pψk (·,0) -almost all initial positions x 0 ∈ R3 . (ii) For all T > 0 and all γ > 0 there exist some R < ∞ and some ε0 > 0 such that ε
Pψk (·,0) for all 0 < ε ε0 .
√
x 0 ∈ R3 max Xε (x 0 , t) − a(t) R ε > 1 − γ t∈[0,T ]
(23)
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For the proof we shall use that the probability that a Bohmian trajectory crosses a certain surface (here the moving sphere SR √ε (a(t))) is bounded by the quantum probability flux ε ε j ψk = v ψk |ψkε |2 = Im[(ψkε )∗ ∇ψkε ] across this surface (Section 5.1). For that we need pointwise estimates on the quantum probability current density, i.e. on ψkε and ∇ψkε . 2
Lemma 1. Let H ε = − ε2 + V (x), D(H ε ) ⊂ L2 (R3 ) with V ∈ GV . For k ∈ N3 let Φkε (x, t) be given by (17) and ψkε (x, t) by (18). Then for all T > 0 there exists some C < ∞ such that 1 max ψkε (·, t) − Φkε (·, t) ∞ Cε − 4
(24)
5 max ∇ψkε (·, t) − ∇Φkε (·, t) ∞ Cε − 4
(25)
t∈[0,T ]
and t∈[0,T ]
where · ∞ = supx∈R3 | · |. (15)
3
(16) 1 ε ε Φk ∞ +
For the proof see Section 5.2. Note that, since Φkε ∞ ∼ ε − 4 resp. ∇Φkε ∞ ∼
5
7
ε√− 4 ∼ ε − 4 , the relative value of the differences ψkε − Φkε ∞ resp. ∇(ψkε − Φkε ) ∞ is of order ε each. 4. What about velocities? The theorem above is a result about a particle’s typical Bohmian position as a function of time. To extend this to velocities, i.e. to show that also ε
Pψk (·,0)
ε √
x 0 ∈ R3 max v ψk Xε (x 0 , t), t − η(t) K ε > 1 − γ t∈[0,T ]
(26)
for some K < ∞ and all ε small enough, one needs to control the probability that the Bohmian ε ∇ψ ε trajectory comes too close to the wave function’s nodes where the velocity field v ψk = ε Im( ψ εk ) k is ill defined. More precisely, since by (16) and Lemma 1 ε∇ψkε − iηψkε ψε v k − η = Im ψkε
√ |ε∇Φkε − iηΦkε | + ε|∇ψkε − ∇Φkε | + η|ψkε − Φkε | ε ∼ 3 , ε |ψk | ε 4 |ψkε |
one needs that there exists some δT ,k (γ ) > 0 such that 3 ε Pψk (·,0) x 0 ∈ R3 ψkε Xε (x 0 , t), t > ε − 4 δT ,k (γ ) for all t ∈ [0, T ] > 1 − γ ε
(27)
for all ε small enough. From the Pψk (·,0) -almost sure global existence of Bohmian mechan-
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
2411
ics [3,13] one has that for all ε > 0 there is some δkε (γ ) > 0 such that ε Pψk (·,0) x 0 ∈ R3 ψkε Xε (x 0 , t), t > δkε (γ ) for all t ∈ R > 1 − γ .
(28)
However, we need more, namely the ε-dependence of δkε (γ ). This may be achieved by scrutinizing the existence proof, in particular the proof of (28) in [3]. We shall not do so here. Instead, we note that for the ground state k = 0 (26) is an easy corollary of our theorem and Lemma 1. This is due to the fact that Φ0ε is just a Gaussian and thus does not possess any nodes. Similarly, also WKB-wave functions do not possess nodes. See [11] for an assertion concerning Bohmian velocities in that case. For the Gaussian ground state (9) (24) ε √ ψ (x, t) Φ ε (x, t) − Cε − 14 Cε − 34 e−CR − ε Cε − 34 0 0
and thus ψε √ v 0 (x, t) − η(t) C ε whenever |x − a(t)|
√ εR and ε small enough. So our theorem gives
Corollary 1. Under the same assumptions as in the theorem for all T > 0 and all γ > 0 there exist some R < ∞, K < ∞ and some ε0 > 0 such that ε
Pψ0 (·,0)
√ x 0 ∈ R3 max Xε (x 0 , t) − a(t) R ε t∈[0,T ]
ε √
∧ max v ψ0 Xε (x 0 , t), t − η(t) K ε > 1 − γ t∈[0,T ]
(29)
for all 0 < ε ε0 . A weaker statement which is true for any ψkε is the following. Since a typical Bohmian trajectory may not deviate too much from its corresponding classical one, the time averaged values of the velocities must be close: For any macroscopic time interval 0 < δt T2 define the timeaveraged Bohmian and classical velocities (t ∈ [δt, T − δt]) ψε
v δtk (x 0 , t) :=
1 2δt
t+δt
ε v ψk Xε (x 0 , s), s ds,
t−δt
1 ηδt (t) := 2δt
t+δt
η(s) ds. t−δt
√ Now suppose x 0 ∈ R3 is such that maxt∈[0,T ] |X ε (x 0 , t) − a(t)| R ε. Then
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t+δt ε ψkε ψk ε v (x 0 , t) − ηδt (t) = 1 v X (x 0 , s), s − η(s) ds δt 2δt t−δt
1 ε X (x 0 , t + δt) − a(t + δt) + Xε (x 0 , t − δt) − a(t − δt) 2δt R√ ε. δt So our theorem gives Corollary 2. Under the same assumptions as in the theorem for all T > 0, γ > 0 there exist some R < ∞ and some ε0 > 0 such that for any 0 < δt T2 ε
Pψk (·,0)
x 0 ∈ R3
ψkε √ v (x 0 , t) − ηδt (t) R ε δt t∈[δt,T −δt] δt max
>1−γ
(30)
for all 0 < ε ε0 . We conclude with a note on the Hamilton–Jacobi form of Bohmian mechanics. Setting i
ψ ε (x, t) = R ε (x, t)e ε S
ε (x,t)
,
the real part of Schrödinger’s equation (3) gives ∂t S ε (x, t) + V (x) −
2 ε 2 R ε (x, t) 1 ε + ∇S (x, t) = 0 ε 2 R (x, t) 2
while (4) reads d ε X (x 0 , t) = ∇S ε Xε (x 0 , t), t . dt Except for the additional “quantum potential” VQε := − ε2 R R ε these are the classical Hamilton– Jacobi equations. This suggests very directly that Bohmian particles behave classically whenever VQε is negligible [4,10,11]. However, due to the occurrence of R1ε a proof along these lines must deal with the nodes problem we discussed above. 2
ε
5. Proof 5.1. Proof of the theorem (i) is a direct consequence of Corollary 3.2 in [3] resp. of Corollary 4 in [13] if we can show that the initial wave function ψkε (·, 0) = Φkε (·, 0) is a C ∞ -vector of H ε , Φkε (·, 0) ∈ C ∞ (H ε ) = ∞ ε ε n 3 ε n=1 D((H ) ). This is the case, since Φk (·, 0) ∈ S(R ) and V ∈ GV guarantees that H maps 3 3 ∞ ε the Schwartz space S(R ) into itself and thus that S(R ) ⊂ C (H ).
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
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(ii) Let γ > 0. For ε > 0 and R < ∞ define √ GεR := x 0 ∈ R3 max Xε (x 0 , t) − a(t) < R ε . t∈[0,T ]
ε
Our task is to show that, for suitable R and ε, Pψk (·,0) ((GεR )c ) < γ . For this we split off the probability that a trajectory already starts too far off from the classical one: c ε Pψk (·,0) GεR √ ε Pψk (·,0) x 0 ∈ R3 x 0 − a(0) R ε √ √ ε + Pψk (·,0) x 0 ∈ R3 x 0 − a(0) < R ε ∧ ∃t ∈ (0, T ]: Xε (x 0 , t) − a(t) R ε c ε ε (31) =: Pψk (·,0) BR √ε a(0) + Pψk (·,0) MTR √ε . Then ψkε (·,0)
P
c BR √ε a(0) =
√ |x−a(0)|R ε
(18)
=
ε ψ (x, 0)2 d 3 x k
ϕk A(0), B(0), ε, a(0), η(0), x 2 d 3 x
√ |x−a(0)|R ε (15)
=
ϕk A(0), B(0), 1, 0, 0, y 2 d 3 y,
yR √ . Since ϕk (A(t), B(t), 1, 0, 0, ·) is square where in the last step we substituted y = x−a(0) ε summable (in fact it is normalized) there is some R > 0 independent of ε such that
c γ ε Pψk (·,0) BR √ε a(0) 2
(32)
for all R > R . Since Xε (x 0 , t) (as a solution of (2)) is continuous in t, x0 ∈ MTR √ε implies that Xε (x 0 , t) crosses the moving sphere SR √ε (a(t)) at least once and outwards in (0, T ]. Therefore ε Pψk (·,0) (MTR √ε ) is bounded from above by the probability that some trajectory crosses SR √ε (a(t)) in any direction in (0, T ]. In Section 2.3.2 of [2] Berndl invoked the probabilistic meaning of the quantum probability current density J ψ := (j ψ , |ψ|2 ) with j ψ := ε Im(ψ ∗ ∇ψ) to prove that the expected number of crossings2 through a smooth surface Σ in configurationspace–time by the random configuration-space–time trajectory (X ε (·, t), t) is given by the modulus of the flux across this surface, 2 This also includes tangential “crossings” in which the trajectory remains on the same side of Σ .
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ψ J (x, t) · U dσ,
Σ
where U denotes the local unit normal vector at (x, t) (see also the argument given in [3, p. 11]). Since any trajectory (Xε (x 0 , t), t) will cross Σ an integral number of times (including 0 and ∞) this expected value gives an upper bound for the probability that (Xε (x 0 , t), t) crosses Σ . So in our case we obtain ε ε (33) Pψk (·,0) MTR √ε J ψk (x, t) · U dσ ΣTε
where ΣTε = (x, t) t ∈ [0, T ], x ∈ SR √ε a(t) and, using spatial polar coordinates centered at a(t), U = √ 1 ( er , − η(t), er ) and 1+ η(t), er 2 dσ = 1 + η(t), er 2 εR 2 dΩ dt. Here er = (cos ϕ sin θ, sin ϕ sin θ, cos θ ) and dΩ = sin θ dϕ dθ . Thus ψε J k (x, t) · U dσ = j ψkε (x, t) − ψ ε (x, t)2 η(t), er εR 2 dΩ k ε 2 j ψk (x, t)−ψkε (x, t) η(t)εR 2 dΩ
(34)
ε
where j ψk (x, t) − |ψkε (x, t)|2 η(t) is evaluated at points (x, t) ∈ ΣTε . By the definition of j ψ and since η(t) is always real ψ ε ε 2 j k −ψ η(t) = Im ψ ε ∗ ε∇ψ ε − iη(t)ψ ε ψ ε ε∇ψ ε − iη(t)ψ ε k k k k k k k Φkε + ψkε − Φkε × ε ∇ψ ε − ∇Φ ε + η(t)ψ ε − Φ ε + ε∇Φ ε − iη(t)Φ ε . k
k
k
k
k
k
Then by (15), (16) and Lemma 1 ψε a(t)| k − 12 C( |x−a(t)| √ )2 − 14 ε j k (x, t) − ψ ε (x, t)2 η(t) Cε − 34 1 + |x − e + Cε √ k ε 1 1 |x − a(t)| k+1 − 12 C( |x−a(t)| √ )2 ε e × Cε − 4 + Cε − 4 1 + √ ε −1 1 1 2 C ε (1 + R)2k+1 e− 2 CR + ε − 2 where we have used that η(t) is continuous and thus bounded on [0, T ] and that (x, t) ∈ ΣTε √ entails |x−a(t)| = R. Plugging this into (34), we see that ε ψε √ J k (x, t) · U dσ C (1 + R)2k+1 e− 12 CR 2 + ε R 2 dΩ.
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
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Thus by (33) ψkε (·,0)
P
T MR √ε
T
2π dt
0
π
√ 1 2 dθ sin(θ )CR 2 (1 + R)2k+1 e− 2 CR + ε
dϕ 0
0
√ γ 1 2 2πT C R 2 (1 + R)2k+1 e− 2 CR + R 2 ε < 2
(35)
for R big and ε small enough. Together (32) and (35) give the desired result: c ε ε Pψk (·,0) GεR = 1 − Pψk (·,0) GεR > 1 − γ for all R big and all ε small enough. 5.2. Proof of Lemma 1 In view of (6) and (21) we have that for all t ∈ [0, T ] ε (t, 0) Φ ε (x, 0) ψkε (x, t) − Φkε (x, t) = U ε (t) − U k i =− ε
t
U ε (t − s)V3 x, a(s) Φkε (x, s) ds,
(36)
0
ε is the third order remainder term of the potential’s Taylor expansion where V3 = H ε − H about a. A priori, equality in (36) holds in the sense of L2 -functions, i.e. for almost every x ∈ R3 , only. In the course of our proof (Lemma 2 below) we shall however see that U ε V3 Φkε is continuously differentiable with respect to x and that U ε V3 Φkε and ∇U ε V3 Φkε are bounded for all s, t ∈ [0, T ] and x ∈ R3 . So by dominated convergence also ψkε − Φkε (and thus ψkε ) is continuously differentiable with t i ε ε ∇ψk (x, t) − ∇Φk (x, t) = − ∇ U ε (t − s)V3 x, a(s) Φkε (x, s) ds ε 0
=−
i ε
t
∇U ε (t − s)V3 x, a(s) Φkε (x, s) ds.
0
Moreover, by continuity (36) and (37) hold in fact pointwise for all x ∈ R3 . Our control on (∇)U ε V3 Φkε is given in Lemma 2. Let V ∈ GV , k ∈ N3 and T > 0. For m ∈ N let Vm (x, a) := V (x) −
m−1 |α|=0
1 α D V (a)(x − a)α α!
denote the mth remainder term of the Taylor expansion of V about a.
(37)
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Then U ε (t − s)Vm (·, a(s))Φkε (·, s) is continuously differentiable for all s, t ∈ [0, T ] and there exists some C < ∞ such that m 3 max U ε (t − s)Vm ·, a(s) Φkε (·, s) ∞ Cε 2 − 4
(38)
m 7 max ∇U ε (t − s)Vm ·, a(s) Φkε (·, s) ∞ Cε 2 − 4 .
(39)
s,t∈[0,T ]
and s,t∈[0,T ]
Then, plugging (38) and (39) into (36) and (37) immediately yields Lemma 1, i.e. T max ψkε (·, t) − Φkε (·, t) ∞ t∈[0,T ] ε
1 max U ε (t − s)V3 ·, a(s) Φkε (·, s) ∞ CT ε − 4
s,t∈[0,T ]
and T max ∇ψkε (·, t) − ∇Φkε (·, t) ∞ t∈[0,T ] ε
max ∇U ε (t − s)V3 ·, a(s) Φkε (·, s) ∞
s,t∈[0,T ] 5
CT ε − 4 . ε (x, t, s):=U ε (t −s)V (x, a(s))Φ ε (x, s) Proof of Lemma 2. First we fix some notation. Let gm,k m k i
ε (x, t, s) := e− ε η(t),x−a(t) g ε (x, t, s). and gm,k m,k We shall use an instance of Gagliardo–Nirenberg’s inequality [7,12]: For every n ∈ N and l > n l,2 n 2 n α 2 there is some C < ∞ such that for every f ∈ W (R ) = {f ∈ L (R ) | max|α|l D f 2 < ∞}
2ln 1− n
f ∞ C max D α f 2 f 2 2l .
(40)
|α|=l
Moreover, f ∈ C r (Rn ) for all 0 r < l − n2 . ε gives Applying (40) with n = 3 and l = 2 to gm,k 3 1 ε g (·, t, s) C max D α g ε (·, t, s) 4 g ε (·, t, s) 4 m,k m,k m,k 2 ∞ 2
(41)
|α|=2
m−|α|
ε = ε −|α| p α U ε V Φ ε ∼ ε 2 for all t, s ∈ [0, T ]. Thus we get (38) if D α gm,k for all 2 m k 2 3 α ∈ N with |α| ∈ {0, 2}. Unfortunately the latter is generally false. This is due to the fact that in order to have pΦkε = −iε∇Φkε ∼ ηΦkε (i.e. part two of (14)), the Φkε ’s must possess an appropriate, fast varying phase √ acts on Φ ε as a combination of factor. Indeed, since, roughly, Vm (x, a) ∼ (x − a)m and x−a k ε lowering and raising operators (cf. (11)), m
Vm Φkε ∼ ε 2
|k −k|m
Φkε
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423 m
m
2417 |α|
m
and thus even D α Vm Φkε 2 ∼ ε 2 −|α| p α Φkε 2 ∼ ε 2 −|α| ηα Φkε 2 ∼ ε 2 −|α| is of order ε − 2 worse than what we need. To account for this, we substract the problematical phase, that is we use Gagliardo–Nirenberg i ε ε = U ε Vm Φkε itself but on gm,k = e− ε η,x−a U ε V3 Φkε . Then instead of (41) we get not on gm,k ε 34 ε 14 ε ε C max D α g (·, t, s) = g g (·, t, s) g (·, t, s) (·, t, s) m,k m,k m,k m,k 2 2 ∞ ∞ |α|=2
with the higher order terms α ε α ε D gm,k (·, t, s) 2 = ε −|α| p − η(t) gm,k (·, t, s) 2 . ε ∼ε So (38) holds if (p − η)α gm,k 2
m+|α| 2
for all α ∈ N3 with |α| ∈ {0, 2}. Analogously one sees
m+|α| 2
ε ∼ε that (39) holds if (p − η)α gm,k for all α ∈ N3 with |α| 3. However, that these 2 ε α estimates for (p − η) gm,k hold true is the content of Lemma 3 below. 2
Remark 1. Instead of the Gagliardo–Nirenberg inequality (40) one could also use canonical Sobolev inequalities. However, then one gets results that are not of optimal order in ε, that is instead of Lemma 2 one only gets ε U Vm Φ ε
k ∞
C
2
2 α ε −|α| p − η(t) U ε Vm Φ ε
1
k 2
|α|=0
2
2 −1 Cε m
and ∇U ε Vm Φ ε k
C ∞
3
2 α ε −|α| p − η(t) U ε Vm Φ ε
|α|=0
k 2
1 2
2 −2 . Cε m
Note that also these weaker results suffice to get convergence to classical behavior in the sense of our theorem – but with a lower rate of convergence. More precisely, instead of (23) one gets
1 ε Pψk (·,0) x ∈ R3 max Xε (x 0 , t) − a(t) Rε 4 > 1 − γ . t∈[0,T ]
Lemma 3. Let V ∈ GV . For every T > 0, m ∈ N and k ∈ N3 there exists some C < ∞ such that α m+|α| max p − η(t) U ε (t − s)Vm ·, a(s) Φkε (·, s) 2 Cε 2
s,t∈[0,T ]
(42)
for all multi-indices 0 |α| 3. Remark 2. Since ψkε (x, t) = U ε (t)Φkε (x, 0), by replacing Vm (x, a(s))Φkε (x, s) with Φkε (x, s) in the proof of Lemma 3 and setting s = 0 one can easily show that also α |α| max p − η(t) ψkε (·, t) 2 Cε 2
t∈[0,T ]
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D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
for some C < ∞ and all 0 |α| 3. So we have, for example, that regarding momentum not √ only the Φkε (x, t)’s but also the ψkε (x, t)’s standard deviation is of order ε. Since the momen√ tum operator p is unbounded this is not a consequence of Hagedorn’s results ψkε − Φkε 2 ∼ ε and (p − η)α Φkε 2 ∼ ε
|α| 2
[8,9].
Proof of Lemma 3. We expand the notation of Lemma 2: For any l m ∈ N let Vl,m (x, a) := Vl (x, a) − Vm+1 (x, a) =
m 1 α D V (a)(x − a)α α!
|α|=l
and ε (x, s) := Vm x, a(s) Φkε (x, s) fm,k ε ε gm,k (x, t, s) = U ε (t − s)fm,k (x, s)
ε resp. f(m,l),k (x, s) := Vm,l x, a(s) Φkε (x, s), ε ε resp. g(m,l),k (x, t, s) := U ε (t − s)f(m,l),k (x, s).
In the following we set · = · 2 . We shall first prove the weaker result (|α| 3) α ε m max p − η(t) gm,k (·, t, s) Cε 2
s,t∈[0,T ]
(43)
and then use a bootstrapping argument to arrive at (42). Since η(t) is bounded on [0, T ], instead of (43) it suffices to prove that m ε max p α gm,k (·, t, s) Cε 2
s,t∈[0,T ]
(44)
for some C < ∞ and all |α| 3. For that we first get rid of the (unitary) time evolution U ε , i.e. ε in terms of f ε , H ε f ε and (H ε )2 f ε . We then mimic the we shall express p α gm,k m,k m,k m,k proof of (2.38) in [9] to find estimates for the latter. Since U ε is unitary ε g (·, t, s) = f ε (·, s) . m,k m,k
(45)
Since p = −iε∇ is self-adjoint, by Schwarz’s inequality and (45)
ε 1 ε ε max pα gm,k (·, t, s) = max gm,k (·, t, s), pj2 gm,k (·, t, s) 2
|α|=1
j
1 ε ε fm,k (·, s) p 2 gm,k (·, t, s) 2 , ε ε max p α gm,k (·, t, s) p 2 gm,k (·, t, s)
|α|=2
and 1 ε ε ε max pα gm,k (·, t, s) p 2 gm,k (·, t, s) p 4 gm,k (·, t, s) 2 .
|α|=3
(46)
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
2419 m
ε , p 2 g ε and p 4 g ε are of order ε 2 . Write Thus we get (44) if we can show that fm,k m,k m,k p 2 = 2(H ε − V ). Since [H ε , U ε ] = 0 and V is bounded by CV (cf. Definition 1),
2 ε p g (·, t, s) = 2 H ε − V g ε (·, t, s) m,k m,k ε ε 2 H ε U ε (t − s)fm,k (·, s) + V ∞ gm,k (·, t, s) ε ε 2 H ε fm,k (·, s) + CV fm,k (·, s) .
(47)
In the same way 4 ε p g (·, t, s) = 4 H ε − V 2 g ε (·, t, s) m,k m,k 2 ε ε 4 H ε fm,k (·, s) + 2 V ∞ H ε fm,k (·, s) ε ε + V 2∞ fm,k (·, s) + H ε , V gm,k (·, t, s) 2 ε ε ε 4 H ε fm,k (·, s) + 2CV H ε fm,k (·, s) + CV2 fm,k (·, s) ε2 ε ε + ε ∇V , p gm,k (·, t, s) + V ∞ fm,k (·, s) . 2 Since V ∈ GV implies that also ∇V and V are bounded by CV , this yields 2 (46) 4 ε p g (·, t, s) 4 H ε 2 f ε (·, s) + 2CV H ε f ε (·, s) + CV CV + ε f ε (·, s) m,k m,k m,k m,k 2 1 ε ε + 3εCV fm,k (·, s) p 2 gm,k (·, t, s) 2 ε ε ε 2 ε ε2 f ε (·, s) 4 H fm,k (·, s) + 2CV H fm,k (·, s) + CV CV + m,k 2 √ 1 1 ε ε ε + 3 2εCV fm,k (·, s) 2 H ε fm,k (·, s) + CV fm,k (·, s) 2 .
(47)
m
ε , H ε f ε and (H ε )2 f ε are of order ε 2 . Thus we get (44) if we can show that fm,k m,k m,k We mimic the proof of (2.38) in [9] and introduce the following splitting (R > 0):
ε f (·, s) 2 = m,k
Vm x, a(s) Φ ε (x, s)2 d 3 x +
k
Vm x, a(s) Φ ε (x, s)2 d 3 x k
|x−a(s)|>R
|x−a(s)|R
=: I + II. Remember that Vm is the remainder Vm (x, a) = V (x) −
m−1 |α|=0
1 1 α D V (a)(x − a)α = Dα V ξ (x, a) (x − a)α α! α! |α|=m
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D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
where ξ (x, a) = a + λ(x − a) for some λ ∈ (0, 1). Thus Vm x, a(s) ⎧ "m−1 " ⎨ V + max α ∞ |α|m−1 |(D V )(a(s))| |α|=l l=0 |x−a(s)|m ⎩ max α m |(D V )(ξ )|3 ξ a(s)+R
l! |x−a(s)|l α! l!
for x ∈ R3 , for |x − a(s)| R
m!
and, since a(s) is continuous in s and V ∈ GV is bounded and C ∞ , there exists some C < ∞ such that Vm x, a(s)
#
" (3|x−a(s)|)l C m−1 Ce3|x−a(s)| l=0 l! C|x − a(s)|m
for all s ∈ [0, T ]. Substituting y :=
x−a(s) √ , ε
for x ∈ R3 , for |x − a(s)| R
with this and (15) we get for all ε small enough
R √ ε
IC
ε m y 2m (1 + y)2k e−Cy dy Cε m 2
0
and ∞ √ − √C II C (1 + y)2k e−y(Cy−6 ε ) dy Ce ε . R √ ε
So ε m (·, s) = O ε 2 . max fm,k
(48)
s∈[0,T ] ε write To estimate H ε fm,k
ε = Vm H ε Φkε + H ε , Vm Φkε . H ε fm,k With Ecl = 12 η2 + V (a) and [H ε , Vm ] = −iε ∇Vm , p −
ε2 2 (Vm )
this gives
ε2 ε H ε fm,k = Ecl Vm Φkε + Vm H ε − Ecl Φkε − iε ∇Vm , p Φkε − (Vm )Φkε 2 1 = Ecl Vm Φkε + Vm p 2 − η2 Φkε + Vm V (x) − V (a) Φkε 2 − iε ∇Vm , η Φkε − iε ∇Vm , p − η Φkε −
ε2 (Vm )Φkε 2
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
2421
1 = Ecl Vm Φkε + Vm (p − η)2 Φkε + Vm η, p − η Φkε + Vm V1 Φkε 2 − iε ∇Vm , η Φkε − iε ∇Vm , p − η Φkε −
ε2 (Vm )Φkε . 2
√ Now, by (12) we see that (p − η)Φkε is ε times a (vector of) linear combination(s) of Φkε ’s with |k − k | = 1 and (p − η)2 Φkε is ε times a linear combination of Φkε ’s with |k − k | ∈ {0, 2}. Thus ε is a sum of terms of the form H ε fm,k ε ε ε C(η)fm,k := C(η)Vm Φk
mε is a wild card for Vm , where C(η) is either a constant or some function of η, |k − k | 2 and V √ 3 2 2 εVm , εVm , Vm V1 , ε(∂j Vm ), ε 2 (∂j Vm ) or ε (∂j Vm ) (j = 1, 2, 3). Note that Dα Vm = Dα V m−|α| , mε is either Vm V1 or of the form ε 2l V is a wild m−r where the new ε-independent “potential” V so V 2 ∈ C∞ card for V , ∂j V or ∂j V and l, r ∈ N are such that l − r 0. Now, since V ∈ GV implies V m+1 ∞ max|α|2 Dα V ∞ CV , not only the proof of Vm V1 Φ ε = O(ε 2 ) but also and V k
m−r Φ ε = O(ε m−r ε is 2 ) is completely analogous to that of (48). Therefore, f that of V k m,k either of order ε
m+1 2
mε = Vm V1 ) or of order ε (if V
m+l−r 2
mε = ε 2 V m−r ), that is we get ε 2 (if V m
l
ε m ε max C(η) fm,k (·, s) max H ε fm,k (·, s) = O ε 2 .
s∈[0,T ]
s∈[0,T ]
(49)
m
ε = O(ε 2 ) clearly follows if we can show that, for each of the above Finally, (H ε )2 fm,k m fε , H ε fε is (at least) of order ε 2 . The proof of the latter, however, is completely analm,k
mk
ogous to that of (49). Just note that this time we get up to fourth order derivatives of V as , which is why in the definition of GV we required that Dα V ∞ CV for new “potentials” V |α| 4. So we have shown that (44) and thus also (43) holds. To get (42) we split off the lowest order term of Vm , Vm = Vm,m + Vm+1 (cf. notation at the beginning of this proof). Then by (43) α ε p − η(t) α g ε (·, t, s) p − η(t) α g ε (m,m),k (·, t, s) + p − η(t) gm+1,k (·, t, s) m,k α ε m+1 p − η(t) g(m,m),k (·, t, s) + Cε 2 . (50) ε To estimate (p − η)α g(m,m),k note that
ε (x, t, s) = U ε (t − s)Vm,m x, a(s) Φkε (x, s) g(m,m),k
1 x − a(s) β ε β D V a(s) = ε U (t − s) Φk (x, s) √ β! ε m 2
ε
|β|=m
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D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
√ )β Φ ε is a finite sum of Φ ε s with |k − k | m and coefficients that are independent and that ( x−a k k ε
of ε and bounded on [0, T ] ((11) and A(s) continuous in s). Since also (Dβ V )(a(s)) is bounded on [0, T ] (V ∈ C ∞ (R3 ) and a(s) continuous in s) it thus suffices to estimate α m ε 2 p − η(t) U ε (t − s)Φkε (x, s) for |k − k | m. Like in (36) ε
U (t
− s)Φkε (x, s) = Φkε (x, t) −
i ε
t
U ε (t − τ )V3 x, a(τ ) Φkε (x, τ ) dτ
s
= Φkε (x, t) −
i ε
t ε g3,k (x, t, τ ) dτ. s 3
ε (·, t, τ ) < Cε 2 , changing the order of differentiation (p = Since by (43) (p − η(t))α g3,k −iε∇) and integration in
t t α α ε ε p − η(t) g3,k (·, t, τ ) dτ g3,k (·, t, τ ) dτ = p − η(t) s
s
is justified by dominated convergence and we thus get (for any s, t ∈ [0, T ]) α α m m ε 2 p − η(t) U ε (t − s)Φkε (·, s) ε 2 p − η(t) Φkε (·, t) m 2 −1
+ε
t
p − η(t) α g ε (·, t, τ ) dτ 3,k
s
α m+|α| p − η(t) m+1 ε ε 2 + ε 2 CT . Φk (·, t) √ ε By (14) this yields m+|α| α m m+1 ε 2 p − η(t) U ε (t − s)Φkε (·, s) C ε 2 + ε 2 and thus also p − η(t) α g ε
m+|α| C ε 2 + ε m+1 2 .
(m,m),k (·, t, s)
Putting this into (50) we see that we can sharpen (43) to m+|α| α ε m+1 m+1 max p − η(t) gm,k (·, t, s) C ε 2 + ε 2 Cε 2 .
s,t∈[0,T ]
Iterating this bootstrapping argument several times we finally arrive at
D. Dürr, S. Römer / Journal of Functional Analysis 259 (2010) 2404–2423
2423
m+|α| α ε m+|α| max p − η(t) gm,k (·, t, s) C ε 2 + ε 2 ,
s,t∈[0,T ]
i.e. at (42).
2
References [1] V. Allori, D. Dürr, S. Goldstein, N. Zanghì, Seven steps towards the classical world, J. Opt. B 4 (2002) 482–488, arXiv:quant-ph/0112005. [2] Karin Berndl, Zur Existenz der Dynamik in Bohmschen Systemen, PhD thesis, Ludwig-Maximilians-Universität München, 1994. [3] K. Berndl, D. Dürr, S. Goldstein, G. Peruzzi, N. Zanghì, On the global existence of Bohmian mechanics, Comm. Math. Phys. 173 (3) (1995) 647–673. [4] David Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variables I, II, Phys. Rev. 85 (1952) 166–179, 180–193. [5] Detlef Dürr, Stefan Teufel, Bohmian Mechanics, Springer, Berlin, 2009; revised translation of D. Dürr, Bohmsche Mechanik als Grundlage der Quantenmechanik, Springer, Berlin, 2001. [6] Detlef Dürr, Sheldon Goldstein, Nino Zanghì, Quantum equilibrium and the origin of absolute uncertainty, J. Stat. Phys. 67 (1992) 843–907. [7] Emilio Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat. 8 (1959) 24–51. [8] George A. Hagedorn, Semiclassical quantum mechanics. IV. Large order asymptotics and more general states in more than one dimension, Ann. Inst. H. Poincaré Phys. Théor. 42 (4) (1985) 363–374. [9] George A. Hagedorn, Raising and lowering operators for semiclassical wave packets, Ann. Physics 269 (1998) 77–104. [10] Peter R. Holland, The Quantum Theory of Motion, Cambridge University Press, Cambridge, 1995. [11] Peter Markowich, Thierry Paul, Christof Sparber, Bohmian measures and their classical limit, J. Funct. Anal. 259 (6) (2010) 1542–1576. [12] L. Nirenberg, On elliptic partial differential equations: Lecture II, Ann. Sc. Norm. Super. Pisa (3) 13 (1959) 115– 162. [13] Stefan Teufel, Roderich Tumulka, A simple proof for global existence of Bohmian trajectories, Comm. Math. Phys. 258 (2) (2005) 349–365.
Journal of Functional Analysis 259 (2010) 2424–2436 www.elsevier.com/locate/jfa
Weighted interpolation in Paley–Wiener spaces and finite-time controllability ✩ Birgit Jacob a,∗,1 , Jonathan R. Partington b , Sandra Pott c a Department of Mathematics, University of Wuppertal, Gaußstraße 20, D-42119 Wuppertal, Germany b School of Mathematics, University of Leeds, Leeds LS2 9JT, UK c Mathematics, Faculty of Science, Lunds Universitet, Postbox 118, 22100 Lund, Sweden
Received 31 March 2010; accepted 22 June 2010 Available online 21 July 2010 Communicated by J. Coron
Abstract This paper considers the solution of weighted interpolation problems in model subspaces of the Hardy space H 2 that are canonically isometric to Paley–Wiener spaces of analytic functions. A new necessary and sufficient condition is given on the set of interpolation points which guarantees that a solution in H 2 can be transferred to a solution in a model space. The techniques used rely on the reproducing kernel thesis for Hankel operators, which is given here with an explicit constant. One of the applications of this work is to the finite-time controllability of diagonal systems specified by a C0 semigroup. © 2010 Elsevier Inc. All rights reserved. Keywords: Interpolation; Model space; Paley–Wiener space; Controllability; Observability; C0 semigroup; Hankel operator; Carleson measure
1. Introduction and notation In this paper we treat two closely-linked themes, one from operator theory and complex analysis, the other from control theory. ✩
The first and third authors were supported by the ESF.
* Corresponding author.
E-mail addresses:
[email protected] (B. Jacob),
[email protected] (J.R. Partington),
[email protected] (S. Pott). 1 The third author was supported by a Heisenberg fellowship of the German Research Foundation (DFG). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.014
B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
2425
The more abstract questions are to do with the solution of interpolation problems in functional model spaces of the form Kθτ := H 2 (C+ ) θτ H 2 (C+ ), where C+ denotes the right-hand halfplane, H 2 (C+ ) is the Hardy space, τ > 0 and θτ is the inner function θτ (s) = e−sτ for s ∈ C+ . By means of the inverse Laplace transform, the space Kθτ is (up to a constant) isometrically isomorphic to the space L2 (0, τ ), and can thus be regarded as a Paley–Wiener space (see, for example, [11,13]). Specifically, a theorem due to McPhail [8] solves the following problem: given a complex nonzero sequence (bn ) and a sequence (λn ) of points in C+ , when does the equation bn f (λn ) = an have a solution f ∈ H 2 for all sequences (an ) ∈ 2 ? The solution is based on the Shapiro–Shields theory of interpolation [20] (see also [9]). The solution to this problem was applied in [5] to provide necessary and sufficient conditions for the exact (infinite-time) controllability of certain ‘diagonal’ linear systems with scalar inputs: we shall discuss this further in Section 2, but we remark here that there is a duality between controllability and observability (see, for example, [21]), which means that results can be stated in either context. In [6], a vectorial version of the interpolation theorem was derived, and applied to analyze controllability with multiple inputs. If we take the same interpolation approach to controllability in finite time τ > 0, then we are led to look at questions of weighted interpolation in Kθτ . More precisely, for (an ) ∈ 2 , we look for solutions to bn f (λn ) = an with f ∈ Kθτ . This is a significantly more difficult question than the corresponding interpolation question in H 2 . For an overview on interpolation results in Paley–Wiener space, see [19] and the references therein. In particular, the existing results in this area (cf. [11]) employ the additional hypothesis that the sequence (λn ) forms a Carleson sequence in C+ . In this case, it is possible to transfer interpolation results from H 2 to Kθτ . However, as we shall show in Section 3, a much weaker hypothesis on the (λn ) is both necessary and sufficient for this transference to be possible for some finite value of τ . This will be expressed in terms of the properties of certain Toeplitz and Hankel operators. An explicit bound on the size of τ is provided: to do this, we derive a definite constant for the so-called ‘reproducing kernel thesis’, which says that boundedness of Hankel operators can be tested on normalized reproducing kernels (cf. [1,11,14]). We conclude with the main application: a result asserting that infinite-time controllability implies finite-time controllability, with an explicit bound on τ depending only on the eigenvalue sequence (λn ). Throughout this article we will use the following notation. For a Blaschke sequence (λn )n1 Re λn in the right-hand half-plane C+ , that is, one satisfying ∞ n=1 1+|λn |2 < ∞, let β denote the infinite Blaschke product with zeroes (λn )n1 . More generally, given N ∈ N, for a Blaschke sequence (λn ) and a sequence of proper subspaces In ⊂ CN , we consider the Blaschke– ∞ n Potapov product β(z) = n=1 PIn ⊕ z−λ P ⊥ , where PIn : CN → In is the orthogonal projecz+λn In tion. For τ > 0, the inner function θτ ∈ H ∞ (C+ ) is defined by θτ (s) = e−τ s . For an inner function φ (which for us may be β or θτ ) we write Kφ = H 2 (C+ ) φH 2 (C+ ). We write P± : L2 (iR) → H 2 (C± ) for the standard orthogonal projections, and note that PKφ f = φP− (φf ) for f ∈ H 2 (C+ ). The reproducing kernel functions for H 2 (C+ ) are denoted by kλ , λ ∈ C+ , where kλ (s) = 1 1 1 2 2 2π s+λ for s ∈ C+ , and satisfy f (λ) = g, kλ for f ∈ H (C+ ). Note that kλ = 4π Re λ .
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B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
2. Controllability We consider systems of the form x(t) ˙ = Ax(t) + Bu(t),
t 0,
x(0) = x0 .
(1)
We suppose that A is the generator of an exponentially stable C0 -semigroup (T (t))t0 on a Hilbert space H and that the eigenvectors of −A form an orthonormal basis (φn )n∈N of H with the corresponding eigenvalues (λn )n∈N forming a Blaschke sequence. The eigenvalues (λn )n∈N then lie in the open right half-plane and are uniformly bounded away from the imaginary axis. Moreover, we assume that the operator B can be represented by Bv =
∞
v, bn φn ,
v ∈ CN ,
n=1
for a sequence (bn )n ⊂ CN . Note that B may not map CN boundedly into H , but it does map boundedly into a suitable extrapolation space in which the sequence (φn )n∈N has dense linear span: for example, we may define HB =
∞ n=1
2 ∞ ∞ |xn |2 <∞ , xn φn : xn φn := n2 (1 + bn 2 ) n=1
n=1
in which case the semigroup (T (t))t0 has a canonical extension to HB and B : CN → HB is bounded. We shall use such spaces below without further comment. The so-called mild solution of (1) is given by x(t) = T (t)x0 +
∞
t
e−λn (t−s) u(s), bn ds φn
n=1 0
for u ∈ L2 (0, ∞; CN ). Throughout this article we assume that the system (1) is infinite-time exactly controllable, that is, for every x ∈ H there exists a control u ∈ L2 (0, ∞; CN ) such that ∞
∞
x=
e−λn s u(s), bn ds φn .
n=1 0
The question we want to study is whether or not infinite-time exact controllability implies finite-time exact controllability. The system (1) is called finite-time controllable in time τ > 0 if for every x ∈ H there exists a control u ∈ L2 (0, τ ; CN ) such that ∞
τ
x=
n=1 0
e−λn (τ −s) u(s), bn ds φn .
B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
2427
We define ∞
τ
B τ u :=
e−λn (τ −s) u(s), bn ds φn
n=1 0
and ∞
∞
B∞ u :=
e−λn s u(s), bn ds φn
n=1 0
for u ∈ L2 (0, τ ; CN ) resp. u ∈ L2 (0, ∞; CN ). Analysis of exact controllability is made simpler if one adopts the supplementary hypothesis of admissibility, which is defined below. However, we shall normally prefer to operate without this assumption. In particular we operate without the assumption that the eigenvalues of −A form a Carleson sequence in C+ (which is sometimes used to ensure the admissibility condition, cf. [11, D.3.3]). Definition 2.1. B is called finite-time admissible for (T (t))t0 , if there exists some τ > 0 such that Bτ u ∈ H for every u ∈ L2 (0, ∞; CN ). Note that admissibility implies that B ∈ L(CN , H−α ) for every α > 1/2, where H−α =
∞
−α
xn φn : xn |λn |
∈
2
n∈N
;
n=1
see Rebarber and Weiss [17, Theorem 1.4]. For exponentially stable systems the notion of finitetime admissibility is equivalent to the notion of infinite-time admissibility, that is, B∞ u ∈ H for every u ∈ L2 (0, ∞; CN ). We thus simply say admissibility instead of finite-time or infinitetime admissibility. Admissibility implies that Bτ , B∞ ∈ L(L2 (0, ∞; CN ), H ) and that the mild solution of (1) corresponding to an initial condition x(0) = x0 ∈ H and to u ∈ L2 (0, ∞; CN ) is a continuous H -valued function of t. For further information on admissibility we refer the reader to the survey [4]. A dual notion to controllability, which we shall discuss very briefly below, is exact observability of the system x(t) ˙ = Ax(t),
t 0,
y(t) = Cx(t), x(0) = x0 , where C is an A-bounded operator from the domain of A into a Hilbert space Y . We have exact observability in time τ if there is a constant K > 0 such that
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B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
τ
CT (t)x 2 dt K x 2
0
for all x in the domain of A. Here, τ may be finite or infinite. Remark 2.2. Assuming that B is admissible for (T (t))t0 , infinite-time exact controllability easily implies the existence of a time τ > 0 such that the system (1) is finite-time exactly controllable in time τ . See, for example, [18, Proposition 2.8]. However, under some circumstances it is possible to drop the assumption of admissibility. Indeed, as a consequence of the interpolation results established in Section 3, we shall provide a sufficient condition in terms of the eigenvalues of the operator A guaranteeing that infinitetime exact controllability implies finite-time exact controllability, independently of the control operator B. The following condition on the scalar- or matrix-valued Blaschke (or Blaschke–Potapov) product β will be central to our considerations. Condition 2.3. There are constants a, δ > 0 such that (β(s))−1 0 < Re s < δ}.
1 a
on the strip Sδ = {s ∈ C:
Remark 2.4. Both in the scalar- and in the matrix-valued case, Condition 2.3 is equivalent to the conditions that c = inf Re λn > 0 and
sup
∞
y∈R n=1
(Re λn
)2
Re λn < ∞. + (y − Im λn )2
(2)
Proof. Note that inf Re λn > 0 is already implied by the exponential stability assumption. In the one-dimensional case, equivalence is shown in [7, Lemma 5.7]. Observe that (2) can be seen as a Carleson measure-type condition on the measure ∞ n=1 Re λn δλn , but tested only on the reproducing kernels kλ with Re λ = c. The matrix-valued case follows easily by observing that for a Blaschke–Potapov product β, det β is a scalar Blaschke product with zeroes λn , up to multiplicity N , and that β(s) −1 det β(s)−1 β(s) −1 N .
2
Remark 2.5. Let c := inf Re λn > 0 and assume that α := sup
∞
y∈R n=1
Re λn < ∞. (Re λn )2 + (y − Im λn )2
Then, by Remark 2.4, Condition 2.3 holds and in particular for every δ ∈ (0, c), one may choose c2 α a := exp(−2δ (c−δ) 2 ) as the following calculation shows:
B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
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x + iy + λ¯ n 2 −1 β (x + iy)2 x + iy − λ n n∈N 4x Re λn 1+ = (x − Re λn )2 + (y − Im λn )2 n∈N 4x Re λn exp (x − Re λn )2 + (y − Im λn )2 n∈N 4x Re λn = exp (x − Re λn )2 + (y − Im λn )2 n∈N (Re λn )2 Re λn exp 4δ (x − Re λn )2 (Re λn )2 + (y − Im λn )2 n∈N c2 α exp 4δ (c − δ)2 for every x + iy ∈ Sδ . The following theorem shows that the property of infinite-time exact controllability implies exact controllability in a finite time τ depending only on the eigenvalues (λn ) and the dimension N , provided that Condition 2.3 holds. Theorem 2.6. Suppose that Condition 2.3 holds and that the system (1) is exactly controllable in infinite time. Then (1) is exactly controllable in any time τ satisfying τ>
m 2 log √ a −1 + 1 log(N + 1) δ π √
for some constant m > 0. For N = 1, m may be chosen as 4 log2e2 . Note that Condition 2.3 is independent of the control operator B. The proof of this theorem can be found at the end of the following section. Remark 2.7. The situation is different if we deal with approximate controllability instead of exact controllability. The system (1) is called infinite-time approximately controllable, if the intersection of the range of B∞ with H is dense in H , and the system (1) is called finite-time approximately controllable in time τ , if the intersection of the range of Bτ with H is dense in H . However, even for bounded control operators B, infinite-time approximate controllability does in general not imply finite-time approximate controllability, as the following example shows. Note that approximate controllability is dual to approximate observability [21], and thus it is sufficient to state the example for observation operators. Our example is a slight modification of Example 4.1.16 in [2]. Let H be the Sobolev space W 1 (0, ∞), A be the generator of the damped left shift on (0, ∞), that is,
T (t)f (x) := e−t f (t + x),
t, x > 0,
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and C ∈ L(H, C) given by Cf = f (0). Now CT (t)f = 0 on [0, τ ] is equivalent to f (t) = 0 on [0, τ ], and thus this system is not finite-time approximately observable. However, it is infinitetime approximately observable. 3. Interpolation results We summarize some results linking Hankel operators, Toeplitz operators and interpolation. Here, the vector-valued case may be considered, with a Blaschke–Potapov product β, Kβ = H 2 (C+ , CN ) βH 2 (C+ , CN ), Kθτ = H 2 (C+ , CN ) θτ H 2 (C+ , CN ). Lemma 3.1. The following conditions are equivalent: 1. the operator PKθτ |Kβ is bounded below; 2. the Toeplitz operator Tβ ∗ θτ is bounded below on H 2 (C+ , CN ); 3. the Hankel operator Γβ ∗ θτ has norm strictly less than 1, i.e., dist(β ∗ θτ , H ∞ (Mat(N × N ))) < 1; F (λn ) = P ⊥ G(λn ) for 4. given F ∈ H 2 (C+ , CN ) there exists a function G ∈ Kθτ such that P ⊥ I˜n I˜n all n, where I˜n = range β(λn ). Proof. The equivalence of (2) and (3) follows directly from the isometry of the multiplication operator Mβ ∗ θτ and the vector Nehari Theorem [12]. As in [11, Lemma D.4.4.4, Corollary D.4.4.5], we see that the restriction PKθτ |Kβ is bounded below if and only if the Toeplitz operator Tβ ∗ θτ is bounded below. The only remaining assertion is the equivalence of condition 4 with the others. Note that condition 4 can be reformulated as stating that H 2 (C+ , CN ) = βH 2 (C+ , CN ) + Kθτ . This is equivalent to the property that PKβ |Kθτ = Kβ , and this is equivalent to condition 1 by Banach’s closed range theorem. 2 We further require the reproducing kernel thesis for Hankel operators [1,14], with an estimate of constants. It will be convenient to work in the disc rather than the half-plane. Let m denote normalized Lebesgue measure on T, and let A be normalized area measure on D. For λ ∈ D we write Kλ for the normalized reproducing kernel for H 2 , that is, Kλ (z) =
(1 − |λ|2 )1/2 1 − λz
(z ∈ D).
For b ∈ H02 , we write Γb for the Hankel operator Γ : H 2 → H02 , defined by Γb f = P− (bf ). Then we have the following quantitative form of the reproducing kernel thesis. Theorem 3.2. For a Hankel operator Γ = Γb we have Γ M sup Γ Kλ , λ∈D
√ where M can be taken to be 4 2e.
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Proof. We retrace the steps in [3], with a careful estimate of constants. For λ ∈ D we have P− (bKλ )2 = bKλ − b(λ)Kλ 2 2 2 = b(t) − b(λ) Kλ (t) dm(t) T
=
b ◦ φλ (s) − b ◦ φλ (0)2 dm(s),
(3)
T
where φλ denotes the Möbius mapping φλ (s) =
s +λ 1 + λs
.
Using the Littlewood–Paley identity as in [3, Lemma VI.3.2], we can bound (3) below and obtain P− (bKλ )2 1 2 1 = 2
(b ◦ φλ ) (z)2 1 − |z|2 dA(z)
D
b (w)2 1 − |w|2 Kλ (w)2 dA(w),
(4)
D
by a standard change of variables. On the other hand, we may calculate the norm of the Hankel operator by taking b ∈ H02 , f ∈ H 2 and g ∈ H02 , so that bf, g = b(t)f (t)g(t) dm(t) T
1 = b (z)(f g) (z) log dA(z), |z| D
by the polarized form of the Littlewood–Paley identity. Differentiating the product f g gives two terms, so we estimate b (z)f (z)g(z) log 1 dA(z) |z| D
D
1/2 2 b (z) g(z)2 log 1 dA(z) |z|
× D
1/2 2 f (z) log 1 dA(z) |z|
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B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
1/2 2 2 2 2 b (z) g(z) 1 − |z| dA(z) f 2 D
1/2 √ √ 2 2 2 b (z) Kλ (z) 1 − |z| dA(z) 2 2e sup f 2 g 2 , λ∈D
D
√ using the reproducing kernel thesis for Carleson embeddings with the constant 2e given in [15]. The other term may be estimated similarly. Now, combining this with the estimate given in (4), and taking the supremum over all f and g of norm 1, we arrive at √ Γ 4 2e sup Γ Kλ , λ∈D
as required.
2
Since Hankel operators on the half-plane are unitarily equivalent to Hankel operators on the disc (cf. [16, Chapter 2]), we have the same constant in the reproducing kernel thesis for Hankel operators on the half-plane and thus Corollary 3.3. √ Γ 4 2e sup Γ kλ / kλ
(5)
λ∈C+
for every Hankel operator Γ : H 2 (C+ ) → H 2 (C− ). The same reasoning can also be applied to the case of vector Hankel operators Γ = ΓB : H 2 (C+ , CN ) → H 2 (C− , CN ), where B is an N × N matrix-valued symbol. In this case, the Carleson embedding constant employed in Theorem 3.2 is known to grow proportionally to log(N + 1) [10]. Remark 3.4. There exists a constant m > 0 such that Γ m log(N + 1)
sup
Γ kλ e / kλ e
(6)
λ∈C+ ,e∈CN , e >0
for each N ∈ N and each vector Hankel operators Γ : H 2 (C+ , CN ) → H 2 (C− , CN ). Here is our main technical result of this section. Theorem 3.5. Condition 2.3 holds if and only if the equivalent conditions of Lemma 3.1 hold for some τ > 0. More precisely, there exists a constant m > 0 such that given a, δ from Condition 2.3, the equivalent conditions of Lemma 3.1 hold for each τ > 2δ log( √mπ (a −1 + 1) log(N + 1)). In case N = 1, m may be chosen as
√ 4 2e log 2 .
B. Jacob et al. / Journal of Functional Analysis 259 (2010) 2424–2436
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Proof. We claim that Γβ ∗ θτ → 0 as τ → ∞. To see this we use the reproducing kernel thesis above and consider σ :=
sup λ∈C+ , h =1
P− β ∗ θτ kλ h k λ
(7)
.
H 2 (C− ,CN )
Flipping half-planes, we see that 1 σ=√ π
sup λ∈C+ , h =1
θ−τ (s)β(s) 1/2 P+ (Re λ) h s −λ
,
H 2 (C+ ,CN )
where s is a dummy variable. Adding on a function in H 2 (C− , CN ) before projecting we see that this equals sup λ∈C+ , h =1
θ−τ (s)(β(s) − β(λ)) 1/2 P+ (Re λ) h s −λ
.
H 2 (C+ ,CN )
h (which lies in H 2 (−δ/2 + C+ )) along the vertical We now estimate the L2 norm of β(s)−β(λ) s−λ strip {s ∈ C: Re s = −δ/2}. First, since β is inner, we have β(s) = β −1 (−s) 1/a, so (β(s) − β(λ))h a −1 + 1, and integrating we obtain β(s) − β(λ)
1 h a −1 + 1 . 2 s −λ (Re λ + δ/2)1/2 H (−δ/2+C+ ,CN ) Thus the inverse Laplace transform of (Re λ)1/2 β(s)−β(λ) h may be written as a function φ : t → s−λ 1 −δt/2 −1 √ gλ,h (t), where gλ,h 2 (a + 1), independently of λ and h. e 2π Finally, since in the time domain multiplication by θ−τ translates into a left shift by τ , we see that ∞ 1/2 √ 2 θ−τ (s)(β(s) − β(λ)) 1/2 P+ φ(t) dt (Re λ) h = 2π 2 s −λ H (C+ ,CN ) τ
√ 2πe−δτ/2 gλ,h 2
e−δτ/2 a −1 + 1 . √ Thus if eδτ/2 > m log(N + 1)(a −1 + 1)/ π , we have that Γβ ∗ θτ < 1, and hence the equivalent conditions of Lemma 3.1 hold. Conversely, suppose that Condition 2.3 does not hold. We want to show that the equivalent conditions of Lemma 3.1 fail for each τ > 0. Let τ > 0 be fixed. We will show that Γβ ∗ θτ = 1. Since Condition 2.3 fails, there exists a sequence (zk ) = (ak + ibk ) in C+ and a sequence (hk ) in CN such that ak → 0, β ∗ (zk )hk → 0, hk = 1 for all k. Thus
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Γβ ∗ θτ Kzk hk 2 = Γβ ∗ Tθτ Kzk hk 2
Γβ ∗ eiτ bk Kzk hk 2 − Γβ ∗ eiτ bk 1 − Tθτ Kzk hk 2 β ∗ (·) − β ∗ (zk ) Kzk hk 2 1/2 τ ∞ −z t 2 2 −z t 2 a τ e k dt + 1 − e k e k dt − 2ak τ
0
1 − β ∗ (zk )hk − 2ak
τ
−z t 2
e k dt + 1 − eak τ 2
1/2 k→∞
→ 1,
0
and hence Γβ ∗ θτ = 1.
2
We summarize our results on tangential interpolation in the Paley–Wiener space P Wτ (CN ) = {u: ˆ u ∈ L2 ([0, τ ]; CN )}. Here, uˆ denotes the Laplace transform of u. For a Blaschke sequence λn in C+ and a sequence (bn ) of nonzero vectors in CN , we write Kn = span{kλj bj , j = n}H
2
for each n. In the following, will always denote the angle in H 2 (C+ , CN ). Note that in case N = 1, sin(| (kλk , Kk )|)2 = |βk (λk )|2 , where βk denotes the Blaschke product for the Blaschke sequence (λn )n=k . Theorem 3.6. Let N ∈ N, (λn ) be a Blaschke sequence satisfying Condition 2.3, τ > 2 −1 + 1) log(N + 1)) with m as in Remark 3.4. For N = 1, we may choose √m δ log( π (a √
m = 4 log2e2 . Let (bn ) be a sequence of nonzero vectors in CN and let the subspaces Kn be as defined above. Finally, let E denote the weighted tangential evaluation operator given by f → Ef = ( f (λn ), bn ). Then the following are equivalent: 1. E(H 2 (C+ , CN )) ⊇ 2 . 2. E(P Wτ (CN )) ⊇ 2 . 3. The measure μ=
∞ n=1
bn
2 |
Re λ2n δλ (kλn bn , Kn )|2 n
is a Carleson measure on C+ . Proof. The equivalence of (1) and (3) is contained in [6, Theorem 2.18]. Obviously (2) ⇒ (1). The implication (1) ⇒ (2) follows from Theorem 3.5, since for the given τ , each solution of the interpolation problem in H 2 (C+ , CN ) can be lifted to a solution in P Wτ (CN ) by condition 4 of Lemma 3.1. 2 We are now in the position to prove Theorem 2.6.
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Proof of Theorem 2.6. Suppose that Condition 2.3 holds with a, δ. By Theorem 3.5, PKθτ |Kβ is bounded below for any time τ>
2 m log √ a −1 + 1 log(N + 1) . δ π
Note that τ is independent of B. Suppose that the system (1) is exactly controllable in infinite time, that is, H ⊂ R(B∞ ). Exact controllability for (1) in infinite-time implies that for every 2 sequence (cn )n1 we can find u ∈ L2 (0, ∞) such that u(λ ˆ n ), bn = cn for each n, where uˆ = Lu, the Laplace transform of u; equivalently, u, ˆ kλn bn = cn (we may clearly exclude the case when any of the bn vanish). Let P : Kβ → Kθτ denote the mapping PKθτ |Kβ , which is bounded below and so has closed range, and let Q : Kθτ → H 2 (C+ , CN ) denote a (bounded) left inverse, so that QP = I on Kβ . Thus u, ˆ QP kλn bn = cn for each n, or Q∗ u, ˆ P kλn bn = cn ; we therefore have exact controllability at time τ using the input L−1 (Q∗ u). ˆ 2 References [1] F.F. Bonsall, Boundedness of Hankel matrices, J. Lond. Math. Soc. (2) 29 (2) (1984) 289–300. [2] R.F. Curtain, H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts Appl. Math., vol. 21, Springer-Verlag, New York, 1995. [3] J.B. Garnett, Bounded Analytic Functions, revised first edition, Grad. Texts in Math., vol. 236, Springer-Verlag, New York, 2007. [4] B. Jacob, J.R. Partington, Admissibility of control and observation operators for semigroups: a survey, in: Current Trends in Operator Theory and Its Applications, in: Oper. Theory Adv. Appl., vol. 149, Birkhäuser, Basel, 2004, pp. 199–221. [5] B. Jacob, J.R. Partington, On controllability of diagonal systems with one-dimensional input space, Systems Control Lett. 55 (4) (2006) 321–328. [6] B. Jacob, J.R. Partington, S. Pott, Interpolation by vector-valued analytic functions, with applications to controllability, J. Funct. Anal. 252 (2) (2007) 517–549. [7] B. Jacob, H. Zwart, Properties of the realization of inner functions, Math. Control Signals Systems 15 (4) (2002) 356–379. [8] J.D. McPhail, A weighted interpolation problem for analytic functions, Studia Math. 96 (2) (1990) 105–116. [9] T. Nakazi, Interpolation of weighted l q sequences by H p functions, Taiwanese J. Math. 9 (3) (2005) 457–467. [10] F. Nazarov, G. Pisier, S. Treil, A. Volberg, Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts, J. Reine Angew. Math. 542 (2002) 147–171. [11] N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz. Vol. 2. Model Operators and Systems, Math. Surveys Monogr., vols. 92–93, Amer. Math. Soc., Providence, RI, 2002, translated from the French by Andreas Hartmann and revised by the author. [12] L.B. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970) 529–539. [13] J.R. Partington, Linear Operators and Linear Systems. An Analytical Approach to Control Theory, London Math. Soc. Stud. Texts, vol. 60, Cambridge University Press, Cambridge, 2004. [14] J.R. Partington, G. Weiss, Admissible observation operators for the right-shift semigroup, Math. Control Signals Systems 13 (3) (2000) 179–192. [15] S. Petermichl, S. Treil, B.D. Wick, Carleson potentials and the reproducing kernel thesis for embedding theorems, Illinois J. Math. 51 (4) (2007) 1249–1263. [16] S.C. Power, Hankel Operators on Hilbert Space, Res. Notes Math., vol. 64, Pitman (Advanced Publishing Program), Boston, MA–London, 1982. [17] R. Rebarber, G. Weiss, Necessary conditions for exact controllability with a finite-dimensional input space, Systems Control Lett. 40 (3) (2000) 217–227. [18] D.L. Russell, G. Weiss, A general necessary condition for exact observability, SIAM J. Control Optim. 32 (1) (1994) 1–23.
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[19] K. Seip, Interpolation and Sampling in Spaces of Analytic Functions, Univ. Lecture Ser., vol. 33, Amer. Math. Soc., Providence, RI, 2004. [20] H.S. Shapiro, A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961) 513– 532. [21] M. Tucsnak, G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Adv. Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009.
Journal of Functional Analysis 259 (2010) 2437–2454 www.elsevier.com/locate/jfa
The dual space of (L(X, Y ), τp ) and the p-approximation property ✩ Yun Sung Choi a , Ju Myung Kim b,∗ a Department of Mathematics, POSTECH, San 31 Hyoja Dong, Nam-Gu, Pohang, Gyeong-Buk, 790-784, Republic of
Korea b Pohang Mathematics Institute, POSTECH, San 31 Hyoja Dong, Nam-Gu, Pohang, Gyeong-Buk, 790-784, Republic of
Korea Received 4 April 2010; accepted 23 June 2010
Communicated by N. Kalton
Abstract We establish a representation of the dual space of L(X, Y ), the space of bounded linear operators from a Banach space X into a Banach space Y , endowed with the topology τp of uniform convergence on p-compact subsets of X. We apply this representation and solve the duality problem for the p-approximation property (p-AP), that is, if the dual space X ∗ has the p-AP, then so does X. However, the converse does not hold in general. We show that given 2 < p < ∞, there exists a subspace of lq which fails to have the p-AP, when q > 2p/(p − 2). This subspace is the Davie space in lq (Davie (1973) [5]) which does not have the approximation property. It follows that for every 2 < p < ∞ there exists a Banach space Yp such that it has the p-AP, but its dual space Yp∗ fails to have the p-AP. We study the relation of the p-AP with the denseness of finite rank operators in the topology τp . Finally we introduce the p-compact approximation property (p-CAP) and show for every 2 < p < ∞ that the Davie space in c0 fails to have the p-CAP, and also that a variant of the Willis space (Willis (1992) [17]) has the p-CAP, but it fails to have the p-AP. © 2010 Elsevier Inc. All rights reserved. Keywords: p-Compact set; p-Compact operator; p-Approximation property; p-Compact approximation property
✩
This work was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0094068). * Corresponding author. E-mail addresses:
[email protected] (Y.S. Choi),
[email protected] (J.M. Kim). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.017
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1. Introduction The approximation property, which was systematically investigated by Grothendieck [9], is one of the most important properties in Banach space theory. A Banach space X is said to have the approximation property (AP) if for every compact set K in X and every ε > 0, there exists a finite rank operator T on X such that supx∈K T x − x ε. He characterized relatively compact sets in Banach spaces in such a way that a subset K of X is relatively compact if and only if there exists a null sequence (xn ) in X such that K ⊂ abco({xn }) = { n αn xn : (αn ) ∈ Bl1 }, where abco({xn }) is the absolutely convex hull of {xn } and BX is the closed unit ball of a Banach space X (see [12, Proposition 1.e.2]). Motivated by this characterization, Sinha and Karn [15,16] introduced and studied pcompacts sets, p-compact operators and the p-approximation property (for short, p-AP) in the following sense. For 1 p ∞, a subset K of X is said to be relatively p-compact if K ⊂ { n αn xn : (αn ) ∈ Blp∗ }, where p1 + p1∗ = 1 and (xn ) ∈ lp (X) (1 p < ∞) ((xn ) ∈ c0 (X) ⊂ l∞ (X) if p = ∞), where lp (X) and c0 (X) are the Banach spaces of X-valued p-summable and null sequences, respectively. Note that the ∞-compact sets are precisely the compact sets and every p-compact set is q-compact for 1 p < q ∞. We can also see that the closed convex hull co(K) of a relatively p-compact set K is also p-compact, because the set { n αn xn : (αn ) ∈ Blp∗ } is closed and convex. The concept of a p-compact set leads naturally to that of the p-AP, which can be considered as a way to weaken the approximation property. For 1 p ∞, a Banach space X is said to have the p-AP if for every p-compact subset K of X and every ε > 0 there exists a finite rank operator T on X such that supx∈K T x − x ε. In fact, the ∞-AP means the AP. We can see easily that if X has the q-AP, then X has the p-AP for 1 p < q ∞. An interesting result on the p-AP is that every Banach space has the 2-AP [15, Theorem 6.4] (hence the p-AP for every 1 p 2). In case of the AP, Enflo [8] showed that there is a separable Banach space which does not have the AP. Grothendieck [9] initiated the study of the variants of the AP and relations between them. The important tools he used were the topology τ on L(X, Y ) of uniform convergence on compact sets in X and the representation of the continuous linear functionals on (L(X, Y ), τ ). They were applied to show the relation of the AP with the denseness of finite rank operators, and also such a basic property of the AP that if X ∗ has the AP, then so does X. However, it was not solved that if X ∗ has the p-AP, then so does X, even though Delgado et al. [6] showed that if X ∗∗ has the p-AP, then so does X. Our main interest in this paper is to find the representation of the continuous linear functionals on (L(X, Y ), τp ), where τp is the topology on L(X, Y ) of uniform convergence on p-compact sets in X. We apply it to solve the above duality problem for the p-AP and study the relation of the p-AP with the denseness of finite rank operators in the topology τp . In general, even though X has the p-AP, X ∗ does not necessarily have the p-AP. Indeed, we show that given 2 < p < ∞, there exists a subspace Bq of lq for every q > 2p/(p − 2) such that Bq does not have the p-AP. This space Bq is the Davie space in lq [5] which does not have the AP. It follows that for every 2 < p < ∞ there exists a Banach space Yp such that it has the p-AP, but its dual space Yp∗ fails to have the p-AP. Let F (X, Y ) (resp. K(X, Y )) be the space of finite rank (resp. compact) operators from X into Y . The concept of a p-compact operator was defined in [15] in the following manner. For 1 p ∞, an operator T : X → Y is said to be p-compact if T (BX ) is relatively p-compact in Y . Let Kp (X, Y ) be the space of p-compact operators from X into Y .
Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
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Sinha and Karn [15] showed that an operator T ∈ L(X, Y ) is p-compact if and only if it is quotiented in lp∗ . In fact, they showed that T ∈ Kp (X, Y ) factors into the composition of a p-compact operator and a weakly p-compact operator (see [15] for its definition) through a quotient space of lp∗ . We show that a p-compact operator factors into the composition of a compact operator and a p-compact operator through a quotient space of l1 . This factorization is also applied to study the relation of the p-AP with the denseness of finite rank operators. It is an immediate result from the definition of the p-AP that (a) X has the p-AP ⇔ (b) F (Y, X)τp = L(Y, X) for every Banach space Y ⇔ (c) F (X, Y )τp = L(X, Y ) for every Banach space Y (see [15]). This result was improved by Delgado et al. [6] in such a way that (a) X has the p-AP ⇔ (b) Kp (Y, X) ⊂ F (Y, X)· for every Banach space Y ⇔ (c) Kp (Y, X) ⊂ F (Y, X)τ for every Banach space Y . In case of the AP we recall that X has the AP ⇔ F (Y, X)τ = L(Y, X) for every Banach space Y ⇔ F (X, Y )τ = L(X, Y ) for every Banach space Y ⇔ F (Y, X)· = K(Y, X) for every Banach space Y (see [2, Theorem 2.5]). Further, it was shown in [4,10] that X has the AP ⇔ K(Y, X) ⊂ F (Y, X)τ for every Banach space Y ⇔ K(X, Y ) ⊂ F (X, Y )τ for every Banach space Y . From this point of view we show that X has the p-AP ⇔ K(Y, X) ⊂ F (Y, X)τp for every Banach space Y ⇔ K(X, Y ) ⊂ F (X, Y )τp for every Banach space Y . Compared with the above result by Delgado et al., we note that τp τ and Kp (X, Y ) ⊂ K(X, Y ). In the last part of the paper we introduce the p-compact approximation property (p-CAP) and show for every 2 < p < ∞ that the Davie space in c0 fails to have the p-CAP, and also that a variant of the Willis space [17] has the p-CAP, but it fails to have the p-AP. 2. The Dual space (L(X, Y ), τp )∗ and the p-AP Let 1 p ∞. For a p-compact subset K of X, δ > 0, and T ∈ L(X, Y ), we put Np (T ; K, δ) = R ∈ L(X, Y ): sup Rx − T x < δ . x∈K
Then the collection of all such sets Np (T ; K, δ)’s forms a topological basis on L(X, Y ) and let τp be the topology generated by this basis, which is the topology on L(X, Y ) of uniform convergence on p-compact sets in X introduced by Sinha and Karn [15]. We observe that the topology τp is a completely regular and locally convex. For a net (Tα ) and T in L(X, Y ) it is easily seen that τp
Tα −→ T
if and only if
sup Tα x − T x → 0 x∈K
for every p-compact subset K of X, and also that for A ⊂ L(X, Y ) and T ∈ L(X, Y ), T ∈ Aτp if and only if for every p-compact subset K of X and every ε > 0 there exists S ∈ A such that supx∈K Sx − T x ε. Thus the topology τ of uniform convergence on compact sets is stronger than the topology τp for every 1 p < ∞, and τq is also stronger than τp for 1 p < q < ∞. We can also find that X has the AP if and only if idX ∈ F (X, X)τ , and that X has the p-AP if and only if idX ∈ F (X, X)τp , where idX is the identity operator on X. We now show that if X is infinite-dimensional, then the topology τp on L(X, Y ) is not metrizable.
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Proposition 2.1. Let 1 p ∞. Then the topology τp on L(X, Y ) is metrizable if and only if X is finite-dimensional. Proof. The proof follows from the same argument as in the proof of [3, Theorems 2.7 and 3.8]. 2 It was shown that (L(X, Y ), τ ) is complete [3, Theorem 3.10(a)], and so is (L(X, Y ), τp ). Proposition 2.2. Let 1 p ∞. Then (L(X, Y ), τp ) is complete. Proof. Suppose that (Tα ) is a τp -Cauchy net in L(X, Y ). Then (Tα x) is also a Cauchy net in Y for every x ∈ X. Define an operator T : X → Y by T x = limα Tα x. Clearly T is linear. If T is in X such that xn < 1/n2 and T xn > n for unbounded, thenthere exists a sequence (xn ) p 1/p n xn n 1/n2 < ∞, we have (xn ) ∈ lp (X). Hence the every n. Since ( n xn ) set {xn } is relatively p-compact, which implies lim sup(Tα − Tβ )xn = 0. α,β
n
It follows that there exists α0 such that supn (Tα − Tβ )xn 1 for all α, β α0 . Since limα Tα x = T x for every x ∈ X, supn (Tα0 − T )xn 1. Thus for every n Tα0 Tα0 xn T xn − (Tα0 − T )xn > n − 1, which contradicts that Tα0 is bounded. Therefore T is bounded. By the similar argument to the τp
above we can show Tα −→ T , hence (L(X, Y ), τp ) is complete.
2
The following is the τp -version of Mazur’s theorem. Recall that the strong operator topology τsto on L(X, Y ) is the topology of uniform convergence on finite subsets of X. Proposition 2.3. Let 1 p < ∞. If C is a τp -compact set in L(X, Y ), then coτp (C) is τp compact. Proof. If C is τp -compact, then C is compact in the strong operator topology τsto . By [3, Theorem 3.13], coτsto (C) = coτ (C) is τ -compact and so is τp -compact. Since coτ (C) ⊂ coτp (C) ⊂ coτsto (C) = coτ (C), we have the conclusion. 2 We now establish a representation of (L(X, Y ), τp )∗ , which is applied to solve the duality problem for the p-AP. Let lpw (X) be the Banach space of X-valued weakly p-summable sequences with the norm ∗ p 1/p . Let lˇw (X) be a subspace of l w (X) such that (xn )w p = supx ∗ ∈BX∗ ( n |x (xn )| ) p p p 1/p ∗ x (xn ) →0 sup
x ∗ ∈BX∗
for (xn ) ∈ lˇpw (X).
nm
as m → ∞
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Lemma 2.4. Let 1 < p < ∞. Then lˇpw (X)∗ consists of all linear functionals ϕ of the form
ϕ((xn )) =
∞ ∞
λn xj∗ (xn ), j
j =1 n=1 ∗ ∗ ∗ where zj = (λn )∞ n=1 in lp for each j ∈ N and (xj ) in X with j
∞
zj p∗ xj∗ < ∞.
j =1
∞ j ∗ ∗ ˇw Proof. If ϕ is of the above form ϕ((xn )) = ∞ j =1 n=1 λn xj (xn ), then we see that ϕ ∈ lp (X) ∞ j and ϕ j =1 (λn )p∗ xj∗ . ˇ X, Recall that the space lˇpw (X) is isometrically isomorphic to the injective tensor product lp ⊗ w ˇ ˇ that is, every element (xn ) in lp (X) is identified with the element n en ⊗ xn in lp ⊗ X (see ˇ X)∗ is isometrically isomorphic to the projective tensor prod[7, Corollary 1.1.12]), and (lp ⊗ ˆ X ∗ (see [14, Theorem 5.33]). Consequently, lˇpw (X)∗ is isometrically isomorphic to uct lp∗ ⊗ ∗ ˆ X . lp ∗ ⊗ j ∗ Now suppose that ϕ ∈ lˇpw (X)∗ . Then there exist zj = (λn )∞ n=1 in lp for each j ∈ N and ∞ ∗ ∞ ∗ ∗ w (xj )j =1 in X with j =1 zj p∗ xj < ∞ such that for every (xn ) ∈ lˇp (X) ϕ((xn )) = ϕ
∞ ∞ ∞ ∞
j (zj )en xj∗ (xn ) = en ⊗ xn = λn xj∗ (xn ). j =1 n=1
n
2
j =1 n=1
We are now ready to obtain a representation of (L(X, Y ), τp )∗ . Recall that a linear functional f on a topological vector space V is continuous if and only if there exists a neighborhood U of 0 in V such that f (U ) is bounded (see [13, Theorem 2.2.16]). Thus f ∈ (L(X, Y ), τp )∗ if and only if there exist M > 0 and a p-compact subset K of X such that |f (T )| M supx∈K T x for every T ∈ L(X, Y ). The proof of the following theorem is based on the argument in the representation of the dual space (L(X, Y ), τ )∗ given by Grothendieck [9] (see [2, Proposition 2.4] or [12, Proposition 1.e.3]). Theorem 2.5. Let 1 < p < ∞. Then (L(X, Y ), τp )∗ consists of all linear functionals f of the form
f (T ) =
∞ ∞
λn yj∗ (T xn ), j
j =1 n=1 ∗ ∗ ∗ where (xn ) ∈ lp (X), zj = (λn )∞ n=1 in lp for each j ∈ N and (yj ) in Y with ∞. j
∞
∗ j =1 zj p ∗ yj <
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Proof. Suppose that f is of the above form f (T ) =
∞ ∞
λn yj∗ (T xn ). j
j =1 n=1
Lemma 2.4 shows that ϕ :=
∞ ∞
j λn yj∗ (·) ∈ lˇpw (Y )∗ .
j =1 n=1
ˇw ), we see that f (T ) = ϕ((T xn )) for every Since (T xn )∞ n=1 ∈ lp (Y ) for every T ∈ L(X, Y T ∈ L(X, Y ). Consider the p-compact set K = { n αn xn : (αn ) ∈ Blp∗ } in X. It is easy to check that (T xn )w p = supx∈K T x for every T ∈ L(X, Y ). For every T ∈ Np (0; K, 1), f (T ) = ϕ((T xn )) ϕ(T xn )w = ϕ sup T x ϕ, p
x∈K
hence f ∈ (L(X, Y ), τp )∗ . Conversely, suppose f ∈ (L(X, Y ), τp )∗ . Then there exist M > 0 and a p-compact set K in X such that |f (T )| M supx∈K T x for every T ∈ L(X, Y ). Let (xn ) ∈ lp (X) such that K⊂
αn xn : (αn ) ∈ Blp∗ .
n
Then |f (T )| M(T xn )w p for every T ∈ L(X, Y ). Consider the subspace {(T xn ): T ∈ L(X, Y )} of lˇpw (Y ) and the functional ϕ on {(T xn ): T ∈ L(X, Y )} given by ϕ((T xn )) = f (T ). If (T xn ) = (Rxn ), then f (T ) = f (R), because |f (T − R)| M((T − R)xn )w p . Thus ϕ is well defined and linear. Since ϕ((T xn )) = f (T ) M (T xn )w , p
ϕ is a bounded linear functional on the subspace {(T xn ): T ∈ L(X, Y )} of lˇpw (Y ). Then there exists a Hahn–Banach extension ϕˆ ∈ lˇpw (Y )∗ of ϕ such that f (T ) = ϕ((T xn )) = ϕ((T ˆ xn )) for ∗ ∗ ∗ every T ∈ L(X, Y ). By Lemma 2.4 there exist zj = (λn )∞ n=1 in lp for each j ∈ N and (yj ) in Y ∞ j ∗ ∞ ˇw with j =1 zj p∗ yj∗ < ∞ such that ϕ((y ˆ n )) = ∞ j =1 n=1 λn yj (yn ) for every (yn ) ∈ lp (Y ). Therefore we conclude that j
f (T ) = ϕ((T ˆ xn )) =
∞ ∞
λn yj∗ (T xn ) j
j =1 n=1
for every T ∈ L(X, Y ).
2
We now apply this representation and solve the duality problem for the p-AP, that is, if X ∗ has the p-AP, the so does X.
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Lemma 2.6. (See [12, Lemma 1.e.17].) For every Banach space X, F (X ∗ , X ∗ ) ⊂ F ∗ (X, X)τ , where F ∗ (X, X) = {T ∗ : T ∈ F (X, X)}. Consequently, X ∗ has the p-AP if and only if idX∗ ∈ F ∗ (X, X)τp . Theorem 2.7. Let 2 < p < ∞. If X ∗ has the p-AP, then X has the p-AP. Proof. It is enough to show that if f ∈ (L(X, X), τp )∗ and f (T ) = 0 for all T ∈ F (X, X), then f (idX ) = 0. j ∗ ∗ ∗ By Theorem 2.5 there exist (xn ) ∈ lp (X), zj = (λn )∞ n=1 in lp and (xj ) in X with ∞ ∗ j =1 zj p ∗ xj < ∞ such that f (T ) =
∞ ∞
λn xj∗ (T xn ) j
j =1 n=1 ∗ for every ∞ T ∈∗L(X, X). We may assume without loss of generality that zj p 1 for every j and j =1 xj < ∞. On the other hand, we can write
f (T ) =
∞ ∞ j ∗ 1∗ − 1 λn xj p jX (xn )T ∗ xj∗ p∗ xj∗ , n=1 j =1
where jX : X → X ∗∗ is the canonical isometry. Define g(S) =
∞ ∞ j ∗ 1∗ − 1 λn xj p jX (xn )S xj∗ p∗ xj∗ ,
S ∈ L X∗ , X∗ .
n=1 j =1
Then an easy computation shows that ∗ − 1∗ ∗ ∞ ∗ x p x j j j =1 ∈ lp X and j ∗ 1∗ ∞ λn xj p j =1 ∈ lp∗ for every n. Moreover, ∞ j ∗ 1∗ ∞ λn x p ∗ jX (xn ) j
j =1 p
n=1
∞ n=1
1 p
xn p
∞ ∞ j p ∗ ∗ λn x j
n=1 j =1
1 p∗
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=
∞
1∗ 1 ∞ ∞ p p ∗ p λjn x ∗ xn p j j =1 n=1
n=1
∞
1 p
xn p
∞ ∗ x
1 p∗
< ∞.
j
j =1
n=1
Thus it follows from Theorem 2.5 that g ∈ (L(X ∗ , X ∗ ), τp )∗ . By the assumption, f (T ) = 0 for all T ∈ F (X, X) ⇔ g(T ∗ ) = 0 for all T ∈ F (X, X). Since ∗ X has the p-AP, we have F (X ∗ , X ∗ )τp = L(X ∗ , X ∗ ), hence F ∗ (X, X)τp = L(X ∗ , X ∗ ) by Lemma 2.6. This implies that g(idX∗ ) = 0, hence f (idX ) = g(idX∗ ) = 0, which completes the proof. 2 However, the converse of Theorem 2.7 is not true in general. We show that for every 2 < p < ∞ there exists a Banach space Yp such that it has the p-AP, but its dual space Yp∗ fails to have the p-AP. We start with the following variant of the Davie space [5] which does not have the AP. Let G0 = {1, 2, 3 · 20 } and for each positive integer k, Gk = max Gk−1 + 1, max Gk−1 + 2, . . . , max Gk−1 + 3 · 2k . Then |Gk | = 3 · 2k for k = 0, 1, 2, . . . , Gk ∩ Gl = φ for each k = l, and N = Let 1 q, r < ∞. We consider the Banach space
∞
k=0 Gk .
Zq,r
r ∞ q q f (g) = f: N→C <∞ , k=0
g∈Gk
with the norm f Zq,r =
r 1 ∞ q q r f (g) , k=0
g∈Gk
3·2k which is isometrically isomorphic to ( ∞ k=0 lq )r . Note Zq,q = lq . We use the same notations in [5]. For each k 0 and 1 j 2k , let ejk : N → C be the function in [5]. Let Bq,r be the closed linear span of {ejk : k 0, j = 1, . . . , 2k } in Zq,r . For each k 0, let β k be the linear functional on L(Bq,r , Bq,r ) and φgk : N → C the function for each g ∈ Gk in [5]. Then as in the proof of [5], we can show that (2.1) there exists Aq,r > 0 such that for every k 0 k φ
g Zq,r
for all g ∈ Gk ,
1
Aq,r (k + 1) 2 2
− k2 + qk
Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
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(2.2) for every T ∈ L(Bq,r , Bq,r ) β(T ) 3 sup T xn Z , q,r
lim β k (T ) = β(T ) exists, k
n
and β(idBq,r ) = 1,
where (xn ) = e10 , φg01 , φg02 , φg03 , (1 + 1)2 φg11 , . . . , (1 + 1)2 φg13·2 , . . . , (k + 1)2 φgk1 , . . . , (k + 1)2 φgk k , . . . . 3·2
Theorem 2.8. Let 1 r < ∞ and 2 < p < ∞. Then Bq,r does not have the p-AP, when q > 2p/(p − 2). Proof. Let q > 2p/(p − 2). It follows from (2.1) that
p p xn Zq,r = e10 Z
n
q,r
+
∞ (k + 1)2 φ k p
g Zq,r
k=0 g∈Gk
p e10 Z
q,r
+ Aq,r
∞
3 · 2k (k + 1)5p/2 2
pk − pk 2 + q
k=0
p = e0
1 Zq,r
∞ (2p+2q−pq)k 2q + 3Aq,r (k + 1)5p/2 2 < ∞, k=0
hence (xn ) ∈ lp (Bq,r ). We also deduce from (2.2) that ∗ β ∈ L(Bq,r , Bq,r ), τp . The proof of [5] and (2.2) show that β(idBq,r ) = 1 and β(T ) = 0 for every T ∈ F (Bq,r , Bq,r ), hence idBq,r ∈ / F (Bq,r , Bq,r )τp . Therefore Bq,r does not have the p-AP. 2 Corollary 2.9. Let 2 < p < ∞. For every 2p/(p − 2) < q < ∞, the Banach space Bq,q is a subspace of lq failing to have the p-AP. Lemma 2.10. (See [2, Proposition 1.3].) If X is a separable Banach space, then there exists a separable Banach space Y such that Y ∗∗ has a basis and Y ∗∗∗ is isomorphic to Y ∗ ⊕ X ∗ . The following theorem follows from Corollary 2.9 and Lemma 2.10. Theorem 2.11. For every 2 < p < ∞ there exists a separable Banach space Xp such that Xp∗∗ has a basis, but Xp∗∗∗ fails to have the p-AP.
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3. A factorization of p-compact operators Sinha and Karn [15] showed that an operator is p-compact if and only if it is quotiented in lp∗ . More precisely, for T ∈ Kp (X, Y ) there is a y = (yn ) ∈ lp (Y ) such that T (BX ) ⊂
αn yn : (αn ) ∈ Blp∗ .
n
Define a bounded operator Ty : lp∗ → Y by Ty (α) = ∞ n=1 αn yn , α = (αn ) ∈ lp ∗ . Define ˆ ˆ ∗ ∗ Ty : lp /ker(T ∞y ) → Y by Ty ([α]) = Ty (α). For each x ∈ X, there exists β = (βn ) ∈ lp such ∗ that T x = n=1 βn yn . Define a bounded operator Qy : X → lp /ker(Ty ) by Qy (x) = [β]. Then we can see easily that T = Tˆy Qy . Here, we notice that Tˆy is p-compact and Qy is weakly pcompact. We show that the operator Ty factors further into V ◦ U through l1 , where U is p-compact and V is compact, hence a p-compact operator T factors into the composition of a compact operator and a p-compact operator through a quotient space of l1 . This factorization is also applied to study the relation of the p-AP with the denseness of finite rank operators. The same notations as in the above are used in the following. Theorem 3.1. Let 1 p < ∞. Then T : X → Y is p-compact if and only if there exist y ∈ lp (Y ), a closed subspace M of l1 , a p-compact operator Uˆ : lp∗ /ker(Ty ) → l1 /M, and a compact operator Vˆ : l1 /M → Y such that T = Vˆ Uˆ Qy . Proof. We only need to prove the “only if” part. Let T : X → Y be a p-compact operator. Then there exists y = (yn ) ∈ lp (Y ) such that T = Tˆy ◦ Qy , that is, T
X
Y. Tˆy
Qy
lp∗ /ker(Ty ) We may assume yn = 0 for every n. Let (βn ) be a sequence of positive numbers with βn → 0 such that
p
yn p /βn < ∞.
n
Let λ = (λn ) = (yn /βn ) and zn = yn /λn for each n. Then λ ∈ lp , zn → 0 as n → ∞, and T (BX ) ⊂
αn λn zn : (αn ) ∈ Blp∗ .
n
Define bounded operators U : lp∗ → l1 and V : l1 → Y by
Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
U (α) = (λn αn ),
2447
α = (αn ) ∈ lp∗
and V (γ ) =
γ = (γn ) ∈ l1 ,
γn zn ,
n
respectively. Then it is easy to see that Ty = V U , that is, Ty
lp ∗
Y. V
U
l1 Since (λn en ) ∈ lp (l1 ) and (zn ) ∈ c0 (Y ), the operator U is p-compact and V is compact. Define Uˆ : lp∗ /ker(Ty ) → l1 /ker(V ) by
Uˆ [α] = (λn αn ) ∈ l1 /ker(V ) for α ∈ lp∗ and define Vˆ : l1 /ker(V ) → Y by Vˆ [γ ] = V (γ ) ∈ Y for γ ∈ l1 . Clearly, Uˆ is p-compact, Vˆ is compact, and Tˆy
lp∗ /ker(Ty )
Y. Uˆ
Vˆ
l1 /ker(V ) Hence T = Tˆy Qy = Vˆ Uˆ Qy .
2
4. Denseness of finite rank operators in the topology τp We apply the representation of the continuous linear functionals on (L(X, Y ), τp ) and also the factorization of p-compact operators to study the relation of the p-AP with the denseness of finite rank operators in the topology τp . We recall that X has the p-AP is equivalent to that if f ∈ (L(X, X), τp )∗ and f (T ) = 0 for all T ∈ F (X, X), then f (idX ) = 0. Since every Banach space has the 2-AP [15, Theorem 6.4] (hence the p-AP for every 1 p 2), the following results in this section are valid for 2 < p ∞.
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Theorem 4.1. Let 2 < p ∞. The following are equivalent. (a) For every Banach space Y , K(X, Y ) ⊂ F (X, Y )τp . (b) For every separable reflexive Banach space Y , K(X, Y ) ⊂ F (X, Y )τp . (c) For every Banach space Y and T ∈ K(X, Y ), T ∈ T S: S ∈ F (X, X) τp . (d) For every separable reflexive Banach space Y and T ∈ K(X, Y ), T ∈ T S: S ∈ F (X, X) τp . (e) X has the p-AP. Proof. (a) ⇒ (b), (c) ⇒ (d), and (e) ⇒ (a) are trivial. The proof of (b) ⇒ (c) follows from that of [10, Theorem (b) ⇒ (c)]. (d) ⇒ (e) Assume that f ∈ (L(X, X), τp )∗ and f (T ) = 0 for all T ∈ F (X, X). By Thej ∗ ∗ ∗ orem 2.5 there exist (xn ) ∈ lp (X), zj = (λn )∞ n=1 in lp for each j ∈ N, (xj ) in X with ∞ ∗ j =1 zj p ∗ xj < ∞ and that f (T ) =
∞ ∞
λn xj∗ (T xn ) j
j =1 n=1
for every T ∈ L(X, X). In order to complete the proof it is enough to show that f (idX ) =
∞ ∞
λn xj∗ (xn ) = 0. j
j =1 n=1
We may assume without loss of generality that xj∗ 1 for every j , xj∗ → 0 as j → ∞, and
zj p∗ < ∞.
j
By the result of [11] there exists a separable reflexive Banach space Z such that the inclusion mapping J : Z → X ∗ is compact and abco({xj∗ }) ⊂ BZ . By the assumption J ∗ jX ∈ J ∗ jX S: S ∈ F (X, X) τp ⊂ L X, Z ∗ , where jX : X → X ∗∗ is the canonical isometry. By Theorem 2.5 we can also define g ∈ (L(X, Z ∗ ), τp )∗ by g(R) =
∞ ∞ j =1 n=1
j λn jZ xj∗ (Rxn ).
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An easy computation and the assumption show that for every S ∈ F (X, X) we have g(J ∗ jX S) = f (S) = 0, which implies that ∞ ∞ ∞ ∞ j j 0 = g J ∗ jX = λn jZ xj∗ J ∗ jX xn = λn xj∗ (xn ) = f (idX ). j =1 n=1
2
j =1 n=1
The following is the dual version of Theorem 4.1. Theorem 4.2. Let 2 < p ∞. The following are equivalent. (a) For every separable reflexive Banach space Y , K(Y, X) ⊂ F (Y, X)τp . (b) For every Banach space Y , K(Y, X) ⊂ F (Y, X)τp . (c) X has the p-AP. Proof. (c) ⇒ (a) is trivial. (a) ⇒ (b) Let Y be a Banach space and let T ∈ K(Y, X), let K be a p-compact set in Y and let ε > 0. By a result of [11] there exist a separable reflexive Banach Z, a compact operator R : Y → Z, and a compact operator S : Z → X such that T = SR. Since R(K) is a p-compact set in Z, by the assumption (a) there exists a U ∈ F (Z, X) so that sup U Ry − T y = sup U Ry − SRy ε. y∈K
y∈K
Since U R ∈ F (Y, X), the conclusion follows. (b) ⇒ (c) By [6, Theorem 2.1] X has the p-AP if and only if for every Banach space Y , Kp (Y, X) ⊂ F (Y, X)· . Let T ∈ Kp (Y, X) and ε > 0. By Theorem 3.1 there exist a Banach space Z, a p-compact operator R : Y → Z, and a compact operator S : Z → X such that T = SR. By the assumption (b) there exists F ∈ F (Z, X) such that T − F R = SR − F R = sup Sz − F z ε. z∈R(BY )
Hence T ∈ F (Y, X)· .
2
For complex Banach spaces X and Y , let HK (X, Y ) be the space of compact holomorphic mappings from X into Y (see [1]) and let H(X) be the space of scalar valued holomorphic mappings on X. In [6, Corollary 2.7] it was shown that, if HK (Y, X) ⊂ H(Y ) ⊗ Xτ for every separable reflexive Banach space Y , then X has the AP. Theorem 4.2 can be applied to show the following. Corollary 4.3. Let 2 < p < ∞. If HK (Y, X) ⊂ H(Y ) ⊗ Xτp for every separable reflexive Banach space Y , then X has the p-AP. 5. The p-compact approximation property For 1 p ∞, a Banach space X is said to have the p-compact approximation property (p-CAP) if idX ∈ K(X, X)τp . Then the ∞-CAP is well known as the compact approximation
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property (CAP) (see [2, Section 8]). In general, the p-CAP (resp. CAP) is weaker than the p-AP (resp. AP). Willis [17] constructed a Banach space having the CAP, which fails to have the AP. We show for every 2 < p < ∞ that a variant of the Willis space has the CAP, hence p-CAP, but it fails to have the p-AP. Further, the Davie space in c0 [5] does not have even the p-CAP. For each k 0, let Gk be the subset of N and for each 1 j 2k , let ejk : N → C be the function in Section 2. Let E be the closed linear span of {ejk : k 0, j = 1, . . . , 2k } in c0 . For every k 0 and for every g ∈ Gk , let β k ∈ L(E, E)∗ and φgk ∈ E be defined in the same way as in Section 2. Then it was shown in [5] that (5.1) there exists an A > 0 such that for every k 0 k φ
1
g ∞
k
A(k + 1) 2 2− 2
for all g ∈ Gk , (5.2) for every T ∈ L(E, E) lim β k (T ) = β(T ) exists, k
β(T ) 3 sup T xn ∞ , n
and β(idE ) = 1,
where (xn ) ⊂ E is the sequence defined in (2.2). Theorem 5.1. The Davie space E fails to have the p-CAP for every 2 < p < ∞. Proof. Let 2 < p < ∞. It follows from (5.1) that n
p xn ∞
∞ 0 p (k + 1)2 φ k p = e1 ∞ + g ∞ k=0 g∈Gk
1 + Ap
∞
5
3 · 2k (k + 1) 2 p 2−
pk 2
k=0
= 1 + 3Ap
∞ (2−p)k 5 (k + 1) 2 p 2 2 < ∞, k=0
hence (xn ) ∈ lp (E). We also deduce from (5.2) that β ∈ (L(E, E), τp )∗ and β(idE ) = 1. It is enough to show that β(T ) = 0 for every T ∈ K(E, E), because it implies idE ∈ / K(E, E)τp . For k each k 0, recall that the linear functional β on L(E, E) is defined by k
2 1 k k β (T ) = k αj T ej 2 k
j =1
and for each y ∈ E k
2 1 k αj (y) → 0 2k j =1
Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
2451
as k → ∞ (see [5]). Now let T ∈ K(E, E) and ε > 0. Since C = ejk : k 0, j = 1, . . . , 2k is bounded in E, T (C) has an ε-net {yi }ni=1 . Fix k 0. For each 1 j 2k there exists ykj ∈ {yi }ni=1 such that T ejk − ykj ∞ ε. Then 2k k 1 k k β (T ) = αj T ej − ykj + ykj 2k j =1 k
k
j =1
j =1
2 2 1 k 1 k k k αj (ykj ) + k αj T ej − ykj 2 2 k
2 n 1 k k αj (yi ) + ε. 2 j =1 i=1
Hence |β(T )| = limk |β k (T )| ε. Since ε > 0 was arbitrary, β(T ) = 0.
2
We now construct a simple variant of the Willis space [17]. Let 2 < p < ∞ and consider the sequence (xn ) ⊂ E defined in (2.2). Then idE cannot be approximated on the set {xn }∞ n=1 by the elements of K(E, E), because 1 = β(T ) − β(idE ) β(T − idE ) 3 sup(T − idE )(xn )∞ n
for every T ∈ K(E, E). This fact is applied in the proof of Theorem 5.2. Without loss of generality we may assume that xn = 0 for every n. Then there exists a decreasing sequence (βn ) such that βn → 0, 0 < βn < 1, and p xn ∞ /βn 1. n
Let λn = xn ∞ /βn and zn = xn /λn for each n. Clearly, (λn ) ∈ Blp , 1 > zn ∞ zn+1 ∞ →n 0 with zn ∞ > 0 for every n. For 0 t < 1, let Ut =
αn zn /zn t∞ :
(αn ) ∈ Bl1 .
n
We can check that xn , zn ∈ Ut for every 0 t < 1 and Us ⊂ Ut for 0 s < t < 1. Let Yt be the linear span of Ut with the norm xt = inf{λ > 0: x ∈ λUt }.
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Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
Then (Yt , · t ) is a Banach space with the closed unit ball Ut , and Ys ⊂ Yt and xt xs for 0 s < t < 1. Let Lt : Yt → E be the inclusion mapping. Then Lt is a compact operator. Let Z = span{yχ(s,t] : 0 s < t 1, y ∈ Ys }, where χ(s,t] is the characteristic function of (s, t]. We define a norm on Z by 1 f =
f (r) dr. r
0
Now let Z be the completion of Z with respect to this norm. Define the operator R : Y 1 → Z by 2
R(y) = 2yχ( 1 ,1] 2
and J : Z → E by 1 J (f ) =
f (r) dr. 0
Then J R = L 1 and J 1. 2
Theorem 5.2. The Banach space Z has the CAP, but it fails to have the p-AP for every 2 < p < ∞. Proof. By the same argument as in [17, Proposition 2] Z has the CAP, in fact, the metric CAP. Suppose that Z has the p-AP for some 2 < p < ∞. Then R ∈ F (Y 1 , Z)τp . Since 2
1/2
zn /zn ∞ ∈ U 1 , 2
we can see 1/2
zn 1 zn ∞ , 2
hence zn 1 → 0 as n → ∞. In particular, 2
n
p
xn 1 = 2
p
λn zn 1 < ∞. 2
n
Therefore {xn }∞ n=1 is a relatively p-compact set in Y 1 . Since 2
R ∈ F (Y 1 , Z)τp , 2
Y.S. Choi, J.M. Kim / Journal of Functional Analysis 259 (2010) 2437–2454
2453
given ε > 0, there exists S ∈ F (Y 1 , Z) such that 2
sup J Sxn − L 1 xn ∞ = sup J Sxn − J Rxn ∞ sup Sxn − Rxn Z ε. 2
n
n
n
We have shown that L 1 can be approximated on the relatively p-compact set {xn }∞ n=1 in Y 12 by 2 the elements of F (Y 1 , E). Now let ε > 0 and let T ∈ F (Y 1 , E) such that 2
2
ε sup L 1 xn − T xn ∞ . 2 2 n ∗ We may write T = N k=1 ψk (·)uk , where ψk ∈ Y 1 , uk ∈ E for each k = 1, . . . , N and 2 N ∗ ∗ ∗ τp k=1 uk ∞ = 1. Since L 1 is injective, Y 1 = L 1 (E ) . Thus for each k = 1, . . . , N there exists u∗k ∈ E ∗ such that
2
2
2
ε supψk (xn ) − L∗1 u∗k (xn ) . 2 2 n Consider
N
∗ k=1 uk (·)uk
∈ F (E, E). Then for every n
N u∗k (xn )uk − xn k=1
∞
N u∗k (xn )uk − T xn k=1
+ T xn − L 1 xn ∞
∞
2
N ∗ u (xn ) − ψk (xn )uk ∞ + ε k 2 k=1
=
N ∗ ∗ L 1 u (xn ) − ψk (xn )uk ∞ + ε ε. k 2 2 k=1
Hence idE can be approximated on the set {xn }∞ n=1 by the elements of F (E, E), which is a contradiction. 2 Acknowledgment The authors thank the anonymous referee whose suggestions have improved the final form of the paper. References [1] R.M. Aron, M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21 (1976) 7–30. [2] P.G. Casazza, Approximation properties, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 1, Elsevier, Amsterdam, 2001, pp. 271–316. [3] C. Choi, J.M. Kim, Locally convex vector topologies on B(X, Y ), J. Korean Math. Soc. 45 (2008) 1677–1703. [4] C. Choi, J.M. Kim, K.Y. Lee, Right and left weak approximation properties in Banach spaces, Canad. Math. Bull. 52 (2009) 28–38.
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[5] A.M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973) 261–266. [6] J.M. Delgado, E. Oja, C. Piñeiro, E. Serrano, The p-approximation property in terms of density of finite rank operators, J. Math. Anal. Appl. 354 (2009) 159–164. [7] J. Diestel, J.H. Fourie, J. Swart, The Metric Theory of Tensor Products, Amer. Math. Soc., Providence, 2008. [8] P. Enflo, A counterexample to the approximation property for Banach spaces, Acta Math. 130 (1973) 309–317. [9] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955). [10] J.M. Kim, New criterion of the approximation property, J. Math. Anal. Appl. 345 (2008) 889–891. [11] Å. Lima, O. Nygaard, E. Oja, Isometric factorization of weakly compact operators and the approximation property, Israel J. Math. 119 (2000) 325–348. [12] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer, Berlin, 1977. [13] R.E. Megginson, An Introduction to Banach Space Theory, Springer, New York, 1998. [14] R.A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer, Berlin, 2002. [15] D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of lp , Studia Math. 150 (2002) 17–33. [16] D.P. Sinha, A.K. Karn, Compact operators which factor through subspaces of lp , Math. Nachr. 281 (2008) 412–423. [17] G.A. Willis, The compact approximation property does not imply the approximation property, Studia Math. 103 (1992) 99–108.
Journal of Functional Analysis 259 (2010) 2455–2456 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Continuity and generators of dynamical semigroups for infinite Bose systems” [J. Funct. Anal. 256 (5) (2009) 1453–1475] Philippe Blanchard a , Mario Hellmich a,∗ , Piotr Ługiewicz b , Robert Olkiewicz b a Faculty of Physics and BiBoS, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany b Institute of Theoretical Physics, University of Wrocław, Pl. M. Borna 9, 50204 Wrocław, Poland
Received 21 June 2010; accepted 12 July 2010
After publication of our paper we discovered an error in the proof of Theorem 4, which holds only under an additional assumption. In fact we have the following. 0 ∗ ∗ Theorem 4. Suppose t → Tt (x) is weak -measurable for all x ∈ X0 . Then {Tt }t0 is weak continuous if and only if t>0 ker Tt = {0}.
Proof. If 0 = x ∈ t>0 ker Tt then there is ϕ ∈ X∗ with x, ϕ = 1, thus 0 = limt↓0 Tt (x), ϕ = x, ϕ = 1, so we obtain a contradiction. Conversely, consider the predual semigroup Tt,∗ : X∗ → X∗ . Since (ran Tt,∗ )⊥ = ker Tt we get ( t>0 ran Tt,∗ )⊥ = t>0 ker Tt = {0}, hence by the bipolar theorem t>0 ran Tt,∗ is weak and thus norm dense in X∗ . Next, since ball X is metrizable in the weak∗ -topology for any x ∈ ball X there is a sequence xn0 ∈ ball X0 with xn0 → x, so t → Tt (x) is weak∗ -measurable and t → Tt,∗ (ϕ) is weakly measurable. By the Pettis theorem we conclude that this map is strongly measurable. Finally we apply the Hille theorem about the strong continuity of semigroups. 2 Since Theorem 4 only holds under an additional assumption we have to modify the proof of Proposition 8. We strengthen the assumptions E1–E6 as follows: Let S be a complex Hilbert space with norm · ; we consider it as a symplectic space via σ (f, g) = Imf, g as well as a real Hilbert space (SR , ·,·R ) with f, gR = Ref, g. DOI of original article: 10.1016/j.jfa.2008.05.013. * Corresponding author.
E-mail address:
[email protected] (M. Hellmich). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.012
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P. Blanchard et al. / Journal of Functional Analysis 259 (2010) 2455–2456
E1 The family of maps f → (F μt )(f ), t 0, is equicontinuous with respect to the nuclear topology τn (SR , · ). I.e. for any > 0 there exists δ > 0 and p ∈ Pn (S, · ) such that |1 − (F μt )(f )| < for any t 0 provided p(f ) < δ. E2 t → (F μt )(f ) is continuous for any f ∈ S. E3 The map S f → Wω (f ) is continuous when S is endowed with the norm topology and M with the strong operator topology. E4–E6 remain unchanged. ∗ group of automorProof of Proposition 8. Since {βt }t0 extends to a weak -continuous phisms on M it remains to consider Vt (x) = S Wω (g)xW (g)∗ dνt (g). By E3 the map g → Wω (g)xWω (g)∗ is weak∗ -continuous for all x ∈ M. The quasimeasure νt is σ -additive by E1 and thus extends to a measure. Now we have f, gR = −σ (if, g), thus eif,gR dνt (g) − 1 = eiσ (−if,g) dνt (g) − 1 = (F μt )(−if ) − 1 < SR
S
provided p(f ) = p(−if ) < δ. Hence the Fourier transforms νˆ t are equicontinuous at 0 with respect to the τn (SR , · )-topology on (SR , ·,·R ). Now by Theorem VI.2.3. of [39] the set {νt }t0 is relatively weakly compact in M1+ (S). By E2 the map t → νˆ t (f ) is continuous for any f ∈ SR , so we obtain that t → νt is weakly continuous by a standard result, i.e. t → ψ dνt is continuous for any bounded · -continuous function ψ . It follows that Vt (x) → x relative to the weak∗ -topology as t ↓ 0 for any x ∈ M. We conclude that t>0 ker Vt = {0}, so Tt = βt ◦ Vt is a weak∗ -continuous semigroup on M by Theorem 4. 2 We remark that it can be shown that with the original assumptions E1–E6 in the paper the conclusion of Proposition 8 holds if we additionally assume that limt↓0 S cω (g + g ) dνt (g) = cω (g ) for any g ∈ S, where cω is the generating functional of ω. With these changes all further results in the paper remain true. References [39] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
Journal of Functional Analysis 259 (2010) 2457–2519 www.elsevier.com/locate/jfa
Well-posedness and regularity of generalized Navier–Stokes equations in some critical Q-spaces ✩ Pengtao Li a,∗ , Zhichun Zhai b a Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira,
Taipa, Macau, China b Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building,
Edmonton, AB T6G 2G1 Canada Received 20 February 2009; accepted 22 July 2010 Available online 4 August 2010 Communicated by C. Kenig
Abstract We study the well-posedness and regularity of the generalized Navier–Stokes equations with initial data β,−1 β in a new critical space Qα;∞ (Rn ) = ∇ · (Qα (Rn ))n , β ∈ ( 12 , 1), which is larger than some known critical β
homogeneous Besov spaces. Here Qα (Rn ) is a space defined as the set of all measurable functions with 2(α+β−1)−n sup l(I )
I I
|f (x) − f (y)|2 dx dy < ∞ |x − y|n+2(α−β+1)
where the supremum is taken over all cubes I with edge length l(I ) and edges parallel to the coordinate axes in Rn . In order to study the well-posedness and regularity, we give a Carleson measure characterization of β β Qα (Rn ) by investigating a new type of tent spaces and an atomic decomposition of the predual for Qα (Rn ). In addition, our regularity results apply to the incompressible Navier–Stokes equations with initial data 1,−1 n (R ). in Qα;∞ © 2010 Elsevier Inc. All rights reserved. Keywords: Navier–Stokes equations; Well-posedness; Regularity; Carleson measures; Tent spaces; Duality; Atomic β decomposition; Qα (Rn )
✩
Project supported in part by the Natural Science and Engineering Research Council of Canada.
* Corresponding author.
E-mail addresses:
[email protected] (P. Li),
[email protected],
[email protected] (Z. Zhai). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.013
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
Contents 1. 2. 3.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . β Carleson measure characterization of Qα (Rn ) . . . . . . . β n 3.1. Basic properties of Qα (R ) . . . . . . . . . . . . . . . 3.2. New tent spaces . . . . . . . . . . . . . . . . . . . . . . . β 3.3. The preduality of Qα (Rn ) . . . . . . . . . . . . . . . . 4. Well-posedness of generalized Navier–Stokes equations . β,−1 4.1. Some properties of Qα;∞ (Rn ) . . . . . . . . . . . . 4.2. Several technical lemmas . . . . . . . . . . . . . . . . . 4.3. Well-posedness . . . . . . . . . . . . . . . . . . . . . . . 5. Regularity of generalized Navier–Stokes equations . . . . 5.1. Several technical lemmas . . . . . . . . . . . . . . . . . 5.2. Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2458 2462 2466 2466 2469 2475 2489 2489 2494 2498 2504 2505 2510 2518 2518
1. Introduction This paper considers the well-posedness and regularity of the Cauchy problem for generalized 1+n Navier–Stokes equations on the half-space R+ = (0, ∞) × Rn , n 2: ⎧ 1+n β ⎪ ⎨ ∂t u + (−) u + (u · ∇)u − ∇p = 0, in R+ ; 1+n ∇ · u = 0, in R+ ; ⎪ ⎩ n u|t=0 = a, in R
(1.1)
with β ∈ (1/2, 1) in some Q-type spaces. Here (−)β is the fractional Laplacian with respect to x defined by β u(t, ξ ) = |ξ |2β (−) u(t, ξ ). The fractional Laplacian operator appears in a wide class of physical systems and engineering problems, including Lévy flights, stochastic interfaces and anomalous diffusion problems. In fluid mechanics, the fractional Laplacian is often applied to describe many complicated phenomenons via partial differential equations. Eqs. (1.1) are generalization of the classical Navier– Stokes equations and two-dimensional quasi-geostrophic equations. The research of these two nonlinear partial differential equations have continued to attract attention recently. Eqs. (1.1) are invariant under a particular change of time and space scaling. More exactly, assume that u(t, x) and p(t, x) solve Eqs. (1.1) with initial a(x), then by simple calculations, we can see that λ4β−1 ∂t u λ2β t, λx + λ4β−1 (−)β u λ2β t, λx + λ4β−1 u λ2β t, λx · ∇ u λ2β t, λx − λ4β−1 ∇p λ2β t, λx = 0.
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2459
Hence the re-scaled functions uλ (t, x) = λ2β−1 u λ2β t, λx ,
pλ (t, x) = λ4β−2 p λ2β t, λx
with initial aλ (x) = λ2β−1 a(λx) also solve Eqs. (1.1). This scaling invariance is particularly significant for (1.1) and leads to the following definition. A function space is critical for (1.1) if it is invariant under the scaling fλ (x) = λ2β−1 f (λx).
(1.2)
When β = 1, Eqs. (1.1) become the classical Navier–Stokes equations. A natural approach in studying the solutions is to iterate the corresponding operator t v → e u0 −
e(t−s) P (u ⊗ v) ds
t
0
and to find a fixed point. This solution is called mild solution. For the classical Navier–Stokes equations, this approach was pioneered by Kato and Fujita, see for example, [14]. Since Kato and Fujita’s pioneering works in the 1960s, many meaningful results about the mild solutions to the classical Navier–Stokes equations have been established. Among them, the well-posedness and regularity of the mild solutions are two basic questions which were greatly concerned about by many mathematicians. If the solution exists, is unique, and depends continuously on the initial data (with respect to a given topology), then we say that the Cauchy problem of the classical Navier–Stokes equations is well posed in that topology. For results of well-posedness of mild solution, see Kato [13], Cannone [3], Giga and Miyakawa [11], Koch and Tataru [16], Xiao [29] and the references therein. Particularly, in 2001, Koch and Tataru [16] established the well-posedness for the classical Navier–Stokes equations in BMO−1 (Rn ) by the fixed point algorithm. Briefly speaking, they constructed a functional space X 0 on Rn+ and proved the bilinear operator B(u, v) associated with the equation is bounded from (X 0 )n × (X 0 )n to (X 0 )n . Then under the assumption that the initial data is small, the existence and uniqueness of the solutions can be obtained by the classical n iteration method. In 2007, Koch–Tataru’s results were generalized by Xiao in [29] to Q−1 α;∞ (R ). For the regularity of Eqs. (1.1) with β = 1, Miura and Sawada in [20] showed the regularizing rate estimates for Koch–Tataru’s solution in the L∞ norm. Germain, Pavlovi´c and Staffilani in [10] established the regularizing estimates for Koch and Tataru’s solution in both the L∞ and the Carleson norms. More precisely, they showed that under certain smallness condition of initial data in BMO−1 (Rn ), the solution u to the classical Naiver–Stokes equations constructed in [16] satisfies the following regularity property: k
t 2 ∇ k u ∈ X0 ,
for all k ∈ N0 := N ∪ {0},
where X 0 denotes the space which was introduced by Koch and Tataru in [16]. To achieve their goal, they modified Koch–Tataru’s arguments. In fact, they used the Oseen kernel estimates, modified maximal regularity estimates and a T T ∗ arguments so that the fixed point algorithm works well. Similar regularizing rate estimates of the Navier–Stokes equation in Lebesgue space
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Ln (Rn ) and homogeneous Sobolev space H˙ n/2−1 (Rn ) were studied by Giga and Sawada [12] and Sawada [22] via an appropriate modification of the corresponding existence results. For Eqs. (1.1), Lions [18] proved the global existence of the classical solutions when β 54 in dimension 3. Similar result holds for general dimension n if β 12 + n4 , see Wu [25]. For the important case β < 12 + n4 , Wu in [26,27] established the global existence for Eqs. (1.1) in the 1+ n −2β
homogeneous Besov spaces B˙ p,q p (Rn ) for 1 q ∞ and for either 1/2 < β and p = 2 or r (Rn ) with r > max{1, 1 + n − 2β}. For the regularity ˙ 1/2 < β 1 and 2 < p < ∞ and in B2,∞ p results, we refer the readers to Katz and Pavlovi´c [15], Dong and Li [7], Wu [28] and Zhai [31]. Our work originates mainly from two observations in mathematics. At first, it is worth pointing out that most of the function spaces listed above are critical spaces. For example, the spaces −1+ pn (Rn ) and BMO−1 (Rn ) are critical spaces for β = 1. For general L˙ 2n (Rn ), Ln (Rn ), B˙ 2 −1
p|p<∞,∞
1+ −2β 1+ −2β positive β, B˙ 2,1 2 (Rn ) and B˙ 2,∞2 (Rn ) are critical spaces. We see that n
n
n −1+ pn R → BMO−1 Rn . L˙ 2n −1 Rn → Ln Rn → B˙ p|p<∞,∞ 2
BMO−1 (Rn ) is the largest critical space for β = 1 among the spaces listed above where such existence results are available. This fact inspires us to find a larger critical space for general 1+ n −2β β > 0 which includes B˙ 2,1 2 (Rn ) and has a structure similar to BMO−1 (Rn ). Moreover, we will obtain the corresponding global existence result with the small initial data in this larger critical space. −1 n n On the other hand, in [29], Xiao introduced a new space Q−1 α;∞ (R ) to replace BMO (R ) in [16] and generalized Koch–Tataru’s global existence result for the classical Navier–Stokes n n n n equations. Here Q−1 α;∞ (R ) is ∇ ·(Qα (R )) and Qα (R ) is the space of all measurable complexn valued functions f on R satisfying
1/2 2α−n |f (x) − f (y)|2 dx dy <∞ f Qα (Rn ) = sup l(I ) |x − y|n+2α I I
(1.3)
I
where the supremum is taken over all cubes I with edge length l(I ) and edges parallel to the −1 n n coordinate axes in Rn . It is easy to see that Q−1 α;∞ (R ) and BMO (R ) are invariant under the scaling f (x) → λf (λx) which is corresponding to the classical Navier–Stokes equations. If we want to generalize the results of Koch and Tataru [16] and Xiao [29] to the fractional case, we should find a class of spaces such that their derivative spaces are invariant under the scaling (1.2). The above two observations suggest us introducing the following spaces. β
Definition 1.1. For α ∈ (−∞, β) and β ∈ (1/2, 1), we define Qα (Rn ) to be the set of all measurable complex-valued functions f on Rn satisfying f Qβ (Rn ) = sup α
I
2(α+β−1)−n l(I ) I
I
|f (x) − f (y)|2 dx dy |x − y|n+2(α−β+1)
1/2 <∞
(1.4)
where the supremum is taken over all cubes I with edge length l(I ) and edges parallel to the coordinate axes in Rn .
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2461
β
Remark 1.2. Obviously, when β = 1 and α < 0, Qα (Rn ) = BMO(Rn ) (see Xiao [29] for details). Thus our well-posedness result generalizes that of Koch and Tataru [16] and Xiao [29] to β the fractional equations. It is easy to show that Qα (Rn ) is invariant in the following sense: for β any f ∈ Qα (Rn ), f (·)
β
Qα (Rn )
= λ2β−2 f (λ · +x0 ) Qβ (Rn ) ,
λ > 0 and x0 ∈ Rn .
α
Xiao in [29] characterized Qα (Rn ) equivalently as r 2 f 2Qα (Rn )
=
sup
r
t
∇e f (y) 2 t −α dt dy < ∞.
2α−n
x∈Rn ,r∈(0,∞)
(1.5)
0 |y−x|
The advantage of this equivalent characterization is the occurrence of et which generates the mild solutions for the classical Navier–Stokes equations. Note that the mild solutions for Eqs. (1.1) can be represented as
u(t, x) = e
−t (−)β
t a(x) −
e−(t−s)(−) P ∇(u ⊗ u) ds, β
0
where e−t (−) f (x) := Kt (x) ∗ f (x) β
β
2β β with Kt (ξ ) = e−t|ξ |
and P is the Helmboltz–Weyl projection: P = {Pj,k }j,k=1,...,n = {δj,k + Rj Rk }j,k=1,...,n with δj,k being the Kronecker symbol and Rj = ∂j (−)−1/2 being the Riesz transform. Thus, β β β we should characterize Qα (Rn ) by the semigroup e−t (−) . In fact, we prove that f ∈ Qα (Rn ) if and only if r 2β sup
r
x∈Rn ,r∈(0,∞)
2α−n+2β−2
2 − α
−t (−)β
∇e f (y) t β dy dt < ∞.
(1.6)
0 |y−x|
It is easy to check that for each j = 1, . . . , n, ∂j K1 (x) := φj (x) is a C ∞ real-valued function on Rn satisfying the properties: β
φj (x) 1 + |x| −(n+1) ,
x . φj (x) dx = 0 and (φj )t (x) = t −n φj t φj ∈ L1 Rn ,
Rn
(1.7)
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This observation leads us to characterize Qα (Rn ) by a general C ∞ real-valued function φ on Rn with the properties (1.7) as β
f ∈ Qβα Rn r ⇔
sup
r
2α−n+2β−2
x∈Rn ,r∈(0,∞)
f ∗ φt (y) 2 t −(1+2(α−β+1)) dt dy < ∞,
(1.8)
0 |y−x|
i.e., dμf,φ,α,β (t, x) = |f ∗ φt (x)|2 t −(1+2(α−β+1)) dt dy is a 1 − 2(α + β − 1)/n-Carleson measure. In order to get (1.8), inspired by Coifman, Meyer and Stein [5] and Dafni and Xiao [6], we 1 and T ∞ , then define a space H H 1 n introduce new tent spaces Tα,β α,β −α,β (R ) as a subspace of disβ tributions in L˙ 2 n (Rn ). Finally, we identify Qα (Rn ) with the dual space of H H 1 (Rn ). −α,β
− 2 +2(β−1)
β
In order to establish the equivalent norm (1.6) of the space Qα (Rn ) in Definition 1.1, we need the notation of Hausdorff capacity (see [1] and [30]). By the definition of Hausdorff capacity, we should assume α + β − 1 0 to guarantee Λ∞ n−2(α+β−1) is meaningful. However if we define β
the space Qα (Rn ) by (1.6) directly, the proofs for the well-posedness and regularity still holds without the restriction that α + β − 1 0. We now give the organization of this paper. In Section 2, we introduce some notation and some facts about homogeneous Besov spaces, Hausdorff capacity and Carleson measures. In Section 3, in order to establish (1.8), we introduce a new type of tent spaces, their atomic decomβ positions and the predual space of Qα (Rn ). The proofs of the main theorems in this section are similar to that of Dafni and Xiao [6]. For the completeness, we provide the details. In Section 4, we establish the well-posedness of the generalized Navier–Stokes equations in a new critical β,−1 β space Qα;∞ (Rn ) which is the derivative spaces of Qα (Rn ) and contains all known critical homogeneous Besov spaces for Eqs. (1.1). In Section 5, we establish the regularity of the global β,−1 solutions to Eqs. (1.1) with the small initial value in Qα,∞ (Rn ) for β ∈ (1/2, 1] by modifying the arguments of Germain, Pavlovi´c and Staffilani [10]. −s,τ/q In [30], D. Yang and W. Yuan introduced two new classes of spaces, i.e. F˙p ,q (Rn ) and s,τ (Rn ), and studied their dual relation. It’s worth mentioning that when dealing with the F H˙ p,q duality relation our method is different from that in [30]. Because when p = q, the atomic de−s,τ/q composition of the tent space F T˙p,q (Rn ) is not completely known, the authors do not invoke 1
α+β−1
α−β+1, 2 − n β it in [30]. For our case, because Qα (Rn ) = F˙2,2 (Rn ), we can apply the atomic de1 and Hardy–Hausdorff space H H 1 n compositions of the tent space Tα,β −α,β (R ) to prove the dual β
1 relation between Qα (Rn ) and H H−α,β (Rn ). See also [30, Remark 5.2, p. 2080].
2. Notation and preliminaries In this paper the symbols C, Z and N denote the sets of all complex numbers, integers and natural numbers, respectively. For n ∈ N, Rn is the n-dimensional Euclidean space, with Eu1+n clidean norm denoted by |x| and the Lebesgue measure by dx. R+ is the upper half-space 1+n {(t, x) ∈ R+ : t > 0, x ∈ Rn } with Lebesgue measure denoted by dt dx.
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
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A ball in Rn with center x and radius r will be denoted by B = B(x, r), its Lebesgue measure by |B|. A cube in Rn will always mean a cube in Rn with sides parallel to the coordinate axes. The sidelength of a cube I will be denoted by l(I ). Similarly, its volume will be denoted by |I |. The symbol U V means that there exists a positive constant C such that U CV . U ≈ V means U V and V U. For convenience, the positive constants C may change from one line to another and usually depend on the dimension n, α, β and other fixed parameters. The characteristic function of a set A will be denoted by 1A . For Ω ⊂ Rn , the space C0∞ (Ω) consists of all smooth functions with compact support in Ω. The Schwartz class of rapidly decreasing functions and its dual will be denoted by S (Rn ) and S (Rn ), respectively. For a function f ∈ S (Rn ), f means the Fourier transform of f. We use S0 (Rn ) to denote the following subset of S (Rn ), n n γ S0 R = φ ∈ S R : ψ(x)x dx = 0, |γ | = 0, 1, 2, . . . , Rn γ1 γ2
γn
where x γ = x1 x2 · · · xn , |γ | = γ1 + γ2 + · · · + γn . Its dual S0 (Rn ) = S (Rn )/S0⊥ (Rn ) = S (Rn )/P(Rn ), where P(Rn ) is the space of multinomials. We introduce a dyadic partition of Rn . For each j ∈ Z, we let Dj = ξ ∈ Rn : 2j −1 < |ξ | 2j +1 . We choose φ0 ∈ S (Rn ) such that supp(φ0 ) = {ξ : 2−1 < |ξ | 2} and φ0 > 0 on D0 . Let φ (ξ ) j (ξ ) = j . and Ψ φj (ξ ) = φ0 2−j ξ j φj (ξ ) j (ξ ) = Ψ 0 (2−j ξ ), supp(Ψ j ) ⊂ Dj , Ψj (x) = 2j n Ψ0 (2j x). Moreover, Then Ψj ∈ S (Rn ) and Ψ ∞
k (ξ ) = Ψ
k=−∞
1, if ξ ∈ Rn \{0}, 0, if ξ = 0.
(2.1)
To define the homogeneous Besov spaces, we let j f = Ψj ∗ f,
j = 0, ±1, ±2, . . . .
s (Rn ) to be the set of For s ∈ Rn and 1 p, q ∞, we define homogeneous Besov spaces B˙ p,q n all f ∈ S0 (R ) with
f B˙ s
p,q (R
f B˙ s
n)
∞ js q 2 j f Lp (Rn )
=
p,q (R
1/q <∞
for q < ∞,
j =−∞ n)
=
sup
−∞<j <∞
2j s j f Lp (Rn ) < ∞ for q = ∞.
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When 0 < s < 1, we have the following equivalent characterization. If 1 p, q < ∞, then f ∈ s (Rn ) is equivalent to B˙ p,q Rn
f (x + y) − f (x) p dx
Rn
q/p
dy < ∞; |y|n+qs
(2.2)
s (Rn ) amounts to if 0 < s < 1 and 1 p < q = ∞, f ∈ B˙ p,q
−s
sup |y|
y∈Rn
f (x + y) − f (x) p
1/p < ∞.
(2.3)
Rn
We refer to Peetre [21] and Triebel [24] for more information. The usual homogeneous Sobolev space L˙ 2s (Rn ) defined by L˙ 2s (Rn ) = (−)−s/2 L2 (Rn ) is a special type of the homogeneous s (Rn ) = L ˙ 2s (Rn ). Besov space. That is, B˙ 2,2 The homogeneous Besov spaces obey the following inclusion relations (see [2]). Theorem 2.1. Let s ∈ R and p, q ∈ [1, ∞]. s s (i) If 1 q1 q2 ∞, then B˙ p,q (Rn ) ⊆ B˙ p,q (Rn ). 1 2 1 1 (ii) If 1 p1 p2 ∞ and s1 = s2 + n( p1 − p2 ), then B˙ ps11 ,q (Rn ) ⊆ B˙ ps22 ,q (Rn ).
We recall the definition of fractional Carleson measures (see Essen, Janson, Peng and Xiao [8]) and their connection with Hausdorff capacity established by Dafni and Xiao in [6]. 1+n Definition 2.2. For p > 0, we say that a Borel measure μ on R+ is a p-Carleson measure provided that
|||μ|||p = sup
μ(S(I )) <∞ (l(I ))np
(2.4)
where the supremum is taken over all Carleson boxes S(I ) = {(t, x): x ∈ I, t ∈ (0, l(I ))}. Obviously, the 1-Carleson measures are the usual Carleson measures. On the other hand, similar to the case p = 1, if we denote by 1+n T (E) = (t, x) ∈ R+ : B(x, t) ⊂ E 1+n is a p-Carleson measure if the tent based on the set E ⊂ Rn , then a Borel measure μ on R+ p n and only if |μ|(T (B)) C|B| holds for all balls B ⊂ R . That is to say p-Carleson measures can be equivalently defined in terms of tents over balls. We recall some definitions and properties about Hausdorff capacity (see Adams [1], Dafni and Xiao [6] and Yang and Yuan [30]).
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
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Definition 2.3. Let d ∈ (0, n] and E ⊂ Rn . (i) The d-dimensional Hausdorff capacity of E is defined by (∞) Λd (E) := inf
rjd : E ⊂
∞
B(xj , rj ) ,
(2.5)
j =1
j
where the infimum is taken over all covers of E by countable families of open (closed) balls with radii rj . (∞) (E) in the sense of Choquet is defined by (ii) The capacity Λ d (∞) (E) := inf Λ d
l(Ij ) : E ⊂ d
∞
o Ij
,
j =1
j
where B o denotes the interior of B and the infimum ranges only over covers of E by dyadic cubes. (iii) For a function f : Rn → [0, ∞], we define
(∞) dΛd
f
∞ :=
Rn
(∞)
Λd
x ∈ Rn : f (x) > λ
dλ.
0
(∞) Remark 2.4. (i) Λd(∞) is not a capacity in the sense of Choquet. But, its dyadic counterpart Λ d is a capacity since it is monotone, vanishes on the empty set, and satisfies the strong subadditivity condition (∞) (E1 ∪ E2 ) + Λ (∞) (E1 ∩ E2 ) Λ (∞) (E1 ) + Λ (∞) (E2 ), Λ d d d d as well as the continuity conditions (see Adams [1]): (∞) Λ d
i
(∞) Λ d
(∞) (Ki ), Ki = lim Λ d i→∞
{Ki } a decreasing sequence of compact sets,
(∞) (Ki ), Ki = lim Λ d i→∞
i
{Ki } an increasing sequence of sets.
(ii) There exist positive constants C1 (n, d) and C2 (n, d) such that (∞)
C1 (n, d)Λd
(∞) (E) C2 (n, d)Λ(∞) (E) (E) Λ d d
(iii) The integral with respect to Λ d
(∞)
Rn
for all E ⊂ Rn .
(2.6)
(E) satisfies Fatou’s lemma
(∞) lim inf fn lim inf fn d Λ d
Rn
dΛ d
(∞)
.
(2.7)
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For x ∈ Rn , let Γ (x) = {(y, t) ∈ Rn+1 + : |y − x| < t} be the cone at x. Define the nontangential maximal function N (f ) of a measurable function on Rn+1 + by N (f )(x) :=
f (y, t) .
sup (y,t)∈Γ (x)
In [6], Dafni and Xiao characterized the fractional Carleson measures as follows. 1+n Theorem 2.5. (See [6, Theorem 4.2].) Let d ∈ (0, n] and μ be a Borel measure on R+ . Then μ is a d/n-Carleson measure if and only if the inequality
f (t, y) d|μ| A
(∞)
(2.8)
N (f ) dΛd
Rn
R1+n +
1+n . If this is the case then in (2.8) the constant holds for all Borel measurable functions f on R+ A ≈ |||μ|||d/n which is defined by (2.4). β
Rn ) 3. Carleson measure characterization of Qα (R In this section, we establish the equivalent characterization (1.8). We first give some basic β properties of Qα (Rn ). Then inspired by Coifman, Meyer and Stein [5] and Dafni and Xiao [6], 1 and T ∞ . Finally, we obtain the predual space of Qβ (Rn ). we introduce new tent spaces Tα,β α α,β β
3.1. Basic properties of Qα (Rn ) β
Lemma 3.1. Let −∞ < α and max{α, 1/2} < β < 1. Then f ∈ Qα (Rn ) if and only if −n+2(α+β−1) sup l(I ) I
f (x + y) − f (x) 2
|y|
dx dy |y|n+2(α−β+1)
< ∞.
(3.1)
Proof. If the double integrals (1.4) and (3.1) are denoted by U1 (I ) and U2 (I ), respectively, √ then by the change of variable y → x + y and simple geometry one obtains U1 (I ) U2 ( nI ) and U2 (I ) U1 (3I ). 2 Theorem 3.2. Let −∞ < α and max{α, 1/2} < β < 1. Then β
(i) Qα (Rn ) is decreasing in α for a fixed β, i.e. Qβα1 Rn ⊆ Qβα2 Rn ,
if α2 α1 ;
(ii) If α ∈ (−∞, β − 1), then β Qβα Rn = Q− n +β−1 Rn := BMOβ Rn . 2
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
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β
Proof. (i) Suppose α1 α2 . If f ∈ Qα2 (Rn ), then for any cube I we have I
I
2(α −α ) |f (x) − f (y)|2 dx dy l(I ) 2 1 n+2α −2β+2 1 |x − y|
I
I
|f (x) − f (y)|2 dx dy |x − y|n+2α2 −2β+2
n−2α1 −2β+2 l(I ) f 2 β
Qα2 (Rn )
.
(ii) We divide the discussion into two cases. β β Case I: − n2 + β − 1 α < β − 1. By (i), we have Qα (Rn ) ⊆ Q− n +β−1 (Rn ) = BMOβ (Rn ). 2
On the other hand, if f ∈ BMOβ (Rn ) and I is a cube, then for every y ∈ Rn with |y| < l(I ),
f (x + y) − f (x) 2 dx
I
f (x + y) − f (2I ) 2 + f (x) − f (2I ) 2 dx
I
4(β−1)
f (x) − f (2I ) 2 dx |I |1− n f 2
BMOβ (Rn )
,
2I
since we can get easily f 2 β Q− n +β−1 (Rn ) 2
with f (I ) = |I |−1 Hence
|y|
I
−1+ 4(β−1) n
≈ sup |I | I
f (x) − f (I ) 2 dx
I
f (x) dx being the mean value of f over the cube I.
n+4−4β |f (x + y) − f (x)|2 dx dy l(I ) f 2BMOβ (Rn ) n+2α−β+2 |y|
|y|
dy |y|n+2α−2β+2
n−2α−2β+2 l(I ) f 2BMOβ (Rn ) . β
This tells us BMOβ (Rn ) ⊆ Qα (Rn ). β β Case II: α ∈ (−∞, − n2 + β − 1]. In this case, BMOβ (Rn ) ⊆ Qα (Rn ). If f ∈ Qα (Rn ), let I be a cube. If x, y ∈ I , then the set {z ∈ I : min(|x − z|, |y − z|) > 18 l(I )} has measure at least 12 |I | and thus for −2α − n + 2β − 2 > 0,
−2α−n+2β−2 n l(I ) min |x − z|−2α−n+2β−2 , |y − z|−2α−n+2β−2 dz C l(I )
I
Hence we can get
−2α+2β−2 C l(I ) .
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−2n+4β−4 l(I )
I
f (x) − f (y) 2 dx dy
I
2α−2n+2β−2 l(I )
I
I
f (x) − f (y) 2
I
× min |x − z|−2α−n+2β−2 , |y − z|−2α−n+2β−2 dx dy dz −n+2(α+β−1) |f (x) − f (y)|2 dx dy f 2 β n . l(I ) Qα (R ) |x − y|n+2(α−β+1) I
I
β
Thus Qα (Rn ) ⊆ BMOβ (Rn ). This completes the proof of Theorem 3.2.
2
β
In the following, we establish the connection between Qα (Rn ) and homogeneous Besov spaces. Theorem 3.3. Let n 2 and max{1/2, α} < β < 1. α−β+1 β (i) If 1 q 2 and α + β − 1 > 0, then B˙ n ,q (Rn ) ⊆ Qα (Rn ). α+β−1
γ1 (Rn ) ⊆ (ii) Let 1 q ∞, γ1 > (α − β + 1) and γ2 > 0. If γ1 − γ2 = 2 − 2β, then B˙ n/γ 2 ,q β
Qα (Rn ). Remark 3.4. Similar results hold for β = 1, see Essen, Janson, Peng and Xiao [8, Theorem 2.7]. α−β+1 α−β+1 Proof of Theorem 3.3. (i) It follows form (i) of Theorem 2.1 that B˙ n ,q (Rn ) ⊆ B˙ n ,2 (Rn ), α+β−1
α+β−1
α−β+1 so we can assume q = 2. For f ∈ B˙ n ,2 (Rn ), Hölder’s inequality implies that for any cube I α+β−1
in Rn ,
f (x + y) − f (x) 2
|y|
|I |(n−2(α+β−1))/n Rn
dy |y|2+2(α−β+1)
n
f (x + y) − f (x) (α+β−1) dx
I β
2(α+β−1)/n
dy |y|n+2(α−β+1)
.
This estimate, Lemma 3.1 and (2.2) imply that f ∈ Qα (Rn ). γ1 θ1 (ii) According to (ii) of Theorem 2.1, we have B˙ n/γ (Rn ) ⊂ B˙ n/θ (Rn ) for γ1 > θ1 , 2 ,q 2 ,q γ2 > θ2 and γ1 − γ2 = θ1 − θ2 . Thus we can suppose that γ1 < 1 and γ2 < 1. Assume that γ1 γ1 f ∈ B˙ n/γ (Rn ) ⊆ B˙ n/γ (Rn ). For any cube I in Rn , Hölder’s inequality and (2.3) tell us 2 ,q 2 ,∞
|y|
f (x + y) − f (x) 2 dx
dy |y|n+2(α−β+1)
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
|I |
(n−2γ2 )/n |y|
n−2γ2 l(I )
f (x + y) − f (x) n/γ2
I
|y|2γ1 −n−2(α−β+1) dy f 2˙ γ1
Bn/γ
|y|
n−2(α+β−1) l(I ) f 2˙ γ1
Bn/γ
β
Thus f ∈ Qα (Rn ).
2γ2 /n
2 ,∞
(Rn )
2 ,∞
2469
dy |y|n+2(α−β+1)
(Rn )
.
2
3.2. New tent spaces We introduce new tent spaces motivated by similar arguments in Dafni and Xiao [6]. ∞ to be the Definition 3.5. For α > 0 and max{1/2, α} < β < 1 with α + β − 1 0, we define Tα,β 1+n with class of all Lebesgue measurable functions f on R+
∞ = sup f Tα,β
1 |B|1−2(α+β−1)/n
B⊂Rn
f (t, y) 2
dt dy
1/2 < ∞,
t 1+2(α−β+1)
T (B)
where B runs over all balls in Rn . 1+n Definition 3.6. For α > 0 and max{1/2, α} < β < 1 with α + β − 1 0, a function a on R+ 1 n is said to be a Tα,β -atom provided there exists a ball B ⊂ R such that a is supported in the tent T (B) and satisfies
a(t, y) 2
T (B)
dt dy 1 . t 1−2(α−β+1) |B|1−2(α+β−1)/n
1 consists Definition 3.7. For α > 0 and max{1/2, α} < β < 1 with α + β − 1 0 the space Tα,β 1+n with of all measurable functions f on R+
f T 1
α,β
f (t, x) 2 ω−1 (t, x) = inf ω
dt dx t 1−2(α−β+1)
1/2 < ∞,
R1+n + 1+n where the infimum is taken over all nonnegative Borel measurable functions ω on R+ with
N ω dΛ∞ n−2(α+β−1) 1
Rn
and with the restriction that ω is allowed to vanish only where f vanishes.
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Lemma 3.8. If
j
gj T 1 < ∞, then g =
α,β
g T 1
α,β
j
1 with gj ∈ Tα,β
C1−1 (n, d)C2 (n, d)
gj T 1 , α,β
j
where C1 (n, d), C2 (n, d) are the constants in (2.6). Proof. The proof of this lemma is similar to that of Dafni and Xiao [6, Lemma 5.3].
2
Theorem 3.9. Let α > 0 and max{1/2, α} < β < 1 with α + β − 1 0, then 1 if and only if there is a sequence of T 1 -atoms a and an l 1 -sequence {λ } such (i) f ∈ Tα,β j j α,β that f = j λj aj . Moreover
|λj |: f = λj aj f T 1 ≈ inf α,β
j
j
1 . The rightwhere the infimum is taken over all possible atomic decompositions of f ∈ Tα,β 1 hand side thus defines a norm on Tα,β which makes it into a Banach space. (ii) The inequality
f (t, y)g(t, y) dt dy C f 1 g T ∞ Tα,β α,β t
(3.2)
R1+n + 1 and g ∈ T ∞ . holds for all f ∈ Tα,β α,β 1 can be identified with T ∞ under the following pairing (iii) The Banach space dual of Tα,β α,β
f, g =
f (t, y)g(t, y)
dt dy . t
R1+n + 1 atom. Then we can find a ball B = B(x , r) ⊂ Rn such that supp(a) ⊂ Proof. (i) Let a be a Tα,β B T (B) and
a(t, y) 2
T (B)
dt dy 1 . t 1−2(α−β+1) |B|1−2(α+β−1)/n
Fix ε > 0 and define ω(t, x) = κr
−n+2(α+β−1)
min 1,
r |x − xB |2 + t 2
n−2(α+β−1)+ε ,
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where |x − xB |2 + t 2 is the distance between (t, x) and (0, xB ). For x ∈ Rn , the distance in 1+n √ B | . So R+ from the cone Γ (x) to (0, xB ) is |x−x 2
Nω(x) =
sup
−n+2(α+β−1)
κr min 1,
r
n−2(α+β−1)+ε
2
|x − xB |2 + t √ n−2(α+β−1)+ε
2r κr −n+2(α+β−1) min 1, . |x − xB | (t,y)∈Γ (x)
Thus
κ
−1
N ω dΛ∞ n−2(α+β−1)
∞
Rn
0
If λ < N ω(x), then |x − xB | so we obtain
κ
−1
Λ∞ n−2(α+β−1) x: N ω(x) > λ dλ.
√ rε 1 2( λ ) n−2(α+β−1)+ε . Meanwhile, λ < N ω(x) κr −n+2(α+β−1) ,
(∞) N ω dΛn−2(α+β−1)
r −n+2(α+β−1)
Rn
rε λ
n−2(α+β−1) n−2(α+β−1)+ε
dλ 1.
0
1 -atom, we get Moreover, on T (B) we have ω−1 (t, x) = r n−2(α+β−1) . By the definition of Tα,β
a(t, y) 2 ω−1 (t, x)
dt dx 1. t 1−2(α−β+1)
T (B) 1 with a Thus a ∈ Tα.β T 1 1. For any sum α,β
j
λj aj with {λj } l1 =
1 |λj | < ∞ and Tα,β
1 with atoms aj , Lemma 3.8 implies that the sum converges in the quasi-norm to f ∈ Tα,β f T 1 j |λj |. α,β
1 . There exists a Borel measurable function ω 0 on R1+n Conversely, suppose that f ∈ Tα,β + such that
f (t, x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
2 f 2T 1 . α,β
R1+n +
For each k ∈ Z, let Ek = {x ∈ Rn : N ω(x) > 2k }. According to Dafni and Xiao [6, Lemma 4.1], there exists a sequence of dyadic cubes {Ij,k } with disjoint interiors such that j
(∞) l(Ij,k )n−2(α+β−1) 2Λ n−2(α+β−1) (Ek )
and T (Ek ) ⊂
j
S ∗ (Ij,k ).
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1+n Here we have used a Carleson box: S ∗ (Ij,k ) = {(t, y) √ ∈ R+ : y ∈ Ij,k , t < 2 diam(Ij,k )} to ∗ ∗ replace the tent T (Ij,k ) over the dilated cube Ij,k = 5 nIj,k . Consequently, if we define Tj,k = ! ! S ∗ (Ij,k )\ m>k l S ∗ (Il,m ), these will have disjoint interiors for different values of j or k. Now K
Tj,k =
k=−K j
" " S ∗ (Ij,−K ) S ∗ (Il.m ) ⊇ T (E−K ) S ∗ (Il.m ).
j
m>K l
m>K l
Similar to the discussion in the proof of Dafni and Xiao [6, Theorem 5.4], we have k
Tj,k ⊇
j
" 1+n T (Ek ) S ∗ (Il,m ) = (t, x) ∈ R+ : ω(t, x) > 0 \T∞
k
k m>k l
(∞)
with Λn−2(α+β−1) (T∞ ) = |T∞ | = 0. Since ω is allowed to vanish only where f vanishes, f = 1+n f 1Tj,k a.e. on R+ . Defining aj,k = f 1Tj,k (λj,k )−1 and
λj,k =
∗ n−2(α+β−1) l Ij,k
f (t, x) 2
1/2
dt dx t 1−2(α−β+1)
,
Tj,k
we get f = j,k λj,k aj,k almost everywhere. Since S ∗ (Ij,k ) ⊂ T (Bj,k ) where Bj,k is the ball ∗ )/2. a with the same center as Ij,k and radius l(Ij,k j,k is supported in T (Bj,k ) and
aj,k (t, y) 2
dt dy t 1−2(α−β+1)
T (Bj,k )
∗ −n+2(α+β−1) l Ij,k
f (t, x) 2
−1
dt dx t 1−2(α−β+1)
Tj,k
f (t, x) 2
dt dx t 1−2(α−β+1)
T (Bj,k )
∗ −n+2(α+β−1) |Bj,k |−1+2(α+β−1)/n . l Ij,k 1 -atom. Thus each aj,k is a Tα,β Next, we prove that {λj,k } is l 1 -summable. Noting that ω 2k+1 on
Tj,k ⊂
c c S ∗ (Il,k+1 ) ⊂ T (Ek+1 )
l
and applying the Cauchy–Schwarz inequality, we obtain
|λj,k |
j,k
n −(α+β−1)
∗ 2
f (t, x) 2 l Ij,k j,k
dt dx
1/2
t 1−2(α−β+1)
Tj,k
sup ω
j,k Tj,k
1/2
∗ n −(α+β−1) l Ij,k 2
Tj,k
f (t, x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
1/2
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2
(k+1)
∗ n−2(α+β−1) l Ij,k
1/2
j,k
f (t, x) 2 ω−1 (t, x)
j,k T
2473
1/2
dt dx t 1−2(α−β+1)
j,k
f T 1
α,β
2k
n−2(α+β−1) 1/2 l(Ij,k )
k
f T 1
j
α,β
1/2
2
k
(∞) Λn−2(α+β−1) (Ek )
k
f T 1
α,β
(∞) N ω dΛn−2(α+β−1)
1/2 f T 1 . α,β
Rn 1 is a Banach space since it is complete in the quasi-norm (Lemma 3.8) and Thus Tα,β
f T 1 ≈ |||f |||T 1 = inf |λj |: f = λj aj α,β
α,β
j
j
1 and ||| · ||| where the infimum is taken over all possible atomic decompositions of f ∈ Tα,β 1 is Tα,β a norm. (∞) 1+n satisfying Rn N ω dΛα,β 1. (ii) Let ω be a nonnegative Borel measurable function on R+ ∞ , dμ 2 −1−2(α−β+1) dt dx is a 1 − 2(α + β − 1)/nFor g ∈ Tα,β g,n−2(α+β−1) (t, x) = |g(t, x)| t Carleson measure. Then (2.8) tells us, with A ≈ |||μg,n−2(α+β−1) |||n−2(α+β−1)/n ≈ g 2T ∞ , α,β
2
ω(t, x) g(t, x)
dt dx t 1+2(α−β+1)
R1+n +
g 2T ∞
(∞)
N ω dΛn−2(α+β−1) g 2T ∞ .
α,β
α,β
Rn
1 , then Thus if f ∈ Tα,β
R1+n +
f (t, x)g(t, x) dt dx t
f (t, x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
1/2 ∞ . g Tα,β
R1+n +
Hence we finish the proof of (ii) by taking the infimum on the right over all admissible ω. ∞ , the pairing (iii) Form (ii), we know that for every g ∈ Tα,β f, g =
f (t, y)g(t, y)
dt dy t
R1+n + 1 . Now we prove the converse. Let L be a bounded defines a bounded linear functional on Tα,β 1 linear functional on Tα,β . Fix a ball B = B(xB , r) ⊂ Rn . If f is supported on T (B) with f ∈ L2 (T (B), t −1 dt dx) then
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
f (t, x) 2
dt dx t 1−2(α−β+1)
r
f (t, x) 2 dt dx t
2(α−β+1)
T (B)
T (B)
1
r n−2(α+β−1)+2(α−β+1) |B|1−2(α+β−1)/n
f (t, x) 2 dt dx t
T (B)
1 r n−4β+4 f 2L2 (T (B),t −1 dt dx) . |B|1−2(α+β−1)/n
1 -atom and L is a bounded linear functional on This tells us that f (t, x) is a multiple of a Tα,β L2 (T (B), t −1 dt dx) which can be represented by the inner-product with some function gB ∈ L2 (T (B), t −1 dt dx). Taking Bj = B(0, j ), j ∈ N, then gBj = gBj +1 on T (Bj ). So we get a 1+n that is locally in L2 (t −1 dt dx) such that single function g on R+
L(f ) =
f (t, x)g(t, x)
dt dx t
R1+n + 1 is supported in some tent T (B). By the atomic decomposition, the subset of whenever f ∈ Tα,β 1 . We only need to prove g ∈ T ∞ with g ∞ L . such f is dense in Tα,β Tα,β α,β For a ball B ⊂ Rn and every ε > 0, we set
fε (t, x) = t −2(α−β+1) g(t, x)1T ε (B) (t, x) where T ε (B) is the truncated tent T (B) ∩ {(t, x): t > ε}. Since g ∈ L2 (T (B)), we have
dt dx dt dx
fε (t, x) 2
g(t, x) 2 = ∞. 1−2(α−β+1) t t 1+2(α−β+1) T ε (B)
T (B)
1 -atom with Hence we can obtain that fε is a multiple of a Tα,β
dt dx 2 n−2(α+β−1)
g(t, x) 2 fε T 1 r . 1+2(α−β+1) α,β t T ε (B)
According to the representation above, we also get 1/2
2
dt dx dt dx n−2(α+β−1)
g(t, x) 2
g(t, x) 1+2(α−β+1) L r . t 1+2(α−β+1) t T ε (B)
T ε (B)
This gives us
r
−n+2(α+β−1)
T ε (B)
g(t, x) 2
dt dx t 1+2(α−β+1)
1/2 L ,
∞ with g ∞ L . This completes the proof of Theorem 3.9. that is, g ∈ Tα,β Tα,β
2
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2475
β
3.3. The preduality of Qα (Rn ) In this subsection, we introduce a new space which can be viewed as the predual space of β Qα (Rn ). Then, we give an atomic decomposition for this space. For this purpose we need the following lemma which is Lemma 1.1 in [9]. Lemma 3.10. Fix N ∈ N. Then there exists a function φ : Rn → Rn such that (1) (2) (3) (4) (5)
supp(φ) ⊂ {x ∈ Rn : |x| 1}; φ is radial; φ ∈ C ∞ (Rn ); γ1 γ2 γn γ n γ n x φ(x) dx = 0 if γ ∈ N , x = x1 x2 · · · xn , |γ | = γ1 + γ2 + · · · + γn ; R ∞ 2 dt n 0 (φ (tξ )) t = 1 if ξ ∈ R \{0}.
For φ satisfying the conditions of Lemma 3.10 and any f ∈ S (Rn ), we have the well-known Calderón reproducing formula ∞ f=
dt = lim f ∗ φt ∗ φt ε→0,N →∞ t
0
N f ∗ φt ∗ φt
dt . t
(3.3)
ε
1 We introduce the notation of H H−α,β (Rn ) in the sense of distributions.
Definition 3.11. For φ as in above lemma, α > 0 and max{1/2, α} < β < 1 with α + β − 1 0, 1 we define the Hardy–Hausdorff space H H−α,β (Rn ) to be the class of all distributions f ∈ L˙ 2− n +2(β−1) (Rn ) with 2
f H H 1
−α,β (R
n)
:= f ∗ φt (·) T 1 < ∞. α,β
1 Theorem 3.12. · H H 1 (Rn ) is a quasi-norm. Furthermore, H H−α,β (Rn ) is complete under −α,β this quasi-norm.
Proof. Obviously, · H H 1 (Rn ) is a quasi-norm according to the linearity of ρφ (t, x) = −α,β f ∗ φt (x) and the corresponding property of · T 1 . Suppose that {fj } is a Cauchy sequence. α,β
β By the Calderón reproducing formula and Theorem 3.3, we get L˙ 2n −2(β−1) (Rn ) → Qα (Rn ) and 2 for every ψ ∈ S (Rn )
fj − fk , ψ ρφ (fj − fk ) 1 φt ∗ ψ T ∞ Tα,β α,β ρφ (fj − fk ) T 1 ψ Qβ (Rn ) α α,β ρφ (fj − fk ) T 1 ψ L˙ 2n α,β
2 −2(β−1)
(Rn ) .
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This deduces that {fj } is a Cauchy sequence in L˙ 2− n +2(β−1) (Rn ). By completeness, f = lim fn 2 exists in L˙ 2− n +2(β−1) (Rn ). Thus there exists a subsequence such that f = f1 + j 1 (fj +1 − fj ) 2 in S (Rn ) with fj +1 − fj H H 1 (Rn ) < ∞. Then we have −α,β
ρφ (f )
1 Tα,β
# $ ρφ (fj +1 − fj ) 1 < ∞ ρφ (f1 ) T 1 + T α,β
α,β
1 1 and so f ∈ H H−α,β (Rn ). Similarly we can prove fj → f in H H−α,β (Rn ).
2
Definition 3.13. Let α > 0 and max{1/2, α} < β < 1 with α + β − 1 0. A tempered distribu1 tion a is called an H H−α,β (Rn ) atom if a is supported in a cube I and satisfies the following two conditions: (i) a local Sobolev-(α − β + 1) condition: for all ψ ∈ S (Rn )
a, ψ diam(I )− n2 +α+β−1
I
I
|ψ(x) − ψ(y)|2 dx dy |x − y|2(α−β+1)
1/2 ;
(ii) a cancellation condition: a, ψ = 0 for any ψ ∈ S (Rn ) which coincides with a polynomial of degree n2 + 1 in a neighborhood of I . In [6], Dafni and Xiao established the following fractional Poincaré inequality which will help us to understand the previous definition. Lemma 3.14. Let ψ ∈ C ∞ (Rn ) and I be a cube. Denote by ψ(I ) the average of ψ over I. If 0 α1 , α2 < β for a fixed β ∈ (1/2, 1), then ψ − ψ(I )
nn/4 diam(I )α1 −β+1 L2 (Rn )
I
nn/4 diam(I )α2 −β+1
I
I
I
|ψ(x) − ψ(y)|2 dx dy |x − y|n+2(α1 −β+1) |ψ(x) − ψ(y)|2 dx dy |x − y|n+2(α2 −β+1)
1/2
1/2
C diam(I ) ∇ψ L2 (I ) with C depending only on the dimension and α2 . If in addition then the quantities above are also bounded by
∂ψ I ∂xk
dx = 0 for all k = 1, . . . , n,
C diam(I ) ∇ψ − (∇ψ)I L2 (I )
1/2 |∇ψ(x) − ∇ψ(y)|2 Cnn/4 diam(I )α1 −β+2 dx dy . |x − y|n+2(α1 −β+1) I
I
∂ψ Here (∇ψ)I denotes the vector whose coordinates are the means of ( ∂x )(I ), k = 1, . . . , n. k
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2477
Remark 3.15. Similar to remark (2) after Lemma 6.2 of Dafni and Xiao [6], we can prove that an 1 (Rn )-atom a belongs to the homogeneous Sobolev spaces L˙ 2−s (Rn ) with α + β − 1 H H−α,β n s 2 + 1. Particularly, we have
n
a, ψ diam(I ) − 2 +α+β−1 ψ ˙ 2
Lα−β+1 (Rn ) .
n
This deduces a L˙ 2
−(α−β+1) (R
n)
(diam(I ))− 2 +α+β−1 . Meanwhile,
a, ψ diam(I ) ψ ˙ 2
L n −2β+3 (Rn ) 2
and so a L˙ 2
−( n 2 −2β+3)
(Rn )
diam(I ).
1 We can obtain the atomic decomposition of H H−α,β (Rn ) as follows.
Theorem 3.16. Let α > 0 and max{1/2, α} < β < 1 with α + β − 1 0. A tempered distribu1 1 tion f on Rn belongs to H H−α,β (Rn ) if and only if there exist H H−α,β (Rn )-atoms {aj } and an l 1 -summable sequence {λj } such that f = j λj aj in the sense of distributions. Moreover, f H H 1
−α,β
|λj |: f = λj aj . (Rn ) ≈ inf j
j
1 Proof. Part 1. “⇐” By the completeness of H H−α,β (Rn ), we only need to prove that if a is an 1 1 H H−α,β (Rn )-atom then a is in H H−α,β (Rn ) with the quasi-norm bounded by a constant. Since 1 n a is an H H−α,β (R )-atom and α + β − 1 n2 − 2(β − 1) n2 + 1, Remark 3.15 implies that a ∈ L˙ 2− n +2(β−1) (Rn ) with norm bounded by a constant. On the other hand, assume that I is the 2 support of a and xI represents its center. For ε ∈ (0, 2), let
−n+2(α+β−1) ω(t, x) = κ l(I ) min 1,
l(I )
n−2(α+β−1)+ε
(x − xI )2 + t 2
where κ is a constant to be chosen later. Similar to the proof of Theorem 3.9, we have √ −n+2(α+β−1) 2l(I ) n−2(α+β−1)+ε Nω(x) κ l(I ) min 1, |x − xI |
and so
Rn
(∞)
Nω dΛn−2(α+β−1) κ 1 by choosing κ small enough.
1+n Now, let BI = B(xI , diam(I )), EI = (0, diam(I )) × BI and EIc = R+ \EI . Suppose Sa is 1+n the support of a ∗ φt (x) in R+ . We have
R1+n +
a ∗ φt (x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
=
+
EI
EIc ∩Sa
a ∗ φt (x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
.
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
1+n By the definition of the cylinder EI in R+ , we can find a half-ball centered at (0, xI ) to −1 n−2(α+β−1) on EI . This fact implies that cover EI . Thus we have ω (l(I ))
a ∗ φt (x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
EI
n−2(α+β−1) l(I )
∞
2
(tξ ) 2 dξ a (ξ ) φ
0 Rn
n−2(α+β−1) l(I )
dt t 1−2(α−β+1)
2
a (ξ ) |ξ |−2(α−β+1) dξ
Rn
∞
φ (t) 2
dt t 1−2(α−β+1)
0
n−2(α+β−1) l(I ) a L˙ 2
−(α−β+1) (R
n)
1.
For the integral on EIc ∩ Sa . If z ∈ I, x ∈ / BI and t |x − xI |/2, then |x − z| |x − xI | − diam(I)/2 |x − xI |/2 t, and a ∗ φt (x) = a(z)φt (x − z) dz = 0. Otherwise, we have
a ∗ φt (x) a ˙ 2
L− n +2β−3 (Rn )
x φ ˙ 2 t
2
diam(I )t −(n−2β+3)
L n −2β+3 (Rn )
2
φ (ξ ) 2 |ξ |n−4β+6 dξ
1/2
Rn
diam(I )t It is easy to check t ≈
−(n−2β+3)
.
|x − xI |2 + t 2 := r(t, x) > diam I . This implies that
n−2(α+β−1) t n−2(α+β−1)+ε −ε l(I ) t n−2(α+β−1)+ε . ω−1 (t, x) ≈ κ −1 l(I ) n−2(α+β)+ε (l(I )) Then we can get EIc ∩Sa
a ∗ φt (x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
2−ε l(I )
EIc ∩Sa
2−ε l(I )
t ε−n−3 dt dx r(t, x)ε−n−3 dt
r(t,x)diam(I )
2−ε+ε−2 1. l(I ) 1 (Rn ). Note that the Calderón reproducing formula (3.3) Part 2. “⇒” Suppose f ∈ H H−α,β holds in the sense of distributions. Since the support of φ is the unit ball, we can denote
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
f
ε,N
(x) =
F (t, y)φt (x − y)
2479
dt dy t
S ε,N 1+n where F (t, y) = f ∗ φt (y) and S ε,N is the strip {(t, x) ∈ R+ : ε t N }. Similar to the proof (∞) 1+n of Theorem 3.9, there exists an ω 0 on R+ such that Rn N ω dΛn−2(α+β−1) 1 and
F (t, x) 2 ω−1 (t, x)
dt dx 2 F T 1 . α,β t 1−2(α−β+1)
R1+n +
Let Tj,k be the corresponding structures over the set Ek = {N ω > 2k } as those in Theorem 3.9(i). 1+n Noting that Tj,k have mutually disjoint interiors and F = F χTj,k a.e. on R+ , we let ε,N gj,k (x) =
F (t, y)φt (x − y)
dt dy . t
S ε,N ∩Tj,k ∗ ), these smooth functions in x is supported in {x: Γ (x) ∩ T ∗ Since Tj,k ⊂ T (Ij,k j,k = ∅} ⊂ Ij,k and have the same number moments as φ. We want to verify that there are distributions gj,k such ε,N → gj,k as ε → 0 and N → ∞ with f = j,k gj,k in S (Rn ). To see this, noting that that gj,k ω 2k+1 on Tj,k , we have
% ε,N &
g ,ψ = j,k
Rn
dt dy ψ(x)dx
F (t, y)φt (x − y) t
S ε,N ∩Tj,k
2
(k+1)/2
F (t, y) 2 ω−1 (t, y)
dt dy
1/2
t 1−2(α−β+1)
S ε,N ∩Tj,k
ψ ∗ φt (y) 2
×
1/2
dt dy t 1+2(α−β+1)
S ε,N ∩Tj,k
2(k+1)/2
F (t, y) 2 ω−1 (t, y)
dt dy t 1−2(α−β+1)
S ε,N ∩Tj,k
× ∗ 3Ij,k
∗ 3Ij,k
|ψ(x) − ψ(y)|2 dt dy |x − y|n+2(α−β+1)
Similarly, we obtain that for ε1 < ε2 and N1 > N2 ,
1/2 .
1/2
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
% ε1 ,N1 &
g − g ε2 ,N2 , ψ j,k
j,k
F (t, y) 2 ω−1 (t, y)
Ck
1/2
dt dy
ψ L˙ 2
α−β+1 (R
t 1−2(α−β+1)
n)
.
(S ε1 ,N1 \S ε2 ,N2 )∩Tj,k ε1 ,N1 ε2 ,N2 This gives us that gj,k − gj,k L˙ 2
−(α−β+1) (R
n)
→ 0 as ε1 , ε2 → 0 and N1 , N2 → ∞. Thus,
ε,N ∗ with gj,k → gj,k ∈ L˙ 2−(α−β+1) (Rn ) in the sense of distributions and gj,k is supported in Ij,k
gj,k L˙ 2
∗ −(α−β+1) (3Ij,k )
2(k+1)/2
F (t, y) 2 ω−1 (t, y)
1/2
dt dy
.
t 1−2(α−β+1)
Tj,k
Let aj,k = gj,k gj,k −1 ˙2 ∗ )) (l(3Ij,k
n 2 −(α+β−1)
n
∗ ) L−(α−β+1) (3Ij,k
∗ ))(α+β−1)− 2 and λ (l(3Ij,k j,k = gj,k L˙ 2
∗ −(α−β+1) (3Ij,k )
×
. Then
aj,k , ψ
1 gj,k L˙ 2
∗ −(α−β+1) (3Ij,k )
∗ (α+β−1)− n 2 l 3Ij,k
F (t, y) 2 w(t, y)−1
×
1/2
dt dy t 1−2(α−β+1)
S ε,N ∩Tj,k
× ∗ 3Ij,k
∗ 3Ij,k
|ψ(x) − ψ(y)|2 dx dy |x − y|n+2(α−β+1)
1/2 .
1 This means that aj,k are H H−α,β (Rn )-atoms. On the other hand, the Cauchy–Schwarz inequality implies that
|λj,k |
j,k
2
k+1
∗ n−2(α+β−1) l 3Ij,k
1/2
j,k
j,k T
∗ 1/2 (∞)
F (t, y) 2 ω−1 (t, y) 2k+1 Λn−2(α+β−1) 3Ij,k
j,k
1/2
t 1−2(α−β+1)
−α,β (R
k E k
(∞)
N ω(x) dΛn−2(α+β−1)
dt dy
1/2
t 1−2(α−β+1)
R1+n +
1/2 (∞) 2k+1 dΛn−2(α+β−1) (Ek ) f H H 1
k E K
dt dy
j,k
j,k
F (t, y) 2 ω−1 (t, y)
n)
1/2 f H H 1
−α,β (R
n)
f H H 1
−α,β (R
n)
.
1 The above estimates tell us that gj,k = λj,k aj,k converges to a distribution g in H H−α,β (Rn ). n We need to verify that g = f. Since for a fixed ψ ∈ S (R ), every 0 < ε < N,
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2481
% ε,N &
g ,ψ j,k
2
F (t, y) 2 w(t, y)−1
(k+1)/2
dt dy
1/2
t 1−2(α−β+1)
∗ 3I ∗ 3Ij,k j,k
Tj,k
2
(k+1)/2
∗ n −(α+β−1) l 3Ij,k 2
F (t, y) 2 w(t, y)−1
|ψ(x) − ψ(y)|2 dt dy |x − y|n+2(α−β+1)
1/2
1/2
dt dy
ψ Qα,β (Rn )
t 1−2(α−β+1)
Tj,k
f H H 1
−α,β (R
n)
ψ Qα,β (Rn ) .
Then, limε→0,N →∞ j,k
j,k g
ε,N
=
j,k gj,k
1S ε,N ∩Tj,k (t, y)F (t, y)φt ∗ ψ(y)
= g. Meanwhile, we can also obtain that dt dy = t
F (t, y)φt ∗ ψ(y)
dt dy % ε,N & = f ,ψ . t
S ε,N
R1+n +
This tells us
ε,N j,k gj,k
= f ε,N → f in S (Rn ). Therefore f = g in S (Rn ).
2
Lemma 3.17. 1 (i) If a is an H H−α,β (Rn )-atom, then there exists a nonnegative function ω on R1+n with + (∞) Rn N ω dΛn−2(α+β−1) 1 and
a ∗ ψt (x − y) − a ∗ ψt (x) 2 ω(t, x)−1
σδ (a, ω) = sup
|y|δ
dt dx
1/2
t 1−2(α−β+1)
→ 0.
R1+n +
1 1 (Rn ) ∩ C0∞ (Rn ) is dense in H H−α,β (Rn ). (ii) H H−α,β
Proof. (i) For a fixed ε ∈ (0, 2), the same ω defined in the proof of Theorem 3.16, y ∈ B(0, δ) x−y and x ∈ Rn , we have a ∗ φt (x − y) − a ∗ φt (x) = a, φt − φtx and
x−y
φ t (ξ ) . t t (ξ ) C min{2, δ|ξ |} φ tx (ξ ) = 1 − e2πiy·ξ
φ −φ
(3.4)
Note that
a ∗ φt (x − y) − a ∗ φt (x) 2 ω−1 (t, x)
sup
|y|<δ
R1+n +
sup
|y|<δ EI
+ sup
|y|<δ EIc ∩Sa,δ
dt dx
1/2
t 1−2(α−β+1)
a ∗ φt (x − y) − a ∗ φt (x) 2 ω−1 (t, x)
dt dx t 1−2(α−β+1)
1/2 ,
where BI is the ball B(xI , 2 diam(I )), and EI = (0, 2 diam(I )) × BI . By Fourier transforms, we can estimate the first term as
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sup
|y|<δ EI
a ∗ φt (x − y) − a ∗ φt (x) 2 ω−1 (t, x)
n−2(α+β−1) l(I ) sup
|y|<δ
n−2(α+β−1) l(I ) sup
∞
y
a ∗ φt − φt (x) 2
0 Rn
|y|<δ Rn
n−2(α+β−1) l(I ) sup
dt dx t 1−2(α−β+1)
|y|<δ Rn
2
2
a (ξ ) min 2, δ|ξ |
dt dx t 1−2(α−β+1)
∞
2
φ t|ξ |
dt t 1−2(α−β+1)
dξ
0
2
a (ξ ) δ 2 |ξ |2 |ξ |−2(α−β+1)
∞
ψ (t) 2
dt t 1−2(α−β+1)
dξ → 0
0
as δ → 0 according to the dominated convergence theorem. For the second term. Since supp(a) = I, when x ∈ / BI and t |x − xI |/4, we obtain |y| < diam(I ) < 12 |x − xI | for y ∈ B(0, δ) with δ < diam(I ). Therefore 3 |x − y − z| > |x − xI | − |z − xI | − |y| |x − xI | t. 4 On the other hand |x − z| 34 |x − xI | > t. These estimates imply that a ∗ [φt (x − y) − φt (x)] = 0. Otherwise, we have
a ∗ φt (x − y) − a ∗ φt (x) a ˙ 2
L−( n −2β+3) (Rn )
x−y φt − φx ˙ 2
2
t
L n −2β+3 (Rn ) 2
1/2
2 n−4β+6
x−y x
diam(I ) dξ φt (ξ ) − φt (ξ ) |ξ | Rn
1/2
2 n−4β+6 2
diam(I ) min 2, δ|ξ | φt (ξ ) |ξ | dξ Rn
diam(I )δ
φ t (ξ ) 2 |ξ |n−4β+8 dξ
1/2
Rn
diam(I )δt 2β−4−n . Using the above estimates and the fact ω−1 t n−2(α+β−1)+ε , we have
a ∗ φt (x − y) − a ∗ φt (x) 2 ω−1 (t, x)
EIc ∩Sa,δ
2−ε δ 2 l(I )
EIc ∩Sa,δ
t −n−5+ε dt dx
dt dx t 1−2(α−β+1)
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2−ε δ l(I )
2483
∞ λε−5 dλ → 0
2
l(I )
as δ → 0. Thus σδ (a, ω) → 0 as δ → 0. 1 (Rn )-atom a, take η ∈ C ∞ (Rn ) with support in B(0, 1) and η = 1. Then (ii) For an H Hα,β a ∗ ηj ∈ C0∞ (Rn ) and ηj = j n η(j x) form an approximate identity, a ∗ ηj → a in S (Rn ) as (∞) 1+n with Rn N ω dΛn−2(α+β−1) 1, we have j → ∞. For any nonnegative function ω on R+
a ∗ ηj ∗ φt (x) − a ∗ φt (x) 2 ω−1 (t, x)
R1+n +
ηj (y)
Rn
dt dx
1/2
t 1−2(α−β+1)
a ∗ φt (x − y) − a ∗ φt (x) 2 ω−1 (t, x)
1/2
dt dx t 1−2(α−β+1)
R1+n +
σ 1 (a, ω). j
From (i), we know that for every ε > 0 there exists an ω such that σ 1 (a, ω) < ε with j large j
enough. Taking the infimum over all ω induces a ∗ ηj − a H H 1
−α,β (R
n)
< ε,
for large j,
1 that is, a ∗ ηj → a in H H−α,β (Rn ). Hence, we can get the desired density from the fact that 1 every f ∈ H H−α,β (Rn ) can be approximated by finite sums of atoms. 2
Lemma 3.18. For α > 0, max{α, 1/2} < β < 1 with α + β − 1 0, f ∈ L2loc (Rn ) and φ ∈ S (Rn ) with Rn φ(x) dx = 0, let
2 dμf,φ,α,β (t, x) = (f ∗ φt )(y) t −1−2(α−β+1) dt dy. Then there is a constant C such that for any cubes I and J in Rn with center x0 and l(J ) 3l(I ), (i) μf,φ,α,β S(I )
J J
|f (x) − f (y)|2 dx dy |x − y|n+2(α−β+1)
n−2(α−β) + l(I )
Rn \ 23 J
|f (x) − f (y)| dx |x − x0 |n+1
2 .
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
(ii) If in addition supp(φ) ⊂ {x ∈ Rn : |x| 1} then μf,φ,α,β S(I ) C
J J
|f (x) − f (y)|2 dx dy. |x − y|n+2(α−β+1) 2
Proof. This lemma is a special case of Dafni and Xiao [6, Lemma 3.2].
Theorem 3.19. Let φ be a function as in Lemma 3.10, α > 0 and max{α, 1/2} < β < 1 with β α + β − 1 0. If f ∈ Qα (Rn ) then dμf,φ,α,β (t, x) = |(f ∗ φt )(x)|2 t −1−2(α−β+1) dt dx is a 1 − 2(α + β − 1)/n-Carleson measure. Proof. The proof follows from (ii) of Lemma 3.18 by taking J = 3I.
2
To establish the equivalent (1.8) we need another theorem which contains the converse of Theorem 3.19. Theorem 3.20. Consider the operator πφ defined by ∞ πφ (F ) =
F (t, ·) ∗ φt
dt . t
(3.5)
0 ∞ to Q (Rn ). More pre(i) The operator πφ is a bounded and surjective operator from Tα,β α ∞ cisely, if F ∈ Tα,β then the right-hand side of the above integral converges to a function β
β
f ∈ Qα (Rn ) and ∞ f Qβ (Rn ) F Tα,β α
and any f ∈ Qα,β (Rn ) can be thus represented. 1 with compact support in R1+n extends to a (ii) The operator πψ initially defined on F ∈ Tα,β + 1 to H H 1 n ). bounded and surjective operator from Tα,β (R −α,β Proof. (i) Taking f = πφ (F ), we only need to prove supI Df,α,β (I ) < ∞ where 2α−n+2β−2 Df,α,β (I ) = l(I )
f (x + y) − f (y) 2
|y|
dx dy |y|n+2(α−β+1)
.
Denote the function x → f (x + y) by fy and note that the integral in (3.5) is valid in S (Rn ) modulo constants, that is, when it acts on test functions of integration zero, we obtain ∞ dt , F (t, ·) ∗ φt y − F (t, ·) ∗ φt fy − f = t 0
in S Rn .
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
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Fix a cube I and y ∈ B(0, l(I )). For any g ∈ C0∞ (I ), we write
fy − f, g
|y| 0 Rn
F (t, x)
φt ∗ (g−y − g)(x) dt dx t
l(I ) + |y| Rn
∞ + l(I ) Rn
F (t, ·) ∗ φt (x + y) − F (t, ·) ∗ φt (x)
g(x) dt dx t
F (t, x)
φt ∗ (g−y − g)(x) dt dx t
:= A1 (g, y) + A2 (g, y) + A3 (g, y). For A1 (g, y), |y| < l(I ) verifies that g−y − g is supported in the dilated cube 3I . Also if t |y| we have that φt ∗ (gy − g) is supported in the large cube J = 5I . Then we can get |y| A1 (g, y) 0
F (t, x) 2 dx
1/2
φt ∗ (g−y − g) 2 dt L t
J
|y| φ L1 (Rn ) g L2 (I )
F (t, x) 2 dx
1/2
dt . t
J
0
For A2 , if |y| < t, by changing variable z − y = z, we get
F (t, ·) ∗ φt (x + y) − F (t, ·) ∗ φt (x) φ t −1 y + z − φ(z)
F (t, x − tz) dz Rn
t
−1
|y| sup ∇φ(ξ ) |ξ |1
Cφ t −1 |y|
F (t, x − tz) dz
|z|2
F (t, x − tz) dz
|z|2
with Cφ = sup |∇φ| < ∞. Fubini’s theorem and the fact that g is supported in I imply that l(I )
A2 (g, y) Cφ |y| |y| |z|2 I
F (t, x − tz)
g(x) dx dz dt t2 l(I )
CCφ g L2 (Rn ) |y| |y| |z|2
where |zt | 2 and C = Vol(B(0, 2)).
I
F (t, x − tzt ) 2 dx
1/2 dz
dt t2
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
For A3 , let Gy (t, x) = φt ∗ (g−y − g)(x)1{(t,x): t|y|} . Then the inequality (3.2) implies that A3 =
F (t, x)Gy (t, x) dt dx F T ∞ Gy 1 Tα,β α,β t
R1+n + 1 . To prove G ∈ T 1 , we follow the proof of Lemma 3.17(i) and if we claim that Gy ∈ Tα,β y α,β choose ω to be the same function as that in Theorem 3.16 with 0 < 2(α +β −1) < ε < 2−4+4β. Note that if Sy := supp(Gy ), then we obtain ω−1 (x) l(I )−ε t n−2(α+β−1)+ε . Hence
Gy (t, x) 2 ω−1 (t, x)
dx dt t 1−2(α−β+1)
Rn+1 + −ε
∞
l(I )
φt ∗ (g−y − g)(x) 2 dx t n−2(α+β−1)+ε
l(I ) Rn
l(I )−ε g 2L1 (Rn )
dt t 1−2(α−β+1)
∞ y φt − φt 2 2 t n−4β+3+ε dt L
l(I )
∞ l(I )n−ε g 2L2 (Rn )
y
φt − φ t (ξ ) 2 dξ t n−4β+3+ε dt
l(I ) Rn
∞ l(I )
n−ε
g 2L2 (Rn ) l(I ) Rn
l(I )
n−ε
g 2L2 (Rn ) Rn
2 dt
1 − e2πiy·ξ 2 φ t|ξ | dξ t n−4β+4+ε t
|1 − e2πiy·ξ |2 dξ |ξ |n−4β+4+ε
∞
φ(t) ' 2 t n−4β+4+ε dt t 0
Cφ l(I )n−ε g 2L2 (Rn ) |y|−4β+4+ε . In the last inequality we have used the fact: Rn
|1 − e2πiyξ |2 dξ |y|ε−4β+4 . |ξ |n−4β+4+ε
In fact, we can write Rn
|1 − e2πiyξ |2 dξ |ξ |n−4β+4+ε
|1 − e2πiyξ |2 n−4β+4+ε d(yξ ) |y| |y|n |yξ |n−4β+4+ε
Rn
|y|
+
ε−4β+4 |z|1
|z>1|
|1 − e2πiz |2 dz |z|n−4β+4+ε
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2487
:= |y|ε−4β+4 (I1 + I2 ). It is easy to see that
|1 − e2πiz |2 dz |z|n+ε−4β+4
I2 = |z|1
|z|n−1
|z|1
d|z| 1,
2
∞ (2πiz)k−1
|z|2
dz
|z|n+ε−4β+4 k!
|1 − e2πiz |2 dz |z|n+ε−4β+4
I1 =
|z|n+ε−4β+4
|z|<1 k=1
|z|<1
|z|1−ε+4β−4 d|z| 1.
|z|<1
Then Gy T 1 g L2 l(I )n−ε |y|ε−4β+4 . Thus we get α,β
fy − f L2 (I )
fy − f, g
sup
g∈C0∞ (I ), g 2 1
|y|
F (t, x) 2 dx
1/2
dt + |y| t
J
0
l(I ) |y|
F (t, x − tzt ) 2 dx
I
(n−ε)/2 ∞ l(I ) + F Tα,β |y|ε/2−2β+2 .
Then, by Hardy’s inequality (see Stein [23]), we have
f (x + y) − f (x) 2
|y|
l(I ) s 0
F (t, x) 2 dx
1/2
dt t
+ s
F (t, x − tzt ) 2 dx
1/2
I
l(I ) + F 2T ∞ l(I )n−ε α,β
s n−1 s ε−4β+4 ds s n+2(α−β+1)
0
l(I ) 0
J
ds s 1+2(α−β+1)
J
0
l(I ) l(I ) 0
dx dy |y|n+2(α−β+1)
F (t, x) 2 t −1−2(α−β+1) dx dt
dt t2
2
ds s 2(α−β+1)−1
1/2
dt t2
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
l(I ) + 0
F (t, x − tzt ) 2 t −1−2(α−β+1) dt dx
I
+ F 2T ∞ l(I )n−ε l(I )ε−2(α+β−1) α,β
l(I )
n−2(α+β−1)
F 2T ∞
α,β
since for each t l(I ), |zt | 2 implies I − tzt ⊂ J = 5I. Then we get supI Df,α,β (I ) β ∞ . F 2T ∞ < ∞, that is f ∈ Qα (Rn ) and f Qβ (Rn ) F Tα,β α
α,β
1 -atom a, the integral in (3.5) converges in L n ˙2 (ii) Firstly, we prove that for a Tα,β −n/2−2+2β (R ) 1 n to a distribution which is a multiple of an H H−α,β (R )-atom. Assume a(x, t) is supported in T (B) for some B. For ε > 0, let
∞ πφε (a) =
a(t, ·) ∗ φt (x)
dx dt t
ε
and T ε (B) be the truncated tent T (B) ∩ {(t, x): t > ε}. The Cauchy–Schwarz inequality and (ii) of Lemma 3.18 imply that
% ε &
π (a), ψ
φ
a(t, x) 2
t 1−2(α−β+1)
T ε (B)
1/2
dx dt
2α−n+2β−2 l(B)
B B
ψ ∗ φt (x) 2
T ε (B)
|ψ(x) − ψ(y)|2 dx dy t n+2(α−β+1)
dx dt
1/2
t 1+2(α−β+1)
1/2
is some fixed dilate of the ball B. Since the right-hand side holds for any ψ ∈ S (Rn ), where B is dominated by ψ Qβ (Rn ) ψ L˙ 2 (Rn ) , the same argument also gives, for 0 < ε1 < ε2 , α
% ε1 &
π (a) − π ε2 (a), ψ φ
n/2+2−2β
φ
a(t, x) 2
T ε1 (B)\T ε2 (B)
1/2
dx dt
ψ L˙ 2
n/2+2−2β (R
t 1−2(α−β+1)
n)
.
and Thus πφ (a) = limε→0 πφε (a) exists in L˙ 2−n/2−2+2β (Rn ). This distribution is supported in B satisfies condition (i) of Definition 3.13 since φ satisfies the Therefore πφ (a) same condition. 1 1 and a test function is a multiple of an H H−α,β (Rn )-atom. For a function F = j λj aj in Tα,β ψ ∈ S (Rn ), by Theorem 3.9, we have Rn+1 +
( ) dx dt = F (t, ·) ∗ φt (x)ψ(x) λj πφ aj , ψ = λj πφ aj , ψ , t j
∞ . So π (F ) = since ρφ (ψ)(t, x) = (φt ∗ ψ)(x) is a function in Tα,β φ
j
j
λj πφ aj ∈ S (Rn ) and
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
πφ (F )
−1 H H−α,β (Rn )
inf
2489
|λj | ≈ F T 1
α,β
j
1 . This finishes the the infimum being taken over all possible atomic decompositions of F in Tα,β proof of Theorem 3.20. 2
By Theorem 3.19, Lemma 3.17 and Theorem 3.20, using a similar argument of Dafni and Xiao [6, Theorem 7.1], we can prove the following duality theorem. β
β
1 (Rn ) is Qα (Rn ) in the following sense: if g ∈ Qα (Rn ) Theorem 3.21. The duality of H H−α,β then the linear functional L(f ) = f (x)g(x) dx, Rn 1 defined initially for f ∈ H H−α,β (Rn ) ∩ C0∞ (Rn ), has a bounded extension to all elements 1 n of H H−α,β (R ) with L C g Qβ (Rn ) . Conversely, if L is a bounded linear functional on α
β
1 (Rn ) then there is a function g ∈ Qα (Rn ) so that g Qβ (Rn ) C L and L can be H H−α,β 1 (Rn ) ∩ C0∞ (Rn ). written in the above form for every f ∈ H H−α,β
α
4. Well-posedness of generalized Navier–Stokes equations In this section, we deal with the well-posedness for the generalized Navier–Stokes system in β the setting of Qα (Rn ). Before stating our main result, we first introduce a new critical spaces, β i.e. the derivative spaces of Qα (Rn ). Then we establish some theorems and lemmas which will be used in the proof of the well-posedness. β,−1
4.1. Some properties of Qα;∞ (Rn ) Definition 4.1. For α > 0 and max{α, 1/2} < β < 1 with α + β − 1 0, we say that a tempered β,−1 distribution f ∈ Qα;∞ (Rn ) if and only if r 2β sup x∈Rn ,r∈(0,∞)
r
2α−n+2β−2
β α
Kt ∗ f (y) 2 t − β dy dt < ∞.
0 |y−x|
1,−1 Remark 4.2. In Definition 4.1, if we take β = 1, the space Qα,∞ (Rn ) becomes the space −1 n Qα,∞ (R ) introduced by Xiao in [29]. β,−1
In the next theorem, we prove a useful characterization of Qα,∞ (Rn ). For this purpose, we need the following lemma. This result generalizes Lemma 2.2 in [29]. Lemma 4.3. For α > 0 and max{α, 12 } < β < 1 with α + β − 1 0, let fj,k = ∂j ∂k (−)−1 f β,−1 β,−1 (j, k = 1, 2, . . . , n). If f ∈ Qα,∞ (Rn ) for β ∈ ( 12 , 1), then fj,k ∈ Qα,∞ (Rn ).
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
Proof. Take φ ∈ C0∞ (Rn ) with supp(φ) ⊂ B(0, 1) = {x ∈ Rn : |x| < 1} and Rn φ(x) dx = 1. β Write φr (x) = r −n φ( xr ) and define gr (t, x) = φr ∗ ∂j ∂k (−)−1 e−t (−) f (x). Then e−t (−) fj,k (x) = ∂j ∂k (−)−1 e−t (−) f (x) = fr (t, x) + gr (t, x). β
β
2β−1 1−2β β,−1 Since B˙ 1,1 (Rn ) is the predual of the homogeneous Besov space B˙ ∞,∞ (Rn ) and Qα,∞ (Rn ) → 1−2β B˙ ∞,∞ (Rn ) (see Remark 4.5 and Proposition 4.6 below), we have
gr (t, ·) ∞ n φ 2β−1 n ∂j ∂k (−)−1 e−t (−)β f 1−2β n Cr 1−2β f 1−2β n . L (R ) B˙ B˙ B˙ (R ) (R ) (R ) ∞,∞
1,1
∞,∞
Therefore r 2β
gr (t, y) 2 t −α/β dy dt r n−2α−2β+2 f 2 1−2β
B˙ ∞,∞ (Rn )
0 |y−x|
r n−2α−2β+2 f 2 β,−1
Qα,∞ (Rn )
.
To estimate fr we take ϕ ∈ C0∞ (Rn ) with ϕ = 1 on B(0, 10) = {x ∈ Rn : |x| < 10} and define ϕr,x = ϕ( y−x r ). Then fr = Fr,x + Gr,x with Gr,x = ∂j ∂k (−)−1 ϕr,x e−t (−) f − φr ∗ ∂j ∂k (−)−1 ϕr,x e−t (−) f. β
β
Using Plancherel’s identity, we have r 2β ∂j ∂k (−)−1 ϕr,x e−t (−)β f 2 2
dt
L (Rn ) t α/β
0
r 2β 0
ξj ξk |ξ |−2 ϕr,x e−t (−)β f (ξ ) 2 dξ
Rn
r 2β β 2 ϕr,x e−t (−) f 2
dt t α/β
dt
L (Rn ) t α/β
0
r 2β β 2 ϕr,x e−t (−) f 2
dt
L (Rn ) t α/β
.
0
Similarly we can prove r 2β r 2β dt dt β 2 β 2 −1 −t (−) φr ∗ ∂j ∂k (−) ϕr,x e 2 n ϕr,x e−t (−) f L2 (Rn ) α/β . L (R ) t α/β t 0
0
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2491
Thus, we obtain r 2β Gr,· (t, ·) 2 2
dt
L (Rn ) t α/β
r 2β β 2 ϕr,x e−t (−) f 2
dt
L (Rn ) t α/β
0
.
0
To bound Fr,x , noting that
Fr,x (t, y) 2 dy r n+1
|y−x|
2
−t (−)β
e f (w) |x − w|−(n+1) dw,
|w−x|r
we establish r 2β
Fr,x (t, y) 2 dy
|y−x|
0
r 2β
2 dt
−t (−)β −(n+1)
e |x − w| f (w) α/β dw t
r n+1 |w−x|r
∞
2
∞
0
−k(n+1)
k=1
dt t α/β
r 2β
|w−x|2k+1 r
2−k(n+1)
2β (2k+1 r)
k=1
0
dw
0
2 dt
−t (−)β
e f (w) α/β t
dw
|w−x|2k+1 r
r n−2α−2β+2 f Qβ,−1 (Rn ) α;∞
2 dt
−t (−)β
e f (w) α/β t
∞
2−k(2α+2β−1) r n−2α−2β+2 f Qβ,−1 (Rn ) .
k=1
α;∞
Now we have proved that r 2β 0 |y−x|
that is, fj,k ∈ Qα;∞ (Rn ).
fr (t, y) 2 dy dt r n−2α−2β+2 f 2 β,−1 , Qα;∞ (Rn ) t α/β
2
Using Lemma 4.2, we can prove the following proposition. By this result, we can regard β,−1 β Qα,∞ (Rn ) as derivatives of Qα (Rn ).
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
Proposition 4.4. For α > 0 and max{α, 1/2} < β < 1 with α + β − 1 0, n β,−1 Qα;∞ Rn = ∇ · Qβα Rn , β
where a tempered distribution f ∈ Rn belongs to ∇ · (Qα (Rn ))n if and only if there are fj ∈ β Qα (Rn ) such that f = nj=1 ∂j fj . β
β
Proof. For any f ∈ ∇ · (Qα (Rn ))n , there exist f1 , f2 , . . . , fn ∈ Qα (Rn ) such that f = n j =1 ∂j fj . We have f Qβ,−1 (Rn ) α;∞
n j =1
∂j fj Qβ,−1 (Rn ) α;∞
n j =1
fj Qβ (Rn ) . α
On the other hand, if f ∈ Qα;∞ (Rn ) and fj,k = ∂j ∂k (−)−1 f, then fj,k ∈ Qα;∞ (Rn ) accordβ,−1
β,−1
ing to Lemma 4.3. Thus we have fk = −∂k (−)−1 f ∈ Qα (Rn ) and β
n n n ∂k fk (ξ ) = − iξk f k (ξ ) = − iξk × iξk |ξ |−2 f (ξ ) = f (ξ ). k=1
k=1
2
k=1
β,−1
β,−1
Remark 4.5. Qα;∞ (Rn ) is critical for Eqs. (1.1) since Qα;∞ (Rn ) is the derivative space of β
β
Qα (Rn ) and Qα (Rn ) is invariant under the scaling f (x) → λ2β−2 f (λx). 0 ” In the following proposition we apply the arguments in the proof of the “minimality of B˙ 1,1 1−2β used by Frazier, Jawerth and Weiss in [9] to prove that B˙ ∞,∞ (Rn ) contains all critical spaces for Eqs. (1.1). The special case β = 1 of this proposition was proved by Cannone in [4].
Proposition 4.6. If a translation invariant Banach space of tempered distributions X is a critical space of the generalized Navier–Stokes equations (1.1), then X is continuously embedded in the 1−2β Besov space B˙ ∞,∞ (Rn ). Proof. It follows from the assumption that X → S (Rn ) and for any f ∈ X f (·) = λ2β−1 f (λ · −x0 ) , X X
λ > 0, x0 ∈ Rn .
X → S (Rn ) implies that there exists a constant C such that
% 2β &
K , f C f X , 1
∀f ∈ X.
According to the transformation invariance of X, we have −(−)β 2β e f L∞ (Rn ) = K1 ∗ f L∞ (Rn ) C f X ,
∀f ∈ X.
(4.1)
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2493
β Using the fact f (λx)(ξ ) = λ−n f (ξ/λ), the definition of e−(−) f (x) and the scaling property (4.1), we obtain that
2β β λ2β−1 e−λ (−) f L∞ (Rn ) C f X . s It follows from Miao, Yuan and Zhang [19, Proposition 2.1] that for s < 0, f ∈ B˙ ∞,∞ (Rn ) if and only if
2β β sup r −s e−r (−) f L∞ (Rn ) < ∞. r>0
Thus X → B˙ ∞,∞ (Rn ). 1−2β
2
Theorem 4.7. Let α > 0 and max{α, 12 } < β < 1 with α + β − 1 0. If 1 q ∞, 2 < p < ∞ and α + β < 1 +
n p
1+ n −2β
< 2β, then B˙ p,q p
n
1+ −2β (Rn ) and B˙ 2,q 2 (Rn ) are continuously embedded in
β,−1
Qα;∞ (Rn ). 1+ n −2β
1+ n −2β
1+ n −2β
(Rn ) → Qα;∞ (Rn ). Since B˙ p,q p (Rn ) ⊂ B˙ p,∞p (Rn ). AsProof. We first prove B˙ p,q p n sume that q = ∞, it follows form 1 + p − 2β < 0 and Proposition 2.1 of [19] that for any 1+ n −2β
f ∈ B˙ p,∞p
β,−1
(Rn ), sup r
−(1+ pn −2β)/2β −r(−)β
e
r>0
f Lp (Rn ) < ∞.
Then we have r 2β
2
−t (−)β
e f (y) t −α/β dy dt
0 |y−x|
r 2β β 2 r n(p−2)/p e−t (−) f p
L (Rn )
t −α/β dt
0
r n(p−2)/p
r 2β# sup t r 2β t 0
r
n−2(α+β−1)
e
t>0
0
r n(p−2)/p
−(1+ pn −2β)/2β −t (−)β
.
(1+ pn −2β)/β −α/β
t
dt
$2 (1+ n −2β)/β −α/β f Lp (Rn ) t p t dt
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 n
1+ −2β β,−1 β,−1 Thus f ∈ Qα;∞ (Rn ). Now we prove B˙ 2,q 2 (Rn ) → Qα;∞ (Rn ). Since 0 < α < β and 1/2 < β < 1, we can find p ∈ (2, ∞) large enough such that α + β < 1 + pn < 2β and 1 + n2 − 2β =
1+
n p
− 2β + n( 12 − p1 ). Then (ii) of Theorem 2.1 implies 1+ n −2β n 1+ n −2β n β,−1 R → B˙ p,q p R → Qα;∞ Rn . B˙ 2,q 2
2
4.2. Several technical lemmas We prove several technical lemmas used in the proof of our well-posedness result. The first one can be regarded as a generalization of Lemma 3.1 in [29]. However since Schur’s lemma is not valid for general β, we can’t apply the method used in [29]. 1+n Lemma 4.8. Given α ∈ (0, 1). For a fixed T ∈ (0, ∞] and a function f (·,·) on R+ , let A(t) = t −(t−s)(−)β β (−) f (s, x) ds. Then 0e
T
A(t, ·) 2 2
dt
L (Rn ) t α/β
T
0
f (t, ·) 2 2
dt
L (Rn ) t α/β
.
(4.2)
0
Proof. It is easy to see that we only need to prove the above inequality for T = ∞. In fact if T < ∞, one could extend f by letting f = 0 on (T , ∞) because Af counts only on the values of f on (0, t) × Rn . Furthermore we may define f = 0 = Af for t ∈ (−∞, 0). β According to the definition of e−(t−s)(−) , by Fubini’s and Plancherel’s theorems, we have ∞ 2 IA = A(t, ·) 2
dt
L (Rn ) t α/β
0
2 ∞ t 2β −(t−s)|ξ |2β f (s, ξ ) dξ = |ξ | e 2 0
L (Rn )
0
∞ t 0
Rn
0
∞ ∞ Rn
0
0
|ξ |2β
ds
f (s, ξ ) exp (t − s)|ξ |2β
1{0st} Rn
Since
t 0
0
2
dξ
dt t α/β
|ξ |2β
ds
f (s, ξ ) 1{0st} exp (t − s)|ξ |2β
∞ ∞
dt t α/β
0
t
|ξ |2β e(t−s)|ξ |
2β
ds 0
2
dt t α/β
2β
2β
dξ
2
dt
f (s, ξ ) ds ds α/β dξ. 2β t e(t−s)|ξ | |ξ |2β
|ξ |2β e−(t−s)|ξ | ds e−t|ξ | (et|ξ | − 1) 1, we have 2β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
∞ t IA Rn
0
0
2
|ξ |2β
f (s, ξ ) ds exp (t − s)|ξ |2β
∞ ∞
2 s|ξ |2β
f (s, ξ ) e Rn
0
Rn
0
s
|ξ |2β exp t|ξ |2β
2495
dt
dξ
t α/β
ds dt α/β s
dξ
∞
2 s|ξ |2β −t|ξ |2β ∞ ds
f (s, ξ ) e −e |s dξ s α/β ∞
f (t, ·) 2 2
dt
L (Rn ) s α/β
2
.
0
The next lemma is a Carleson-type estimate. Such an estimate was originally proved by Koch and Tataru [16] for α = 0, β = 1 and k = 0. Then Germain, Pavlovi´c and Staffilani [10] proved a similar result for α = 0, β = 1 and k > 0. In our result, we consider more general case β ∈ ( 12 , 1). Lemma 4.9. For β ∈ (1/2, 1) and N (t, x) defined on (0, 1) × Rn , let A(N ) be the quantity r 2β A(α, β, N) =
sup
r
2α−n+2β−2
x∈Rn ,r∈(0,1)
0 |y−x|
N (t, x) dx dt . t α/β
Then for each k ∈ N0 := N ∪ {0} there exists a constant b(k) such that the following inequality holds: 2 1 t k kβ+1 t β 2 − (−) N (s, ·) ds t (−) 2 e 2 2 0
L (Rn )
0
1 b(k)A(α, β, N) 0
Rn
dt t α/β
N (s, x) dx ds . s α/β
(4.3)
Proof. The main tool used in the proof is a T T ∗ type argument which was used first in [16] and was after that used in [10]. We follow their ideas. Using the inner-product ·,· in L2 with respect to the spatial variable x ∈ Rn , we obtain 2 1 t k kβ+1 t β 2 − (−) I = t (−) 2 e 2 N (s, ·) ds 2 0
=
k 2
t (−) 0
L (Rn )
0
1 * t 0
kβ+1 2
e
− 2t (−)β
dt
t N (s, ·) ds,
t α/β +
k 2
t (−) 0
kβ+1 2
e
− 2t (−)β
dt
N (h, ·) dh L2
t α/β
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
*
= 2Re
N (s, ·),
kβ+1 −t (−)β
k
t (−)
e
N (h, ·) dt L2
s
0
+
1
*
= 2Re
1 N (s, ·), (−)
1−β
dh ds s α/β
+ β k −t (−)β β t (−) e N (h, ·) d t (−) L2
s
0
+ 1
* s
1−β Lk (1) − Lk (s) N (h, ·) dh N (s, ·), (−)
0
0
dh ds s α/β
ds
α/β
s 2
L
β where Lk (t) = km=0 bm (k)t m (−)mβ e−t (−) . β We consider the ν-th derivative of the kernel Kt (x) and let β ν β K1 (x) = (−)ν/2 K1 (x)
and
β ν β Kt (x) = (−)ν/2 Kt (x).
Using the estimates β ν K1 (x)
β ν − ν − n β ν and Kt (x) = t 2β t 2β K1
1 (1 + |x|)n+ν
x
t 1/2β
(see Miao, Yuan and Zhang [19, Lemma 2.2 and Remark 2.1]), we get the kernel of the above operator satisfies the estimate:
(−)1−β Lk (t)(x, y)
k
t
m− 2mβ+n+2−2β 2β
m=0
t
− n+2−2β 2β
k m=0
bm (k) − y|)n+2mβ+2−2β
(1 + t −1/2β |x
bm (k) , − y|)n+2mβ+2−2β
(1 + t −1/2β |x
then we have
s
1−β Lk (s)N (h, x) dh
(−)
0
s
− n+2−2β 2β
s k
bm (k)
0 Rn m=0
s
− n+2−2β 2β
k m=0
bm (k)
|N (h, y)| dy dh (1 + s −1/2β |x − y|)n+2mβ+2−2β
s k∈Zn 0
x−y t 1/2β
∈k+[0,1]n
|N (h, y)| dy dh (1 + s −1/2β |x − y|)n+2mβ+2−2β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
b(k) sup sup t x∈Rn
− n+2−2β 2β
0
t
2497
N (h, y) dy dh
0 |x−y|
ρ
2β
b(k) sup sup ρ 2α−n+2β−2 x∈Rn 0<ρ<1
0 |x−y|<ρ
N (h, y) dy dh . hα/β
Hence we can get 1 I b(k) 0 Rn
N (s, x) ds dx A(α, β, N ). s α/β
2
This completes the proof.
Remark 4.10. Similarly when k = 0, we can prove the following inequality: 2 1 t 1 β N (s, ·) ds (−) 2 e−t (−) 2 0
dt
L (Rn )
0
t α/β
1 A(α, β, N ) 0 Rn β
N (s, x) dx ds . s α/β
1 −t (−)β ∂j ∂k e β smooth function Kj,k
Lemma 4.11. For 1 j, k n and t > 0, the operator Qj,k,t = operator with the kernel α ∈ Nn
β β 1 x Kj,k ( t 1/2β ) Kj,k,t (x) = t n/2β
for a
(4.4)
is a convolution such that for all
n+|α| α β 1 + |x| ∂ Kj,k ∈ L∞ Rn . 2β ξ ξ ' β Proof. Since Kj,k (ξ ) = |ξj |2k e−|ξ | , we have β β ∂ α Kj,k (ξ ) dξ < ∞. Thus ∂ α Kj,k (x) ∈ L∞ (Rn ). For |x| 1, we have
2β ξ ξ β ∂ α Kj,k (ξ ) |ξ ||α| |ξj |2k e−|ξ |
and
1 + |x| n+|α| ∂ α K β (x) ∂ α Kj,k (x) 1. j,k β β β β β For |x| > 1, we write Kj,k = (I −S0 )Kj,k + l<0 l Kj,k where (I −S0 )Kj,k ∈ S and l Kj,k = l 2β ξ ξ β β β 2ln ωj,k,l (2l x) with ωj,k,l = ψ(ξ ) |ξj |2k e−|2 ξ | ∈ L1 . Then the set {ωj,k,l : l < 0} is bounded in S and there exists a uniform constant CN such that
N β 1 + 2l |x| 2l(n+|α|) ∂ α l Kj,k (x) CN . Thus
α
∂ S0 Kj,k (x) 2l(n+|α|) + 2l(n+|α|−N ) |x|−N |x|−n−|α| . 2l |x|1
2l |x|>1
2
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
Remark 4.12. Taking β = 1, the above result comes back to [17, Proposition 11.1]. 4.3. Well-posedness In this subsection, we establish the well-posedness result for the solutions to Eqs. (1.1). Throughout this subsection, we always assume β ∈ ( 12 , 1). In fact our results can be also applied to the case β = 1, that is, the classical Naiver–Stokes equations. Hence our results can be regarded as a generalization of the result of Koch and Tataru [16] when α = 0, β = 1 and that of Xiao [29] when α ∈ (0, 1), β = 1. Definition 4.13. Let α > 0 and max{1/2, α} < β < 1 with α + β − 1 0. β,−1
(i) A tempered distribution f on Rn belongs to Qα;T (Rn ) provided f Qβ,−1 (Rn ) = α;T
r 2β
β α
Kt ∗ f (y) 2 t − β dy dt
r 2α−n+2β−2
sup x∈Rn ,r∈(0,T )
1/2 < ∞.
0 |y−x|
(ii) A tempered distribution limT →0 f Qβ,−1 (Rn ) = 0.
on
f
Rn
belongs
to
β,−1
V Qα
(Rn )
provided
α;T
β
1+n belongs to the space Xα;T (Rn ) provided (iii) A function g on R+
g Xβ
n α;T (R )
= sup t
g(t, ·) L∞ (Rn )
1 1− 2β
t∈(0,T )
+
r 2β
g(t, y) 2 t −α/β dy dt
r 2α−n+2β−2
sup x∈Rn ,r 2β ∈(0,T )
1/2 < ∞.
0 |y−x|
Theorem 4.14. Let n 2, α > 0 and max{α, 1/2} < β < 1 with α + β − 1 0. Then (i) The generalized Navier–Stokes system (1.1) has a unique small global mild solution in β (Xα;∞ )n for all initial data a with ∇ · a = 0 and a (Qβ,−1 (Rn ))n being small. α;∞
(ii) For any T ∈ (0, ∞) there is an ε > 0 such that the generalized Navier–Stokes system (1.1) β has a unique small mild solution in (Xα,T )n on (0, T ) × Rn when the initial data a satisfies β,−1
∇ · a = 0 and a (Qβ,−1 (Rn ))n ε. In particular for all a ∈ (V Qα α;T
(Rn ))n with ∇ · a = 0
β
there exists a unique small local mild solution in (Xα;T )n on (0, T ) × Rn . Proof. By Picard’s contraction principle, it is sufficient to verify the bilinear operator t B(u, v) = 0
e−(t−s)(−) P ∇ · (u ⊗ v) ds β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 β
β
2499
β
is bounded from (Xα;T )n × (Xα;T )n to (Xα;T )n . Our proof follows the idea introduced by [16] and used in [17] and [29]. We need to estimate the L2 -bound and the L∞ -bound, respectively. Part 1. L2 -bound. We want to establish that if x ∈ Rn and r 2β ∈ (0, T ) then r 2β r 2α−n+2β−2 0 |y−x|
B(u, v) 2 dy ds u 2 β v 2 β . (Xα;T )n (Xα;T )n s α/β
(4.5)
To this aim, define 1r,x (y) = 1|y−x|<10r (y), i.e., the indicate function on the ball {y ∈ Rn : |y − x| < 10r}. We divide B(u, v) into three parts: B(u, v) = B1 (u, v) + B2 (u, v) + B3 (u, v), where s B1 (u, v) =
β e−(s−h)(−) P ∇ · (1 − 1r,x )u ⊗ v dh,
0 −1/2
B2 (u, v) = (−)
s P∇ ·
β β e−(s−h)(−) (−) (−)−1/2 I − e−h(−) (1r,x )u ⊗ v dh,
0 −1/2
B3 (u, v) = (−)
1/2 −s(−)β
P ∇ · (−)
s
e
(1r,x )u ⊗ v dh.
0
At first, we estimate B2 (u, v) as r 2β 2 dt I = B2 (u, v) L2 (Rn ) α/β t 0
2 r 2β s −(s−h)(−)β −1/2 −h(−)β I −e (1r,x )u ⊗ v dh e (−) (−) 2 0
dt
L (Rn )
0
2 r 2β s β β e−(s−h)(−) (−)β (−)1/2−β I − e−h(−) (1r,x )u ⊗ v dh 2 0
t α/β
L (Rn )
0
dt t α/β
r 2β 2 dt β (−)1/2−β I − e−h(−) (1r,x )u ⊗ v dh L2 (Rn ) α/β t 0
where the L2 boundedness of Riesz transform is used in the second inequality and in the last one, we have used Lemma 4.8. 2β Since sups∈(0,∞) s 1−2β (1 − e−s ) < ∞ for 12 < β < 1, we can obtain that (−)1/2−β (I − 1
e−s(−) ) is bounded on L2 with operator norm s 2β . Write (1r,x )u(s, x) ⊗ v(s, x) = M(s, x). Thus, using the Cauchy–Schwarz inequality, we have β
1−
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
r 2β 2 ds 2− 1 I s β M(s, ·) L2 (Rn ) α/β s 0
r 2β 2− 1 s β
|y−x|
0
#
sup s
u(s, y)v(s, y) 2 dy ds s α/β $# $ 1− 1 u(s, y) L∞ (Rn ) sup s 2β v(s, y) L∞ (Rn )
1 1− 2β
s∈(0,T )
s∈(0,T )
r 2β × 0 |y−x|
u(s, y) 2 dy ds s α/β
r n−2α−2(β−1) u 2
β
(Xα;T )n
v 2
1/2 r 2β 0 |y−x|
β
(Xα;T )n
v(s, y) 2 dy ds s α/β
1/2
.
Now by Lemma 4.9 with k = 0, we estimate the term B3 as follows. r 2β B3 (u, v) 2 2
dt
L (Rn ) t α/β
0
2 t r 2β 1/2 −t (−)β (−) e M(s, ·) ds 2 0
L (Rn )
0
dt t α/β
2 τ 1 β r n−2α+6β−2 (−)1/2 e−τ (−) M r 2β θ, r· dθ 2 0
1 r
n−2α+6β−2
=r
n−2α+6β−2
L (Rn )
0
2β M r s, r·
L1 (Rn )
dτ τ α/β
ds C(α, β; f ) s α/β
0
× II × A α, β; M r 2β s, ry .
For II, we have r 2β II = r 2α−n−2β 0 |z−x|
M(t, z) dz dt r 2−4β u β n v β n . (Xα;T ) (Xα;T ) t α/β
For C(α, β; M(r 2β s, ry)), we have C α, β; M r 2β s, ry ρ 2α−n+2(β−1)
ρ
2β
0 |y−x|<ρ
2β
M r s, ry dy ds s α/β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 2β (rρ)
ρ
2α−n+2(β−1) 2α−n−2β
r
|z−x|
0 2β (rρ)
r
2−4β
2α−n+2(β−1)
(rρ)
|z−x|
0
r 2−4β u (Xβ
α;T )
n
v (Xβ
α;T )
n
2501
M(t, z) dz dt t α/β
M(t, z) dz dt t α/β
.
Therefore we get r 2β B3 (u, v) 2 2
dt
L (Rn ) t α/β
r n−2α+6β−2 r 2−4β r 2−4β u 2
β
(Xα;T )n
v 2
β
(Xα;T )n
0
= r n−2α−2β+2 u 2
β
(Xα,T )n
v 2
β
(Xα;T )n
,
that is, r 2β 2 r 2α−n+2(β−1) B3 (u, v) 2
dt
L (Rn ) t α/β
u 2
β
(Xα,T )n
v 2
β
(Xα;T )n
.
0
For the estimate of B1 , according to Lemma 4.11, we have e−t (−) P ∇ · f (x) = β
β
∇Kj,k,t (x − y)f (y) dy
and 1
1
β
∇Kj,k,t (x − y)
t
n 1 2β + 2β
(1 + t −1/2β |x
− y|)n+1
(t 1/2β
1 . + |x − y|)n+1
Thus
s
B1 (u, v)
e−(s−h)(−)β P ∇ · (1 − 1r,x )u ⊗ v dh
0
s
0 |z−x|10r
|u(h, z)||v(h, z)| dz dh. ((s − h)1/2β + |z − y|)n+1
When |z − x| 10r, 0 < s < r 2β and |y − x| < r, we have |y − z| |z − x| − |y − x| 9r > 9|y − x|.
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
Thus 1 10 |x − z| |x − y| + |y − z| |y − z| + |y − z| = |y − z|. 9 9 This gives us
B1 (u, v)
r 2β
0 |z−x|10r
|u(h, z)||v(h, z)| dz dh = I1 × I2 , |x − z|n+1
where r 2β
I1 = 0 |z−x|10r
∞
1/2 |u(h, z)|2 dz dh |x − z|n+1
r 2β
j =3 0 j 2 r|z−x|2j +1 r
∞ j =3
1/2 |u(h, z)|2 dz dh (2j r)n+1
2β α/β j 2β−2 j 2−2β 1 r 2 r 2 r (2j r)n+1
r 2β
r 2β−1
0 |z−x|2j +1 r
1/2
1
0 2j r|z−x|2j +1 r
∞ j 2α−n j −1 j 2β−2 j 2−2β 2 r 2 r 2 r 2 r j =3
r 2β
u (Xβ
α;T )
u(h, z) 2 dz dh hα/β
u(h, z) 2 dz dh hα/β
α;T )
n
. Thus |B1 (u, v)|
1 u (Xβ )n v (Xβ )n . r 2β−1 α;T α;T
When 0 < α < β, we have
0 |y−x|
1/2
n
where we have used the condition β > 1/2. 1 Similarly, we obtain I2 ( r 2β−1 )1/2 v (Xβ r 2β
1/2
B1 (u, v) 2 dy dt 1 r n t α/β r 4β−2
r 2β
dt u 2 β n v 2 β n (Xα;T ) (Xα;T ) t α/β
0
r n−2α−2β+2 u 2
β
(Xα;T )n
v 2
β
(Xα;T )n
.
This implies that r 2β r 2α−n+2(β−1) 0 |y−x|
B1 (u, v) 2 dy dt u 2 β v 2 β . (Xα;T )n (Xα;T )n t α/β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2503
Part 2. L∞ -bound. The aim of this part is to prove B(u, v) If
t 2
1
L∞ (Rn )
t 2β
−1
u (Xβ
α;T )
n
v (Xβ
α;T )
n
∀t ∈ (0, T ).
,
s < t then −(t−s)(−)β e P ∇ · (u ⊗ v)
L∞ (Rn )
u L∞ (Rn ) v L∞ (Rn )
1
(t − s) 2β 1 − 2β
(t − s) If 0 < s <
t 2
1
−2
sβ
u (Xβ
α;T )
n
v (Xβ
α;T )
n
.
then t − s ≈ t and so
−(t−s)(−)β
e P ∇ · (u ⊗ v)
1
Rn
|u(s, y)||v(s, y)| ((t − s) 2β + |x − y|)n+1 |u(s, y)||v(s, y)| 1
(t 2β + |x − y|)n+1
dy
dy
Rn
|u(s, y)||v(s, y)| 1
k∈Zn
(t 2β (1 + |k|))(n+1)
1
dy.
x−y∈t 2β (k+[0,1]n )
This gives us
B(u, v)
t/2 t
−(t−s)(−)β
β
e P ∇ · (u ⊗ v) ds + e−(t−s)(−) P ∇ · (u ⊗ v) ds 0
t/2
1 −(n+1) t 2β 1 + |k|
t/2
k∈Zn
0
t +
1 − 2β
(t − s)
1
−2
sβ
u(s, y)
v(s, y) dy
1
x−y∈t 2β (k+[0,1]n )
ds u (Xβ
α;T )
n
v (Xβ
α;T )
n
t/2
:= I3 + I4 . Here, t I4 t/2
1 − 2β
(t − s)
1
sβ
−2
ds u (Xβ
α;T )
n
v (Xβ
α;T )
n
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 1
tβ
1 −2 1− 2β
1
t
t 2β
−1
u (Xβ
α;T )
u (Xβ
α;T )
n
n
v (Xβ
α;T )
v (Xβ
α;T )
n
.
n
On the other hand, we have t/2 1 −(n+1) I3 t 2β 1 + |k| k∈Zn
0
t/2
:=
u(s, y) 2 dy ds
1/2
1
|x−y|t 2β
v(s, y) 2 dy ds
× 0
1/2
1
|x−y|t 2β
1 −(n+1) t 2β 1 + |k| I3,1 × I3,2 . k∈Zn
Here, t/2
u(s, y) 2 dy ds
I3,1 = 0
1
|x−y|t 2β
= t
1/2
1 1 2β (n−2β+2) 2β (2α−n+2β−2)
t/2
1
(n−2β+2)
1
Thus t
1 1− 2β
1 − 2β (n+1)
u (Xβ
α;T )
Similarly, we get I3,2 t 4β I3 t
u(s, y) 2 dy ds s α/β
t
0
t 4β
1
t 2β
(n−2β+2)
(n−2β+2)
n
α;T )
n α;T )
B(u, v) L∞ (Rn ) u (Xβ
α;T )
n
1
|x−y|t 2β
.
v (Xβ
u (Xβ
1/2
n
. These estimates about I3,1 and I3,2 imply that
v (Xβ
n α;T )
v (Xβ
α;T )
n
1
t 2β
−1
u (Xβ
α;T )
n
v (Xβ
α;T )
n
.
.
Therefore, we establish the boundedness of B(u, v) and finish the proof of (i) and (ii) by taking T = ∞ and T ∈ (0, ∞), respectively. 2 5. Regularity of generalized Navier–Stokes equations In this section, we study the regularity of the solutions to Eqs. (1.1) with β ∈ (1/2, 1). For β = 1, that is, the classical Navier–Stokes equations, the regularity has been studied by sev-
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2505
eral authors. In [10], Germain, Pavlovi´c and Staffilani analyzed the regularity properties of the solutions constructed by Koch and Tataru. In this section, we establish a similar result for the solutions of Eqs. (1.1) evolving initial data β,−1 in Qα;∞ (Rn ) with β ∈ (1/2, 1]. In fact we get the solution u to Eqs. (1.1) satisfies: k
t 2β ∇ k u ∈ Xαβ,0 , β,0
where Xα
∀k ∈ N0 ,
β
1 is the space Xα;∞ constructed in (iii) of Definition 4.13 for β ∈ (1/2, 1) and Xα;∞ β,k
in Xiao [29] for β = 1. For convenience of the study, we introduce a class of spaces Xα follows.
as
β,k
Definition 5.1. For a nonnegative integer k and β ∈ (1/2, 1], we introduce the space Xα which is equipped with the following norm: u Xβ,k = u N β,k + u N β,k α
α,∞
α,C
where u N β,k = α,∞
sup
sup t
2β−1+k 2β
α1 +···+αn =k t>0
α ∂ 1 · · · ∂ αn u(·, t) x1
u N β,k = α,C
sup
xn
r 2β
sup r
2α−n+2β−2
α1 +···+αn =k x0 ,r
0 |y−x0 |
L∞ (Rn )
,
k α
t 2β ∂ 1 · · · ∂ αn u(t, y) 2 dy dt x1 xn t α/β
1/2 .
In the following, we will denote ∇ k u = ∂xα11 · · · ∂xαnn u with (α1 , α2 , . . . , αn ) ∈ Nn0 and k = α1 + · · · + αn . 5.1. Several technical lemmas Before stating the main result of this section, we prove several preliminary lemmas associated β β β β with the fractional heat semigroup e−t (−) . Recall that e−t (−) f (x) = Kt ∗ f (x) where Kt is 2β β defined by (Kt )(ξ ) = e−t|ξ | and P is the Helmboltz–Weyl projection. In the next lemma, we get an estimate about the fractional heat semigroup. The special case β = 1 was proved in [20, Lemma 2.5] and was stated and used in [10, Proposition 3.6]. Lemma 5.2. Let β ∈ (1/2, 1). There exists a constant C > 0 depending only on n such that
k 1 1
∂ P ∇Ktβ (x) C k k k/2β t −k/2β k − 2β t 2β + |x| −n−1 x for all t > 0, x ∈ Rn and k ∈ N. Proof. By a dilation argument, we have β
∂xk P ∇Kt (x) = t
−k−1 2β
t
n 1 β − 2β ∂xk P ∇K1 x/t 2β .
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 β
If we could prove |∂xk P ∇K1 (x)| C k k k/2β (k
1 − 2β
+ |x|)−n−1 , then we have
−n−1
k k+1 n 1
∂ P ∇Ktβ (x) t − 2β t − 2β C k k k/2β k − 2β + x x
t 1/2β −n−1 − k 1 − 1 ck k k/2β t 2β t 2β k 2β + |x| . Hence we obtain the desired. β β β By the semigroup property, it is easy to see that ∂xk P ∇K1 = P ∇K1/2 ∗ ∂xk K1/2 . So we need to prove the following two estimates:
P ∇K β (x) C 1 + |x| −n−1 , 1/2
k β
k−1 1
∂ K (x) C k−1 k 2β k − 2β + |x| −n−1 . x 1/2 For (5.1), taking α = 1 in the Lemma 4.11, we have
n+1
P ∇K β (x) C, 1 + |x| 1/2 that is, (5.1) is obvious. β For (5.2), we claim that |∂i K1/2 (x)| C(1 + |x|)−n−1 . In fact when |x| < 1,
n+1
∂i K β (x) 2n+1 1 + |x| 1/2
|iξi |e−|ξ |
2β /2
dξ C.
Rn
When |x| > 1, we define the operator L(x, D) =
x · ∇ξ , i|x|2
that is,
L(x, D)eix·ξ = eix·ξ
and choose a Cc∞ (Rn )-function ρ(x) satisfying: ρ(ξ ) =
1, |ξ | 1, 0, |ξ | > 2,
we have
∂i K β (x) ρ ξ iξi e−|ξ |2β /2 eix·ξ dξ 1/2
δ Rn
,
ξ −|ξ |2β /2 ix·ξ
+ 1−ρ iξi e e dξ
δ Rn
:= I3 + I4 . For I3 , we have
(5.1) (5.2)
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
I3
ρ
ξ 2β |ξ |e−|ξ | /2 dξ δ
2507
δ dξ δ n+1 .
|ξ |<2δ
Rn
For I4 , using the integration by parts and L∗ = − i|x|2ξ , we have x·∇
,
∗ N
ξ −|ξ |2β /2 ix·ξ
1−ρ e I4 = iξi e L dξ
δ Rn
−N
CN |x|
,
N ξ
N −k k k −|ξ |2β /2 ∇ξ iξ e C N ∇ξ 1 − ρ
dξ
δ
Rn
k=0
−N
N
CN |x|
|ξ |2βk−N +1 e−|ξ |
2β /2
dξ
|ξ |>δ k=1
+ CN |x|−N
−N
N
k −k CN δ
δ|ξ |2δ k=1
CN |x|
|ξ |
1−N
l 2βl−N +k+1 −|ξ | CN e −k |ξ |
2β /2
dξ
l=0
−N
dξ + CN |x|
|ξ |>δ
δ −k |ξ |1−N +k dξ
δ<|ξ |<2δ
−N n+1−N
CN |x|
N −k
δ
.
So we get, taking δ = |x|−1 ,
∂i K β (x) δ n+1 + CN |x|−N δ n+1−N CN |x|−(n+1) CN 1 + |x| −(n+1) . 1/2 Then we have
1 n 1
∂i K β1 (x) k 2β k 2β ∂i K β k 1/2β x C k − 2β + |x| −n−1 . 1/2 2k
Because the following integral inequality (see [20]): −n−1 −n−1 −n−1 a + |x − y| b + |y| dy ca −1 a + |x|
for 0 < a b,
Rn
we have
2 β
∂ K 1 (x) = ∂i K β1 (x − y)∂j K β1 (y) dy i,j
k
Rn
Rn
2k
2k
−1 −n−1 − 1 −n−1 k 2β + |x − y| k 2β + |y| dy
1 −n−1 − 1 k 2β k 2β + |x| .
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 β
Operating the process above k − 1 times, we get |∂xk K1/2 (x)| k
k−1 2β
(k
1 − 2β
+ |x|)−n−1 and
k
∂ P ∇K β (x) = P ∇K1/2 (x − y)∂ k K β (y) dy x x 1/2 1
Rn
−n−1 k−1 k−1 − 1 −n−1 c 1 + |x − y| c k 2β k 2β + |y| dy
Rn
1 −n−1 k−1 − 1 C k k 2β k 2β + |x| k 2β k −n−1 − 1 C k k 2β k 2β + |x| .
This completes the proof of this lemma.
2
The following lemma can be regarded as a generalization of Proposition 3.2 of [10]. Lemma 5.3. If r is a natural number, α ∈ (0, 1) and max{α, 12 } < β 1, the operator t Prβ f (t, x) =
1 r β 1 e−(t−s)(−) t 2β − s 2β ∇ r+2β f (s, x) ds
0 dt is bounded on L2 ([0, T ], L2 (Rn , dx), t α/β ) for any T ∈ [0, ∞] with constants p(r) and q(r).
Proof. By Plancherel’s theorem and Hölder’s inequality, we have ∞ β P f (t, ·) 2 2 r
dt
L (Rn ) t α/β
0
∞ t 0
Rn
e
−(t−s)|ξ |2β
e
−(t−s)|ξ |2β
2
dt t α/β
dξ
0
∞ t 0
1 r 1 t 2β − s 2β |ξ |r+2β f (s, ξ ) ds
Rn
1 r 1 t 2β − s 2β |ξ |r+2β ds
0
t
×
e
−(t−s)|ξ |2β
2 1 r 1 t 2β − s 2β |ξ |r+2β f (s, ξ ) ds
dt dξ. t α/β
0
Because t 1/2β − s 1/2β (t − s)1/2β for 2β > 1 and 0 < s < t, it is easy to see that t e 0
−(t−s)|ξ |2β
1 r 1 t 2β − s 2β |ξ |r+2β ds
t 0
r
e−(t−s)|ξ | (t − s) 2β |ξ |r+2β ds 2β
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
∞
2509
e−v v r/2β dv 1.
0
Then we have, by t 1/2β − s 1/2β (t − s)1/2β for 2β > 1 and 0 < s < t again, ∞ β P f (t, ·) 2 2 r
dt
L (Rn ) t α/β
∞ ∞
2 r ds dξ 2β
f (s, ξ ) e−(t−s)|ξ | (t − s) 2β |ξ |r+2β dt s α/β Rn 0
0
s
∞ ∞
2 r ds dξ −u|ξ |2β 2β r+2β
e u |ξ | du f (s, ξ ) s α/β Rn 0
∞ 0 Rn
0
f (s, ξ ) 2 ds dξ. s α/β 2
This completes the proof of this lemma.
β,−1
In the next lemma we prove the fractional heat semigroup is bounded from Qα (Rn ) β,k to Xα . Our result generalizes the related result for β = 1 which was first proved in [10]. Lemma 5.4. For any k 0, α > 0 and max{α, 1/2} < β < 1 with α + β − 1 0, there exists a constant C(k) such that −t (−)β e u
β,k
Xα
C(k) u Qβ,−1 (Rn ) . α;∞
Proof. Because u Xβ,k = u N β,k + u N β,k , we split the proof into two parts. α
α,∞
α,C
β,−1 1−2β 1−2β L∞ part of the norm. Because Qα;∞ (Rn ) → B˙ ∞,∞ (Rn ) and ∇ k : B˙ ∞,∞ (Rn ) →
1−2β−k B˙ ∞,∞ (Rn ), we have
k −t (−)β ∇ e u
L∞ (Rn )
t t
Then we can get t
2β−1+k 2β
1−2β−k 2β
1−2β−k 2β
k ∇ u
1−2β−k B˙ ∞,∞ (Rn )
t
1−2β−k 2β
u B˙ 1−2β (Rn ) ∞,∞
u Qβ,−1 (Rn ) . α;∞
∇ k e−t (−) u L∞ (Rn ) u Qβ,−1 (Rn ) . β
α;∞
β,−1
β
Carleson part. Because u ∈ Qα;∞ (Rn ) = ∇ · (Qα (Rn ))n , there exists a sequence {fj } ⊂ β Qα (Rn ) such that u = j ∂j fj . We only need to prove
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
r 2β sup
r
2α−n+2β−2
x∈Rn ,r>0
0 |y−x|
2 dy dt
k k −t (−)β
t 2β ∇ e ∂j fj (x) α/β C(k) ∂j fj 2 β,−1 n Qα;∞ (R ) t = C(k) fj 2 β
Qα (Rn )
.
(5.3)
β 2β Taking ψ(x) = ∇ k ∂j e−(−) (x) = Rn (iξ )k iξj e−|ξ | e2πix·ξ dξ, we can justify the function t (ξ ) = (itξ )k (itξj )e−t 2β |ξ |2β satisfying the conditions in (1.7): ψ(x) with ψ
ψ(x) 1 + |x| −n−1 ,
ψ(x) ∈ L1
ψ(x) dx = 0.
and Rn
β
By the equivalent characterization of Qα (Rn ) (see (1.8)), we have r
2
k k −s 2β (−)β
s ∇ e (s∂j )fj (x)
sup r 2α−n+2β−2 x,r
0 |y−x|
dy ds C(k) fj 2 β n . Qα (R ) s 1+2α−2(β−1)
By a change of variable: t = s 2β , we get the desired result (5.3).
2
5.2. Regularity Now we state the main theorem of this section. Our regularity result for the mild solutions to Eqs. (1.1) is a generalization of the results obtained in [10]. We apply the author’s idea. Briefly, β,k β,k we estimate the Nα,∞ norm and the Nα,C norm of the nonlinear terms, separately. Let α = 0 and β,k
β,k
k and N k which were introduced in Section 4.3 β = 1, our spaces Nα,∞ and Nα,C retreat to N∞ C and Section 4.4 of [10].
Theorem 5.5. Let α > 0 and max{α, 1/2} < β < 1 with α + β − 1 0. There exists an ε = ε(n) such that if u0 Qβ,−1 (Rn ) < ε, the solution u to Eqs. (1.1) verifies: α;∞
k
t 2β ∇ k u ∈ Xαβ,0 ,
∀k 0.
Proof. We can see that the solution to Eqs. (1.1) can be represented as u(t, x) = e−t (−) u(0, x) − B(u, u)(t, x), β
where t B(u, v)(t, x) = 0
β e−(t−s)(−) P ∇ · u(s, x) ⊗ v(s, x) ds.
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2511
Here u ⊗ v denotes the tensor product of u and v. For the linear term e−t (−) u(0, x) := β β e−t (−) u0 , by Proposition 5.4, we have e−t (−) u0 Xβ,k C(k) u0 Qβ,−1 (Rn ) . Now we estiα α . αβ,k = kl=0 Xαβ,k equipped with the norm kl=0 · β,l . We mate the nonlinear term. We write X β
Xα
shall prove that the bilinear operator maps αβ,k → X αβ,k . αβ,k × X B(u, v) : X β,k
Part 1. Nα,∞ norm. Here we shall prove that B(u, v)
C0 (k) u Xβ,0 v Xβ,0 + C(k)
β,k Nα,∞
α
k−1
α
u N β,l v N β,k−l α,∞
α,∞
l=1
+ C1 u Xβ,0 v Xβ,k + C1 u Xβ,0 v Xβ,k . α
α
If 0 < s < t (1 −
α
α
< t − s < t, by Lemma 5.2, we have
1 t m ), m
1 t (1− m )
k −(t−s)(−)β
∇ e P ∇ · u(s, x) ⊗ v(s, x) ds
I= 0
1 t (1− m )
=C k k
k 2β
Ck k
Because
q∈Zn
m t
1 − 2β
I C k k k/2β
m t
C k
m t
x−y t 1/2β
0
1 − 2β
+
k+1 2β
:= C0 (k)t where C0 (k) = C k k
m
1 t (1− m )
(n+k+1)/2β k
(n+k+1)/2β k
n+k+1 2β
−k−2β+1 2β
t
−k−2β+1 2β
1 2β
1 2β
m
[k
1 − 2β
+ |q|]n+1
dy ds.
1
t t
u Xβ,0 v Xβ,0
u Xβ,0 v Xβ,0
n+k+1 2β
.
α
u(s, y)
v(s, y) dy ds
x−y∈t 2β (q+[0,1]n )
α/β
α
α
k+1 2β
dy ds
|u(s, y)||v(s, y)|
∈q+[0,1]n
0 |x−y|
Ck k
|x−y| n+1 ] (t−s)1/2β
≈ k 1/2β , we have
0
k k/2β
[k
q∈Zn
+|q|]n+1
n+1 2β
1 t (1− m )
(n+k+1)/2β
1 [k
(t − s) (t − s)
Rn
0
k 2β
|u(s, y)||v(s, y)| k 2β
α
u(s, y)
v(s, y) dy ds s α/β
2512
If t (1 −
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519 1 m) s
t, by Young’s inequality, we have
k −(t−s)(−)β
∇ e P ∇ · u(s, x) ⊗ v(s, x)
β = P ∇e−(t−s)(−) ∇ k u(s, x) ⊗ v(s, x) β P ∇e−(t−s)(−) (x) L1 (Rn ) ∇ k u(s, x) ⊗ v(s, x) L∞ (Rn ) . By the estimate for the generalized Oseen kernel:
P ∇ l+1 e−(−)β (x) P∇
l+1 −u(−)β
e
1 (1 + |x|)n+1+l
− n+1+l 2β
P∇
(1 +
|x|
=u
l+1 −(−)β
e
and
, u1/2β x
we have
n+l+1
P ∇ l+1 e−u(−)β u− 2β
1 u1/2β
)l+n+1
(u1/2β
1 . + |x|)l+n+1
Then we take l = 0 and have P ∇e−u(−)β
∞ L1 (Rn )
|x|n−1
0
(u
1 2β
+ |x|)n+1
d|x|
1 . u1/2β
Hence we can get
k −(t−s)(−)β
∇ e P ∇ u(s, x) ⊗ v(s, x) k 1 k ∇ l u(s, ·) ∞ n ∇ k−l v(s, ·) ∞ n L (R ) L (R ) l (t − s)1/2β l=0
k u N β,l v N β,k−l 1 k α,∞ α,∞ . 1/2β (2β−1+l)/2β l s (t − s) s (2β−1+k−l)/2β l=0
So we have
t
∇e
−(t−s)(−)β
P ∇ k+1 u(s, x) ⊗ v(s, x) ds
1 t (1− m )
k k l=0
l
t u N β,l v N β,k−l α,∞
α,∞
1 t (1− m )
1 1 ds. 1/2β (4β−2+k)/2β (t − s) s
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2513
For the integral in the last inequality, we make the change of variable: s = zt. Because t (1− m1 ) < s < t implies (1 − m1 ) < z < 1, we have t
1 1 ds (t − s)1/2β s 4β−2+k 2β
II = 1 t (1− m )
t
1−2β−k 2β
1−
1 m
− 4β−2+k 1 2β
1 − 2β
(1 − z)
dz
1 1− m
=t
Denote g(m) = (1 −
1−2β−k 2β
1 1− m
4β−2+k 1− 1 1 − 2β ( m1 ) 2β m)
− 4β−2+k
2β
∇e
−(t−s)(−)β
P∇
k+1
1−
1 2β
.
k−3
and take m = m(k) = k n+k+1 . We can prove that
g(m) → 0 as k → ∞. Then we have II Ct Therefore we have
t
1 m
1−2β−k 2β
for k 1.
u(s, x) × v(s, x) ds
1 t (1− m )
/ Ct
1−2β−k 2β
k k l=1
l
0 u N β,l v N β,k−l + u N β,0 v N β,k + u N β,k v N β,0 . α,∞
α,∞
α,∞
α,∞
α,∞
α,∞
β,k
Part 2. Nα,C norm. We split B(u, v) as follows: B(u, v) = B1 (u, v) + B2 (u, v) with t B1 (u, v)(t, x) =
e
−(t−s)(−)β
,
x − x0 u(s, x) ⊗ v(s, x) ds, P∇ 1 −φ R 1/2β
0
t B2 (u, v)(t, x) =
e−(t−s)(−) P ∇φ β
x − x0 u(s, x) ⊗ v(s, x) ds R 1/2β
0 1
where φ
1 R 2β
,x0
= φ((x − x0 )/R 2β ) for a smooth function φ supported in B(0, 15) and equals
to 1 on B(0, 10). β For the estimate for B1 , because |P ∇ k+1 e−(t−s)(−) (x)| t < R, we have
K(k) [(t−s)1/2β +|x−y|]n+k+1
and 0 <
2514
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
k k
t 2β ∇ B1 (u, v)(t, x) t
k 2β
t
k
∇
e
−(t−s)(−)β
0 |y−x0 |10R 1/2β
K(k)t
k 2β
t
|u(s, y)||v(s, y)| dy ds [(t − s)1/2β + |x − y|]n+k+1
0 |y−x0 |10R 1/2β
K(k)R
k 2β
R
|u(s, y)||v(s, y)|
0 |y−x0 |10R 1/2β
K(k)R
k n+k+1 2β − 2β
q∈Zn
K(k)D(k)R
1−2β 2β
P ∇(x − y)u(s, y)v(s, y) dy ds
R
n+k+1 2β
1
2β + [( t−s R )
R
1
R |q|n+k+1
|x−y| n+k+1 ] R 1/2β
α/β
dy ds
u(s, y)
v(s, y) dy ds s α/β
0 |y−x0 |
u Xβ,0 v Xβ,0 . α
α
Then we have, taking R = r 2β , r 2β r
2α−n+2β−2 0 |y−x|
k k
t 2β ∇ B1 (u, v)(t, y) 2 dy dt t α/β
2 K(k)D(k) r 2α−n+2β−2
r 2β r 2−4β u 2 β,0 v 2 β,0 0 |y−x|
Xα
Xα
dy dt t α/β
2 K(k)D(k) u 2 β,0 v 2 β,0 . Xα
Xα
For the estimate for B2 , we further split B2 as B2 = B21 + B22 with
B21
1 P∇ =√ − B22
t 0
β I − e−s(−) φ 1 u(s, x) ⊗ v(s, x) ds, e−(t−s)(−) √ 2β R ,x0 −
1 β P ∇e−t (−) =√ − k
t φ 0
1
R 2β ,x0
u(s, x) ⊗ v(s, x) ds.
At first we estimate the term t 2β ∇ k B21 . Without loss of generality we assume k is odd. The proof of the case that k is even is similar. If k is odd, we have k = 2K + 1 for K ∈ Z+ . Because 1 2 < β < 1, we have
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
2515
k 1 1 2K 1 1 1 1 t 2β − s 2β + s 2β t 2β = t 2β − s 2β + s 2β 2K
1 2K−l l 1 1 1 2K 2β1 t − s 2β = s 2β t 2β − s 2β + s 2β l
l=0
=
2K
2K
1 2K−l+1 l 1 2K−l l+1 2K 2β1 2K 2β1 t − s 2β t − s 2β s 2β + s 2β . l l l=0
l=0
Then we have, setting M(s, x) = φ
1
R 2β ,x0
t
k 2β
∇
k
B21
=
2K−1
l=0
2K l
(x)u(s, x) ⊗ v(s, x),
1 P∇ β β l P2K−l+1 (−) 2 −β I − e−s(−) s 2β ∇ l M(s, x) √ −
1 P ∇ β β 2K P1 (−) 2 −β I − e−s(−) s 2β ∇ 2K M(s, x) +√ − 2K−1 2K P ∇ β 1 β l+1 P2K−l (−) 2 −β I − e−s(−) s 2β ∇ l+1 M(s, x) + √ l − l=0 1 P ∇ β β 2K+1 P0 (−) 2 −β I − e−s(−) s 2β ∇ 2K+1 M(s, x) . +√ − Since sups∈(0,∞) s 1−2β (1 − e−s ) < ∞ for 2β
1 2
< β < 1, we can obtain that (−)1/2−β (I −
−s(−)β
) is bounded on L2 with operator norm s e boundedness of Riesz transform, we have r 2β r
2α−n+2β−2 0 |x−x0 |
. By Lemma 5.3 and the L2 -
2
P∇ β
dx ds 1 β l+1 −β −s(−) l+1
√ 2β I −e s ∇ M(s, x)
α/β
− P2K−l (−) 2 s r 2β
p(2K − l)r 2α−n+2β−2 0 Rn
r 2β p(2K − l)r
1 1− 2β
l+1
(−) 12 −β I − e−s(−)β s 2β ∇ l+1 M(s, x) 2 dx ds s α/β
2α−n+2β−2 0 |x−x0 |
1− 1 l+1 l+1
s 2β s 2β ∇ M(s, x) 2 dx ds . s α/β
Because 0 < s < r 2β and l
s 2β
+1
l
∇ l+1 M(s, x) = s 2β =
+1
∇ l+1 φ
1
R 2β ,x0
u(s, x) ⊗ v(s, x)
2β−1+m m η s 2β ∇ u(s, x) s 2β ∇ η v(s, x)
m+ηl+1
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
l +1− 2β−1+m − η l+1−m−η 2β 2β ∇ × s 2β φ
1 R 2β
, ,x0
then we can get, taking R = r 2β , r 2β r
2α−n+2β−2 0 |x−x0 |
1− 1 l+1 l+1
s 2β s 2β ∇ M(s, x) 2 dx ds s α/β
u 2 β,m v 2 β,η .
m+ηl+1
Nα,∞
Nα,C
In a similar way, we have r 2β r
P ∇ β 2K+1 2K+1 2 dx ds 1 −β −s(−)β
√ 2β 2 I −e s ∇ M(s, x)
α/β
− P0 (−) s
2α−n+2β−2 0 |x−x0 |
r 2β p(0)r 2α−n+2β−2 0 Rn
r 2β p(0)r
2 dx ds
1− 1 2K+1 2K+1
s 2β s 2β ∇ M(s, x) α/β s
2K +1
s 2β
2α−n+2β−2 0 |x−x0 |
∇ u∇ v∇ m
η
2K+1−m−η
m+η2K+1
2
dx ds φ 1
α/β 2β s R ,x0
p(0) u 2 β,0 v 2 β,2K+1 + v 2 β,0 u 2 β,2K+1 + r(K) u 2β,2K v 2β,2K . Nα,∞
Nα,C
Nα,∞
Xα
Nα,C
β
Xα
β
Similarly we can estimate the terms associated with P1 and P2K−l+1 . Combining all the estimates together, we can prove
r 2β
r
2α−n+2β−2 0 |y−x0 |
k k 1
t 2β ∇ B (u, v)(t, x) 2 dx dt 2 t α/β
1/2
C1 u Xβ,0 v X β,k + C1 v X β,0 u X β,k + C(k) u X β,k−1 u X β,k−1 . α
α
α
α
α
α
Now we estimate the term B22 . Taking the change of variables: s = r 2β θ , x = rz and t = r 2β τ, we have r 2β I =r
2α−n+2β−2 0 |y−x0 |
r 2β =r
2α−n+2β−2 0 |y−x0 |
k k 2
t 2β ∇ B (u, v)(t, x) 2 dx dt 2 t α/β
2 t
k
dx dt
2β k P ∇ √
−t (−)β −e M(s, x) ds α/β
t ∇ √
t − 0
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
r
2α−n+2β−2
2517
2 r 2β
t
dx dt
2βk k+1 −t (−)β
M(s, x) ds α/β
t ∇ e
t 0 Rn
0
2 1
τ
r n+2β dz dτ k 1 β
= r 2α−n+2β−2 M r 2β θ, rz r 2β dθ
τ r 2β 2β k+1 ∇zk+1 e−τ (−z )
r 2α τ α/β r 0 Rn
0
2 1
τ
dz dτ β
= r 8β−4 M r 2β θ, rz α/β .
τ k/2β ∇zk+1 e−τ (−z )
τ 0 Rn
Denote by ∇zν e−τ (−z ) β 1, we have
0 β/2
(x, y) the kernel of the operator ∇zν e−τ (−z )
k(1−β)
k(1−β) k(1−β) −τ (− )β /2
z
τ 2β ∇ e (x, y) τ 2β
1
z
(τ/2)
β /2
, ν > 0. Because
1
k(1−β)+n 2β
(1 +
|x−y| n+k(1−β) ) τ 1/2β
1 2
∈ L1 R n
uniformly in τ. By Young’s inequality and Lemma 4.9, we have
I =r
8β−4
2 1
k(1−β) τ 2β
dz dτ
2β k(1−β) −τ (−z )β /2 k2 kβ+1 −τ (−z )β /2 ∇z e τ ∇z e M r θ, rz α/β
τ
τ 0 Rn
0
2 1
τ
dz dτ k β
r 8β−4
τ 2 ∇zkβ+1 e−τ (−z ) /2 M r 2β θ, rz α/β
τ 0 Rn
0
1 r
8β−4
b(k)A(α, β, M) 0 Rn
2β
M r θ, rz dz dθ θ α/β
:= r 8β−4 b(k)A(α, β, M)IM . For A(α, β, M), we have ρ
2β
A(α, β, M) = ρ 2α−n+2β−2 0 |y−x|<ρ
2β
M r s, ry ds dy s α/β
2β (rρ)
r 2−4β (rρ)2α−n+2β−2 0
r 2−4β u Xβ,0 v Xβ,0 . α
For IM , we have
α
|z−rx|
M(t, z) dz dt t α/β
<
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P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
1 0 Rn
2β
M r θ, rz dz dθ θ α/β
r 2β 0 Rn
−2β−n dt dz
M(t, z) r r 2−4β u Xβ,0 v Xβ,0 . α α r −2α t α/β
Then we get r 2β r
2α−n+2β−2 0 |y−x0 |
k k 2
t 2β ∇ B (u, v)(t, x) 2 dx dt b(k) u 2 β,0 v 2 β,0 . 2 Xα Xα t α/β
Now we have proved that B(u, v)
β,k
Xα
C0 (k) u Xβ,0 v Xβ,0 + C(k) u X β,k−1 v X β,k−1 α
α
α
α
+ C1 u Xβ,0 v Xβ,k + C1 u Xβ,k v Xβ,0 . α
α
α
α
Similar to the method applied in Lemma 4.3 of [10], if we construct the approximating sequence uj by u−1 = 0,
u0 = e−t (−) u0 , β
uj +1 = u0 + B uj , uj ,
we can get the following lemma and hence complete the proof of Theorem 5.5. Lemma 5.6. Let α > 0 and max{ 12 , α} < β < 1 with α + β − 1 0. Suppose u0 to be small β,−1 enough in Qα;∞ (Rn ). Then for any k 0, there exist constants Dk and Ek such that j u
αβ,k X
Dk
j j +1 2 j u and −u X . αβ,k Ek 3
αβ,k . In particular, for any k 0, uj converges in X
2
Acknowledgments We would like to thank our supervisor Professor Jie Xiao for suggesting the problem and kind encouragement. The authors also thank the referee for reading carefully and suggesting some improvements. References [1] D.R. Adams, A note on Choquet integral with respect to Hausdorff capacity, in: M. Lwikel, J. Peetre, Y. Sagher, H. Wallin (Eds.), Function Spaces and Applications, Lund, 1986, in: Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 115–124. [2] J. Bergh, J. Löfström, Interpolation Spaces: An Introduction, Springer, Heidelberg, 1976. [3] M. Cannone, A generalization of a theorem by Kato on Navier–Stokes equations, Rev. Mat. Iberoamericana 13 (1997) 673–697. [4] M. Cannone, Harmonic analysis tools for solving the incompressible Navier–Stokes equations, in: S. Friedlander, D. Serre (Eds.), Handbook of Mathematical Fluid Dynamics, vol. 3, Elsevier, 2004, pp. 161–244.
P. Li, Z. Zhai / Journal of Functional Analysis 259 (2010) 2457–2519
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[5] R.R. Coifman, Y. Meyer, E.M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985) 304–335. [6] G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα (Rn ), J. Funct. Anal. 208 (2004) 377–422. [7] H. Dong, D. Li, Optimal local smoothing and analyticity rate estimates for the generalized Navier–Stokes equations, Commun. Math. Sci. 7 (2009) 67–80. [8] M. Essen, S. Janson, L. Peng, J. Xiao, Q space of several real variables, Indiana Univ. Math. J. 49 (2000) 575–615. [9] M. Frazier, B. Jawerth, G. Weiss, Littlewood–Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math., vol. 79, Amer. Math. Soc., Providence, RI, 1991. [10] P. Germain, N. Pavlovi´c, G. Staffilani, Regularity of solutions to the Naiver–Stokes equations evolving from small data in BMO−1 , Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm087. [11] Y. Giga, T. Miyakawa, Navier–Stokes flow in R3 with measures as initial vorticity and Morry spaces, Comm. Partial Differential Equations 14 (1989) 577–618. [12] Y. Giga, O. Sawada, On regularizing-decay rate estimates for solutions to the Navier–Stokes initial value problem, Nonlinear Anal. Appl. 1 (2002) 549–562. [13] T. Kato, Strong Lp -solutions of the Navier–Stokes in Rn with applications to weak solutions, Math. Z. 187 (1984) 471–480. [14] T. Kato, H. Fujita, On the non-stationary Navier–Stokes system, Rend. Semin. Mat. Univ. Padova 30 (1962) 243– 260. [15] N.H. Katz, N. Pavlovi´c, A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyperdissipation, Geom. Funct. Anal. 12 (2002) 355–379. [16] H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001) 22–35. [17] P.G. Lemarié-Rieusset, Recent Development in the Navier–Stokes Problem, Chapman & Hall/CRC Press, Boca Raton, 2002. [18] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod/Gauthier–Villars, Paris, 1969 (in French). [19] C. Miao, B. Yuan, B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. 68 (2008) 461–484. [20] H. Miura, O. Sawada, On the regularizing rate estimates of Koch–Tataru’s solution to the Naiver–Stokes equations, Asymptot. Anal. 49 (2006) 1–15. [21] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser., Duke Univ. Press, Durham, 1976. [22] O. Sawada, On analyticity rate estimates of the solution to the Navier–Stokes equations in Bessel-potential spaces, J. Math. Anal. Appl. 312 (2005) 1–13. [23] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. [24] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992. [25] J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003) 284–312. [26] J. Wu, The generalized incompressible Navier–Stokes equations in Besov spaces, Dyn. Partial Differ. Equ. 1 (2004) 381–400. [27] J. Wu, Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces, Comm. Math. Phys. 263 (2005) 803–831. [28] J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations 33 (2008) 285–306. [29] J. Xiao, Homothetic variant of fractional Sobolev space with application to Navier–Stokes system, Dyn. Partial Differ. Equ. 2 (2007) 227–245. [30] D. Yang, W. Yuan, A new class of function spaces connecting Triebel–Lizorkin spaces and Q spaces, J. Funct. Anal. 255 (2008) 2760–2809. [31] Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl. 356 (2009) 642–658.
Journal of Functional Analysis 259 (2010) 2520–2556 www.elsevier.com/locate/jfa
Some nonlinear Brascamp–Lieb inequalities and applications to harmonic analysis ✩ Jonathan Bennett, Neal Bez ∗ School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England, United Kingdom Received 11 June 2009; accepted 26 July 2010 Available online 7 August 2010 Communicated by K. Ball
Abstract We use the method of induction-on-scales to prove certain diffeomorphism-invariant nonlinear Brascamp– Lieb inequalities. We provide applications to multilinear convolution inequalities and the restriction theory for the Fourier transform, extending to higher dimensions recent work of Bejenaru–Herr–Tataru and Bennett–Carbery–Wright. © 2010 Elsevier Inc. All rights reserved. Keywords: Brascamp–Lieb inequalities; Induction-on-scales; Fourier extension estimates
1. Introduction The purpose of this paper is to obtain nonlinear generalisations of certain Brascamp–Lieb inequalities and apply them to some well-known problems in euclidean harmonic analysis. Our particular approach to such inequalities is by induction-on-scales, and builds on the recent work of Bejenaru, Herr and Tataru [4]. The Brascamp–Lieb inequalities simultaneously generalise important classical inequalities such as the multilinear Hölder, sharp Young convolution and Loomis–Whitney inequalities. They may be formulated as follows. Suppose m 2 and d, d1 , . . . , dm are positive integers, and for ✩
Both authors were supported by EPSRC grant EP/E022340/1.
* Corresponding author.
E-mail addresses:
[email protected] (J. Bennett),
[email protected] (N. Bez). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.015
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2521
each 1 j m, Bj : Rd → Rdj is a linear surjection and pj ∈ [0, 1]. The Brascamp–Lieb inequality associated with these objects takes the form m Rd
(fj ◦ Bj )
pj
C
j =1
m j =1
pj (1)
fj
d Rj
for all nonnegative fj ∈ L1 (Rdj ), 1 j m. Here C denotes a constant depending on the datum (B, p) := ((Bj ), (pj )), which at this level of generality may of course be infinite. For nonnegative functions fj ∈ L1 (Rdj ) satisfying 0 < fj < ∞, we define the quantity BL(B, p; f) =
Rd
m
◦ Bj )pj pj , j =1 ( Rdj fj )
m
j =1 (fj
where f := (fj ). We may then define the Brascamp–Lieb constant 0 < BL(B, p) ∞ to be the supremum of BL(B, p; f) over all such inputs f. The quantity BL(B, p) is of course the smallest 0 < C ∞ for which (1) holds. It should be noted here that there is a natural equivalence relation on Brascamp–Lieb data, where (B, p) ∼ (B , p ) if p = p and there exist invertible linear transformations C : Rd → Rd and Cj : Rdj → Rdj such that Bj = Cj−1 Bj C for all j ; we refer to C and Cj as the intertwining transformations. In this case, simple changes of variables show that m pj j =1 |det Cj | BL(B, p), BL B , p = |det C| and thus BL(B, p) < ∞ if and only if BL(B , p ) < ∞. This terminology is taken from [5]. The generality of this setup of course raises questions, many of which have been addressed in the literature. In [15] Lieb showed that the supremum above is exhausted by centred gaussian inputs, prompting further investigation into issues including the finiteness of BL(B, p) and the extremisability/gaussian-extremisability of BL(B, p; f). A fuller description of the literature is not appropriate for the purposes of this paper. The reader is referred to the survey article [2] and the references there. A large number of problems in harmonic analysis require nonlinear versions of inequalities belonging to this family; see [3,4,7,14,18,23] for instance. The generalisations we seek here are local in nature, and amount to allowing the maps Bj to be nonlinear submersions in a neighbourhood of a point x0 ∈ Rd , and then looking for a neighbourhood U of x0 such that if ψ is a cutoff function supported in U , there exists a constant C > 0 for which m Rd
j =1
m p fj Bj (x) j ψ(x) dx C j =1
pj fj
(2)
d Rj
for all nonnegative fj ∈ L1 (Rdj ), 1 j m. The applications of such inequalities invariably require more quantitative statements involving the sizes of the neighbourhood U and constant C, and also the nature of any smoothness/non-degeneracy conditions imposed on the nonlinear maps (Bj ).
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J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
Notice that if dj = d for each j , then the nonlinear Bj are of course local diffeomorphisms. In this situation necessarily p1 + · · · + pm = 1 and (2) follows from the m-linear Hölder inequality. Similar considerations allow to reduce matters to the case where dj < d for all j . It is perhaps reasonable to expect to obtain an inequality of the form (2) for smooth nonlinear maps (Bj ) and exponents (pj ) for which BL((dBj (x0 )), (pj )) < ∞. Here dBj (x0 ) denotes the derivative map of Bj at x0 . However, the techniques that we employ in this paper appear to require additional structural hypotheses on the maps dBj (x0 ), and so instead we seek to identify a natural class
C ⊆ (B, p): each Bj is linear and BL(B, p) < ∞ such that (2) holds for nonlinear (Bj ) with ((dBj (x0 )), (pj )) ∈ C. As will become clear in Section 2, a natural choice for consideration is C = (B, p):
m
j =1
1 ker Bj = R , p1 = · · · = pm = . m−1 d
(3)
This class contains the classical Loomis–Whitney datum [16], whereby m = d, dj = d − 1, pj = 1/(d − 1) and Bj (x1 , . . . , xd ) = (x1 , . . . , xj , . . . , xd ) for all 1 j d. Here denotes omission. The purpose of this paper is two-fold. Firstly, we establish an inequality of the form (2) whenever ((dBj (x0 )), (pj )) ∈ C, where C is defined in (3). Secondly, we use these inequalities to deduce certain sharp multilinear convolution estimates, which in turn yield progress on the multilinear restriction conjecture for the Fourier transform. These applications can be found in Section 7. Before stating our nonlinear Brascamp–Lieb inequalities, it is important that we discuss further the class C given in (3). Notice that the transversality hypothesis m
ker Bj = Rd
(4)
j =1
is preserved under the equivalence relation on Brascamp–Lieb data; that is, it is invariant under Bj → Cj−1 Bj C for invertible linear transformations C : Rd → Rd and Cj : Rdj → Rdj . By choosing appropriate intertwining transformations C and Cj , an elementary calculation shows that if (B, p) ∈ C then (B, p) ∼ (Π, p), where Π = (Πj )m j =1 are certain coordinate projections. In order to define Πj we let Kj ⊆ {1, . . . , d} be given by
Kj = d1 + · · · + dj −1 + 1, . . . , d1 + · · · + dj −1 + dj , where dj = d − dj denotes the dimension of the kernel of Bj , so that K1 , . . . , Km form a partition of {1, . . . , d}. Then we let Πj : Rd → Rdj be given by Πj (x) = (xk )k∈Kjc .
(5)
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2523
1 1 Proposition 1.1. (See [13].) If p = ( m−1 , . . . , m−1 ) then BL(Π, p) = 1, and thus
m Rd
1
fj (Πj x) m−1 dx
j =1
m j =1
R
1 m−1
fj
(6)
dj
holds for all nonnegative fj ∈ L1 (Rdj ), 1 j m. Proposition 1.1 follows from work of Finner [13] where a stronger result was established for Π consisting of more general coordinate projections and in the broader setting of product measure spaces. In particular, this includes the discrete inequality m
fj (Πj n)
n∈Nd j =1
1 m−1
m
j =1
1 m−1 fj ()
(7)
d ∈N j
which holds for all nonnegative fj ∈ 1 (Ndj ), 1 j m. We mention this case specifically as it will be important later in the paper. We remark that (6) is a generalisation of the classical Loomis–Whitney inequality [16] whereby m = d and Kj = {j } for 1 j d. 1 1 In order for BL(Π, p) to be finite it is necessary that p = ( m−1 , . . . , m−1 ), and this follows by a straightforward scaling argument. The standard proof of Proposition 1.1 proceeds via the multilinear Hölder inequality and induction (see [13]). This proof and, to the best of our knowledge, other established proofs of Proposition 1.1 rely heavily on the linearity of the Πj and break down completely in the nonlinear setting. Since we would like to state our main theorem regarding nonlinear Bj in a diffeomorphisminvariant way, it is appropriate that we first formulate an affine-invariant version of Proposition 1.1. In order to state this it is natural to use language from exterior algebra; the relevant concepts and terminology can be found in standard texts such as [12]. In particular, Λn (Rd ) will denote the nth exterior algebra of Rd and : Λn (Rd ) → Λd−n (Rd ) will denote the Hodge star operator. (It is worth pointing out here that if the reader is prepared to sacrifice the explicit diffeomorphism-invariance that we seek, then they may effectively dispense with these exterior algebraic considerations.) Given (B, p) ∈ C define Xj (Bj ) ∈ Λdj (Rd ) to be the wedge product of the rows of the dj × d matrix Bj . By (4) it follows that
m
Xj (Bj ) ∈ R\{0}.
(8)
j =1
The quantity in (8) is a certain determinant and should be viewed as a means of quantifying the transversality hypothesis (4). Proposition 1.2. If (B, p) ∈ C then m − 1 m−1 BL(B, p) = Xj (Bj ) , j =1
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and thus m Rd
fj (Bj x)
j =1
1 m−1
m − 1 m 1 m−1 m−1 dx Xj (Bj ) fj j =1
j =1
R
(9)
dj
for all nonnegative fj ∈ L1 (Rdj ), 1 j m. One may reduce Proposition 1.2 to Proposition 1.1 by appropriate linear changes of variables; see Appendix A for full details of this argument which will be of further use in Section 4 for the nonlinear case. Since the inequality (9) is affine-invariant, one should expect it to have a diffeomorphisminvariant nonlinear version. This is our main result with regard to nonlinear generalisations of Brascamp–Lieb inequalities. Theorem 1.3. Let β, ε, κ > 0 be given. Suppose that Bj : Rd → Rdj is a C 1,β submersion satisfying Bj C 1,β κ in a neighbourhood of a point x0 ∈ Rd for each 1 j m. Suppose further that m
ker dBj (x0 ) = Rd
(10)
j =1
and m Xj dBj (x0 ) ε. j =1
Then there exists a neighbourhood U of x0 depending on at most β, ε, κ and d, such that for all cutoff functions ψ supported in U , there is a constant C depending only on d and ψ such that m Rd
1 m m−1 1 1 − m−1 m−1 fj Bj (x) ψ(x) dx Cε fj
j =1
j =1
R
(11)
dj
for all nonnegative fj ∈ L1 (Rdj ), 1 j m. Inequality (11) may be interpreted as a multilinear “Radon-like” transform estimate. This is made explicit in the following corollary, upon which our applications in Section 7 depend. Corollary 1.4. Let β, ε, κ > 0 be given. If F : (Rd−1 )d−1 → R is such that F C 1,β κ and det ∇u F (0), . . . , ∇u F (0) ε, 1 d−1 then there exists a neighbourhood V of the origin in (Rd−1 )d−1 , depending only on β, ε, κ and d, and a constant C depending only on d, such that
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d 1 f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F (u) du Cε − d−1 fj (d−1)
(12)
j =1
V
for all nonnegative fj ∈ L(d−1) (Rd−1 ), 1 j m. The case d = 3 of Corollary 1.4 was proved in [7] as a consequence of the nonlinear Loomis– Whitney inequality. It is perhaps interesting to view Corollary 1.4 in the light of the theory of multilinear weighted convolution inequalities for L2 functions developed in [19]. Inequality (12) is an example of such a convolution inequality in an Lp setting and with a singular (distributional) weight. We conclude this section with a number of remarks on Theorem 1.3. As in the reduction of Proposition 1.2 to Proposition 1.1, a linear change of variables argument shows that Theorem 1.3 may be reduced to the case where each linear mapping dBj (x0 ) is equal to the coordinate projection Πj given by (5), in which case
m
Xj dBj (x0 ) = 1.
j =1
Although this reduction is not essential, it does lead to some conceptual and notational simplification in the subsequent analysis. The details of this reduction may be found in Section 4. The core component of the proof of Theorem 1.3 that we present is based on [4] and uses the idea of induction-on-scales. This approach provides additional information about the sizes of the neighbourhood U and constant C appearing in its statement; see Section 4 for further details of this. In Section 2 we offer an explanation of why the induction-on-scales approach is natural in the context of Brascamp–Lieb inequalities and why the class C given in (3) is a natural class for consideration. In Section 3, we provide an outline of the proof of Theorem 1.3 which should guide the reader through the full proof which is contained in Sections 4 and 5. In the case where dj = d − 1 for all j , Theorem 1.3 reduces to the nonlinear Loomis–Whitney inequality in [7] except that the stronger hypothesis Bj ∈ C 3 is assumed in [7]. The proof of the result in [7] is quite different from the proof we give here, and is based on the so-called method of refinements of M. Christ [11]. We make some further remarks on the role of the smoothness of the mappings Bj at the end of Section 5. The condition (10) is somewhat less restrictive than it may appear. For example, consider smooth mappings Bj : R5 → R2 satisfying ker dBj (x0 ) = {ej , e(j +1) mod 5 , e(j +2) mod 5 } for each 1 j 5, where ej denotes the j th standard basis vector in R5 . Evidently the condition (10) is not satisfied. However we may write 5
(fj ◦ Bj )1/2 =
j =1
5 j =1
j )1/4 , (fj ◦ B
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j := (Bj , B(j +2) mod 5 ) : R5 → R4 . Since where fj := fj ⊗ f(j +2) mod 5 : R4 → [0, ∞) and B j do satisfy the condition (10), j (x0 ) = {e(j +2) mod 5 } for each 1 j 5, the mappings B ker dB and so by Theorem 1.3 5 R5
(fj ◦ Bj )1/2 ψ =
j =1
5 R5
C
j =1 5 j =1
=C
j )1/4 ψ (fj ◦ B
R4
5 j =1
fj
1/4
1/2 fj
.
R2
Here the cutoff function ψ and constant C are as in the statement of Theorem 1.3. This inequality is optimal in the sense that BL((dBj (x0 )), (pj )) < ∞ if and only if p1 = · · · = p5 = 1/2 – see [13]. Similar considerations form an important part of the proof of Corollary 1.4 in dimensions d 4. Very recently, Stovall [18] considered inequalities of the type (2) for the case dj = d − 1 for all j where one does not necessarily have the transversality hypothesis (10). Here, curvature of the fibres of the Bj plays a crucial role. In [18], Stovall determined completely all data (B, p), up to endpoints in p, for which inequality (2) holds when each Bj : Rd → Rd−1 is a smooth submersion. The work in [18] generalised work of Tao and Wright [23] for the bilinear case m = 2, and both approaches are based on Christ’s method of refinements. It would be interesting to complete the picture further and understand the case where one does not necessarily have transversality and each dj is not necessarily equal to d − 1. We do not pursue this matter here. Given that Theorem 1.3 is a local result it is natural to ask whether one may obtain global versions based on the assumption that hypothesis (4) holds at every point x0 ∈ Rd , possibly with the insertion of a suitable weight factor. Simple examples show that naive versions, involving weights which are powers of the quantity m j =1 Xj (dBj (x)) cannot hold; see [7] for an explicit example. Organisation of the paper. To recap, in the next section we give some justification for our choice of proof of Theorem 1.3 and the class C. In Section 3 we give an outline of the proof of Theorem 1.3 by considering the special case of the nonlinear Loomis–Whitney inequality in three dimensions. The full proof begins in Section 4 where we make the reduction to the coordinate projection case. The proof for this case rests on the induction-on-scales argument which appears in Section 5. In Section 6 we give a proof of Corollary 1.4, and in Section 7 we provide applications to two closely related problems in harmonic analysis. 2. Induction-on-scales and the class C The Brascamp–Lieb inequalities (1) possess a certain self-similar structure that strongly suggests an approach to the corresponding nonlinear statements by induction-on-scales. Inductionon-scales arguments have been used with great success in harmonic analysis in recent years. Very
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closely related to the forthcoming discussion is the induction-on-scales approach to the Fourier restriction and Kakeya conjectures originating in work of Bourgain [8], and developed further by Wolff [24] and Tao [20]; see also the survey article [21]. This self-similarity manifests itself most elegantly in an elementary convolution inequality due to Ball [1] (see also [5]), which we now describe. Let (B, p) be a Brascamp–Lieb datum where each Bj is linear. Let f and f be two inputs and we assume, for clarity of exposition, that these inputs are L1 -normalised. For each x ∈ Rd and 1 j m let gjx : Rdj → [0, ∞) be given by gjx (y) = fj (Bj x − y)fj (y). By Fubini’s theorem and elementary considerations we have that BL(B, p; f) BL B, p; f =
m Rd
(fj ◦ Bj )pj ∗
j =1
m p fj ◦ Bj j j =1
m x pj gj ◦ Bj = dx Rd
Rd
j =1
pj m x x gj (y) dy BL B, p; gj dx j =1
Rd
R
dj
m pj x fj ∗ fj (Bj x) BL B, p; gj dx = j =1
Rd
and therefore BL(B, p; f) BL B, p; f sup BL B, p; gjx BL B, p; f ∗ f ,
(13)
x∈Rd
where f ∗ f := (fj ∗ fj ). Notice that if f is an extremiser to (1), i.e. BL B, p; f = BL(B, p), then since BL B, p; f ∗ f BL(B, p), we may deduce that BL(B, p; f) sup BL B, p; gjx .
(14)
x∈Rd
In particular, in the presence of an appropriately “localising” extremiser f (such as of compact support), (14) suggests the viability of a proof of nonlinear inequalities such as (2) by induction
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on the “scale of the support” of f. The point is that gjx may be thought of as the function fj localised by fj to a neighbourhood of the general point Bj x. With the above discussion in mind it is natural to restrict attention to data (B, p) for which (1) has extremisers of the form f = (χEj ), where for each j , Ej is a subset of Rdj which tiles by translation. Furthermore, given our aspirations, it is natural to choose a class of data which is affine-invariant and stable under linear perturbations of B. These requirements lead us to the transversality hypothesis in (4). Indeed, as there are linear changes of variables which show that Proposition 1.2 follows from Proposition 1.1 (see Appendix A), it is straightforward to observe that characteristic functions of certain parallelepipeds are extremisers for (9). Such sets of course tile by translation. We remark that there are other hypotheses on the datum B which fulfill our requirements. For example, one may replace (4) by m
coker Bj = Rd .
j =1
However, after appropriate changes of variables, the corresponding nonlinear inequality (2) merely reduces to a statement of Fubini’s theorem, and in particular, pj = 1 for all j . There are further alternatives which are hybrids of these and are similarly degenerate. Remark 2.1. Notice that if f is an extremiser to (1) then we may also deduce from (13) that BL(B, p; f) BL B, p; f ∗ f .
(15)
This inequality suggests the viability of a proof of nonlinear inequalities such as (2) by induction on the “scale of constancy” of f . Certain weak versions of inequality (2), where the resulting constant C has a mild dependence on the smoothness of the input f, have already been treated in this way in [6] (see Remarks 6.3 and 6.6). In certain situations, (15) leads to the monotonicity of BL(B, p; f) under the action of convolution semigroups on the input f. In the context of heat-flow, this observation originates in [10] and [5]; see the latter for further discussion of this perspective. 3. An outline of the proof of Theorem 1.3 The purpose of this section is to bring out the key ideas in the proof of Theorem 1.3. It is also an opportunity to introduce some notation which will be adopted (modulo small modifications) in the full proof in Section 5. As it is an outline we will sometimes compromise rigour for the sake of clarity. Our approach is based on [4]. Since the induction-on-scales argument we use to prove Theorem 1.3 is guided by the underlying geometry, in this outline we will consider the Loomis–Whitney case where d = 3, m = 3 and dBj (x0 ) = Πj
(16)
for j = 1, 2, 3. In particular, we have ker dBj (x0 ) = ej where ej denotes the j th standard basis vector in R3 .
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
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We shall use Q(x, δ) to denote the axis-parallel cube centred at x with sidelength equal to δ. Fix a small sidelength δ0 > 0 which, in terms of the induction-on-scales argument, represents the largest or “global” scale. For δ, M > 0 we let C(δ, M) denote the best constant in the inequality
1 1 1 f1 B1 (x) 2 f2 B2 (x) 2 f3 B3 (x) 2 dx C
f1 R2
Q
1 2
1 2 f2
R2
1 2
f3 R2
over all axis-parallel subcubes Q of Q(x0 , δ0 ) of sidelength δ and all inputs f1 , f2 , f3 ∈ L1 (R2 ) which are “constant” at the scale M −1 . The goal is to prove that C(δ0 , M) is bounded above by a constant independent of M, allowing the use of a density argument to pass to general f1 , f2 , f3 ∈ L1 (R2 ). As our proof proceeds by induction it consists of two distinct parts. (i) The base case: For each M > 0, C(δ, M) is bounded by an absolute constant for all δ sufficiently small. (ii) The inductive step: There exist γ > 0 and α > 1 such that C(δ, M) 1 + O δ γ C 2δ α , M
(17)
uniformly in δ δ0 and M > 0. Claims (i) and (ii) quickly lead to the desired conclusion since on iterating (17) we find that C(δ0 , M) is bounded by a convergent product of factors of the form (1 + O(δ γ )) with δ δ0 . To see why the base case is true, let Q be any axis-parallel cube contained in Q(x0 , δ0 ) with centre xQ and sidelength δ, and let f1 , f2 , f3 ∈ L1 (R2 ) be constant at scale M −1 . Observe that if δ is sufficiently small then each fj does not “see” the difference between Bj (x) and dBj (xQ )x for x ∈ Q in the sense that fj ◦ Bj ∼ fj ◦ dBj (xQ ) (up to harmless translations) on Q. Now, by (16) and the smoothness of the Bj we know that Xj dBj (xQ ) − ej = Xj dBj (xQ ) − Xj (Πj ) 1/10 if δ0 is sufficiently small. Hence by Proposition 1.2 it follows that C(δ, M) is bounded above by an absolute constant for such δ. Turning to the inductive step, fix any axis-parallel cube Q contained in Q(x0 , δ0 ) with centre 1 2 −1 xQ and decompose sidelength δ, and let f1 , f2 , f3 ∈ L (R ) be constant at scale M . First we Q = P (n), where the P (n) are axis-parallel subcubes with equal sidelength δ α , and α > 1. We choose the natural indexing of the P (n) by n ∈ N3 . Unfortunately this decomposition is too naive to prove the inductive step but nevertheless it is instructive to see where the proof breaks down.
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Fig. 1. Subcubes P (n) parametrised by n ∈ N3 and tubes T3 () parametrised by ∈ N2 with direction e3 .
Observe that
1 1 1 f1 B1 (x) 2 f2 B2 (x) 2 f3 B3 (x) 2 dx
Q
=
1 1 1 f1 B1 (x) 2 f2 B2 (x) 2 f3 B3 (x) 2 dx
n∈N3 P (n)
C δα , M n∈N3
If n = (n1 , n2 , n3 ) then and equal to Π1 then
1
2
B2 (P (n))
1
f2
B1 (P (n))
B1 (P (n)) f1
1
2
f1
2
f3
.
(18)
B3 (P (n))
is “almost” a function of n2 and n3 . Indeed, if B1 is linear
B1 P (n) = B1 T1 (n2 , n3 ) where T1 (n2 , n3 ) is a cuboid (or “tube”) withlong side in the direction of e1 and containing P (n). A similar remark holds for B2 (P (n)) f2 and B3 (P (n)) f3 . For j = 1, 2, 3 this leads us to define cuboids Tj () =
P (n)
n∈N3 :
Πj n=
for ∈ N2 . Note that Tj () has direction ej and its location is determined by ∈ N2 . In particular, for each n ∈ N3 , Tj (Πj n) is a cuboid in the direction ej which passes through P (n). See Fig. 1. Accordingly, we define
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2531
Fj () =
fj
Bj (Tj ())
for j = 1, 2, 3 and ∈ N2 . Then by (18) and the discrete inequality (7),
1 1 1 1 1 1 f1 B1 (x) 2 f2 B2 (x) 2 f3 B3 (x) 2 dx C δ α , M F1 (Π1 n) 2 F2 (Π2 n) 2 F3 (Π3 n) 2 n∈N3
Q
1 1 1 C δ α , M F1 21 (N2 ) F2 21 (N2 ) F3 21 (N2 ) .
If we had disjointness in the sense that Bj Tj () ∩ Bj Tj = ∅ whenever = ,
(19)
then Fj 1 (N2 )
fj R2
would hold for each j = 1, 2, 3, and hence
1 1 1 f1 B1 (x) 2 f2 B2 (x) 2 f3 B3 (x) 2 dx
Q
1 1 1 2 2 2 C δα , M f1 f2 f3 R2
R2
(20)
R2
would follow immediately. If each Bj is linear and equal to Πj then (19) is of course true, although otherwise it is not. In order to achieve a version of (19) in general, it is necessary to modify our decomposition of Q. To better understand the location of each image Bj (Tj (Πj n)) the P (n) should in fact be parallelepipeds whose faces are given by pull-backs of certain lines in R2 under the linear maps dBj (xQ ). However, we still need to fully accommodate for the nonlinearity and in particular the difference between Bj (Tj ()) and dBj (xQ )(Tj ()). Following the approach in [4] it is natural to insert relatively narrow “buffer zones” between the P (n) to provide sufficient separation in order to guarantee the sought after disjointness property (19). Clearly this depends on the smoothness of the Bj and, since we assume C 1,β regularity, we take the P (n) to have sidelengths approximately δ α0 and the buffer zones to have width approximately δ α1 where 1 < α0 < α1 < 1 + β. The decomposition of Q now has a “main component” from the P (n) and an “error component” from the buffer zones. We would like to use the above argument which led to (20) on each
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Fig. 2. The modified decomposition of Q.
component. However, in order for the error component to genuinely contribute an acceptable error term, we need to relax the regular decomposition (into equally sized P (n)) since a “large” amount of mass of the fj ◦ Bj may lie on the buffer zones. Again following ideas from [4] we use a simple pigeonholing argument to position the buffer zones in an efficient location given the constraint that the P (n) should have essentially the same sidelengths. See Fig. 2. Putting the resulting estimates together yields the desired recursive inequality (17) with α = α0 and some γ > 0. See Section 5 for the complete details of this induction-on-scales argument in the full generality of Theorem 1.3. 4. Preparation and reduction to the orthogonal projection case Recall the definition of Πj : Rd → Rdj given by (5). In this section we shall prove that Theorem 1.3 is a consequence of the following nonlinear version of Proposition 1.1. Proposition 4.1. Suppose β, κ > 0 are given and α0 , α1 satisfy 1 < α0 < α1 < 1 + β. Let δ0 = min
cd κ
1 1+β−α1
1 1 min{α0 −1,α1 −α0 } . , 4
(21)
Suppose that Bj : Rd → Rdj is a C 1,β submersion satisfying Bj C 1,β κ in Q(x0 , δ0 ) and dBj (x0 ) = Πj for each 1 j m. Then for cd ∈ (0, κ) sufficiently small, Q(x0 ,δ0 )
α1 −α0 1 m d δ m−1 m−1 1 10 0 d m−1 fj Bj (x) dx 10 exp fj α1 −α0 1 − 2− m−1 j =1 dj j =1
m
for all nonnegative fj ∈ L1 (Rdj ), 1 j m.
R
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2533
As mentioned already in the previous section, the proof of Proposition 4.1 will proceed by an induction-on-scales argument. For a cube at scale δ, we decompose into parallelepipeds of sidelength approximately δ α0 and the buffer zones will have thickness approximately δ α1 . We have stated Proposition 4.1 with this in mind and we have provided explicit information on how the size of the neighbourhood and the constant depend on the relevant parameters. Deduction of Theorem 1.3 from Proposition 4.1. The argument which follows is similar to the argument given in Appendix A for the corresponding claim in the linear case. A little extra work is required to verify the uniformity claims in Theorem 1.3 concerning the neighbourhood and the constant. Select any set of vectors {ak : k ∈ Kj } forming an orthonormal basis for ker dBj (x0 ). By definition of the Hodge star and orthogonality we get Xj dBj (x0 ) = Xj dBj (x0 ) Λdj (Rd ) ak .
(22)
k∈Kj
Let A be the d × d matrix whose ith column is equal to ai for each 1 i d. Finally, let Cj be the dj × dj matrix given by Cj = dBj (x0 )Aj , where Aj is the d × dj matrix obtained by deleting from A the columns ak for each k ∈ Kj . j : Rd → Rdj given by Then, by construction, the map B j (x) = C −1 Bj (Ax) B j satisfies j ( dB x0 ) = Cj−1 dBj (x0 ) A = Πj ,
(23)
where x0 = A−1 x0 . Since we are assuming (4) and since Bj is a submersion at x0 we know that the matrices A and Cj are invertible. Let U be some neighbourhood of x0 and ψ a cutoff function supported in U . Using A to change variables one obtains m Rd
j =1
1 fj Bj (x) m−1 ψ(x) dx = det(A)
m
Rd
1 j (x) m−1 ψ (x) dx, fj B
(24)
j =1
= ψ ◦ A is a cutoff function supported in A−1 U and fj = fj ◦ Cj , 1 j m. Of where ψ j ( x0 ) = Πj by (23). Notice also that course, we know that dB dB j (x) − dB j (y) = C −1 dBj (Ax) − dBj (Ay) A Cκ C −1 |x − y|β , j j where the constant C depends on at most d. To show that we may choose the neighbourhood U and the constant in the claimed uniform manner we need to show that suitable upper bounds hold for the norms of A−1 and each Cj−1 .
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For A−1 , we note that
m j =1
m m Xj dBj (x0 ) dj d Xj dBj (x0 ) = ak Λ (R ) j =1 k∈Kj
j =1
by (22) and therefore
m j =1
m Xj dBj (x0 ) dj d . Xj dBj (x0 ) = det(A) Λ (R )
(25)
j =1
Since Bj C 1,β κ it follows that m Xj dBj (x0 ) C det(A) j =1
for some constant C depending on κ and d. Since each column of A is a unit vector, it follows that the norm of A−1 is bounded above by a constant depending on ε, κ and d. For Cj−1 , from (22) we get det(Cj ) = Xj dBj (x0 )
d Λ j (Rd )
det(A).
(26)
By (25), ε C Xj dBj (x0 ) Λdj (Rdj ) det(A), for some constant C depending on κ and d. It follows that the norm of Cj−1 is also bounded above by a constant depending on ε, κ and d. Applying Proposition 4.1 it follows that there exists a neighbourhood U of x0 depending on at most β, ε, κ and d such that m Rd
m 1 fj Bj (x) m−1 ψ(x) dx C det(A)
j =1
j =1
fj
1 m−1
,
d Rj
where C depends on at most d and ψ . Thus 1 m m−1 1 |det(A)| m−1 fj Bj (x) ψ(x) dx C fj 1 m−1 ( m j =1 j =1 j =1 |det(Cj )|) d
m Rd
R
j
m − 1 m 1 m−1 m−1 = C Xj dBj (x0 ) fj , j =1
j =1
R
dj
where the equality holds because of (25) and (26). Theorem 1.3 now follows.
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For the various constants appearing in the above proof, one may easily obtain some explicit dependence in terms of the relevant parameters. Combined with Proposition 4.1, this gives additional information on the sizes of the neighbourhood U and constant C appearing in the statement of Theorem 1.3. We do not pursue this matter further here. 5. Proof of Proposition 4.1: Induction-on-scales Before stating the main induction lemma we use to prove Proposition 4.1, we need to fix some further notation. For each 1 j m and M > 0, let L1M (Rdj ) denote those nonnegative f ∈ L1 (Rdj ) satisfying f (y1 ) 2f (y2 ) whenever y1 and y2 are in the support of f and |y1 − y2 | M −1 ; that is, those f which are effectively constant at the scale M −1 . One may (dj ) (dj ) easily check that if μ is a finite measure on Rdj then Pc/M ∗ μ ∈ L1M (Rdj ), where Pc/M ded j notes the Poisson kernel on R at height c/M. Here c is a suitably large constant depending only on dj . By an elementary density argument, it will be enough to prove Proposition 4.1 for fj ∈ L1M (Rdj ), 1 j m, with neighbourhood U and constant C independent of M. As we shall shortly see, we consider such a subclass of functions in order to provide a “base case” for the inductive argument. For β, κ > 0, 1 < α0 < α1 < 1 + β and x0 ∈ Rd we let B(β, κ, α0 , α1 , x0 ) be the family of data B such that Bj belongs to C 1,β (Q(x0 , δ0 )) with Bj C 1,β κ and satisfies dBj (x0 ) = Πj , 1 j m. Here, δ0 is given by (21). Now let C(δ, M) denote the best constant in the inequality m
m 1 fj Bj (x) m−1 dx C
Q j =1
j =1
1 m−1
fj R
dj
over all B ∈ B(β, κ, α0 , α1 , x0 ), all axis-parallel subcubes Q of Q(x0 , δ0 ) with sidelength equal to δ and all inputs f such that fj belongs to L1M (Rdj ), 1 j m. We note that the constant C(δ, M) also depends on the parameters β, κ, α0 and α1 , although there is little to be gained in what follows from making this dependence explicit. The main induction-on-scales lemma is the following. Lemma 5.1. For all 0 < δ δ0 we have α1 −α0 C(δ, M) 1 + 10d δ m−1 C 2δ α0 , M .
The proof of Lemma 5.1 is a little lengthy. Before giving the proof we show how Lemma 5.1 implies Proposition 4.1. Deduction of Proposition 4.1 from Lemma 5.1. Firstly we claim that the “base case” inequality C δ0 /2N , M 10d
(27)
holds for sufficiently large N . To see (27), suppose B ∈ B(β, κ, α0 , α1 , x0 ), Q is a subcube of Q(x0 , δ0 ) with centre xQ and sidelength δ0 /2N , and the input f is such that fj belongs to L1M (Rdj ), 1 j m. For any x ∈ Q,
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Bj (x) − Bj (xQ ) + dBj (xQ )(x − xQ ) κ|x − xQ |1+β 1/M if N is sufficiently large (depending on β, κ, d and M). Since fj ∈ L1M (Rdj ) it follows that m
1 fj Bj (x) m−1 dx 2m
Q j =1
m
1 fj · + Bj (xQ ) dBj (xQ ) x m−1 dx.
Q−{xQ } j =1
Now dBj (xQ ) − Πj = dBj (xQ ) − dBj (x0 )
1 , 100d
which implies that
m j =1
1 Xj dBj (xQ ) , 2
and therefore m
1 m m−1 1 fj Bj (x) m−1 dx 10d fj
Q j =1
j =1
R
dj
by Proposition 1.2. Hence, (27) holds. For 0 < δ δ0 (1/4)1/α0 −1 it follows from Lemma 5.1 that α1 −α0 C(δ, M) 1 + 10d δ m−1 C(δ/2, M).
Applying (28) iteratively N times we see that −1 α1 −α0 N C(δ0 , M) C δ0 /2N , M 1 + 10d δ0 /2r m−1 . r=0
The product term is under control uniformly in N because
log
N −1
−1 α1 −α0 N α1 −α0 d r m−1 1 + 10 δ0 /2 = log 1 + 10d δ0 /2r m−1
r=0
r=0 α1 −α0
10d δ0 m−1
∞ r=0
α1 −α0 m−1
10d δ0
1 − 2−
α1 −α0 m−1
.
2−
α1 −α0 m−1 r
(28)
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From the base case (27) it follows that α1 −α0 d δ m−1 10 0 ; C(δ0 , M) 10d exp α1 −α0 1 − 2− m−1
that is, Q(x0 ,δ0 )
α1 −α0 1 m d δ m−1 m−1 1 10 0 d m−1 fj Bj (x) dx 10 exp fj α1 −α0 − m−1 1−2 j =1 j =1 dj
m
(29)
R
for all fj ∈ L1M (Rdj ), 1 j m. Since the constant in (29) is independent of M, it follows that the inequality is valid for all fj ∈ L1 (Rdj ). This completes our proof of Proposition 4.1. Proof of Lemma 5.1. Suppose B = (Bj ) ∈ B(β, κ, α0 , α1 , x0 ), Q is an axis-parallel subcube of Q(x0 , δ0 ) with sidelength equal to δ and centre xQ , and suppose f = (fj ) is such that fj belongs to L1M (Rdj ), 1 j m. Notice that the desired inequality m
m α1 −α0 1 fj Bj (x) m−1 dx 1 + 10d δ m−1 C 2δ α0 , M
Q j =1
j =1
fj
R
1 m−1
(30)
dj
where B j = Bj (· + xQ ) − Bj (xQ ), is invariant under the transformation (B, f, Q) → ( B, f, Q) Q = Q − {xQ } and fj = fj (· + Bj (xQ )). Hence, without loss of generality, Q = Q(0, δ) and Bj (0) = 0 for 1 j m. This reduction is merely for notational convenience; in particular, it ensures Bj (x) − dBj (0)x κ|x|1+β . By the smoothness hypothesis, we have that dBj (0) − Πj
1 100d
(31)
for sufficiently small cd . Since ker Πj = {ek : k ∈ Kj } , it follows that for each 1 k d there exist ak ∈ Rd such that |ak − ek |
1 , 10d
and ker dBj (0) = {ak : k ∈ Kj } for each 1 j m. Here, ek denotes the kth standard basis vector in Rd .
(32)
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The proof of Lemma 5.1 naturally divides into four steps. Step I: Foliations of Rd . For each 1 i d consider the one-parameter family of hypersurfaces {ak : k = i} + s ak
(33)
k=i
where s ∈ R. We point out that k=i ak is simply the cross product of the vectors {ak : k = i}, yielding a vector normal to {ak : k = i}. The set of vectors { k=i ak : 1 i d} in Rd is linearly independent since the same is true of {ai : 1 i d}. Consequently, we may decompose Rd into parallelepipeds whose faces are contained in hyperplanes of the form (33), 1 i d. We will use this to decompose the cube Q. As we shall see in the steps that follow, an important feature of these hypersurfaces is that they may be expressed as inverse images of hypersurfaces under the mappings dBj (0). To this end, let σ : {1, . . . , d} → {1, . . . , m} be the map given by σ (i) = (j + 1) mod m for i ∈ Kj . As will become apparent under closer inspection, there is some freedom in our choice of this map; all that we require of σ is that j → σ (Kj ) is a permutation of {1, 2, . . . , m} with no fixed points. For each 1 i d and J ⊂ R we define the set Σ(i, J ) = dBσ (i) (0) {ak : k = i} + s dBσ (i) (0) ak : s ∈ J .
(34)
k=i
If J = {s} is a singleton set then Σ i, {s} = dBσ (i) (0) {ak : k = i} + s dBσ (i) (0) ak k=i
is a hyperplane in Rdσ (i) since ker dBσ (i) (0) ⊆ {ak : k = i}. Similarly, ak dBσ (i) (0)−1 Σ i, {s} = {ak : k = i} + s
(35)
k=i
which is of course the hyperplane (33). As outlined in Section 3, a regular decomposition of Rd into parallelepipeds of equal size and adapted to a lattice (where for each i, the sequence of parameters s (i) that we choose is in arithmetic progression) will not suffice to prove Lemma 5.1. Moreover, our decomposition will need to incorporate certain “buffer zones” between the parallelepipeds to create separation. In Step II below we determine the location of the buffer zones and thus the desired decomposition of Q. Step II: The decomposition of Q. For each 1 i d we claim that there exists a sequence (i) (sn )n1 such that
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
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1 (i) sn(i) + δ α0 sn+1 sn(i) + δ α0 2 and
fσ (i) χQ 4δ (i) (i) Σ(i,[sn+1 ,sn+1 +δ α1 ])
(36)
α1 −α0
(37)
fσ (i) χQ . (i) (i) Σ(i,[sn + 12 δ α0 ,sn +δ α0 ])
To prove this, we shall choose the sequence (sn(i) )n1 iteratively. We begin by choosing s1(i) to be (i) (i) (i) any real number such that Bσ (i) (Q) ⊆ Σ(i, [s1 , ∞)). Suppose that we have chosen s1 , . . . , sn for some n 1. Now let N be the largest integer which is less than or equal to 12 δ α0 −α1 . Set (i) (i) (i) (i) ζ0 = sn + 12 δ α0 and then define ζr = ζr−1 + δ α1 iteratively for 1 r N so that
N ! 1 1 1 (i) sn(i) + δ α0 , sn(i) + δ α0 ⊇ sn(i) + δ α0 , sn(i) + δ α0 + N δ α1 = ζr−1 , ζr(i) . 2 2 2 r=1
Then, fσ (i) χQ (i)
(i)
Σ(i,[sn + 12 δ α0 ,sn +δ α0 ])
N
fσ (i) χQ ,
r=1 (i) (i) Σ(i,[ζr−1 ,ζr ]) (i)
and therefore by the choice of δ0 in (21) and the pigeonhole principle, there exists sn+1 such that (36) holds and 1 fσ (i) χQ δ α0 −α1 fσ (i) χQ ; 4 (i)
(i)
Σ(i,[sn + 12 δ α0 ,sn +δ α0 ])
(i)
(i)
Σ(i,[sn+1 ,sn+1 +δ α1 ])
that is, (37) also holds. We shall use the notation J (i, n, 0) and J (i, n, 1) for the intervals given by 2 α1 (i) 1 α1 (i) J (i, n, 0) = sn + δ , sn+1 + δ 3 3 and
1 2 J (i, n, 1) = sn(i) + δ α1 , sn(i) + δ α1 . 3 3
(38)
(39)
Notice that the lengths of J (i, n, 0) and J (i, n, 1) are comparable to δ α0 and δ α1 respectively. By construction, the sets Σ(i, J (i, n, 1)) contain a relatively small amount of the mass of the function fσ (i) in the sense of (37). Furthermore, the inverse images of these sets, dBσ (i) (0)−1 Σ i, J (i, n, 1) ,
(40)
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are O(δ α1 ) neighbourhoods of hyperplanes in Rd , which as n varies are separated by O(δ α0 ). We refer to the sets (40) as buffer zones. The decomposition of Q we use is given by Q=
P (n, χ)
(41)
χ∈{0,1}d n∈Nd
where P (n, χ) =
d "
dBσ (i) (0)−1 Σ i, J (i, ni , χi ) ∩ Q.
(42)
i=1
When χ = 0, the P (n, χ) are large parallelepipeds (intersected with Q) with sidelength approximately δ α0 which form the main part of our decomposition. For χ = 0, the P (n, χ) are small parallelepipeds (intersected with Q) with at least one sidelength approximately δ α1 , which decompose the buffer zones. Step III: Disjointness. In this step we make precise the role of the buffer zones. For each 1 j m, ∈ Ndj and χ ∈ {0, 1}d let Tj (, χ) =
P (n, χ).
n∈Nd : Πj n=
It is the disjointness of the images of such sets under the mapping Bj that is crucial to the induction-on-scales argument which follows in Step IV. Proposition 5.2. Fix j with 1 j m and χ ∈ {0, 1}d . If , ∈ Ndj are distinct then Bj Tj (, χ) ∩ Bj Tj , χ = ∅.
(43)
To prove Proposition 5.2 we use the following. Lemma 5.3. For each 1 j m there exists a map Φj : Rd → Rd such that (i) (ii) (iii) (iv)
Φj (0) = 0 and dΦj (0) is equal to the identity matrix Id , Bj = dBj (0) ◦ Φj , dΦj (x) − dΦj (y) 2κ|x − y|β for each x, y ∈ Q, |x − Φj (x)| 2dκδ 1+β for each x ∈ Q.
Proof. Let Idj be the invertible dj × dj matrix obtained by deleting the kth column of dBj (0) for each k ∈ Kj . For k ∈ Kj define the kth component of Φj (x) to be xk . Define the remaining dj components of Φj (x) by stipulating that the element of Rdj obtained by deleting the kth components of Φj (x) for k ∈ Kj is equal to B (x) − x dB (0) e Id−1 j k j k . j k∈Kj
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Then a direct computation verifies that properties (i) and (ii) hold for Φj . Also, −1 dΦj (x) − dΦj (y) = Idj dBj (x) − dBj (y) 2κ|x − y|β , since Idj − Idj 1/10, and therefore (iii) holds. Finally, property (iv) follows from properties (i) and (iii), and the mean value theorem. 2 Proof of Proposition 5.2. Suppose = and, for a contradiction, suppose that z = Bj (x) = Bj (y) where x ∈ Tj (, χ) and y ∈ Tj ( , χ). Then x ∈ P (n, χ) and y ∈ P (n , χ) for some n, n ∈ Nd satisfying Πj n = and Πj n = . Since Πj n = Πj n there exists i ∈ Kjc such that ni = ni . By (42) and (34) it follows that there exist s(x) ∈ J (i, ni , χi ) and s(y) ∈ J (i, ni , χi ) such that # x,
k=i
$ 2 ak = s(x) ak
and
k=i
# $ 2 y, ak = s(y) ak . k=i
k=i
Therefore # $ 2 1 α 2 1 x − y, ak = s(x) − s(y) a k δ ak 3 k=i
k=i
k=i
where the inequality follows from (36), (38) and (39) since ni = ni . On the other hand, since x and y belong to the fibre Bj−1 (z), it follows from Lemma 5.3(ii) that Φj (x) and Φj (y) belong to dBj (0)−1 (z) and thus Φj (x) − Φj (y) ∈ ker dBj (0). Since i ∈ Kjc and ker dBj (0) = {ar : r ∈ Kj } the vector k=i ak belongs to the orthogonal complement of ker dBj (0). Therefore, # $ # $ # $ x − y, ak = x − Φj (x), ak − y − Φj (y), ak , k=i
k=i
k=i
and so by the Cauchy–Schwarz inequality and Lemma 5.3(iv) it follows that # $ 1+β x − y, ak 4dκδ ak . k=i
k=i
Since | k=i ak | 1/2 we conclude that 24dκδ 1+β δ α1 . For a sufficiently small choice of cd , this is our desired contradiction. 2 Step IV: The conclusion via the discrete inequality. Using the decomposition in Step II, m Q j =1
1 fj Bj (x) m−1 dx =
m
χ∈{0,1}d n∈Nd P (n,χ) j =1
1 fj Bj (x) m−1 dx.
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By (32), 1 a − e , k i 10 k=i
and thus each P (n, χ) is contained in an axis-parallel cube with sidelength equal to 2δ α0 . The main term: χ = 0. It follows that
m
m 1 α m−1 0 fj Bj (x) dx C 2δ , M
n∈Nd P (n,0) j =1
n∈Nd j =1
C 2δ α0 , M
m
1 m−1
fj Bj (P (n,0)) 1
Fj (Πj n) m−1
n∈Nd j =1
where Fj : Ndj → [0, ∞) is given by Fj () =
fj .
Bj (Tj (,0))
Hence, by (7),
m
n∈Nd P (n,0) j =1
m 1 1 fj Bj (x) m−1 dx C 2δ α0 , M Fj m−1 . 1 dj (N )
j =1
Consequently, by Proposition 5.2,
m
m 1 fj Bj (x) m−1 dx C 2δ α0 , M
n∈Nd P (n,0) j =1
j =1
fj
R
1 m−1
.
(44)
dj
The remaining terms: χ = 0. To allow us to capitalise on the pigeonholing in Step II we need the following. Lemma 5.4. For each 1 i d we have ! (i) (i) α1 dBσ (i) (0)−1 Σ i, J (i, ni , 1) ∩ Q ⊆ Bσ−1 ∩ Q. (i) Σ i, sni , sni + δ (i)
(i)
Note here that [sni , sni + δ α1 ] is simply the “concentric triple” of J (i, ni , 1). Proof. Suppose x ∈ Q satisfies dBσ (i) (0)x ∈ Σ(i, J (i, ni , 1)) so that
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
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dBσ (i) (0)x = dBσ (i) (0)y + s dBσ (i) (0) ak
(45)
k=i
for some s ∈ [sn(i)i + 13 δ α1 , sn(i)i + 23 δ α1 ] and y ∈ {ak : k = i}, by (39) and (34). By Lemma 5.3(ii), Bσ (i) (x) = dBσ (i) (0)x + dBσ (i) (0) Φσ (i) (x) − x . Now Φσ (i) (x) − x = y + s
k=i ak
(46)
for some s ∈ R and y ∈ {ak : k = i}, and thus
# $ 2 Φσ (i) (x) − x, ak = s ak . k=i
k=i
k=i ak | 1/2, and by the Cauchy–Schwarz inequality and Lemma 5.3(iv), it follows (i) (i) that |s | 4dκδ 1+β . Now s + s ∈ [sni , sni + δ α1 ] for a sufficiently small choice of cd . Therefore, (i) (i) by (45) and (46), Bσ (i) (x) ∈ Σ(i, [sni , sni + δ α1 ]) as required. 2
Since |
Fix χ = 0 and any i such that χi = 1. As above for the main term, it follows from (7) that
m
n∈Nd P (n,χ) j =1
m 1 1 fj Bj (x) m−1 dx C 2δ α0 , M Fj m−1 1 dj
(N )
j =1
where now Fj () =
fj .
Bj (Tj (,χ))
By Proposition 5.2 it follows that
m
n∈Nd P (n,χ) j =1
1 1 fj Bj (x) m−1 dx C 2δ α0 , M Fσ (i) m−1 1 dσ (i)
(N
)
j =σ (i)
1 m−1
fj
d Rj
and thus it suffices to show that Fσ (i) 1 (Ndσ (i) ) 4δ α1 −α0 .
(47)
To see (47), first set j = σ (i). Given the choice of notation in Step II, it is convenient to write Fj 1 (Ndj ) =
d ∈N j
fj = Bj (Tj (,χ))
nk : k∈Kjc Bj (Tj (Πj n,χ))
fj .
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Now, since i ∈ Kjc we may write Fj 1 (Ndj ) =
ni
fj .
nk : k∈Kjc \{i} Bj (Tj (Πj n,χ))
By Lemma 5.4 it follows that nk : k∈Kjc \{i}
! Bj Tj (Πj n, χ) ⊆ Σ i, sn(i)i , sn(i)i + δ α1 ∩ Q.
Therefore, by Proposition 5.2 and (37),
fj
nk : k∈Kjc \{i} Bj (Tj (Πj n,χ))
fj χQ (i)
(i)
Σ(i,[sni ,sni +δ α1 ])
4δ α1 −α0
fj χQ ,
(i) (i) + 1 δ α0 ,sn −1 +δ α0 ]) i −1 2 i
Σ(i,[sn
from which (47) follows by summing in ni and disjointness. This completes the proof of Lemma 5.1. 2 Remark 5.5. In Theorem 1.3, the smoothness assumption that each mapping Bj belongs to C 1,β may be weakened. Suppose that each Bj is a C 1 submersion in a neighbourhood of x0 such that the modulus of continuity of dBj , which we denote by ωdBj , satisfies ωdBj (δ) κΩ(δ), where, for some 0 < η < 1, Ω satisfies the summability condition ∞
1−η Ω 2−r <∞
(48)
r=0
and κ is a positive constant. Without significantly altering the above proof, one can show that Theorem 1.3 holds under such a smoothness hypothesis. Of course, Theorem 1.3 corresponds to Ω(δ) = δ β with β > 0. It is of course easy to choose Ω satisfying δ β = o(Ω(δ)) as δ → 0 for all β > 0, and still satisfying (48); for example, Ω(δ) = (log 1/δ)−2 . Naturally, one pays for allowing a lower level of smoothness in the size of the neighbourhood on which the estimate in (11) holds.
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2545
6. Proof of Corollary 1.4 Without loss of generality we may suppose that there is a point a belonging to a sufficiently small neighbourhood of the origin in (Rd−1 )d−1 (depending on at most d, β, ε and κ) such that F (a) = 0; otherwise the neighbourhood V in the statement of the corollary could be chosen so that the left-hand side of (12) vanishes. By considering a translation taking a to the origin, we may suppose that a = 0. (Here we are using the uniformity claim relating to the neighbourhood V .) Furthermore, we may assume that ∇uj F (0) = ej ,
(49)
the j th standard basis vector in Rd−1 , for each 1 j d − 1. We shall see that the full generality of Corollary 1.4 follows from this case by a change of variables. Fix nonnegative fj ∈ L(d−1) (Rd−1 ), 1 j m. We proceed in a similar way to the proof of Proposition 7 of [7]. Since ∂(ud−1 )d−1 F (0) = 1 it follows that there exists a neighbourhood W of the origin in Rd(d−2) and a mapping η : W → R such that for each x = u1 , . . . , ud−2 , (ud−1 )1 , . . . , (ud−1 )d−2 ∈ W we have F x, η(x) = 0.
(50)
The neighbourhood W depends only on β and κ, and the mapping η satisfies η C 1,β κ for some constant κ which depends only on d, β and κ. Our claims follow from the implicit function theorem in quantitative form. For completeness we have included an adequate version in Appendix B. Let Bj : W → Rd−1 be given by Bj (x) = (x(d−1)j −d+2 , . . . , x(d−1)j ) for 1 j d − 2, Bd−1 (x) = x(d−1)2 −d+2 , . . . , x(d−1)2 −1 , η(x) , and Bd = B1 + · · · + Bd−1 . We claim that there exists a neighbourhood U of the origin, with U ⊂ W , depending only on d, β and κ, and a constant C depending on d, such that d U j =1
d fj Bj (x) dx C fj (d−1) . j =1
(51)
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Since the subspaces ker dB1 (0), . . . , ker dBd (0) are such that at least one pair has a nontrivial intersection, we cannot directly apply Theorem 1.3 to B = (Bj ) in order to prove (51) (except in the special case d = 3 – see [7]). It is, however, possible to construct mappings Bj⊕ : Rd(d−2) → R(d−1)(d−2) for 1 j d in block form so that d
ker dBj⊕ (0) = Rd(d−2) .
(52)
j =1
We fix 1 j d and define Bj⊕ : Rd(d−2) → R(d−1)(d−2) as follows. Let S (j ) be the (d − 2)tuple obtained by deleting j and j + 1 (mod d) from the d-tuple (1, . . . , d).1 Then define Bj⊕ : Rd(d−2) → R(d−1)(d−2) by Bj⊕ (x) = BS (j ) (x), . . . , BS (j ) (x) . 1
d−2
To see that (52) holds, we compute the required kernels using the fact that ker dBj⊕ (0) =
d−2 " l=1
ker dBS (j ) (0) l
and using straightforward considerations. In order to write these down we write elements of Rd(d−2) as ud−1 ) (u1 , u2 , . . . , ud−3 , ud−2 ; ud−1 ∈ Rd−2 . Then, using (49) and (50), we have where each uj ∈ Rd−1 and
ker dB1⊕ (0) = (u, −u, 0, 0, . . . , 0, 0, 0; 0): u ∈ e1 − e2 ⊥ ,
ker dB2⊕ (0) = (0, u, −u, 0, . . . , 0, 0, 0; 0): u ∈ e2 − e3 ⊥ , .. .
⊕ ker dBd−3 (0) = (0, 0, 0, 0, . . . , 0, u, −u; 0): u ∈ ed−3 − ed−2 ⊥ ,
⊕ ker dBd−2 (0) = 0, 0, 0, 0, . . . , 0, 0, u; (−u1 , . . . , −ud−2 ) : u ∈ ed−2 − ed−1 ⊥ ,
⊕ ker dBd−1 (0) = (0, 0, 0, 0, . . . , 0, 0, 0; u): u ∈ Rd−2 ,
ker dBd⊕ (0) = (u, 0, 0, 0, . . . , 0, 0, 0; 0): u ∈ e1 ⊥ . An elementary calculation now shows that (52) holds. Consequently, it follows from Theorem 1.3 that there exists a neighbourhood U of the origin, depending on d, β and κ, and a constant C depending on d, such that 1 There is some freedom in the choice of the S (j ) ; we only require that the components of each S (j ) are distinct and that for each fixed k ∈ {1, . . . , d} there are exactly d − 2 occurrences of k over all the components of S (1) , . . . , S (d) .
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
d
d gj Bj⊕ (x) dx C gj d−1
U j =1
2547
(53)
j =1
for all gj ∈ Ld−1 (R(d−1)(d−2) ). Now, if fj⊗ ∈ Ld−1 (R(d−1)(d−2) ) is given by fj⊗ =
d−2 %
f
l=1
1/(d−2) (j )
Sl
then by construction, d
fj⊗ Bj⊕ (x) dx =
U j =1
d
fj Bj (x) dx
U j =1
and d ⊗ f
d
j =1
j =1
j
= d−1
fj (d−1) .
Thus, (51) follows immediately from (53). Finally, by the mean value theorem, it is easy to see that there is a neighbourhood V of the origin in (Rd−1 )d−1 , depending only on d, β and κ, such that
f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F (u) du 2
d
fj Bj (x) dx.
U j =1
V
Hence, whenever ∇uj F (0) = ej and F C 1,β κ there exists a neighbourhood V of the origin in (Rd−1 )d−1 , depending only on d, β and κ, and a constant C depending only on d, such that
d f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F (u) du C fj (d−1)
(54)
j =1
V
for all fj ∈ L(d−1) (Rd−1 ). Now suppose that F : (Rd−1 )d−1 → R is such that F C 1,β κ and det ∇u F (0), . . . , ∇u F (0) > ε. 1 d−1 Let A⊕ be the block diagonal (d − 1)2 × (d − 1)2 matrix with d − 1 copies of the matrix T A = ∇u1 F (0), . . . , ∇ud−1 F (0) along the diagonal. Then, by the change of variables u → A⊕ u it follows that
(55)
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f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F (u) du
V
−(d−1) = det(A)
(u) du f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F
A⊕ (V )
= F ◦ (A⊕ )−1 . The neighbourhood V of the origin shall be chosen where fj = fj ◦ A−1 and F momentarily. By (55) it follows that the norm of A−1 is bounded above by a constant depending on only d, . Since, by construction, ε and κ. It follows that the same conclusion holds for the C 1,β norm of F ∇uj F (0) = ej , and by (54), it follows that there exists a neighbourhood V , depending on only d, β, ε and κ, and a constant C depending only on d, such that
d (u) du C fj (d−1) . f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F j =1
A⊕ (V )
Therefore, by (55),
f1 (u1 ) · · · fd−1 (ud−1 )fd (u1 + · · · + ud−1 )δ F (u) du
V d d −1/(d−1) C det(A) fj (d−1) Cε −1/(d−1) fj (d−1) . j =1
j =1
This concludes the proof. 7. Applications to harmonic analysis 7.1. Multilinear singular convolution inequalities Given three transversal and sufficiently regular hypersurfaces in R3 , the convolution of two L2 functions supported on the first and second hypersurface, respectively, restricts to a well-defined L2 function on the third. Under a C 1,β regularity hypothesis and further scaleable assumptions, this was proved by Bejenaru, Herr and Tataru in [4]. We note that the inequality underlying this restriction phenomenon also follows from the nonlinear Loomis–Whitney inequality in [7]; the precise versions of the underlying inequalities differ in [4,7] because a stronger regularity assumption is made in [7] and a uniform transversality assumption is made in [4]. Here we show that natural higher-dimensional analogues of this phenomenon may be deduced from Corollary 1.4. For d 2 and 1 j d, let Uj be a compact subset of Rd−1 and Σj : Uj → Rd parametrise a C 1,β codimension-one submanifold Sj of Rd . Let the measure dσj on Rd supported on Sj be given by
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
ψ(x) dσj (x) = Rd
2549
ψ Σj x dx ,
Uj
where ψ denotes an arbitrary Borel measurable function on Rd . Theorem 7.1. Suppose that the submanifolds S1 , . . . , Sd are transversal in a neighbourhood of the origin, 1 q ∞ and p (d − 1)q . Then there exists a constant C such that f1 dσ1 ∗ · · · ∗ fd dσd Lq (Rd ) C
d
fj Lp (dσj )
(56)
j =1
for all fj ∈ Lp (dσj ) with support in a sufficiently small neighbourhood of the origin. Remark 7.2. (i) By Hölder’s inequality it suffices to prove Theorem 7.1 when p = (d − 1)q . One can also verify that the exponents in Theorem 7.1 are optimal, as may be seen by taking fj to be the characteristic function of a small cap on Sj . As such examples illustrate, at this level of multilinearity, the transversality hypothesis prevents any additional curvature hypotheses on the submanifolds Sj from giving rise to further improvement. See [6] for further discussion of such matters. (ii) Certain bilinear versions of Theorem 7.1 are well known and discussed in detail in [21]. In particular, it follows from [22] that for transversal S1 and S2 (as above), which are smooth with nonvanishing gaussian curvature, there is a constant C for which f1 dσ1 ∗ f2 dσ2 L2 (Rd ) C f1
4d
L 3d−2 (dσ1 )
f2
4d
L 3d−2 (dσ2 )
.
4d here is optimal given the L2 norm on the left-hand side. The case d = 3 The exponent 3d−2 of this inequality was obtained previously in [17]. See for instance [9] for earlier manifestations of such inequalities. (iii) In particular, when q = ∞ inequality (56) implies that
f1 dσ1 ∗ · · · ∗ fd dσd (0) C
d j =1
fj L(d−1) (dσj ) .
By duality, this is equivalent to the statement that, provided f1 , . . . , fd−1 have support restricted to a sufficiently small fixed neighbourhood of the origin, then the multilinear operator (f1 , . . . , fd−1 ) → f1 dσ1 ∗ · · · ∗ fd−1 dσd−1 |Sd
is bounded from L(d−1) (dσ1 ) × · · · × L(d−1) (dσd−1 ) to Ld−1 (dσd ). For d = 3 this is a local variant of the result in [4].
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(iv) The proof of Theorem 7.1 (below) leads to a stronger uniform statement, whereby the sizes of the constant C and neighbourhood of the origin may be taken to depend only on natural transversality and smoothness parameters. We omit the details of this. Proof of Theorem 7.1. By multilinear interpolation and the trivial estimate f1 dσ1 ∗ · · · ∗ fd dσd L1 (Rd )
d
fj L1 (dσj ) ,
j =1
it suffices to prove Theorem 7.1 for q = ∞. By considering a rotation in Rd , we may assume without loss of generality that the submanifolds Sj are hypersurfaces; i.e. given by Σj (x ) = (x , φj (x )) for C 1,β functions φj : Uj → R. Now, for fj supported on Sj for each 1 j d, and any y ∈ Rd we may write f1 dσ1 ∗ · · · ∗ fd dσd (y) d = fj (xj )δ xj d − φj xj δ(x1 + · · · + xd − y) dx1 · · · dxd j =1
(Rd )d
=
d
fj xj , φj xj δ x1 + · · · + xd − y
U1 ×···×Ud j =1
× δ φ1 x1 + · · · + φd xd − yd dx1 · · · dxd d = gj xj δ x1 + · · · + xd − y δ φ1 x1 + · · · + φd xd − yd dx1 · · · dxd U1 ×···×Ud j =1
=
d−1
gd x1 + · · · + xd−1 δ F x1 , . . . , xd−1 dx1 · · · dxd−1 gj xj
U1 ×···×Ud−1 j =1
where gj xj := fj xj , φj xj ,
gd (u) := gd y − u
and = φ1 x1 + · · · + φd−1 xd−1 + φd y − x1 + · · · + xd−1 − yd . F x1 , . . . , xd−1 Observe that F ∈ C 1,β uniformly in y belonging to a sufficiently small neighbourhood of the origin, and that by the transversality hypothesis (combined with the smoothness hypothesis), F (0) = det det ∇x1 F (0), . . . , ∇xd−1
1 ∇φ1 (0)
··· 1 · · · ∇φd−1 (0)
similarly uniformly. Theorem 7.1 now follows by Corollary 1.4.
2
1 ∇φd (y )
= 0
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2551
Estimates of the type (56) are intimately related to the multilinear restriction theory for the Fourier transform, to which we now turn. 7.2. A multilinear Fourier extension inequality Very much as before, let U be a compact neighbourhood of the origin in Rd−1 and Σ : U → Rd parametrise a C 1,β codimension-one submanifold S of Rd . To the mapping Σ we associate the operator E, given by Eg(ξ ) =
g(x)eiξ,Σ(x) dx; U
here g ∈ L1 (U ) and ξ ∈ Rd . We note that the formal adjoint E ∗ is given by the restriction E ∗ f = f◦ Σ , where denotes the Fourier transform on Rd . The operator E is thus referred to as an adjoint Fourier restriction operator or Fourier extension operator. Suppose that we have d such extension operators E1 , . . . , Ed , associated with mappings Σ1 : U1 → Rd , . . . , Σd : Ud → Rd and submanifolds S1 , . . . , Sd . Conjecture 7.3 (Multilinear restriction). (See [7,6].) Suppose that the submanifolds S1 , . . . , Sd 2d are transversal in a neighbourhood of the origin, q d−1 and p d−1 d q. Then there exists a constant C for which d E j gj j =1
C
d
gj Lp (Uj )
(57)
j =1
Lq/d (Rd )
for all g1 , . . . , gd supported in a sufficiently small neighbourhood of the origin. Remark 7.4. Conjecture 7.3 implies Theorem 7.1. To see this we first observe that for any func tion fj on Sj , f j dσj = Ej gj where gj = fj ◦ Σj . Now, if 2 q ∞ and p = (d − 1)q , then by the Hausdorff–Young inequality followed by Conjecture 7.3, d f1 dσ1 ∗ · · · ∗ fd dσd Lq (Rd ) E j gj j =1
C
d
Lq (Rd )
gj Lp (Uj ) = C
j =1
d
fj Lp (dσj ) .
j =1
This link was observed for d = 3 in [4]. In [6] a local form of Conjecture 7.3 was proved with an ε-loss; namely for each ε > 0 the above conjecture was obtained with (57) replaced by d E j gj j =1
Lq/d (B(0,R))
Cε R ε
d j =1
gj Lp (Uj ) ,
(58)
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J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
for all R > 0. In [7] the global estimate (57) was obtained for d = 3 and q = 6. Here we extend this global result to all dimensions. Theorem 7.5. If S1 , . . . , Sd are transversal in a neighbourhood of the origin then there exists a constant C such that d d E j gj C gj 2d−2 (59) 2 d L 2d−3 (Uj ) j =1
L (R )
j =1
for all g1 , . . . , gd supported in a sufficiently small neighbourhood of the origin. Proof. By Plancherel’s theorem, (59) is equivalent to the estimate f1 dσ1 ∗ · · · ∗ fd dσd L2 (Rd ) C
d
fj
j =1
2d−2
L 2d−3 (dσj )
,
where as before we are identifying fj with gj by gj = fj ◦ Σj . Theorem 7.5 now follows immediately from Theorem 7.1. 2 Remark 7.6. The Lebesgue exponent 2d−2 2d−3 on the right-hand side of (59) is best-possible given the L2 norm on the left. Again, at this level of multilinearity, the transversality hypothesis prevents any additional curvature hypotheses from giving rise to further improvement. See [6] for further discussion. Acknowledgments The authors would like to express gratitude to the anonymous referee for their careful reading of the manuscript and extremely helpful recommendations, and also to Steve Roper at the University of Glasgow for creating the figures in Section 3. Appendix A. Proposition 1.1 implies Proposition 1.2 Assume that, for each 1 j m, Bj : Rd → Rdj is a linear surjection and (4) holds. Let Πj : Rd → Rdj be given by (5) where dj is the dimension of ker Bj . Select any set of vectors {ak : k ∈ Kj } forming an orthonormal basis for ker Bj ; that is, the orthogonal complement of the subspace spanned by the rows of Bj . By definition of the Hodge star and orthogonality considerations it follows that ak . Xj (Bj ) = Xj (Bj ) Λdj (Rd )
(A.1)
k∈Kj
Here, · Λdj (Rd ) : Λdj (Rd ) → [0, ∞) is the norm induced by the standard inner product
·,·Λdj (Rd ) : Λdj (Rd ) × Λdj (Rd ) → R given by
u1 ∧ · · · ∧ udj , v1 ∧ · · · ∧ vdj Λdj (Rd ) = det uk , v 1k,d . j
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2553
Let A be the d × d matrix whose ith column is equal to ai for each 1 i d and let Cj be the dj × dj matrix given by Cj = Bj Aj , where Aj is the d × dj matrix obtained by deleting from A the columns ak for each k ∈ Kj . Then, by construction, Πj = Cj−1 Bj A. The matrices A and Cj are invertible by the hypothesis (4). Using A to change variables one obtains m
fj (Bj x)
dx = det(A)
1 m−1
j =1
Rd
m
Rd
1
fj (Πj x) m−1 dx,
j =1
where fj = fj ◦ Cj , 1 j m. By Proposition 1.1 it follows that m Rd
m 1 fj (Bj x) m−1 dx det(A)
j =1
j =1
R
fj dx
1 m−1
dj
1 m m−1 |det(A)| = f j 1 m−1 ( m j =1 j =1 |det(Cj )|) d R
j
and it remains to check that |det(A)|
1 m−1 ( m j =1 |det(Cj )|)
m − 1 m−1 = Xj (Bj ) .
(A.2)
j =1
To this end, note that
m
Xj (Bj ) =
j =1
m Xj (Bj ) j =1
d Λ j (Rd )
m
ak
j =1 k∈Kj
by (A.1) and therefore
m
Xj (Bj ) = det(A)
j =1
since K1 , . . . , Km partitions {1, . . . , d}.
m Xj (Bj ) j =1
d
Λ j (Rd )
(A.3)
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Again use (A.1) to write # $ det(Cj ) = Xj (Bj ), al l∈ / Kj
d
Λ j (Rd )
# $ ak , al d Λ j (Rd )
= Xj (Bj )
k∈Kj
l∈ / Kj
d
Λ j (Rd )
and therefore, by definition of the Hodge star, det(Cj ) = Xj (Bj )
d
Λ j (Rd )
det(A).
(A.4)
Now (A.2) follows from (A.3) and (A.4). This completes the reduction of Proposition 1.2 to Proposition 1.1. Appendix B. A quantitative version of the implicit function theorem We provide a quantitative version of the implicit function theorem for C 1,β functions which we used in the proof of Proposition 1.4. Below we use the notation B(0, R) to denote the open euclidean ball centred at the origin with radius R > 0 in either Rn or R; the dimension of the ball will be clear from the context. Similarly, we denote by B(0, R) the closed euclidean ball centred at the origin with radius R > 0. Theorem B.1. Suppose n ∈ N and β, κ > 0 are given. Let R1 , R2 > 0 be given by R1 =
1 1 min 1, 10κ (100κ)1/β
and R2 =
1 . (100κ)1/β
(B.1)
If F : Rn × R → R is such that F C 1,β κ, F (0, 0) = 0 and ∂n+1 F (0, 0) = 1 then there exists a function η : B(0, R1 ) → B(0, R2 ) such that F x, η(x) = 0 for each x belonging to B(0, R1 ), and a constant κ , depending on at most n, β, and κ, such that η C 1,β κ. Proof. The proof proceeds via a standard fixed point argument applied to the map Ψx : B(0, R2 ) → R given by Ψx (η) = η − F (x, η) for fixed x ∈ B(0, R1 ). We shall prove that Ψx is a contraction which maps B(0, R2 ) to itself. Let Φ : (Rn × R)2 → R be the map given by F (x2 , η2 ) − F (x1 , η1 ) − dF (x1 , η1 )(x2 − x1 , η2 − η1 ) Φ (x1 , η1 ), (x2 , η2 ) = |(x2 − x1 , η2 − η1 )|
J. Bennett, N. Bez / Journal of Functional Analysis 259 (2010) 2520–2556
2555
whenever (x1 , η1 ), (x2 , η2 ) ∈ Rn × R are distinct, and zero otherwise. By the mean value theorem and the fact that F C 1,β κ it follows that Φ is everywhere continuous and Φ (x1 , η1 ), (x2 , η2 ) 1/4 for all (xj , ηj ) ∈ B(0, R2 ) × B(0, R2 ).
(B.2)
For each η1 , η2 ∈ B(0, R2 ) we have Ψx (η1 ) − Ψx (η2 ) = 1 − ∂n+1 F (x, η1 ) (η1 − η2 ) + Φ (x, η1 ), (x, η2 ) |η1 − η2 |. Since ∂n+1 F (0, 0) = 1 and F C 1,β κ it follows that 1 − ∂n+1 F (x, η) 1/4 whenever (x, η) ∈ B(0, R2 ) × B(0, R2 ).
(B.3)
Hence, by (B.2) and (B.3) it follows that Ψx (η1 ) − Ψx (η2 ) 1 |η1 − η2 | 2
(B.4)
and Ψx is a contraction. Now let η ∈ B(0, R2 ). Using the hypothesis F C 1,β κ, along with (B.4) and (B.1), it follows that Ψx (η) Ψx (η) − Ψx (0) + Ψx (0) R2 . Hence Ψx (B(0, R2 )) ⊆ B(0, R2 ). By the Banach fixed point theorem, there exists a mapping η : B(0, R1 ) → B(0, R2 ) such that Ψx (η(x)) = η(x), or equivalently F (x, η(x)) = 0, for each x ∈ B(0, R1 ). It remains to show that η belongs to C 1,β and η C 1,β κ for some constant κ depending on at most n, β and κ. To see that η is differentiable, fix x, h ∈ B(0, R1 ) such that x + h ∈ B(0, R1 ). Since F (x + h, η(x + h)) = F (x, η(x)) it follows that dF x, η(x) h, η(x + h) − η(x) + Φ x, η(x) , x + h, η(x + h) h, η(x + h) − η(x) =0 and therefore ∂n+1 F x, η(x) η(x + h) − η(x) = − ∇x F x, η(x) , h − Φ x, η(x) , x + h, η(x + h) h, η(x + h) − η(x) . Note that by (B.2) and (B.3) it follows that η(x + h) − η(x) C|h| for some finite constant C independent of h. Moreover, Φ is continuous and vanishes along the
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diagonal. It follows that η is differentiable at x and ∇η(x) = −
∇x F (x, η(x)) . ∂n+1 F (x, η(x))
Using F C 1,β κ and (B.3) one quickly obtains the inequality η C 1,β κ for some constant κ depending only on n, β and κ. 2 References [1] K. Ball, Volumes of sections of cubes and related problems, in: J. Lindenstrauss, V.D. Milman (Eds.), Geometric Aspects of Functional Analysis, in: Springer Lecture Notes in Math., vol. 1376, 1989, pp. 251–260. [2] F. Barthe, The Brunn–Minkowski theorem and related geometric and functional inequalities, in: International Congress of Mathematicians, vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1529–1546. [3] I. Bejenaru, S. Herr, J. Holmer, D. Tataru, On the 2d Zakharov system with L2 Schrödinger data, Nonlinearity 22 (2009) 1063–1089. [4] I. Bejenaru, S. Herr, D. Tataru, A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoamericana 26 (2) (2010) 707–728, arXiv:0809.5091. [5] J. Bennett, A. Carbery, M. Christ, T. Tao, The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2007) 1343–1415. [6] J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006) 261– 302. [7] J. Bennett, A. Carbery, J. Wright, A nonlinear generalisation of the Loomis–Whitney inequality and applications, Math. Res. Lett. 12 (2005) 443–457. [8] J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 22 (1991) 147–214. [9] J. Bourgain, On the Restriction and Multiplier Problem in R3 , Lecture Notes in Math., vol. 1469, Springer-Verlag, 1991. [10] E.A. Carlen, E.H. Lieb, M. Loss, A sharp analog of Young’s inequality on S N and related entropy inequalities, J. Geom. Anal. 14 (2004) 487–520. [11] M. Christ, Convolution, curvature, and combinatorics: a case study, Int. Math. Res. Not. 19 (1998) 1033–1048. [12] R.W.R. Darling, Differential Forms and Connections, Cambridge University Press, 1999. [13] H. Finner, A generalization of Hölder’s inequality and some probability inequalities, Ann. Probab. 20 (1992) 1893– 1901. [14] P.T. Gressman, Lp -improving properties of averages on polynomial curves and related integral estimates, Math. Res. Lett. 16 (2009) 971–989. [15] E.H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990) 179–208. [16] L.H. Loomis, H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. (N.S.) 55 (1949) 961–962. [17] A. Moyua, A. Vargas, L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in R3 , Duke Math. J. 96 (1999) 547–574. [18] B. Stovall, Lp improving multilinear Radon-like transforms, preprint. [19] T. Tao, Multilinear weighted convolution of L2 -functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001) 839–908. [20] T. Tao, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003) 1359–1384. [21] T. Tao, Recent progress on the restriction conjecture, in: Park City Proceedings, arXiv:math/0311181. [22] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998) 967–1000. [23] T. Tao, J. Wright, Lp improving bounds for averages along curves, J. Amer. Math. Soc. 16 (2003) 605–638. [24] T.H. Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. 153 (2001) 661–698.
Journal of Functional Analysis 259 (2010) 2557–2586 www.elsevier.com/locate/jfa
A Schur type analysis of the minimal weak unitary Hilbert space dilations of a Kre˘ın space bicontraction and the Relaxed Commutant Lifting Theorem in a Kre˘ın space setting S.A.M. Marcantognini a,∗ , M.D. Morán b a Department of Mathematics, Instituto Venezolano de Investigaciones Científicas, Km. 11 Carretera Panamericana,
Caracas, Venezuela b Escuela de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela, Apartado Postal 20513,
Caracas 1020A, Venezuela Received 3 February 2010; accepted 15 July 2010 Available online 10 August 2010 Communicated by D. Voiculescu
Abstract A parameterization of the minimal weak unitary Hilbert space dilations of a given continuous bicontractive operator on a Kre˘ın space by means of operator-valued Schur functions is obtained from the Arov–Grossman functional model. The result is combined with the coupling method to give a parametric description of the interpolants in a Kre˘ın space version of the Relaxed Commutant Lifting Theorem. © 2010 Elsevier Inc. All rights reserved. Keywords: Kre˘ın space bicontraction; Weak unitary dilation; Schur analysis; Relaxed Commutant Lifting Theorem; Interpolant; Arov–Grossman model; Coupling method
1. Introduction A recent development in lifting theory is the relaxation of the Commutant Lifting Theorem by C. Foias, A.E. Frazho and M.A. Kaashoek [12]. In the relaxed commutant lifting framework we are given a data set {C, T , VT , R, Q} consisting of five Hilbert space operators: a contrac* Corresponding author.
E-mail addresses:
[email protected] (S.A.M. Marcantognini),
[email protected] (M.D. Morán). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.008
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tion C : E → H, a contraction T : H → H with minimal isometric dilation VT : KT → KT and bounded linear operators R, Q : E0 → E. The given operators are assumed to satisfy the relations T CR = CQ
and R ∗ R Q∗ Q.
The relaxed commutant lifting problem is to find all contractions D : E → KT such that KT PH D=C
and VT DR = DQ
KT is the orthogonal projection from KT onto H. The operators D satisfying the above where PH constraints are said to be the interpolants for {C, T , VT , R, Q}. From [12] it is known that an interpolant D exists for any given data set {C, T , VT , R, Q}. Descriptions of the interpolants in the relaxed version of the Commutant Lifting Theorem were given by A.E. Frazho, S. ter Host and M.A. Kaashoek in [13] and [14], by W.S. Li and D. Timotin in [15], and by the authors of the present note in [17]. The classical Commutant Lifting Theorem by D. Sarason [19] and B. Sz.-Nagy and C. Foia¸s [20] appears in the relaxed setting when R = 1 (the identity on E, so that E0 = E) and Q is an isometry on E (in this case, R ∗ R = 1 = Q∗ Q). Extensions of this celebrated theorem were proved in various Kre˘ın space settings by several authors and in full generality by M.A. Dritschel (PhD Dissertation, University of Virginia, 1989). The reader is referred to [10] for Dritschel’s proof of the Commutant Lifting Theorem for contraction operators on Kre˘ın spaces and for the references about previous results. Different proofs, based on the coupling method, were given later on in [16] and [8]. In [9] the approach adopted in [8] combined with a revision of the Arov–Grossman model in the Kre˘ın space framework yielded a labeling of the interpolants in the Commutant Lifting Theorem for contraction operators on Kre˘ın spaces. Kre˘ın space contractions show significant differences from their counterparts in the Hilbert space case. For instance, a contraction operator T from a Kre˘ın space H into a Kre˘ın space K need not be continuous and its adjoint T ∗ may not be a contraction. When both T and T ∗ are contractions, we call T a bicontraction. We may well say that in the Kre˘ın space setting the visà-vis of the Hilbert space notion of contraction is that of continuous bicontraction. So it comes as no surprise that the first versions of the Commutant Lifting Theorem in the Kre˘ın space case were dealing with continuous bicontractions. We pursue the same course of action when extending the Relaxed Commutant Lifting Theorem to Kre˘ın spaces and consider continuous bicontractions as well. We follow the approach adopted in [17]: first we present a version of the Arov–Grossman model which provides a parameterization of the minimal unitary Hilbert space extensions of a given Kre˘ın space isometry whose defect subspaces are Hilbert spaces, then we use it to parameterize the minimal weak unitary Hilbert space dilations of a given continuous bicontraction X defined on a regular subspace B of a Kre˘ın space H such that H B is a Hilbert subspace of H, finally we tackle the relaxed commutant lifting problem with a given data set {C, T , VT , R, Q} of five Kre˘ın space operators, where T is a continuous bicontraction, and give a parametric descriptions of the interpolants in this setting. In addressing the lifting problem we apply the coupling method to get a continuous bicontraction X from the data set {C, T , VT , R, Q} and we show that the interpolants can be obtained from a subclass of the minimal weak unitary Hilbert space dilations of X. Then a parametric description of the interpolants is given by a map from certain set of operator-valued Schur functions (the parameters). The map, however, does not establish a one-to-one correspondence between the parameter and the interpolant, as it may happen that
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2559
different parameters yield the same interpolant. The special case of the Commutant Lifting Theorem for Kre˘ın space bicontractions is discussed in this framework. For this case we get that the map does provide a proper parameterization. The paper is organized in five sections. Section 1, this one, serves as introduction. In Section 2 we fix the notation and review some notions and results needed in the following sections. Section 3 contains the Kre˘ın space extension of the Arov–Grossman model. In Section 4 we introduce the class of unitary operators related to a continuous bicontraction as its minimal weak unitary Hilbert space dilations and use the Kre˘ın space extension of the Arov–Grossman model to give a labeling of the class by means of operator-valued Schur functions. Section 5 is devoted to the Relaxed Commutant Lifting Theorem in the above outlined Kre˘ın space setting. 2. Notation and preliminaries The sets of natural, integral and complex numbers are denoted by N, Z and C, respectively; D stands for the open unit disk in the complex plane and T for its boundary. For any separable Hilbert space H over the complex numbers (in the sequel all Hilbert spaces are supposed to be complex and separable) we denote by L2 (H) the class of all functions f : T → H which are measurable (strongly or weakly, which turns out to be the same due to the separability of H) and such that 1 2π
2π it 2 f e dt < ∞. H 0
Under the interpretation that two functions in L2 (H) are identical if they differ on a set of measure zero, L2 (H) becomes a Hilbert space with the scalar product 1 f, gL2 (H) := 2π
2π it it f e , g e H dt
f, g ∈ L2 (H) .
0
For each n ∈Z, Gn (H) := {f ∈ L2 (H): f (eit ) = eint x for some x ∈ H} is a closed subspace of L2 (H) and n∈Z Gn (H) = L2 (H). The elements of H 2 (H) are all the analytic functions u : D → H whose coefficients Taylor n {un } are square summable, that is, if u ∼ {un }, in the sense that u(z) = ∞ n=0 z un , z ∈ D and {un } ⊆ H, then ∞
un 2H < ∞.
n=0
We recall that H 2 (H) is a Hilbert space with the scalar product u, vH 2 (H) :=
∞
un , vn H n=0
u, v ∈ H 2 (H), u ∼ {un }, v ∼ {vn } .
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As a consequence of Fatou’s Theorem, the radial limit limr↑1 u(reit ) exists almost everywhere. The application that maps each u(z) ∈ H 2 (H) into its radial limit provides an embedding of Via the Poisson integral it can be H 2 (H) into L2 (H) preserving the Hilbert space structures. ∞ 2 (H) onto the subspace 2 shown that the application maps H n=0 Gn (H) of L (H). Therefore ∞ 2 we may consider that H (H) and n=0 Gn (H) amount to the same Hilbert space. If N and M are given Hilbert spaces then S(N , M) stands for the L(N , M)-Schur class, so ϑ ∈ S(N , M) if and only if ϑ : D → L(N , M) is an analytic function such that supz∈D ϑ(z) 1. If ϑ ∈ S(N , M) then limr↑1 ϑ(reit ) exists almost everywhere as a strong limit of operators and determines a contraction operator in L(N , M). With each ϑ ∈ S(N , M) we associate a contraction operator from L2 (N ) into L2 (M) defined by it f e ∈ L2 (N ) f eit → ϑ eit f eit and a contraction operator from H 2 (N ) into H 2 (M) defined by u(z) → ϑ(z)u(z) u(z) ∈ H 2 (N ) . ∞ 2 2 Due n=0 Gn (N ) and ∞ to the identification of H (N ) and H (M) with the subspaces n=0 Gn (M), respectively, the latter operator may be considered as a restriction of the former one. We denote both of them by ϑ . When N = M = H and ϑ(z) ≡ z (z times the identity operator on H) the associated operator is the (forward) shift S. 1 Given ϑ ∈ S(N , M) we write ϑ to denote the operator (1 − ϑ ∗ ϑ) 2 on L2 (N ). The basic reference for vector and operator-valued analytic functions is [20]. We refer the reader to the detailed exposition given therein. Although familiarity with operator theory on Kre˘ın spaces is presumed, we hereafter include some basic notions. We emphasize that the common Hilbert space notation is carried over into the Kre˘ın space setting. Standard references on Kre˘ın spaces and operators on them are [1,5,6]. We also refer to [10] and [11] as authoritative accounts of the subject, in particular, for a detailed exposition on the Kre˘ın space extensions of the Hilbert space concepts of defect operator, Julia operator, minimal isometric dilation and minimal unitary dilation. We recall that a Kre˘ın space is a linear space H (over C) equipped with an inner product (a hermitian sesquilinear form) ·,·H such that there exist two subspaces H+ and H− with the following properties: (a) H is the direct algebraic sum of H+ and H− . (b) H+ , H− H = {0}. (c) (H+ , ·,·H ) and (H− , −·,·H ) are (separable) Hilbert spaces. A fundamental decomposition of a Kre˘ın space (H, ·,·H ) is an orthogonal direct sum H = H+ ⊕H− , where H+ and H− are subspaces as those in the above definition. A fundamental decomposition H = H+ ⊕ H− induces a Hilbert space inner product. Namely, if J x = x + − x − whenever x = x + + x − with x ± ∈ H± , then the Hilbert space inner product of x, y ∈ H is given by J x, yH . The operator J is called a signature operator or fundamental symmetry for H. In general, a Kre˘ın space has infinitely many fundamental decompositions. Nevertheless, the quadratic norms associated with any two fundamental decompositions are equivalent and provide the topology of H.
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Given a fundamental decomposition H = H+ ⊕ H− of the Kre˘ın space H with corresponding signature operator J , we write |H| for H viewed as the Hilbert space relative to the fundamental decomposition. Therefore, x, y|H| = J x, yH and x, yH = J x, y|H| for all x, y ∈ H. The symbol |H− | is used to denote the Hilbert space (H− , −·,·H ). In this notation, |H| = H+ ⊕ |H− |. As a matter of notation, if {Gι }ι∈I is a collection of linear subspaces of a Kre˘ın space H, we will denote by ι∈I Gι the least closed subspace of K containing all the subspaces Gι . Also, if C, D are Kre˘ın spaces and E = C ⊕ D, we will write the elements of E as sums c + d, pairs (c, d) c or columns d with no distinctness. A subspace E of H is negative if x, xH 0 for all x ∈ E. A negative subspace E of H is uniformly negative if, for some (hence, all) Hilbert space norm · |H| on H, there exists a constant ε > 0 such that x, xH −ε x 2|H| for all x ∈ E. We obtain parallel definitions for a subspace to be positive or uniformly positive. A maximal subspace with respect to some property is a subspace which has the property and is not properly contained in another subspace with the same property. A regular subspace of a Kre˘ın space H is a closed subspace F of H which is itself a Kre˘ın space in the inner product inherited from H. For a closed subspace F of H to be regular it is necessary and sufficient that H = F ⊕ F ⊥ . Therefore, F is regular if and only if F ⊥ is regular. If F is a regular subspace of H, its orthogonal companion F ⊥ will be also denoted by H F . By L(H, K) we mean the space of all everywhere defined continuous linear operators on the Kre˘ın space H to the Kre˘ın space K. We set L(H) for L(H, H). The space L(H, K) has the structure of a Banach space depending on choices of fundamental decompositions H = H+ ⊕H− and K = K+ ⊕ K− and associated Hilbert spaces |H| = H+ ⊕ |H− | and |K| = K+ ⊕ |K− |. The corresponding operator norm for L(H, K) is the norm · of L(|H|, |K|). Any two operator norms for L(H, K) are equivalent and provide a topology for L(H, K). By 1 we indicate either the scalar unit or the identity operator, depending on the context. If A : H → K is any linear mapping, then ker A ⊆ H is the null space of A and ran A ⊆ K is its range. For each A ∈ L(H, K) there is a unique A∗ ∈ L(K, H) such that Ax, yK = x, A∗ yH for all x ∈ H and y ∈ K. We say that (i) A ∈ L(H) is selfadjoint if A∗ = A; (ii) P ∈ L(H) is a projection if P is selfadjoint and P 2 = P ; (iii) V ∈ L(H, K) is an isometry if V ∗ V = 1; (iv) V ∈ L(H, K) is an isometric isomorphism or unitary operator if both V and V ∗ are isometries. The regular subspaces of H are those that are the ranges of projections. If F is a regular subspace of H we write PFH to indicate the orthogonal projection from H onto F . According with the Bognár–Krámli factorization, any selfadjoint operator A can be written in the form A = DD ∗ where D ∈ L(D, H) for some Kre˘ın space D and ker D = {0}. T , H), with D T a Kre˘ın space, T ∈ L(D A defect operator for T ∈ L(H, K) is any operator D ∗ ∗ T a defect space for T . T = {0} and 1 − T T = D T D . We call D such that ker D T T , H) is a given defect operator for T ∈ L(H, K), then there exists a unitary T ∈ L(D If D operator U having the form T ∗ T D T ∗ → K ⊕ D T :H ⊕ D U = ∗ L DT T ∗ , K) is a defect operator for T ∗ and L ∈ L(D T ∗ , D T ). Such a unitary operaT ∗ ∈ L(D where D tor is a Julia operator for T .
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We say that (i) T ∈ L(H, K) is a contraction if, for all x ∈ H, T x, T xK x, xH ; (ii) T ∈ L(H, K) is a bicontraction if both T and T ∗ are contractions. T , H) be any defect operator for T ∈ L(H, K). It can be seen that if T is T ∈ L(D Let D T is a Hilbert space. Moreover, if T is a bicontraction then the kernel a contraction then D ∗ T . The deT → H is a Hilbert space in the inner product of K ⊕ D of [ T DT ] : K ⊕ D fect operators for a given contraction T are abstractly undistinguishable in the sense that if T , H) and D , H) are two defect operators for T then there exists a unitary ∈ L(D T ∈ L(D D T T such that D T → D T = D Φ. operator Φ : D T T An isometric linear mapping τ from a subspace D of a Kre˘ın space H onto a subspace R of a Kre˘ın space K is said to be a weak isomorphism if D is dense in H and R is dense in K. A weak isomorphism can be extended to a unitary operator if either D or R contains a maximal uniformly negative subspace. More generally, if T0 is a densely defined linear mapping from a Kre˘ın space H to a Kre˘ın space K such that T0 x, T0 xK x, xH for all x in the domain of T0 and we assume that the domain of T0 contains a maximal uniformly negative subspace E of H and that T0 E is a maximal uniformly negative subspace of K, then T0 has an extension by continuity to a bicontractive operator T ∈ L(H, K). Let H be a Kre˘ın space and let B be a regular subspace of H. If V ∈ L(B, H) is an isometry, we say that a unitary operator U on a Kre˘ın space F is a minimal unitary Hilbert space extension of V if H is a regular subspace of F such that F H is a Hilbert space, U |B = V and F = n∈Z U n H. Two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), are regarded as identical whenever there exists an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ . In the sequel we denote by U(V ) the set of the undistinguishable minimal unitary Hilbert space extensions of V . If V ∈ L(B, H) is an isometry as above then ran V is a regular subspace of H. The defect subspaces of V are N := H B and M := H ran V . If N and M are Hilbert subspaces of H, then U(V ) = ∅ and with each minimal unitary Hilbert space extension U of V acting on F we may associate the L(N , M)-valued function ϑU (z) given by −1 F . ϑU (z) := PM U 1 − zPFFH U N It turns out that ϑU ∈ S(N , M). Conversely, to each ϑ ∈ S(N , M) there corresponds a model unitary operator Uϑ ∈ L(Fϑ ) which is a minimal unitary Hilbert space extension of V . The outlined map ϑ → Uϑ ∈ L(Fϑ ) gives a labeling of U(V ) by S(N , M)-functions. In the next section we state and prove the result encompassing such a description. The model is a Kre˘ın space extension of the one given by D.Z. Arov and L.Z. Grossman [2,3] in the Hilbert space framework (see also [17]). A more general result for continuous isometric operators with regular domain and range is given in [7].
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3. A Kre˘ın space extension of the Arov–Grossman functional model Theorem 3.1. Let H be a Kre˘ın space and let B be a regular subspace of H. Let V ∈ L(B, H) be an isometry and assume that its defect subspaces N and M are Hilbert subspaces of H. Given ϑ ∈ S(N , M), set ⊥ Eϑ := H 2 (M) ⊕ ϑ L2 (N ) ∩ (ϑχ, ϑ χ): χ ∈ H 2 (N ) . Define Fϑ := H ⊕ Eϑ and Uϑ : Fϑ → Fϑ by Uϑ
h φ ψ
⎡ ⎢ := ⎣
H h + φ(0) V PBH h + ϑ(0)PN H h) S ∗ (φ + ϑPN
⎤ ⎥ ⎦
h ∈ H,
H h) S ∗ (ψ + ϑ PN
φ ∈ Eϑ ψ
where S is the shift on either H 2 (M) or L2 (N ), depending on the context. Then: (i) Uϑ ∈ L(Fϑ ) is a minimal unitary Hilbert space extension of V such that −1 Fϑ = ϑ(z) Uϑ 1 − zPEFϑϑ Uϑ PM N for all z ∈ D. (ii) For any minimal unitary Hilbert space extension U of V on F , the function −1 F ϑU (z) := PM U 1 − zPFFH U N belongs to S(N , M). (iii) Two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), are identified under an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ if and only if −1 −1 F F U 1 − zPFF H U N = PM U 1 − zPFFH U PM N for all z ∈ D. Therefore, the map ϑ → Uϑ ∈ L(Fϑ ) establishes a bijective correspondence between S(N , M) and U(V ), up to isometric isomorphisms as far as U(V ) is concerned.
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The proof can be seen in [18]. From it we mention the following fact. Given any ϑ ∈ S(N , M), let Uϑ ∈ L(Fϑ ) be the corresponding element in U(V ) and set H Tϑ (z) := V PBH + ϑ(z)PN
(z ∈ D).
If z ∈ D is such that (1 − zUϑ )−1 ∈ L(Fϑ ) then (1 − zTϑ (z))−1 ∈ L(H) and −1 −1 F PH Uϑ (1 − zUϑ )−1 H = Tϑ (z) 1 − zTϑ (z) = 1 − zTϑ (z) Tϑ (z).
(3.1)
4. The minimal weak unitary Hilbert space dilations of a continuous bicontraction Given a Kre˘ın space A, let B be a regular subspace of A such that A B is a Hilbert subspace is said to of A and let X ∈ L(B, A) be a bicontraction. A unitary operator W on a Kre˘ın space A such be a minimal weak unitary Hilbert space dilation of X if A is a regular subspace of A A n that A A is a Hilbert space, PA W |B = X and A = n∈Z W A. and W ∈ L(A ), Two minimal weak unitary Hilbert space dilations of X, say W ∈ L(A) such that →A are regarded as identical whenever there exists an isometric isomorphism τ : A τ |A = 1 and τ W = W τ . Under the above identification, WUD(X) denotes the set of the minimal weak unitary Hilbert space dilations of X. X , B) for the given bicontraction X and set X ∈ L(D Fix a defect operator D X . H1 := A ⊕ D
(4.1)
View B as a regular subspace of H1 and define V1 :=
X ∗ D X
: B → H1 .
Then V1 ∈ L(B, H1 ) is an isometry whose defect subspaces are X and M1 = ker X ∗ N1 = (A B) ⊕ D
(4.2)
X . D
As N1 and M1 are Hilbert subspaces of H1 , it is granted that U(V1 ) = ∅. If U ∈ L(F ) is any X is a Hilbert subspace of F and X = P AV1 = element of U(V1 ), then F A = (F H1 ) ⊕ D A A ∗ b for all b ∈ B, it comes that F = n∈Z U n A. Whence U PA U |B . Also, since (U − X)b = D X belongs to WUD(X). Thus U(V1 ) ⊆ WUD(X) and, accordingly, WUD(X) = ∅. belong to WUD(X). Consider the decompositions Let W ∈ L(A) = B ⊕ (A B) = A ⊕ (A A) A and write W=
X W21
W12 W22
B) → A ⊕ (A A). : B ⊕ (A
Clearly, for all b ∈ B, b, bA = W b, W bA = Xb, XbA + W21 b, W21 bA
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∗ b, D ∗ b = W21 b, W21 b . Set Y D ∗ b := W21 b (b ∈ B). Then Y is a densely so that D X X DX X A X (we keep defined isometry from DX into A A and its extension by continuity to all of D denoting it by Y ) is an isometry from DX into A A. Likewise, for all a ∈ A, ∗ ∗ a, aA = W ∗ a, W ∗ a A = X ∗ a, X ∗ a A + W12 a, W12 a A X∗ , A) a defect operator for X ∗ . ∗ ∗ a, D ∗ ∗ a = W ∗ a, W ∗ a , with D X∗ ∈ L(D hence D X X DX ∗ 12 12 A X∗ (a Hilbert space) ∗ ∗ a := W ∗ a (a ∈ A). Then E is a densely defined isometry from D Put E D X 12 B (= (A A) ⊕ (A B) a Hilbert space as well). If Z is the adjoint of the extension into A X∗ is a coisometry, meaning that Z ∗ is an isometry, and ∗ of E to all DX , then Z : A B → D ∗ W12 = DX Z. Therefore W can be represented as W=
X ∗ YD
X
X∗ Z D W22
B) → A ⊕ (A A). : B ⊕ (A
(4.3)
Since W is unitary, it should be 1 = W ∗W =
1 ∗ ∗ YD ∗ ∗ Z DX∗ X + W22 X
X Y ∗ W22 X∗ Z + D X∗ D ∗ ∗ D X∗ Z + W ∗ W22 Z∗D X 22
1 ∗ ∗ ∗ ∗ Y DX X + W22 Z ∗ D X
X∗ ZW ∗ X Y ∗ + D XD 22 X∗ D X Y ∗ + W22 W ∗ , YD 22
and 1 = WW∗ =
that is, X Y ∗ W22 = 0, X∗ Z + D X∗ D
(4.4)
∗ ∗ ∗ X Z∗D ∗ DX Z + W22 W22 = 1,
(4.5)
∗ X∗ ZW22 X Y + D = 0, XD ∗ X Y ∗ + W22 W22 X∗ D = 1. YD ∗
(4.6) (4.7)
X , B) for X we associate a Julia operator X ∈ L(D With the chosen defect operator D
X ∗ DX
X∗ D L
X∗ → A ⊕ D X :B ⊕ D
X∗ , A) is the defect operator for X ∗ appearing in the 2 × 2 block matrix X∗ ∈ L(D so that D representation (4.3) of W . Then X L, X∗ = −D X∗ D ∗ ∗ ∗ X D ∗ DX + L L = 1,
X = −D X∗ L∗ , XD ∗ X D DX + LL∗ = 1.
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Making use of the above relations, we get that (4.4) is equivalent to Y ∗ W22 = LZ and (4.6) to ∗ = L∗ Y ∗ , whence ZW22 ∗ ∗ ∗ ∗ X W22 W22 Z ∗ Z = W22 Y LZ = Z ∗ L∗ LZ = Z ∗ 1 − D ∗ DX Z and, similarly, ∗ ∗ X W22 W22 YY∗ = Y 1 − D DX Y ∗ . Thus ∗ ∗ ∗ ∗ ∗ ∗ X Z∗ D ∗ DX Z Z Z + W22 W22 Z Z = Z Z and ∗ ∗ ∗ X Y ∗D Y Y = Y Y ∗. DX Y ∗ Y Y ∗ + W22 W22 ∗ W (1 − Z ∗ Z) = 1 − Z ∗ Z and (4.7) to W W ∗ (1 − So (4.5) turns out to be equivalent to W22 22 22 22 ∗ ∗ YY ) = 1 − YY . We then conclude that a generic W in WUD(X) has a 2 × 2 block matrix representation in the form X∗ Z X D B) → A ⊕ (A A), : B ⊕ (A (4.8) W= ∗ YD W22 X
X → A A is an isometry, Z : A B → D X∗ is a coisometry, W22 : A B → A A where Y : D is a contraction and Y ∗ W22 = LZ, ∗ W22 W22 1 − Z ∗ Z = 1 − Z ∗ Z, ∗ ZW22 = L∗ Y ∗ , ∗ W22 W22 1 − Y Y ∗ = 1 − Y Y ∗,
with
X∗ X D ∗ ∗ L : B ⊕ DX D X
X a fixed Julia operator for X. →A⊕D
belonging to WUD(X), set HY := A ⊕ Y D X and VY := W |B . Then Now, given W ∈ L(A) VY ∈ L(B, HY ) is an isometry and W ∈ U(VY ). The defect subspaces of VY can be easily computed to yield
X and MY = ker X ∗ D X Y ∗ . NY = (A B) ⊕ Y D Let V1 be the isometry on the Kre˘ın space H1 given in (4.1) and (4.2), in which case V1 = VY X . We already found out that the defect subspaces of V1 are the with Y the identity operator on D X and M1 = ker [ X ∗ D X ]. Particularly, the Arov–Grossman Hilbert spaces N1 = (A B) ⊕ D model (Theorem 3.1) can be used to parameterize U(V1 ). The following lemma grants that the Arov–Grossman model also applies to U(VY ) for each X D X∗ Z W = YD ∈ WUD(X). ∗ W X
22
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Lemma 4.1. NY and MY are Hilbert subspaces of HY . Proof. It is clear that NY is a Hilbert space. Let γY : HY → H1 be defined as the identity operator on A and Y ∗ on Y DX . Then γY is a unitary operator such that γY MY = M1 . Since M1 is a Hilbert space, MY is a Hilbert space too. 2 Notice that γY VY = V1 and γY NY = N1 . Theorem 4.2 (Labeling of WUD(X)). Let A be a Kre˘ın space and let B be a regular subspace of A such that A B is a Hilbert subspace of A. Given a bicontraction X ∈ L(B, A), fix a defect X , B) for X and set X ∈ L(D operator D X N1 := (A B) ⊕ D
and M1 := ker X ∗
X . D
Given ϑ ∈ S(N1 , M1 ), let Uϑ ∈ L(Fϑ ) be the corresponding unitary operator as in Theorem 3.1. Then the map ϑ → Uϑ ∈ L(Fϑ ) establishes a one-to-one correspondence between S(N1 , M1 ) and WUD(X), up to isometric isomorphisms as far as WUD(X) is concerned. Proof. First we show that the map ϑ → Uϑ ∈ L(Fϑ ) is one-to-one. Consider α, β ∈ S(N1 , M1 ) and let Uα ∈ L(Fα ), Uβ ∈ L(Fβ ) be the corresponding unitary operators. Suppose that they are undistinguishable elements of WUD(X), so that there exists an isometric isomorphism σ : Fα → Fβ such that σ |A = 1 and σ Uα = Uβ σ . Then, for all a, a ∈ A and all n ∈ Z, Uαn a, a F = Uβn a, a F . α β
Set Ω := {z ∈ D: (1 − zUα )−1 ∈ L(Fα ) and (1 − zUβ )−1 ∈ L(Fβ )}. Then F Fα (1 − zUα )−1 A = PA β (1 − zUβ )−1 A , PA
z ∈ Ω.
(4.9)
Conversely, if (4.9) holds then F Fα n Uα A = PA β Uβn A , PA
n ∈ N.
It follows that Uαn a → Uβn a (a ∈ A, n ∈ Z) defines a weak isomorphism from n∈Z Uαn A = Fα to n∈Z Uβn A = Fβ . Since any maximal uniformly negative subspace of A is also maximal uniformly negative in Fα (and Fβ as well), the weak isomorphism can be extended to a unitary operator σ : Fα → Fβ such that σ |A = 1 and σ Uα = Uβ σ . Therefore Uα ∈ L(Fα ) and Uβ ∈ L(Fβ ) are undistinguishable elements of WUD(X) if and only if (4.9) holds. Let Uϑ ∈ L(Fϑ ) be the unitary operator associated with a given ϑ ∈ S(N1 , M1 ) and define Ωϑ := {z ∈ D: (1 − zUϑ )−1 ∈ L(Fϑ )}. We claim that, for all z ∈ Ωϑ ,
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−1 H1 H1 Fϑ PA (1 − zUϑ )−1 A = 1 − z XPBH1 + PA ϑ(z) 1 − P ϑ(z) A ∗ A −1 . × PA B + zDX PB
DX
If z ∈ Ωϑ is given and a ∈ A then (1 − zUϑ )−1 a = fϑ for some fϑ = φ ψ ∈ Eϑ , if and only if
h1 φ ψ
, with h1 ∈ H1 and
⎡ h1 − z(XP H1 h1 + D ∗ P H1 h1 + ϑ(0)P H1 h1 + φ(0)) ⎤ a X B B N1 ⎦ 0 =⎣ φ − zS ∗ (φ + ϑPN1 h1 ) 0 ψ − zS ∗ (ψ + ϑ PN h1 ) 1
or, equivalently, H1 H1 H1 a = PA ϑ(0)PN h1 − z XPBH1 h1 + PA h1 + φ(0) , 1 ∗ H1 X P h1 + P H1 ϑ(0)P H1 h1 + φ(0) , 0 = P H1 h1 − z D X D
X D
B
N1
H1 0 = φ − zS ∗ φ + ϑPN h1 , 1 H1 0 = ψ − zS ∗ ψ + ϑ PN h1 . 1
(4.10) (4.11) (4.12) (4.13)
H1 As (4.12) holds true if and only if φ = (1 − zS ∗ )−1 zS ∗ ϑPN h1 , it happens that φ(0) = (ϑ(z) − 1
H1 ϑ(0))PN h1 . From this relation and a straightforward computation, it follows that (4.11) holds 1 if and only if
−1 ∗ H1 H1 H1 X P h1 + P H1 ϑ(z)P H1 h1 . P D h1 = z 1 − zP ϑ(z) B A B DX
DX
DX
Therefore −1 H1 H1 H1 H1 H1 ∗ H1 X PN PAB h1 + zD h1 = PA PB h1 B h1 + P h1 = 1 − zP ϑ(z) 1 DX
DX
and (4.10) holds if and only if H1 H1 H1 a = PA h1 − z XPBH1 h1 + PA ϑ(z)PN h1 1 −1 H1 H1 H1 H1 ∗ H1 X = PA h1 − zXPBH1 h1 − zPA ϑ(z) 1 − zP ϑ(z) × PAB h1 + zD PB h1 DX
−1 A H1 H1 H1 ∗ A X PAB + zD × PA ϑ(z) 1 − zP ϑ(z) PB h1 = 1 − z XPBH1 + PA DX
−1 A H1 H1 Fϑ ∗ A X PAB + zD × PA = 1 − z XPBH1 + PA ϑ(z) 1 − zP ϑ(z) PB (1 − zUϑ )−1 a. DX
The result is now obtained since the last relation implies that
−1 −1 A H1 H1 ∗ A X 1 − z XPBH1 + PA PAB + zD ϑ(z) 1 − zP ϑ(z) PB ∈ L(A) DX
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and, furthermore,
−1 −1 A H1 H1 Fϑ ∗ A X 1 − z XPBH1 + PA PAB + zD ϑ(z) 1 − zP ϑ(z) PB = PA (1 − zUϑ )−1 A . DX
In particular, we get that (4.9) holds if and only if, for all z ∈ Ω (= Ωα ∩ Ωβ ), −1 −1 A H1 H1 ∗ A X PAB + zD α(z) 1 − zP α(z) PB 1 − z XPBH1 + PA DX
−1 −1 A H1 H1 ∗ A X PAB + zD = 1 − z XPBH1 + PA β(z) 1 − zP β(z) PB .
DX
H1 H1 α(z))−1 , (1 − zP β(z))−1 ∈ L(N1 ) for all z ∈ D, we may conclude that Uα ∈ As (1 − zP DX DX L(Fα ) and Uβ ∈ L(Fβ ) are undistinguishable elements of WUD(X) if and only if
−1 −1 H1 H1 H1 H1 α(z) 1 − zP α(z) = PA β(z) 1 − zP β(z) , PA DX
DX
z ∈ D.
(4.14)
Note that (4.14) holds if and only if
H1 −1 H1 H1 −1 H1 PA 1 − zα(z)P α(z) − β(z) PA 1 − zP β(z) ≡ 0. DX
DX
For any given u ∈ N1 and all z ∈ D, H1 1 − zP β(z) u ∈ N1 DX
and
H1 −1 α(z) − β(z) u ∈ M1 . 1 − zα(z)P DX
Therefore (4.14) holds whenever, for any u ∈ N1 , the M1 -valued function v(z) := (1 − H1 −1 H1 X ], v(z) is ) [α(z) − β(z)]u (z ∈ D) satisfies PA v(z) ≡ 0. Since M1 = ker [ X ∗ D zα(z)P DX
X P H1 v(z) ≡ 0. As ker D X = {0}, we get that v(z) ≡ 0 and, therefore, α(z) = β(z) to verify D X D for all z ∈ D. This shows that the map ϑ → Uϑ ∈ L(Fϑ )
is one-to-one. belonging to WUD(X), there exists ϑ ∈ S(N1 , M1 ) It remains to see that, given W ∈ L(A) such that W ∈ L(A) is undistinguishable from Uϑ ∈ L(Fϑ ). We recall that X∗ Z X D B) → A ⊕ (A A), : B ⊕ (A W= ∗ YD W22 X X → A A is an isometry, Z : A B → D X∗ is a coisometry and W22 : A B → where Y : D X AA is a contraction. We also recall that VY is the isometry on the Kre˘ın space HY := A⊕Y D X defined in B by VY := Y D ∗ . Let γY : HY → H1 be the unitary operator defined in the proof of X X Lemma 4.1, so that γY VY = V1 , γY NY = N1 and γY MY = M1 , where NY = (A B) ⊕ Y D ∗ ∗ and MY = ker [ X DX Y ].
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Let ρ ∈ S(NY , MY ) correspond to W ∈ WUD(X) when W is viewed as an element of U(VY ) onto Fρ := HY ⊕ Eρ satisfying like in Theorem 3.1. Call σ the isometric isomorphism from A σ |HY = 1 and σ W = Uρ σ . Here ⊥ Eρ := H 2 (MY ) ⊕ ρ L2 (NY ) ∩ (ρχ, ρ χ): χ ∈ H 2 (NY ) and Uρ
hY φ ψ
⎡ ⎢ := ⎢ ⎣
HY hY + φ(0) VY PBHY hY + ρ(0)PN Y HY + ρPN hY ) Y HY + ρ PNY hY )
S ∗ (φ S ∗ (ψ
⎤ ⎥ ⎥ ⎦
hY ∈ HY ,
φ ∈ Eρ . ψ
Define ϑ ∈ S(N1 , M1 ) by ϑ(z) := γY ρ(z)γY∗ |N1 (z ∈ D). Let Uϑ ∈ L(Fϑ ) be the element and Uϑ ∈ in U(V1 ) ⊆ WUD(X) associated with ϑ as in Theorem 3.1. We claim that W ∈ L(A) L(Fϑ ) are undistinguishable elements of WUD(X). The proof is obtained by mimicking the arguments in [17]. We extend γY to a unitary operator from Fρ to Fϑ by setting γY φ(z) := γY φ(z)
φ ∈ H 2 (MY ), z ∈ D
and ψ ∈ ρ L2 (NY ), ζ ∈ T . γY ψ(ζ ) := γY ψ(ζ ) → Fϑ as τ := γY σ . Finally we show that τ |A = 1 and τ W = Uϑ τ . Then we define τ : A
2
5. A Kre˘ın space version of the Relaxed Commutant Lifting Theorem We are given three Kre˘ın spaces E , E0 and H and four operators: • a contraction C ∈ L(E, H), • a bicontraction T ∈ L(H), and • R, Q ∈ L(E0 , E). The operators satisfy the relations T CR = CQ
and R ∗ R Q∗ Q.
Here R ∗ R Q∗ Q means that Re0 , Re0 E Qe0 , Qe0 E ,
e0 ∈ E 0 ,
where ·,·E is the indefinite inner product which E is endowed with. It is known (see [10,11]) that any continuous linear operator T on a Kre˘ın space H has a minimal isometric dilation VT ∈ L(KT ), meaning that KT is a Kre˘ın space containing H as KT n VT |H regular subspace and VT is an isometry everywhere defined on KT such that T n = PH
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for all n ∈ N and KT is the least Kre˘ın space containing {VTn H}∞ n=0 . As a matter of fact, if T ∈ L(H) is a contraction, then KT can be chosen to be T ) KT := H ⊕ H 2 (D
(5.1)
T , H) a defect operator for T , and VT can be defined as the 2 × 2 block matrix T ∈ L(D with D VT :=
0 S
T ∗ D T
: KT → KT
(5.2)
T ). We call VT ∈ L(KT ) as in (5.1) and (5.2) the canonical with S the shift operator on H 2 (D minimal isometric dilation of T ∈ L(H). The canonical minimal isometric dilation has special properties: (a) (b) (c) (d)
H is VT∗ -invariant and VT∗ |H = T ∗ . KT H is a Hilbert subspace of KT . T of {(VT − T )h: h ∈ H} is a Hilbert subspace of KT . The closure L T is the wandering subspace for VT , meaning that L T ⊕ V 2 L T ⊕ VT L KT = H ⊕ L T T ⊕ ···.
It happens that if V ∈ L(K) is a minimal isometric dilation of the given contraction T ∈ L(H), then the map V n h → VTn h (h ∈ H, n ∈ N ∪ {0}) gives rise to an isometric isomorphism τ : K → KT such that τ |H = 1 and τ V = VT τ . Therefore any minimal isometric dilation of T ∈ L(H) is undistinguishable from the canonical one. In the context of the Relaxed Commutant Lifting Theorem, any contraction operator D ∈ L(E, KT ) such that KT and PH D=C
VT DR = DQ
is said to be an interpolant for {C, T , VT , R, Q}. In the Kre˘ın space framework we have set up, the problem we address is to find the interpolants for {C, T , VT , R, Q}. Since T ∈ L(H) is assumed to be a bicontraction, we can likewise consider the canonical minimal isometric dilation WT ∈ L(GT ) of T ∗ ∈ L(H). Both VT ∈ L(KT ) and WT ∈ L(GT ) can be expressed in terms of a distinguished unitary operator built up from T as follows. T , H) and D T ∗ , H) are defect operators for T and T ∗ , respectively, and T ∈ L(D T ∗ ∈ L(D If D T D T ∗ T := H 2 (D T ∗ ) ⊕ H ⊕ H 2 (D T ) (take into is the associated Julia operator for T , set H ∗ D T
L
T ∗ is a Hilbert space) and define account that D ⎡
S∗ T ∗ P0 UT := ⎣ D LP0
0 T ∗ D T
⎤ 0 T → H T , 0 ⎦ :H S
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with P0
∞
z vn := v0 n
n=0
∞
! T ∗ ) z vn ∈ H (D n
2
n=0
T ) or H 2 (D T ∗ ) depending on the context. Then H T is a Kre˘ın space and S the shift on H 2 (D containing H as regular subspace and UT is an everywhere defined continuous linear operator T . It can be seen that UT is a unitary operator such that, for all h ∈ H and n ∈ N, on H
HT n HT ∗n UT h = T n h and PH UT h = T ∗n h. PH
(5.3)
Furthermore ∞ " n=0
T ) UTn H = H ⊕ H 2 (D
and
∞ "
T ∗ ) ⊕ H. UT∗n H = H 2 (D
n=0
A minimal unitary dilation of T ∈ L(H) is a unitary operatornUT ∈ L(HT ) such that HT is a T = ∞ Kre˘ın space containing H as regular subspace, H U H and the relations (5.3) hold n=−∞ T for all h ∈ H and n ∈ N. Two methods for constructing minimal unitary dilations of a generic continuous linear operator on a Kre˘ın space are given in [10]. If T ∈ L(H) is a bicontraction, which is the case, then any two minimal unitary dilations of T are isometrically isomorphic. The minimal unitary dilation in the above construction is in the sequel the canonical one. It has special properties, in particular: T and (a) If VT ∈ L(KT ) is the canonical minimal isometric dilation of T then KT ⊆ H VT = UT |KT . T ∗ ) ⊕ H) ⊆ H 2 (D T ∗ ) ⊕ H and U ∗ | 2 (b) UT∗ (H 2 (D T H (DT ∗ )⊕H is a minimal isometric dilation ∗ of T . T ∗ ) ⊕ H and WT := U ∗ |G to get Whence, according with (b) above, we may take GT := H 2 (D T T ∗ a minimal isometric dilation WT ∈ L(GT ) of T (as a matter of fact, the canonical one). In what follows we fix this minimal isometric dilation of T ∗ . In E × GT consider the hermitian sesquilinear form e e , := e, e E + g, Ce G + Ce, g G + g, g G . T T T g g C C , E) is a fixed defect operator for C, consider the Kre˘ın space A := D C ⊕ GT C ∈ L(D If D with the standard inner product. Then, for all e, e ∈ E and all g, g ∈ GT , ∗ e e C e, D C∗ e + Ce + g, Ce + g = D , D GT C g g C ∗ C e + (Ce + g), D C∗ e + Ce + g . = D A
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Hence, by setting e C∗ e + (Ce + g) σ := D g
e ∈ E × GT , g
we get an isometric map from (E × GT , [·,·]C ) into (A, ·,·A ). C , let {e(n)} ⊆ E be a sequence such that limn→∞ D ∗ e(n) = x. Then, for any g ∈ GT , If x ∈ D C
lim σ
n→∞
e(n) ∗ e(n) + g = x + g. = lim D n→∞ C −Ce(n) + g
Thus σ (E × GT ) is a dense subspace of A. We call A the coupling Kre˘ın space associated with {C, T }, when C is viewed as a linear operator from E to GT ⊇ H. Define B := σ (QE0 × GT ). Set P (x + g) := P
C D
∗ QE DC 0
x +g
C , g ∈ GT ). (x ∈ D
∗ QE0 ⊕ GT , a regular It can be easily seen that P ∈ L(A) is a projection whose range is D C subspace of A. If e0 ∈ E0 and g ∈ GT then C∗ Qe0 + g = σ D
Qe0 ∈ σ (QE0 × GT ) −CQe0 + g
and
Qe0 C∗ Qe0 + (CQe0 + g) ∈ D C∗ QE0 ⊕ GT . σ =D g ∗ QE0 ⊕ GT and B = D ∗ QE0 ⊕ GT . It follows that B is a regular subThus σ (QE0 × GT ) = D C C C D ∗ QE0 is a Hilbert subspace of A. space of A. Moreover, A B = D C Define the map X0 σ
Qe0 Re0 := σ g WT g
(e0 ∈ E0 , g ∈ GT ).
For all e0 ∈ E0 and g ∈ GT , # $ $ # Re0 Qe0 Qe0 Re0 X0 σ , X0 σ ,σ = σ g g WT g WT g A A = Re0 , Re0 E + 2 ReCRe0 , WT gGT + WT g, WT gGT = Re0 , Re0 E + 2 ReCQe0 , gGT + g, gGT Qe0 , Qe0 E + 2 ReCQe0 , gGT + g, gGT $ # Qe0 , Qe0 , ,σ = σ . g g A
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So X0 is a densely defined contraction from B to A. Note that GT ⊆ σ (QE0 × GT ) and X0 GT = WT GT = UT∗ GT . If GT− is a maximal uniformly negative subspace of GT then GT− is also maximal uniformly negative in B. Since UT∗ is unitary and UT∗ GT ⊆ GT , it follows that X0 GT− is maximal uniformly negative in GT , hence, in A. Therefore, X0 has an extension by continuity to a bicontractive operator X ∈ L(B, A). We term X : B ⊆ A → A the coupling bicontraction underlying the lifting data set {C, T , VT , R, Q}. Write Q∗ Q − R ∗ R = D0 D0∗ with D0 ∈ L(D0 , E0 ) such that ker D0 = {0}. Since R ∗ R Q∗ Q, we have that D0 is a Hilbert space. Notice that # $ $ # Qe0 Qe0 Qe0 Qe0 Xσ , Xσ ,σ = σ − D0∗ e0 , D0∗ e0 D 0 g g g g A A X , B) is a defect operator for X, X ∈ L(D for all e0 ∈ E0 and g ∈ GT . Hence, if D # $ Qe0 ∗ Qe0 ∗ X , DX σ σ = D0∗ e0 , D0∗ e0 D D 0 g g D X
for all e0 ∈ E0 and g ∈ GT . Now we consider the set WUD(X) as described in Theorem 4.2. Given ϑ ∈ S(N1 , M1 ), let Uϑ ∈ L(Fϑ ) be the minimal weak unitary Hilbert space dilation of X associated with ϑ . Define φϑ VTn h := Uϑ−n h
(h ∈ H, n = 0, 1, 2, . . .)
to get a linear map from the linear span of {VTn H}∞ n=0 into Fϑ . Note that, for all h ∈ H and n = 0, 1, 2, . . . , Uϑn h = WTn h. Therefore, for all h, h ∈ H and all n = 0, 1, 2, . . . , φϑ VTn h, φϑ h F = Uϑ−n h, h F = h, Uϑn h F ϑ ϑ ϑ n ∗n = h, WT h G = h, T h H T = T n h, h H = VTn h, h K .
T
Since KT H is a Hilbert space and the domain of φϑ contains H, φϑ can be extended to an Φϑ ∈ L(KT , Fϑ ). It is clear that Φϑ is a unitary operator from KT onto Kϑ := ∞isometry −n n=0 Uϑ H. So, as a by product, we get that Kϑ is a regular subspace of Fϑ . Moreover, since C ⊕ (GT H) ⊕ (Fϑ A) is a Hilbert space, it comes that Fϑ Kϑ is H ⊆ Kϑ and Fϑ H = D a Hilbert space. Also, Φϑ |H = 1, Φϑ VT = Uϑ−1 Φϑ and Φϑ∗ is a continuous contraction from Fϑ to KT . Now, define D : E → KT by De := Φϑ∗ σ
e 0
(e ∈ E).
(5.4)
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Let |E| and |H| be the Hilbert spaces arising from given fundamental decompositions of E 1 and H, and let C be computed with respect to · |E | and · |H| . Then D (1 − 2 C 2 ) 2 when D is viewed in L(|E|, |KT |) with |KT | := |H| ⊕ (KT H). For all e ∈ E, # $ # $ e e e e De, DeKT = Φϑ∗ σ , Φϑ∗ σ σ ,σ = e, eE 0 0 K 0 0 A T
and, for all h ∈ H, # $ # $ e e De, hKT = Φϑ∗ σ ,h = σ , h = Ce, hH . 0 0 K A T
KT Whence D ∈ L(E, KT ) is a contraction such that PH D = C. To get that D is an interpolant for {C, T , VT , R, Q} it remains to see whether VT DR = DQ. Let e0 ∈ E0 and h ∈ H be given. Then
VT DRe0 , hKT = DRe0 , T ∗ h K = CRe0 , T ∗ h H T
= CQe0 , hH = DQe0 , hKT and VT DRe0 , VT hKT = DRe0 , hKT = CRe0 , hH $ $ # # Re0 Qe0 , h = Xσ ,h = σ 0 0 A A $ # # $ Qe0 Qe0 ,h , Uϑ−1 h = Uϑ σ = σ 0 0 Fϑ Fϑ # $ # $ Qe0 Qe0 , Φϑ VT h , VT h = σ = Φϑ∗ σ 0 0 Fϑ K
T
= DQe0 , VT hKT , so that
VT DRe0 , (VT − T )h K = DQe0 , (VT − T )h K . T
T
T ⊕ V 2 L T ⊕ VT L Since KT = H ⊕ L T T ⊕ · · · , LT := {(VT − T )h: h ∈ H}, it is clear that VT DR = DQ as far as
VT DRe0 , VTn (VT − T )h K = DQe0 , VTn (VT − T )h K T
for all e0 ∈ E0 , h ∈ H and n ∈ N. Note that Φϑ VTn (VT − T )h = Uϑ−n−1 (1 − WT T )h,
h ∈ H,
T
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and Uϑ σ
Re0 Qe0 Qe0 ∗ X =σ +D , σ 0 0 0
e0 ∈ E 0 .
Therefore, for all e0 ∈ E0 , h ∈ H and n ∈ N, # $ Qe0 , Uϑ−n−1 (1 − WT T )h DQe0 , VTn (VT − T )h K = σ T 0 Fϑ # $ Qe0 , Uϑ−n (1 − WT T )h = Uϑ σ 0 Fϑ # $ Re0 , Uϑ−n (1 − WT T )h = σ 0 Fϑ # $ Qe0 ∗ X , Uϑ−n (1 − WT T )h + D σ 0 Fϑ
and
VT DRe0 , VTn (VT − T )h K = DRe0 , VTn−1 (VT − T )h K T T # $ Re0 , Uϑ−n (1 − WT T )h = σ . 0 Fϑ
Thus VT DR = DQ if and only if # $ Qe0 ∗ X Uϑn D , (1 − WT T )h σ = 0, 0 Fϑ
e0 ∈ E0 , h ∈ H, n ∈ N.
Consequently, ϑ ∈ S(N1 , M1 ) yields a D such that VT DR = DQ if and only if the corresponding element in WUD(X), Uϑ ∈ L(Fϑ ), satisfies
(1 − zUϑ )−1 Uϑ u, (1 − WT T )h F = 0, ϑ
X , h ∈ H. z ∈ Ωϑ , u ∈ D
(5.5)
Here, like in the proof of Theorem 4.2, Ωϑ := {z ∈ D: (1 − zUϑ )−1 ∈ L(Fϑ )}. Let JT be the closure of {(1 − WT T )h: h ∈ H}. Since (1 − WT T )h = Uϑ Φϑ (VT − T )h,
h ∈ H,
T . As L T is a Hilbert subspace of KT and both Uϑ ∈ L(Fϑ ) and it follows that JT ⊆ Uϑ Φϑ L C ⊕ Φϑ ∈ L(KT , Kϑ ) are unitary operators, it turns out that JT is a Hilbert subspace of Fϑ = D ∗ GT ⊕ (Fϑ A), hence, of GT . Besides, note that JT = ker WT . With the introduction of JT , we see that (5.5) is equivalent to PJFTϑ (1 − zUϑ )−1 Uϑ D = 0, X
z ∈ Ωϑ .
(5.6)
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X and all z ∈ Ωϑ , For any given u ∈ D Fϑ Uϑ (1 − zUϑ )−1 u + PJFTϑ Uϑ (1 − zUϑ )−1 u PGFTϑ Uϑ (1 − zUϑ )−1 u = PW T GT
= WT PGFTϑ (1 − zUϑ )−1 u + PJFTϑ Uϑ (1 − zUϑ )−1 u
= WT PGFTϑ u + zUϑ (1 − zUϑ )−1 u + PJFTϑ Uϑ (1 − zUϑ )−1 u = zWT PGFTϑ Uϑ (1 − zUϑ )−1 u + PJFTϑ Uϑ (1 − zUϑ )−1 u. Therefore (1 − zWT )PGFTϑ Uϑ (1 − zUϑ )−1 u = PJFTϑ Uϑ (1 − zUϑ )−1 u. Since 1 − zWT = (1 − zUϑ )|GT and 1 − zUϑ is invertible for all z ∈ Ωϑ , the above arguments lead to conclude that (5.6) is equivalent to PGFTϑ Uϑ (1 − zUϑ )−1 D = 0, X
z ∈ Ωϑ .
(5.7)
Hereafter it could be helpful to recall that N1 and M1 can be interpreted as the defect X defined in B by V1 := X∗ and that each subspaces of the isometry V1 on H1 := A ⊕ D DX Uϑ ∈ L(Fϑ ) belongs indeed to U(V1 ). Additionally, under this viewpoint, we handle a more compact notation. In particular we get that, for all z ∈ Ωϑ , Fϑ −1 = 1 − z V1 P H1 + ϑ(z)P H1 −1 V1 P H1 + ϑ(z)P H1 PH (1 − zU ) U ϑ ϑ B N1 B N1 D X DX 1 H1 −1 = 1 − z V1 PBH1 + ϑ(z)PN ϑ(z) D 1 X
(see (3.1)). The above discussion can be summarized in the following: Proposition 5.1. The following statements are equivalent: (i) The operator D associated with a given ϑ ∈ S(N1 , M1 ) by means of (5.4) satisfies VT DR = DQ and, hence, is an interpolant for {C, T , VT , R, Q}. H1 −1 (ii) PJHT1 (1 − zV1 PBH1 − zϑ(z)PN ) ϑ(z)|D X = 0 for all z ∈ Ωϑ .
H1 −1 (iii) PGHT 1 (1 − zV1 PBH1 − zϑ(z)PN ) ϑ(z)|D X = 0 for all z ∈ Ωϑ .
From the proposition it results that the operator D associated with a given ϑ ∈ S(N1 , M1 ) by means of (5.4) satisfies VT DR = DQ and, hence, is an interpolant for {C, T , VT , R, Q} if, for instance, ϑ(z)|D X = 0 for all z ∈ D. Before embarking on analyzing the map (5.4), let us establish a closed formula for the direct connection between ϑ and D.
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T ⊕ · · · and P KT D = C, it is clear that D = C + T ⊕ VT L Since KT = H ⊕ L H
∞
KT
n=0 PV n L T D. T
Therefore D is determined by the sequence of operators {P KnT D}∞ n=0 (cf. [4]). With each D VT LT
T )-valued function SD (z) defined around 0 by the power series we associate an L(E, L SD (z) :=
∞
zn S% D (n)
n=0
where ∗n KT T ) S% D (n) := VT P n D ∈ L(E, L VT LT
(n = 0, 1, 2, . . .).
For all e ∈ E and h ∈ H, C∗ e, (1 − WT T )h SD (z)e, (VT − T )h K = PJFTϑ (1 − zUϑ )−1 Uϑ D F
T
ϑ
whenever z ∈ Ωϑ . Define G(1 − WT T )h := (VT − T )h (h ∈ H) to get a unitary operator T . Then G : JT → L C∗ , SD (z) = GPJFTϑ (1 − zUϑ )−1 Uϑ D
z ∈ Ωϑ .
(5.8)
∗ where ∗ = P H1 [P Fϑ (1 − zUϑ )−1 Uϑ |H ]D Note that PJFTϑ (1 − zUϑ )−1 Uϑ D 1 C C JT H1 Fϑ −1 = Tϑ (z) 1 − zTϑ (z) −1 = 1 − zTϑ (z) −1 Tϑ (z) (1 − zU ) U PH ϑ ϑ H 1 1
H1 (see (3.1)). with Tϑ (z) := V1 PBH1 + ϑ(z)PN 1 Fix the Hilbert space
C ⊕ JT ⊕ WT (GT H) ⊕ WT H+ ⊕ WT H− |A| := D to get X |H1 | := |A| ⊕ D and, for any ϑ ∈ S(N1 , M1 ), |Fϑ | = |H1 | ⊕ Eϑ . Observe that any C ⊆ H1+ satisfies that ·,·H1 |C = ·,·|H1 | |C . It is clear that JT , N1 ⊆ H1+ . As PGHT 1 M1 ⊆ JT , it turns out that M1 ⊆ H1+ as well. Since JT ⊆ H1+ , g 2|H1 | V1 g 2|H1 | for all g ∈ JT . Hence V1 1 when the norm is computed in L(|H1 |). Let ϑ ∈ S(N1 , M1 ) be given. As N1 ⊕ Eϑ , M1 ⊕ Eϑ ⊆ Fϑ+ , it follows that, for all fϑ ∈ Fϑ ,
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2 Fϑ Uϑ fϑ 2|Fϑ | = V1 PBFϑ fϑ + Uϑ PN f ϑ ⊕ E |Fϑ | ϑ 1 2 2 Fϑ = V1 PBFϑ fϑ |H | + Uϑ PN fϑ |F | 1 ⊕ Eϑ ϑ 1 Fϑ 2 2 2 Fϑ V1 P fϑ + P fϑ B
|H1 |
N 1 ⊕ Eϑ
|Fϑ |
V1 2 fϑ 2|Fϑ | . Therefore Uϑ = V1 when Uϑ is regarded in L(|Fϑ |). Set r := V1 PBH1 −1 . Then (1 − zV1 PBH1 )−1 ∈ L(|H1 |) for all |z| < r. Also, for any ϑ ∈ S(N1 , M1 ), rUϑ < 1. Whence (1 − zUϑ )−1 ∈ L(|Fϑ |) for all |z| < r. Therefore, for all |z| < r, (1 − zTϑ (z))−1 ∈ L(|H1 |) and −1 −1 −1 H1 −1 1 − zTϑ (z) 1 − zV1 PBH1 = 1 − z 1 − zV1 PBH1 ϑ(z)PN 1 −1 −1 −1 H1 1 − zϑ(z)PN 1 − zV1 PBH1 = 1 − zV1 PBH1 . 1 Note that the inverse operators of 1 − zV1 PBH1 on |H1 | and 1 − zUϑ on |Fϑ | can be computed by means of Neumann series. From the above discussion we get the following: Proposition 5.2. For any ϑ ∈ S(N1 , M1 ) the following assertions hold: Fϑ (i) For each h1 ∈ H1 , z ∈ D → PH (1 − rzUϑ )−1 h1 is an H 2 (|H1 |)-function. 1 (ii) For all |z| < r,
−1 Fϑ (1 − zUϑ )−1 Uϑ H = A(z) + B(z)ϑ(z) 1 − C(z)ϑ(z) D(z) PH 1 1
where −1 A(z) := V1 PBH1 1 − zV1 PBH1 ∈ L |H1 | , −1 B(z) := 1 − zV1 PBH1 M1 ∈ L M1 , |H1 | , −1 H1 1 − zV1 PBH1 C(z) := zPN M1 ∈ L(M1 , N1 ), 1 −1 H1 1 − zV1 PBH1 ∈ L |H1 |, N1 . D(z) := PN 1 n In computing PJFTϑ (1 − zUϑ )−1 Uϑ |D C it is relevant to consider that V1 JT ⊆ B and
V1n+1 JT ⊥ JT for all n ∈ N ∪ {0}. Also it is useful to take into account that [1 − z(1 −
PJHT1 )V1 PBH1 ]−1 ∈ L(|H1 |) if and only if [1 − z(1 − PJAT )XPBA ]−1 ∈ L(|A|) and that
−1 = 1 − z 1 − PJHT1 V1 PBH1
[1 − z(1 − PJAT )XPBA ]−1
0
∗ P A [1 − z(1 − P A )XP A ]−1 zD X B JT B
1
.
The first assertion is plain consequence of the definitions of the isometry V1 and the subspace JT . As for the proof of the second assertion, it is enough to write 1 − z(1 − PJHT1 )V1 PBH1 ∈ L(|H1 |) X . as a 2 × 2 block matrix with respect to the decomposition |H1 | := |A| ⊕ D
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Since JT ⊆ H1+ , it comes that (1 − PJHT1 )V1 PBH1 V1 PBH1 when both operators are
regarded in L(|H1 |). Thus [1 − z(1 − PJHT1 )V1 PBH1 ]−1 ∈ L(|H1 |) for all |z| < r. Then a straightforward computation we omit in the present discussion gives the following: Proposition 5.3. For all ϑ ∈ S(N1 , M1 ) and all |z| < r, −1 d(z), PJFTϑ (1 − zUϑ )−1 Uϑ D = a(z) + b(z)ϑ(z) 1 − c(z)ϑ(z) C
where
−1 a(z) := PJAT XPBA 1 − z 1 − PJAT XPBA D ∈ L(DC , JT ), C
−1 H1 b(z) := PJHT1 1 + zXPBA 1 − z 1 − PJAT XPBA PA
−1 H1 c(z) := zPN 1 − z 1 − PJHT1 V1 PBH1 M1 1
× 1 − PJHT1 M ∈ L(M1 , JT ), 1
∈ L(M1 , N1 ),
−1 H1 ∈ L(D C , N1 ). d(z) := PN 1 − z 1 − PJHT1 V1 PBH1 D 1 C
Therefore, for all |z| < r, −1 ∗ C . SD (z) = G a(z) + b(z)ϑ(z) 1 − c(z)ϑ(z) d(z) D T is a Hilbert space with the inner product inherited from KT . So we consider Recall that L that it is endowed with the quadratic norm · 2 = ·,·KT |LT . From (5.8) and Propositions 5.2 LT T )-function for each e ∈ E . and 5.3 it follows that z ∈ D → SD (rz)e is an H 2 (L 2 2 So if Γ : H (LT ) → LT ⊕ VT LT ⊕ VT LT ⊕ · · · is the unitary operator given by Γ
∞
n=0
z xn := n
∞
n=0
VTn xn
x(z) =
∞
! z xn ∈ H (LT ) n
2
n=0
then ∞
−1 ∗ C e P KnT De = Γ G a(rz) + b(rz)ϑ(rz) 1 − c(rz)ϑ(rz) d(rz) D n=0
VT LT
for all e ∈ E. Theorem 5.4 (Description of the interpolants in the Relaxed Commutant Lifting Theorem). Consider the lifting data set {C, T , VT , R, Q}. Let WT ∈ L(GT ) be the canonical minimal isometric dilation of T ∗ . Let A be the coupling Kre˘ın space associated with {C, T }, when C is viewed as a linear operator from E into GT ⊇ H, and let X : B ⊆ A → A be the coupling X , B) be a fixed X ∈ L(D bicontraction underlying the lifting data set {C, T , VT , R, Q}. Let D
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T for the wandering subspace of VT . Define Γ : H 2 (L T ) → defect operator for X. Write L 2 LT ⊕ VT (LT ) ⊕ VT (LT ) ⊕ · · · to be the unitary operator given by Γ
∞
n=0
z xn := n
∞
n=0
VTn xn
x(z) =
∞
! T ) . z xn ∈ H (L 2
n
n=0
T , where JT := Set G(1 − WT T )h := (VT − T )h (h ∈ H) to get a unitary operator G : JT → L ∗ {(1 − WT T )h: h ∈ H} is the null space of WT . Define X N1 := (A B) ⊕ D and X X X : X ∗ P A⊕D X P A⊕D M1 := x ∈ A ⊕ D x +D x=0 . A DX
Then there exists 0 < r < 1 such that the following assertions hold: T ), T ), C , N1 ) C , L b : D → L(M1 , L c : D → L(M1 , N1 ) and d: D → L(D (i) If a : D → L(D are defined by
−1 a (z) := PJAT XPBA 1 − rz 1 − PJAT XPBA D , C
X A⊕ D
−1 A⊕D X 1 + rzXPBA 1 − rz 1 − PJAT XPBA PA × 1 − PJHT1 M , b(z) := PJT 1 A A A −1 0 PAB [1 − rz(1 − PJT )XPB ] , c(z) := rz ∗ A A A −1 P [1 − rz(1 − P )XP ] rzD 1 M1 X B JT B A A A −1 PAB [1 − rz(1 − PJT )XPB ] , := d(z) ∗ A A A −1 P [1 − rz(1 − P )XP ] rzD DC X B JT B
then −1 ∗ D C e z ∈ D → Λϑ (e) := G a (z) + b(z)ϑ(rz) 1 − c(z)ϑ(rz) d(z) T )-function for any ϑ ∈ S(N1 , M1 ) and each e ∈ E . is an H 2 (L (ii) Given any ϑ ∈ S(N1 , M1 ) satisfying ϑ(z)|D X ≡ 0, define D : E → KT by De := Ce + Γ Λϑ (e)
(e ∈ E).
(5.9)
Then D is an interpolant for {C, T , VT , R, Q}. Moreover, in this way all interpolants for {C, T , VT , R, Q} are obtained. A function ϑ ∈ S(N1 , M1 ) satisfying ϑ(z)|DX ≡ 0 always exists. For instance, one can take ϑ ≡ 0. This choice for ϑ yields the so-called central interpolant.
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Proof. The proof is complete if we see that all interpolants for {C, T , VT , R, Q} are obtained as indicated. Thus, given a contraction D ∈ L(E, KT ) satisfying KT PH D=C
and VT DR = DQ,
let us show that D can be obtained from a function ϑ ∈ S(N , M) as in (5.9). In what follows we consider all the elements in the construction of the Kre˘ın space A and X and the isometry V1 := B∗ ∈ the bicontraction X as well as the Kre˘ın space H1 := A ⊕ D DX L(B, H1 ). We also use that each undistinguishable element of WUD(X) belongs indeed to U(V1 ). T ) be the canonical minimal unitary dilation of T (recall that KT , GT ⊆ H T , Let UT ∈ L(H T and let D D ∈ VT = UT |KT and WT = UT∗ |GT ). View D as a linear operator from E into H D ⊕ H T . D , E) be a defect for D. Then set AD := D L(D In E × HT consider the hermitian sesquilinear form e e , := e, e E + g, De H T + De, f H T + f, f H T . f f D Set σD
e ∗ D := D e + (De + f ) f
e T . ∈E ×H f
T , ·,·D ) into AD with dense range and such that Then σD is an isometry from (E × H e 0 ∗ σD = DD e and σD =f −De f T T T H H T . Moreover, since P H for all e ∈ E and f ∈ H GT |KT = PH |KT and PH D = C, then, for all e, e ∈ E and g, g ∈ GT ,
# $ e e σD , σD = e, e E + De, g H T + g, De H T + g, g H T g g A D = e, e E + Ce, g G + g, Ce G + g, g G T T T # $ e e = σ ,σ . g g A So ρ0 σ := σD |E ×GT is a densely defined isometry from A to AD . Since any maximal uniformly negative subspace of H is also maximal uniformly negative in both A and AD and H = σ ({0} × H) = σD ({0} × H), ρ0 can be extended by continuity to a bicontractive isometry ρ ∈ L(A, AD ). T ) and define Set BD := σD (QE0 × H Qe0 Re0 Qe0 (5.10) : → σD ∈ Q(E0 ) × HT . σD f UT∗ f f As VT DR = DQ, VT = UT |KT and R ∗ R Q∗ Q, the map (5.10) yields a densely defined contraction from BD to AD . Similar arguments as those exhibited to get B and show that
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T and ∗ QE0 ⊕ H X : B ⊆ A → A is a continuous bicontraction can be used to see that BD = D D T and that (5.10) gives rise to a continuous bicontraction XD : BD ⊆ AD → AD . Since GT ⊆ H ∗ WT = UT |GT , it holds that ρB ⊆ BD and ρX = XD ρ|B . Note also that ρB is a regular subspace of BD such that BD ρB is a Hilbert space. XD of XD is determined by D0 ∈ L(D0 , E0 ) (recall that As before for X, a defect operator D ∗ ∗ D0 is a Hilbert space, Q Q − R R = D0 D0∗ and ker D0 = {0}). To wit, # $ Qe0 ∗ Qe0 ∗ X , D σ σ = D0∗ e0 , D0∗ e0 D D X D D 0 f f D X
T . for all e0 ∈ E0 and f ∈ H Set HD := AD ⊕ DXD and define
XD VD := ∗ D
: BD → HD .
XD
Then VD ∈ L(BD , HD ) is an isometry whose defect subspaces are the Hilbert spaces
XD and MD := ker X ∗ D XD . ND := (AD BD ) ⊕ D D ∗ := D ∗ ρ|B . Then ρ ∈ L(H1 , HD ) is an isometry Extend ρ from A to all of H1 by setting ρ D X XD such that ρV1 = VD ρ|B . Fix ν ∈ S(ND , MD ) and let Uν ∈ L(Fν ) be the corresponding minimal weak unitary Hilbert space dilation of XD . For the sake of simplicity we consider ν ≡ 0 and call U0 the corresponding unitary operator on F0 . Set F1 := n∈Z U0n ρH1 . Then F1 is a closed linear subspace of F0 such that H ⊆ ρH1 ⊆ F1 . Notice that ρH1 is a regular subspace of HD such that HD ρH1 is a Hilbert space and that any maximal uniformly negative subspace of H is also maximal uniformly negative in ρH1 , hence, in both HD and F0 . Therefore F1 is a regular subspace of F0 such that F1 ρH1 is a Hilbert space. Since U0 ∈ U(VD ) and ρV1 = VD ρ|B , it follows that U1 := U0 |F1 is a minimal unitary Hilbert space extension of the isometry ρV1 ρ ∗ ∈ L(ρB, ρH1 ). The defect subspaces of ρV1 ρ ∗ are ρN1 and ρM1 , both Hilbert subspaces of ρH1 . Thus ϑ1 (z) := 1 U1 (1 − zPFF11ρ H1 U1 )−1 |ρ N1 belongs to S(ρN1 , ρM1 ). So ϑ(z) := ρ ∗ ϑ1 (z)ρ belongs to PρFM 1 S(N1 , M1 ). Let Uϑ ∈ L(Fϑ ) be the corresponding minimal weak unitary Hilbert space dilation of X. It can be computed ρ ∗ PρFH1 1 U1 (1 − zU1 )−1 |ρ H1 to yield Fϑ ρ ∗ PρFH1 1 U1 (1 − zU1 )−1 ρ H = PH Uϑ (1 − zUϑ )−1 H . 1 1 1
(5.11)
Notice that, for all h ∈ H and n ∈ N, U0n UTn h = VDn UTn h = h so that, for all e ∈ E,
De, VTn h K T
$ e −n , U0 h = = σD 0 F0 # $ # $ e e −n n = ρσ , U0 ρh = U1 ρσ , ρh . 0 0 F F
#
De, UTn h H T
0
1
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From this and (5.11) it follows that, for all e ∈ E, h ∈ H and n ∈ N, # $ e De, VTn h K = σ , Uϑ−n h . T 0 Fϑ
Therefore D is given by ϑ as in (5.4), hence, by (5.9). It remains to see that ϑ(z)|D X = 0 for all z ∈ D. Notice that ϑ can also be written as −1 F F F 1 ϑ(z) = ρ ∗ PρFM 1 − zVD PBD0 ρ B PF10ρ H1 VD PBD0 ∩(F1 ρ H1 ) ρ N . 1 1
X ⊥ BD the result follows and the proof is finished. As ρ D
2
It may happen that different parameters ϑ ’s in S(N1 , M1 ), constrained to satisfy ϑ(z)|D X ≡ 0, provide the same interpolant D via (5.9). As for a concrete example, consider C = 0, an isometry T and R = Q such that ker Q∗ = {0} is a Hilbert subspace of E. In this particular case, the coupling Kre˘ın space is A = E ⊕ GT and the coupling bicontraction X is an isometry. Both the defect subspaces of the isometry X equal ker Q∗ . Since T is isometric, VT = T . Hence, there is only one interpolant D for the data set {0, T , T , Q, Q}, namely, D = C = 0. Also, as T is isometric, JT = {0}. Therefore, it comes that a (z), b(z) ≡ 0 in formula (5.9). Hence, when applying (5.9), any ϑ ∈ S(ker Q∗ , ker Q∗ ) gives D. In the present discussion, the Commutant Lifting Theorem for T bicontractive occurs when R is the identity on E (thus E0 = E) and Q is an isometry on E . In this case, R ∗ R = 1 = Q∗ Q and the underlying bicontraction X is an isometry. In the framework of the Commutant Lifting Theorem for T bicontractive it turns out that Theorem 5.4 does give a proper parameterization of the interpolants. Even when T is supposed to be just a contraction (although the construction of the interpolants differs from the one given herein) there is a bijective correspondence between certain class of operator-valued Schur functions and the set of interpolants (see [9]). The Commutant Lifting Theorem for T bicontractive is included as a particular case in the following: ∗ RE0 = DC . Then the map (5.9) in Theorem 5.4 Theorem 5.5. Assume that R ∗ R = Q∗ Q and D C gives a bijective correspondence between the functions ϑ ∈ S(N1 , M1 ) and the interpolants D for {C, T , VT , R, Q}, with N1 and M1 being given by C D ∗ QE0 N1 := D C
(5.12)
∗ C ⊕ JT D Re0 ⊕ (1 − WT T )CRe0 : e0 ∈ E0 . M1 := D C
(5.13)
and
Proof. From the assumption that R ∗ R = Q∗ Q it readily follows that X is an isometry. In particular, N1 = A B and M1 = ker X ∗ . We then get (5.12) and (5.13). From the fact that X is isometric we also get that b, c and d take on the new forms
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2585
−1 1 − PJAT M , b(z) := PJAT 1 + rzXPBA 1 − rz 1 − PJAT XPBA 1 A A A −1 c(z) := c(z) := rzPAB 1 − rz 1 − PJT XPB M1 ,
:= P A 1 − rz 1 − P A XP A −1 , d(z) A B JT B D C
while a remains as set in Theorem 5.4(i). If α is any given function in S(N1 , M1 ) and D is the corresponding interpolant, then β ∈ S(N1 , M1 ) yields the same D via (5.9), hence, by (5.4), if and only if, according with (5.8) and Proposition 5.2, F Fα −1 , PJTβ (1 − zUβ )−1 Uβ D C = PJT (1 − zUα ) Uα D C
|z| < r.
Observe that PJFTϑ (1 − zUϑ )−1 Uϑ |GT = PJGTT (1 − zWT )−1 WT = 0 for all ϑ ∈ S(N1 , M1 ) and all |z| < r. Thus β yields the same interpolant as α if and only if F PJTβ (1 − zUβ )−1 Uβ A = PJFTα (1 − zUα )−1 Uα A ,
|z| < r.
The same arguments that led to Proposition 5.1 can be used to see that the above condition is granted if and only if, for all |z| < r, A −1 A −1 PGAT 1 − zXPBA − zβ(z)PA = PGAT 1 − zXPBA − zα(z)PA , B B that is, A −1 PGAT 1 − zXPBA − zα(z)PA α(z) − β(z) = 0. B
(5.14)
C )-valued function μ(z) defined We see that (5.14) holds whenever there exists an L(A B, D and analytic on |z| < r such that A A |z| < r. β(z) = α(z) 1 − zPA B μ(z) + 1 − zXPB μ(z), Note that, for each |z| < r, 0 = X ∗ 1 − zXPBA μ(z) = X ∗ μ(z) − zPBA μ(z). C , it follows that μ(0) = 0. Hence, μ(z) ≡ zμ1 (z) ∗ RE0 = D In particular, X ∗ μ(0) = 0. As D C C ). The same argument as before yields for some analytic function μ1 on |z| < r to L(A B, D μ1 (0) = 0. By iteration we get μ ≡ 0 and β ≡ α. 2 References [1] T. Ando, Linear Operators in Kre˘ın Spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979. [2] D.Z. Arov, L.Z. Grossman, Scattering matrices in the theory of dilations of isometric operators, Soviet Math. Dokl. 27 (1983) 518–522. [3] D.Z. Arov, L.Z. Grossman, Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr. 157 (1992) 105–123.
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[4] G. Arsene, Z. Ceausescu, C. Foias, On intertwining dilations VIII, in: Oper. Theory Adv. Appl., vol. 4, Birkhäuser, Basel, 1980, pp. 55–91. [5] T. Azizov, I.S. Iokhividov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric, Nauka, Moscow, 1986, English transl.: Linear Operators in Spaces with Indefinite Metric, Wiley, New York, 1989. [6] J. Bognar, Indefinite Inner Product Spaces, Springer, Berlin, 1974. [7] A. Dijksma, H. Langer, H.S.V. de Snoo, Generalized coresolvents of standard isometric operators and generalized resolvents of standard symmetric relations in Krein spaces, in: Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 261–274. [8] A. Dijksma, M.A. Dritschel, S.A.M. Marcantognini, H.S.V. de Snoo, The commutant lifting theorem for contractions on Kre˘ın spaces, in: Oper. Theory Adv. Appl., vol. 61, Birkhäuser, Basel, 1993, pp. 65–83. [9] A. Dijksma, S.A.M. Marcantognini, H.S.V. de Snoo, A Schur type analysis of the minimal unitary Hilbert space extensions of a Kre˘ın space isometry whose defect subspaces are Hilbert spaces, Z. Anal. Anwend. 13 (2) (1994) 233–260. [10] M.A. Dritschel, J. Rovnyak, Extension theorems for contraction operators on Kre˘ın spaces, in: Oper. Theory Adv. Appl., vol. 47, Birkhäuser, Basel, 1990, pp. 221–305. [11] M.A. Dritschel, J. Rovnyak, Operators on indefinite inner product spaces, in: Lectures on Operator Theory and Its Applications, Waterloo, ON, 1994, in: Fields Inst. Monogr., vol. 3, Amer. Math. Soc., Providence, RI, 1996, pp. 141–232. [12] C. Foias, A.E. Frazho, M.A. Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations Operator Theory 42 (2002) 253–310. [13] A.E. Frazho, S. ter Horst, M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations Operator Theory 54 (2006) 33–67. [14] A.E. Frazho, S. ter Horst, M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (1–2) (2006) 299–318. [15] W.S. Li, D. Timotin, The relaxed intertwining lifting in the coupling approach, Integral Equations Operator Theory 54 (2006) 97–111. [16] S.A.M. Marcantognini, The commutant lifting theorem in the Kre˘ın space setting: a proof based on the coupling method, Indiana Univ. Math. J. 41 (4) (1992) 1303–1314. [17] S.A.M. Marcantognini, M.D. Morán, A Schur analysis of the minimal weak unitary dilations of a contraction operator and the Relaxed Commutant Lifting Theorem, Integral Equations Operator Theory 64 (2009) 273–299. [18] S.A.M. Marcantognini, M.D. Morán, Abstract Hankel operators in Kre˘ın spaces, Integral Equations Operator Theory 66 (2010) 397–424. [19] D. Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967) 179–203. [20] B. Sz.-Nagy, C. Foia¸s, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., Amsterdam, London, 1970.
Journal of Functional Analysis 259 (2010) 2587–2612 www.elsevier.com/locate/jfa
Support theorems on Rn and non-compact symmetric spaces ✩ E.K. Narayanan ∗ , Amit Samanta Department of Mathematics, Indian Institute of Science, Banaglore 560012, India Received 17 February 2010; accepted 30 July 2010 Available online 10 August 2010 Communicated by L. Gross
Abstract We consider convolution equations of the type f ∗ T = g, where f, g ∈ Lp (Rn ) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T , we show that f is compactly supported, provided g is. Similar results are proved for non-compact symmetric spaces as well. © 2010 Elsevier Inc. All rights reserved. Keywords: Support theorems; Paley–Wiener theorems; Analytic sets; Symmetric spaces; Spherical Fourier transform
1. Introduction Support theorems have attracted a lot of attention in the past. We recall two such results. First, the famous result due to Helgason [8] (see page 107). This result states the following: If a measurable function f on Rn satisfies (1 + |x|)N f ∈ L1 (Rn ), for each integer N > 0 and f integrates to zero over all spheres enclosing a fixed ball of radius R > 0, then f is supported in BR , where BR is the ball of radius R centered at the origin. An analogue holds also for rank one symmetric spaces of non-compact type [4]. The second is a result by A. Sitaram. In [17], he proved the following support theorem: If f ∈ L1 (Rn ) is such that f ∗ χBr = g, where χBr is the indicator function of Br and g is supported in BR , then supp f ⊆ BR+r . ✩
The first author was supported in part by a grant from UGC via DSA-SAP and the second author was supported by Research Fellowship of Indian Institute of Science, Bangalore. * Corresponding author. E-mail addresses:
[email protected] (E.K. Narayanan),
[email protected] (A. Samanta). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.019
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In this paper we are interested in the second result. We consider convolution equations of the form f ∗ T = g, where T is a compactly supported distribution on Rn and f ∈ Lp (Rn ). The question we are interested in is: can we conclude that f is compactly supported, if g is compactly supported? Combining methods from several complex variables and harmonic analysis we prove general support theorems under natural assumptions on the zero set of the entire function Tˆ (Fourier transform of T ). When T = χBr or μr (the normalized surface measure on the sphere of radius r on Rn ), this problem was studied by Sitaram [17], Volchkov [19], etc. When T is a distribution supported at the origin, this becomes a problem in PDE. In [18], Trèves proved that, if P (D)u = v and v is compactly supported, then u is also compactly supported, provided u ∈ S(Rn ) (Schwartz space) and the variety of zeros of each irreducible factor of P in Cn intersects Rn . These questions were later taken up by Littman in [12] and [13]. Considering the principal value integral Rn
v(y) ˆ eix·y dy P (y)
he was able to show that u is compactly supported with the assumption that {x ∈ Rn : P (x) = 0} has dimension (n − 1). Hörmander strengthened these results in [11]. Our results may be viewed as generalizations of these results. We end this section with the following theorems which will be needed later. Theorem 1.1. If f ∈ Lp (Rn ) and supp fˆ is carried by a C 1 manifold of dimension d < n then f = 0 provided 1 p 2n d and d > 0. If d = 0 then f = 0 for 1 p < ∞. (See [2].) Theorem 1.2. Let f and g be entire functions of exponential type defined on Cn such that h = f/g is entire, then h is of exponential type. This result is due to Malgrange. See [14]. 2. Support theorems on R n In this section we prove support theorems on Rn under natural assumptions on the zero set of the Fourier transform of the distribution T . Before we state our results we recall some notation from several complex variables which will be used throughout. Let F be an entire function on Cn . Then ZF will denote the zero set of F , i.e. ZF = {z ∈ Cn : F (z) = 0}. The set ZF is a complex analytic set and it can be written as a union of irreducible complex analytic sets, where, by an irreducible complex analytic set we mean a complex analytic set which cannot be written as a union of two non-empty complex analytic sets. For more details on complex analytic sets we refer to [5]. Let Reg(ZF ) denote the regular points of ZF . If z ∈ ZF then Ordz F will denote the order of F at z (see [5, page 16]). We also recall that the order is a constant on each connected component of Reg(ZF ). If A is a complex analytic set, Sing A will denote the singular points. That is, Sing A = A − Reg A. We start with the following general result.
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Theorem 2.1. Let T be a compactly supported distribution on Rn and f ∈ Lp (Rn ) for some p 2n . Assume the following: with 1 p n−1 (a) If V is any irreducible component of ZTˆ , then dimR (V ∩ Rn ) = n − 1. (b) grad Tˆ = 0 on Reg(ZTˆ ) ∩ Rn . Suppose f ∗ T = g, where g is compactly supported, then f is also compactly supported. We need several lemmas for the proof of this theorem. Lemma 2.2. If f ∈ Lp (Rn ), p = we have
2n n−1,
then ∃rk → ∞ such that, for any fixed constants s1 , s2 > 0
f (x)2 dx → 0
rk −s1 |x|rk +s2
as k → ∞. Proof. By contrary, assume that ∃a > 0 and R > 0 such that f (x)2 dx a, ∀r R.
(2.1)
r−s1 |x|r+s2
By Hölder’s inequality we have
f (x)2 dx
r−s1 |x|r+s2
2n f (x) n−1 dx
n−1 n
r−s1 |x|r+s2
1
n
dx
.
r−s1 |x|r+s2
From (2.1) and the above it follows that for some constant c > 0 2n f (x) n−1 dx c , ∀r > R. r r−s1 |x|r+s2
In particular, for each integer k > R, the above inequality is true for r = s1 + k(s1 + s2 ). Now, 2n . summing all these inequalities we get a contradiction to the fact that f ∈ Lp (Rn ), p = n−1 Hence the lemma is proved. 2 Lemma 2.3. Let F and G be two entire functions on Cn such that: (a) The intersection with Rn of each connected component of Reg(ZF ) has real dimension (n − 1). (b) (Reg ZF ) ∩ Rn ⊆ ZG ∩ Rn . (c) Ordx F Ordx G ∀x ∈ Rn ∩ Reg ZF . Then
G F
is an entire function.
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Proof. Let Reg ZF = j ∈J Sj be the decomposition of Reg ZF into connected components. Then ZF = j ∈J Aj where Aj = S¯j gives the decomposition of ZF into irreducible components. If the complex dimension dimC (Aj ∩ ZG ) (n − 2), then dimR (Aj ∩ ZG ∩ Rn ) (n − 2) which contradicts (a) due to (b) in the assumptions. It follows that dimC (Aj ∩ ZG ) = (n − 1). Since Aj is an irreducible analytic set in Cn , this will force Aj to be an irreducible component of ZG (see [5]). It follows that Reg(ZF ) ⊆ Reg(ZG ). Since the order is a constant on the regular part of an analytic set we also have Ordz F Ordz G ∀z ∈ Reg ZF . Consequently G F is holomorphic in Cn − Sing(ZF ). However, the (2n − 2) Hausdorff measure of (Sing ZF ) is zero (see [5, page 22]) and so by Proposition 2, page 298, in [5], G F extends to an entire function. 2 2n Lemma 2.4. Let f ∈ Lp (Rn ), 1 p n−1 . Let T be a compactly supported distribution on Rn and f ∗ T = g, where g is compactly supported. If Tˆ is zero on a smooth (n − 1)-dimensional ˆ = 0 ∀x ∈ M. manifold M ⊆ Rn , then g(x)
Proof. By convolving with radial approximate identities we may assume that f ∈ Lp0 (Rn ) ∩ 2n and T ∈ L1 (Rn ). Let supp T ⊆ BR1 and supp g ⊆ BR2 . For r > 0 C ∞ (Rn ) where p0 = n−1 define fr (x) = χ|x|r (x)f (x) and write f r ∗ T = g + gr .
(2.2)
If r is very large, then supp gr ⊆ {x: r − R1 |x| r + R1 } and for r − R1 |x| r + R1 we have gr (x) = T ∗ fr−2R1 ,r (x)
(2.3)
where fr−2R1 ,r (x) = χr−2R1 |x|r (x)f (x). Next, let φ ∈ Cc∞ (Rn ) and consider the measure μ defined by dμ = φ(x) dxM where dxM is the surface measure on M. Then μ is a compactly supported measure on M. Since Tˆ is zero on M, it is easy to see by taking the Fourier transform that T ∗ fr ∗ μˆ vanishes identically. From (2.2) it follows that g ∗ μˆ + gr ∗ μˆ ≡ 0. ˆ goes to zero ∀x ∈ Rn as r → ∞, which implies that g ∗ μˆ vanishes We will show that gr ∗ μ(x) identically. Taking the Fourier transform again we obtain that gˆ vanishes on (supp φ) ∩ M. Since φ was arbitrary this proves the lemma. ˆ 0 ). We have, by (2.3) Fix x0 ∈ Rn and consider gr ∗ μ(x T ∗ fr−2R ,r (y)μ(x gr ∗ μ(x ˆ 0 − y) dy. (2.4) ˆ 0) 1 r−R1 |y|r+R1
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Now if ν is a compactly supported smooth measure on M then ∃c > 0 such that
νˆ (sω)2 dω
1 2
S n−1
c s
n−1 2
s > 0.
,
(See [3, Proposition 1, page 2563].) Apply the above to the measure eix0 ·y φ(y) dyM on M to obtain
2 μ(x ˆ 0 − sω) dω
1 2
c(x0 ) s
S n−1
n−1 2
,
implying
2 μ(x ˆ 0 − y) dy
1 2
C(x0 ),
r−R1 |y|r+R1
where c(x0 ) and C(x0 ) are some constants depending on x0 . A simple application of the Cauchy– Schwarz inequality to (2.4) along with the above estimates gives us gr ∗ μ(x ˆ 0 ) C(x0 )T ∗ fr−2R1 ,r 2 . Therefore, by Young’s inequality we get gr ∗ μ(x ˆ 0 ) C(x0 ) T 1
f (y)2 dy
1 2
.
r−2R1 |y|r
Choosing {rk } as in Lemma 2.2 we finish the proof.
2
2n Proof of Theorem 2.1. Without loss of generality we may assume that f ∈ Lp0 (Rn ), p0 = n−1 . n Since f ∗ T = g and (Reg ZTˆ ) ∩ R is a smooth (n − 1)-dimensional manifold, Lemma 2.4 implies that g(x) ˆ = 0 if Tˆ (x) = 0. Since grad Tˆ is non-zero on Reg ZTˆ we have Ordx Tˆ = 1 if x ∈ ˆ = 0 ∀x ∈ (Reg ZTˆ ) ∩ Rn it follows that Ordx gˆ Ordx Tˆ ∀x ∈ Reg ZTˆ ∩ Rn . Reg ZTˆ . Since g(x)
By Lemma 2.3 we have that
gˆ Tˆ
is an entire function. Hence we have gˆ fˆ = + δ, Tˆ
(2.5)
where δ is a distribution supported on ZTˆ ∩ Rn . We will show that δ ≡ 0. Let φ ∈ Cc∞ (Rn ). Multiplying (2.5) with φ and taking the inverse Fourier transform we obtain (φδ)ˇ = φˇ ∗ f − h
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2n where h ∈ S(Rn ). Notice that φˇ ∗ f ∈ Lp0 (Rn ), p0 = n−1 . From Theorem 1.1 it follows that g ˆ φδ = 0. Since φ was arbitrary it follows that fˆ = . By Malgrange’s theorem fˆ is an entire Tˆ
function of exponential type. If Tˆ is slowly decreasing this readily implies that f is compactly supported. However, this extra assumption is not needed as can be seen below. Let ψ ∈ S(Rn ) be such that ψˆ is compactly supported. Then ˆ )(x) = ψˆ ∗ fˆ(x) (ψf ˆ fˆ(x − t) dt = ψ(t) Rn
ˆ ) is bounded clearly extends to an entire function of exponential type. Since ψf ∈ L1 (Rn ), (ψf n on R . By the Paley–Wiener theorem we obtain that ψf is compactly supported which finishes the proof. 2 Remark 2.5. It is possible to weaken the condition grad Tˆ = 0 on Reg ZTˆ ∩ Rn as follows. Let V be any global irreducible component of ZTˆ . Then there exists an entire function fV whose zero locus is exactly V and there exists a positive integer k such that Cn .
Tˆ fVk
is non-zero on V . This
is an application of the Cousin II problem on See [7]. This function fV is unique up to multiplication by units. A close examination of the proof shows that it suffices to assume that grad fV = 0 on V ∩ Rn for all V . In particular when Tˆ = f1m1 f2m2 · · · fkmk where f1 , f2 , . . . , fk are irreducible entire functions then it suffices to assume that grad fj = 0 on Zfj ∩ Rn . Also see Hörmander [11, Theorem 3.1]. Next we show that if 1 p 2 or T is a radial distribution then the condition on grad Tˆ is not needed in Theorem 2.1. Theorem 2.6. Let T be a compactly supported distribution on Rn and f ∈ Lp (Rn ) for some p with 1 p 2. If f ∗ T is compactly supported and condition (a) of the previous theorem is satisfied then f is compactly supported. Proof. Let f ∗ T = g. Convolving with compactly supported approximate identities we may as sume that f ∈ L2 (Rn ) and g ∈ Cc∞ (Rn ). Since Tˆ fˆ = gˆ and f ∈ L2 (Rn ) we have Rn | gˆˆ |2 < ∞. T We will show that, if x0 ∈ Reg(Z ˆ ) ∩ Rn then Ordx0 (Tˆ ) Ordx0 (g). ˆ Then we may argue as in Theorem 2.1 to conclude that
gˆ Tˆ
T
is entire which will prove the theorem. As in the proof of Theorem 2.1 we have ZTˆ ⊂ Zgˆ . Without loss of generality we can assume x0 = 0. If Ordx0 (Tˆ ) = m1 and Ordx0 (g) ˆ = m2 then there exist holomorphic functions ϕ, ψ1 and ψ2 such that
m Tˆ (z) = zn − ϕ z 1 ψ1 (z) and
m g(z) ˆ = zn − ϕ z 2 ψ2 (z)
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in a neighborhood V (in Cn ) of the origin, where ψ1 and ψ2 are zero free in V . Here z = (z1 , z2 , . . . , zn−1 ) ∈ Cn−1 . Since g/ ˆ Tˆ ∈ L2 , the above implies that,
1 dx < ∞, |xn − ϕ(x )|2(m1 −m2 )
[−a,a]n
for some a > 0. By a change of variables we get
a−ϕ(x )
−a−ϕ(x )
[−a,a]n−1
1 r 2(m1 −m2 )
dr dx < ∞.
Now, since ϕ(0) = 0, if we choose 0 < ε < a, then there exists 0 < δ < a such that |ϕ(x )| < ε ∀x ∈ [−δ, δ]n−1 . Therefore, a−ε
[−δ,δ]n−1
−a+ε
1 r 2(m1 −m2 )
dr dx < ∞
implying that a−ε −a+ε
1 dr < ∞. r 2(m1 −m2 )
Hence m2 m1 , which finishes the proof.
2
1 Next, suppose that T is a radial distribution on Rn . Then Tˆ is a function of (z12 +z22 +· · ·+zn2 ) 2 and the assignment
Tˆ (z1 , z2 , . . . , zn ) = GT (s), where s 2 = z12 + z22 + · · · + zn2 , defines an even entire function GT on the complex plane C of exponential type and at most polynomial growth on R. The converse also holds. If the entire function GT has only real zeros then ZTˆ (in Cn ) is a disjoint union of sets of the form {(z1 , z2 , . . . , zn ): z12 + z22 + · · · + zn2 = a} for a > 0. It is easy to see that such T satisfies the condition (a) of Theorem 2.1. Our next theorem shows that condition (b) of Theorem 2.1 is not necessary if we are dealing with radial distributions of the above kind. Theorem 2.7. Let T be a compactly supported radial distribution on Rn such that the zeros of 2n the entire function GT (s) are contained in R − {0}. If f ∈ Lp (Rn ), 1 p n−1 and f ∗ T is compactly supported then f is compactly supported.
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Proof. Let f ∗ T = g and let 0 < λ1 < λ2 < λ3 < · · · be the positive zeros of GT (s) with ˆ As in the previous case we will show that gˆˆ is multiplicities m1 , m2 , . . . . We have Tˆ fˆ = g. ˆ Now entire. It clearly suffices to show that (z12 + z22 + · · · + zn2 − λ2k )mk divides g.
GT (s) s 2 −λ2k
T
is an
even entire function of exponential type on C and is of at most polynomial growth on R. It follows that there exists a compactly supported radial distribution V on Rn such that GV (s) =
GT (s) . s 2 − λ2k
Now, 2
z1 + z22 + · · · + zn2 − λ2k
Tˆ fˆ = gˆ (z12 + z22 + · · · + zn2 − λ2k )
implies that
− − λ2k (V ∗ f ) = g.
(2.6)
2n Convolving f with a radial Cc∞ function we may assume that V ∗ f ∈ Lp (Rn ), 1 p n−1 . 2 Note that − − λk is a distribution supported at the origin and satisfies the conditions in Theorem 2.1. It follows that V ∗ f is compactly supported. Taking Fourier transform in (2.6) we ˆ This surely can be repeated to prove that gˆˆ is obtain that (z12 + z22 + · · · + zn2 − λ2k ) divides g. T entire. The proof now can be completed as in the previous case. 2
In our next result we show that assuming T is a compactly supported positive distribution (i.e. T (φ) 0 if φ 0) gives us precise information about the support of the function f . Recall that a positive distribution is a positive measure. Theorem 2.8. Let T be a compactly supported radial positive measure with supp T = B¯R1 . 2n and Assume that the entire function GT (s) has only real zeros. If f ∈ Lp (Rn ), 1 p n−1 f ∗ T = g with supp g ⊆ BR2 then f is compactly supported and supp f ⊆ BR2 −R1 . We start with the following lemma which is a simple application of the Phragmén–Lindelöf theorem. Lemma 2.9. Let A(s) be an entire function of exponential type on C and 0 < R1 < R2 < ∞. Suppose that |A(s)| eR2 |s| ∀s ∈ C and (a) |A(is)| e(R2 −R1 )|s| ∀s ∈ R. (b) |A(s)| e(R2 −R1 )|s| ∀s ∈ R. Then |A(s)| e(R2 −R1 )|s| ∀s ∈ C. Proof. Define H (s) =
A(s) , e(R2 −R1 )(s)
s ∈ C.
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By the given condition H is an entire function of exponential type on C. Also H is bounded on real and imaginary axis. Now consider the region Ω = {s: Im s > 0 and Re s > 0} which is a sector of angle π2 . Then H is bounded on ∂Ω and we can find P > 0 and b < 2 such that b H (s) P e|s| ∀z ∈ Ω. By the Phragmén–Lindelöf theorem H is bounded on Ω. We can repeat the argument in other quadrants. Hence the lemma follows. 2 Proof of Theorem 2.8. Let μ be the compactly supported radial positive measure which defines the distribution T . Then f ∗ μ = g. By Theorem 2.7 we already know that f is compactly supported. In particular f ∈ L1 (Rn ). Also fˆ = μgˆˆ is an entire function of exponential type (by Malgrange’s theorem). Proof will be completed by Lemma 2.9 and the Paley–Wiener theorem once we prove that for each > 0, there exists c > 0 such that c e(R1 −)|y| μ(iy) ˆ
∀y ∈ Rn .
Now, μ(iy) ˆ =
ex·y dμ(x).
|x|R1
Given > 0, it is possible to choose a fixed radius δ > 0 such that x · y (R1 − )|y| y for all x in a δ-neighborhood Bδ of R1 |y| . Hence
μ(iy) ˆ
ex·y dμ(x)
x∈Bδ
c e(R1 −)|y| , for some constant c . Notice that we need supp μ = B¯R1 here. This finishes the proof.
2
Remark 2.10. When T = χBr or μr this improves the result of Sitaram in [17]. Theorem 2.8 is also proved by Volchkov in [19] in a different way. See also [1]. The following theorem shows that the class of distributions which satisfies the conditions in Theorem 2.7 is large. Notice that if G is an even entire function of exponential type on C whose zeros are all non-zero reals and T is a radial, compactly supported distribution on Rn defined by
1
Tˆ (z1 , z2 , . . . , zn ) = G z12 + z22 + · · · + zn2 2 then T satisfies the conditions of Theorem 2.7.
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Theorem 2.11. Let φ : R → R be a positive even C 2 function. Assume that φ is increasing on [0, 1]. Then the entire function (on C) 1 G(z) :=
φ(t)e−itz dt
−1
has only real zeros. Proof of the above requires several lemmas. Lemma 2.12. (I) Let g be a positive C 1 function on [0, a] such that both g and g are strictly increasing on [0, a]. Then a I=
g(t) cos t dt 0
is non-zero if a = 2nπ + θ or 2nπ + π + θ , 0 θ π2 . (II) Let g be as above with g(0) = 0. Then a J=
g(t) sin t dt 0
is non-zero if a = 2nπ +
π 2
+ θ or 2nπ +
3π 2
+ θ , 0 θ π2 .
Proof. (I) Case 1: Let a = 2nπ + θ , 0 θ π2 . Then 2nπ n−1 I g(t) cos t dt = Ik k=0
0
where 2kπ+2π
Ik =
2π
g(t) cos t dt = 2kπ
g(2kπ + t) cos t dt. 0
First, consider I0 . π
2 I0 =
G0 (t) cos t dt 0
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where G0 (t) = g(2π − t) − g(π + t) − g(π − t) + g(t). Now, G0 ( π2 ) = 0 and G0 (t) = −g (2π − t) − g (π + t) + g (π − t) + g (t) is negative by the assumption on g. It follows that G0 (t) > 0 for t ∈ [0, π2 ). Hence I0 > 0. Notice 2π that each Ik is given by an integral 0 Gk (t) dt where Gk is just G0 translated by a multiple of π. Hence each Ik > 0 which implies that I is non-zero. Case 2: Let a = 2nπ + π + θ , 0 θ π2 . Then 2nπ+π
g(t) cos t dt = I¯ +
−I − π 0
I¯k
k=0
0
where I¯ = −
n−1
g(t) cos t dt and
I¯k = −
(2k+1)π+2π
2π
g(t) cos t dt =
g (2k + 1)π + t cos t dt.
0
(2k+1)π
π Now I¯ = 02 [g(π − t) − g(t)] cos t dt > 0. Also as in the previous case I¯k > 0. Therefore I is non-zero. (II) Case 1: Let a = 2nπ + π2 θ , 0 θ π2 . Then 2nπ+ π2
J
g(t) sin t dt =
n−1
Jk
k=0
0
where 2kπ+ π2 +2π
Jk =
π 2
+2π g(t) sin t dt = g(2kπ + t) sin t dt.
2kπ+ π2
π 2
First consider J0 . π
2 J0 =
E0 (t) sin t dt 0
where
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E0 (t) = g(2π + t) − g(2π − t) − g(π + t) + g(π − t). Now, E0 (0) = 0 and E0 (t) = g (2π + t) + g (2π − t) − g (π + t) − g (π − t) is positive by assumption on g. It follows that E0 (t) > 0 for t ∈ (0, π2 ]. Hence J0 > 0. Similarly each Jk > 0 which implies that J is non-zero. π Case 2: Let a = 2nπ + 3π 2 + θ , 0 θ 2 . Then 2nπ+ 3π 2
−J −
g(t) sin t dt = J¯ +
3π 2
0
g(t) sin t dt and 2kπ+ 3π 2 +2π
π 2
+2π
g(t) sin t dt = g (2k + 1)π + t sin t dt.
J¯k = −
π 2
2kπ+ 3π 2
J¯ =
π 2
0
J¯k
k=0
0
where J¯ = −
n−1
E(t) sin t dt where E(t) = g(π + t) − g(π − t) − g(t).
Now E(0) = 0 and E (t) = g (π + t) + g (π − t) − g (t) is positive by assumptions on g. It follows that E(t) > 0 for t ∈ (0, π2 ]. Hence J¯ > 0. Also as in the previous case J¯k > 0. Therefore J is non-zero. 2 Lemma 2.13. (I) Let g be a non-negative continuous strictly increasing function on [0, a]. Then, a I :=
g(t) cos t dt 0
is non-zero if a = + kπ for some non-negative integer k. (II) Let g be as above. Then, π 2
a J :=
g(t) sin t dt 0
is non-zero if a = kπ for some positive integer k.
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Proof. Let a =
π 2
2599
+ kπ for some non-negative integer k. Then, π
2
g(t) cos t +
I=
(k−1)
Ij
j =0
0
where π 2 +(j +1)π
Ij =
g(t) cos t dt. π 2
+j π
If k is even we can write π
2 I=
k−2
2 g(t) cos t dt + (I2j + I2j +1 ).
j =0
0
By a change of variable we get π π π g π + +t −g + t sin t dt I0 + I1 = 2 2 0
which is positive since g is strictly increasing. Similarly each I2j + I2j +1 is positive. Hence I is positive. If k is odd then we can write 3π
2 I=
k−1
2 g(t) cos t dt + (I2j −1 + I2j ).
j =1
0
Again using a change of variable we get π π π g +π +t −g + 2π + t sin t dt I1 + I 2 = 2 2 0
which is negative since g is strictly increasing. Similarly each I2j −1 + I2j is negative. Also 3π
π
2
π+ π2
2
g(t) cos t dt +
g(t) cos t dt < 0
g(t) cos t dt π
0 π
2 < 0
g(t) − g(π + t) cos t dt
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is negative. Therefore I is negative. Hence (I) is proved. (II) can be proved using a similar type of argument. 2 Lemma 2.14. (I) Let g be a non-negative increasing C 2 function on [0, 1] such that for some M > 1, Mg(t) + g (t) 0 ∀t ∈ [0, 1]. Then, for each fixed y > M the function 1 Fy (x) :=
g(t) eyt + e−yt cos(xt) dt
0
can vanish at most once in each of the intervals [ π2 + kπ, π2 + (k + 1)π], where k is a non-negative integer. (II) Let g be as above. Then, for each fixed y > M the function 1 Gy (x) =
g(t) eyt + e−yt sin(xt) dt
0
can vanish at most once in each of the intervals [kπ, (k + 1)π], where k is a non-negative integer. Proof. To prove (I) first note that we can write Fy (x) and Fy (x) in the following way: 1 Fy (x) = x
x y
t ty e x + e−t x cos t dt g x 0
and 1 Fy (x) = −
x
x
y
t ty t g e x + e−t x sin t dt. x x
0
Now, if possible assume that there exists y0 > M and a non-negative integer k0 such that the interval [ π2 + k0 π, π2 + (k0 + 1)π] contains at least two zeros of the function Fy0 (x). Because of the given conditions an easy calculation shows that the functions g( xt )(et y0 x
y0 x
y0 x
+ et
y0 x
) and
+ e ) on the interval [0, x] satisfy the conditions of (I) and (II) of Lemma 2.11 respectively. Hence, Fy0 (x) and Fy 0 (x) cannot vanish in the intervals [ π2 + kπ + π2 , π2 + (k + 1)π] and [ π2 + kπ, π2 + kπ + π2 ] respectively. Therefore, Fy0 (x) vanishes at least twice in the interval [ π2 + kπ, π2 + kπ + π2 ] which implies, by Rolle’s theorem, that Fy 0 (x) has at least one zero in the same interval, which is a contradiction. This finishes the proof of (I). Using a similar type of argument, we can also prove (II). 2 t t t x g( x )(e
t
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Lemma 2.15. Let g be an even or odd continuous function on [−1, 1] such that on [0, 1] it is non-negative, increasing and C 2 . Assume that for some M > 1, Mg(t) + g (t) 0 ∀t ∈ [0, 1]. Let the entire function 1
g(t)e−izt dt
H1 (z) := −1
have a non-real zero. Then the entire function 1 H2 (z) :=
tg(t)e−izt dt
−1
also has a non-real zero. Proof. First assume that g is even. Since g is also real valued, there exist x0 > 0 and y0 > 0 such that H1 is zero at z0 = x0 + iy0 . Now, if possible assume that H2 has only real zeros, i.e. for any z = x + iy, y = 0, 1 Re H2 (z) =
tg(t) eyt − e−yt cos(xt) dt,
0
and 1 Im H2 (z) = −
tg(t) eyt + e−yt sin(xt) dt
0
cannot vanish simultaneously. But this implies that, if we define the smooth function F : R2 → R by 1 F (x, y) = Re H1 (x + iy) =
g(t) eyt + e−yt cos(xt) dt
0
then the gradient vector 1 1
yt ∇F (x, y) = − tg(t) e + e−yt sin(xt) dt, tg(t) eyt − e−yt cos(xt) dt = 0 0
0
whenever z = x + iy is not real i.e. y = 0. Therefore, the zero set of F is closed and except on the real axis it defines a smooth one-dimensional manifold.
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By (I) of Lemma 2.13, the connected component of the zero set through (x0 , y0 ) (call it C) is contained in the region R := {(x, y): π2 + kπ < x < π2 + (k + 1)π} for some non-negative integer k. Since the curve C is closed and ∇F = 0 on the non-real points of C, there are three possibilities: (a) C intersects the real axis, (b) C is a smooth closed loop in the region {(x, y) ∈ R: y > 0}, (c) C is a smooth curve in the region {(x, y) ∈ R: y > 0}, with both ends going upwards to infinity along the direction of y-axis. Notice that F is a harmonic function, hence (b) cannot occur. By Lemma 2.14, (c) is also ruled out. Consider the first case. Parametrize a portion of C by a continuous function γ : [0, 1] → C such that γ (0) = (x0 , y0 ), γ (1) = (u0 , 0) ( π2 + kπ < u0 < π2 + (k + 1)π ), and for all s ∈ (0, 1) γ is smooth, γ (s) ∈ / R, γ (s) = 0. Now, identifying R2 with C, consider the function H1 ◦ γ . It is easy to see that, this is a purely imaginary-valued continuous function on [0, 1], smooth on (0, 1), which vanishes at 0 and 1. Since γ is non-zero on (0, 1), applying Rolle’s theorem 1 to the function i(H1 ◦ γ ) we get that −1 tg(t)e−iγ (s0 )t dt = 0 for some s0 ∈ (0, 1), which is a contradiction, because γ (s0 ) is not real. This finishes the proof when g is even. When g is odd the proof is almost similar except the fact that instead of finding a path (C) on which H1 is purely imaginary (0 included) we find a path on which H1 is real. 2 Proof of Theorem 2.11. If possible assume that G has a non-real zero. Now, from the given conditions it is easy to see that for some large M > 0 Mφ(t) + φ (t) 0 and hence for any positive integer n, M(t n φ(t)) + (t n φ(t)) 0, for all t ∈ [0, 1]. By Lemma 2.15 and using induction we can say that for each positive integer n the entire function 1 Gn (s) :=
φn (t)e−its dt
−1
has a non-real zero, where φn (t) := t n φ(t)
∀t ∈ R.
Since φn (t) = nt (n−1) φ(t) + t n φ (t) and φn (t) = n(n − 1)t (n−2) φ(t) + 2nt n−1 φ (t) + t n φ (t) = t (n−2) n(n − 1)φ(t) + t 2 φ (t) + 2nt (n−1) φ (t), by the given conditions it follows that, for some large positive integer N (we can take N to be (t) 0 and φ (t) 0 for all t ∈ [0, 1], i.e. φ and φ both are increasing on [0, 1]. even) φN N N N
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Now, since φN is even and real valued, we will get a contradiction if we can prove that GN (s) has no zero in {s ∈ C: s = x + iy, x > 0, y > 0}. Now, 1 GN (s) = 2
φN (t) e−its + eits dt
0
1 =
φN (t) e−itx ety + eitx e−ty dt
0
2 = x
y
t −it t y e e x + eit e−t x dt. φN x
x 0
Therefore, 2 Re GN (s) = x
x φN
y
t ty e x + e−t x cos t dt x
0
and 2 −Im GN (s) = x
x
y
t ty e x − e−t x sin t dt. φN x
0 y
both are increasing on [0, 1], it is easy to see that the functions φ ( t )(et x + Since φN and φN N x y y t −t yx ) and φN ( x )(et x − e−t x ) on the interval [0, x] satisfy the assumptions in Lemma 2.12. e Therefore, both Re GN (s) and Im GN (s) cannot be simultaneously zero in the first quadrant which finishes the proof. 2
3. Support theorems on non-compact symmetric spaces In this section we prove support theorems on non-compact symmetric spaces. Let G be a connected, non-compact semisimple Lie group with finite center. Let K ⊆ G be a fixed maximal compact subgroup and X = G/K, the associated Riemannian space of non-compact type. Endow X with the G-invariant Riemannian structure induced from the Killing form. Let dx denote the Riemannian volume element on X. We study convolution equations of the form f ∗ T = g, where f ∈ C ∞ (X) ∩ Lp (X), T is a K-biinvariant compactly supported distribution on X and g ∈ Cc∞ (X). We show that under natural assumptions on the zero set of the spherical Fourier transform of T , f turns out to be compactly supported. (The function f is assumed to be smooth only to make sure that the convolution f ∗ T is well defined.) Before we state our results we recall necessary details. We follow the notation in [8] and [9]. Let G = KAN be an Iwasawa decomposition of G and a be the Lie algebra of A. Let a∗ be the real dual of a and a∗C its complexification. Then for any g ∈ G, g = k(g) exp H (g)n(g) where
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k(g) ∈ K, H (g) ∈ a, n(g) ∈ N . Let M be the centralizer of A in K. For a suitable function f on X, the Helgason–Fourier transform is defined by f˜(λ, k) =
f (x)e(iλ−ρ)H (x
−1 k)
dx,
G
where ρ is the half sum of positive roots and λ ∈ a∗ . We note that f˜(λ, k) = f˜(λ, kM) and so sometimes we will write f˜(λ, b) where b = kM. For each λ ∈ a∗C , let φλ be the elementary spherical function given by: φλ (x) =
e(iλ−ρ)H (x
−1 k)
dk.
K
They are the matrix elements of the spherical principal representations πλ of G defined for λ ∈ a∗C on L2 (K/M) by
−1 πλ (x)v (b) = e(iλ−ρ)H (x b) v k x −1 b , where v ∈ L2 (K/M). The representations πλ are unitary if and only if λ ∈ a∗ . They are also irreducible if λ ∈ a∗ . For f ∈ L1 (X), the group Fourier transform πλ (f ), defined by πλ (f ) =
f (xK)πλ (x) dx G
is a bounded linear operator on L2 (K/M). Its action is given by
πλ (f )v (b) =
v(k) dk f˜(λ, b).
K/M
We also have the Plancherel formula which says that f → f˜(λ, b) is an isometry from L2 (X) onto L2 (a∗ × K/M, |c(λ)|−2 dλ) where c(λ) is the Harish-Chandra c-function. In particular, X
f (x)2 dx = |W |−1
f˜(λ, w)2 c(λ)−2 dλ dk.
a∗ K/M
Next we comment on the pointwise existence of the Helgason–Fourier transform. For 1 p 2, p p define Sp = a∗ + iCρ , where Cρ is the convex hull of {s( p2 − 1)ρ: s ∈ W }, W being the Weyl 0
group. Let Sp be the interior of Sp . The following result from [15] proves the existence of Helgason–Fourier transform pointwise. Theorem 3.1. Let f ∈ Lp (X), 1 p 2. Then ∃ a subset B(f ) ⊆ K, of full measure such that f˜(λ, b) exists ∀b ∈ B and λ ∈ Sp0 . Moreover, for every b ∈ B(f ) fixed, λ → f˜(λ, b) is holomorphic on Sp0 and f˜(λ, ·) L1 (K) → 0 as |λ| → ∞ in Sp0 .
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Remark 3.2. (1) When p = 1 we have f˜(λ, ·) L1 (K) f 1 ∀λ ∈ S1 . (2) When p = 2, existence of f˜(λ, b) is provided by the Plancherel theorem. We also have the Paley–Wiener theorem for compactly supported functions and distributions. Theorem 3.3. The Fourier transform is a bijection from Cc∞ (X) to C ∞ functions ψ on a∗C × K/M satisfying (a) ψ(λ, b) is holomorphic as a function of λ. (b) There is a constant R 0 such that ∀N > 0 sup
λ∈a∗C , b∈K/M
N e−R|Im λ| 1 + |λ| ψ(λ, b) < ∞.
(c) For any σ in the Weyl group and g ∈ G −1 −1 e−(iσ λ+ρ)H (g k) ψ(σ λ, kM) dk = e−(iλ+ρ)H (g k) ψ(λ, kM) dk. K/M
K/M
See [9, page 270]. We restate the above as in [16]. Let vj , j = 0, 1, 2, . . . be an orthonormal basis for L2 (K/M) where each vj transforms according to some irreducible unitary representation of K and v0 is the constant function 1 on K/M. (Note that, for any λ ∈ a∗ πλ (k)v0 = v0 and v0 is the essentially unique vector with this property.) Let KˆM consist of all unitary irreducible representations of K which have an M fixed vector. For δ ∈ KˆM let χδ be its character and d(δ) its dimension. If f ∈ C ∞ (X), then f=
d(δ)χδ ∗ f,
δ∈KˆM
where the convergence is absolute (see [8, page 532]). It follows that f is compactly supported if and only if χδ ∗ f is compactly supported for all δ. We now state the Paley–Wiener theorem in the following form: Theorem 3.4. Let f ∈ Lp (X), 1 p 2 and f = χδ ∗ f for some δ ∈ KˆM . Then f˜(λ, b) = a1 (λ)vi1 (b) + a2 (λ)vi2 (b) + · · · + an (λ)vin (b). (a) If supp f ⊆ BR , then each ai (λ) extends to an entire function on a∗C of exponential type R. (b) Conversely, if each ai extends to an entire function of exponential type R then supp f ⊆ BR . Remark 3.5. In [16] the above theorem is stated only for f ∈ L1 (X). But, this clearly extends to f ∈ Lp (X), 1 p 2. We also recall that if f is K-biinvariant, then the Helgason–Fourier transform is independent of b, and it reduces to the spherical Fourier transform of f defined by f˜(λ) = f (x)φλ (x) dx.
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If T is a K-biinvariant compactly supported distribution, then T˜ (λ) is defined by T˜ (λ) = T (φλ ). We also have a Paley–Wiener theorem for distributions. See [6]. Theorem 3.6. The spherical Fourier transform is a bijection from the space of K-biinvariant compactly supported distributions on X onto the space of Weyl group invariant entire functions of exponential type on a∗C which are of at most polynomial growth on a∗ . We start with the following proposition. Proposition 3.7. Let f ∈ Lp (X) ∩ C ∞ (X), 1 p 2 and T be a compactly supported Kbiinvariant distribution such that f ∗ T is compactly supported. Then (f ∗ T )˜(λ, b) = f˜(λ, b)T˜ (λ). Proof. Since Lp ⊆ L1 + L2 , it suffices to prove this for L1 and L2 . If φ ∈ Cc∞ (K\G/K) then T ∗ φ = φ ∗ T ∈ Cc∞ (K\G/K) and ˜ (T ∗ φ)˜(λ, b) = T˜ (λ)φ(λ). Also if f ∈ L1 or L2 and g ∈ Cc∞ (K\G/K) then (f ∗ g)˜(λ, b) = f˜(λ, b)g(λ). ˜ Now, by assumption f ∗ T ∈ Cc∞ (X). So
˜ ˜ (f ∗ T ) ∗ φ (λ, b) = (f ∗ T )˜(λ, b)φ(λ). But (f ∗ T ) ∗ φ = f ∗ (T ∗ φ) and
˜ ˜ f ∗ (T ∗ φ) (λ, b) = f˜(λ, b)T˜ (λ)φ(λ) which proves the proposition.
2
Now we are in a position to state the analogue of Theorem 2.6 in the previous section. We first deal with the case 1 p < 2. Theorem 3.8. Let f ∈ Lp (X) ∩ C ∞ (X), 1 p < 2 and T be a compactly supported Kbiinvariant distribution. Assume that f ∗ T is compactly supported. If all irreducible components of ZT˜ intersect Sp0 , then f is compactly supported. Proof. Let f ∗ T = g, for g ∈ Cc∞ (X). We may assume that f = χδ ∗ f and so g = χδ ∗ g as T is K-biinvariant. We have g(λ, ˜ b) = a1 (λ)vi1 + a2 (λ)vi2 + · · · + an (λ)vin , where each ai (λ) extends to an entire function on a∗C of exponential type R (for some R > 0), whose restriction to a∗ is bounded. Next, by Proposition 3.7
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(f ∗ T )˜(λ, b) = f˜(λ, b)T˜ (λ). It follows that f˜(λ, b) = b1 (λ)vi1 (b) + b2 (λ)vi2 (b) + · · · + bn (λ)vin (b), where aj (λ) = T˜ (λ)bj (λ). Now, bj (λ) = aj (λ)/T˜ (λ) are holomorphic functions in the open set Sp0 and all the irreducible components of ZT˜ intersect Sp0 . Hence, in the open set Sp0 , all the irreducible components of ZT˜ intersected with Sp0 are contained in the zero set of aj (λ). By irreducibility, this will force all a the components of ZT˜ to be contained in the zero set of aj . It immediately follows that ˜j is an T entire function of exponential type. This finishes the proof. 2 To prove the L2 case we need to recall details about the δ-spherical transform and analyze the c-function in detail. If f ∈ C ∞ (X) then we have f=
d(δ)χδ ∗ f,
δ∈KˆM
where KˆM consists of all unitary irreducible representations of K which have M-fixed vector. We also have L2 (K/M) = δ∈KˆM Vδ , where Vδ consists of the vectors in L2 (K/M) that transform according to the representation δ under the K-action. Let VδM = {v ∈ Vδ : δ(m)v = v ∀m ∈ M}. For δ ∈ KˆM define spherical functions of type δ by Φλδ (x) =
e−(iλ+ρ)(H (x
−1 k))
δ(k) dk,
λ ∈ a∗C , x ∈ X.
K
Then, Φλ,δ (kx) = δ(k)Φλ,δ (x), and Φλ,δ (x)δ(m) = Φλ,δ (x),
m ∈ M.
If f = d(δ)χδ ∗ f , define its δ-spherical Fourier transform by f˜(λ) = d(δ)
X
∗ f (x)Φλ,δ (x) dx,
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where ∗ denotes the adjoint. If δ is the trivial representation then f → f˜ is the spherical Fourier transform. In general δ(m)f˜(λ) = f˜(λ) and so f˜(λ) ∈ Hom(Vδ , VδM ). If f˜(λ, kM) is the Helgason–Fourier transform of f then we have f˜(λ) = d(δ)
f˜(λ, kM)δ k −1 dk,
f˜(λ, kM) = Trace δ(k)f˜(λ) .
K
The δ-spherical Fourier transform is inverted by 1 f (x) = Trace |W |
−2 ˜ Φλ,δ (x)f (λ) c(λ) dλ .
a∗
For each δ ∈ KˆM , we also have the Qδ (λ) matrices which are l(δ) × l(δ) matrices whose entries are polynomial factors in λ (see [9, page 238]). Here l(δ) = dim VδM . Let δˇ denote the contragredient representation of K on the dual space of Vδ . Then, the Paley–Wiener theorem for the δ-spherical transform (see [9, page 285]) says the following: Let H δ (a∗ ) stand for all the functions F : a∗C → Hom(Vδ , VδM ) such that (i) F is holomorphic and is of exponential type, F is holomorphic and Weyl group invariant. (ii) Q−1 ˇ δ
Theorem 3.9. The δ-spherical transform f → f˜ is a bijection from {f ∈ Cc∞ (X): f = d(δ)χδ ∗ f } onto H δ (a∗ ). We are now in a position to state the L2 version of Theorem 3.8. Also recall that if G is a real rank one group then a and a∗ may be identified with R and a∗C with C. Theorem 3.10. (1) Let G be a real rank one group and T be a compactly supported K-biinvariant distribution such that all the zeros of T˜ (λ) are real. If f ∈ L2 ∩ C ∞ (G/K) and f ∗ T is compactly supported then f is compactly supported. (2) Let G have only one conjugacy class of Cartan subgroups. Let T be a K-biinvariant compactly supported distribution such that any irreducible component of ZT˜ intersected with a∗ has real dimension (n − 1). If f ∈ L2 ∩ C ∞ (X) and f ∗ T is compactly supported then f is compactly supported. Proof. (1) In the rank one case it is known that λ → c(λ) is a meromorphic function on C with simple poles, all lying on the imaginary axis. In particular, λ = 0 is a simple pole. It follows that |c(λ)|−2 = c(λ)c(−λ) is a holomorphic function in a small strip containing the real line and the only zero of |c(λ)|−2 in that strip is λ = 0, of order 2. As in the previous theorem we assume that f = d(δ)χδ ∗ f and so g = d(δ)χδ ∗ g. Applying the δ-spherical transform to f ∗ T = g we obtain T˜ (λ)f˜(λ) = g(λ). ˜
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Since l(δ) = dim VδM = 1, both f˜(λ) and g(λ) ˜ are 1 × d(δ) vectors. So, to be consistent with the previous notation we write
g(λ) ˜ = a1 (λ), a2 (λ), . . . , ad(δ) (λ) , and
f˜(λ) = b1 (λ), b2 (λ), . . . , bd(δ) (λ) , where bj (λ) =
aj (λ) . T˜ (λ)
By the Paley–Wiener theorem (Theorem 3.4) λ → aj (λ) is an entire function of exponential type a (λ) and Qj (λ) is an even entire function on C. We also have δˇ
−2 aj (λ) 2 T˜ (λ) c(λ) dλ < ∞. ∗
(3.1)
a
Now if 0 = λ0 is a zero of T˜ (λ) of order k, since |c(λ0 )|−2 = 0 it readily follows from (3.1) that λ0 is a zero of aj (λ) of order at least k. Next, suppose that λ = 0 is a zero T˜ (λ). Since T˜ (λ) is even it follows that ∃ a positive integer l such that T˜ (λ) ∼ λ2l in a neighborhood of λ = 0. Recall a (λ) that Qδˇ (λ) = 0 on a∗ and h(λ) = Qj (λ) is even, holomorphic. Now (3.1) implies that δˇ
−2 h(λ) 2 T˜ (λ) c(λ) dλ < ∞,
(3.2)
|λ|ε
for some ε > 0. Since |c(λ)|−2 ∼ λ2 near zero (3.2) implies that h(λ) = 0 if λ = 0. Since h(λ) is even h(λ) ∼ λ2m in a neighborhood of λ = 0. Then (3.2) implies that m l which in turn implies a (λ) that ˜j is entire which is of exponential type by Malgrange’s theorem. This finishes the proof. T (λ)
(2) If G has only one conjugacy class of Cartan subgroups then the Plancherel density |c(λ)|−2 is given by a polynomial which we describe now. Let Σ0+ be the set of positive indivisible roots. If α ∈ Σ0+ then the multiplicity mα is even ∀α and m2α = 0. For α ∈ Σ0+ define λα =
λ, α , α, α
λ ∈ a∗C .
With the convention that the product over an empty set is 1 the explicit expression for |c(λ)|−2 is given by /2−1 mα 2
c(λ)−2 = c λ2α λα + k 2 , α∈
(see [10]) where c is a positive constant.
+ 0
k=1
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Proceeding as in the previous case we obtain that g(λ) ˜ . f˜(λ) = T˜ (λ) ˜ = Notice that both f˜(λ) and g(λ) ˜ belong to Hom(Vδ , VδM ). Write f˜(λ) = (f˜ij (λ)) and g(λ) (g˜ij (λ)). By the Plancherel theorem we have −2 gij (λ) 2 T˜ (λ) c(λ) dλ < ∞. ∗
a
From the above and the expression for |c(λ)|−2 we also have p(λ)gij (λ) 2 T˜ (λ) dλ < ∞, ∗
(3.3)
a
where p(λ) is the polynomial given by p(λ) =
λα .
α∈Σ0+
Let dim a∗ = l. Since λ → p(λ)gij (λ) is an entire function of exponential type with rapid decay on a∗ , we have H ∈ Cc∞ (Rl ) such that the Euclidean Fourier transform of H , Hˆ (λ) = ˆ p(λ)gij (λ). Similarly, let S be the compactly supported distribution on Rl such that S(λ) = T˜ (λ). 2 l ˆ From (3.3) it follows that there exists F ∈ L (R ) such that F ∗Rl S = H . Since S(λ) = T˜ (λ) p(λ)g (λ) satisfies the conditions in Theorem 2.6 we obtain that F ∈ Cc∞ (Rl ). It follows that ˜ ij is an T (λ)
entire function of exponential type with rapid decay on
a∗ .
However we need to show that
is entire. This follows from applying the following lemma to matrix entries of
Qδˇ (λ)−1 g(λ) ˜ . T˜ (λ)
gij (λ) T˜ (λ)
2
Lemma 3.11. Let p(λ) be as above and ψ(λ) be a holomorphic function defined on a∗C − {λ: p(λ) = 0} such that p(λ)ψ(λ) has an entire extension. If ψ(λ) is Weyl group invariant then ψ(λ) is an entire function. Proof. Since p(λ) is a product of irreducibles it suffices to show that R(λ) = p(λ)ψ(λ) vanishes on {λ ∈ a∗C : p(λ) = 0}. This will follow if we show that R(λ) vanishes on {λ ∈ a∗ : p(λ) = 0}. Fix α ∈ Σ0+ and let 0 = λ0 ∈ a∗ be such that α, λ0 = 0 and β, λ0 = 0 if β = α. It is easy to see that, in a small enough neighborhood of λ0 , α, λ takes both positive and negative values while sgn(β, λ) is constant ∀β ∈ Σ0+ , β = α. Since ψ(λ) is Weyl group invariant this will force R(λ) = 0 if λ = λ0 . This proves that R(λ) is zero on (real) (n − 1)-dimensional strata of the set {λ ∈ a∗ : p(λ) = 0}. This clearly implies that R(λ) = 0 whenever p(λ) = 0. This finishes the proof. 2
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Remark 3.12. The Paley–Wiener theorem (Theorem 3.6) and Theorem 2.11 provide a large class of compactly supported distributions which satisfy the assumptions in Theorem 3.10. Also, the main result in [16] may be improved as in Theorem 2.8. Our proof works well for many other cases as well. To explain this, first we reproduce the computation of the c-function from [10]. The Plancherel density |c(λ)|−2 is given by the product formula cα (λ)−2 , c(λ)−2 = c α∈Σ0+
where cα (λ) =
2−iλα Γ (iλα ) Γ ( iλ2α +
mα 4
+ 12 )Γ ( iλ2α +
mα 4
+
m2α 2 )
.
Recall that if both α and 2α are roots, then mα is even and m2α is odd. Consider the following cases: (a) (b) (c) (d)
mα even, m2α = 0, mα odd, m2α = 0, mα /2 even, m2α odd, mα /2 odd, m2α odd.
If λα = λ,α α,α , with the convention that product over an empty set is 1, the explicit expression for |cα (λ)|−2 is given (up to a constant) by λα pα (λ)qα (λ) where pα and qα are the following, in the four cases listed above, respectively: m2α −1 2 (a) pα (λ) = k=1 [λα + k 2 ], qα (λ) = 1. mα2−3 2 (b) pα (λ) = k=0 [λα + (k + 12 )2 ], qα (λ) = tanh πλα . m4α −1 λα 2 m4α + mα2−1 −1 λα 2 (c) pα (λ) = k=0 [( 2 ) + (k + 12 )2 ] k=0 [( 2 ) + (k + 12 )2 ], πλα qα (λ) = tanh 2 . mα4−2 λα 2 mα +2m2α −1 λα 2 (d) pα (λ) = k=0 [( 2 ) + k 2 ] k=1 4 [( 2 ) + k 2 ], πλα qα (λ) = coth 2 . The case (a) corresponds to the case dealt with in Theorem 3.9. It is clear from the above expression that if mα is large enough ∀α ∈ + 0 then λα pα (λ)qα (λ) λ2α ,
∀α ∈ Σ0+
and consequently we obtain (3.3). Hence the theorem holds for all groups with this property. Simple Lie groups with this property can be read off from the list in [20] (see pages 30–32).
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Acknowledgments We are grateful to the referee who read the manuscript very carefully and pointed out several errors which improved the presentation greatly. We also thank Kaushal Verma, A. Sitaram and Sushil Gorai for many useful discussions. References [1] M.L. Agranovsky, P. Kuchment, The support theorem for the single radius spherical transform, preprint, http://arxiv.org/abs/0905.1310. [2] M.L. Agranovsky, E.K. Narayanan, Lp -integrability, supports of Fourier transforms and uniqueness for convolution equations, J. Fourier Anal. Appl. 10 (3) (2004) 315–324. [3] L. Brandolini, L. Greenleaf, G. Travaglini, Lp − Lp estimates for overdetermined Radon transforms, Trans. Amer. Math. Soc. 359 (6) (2007) 2559–2575. [4] W.O. Bray, Generalized spectral projections on symmetric spaces of non compact type: Paley–Wiener theorems, J. Funct. Anal. 135 (1) (1996) 206–232. [5] E.M. Chirka, Complex Analytic Sets, Math. Appl., vol. 46, Kluwer Acad. Publ., 1989. [6] M. Eguchi, M. Hashizume, K. Okamoto, The Paley–Wiener theorem for distributions on symmetric spaces, Hiroshima Math. J. 3 (1973) 109–120. [7] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley Classics Lib., John Wiley and Sons, Inc., New York, 1994, reprint of the 1978 original. [8] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. [9] S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr., vol. 39, Amer. Math. Soc., Providence, RI, 1994. [10] J. Hilgert, A. Pasquale, Resonances and residue operators for symmetric spaces of rank one, J. Math. Pures Appl. (9) 91 (5) (2009) 495–507. [11] L. Hörmander, Lower bounds at infinity for solutions to partial differential equations with constant coefficients, Israel J. Math. 16 (1973) 103–116. [12] W. Littman, Decay at infinity of solutions to partial differential equations with constant coefficients, Trans. Amer. Math. Soc. 123 (1966) 449–459. [13] W. Littman, Decay at infinity of solutions to higher order partial differential equations: removal of the curvature assumption, Israel J. Math. 8 (1970) 11–20. [14] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955–1956) 271–355. [15] P. Mohanty, S.K. Ray, R.P. Sarkar, A. Sitaram, The Helgason–Fourier transform for symmetric spaces. II, J. Lie Theory 14 (1) (2004) 227–242. [16] M. Shahshahani, A. Sitaram, The Pompeiu problem in exterior domains in symmetric spaces, in: Integral Geometry, Brunswick, Maine, 1984, in: Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 267–277. [17] A. Sitaram, Fourier analysis and determining sets for Radon measures on Rn , Illinois J. Math. 28 (2) (1984) 339– 347. [18] F. Trèves, Differential polynomials and decay at infinity, Bull. Amer. Math. Soc. 66 (1960) 184–186. [19] V.V. Volchkov, Integral Geometry and Convolution Equations, Kluwer Acad. Publ., Dordrecht–Boston–London, 2003. [20] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer-Verlag, New York–Heidelberg, 1972.
Journal of Functional Analysis 259 (2010) 2613–2641 www.elsevier.com/locate/jfa
Comparison inequalities for heat semigroups and heat kernels on metric measure spaces ✩ Alexander Grigor’yan a , Jiaxin Hu b,∗ , Ka-Sing Lau c a Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China c Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Received 7 April 2010; accepted 19 July 2010 Available online 2 August 2010 Communicated by L. Gross
Abstract We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates. © 2010 Elsevier Inc. All rights reserved. Keywords: Dirichlet form; Heat semigroup; Heat kernel; Maximum principle
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . Preliminaries on Dirichlet forms . . . . . . . . Basic comparison theorem . . . . . . . . . . . . Comparison results for the heat semigroups
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A.G. was supported by SFB 701 of the German Research Council (DFG) and the Grants from the Department of Mathematics and IMS in CUHK. J.H. was supported by NSFC (Grant No. 10631040), SFB 701 and the HKRGC Grant in CUHK. K.S.L. was supported by the HKRGC Grant in CUHK. * Corresponding author. E-mail addresses:
[email protected] (A. Grigor’yan),
[email protected] (J. Hu),
[email protected] (K.-S. Lau). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.010
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4.1. General regular Dirichlet forms . . . . . . 4.2. Quasi-local Dirichlet forms . . . . . . . . . 5. Comparison results for heat kernels . . . . . . . . . 6. Pointwise off-diagonal estimates of heat kernels Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this paper, we are concerned with certain inequalities involving heat kernels on arbitrary metric measure spaces. The motivation comes from the following three results. 1. Let M be a Riemannian manifold and pt (x, y) be the heat kernel on M associated with the Laplace–Beltrami operator . Let {Xt }t0 be the diffusion process generated by . For any open set Ω, denote by ψΩ (t, x) the probability that Xt exits from Ω before the time t, provided X0 = x. It was proved in [8] that, for any two disjoint open subsets U and V of M and for all x ∈ U , y ∈ V , t, s > 0, pt+s (x, y) ψU (t, x)
sup
st t+s u∈∂U
pt (u, y) + ψV (s, y)
sup
tt t+s v∈∂V
pt (v, x)
(1.1)
(see Fig. 1). Similarly, if U ⊂ V then, for all x ∈ U and y ∈ V , V pt+s (x, y) pt+s (x, y) + ψU (t, x)
sup
st t+s u∈∂U
pt (u, y) + ψV (s, y)
sup
tt t+s v∈∂V
pt (v, x), (1.2)
where ptV (x, y) is the heat kernel in V with the Dirichlet boundary condition in ∂V (see Fig. 2). The estimates (1.1) and (1.2) were used in [8] to obtain heat kernel bounds on manifolds with ends. 2. Let now {Xt }t0 be a diffusion process on a metric measure space (M, d, μ), and assume that {Xt } possesses a continuous transition density pt (x, y) that will be called the heat kernel. It was proved in [11] that, for any open set V ⊂ M and for all x ∈ V , t > 0, V (x, x) + 2ψV (t, x) sup pt (v, v). p2t (x, x) p2t
(1.3)
v∈V
Fig. 1. Any sample path, connecting x and y, either exits from the set U before time t when starting at x, or exits from the set V before time s when starting at y.
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Fig. 2. Any sample path, connecting x and y, either stays in V , or exits from the set U before time t when starting at x, or exits from the set V before time s when starting at y.
In the setting of manifolds, one sees that (1.3) is a particular case of (1.2) where U = V and x = y since sup pt (v, v) sup pt (v, v).
tt 2t v∈∂V
v∈V
Kigami used (1.3) in [11] to develop a technique for obtaining an upper bound of pt (x, x), given a certain estimate of the Dirichlet heat kernel ptV (x, x). He then applied this technique to obtain heat kernel estimates on post-critically finite self-similar fractals. 3. In the previous setting, but without the continuity of the heat kernel, the authors proved in [6] the following inequality: esup pt+s (x, y) esup ptV (x, y) + ψV (t, x) esup ps (y, z) y∈V
y∈V
(1.4)
y,z∈V
for all t, s > 0 and almost all x ∈ V , where esup stands for the essential supremum. We refer to the estimates of types (1.1)–(1.4) as comparison inequalities for heat kernels. The purpose of this paper is to prove such inequalities in the most general setting, where the heat semigroups are determined by regular Dirichlet forms, under minimal a priori assumptions about the underlying space and the Dirichlet form. Our method applies to local as well as to nonlocal regular Dirichlet forms, that is, the associated Hunt process can be a diffusion or not. We prove the comparison inequalities for the heat semigroups without assuming the existence of the heat kernels. If the heat kernels do exist, then we obtain the comparison inequalities for the heat kernels without assuming their continuity. We hope that this level of generality for comparison inequalities will find applications in diverse settings of both diffusion and jump processes on abstract metric measure spaces. Despite the probabilistic motivation, all the proofs in this paper are entirely analytic and are based on the version of the parabolic maximum principle, developed by the authors [5,7] in the abstract setting. Our basic result is the inequality (3.3) of Theorem 3.1, which holds true for a weak subsolution of the heat equation associated with any regular Dirichlet form. A refinement of Theorem 3.1 for quasi-local Dirichlet forms is given in Theorem 4.3. It turns out that this basic inequality (3.3) (and its version (4.4) for quasi-local forms) is a source of various interesting comparison inequalities for heat semigroups and heat kernels. For example, the inequality (5.13) of Theorem 5.1 contains (1.1), and the inequality (5.12) contains (1.2) and (1.3). General comparison estimates for heat semigroups are given by Propo-
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sition 4.1 for arbitrary regular Dirichlet forms and by Corollary 4.8 for quasi-local Dirichlet forms. The structure of this paper is as follows. In Section 2 we give some preliminaries on Dirichlet forms and weak solutions of the associated heat equation. In Section 3, we prove the basic Theorem 3.1. The consequences of Theorem 3.1 – various comparison inequalities, are proved in Section 4 for the heat semigroups and in Section 5 for the heat kernels. Finally, in Section 6, we give an example of application of the comparison inequalities, that is, deducing the off-diagonal sub-Gaussian upper bound of the heat kernel from the on-diagonal bound and the tail estimate. 2. Preliminaries on Dirichlet forms In this section, we first recall some terminology from the theory of Dirichlet form (cf. [4]) and prove some further properties of Dirichlet forms, which are of independent interest for their own right. Let (M, d, μ) be a metric measure space, that is, the pair (M, d) is a locally compact separable metric space and μ is a Radon measure on M with a full support, that is, μ(Ω) > 0 for any nonempty open subset Ω of M. Let (E, F ) be a Dirichlet form in L2 := L2 (M, μ), that is, F is a dense subspace of L2 and E(f, g) is a bilinear, symmetric, non-negative definite, closed, and Markovian functional on F × F . The closedness of (E, F ) means that F is a Hilbert space with the norm (f 22 + E(f ))1/2 , where · 2 is the norm of L2 (M, μ) and E(f ) := E(f, f ). The Markovian property means that f ∈ F implies f˜ := (f ∨ 0) ∧ 1 ∈ F and E(f˜) E(f ). Let be the generator of (E, F ), that is, an operator in L2 with the maximal domain dom() ⊂ F such that E(f, g) = −(f, g) for all f ∈ dom(), g ∈ F . Then is a non-positive definite self-adjoint operator in L2 . Let {Pt }{t0} be the heat semigroup associated with the form (E, F ), that is, Pt = exp(t). It follows that, for any t 0, Pt is a bounded self-adjoint operator in L2 . The relation between Pt and is given also by the identity 1 f = L2 − lim (Pt f − f ), t→0 t where the limit exists if and only if f ∈ dom(). A similar relation takes place between Pt and E : 1 E(f, g) = lim (f − Pt f, g), t→0 t for all f, g ∈ F . The heat semigroup {Pt } of a Dirichlet form is always Markovian, that is, for any 0 f 1 a.e. in M, we have that 0 Pt f 1 a.e. in M for any t > 0. A family {pt }t>0 of μ × μ-measurable functions on M × M is called the heat kernel of the Dirichlet form (E, F ) if pt is the integral kernel of the operator Pt , that is, for any t > 0 and for any f ∈ L2 (M, μ), (2.1) Pt f (x) = pt (x, y)f (y) dμ(y) M
for μ-almost all x ∈ M.
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The form (E, F ) is regular if the space F ∩ C0 (M) is dense both in F and in C0 (M), where C0 (M) is the space of all real-valued continuous functions in M with compact support. For any two subsets U, Ω (U Ω) of M, a cut-off function φ for the pair (U, Ω) is a function in F ∩ C0 (M) such that 0 φ 1 in M, φ = 1 in an open neighborhood of U , and supp(φ) ⊂ Ω. If (E, F ) is a regular Dirichlet form, then a cut-off function exists for any pair (U, Ω) provided that Ω is open and U is a non-empty compact subset of Ω (cf. [4, p. 27]). Let Ω be a non-empty open subset of M. We identify the space L2 (Ω) as a subspace of 2 L (M) by extending any function f ∈ L2 (Ω) to M by setting f = 0 outside Ω. Denote by F (Ω) the closure of F ∩ C0 (Ω) in F -norm. It is known that if (E, F ) is regular, then (E, F (Ω)) is a regular Dirichlet form in L2 (Ω) (cf. [4]). We refer to (E, F (Ω)) as a restricted Dirichlet form. Denote by {PtΩ }t0 the heat semigroup of (E, F (Ω)). It is known that, for any two open subsets Ω1 ⊂ Ω2 of M, for any 0 f ∈ L2 , and for any t > 0, PtΩ1 f PtΩ2 f
a.e. in M.
Also, if {Ωk }∞ k=1 is an increasing sequence of open sets and Ω = Ω
Pt k f → PtΩ f
∞
k=1 Ωk
then, for any t > 0,
a.e. in M as k → ∞
(see [5, Lemma 4.17]). The form (E, F ) is called local if E(f, g) = 0 for any f, g ∈ F with disjoint compact supports in M. For 0 ρ < ∞, the form (E, F ) is said to be ρ-local if E(f, g) = 0 for any f, g ∈ F with compact supports in M and such that dist supp(f ), supp(g) > ρ. In particular, if ρ = 0 then the ρ-local is the same as the local. We say that the form (E, F ) is quasi-local if it is ρ-local for some ρ 0. Let Ω be an open subset of M and I be an open interval in R. A path u : I → L2 (Ω) is said to be weakly differentiable at t ∈ I if, for any ϕ ∈ L2 (Ω), the function (u(·), ϕ) is differentiable at t, that is, the limit
u(t + ε) − u(t) ,ϕ lim ε→0 ε
exists. If this is the case then it follows from the principle of uniform boundedness that there is a (unique) function w ∈ L2 (Ω) such that lim
ε→0
u(t + ε) − u(t) , ϕ = (w, ϕ), ε
for all ϕ ∈ L2 (Ω). We refer to the function w as the weak derivative of u at t and write w = ∂u ∂t . A path u : I → F is called a weak subsolution of the heat equation in I × Ω, if the following two conditions are fulfilled: • the path t → u(t)|Ω is weakly differentiable in L2 (Ω) at any t ∈ I ;
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• for any non-negative ϕ ∈ F (Ω), we have
∂u , ϕ + E(u, ϕ) 0. ∂t
(2.2)
Similarly one can define the notions of weak supersolution and weak solution of the heat equation. Remark 2.1. Note that, for any f ∈ L2 (Ω), the function PtΩ f is a weak solution in (0, ∞) × Ω (cf. [5, Example 4.10]), and hence, in (0, +∞) × U for any open subset U ⊂ Ω. We use the following notation: f+ := f ∨ 0 and f− = −(f ∧ 0). H
H
Denote by the sign a weak convergence in a Hilbert space H and by → the strong (norm) convergence in H. The following statements will be used in this paper. Proposition 2.2. (See [6, Proposition 4.9].) Let {uk } be a sequence of functions in F such that F
L2
uk u ∈ F as k → ∞. If in addition the sequence {E(uk )} is bounded, then uk u as k → ∞. Proposition 2.3. (See [4, Theorem 1.4.2].) Any Dirichlet form (E, F ) possesses the following properties: • If u, v ∈ F , then all the functions u ∧ v, u ∨ v, u ∧ 1, u+ , u− , |u| also belong to F . • If u, v ∈ F ∩ L∞ (M), then uv ∈ F . F
• If 0 u ∈ F , then u ∧ n → u as n → ∞. • Let φ(s) be a Lipschitz function on R such that φ(0) = 0. Then, for any u ∈ F , φ(u) ∈ F F
also. Moreover, if {un }∞ n=1 is a sequence of functions from F and un → u ∈ F as n → ∞, F
F
then φ(un ) φ(u). Furthermore, if φ(u) = u then φ(un ) → φ(u). Proposition 2.4. (See [5, Lemma 4.4].) Let (E, F ) be a regular Dirichlet form, and let u ∈ F and Ω be an open subset of M. Then the following are equivalent: (1) u+ ∈ F (Ω). (2) u v in M for some function v ∈ F (Ω). Proposition 2.5 (Parabolic maximum principle). (See [7, Proposition 5.2].) Assume that (E, F ) is a regular Dirichlet form in L2 . For T ∈ (0, +∞] and for an open subset Ω of M, let u be a weak subsolution of the heat equation in (0, T ) × Ω satisfying the following boundary and initial conditions: • u+ (t, ·) ∈ F (Ω) for any t ∈ (0, T ); L2 (Ω)
• u+ (t, ·) −→ 0 as t → 0.
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Then u(t, x) 0 for any t ∈ (0, T ) and μ-almost all x ∈ Ω. Next we prove further some general results on Dirichlet forms that will be used later on and are of independent interest. Proposition 2.6. Let Ω be a non-empty open subset of M. Then, for any non-negative f ∈ L2 (Ω), the path u(t) = PtΩ f is a weak subsolution of the heat equation in (0, ∞) × M. Proof. We know that u(t) is weakly differentiable in t in L2 (Ω). Let us show that u(t) is weakly differentiable also in L2 (M). Indeed, for any function ϕ ∈ L2 (M), we have
u(t + s) − u(t) u(t + s) − u(t) u(t + s) − u(t) c ,ϕ = , ϕ1Ω + , ϕ1Ω . s s s
(2.3)
Since ϕ1Ω ∈ L2 (Ω), the first term in the right-hand side of (2.3) converges to ( ∂u ∂t , ϕ1Ω ) where ∂u 2 ∂t is the weak derivative in L (Ω). The second term is obviously 0, whence the convergence of the whole sum to ( ∂u ∂t , ϕ) follows. Next, let us show that, for any non-negative ψ ∈ F ,
∂u , ψ + E(u, ψ) 0 ∂t
for any t > 0.
(2.4)
Indeed, noting that Ps u(t) PsΩ u(t) = u(t + s), we obtain as s → 0+ that 1 ∂u 1 1 Ω Es (u, ψ) = (u − Ps u, ψ) u − Ps u, ψ = u(t) − u(t + s), ψ → − , ψ . s s s ∂t Since Es (u, ψ) → E(u, ψ) as s → 0, the desired inequality (2.4) follows.
2
The following proposition will be used to prove Proposition 2.9. Proposition 2.7. Let Ω1 , Ω2 be two non-empty open subsets of M. Then F (Ω1 ) ∩ F (Ω2 ) = F (Ω1 ∩ Ω2 ).
(2.5)
Proof. Since F (Ω1 ∩ Ω2 ) ⊂ F (Ωi ) for i = 1, 2, we see that F (Ω1 ∩ Ω2 ) ⊂ F (Ω1 ) ∩ F (Ω2 ). To prove the opposite inclusion, we need to verify that f ∈ F (Ω1 ) ∩ F (Ω2 ) implies f ∈ F (Ω1 ∩ ∞ Ω2 ). Assume first that f 0. Let {fk }∞ k=1 and {gk }k=1 be two sequences from F ∩ C0 (Ω1 ) and F ∩ C0 (Ω2 ), respectively, that both converge to f in F -norm. As f 0 and, hence, f+ = f , it follows from Proposition 2.3 that F
(fk )+ → f
F
and (gk )+ → f
as k → ∞.
(2.6)
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Since (fk )+ ∈ F ∩ C0 (Ω1 ) and (gk )+ ∈ F ∩ C0 (Ω2 ), we see that hk := (fk )+ ∧ (gk )+ ∈ F ∩ C0 (Ω1 ∩ Ω2 ) ⊂ F (Ω1 ∩ Ω2 ). F
Setting uk = (fk )+ − (gk )+ and noticing that uk → 0 as k → ∞, we obtain by Proposition 2.3 F
that |uk | 0 as k → ∞. It follows that hk =
F 1 (fk )+ + (gk )+ − (fk )+ − (gk )+ f 2
as k → ∞.
Since F (Ω1 ∩ Ω2 ) is a closed and, hence, weakly closed subspace of F , we conclude that f ∈ F (Ω1 ∩ Ω2 ). For a signed function f ∈ F (Ω1 ) ∩ F (Ω2 ), we have f+ , f− ∈ F (Ω1 ) ∩ F (Ω2 ), whence, by the first part of the proof, f+ , f− ∈ F (Ω1 ∩ Ω2 ) and f = f+ − f− ∈ F (Ω1 ∩ Ω2 ), which finishes the proof. 2 Proposition 2.8. Let U be a non-empty open subset of M, and let u ∈ F such that supp(u) ⊂ U and is compact. Then u ∈ F (U ). Proof. We can assume that u 0 because a signed u follows from the decomposition u = u+ − u− . Next, we can assume that u is bounded because otherwise consider a sequence uk := u ∧ k that tends to u in F -norm as k → ∞ by Proposition 2.3; if we already know that uk ∈ F (U ) then we can conclude that also u ∈ F (U ). Hence, we can assume in the sequel that u is non-negative and bounded in M, say 0 u 1. Let ϕ be a cut-off function for the pair (supp(u), U ). Let {uk }∞ k=1 be a sequence from F ∩ F
C0 (M) such that uk → u as k → ∞. As u 0, we have by the last results in Proposition 2.3 that F
F
(uk )+ → u as k → ∞ and |(uk )+ − ϕ| |u − ϕ| as k → ∞. It follows that (uk )+ ∧ ϕ =
F 1
1 (uk )+ + ϕ − (uk )+ − ϕ u + ϕ − |u − ϕ| = u ∧ ϕ = u as k → ∞. 2 2
Since (uk )+ ∧ ϕ ∈ F ∩ C0 (U ), we conclude that u ∈ F (U ).
2
Proposition 2.9. Let Ω be a precompact open subset of M and U be an open subset of M, and let K be a closed subset of M such that K ⊂ U (see Fig. 3). Let u ∈ F be a function such that u+ ∈ F (Ω) and u ψ in Ω \ K for some 0 ψ ∈ F . Then (u − ψ)+ ∈ F (Ω ∩ U ).
(2.7)
Proof. Since u − ψ u+ ∈ F (Ω), it follows by Proposition 2.4 that (u − ψ)+ ∈ F (Ω). Let us verify that (u − ψ)+ ∈ F (U ),
(2.8)
which will then imply (2.7) by Proposition 2.7. Indeed, noticing that (u − ψ)+ = 0 in Ω \ K and in Ω c , we see that
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Fig. 3. Domains Ω, U and K.
supp (u − ψ)+ ⊂ K ∩ Ω ⊂ K ∩ Ω. On the other hand, the set K ∩ Ω is compact and is contained in U , so that (2.8) follows from Proposition 2.8. 2 Remark 2.10. The statement of Proposition 2.9 was proved in [6, Proposition 4.10] under additional condition that u ∈ L∞ (M) ∩ F (Ω) and supp(ψ) is compact. The present proof is also shorter than the one from [6]. 3. Basic comparison theorem The next theorem is the basic technical result of this paper. Theorem 3.1. Let (M, d, μ) be a metric measure space and let (E, F ) be a regular Dirichlet form in L2 (M, μ). Let Ω ⊂ M be a precompact open set and U ⊂ M be an open such that μ(U ) < ∞. Let u be a weak subsolution of the heat equation in (0, T0 ) × (Ω ∩ U ) where T0 ∈ (0, +∞], such that u+ (t, ·) ∈ F (Ω)
for any t ∈ (0, T0 ),
L2 (Ω∩U )
u+ (t, ·) −→ 0 as t → 0.
(3.1) (3.2)
Let K be a closed subset of M such that K ⊂ U . Then, for any t ∈ (0, T0 ) and for almost all x ∈ M, u(t, x) 1 − PtU 1U (x) sup u+ (s, ·) L∞ (Ω\K) ,
(3.3)
0<st
provided that sup0<st u+ (s, ·)L∞ (Ω\K) < ∞. Remark 3.2. If Ω ⊂ U , then all the conditions of Proposition 2.5 are satisfied, so that we conclude u 0 in (0, T0 ) × Ω. Hence, in this case the inequality (3.3) is trivially satisfied. Remark 3.3. If U, Ω are open domains in Rn with smooth boundaries, then one can rephrase the statement of Theorem 3.1 for strong solutions as follows: if u solves the heat equation in (0, T0 ) × (Ω ∩ U ) and satisfies the initial and boundary conditions
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Fig. 4. Illustration to Theorem 3.1 in the classical case.
u 0 on ∂Ω ∩ U instead of u+ ∈ F (Ω) , um
on ∂U ∩ Ω for some m 0 (instead of u m on Ω \ K), u(t, ·) → 0 as t → 0 in Ω ∩ U,
then u (1 − PtU 1U )m in (0, T0 ) × (Ω ∩ U ) (see Fig. 4). Indeed, the function v = (1 − PtU 1)m satisfies the heat equation in (0, ∞) × U , the boundary conditions v 0 on ∂Ω, v = m on ∂U , and the initial condition v(t, ·) → 0 as t → 0 in U . Applying the classical parabolic maximum principle in Ω ∩ U , we obtain u v. Proof of Theorem 3.1. Outside Ω the inequality (3.3) is trivial because u 0 by (3.1). In Ω \ U (3.3) is also obvious because PtU 1U = 0 and K ⊂ U . It remains to prove (3.3) in Ω ∩ U . Fix a number T ∈ (0, T0 ) and define m by m = sup u+ (t, ·) L∞ (Ω\K) .
(3.4)
0
Let us first prove that, for any t ∈ (0, T ) and for μ-almost all x ∈ Ω ∩ U , u(t, x) m.
(3.5)
Let φ be a cut-off function for the pair (Ω, M) and consider the function w = u − mφ.
(3.6)
Then (3.5) will follow if we show that w 0 in (0, T ) × (Ω ∩ U ). The latter will be proved by using the maximum principle of Proposition 2.5. We need to verify the following conditions. • The function w is a weak subsolution of the heat equation in (0, T ) × (Ω ∩ U ). Indeed, the function φ, considered as a function of (t, x), is a weak supersolution of the heat equation in (0, ∞) × Ω, since for any non-negative function ψ ∈ F (Ω), E(φ, ψ) = lim t −1 (φ − Pt φ, ψ) = lim t −1 (1 − Pt φ, ψ) 0. t→0
t→0
Since u is a weak subsolution in (0, T ) × (Ω ∩ U ), we see from (3.6) that so is w.
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Fig. 5. Illustration to the proof of (3.7) in the case U ⊂ Ω.
• For any t ∈ (0, T ), we have w+ (t, ·) ∈ F (Ω ∩ U ). Indeed, using the facts that u+ (t, ·) ∈ F (Ω) and u m = mφ in Ω \ K (which is true by (3.4)), we obtain from Proposition 2.9 that w+ (t, ·) = u(t, ·) − mφ + ∈ F (Ω ∩ U ). L2 (Ω∩U )
• The initial condition w+ (t, ·) −→ 0 as t → 0 follows from w+ (t, ·) u+ (t, ·) and (3.2). Therefore, by the parabolic maximum principle of Proposition 2.5, we conclude that w 0 in (0, T ) × (Ω ∩ U ), thus proving (3.5). We are now in a position to prove the following improvement of (3.5): u 1 − PtU 1U m in (0, T ) × (Ω ∩ U )
(3.7)
(see Fig. 5 where the case U ⊂ Ω is shown). The path t → u(t, ·) is weakly differentiable in L2 (Ω ∩ U ) and, hence, is strongly continuous in L2 (Ω ∩ U ) (see [7, Lemma 5.1]). The same applies to the path t → PtU 1U so that the inequality (3.7) extends to t = T by continuity. Hence, (3.7) implies (3.3). Consider the function v = u − mφ 1 − PtU 1U ,
(3.8)
where m and φ are the same as above. As μ(U ) < ∞, we have 1U ∈ L2 (U, μ) and, hence, PtU 1U ∈ F (U ). We claim that v is a weak subsolution of the heat equation in (0, T ) × (Ω ∩ U ). Since u is a weak subsolution, it suffices to show that the function f := φ 1 − PtU 1U is a weak supersolution in (0, T ) × (Ω ∩ U ). Since the both functions φ and PtU 1U belong to L∞ (M) ∩ F , so does the product φPtU 1U , whence
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f = φ − φPtU 1U ∈ L∞ (M) ∩ F . For any t, s ∈ (0, T ), we have that in Ω ∩ U , f − Ps f = φ 1 − PtU 1U − Ps φ 1 − PtU 1U 1 − PtU 1U − Ps 1 − PtU 1U U = (1 − Ps 1) − PtU 1U + Ps PtU 1U Pt+s 1U − PtU 1U , which yields that, for any 0 ψ ∈ F (Ω ∩ U ), 1 1 U E(f, ψ) = lim (f − Ps f, ψ) lim Pt+s 1U − PtU 1U , ψ = s→0 s s→0 s
∂ U P 1U , ψ . ∂t t
On the other hand,
∂ ∂f ∂ U , ψ = −φ PtU 1U , ψ = − Pt 1U , ψ . ∂t ∂t ∂t
Therefore,
∂f , ψ + E(f, ψ) 0, ∂t
showing that f is a weak supersolution. Hence, we have proved that v is a weak subsolution. Since v u, it follows from (3.2) that L2 (U ∩Ω)
v+ (t, ·) −→ 0
as t → 0.
It remains to verify the boundary condition: v+ (t, ·) ∈ F (Ω ∩ U ) for any t ∈ (0, T ). Observe that u − mφ 0
in M
(3.9)
because we have • u − mφ 0 in M \ Ω by (3.1), • u − mφ = u − m 0 in Ω \ U by (3.4), • u − mφ = u − m 0 in Ω ∩ U by (3.5). Using (3.9), we obtain that in M v = u − mφ 1 − PtU 1U mφPtU 1U mPtU 1U . Since the function PtU 1U belongs to F (U ), we conclude by using Proposition 2.4 that also v+ ∈ F (U ). On the other hand, we have v = u − mφ 1 − PtU 1U u u+ ∈ F (Ω),
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whence it follows that v+ ∈ F (Ω). Therefore, by Proposition 2.7 we obtain that v+ ∈ F (U ∩ Ω), thus proving the boundary condition. Finally, we conclude by the maximum principle of Proposition 2.5 that v 0 in (0, T ) × (Ω ∩ U ), whence (3.7) follows. 2 Remark 3.4. The boundary condition (3.1) in Theorem 3.1 can be relaxed as follows: u+ (t, ·) ∈ F (Ω)
for any t ∈ (0, T0 ) ∩ Q,
(3.10)
provided one assumes in addition that t → u(t, ·)
is weakly continuous in L2 (Ω), t → E u(t, ·) is locally bounded,
(3.11) (3.12)
for t ∈ (0, T0 ). Under the hypotheses (3.10)–(3.12), the inequality (3.3) can be replaced by a stronger one: u(t, x) 1 − PtU 1U (x) sup u+ (s, ·) L∞ (Ω\K) .
(3.13)
0<st s∈Q
The proof goes exactly as the above except that the supremum for defining the constant m in (3.4) is taken only over rational t ∈ (0, T ]. (The reason for taking the supremum over the rational, instead of over the real, is that such a function is measurable, see Appendix A.) Then we need to verify that the functions w and v, defined by (3.6), (3.8), respectively, satisfy the boundary condition (3.1) for all real t ∈ (0, T ) in order to be able to use the maximum principle of Proposition 2.5. Indeed, for any t ∈ (0, T ), let {tk }∞ k=1 be a sequence of rationals such that tk → t as k → ∞. By (3.6) and (3.11), we have L2 (Ω)
w(tk , ·) − w(t, ·) = u(tk , ·) − u(t, ·) 0, and thus L2 (Ω)
w+ (tk , ·) w+ (t, ·). By (3.12), E(w(tk , ·)) is bounded as k → ∞. Hence, we obtain by Proposition 2.2 that F
w+ (tk , ·) w+ (t, ·). Since w+ (tk , ·) ∈ F (Ω) by (3.10), we conclude that w+ (t, ·) ∈ F (Ω). Similarly, one has v+ (t, ·) ∈ F (Ω) for all real t ∈ (0, T ). The inequality (3.3) gives a rise to various interesting comparison inequalities for heat semigroups and heat kernels that will be presented in the next sections. Before that, let us state a useful particular case of Theorem 3.1 when U ⊂ Ω (cf. Fig. 5).
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Corollary 3.5. Let (M, d, μ) be a metric measure space and let (E, F ) be a regular Dirichlet form in L2 (M, μ). Let Ω ⊂ M be a precompact open set and U be an open subset of Ω. Let u be a weak subsolution of the heat equation in (0, T0 ) × U where T0 ∈ (0, +∞], such that u+ (t, ·) ∈ F (Ω)
for any t ∈ (0, T0 ),
L2 (U )
u+ (t, ·) −→ 0 as t → 0.
(3.14)
Then the conclusion of Theorem 3.1 holds for any compact subset K of U , any t ∈ (0, T0 ) and almost all x ∈ M. 4. Comparison results for the heat semigroups In this section, we give various applications of Theorem 3.1 to the semigroup solutions, including a specific case of quasi-local Dirichlet form. 4.1. General regular Dirichlet forms Proposition 4.1. Let (E, F ) be a regular Dirichlet form in L2 (M, μ), and let Ω, U be two nonempty open subsets of M such that μ(U ) < ∞. Let K be any closed subset of M such that K ⊂ U . Then, for any 0 f ∈ L2 (Ω), PtΩ f (x) − PtU f (x) 1 − PtU 1U (x) sup PsΩ f L∞ (Ω\K) ,
(4.1)
0<st
for all t > 0 and almost all x ∈ M. Proof. Without loss of generality, assume that 0 f ∈ L∞ (Ω) (otherwise, apply (4.1) to the function fk = f ∧ k and then pass to the limit as k → ∞). Let {Ωi } be a sequence of precompact open subsets exhausting Ω. Consider the function u(t, ·) := PtΩi f − PtΩi ∩U f and we shall verify that u satisfies all the hypothesis of Theorem 3.1 with the sets Ωi and U . Ω Indeed, u is a weak subsolution of the heat equation in (0, ∞) × (Ωi ∩ U ) because so are Pt i f Ωi ∩U Ωi Ωi ∩U f (cf. Remark 2.1). Next, u(t, ·) ∈ F (Ωi ) because both Pt f and Pt f belong and Pt Ωi Ωi ∩U 2 f converge to f as t → 0 in L (Ωi ∩ U ), it follows that to F (Ωi ). Since both Pt f and Pt u(t, ·)
L2 (Ωi ∩U )
−→
0 as t → 0. By Theorem 3.1, we obtain that
PtΩi f − PtΩi ∩U f 1 − PtU 1U sup PsΩi f − PsΩi ∩U f L∞ (Ω \K) 0<st
i
1 − PtU 1U sup PsΩ f L∞ (Ω\K) . 0<st
Ωi ∩U
Noticing that Pt desired. 2
f PtU f and then passing to the limit as i → ∞, we obtain (4.1), as
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Remark 4.2. Let us mention for comparison that the following inequality was proved in [6, Proposition 4.7]: PtΩ f (x) − PtU f (x) sup PsΩ f L∞ (Ω\K) .
(4.2)
0<st
Obviously, (4.1) is an improvement of (4.2). On the other hand, the estimate (4.2) was proved in [6] for arbitrary open set U without the hypotheses of the finiteness of its measure. For applications of (4.2) see [6, Theorem 5.12]. 4.2. Quasi-local Dirichlet forms Given an open set U ⊂ M and non-negative number ρ, define the ρ-neighborhood Uρ of U as follows:
Uρ = x ∈ M: d(x, U ) < ρ Uρ = U
if ρ > 0,
if ρ = 0,
where d(x, U ) = infy∈U d(x, y). Theorem 4.3. Assume that (E, F ) is a ρ-local regular Dirichlet form in L2 (M, μ) where ρ 0. Let U be an open subset of M such that Uρ is precompact, and let u be a weak subsolution of the heat equation in (0, T0 ) × U where T0 ∈ (0, +∞]. Assume that, for any t ∈ (0, T0 ), u(t, ·) ∈ L∞ (M) and L2 (U )
u+ (t, ·) −→ 0 as t → 0.
(4.3)
Then for any compact subset K of U , for all t ∈ (0, T0 ), and almost all x ∈ Uρ , u(t, x) 1 − PtU 1U (x) sup u+ (s, ·) L∞ (U 0<st
ρ \K)
,
(4.4)
provided sup0<st u+ (s, ·)L∞ (Uρ \K) < ∞. Proof. Since PtU 1U = 0 outside U , the inequality (4.4) is trivially satisfied if x ∈ Uρ \ U . Hence, it suffices to prove (4.4) for x ∈ U . Fix an open subset W of U such that W ⊂ U . Then Wρ ⊂ Uρ so that Wρ is precompact. Let φ be a cut-off function for the pair (Wρ , Uρ ). Let us show that the function w = uφ satisfies all the hypothesis of Corollary 3.5 where the domains Ω, U are replaced by Uρ , W respectively. Note that the function u may not satisfy the condition (3.14) so that we have to use w instead. Let us first show that w is a weak subsolution of the heat equation in (0, T0 ) × W . Indeed, since u(t, ·), φ ∈ F ∩ L∞ (M) for any t ∈ (0, T0 ) × W , it follows that w(t, ·) ∈ F . Since u is a subsolution in (0, T0 ) × W and φ ≡ 1 in W , we have, for any non-negative function ψ ∈ F (W ),
∂w ∂u ∂u ,ψ = φ ,ψ = , ψ −E(u, ψ) ∂t ∂t ∂t = −E(w, ψ) + E (φ − 1)u, ψ = −E(w, ψ),
(4.5)
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where we have used the fact that E((φ − 1)u, ψ) = 0 by the ρ-locality of E, because supp(ψ) ⊂ W , and the function (φ − 1)u is compactly supported outside Wρ , so that the distance between the supports of ψ and (φ − 1)u is larger than ρ. Since supp ϕ ⊂ Uρ , we see that supp w(t, ·) ⊂ Uρ , and hence, w(t.·) ∈ F (Uρ ) and, w+ (t, ·) ∈ F (Uρ ). Moreover, it follows from (4.3) that L2 (W )
w+ (t, ·) = φu+ (t, ·) −→ 0 as t → 0. Hence, w satisfied the required boundary and initial conditions, and by Corollary 3.5 we obtain that in (0, T0 ) × W , u(t, x) = w(t, x) 1 − PtW 1W (x) sup w+ (s, ·) L∞ (U 0<st
1 − PtW 1W (x) sup u+ (s, ·) L∞ (U 0<st
ρ \K)
ρ \K)
.
Taking an exhaustion of U by sets like W and then passing to the limit as W → U , we obtain (4.4). 2 Remark 4.4. If function u in Theorem 4.3 further satisfies (3.11) and (3.12) with Ω = Uρ , then we conclude from Remark 3.4 that the inequality (4.4) can be replaced by a stronger one: u(t, x) 1 − PtU 1U (x) sup u+ (s, ·) L∞ (U 0<st s∈Q
ρ \K)
.
(4.6)
For the case of local Dirichlet forms, we obtain the following improvement of Theorem 4.3 where the condition of the compactness of Uρ is dropped. Corollary 4.5. Assume that (E, F ) is a local regular Dirichlet form in L2 (M, μ). Let U be an open subset of M and let u be a weak subsolution of the heat equation in (0, T0 ) × U where T0 ∈ (0, +∞]. Assume that, for any t ∈ (0, T0 ), the function u(t, ·) ∈ L∞ (M) and L2 (U )
u+ (t, ·) −→ 0 as t → 0. Then, for any compact subset K of U , for all t ∈ (0, T0 ), and almost all x ∈ U , u(t, x) 1 − PtU 1U (x) sup u+ (s, ·) L∞ (U \K) ,
(4.7)
0<st
provided sup0<st u+ (s, ·)L∞ (U \K) < ∞. Proof. Let {Ui }∞ i=1 be an exhaustion of U , each Ui being precompact and K ⊂ Ui for all i. By Theorem 4.3, we obtain the estimate (4.7) for Ui instead of U , and then pass to the limit as i → ∞. 2
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Remark 4.6. A particular case of the estimate (4.7) with K = ∅ was proved in [6, Lemma 4.3]. However, having an arbitrary compact K can be an advantage in certain situations. For example, if U is precompact and u(t, ·) is continuous in U , then taking exhaustion of U by compact sets K ⊂ U , one can replace the L∞ -norm in (4.7) by sup∂U u+ . Remark 4.7. If (E, F ) is ρ-local with ρ > 0 and in addition all metric balls in M are precompact then the hypothesis of the compactness of Uρ in Theorem 4.3 can also be dropped. Indeed, firstly, it suffices to assume that U is precompact, since it implies that Uρ is precompact. Then one extends the result to all open sets U as in the proof of Corollary 4.5. As an another consequence of Theorem 4.3, we obtain the following useful comparison inequality for heat semigroups. Corollary 4.8. Assume that (E, F ) is a ρ-local regular Dirichlet form in L2 (M, μ) where ρ 0. Let U, Ω be two open subsets of M such that Uρ is precompact and Uρ ⊂ Ω. Then for any 0 f ∈ L∞ (M), for all t > 0 and almost all x ∈ Uρ , PtΩ f (x) − PtU f (x) 1 − PtU 1U (x) sup PsΩ f L∞ (U 0<st
(4.8)
ρ \K)
for any compact subset K of U . Moreover, if ρ = 0, that is, (E, F ) is local then the same is true without assuming that Uρ is precompact. In this case, (4.8) becomes PtΩ f (x) − PtU f (x) 1 − PtU 1U (x) sup PsΩ f L∞ (U \K) .
(4.9)
0<st
Proof. Consider the function u(t, ·) = PtΩ f (·) − PtU f (·), that is bounded on M for any t > 0, is a weak subsolution of the heat equation in (0, ∞) × U , and satisfies the initial condition (4.3). Hence, it follows from (4.4) that, for all t > 0 and almost all x ∈ Uρ , PtΩ f (x) − PtU f (x) 1 − PtU 1U (x) sup PsΩ f − PsU f L∞ (U 0<st
ρ \K)
,
whence (4.8) follows. In the case of a local form, one passes from precompact U to arbitrary U as in the proof of Corollary 4.5. 2 Remark 4.9. In fact, the inequality (4.8) can be improved as follows: PtΩ f (x) − PtU f (x) 1 − PtU 1U (x) sup PsΩ f L∞ (U 0<st s∈Q
ρ \K)
,
(4.10)
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because the function u = PtΩ f − PtU f automatically satisfies conditions (3.11) and (3.12). Since U ⊂ Ω, it suffices to verify that the function u = PtΩ f satisfies (3.11) and (3.12). Indeed, (3.11) follows from the strong continuity of the semigroup {PtΩ } in L2 (Ω) whilst (3.12) follows from the fact that E(PtΩ f ) is a decreasing function of t, the latter being a consequence of the identity E PtΩ f =
∞
λe−2λt d(Eλ f, f ),
0
where {Eλ } is the spectral resolution of the operator Ω , the generator of (E, F (Ω)). Hence, (4.10) follows from (4.6). Remark 4.10. The estimate (4.9) with K = ∅ was proved also in [6, (4.10) in Corollary 4.4]. A useful particular case of (4.9) is when the function f vanishes in U . In this case, (4.8) becomes PtΩ f (x) 1 − PtU 1U (x) sup PsΩ f L∞ (U 0<st
ρ \K)
.
(4.11)
5. Comparison results for heat kernels In this section we will prove a symmetric comparison inequality for the heat kernel of a ρlocal Dirichlet form. The motivation is as follows. Let (E, F ) be an arbitrary regular Dirichlet form and let U, V ⊂ Ω be three open subsets of M such that U ∩ V = ∅. We claim that, for all t, s > 0 and μ-almost all x ∈ U , y ∈ V ,
Ω pt+s (x, y) 1 − PtU 1U (x) psΩ (·, y) L∞ (Ω\U )
+ 1 − PsV 1V (y) ptΩ (·, x) L∞ (Ω\V ) .
(5.1)
Indeed, noticing that ptΩ (x, z) dμ(z) 1 − PtΩ 1U (x) 1 − PtU 1U (x), Ω\U
we obtain that
ptΩ (x, z)psΩ (z, y) dμ(z) psΩ (·, y) L∞ (Ω\U )
Ω\U
ptΩ (x, z) dμ(z)
Ω\U
1 − PtU 1U (x) psΩ (·, y) L∞ (Ω\U ) .
(5.2)
In a similar way, we have Ω\V
ptΩ (x, z)psΩ (z, y) dμ(z) 1 − PsV 1V (y) ptΩ (·, x) L∞ (Ω\V ) .
(5.3)
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Therefore, by the semigroup property, Ω (x, y) = pt+s
ptΩ (x, z)psΩ (z, y) dμ(z) Ω
ptΩ (x, z)psΩ (z, y) dμ(z) +
Ω\U
ptΩ (x, z)psΩ (z, y) dμ(z),
Ω\V
which together with (5.2) and (5.3) yields (5.1). The purpose of the next theorem is to use the ρ-locality in order to replace in (5.1) the L∞ norms in Ω \ U , Ω \ V by those in smaller sets, which is frequently critical for applications. Theorem 5.1. Let (E, F ) be a ρ-local regular Dirichlet form in L2 (M, μ) where ρ 0, and let U, V , Ω be three open subsets of M such that Uρ , Vρ are precompact and Uρ , Vρ ⊂ Ω. Assume that all the Dirichlet heat kernels ptU , ptV , ptΩ exist and that ptΩ (x, y) is locally bounded in R+ × Ω × Ω. Then, for all t, s > 0 and μ-almost all x ∈ U , y ∈ V , Ω (x, y) pt+s
ptU (x, z)psV (z, y) dμ(z) + 1 − PtU 1U (x)
Ω
+ 1 − PsV 1V (y)
sup ptΩ (·, x) L∞ (V
t
sup ptΩ (·, y) L∞ (U
s
ρ \K2 )
ρ \K1 )
(5.4)
,
where K1 , K2 are any compact subsets of U and V respectively. In the case ρ = 0, that is, when (E, F ) is local, the assumption of the compactness of Uρ , Vρ can be dropped. Proof. Let v be a non-negative function from L∞ ∩ L1 (V ). Setting f = PsΩ v and noticing that all the hypotheses of Corollary 4.8 are satisfied, we obtain by (4.10) that the following inequality is true in U for all t > 0:
Ω v PtU PsΩ v + 1 − PtU 1U sup PtΩ +s v L∞ (U Pt+s
= PtU PsΩ v + 1 − PtU 1U
0
sup
s
Ω P v t
ρ \K1 )
L∞ (Uρ \K1 )
,
(5.5)
Ω v. Consider the function where we have used that PtΩ f = Pt+s
F (y) :=
sup
esup ptΩ (z, y),
s
which is bounded in V . Note that F (y) is measurable as the supremum of a countable family of measurable functions of y since
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y → esup ptΩ (z, y) z∈Uρ \K1
is measurable by Proposition A.1, and t varies in Q. We have then sup
s
Ω P v
L∞ (Uρ \K1 )
t
=
sup
esup
s
ptΩ (z, y)v(y) dμ(y)
(5.6)
F (y)v(y) dμ(y). V
Multiplying (5.5) by a non-negative function u ∈ L∞ ∩ L1 (U ) and integrating over U , we obtain
Ω Pt+s v, u PtU PsΩ v , u +
1 − PtU 1U (x) F (y)u(x)v(y) dμ(x) dμ(y). (5.7)
U ×V
On the other hand, observe that U Ω Ω Pt Ps v , u = Ps v, PtU u = v, PsΩ PtU u .
(5.8)
Using (4.10) again, now with f = PtU u and with V in place of U , we obtain the following inequality in V :
PsΩ PtU u = PsΩ f PsV f + 1 − PsV 1V sup PtΩ f L∞ (V 0
ρ \K2 )
.
(5.9)
Observing that PtU u PtΩ u, we obtain that PtΩ f = PtΩ PtU u PtΩ PtΩ u = PtΩ +t u. Similarly to (5.6), we have sup
t
Ω P u t
L∞ (Vρ \K2
)
G(x)u(x) dμ(x) U
where G(x) :=
sup
t
esup ptΩ (z, x)
z∈Vρ \K2
is a bounded measurable function on U . Substituting into (5.9), we obtain in V
PsΩ PtU u PsV PtU u + 1 − PsV 1V
G(x)u(x) dμ(x). U
(5.10)
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Multiplying (5.10) by v and integrating over V , we obtain v, PsΩ PtU u v, PsV PtU u +
1 − PsV 1V (y) G(x)u(x)v(y) dμ(x) dμ(y).
U ×V
Combining this with (5.7) and (5.8), we obtain Ω Pt+s v, u v, PsV PtU u
1 − PtU 1U (x) F (y)u(x)v(y) dμ(x) dμ(y) + U ×V
+
1 − PsV 1V (y) G(x)u(x)v(y) dμ(x) dμ(y).
U ×V
Since v, PsV PtU u =
ptU (x, z)psV (z, y) dμ(z)
U ×V
u(x)v(y) dμ(x) dμ(y),
Ω
we can rewrite the previous inequality in the form
Ω pt+s (x, y)u(x)v(y) dμ(x) dμ(y)
U ×V
Φ(x, y)u(x)v(y) dμ(x) dμ(y), (5.11) U ×V
where Φ(x, y) =
ptU (x, z)psV (z, y) dμ(z) + 1 − PtU 1U (x) F (y) + 1 − PsV 1V (y) G(x).
U ∩V
Obviously, Φ(x, y) is a bounded measurable function on U × V . By [6, Lemma 3.4], the inequality (5.11) implies Ω (x, y) Φ(x, y) pt+s
for almost all x ∈ U and y ∈ V , which proves (5.4). In the case of a local form (E, F ), one obtains the claim for arbitrary open sets U, V by passing to the limit when exhausting U and V by precompact open sets. 2 Remark 5.2. If U ⊂ V , it follows that V ptU (x, z)psV (z, y) dμ(z) ptV (x, z)psV (z, y) dμ(z) = ps+t (x, y). M
Therefore, we obtain from (5.4) that
M
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Ω V pt+s (x, y) ps+t (x, y) + 1 − PtU 1U (x)
+ 1 − PsV 1V (y)
sup ptΩ (·, y) L∞ (U
s
sup ptΩ (·, x) L∞ (V
t
ρ \K2 )
ρ \K1 )
(5.12)
.
On the other hand, if U ∩ V = ∅, then using the fact that psV (z, y) = 0 for μ-almost all z ∈ U , we obtain that U V pt (x, z)ps (z, y) dμ(z) = ptU (x, z)psV (z, y) dμ(z) = 0, M
U
so that (5.4) becomes
Ω (x, y) 1 − PtU 1U (x) pt+s
sup ptΩ (·, y) L∞ (U
s
+ 1 − PsV 1V (y)
ρ \K1 )
sup ptΩ (·, x) L∞ (V
t
ρ \K2 )
.
(5.13)
6. Pointwise off-diagonal estimates of heat kernels In this section we introduce a technique for self-improvement of pointwise upper estimates of the heat kernel of a local, conservative, regular Dirichlet form. This issue was addressed in [10,11,5,6] on abstract metric measure spaces, and in [1,2,9] on some fractal sets. Motivated by the application of symmetric comparison inequalities for the heat kernels in [8], we here present an alternative approach to such results, which is based on Theorem 5.1. Let {Pt }t0 , {PtΩ }t0 be the semigroups of the Dirichlet forms (E, F ), (E, F (Ω)) respectively as before. For any x ∈ M and r > 0, define the metric ball
B(x, r) = y ∈ M: d(x, y) < r . For any ball B = B(x, r) and any positive constant λ, denote by λB the ball B(x, λr). Recall that a Dirichlet form (E, F ) in L2 (M, μ) is called conservative if the heat semigroup {Pt }t0 of (E, F ) satisfies the following property: Pt 1 = 1 in M for any t > 0. Lemma 6.1. Assume that (E, F ) is a conservative, regular Dirichlet form in L2 (M, μ), and let {Pt }t0 be the heat semigroup of (E, F ). Assume that φ(r, t) is a non-negative function on (0, ∞) × (0, ∞) such that φ(r, ·) is increasing in (0, ∞) for every r > 0. If, for any t > 0 and any ball B in M of radius r, Pt 1B c φ(r, t)
1 in B, 4
(6.1)
then
1 − PtB 1B
r 2φ , t 4
1 in B. 4
(6.2)
A. Grigor’yan et al. / Journal of Functional Analysis 259 (2010) 2613–2641
2635
Fig. 6. Illustration to the proof of Lemma 6.1.
Remark 6.2. A version of this statement appeared in [1, proof of Lemma 3.9] where a probabilistic proof was given. We follow the argument of [5, Theorem 3.1], [6, Theorem 5.13] where this statement was proved with some additional restrictions. Proof of Lemma 6.1. Applying the estimate (4.2) with Ω = M, U = B, K = 34 B and f = 1 1 B , 2 we obtain that, for any t > 0 and almost everywhere in M, PtB 1 1 B Pt 1 1 B − sup Ps 1 1 B L∞ (( 3 B)c ) . 2
2
0<st
2
4
(6.3)
For any x ∈ 14 B, we have that B(x, r/4) ⊂ 12 B (see Fig. 6). Using the identity Pt 1 = 1, we have that, for any x ∈ 14 B, Pt 1 1 B = 1 − Pt 1( 1 B)c 1 − Pt 1B(x,r/4)c . 2
2
Applying (6.1) for the ball B(x, r/4), we see that Pt 1B(x,r/4)c φ(r/4, t) It follows that, for any x ∈ 14 B,
in B(x, r/16).
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Pt 1 1 B 1 − φ(r/4, t)
in B(x, r/16).
2
Covering 14 B by a countable family of balls B(xk , r/16) where xk ∈ 14 B, we obtain that 1 in B. 4
Pt 1 1 B 1 − φ(r/4, t) 2
(6.4)
On the other hand, for any y ∈ ( 34 B)c , we have that 12 B ⊂ B(y, r/4)c , and so Ps 1 1 B Ps 1B(y,r/4)c . 2
Applying (6.1) for the ball B(y, r/4) at time s and using the monotonicity of φ(r, s) in s, we obtain that, for any 0 < s t, Ps 1B(y,r/4)c φ(r/4, s) φ(r/4, t)
in B(y, r/16).
It follows that, for any y ∈ ( 34 B)c and any 0 < s t, Ps 1 1 B φ(r/4, t) 2
in B(y, r/16),
which implies that Ps 1 1 B φ(r/4, t) 2
in
3 B 4
c .
(6.5)
1 in B, 4
(6.6)
Combining (6.3), (6.4) and (6.5), we obtain that, for any t > 0, PtB 1B PtB 1 1 B 1 − 2φ(r/4, t) 2
which was to be proved.
2
In the next statement, we use a function F : M × M × (0, ∞) → (0, ∞) with the following properties: (F1) F (x, y, s) = F (y, x, s) for all x, y ∈ M and s > 0; (F2) F (x, y, s) is decreasing in s for any x, y ∈ M; (F3) there exist α, C > 0 such that d(x, z) α F (z, y, s) C 1+ F (x, y, s) s
(6.7)
for all x, y, z ∈ M and s > 0. Theorem 6.3. Let (E, F ) be a conservative, local, regular Dirichlet form in L2 (M, μ). Let h be a positive increasing function on (0, +∞). Assume in addition that the following two conditions hold:
A. Grigor’yan et al. / Journal of Functional Analysis 259 (2010) 2613–2641
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(1) The heat kernel pt of (E, F ) exists and satisfies the inequality pt (x, y) F x, y, h(t) ,
(6.8)
for all t > 0, μ-almost all x, y ∈ M, where F is a function that satisfies the conditions (F1)–(F3) above. (2) There exist ε ∈ (0, 12 ) and δ > 0 such that, for any ball B of radius r > 0 and for any t > 0, we have 1 in B 4
Pt 1B c ε
(6.9)
whenever h(t) δr. Then, for all λ, t > 0 and μ-almost all x, y ∈ M, t cr pt (x, y) CF x, y, h exp −c tΨ 2 t
(6.10)
where r = d(x, y), the constant C > 0, and Ψ is defined by
s Ψ (s) = sup −λ . λ>0 h(1/λ)
(6.11)
Proof. Fix t > 0, two distinct points x0 , y0 ∈ M and set r = 12 d(x0 , y0 ). Applying (5.13) with U = B(x0 , r), V = B(y0 , r), Ω = M and ρ = 0, we obtain that, for μ-almost all x ∈ B(x0 , r) and y ∈ B(y0 , r),
U pt (x, y) 1 − Pt/2 1U (x) sup
esup ps (z, y)
(6.12)
t/2<st z∈B(x0 ,r)
V + 1 − Pt/2 1V (y) sup
esup ps (z, x).
(6.13)
t/2<st z∈B(y0 ,r)
In what follows, we estimate the term on the right-hand side of (6.12), while the term in (6.13) can be treated similarly. We claim that, for all λ > 0, U 1 − Pt/2 1U
C exp c λt −
cr h(1/λ)
1 in U. 4
Indeed, we see from (6.9) that the hypothesis (6.1) of Lemma 6.1 is satisfied with φ(r, t) =
ε, 1,
if h(t) δr, otherwise.
Therefore, by Lemma 6.1, we obtain that, for all balls B of radius r, 1 − PtB 1B 2φ
r , t 2ε 4
1 in B, 4
(6.14)
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provided that h(t) δr/4. It follows from [5, Theorem 3.4] (see also [6, Theorem 5.7]) that, for any ball B of radius r and for any λ > 0, Pt 1B c
C exp c λt −
cr h(1/λ)
1 in B. 2
Using Lemma 6.1 again, this time with the function φ(r, t) = C exp c λt −
cr , h(1/λ)
we obtain 1 − PtB 1B
1 cr/4 in B, 2C exp c λt − h(1/λ) 4
which proves (6.14). On the other hand, for all z ∈ B(x0 , r) and x ∈ B(x0 , r), we have that z ∈ B(x, 2r), whence by condition (F3) α α F (z, y, h(t/2)) 2r r C 1+ 2α C 1 + . F (x, y, h(t/2)) h(t/2) h(t/2) Noting that h is increasing and F (x, y, ·) is decreasing, we have from (6.8) that, for all and for μ-almost all z ∈ B(x0 , r) and y ∈ B(y0 , r),
t 2
st
ps (z, y) F z, y, h(s) F z, y, h(t/2)
α F (z, y, h(t/2)) r α 2 CF x, y, h(t/2) 1 + = F x, y, h(t/2) . F (x, y, h(t/2)) h(t/2)
Therefore, we have, for almost all y ∈ B(y0 , r), sup
esup ps (z, y) CF x, y, h(t/2) 1 +
t/2<st z∈B(x0 ,r)
r h(t/2)
α .
(6.15)
Combining (6.14) and (6.15) and a similar estimate for the term in (6.13), we obtain from (6.12) and (6.13) that, for μ-almost all x ∈ B(x0 , 14 r), y ∈ B(y0 , 14 r), pt (x, y) CF x, y, h(t/2) 1 +
r h(t/2)
α
exp c λt −
cr . h(1/λ)
(6.16)
In order to absorb the middle term to the exponential on the right-hand side in (6.16), fix r, t and consider the function G(λ) :=
cr − c λt, h(1/λ)
where c, c are the same as in (6.16). Using this with λ = 2/t and the elementary inequality
A. Grigor’yan et al. / Journal of Functional Analysis 259 (2010) 2613–2641
c α log(1 + s) s + c , 2
2639
s 0,
where c is as above and c = c (c, α) is large enough, we obtain that α log 1 +
r h(t/2)
1 cr + c 2 h(t/2)
1 1 = G(2/t) + c + c sup G(λ) + c + c . 2 2 λ>0 Therefore, 1+
r h(t/2)
α
1 exp − sup G(λ) exp − sup G(λ) + c + c 2 λ>0 λ>0 1 C exp − sup G(λ) 2 λ>0 1 C exp − G(λ) . 2
Therefore, we obtain from (6.16) that, for any λ > 0 and μ-almost all x ∈ B(x0 , 14 r), y ∈ B(y0 , 14 r), 1 pt (x, y) CF x, y, h(t/2) exp − G(λ) . 2
(6.17)
Since M × M \ diag can be covered by a countable family of sets B(x0 , 14 r) × B(y0 , 14 r) as above, it follows that (6.17) holds for μ-almost all x, y ∈ M. Taking sup in λ > 0, we obtain (6.10). 2 Let us give an example to illustrate Theorem 6.3. Set V (x, r) := μ B(x, r) and assume in the sequel that the following volume doubling condition (VD) is satisfied: there is a constant CD 1 such that V (x, 2r) CD V (x, r)
(6.18)
for all x ∈ M and r > 0. It is known that (VD) implies the existence of a constant α > 0 such that V (x, R) d(x, y) + R α CD V (y, r) r for all x, y ∈ M and 0 < r R (see, for example, [6]). Define functions h and F as follows: h(t) = t 1/β
(6.19)
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and F (x, y, s) = √
C , V (x, h(s))V (y, h(s))
for all t, s > 0 and x, y ∈ M, where β > 1 is some constant. It follows from (6.19) that F satisfies β
conditions (F1)–(F3). It is easy to see that the supremum in (6.11) is attained at λ = cs β−1 so that β
Ψ (s) = cs β−1 . The estimate (6.10) becomes β d(x, y) β−1 , pt (x, y) exp −c t 1/β V (x, t 1/β )V (y, t 1/β ) C
for all t > 0 and almost all x, y ∈ M. Using (6.19) again and applying the same argument as in the proof of Theorem 6.3, we obtain that β d(x, y) β−1 C pt (x, y) exp −c . V (x, t 1/β ) t 1/β
(6.20)
In particular, if V (x, r) r α for some α > 0, then (6.20) becomes pt (x, y)
C t α/β
β d(x, y) β−1 . exp −c t 1/β
(6.21)
Remark 6.4. The estimate of type (6.21) was obtained in [3] for the Sierpinski gasket, and in [2] for the Sierpinski carpet, and in [9] for a certain class of post-critically finite self-similar sets. The estimate (6.20) with β = 2 was obtained by Li and Yau [13] for Riemannian manifolds of non-negative curvature, and with any β > 1 by Kigami [12] for some general class of self-similar sets. Appendix A Proposition A.1. Let F (x, y) be a non-negative μ-measurable function of x, y ∈ M. Then the function f (x) = esup F (x, y) y
is measurable. Proof. Fix a pointwise realization of F . Assume first that F is bounded. For any x ∈ M, consider the mapping
A. Grigor’yan et al. / Journal of Functional Analysis 259 (2010) 2613–2641
2641
L ϕ → T ϕ(x) := 1
F (x, y)ϕ(y) dμ(y) M
which is a bounded linear functional on L1 . We have f (x) = sup T ϕ(x). ϕ1 1
Since T is continuous in ϕ, the supremum can be replaced by the one over a dense subset S ⊂ L1 , that is, f (x) =
sup
ϕ1 1, ϕ∈S
T ϕ(x).
Since T ϕ is a measurable function, the supremum over a countable family is also measurable, and hence, the function f is measurable. For an arbitrary F , consider Fk = F ∧ k, we have from above that fk (x) := esupy Fk (x, y) is measurable. Note that the sequence {fk }∞ k=1 increases and converges to f pointwise as k → ∞. Hence, the function f is measurable. 2 References [1] M.T. Barlow, Diffusions on Fractals, Lecture Notes in Math., vol. 1690, Springer, 1998, pp. 1–121. [2] M.T. Barlow, R.F. Bass, Brownian motion and harmonic analysis on Sierpínski carpets, Canad. J. Math. (4) 51 (1999) 673–744. [3] M.T. Barlow, E.A. Perkins, Brownian motion on the Sierpínski gasket, Probab. Theory Related Fields 79 (1988) 543–623. [4] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Stud. Math., De Gruyter, 1994. [5] A. Grigor’yan, J. Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math. 174 (2008) 81–126. [6] A. Grigor’yan, J. Hu, Upper bounds of heat kernels on doubling spaces, preprint, 2008. [7] A. Grigor’yan, J. Hu, K.-S. Lau, Heat kernels on metric spaces with doubling measure, in: Proceedings of Conference on Fractal Geometry in Greifswald IV, Birkhäuser, 2009, pp. 3–44. [8] A. Grigor’yan, L. Saloff-Coste, Heat kernel on manifolds with ends, Ann. Inst. Fourier (Grenoble) 59 (2009). [9] B.M. Hambly, T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. Lond. Math. Soc. (3) 79 (1999) 431–458. [10] W. Hebisch, L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001) 1437–1481. [11] J. Kigami, Local Nash inequality and inhomogeneous of heat kernels, Proc. Lond. Math. Soc. 89 (2004) 525–544. [12] J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 199 (932) (2009). [13] P. Li, S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (3–4) (1986) 153–201.
Journal of Functional Analysis 259 (2010) 2642–2672 www.elsevier.com/locate/jfa
On the Dirichlet semigroup for Ornstein–Uhlenbeck operators in subsets of Hilbert spaces Giuseppe Da Prato a , Alessandra Lunardi b,∗ a Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy b Dipartimento di Matematica, Università di Parma, Viale G.P. Usberti, 53/A, 43124 Parma, Italy
Received 15 April 2010; accepted 18 April 2010
Communicated by Paul Malliavin
Abstract We consider a family of self-adjoint Ornstein–Uhlenbeck operators Lα in an infinite dimensional Hilbert space H having the same gaussian invariant measure μ for all α ∈ [0, 1]. We study the Dirichlet problem for the equation λϕ − Lα ϕ = f in a closed set K, with f ∈ L2 (K, μ). We first prove that the variational solution, trivially provided by the Lax–Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution ϕ (which is by definition in a Sobolev space Wα1,2 (K, μ)) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior Wα2,2 regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to Wα1,2 (H, μ). In the second case we exploit the Malliavin’s theory of surface integrals which is recalled in Appendix A of the paper, then we are able to give a meaning to the trace of ϕ at ∂K and to show that it vanishes, as it is natural. © 2010 Elsevier Inc. All rights reserved. Keywords: Ornstein–Uhlenbeck operators; Invariant measures; Dirichlet problems
* Corresponding author.
E-mail addresses:
[email protected] (G. Da Prato),
[email protected] (A. Lunardi). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.04.016
G. Da Prato, A. Lunardi / Journal of Functional Analysis 259 (2010) 2642–2672
2643
1. Introduction and setting of the problem In this paper we present some results on second order elliptic and parabolic equations with Dirichlet boundary conditions in a closed set of a separable real Hilbert space H (norm | · |, inner product ·,·). A motivation for the study of Dirichlet problems in proper subsets of H is to provide a natural development of the potential theory in infinite dimensions started in [9]. Only a few results seem to be available in this field, see e.g. [5] and the references therein. The finite dimensional theory in spaces of continuous functions is hardly extendable to the infinite dimensional setting. While in finite dimensions smooth boundaries consist only of regular points in the sense of Wiener, in infinite dimensions this is not true: for instance, certain hyperplanes and the boundary of the unit ball contain dense subsets of irregular points for suitable Ornstein–Uhlenbeck operators [4]. This leads to the lack of regularity results up to the boundary. Here we avoid a part of such difficulties working in suitable L2 spaces. To begin with, we consider a class of Ornstein–Uhlenbeck operators of the type Lα ϕ(x) =
1 1 1−α 2 Tr Q D ϕ(x) − x, Q−α Dϕ(x) , 2 2
(1.1)
where Q ∈ L(H ) is a symmetric positive operator with finite trace, and 0 α 1. The most popular among such operators are L0 and L1 : L0 ϕ(x) =
1 1 Tr QD 2 ϕ(x) − x, Dϕ(x) , 2 2
is the operator that arises in the Malliavin calculus (e.g., [14]), while L1 ϕ(x) =
1 1 2 Tr D ϕ(x) − x, ADϕ(x) 2 2
(with A = Q−1 ) is the generator of the Ornstein–Uhlenbeck semigroup with the best smoothing properties. See e.g. [5]. The operators Lα exhibit an important common feature: the associated Ornstein–Uhlenbeck semigroups Tα (t) in Cb (H ) have the same invariant measure μ = NQ , the Gaussian measure of mean 0 and covariance Q. In this paper we shall consider realizations of the operators Lα in the ˚ space L2 (K, μ), where K is a closed set in H with nonempty interior part K. A unique weak solution to the Dirichlet problem
λϕ(x) − Lα ϕ(x) = f (x), ϕ(x) = 0,
in K, on ∂K
(1.2)
with λ > 0 and f ∈ L2 (K, μ) is easily obtained via the Lax–Milgram Theorem, applied in a Hilbert space W˚ α1,2 (K, μ) “naturally” associated to Lα (see next section). This allows to define a dissipative self-adjoint operator Mα in L2 (K, μ) such that ϕ = R(λ, Mα )f . As all dissipative self-adjoint operators in Hilbert spaces, Mα is the infinitesimal generator of an analytic contraction semigroup. We give an explicit expression of the semigroup generated by Mα . Precisely, we identify it with the natural extension to L2 (K, μ) of the so-called stopped semigroup TαK (t). In analogy
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with the finite dimensional case (e.g., [8]), it is defined in Bb (K) (the space of the bounded and Borel measurable functions defined in K) by TαK (t)ϕ(x) = E ϕ Xα (t, x) 1τx t = ϕ Xα (t, x) dP,
∀x ∈ K,
(1.3)
{τx t}
where τx is the entrance time in the complement of K,
τx := inf t 0: Xα (t, x) ∈ K c ,
∀x ∈ K,
(1.4)
and Xα (t, x) is the solution to 1 dXα (t, x) = − Aα Xα (t, x) dt + A(α−1)/2 dW (t), 2
X(0, x) = x.
(1.5)
Here W (t) is a standard cylindrical Wiener process in H , defined in a filtered probability space (Ω, F , (Ft )t0 , P). ˚ τ˜x := The definition of TαK (t) is similar to the one in [15], where the exit time from K, c ˚ inf{t 0: Xα (t, x) ∈ K } was used instead of our τx . In finite dimensions, if K is the closure of a bounded open set with smooth boundary the two definitions are equivalent, and TαK (t) is the semigroup associated to the realization of Lα with Dirichlet boundary condition [8, §6.5]. Therefore, a lot of regularity results, both interior and up to the boundary, are well known. In infinite dimensions, interior regularity results were given in [15] for α > 0 that extended to infinite dimensions a result of [7]. We do not know regularity results up to the boundary, even in the case of very smooth bounded sets such as balls. Here we prove that μ is a sub-invariant measure for TαK (t). Therefore, TαK (t) has a natural extension (still called TαK (t)) to a contraction semigroup in L2 (K, μ). The domain of its generator LK α consists of the range of the resolvent operator, R λ, LK α f =
∞
e−λt TαK (t)f dt,
f ∈ L2 (K, μ),
(1.6)
0
which is well defined for λ > 0 since TαK (t) is a contraction semigroup. We prove that for each λ > 0 and f ∈ L2 (K, μ), the function ϕ := R(λ, LK α )f belongs to the above mentioned space W˚ α1,2 (K, μ), and satisfies the weak formulation of (1.2). Therefore, LK α = Mα . Our main tool in the proof is the approximating Feynman–Kac semigroup 1 t Pαε (t)ϕ(x) = E ϕ Xα (t, x) e− ε 0 V (Xα (s,x)) ds ,
(1.7)
where V is a (fixed) bounded continuous function that vanishes in K and has positive values in K c . Its infinitesimal generator in L2 (H, μ) is the operator Mαε : D(Mαε ) = D(Lα ) → L2 (H, μ), Mαε ϕ = Lα ϕε − 1ε V ϕ, and we prove that for each ϕ ∈ L2 (K, μ), t > 0, λ > 0 we have
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TαK (t)ϕ = lim Pαε (t)ϕ˜ |K , ε→0
2645
ε R λ, LK α ϕ = lim R λ, Mα ϕ˜ |K ε→0
in L2 (K, μ), where ϕ˜ is the null extension of ϕ to the whole H . Problem (1.2) is of interest for λ = 0 too. Using the fact that D(LK α ) is compactly embedded in L2 (K, μ), in Section 3.3 we prove that 0 ∈ ρ(LK ) and that a Poincaré estimate holds in α W˚ α1,2 (K, μ), for α ∈ (0, 1]. Therefore, the supremum of σ (LK ) is negative. α These results are proved without additional assumptions on K. In particular, we do not require that K is bounded, or that its boundary is smooth. If the boundary of K is suitably smooth, it is possible to define surface integrals and traces at the boundary of functions in the Sobolev spaces Wα1,2 (K, μ). Then we prove that the traces of the functions in W˚ α1,2 (K, μ) vanish. Therefore, the Dirichlet boundary condition in (1.2) is satisfied in the sense of the trace, and TαK (t)ϕ has null trace at the boundary for every t > 0 and ϕ ∈ L2 (K, μ). Surface integrals for gaussian measures in Hilbert spaces are not a straightforward extension of the finite dimensional theory. To our knowledge the best reference is [2, §6.10], where the Malliavin theory is presented. It deals with level surfaces of smooth functions g in a more general context than ours, since Souslin spaces X are considered instead of Hilbert spaces. A part of the theory may be simplified in our Hilbert setting, and moreover some of the smoothness assumptions on g can be weakened. Therefore, we end the paper with Appendix A describing surface measures for level surfaces of suitably regular functions g : H → R. Several related important problems remain open, even for bounded K with smooth boundary. Among them: (a) While in finite dimensions ϕ = R(λ, LK α )f is a strong solution to (1.2) and it belongs to W 2,2 (K, μ) under reasonable assumptions on the boundary ∂K [13], in infinite dimensions we do not know whether ϕ possesses second order derivatives in L2 (K, μ), even if K is the closed unit ball. In fact, even in the case α = 1, the estimates found in [4,15] are very bad both near the boundary and near t = 0, and it is not clear how to use them to get informations on the resolvent. (b) We do not know whether TαK (t) is strong Feller in K (i.e., it maps Bb (K), the space of the bounded Borel functions in K, to Cb (K)). This problem is open even for K = {x ∈ H : |x| 1}. (c) In finite dimensions, if ∂K is regular enough there are several characterizations of the space W˚ α1,2 (K, μ), that coincides with W˚ 11,2 (K, μ) for every α ∈ [0, 1]. The most obvious is the following: since μ is locally equivalent to the Lebesgue measure, W˚ 11,2 (K, μ) coincides with the space of the functions f ∈ W11,2 (K, μ) whose trace at the boundary vanishes. We do not know whether a similar characterization holds in infinite dimensions. Referring to problem (a), in the recent paper [1] a self-adjoint realization L of L1 in L2 (K, μ) with Neumann boundary condition has been studied, in the case that K is a convex set with regular boundary. By means of a different (and better) approximation procedure, it has been proved that the resolvent R(λ, L) maps L2 (K, μ) into W12,2 (K, μ). Here we prove interior Wα2,2 regularity, for those α such that Tr[Q1−α ] < ∞. In this case we show that for every ball B ⊂ K with positive distance from ∂K and for every ϕ ∈ D(LK α ), the restriction ϕ|B belongs to Wα2,2 (B, μ).
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2. Notation and preliminaries We denote by ·,· and by | · | the scalar product and the norm in H . L(H ) is the space of the linear bounded operators in H . Let Q be a symmetric (strictly) positive operator in L(H ) with finite trace, and let A := Q−1 . Accordingly, let {ek } be an orthonormal basis in H consisting of eigenfunctions of Q, i.e. Qek = λk ek ,
Aek =
1 ek , λk
∀k ∈ N.
We denote by Dk the derivative in the direction of ek and by D the gradient of any differentiable function. Moreover we set xk = x, ek for all x ∈ H , k ∈ N. Throughout the paper we consider the σ -algebra B(H ) of the Borel subsets of H and the Gaussian measure with center 0 and covariance Q in B(H ), denoting it by μ. An orthonormal basis of L2 (H, μ) consists of the Hermite polynomials. More precisely, for each n ∈ N ∪ {0} let Hn (ξ ) := (−1)n n!−1/2 eξ
2 /2
2 D n e−ξ /2 ,
ξ ∈ R,
be the usual
normalized n-th Hermite polynomial. We denote by Γ the set of all γ : N → N ∪ {0} such that ∞ k=1 γ (k) < ∞. For each γ ∈ Γ let ∞
Hγ (x) :=
Hγ (k)
k=1
xk , √ λk
x ∈ H,
be the corresponding Hermite polynomial in H . Then, the linear span H of all the Hermite polynomials Hγ is dense in L2 (H, μ), and the linear span Λ0 of the functions Hγ ⊗ eh , with γ ∈ Γ 2 and h ∈ N, is dense in the space H ) of all the (equivalence classes of) measurable L (H, μ; functions F : H → H such that H |F (x)|2 μ(dx) < ∞. Other important dense subspaces of L2 (H, μ) are the spaces Eα (H ), the linear spans of the real and imaginary parts of the functions x → eix,h , with h ∈ D(Aα ), 0 α 1. 2.1. Sobolev spaces over H We have the following integration formula, Dk ϕ dμ = H
1 λk
xk ϕ dμ,
ϕ ∈ Eα (H ), k ∈ N.
(2.1)
H
It may be extended to
Dϕ, G dμ + H
ϕ div G dμ =
H
H
ϕ x, AG(x) dμ,
ϕ ∈ Cb1 (H ), G ∈ Λ0 ,
(2.2)
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(1−α)/2 D is well defined from where div G(x) = ∞ k=1 DG(x), ek . The linear operator Q 2 2 Eα (H ) ⊂ L (H, μ) to L (H, μ; H ), by Q(1−α)/2 Dϕ =
∞
(1−α)/2
λk
Dk ϕ ek .
k=1
Using formula (2.2) it is easy to see that Q(1−α)/2 D is closable. We still denote by Q(1−α)/2 D its closure, and by Wα1,2 (H, μ) the domain of the closure. (Note that for α = 0, Q1/2 D is nothing but the Malliavin derivative.) Wα1,2 (H, μ) is endowed with the inner product ϕ, ψW 1,2 (H,μ) =
ϕψ dμ +
α
H
Q(1−α)/2 Dϕ, Q(1−α)/2 Dϕ dμ
H
=
ϕψ dμ +
∞
λk1−α Dk ϕDk ψ dμ.
(2.3)
k=1 H
H
So, Wα1,2 (H, μ) is the completion of Eα (H ) in the norm associated to the scalar product (2.3). It is also possible to characterize it through the Hermite polynomials. We have ϕ ∈ Wα1,2 (H, μ) iff ∞
2 γh λ−α h ϕγ < ∞
γ ∈Γ h=1
in which case the above sum is equal to H |Q(1−α)/2 Dϕ|2 dμ. Indeed, the proof in [5, Sect. 9.2.3] for α = 1 works as well for any α ∈ [0, 1). From this characterization it is clear that Wα1,2 (H, μ) ⊂ W01,2 (H, μ) for every α ∈ (0, 1], with continuous embedding. Similarly, Wα2,2 (H, μ) is the completion of Eα (H ) in the norm associated to the scalar product ϕ, ψW 2,2 (H,μ) = ϕ, ψW 1,2 (H,μ) + α
α
Tr Q2−2α D 2 ϕD 2 ψ dμ
H
= ϕ, ψW 1,2 (H,μ) +
∞
α
1−α λ1−α h λk Dh,k ϕDh,k ψ dμ.
h,k=1 H
Next lemma is a consequence of [2, Lemma 5.1.12] or [5, Lemma 9.2.7]. Lemma 2.1. There is C > 0 such that |x|2 ϕ(x)2 dμ Cϕ2
W01,2 (H,μ)
H
,
ϕ ∈ W01,2 (H, μ).
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Lemma 2.1, together with (2.1), yields the integration by parts formula in W01,2 (H, μ) (and hence, in all spaces Wα1,2 (H, μ)),
Dk ϕψ dμ = − H
ϕDk ψ dμ + H
1 λk
xk ϕψ dμ,
ϕ, ψ ∈ W01,2 (H, μ), k ∈ N.
(2.4)
H
For 0 α 1 let Tα (t) be the Ornstein–Uhlenbeck semigroup Tα (t)ϕ(x) := ϕ(y)Ne−tAα /2 x,Qt (dy), t > 0,
(2.5)
H
with t Qt :=
α α e−sA Q1−α ds = Q I − e−tA .
0
Tα (t) is a Markov semigroup in Cb (H ), whose unique invariant measure is μ. Its extension to L2 (H, μ) is a strongly continuous contraction semigroup, still denoted by Tα (t), whose infinitesimal generator Lα is the closure of Lα : Eα (H ) → L2 (H, μ). The domain of Lα is continuously embedded in Wα2,2 (H, μ). Moreover, for any ϕ, ψ ∈ D(Lα ) we have 1 (1−α)/2 Q Lα ϕψ dμ = − Dϕ, Q(1−α)/2 Dψ dμ. (2.6) 2 H
H
We refer to [5, Ch. 9, 10] for the proofs of the above statements, and we add further properties of the spaces Wα1,2 (H, μ) that will be used later. For each ϕ ∈ L1 (H, μ) we denote by ϕ the mean value of ϕ, ϕ := ϕ dμ. H
Proposition 2.2. Let 0 α 1. Then (a) A Poincaré estimate holds in Wα1,2 (H, μ), and precisely
(ϕ − ϕ)2 dμ λα1
H
2 (1−α)/2 Q Dϕ dμ,
(2.7)
H
where λ1 is the maximum eigenvalue of Q. (b) The space Wα1,2 (H, μ) is compactly embedded in L2 (H, μ) for α > 0. Proof. A proof of statement (a) that follows the approach of Deuschel and Strook [6] is in [5, Ch. 10] for α = 1. The same procedure works for α ∈ [0, 1), since the key points of the proof still hold. Precisely, we have
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(i) |Q(1−α)/2 DT α (t)ϕ|2 e−t/λ1 T α (t)(|Q(1−α)/2 Dϕ|2 ), ϕ ∈ Cb1 (H ), t > 0; (ii) H ϕLα ϕ dμ = − 12 H |Q(1−α)/2 Dϕ|2 dμ, ϕ ∈ D(Lα ); (iii) limt→∞ T α (t)ϕ(x) = ϕ, ϕ ∈ Eα (H ), x ∈ H . α
Once (i), (ii), (iii) are satisfied one can follow the proof of [5, Prop. 10.5.2] step by step. (ii) and (iii) follow from [5, Prop. 10.2.3, Prop. 10.1.1]. To check that (i) holds is easy and it is left to the reader. Statement (b) should be well known, however we give here a simple proof following [3, Thm. ϕ of L2 (H, μ) as
10.16] that concerns the case α = 1. We write every element 1,2 ϕ = γ ∈Γ ϕγ Hγ , with ϕγ = ϕ, Hγ . We already remarked that ϕ ∈ Wα (H, μ) iff ∞
2 γh λ−α h ϕγ < ∞.
γ ∈Γ h=1
If a sequence (ϕ (n) ) is bounded in Wα1,2 (H, μ), say ϕ (n) W 1,2 (H,μ) K for each n ∈ N, a subseα
quence (ϕ (nk ) ) converges weakly in Wα1,2 (H, μ) to a limit ϕ, that still satisfies ϕW 1,2 (H,μ) K. We shall show that limk→∞ ϕ (nk ) − ϕL2 (H,μ) = 0.
−α For each N ∈ N, let ΓN = {γ ∈ Γ : ∞ h=1 γh λh < ∞}. Then
ϕ (nk ) − ϕ
2
dμ =
2 2 ϕγ(nk ) − ϕγ + ϕγ(nk ) − ϕγ γ ∈ΓN
H
γ ∈ΓNc
∞ 2 (n ) 2 1 k −ϕ ϕγ(nk ) − ϕγ + γh λ−α γ h ϕγ N
γ ∈ΓN
α
γ ∈Γ h=1
2 (2K)2 . ϕγ(nk ) − ϕγ + N
γ ∈ΓN
For ε > 0 fix N ∈ N such that 4K 2 /N ε. Since α > 0, then limh→∞ λ−α h = +∞, so that the (n ) k set ΓN has a finite number of elements. Since ϕ converges weakly to ϕ in Wα1,2 (H, μ), it con(n ) 2 verges weakly to ϕ in L (H, μ); in particular limh→∞ ϕγ k = ϕγ for each γ ∈ ΓN . Therefore,
(nk ) for k large enough we have γ ∈ΓN (ϕγ − ϕγ )2 ε, and the statement follows. 2 2.2. Sobolev spaces over K Throughout the paper we assume that K ⊂ H is a closed set with positive measure. To avoid trivialities, we assume that also K c has positive measure. To treat the Dirichlet problem (1.2) we introduce Sobolev spaces over K. We denote by Wα1,2 (K, μ) the space of the functions u : K → R that have an extension belonging to Wα1,2 (H, μ), endowed with the standard inf norm. Moreover we denote by W˚ α1,2 (K, μ) the sub-
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space of Wα1,2 (K, μ) consisting of the functions u : K → R whose null extension to the whole H belongs to the Sobolev space Wα1,2 (H, μ). Therefore, 2 = u2 dμ + Q(1−α)/2 Du dμ, u ∈ W˚ α1,2 (K, μ), u2 1,2 Wα (K,μ)
K
K
so that the Wα1,2 (K, μ)-norm in W˚ α1,2 (K, μ) is associated to the inner product u, vW 1,2 (K,μ) =
uv dμ +
α
K
Q(1−α)/2 Du, Q(1−α)/2 Dv dμ.
(2.8)
K
From the results of the next section it will be clear that such a space is not trivial, since it coincides 1/2 . Moreover, since W 1,2 (H, μ) is continuously embedded in with the domain of (I − LK α) α 1,2 1,2 ˚ W0 (H, μ), then Wα (K, μ) is continuously embedded in W˚ 01,2 (K, μ), for every α ∈ (0, 1]. 2.3. The weak solution to (1.2) The quadratic form Qα associated to Lα , 1 (1−α)/2 Q Qα (u, v) := Du, Q(1−α)/2 Dv dμ, 2
u, v ∈ W˚ α1,2 (K, μ),
(2.9)
K
is continuous, nonnegative, and symmetric. Therefore, for every λ > 0 and f ∈ L2 (K, μ) there exists a unique ϕ ∈ W˚ α1,2 (K, μ) such that ϕv dμ +
λ K
1 2
Q(1−α)/2 Dϕ, Q(1−α)/2 Dv dμ =
K
f v dμ,
∀v ∈ W˚ α1,2 (K, μ). (2.10)
K
The function ϕ may be considered as a weak solution to (1.2). Moreover, there exists a dissipative self-adjoint operator Mα in L2 (K, μ) such that ϕ = R(λ, Mα )f . Like all dissipative self-adjoint operators in Hilbert spaces, Mα is the infinitesimal generator of an analytic contraction semigroup, and several properties of Mα follow. See e.g. [11, Ch. 6]. 3. The Dirichlet semigroup In this section we give an explicit representation formula for the semigroup generated by the operator Mα defined in Section 2.3, through the approximation procedure described in the Introduction. Moreover we show some properties of the semigroup and of its generator. 3.1. The approximating semigroups We fix once and for all a function V ∈ Cb (H ) such that V (x) = 0,
x ∈ K,
For ε > 0 let Pαε (t) be defined by (1.7).
V (x) > 0,
x ∈ Kc.
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Proposition 3.1. For any ϕ ∈ Cb (H ) we have
ε 2 Pα (t)ϕ(x) μ(dx)
H
ϕ 2 (x) μ(dx).
(3.1)
H
Consequently, Pαε (t) is uniquely extendable to a C0 -semigroup in L2 (H, μ) which we shall denote by the same symbol. Proof. We have in fact, by the Hölder inequality ε 2 2 t Pα (t)ϕ(x) E ϕ 2 Xα (t, x) e− ε 0 V (Xα (s,x)) ds Tα (t) ϕ 2 (x), where Tα (t) is the Ornstein–Uhlenbeck semigroup defined in (2.5). Since μ is invariant for Tα (t), then
ε 2 Pα (t)ϕ(x) μ(dx)
H
Tα (t) ϕ 2 (x) μ(dx) =
H
ϕ 2 (x) μ(dx).
2
(3.2)
H
We denote by Mαε the infinitesimal generator of Pαε (t) in L2 (H, μ) and we want to show that = Lα − 1ε V . To this aim, for λ > 0 and f ∈ L2 (H, μ) we consider the resolvent equation
Mαε
1 λϕε − Lα ϕε + V ϕε = f. ε
(3.3)
Proposition 3.2. Let λ > 0, ε > 0, and f ∈ L2 (H, μ). Then Eq. (3.3) has a unique solution ϕε ∈ D(Lα ), and the following estimates hold ϕε2 dμ H
H
1 λ2
f 2 dμ, H
(1−α)/2 2 2 Q Dϕε dμ λ Kc
ε V ϕε2 dμ λ
(3.4)
f 2 dμ,
(3.5)
H
f 2 dμ.
(3.6)
H
Proof. Fix λ > 0 and ε > 0. Since Lα is maximal dissipative and ϕ → 1ε V ϕ is bounded and monotone increasing in L2 (H, μ), it follows by standard arguments that the operator D(Lα ) → L2 (H, μ),
1 ϕ → Lα ϕ − V ϕ, ε
is maximal dissipative. So, Eq. (3.3) has a unique solution ϕε ∈ D(Lα ), that satisfies (3.4).
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Multiplying both sides of (3.3) by ϕε , integrating over H and taking into account (2.6) yields ϕε2 dμ +
λ H
The inequality λ
1 2
(1−α)/2 2 1 Q Dϕε dμ + ε
H
H
|ϕε |2 dμ 1 2
H
V ϕε2 dμ = Kc
(3.7)
f ϕε dμ. H
f ϕε dμ yields again (3.4). The inequality
(1−α)/2 2 Q Dϕε dμ
H
f ϕε dμ H
implies (3.5), using the Hölder inequality in the right-hand side and then (3.4). The inequality 1 ε
V ϕε2 dμ Kc
f ϕε dμ H
implies (3.6), using again the Hölder inequality in the right-hand side and then (3.4).
2
Proposition 3.3. Let Mαε be the infinitesimal generator of Pαε (t). Then D(Mαε ) = D(Lα ) and 1 Mαε ϕ = Lα ϕ − V ϕ, ε
∀ϕ ∈ D(Lα ).
(3.8)
Proof. Let us show that D(Lα ) ⊂ D(Mαε ), and that (3.8) holds. First, let ϕ ∈ D(Lα ) ∩ Cb (H ). For x ∈ H , h > 0 we have 1 h Phε ϕ(x) − ϕ(x) = Tα (h)ϕ(x) − ϕ(x) + E e− ε 0 V (Xα (r,x)) dr − 1 ϕ Xα (h, x) .
(3.9)
We recall that, since Aα is self-adjoint, Xα (·, x) possesses a.s. continuous paths [12,16]. Therefore the functions r → ϕ(Xα (r, x)) and r → V (Xα (r, x)) are continuous a.s. Dividing both sides of (3.9) by h and letting h → 0, we obtain limh→0 (Phε ϕ − ϕ)/ h = Lα ϕ − V ϕ/ε pointwise and (by dominated convergence) in L2 (H, μ), so that ϕ ∈ D(Mαε ) and (3.8) holds. Let now ϕ ∈ D(Lα ), and let (ϕn ) be a sequence of functions in Eα (H ) that converges to ϕ in D(Lα ). Then, ϕn → ϕ in L2 (H, μ), so that 1ε V ϕn → 1ε V ϕ in L2 (H, μ), moreover Lα ϕn → Lα ϕ in L2 (H, μ). It follows that Mαε ϕn → Mαε ϕ in L2 (H, μ), and since Mαε is closed, then ϕ ∈ D(Mαε ) and (3.8) holds. The other inclusion D(Mαε ) ⊂ D(Lα ) is immediate. Indeed, for any ϕ ∈ D(Mαε ) set f = λϕ − ε Mα ϕ, and let ϕε be the solution of (3.3). Then ϕε ∈ D(Lα ) ⊂ D(Mαε ), so that (λ − Mαε )−1 f = ϕε = ϕ which implies that ϕ ∈ D(Lα ). 2 Remark 3.4. From the very beginning, one would be tempted to replace the continuous function V by 1K c in the definition of Mε . But with this choice the proof of Proposition 3.3 does not work. Indeed, it is not obvious that (Phε ϕ − ϕ)/ h converges as h → 0 for any ϕ ∈ Cb (H ) ∩ D(Lα ), if x ∈ ∂K, because the function r → 1K c (Xα (r, x)) could be discontinuous
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at r = 0. If μ(∂K) = 0 this difficulty is not relevant, since we are interested in L2 convergence rather than in pointwise convergence. However, we prefer to make no further assumptions on ∂K in this first part of the paper. 3.2. Identification of TαK (t) Let TαK (t), Pαε (t) be defined by (1.3), (1.7) respectively. Proposition 3.5. For any ϕ ∈ Bb (H ), t > 0, and for any x ∈ K we have lim Pαε (t)ϕ(x) = TαK (t)ϕ|K (x).
ε→0
(3.10)
Moreover TαK (t) is a semigroup of linear bounded operators in Bb (K). Proof. Let t > 0, x ∈ K. Then
K
τx t = ω ∈ Ω: Xα (s, x) ∈ K, ∀s ∈ [0, t) and
K τx < t = ω ∈ Ω: ∃s0 ∈ (0, t): Xα (s0 , x) ∈ K c . Then we have
ϕ Xα (t, x) dP +
Pαε (t)ϕ(x) = {τxK t}
1 t ϕ Xα (t, x) e− ε 0 V (Xα (s,x)) ds dP.
{τxK
In view of the dominated convergence theorem, to prove the statement it is enough to show that 1
lim e− ε
t 0
V (Xα (s,x)) ds
ε→0
= 0,
(3.11)
for a.s. ω such that τxK (ω) < t. We already mentioned that Xα (·, x) possesses a.s. continuous paths. Let ω ∈ Ω be such that Xα (·, x)(ω) is continuous. If τxK (ω) < t, there exist s0 < t, δ > 0 (depending on ω) such that Xα (s, x) ∈ K c ,
∀s ∈ [s0 − δ, s0 + δ].
Since V is continuous and it has positive values in K c , then
c := inf V X(s, x) : s ∈ [s0 − δ, s0 + δ] > 0. It follows that 1
e− ε
t 0
V (Xα (s,x)) ds
2c
e− ε δ → 0,
So, (3.11) holds. The last statement is straightforward.
2
as ε → 0.
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In the next proposition we show that μ is sub-invariant for TαK (t). We use the following notation. For each ϕ ∈ Bb (K) we set ϕ(x) ˜ =
if x ∈ K, if x ∈ / K.
ϕ(x), 0,
Proposition 3.6. For any ϕ ∈ Bb (K), t > 0, we have
K 2 Tα (t)ϕ(x) μ(dx)
K
ϕ 2 (x) μ(dx).
(3.12)
K
Consequently, TαK (t) can be uniquely extended to a C0 semigroup of contractions in L2 (K, μ). Proof. By the Hölder inequality we have for all x ∈ K K 2 Tα (t)ϕ(x) E ϕ 2 Xα (t, x) 1τxK t E ϕ˜ 2 Xα (t, x) 1τxK t Tα (t) ϕ˜ 2 (x). Since μ is invariant for Tα (t), it follows that
K 2 Tα (t)ϕ(x) dμ
K
Tα (t) ϕ˜ 2 (x) dμ
K
H
The conclusion follows.
Tα (t) ϕ˜ 2 dμ
Tα (t) ϕ˜ 2 dμ
H
ϕ˜ dμ = 2
H
ϕ(x)2 dμ. K
2
K 2 We shall denote by LK α the infinitesimal generator of Tα (t) in L (K, μ).
Proposition 3.7. For any f ∈ L2 (K, μ) and t > 0 we have lim Pαε (t)f˜ |K = TαK (t)f,
→0
in L2 (K, μ)
(3.13)
and, for λ > 0, −1 lim R λ, Mαε f˜ |K = λ − LK f, α
→0
in L2 (K, μ).
(3.14)
Proof. Let f ∈ Cb (H ). By Proposition 3.5, Pαε f converges pointwise to TαK (t)f in K. Moreover, |(Pαε (t)f )(x)| f ∞ , |(TαK (t)f )(x)| f ∞ for each x ∈ K and t > 0. By dominated convergence, limε→0 Pαε (t)f − TαK (t)f L2 (K,μ) = 0. Let now f ∈ L2 (K, μ). Since Cb (H ) is dense in L2 (H, μ), there is a sequence (fn ) ⊂ Cb (H ) such that 1 f˜ − fn L2 (H,μ) , n
∀n ∈ N.
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Then we have K T (t)f − P ε (t)f˜ 2 TαK (t)(f − fn )L2 (K,μ) α α L (K,μ) + TαK (t)fn − Pαε (t)fn L2 (K,μ) + Pαε (t)(fn − f˜)L2 (K,μ)
2 + TαK (t)fn − Pαε (t)f˜n L2 (K,μ) , n
∀n ∈ N,
and (3.13) follows. To prove (3.14), we use the identity (in L2 (H, μ)) R λ, Mαε f˜ =
∞
e−λt Pαε (t)f˜ dt.
0
Taking the restrictions to K of both sides and using (3.13) we obtain lim R λ, Mαε f˜ |K =
∞
→0
e−λt TαK (t)f dt,
0
which coincides with (3.14).
2
Theorem 3.8. For every λ > 0 and f ∈ L2 (K, μ), the function ϕ := R(λ, LK α )f belongs to W˚ α1,2 (K, μ) and satisfies (2.10). Therefore, TαK (t) is the semigroup generated by Mα in L2 (K, μ). Proof. For ε > 0 define ϕε := R(λ, Mαε )f˜. By Proposition 3.3, ϕε is the solution to (3.3), with f replaced by f˜. By Proposition 3.2, the Wα1,2 (H, μ)-norm of ϕε is bounded by a constant independent of ε. Therefore, there is a sequence εk → 0 such that ϕεk converges weakly in Wα1,2 (H, μ) to a function Φ. Let us prove that Φ = ϕ. ˜ For every ψ ∈ L2 (K, μ) we have Φψ dμ = lim ϕεk ψ˜ dμ = lim ϕεk ψ dμ = ϕψ dμ k→∞
K
k→∞
H
K
K
since, by Proposition 3.7, limε→0 ϕε|K − ϕL2 (K,μ) = 0. Then, Φ|K = ϕ. Moreover, 2 Φ V dμ = Φ · ΦV 1K c dμ = lim ϕεk ΦV 1K c dμ, k→∞
Kc
H
H
and by estimate (3.6) and the Hölder inequality we have 1/2 1/2 2 2 ϕε ΦV 1K c dμ ϕ V dμ Φ V dμ → 0 as k → ∞. εk k H
Kc
Kc
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It follows that Φ|K c = 0. Therefore, Φ = ϕ˜ ∈ Wα1,2 (H, μ), that is ϕ ∈ W˚ α1,2 (K, μ). For every v ∈ W˚ α1,2 (K, μ) and k ∈ N we have (since H V ϕεk v˜ dμ = 0) ϕεk v˜ dμ +
λ
1 2
H
Q(1−α)/2 Dϕεk , Q(1−α)/2 D v˜ dμ =
H
f v dμ, H
and letting k → ∞ we obtain ϕ˜ v˜ dμ +
λ H
1 2
Q(1−α)/2 D ϕ, ˜ Q(1−α)/2 D v˜ dμ =
H
f v˜ dμ, H
so that ϕ satisfies (2.10), and the statement follows.
2
3.3. Consequences We list here some consequences of the results of this section, that hold for every α ∈ [0, 1]. (i) TαK (t) is an analytic semigroup in Lp (K, μ) for every p ∈ (1, ∞). 1/2 . (ii) The space W˚ α1,2 (K, μ) coincides with the domain of (I − LK α) 2 (iii) For each f ∈ L (K, μ) we have 2 (1−α)/2 1 Q DTαK (t)f μ(dx) √ f 2 (x) μ(dx), t K
t > 0.
K
These statements follow in a standard way from the fact that the infinitesimal generator LK α of TαK (t) is the operator associated to the symmetric quadratic form Qα defined in (2.9), and that it is dissipative. Less standard consequences are a Poincaré inequality in the space W˚ α1,2 (K, μ) and the invertibility of LK α for α > 0, proved in the next proposition. Proposition 3.9. For α ∈ (0, 1] the spaces W˚ α1,2 (K, μ) and D(LK α ) are compactly embedded in L2 (K, μ). Moreover 0 ∈ ρ(LK ), and a Poincaré inequality holds in W˚ α1,2 (K, μ), α uL2 (K,μ) C
(1−α)/2 2 Q Du dμ,
u ∈ W˚ α1,2 (K, μ).
K
Proof. Since the embedding Wα1,2 (H, μ) ⊂ L2 (H, μ) is compact by Proposition 2.2(b), the embedding W˚ α1,2 (K, μ) ⊂ L2 (K, μ) is compact too. Indeed, a sequence un is bounded in W˚ α1,2 (K, μ) iff the sequence u˜ n is bounded in Wα1,2 (H, μ). In this case, there is a subsequence of u˜ n that converges to a function v ∈ L2 (H, μ). Therefore, a subsequence of un converges to the restriction v|K , in L2 (K, μ). ˚ 1,2 Since the domain D(LK α ) is continuously embedded in Wα (K, μ), it is compactly embedded 2 K in L (K, μ). Therefore, the spectrum of Lα consists of (nonpositive) eigenvalues. Let us prove that 0 is not an eigenvalue.
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K Let u ∈ D(LK α ) be such that Lα u = 0. Then
0=
uLK α u dμ = − K
1 2
(1−α)/2 2 1 Q Du dμ = − 2
K
(1−α)/2 2 Q D u˜ dμ,
H
and by the Poincaré inequality in Wα1,2 (H, μ) (Proposition 2.2(a)) we have H
2
u˜ −
u˜ dμ
dμ = 0.
H
So, u˜ is constant a.e. in H , but since it vanishes in K c , whose measure is positive, then it vanishes a.e. in H . Therefore, u = 0. This implies that the seminorm u → ( K |Q(1−α)/2 Du|2 dμ)1/2 is in fact an equivalent norm in W˚ α1,2 (K, μ), that is, a Poincaré inequality holds in W˚ α1,2 (K, μ). Indeed, since −LK α is invertible, also (−LK )1/2 is invertible, so that the seminorm u → (−LK )1/2 uL2 (K,μ) = α α 1 (1−α)/2 Du|2 dμ is an equivalent norm in D((−LK )1/2 ) = W ˚ α1,2 (K, μ); in other words α 2 K |Q (1−α)/2 2 there is C > 0 such that uL2 (K,μ) C K |Q Du| dμ for every u ∈ W˚ α1,2 (K, μ). 2 4. Interior regularity In this section we prove an interior regularity result for the solution to (1.2) for α < 1. We use the following lemma. Lemma 4.1. For every ϕ ∈ D(Lα ) and for every β ∈ Eα (H ), the product ϕβ belongs to the domain of Lα , and Lα (ϕβ) = βLα ϕ + ϕLα β + Q1−α Dϕ, Dβ . Proof. Since Eα (H ) is dense in D(Lα ), there is a sequence (ϕn ) ⊂ Eα (H ) that converges to ϕ in D(Lα ). For every n, βϕn is still in Eα (H ), hence it belongs to D(Lα ) and the statement follows easily. 2 Proposition 4.2. Assume that Tr Q1−α =
∞
λk1−α < ∞.
k=1
Then for every y ∈ K˚ and r > 0 such that dist(B(y, r), ∂K) > 0, the restriction to B(y, r) of the solution ϕ to (1.2) belongs to Wα2,2 (B(y, r), μ). Proof. It is enough to prove that the statement holds for y ∈ D(Aα/2 ). Indeed, since D(Aα/2 ) is dense in H , for each y ∈ K˚ and r > 0 such that dist(B(y, r), ∂K) > 0 there are y1 ∈ K˚ ∩D(Aα/2 ) and r1 > r such that B(y, r) ⊂ B(y1 , r1 ) and dist(B(y1 , r), ∂K) > 0.
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˚ Let So, let y ∈ D(Aα/2 ) and let r1 > r be such that the ball B(y, r1 ) is contained in K. ρ : R → [0, 1] be a C 2 function such that ρ(ξ ) = 1,
ξ r 2,
ρ(ξ ) = 0,
ξ r12 ,
and define a cutoff function θ by θ (x) := ρ |x − y|2 ,
x ∈ H.
Our aim is to show that the product ϕθ ˜ belongs to Wα2,2 (H, μ). Since the restriction to B(y, r) of ϕθ ˜ coincides with the restriction to B(y, r) of ϕ, the statement will follow. The proof is in three steps. As a first step, we show that θ ∈ D(Lα ). Then we show that ˜ ∈ ϕε θ belongs to D(Lα ) for every ε > 0, where ϕε = R(λ, Mαε )f˜. Eventually, we prove that ϕθ Wα2,2 (H, μ). First step: θ ∈ D(Lα ). We approach each x ∈ H by the sequence xn = consider the sequence of functions θn (x) := ρ |xn − yn |2 ,
n
k=1 x, ek ek ,
and we
x ∈ H, n ∈ N.
Each of them belongs to D(Lα ). This is because it depends only on the first n coordinates, it is bounded and it has bounded first and second order derivatives, and in finite dimensions the inclusion Cb2 (H ) ⊂ D(Lα ) holds. Therefore, it is easy to see that there exists the limit limt→0 (Tα (t)θn − θn )/t = Lα θn in L2 (H, μ), where n n Lα θn (x) = ρ |xn − yn |2 λk1−α − λ−α k x, ek x − y, ek
k=1
k=1
+ 2ρ |xn − yn |2 Q1−α (xn − yn ), xn − yn .
(4.1)
Letting n → ∞, ρ (|xn − yn |2 ) and ρ (|xn − yn |2 )Q1−α (xn − yn ), xn − yn converge in to ρ (|x − y|2 ) and to ρ (|x − y|2 )Q1−α (x − y), x − y, respectively, by dominated convergence. The sum nk=1 λ−α k x, ek x − y, ek converges too. Indeed, for p < q ∈ N we have q q −α/2 −α/2 −α λ λk x, ek x − y, ek x, ek L2 (H,μ) λk x − y, ek L2 (H,μ) k 2 L2 (H, μ)
k=p
k=p
L (H,μ)
=
q
(1−α)/2 −α/2 λk λk
2 1/2 + y, ek
(1−α)/2 (1−α)/2 λk
+ λk
λk
k=p
q
λk
−α/2
y, ek
k=p
q k=p
λk1−α +
2 1 1−α , λ + λ−α k y, ek 2 k
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1−α −α 2 α/2 ). where ∞ < ∞ by assumption, and ∞ k=1 λk k=1 λk |y, ek | < ∞ because y ∈ D(A Therefore, ∃L2 (H, μ) − lim
n→∞
n
α λ−α k x, ek x − y, ek := x, A (x − y) .
k=1
(Note x, Aα (x − y) is not defined pointwise.) It follows that ρ (|xn − yn |2 ) ·
n that −α 2 α 2 k=1 λk x, ek x − y, ek converges to ρ (|x − y| )x, A (x − y) in L (H, μ). Since Lα is closed, θ ∈ D(Lα ). Second step: ϕε θ belongs to D(Lα ). Since ϕε ∈ D(Lα ) and Eα (H ) is a core of Lα , there is a sequence of exponential functions βn that converges to ϕε in D(Lα ). Since θ is bounded, βn θ converges to ϕε θ in L2 (H, μ). By Lemma 4.1, βn θ belongs to D(Lα ) for every n, and we have Lα (βn θ ) = βn Lα θ + θ Lα βn + Q1−α Dβn , Dθ . As n → ∞, βn converges to ϕε , Lα βn converges to Lα ϕε , and Q1−α Dβn , Dθ = Q(1−α)/2 Dβn , Q(1−α)/2 Dθ converges to Q(1−α)/2 Dϕε , Q(1−α)/2 Dθ in L2 (H, μ) since D(Lα ) ⊂ Wα1,2 (H, μ) and Q(1−α)/2 Dθ is bounded. Therefore, Lα (βn θ ) converges in L2 (H, μ), and since Lα is closed, ϕε θ belongs to D(Lα ) and Lα (θ ϕε ) = (Lα θ )ϕε + Q(1−α)/2 Dθ, Q(1−α)/2 Dϕε + θ Lα ϕε .
(4.2)
Third step: ϕθ ˜ belongs to Wα2,2 (H, μ). Using (4.2) and (3.3) we get λθ ϕε − Lα (θ ϕε ) = θ f˜ − (Lα θ )ϕε − Q(1−α)/2 Dθ, Q(1−α)/2 Dϕε := f1,ε . The L2 norm of the right-hand side f1,ε is bounded by a constant independent of ε. Therefore, θ ϕε D(Lα ) is bounded by a constant independent of ε, and since D(Lα ) is continuously embedded in Wα2,2 (H, μ), also θ ϕε W 2,2 (H,μ) is. α Let {εk } be the sequence used in the proof of Proposition 3.8, so that ϕεk converges weakly in ˜ Possibly taking a further subsequence, (θ ϕεk ) converges weakly in Wα2,2 (H, μ) Wα1,2 (H, μ) to ϕ. ˜ indeed, for each ψ ∈ L2 (H, μ) we have to a function u that belongs to Wα2,2 (H, μ). Then u = θ ϕ;
uψ dμ = lim
k→∞
H
So, θ ϕ˜ ∈ Wα2,2 (H, μ).
θ ϕεk ψ dμ = lim
θ ϕψ ˜ dμ.
k→∞
H
2
5. Domains with smooth boundaries In this section we assume that
K = x ∈ H : g(x) 1
H
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where g : H → R is a C 1 function that belongs to D(L0 ) and satisfies (A.8). Moreover we assume that sup g > 1, so that K is a proper subset of H , and inf g < 1, so that the interior part of K is not empty and the surface measure dσ is well defined in the boundary Σ of K, Σ = {x ∈ H : g(x) = 1}. See Appendix A, to which we refer for the definition and properties of surface measures. The aim of this section is to give a reasonable definition of the trace at ∂K of any function in Wα1,2 (H, μ), and to show that the functions in W˚ α1,2 (H, μ) have null trace at ∂K. This implies that R(λ, LK α )f satisfies the Dirichlet boundary condition in (1.2) in the sense of the trace for every f ∈ L2 (K, μ), and that TαK (t)f has null trace at the boundary for every t > 0 and f ∈ L2 (K, μ). As a first step we prove integration formulas for functions in the core E0 (H ). Proposition 5.1. Let k ∈ N be such that Dk g/|Q1/2 Dg| ∈ W02,2 (H, μ). Then for every ϕ ∈ E0 (H ) we have Dk g 1 ϕ dσ. (5.1) Dk ϕ dμ = xk ϕ dμ + λk |Q1/2 Dg| K
Σ
K
If |Q1/2 Dg| ∈ W02,2 (H, μ), then for every ϕ ∈ E0 (H ) we have
2
1/2
ϕ Q
Dg dσ1 =
1/2 Dϕ, Q1/2 Dg dμ + L0 gϕ 2 dμ, K ϕQ K − K c ϕQ1/2 Dϕ, Q1/2 Dg dμ − K c L0 gϕ 2 dμ.
Σ
(a) (b)
(5.2)
Proof. For small ε > 0 define the pathwise linear function θε by ⎧ ⎨ 2, θε (ξ ) := 1ε (1 − ξ ) + 1, ⎩ 0,
ξ 1 − ε, 1 − ε < ξ < 1 + ε, ξ 1 + ε,
and set ρε (x) := θε g(x) ,
x ∈ H.
Since θε is Lipschitz continuous, then ρε ∈ W01,2 (H, μ) [2, Rem. 5.2.1]. Then the product ρε ϕ belongs to W01,2 (H, μ) and Dk (ρε ϕ) = θε (g(x))Dk g(x)ϕ(x) + ρε (x)Dk ϕ(x), so that (Dk ϕ)ρε dμ − H
1 ε
ϕDk g dμ =
1 λk
xk ϕρε dμ,
k ∈ N.
(5.3)
H
1−ε
Let us prove (5.1). Letting ε → 0, ρε converges pointwise to 21K in H \ Σ , whose measure is 1. Since ρε 2, by dominated convergence we get 1 1 ∃ lim ϕDk g dμ = Dk ϕ dμ − xk ϕ dμ. ε→0 2ε λk 1−ε
K
K
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Let us identify the limit in the left-hand side as a boundary integral. Since ϕDk g|Q1/2 Dg|−1 ∈ W02,2 (H, μ), by Remark A.7 we have
1 ε→0 2ε
ϕDk g dμ =
lim
Σ
1−ε
Dk g ϕ dσ |Q1/2 Dg|
and (5.1) follows. Let us prove (5.2)(a). For every ε > 0 and k ∈ N, the function ρε ϕ 2 Dk g still belongs to 1,2 W0 (H, μ). Therefore we may replace ϕ in (5.3) by λk ϕ 2 Dk g, and summing over k (recall Lemma 2.1), we obtain
1/2
2ϕ Q
1/2
Dϕ, Q
Dg ρε dμ +
H
1 2L0 gϕ ρε dμ = ε
2
H
2 ϕ 2 Q1/2 Dg dμ.
1−ε
Letting ε → 0 as before, by dominated convergence we get lim
ε→0
ϕ Q1/2 Dϕ, Q1/2 Dg ρε dμ =
H
K
L0 gϕ ρε dμ = 2
lim
ε→0
ϕ Q1/2 Dϕ, Q1/2 Dg dμ,
H
L0 g ϕ 2 dμ. K
Therefore, there exists the limit 2 1 lim ϕ 2 Q1/2 Dg dμ = ϕ Q1/2 Dϕ, Q1/2 Dg dμ + L0 g ϕ 2 dμ ε→0 2ε K
1−ε
K
that we identify as a boundary integral. Indeed, since ϕ 2 |Q1/2 Dg| ∈ W02,2 (H, μ), by Remark A.7 we have 2 1 lim ϕ 2 Q1/2 Dg dμ = ϕ 2 Q1/2 Dg dσ. ε→0 2ε Σ
1−ε
So, (5.2)(a) holds. To prove (5.2)(b), we may follow the same procedure replacing K by K c and θε by ⎧ ξ 1 − ε, ⎨ 0, θ˜ε (ξ ) := 1ε (ξ − 1) + 1, 1 − ε < ξ < 1 + ε, ⎩ 2, ξ 1 + ε, or else, we may use the equality Kc
ϕ Q1/2 Dϕ, Q1/2 Dg dμ +
Kc
L0 gϕ 2 dμ = − K
ϕ Q1/2 Dϕ, Q1/2 Dg dμ −
L0 gϕ 2 dμ K
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that follows from 1 1/2 Q Dg, Q1/2 D ϕ 2 dμ = − Q1/2 Dg, Q1/2 Dϕ ϕ dμ L0 g ϕ 2 dμ = − 2 H
(see formula (2.6)).
H
H
2
As a second step, with the aid of Proposition 5.1 we prove an integration by parts formula in W01,2 (H, μ) and we define the trace ϕ|Σ at the boundary Σ of any function in W01,2 (H, μ). Corollary 5.2. Assume that |Q1/2 Dg| ∈ W02,2 (H, μ), and that |Q1/2 Dg| is bounded and L0 g has at most linear growth either on K or on K c . Then for every ϕ ∈ W01,2 (H, μ) there exists ψ ∈ L2 (Σ, σ ) with the following property: for each sequence (ϕn ) ∈ E0 (H ) such that limn→∞ ϕn − 1/2 ϕW 1,2 (H,μ) = 0, the sequence (ϕn |Q1/2 Dg||Σ ) converges to ψ in L2 (Σ, σ ). 0
Proof. It is sufficient to recall formula (5.2) and Lemma 2.1.
2
Note that the assumptions of Corollary 5.2 are satisfied by the functions g in Example A.3 of Appendix A. Definition 5.3. Under the assumptions of Corollary 5.2, for each ϕ ∈ W01,2 (H, μ) the trace of ϕ at Σ is defined by ϕ|Σ =
ψ , |Q1/2 Dg|1/2
where ψ is given by Corollary 5.2. Note that in general ϕ|Σ does not belong to L2 (Σ, σ ), because |Q1/2 Dg|−1/2 may be unbounded in Σ. Of course, if |Q1/2 Dg|−1/2 is bounded in Σ (that is, if infΣ |Q1/2 Dg| > 0), then ϕ|Σ ∈ L2 (Σ, σ ) for every ϕ ∈ W01,2 (H, μ) and the mapping W01,2 (H, μ) → L2 (Σ, σ ), ϕ → ϕ|Σ is continuous. In general, we have the following lemma. Lemma 5.4. Under the assumptions of Corollary 5.2, for every ϕ ∈ W01,2 (H, μ), ϕ|Σ ∈ L1 (Σ, σ ) and the mapping W01,2 (H, μ) → L1 (Σ, σ ), ϕ → ϕ|Σ is continuous. Proof. Since ϕ|Σ = ψ|Q1/2 Dg|−1/2 with ψ ∈ L2 (Σ, σ ), it is sufficient to prove that |Q1/2 Dg|−1/2 ∈ L2 (Σ, σ ). The assumptions Q1/2 D 2 g Q1/2 L(H ) /|Q1/2 Dg|2 ∈ L2 (H, μ) and |Q1/2 Dg|−1 ∈ L4 (H, μ), that are contained in assumption (A.8), imply that the function ˜ 1/2 Dg|1/2 = |Q1/2 Dg|−1/2 has ϕ˜ := |Q1/2 Dg|−1 belongs to W01,2 (H, μ). By Corollary 5.2, ϕ|Q 2 trace in L (Σ, σ ). 2 Corollary 5.5. Let the assumptions of Corollary 5.2 be satisfied. The following statements hold for every α ∈ [0, 1].
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(i) If Dk g/|Q1/2 Dg| ∈ W02,2 (H, μ), for every ϕ ∈ Wα1,2 (H, μ) the integration by parts formula (5.1) holds. (ii) If ϕ ∈ W˚ α1,2 (K, μ), its trace at Σ1 vanishes. Proof. Since Wα1,2 (H, μ) ⊂ W01,2 (H, μ), and W˚ α1,2 (K, μ) ⊂ W˚ 01,2 (K, μ), it is enough to prove that the statements hold for α = 0. (i) It is sufficient to approach every ϕ ∈ W01,2 (H, μ) by a sequence (ϕn ) ⊂ E0 (H ), and to recall Lemma 5.4. (ii) If ϕ ∈ W˚ 01,2 (K, μ), it vanishes a.e. in K c , and formula (5.2)(b) yields the statement. 2 Appendix A. Surface integrals We consider level surfaces of smooth functions g. We refer to [2, §6.10], where the functions g under consideration belong to the space W ∞ (H, μ) defined by
W ∞ (H, μ) :=
W k,p (H, μ)
k∈N,p>1
and W k,p (H, μ) is the completion of the smooth cylindrical functions1 in the norm
f k,p := f Lp (H,μ) +
k j =1
H
1/p 2 p/2 λi1 · · · · · λik Di1 . . . Dik f (x) μ(dx) . i1 ,...,ij 1
(In particular, the spaces W k,2 (H, μ) coincide with our W0k,2 (H, μ) for k = 1, 2.) Another assumption is 1/2 −1 p Q Dg ∈ L (H, μ). p>1
Our aim here is to give a simplified presentation of surface measures in the case of a Hilbert space setting, under less heavy (although less elegant) assumptions on g. For any continuous g : H → R and r in the range of g let us define the level sets
Σr := x ∈ H : g(x) = r . We shall define probability measures on the surfaces Σr with r in the interior part of the range of g. To this aim, a first step is the study of the image of μ on R under the mapping g, defined by μ ◦ g −1 (I ) := μ g −1 (I ) ,
I ∈ B(R).
1 That is, functions of the type f (x) = ϕ(x, x , . . . , x, x ) with x , . . . , x ∈ H and ϕ ∈ C ∞ (Rn ). n n 1 1 b
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We shall give sufficient conditions for μ ◦ g −1 to have continuous (in fact, W 1,2 ) density k with respect to the Lebesgue measure. Similarly, for ρ ∈ L1 (H, μ) we shall consider the signed measure (ρμ)(B) := ρ(x) μ(dx), B ∈ B(H ) B
and its image under the mapping g, ρμ ◦ g −1 (I ) := (ρμ) g −1 (I ) ,
I ∈ B(R),
and we shall give sufficient conditions for ρμ ◦ g −1 to have continuous density kρ with respect to the Lebesgue measure. A key role will be played by the function ψ defined by ψ :=
Q1/2 D 2 g Q1/2 · Q1/2 Dg, Q1/2 Dg L0 g − , |Q1/2 Dg|2 |Q1/2 Dg|4
(A.1)
if g ∈ D(L0 ). We shall use the following proposition. Proposition A.1. Let g ∈ D(L0 ) be such that |Q1/2 Dg|−1 ∈ L4 (H, μ). Then (a) μ ◦ g −1 is absolutely continuous with respect to the Lebesgue measure. (b) If a function ρ ∈ W01,1 (H, μ) is such that ψρ ∈ L1 (H, μ),
|Q1/2 Dρ| ∈ L1 (H, μ), |Q1/2 Dg|
(A.2)
where ψ is defined in (A.1), then ρμ ◦ g −1 is absolutely continuous with respect to the Lebesgue measure. Proof. To prove statement (a) we shall show that there exists C > 0 such that ϕ (r) μ ◦ g −1 (dr) Cϕ∞ , ϕ ∈ C 1 (R). b
(A.3)
R
For each k ∈ N we have Dk (ϕ ◦ g)(x) = ϕ g(x) Dk g(x),
x ∈ H,
(A.4)
so that 2 D(ϕ ◦ g)(x), QDg(x) = ϕ ◦ g (x)Q1/2 Dg(x) ,
x ∈ H,
(A.5)
a.e. x ∈ H.
(A.6)
i.e. Q1/2 D(ϕ ◦ g)(x), Q1/2 Dg(x) , ϕ ◦ g (x) = |Q1/2 Dg(x)|2
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Therefore,
ϕ (r) μ ◦ g −1 (dr) =
R
k λk Dk (ϕ ◦ g)(x)Dk g(x) |Q1/2 Dg(x)|2
ϕ ◦ g dμ = H
H
dμ.
Integrating by parts and recalling that Dk
1 |Q1/2 Dg|2
= −2
i λi Di gDik g |Q1/2 Dg|4
(A.7)
we obtain
ϕ ◦ g dμ = − H
ϕ◦g
ϕ◦g
ϕ◦g
λk
k
H
= −2
k
H
+
λk Dk
k
H
=−
xk Dk g Dk g dμ + ϕ◦g dμ 1/2 2 |Q Dg| |Q1/2 Dg|2 H
Dkk g − 2Dk g |Q1/2 Dg|2
k
λi Di gDik g dμ |Q1/2 Dg|4 i
xk Dk g dμ |Q1/2 Dg|2
(ϕ ◦ g)(x)ψ(x) dμ, H
where the function ψ is defined in (A.1). The first addendum in ψ, L0 g/|Q1/2 Dg|2 , belongs to L1 (H, μ) since both L0 g and 1/|Q1/2 Dg|2 are in L2 (H, μ). Concerning the second addendum we have |Q1/2 D 2 g Q1/2 · Q1/2 Dg, Q1/2 Dg| Q1/2 D 2 g Q1/2 L(H ) . |Q1/2 Dg|4 |Q1/2 Dg|2 Recalling that there exists C0 > 0 such that [2, Thm. 5.7.1] x → Q1/2 D 2 gQ1/2
L(H ) L2 (H,μ)
C0 gD(L0 ) ,
it follows that the second addendum in ψ belongs to L1 (H, μ). Then formula (A.3) follows, with C = ψL1 (H,μ) const (gD(L0 ) + |Q1/2 Dg|−1 L4 (H.μ) ). We prove statement (b) by the same procedure, replacing μ by ρμ. For every ϕ ∈ Cb1 (R) we have ρ(x) ϕ ◦ g ρ dμ = λk Dk (ϕ ◦ g)(x)Dk g(x) 1/2 dμ |Q Dg(x)|2 H
H
= H
k
Q1/2 Dg, Q1/2 Dρ dμ ϕ ◦ g −2ψρ − |Q1/2 Dg(x)|2
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where ψ is the function defined in (A.1). Assumption (A.2) implies that the functions ψρ and Q1/2 Dg, Q1/2 Dρ/|Q1/2 Dg(x)|2 belong to L1 (H, μ). Then, ϕ (r) μ ◦ g −1 (dr) = ϕ ◦ g ρ dμ Cϕ∞ , R
ϕ ∈ Cb1 (R)
H
|Q Dρ| with C = 2ψρL1 + |Q 1/2 Dg| L1 . The statement follows. 1/2
2
Proposition A.2. Let the assumptions of Proposition A.1 hold. Then: (a) If the function ψ defined in (A.1) belongs to W01,2 (H, μ), then the density k of μ ◦ g −1 belongs to W 1,1 (R). (b) If ρ ∈ W01,1 (H, μ) satisfies (A.2) and moreover, setting ρ1 := 2ψρ +
Q1/2 Dg, Q1/2 Dρ |Q1/2 Dg|2
we have ρ1 ∈ W01,1 (H, μ), ψρ1 ∈ L1 (H, μ),
|Q1/2 Dρ1 | |Q1/2 Dg|
∈ L1 (H, μ), then kρ ∈ W 1,1 (R).
Proof. To prove statement (a) we shall show that there is C1 > 0 such that ϕ (r) μ ◦ g −1 (dr) C1 ϕ∞ ,
ϕ ∈ Cb2 (R).
R
Indeed, this implies that k is weakly differentiable with k ∈ L1 (R). Differentiating (A.4) we get 2 Dkk (ϕ ◦ g)(x) = ϕ g(x) Dk g(x) + ϕ g(x) Dkk g(x),
x ∈ H,
and summing over k 2 Tr QD 2 (g ◦ ϕ) = ϕ g(x) Q1/2 Dg(x) + ϕ g(x) Tr QD 2 g(x) so that Tr(QD 2 (ϕ ◦ g)) Tr(QD 2 g) − ϕ ◦g |Q1/2 Dg|2 |Q1/2 Dg|2 2L0 (ϕ ◦ g) + x, D(ϕ ◦ g) 2L0 g + x, Dg = − ϕ ◦g |Q1/2 Dg|2 |Q1/2 Dg|2 2L0 (ϕ ◦ g) 2L0 g = 1/2 − ϕ ◦g . |Q Dg|2 |Q1/2 Dg|2
ϕ ◦ g =
Using again (A.7) we get
G. Da Prato, A. Lunardi / Journal of Functional Analysis 259 (2010) 2642–2672
ϕ ◦ g dμ =
H
−2 − Q1/2 D(ϕ ◦ g), Q1/2 D Q1/2 Dg − 2 ϕ ◦ g H
=
ϕ ◦g H
= −2
1/2
Q
2667
L0 g dμ |Q1/2 Dg|2
L0 g Q1/2 D 2 gQ1/2 · Q1/2 Dg − 2 1/2 dμ Dg, 2 |Q1/2 Dg|4 |Q Dg|2
ϕ ◦ g ψ dμ,
H
where ψ is defined in (A.1). Then we may use Proposition A.1, with ρ = ψ . By assump1/2 Dψ| 1 tion, ψ ∈ W01,2 (H, μ) ⊂ W01,1 (H, μ), moreover ψ 2 ∈ L1 (H, μ) and |Q 1/2 Dg| ∈ L (H, μ) |Q since |Q1/2 Dψ| ∈ L2 (H, μ), |Q1/2 Dg|−1 ∈ L2 (H, μ). We get | H (ϕ ◦ g)ψ dμ| CψW 1,2 (H,μ) ϕ∞ , and statement (a) follows. 0
Concerning statement (b), the proof is similar, replacing μ by ρμ. For every ϕ ∈ Cb2 (R) we have
ϕ ◦ g ρ dμ =
H
H
ρL0 g 2ρL0 (ϕ ◦ g) dμ − 2 ϕ ◦ g |Q1/2 Dg|2 |Q1/2 Dg|2
1/2 1/2 = − Q D(ϕ ◦ g), Q D H
ϕ ◦ g
= H
− H
=− H
ρ 1/2 |Q Dg|2
− 2 ϕ ◦ g
ρL0 g dμ |Q1/2 Dg|2
L0 g Q1/2 D 2 gQ1/2 · Q1/2 Dg − 2 ρ dμ Q1/2 Dg, 2 |Q1/2 Dg|4 |Q1/2 Dg|2
ϕ ◦ g
Q1/2 Dg, Q1/2 Dρ dμ |Q1/2 Dg|2
Q1/2 Dg, Q1/2 Dρ dμ = − ϕ ◦ gρ1 dμ, ϕ ◦ g 2ψ ρ + |Q1/2 Dg|2
H
where the function ρ1 satisfies the assumptions of Proposition A.1(b). We obtain | H (ϕ ◦ g)ρ1 dμ| Cϕ∞ and the statement follows. 2 One can play with ρ and g in order that the assumptions of Proposition A.2(b) are satisfied. In the next proposition we give sufficient conditions that are useful for the sequel. Proposition A.3. The assumptions of Proposition A.2(b) are satisfied by every ρ ∈ W02,2 (H, μ) provided g ∈ D(L0 ) is such that ⎧ 1/2 −1 4 ⎪ ⎨ |Q Dg| ∈ L (H, μ), 1/2 D 2 g Q1/2
Q ⎪ ⎩
|Q1/2 Dg|2
L(H )
ψ ∈ W01,4 (H, μ),
∈ L2 (H, μ),
Q1/2 D 2 g Q1/2 L(H ) ∈ L2 (H, μ). |Q1/2 Dg|3
(A.8)
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In this case there exists C2 > 0, depending only on g, such that ϕ (r) ρμ ◦ g −1 (dr) C2 ρ 2,2 ϕ∞ , W0 (H,μ)
ϕ ∈ Cb2 (R).
R
Consequently, if ρn → ρ in W02,2 (H, μ) then kρn → kρ in W 1,1 (R), hence kρn → kρ in L∞ (R). Proof. Since ψ ∈ L2 and |Q1/2 Dg|−1 ∈ L4 , then ρ1 ∈ L1 . Computing Q1/2 Dρ1 we obtain Q1/2 Dρ1 = ρQ1/2 Dψ + ψQ1/2 Dρ −
Q1/2 D 2 g Q1/2 · Q1/2 Dρ + Q1/2 D 2 ρQ1/2 · Q1/2 Dg |Q1/2 Dg|2
Q1/2 D 2 g Q1/2 · Q1/2 Dg + 2 Q1/2 Dg, Q1/2 Dρ . |Q1/2 Dg|4 Estimating each addendum we get • ρ|Q1/2 Dψ| ∈ L1 , since |Q1/2 Dψ| ∈ L2 ; • ψ|Q1/2 Dρ| ∈ L1 , since ψ ∈ L2 ; • • •
Q1/2 D 2 g Q1/2 L(H ) |Q1/2 Dρ| Q1/2 D 2 g Q1/2 L(H ) ∈ L1 , since |Q1/2 Dg|2 |Q1/2 Dg|2 Q1/2 D 2 ρQ1/2 L(H ) 1 1 2 ∈ L , since |Q1/2 Dg| ∈ L ; |Q1/2 Dg| 1/2 D 2 g Q1/2 Q L(H ) |Q1/2 Dρ| ∈ L1 , as above. |Q1/2 Dg|2
∈ L2 ;
Therefore ρ1 ∈ L1 , and ρ1 L1 (H,μ) cρW 2,2 (H,μ) . The assumptions ψ ∈ L4 , To check that We get
|Q1/2 Dρ1 | |Q1/2 Dg|
1
|Q1/2 Dg|
0
∈ L4 imply that ψρ1 ∈ L1 .
∈ L1 we redo the estimates above, dividing each term by |Q1/2 Dg|.
Dψ| • ρ |Q ∈ L1 , since |Q1/2 Dψ| ∈ L4 and |Q1/2 Dg| 1/2
|Q1/2 Dρ|
• ψ |Q1/2 Dg| ∈ L1 , since ψ ∈ L4 and • • •
1 |Q1/2 Dg| ∈ L4 ;
∈ L4 ;
1 |Q1/2 Dg| Q1/2 D 2 g Q1/2 L(H ) |Q1/2 Dρ| Q1/2 D 2 g Q1/2 L(H ) ∈ L1 , since |Q1/2 Dg|3 |Q1/2 Dg|3 Q1/2 D 2 ρQ1/2 L(H ) ∈ L1 , since |Q1/21 Dg| ∈ L4 ; |Q1/2 Dg|2 Q1/2 D 2 gQ1/2 L(H ) |Q1/2 Dρ| ∈ L1 , as above. |Q1/2 Dg|3
∈ L2 ;
Dρ1 | 1 are bounded by cρW 2,2 (H,μ) . Applying Therefore, the norms ψρ1 L1 and |Q |Q1/2 Dg| L 0 Proposition A.2(b) the statement follows. 2 1/2
Example. Let us consider some simple examples. (a) g(x) = b, x, with |b| = 1,
G. Da Prato, A. Lunardi / Journal of Functional Analysis 259 (2010) 2642–2672
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(b) g(x) = T
x, x,2with T ∈ L(H ), T ek = tk ek for each k ∈ N and tk = 0 for infinitely many k, (c) g(x) = 13 k=1 xk . In all these cases g satisfies the conditions of Proposition A.3. Proof. In the case (a) we have Dg = b, D 2 g = 0 so that L0 g = −b, x/2 = −g/2 and ψ =−
b, x 2|Q1/2 b|2
which belongs to W01,4 (H, μ). The other conditions of Proposition A.3 are obviously satisfied. In the case (b) we have Dg(x) = 2T x, D 2 g(x) = 2T so that L0 g = Tr[QT ] − g and
ψ(x) =
Tr[QT ] − T x, x Q2 T 3 x, x − . 2|Q1/2 T x|2 |Q1/2 T x|2
(A.9)
many k, then x → |Q1/2 Dg(x)|−1 belongs to all spaces Lp (H, μ). Since tk = 0 for infinitely 2 2 Indeed, |Q1/2 Dg(x)|2 4 N k=1 λi tk xk where N is so large that at least [p] + 1 addenda do not vanish. The other assumptions of Remark A.3 are easily seen to be satisfied.
In the case (c) we still have g(x) = T x, x with T ∈ L(H ), T x = 13 k=1 xk ek , so
13 2 1/2 −1 −1/2 that tk = 0 only for k = 1, . . . , 13. However, |Q Dg(x)| c0 ( k=1 xk ) with c0 = 1/2 1/2 −1 p 1/ min{λk : k = 1, . . . , 13} so that |Q Dg| ∈ L (H, μ) for every p < 13. The function ψ is still given by (A.9) on span{e1 , . . . , e13 } and it belongs to Lp (H, μ) for every p < 13/3, in particular it belongs to L4 (H, μ), as well as |Q1/2 Dψ|−1 . The other conditions of Proposition A.3 are easily seen to be satisfied. 2 In the cases (a) and (b) with T = I it is possible to give a representation formula for k that 2 −1/2 with shows that k ∈ C ∞ , see [10]. In the case (c) we have |Q1/2 Dg(x)|−1 c1 ( 13 k=1 xk ) 1/2 c1 = 1/ max{λk : k = 1, . . . , 13} so that |Q1/2 Dg|−1 ∈ / Lp (H, μ) for p 13. The construction of the surface measures goes as follows. First, one constructs surface measures depending explicitly on g by an approximation procedure. One fixes once and for all a convex compact set K which is symmetric with respect to the origin and has positive measure, say μ(K) > 1/2. Such a K does exist. Indeed, it is well known that there are compact sets K with positive (arbitrarily close to 1) measure (a simple proof is e.g. in [3, Thm. 6.2]). The absolute convex hull K of K is compact, symmetric with respect to the origin and contains K, so that μ(K) μ(K). Then we need a regular cutoff function. The proof of its existence follows closely [2, Prop. 5.4.12], with a few simplifications due to our Hilbert space setting. Lemma A.4. Let K ⊂ H be compact, convex, symmetric with respect to the origin, with μ(K) > 1/2. Then there exists a function θ ∈ W ∞ (H, μ) such that θ ≡ 1 on K, θ = 0 a.e. outside 2K and 0 θ (x) 1 for all x ∈ H .
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Proof. By the 0−1 law (e.g., [2, Thm. 2.5.5]), the vector space E spanned by K has measure 1. Consequently, limm→∞ μ(mK) = 1. Fix m ∈ N such that 8 μ(mK) > . 9 Let us consider the Minkowski functional defined by pK (x) := inf{α > 0: x ∈ αK} for x ∈ E, and the function d(x) := inf{pK (x − y): y ∈ K} if x ∈ E, d(x) = 1 if x ∈ / E. We modify it setting ϕ(x) = 1 − h d(x) ,
x ∈ H,
where h(t) = t for t 1 and h(t) = 1 for t 1. The function ϕ is Borel measurable, has values between 0 and 1, ϕ ≡ 1 on K and ϕ ≡ 0 outside E and outside 2K. We regularize it applying T0 (t), where t > 0 is chosen such that 1 1 − e−t/2 < , 8
! 1 m 1 − e−t < . 8
Since ϕ ∈ Bb (H ), then T0 (t)ϕ ∈ W ∞ (H, μ) (e.g., [2, Prop. 5.4.8]). Moreover, 2 T0 (t)ϕ(x) , 3
∀x ∈ K,
3 T0 (t)ϕ(x) , 5
∀x ∈ E \ 2K.
(A.10)
√ Indeed, let x ∈√K. Then e−t/2 x ∈ K, and for each y ∈ mK we have 1 − e−t y ∈ K/8. The √ −t/2 x + 1 − e−t y) 1/8 and therefore −t sum e−t/2 x + √ 1 − e y belongs to 9K/8, so that d(e −t/2 −t x + 1 − e y) 7/8. Since μ(H \ mK) 1/9, we get T0 (t)ϕ(x) 7/8 − 1/9 > 2/3. ϕ(e −t/2 > 7/8, e−t/2 x ∈ Let now x ∈ E \ 2K. / 7K/4 and consequently for every y ∈ mK √ Since e −t/2 −t x + 1 − e y does not belong to 7K/4 − K/8 = 13K/8. Therefore, d(e−t/2 x + the sum e √ √ −t/2 −t −t 1 − e y) 5/8, so that ϕ(e x + 1 − e y) 3/8. Again since μ(H \ mK) 1/9, we get T0 (t)ϕ(x) 3/8 + 1/9 = 35/72 < 3/5, and (A.10) is proved. Now fix a function η ∈ C ∞ (R) such that 0 η 1, η(t) = 0 for t 3/5, η(t) = 1 for t 2/3, and set θ (x) = η T0 (t)ϕ(x) ,
x ∈ H.
The function θ is what we were looking for. It has values between 0 and 1, it belongs to W ∞ (H, μ), θ (x) = 1 for x ∈ K and θ (x) = 0 for x ∈ E \ 2K. Since μ(E) = 1, then θ (x) = 0 for almost all x ∈ H \ 2K. The statement follows. 2 Now we fix ϕ0 ∈ Cc∞ (R) with 0 ϕ0 1, R ϕ0 (t) dt = 1 and ϕ0 ≡ 1 in a neighborhood of 0, ϕ0 ≡ 0 outside (−1, 1). Then for each r ∈ R the sequence {ϕ0 (j (t − r)) dt/j } converges weakly to the Dirac measure δr . For each r in the interior part of g(H ) we set θn (x) = θ
x , n
x ∈ H;
ϕj (t) =
ϕ0 (j (t − r)) , j
j ∈ N, t ∈ R.
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The following proposition is proved in [2]. Since there are not simplifications with respect to the general setting of [2] in the Hilbert space case, we refer to [2, Lemma 6.10.1, Thm. 6.10.2] for the proof. Proposition A.5. (a) For each n ∈ N, the sequence of measures νn,j (dx) = θn (x)
ϕj (g(x)) μ(dx) k(g(x))
converges weakly to a measure νn concentrated on Σr := g −1 (r). Moreover, for each continuous f ∈ W02,2 (H ) we have
f dνn =
f dνn =
kf θn (r) . k(r)
(A.11)
Σr
H
(g)
(b) In its turn, the sequence νn converges weakly to a probability measure σr on Σr , such that for each continuous f ∈ W02,2 (H ) we have f
(g) dσr
H
=
(g)
f dσr
=
kf (r) . k(r)
concentrated
(A.12)
Σr
Definition A.6. For every Borel bounded function ϕ : H → R and for every r in the interior part of g(H ) we set (g) ϕ dσr := k(r) ϕ Q1/2 Dg dσr . Σr
Σr
Remark A.7. It is easy to see that for every f : H → R such that f |Q1/2 Dg| ∈ W02,2 (H, μ) ∩ C(H ) we have 1 f dσr = lim f Q1/2 Dg dμ. ε→0 2ε Σr
r−εg(x)r+ε
Indeed, applying Proposition A.2 we get 1 lim ε→0 2ε
1 f Q1/2 Dg dμ = lim ε→0 2ε
r+ε d f Q1/2 Dg ◦ μ r−ε
r−εg(x)r+ε
1 = lim ε→0 2ε
r+ε kf |Q1/2 Dg| (t) dt = kf |Q1/2 Dg| (r). r−ε
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On the other hand, by Proposition A.5(b) we have (g) kf |Q1/2 Dg| (r) = k(r) f Q1/2 Dg dσr Σr
and the right-hand side is just
Σr
f dσr by definition.
References [1] V. Barbu, G. Da Prato, L. Tubaro, Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space, Ann. Probab. 37 (2009) 1427–1458. [2] V.I. Bogachev, Gaussian Measures, Amer. Math. Soc., Providence, 1998. [3] G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer-Verlag, Berlin, 2006. [4] G. Da Prato, B. Goldys, J. Zabczyk, Ornstein–Uhlenbeck semigroups in open sets of Hilbert spaces, C. R. Math. Acad. Sci. Paris 325 (1997) 433–438. [5] G. Da Prato, J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Notes Ser., vol. 293, Cambridge University Press, Cambridge, 2002. [6] J.D. Deuschel, D. Stroock, Large Deviations, Academic Press, San Diego, 1984. [7] E.B. Dynkin, Markov Processes, vol. I, Springer-Verlag, Berlin, 1965. [8] A. Friedman, Stochastic Differential Equations and Applications, vol. 1, Academic Press, New York, 1975. [9] L. Gross, Potential theory in Hilbert spaces, J. Funct. Anal. 1 (1965) 123–189. [10] A. Hertle, Gaussian surface measures and the Radon transform on separable Banach spaces, in: Measure Theory, Proc. Conf., Oberwolfach, 1979, in: Lecture Notes in Math., vol. 794, Springer-Verlag, Berlin, 1980, pp. 513–531. [11] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. [12] P. Kotelenez, A submartingale type inequality with applications to stochastic evolution equations, Stochastics 8 (1982) 139–151. [13] A. Lunardi, G. Metafune, D. Pallara, Dirichlet boundary conditions for elliptic operators with unbounded drift, Proc. Amer. Math. Soc. 133 (2005) 2625–2635. [14] P. Malliavin, Stochastic Analysis, Springer-Verlag, Berlin, 1997. [15] A. Talarczyk, Dirichlet problem for parabolic equations on Hilbert spaces, Studia Math. 141 (2000) 109–142. [16] L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral, Stoch. Anal. Appl. 2 (1984) 187–192.
Journal of Functional Analysis 259 (2010) 2673–2701 www.elsevier.com/locate/jfa
Bounded symbols and Reproducing Kernel Thesis for truncated Toeplitz operators ✩ Anton Baranov a , Isabelle Chalendar b , Emmanuel Fricain b,∗ , Javad Mashreghi c , Dan Timotin d a Department of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetskii pr.,
St. Petersburg, 198504, Russia b Université de Lyon; Université Lyon 1; INSA de Lyon; Ecole Centrale de Lyon; CNRS, UMR5208,
Institut Camille Jordan; 43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France c Département de Mathématiques et de Statistique, Université Laval, Québec, QC, Canada G1K 7P4 d Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 014700, Romania
Received 20 April 2010; accepted 12 May 2010 Available online 8 June 2010 Communicated by D. Voiculescu
Abstract Compressions of Toeplitz operators to coinvariant subspaces of H 2 are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the Reproducing Kernel Thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive. © 2010 Elsevier Inc. All rights reserved. Keywords: Toeplitz operators; Reproducing Kernel Thesis; Model spaces
✩
This work was partially supported by funds from NSERC (Canada), Centre International de Rencontres Mathématiques (France) and by the grants RFBR 08-01-00723 and MK 7656.2010.1 (Russia). * Corresponding author. E-mail addresses:
[email protected] (A. Baranov),
[email protected] (I. Chalendar),
[email protected] (E. Fricain),
[email protected] (J. Mashreghi),
[email protected] (D. Timotin). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.005
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1. Introduction Truncated Toeplitz operators on model spaces have been formally introduced by Sarason in [29], although special cases have long ago appeared in literature, most notably as model operators for contractions with defect numbers one and for their commutant. They are naturally related to the classical Toeplitz and Hankel operators on the Hardy space. This is a new area of study, and it is remarkable that many simple questions remain still unsolved. As a basic reference for their main properties [29] is invaluable; further study can be found in [8,9,17] and in [30, Section 7]. The truncated Toeplitz operators live on the model spaces KΘ . These are subspaces of H 2 (see Section 2 for precise definitions) that have attracted attention in the last decades; they are relevant in various subjects such as for instance spectral theory for general linear operators [25], control theory [26], and Nevanlinna domains connected to rational approximation [15]. Given a model space KΘ and a function ϕ ∈ L2 , the truncated Toeplitz operator AΘ ϕ is defined on a dense subspace of KΘ as the compression to KΘ of multiplication by ϕ. The function ϕ is then called a symbol of the operator, and it is never uniquely defined. In the particular case where ϕ ∈ L∞ the operator AΘ ϕ is bounded. In view of well-known facts about classical Toeplitz and Hankel operators, it is natural to ask whether the converse is true, that is, if a bounded truncated Toeplitz operator has necessarily a bounded symbol. This question has been posed in [29], where it is noticed that it is nontrivial even for rank one operators. In the present paper we will provide a class of inner functions Θ for which there exist rank one truncated Toeplitz operators on KΘ without bounded symbols. On the other hand, we obtain positive results for some basic examples of model spaces. Therefore the situation is quite different from the classical Toeplitz and Hankel operators. The other natural question that we address is the Reproducing Kernel Thesis for truncated Toeplitz operators. Recall that an operator on a reproducing kernel Hilbert space is said to satisfy the Reproducing Kernel Thesis (RKT) if its boundedness is determined by its behaviour on the reproducing kernels. One of the first examples of (RKT) is the proof of Carleson embedding theorem by S.A. Vinogradov (see [25]). This property has been studied for several classes of operators: Hankel and Toeplitz operators on the Hardy space of the unit disk [6,21, 32], Toeplitz operators on the Paley–Wiener space [31], semicommutators of Toeplitz operators [25], Hankel operators on the Bergman space [3,19], and Hankel operators on the Hardy space of the bidisk [16,27]. Though there were many results of this type before, philosophically the idea to study (RKT) for classes of operators in general reproducing kernel Hilbert spaces comes from [20]. It appears thus natural to ask the corresponding question for truncated Toeplitz operators. We will show that in this case it is more appropriate to assume the boundedness of the operator on the reproducing kernels as well as on a related “dual” family, and will discuss further its validity for certain model spaces. The paper is organized as follows. The next two sections contain preliminary material concerning model spaces and truncated Toeplitz operators. Section 4 introduces the main two problems we are concerned with: existence of bounded symbols and the Reproducing Kernel Thesis. The counterexamples are presented in Section 5; in particular, Sarason’s question on the general existence of bounded symbols is answered in the negative. Section 6 exhibits some classes of model spaces for which the answers to both questions are positive. Finally, in Section 7 we present another class of well behaved truncated Toeplitz operators, namely operators with positive symbols.
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2. Preliminaries on model spaces Basic references for the content of this section are [14,18] for general facts about Hardy spaces and [25] for model spaces and operators. 2.1. Hardy spaces The Hardy space H p of the unit disk D = {z ∈ C: |z| < 1} is the space of analytic functions f on D satisfying f p < +∞, where 2π 1/p iθ p dθ f re f p = sup , 2π 0r<1
1 p < +∞.
0
The algebra of bounded analytic functions on D is denoted by H ∞ . We denote also H0 = zH p . Alternatively, H p can be identified (via radial limits) with the subspace of functions f ∈ Lp = Lp (T) for which fˆ(n) = 0 for all n < 0. Here T denotes the unit circle with normalized Lebesgue measure m. For any ϕ ∈ L∞ , we denote by Mϕ f = ϕf the multiplication operator on L2 ; we have Mϕ = ϕ∞ . The Toeplitz and Hankel operators on H 2 are given by the formulas p
Tϕ = P+ Mϕ ,
Tϕ : H 2 → H 2 ,
Hϕ = P− Mϕ ,
Hϕ : H 2 → H−2 ,
where P+ is the Riesz projection from L2 onto H 2 and P− = I − P+ is the orthogonal projection from L2 onto H−2 = L2 H 2 . In case where ϕ is analytic, Tϕ is just the restriction of Mϕ to H 2 . We have Tϕ∗ = Tϕ and Hϕ∗ = P+ Mϕ P− ; we also denote S = Tz the usual shift operator on H 2 . Evaluations at points λ ∈ D are bounded functionals on H 2 and the corresponding reproducing kernel is kλ (z) = (1 − λz)−1 ; thus, f (λ) = f, kλ , for every function f in H 2 . If ϕ ∈ H ∞ , then kλ is an eigenvector for Tϕ∗ , and Tϕ∗ kλ = ϕ(λ)kλ . By normalizing kλ we obtain hλ = kkλλ2 = 1 − |λ|2 kλ . 2.2. Model spaces Suppose now Θ is an inner function, that is, a function in H ∞ whose radial limits are of modulus one almost everywhere on T. In what follows we consider only nonconstant inner functions. We define the corresponding shift-coinvariant subspace generated by Θ (also called model p p space) by the formula KΘ = H p ∩ ΘH0 , 1 p < +∞. We will be especially interested in the 2 ; it is easy to see that K is Hilbert case, that is, when p = 2. In this case we write KΘ = KΘ Θ also given by
KΘ = H 2 ΘH 2 = f ∈ H 2 : f, Θg = 0, ∀g ∈ H 2 . The orthogonal projection of L2 onto KΘ is denoted by PΘ ; we have PΘ = P+ − ΘP+ Θ. Since the Riesz projection P+ acts boundedly on Lp , 1 < p < ∞, this formula shows that PΘ can also p be regarded as a bounded operator from Lp onto KΘ , 1 < p < ∞.
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The reproducing kernel in KΘ for a point λ ∈ D is the function kλΘ (z) = (PΘ kλ )(z) =
1 − Θ(λ)Θ(z) 1 − λz
;
(2.1)
we denote by hΘ λ the normalized reproducing kernel, hΘ λ (z) =
1 − |λ|2 Θ k (z). 1 − |Θ(λ)|2 λ
(2.2)
Note that, according to (2.1), we have the orthogonal decomposition kλ = kλΘ + ΘΘ(λ)kλ .
(2.3)
We will use the antilinear isometry J : L2 → L2 , given by J (f )(ζ ) = ζf (ζ ); it maps H 2 into H02 = L2 H 2 = H−2 and conversely. More often another antilinear isometry ω = ΘJ will appear, whose main properties are summarized below. Lemma 2.1. Define, for f ∈ L2 , ω(f )(ζ ) = ζf (ζ )Θ(ζ ). Then: (i) ω is antilinear, isometric, onto; (ii) ω2 = Id; (iii) ωPΘ = PΘ ω (and therefore KΘ reduces ω), ω(ΘH 2 ) = H−2 and ω(H−2 ) = ΘH 2 . Θ We define the difference quotient k˜λΘ = ω(kλΘ ) and h˜ Θ λ = ω(hλ ); thus
k˜λΘ (z) =
Θ(z) − Θ(λ) , z−λ
h˜ Θ λ (z) =
1 − |λ|2 Θ(z) − Θ(λ) . z−λ 1 − |Θ(λ)|2
(2.4)
In the sequel we will use the following simple lemmas. Lemma 2.2. Suppose Θ1 , Θ2 are two inner functions, f1 ∈ KΘ1 , f2 ∈ KΘ2 ∩ H ∞ . Then f1 f2 , zf1 f2 ∈ KΘ1 Θ2 . Proof. Obviously zf1 f2 ∈ H 2 . On the other side, f1 ∈ KΘ1 implies f1 = Θ1 zg1 , with g1 ∈ H 2 , and similarly f2 = Θ2 zg2 , g2 ∈ H ∞ . Thus zf1 f2 ∈ Θ1 Θ2 zH 2 . Therefore zf1 f2 ∈ H 2 ∩ Θ1 Θ2 H02 = KΘ1 Θ2 . The claim about f1 f2 is an immediate consequence, since the model spaces are invariant under the backward shift operator S ∗ . 2 Recall that, given two inner functions θ1 , θ2 , we say that θ2 divides θ1 if there exists an inner function θ3 such that θ1 = θ2 θ3 . Lemma 2.3. Suppose that θ and Θ are two inner functions such that θ 3 divides zΘ. Then: (a) θ Kθ ⊂ Kθ 2 ⊂ KΘ ; (b) if f ∈ H ∞ ∩ θ Kθ and ϕ ∈ Kθ + Kθ , then the functions ϕf and ϕf belong to KΘ .
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Proof. Since θ 3 divides zΘ, there exists an inner function θ1 such that zΘ = θ 3 θ1 . In particular it follows from this factorization that θ (0)θ1 (0) = 0, which implies that θ θ1 H 2 ⊂ zH 2 . Using Kθ = H 2 ∩ θ zH 2 , we have θ Kθ = θ H 2 ∩ θ 2 zH 2 ⊂ H 2 ∩ θ 2 zH 2 = Kθ 2 . Further, Kθ 2 = H 2 ∩ θ 2 zH 2 = H 2 ∩ ΘzΘθ 2 H 2 = H 2 ∩ Θθ θ1 H 2 ⊂ H 2 ∩ ΘzH 2 = KΘ , because θ θ1 H 2 ⊂ zH 2 ; thus (a) is proved. Let now f = θf1 and ϕ = ϕ1 + ϕ2 , with f1 ∈ H ∞ ∩ Kθ and ϕ1 , ϕ2 ∈ Kθ . Since ϕ2 ∈ Kθ , using Lemma 2.1, we have ϕ2 = θ zϕ˜2 , with ϕ˜2 ∈ Kθ , which implies that ϕf = θf1 (ϕ1 + ϕ2 ) = θf1 ϕ1 + zf1 ϕ˜ 2 . But it follows from Lemma 2.2 that zf1 ϕ˜ 2 ∈ Kθ 2 ; by (a), we obtain zf1 ϕ˜ 2 ∈ KΘ . So it remains to prove that θf1 ϕ1 ∈ KΘ . Obviously θf1 ϕ1 ∈ H 2 ; moreover, for every function h ∈ H 2 , we have
θf1 ϕ1 , Θh = zθf1 ϕ1 , zΘh = zθf1 ϕ1 , θ 3 θ1 h = zf1 ϕ1 , θ 2 θ1 h = 0, because another application of Lemma 2.2 yields zf1 ϕ1 ∈ Kθ 2 . That proves that θf1 ϕ1 ∈ KΘ and thus ϕf ∈ KΘ . Since KΘ + KΘ is invariant under the conjugation, we obtain also the result for ϕf . 2 2.3. Angular derivatives and evaluation on the boundary The inner function Θ is said to have an angular derivative in the sense of Carathéodory at ζ ∈ T if Θ and Θ have a non-tangential limit at ζ and |Θ(ζ )| = 1. Then it is known [1] that evaluation at ζ is continuous on KΘ , and the function kζΘ , defined by kζΘ (z) :=
1 − Θ(ζ )Θ(z) 1 − ζz
,
z ∈ D,
belongs to KΘ and is the corresponding reproducing kernel. Replacing λ by ζ in the formula (2.4) gives a function k˜ζΘ which also belongs to KΘ and ω(kζΘ ) = k˜ζΘ = ζ Θ(ζ )kζΘ . Moreover we have kζΘ 2 = |Θ (ζ )|1/2 . We denote by E(Θ) the set of points ζ ∈ T where Θ has an angular derivative in the sense of Carathéodory. In [1] and [12] precise conditions are given for the inclusion of kζΘ into Lp (for 1 < p < ∞); namely, if (ak ) are the zeros of Θ in D and σ is the singular measure on T corresponding to the singular part of Θ, then kζΘ ∈ Lp if and only if 1 − |ak |2 dσ (τ ) + < +∞. |ζ − ak |p |ζ − τ |p k
T
We will use in the sequel the following easy result.
(2.5)
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Lemma 2.4. Let 1 < p < +∞ and let Θ be an inner function. Then we have: (a) E(Θ 2 ) = E(Θ); (b) infλ∈D∪E(Θ) kλΘ 2 > 0; (c) for λ ∈ D, we have 2 C kλΘ p kλΘ p 2kλΘ p ,
(2.6)
where C = PΘ −1 Lp →Lp is a constant which depends only on Θ and p. Also, if ζ ∈ E(Θ), 2 Θ p then kζ ∈ L if and only if kζΘ ∈ Lp , and (2.6) holds for λ = ζ . Proof. The proof of (a) is immediate using the definition. For the proof of (b) note that, for λ ∈ D ∪ E(Θ), we have 1 − Θ(0)Θ(λ) = k Θ (λ) k Θ k Θ = 1 − Θ(0)2 1/2 k Θ , λ 2 0 0 2 λ 2 1/2 . which implies kλΘ 2 ( 1−|Θ(0)| 1+|Θ(0)| ) 2
2
It remains to prove (c). We have kλΘ = (1+Θ(λ)Θ)kλΘ , whence PΘ kλΘ = kλΘ . Thus the result follows from the fact that PΘ is bounded on Lp and from the trivial estimate |1 + Θ(λ)Θ(z)| 2, z ∈ T. 2 2.4. The continuous case It is useful to remember the connection with the “continuous” case, for which we refer to [14,22]. If u(w) = w−i w+i , then u is a conformal homeomorphism of the Riemann sphere. It maps −i to ∞, ∞ to 1, R onto T and C+ to D (here C+ = {z ∈ C: Im z > 0}). The operator 1 (Uf )(t) = √ f u(t) π(t + i) maps L2 (T) unitarily onto L2 (R) and H 2 unitarily onto H 2 (C+ ), the Hardy space of the upper half-plane. The corresponding transformation for functions in L∞ is ˜ U(ϕ) = ϕ ◦ u;
(2.7)
it maps L∞ (T) isometrically onto L∞ (R), H ∞ isometrically onto H ∞ (C+ ) and inner functions in D into inner functions in C+ . Now if Θ is an inner function in D, we have UPΘ = P Θ U and then UKΘ = K Θ , where Θ = Θ ◦ u, K Θ = H 2 (C+ ) ΘH 2 (C+ ) and P Θ is the orthogonal projection onto K Θ . Moreover Θ UhΘ λ = cμ hμ
where μ = u−1 (λ) ∈ C+ , cμ =
μ−i |μ+i|
Θ
˜ and U h˜ Θ λ = cμ hμ ,
is a constant of modulus one,
(2.8)
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hΘ μ (ω) =
i √ π
Im μ 1 − Θ(μ)Θ(ω) , 2 ω−μ 1 − |Θ(μ)|
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ω ∈ C+ ,
is the normalized reproducing kernel for K Θ , while Θ h˜ μ (ω) =
1 √
i π
Im μ Θ(ω) − Θ(μ) , 2 ω−μ 1 − |Θ(μ)|
ω ∈ C+ ,
is the normalized difference quotient in K Θ . 3. Truncated Toeplitz operators In [29], D. Sarason studied the class of truncated Toeplitz operators which are defined as the compression of Toeplitz operators to coinvariant subspaces of H 2 . Note first that we can extend the definitions of Mϕ , Tϕ , and Hϕ in Section 2 to the case when the symbol is only in L2 instead of L∞ , obtaining (possibly unbounded) densely defined operators. Then Mϕ and Tϕ are bounded if and only if ϕ ∈ L∞ (and Mϕ = Tϕ = ϕ∞ ), while Hϕ is bounded if and only if P− ϕ ∈ BMO (and Hϕ is equivalent to P− ϕBMO ). In [29], D. Sarason defines an analogous operator on KΘ . Suppose Θ is an inner function and ϕ ∈ L2 ; the truncated Toeplitz operator AΘ ϕ will in general be a densely defined, possibly unbounded, operator on KΘ . Its domain is KΘ ∩ H ∞ , on which it acts by the formula AΘ ϕ f = PΘ (ϕf ),
f ∈ KΘ ∩ H ∞ .
In particular, KΘ ∩ H ∞ contains all reproducing kernels kλΘ , λ ∈ D, and their linear combinations, and is therefore dense in KΘ . We will denote by T (KΘ ) the space of all bounded truncated Toeplitz operators on KΘ . It follows from [29, Theorem 4.2] that T (KΘ ) is a Banach space in the operator norm. Using Lemma 2.1 and the fact that ωMϕ ω = Mϕ , it is easy to check the useful formula Θ ∗ Θ ωAΘ ϕ ω = Aϕ = Aϕ .
(3.1)
Θ We call ϕ a symbol of the operator AΘ ϕ . It is not unique; in [29], it is shown that Aϕ = 0 if and
only if ϕ ∈ ΘH 2 + ΘH 2 . Let us denote SΘ = L2 (ΘH 2 + ΘH 2 ) and PSΘ the corresponding orthogonal projection. Two spaces that contain SΘ up to a subspace of dimension at most 1 admit a direct description, and we will gather their properties in the next two lemmas. Lemma 3.1. Denote by QΘ the orthogonal projection onto KΘ ⊕ zKΘ . Then: (a) QΘ (Θ) = Θ − Θ(0)2 Θ; (b) we have KΘ ⊕ zKΘ = SΘ ⊕ CqΘ , where qΘ = QΘ (Θ)−1 2 QΘ (Θ); (c) QΘ and PSΘ are bounded on Lp for 1 < p < ∞.
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Proof. Since by Lemma 2.1 zKΘ = ΘKΘ , we have KΘ ⊕ zKΘ = KΘ ⊕ ΘKΘ , and therefore QΘ = PΘ + MΘ PΘ MΘ . Thus QΘ is bounded on Lp for all p > 1. Further, if we denote by 1 the constant function equal to 1, then QΘ (Θ) = PΘ (Θ) + MΘ PΘ MΘ (Θ) = PΘ Θ(0)1 + MΘ PΘ 1 = Θ(0) + Θ 1 − Θ(0)Θ = Θ − Θ(0)2 Θ. Thus (a) is proved. Since L2 = ΘH 2 ⊕ ΘH02 ⊕ KΘ ⊕ zKΘ , it follows that SΘ ⊂ KΘ ⊕ zKΘ and thus KΘ ⊕ zKΘ = QΘ SΘ + ΘH 2 + ΘH02 + CΘ = SΘ ⊕ CQΘ (Θ),
(3.2)
which proves (b). Note that according to (a), one easily see that QΘ (Θ) ≡ 0. Now we have for f ∈ Lp , PSΘ f = QΘ f − f, qΘ qΘ ,
(3.3)
and the second term is bounded in Lp , since qΘ belongs to L∞ . This concludes the proof of (c). 2 Lemma 3.2. We have SΘ ⊂ KΘ + KΘ . Each truncated Toeplitz operator has a symbol ϕ of the form ϕ = ϕ+ + ϕ− with ϕ± ∈ KΘ ; any other such decomposition corresponds to ϕ+ + ck0Θ , ϕ− − ck0Θ for some c ∈ C. In particular, ϕ± are uniquely determined if we fix (arbitrarily) the value of one of them in a point of D. Proof. See [29, Section 3].
2
The formulas ψ = limn→∞ zn Tψ (zn ) and P− ψ = Hψ (1) allow one to recapture simply the unique symbol of a Toeplitz operator as well as the unique symbol in H−2 of a Hankel operator. It is interesting to obtain a similar direct formula for the symbol of a truncated Toeplitz operator. Lemma 3.2 says that the symbol is unique if we assume, for instance, that ϕ = ϕ+ + ϕ− , with Θ ˜Θ ϕ± ∈ KΘ and ϕ− (0) = 0. We can then recapture ϕ from the action of AΘ ϕ on kλ and kλ . Indeed, one can check that Θ AΘ ϕ k0 = ϕ+ − Θ(0)Θϕ− , ˜Θ AΘ ϕ k0 = ω ϕ− + ϕ+ (0) − Θ(0)Θϕ+ . Θ Θ From the first equation we obtain ϕ+ (0) = AΘ ϕ k0 , k0 . Then (3.4) imply, for any λ ∈ D,
Θ Θ ϕ+ (λ) − Θ(0)Θ(λ)ϕ− (λ) = AΘ ϕ k0 , kλ ,
Θ Θ Θ ˜Θ ˜Θ ϕ− (λ) − Θ(0)Θ(λ)ϕ+ (λ) = AΘ ϕ k0 , kλ − Aϕ k0 , k0 .
(3.4)
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This is a linear system in ϕ+ (λ) and ϕ− (λ), whose determinant is 1 − |Θ(0)Θ(λ)|2 > 0; therefore, ϕ± can be made explicit in terms of the products in the right-hand side. Note, however, that AΘ ϕ is completely determined by its action on reproducing kernels, so Θ one should be able to recapture the values of the symbol only from AΘ ϕ kλ . The next proposition shows how one can achieve this goal; moreover, one can also obtain an estimate of the L2 -norm of the symbol. Namely, for an inner function Θ and any (not necessarily bounded) linear operator T whose domain contains KΘ ∩ H ∞ , define ρr (T ) := supT hΘ λ 2.
(3.5)
λ∈D
We will have the occasion to come back to the quantity ρr in the next section. To simplify the next statement, denote Θ Θ Θ ∗ Fλ,μ = I − λS ∗ ω AΘ ϕ kλ − I − μS ω Aϕ kμ ,
λ, μ ∈ D.
(3.6)
Proposition 3.3. Let Θ be an inner function, AΘ ϕ a truncated Toeplitz operator, and μ ∈ D such that Θ(μ) = 0. Suppose ϕ = ϕ+ + ϕ− is the unique decomposition of the symbol with ϕ± ∈ KΘ , ϕ− (μ) = 0. Then (S − μ)(I − μS ∗ )−1 Fλ,μ , kμΘ
ϕ− (λ) =
Θ(μ)(Θ(0)Θ(μ) − 1)
,
λ ∈ D,
(3.7)
and ϕ+ = ω(ψ+ ), where Θ ∗ ψ+ = I − μS ∗ ω AΘ ϕ kμ + Θ(μ)S ϕ− .
(3.8)
Moreover, there exists a constant C depending only on Θ and μ such that
max ϕ− 2 , ϕ+ 2 Cρr AΘ ϕ .
(3.9)
Proof. First note that for any λ ∈ D, we have Θ ∗ ∗ I − λS ∗ ω AΘ ϕ kλ = ψ+ + ϕ− (λ)S Θ − Θ(λ)S ϕ− .
(3.10)
Indeed, Θ PΘ ϕ+ kλ = PΘ ϕ+
1 1 − λz
= ϕ+ + λPΘ
Θzψ+ z−λ
ψ+ − ψ+ (λ) . = ϕ+ + λΘz z−λ
Thus, ψ+ − ψ+ (λ) zψ+ − λψ+ (λ) Θ = . ω AΘ ϕ+ kλ = ψ+ + λ z−λ z−λ
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One can easily check that −1 f − f (λ) I − λS ∗ S ∗ f = , z−λ
(3.11)
for every function f ∈ H 2 ; then we obtain
Θ I − λS ∗ ω AΘ ϕ+ kλ = ψ+ .
(3.12)
On the other hand, ϕ− − ϕ− (λ) ϕ− (λ) ϕ− − ϕ− (λ) ϕ− (λ) + − Θ(λ)zΘ − Θ(λ)Θ PΘ ϕ− kλΘ = PΘ z z−λ 1 − λz z−λ 1 − λz ϕ− − ϕ− (λ) . = ϕ− (λ)kλΘ − Θ(λ)zΘ z−λ Hence, ϕ− − ϕ− (λ) Θ − Θ(λ) Θ − Θ(λ) ω AΘ ϕ− kλ = ϕ− (λ) z−λ z−λ and Θ ∗ ∗ I − λS ∗ ω AΘ ϕ− kλ = ϕ− (λ)S Θ − Θ(λ)S ϕ− .
(3.13)
Thus (3.10) follows immediately from (3.12) and (3.13). If we take λ = μ in (3.10), we get (remembering that ϕ− (μ) = 0) Θ ∗ ψ+ = I − μS ∗ ω AΘ ϕ kμ + Θ(μ)S ϕ− .
(3.14)
Now plugging (3.14) into (3.10) yields ϕ− (λ)S ∗ Θ + Θ(μ) − Θ(λ) S ∗ ϕ− = Fλ,μ . Therefore, applying (S − μ)(I − μS ∗ )−1 and using ϕ− (μ) = 0 and (3.11), we obtain −1 ϕ− (λ) Θ − Θ(μ) + Θ(μ) − Θ(λ) ϕ− = (S − μ) I − μS ∗ Fλ,μ .
(3.15)
Finally, we take the scalar product of both sides with kμΘ and use the fact that Θ ⊥ KΘ , PΘ 1 = 1 − Θ(0)Θ, and again ϕ− (μ) = 0. Therefore −1
−ϕ− (λ)Θ(μ) 1 − Θ(0)Θ(μ) = (S − μ) I − μS ∗ Fλ,μ , kμΘ , which immediately implies (3.7).
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To obtain the boundedness of the L2 norms, fix now λ ∈ D such that Θ(λ) = Θ(μ). Since I − μS ∗ ω AΘ k Θ 2AΘ k Θ 2k Θ ρr AΘ ϕ μ ϕ μ 2 μ 2 ϕ 2 Θ Θ and a similar estimate holds for (I − λS ∗ )ω(AΘ ϕ kλ )2 , we have Fλ,μ 2 C1 ρ(Aϕ ), where C1 , as well as the next constants appearing in this proof, depends only on Θ, λ, μ. By (3.15), it follows that
ϕ− (λ) Θ − Θ(μ) + Θ(μ) − Θ(λ) ϕ− C2 ρr AΘ . ϕ 2 Projecting onto KΘ decreases the norm; since PΘ (ϕ− (λ)Θ) = 0 and PΘ (1) = k0Θ , we obtain −Θ(μ)ϕ− (λ)k Θ + Θ(μ) − Θ(λ) ϕ− C2 ρr AΘ . ϕ 0 2 Write now ϕ− = h + ck0Θ with h ⊥ k0Θ . Then (Θ(μ) − Θ(λ))h2 C2 ρr (AΘ ϕ ), whence Θ (μ) = 0, which implies that h2 C3 ρr (AΘ ). Since ϕ (μ) = 0, we have h(μ) + ck − ϕ 0 −1 |c| = k0Θ (μ) h(μ) C4 ρr AΘ ϕ . Therefore we have ϕ− 2 C5 ρr (AΘ ϕ ). Finally, (3.8) yields a similar estimate for ψ+ and then for ϕ+ . 2 The following proposition yields a relation between truncated Toeplitz operators and usual Hankel operators. Proposition 3.4. With respect to the decompositions H−2 = ΘKΘ ⊕ ΘH−2 , H 2 = KΘ ⊕ ΘH 2 , the operator HΘ∗ HΘϕ HΘ∗ : H−2 → H 2 has the matrix
AΘ ϕ MΘ 0
0 . 0
(3.16)
Proof. If f ∈ ΘH−2 , then HΘ∗ f = 0. If f ∈ ΘKΘ , then HΘ∗ f = Θf ∈ KΘ . Since PΘ = P+ MΘ P− MΘ , it follows that, for f ∈ KΘ , ∗ AΘ ϕ f = PΘ Mϕ f = P+ MΘ P− MΘ Mϕ f = HΘ HΘϕ f, ∗ ∗ and therefore, if f ∈ ΘKΘ , then AΘ ϕ Θf = HΘ HΘϕ HΘ f as required.
2
The non-zero entry in (3.16) consists in the isometry MΘ : ΘKΘ → KΘ , followed by AΘ ϕ acting on KΘ . There is therefore a close connection between properties of AΘ and properties ϕ of the corresponding product of three Hankel operators. Such products of Hankel operators have been studied for instance in [4,7,33].
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Remark 3.5. Truncated Toeplitz operators can be defined also on model spaces of H 2 (C+ ), that is, K Θ = H 2 (C+ ) ΘH 2 (C+ ) for an inner function Θ in the upper half-plane C+ . We start then with a symbol ϕ ∈ (t + i)L2 (R) (which contains L∞ (R)) and define (for f ∈ K Θ ∩ (z + i)−1 H ∞ (C+ ), a dense subspace of K Θ ) the truncated Toeplitz operator AΘ ϕ f = PΘ (ϕf ). Let us briefly explain the relations between the truncated Toeplitz operators corresponding to model spaces on the upper half-plane and those corresponding to model spaces on the unit disk. If Θ = Θ ◦ u−1 and ψ = ϕ ◦ u−1 , using the fact that UPΘ U ∗ = PΘ and UMψ = Mϕ U , we easily obtain Θ ∗ AΘ ϕ = UAψ U .
In particular, if A is a linear operator on K Θ , then A is a truncated Toeplitz operator on K Θ if and only if A = U ∗ AU is a truncated Toeplitz operator on KΘ , and ϕ is a symbol for A if and only if ψ := ϕ ◦ u−1 is a symbol for A. It follows that A is bounded (or has a bounded symbol) if and only if A is bounded (respectively, has a bounded symbol). Moreover we easily deduce from (2.8) that Θ Θ A h = AΘ hΘ ϕ μ 2 ψ λ 2
Θ Θ ˜Θ ˜ and AΘ ϕ hμ 2 = Aψ hλ 2 ,
for every μ ∈ C+ and λ = u(μ). Finally, the truncated Toeplitz operator AΘ ϕ = 0 if and only if the symbol ϕ ∈ (t + i)(ΘH 2 (C+ ) ⊕ ΘH 2 (C+ )) (note that the sum is in this case orthogonal, since H 2 (C+ ) ⊥ H 2 (C+ )). 4. Existence of bounded symbols and the Reproducing Kernel Thesis As noted in Section 3, a Toeplitz operator Tϕ has a unique symbol, Tϕ is bounded if and only if this symbol is in L∞ , and the map ϕ → Tϕ is isometric from L∞ onto the space of bounded Toeplitz operators on H 2 . The situation is more complicated for Hankel operators: there is no uniqueness of the symbol, while the map ϕ → Hϕ is contractive and onto from L∞ to the space of bounded Hankel operators (the boundedness condition P− ϕ ∈ BMO is equivalent to the fact that any bounded Hankel operator has a symbol in L∞ ). ∞ In the case of truncated Toeplitz operators, the map ϕ → AΘ ϕ is again contractive from L to T (KΘ ). It is then natural to ask whether it is onto, that is, whether any bounded truncated Toeplitz operator is a compression of a bounded Toeplitz operator in H 2 . This question has been asked by Sarason in [29]. Question 1. Does every bounded truncated Toeplitz operator on KΘ possess an L∞ symbol? One may expect the answer to depend on the function Θ, and indeed we show below that it is the case. Assume that for some inner function Θ, any operator in T (KΘ ) has a bounded symbol. Then if follows from the open mapping theorem that there exists a constant C such that for any A ∈ T (KΘ ) one can find ϕ ∈ L∞ with A = AΘ ϕ and ϕ∞ CA. A second natural question that may be asked about truncated Toeplitz operators is the Reproducing Kernel Thesis (RKT). This is related to the quantity ρr defined in (3.5). The functions hΘ λ Θ Θ have all norm 1, so if AΘ ϕ is bounded then obviously ρr (Aϕ ) Aϕ . The following question is then natural:
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Question 2 (RKT for truncated Toeplitz operators). let Θ be an inner function and ϕ ∈ L2 . Θ Assume that ρr (AΘ ϕ ) < +∞. Is Aϕ bounded on KΘ ? As we have seen in the introduction, the RKT is true for various classes of operators related to the truncated Toeplitz operators, and it seems natural to investigate it for this class. We will see in Section 5 that the answer to this question is in general negative. As we will show below, it is more natural to restate the RKT by including in the hypothesis ∞ also the functions h˜ Θ λ . Thus, for any linear operator T whose domain contains KΘ ∩ H , define ρd (T ) = supT h˜ Θ λ 2, λ∈D
and ρ(T ) = max{ρr (T ), ρd (T )}. The indices r and d in notation ρr and ρd stand for “reproducing kernels” and “difference quotients”. Θ Θ ∗ Note that if AΘ ϕ is a truncated Toeplitz operator, then by (3.1), we have ρd (Aϕ ) = ρr ((Aϕ ) ), and then Θ Θ ∗
. ρ AΘ ϕ = max ρr Aϕ , ρr Aϕ Θ Question 3. Let Θ be an inner function and ϕ ∈ L2 . Assume that ρ(AΘ ϕ ) < ∞. Is Aϕ bounded on KΘ ?
In Section 5, we will show that the answer to Questions 1 and 2 may be negative. Question 3 remains in general open. In Section 6, we will give some examples of spaces KΘ on which the answers to Questions 1 and 3 are positive. In the rest of this section we will discuss the existence of bounded symbols and the RKT for some simple cases. First, it is easy to deal with analytic or antianalytic symbols. The next proposition is a straightforward consequence of Bonsall’s theorem [6] and the commutant lifting theorem. The equivalence between (i) and (ii) has already been noticed in [29]. Proposition 4.1. Let ϕ ∈ H 2 and let AΘ ϕ be a truncated Toeplitz operator. Then the following assertions are equivalent: (i) AΘ ϕ has a bounded symbol; (ii) AΘ ϕ is bounded; (iii) ρr (AΘ ϕ ) < +∞. More precisely there exists a universal constant C > 0 such that any bounded truncated Toeplitz 2 Θ operator AΘ ϕ (ϕ ∈ H ) has a bounded analytic symbol ϕ0 with ϕ0 ∞ Cρr (Aϕ ). Proof. It is immediate that (i) ⇒ (ii) ⇒ (iii). The implication (ii) ⇒ (i) has already been noted Θ Θ in [29]; indeed if ϕ ∈ H 2 and AΘ ϕ is bounded, then Aϕ commutes with SΘ := Az and then, by a ∞ symbol with norm equal to the norm corollary of the commutant lifting theorem, AΘ ϕ has an H Θ of Aϕ .
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Θ So it remains to prove that there exists a constant C > 0 such that AΘ ϕ Cρr (Aϕ ). If ∞ 2 Θ f ∈ KΘ ∩ H , then ϕf ∈ H . Therefore PΘ (ϕf ) = ΘP− (Θϕf ), or, in other words, Aϕ (f ) = ΘHΘϕ f . On the other hand, ΘH 2 ⊂ ker HΘϕ , and therefore, with respect to the decompositions H 2 = KΘ ⊕ ΘH 2 , H−2 = ΘKΘ ⊕ ΘH−2 , one can write
HΘϕ =
ΘAΘ ϕ 0
0 . 0
(4.1)
It follows that AΘ ϕ is bounded if and only if HΘϕ is. By Bonsall’s Theorem [6], there exists a universal constant C (independent of ϕ and Θ) such that the boundedness of HΘϕ is equivalent to supλ∈D HΘϕ hλ 2 < ∞, and HΘϕ C sup HΘϕ hλ 2 . λ∈D
But, again by (4.1) and using (2.1) and (2.2), we have 2 1/2 Θ Θ HΘϕ hλ = ΘAΘ Aϕ hλ , ϕ PΘ hλ = Θ 1 − Θ(λ) Θ Θ and thus supλ∈D HΘϕ hλ 2 supλ∈D AΘ ϕ hλ 2 = ρr (Aϕ ). The proposition is proved.
2
A similar result is valid for antianalytic symbols. Proposition 4.2. Let ϕ ∈ H 2 and let AΘ ϕ be a truncated Toeplitz operator. Then the following assertions are equivalent: (i) AΘ ϕ has a bounded symbol; (ii) AΘ ϕ is bounded; (iii) ρd (AΘ ϕ ) < +∞. More precisely there exists a universal constant C > 0 such that any bounded truncated Toeplitz Θ 2 operator AΘ ϕ (ϕ ∈ H ) has a bounded antianalytic symbol ϕ0 with ϕ0 ∞ Cρd (Aϕ ). Θ Θ ∗ 2 Proof. Suppose ϕ ∈ H 2 . Since AΘ ϕ = (Aϕ ) = Aϕ , and ϕ ∈ H , we may apply Proposition 4.1 to AΘ ϕ because by (3.1), we have
Θ Θ Θ Θ Θ Θ Θ ˜ ρr AΘ ϕ = sup Aϕ hλ 2 = sup Aϕ ωhλ 2 = sup Aϕ hλ 2 = ρd Aϕ . λ∈D
λ∈D
2
λ∈D
As we have seen, if ϕ is bounded, then obviously the truncated Toeplitz operator AΘ ϕ is bounded. We will see now that one can get a slightly more general result. It involves the socalled Carleson curves associated with an inner function (see for instance [18]). Recall that if Θ is an inner function and α ∈ (0, 1), then the system of Carleson curves Γα associated to Θ and α is the countable union of closed simple and rectifiable curves in clos D such that:
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• The interior of curves in Γα are pairwise disjoint. • There is a constant η(α) > 0 such that for every z ∈ Γα ∩ D we have η(α) Θ(z) α.
(4.2)
• Arclength |dz| on Γα is a Carleson measure, which means that there is a constant C > 0 such that f (z)2 |dz| Cf 2 , 2 Γα
for every function f ∈ H 2 . • For every function ϕ ∈ H 1 , we have ϕ(z) ϕ(z) dz = dz. Θ(z) Θ(z) T
(4.3)
Γα
Proposition 4.3. Let ϕ ∈ H 2 and assume that |ϕ||dz| is a Carleson measure on Γα . Then AΘ ϕ is a bounded truncated Toeplitz operator on KΘ and it has a bounded symbol. Proof. Let f, g ∈ KΘ and assume further that f ∈ H ∞ . Then we have
Θ Aϕ f, g = ϕf, g = ϕ(z)f (z)g(z) dz. T
Since g ∈ KΘ , we can write (on T), g(z) = zh(z)Θ(z), with h ∈ KΘ . Therefore
Θ zϕ(z)f (z)h(z) dz. Aϕ f, g = Θ(z) T
But zf (z)ϕ(z)h(z) ∈ H 1 and using (4.3), we can write Θ
zϕ(z)f (z)h(z) dz. Aϕ f, g = Θ(z) Γα
Therefore, according to (4.2), we have Θ
A f, g |zϕ(z)f (z)h(z)| |dz| 1 f (z)h(z)ϕ(z)|dz|. ϕ |Θ(z)| η(α) Γα
Γα
Hence, by the Cauchy–Schwarz inequality and using the fact that |ϕ||dz| is a Carleson measure on Γα , we have Θ
A f, g C 1 f 2 g2 . ϕ η(α)
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Θ Finally, we get that AΘ ϕ is bounded. Since ϕ is analytic it follows from Proposition 4.1 that Aϕ has a bounded symbol. 2
Corollary 4.4. Let ϕ = ϕ1 + ϕ2 , with ϕi ∈ H 2 , i = 1, 2. Assume that |ϕi ||dz| are Carleson measures on Γα for i = 1, 2. Then AΘ ϕ is bounded and has a bounded symbol. Proof. Using Proposition 4.3, we get immediately that AΘ ϕi is bounded and has a bounded symbol Θ Θ ∗ ϕ˜i , for i = 1, 2. Therefore, Aϕ2 = (Aϕ2 ) is also bounded and has a bounded symbol ϕ˜2 . Hence Θ Θ we get that AΘ ϕ = Aϕ1 + Aϕ2 is bounded and it has a bounded symbol, say ϕ˜ 1 + ϕ˜ 2 .
2
Remark 4.5. By the construction of the Carleson curves Γα associated to an inner function Θ, we know that |dz| is a Carleson measure on Γα . Therefore, Proposition 4.3 can be applied if ϕ is bounded on Γα and Corollary 4.4 can be applied if ϕ1 , ϕ2 are bounded on Γα . 5. Counterexamples We will show that under certain conditions on the inner function Θ there exist rank one bounded truncated Toeplitz operators that have no bounded symbol. It is proven in [29, Theorem 5.1] that any rank one truncated Toeplitz operator is either of the form kλΘ ⊗ k˜λΘ or k˜λΘ ⊗ kλΘ for λ ∈ D, or of the form kζΘ ⊗ kζΘ where ζ ∈ T and Θ has an angular derivative at ζ . In what follows we will use a representation of the symbol of a rank one operator which differs slightly from the one given in [29]. 2 Lemma 5.1. If λ ∈ D ∪ E(Θ), then ϕλ = ΘzkλΘ ∈ KΘ ⊕ zKΘ is a symbol for k˜λΘ ⊗ kλΘ . In 2
particular, if ζ ∈ E(Θ), then ϕζ = ΘzkζΘ is a symbol for Θ(ζ )ζ kζΘ ⊗ kζΘ . 2
Proof. If ζ ∈ E(Θ), then by Lemma 2.4, Θ 2 has an angular derivative at ζ , and so kζΘ ∈ KΘ 2 = 2
KΘ ⊕ ΘKΘ . It follows from Lemma 2.1 that ΘzkλΘ ∈ KΘ ⊕ zKΘ for λ ∈ D ∪ E(Θ). Take g, h ∈ KΘ , and, moreover, let g ∈ L∞ . Then
Θ Aϕλ g, h = ϕλ g, h =
2
ΘzkλΘ gh dm. T
But Θzh = ω(h) ∈ KΘ , g ∈ KΘ ∩ L∞ , and so by Lemma 2.2 gΘzh ∈ KΘ 2 . Therefore T
2 2 ΘzkλΘ gh dm = gω(h), kλΘ = g(λ) ω(h) (λ) = g, kλΘ ω(h), kλΘ
= g, kλΘ h, ω kλΘ = g, kλΘ h, k˜λΘ = k˜λΘ ⊗ kλΘ g, h .
Θ Θ ˜Θ ˜Θ Therefore AΘ ϕλ = kλ ⊗ kλ as claimed. Finally, recall that, for ζ ∈ E(Θ), we have kζ = ω(kζ ) = Θ(ζ )ζ kζΘ . 2
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The construction of bounded truncated Toeplitz operators that have no bounded symbol is based on the next lemma. Lemma 5.2. Let Θ be an inner function and 1 < p < ∞. There exists a constant C depending only on Θ and p such that, if ϕ, ψ ∈ L2 are two symbols for the same truncated Toeplitz operator, with ϕ ∈ KΘ ⊕ zKΘ , then ϕp C ψp + ϕ2 . In particular, if ψ ∈ Lp , then ϕ ∈ Lp . Proof. By hypothesis PSΘ ϕ = PSΘ ψ ; therefore, using (3.3), ϕ = QΘ ϕ = PSΘ ϕ + ϕ, qΘ qΘ = PSΘ ψ + ϕ, qΘ qΘ . By Lemma 3.1 we have PSΘ ψp C1 ψp , while ϕ, qΘ qΘ ϕ2 · qΘ p , p whence the lemma follows.
2
If Θ is an inner function and ζ ∈ E(Θ), then, as noted above, kζΘ ⊗ kζΘ is a rank one operator in T (KΘ ). In [29] Sarason has asked specifically whether this operator has a bounded symbol. We can now show that in general this question has a negative answer. Theorem 5.3. Suppose that Θ is an inner function which has an angular derivative at ζ ∈ T. Let p ∈ (2, +∞). Then the following are equivalent: (1) the bounded truncated Toeplitz operator kζΘ ⊗ kζΘ has a symbol ψ ∈ Lp ; (2) kζΘ ∈ Lp . / Lp for some p ∈ (2, ∞), then kζΘ ⊗ kζΘ is a bounded truncated Toeplitz In particular, if kζΘ ∈ operator with no bounded symbol. 2
Proof. A symbol for the operator kζΘ ⊗ kζΘ is, by Lemma 5.1, ϕ = Θ(ζ )ζ ΘzkζΘ . Since by Lemma 2.4 ϕ ∈ Lp if and only if kζΘ ∈ Lp , we obtain that (2) implies (1). Conversely, assume that ψ ∈ Lp is a symbol for kζΘ ⊗ kζΘ . We may then apply Lemma 5.2 and obtain that ϕ ∈ Lp . Once again according to Lemma 2.4, we get that kζΘ ∈ Lp , which proves that (1) implies (2). 2 To obtain a bounded truncated Toeplitz operator with no bounded symbol, it is sufficient to have a point ζ ∈ T such that (2.5) is true for p = 2 but not for some strictly larger value of p. It is now easy to give concrete examples, as, for instance:
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(1) a Blaschke product with zeros ak accumulating to the point 1, and such that 1 − |ak |2 k
|1 − ak
(2) a singular function σ = k
1 − |ak |2 = +∞ |1 − ak |p
< +∞,
|2
for some p > 2;
k
with
k c k δζ k
k ck
ck < +∞, |1 − ζk |2
k
< +∞, ζk → 1, and
ck = +∞ |1 − ζk |p
for some p > 2.
Remark 5.4. A related question raised in [29] remains open. Let μ be a positive measure on T such that the support of the singular part of μ (with respect to the Lebesgue measure) is contained in T \ σ (Θ), where σ (Θ) is the spectrum of the inner function Θ. Then we say that μ is a Carleson measure for KΘ if there is a constant c > 0 such that |f |2 dμ cf 22 ,
f ∈ KΘ .
(5.1)
T
It is easy to see (and had already been noticed in [11]) that (5.1) is equivalent to the boundedness of the operator AΘ μ defined by the formula
AΘ μ f, g
=
f g dμ,
f, g ∈ KΘ ;
(5.2)
T
it is shown in [29] that AΘ μ is a truncated Toeplitz operator. More generally, a complex measure ν on T is called a Carleson measure for KΘ if its total variation |ν| is a Carleson measure for KΘ . In this case there is a corresponding operator AΘ ν , defined also by formula (5.2), which ∞ belongs to T (KΘ ). Now if a truncated Toeplitz operator AΘ ϕ has a bounded symbol ψ ∈ L Θ then the measure dμ = ψ dm is a Carleson measure for KΘ and AΘ ϕ = Aμ . The natural question Θ whether every operator in T (KΘ ) is of the form Aμ (for some Carleson measure μ for KΘ ) is not answered by our counterexample; indeed (as already noticed in [29]) if Θ has an angular derivative in the sense of Carathéodory at ζ ∈ T, then δζ is a Carleson measure for KΘ and kζΘ ⊗ kζΘ = AΘ δζ . Remark 5.5. We arrive at the same class of counterexamples as in Theorem 5.3 if we follow an idea due to Sarason [29] (we would like to emphasize that our first counterexample was obtained in this way). It is shown in [29, Section 5] that, for an inner function Θ which has an angular derivative at the point ζ ∈ T, the rank one operator kζΘ ⊗ kζΘ has a bounded symbol if and only if there exists a function h ∈ H 2 such that Re
Θ(ζ ) Θ 1 − ζz
+ Θh ∈ L∞ .
(5.3)
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Since Re(1 − ζ z)−1 = 1/2 a.e. on T, condition (5.3) is, obviously, equivalent to Re kζΘ + Θh ∈ L∞ . Then, by the M. Riesz theorem, kζΘ + Θh ∈ Lp for any p ∈ (2, ∞) and the boundedness of the projection PΘ in Lp implies that kζΘ ∈ Lp . The next theorem provides a wider class of examples. Theorem 5.6. Suppose that Θ is an inner function with the property that each bounded operator in T (KΘ ) has a bounded symbol. Then for each p > 2 we have sup λ∈D
kλΘ p kλΘ 22
< ∞.
(5.4)
Proof. As mentioned in the previous section, it follows from the open mapping theorem that there exists a constant C > 0 such that for any operator A ∈ T (KΘ ) one can always find a symbol ψ ∈ L∞ with ψ∞ CA. Fix λ ∈ D, and consider the rank one operator k˜λΘ ⊗ kλΘ , which has operator norm kλΘ 22 . Θ ˜Θ Therefore there exists ψλ ∈ L∞ with AΘ ψλ = kλ ⊗ kλ and 2 ψλ p ψλ ∞ C kλΘ 2 .
(5.5)
2 On the other hand, ϕλ = ΘzkλΘ ∈ KΘ ⊕ zKΘ is also a symbol for k˜λΘ ⊗ kλΘ by Lemma 5.1. Applying Lemma 5.2, it follows that there exists a constant C1 > 0 such that
ϕλ p C1 ψλ p + ϕλ 2 . By (2.6) and Lemma 2.4 (b), we have 2 2 ϕλ 2 = kλΘ 2 2kλΘ 2 C2 kλΘ 2 .
(5.6)
Therefore (5.5) and (5.6) yield 2 ϕλ p C1 (C + C2 )kλΘ 2 . 2
Since ϕλ p = kλΘ p , using once more (2.6) concludes the proof.
2
It is easy to see that if there exists ζ ∈ E(Θ) such that kζΘ ∈ / Lp , then sup r<1
Θ krζ p Θ 2 krζ 2
= ∞.
Therefore the existence of operators in T (KΘ ) without bounded symbol, under the hypothesis of Theorem 5.3, is also a consequence of Theorem 5.6. Note however that Theorem 5.6 does
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not show that the particular operator kζΘ ⊗ kζΘ is a bounded truncated Toeplitz operator without bounded symbol. A larger class of examples is described below. Example 5.7. Let Θ be a Blaschke product such that for some sequence of its zeros zn and some points ζn ∈ T (which are “close to zn ”), we have, for some p ∈ (2, ∞), Θ (ζn ) = k Θ 2 1 − |zn | , ζn 2 |ζn − zn |2
Θ p k 1 − |zn | ζn p |ζn − zn |p
(5.7)
(notation X Y means that the fraction X/Y is bounded above and below by some positive constants), and 1− p1
(1 − |zn |) lim n→+∞ |ζn − zn |
= 0.
(5.8)
Condition (5.7) means that the main contribution to the norms of kζΘn is due to the closest zero zn . Then, by Theorem 5.6, there exists a bounded truncated Toeplitz operator without bounded symbol. Such examples may be easily constructed. Take a sequence wk ∈ D such that wk → ζ and (1 − |wk |)γ =0 k→+∞ |wk − ζ | lim
for some ζ ∈ T and γ ∈ (0, 1). Then it is not difficult to see that for any p > max(2, (1 − γ )−1 ) one can construct recurrently a subsequence zn = wkn of wk and a sequence ζn ∈ T with the properties (5.7) and (5.8). Although related to the examples of Theorem 5.3, this class of examples may be different. Indeed, it is possible that Θ has no angular derivative at ζ , e.g., if 1 − |zn | = |ζ − zn |2 . Also, if the zeros tend to ζ “very tangentially”, it is possible that kζΘ is in Lp for any p ∈ (2, ∞), but there exists a bounded truncated Toeplitz operator without a bounded symbol. We pass now to the Reproducing Kernel Thesis. The next example shows that in general Question 2 has a negative answer. Example 5.8. Suppose Θ is a singular inner function and s ∈ [0, 1). Then AΘ k Θ = PΘ Θs λ = PΘ
Θ s − Θ(λ)Θ 1−s 1 − λz
Θ s − Θ(λ)s + Θ(λ)s (1 − Θ(λ)1−s Θ 1−s )
= PΘ z
Θs
= PΘ zk˜λ
Θs
− Θ s (λ)
1 − λz + Θ(λ)s PΘ
z−λ 1−s + Θ(λ)s PΘ kλΘ .
1 − Θ(λ)1−s Θ 1−s 1 − λz
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The first term zk˜λΘ is in zH 2 , which is orthogonal to KΘ , while the second kλΘ in KΘ 1−s ⊂ KΘ . Therefore we have s
k Θ = Θ(λ)s kλΘ AΘ Θs λ
1−s
2693 1−s
is contained
,
and 2−2s Θ Θ 2 A s k = Θ(λ)2s 1 − |Θ(λ)| , λ 2 Θ 1 − |λ|2
It is easy to see that supy∈[0,1)
y s −y 1−y
Θ Θ 2 |Θ(λ)|2s (1 − |Θ(λ)|2−2s ) A s h = . Θ λ 2 1 − |Θ(λ)|2
1 − s → 0 when s → 1, and therefore
Θ Θ 2 A s h → 0 for s → 1. = sup ρr AΘ s Θ Θ λ 2 λ∈D
On the other hand, Θ s KΘ 1−s ⊂ KΘ and Θ s (Θ s KΘ 1−s ) = KΘ 1−s ⊂ KΘ ; therefore AΘ acts Θs isometrically on Θ s KΘ 1−s , so it has norm 1. Thus there is no constant M such that Θ A M sup ρr AΘ ϕ
λ∈D
ϕ
for all ϕ. It seems natural to deduce that in the previous example we may actually have a truncated Toeplitz operator which is uniformly bounded on reproducing kernels but not bounded. This is indeed true, by an abstract argument based on Proposition 3.3. Note that the quantity ρr introduced in (3.5) is a norm, and ρr (T ) T , for every linear operator T whose domain contains H ∞ ∩ KΘ . Proposition 5.9. Assume that for any (not necessarily bounded) truncated Toeplitz operator A on KΘ the inequality ρr (A) < ∞ implies that A is bounded. Then T (KΘ ) is complete with respect to ρr , and ρr is equivalent to the operator norm on T (KΘ ). Proof. Fix μ ∈ D such that Θ(μ) = 0. Let AΘ ϕn be a ρr -Cauchy sequence in T (KΘ ). Suppose all ϕn are written as ϕn = ϕn,+ + ϕn,− , with ϕn,+ , ϕn,− ∈ KΘ , and ϕn,− (μ) = 0. According to (3.9), the sequences ϕn,± are Cauchy sequences in KΘ and thus converge to functions ϕ± ∈ KΘ ; moreover we also have ϕ− (μ) = 0 (because norm convergence in H 2 implies pointwise convergence). Define then ϕ = ϕ+ + ϕ− ∈ L2 . By (3.10), we have Θ ∗ −1 AΘ ω(ϕn,+ ) + ϕn,− (λ)S ∗ Θ − Θ(λ)S ∗ ϕn,− , ϕn kλ = ω I − λS Θ Θ Θ Θ so the sequence AΘ ϕn kλ tends (in KΘ ) to Aϕ kλ , for all λ ∈ D. In particular, we have ρr (Aϕ ) < Θ Θ Θ +∞, whence Aϕ ∈ T (KΘ ). Now it is easy to see that Aϕn → Aϕ in the ρr -norm. Thus T (KΘ ) is indeed complete with respect to the ρr -norm. The equivalence of the norms is then a consequence of the open mapping theorem. 2
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Proposition 5.9 and Example 5.8 imply that, if Θ is a singular inner function, then there exist Θ Θ truncated Toeplitz operators AΘ ϕ with ρr (Aϕ ) finite, but Aϕ unbounded. Therefore Question 2 has a negative answer for a rather large class of inner functions Θ. If we consider such a truncated Toeplitz operator, then its adjoint, AΘ ϕ , is an unbounded truncated Toeplitz operator with Θ Θ ρd (Aϕ ) = ρr (Aϕ ) < +∞. It is easy to see, however, that in Example 5.8 ρd (AΘ ) = 1 for all s < 1. This suggests that we Θs should rather consider boundedness of the action of the operator on both the reproducing kernels and the difference quotients, and that the quantity ρ might be a better estimate for the norm of a truncated Toeplitz operator than either ρr or ρd . We have been thus lead to formulate Question 3 as a more relevant variant of the RKT; further arguments will appear in the next section. 6. Positive results There are essentially two cases in which one can give positive answers to Questions 1 and 3. There are similarities between them: in both one obtains a convenient decomposition of the symbol in three parts: one analytic, one coanalytic, and one that is neither analytic nor coanalytic, but well controlled. 6.1. A general result As we have seen in Proposition 4.1 and 4.2, the answers to Questions 1 and 3 are positive for classes of truncated Toeplitz operators corresponding to analytic and coanalytic symbols. We complete these propositions with a different boundedness result, which covers certain cases when the symbol is neither analytic nor coanalytic. The proof is based on an idea of Cohn [13]. Theorem 6.1. Suppose θ and Θ are two inner functions such that θ 3 divides zΘ and Θ divides θ 4 . If ϕ ∈ Kθ + Kθ then ϕ∞ 2ρr (AΘ ϕ ). Proof. Using Lemma 2.3, if f ∈ L∞ ∩ θ Kθ , then f ∈ KΘ and ϕf ∈ KΘ ; thus AΘ ϕ f = ϕf . If Θ 2 we write f = θf1 , ϕ1 = θ ϕ, then ϕ1 ∈ H , f1 ∈ Kθ , and ϕ1 f1 = ϕf = Aϕ f ∈ KΘ . Therefore, for λ ∈ D,
ϕ1 (λ)f1 (λ) = ϕ1 f1 , k Θ = θf1 , ϕk Θ = θf1 , AΘ k Θ ϕ λ λ λ Θ Θ Θ f1 AΘ ϕ kλ 2 f1 kλ 2 ρr Aϕ , where we used the fact that θf1 ∈ KΘ . For a fixed λ ∈ D, sup
f1 ∈Kθ ∩L∞ f1 2 1
f1 (λ) =
sup
f1 ∈Kθ ∩L∞ f1 2 1
f1 , k θ = k θ , λ λ 2
and thus 2 1/2 Θ ϕ1 (λ) ρr AΘ kλ 2 = ρr AΘ (1 − |Θ(λ)| ) . ϕ ϕ (1 − |θ (λ)|2 )1/2 kλθ 2
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If Θ divides θ 4 , then |Θ(λ)| |θ (λ)|4 , and therefore 8 2 2 1 − Θ(λ) 1 − θ (λ) 4 1 − θ (λ) . Θ It follows that |ϕ1 (λ)| 2ρr (AΘ ϕ ) for all λ ∈ D, and thus ϕ1 ∞ 2ρr (Aϕ ). The proof is finished by noting that ϕ∞ = ϕ1 ∞ . 2
As a consequence, we obtain a general result for the existence of bounded symbols and Reproducing Kernel Thesis. Corollary 6.2. Let Θ be an inner function and assume that there is another inner function θ such that θ 3 divides zΘ and Θ divides θ 4 . Suppose also there are constants Ci > 0, i = 1, 2, 3 such that any ϕ ∈ L2 can be written as ϕ = ϕ1 + ϕ2 + ϕ3 , with: (a) ϕ1 ∈ Kθ + Kθ , ϕ2 ∈ H 2 , and ϕ3 ∈ H 2 ; Θ (b) ρ(AΘ ϕi ) Ci ρ(Aϕ ) for i = 1, 2, 3. Then the following are equivalent: (i) AΘ ϕ has a bounded symbol; (ii) AΘ ϕ is bounded; (iii) ρ(AΘ ϕ ) < +∞. More precisely, there exists a constant C > 0 such that any truncated Toeplitz operator AΘ ϕ has a symbol ϕ0 with ϕ0 ∞ Cρ(AΘ ). ϕ There are of course many decompositions of ϕ as in (a); the difficulty consists in finding one that satisfies (b). Proof. It is immediate that (i) ⇒ (ii) ⇒ (iii), so it remains to prove (iii) ⇒ (i). Since ρ(AΘ ϕi ) < Θ +∞, i = 2, 3, Proposition 4.1 and 4.2 imply that Aϕi have bounded symbols ϕ˜ i with ϕ˜ i ∞ Θ ) CC ˜ ˜ i ρ(AΘ Cρ(A ϕi ϕ ). As for ϕ1 , we can apply Theorem 6.1 which gives that ϕ1 is bounded Θ Θ with ϕ1 ∞ 2ρr (AΘ ϕ1 ) 2C1 ρ(Aϕ ). Finally Aϕ has the bounded symbol ϕ0 = ϕ1 + ϕ˜ 2 + ϕ˜ 3 Θ ˜ 2 + C3 ))ρ(Aϕ ). 2 whose norm is at most (2C1 + C(C 6.2. Classical Toeplitz matrices Suppose Θ(z) = zN ; the space KΘ is then an N -dimensional space with orthonormal basis formed by monomials, and truncated Toeplitz operators have a (usual) Toeplitz matrix with respect of this basis. Of course every truncated Toeplitz operator has a bounded symbol; it is however interesting that there exists a universal estimate of this bound. The question had been raised in [29, Section 7]; the positive answer had actually been already independently obtained in [5] and [24]. The following result is stronger, giving a universal estimate for the symbols in terms of the action on the reproducing kernels.
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Theorem 6.3. Suppose Θ(z) = zN . There exists a constant C > 0, independent of N , such that ∞ Θ any truncated Toeplitz operator AΘ ϕ has a symbol ϕ0 ∈ L such that ϕ0 ∞ Cρ(Aϕ ). Proof. Consider a smooth function ηk on T, and the convolution (on T) ϕk = ηk ∗ ϕ, that is, 1 ϕk eis = 2π
π
ηk eit ϕ ei(s−t) dt.
−π
ˆ n ∈ Z. We have then ϕˆ k (n) = ηˆ k (n)ϕ(n), The map τt defined by τt : f (z) → f (eit z) is a unitary on KΘ and straightforward computations show that Θ τt hΘ λ = he−it λ
i(N −1)t ˜ Θ and τt h˜ Θ he−it λ , λ =e
(6.1)
for every λ ∈ D. By Fubini’s Theorem and a change of variables we have
1 AΘ ϕk f, g = 2π
π
ηk eit AΘ ϕ τt (f ), τt (g) dt,
−π
for every f, g ∈ KΘ . That implies that Θ Θ
A h = sup AΘ hΘ , g sup ϕk λ ϕk λ g∈KΘ g2 1
g∈KΘ g2 1
1 2π
π
it Θ Θ
ηk e A τt h , τt (g) dt, ϕ
λ
−π
and using (6.1), we obtain Θ Θ A h ηk 1 ρr AΘ ηk 1 ρ AΘ . ϕk λ
ϕ
ϕ
A similar argument shows that Θ Θ A h˜ ηk 1 ρ AΘ ϕk λ
ϕ
and thus Θ ρ AΘ ϕk ηk 1 ρ Aϕ .
(6.2)
Now consider the Fejér kernel Fm , defined by the formula Fˆm (n) = 1 − |n| m for |n| m and Fˆm (n) = 0 otherwise. It is well known that Fm 1 = 1 for all m ∈ N. If we take M = [ N 3+1 ] and define ηi (i = 1, 2, 3) by η1 = F M ,
η2 = 2e2iMt F2M − e2iMt FM ,
η3 = η 2 ,
then ηˆ 2 (n) = 0 for n < 0, ηˆ 3 (n) = 0 for n > 0, ηˆ 1 (n) + ηˆ 2 (n) + ηˆ 3 (n) = 1 for |n| N , and η1 1 = 1, ηi 1 3 for i = 2, 3. If we denote ϕi = ηi ∗ ϕ, then ϕ = ϕ1 + ϕ2 + ϕ3 , ϕ1 ∈
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KzM + KzM , ϕ2 is analytic and ϕ3 is coanalytic. Moreover z3M divides zN +1 and zN divides z4M . According to (6.2), we can apply Corollary 6.2 to obtain that there exists a universal constant Θ C > 0 such that AΘ ϕ has a bounded symbol ϕ0 with ϕ0 ∞ Cρ(Aϕ ). 2 N
In particular, it follows from Theorem 6.3 that any (classical) Toeplitz matrix Azϕ has a N
symbol ϕ0 such that ϕ0 ∞ CAzϕ . The similar statement is proved with explicit estimates N
N
ϕ0 ∞ 4Azϕ in [5] and ϕ0 ∞ 3Azϕ in [24]. We can obtain a slightly more general result (in the choice of the function Θ). α−z Corollary 6.4. Suppose Θ = bαN , with bα (z) = 1−αz a Blaschke factor. There exists a universal ∞ constant C > 0 such that any truncated Toeplitz operator AΘ ϕ has a symbol ϕ0 ∈ L such that Θ ϕ0 ∞ Cρ(Aϕ ).
Proof. The mapping U defined by (1 − |α|2 )1/2 f bα (z) , U (f ) (z) := 1 − αz
z ∈ D, f ∈ H 2 ,
is unitary on H 2 and one easily checks that U PzN = PΘ U . In particular, it implies that U (KzN ) = KΘ ; straightforward computations show that N
U hzλ = cλ hΘ bα (λ)
and U h˜ zλ = −cλ h˜ Θ bα (λ) , N
(6.3)
for every λ ∈ D, where cλ := |1 − λα|(1 − λα)−1 is a constant of modulus one. Suppose AΘ ϕ is a (bounded) truncated Toeplitz operator; if Φ = ϕ ◦ bα , then the relation N
U PzN = PΘ U yields AzΦ = U ∗ AΘ ϕ U . Thus, using (6.3), we obtain zN zN A h = U ∗ AΘ U hzN = AΘ hΘ ϕ ϕ bα (λ) 2 Φ λ 2 λ 2 and zN zN A h˜ = U ∗ AΘ U h˜ zN = AΘ h˜ Θ , ϕ ϕ bα (λ) 2 Φ λ 2 λ 2 which implies that N ρ AzΦ = ρ AΘ ϕ . Now it remains to apply Theorem 6.3 to complete the proof.
(6.4) 2
6.3. Elementary singular inner functions Let us now take Θ(z) = exp( z+1 z−1 ). A positive answer to Questions 1 and 3 is a consequence of results obtained by Rochberg [28] and Smith [31] on the Paley–Wiener space. We sketch the proof for completeness, without entering into details.
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Θ Theorem 6.5. If Θ(z) = exp( z+1 z−1 ) and Aϕ is a truncated Toeplitz operator, then the following are equivalent:
(i) AΘ ϕ has a bounded symbol; (ii) AΘ ϕ is bounded; (iii) ρ(AΘ ϕ ) < ∞. More precisely, there exists a constant C > 0 such that any truncated Toeplitz operator AΘ ϕ has a symbol ϕ0 with ϕ0 ∞ Cρ(AΘ ). ϕ Proof. By Remark 3.5 it is enough to prove the corresponding result for the space K Θ , where Θ(w) = eiw , and ρ is the analogue of ρ for operators on K Θ . If F denotes the Fourier transform on R, then K Θ = F −1 (L2 ([0, 1])), and we may suppose that the symbol ϕ ∈ (t + i)F −1 (L2 ([−1, 1])). For a rapidly decreasing function η on R, define Ψ (s) =
η(t)ϕ(s − t) dt.
(6.5)
R Θ We have then Ψˆ = ηˆ ϕˆ and ρ(AΘ ψ ) η1 · ρ(Aϕ ). Take now ψi , i = 1, 2, 3, such that supp ψˆ 1 ⊂ [−1/3, 1/3], supp ψˆ 2 ⊂ [0, 2], supp ψˆ 3 ⊂ [−2, 0], and ψˆ 1 + ψˆ 2 + ψˆ 3 = 1 on [−1, 1]. If we define ϕ i by replacing η with ψi in (6.5), Θ then there is a constant C1 > 0 such that ρ(AΘ ϕ i ) C1 ρ(Aϕ ) for i = 1, 2, 3. On the other hand, ϕ = ϕ 1 + ϕ 2 + ϕ 3 , ϕ 1 ∈ K Θ 1/3 + K Θ 1/3 , ϕ 2 is analytic, ϕ 3 is antianalytic. We may then apply the analogue of Corollary 6.2 for the upper half-plane which completes the proof. 2
One can see easily that a similar result is valid for any elementary singular function Θ(z) = exp(a z+ζ z−ζ ), for ζ ∈ T, a > 0. Remark 6.6. Truncated Toeplitz operators on the model space K Θ with Θ(w) = eiaw are closely connected with the so-called truncated Wiener–Hopf operators. Let ϕ ∈ L1 (R) and let a (Wϕ f )(x) =
f (t)ϕ(x − t) dt,
x ∈ (0, a),
0
for f ∈ L2 (0, a) ∩ L∞ (0, a). If W extends to a bounded operator on L2 (0, a), then it is called a truncated Wiener–Hopf operator. If ϕ = ψˆ with ψ ∈ (t + i)L2 (R) (the Fourier transform may be understood in the distributional sense), then Wϕ f = F PΘ (ψg) for g = fˇ ∈ K Θ . Thus, the Wiener–Hopf operator Wϕ is unitarily equivalent to AΘ ψ.
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7. Truncated Toeplitz operators with positive symbols As noted in Remark 5.4, if ϕ ∈ L2 is a positive function, then AΘ ϕ is bounded if and only if ϕ dm is a Carleson measure for KΘ . As a consequence mainly of results of Cohn [10,11], one can say more for positive symbols ϕ for a special class of model spaces. Recall that Θ is said to satisfy the connected level set condition (and we write Θ ∈ (CLS)) if there is ε ∈ (0, 1) such that the level set
Ω(Θ, ε) := z ∈ D: Θ(z) < ε is connected. Such inner functions are also referred to as one-component inner functions. Theorem 7.1. Let Θ be an inner function such that Θ ∈ (CLS). If ϕ is a positive function in L2 , then the following conditions are equivalent: (1) (2) (3) (4)
2 AΘ ϕ is a bounded operator on KΘ ; Θ supλ∈D AΘ ϕ hλ 2 < +∞; Θ Θ supλ∈D |Aϕ hΘ λ , hλ | < +∞; AΘ ϕ has a bounded symbol.
Proof. The implications (4) ⇒ (1) ⇒ (2) ⇒ (3) are obvious. We have
2
dm = ϕhΘ , hΘ = PΘ ϕhΘ , hΘ = AΘ hΘ , hΘ . ϕ hΘ ϕ λ λ λ λ λ λ λ
(7.1)
T
It is shown in [10] that, for Θ ∈ (CLS), a positive μ satisfies supλ∈D hΘ λ L2 (μ) < ∞ if and only if it is a Carleson measure for KΘ . Thus (3) implies that ϕ dm is a Carleson measure for KΘ , which has been noted above to be equivalent to AΘ ϕ bounded; so (1) ⇔ (3). On the other hand, it is proved in [11] that if AΘ ϕ is bounded, then there are functions v ∈ ∞ 2 L (T) and h ∈ H such that ϕ = Re(v + Θh). Write then 1 ϕ = Re v + (Θh + Θh), 2 Θ ∞ which implies that ϕ − Re v ∈ ΘH 2 + ΘH 2 . Therefore AΘ ϕ = ARe v and Re v ∈ L (T). Thus the last remaining implication (1) ⇒ (4) is proved. 2
Remark 7.2. In [10], Cohn asked the following question: let Θ be an inner function and let μ be a positive measure on T such that the singular part of μ is supported on a subset of T \ σ (Θ); is it sufficient to have sup λ∈D
Θ 2 h dμ < +∞, λ
T
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to deduce that μ is a Carleson measure for KΘ ? In [23] Nazarov and Volberg construct a counterexample to this question with a measure μ of the form dμ = ϕ dm where ϕ is some positive function in L2 . In our context, this means that they provide an inner function Θ and a positive function ϕ ∈ L2 such that
Θ Θ sup AΘ ϕ hλ , hλ < +∞,
(7.2)
λ∈D
Θ while AΘ ϕ is not bounded. But the condition (7.2) is obviously weaker than ρr (Aϕ ) < +∞ (note Θ that since ϕ is positive, the truncated Toeplitz operator is positive and ρr (Aϕ ) = ρ(AΘ ϕ )). Thus an answer to Question 3 does not follow from the Nazarov–Volberg result.
Remark 7.3. It is shown by Aleksandrov [2, Theorem 1.2] that the condition sup λ∈D
kλΘ ∞ kλΘ 22
< +∞
is equivalent to Θ ∈ (CLS). On the other hand, as we have seen in Theorem 5.6, the condition sup λ∈D
kλΘ p kλΘ 22
= +∞
for some p ∈ (2, ∞) implies that there exists a bounded operator in T (KΘ ) without a bounded symbol. Therefore, based on Theorem 7.1 and Theorem 5.6, it seems reasonable to state the following conjecture. Conjecture. Let Θ be an inner function. Then any bounded truncated Toeplitz operator has a bounded symbol if and only if Θ ∈ (CLS). References [1] P.R. Ahern, D.N. Clark, Radial limits and invariant subspaces, Amer. J. Math. 92 (1970) 332–342. [2] A. Aleksandrov, On embedding theorems for coinvariant subspaces of the shift operator. II, J. Math. Sci. 110 (5) (2002) 2907–2929. [3] S. Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (2) (1986) 315–332. [4] S. Axler, S.-Y. Chang, D. Sarason, Products of Toeplitz operators, Integral Equations Operator Theory 1 (3) (1978) 285–309. [5] M. Bakonyi, D. Timotin, On an extension problem for polynomials, Bull. London Math. Soc. 33 (5) (2001) 599–605. [6] F.F. Bonsall, Boundedness of Hankel matrices, J. London Math. Soc. (2) 29 (2) (1984) 289–300. [7] A. Brown, P. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963) 89–102. [8] J. Cima, S. Garcia, W. Ross, W. Wogen, Truncated Toeplitz operators: spatial isomorphism, unitary equivalence, and similarity, Indiana Univ. Math. J., in press. [9] J. Cima, W. Ross, W. Wogen, Truncated Toeplitz operators on finite dimensional spaces, Oper. Matrices 3 (2) (2008) 357–369. [10] B. Cohn, Carleson measures for functions orthogonal to invariant subspaces, Pacific J. Math. 103 (2) (1982) 347– 364. [11] W.S. Cohn, Carleson measures and operators on star-invariant subspaces, J. Operator Theory 15 (1) (1986) 181–202. [12] W.S. Cohn, Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J. Math. 108 (3) (1986) 719–749.
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[13] W.S. Cohn, A maximum principle for star invariant subspaces, Houston J. Math. 14 (1) (1988) 23–37. [14] P. Duren, Theory of H p Spaces, Pure Appl. Math., vol. 38, Academic Press, New York, 1970. [15] K. Fedorovskii, On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math. 253 (2006) 186–194. [16] S. Ferguson, C. Sadosky, Characterizations of bounded mean oscillation on the polydisk in terms of Hankel operators and Carleson measures, J. Anal. Math. 81 (2000) 239–267. [17] S. Garcia, W. Ross, The norm of a truncated Toeplitz operator, CRM Proc. Lecture Notes 51 (2010) 59–64. [18] J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. [19] Z. Harper, M. Smith, Testing Schatten class Hankel operators, Carleson embeddings and weighted composition operators on reproducing kernels, J. Operator Theory 55 (2) (2006) 349–371. [20] V.P. Havin, N.K. Nikolski, Stanislav Aleksandrovich Vinogradov, his life and mathematics, in: V. Havin, N. Nikolski (Eds.), Complex Analysis, Operators, and Related Topics: S.A. Vinogradov–In Memoriam, in: Oper. Theory Adv. Appl., vol. 113, Birkhäuser Verlag, Basel, 2000, pp. 1–18. [21] F. Holland, D. Walsh, Hankel operators in von Neumann–Schatten classes, Illinois J. Math. 32 (1) (1988) 1–22. [22] S. Hrušˇcëv, N. Nikol’ski˘ı, B. Pavlov, Unconditional Bases of Exponentials and of Reproducing Kernels, Lecture Notes in Math., vol. 864, 1981, pp. 214–335. [23] F. Nazarov, A. Volberg, The Bellman function, the two-weight Hilbert transform, and embeddings of the model spaces Kθ , J. Anal. Math. 87 (2002) 385–414. [24] L.N. Nikol’skaya, Y.B. Farforovskaya, Toeplitz and Hankel matrices as Hadamard–Schur multipliers, Algebra i Analiz 15 (6) (2003) 141–160; English translation in St. Petersburg Math. J. 15 (6) (2004) 915–928. [25] N.K. Nikolski, Treatise on the Shift Operator, Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin, 1986. [26] N.K. Nikolski, Operators, Functions, and Systems: an Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, American Mathematical Society, 2002. [27] S. Pott, C. Sadosky, Bounded mean oscillation on the bidisk and operator BMO, J. Funct. Anal. 189 (2) (2002) 475–495. [28] R. Rochberg, Toeplitz and Hankel operators on the Paley–Wiener space, Integral Equations Operator Theory 10 (2) (1987) 187–235. [29] D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (4) (2007) 491–526. [30] D. Sarason, Unbounded Toeplitz operators, Integral Equations Operator Theory 61 (2) (2008) 281–298. [31] M. Smith, The reproducing kernel thesis for Toeplitz operators on the Paley–Wiener space, Integral Equations Operator Theory 49 (1) (2004) 111–122. [32] M.P. Smith, Testing Schatten class Hankel operators and Carleson embeddings via reproducing kernels, J. London Math. Soc. (2) 71 (1) (2005) 172–186. [33] D. Xia, D. Zheng, Products of Hankel operators, Integral Equations Operator Theory 29 (1997) 339–363.
Journal of Functional Analysis 259 (2010) 2702–2726 www.elsevier.com/locate/jfa
On fractional powers of generators of fractional resolvent families ✩ Miao Li, Chuang Chen, Fu-Bo Li ∗ Department of Mathematics, Sichuan University, Chengdu 610064, PR China Received 23 April 2010; accepted 13 July 2010 Available online 31 July 2010 Communicated by N. Kalton
Abstract We show that if −A generates a bounded α-times resolvent family for some α ∈ (0, 2], then −Aβ generates an analytic γ -times resolvent family for β ∈ (0, 2π−πγ 2π−πα ) and γ ∈ (0, 2). And a generalized subordination principle is derived. In particular, if −A generates a bounded α-times resolvent family for some α ∈ (1, 2], then −A1/α generates an analytic C0 -semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order. © 2010 Elsevier Inc. All rights reserved. Keywords: α-Times resolvent families; C0 -semigroups; Generators; Fractional powers; Subordination principle; Fractional Cauchy problems
1. Introduction Let A be a closed densely defined linear operator on a Banach space X. The resolvent families were introduced by Da Prato [10] to study Volterra integral equations of the form t u(t) = f (t) + A
a(t − s)u(s) ds. 0
✩
This project is supported by the NSF of China (No. 10971146).
* Corresponding author.
E-mail addresses:
[email protected] (M. Li),
[email protected] (C. Chen),
[email protected] (F.-B. Li). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.007
(1.1)
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A family {R(t)}t0 ⊂ B(X) is called a resolvent family for A with kernel a if (a) R(0) = I and R(t) is strongly continuous; (b) AR(t) ⊆ R(t)A for every t 0; (c) for every x ∈ D(A), t R(t)x = x +
a(t − s)R(s)Ax ds. 0
It is shown that the problem (1.1) is well-posed (in the sense of [28]) if and only if there is a resolvent family for A. Since a C0 -semigroup is a resolvent family for its generator with kernel a1 (t) ≡ 1, and a cosine operator function is a resolvent family for its generator with kernel α−1 a2 (t) = t, it is natural to consider the resolvent family with kernel aα (t) = Γt (α) . Also note the following facts: if A generates a C0 -semigroup, then the Cauchy problem of first order u (t) = Au(t),
t 0;
u(0) = x
is well-posed; and if A generates a cosine operator function, then the second order Cauchy problem u (t) = Au(t),
t 0;
u(0) = x,
u (0) = y,
is also well-posed. This motivates one to consider the relations between the existence of resolvent family for A with kernel aα (t) and the well-posedness of some kind of fractional Cauchy problem Dtα u(t) = Au(t) with proper initial values. Such relation was proved by Bajlekova [3] in 2001. The resolvent family for A with kernel aα was therefore called α-times resolvent family. For more general resolvent families see [24,25]. On the other hand, it is well known that if −A generates a bounded cosine function operator, then −A1/2 generates an analytic C0 -semigroup of angle π/2 (cf. [18]). And it was proved by Yosida in 1960 (cf. [17,32]) that if T is a bounded C0 -semigroup on a complex Banach space X, with the generator A, then −Aα , 0 < α < 1, generates an analytic semigroup Tα on X, and Tα is subordinated to T through the Lévy stable density function. For α-times resolvent family, the questions of interest are: (Q1 ) If −A generates a bounded C0 -semigroup, does −Aα generate an α-times resolvent family? (Q2 ) If −A generates a bounded α-times resolvent family, does −A1/α generate a C0 semigroup? (Q3 ) If −A generates a bounded α-times resolvent family, does −A1/2 generate an α/2-times resolvent family? (Q4 ) If −A generates a bounded α-times resolvent family, does −Aβ also generate an α-times resolvent family for some suitable β? (Q5 ) If −A generates a bounded α-times resolvent family, does −Aβ generate a γ -times resolvent family for some suitable β and γ ?
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Our first aim in this paper is trying to give answers to the above questions in a unified way. We first note the fact: if −A is the generator of a bounded α-times resolvent family, then A is a sectorial operator (see Section 2 for details). Therefore, it is possible to define the fractional power Ab for b > 0. By using the theory of functional calculus for sectorial operators (see [4,14, 21,26]), we are able to give positive answers to the questions above. These relations are clarified in Section 3. The second purpose of this paper is to establish connections between solutions of fractional Cauchy problems and Cauchy problems of first order. Obverse that many phenomena in the theory of stochastic processes, finance and hydrology are recently described through fractional evolution equations, see [6,7,30,33] and the references therein. For example, Zaslavsky [33] introduced the fractional kinetic equation Dtα u(t, x) + Lx u(t, x) = 0,
t > 0,
u(0, x) = f (x),
(1.2)
for Hamiltonian chaos, where α ∈ (0, 1), −Lx is the generator of some continuous Markov process, and Dtα is understood the Caputo fractional derivative in time (see Section 2). Baeumer and Meerschaert [5], and Meerschaert and Scheffler [27] showed that the fractional Cauchy problem (1.2) is related to a certain class of subordinated stochastic processes. More precisely, Theorem 3.1 in [5] shows that the formula ∞ u(t, x) =
v (t/s)α , x bα (s) ds,
(1.3)
0
yields a unique strong solution of (1.2), where of the stable subordinator ∞ bα is the smooth density α such that the Laplace transform bα (λ) = 0 e−λt bα (t) dt = e−λ and v is the solution of vt (t, x) + Lx v(t, x) = 0,
t > 0,
v(0, x) = f (x).
(1.4)
The formula (1.4) can also be explained by the subordination principle for fractional resolvent family, see Theorem 3.1 in [3] or Lemma 2.9. If the fractional power of Lx , Lαx , is defined, it is also of interest to know the relations between the solution of (1.4) and that of Dtα u(t, x) + Lαx u(t, x) = 0,
t > 0,
u(0, x) = f (x).
(1.5)
In Section 4 we will give this connection. Moreover, Baeumer, Meerschaert and Nane [6] proved that Eq. (1.2) with α = 1/2 and the initial value problem ut (t, x) − L2x u(t, x) +
t −1/2 Lx f (x) = 0, Γ (1/2)
u(0, x) = f (x),
t > 0, (1.6)
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have the same solution; and (1.2) with α = 1/3 and ut (t, x) + L3x u(t, x) +
t −2/3 t −1/3 2 Lx f (x) − L f (x) = 0, Γ (1/3) Γ (2/3) x u(0, x) = f (x),
t > 0, (1.7)
have the same solution, respectively. Another example is given by Allouba and Zheng [1] and DeBlassie [11], they consider the case that Lx = −, the Laplace operator. Keyantuo and Lizama [19] gave the connections between (1.2) with α = 1/m and ordinary non-homogeneous equations. In Section 4, by analysing the solutions of fractional Cauchy problems directly we can recover the result in [19]. Moreover, we will consider more general fractional Cauchy problem with the fractional order not necessarily a rational number. Our work is organized as follows. We provide in Section 2 some preliminaries of fractional resolvent families and fractional powers of sectorial operators. And then give positive answers to the questions (Q1 )–(Q5 ) in Section 3 and more results of fractional generations are obtained as well. Finally, we discuss the relations of solutions of fractional Cauchy problems and Cauchy problems of first order in Section 4. 2. Preliminaries Throughout the paper, (X, · ) is a complex Banach space, and B(X) is the space of all bounded linear operators on X. A is a closed linear operator on X. We assume throughout this paper that A is densely defined. By D(A), R(A), ρ(A), σ (A) and R(λ, A) (λ ∈ ρ(A)) we denote the domain, range, resolvent set, spectrum set and resolvent of the operator A, respectively. Recall the Caputo fractional derivative of order α > 0 Dtα f (t) := Jtm−α
dm f (t), dt m
where m is the smallest integer greater than or equal to α, and the Riemann–Liouville fractional integral of order β > 0
β Jt f (t) = gβ
t ∗ f (t) :=
gβ (t − s)f (s) ds, 0
where gβ (t) :=
t β−1 Γ (β) ,
t > 0,
0,
t 0.
Set moreover g0 (t) := δ(t), the Dirac delta-function. For details in fractional calculus, we refer the reader to [20,29].
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The Mittag–Leffler function is defined by Eα,β (z) :=
∞
zn 1 = Γ (αn + β) 2πi
n=0
C
μα−β eμ dμ, μα − z
α, β > 0, z ∈ C,
(2.1)
where the path C is a loop which starts and ends at −∞, and encircles the disc |t| |z|1/α in the positive sense. Eα (z) := Eα,1 (z). The Mittag–Leffler function Eα (t) satisfies the fractional differential equation Dtα Eα ωt α = ωEα ωt α . The most interesting properties of the Mittag–Leffler functions are associated with their Laplace integral ∞ 0
λα−β , e−λt t β−1 Eα,β ωt α dt = α λ −ω
Re λ > ω1/α , ω > 0
(2.2)
and with their asymptotic expansion as z → ∞. If 0 < α < 2, β > 0, then 1 (1−β)/α 1 z exp z1/α + εα,β (z), | arg z| απ, α 2
1 Eα,β (z) = εα,β (z), arg(−z) < 1 − α π, 2
Eα,β (z) =
(2.3) (2.4)
where εα,β (z) = −
N −1 n=1
z−n + O |z|−N Γ (β − αn)
as z → ∞, and the O-term is uniform in arg z if | arg(−z)| (1 − α/2 − )π . It is also of interest to know the relations between the Mittag–Leffler function and function of Wright type: ∞ Eγ (z) =
Ψγ (t)ezt dt,
z ∈ C, 0 < γ < 1,
0
where Ψγ (z) :=
∞ n=0
1 (−z)n = n!Γ (−γ n + 1 − γ ) 2πi
μγ −1 exp μ − zμγ dμ
(2.5)
Γ
with Γ a contour which starts and ends at −∞ and encircles the origin once counterclockwise. For more properties of the Mittag–Leffler function and function of Wright type, we refer to [12,13].
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We now turn to a short introduction to fractional powers of sectorial operators. Let A be a densely defined closed linear operator on Banach space X. Definition 2.1. The operator A is called sectorial of angle ω ∈ [0, π) (A ∈ Sect(ω), in short) if (1) σ (A) is contained in the closure of the sector
Σω := z ∈ C: z = 0 and | arg z| < ω , for ω > 0 or Σ0 := (0, ∞). (2) For every ω ∈ (ω, π), sup{ zR(z, A) : z ∈ C\Σω } < ∞. A family of operators (Aτ )τ ∈Λ is called uniformly sectorial of angle ω ∈ [0, π) if Aτ ∈ Sect(ω) for each τ , and sup{ zR(z, Aτ ) : τ ∈ Λ, z ∈ C\Σω } < ∞. If 0 ∈ ρ(A) for a sectorial operator A, then we can define its fractional powers as follows. For b > 0, define A−b by A−b := −
1 2πi
λ−b R(λ, A) dλ,
(2.6)
Γ (ζ )
where the path Γ (ζ ) runs in the resolvent set of A from ∞e−iζ to ∞eiζ , while avoiding the negative real axis and the origin, and λb is taken as the principle branch. Noticing that A−b ∈ B(X) is injective for all b > 0, we can define Ab := (A−b )−1 and A0 := I . On the other hand, for a sectorial operator A without the assumption that 0 ∈ ρ(A), since A + is sectorial and 0 ∈ ρ(A + ), it makes sense to consider the operator (A + )b and define the fractional powers of A by Ab := s − lim (A + )b →0+
for b > 0 and so corresponding results for such fractional powers can be obtained by similar argument (cf. [14,26]). We collect some basic properties of fractional powers in the following lemma. Lemma 2.2. (See [14].) Let b > 0 and A−b is defined as above. The following assertions hold. (a) Ab is closed and D(Ab ) ⊂ D(Ac ) for b > c > 0. (b) Ab x = Ab−n An x for all x ∈ D(An ) and n > b, n ∈ N. (c) Let d > b > 0. If B ⊂ Ab and D(B) = D(Ad ), then B is closable and B = Ab , where B is the closure of B. (d) σ (Ab ) = (σ (A))b . (e) If A ∈ Sect(ω) for some ω ∈ (0, π), then for every β ∈ (0, π/ω) the operator Aβ is sectorial of angle βω. (f) If A ∈ Sect(ω) for some ω ∈ (0, π), then the family (A + ε)ε0 is uniformly sectorial of angle ω.
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Finally we recall the notion of α-times resolvent families. Also here we suppose that A is a densely defined closed linear operator on X. Definition 2.3. Let α > 0. A family {Sα (t)}t0 ⊂ B(X) is called an α-times resolvent family generated by A if the following conditions are satisfied: (a) Sα (t) is strongly continuous for t 0 and Sα (0) = I ; (b) Sα (t)A ⊂ ASα (t) for t 0; (c) for x ∈ D(A), the resolvent equation t Sα (t)x = x +
gα (t − s)Sα (s)Ax ds
(2.7)
0
holds for all t 0. Remark 2.4. Since A is densely defined and closed, t t it is easy to show that for all x ∈ X, g (t − s)S (s)x ds ∈ D(A) and S (t)x = x + A( α α 0 α 0 gα (t − s)Sα (s)x ds). Definition 2.5. (a) An α-times resolvent family {Sα (t)}t0 is said to be bounded if there exist constants M 1 such that Sα (t) M for all t 0. If A generates a bounded α-times resolvent family Sα , we will write (A, Sα ) ∈ Cα (0) or A ∈ Cα (0) for short. (b) Let θ0 ∈ (0, π/2] and ω0 0. An α-times resolvent family {Sα (t)}t0 is called analytic of angle θ0 for some θ0 ∈ (0, π/2] if Sα (t) admits an analytic extension to the sector Σθ0 . An analytic α-times resolvent family {Sα (z)}z∈Σθ0 is said to be bounded if for each θ ∈ (0, θ0 ) there exists a constant Mθ such that Sα (z) Mθ ,
z ∈ Σθ .
If A generates a bounded analytic α-times resolvent family Sα of angle θ0 , we will write (A, Sα ) ∈ Aα (θ0 ) or A ∈ Aα (θ0 ) for short. Lemma 2.6. (See [3].) Let 0 < α 2. A ∈ Cα (0) if and only if Σπα/2 ⊂ ρ(A) and there exists a strongly continuous function Sα : R+ → B(X) such that Sα (t) M for all t 0 and
λ
α−1
α −1 λ −A x =
∞
e−λt Sα (t)x dt,
λ ∈ Σπ/2
(2.8)
0
for all x ∈ X. Furthermore, {Sα (t)}t0 is the α-times resolvent family generated by A. In the sequel we need the following important lemma on analyticity criteria for α-times resolvent families.
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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Lemma 2.7. Let α ∈ (0, 2) and θ0 ∈ (0, min{ π2 , πα − π2 }]. The following assertions are equivalent. (a) (A, Sα ) ∈ Aα (θ0 ). (b) Σα( π2 +θ0 ) ∈ ρ(A), and for each θ ∈ (0, θ0 ), there exists a constant Mθ such that λ(λ − A)−1 Mθ ,
λ ∈ Σα( π2 +θ) .
(c) −A ∈ Sect(π − ( π2 + θ0 )α). The equivalence of (a) with (b) is given in [3]. (b) is equivalent to (c) by the definition of sectorial operators, which is also mentioned in Remark 3 of [16]. Remark 2.8. (a) By Lemma 2.7, −A generates a bounded analytic α-times resolvent family if and only if A is sectorial of angle ϕ < π − πα/2. (b) If −A generates a bounded α-times resolvent family, then A is sectorial of angle π − πα/2. Recall that if {Sα (z)}z∈Σθ is a bounded analytic α-times resolvent family with generator A, then for t > 0, −1 1 eλt λα−1 λα − A dλ, (2.9) Sα (t) = 2πi Γθ0
where Γθ0 is any piecewise smooth curve in Σπ/2+θ going from ∞e−i(π/2+θ0 ) to ∞ei(π/2+θ0 ) for some 0 < θ0 < θ (cf. [3,8]). The following subordination principle is important in the theory of fractional resolvent families, which will be extended to more general cases in Theorem 3.1. Lemma 2.9. (See [3].) Let 0 < β < α 2, γ = β/α and ω 0. If A ∈ Cα (0) then A ∈ Cβ (0) and the following representation holds ∞ Sβ (t) =
ϕγ (t, s)Sα (s) ds,
t > 0,
0
where ϕγ (t, s) = t −γ Φγ (st −γ ) with Φγ defined by (2.5), in the strong sense. 3. Fractional powers of generators of fractional resolvent families In this section we consider the fractional generations for bounded analytic fractional resolvent families. The following theorem is our main result, which gives the answer to question (Q5 ) in the Introduction. Theorem 3.1. Let α ∈ (0, 2] and A be sectorial of angle π − α2 π on a Banach space X, and let 0 < γ < 2.
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β π π β (a) For each β ∈ (0, 2π−πγ 2π−πα ), −A ∈ Aγ (ϕ0 ) with ϕ0 = min{ 2 , − γ (π − 2 α) +
π γ
− π2 }.
β
(b) If 0 ∈ ρ(A), then the γ -times resolvent family generated by −Aβ , Sγ , can be represented by Sγβ (t) =
1 2πi
Eγ −μβ t γ (A − μ)−1 dμ,
t > 0,
(3.1)
Γω
where Γω is a smooth path in the resolvent of A from ∞e−iω to ∞eiω , avoiding the negative axis and zero, with ω ∈ (π − α2 π, β1 (π − γ2 π)). (c) If in addition −A generates a bounded α-times resolvent family Sα , then the following generalized subordination principle ∞ Sγβ (t)x
=
β fα,γ (t, s)Sα (s)x ds,
t > 0,
(3.2)
0
holds for x ∈ X, where β fα,γ (t, s) =
1 2πi
1/α Eγ −μβ t γ (−μ)1/α−1 e−(−μ) s dμ
(3.3)
∂Σω
with ω as in (b), ∂Σω is the two rays {ρe±iω : ρ 0} and (−ρe±iω )1/α = ρ 1/α e∓i(π−ω)/α . Proof. (a) Since A is sectorial of angle π − π2 α, by Lemma 2.2(e), Aβ is sectorial of an2π gle β(π − π2 α) for β ∈ (0, 2π−πα ). By Lemma 2.7, −Aβ ∈ Aγ (ϕ0 ) if and only if Aβ ∈
Sect(π − ( π2 + ϕ0 )γ ). To guarantee that ϕ0 > 0, we need β < 2π−πγ 2π−πα . (b) Since Aβ ∈ Sect(β(π − π2 α)), ρ(Aβ ) ⊃ C − Σβ(π− π2 α) . Thus (λ + Aβ )−1 exists and belongs to B(X) for λ ∈ Σπ−β(π− π2 α) . Let ω > π − π2 α. Since 0 ∈ ρ(A), we can find d > 0 such that {z ∈ C: |z| < d} ⊂ ρ(A) and then choose Γω as the union of Γω1 , Γω2 and Γω3 , where
Γω1 = reiω : r > d ,
Γω2 = deiθ : −ω < θ < ω ,
Γω3 = re−iω : r > d .
For λ ∈ Σπ−βω , the function f (μ) = operator f (A) as f (A) =
1 λ+μβ
1 2πi
is analytic on Σω , we can therefore define a bounded
f (μ)(A − μ)−1 dμ.
Γω
Since β(π − ω) < β(π − π2 α), (λ + Aβ )−1 ∈ B(X) for λ ∈ Σπ−βω . It is routine to show that for
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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such λ ∈ Σπ−βω ,
−1 1 λ + Aβ = f (A) = 2πi
−1 λ + μβ (A − μ)−1 dμ.
(3.4)
Γω
Now take β and ϕ0 as in (a). Since −Aβ ∈ Aγ (ϕ0 ), for 0 < δ < ϕ0 , choose ω < β1 [π − ( π2 + δ)γ ] such that when λ ∈ Γ π2 +δ then λγ ∈ Σπ−βω . Thus by (2.9), (3.4) and Fubini’s theorem we have Sγβ (t) =
1 2πi
−1 eλt λγ −1 λγ + Aβ dλ
Γ π +δ 2
1 = 2πi
λt γ −1
e λ Γ π +δ
1 2πi
1 2πi
Γω
=
1 2πi
1 2πi
γ β −1 −1 λ +μ (A − μ) dμ dλ
Γω
2
=
−1 eλt λγ −1 λγ + μβ dλ (A − μ)−1 dμ
Γ π +δ 2
Eγ −μβ t γ (A − μ)−1 dμ.
Γω
(c) We first assume that 0 ∈ ρ(A) and A ∈ Cα (0). Let Γω be as in (b). By (b), (2.8) and Fubini’s theorem, for x ∈ X, Sγβ (t)x
1 = 2πi
Eγ −μβ t γ (A − μ)−1 x dμ
Γω
∞ β γ 1/α−1 −(−μ)1/α s −(−μ) Eγ −μ t e Sα (s)x ds dμ
Γω1 ∪Γω3
0
1 = 2πi +
1 2πi
Eγ −μβ t γ (A − μ)−1 x dμ
Γω2
∞ −
=
1 2πi
0
+
1 2πi
1/α Eγ −μβ t γ (−μ)1/α−1 e−(−μ) s dμ Sα (s)x ds
Γω1 ∪Γω3
Eγ −μβ t γ (A − μ)−1 dμ.
Γω2
The integration on Γω2 converges to 0 if d → 0 since 0 ∈ ρ(A). Moreover, since | arg(μβ t γ )| < π − γ2 π , by (2.4) the integration on Γω1 ∪ Γω3 is absolutely convergent if d → 0. So letting d → 0,
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we get Sγ (t)x =
∞ 0
β
fα,γ (t, s)Sα (s)x ds with
β fα,γ (t, s) = −
1 2πi
1 1−α α Eγ −μβ t γ (−μ) α e−(−μ) s dμ.
∂Σω
Thus (3.2) holds for −A ∈ Cα (0) with 0 ∈ ρ(A). Next we show that (3.2) holds when 0 ∈ / ρ(A) and −A generates a bounded analytic α-times resolvent family. For ε > 0, 0 ∈ ρ(A + ε) and (A + ε)ε0 is uniformly sectorial of angle π − α2 π β by Lemma 2.2(f). By (b), the γ -times resolvent family, ε Sγ , generated by −(A + ε)β is given by β ε Sγ (t) =
1 2πi
Eγ −μβ t γ (A + ε − μ)−1 dμ,
t > 0,
(3.5)
Γω
since (A + ε − μ)−1 → (A − μ)−1 as ε → 0, by (2.4) and Lebesgue’s dominated convergence β β theorem ε Sγ (t) → Sγ (t) as ε → 0 for every t 0. On the other hand, by the first step we can β represent ε Sγ (t) by ∞ β ε Sγ (t)x
=
β fα,γ (t, s)ε Sα (s)x ds,
t > 0,
(3.6)
0
where ε Sα (s) is the α-times resolvent family generated by −(A + ε). Since ε Sα (s) is uniformly bounded and (A + ε − μ)−1 → (A − μ)−1 as ε → 0, by the approximation theorem for αtimes resolvent family (Theorem 4.2 in [23]) one has ε Sα (s) → Sα (s) in strong sense for every β s 0. Note that fα,γ (t, ·) is absolutely integrable by (2.4), by letting ε to 0 in (3.6) we obtain (3.2). Finally we show that (3.2) holds when 0 ∈ / ρ(A) and −A ∈ Cα (0). For every α < α, −A gen erates a bounded analytic α -times resolvent family by (a) or Lemma 2.9, so by our second step we have for every x ∈ X, ∞ Sγβ (t)x
=
β
fα ,γ (t, s)Sα (s)x ds,
t > 0,
0
where Sα is the α -times resolvent family generated by −A. Since Sα (t) → Sα (t) strongly by β β Theorem 4.5 in [23] and fα ,γ (t, s) → fα,γ (t, s), (3.2) is obtained by letting α to α. 2 Remark 3.2. (a) Note that by Remark 2.8(b), if A ∈ Cα (0), then A is sectorial of angle π − απ/2. (b) If α = 1, we can shift the contour in (3.3) to Γω , that is, 1 β Eγ −μβ t γ eμs dμ. f1,γ (t, s) = 2πi Γω
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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(c) If β = 1, then in the proof of (b) we do not need the assumption that 0 ∈ ρ(A). Indeed, in this case we can replace the contour Γω by Γ˜ω := Γω1 ∪ Γω3 ∪ Γ˜ω2 , where Γ˜ω2 = {deiθ : ω < θ < 2π − ω}, and then 1 1 Eγ −μt γ (A − μ)−1 dμ, t > 0, Sγ (t) = 2πi Γ˜ω
and 1 (t, s) = − fα,γ
1 2πi
1 1−α α Eγ −μt γ (−μ) α e−(−μ) s dμ.
Γ˜ω
In particular, if (A, Sα ) ∈ Aα (θ0 ) with θ0 > 0, then for each θ ∈ (0, θ0 ) and z ∈ Σθ , Sα (z) has the following integrated representation: 1 Eα μzα (μ − A)−1 dμ, (3.7) Sα (z) = 2πi Γ˜ω
where θ ∈ (πα/2, (π/2 + θ0 )α). Note that (3.7) is a Dunford integral, sometimes it will be more convenient than the identity (2.9). (d) If γ = 1, by changing the variable μ in (3.3) to ρeiω and ρe−iω , 0 < ρ < ∞, one gets β fα,1 (t, s) =
1 π
∞
ρ (1−α)/α exp −sρ 1/α cos(π − ω)/α − tρ β sin βω
0
· sin tρ β sin βω − sρ 1/α sin(π − ω)/α + (π − ω)/α dρ. As consequences of Theorem 3.1 and Remark 3.2 we have the following results, which give positive answers to questions (Q1 )–(Q4 ). Corollary 3.3. The following assertions hold. (a) If (−A, S1 ) ∈ C1 (0) then −Aα ∈ A1 ( π2 (1 − α)) for each α ∈ (0, 1). Moreover, the C0 semigroup generated by −Aα is given by ∞ pα (t, s)S1 (s) ds,
t >0
0
where p α (t, ·)(λ) := 1 pα (t, s) = π
∞ 0
∞ 0
for π/2 < θ < π .
e−λs pα (t, s) ds = e−λ
αt
and
exp sρ cos θ − tρ α cos αθ · sin(sρ sin θ − tρ sin αθ + θ ) dρ,
(3.8)
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(b) If (−A, Sα ) ∈ Cα (0) then −A ∈ Aβ (min{π, (α/β − 1)π/2}) for each β ∈ (0, α). Moreover, the β-times resolvent family generated by −A is given by ∞ ϕβ/α (t, s)Sα (s) ds,
t >0
0 γ −1 e−λ s , ϕ γ where ϕ γ (·, s)(λ) = λ γ (t, ·)(λ) = Eγ (−λt ) for 0 < γ < 1 and γ
1 ϕγ (t, s) = π
∞
ρ γ −1 exp −sρ γ cos γ (π − θ ) − tρ cos θ
0
· sin tρ sin θ − sρ γ sin γ (π − θ ) + γ (π − θ ) dρ π for θ ∈ (π − 2γ , π/2). (c) If (−A, S1 ) ∈ C1 (0) then −Aα ∈ Aα (min{ πα − π, π/2}) for each α ∈ (0, 1). Moreover, the α-times resolvent family generated by −Aα is given by
∞ α f1,α (t, s)S1 (s) ds,
t >0
0
α (t, ·)(λ) = E (−λα t α ) and f α (t, s) = ∞ ϕ (t, τ )p (τ, s) dτ . where f1,α α α α 1,α 0 (d) If (−A, Sα ) ∈ Cα (0) for some α ∈ (1, 2] then −A1/α ∈ A1 (π − π/α). Moreover, the C0 semigroup generated by −A1/α is given by ∞
1/α
fα,1 (t, s)Sα (s) ds,
t >0
0
where 1/α fα,1 (t, s) =
α π
∞
exp −sρ cos(π − θ )/α − tρ cos θ/α
0
· sin tρ sin θ/α − tρ sin(π − θ )/α + (π − θ )/α dρ ∞ 1/α for θ ∈ (π − απ 2 , απ/2) and fα,1 (t, s) = 0 p1/α (t, τ )ϕ1/α (τ, s) dτ . (e) If (−A, Sα ) ∈ Cα (0) for some α ∈ (0, 2] then −A1/2 ∈ Aα/2 (π/2). Moreover, the α/2-times resolvent family generated by −A1/2 is given by α α t2 π
∞ 0
α
s 2 −1 Sα (s) ds, sα + t α
t > 0.
(3.9)
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
(f) If (−A, Sα ) ∈ Cα (0) for some α ∈ (0, 2) then −Aβ ∈ Aα (min{( πα − β ∈ (0, 1).
π 2 )(1
2715
− β), π/2}) for
Proof. (a) follows from Remark 3.2(a), (b) and (d). (b) By Remark 3.2(c), (2.1), Fubini’s theorem and Cauchy’s integral formula,
1 (t, s) = − fα,β
1 2πi
= Γ˜ω
1 = 2πi
1 1−α α Eβ −μt β (−μ) α e−(−μ) s dμ
Γ˜ω
1 2πi
λt β−1
e λ Γ π +δ
2
λt β−1
e λ 2
1 2πi
1 β −1 1−α αs −(−μ) λ + μ (−μ) α e dμ dλ
Γ˜ω
Γ π +δ
=
1 β −1 1−α α λ +μ dλ (−μ) α e−(−μ) s dμ
eλt λβ−1 λβ(
1−α α )
e−λ
β/α s
dλ
Γ π +δ 2
= ϕβ/α (t, s). 1 And the last identity follows from Remark 3.2(d) and by noting that ϕγ (t, s) = f1/γ ,1 (t, s). (c) By Remark 3.2(b), for λ > 0,
∞ e
−λs
∞ α f1,α (t, s) ds
=
0
e
−λs
Γω
=
1 2πi
1 2πi
α α μs Eα −μ t e dμ ds
Γω
0
1 = 2πi
Γω
Eα −μα t α
∞
e
−λs μs
e
ds dμ
0
Eα (−μα t α ) dμ = Eα −λα t α λ−μ
holds by Cauchy’s integral formula and (2.4). The last statement follows from the calculation of ∞ the Laplace transform of the function 0 ϕα (t, τ )pα (τ, ·) dτ . 1/α (d) The representation of fα,1 follows from Remark 3.2(d). By (b), the C0 -semigroup gener∞ ated by −A is given by T (t) = 0 ϕ1/α (t, s)Sα (s) ds; and then by (a), the (1/α)-times resolvent ∞ family generated by −A1/α is given by 0 p1/α (t, s) T (s) ds. (f) and the first part of (e) are immediate consequences of Theorem 3.1. It remains to prove the subordination formulas (3.9). Indeed, let Sα/2 be the α/2-times resolvent family generated
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M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
by −A1/2 . Since α/2 < 1, by (5.24) in [26], α/2 −1 1 λ + A1/2 = π
∞ 0
μ1/2 (μ + A)−1 dμ. μ + λα
(3.10)
Therefore, it follows from (2.8), (2.9) (3.10), and Fubini’s theorem that 1 Sα/2 (t) = 2πi
−1 eλt λα/2−1 λα/2 + A1/2 dλ
Γθ0
1 = 2πi
λt α/2−1
e λ
1 π
Γθ0
1 = 2πi
0
λt α/2−1
e λ
1 π
Γθ0
1 = 2πi
eλt
α π
−1 μ1/2 1/α α μ +A dμ dλ μ + λα μ1/α−1/2 μ + λα
∞
∞ Sα (s) ds
0
∞
∞ Sα (s) ds
0
α = t α/2 π
∞ 0
Γθ0
α = π
∞
0
e−sν
1 2πi
∞ 0
e
−sμ1/α
Sα (s) ds dμ dλ
0
α/2−1 ν α/2 λ e−sν α dν dλ ν + λα
eλt Γθ0
0
∞
λα/2−1 ν α/2 dλ dν ν α + λα
α
s 2 −1 Sα (s) ds, sα + t α
t > 0.
2
Remark 3.4. −t 2 /4s
(a) (3.8) is the formula (11) in [32]. Note that p1/2 (t, s) = 2te√π s 3/2 (see Lemma 1.6.7 in [2]). (b) By Corollary 3.3(b), we obtain the subordinate principle (Theorem 3.1 in [3]) for bounded fractional resolvent families. The formula is also applied to exponentially bounded fractional resolvent families by small modification, since we do not need the fractional power here. By −s 2 /4t
Lemma 1.6.7 in [2] ϕ1/2 (t, s) = e √πt . (c) By Corollary 3.3(e), if A generates a bounded C0√-semigroup T (t), then the 1/2-times resol ∞ T (s) vent family generated by −(−A)1/2 is given by πt 0 (t+s)s 1/2 ds. (d) By Corollary 3.3(e), if A generates a bounded cosine function C(t), then the C0 -semigroup ∞ C(s) generated by −(−A)1/2 is given by 2t ds. See also Lemma 2.1 in [9]. 2 2 π 0 t +s The following results for generators of analytic fractional resolvent families can be proved similarly as the proof of Theorem 3.1(a) by using Lemma 2.7.
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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Proposition 3.5. The following assertions hold: (a) If −A ∈ A1 (θ0 ) for some θ0 ∈ (0, π2 ], then −Aα ∈ A1 ( π2 − ( π2 − θ0 )α) for each α ∈ π (0, π−2θ ). 0 (b) If −A ∈ A1 (θ0 ) for some θ0 ∈ (0, π2 ], then −Aα ∈ Aα (min{ πα + θ0 − π, π/2}) for each π α ∈ (0, π−θ ). 0 (c) If −A ∈ Aα (θ0 ) for some α ∈ (0, 2) and θ0 ∈ (0, min{ π2 , πα − π2 }], then −A ∈ 0 )α ). Aγ (min{ γα ( π2 + θ0 ) − π2 , π2 }) for each γ ∈ (0, (π+2θ π (d) If −A ∈ Aα (θ0 ) for some α ∈ (0, 2) and θ0 ∈ (0, min{ π2 , πα − π2 }], then −A1/α ∈ π A1 (− πα + θ0 + π) if α ∈ ( π+θ , 2). 0 (e) If −A ∈ Aα (θ0 ) for some α ∈ (0, 2) and θ0 ∈ (0, min{ π2 , πα − π2 }], then −Aβ ∈ β (2−α)π π + βθ0 + πα − π2 , π/2}) if β ∈ (0, 2π−(π+θ ). Aα (min{− βα π + 2α 0 )α Remark 3.6. Proposition 3.5(a) improves Theorem 3.1 in [15] in that we do not need 0 ∈ ρ(A). Example 3.7. Let α ∈ (0, 2) and k > 0. Set X := Lp (R), A := −kDx2 with D(A) = W 2,p (R). It is well known that −A generates a bounded analytic semigroup of angle π2 . Thus, by Proposition 3.5 one has (a) −A ∈ Aα (min{ πα − π2 , π2 }) for all α ∈ (0, 2); (b) −Aα ∈ Aα (min{ πα − π2 , π2 }) for all α ∈ (0, 2); (c) −Aα ∈ A1 (π/2) for all α ∈ (0, ∞). Example 3.8. Let α ∈ (0, 2) and θ ∈ [0, π). Set X := L2 (0, 1), Bθ := −eiθ Dx2 with D(Bθ ) = {f ∈ W 2,2 (0, 1): f (0) = f (1) = 0}. It is proved that for π2 < θ (1 − α2 )π , −Bθ ∈ Aα (θ0 ) with θ0 = min{ πα − αθ − π2 , π2 }, but does not generate any C0 -semigroup (see Example 2.20 in [3]). 1/α However, by Corollary 3.3(d), −Bθ ∈ A1 ( π2 − αθ ) for α ∈ (1, 2). 4. Solutions to fractional Cauchy problems In this section we will consider the solutions of fractional Cauchy problems. First we give the definitions of solutions to the inhomogeneous initial value problem Dtα u(t) = Au(t) + f (t), u(k) (0) = xk ,
t ∈ (0, τ ),
k = 0, 1, . . . , m − 1,
(4.1)
where τ ∈ (0, +∞], f ∈ L1loc ([0, τ ); X) and A is a closed densely defined operator on Banach space X. Definition 4.1. A function u(t) ∈ C([0, τ ); X) is called a strong solution (or simply solution) of (4.1) if u(t) satisfies: (a) u(t) ∈ C([0, τ ); D(A)) ∩ C m−1 ([0, τ ); X).
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(b) gm−α ∗ (u −
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
m−1 k=0
gk+1 xk ) ∈ C m ([0, τ ); X).
(c) u(t) satisfies Eq. (4.1). u(t) ∈ C([0, τ ); X) is called a mild solution of (4.1) if gα ∗ u ∈ D(A) and u(t) =
m−1
gk+1 (t)xk + A(gα ∗ u)(t) + (gα ∗ f )(t),
t 0.
k=0
Suppose that A generates an α-times resolvent family Sα (t), then the strong solution of (4.1) with f = 0 and xk ∈ D(A) is given by u(t) =
m−1
(gk ∗ Sα )(t)xk ,
k=0
see [3] for more details. So we now turn to the following problem Dtα u(t) = Au(t) + f (t), u(k) (0) = 0,
t ∈ (0, τ ),
k = 0, 1, . . . , m − 1.
(4.2)
If u is a mild solution of (4.2), then gα ∗ u ∈ D(A) and u = A(gα ∗ u) + gα ∗ f . By Remark 2.4, 1 ∗ u = Sα − A(gα ∗ Sα ) ∗ u = Sα ∗ u − Sα ∗ A(gα ∗ u) = Sα ∗ u − Sα ∗ u + Sα ∗ gα ∗ f = gα ∗ Sα ∗ f, which means that gα ∗ Sα ∗ f is differentiable and the mild solution is given by u(t) =
d (gα ∗ Sα ∗ f )(t), dt
t 0.
(4.3)
Consequently we have Proposition 4.2. Let A be the generator of an α-times resolvent family Sα and let f ∈ L1loc ([0, τ ); X). If (4.2) has a mild solution, then it is given by (4.3). And the mild solution of (4.1) is given by u(t) =
m−1
(gk ∗ Sα )(t)xk +
k=0
d (gα ∗ Sα ∗ f )(t), dt
t 0.
For the strong solutions of (4.2), we have Proposition 4.3. Let α ∈ (0, 2]. Suppose that A is the generator of an α-times resolvent family Sα and let f ∈ C([0, τ ); X). Then the following statements are equivalent: (a) (4.2) has a strong solution on [0, τ ).
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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(b) Sα ∗ f is differentiable on [0, τ ). d d (gα ∗ Sα ∗ f )(t) ∈ D(A) for t ∈ [0, τ ) and A( dt (gα ∗ Sα ∗ f )(t)) is continuous on [0, τ ). (c) dt In the case α ∈ [1, 2], the condition (c) can be replaced by (c) (gα−1 ∗ Sα ∗ f )(t) ∈ D(A) for t ∈ [0, τ ) and A(gα−1 ∗ Sα ∗ f )(t) is continuous on [0, τ ). Proof. The equivalence of (a), (b) and (c) was given in [22] for the case α ∈ [1, 2]. The case α ∈ (0, 1) can be proved similarly. 2 As a corollary we have Corollary 4.4. Let α ∈ (0, 2]. Suppose that A is the generator of an α-times resolvent family. Then (4.2) has a strong solution on [0, τ ) if one of the following conditions is satisfied: (a) f is continuously differentiable on [0, τ ). (b) α ∈ [1, 2], f (t) ∈ D(A) for t ∈ [0, τ ) and Af (t) ∈ L1loc ([0, τ ); X). (c) α ∈ (0, 1), f (t) ∈ D(A) for t ∈ [0, τ ) and gα ∗ f is continuously differentiable on [0, τ ). If A generates an α-times resolvent family Sα , then for x ∈ D(An ) by using (2.7) several times we have Sα (t)x = x + (gα ∗ Sα )(t)Ax
= x + (gα ∗ 1)(t)Ax + gα ∗ (gα ∗ Sα ) (t)A2 x
= x + gα+1 (t)Ax + (g2α ∗ Sα )(t)A2 x = ··· = x + gα+1 (t)Ax + · · · + g(n−1)α+1 (t)An−1 x + (gnα ∗ Sα )(t)An x,
(4.4)
which leads to Lemma 4.5. If A generates an α-times resolvent family Sα , then for x ∈ D(An ) with nα 1, Sα (t)x is differentiable and d Sα (t)x = gkα (t)Ak x + (gnα−1 ∗ Sα )(t)An x, dt n−1
t > 0.
k=1
In particular, let α = 1/m with m ∈ N, we obtain Proposition 4.6. Let m ∈ N. Suppose that A generates a (1/m)-times resolvent family S1/m . Then for each x ∈ D(Am ), S1/m (·)x solves the fractional Cauchy problem 1/m
Dt
u(t) = Au(t), u(0) = x,
t > 0, (4.5)
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and the initial value problem
v (t) = Am v(t) +
m−1
gk/m (t)Ak x,
t > 0,
k=1
v(0) = x.
(4.6)
Remark 4.7. (a) If A generates a C0 -semigroup, then by the subordination principle A generates an (analytic) (1/m)-times resolvent family. So Proposition 4.6 gives Theorem 3.3 in [19] immediately. (b) Note that Am does not necessarily generate a C0 -semigroup when A generates a 1/mresolvent family, we cannot obtain the uniqueness of solution of (4.6) without any further assumption on the operator A and a counterexample was given in [6]. For the corresponding inhomogeneous problems, we have Proposition 4.8. Let m 2 be fixed. Assume that A is the generator of a (1/m)-times resolvent family S1/m , then for x ∈ D(Am ), f (t) ∈ C(R+ , D(Am )), the function S1/m (t)x + (S1/m ∗ f )(t) solves the two equations: 1/m
Dt
u(t) = Au(t) + (g(1−1/m) ∗ f )(t),
t > 0,
u(0) = x
(4.7)
and
v (t) = Am v(t) +
m−1
gk/m (t)Ak x +
k=1
m−1
gk/m ∗ Ak f (t),
t > 0,
k=0
v(0) = x.
(4.8)
Proof. Since g1/m ∗ (g(1−1/m) ∗ f ) = g1 ∗ f is differentiable and f (t) ∈ D(A) for all t 0, by Proposition 4.2 and Corollary 4.4(c), S1/m (t)x + (S1/m ∗ f )(t) solves (4.7). It remains to show that it is also a solution of (4.8). By Proposition 4.6, we only need to show that S1/m ∗ f is differentiable, (S1/m ∗ f )(t) ∈ D(Am ) and
(S1/m ∗ f ) (t) = Am (S1/m ∗ f )(t) +
m−1
gk/m ∗ Ak f (t),
k=0
This follows from (4.4) since
t > 0.
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
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t (S1/m ∗ f )(t) =
S1/m (t − s)f (s) ds 0
t =
f (s) + 0
m−1 k=1
g k +1 (t − s)A f (s) + (g1 ∗ S1/m )(t − s)A f (s) ds. k
m
m
2
Next we will discuss the connections between some pairs of the Cauchy problems of fractional order (not necessarily a rational number) and first order. First, we have the following direct consequences of Corollary 3.3. Theorem 4.9. (a) Let α ∈ (0, 1) and −A ∈ C1 (0). The fractional Cauchy problem Dtα v(t) = −Av(t),
t > 0,
v(0) = x,
(4.9)
is well-posed and its unique solution is given by ∞ v(t) =
ϕα (t, s)u(s) ds,
t > 0,
0
for each x ∈ D(A), where ϕα is given as in Corollary 3.3 and u is the solution to the Cauchy problem u (t) = −Au(t),
t > 0,
u(0) = x.
(4.10)
(b) Let α ∈ (0, 1) and −A ∈ C1 (0). The fractional Cauchy problem v (t) = −Aα v(t),
t > 0,
v(0) = x,
(4.11)
is well-posed and its unique solution is given by ∞ v(t) =
pα (t, s)u(s) ds,
t > 0,
0
for each x ∈ D(A), where pα is given as in Corollary 3.3 and u is the solution to the Cauchy problem (4.10).
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(c) Let α ∈ (0, 1) and −A ∈ C1 (0). The fractional Cauchy problem Dtα v(t) = −Aα v(t),
t > 0,
v(0) = x,
(4.12)
is well-posed and its unique solution is given by ∞ v(t) =
α f1,α (t, s)u(s) ds,
t > 0,
0 α is given as in Corollary 3.3 and u is the solution to the for each x ∈ D(Aα ), where f1,α Cauchy problem (4.10). (d) Let β ∈ (1, 2] and −A ∈ Cβ (0). The Cauchy problem (4.10) is well-posed and its unique solution is given by
∞ u(t) =
ϕ1/β (t, s)v(s) ds,
t > 0,
0
for each x ∈ D(A), where v is the solution to the fractional Cauchy problem β
Dt v(t) = −Av(t), v(0) = x,
t > 0,
v (0) = 0.
(4.13)
(e) Let β ∈ (1, 2] and −A ∈ Cβ (0). The Cauchy problem u (t) = −A1/β u(t),
t > 0,
u(0) = x,
(4.14)
is well-posed and its unique solution is given by ∞ u(t) =
1/β
fβ,1 (t, s)v(s) ds,
t > 0,
0 1/β
for each x ∈ D(A1/β ), where fβ,1 is given as in Corollary 3.3 and v is the solution to the fractional Cauchy problem (4.13). Remark 4.10. (a) In Theorem 4.9(a) and (c), if A generates an analytic C0 -semigroup, then the restriction on α can be relaxed by using Proposition 3.5. (b) By using the generalized subordination principle in Theorem 3.1 and Proposition 4.2 one can also consider the inhomogeneous fractional Cauchy problems.
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
2723
Remark 4.11. The results in Theorem 4.9 can be interpreted in terms of stochastic solutions. Let 0 < α < 1 and X be a Markov process with a semigroup T (t)f (x) = E(f (X(t))) generated by −A and let E(t) := inf{x > 0: D(t) > t} be the inverse or hitting time process of the stable α subordinator D(t), independent of X, with E(e−sD(t) ) = e−ts . If u is a solution to the problem u (t) = −Au(t);
u(0) = f (x),
(4.15)
then (a) the problem Dtα v(t) = −Av(t);
v(0) = f (x),
has a unique solution given by v(t) = E f X E(t) =
∞ u(s)fE(t) (s) ds, 0
where fE(t) (s) is the density of the inverse stable subordinator of index α (see also Theorem 3.3 in [7]); (b) the problem W (t) = −Aα W (t);
W (0) = f (x),
has a unique solution given by W (t) = E f X D(t) =
∞ u(s)fD(t) (s) ds, 0
where fD(t) (s) is the density of the stable subordinator of index α; (c) the problem Dtα v(t) = −Aα v(t);
v(0) = f (x),
has a unique solution given by v(t) = E f X D E(t) = ∞ ∞ =
W (s)fE(t) (s) ds 0
u(r)fD(s) (r) dr fE(t) (s) ds
0
with W given in (b);
∞
0
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M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
(d) if in addition that for some β ∈ (1, 2] the fractional Cauchy problem D β V (t) = −AV (t);
V (0) = f (x),
V (0) = 0
is well-posed, then the solution of (4.15), u, is subordinated to V by ∞ u(t) =
V (s)hE(t) (s) ds, 0
where hE(t) (s) is the density of the inverse stable subordinator of index 1/β; (e) if the assumptions of (d) hold, then the solution to the Cauchy problem v (t) = −A1/β v(t);
v(0) = f (x),
is connected to V by v(t) =
∞ ∞
∞ u(s)hD(t) (s) ds = 0
V (r)hE(s) (r) dr hD(t) (s) ds 0
0
where hE(t) is as in (d) and hD(t) (s) is the density of the stable subordinator of index 1/β. We end this paper with two examples. Example 4.12. Let ρ > 0 and m ∈ N. Consider the fractional relaxation equation (cf. [12]) 1/m
Dt
u(t) = −ρu(t),
t > 0,
u(0) = x.
(4.16)
The solution of (4.16) is given by u(t) = xE1/m (−ρt 1/m ). By Proposition 4.6, u(t) also solves v (t) = (−ρ)m v(t) +
m−1 k=1
g k (t)(−ρ)k x, m
v(0) = x.
t > 0, (4.17)
Note that the solution of (4.17) is unique. Therefore, the problem (4.16) is equivalent to the problem (4.17). Example 4.13. By Theorem 4.9, the solution of the fractional diffusion equation of order 0 < α1 Dtα u(t, x) = u(t, x), u(0, x) = f0 (x)
t > 0, (4.18)
M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
is given by u(t, x) = by . Since
∞ 0
2725
ϕα (t, s)(T (s)f0 )(x) ds, where T is the Gaussian semigroup generated
−n/2
T (s)f (x) = (ks ∗ f )(x) = (4πs)
e−|x−y|
2 /4s
f0 (y) dy,
Rn
we have ∞ u(t, x) = Rn
ϕα (t, s)(4πs)−n/2 e
−|x−y|2 /4s
ds f0 (y) dy.
0
See also [31]. Acknowledgment The authors are grateful to the referee for the valuable comments and suggestions, and especially for the suggestion of giving the interpretation of our results for stochastic solutions. Remark 4.11 is in fact suggested by the referee. References [1] H. Allouba, W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab. 29 (2001)(2) 1780–1795. [2] W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monogr. Math., vol. 96, Birkhäuser, Basel, 2001. [3] E.G. Bajlekova, Fractional evolution equations in Banach spaces, PhD thesis, Department of Mathematics, Eindhoven University of Technology, 2001. [4] A.V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960) 419–437. [5] B. Baeumer, M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001) 481–500. [6] B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc. 361 (2009) 3915–3930. [7] D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resources Res. 36 (6) (2000) 1413–1424. [8] C. Chen, M. Li, On fractional resolvent operator functions, Semigroup Forum 80 (2010) 121–142. [9] I. Cioranescu, V. Keyantuo, On operator cosine functions in UMD spaces, Semigroup Forum 63 (2001) 429–440. [10] G. Da Prato, M. Iannelli, Linear integro-differential equations in Banach space, Rend. Sem. Mat. Univ. Padova 62 (1980) 207–219. [11] R.D. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab. 14 (3) (2004) 1529–1558. [12] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien/New York, 1997, pp. 223–276. [13] R. Gorenflo, Y. Luchko, F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal. 2 (4) (1999) 383–414. [14] M. Haase, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., vol. 169, Birkhäuser-Verlag, Basel, 2006. [15] Y.Z. Huang, Q. Zheng, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. Differential Equations 203 (2004) 38–54. [16] A. Karczewska, C. Lizama, Stochastic Volterra equations driven by cylindrical Wiener process, J. Evol. Equ. 7 (2007) 373–386.
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M. Li et al. / Journal of Functional Analysis 259 (2010) 2702–2726
[17] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad. 36 (1960) 94–96. [18] V. Keyantuo, On analytic semigroups and Cosine operator functions in Banach spaces, Studia Math. 129 (2) (1998) 137–156. [19] V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems, http://netlizama.usach.cl/Keyantuo-Lizama(AMPA)(2009).pdf. [20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Math. Stud., vol. 204, North-Holland, Amsterdam, 2006. [21] H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966) 285–346. [22] F.B. Li, M. Li, On maximal regularity and semivariation of α-times resolvent families, arXiv:1007.4455 [math.FA]. [23] M. Li, Q. Zheng, On spectral inclusions and approximations of α-times resolvent families, Semigroup Forum 69 (2004) 356–368. [24] C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl. 243 (2000) 278–292. [25] C. Lizama, On approximation and representation of k-regularized resolvent families, Integral Equations Operator Theory 41 (2001) 223–229. [26] C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Stud., vol. 187, NorthHolland, Amsterdam, 2001. [27] M.M. Meerschaert, H.P. Scheffler, Limit Distributions for Sum of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley Ser. Probab. Stat., Wiley Interscience, New York, 2001. [28] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993. [29] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., vol. 198, Academic Press, San Diego, 1999. [30] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000) 376–384. [31] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134–144. [32] K. Yosida, Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them, Proc. Japan Acad. 36 (1960) 86–89. [33] G.M. Zaslavsky, Fractional kinetic equations for Hamiltonian chaos, Chaotic advection, tracer dynamics and turbulent dispersion, Phys. D 76 (1994) 110–122.
Journal of Functional Analysis 259 (2010) 2727–2756 www.elsevier.com/locate/jfa
Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights Jordi Pau a,1 , José Ángel Peláez b,∗,1 a Departament de Matemàtica Aplicada i Analisi, Universitat de Barcelona, 08007 Barcelona, Spain b Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
Received 23 April 2010; accepted 24 June 2010 Available online 13 July 2010 Communicated by N. Kalton
Abstract We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap (w) ⊂ Lq (μ), 0 < p, q < ∞, where w belongs to a large class W of rapidly decreasing weights which −1 ), α > 0, and some double exponential type weights. includes the exponential weights wα (r) = exp( (1−r) α As an application of that result, we characterize the boundedness and the compactness of Tg : Ap (w) → Aq (w), 0 < p, q < ∞, w ∈ W, where Tg is the integration operator z (Tg f )(z) =
f (ζ )g (ζ ) dζ.
0
The particular choice of the weight wα (r) answers an open question posed by A. Aleman and A. Siskakis. We also describe those analytic functions in D for which Tg belongs to the Schatten p-class of A2 (w), 0 < p < ∞. © 2010 Elsevier Inc. All rights reserved. Keywords: Integration operators; Bergman spaces; Carleson measures; Schatten classes
* Corresponding author.
E-mail addresses:
[email protected] (J. Pau),
[email protected] (J.Á. Peláez). 1 The first author is supported by the grant 2009 SGR 420 and DGICYT grant MTM2008-05561-C02-01
(MCyT/MEC), while the second author is supported by MTM2007-60854 (MCyT/MEC), FQM210 (Junta de Andalucía) and MTM2006-26627-E (“Acciones Complementarias”, MEC) and by the Ramón y Cajal program. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.06.019
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1. Introduction and main results Let D be the unit disc in the complex plane, dm(z) = dxπdy be the normalized area measure on D, and denote by H (D) the space of all analytic functions in D. A positive function w(r), 0 r < 1, which is integrable in (0, 1), will be called a weight function. We extend w to D setting w(z) = w(|z|), z ∈ D. For 0 < p < ∞, the weighted Bergman space Ap (w) is the space of functions f ∈ H (D) such that
p
f Ap (w) =
f (z)p w(z) dm(z) < ∞.
D
Note that Ap (w) is the closed subspace of Lp (D, w dm) consisting of analytic functions. If p w(r) = (1 − r)α , α > −1, the standard Bergman spaces Aα are obtained. In this work we are going to study Carleson measures and integration operators on Bergman spaces with rapidly decreasing weights, that is, weights that are going to decrease faster than any standard weight (1 − r)α , α > 0. Concretely, we consider a decreasing weight of the form w(z) = e−ϕ(z) , where ϕ ∈ C 2 (D) is a radial function such that ϕ(z) Bϕ > 0 for some positive constant Bϕ depending only on the function ϕ. Here denotes the standard Laplace operator. We will assume that (ϕ(z))−1/2 τ (z), where τ (z) is a radial positive function that decreases to 0 as |z| → 1− , and limr→1− τ (r) = 0. Furthermore, we shall also suppose that either there exists a constant C > 0 such that τ (r)(1 − r)−C increases for r close to 1 or lim τ (r) log
r→1−
1 = 0. τ (r)
If the weight w satisfies all the previous conditions, we shall say that the weight w belongs to the class W. The class W includes (see Section 7) the exponential type weights wγ ,α (r) = (1 − r) exp γ
−c , (1 − r)α
γ 0, α > 0, c > 0,
and the double exponential weights w(r) = exp −γ exp
β (1 − r)α
,
α, β, γ > 0.
For the weights w considered in this paper, the point evaluations La at the point a ∈ D, are bounded linear functionals on A2 (w). Therefore, there are reproducing kernels Ka ∈ A2 (w) with La = Ka A2 (w) such that La f = f, Ka =
f (z)Ka (z)w(z) dm(z),
f ∈ A2 (w).
D
Some basic properties of the Bergman spaces Ap (w), w ∈ W, are not yet well understood and have attracted some attention in recent years. The interest in such spaces arises from the fact that
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the usual techniques for the standard Bergman spaces fail to work in this context. For example, the natural Bergman projection Pf (a) = f (z)Ka (z)w(z) dm(z), a ∈ D, D
is not necessarily bounded on Lp (D, w dm) if p = 2 (see [9]). This carries that the dual spaces of Ap (w), w ∈ W have not been identified. Let X be a space of analytic functions on the unit disc D. A positive Borel measure μ in D is said to be a q-Carleson measure for X if the embedding X ⊂ Lq (μ), 0 < q < ∞, is continuous. After the pioneering works of L. Carleson (see [7] and [8]), there has been a great amount of research on this topic, and these measures have found many applications in other related areas. A description of those measures have been obtained for several spaces of analytic functions (see e.g. [5,11,14,19]). Here we obtain a complete description of the q-Carleson measures for Ap (w), 0 < p, q < ∞, for weights w in the class W. Let D(a, r) be the Euclidean disc centered at a with radius r > 0, and for easy of notation, for any δ > 0 we write D(δτ (a)) for the disc D(a, δτ (a)). Theorem 1. Let w ∈ W and let μ be a finite positive Borel measure on D. (I) Let 0 < p q < ∞. (a) The embedding Id : Ap (w) → Lq (μ) is bounded if and only if for each sufficiently small δ > 0 we have 1 Kμ,w := sup w(z)−q/p dμ(z) < ∞. (1) 2q/p a∈D τ (a) D(δτ (a))
Moreover, if any of the two equivalent conditions holds, then q
Kμ,w Id Ap (w)→Lq (μ) . (b) The embedding Id : Ap (w) → Lq (μ) is compact if and only if for each sufficiently small δ > 0 we have 1 lim sup w(z)−q/p dμ(z) = 0. (2) r→1− |a|>r τ (a)2q/p D(δτ (a))
(II) Let 0 < q < p < ∞. The following conditions are equivalent: (a) Id : Ap (w) → Lq (μ) is compact; (b) Id : Ap (w) → Lq (μ) is bounded; (c) For each sufficiently small δ > 0, the function 1 z → w(ζ )−q/p dμ(ζ ) τ (z)2 D(δτ (z)) p
belongs to L p−q (D, dm).
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For the standard Bergman spaces Aα , the statement of Theorem 1 also holds, and in that case the condition can be simplified (see [17, Theorem 2.2] for the case 0 < p q < ∞, and [20] for the case 0 < q < p < ∞). Related results can be found in [21] and [22]. The above result can be used in order to study several related questions. Here we put our attention on the operators Tg defined by z (Tg f )(z) =
f (ζ )g (ζ ) dζ,
0
where g is an analytic function on D. The boundedness and compactness of Tg on classical spaces has attracted a lot of attention in recent years (see [2,3] for Hardy spaces, [4,10,25] for weighted Bergman spaces, and [13,14] for Dirichlet-type spaces). We also mention the surveys [1] and [27] for an account of results and open questions on the operator Tg . Note that as special choices of the symbol g one gets several important operators: the Volterra operator (Tg with g(z) = z) and the Cesáro operator (Tg with g(z) = log(1/(1 − z))). In particular, A. Aleman and A. Siskakis proved in [4] the following result. Theorem A. Suppose that w is a weight satisfying the following conditions: (P 1) There is a positive constant C such that 1 w(r)
1 w(s) ds C(1 − r). r
(P 2) There are s ∈ (0, 1) and a positive constant C such that w(sr + 1 − s) Cw(r),
0 < r < 1.
Then for 1 p < ∞ Tg is bounded (compact) on Ap (w) if and only if g belongs to the Bloch space (little Bloch space). Recall that an analytic function f in D belongs to the Bloch space if supz∈D (1 − |z|)|f (z)| < ∞, and f belongs to the little Bloch space if lim|z|→1− (1 − |z|)|f (z)| = 0. The large class of radial weights w satisfying conditions (P 1) and (P 2) of Theorem A includes the standard weights w(r) = (1 − r)α , α > −1, but excludes the exponential ones wα (r) = exp
−c , (1 − r)α
c > 0, α > 0
(3)
(they do no satisfy condition (P 2)). In relation with the exponential weights, A. Aleman and A. Siskakis in [4] proved that, for 1 p < ∞, Tg is bounded on Ap (wα ) if α+1 g (z) = O(1), 1 − |z|
as |z| → 1,
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and the operator Tg is compact on Ap (wα ) whenever the corresponding “little oh” condition holds, and raised the open problem of whether this condition is necessary for the boundedness (compactness) of Tg . A positive answer for p = 2, c > 0 and α ∈ (0, 1] has been given recently by Dostani´c in [10] by using precise and specific techniques which involve the exponential weight defined in (3). In this paper we shall give a positive answer to that question (see Theorem 2 below). Indeed, we completely describe the boundedness and compactness of Tg : Ap (w) → Aq (w), 0 < p, q < ∞, for weights w ∈ W. In the proof of Theorem A given in [4], two facts play an essential role. First, for some 0 < s < 1 the composition operator induced by the symbol ψs (z) = sz − s + 1 is bounded on Ap (w). Since this does not remain true for rapidly decreasing weights such that wα (see [15, Theorem 1.1]) their method cannot be applied in our case. The second key step in the proof of Theorem A consists of proving that 1 − |z| f (z) ∈ Lp (w), (4) f ∈ Ap (w) ⇔ consequently it will be useful for our class of weights to establish a result analogous to (4) replacing the quantity (1 − |z|) for a suitable distortion function. So, following Siskakis [26], for a given weight w, we define the distortion function of w by 1 ψw (r) = w(r)
1 w(u) du,
0 r < 1.
r
The next result is the case q = p of Theorem 1.1 of [24]. Theorem B. Suppose that w is a differentiable weight, and there is L > 0 such that w (r) sup 2 0
1 w(x) dx L,
(5)
r
then for each p ∈ (0, ∞) and g ∈ H (D) g(z)p w(z) dm(z) g(0)p + g (z)p ψw (z)p w(z) dm(z). D
D
It is clear that condition (5) is satisfied for any decreasing differentiable weight. Finally, we are going to state the promised result about the description of the boundedness and compactness of Tg : Ap (w) → Aq (w), 0 < p, q < ∞, w ∈ W. Theorem 2. Let 0 < p, q < ∞, g ∈ H (D), and w ∈ W with ϕ(z)
t −1 1 − |z| ψw (z) ,
z ∈ D, for some t 1.
(I) (a) Tg is bounded on Ap (w) if and only if sup ψw (z)g (z) < ∞. z∈D
(6)
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(b) Tg is compact on Ap (w) if and only if lim sup ψw (z)g (z) = 0.
r→1− |z|>r
(II) Let 0 < p < q < ∞. Then Tg : Ap (w) → Aq (w) is bounded if and only if g is constant. (III) Let 0 < q < p < ∞. The following conditions are equivalent: (a) Tg : Ap (w) → Aq (w) is compact; (b) Tg : Ap (w) → Aq (w) is bounded; pq (c) The function g ∈ Ar (w), where r = p−q . p
The corresponding result for the standard Bergman spaces Aα can be found in [4] and [25]. The particular choice of the exponential weight wα defined in (3) solves the problem posed in [4, p. 353], since the distorsion function of wα is comparable to (1 − |z|)1+α (see e.g. [26, Example 3.2]). If T is a compact operator on a separable Hilbert space H , then there exist orthonormal sets {en } and {σn } in H such that Tx =
λn x, en σn ,
x ∈ H,
n
where λn is the nth singular value of T . Given 0 < p < ∞, let Sp (H ) denote the Schatten p-class of operators on H . The class Sp (H ) consists of those compact operators T on H with its sequence of singular numbers λn belonging to p , the p-summable sequence space. It is also well known that, if λn are the singular numbers of an operator T , then
λn = λn (T ) = inf T − K: rank K n . Thus finite rank operators belong to every Sp (H ), and the membership of an operator in Sp (H ) measures in some sense the size of the operator. It is also clear that the use of two equivalent norms in H does not change the class Sp (H ). In the case when 1 p < ∞, Sp (H ) is a Banach space with the norm T p =
1/p |λn |
p
,
n
while for 0 < p < 1 we have the following inequality p
p
p
T + Sp T p + Sp . We refer to [28] for more information about Sp (H ). In [3], using a result D. Luecking [18] concerning the Schatten classes of certain Toeplitz operators (see also [2] and [16] for related results), a description of those g ∈ H (D) for which Tg ∈ Sp (A2α ) is obtained. In this paper we give a direct proof of the following result.
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Theorem 3. Let g ∈ H (D) and w ∈ W satisfying (6). (a) If 1 < p < ∞ then Tg ∈ Sp (A2 (w)) if and only if ψw g ∈ Lp (D, ϕ dm). (b) If 0 < p 1 then Tg ∈ Sp (A2 (w)) if and only if g is constant. The paper is organized as follows: Section 2 is devoted to some preliminaries needed for the proofs of the main results. In Section 3 we construct the test functions which are used in the proof of Theorem 1 in Section 4. We prove Theorem 2 in Section 5 and Theorem 3 in Section 6. Finally, in Section 7 several examples of weights w in the class W are given. Throughout the paper, the letter C will denote an absolute constant whose value may change at different occurrences. We also use the notation a b to indicate that there is a constant C > 0 with a Cb, and the notation a b means that a b and b a. 2. Preliminaries In this section we present some previous results, that can be of independent interest, which are needed to prove the main results. Let τ be a positive function on D. We say that τ ∈ L if there exist constants c0 > 0 and c1 > 0 such that τ (z) c0 1 − |z| for z ∈ D; τ (z) − τ (ζ ) c1 |z − ζ | for z, ζ ∈ D.
(7) (8)
Throughout this paper, we will always use the notation mτ =
min(1, c0−1 , c1−1 ) , 4
where c0 and c1 are the constants in (7) and (8). Lemma 2.1. Suppose that τ ∈ L, 0 < δ mτ and a ∈ D. Then, 1 τ (a) τ (z) 2τ (a) 2
if z ∈ D δτ (a) .
Proof. Note that, by condition (8) we have 1 τ (a) τ (z) + c1 |z − a| τ (z) + τ (a) 4
if |z − a| δτ (a).
Therefore τ (a) 2τ (z) if |z − a| δτ (a). Similarly it can be proved that τ (z) 2τ (a).
2
The following result, where the fact that |f (z)|p w(z) verifies a certain sub-mean-value property is proved, will play an essential role in the proof of the main theorems of this paper.
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Lemma 2.2. Let ϕ be a subharmonic function, w = e−ϕ , and let τ ∈ L such that τ (z)2 ϕ(z) c2 for some constant c2 > 0. If β ∈ R, there exists a constant M 1 such that f (a)p w(a)β M f (z)p w(z)β dm(z), 2 2 δ τ (a) D(δτ (a))
for all 0 < δ mτ and f ∈ H (D). Proof. By Green’s formula we have 1 2π
|ζ −a|=r
r ϕ(ζ ) dζ = rϕ(a) + 2
ϕ(z) log |z−a|r
r dm(z). |z − a|
Integrating this identity between zero and δτ (a), using that τ (z)2 ϕ(z) c2 , and Lemma 2.1, we obtain 1 τ (a)2
c2 ϕ(ζ ) dm(ζ ) δ ϕ(a) + τ (a)2
δτ (a)
2
0
D(δτ (a))
4c2 δ ϕ(a) + τ (a)4
|z−a|r
δτ (a)
2
log 0
2c2 = δ ϕ(a) + τ (a)4
τ (z)−2 log
|z−a|r
r dm(z) r dr |z − a|
r dm(z) r dr |z − a|
δτ (a)
2
r 3 dr 0
= δ 2 ϕ(a) +
c2 δ 4 . 2
So if β > 0, rewriting the previous equation in terms of the weight w, and multiplying by β, we obtain 1 log w(a)β 2 log w(ζ )β dm(ζ ) + log M, (9) δ τ (a)2 D(δτ (a))
with M = exp(βc2 δ 2 /2) 1. Also, the subharmonicity of log|f | gives p p 1 logf (z) dm(z). log f (a) 2 2 δ τ (a)
(10)
D(δτ (a))
Now, adding Eqs. (10) and (9), and using the arithmetic–geometric inequality we get the desired result. If β 0, then w β is a logarithmic subharmonic function, so (9) holds with M = 1, and arguing as in the previous case the conclusion is obtained. This finishes the proof. 2
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We note that if a weight w belongs to the class W, then its associated function τ (z) belongs to the class L. Thus Lemma 2.2 proves that for weights w in the class W, the point evaluations La are bounded linear functionals on Ap (w). Another consequence is that norm convergence implies uniform convergence on compact subsets of D. It follows that for w ∈ W, the space Ap (w) is complete. The next result proves that the weights in the class W decrease faster than the standard weights (1 − |z|)α , α > 0. Lemma 2.3. Consider a weight of the form w(z) = e−ϕ(z) , where ϕ ∈ C 2 (D) is a radial function with (ϕ(z))−1/2 τ (z), and τ (z) is a radial positive differentiable function that decreases to 0 as |z| → 1− . If limr→1 τ (r) = 0, then w(z) = 0, τ (z)α
lim
|z|→1−
for each α > 0.
Proof. We may assume without loss of generality that ϕ(0) = ϕ (0) = 0. Since w(z) = e−ϕ(z)−α log τ (z) , τ (z)α it is enough to show that lim
ϕ(z)
|z|→1−
1 log τ (z)
= +∞.
Write r = |z|. Use the fact that τ (r) is negative in some interval (r0 , 1), the formula 1 ϕ (r) = r
r sϕ(s) ds, 0
and the assumption limr→1 τ (r) = limr→1 τ (r) = 0, to obtain ϕ (r) C
r
s ds C τ (s)2
r0
r
(−τ (s)) ds C τ (r) τ (s)2
r0
for r close to 1. This, together with Bernouilli–l’Hôpital theorem gives lim
r→1−
This finishes the proof.
ϕ(r) log
1 τ (r)
= lim
r→1−
ϕ (r)τ (r) C lim = +∞. −τ (r) r→1− −τ (r)
2
The following lemma on coverings is due to Oleinik, see [21]. Lemma A. Let τ be a positive function in D in the class L, and let δ ∈ (0, mτ ). Then there exists a sequence of points {zj } ⊂ D, such that the following conditions are satisfied:
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zj ∈ / D(δτ (zk )), j = k. j D(δτ (zj )) = D. ˜ ˜ D(δτ (zj )) ⊂ D(3δτ (zj )), where D(δτ (zj )) = z∈D(δτ (zj )) D(δτ (z)), j = 1, 2, . . . . {D(3δτ (zj ))} is a covering of D of finite multiplicity N .
3. Test functions It is known that having an appropriate family of test functions in a space of analytic functions X can be a good help in order to characterize the q-Carleson measures for X. In this section we will do the job for the spaces Ap (w). The following result, partially proved in [6], will be a key in the proof of Theorem 1. Lemma 3.1. Assume that 0 < p < ∞, n ∈ N \ {0} with np 1 and w ∈ W. Then, there is a number ρ0 ∈ (0, 1) such that for each a ∈ D with |a| ρ0 , there is a function Fa,n,p analytic in D with Fa,n,p (z)p w(z) 1 if |z − a| < τ (a),
(11)
and 3n Fa,n,p (z)w(z)1/p min 1, min(τ (a), τ (z)) , |z − a|
z ∈ D.
(12)
Proof. If 1 p < ∞, and n = 1 the result follows directly from Lemma 3.3 in [6]. Now, if 0 < p < ∞ and np 1 applying the previous case, we have Fa,1,np (z)np w(z) 1
if |z − a| < τ (a),
and 3 1 Fa,1,np (z)w(z) np min 1, min(τ (a), τ (z)) , |z − a|
z ∈ D.
n That is, if we choose Fa,n,p = Fa,1,np
Fa,n,p (z)p w(z) 1 if |z − a| < τ (a), and 3n Fa,n,p (z)w(z)1/p min 1, min(τ (a), τ (z)) , |z − a| This finishes the proof.
z ∈ D.
2
As an immediate consequence of that lemma, as it is noticed also in [6], we get an estimate for the reproducing kernels of A2 (w).
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Corollary 1. Let w ∈ W. There is a number ρ0 ∈ (0, 1) such that (i) for 0 < p < ∞ and n ∈ N \ {0} with np 1 the function Fa,n,p in Lemma 3.1 belongs to Ap (w) with p
Fa,n,p Ap (w) τ (a)2 ,
ρ0 |a| < 1;
(ii) the reproducing kernel Ka of A2 (w) satisfies the estimate Ka 2A2 (w) w(a) τ (a)−2 ,
ρ0 |a| < 1.
Proof. Let a ∈ D with ρ0 |a| < 1, and consider the functions Fa,n,p obtained in Lemma 3.1. Write
Rk (a) = z ∈ D: 2k−1 τ (a) < |z − a| 2k τ (a) ,
k = 1, 2 . . . .
Note that (11) gives
Fa,n,p (z)p w(z) dm(z) τ (a)2 ,
|z−a|<τ (a)
and, by (12) and the fact that 3np > 2,
∞ Fa,n,p (z)p w(z) dm(z)
Fa,n,p (z)p w(z) dm(z)
k=1 R (a) k
|z−a|>τ (a)
τ (a)3np
∞ k=1 R (a) k
∞
dm(z) |z − a|3np
2−3npk m Rk (a)
k=1
τ (a)2 . p
Therefore Fa,n,p ∈ Ap (w) with Fa,n,p Ap (w) τ (a)2 , which gives (i). The use of Lemma 2.2 (with β = 1) gives the upper estimate of (ii), Ka 2A2 (w) w(a) τ (a)−2 . On the other hand, the functions Fa,1,2 obtained from the previous lemma satisfy (by (i)) that Fa,1,2 ∈ A2 (w) with Fa,1,2 2A2 (w) τ (a)2 , and by (11) this gives Fa,1,2 (a)2 w(a)−1 w(a)τ (a)2 −1 Fa,1,2 2 2
A (w)
.
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Since Ka A2 (w) = La , where La is the point evaluation functional at the point a, this proves the lower estimate of (ii). 2 Proposition 2. Let w ∈ W, 0 < p < ∞ and n ∈ N such that n max {1/p, p}. If ρ0 is the number given in Lemma 3.1 and {zk } ⊂ D is the sequence from Lemma A, the function
F (z) =
ak
zk : |zk |ρ0
Fzk ,n,p (z) τ (zk )2/p
belongs to Ap (w) for every sequence {ak } ∈ p . Moreover, F Ap (w)
1/p |ak |
p
.
k
Proof. In what follows, we shall write
F (z) =
ak
zk : |zk |ρ0
Fzk ,n,p (z) Fzk ,n,p (z) = ak . τ (zk )2/p τ (zk )2/p k
If 0 < p 1, then bearing in mind Corollary 1, we have that p
F Ap (w) =
Fzk ,n,p (z) p a w(z) dm(z) k τ (z )2/p k
D
k
|ak |p p Fzk ,n,p Ap (w) τ (zk )2 k C |ak |p .
k
If p > 1, an application of Hölder’s inequality yields |ak |p p(n−p+1) p/n p−1 2 F (z)p Fz ,n,p (z) n F τ (z ) (z) . k zk ,n,p τ (zk )2p k k
(13)
k
Now, we claim that k
p/n τ (z)2 τ (zk )2 Fzk ,n,p (z) . w(z)1/n
(14)
In order to prove (14), note first that using the estimate (11), Lemma 2.1 and (iv) of Lemma A, we deduce that {zk ∈D(z,δ0 τ (z))}
p/n τ (zk )2 Fzk ,n,p (z)
1 w(z)1/n
{zk ∈D(z,δ0 τ (z))}
τ (zk )2
τ (z)2 . w(z)1/n
(15)
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On the other hand, an application of (12) gives {zk ∈D(z,δ / 0 τ (z))}
p/n τ (z)3p τ (zk )2 Fzk ,n,p (z) w(z)1/n =
{zk ∈D(z,δ / 0 τ (z))}
∞ τ (z)3p w(z)1/n
τ (zk )2 |z − zk |3p
τ (zk )2 , |z − zk |3p
j =0 zk ∈Rj (z)
where
Rj (z) = ζ ∈ D: 2j δ0 τ (z) < |ζ − z| 2j +1 δ0 τ (z) ,
j = 0, 1, 2 . . . .
Now observe that, using (8), it is easy to see that, for j = 0, 1, 2, . . . , D zk , δ0 τ (zk ) ⊂ D z, 5δ0 2j τ (z)
if zk ∈ D z, 2j +1 δ0 τ (z) .
This fact together with the finite multiplicity of the covering (see Lemma A) gives
τ (zk )2 m D z, 5δ0 2j τ (z) 22j τ (z)2 .
zk ∈Rj (z)
Therefore {zk ∈D(z,δ / 0 τ (z))}
∞ p/n τ (z)3p τ (zk )2 Fzk ,n,p (z) w(z)1/n
j =0 zk ∈Rj (z)
∞ 1 2−3pj w(z)1/n j =0
τ (zk )2 |z − zk |3p
τ (zk )2
zk ∈Rj (z)
∞ τ (z)2 (2−3p)j 2 w(z)1/n j =0
τ (z)2 , w(z)1/n
which together with (15), proves (14). Now, joining (13) and (14), we obtain |ak |p p(n−p+1) n−p+1 p Fz ,n,p (z) n τ (z)2p−2 w(z) n dm(z). F Ap (w) k 2p τ (zk ) k
D
So, it is enough to show that p(n−p+1) n−p+1 Fz ,n,p (z) n τ (z)2p−2 w(z) n dm(z) τ (zk )2p k D
(16)
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to obtain the desired estimate p
F Ap (w)
|ak |p .
k
It follows from (11) that
F z
k ,n,p
p(n−p+1) n−p+1 (z) n τ (z)2p−2 w(z) n dm(z)
|z−zk |<τ (zk )
τ (z)2p−2 dm(z) τ (zk )2p .
(17)
|z−zk |<τ (zk )
On the other hand, using (8), it follows that τ (z) C2j τ (zk )
if |z − zk | < 2j τ (zk ).
Thus, since n p, bearing in mind (12), we deduce that
F z
k ,n,p
p(n−p+1) n−p+1 (z) n τ (z)2p−2 w(z) n dm(z)
|z−zk |τ (zk )
τ (zk )3p(n−p+1) |z−zk |τ (zk )
τ (zk )
3p(n−p+1)
∞
∞
j =0
2j τ (zk )|z−zk |<2j +1 τ (zk )
2−3jp(n−p+1)
j =0
τ (zk )2p
τ (z)2p−2 dm(z) |z − zk |3p(n−p+1) τ (z)2p−2 dm(z) |z − zk |3p(n−p+1)
τ (z)2p−2 dm(z)
2j τ (zk )|z−zk |<2j +1 τ (zk ) ∞
2−jp(3n−3p+1) τ (zk )2p ,
j =0
which together with (17) gives (16). This finishes the proof.
2
4. Proof of Theorem 1 4.1. Proof of (I): boundedness Suppose first that Id : Ap (w) → Lq (μ) is bounded. It is enough to deal with the case |a| ρ0 , where ρ0 ∈ (0, 1) is the number given in Lemma 3.1. For a ∈ D with |a| ρ0 , consider the function Fa,n0 ,p obtained in Lemma 3.1, where n0 is the smallest natural number such that n0 p 1. p By Corollary 1, we have Fa,n0 ,p Ap (w) τ (a)2 . Then, using the estimate (11) of Lemma 3.1,
J. Pau, J.Á. Peláez / Journal of Functional Analysis 259 (2010) 2727–2756
w(z)−q/p dμ(z)
D(δτ (a))
Fa,n
0 ,p
2741
q (z) dμ(z)
D(δτ (a))
Fa,n
0 ,p
q (z) dμ(z)
D q
q
Id Ap (w)→Lq (μ) Fa,n0 ,p Ap (w) 2q
q
Id Ap (w)→Lq (μ) τ (a) p , q
proving that Kμ,w CId Ap (w)→Lq (μ) . Conversely, suppose that (1) holds. This implication was proved by Oleinik in [21], but we give a proof here for the sake of completeness. Using Lemma A, and Lemmas 2.2 and 2.1, it follows that
f (z)q dμ(z)
f (z)q dμ(z)
j D(δτ (z )) j
D
j D(δτ (z )) j
j
q p q f (ζ )p w(ζ ) dm(ζ ) w(z)− p dμ(z)
1 τ (z)2
D(δτ (z))
q p p f (ζ ) w(ζ ) dm(ζ )
D(3δτ (zj ))
Kμ,w
j
− pq
w(z)
2q
D(δτ (zj ))
τ (z) p
q p p f (ζ ) w(ζ ) dm(ζ ) .
dμ(z)
(18)
D(3δτ (zj ))
Now, using Minkowski inequality and the finite multiplicity N of the covering {D(3δτ (zj ))}, we have
f (z)q dμ(z) Kμ,w
D
q/p f (ζ )p w(ζ ) dm(ζ )
j D(3δτ (z )) j q
Kμ,w N q/p f Ap (w) , q
proving that Id : Ap (w) → Lq (μ) is continuous with Id Ap (w)→Lq (μ) Kμ,w . 4.2. Proof of (I): compactness Suppose that (2) holds. Fixed δ ∈ (0, mτ ), consider the covering {D(δτ (zj ))} given by Lemma A, and let {fn } be a bounded sequence in Ap (w). By Lemma 2.2, {fn } is uniformly
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bounded on compact sets, and by Montel’s theorem {fn } is a normal family. Then we may extract a subsequence {fnk } converging uniformly on compact sets of D to some function f . Using Fatou’s lemma, it is easy to see that f must be in Ap (w). Given ε > 0, fix 0 < r0 < 1 with 1 sup 2q/p τ (a) |a|>r0
w(z)−q/p dμ(z) < ε.
(19)
D(δτ (a))
Observe that there is r0 < 1 with r0 r0 such that if a point zk of the sequence {zj } belongs to {|z| r0 }, then D(δτ (zk )) ⊂ {|z| r0 }. So, take nk big enough such that sup|z|r |fnk (z) − 0 f (z)| < ε. Then, setting gnk = fnk − f , and arguing as in (18), it follows that
q
gn (z)q dμ(z) + k
gnk Lq (μ)
gn (z)q dμ(z) k
|zj |>r0 D(δτ (z )) j
|z|r0
q sup gnk (z) μ(D) + |z|r0
gn (z)q dμ(z) k
|zj |>r0 D(δτ (z )) j q
Cε + Cgnk Ap (w) sup
|zj |>r0
1 τ (zj )2q/p
w(z)−q/p dμ(z)
D(δτ (zj ))
< Cε. In the last inequality we use (19) and the fact that {fnk − f } is also a bounded sequence in Ap (w). This proves that Id : Ap (w) → Lq (μ) is compact. Conversely, suppose that Id : Ap (w) → Lq (μ) is compact. Take the smallest natural number n0 such that n0 p 1 and let fa,n0 ,p (z) =
Fa,n0 ,p (z) , τ (a)2/p
ρ0 |a| < 1,
where ρ0 ∈ (0, 1) and Fa,n0 ,p are obtained from Lemma 3.1. By Corollary 1, sup fa,n0 ,p Ap (w) C,
|a|ρ0
and the compactness of the identity operator implies that {fa,n0 ,p : ρ0 |a| < 1} is a compact set in Lq (μ). Thus lim
r→1 r<|z|<1
fa,n ,p (z)q dμ(z) = 0 uniformly in a. 0
On the other hand, the estimate (12) gives fa,n
0 ,p
p τ (a)3n0 p−2 (z) w(z) , (1 − r)3n0 p
|z| r, |a|
1+r . 2
(20)
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2743
Thus fa,n0 ,p → 0 as |a| → 1 uniformly on compact subsets of D, which together with (20) implies that lim|a|→1− fa,n0 ,p Lq (μ) = 0. Therefore, using the estimate (11) of Lemma 3.1, 1 sup 2q/p τ (a) |a|>r
−q/p
w(z)
dμ(z) sup
|a|>r D(δτ (a))
D(δτ (a))
fa,n ,p (z)q dμ(z) 0 q
sup fa,n0 ,p Lq (μ) , |a|>r
and this tends to zero as r → 1− , completing the proof. 4.3. Proof of (II) The implication (a) ⇒ (b) is obvious. To prove that (b) ⇒ (c), we use an argument of Luecking (see [20]). Let {zk } be the sequence of points in D from Lemma A. Let n be a positive integer such that n max(1/p, p), and for an arbitrary sequence {ak } ∈ p , consider the function Gt (z) =
ak rk (t)
zk : |zk |ρ0
Fzk ,n,p (z) , τ (zk )2/p
0 < t < 1,
where rk (t) is a sequence of Rademacher functions (see p. 336 of [20], or Appendix A of [12]). By Proposition 2, the function Gt belongs to Ap (w) with Gt Ap (w) C
1/p |ak |
p
.
k
Thus, condition (b) gives
q/p Gt (z)q dμ(z) C |ak |p ,
0 < t < 1.
k
D
Integrating with respect to t from 0 to 1, applying Fubini’s theorem, and invoking Khinchine’s inequality (see [20]), we obtain D
zk : |zk |ρ0
|ak |2
|Fzk ,n,p (z)|2 τ (zk )4/p
q/2 dμ(z) C
q/p |ak |p
.
(21)
k
Let δ ∈ (0, mτ ). If χE (z) denotes the characteristic function of a set E, bearing in mind the estimate (11), and the finite multiplicity N of the covering {D(3δτ (zk ))} (see (iv) of Lemma A), we have |ak |q w(z)−q/p dμ(z) 2q zk : |zk |ρ0 τ (zk ) p D(3δτ (z )) k |ak |q Fz ,n,p (z)q dμ(z) k 2q zk : |zk |ρ0 τ (zk ) p D(3δτ (z )) k
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=
D zk : |zk |ρ0
q |ak |q Fzk ,n,p (z) χD(3δτ (zk )) (z) dμ(z) 2q τ (zk ) p
max 1, N 1−q/2
|ak |2
zk : |zk |ρ0
D
|Fzk ,n,p (z)|2 τ (zk )4/p
q/2 dμ(z).
This, together with (21) yields
|ak |q
|zk |ρ0
τ (zk ) p
−q/p
w(z)
2q
between and
p p−q
q/p |ak |
p
.
k
D(3δτ (zk ))
Since the sequence {ak } ∈ p is arbitrary and p q
dμ(z) C
p q
> 1, if we put bk = |ak |q , then using the duality
we conclude that the sequence
w(z)−q/p dμ(z)
1 2q
τ (zk ) p
|zk |ρ0
D(3δτ (zk ))
p
belongs to p−q , that is |zk |ρ0
p p−q w(z)−q/p dμ(z) τ (zk )2 < ∞.
1 τ (zk )2
(22)
D(3ατ (zk ))
Note that there is ρ1 < 1, with ρ0 ρ1 such that if a point zk of the sequence {zj } belongs to {|z| < ρ0 }, then D(δτ (zk )) ⊂ {|z| < ρ1 }. Therefore, using Lemma 2.1, (ii) and (iii) of Lemma A, and (22) we deduce that |z|ρ1
1 τ (z)2
−q/p
w(ζ ) D(δτ (z))
|zk |ρ0 D(δτ (z )) k
|zk |ρ0
p p−q dμ(ζ ) dm(z)
|zk |ρ0
1 τ (zk )2
1 τ (z)2
−q/p
w(ζ ) D(δτ (z))
p p−q
−q/p
w(ζ ) D(δτ (zk ))
1 τ (zk )2
p p−q dμ(ζ ) dm(z)
D(δτ (z)) −q/p
w(ζ ) D(3δτ (zk ))
p p−q dμ(ζ ) dm(z)
p p−q dμ(ζ ) τ (zk )2 < ∞.
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2745
This, together with the fact that the integral
|z|<ρ1
p p−q w(ζ )−q/p dμ(ζ ) dm(z)
1 τ (z)2
D(δτ (z))
is clearly finite, proves that (c) holds. Finally, we are going to prove that (c) implies (a). The argument is standard, but it is included for the sake of completeness. It is enough to prove that if {fn } is a bounded sequence in Ap (w) that converges to 0 uniformly on compact subsets of D then limn→∞ fn Lq (dμ) = 0. Let δ ∈ (0, mτ ). Bearing in mind (7), we deduce that for any r > 13 D
δ r , τ (z) ⊂ ζ ∈ D: |ζ | > 2 2
if |z| > r.
(23)
On the other hand, it follows from Lemma 2.2 that
−q/p fn (z)q C w(z) τ (z)2
fn (ζ )q w(ζ )q/p dm(ζ ).
D( 2δ τ (z))
Integrate respect to dμ, apply Fubini’s theorem, use (23) and Lemma 2.1 to obtain
fn (z)q dμ(z)
{z∈D: |z|>r}
C {z∈D: |z|>r}
C
1 τ (z)2
fn (ζ )q w(ζ )q/p dm(ζ ) w(z)−q/p dμ(z)
D( 2δ τ (z))
fn (ζ )q w(ζ )q/p
{ζ ∈D: |ζ |> 2r }
C {ζ ∈D:
w(z)−q/p dμ(z) dm(ζ ) τ (z)2
D(δτ (ζ ))
fn (ζ )q w(ζ )q/p
|ζ |> 2r }
1 τ (ζ )2
−q/p
w(z)
dμ(z) dm(ζ ).
D(δτ (ζ ))
If condition (c) holds, then for any fixed ε > 0, there is r0 ∈ (0, 1), such that {ζ ∈D:
r |ζ |> 20 }
1 τ (ζ )2
−q/p
w(z)
p p−q p dμ(z) dm(ζ ) < ε p−q .
D(δτ (ζ ))
Then (24) and an application of Hölder’s inequality yields
(24)
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fn (z)q dμ(z)
{ζ ∈D: |z|>r0 }
q Cfn Ap (w) {ζ ∈D:
r |ζ |> 20 }
1 τ (ζ )2
−q/p
w(z)
p p−q p−q p dμ(z) dm(ζ )
D(δτ (ζ ))
q Cε supfn Ap (w)
Cε.
(25)
Moreover, since {|z| r0 } is a compact subset of D, we have lim
n→∞ |z|r0
fn (z)q dμ(z) = 0,
which together with (25), gives lim fn Lq (dμ) = 0.
n→∞
This completes the proof of Theorem 1. 5. Proof of Theorem 2 5.1. Proof of (I) and (II) Let 0 < p q < ∞, and let δ ∈ (0, mτ ). Since Tg f (0) = 0 and (Tg f ) (z) = f (z)g (z), Theorem B gives q Tg f Aq (w)
f (z)q g (z)q ψw (z)q w(z) dm(z).
D
Therefore, the boundedness of the integration operator Tg : Ap (w) → Aq (w) is equivalent to the continuity of the embedding Id : Ap (w) → Lq (μg,w ) with q dμg,w (z) = g (z) ψw (z)q w(z) dm(z).
(26)
By Theorem 1, this holds if and only if sup I (q, p, a) < ∞, a∈D
where I (q, p, a) =
1 τ (a)2q/p
D(δτ (a))
q g (z) ψw (z)q w(z)1−q/p dm(z).
(27)
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2747
Suppose that Tg : Ap (w) → Aq (w) is bounded. Note that, bearing in mind that τ (z)2 (ϕ(z))−1 and using Lemma 2.1, condition (6) gives ψw (z) Write s =
1 p
if z ∈ D δτ (a) .
τ (z)2 τ (a)2 ψw (a) (1 − |z|)t (1 − |a|)t
(28)
− q1 . Then, using Lemma 2.2 (with β = 1 − pq ) and Corollary 1 we obtain
2qs
q ψw (a)q Ka A2 (w) g (a) ψw (a)Ka 2s A2 (w) 1− q τ (a)2 w(a) p
D(δτ (a))
2qs
Ka A2 (w)
− pq
w(z)
1− pq
τ (a)2 w(a)
1 τ (a)2q/p
q q g (z) w(z)1− p dm(z)
dμg,w (z)
D(δτ (a)) − pq
w(z)
dμg,w (z) = I (q, p, a).
D(δτ (a))
Thus, if Tg : Ap (w) → Aq (w) is bounded, it follows from (27) that g (a) < ∞. sup ψw (a)Ka 2s A2 (w)
(29)
a∈D
If q = p then s = 0, and (29) proves that if Tg is bounded on Ap (w), then sup ψw (a)g (a) < ∞.
(30)
a∈D
Conversely, if (30) holds, then it follows directly that supa∈D I (p, p, a) < ∞. Thus Tg is bounded on Ap (w), and the proof of part (a) of (I) is complete. If 0 < p < q < ∞, we are going to show that condition (29) implies g ≡ 0. To prove this, it goes to infinity as |a| → 1− . By Corollary 1 and condiis enough to see that ψw (a)Ka 2s A2 (w) tion (6) ψw (a)Ka 2s A2 (w)
τ (a)2−2s , (1 − |a|)t w(a)s
and this tends to infinity as |a| → 1− by Lemma 2.3. This finishes the proof of (II), since the other implication is trivial. Concerning the compactness part (b) of (I), note that using Theorem B, the compactness of the operator Tg on Ap (w) is equivalent to the compactness of the embedding Id : Ap (w) → Lp (μg,w ), where μg,w is the measure defined by (26) with q = p. By part (I) of Theorem 1, this holds if and only if lim sup
r→1−
|a|>r
1 τ (a)2
D(δτ (a))
p g (z) ψw (z)p dm(z) = 0,
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and proceeding as in the boundedness part, we see that this is equivalent to lim sup ψw (a)g (a) = 0.
r→1− |a|>r
5.2. Proof of (III) By Theorem B one has q Tg f Aq (w)
f (z)q g (z)q ψw (z)q w(z) dm(z).
(31)
D
Using (31), we have that Tg : Ap (w) → Aq (w) is bounded if and only the embedding Id : Ap (w) → Lq (μg,w ) is bounded, where μg,w is the measure defined by (26). Thus the equivalence (a) ⇔ (b) follows from part (II) of Theorem 1. (b) ⇒ (c). We also use part (II) of Theorem 1 to assert that (b) is equivalent to Cg,w := D
1 τ (z)2
p p−q p−q q g (ζ ) ψw (ζ )q w(ζ ) p dm(ζ ) dm(z) < ∞.
D(τ (z))
Now, using Theorem B, Lemma 2.2 and (28), we obtain grAr (w)
r g(0) +
q g (z) ψw (z)q w(z)q/r r/q dm(z)
D
r g(0) + Cg,w . (c) ⇒ (b). If g ∈ Ar (w), then (31), Hölder’s inequality and Theorem B gives q
Tg f Aq (w)
q/p q/r r f (z)p w(z) dm(z) g (z) ψw (z)r w(z) dm(z)
D
D
q q gAr (w) f Ap (w) .
Thus Tg : Ap (w) → Aq (w) is bounded with Tg gAr (w) . This finishes the proof. 6. Schatten classes on A2 (w) If {en } is an orthonormal basis of a Hilbert space H of analytic functions in D with reproducing kernel Kz , then Kz (ζ ) =
n
en (ζ )en (z)
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2749
for all z and ζ in D (see e.g. [28, Theorem 4.19]). It follows that en (z)2 Kz 2
(32)
H
n
for any orthonormal set {en } of H . Also, by (6) we have ∂ Kz (ζ ) = en (z)en (ζ ), ∂ z¯ n
z, ζ ∈ D.
Thus Parseval’s identity gives 2 ∂ Kz = e (z)2 . n ∂ z¯ H
(33)
n
Now, we are going to give the proof of Theorem 3 on the description of the Schatten classes Sp := Sp (A2 (w)). First we consider the sufficiency part of the case 1 < p < ∞. We need the following lemma. Lemma 6.1. Let w ∈ W satisfying (6). Then ∂ Kz ∂ z¯
A2 (w)
=O
Kz A2 (w) ψw (r)
,
|z| = r.
Proof. Let {en } be the orthonormal basis of A2 (w) given by en (z) = zn δn−1 , where δn2 = 2
1 0
n ∈ N,
r 2n+1 w(r) dr. Using Corollary 1 and (6), we have that
∞
∞ en (z)2 = Kz 2 2 r 2n δ −2 = n
n=0
A (w)
(1 − r)−t
−1
1
n=0
w(s) ds r
for some t 1. So, if we consider the analytic function in D defined by f (z) =
∞
zn δn−1 ,
n=0 1 then M2 (r, f ) = ( 2π
π
−π
1
|f (reiθ )|2 dθ ) 2 Φ(r), as r → 1− , where −t/2
−1/2
1
Φ(r) = (1 − r)
w(s) ds r
.
,
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Now, bearing in mind that ψw (r) (1 − r) for a decreasing weight w, a calculation shows that Φ (r)
Φ(r) , ψw (r)
r → 1− .
Moreover, it is easy to see that for w ∈ W lim sup r→1−
Φ (r)Φ(r) < ∞. (Φ (r))2
(34)
Thus we can apply Theorem 2.1 of [23], which says that M2 r, f = O Φ (r)
if M2 (r, f ) = O Φ(r) ,
when (34) holds (see condition (3.3) of [23]). Finally, since for r = |z|, 2 ∂ Kz ∂ z¯ 2
A (w)
=
∞ ∞ 2 e (z) = n2 r 2n−2 δ −2 = M 2 r, f n
n
n=0
2
n=1
we obtain ∂ Kz ∂ z¯
Kz Φ(r) , = M2 r, f = O Φ (r) ψ (r) ψ w w (z) A2 (w)
r → 1− .
2
Proposition 3. Let g ∈ H (D), 1 < p < ∞ and w ∈ W satisfying (6). If ψw |g | ∈ Lp (D, ϕ dm) then Tg ∈ Sp (A2 (w)). Proof. By Theorem B, the inner product
f, g ∗ = f (0)g(0) +
f (z)g (z)ψw (z)2 w(z) dm(z)
D
gives a norm on A2 (w) equivalent to the usual one. If 1 < p < ∞, the operator Tg belongs to the Schatten p-class Sp if and only if Tg en , en ∗ p < ∞ n
for any orthonormal set {en } (see [28, Theorem 1.27]). Let {en } be an orthonormal set of (A2 (w), , ∗ ). Note that Theorem B gives
en (z)e (z)ψw (z)w(z) dm(z) e2 n
D
n A1 (w)
en 2A2 (w) = 1.
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2751
This together with Hölder’s inequality yields p Tg en , en ∗ p g (z)en (z)e (z)ψw (z)2 w(z) dm(z) n n
n
D
g (z)p en (z)e (z)ψw (z)p+1 w(z) dm(z) n
n
=
D
p g (z) en (z)e (z) ψw (z)p+1 w(z) dm(z), n n
D
and since Kz 2A2 (w) w(z) ϕ(z) (see Corollary 1), the result will be proved if we are able to show that 2 Kz A2 (w) en (z)e (z) . n ψw (z) n
(35)
To prove (35), we use Cauchy–Schwarz inequality together with (32) and (33) to obtain 1/2 2 1/2 en (z)e (z) en (z)2 e (z) n
n
n
n
n
∂ K Kz A2 (w) ∂ z¯ z 2 . A (w)
Now, the inequality (35) follows from Lemma 6.1. This completes the proof of the proposition. 2 Now we turn to show the necessity for the case 0 < p < ∞. Proposition 4. Let g ∈ H (D), 0 < p < ∞ and w ∈ W satisfying (6). If Tg ∈ Sp (A2 (w)), then ψw |g | ∈ Lp (D, ϕ dm). Proof. We split the proof in two cases. Case 2 p < ∞. Suppose that Tg is in Sp , and let {ek } be an orthonormal set in A2 (w) and n max {1/p, p}. Let {zk } be the sequence from Lemma 3.1, and consider the operator A taking ek (z) to fzk (z) = Fzk ,n,p (z)/τ (zk ). It follows from Proposition 2 that the operator A is bounded on A2 (w). Since Sp is a two-sided ideal in the space of bounded linear operators on A2 (w), then Tg A belongs to Sp (see [28, p. 27]). Thus, by [28, Theorem 1.33] p Tg (fz )p 2 = Tg Aek A2 (w) < ∞. k A (w) k
k
This together with Lemma 3.1 and Theorem B gives
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k
1 τ (zk )p
p/2 2 g (z) ψw (z)2 dm(z)
D(τ (zk ))
k
p/2 fz (z)2 g (z)2 ψw (z)2 w(z) dm(z) k
D(τ (zk ))
k
p/2 fz (z)2 g (z)2 ψw (z)2 w(z) dm(z) k
D
Tg (fz )p 2 < ∞. k A (w)
(36)
k
On the other hand, if δ is sufficiently small, applying Lemmas 2.2, 2.1 and Lemma A and arguing as in (28), it follows that
p g (z) ψw (z)p ϕ(z) dm(z)
D
k D(δτ (z )) k
k
k
D(δτ (z))
1 τ (zk )p
D(δτ (zk ))
1 τ (zk )p
p/2 2 dm(z) g (ζ ) dm(ζ ) ψw (z)p τ (z)2
1 τ (z)2
˜ D(δτ (zk ))
p/2 2 dm(z) g (ζ ) ψw (ζ )2 dm(ζ ) τ (z)2
p/2 2 g (ζ ) ψw (ζ )2 dm(ζ ) .
D(3δτ (zk ))
This together with (36) concludes the proof. Case 0 < p < 2. If Tg ∈ Sp then the positive operator Tg∗ Tg belongs to Sp/2 . Without loss of generality we may assume that g = 0. Suppose Tg∗ Tg f =
λn f, en en
n
is the canonical decomposition of Tg∗ Tg . Then not only is {en } an orthonormal set, it is also an orthonormal basis. Indeed, if there is a unit vector e ∈ A2 (w) such that e ⊥ en for all n 1, then
2 g (z) e(z)2 ψw (z)2 w(z) dm(z) Tg e2 2
A (w)
= Tg∗ Tg e, e = 0
D
because Tg∗ Tg is a linear combination of the vectors en . This would give g ≡ 0.
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2753
Since {en } is an orthonormal basis of A2 (w), then Kz 2 =
en (z)2 . n
This identity together with Corollary 1 and Hölder’s inequality yields
p g (z) ψw (z)p ϕ(z) dm(z)
D
p g (z) ψw (z)p Kz 2 w(z) dm(z)
D
=
g (z)p ψw (z)p en (z)2 w(z) dm(z) n
D
p/2 g (z)2 ψw (z)2 en (z)2 w(z) dm(z) n
D
Tg∗ Tg en , en
n
This completes the proof.
p/2
=
p/2
λn
n
p/2 = Tg∗ Tg S . p/2
2
Finally, we shall prove the main result in this section. Proof of Theorem 3. Part (a) follows directly from Propositions 3 and 4. Moreover, if 0 < p 1 and Tg ∈ Sp (A2 (w)), Proposition 4, (7) and (6) imply that D
|g (z)|p dm(z) (1 − |z|)tp (1 − |z|)2(1−p)
D
D
|g (z)|p dm(z) (1 − |z|)tp τ (z)2(1−p) p ϕ(z)1−p g (z) dm(z) (1 − |z|)tp p g (z) ψω (z)p ϕ(z) dm(z) < ∞.
D
Therefore, it follows that (t − 2)p + 2 1, and consequently g ≡ 0, which gives (b). The proof is complete. 2 7. Some examples of weights in the class W In this section, several examples of weights in the class W are given. We check that they satisfy the condition (6) in Theorem 2, and by computing the distorsion functions, we also offer the corresponding description for the boundedness and compactness of the integration operator Tg in each case.
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Example 1. The weights wγ ,α (r) = (1 − r) exp γ
−c , (1 − r)α
γ 0, α > 0, c > 0,
are in the class W with associated subharmonic function −α ϕγ ,α (z) = −γ log 1 − |z| + c 1 − |z| . We have that −1 2+α ϕγ ,α (z) τ (z)2 = 1 − |z| , and it is easy to see that τ (z) satisfies the conditions in the definition of the class W. Also, since ψwγ ,α (r) (1 − r)1+α (see e.g. [26, Example 3.2]), (6) is satisfied with t = 1. In particular, the case q = p of Theorem 2 says that Tg is bounded on Ap (wγ ,α ) if and only if 1+α g (z) < ∞, sup 1 − |z| z∈D
and Tg is compact on Ap (wγ ,α ) whenever 1+α g (z) = 0. lim 1 − |z|
|z|→1−
As mentioned above, this answers a question raised in [4, p. 353]. We also note that, by Theorem 3, the operator Tg belongs to the Schatten p-class of A2 (wγ ,α ) if and only if g is in the p Dirichlet-type space Dβ with β = p − 2 + α(p − 1), that is, the space of those g ∈ H (D) with
p g (z) 1 − |z|2 β dm(z) < ∞.
D
Example 2. For α > 1 and A > 0 the weights w(r) = exp −A log
e 1−r
α ,
e with associated subharmonic function ϕ(z) = A(log 1−|z| )α , belong to the class W. Indeed, it is easy to see that
−2 log ϕ(z) 1 − |z| e so τ (z) = (1 − |z|)(log 1−|z| )
1−α 2
α−1
, and since α > 1
τ (r) log
e 1 − |z|
e 1−r
−α+1 2
,
r → 1− ,
,
(37)
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which implies that limr→1− τ (r) = limr→1− τ (r) = 0. Moreover, the function τ (r)(1 − r)−2 increases for r close to 1. This proves that w ∈ W. On the other hand, since w has distorsion function (see [26, Example 3.4]) ψw (r)
1−r e α−1 , (log 1−r )
then (37) gives (ϕ(z))−1 (1 − |z|)ψw (z). Therefore, (6) is satisfied with t = 1. For this weight, Theorem 2 says that Tg is bounded on Ap (w) if and only if sup 1 − |z| log z∈D
e 1 − |z|
1−α
g (z) < ∞,
and Tg is compact on Ap (w) whenever the corresponding “little oh” condition holds. Example 3. For α, β, γ > 0, the double exponential weight w(r) = exp −γ exp
β (1 − r)α
β belongs to W. Indeed, the associated subharmonic function is ϕ(z) = γ exp( (1−|z|) α ), and a straightforward computation gives
−2α−2 ϕ(z) 1 − |z| exp
β . (1 − |z|)α
(38)
−β/2 Then we can take τ (z) = (1 − |z|)α+1 exp( (1−|z|) α ). Since
−β/2 , τ (r) exp (1 − r)α
r → 1− ,
1 = we obtain limr→1− τ (r) = limr→1− τ (r) = 0. Also, it is easy to see that limr→1− τ (r) log τ (r) 0. This proves that w ∈ W. Moreover, since w has distorsion function (see [26, Example 3.3])
ψw (r) (1 − r)
1+α
exp −
β , (1 − r)α
then (38) gives (ϕ(z))−1 (1 − |z|)1+α ψw (z). So (6) is satisfied with t = 1 + α. In this example, the case q = p of Theorem 2 says that Tg is bounded on Ap (w) if and only if 1+α sup 1 − |z| exp − z∈D
β g (z) < ∞, (1 − |z|)α
and Tg is compact on Ap (w) if the “little oh” condition holds.
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References [1] A. Aleman, A class of integral operators on spaces of analytic functions, in: Topics in Complex Analysis and Operator Theory, Univ. Málaga, Málaga, 2007, pp. 3–30. [2] A. Aleman, J.A. Cima, An integral operator on H p and Hardy’s inequality, J. Anal. Math. 85 (2001) 157–176. [3] A. Aleman, A. Siskakis, An integral operator on H p , Complex Var. 28 (1995) 149–158. [4] A. Aleman, A. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997) 337–356. [5] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002) 443–510. [6] A. Borichev, R. Dhuez, K. Kellay, Sampling and interpolation in large Bergman and Fock spaces, J. Funct. Anal. 242 (2007) 563–606. [7] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958) 921–930. [8] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962) 547–559. [9] M. Dostani´c, Unboundedness of the Bergman projections on Lp spaces with exponential weights, Proc. Edinb. Math. Soc. 47 (2004) 111–117. [10] M. Dostani´c, Integration operators on Bergman spaces with exponential weights, Rev. Mat. Iberoamericana 23 (2007) 421–436. [11] P.L. Duren, Extension of a theorem of Carleson, Bull. Amer. Math. Soc. 75 (1969) 143–146. [12] P.L. Duren, Theory of H p Spaces, Academic Press, New York, London, 1970, reprint, Dover, Mineola, New York, 2000. [13] P. Galanopoulos, D. Girela, J.A. Peláez, Multipliers and integration operators on Dirichlet spaces, Trans. Amer. Math. Soc., in press, available at http://webpersonal.uma.es/~JAPELAEZ/preprints.html. [14] D. Girela, J.A. Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006) 334–358. [15] T.L. Kriete, B.M. MacCluer, Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J. 41 (1992) 755–788. [16] P. Lin, R. Rochberg, Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergman spaces with exponential type weights, Pacific J. Math. 173 (1996) 127–146. [17] D.H. Luecking, Forward and reverse inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985) 85–111. [18] D.H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987) 345–368. [19] D.H. Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. Lond. Math. Soc. 63 (1991) 595–619. [20] D.H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Michigan Math. J. 40 (1993) 333–358. [21] V.L. Oleinik, Embedding theorems for weighted classes of harmonic and analytic functions, J. Soviet. Math. 9 (1978) 228–243. [22] V.L. Oleinik, G.S. Perelman, Carleson’s imbedding theorem for weighted Bergman space, Mat. Zametki 47 (1990) 74–79 (in Russian); translation in Math. Notes 47 (1990) 577–581. [23] M. Pavlovi´c, On harmonic conjugates with exponential mean growth, Czechoslovak Math. J. 49 (124) (1999) 733– 742. [24] M. Pavlovi´c, J.A. Peláez, An equivalence for weighted integrals of an analytic function and its derivative, Math. Nachr. 281 (2008) 1612–1623. [25] J. Rättyä, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Austral. Math. Soc. 75 (2007) 431–446. [26] A. Siskakis, Weighted integrals and conjugate functions in the unit disk, Acta Sci. Math. (Szeged) 66 (2000) 651– 664. [27] A. Siskakis, Volterra operators on spaces of analytic functions – a survey, in: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 51–68. [28] K. Zhu, Operator Theory in Function Spaces, second ed., Math. Surveys Monogr., vol. 138, American Mathematical Society, Providence, RI, 2007.
Journal of Functional Analysis 259 (2010) 2757–2758 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “The role of BMOA in the boundedness of weighted composition operators” [ J. Funct. Anal. 258 (11) (2010) 3593–3603] Eva A. Gallardo-Gutiérrez a,∗ , Jonathan R. Partington b a Departamento de Matemáticas, Universidad de Zaragoza e IUMA, Plaza San Francisco s/n, 50009 Zaragoza, Spain b School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
We regret that it is necessary to make a correction to the statement of Theorem 2.1 of [1] since condition (c) can be empty whenever ϕ∞ < 1. In that case, it does not hold that boundedness of Wh,ϕ is equivalent to the fact that h ∈ BMOA (see [2, Theorem 2.2] for the characterization of the weights h which induce bounded Wh,ϕ in H2 when ϕ∞ < 1). As can be seen from the proof, in Theorem 2.1 condition (a) should be replaced by the condition that expression (4) remains uniformly bounded in w (which, in particular, is implied by the BMOA condition). Analogous modifications are needed in the statement of the other theorems of the paper, where in each case (a) must be replaced by an apparently weaker condition. For the sake of completeness, and as it was stated in the abstract, we point out that the following characterization of bounded (resp. compact) weighted composition operators Wh,ϕ acting on the classical Hardy space H2 in terms of a Nevanlinna counting function associated to the symbols h and ϕ whenever h ∈ BMOA (resp. VMOA) is proved in the paper: Theorem 0.1. Let h ∈ H2 and ϕ ∈ H∞ such that ϕ(D) ⊆ D. Suppose that h ∈ BMOA (resp. VMOA). Then Wh,ϕ is a bounded (resp. compact) operator on H2 if and only if h(z)2 log 1 dA(u) = O(r) resp. o(r) sup |z| ζ ∈T D(ζ,r)
ϕ(z)=u
where dA denotes normalized Lebesgue measure on D. DOI of original article: 10.1016/j.jfa.2010.02.019. * Corresponding author.
E-mail addresses:
[email protected] (E.A. Gallardo-Gutiérrez),
[email protected] (J.R. Partington). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.021
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References [1] E.A. Gallardo-Gutiérrez, J.R. Partington, The role of BMOA in the boundedness of weighted composition operators, J. Funct. Anal. 258 (11) (2010) 3593–3603. [2] E.A. Gallardo-Gutiérrez, Romesh Kumar, J.R. Partington, Boundedness, compactness and Schatten-class membership of weighted composition operators, Integral Equations Operator Theory (2010), in press.
Journal of Functional Analysis 259 (2010) 2759 www.elsevier.com/locate/jfa
Message from the Editors
The end of an era Paul George Malliavin died on 3 June, 2010. Obviously his absence will be felt most keenly by his family and friends, but his death also marks the end of an era for this journal. When Irving M. Segal conceived the idea of creating the Journal of Functional Analysis, he first enlisted the support of Ralph S. Phillips and the two of them approached Malliavin at the ICM held in 1962. JFA came into existence shortly thereafter. Now all three of its founders are dead. Both Segal and Phillips died in 1998. Until his death, Segal was the central figure in the management of the journal. He oversaw the business arrangements and played a decisive role in the selection of editors. Fortunately for JFA, Malliavin was ready and able to pick up the loose ends which Segal had left behind and to assume the role which he had played. For the last twelve years, the most important decisions about the journal’s future were made by Malliavin, sitting in the tiny office next to the dining room in his apartment. During those years, Malliavin guided JFA through some difficult times when, without his guidance, the journal’s very existence might have been in jeopardy. Because of his dedication and wisdom, JFA continues to play an important role in the international mathematics community and promises to be a strong presence in the future. Losing Malliavin is a blow to all of us, but we can take solace and pride in the fact that JFA will live on. Yours, Alain Connes Cédric Villani Daniel Stroock
0022-1236/$ – see front matter © 2010 Published by Elsevier Inc. doi:10.1016/j.jfa.2010.08.001
Journal of Functional Analysis 259 (2010) 2760–2775 www.elsevier.com/locate/jfa
¯ Regularity of the ∂-Neumann problem at point of infinite type Tran Vu Khanh, Giuseppe Zampieri ∗ Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova, Italy Received 8 November 2008; accepted 6 August 2010
Communicated by I. Rodnianski
Abstract ¯ We introduce general estimates for “gain of regularity” of solutions of the ∂-Neumann problem and relate it to the existence of weights with large Levi form at the boundary. This enables us to discuss in a unified framework the classical results on fractional ellipticity (= subellipticity), superlogarithmic ellipticity and compactness. For each case, we exhibit a corresponding class of domains. © 2010 Elsevier Inc. All rights reserved. ¯ Keywords: ∂-Neumann problem
1. Introduction ¯ Let D be a bounded domain of Cn defined by r < 0 for ∂r = 0. The ∂-Neumann problem ¯ = v which is orthogonal to ker ∂¯ under the compatibility consists in finding the solution of ∂u ¯ = 0. Here v and u are forms of degree k and k − 1 respectively. Related to this, is condition ∂v the equation 2u = v, with u and v of the same degree k, where 2 := ∂¯ ∗ ∂¯ + ∂¯ ∂¯ ∗ . If 2 is invertible, in a suitable Hilbert space, there is well-defined a Neumann operator N := 2−1 and the solution ¯ ∂¯ ∗ ) and for 2 which to the first problem is produced by u := ∂¯ ∗ N v. We discuss estimates for (∂, s ∞ assure continuity of N in the spaces H and C up to the boundary ∂D. We wish to recall the theory by Catlin of [2]. Assume that, in a neighborhood of a boundary point zo ∈ ∂D, there is a * Corresponding author.
E-mail addresses:
[email protected] (T.V. Khanh),
[email protected] (G. Zampieri). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.004
T.V. Khanh, G. Zampieri / Journal of Functional Analysis 259 (2010) 2760–2775
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family of weights ϕ = ϕδ for δ → 0, which are plurisubharmonic, have bound |ϕ| 1 over the strip Sδ := {z ∈ D : dist(z, ∂D) < δ} and whose Levi form ϕij satisfies ϕij (z) δ −2
for any z ∈ Sδ
(1.1)
(in the sense that the lowest eigenvalue of ϕij is δ −2 ). Here and in what follows, or denote inequality up to a constant. Note that Catlin requires in addition |ϕ| < 1 on the whole D; it is clear from the proof of Theorem 1.4 that the ϕ’s may be arranged so that this last condition is fulfilled. In [2] Catlin proves that finite type of ∂D in the sense of D’Angelo [4] yields a family of weights satisfying (1.1). In turn, he proves that these weights give subelliptic estimates for the ¯ ∂-Neumann problem (which were already obtained by Kohn [13] for real analytic boundaries). Theorem 1.1. (See Catlin [2], Theorem 2.2.) Let D be pseudoconvex; then the existence of a family of weights {ϕ δ } which satisfy (1.1) in a neighborhood of zo ∈ ∂D implies, for a smaller neighborhood V of zo , ¯ 20 + ∂¯ ∗ u2 + u20 |||u|||2 ∂u 0
for u ∈ Cc∞ (D¯ ∩ V ) ∩ Dom∂¯ ∗ of degree k 1.
(1.2)
Here ||| · ||| is the tangential -Sobolev norm. We want to generalize this result in two directions. The first consists in considering more general q-pseudoconvex (or q-pseudoconcave), instead of merely pseudoconvex, domains and prove (1.2) for forms of related degree k q (or k q); this was already achieved in [10–12,20]. The second, consists in considering estimates with a weaker gain of regularity than the tangential fractional -Sobolev. This is the specific novelty of the present paper. To introduce q-pseudoconvexity/concavity we need to de+ , velop some notations and terminology: L∂D = (rij )|T C ∂D is the Levi form of the boundary, s∂D − 0 s∂D , s∂D are the numbers of eigenvalues of L∂D which are > 0, < 0, = 0 respectively and finally ∂D ∂D λ∂D 1 λ2 · · · λn−1 are its ordered eigenvalues. We take a pair of indices 1 q n − 1 and 0 qo n − 1 such that q = qo . We assume that there is a bundle V qo ⊂ T 1,0 (∂D) of rank qo with smooth coefficients in a neighborhood V of zo , say the bundle of the first qo coordinate tangential vector fields ∂ω1 , . . . ∂ωqo , such that q j =1
λ∂D j −
qo
rjj 0
on ∂D ∩ V .
(1.3)
j =1
Definition 1.2. (i) If q > qo we say that D is q-pseudoconvex at zo . (ii) If q < qo we say that D is q-pseudoconcave at zo . This condition contains, as a particular case, the classical q-pseudoconvexity (resp. q-pseudoconcavity) to the choice qo = 0 (resp. qo = n − 1), q“by compensation” which n−1 corresponds ∂D 0) (cf. [7] and more recent developments by that is, j =1 λ∂D 0 (resp. − λ j =q+1 j j [19] and [18]). We write k-forms as u = (uJ )J where J = j1 < j2 < · · · < jk are ordered multiindices. When the multiindices are not ordered, the coefficients are assumed to be alter nating. Thus, if J decomposes as J = j K, then uj K = sign jJK uJ . We take an orthonormal
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basis of (1, 0) forms ω1 , . . . , ωn = ∂r and the dual basis of (1, 0) vector fields ∂ω1 , . . . , ∂ωn ; thus ∂ω1 , . . . , ∂ωn−1 generate T 1,0 (∂D). Under the choice of such basis, we check readily that u ∈ Dom∂¯ ∗ if and only if unK |∂D ≡ 0 for any K. We use the notation rj := ∂ωj r. Integration by parts and use of the tangentiality conditions unK |∂D ≡ 0, as well of the vanishing rj |∂D ≡ 0 for j n − 1 which follows from the choice of the orthonormal basis adapted to the boundary, yields the “basic” estimates [8,9,21] ¯ 2 0 + ∂¯ϕ∗ u2 0 + Cu2 0 ∂u H H H ϕ
ϕ
n
ϕ
e−ϕ ϕij uiK u¯ j K dv −
+
n−1
e
−ϕ
rij uiK u¯ j K ds −
1 + 2
q0 j =1
qo
e−ϕ rjj |uJ |2 ds
|J |=q j =1 ∂D
|K|=k−1 i,j =1∂D
e−ϕ ϕjj |uJ |2 dv
|J |=k j =1 D
|K|=k−1 i,j =1 D
qo
ϕ 2 δ u 0 + ωj H ϕ
n j =qo +1
∂ω¯ j u2H 0 ϕ
for u ∈ Cc∞ (D¯ ∩ V )k ∩ Dom∂¯ ∗ .
(1.4)
ϕ
Here the δωj ’s are the adjoints to the −∂ω¯ j ’s and dv and ds are the elements of volume in D and of area on ∂D respectively. We refer for instance to [21] for the proof (1.4). By choosing ϕ so that e−ϕ is bounded, we may remove the weight functions in (1.4). We note that there is no relation between k and qo in the above inequality and that C is independent of ϕ (and u). However, if we assume that D is q-pseudoconvex (resp. q-pseudoconcave) and restrain the degree k of u to k q (resp. k q), then the third line of (1.4) can be discarded since it is positive and we get an estimate which does not involve boundary integrals. Now the crucial point has become to make the right choice of the weight ϕ in order to get full advantage of the second line of (1.4). Also, we wish to treat lower bounds for the Levi form, smaller than δ −2 . Let f be a smooth monontonic 1 increasing function f : R+ → R+ with f (t) t 2 . We consider weights ϕ = ϕ δ in C 2 (D¯ ∩ V ), ϕ ϕ absolutely bounded in S¯δ ∩ V and with the property that, if λ1 (z) λ2 (z) · · · are the ordered eigenvalues of the form ϕijδ (z), we have q j =1
ϕ
λj (z) −
qo j =1
ϕjj (z) f
2 qo 1 ϕj (z) 2 , + δ
for any z ∈ S¯δ ∩ V .
(1.5)
j =1
According to the point (a) of the proof of Theorem 1.4 which follows, we can modify ϕ to a new weight for which (1.5) holds in the whole D ∩ V , instead of the only Sδ ∩ V , but with the term q in the right reduced to j o=1 |ϕj (z)|2 . In the same way as (1.3) says that the second line of (1.4) is positive, we can see that this modified version of (1.5) gives a good lower bound for the first line over forms in degree k q. Definition 1.3. If ∂D is q-pseudoconvex/concave and there is a family of weights {ϕ} = {ϕ δ } which are absolutely bounded on S¯δ ∩ V and satisfy (1.5), we say that D satisfies (f -P -q). We introduce special coordinates (a, r) ∈ R2n−1 × R, denote by ξ the dual coordinates to a and by Fτ the tangential Fourier transform, that is, the partial Fourier transform with respect to a.
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1
We denote by Λξ = (1 + |ξ |2 ) 2 the standard elliptic symbol of order 1 and by Λ∂ the operator 1 with symbol Λξ . For a smooth monotonic increasing function f : R+ → R+ with f (t) t 2 , we consider the symbol f (Λξ ) and the associated tangential pseudodifferential operator f (Λ∂ ). This is defined by f (Λ∂ )u = Fτ−1 f (Λξ )Fτ (u) . Here is the main result of the present section. Theorem 1.4. Let D be q-pseudoconvex (resp. q-pseudoconcave), assume that ∂D satisfies (f -P -q) for q > qo (resp. q < qo ) and let k q (resp. k q). Then f (Λ∂ )u2 ∂u ¯ 2 + ∂¯ ∗ u2 + u2
for u ∈ C ∞ (D¯ ∩ V ) ∩ Dom∂¯ ∗ of degree k. (1.6)
Before the proof, some remarks are in order. Remark 1.5. We point our attention to the rate of f as t → ∞ in three relevant cases: (i) f t , (ii) f k log t for any k, (iii) f k for any k. It is obvious that (i) implies (ii) and (ii) implies (iii). The estimates (1.6) are said subelliptic, superlogarithmic and of compactness, when f satisfies (i), (ii) and (iii) respectively. For the case of pseudoconvex domains, the first are discussed, as it has already been said, by Catlin in [2], the second by Kohn in [16] and the third by Catlin [1], Straube [19], Mc Neal [17], Harrington [6] and others. Remark 1.6. Classically, superlogarithmic estimates are defined by log(Λ∂ )u2 ∂u ¯ 2 + ∂¯ ∗ u2 + C u2 0
0
0
−1
for any > 0.
(1.7)
But (1.6) for f satisfying (ii), that is, f k log t for any k, implies (1.7). In fact, under the substitution t = |ξa |, we have f −1 log(|ξa |) for any ξa outside a suitable compact K R2n−1 . It follows log(Λ∂ )u2 f 2 Λ(ξ ) |Fτ u|2 dξ dr + sup f 2 (Λξ ) |Fτ u|2 dξ dr (R2n−1 \K )×R
ξ ∈K
K ×R
f 2 Λ(ξ ) |Fτ u|2 dξ dr + C u2−1 .
(R2n−1 \K )×R
This, combined with (1.6), yields (1.7). ¯ 2 + ∂¯ ∗ u2 ) + C u2 for Similarly, compactness is classically defined by u20 (∂u 0 0 −1 any ; again, this estimate is a consequence of (1.6) for f satisfying (iii).
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¯ Remark 1.7. Let 2 = ∂¯ ∂¯ ∗ + ∂¯ ∗ ∂¯ be the ∂-Neumann Laplacian. It is well known (cf. [5] and [16]) that subelliptic and superlogarithmic estimates imply local hypoellipticity of 2: 2u ∈ C ∞ (D¯ ∩ V ) implies u ∈ C ∞ (D¯ ∩ V ). On the other hand, compactness over a covering ¯ implies u ∈ C ∞ (D). ¯ We have another {D} of ∂D implies global hypoellipticity: 2u ∈ C ∞ (D) ¯ = v with ∂v ¯ = 0, we define the “canonical” version of these two statements. For the equation ∂u solution by u := ∂¯ ∗ N v where N is the H 0 inverse to 2. Thus local (global) hypoellipticity of 2 implies that the canonical solution u inherits local (global) smoothness from v at ∂D (it surely does in the interior). Proof of Theorem 1.4. (a) We “globalize” ϕ by multiplication for a cut-off χ and next deform by composition with a convex function ψ so that the resulting function ψ ◦ (χϕ), that we still denote by ϕ, satisfies
n−1
ϕij uτiK u¯ τj K
|K|=k−1 ij =1
−
qo j =1
0 f (δ −1 )2
2 ∂ϕ · uτ 2 ϕjj uτ − 2 ·K |K|=k−1
in D ∩ V , in S δ ∩ V ,
(1.8)
2
for uτ tangential, that is, satisfying uτJ ≡ 0 if n ∈ J even for z ∈ / ∂D, of degree k q (resp. k q). For this, we take a smooth decreasing cut-off function satisfying χ ≡ 1 on [0, 12 ] and χ ≡ 0 on [ 23 , 1], and define ϕ˜ = ϕ˜ δ by ϕ˜ δ := χ(− δr )ϕ δ . Recall that rj = 0 for j n − 1 and uτnK ≡ 0. Then, over such forms we have
n−1 i,j =1
ϕ˜ ij uτiK u¯ τj K
−
qo j =1
n qo τ 2 τ 2 τ τ ϕ˜jj uJ χ · ϕij uiK u¯ j K − ϕjj uJ . i,j =1
(1.9)
j =1
In fact, if i and j denote derivation in ∂ωi and ∂ω¯ j respectively, we have
ri ϕj ri rj rj ϕi r ϕ χ − ϕ − χ˙ + χϕij = χ¨ 2 ϕ − χ˙ rij − χ˙ δ δ δ δ δ ij ϕ = −χ˙ rij + χϕij over tangential forms uτ δ
(1.10)
(where we have to remember that rj ≡ 0 for any j n − 1). Since −χ˙ 0, then (1.10) implies (1.9). Note that ∂ ϕ˜ = χ˙ ∂rϕ + χ∂ϕ and recall that ∂r · uτ ≡ 0 (and that u has degree q). It follows that ϕ˜ satisfies (1.8) with the constant 2 replaced by a more general c > 0 according to (f -P -q). To get (1.8) for the precise constant 2 we have to compose ψ ◦ ϕ. ˜ From ¯ ◦ ϕ) ˙ ∂¯ ϕ˜ + ψ∂ ¨ ϕ˜ ⊗ ∂¯ ϕ, ∂ ∂(ψ ˜ = ψ∂ ˜ we get that ψ ◦ ϕ˜ satisfies (1.8) as soon as ⎧ ⎨ ψ¨ 2ψ˙ 2 , c ˙ ⎩ ψ¨ ψ. 2
(1.11)
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c
A choice for such a function is ψ = 12 e 2 (t−1) ; we still denote by ϕ this new weight ψ ◦ ϕ˜ which satisfies (1.8). We wish to remove now the weight from the adjunction ∂¯ϕ∗ . We note that ∗ τ 2 1 ∗ τ 2 ∂¯ u ∂¯ u − ∂ϕ · uτ 2 . ϕ ·K 2
(1.12)
|K|=k−1
It follows 2 2 τ 2 1 ¯ − QD uτ , uτ + C uτ D ∂¯ϕ∗ uτ D + ∂u ∂ϕ · u·K 2D D 2
1 2 D
−
n−1
ϕij uτiK u¯ τj K
|K|=k−1 ij =1
−
qo
τ 2 ϕjj u dv
j =1
∂ϕ · u·K 2D
|K|=k−1
·−
Sδ
|K|=k−1
|K|=k−1
· 2S δ
2
2
2 2 f δ −1 uτ S , δ 2
(1.13)
where the first inequality follows from (1.12), the second from (1.4) in addition to qpseudoconvexity/concavity, the third from the first occurrence of (1.8) and the fourth from the second of (1.8). (b) We will prove in (c) and (d) which follow that (1.13) implies (1.6) for tangential forms uτ . We prove now that an estimate for uτ entails an estimate for the full u: Lemma 1.8. The estimate f (Λ∂ )uτ 2 Q(uτ , uτ ) for any uτ implies f (Λ∂ )u2 Q(u, u) for any u. Proof. We decompose u as u = uτ + uν where uτ is the tangential part which collects the / J and uν is the normal part, that is, the complementary component. coefficients uJ of u with n ∈ ν Since u |∂D ≡ 0, we then have by Garding inequality ν ν ν 2 Q u , u u 1 Q(u, u) + u2 , Q uτ , uτ Q(u, u) + Q uν , uν Q(u, u) + u2 . We then have f (Λ∂ )u2 f (Λ∂ )uτ 2 + uν 2 1 τ τ ν 2 Q u ,u + u 1 Q(u, u) + u2 ,
(1.14)
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where the first inequality is obvious, the second follows from (1.6) for uτ and the third from (1.14). 2 (c) The rest of the proof is devoted to prove that (1.8) implies (1.6) for uτ . To begin with, we need the following generalization of [5], Theorem 2.4.5. The generalization consists in passing from the system {∂ω¯ j }j =1,...,n to any elliptic system {Mj }j =1,...,N such as {∂ωj }j =1,...,qo ∪ {∂ω¯ j }j =qo +1,...,n . Proposition 1.9. Let {Mj }j =1,...,N be a elliptic system of vector fields, that is, the symbols σ (Mj ) have no common zeroes in R2n \ {0}. We then have 2n N −1 −1 Λ f (Λ∂ )Di u2 Λ f (Λ∂ )Mj u2 + f (Λ∂ )ub 2 ∂
− 12
∂
j =1
i=1
for u ∈ Cc∞ (D¯ ∩ V ),
(1.15)
where Di denote all coordinate derivatives and ub the restriction of u to M. Proof. (i) It is not restrictive to assume that the Mj ’s have constant coefficients, that is, Mj = i aij Di for aij ≡ aij (zo ). In fact, if |aij (z) − aij (zo )| < in a neighborhood of zo , then, if u 2 is supported by such neighborhood, each Λ−1 ∂ f (Λ∂ )(Mj − Mj (zo ))u can be absorbed in the left of (1.15). (ii) We define 1 2 2 w := Fτ−1 e(1+|ξ | ) r Fτ u(ξ, 0) ,
and set v := u − w. Since v|∂D ≡ 0, then N
Mj v2 ∼
j =1
N
Mj v2 +
j =1
N
M¯ j v2 ∼
j =1
2n
Di v2 .
i=1
2n 2 2 Similarly, N j =1 f (Λ∂ )Mj v ∼ i=1 f (Λ∂ )Di v . Combination of these estimates yields (1.15) for v without boundary integral. It is useful for the following to notice that it is not made any assumption compactness of the support of v. (iii) To carry out the proof of the proposition, we need to prove that 1 −1 Λ f (Λ∂ )Di w 2 Λ− 2 f (Λ∂ )wb 2 .
∂
∂
We distinguish now the case Di = Dai (tangential derivative) from the case Di = Dr (normal derivative). In the first case we have −1 Λ f (Λ∂ )Da w 2 = i ∂
0
R2n−1 −∞
1 + |ξ |2
−1
1 2 |ξai |2 f 1 + |ξ |2 2
1 × exp 2 1 + |ξ |2 2 r |Fτ u|2 (ξ, 0) dr dξ
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− 1 1 2 1 + |ξ |2 2 f 1 + |ξ |2 2
2767
0 e
2ρ
dρ |Fτ u|2 (ξ, 0) dξ
−∞
R2n−1
2 1 = f (Λ∂ )ub − 1 . 2 2 In the second case −1 Λ f (Λ∂ )Dr w 2 = ∂
0
−1 1 2 1 2 1 + |ξ |2 f 1 + |ξ |2 2 2 1 + |ξ |2 2
R2n−1 −∞
1 × exp 2 + |ξ |2 2 r |Fτ u|2 (ξ, 0) drdξ 0 1 1 2 2 2 2 −2 2ρ = f 1 + |ξ | 1 + |ξ | e dρ |Fτ u|2 (ξ, 0) dξ −∞
R2n−1
2 1 = f (Λ∂ )ub − 1 . 2 2 2
This completes the proof of Proposition 1.9.
(d) We complete the proof of (1.6) for uτ . We begin by noticing that the term N −1 Λ f (Λ∂ )Mj uτ 2 ∂
j =1
of (1.15) can be estimated by Q(uτ , uτ ); thus what is left to prove is that f (Λ∂ )ub 2
− 12
2 Q uτ , uτ + uτ .
The first part of the discussion holds for general u, not necessarily tangential. We recall the microlocalization procedure of Catlin. Let {pk }k be a sequence of C ∞ functions in R+ such that
pk2 = 1,
supp(pk ) ⊂ 2k−1 , 2k+1 ,
supp(p0 ) ⊂ (0, 2),
p 2−k . k
k
By the aid of the pk ’s we introduce, following Catlin, the pseudodifferential operators Pkτ u = Fτ−1 pk (Λξ )Fτ u . We can then show that 2 s 2 Λ f (Λ∂ )u2 ∼ 22ks f 2k P τ u . ∂
k
0
k
0
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This is a standard result granted that 1 2ks f 2k 1 + |ξ |2 2 f |ξ | 2(k+1)s f 2k+1
for |ξ | ∈ 2k−1 , 2k+1 .
We apply this result for s = 12 . We follow now step by step the procedure of the proof of Theorem 2.2 of [2]. We use the elementary inequality k g(0) 2 2 η
0
g(r) 2 dr + 2−k η
−2−k
0
2 g (r) dr,
−2−k
which holds for any g such that g(−2−k ) = 0. If we apply it for g(r) = χk (r)Pk u(·, r), where χk ∈ Cc∞ (−2−k , 0] with 0 χk 1 and χk (0) = 1, we get ∞ 2 2 f (Λ∂ )ub 2 1 ∼ f 2k 2−k χk (0)Pk ub − 2
k=0
η
−1
0 ∞ k 2 χk Pk u(., r)2 dr f 2 k=0
+η
−2−k
0 ∞ 2 Dr χk Pk u(., r) 2 dr. f 2k 2−2k k=0
−2−k
We specify now u = uτ and denote by (I) and (II) the two sums in the second line of the above estimate. Now, (I) η−1
∞ Q Pk uτ , Pk uτ k=o
2 2 η−1 Q uτ , uτ + uτ + Dr uτ −1 ,
(1.16)
where the first inequality follows from (1.13) and the second from the estimates of the commu¯ Pk ] and [∂¯ ∗ , Pk ]. We also have the estimate tators [∂, 2 (II) ηQ uτ , uτ + ηf (Λ∂ )uτ .
(1.17)
By combining (1.16) and (1.17) and by absorbing ηf (Λ∂ )u2 in the left-hand side of the estimate we get f (Λ∂ )ub 2
− 12
The proof of Theorem 1.4 is complete.
2
2 Q uτ , uτ + uτ .
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2. A geometric criterion for (f -P -q) property When the Levi form of the boundary is nondegenerate one has the strongest estimates for the ¯ ∂-Neumann problem, that is, 12 -subelliptic ones. When the Levi form decreases with a certain rate in correspondence to a submanifold S ⊂ ∂D, with dimCR S q − 1, then we can prove (f -P -q) for f related to the inverse of the Levi vanishing rate. Let ∂D be q-pseudoconvex (resp. q-pseudoconcave) in a neighborhood of zo and let S ⊂ ∂D, be a submanifold containing zo and with the properties
T C S ⊃ V qo |S resp. T C S ⊂ V qo |S , (ii) dim TzC S q − 1 resp. dim TzC S q − 1 for any z close to zo . (i)
(2.1)
We denote by dS the distance-function to S, consider a real function F = F (δ), δ ∈ R + such ∗ ∗ −1 ))−1 . With that Fδ(δ) 2 +∞ as δ 0, denote by F the inverse to F and define f (t) := (F (t these notations we have Theorem 2.1. Let ∂D be q-pseudoconvex (resp. q-pseudoconcave) and let S ⊂ ∂D be a submanifold satisfying (2.1). Suppose that q j =1
λ∂D j −
qo j =1
rjj
F (dS ) . dS2
(2.2)
Then (f -P -q) property holds for q > qo (resp. q < qo ) where f (t) := (F ∗ (t −1 ))−1 . Proof. We first consider the case q-pseudoconvex. We take χ = χ(t) in C ∞ with χ ≡ 1 for 0 t 1 and χ ≡ 0 for t 2 and define our family of weights ϕ = ϕ δ by
dS2 dS2 −r + 1 + cχ ϕ = − log log +1 , δ 2f −2 (δ −1 ) 2f −2 (δ −1 ) δ
(2.3)
where c is a small constant to be specified later. Note that the ϕ’s are absolutely bounded on S¯δ ∩ V . We observe that ¯ S2 = 2∂dS ⊗ ∂d ¯ S + 2dS ∂ ∂d ¯ S. ∂ ∂d
(2.4)
When we compose with log we get ¯ S ¯ S + 2dS ∂ ∂d ¯ S 2∂dS ⊗ ∂d dS2 ∂dS ⊗ ∂d − 4 ∂ ∂¯ log dS2 + 2f −2 = (dS2 + 2f −2 ) (dS2 + 2f −2 )2 =
¯ S (2d 2 + 4f −2 − 4d 2 ) + 2dS ∂ ∂d ¯ S (d 2 + 2f −2 ) ∂dS ⊗ ∂d S S S (dS2 + 2f −2 )2
.
(2.5)
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Now, if dS2 < f −2 , then (2.5) can be continued by
¯ S f −2 + 2dS ∂ ∂d ¯ S (d 2 + 2f −2 ) 2∂dS ⊗ ∂d S (dS2
+ 2f −2 )2
2 3 ∂dS
¯ S + 2dS ∂ ∂d ¯ S ⊗ ∂d (dS2 + 2f −2 )
2 3 ∂dS
¯ S + 2dS ∂ ∂d ¯ S ⊗ ∂d . 3f −2
(2.6)
¯ S respectively in the basis of We denote by (dS )j and (dS )ij the components of ∂dS and ∂ ∂d forms {ωj }. We notice that by (2.4) implies for forms u of degree k q, 2 (dS )j uj K |u|2 .
|K|=k−1
(2.7)
j
We also notice that dS
(dS )ij uiK u¯ j K −|u|2 .
(2.8)
|K|=k−1 ij d2
S ¯ We introduce the notation (Bij ) := ∂ ∂(log( + 1)). In conclusion, if dS2 < f −2 , combination 2f −2 of (2.5), (2.6), (2.7) and (2.8) yields
n
Bij uiK u¯ j K −
ij =1
qo
Bjj |uJ |2 f −2 |u|2 .
(2.9)
j =1
¯ and the similar property for (Aij ) := ∂ ∂(− ¯ log( −r + 1)), we have that Because of (1.3) for ∂ ∂r δ ¯ δ. (2.9) is true not only for (Bij ) but also for ∂ ∂ϕ We suppose now dS2 f −2 ; then F (dS ) F (f −1 (δ −1 )) δ = −2 −1 . 2 −2 −1 f (δ ) δ f dS
(2.10)
It follows that n ij =1
Aij uiK u¯ j K −
qo
Ajj |uJ |2 f 2 δ −1 |u|2 .
(2.11)
j =1
Now, because of the cut-off χ , the contribution of (Bij ) can get negative when dS2 f −2 and therefore χ˙ = 0 or χ¨ = 0. However, (Bij ) −cf 2 (δ −1 ) and hence (Aij ) controls this negative ¯ δ. term by a suitable choice of c; thus (2.11) implies the similar estimate for ∂ ∂ϕ q δ ¯ The family of weights {ϕ } satisfies (1.5) on Sδ ∩ V without the term j o=1 |ϕj (z)|2 in the right-hand side. As for this term, in the q-pseudoconvex case, it vanishes and so there is nothing else to prove. Instead, in the q-pseudoconcave case, it is not 0; also, the definition of ϕ needs to
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be modified by multiplying the second term in the right of (2.3) by −1. The proof goes through with a slight modification such as in [10] formulas (5.9)–(5.13). 2 3. Domains which have subelliptic, superlogarithmic and compactness estimates We introduce fairly general classes of domains D for which we are able to prove the hypotheses of Theorem 2.1; this implies (f -P -q) property according to Section 2 and then (f -q) estimates by Section 1. First, we treat the case q-decoupled-pseudoconvex domains; these are defined near zo = 0 by r < 0 for r in the form
r = 2Re zn − h(z1 , . . . , zqo ) +
n−1
hj (zj )
for q qo + 1,
(3.1)
j =q
¯ 0 and the hj ’s are subharmonic, non-harmonic, functions vanishing at zj = 0. where ∂ ∂h Decoupled domains are treated, among others, by Mc Neal [17]. Similarly, we consider qpseudoconcave domains whose defining function r is of the type
r = 2Re zn + h(zqo +1 , . . . , zn−1 ) −
q+1
hj (zj )
for q qo − 1
(3.2)
j =1
¯ 0 and the hj ’s subharmonic, non-harmonic and vanishing at zj = 0. It is obvious that with ∂ ∂h a domain D endowed with such a defining function r is q-pseudoconvex or q-pseudoconcave in the two respective cases of (3.1) and (3.2). When the hj ’s have finite vanishing order 2mj , these domains are treated in [10]. This leads to subelliptic estimates, that is (1.6) for f satisfying (i) for < 2 max1j mj . We recall briefly the argument of the proof. We choose the weights ϕ := − log(−r + δ) +
1 log |zj |2 + δ mj ,
j
and normalize them by a factor c| log δ|−1 . Thus they are absolutely bounded and their Levi form ϕij satisfies (1.5) for f ( 1δ )2 = δ −2 for any < 2 max1j mj . Thus the conclusion is a consequence of Theorem 1.1. We introduce two new cases. Before, we notice that it is not restrictive to assume 2 log ∂
zj z¯ j hj
+∞ as |zj | 0.
(3.3)
Otherwise, we would have ∂z¯2j z¯ j hj (zo ) = 0 for some j and therefore − log(−r + δ) would have
a Levi form that, applied to uj K would be δ −1 |uj K |2 . But this is the 12 -subelliptic estimate which is the best we can expect in a neighborhood of the boundary. Thus we assume (3.3) in what follows.
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Proposition 3.1. Let D be a q-pseudoconvex (resp. q-pseudoconcave) domain defined by (3.1) (resp. (3.2)) and suppose that |zj |α log ∂z2j z¯ j hj (zj ) 0 as |zj | 0.
(3.4)
(a) If 0 < α 1 then we have (1.6) with f satisfying (ii), that is, superlogarithmic estimates for ¯ the ∂-Neumann problem over forms of degree k q (resp. k q). ¯ (b) If α > 1 then we have (1.6) with f satisfying (iii), that is, compactness of the ∂-Neumann problem for forms of degree k q (resp. k q). Before the proof, an example is in order. Example 3.2. For the case (a), we can choose hj = e
− |z 1|α j
for α < 1 or else hj = e
1 j || log |zj ||
− |z
.
− |z 1|α
For (b) we take hj = e j for α 1. Now, there is no doubt that the two above choices of hj fulfill (3.3) for α satisfying (a) and (b) respectively. We then consider the domains defined by 2Re zn − h +
hj < 0,
j
or 2Re zn + h −
hj < 0,
j
A few words are maybe needed to show that the above domains are q-pseudoconvex and q−1
pseudoconcave respectively. In fact, if we write the exponentials as e g it suffices to prove that these are subharmonic. Here g = |zj || log |zj || or g = |zj |α ; thus g is subharmonic for zj = 0. But then e
− g1
itself is subharmonic, including at zj = 0 because of the identity
¯ −1 |∂g|2 |∂g|2 − 1 ∂ ∂g . ∂ ∂¯ e g = e g − 2 + g2 g3 g4
Thus, the above domains have superlogarithmic or compactness estimates according to the cases (a) and (b) (and in degree k q and k q in the case q-pseudoconvex and qpseudoconcave respectively). Proof of Proposition 3.1. We define 2aj−1 (|zj |) = |zj |α | log(∂z2j z¯ j (hj (zj ))|zj |2 )|; we note that we have aj−1 0 as |zj | 0. Referring to the terminology of Theorem 2.1, we denote by Sj the origin in the zj -plane; thus dSj = |zj |. We have
∂z2j z¯ j hj
=
e
2 α j | aj (zj )
− |z
|zj
|2
=
Fj (|zj |) , |zj |2
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1
2773
for Fj = e δ aj (δ) . Setting fj−1 (δ) = c(| log δ|aj (| log δ|− α ))− α and choosing a cut-off χ with χ ≡ 1 for 0 t 1 and χ ≡ 0 for t 2, we define α
1
1
n−1
|zj |2 |zj |2 −r +1 + χ log + 1 . ϕ δ = − log δ 2fj−2 (δ) 2fj−2 (δ)
(3.5)
j =q
We also set f := minj fj . When D is q-pseudoconvex, q the family of weights ϕ satisfies (1.5) for the above defined f on S¯δ ∩ V without the term j o=1 |ϕj (z)|2 in the right-hand side. However, this term vanishes and so there is nothing else to prove. The variant for the q-pseudoconcave case follows the lines of the similar variant in Theorem 2.1 (in particular by inserting a crucial factor −1 in the second log of (3.5)). 2 4. The tangential system We consider a hypersurface M ⊂ Cn and denote by D ± the two sides of M. We suppose all through this section that D + is pseudoconvex. We parametrize M over R2n−1 with variable a by a diffeomorphism Φ, so that ∂a2n−1 corresponds to the totally real vector field tangential to M that we also denote by T . We denote by ∂¯b = Φ∗−1 ∂¯ the induced complex, by ∂¯b∗ the adjoint to ∂¯b , set 2b := ∂¯b∗ ∂¯b + ∂¯b ∂¯b∗ and put Qb (ub , ub ) = ∂¯b ub 2 + ∂¯b∗ ub 2 for any form ub on M. We denote by ξ the coordinates dual to the a’s. We consider a conic partition of the unity 1 ≡ ψ + + ψ − + ψ 0 on R2n−1 for |ξ | 1 such that ψ + ≡ 1 for ξa2n−1 > |ξ |, ψ − ≡ 1 for ξa2n−1 < −|ξ | and ψ 0 ≡ 1 in a conic neighborhood of the plane ξa2n−1 = 0. We introduce briefly the conclusions of the microlocalization method by Kohn. Let ub + = ψ + (Λ∂ )ub where ψ + (Λ∂ ) is the pseudodifferential operator with symbol ψ + (Λξ ) and similarly define u− b and − 0 . We then say that u is C ∞ in + u + u u0b . This yields a microlocal decomposition ub = u+ b b b b − ∞ ± direction +da2n−1 (resp. −da2n−1 ) when u+ b (resp. ub ) is C . For u ∈ Dom∂¯ ∗ (D ), we set 2 2 ∗ ¯ ¯ QD ± (u, u) := ∂uD ± + ∂ uD ± . The operation of restriction of forms of Dom∂¯ ∗ from D ± to M and that of (harmonic) extension from M to D ± yields Theorem 4.1. (See Kohn [16].) We have for forms of degree k with 1 k n − 2, f (Λ∂ )u2 ± QD ± (u, u) + u2 ± D D
for any u ∈ Dom∂¯ ∗ D ± ,
if and only if f (Λ∂ )ζ u± 2 Qb u± , u± + u± 2 ± b
b
b
b
b
D
for any u± b,
where ζ denotes a cut-off function. The result of Kohn is stated only for f (Λ∂ ) = Λτ ; but the extension to general f is straightfor 0 2 0 2 ward. We observe now that T u0b 2b n−1 j =1 (∂ωj ub b + ∂ω¯ j ub b ) because the characteristic 0 variety of T is transversal to supp(ψ ). We also have n−1 n−1 ∂ω u0 2 ∂ω¯ u0 2 + T u0 2 + C u0 2 , j b b j b b b b b b j =1
j =1
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which is readily proved by integration by parts. It follows 0 2 u b
H 1 (M)
n−1 ∂ω¯ u0 2 + u0 2 j b b b b j =1
2 Qb u0b , u0b + u0b b .
(4.1)
By combining (4.1) with Theorem 4.1 we get Corollary 4.2. We have for forms of degree k with 1 k n − 2, f (Λ∂ )u2 QD (u, u) + u2 D D
for both D = D + and D = D −
(4.2)
if and only if f (Λ∂ )ub 2 Qb (ub , ub ) + ub 2 . b
(4.3)
Corollary 4.2 provides a tool for transferring estimates from D + and D − to ∂D and conversely; in this way, when f (t) k log t for any k, for a suitable ck and for any t ck , then the related hypoellipticity, is also transferred. In particular, let M be graphed as xn = g where g=
n−1 j =1
e
− |z 1|α j
or g =
n−1
e
− |x 1|α j
;
(4.4)
j =1
note that D + is pseudoconvex (and D − is (n − 1)-pseudoconcave). If α < 1 or, for α = 1, if we replace |zj | by aj (zj )|zj | for aj +∞ when zj 0, then we have superlogarithmic estimates in D ± ; in particular 2 is hypoelliptic. Using Corollary 4.2 we have, for forms in degree k with 1 k n − 2, Proposition 4.3. (i) If α < 1, then 2b has superlogarithmic estimates; in particular, it is hypoelliptic. (ii) Is α = 1 and we replace |zj | by aj (zj )|zj | (or |xj | by aj (xj )|xj |) for aj +∞, then the same conclusion as in (i) holds. If ∂¯b has closed range over functions (resp. ∂¯b∗ has closed range over (n − 1)-forms), Kohn has a result also in the critical degree k = 0 (resp. k = n − 1) [16], Theorem 1.6. There are superlogarithmic estimates which imply that ∂¯b (resp. ∂¯b∗ ) is hypoelliptic on the orthogonal complement of Ker ∂¯b over functions (resp. Ker ∂¯b∗ over (n − 1)-forms). We still keep the structure (4.4) for the equation of the domain but restrict to dimension n = 2. We also point our attention to the action of ∂¯b over functions and disregard ∂¯b∗ over (n − 1)-forms. Now, when α 1 the domain defined by the first occurrence of (4.4), in which g depends on |z1 |, stays hypoelliptic. Instead, in the “tube domain”, in which g depends on the only |x1 |, is not. The first follows from Kohn [15] combined with the argument of Kohn [14] Theorem 2.6, whereas the second is proved by Christ in [3]. Here is the geometric explanation. In the first case, the
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set of the points where the system of complex tangential vector fields fails to have finite type is confined to the real curve {0} × Ry2 transversal to T C M. Instead, for the tube, the points of non-finite type are the two-dimensional plane R2y . References ¯ [1] D. Catlin, Global regularity of the ∂-Neumann problem, in: Complex Analysis of Several Variables, Madison, Wis., 1982, in: Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984. ¯ [2] D. Catlin, Subelliptic estimates for the ∂-Neumann problem on pseudoconvex domains, Ann. of Math. 126 (1987) 131–191. [3] M. Christ, Hypoellipticity of the Kohn Laplacian for three-dimensional tubular Cauchy–Riemann structures, J. Inst. Math. Jussieu 1 (2002) 279–291. [4] J. D’Angelo, Real hypersurfaces, order of contact, and applications, Ann. of Math. 115 (1982) 615–637. [5] G.B. Folland, J.J. Kohn, The Neumann Problem for the Cauchy–Riemann Complex, Ann. of Math. Stud., vol. 75, Princeton Univ. Press, Princeton, NJ, 1972. [6] P.S. Harrington, A quantitative analysis of Oka’s lemma, Math. Z. 256 (1) (2007) 113–138. ¯ [7] L.H. Ho, Subellipticity of the ∂-Neumman problem for n − 1 forms, Trans. Amer. Math. Soc. 325 (1991) 171–185. [8] L. Hormander, L2 estimates and existence theorems for the ∂¯ operator, Acta Math. 113 (1965) 89–152. [9] L. Hörmander, An Intoduction to Complex Analysis in Several Complex Variables, Van Nostrand, Princeton, NJ, 1973. ¯ [10] T.V. Khanh, G. Zampieri, Subellipticity of the ∂-Neumann problem on a weakly q-pseudoconvex/concave domain, arXiv:0804.3112v1, 2008. ¯ [11] T.V. Khanh, G. Zampieri, Compactness of the ∂-Neumann operator on a q-pseudoconvex domain, Complex Var. Elliptic Equ. (2009). ¯ [12] T.V. Khanh, G. Zampieri, Pseudodifferential gain of regularity for solutions of the tangential ∂-system, 2009. ¯ [13] J.J. Kohn, Subellipticity of the ∂-Neumann problem on pseudoconvex domains: sufficient conditions, Acta Math. 142 (1979) 79–122. [14] J.J. Kohn, Estimates for ∂b on pseudoconvex CR manifolds, in: Proc. Sympos. Pure Math., vol. 43, 1985, pp. 207– 217. [15] J.J. Kohn, Hypoellipticity at points of infinite type, in: Contemp. Math., vol. 251, 2000, pp. 393–398. [16] J.J. Kohn, Superlogarithmic estimates on pseudoconvex domains and CR manifolds, Ann. of Math. 156 (2002) 213–248. [17] J.D. Mc Neal, Estimates of the Bergman kernels of convex domains, Adv. Math. 109 (1994) 108–139. [18] A. Raich, Compactness of the complex Green operator on CR-manifolds of hypersurface type, arXiv:0810.2553v2, 2008. ¯ [19] E. Straube, A sufficient condition for global regularity of the ∂-Neumann operator, Adv. Math. 217 (2008) 1072– 1095. ¯ [20] G. Zampieri, q-Pseudoconvexity and regularity at the boundary for solutions of the ∂-problem, Compos. Math. 121 (2) (2000) 155–162. [21] G. Zampieri, Complex Analysis and CR Geometry, Univ. Lecture Ser., vol. 43, Amer. Math. Soc., 2008.
Journal of Functional Analysis 259 (2010) 2776–2792 www.elsevier.com/locate/jfa
The McShane integral in weakly compactly generated spaces ✩ A. Avilés a , G. Plebanek b , J. Rodríguez c,∗ a Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain b Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland c Departamento de Matemática Aplicada, Facultad de Informática, Universidad de Murcia, 30100 Espinardo (Murcia),
Spain Received 2 September 2009; accepted 10 August 2010 Available online 21 August 2010 Communicated by N. Kalton
Abstract Di Piazza and Preiss asked whether every Pettis integrable function defined on [0, 1] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative. Moreover, we give a counterexample where the target Banach space is reflexive. © 2010 Elsevier Inc. All rights reserved. Keywords: Pettis integral; McShane integral; Scalarly null function; Filling family
1. Introduction The classical Pettis’ measurability theorem [15] ensures that scalar and strong measurability are equivalent for functions taking values in separable Banach spaces. This fact has many interesting consequences in vector integration. For instance, it is a basic tool to prove that Pettis and McShane integrability coincide in separable Banach spaces [10,12,13]. However, for non✩ A. Avilés and J. Rodríguez were supported by MEC and FEDER (Project MTM2008-05396) and Fundación Séneca (Project 08848/PI/08). A. Avilés was supported by Ramon y Cajal contract (RYC-2008-02051). G. Plebanek wishes to thank A. Avilés, B. Cascales and J. Rodríguez for their hospitality during his stay in Murcia in February 2009; the visit was supported by Departamento de Matemáticas, Universidad de Murcia. * Corresponding author. E-mail addresses:
[email protected] (A. Avilés),
[email protected] (G. Plebanek),
[email protected] (J. Rodríguez).
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.007
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separable Banach spaces the notions of scalar and strong measurability are different in general. This leads to subtle problems when trying to compare different types of integrals. In this paper we deal with the Pettis and McShane integrals. Di Piazza and Preiss [2] asked whether every Pettis integrable function f : [0, 1] → X is McShane integrable if X is a weakly compactly generated (WCG) Banach space. Recently, Deville and the third author [1] have proved that the answer is affirmative when X is Hilbert generated, thus improving the previous results obtained in [2,17]. Our main purpose here is to show that the question of Di Piazza and Preiss has negative answer in general. The paper is organized as follows. In Section 2 we introduce the MC-integral for Banach space-valued functions defined on probability spaces. This auxiliary tool is used as a substitute of the McShane integral at some stages. We prove that, for functions defined on quasi-Radon probability spaces, MC-integrability always implies McShane integrability (Proposition 2.2), while the converse holds if the topology on the domain has a countable basis (Proposition 2.3). This approach allows us to give a partial answer (Corollary 2.4) to a question posed by Fremlin in [10, 4G(a)]. In Section 3 we show that the existence of scalarly null (hence Pettis integrable) WCG-valued functions which are not McShane integrable is strongly related to the existence of families of finite sets which are “measure filling” in the sense of the following definition. Throughout the paper (Ω, Σ, μ) is a probability space and we use the symbol [S]<ω to denote the family of all finite subsets of a given set S. Definition 1.1. A family F ⊂ [Ω]<ω is called MC-filling on Ω if it is hereditary and there exists ε > 0 such that for every countable partition (Ωm ) of Ω there is F ∈ F such that μ∗ {Ωm : F ∩ Ωm = ∅} > ε, where μ∗ is the outer measure induced by μ. This concept should be viewed as a measure-theoretic analogue of the notion of ε-filling families, defined as follows: Definition 1.2. Let ε > 0. A family F ⊂ [S]<ω is called ε-filling on the set S if it is hereditary and for every H ∈ [S]<ω there is F ∈ F with F ⊂ H and |F | ε|H |. The following problem by Fremlin [8] remains open: Problem DU. Are there compact ε-filling families on uncountable sets? However, we show that compact MC-filling families on [0, 1] can be constructed from some weaker versions of filling families that Fremlin proved to exist (Theorem 3.5). This leads to our main result: Theorem 3.6. There exist a WCG Banach space X and a scalarly null function f : [0, 1] → X which is not McShane integrable. In fact, the space X can be taken reflexive (Theorem 3.7). Observe that Theorem 3.6 also answers in the negative the question (attributed to Musiał in [2]) whether every scalarly null
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Banach space-valued function on [0, 1] is McShane integrable. In [2,17] two counterexamples had been constructed under the Continuum Hypothesis (where the target spaces are non-WCG spaces). We emphasize that all results in this paper are in ZFC. In Section 4 we prove that if a family F ⊂ [A]<ω is ε-filling on a set A ⊂ Ω of positive outer measure then it is MC-filling on Ω. Finally, in Section 5 we provide an example of a McShane integrable function which is not MC-integrable (Theorem 5.5). Our example also makes clear that, in general, the results on the coincidence of Pettis and McShane integrability of [1,2] do not hold when McShane integrability is replaced by MC-integrability. 1.1. Terminology Our standard references are [4,19] (vector integration) and [11] (topological measure theory). By a partition of a set S we mean a collection of pairwise disjoint (maybe empty) subsets whose union is S. We say that a set is countable if it is either finite or countably infinite. The symbol |S| stands for the cardinality of a set S. A family F ⊂ [S]<ω is called hereditary if G ∈ F whenever G ⊂ F ∈ F . A family F ⊂ [S]<ω is called compact if the set {1A : A ∈ F } is compact in 2S equipped with the product topology. Here we write 1A to denote the function on S defined by 1A (s) = 1 if s ∈ A and 1A (s) = 0 if s ∈ / A. It is well known that a hereditary family F ⊂ [S]<ω is compact if and only if every set A ⊂ S with [A]<ω ⊂ F is finite. We say that a family E ⊂ Σ is η-thick (for some η > 0) if μ(Ω \ E) η. Throughout the paper X is a (real) Banach space. The norm of X is denoted by · if it is needed explicitly. We denote by X ∗ the topological dual of X and put BX = {x ∈ X: x 1}. The space X is called WCG if there is a weakly compact subset of X whose linear span is dense in X. A function f : Ω → X is called scalarly null if, for each x ∗ ∈ X ∗ , the composition x ∗ f : Ω → R vanishes μ-a.e. (the exceptional set depending on x ∗ ). If T ⊂ Σ is a topology on Ω, we say that (Ω, T, Σ, μ) is a quasi-Radon probability space (following [11, Chapter 41]) if μ is complete, inner regular with respect to closed sets, and μ( G) = sup{μ(G): G ∈ G} for every upwards directed family G ⊂ T. A gauge on Ω is a function δ : Ω → T such that t ∈ δ(t) for all t ∈ Ω. Every Radon probability space is quasiRadon [11, 416A]. The vector-valued McShane integral was first studied in [12,13] for functions defined on [0, 1] equipped with the Lebesgue measure. Fremlin [10] extended the theory to deal with functions defined on arbitrary quasi-Radon probability spaces. We next recall an alternative way of defining the McShane integral taken from [9, Proposition 3]. Definition 1.3. Suppose (Ω, T, Σ, μ) is quasi-Radon. A function f : Ω → X is said to be McShane integrable, with integral x ∈ X, if for every ε > 0 there exist η > 0 and a gauge δ on Ω such that: for every η-thick finite family (Ei ) of pairwise disjoint measurable sets and every choice of points ti ∈ Ω with Ei ⊂ δ(ti ), we have ε. μ(E )f (t ) − x i i i
Every McShane integrable function is also Pettis integrable (and the corresponding integrals coincide) [10, 1Q]. The converse does not hold in general, see [1,2,12,17] for examples.
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2. Another look at the McShane integral We next introduce a variant of the McShane integral that is defined in terms of the measure space only, without any reference to a topology. Definition 2.1. A function f : Ω → X is called MC-integrable, with integral x ∈ X, if for every ε > 0 there exist η > 0, a countable partition (Ωm ) of Ω and sets Am ∈ Σ with Ωm ⊂ Am , such that: for every η-thick finite family (Ei ) of pairwise disjoint elements of Σ with Ei ⊂ Am(i) and every choice of points ti ∈ Ωm(i) , we have μ(Ei )f (ti ) − x ε. i
Clearly, given η > 0, a countable partition (Ωm ) of Ω and sets Ωm ⊂ Am ∈ Σ, we can always find families (Ei ) as in Definition 2.1. It is routine to check that the vector x in Definition 2.1 is unique. The relationship between the MC-integral and the McShane integral is analyzed in the following two propositions. Proposition 2.2. Suppose (Ω, T, Σ, μ) is quasi-Radon. If f : Ω → X is MC-integrable, then it is McShane integrable (and the corresponding integrals coincide). Proof. Let x ∈ X be the MC-integral of f and fix ε > 0. Since f is MC-integrable, there exist η > 0, a countable partition (Ωm ) of Ω and measurable sets Am ⊃ Ωm satisfying the condition of Definition 2.1. For each m, n ∈ N, set Ωm,n := {t ∈ Ωm : n − 1 f (t) < n} and choose open Um,n ⊃ Am such that μ(Um,n \ Am )
1 2m+n
ε η . min , n 2
Clearly, (Ωm,n ) is a partition of Ω. Define a gauge δ : Ω → T by δ(t) := Um,n if t ∈ Ωm,n . Let (Ei ) be a η2 -thick finite family of pairwise disjoint measurable sets and let ti ∈ Ω be points such that Ei ⊂ δ(ti ). We will check that 2ε. μ(E )f (t ) − x i i
(1)
i
For each i, let m(i), n(i) ∈ N be such that ti ∈ Ωm(i),n(i) . The set Fi := Ei ∩ Am(i) is measurable, Fi ⊂ Am(i) and ti ∈ Ωm(i) . The Fi ’s are pairwise disjoint. Since Ei \ Fi = Ei \ Am(i) ⊂ δ(ti ) \ Am(i) = Um(i),n(i) \ Am(i) we have
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μ Ω\ Fi = μ Ω \ Ei + μ Ei \ Fi i
η +μ 2
i
i ∞
i
Um,n \ Am
m,n=1
∞ η η = η, + 2 2m+n+1 m,n=1
and so the family (Fi ) is η-thick. From the MC-integrability condition it follows that μ(Fi )f (ti ) − x ε.
(2)
i
For each m, ˜ n˜ ∈ N, let I (m, ˜ n) ˜ be the (maybe empty) set of all indexes i for which m(i) = m ˜ and n(i) = n. ˜ Observe that
μ(Ei \ Fi )f (ti )
i∈I (m, ˜ n) ˜
μ(Ei \ Fi )n˜ =
i∈I (m, ˜ n) ˜
=μ
Ei \ Fi n˜ μ(Um, ˜ ˜ n˜ \ Am ˜ )n
i∈I (m, ˜ n) ˜
ε ˜ n˜ 2m+
.
Therefore μ(Ei )f (ti ) − μ(Fi )f (ti ) μ(Ei \ Fi )f (ti ) i
i
i
=
∞
μ(Ei \ Fi )f (ti )
m, ˜ n=1 ˜ i∈I (m, ˜ n) ˜
∞ m, ˜ n=1 ˜
ε = ε. ˜ n˜ 2m+
(3)
Inequality (1) now follows from (2) and (3). This shows that f is McShane integrable, with McShane integral x. 2 The converse of Proposition 2.2 does not hold in general (see Theorem 5.5 below), although it is true for certain quasi-Radon spaces like [0, 1], as we next prove. Proposition 2.3. Suppose (Ω, T, Σ, μ) is quasi-Radon and T has a countable basis. Then f : Ω → X is McShane integrable (if and) only if it is MC-integrable. Proof. It only remains to prove the “only if”. Assume that f is McShane integrable, with McShane integral x ∈ X. Let {Um : m ∈ N} be a countable basis for T consisting of mutually distinct elements. Fix ε > 0. Find η > 0 and a gauge δ on Ω fulfilling the condition of Definition 1.3. We can suppose without loss of generality that δ(t) ∈ {Um : m ∈ N} for every t ∈ Ω. Set and Am := Um for all m ∈ N. Ωm := t ∈ Ω: δ(t) = Um
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Clearly, (Ωm ) is a partition of Ω and Ωm ⊂ Am ∈ Σ . Now let (Ei ) be an η-thick finite family of pairwise disjoint measurable sets with Ei ⊂ Am(i) and let ti ∈ Ωm(i) . Then δ(ti ) = Um(i) = A m(i) , hence Ei ⊂ δ(ti ) for all i. From the McShane integrability condition it follows that i μ(Ei )f (ti ) − x ε. This shows that f is MC-integrable. 2 Fremlin raised in [10, 4G(a)] the following question: Does any topology on Ω for which μ is quasi-Radon yield the same collection of McShane integrable X-valued functions? In view of Propositions 2.2 and 2.3, we get a partial answer: Corollary 2.4. Let T1 and T2 be two topologies on Ω for which μ is quasi-Radon. Suppose T1 has a countable basis. If f : Ω → X is McShane integrable with respect to T1 , then it is also McShane integrable with respect to T2 (and the corresponding integrals coincide). 3. MC-filling families versus the McShane integral The connection between MC-filling families (Definition 1.1) and the MC-integral is explained in Proposition 3.2 below. First, it is convenient to characterize MC-filling families as follows: Lemma 3.1. A hereditary family F ⊂ [Ω]<ω is MC-filling on Ω if (and only if ) there exists ε > 0 such that for every countable partition (Ωm ) of Ω and sets Am ∈ Σ with Ωm ⊂ Am , there is F ∈ F such that μ
{Am : F ∩ Ωm = ∅} > ε.
Proof. The “only if” is obvious. For the converse, we will prove that the condition of Definition 1.1 holds for 0 < η < ε. Suppose we are givena countable partition (Ωm )of Ω. For every fi∗ nite set I ⊂ N, we choose BI ∈ Σ such that BI ⊃ m∈I Ωm and μ(BI )−μ ( m∈I Ωm ) < ε −η. For each m ∈ N, we define Am := {BI : m ∈ I }. We have Ωm ⊂ Am ∈ Σ , so we can apply the hypothesis to find F ∈ F such that μ
{Am : F ∩ Ωm = ∅} > ε.
Consider the (finite) set I := {m ∈ N: F ∩ Ωm = ∅}. Since μ∗
m∈I
Am ⊂ BI , we have
Ωm > μ(BI ) − (ε − η) μ Am − (ε − η) > η.
m∈I
This proves that F is MC-filling.
m∈I
2
A set Λ ⊂ BX∗ is called norming if x = sup{|x ∗ (x)|: x ∗ ∈ Λ} for all x ∈ X. Proposition 3.2. Let f : Ω → X be a function for which there exist a norming set Λ ⊂ BX∗ and a family (Cx ∗ )x ∗ ∈Λ of subsets of Ω such that x ∗ f = 1Cx ∗ and μ∗ (Cx ∗ ) = 0 for every x ∗ ∈ Λ. The following statements are equivalent:
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(i) f is not MC-integrable; (ii) x ∗ ∈Λ [Cx ∗ ]<ω is MC-filling on Ω. Proof. Observe first that for every finite family (Ei ) of pairwise disjoint elements of Σ and every choice of points ti ∈ Ω, we have ∗ μ(Ei )f (ti ) = sup x μ(Ei )f (ti ) i
x ∗ ∈Λ
= sup
x ∗ ∈Λ i
i
μ(Ei )1Cx ∗ (ti ) = sup μ x ∗ ∈Λ
{Ei : ti ∈ Cx ∗ } .
(4)
Since Λ separates the points of X and x ∗ f vanishes μ-a.e. for each x ∗ ∈ Λ, the MC-integral of f is 0 ∈ X whenever f is MC-integrable. Bearing in mind (4), statement (i) is equivalent to: (iii) There exists ε > 0 such that for every η > 0, every countable partition (Ωm ) of Ω and every sets Am ∈ Σ with Ωm ⊂ Am , there exist an η-thick finite family (Ei ) of pairwise ∗ disjoint elements of Σ with Ei ⊂ Am(i) , points ti ∈ Ωm(i) and a functional x ∈ Λ such that μ( {Ei : ti ∈ Cx ∗ }) > ε. Let us turn to the proof of (iii) ⇔ (ii). Assume first that (iii) holds and take a countable partition (Ωm ) of Ω and sets Ωm ⊂ Am ∈ Σ . Choose η > 0 arbitrary and let (Ei ), (ti ) and x ∗ be as in (iii). Observe that the set F made up of all ti ’s belonging to Cx ∗ satisfies {Am : F ∩ Ωm = ∅} ⊃ {Ei : ti ∈ Cx ∗ } and so μ( {Am : F ∩ Ωm = ∅}) > ε. According to Lemma 3.1, this proves that the family <ω is MC-filling on Ω. x ∗ ∈Λ [Cx ∗ ] assume that (ii) holds. Let ε > 0 be as in Lemma 3.1 applied to the family Conversely, ∗ ]<ω . Fix η > 0, a countable partition (Ωm ) of Ω and sets Am ∈ Σ with Ωm ⊂ Am . [C ∗ x x ∈Λ There exist x ∗ ∈ Λ and finite F ⊂ Cx ∗ such that μ( m∈I Am ) > ε, where I := {m ∈ N: F ∩ Ωm = ∅}. Now take a finite set J ⊂ N disjoint from I such that (Am )m∈I ∪J is ηthick. Enumerate I = {m(1), . . . , m(n)} and J = {m(n + 1), . . . , m(k)}. Set E1 := Am(1) and (Ei ) is an η-thick finite family of pairwise Ei := Am(i) \ i−1 j =1 Am(j ) for i = 2, . . . , k. Then n disjoint elements of Σ with Ei ⊂ Am(i) and i=1 Ei = m∈I Am . Choose ti ∈ F ∩ Ωm(i) for i = 1, . . . , n and choose ti ∈ Ωm(i) arbitrary for i = n + 1, . . . , k. Then n
μ Ei = μ Am > ε. {Ei : ti ∈ Cx ∗ } μ i=1
This shows that (iii) holds, that is, f is not MC-integrable.
m∈I
2
Given a compact Hausdorff topological space K, we write C(K) to denote the Banach space of all real-valued continuous functions on K with the sup norm. Proposition 3.3. Let F ⊂ [Ω]<ω be a compact hereditary family made up of sets having outer measure 0. Let f : Ω → C(F ) be defined by f (t)(F ) := 1F (t), t ∈ Ω, F ∈ F . Then:
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(i) f is scalarly null; (ii) f is not MC-integrable if and only if F is MC-filling on Ω. Proof. Part (i) follows from a standard argument which we include for the sake of completeness. Since F is an Eberlein compact (i.e., it is homeomorphic to a weakly compact subset of a Banach space), the space C(F ) is WCG and BC(F )∗ is an Eberlein compact when equipped with the w ∗ topology, cf. [3, Theorem 4, p. 152]. Set Λ := {δF : F ∈ F } ⊂ BC(F )∗ , where δF denotes the “evaluation functional” at F . Since Λ is norming, its absolutely convex hull aco(Λ) is w ∗ -dense in BC(F )∗ . Bearing in mind that (BC(F )∗ , w ∗ ) is homeomorphic to a weakly compact subset of a Banach space, the Eberlein–Smulyan theorem (cf. [7, Theorem 3.10]) ensures that aco(Λ) is w ∗ -sequentially dense in BC(F )∗ . Since the composition δF f = 1F vanishes μ-a.e. for every F ∈ F , we conclude that f is scalarly null. Part (ii) follows from Proposition 3.2 applied to f and to the norming set Λ defined above. 2 On the other hand, it turns out that we can find compact MC-filling families on [0, 1]. In order to establish this, we need the following lemma due to Fremlin [8, 4B]. Fremlin’s result is more general, but this statement is enough for our purposes. We include an elegant proof of it that has been communicated to us by Jordi López-Abad. Let c denote the cardinality of the continuum. Lemma 3.4 (Fremlin). There exists a compact hereditary family D ⊂ [c]<ω such that for every infinite P ⊂ c and every n ∈ N there is D ∈ D such that D ⊂ P and |D| = n. Proof. The so-called Schreier family S = {S ⊂ N: |S| min(S)} is a compact 12 -filling family on N (notice that for any F ∈ [N]<ω with |F | even (resp. odd), the last |F2 | (resp. [ |F2 | ] + 1) elements of F form a member of S). Let T = 2<ω be the dyadic tree, the set of all finite sequences of 0’s and 1’s, endowed with the tree order: (x1 , . . . , xn ) (y1 , . . . , ym ) if n m and xi = yi for i n. Let S be a compact 12 -filling family on T , that we can obtain by transferring the Schreier family S through a bijection between N and T . Given two different x = (x1 , x2 , . . .) and y = (y1 , y2 , . . .) in 2N , we define m(x, y) := min{k: xk = yk } and v(x, y) := (x1 , . . . , xm(x,y)−1 ) = (y1 , . . . , ym(x,y)−1 ) ∈ T . Given a set D ⊂ 2N , we consider v(D) := {v(x, y): x, y ∈ D, x = y}. The family <ω D := D ∈ 2N : v(D) ∈ S is obviously hereditary and compact. Indeed, take any set A ⊂ 2N with [A]<ω ⊂ D. Then [v(A)]<ω ⊂ S and the compactness of S ensures that v(A) is finite, and hence, so is A. We shall prove that for every finite set A ⊂ 2N with |A| 2 there is D ∈ D such that D ⊂ A and |D| > 12 log2 (|A| − 1) + 1. This implies the property of D stated in the lemma. To this end, we first prove by induction on k ∈ N ∪ {0} that: (*) If C ⊂ T is a finite set satisfying |C| 2k and inf{s, s } ∈ C whenever s, s ∈ C, then C contains a totally ordered subset of cardinality k + 1. Indeed, let C ⊂ T be a finite set such that |C| 2k+1 and that inf{s, s } ∈ C whenever s, s ∈ C. If we denote t := min C, then C \{t} = B0 ∪B1 , where Bi consists of those elements of C extending
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t i for i = 0, 1. Clearly, we have inf{s, s } ∈ Bi whenever s, s ∈ Bi . For some j ∈ {0, 1} we have |Bj | 2k , so by the inductive assumption Bj contains a totally ordered subset S with |S| = k + 1. Therefore, {t} ∪ S is a totally ordered subset of C with cardinality k + 2. This finishes the proof of (*). Now, fix a finite set A ⊂ 2N with |A| 2. It is easy to check that inf{s, s } ∈ v(A) for every s, s ∈ v(A) and that |v(A)| = |A| − 1. Let k be the largest integer less than or equal to log2 (|v(A)|). Property (*) ensures the existence of a totally ordered set U ⊂ v(A) with |U | = k + 1, hence |U | > log2 v(A) = log2 |A| − 1 . Since S is 12 -filling, there exists W ⊂ U such that W ∈ S and 1 |W | |U |. 2 Let us write W = {w1 ≺ w2 ≺ · · · ≺ wm }. For each i = 1, . . . , m − 1, since wi ∈ v(A), we can choose ai ∈ A such that wi is an initial segment of ai but wi+1 is not. We can also choose am , am+1 ∈ A with v(am , am+1 ) = wm . Set D := {a1 , . . . , am+1 }. Notice that v(D) = W ∈ S , hence D ∈ D. Finally, 1 1 |D| = |W | + 1 |U | + 1 > log2 |A| − 1 + 1. 2 2 The proof is over.
2
Theorem 3.5. There exists a compact MC-filling family on [0, 1] equipped with the Lebesgue measure. Proof. We denote by λ the Lebesgue measure on [0, 1]. Fix a partition {Zα : α < c} of [0, 1] made up of sets of outer measure one (cf. [11, 419I]). Let ϕ : [0, 1] → c be the function defined by ϕ(t) = α whenever t ∈ Zα . Let D ⊂ [c]<ω be the family provided by Lemma 3.4 and define F := F ⊂ [0, 1] finite: ϕ is one-to-one on F and ϕ(F ) ∈ D . We claim that F is compact. Indeed, take a set A ⊂ [0, 1] with [A]<ω ⊂ F . Observe that ϕ is one-to-one on A. Given any C ∈ [ϕ(A)]<ω , we have C = ϕ(B) for some B ∈ [A]<ω ⊂ F and so C ∈ D. Hence [ϕ(A)]<ω ⊂ D and the compactness of D ensures that ϕ(A) is finite. Since ϕ is one-to-one on A, we conclude that A is finite. This shows that F is compact, as claimed. We shall check that F is ε-MC-filling on [0, 1] with an arbitrary constant 0 < ε < 1. Fix a countable partition (Ωm ) of [0, 1]. For each α < c we have ∗
1 = λ (Zα ) = lim λ n→∞
∗
Zα ∩
n m=1
Ωm ,
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so we can pick k(α) ∈ N such that λ
∗
Zα ∩
k(α)
(5)
Ωm > ε.
m=1
Fix n ∈ N such that Pn := {α < c: k(α) = n} is infinite. By Lemma 3.4, there is D ∈ D such that D ⊂ Pn and |D| = n. Write D = {α1 , . . . , αn }. We next define tj ∈ Zαj and = i) inductively as mj ∈ {1, . . . , n} (with mj = mi whenever j follows. By (5) the set Zα1 ∩ nm=1 Ωm is nonempty. Pick then any t1 ∈ Zα1 ∩ nm=1 Ωm . Choose m1 ∈ {1, . . . , n} so that t1 ∈ Ωm1 . Now suppose we have already constructed a set {m1 , . . . , ml } ⊂ {1, . . . , n} and points tj ∈ Zαj ∩ Ωmj for j = 1, . . . , l. If λ∗ ( lj =1 Ωmj ) > ε, the construction stops. Otherwise λ∗ ( lj =1 Ωmj ) ε and therefore l < n (bear in mind that λ∗ ( nm=1 Ωm ) > ε by (5)). Writing N := {1, . . . , n} \ {m1 , . . . , ml }, another appeal to (5) yields
n l ∗ ∗ λ Zαl+1 ∩ Ωm λ Zαl+1 ∩ Ωm − λ Zαl+1 ∩ Ωmj > 0, ∗
m∈N
j =1
m=1
so we can find tl+1 ∈ Zαl+1 ∩ Ωml+1 for some ml+1 ∈ N . Repeating the process, the construction stops for some l ∈ {1, . . . , n}. After that, we obtain a set {m1 , . . . , ml } ⊂ {1, . . . , n} with λ∗ ( lj =1 Ωmj ) > ε and points tj ∈ Zαj ∩ Ωmj for all j = 1, . . . , l. Putting F := {t1 , . . . , tl } we have l
∗ Ωmj > ε. λ {Ωm : F ∩ Ωm = ∅} = λ ∗
j =1
Since ϕ(tj ) = αj for all j , it follows that ϕ is one-to-one on F and ϕ(F ) ⊂ D, thus ϕ(F ) ∈ D and so F ∈ F . We proved that the family F is ε-MC-filling. 2 It was kindly communicated to us by Marian Fabian (during the 25th Summer Conference on Topology and its Applications, Kielce, July 2010) that the proof of Theorem 3.5 still works if we assume that D has the following property, weaker than the one stated in Lemma 3.4: • D is a compact hereditary family of countable subsets of c such that, for every countable decomposition c = n Γn , there exist D ∈ D and n ∈ N such that |D ∩ Γn | > n. A family D with this property is the same as an Eberlein compact which is not uniform Eberlein compact subset of the Σ-product Σ(2c ), cf. [6] for further reference. We now arrive at our main result (valid in ZFC): Theorem 3.6. There exist a WCG Banach space X and a scalarly null function f : [0, 1] → X which is not McShane integrable.
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Proof. By Theorem 3.5, there is a compact MC-filling family F on [0, 1]. As we observed in the proof of Proposition 3.3, the space X := C(F ) is WCG. The function f : [0, 1] → C(F ) defined by f (t)(F ) := 1F (t),
t ∈ [0, 1], F ∈ F ,
from Proposition 3.3 is scalarly null and fails to be MC-integrable. According to Proposition 2.3, f is not McShane integrable. 2 Moreover, the target space can be taken reflexive: Theorem 3.7. There exist a reflexive Banach space Y and a scalarly null function g : [0, 1] → Y which is not McShane integrable. Proof. Let F and f be as in the proof of Theorem 3.6. Observe first that f ([0, 1]) is relatively weakly compact in C(F ). Indeed, by the Eberlein–Smulyan theorem (cf. [7, 3.10]), it is enough to check that (f (tn )) converges weakly to 0 whenever (tn ) is a sequence of distinct points of [0, 1]. But this follows directly from Grothendieck’s theorem (cf. [7, 4.2]) just bearing in mind that for each F ∈ F (finite!) we have f (tn )(F ) = 1F (tn ) = 0 for n large enough. Then, by the Davis–Figiel–Johnson–Pelczynski theorem (cf. [3, Chapter 5, §4]), there exist a reflexive Banach space Y and a one-to-one linear continuous mapping T : Y → C(F ) such that f ([0, 1]) ⊂ T (BY ). The set of compositions V := φ ◦ T : φ ∈ C(F )∗ is a linear subspace of Y ∗ which separates the points of Y (because T is one-to-one). Since Y is reflexive, V is norm dense in Y ∗ . Let g : [0, 1] → Y be the function satisfying T ◦ g = f . For each y ∗ ∈ V the composition y ∗ g vanishes a.e. (f is scalarly null). This fact and the norm density of V imply that g is scalarly null. Moreover, since f is not McShane integrable and T is linear and continuous, g is not McShane integrable either. 2 Remark 3.8. A glance at the proof of Proposition 3.3 reveals that the function f from Theorem 3.6 satisfies that, for each x ∗ ∈ X ∗ , the composition x ∗ f vanishes up to a countable set. This property and the boundedness of f ensure that f is universally Pettis integrable, that is, Pettis integrable with respect to any Radon probability on [0, 1]. The same holds true for the function g from Theorem 3.7. Thus, we answer Question 2.2 in [18]: there exist ZFC examples of universally Pettis integrable functions which are not universally McShane integrable. 4. Filling versus MC-filling families In this section we prove that ε-filling families (Definition 1.2) on sets of positive outer measure are MC-filling. This result is less powerful than Theorem 3.5, in the sense that the existence of compact ε-filling families on uncountable sets is unknown while Theorem 3.5 is a ZFC result. Yet, we have decided to include it as it may have some interest in relation with problem DU. Theorem 4.1. Suppose μ is atomless. Let A ⊂ Ω with μ∗ (A) > 0 and F ⊂ [A]<ω be a family which is ε-filling on A for some ε > 0. Then F is MC-filling on Ω.
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Proof. Denote η := μ∗ (A) and fix η > η1 > η2 > 0. Take any countable partition (Ωm ) of Ω and sets Am ⊃ Ωm with Am ∈ Σ . We will prove that there is F ∈ F such that μ
{Am : F ∩ Ωm = ∅} > ε(η − η1 ).
According to Lemma 3.1, this means that F is MC-filling on Ω. To this end, take m0 ∈ N large enough such that ∗
μ (A) − μ
∗
m0
A∩
(6)
Ω m < η2 .
m=1
Since μ is atomless, every finite subset of Ω has outer measure 0, so we can assume without loss of generality that A ∩ Ωm is infinite for all m = 1, . . . , m0 . Take 0 < 2 )/m0 . ηm30 < (η1 − η m0 We can find pairwise disjoint B1 , . . . , Bm0 ∈ Σ such that m=1 Bm = m=1 Am and Bm ⊂ Am . Let M be the set of all m ∈ {1, . . . , m0 } for which μ(Bm ) > 0. For each m ∈ M, choose a positive rational αm such that μ(Bm ) > αm > μ(Bm ) − η3 . We can write αm = pm /q for some pm ∈ N and q ∈ N, for m = 1, . . . , m0 . Set θ := 1/q. Since μ is atomless, for each m ∈ M we can find pairwise disjoint E1m , . . . , Epmm ∈ Σ contained in Bm with μ(Eim ) = θ . Then μ Bm \
pm
Eim
< η3
i=1
and we have μ
∗
A∩
m0
Ωm μ
∗
m=1
A∩
m0
Am = μ
m=1
μ Bm \
m∈M
pm i=1
|M|η3 +
∗
A∩
Bm
m∈M
Eim
+
pm μ Eim m∈M i=1
pm θ
m∈M
m0 η3 +
pm θ < (η1 − η2 ) + pm θ.
m∈M
m∈M
From these inequalities and (6) we obtain η = μ∗ (A) < η1 +
m∈M
pm θ.
(7)
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For each m ∈ M and i = 1, . . . , pm we pick a point t(m,i) ∈ A ∩ Ωm . This can be done in such a way that the points t(m,i) ’s are different, since A ∩ Ωm is infinite for all m ∈ M. Now H := {t(m,i) : m ∈ M, i = 1, . . . , pm } is a subset of A with cardinality m∈M pm . Since F is ε-filling on A, there exists F ⊂ H with F ∈ F such that |F | ε m∈M pm . By (7), we get
m μ {Am : F ∩ Ωm = ∅} μ Ei : t(m,i) ∈ F = |F |θ ε pm θ > ε(η − η1 ). m∈M
The proof is over.
2
5. McShane integrability versus MC-integrability This section is devoted to ensure the existence of McShane integrable functions which are not MC-integrable (Theorem 5.5). The proof is divided into several auxiliary lemmas. The first one translates the problem into the language of MC-filling families. Lemma 5.1. Let Γ be a set. The following statements are equivalent: (i) there exists a scalarly null function f : Ω → c0 (Γ ) which is not MC-integrable and satisfies f (Ω) ⊂ {eγ : γ ∈ Γ }, where eγ (γ ) = δγ ,γ (the Kronecker symbol) for all γ , γ ∈ Γ ; (ii) there exists a partition (Cγ )γ ∈Γ of Ω into sets having outer measure 0 such that the family <ω is MC-filling on Ω. γ ∈Γ [Cγ ] Proof. The set Λ := {eγ∗ : γ ∈ Γ } ⊂ Bc0 (Γ )∗ is norming, where eγ∗ (x) = x(γ ) for all x ∈ c0 (Γ ) and γ ∈ Γ . (i) ⇒ (ii) For each γ ∈ Γ we have eγ∗ f = 1Cγ , where Cγ := {t ∈ Ω: f (t) = eγ } has outer measure 0 (because f is scalarly null). Clearly, (Cγ )γ ∈Γ is a partition of Ω. Since f is not MCintegrable, we can apply Proposition 3.2 to conclude that the family γ ∈Γ [Cγ ]<ω is MC-filling on Ω. (ii) ⇒ (i) Define f : Ω → c0 (Γ ) by f (t) := eγ whenever t ∈ Cγ , γ ∈ Γ . Then eγ∗ f = 1Cγ for all γ ∈ Γ and f is scalarly null, because μ∗ (Cγ ) = 0 for all γ ∈ Γ andthe linear span of {eγ∗ : γ ∈ Γ } is norm dense in c0 (Γ )∗ = 1 (Γ ). By Proposition 3.2, since γ ∈Γ [Cγ ]<ω is MC-filling on Ω, the function f is not MC-integrable. 2 Thus, bearing in mind that Pettis and McShane integrability are equivalent for c0 (Γ )-valued functions [2], in order to find McShane integrable functions which are not MC-integrable we will look for MC-filling families like in condition (ii) of Lemma 5.1. The following sufficient condition will be helpful. Lemma 5.2. Let (Cγ )γ ∈Γ be a partition of Ω and (ΓA )A⊂N is a ε > 0 be such that, whenever partition of Γ , there is some A ⊂ N such that μ∗ ( γ ∈ΓA Cγ ) > ε. Then the family γ ∈Γ [Cγ ]<ω is MC-filling on Ω. Proof. Fix a countable partition (Ωm ) of Ω. For each A ⊂ N, set ΓA := γ ∈ Γ : {m ∈ N: Cγ ∩ Ωm = ∅} = A .
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Then (ΓA )A⊂N is a partition of Γ and so there is A ⊂ N such that μ∗ ( γ ∈ΓA Cγ ) > ε. Observe that Ωm ⊃ Cγ , m∈A
γ ∈ΓA
hence μ∗ ( m∈A Ωm ) > ε. Choose B ⊂ A finite with μ∗ ( m∈B Ωm ) > ε. Take γ ∈ ΓA . We can find a finite set F ⊂ Cγ such that F ∩ Ωm = ∅ for every m ∈ B, hence μ∗
{Ωm : F ∩ Ωm = ∅} μ∗ Ωm > ε. m∈B
This shows that
<ω γ ∈Γ [Cγ ]
is MC-filling on Ω.
2
We now focus on 2κ (for a cardinal κ), which is a Radon probability space when equipped with (the completion of) the usual product probability, cf. [11, 416U]. Lemma 5.3. Let κ be an uncountable cardinal, (Aα )α<κ a partition of κ into infinite sets and consider, for each α < κ, the sets Dα := x ∈ 2κ : x(γ ) = 0 for all γ ∈ Aα
and Eα := Dα \
Dβ .
β<α
Then
α∈I
Eα has outer measure 1 for every uncountable set I ⊂ κ.
Proof. It suffices to check that Z ∩ ( α∈I Eα ) = ∅ whenever Z belongs to the product σ -algebra of 2κ and has positive measure. Fix a countable set A ⊂ κ such that, for any z ∈ Z, we have x ∈ 2κ : x(γ ) = z(γ ) for all γ ∈ A ⊂ Z.
(8)
Since the Aα ’s are disjoint, the set J := {α < κ: A ∩ Aα = ∅} is countable. Clearly, the Dα ’s have measure zero (because Aα is infinite) and so Z \ α∈J Dα has positive measure. In particular, we can choose z ∈ Z \ α∈J Dα . Since J is countable and I is not, there is β ∈ I \ J . We now define an element x ∈ 2κ by declaring ⎧ ⎨ z(γ ) if γ ∈ α∈J Aα , x(γ ) := 0 if γ ∈ Aβ , ⎩ 1 otherwise. We claim that x ∈ Z ∩ Eβ . Indeed, we have x ∈ Z by (8) (bear in mind that A ⊂ α∈J Aα ). On / Dα as well. If α ∈ / J, the other hand, take any α < κ with α = β. If α ∈ J then z ∈ / Dα and so x ∈ and so x ∈ / D . It follows that x ∈ Z ∩ E , as claimed. Therefore then x(γ ) = 1 for all γ ∈ A α α β Z ∩ ( α∈I Eα ) = ∅. 2 Lemma 5.4. Let κ be a cardinal with κ > c. Then there is a partition (Cγ )γ ∈Γ of 2κ into sets of measure zero such that, whenever (ΓA )A⊂N is a partition of Γ , there is some A ⊂ N such that γ ∈ΓA Cγ has outer measure 1.
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Proof. Let (Aα )α<κ be a partition of κ into infinite sets. Clearly, the Eα ’s of Lemma 5.3 are pairwise disjoint and have measure zero (since Aα is infinite). We claim that the partition κ C := {Eα : α < κ} ∪ {x}: x ∈ 2 \ Eα α<κ
of 2κ satisfies the desired property. Indeed, let (CA )A⊂N be any partition of C. Since κ > c, there is some A ⊂ N such that CA contains uncountably many Eα ’s. By Lemma 5.3, the outer measure of CA is 1, as required. 2 We can now state the aforementioned result: Theorem 5.5. Let κ be a cardinal with κ > c. Then there is a McShane integrable function f : 2κ → c0 (Γ ) ( for some set Γ ) which is not MC-integrable. Proof. By Lemmas 5.1, 5.2 and 5.4, there is a scalarly null function f : 2κ → c0 (Γ ) (for some set Γ ) which is not MC-integrable. Since f is Pettis integrable, it is also McShane integrable [2]. 2 In [1] it is proved that Pettis and McShane integrability are equivalent for X-valued functions defined on quasi-Radon probability spaces whenever X is Hilbert generated (i.e., there exist a Hilbert space Y and a linear continuous map T : Y → X such that T (Y ) is dense in X). Clearly, every Hilbert generated space is WCG. Typical examples of Hilbert generated spaces are the separable ones, c0 (Γ ) (for any set Γ ), L1 (ν) (for any probability measure ν) and C(K) where K is a uniform Eberlein compact space. Moreover, any super-reflexive space embeds into a Hilbert generated space. For more information on this class of spaces, we refer the reader to [5,6] and [14, Chapter 6]. In view of our Theorem 5.5, we cannot replace McShane integrability by MC-integrability in the results of [1]. However, something can be said for a particular class of functions. The following proposition is inspired by [1, Lemma 3.3]. Proposition 5.6. Suppose μ is atomless and X is a subspace of a Hilbert generated Banach space such that |Ω| dens(X). Let I ⊂ BX be a set such that: • the linear span of I is dense in X; • for each x ∗ ∈ X ∗ , the set {x ∈ I : x ∗ (x) = 0} is countable. Then any one-to-one function f : Ω → I ⊂ X is scalarly null and MC-integrable. Proof. For each x ∗ ∈ X ∗ the composition x ∗ f vanishes up to a countable set, which has outer measure 0 (because μ is atomless). So, f is scalarly null. Fix ε > 0. Since X embeds into a Hilbert generated space, there is a partition I = m∈N Im such that (9) for all x ∗ ∈ BX∗ and all m ∈ N, x ∈ Im : x ∗ (x) > ε m, see [6, Theorem 6] (cf. [14, Theorem 6.30]).
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For each m ∈ N, define Ωm := ϕ −1 (Im ) and choose pairwise disjoint sets A1,m , . . . , AN (m),m ∈ Σ with μ(An,m )
ε , 2m m
n = 1, . . . , N (m),
(m) and Ωm ⊂ N n=1 An,m . Set Ωn,m := Ωm ∩ An,m for all m ∈ N and n = 1, . . . , N (m), so that (Ωn,m ) is a countable partition of Ω. Fix a finite family (Ej )j ∈J of pairwise disjoint elements of Σ with Ej ⊂ An(j ),m(j ) and choose any points tj ∈ Ωn(j ),m(j ) . We can and do assume without loss of generality that tj = tj whenever j = j . Fix x ∗ ∈ BX∗ . Define C := j ∈ J : x ∗ f (tj ) ε and Bm := j ∈ J : tj ∈ Ωm and x ∗ f (tj ) > ε for all m ∈ N. Observe that |Bm | m for all m ∈ N (this follows from (9), the injectivity of ϕ and the fact that tj = tj whenever j = j ). We can write
μ(Ej )f (tj ) =
j ∈J
μ(Ej )f (tj ) +
j ∈C
μ(Ej )f (tj ).
(10)
m∈N j ∈Bm
On one hand x∗
μ(Ej )f (tj ) μ Ej ε ε.
j ∈C
(11)
j ∈C
On the other hand, take any m ∈ N and any j ∈ Bm . Then tj ∈ Ωm ∩ Ωn(j ),m(j ) and so m(j ) = m. Therefore Ej ⊂ An(j ),m and so μ(Ej ) ε/(2m m). Thus x
∗
j ∈Bm
ε ε μ(Ej )f (tj ) μ(Ej ) |Bm | m m . 2 m 2
(12)
j ∈Bm
From (10), (11) and (12) it follows that x
∗
μ(Ej )f (tj ) 2ε.
j ∈J
As x ∗ ∈ BX∗ is arbitrary, we have MC-integral 0 ∈ X. 2
j ∈J
μ(Ej )f (tj ) 2ε. Hence f is MC-integrable, with
A set I satisfying the requirements of Proposition 5.6 is constructed for instance in [6, Theorem 2]. Also, for I we can take any Markushevich basis in X (cf. [14, Lemma 5.35]). Similar ideas yield the following result (cf. [16, Proposition 4.14]):
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Proposition 5.7. Suppose μ is atomless and X is weakly Lindelöf determined such that |Ω| = dens(X). Then there exists a one-to-one scalarly null function f : Ω → X such that the linear span of f (Ω) is dense in X. Acknowledgments We thank the referees for valuable suggestions which improved the presentation of the paper. Thanks are also due to Marian Fabian, David Fremlin and Jordi López-Abad for useful comments. References [1] R. Deville, J. Rodríguez, Integration in Hilbert generated Banach spaces, Israel J. Math. 177 (2010) 285–306. [2] L. Di Piazza, D. Preiss, When do McShane and Pettis integrals coincide? Illinois J. Math. 47 (4) (2003) 1177–1187, MR 2036997 (2005a:28023). [3] J. Diestel, Geometry of Banach Spaces—Selected Topics, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin, 1975, MR 0461094 (57 #1079). [4] J. Diestel, J.J. Uhl Jr., Vector Measures, Math. Surveys, vol. 15, American Mathematical Society, Providence, RI, 1977, MR 0453964 (56 #12216). [5] M. Fabian, G. Godefroy, P. Hájek, V. Zizler, Hilbert-generated spaces, J. Funct. Anal. 200 (2) (2003) 301–323, MR 1979014 (2004b:46011). [6] M. Fabian, G. Godefroy, V. Montesinos, V. Zizler, Inner characterizations of weakly compactly generated Banach spaces and their relatives, J. Math. Anal. Appl. 297 (2) (2004) 419–455, special issue dedicated to John Horváth, MR 2088670 (2005g:46046). [7] K. Floret, Weakly Compact Sets, Lecture Notes in Math., vol. 801, Springer, Berlin, 1980, lectures held at S.U.N.Y., Buffalo, in Spring 1978, MR 576235 (82b:46001). [8] D.H. Fremlin, Problem DU, note of 30.9.2009, available at http://www.essex.ac.uk/maths/staff/fremlin/ problems.htm. [9] D.H. Fremlin, Problem ET, note of 27.10.2004, available at http://www.essex.ac.uk/maths/staff/fremlin/ problems.htm. [10] D.H. Fremlin, The generalized McShane integral, Illinois J. Math. 39 (1) (1995) 39–67, MR 1299648 (95j:28008). [11] D.H. Fremlin, Measure Theory, vol. 4. Topological Measure Spaces. Part I, II, Torres Fremlin, Colchester, 2006, corrected second printing of the 2003 original, MR 2462372. [12] D.H. Fremlin, J. Mendoza, On the integration of vector-valued functions, Illinois J. Math. 38 (1) (1994) 127–147, MR 1245838 (94k:46083). [13] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. Math. 34 (3) (1990) 557–567, MR 1053562 (91m:26010). [14] P. Hájek, V. Montesinos Santalucía, J. Vanderwerff, V. Zizler, Biorthogonal Systems in Banach Spaces, CMS Books Math./Ouvrages Math. SMC, vol. 26, Springer, New York, 2008, MR 2359536 (2008k:46002). [15] B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (2) (1938) 277–304, MR 1501970. [16] J. Rodríguez, The Bourgain property and convex hulls, Math. Nachr. 280 (11) (2007) 1302–1309, MR 2337347 (2009b:46094). [17] J. Rodríguez, On the equivalence of McShane and Pettis integrability in non-separable Banach spaces, J. Math. Anal. Appl. 341 (1) (2008) 80–90, MR 2394066 (2009b:46095). [18] J. Rodríguez, Some examples in vector integration, Bull. Aust. Math. Soc. 80 (3) (2009) 384–392, MR 2569913. [19] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (307) (1984), ix+224, MR 756174 (86j:46042).
Journal of Functional Analysis 259 (2010) 2793–2813 www.elsevier.com/locate/jfa
Left inverses of matrices with polynomial decay Romain Tessera 1 École Normale Supérieure de Lyon, CNRS UMR 5669, 46, allée d’Italie, Lyon, France Received 26 November 2009; accepted 23 July 2010 Available online 17 August 2010 Communicated by N. Kalton
Abstract It is known that the algebra of Schur operators on 2 (namely operators bounded on both 1 and ∞ ) is not inverse-closed. When 2 = 2 (X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω (X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A ∈ Aω (X) satisfies Af p f p , for some 1 p ∞, then it admits a left-inverse in Aω (X). The main difficulty here is to obtain the above inequality in 2 . The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X = Zd , under additional conditions on the decay. © 2010 Elsevier Inc. All rights reserved. Keywords: Stability of Schur operators; Left inverse for infinite matrices with off-diagonal decay
1. Introduction In this paper, we study the left-invertibility of certain classes of bounded linear operators A : p (X) → p (Y ) where X is a metric space and Y is any set. E-mail address:
[email protected]. 1 This work was conducted in June 2007, while the author was visiting the Bernoulli center in Lausanne. The author is
supported by the NSF grant DMS-0706486. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.014
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We say that such an operator is bounded below in p if λp (A) := inf
f =0
Af p > 0. f p
If A is left invertible in p , i.e. if there exists a bounded linear map B : p (Y ) → p (X) such that BA = I , then A is clearly bounded below in p . But unless p = 2, the converse is not true in general. Our main concern in this article will be to prove the converse in certain situations, namely when the matrix satisfies some decay condition. The first results of this kind were obtained in [1]. This type of problem arises naturally in frame theory and in sampling theory [1]. More generally matrices with certain decay far from the diagonal have been extensively studied over the last 20 years (see for instance [3,17,8,9,23]). It has applications in various fields of analysis, such as pseudo-differential operators [20,12], numerical analysis [6,21,22], wavelet analysis [17], time-frequency analysis [13,10,11], sampling [1,7,11], and Gabor frames [2,7,20]. 1.1. Left-invertibility of thin–sparse operators Recall that a discrete metric space X is called doubling with doubling constant D if for all r > 0 and x ∈ X, V (x, 2r) DV (x, r), where V (x, r) denotes the cardinality of the closed ball of radius r. Examples of doubling metric spaces are Zn , and more generally groups with polynomial growth. Recall that a countable group G has polynomial growth if for every finite subset U ⊂ G, there exist C = C(U ) and d = d(U ) such that |U n | Cnd . By a deep theorem of Gromov [15], a finitely generated group G has polynomial growth if and only if has a nilpotent normal subgroup of finite index. It then follows from [16] that there exists an integer d = d(G) such that for all finite symmetric generating subset U of G, there exists C = C(U ) such that C −1 nd U n Cnd . As a result, the group G, equipped with the word metric dU (g, h) = inf{n ∈ N, g −1 h ∈ U n } is a doubling metric space. Given a doubling metric space X and a countable set Y , we consider an operator A = (ay,x )(y,x)∈Y ×X , bounded on 2 , whose rows are supported in balls of bounded radius (i.e. are thin), and whose columns have only a bounded number of non-zero entries (i.e. are sparse): we call such a matrix thin–sparse. Our first main result states that if A is bounded below in p for some 1 p ∞, then, B = (A∗ A)−1 A∗ defines a left-inverse for A, which is uniformly bounded on q for q ∈ [1, ∞]. Theorem 1.1. Let X be a doubling metric space and let A = (ay,x )(y,x)∈Y ×X be thin–sparse matrix with bounded coefficients. Then, • either λp (A) = 0, for all 1 p ∞,
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• or there exists C < ∞, such that B = (A∗ A)−1 A∗ satisfies Bp→p C, for all 1 p ∞, and hence defines a left-inverse for A. Remark 1.2. Note that for a matrix A whose rows have bounded support, a uniform bound on the coefficients is equivalent to the fact that A is bounded in ∞ . So, if A is bounded in p for some 1 p ∞, as in particular its coefficients are bounded, it is also bounded in ∞ . Hence by interpolation, it is bounded for all p q ∞. We shall discuss the optimality of this result latter in Section 1.4. One can actually drop the assumption of sparseness on the columns of A, and obtain the following stronger statement (indeed Theorem 1.1 follows by taking p < 1 in the following theorem). Say that a matrix (ay,x )(y,x)∈Y ×X is thin-Ø if rows are thin, i.e. supported on balls of bounded radius (and no assumption is made on columns). Theorem 1.3. Let A = (ay,x )(y,x)∈Y ×X be a thin-Ø matrix. Assume moreover that A is bounded as an operator p (X) → p (Y ) for some 0 < p < ∞ (equivalently bounded on q for all p q ∞). Then, • either λq (A) := inf
f =0
Af q =0 f q
whenever p < q ∞ and q 1; • or there exists c > 0, such that λq (A) c, if max(p, 1) q ∞. In the latter case, if p 2, then B = (A∗ A)−1 A∗ defines a leftinverse for A, which is uniformly bounded on q for max(p, 1) q p/ max(p, 1) − 1 . The conclusion of Theorem 1.3 is optimal as one can easily construct for every 1 p ∞ a matrix A = (ay,x )y,x∈N with one non-zero coefficient in each row and such that • A is bounded in q , for q p, • λp (A) > 0, • λq (A) = 0 for all p < q ∞. To see this, consider a matrix such that the nth column contains exactly n non-zero coefficients equal to n−1/p , such that the columns are piecewise orthogonal (i.e. have disjoint supports).
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Remark 1.4. Theorem 1.1 has been proved recently [1] for slanted matrices: let α ∈ R∗ , a matrix (ay,z )y,z∈Zd is called α-slanted if its support in Zd × Zd lies at bounded distance from the subspace of Rd × Rd defined by {(x, y) ∈ Rd × Rd , y = αz}. Although our proof is clearly different from the one of [1], both approaches share an important idea which consists in restricting A to functions supported in balls of radius L. This reduces the problem to dimension Ld , which enables us to use quantitative comparisons between p norms, before letting L go to infinity. Precisely, we prove the following fact which might be of independent interest (see Theorem 4.1 for a more general statement). Theorem 1.5. Let X be a doubling metric space, and let A = (ay,x )(y,x)∈Y ×X be a thin-Ø matrix. Assume that the matrix |A| = (|ay,x |)y∈Y,x∈X defines a bounded operator p (X) → p (Y ), for some 1 p ∞. Then, there exist C1 and C2 such that for all L 1, there is a non-zero function h supported in a ball of radius L such that for all p q ∞, Ahq C2 C1 λq (A) + . hq L (C1 only depends on the space X, and for X = Z, we can take C1 = 6. But C2 also depends on |A|p→p .) The estimate in O(1/L) for the error term is optimal as one can easily check with A = 1 − P , where P is2 the convolution by the normalized characteristic function of {−1, 1}, acting on p (Z). 1.2. Application to Schur operators We are able (see Theorem 6.2) to extend Theorem 1.1 in a way to include all matrices which can be approximated in a suitable sense by thin–sparse matrices. Here, we only focus on a special case, i.e. where X = Y and where the matrices can be approximated by banded ones. We will say that a matrix (ax,y ) indexed by a metric space X is N -banded (or has propagation N ) if ax,y = 0 as soon as d(x, y) > N . We will denote by A the algebra of Schur operators. Recall a Schur operator on 2 is an 1 ∞ operator which is bounded both on and on , its Schur norm being defined as AA = A1→1 + A∞→∞ = supi j |ai,j | + supj i |ai,j |. Theorem 1.6. Let X be a doubling metric space, and let A = (ax,y ) be a Schur matrix indexed by X such that there exists a sequence of r-banded matrices Ar such that r→∞
r t · A − Ar A −→ 0, for some t > 0. Then the following are equivalent: • A is bounded below for some 1 p ∞, 2 Note that P is the diffusion operator associated with the simple random walk on Z.
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• A is bounded below for all such p, • B = (A∗ A)−1 A∗ defines a left-inverse of A lying in A. The first notion of weighted Schur algebra has been introduced in [14], and then generalized in [24]. Following [24, Section 2.2], if X is a metric space and ω : X × X → [1, ∞) is an admissible weight in the sense of [14] or of [24], then we can define the weighted Schur algebra Aω (X) as the space of operators which are bounded for the norm AA,ω = sup x
ω(x, y)|ax,y | + sup y
y
ω(x, y)|ax,y |.
x
Typical admissible weights are ω(x, y) = 1 + d(x, y)α , for α 0, and ω(x, y) = exp Cd(x, y)δ , for some C > 0, and 0 < δ < 1. Since the notion of admissible weight is very technical, and will never be used here, we will not recall it (or else, we suggest the reader to consider the two previous typical examples as a definition of admissible weights since they both satisfy the conditions of [14] and [24]). Corollary 1.7. Let X be a doubling metric space, and let ω be an admissible weight such that ω(x, y) d(x, y)α for some α > 0. Then the following are equivalent: • A is bounded below for some 1 p ∞, • A is bounded below for all such p, • B = (A∗ A)−1 A∗ defines a left-inverse of A lying in Aω (X). Proof. First an easy observation shows that the matrices AN obtained naïvely by replacing all coefficients ax,y , where d(x, y) > N by zeros satisfy the hypothesis of Theorem 1.6. The last statement follows from Theorem 1.6, together with the facts that Aω (X) is an involutive algebra, and is spectral (or inverse-closed), which are both proved in [14,24] (for different types of weights). 2 1.3. Application to the class of convolution-dominated operators Let G be a discrete group. Recall the Gohberg–Baskakov–Sjöstrand class [24] (also called the convolution dominated operators class [9]) C(G) is the set of all operators on 2 (G) which are bounded for the following norm AC (G) =
sup |ag,h |.
−1 k∈G g h=k
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Let ω be an admissible weight. We shall also suppose that ω is left-invariant, i.e. satisfies3 ω(gk, gh) = ω(k, h) for all g, h, k ∈ G. Following [9], one can define the weighted convolution dominated algebra, comprising all matrices A which are bounded for the following norm ACω (G) =
sup ω(g, h)|ag,h |.
−1 k∈G g h=k
Theorem 1.8. Let G be a group with polynomial growth, and let ω be an admissible left-invariant weight such that ω(g, h) d(g, h)α for some α > 0. Then the following are equivalent: • A is bounded below for some 1 p ∞, • A is bounded below for all such p, • B = (A∗ A)−1 A∗ defines a left-inverse of A lying in Cω (G). The proof is completely similar to that of Theorem 1.7 using the fact, proved in [9] (see also [24] for a weaker statement) that Cω (G) is a spectral involutive algebra for all admissible weight. In [1, Theorem 2.3], this theorem was proved for G = Zd , under the stronger assumption that α > (d +1)2 . It turns out that our condition on the weight, namely α > 0 is not optimal. Indeed, in a very recent paper, Shin and Sun managed to prove the above theorem for any admissible weight when G = Zn [19]. We believe that their proof should also work for a group with polynomial growth, although this remains to be checked carefully. Finally, let us mention that even in the context of convolution operators on a group of polynomial growth, the above theorem is new, and has the following application. In view of [5, Theorem 4.3], we obtain: Corollary 1.9. Let G be a group with polynomial group, and suppose that an element A ∈ CG is bounded below in p for some 1 p ∞, then A is invertible in B(q (G)) for all 1 q ∞. 1.4. Optimality of the assumptions of Theorem 1.1 and Corollary 1.7 There are two natural questions arising from Corollary 1.7. Namely, can we relax, or simply drop one of the two main assumptions: the doubling condition on the space X, and the strict polynomial decay of the coefficients? First, Corollary 1.7 cannot be extended to the unweighted Schur algebra A since we exhibited in [25] a matrix in A which is bounded below in 2 but not in ∞ . As Nigel Kalton pointed to me, this fact is actually well-known amongst interpolation theoretists. An easy example is A = I − D, where D is the dilation operator on 2 (N), i.e. D(a0 , a1 , . . .) = (a0 /2, a0 /2, a1 /2, a1 /2, . . .). Note that the operator A∗ = 1 − D ∗ is invertible in 2 but not left-invertible in 1 . Indeed, the sequence of normalized characteristic functions φn = 1[0,n−1] /n satisfies A∗ φn 1 → 0. One can extend this idea to get examples which are not left-invertible in p for 1 < p < 2, by replacing √ D by λD, where 1 < λ < 2. 3 Observe that the two typical classes of weights defined at the previous subsection are indeed left-invariant, when defined with a left-invariant metric.
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Note that these examples do not exhibit any decay at infinity. On the other hand, the example given in [25] is a banded matrix4 indexed by the vertex set of the 3-regular tree T . Therefore it belongs to Aω (T ) for any weight ω on T . Hence, it gives a partial answer to the question of whether the metric space is required to be doubling or not. Actually, it is easy to see that T has exponential growth, and therefore does not satisfy the doubling condition. Moreover, as we will see below, T is a key example among those spaces.5 Note that a matrix indexed by T can be easily “extended” to a matrix indexed by X still satisfying the properties we are interested in. This provides a wide class of examples of metric spaces for which Corollary 1.7 (and actually even Theorem 1.1 for banded matrices) fails to be true. For instance, this excludes any metric space which is the vertex set of some non-amenable k-regular graphs. Those are graphs satisfying an isoperimetric inequality |∂A| c|A|, for every finite subset A of vertices of the graph, where c is some positive constant. The boundary ∂A denotes the set of edges joining vertices of A to its complement. Indeed, by the main result of [4], such a graph admits a bi-Lipschitz embedded 3-regular tree. Most known finitely generated groups have exponential growth, and among them, a large class have been shown to admit a Lipschitz embedded copy of T : this comprises by the previously mentioned result the huge class of non-amenable groups, while for instance Rosenblatt [18] proved it for non-virtually nilpotent solvable groups, which form a large class of amenable groups with exponential growth. However, there is still an interesting question which remains open: sticking to matrices indexed by Z for instance, does the conclusion of Corollary 1.7 hold for – say – logarithmic decay? 1.5. About the proofs The proofs of Theorem 1.1 and of its variants split into two main parts. First, we need to show that if A is bounded below for some p, then it is uniformly bounded below in q for all q’s. The second part of the proof consists in showing that the left-inverse exists and is uniformly bounded in p for all p’s. Let us now explain how the second part follows from the first one. We will deduce it from the following elementary observation. Proposition 1.10. Let X and Y be two sets, and let A be an operator 2 (X) → 2 (Y ) such that A and A∗ are uniformly bounded in p for all 1 p ∞. We have • λ2 (A∗ A) = λ2 (A)2 , • if A is self-adjoint and λp (A) > 0, for all 1 p ∞, then A is invertible in p , and A−1 p = 1/λp (A). Proof. The first statement simply follows from λ2 A∗ A = inf A∗ Af, f = inf Af 22 = λ2 (A)2 . f 2 =1
f 2 =1
4 Indeed, the operator considered in [25] is a symmetric element of the group algebra of the free group with two generators F2 seen as a convolution operator on p (F2 ). 5 Indeed, it is an open question whether a discrete metric space X with exponential growth admits a Lipschitz embedded copy of T .
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To show the second statement, observe that since A is self-adjoint, λ2 (A) > 0 implies that A is invertible in 2 . Hence, A−1 is defined on p (Y ) ∩ 2 (Y ) which is dense in p (Y ) for all p. But then λp (A) = =
inf
Af p f p
inf
f p A−1 f p
f ∈p (Y )∩2 (Y )
f ∈p (Y )∩2 (Y )
= 1/ A−1 p→p . So the proposition is proved.
2
To fix the ideas, let us focus on the second statement of Theorem 1.1, assuming the first statement. If λp (A) c > 0 for all 1 p ∞, then in particular, this is true for p = 2. So λ2 (A∗ A) c2 , which implies that A∗ A is invertible. But λ2 (A∗ A) > c2 , and by Proposition 3.2, A∗ A is banded. So by the first statement of Theorem 1.1 applied to A∗ A, there exists c > 0 such that λp (A∗ A) c for all 1 p ∞. Finally as (A∗ A)−1 p = 1/λp (A∗ A) 1/c , we conclude that B = (A∗ A)−1 A∗ satisfies Bp A∗ p /c , which is bounded independently of p. Remark 1.11. Note that the fact that the left-inverse A∗ (A∗ A)−1 is uniformly bounded in p for all p is also an immediate consequence of the fact that (A∗ A)−1 lies in the Schur algebra [14,24]. Let us now summarize the first part of the proof of Theorems 1.1, 1.3. Let us assume that λp0 > 0 for some 1 p0 ∞. In views of Proposition 1.10, we only need to show that λp > 0 for all p. 1. The first step, Theorem 1.5, is the central part of this paper (see Section 4). We show that the doubling property can be used to approximate the p -norm of a function f by taking the norm of its projection over a subset consisting of a union of distant balls of fixed radius. However, the naive idea consisting in applying A directly to this projection would only yield an error term in L1/p , which would not enable us to deduce anything from the statement that λ∞ (A) > 0 (but would work for any p < ∞). Instead, we multiply f by a certain Lipschitz function which is also supported on a union of distant balls. 2. To obtain the uniform lower bound for λq (A), using Theorem 1.5 is quite technical but the general idea is easy to understand: Theorem 1.5 says that we can approximate λq (A) by Ah quotients of the form hqq , where h are supported in balls of radius L (hence, restricting to subspaces of dimension ≈ v(L) which is roughly less than Ld for some d), and the error that we make is roughly in 1/L. Comparing these quotients for different values of q (and the same function h), we multiply our error term by Ld|1/p−1/q| . The resulting error term will therefore go to zero if p and q are close enough, namely if d|1/p − 1/q| < 1. Then, we just need to “propagate” the comparison that we get between λp (A) and λq (A) to obtain a uniform lower bound. Note that similar ideas are used in [1,19].
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3. Then, we extend Theorem 1.1 to operators that are somehow “polynomially approximated” by thin–sparse operators: we call them almost thin–sparse operators (see Section 6). The idea of the proof is very similar to step 2 (see Lemma 6.4). 4. The proof of Theorem 1.3 essentially consists in showing that a thin-Ø operator which is bounded in p , is almost thin–sparse in q for all q > p, which is easily checked. 2. Notation for thin–sparse operators In all the sequel, X and Y are discrete metric spaces with bounded geometry (balls of radius r have less than v(r) elements, for a given function v). However, in the definition of thin–sparse operators, only X needs a structure of metric space (Y can be any set). Let Cc (X) be the space of finitely supported real-valued functions on X. Let A be a linear map from Cc (X) to R Y . The kernel (also called the matrix) of A, (ay,x )(y,x)∈Y ×X is defined by the relation Af (y) =
ay,x f (x),
x∈X
for every f ∈ Cc (X). Conversely a matrix, i.e. a family of reals (ay,x )(y,x)∈Y ×X defines a linear morphism by the same formula. The row of index y ∈ Y of A is the vector (ay,x )x∈X of RX . The column of index x ∈ X of A is the vector (ay,x )y∈Y of RY . The support of A is the subset of Y × X on which ay,x = 0. We define similarly the support of a row or of a column of A. Notation 2.1. If the rows of a matrix A = (ay,x )y∈Y satisfy some property “P”, and if its columns satisfy some property “Q”, we will say that “A is P-Q”. If we make no assumption on the columns, we will say that A is P-Ø, and so on. We will consider two properties for the rows or the columns: • We say that the rows (or the column) of A are thin, of thickness at most r if their support are contained in balls of radius r. • We say that the rows (or the columns) are sparse, of sparseness at most v if their support has cardinality at most v. • We denote by T S(X, Y ) (resp. ST (X, Y ), T (X, Y ), ØT (X, Y ) and T Ø(X, Y )) the space of thin–sparse (resp. sparse–thin, thin–thin, Ø-thin and thin-Ø) operators. As the spaces have bounded geometry, sparse is a weaker condition than thin. Hence sparse– sparse is weaker than thin–sparse, which is weaker than thin–thin, etc. Remark 2.2. A particular case of thin–thin matrices (when X = Y ) are matrices for which the support is contained in {(y, x) ∈ X 2 , d(x, y) r} for some r > 0. Such matrices are sometimes called banded, or with finite propagation. Notation 2.3. • For all 1 p ∞, the norm of an operator A : p (X) → p (Y ) is called the p -norm of A and is denoted by Ap→p .
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• Let A = (ay,x )(y,x)∈Y ×X . The absolute value of A is operator |A| = (|ay,x |)(y,x)∈Y ×X . • We say that A is absolutely uniformly bounded if sup |A|p→p < ∞. 1p∞
3. Preliminary remarks about thin–sparse operators 3.1. Combinatorial properties The following easy fact is a crucial property of TS operators. We say that two subsets U and V of a metric space are t-disjoint if d(x, y) > t for all (x, y) ∈ U × V . Proposition 3.1. Let X be a metric space, and Y be a set. Let A be a thin-Ø operator of thickness r and let v and u be two functions on X whose supports are 2r-disjoint. Then, Au and Av (which are well-defined functions) have disjoint support. Proof. We just have to consider a row L of A and to prove that L, u = 0 implies L, v = 0. But this is a trivial consequence of the fact that L is supported in a ball of radius r, which has diameter 2r, and that the supports of u and v are at distance > 2r. 2 The following proposition is straightforward and left as an exercise. Proposition 3.2. Let X be a metric space and let Y be a set. If A ∈ T Ø(X, Y ) then A∗ A (when it exists) is banded. 3.2. Norms of sparse–sparse operators are equivalent Proposition 3.3. A sparse–sparse operator A is absolutely uniformly bounded, if and only if it is bounded in p for some 1 p ∞, if and only if it has bounded coefficients. Proof. Let X and Y be two sets and let A = (ay,x )(x,y)∈X×Y be a sparse–sparse operator of sparseness v. Note that the norm A∞ = sup(y,x)∈Y ×X |a(y, x)| is trivially less than all operator norms. Hence it is enough to prove that for every 1 p ∞, Ap→p CA∞ for some C depending only on v. Fix y ∈ Y , and let Sy be the support of the corresponding row (ay,x )x∈X . For every f ∈ Cc (X), Af (y) = A∞ f (x). a f (x) y,x x∈X
x∈Sy
Hence, using Hölder’s inequality and the majoration |Sy | v for all y ∈ Y , we obtain p
p
Af p A∞
p f (x) y∈Y
p
A∞
y∈Y
x∈Sy
v p−1
f (x)p .
x∈Sy
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Now, note that for every x ∈ X and every k ∈ N, f (x) appears k times in the sum above if there are k distinct elements of Y , y1 , . . . , yk such that x ∈ Sy1 ∩ · · · ∩ Syk , hence if y1 , . . . , yk lie in the support of the column (ay,x )y∈Y . But as the sparseness of A is at most v, this implies that k v. Therefore, we have p
A∞
v p−1
y∈Y
f (x)p v p Ap∞ f (x)p x∈Sy
x∈X
=v
p
p p A∞ f p .
2
4. Proof of the approximation property Recall that a discrete metric space X is said to be doubling of doubling constant C < ∞ if for all x ∈ X and every r > 0, B(x, 2r) C B(x, r). Our purpose in this section is to prove the following theorem Theorem 4.1. Assume that X is a doubling metric space and let A ∈ T Ø(X, Y ) of thickness r, such that |A|p→p 1 for some 1 p < ∞. There exists C such that for every f ∈ Lp (X), and every L r, there exists a function h ∈ Lp (X) supported in a ball of radius 2L such that
Ahp Af p r , C + hp f p L where, the quantity C only depends on the doubling constant of X. 4.1. Coloring of a family of balls Recall that a d-coloring of a set P of subsets of X is a map j : P → {1, 2, . . . , d + 1} such that every two elements in P with the same color (i.e. same image by j ) are disjoint. Also classical is the notion of coloring of a graph: a d-coloring of a graph G is a map j : V (G) → {1, 2, . . . , d + 1}, where V (G) is the vertex set of G, such that any two adjacent vertices have distinct colors. A classical result of graph theory, known as Brooks’ theorem says that any graph of degree at most d admits a d-coloring. It tuns out that these two definitions of coloring are related via the notion of dual graph. Recall that the dual graph G of P is defined as follows: the set of vertices V (G) is P, and two vertices are adjacent if and only if they have a non-empty intersection. Clearly, a d-coloring of G yields a d-coloring of P and conversely. We will need the following lemma.
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Lemma 4.2. Let X be a doubling metric space and let α 1. There exists an integer d such that for every L > 0, there exists a covering of X by balls of radius L admitting a d-coloring such that the centers of two balls of same color are at distance αL from one another. Proof. Consider a minimal covering B = (B(xi , L))i of X (which exists since X is doubling). By minimality, the balls B(xi , L/4) are piecewise disjoint. Now, consider the covering B = (B(xi , αL))i . It is easy to see that the doubling property implies that the dual graph of B has degree less than a certain constant d. Indeed, for every i, let di be degree at the vertex i of the dual graph. In other words, di is the number of balls B(xj , αL) with j = i, intersecting B(xi , αL). Let Ji be the set of such indices. Note that the disjoint union j ∈Ji B(xj , L/4) is contained in B(xi , 4αL). On the other hand, by the doubling property, there exits c > 0 only depending on α such that infj ∈Ji V (xj , L/4)/V (xi , 4αL) c. But since di inf V (xj , L/4) V (xi , 4αL), j ∈Ji
we deduce that di 1/c, so that we can set d = [1/c]. Hence, by Brooks’ theorem, this graph admits a d-coloring, which means that B has a dcoloring. Inducing this coloring to B yields the desired d-coloring. 2 4.2. Approximating a function by a function supported by a disjoint union of balls of fixed radius In the following lemma we characterize the doubling condition in terms of approximation of functions by functions supported by disjoint unions of balls of fixed radius. For every subset Ω of a metric space X and every L > 0, we denote
[Ω]L = x ∈ X, d(x, Ω) L . We also denote the characteristic function of a subset Ω by 1Ω . Finally, a K-separated subset of X is a subset whose elements are pairwise at distance at least K. Lemma 4.3. A metric space X is doubling if and only if for every α 1, there exists a constant c > 0 such that for every 1 p ∞, every f ∈ p (X) and every L > 0, one can find an αLseparated subset P of X such that 1[P ]L f p cf p . Proof. Consider the covering B of the previous lemma and for every 1 k d + 1, let Pk be the set of centers of balls of B with same color k. Since X = d+1 k=1 [Pk ]L , we have 1 |f | 1[Pk ]L f p (d + 1) max 1[Pk ]L f p . f p [Pk ]L k
p
k
k
So Lemma 4.3 follows taking P = Pk with a k for which the max is attained. The converse follows by taking f to be the characteristic function of a ball of radius 2L and α 6, so that the intersection between [P ]L and our ball of radius 2L is contained in a single ball of radius L. 2
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In the sequel, we fix α = 6. The following lemma is trivial and left to the reader. Lemma 4.4. For each P like in the previous lemma, the function P , defined by
P (x) = max 0, 1 − d(x, P )/(2L) , satisfies: 1. 2. 3. 4.
P = 0 outside of [P ]2L .
P 1/2 on [P ]L .
P is 1/(2L)-Lipschitz. 0 P 1.
Remark 4.5. Keeping the notation of the previous lemmas, the function g = P f satisfies, thanks to the second property of P and to Lemma 4.3, gp cf p . On the other hand, the support of g is contained in a union of 4L-disjoint balls of radius 2L. Write g = i gi , where each gi is supported in one of those balls. Assume that 4L 2r. Then by Proposition 3.1, p p Agi p . Agp = i
So we have inf i
Agp Agi p . gi p gp
Proof of Theorem 4.1. Thanks to the previous remark, we just need to prove a weaker version of the theorem where in the conclusion, the function h is replaced by a function g supported in a union of 2r-disjoint balls of radius 2L. We consider g = P f , which has this property since L r. Let us start with a pointwise estimate. Fix some y0 ∈ Y . For every x, z ∈ X, g(x) = P (x)f (x) = P (z)f (x) + P (x) − P (z) f (x). We now specify z = x0 , such that the support of the row (ay0 ,x )x is contained in B(x0 , r). We have Ag(y0 ) = P (x0 ) ay0 ,x f (x) + ay0 ,x P (x) − P (x0 ) f (x). x
x
So by property 4 of P , Ag(y0 ) Af (y0 ) + |ay
0 ,x
x
| P (x) − P (x0 )f (x).
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By property 3 of P , Ag(y0 ) Af (y0 ) + r|A||f |(y0 ) . L Now, taking the p norm and applying the triangular inequality, we obtain Agp Af p +
r|A||f |p . L
We finally divide by gp , and conclude thanks to the inequality gp cf p .
2
Remark 4.6. For X = Z, we have v(k) = 2k + 1, and the doubling constant is less than 2. Note that we can take P = {x0 + 6kL, k ∈ Z} for some x0 . Moreover, one checks easily that a good choice of x0 gives 1P f p f p /3. Now assume that A is thin–thin of thickness r. By the proof of Proposition 3.3, we have |A| v(r)A∞ . Hence, we obtain that there exists a function h supported on a ball of radius r such that
Ahp Af p 3r 2 A∞ . 3 + hp f p L 5. p -Stability of thin–sparse operators Here is a more general version of Theorem 1.1, with some precisions that we omitted in the introduction. Theorem 5.1. Let X be a metric space of doubling constant D < ∞ and let Y be any set. Fix some r, v > 0. Let A ∈ T S(X, Y ) be of thickness at most r, sparseness at most v. Assume moreover that |A|p→p 1 for all 1 p ∞. Then there exist c = c(r, v, D) > 0 and δ = δ(D) > 0 for all 1 p, q ∞, λp (A) cλq (A)δ . In Section 6, we prove that the conclusion Theorem 5.1 is true for more general operators which are “well” approximated by thin–sparse and thin-Ø operators respectively. Theorem 5.1 (and the remark following Theorem 7.1) result from the following more precise results. Let λ=
inf
λp (A),
sup
λp (A),
p0 p∞
and let pm be such that λpm λ/2. Let Λ= and let pM be such that λpM 2Λ.
p0 p∞
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Note that since X is doubling, there exist d and K such that V (x, R) KR d for all x ∈ X and R > 0. Theorem 5.2. Let A ∈ T S(X, Y ) of thickness r, sparseness v and such that |A|p→p 1 for all p0 p ∞. Then there exists k = k(v, r, d) > 0 such that λ kΛ4d . Theorem 5.3. Let A ∈ T Ø(X, Y ) of thickness r and such that |A|p→p 1 for all p0 p ∞. Then there exists k = k(r, d) > 0 such that for all p0 p q ∞, λp kλ4d q . These theorems will be proved after a series of lemmas. Lemma 5.4. Fix some 1 p0 < ∞. Let A ∈ T Ø(X, Y ) of thickness r and such that |A|p→p 1 for all p0 p ∞. (i) there exist d > 0 and C (depending on the doubling constant) such that for all p0 p q ∞ and all L r, λq (A) C L
| pd − dq |
λp (A) + r/L .
(ii) if moreover, A ∈ T S(X, Y ) of sparseness v, then for all p0 q p ∞, 1 1 d d |p−q | |p−q |
λq (A) C v
L
r λp (A) + . L
Proof. Theorem 4.1 implies
Ahp r . inf C λp (A) + Supp(h)⊂B(x,2L) hp L On the other hand, if h is supported in a subset of size N , then for p q, hq hp N
| p1 − q1 |
hq .
(5.1)
The power in L appearing in the inequalities now comes from the inequality V (x, L) KLd . Indeed, if p q, then we obtain (i) applying the left inequality of (5.1) to Ah (where the support of Ah does not play any role) and the right inequality to h, whose support has cardinality at most KLd . So take C = CK. If p q, then we apply the right inequality of (5.1) to Ah for which we control the support thanks to the sparseness of A’s columns. Namely, the cardinality of the support of Ah is at most v times the cardinality of h’s support. This explains the corresponding power of v in (ii). 2 Lemma 5.5. Let A ∈ T Ø(X, Y ) of thickness r and such that |A|p→p 1 for all p0 p ∞.
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(i) For all p0 p ∞, λp (A) = 0 implies λq (A) = 0 for all q p. (ii) Let K be twice the constant C of Lemma 5.4. Then, for all p0 p q ∞, λq (A) Kr
| pd − dq |
1−| pd − dq |
λp (A)
.
Lemma 5.6. Let A ∈ T S(X, Y ) of sparseness v and thickness r, and such that |A|p→p 1. Then, for all p0 p ∞: (i) For every p0 p, q ∞, λp (A) = 0 if and only λq (A) = 0. (ii) Let K be twice the constant C of Lemma 5.4. For all p0 p, q ∞, λq (A) Kv
| p1 − q1 | | pd − dq |
r
1−| pd − dq |
λp (A)
.
Proof. Both lemmas are proved in the same way: so let us show Lemma 5.6. To obtain (ii), take L = r/λp (A) in Lemma 5.4. To prove (i), we just have to note that the vanishing of λp (A) “propagates” thanks to Lemma 5.4: λp (A) = 0 ⇒ λq (A) = 0 if | dq − pd | 1/2 (let L → ∞). 2 Proof. To show Theorems 5.2 and 5.3, we “propagate” the inequalities (ii) of Lemmas 5.5 and 5.6. As the proofs are the same for both theorems, let us focus on the first one. If | pd − dq | 1/2, the inequality (ii) of Lemma 5.6 yields λp (A) C(v, r, d)λq (A)2 . Now, as | pdm −
d pM | d,
we just need to iterate this 2d times, which gives the theorem.
2
Remark 5.7. Here, assume that X = Y = Z, and that A is thin–thin of thickness r. Instead of assuming that |A| = 1, we prefer to write Lemma 5.6 with respect to A∞ (which is easier to compute in general): a consequence is that we have to replace r by 3r 3 A∞ . From Remark 4.6 that we can take C = 9 in Lemma 5.4 (as v(r) 3r). Hence we can take K = 18. Directly from Lemma 5.6(ii), we obtain that λ2 (A)
Λ2 . 162r 3 A∞
6. Extension to (t, s)-almost thin–sparse operators Definition 6.1. Fix some t, s > 0 and some 1 p ∞. An operator is (t, s)-almost thin–sparse for in q for all q p if there exists K < ∞ such that for all r, v > 0, there is an element Ar,v ∈ T S(X, Y ) of thickness r and sparseness v such that |A−Ar,v |q→q K(r −t +v −s ) for all q p. This section is devoted to the proof of the following result. Theorem 6.2. Fix some t, s > 0 and some 1 p0 ∞. Let X be a metric space with the doubling property, and let Y be any set. Let A be (t, s)-almost thin–sparse in p for all p p0 . Then either λp (A) = 0 for all 1 p0 p ∞, or there exists c > 0 such that λp (A) > c for all p0 p ∞.
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This will result from the following analogue of Theorem 5.2 for (t, s)-almost thin–sparse operator. Theorem 6.3 will also be used in the proof of Theorem 7.1. Theorem 6.3. Fix some t, s > 0 and some 1 p0 ∞. Let X be a doubling metric space with doubling constant D, and let Y be any set. Let A be (t, s)-almost thin–sparse in p for all p p0 . Then there is c = c(D, t, s) > 0, and δ = δ(D, t, s) > 0 such that for all p0 p, q ∞, λp (A) cλq (A)δ . In the sequel, a b will mean a Cb, where C = C(D, t, s). Proof. First, we need the analogue of Theorem 4.1. Lemma 6.4. For all p0 p ∞, f ∈ Lp (X), all L 1 and r, v > 0, there exists a function h ∈ Lp (X) supported in a ball of radius 2L such that Ahp Af p r + + r −t + v −s . hp f p L Proof. This is immediate, writing A = Ar,v + (A − Ar,v ) where Ar,v is thin–sparse of thickness r and sparseness v, and using |A − Ar,v |p K(r −t + v −s ). 2 Then we need the analogues of Lemmas 5.4, 5.5 and 5.6. Lemma 6.5. For all p0 p, q ∞, and for all L 1 and r, v > 0, λq (A) v
| p1 − q1 | | pd − dq |
L
r −t −s λp (A) + + r + v . L
Proof. This is proved exactly as we proved Lemma 5.4.
2
Lemma 6.6. There exists u = u(D, s, t) such that for all p0 p, q ∞, 2d 2d 1−| up − uq |
λq (A) λp (A)
. −1/u
Proof. The proof follows by choosing in the previous lemma, r = L1/2 , v = Ld , and L = λp where u = min{1/2, t/2, sd}. 2
,
The proof of Theorem 6.3 now relies on an argument of propagation similar to the one used in the proof of Theorem 5.2. 2 7. Left-invertibility of thin-Ø-operators Theorem 7.1. Let X be a metric space of doubling constant D < ∞ and let Y be any set. Let A = (ay,x )(y,x)∈Y ×X be a thin-Ø matrix. Assume moreover that A is bounded as an op-
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erator p0 (X) → p0 (Y ) for some 0 < p0 ∞. Then for every p1 > p0 , there exist c = c(p1 − p0 , r, D) > 0 and δ = δ(p1 − p0 , D) > 0 such that for all max{1, p1 } p, q ∞, λp (A) cλq (A)δ . Remark 7.2. Before proving the theorem, we point out that one cannot improve the theorem to have p0 = p1 . Indeed, in the spirit of the example explained in the introduction, for r = 1 and X = Y = Z, we can find a sequence of thin–sparse operators An = (ay,x )(y,x)∈Y ×X of thickness 1, sparseness n, and such that • An p0 →p0 = λp0 (An ) = 1 for all n ∈ N, • and λp (An ) → 0 when n → 0 for all p > p0 . On the other hand, it is interesting to note that (in virtue of Theorem 5.3) there exist c = c(r, D) > 0 and δ = δ (D) > 0 such that for all p0 p q ∞,
λp (A) c λq (A)δ . Theorem 7.1 results from Theorem 6.3 and from the fact that thin-Ø operators that are bounded in p are (1, 1/p − 1/q)-almost thin–sparse in q for all q > p. This is a consequence of the following proposition. Proposition 7.3. Let X = (X, d) be a metric space such that balls of radius r have cardinality at most v(r), and let Y be a set. Fix some ε > 0 and some r 1. Let A = (ay,x )(y,x)∈Y ×X be a thin-Ø operator of thickness r such that Ap→p = 1 for some 0 < p < ∞. Then, there is C = C(ε) such that for every q p + ε and every m ∈ N, there exists a thin–sparse operator Am of thickness r, sparseness m such that |A − Am |q→q
Cv(r)1−1/q . m1/p−1/q
Proof. First, let us prove the following lemma. Lemma 7.4. Let n be a positive integer, and 0 < an · · · a1 such that all 0 m n, and q p,
n
1/q q ai
i=m+1
n
p i=1 ai
(p/q)1/q (1 − p/q)1/p−1/q . m1/p−1/q
In particular, for every ε > 0 there exists C = C(ε) such that for all q p + ε,
n i=m+1
1/q q ai
C . m1/p−1/q
= 1, then for
(7.1)
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Proof of Lemma 7.4. Let us find the maximum of the function θm,q (a1 , . . . , an ) =
n
q
ai ,
m+1
under the conditions n
p
ai = 1,
i=1
and for all 1 i n − 1, ai+1 − ai 0. Claim 7.5. The maximum of θm,q is attained at (a1 , . . . , an ) such that ai = 0 for i k and ai = 1/k 1/p for i < k, where k is an integer m + 1. Proof of Claim 7.5. First, note that since (ai ) is non-increasing, the maximum will be attained when ai = aj for all i j m. On the other hand, a straightforward application of Lagrange multipliers shows that θm,q cannot reach its maximum at a point (a1 , . . . , an ) such that 0 < ai+1 < ai for some 1 i n − 1. Hence, if ai+1 < ai , then ai+1 = 0. There exists therefore only one such i. Let k := i + 1. Note that θm,q is not identically zero: hence, since the sequence (aj ) corresponds to a maximum of θm,q , k has to be m + 1. Summarizing this discussion, there exists k m + 1 such that the sequence ai = 0 for i k and ai = 1/k 1/p for i < k. 2 With the notation of the claim, we have max θm,q =
k−m . k q/p
(7.2)
with respect to k vanishes To finish the proof of Lemma 7.4, note that the derivative of k−m k q/p exactly at the value m/(1 − p/q), which corresponds to a maximum. Replacing k by this value in (7.2) yields (7.1). 2 Now, let us prove the proposition. As Ap→p = 1, for every x ∈ X, the column Cx = (ay,x )y∈Y has p -norm at most 1. By Lemma 7.4, there exists a subset Sx of Y of cardinality m such that
|ay,x |q C q /mq/p−1 .
y∈Y Sx
Now, we define Am from A by replacing the coefficient ay,x by 0 whenever y ∈ Y Sx . By construction, Am is thin–sparse of thickness r and sparseness m. Let f ∈ q (X). Denote by Cm = |A − Am | = (cy,x )(y,x)∈Y ×X . Using Hölder inequality (which is possible since q 1), we obtain
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q |A − Am |f q = c f (x) y,x q y∈Y
x∈X
v(r)q−1
y∈Y
q q cy,x f (x)
x∈X
f (x)q |ay,x |q = v(r)q−1 y∈Y Sx
x∈X
C q v(r)q−1 mq/p−1
q
f q .
2
Acknowledgments I am grateful to Akram Aldroubi, Ilia Krishtal, Qiyu Sun, Karlheinz Gröchenig for valuable discussions. I also thank Nigel Kalton for telling me his example of a matrix in A(N) which is invertible in 2 and not in 1 . I also thank Yemon Chu for pointing me his interesting paper [5], and for his remarks and corrections. References [1] A. Aldroubi, A. Baskarov, I. Krishtal, Slanted matrices, Banach frames and sampling, J. Funct. Anal. 255 (2008) 1667–1691. [2] R. Balan, P.G. Cassazza, C. Heil, Z. Landau, Density, overcompleteness and localization of frames, I. Theory; II. Gabor system, preprint, 2004. [3] A. Baskarov, Asymptotic estimates for elements of matrices of inverse operators, Siberian Math. J. 38 (1) (1997) 10–22. [4] I. Benjamini, O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant, Geom. Funct. Anal. 7 (1997) 403–419. [5] Y. Choi, Group representations with empty residual spectrum, Integral Equations Operator Theory 67 (1) (2010) 95–107, doi:10.1007/s00020-010-1772-0. [6] O. Christensen, T. Strohmer, The finite section method and problems in frame theory, J. Approx. Theory 133 (2005) 221–237. [7] E. Cordero, K. Gröchenig, M. Leinert, Localization of frames, II, Appl. Comput. Harmon. Anal. 17 (2004) 29–47. [8] G. Fendler, K. Gröchenig, M. Leinert, Symmetry of weighted L1 -algebras and the GRS-condition, Bull. Lond. Math. Soc. 38 (4) (2006) 625–635. [9] G. Fendler, K. Gröchenig, M. Leinert, Convolution-dominated operators on discrete groups, Integral Equations Operator Theory 61 (2008) 493–500. [10] K. Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003) 149–157. [11] K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004) 105–132. [12] K. Gröchenig, Time-frequency analysis of Sjöstrand’s class, Rev. Mat. Iberoamericana 22 (2) (2006) 703–724. [13] K. Gröchenig, M. Leinert, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, 2001. [14] K. Gröchenig, M. Leinert, Symmetry of matrix algebras and symbolic calculus for infinite matrices, Trans. Amer. Math. Soc. 358 (2006) 2695–2711. [15] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Etudes Sci. 53 (1981) 53–73. [16] Y. Guivarc’h, Croissance polynômiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973) 333–379. [17] S. Jaffard, Propriétés des matrices “bien localisée” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Lineaire 7 (5) (1990) 461–473. [18] J. Rosenblatt, Invariant measures and growth conditions, Trans. Amer. Math. Soc. 193 (1974) 33–53. [19] C.E. Shin, Q. Sun, Stability of localized operators, J. Funct. Anal. 256 (2009) 2417–2439.
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[20] J. Sjöstrand, Wiener type algebra of pseudodifferential operators, Centre de Mathematiques, Ecole Polytechnique, Palaiseau France, Seminaire 1994–1995, December 1994. [21] J. Stohmer, Rates of convergence for the approximation of shift-invariant systems in 2 (Z), J. Fourier Anal. Appl. 5 (2000) 519–616. [22] J. Stohmer, Four short stories about Toeplitz matrix calculations, Linear Algebra Appl. 343/344 (2002) 321–344. [23] J. Stohmer, Pseudo-differential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2) (2006) 237–249. [24] Q. Sun, Wiener’s lemma for infinite matrices, Trans. Amer. Math. Soc. 359 (7) (2007) 3099–3123 (electronic). [25] R. Tessera, The inclusion of the Schur algebra in B(2 ) is not inverse-closed, preprint, 2009.
Journal of Functional Analysis 259 (2010) 2814–2855 www.elsevier.com/locate/jfa
On differentiable vectors for representations of infinite dimensional Lie groups Karl-Hermann Neeb 1 Department Mathematik, FAU Erlangen–Nürnberg, Bismarckstrasse 1 1/2, 91054-Erlangen, Germany Received 9 February 2010; accepted 30 July 2010 Available online 17 August 2010 Communicated by P. Delorme
Abstract In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π : G → GL(V ) of an infinite dimensional Lie group G on a locally convex space V . The first class of results concerns the space V ∞ of smooth vectors. If G is a Banach–Lie group, we define a topology on the space V ∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V ∞ is a Fréchet space. This applies in particular to C ∗ -dynamical systems (A, G, α), where G is a Banach–Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function π(g)v, v is smooth. The second class of results concerns criteria for C k -vectors in terms of operators of the derived representation for a Banach–Lie group G acting on a Banach space V . In particular, we provide for each k ∈ N examples of continuous unitary representations for which the space of C k+1 -vectors is trivial and the space of C k -vectors is dense. © 2010 Elsevier Inc. All rights reserved. Keywords: Infinite dimensional Lie group; Representation; Differentiable vector; Smooth vector; Derived representation
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . Locally convex Lie groups . . . . . . . . . . . . Basic facts and definitions . . . . . . . . . . . . A topology on the space of smooth vectors .
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E-mail address:
[email protected]. 1 Supported by DFG-grant NE 413/7-1, Schwerpunktprogramm “Darstellungstheorie”.
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.020
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4.1. Distribution vectors . . . . . . . . . . . 5. Smooth vectors in Banach spaces . . . . . . . 6. An applications to C ∗ -dynamical systems . 7. Smooth vectors for unitary representations . 8. C 1 -vectors for Banach representations . . . . 9. C k -vectors for Banach representations . . . . 10. A family of interesting examples . . . . . . . 11. Smooth vectors for direct limits . . . . . . . . 12. Smooth vectors for projective limits . . . . . 13. Perspectives . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction Let G be a Lie group modeled on a locally convex space (cf. [43] for a survey on locally convex Lie theory). A representation π : G → GL(V ) of G on the locally convex space V is called continuous if it defines a continuous action of G on V . We call an element v ∈ V a C k -vector (k ∈ N0 ∪ {∞}) if the orbit map π v : G → V , g → π(g)v, is a C k -map and write V k = V k (π) for the space of C k -vectors (cf. Section 2 for precise definitions). It is a fundamental problem in the representation theory of Lie groups to understand the interplay between differentiability properties of group representations and representations of the Lie algebra g = L(G) of G. In particular, the following questions are of interest: (1) How to find C k -vectors? Can they be characterized in term of the Lie algebra? (2) Does the space V ∞ of smooth vectors carry a natural topology for which the action of G on this space is smooth? (3) Under which circumstances do differentiable, resp., smooth vectors exist, resp., form a dense subspace? In this note we provide answers to some of these questions for infinite dimensional Lie groups G. For finite dimensional groups most of these results are either trivial or well known (see [14] for (1), [32] for (2) and [10] for (3)), so that the main issue is to identify and deal with the subtleties of infinite dimensional Lie groups. After discussing some preliminaries on Lie groups and their representations in Sections 2 and 3, we turn in Section 4 to the space V ∞ of smooth vectors of the representation of a Banach–Lie group G on a locally convex space V . Here our main result is that, by embedding V ∞ into a product of the Banach spaces Multp (g, V ) of continuous p-linear maps gp → V , we obtain a locally convex topology on V ∞ for which the G-action on V ∞ is smooth (Theorem 4.4). For finite dimensional groups, this follows easily from the smoothness of the translation action of G on the space C ∞ (G, V ) (cf. Proposition 4.6), but for infinite dimensional groups there seems to be no “good” topology on the space C ∞ (G, V ). The smooth compact open topology is too coarse to ensure continuity of the action. For the Fréchet–Lie group G = RN and the unitary representation on H = 2 (N, C) defined by (π(x)y)n = eixn yn , there exists no locally convex topology on H∞ = C(N) for which the G-action is smooth. In view of this example, Theorem 4.4 on the smoothness of the action on V ∞ does not extend to unitary representations of
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Fréchet–Lie groups. The locally convex space V ∞ has a topological dual space V −∞ := (V ∞ ) , whose elements are called distribution vectors. For finite dimensional groups they correspond to equivariant embeddings of V into the space of distributions on G, and in our context they still parametrize equivariant embeddings V ∞ → C ∞ (G), where C ∞ (G) carries the left regular representation. In this sense the spaces V −∞ provide natural candidates for spaces of distributions on Banach–Lie groups. For discussions of distributions on infinite dimensional vector spaces, we refer to [27,26,25]. We continue the discussion of V ∞ in Section 5, where we show that if V is a Banach space, then V ∞ is complete, hence a Fréchet space. In Section 6 we briefly discuss an application of these results to C ∗ -dynamical systems (A, G, α), where G is a Banach–Lie group and A is a unital Banach algebra. Then the Fréchet space A∞ is shown to be a continuous inverse algebra, i.e., its unit group is open and the inversion is a continuous map. The main result of Section 7 is that, for a unitary representation (π, H), a vector v ∈ H is smooth if and only if the corresponding matrix coefficient π v,v (g) = π(g)v, v is smooth. Although smooth vectors form the natural domain for the derived representation of the Lie algebra g of G, in many situations it is desirable to have some information on C k -vectors for k < ∞. This motivates the discussion of C 1 -vectors for continuous representations of Banach– Lie groups on Banach spaces in Section 8. The main result is that the intersection Dg ⊆ V of the domains of the generators d π(x) of the one-parameter groups t → π(expG (tx)) on V coincides with the space of C 1 -vectors (Theorem 8.5). Here the main difficulty lies in the verification of the additivity of the map ωv : g → V , x → d π(x)v, for which we use quite recent refinements of Chernoff’s Theorem by Neklyudov [52]. If ωv is assumed to be continuous, its linearity is easily verified (Lemma 3.3). It is also easily verified for finite dimensional groups [15, p. 221]. All this is applied in Section 9 to C k -vectors, which are shown to coincide (for a representation of a Banach–Lie group on a Banach space) with the space Dgk , the common domain of k-fold products of the operators d π(x), x ∈ g (Theorem 9.4). Here an interesting point is that the C k concept we use is weaker than the C k -concept used in the context of Fréchet differentiable maps between Banach spaces. The discussion of the examples in Section 10 shows that, in general, the space of Fréchet-C k -vectors is a proper subspace of Dgk which can be trivial. A crucial problem one has to face in the representation theory of infinite dimensional Lie groups is that a continuous representation need not have any differentiable vector, as is the case for finite dimensional Lie groups [10]. Refining a construction from [3], we discuss in Section 10 the continuous unitary representation of the Banach–Lie group G = (Lp ([0, 1], R), +) on the Hilbert space L2 ([0, 1], C) by π(g)f = eig f and determine for every k ∈ N the space of C k vectors. For p = 1 we thus obtain a continuous representation with no non-zero C 1 -vector, and for p = 2k, the space of C k -vectors is non-zero, but there is no non-zero C k+1 -vector. The situation is much better for Lie groups which are direct limits of finite dimensional ones. For these groups strong existence results on smooth vectors are available ([62,7] for unitary representations and [65] for Banach representations), and we explain in Section 11 how they fit into our general framework. The density of H∞ for particular unitary representations of diffeomorphism groups is shown in [64]. In Section 12 we show that, for rather trivial reasons, the space of smooth vectors of a continuous unitary representation is also dense for projective limits of finite dimensional Lie groups. We conclude this paper in Section 13 with a discussion of some open problems. The class of groups for which the theory developed in this article has the strongest impact is the class of Banach–Lie groups and we hope in particular that our results on smooth vectors will lead to a better understanding of their unitary representations. To make this more concrete,
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we recall some of the major classes of Banach–Lie groups and what is known on their unitary representation theory. For a Banach–Lie group G, the most regular class of unitary representations are the bounded ones, i.e., those for which π : G → U(H) is a morphism of Banach–Lie groups with respect to the norm topology on U(H). It is easy to see that faithful representations of this kind exist only when L(G) carries an Ad(G)-invariant norm, a property which for finite dimensional Lie algebras is equivalent to compactness. If G has a universal complexification ηG : G → GC (see [16] for criteria on the existence), then every bounded unitary representation extends to a holomorphic representation π : GC → GL(H). Holomorphic representations of Banach–Lie groups have been studied systematically in the context of the large class of root graded Lie groups in [35] and [49]. The class of root graded Lie groups contains in particular analogs of classical groups over unital Banach algebras and the most typical example is GLn (A) for a unital Banach algebra A. In [35] we developed a Borel–Weil theory for such groups, showing in particular that representations in spaces of holomorphic sections of bundles over the natural analogs G/P of flag manifolds are always holomorphic representation by bounded operators and that all holomorphic irreducible representations are of this form. In this context the most intriguing problem is to decide for which holomorphic representations ρ : P → GL(V ), the associated holomorphic vector bundle has nonzero holomorphic sections. For root graded groups whose Lie algebra has the form g ⊗ A, g complex semisimple and A a unital commutative Banach algebra, this question has been answered completely in [49] for the special case of line bundles, i.e., ρ : P → C× is a holomorphic character. From this classification result it follows in particular that the spaces of holomorphic sections of these line bundles are always finite dimensional. As a special case, this theory leads to a complete classification of the bounded unitary representations of the unitary groups Un (A), where A ∼ = C(X) is a commutative unital C ∗ -algebra and all these representation factor through some quotient group Un (C)d . In [4] the Schur–Weyl theory of representations of Un (C) is extended to unitary groups U(A) of a C ∗ -algebra. It is shown that tensor products of irreducible representations of A and their duals decompose into finitely many bounded unitary representations of U(A) according to the classical Schur–Weyl pattern. If A is commutative, then the aforementioned results from [49] imply that we thus obtain all irreducible unitary representations of SUn (A) ⊆ U(Mn (A)), but if A is non-commutative the classification problem for irreducible unitary representations of Un (A) is open. Thanks to the existence of group complexifications, the main tool in the analysis of bounded unitary representations of Banach–Lie groups is complex analysis, resp., holomorphic extension. This method is also exploited extensively in the classification of bounded unitary representations of the operator groups Up (H) := U(H) ∩ 1 + Lp (H) ,
1 p ∞,
where Lp (H) is the p-th Schatten ideal ([36], see also [42]). Here a remarkable result is that the classification does not depend on p in the range 1 < p ∞, so that the picture is the same as for p = ∞, where L∞ (H) = K(H) is the C ∗ -algebra of compact operators, and the irreducible representations are obtained from Schur–Weyl theory by decomposing tensor products of the form H⊗n ⊗ H⊗m . For p = 1, the bounded representation theory of U1 (H) is much richer but not of type I . A classification of the bounded irreducible, resp., factor representations of this group is still an interesting open problem. In this context a remarkable result of D. Pickrell asserts that for the full unitary group U(H) of an infinite dimensional separable Hilbert space over R, C or H, every separable continu-
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ous unitary representation is automatically bounded, a direct sum of irreducible representations, and these are obtained by Schur–Weyl theory from the decomposition of the tensor products ⊗n [57]. Basically this means that, as far as separable representations are concerned, the full HC unitary groups and finite products thereof behave exactly like compact groups. It is a rule of thumb that complex analytic methods apply well to bounded and semibounded unitary representations, where the latter class is defined by the condition that the operators id π(x) are uniformly bounded above on some open subset of the Lie algebra (see [45] for a survey on semibounded representations). Beyond semibounded representations one has to rely on real analytic techniques. Typical analogs of finite dimensional non-compact semisimple groups are the automorphism groups of Hilbert–Riemannian symmetric spaces such as restricted Graßmannians and symmetric Hilbert domains [48,40,42]. For the same reason that finite dimensional groups with non-compact Lie algebra have no faithful finite dimensional unitary representation, these groups have no faithful bounded unitary representation. An important example is the symplectic group Sp(H) of all real linear symplectic automorphisms of (H, ω), where H is a complex Hilbert space and ω(v, w) = Imv, w. Even more important is the restricted symplectic group Spres (H) consisting of all elements for which g g − 1 is Hilbert–Schmidt (cf. [45]). The latter group has a by far richer (projective) representation theory than the former. It is the prototype of a hermitian Lie groups, i.e., an automorphism group of an infinite dimensional hermitian symmetric space. For hermitian Lie groups and their central extensions, the irreducible semibounded unitary representations have recently been classified in [46] by a combination of complex analytic methods with Pickrell’s Theorem [57] and some results on unitarity of highest weight modules [48]. For Spres (H) it turns out that all separable semibounded representations are trivial, but it has a central extension with a rich semibounded representation theory. For the full symplectic group Sp(H) it seems quite likely that all its unitary representation are trivial. So it becomes an interesting issue which structural features of a Banach–Lie group lead to obstructions for the existence of non-trivial unitary representations. An area which is still largely unexplored is the theory of (g, K)-modules for pairs such as (spres (H), U(H)). Related problems have been addressed by G. Olshanski from the point of view of topological groups in terms of (G, K)-pairs, where K is a subgroup of G, such as (Spres (H), U(H)) (cf. [54,58]), and we hope that, eventually, a better understanding of differentiability properties as discussed in the present paper and integration techniques as in [44] and [31] will lead to a more transparent Lie theoretic understanding of the representations of such groups. The groups G = C k (M, K) of C k -maps, k 1, on a compact manifold M with values in a finite dimensional Lie group K form a another class of interesting Banach–Lie groups; more generally one considers C k -gauge transformations of principal bundles. These groups have in [47], and for the case M = S1 , where G is a loop group, the teresting central extensions G central extension G has an interesting family of unitary representations extending to a semidi T, where T acts on S1 by rigid rotations (cf. [60,39,1]). These groups are rect product G Banach manifolds and topological groups, but their multiplication is not smooth. We expect that a thorough analysis of such semidirect products N H , where H is a Banach–Lie group acting continuously on the Banach–Lie group N , can be based on the results of the present paper. For the case where (A, H, α) is a C ∗ -dynamical system, the corresponding covariant representations lead in particular to unitary representations of the groups U(A) α H which are of this type (cf. [56]). Another interesting class of such groups providing testing cases for a general theory
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are the semidirect products Heis(H) α R, where H is a complex Hilbert space and α(t) = eitH a strongly continuous unitary one-parameter group (cf. [51]). 2. Locally convex Lie groups In this section we briefly recall the basic concepts related to infinite dimensional Lie groups. Definition 2.1. Let E and F be locally convex spaces, U ⊆ E open and f : U → F a map. Then the derivative of f at x in the direction h is defined as 1 d df (x)(h) := (∂h f )(x) := f (x + th) = lim f (x + th) − f (x) t→0 t dt t=0 whenever it exists. The function f is called differentiable at x if df (x)(h) exists for all h ∈ E. It is called continuously differentiable, if it is differentiable at all points of U and df : U × E → F,
(x, h) → df (x)(h)
is a continuous map. Note that this implies that the maps df (x) are linear (cf. [13, Lemma 2.2.14]). The map f is called a C k -map, k ∈ N ∪ {∞}, if it is continuous, the iterated directional derivatives dj f (x)(h1 , . . . , hj ) := (∂hj · · · ∂h1 f )(x) exist for all integers j k, x ∈ U and h1 , . . . , hj ∈ E, and all maps dj f : U × E j → F are continuous. As usual, C ∞ -maps are called smooth. Once the concept of a smooth function between open subsets of locally convex spaces is established (cf. [43,33,13]), it is clear how to define a locally convex smooth manifold. A (locally convex) Lie group G is a group equipped with a smooth manifold structure modeled on a locally convex space for which the group multiplication and the inversion are smooth maps. We write 1 ∈ G for the identity element and λg (x) = gx, resp., ρg (x) = xg for the left, resp., right multiplication on G. Then each x ∈ T1 (G) corresponds to a unique left invariant vector field xl with xl (g) := T1 (λg )x, g ∈ G. The space of left invariant vector fields is closed under the Lie bracket of vector fields, hence inherits a Lie algebra structure. In this sense we obtain on g := T1 (G) a continuous Lie bracket which is uniquely determined by [x, y]l = [xl , yl ] for x, y ∈ g. We shall also use the functorial notation L(G) := (g, [·,·]) for the Lie algebra of G and, accordingly, L(ϕ) = T1 (ϕ) : L(G1 ) → L(G2 ) for the Lie algebra morphism associated to a morphism ϕ : G1 → G2 of Lie groups. Then L defines a functor from the category of locally convex Lie groups to the category of locally convex topological Lie algebras. The adjoint action of G on g is defined by Ad(g) := L(cg ), where cg (x) = gxg −1 is the conjugation map. The adjoint action is smooth and each Ad(g) is a topological isomorphism of g. If g is a Fréchet, resp., a Banach space, then G is called a Fréchet-, resp., a Banach–Lie group. For every Lie group G, the tangent bundle T G is a Lie group with respect to the tangent map T (mG ) of the multiplication mG : G × G → G on G. It contains G as the zero section, which is a Lie subgroup, and the projection T G → G is a morphism of Lie groups whose kernel is
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the additive group of g ∼ = T1 (G). In this sense we write g.x = T(g,1) (mG )(0, x) = T1 (λg )x and x.g = T(1,g) (mG )(x, 0) = T1 (ρg )x for g ∈ G and x ∈ g. The two maps G × g → T G,
(g, x) → g.x
and G × g → T G,
(g, x) → x.g
(1)
trivialize the tangent bundle. A smooth map expG : g → G is called an exponential function if each curve γx (t) := expG (tx) is a one-parameter group with γx (0) = x. The Lie group G is said to be locally exponential if it has an exponential function for which there is an open 0-neighborhood U in g mapped diffeomorphically by expG onto an open subset of G. All Banach–Lie groups are locally exponential [43, Prop. IV.1.2]. Not every infinite dimensional Lie group has an exponential function [43, Ex. II.5.5], but exponential functions are unique whenever they exist. If π : G → GL(V ) is a representation of G on a locally convex space V , the exponential function permits us to associate to each element x of the Lie algebra a one-parameter group πx (t) := π(expG tx). We therefore assume in the following that G has an exponential function. 3. Basic facts and definitions In this section we introduce some basic notation and derive some general results for representations of Lie groups on locally convex spaces. Definition 3.1. Let (π, V ) be a representation of the Lie group G (with a smooth exponential function) on the locally convex space V . (a) We say that π is continuous if the action of G on V defined by (g, v) → π(g)v is continuous. (b) An element v ∈ V is a C k -vector, k ∈ N0 ∪ {∞}, if the orbit map π v : G → V , g → π(g)v is a C k -map. We write V k := V k (π) for the linear subspace of C k -vectors and we say that the representation π is smooth if the space V ∞ of smooth vectors is dense. If G is a Banach–Lie group and V a Banach space, then we write F V k ⊆ V k for the subspace of those C k -vectors whose orbit map is also C k in the Fréchet sense. (c) For each x ∈ g, we write d π(expG tx)v exists Dx := v ∈ V : dt t=0 for the domain of the infinitesimal generator d d π(x)v := π expG (tx) v dt t=0 of the one-parameter group π(expG (tx)). We write Dg := x∈g Dx and ωv (x) := d π(x)v for v ∈ Dg . Each Dx and therefore also Dg are linear subspaces of V , but at this point we do not know whether ωv is linear (cf. Theorem 8.2 for a positive answer for Banach–Lie groups). (d) We define inductively Dg1 := Dg and
Dgn := v ∈ Dg : (∀x ∈ g)d π (x)v ∈ Dgn−1
for n > 1,
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so that ωvn (x1 , . . . , xn ) := d π(x1 ) · · · d π(xn )v is defined for v ∈ Dgn and x1 , . . . , xn ∈ g. We further put Dg∞ := n∈N Dgn . (e) Let A be an operator with domain D(A) on the locally convex space V . An element in the n space D∞ (A) := ∞ n=1 D(A ) is called a smooth vector for A. (f) If V is a Banach space, then we say that (π, V ) is locally bounded if there exists a 1neighborhood U ⊆ G for which π(U ) is a bounded set of linear operators on V . It is easy to see that this requirement implies boundedness of π on a neighborhood of any compact subset of G. Remark 3.2. (a) For every representation (π, V ) we have V 1 (π) ⊆ Dg
and V k (π) ⊆ Dgk
for k ∈ N.
Note that ωvk is continuous and k-linear for every v ∈ V k (π). (b) By definition, we have Dg2 ⊆ Dg1 , so that we obtain by induction that Dgn+1 ⊆ Dgn for every n ∈ N. (c) If v ∈ Dx , then γ (t) := π(expG (tx))v is a C 1 -curve in V with γ (t) = π expG (tx) d π(x)v, so that 1 π(expG x)v − v =
π(expG tx)d π (x)v dt.
(2)
0
Lemma 3.3. Suppose that (π, V ) is a continuous representation of the Lie group G on V . Then a vector v ∈ Dg is a C 1 -vector if and only if the following two conditions are satisfied: (i) For every smooth curve γ : [−ε, ε] → G with γ (0) = 1 and γ (0) = x, the derivative d dt |t=0 π(γ (t))v exists and equals d π(x)v. (ii) ωv : g → V , x → d π(x)v is continuous. If G is locally exponential, then (i) follows from (ii), and if G is finite dimensional, then (i) and (ii) hold for every v ∈ Dg . Proof. If π v is a C 1 -map, then ωv = T1 (π v ) is a continuous linear map satisfying d
dt |t=0 π(γ (t))v = ωv (x) for each smooth curve in G with γ (0) = 1 and γ (0) = x. Suppose, conversely, that (i) and (ii) are satisfied. Then the relation π γ (t + h) = π γ (t) π γ (t)−1 γ (t + h) implies that for each smooth curve γ in G, we have d π γ (t) v = π γ (t) ωv δ(γ )t , dt where δ(γ )t = γ (t)−1 .γ (t) is the left logarithmic derivative of γ (cf. (1) in Section 2).
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We conclude that the orbit map π v has directional derivatives in each point g ∈ G, and that its tangent map is given by Tg π v (g.x) = π(g)ωv (x) (cf. (1)). Since ωv and the action of G on V are continuous, the map T (π v ) : T G ∼ =G×g→V is continuous, i.e., π v ∈ C 1 (G, V ). From [13, Lemma 2.2.14] it now follows that ωv is linear. Now we assume, in addition, that G is locally exponential and claim that (ii) implies (i). Any smooth curve γ with γ (0) = 1 and γ (0) = x can be written for sufficiently small values of t ∈ R as expG (η(t)) with a smooth curve η : [−ε, ε] → g satisfying η(0) = 0 and η (0) = x. Therefore it suffices to show that for any such curve η, we have d π expG η(s) v = ωv (x). ds s=0 First we note that η(0) = 0 implies that η(s) 1 = s s
1
1
η (ts)s dt = 0
η (st) dt
0
extends by the value x = η (0) to a smooth curve on [−ε, ε]. Next, we derive for v ∈ Dg from (2) the relation π expG η(s) v − v =
1
π expG tη(s) ωv η(s) dt,
(3)
0
which leads to 1 d π expG η(s) v = lim π expG tη(s) ωv η(s)/s dt s→0 ds s=0 0
1 ωv (x) dt = ωv (x)
= 0
because, in view of (ii), the integrand in (3) is a continuous function of (s, t), so that we may exchange integration and the limit. This means that (i) is satisfied. If G is finite dimensional, then Goodman’s argument in [15, p. 221] implies that every v ∈ Dg is a C 1 -vector, so that (i) and (ii) are satisfied. 2 Lemma 3.4. If G is locally exponential, then a vector v ∈ V is a C k -vector if and only if v ∈ Dgk and the maps ωvn , n k, are continuous and n-linear. In particular, v is a smooth vector if and only if v ∈ Dg∞ and all the maps ωvn are continuous and n-linear.
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Proof. If v is a C k -vector, then ωvn (x1 , . . . , xk ) =
∂n π expG (t1 x1 ) · · · expG (tn xn ) v ∂t1 · · · ∂tn t1 =···=tn =0
is a continuous n-linear map for any n k. Suppose, conversely, that this is the case for some k 1. Lemma 3.3 takes care of the case k = 1. We may therefore assume that k > 1 and that the assertion holds for k − 1. From Lemma 3.3 we derive that v is a C 1 -vector with T (π v )(g.x) = π(g)ωv1 (x). We have to show that T (π v ) : T G ∼ = G × g → V is a C k−1 -map. For each fixed x ∈ g, the element ωv1 (x) = d π(x)v is contained in Dgk−1 , so that, in view of ωωn 1 (x) (x1 , . . . , xn ) = ωvn+1 (x1 , . . . , xn , x) v
for n k − 1,
our induction hypothesis implies that ωv1 (x) is a C k−1 -vector, hence that T (π v ) has directional derivatives of all orders k − 1, and that they are sums of terms of the form π(g)d π (x1 ) · · · d π(xj )v = π(g)ωvj (x1 , . . . , xj ), which are continuous functions on G × gj , j k − 1. This proves that T (π v ) is a C k−1 -map, and hence that v is a C k -vector. 2 4. A topology on the space of smooth vectors Throughout this section, we assume that G is a Banach–Lie group and that · is a compatible norm on g = L(G) satisfying [x, y] x · y,
x, y ∈ g.
We shall define a topology on V ∞ for which the action σ : G × V ∞ → V ∞,
(g, v) → π(g)v
is smooth. Let d π : g → End V ∞ ,
d π(x)v :=
d π(exp tx)v dt t=0
denote the derived action of g on V ∞ . That this is indeed a representation of g follows by observing that the map V ∞ → C ∞ (G, V ), v → π v intertwines the action of G with the right translation action on C ∞ (G, V ), and this implies that the derived action corresponds to the action of g on C ∞ (G, V ) by left invariant vector fields (cf. [38, Rem. IV.2] for details). Definition 4.1. (a) For n ∈ N0 , let Multn (g, V ) be the space of continuous n-linear maps gn → V (for n = 0 we interpret this as the constant maps) and write P(V ) for the set of continuous
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seminorms on V . The space Multn (g, V ) carries a natural locally convex topology defined by the seminorms
p(ω) := sup p ω(x1 , . . . , xn ) : x1 , . . . , xn 1 , p ∈ P(V ). Note that p(ω) is the smallest constant c 0 for which we have the estimate p ω(x1 , . . . , xn ) cx1 · · · xn for x1 , . . . , xn ∈ g.
(4)
(b) To topologize V ∞ , we define for each n ∈ N0 a map Ψn : V ∞ → Multn (g, V ),
Ψn (v)(x1 , . . . , xn ) := d π(x1 ) · · · d π(xn )v.
That Ψn (v) defines a continuous n-linear map follows from the smoothness of the orbit map π v : G → V , g → π(g)v and the fact that Ψn (v) is obtained by n-fold partial derivatives in 1. We thus obtain an injective linear map Multn (g, V ), v → Ψn (v) n∈N , Ψ :V∞ → 0
n∈N0
and define the topology on V ∞ such that Ψ is a topological embedding. This means that the topology on V ∞ is defined by the seminorms
pn (v) := sup p d π(x1 ) · · · d π(xn )v : xi ∈ g, xi 1 , p ∈ P(V ), n ∈ N0 . Note that pn (v) = p(Ψn (v)) is the smallest constant c 0 for which we have the estimate p d π(x1 ) · · · d π(xn )v cx1 · · · xn , x1 , . . . , xn ∈ g.
(5)
We endow V ∞ with the locally convex topology defined by the seminorms pn , n ∈ N0 , p ∈ P(V ). Lemma 4.2. The linear map d π : g × V ∞ → V ∞ , (x, v) → d π(x)v is continuous. Proof. This follows from pn (d π(x)v) pn+1 (v)x, which is a consequence of (5).
2
Lemma 4.3. The group G acts by continuous linear operators on V ∞ . More precisely, we have
n pn π(g)v p ◦ π(g) n (v) Ad(g)−1 . (6) Proof. For x1 , . . . , xn ∈ g, we have d π(x1 ) · · · d π(xn )π(g)v = π(g)d π Ad g −1 x1 · · · d π Ad g −1 xn v, and, in view of (5), this implies (6).
2
Theorem 4.4. If (π, V ) is a representation of the Banach–Lie group G on the locally convex space V defining a continuous action of G on V , then the action σ (g, v) := π(g)v of G on V ∞ is smooth.
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Proof. First we show that σ is continuous. In view of Lemma 4.3, it suffices to show that σ is continuous in (1, v) for each v ∈ V ∞ . For w ∈ V ∞ , the relation π(g)w − v = π(g)(w − v) + π(g)v − v permits us to break the argument into a proof for the continuity of σ in (1, 0) and the continuity of the orbit map π v : G → V ∞ . First we use (6) to obtain
n pn π(g)w p ◦ π(g) n (w) Ad(g)−1 . Since U p := {w ∈ V : p(w) 1} is a 0-neighborhood in V , the continuity of the action of G on V implies the existence of a 1-neighborhood UG ⊆ G and a continuous seminorm q on V with π(UG )U q ⊆ U p . This means that p(π(g)v) q(v) for v ∈ V and g ∈ UG , i.e., p ◦ π(g) q. We thus obtain for g ∈ UG the estimate
n pn π(g)(w) qn (w) Ad(g)−1 , and since Ad : G → GL(g) is locally bounded, σ is continuous in (1, 0). To verify the continuity of the orbit maps π v , v ∈ V ∞ , we consider the smooth function f : G × G → V , (g, h) → π(g)π(h)v and observe that f h (g) := f (g, h) = π π(h)v (g). Using the family P(V ) of all continuous seminorms on V , we embed V into the−1topological product p∈P (V ) Vp , where Vp is the Banach space obtained by completing V /p (0) with respect to the norm on this space induced by p. We write v → [v]p for the corresponding quotient map. Then the smoothness of the map f is equivalent to the smoothness of the component mappings fp : G × G → Vp , (g, h) → [f (g, h)]p , and since smoothness for Banach space-valued maps implies smooth dependence of higher derivatives (cf. [38, Thm. I.7], [29,13]), it follows that the Banach space-valued maps fpn : G → Multn (g, Vp ),
h → Ψn π(h)v p
are smooth. Here we use that for each open subset U in a Banach space E and a smooth map F : U → Vp , the map F n : U → Multn (E, Vp ), defined by F n (x)(v1 , . . . , vn ) := (∂v1 · · · ∂vn F )(x), is smooth. This implies that the corresponding map fp : G →
Multn (g, Vp ),
h → Ψ π(h)v p = Ψ π v (h) p
n∈N0
is smooth, and this in turn means that the orbit map π v : G → V ∞ is smooth, hence in particular continuous. This completes the proof of the continuity of σ . The preceding argument already implies that the partial derivatives of the action map σ exist and that they are given by T(g,v) (σ )(g.x, w) = π(g)d π(x)v + π(g)w.
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From the continuity of σ and the action of g on V ∞ (Lemma 4.2), it now follows that T σ is continuous, so that σ is actually C 1 . Iterating this argument, we see that whenever σ is C n , then T σ also is C n , and this shows that σ is smooth. 2 Corollary 4.5. If G is a Banach–Lie group and (π, H) a continuous unitary representation of G on H, then the induced action of G on H∞ is smooth. It is instructive to compare our topology on V ∞ with the standard construction for finite dimensional Lie groups: Proposition 4.6. Suppose that G is finite dimensional, V is a locally convex space, and C ∞ (G, V ) is endowed with the smooth compact open topology, i.e., the topology of uniform convergence of all derivatives on compact subsets. Then for any continuous representation (π, V ) of G, the injection η : V ∞ → C ∞ (G, V ),
v → π v
is a topological embedding whose range is the subspace of smooth maps f : G → V which are equivariant in the sense that f (gh) = π(g)f (h)
for g, h ∈ G.
Proof. We recall that the smooth compact open topology is defined by the property that the embedding C ∞ (G, V ) → n∈N0 C(T n G, T n V ), f → (T n (f ))n∈N0 , defined by the tangent maps, is a topological embedding, where the factors on the right are endowed with the compact open topology. It is easy to see that this is a locally convex topology for which the action of G on C ∞ (G, V ) by right translations is smooth (cf. [38, Sect. III]). Here the key observation is that the smoothness of the map G × C ∞ (G, V ) → C ∞ (G, V ),
(g, f ) → f ◦ ρg
follows from the smoothness of the corresponding map G × C ∞ (G, V ) × G → V ,
(g, f, x) → f (xg)
[50, Lemma A.2]. This in turn follows from the smoothness of the evaluation map ev: C ∞ (G, V ) × G → V , whose smoothness follows from the smoothness of idC ∞ (G,V ) (cf. [50, Lemma A.3(1)]; resp., [12, Prop. 12.2]).2 If (π, V ) is a continuous representation of G on V , then η(v) := π v injects V ∞ into ∞ C (G, V ) and the image of η clearly is the subspace of equivariant map because any such map f satisfies f = π f (1) . Since im(η) is a closed subspace invariant under right translation, we obtain by restriction a smooth action on this space. 2 In the setting of convenient calculus, one finds similar arguments for the smoothness of the G-action on V ∞ in [32, Thm. 5.2].
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For d = dim g, we also have a topological isomorphism Multp (g, V ) ∼ = V pd by evaluating p-linear maps on p-tuples of elements of a fixed basis of g. Therefore the topology on V ∞ is defined by the seminorms v → p d π(x1 ) · · · d π(xn )v ,
p ∈ P(V ),
where x1 , . . . , xn are n elements of a fixed basis of g (Definition 4.1). Since g acts on C ∞ (G, V ) by continuous linear maps, all the operators d π(x) are continuous with respect to the smooth compact open topology, and therefore the evaluation map ev1 : im(η) → V ∞ is a continuous bijection. Conversely, the smoothness of the action of G on V ∞ (Theorem 4.4) implies that the linear map η is continuous [50, Lemma A.2], hence a topological embedding. 2 Remark 4.7. (a) If G is finite dimensional with countably many connected components, which is equivalent to being a countable union of compact subsets, and V is a Fréchet space, then C ∞ (G, V ) is also Fréchet, and therefore V ∞ is a Fréchet space. (b) For a unitary representation (π, H) of a finite dimensional Lie group G, Goodman uses in [14] Sobolev space techniques to show that the space H∞ of smooth vectors is the intersection of the spaces D∞ (d π (xj )), j = 1, . . . , n, where x1 , . . . , xn is a basis for g ([14, Thm. 1.1]; see also [63, Thm. 10.1.9]). This implies in particular that H∞ is complete and that the topology on this space is compatible with our construction (cf. Proposition 4.6). It seems that the chances that the results in this section extend to some classes of Fréchet–Lie groups are not very high, as the following two examples show. Example 4.8. For the Fréchet–Lie group G = (RN , +) (endowed with the product topology), we consider the continuous unitary representation on H = 2 (N, C) given by (π(g)x)n = eign xn . Then
Dg = x ∈ H: ∀y ∈ RN (xn yn ) ∈ H = span{en : n ∈ N} = C(N) is a countably dimensional vector space. Since Dg is spanned by eigenvectors, we see that Dg = H∞ . We claim that for no locally convex topology on H∞ the bilinear map β : g × H∞ → H,
(x, v) → d π(x)v
is continuous, which implies in particular that the action of G on H∞ is not C 1 , hence in particular not smooth. To verify our claim, let B ⊆ H denote the closed unit ball. If β is continuous, then there exist 0-neighborhoods Ug ⊆ g and U ⊆ H∞ with β(Ug × U ) ⊆ B. Next we observe that Ug contains a subspace of the form RM , M := N \ {1, . . . , n}. In particular Ren+1 ⊆ Ug . Since we also have εen+1 ∈ U for some ε > 0, we arrive at the contradiction R · εen+1 = Ren+1 ⊆ B. This proves that Theorem 4.4 does not generalize to unitary representations of general Fréchet–Lie groups.
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Example 4.9. We have seen in Proposition 4.6 that, for finite dimensional Lie groups, we obtain the natural topology on V ∞ by embedding it into the space C ∞ (G, V ), endowed with the smooth compact open topology. In this case the smoothness of the G-action on C ∞ (G, V ) yields the smoothness on the invariant subspace V ∞ . For any infinite dimensional Lie group G, the smooth compact open topology still makes sense, but, as the following example shows, this topology is too weak to guarantee the continuity of the G-action. To substantiate this claim, we consider a locally convex space G, considered as a Lie group, and the space E := Aff(G, R) of affine real-valued functions on G. Then G acts on E by (π(g)f )(x) := f (x + g). Identifying E with R × G , we see that for f = c + α (c ∈ R, α ∈ G ) we have π(g)(c + α) = α(g) + c + α. Therefore π is continuous with respect to a locally convex topology on E, resp., G , if and only if the evaluation map G × G → R,
(α, x) → α(x)
is continuous. The smooth compact open topology on C ∞ (G, R) corresponds on G to the topology of uniform convergence on compact subsets of G. The continuity of the evaluation map with respect to this topology is equivalent to the existence of a compact subset C ⊆ G for which the closed convex hull conv(C) is a 0-neighborhood. If V is complete, then the precompactness of conv(C) [5, Ch. II, §4, no. 2, Prop. 3] implies that G has a compact 0-neighborhood, hence that it is finite dimensional. A similar argument shows that, for the finer topology of uniform convergence on bounded subsets of G , the evaluation map is continuous if and only if G has a bounded 0-neighborhood which means that it is a normed space. 4.1. Distribution vectors An element of the topological dual space V −∞ := (V ∞ ) is called a distribution vector. The main property of these functionals is that they correspond to G-morphisms V ∞ → C ∞ (G). Note that (g.α) := α ◦ π(g)−1 defines a natural action of G on V −∞ . Lemma 4.10. We have an injective map Φ : V −∞ → HomG V ∞ , C ∞ (G) ,
Φ(α)(v)(g) := α π(g)−1 v ,
where G acts on C ∞ (G) by (g.f )(h) := f (g −1 h). The range of Φ consists of all those Gmorphisms ϕ : V ∞ → C ∞ (G) for which the composition ev1 ◦ϕ is continuous, i.e., an element of V −∞ . Proof. The smoothness of the functions Φ(α)(v) follows from the smoothness of the G-action on V ∞ , and the equivariance of Φ(α) is immediate from the definition.
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For ϕ := Φ(α) we have α = ev1 ◦ϕ ∈ V −∞ . If, conversely, ϕ ∈ HomG (V ∞ , C ∞ (G)) is such that α := ev1 ◦ϕ is continuous, we have ϕ(v)(g) = g −1 .ϕ(v) (1) = ϕ π(g)−1 v (1) = (ev1 ◦ϕ) π(g)−1 v = Φ(α)(v)(g), i.e., ϕ = Φ(α).
2
Remark 4.11. (a) For finite dimensional Lie groups, compactly supported distributions are defined as the elements of the topological dual C ∞ (G) of the space of smooth functions. Since the G-action on C ∞ (G) is smooth (cf. the proof of Proposition 4.6), compactly supported distributions are distribution vectors in the sense defined above. (b) We do not know if there exists a locally convex topology on C ∞ (G) for which the left or right translation action is smooth and therefore it is not clear what a good concept of a distribution on a Banach–Lie group is. However, the topology on V ∞ for a continuous representation (π, V ) provides a natural concept of a distribution vector. In harmonic analysis on homogeneous spaces G/H , distribution vectors of unitary representations play an important role. If H ⊆ G is a Lie subgroup, so that the quotient G/H carries a manifold structure for which the quotient map q : G → G/H , g → gH, is a submersion, then we may identify C ∞ (G/H ) with the subspace of C ∞ (G) consisting of all functions constant on the cosets gH . For the map Φ in Lemma 4.10 we immediately see that Φ(α) V ∞ ⊆ C ∞ (G/H ) is equivalent to the invariance of α under H . Therefore the space (V ∞ )H of H -invariant distribution vectors parametrizes the G-morphisms ϕ : V ∞ → C ∞ (G/H ) which are continuous in the very weak sense that their composition with point evaluations is continuous (Lemma 4.10, see also [17, p. 137]). This situation is of particular interest if these embeddings are essentially unique: A pair (G, H ) of a Banach–Lie group G and a subgroup H is called a generalized Gelfand pair if H dim H−∞ 1 holds for every irreducible continuous unitary representation (π, H) of G. For finite dimensional Lie groups there exists a rich theory of generalized Gelfand pairs with many applications in harmonic analysis (cf. [8]). It is a very worthwhile project to explore this concept also for Banach–Lie groups, e.g., for automorphism groups of infinite dimensional semi-Riemannian symmetric spaces such as hyperboloids
X = (t, v) ∈ R × H: v2 − t 2 = 1 , where H is a real Hilbert space (cf. [9] for the case dim H < ∞). 5. Smooth vectors in Banach spaces In this section we assume that G is a Banach–Lie group and that V is a Banach space. Our goal is to show that the space V ∞ is complete, hence a Fréchet space.
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The following proposition generalizes the well-known fact that separately continuous bilinear maps on Fréchet spaces are continuous. Proposition 5.1 (Continuity criterion). Let X be a first countable topological space, F a Fréchet space and V a topological vector space. Let α : X → L(F, V ) (the space of continuous linear operators F → V ) be a map such that, for each f ∈ F , the map αf : X → V ,
x → α(x)f
is continuous. Then the map α : X × F → V , (x, f ) → α(x)f is continuous. Proof. Since X is first countable, the same holds for the product space X × F , so that we only have to show that, for any sequence (xn , fn ) → (x0 , f0 ) in X × F , we have α(xn )fn → α(x0 )f0 . Our assumption implies that the sequence α(xn ) of linear maps converges pointwise to α(x0 ). In particular, for each f ∈ F , the sequence α(xn )f in V is bounded. Now the Banach–Steinhaus Theorem [61, Thm. 2.6] implies that the sequence α(xn ) is equicontinuous, which leads to α(xn )(fn − f0 ) → 0. We thus obtain α(xn )fn − α(x0 )f0 = α(xn )fn − α(xn )f0 + α(xn )f0 − α(x0 )f0 → 0. This proves the continuity of α.
2
Lemma 5.2. Let V be a Banach space, G be a topological group and π : G → GL(V ) be a homomorphism. Then the following are equivalent: (i) The linear action σ : G × V → V , (g, v) → π(g)v is continuous. (ii) σ is continuous in (1, 0). (iii) π is locally bounded and π is strongly continuous, i.e., all orbit maps π v (g) = σ (g, v) are continuous. If, in addition, G is metrizable, then (i)–(iii) are equivalent to (iv) π is strongly continuous. Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): If σ is continuous in (1, 0), then there exists a 1-neighborhood U ⊆ G and a ball Bε (0) in V with π(U )Bε (0) ⊆ B1 (0). This implies that π(g) ε −1 for every g ∈ U . (iii) ⇒ (i) follows from π(g)v − π(h)w π(g)v − w + π(g)w − π(h)w. (iv) ⇒ (i): If, in addition, G is metrizable, then it is first countable, so that Proposition 5.1 shows that (iv) implies (i). 2 Remark 5.3. If π : G → GL(V ) is a representation of the topological group G on the Banach space V which is locally bounded, then the subspace V 0 of continuous vectors, i.e., of those v ∈ V for which the orbit map π v : G → V , g → π(g)v, is continuous is closed. In fact, suppose that vn → v holds for some sequence vn ∈ V 0 and that U is a neighborhood of g0 ∈ G on which π is bounded. Then π vn → π v holds uniformly on U . Therefore the continuity of the maps π vn implies the continuity of π v on U . Since g0 was arbitrary, it follows that v ∈ V 0 , and
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hence that V 0 is closed. Since V 0 is G-invariant, we obtain a locally bounded representation π 0 : G → GL(V 0 ), and its continuity follows from Lemma 5.2. In view of the preceding lemma, the continuity of the action of G on V implies that π is locally bounded. The following proposition can also be derived from the results in Section 9 (cf. Remark 9.6). Proposition 5.4. If (π, V ) is a continuous representation of the Banach–Lie group G on the Banach space V , then V ∞ is complete, i.e., a Fréchet space. Proof. First we note that n∈N0 Multn (g, V ) is a countable product of Banach spaces, hence a ∞ of the topological vector space V ∞ can be identiFréchet space. Therefore the completion V fied with the closure of the subspace Ψ (V ∞ ). Since V is complete, we have a continuous linear ∞ → V , extending the inclusion V ∞ → V , and, with respect to the realization of V ∞ map ι : V as the closure of the image of Ψ , this map is given by ι((αn )n∈N0 ) = α0 . In the course of the ∞ proof we shall see that ι is injective and that its range coincides with V ∞ . This implies that V ∞ ∞ is not larger than V , so that V is complete. Let (vn )n∈N be a sequence in V ∞ for which Ψk (vn ) converges to some ωk ∈ Multk (g, V ) for each k ∈ N0 . We have to show that v := ω0 is a smooth vector and that ωk = Ψk (v) for each k. Since the continuous linear maps d π(x) : V ∞ → V ∞ extend to continuous linear maps ∞ → V ∞ d π(x) : V ∞ implies that on the completion, the convergence vn → v in V d π(x1 ) · · · d π(xk )vn → d π(x1 ) · · · d π(xk )v, and hence that ∞ . ωk (x1 , . . . , xk ) = d π(x1 ) · · · d π(xk )v ∈ V
(7)
This proves in particular that v = 0 implies ωk = 0 for each k > 0, and hence that the map ∞ → V is injective. ι:V For x ∈ g, we obtain with (2) that 1 π(expG x)vn − vn =
π(expG tx)d π(x)vn dt 0
for each n. Since d π(x)vn → ω1 (x) and the linear map w → (π is locally bounded), we obtain
1 0
π(expG (tx))w dt is continuous
1 π(expG x)v − v =
π(expG tx)ω1 (x) dt, 0
(8)
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which leads to 1 1 d 1 π(expG sx)v = lim π(expG stx)ω (x) dt = ω1 (x) dt = ω1 (x). s→0 ds s=0 0
0
This implies that v ∈ Dg with ωv = ω1 . In particular, ωv is continuous and linear. Since Banach– Lie groups are locally exponential, Lemma 3.3 now implies that v is a C 1 -vector. To see that π v is C 2 , we have to show that T (π v ) is C 1 . From (7) we recall that, for each ∞ is a C 1 -vector by the preceding argument. Therefore d π(x)v ∈ V x ∈ g, ω1 (x) = T πv : T G ∼ = G × g → V,
(g, x) → π(g) d π(x)v
has directional derivatives given by Tg,x T π v (g.y, w) = π(g) d π(y) d π(x)v + π(g) d π(y)v. We have already seen above that the second term is continuous, and the continuity of the first term follows from the continuity of the action of G on V and the continuity of the bilinear map ω2 (y, x) = d π(y) d π(x)v. This proves that each π v is C 2 . Iterating this argument implies that π v has directional derivatives of any order k, and that they are sums of terms of the form π(g) d π(x1 ) · · · d π(xj ) = π(g)ωj (x1 , . . . , xj ), which are continuous on G × gj . Therefore v is a C k -vector for any k, hence smooth, and thus ∞ ⊆ V ∞ implies that V ∞ is complete. 2 V 6. An applications to C ∗ -dynamical systems In this section we show that in the special situation where a Banach–Lie group G acts by automorphisms on a unital C ∗ -algebra A, i.e., for C ∗ -dynamical systems, the Fréchet space A∞ is a continuous inverse algebra, i.e., its unit group is open and the inversion is a continuous map. Definition 6.1. (a) A locally convex unital algebra A is called a continuous inverse algebra if its group of units A× is open and the inversion map η : A× → A, a → a −1 is continuous. (b) Let G be a topological group and A be a C ∗ -algebra. A C ∗ -dynamical system is a triple (A, G, α), where α : G → Aut(A), g → αg , is a homomorphism defining a continuous action of G on A. Theorem 6.2. If G is a Banach–Lie group and (A, G, α) a C ∗ -dynamical system with a unital C ∗ -algebra A, then the space A∞ of smooth vectors is a subalgebra, which is a topological algebra with respect to its natural topology, and the action of G on the Fréchet space A∞ is smooth. If, in addition, A is unital, then A∞ is a continuous inverse algebra.
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Proof. In view of Theorem 4.4, the action of G on A∞ is smooth, and it is a Fréchet space by Proposition 5.4. It therefore remains to verify that A∞ is a topological algebra, i.e., a subalgebra on which the product and the involution are continuous. If a, b ∈ A∞ , then the orbit map of ab is given by αg (ab) = αg (a)αg (b), and since the multiplication in A is continuous bilinear, hence smooth, ab ∈ A∞ . We also derive from the smoothness of the inversion in A× that if a ∈ A× ∩ A∞ , then also a −1 ∈ A∞ . The continuity of the multiplication in A∞ follows from the fact that the operators d π(x) : A∞ → A∞ are derivations. If we write for a subset S = {i1 < i2 < · · · < ik } ⊆ {1, . . . , n}, d π(xS ) := d π(xi1 ) · · · d π(xik ), then d π(x1 ) · · · d π(xn )(ab) =
d π(xS )(a)d π(xS c )(b).
(9)
S
In view of the submultiplicativity of the norm, this leads to
d π(x1 ) · · · d π(xn )(ab)
dπ(xS )(a) · d π(xS c )(b) , S
so that the seminorm p(a) := a satisfies pn (ab) S p|S| (a)p|S c | (b). Since the sequence (pn )n∈N of seminorms defines the topology on A∞ , it follows that the multiplication in A is continuous. The continuity of the involution on A∞ follows from the fact that the operators d π(x) commute with ∗: d π(x1 ) · · · d π(xn ) a ∗ = d π(x1 ) · · · d π(xn )(a)∗ , which leads to pn (a ∗ ) = pn (a) for a ∈ A∞ . Since the inclusion A∞ → A is continuous, the unit group ∞ × = A∞ ∩ A× A is open. To prove the continuity of the inversion, we show by induction that it is continuous with respect to the seminorms pn , n N . For N = 0, the assertion follows from the continuity of the inversion on A× . For N > 0, we apply (9) to b = a −1 to obtain 0=
d π(xS )(a)d π(xS c ) a −1 ,
S
where the sum is extended over all subsets of {1, . . . , N}. This leads to d π(x1 ) · · · d π(xn ) a −1 = −a −1 d π(xS )(a)d π(xS c ) a −1 . S=∅
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That this sum depends, as an element of MultN (g, A), continuously on a follows by induction because |S c | < N whenever S = ∅. 2 7. Smooth vectors for unitary representations In this section we provide a remarkably effective criterion for the smoothness of vectors in unitary representations of infinite dimensional Lie groups, namely that v ∈ H∞ if and only if the matrix coefficient π v,v (g) = π(g)v, v is smooth in an identity neighborhood. This will be a simple consequence of the following observation.3 Theorem 7.1. Let M be a smooth manifold, H a Hilbert space and γ : M → H be a map. Then γ is a smooth map if and only if the kernel K(x, y) := γ (x), γ (y) is a smooth function on M × M. Proof. Since the smoothness only refers to the real structure on H, we may assume that H is a real Hilbert space. If γ is smooth, then K is also smooth because the scalar product on H is real bilinear and continuous, hence smooth. Step 1: γ is continuous: This follows from
γ (x) − γ (y) 2 = K(x, x) + K(y, y) − 2K(x, y). Step 2: Now we consider the case where M ⊆ R is an open interval, so that γ : M → H is a curve in H. From Step 1 we know that γ is continuous. For a fixed t ∈ M we now consider on M − t the function
2
f (h) := γ (t + h) − γ (t) = K(t + h, t + h) + K(t, t) − 2K(t + h, t). It is smooth with f (0) = 0 and f (0) = (∂1 K)(t, t) + (∂2 K)(t, t) − 2(∂1 K)(t, t) = 0 because K is symmetric. This implies that lim
h→0
2 f (h) = f
(0) h2
exists, and from the Chain Rule and the symmetry of K we obtain f
(0) = ∂12 K (t, t) + ∂22 K (t, t) + 2(∂1 ∂2 K)(t, t) − 2 ∂12 K (t, t) = 2(∂1 ∂2 K)(t, t). We conclude that
1 2
= lim 1 f (h) = 1 f
(0) = (∂1 ∂2 K)(t, t) γ (t + h) − γ (t) lim
h→0 h h→0 h2 2
(10)
3 A finer analysis of the situation shows that if K has continuous k-fold derivatives in both variables separately in some neighborhood of the diagonal, then γ is a C k -map, k ∈ N ∪ {∞} (cf. [28, p. 78] for the case where M is a real interval).
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exists. We also note that, for each s ∈ M, d γ (t), γ (s) = (∂1 K)(t, s) dt d exists. Since h1 (γ (t + h) − γ (t)) is bounded by (10), it follows that dt γ (t), v exists for each v ⊥ in the closed subspace generated by γ (M). For v ∈ γ (M) , the expression vanishes anyway, so that γ (t) exists weakly and satisfies
γ (t), γ (s) = (∂1 K)(t, s).
In particular, we have
2
γ (t) = lim γ (t), γ (t + h) − γ (t) = lim 1 (∂1 K)(t, t + h) − (∂1 K)(t, t) h→0 h→0 h h
1 2
, γ (t + h) − γ (t) = (∂1 ∂2 K)(t, t) = lim
h→0 h and this implies that lim
h→0
1 γ (t + h) − γ (t) = γ (t) h
holds in the norm topology of H. We conclude that γ is a C 1 -curve. Next we observe that
γ (t), γ (s) = ∂1 ∂2 γ (t), γ (s) = (∂1 ∂2 K)(t, s)
is also a smooth kernel. Therefore the argument above implies that γ is C 1 , so that γ is C 2 . Repeating this argument shows that γ is C k for every k ∈ N, hence smooth. Step 3: Now we consider a general locally convex manifold M. As the assertion of the proposition is local, we may assume that M is an open subset of a locally convex space. From Step 2 we derive that the map γ has directional derivatives in all directions, which leads to a “tangent map” d dγ : T M → H, (dγ )(x, v) := γ (x + tv). dt t=0 Since the kernel d d (dγ )(x, v), (dγ )(y, w) = γ (x + tv), γ (y + sw) dt t=0 ds s=0 d d γ (x + tv), γ (y + sw) = dt t=0 ds s=0 d d K(x + tv, y + sw) = (∂(v,0) ∂(0,w) K)(x, y), = dt t=0 ds s=0
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is smooth, Step 1 shows that dγ is continuous, so that γ is a C 1 -map. Applying the same argument to dγ instead of γ , it follows that dγ is C 1 and hence that γ is C 2 . Iterating this argument implies that γ is smooth. 2 The following theorem substantially sharpens the well-known criterion (for finite dimensional groups) that v ∈ H∞ if its orbit map π v is weakly smooth, i.e., all matrix coefficients π v,w (g) = π(g)v, w are smooth (cf. [63, Cor. 10.1.3]). In [59, p. 278] one finds a remark suggesting its validity for finite dimensional groups, which is proved in [37, Prop. X.6.4] by using Goodman’s characterization of smooth vectors [14]. Theorem 7.2. If (π, H) is a unitary representation of a Lie group G, then v ∈ H is a smooth vector if and only if the corresponding matrix coefficient π v,v (g) := π(g)v, v is smooth on a 1-neighborhood in G. Proof. Clearly, π v,v is smooth if v is a smooth vector. Suppose, conversely, that π v,v is smooth in a 1-neighborhood U ⊆ G and let U be a 1-neighborhood with h−1 g ∈ U for g, h ∈ U . In view of Theorem 7.1, the smoothness of π v on xU , x ∈ G, is equivalent to the smoothness of the function (g, h) → π(xg)v, π(xh)v = π v,v h−1 g on U × U , which follows from the smoothness of π v,v on U .
2
Corollary 7.3. If the continuous unitary representation (π, H) of the Lie group G has a cyclic vector v for which the function π v,v (g) := π(g)v, v is smooth on some identity neighborhood, then the representation (π, H) is smooth, i.e., H∞ is dense. Proof. The preceding theorem implies that v is a smooth vector, and therefore span π(G)v consists of smooth vectors. Hence H∞ is dense. 2 Corollary 7.4. If ϕ is a positive definite function on a Lie group G which is smooth in a 1neighborhood, then ϕ is smooth. Proof. Since ϕ is positive definite, the GNS construction provides a unitary representation (π, H) of G and a vector v ∈ H with ϕ = π v,v . Now Theorem 7.2 implies that v ∈ H∞ , but this implies that ϕ is smooth on all of G. 2 8. C 1 -vectors for Banach representations In this section we consider a continuous representation (π, V ) of the Banach–Lie group G on the Banach space V , which implies in particular that π is locally bounded (Lemma 5.2). Since Banach–Lie groups are locally exponential, Lemma 3.3 implies v ∈ V is a C 1 -vector if and only if v ∈ Dg and the map ωv : g → V , x → d π(x) is continuous, which implies in particular that it is linear. The goal of this section is to see that the space Dg coincides with the space of C 1 vectors for the action of G on V (Theorem 8.5). In view of what we know already, the main point is that, for every v ∈ Dg , the map ωv is continuous. Surprisingly (for us), the most difficult part in our argument is to see that the maps ωv : g → V , x → d π(x)v, are additive for each v ∈ Dg .
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Lemma 8.1. Let F : [0, ∞[ → L(V ) be a curve with F (0) = 1 and M > 0 with
F (t) Meat
for t 0
and F (t)n M
for tn 1.
Then there exists an ω ∈ R with
F (t)n Meωnt
for t 0, n ∈ N.
Proof. We put ω := 2 max(a, log M) and observe that M F (0) = 1 implies ω 0. The assertion clearly holds for t = 0, so that we may w.l.o.g. assume that t > 0. 1 and write n as First we discuss the case where t 1. Then we define k ∈ N by 21k < t 2k−1 k−1 k−1 k−1 n = q2 + r with q, r ∈ N0 and r < 2 . Now rt < 2 t 1 leads to
F (t)n F (t)2k−1 q F (t)r M q · M. On the other hand, q
n 2k−1
< 2nt, so that
F (t)n Meq log M Ment2 log M Mentω . Now we consider the case where t 1 and observe that
F (t)n M n enat = en log M+nat ent log M+nta = ent (log M+a) entω Mentω .
2
Theorem 8.2. Let (π, V ) be a representation of the Banach–Lie group G on the Banach space V for which the corresponding action is continuous and v ∈ Dg . Then the map ωv : g → V ,
v → d π(x)v
is linear. Proof. Step 1: Since G is a Banach–Lie group, the Trotter Product Formula n lim expG (tx/n) expG (ty/n) = expG t (x + y)
n→∞
holds uniformly for |t| N and any N ∈ N. Below we need a slight refinement which can be obtained as follows. First we observe that in a Banach–Lie algebra, we have for each real sequence αn → s > 0 and the Baker–Campbell– Hausdorff multiplication ∗ the relation n
n αn αn αn αn x ∗ y = αn x ∗ y → s(x + y), n n αn n n
so that applying the exponential function on both sides leads to n expG (αn x/n) expG (αn y/n) → expG s(x + y) .
(11)
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Step 2: For x, y ∈ g, we consider the strongly continuous curve F : R → GL(V ),
F (t) := π(expG tx)π(expG ty).
Since π is locally bounded, it is bounded in a neighborhood of expG ([0, 1](x + y)), so that we find with Step 1 a M0 > 0 with
F (t/n)n M0 for 0 t 1. The strongly continuous one-parameter semigroups π(expG (tx)) and π(expG (ty)) satisfy
π exp (ty) M2 etω2
π exp (tx) M1 etω1 , G G for suitable M1 , M2 > 0, ω1 , ω2 ∈ R and all t 0 [55, Thm. 2.2]. This leads to the estimate
F (t) M1 M2 et (ω1 +ω2 ) . Applying Lemma 8.1, we now find a constant M > 0 and ω ∈ R with
F (t)n Meωnt for t 0, n ∈ N.
(12)
Step 3: From π(g)Dx = DAd(g)x for g ∈ G and x ∈ g it follows that Dg is a G-invariant subspace of V . Since we are only interested in elements of Dg , we may w.l.o.g. assume that Dg is dense in V . From the definition it immediately follows that d π(λx)v = λd π(x)v for λ ∈ R. For the operators C := d π(x) and D := d π(y) we then find that Dg ⊆ D(C) ∩ D(D) is dense in V . Further, (12) is the estimate required in [52, Cor. 5.2(ii)], which asserts that the unbounded operator C + D, defined on D(C) ∩ D(D), is closable and its closure generates a C0 -semigroup if and only if there exists a dense linear subspace A ⊆ D(C) ∩ D(D) such that for all f ∈ A and s 0 there exist relatively compact sequences (fns )n∈N in D(C) ∩ D(D) such that (a) The set {(C + D)fns : n ∈ N} is precompact for every s 0. (b) limn→∞ F (t/n)[ns] f − fns = 0. We put A := Dg , and for f ∈ Dg , we put fns := π(expG (ts(x + y)))f . Then (a) is trivially satisfied. To verify (b), we note that for αn := [ns] n the relation ns − 1 [ns] ns implies αn = [ns]/n → s. Hence (11) leads to αn t n F f → π expG st (x + y) f for t ∈ R, f ∈ Dg . n We thus obtain [ns] f = π expG st (x + y) f = fns . lim F (t/n)[ns] f = lim F αn t/[ns]
n→∞
n→∞
Now [52, Cor. 5.2] (see also the concluding Remark in [53]) applies and yields d π(x) + d π(y) = C + D = d π(x + y). In particular, we obtain ωv (x + y) = ωv (x) + ωv (y) for v ∈ Dg .
2
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Remark 8.3. If π(G) consists of isometries, there are more direct arguments for the additivity of ωv . For the strongly continuous curve F : R → GL(V ),
F (t) := π(expG tx)π(expG ty),
we obtain for each v ∈ Dg the relation 1 1 1 F (t)v − v = π(expG tx) π expG (ty) v − v + π expG (tx) v − v t t t → d π(y)v + d π(x)v, so that the unbounded operator F (0) is defined on Dg , where it coincides with d π(x) + d π(y). On the other hand, the Trotter–Product Formula in G yields n lim F (t/n)n = lim π expG (x/t) expG (y/t) = π expG t (x + y)
n→∞
n→∞
in the strong operator topology. Now [6, Theorem 3.1] implies that d π(x + y) extends the operator F (0) on Dg . In particular, we obtain for v ∈ Dg that d π(x + y)v = d π(x)v + d π(y)v. We now take a closer look at the continuity of the linear maps ωv from Theorem 8.2. Lemma 8.4. Let (π, V ) be a continuous representation of the Fréchet–Lie group G on the Fréchet spaceV . Then, for each v ∈ Dg for which ωv is linear, it is continuous. Proof. (S. Merigon) Assume that ωv is a linear map. In view of the Closed Graph Theorem [61, Thm. 2.15], it suffices to show that the graph of ωv is closed. Suppose that xn → x in g such that ωv (xn ) → w. Then we obtain from (2) the relation π expG (txn ) v − v = t
1
π expG (stxn ) ωv (xn ) ds.
0
Since the function [0, 1]2 × g × V → V ,
(s, t, x, v) → π expG (stx) v
is continuous, integration over s ∈ [0, 1] leads to a continuous function 1 [0, 1] × g × V → V ,
(t, x, v) →
π expG (stx) v ds.
0
We thus obtain in the limit n → ∞: π expG (tx) v − v = t
1 0
π expG (stx) w ds.
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For the derivative in t = 0 this leads to 1 ωv (x) = d π(x)v =
π expG (0x) w ds =
0
1 w ds = w.
2
0
For continuous actions of Banach–Lie groups on Banach spaces we thus obtain with Theorem 8.2 and Lemmas 8.4 and 3.3. Theorem 8.5. Let (π, V ) be a representation of the Banach–Lie group G on the Banach space V for which the corresponding action is continuous. Then Dg coincides with the space of C 1 vectors. 9. C k -vectors for Banach representations We continue our discussion of differentiable vectors for a continuous representation (π, V ) of a Banach–Lie group G on a Banach space V . We know already that for each v ∈ Dg the map ωv : g → V is continuous and linear (Theorems 8.2, Lemma 8.4). We thus obtain a norm on Dg by
v1 := v + ωv = v + sup ωv (x) x1
(cf. [15] and [23] for similar constructions for finite dimensional Lie algebras). With respect to this norm on Dg , the bilinear map g × Dg → V ,
(x, v) → ωv (x) = d π(x)v
satisfies ωv (x) ωv x v1 x, so that it is continuous. For the sake of completeness, we recall the following variant of [41, Lemmas A.1/2]: Lemma 9.1. Let X1 , . . . , Xn , Y and Z be Banach spaces and η : Y → Z a continuous injection. Suppose that A : X1 × · · · × Xn → Z is a continuous n-linear map with im(A) ⊆ im(η). Then the induced n-linear map : X1 × · · · × X n → Y A
= A with η ◦ A
is continuous. Proof. We argue by induction on n. First we consider the case n = 1. By the Closed Graph : X1 → Y is closed. Assume that Theorem, it suffices to show that the graph of A n ) → (x, y) ∈ X × Y. (xn , Ax and therefore Ax = y. Thus A is n ) = Axn → Ax implies that η(y) = Ax = η(Ax), Then η(Ax continuous.
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Now we consider the general case n > 1. From the case n = 1, we know that for fixed elements xj ∈ Xj , j < n, the map Xn → Y,
1 , · · · , xn−1 , x) x → A(x
is continuous. By the induction hypothesis, applied to the continuous inclusion L(Xn , Y ) → L(Xn , Z),
B → η ◦ B,
the (n − 1)-linear map 1 , . . . , xn−1 , ·) (x1 , . . . , xn−1 ) → A(x
: X1 × · · · × Xn−1 → L(Xn , Y ), C is continuous because the corresponding map C : X1 × · · · × Xn−1 → L(Xn , Z),
(x1 , . . . , xn−1 ) → A(x1 , . . . , xn−1 , ·)
is continuous. Since the bilinear evaluation map L(Xn , Z) × Xn → Z,
(ϕ, x) → ϕ(x)
is a continuous n-linear map. is continuous, it follows that A
2
Lemma 9.2. Dg is complete with respect to · 1 . Proof. The map ι : Dg → V × L(g, V ), v → (v, ω(v)) is isometric with respect to · 1 on Dg and the norm (v, α) := v + α on the product Banach space V × L(g, V ). Therefore the completeness of Dg is equivalent to the closedness of the graph Γ (ω) = im(ι) of the linear map ω : Dg → L(g, V ), v → ωv . We now show that Γ (ω) is closed. Assume that (vn , ωvn ) → (v, α). Then (2) yields for each x ∈ g the relation π expG (tx) vn − vn = t
1
π expG (stx) ωvn (x) ds.
0
Passing to the limit on both sides yields 1 π(expG (tx)v − v = t
π expG (stx) α(x) ds.
0
This implies that v ∈ Dx with d π(x)v = α(x). We conclude that v ∈ Dg with α = ωv , and hence that Γ (ω) is closed. 2 Lemma 9.3. For each v ∈ Dgn , the corresponding n-linear map ωvn : gn → V , is continuous.
(x1 , . . . , xn ) → d π(x1 )d π (x2 ) · · · dπ(xn )v
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Proof. First we observe that Theorem 8.2 implies that ωvn is n-linear. We argue by induction on n to show that it is continuous. For n = 1 this follows from Lemma 8.4. Now we assume n > 1 and that ωvn−1 : gn−1 → V is a continuous (n − 1)-linear map. From v ∈ Dgn we derive that im(ωvn−1 ) ⊆ Dg . Since Dg is a Banach space with a continuous injection into V (Lemma 9.2), Lemma 9.1 implies that the (n − 1)-linear map ωvn−1 : gn−1 → Dg is continuous. Since the bilinear evaluation map g × Dg → V , (x, w) → ωw (x) is continuous, we see that ωvn (x1 , . . . , xn ) = ωωvn−1 (x2 ,...,xn ) (x1 ) is a continuous n-linear map.
2
Theorem 9.4. For a continuous representation of the Banach–Lie group G on the Banach space V , the space Dgk coincides with the space V k of C k -vectors. This space is complete with respect to the norm vk := v +
k
j
ω ,
(13)
v
j =1
hence a Banach space, and the bilinear map ξ : g × Dgk+1 → Dgk ,
(x, v) → d π(x)v j
is continuous with respect to the norms · j on Dg , j = k, k + 1. Proof. Combining the preceding lemma with the general Lemma 3.4, it follows that V k = Dgk . To verify the completeness of Dgk , we argue by induction. For k = 1, this is Lemma 9.2. Assume that k > 1. We have to show that the graph of the linear map ω : Dgk →
k
Multj (g, V ),
v → ωv1 , . . . , ωvk
j =1
is closed in V ×
k
j =1 Mult
j
(g, V ). Suppose that
k Multj (g, V ). vn , ωv1n , . . . , ωvkn → (v, α1 , . . . , αk ) ∈ V × j =1 j
Our induction hypothesis implies that v ∈ Dgk−1 with αj = ωv for j k − 1. For x, x2 , . . . , xk ∈ g we further have d π(x)ωvk−1 (x2 , . . . , xk ) = ωvkn (x, x2 , . . . , xk ) → α(x, x2 , . . . , xk ) n (x2 , . . . , xk ) → ωvk−1 (x2 , . . . , xk ). Since the graph of d π(x) is closed (apply Lemma 9.2 and ωvk−1 n with g = R), we obtain ωvk−1 (x2 , . . . , xk ) ∈ Dx
and
This implies that v ∈ Dgk with α = ωvk .
d π(x)ωvk−1 (x2 , . . . , xk ) = α(x, x2 , . . . , xk ).
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To show that the bilinear map ξ is continuous, we first note that, for v ∈ Dgk+1 , we have for j k the estimate
j
ω
ξ(x,v)
j
= ω
d π (x)v
j +1
ω x, v
and this implies that k k+1
j
j
ξ(x, v) =
ω
ω x vk+1 x. v ξ(x,v) k j =0
Therefore ξ is continuous.
j =1
2
Remark 9.5. Note that we have for each n an isometric embedding Dgn → Dgn−1 × Multn (g, V ), where the norm on the product space is (v, α) := vn−1 + α. Remark 9.6. The preceding discussion leads in particular to a Fréchet structure on Dg∞ , consid n ered as a subspace of ∞ n=0 Mult (g, V ). It coincides with the one from Definition 4.1, so that we obtain a second proof of Proposition 5.4. Now that we know that each space Dgk is complete, it is natural to ask for the extent to which the G-action on this space is continuous. Proposition 9.7. The representation π k of G on the Banach space V k of C k -vectors has the following properties: (i) π k is locally bounded. (ii) For an element v ∈ V k , the following are equivalent: j (a) The maps G → Multj (g, V ), g → π(g) ◦ ωv are continuous for j k. j j (b) The maps G → Mult (g, V ), g → π(g) ◦ ωv ◦ (Ad(g)−1 )×j are continuous for j k. k 0 k (c) v ∈ (V ) , i.e., the orbit map G → V , g → π(g)v is continuous. (d) The orbit map π v : G → V is Fréchet-C k , i.e., v ∈ F V k . (iii) V k+1 ⊆ F V k . (iv) The subspace F V k of V k is closed and the G-action on F V k is continuous. It is the maximal G-invariant subspace of V k for which this is the case. In particular, the action of G on V k is continuous if and only if V k = F V k . Proof. (i) The group G acts naturally by continuous linear operators on Multn (g, V ) via ×n , g.ω := π(g) ◦ ω ◦ Ad g −1 × · · · × Ad g −1 = π(g) ◦ ω ◦ (Ad g −1 and we have
n g.ω Ad g −1 π(g) ω,
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i.e., the corresponding representation is locally bounded. Since the topological embedding ω:Vk →V ×
k
Multj (g, V ),
v → v, ωv1 , . . . , ωvk
(14)
j =1
is G-equivariant, the local boundedness of π k follows. (ii) The equivalence of (a) and (c) follows from the embedding (14). Further, the equivalence of (a) and (b) follows from the fact that the action of G on each space Multj (g, V ) by g ∗ ω := ω ◦ (Ad(g)−1 )×j defines a morphism of Banach–Lie groups G → GL(Multj (g, V )) (cf. [42, Exer. IV.6]). For v ∈ Dg , the orbit map π v : G → V is C 1 with ∂g.x π v (g) := Tg π v (g.x) = π(g)d π (x)v = π(g)ωv (x), and, by iterating this argument, we obtain for v ∈ V k and j k: ∂g.x1 · · · ∂g.xj π v (g) = π(g)ωvj (x1 , . . . , xj ).
Ck
With similar arguments as in the proof of [38, Thm. I.7], we now see that the C k -map π v is in the Fréchet sense if and only if, for j k, the maps G → Multj (g, V ),
g → (x1 , . . . , xj ) → ∂g.x1 · · · ∂g.xj π v (g)
are continuous. In view of the preceding calculations, this means that (a) is equivalent to (d). (iii) Follows from the general fact that each C k+1 -map is C k in the Fréchet sense (cf. [38, Thm. I.7(ii)], [13]). (iv) Since the G-representation on V k ⊆ kj =0 Multj (g, V ) is locally bounded, the subspace of elements with continuous orbit maps is closed. As we have seen in (iii) above, this subspace coincides with F V k . It is clearly invariant and Lemma 5.2 implies that G acts continuously on F V k . 2 Remark 9.8. If G is finite dimensional, then each C k -map is also Fréchet-C k , so that the action of G on Dgk is again a continuous action (Remark 5.3). In sharp contrast to this situation is the fact that, for an infinite dimensional Banach–Lie group, Dg need not contain any non-zero continuous vector (cf. Remark 10.7 below). Example 9.9. For the examples discussed in Section 10 below, it is shown that the action of G on Dg is continuous for p 4, because in this case Dg2 contains the dense subspace L∞ ([0, 1]) (cf. Remarks 5.3 and 10.7). Example 9.10. We consider the action of the one-dimensional Lie group G = R on the Banach space V = C(R, R)per of 1-periodic functions by (π(g)f )(x) := f (g + x). Then the uniform continuity of each element of V implies that the orbit maps are continuous, and since G acts by isometries, Lemma 5.2 implies the continuity of the G-action on V .
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Since the point evaluations are continuous, the space Dgk of C k -vectors consists of C k functions on R. If, conversely, f is a C 1 -function, then f (x + h) − f (x) = h
1
f (x + sh) ds
0
converges uniformly to f (x) for h → 0, so that f is a C 1 -vector. A similar argument, based on the integral form of the remainder term in the Taylor formula implies that Dgk = C k (R, R)per . 10. A family of interesting examples We take a closer look at the unitary representation of the Banach–Lie group G := (Lp ([0, 1], R), +), p ∈ [1, ∞[, on the Hilbert space H = L2 ([0, 1], C) by π(g)f := eig f . In [3] it is shown that, for p = 2, this representation is continuous with H∞ = {0}. Here we shall see that it is always continuous and determine the space of C k -vectors. In the following we write g := Lp ([0, 1], R) for the Lie algebra of G and abbreviate Lp ([0, 1]) := Lp ([0, 1], C). We start with a general observation on the inclusions between Lp -spaces. Remark 10.1. (a) If (X, μ) is a finite measure space, then Lq (X, μ) ⊆ Lp (X, μ)
for 1 p < q,
where the inclusion is continuous. In fact, for any measurable function f : X → C, we have
|f |p =
|f |p +
{|f |1}
|f |p μ(X) +
{|f |>1}
|f |q .
{|f |>1}
This implies that · p is bounded on the unit ball of Lq (X, μ). (b) Assume that X has a decomposition into a sequence (Xn ) of pairwise disjoint subsets with 0 < μ(Xn ) 2−n μ(X). We claim that Lq (X, μ) = Lp (X, μ)
for 1 p < q.
We consider the function f whose value on Xn is constant μ(Xn )−1/q . Then q f q = n μ(Xn )−1 μ(Xn ) = ∞ and p
f p =
n
μ(Xn )1−p/q μ(X)1−p/q
n 2p/q−1 < ∞. n
(c) If X = [0, 1] is the unit interval and μ is Lebesgue measure, then the property under (b) is satisfied for every subset Y ⊆ X of positive measure. Lemma 10.2. The representation (π, H) is continuous for every p 1.
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Proof. (Cf. [3, Prop. 2.1] for p = 2.) Let (gn ) be a sequence in G = Lp ([0, 1], R) converging to 0. Then there exists a subsequence (gnk )k∈N converging almost everywhere to 0. Then ignk − 1|2 → 0 almost everywhere, so that the Dominated Convergence Theorem implies that |e ignk − 1|2 → 0. For every ξ ∈ L∞ ([0, 1], C), we thus obtain [0,1] |e
π(gn )ξ − ξ 2 = k 2
ig e nk − 12 |ξ |2 ξ 2 ∞
[0,1]
ig e nk − 12 → 0.
[0,1]
We conclude that π(gnk ) → 1 with respect to the strong topology. If π is not continuous, then there exists a sequence hn → 0 in G with π(hn ) → π(0). This means that there exist a 1-neighborhood U in U(H) with respect to the strong operator topology / U } is infinite. This leads to a sequence gn in G with gn → 0 and π(gn ) ∈ / for which {n: π(hn ) ∈ U for every n, and we thus obtain a contradiction to the argument in the preceding paragraph. We conclude that π is continuous. 2 Since the infinitesimal generator of the one-parameter group t → π(tf ) is the multiplication with the function f , it follows that
Dg = Dg1 = ξ ∈ H: ∀f ∈ Lp [0, 1], R f ξ ∈ H . Lemma 10.3. We have ⎧ ⎪ ⎨ {0} ∞ Dg = L ([0, 1]) ⎪ 2p ⎩ p−2 L ([0, 1])
for p < 2, for p = 2, for p > 2.
Proof. Let ξ ∈ Dg . If ξ = 0, then there exists an ε > 0 for which Xε := {|ξ | > ε} has positive measure. Now Lp ([0, 1]) · ξ ⊆ L2 ([0, 1]) implies that Lp (Xε ) ⊆ L2 (Xε ), and in view of Remark 10.1(b/c), this leads to p 2. For p = 2, the multiplication with ξ defines a bounded operator on L2 ([0, 1]), so that ξ ∈ ∞ L ([0, 1]). This proves that Dg = L∞ ([0, 1]) in this case. Now we assume that p > 2. The condition ξ · Lp ([0, 1]) ⊆ L2 ([0, 1]) is equivalent to ξ 2 · p p/2 . L ([0, 1]) ⊆ L1 ([0, 1]), which is equivalent to ξ 2 ∈ Lq ([0, 1]) for q1 + p2 = 1, i.e., q = p−2 2p
This proves that Dg = L p−2 ([0, 1]).
2
Next we note that span{f k : f ∈ g} = span{f1 · · · fk : f1 , . . . , fk ∈ g} shows that
Dgk = ξ ∈ H: ∀f ∈ Lp [0, 1], R f k ξ ∈ H . Now
f k : f ∈ g = Lp [0, 1] = Lp/k [0, 1]
for p k,
leads for p k to
Dgk = ξ ∈ H: ∀f ∈ Lp/k [0, 1], R f ξ ∈ H = DLp/k .
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Proposition 10.4. For p ∈ N and k p we have
Dgk = DLp/k
⎧ {0} ⎪ ⎨ ∞ = L ([0, 1]) ⎪ 2p ⎩ p−2k L ([0, 1])
for k > p2 , for k = p2 , for k < p2 .
For p = 2 we obtain in particular Dg2 = {0} and Dg1 = L∞ ([0, 1]) which refines the observations in [3]. Remark 10.5. The preceding proposition also shows that there exists for every n ∈ N a continuous unitary representation (π, H) of a Lie group G = (L2n ([0, 1], R), +) with Dgn+1 = {0} and Dgn = {0}. Remark 10.6. It is easy to see that the norm · k on the space Dgk is equivalent to the natural norm suggested by Proposition 10.4. For p = 4 and Dg = L4 ([0, 1]), the density of D2 = L∞ ([0, 1]) in Dg implies the continuity of the isometric action of G on Dg (cf. Remark 5.3). Remark 10.7. For maps between Fréchet spaces, we also have the stronger notion of C k -maps in the Fréchet sense. If ξ ∈ H = L2 ([0, 1], C) is a C 1 -vector for G = (Lp ([0, 1], R), +), then the orbit map π ξ (g) = eig ξ has in g the differential Tg π ξ f = eig f ξ. Therefore ξ is a Fréchet-C 1 -vector if and only if the map F : G → L(g, H),
g → Meig ξ ,
Mh f = hf,
is continuous. We consider the case p = 2. Then L(g, H) ∼ = L(H), so that we may consider F as a map F : G → L∞ [0, 1], C ,
g → eig ξ,
and the question is when this map is continuous. First we consider the case ξ = 1. Then F is a homomorphism of Banach–Lie groups. If it is continuous, it is smooth, which implies that L(F ) : g → L∞ ([0, 1], C), g → ig, is a continuous liner map, which is not the case. For a general ξ ∈ L∞ ([0, 1], C), we consider for ε > 0 the subsets Xε := {|ξ | ε}. On each of these sets, ξ |Xε is invertible in the Banach algebra L∞ ([0, 1], C), so that multiplication with ξ −1 leads to the situation of the previous paragraph. Therefore Xε has measure zero, and since ε > 0 was arbitrary, ξ = 0. Hence all Fréchet-C 1 -vectors for G = L2 ([0, 1], R) are trivial. Another way to put this is to say that the action of G on the Banach space Dg = L∞ ([0, 1], C) has no non-zero continuous vector. On the other hand, we know that every C 2 -vector is a Fréchet-C 1 -vector [38, Thm. I.7], so that we obtain non-trivial Fréchet-C 1 -vectors for G = L4 ([0, 1], R).
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11. Smooth vectors for direct limits Definition 11.1. If (Gn )n∈N is a sequence of finite dimensional Lie groups with homomorphisms ϕn : Gn → Gn+1 , then Glöckner has shown in [11] that the corresponding direct limit group G := lim −→ Gn , endowed with the direct limit topology carries a compatible Lie group structure with Lie algebra g := lim −→ L(Gn ), endowed with the direct limit topology. We call G a direct limit Lie group. Remark 11.2. The Lie groups obtained by this construction are precisely the Lie groups G with a smooth exponential function whose Lie algebra g is a countable union of finite dimensional subalgebras, endowed with the direct limit topology. In fact, writing g = n∈N gn with finite dimensional Lie algebras gn ⊆ gn+1 , we obtain a corresponding sequence of Lie groups Gn injecting into G, and then it is not hard to verify that the corresponding morphism of Lie groups lim −→ Gn → G is an isomorphism (cf. [13]). Theorem 11.3. For each continuous unitary representation (π, H) of a direct limit Lie group G, the space of smooth vectors is dense. Proof. Since H is a direct sum of subspaces on which the representation if cyclic, we may w.l.o.g. assume that the representation is cyclic. Since smoothness of a vector in H is equivalent to smoothness for the identity component, we may also assume that G is connected. Then G is a countable direct limit of connected finite dimensional Lie groups, hence separable, and therefore the cyclicity of (π, H) implies that H is separable. Danilenko shows in [7] that there exists a dense subspace D ⊆ H satisfying (a) D is π(G)-invariant. (b) D ⊆ Dg . (c) d π(x)D ⊆ D for every x ∈ g. Since each group Gn is in particular a Banach–Lie group, conditions (b) and (c) imply that D ⊆ Dg∞n , so that Lemma 3.4 implies that D consists of smooth vectors for each Gn . Since G is also the direct limit of the Gn in the category of smooth manifolds [11], D ⊆ H∞ consists of smooth vectors for G. 2 Remark 11.4. Typical examples of Lie algebras g which are locally finite in the sense that every finite subset generates a finite dimensional subalgebra are nilpotent Lie algebras. These Lie algebras are countable direct limits of finite dimensional ones if they are countably dimensional. This condition is very restrictive, so that one is also interested in situations, where the Lie algebra g is not of countable dimension but carries a locally convex topology which is coarser than the direct limit topology. An important class of corresponding groups are the Heisenberg groups Heis(V ) of a locally convex space V , endowed with a continuous scalar product (a locally convex Euclidean space). More precisely, we have Heis(V ) = R × V × V , (z, v, w) z , v , w := z + z + 1/2 v, w − v , w , v + v , w + w .
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A unitary representation (π, H) of this group with π(z, 0, 0) = eiz for z ∈ R provides a representation of the canonical commutation relations in the sense that the restriction to the subgroups {0} × V × {0} and {(0, 0)} × V defines a unitary representation, and for v, w ∈ V we have π(0, v, 0)π(0, 0, w) = eiv,w π(0, v, w) = e2iv,w π(0, 0, w)π(0, v, 0). In [18] Hegerfeldt shows that, if V is separable, barreled and nuclear, for any continuous unitary representation of G = Heis(V ) there exists a dense subspace D with the following properties: (a) (b) (c) (d)
D is π(G)-invariant. D ⊆ Dg . d π(x)D ⊆ D for every x ∈ g, in particular D ⊆ Dg∞ . For each v ∈ D the map ωvn : gn → H is continuous.
In view of Lemma 3.4 and the local exponentiality of G, (c) and (d) imply that D consists of smooth vectors for G. Actually Hegerfeldt shows that the elements of Dg even have analyticity properties that lead to holomorphic extensions of their orbit maps to the complexified group GC . For a detailed discussion of this aspect we refer to [44]. 12. Smooth vectors for projective limits Structurally direct limits of finite dimensional Lie groups are groups with a relatively simple structure, but they have many continuous unitary representation because they carry a very fine topology. This situation is the opposite of what we find for projective limits of finite dimensional Lie groups. These groups are also called pro-Lie groups, and the Lie groups among the pro–Lie groups have been characterized recently in [19]. For any such Lie group G = lim ←− Gj , it makes sense to ask for smooth vectors in unitary representations. As we shall see in this section, for this class of Lie groups the space H∞ is always dense. Here the main point is a quite general argument concerning unitary representations of projective limits of topological groups. Let G be a topological group and N := (Ni )i∈I be a filter basis of closed normal subgroups Ni P G with lim N = {1}, i.e., for each 1-neighborhood U in G there exists some i ∈ I with Ni ⊆ U . Lemma 12.1. Let (π, H) be a continuous unitary representation of G. Then the union of the closed invariant subspaces HNi of Ni -fixed vectors is dense in H.
i∈I
H Ni
Proof. Let v ∈ H and pick ε > 0. Then there exists some i with π(Ni )v ⊆ Bε (v). Then conv(Ni .v) is a bounded closed convex Ni -invariant subset of H, hence contains a Ni -fixed point by the Bruhat–Tits Fixed Point Theorem [30]. Therefore dist(v, HNi ) ε. 2 Theorem 12.2. Any continuous representation (π, H) of G is a direct sum of representations on which some Ni acts trivially.
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Proof. Let F be the set of all sets F of pairwise orthogonal G-invariant closed subspaces of H on which some Ni acts trivially. We order F by set inclusion. Then F is inductively ordered, so that Zorn’s Lemma implies the existence of a maximal element Fm . Then K :=
Hj
Hj ∈Fm
is a closed G-invariant subspace of H. We claim that K = H, which implies the assertion. If this is not the case, K⊥ is non-zero, and Lemma 12.1 implies that for some i the set (K⊥ )Ni is non-zero, contradicting the maximality of Fm . 2 Corollary 12.3. Each irreducible continuous unitary representation of G factors through some G/Ni , i.e., the set G of equivalence classes of irreducible continuous unitary representations satisfies G = i∈I (G/Ni ). Theorem 12.4. If G = lim ←− Gj is a Lie group which, as a topological group, is a projective limit of finite dimensional Lie groups Gj , then for each continuous unitary representation (π, H) of G the space H∞ of smooth vectors is dense. Proof. Let qj : G → Gj be the natural projections and apply Theorem 12.2 to the family Nj = ker qj . This reduces the problem to the case where some Nj acts trivially on H, so that we actually have a representation of the finite dimensional quotient Lie group Gj ∼ = G/Nj . Now the assertion follows from the density of smooth vectors for Gj in HNj [10]. 2 Remark 12.5. The group G = RN is a projective limit of the Lie group Gn = Rn , where qn : G → Gn is the projection onto the first n factors. In this sense Example 4.8 is a continuous unitary representation of a pro-Lie group. Remark 12.6. Let G = (V , +) be the additive group of a locally convex space V . For each continuous seminorm p ∈ P(V ), we have a closed subspace Np := p −1 (0) for which p induces a norm on the quotient space V /Np . Now N = {Np : p ∈ P(V )} is a filter basis of closed subgroups with lim N = {1}, so that Theorem 12.2 applies. We conclude that every continuous unitary representation (π, H) of V is a direct sum of representations (π, Hi ) on which some Np acts trivially, so that the representation πi factors through a representation of the normed space V /Np . 13. Perspectives There are several interesting problems concerning representations of infinite dimensional Lie groups G on Banach spaces V . Problem 13.1 (Integrability). Suppose that V is a locally convex space, D ⊆ V a dense subspace and ρ : g → End(D) a representation of the Lie algebra g on D. We thus obtain for each x ∈ g an unbounded operator ρ(x) on V . We assume that all these operators are closable and that each closure ρ(x) generates a strongly continuous one-parameter group. A characterization of such operators on Banach spaces is given by the Hille–Yoshida Theorem, and [24] contains some generalization to locally convex spaces.
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Suppose that g is a Banach–Lie algebra. Then we obtain a map F : g → GL(V ),
x → eρ(x) .
When does this map define a local group homomorphism, hence a representation of any corresponding simply connected Lie group G? More precisely, under which conditions on ρ do we have F (x ∗ y) = F (x)F (y) for x and y in a small ball centered in 0? For finite dimensional Lie algebras, problems of this kind are studied in [20, Thms. 1.1] and Chapter 8 of [23]. Maybe some of these results, such as Theorem A.1–3 in [21] can be extended to Banach–Lie algebras. It would also be interesting to have a version of the integrability result [23, Thm. 8.1] for representations on locally convex space or [23, Thm. 8.6] for representations on Banach spaces (cf. also [34]). A first step in this direction is taken by S. Merigon in [31], where he obtains such a result for representation on Hilbert spaces. Problem 13.2 (Smoothness). We have seen above that for every continuous representation (π, V ) of the Banach–Lie group G on the Banach space V , we obtain a sequence (V k , · k ) of Banach spaces, where V k = Dgk is the space of C k -vectors, endowed with its natural norm (Theorem 9.4), and in this picture V ∞ is the projective limit of the Banach spaces V k (in the category of locally convex spaces) (cf. [23, p. 20] for finite dimensional Lie algebras). The C k -variant of the derived action is given by the sequence g × V k → V k−1 ,
(x, v) → dπ(x)v,
k ∈ N,
(16)
of continuous bilinear maps. Interesting questions in this context are: (a) Does the density of V ∞ in V imply the density in each of the Banach spaces V k ? (b) In [23] the continuity of the action of G on the Banach space (Dg , · 1 ) (graph density) plays an important role. As follows from Proposition 9.7 and Remark 5.3, this is equivalent to the density of the continuous vectors in Dg , which in turn follows from the density of Dg2 in Dg1 . Maybe these conditions can also be exploited for Banach–Lie groups. (c) Suppose that we are given a Lie algebra representation ρ : g → End(D), where D ⊆ V is a dense subspace. Suppose that all the k-linear maps ωvk : gk → V are continuous and write V k for the completion of D with respect to the norm vk := v +
k
j
ω . v
j =1
Then V k can be realized as subspaces of V . Is it possible to characterize in this context representations which are integrable to continuous representations of G on V (cf. [23] for a discussion of similar problems for finite dimensional Lie algebras). Natural assumptions in this context are that the closures of the operators ρ(x) generate one-parameter groups preserving D (and all the spaces V k ).
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Problem 13.3. We have seen in Sections 4 and 5 that for every unitary representation (π, H) of a Banach–Lie group G, the space H∞ carries a natural Fréchet topology with respect to which G acts smoothly. Can this information be used to show in certain situations that a direct integral decomposition ⊕ (π, H) = (πx , Hx ) dμ(x) X
also yields a “direct integral decomposition” H
∞
⊕ =
Hx∞ dμ(x)?
X
This would be extremely useful for the analysis of smooth unitary representations. For results of this type for finite dimensional groups we refer to [2]. Problem 13.4. The argument in the proof of Theorem 8.2 touches on an interesting question concerning the differentiability of functions f : G → R on a Banach–Lie group. Suppose that for every g ∈ G and x ∈ g the derivative d df (g)(g.x) := f g expG (tx) dt t=0 exists. When are the maps df (g) : g → R linear? They clearly satisfy df (g)(λx) = λd f (g)x for λ ∈ R, so that the additivity is the crucial issue. The connection to Theorem 8.2 is given by functions of the form f (g) = α(π(g)v) with α ∈ V and v ∈ Dg , because in this case we have df (1)(x) = α(d π (x)v) = α(ωv (x)), and the additivity of every α is equivalent to the additivity of ωv . In Lemma 3.3 we have already seen that, for v ∈ Dg , the continuity of the map ωv : g → V implies its linearity. Any more direct proof of the continuity of ωv for v ∈ Dg would therefore lead to a more direct proof of Theorem 8.2. Problem 13.5. For “selfadjoint” representations of the complex enveloping algebra U(g)C of a finite dimensional Lie algebra g on the dense subspace D of the Hilbert space H, there exists an integrability criterion due to Powers [59, Thm. 4.5]. The requirement is that the map π : U(g)C → End(D) is “completely strongly positive” with respect to a certain convex cone Q ⊆ U(g)C . It would be very interesting to see if this result extends to Banach–Lie groups. Problem 13.6. As we have seen in Corollary 7.4, a positive definite function on a Lie group G is smooth if it is smooth in some identity neighborhood. In some case one may even expect that the whole function can be reconstructed from the restriction to some identity neighborhood, even if it is not analytic. Theorem 2.1 in [22] contains a criterion for the extendability of a “local” positive definite function to the whole group for finite dimensional unimodular Lie groups. It would be very interesting to understand if there are variants of this result for more general topological groups and in particular for infinite dimensional Lie groups. Here the key point is to find appropriate positivity conditions, such as the complete strong positivity used in [22, Cor. 4.1].
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Acknowledgments We thank Stéphane Merigon and Christoph Zellner for a careful reading of this paper and for several discussions on its subject matter. References [1] S. Albeverio, R.J. Høegh-Krohn, J.A. Marion, D.H. Testard, B.S. Torresani, Noncommutative Distributions – Unitary Representations of Gauge Groups and Algebras, Pure Appl. Math., vol. 175, Marcel Dekker, New York, 1993. [2] D. Arnal, Symmetric nonself-adjoint operators in an enveloping algebra, J. Funct. Anal. 21 (1976) 432–447. [3] D. Beltita, K.-H. Neeb, A non-smooth continuous unitary representation of a Banach–Lie group, J. Lie Theory 18 (2008) 933–936. [4] D. Beltita, K.-H. Neeb, Schur–Weyl theory for C ∗ -algebras, in preparation. [5] N. Bourbaki, Espaces vectoriels topologiques, Chap.1 à 5, Springer-Verlag, Berlin, 2007. [6] P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, Mem. Amer. Math. Soc., vol. 140, Amer. Math. Soc., Providence, RI, 1974. [7] A.I. Danilenko, Gårding domains for unitary representations of countable inductive limits of locally compact groups, Mat. Fiz. Anal. Geom. 3 (3–4) (1996) 231–260. [8] G. van Dijk, Introduction to Harmonic Analysis and Generalized Gelfand Pairs, Stud. Math., vol. 36, de Gruyter, Berlin, 2009. [9] J. Faraut, Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. 5 (1979) 369–444. [10] L. Gårding, Note on continuous representations of Lie groups, Proc. Natl. Acad. Sci. USA 33 (1947) 331–332. [11] H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Univ. 43 (2003) 1–26. [12] H. Glöckner, Lie groups over non-discrete topological fields, arXiv:math.GR/0408008. [13] H. Glöckner, K.-H. Neeb, Infinite Dimensional Lie Groups, vol. I, Basic Theory and Main Examples, in preparation. [14] R.W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969) 55–76. [15] R.W. Goodman, One-parameter groups generated by operators in an enveloping algebra, J. Funct. Anal. 6 (1970) 218–236. [16] H. Glöckner, K.-H. Neeb, Banach–Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math. 560 (2003) 1–28. [17] G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspect. Math., Acad. Press, 1994. [18] G.C. Hegerfeldt, Gårding domains and analytic vectors for quantum fields, J. Math. Phys. 13 (1972) 821–827. [19] K.H. Hofmann, K.-H. Neeb, Pro-Lie groups which are infinite dimensional Lie groups, Math. Proc. Cambridge Philos. Soc. 146 (2009) 351–378. [20] P.E.T. Jørgensen, The integrability problem for infinite-dimensional representations of finite-dimensional Lie algebras, Exp. Math. 4 (1983) 289–306. [21] P.E.T. Jørgensen, Operators and Representation Theory, Math. Stud., vol. 147, North-Holland, 1988. [22] P.E.T. Jørgensen, Integral representations for locally defined positive definite functions on Lie groups, Internat. J. Math. 2 (3) (1991) 257–286. [23] P.E.T. Jørgensen, R.T. Moore, Operator Commutation Relations, Math. Appl., D. Reidel Publ. Co., Dordrecht– Boston–Lancaster, 1984. [24] T. Komura, Semigroups of operators on locally convex spaces, J. Funct. Anal. 2 (1968) 258–296. [25] Y.G. Kondratiev, Nuclear spaces of entire functions in problems of infinite-dimensional analysis, Sov. Math. Dokl. 22 (2) (1980) 588–592. [26] Y.G. Kondratiev, Ju.S. Samo˘ilenko, Integral representations of generalized positive definite kernels of an infinite number of variables, Sov. Math. Dokl. 17 (2) (1976) 517–521. [27] Y.G. Kondratiev, L. Streit, W. Westerkamp, J. Yan, Generalized functions in infinite dimensional analysis, arXiv:math.FA/0211196v1, 13 November 2002. [28] M.G. Krein, Hermitian positive definite kernels on homogeneous spaces I, II, Ukrainian Math. J. 1 (1949) 64–98, Ukrainian Math. J. 2 (1950) 10–59, English transl. in: Amer. Math. Soc. Transl. Ser. 2 34 (1963) 69–108, 109– 164.
2854
K.-H. Neeb / Journal of Functional Analysis 259 (2010) 2814–2855
[29] A. Kriegl, P. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr., vol. 53, Amer. Math. Soc., 1997. [30] S. Lang, Math Talks for Undergraduates, Springer-Verlag, 1999. [31] S. Merigon, Integrating representations of Banach–Lie algebras, arXiv:1003.0999v1 [math.RT], 4 March 2010. [32] P. Michor, The moment mapping for unitary representations, Ann. Global Anal. Geom. 8 (3) (1990) 299–313. [33] J. Milnor, Remarks on infinite-dimensional Lie groups, in: B. DeWitt, R. Stora (Eds.), Relativité, groupes et topologie II, Les Houches, 1983, North-Holland, Amsterdam, 1984, pp. 1007–1057. [34] R.T. Moore, Exponentiation of operator Lie algebras on Banach spaces, Bull. Amer. Math. Soc. 71 (1965) 903–908. [35] C. Müller, K.-H. Neeb, H. Seppänen, Borel–Weil theory for root graded Banach–Lie groups, Int. Math. Res. Not. IMRN 5 (2010) 783–823. [36] K.-H. Neeb, Holomorphic highest weight representations of infinite dimensional complex classical groups, J. Reine Angew. Math. 497 (1998) 171–222. [37] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expo. Math., vol. 28, de Gruyter Verlag, Berlin, 1999. [38] K.-H. Neeb, Infinite dimensional groups and their representations, in: A. Huckleberry, T. Wurzbacher (Eds.), Infinite Dimensional Kähler Manifolds, in: DMV-Seminar, vol. 31, Birkhäuser Verlag, 2001, pp. 131–178. [39] K.-H. Neeb, Borel–Weil theory for loop groups, in: A. Huckleberry, T. Wurzbacher (Eds.), Infinite Dimensional Kähler Manifolds, in: DMV-Seminar, vol. 31, Birkhäuser Verlag, 2001, pp. 179–229. [40] K.-H. Neeb, Compressions of infinite dimensional bounded symmetric domains, Semigroup Forum 63 (1) (2001) 71–105. [41] K.-H. Neeb, Classical Hilbert–Lie groups, their extensions and their homotopy groups, in: A. Strasburger, et al. (Eds.), Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, in: Banach Center Publ., vol. 55, Polish Acad. Sci., Warszawa, 2002, pp. 87–151. [42] K.-H. Neeb, Infinite-dimensional Lie groups and their representations, in: J.P. Anker, B. Ørsted (Eds.), Lie Theory: Lie Algebras and Representations, in: Progr. Math., vol. 228, Birkhäuser Verlag, 2004, pp. 213–328. [43] K.-H. Neeb, Towards a Lie theory of locally convex groups, Jpn. J. Math. 1 (2) (2006) 291–468. [44] K.-H. Neeb, On analytic vectors for unitary representations of infinite dimensional Lie groups, arXiv:1002.4792v1 [math.RT], 25 February 2010. [45] K.-H. Neeb, Semibounded representations and invariant cones in infinite dimensional Lie algebras, Confluentes Math. 2 (1) (2010) 37–134. [46] K.-H. Neeb, Semibounded representations of infinite dimensional hermitian groups, in preparation. [47] K.-H. Neeb, Lie groups of bundle automorphisms and their extensions, in: K.-H. Neeb, A. Pianzola (Eds.), Developments and Trends in Infinite Dimensional Lie Theory, in: Prog. Math., Birkhäuser, in press, arXiv:0709.1063 [math.DG] 7 September 2007. [48] K.–H. Neeb, B. Ørsted, Unitary highest weight representations in Hilbert spaces of holomorphic functions on infinite dimensional domains, J. Funct. Anal. 156 (1998) 263–300. [49] K.-H. Neeb, H. Seppänen, Borel–Weil theory for groups over commutative Banach algebras, J. Reine Angew. Math., in press, arXiv:0909.1697 [math.RT] 11 September 2009. [50] K.-H. Neeb, F. Wagemann, Lie group structures on groups of maps on non-compact manifolds, Geom. Dedicata 134 (2008) 17–60. [51] K.-H. Neeb, Chr. Zellner, Oscillator algebras with semi-equicontinuous coadjoint orbits, in preparation. [52] A. Neklyudov, Chernoff and Trotter type product formulas, arXiv:0803.1283v1 [math.FA]. [53] A. Neklyudov, Inversion of Chernoff’s theorem, Math. Notes 83 (4) (2008) 530–538. [54] G.I. Olshanski, Unitary representations of infinite dimensional (G, K)-pairs and the formalism of R. Howe, in: A.M. Vershik, D.P. Zhelobenko (Eds.), Representations of Lie Groups and Related Topics, in: Adv. Stud. Contemp. Math., vol. 7, Gordon and Breach Science Publ., 1990. [55] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983. [56] G.K. Pedersen, C∗ -Algebras and Their Automorphism Groups, Academic Press, London, 1989. [57] D. Pickrell, The separable representations of U (H), Proc. Amer. Math. Soc. 102 (1988) 416–420. [58] D. Pickrell, Separable representations for automorphism groups of infinite symmetric spaces, J. Funct. Anal. 90 (1) (1990) 1–26. [59] R.T. Powers, Selfadjoint algebras of unbounded operators, II, Trans. Amer. Math. Soc. 187 (1) (1974) 261–293. [60] A. Pressley, G. Segal, Loop Groups, Oxford University Press, Oxford, 1986. [61] W. Rudin, Functional Analysis, McGraw–Hill, 1973. [62] Y.S. Samoilenko, Spectral Theory of Families of Self-Adjoint Operators, Math. Appl. (Sov. Ser.), Kluwer Acad. Publ., 1991.
K.-H. Neeb / Journal of Functional Analysis 259 (2010) 2814–2855
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[63] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Oper. Theory Adv. Appl., vol. 37, Birkhäuser Verlag, Basel, 1990. [64] H. Shimomura, Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations, J. Funct. Anal. 187 (2001) 406–441. [65] J. Simon, Gårding domain for representations of some Hilbert–Lie groups, Lett. Math. Phys. 1 (1975) 23–29.
Journal of Functional Analysis 259 (2010) 2856–2885 www.elsevier.com/locate/jfa
Existence of solutions for the equations modeling the motion of rigid bodies in an ideal fluid Jean Gabriel Houot a , Jorge San Martin b , Marius Tucsnak a,∗ a Institut Élie Cartan, Nancy Université/CNRS/INRIA, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France b Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile,
Casilla 170/3 – Correo 3, Santiago, Chile Received 26 April 2010; accepted 13 July 2010 Available online 21 July 2010 Communicated by J. Coron
Abstract In this paper, we study the motion of rigid bodies in a perfect incompressible fluid. The rigid-fluid system fills a bounded domain in R3 . Adapting the strategy from Bourguignon and Brezis (1974) [1], we use the stream lines of the fluid and we eliminate the pressure by solving a Neumann problem. In this way, the system is reduced to an ordinary differential equation on a closed infinite-dimensional manifold. Using this formulation, we prove the local in time existence and uniqueness of strong solutions. © 2010 Elsevier Inc. All rights reserved. Keywords: Euler equations; Rigid body-fluid interaction
Notation. Throughout this paper Ω denotes an open bounded and connected subset of R3 and S0 is a closed set with nonempty interior and with smooth boundary such that S0 ⊂ Ω. We denote as usual by SO3 (R) the special orthogonal group on R3 . We will often use functions defined from a time interval to R3 or to SO3 (R). These functions will be denoted using bold characters, such as h : [0, T ] → R3 or R : [0, T ] → SO3 (R). The same kind of notation will be used for three other time dependent vector fields k, ω, η and ξ which will be defined in the sequel. The five time dependent fields mentioned above will define the state z of the fluid-solid system. A vector from R3 or a matrix from SO3 (R) will be denoted by h or by R, respectively. The transposed * Corresponding author.
E-mail addresses:
[email protected] (J.G. Houot),
[email protected] (J.S. Martin),
[email protected] (M. Tucsnak). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.006
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∗ of a matrix a will be denoted by so that the column vector of components a and b is denoted either b or by (a, b)∗ . Differentiation with respect to time is often denoted a dot. The vector, respectively the inner, product of v, w ∈ R3 will be denoted by v ∧ w and v · w, respectively. The Jacobian matrix of a vector field y → f (y) defined on an open subset of R3 will be denoted by Dy f or simply by Df .
1. Introduction The interaction of rigid bodies and of ideal fluids is a topic which was probably first tackled by d’Alembert, Kelvin and Kirchhoff, who considered the case of a potential fluid (sometimes called inviscid fluid), with the solid-fluid system filling the whole space. In this case the governing equations can be reduced to system of ODE’s on a finite-dimensional manifold. We refer to the book of Lamb [9, Chapter 6] for a detailed presentation of these early contributions and to Kanso, Marsden, Rowley and Melli-Huber [8] for the application of the above theory to selfpropelled motions of solids in an inviscid fluid. Recently Houot and Munnier in [7] used shape sensitivity analysis techniques to deal with either bounded or unbounded domains. They also tackled the special case of a cylinder in a half space. They showed in particular that, unlike the case of a viscous fluid (see San Martín, Starovoitov and Tucsnak [15], Hillairet [6], Hesla [5]), the cylinder can touch the wall in finite time with nonzero velocity. The damping effect of the wall on the cylinder is also studied. In the general case the system is genuinely infinite-dimensional, so it cannot be reduced to ODE’s on finite-dimensional manifolds. As usual in fluid-solid interaction problems, a major difficulty comes from the fact that the equations for the fluid (Euler’s equations in our case) hold in a time dependent domain, so that we have a free boundary value problem. As far as we know, the first papers tackling the case of a non-potential flow are Ortega, Rosier and Takahashi [12] and [13]. The main result in these works asserts the existence and uniqueness of classical solutions in two space dimensions and with the rigid-fluid system filling the whole space. More recently, Rosier and Rosier in [14] proved the existence of strong solutions in the case in which the solid is a ball, with the fluid-rigid system filling Rn , with n 2. The aim of the present work is to prove the existence and uniqueness of strong solutions in three space dimensions, with a bounded fluid-rigid domain and with the possibility of considering more than one solid. An idea which seems attractive, since it yields a transformed problem written in a fixed domain, is the use of groups of diffeomorphisms as proposed in Ebin and Marsden [3]. Our approach, based on this idea, follows more closely Bourguignon and Brezis [1]. The first new difficulty we need to tackle is that, the fluid domain being variable and the normal velocity of the fluid being different from zero on the fluid-solid interface we are not able to apply the Leray projector. Therefore, in order to eliminate the pressure we need to solve non-homogeneous Neumann problems for the Laplacian. The second difficulty consists in the fact that we need to compare solutions of these Neumann problems in different domains and to show that they depend smoothly on some geometric parameters. To be more precise, the motion of the fluid is described by the classical Euler equations, whereas the motion of the rigid bodies is governed by the balance equations for linear and angular momentum (Newton’s laws). For the sake of simplicity we state and prove our results in the case of a single rigid body, but our methods can be easily adapted to the case of several rigid bodies. Assume that the system fluid-rigid body fills the domain Ω in R3 and that at t = 0 the solid is located at S0 (see the paragraph on notation from the beginning of the paper for the properties of Ω and S0 ). The position of the solid at instant t 0 is denoted by S(t). We assume that
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the solid is surrounded by a perfect homogeneous incompressible fluid filling, for each t 0, the domain F (t) = Ω \ S(t). In this work we study the following initial and boundary value problem: ρF
∂u + ρF (u · ∇)u + ∇p = 0, x ∈ F (t), t 0, ∂t div u = 0, x ∈ F (t), t 0,
u · n = 0, x ∈ ∂Ω, t 0, u · n = h˙ + ω ∧ (x − h) · n, x ∈ ∂S(t), t 0, ¨ pn dx, t 0, ms h =
(1.1a) (1.1b) (1.1c) (1.1d) (1.1e)
∂S(t)
d (J ω) = dt
p(x − h) ∧ n dx,
t 0,
(1.1f)
∂S(t)
˙ R(t) = A ω(t) R(t), u(0, x) = u0 (x), h(0) = h0 ,
˙ h(0) = k0 ,
t 0,
(1.1g)
x ∈ F (0),
(1.1h)
R(0) = IdM3 ,
ω(0) = ω0 ,
(1.1i)
where the unknowns are u (the Eulerian velocity field of the fluid), p (the pressure of the fluid), h (the trajectory of the mass center of the rigid body), R (the time variation of the orthogonal matrix giving the orientation of the solid) and ω (the time variation of the angular velocity of the rigid body). The density of the fluid ρF is supposed to be a constant. The fluid occupies, at t = 0, the domain F0 = Ω \ S0 . The domain S(t) is defined by S(t) = h(t) + R(t)(y − h0 ) y ∈ S0 , t 0 . The skew-symmetric matrix A(ω) is given by A(ω) =
0 ω3 −ω2
−ω3 0 ω1
ω2 −ω1 0
ω ∈ R3 .
(1.2)
The notation ms stands for the mass of the solid and J (t) designs its inertia matrix defined by Ji,j (t) = ρs
x − h(t) ∧ ei · x − h(t) ∧ ej dx
i, j ∈ {1, 2, 3} ,
(1.3)
S(t)
where the constant ρs stands for the density of the solid and (ek )k=1,2,3 is the canonical basis in R3 . It is easy to check that J (t) = R(t)J0 R∗ (t) for every t 0, where J0 is the matrix defined by (J0 )i,j = ρS S0
(y − h0 ) ∧ ei · (y − h0 ) ∧ ej dy,
(1.4)
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for every i, j ∈ {1, 2, 3}. Notice that the matrix J0 does not depend on the position of the solid and that the last formula easily implies that d ˙ (J ω) = ω ∧ (J ω) + J ω. dt
(1.5)
Moreover, we have denoted by ∂S(t) the boundary of the rigid body at instant t and by n(t, x) the unit normal to ∂S(t) at the point x directed to the interior of the rigid body. Throughout this paper we assume that the considered boundaries are smooth in the sense that there exist the functions δ0 , δ1 ∈ C ∞ (R3 , R) such that ∂Ω = x ∈ R3 δ0 (x) = 0 , n(x) = −∇δ0 (x),
x ∈ ∂Ω,
∂S(0) = x ∈ R3 δ1 (x) = 0 ,
(1.6)
n(x) = −∇δ1 (x),
(1.7)
x ∈ ∂S(0).
An important role in this work will be played by the set P(Ω, S0 ), defined as follows: Definition 1.1. The set of all admissible solid configurations from the solid position S0 , denoted P(Ω, S0 ), is the set of all pairs Rh11 ∈ R3 × SO3 (R) such that there exist functions h ∈ C [0, 1]; R3 ,
R ∈ C [0, 1]; SO3 (R) ,
with h(0) = h0 ,
h(1) = h1 ,
R(0) = Id3 , R(1) = R1 , h(t) + R(t)(y − h0 ) ∈ Ω t ∈ [0, 1], y ∈ S0 .
Remark 1.2. For each t 0 the position of the solid and the domain filled by the fluid are completely described by the pair (h(t), R(t))∗ ∈ P(Ω, S0 ). Therefore, the evolution of the domains F (t) and S(t) is totally described by the function q ∈ C 2 ([0, T ], P(Ω, S0 )) defined by q(t) = (h(t), R(t))∗ . Consequently, in the remaining part of this work, we use the notation Fq(t) and Sq(t) instead of F (t) and S(t). We also denote q0 = (h0 , IdM3 )∗ = q(0). More generally, for every q = (h, R)∗ ∈ P(Ω, S0 ) we denote Sq = h + R(y − h0 ) y ∈ S0 ,
Fq = Ω \ S q .
(1.8)
In order to give a precise statement of our main result we first introduce some spaces. For an open set O ⊂ R3 we denote
N m (O) = q ∈ H m (O) q(x) dx = 0 .
(1.9)
O
We next define some spaces of functions defined on time variable domains. Let q ∈ C 2 ([0, ∞), P(Ω, S0 )) and let Ψ : [0, ∞) × R3 → R3 be a C 2 function such that for every t ∈ [0, T ], the map x → Ψ (t, x) is a C ∞ diffeomorphism from F0 to Fq(t) .
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Let v(t, ·), t 0 be a family of functions with v(t, ·) : Fq(t) → R3 for every t 0. Denote vΨ (t, y) = v(t, Ψ (t, y)), for all t 0 and for all y ∈ F0 . With the above notation we introduce the following function spaces: C k [0, T ], H m (Fq ) = v vΨ ∈ C k [0, T ], H m (F0 ) , C k [0, T ], N m (Fq ) = v vΨ ∈ C k [0, T ], N m (F0 ) , where k ∈ {0, 1}, m 0 is an integer and H m are the usual Sobolev spaces. It is not difficult to check that the above definitions are independent of the choice of the diffeomorphism Ψ . We can now state the main result in this paper. Theorem 1.3. Let m 3 be an integer. Let S0 ⊂ Ω be as in the notational preamble of this work 1 x dx. Let k0 ∈ R3 , ω0 ∈ R3 and u0 ∈ H m (F0 , R3 ) satisfy: and let h0 = vol(S 0 ) S0 div u0 (x) = 0, x ∈ F0 , u0 (x) · n(x) = k0 + ω0 ∧ (x − h0 ) · n(x), u0 (x) · n(x) = 0,
x ∈ ∂S0 ,
x ∈ ∂Ω.
Then there exists T0 > 0 such that (1.1) admits a unique solution (q, u, p) with q ∈ C 2 [0, T0 ), P(Ω, S0 ) , u ∈ C [0, T0 ), H m (Fq ) ∩ C 1 [0, T0 ), H m−1 (Fq ) , p ∈ C [0, T0 ), N m+1 (Fq ) .
(1.10) (1.11) (1.12)
2. Idea of the proof of Theorem 1.3 As already mentioned, the basic idea of the proof, borrowed from Bourguignon and Brezis [1], consists in reducing (1.1) to an ODE on an infinite-dimensional manifold. In this section we briefly describe this reduction process and we give the main steps of the proof of Theorem 1.3. Let q = Rh and u be functions satisfying (1.10) and (1.11) for some T0 > 0, with div u = 0. We introduce the flow η associated to u, which is defined as the solution of ∂η (t, y) = u t, η(t, y) , ∂t
η(0, y) = y
for all y ∈ F0 .
(2.1)
By the Cauchy–Lipschitz Theorem η(t, ·) is a diffeomorphism from F0 onto Fq(t) . Moreover, since div u = 0, by Liouville’s Theorem (see, for instance, Hartman [4, p. 96]), we have
det Dy η(t, y) = 1 t ∈ [0, T0 ), y ∈ F0 . Moreover, we set ∂η (t, y) = ξ (t, y) ∂t
(t 0, y ∈ F0 ).
(2.2)
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Notice that u can be expressed in terms of η and ξ by t ∈ [0, T0 ), y ∈ F0 .
u(t, x) = ξ t, η−1 (t, x)
(2.3)
In order to express (1.1) as a first-order ordinary differential equation we note that from the formula ∂u ∂ξ (t, y) = t, η(t, y) + (u · ∇)u t, η(t, y) , ∂t ∂t it follows that u satisfies (1.1a) iff ∂ξ (t, y) = −∇p t, η(t, y) ∂t
y ∈ F0 , t ∈ [0, T0 ) .
(2.4)
Consider the function k ∈ C 1 ([0, T0 ), R3 ) defined by ˙ = k(t) h(t)
t ∈ [0, T0 ) .
(2.5)
Define ω ∈ C 1 ([0, T0 ), R3 ) by ˙ R(t) = A ω(t) R(t) t ∈ [0, T0 ) .
(2.6)
As it will be shown in Sections 3 and 4, by solving appropriate Neumann problems, the pressure p can be expressed, for each t ∈ [0, T0 ) as a function of z = (η, q, ξ , k, ω)∗ , so that, using (2.4)–(2.6), the system (1.1) can be written in the equivalent form z˙ (t) = L z(t) ,
z(0) = z0 ,
where L : F m → E m , with E m = H m F0 , R3 × R3 × M3 (R) × H m F0 , R3 × R3 × R3 ,
(2.7)
and F m is a closed subset of E m . The norm of z ∈ E m is defined by z 2E m = η 2H m (F
0 ,R
3)
+ h 2 + R 2 + ξ 2H m (F
0 ,R
3)
+ k 2 + ω 2 ,
where · stand for the Euclidean norm on Rn . Endowed with this norm E m is a Hilbert space. For (q, u, p) satisfying (1.10)–(1.12) we define ⎛
⎞ η(t, ·) ⎜ q(t) ⎟ ⎜ ⎟ z(t) = ⎜ ξ (t, ·) ⎟ , ⎝ ⎠ k(t) ω(t) where ξ (t, ·), η(t, ·), k(t) and ω(t) are defined by (2.1), (2.2), (2.5) and (2.6), respectively.
(2.8)
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To define F m we introduce, for every q ∈ P(Ω, S0 ), the sets Diff m (F0 , Fq ) = η : F0 → Fq η invertible, η ∈ H m F0 , R3 ,
η−1 ∈ H m Fq , R3 and det Dy η(y) = 1 ,
η Σ m (Ω, S0 ) = σ = q ∈ P(Ω, S0 ) and η ∈ Diff m (F0 , Fq ) , q
(2.9) (2.10)
where P(Ω, S0 ) has been defined in Definition 1.1 and Fq is given in (1.8). The set Σ m (Ω, S0 ), simply denoted by Σ m in the sequel, is formed by admissible positions of the system. The ηthe set of admissible velocities from a position σ = q describes the tangent space to Σ m at the point σ , which is given by Tσ Σ m = (ξ, k, ω)∗ ∈ H m F0 , R3 × R3 × R3 u = ξ ◦ η−1 ∈ H m Fq , R3 ,
div u = 0 in Fq , u · n = 0 on ∂Ω, u · n = k + ω ∧ (x − h) · n on ∂Sq .
(2.11)
The subset F m of E m is defined by F m = z ∈ E m σ = (η, q)∗ ∈ Σ m , (ξ, k, ω)∗ ∈ Tσ Σ m .
(2.12)
It is not difficult to check that F m is a locally closed subset of E m , in the sense that for every z ∈ F m there exists a closed ball B of F m centered at z such that F m ∩ B is a closed subset of E m . Moreover, as it will be shown in Section 6, Σ m is an infinite-dimensional manifold and F m is its tangent bundle. The precise definition of L requires some preparation, so it is postponed to Sections 3–6. In order to prove the main result we show in Section 5 that L satisfies the assumptions of the following version of the Cauchy–Lipschitz Theorem, which is a particular case of Theorem 2 from Martin [11]. Proposition 2.1. Let F be a locally closed subset of a Hilbert space E and let L : [0, T ) × F → E be such that a) L is a locally Lipschitz in z and continuous in t; b) lims→0+ 1s dist(z + sL zt ; F ) = 0 ( zt ∈ [0, T ) × F ). Then for every z0 ∈ F there exists T0 > 0 such that the equation z˙ (t) = L t, z(t) ,
z(0) = z0
admits a unique solution z ∈ C 1 ([0, T0 ), F ). 3. Study of the pressure The study of the pressure p is the key point in order to reduce (1.1) to a system of ordinary differential equations. In this section we write the pressure as the sum of two terms, each of them satisfying a Neumann problem for the Laplacian. We first introduce some function spaces
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and we recall classical results related to Neumann problems. Let O be a bounded domain in R3 with smooth boundary ∂O, m ∈ N and n ∈ N. Recall the definition of N m (O) from (1.9) and let V m (O) the space defined by
V m (O) = (f, g)∗ ∈ H m (O) × H m+1/2 (∂O) f (x) dx + g(x) dσx = 0 . O
(3.1)
∂O
The following classical result on the well-posedness of the Neumann problem for the Laplace operator can be found in the book of Lions and Magenes [10, Chapter 5]. Theorem 3.1. Let m ∈ N. Then, for every
f g
∈ V m (O), the boundary value problem
− ϕ(x) = f (x), ∂ϕ (x) = g(x), ∂n
x ∈ O, x ∈ ∂O,
admits a unique solution ϕ ∈ N m+2 (O). Moreover, ϕ satisfies
∇ϕ · ∇ψ dx = O
f ψ dx +
O
gψ dσx
ψ ∈ H m+2 (O) ,
(3.2)
∂O
and there exists a constant C (depending only on O and m) such that ∇ϕ H m+1 (O) C f H m (O) + g H m+1/2 (∂ O)
f ∈ V m (O) . g
In order to prove that the boundary value problem for the pressure is well posed, we need several technical results. Let q ∈ P(Ω, S0 ). We first note that, thanks to the smoothness of ∂Fq , the map x → n(x), defined on ∂Fq , can be extended to Fq by a function in H m (Fq ). This extension is not unique so that the partial derivatives of n on ∂Fq are not uniquely determined. However, it can be easily checked that for every vector field τ which is tangent to ∂Fq , the ∂ni quantity 3j =1 τj ∂x , with i ∈ {1, 2, 3} does not depend on the choice of the extension. j Proposition 3.2. Let m 3 be an integer, let q ∈ P(Ω, S0 ) and assume that w ∈ H m (Fq , R3 ), ∂w m−1 (F ) whereas the function i w · n = 0 on ∂Fq . Then the function x → i,j ∂xij ∂w q ∂xj , is in H ∂ni m−1/2 x → i,j wi wj ∂xj , is in H (Γ ), where Γ is either ∂Ω or ∂Sq . Proof. The first property follows from the fact that, under our assumptions, H m−1 (Fq ) is an algebra. To prove the second property we notice that it suffices to use the fact that H m (Fq ) is an ∂ni (x) defined on Fq and the trace theorem. 2 algebra, the smoothness of the map x → ∂x j
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The above lemma allows us to introduce, for every q ∈ P(Ω, S0 ), the operators: Fq : H m Fq , R3 → H m−1 (Fq ), Gq : H m Fq , R3 → H m−1/2 (Γ ),
∂uj ∂ui , ∂xi ∂xj i,j ∂ni Gq (u) = ui uj , ∂xj Γ Fq (u) =
(3.3)
(3.4)
i,j
where Γ is either ∂Ω or ∂Sq . An important ingredient allowing to write (1.1) as an ordinary differential equation is the following result: Proposition 3.3. Let T0 > 0, let m 3 be an integer, let h = q ∈ C 2 [0, T0 ], P(Ω, S0 ) , R u ∈ C [0, T0 ), H m (Fq ) ∩ C 1 [0, T0 ), H m−1 (Fq ) , p ∈ C [0, T0 ), N m+1 (Fq ) . Assume that u satisfies (div u)(t, x) = 0,
x ∈ Fq(t) , t ∈ [0, T0 ),
(u · n)(t, x) = 0,
x ∈ ∂Ω, t ∈ [0, T0 ),
(u · n)(t, x) = v(t, x) · n(t, x),
x ∈ ∂Sq(t) , t ∈ [0, T0 ),
where ˙ + ω(t) ∧ x − h(t) , v(t, x) = h(t)
for all x ∈ Fq(t) , t ∈ [0, +∞).
(3.5)
Moreover, assume that u, p and q satisfy (1.1a). Then, for very t ∈ [0, T0 ), we have − p(t, x) = ρF Fq(t) (u)(t, x)
(x ∈ Fq(t) ),
∂p (t, x) = ρF Gq(t) (u)(t, x) (x ∈ ∂Ω), ∂n ∂p (t, x) = ρF Gq(t) (u − v)(t, x) + 2ρF (u − v) · ω(t) ∧ n(t, x) ∂n
¨ + ω(t) ˙ ∧ x − h(t) − ρF h(t) + ω(t) ∧ ω(t) ∧ x − h(t) · n(t, x) (x ∈ ∂Sq(t) ),
(3.6) (3.7)
(3.8)
where Fq and Gq are defined by (3.3) and (3.4) and v stands for the velocity of the solid defined by (3.5).
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Proof. Assume that u, p, q satisfy (1.1a). By applying the div operator to (1.1a) we get that p satisfies, for every t ∈ [0, T0 ), ∂u (t, x) + u(t, x) · ∇ u(t, x) − p(t, x) = ρF div ∂t
(x ∈ Fq(t) ).
(3.9)
By using the fact that div u ≡ 0, the right-hand side of the above relation can be expressed as
∂uj ∂ui ∂u div (t, x) + (u · ∇)u(t, x) = (t, x) + u.∇ div(u)(t, x) ∂t ∂xi ∂xj i,j
∂uj ∂ui = (t, x). ∂xi ∂xj i,j
The above formula and (3.9) imply (3.6). On the other hand, by taking normal traces of all the terms in (1.1a) we obtain ∂u ∂p (t, x) = ρF − (t, x) − (u · ∇)u(t, x) · n(t, x) ∂n ∂t
(x ∈ Γ ),
(3.10)
where Γ = ∂Ω or Γ = Sq(t) . The above boundary conditions can be expressed in terms of the velocity and of the position of the solid. First note that n(t, x) = n(0, x)
x ∈ ∂Ω, t ∈ [0, T0 ) .
Additionally, note that, for every t ∈ [0, T0 ) and y ∈ ∂S0 , we have n t, Ψ (t, y) = R(t)n(0, y), and u t, Ψ (t, y) · R(t)n(0, y) = v t, Ψ (t, y) · R(t)n(0, y) ,
(3.11)
where x = Ψ (t, y) = h(t) + R(t)(y − h0 )
(y ∈ ∂S0 ),
and v is the solid velocity given in (3.5). By taking the derivative with respect to t of the two sides of (3.11), we obtain that for every t ∈ [0, T0 ) and x ∈ ∂Sq(t) we have
∂u (t, x) + (v · ∇)u(t, x) · n(t, x) + u(t, x) · ω(t) ∧ n(t, x) ∂t
∂v (t, x) + (v · ∇)v(t, x) · n(t, x) + v(t, x) · ω(t) ∧ n(t, x) . = ∂t
Using in the above formula the fact (easy to check) that
(3.12)
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v(t, x) · ∇ v(t, x) · n(t, x) = −v(t, x) · ω(t) ∧ n(t, x) , we obtain that for every t ∈ [0, T0 ) and x ∈ ∂Sq(t) we have
∂u + (u · ∇)u · n (t, x) = (u − v) · ∇ (u − v) · n(t, x) − 2(u − v) · ω ∧ n(t, x) ∂t ∂v + (v · ∇)v + · n(t, x). (3.13) ∂t
Using again the relation (u − v) · n = 0 on Sq(t) , we have
(u − v) · ∇ (u − v) · n(t, x) = −Gq(t) (u − v)(t, x). By combining (3.13) and (3.14) and (3.10) we obtain (3.8). To obtain (3.7) it suffices to apply (3.10) with v = 0 (so that ω = 0).
(3.14)
2
From Proposition 3.3 (more precisely from (3.8)) we note that the pressure depends on h¨ and ˙ In order to make this dependence more precise we introduce, for every q ∈ P(Ω, S0 ), the on ω. potential functions Φi for i = 1, . . . , 6 which are solutions of the Neumann problems: − Φi (q; x) = 0,
x ∈ Fq ,
∂Φi (q; x) = 0, x ∈ ∂Ω, ∂n ∂Φi (q; x) = Ki (q; x), x ∈ ∂Sq , ∂n
(3.15a) (3.15b) (3.15c)
where Ki (q; x) = ni (x) for i = 1, 2, 3,
Ki (q; x) = (x − h) ∧ n(x) i−3 for i = 4, 5, 6.
(3.16)
Denote Φ = (Φ1 , . . . , Φ6 )∗ . These functions have been introduced in the book of Lamb [9] and they were used, in particular, in the work of Houot and Munnier [7] to describe the motion of rigid bodies in a perfect fluid undergoing a potential flow. From Theorem 3.1 on the Neumann problem, it is easy to check that Φ ∈ C ∞ (Fq ; R6 ). Moreover, the following properties are proved in [7]. Proposition 3.4. For every q0 ∈ P(Ω, S0 ), there exists a neighborhood O of q0 such that • the mapping q → Φ(q; ·) from O to C ∞ (Fq ; R6 ) is of class C 2 ; • for all i, j ∈ {1, . . . , 6} the mappings q → Ii,j (q) = ∇Φi (q; x) · ∇Φj (q; x) dx, Fq
are of class C 2 on O.
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We also need a potential μ, defined as follows. For z = (η, q, ξ, k, ω)∗ ∈ F m , where F m has been defined in (2.12), we set u(z; x) = ξ(η−1 (x)), with x ∈ Fq . The potential μ is defined as the solution of the boundary value problem − μ(z; x) = Fq u(z; x)
(x ∈ Fq ),
(3.17a)
∂μ (z; x) = Gq u(z; x) (x ∈ ∂Ω), ∂n ∂μ (z; x) = Gq (u − v)(x) + 2 u(z; x) − v(z; x) · ω ∧ n(x) ∂n
− ω ∧ ω ∧ (x − h) · n(x) (x ∈ ∂Sq ),
(3.17b)
(3.17c)
where v(z; x) = k + ω ∧ (x − h), and Fq , Gq are defined in (3.3), (3.4). Remark 3.5. With the above notation for Φ and μ, if (u, p, q) satisfy (3.6)–(3.8) and z(t) is defined by (2.8), then the pressure can be written ∗ ¨ ˙ p z(t); x = ρF μ z(t); x − ρF Φ q(t); x · h(t), ω(t) ,
(3.18)
where · stands for the inner product in R6 . 4. An equivalent form of the governing equations Throughout this section we assume that m 3 and q ∈ C 2 [0, T ), P(Ω, S0 ) , u ∈ C [0, T ), H m (Fq(·) ) ∩ C 1 [0, T ), H m−1 (Fq(·) ) , p ∈ C [0, T ), N m+1 (Fq(t) ) . At this point we need the virtual mass of the solid (see, for instance, [7]) which is the six by six matrix K(q) defined, for every q ∈ P(Ω, S0 ), by
ms Id3 0 KF (q) = ρF ∇Φi (q; x) · ∇Φj (q; x) dx
K(q) = KS (q) + KF (q),
Fq
KS (q) =
0 J
, ,
(4.1)
1i,j 6
where J = J (q) is the inertia matrix of the solid (1.3). It is easy to check that KS (q) is strictly positive and KF (q) is positive so that K(q) is invertible. ¨ The result below shows that Eqs. (1.1e) and (1.1f) can be rewritten as equations giving h(t) ˙ and ω(t), in terms of z(t) defined in (2.8).
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Proposition 4.1. Assume that u, p, q satisfy (1.1). Then
¨ h(t) ˙ ω(t)
−1 = K q(t)
0 (J (t)ω(t)) ∧ ω(t)
+ ρF
∇μ z(t); x · ∇Φ q(t); x dx .
Fq(t)
(4.2) In the above formula, the notation ∇μ · ∇Φ stands for the six-dimensional vector of components (∇μ · ∇Φi )1i6 , where μ is the solution of (3.17) and (Φi )1i6 are defined by (3.15). Proof. The decomposition of the pressure (3.18), the formulas (1.1e) and (1.1f) imply that, for every t ∈ [0, T0 ) we have ms h¨ j (t) = ρF
μ z(t); x Kj q(t); x dσx
∂Sq(t)
− ρF
3 i=1
− ρF
3
Φi q(t); x Kj q(t); x dσx
∂Sq(t)
Φi+3 q(t); x Kj q(t); x dσx ,
ω˙ i (t)
i=1 3
h¨ i (t)
(4.3)
∂Sq(t)
Ji,j (t)ω˙i (t) = J (t)ω(t) ∧ ω(t) j + ρF
i=1
μ z(t); x Kj +3 q(t); x dσx
∂Sq(t)
− ρF
3 i=1
− ρF
3 i=1
h¨ i (t)
Φi q(t); x Kj +3 q(t); x dσx
∂Sq(t)
Φi+3 q(t); x Kj +3 q(t); x dσx ,
ω¨ i (t) ∂Sq(t)
where Kj have been defined in (3.16). On the other hand, using (3.15), (3.17) and Green’s formula we get ∂Sq(t)
Φi q(t); x Kj q(t); x dσx =
∂Φj (x) dσx Φi q(t); x ∂n
∂Sq(t)
= Fq(t)
∇Φi q(t); x · ∇Φj (t, x) dx,
(4.4)
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μ z(t); x Kj q(t); x dσx =
∂Sq(t)
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∂Φj μ z(t); x (x) dσx ∂n
∂Sq(t)
=
∇μ z(t); x · ∇Φj q(t); x dx.
Fq(t)
Using the last two formulas in (4.3) and (4.4) we obtain the conclusion (4.2).
2
Recall the definition of E m and F m from (2.7) and (2.12), respectively, and let LF : F m → be defined by
H m (Ω, R3 )
LF (z)(y) = ρF ∇Φ(q; η) · LS (z)(y) − ρF ∇μ z; η(y)
(y ∈ F0 ),
(4.5)
for every z = (η, q, ξ, k, ω)∗ ∈ F m , where q = (h, R), the function Φ is the solution of the Neumann problem (3.15), μ is solution of (3.17) and
−1 LS (z) = K(q)
0 J (ω) ∧ ω
+ ρF
∇μ(z; x) · ∇Φ(q; x) dx .
(4.6)
Fq
Let L : F m → E m be defined by ⎛
⎞ ξ ⎜ k ⎟ ⎜ ⎟ L(z) = ⎜ A(ω)R ⎟ . ⎝ ⎠ LF (z) LS (z)
(4.7)
In the last part of this section we show that the system (1.1) is equivalent to the ordinary differential equation dz (t) = L z(t) , dt
z(0) = (IdFq0 , h0 , Id3 , u0 , k0 , ω0 )∗ .
(4.8)
In the following proposition we prove that every solution of (1.1) generates a solution of (4.8). Proposition 4.2. Let m 3 an integer, assume that (h0 , Id3 )∗ ∈ P(Ω, S0 ) and (u0 , k0 , ω0 )∗ ∈ Tσ0 Σ where σ0 = (IdFq0 , h0 , Id3 ). Moreover, assume that q ∈ C 2 [0, T ), P(Ω, S0 ) , u ∈ C [0, T ), H m (Fq ) ∩ C 1 [0, T ), H m−1 (Fq ) , p ∈ C [0, T ), N m+1 (Fq ) satisfy the system (1.1). Then z defined by (2.8) satisfies (4.8).
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Proof. The equations for η, h, R in (4.8) are nothing else but the definitions of ξ , k and ω from (2.2), (2.5) and (2.6), respectively. The fact that the equations for ξ , k and ω hold follows from (2.4), Proposition 4.1 and from (3.18). 2 We still have to show that every solution of (4.8) generates a strong solution of (1.1). Proposition 4.3. Let m 3 an integer, assume that (h0 , Id3 )∗ ∈ P(Ω, S0 ) and (u0 , k0 , ω0 )∗ ∈ Tσ0 Σ where σ0 = (IdFq0 , h0 , Id3 )∗ . Moreover, assume that z = (η, h, R, ξ , k, ω)∗ ∈ C [0, T0 ); F m ∩ C 1 [0, T0 ); E m , is a solution of (4.8). Let q, u, p be defined by q = (h, R)∗ , u(t, x) = ξ t, η−1 (t, x) ,
t ∈ [0, T0 ), x ∈ Fq(t) ,
and let the pressure p be defined by (3.18). Then q, u, p satisfy the smoothness conditions (1.10)– (1.12) and the system (1.1). Proof. First remark that, since z ∈ C([0, T ); F m ) ∩ C 1 ([0, T ); E m ), we have t ∈ [0, T0 ), x ∈ Fq(t) , u(t, x) · n(t, x) = 0 t ∈ [0, T0 ), x ∈ ∂Ω , u(t, x) · n(t, x) = v(t, x) · n(t, x) t ∈ [0, T0 ), x ∈ ∂Sq(t) , (div u)(t, x) = 0
so that Eqs. (1.1b), (1.1c), (1.1d) are satisfied. From the definition (4.8) of L we obtain that ˙ = A(ω)R and R
¨ h(t) ˙ ω(t)
−1 = K q(t)
0 (J (t)ω(t)) ∧ ω(t)
+ ρF
∇μ(t, x) · ∇Φ(t, x) dx , Fq (t)
ξ˙ (t) = ρF ∇Φ t, η(t, y) · LS t, z(t) − ρF ∇μ t, η(t, y) , where K(q) is given by (4.1). The Newton’s laws (1.1e) and (1.1f) come from the definition of the pressure (3.18) in the same way as in the proof of Proposition 4.1. Finally, using the relation ξ = u ◦ η and (3.18), we obtain that (1.1a) also holds. 2 5. Locally Lipschitz property of L In this section we tackle a key point of our approach, which consists in proving that the map L is locally Lipschitz. We frequently use below results and methods from [1] such as the following two results (stated here in the particular case needed in the present work). Lemma 5.1. Let Ω, Ω ∗ ⊂ R3 be open bounded sets with smooth boundary. Let s > 3/2, f ∈ C s+1 (Ω ∗ ), u ∈ H s (Ω, Ω ∗ ) and v ∈ H s (Ω, Ω ∗ ) then we have
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f ◦ u − f ◦ v H s (Ω) K f C s+1 u − v H s (Ω) u sH s (Ω) + v sH s (Ω) + 1 where K depends only on s, Ω and Ω ∗ . Lemma 5.2. Let Ω, Ω ∗ ⊂ R3 be open bounded sets with smooth boundary. Let s > 5/2, f ∈ H s (Ω ∗ ) and u ∈ H s (Ω, Ω ∗ ) be a C 1 diffeomorphism. Then f ◦ u ∈ H s (Ω) and it satisfies f ◦ u H s (Ω) K f H s (Ω ∗ )
1 . u sH s (Ω) + 1 1/2 inf |Jac(u)|
(5.1)
where K depends on s, Ω and Ω ∗ . The above results will be combined with techniques specific to our problem, which require to compare functions defined on two different open sets. Recall that the manifold F m is defined by
ξ η σ ∈ Σ m , ν = k ∈ Tσ Σ m , Fm = z = ∈ Em σ = q ν ω where Σ m is defined by (2.10) and Tσ Σ m by (2.11). For an element z ∈ F m , the first three components σ = (η, h, R)∗ ∈ Σ m define the “position” of the system whereas ν = (ξ, k, ω)∗ ∈ Tσ Σ m defines the velocity. The key point in this section is the study of the application μ from (3.17). Recall the notation q = Rh . The main new issue we need to tackle is the study the dependence of the solution μ of (3.17) with respect to the geometric parameter q. The dependence of μ with respect to ξ and η is studied using the ideas in [1]. We first introduce several functions which are useful for the remaining part of this section. Let α, β0 , β and τ be the mappings on F m defined by α(z; y) = Fq ξ ◦ η−1 η(y) (y ∈ F0 ), β0 (z; y) = Gq ξ ◦ η−1 η(y) (y ∈ ∂Ω), β(z; y) = Gq ξ ◦ η−1 − v η(y) (y ∈ ∂S0 ), τ (z; y) = 2 ξ(y) − v η(y) · ω ∧ RN ( y)
− (v · ∇)v η(y) − ω ∧ k · RN ( y)
(5.2) (5.3) (5.4)
(y ∈ ∂S0 ),
(5.5)
where y = h0 + R ∗ (η(y) − h), Fq and Gq have been defined in (3.3) and (3.4), whereas N is a smooth extension of the unit normal vector of ∂F0 to F0 . Remark 5.3. According to a result from Takahashi [17] and Cumsille and Tucsnak [2], for every q = (h, R)∗ ∈ P(Ω, S0 ) and ε > 0 small enough there exists a C ∞ diffeomorphism Ψq : F0 → Fq such that det[DΨq (y)] = 1 for all y ∈ F0 and Ψq (y) = y
if d(y, ∂Ω) ε,
Ψq (y) = h + R(y − h0 )
if d(y, ∂S0 ) ε.
(5.6)
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By setting n(Ψq (y)) = DΨq (y)N (y) for every y ∈ F0 we obtain a vector field n on Fq , extending the unit normal vector field on ∂Fq , such that Dn(x) = DN (x), x ∈ ∂Ω, Dn(x) = RDN h0 + R ∗ (x − h) R ∗ ,
(5.7) x ∈ ∂Sq .
(5.8)
Moreover, the construction of Ψq in [17] shows that, for every y ∈ F0 , the map q → Ψq (y) is C ∞ . Proposition 5.4. Let m 3 be an integer. Then the mappings α, β0 , β and τ are locally Lipschitz (with respect to z) from F m to H m−1 (F0 ), H m−1/2 (∂Ω), H m−1/2 (∂S0 ) and H m−1/2 (∂S0 ), respectively. Proof. Let z0 = (σ0 , ν0 )∗ ∈ F m with σ0 =
IdF0 h0 Id3
∈Σ , m
ν0 =
u0 k0 ω0
∈ Tσ 0 Σ m .
For r > 0 we define B m (r) ⊂ F m by B m (r) = z ∈ F m z − z0 E m r . We first note that, by the chain rule, we have
−1 2 α(z; y) = tr Dξ(y) Dη(y)
(y ∈ F0 ).
Since H m−1 (F0 ) is a Banach algebra, to show that α is Lipschitz on B m (r) it suffices to check that the maps z → Dξ,
z → [Dη]−1 ,
are Lipschitz from B m (r) to [H m−1 (F0 )]9 . The first map above is obviously Lipschitz whereas for the second one it suffices to use the fact that for every 3 × 3 matrices A, B of determinant equal to 1 we have A−1 − B −1 = A−1 (B − A)B −1 ,
A−1 = cof(A)t ,
where cof(A) is the signed cofactors matrix of A. For β0 we remark that, using again the chain rule combined with (5.7), we have
β0 (z; y) = DN η(y) ξ(y) · ξ(y)
(y ∈ ∂Ω).
By applying Lemma 5.1 it follows that the mapping z → DN ◦ η is Lipschitz B m (r) to [H m (F0 )]9 . Using again the fact that H m (F0 ) is a Banach algebra it follows that β0 is Lipschitz from B m (r) to H m (F0 ). By the trace theorem it follows that β0 is Lipschitz from B m (r) to H m−1/2 (∂Ω).
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For β we note that
β(z; y) = R DN η(y) R ∗ ξ(y) − v η(y) · ξ(y) − v η(y)
(y ∈ F0 ).
The fact that β is Lipschitz from B m (r) to H m−1/2 (∂Ω) can now be proved in the same way as for β0 . Finally for τ we notice that for every x ∈ Fq we have
(v · ∇)v(x) − ω ∧ k = ω ∧ ω ∧ (x − h) . Inserting the above formula in (5.5) and applying again Lemma 5.1, the claimed Lipschitz property of τ easily follows. 2 We also need the following classical result (see, for instance, [1, Lemma 5]): Proposition 5.5. Let m 3 be an integer and let Ω be bounded domain of R3 with smooth boundary. Then for every u ∈ H m (Ω, R3 ) there exists a constant K, which depends on m and on Ω, such that
u H m (Ω,R3 ) K div u H m−1 (Ω,R3 ) + curl(u) H m−1 (Ω,R3 ) + u · n H m−1/2 (∂Ω,R3 ) + u H m−1 (Ω,R3 ) , where (curl u)i,j =
∂uj ∂ui − ∂xj ∂xi
i, j ∈ {1, 2, 3} .
Moreover if ||| · ||| is a norm on H m−1 (Ω, R3 ) such that |||u||| C u H m−1 (F0 ,R3 )
u ∈ H m−1 F0 , R3
(5.9)
for some C > 0 then there exists a constant K > 0 such that
u H m (Ω,R3 ) K div u H m−1 (Ω,R3 ) + curl u H m−1 (Ω,R3 ) + u · n H m−1/2 (∂Ω,R3 ) + |||u||| . We are now in a position to give the main ingredient needed to prove that L is locally Lipschitz. This result concerns the potential μ introduced in (3.17). Proposition 5.6. For every integer m 3, the function χ defined on F m by χ(z)(y) = ∇μ(z) η(y) is locally Lipschitz from F m to H m−1 (F0 , R3 ).
(y ∈ F0 ),
(5.10)
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Proof. Let z0 = (σ0 , ν0 )∗ ∈ F m with σ0 = (IdF0 , h0 , Id3 )∗ ∈ Σ m ,
ν0 = (u0 , k0 , w0 )∗ ∈ Tσ0 Σ m .
We use again the notation from the proof of Proposition 5.4, i.e., for r > 0 we set B m (r) = z ∈ F m z − z0 E m r . In the remaining part of this proof, z1 and z2 are generic points in B m (r) and we denote by K(r) any Lipschitz constant obtained in Proposition 5.4. With the notation from this section, it is not difficult to check that the Neumann problem (3.17) can be rewritten as: − μ(z; x) = α z; η−1 (x)
(x ∈ Fq ),
(5.11a)
∂μ (z; x) = β0 z; η−1 (x) (x ∈ ∂Ω), ∂n ∂μ (z; x) = β z; η−1 (x) + τ z; η−1 (x) ∂n
(5.11b) (x ∈ ∂Sq ),
(5.11c)
where α, β0 , β and τ have been defined in (5.2), (5.3), (5.4) and (5.5). The main difficulty consists in the fact that the functions μ1 = μ(z1 ; ·) and μ2 = μ(z2 ; ·) are not defined on the same domain. Using Lemma 5.2 we have χ(z1 ) − χ(z2 )
H m (F0 ,R3 )
= ∇μ1 ◦ η1 − ∇μ2 ◦ η2 H m (F0 ,R3 ) K(r) ∇μ1 ◦ η − ∇μ2 H m (Fq
2 ,R
3)
,
(5.12)
where η = η1 ◦ η2−1 . By applying Proposition 5.5 we obtain ∇μ1 ◦ η − ∇μ2 H m (Fq
2 ,R
3)
K(r)(I1 + I2 + I3 + I4 ),
(5.13)
where Ii , with i ∈ {1, 2, 3, 4}, are given by I1 = div(∇μ1 ◦ η − ∇μ2 ) H m−1 (F ) , q2 I2 = curl(∇μ1 ◦ η − ∇μ2 ) m−1 H
(Fq2 ,M3 (R))
I3 = (∇μ1 ◦ η − ∇μ2 ) · n H m−1/2 (∂F
q2 )
,
,
I4 = |||∇μ1 ◦ η − ∇μ2 |||, where ||| · ||| is the norm on H m−1 (Fq2 , R3 ) defined by |||u||| = sup
u(x) · γ (x) dx γ ∈ C m Fq2 , R3 ,
Fq2
γ (x) = 0 for all x ∈ ∂Fq2 , γ C m (Fq
2
,R3 )
1 ,
(5.14)
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which clearly satisfies (5.9). Using Lemma 5.2 from [1] and (5.11) we have I1 K(r) div(∇μ1 ◦ η) − μ1 ◦ η H m−1 (F
q2
,R3 )
+ α(z1 ) − α(z2 ) H m−1 (F
q0 ,R
3)
.
Using Lemma 4 from [1] and Lemma 5.2 we obtain div(∇μ1 ◦ η) − μ1 ◦ η
H m−1 (Fq2 )
K(r) η − IdFq2 H m (Fq2 ) ∇μ1 H m (Fq1 ) K(r) η1 − η2 H m (F0 ) ∇μ1 H m (Fq1 ) .
On the other hand, using again Lemma 5.2 and Theorem 3.1 we have ∇μ1 H m (Fq1 ) K(r) α(z1 ) H m−1 (F ) + β0 (z1 ) H m−1/2 (∂Ω) 0 z1 ∈ B m (r) . + β(z1 ) + τ (z1 ) H m−1/2 (∂S ) K(r) q0
(5.15)
The last two estimates and the locally Lipschitz property of α proved in Proposition 5.4 imply that I1 K(r) z1 − z2 E m
z1 , z2 ∈ B m (r) .
(5.16)
Using the fact that curl(∇f ) = 0 together with arguments completely similar to those used for I1 , we obtain a constant K(r) such that I2 K(r) z1 − z2 E m
z1 , z2 ∈ B m (r) .
(5.17)
To tackle I3 , let ni the unit normal vector to ∂Fqi , with i ∈ {1, 2}. We have ∂μ1 2 ∂μ2 1 . I3 (∇μ1 ◦ η) · n − n ◦ η H m−1/2 (∂F ) + 1 ◦ η − 2 q2 ∂n ∂n H m−1/2 (∂Fq ) 2
Using trace inequalities, estimate (5.15) and Lemma 5.2 we obtain that ∂μ1 2 ∂μ2 1 I3 K(r) n (η2 ) − n (η1 ) H m−1/2 (∂F ) + 1 ◦ η1 − 2 ◦ η2 0 ∂n ∂n H m−1/2 (∂F0 )
2 1 = K(r) n (η2 ) − n (η1 ) H m−1/2 (∂F ) + β0 (z1 ) − β0 (z2 ) H m−1/2 (∂Ω) 0 + β(z1 ) + τ (z1 ) − β(z2 ) − τ (z2 ) H m−1/2 (∂S ) . 0
Applying Lemma 5.1 to the extensions of ni to Fqi (these extensions have been defined in Remark 5.3), it follows that 2 n (η2 ) − n1 (η1 )
H m−1/2 (∂F0 ,R3 )
K(r) z1 − z2 E m
z1 , z2 ∈ B m (r) .
The last two estimates and the Lipschitz properties of α, β0 , β and τ imply that I3 K(r) z1 − z2 E m
z1 , z2 ∈ B m (r) .
(5.18)
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To study I4 we first note that, for every γ ∈ C m (Fq2 , R3 ) with γ = 0 of ∂Fq2 we have
∇μ1 ◦ η(x) − ∇μ2 (x) · γ (x) dx =
Fq2
∇μ1 · γ ◦ η−1 (x) dx
Fq1
−
∇μ2 (x) · γ (x) dx.
(5.19)
Fq2
Consider the functions ψk : Fqk → R, defined as the solutions of the Neumann problems: − ψ1 = − div γ ◦ η−1
in Fq1 ,
(5.20a)
∂ψ1 = 0 on ∂Fq1 , ∂n − ψ2 = − div γ
(5.20b) in Fq2 ,
(5.21a)
∂ψ2 = 0 on ∂Fq2 . ∂n
(5.21b)
Taking the inner product in L2 (Fq1 ) (respectively in L2 (Fq2 )) of the first equation in (5.20) (respectively in (5.21)) by μ1 (respectively by μ2 ) and the subtracting side by side, we obtain that
∇ψ1 · ∇μ1 dx −
Fq1
∇ψ2 · ∇μ2 dx =
Fq2
∇μ1 · γ ◦ η
−1
(x) dx −
Fq1
∇μ2 (x) · γ (x) dx.
Fq2
The above formula and (5.19) yield that
∇μ1 ◦ η(x) − ∇μ2 (x) · γ (x) dx =
Fq2
∇ψ1 · ∇μ1 dx −
Fq1
∇ψ2 · ∇μ2 dx.
Fq2
Using the variational formulation of the Neumann problem (5.11) we obtain that, for i ∈ {1, 2}, we have
α zi ; ηi−1 (x) ψi (x) dx +
∇μi · ∇ψi dx = Fqi
Fqi
β0 zi ; ηi−1 (x) ψi (x) dσx
∂Ω
+ ∂Sqi
The last two formulas imply that
β zi ; ηi−1 (x) + τ zi ; ηi−1 (x) ψi (x) dσx .
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∇μ1 ◦ η(x) − ∇μ2 (x) · γ (x) dx
Fq2
α z1 ; η1−1 (x) ψ1 (x) dx −
= Fq1
+ +
β0 z1 ; η1−1 (x) ψ1 (x) dσx − β z1 ; η1−1 (x) dσx −
∂Sq1
+
α z2 ; η2−1 (x) ψ2 (x) dx
Fq2
∂Ω
β0 z2 ; η2−1 (x) ψ2 (x) dσx
∂Ω
β z2 ; η2−1 (x) dσx
∂Sq2
τ z1 ; η1−1 (x) ψ1 (x) dσx −
∂Sq1
τ z2 ; η2−1 (x) ψ2 (x) dσx .
(5.22)
∂Sq2
To estimate the difference of the first two terms in the right-hand side of the above formula, we note that
α z1 ; η1−1 (x) ψ1 (x) dx −
Fq1
=
α z2 ; η2−1 (x) ψ2 (x) dx
Fq2
α(z1 ; y)(ψ1 ◦ η1 )(y) − α(z2 ; y)(ψ2 ◦ η2 )(y) dy
Fq0
α(z1 ; ·) − α(z2 ; ·) ψ1 ◦ η1 + α(z2 ; ·) ψ1 ◦ η1 − ψ2 ◦ η2 ,
(5.23)
where all the norms above are in L2 (Fq0 ). The first term in the right-hand side of the above relation is readily estimated by using Proposition 5.4 to get α(z1 ; ·) − α(z2 ; ·) ψ1 ◦ η1 K(r) z1 − z2 E m γ
C m (Fq2 ;R3 ) ,
(5.24)
for every z1 , z2 ∈ B m (r). To estimate the second term in the right-hand side of (5.23) we remark that, using the variational formulations of (5.20) and (5.21) and a simple change of variables we have, for k ∈ {1, 2},
(∇ψk ◦ ηk ) · (∇ϕk ◦ ηk ) dy =
Fq0
(γ ◦ η2 ) · (∇ϕk ◦ ηk ) dy
ϕk ∈ H m (Fqk ) .
Fq0
k = ψk ◦ ηk the last formula becomes Denoting ψ Fq0
−1 −1 ∗ k · ∇ϕ dy = Dηk Dηk ∇ψ
Fq0
−1 Dηk (γ ◦ η2 ) · ∇ϕ dy
ϕ ∈ H m (Fq0 ) .
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Subtracting side by side the formulas corresponding to k = 1 and k = 2 it is not difficult to see that, for every γ ∈ C m Fq2 , R3 ,
γ = 0 on ∂Fq2 ,
we have
1 − ∇ ψ 2 |2 dy K(r) z1 − z2 2E m |∇ ψ
z1 , z2 ∈ B m (r) .
Fq0
The above estimate, combined to (5.23) and (5.24) imply that α
z1 ; η1−1 (x) ψ1 (x) dx
Fq1
−
α z2 ; η2−1 (x) ψ2 (x) dx
Fq2
K(r) z1 − z2 2E m
z1 , z2 ∈ B m (r) .
The other terms in the right-hand side of (5.22) can be estimated in a similar way, yielding that I4 K(r) z1 − z2 2E m
z1 , z2 ∈ B m (r) .
(5.25)
By combining (5.12), (5.13), (5.16)–(5.18) and (5.25) we obtain the conclusion that χ is locally Lipschitz from F m to H m−1 (F0 , R3 ). 2 We are now in position to prove that L is locally Lipschitz. Proposition 5.7. The mappings LS , LF and L are locally Lipschitz on F m . Proof. We begin by showing that LS is locally Lipschitz in B m (r) for a given r. From Proposition 3.4 the mapping q → K(q) is C 2 from P(Ω, S) to M6 (R) (recall that K(q) is the virtual mass matrix defined in (4.1)). Using Proposition 3.4 together with Lemmas A.2 from [1] and Lemma 5.1, it follows that the mapping z → ∇Φ ◦ η is Lipschitz from B m (r) to H m (F0 , M3×6 (R)), where (Φk )k∈{1,...,6} satisfy (3.15). Moreover, the mapping z → (03 , (J ω) ∧ ω)∗ is Lipschitz from B m (r) to R6 . Using the notation in Proposition 4.1, the last term in the right-hand side of (4.2) writes
∇μ(x) · ∇Φ(x) dx = Fq
∇μ η(y) · ∇Φ η(y) dy,
F0
so that, using Propositions 5.6 and 3.4, we obtain that this term defines a Lipschitz function from B m (r) to R6 . Using next the smoothness of the map q → K(q), it follows that LS is Lipschitz from B m (r) to R6 . Finally, the fact that LF is locally Lipschitz readily follows from (4.5) and the corresponding properties of Φ, LS and μ. 2
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6. L is tangent to F m In this section we show that the vector field defined by the operator L from (4.7) is tangent to the closed set F m which has been defined in (2.12). More precisely, the main result of this section is Proposition 6.1. Let m 3 an integer and let z0 ∈ F m . Then 1 dist z0 + rL(z0 ); F m = 0. r→0 r lim
In order to prove the above proposition we need some notation and several auxiliary results. σ Throughout this section e0 denotes the identity map on F0 and q0 = Idh03 . Moreover z0 = ν00 η0 ! ξ0 ! denotes a generic element of F m , where σ0 = h0 ∈ Σ m and ν0 = k0 ∈ Tσ0 Σ m . Let R0
η h R
σ=
ω0
∈ H m F0 , R3 × P(Ω, S0 ),
and recall the properties (1.6) and (1.7) of ∂Ω. We define the map ϑ(σ ) =
ϑ1 ϑ2
(σ )
σ ∈ H m F0 , R3 × P(Ω, S0 ) ,
(6.1)
where 1 ϑ1 (σ )(y) = det(Dη)(y) − |F0 | −
1 |F0 |
F0
1 det Dη(x) dx − |F0 |
δ h0 + R ∗ η(y) − h dσx
δ0 η(x) dσx
∂Ω
σ ∈ H m F0 , R3 × P(Ω, S0 ), y ∈ F0 ,
∂S0
(6.2) ϑ2 (σ )(y) =
δ0 (η(y)) δ(h0 + R ∗ (η(x) − h))
(σ ∈ H m (F0 , R3 ) × P(Ω, S0 ), y ∈ ∂Ω), (σ ∈ H m (F0 , R3 ) × P(Ω, S0 ), y ∈ ∂S0 ).
(6.3)
Since we obviously have
ϑ1 dy + F0
ϑ2 dσy = 0,
∂F0
it follows that ϑ maps H m (F0 , R3 ) × P(Ω, S0 ) into V m−1 (F0 ) (see (3.1) for the definition of this space).
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Lemma 6.2. The function ϑ defined above is of class C k for all integer k 1 and we have ∂η ϑ Moreover, ∂η ϑ
e0 q0
e0 q0
(V ) =
div V −V · n|∂F0
V ∈ H m F0 , R 3 .
(6.4)
maps H m (F0 , R3 ) onto V m−1 (F0 ).
Proof. Since m 3, it follows that H m−1 is an algebra so that the map η → det(Dη) is of class C k from H m (F0 , R3 ) to H m−1 (F0 , R) for every k 1. It is easy to check that the other terms in the definition of ϑ2 and are smooth functions so that ϑ is of class C k for every k 1. Using (1.7) it follows that, for every σ ∈ H m (F0 , R3 ) × P(Ω, S0 ), V ∈ H m (F0 , R3 ), we have (∂η ϑ2 )(σ )(V )(y) =
−V (y) · n(η(y)) −R ∗ V (y) · n(h0 + R ∗ (η(y) − h))
(y ∈ ∂Ω), (y ∈ ∂S0 ).
(6.5)
On the other hand, using the fact that the differential of A → det(A) is the linear map H → tr(cof(A)H ), where cof(A) is the signed cofactors matrix of A, we obtain that, for every σ ∈ H m (F0 , R3 ) × P(Ω, S0 ), V ∈ H m (F0 , R3 ), we have 1 ∂η ϑ1 (σ )(V )(y) = tr cof(Dη)DV (y) − |F0 | + +
1 |F0 | 1 |F0 |
tr cof(Dη)DV (y) dy
F0
V (y) · n η(y) dσy
∂Ω
R ∗ V (y) · n h0 + R ∗ η(y) − h dσy .
∂S0
Taking h = h0 , R = Id3 and η = e0 in the above formula and by using (6.5) we obtain (6.4). Finally, the fact that the right-hand side of (6.4) defines a map from H m (F0 , R3 ) onto V m−1 (F0 ) is classical, see for instance, Lemma 2.4.1 in Sohr [16, p. 79]. 2 The above lemma can be used, in particular, to show that Σ m is an infinite-dimensional manifold over H m (F0 , R3 ) × R3 × M3 (R) and to compute its tangent space at σ0 . Proposition 6.3. We have Σm =
q η =0 . ∈ H m F0 , R3 × P(Ω, S0 ) ϑ η q
Moreover, the tangent space to Σ m at every σ = defined in (2.11).
η q
(6.6)
∈ O ∩ Σ m (Ω, S0 ) is the space Tσ Σ m
Proof. The fact that the set in the left-hand side of (6.6) is a subset of the set int the righthand side is obvious. To prove the converse inclusion, we first note from ϑ1 (q, η) = 0 it follows that det(Dη) is constant in F0 . On the other hand, from ϑ2 (q, η) = 0 it follows that
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η(q)(∂F0 ) ⊂ ∂Fq . These assertions, combined to the fact that F0 and Fq have the same volume, imply that det(Dη) = 1 in F0 . Moreover, the above properties enable us to apply the global inverse mapping theorem of Caccioppoli (see, for instance, Zeidler [18, Theorem 4.G, p. 174]) to obtain that η ∈ Diff m (F0 , Fq ). This concludes the proof of (6.6). In order to prove the second assertion in the proposition, we first note that for every σ = (η, h, R)∗ ∈ Σ m and every (ξ, kω)∗ ∈ H m (F0 , R3 ) × R3 × R3 , we have ξ Dϑ1 (σ ) k (y) = tr Dη−1 Dξ (y ∈ F0 ), ω
ξ −ξ · n(η(y)) Dϑ2 (σ ) k (y) = −R ∗ [ξ(y) − k − A(ω)(η(y) − h)] · n(h0 + R ∗ (η(y) − h)) ω
(y ∈ ∂Ω), (y ∈ ∂S0 ),
where A(ω) has been defined in (1.2). Since η is a diffeomorphism from F0 to Fq , denoting u = ξ ◦ η−1 and making the change of variable x = η(y), we obtain ξ Dϑ1 (σ ) k η−1 (x) = (div u)(x) ω
(x ∈ Fq ),
ξ −u · n(x) (x ∈ ∂Ω), Dϑ2 (σ ) k η−1 (x) = −[u(x) − k − ω ∧ (x − h)] · n(x) (x ∈ ∂Sq ). ω From the above formulas it follows that the kernel of Dφ(σ ) is Tσ Σ m so that we obtain the second assertion in the proposition. 2 Proposition 6.4. Let σ0 =
e0 ! h0 ∈ Σ m , ν0 Id3
γ0 =
Γ L M
=
u0 ! k0 ∈ Tσ Σ m 0 ω0
and let
∈ H m F (0), R3 × R3 × R3
such that div(Γ )(y) = F u0 (y) (y ∈ F0 ), Γ · n(y) = −G u0 (y) (y ∈ ∂Ω), Γ · n(y) = −G u0 (y) − v0 (y) − 2(u0 − v0 ) · (ω0 ∧ n)
+ L + M ∧ (y − h0 ) + ω0 ∧ w0 ∧ (y − h0 ) · n where v0 (y) = k0 + ω0 ∧ (y − h0 ). Then there exists ε > 0 and a curve η σ = h ∈ C 2 [0, ε]; Σ m (Ω, S0 ) R
(6.7a) (6.7b) (y ∈ ∂S0 ),
(6.7c)
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satisfying σ˙ (0) =
σ (0) = σ0 ,
u0 , k0 A(ω0 )
σ¨ (0) =
Γ L A(M) + [A(ω0 )]2
.
(6.8)
Proof. According to Proposition 6.3 the curve σ is contained in Σ m iff ϑ(σ (t)) = 0 for every t ∈ [0, ε]. We begin by constructing the “rigid displacement part” Rh of the curve σ . To do that, we define h, ω : R → R3 h(t) = h0 + tk0 +
t2 L, 2
ω(t) = ω0 + tM,
t ∈ R,
(6.9)
and R : R → SO3 (R) is defined as the solution of the initial value problem ˙ R(t) = A ω(t) R(t),
R(0) = Id3 ,
(6.10)
where A(ω) is the skew-adjoint matrix defined in (1.2). Note that
2 ¨ R(0) = A(M) + A(ω0 ) .
˙ R(0) = A(ω0 ),
(6.11)
The above functions being continuous, it follows that there exists ε > 0 such that q(t) =
h(t) R(t)
∈ P(Ω, S0 ),
t ∈ 0, ε .
e0 In order to construct the “fluid part” η of the curve σ we first note that, since q(0) ∈ Σ m , we e0 have ϑ q(0) = 0. Therefore, by combining Lemma 6.2 with a version of the implicit function theorem (see, for instance, Zeidler [18, Theorem 4.H, p. 171]) it follows that there exists ε ∈ (0, ε ] and a function η : [0, ε] → H m (F0 , R3 ) such that η(0) = e0 and ϑ
η(t) q(t)
= 0,
t2 P η(t) − tu0 − Γ = 0 2
t ∈ [0, ε] ,
where P is the orthogonal projector from H m (F0 , R3 ) onto Ker ∂n ϑ to Lemma 6.2, we have Ker ∂n ϑ
e0 q(0)
e0 q(0) .
(6.12)
Note that, according
= u ∈ H m F0 , R3 div u = 0, u · n = 0 on ∂F0 .
(6.13)
In the remaining part of the proof we show that, with the above choice of η, h and R, the curve σ (t) =
η(t) h(t) R(t)
t ∈ [0, ε]
J.G. Houot et al. / Journal of Functional Analysis 259 (2010) 2856–2885
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satisfies (6.8). We first note that from (6.9) it follows that h(0) = h0 ,
˙ h(0) = k0 ,
¨ h(0) = L.
From the above formula combined with (6.10) and (6.11) we see that, in order to prove (6.8), we have only to check that ˙ η(0) = u0 ,
¨ η(0) = Γ.
(6.14)
Taking the derivative with respect to t of the formula det(Dη(t)) = 1 we obtain that −1 ˙ tr Dη(t) Dη(t) = 0 t ∈ [0, ε] .
(6.15)
Using next the fact that δ0 (η(t)) = 0 on ∂Ω it follows that ˙ · n η(t) = 0 η(t)
(6.16)
(on ∂Ω).
Moreover, since δ(h0 + R∗ (t)(η(t) − h(t))) = 0 on ∂S0 , we have ˙ − ω(t) ∧ η(t) − h(t) · R(t)n h0 + R∗ (t) η(t) − h(t) = 0 (on ∂S0 ). ˙ − h(t) η(t) (6.17) On the other hand, taking the derivative of the second formula in (6.12) with respect to t we obtain ˙ − u0 − tΓ = 0 t ∈ [0, ε] . P η(t)
(6.18)
Taking t = 0 in (6.15)–(6.18) we obtain ˙ div η(0) = 0 (in F0 ), ˙ η(0) · n = 0 (on ∂Ω), ˙ η(0) · n = k0 + ω0 ∧ (y − h0 ) ˙ P η(0) = P(u0 ).
(y ∈ ∂S0 ),
The above relations clearly imply that the first equality in (6.14) holds. In order to prove the second equality in (6.14) we take the derivative of (6.15)–(6.18) and then we make t = 0. In this way we obtain ¨ div η(0) = F (u0 ) ¨ η(0) · n = −G(u0 )
(in F0 ), (on ∂Ω),
¨ η(0) · n = −G(u0 − v0 ) − 2(u0 − v0 ) · (ω0 ∧ n)
+ L + M ∧ (y − h0 ) + ω0 ∧ ω0 ∧ (y − h0 ) · n ¨ P η(0) = P(Γ ),
(y ∈ ∂S0 ),
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where v0 has been defined in the statement of this proposition. Using (6.7) it follows that the second equality in (6.14) also holds. 2 We are now in a position to prove the main result of this section. Proof of Proposition 6.1. Recall the notation for z0 from the beginning of this section. We first note (by using an appropriate change of variables) that it suffices to prove the result for η0 = e0 and R0 = Id3 . This will be done by constructing a curve Z(·) in F m such that 1 dist z0 + rL(z0 ); Z(r) = 0. r→0 r lim
(6.19)
The main tool of the proof is Proposition 6.4, with an appropriate choice of Γ , L and M. More precisely, u0 , k0 and ω0 are chosen to be those in (1.1) and we take Γ = LF (z0 ),
L M
= LS (z0 ),
(6.20)
where LF and LS have been defined in (4.5) and (4.6), respectively. The fact that Γ , L and M chosen above satisfy the assumptions in Proposition 6.4 follows from (3.15) and (3.17). Define σ (t) ⎞ ˙ ⎟ ⎜ η(t) Z(t) = ⎝ ˙ ⎠, h(t) ω0 + tM ⎛
where σ (t) = (η(t), h(t), R(t))∗ is the curve constructed in Proposition 6.4. By combining (6.8) and (6.20) it follows that Z(0) = z0 , which imply (6.19).
˙ Z(0) = L(z0 ),
2
The proof of our main result in Theorem 1.3 can be now written as follows. Proof of Theorem 1.3. The assumptions in Theorem 1.3 imply that z0 = (e0 , h0 , Id3 , u0 , k0 , ω0 )∗ ∈ F m . Therefore, we can combine Propositions 5.7, 6.1 and 2.1 to obtain that the initial value problem z˙ = L(z),
z(0) = z0 ,
admits a unique solution z = (η, h, R, ξ , k, ω)∗ ∈ C 0 [0, T0 ); F m ∩ C 1 [0, T0 ); E m . According to Proposition 4.3, q, u defined by q = (h, R)∗ ,
(6.21)
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u(t, x) = ξ t, η−1 (t, x)
2885
t ∈ [0, T0 ), x ∈ Fq(t) ,
and the pressure p defined by (3.18) define a strong solution of (1.1). We have thus shown the announced existence result. To prove the uniqueness, it suffices to note that, according to Proposition 4.2, any strong solution of (1.1) defines a solution of (6.21) and to apply Proposition 2.1. 2 Acknowledgments The second author was partially supported by Grant Fondecyt 1090239 and BASAL-CMM Project. References [1] J.P. Bourguignon, H. Brezis, Remarks on the Euler equation, J. Funct. Anal. 15 (1974) 341–363. [2] P. Cumsille, M. Tucsnak, Wellposedness for the Navier–Stokes flow in the exterior of a rotating obstacle, Math. Methods Appl. Sci. 29 (2006) 595–623. [3] D.G. Ebin, J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2) 92 (1970) 102–163. [4] P. Hartman, Ordinary Differential Equations, John Wiley & Sons Inc., New York, 1964. [5] T.I. Hesla, Collisions of smooth bodies in viscous fluids, a mathematical investigation, PhD thesis, University of Minnesota, May 2005. [6] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations 32 (2007) 1345–1371. [7] J.-G. Houot, A. Munnier, On the motion and collisions of rigid bodies in an ideal fluid, private communication, 2006. [8] E. Kanso, J.E. Marsden, C.W. Rowley, J.B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Sci. 15 (2005) 255–289. [9] H. Lamb, Hydrodynamics, sixth ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 1993, with a foreword by R.A. Caflisch [Russel E. Caflisch]. [10] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes. II, Ann. Inst. Fourier (Grenoble) 11 (1961) 137– 178. [11] R.H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973) 399–414. [12] J.H. Ortega, L. Rosier, T. Takahashi, Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid, M2AN Math. Model. Numer. Anal. 39 (2005) 79–108. [13] J.H. Ortega, L. Rosier, T. Takahashi, On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) 139–165. [14] C. Rosier, L. Rosier, Smooth solutions for the motion of a ball in an incompressible perfect fluid, J. Funct. Anal. 256 (2009) 1618–1641. [15] J.A. San Martín, V. Starovoitov, M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal. 161 (2002) 113–147. [16] H. Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basel Lehrbücher (Birkhäuser Advanced Texts: Basel Textbooks), Birkhäuser Verlag, Basel, 2001. [17] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003) 1499–1532. [18] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986, translated from the German by Peter R. Wadsack.
Journal of Functional Analysis 259 (2010) 2886–2901 www.elsevier.com/locate/jfa
Absolutely summing operators on C[0, 1] as a tree space and the bounded approximation property ✩ Åsvald Lima a,∗ , Vegard Lima b , Eve Oja c a Department of Mathematics, University of Agder, Serviceboks 422, N-4604 Kristiansand, Norway b Aalesund University College, Postboks 1517, N-6025 Ålesund, Norway c Faculty of Mathematics and Computer Science, University of Tartu, J. Liivi 2, EE-50409 Tartu, Estonia
Received 3 May 2010; accepted 29 July 2010 Available online 12 August 2010 Communicated by N. Kalton
Abstract Let X be a Banach space. For describing the space P(C[0, 1], X) of absolutely summing operators from C[0, 1] to X in terms of the space X itself, we construct a tree space tree 1 (X) on X. It consists of special trees in X which we call two-trunk trees. We prove that P(C[0, 1], X) is isometrically isomorphic to tree 1 (X). As an application, we characterize the bounded approximation property (BAP) and the weak BAP in terms of X ∗ -valued sequence spaces. © 2010 Elsevier Inc. All rights reserved. Keywords: Banach spaces; Absolutely summing operators; Two-trunk trees; Linear B-splines; Continuous functions on [0, 1]; Bounded approximation properties
✩
The research of Eve Oja was partially supported by Estonian Science Foundation Grant 7308 and Estonian Targeted Financing Project SF0180039s08. * Corresponding author. E-mail addresses:
[email protected] (Å. Lima),
[email protected] (V. Lima),
[email protected] (E. Oja). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.017
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1. Introduction Let X and Y be Banach spaces. Recall that a linear operator T : X → Y is said to be absolutely summing if there exists a constant C 0 such that n k=1
n ∗ x (xk ): x ∗ ∈ X ∗ , x ∗ 1 T xk C sup
k=1
for every choice of elements x1 , . . . , xn in X. The minimum value of the constant C is called the absolutely summing norm of T and is denoted by T P . The Banach space of absolutely summing operators from X to Y , equipped with the norm · P , is denoted by P(X, Y ). It is well known that every absolutely summing operator factors through some Banach space C(K) of continuous functions on a compact Hausdorff space K. There is a vast literature on absolutely summing operators from C(K)-spaces to Banach spaces; see, e.g., [5] and [6] for results and references. In this paper, we are interested in the classical case K = [0, 1]. The main aim of this paper is to describe the space P(C[0, 1], X) in terms of the space X itself. This aim is motivated by our recent investigations [8–15] on the classical bounded approximation property (BAP) and its weak counterpart (see Section 5 for these notions). We construct a tree space on X (see Section 2) and we prove that P(C[0, 1], X) is isometrically isomorphic to this tree space (see Section 3). Our tree space will be called the 1 -tree space on X and denoted by tree 1 (X). It consists of special trees in X which will be called two-trunk trees. The represen∗ tation theorem for P(C[0, 1], X) from Section 3 is applied in Sections 4 and 5 to study tree 1 (X ) as a dual Banach space, to derive a representation theorem for the Banach space CX [0, 1] of continuous X-valued functions on [0, 1], and to characterize the weak BAP and the BAP of X in terms of X ∗ -valued sequence spaces. Among others, we prove that the Banach space 1 (X) of absolutely summable X-valued sequences nicely embeds in tree 1 (X). Our representation of P(C[0, 1], X) relies on linear B-splines. Since the linear splines are known to be efficient for both computational and implementation purposes, our representation of absolutely summing operators might also be useful in Numerical Analysis. Our notation is standard. We consider Banach spaces over the real field R. We denote by L(X, Y ) the Banach space of all bounded linear operators from X to Y , and we write L(X) for L(X, X). Besides the operator ideal P of absolutely summing operators, we also need the ideals I and N of integral operators and of nuclear operators. Integral and nuclear norms of operators are denoted by · I and · N , respectively. For P, I, and N , we refer to the books by Diestel, Jarchow, and Tonge [5], Pietsch [17], and Ryan [18]. And we refer to the books by Diestel and Uhl [6] and Ryan [18] for the classical approximation properties and tensor products. 2. Two-trunk trees and the 1 -tree space Let X be a Banach space. Recall that a (standard) tree in X is a system ((xk,2n )2k=1 )∞ n=0 of elements of X with the property that n
1 1 xk,2n = x2k−1,2n+1 + x2k,2n+1 2 2
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for all n = 0, 1, . . . and k = 1, 2, . . . , 2n . Hence, a tree looks as follows: x1,1 x1,2 x1,4
x2,2 x2,4
x3,4
x4,4
where, for each connecting line, its connecting coefficient is 1/2. The first element x1,1 is called the trunk of the tree. The elements x1,2n , x2,2n , . . . , x2n ,2n are said to be on the same (n-th) level, or to form the n-th (year) growth. The study of trees in Banach spaces was initiated by James in his seminal paper [7]. By now, there is a vast literature on various variants of trees and related tree spaces. For our purpose, we need to introduce the following version of tree which will be called a two-trunk tree (not to be pronounced “too drunk tree”). Definition 2.1. Let X be a Banach space. We say that a system ((xk,2n )2k=0 )∞ n=0 of elements of X is a two-trunk tree in X if for all n = 0, 1, . . . and k = 1, 2, . . . , 2n − 1 n
1 1 xk,2n = x2k−1,2n+1 + x2k,2n+1 + x2k+1,2n+1 , 2 2 1 x0,2n = x0,2n+1 + x1,2n+1 , 2 1 x2n ,2n = x2n+1 −1,2n+1 + x2n+1 ,2n+1 . 2 A two-trunk tree looks as follows: x0,1
x1,1
x0,2 x0,4
x1,2 x1,4
x2,4
x2,2 x3,4
x4,4
where connecting coefficients are 1 for the vertical lines and 1/2 for the sloping lines. Compared to a standard tree which has 2n elements on its n-th level, a two-trunk tree has n 2 + 1 elements on its n-th level. Our basic example, which is also a prototype of our concept of a two-trunk tree, is the twotrunk tree in C[0, 1] consisting of linear B-splines. Example 2.2. For n = 0, 1, . . . , let Sn denote the space of all linear splines on the interval [0,1] with the knots {k/2n : k = 0, 1, . . . , 2n }. The spline space Sn , equipped with the maximum norm
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from C[0, 1], is a (2n + 1)-dimensional subspace of the space C[0, 1]. The space Sn has a natural n basis (gk,2n )2k=0 where gk,2n ∈ Sn , k = 0, . . . , 2n , are defined by the conditions gk,2n
k 2n
= 1 and gk,2n
j 2n
= 0 if j = k.
If g ∈ Sn , then clearly 2n k g= g n gk,2n . 2 k=0
The functions gk,2n , n = 0, 1, . . . , k = 0, 1, . . . , 2n , are called linear B-splines (“B” comes from “basis”). Sometimes they are also called “the second order cardinal B-spline functions” (see, e.g., [4]). (The functions gk,2n are generated by the scaling function ϕ : R → R, ϕ(t) = 1 + t for t ∈ [−1, 0], ϕ(t) = 1 − t for t ∈ [0, 1], and ϕ(t) = 0 for t ∈ / [−1, 1]; so that gk,2n (t) = ϕ(2n t − k), t ∈ [0, 1]. But we shall not need this description of the functions gk,2n .) Since g
k,2n
=
n+1 2
g
k,2n
j =0
j 2n+1
gj,2n+1
and, by the definition of the gk,2n , g
k,2n
j 2n+1
⎧ ⎨ 0, = 1, ⎩1 2,
for j ∈ / {2k − 1, 2k, 2k + 1}, for j = 2k, for j ∈ {2k − 1, 2k + 1},
it is immediate that ((gk,2n )2k=0 )∞ n=0 is a two-trunk tree in C[0, 1]. Its trunks are g0,1 = g0,1 (t) = 1 − t and g1,1 = g1,1 (t) = t, t ∈ [0, 1]. n
Definition 2.3. Let X be a Banach space. The 1 -tree space tree 1 (X) consists of all two-trunk n 2 ∞ n n trees x = (xk,2 ) = ((xk,2 )k=0 )n=0 in X such that n
x := sup n
2
xk,2n < ∞.
k=0
2 +1 Thus, the space tree (X)). In the next 1 (X) is defined as a subspace of the space ∞ (1 tree section, we shall prove that 1 (X) is isometrically isomorphic to P(C[0, 1], X). Hence, in particular, tree 1 (X) is a Banach space. Before proceeding to the description of P(C[0, 1], X), let us reformulate the notion of a twotrunk tree in terms of connecting matrices. Looking at Definition 2.1, for n = 0, 1, . . . , denote by Mn the matrix whose k-th row is formed by the coefficients of xk,2n in (x0,2n+1 , x1,2n+1 , . . . , x2n+1 ,2n+1 ). The matrix Mn is of order (2n + 1) × (2n+1 + 1), and we have n
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M0 = M1 = ⎛
1 ⎜0 ⎜ M2 = ⎜ 0 ⎝ 0 0
1/2 1/2 0 0 0
1 0 0
1 1/2 0 , 0 1/2 1
1/2 0 0 0 1/2 1 1/2 0 , 0 0 1/2 1
0 0 0 0 0 1 1/2 0 0 0 0 1/2 1 1/2 0 0 0 0 1/2 1 0 0 0 0 0
⎞ 0 0⎟ ⎟ 0⎟, ⎠ 0 1
0 0 0 1/2 1/2
j etc. By Example 2.2, we can write that Mn = (gk,2n ( 2n+1 ))k=0,1,...,2n ; j =0,1,...,2n+1 . n+1 +1
Considering Mn = (mnkj ) as the matrix operator from X 2 way, i.e., Mn (x0 , x1 , . . . , x2n+1 ) =
2n+1
n +1
to X 2
defined in a usual
2n mnkj xj
j =0
, k=0
we have the following reformulations. Proposition 2.4. Let X be a Banach space. A system ((xk,2n )2k=0 )∞ n=0 of elements of X is a twotrunk tree in X if and only if for all n = 0, 1, . . . n
n n+1 (xk,2n )2k=0 = Mn (xj,2n+1 )2j =0 , and 2n +1 ∞ tree (X) : zn = Mn zn+1 . 1 (X) = (zn )n=0 ∈ ∞ 1 It is easily computed that for Mn : 12 +1 (X) → 12 +1 (X) one has Mn = 1. Therefore, in Proposition 2.4, zn zn+1 , n = 0, 1, . . . , and (zn )∞ = limzn . n=0 n+1
n
n
3. A representation theorem for the absolutely summing operators on C[0, 1] Let X and Y be Banach spaces. Let Tn ∈ L(Y, X) be such that the limit T y := limn Tn y exists for every y ∈ Y . It is a well-known result from Banach’s thesis [1, p. 157] that then (Tn ) is bounded in L(Y, X), T ∈ L(Y, X), and T supn Tn . The following version of Banach’s result is now immediate from the definition of absolutely summing operators (see the Introduction). Lemma 3.1. Let X and Y be Banach spaces, and let Tn ∈ P(Y, X). If the sequence (Tn ) is bounded in P(Y, X), and for every y ∈ Y the limit T y := limn Tn y exists, then T ∈ P(Y, X) and T P supn Tn P .
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Theorem 3.2. Let X be a Banach space. Then P(C[0, 1], X) is isometrically isomorphic to tree 1 (X), by the mapping n ∞ T → (T gk,2n )2k=0 n=0 ,
T ∈ P C[0, 1], X ,
where gk,2n , k = 0, 1, . . . , 2n , are the linear B-splines on [0, 1] with knots 0/2n , 1/2n , . . . , 2n /2n . The inverse mapping n ∞ (xk,2n )2k=0 n=0 → T is given by 2n k Tf = lim f n xk,2n , n 2
f ∈ C[0, 1].
k=0
Proof. 1) Let us start with some preparation. For n = 0, 1, . . . , define projections Pn from C[0, 1] onto its subspace of linear splines Sn (see Example 2.2) by 2n k Pn f = f n gk,2n , 2
f ∈ C[0, 1].
k=0
Since Pn f ∈ Sn and (Pn f )(k/2n ) = f (k/2n ), k = 0, 1, . . . , 2n , meaning that Pn f is the piecewise linear interpolant of f (Pn f agrees with f at the knots and interpolates linearly in between), Pm Pn = Pn for m n, k Pn f = max f n : k = 0, 1, . . . , 2n , 2
f ∈ C[0, 1],
and therefore Pn = 1. Since Pn 1 = 1, we have n
2
gk,2n = 1.
k=0
Using the uniform continuity of f on [0, 1], it follows that Pn f − → n f in C[0, 1]. n is a two-trunk tree in C[0, 1] and 2) Let T ∈ P(C[0, 1], X) be arbitrary. Since ((gk,2n )2k=0 )∞ n=0 n 2 ∞ T is a linear operator, zT := ((T gk,n )k=0 )n=0 is a two-trunk tree in X. Recall that [0, 1] can be identified with a weak* compact norming subset of B(C[0,1])∗ (by the correspondence t → δt , where δt (f ) = f (t), f ∈ C[0, 1]). Therefore, by the Pietsch domination theorem (see, e.g., [5, p. 44]), there exists a regular Borel probability measure μ on [0, 1] such that for each gk,2n 1 T gk,2n T P 0
gk,2n (t) dμ(t).
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Hence for each n n
2
T gk,2n T P
k=0
1 2n
gk,2n (t) dμ(t) = T P ,
(1)
0 k=0
showing that zT ∈ ∞ (12 +1 (X)), hence zT ∈ tree 1 (X) as desired, and zT T P . Moreover, the mapping T → zT is clearly linear. To show that it is also an isometric mapping, it suffices to prove that T P zT . Since Pn f → f for every f ∈ C[0, 1], also T Pn f → Tf for every f ∈ C[0, 1]. The sequence (T Pn ) is bounded in P(C[0, 1], X) because T Pn P T P Pn = T P for all n. Hence, by Lemma 3.1, n
T P supT Pn P .
(2)
n
In fact, T P = supT Pn P = limT Pn P n
n
because, as we saw, T Pn P T P , and T Pn P = T Pn+1 Pn P T Pn+1 P . 2n +1 be defined by I g = We shall now estimate T Pn P from above. Let In : Sn → ∞ n n 2 n (g(k/2 ))k=0 . Then k n In g = max g n : k = 0, 1, . . . , 2 = g, 2 and In is a linear isometry, whose inverse mapping In−1 is given by n
In−1 α
=
2
ek (α)gk,2n ,
2 +1 α ∈ ∞ , n
k=0 2 +1 )∗ = 2 +1 . where e0 = (1, 0, . . . , 0), . . . , e2n = (0, . . . , 0, 1) are the unit vectors in (∞ 1 The operator T Pn being of finite-rank, we can look at its nuclear norm T Pn N . We have n
n
T Pn P T Pn N = T In−1 In Pn N T In−1 N 2n 2n = ek ⊗ T gk,2n ek T gk,2n k=0
N
k=0
2n
=
k=0
T gk,2n .
(3)
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In fact, n
T Pn P =
2
T gk,2n
(4)
k=0
because replacing T with T Pn in (1) yields that n
2
n
T gk,2n =
k=0
2
T Pn gk,2n T Pn P .
k=0
From (2) and (3), it is clear that T P zT . This proves that the mapping T → zT is an isometric isomorphism of P(C[0, 1], X) into tree 1 (X). n tree ∞ n 2 3) Let now z := (zn )∞ n=0 := ((xk,2 )k=0 )n=0 ∈ 1 (X) be arbitrary. Set 2n k Tn f = f n xk,2n , 2
f ∈ C[0, 1].
k=0
Then Tn ∈ F (C[0, 1], X) and Tn gk,2n = xk,2n for k = 0, 1, . . . , 2n , because gk,2n (j/2n ) = δkj . Hence, Tn Pn f = Tn f, f ∈ C[0, 1], and, since Tn ∈ P(C[0, 1], X), using (4), we have n
Tn P = Tn Pn P =
2
n
Tn gk,2n =
k=0
2
xk,2n z.
k=0
Next we show that the sequence (Tn ) converges pointwise in L(C[0, 1], X). Since the functions gk,2n , n = 0, 1, . . . , k = 0, 1, . . . , 2n , span a dense subset of C[0, 1] and the sequence (Tn ) is bounded in L(C[0, 1], X), it suffices to prove that the limit limn Tn gj,2m exists for every gj,2m , m = 0, 1, . . . , j = 0, 1, . . . , 2m . We know already that Tm gj,2m = xj,2m . Recalling about matrices Mn and Proposition 2.4, we have m (Tm+1 gj,2m )2j =0
=
2m+1
gj,2m
k=0
k
2m
2m+1
= Mm zm+1 = zm ,
xk,2m+1 j =0
in particular, Tm+1 gj,2m = xj,2m for each j = 0, 1, . . . , 2m . Consequently, Tm+2 gk,2m+1 = xk,2m+1 for each k = 0, 1, . . . , 2m+1 , and therefore
Tm+2 gj,2m = Tm+2
2m+1 k=0
gj,2m
gk,2m+1 m+1 k
2
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=
m+1 2
g
j,2m
k 2m+1
k=0
xk,2m+1 = Tm+1 gj,2m
= xj,2m . Continuing similarly, we get that for each n m Tn gj,2m = xj,2m ,
j = 0, 1, . . . , 2m .
Hence limn Tn gj,2m = xj,2m for each m = 0, 1, . . . and j = 0, . . . , 2m . It follows that (Tn ) converges pointwise. Thus the operator T is well defined, T ∈ P(C[0, 1], X) by Lemma 3.1, and T → z because T gj,2m = xj,2m . 2 Remark 3.1. Let X be a Banach space, let Σ be the σ -field of Borel sets in [0, 1], and let G be a countably additive vector measure of bounded variation with values in X. Then we can define 1 T ∈ P(C[0, 1], X) by Tf = 0 f dG for f ∈ C[0, 1] (see [6, p. 162]). The formula Tf = lim n
2n k f n xk,2n 2 k=0
in Theorem 3.2 can be rewritten as follows: 1 0
1 2n k f dG = lim f n gk,2n dG. n 2 k=0
0
In the special case when X = R and G = m, the Lebesgue measure on [0, 1], this formula looks as follows: 1 f dm = lim n
n 1 2 2 −1 f (0) + 2f + 2f + · · · + 2f + f (1) , 2n 2n 2n 2n+1 1
0
which is the Trapezoidal Rule for numerical integration. tree ∗ Let us denote tree 1 = 1 (R). It is essentially well known that P(X, R) = X and f P = ∗ ∗ f , for all f ∈ X . This easily follows from the fact that if f ∈ X , then n n n f f (xk ) = f x ∗ (xk ): x ∗ ∈ X ∗ , x ∗ 1 f (xk ) f sup k=1
k=1
k=1
for every choice of elements x1 , . . . , xn in X. The following is now immediate from Theorem 3.2. Corollary 3.3. Through the canonical isometric isomorphism μ → (μ(gk,2n )) ∈ tree 1 , μ ∈ C[0, 1]∗ , one has the identification C[0, 1]∗ = tree 1 .
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Corollary 3.3 enables us to obtain a new equivalent formulation of the Radon–Nikodým property of Banach spaces; see [6, pp. 217–218] for a summary containing more than twenty ˆ =⊗ ˆ π stands for the (comequivalent formulations of the Radon–Nikodým property. Below, ⊗ pleted) projective tensor product. Corollary 3.4. A Banach space X has the Radon–Nikodým property if and only if tree ˆ tree 1 ⊗ X = 1 (X),
where the canonical isometric isomorphism is given by the mapping (ak,2n ) ⊗ x → (ak,2n x), (ak,2n ) ∈ tree 1 , x ∈ X. ∗ ˆ ˆ Proof. By the above, tree 1 ⊗ X = C[0, 1] ⊗ X = N (C[0, 1], X) (the latter equality is well ∗ known and it holds because C[0, 1] has the approximation property (see, e.g., [18, p. 76])) and, under this identification, the elementary tensor (ak,2n ) ⊗ x corresponds to the operator μ ⊗ x, where ak,2n = μ(gk,2n ), μ ∈ C[0, 1]∗ . It is known (see, [6, pp. 174–175]) that X has the Radon– Nikodým property if and only if N (C[0, 1], X) = P(C[0, 1], X) with the equality of nuclear and absolutely summing norms. By Theorem 3.2, P(C[0, 1], X) = tree 1 (X) and the operator μ ⊗ x corresponds to the two-trunk tree ((μ ⊗ x)(gk,2n )) = (μ(gk,2n )x) = (ak,2n x) ∈ tree 1 (X). 2
Remark 3.2. It is well known (see, e.g., [18, p. 19]) that for every Banach space X, one has ˆ X = 1 (X), 1 ⊗ where the canonical isometric isomorphism is given by the same mapping, as in Corollary 3.4, namely, (an ) ⊗ x → (an x), (an ) ∈ 1 , x ∈ X. ∗ 4. Preduals of tree 1 (X ) and a representation theorem for CX [0, 1] ∗ An easy application of Theorem 3.2 shows that tree 1 (X ) is a dual Banach space. Indeed, let X be a Banach space. By the well-known results of Grothendieck (see, e.g., [5, p. 99] and [18, p. 67]), ˇ X ∗ P C[0, 1], X ∗ = I C[0, 1], X ∗ = C[0, 1] ⊗
ˇ =⊗ ˆ ε stands for the as Banach spaces, where I denotes the ideal of integral operators and ⊗ ∗ ) is isometrically isomor(completed) injective tensor product. Hence, by Theorem 3.2, tree (X 1 ∗ ˇ X is a predual of tree ˇ X)∗ , so that C[0, 1] ⊗ phic to (C[0, 1] ⊗ 1 (X ). tree ∗ The fact, that 1 (X ) is a dual Banach space, can also be seen in a straightforward ∗ manner, without having recourse to Theorem 3.2. Indeed, tree 1 (X ) is a closed subspace of n +1 n +1 2n +1 ∗ 2 ∗ 2 ∗ ∗ ⊥ ∞ (1 (X )) = ∞ ((∞ (X)) ) = (1 (∞ (X))) . We shall show that tree 1 (X ) = Z , the n +1 2 annihilator of some closed subspace Z of 1 (∞ (X)), to be specified below. In particular, ∗ 2n +1 ∗ tree 1 (X ) is a weakly* closed subspace of (1 (∞ (X))) and a dual Banach space. Definition 4.1. Let X be a Banach space. Let us define Z to be the closed subspace of 2n +1 (X)) spanned by the sequences of the form 1 (∞ 0, . . . , 0, zn , −Mn∗ zn , 0, 0, . . . , n = 0, 1, . . . ,
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2 +1 (X) and M ∗ : X 2 +1 → X 2 where zn ∈ ∞ n transpose of the matrix Mn (see Section 2). n
n
n+1 +1
denotes the matrix operator defined by the
∗ ⊥ 2 +1 ∗ Proposition 4.2. Let X be a Banach space. Then tree 1 (X ) = Z in (1 (∞ (X))) . n
2 +1 tree ∗ 2 ∗ ∗ 2 +1 (X))∗ . Let Proof. Let z∗ = (zn∗ )∞ (X ∗ ) = (∞ n=0 ∈ 1 (X ), where zn = (xk,2n )k=0 ∈ 1 n n 2 +1 (X). Then z = (0, . . . , 0, zn , −Mn∗ zn , 0, 0, . . .), for some fixed n, with zn = (xk,2n )2k=0 ∈ ∞ n
n
n
∗ ∗ ∗ z∗ (z) = zn∗ (zn ) − zn+1 Mn zn = zn∗ (zn ) − Mn zn+1 (zn ) = zn∗ (zn ) − zn∗ (zn ) = 0 ∗ ⊥ (see Proposition 2.4). Hence, tree 1 (X ) ⊂ Z . 2n +1 ∗ ∗ ∗ (X ∗ ))\tree On the other hand, if z∗ := (zn∗ )∞ n=0 ∈ ∞ (1 1 (X ), then zn = Mn zn+1 in n n 2n +1 2 +1 (X))∗ for some fixed n. Hence, there is z ∈ 2 +1 (X) such that z∗ (z ) = 1 (X ∗ ) = (∞ n ∞ n n ∗ )(z ). Using the above notation, we get by the above that z∗ (z) = 0. Hence, (Mn zn+1 n z∗ ∈ / Z⊥ . 2 2 +1 (X))Z)∗ , we immediately have the Since Z ⊥ can be canonically identified with (1 (∞ following result. n
∗ Corollary 4.3. Let X be a Banach space. Then tree 1 (X ) is isometrically isomorphic to n 2 +1 (X))Z)∗ . (1 (∞ ∗ 2 +1 ˇ Thus, tree 1 (X ) has two preduals: C[0, 1] ⊗ X and 1 (∞ (X))Z. Theorem 4.4 below ˇ X, which will show, in particular, that these preduals will provide a description of C[0, 1] ⊗ are isometrically isomorphic. For completeness, let us recall that two preduals of a dual Banach space need not be isomorphic, in general. For instance, 1 = c0∗ and also 1 = (C(ωω ))∗ , where C(ωω ) denotes the Banach space of continuous functions on the compact Hausdorff space ωω of ordinal numbers ωω , but c0 is not isomorphic to C(ωω ) (see [2]). n For a Banach space X, let us denote by Φ : P(C[0, 1], X ∗ ) → ∞ (12 +1 (X ∗ )) the into isom∗ ∗ etry given by Theorem 3.2. Recall that ran Φ = tree 1 (X ). Identifying P(C[0, 1], X ) with n +1 n 2 2 +1 (X)))∗ , we have ˇ X)∗ and ∞ (1 (X ∗ )) with (1 (∞ (C[0, 1] ⊗ n
2n +1 ∗ ˇ X ∗ → 1 ∞ Φ : C[0, 1] ⊗ (X) . 2 +1 (X)) → C[0, 1] ⊗ ˇ X by Theorem 4.4. Let X be a Banach space. Define Ψ : 1 (∞ n
∞ 2 n ∞ gk,2n ⊗ xk,2n , Ψ (xk,2n )2k=0 n=0 = n
n=0 k=0
where gk,2n , k = 0, 1, . . . , 2n , are the linear B-splines on [0, 1] with knots 0/2n , 1/2n , . . . , 2n /2n . Then Ψ is a quotient mapping, ker Ψ = Z, and Φ = Ψ ∗ . Proof. First of all, Ψ is correctly defined, because the defining series converges absolutely in n ˇ X. Indeed, let ε = · ε denote the injective tensor norm. Denoting zn = (xk,2n )2k=0 ∈ C[0, 1] ⊗
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2 +1 (X) and recalling that C[0, 1] ⊗ X can be identified with a subspace of L(X ∗ , C[0, 1]), ∞ ε we have n
2n 2n gk,2n ⊗ xk,2n = sup x ∗ (xk,2n )gk,2n ∗ x 1 k=0 k=0 ε C[0,1] ∗ = sup max n x (xk,2n ) = max n xk,2n = zn , x ∗ 1 0k2
0k2
where the second equality holds because a linear spline attains its extremal value at some of its knots. Since (zn ) = n zn < ∞, Ψ is correctly defined and linear. We also clearly have that Ψ 1. n 2n +1 (X)). Then ˇ X)∗ and (zn ) = ((xk,2n )2k=0 ) ∈ 1 (∞ Let now T ∈ P(C[0, 1], X ∗ ) = (C[0, 1] ⊗ ∞ 2 n (T gk,2n )(xk,2n ) (ΦT )(zn ) = (T gk,2n )2k=0 (zn ) = n
n=0 k=0
=
∞ 2n
T , gk,2n ⊗ xk,2n = T , Ψ (zn ) = Ψ ∗ T (zn ),
n=0 k=0
meaning that Φ = Ψ ∗ . Since Ψ ∗ is an into isometry, it is well known (see, e.g., [17, B.3.9]) that Ψ is a quotient mapping. Using Proposition 4.2, we also have ∗ X ⊥ = Z ⊥ ⊥ = Z. ker Ψ = ran Ψ ∗ ⊥ = tree 1
2
It is well known (see, e.g., [18, p. 49]) that the Banach space CX [0, 1] can be identified with ˇ X. Since Ψ is a quotient mapping with ker Ψ = Z, the following representation result C[0, 1] ⊗ is immediate from Theorem 4.4. ˇ X and CX [0, 1] are both isometrically isomorphic to the quotient space Theorem 4.5. C[0, 1] ⊗ 2n +1 (X))Z. 1 (∞ 5. Bounded approximation properties and embedding 1 (X) in tree 1 (X) Let X be a Banach space. We denote by F (X) the subspace of L(X) of finite-rank operators. Let IX denote the identity operator on X. Recall that a Banach space X is said to have the approximation property (AP) if there exists a net (Sα ) ⊂ F (X) such that Sα → IX uniformly on compact subsets of X. If (Sα ) can be chosen with supα Sα λ for some λ 1, then X is said to have the λ-bounded approximation property (λ-BAP). Recently, the weak bounded approximation property was introduced in [10]: X has the weak λ-bounded approximation property (weak λ-BAP) if for every Banach space Y and for every weakly compact operator T : X → Y there exists a net (Sα ) ⊂ F (X) such that Sα → IX uniformly on compact subsets of X and supα T Sα λT . By [12] (see [15] for a simpler proof), the weak λ-BAP and the λ-BAP are equivalent for a Banach space X whenever X ∗ or X ∗∗ has the Radon–Nikodým property. It remains open whether the weak λ-BAP is strictly weaker than the λ-BAP. If they were equivalent, then, by [10], the
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answer to the long-standing famous open problem (Problem 3.8 in [3]), whether the AP of a dual Banach space implies the 1-BAP, would be “yes”. For a recent survey on bounded approximation properties, see [16]. Recall that I denotes the ideal of integral operators. In [9, Theorem 1.3 and Corollary 3.4] we proved that X has the λ-BAP if and only if for every T ∈ I(C[0, 1], X ∗ ) there exists a net (Sα ) ⊂ F (X) such that Sα → IX pointwise and lim supα Sα∗ T I λT I . It is well known that I(C[0, 1], X ∗ ) = P(C[0, 1], X ∗ ) with equality of norms (see, e.g., [5, p. 99]). By Theorem 3.2, ∗ )2n )∞ ∈ tree (X ∗ ) and S ∈ F (X), if T ∈ P(C[0, 1], X ∗ ) is canonically identified with ((xk,2 n k=0 n=0 1 ∗ )2n )∞ . Hence, the BAP of X can be characthen Sα∗ T is canonically identified with ((S ∗ xk,2 n k=0 n=0 ∗ terized in terms of the X ∗ -valued sequence space tree 1 (X ) as follows. Theorem 5.1. Let X be a Banach space and let 1 λ < ∞. The following statements are equivalent. (a) X has the λ-BAP. ∗ ) = ((x ∗ )2n )∞ ∈ tree (X ∗ ) there exists a net (S ) ⊂ F (X) such that (b) For every (xk,2 n α k,2n k=0 n=0 1 ∗ ) λ(x ∗ ). Sα → IX pointwise and lim supα (Sα∗ xk,2 n n k,2 Recall that N denotes the ideal of nuclear operators. In [8] we proved that X has the weak λ-BAP if and only if for every T ∈ N (X, 1 ) there exists a net (Sα ) ⊂ F (X) such that Sα → IX pointwise and lim supα T Sα N λT N . It is well known that N (X, 1 ) is isometrically isomorphic to 1 (X ∗ ) since both spaces can be identified with the projective tensor product ˆ 1 (see, e.g., [18, pp. 19–20, 76]). It can be easily verified that a linear isometry between X∗ ⊗ N (X, 1 ) and 1 (X ∗ ) is explicitly given by the mapping T → (T ∗ en ), T ∈ N (X, 1 ), where (en ) is the unit vector basis of c0 . And the inverse mapping (xn∗ ) → T , (xn∗ ) ∈ 1 (X ∗ ), is given by ∗ ∗ ∗ Tx = ∞ n=1 xn (x)en , x ∈ X, where (en ) is the unit vector basis of 1 . Therefore, if T ∈ N (X, 1 ) ∗ is canonically identified with (xn ) ∈ 1 (X ∗ ) and S ∈ F (X), then T S is canonically identified with (S ∗ xn∗ ). Hence, the weak BAP of X can be characterized as follows. Theorem 5.2. Let X be a Banach space and let 1 λ < ∞. The following statements are equivalent. (a) X has the weak λ-BAP. ∗ (b) For every (xn∗ ) = (xn∗ )∞ n=1 ∈ 1 (X ) there exists a net (Sα ) ⊂ F (X) such that Sα → IX point∗ ∗ wise and lim supα (Sα xn ) λ(xn∗ ). Since the λ-BAP of X implies the weak λ-BAP of X, condition (b) of Theorem 5.1 implies condition (b) of Theorem 5.2. However, to see the latter implication, there is no need to have recourse to the approximation properties (i.e. conditions (a) of Theorems 5.1 and 5.2): our final ∗ result Theorem 5.3 shows that 1 (X ∗ ) embeds in tree 1 (X ) in the way that makes the implication hold. ∗ It is an easy exercise to show that 1 embeds isometrically in C[0, 1]∗ . Since tree 1 = C[0, 1] tree tree (see Corollary 3.3), 1 = 1 (R) embeds in 1 = 1 (R). It is not clear a priori that 1 (X) embeds in tree 1 (X) for an arbitrary Banach space X. Our final purpose is to prove that this is indeed the case, and moreover, 1 (X) embeds in tree 1 (X) in such a way that the embedding respects all bounded linear operators on X. To make this precise, we need some notation.
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Let S ∈ L(X). Define S(xn ) = (Sxn ), ˜ k,2n ) = (Sxk,2n ), S(x
(xn ) = (xn )∞ n=1 ∈ 1 (X), n ∞ (xk,2n ) = (xk,2n )2k=0 n=0 ∈ tree 1 (X).
˜ It is straightforward to verify that S ∈ L(1 (X)), S˜ ∈ L(tree 1 (X)), S = S = S, and the mappings S → S and S → S˜ are linear isometries from L(X) into L(1 (X)) and into L(tree 1 (X)), respectively. Thus, one can naturally embed L(X) both in L(1 (X)) and (X)). The following result shows that there exists a linear isometry from 1 (X) into L(tree 1 tree 1 (X) which identifies S with S˜ for all S ∈ L(X). Theorem 5.3. Let X be a Banach space. Then there exists a linear isometry ϕ from 1 (X) into ˜ tree 1 (X) such that ϕS = Sϕ for all S ∈ L(X). Proof. We shall construct ϕ as the pointwise limit of a sequence ϕm ∈ L(1 (X), tree 1 (X)). 1 )∞ ∈ tree (X) as follows. We first put z1 = Let x = (xn )∞ ∈ (X). We define ϕ x = (z 1 1 n n=0 n=1 1 1 (x1 , x2 , x3 ), and define z01 = M0 z11 . Then, departing again from z11 , we put z21 = (x1 , 0, x2 , 0, x3 ), 1 z31 = (x1 , 0, 0, 0, x2 , 0, 0, 0, x3 ), . . . . Thus, zn+1 is obtained from zn1 , n = 1, 2, . . . , by inserting tree zeros between the components of the previous vector zn1 . To define ϕ2 x = (zn2 )∞ n=0 ∈ 1 (X), we first put z22 = z21 + (0, x4 , 0, x5 , 0) = (x1 , x4 , x2 , x5 , x3 ). Then, departing from z22 , we define z12 = M1 z22 , z02 = M0 z12 , and we define z32 , z42 , . . . as above, by inserting zeros between the components of z22 , z32 , . . . . In general, to define ϕm x = (znm )∞ n=0 ∈ (X), we first put tree 1 m m−1 zm = zm + (0, x2m−1 +2 , 0, x2m−1 +3 , . . . , 0, x2m +1 , 0). m , zm , . . . , zm are defined by Then zm−1 m−2 0 m znm = Mn zn+1 ,
n = m − 1, m − 2, . . . , 0,
m , zm , . . . are defined by inserting zeros between the components of the previous vecand zm+1 m+2 tor. This latter procedure can be formalized by introducing the (2n+1 + 1) × (2n + 1) matrices An with
⎛
⎛
1 ⎜0 ⎜ A1 = ⎜ 0 ⎝ 0 0
0 0 1 0 0
⎞ 0 0⎟ ⎟ 0⎟, ⎠ 0 1
1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ A2 = ⎜ 0 ⎜ ⎜0 ⎜ ⎜0 ⎝ 0 0
0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0
⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟, ⎟ 0⎟ ⎟ 0⎟ ⎠ 0 1
...,
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m = A zm so that Mn An = I2n +1 , the identity matrix of order 2n +1. Then the recurring rule is zn+1 n n for n = m, m + 1, . . . , m = 1, 2, . . . . m for m = 1, 2, . . . , n = 0, 1, . . . , so that ϕ x = This construction guarantees that znm = Mn zn+1 m tree (X) (see Proposition 2.4). We also clearly have that ϕ is linear and ϕ S = Sϕ ˜ m, (znm )∞ ∈ m m n=0 1 m = 1, 2, . . . , S ∈ L(X). To estimate the norm of ϕm x, set s = x = x1 + x2 + · · · , and sn = x1 + · · · + xn . If n = m, m + 1, . . . , then, by construction,
m m z = z = s2m +1 . n
m
Hence (see Proposition 2.4), ϕm x = limznm = s2m +1 s = x, n
so that ϕm ∈ L(1 (X), tree 1 (X)), m = 1, 2, . . . , and ϕm x − → m x,
x ∈ 1 (X).
Similarly, m m−1 = (0, x2m−1 +2 , . . . , 0, x2m +1 , 0) = s2m +1 − s2m−1 +1 , − zm ϕm x − ϕm−1 x = zm so that for m > n ϕm x − ϕn x s2m +1 − s2m−1 +1 + s2m−1 +1 − · · · + s2n+1 +1 − s2n +1 = s2m +1 − s2n +1 − −→ 0. m,n Hence, the limit ϕx := lim ϕm x m
exists for all x ∈ 1 (X), and ϕ ∈ L(1 (X), tree 1 (X)). Moreover, ϕx = limϕm x = x, m
x ∈ 1 (X),
meaning that ϕ is isometric, and for all S ∈ L(X), ˜ ˜ m x = Sϕx, ϕSx = lim ϕm Sx = lim Sϕ m
m
x ∈ 1 (X).
2
References [1] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133–181. [2] C. Bessaga, A. Pełczy´nski, Spaces of continuous functions. IV. (On isomorphical characterizations of spaces C(S)), Studia Math. 19 (1960) 53–62. [3] P.G. Casazza, Approximation properties, in: W.B. Johnson, J. Lindenstrauss (Eds.), Handbook of the Geometry of Banach Spaces, vol. 1, Elsevier, 2001, pp. 271–316. [4] C.K. Chui, An Introduction to Wavelets, Academic Press, Inc., London, 1992.
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[5] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge University Press, 1995. [6] J. Diestel, J.J. Uhl Jr., Vector Measures, Math. Surveys, vol. 15, Amer. Math. Soc., Providence, RI, 1977. [7] R.C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 (1974) 738–743. [8] Å. Lima, V. Lima, E. Oja, Bounded approximation properties via integral and nuclear operators, Proc. Amer. Math. Soc. 138 (2010) 287–297. [9] Å. Lima, V. Lima, E. Oja, Bounded approximation properties in terms of C[0, 1], Preprint, 2010. [10] Å. Lima, E. Oja, The weak metric approximation property, Math. Ann. 333 (2005) 471–484. [11] V. Lima, The weak metric approximation property and ideals of operators, J. Math. Anal. Appl. 334 (2007) 593–603. [12] E. Oja, The impact of the Radon–Nikodým property on the weak bounded approximation property, Rev. R. Acad. Cienc. Ser. A Mat. 100 (2006) 325–331. [13] E. Oja, Lifting of the approximation property from Banach spaces to their dual spaces, Proc. Amer. Math. Soc. 135 (2007) 3581–3587. [14] E. Oja, The strong approximation property, J. Math. Anal. Appl. 338 (2008) 407–415. [15] E. Oja, Inner and outer inequalities with applications to approximation properties, Trans. Amer. Math. Soc., in press. [16] E. Oja, On Bounded Approximation Properties of Banach Spaces, Banach Center Publ., in press. [17] A. Pietsch, Operator Ideals, North-Holland Publishing Company, Amsterdam, New York, Oxford, 1980. [18] R.A. Ryan, Introduction to Tensor Products of Banach Spaces, Springer Monogr. Math., Springer-Verlag, London, 2002.
Journal of Functional Analysis 259 (2010) 2902–2922 www.elsevier.com/locate/jfa
SPDE in Hilbert space with locally monotone coefficients ✩ Wei Liu a,∗ , Michael Röckner a,b a Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany b Department of Mathematics and Statistics, Purdue University, West Lafayette, IN 47906, USA
Received 10 May 2010; accepted 20 May 2010 Available online 1 June 2010 Communicated by Paul Malliavin
Abstract The aim of this paper is to extend the usual framework of SPDE with monotone coefficients to include a large class of cases with merely locally monotone coefficients. This new framework is conceptually not more involved than the classical one, but includes many more fundamental examples not included previously. Thus our main result can be applied to various types of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier–Stokes equation, stochastic p-Laplace equation and stochastic porous media equation with non-monotone perturbations. © 2010 Elsevier Inc. All rights reserved. Keywords: Stochastic evolution equation; Locally monotone; Coercivity; Navier–Stokes equation; Variational approach
1. Introduction Let V ⊂ H ≡ H∗ ⊂ V ∗ be a Gelfand triple, i.e. (H, ·,·H ) is a separable Hilbert space and identified with its dual space by the Riesz isomorphism, V is a reflexive Banach space such that it is continuously and densely ✩
Supported in part by the SFB-701, the BiBoS-Research Center and NNSFC (10721091).
* Corresponding author.
E-mail address:
[email protected] (W. Liu). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.012
W. Liu, M. Röckner / Journal of Functional Analysis 259 (2010) 2902–2922
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embedded into H . If V ∗ ·,·V denotes the dualization between V and its dual space V ∗ , then it follows that V ∗ u, vV
= u, vH ,
u ∈ H, v ∈ V .
Let {Wt }t0 be a cylindrical Wiener process on a separable Hilbert space U w.r.t. a complete filtered probability space (Ω, F , Ft , P) and (L2 (U ; H ), · 2 ) denotes the space of all Hilbert– Schmidt operators from U to H . We consider the following stochastic evolution equation dXt = A(t, Xt ) dt + B(t, Xt ) dWt ,
(1.1)
where for some fixed time T A : [0, T ] × V × Ω → V ∗ ;
B : [0, T ] × V × Ω → L2 (U ; H )
are progressively measurable, i.e. for every t ∈ [0, T ], these maps restricted to [0, t] × V × Ω are B([0, t]) ⊗ B(V ) ⊗ Ft -measurable (where B denotes the corresponding Borel σ algebra). It is well known that (1.1) has a unique solution if A, B satisfy the classical monotone and coercivity conditions (cf. [15,25]), which we recall in Appendix A below. The theory of monotone operators starts from substantial work of Minty [22,23] and Browder [6,7] for PDE. We refer to [35,32] for a detailed exposition and references. In recent years, this variational approach has been also used intensively for analyzing SPDE driven by infinite-dimensional Wiener process. Unlike the semigroup approach (cf. [10]), it is not necessary to have a linear operator in the drift part which has to generate a semigroup. Hence the variational approach can be used to investigate nonlinear SPDE which are not necessarily of semilinear type. For general results on the existence and uniqueness of solutions to SPDE we refer to [24,15,12,26,34]. Within this framework many different types of properties have already been established, e.g. see [8,18,27,31] for the small noise large deviation principle, [13,14] for discretization approximation schemes to the solutions of SPDE, [33,16,17] for the dimension-free Harnack inequality and resulting ergodicity, compactness and contractivity properties of the associated transition semigroups, and [19,5,11] for the invariance of subspaces and existence of random attractors for corresponding random dynamical systems. As one typical example of SPDE in this framework, the stochastic porous media equation has been extensively studied in [3,1,2,4,9,30]. The main aim of this paper is to provide a more general framework for the variational approach, being conceptually not more complicated than the classical one (cf. [15]), but including a large number of new applications as e.g. fundamental SPDE as the stochastic 2-D Navier– Stokes equation and stochastic Burgers type equation. The main changes consist of localizing the monotonicity condition and relaxing the growth condition. This new framework is, in addition, more stable with respect to perturbations. We refer to Section 3 below for details. In particular, we can simplify the related approach to the stochastic 2-D Navier–Stokes equation in the nice paper [21], which inspired us a lot to start this work. However, our approach also easily covers the case of arbitrary multiplicative noise, whereas in [21] only additive noise was considered. It is also straight forward to extend our new framework to more noise terms, e.g. Levy noise (cf. [12] for the classical case). This and further new applications will be the subject of future work.
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Let us now state the precise conditions on the coefficients of (1.1): Suppose there exist constants α > 1, β 0, θ > 0, K and a positive adapted process f ∈ L1 ([0, T ] × Ω; dt × P) such that the following conditions hold for all v, v1 , v2 ∈ V and (t, ω) ∈ [0, T ] × Ω. (H 1) (Hemicontinuity) The map s → V ∗ A(t, v1 + sv2 ), vV is continuous on R. (H 2) (Local monotonicity) 2 2V ∗ A(t, v1 ) − A(t, v2 ), v1 − v2 V + B(t, v1 ) − B(t, v2 )2 K + ρ(v2 ) v1 − v2 2H , where ρ : V → [0, +∞) is a measurable function and locally bounded in V . (H 3) (Coercivity) 2 2V ∗ A(t, v), v V + B(t, v)2 + θ vαV ft + Kv2H . (H 4) (Growth) α β α A(t, v) α−1 V ∗ ft + KvV 1 + vH . Remark 1.1. (1) (H 2) is essentially weaker than the standard monotonicity (A2) (i.e. ρ ≡ 0). One typical form of (H 2) in applications is ρ(v) = Cvγ , where · is some norm on V and C, γ are some constants. One typical example is the stochastic 2-D Navier–Stokes equation on a bounded or unbounded domain, which satisfies (H 2) but does not satisfy (A2) (see Section 3). In fact, if A(t, v) = νPH v − PH [(v · ∇)v], we have ν 16 2 4 2V ∗ A(t, v1 ) − A(t, v2 ), v1 − v2 V − v1 − v2 V + ν + 3 v2 L4 v1 − v2 2H . 2 ν (2) If the noise is zero or additive type in (1.1), then the existence and uniqueness of solutions to (1.1) can be established by replacing (H 2) with the following more general type of local monotonicity: V∗
A(t, v1 ) − A(t, v2 ), v1 − v2
V
K + η(v1 ) + ρ(v2 ) v1 − v2 2H ,
where η, ρ : V → [0, +∞) are measurable functions and locally bounded in V . This will be investigated in a seperated paper [20]. (3) (H 4) is also weaker than the standard growth condition (A4) (see Appendix A) assumed in the literature (cf. [15,35,25]). The advantage of (H 4) is, e.g., to include many semilinear type equations with nonlinear perturbation terms. For example, if we consider a reaction–diffusion type equation, i.e. A(u) = u + F (u), then for verifying (H 3) we have α = 2. Hence (A4) would imply that F has at most linear growth. However, we can allow F to have some polynomial growth by using the weaker condition (H 4) here. We refer to Section 3 for more details.
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Definition 1.1 (Solution of SEE). A continuous H -valued (Ft )-adapted process {Xt }t∈[0,T ] is called a solution of (1.1), if for its dt ⊗ P-equivalent class X¯ we have X¯ ∈ Lα [0, T ] × Ω, dt ⊗ P; V ∩ L2 [0, T ] × Ω, dt ⊗ P; H and P-a.s., t Xt = X0 +
A(s, X¯ s ) ds +
0
t
B(s, X¯ s ) dWs ,
t ∈ [0, T ].
0
Now we can state the main result. Theorem 1.1. Suppose (H 1)–(H 4) hold for f ∈ Lp/2 ([0, T ] × Ω; dt × P) with some p β + 2, and there exists a constant C such that B(t, v)2 C ft + v2 , t ∈ [0, T ], v ∈ V ; H 2 β ρ(v) C 1 + vαV 1 + vH , v ∈ V .
(1.2)
Then for any X0 ∈ Lp (Ω → H ; F0 ; P) (1.1) has a unique solution {Xt }t∈[0,T ] and satisfies
E
sup
t∈[0,T ]
p Xt H
T Xt αV
+
dt < ∞.
0
Moreover, if A(t, ·)(ω), B(t, ·)(ω) are independent of t ∈ [0, T ] and ω ∈ Ω, then the solution {Xt }t∈[0,T ] of (1.1) is a Markov process. 2. Proof of the main theorem The first step of the proof is mainly based on the Galerkin approximation. Let {e1 , e2 , . . .} ⊂ V be an orthonormal basis of H and let Hn := span{e1 , . . . , en } such that span{e1 , e2 , . . .} is dense in V . Let Pn : V ∗ → Hn be defined by Pn y :=
n
V ∗ y, ei V ei ,
y ∈ V ∗.
i=1
Obviously, Pn |H is just the orthogonal projection onto Hn in H and we have V∗
Pn A(t, u), v
V
= Pn A(t, u), v H = V ∗ A(t, u), v V ,
Let {g1 , g2 , . . .} be an orthonormal basis of U and
u ∈ V , v ∈ Hn .
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(n)
Wt
:=
n Wt , gi U gi = P˜n Wt , i=1
where P˜n is the orthogonal projection onto span{g1 , . . . , gn } in U . Then for each finite n ∈ N we consider the following stochastic equation on Hn (n)
dXt
(n) (n) (n) = Pn A t, Xt dt + Pn B t, Xt dWt ,
(n)
X0 = Pn X0 .
(2.1)
By the classical result for the solvability of SDE in finite-dimensional space (cf. [15,25]) we know that (2.1) has a unique strong solution. In order to construct the solution of (1.1), we need some a priori estimates for X (n) . For convenience we use following notations: K = Lα [0, T ] × Ω → V ; dt × P ; α K ∗ = L α−1 [0, T ] × Ω → V ∗ ; dt × P ; J = L2 [0, T ] × Ω → L2 (U ; H ); dt × P . Lemma 2.1. Under the assumptions in Theorem 1.1, there exists C > 0 such that for all n ∈ N (n) X + sup EXt(n) 2 C. H K t∈[0,T ]
Proof. The conclusion follows from (H 3) by using the same argument as in [25, Lemma 4.2.9]. Hence we omit the details here. 2 Lemma 2.2. Under the assumptions in Theorem 1.1, there exists C > 0 such that for all n ∈ N we have (n) p E sup Xt H + E t∈[0,T ]
T
T (n) p−2 (n) α p p/2 Xt Xt dt C EX0 + E ft dt . H H V
0
0
In particular, there exists C > 0 such that for all n ∈ N A ·, X (n)
K∗
C.
Proof. By Itô’s formula, Young’s inequality and (1.2) we have (n) p (n) p Xt = X + p(p − 2) 0 H H
t
(n) p−4 X Pn B s, X (n) P˜n ∗ X (n) 2 ds s s s H H
0
+
p 2
t 0
(n) p−2 2 X 2V ∗ A s, Xs(n) , Xs(n) V + Pn B s, Xs(n) P˜n 2 ds s H
(2.2)
W. Liu, M. Röckner / Journal of Functional Analysis 259 (2010) 2902–2922
t +p
2907
(n) p−2 (n) X X , Pn B s, X (n) dW (n) s
s
H
s
s
H
0 p X0 H
pθ − 2
t
(n) p−2 (n) α X X ds + C s s H V
0
t +p
t
(n) p X + fs · X (n) p−2 ds s s H H
0
(n) p−2 (n) X X , Pn B s, X (n) dW (n) s
s
H
s
s
H
0 p X0 H
pθ − 2
t
(n) p−2 (n) α X X ds + C s s H V
0
t +p
t
(n) p X + fsp/2 ds s H
0
(n) p−2 (n) X Xs , Pn B s, Xs(n) dWs(n) H , s H
t ∈ [0, T ],
0
where C is a generic constant (independent of n) and may change from line to line. For any given n we define the stopping time (n)
(n) τR = inf t ∈ [0, T ]: Xt H > R ∧ T , Here we take inf ∅ = ∞. It’s obvious that (n) lim τ R→∞ R
= T,
P-a.s., n ∈ N.
Then by the Burkholder–Davis–Gundy inequality we have r (n) p−2 (n) Xs H Xs , Pn B s, Xs(n) dWs(n) H E sup r∈[0,t] 0
t 3E
(n) 2p−2 X B s, X (n) 2 ds s
s
H
1/2
2
0
3E
2p−2 sup Xs(n) H · C
s∈[0,t]
p 3E ε sup Xs(n) H + Cε s∈[0,t]
t
(n) 2 X + fs ds s H
1/2
0
t 0
(n) 2 X + fs ds s H
p/2
R > 0.
(2.3)
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p 3εE sup Xs(n) H + 3 · (2T )p/2−1 Cε E s∈[0,t]
t
(n) p X + fsp/2 ds, s H
(n) t ∈ 0, τR ,
0
(2.4) where ε > 0 is a small constant and Cε comes from Young’s inequality. Then by (2.3), (2.4) and Gronwall’s lemma we have (n)
E
(n) p sup Xt H + E
τR
(n)
t∈[0,τR ]
0
C
(n) p−2 (n) α X X ds s s H V
p EX0 H
T +E
p/2 fs ds
n 1,
,
0
where C is a constant independent of n. For R → ∞, (2.2) follows from the monotone convergence theorem. Moreover, by (H 4) and p β + 2 we have A ·, X (n) where C is a constant independent of n.
K∗
C,
n 1,
2
Proof of Theorem 1.1. (1) Existence: By Lemmas 2.1 and 2.2 there exists a subsequence nk → ∞ such that (i) X (nk ) → X¯ weakly in K and weakly star in Lp (Ω; L∞ ([0, T ]; H )). (ii) Y (nk ) := A(·, X (nk ) ) → Y weakly in K ∗ . (iii) Z (nk ) := Pnk B(·, X (nk ) ) → Z weakly in J and hence · Pnk B
s, Xs(nk )
· dWs(nk )
→
0
Zs dWs 0
weakly in L∞ ([0, T ], dt; L2 (Ω, P; H )). Now we define t Xt := X0 +
t Ys ds +
0
it is easy to show that X = X¯ dt ⊗ P-a.e.
Zs dWs , 0
t ∈ [0, T ],
(2.5)
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Then by [25, Theorem 4.2.5] we know that X is an H -valued continuous (Ft )-adapted process and
E
sup
t∈[0,T ]
p Xt H
T Xt αV
+
dt < ∞.
0
Therefore, it remains to verify that ¯ = Y, A(·, X)
¯ = Z dt ⊗ P-a.e. B(·, X)
Define
T
M = φ: φ is V -valued (Ft )-adapted process such that E
ρ(φs ) ds < ∞ . 0
For φ ∈ K ∩ M ∩ Lp (Ω; L∞ ([0, T ]; H )), (n ) 2 t (n ) 2 E e− 0 (K+ρ(φs )) ds Xt k H − E X0 k H t s 2 e− 0 (K+ρ(φr )) dr 2V ∗ A s, Xs(nk ) , Xs(nk ) V + Pnk B s, Xs(nk ) P˜nk 2 =E 0
2 − K + ρ(φs ) Xs(nk ) H ds t E
e−
s
0 (K+ρ(φr )) dr
2V ∗ A s, Xs(nk ) , Xs(nk ) V
0
2 2 + B s, Xs(nk ) 2 − K + ρ(φs ) Xs(nk ) H ds t =E
e−
s
0 (K+ρ(φr )) dr
2V ∗ A s, Xs(nk ) − A(s, φs ), Xs(nk ) − φs V
0
2 2 + B s, Xs(nk ) − B(s, φs )2 − K + ρ(φs ) Xs(nk ) − φs H ds t +E 0
e−
s
0 (K+ρ(φr )) dr
2V ∗ A s, Xs(nk ) − A(s, φs ), φs V + 2V ∗ A(s, φs ), Xs(nk ) V
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2 − B(s, φs )2 + 2 B s, Xs(nk ) , B(s, φs ) L
2 (U,H )
(n ) − 2 K + ρ(φs ) Xs k , φs H + K + ρ(φs ) φs 2H ds .
(2.6)
Let k → ∞, by (H 2) and the lower semicontinuity (cf. e.g. [25, (4.2.27)] for details) we have for every nonnegative ψ ∈ L∞ ([0, T ]; dt), T E
− t (K+ρ(φ )) ds 2 2 s ψt e 0 Xt H − X0 H
0
T lim inf E k→∞
(nk ) 2 − t (K+ρ(φ )) ds (nk ) 2 s Xt − X ψt e 0 0 H H
0
T E
t ψt
0
e−
s
0 (K+ρ(φr )) dr
0
2V ∗ Ys − A(s, φs ), φs V
2 − B(s, φs )2 + 2 Zs , B(s, φs ) L (U,H ) 2 − 2 K + ρ(φs ) Xs , φs H + K + ρ(φs ) φs 2H ds dt .
+ 2V ∗ A(s, φs ), X¯ s
V
(2.7)
By Itô’s formula we have for φ ∈ K ∩ M ∩ Lp (Ω; L∞ ([0, T ]; H )), t E e− 0 (K+ρ(φs )) ds Xt 2H − E X0 2H t s =E e− 0 (K+ρ(φr )) dr 2V ∗ Ys , X¯ s V + Zs 22 − K + ρ(φs ) Xs 2H ds .
(2.8)
0
By inserting (2.8) into (2.7) we obtain T 0E
t ψt
0
e−
s
0 (K+ρ(φr )) dr
2V ∗ Ys − A(s, φs ), X¯ s − φs V
0
2 2 + B(s, φs ) − Zs 2 − K + ρ(φs ) Xs − φs H ds dt .
(2.9)
Note that (1.2), Lemmas 2.1 and 2.2 imply that X¯ ∈ K ∩ M ∩ Lp Ω; L∞ [0, T ]; H . ¯ Next, we first take φ = X¯ − ε φv ˜ for By taking φ = X¯ we obtain that Z = B(·, X).
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φ˜ ∈ L∞ ([0, T ] × Ω; dt ⊗ P; R) and v ∈ V , then we divide by ε and let ε → 0 to derive that T 0E
t ψt
e
0
−
s
¯ 0 (K+ρ(Xr )) dr
φ˜ s V ∗ Ys − A(s, X¯ s ), v
V
ds dt .
(2.10)
0
¯ ˜ we conclude that Y = A(·, X). By the arbitrariness of ψ and φ, ¯ Hence X is a solution of (1.1). (2) Uniqueness: Suppose Xt , Yt are the solutions of (1.1) with initial conditions X0 , Y0 respectively, i.e. t Xt = X0 +
t A(s, Xs ) ds +
0
0
t
t
Yt = Y0 +
A(s, Ys ) ds + 0
B(s, Xs ) dWs ,
t ∈ [0, T ];
B(s, Ys ) dWs ,
t ∈ [0, T ].
(2.11)
0
Then by the product rule, Itô’s formula and (H 2) we have
e
−
t
t
0 (K+ρ(Ys )) ds
Xt − Yt 2H
X0 − Y0 2H
+2
e−
s
0 (K+ρ(Yr )) dr
0
× Xs − Ys , B(s, Xs ) dWs − B(s, Ys ) dWs
H
,
t ∈ [0, T ].
By a standard localization argument we have t E e− 0 (K+ρ(Ys )) ds Xt − Yt 2H EX0 − Y0 2H ,
t ∈ [0, T ].
If X0 = Y0 , P-a.s., then t E e− 0 (K+ρ(Ys )) ds Xt − Yt 2H = 0,
t ∈ [0, T ].
Since (1.2) and Lemmas 2.1, 2.2 imply that t
K + ρ(Ys ) ds < ∞,
P-a.s., t ∈ [0, T ],
0
we have Xt = Yt ,
P-a.s., t ∈ [0, T ].
Therefore, the pathwise uniqueness follows from the path continuity of X, Y in H . (3) Markov property: The proof of Markov property is standard, we refer to [25, Proposition 4.3.5] or [15, Theorem II.2.4]. 2
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3. Application to examples Obviously, the main result can be applied to stochastic evolution equations with monotone coefficients (cf. [25] for the stochastic porous medium equation and stochastic p-Laplace equation) and non-monotone perturbations (e.g. some locally Lipschitz perturbation) in the drift. Below we present some examples where the coefficients are only locally monotone, hence the classical result of monotone operators cannot be applied. In this section we use the notation Di to denote the spatial derivative ∂x∂ i , Λ ⊆ Rd is an open 1,p
bounded domain with smooth boundary. For the standard Sobolev space W0 (Λ) (p 2) we always use the following (equivalent) Sobolev norm: u1,p :=
∇u(x)p dx
1/p .
Λ
For simplicity we only consider examples where the coefficients are time independent, but one can easily adapt those examples to the time dependent case. Lemma 3.1. Consider the Gelfand triple V := W01,2 (Λ) ⊆ H := L2 (Λ) ⊆ W −1,2 (Λ) and the operator A(u) = u +
d
fi (u)Di u,
i=1
where fi (i = 1, . . . , d) are bounded Lipschitz functions on R. (1) If d < 3, then there exists a constant K such that 2V ∗ A(u) − A(v), u − v V −u − v2V + K + Kv2V u − v2H ,
u, v ∈ V .
(2) If d = 3, then there exists a constant K such that 2V ∗ A(u) − A(v), u − v V −u − v2V + K + Kv4V u − v2H ,
u, v ∈ V .
(3) If fi are independent of u for i = 1, . . . , d, i.e. A(u) = u +
d
fi · Di u,
i=1
then for any d 1 we have 2V ∗ A(u) − A(v), u − v V −u − v2V + Ku − v2H ,
u, v ∈ V .
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Proof. (1) Since all fi are bounded and Lipschitz, we have V∗
A(u) − A(v), u − v
= −u − v2V +
V
d
fi (u)Di u − fi (v)Di v (u − v) dx
i=1 Λ
= −u − v2V
d fi (u)(Di u − Di v) + Di v fi (u) − fi (v) (u − v) dx + i=1 Λ
−u − v2V +
d i=1
Λ
2 fi (u) − fi (v) (u − v)2 dx
2
(Di v) dx Λ
−u − v2V
Λ
1/2 fi2 (u)(u − v)2 dx
Λ
1/2
+
1/2 (Di u − Di v)2 dx
+ Ku − vV
1/2
1/2 (u − v) dx
+ KvV
2
Λ
1/2 (u − v) dx 4
Λ
3 − u − v2V + Ku − v2H + KvV u − v2L4 , 4
u, v ∈ V ,
(3.1)
where K is a generic constant that may change from line to line. For d < 3, we have the following well-known estimate on R2 (see [21, Lemma 2.1]) u4L4 2u2L2 ∇u2L2 ,
u ∈ W01,2 (Λ).
(3.2)
Hence combining with (3.1) we have V∗
A(u) − A(v), u − v
V
1 − u − v2V + K + Kv2V u − v2H , 2
u, v ∈ V .
(2) For d = 3 we use the following estimate (cf. [21]) u4L4 4uL2 ∇u3L2 ,
u ∈ W01,2 (Λ),
(3.3)
then the second assertion can be derived similarly from (3.1) and Young’s inequality. (3) This assertion can be easily derived as (3.1). 2 Example 3.2 (Semilinear stochastic equations). Let Λ be an open bounded domain in Rd with smooth boundary. We consider the following triple ∗ V := W01,2 (Λ) ⊆ H := L2 (Λ) ⊆ W01,2 (Λ) and the semilinear stochastic equation
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dXt = Xt +
d
fi (Xt )Di Xt + g(Xt ) dt + B(Xt ) dWt ,
(3.4)
i=1
where Wt is a Wiener process on L2 (Λ) and fi , g, B satisfy the following conditions: (i) fi are bounded Lipschitz functions on R for i = 1, . . . , d; (ii) g is a continuous function on R such that g(x) C |x|r + 1 , x ∈ R; g(x) − g(y) (x − y) C 1 + |y|s (x − y)2 ,
x, y ∈ R,
(3.5)
where C, r, s are some positive constants. (iii) B : V → L2 (L2 (Λ)) is Lipschitz. Then we have the following result: (1) If d = 1, r = 3, s = 2, then for any X0 ∈ L6 (Ω, F0 , P; H ), (3.4) has a unique solution {Xt }t∈[0,T ] and this solution satisfies
E
T sup
t∈[0,T ]
Xt 6H
Xt 2V
+
dt < ∞.
(3.6)
0
(2) If d = 2, r = 73 , s = 2, then for any X0 ∈ L6 (Ω, F0 , P; H ), (3.4) has a unique solution {Xt }t∈[0,T ] and this solution satisfies (3.6). (3) If d = 3, r = 73 , s = 43 , fi , i = 1, . . . , d are bounded measurable functions and independent of u, then for any X0 ∈ L6 (Ω, F0 , P; H ), (3.4) has a unique solution {Xt }t∈[0,T ] and this solution satisfies (3.6). Proof. (1) We define the operator
A(u) = u +
d
fi (u)Di u + g(u),
u ∈ V.
i=1
The hemicontinuity (H 1) follows easily from the continuity of f and g. Note that (3.5) and (3.2) imply V∗
g(u) − g(v), u − v
V
C 1 + vsL2s u − v2L4 1 2 u − v2V + C 1 + v2s 2s u − vH , L 4
Therefore, by Lemma 3.1 we have for d < 3
u, v ∈ V .
(3.7)
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2V ∗ A(u) − A(v), u − v V 1 2 − u − v2V + C 1 + v2V + v2s 2s u − vH , L 2
u, v ∈ V ,
i.e. (H 2), (H 3) hold with ρ(v) = v2V + v2s and α = 2. L2s For d = 1, r = 3, by the Sobolev embedding theorem we have g(u)
V∗
C 1 + u3L3 C 1 + uV u2H ,
u ∈ V.
Then it is easy to show that A(u)
V∗
C 1 + uV + uV u2H ,
u ∈ V.
Hence (H 4) holds with β = 4. Therefore, all assertions follow from Theorem 1.1 by taking p = 6. (2) For d = 2, 3 we have g(u)
V∗
C 1 + urL6r/5 ,
u ∈ V.
For r = 73 , by the interpolation theorem we have 4/7
3/7
uL6r/5 uL2 uL6 ,
u ∈ W01,2 (Λ) ⊆ L6 (Λ).
Then g(u)
V∗
4/3 C 1 + urL6r/5 C 1 + uH uV ,
u ∈ V.
(3.8)
Hence (H 4) holds for d = 2, 3 with β = 8/3. Therefore, for d = 2, all assertions follow from Theorem 1.1 by taking p = 6 (in fact, p = 14/3 is enough). (3) If d = 3 and fi , i = 1, 2, 3 are bounded measurable functions and independent of u, then by Lemma 3.1 and (3.3) we have 1 u − v2H , 2V ∗ A(u) − A(v), u − v V − u − v2V + K 1 + v4s L2s 2 Hence (H 2), (H 3) hold with ρ(v) = v4s and α = 2. L2s Since s = 43 , by the interpolation inequality we have 5/8
3/8
uL2s uL2 uL6 ,
u ∈ V.
Therefore, 10/3
u4s CuH u2V , L2s i.e. (1.2) holds with β = 10/3.
u ∈ V,
u, v ∈ V .
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Hence combining with (3.8) we can take p = 6 (in fact, p 16/3 is enough). Then all assertions follow from Theorem 1.1. 2 Remark 3.1. (1) For some specific examples, one might derive the local monotonicity (H 2) without assuming the boundedness of fi , i = 1, . . . , d. For instance, Wilhelm Stannat (whom we like to thank for this at this point) pointed out to us that our local monotonicity condition is also fulfilled by the classical stochastic Burgers equation. Since the remaining conditions hold anyway in this case, all our results apply to the classical stochastic Burgers equation as well. More precisely, for the classical stochastic Burgers equation we have d = 1,
Λ = [0, 1],
A(u) = u + u
∂u , ∂x
then we can derive the following local monotonicity: V∗
A(u) − A(v), u − v V ∂u ∂v 2 u −v (u − v) dx = −u − vV + ∂x ∂x Λ
= −u − v2V −
1 2
(u − v + 2v)(u − v)
∂ (u − v) dx ∂x
Λ
= −u − v2V −
v(u − v)
∂ (u − v) dx ∂x
Λ
≤ −u − v2V
+ vL4 u − vL4 u − vV 1/2
3/2
≤ −u − v2V + KvL4 u − vH u − vV 3 ≤ − u − v2V + Kv4L4 u − v2H , 4
u, v ∈ V ,
where K is some constant that may change from line to line. (2) One obvious generalization is that one can replace in (3.4) by the p-Laplace operator div(|∇u|p−2 ∇u) or the more general quasi-linear differential operator
(−1)|α| Dα Aα (Du),
|α|m
where Du = (Dβ u)|β|m . Under certain assumptions (cf. [35, Proposition 30.10]) this operator satisfies the monotonicity and coercivity condition. Then, according to Theorem 1.1, we can obtain the existence and uniqueness of solutions to this type of quasi-linear SPDE with nonmonotone perturbations (e.g. some locally Lipschitz lower order terms). Now we apply Theorem 1.1 to the stochastic 2-D Navier–Stokes equation.
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Let Λ be a bounded domain in R2 with smooth boundary. Define
V = v ∈ W01,2 Λ, R2 : ∇ · v = 0 a.e. in Λ ,
vV :=
1/2 |∇v|2 dx
,
Λ
and H is the closure of V in the following norm vH :=
1/2 |v|2 dx
.
Λ
The linear operator PH (Helmhotz–Hodge projection) and A (Stokes operator with viscosity constant ν) are defined by PH : L2 Λ, R2 → H orthogonal projection; A : W 2,2 Λ, R2 ∩ V → H, Au = νPH u. It is well known then the Navier–Stokes equation can be reformulated as follows u = Au + F (u) + f,
u(0) = u0 ∈ H,
(3.9)
where f ∈ L2 (0, T ; V ∗ ) denotes some external force and F : DF ⊂ H × V → H,
F (u, v) = −PH (u · ∇)v ,
F (u) = F (u, u).
It is standard that using the Gelfand triple V ⊆ H ≡ H ∗ ⊆ V ∗, we see that the following mappings A : V → V ∗,
F :V ×V →V∗
are well defined. In particular, we have V∗
F (u, v), w V = −V ∗ F (u, w), v V ,
V∗
F (u, v), v
V
= 0,
u, v, w ∈ V .
Now we consider the stochastic 2-D Navier–Stokes equation dXt = AXt + F (Xt ) + ft dt + B(Xt ) dWt , where Wt is a Wiener process on H .
(3.10)
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Example 3.3 (Stochastic 2-D Navier–Stokes equation). Suppose that X0 ∈ L4 (Ω, F0 , P; H ) and B : V → L2 (H ) satisfies B(v1 ) − B(v2 )2 K 1 + v2 4 4
L (Λ;R2 )
2
v1 − v2 2H ,
v1 , v2 ∈ V ,
where K is some constant. Then (3.10) has a unique solution {Xt }t∈[0,T ] and this solution satisfies
E
T sup Xt 4H +
t∈[0,T ]
Xt 2V dt < ∞. 0
Proof. The hemicontinuity (H 1) is obvious since F is a bilinear map. Note that V ∗ F (v), vV = 0, it is also easy to show (H 3) with α = 2: V∗
ν −νv2V + ft V ∗ vV − v2V + Cft 2V ∗ , 2 2 2 B(v) 2Kv2 + 2B(0) , v ∈ V . H 2 2
Av + F (v) + ft , v
V
v ∈ V,
Recall the following estimates (cf. e.g. [21, Lemmas 2.1, 2.2]) V ∗ F (w), v 2w 4 L (Λ;R2 ) vV ; V V ∗ F (w), v 2w3/2 w1/2 v 4 L (Λ;R2 ) , v, w ∈ V . V H V
(3.11)
Then we have V∗
F (u) − F (v), u − v
V
= −V ∗ F (u, u − v), v V + V ∗ F (v, u − v), v V = −V ∗ F (u − v), v V 3/2
1/2
2u − vV u − vH vL4 (Λ;R2 ) ν 32 u − v2V + 3 v4L4 (Λ;R2 ) u − v2H , 2 ν
u, v ∈ V . (3.12)
Hence we have the local monotonicity (H 2) with ρ(v) = v4L4 (Λ;R2 ) : V∗
Au + F (u) − Av − F (v), u − v
V
ν 32 − u − v2V + 3 v4L4 (Λ;R2 ) u − v2H . 2 ν
(3.11) and (3.2) imply that (H 4) and (1.2) hold with β = 2. Therefore, the existence and uniqueness of solutions to (3.10) follow from Theorem 1.1 by taking p = 4. 2
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Remark 3.2. (1) If the noise in (3.10) is additive type, then the existence and uniqueness of solutions to (3.10) have been established in [21]. Here we can conclude the same result for (3.10) with general multiplicative noise by a direct application of our main result. (2) For the 3-D Navier–Stokes equation, we recall the following well-known estimate (cf. e.g. [21, (2.5)]) ψ4L4 4ψL2 ∇ψ3L2 ,
ψ ∈ W01,2 Λ; R3 .
Then one can show that V∗
F (u) − F (v), u − v
V
= −V ∗ F (u − v), v V 7/4
1/4
2u − vV u − vH vL4 (Λ;R3 ) ν 212 u − v2V + 7 v8L4 (Λ;R3 ) u − v2H , 2 ν
u, v ∈ V .
Hence we have the following local monotonicity (H 2): V∗
Au + F (u) − Av − F (v), u − v
V
ν 212 − u − v2V + 7 v8L4 (Λ;R3 ) u − v2H . 2 ν
Another form of local monotonicity can be derived similarly: V∗
F (u) − F (v), u − v
V
= −V ∗ F (u − v), v V 3/2
1/2
2u − vV u − vH vL6 (Λ;R3 ) ν 32 u − v2V + 3 v4L6 (Λ;R3 ) u − v2H , 2 ν
u, v ∈ V .
(3) Concerning the growth condition, we have in the 3-D case that F (u)
V∗
1/2
3/2
2u2L4 (Λ;R3 ) 4uH uV ,
u ∈ V.
Unfortunately, this is not enough to verify (H 4) in Theorem 1.1. (4) One should note that the only role of (H 4) is to assure that A(·, X (n) )K ∗ is uniformly bounded for all n (see Lemma 2.2). Therefore, one can replace (H 4) by some weaker growth condition once we can derive some stronger a priori estimate for X (n) in the Galerkin approximation (e.g. as in [29]). One good example of a further generalization of our main result is that we can apply Theorem 1.1 (with a revised version of (H 4)) to derive the existence and uniqueness of solutions to the following stochastic tamed 3-D Navier–Stokes equation with smooth enough initial condition: dXt = AXt + F (Xt ) + ft − PH gN |Xt |2 Xt dt + B(Xt ) dWt , where the taming function gN : R+ → R+ is smooth and satisfies for some N > 0
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⎧ g (r) = 0, ⎨ N gN (r) = (r − N )/ν, ⎩ 0 gN (r) C,
if r N, if r N + 1, r 0.
We refer to [28,29] for more details on the stochastic tamed 3-D Navier–Stokes equation. Appendix A. The classical monotone and coercive conditions For the existence and uniqueness of the solution to (1.1) we recall the following classical monotone and coercive conditions on A and B. Suppose there exist constants α > 1, θ > 0, K and a positive adapted process f ∈ L1 ([0, T ] × Ω; dt × P) such that the following conditions hold for all v, v1 , v2 ∈ V and (t, ω) ∈ [0, T ] × Ω. (A1) (Hemicontinuity) The map s → V ∗ V ∗ A(t, v1 + sv2 ), vV is continuous on R. (A2) (Monotonicity) 2 2V ∗ A(t, v1 ) − A(t, v2 ), v1 − v2 V + B(t, v1 ) − B(t, v2 )2 Kv1 − v2 2H . (A3) (Coercivity) 2 2V ∗ A(t, v), v V + B(t, v)2 + θ vαV ft + Kv2H . (A4) (Growth) A(t, v)
V∗
(α−1)/α
ft
+ Kvα−1 V .
Theorem 4.1. ([15, Theorems II.2.1, II.2.2].) Suppose (A1)–(A4) hold, then for any X0 ∈ L2 (Ω, F0 , P; H ) (1.1) has a unique solution {Xt }t∈[0,T ] and this solution satisfies E sup Xt 2H < ∞. t∈[0,T ]
Moreover, we have the following Itô formula t Xt 2H
= X0 2H
+
2 2V ∗ A(s, Xs ), Xs V + B(s, Xs )2 ds
0
t +2
Xs , B(s, Xs ) dWs
0
H
,
t ∈ [0, T ], P-a.s.
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References [1] V. Barbu, G. Da Prato, M. Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equation, Indiana Univ. Math. J. 57 (1) (2008) 187–212. [2] V. Barbu, G. Da Prato, M. Röckner, Some results on stochastic porous media equations, Boll. Unione Mat. Ital. 9 (2008) 1–15. [3] V. Barbu, G. Da Prato, M. Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab. 37 (2009) 428–452. [4] V. Barbu, G. Da Prato, M. Röckner, Stochastic porous media equation and self-organized criticality, Comm. Math. Phys. 285 (2009) 901–923. [5] W. Beyn, B. Gess, P. Lescot, M. Röckner, The global random attractor for a class of stochastic porous media equations, preprint. [6] F.E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc. 69 (1963) 862–874. [7] F.E. Browder, Non-linear equations of evolution, Ann. Math. 80 (1964) 485–523. [8] P.L. Chow, Large deviation problem for some parabolic Itô equations, Comm. Pure Appl. Math. 45 (1992) 97–120. [9] G. Da Prato, M. Röckner, B.L. Rozovskii, F.-Y. Wang, Strong solutions to stochastic generalized porous media equations: existence, uniqueness and ergodicity, Comm. Partial Differential Equations 31 (2006) 277–291. [10] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., Cambridge University Press, 1992. [11] B. Gess, W. Liu, M. Röckner, Random attractor for SPDE driven by general additive noise, preprint. [12] I. Gyöngy, On stochastic equations with respect to semimartingale III, Stochastics 7 (1982) 231–254. [13] I. Gyöngy, A. Millet, Rate of convergence of implicit approximations for stochastic evolution equations, in: Interdiscip. Math. Sci., vol. 2, World Sci. Publ., Hackensack, 2007, pp. 281–310. [14] I. Gyöngy, A. Millet, Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal. 30 (2009) 29–64. [15] N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations, Ser. Sovrem. Probl. Mat. 14 (1979) 71–146, translated from Itogi Nauki i Tekhniki. [16] W. Liu, F.-Y. Wang, Harnack inequality and strong Feller property for stochastic fast diffusion equations, J. Math. Anal. Appl. 342 (2008) 651–662. [17] W. Liu, Harnack inequality and applications for stochastic evolution equations with monotone drifts, J. Evol. Equ. 9 (2009) 747–770. [18] W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim. 61 (2010) 27–56. [19] W. Liu, Invariance of subspaces under the solution flow of SPDE, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010) 87–98. [20] W. Liu, Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators, preprint. [21] J.-L. Menaldi, S.S. Sritharan, Stochastic 2-D Navier–Stokes equation, Appl. Math. Optim. 46 (2002) 31–53. [22] G.J. Minty, Monotone (non-linear) operators in Hilbert space, Duke Math. J. 29 (1962) 341–346. [23] G.J. Minty, On a monotonicity method for the solution of non-linear equations in Banach space, Proc. Natl. Acad. Sci. USA 50 (1963) 1038–1041. [24] E. Pardoux, Equations aux dérivées partielles stochastiques non linéaires monotones, PhD thesis, Université Paris XI, 1975. [25] C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1905, Springer, 2007. [26] J. Ren, M. Röckner, F.-Y. Wang, Stochastic generalized porous media and fast diffusion equations, J. Differential Equations 238 (2007) 118–152. [27] J. Ren, X. Zhang, Fredlin–Wentzell’s large deviation principle for stochastic evolution equations, J. Funct. Anal. 254 (2008) 3148–3172. [28] M. Röckner, X. Zhang, Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity, Probab. Theory Related Fields 145 (1–2) (2009) 211–267. [29] M. Röckner, T. Zhang, Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and small time large deviation principles, BiBoS-preprint 10-03-339. [30] M. Röckner, F.-Y. Wang, Non-monotone stochastic porous media equation, J. Differential Equations 245 (2008) 3898–3935.
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[31] M. Röckner, F.-Y. Wang, L. Wu, Large deviations for stochastic generalized porous media equations, Stochastic Process. Appl. 116 (2006) 1677–1689. [32] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Math. Surveys Monogr., vol. 49, American Mathematical Society, Providence, 1997. [33] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333–1350. [34] X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stoch. Dyn. 9 (2009) 549–595. [35] E. Zeidler, Nonlinear Functional Analysis and Its Applications II/B: Nonlinear Monotone Operators, SpringerVerlag, New York, 1990.
Journal of Functional Analysis 259 (2010) 2923–2938 www.elsevier.com/locate/jfa
A generalization of Chernoff’s product formula for time-dependent operators Pierre-A. Vuillermot UMR–CNRS 7502, Institut Élie Cartan, Nancy, France Received 16 May 2010; accepted 28 July 2010 Available online 7 August 2010 Communicated by L. Gross
Abstract In this article we provide a set of sufficient conditions that allow a natural extension of Chernoff’s product formula to the case of certain one-parameter family of functions taking values in the algebra L(B) of all bounded linear operators defined on a complex Banach space B. Those functions need not be contractionvalued and are intimately related to certain evolution operators U (t, s)0stT on B. The most direct consequences of our main result are new formulae of the Trotter–Kato type which involve either semigroups with time-dependent generators, or the resolvent operators associated with these generators. In the general case we can apply such formulae to evolution problems of parabolic type, as well as to Schrödinger evolution equations albeit in some very special cases. The formulae we prove may also be relevant to the numerical analysis of non-autonomous ordinary and partial differential equations. © 2010 Elsevier Inc. All rights reserved. Keywords: Evolution equations; Trotter–Kato formulae
1. Introduction It is well known that strongly convergent product formulae of the form n τ τ exp − A exp − B , n→+∞ n n
exp −τ (A + B) = lim
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.018
(1)
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with τ ∈ [0, +∞) and A, B linear transformations operating in a complex Banach space B and satisfying certain conditions, are quite relevant to the understanding of certain basic questions in mathematics or mathematical physics. For instance, while (1) makes it possible to relate the solutions of certain evolution problems to the theory of Wiener integrals through the celebrated Feynman–Kac formula, a slightly modified version of it allows a rigorous construction of the so-called Feynman path integral representation of solutions to Schrödinger equations with timeindependent potentials in quantum mechanics (we refer the reader for instance to [6,12,18] and to the references therein for a comprehensive analysis of such results). Along a different line, when B is finite-dimensional formulae such as (1) together with the related Baker–Campbell– Hausdorff formulae of Lie group theory also play an important rôle in numerical analysis (see for instance [5,10]). Ever since the publication of Trotter’s seminal contribution in [20], there have been numerous extensions of (1) in various directions such as those discussed in [2–4,8,11,13,14,12,15,16], to name only a few. Of particular significance among these works is Chernoff’s paper [2], where the author develops a compact and self-contained presentation of earlier results that allows him to reproduce many of the exponential formulae of semigroup theory, particularly those involving resolvent operators. Furthermore a generalization of (1) relative to operators A(t) and B(t) that may depend explicitly on time, and which are defined on time-dependent domains in B, is proved in [21] within the framework of the Kato–Tanabe theory, thus extending the results of [9] which had been previously obtained under the very restrictive condition that the domains of A(t) and B(t) be time-independent (we refer the reader to [14,19] for very nice accounts of the Kato– Tanabe theory). In this article we provide a set of sufficient conditions that allow a generalization of Chernoff’s result to the case of time-dependent operators, thereby developing a unified treatment of the non-autonomous case. Accordingly, we organize the remaining part of this article in the following way: in Section 2 we state our main result of which we give a very detailed but concise proof. The validity of our arguments there rests on the existence of an evolution system U (t, s)0stT in B possessing strong smoothing properties that make our result rather suitable for applications to evolution problems of parabolic type, as well as to Schrödinger evolution equations in some very particular cases. The reason why restrictions indeed do come about in the latter case simply lies in the fact that Schrödinger propagators do not share the typical holomorphic regularization properties of parabolic propagators. Finally, we illustrate our result in Section 3 by means of several examples ranging from finite-dimensional problems to an infinitedimensional one involving non-autonomous linear parabolic equations and a question related to quantum mechanics. 2. Statement and proof of the main result In what follows we write c for all the irrelevant constants that occur in the various estimates unless we specify these constants otherwise. Furthermore, by an evolution system U (t, s)0stT in B we mean a two-parameter family of bounded linear operators satisfying the usual strong continuity properties and composition laws (see for instance [17] or [19]). Finally, we denote by . the norm of B and by .∞ the usual supremum-norm of L(B), the algebra of all bounded linear operators defined on B the identity element of which we denote by I. For any T ∈ (0, +∞) and each t ∈ [0, T ] we consider functions Ft : [0, +∞) → L(B) satisfying the following hypothesis:
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
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(F1) We have Ft (0) = I for every t ∈ [0, T ] and there exists a constant c ∈ [0, +∞) such that the inequality sup Ft (τ )∞ exp[cτ ]
(2)
t∈[0,T ]
holds for all τ ∈ [0, +∞). Moreover, for every such τ the function t → Ft (τ ) is continuous on [0, T ] in the strong operator topology of L(B). Furthermore, let D(Ft (0)) be the linear set of all v ∈ B such that the strong limit lim
τ →0+
Ft (τ )v − v := Ft (0)v τ
exists in B for every t ∈ [0, T ]. The generalization of Chernoff’s result we have in mind requires Ft (0) to be intimately related to a family of evolution operators possessing strong regularity properties. In order to make this requirement precise, we consider the non-autonomous initialvalue problem du(t) = Ft (0)u(t), t ∈ (s, T ], dt u(s) = v
(3)
and assume that the following hypothesis holds: (F2) There exists an evolution system U (t, s)0stT in B and a dense set D ⊆ B such that for all s, t with 0 s < t T and for every v ∈ D the following conditions are valid: (a) We have U (t, s)v ∈ D(Ft (0)) and lim
Ft (τ )U (t, s)v − U (t, s)v − Ft (0)U (t, s)v sup = 0. τ
τ →0+ t∈(s,T )
(4)
(b) The function t → u(t) := U (t, s)v is strongly once continuously differentiable in B and satisfies (3). Under these conditions our main result is the following. Theorem. Assume that hypotheses (F1) and (F2) hold. Then for all 0 s t T we have
U (t, s) = lim
n→+∞
in the strong operator topology of L(B).
0 γ =n−1
Fs+ γ (t−s) n
t −s n
(5)
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Proof. We may assume that s < t. For every n ∈ N+ sufficiently large we set h = the sequence of products (Pn (t, s)) ⊂ L(B) by Pn (t, s) := U (t, s) −
1
t−s n
and define
(6)
Fs+(γ −1)h (h).
γ =n
Since (2) holds and since nh remains bounded the sequence (Pn (t, s)) is bounded in L(B), so that in order to prove (5) it is sufficient to show that Pn (t, s)v → 0 as n → +∞ in the strong topology of B for every v ∈ D, the dense set of hypothesis (F2). We first remark that we may write (6) as Pn (t, s) =
1
1
U s + γ h, s + (γ − 1)h − Fs+(γ −1)h (h)
γ =n
=
1
γ =n
Uγ ,n (t, s) −
γ =n
1
Fγ ,n (t, s)
γ =n
by virtue of the basic composition law of the operators U (t, s), where we have introduced the shorthand notation
Uγ ,n (t, s) := U s + γ h, s + (γ − 1)h
(7)
Fγ ,n (t, s) := Fs+(γ −1)h (h).
(8)
and
Furthermore, for n 3 we have 1
Uγ ,n (t, s) −
γ =n
=
1
Fγ ,n (t, s)
γ =n 2
Fα,n (t, s) × U1,n (t, s) − F1,n (t, s)
α=n
+
+1 n−1 γ
1
Fα,n (t, s) × Uγ ,n (t, s) − Fγ ,n (t, s) × Uβ,n (t, s)
γ =2 α=n
β=γ −1
1
+ Un,n (t, s) − Fn,n (t, s) × Uβ,n (t, s), β=n−1
which can easily be checked directly. Therefore, by using (7), (8) and the fundamental properties of the operators U (t, s) once again we obtain
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
Pn (t, s) =
+1 n−1 γ
2927
Fα,n (t, s) × Uγ ,n (t, s) − Fγ ,n (t, s) U s + (γ − 1)h, s
γ =1 α=n
+ Un,n (t, s) − Fn,n (t, s) U s + (n − 1)h, s ,
so that by invoking (2) we get the simple inequality n
Pn (t, s)v c Uγ ,n (t, s) − Fγ ,n (t, s) U s + (γ − 1)h, s v
(9)
γ =1
for every v ∈ D, again because nh remains bounded. We now substitute (7) along with (8) back into (9) and then set rγ := s + (γ − 1)h; this leads to the estimates Pn (t, s)v cn
max U (rγ + h, s)v − Frγ (h)U (rγ , s)v
γ ∈{1,...,n}
cn
U (r + h, s)v − Fr (h)U (r, s)v
sup
r∈[s,s+(n−1)h]
cn = cn
sup
U (r + h, s)v − Fr (h)U (r, s)v
sup
U (r + h, s)v − Fr (h)U (r, s)v
r∈[s,T −h]
r∈(s,T −h]
(10)
where the last equality in (10) follows from the strong continuity of the function r → U (r + h, s)v − Fr (h)U (r, s)v on [s, T − h]. This property is indeed an immediate consequence of the strong continuity of the evolution system and of the very last requirement of hypothesis (F1). Next, we proceed from (10) by introducing the two families of linear operators
L(h, r) := h−1 I−Fr (h) + Fr (0)
(11)
M(h, r) := h−1 I−U (r + h, r) + Fr (0)
(12)
and
defined on the domain D(Fr (0)). Since nh remains bounded and since U (r, s)v ∈ D(Fr (0)) for every r ∈ (s, T − h] according to the first part of (a) in hypothesis (F2), the substitution of (11) and (12) into (10) then gives Pn (t, s)v c
sup
L(h, r) − M(h, r) U (r, s)v
r∈(s,T −h]
so that it is sufficient to prove the relations
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lim
sup
L(h, r)U (r, s)v = 0
(13)
lim
sup
M(h, r)U (r, s)v = 0
(14)
h→0+ r∈(s,T −h]
and
h→0+ r∈(s,T −h]
to get the desired result. The fact that (13) holds is a consequence of (4) and (11). As for the proof of (14) we write −1
M(h, r)U (r, s)v = h
r+h
dk Fr (0)U (r, s)v − Fk (0)U (k, s)v
(15)
r
which follows from (12) and (b) of hypothesis (F2), and remark that the function k → Fk (0)U (k, s)v is uniformly strongly continuous on [r, r + h]; therefore, for every ∈ (0, +∞) there exists h ∈ (0, +∞) such that the inequalities 0 k − r h h along with (15) imply the estimate M(h, r)U (r, s)v uniformly in r.
2
In the next section we display some of the most direct consequences of the above theorem in the form of new product formulae of the Trotter–Kato type. 3. Some examples In the first two examples and for the sake of simplicity we consider the case of bounded, time-dependent generators which, of course, includes the matrix case. Example 1. Let us consider the functions Ft : [0, +∞) → L(B) given by N
Ft (τ ) =
exp −τ Aj (t)
(16)
j =1
where the Aj : [0, T ] → L(B) are continuous functions relative to the uniform operator topology. By expressing each of the factors in (16) by means of the exponential series we can easily verify that (F1) and (F2) hold, the corresponding evolution system UA1 +···+AN (t, s)0stT being associated with the initial-value problem du(t) Aj (t)u(t), =− dt N
t ∈ (s, T ],
j =1
u(s) = v.
(17)
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
2929
In particular, we may choose D = B in (F2) and the uniformity requirements in (2) and (4) are simple consequences of the bound sup Aj (t)∞ < +∞
max
(18)
j ∈{1,...,N } t∈[0,T ]
and of the boundedness of t → UA1 +···+AN (t, s)v. From (5) we then obtain UA1 +···+AN (t, s) = lim
N 0
n→+∞
γ =n−1 j =1
γ t −s Aj s + (t − s) exp − n n
(19)
in the strong operator topology of L(B), which gives UA+B (t, s) = lim
0
n→+∞
γ =n−1
γ t −s γ t −s A s + (t − s) exp − B s + (t − s) exp − n n n n (20)
in the case of only two generators A(t) and B(t). This last relation obviously reduces to (1) when A and B are time-independent. Another particular case of the above formula is UA (t, s) = lim
n→+∞
0 γ =n−1
γ t −s A s + (t − s) , exp − n n
(21)
which allows the reconstruction of UA (t, s) in a simple way in terms of the basic semigroups exp[−τ A(t)]τ 0 . Example 2. The operators Aj (t) are still the same as in the preceding example but we now impose the additional restriction that the semigroups exp[−τ Aj (t)]τ 0 be contractive. From a standard Laplace transform argument for the resolvents of their generators we then obtain the estimate max
sup
j ∈{1,...,N } (τ,t)∈[0,+∞)×[0,T ]
I + τ Aj (t) −1
∞
1
(22)
so that the functions Ft : [0, +∞) → L(B) given by Ft (τ ) =
N
−1 I + τ Aj (t)
(23)
j =1
again satisfy (F1). Furthermore, we may choose D = B anew in (F2) and a simple calculation gives Ft (0) = −
N j =1
Aj (t),
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while the uniformity of (4) with U (t, s) = UA1 +···+AN (t, s) follows from (18) and (22) by writing each resolvent factor in (23) as the Laplace transform of the associated semigroup. From (5) we thus get
UA1 +···+AN (t, s) = lim
n→+∞
0 N γ =n−1 j =1
−1 t −s γ I+ Aj s + (t − s) n n
(24)
in the strong operator topology of L(B), a particular case of which being this time UA+B (t, s) = lim
n→+∞
0 γ =n−1
I+
−1 t −s γ A s + (t − s) n n
−1 γ t −s B s + (t − s) × I+ . n n
(25)
Remark. Both (20) and (25) are related to the solution of the initial-value problem
du(t) = − A(t) + B(t) u(t), dt u(s) = v.
t ∈ (s, T ], (26)
One reason why one might prefer one approximation over the other regarding the numerical resolution of (26) lies in the fact that one might converge faster and be more stable than the other depending on the nature of the operators A(t) and B(t). We refer the reader for instance to [5,10] for further discussions of related questions and much more concerning the autonomous case. The situation is no longer that simple when at least one operator is unbounded. In the next example one generator depends on time and is defined on a time-dependent domain. There and further below we use the standard notations for the usual spaces of continuous functions, of Lebesgue integrable functions and for the corresponding Sobolev spaces defined on regions of Euclidean space (see for instance [1]). Example 3. Let D ⊂ Rd be an open bounded domain the boundary of which we denote by ∂D. We consider parabolic initial–boundary value problems of the form
∂u(x, t) = divx k(x, t)∇x u(x, t) − m(x, t)u(x, t), ∂t u(x, 0) = u0 (x), x ∈ D, ∂u(x, t) = 0, ∂n(k)
(x, t) ∈ D × (0, T ],
(x, t) ∈ ∂D × (0, T ]
(27)
where the last relation stands for the conormal derivative of u with respect to the matrix-valued function k. We assume that the following hypotheses hold:
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
2931
(K) We have k : D × [0, T ] → Rd and ki,j = kj,i ∈ L∞ (D × (0, T ), R) for every i, j ∈ {1, . . . , d}. Moreover, there exists a constant k ∈ (0, +∞) such that the inequality 2
inf
k(x, t)q, q Rd k|q|2
(x,t)∈D×[0,T ]
holds for all q ∈ Rd , where (.,.)Rd and |.| denote the Euclidean inner product and the induced norm in Rd , respectively. Finally, there exist constants c∗ ∈ (0, +∞) and σ ∈ ( 12 , 1] such that the Hölder continuity estimate max
sup ki,j (x, t) − ki,j (x, s) c∗ |t − s|σ
i,j ∈{1,...,d} x∈D
is valid for all s, t ∈ [0, T ]. (M) We have m ∈ L∞ (D × (0, T ), R+ ) along with sup m(x, t) − m(x, s) c∗ |t − s|σ
x∈D
for all s, t ∈ [0, T ]. Assuming furthermore that u0 ∈ L2 (D, R), we choose in this example B = L2 (D, C) endowed with its canonical inner product (.,.)2 and the related norm .2 . We then define the self-adjoint, positive operators A(t) and B(t) by
A(t)v := − div k(., t)∇v
(28)
B(t)v := m(., t)v
(29)
D A(t) = v ∈ H 1 (D, C): A(t)v ∈ L2 (D, C), A(t)v, w 2 = a(t, v, w)
(30)
and
on the domains
and D(B(t)) = L2 (D, C), respectively. Thus B(t) remains bounded, which will simplify the matter a bit. In (30) the last relation holds for every w ∈ H 1 (D, C), with the sesquilinear form a :[0, T ] × H 1 (D, C) × H 1 (D, C) given by
a(t, v, w) =
dx k(x, t)∇x v(x), ∇x w(x) Cd
D
where (.,.)Cd denotes the standard Hermitian inner product in Cd . The conormal boundary condition we referred to above is evidently encoded in (30), and under these conditions it follows that there exists an evolution system UA+B (t, s)0stT in L2 (D, C) which solves the initial-value problem
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du(t) = − A(t) + B(t) u(t), dt u(0) = u0
t ∈ (0, T ], (31)
associated with (27) (see for instance Theorem 5.3.3 in [19], which can be applied to the operator A(t) + B(t) in this case, or more specifically the very first statement of the main theorem in [21]). If we require in addition that the boundary ∂D be sufficiently smooth and that the ki,j be sufficiently regular on D = D ∪ ∂D, standard elliptic regularity theory allows us to rewrite (30) as
D A(t) = v ∈ H 2 (D, C): ∇x v(x), k(x, t)n(x) Cd = 0, x ∈ ∂D where n(x) stands for the outer unit normal vector at x (see for instance [17,19,21] and the references therein). Moreover, in such a case we have sup A(t)v 2 < +∞
(32)
sup B(t)v 2 < +∞
(33)
t∈(0,T ]
and
t∈(0,T ]
for all v ∈ C02 (D, C), the set of all complex-valued, twice continuously differentiable functions with compact support in D. Under these conditions our contention is now that both (20) and (25) hold true for UA+B (t, s) with A(t) and B(t) given by (28) and (29), respectively. Our proof of this statement will show that (32) and (33) are the natural substitutes for (18) in this case. We consider again the two natural choices for Ft : [0, +∞) → L(L2 (D, C)), namely, Ft (τ ) = exp −τ A(t) exp −τ B(t)
(34)
−1
−1
Ft (τ ) = I + τ A(t) I + τ B(t) ,
(35)
and
and show that (F1) and (F2) hold in either case. From the hypotheses regarding (28) and (29) we first infer that those operators generate holomorphic contraction semigroups on L2 (D, C), and that the Laplace transforms of these give
−1 I + τ A(t) =
+∞ dσ exp[−σ ] exp −σ τ A(t) 0
and
(36)
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
−1 I + τ B(t) =
+∞ dσ exp[−σ ] exp −σ τ B(t)
2933
(37)
0
for every τ ∈ [0, +∞), respectively. Therefore, as in (22) we obtain sup
I + τ A(t) −1
∞
1
(38)
sup
I + τ B(t) −1
∞
1.
(39)
(τ,t)∈[0,+∞)×[0,T ]
and
(τ,t)∈[0,+∞)×[0,T ]
Consequently we have (2) in both cases, as well as the continuity of t → Ft (τ ) in the strong operator topology of L(L2 (D, C)). In order to verify this last point it is sufficient to observe that this property for t → exp[−τ A(t)] follows for instance from a suitable modification of the proof of Lemma 5.3.1 in [19], while that of t → exp[−τ B(t)] is immediate from the continuity of m relative to the time variable, (29) and a simple dominated convergence argument in L2 (D, C). The corresponding property for each of the factors in (35) then follows from (36), (37) and dominated convergence again, this time on [0, +∞) with respect to the measure dσ exp[−σ ]. Therefore, hypothesis (F1) does hold for both (34) and (35). As for the verification of (F2) we remark that in both cases we have
Ft (0) = − A(t) + B(t) with D(Ft (0)) = D(A(t)) for every t ∈ [0, T ], so that (31) is indeed of the form (3). Let us now choose D = C02 (D, C); we then infer from standard parabolic theory that UA+B (t, s)v ∈ D(A(t)) for all v ∈ C02 (D, C) whenever 0 s < t T , and that part (b) of hypothesis (F2) holds for every such v (see, for instance, relation (31) of the main theorem in [21]). Consequently, it remains to verify the uniformity of the limit in (4). For Ft of the form (34) we start by breaking up the operator (11) that lurks in (4) into two pieces, namely, L(τ, t) = L1 (τ, t) + L2 (τ, t) where
L1 (τ, t) := τ −1 I− exp −τ A(t) − A(t)
(40)
L2 (τ, t) := τ −1 exp −τ A(t) I − exp −τ B(t) − B(t).
(41)
and
In order to obtain the desired uniformity it is then sufficient to prove that lim
sup L1 (τ, t)UA+B (t, s)v 2 = 0
τ →0+ t∈(s,T )
(42)
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and lim
sup L2 (τ, t)UA+B (t, s)v 2 = 0
τ →0+ t∈(s,T )
(43)
for every v ∈ C02 (D, C). In a similar way, for Ft of the form (35) we split (11) as 2 (τ, t) 1 (τ, t) + L L(τ, t) = L where
1 (τ, t) := τ −1 I− I + τ A(t) −1 − A(t) L
(44)
2 (τ, t) := τ −1 I + τ A(t) −1 I − I + τ B(t) −1 − B(t). L
(45)
and
As before the problem is reduced to proving that lim
sup L1 (τ, t)UA+B (t, s)v 2 = 0
(46)
lim
sup L2 (τ, t)UA+B (t, s)v 2 = 0.
(47)
τ →0+ t∈(s,T )
and τ →0+ t∈(s,T )
In order to illustrate our method of proof of these relations we shall now focus on (46), as we can handle the remaining cases (42), (43) and (47) in a similar way. Thus, in order to get (46) it is sufficient to show that the limit lim
L1 (τ, t)UA+B (t, s)v 2 = 0
sup
τ →0+ t∈[s+μ,T ]
(48)
holds uniformly in μ ∈ (0, T − s). The starting point for this is the relation 1 (τ, t)UA+B (t, s)v = τ −1 L
τ dσ
−2 I + σ A(t) − I A(t)UA+B (t, s)v,
(49)
0
which follows from (44) and the continuous differentiability of the mapping τ → (I + τ A(t))−1 with respect to the strong operator topology, since UA+B (t, s)v ∈ D(A(t)) for all v ∈ C02 (D, C) whenever 0 s < t T . From (49) we thus obtain
−2 L1 (τ, t)UA+B (t, s)v 2 sup I + σ A(t) − I A(t)UA+B (t, s)v 2 σ ∈[0,τ ]
so that in order to establish (48) it is sufficient to have
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
2935
I + τ A(t) −2 − I A(t)UA+B (t, s)v = 0 2
(50)
lim
sup
τ →0+ t∈[s+μ,T ]
for every v ∈ C02 (D, C), uniformly in μ. Let us define R(τ, t) ∈ L(L2 (D, C)) by
−2 R(τ, t) := I + τ A(t) − I. We have R(τ, t)
sup
(τ,t)∈[0,+∞)×(s,T )
∞
< +∞
by virtue of (38) and furthermore we claim that lim
sup
R(τ, t)v = 0 2
τ →0+ t∈[s+μ,T ]
(51)
for every v ∈ L2 (D, C) uniformly in μ. Indeed, first for v ∈ C02 (D, C) we have
−1
−1 R(τ, t)v = I+ I + τ A(t) I + τ A(t) v − v
−1 = − I+ I + τ A(t)
τ
−2 dσ I + σ A(t) A(t)v
0
since C02 (D, C) ⊂ D(A(t)), and consequently R(τ, t)v cτ sup A(t)v → 0 2 2 t∈(s,T ]
as τ → 0+ for every μ ∈ (0, T − s) because of (32), as desired. Statement (51) for an arbitrary v ∈ L2 (D, C) then follows from the density of C02 (D, C) in L2 (D, C). We conclude the argument by showing that (50) follows from (51). To this end let us consider the set Kμ = v ∈ L2 (D, C): v = A(t)UA+B (t, s)w, t ∈ [s + μ, T ] where w ∈ C02 (D, C); since our hypotheses imply that the mapping t →
∂UA+B (t, s)w = − A(t) + B(t) UA+B (t, s)w ∂t
is continuous in the strong topology of L2 (D, C) for t ∈ [s + μ, T ], the same is true of the mapping
t → A(t)UA+B (t, s)w = A(t) + B(t) UA+B (t, s)w − B(t)UA+B (t, s)w.
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Consequently the set Kμ is compact, which implies that (51) remains true for the time-dependent v ∈ Kμ since the limit in (51) is now uniform in these v. Thus, (50) is indeed a consequence of (51). As already mentioned we omit the proofs of (42), (43) and (47), which follow from similar arguments based on (32) and (33). To summarize briefly, if hypotheses (K) and (M) hold and if ∂D and ki,j are sufficiently smooth then we have simultaneously UA+B (t, s) = lim
n→+∞
= lim
n→+∞
0 γ =n−1
γ t −s γ t −s A s + (t − s) exp − B s + (t − s) exp − n n n n
0 γ =n−1
−1 −1 t −s t −s γ γ I+ I+ A s + (t − s) B s + (t − s) n n n n
in the strong operator topology of L(L2 (D, C)). To the best of our knowledge these two product formulae are new and might provide a way to analyze the solutions to (27) numerically. Remark. The results of this article generalize those of [21] in several directions; for instance our theorem provides a large class of approximations for the given evolution system U (t, s)0stT through the one-parameter family of functions Ft∈[0,T ] , in contrast to the sole exponential approximations of [21]. Moreover, the operators B(t) in the above examples are not viewed as small perturbations of the A(t)’s as they are in [21]. Finally, as long as the generators involved remain bounded the considerations of Section 2 also apply to evolution problems of Schrödingertype without any changes in the arguments. A case in point is the following elementary example, where we have switched to the more conventional notation and terminology of quantum mechanics. Example 4. Let B = H be a complex Hilbert space and let us consider the functions Ft : [0, +∞) → L(H) given by Ft (τ ) = exp −iτ H (t) exp −iτ V (t) and −1
−1
Ft (τ ) = I+iτ H (t) I+iτ V (t) where the functions H , V : [0, T ] → L(H) are continuous in the uniform operator topology with H (t) and V (t) self-adjoint on H. As in the first two examples it is easily verified that both hypotheses (F1) and (F2) hold; in particular, the initial-value problem
du(t) = −i H (t) + V (t) u(t), dt u(s) = v
t ∈ (s, T ],
is of the form (3), and there exists a unitary evolution system UH +V (t, s)s,t∈[0,T ] given by
(52)
P.-A. Vuillermot / Journal of Functional Analysis 259 (2010) 2923–2938
2937
UH +V (t, s) = I+
+∞
t γ
(−i)
γ =1
= lim
n→+∞
= lim
n→+∞
τ1
γ =n−1
s
s
γ t −s γ t −s H s + (t − s) exp −i V s + (t − s) exp −i n n n n
0 γ =n−1
dτγ H (τ1 ) + V (τ1 ) · · · H (τγ ) + V (τγ )
dτ2 · · ·
dτ1 s
0
τ γ −1
I+i
−1 −1 t −s t −s γ γ I+i H s + (t − s) V s + (t − s) (53) n n n n
in the strong operator topology of L(H) that solves (52). Indeed, the first right-hand side in (53) is the so-called Dyson expansion of UH +V (t, s) (see for instance [18]), while the other two equalities follow from our theorem. Moreover, the fact that all three expressions coincide reflects the uniqueness of the solution to (52). We omit the details of the corresponding elementary arguments. Remark. If the self-adjoint, time-dependent operators in (52) are unbounded the conditions of hypothesis (F2) are seldom satisfied, a fact we have already alluded to and very briefly explained in the introduction, since we cannot expect in general that UH +V (t, s)v belongs to the domain of the Hamiltonian H (t) + V (t) when t > s even for very smooth initial data. For what regards the proof of suitable Trotter–Kato formulae in such cases, this leads to a substantial number of qualitatively new difficulties when the domains of the H (t) depend explicitly on time, and more particularly when the domains of the associated quadratic forms are time-dependent as well. As one can easily infer for instance from [7] and the references therein, the former situation can occur when considering the motion of a single quantum particle in three-dimensional Euclidean space subjected to a finite number of zero-range interactions, while the latter case can manifest itself whenever those interactions are themselves in motion on prescribed, classical and nonintersecting trajectories. Furthermore, in such and in many other cases equations of the form (52) can only be satisfied in a very weak sense. Therefore, the proof of Trotter–Kato formulae for their solutions remains a challenging open problem. Acknowledgments The author would like to thank the referee for his or her very careful reading of the manuscript. He is also indebted to Professor W.P. Petersen for stimulating discussions regarding the possibility of applying the results of this paper to numerical analysis, and for having pointed out to him reference [10]. This work was carried out while the author was visiting the Institute for Theoretical Physics of the ETH in Zurich, the generous financial support and the warm hospitality of which he gratefully acknowledges. He is particularly indebted to Professors J. Fröhlich and G.M. Graf for having made his visit possible. References [1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Academic Press, New York, 2003. [2] P.R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2 (1968) 238–242.
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[3] P.R. Chernoff, Semigroup product formulas and addition of unbounded operators, Bull. Amer. Math. Soc. 76 (1970) 395–398. [4] P.R. Chernoff, Product formulas, nonlinear semigroups, and addition of unbounded operators, Mem. Amer. Math. Soc. 140 (1974). [5] A.J. Chorin, T.J.R. Hughes, M.F. McCracken, J.E. Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math. 31 (1978) 205–256. [6] E.B. Davies, One-Parameter Semigroups, Academic Press, London, 1980. [7] G.F. Dell’Antonio, R. Figari, A. Teta, The Schrödinger equation with moving point interactions in three dimensions, Canad. Math. Soc. Conf. Proc. 28 (2000) 99–113. [8] W.G. Faris, The product formula for semigroups defined by Friedrichs extensions, Pacific J. Math. 22 (1967) 47–70. [9] W.G. Faris, Product formulas for perturbations of linear propagators, J. Funct. Anal. 1 (1967) 93–108. [10] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, New York, 2006. [11] T. Ichinose, H. Tamura, Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations, Osaka J. Math. 35 (1998) 751–770. [12] G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford University Press, Oxford, 2000. [13] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in: I. Gohberg, M. Kac (Eds.), Topics in Functional Analysis, Academic Press, London, 1978. [14] T. Kato, H. Tanabe, On the abstract evolution equation, Osaka J. Math. 14 (1962) 107–133. [15] H. Neidhardt, V.A. Zagrebnov, Trotter–Kato product formula and operator-norm convergence, Comm. Math. Phys. 205 (1999) 129–159. [16] G. Nickel, Evolution semigroups and product formulas for nonautonomous Cauchy problems, Math. Nachr. 212 (2000) 101–116. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. [18] M. Reed, B. Simon, Methods of Modern Mathematical Physics I, II, Academic Press, London, 1975. [19] H. Tanabe, Equations of Evolution, Pitman, London, 1979. [20] H. Trotter, On the product of semigroups of operators, Proc. Amer. Math. Soc. 10 (1959) 545–551. [21] P.A. Vuillermot, W.F. Wreszinski, V.A. Zagrebnov, A general Trotter–Kato formula for a class of evolution operators, J. Funct. Anal. 257 (2009) 2246–2290.
Journal of Functional Analysis 259 (2010) 2939–2960 www.elsevier.com/locate/jfa
Tracial algebras and an embedding theorem Tim Netzer, Andreas Thom ∗ Universität Leipzig, Germany Received 18 May 2010; accepted 12 August 2010 Available online 21 August 2010 Communicated by S. Vaes
Abstract We prove that every positive trace on a countably generated ∗-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial ∗-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial ∗-algebras, which as ∗-algebras embed into a matrix-ring over a commutative algebra. © 2010 Elsevier Inc. All rights reserved. Keywords: Tracial algebras; Connes embedding problem; Ultraproduct; Von Neumann algebra; Convex geometry
Contents 0. 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Convex geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The algebra of non-commutative polynomials . . . . . . . . . . . 1.3. Polynomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Approximation of traces . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hölder and Minkowski type inequalities in tracial ∗-algebras 2.3. Metric ultraproducts of tracial ∗-algebras . . . . . . . . . . . . . . 3. Embedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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* Corresponding author.
E-mail addresses:
[email protected] (T. Netzer),
[email protected] (A. Thom). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.010
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959
0. Introduction The theory of Rings of operators was founded by F. Murray and J. von Neumann in the first half of the last century, see [13–15]. Later, the term von Neumann algebra was coined to emphasize the seminal contributions of John von Neumann. We will freely use standard results in the theory of von Neumann algebras. For those results and all necessary definitions we refer to [23] and the references therein. An early achievement of Murray and von Neumann was a classification of von Neumann algebras into types. In this note we are concerned with finite von Neumann algebras with separable pre-dual, which decompose as products of algebras of the types In for 1 n < ∞ and type II1 . The classification of algebras of type In is complete as they are all isomorphic to algebras L∞ (X, C) ⊗ Mn C for some probability measure space (X, μ). Hence, the key objects of study are II1 -factors with separable pre-dual, i.e. infinite-dimensional and weakly closed ∗-subalgebras of B(H ), which carry a faithful trace, have trivial center and admit a countable weakly dense subset. Here, we denote the space of bounded operators on the Hilbert space H by B(H ) as usual. There are various constructions of II1 -factors, e.g. from groups or group actions, but also via internal constructions like free products, amplification etc. An important question right from the beginning was to which extend general II1 -factors are close to matrix algebras. A main result in the work of Murray and von Neumann was the construction of the hyperfinite II1 -factor R and the proof of its uniqueness with respect to some strong form of approximation with matrices. Murray and von Neumann also gave examples of II1 -factors, namely free group factors, which were not hyperfinite. The embedding conjecture of Alain Connes states that every type II1 -factor with separable pre-dual can be embedded into an ultrapower of the hyperfinite II1 -factor. This assertion is equivalent some weak form of approximation by matrices which ought to hold always. The conjecture dates back to Connes’ seminal work on injective von Neumann algebras, [7, p. 105]. Although it is well known that many II1 -factors, including free group factors, do embed into an ultraproduct of the hyperfinite II1 -factor, this conjecture remains open and has triggered a lot of interesting and deepgoing research. There are various ways of reformulating the Connes embedding problem, and one way of putting it is to ask for an embedding into a metric ultraproduct of a sequence of finite von Neumann algebras of type I. We will prove that such an embedding always exists if one allows to approximate with a more general class of tracial ∗-algebras of type I. Motivated by the work of D. Hadwin, see [9], F. R˘adulescu established a relationship between the Connes embedding conjecture and some analogue of Hilbert’s 17th problem on positive polynomials, see [20]. This approach was carried further in a more algebraic setup by I. Klep and M. Schweighofer, see [11]. In this note we want to follow this line of approach, and instead of giving yet another reformulation of the original problem we will obtain an affirmative answer to a different but analogous problem. Indeed, we will enlarge the realm and consider general ∗-algebras with a positive, faithful and unital trace. More precisely: Definition. A tracial ∗-algebra is a unital complex algebra A with involution and a complexlinear functional τ : A → C such that: (1) τ (1) = 1,
T. Netzer, A. Thom / Journal of Functional Analysis 259 (2010) 2939–2960
2941
(2) τ (a ∗ a) 0 for all a ∈ A, and (3) τ (ab) = τ (ba), for all a, b ∈ A. τ is called a (positive) trace. Note that τ (a ∗ ) = τ (a) for all a ∈ A is an automatic consequence, since τ ((1 + a ∗ )(1 + a)) 0 and τ ((1 − ia ∗ )(1 + ia)) 0. In the context of this definition, the trace τ is said to be faithful if τ (a ∗ a) = 0 holds only if a = 0; the tracial ∗-algebra (A, τ ) is called trace-reduced then. We say that a ∈ A is τ -bounded, if τ ((a ∗ a)p ) C 2p for some constant C > 0, and all p ∈ N. Moreover, (A, τ ) is called bounded if every element in A is τ -bounded. It is clear that every finite von Neumann algebra with a specified trace gives rise to a bounded and trace-reduced tracial ∗-algebra. Conversely, the GNSconstruction allows to construct a trace-preserving embedding of any bounded and trace-reduced tracial ∗-algebra into a finite von Neumann algebra (this is for example demonstrated in [16]). The lack of bounded elements in a general tracial ∗-algebra causes many pathologies and gives rise to phenomena that makes the study of general tracial ∗-algebras quite different compared to the study of finite von Neumann algebras. However, in this more general setup, we may still talk about algebras of various types. Indeed, a finite von Neumann algebra is a sum of algebras of the form L∞ (Xk , C) ⊗C Mk C for k n if and only if it satisfies a certain universal polynomial identity, similar to the commutator relation which characterizes type I1 . Hence, we will say that a tracial ∗-algebra is of type In if it satisfies this identity (Definition 1.17). Equivalently, we could say that a tracial ∗-algebra is of type In if and only if it embeds (not necessarily preserving the unit) into a ring of n × n-matrices over a commutative C-algebra. Algebras which satisfy a polynomial identity are called PI-algebras and have been studied in detail over the last decades. We will recall various results in Section 1.3 and refer for the general theory to [19]. We are going to define a suitable notion of metric ultraproduct for arbitrary tracial ∗-algebras – which contains the usual ultraproduct of von Neumann algebras – and prove a general embedding theorem. Indeed, the main application of our results is the following theorem (Corollary 3.3 combined with Definition 1.17 and Corollary 2.15): Theorem. Let (M, τ ) be a finite von Neumann algebra with separable pre-dual. Then there exists a sequence (An , τn ) of trace-reduced tracial ∗-algebras such that: (1) For every n ∈ N, the complex algebra An embeds as a complex algebra into the ring of n × n-matrices over a commutative complex algebra, and (2) there exists a trace-preserving embedding ι : (M, τ ) →
(An , τn ).
n→ω
Our results (Theorems 3.1 and 3.2) cover more generally all countably generated tracereduced tracial ∗-algebras, but the case of finite von Neumann algebras is certainly the most interesting one. This article is organized as follows: It starts with an introduction which explains the motivation. In Section 1 we explain basic notions of convex geometry, introduce the algebra of non-commutative polynomials and study basic properties of polynomial identities. We identify various cones inside the algebra of non-commutative polynomials. In particular, we study the h formed by selfcone formed by sums of hermitian squares Q(CX), the linear subspaces Ccyc
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adjoint commutators, and the subspace Jkh formed by identities which are satisfied in the algebra of k × k-matrices over C. h + J h is a closed cone in CX – this is the In Section 2, we show that Q(CX) + Ccyc k main technical result. Moreover, on polynomials of degree at most k−1 2 , we show that this cone h . A trace on CX is nothing but a functional which is noncoincides with Q(CX) + Ccyc h . The idea is now to use that the restriction of a trace negative on the cone Q(CX) + Ccyc k−1 to polynomials of degree at most 2 is positive with respect to the intersection of the cone h + J h with the space of those polynomials. Hence, we may extend a small Q(CX) + Ccyc k h + perturbation of the restriction to a functional on CX, which is positive on Q(CX) + Ccyc Jkh . Since Jkh + i · Jkh is a two-sided ideal, this functional is well-defined on the quotient algebra CX/(Jkh + i · Jkh ), which is identified with the ring of generic matrices. As a consequence, every trace is a pointwise limit of traces which are well-defined on rings of generic matrices. We now proceed in Section 3 and reformulate this approximation result as an embedding statement. We prove that any trace-reduced tracial ∗-algebra embeds into a metric ultra-product of algebras of generic matrices. In order to carry this out, we need to define a suitable notion of metric ultraproduct which extends the usual von Neumann algebra ultraproduct. It turns out to be rather tricky to show that the naive definition works. For this, we need to prove analogues of the Hölder inequality and the Minkowski inequality in the context of tracial ∗-algebras. This is carried out at the end of Section 2. 1. Preliminaries 1.1. Convex geometry We first recall some basic concepts and results from convex geometry. See for example [4] for details and proofs. Definition 1.1. Let V be a real vector space. A subset C ⊂ V is said to be a convex cone, if C + C ⊂ C,
and R0 · C ⊂ C.
A linear functional φ : V → R is said to be positive with respect to the cone C, if φ(c) 0 for all c ∈ C. We will frequently apply this to the situation where C contains a linear subspace L, where we get L ⊆ C ∩ (−C) ⊂ ker φ. For any subset C ⊆ V we denote by C ∨ its dual, which is by definition: C ∨ = φ : V → R linear φ(c) 0, ∀c ∈ C . The double dual of C in V is C ∨∨ = v ∈ V φ(v) 0, ∀φ ∈ C ∨ . A classical result following from the Hahn–Banach Theorem is C ∨∨ = conv(C),
T. Netzer, A. Thom / Journal of Functional Analysis 259 (2010) 2939–2960
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i.e. the double dual is nothing but the closed convex hull of C, in the finest locally convex topology on V . Another useful property is that (A + B)∨ = A∨ ∩ B ∨ . We record some facts about the finest locally convex topology. It can be defined as the coarsest vector space topology making all seminorms continuous. All subspaces are then closed and inherit again the finest locally convex topology. So every finite-dimensional subspace inherits the Euclidean topology. All linear mappings into a vector space with any locally convex topology are continuous. The following result is well known and can for example be found as Proposition 1 in [5]: Lemma 1.2. A subset B ⊂ V in a countable dimensional real vector space is closed with respect to the finest locally convex topology if and only if B ∩ W is closed for all finite-dimensional subspaces W of V . A somewhat subtle consequence, which we can derive is the following result: Proposition 1.3. Let V be a countable dimensional real vector space, equipped with the finest locally convex topology, and let F ⊂ V be a finite-dimensional subspace, with a fixed norm. Let B ⊂ V be a closed convex cone, and let ε > 0 be arbitrary. If φ : F → R is a linear functional which satisfies φ(b) 0, for all b ∈ F ∩ B, then there exists a linear functional ψ : V → R with ψ(b) 0, for all b ∈ B, and ψ|F − φ ε. Proof. First assume that also V is finite-dimensional. Let F ⊂ V have codimension k. Let V be the dual space of V and F ∨ the k-dimensional linear space of functionals which vanish on F . Denote by B ∨ the dual cone of B. The space of all extensions of φ is an affine space of dimension k of the form φ + F ∨ . Our assumption on the positivity of φ translates simply into φ + F ∨ ⊂ B ∨ + F ∨.
(1)
Indeed the dual cone of F ∩ B is just the closure of B ∨ + F ∨ , and every linear extension of φ will be positive for the cone F ∩ B. Note, at this point we are using that B is a closed convex cone and V is finite-dimensional. We see from (1) that every neighborhood of φ + F ∨ has to intersect B ∨ + F ∨ . Hence we can perturb φ as little as we wish to some functional φ such that (φ + F ∨ ) ∩ B ∨ = ∅. Any element ψ in the intersection of φ + F ∨ and B ∨ gives a positive extension of φ . Now let V be countable dimensional and let (Vi )i1 be an increasing sequence of finite dimensional subspaces, with V0 := F ⊆ Vi for all i 1, and V = i Vi . Equip the Vi with compatible norms. Now choose inductively for each i a linear functional ψi : Vi → R with ψi 0 on B ∩ Vi and ψi |Vi−1 − ψi−1
ε . 2i
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This can be done by the already proven result for finite dimension. Then for all 0 j < i ψi |Vj
i ε − ψj . 2k k=j +1
This shows that the sequence (ψi )i converges pointwise on V to some linear functional ψ : V → R, which is nonnegative on B and fulfills ψ|F − φ ε. 2 1.2. The algebra of non-commutative polynomials Throughout X denotes the finite set of letters X1 , . . . , Xn . We denote by CX the algebra of non-commutative polynomials in X with coefficients in C, i.e. CX consists of all C-linear combinations of words ω in the letters X. A monomial is an expression λω with a word ω and λ ∈ C. Each monomial in CX has a natural length which we call its degree. In this way the algebra CX becomes a graded algebra; the degree of a polynomial is the highest degree of a monomial occurring with a non-zero coefficient. CX is equipped with an involution P → P ∗ such that Xi∗ = Xi for all i and λ∗ = λ for λ ∈ C. The set of hermitian elements CXh := P ∈ CX P ∗ = P carries the structure of a real vector space. It is also a homogeneous subset of CX, i.e. a polynomial is hermitian if and only if all of its homogeneous parts are hermitian. We will always equip this space with the finest locally convex topology. We denote by CXk and CXhk the subspaces of elements of degree at most k. Definition 1.4. We denote by Q(CX) the convex cone of sums of hermitian squares:
m ∗ Q CX = Pi Pi m ∈ N, Pi ∈ CX . i=1
Clearly, Q(CX) ⊆ CXh . A linear functional is said to be positive if it is positive with respect to the convex cone of hermitian squares. Definition 1.5. Denote by Ccyc ⊂ CX the C-subspace which is spanned by all linear commutators. Two elements P1 , P2 ∈ CX are said to be cyclically equivalent if their difference is in h := C h Ccyc , i.e. it is a sum of commutators. Write Ccyc cyc ∩ CX . Remark 1.6. Note that Ccyc is a homogeneous subspace of CX, i.e. the homogeneous parts of h is also a homogeneous subspace a sum of commutators are again sums of commutators. So Ccyc h of CX . h + i · C h = C . Thus C h is the real vector space spanned by all Lemma 1.7. We have Ccyc cyc cyc cyc hermitian commutators.
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Proof. Let P , Q ∈ A := CX. Write P = a + ib, Q = c + id with a, b, c, d ∈ Ah . Then P Q − QP = (a + ib)(c + id) − (c + id)(a + ib) = ac − ca + db − bd + i(bc − cb) + i(ad − da) = i(bc − cb) + i(ad − da) + i −i(ac − ca) − i(db − bd) . In the last line, all the elements i(bc − cb), i(ad − da), −i(ac − ca) and −i(db − bd) are hermitian commutators. This establishes the first claim. The second claim follows immediately from the first. 2 Remark 1.8. Note that the intersection between Q(CX) and Ccyc consists only of 0. There are many ways to see this; the most intuitive way is to say that identities in CX can be checked by checking that they hold whenever one specializes to self-adjoint matrices. This is justified by Theorem 1.15 in combination with Lemma 1.13. In matrices, a commutator has always vanishing trace whereas a sum of squares with vanishing trace is zero. The following lemma is an obvious consequence of Carathéodory’s lemma. Lemma 1.9. Let P ∈ CXm ∩ Q(CX). Then, there exist elements P1 , . . . , P(n+1)m ∈ CXm/2 , such that P=
m (n+1)
Pi∗ Pi .
i=1
It can be used to show that Q(CX) is closed, see Theorem 4.2 in [21]. We are going to prove h is a closed convex cone in CXh . This is analogous in Corollary 2.3 that also Q(CX) + Ccyc to the statement of Proposition 5.1 in [11]. Remark 1.10. Assume τ is a normalized R-linear functional on CXh (normalized means h . Then the complex τ (1) = 1), which is positive with respect to the convex cone Q(CX) + Ccyc h linear extension on CX is nothing but a positive trace. Indeed, since Ccyc is a linear subspace on which τ is positive, it has to vanish on it. Hence the complex linear extension vanishes on Ccyc , by Lemma 1.7, and is thus a trace. We will later also be interested in the algebra generated by countably many non-commuting variables X1 , X2 , . . . . We denote this algebra by CX∞ . The involution and the grading extend h , . . . in the obvious way for this bigger algebra. naturally. We define all notions as Ccyc,∞ , Ccyc,∞ Note however that CX∞ m is not a finite-dimensional space any more. 1.3. Polynomial identities Recall that a complex algebra A satisfies a polynomial identity if there is a non-zero polynomial P ∈ CX1 , . . . , Xn , for some n ∈ N, such that P (a1 , . . . , an ) = 0 holds for all a1 , . . . , an ∈ A. Such an algebra A is then called a PI-algebra. Note that if A satisfies a polynomial identity, then it satisfies a polynomial identity in 2 variables already. Indeed, replacing Xi by U i V in a polynomial identity for A results in a nontrivial polynomial identity in the two variables U, V .
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Let us recall the following facts about PI-algebras. Finitely generated PI-algebras are Jacobson rings, i.e. every prime ideal is an intersection of maximal ideals (see [3, Corollary 1.3]). The Jacobson radical of a finitely generated PI-algebra is nilpotent (see [6, Theorem B]). Every prime PI-ring is a subring of the r × r-matrices, for some r, over a division ring which is finitedimensional over its center, and it has a two-sided ring of quotients which is all of the matrix ring. This is Posner’s Theorem, see [18]. In particular, every prime PI-ring is a subring of the r × r-matrices over some (commutative) field. Denote by Jk the two-sided ∗-ideal in CX of all expressions which are zero, whenever we specify the X to self-adjoint complex k × k-matrices. We let Jkh be the real vector space of selfh in CX , but we will consider adjoint elements in Jk . In the same way we define Jk,∞ and Jk,∞ ∞ these sets only later. Proposition 1.11. Let k 1. The ideal Jk is homogeneous, i.e. if P is in Jk , then so is each of its homogeneous parts. So also Jkh is homogeneous in CXh . Further, Jkh + i · Jkh = Jk holds. Proof. We define the number operator N : CX → CX to be the linear extension of multiplication by 2degree(w) on a monomial w. Clearly, P (2a1 , . . . , 2an ) = (N P )(a1 , . . . , an ), for self-adjoint matrices a1 , . . . , an ∈ Mk C. Hence Jk is closed under the number operator. It is easy to conclude from this that Jk is indeed homogeneous. Then clearly Jkh is homogeneous in CXh . Finally let P ∈ Jk . Write P = P1 + iP2 with P1 , P2 ∈ CXh . It follows easily that P1 , P2 ∈ Jk , which completes the proof. 2 Proposition 1.12. Let k 1. Let P1 , . . . , Pm ∈ CX. If m
Pi∗ Pi ∈ Jk + Ccyc ,
(2)
i=1
then Pi ∈ Jk , for all 1 i m. Proof. Condition 2 says that m tr Pi (a1 , . . . , an )∗ Pi (a1 , . . . , an ) = 0 i=1
for all self-adjoint k × k-matrices a1 , . . . , an . Since this is just the sum of the squares of the Hilbert–Schmidt norms we can conclude that Pi (a1 , . . . , an ) = 0. Hence, Pi ∈ Jk as desired.
2
A priori, there is a little ambiguity in the definition of Jk , since we could have taken all complex matrices instead of self-adjoint matrices. The following lemma shows that this makes no difference:
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Lemma 1.13. Let P ∈ Jk and a1 , . . . , an be complex k × k-matrices. Then, P (a1 , . . . , an ) = 0. Proof. We first claim that P can be derived from a multi-linear relation with the same degree but in possibly more variables. Using a similar argument as in the proof of Proposition 1.11, we may assume that P is homogeneous of multi-degree (k1 , . . . , kn ). We set m = ni=1 ki and let: Q(x1 , . . . , xm ) = P (x1 + · · · + xk1 , xk1 +1 + · · · + xk1 +k2 , · · · , xm−kn +1 + · · · + xm ). Then Q = 0 is an identity that holds for all self-adjoint matrices of size k. Let Q be its homogeneous part of multi-degree (1, . . . , 1), which is also an identity for all self-adjoint matrices of size k. Clearly, P (x1 , . . . , xn ) =
1 Q (x1 , . . . , x1 , x2 , . . . , x2 , . . . , xn , . . . , xn ),
k1 !k2 ! · · · kn ! k1
k2
kn
and Q is multilinear. Just by linearity of Q , the relation Q = 0 does also hold for complex matrices. Hence, P = 0 holds for all complex matrices. 2 Remark 1.14. We see from Lemma 1.13 that for P ∈ Jk and Q1 , . . . , Qn ∈ CX, the polynomial P (Q1 , . . . , Qn ) again belongs to Jk . We define jk =
(−1)sg(σ ) Xσ (1) Xσ (2) · · · Xσ (2k) ∈ CX1 , . . . , X2k .
σ ∈S2k
The most striking result about the ideal Jk is a consequence of a theorem of Amitsur and Levitzki, which can be found in [1,2]. Theorem 1.15 (Amitsur–Levitzki). Any polynomial identity which holds in the ring of complex k ×k-matrices has degree 2k. Moreover, every such polynomial identity of degree 2k is a scalar multiple of jk . The first useful consequence of the Amitsur–Levitzki theorem to our set of questions is the following corollary, which will be crucial in the proof of Theorem 2.2. Corollary 1.16. Any homogeneous part of an element in Jk has degree 2k. Proof. Since Jk is homogeneous, any homogeneous part of an element from Jk again belongs to Jk . Using Lemma 1.13, the corresponding relation holds for all complex matrices. By Theorem 1.15, no such element can exist of degree < 2k. This finishes the proof. 2 Since Mk (C) satisfies the identity jk , it satisfies also an identity in two variables. So all Jk are nontrivial ideals in CX, for n 2. The quotient algebra An,k := CX/Jk is a PI-algebra,
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fulfilling every identity from Jk . This can for example be seen from Remark 1.14. We call this quotient the algebra of n generic self-adjoint k × k-matrices. (l) This notion can be justified as follows. Let ξij be commuting variables, for i, j = 1 . . . k (l)
and l = 1, . . . , n. Let Cn,k = C[ξij | i, j = 1, . . . , k, l = 1, . . . , n] be the commutative algebra generated by these variables, and extend the involution from C by defining (ξij )∗ = ξj i . Now let Mk (Cn,k ) be the algebra of k × k-matrices over Cn,k , equipped with the canonical extended (l) involution. For l = 1, . . . , n write Yl = (ξij )i,j and let Gn,k be the subalgebra of Mk (Cn,k ) generated by Y1 , . . . , Yn . Now one checks that sending Xi to Yi induces a well-defined ∗-algebra isomorphism between An,k and Gn,k . There is a large amount of literature on PI-algebras and especially algebras of generic matrices, see for example [10,19]. Note however that algebras of generic matrices are often defined to be quotients with respect to the ideal Jk,∞ of the polynomial algebra CX∞ in countably many variables. So we denote by Ak the quotient CX∞ /Jk,∞ and call it the algebra of generic self-adjoint k × k-matrices. Clearly each An,k is a subalgebra of Ak . The algebras Ak are very nice structures. For example they are domains (see for example [10, Section II.4]), and by Posner’s Theorem [18] they are subrings of matrices over a division ring that is finite-dimensional over its center. The same is thus true for the algebras An,k . We will use all these algebras below to approximate an arbitrary finitely generated tracial ∗-algebra. We end this section with a definition of type in the realm of tracial ∗-algebras. We have argued in the introduction that this extends the usual classification of finite von Neumann algebras into types. (k)
(k)
Definition 1.17. A tracial ∗-algebra is said to be of type Ik if it satisfies the polynomial identity jk =
(−1)sg(σ ) Xσ (1) Xσ (2) · · · Xσ (2k) ∈ CX1 , . . . , X2k .
σ ∈S2k
2. The main theorems 2.1. Approximation of traces We will deduce our main results as a consequence of rather elementary results in convex geometry, for example Proposition 1.3, and the main work is needed to show that those results can be applied. Therefore, we will spend some time to show that certain cones in CX under consideration are closed in the finest locally convex topology. h + J h is a closed convex cone in CXh . Theorem 2.1. Q(CX) + Ccyc k h + J h ) is closed, for all m. First take Proof. We have to show that CXhm ∩ (Q(CX) + Ccyc k h h a ∈ Q(CX) + Ccyc + Jk of degree at most m. We write: a = q + c + j with q ∈ Q(CX), c ∈ ∗ h and j ∈ J h . Assume degree(q) = 2l > m, write q = Ccyc i Pi Pi and let Qi denote the degree k l homogeneous part of Pi . Then q = i Q∗i Qi is the highest degree part of q, and from Propoh and J h at this point. sition 1.12 we see Qi ∈ Jk for all i. Note that we use homogeneity of Ccyc k So we can replace Pi by Pi − Qi , and add the counterterms Pi∗ Qi + Q∗i Pi − Q∗i Qi ∈ Jkh to the
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element j . This shows that we can assume degree(q) m in the representation of a. Again by h and J h we can thus also assume degree(c) m, degree(j ) m. homogeneity of Ccyc k h + J h ). There exists a seNow take any y from the closure of CXhm ∩ (Q(CX) + Ccyc k h + J h ) with limit point y ∈ CXh . We write quence (yi )i∈N in CXhm ∩ (Q(CX) + Ccyc k m yi = qi + ci + ji as above, where the degrees of qi , ci and ji do not exceed m. We claim that, up to a modification, we can choose a convergent subsequence of the sequence (qi )i∈N . Indeed, let us consider a rapidly decreasing, nowhere vanishing and positive function ψ on Yk = (Mk Ch )n , the n-fold Cartesian product of the space of self-adjoint k × k-matrices. We may assume that ψ(x) dλ(x) = 1. Yk
Obviously, the kernel of the tautological homomorphism φk : CX → C(Yk , Mk C) is precisely Jk . We define the ψ -normalized trace τ (P ) = tr φk P (x) ψ(x) dλ(x), Yk
which clearly vanishes on Ccyc + Jk . By Lemma 1.9, we can write qi =
m (n+1)
xij∗ xij ,
j =1
for some sequences (xij )i∈N in CXm/2 and 1 j (n + 1)m . We get τ
xij∗ xij
→ τ (y),
for i → ∞
j
by continuity. Hence, the sequences (xij )i∈N are bounded with respect to the associated 2-norm on CXm/2 /(CXm/2 ∩Jk ). Therefore, we can lift them to bounded sequences (xij +yij )i∈N in CX m/2 with yij ∈ CXm/2 ∩ Jk . After a modification in CXm ∩ Jk , i.e. replacing qi by j (xij + yij )∗ (xij + yij ), ji by ji − j (xij∗ yij + yij∗ xij + yij∗ yij ) and passing to a subsequence, we may thus assume that qi converges to some q ∈ CXm . Since Q(CX) is closed, h and we get q ∈ Q(CX) ∩ CXm . Note, that now also ci + ji → c + j for some c ∈ Ccyc h h h h h j ∈ Jk , since Ccyc + Jk is closed. Hence y = q + c + j ∈ Q(CX) + Ccyc + Jk , as desired. 2 Theorem 2.2. Let 2k > m be natural numbers. Then h h CXhm ∩ Q CX + Ccyc + Jkh = CXhm ∩ Q CX + Ccyc Proof. The inclusion of the right side in the left side is obvious. For the other direction let a ∈ h + J h be of degree at most m. Write a = q + c + j with q ∈ Q(CX), c ∈ C h Q(CX) + Ccyc cyc k and j ∈ Jkh . As in the proof of Theorem 2.1 we can assume that q, c, j have degree at most m.
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By Corollary 1.15, for 2k > m, the least degree where a non-vanishing coefficient of j appears is bigger than m. Hence j = 0 and the proof is finished. 2 h is a closed convex cone in CXh . Corollary 2.3. Q(CX) + Ccyc
Proof. Clear from Theorem 2.1, Theorem 2.2 and Lemma 1.2.
2
Theorem 2.4. Let τ be a positive trace on CX. Then there is a sequence of positive traces (τk )k∈N , converging to τ pointwise on CX, such that each τk vanishes on Jk . Proof. Consider τ as an R-linear functional on CXhk . It is clearly nonnegative on the convex h ) ∩ CX . We can apply Theorem 2.2 and see that τ is nonnegative on cone (Q(CX) + Ccyc k h h (Q(CX) + Ccyc + Jk ) ∩ CXk . Now we apply Proposition 1.3 and find a linear functional h + J h and coincides on CXh with τ up τk on CXh that is nonnegative on Q(CX) + Ccyc k k
to k1 say, in the operator norm. Note that we have also used Theorem 2.1 here. The sequence of complex linear extension of the τk now converges pointwise on CX to τ . So in particular τk (1) → τ (1) = 1, and we can scale each τk with the positive factor τk1(1) without destroying the convergence. The sequence we obtain in this way consists of positive traces as desired, see Remark 1.10. 2 Remark 2.5. Our setup is such that X is always a finite set. If one considers a countable set of letters as we did already, a straightforward diagonalization procedure shows that the preceding theorem extends to traces on CX∞ . Indeed one can approximate the trace τ on the subalgebra CX1 , . . . , Xn by a trace τn , and use the canonical projection CX∞ → CX1 , . . . , Xn to pull τn back to a trace on CX∞ . 2.2. Hölder and Minkowski type inequalities in tracial ∗-algebras Let A be a unital ∗-algebra over C. Recall that a positive trace (or just a trace) on A is a C-linear mapping τ : A → C fulfilling the following conditions: (1) τ (1) = 1, (2) τ (a ∗ a) 0 for all a ∈ A, and (3) τ (ab) = τ (ba) for all a, b ∈ A. Now for a fixed trace τ , an even number p and a ∈ A we define p 1 ap := τ a ∗ a 2 p . This establishes a well defined mapping · p : A → R0 , which clearly fulfills λap = |λ| · ap and ap = a ∗ p for a ∈ A and λ ∈ C. In [8], T. Fack and H. Kosaki showed that in a finite von Neumann algebra with a specified trace, one has a + bp ap + bp ,
and abp ar bs
for all a, b ∈ A, and positive p, r, s satisfying p1 = 1r + 1s . The first inequality is a generalization of the classical Minkowski inequality whereas the second inequality generalizes Hölder’s
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inequality. It is natural to expect that similar inequalities hold in the context of tracial ∗-algebras. We were not able to obtain the precise analogues of those inequalities, but we can establish the following weaker result. Partial results in this direction where already obtained in the work of R. Kunze [12] and D. Hadwin [9]. Theorem 2.6. Let (A, τ ) be a tracial ∗-algebra and let p be an even positive integer. We have a + bp aq + bq , p
with q = 22 4
−1
+1
and abp aq bq
p
and q = 2 2 +1 .
The preceding theorem is a consequence of Propositions 2.9 and 2.10, which we are going to prove in the sequel. We do not claim that the dependence of q and q on p is optimal. Indeed, we were not able to exclude the possibility that the inequalities of Minkowski and Hölder have an extension in their classical form to the realm of tracial ∗-algebras. One can easily see that for any trace which is a point-wise limit of bounded traces, the inequalities hold in their classical form. n For the whole section we will use the abbreviation ϕ(n) = 2 2 −1 for even n ∈ N, and ϕ(n) = ϕ(n + 1) for n odd. We start with the following generalized Cauchy–Schwarz inequality: Lemma 2.7. Let τ be a trace on A. For all n ∈ N and all a1 , . . . , an ∈ A we have n ϕ(n) 1 2ϕ(n) . τ (a1 · · · an ) τ a ∗ ai i
i=1
Proof. The proof is by induction on n, first for n even. For n = 2 the statement follows immediately from the Cauchy–Schwarz inequality, using τ (a ∗ a) = τ (aa ∗ ) for all a ∈ A. For arbitrary even n 4 we find 1 1 τ (a1 · · · an ) τ a ∗n · · · a ∗ a1 · · · a n 2 · τ a ∗ · · · a ∗n a n +1 · · · an 2 n 1 +1 2 2
2
= τ a n2 a ∗n a ∗n −1 · · · a1∗ a1 · · · a n2 −1 2
2
1 2
2
1 ∗ · τ an an∗ an−1 · · · a ∗n +1 a n2 +1 · · · an−1 2 , 2
where we used the Cauchy–Schwarz inequality in the first step, and condition (3) of the trace τ in the second. Now we apply the induction hypothesis for n − 2 to the two terms of the product. Indeed we have τ
∗ ∗
a n2 a n a n −1 · · · a1∗ a1 · · · a n2 −1 2 2
2ϕ(n−2)
2 ϕ(n−2) ϕ(n−2) · τ a n2 a ∗n a n2 a ∗n τ ai ai∗ 2
2
i= n2 −1 n
−1 ∗ ϕ(n−2) 2 ∗ ϕ(n−2) ∗ · τ a1 a1 a 1 a1 · τ ai ai i=2 n 2 −1
2ϕ(n−2) ∗ ϕ(n−2) 2 = τ a1∗ a1 · τ ai ai i=2
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2ϕ(n−2) · τ a ∗n a n2 2
n 2
2ϕ(n−2) . τ ai∗ ai i=1
Note that for the first step we have used the induction hypothesis, for the second step property (3) of τ again, and in the third step the Cauchy–Schwarz inequality to see ϕ(n−2) 2 2ϕ(n−2) τ ai∗ ai . τ ai∗ ai The same procedure applies to the second term in the above inequality and all in all yields n 2ϕ(n−2) 1 · 1 2ϕ(n−2) 2 . τ (a1 · · · an ) τ a ∗ ai i
i=1
Since 2ϕ(n − 2) = ϕ(n), the claim is proven for even n. For n odd it now follows easily by setting an+1 = 1 and applying the already established result to a1 , . . . , an+1 . 2 The result can be used to derive the following inequality, which will be used below: Lemma 2.8. Let τ be a trace on A. For all r, n ∈ N and all a1 , . . . , an ∈ A we have τ (a1 + · · · + an )r
n ϕ(r) 1 2ϕ(r) τ ai∗ ai
r .
i=1
Proof. Let Wnr denote the set of all words in a1 , . . . , an of length exactly r. For ω ∈ Wnr denote by ω(i) the number of occurences of ai in ω. We have τ (a1 + · · · + an )r τ (ω) ω∈Wnr n ϕ(r) ω(i) 2ϕ(r) τ ai∗ ai ω∈Wnr i=1
=
n ϕ(r) 1 2ϕ(r) τ ai∗ ai i=1
Note that we have used Lemma 2.7 for the second inequality. The following is a Hölder type inequality for · p : Proposition 2.9. For any even p and any a, b ∈ A we have abp aϕ(p+4) bϕ(p+4) .
2
r .
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Proof. We find 1 p 1 abp = τ b∗ a ∗ ab 2 p = τ a ∗ abb∗ · · · b∗ a ∗ abb∗ p ϕ(p) 1 ∗ ∗ ϕ(p) 1 2pϕ(p) τ 2pϕ(p) · · · τ a ∗ aa ∗ a bb bb ϕ(p) 1 p ∗ ∗ ϕ(p) 1 p 2pϕ(p) 2 τ 2pϕ(p) 2 = τ a ∗ aa ∗ a b bb b 2ϕ(p) 1 ∗ 2ϕ(p) 1 4ϕ(p) τ 4ϕ(p) = τ a∗a b b = aϕ(p+4) bϕ(p+4) , where we used property (3) of τ several times, and Lemma 2.7 for the inequality in the third step. 2 We also get a Minkowski type inequality: Proposition 2.10. For any even p and any a, b ∈ A we have a + bp aψ(p) + bψ(p) , where ψ(p) := ϕ(2ϕ( p2 ) + 4). Proof. We use Proposition 2.8 to get p 2 a + b2p = τ a ∗ a + a ∗ b + b∗ a + b∗ b 2 p ϕ( p ) 1p ϕ( p ) 1p τ a ∗ aa ∗ a 2 2ϕ( 2 ) + τ b∗ aa ∗ b 2 2ϕ( 2 ) ϕ( p ) 1p ϕ( p ) 1p + τ a ∗ bb∗ a 2 2ϕ( 2 ) + τ b∗ bb∗ b 2 2ϕ( 2 ) = a ∗ a 2ϕ( p ) + a ∗ b2ϕ( p ) + b∗ a 2ϕ( p ) + b∗ b2ϕ( p ) . 2
2
2
2
If we now apply Proposition 2.9 to each term in this last sum we get 2 a + b2p a2ψ(p) + 2aψ(p) bψ(p) + b2ψ(p) = aψ(p) + bψ(p) , and from this the desired result.
2
We can now give an easy proof of the following well-known corollary: Corollary 2.11. Let (A, τ ) be a tracial ∗-algebra. Then Zτ := a ∈ A a2 = 0 = a ∈ A ap = 0 for all even p is a two-sided ∗-ideal in A. The trace τ vanishes on Zτ and thus induces a faithful trace on the ∗-algebra A/Zτ .
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Proof. For any even p and any a ∈ A we have as a consequence of the usual Cauchy–Schwarz inequality 1 p p · · · a ∗ a ∗ a · · · a ∗ 2 a2 . ap = τ a ∗ a 2 τ a ∗ a p−1
p−1
This shows that Zτ is the set of elements with ap = 0 for all even p, and with the above results Zτ is then easily seen to be a two-sided ∗-ideal. From the usual Cauchy–Schwarz inequality we see that τ vanishes on Zτ . The rest is clear. 2 Definition 2.12. We call the algebra A/Zτ , with trace induced by τ , the trace-reduction of (A, τ ). Remark 2.13. It is clear that the trace-reduction A/Zτ of a tracial ∗-algebra (A, τ ) is tracereduced. Note that on a trace-reduced tracial ∗-algebra, a, b := τ (a ∗ b) defines an inner product. Thus · 2 is a norm. Let ϕ : (A, τ ) → (B, ρ) be a homomorphism of tracial ∗-algebras, i.e. a ∗-algebra homomorphism that fulfills ρ(ϕ(a)) = τ (a) for all a ∈ A. Then ϕ(a)p = ap holds for all a ∈ A. So ker ϕ ⊆ Zτ , and if (A, τ ) is trace-reduced, then ϕ is injective. Any ϕ induces a tracial ∗-algebra embedding on the trace-reductions A/Zτ → B/Zρ . Let (A, τ ) be a trace-reduced tracial ∗-algebra. It is clear that Theorem 2.6 is telling us that the topology which is induced by the p-norms is compatible with addition and multiplication. Indeed, the sets 1 Vn,p := a ∈ A ap < n form a countable subbasis of a vector space topology on A. Hence, we obtain a metrizable topology on A which endows A with the structure of a metrizable topological algebra. We say that a sequence (an )n∈N is a Cauchy sequence in A if an − am p → 0 for all even p as n, m → ∞. Moreover, a trace-reduced tracial ∗-algebra A is complete if every Cauchy sequence in A has a limit in A. It is easy to check that Theorem 2.6 implies that every trace-reduced tracial ∗-algebra has a natural completion to a complete trace-reduced tracial ∗-algebra, which is defined as usual to be the space of Cauchy-sequences modulo null-sequences. We will also need the following observations: Lemma 2.14. A trace-reduced tracial ∗-algebra A does not contain nilpotent left-ideals other than {0}. Proof. Let I be a nilpotent left-ideal and x ∈ I . Then x ∗ x ∈ I , and thus (x ∗ x)2 = 0 for some n n 1. So τ ((x ∗ x)2 ) = 0, and since τ is faithful, this immediately implies x = 0. 2 n
Corollary 2.15. Let (A, τ ) be a trace-reduced tracial ∗-algebra which satisfies the polynomial identity jk . Then A embeds into a k × k-matrix algebra over a commutative ring.
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Proof. We already recalled the fact that the Jacobson radical of A is nilpotent. By Lemma 2.14 it is trivial. The embedding result is now [19, Corollary 1.6.7]. Concretely, there exists an embedding A/m, ι : A → m⊂A
where m runs through all maximal ideals. Indeed, the Jacobson ideal agrees with the intersection of all maximal ideals of A and hence ι is injective. Now, A/m is simple and satisfies the polysubring of the k × k-matrices over a field km . nomial identity jk . Hence, A/m is isomorphic to a We may consider the commutative C-algebra B = m⊂A km and see that A is now realized as a ∗-subalgebra of Mk (B). This finishes the proof. 2 2.3. Metric ultraproducts of tracial ∗-algebras We are now concerned with the notion of ultraproduct of tracial ∗-algebras. We will see that even though the definition is reasonable and natural, it has its pathologies. One of the unusual features is that the tracial ultraproduct of a sequence of tracial ∗-algebras may consist only of multiples of identity. Let I be an index set and (Ai , τi ) a tracial ∗-algebra, for each i ∈ I . Most of the time, I will be just the set N. Define (Ai , τi ) = (ai )i∈I ∈ Ai (ai p )i∈I is bounded, for all even p . i∈I
i∈I p
1
Here, ai p = τi ((ai∗ ai ) 2 ) p , as in the last section. Using the Hölder and Minkowski type in equalities from the last section, i∈I (Ai , τi ) is easily seen to be a ∗-subalgebra of the product algebra. Now let ω be a free ultrafilter on I . For x = (ai )i∈I ∈ i∈I (Ai , τi ) we define τω (x) := lim τi (ai ). i→ω
Since |τi (ai )| ai 2 , and by the definition of i∈I (Ai , τi ), this is a well defined mapping τω :
i∈I
(Ai , τi ) → C.
One easily verifies that τω is even a trace. For the induced p-norms we find p 1 p 1 xp = τω x ∗ x 2 p = lim τi ai∗ ai 2 p = lim ai p . i→ω
i→ω
So consider Zτω := x ∈ (Ai , τi ) x2 = 0 , i∈I
which is a two-sided ∗-ideal in i∈I (Ai , τi ) by Corollary 2.11.
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Definition 2.16. We call the quotient ∗-algebra (Ai , τi ) := (Ai , τi ) Zτω
i→ω
i∈I
the metric ultraproduct with respect to the ultrafilter ω of the family ((Ai , τi ))i∈I . We equip it with the trace induced by τω . Remark 2.17. The construction is functorial in the following sense: if ϕi : (Ai , τi ) → (Bi , ρi ) are homomorphism of tracial ∗-algebras for all i∈ I , then they induce a canonical homomor (A , τi ) → i→ω (Bi , ρi ). The usual functorial properties phism of tracial ∗-algebras ϕ : i i→ω are fulfilled. ϕ is injective, since i→ω (Ai , τi ) is trace-reduced, see Remark 2.13. A sufficient condition for ϕ being surjective is for example the following: {i ∈ I | ϕi is surjective} ∈ ω. This is easily checked. So for example replacing each Ai by its trace-reduction Ai /Zτi does not change the metric ultraproduct. Remark 2.18. If the algebras (Ai , τi ) are finite von Neumann algebras with specified traces, it is easy to see that the von Neumann algebraic ultraproduct is identified with the algebra of bounded elements inside the metric ultraproduct of the tracial ∗-algebras (Ai , τi ). In this sense, our definition of metric ultraproduct is a variant of the usual von Neumann algebraic ultraproduct. Lemma 2.19. The metric ultraproduct of a sequence of tracial ∗-algebras is always tracereduced and complete. Proof. It is trace-reduced since we defined it to be the trace-reduction of a tracial ∗-algebra. A standard diagonalization argument shows that the metric ultraproduct is also complete. 2 Example 2.20. Let A be a ∗-algebra and let τ, τi for i ∈ N be traces on A. Assume limi→∞ τi = τ , pointwise on A. Consider the diagonal embedding ι:A→
(A, τi );
a → (a)i∈I .
i∈I p
p
Since τi ((a ∗ a) 2 ) → τ ((a ∗ a) 2 ) for fixed p and a, each diagonal sequence is indeed bounded with respect to · p . Fix an ultrafilter ω on N which is not principal. Then ι is a tracial ∗-algebra homomorphism, when A is equipped with τ . Thus we get a morphism of tracial ∗-algebras (A, τ ) →
(A, τi ).
i→ω
3. Embedding theorems We can now state the approximation results from Theorems 2.4 in a more conceptual way, as embedding results.
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Theorem 3.1. Let (A, τ ) be a tracial ∗-algebra, generated by n hermitian elements. Then there is a tracial ∗-algebra homomorphism into a metric ultraproduct ϕ : (A, τ ) →
(An,k , τk )
k→ω
on the index set N, where An,k is the algebra of n generic self-adjoint k × k-matrices and τk : An,k → C is a complex-linear functional which makes (An,k , τk ) into a tracial ∗-algebra for each k ∈ N. Proof. First note that we can assume that A = CX for X = {X1 , . . . , Xn }. Indeed choose a surjective ∗-homomorphism π : CX → A, and pull back the trace τ to a trace τ on CX. Then ker π ⊆ Zτ , see Remark 2.13. So we obtain a surjection A CX/Zτ , and any tracepreserving homomorphism from CX to an ultraproduct factors through CX/Zτ , again by Remark 2.13. But for A = CX we have shown that τ can be approximated by traces τk that vanish on Jk , see Theorem 2.4. So as described in Example 2.20 we get a tracial ∗-algebra homomorphism CX, τk . CX, τ → k→ω
But since τk vanishes on Jk we can replace (CX, τk ) by An,k = CX/Jk with the induced trace, without changing the ultraproduct (see Remark 2.17). This finishes the proof. 2 Theorem 3.2. Let (A, τ ) be a countably generated tracial ∗-algebra. Then there is a tracial ∗-algebra homomorphism into a metric ultraproduct ϕ : (A, τ ) →
(Ak , τk ),
k→ω
where Ak is the algebra of generic self-adjoint k × k-matrices. Proof. The proof is exactly the same as before, reducing to A = CX∞ first and applying Remark 2.5. 2 Corollary 3.3. Let (A, τ ) be a trace-reduced tracial ∗-algebra and suppose that A admits a countably generated dense subalgebra. Then there exist trace-reduced tracial ∗-algebras (An , τn ) of type In and a trace-preserving embedding into a metric ultraproduct ι : (A, τ ) →
(An , τn ).
n→ω
Proof. Apply Theorem 3.2 to the countably generated dense subalgebra and replace the rings of generic matrices with their reductions modulo the trace ideals. Now use that the ultra-product is complete and observe that the embedding extends uniquely to an embedding of the completion. This finishes the proof 2
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In order to apply the preceding corollary to finite von Neumann algebras with a separable predual, we should convince ourselves that a finite von Neumann algebra with a separable pre-dual admits indeed a countably generated dense subalgebra with respect to the topology induced by all the p-norms. Proposition 3.4. Let (M, τ ) be a finite von Neumann algebra with a separable pre-dual and a specified trace τ : M → C. Then, there exists a countably generated ultra-weakly dense Csubalgebra A ⊂ M, and for any such algebra (A, τ |A ), the completion of (A, τ |p ) with respect to the p-norms is canonically contained in the GNS Hilbert space L2 (M, τ ) and contains M as the sub-algebra of bounded elements. Proof. It is well known, that if M has a separable pre-dual, then M admits a countable ultraweakly dense set and hence a countably generated C-subalgebra A, which is dense in the ultraweak topology. Since any Cauchy sequence with respect to the p-norms is a Cauchy sequence with respect to the 2-norm, we may identify the completion with a subspace of the GNS space L2 (A, τ ). We denote the norm-completion of A by C ∗ (A, τ ). It is clear that any norm-limit is also a limit with respect to the p-norms since x = sup xp , p
∀x ∈ M.
Hence, C ∗ (A, τ ) is contained in the completion with respect to the p-norms. Pedersen’s Theorem [17, Thm. 2.7.3] states that for every x ∈ M and n ∈ N, there exists a projection qn ∈ M with τ (qn ) 1 − n1 and yn ∈ C ∗ (A, τ ) with yn x, such that qn (x − yn ) n1 . We conclude from this 1 x − yn p qn (x − yn )p + (1 − qn )(x − yn )p + 1 − qn p x − yn n 1 1 + 1/p · 2x. n n 1 Here, we used 1 − qn p = τ ((1 − qn )p )1/p n1/p . In particular, we conclude that yn → x in the topology induced by the p-norms. Hence M is contained in the completion and it is easy to see that M consists precisely of those elements which are bounded with respect to the trace. 2
Remark 3.5. We note that a finite von Neumann algebra with a specified trace is not complete with respect to the p-norms unless it is finite-dimensional. Acknowledgments The results in Section 2.1 were obtained in 2007 by the second author in an unsuccessful attempt to prove Connes’ embedding conjecture. He wants to thank Eberhard Kirchberg for pointing out the crucial omission of the boundedness issue. The occurring pathologies in GNSrepresentations with respect to unbounded traces are astonishing and in harsh contrast to the widely experienced automatic regularity of the analytic behavior in the presence of traces – for example when working with the algebra of unbounded operators affiliated with a finite von
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Neumann algebra. Sufficient conditions for unbounded traces to lead to a well-behaved representation of a finite von Neumann algebra have been studied for example in [24]. However, it is well known in the context of Real Algebraic Geometry that positive traces on the commutative algebra C[X1 , X2 ] = CX1 , X2 /J1 can already be rather pathological and take negative values on polynomials which are strictly positive on the plane R2 but are not a sums of squares of polynomials. Hence, the GNS-representation of such functionals cannot yield a measure on R2 and must necessarily have its pathologies. It is also clear that such a trace cannot be the pointwise limit of bounded traces. Over the years, the insight grew that the results in Section 2.1, and the consequences which we discussed in this paper, have indeed only very little to do with Connes’ original conjecture. The second author wants to thank Konrad Schmüdgen for encouragement and helpful discussions about concepts of Non-commutative Real Algebraic Geometry [22]. We want to thank the unknown referee for a detailed report which has led to improvements in the presentation of our work. References [1] A.S. Amitsur, J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. (ISSN 0002-9939) 1 (1950) 449–463. [2] A.S. Amitsur, J. Levitzki, Remarks on minimal identities for algebras, Proc. Amer. Math. Soc. (ISSN 0002-9939) 2 (1951) 320–327. [3] A.S. Amitsur, C. Procesi, Jacobson-rings and Hilbert algebras with polynomial identities, Ann. Mat. Pura Appl. (4) (ISSN 0003-4622) 71 (1966) 61–72. [4] A. Barvinok, A Course in Convexity, Grad. Stud. Math., vol. 54, Amer. Math. Soc., Providence, RI, ISBN 0-82182968-8, 2002. [5] T.M. Bisgaard, The topology of finitely open sets is not a vector space topology, Arch. Math. (Basel) (ISSN 0003889X) 60 (6) (1993) 546–552. [6] A. Braun, The radical in a finitely generated P.I. algebra, Bull. Amer. Math. Soc. (N.S.) (ISSN 0273-0979) 7 (2) (1982) 385–386. [7] A. Connes, Classification of injective factors. Cases II 1 , II ∞ , III λ , λ = 1, Ann. of Math. (2) (ISSN 0003486X) 104 (1) (1976) 73–115. [8] T. Fack, H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. (ISSN 0030-8730) 123 (2) (1986) 269–300. [9] D. Hadwin, A noncommutative moment problem, Proc. Amer. Math. Soc. (ISSN 0002-9939) 129 (6) (2001) 1785– 1791 (electronic). [10] N. Jacobson, PI-Algebras, An Introduction, Lecture Notes in Math., vol. 441, Springer-Verlag, Berlin, 1975, MR0369421 (51 #5654). [11] I. Klep, M. Schweighofer, Connes’ embedding conjecture and sums of Hermitian squares, Adv. Math. (ISSN 00018708) 217 (4) (2008) 1816–1837. [12] R.A. Kunze, Lp Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. (ISSN 00029947) 89 (1958) 519–540. [13] F.J. Murray, J. von Neumann, On rings of operators, Ann. of Math. (2) (ISSN 0003-486X) 37 (1) (1936) 116–229. [14] F.J. Murray, J. von Neumann, On rings of operators. II, Trans. Amer. Math. Soc. (ISSN 0002-9947) 41 (2) (1937) 208–248. [15] F.J. Murray, J. von Neumann, On rings of operators. IV, Ann. of Math. (2) (ISSN 0003-486X) 44 (1943) 716–808. [16] W.L. Paschke, L2 -homology over traced ∗-algebras, Trans. Amer. Math. Soc. (ISSN 0002-9947) 349 (6) (1997) 2229–2251. [17] G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, London Math. Soc. Monogr., vol. 14, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, ISBN 0-12-549450-5, 1979. [18] E.C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. (ISSN 0002-9939) 11 (1960) 180–183. [19] L.H. Rowen, Polynomial Identities in Ring Theory, Pure Appl. Math. (N.Y.), vol. 84, Academic Press Inc., New York, ISBN 0-12-599850-3, 1980.
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[20] F. R˘adulescu, Convex sets associated with von Neumann algebras and Connes’ approximate embedding problem, Math. Res. Lett. (ISSN 1073-2780) 6 (2) (1999) 229–236. [21] K. Schmüdgen, Graded and filtrated topological ∗ -algebras. II. The closure of the positive cone, Rev. Roumaine Math. Pures Appl. (ISSN 0035-3965) 29 (1) (1984) 89–96. [22] K. Schmüdgen, Noncommutative real algebraic geometry—some basic concepts and first ideas, in: Emerging Applications of Algebraic Geometry, in: IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 325–350. [23] M. Takesaki, Theory of Operator Algebras. II, Encyclopaedia Math. Sci., vol. 125, Springer-Verlag, Berlin, ISBN 3540-42914-X, 2003, Operator Algebras and Non-commutative Geometry, 6. [24] K. Takesue, Standard representations induced by positive linear functionals, Mem. Fac. Sci. Kyushu Univ. Ser. A (ISSN 0373-6385) 37 (2) (1983) 211–225.
Journal of Functional Analysis 259 (2010) 2961–2998 www.elsevier.com/locate/jfa
Gradient estimates via linear and nonlinear potentials Frank Duzaar a , Giuseppe Mingione b,∗ a Department Mathematik, Universität Erlangen–Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Germany b Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy
Received 20 May 2010; accepted 10 August 2010 Available online 21 August 2010 Communicated by N. Kalton
Abstract We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by −p u = μ. In particular, no matter the nonlinearity of the equations considered, we show that in the case p 2 a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case p 2. In the case p > 2 we prove a new gradient estimate employing nonlinear potentials of Wolff type. © 2010 Elsevier Inc. All rights reserved. Keywords: Nonlinear potential theory; p-Laplacian; Regularity
1. Introduction and results In this paper we are considering possibly degenerate quasilinear equations in divergence form with p-growth of the type − div a(x, Du) = μ,
(1.1)
in a bounded domain Ω ⊂ Rn with n 2, where μ is a Radon measure defined on Ω with finite total mass. Eventually letting μ(Rn \ Ω) = 0 we shall assume that μ is defined on the * Corresponding author.
E-mail addresses:
[email protected] (F. Duzaar),
[email protected] (G. Mingione). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.006
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whole Rn . The continuous vector field a : Ω × Rn → Rn is assumed to be C 1 -regular in the gradient variable z, with the partial derivative with respect to the gradient variable az (·) being itself continuous, and satisfying the following growth, ellipticity and continuity assumptions: ⎧ 1 p−1 a(x, z) + az (x, z) |z|2 + s 2 2 L |z|2 + s 2 2 , ⎪ ⎪ ⎪ ⎪ ⎨ p−2
ν −1 |z|2 + s 2 2 |λ|2 az (x, z)λ, λ , ⎪ ⎪ ⎪ ⎪ p−1 ⎩ a(x, z) − a(x0 , z) L1 ω |x − x0 | |z|2 + s 2 2
(1.2)
whenever x, x0 ∈ Ω and z, λ ∈ Rn . Here and in the rest of the paper we are assuming that ν, L, s, L1 are fixed parameters such that 0 < ν L and s 0, L1 1, while the function ω : [0, ∞) → [0, 1] is a modulus of continuity i.e., a non-decreasing subadditive function such that ω(0) = 0 = lim ω() ↓0
and ω(·) 1. On such a function we impose a natural decay property, which is essentially optimal for the result we are going to have, and prescribes a Dini-continuous dependence of the partial map x → a(x, z)/(|z| + s)p−1 ; specifically, we assume R ω()
d := d(R) < ∞,
(1.3)
0
whenever R < ∞. The prototype of (1.1) is – choosing s = 0 – clearly given by the p-Laplacian equation with coefficients (1.4) − div γ (x)|Du|p−2 Du = μ. In this case ω(·) represents the modulus of continuity of the function γ (·), which is in fact assumed to be Dini continuous and satisfying the “ellipticity” condition 0 < ν γ (x) L1 = L. See Section 2 for more notation. 1,p−1 By a weak (distributional) solution to Eq. (1.1) we mean a function u ∈ Wloc (Ω) such that the distributional relation
a(x, Du), Dϕ dx = ϕ dμ Ω
Ω
holds whenever ϕ ∈ C0∞ (Ω) has compact support in Ω. This definition is well known to be problematic in the sense that such kind of solutions are in general not unique when considered in connection for instance to a Dirichlet data. Their distinguishing feature is that they in general do not belong to the natural Sobolev space W 1,p , and for this reason are called very weak solutions. We refer to for instance to the surveys [28,29] for a discussion of the problem and a suitable list of references. Such solutions are usually obtained via approximations methods through the development of suitable a priori estimates coupled with proper monotonicity-based convergence methods; for this reason they are often called SOLA (Solutions Obtained by Limiting Approximation). The relevant existence theory is developed in the papers of Boccardo and Gallöuet
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[2–4] to which we refer for more details. An alternative existence theory for equations with non-negative measure data, featuring a suitable notion of solution – so-called p-superharmonic functions – is developed in [17,20,24]. In this paper we want to give new pointwise estimates for the gradient of solutions in term of suitable linear and nonlinear potentials of the right-hand side measure μ. Such estimates are bounded to extend to the nonlinear setting the classical ones valid for solutions to the Poisson equation − u = μ, where Du can be estimated in terms of the Riesz potential of the right-hand side I1 (μ). The story starts with a fundamental paper of Kilpeläinen and Malý [20] in which the authors were able to prove pointwise estimates for u in terms of the (truncated) Wolff potential μ Wβ,p (x, R) defined by
μ Wβ,p (x, R) :=
R
|μ|(B(x, )) n−βp
1 p−1
d
β ∈ (0, n/p].
(1.5)
0
More precisely, in [20] – and in [35,22], where a different and interesting approach was later developed – we can find the estimate u(x) c
1/γ γ μ – |u| + Rs dx + cW1,p (x, R),
γ > p − 1,
(1.6)
B(x,R)
valid whenever B(x, R) Ω, with x being a Lebesgue point of u; the constant only depends on n, p, ν, L. Estimate (1.6) has proved to play an essential role in the nonlinear potential theory. For this we refer to the recent, basic work of Phuc and Verbitsky [30–32], where solvability conditions for supercritical Lane–Emden type equations involving the p-Laplacian operator are given using estimate (1.6) as a replacement for the Green’s function – see the work of Kalton and Verbitsky [18] for the linear case involving the Laplacian. For more on Wolff potentials we refer to [16,19]. More recent developments have been given by the authors of the present paper – first in [27] for the case p = 2, and then in [8,9,11] for the case p 2 – in that a pointwise estimate similar to (1.6) been shown to hold at the gradient level. In fact – considering SOLA – in [8] the authors have proved that the pointwise a priori estimate Du(x) c
μ – |Du| + s dy + cW 1 (x, R) p ,p
B(x,R)
=c
1 R |μ|(B(x, )) p−1 d – |Du| + s dy + n−1 B(x,R)
(1.7)
0
holds at every Lebesgue point x of Du when p 2. The constant c depends this time upon n, p, ν, L, L1 , ω(·); an extension of (1.7) to a class of anisotropic operators has been later given in [5]. Estimate (1.7) holds in particular for W 1,p -solutions to (1.1).
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The aim of this paper is now twofold. We first give an extension to the a priori estimate (1.7) when p 2, and in the case solutions to measure data problems like (1.1) belong to the Sobolev space W 1,1 ; this is known to happen in the case 2−
1 < p. n
(1.8)
The previous bound is optimal to have W 1,1 -solutions, as revealed by the analysis of the so-called nonlinear fundamental solution Gp to the problem
− p Gp = δ Gp = 0
in B1 , on ∂B1 ,
(1.9)
where δ is the Dirac measure charging the origin, and B1 is the ball centered at the origin with radius equal one. In this case we have Gp (x) ≈
p−n
(|x| p−1 − 1) − log |x|
if 1 < p = n, if p = n.
We refer to [28] for a proof of the uniqueness of solutions to the problem (1.9) in the framework of SOLA. It follows that DGp ∈ L1loc iff (1.8) holds. Before stating the first result of the paper let us recall that truncated linear Riesz potentials are defined as
μ Iβ (x, R) :=
R
μ(B(x, )) d , n−β
β ∈ (0, n].
0
Theorem 1.1 (Linear potential gradient bound). Let u ∈ C 1 (Ω) be a weak solution to (1.1) with μ ∈ L1 (Ω), under the assumptions (1.2) with 2 − 1/n < p 2. Then there exists a constant c ≡ c(n, p, ν, L, L1 , ω(·), diam(Ω)) > 0 such that the pointwise estimate Du(x) c
|μ| 1 – |Du| + s dy + c I1 (x, R) p−1
(1.10)
B(x,R)
holds whenever B(x, R) ⊆ Ω. Remark 1.1. In Theorem 1.1 we are restricting ourselves to the case the solution is a priori considered to the be of class C 1 , and therefore, in particular if class W 1,p ; i.e. very weak solutions are not considered in Theorem 1.1. Moreover we assume that the measure μ is an integrable function. Such additional regularity assumptions are by no mean restrictive in that by coupling estimate (1.10) with the convergence methods developed in [2–4] estimate (1.10) turns to hold for general solutions to measure data problems, and in particular for SOLA, provided x is a Lebesgue point of Du. We refer to [8,27] for more details on this aspect and for a detailed description of the required approximation methods.
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Theorem 1.1 presents in our opinion two interesting features: first of all it says that, at the least for the considered range of p, the gradient of solutions can be pointwise estimated by Riesz potentials exactly as in the case of the standard Poisson equation, provided of course the scaling |μ| |μ| of the equation is taken into account, i.e. one takes [I1 (x, R)]1/(p−1) rather than I1 (x, R). We notice that, in fact, the operator 1 |μ| μ → I1 (·, R) p−1
(1.11)
defines a new nonlinear potential which has the same scaling and homogeneity properties μ of W 1 (·, R). The appearance of the Riesz based potential (1.11) was already noted in [26, p ,p
Section 1.4] when proving a sort of “level set version” of (1.10); see also [29, Theorem 3.6, Remark 3.5]. Second, in Theorem 1.1 we have that the Dini modulus of continuity assumed on the coefficients in (1.3), and known to be sharp for linear elliptic equations – see [14] for counterexamples – is now found to apply to the nonlinear case too. Finally, we remark that the constant c involved in estimate (1.10) is stable when p approaches 2 i.e. letting p 2 in (1.10) we recover the usual I1 estimate valid for the case of the Poisson equation − u = μ, that is Du(x) c
|μ| – |Du| + s dy + cI1 (x, R).
(1.12)
B(x,R)
The last estimate has been proved in [27,8] for general nonlinear equations. See also Remark 5.1 below. As a matter of fact, we can directly take p = 2 in Theorem 1.1 in order to obtain (1.12). The second aim of this paper is to present a refinement of the main result of [8] in the case p > 2, that is we replace the Wolff potential in the right-hand side of (1.7) with another, slightly μ μ smaller nonlinear potential of Wolff type, namely we employ W p 3p−2 instead of W 1 . The 3p−2 ,
p
p ,p
relation between the two potentials is clarified in (1.16) below; moreover, observe that the two |μ| potentials still coincide with the Riesz potential I1 when p = 2. To this aim we shall consider a more restricted class of equations, with vector fields a(·) satisfying an additional assumption of Hölder continuity type with respect to the gradient variable z. Namely, we shall assume that there exists a positive number α satisfying 0 < α < min 1, 4/(n − 2), p − 2
(1.13)
such that the renormalized Hölder continuity property p−2−α az (x, z2 ) − az (x, z1 ) L |z1 |2 + |z2 |2 + s 2 2 |z2 − z1 |α
(1.14)
holds whenever z1 , z2 ∈ Rn and x ∈ Ω. This assumption is obviously satisfied by the model example in (1.4).
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Theorem 1.2 (Nonlinear potential gradient bound). Let u ∈ C 1 (Ω) be a weak solution to (1.1) with μ ∈ L1 (Ω), under the assumptions (1.2) and (1.14), with p > 2. Then there exists a constant c ≡ c(n, p, ν, L, L1 , α, ω(·)) > 0 such that the pointwise estimate Du(x) c
2 R
p
2 p p |μ|(B(x, )) 2(p−1) d p 2 – |Du| + s dy +c n−1
(1.15)
0
B(x,R)
holds whenever B(x, R) ⊆ Ω. We remark that the case p = 2 has been treated in [8,27]. The previous theorem refines the main result of [8] – that is (1.7) – in two respects. First we observe that when formulating condition (1.3) in [8] we replaced ω(·) by [ω(·)]2/p , thereby considering a slightly stronger continuity condition, still of Dini type. As already mentioned in the case p < 2, we find that the same optimal conditions valid for linear equations actually works in the general degenerate case p = 2; see again [14]. The second and more substantial improvement has already been anticipated above, and concerns the right-hand side nonlinear potential employed in the pointwise estimate (1.15), in the sense that the following inequality holds true: μ W p
3p−2 ,
2 p = 3p−2 (x, R)
R
p
|μ|(B(x, )) n−1
p 2(p−1)
d
2
p
0
2R
|μ|(B(x, )) n−1
1 p−1
d μ = W 1 (x, 2R). p ,p
(1.16)
0
The previous estimate is indeed a consequence of the elementary inequality ∞
q ak
k=0
∞
q ak
,
q 1, ak 0,
∀k ∈ N
(1.17)
k=0
applied with q = p/2 to perform the following standard computation: R
|μ|(B(x, )) n−1
p 2(p−1)
∞
d =
k R/2
k=0 R/2k+1
0
|μ|(B(x, )) n−1
p 2(p−1)
k=0
1 p ∞ |μ|(B(x, R/2k )) p−1 2 k=0
2R 0
(R/2k+1 )n−1 |μ|(B(x, )) n−1
1 p−1
∞ |μ|(B(x, R/2k )) 2(p−1) p
d
p 2
.
(R/2k+1 )n−1
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Remark 1.2 (Comparison between (1.7) and (1.10)). The rate of improvement of (1.15) with respect to (1.7) can be observed in certain borderline situations. We mention two. When considering measures belonging to function spaces we have that inequality (1.7) is strong enough to essentially imply all sharp integrability results for Du below L∞ . For instance, let us consider the case of Lorentz spaces L(q, γ ) – we refer to [34] for an introduction to Lorentz spaces and to [26] for a treatment within the context of interest here. Both estimate (1.7) and (1.15) allow to get the sharp integrability result
μ ∈ L(q, γ )
⇒
Nq Du ∈ L ,γ N −q
for 1 < q < N and 0 < γ < ∞.
(1.18)
When instead asking for the limiting case Du ∈ L∞ , an improvement in terms of the second index in the Lorentz scale is allowed by (1.15) with respect to (1.7). Indeed, while (1.15) allows to conclude that μ ∈ L(n, p/(2p − 2)) implies the local boundedness of Du, inequality (1.7) requires that μ ∈ L(n, 1/(p − 1)), which is a stronger condition for p > 2. Turning our attention to the case when μ is genuinely a measure, we have that the potentials in the two sides of (1.16) become essentially equivalent when for instance the measure uniformly concentrates on a set with dimension that can be described via ordinary Hausdorff measures. This is for instance the case when the measure concentrates uniformly on a σ -Alfhors regular set S μ = Hσ S,
μ(BR ) ≈ R σ ,
σ ∈ [0, n],
which holds whenever BR is centered on S. Here Hσ denotes the σ -dimensional Hausdorff measure. Relevant examples are given by surface measures related to manifolds, where the quantities appearing in the two sides of (1.16) still become equivalent. A strict inequality occurs for instance in the case of those measures uniformly concentrated on sets whose Hausdorff dimension can be described only using in terms of a Gauge function γ (·) of non-power type: μ = Hγ (·) S where μ(BR ) ≈ γ (R). For this we refer for instance to [33]. Remark 1.3. A careful inspection of the proofs will reveal that both in estimate (1.10) and in (1.15) the constant turns out to be independent of diam(Ω) when the vector field a(·) is independent of x. See Step 4 of the proof of Theorem 1.1 below. 1.1. Technical novelties of the paper Although we shall take the strategy adopted in [8] as a guideline, there are here a number of new non-trivial points. We start by the case p 2, that is Theorem 1.1. Usually called the singular case since the modulus of ellipticity tends to infinity when |Du| → 0, the case p < 2 is for our ultimate purposes to be considered as a degenerate one. In fact, since estimates of the type (1.7) and (1.10) are estimates on the size of the gradient, the difficult case for us is when |Du| gets large. In this situation there is a loss of ellipticity in the equation, and estimates become harder to get. Instead a pointwise gradient estimate is in principle easier to get when p gets larger since the coercivity of the operator increases. A manifestation of these difficulties in the case p < 2 is the following major technical difference with respect to the case p 2. In [8] we developed an iteration scheme based on the comparison between the original solution u of (1.1) and solutions
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to homogeneous equations of the type div a(x, Dw) = 0 in a ball BR with radius R, subject to the boundary condition u ≡ w on ∂BR . The outcome was the inequality 1 |μ|(BR ) p−1 . – |Du − Dw| dx c R n−1
(1.19)
BR μ
The quantity on the right-hand side of (1.19) is in turn the density of the potential W 1 , and p ,p
this is a key point in the proof; indeed, a suitable summation process involving (1.19) finally leads to (1.7). As a consequence of the fact that p 2 an estimate of the type (1.19) is no longer possible since the coercivity of the operator is too weak. One of the main challenges here is to find the correct replacement for the quantity appearing in the right-hand side of (1.19) when p 2, which allows to rebalance the weak ellipticity. It turns out that the mixed quantity
|μ|(BR ) R n−1
1 p−1
+
|μ|(BR ) R n−1
2−p – |Du| + s dx
(1.20)
BR
which depends also on the gradient average, and that for this reason cannot represent the density of a potential, is the right one. In fact, the presence of the gradient in (1.20), coupled with the measure, encodes in an optimal way the weaker ellipticity of the problem. It is indeed one of the delicate points of the proof to identify the right form of local comparison estimates that on one hand encode the degenerate character of the case p < 2 in case of large gradient, and on the other allows for the suitable iteration finally leading to desired gradient estimate (1.10). For the case p 2 the key to the improved nonlinear potential estimate (1.15) is the use of the map V (Du) in the estimates, rather than the plain gradient Du. Here it is p−2 V (Du) ≡ Vs (Du) = |Du|2 + s 2 4 Du.
(1.21)
The use of V (Du) rather than Du allows to get better estimates as it allows to incorporate many of the degenerate features of the operator in question in the considered map, allowing for a better potential on the right-hand side. In turn, working with the quantity defined in (1.21) poses additional problems, and in particular a few delicate estimates below the natural growth exponents must be worked out. 2. Notations In what follows we denote by c a general constant larger (or equal) than one, possibly varying from line to line; special occurrences will be denoted by c1 etc; relevant dependences on parameters will be emphasized using parentheses. We also denote by B(x0 , R) := {x ∈ Rn : |x − x0 | < R} the open ball with center x0 and radius R > 0; when not important, or clear from the context, we shall omit denoting the center as follows: BR ≡ B(x0 , R). Unless otherwise stated, different balls in the same context will have the same center. We shall also denote B ≡ B1 = B(0, 1). With
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A being a measurable subset with positive measure, and with g : A → Rk being a measurable map, we shall denote 1 g(x) dx – g(x) dx := |A|
A
A
its integral average. According to what we have stated in the Introduction, when considering an L1 -function μ we shall denote |μ|(A) :=
μ(x) dx.
A
In other words, in this paper we shall deal with L1 -data, but “thinking of the case the datum is a measure”. Indeed, when considering equations as (1.1) in order to get the results we are bounded to present, it is sufficient to consider the case μ ∈ L1 (Ω), the case when μ is a general Borel measure with finite total mass can be obtained via approximation [8,27]. In the following, as it often happens with p-Laplacian type operators, it will be convenient, rather than working with the gradient, to work with a nonlinear quantity involving the gradient, and taking into account the structure properties of the operator in question. With s 0, we define p−2 V (z) = Vs (z) := s 2 + |z|2 4 z,
z ∈ Rn ,
(2.1)
which is easily seen to be a locally bi-Lipschitz bijection of Rn . A basic property of the map V (·), whose proof can be found in [15, Lemma 2.1], is the following: For any z1 , z2 ∈ Rn , and any s 0, it holds p−2 |V (z2 ) − V (z1 )|2 p−2 2 2 2 2 c−1 s 2 + |z1 |2 + |z2 |2 2 c s + |z | + |z | , 1 2 |z2 − z1 |2
(2.2)
where c ≡ c(n, p), is independent of s. The strict monotonicity properties of the vector field a(·) implied by the left-hand side in (1.2)2 can be recast using the map V (·). Indeed – see also [25] – combining (1.2)2 and (2.2) yields, for c ≡ c(n, p, ν) > 0, and whenever z1 , z2 ∈ Rn 2
c−1 V (z2 ) − V (z1 ) a(x, z2 ) − a(x, z1 ), z2 − z1 .
(2.3)
Moreover, when p 2, assumption (1.2)2 – via (2.2)–(2.3) – immediately implies
c−1 |z2 − z1 |p a(x, z2 ) − a(x, z1 ), z2 − z1 .
(2.4)
3. Decay estimates for a0 -harmonic functions The aim of this chapter is to recall a few decay estimates valid for solutions to homogeneous equations of the type div a0 (Dv) = 0,
(3.1)
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where the vector field a0 : Rn → Rn satisfies assumptions (1.2)1,2 and (1.14), with the obvious understanding that now no x-dependence is involved. Such functions are indeed called a0 -harmonic functions. The peculiar point of the results we are going to present is that a few of the decay estimates presented are not formulated in terms of the gradient Du, but rather in terms of the nonlinear quantity V (Du). The decay estimates for solutions found here differ from the usual ones in the fact that the exponents involved are smaller than those typically used, and this will require to employ certain rarely used facts from regularity theory of p-Laplacian type operators. 3.1. A decay estimate involving the V (·)-map In this section we outline the proof of the following: Theorem 3.1. Let v ∈ W 1,p (Ω) be a weak solution to (3.1) under the assumptions (1.2)1,2 with p > 2 and (1.14). Then there exist constants β ∈ (0, 1] and c 1, both depending only on n, p, ν, L, and α such that the following estimate: 2β 2 2 – V (Dv) − V (Dv) B dx c – V (Dv) − V (Dv) B dx R R B
(3.2)
BR
holds whenever B ⊆ BR ⊆ Ω are concentric balls. This result is standard in the case of the p-Laplacian equation – see for instance [7] and references therein – and it is more in general known to hold for minima of certain functionals of the Calculus of Variations with p-growth; moreover, it extends to minimizers of the p-Dirichlet functional (4.31) below in the vectorial case. Here we shall present the necessary modifications to the known proofs in order to prove the result in the context of Theorem 3.1. Although we shall often refer to other papers where a similar estimate is developed in different settings, we shall as much as possible try to give a self-contained proof of (3.2). Proof of Theorem 3.1. We divide the proof in several steps. Step 1: The degenerate case. Here we see that we may reduce to the nondegenerate case s > 0 via an approximation that the reader may for instance find in [12,10]. Let us fix a family {φε }ε>0 of standard mollifiers in Rn and obtained in the following way: φε (z) := ε −n φ(z/ε). Here φ ∈ C ∞ (Rn ) and it is such that φ(z) dz = 1. (3.3) supp φ = B1 and Rn
We define the regularized vector fields aε (z) := (a ∗ φε )(z),
ε > 0.
(3.4)
It obviously follows that aε (·) ∈ C ∞ (Rn ) and moreover, as in [12, Lemma 3.1] – whose arguments apply here since (3.3) is assumed – we have that the assumptions (1.2) and (1.14) are satisfied for new values of ν, L, depending only on the original ones ν, L and moreover on n, p.
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The approximation scheme in question is completely standard and we omit it here (see for in1,p stance [1, Section 4] and [8]). We then define vε ∈ v + W0 (Ω ) as the unique solution to the Dirichlet problem
div aε (Dvε ) = 0 in Ω , vε = v on ∂Ω ,
for a Lipschitz-regular subdomain Ω Ω. The final outcome is that – up to choosing a suitable subsequence ε ≡ εn → 0 – we have that vε → v strongly in W 1,p (Ω ). Needless to say this is sufficient to pass ε → 0 in an estimate like (3.2). Therefore in the rest of the proof we shall with no loss of generality assume that s > 0, catching the case s = 0 by passing to the limit the uniform decay estimates obtained in a standard way. Moreover we shall obviously replace Ω by Ω since the result we are going to prove is local in nature. Step 2: L∞ -estimate. Let us denote 2 2−p a˜ i,j (x) := Dv(x) + s 2 2 ∂zj a0i Dv(x) , and p/2 H ≡ H (Dv) := |Dv|2 + s 2 . 1,2 (Ω) ∩ L∞ It then follows – see for instance the approach in [10, Section 3] – that H ∈ Wloc loc (Ω) and that H is a subsolution of a uniformly elliptic equation with measurable coefficients, that is
a˜ i,j Di H Dj ϕ dx 0,
ϕ0
(3.5)
Ω
holds with ϕ ∈ C0∞ (Ω). In turn this fact implies that H ∈ L∞ loc (Ω) and the quantitative estimate sup H c – H dx BR/2
∀BR ⊆ Ω,
(3.6)
BR
holds for a constant depending only on n, p, ν, L. Step 3: A first oscillation estimate. Denoting 2 φ(R) := – V (Dv) − V (Dv) B dx, R
BR
M(R) := sup H BR
for a fixed ball BR ≡ B(x0 , R). As a consequence of the weak Harnack type inequality valid for subsolutions of (3.5) we have – see [15, Section 4.6] – φ(R) c M(R) − M(R/2) ,
(3.7)
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for c ≡ c(n, p, ν, L). Moreover, a standard difference quotient method asserts that V (Dv) ∈ 1,2 (Ω, Rn ), while the next reverse Hölder’s inequality is just a consequence of the fact that v Wloc solves (3.1), together with the higher differentiability of V (Dv): – V (Dv) − V (Dv) B
R/2
2χ dx
1/χ
2 c – V (Dv) − V (Dv) B dx, R
BR/2
BR
where χ = n/(n − 2) when n > 2 and χ can be chosen arbitrarily large when n = 2. Remark 3.1. The content of the previous two steps has been reported to emphasize that such parts of the proof of estimate (3.2) do not rely on the fact that v minimizes a certain integral functionals, but rather on the fact that v is a solution to a certain equation. Moreover, in Step 2 we emphasize that no particular structure – i.e. dependence on Dv via its modulus – is actually needed in the proof. Step 4: Conclusion. Here we prove Lemma 3.1. Assume that 2 p/2 φ(R) c1 (Dv)BR + s 2 ,
(3.8)
holds for a constant c1 . Then there exists another constant c2 , depending on n, p, ν, L and c1 , such that φ() c2
2 n+2 α R φ(R) 1+ φ(R) R (|(Dv)BR |2 + s 2 )p/2
holds whenever 0 < R. Once the previous lemma is proved, the proof follows along the lines of [15, Section 4.8] – keep in mind that general differential forms are used there. Indeed a delicate but by now standard iteration argument allows to deduce (3.2) from Lemma 3.1 and the content of Step 3. It therefore remains to prove Lemma 3.1, to which we dedicate in the rest of the proof. We again follow the lines of [15], but at several stages we shall use a different argument since we are not dealing with minimizers of integral functionals. Let us set z0 := (Dv)BR . Assumption (3.8) used together with (3.6) yields p/2 sup |Dv − z0 |p c |z0 |2 + s 2 , BR/2
where c depends on p, c1 . We now introduce the frozen matrix (A0 )ij := a0i z (z0 ), j
(3.9)
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which is an elliptic matrix with constant coefficients in the sense that it satisfies the following ellipticity and growth conditions p−2 c−1 |z0 |2 + s 2 2 |λ|2 A0 λ, λ,
p−2 |A0 | c |z0 |2 + s 2 2 ,
(3.10)
whenever λ ∈ Rn , and with c ≡ c(n, p, ν, L). Accordingly, we define v˜ ∈ v + W01,2 (BR/2 ) as the unique solution to the following Dirichlet problem:
˜ = 0 in BR/2 , div(A0 D v) v˜ = v on ∂BR/2 .
This means that the ratio between the highest and the lowest eigenvalue of A0 is bounded by a constant depending only on n, p, ν, L and therefore classical estimates for solutions to linear elliptic equations apply (see for instance [15, Proposition 4.2]). In particular, for c ≡ c(n, p, ν, L) 1 it holds that – |D v˜ − z0 |2+α + |D v˜ − z0 |p dx c – |Dv − z0 |2+α + |Dv − z0 |p dx. BR/2
(3.11)
BR/2
Again as in [15, Proposition 4.3, (4.48)] we arrive at φ() c
2 p−2 R n φ(R/2) + c |z0 |2 + s 2 2 ˜ 2 dx, – |Dv − D v| R
(3.12)
BR/2
for every R/2. We have to estimate the last integral in (3.12): denoting w := v − v˜ we have, by mean of the first inequality in (3.10) p−2 2 |z0 | + s 2 2
– |Dw|2 dx c – A0 Dw, Dw dx. BR/2
(3.13)
BR/2
Now, notice that by writing 1 a0 (z) − a0 (z0 ) =
(a0 )z tz + (1 − t)z0 dt (z − z0 ),
0
and applying (1.14), we obtain p−2−α a0 (z) − a0 (z0 ) − A0 (z − z0 ) c |z0 |2 + s 2 2 |z − z0 |1+α + c|z − z0 |p−1 . (3.14) In turn, using Young’s inequality repeatedly, and the fact that both v and v˜ are solutions, and of course using (3.14), it holds that
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– A0 Dw, Dw dx = – A0 Dv, Dw dx BR/2
BR/2
= − – a0 (Dv) − a0 (z0 ) − A0 (Dv − z0 ), Dw dx BR/2
p−2−α c – |z0 |2 + s 2 2 |Dv − z0 |1+α + |Dv − z0 |p−1 |Dv − z0 | dx BR/2
p−2−α + c – |z0 |2 + s 2 2 |Dv − z0 |1+α + |Dv − z0 |p−1 |D v˜ − z0 | dx BR/2
p−2−α c – |z0 |2 + s 2 2 |Dv − z0 |2+α + |Dv − z0 |p dx BR/2
p−2−α + c – |z0 |2 + s 2 2 |D v˜ − z0 |2+α + |D v˜ − z0 |p dx BR/2
p−2−α c – |z0 |2 + s 2 2 |Dv − z0 |2+α + |Dv − z0 |p dx BR/2
p−2−α c – |z0 |2 + s 2 2 |Dv − z0 |2+α dx. BR/2
In the last two lines we used (3.11) and then (3.9). Combining the last inequality with (3.13) yields p−2 2 |z0 | + s 2 2
p−2−α ˜ 2 dx c – |z0 |2 + s 2 2 |Dv − z0 |2+α dx. – |Dv − D v| BR/2
BR/2
This last estimate is the analogue of the last inequality at page 38 of [15] and from this point the proof of the Lemma follows as in [15, Proposition 4.2]. 2 3.2. Estimates below the natural growth exponent The aim of this section is to give a version of Theorem 3.1 below the natural growth exponent. Indeed, instead of considering V (Du) in L2 , it will be considered in L1 . To begin with we recall a preliminary result on reverse Hölder inequalities. Lemma 3.2. Let g : A → Rk be a integrable map such that
1/χ0 χ0 c∗ – |g| dx – |g| dx BR
B2R
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holds whenever B2R ⊆ A, where A ⊆ Rn is an open subset, and χ0 > 1, c∗ 0. Then, for every t ∈ (0, 1] and χ ∈ (0, χ0 ] there exists a constant c0 ≡ c0 (n, c∗ , t) such that, for every B2R A it holds that
1/χ
1/t – |g|χ dx c0 – |g|t dx . BR
(3.15)
B2R
The proof of the previous result, which is based on a by now standard interpolation/covering argument, can be obtained with minor modifications from [13, Remark 6.12]. Next, a result which can be inferred from [25, Lemma 3.2]; see also [26]. Lemma 3.3. Let v ∈ W 1,p (Ω) be a weak solution to (3.1) under the assumptions (1.2), and fix z0 ∈ Rn . For every t ∈ (0, 1] there exists c ≡ c(n, N, p, ν, L, t) 1, but independent of z0 ∈ Rn , such that
1/2
1/(2t) 2 2t c – V (Dv) − z0 dx , – V (Dv) − z0 dx BR/2
(3.16)
BR
holds whenever BR ⊆ Ω. We now come to the decay estimate below the natural growth exponent. Theorem 3.2. Let v ∈ W 1,p (Ω) be a weak solution to (3.1) under the assumptions (1.2) with p > 2 and (1.14). Then there exist constants β ∈ (0, 1] and c 1, both depending only on n, p, ν, L, and α such that the following estimate: β – V (Dv) − V (Dv) B dx c – V (Dv) − V (Dv) B dx R R B
(3.17)
BR
holds whenever B ⊆ BR ⊆ Ω are concentric balls. Proof. Using estimate (3.2) and Hölder’s inequality we deduce
1/2 2 – V (Dv) − V (Dv) B dx – V (Dv) − V (Dv) B dx
B
B
2β 1/2 2 c dx – V (Dv) − V (Dv) B R/2 R BR/2
2β 1/2 2 c , (3.18) – V (Dv) − V (Dv) B dx R R BR/2
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whenever 0 < R/2. On the other hand, we may apply estimate (3.16) with z0 = (V (Dv))BR getting
1/2 2 c – V (Dv) − V (Dv) B dx. – V (Dv) − V (Dv) B dx R
BR/2
R
BR
Merging this last estimate with (3.18) yields the assertion for the case 0 < R/2; on the other hand estimate (3.17) trivially holds when R/2 R and the proof is complete. 2 The next result has been proved for the case p 2 in [8, Theorem 3.1]; the proof for the case 1 < p < 2 can be obtained with minor modifications. Theorem 3.3. Let v ∈ W 1,p (Ω) be a weak solution to (3.1) under the assumptions (1.2)1,2 with p > 1. Then there exist constants β ∈ (0, 1] and c 1, both depending only on n, p, ν, L, such that the estimate β – Dv − (Dv)B dx c – Dv − (Dv)BR dx R B
(3.19)
BR
holds whenever B ⊆ BR ⊆ A are concentric balls. Estimates of this type, with different exponents involved, have been originally developed in [6,21,23]. Remark 3.2 (Stabilization of the constants I). A very careful analysis of the estimates involved in the proof of (3.19) reveals a continuous dependence of the constants β > 0 and c < ∞ appearing in (3.19). This means that whenever p lies in a compact subset of (1, ∞) then β and c vary in a compact subset of (0, 1) and [1, ∞), respectively. 4. Decay and comparison estimates We now fix, for the rest of the section, a ball B(x0 , 2R) ⊆ Ω that will be shortly denoted by B2R . Unless otherwise stated all the ball considered will concentric to B2R . Moreover, the solution of (1.1) will be always considered under the assumptions of Theorem 1.1, that is of class C 1 . In the rest of the sections u will be the solution considered in Theorems 1.1 and 1.2. 4.1. Two comparison estimates In this section we derive a few crucial comparison estimates between the original solution of (1.1) and solutions to suitably homogeneous boundary value problems. In the case p > 2 the main point is the use of the function V (Du) replacing the gradient Du, while in the second case p 2 the main point is that the mixed quantity in (1.20) involving the right-hand side measure μ and the gradient average will be come into the play.
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1,p
We start defining w ∈ u + W0 (B2R ) as the unique solution to the homogeneous Dirichlet problem div a(x, Dw) = 0 in B2R , (4.1) w=u on ∂B2R . Remark 4.1 (Scaling). Before going on with the proofs, let us recall a few basic properties of equations of the type (1.1) under the assumptions (1.2), where μ ∈ L1 (Ω). Let us consider the ball B2R ≡ B(x0 , 2R) ⊂ Ω and a positive number A > 0, and let us define the new functions u(y) ˜ :=
u(x0 + 2Ry) 2AR
and μ(y) ˜ :=
2Rμ(x0 + 2Ry) , Ap−1
(4.2)
and the new vector field a(y, ˜ z) :=
a(x0 + 2Ry, Az) , Ap−1
for y ∈ B1 and z ∈ Rn . It is now easy to see that u˜ solves the equation − div a(y, ˜ D u) ˜ = μ. ˜ Moreover the new vector field a(·) ˜ satisfies assumptions (1.2) with s replaced by s/A (and ω(·) replaced by ωR (·) := ω(2R·), but in what follows the properties of ω(·) will not be important). This observation will be useful in a few lines, when reducing estimates on general balls to the case the ball in question is B1 . 1,p
Lemma 4.1. Under the assumptions of Theorem 1.1 let w ∈ u + W0 (B2R ) be as in (4.1); assume that p 2. Then the following inequality holds for a constant c ≡ c(n, p, ν): p |μ|(B2R ) 2(p−1) – V (Du) − V (Dw) dx c . R n−1
(4.3)
B2R
Proof. We start observing that by Remark 4.1, with x0 being the center of B2R , by taking A :=
|μ|(B2R ) R n−1
1 p−1
and w(y) ˜ :=
w(x0 + 2Ry) AR
(4.4)
it follows that div a(x, ˜ D w) ˜ = 0 and we may reduce ourselves to the case in which the following holds: B2R ≡ B1
and in turn
|μ|(B1 ) 1,
(4.5)
thereby proving that ˜ − Vs/A (D w) ˜ dy c(n, p, ν). – Vs/A (D u) B1
(4.6)
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Inequality (4.3) then follows by (4.6) scaling back to u and w and noting that ˜ − Vs/A (D w) ˜ dy cAp/2 . – Vs (Du) − Vs (Dw) dx = Ap/2 – Vs/A (D u) B2R
B1
Therefore, from now on we shall argue under the additional assumptions (4.5); it is here needless to remark that we may assume A > 0, otherwise the proof trivializes by the strict monotonicity of the vector filed a(·). For any integer k 0 we define the truncation operators Tk (t) := max −k, min{k, t} , Φk (t) := T1 t − Tk (t) , t ∈ R. (4.7) Since both u and v are solutions agreeing on ∂B1 , we test the weak formulation
a(x, Du) − a(x, Dw), Dϕ dx = ϕ dμ B1
(4.8)
B1
by ϕ ≡ Φk (u − w); using (2.3)–(2.4) and the bound in (4.5), we obtain V (Du) − V (Dw)2 + |Du − Dw|p dx c|μ|(B1 ) c,
(4.9)
Ck
where Ck := x ∈ B1 : k < u(x) − w(x) k + 1 ,
(4.10)
and c ≡ c(n, p, ν). By Hölder’s inequality, and the very definition of Ck , for k > 0 we find
V (Du) − V (Dw) + |Du − Dw|p/2 dx
Ck
c|Ck |
1 2
V (Du) − V (Dw)2 + |Du − Dw|p dx
1 2
Ck (4.9)
1 2
c|Ck |
c k
1 |u − w| dx q
q 2
2
,
(4.11)
Ck
where we choose q in order to satisfy 1
np . 2(n − 1)
Notice that this is possible since p 2. Still, again by Hölder’s inequality we have V (Du) − V (Dw) + |Du − Dw|p/2 dx c(n, p, ν). C0
(4.12)
(4.13)
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Using (4.11), (4.13), and (4.12), and finally Sobolev’s embedding theorem, we have V (Du) − V (Dw) + |Du − Dw|p/2 dx B1
=
V (Du) − V (Dw) + |Du − Dw|p/2 dx
C0
+
∞ V (Du) − V (Dw) + |Du − Dw|p/2 dx k=1C k
c+c
∞ 1 k=1
k
q 2
1
Ck
∞ 1 c+c kq
1
1
2
|u − w| dx q
k=1
2
B1
c+c
2
|u − w| dx q
q |Du − Dw|p/2 dx
p
(4.14)
.
B1
The constant c in the last line also depends on q. Observe now that by (4.12) it follows q/p < 1 and therefore applying Young’s inequality in (4.14) yields V (Du) − V (Dw) + |Du − Dw|p/2 dx c(n, p, ν) B1
from which (4.6) follows. The proof is complete by making a suitable choice of q in (4.12).
2
We now switch to the subquadratic case p 2, which involves a more delicate argument, and a scaling procedure with some non-standard quantities reflecting the behavior of p-Laplacian type operators for p 2. 1,p
Lemma 4.2. Under the assumptions of Theorem 1.2, let w ∈ u + W0 (B2R ) be as in (4.1); assume that 2 − 1/n < p 2. Then the following inequality holds for a constant c ≡ c(n, p, ν): 1
2−p |μ|(B2R ) p−1 |μ|(B2R ) – |Du − Dw| dx c |Du| + s dx + c . (4.15) – R n−1 R n−1 B2R
B2R
Proof. As in the proof of Lemma 4.1 we start by a preliminary reduction appealing to Remark 4.1. In this case we set
|μ|(B2R ) A := R n−1
1 p−1
|μ|(B2R ) + R n−1
2−p . – (|Du| + s) dx B2R
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Scaling as in Remark 4.1, and in particular using the notation established in (4.2), we observe that
2−p 1 ˜ + (s/A) dy |μ|(B ˜ 1 ) p−1 + |μ|(B ˜ 1 ) – |D u| B1
c(n, p) |μ|(B2R ) = A R n−1
1 p−1
2−p c(n, p) |μ|(B2R ) + – |Du| + s dx A R n−1 B2R
c(n, p). Therefore, up to scaling as in Remark 4.1, we may reduce the proof to the case in which B2R ≡ B1 and
2−p |Du| + s dy c |μ|(B1 ) + |μ|(B1 )
(4.16)
B1
holds for a constant c depending only on n and p, thereby ultimately reducing ourselves to prove that (4.17) – |Du − Dw| dx c B1
in turn holds for a new constant c depending only on n, p and ν. We start observing that the assumed lower bound p > 2 − 1/n allows to determine γ ∈ (0, 1) such that p > 2 − γ /n and therefore n(p − 1) > 1. n−γ
(4.18)
As for the proof of Lemma 4.1, we obtain V (Du) − V (Dw)2 dx c(n, p, ν) |μ|(B1 ) ,
(4.19)
Ck
where Ck is defined as in (4.10). For every integer k > 0 we have
p−1 V (Du) − V (Dw)2/p dx c|Ck | p
Ck
V (Du) − V (Dw)2 dx
1
p
Ck
c|Ck | c
k
1 |μ|(B1 ) p
p−1 p n 1 n−γ |μ|(B1 ) p |u − w| dx
p−1 p
n(p−1) p(n−γ )
Ck
(4.20)
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and, again by Hölder’s inequality
1 V (Du) − V (Dw)2/p dx c(n, p, ν) |μ|(B1 ) p .
C0
Therefore, keeping (4.18) in mind, we have
V (Du) − V (Dw)2/p dx
B1
=
∞ V (Du) − V (Dw)2/p dx +
V (Du) − V (Dw)2/p dx
k=1C k
C0 ∞ 1 c |μ|(B1 ) p + c k=1
k
n(p−1) p(n−γ )
∞ 1 p c |μ|(B1 ) + c k=1
1 k
∞ 1 p c |μ|(B1 ) + c k=1
1 c |μ|(B1 ) p + c
1
p−1 p
dx
1 |μ|(B1 ) p
Ck
1 p
n(p−1) n−γ
1 k
|u − w|
n n−γ
∞
p−1 |u − w|
n n−γ
p
dx
1 |μ|(B1 ) p
k=1C k
1 p
|u − w|
n(p−1) n−γ
n n−γ
p−1 p
dx
1 |μ|(B1 ) p
B1
n(p−1) |Du − Dw| dx
p(n−γ )
1 |μ|(B1 ) p .
(4.21)
B1
In the last estimate the constant obviously depends on γ too. In turn, let us write |Du − Dw| =
p−2 1 |Du|2 + |Dw|2 + s 2 2 |Du − Dw|2 2 2−p · |Du|2 + |Dw|2 + s 2 4
2−p cV (Du) − V (Dw) · |Du|2 + |Dw|2 + s 2 4 2−p 2−p 2−p cV (Du) − V (Dw) · |Du − Dw| 2 + |Du| 2 + s 2 ,
(4.22)
where in the second-last line we used (2.2). Therefore, when p = 2, using Young’s in the form
ab we gain
2−p 2
pε
p−2 p
2
a 2/p
+
(2 − p)εb , 2
ε ∈ (0, 1)
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2/p |Du − Dw| cV (Du) − V (Dw) + (1/2)|Du − Dw| 2−p + cV (Du) − V (Dw) |Du| + s 2
(4.23)
and therefore 2/p 2−p |Du − Dw| cV (Du) − V (Dw) + cV (Du) − V (Dw) |Du| + s 2 . By using this last estimate together with Hölder’s inequality we get
V (Du) − V (Dw)2/p dx
|Du − Dw| dx c B1
B1
+c
V (Du) − V (Dw)2/p dx
p
|Du| + s dx
2
B1
2−p 2
.
(4.24)
B1
In turn, combining (4.24) with (4.21) yields
1 |Du − Dw| dx c |μ|(B1 ) p + c
B1
n(p−1)
p(n−γ )
|Du − Dw| dx
1 |μ|(B1 ) p
B1
+c
2−p 1 2 |Du| + s dx |μ|(B1 ) B1
2−p 1
n(p−1) 2 2(n−γ ) |Du| + s dx + c |μ|(B1 ) |Du − Dw| dx B1
B1
c+c
n(p−1)
p(n−γ )
|Du − Dw| dx
+c
B1
n(p−1) |Du − Dw| dx
2(n−γ )
(4.25)
B1
and, keeping in mind (4.16) and the fact that p 2, ultimately
|Du − Dw| dx c + c B1
n(p−1) |Du − Dw| dx
p(n−γ )
.
(4.26)
B1
Now observe that since p 2 n we have n(p − 1) n(p − 1) < 1 p(n − γ ) p(n − 1) so that (4.17) follows from (4.26) applying Young’s inequality. The proof is complete.
(4.27) 2
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Remark 4.2 (Stabilization of the constants II). The dependence of the constants in (4.2) is stable in the sense that letting p 2 in (4.2) we obtain the estimate c|μ|(B2R ) – |Du − Dw| dx , R n−1 B2R
and in fact the proof is such that the previous estimate can be obtained taking p = 2. The stability of the constant follows in particular by the use of Young’s inequality in (4.23) and (4.26). 4.2. Decay estimates Here we prove a decay estimate for solutions to (1.1) which is obtained using the comparison estimates of the previous section. With w been defined in (4.1) – and keeping the ball B2R ⊂ Ω fixed as specified at the begin1,p ning of the section – we define v ∈ w + W0 (BR ) as the unique solution to the homogeneous Dirichlet problem
div a(x0 , Dv) = 0 in BR , v=w on ∂BR ,
(4.28)
and prove yet another comparison estimate. We remark that BR is concentric to B2R . This time we start by the case p 2. Lemma 4.3. Under the assumptions of Theorem 1.1, with w as in (4.1) and v as in (4.28), there exists a constant c ≡ c(n, p, ν, L, L1 ) such that the following inequality holds: 1
2−p |μ|(B2R ) p−1 |μ|(B2R ) |Du| + s dx + c – |Du − Dv| dx c – R n−1 R n−1 BR
+ cω(R) – |Du| + s dx.
B2R
(4.29)
B2R
Proof. We start proving that the inequality – |Dw − Dv| dx cω(R) – |Dw| + s dx BR
(4.30)
B2R
holds for a constant c depending only on n, p, ν, L, L1 . Indeed by [13, Theorem 6.1] and assumptions (1.2) we have that v is a Q-minimizer of the functional z ∈ W 1,p (BR ) → BR
p |Dz| + s dx
(4.31)
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for some Q ≡ Q(n, p, ν, L) 1, and therefore p p |Dw| + s dx. |Dv| dx Q BR
(4.32)
BR
Moreover, a well-known version of Gehring’s lemma applies to w here – see for instance the version presented in [13, Theorem 6.7] – and leads to find a constant χ0 ≡ χ0 (n, p, ν, L) > 1 such that the reverse Hölder type inequality
1/χ0 χ0 p p dx c – |Dw| + s dx – |Dw| + s B/2
B
holds whenever B ⊆ B2R (this time not necessarily concentric to BR ) for a constant c depending only on n, p, ν, L. In turn, applying Lemma 3.2 with g ≡ (|Dw| + s)p , leads to establish that also
1/p p (4.33) c – |Dw| + s dx – |Dw| + s dx BR
B2R
holds. Now, using (2.2) and eventually (2.3), the fact that both v and w are solutions, (1.2)3 and again Young’s inequality, we have p−2 |Dv|2 + |Dw|2 + s 2 2 |Dw − Dv|2 dx BR
c
V (Dw) − V (Dv)2 dx
BR
c
a(x0 , Dw) − a(x0 , Dv), Dw − Dv dx
BR
=c
a(x0 , Dw) − a(x, Dw), Dw − Dv dx
BR
cL1 ω(R)
p−1 |Dw|2 + s 2 2 |Dw − Dv| dx
BR
cL1 ω(R)
1 2
p−1 |Dv|2 + |Dw|2 + s 2 2 |Dw − Dv| dx
BR
p−2 |Dv|2 + |Dw|2 + s 2 2 |Dw − Dv|2 dx
BR
2 + c L1 ω(R)
BR
p |Dv|2 + |Dw|2 + s 2 2 dx.
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Therefore, using again (2.2) we gain p V (Dw) − V (Dv)2 dx c ω(R) 2 |Dv|2 + |Dw|2 + s 2 dx, BR
BR
and by (4.32) also
V (Dw) − V (Dv)2 dx c ω(R) 2
BR
p |Dw| + s dx
(4.34)
BR
for c ≡ c(n, p, ν, L, L1 ). Similarly to (4.22) we now have p p(2−p) |Dw − Dv|p cV (Dw) − V (Dv) |Dv|2 + |Dw|2 + s 2 4 and therefore using the last estimate, (4.32) and Hölder’s inequality in (4.34) yields
p
2−p 2 2 2 p – |Dw − Dv|p dx c – V (Dw) − V (Dv) dx – |Dv|2 + |Dw|2 + s 2 2 dx BR
BR
p p c ω(R) – |Dw| + s dx.
BR
(4.35)
BR
In turn, using first Hölder’s inequality, (4.35) and finally (4.33) we have
1/p p – |Dw − Dv| dx c – |Dw − Dv| dx BR
BR
1/p p cω(R) – |Dw| + s dx BR
cω(R) – |Dw| + s dx, B2R
so that the proof of (4.30) follows. Using (4.30) together with (4.15) we have 1
2−p |μ|(B2R ) p−1 |μ|(B2R ) – |Du − Dv| dx c +c – |Du| + s dx R n−1 R n−1 BR
+ cω(R) – |Dw| + s dx
B2R
B2R
and using again (4.15) to estimate the last integral in the previous inequality (and recalling that ω(R) 1) we finally conclude with (4.29). 2
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We proceed with the case p 2. Lemma 4.4. Let u ∈ C 1 (Ω) be as in Theorem 1.1, then there exist constants β ∈ (0, 1], c1 1, depending on n, p, ν, L and c 1 depending on n, p, ν, L, L1 such that the following estimate holds whenever B ⊆ BR ⊆ B2R ⊆ Ω are concentric balls: n β 1 R |μ|(B2R ) p−1 – Du − (Du)B dx c1 – Du − (Du)B2R dx + c R R n−1 B
B2R
2−p n |μ|(B2R ) R +c – |Du| + s dx R n−1 B2R
n R +c ω(R) – |Du| + s dx .
(4.36)
B2R
Proof. We report the simple proof for the sake of completeness. Starting by B2R we define the comparison functions v and w as in (4.28) and (4.1), respectively. Then we compare Du and Dv by mean of (4.29), using (3.19) as basic reference estimate for v, that we eventually transfer to u: – Du − (Du)B dx 2 – Du − (Dv)B dx B
B
2 – Dv − (Dv)B dx + 2 – |Du − Dv| dx B
B
n β R c – Dv − (Dv)BR dx + c – |Du − Dv| dx R BR
BR
n β R c – Du − (Du)B2R dx + c – |Du − Dv| dx. R B2R
BR
In order to get (4.36) it is now sufficient to estimate the last integral in the previous inequality by mean of (4.29). 2 We now give the suitable version of the last two lemmata in the case p 2; this involves the use of the V (·)-map. Lemma 4.5. Under the assumptions of Theorem 1.2, with v as in (4.28) and w as in (4.1), there exists a constant c ≡ c(n, p, ν, L, L1 ) such that the following inequality holds: p |μ|(B2R ) 2(p−1) V (Du) + s p/2 dx. + cω(R) – – V (Du) − V (Dv) dx c R n−1 BR
B2R
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Proof. The proof is a modification of that of Lemma 4.3. We restart from (4.34) – that holds for p 2 as well – then, using Hölder’s inequality we have
1/2 2 – V (Dw) − V (Dv) dx c – V (Dw) − V (Dv) dx BR
BR
1/2 p cω(R) – |Dw| + s dx . BR
Applying Lemma 3.2 with g ≡ (|Dw| + s)p , leads to
1/p
2/p p p/2 c – |Dw| + s dx – |Dw| + s dx BR
B2R
holds. Combining the last two inequalities we obtain p/2 dx. – V (Dw) − V (Dv) dx cω(R) – |Dw| + s BR
(4.37)
B2R
In turn, using Young’s inequality we observe that when p 2 it holds that |z|p/2 V (z)
(4.38)
and therefore (4.3) yields – |Dw|p/2 dx – V (Dw) − V (Du) dx + – V (Du) dx B2R
B2R
c
|μ|(B2R ) R n−1
B2R
p 2(p−1)
+ – V (Du) dx. B2R
Combining the last estimate with (4.37) yields p |μ|(B2R ) 2(p−1) – V (Dw) − V (Dv) dx c + cω(R) – V (Du) + s p/2 dx, R n−1 BR
and the proof is complete.
B2R
2
The next lemma can be now obtained as Lemma 4.4 using Lemma 4.5 in place of Lemma 4.3, and the decay estimate (3.17) in place of (3.19).
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Lemma 4.6. Let u ∈ C 1 (Ω) be as in Theorem 1.2, then there exists constants β ∈ (0, 1], c1 1, depending on n, p, ν, L, α and c 1 depending on n, p, ν, L, α, L1 such that the following estimate holds whenever B ⊆ BR ⊆ B2R ⊆ Ω are concentric balls it holds that: – V (Du) − V (Du) B dx
B
c1
β – V (Du) − V (Du) B dx 2R R B2R
n 1 n R |μ|(B2R ) p−1 R +c +c ω(R) – V (Du) + s p/2 dx. R n−1
(4.39)
B2R
5. Proof of Theorem 1.1 In the rest of the proof all the balls will be concentric and centered at the point x ∈ Ω identified by the statement of the theorem; all of them will be contained in Ω. In particular we start with a ball B(x, 2R) ≡ B2R ⊂ Ω as in the statement of the Theorem. All the radii R will be such ˜ where the quantity R˜ > 0 will be chosen along the proof in dependence of the data that R R, n, p, ν, L, L1 , ω(·). The main point of the proof is to show how the peculiar quantity (1.20) appearing in the right-hand side of (4.15) can be reabsorbed in a way that make the Riesz potential appear, along the iteration/summation procedure. Step 1: A preliminary estimate. Referring to estimate (4.36), we select an integer H ≡ H (n, p, ν, L) 1 large enough to have c1
1 H
β
1 . 4
(5.1)
Applying (4.36) on arbitrary balls B ≡ BR/2H ⊆ BR/2 ⊂ BR and using the fact that ω(·) is non-decreasing we gain 1 – Du − (Du)BR/2H dy – Du − (Du)BR dy 4 BR/2H
BR
+ c2
|μ|(BR ) R n−1
1 p−1
+ c2
|μ|(BR ) R n−1
+ c2 ω(R) – |Du| + s dy,
2−p – |Du| + s dy BR
(5.2)
BR
where c2 depends only on n, p, ν, L, L1 , H and therefore ultimately on n, p, ν, L, L1 . We reduce the value of R˜ – in a way that makes it depending only on n, p, ν, L, L1 and ω(·) – to get ˜ 1, c2 ω(R) 4
(5.3)
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and using some further elementary estimates we gain 1 – Du − (Du)BR/2H dy – Du − (Du)BR dy 2 BR/2H
BR
+ c2
|μ|(BR ) R n−1
1 p−1
+ c2
|μ|(BR ) R n−1
2−p – |Du| + s dy BR
+ c2 ω(R) (Du)BR + s .
(5.4)
With the ball B(x, 2R) ⊆ Ω being fixed at the beginning and in the statement of Theorem 1.2, for i ∈ {0, 1, 2, . . .}, let us define Bi := B x, R/(2H )i =: B(x, Ri ) and ki := (Du)Bi ,
(5.5)
Ai := – Du − (Du)Bi dy.
(5.6)
and
Bi
For every integer m ∈ N we define and estimate km+1 =
m m (ki+1 − ki ) + k0 – Du − (Du)Bi dy + k0 i=0
i=0 B
i+1
(2H )n
m
Ai + k0 .
(5.7)
i=0
To estimate the right-hand side in (5.7) we observe that (5.4) used with R ≡ Ri−1 yields, whenever i 1 1
2−p |μ|(Bi−1 ) p−1 |μ|(Bi−1 ) 1 Ai Ai−1 + c2 + c2 – |Du| + s dy n−1 n−1 2 Ri−1 Ri−1 Bi−1
+ c2 ω(Ri−1 )(ki−1 + s). Summing up over i ∈ {1, . . . , m} the previous inequality yields m i=1
m−1 m−1 |μ|(Bi ) p−1 1 Ai + c2 2 Rin−1 1
Ai
i=0
+ c2
i=0
m−1 i=0
|μ|(Bi ) Rin−1
2−p m−1 + c2 ω(Ri )(ki + s), – |Du| + s dy Bi
i=0
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and therefore m
Ai A0 + 2c2
i=1
m−1
|μ|(Bi )
m−1 i=0
1 p−1
Rin−1
i=0
+ 2c2
|μ|(Bi ) Rin−1
2−p m−1 + 2c2 ω(Ri )(ki + s). – |Du| + s dy
(5.8)
i=0
Bi
Using the last inequality in (5.7) yields, for every integer m 1 km+1 cA0 + ck0 + c
m−1 i=0
+c
m−1 i=0
|μ|(Bi ) Rin−1
|μ|(Bi )
1 p−1
Rin−1
2−p m−1 +c ω(Ri )(ki + s), – (|Du| + s) dy Bi
(5.9)
i=0
and the constant c depends only on n, p, ν, L, L1 – keep in mind the dependence of H . Step 2: A conditional estimate. This Step is dedicated to the proof of an estimate that holds provided in turn a certain pointwise bound holds as well, and the radius R˜ is further reduced. This is in the following: Lemma 5.1. Assume that there exists an integer m ˜ ∈ N ∪ {∞} such that m ˜ 1 and ˜ − 1. – |Du| dy Du(x) holds whenever 0 i m
(5.10)
Bi
Then for every ε ∈ (0, 1) there exists a constant c˜ ≡ c(ε) ˜ 1 such that km 2c4 M + 2c3 ε Du(x)
(5.11)
˜ Here c3 , c4 1 and R˜ > 0 are constants holds whenever m m ˜ + 1 and provided R R. depending only on n, p, ν, L, L1 , while |μ| 1 ˜ I1 (x, 2R) p−1 . (5.12) M ≡ M(ε) := – |Du| + s dy + 1 + c3 c(ε) BR
Proof. By (5.10), whenever 1 m m ˜ it holds 1 m−1 |μ|(Bi ) p−1 km+1 c A0 + k0 + Rin−1 i=0
m−1 |μ|(Bi ) 2−p m−1 + c + c Du(x) + s 2−p ω(Ri )(ki + s). n−1 Ri i=0 i=0
(5.13)
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We now notice that m−1
|μ|(Bi )
i=0
Rin−1
∞ |μ|(Bi ) i=0
2n−1 log 2
Rin−1 2R
|μ|(B(x0 , )) d n−1
R ∞ Ri |μ|(B(x0 , )) d (2H )n−1 + log 2H n−1 i=0R i+1
|μ|
c(H )I1 (x, 2R).
(5.14)
Moreover, using the elementary inequality (1.17) with q = 1/(p − 1) – notice that q 1 as we are here assuming that p 2 – together with (5.14), we also have that m−1 i=0
|μ|(Bi )
1 p−1
Rin−1
∞ |μ|(Bi )
1 p−1
Rin−1 i=0
|μ| 1 c(H ) I1 (x, 2R) p−1
(5.15)
holds. Finally, as ω(·) is non-decreasing, we have m−1 i=0
ω(Ri )
∞ i=0
2R d = c(H )d(2R), ω(Ri ) c(H ) ω()
(5.16)
0
where the quantity d(·) has been defined in (1.3). We now further reduce R˜ in order to have that ˜ 1 so that d(2R) d(2R) 1.
(5.17)
Moreover, we record the elementary estimates A0 + k0 + d(2R)s 3 – |Du| + s dy,
(5.18)
BR
k1 2n H n
– |Du| dy cM
(5.19)
BR
where the constant c again depends only on n, p, ν, L since H depends on such quantities. Notice that although M in (5.12) has not been fully defined, (5.19) holds for any M having the structure in (5.12). Connecting (5.14)–(5.19) to (5.13) now yields
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|μ| 1 km+1 c – |Du| + s dy + c I1 (x, 2R) p−1 BR m−1 2−p |μ| + c3 I1 (x, 2R) Du(x) + s 2−p + c ω(Ri )(ki + s),
(5.20)
i=0
that holds whenever 1 m m ˜ and for constants c, c3 depending on n, p, ν, L and L1 . In order to estimate the terms appearing in the right-hand side of (5.20), when p < 2 we apply Young’s inequality with conjugate exponents 1/(2 − p) and 1/(p − 1), and with ε ∈ (0, 1) (to be chosen towards the end of the proof) we gain 2−p |μ| 1 |μ| c(ε) ˜ I1 (x, 2R) p−1 + (2 − p)ε Du(x) I1 (x, 2R)Du(x)
(5.21)
where p−2
c(ε) ˜ := (p − 1)ε p−1
(5.22)
|μ| 1 |μ| I1 (x, 2R)s 2−p c I1 (x, 2R) p−1 + s.
(5.23)
and similarly
At this stage ε is still a free parameter to be chosen later and affecting the constant c(ε) ˜ in (5.22). Now, with M defined as in (5.12), incorporating c(ε) ˜ introduced in (5.22), using (5.18), we have that (5.20) gives that km+1 c4 M + c5
m−1
ω(Ri )ki + c3 ε Du(x)
(5.24)
i=0
holds whenever 1 m m, ˜ where the new constants c4 , c5 1 also depend only on n, p, ν, L, L1 . Now we come to the induction argument and we determine the value of R˜ by further reducing it; indeed we take R˜ such that ˜ min 1/(8c5 ), 1/(8c4 ), 1/(8c3 ) . (5.25) d(2R) ˜ that nevertheless Notice that the previous choice determines a smaller value of the radius R, can be chosen in a way that makes it depending only on n, p, ν, L, L1 and ω(·) since c3 , c4 , c5 depends only on n, p, ν, L, L1 . In order to complete the proof of the lemma, we recall (5.18) and then prove that the following inequality holds whenever 0 i m ˜ + 1: (5.26) ki 2c4 M + 2c3 ε Du(x), where c3 , c4 are again the constants appearing in (5.24) and depending only on n, p, ν, L, L1 . The proof is of course by induction. The cases i = 0, 1 simply follow from (5.18)–(5.19). Next, we assume the validity of (5.26) for every i m with 1 m m, ˜ and prove it for i = m + 1. By
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using estimate (5.24) and the induction assumption (5.26) for i m − 1, and finally using also (5.25), we have km+1 c4 M + 2c4 c5 M
m−1
ω(Ri ) + c3 ε + 2c5 c3 ε
i=0
m−1
ω(Ri ) Du(x)
i=0
c4 M + 2c4 c5 d(2R)M + c3 ε 1 + 2c4 c5 d(2R) Du(x) 2c4 M + 2c3 ε Du(x).
(5.27)
This is (5.26) for i = m + 1 so that by induction (5.26) holds whenever i m ˜ + 1. The proof of Lemma 5.1 is now complete. 2 Step 3: Alternatives. We define the set S := i ∈ N: Du(x) – |Du| dy , Bi
and distinguish two cases. Case 1: S = N. In this case we have that – |Du| dy Du(x)
for every i ∈ N
Bi
and therefore we may apply Lemma 5.1 with m ˜ = ∞. In particular this gives that km 2c4 M + 2c3 ε Du(x), holds whenever m ∈ N, where M is defined in (5.12) and ε ∈ (0, 1) is still a free parameter affecting M via the constant c(ε) ˜ defined in (5.22). Now, letting m → ∞ in the previous inequality, and recalling that Du is here assumed to be continuous, yields Du(x) = lim km 2c4 M + 2c3 ε Du(x). (5.28) m→∞
We now choose ε = 1/(4c3 ) so the previous inequality gives Du(x) 4c4 M.
(5.29)
We now notice that since c3 only depends on n, p, ν, L1 we have that so is the dependence of ε and therefore of the (large) constant c(ε) ˜ appearing in the definition of the quantity M in (5.12) and in (5.22). All in all, using (5.12) in (5.29) we have proved that 1 Du(x) c – |Du| + s dy + c I|μ| (x, 2R) p−1 (5.30) 1 B(x,R)
˜ where R˜ in turn holds for a constant c depending only on n, p, ν, L, L1 , whenever R R, depends only on n, p, ν, L, L1 , ω(·). By obviously changing the radius – see Step 4 below –
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˜ We shall remove this restriction later, finally obtainestimate (5.30) implies (1.10) when R R. ing the validity of (1.10) for every ball B(x, 2R) ⊂ Ω, but with a new constant that depends on n, p, ν, L, L1 and ω(·), as prescribed in the statement of the theorem. This will be done in Step 4, at the end of the proof. We now proceed with the proof; our aim is to prove (5.30) in the case S = N. Case 2: S = N. Then we let m ˜ := min(N \ S) 0; this means that Du(x) < – |Du| dy (5.31) Bm˜
and – |Du| dy Du(x),
whenever 0 i m ˜ − 1,
(5.32)
Bi
with the last estimate that holds whenever m ˜ > 0. We further distinguish two cases; the first is when m ˜ = 0; this means that |Du(x)| < (|Du|)B0 and therefore (5.30) trivially follows. The other case is when m ˜ 1; we then use (5.31) as follows: Du(x) < – |Du| dy Bm˜
– Du − (Du)Bm˜ dy + (Du)Bm˜ Bm˜
= Am˜ + km˜ .
(5.33)
Next, we use Lemma 5.1 that gives (5.11) and in particular km˜ 2c4 M + 2c3 ε Du(x),
(5.34)
with ε ∈ (0, 1) free to be chosen, affecting M in the way described in (5.12). On the other hand combining (5.8) and (5.32) and again using (5.11) gives Am˜ A0 + c
m−1 ˜ i=0
|μ|(Bi ) Rin−1
1 p−1
˜ 2−p m−1 |μ|(Bi ) + c Du(x) + s 2−p Rin−1 i=0
˜ m−1 + c 2c4 M + 2c3 ε Du(x) + s ω(Ri ). i=0
Again using (5.14)–(5.16) and (5.18) in the previous estimate yields 2−p μ 1 μ Am˜ c – |Du| + s dy + c I1 (x, 2R) p−1 + cI1 (x, 2R) Du(x) + s 2−p BR
+ cd(2R) 2c4 M + 2c3 ε Du(x) + s ,
(5.35)
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where c ≡ c(n, p, ν, L, L1 ). Estimating as in (5.21)–(5.23) in (5.35), and using (5.25), we have Am˜ cM + cε Du(x)
(5.36)
for yet a new constant c depending only on n, p, ν, L, L1 and where M is defined accordingly to (5.29). Using (5.36) and (5.34) in (5.33) finally gives Du(x) cM + cε Du(x) for c ≡ c(n, p, ν, L, L1 ). Choosing ε = 1/(2c) again gives |Du(x)| cM and recalling the definition of M in (5.12) we once again obtain (5.30), which is valid under the same conditions of ˜ the Case 1, that is, provided R R. ˜ We finally prove estimate (1.10) also in the case Step 4: Getting rid of the condition R R. ˜ Take a ball BR ≡ B(x, R) ⊂ Ω, with R > R; ˜ (5.30) gives R > R. Du(x) c
–
|μ| 1 ˜ p−1 |Du| + s dy + c I1 (x, R)
˜ B(x,R/2)
and then we estimate
c
– ˜ B(x,R/2)
n n R |Du| + s dy c2 R˜
– |Du| + s dy B(x,R)
diam(Ω) c R˜
n
– |Du| + s dy B(x,R)
and, trivially |μ|
|μ|
˜ I (x, R), I1 (x, R) 1 so that (1.10) follows with a new constant – c/R˜ n instead of c – which depends on ˜ which has been n, p, ν, L, L1 , diam(Ω), and additionally on ω(·) due to the presence of R, ˜ previously determined by choosing ω(R) suitably small. The proof is complete. Remark 5.1 (Stability of the constants in (1.10)). The proof of Theorem 1.1 catches the case p = 2, and therefore it reduces to the classical estimate known for solutions to the Poisson equation − u = μ, that is (1.12), an estimate that has been proved in [27,8] for nonlinear equations. The constant c in estimate (1.10) for p < 2 remains bounded when p 2. This is a consequence of the stability of the constants observed in Remarks 3.2 and 4.2 and, as far as the proof of Theorem 1.1 is concerned, of the dependence on p of the constant c(ε) ˜ in (5.22), which is stable as p 2.
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6. Proof of Theorem 1.2 The proof is similar to that of Theorem 1.1, being actually much simpler as no mixed quantity of the type (1.20) shows up in the right-hand side of (4.39) and consequently no alternative – i.e. Case 1 and Case 2 in the proof of Theorem 1.2 – is needed. A minor difference will occur in that the constants involved will exhibit an additional dependence on the number α introduced in (1.13). We shall therefore confine ourselves to give just a sketch of the proof. After choosing H exactly as in (5.1) – but this time referring to Lemma 4.6 for the constants c1 and β – we everywhere consider the quantity dy – V (Du) − V (Du) B R/2H
BR/2H
in place of the “linear excess” – Du − (Du)BR/2H dy. BR/2H
We choose the balls as in (5.5) while now we define ki := V (Du) B and Ai := – V (Du) − V (Du) B dy. i
i
(6.1)
Bi
In other words, instead of dealing with averages of Du as in the proof of Theorem 1.1, we deal with averages of V (Du). Proceeding as in the proof of Theorem 1.1 we arrive at the following analog of (5.9): km+1 cA0 + ck0 + c
m−1
|μ|(Bi )
p 2(p−1)
Rin−1
i=0
+c
m−1
ω(Ri )(ki + s).
(6.2)
i=0
Exactly as in (5.14) we observe that m−1
|μ|(Bi )
i=0
Rin−1
p 2(p−1)
p(n−1)
2 2(p−1) log 2
2R
|μ|(B(x0 , )) n−1
p 2(p−1)
d
R p(n−1) p ∞ Ri |μ|(B(x0 , )) 2(p−1) d (2H ) 2(p−1) + log 2H n−1
i=0R i+1
2R c(H ) 0
Defining this time
|μ|(B(x, )) n−1
p 2(p−1)
d .
(6.3)
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p 2R |μ|(B(x, )) 2(p−1) d p/2 dy + M := – V (Du) + s n−1 BR
0
p/2 dy + c – |Du| + s BR
2R
|μ|(B(x, )) n−1
p 2(p−1)
d ,
0
and taking (6.3) into account, we have that (6.2) implies km+1 c4 M + c5
m−1
ω(Ri )ki
(6.4)
i=0
for constants c4 , c5 depending only on n, p, ν, L, α, L1 . Arguing as in the proof of Theorem 1.1 we then prove by induction that km 2c4 M holds for every m ∈ N; indeed, observe that this time no alternative as in Step 3 of Theorem 1.1 occurs and the induction of Lemma 5.1 can be performed without assuming (5.10). Therefore, by (4.38), we conclude observing that Du(x)p/2 |V Du(x) | lim km 2c4 M. m→∞
The previous relation proves (1.15) whenever R R˜ and R˜ is a fixed radius depending on n, p, ν, L, L1 , and found as in the proof of Theorem 1.1. The general case R > 0 follows as in Step 4 of Theorem 1.1. Acknowledgments This paper is supported by the ERC grant 207573 “Vectorial Problems”. The authors kindly thank the referee for the very careful reading of a preliminary version of the manuscript. References [1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259. [2] L. Boccardo, Elliptic and parabolic differential problems with measure data, Boll. Unione. Mat. Ital. Sez. A (7) 11 (1997) 439–461. [3] L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989) 149–169. [4] L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations 17 (1992) 641–655. [5] V. Bögelein, J. Habermann, Gradient potential estimates for problems with non-standard growth, Ann. Acad. Sci. Fenn. Ser. A I Math., in press. [6] E. DiBenedetto, J.J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993) 1107–1134. [7] F. Duzaar, G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differential Equations 20 (2004) 235–256. [8] F. Duzaar, G. Mingione, Gradient estimates via nonlinear potentials, Amer. J. Math., in press. [9] F. Duzaar, G. Mingione, Gradient estimates in nonlinear potential theory, Rend. Lincei Mat. Appl. 20 (2009) 179– 190. [10] F. Duzaar, G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linèaire, DOI:10.1016/j.anihpc.2010.07.002.
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[11] F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differential Equations, DOI:10.1007/s00526-010-0314-6. [12] L. Esposito, F. Leonetti, G. Mingione, Regularity results for minimizers of irregular integrals with (p, q) growth, Forum Math. 14 (2002) 245–272. [13] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. [14] T. Jin, V. Mazya, J. Van Schaftingen, Pathological solutions to elliptic problems in divergence form with continuous coefficients, Compt. Rend. Math. 347 (2009) 773–778. [15] C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. (Crelles J.) 431 (1992) 7–64. [16] L.I. Hedberg, T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (1983) 161–187. [17] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford. Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1993. [18] N.J. Kalton, I.E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999) 3441–3497. [19] T. Kilpeläinen, Hölder continuity of solutions to quasilinear elliptic equations involving measures, Potential Anal. 3 (1994) 265–272. [20] T. Kilpeläinen, J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172 (1994) 137–161. [21] J. Kinnunen, S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999) 2043–2068. [22] D.A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J. 111 (2002) 1–49. [23] G.M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991) 311–361. [24] P. Lindqvist, On the definition and properties of p-superharmonic functions, J. Reine Angew. Math. (Crelles J.) 365 (1986) 67–79. [25] G. Mingione, The Calderón–Zygmund theory for elliptic problems with measure data, Ann Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007) 195–261. [26] G. Mingione, Gradient estimates below the duality exponent, Math. Ann. 346 (2010) 571–627. [27] G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., in press. [28] G. Mingione, Towards a nonlinear Calderón–Zygmund theory, Quad. Mat. 23 (2009) 371–458. [29] G. Mingione, Nonlinear aspects of Calderón–Zygmund theory, Jahresber. Deutsch. Math.-Verein., DOI:10.1365/s13291-010-0004-5. [30] N.C. Phuc, I.E. Verbitsky, Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, Comm. Partial Differential Equations 31 (2006) 1779–1791. [31] N.C. Phuc, I.E. Verbitsky, Quasilinear and Hessian equations of Lane–Emden type, Ann. of Math. (2) 168 (2008) 859–914. [32] N.C. Phuc, I.E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal. 256 (2009) 1875–1906. [33] C.A. Rogers, Hausdorff Measures, Cambridge University Press, London–New York, 1970. [34] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton Math. Ser., vol. 32, 1971. [35] N.S. Trudinger, X.J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002) 369–410.
Journal of Functional Analysis 259 (2010) 2999–3024 www.elsevier.com/locate/jfa
The chain rule as a functional equation Shiri Artstein-Avidan a,1 , Hermann König b,∗ , Vitali Milman a,2 a School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel b Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
Received 18 June 2010; accepted 7 July 2010 Available online 17 July 2010 Communicated by J. Bourgain
Abstract We consider operators T from C 1 (R) to C(R) satisfying the “chain rule” T (f ◦ g) = (Tf ) ◦ g · T g,
f, g ∈ C 1 (R),
and study under which conditions this functional equation admits only the derivative or its powers as solutions. We also consider T operating on other domains like C k (R) for k ∈ N0 or k = ∞ and study the more general equation T (f ◦ g) = (Tf ) ◦ g · Ag, f, g ∈ C 1 (R) where both T and A map C 1 (R) to C(R). © 2010 Elsevier Inc. All rights reserved. Keywords: Chain rule; Functional equation
1. Introduction and results The derivative D : C 1 (R) → C(R) satisfies the chain rule D(f ◦ g) = (Df ) ◦ g · Dg,
f, g ∈ C 1 (R).
* Corresponding author.
E-mail address:
[email protected] (H. König). 1 The author is supported by ISF grant 865-07 and by BSF grant 2006079. 2 The author is supported by the Alexander von Humboldt Foundation, by ISF grant 387/09 and by BSF grant 2006079.
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.002
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We show in this paper that the chain rule functional equation T (f ◦ g) = (Tf ) ◦ g · T g,
f, g, f ◦ g ∈ D(T )
(1)
characterizes the derivative, or a power of the derivative, under very mild conditions of nondegeneracy. Here D(T ) denotes the domain of the operation T and Im(T ) will denote its image. Various assumptions can be made on both and the answer depends on the choice of D(T ) and Im(T ). We mainly consider the natural case D(T ) = C 1 (R) and Im(T ) ⊂ C(R). Let us emphasize that no continuity assumptions on T are made; however, the continuity of T follows from (1). Also the linearity of T in the derivative is a consequence of (1) and some initial condition; it is not assumed a priori. In contrast, the Leibniz product rule (for a derivation) does not characterize the derivative; it is known that without conditions of continuity or linearity, a derivation does not necessarily give the derivative. The results of the paper may be seen in the light of some recent results on classical operations or transforms in Geometry and Analysis which were shown to be characterized by some very elementary and natural conditions like e.g. anti-monotonicity or multiplicativity. The Fourier transform was studied in [1,2], geometry duality and the Legendre transform in [3,4], mixed volumes in [7]. This paper is concerned with characterizations of the derivative by the chain rule. Let us remark that there are no non-trivial examples with (1) where D(T ) and Im(T ) are both equal to C(R); this will be shown at the end. In this paper, we mainly consider the “natural” setting D(T ) = C 1 (R), Im(T ) ⊂ C(R). In this case, the usual derivative D is surjective onto C(R), of course. In chapter 5 we give modifications for the setting D(T ) = C k (R) and C ∞ (R). Without further restrictions on T , there are examples very different from D which satisfy the functional equation (1). Consider a continuous positive function H : R → R>0 and define Tf :=
H ◦f , H
f ∈ C(R).
(2)
This defines a map of C 1 (R) or C(R) into C(R) satisfying (1). We are grateful to L. Polterovich for pointing out this example to us. Since H needs to be never zero to make (2) a valid example, this map T : C(R) → C(R) is not onto since no functions in its image have zeros. Another example of a map T : C 1 (R) → C(R) satisfying (1) is given by (Tf ) :=
f ,
f ∈ C 1 (R) bijective,
0,
f ∈ C 1 (R) not bijective
.
(3)
Checking (1), note that for f, g ∈ C 1 (R), f ◦ g : R → R is bijective if and only if f and g : R → R are bijective (f ◦ g being bijective implies first that f : Im(g) → R is bijective and thus strictly monotone; then f ∈ C 1 (R) yields Im(g) = R). Let Cb1 (R) = {f ∈ C 1 (R) | f bounded from above or from below (or both)} be the class of half-bounded (or bounded) functions. We call T : D(T ) = C 1 (R) → Im(T ) ⊂ C(R)
S. Artstein-Avidan et al. / Journal of Functional Analysis 259 (2010) 2999–3024
3001
non-degenerate if T |C 1 (R) = 0. This means explicitly b
∃x1 ∈ R, h1 ∈ Cb1 (R),
(T h1 )(x1 ) = 0.
(4)
We also consider two additional non-degeneracy conditions for T : ∃x0 ∈ R, h0 ∈ C 1 (R),
(T h0 )(x0 ) = 0
(4.1)
(T h− )(x− ) < 0.
(4.2)
and ∃x− ∈ R, h− ∈ C 1 (R),
Note that (4) is not satisfied in example (3) and that (4.1) does not hold in example (2). The map T : C 1 (R) → C(R), Tf := |f | also fulfills (1) but does not satisfy (4.2). Our first main result states Theorem 1. Let T : D(T ) := C 1 (R) → Im(T ) ⊂ C(R) be an operation satisfying the chain rule T (f ◦ g) = (Tf ) ◦ g · T g,
f, g ∈ D(T ).
(1)
Assume that T is non-degenerate in the sense of (4). Then there exists some p 0 and a positive continuous function H ∈ C(R) such that either ⎫ H ◦ f p ⎪ ⎪ f Tf = ⎪ ⎬ H or in the case of p > 0, (5) ⎪ ⎪ H ◦ f p ⎪ f sgn f . ⎭ Tf = H If in addition to (4) the non-degeneracy condition (4.1) is satisfied, we have p > 0, i.e. solutions of the type Tf = H ◦ f/H are excluded. If in addition to (4), T (2 Id) = c is a constant function, H is constant and therefore Tf = |f |p or Tf = |f |p sgn(f ), where p = log2 (c). Assuming also (4.2) and c = 2, the only solution to the chain rule equation is Tf = f . If (4.2) holds and only T (2 Id)(0) = 2 is satisfied, T is of the form Tf = HH◦f · f . Any map with (5) satisfies (1), of course. Remarks. (i) Take p > 0 and let G be the anti-derivative of H 1/p > 0, G = H 1/p . Then G is a continuously differentiable strictly monotone function, and we get the following alternative formulation of the result (G ◦ f ) p d(G ◦ f ) p Tf = = G dG or (G ◦ f ) p d(G ◦ f ) p sgn d(G ◦ f ) . Tf = sgn(f ) = dG G dG
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In this sense, all solutions of the chain rule for non-degenerate maps T are p-th powers of some derivative, up to signs. Under surjectivity conditions and the normalization T (2 Id) = 2, the usual derivative is the only solution. We would like to emphasize that no linearity or continuity conditions were imposed on T , but that they are a consequence of the answer. (ii) Note that if T1 and T2 are operations satisfying (1), also T1 · T2 is a solution of (1), and actually for any p > 0 also f → |T1 f |p ,
f → |T1 f |p sgn(T1 f ).
The first of these solutions, also in (5), is not (pointwise) surjective. (iii) Obviously, the non-degeneracy condition is necessary for (5) to hold. (iv) The form of an operator T : C 1 (R) → C(R) satisfying the chain rule (1) and the nondegeneracy conditions (4), (4.1) and (4.2) is actually determined completely by the function T (2 Id): as a restatement of Theorem 1 and its proof below let p := log 2 T (2 Id)(0) and x ϕ(x) := T (2 Id)(x)/T (2 Id)(0). For this ϕ ∈ C(R), the product H (x) := ∞ n=1 ϕ( 2n ) converges uniformly on compact subsets to a continuous function H with H (0) = 1 such that T is given by the formula p Tf = H ◦ f/H · f sgn f not only for f = 2 Id, but for all f ∈ C 1 (R). An analogue of Theorem 1 for C k (R) or C ∞ (R) functions will be given in the last chapter. L. Polterovich mentioned to us that there is a cohomological interpretation of Theorem 1. We give this as a remark after the proof of Theorem 1. As a matter of fact, the standard chain rule is to a certain extent enforced by a much weaker version: One may ask to what extent does the functional equation T (f ◦ g) = (Tf ) ◦ g · Ag,
f, f ◦ g ∈ D(T ), g ∈ D(A)
(6)
characterize the derivative if T : D(T ) = C 1 (R) → Im(T ) ⊂ C(R) and A : D(A) = C 1 (R) → Im(A) ⊂ C(R) are operations connected by (6). If T is non-degenerate, (6) implies that A(Id) = 1. The setting A := 1 would allow solutions not associated with derivatives like Tf := H ◦ f for some H ∈ C(R). Let us introduce the following strong non-degeneracy condition for T , (i)
∀x1 ∈ R, ∃h1 ∈ Cb1 (R), y ∈ R,
h1 (y) = x1 ∧ (T h1 )(y) = 0 ,
(ii)
∃x0 ∈ R, h0 ∈ C 1 (R),
(T h0 )(x0 ) = 0.
(7)
If T satisfies (7) and the operators T and A fulfill (6), we will show that A has the form Ag = T g · A1 ◦ g, g ∈ D(A) where A1 is a suitable function on R. If G ∈ C 1 (R) is strictly monotone and H ∈ C(R) is positive, for any p > 0, p Tf := (G ◦ f ) /H and
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p Tf := (G ◦ f ) /H · sgn f provide examples of operations T satisfying (6) if Ag = T g · A1 ◦ g
p where A1 = H /G .
In the case that H = |G |p , we come back to the solutions d(G ◦ f ) p , Tf = dG
d(G ◦ f ) p sgn dG ◦ f Tf = dG dG
of (1) mentioned previously. Differently from equation (1), the functional equation (6) is not stable under taking products of maps T1 , T2 satisfying (6) for fixed A. Our second main result is: Theorem 2. Let T : D(T ) = C 1 (R) → Im(T ) ⊂ C(R) and A : D(A) = C 1 (R) → Im(A) ⊂ C(R) be operations satisfying the functional equation T (f ◦ g) = (Tf ) ◦ g · Ag,
f, f ◦ g ∈ D(T ), g ∈ D(A).
(6)
Assume that T is strongly non-degenerate in the sense of (7). Then there exists p > 0 and there are positive functions G1 , G2 ∈ C(R) such that Tf = G1 ◦ f ·
G2 p · f G1
G2 p · f sgn f , G1 H ◦f ·K f , Af = Tf/G2 ◦ f = H or Tf = G1 ◦ f ·
p p 1 where H := G G2 and K(u) = |u| or K(u) = |u| sgn(u). Conversely, any such operations T and A satisfy (6).
Therefore the weak form (6) of the chain rule is actually only a small modification of the chain rule in the sense that T (f ◦ g) = (Tf ) ◦ g · T g/G2 ◦ g,
f, g ∈ C 1 (R).
Let us note that condition (7) of strong non-degeneracy of T is necessary for T and A to be of the above form. The example Tf := H ◦ f , H ∈ C(R) and A := 1 (in particular T = Id, A = 1) satisfies (6) but is not of the form given: this T does not fulfill (7). Namely (7)(i) would require that H has no zeros while (7)(ii) only holds if H has at least one zero. The condition (7) of strong non-degeneracy of T implies that A is non-degenerate in the sense of (4) if T and A are intertwined by satisfying (6): Applying (6) to h1 = Id ◦h1 for the h1
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in (7)(i), one finds (Ah1 )(y) = 0 for some y ∈ R and T (Id)(x) = 0 for all x ∈ R. Applying (6) then to h0 = Id ◦h0 for the h0 in (7)(ii) yields (Ah0 )(x0 ) = 0 since T (Id)(h(x0 )) = 0 holds. The following result was already indicated in the beginning of the introduction, concerning operators T on all continuous functions. Proposition 3. Assume that T : C(R) → C(R) satisfies the functional equation T (f ◦ g) = (Tf ) ◦ g · T g for all f, g ∈ C(R) and that there exist g0 ∈ C(R) and x0 ∈ R with (T g0 )(x0 ) = 0. Then T is zero on the class of half-bounded continuous functions, T |Cb (R) = 0. Remarks. Clearly, for any continuous function H : R → R>0 , T (f ) = HH◦f defines a non-trivial operation T : C(R) → C(R) with T (f ◦ g) = (Tf ) ◦ g · T g. However, Tf (x) = 0 for all f ∈ C(R) and all x ∈ R. Also, the example of T : C(R) → C(R) analogous to (3) 1, Tf := 0,
f ∈ C(R), f bijective, f ∈ C(R), f not bijective
shows that there are operations T with T (f ◦ g) = (Tf ) ◦ g · T g and T |C(R) = 0, T |Cb (R) = 0. 2. Localization results The derivative is a local operator. In order to prove our main results, we need to show a similar property for T . These localization results are the subject of this section. We assume throughout that the operator T satisfies the chain rule (1) and the non-degeneracy condition (4) and that D(T ) = C 1 (R),
Im(T ) ⊂ C(R).
We start with a strengthening of (4). Lemma 4. (i) For any x1 ∈ R there exists h1 ∈ Cb1 (R) with (T h1 )(x1 ) = 0. (ii) If additionally (4.1) holds, for any x0 ∈ R there exists h0 ∈ C 1 (R) with (T h0 )(x0 ) = 0. Proof. (i) Let x1 ∈ R. By (4) there exists x2 ∈ R and h2 ∈ Cb1 (R) with (T h2 )(x2 ) = 0. Define g, h1 ∈ Cb1 (R) by g(s) = s + x1 − x2 ,
h1 (s) = h2 (s + x2 − x1 ).
Then g(x2 ) = x1 , h2 = h1 ◦ g and by (1), 0 = (T h2 )(x2 ) = (T h1 )(x1 )(T g)(x2 ), and hence (T h1 )(x1 ) = 0. If h2 is actually bounded, also h1 is. (ii) Let x0 ∈ R. By (4.1) there exists x2 ∈ R and h2 ∈ C 1 (R) with (T h2 )(x2 ) = 0. Define g, h0 ∈ C 1 (R) by g(s) = s + x2 − x0 ,
h0 (s) = h2 (s + x2 − x0 ).
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Then g(x0 ) = x2 , h0 = h2 ◦ g and by (1), (T h0 )(x0 ) = (T h2 )(x2 )(T g)(x0 ) = 0.
2
The following lemmas and proofs contain statements about different sets of open intervals in R depending on whether T h = 0 for all bounded functions h or not. If – case (a) – there is a bounded function h ∈ C 1 (R) and x1 ∈ R with (T h)(x1 ) = 0, we allow all open intervals. In this case, let (a) I := {J ⊂ R | J open interval}. If – case (b) – there is only a half-bounded function, h ∈ C 1 (R), but no bounded function, and x1 ∈ R with (T h)(x1 ) = 0, the statements only hold for the smaller class of half-infinite open intervals (b) I := {J ⊂ R | J = (c, ∞) or (−∞, c) for some c ∈ R}. We use this class I as just defined throughout the rest of this section. Lemma 5. For any interval J ∈ I, any y ∈ J and any x ∈ R there exists g ∈ D(T ) such that g(x) = y, Im(g) ⊂ J and (T g)(x) = 0. Proof. Let J ∈ I, y ∈ J and x ∈ R. By (4) and Lemma 4, there exists a bounded [case (a)] or half-bounded but not bounded [case (b)] function h ∈ C 1 (R) with (T h)(x) = 0. Pick some open interval I ∈ I for which Im(h) ⊂ I (bounded in case (a), half-infinite in case (b)). Choose any bijective C 1 -map f : J → I with f (y) = h(x) [note h(x) ∈ I ]. Clearly, this may be done in such way that f is extendable to a C 1 -map f˜ on R, f˜|J = f . Let g := f −1 ◦ h : R → J ⊂ R. Then g ∈ D(T ), g(x) = y and Im(g) ⊂ J . Since h = f ◦ g = f˜ ◦ g, we find using (1), 0 = (T h)(x) = T (f˜ ◦ g)(x) = (T f˜)(y) · (T g)(x). Hence (T g)(x) = 0, g(x) = y and Im(g) ⊂ J are satisfied.
2
The next lemma is a localized version of the claim that under our conditions, the identity function is mapped to the constant function 1, and that the constant functions are mapped onto the function 0. Lemma 6. For any interval J ∈ I and f ∈ D(T ), c ∈ R the following holds: (i) If (4.1) is satisfied and f |J = c, we have (Tf )|J¯ = 0. (ii) If f |J = IdJ , we have (Tf )|J¯ = 1.
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Proof. (i) Take c ∈ R. We claim that T c = 0 for the constant function given by c. Indeed, if there were x ∈ R with (T c)(x) = 0, then c ◦ g = c for any g ∈ D(T ) and by (1), 0 = (T c)(x) = T (c ◦ g)(x) = (T c) g(x) (T g)(x). Thus (T g)(x) = 0 for all g ∈ D(T ) which contradicts (4.1). Hence T c = 0. For the local version of this, let J ∈ I, y ∈ J and x ∈ R. By Lemma 5 there is a g ∈ D(T ) with Im(g) ⊂ J , g(x) = y and (T g)(x) = 0. From f |J = c we get that f ◦ g = c, so we have by the preceding and (1) that 0 = (T c)(x) = T (f ◦ g)(x) = (Tf )(y)(T g)(x). Since (T g)(x) = 0, we must have (Tf )(y) = 0. Since y was arbitrary in J , we get (Tf )|J = 0. Since Tf is continuous, this also holds on the closure of J , (Tf )|J¯ = 0. (ii) Assume now that f |J = Id |J . Take any y ∈ J , x ∈ R. By Lemma 5 choose g ∈ D(T ) with Im(g) ⊂ J , g(x) = y and (T g)(x) = 0. Then f ◦ g = g so that by (1), 0 = (T g)(x) = (Tf )(y) · (T g)(x). We conclude that (Tf )(y) = 1. Hence (Tf )|J = 1, and by the continuity of Tf , (Tf )|J¯ = 1.
2
Lemma 7. In case (a) let J ∈ I be a bounded open interval and assume that f ∈ D(T ) is such that f |J is strictly monotone. Then (Tf )(x) = 0 for all x ∈ J . Proof. For I := f (J ), f |J : J → I is bijective. Restricting to subintervals J ∈ I if necessary, we may assume that both I, J are bounded. Then the inverse map g := (f |J )−1 : I → J may be extended to some g˜ ∈ D(T ). Then h := g ◦ f satisfies h|J = Id |J , and thus by Lemma 6, 1 = (T h)(x) = (T g) ˜ f (x) (Tf )(x) for any x ∈ J . Therefore (Tf )(x) = 0.
2
The following is the localization of the operation T on an interval J ∈ I. Lemma 8. Let J ∈ I and assume that f1 , f2 ∈ D(T ) satisfy f1 |J = f2 |J . Then (Tf1 )|J¯ = (Tf2 )|J¯ . Proof. Let x ∈ J be arbitrary. Choose a smaller interval J1 ⊂ J , J1 ∈ I and a function g ∈ C 1 (R) such that x ∈ J1 , Im g ⊂ J and g|J1 = Id |J1 . We then have f1 ◦ g = f2 ◦ g, and by Lemma 6, (T g)|J1 = 1. In particular, g(x) = x and (T g)(x) = 1. Hence, using (1), (Tf1 )(x) = (Tf1 ) g(x) = (Tf1 ) g(x) · (T g)(x) = T (f1 ◦ g)(x) = T (f2 ◦ g)(x) = (Tf2 ) g(x) · (T g)(x) = (Tf2 )(x). This shows that (Tf1 )|J = (Tf2 )|J holds. By the continuity of Tf1 and Tf2 , also (Tf1 )|J¯ = (Tf2 )|J¯ is true. 2
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The following lemma gives a converse result to case (i) of Lemma 6. Lemma 9. Let J ∈ I and assume that (4.1) holds and that g ∈ D(T ) satisfies (T g)|J = 0. Then g is constant on J : there exists c ∈ R such that g|J = c. We omit the proof here since it follows directly from Lemma 11 below. We now prove a “pure” localization result. Proposition 10. Assume that T : C 1 (R) → C(R) satisfies the chain rule (1) and is nondegenerate in the sense of (4). Then there is a function F : R3 → R such that Tf (x) = F x, f (x), f (x)
(8)
holds for any f ∈ C 1 (R) and any x ∈ R. Proof. Fix x0 ∈ R and consider f ∈ C 1 (R) with given values f (x0 ) and f (x0 ). Let J1 := (x0 , ∞) and J2 := (−∞, x0 ). Then J1 , J2 ∈ I. Consider the tangent of f at x0 , g(x) := f (x0 ) + (x − x0 )f (x0 ), x ∈ R. It suffices to prove that (Tf )(x0 ) = (T g)(x0 ). Define h ∈ C 1 (R) by h(x) :=
g(x), f (x),
x ∈ J1 , . x ∈ J¯2
Then h|J1 = g|J1 and h|J2 = f |J2 . Hence by Lemma 8, (T g)|J¯1 = (T h)|J¯1
and (T h)|J¯2 = (Tf )|J¯2 .
Since x0 ∈ J¯1 ∩ J¯2 , we conclude that (T g)(x0 ) = (T h)(x0 ) = (Tf )(x0 ). Therefore the value (Tf )(x0 ) depends only on the two parameters f (x0 ) and f (x0 ), for any fixed x0 ∈ R. We encode this information by letting (Tf )(x0 ) = Fx0 (f (x0 ), f (x0 )), where Fx0 : R2 → R is a fixed function for any x0 ∈ R. Finally denoting F (x, y, z) := Fx (y, z), we have that for any x ∈ R and f ∈ D(T ), Tf (x) = F x, f (x), f (x) .
2
3. The strong form of the chain rule In this section we give the proof of Theorem 1. We generally assume that D(T ) = C 1 (R), Im(T ) ⊂ C(R) and that T satisfies the chain rule and the condition (4) of non-degeneracy. Lemma 11. If in addition to (4) also (4.1) holds, for any g ∈ D(T ) and x ∈ R we have: (T g)(x) = 0 holds if and only if g (x) = 0. Remark. Lemma 9 is a direct corollary of this result.
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Proof. Let g ∈ D(T ) and x0 ∈ R. By (8) of Proposition 10, (Tf )(x) = F x, f (x), f (x) ,
x ∈ R, f ∈ D(T ).
Inserting a constant function f = c and using Lemma 6, we see that 0 = (Tf )(x) = F (x, c, 0) for any x, c ∈ R. This shows that g (x0 ) = 0 implies (T g)(x0 ) = 0. Assume now that (T g)(x0 ) = 0. We then have by the chain rule (1) for any f ∈ D(T ), F x0 , f g(x0 ) , f g(x0 ) g (x0 ) = T (f ◦ g)(x0 ) = (Tf ) g(x0 ) (T g)(x0 ) = 0. This implies that for any two numbers b, c ∈ R, F x0 , b, cg (x0 ) = 0. If g (x0 ) = 0 would hold, F (x0 , ·,·) : R2 → R would be the zero function, and hence (Tf )(x0 ) = 0 would follow for any f ∈ D(T ), contradicting assumption (4.1). Hence (Tf )(x0 ) = 0 if and only if f (x0 ) = 0 is true. 2 Remark. If for some x0 , (T g)(x0 ) = 0 holds for all bounded functions g ∈ D(T ), this means that F (x0 , ·,·) : R2 → R is zero and hence (T g)(x0 ) = 0 for all functions g ∈ D(T ), in particular for all half-bounded functions g ∈ D(T ). Thus case (b) in the definition of the set I of intervals (after Lemma 4) is impossible and hence Lemmas 5 to 9 hold for all open intervals. Proposition 12. The function F with (8) representing T has the form F (x, y, z) = H (y)/H (x) · K(z),
x, y, z ∈ R
where H : R → R =0 and K : R → R are suitable functions and where K is multiplicative and zero exactly in zero, K(uv) = K(u)K(v),
u, v ∈ R.
Moreover, either K ≡ 1 or (K(u) = 0 if and only if u = 0). In the case of (4.1) the second case applies. Proof. (i) Consider x0 , y0 ∈ R and f, g ∈ D(T ) such that g(x0 ) = y0 , f (y0 ) = x0 . We have by (1), T (f ◦ g)(x0 ) = (Tf )(y0 )(T g)(x0 ) = (T g)(x0 )(Tf )(y0 ) = T (g ◦ f )(y0 ). Since (f ◦ g)(x0 ) = x0 , (g ◦ f )(y0 ) = y0 , this means, using (8) and the chain rule, F x0 , x0 , f (y0 )g (x0 ) = F y0 , y0 , g (x0 )f (y0 ) .
(9)
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Since there are such functions f, g for which the derivatives f (y0 ) and g (x0 ) attain arbitrary values, we conclude that for all x0 , y0 , u ∈ R, F (x0 , x0 , u) = F (y0 , y0 , u). We may thus define K : R → R by K(u) := F (x0 , x0 , u) independently of x0 ∈ R. Expressing (9) in terms of F , we find K f (y0 )g (x0 ) = F x0 , y0 , g (x0 ) F y0 , x0 , f (x0 ) , and hence for any u, v ∈ R and x0 , y0 ∈ R K(uv) = F (x0 , y0 , u)F (y0 , x0 , v).
(10)
Moreover, K(1) = F (x0 , x0 , 1) = T (Id)(x0 ) = 1 by Lemma 6. Therefore F (x0 , y0 , u)F y0 , x0 , u−1 = 1
(11)
for any u = 0; in particular F (x0 , y0 , u) = 0. Hence F (x0 , y0 , u) =
K(uv) , F (y0 , x0 , v)
v = 0
where the right side is independent of v = 0. Taking v = 1, we define G : R2 → R =0 by G(x0 , y0 ) := 1/F (y0 , x0 , 1), so that G(x0 , x0 ) = 1 and F (x0 , y0 , u) = G(x0 , y0 )K(u),
x 0 , y0 , u ∈ R
and by (11), G(x0 , y0 )G(y0 , x0 ) = 1. Putting x0 = y0 in (10), we also have that K(uv) = K(u)K(v). Since for u = 0, F (x0 , x0 , u) = 0, K(u) = 0 holds for u = 0. For the constant function f = x0 , K(0) = F (x0 , x0 , 0) = (Tf )(x0 ). Assuming (4.1), this is zero by Lemma 6. If (4.1) does not hold, K(0) = 0 is possible. But then K(0)2 = K(0 · 0) = K(0) yields K(0) = 1 and K(0) = K(0 · v) = K(0) · K(v) implies that K(v) = 1 for all v ∈ R, i.e. K = 1. (ii) To finish the proof, we will show that there exists a function H : R → R =0 so that G(x, y) = H (y)/H (x) for all x, y ∈ R. Using the functional equation (1) for f, g ∈ C 1 (R), T (f ◦ g)(x) = (Tf ) g(x) (T g)(x), this is expressed in terms of G and K by G x, f g(x) · K f g(x) g (x) = G g(x), f g(x) · K f g(x) · G x, g(x) · K g (x) ,
x ∈ R.
Together with K(uv) = K(u)K(v), we find choosing f, g with f (g(x)) = 0, g (x) = 0, G x, f g(x) = G x, g(x) G g(x), f g(x) .
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Since the function values g(x), f (g(x)) can be chosen arbitrarily, we conclude that G(x, y) = G(x, w)G(w, y) holds for any x, y, w ∈ R. Further, G(w, x) = 1/G(x, w). We find, defining H : R → R =0 by H (y) := G(0, y) that G(x, y) = G(x, 0)G(0, y) = G(0, y)/G(0, x) = H (y)/H (x).
2
Remarks. 1. H (x0 ) is just the value of the image of the shift by x0 in 0, H (y0 ) = G(0, y0 ) = F (0, y0 , 1) = T (Id +y0 )(0). 2. The conclusion of Proposition 10 and 12 is that (Tf )(x) =
(H ◦ f )(x) · K f (x) , H (x)
x ∈ R, f ∈ C 1 (R)
(12)
with H being never zero, K being multiplicative and H ·, f (·) K f (·) being continuous for any f ∈ C 1 (R). Lemma 13. Assume that K : R → R is measurable, not identically zero and satisfies for all u, v ∈ R that K(uv) = K(u)K(v). Then there exists some p 0 such that K(u) = |u|p
or
K(u) = |u|p sgn(u).
Proof. For u = 0, K(u) = 0 since otherwise K is zero identically. Therefore we can define F : R → R, F (t) := log |K(et )|. Then for s, t ∈ R, F (t + s) = logK et+s = logK et + logK es = F (t) + F (s). Since F is measurable, by a result of Sierpinski [8] and Banach [5], F is linear, i.e. there is p ∈ R such that F (t) = pt, t ∈ R. Thus |K(et )| = ept . Since K(1) = K(1)2 > 0, it follows that K(u) = up for any u > 0. Since K(−1)2 = K(1) = 1, K(−1) ∈ {±1}. If K(−1) = 1, K(u) = |u|p for any u ∈ R, and if K(−1) = −1, K(u) = |u|p sgn(u) for any u ∈ R. Since K(0) = K(0)2 = 0 or 1, p has to be non-negative. 2 Proof of Theorem 1. Under the assumptions of Theorem 1, (12) gives the general form of T . (a) We will show that the function K in (12) is measurable and apply Lemma 13. Applying (12) first to f (x) = 2x, we get that (Tf )(x) = HH(2x) (x) K(2). Since K(2) = 0 and Tf ∈ C(R), ϕ : R → R, ϕ(x) :=
H (2x) H (x)
is continuous on R.
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Then H (2x)/(ϕ(x)H (x)) = 1. Take any a, b ∈ R. Then H (b) ϕ( b2 )H ( b2 )
=1=
H (a) , ϕ( a2 )H ( a2 )
H (b) H ( b2 ) ϕ( b2 ) = · . H (a) H ( a2 ) ϕ( a2 ) Using this iteratively with (b/2i , a/2i ) instead of (b, a), we find for any k ∈ N, k b b H (b) H ( 2k ) ϕ( 2i ) . = H (a) H ( 2ak ) ϕ( 2ai )
(13)
i=1
Choose a = 1 and apply (12) to g(x) = bx to get (T g)(x) =
H (bx) K(b) H (x)
Since T g is continuous in 0, we find that H ( 2bk ) 1 K(b) K(b) = (T g)(0) = lim (T g)(x) = lim (T g) k = lim k→∞ k→∞ H ( 1k ) x→0 2 2
exists for any b; using this and (13) yields 1 = lim
k→∞
H ( 2bk ) H ( 21k )
= lim
H (b) = H (1) · lim
k→∞
k→∞
k ϕ( 1i ) 2 b ϕ( ) i=1 2i
k ϕ( bi ) 2 1 ϕ( ) i=1 2i
·
H (b) , H (1)
.
As a function of b, the right side is a pointwise limit of continuous functions in b since ϕ is continuous. Hence H is measurable as a function of b. (b) Next, choose h(x) = x 2 /2 in (12) to conclude that (T h)(x) =
H (x 2 /2) K(x), H (x)
so that K(x) =
H (x) (T h)(x). H (x 2 /2)
Since H is measurable and (T h) is continuous, the right side and therefore K is measurable as a function of x. Moreover, K is multiplicative and by Lemma 13,
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K(u) = |u|p
or
K(u) = |u|p sgn(u)
for a suitable p 0. Hence (Tf )(x) =
H (f (x)) p f (x) H (x)
or (Tf )(x) =
H (f (x)) p f (x) sgn f (x) . H (x)
We conclude that for any f ∈ C 1 (R), H (f (x))/H (x) is continuous in x (if f (x) = 0). To conclude from this that H is continuous, it suffices to prove that H is continuous in one point, say x0 : the continuity in any other point x1 follows from considering f (x) = x + x1 − x0 and the continuity of H ◦ f/H . (c) For any c ∈ R, let b(c) := limy→c H (y) and a(c) := limx→c H (x). We claim that b(c)/H (c) and a(c)/H (c) are constant functions of c; in the case that for some c0 , b(c0 ) or a(c0 ) are infinity or zero, this should mean that all other values b(c) or a(c) are infinity or zero, too. Assume to the contrary that there are c0 and c1 such that b(c1 )/H (c1 ) < b(c0 )/H (c0 ). Choose any maximizing sequence yn , limn yn = c0 with limn H (yn ) = b(c0 ). Since for f (t) = t + c1 − c0 , H ◦ f/H is continuous, limn H (yn + c1 − c0 )/H (yn ) = H (c1 )/H (c0 ) exists and limn (yn + c1 − c0 ) = c1 implies that limn H (yn + c1 − c0 ) b(c1 ), we arrive at the contradiction b(c0 ) H (c1 ) b(c0 ) = H (c0 ) H (c0 ) H (c1 ) = lim n
H (yn + c1 − c0 ) H (yn ) H (yn ) H (c1 )
limn H (yn + c1 − c0 ) H (c1 )
=
b(c1 ) b(c0 ) < . H (c1 ) H (c0 )
This argument is also valid assuming b(c1 ) < b(c0 ) = ∞. The proof for a(c) is similar. We now use a similar reasoning as in [2]. If H would be discontinuous at some point, it would be discontinuous anywhere and could not be extended to be continuous. Assume that this is the case and take any sequence (cn )n∈N of pairwise disjoint numbers (cn = cm for n = m) with limn cn = 0. Let δn := 14 min{|cn − cm | | m = n} and choose 0 < εn < δn such that n∈N εn /δn < ∞. Since H is discontinuous in any point cn , b(cn )/a(cn ) > 1 holds. By the above argument, this value is independent of n, 1<
b(cn ) b := a a(cn )
for all n ∈ N.
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Hence by definition of b(cn ) and a(cn ) we may find yn , xn ∈ R with |yn −cn | < εn , |xn −cn | < εn and H (yn )/H (xn ) > b+a 2a > 1 (if b/a = ∞, choose them with H (yn )/H (xn ) > 2). Let ψ be a smooth cut-off function like
exp(1 − ψ(x) = 0,
1 ), 1−x 2
|x| < 1, |x| 1
n and put gn (x) := (yn − xn )ψ( x−x δn ). The functions gn are supported disjointly since |xn − xm | |cn − cm | − 2εn 4δn − 2εn 2δn for any m = n. Hence gn (xm ) = (yn − xn )δnm . Since
g n
n ∞
n
|yn − xn |/δn ψ ∞ 2 εn /δn ψ ∞ < ∞ n
holds, f (x) := x +
gn (x),
x∈R
n
defines a differentiable function f ∈ C 1 (R) with f (xn ) = xn + (yn − xn ) = yn , f (0) = 0 and f (0) = 1 = 0. Since xn → 0, yn = f (xn ) → 0, the continuity of H ◦ f/H yields the contradiction 1=
H (0) H (yn ) b + a = lim > > 1. n H (xn ) H (0) 2a
This proves that H is continuous. (d) In the case that T (2 Id) = c is constant, the function ϕ in part (a) is constant and consequently H is constant, giving that Tf = |f |p or Tf = |f |p sgn(f ). Clearly T (2 Id) = 2 yields p = 1, and condition (4.2) excludes the first possibility, implying Tf = f . If only T (2 Id)(0) = 2, H may be non-constant, but p = 1 follows directly from the form of T .
Choosing a = 0 in part (a) gives H (b) = H (0) i∈N ϕ( 2bi ). This, the form of K and the continuity of H justify Remark (iv) after the statement of Theorem 1. A direct calculation shows that any operator T given by (5) satisfies the chain rule functional equation (1). 2 Remark. We now mention a cohomological interpretation of Theorem 1. The semigroup G = (C 1 (R), 0) with the operation of composition acts on the abelian semigroup M = (C(R), ·) with the operation of pointwise multiplication by composition from the right, G × M → M, f H := H ◦ f . Thus M is a module over G. We denote the functions from Gn to M by F n (G, M) and define the coboundary operators d n : F n (G, M) → F n+1 (G, M), using additive notation + for the operation · on M by
n ∈ N0
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d n ϕ(g1 , . . . , gn+1 ) = g1 ϕ(g2 , . . . , gn+1 ) +
n (−1)i ϕ(g1 , . . . , gi−1 , gi gi+1 , gi+2 , . . . , gn+1 ) i=1
+ (−1)n+1 ϕ(g1 , . . . , gn ) where ϕ maps Gn into M and g1 , . . . , gn ∈ G. Theorem 1 characterizes the cocycles in Ker(d 1 ) for n = 1; then ϕ = T : G = C 1 (R) → M = C(R) has coboundaries d 1 T (g1 , g2 ) = g1 T (g2 ) − T (g1 g2 ) + T (g1 ),
g1 , g2 ∈ C 1 (R).
As for cocycles T , d 1 T = 0 means in multiplicative notation that T (g2 ◦ g1 ) = T (g2 ) ◦ g1 · T g1 . The cocycles are just the solutions of the chain rule equation, T g = H ◦ g/H · |g |p [sgn(g )]. For n = 0, ϕ ∈ F 0 (G, M) can be identified with ϕ = H ∈ M = C(R) and thus the coboundaries are d 0 H (g) = gH − H,
g ∈ C 1 (R)
and, in multiplicative notation, for functions H which are never zero, d 0 H (g) = H ◦ g/H. The cohomology group H 1 (G, M) = Ker(d 1 )/ Im(d 0 ) is thus represented by the maps g → |g |p and g → |g |p sgn(g ) from G to M. 4. The weak form of the chain rule We now turn to the proof of Theorem 2 concerning the equation T (f ◦ g) = T (f ) ◦ g · Ag,
f, g ∈ C 1 (R).
(6)
We assume throughout this section that D(T ) = D(A) = C 1 (R), Im(T ), Im(A) ⊂ C(R) and that T is strongly non-degenerate as defined by (7), i.e. that the assumptions of Theorem 2 hold. By (7)(i), for any x ∈ R there is g ∈ Cb1 (R) and y ∈ R such that g(y) = x and (T g)(y) = 0. Since Id ◦g = g, by (6) 0 = (T g)(y) = T (Id) g(y) (Ag)(y) = T (Id)(x)(Ag)(y). Hence for any x ∈ R, T (Id)(x) = 0 holds. We will reduce the proof of Theorem 2 to the proof of Theorem 1. For this, we need two localization results. Lemma 14. Let c ∈ R be arbitrary and J := (c, ∞) or J = (−∞, c). Then for any x ∈ J there exists y ∈ R and g ∈ Cb1 (R) with g(y) = x, (Ag)(y) = 0 and Im(g) ⊂ J .
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This is a stronger condition than (7)(i), but now for A, since additionally Im(g) ⊂ J holds. Proof. Let c ∈ R be arbitrary. We consider the case J = (c, ∞). Take any x ∈ J , thus c < x. By (7) there exists h1 ∈ Cb1 (R) and y ∈ R with h1 (y) = x and (T h1 )(y) = 0. (i) We may assume that h1 is bounded from below: If h1 were bounded from above, consider h2 := 2x − h1 . Then h2 is bounded from below, h2 (y) = x = h1 (y). Let f (s) := 2x − s. Then h1 = f ◦ h2 and h2 = f ◦ h1 . By (6), 0 = (T h1 )(y) = (Tf )(x)(Ah2 )(y), implying (Tf )(x) = 0. Using h1 = Id ◦h1 and (6), we get 0 = (T h1 )(y) = T (Id)(x)(Ah1 )(y), and hence (Ah1 )(y) = 0. Finally, (6) and h2 = f ◦ h1 yield (T h2 )(y) = (Tf )(x)(Ah1 )(y) = 0. (ii) Assume therefore that h1 (y) = x, (T h1 )(y) = 0 and h1 ∈ C 1 (R) is bounded from below. Then Im(h1 ) ⊂ (c1 , ∞) for some c1 ∈ R. Since x ∈ Im(h1 ), c1 < x. If c1 can be chosen such that c1 c, Im(h1 ) ⊂ (c1 , ∞) ⊂ (c, ∞) = J1 . Using (6) for h1 = Id ◦h1 yields (Ah1 )(y) = 0 as noted already. Taking g = h1 , the proof is finished in this case. If c1 is such that c1 < c, we may find λ ∈ (0, 1) such that (1 − λ)x + λc1 > c since x > c (with a possibly small λ > 0). Let g(s) := (1 − λ)x + λh1 (s). Then g(y) = x, Im(g) ⊂ (c, ∞) = J since for any s ∈ R, (1 − λ)x + λh1 (s) > (1 − λ)x + λc1 > c. Note that h1 (s) = γ g(s) − (γ − 1)x for γ = 1/λ. Let f (t) := γ t − (γ − 1)x. Then h1 = f ◦ g and by (6), 0 = (T h1 )(y) = T (f ◦ g)(y) = (Tf )(x)(Ag)(y). This means that g(y) = x, (Ag)(y) = 0 and Im(g) ⊂ J holds as required.
2
Lemma 15. Let c ∈ R be arbitrary and J = (c, ∞) or J = (−∞, c). Assume that f1 , f2 ∈ C 1 (R) satisfy f1 |J = f2 |J . Then (Tf1 )|J¯ = (Tf2 )|J¯ . Proof. Let x ∈ J be arbitrary. By Lemma 14, we can find y ∈ R and g ∈ Cb1 (R) with g(y) = x, (Ag)(y) = 0 and Im(g) ⊂ J . If f1 , f2 ∈ C 1 (R) satisfy f1 |J = f2 |J , f1 ◦ g = f2 ◦ g and (6) implies (Tf1 )(x)(Ag)(y) = T (f1 ◦ g)(y) = T (f2 ◦ g)(y) = (Tf2 )(x)(Ag)(y). Since (Ag)(y) = 0, we find (Tf1 )(x) = (Tf2 )(x). This shows that (Tf1 )|J = (Tf2 )|J , and by continuity of Tf1 , Tf2 , (Tf1 )|J¯ = (Tf2 )|J¯ holds. 2
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The “pure” localization result, the analogue of Proposition 10, states: Proposition 16. Assume (6) holds for T and A and that T is strongly non-degenerate. Then there is a function F : R3 → R such that (Tf )(x) = F x, f (x), f (x) holds for any f ∈ C 1 (R) and x ∈ R. Proof. The proof is the same as the one of Proposition 10, only replacing Lemma 8 by Lemma 15. Note that Lemma 8 was only applied to open intervals J1 = (x0 , ∞) and J2 = (−∞, x0 ) which are just those intervals allowed in Lemma 15. 2 Proposition 17. Assume that T and A satisfy (6), T (f ◦ g) = (Tf ) ◦ g · Ag,
f, g ∈ C 1 (R)
and that T is strongly non-degenerate (7). Then there is a function G : R → R =0 such that Af = Tf · G ◦ f ; f ∈ C 1 (R). Moreover, there are functions G1 , G2 : R → R =0 and K : R → R such that the function F in Proposition 16 has the form F (x, y, z) =
G1 (y) G2 (x)K(z), G1 (x)
x, y, z ∈ R
where G1 is continuous and K(u) = |u|p or K(u) = |u|p sgn(u) for a suitable p > 0. Proof. Choose any g ∈ C 1 (R) and x ∈ R. The chain rule (6) yields for any f ∈ C 1 (R), T (f ◦ g)(x) = (Tf ) g(x) (Ag)(x), which by Proposition 16 means F x, f g(x) , f g(x) g (x) = F g(x), f g(x) , f g(x) (Ag)(x). Now choose f such that f (g(x)) = g(x) and f (g(x)) = 1. Then F x, g(x), g (x) = F g(x), g(x), 1 (Ag)(x). We noted in the introduction of this section that T (Id)(u) = 0 for all u ∈ R. Hence F g(x), g(x), 1 = T (Id) g(x) = 0, and hence we have (Ag)(x) =
F (x, g(x), g (x)) , F (g(x), g(x), 1)
x ∈ R, g ∈ C 1 (R).
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This means that A is also a local operator, and in fact (Ag)(x) = (T g)(x)/G2 g(x) , if we define G2 : R → R =0 by G2 (u) := F (u, u, 1) = T (Id)(u) = 0. Insert this into (6) to find T (f ◦ g)(x) = (Tf ) g(x) (T g)(x)/G2 g(x) . We introduce the operator T : C 1 (R) → C(R) by T f (x) := (Tf )(x)/G2 (x). Then for f, g ∈ C 1 (R), T (f ◦ g)(x) = T (f ◦ g)(x)/G2 (x) = (Tf ) g(x) · (T g)(x)/ G2 g(x) · G2 (x) = T f g(x) · T g (x). Hence T : C 1 (R) → C(R) satisfies the strong chain rule (1), and the (strong) non-degeneracy assumption on T implies that also T is non-degenerate in the sense of (4). By Theorem 1, applied to T , there are continuous functions G1 : R → R =0 and K : R → R, K(u) = |u|p or K(u) = |u|p sgn(u) for some p > 0 such that F x, f (x), f (x) /G2 (x) = (Tf )(x)/G2 (x) = T f (x) =
(G1 ◦ f )(x) K f (x) , G1 (x)
x ∈ R, f ∈ C 1 (R).
Since the values of f (x) and f (x) can be chosen arbitrarily, we find F (x, y, z) = for all x, y, z ∈ R.
G1 (y) G2 (x)K(z) G1 (x)
2
Proof of Theorem 2. The conclusion of Propositions 16 and 17 is that T and A have the form G1 (f (x)) G2 (x)K f (x) , G1 (x) (Af )(x) = (Tf )(x)/G2 f (x) , x ∈ R, f ∈ C 1 (R) (Tf )(x) =
where G1 is continuous and K(u) = |u|p or K(u) = |u|p sgn(u). Since T (f ) is continuous, also G2 has to be continuous. 2
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R) for k = 1 5. The chain rule in C k (R In this chapter we consider the chain rule (1) for functions in D(T ) = C k (R) for k = 1. We first prove Proposition 3 showing that there are no non-trivial examples of solutions to (1) if k = 0 and then consider the chain rule operation as maps T : C ∞ → C,
T : C k → C,
T : C k → C k−1 ,
T : C∞ → C∞
for k 2. We start with the Proof of Proposition 3. Assume that T |Cb (R) = 0 is false. Then there are g1 ∈ Cb (R) and x1 ∈ R with (T g1 )(x1 ) = 0. By assumption, there are also g0 ∈ C(R) and x0 ∈ R with (T g)(x0 ) = 0. Using this, the analogues of Lemmas 4, 5, 6 and 8 may be proved in the same way for continuous functions as for differentiable functions, using T (f ◦ g) = (Tf ) ◦ g · T g for all continuous functions. In particular, for all open half-infinite intervals J ⊂ R and f1 , f2 ∈ C(R) with f1 |J = f2 |J , we have that (Tf1 )|J¯ = (Tf2 )|J¯ . Similarly as in the proof of Proposition 10, this can be used to prove that there is a function F : R2 → R such that Tf (x) = F x, f (x) ,
x ∈ R, f ∈ C(R).
Namely, fix x0 ∈ R and f ∈ C(R) with a given value f (x0 ). Take any other function g ∈ C(R) with g(x0 ) = f (x0 ) and define h ∈ C(R) by g(x), h(x) := f (x),
x ∈ J1 , , x ∈ J¯2
where J1 = (x0 , ∞), J2 = (−∞, x0 ). Then h|J1 = g|J1 and h|J2 = f |J2 . Hence (T g)|J¯1 = (T h)|J¯1 and (T h)|J¯2 = (Tf )|J¯2 . Since x0 ∈ J¯1 ∩ J¯2 , we conclude that (T g)(x0 ) = (T h)(x0 ) = (Tf )(x0 ). Therefore the value (Tf )(x0 ) depends only on x0 and f (x0 ), i.e. Tf (x0 ) = F (x0 , f (x0 )) for a suitable function F : R2 → R and for all x0 ∈ R, f ∈ C(R). By assumption, we may choose x0 ∈ R and g0 ∈ C(R) with (T g0 )(x0 ) = 0. Take any x ∈ R and consider h ∈ C(R), h(s) = s + x0 − x. Then h(x) = x0 and g := g0 ◦ h satisfies by the chain rule (T g)(x) = T (g0 ◦ h)(x) = (T g0 )(x0 )(T h)(x) = 0. Hence for any x there is g ∈ C(R) with (T g)(x) = 0, i.e. F (x, g(x)) = 0. For any f ∈ C(R), again by the chain rule T (f ◦ g)(x) = (Tf ) g(x) (T g)(x), i.e. F (x, f (g(x))) = F (g(x), f (g(x)))F (x, g(x)) = 0. Since f may attain arbitrary values on g(x), we conclude that F (x, y) = 0 for all x, y ∈ R. Hence Tf = 0 for all f ∈ C(R), contradicting the assumption T |Cb (R) = 0. 2 To formulate an analogue of Theorem 1 for C k (R) and C ∞ (R) functions, we need corresponding non-degeneracy conditions for T satisfying the chain rule (1). For k 2, let Cbk (R) :=
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Cb1 (R) ∩ C k (R) and Cb∞ (R) := Cb1 (R) ∩ C ∞ (R). T is non-degenerate on D(T ) = C k (R) provided that ∃x1 ∈ R, h1 ∈ Cbk (R)
(T h1 )(x1 ) = 0.
(4k )
Similarly (4∞ ) denotes this if h1 ∈ Cb∞ (R). The analogous conditions to (4.1) and (4.2) will be denoted by (4.1k ), (4.2k ). We then have: Proposition 18. Assume that k ∈ N2 and that T : D(T ) = C k (R) → Im(T ) ⊂ C(R) satisfies the chain rule T (f ◦ g) = (Tf ) ◦ g · T g,
f, g ∈ D(T ).
(1)
If T is non-degenerate in the sense of (4k ), there exists p 0 and a positive continuous function H ∈ C(R) such that Tf =
H ◦ f p · f H
or in the case of p > 0, Tf =
H ◦ f p · f sgn f . H
This also holds for k = ∞. If (4.1k ) holds, we have p > 0. If (4.2k ) is valid and T (2 Id)(0) = 2, p = 1 and T has the form Tf = HH◦f · f ; the stronger assumption T (2 Id) = 2 implies even Tf = f . If the image of T consists of smooth functions, i.e. if T : C k (R) → C k−1 (R) or T : C ∞ (R) → ∞ C (R) satisfies (1) and (4k ) or (4∞ ), the function H is in C k−1 (R) or C ∞ (R) and p satisfies p ∈ {0, . . . , k − 1} or p > k − 1 for k ∈ N
or (p ∈ N0 for k = ∞).
We need the following lemma for the Proof of Proposition 18. Lemma 19. Let 0 < a 1 and L ∈ C(R) be a continuous function such that for any fixed x0 ∈ R, x , ψ(x) := L(x0 + x) − aL x0 + 2
x∈R
defines a C 1 -function ψ ∈ C 1 (R). Then L is a C 1 -function, L ∈ C 1 (R). Proof. (a) We first prove the differentiability of L in x0 where we may assume that x0 = 0. Then ψ(x) = L(x) − aL( x2 ). Since ψ ∈ C 1 (R) and ψ(0) = (1 − a)L(0), given ε > 0, there is δ > 0 such that for all |x| δ, we have that ψ(x) − ψ(0) − ψ (0) ε. x
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We use the following iteration procedure: for any n ∈ N, n−1
aj ψ
j =0
x 2j
= L(x) − a n L
x , 2n
and replacing x successively by x/2j , we find n−1 (L(x) − a n L( xn )) − (n−1 a j )(1 − a)L(0) j ε a j =0 2 − (0) ψ j x 2 1 − a/2 j =0
aj 1 x n using ∞ j =0 2j = 1−a/2 . Since L is continuous in x0 = 0, for a < 1, a L( 2n ) → 0. We may then take the limit for n → ∞, L(x) − L(0) ε ψ (0) − 2ε. x 1 − a/2 1 − a/2 This also holds if a = 1, then L( 2xn ) → L(0) and the term with (1 − a) disappears, yielding the same estimate. This shows that L (0) = ψ (0)/(1 − a/2) exists. Similarly, L is differentiable in any point x0 ∈ R, using the function ψ defined above in terms of x0 . (b) We next prove the continuity of L in x0 = 0. By assumption, ψ (x) = L (x) − a2 L ( x2 ) is continuous in x0 = 0. Hence for ε > 0, there is δ > 0 such that for |x| δ, L (x) − a L x − 1 − a L (0) ε. 2 2 2 Using a similar iteration technique as before, n−1 j a j =0
2
ψ
x 2j
n x a = L (x) − L n , 2 2
we find n n−1 n ε a a L (x) − a L x − 1− L (0) 2ε. n 2 2 2 2 1 − a/2 j =0
The argument in (a) yields that L remains bounded in a small neighborhood of 0 since ψ ∈ C 1 (R), and hence ( a2 )n L ( 2xn ) → 0 as n → ∞. In the limit, we get for sufficiently small |x| δ2 δ, L (x) − L (0) 2ε and hence L is continuously differentiable in x0 = 0 and any other x0 ∈ R as well.
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Proof of Proposition 18. The proof is basically similar to the one of Theorem 1. We only stress and outline the modifications which are needed. (a) Using the non-degeneracy condition (4k ) for C k (R) functions, the localization on intervals in Lemmas 4, 5, 6 and 8 works the same way; for the pointwise localization in the proof of Proposition 10 one has to use a function h ∈ C k (R) defined by f on one side of x0 and by a Taylor approximation of the k-th order g ∈ C k (R) on the other side. This way, we may conclude that Tf (x) = F x, f (x), f (x), f (x), . . . , f (k) (x) ,
f ∈ C k (R), x ∈ R
(14)
holds for a suitable function F : Rk+2 → R. In the case of C ∞ (R), k = ∞, Tf (x) depends on x, f (x) and all derivatives f (j ) (x), j ∈ N. We claim that Tf (x) actually does not depend on the values of the derivatives of order 2, i.e. that Tf (x) = F˜ (x, f (x), f (x)). The further proof in the case of T : C k (R) → C(R) then proceeds exactly as the one for Theorem 1. For f, g ∈ C k (R), the derivatives of f ◦ g have the form (f ◦ g)(k) = f (k) ◦ g · g k + ϕk f ◦ g, . . . , f (k−1) ◦ g, g , . . . , g (k−1) + f ◦ g · g (k)
(15)
where ϕk depends only on the lower order derivatives of f and g up to the order (k − 1); e.g. ϕ2 = 0, ϕ3 (f ◦ g, f ◦ g, g , g ) = 3f ◦ g · g · g . Let us also remark that for any x0 , y0 ∈ R and any sequence (tn )n∈N of real numbers there is g ∈ C ∞ (R) with g(x0 ) = y0 and g (n) (x0 ) = tn for any n ∈ N. This may be shown by adding infinitely many small bump functions, see [6], p. 16. We now show successively that the function F with (14) does not depend on the variables with values f (x), . . . , f (k) (x). Starting with f (x), to simplify notation, we will not always write the further variables f (x) etc. in detail, but just put . . . for them. We explain below how to work with f (x) etc. If T : C k (R) → C(R) satisfies (1), (14) means for functions f, g ∈ C k (R) with g(x0 ) = y0 , f (y0 ) = z0 , where x0 , y0 , z0 ∈ R are arbitrary, T (f ◦ g)(x0 ) = F x0 , z0 , f (y0 )g (x0 ), f (y0 )g (x0 )2 + g (x0 )f (y0 ), . . . = (Tf )(y0 )(T g)(x0 ) = F y0 , z0 , f (y0 ), f (y0 ), . . . F x0 , y0 , g (x0 ), g (x0 ), . . . , also for k = ∞. If x0 = z0 , also (g ◦ f )(y0 ) is defined and T (f ◦ g)(x0 ) = (Tf )(y0 )(T g)(x0 ) = (T g)(x0 )(Tf )(y0 ) = T (g ◦ f )(y0 ), i.e. F x0 , x0 , f (y0 )g (x0 ), f (y0 )g (x0 )2 + g (x0 )f (y0 ), . . . = F y0 , y0 , g (x0 )f (y0 ), f (y0 )g (x0 ) + g (x0 )f (y0 )2 , . . . .
(16)
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Choosing arbitrary real sequences (tn )n∈N , (sn )n∈N , we find functions f, g ∈ C ∞ (R) with g(x0 ) = y0 , f (y0 ) = x0 , g (n) (x0 ) = tn , f (n) (y0 ) = sn for all n ∈ N. Therefore F x0 , x0 , s1 t1 , s2 t12 + s1 t2 , . . . = F y0 , y0 , s1 t1 , s2 t1 + s12 t2 , . . . .
(17)
Since the (tn ) and (sn ) were chosen independently of x0 and y0 , the left and right side of (17) does not depend on x0 and y0 . Let K(u1 , u2 , . . .) := F (x0 , x0 , u1 , u2 , . . .). Then (17) means that K s1 t1 , t12 s2 + s1 t2 , . . . = K s1 t1 , t1 s2 + s12 t2 , . . . . For k 3 or k = ∞, the structure of the k-th derivative of f ◦ g given by (15) means that in more precise notation K s1 t1 , t12 s2 + s1 t2 , . . . , t1k sk + s1 tk + ϕk1 , . . . = K s1 t1 , t1 s2 + s12 t2 , . . . , t1 sk + s1k tk + ϕk2 , . . . ,
(18)
where ϕk1 , ϕk2 ∈ R depend only on the values of s1 , . . . , sk−1 and t1 , . . . , tk−1 . The notation here is for the C ∞ (R) case; for C k (R) the last dots in (18) and also below should be dropped. Assume that s1 t1 = +1, −1 and s1 t1 = 0. We claim that, given arbitrary values of a2 , . . . , ak , . . . and b2 , . . . , bk , . . . , we may choose s2 , t2 , . . . , sk , tk , . . . such that (18) yields K(s1 t1 , a2 , . . . , ak , . . .) = K(s1 t1 , b2 , . . . , bk , . . .),
(19)
tk i.e. that K only depends on the first variable u = s1 t1 (if u = 0, 1, −1). Note that det 1
s1
t1 s1k
=
(s1 t1 )((s1 t1 )k−1 − 1) = 0. Thus successively, we may solve uniquely the linear equations for s2 , t2 , . . . , sk , tk , starting with j = 2 and continuing up to j = k (and further in the C ∞ -case), and using the obtained values of s2 , t2 , . . . in the later equations to determine the values of ϕk1 , ϕk2 , t12 s2 + s1 t2 = a2 ,
t1 s2 + s12 t2 = b2 ,
t1k sk + s1 tk = ak − ϕk1 ,
...,
t1 sk + s1k tk = bk − ϕk2 .
This means that (18) implies (19) and for u1 = 0, 1, −1, K(u1 , u2 , . . .) is independent of all ˜ 1 ). variables (uj ), j 2, and we write it as K(u In the case of s1 t1 = 1, putting x0 = y0 = z0 in (16), we find that for any s2 , . . . , sk , t2 , . . . , tk (and choosing t1 = 2, s1 = 1/2), ˜ K(1/2). ˜ K 1, 4s2 + t2 /2, . . . , 2k sk + tk /2k + ϕk , . . . = K(2)
(20)
Again for arbitrary values of a2 , . . . , ak , we find successively s2 , t2 , . . . , sk , tk such that the left ˜ = K(1, . . .) is also independent of side of (20) equals K(1, a2 , . . . , ak , . . .) and therefore K(1) the variables (uj ), j 2. Similarly for K(−1, . . .). To show that also K(0, . . .) is independent of the further variables, choose t1 = a, s1 = 0 (thus s1 t1 = 0) in (18) to conclude
S. Artstein-Avidan et al. / Journal of Functional Analysis 259 (2010) 2999–3024
3023
K 0, a 2 s2 , . . . , a k sk + ϕk1 , . . . = K(0, as2 , . . . , ask + ϕk2 , . . .) which again implies independence of the further variables. We now write again K instead of ˜ we know that K(u1 ) = F (x0 , x0 , u1 , u2 , . . .) is independent of x0 and (u2 , . . .). For values K: y0 = x0 , we have by (16), F (x0 , y0 , t1 , t2 , . . .) =
K(s1 t1 ) . F (y0 , x0 , s1 , s2 , . . .)
Since the left side is independent of s1 , s2 , . . . and the right side is independent of t2 , t3 , . . . , the equation is of the form F (x0 , y0 , t) = K(t)/F (y0 , x0 , 1). From here, the same arguments as in the proof of Theorem 1 yield that F (x0 , y0 , t) = H (y0 )/H (x0 )t p sgn(t) with p 0 and H ∈ C(R), i.e. that T has the form given in Proposition 18. (b) If H ∈ C k−1 (R) and p ∈ {0, . . . , k − 1} or p > k − 1, the chain rule (1) defines a map T : C k (R) → C k−1 (R) into smooth functions; the condition on p is obviously needed to have continuous derivatives in zeros of f . Let us conversely assume that T maps C k (R) into C k−1 (R) and verifies (1) where k ∈ N, k 2. We know that T has the form p Tf = H ◦ f/H f sgn f where H ∈ C(R), H > 0. We claim that H is smooth, namely H ∈ C k−1 (R). Let L := − log H . Obviously L ∈ C k−1 (R) if and only if H ∈ C k−1 (R). Take f (x) = x2 . By assumption Tf ∈ C k−1 (R) and hence ϕ(x) := L(x) − L( x2 ) defines a function ϕ ∈ C k−1 (R). We prove by induction on k 2 that ϕ ∈ C k−1 (R) and L ∈ C k−2 (R) implies that L ∈ C k−1 (R). For k = 2, T : C 2 (R) → C 1 (R), ϕ ∈ C 1 (R). Since H ∈ C(R), also L ∈ C(R). By Lemma 19, with ψ = ϕ and a = 1, L ∈ C 1 (R). To prove the induction step, assume that k 3 and ϕ ∈ C k−1 (R) as well as L ∈ C k−2 (R) holds. We have to prove that L ∈ C k−1 (R). 1 L(k−2) ( x2 ). Then ψ ∈ C 1 (R) and L(k−2) ∈ Define ψ(x) := ϕ (k−2) (x) = L(k−2) (x) − 2k−2 C(R). By Lemma 19, with a = 1/2k−2 , L(k−2) ∈ C 1 (R), i.e. L ∈ C k−1 (R). Hence H ∈ C k−1 (R) is true if T : C k (R) → C k−1 (R) satisfies (1). 2 References [1] S. Alesker, S. Artstein-Avidan, V. Milman, A characterization of the Fourier transform and related topics, in: A. Alexandrov, et al. (Eds.), Linear and Complex Analysis, in: Adv. Math. Sci., vol. 63, 2009, pp. 11–26, dedicated to V.P. Havin, Amer. Math. Soc. Transl., vol. 226. [2] S. Alesker, S. Artstein-Avidan, D. Faifman, V. Milman, A characterization of product preserving maps with applications to a characterization of the Fourier transform, in press. [3] S. Artstein-Avidan, V. Milman, The concept of duality for measure projections of convex bodies, J. Funct. Anal. 254 (2008) 2648–2666.
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[4] S. Artstein-Avidan, V. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. 169 (2009) 661–674. [5] S. Banach, Sur l’équation fonctionelle f (x + y) = f (x) + f (y), Fund. Math. 1 (1920) 123–124. [6] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 1983. [7] V. Milman, R. Schneider, Characterizing the mixed volume, submitted for publication. [8] W. Sierpinski, Sur l’équation fonctionelle f (x + y) = f (x) + f (y), Fund. Math. 1 (1920) 116–122.
Journal of Functional Analysis 259 (2010) 3025–3035 www.elsevier.com/locate/jfa
Some critical minimization problems for functions of bounded variations Thomas Bartsch a , Michel Willem b,∗ a Mathematisches Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany b Département Mathématique, Université Catholique de Louvain, 2 chemin du Cyclotron,
1348 Louvain-la-Neuve, Belgium Received 3 July 2010; accepted 18 July 2010
Communicated by H. Brezis
Abstract Using a new elementary method, we prove the existence of minimizers for various critical problems in BV(Ω) and also in W 1,p (Ω), 1 < p < ∞. © 2010 Elsevier Inc. All rights reserved. Keywords: Functions of bounded variation; Critical exponents; Gagliardo–Nirenberg inequality; Isoperimetric inequalities
1. Introduction After the classical results due to Brezis and Nirenberg (see [2]), many papers were devoted to critical minimization problems on W 1,p (Ω) (1 < p < ∞) or on some subspaces. See e.g. the list of references in [8]. When p = 1, it is necessary to replace W 1,1 (Ω) by BV(Ω), the space of integrable functions with bounded variations on Ω. We know only 3 papers devoted to critical minimization problems on BV(Ω): [1,5,7]. (The critical trace problem in BV(Ω) is different since it is convex. We exclude this problem.) * Corresponding author.
E-mail addresses:
[email protected] (T. Bartsch),
[email protected] (M. Willem). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.07.009
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The existence of optimal functions for the sharp Poincaré inequality 1 u dx DuΩ cu − N/(N−1) m(Ω) L (Ω) Ω
is proved in [5] when Ω is a ball or a sphere and in [1] when Ω is a bounded domain with C 2 boundary. The proof in [5] uses a specific isoperimetric inequality and in [1] the concentrationcompactness principle in BV(Ω). When Ω ⊂ R2 , the results in [1] solve a problem of [3]. The minimization problem ⎧ ⎪ ⎨ inf DuΩ + a|u| dx + |u| dσ , ⎪ ⎩
Ω
u ∈ BV(Ω),
∂Ω
uLN/(N−1) (Ω) = 1
is treated in [7] using approximation by subcritical problems and the concentration-compactness
principle in BV(Ω). The penalization term ∂Ω |u| dσ replaces the Dirichlet boundary condition (see [7] and [11]). See also [6] and [14] for the existence of critical points. A general existence theorem for subcritical minimization problems on BV(Ω) is contained in [11]. In this paper, we solve critical minimization problems on BV(Ω) by using a new elementary lemma (Lemma 3.2) or a variant (Lemma 4.1). This method is also applicable to critical minimization problems on W 1,p (Ω) (1 < p < ∞) (see Lemma 5.1), is rather simple and avoid any concentration-compactness type argument. In Section 2 we recall some basic properties of functions of bounded variations (see [9] and [15]). 2. Functions of bounded variations Let Ω be an open subset of RN . The variation of u ∈ L1loc (Ω) is defined by
DuΩ = sup u div v dx: v ∈ D Ω, RN , v∞ 1 Ω
where
N
2 v∞ = sup vk (x) x∈Ω
1/2 .
k=1
The variation is lower semi-continuous L1loc (Ω)
un −→ u
⇒
DuΩ lim Dun Ω . n→∞
On BV(Ω) = u ∈ L1 (Ω): DuΩ < ∞
T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035
3027
we define the norm uBV(Ω) = DuΩ + uL1 (Ω) and the distance of strict convergence d(f, g) = Df Ω − DgΩ + f − gL1 (Ω) . The sequence (un ) converges weakly to u in BV(Ω) (written un u) if un − uL1 → 0, n → ∞, ∗ ∂k un ∂k u in C0 (Ω) , n → ∞, 1 k N, where [C0 (Ω)]∗ denotes the space of finite measures on Ω. It is clear that norm convergence
⇒
strict convergence
⇒
weak convergence.
We now assume that Ω is a bounded domain of RN (N 2) with Lipschitz boundary. Let us recall (see [9]) that, for every u ∈ BV(Ω), the trace of u, γ0 (u), belongs to L1 (∂Ω) and that the extension by 0:
u0 (x) = u(x), = 0,
x ∈ Ω, x ∈ RN \{0},
belongs to BV(RN ). Moreover, Du0 RN = DuΩ +
γ0 (u) dσ
∂Ω
defines an equivalent norm on BV(Ω). The space W 1,1 (Ω) is dense in BV(Ω) with respect to the strict convergence (not the norm convergence!) and the trace operator γ0 : BV(Ω) → L1 (∂Ω) is continuous with respect to the strict convergence (not the weak convergence!). We will also denote by u the trace of u and the extension of u by 0. Let us denote by 1∗ the critical exponent N/(N − 1) and by VN the volume of the unit ball in RN . The following inequality is due to Cherrier [4]. Theorem 2.1. For every ε > 0 there exists cε > 0 such that, for all u ∈ BV(Ω),
N (VN /2)1/N − ε uL1∗ (Ω) DuΩ + cε uL1 (Ω) . Let us recall that, for 1 p < 1∗ , the embedding BV(Ω) ⊂ Lp (Ω) is compact and that the ∗ embedding BV(Ω) ⊂ L1 (Ω) is continuous (but not compact!). We will also need the sharp Gagliardo–Nirenberg inequality due to Maz’ya and Federer and Fleming (see [13]):
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T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035 ∗
Theorem 2.2. For every u ∈ L1 (RN ), 1/N
N VN
uL1∗ (RN ) DuRN .
Moreover equality holds if and only if u is the characteristic function of a ball. We will use truncation as a basic tool. We define, for h > 0,
Th (s) = min max(s, −h), h ,
Rh (s) = s − Th (s).
Proposition 2.3. For every u ∈ BV(Ω), DuΩ = DTh uΩ + DRh uΩ . Proof. It is clear that DuΩ DTh uΩ + DRh uΩ . Let (un ) ⊂ W 1,1 (Ω) be such that un → u strictly in BV(Ω). Then, by lower semi-continuity, DTh uΩ + DRh uΩ lim ∇Th un L1 (Ω) + lim ∇Rh un L1 (Ω) n→∞
n→∞
lim ∇un L1 (Ω) = DuΩ . n→∞
2
The proof of Proposition 2.3 was communicated to us by J. Van Schaftingen. 3. Critical minimizations problems in BV(Ω) The following result is due to Degiovanni and Magrone in the case p = 1∗ (see [6, p. 603]). We give the proof for the sake of completeness. Lemma 3.1. Let Ω be a bounded domain in RN and let 1 p < ∞ and (un ) ⊂ Lp (Ω) be such that (a) sup un p < ∞; (b) (un ) converges to u almost everywhere on Ω. Then p p p p lim un p − Rh un p = up − Rh up .
n→∞
Proof. Let us define p f (s) = |s|p − Rh (s) . For every ε > 0, there exists Cε > 0 such that
f (s) − f (t) ε |s|p + |t|p + Cε .
T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035
3029
It follows from Fatou’s lemma that 2ε |u|p dx + Cε m(Ω) Ω
lim
n→∞
ε |un |p + |u|p + Cε − f (un ) − f (u) dx
Ω
ε sup
|un |p dx + ε
n
Ω
|u|p dx + Cε m(Ω) − lim
n→∞
Ω
f (un ) − f (u) dx.
Ω
Hence lim
n→∞
f (un ) − f (u) dx ε sup
|un |p dx.
n
Ω
Ω
Since ε > 0 is arbitrary, the proof is complete.
2
In this section, we assume that Ω is a bounded domain of RN (N 2) with Lipschitz boundary. ¯ and b ∈ C(∂Ω) be such that ϕ defined on BV(Ω) by Lemma 3.2. Let a ∈ C(Ω) ϕ(u) = DuΩ + a|u| dx + b|u| dσ Ω
∂Ω
satisfies c = inf ϕ(u)/uL1∗ (Ω) : u ∈ BV(Ω)\{0} > 0. Let (un ) ⊂ BV(Ω) be such that un L1∗ (Ω) = 1, ϕ(un ) → c, n → ∞, and un u in BV(Ω). Then either uL1∗ (Ω) = 0 or uL1∗ (Ω) = 1. Proof. By going if necessary to a subsequence, we can assume that un → u a.e. on Ω. We have, using the preceding lemma, c = lim ϕ(Th un ) + ϕ(Rh un ) n→∞ c lim Th un 1∗ + Rh un 1∗ n→∞
∗ ∗ 1/1∗ = c Th u1∗ + 1 + Rh u11∗ − u11∗ . When h → ∞, we obtain ∗ 1/1∗ ∗ 1/1∗ + 1 − u11∗ , 1 u11∗ so that u1∗ = 0 or u1∗ = 1.
2
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¯ and We consider first the case when b = 0. We assume that a ∈ C(Ω) ∗ (A1) 0 < S0 (a, Ω) = inf DuΩ + a|u| dx /uL1 (Ω) : u ∈ BV(Ω)\{0} , Ω
(A2) S0 (A, Ω) < N (VN /2)1/N . Theorem 3.3. Under assumptions (A1), (A2), there exists u ∈ BV(Ω)\{0} such that u 0 and S0 (a, Ω)uL1∗ (Ω) = DuΩ + a|u| dx. Ω
Proof. Let (un ) ⊂ BV(Ω) be such that un 1∗ = 1 and Dun Ω + a|un | dx → S0 (a, Ω). Ω
Since (un ) is bounded in BV(Ω), we can assume that un u in BV(Ω). Let 0 < ε < N(VN /2)1/N − S0 (a, Ω). It follows from Theorem 2.1 that, for some cε > 0, S0 (a, Ω) = lim Dun Ω + a|un | dx n→∞
Ω
N (VN /2)1/N − ε − cε
|u| dx + Ω
a|u| dx. Ω
Hence, u = 0. The preceding lemma implies that u1∗ = 1. Since, by lower semi-continuity, DuΩ + a|u| dx S0 (a, Ω), Ω
u is a minimizer for S0 (a, Ω). Since D|u|Ω DuΩ , we can replace u by |u|.
2
The following result gives a concrete sufficient condition for (A2). ¯ be such that, for Theorem 3.4. Let Ω be a bounded domain with C 2 boundary and let a ∈ C(Ω) some y ∈ ∂Ω, 2
N − 1 VN −1 H (y) > a(y), N + 1 VN
where H (y) denotes the mean curvature of ∂Ω at y. Then (A2) is satisfied. Proof. We can assume that y = 0. For r > 0 small enough, we have δ=
N −1 VN VN −1 H (0) − A > 0, N +1 2
T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035
3031
where A = sup{a(x): x ∈ Ω ∩ B(0, r)}. Let us define uε = χΩ∩B(0,ε) . By formula (1) in [10], we have, for ε → 0,
VN N N − 1 VN −1
∗ ε − H (0)ε N +1 + o ε N +1 , uε 11∗ = m Ω ∩ B(0, ε) = 2 N +1 2 VN N −1 VN −1 − (N − 1) Duε Ω = N ε H (0)ε N + o ε N , 2 2 N VN N ε +o ε . a uε dx A 2 Ω
It follows that, for ε → 0, Duε Ω + a uε dx /uε 1∗ Ω
VN 2
1−N N
VN N −1 VN N − VN −1 H (0)ε + Aε + o(ε) 2 N +1 2
= N (VN /2)1/N − δ(VN /2) so that S0 (a, Ω) < N (VN /2)1/N .
1−N N
ε + o(ε),
2
We consider now the case when b = 1. The following result is due to Demengel [7], but our proof, using Lemma 3.1, is simpler. Let us recall that, for u ∈ BV(Ω), DuRN = DuΩ +
|u| dσ.
∂Ω
¯ and We assume that a ∈ C(Ω) (B1) 0 < S1 (a, Ω) = inf DuRN + a|u| dx /uL1∗ (Ω) : u ∈ BV(Ω)\{0} , Ω 1/N
(B2) S1 (a, Ω) < N VN
.
Theorem 3.5. Under assumptions (B1), (B2), there exists u ∈ BV(Ω)\{0} such that u 0 and S1 (a, Ω)uL1∗ (Ω) = DuRN +
a|u| dx. Ω
Proof. Let (un ) ⊂ BV(Ω) be such that un 1∗ = 1 and Dun RN +
a|un | dx → S1 (a, Ω). Ω
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The sequence (un ) is bounded in BV(RN ). We can assume that un u in BV(RN ). It follows from Theorem 2.2 that S1 (a, Ω) = lim DuRN + a|un | dx n→∞
1/N
N VN
+
Ω
a|u| dx. Ω
By assumption (B2), u = 0. Lemma 3.2 implies that u1∗ = 1. Since, by lower semi-continuity, DuRN + a|u| dx S1 (a, Ω), Ω
u is a minimizer for S1 (a, Ω). Since D|u|RN DuRN , we can replace u by |u|.
2
4. Poincaré inequality Let us recall the general Poincaré inequality in BV(Ω) due to Meyers and Ziemer [12]. Let Ω be a bounded domain of RN (N 2) with Lipschitz boundary and let f ∈ LN (Ω) be such that Ω f dx = 1. Then S2 (f, Ω) = inf DuΩ /uL1∗ (Ω) : u ∈ BV(Ω)\{0}, f u dx = 0 > 0. Ω
When f ≡ 1/m(Ω), this is the Poincaré inequality.
Lemma 4.1. Let f ∈ LN (Ω) be such that Ω f dx = 1 and let (un ) ⊂ BV(Ω) be such that un L1∗ (Ω) = 1, Ω f un dx = 0, Dun Ω → S2 (f, Ω), n → ∞, and un u in BV(Ω). Then either uL1∗ (Ω) = 0 or uL1∗ (Ω) = 1. Proof. By going if necessary to a subsequence, we can assume that un → u a.e. on Ω. Let us define, for h > 0 and n ∈ N, ch,n = f Th un dx, ch = f Th u dx. Ω
Ω
Using Lemma 3.1, we obtain S2 (f, Ω) = lim DTh un Ω + DRh un Ω n→∞ S2 (f, Ω) lim Th un − ch,n 1∗ + Rh un + ch,n 1∗ n→∞
S2 (f, Ω) lim Th un 1∗ + Rh un 1∗ − 2ch,n 1∗ n→∞
∗ ∗ 1/1∗ = S2 (f, Ω) Th u1∗ + 1 + Rh u11∗ − u11∗ − 2ch 1∗ .
T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035
Since limh→∞ ch = limh→∞
Ω
f Th u dx =
Ω
3033
f u dx = 0, we have
∗ 1/1∗ ∗ 1/1∗ + 1 − u11∗ , 1 u11∗ so that u1∗ = 0 or u1∗ = 1.
2
The following theorem was proved by Bouchez and Van Schaftingen in the case f ≡ 1/m(Ω) (see [1]). and let f ∈ LN (Ω) be such Theorem 4.2. Let Ω be a bounded domain of RN with C 2 boundary
that Ω f dx = 1. Then there exists u ∈ BV(Ω)\{0} such that Ω f u dx = 0 and S2 (f, Ω)uL1∗ (Ω) = DuΩ . Proof. (1) Let us first prove that S2 (f, Ω) < N (VN /2)1/N .
(∗)
We can assume that 0 ∈ ∂Ω and that H (0), the mean curvature of ∂Ω at 0, is positive. Let us define, as in [1], for ε > 0 small enough, uε = χΩ∩B(0,ε) − λε χΩ\B(0,ε) , λε = f dx/ f dx. Ω∩B(0,ε)
Ω\B(0,ε)
Hölder inequality implies that λε = o(ε N −1 ). By formula (1) in [10], we have, for ε → 0,
VN N N − 1 VN −1 ∗ ε − H (0)ε N +1 + o ε N +1 , uε 11∗ m Ω ∩ B(0, ε) = 2 N +1 2 VN N −1 VN −1 ε H (0)ε N + o ε N . − (N − 1) Duε Ω = (1 + λε )DχΩ∩B(0,ε) = N 2 2 It follows that, for ε → 0, Duε Ω /uε 1∗
VN 2
1−N N
VN N −1 N − VN −1 H (0)ε + o(ε) , 2 N +1
so that (∗) is satisfied.
(2) Let (un ) ⊂ BV(Ω) be such that un 1∗ = 1, Ω f un dx = 0 and Dun Ω → S2 (f, Ω),
n → ∞.
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We can assume that un u in BV(Ω). Let 0 < ε < N (VN /2)1/N − S2 (f, Ω). It follows from Theorem 2.1 that, for some cε > 0, S2 (f, Ω) = lim Dun Ω N (VN /2)1/N − ε − cε |u| dx. n→∞
Ω
Hence u = 0. The preceding lemma implies that u1∗ = 1. Since semi-continuity,
Ω
f u dx = 0 and, by lower
DuΩ S2 (f, Ω), u is a minimizer for S2 (f, Ω).
2
5. Critical minimization problems in W 1,p (Ω) In this section, we assume that Ω is a smooth bounded domain of RN . We define, for 1 < p < N , the critical exponent p ∗ = Np/(N − p) and X0 = W 1,p (Ω), 1,p
X1 = W0 (Ω), 1,p X2 = u ∈ W (Ω) : f u dx = 0 Ω ∗
where f ∈ Lp (Ω) and Ω f dx = 1. The following lemma is a variant of Lemmas 3.2 and 4.1 with a similar proof. ¯ be such that ϕ defined on Xj (where j = 0, 1 or 2) by Lemma 5.1. Let a ∈ C(Ω) ϕ(u) =
|∇u| dx + p
Ω
a|u|p dx Ω
satisfies p cj = inf ϕ(u)/uLp∗ (Ω) : u ∈ Xj \{0} > 0. Let (un ) ⊂ Xj be such that un Lp∗ (Ω) = 1, ϕ(un ) → cj , n → ∞, and un u in Xj . Then either uLp∗ (Ω) = 0 or uLp∗ (Ω) = 1. The preceding lemma is applicable to many quasilinear critical problems as considered e.g. in [8]. Let us define
N p N p S p, R = inf |∇u| dx/uLp∗ (RN ) : u ∈ D R \{0} . RN
T. Bartsch, M. Willem / Journal of Functional Analysis 259 (2010) 3025–3035
3035
The following theorem is a variant of Theorems 3.3, 3.5 and 4.2. Theorem 5.2. (a) If 0 < c0 < S(p, RN )/2p/N , then c0 is achieved. (b) If 0 < c1 < S(p, RN ), then c1 is achieved. (c) If 0 < c2 < S(p, RN )/2p/N , then c2 is achieved. References [1] V. Bouchez, J. Van Schaftingen, Extremal functions in Poincaré–Sobolev inequalities for functions of bounded variation, in press. [2] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983) 437–477. [3] H. Brezis, J. Van Schaftingen, Circulation integrals and critical Sobolev spaces: problems of optimal constants, in: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, in: Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 33–47. [4] P. Cherrier, Meilleures constantes dans des inégalités relatives aux espaces de Sobolev, Bull. Sci. Math. (2) 108 (1984) 225–262. [5] A. Cianchi, A sharp form of Poincaré type inequalities on balls and spheres, Z. Angew. Math. Phys. 40 (1989) 558–569. [6] M. Degiovanni, P. Magrone, Linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator, Calc. Var. Partial Differential Equations 36 (2009) 591–609. [7] F. Demengel, On some nonlinear partial differential equations involving the “1”-Laplacian and critical Sobolev exponent, ESAIM Control Optim. Calc. Var. 4 (1999) 667–686. [8] S. de Valeriola, M. Willem, On some quasilinear critical problems, Adv. Nonlinear Stud. 9 (2009) 825–836. [9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math., vol. 80, Birkhäuser Verlag, Basel, 1984. [10] D. Hulin, M. Troyanov, Mean curvature and asymptotic volume of small balls, Amer. Math. Monthly 110 (2003) 947–950. [11] B. Kawohl, F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Commun. Contemp. Math. 9 (2007) 515–543. [12] N.G. Meyers, W.P. Ziemer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math. 99 (1977) 1345–1360. [13] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976) 353–372. [14] J. Wigniolle, On some nonlinear equations involving the 1-Laplacian with critical Sobolev growth and perturbation terms, Asymptot. Anal. 35 (2003) 207–234. [15] M. Willem, Principes d’analyse fonctionnelle, Cassini, Paris, 2007.
Journal of Functional Analysis 259 (2010) 3037–3079 www.elsevier.com/locate/jfa
Brownian measures on Jordan–Virasoro curves associated to the Weil–Petersson metric Hélène Airault a,b , Paul Malliavin c , Anton Thalmaier d,∗ a INSSET, Université de Picardie, 48 rue Raspail, 02100 Saint Quentin (Aisne), France b Laboratoire CNRS UMR 6140, LAMFA, 33 rue Saint-Leu, 80039 Amiens, France c 10 rue Saint Louis en l’Isle, 75004 Paris, France d Unité de Recherche en Mathématiques, FSTC, Université du Luxembourg, 6 rue Richard Coudenhove-Kalergi,
L-1359 Luxembourg Received 26 April 2010; accepted 5 August 2010 Available online 1 September 2010 Communicated by Daniel W. Stroock Dedicated to Len Gross for his friendship
Abstract In this paper existence of the Brownian measure on Jordan curves with respect to the Weil–Petersson metric is established. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in C requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S 1 ). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales, together with a Douady–Earle type conformal extension of vector fields on the circle to the disk. © 2010 Elsevier Inc. All rights reserved. Keywords: Jordan curves; Conformal welding; Weil–Petersson metric; Douady–Earle extension; Group of diffeomorphisms; Kähler Brownian motion; Shape representation
Contents I.
Kählerian Geometry on the space of C ∞ Jordan curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3038
* Corresponding author.
E-mail addresses:
[email protected] (H. Airault),
[email protected] (A. Thalmaier). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.002
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H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
1. Structure of homogeneous Kähler manifold on C ∞ Jordan curves 2. Douady–Earle infinitesimal extension . . . . . . . . . . . . . . . . . . . 3. Loewner type equation of a conformal welding flow . . . . . . . . . II. Kählerian Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Canonical Brownian on the diffeomorphism group of the disk . . . 5. Regularized welding process, its holomorphy . . . . . . . . . . . . . . 6. Covariance for the ∂¯ of Douady–Earle extension . . . . . . . . . . . . 7. A priori Hölderian estimates for the regularized welding process . 8. Moduli of continuity of regularized welding . . . . . . . . . . . . . . . 9. Welding Brownian measures to Hölderian Jordan curves . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3038 3042 3045 3049 3049 3053 3056 3057 3064 3076 3078 3078
I. Kählerian Geometry on the space of C ∞ Jordan curves A similar topic has been discussed in [5]. In August 2009 Antti Kupiainen pointed out to us that the Hölderianity stated in Section 6 of [5] is in fact not proved there. In the present paper we establish from scratch existence of the Brownian measure on Jordan curves for the Weil– Petersson metric. New key steps of our paper are: i) the use of a Kählerian diffusion on the space of Jordan curves for which the welding functionals give rise to conformal martingales; ii) the construction of a Douady–Earle type extension of vector fields from the circle to the disk. We thank Antti Kupiainen cordially for his careful reading of [5], which has been at the origin of the present work. We also like to mention the interesting paper [8] which constructs probability measures on Jordan curves by a global approach; this method is quite different from the infinitesimal approach based on a stochastic Loewner equation which is used here. Our work is contiguous to several branches of Mathematics: SLE theory (see for instance [25]); Mumford’s theory of vision [34]; representations of Virasoro algebra [21,2,3,22]; Stochastic Differential Geometry on infinite dimensional homogeneous spaces [12,4,16]; stochastic flows under low regularity assumptions [28,26,6,13,14,33]; stochastic PDE theory as developed further in this paper has been started in [5], as resolution of the non-linear Beltrami PDE by a continuity method along a stochastic flow. Our paper is limited to a short and self-contained proof of the result indicated in the title. 1. Structure of homogeneous Kähler manifold on C ∞ Jordan curves A Jordan curve in the complex plane C is a closed subset Γ ⊂ C for which there exists a continuous injective map φ : S 1 → C of the circle S 1 satisfying φ(S 1 ) = Γ . Such a parametrization φ is not unique: given two parametrizations φ1 , φ2 of the same Jordan curve, there exists a homeomorphism h of S 1 such that φ2 = φ1 ◦ h. Two parametrizations define the same orientation of Γ if h is an orientation preserving homeomorphism of the circle S 1 . The inconvenience of this point of view is that indeterminacy in the parametrization depends on an element of an infinite dimensional group, namely the group of homeomorphisms of the circle.
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
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The holomorphic parametrization is constructed in the following way: the complement of Γ in C is the union of two connected open subsets Γ + and Γ − where Γ + is bounded and Γ − is unbounded. The Riemann mapping theorem gives the existence of a conformal map fΓ of the open unit disk D onto Γ + , defined up to composition with an element of the form z →
az + b , ¯ + a¯ bz
a, b ∈ C,
|a|2 − |b|2 = 1.
(1.1)
By a theorem of Caratheodory [10], see also [31], fΓ has a continuous injective extension to the closure D¯ of D, also denoted fΓ , and fΓ |S 1 gives a parametrization of Γ defined canonically up to a transformation of the form (1.1). The advantage of the holomorphic parametrization is that its indeterminacy corresponds to the finite dimensional Poincaré group H of Möbius ¯ + a). transformations z → (az + b)/(bz ¯ Now consider Γ − and let D − = {z: |z| > 1} be the open exterior of the unit disk. There exists a univalent function hΓ : D − → Γ − ,
hΓ (∞) = ∞,
(1.2)
being uniquely defined up to a Möbius transformation of D − preserving ∞, that is up to a rotation. We eliminate this ambiguity by the extra normalization that lim
z→∞
hΓ (z) is a positive number. z
(1.3)
As above, by the theorem of Caratheodory, hΓ extends to the closures and hΓ |S 1 also provides a parametrization of Γ . Let G be the group of orientation preserving homeomorphisms of the circle S 1 = ∂D; further ¯ + a) let H be the group of Möbius transformation z → (az + b)/(bz ¯ of the unit disk D restricted to the boundary ∂D. Then H is a subgroup of G; we consider the homogeneous space M := H \G.
(1.4)
Theorem 1.1. Let fˆΓ , hˆ Γ be the restrictions of fΓ , hΓ to ∂D. The correspondence Γ → fˆΓ−1 ◦ hˆ Γ defines a map Θ : J → M
(1.5)
where J denotes the set of Jordan curves. Proof. The indeterminacy on fΓ through a Möbius transformation appearing on the right is equivalent to the indeterminacy on fΓ−1 through a Möbius appearing on the left. 2 Remark 1.2. Following Sharon and Mumford [34] we introduce the space of shapes S as the orbit space of J under the action of the affine group
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z → αz + β,
α, β ∈ C, α = 0.
(1.6)
We shall use the following result proved in [34]: Θ realizes an injection of S into M ,
(1.7)
which means that Θ(Γ ) = Θ(Γ ) if and only Γ = αΓ + β. We shall say that a Jordan curve is C ∞ if its holomorphic parametrization together with its inverse are C ∞ ; we denote by J ∞ the set of C ∞ Jordan curves. We say that Γ is Hölderian if fΓ is Hölderian together with its inverse; we denote by J h the set of Hölderian Jordan curves. In the same way we denote by G∞ , Gh the groups of C ∞ diffeomorphisms of S 1 , respectively Hölderian homeomorphisms of S 1 . Letting Θ ∞ , Θ h be the restrictions of Θ to J ∞ , respectively J h , then Θ ∞ : J ∞ → G∞ ,
Θ h : J h → Gh .
(1.8)
Problem 1.3. The C ∞ -welding problem is the following problem: given g ∈ G∞ , find univalent functions f, h defined on the closures of D, resp. D − , such that fˆ−1 ◦ hˆ = g.
(1.9)
It is a classical fact that the C ∞ -welding problem has a solution (see [1] and also the examples given in [17, Sections 5 and 6]); therefore the map Θ ∞ is surjective and Θ ∞ realizes a bijection of S ∞ onto M ∞ := H \G∞ .
(1.10)
Denote by g = diff(S 1 ) the right invariant Lie algebra of G∞ constituted by smooth vector fields on S 1 . The identification of smooth vector fields and smooth functions on S 1 by the formula d identifies g and C ∞ (S 1 ). In terms of this identification the Lie bracket is transferred u → u(θ ) dθ to the following expressions: [u, v](θ ) = u(θ )v (θ ) − u (θ )v(θ ).
(1.11)
In the trigonometric basis the bracket has an easy expression: for instance 2[cos kθ, cos pθ ] = (k − p) sin(k + p)θ + (k + p) sin(k − p)θ. The Lie algebra h of H has the basis 1, cos θ, sin θ.
(1.12)
The Weil–Petersson metric is the unique Hilbertian metric on g invariant under the adjoint action of h. The system cos kθ , √ k3 − k
sin kθ , √ k3 − k
k > 1,
(1.13)
is orthonormal for the Weil–Petersson metric and induces on M ∞ the structure of an infinite dimensional Riemannian manifold. The Hilbert transform is defined by
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
J cos kθ = sin kθ,
J sin kθ = − cos kθ
for k 1.
3041
(1.14)
As usual in harmonic analysis, we take the Hilbert transform of a constant function equal to zero. The Hilbert transform possesses the Nijenhuis property with respect to the Lie bracket (1.11): for u, v ∈ g, [J u, J v] − [u, v] = J [u, J v] + [J u, v] .
(1.15)
As J 2 = −1, the Hilbert transform defines on g0 (= the quotient of g by the constant functions) a completely integrable complex structure. It has been proved (see [4]) that M ∞ has the structure of a complex Kähler manifold.
(1.16)
Set :=
2 1 1 2 ∂cos kθ + ∂sin ; √ kθ 2 k3 − k
(1.17)
k>1
then is an elliptic operator on M ∞ . We regularize by introducing r =
k>1
rk 2 2 ∂cos kθ + ∂sin kθ , 3 k −k
r ∈ ]0, 1].
(1.18)
Theorem 1.4. The operators r have the following properties: 1. In the exponential chart, r do not involve first order derivative terms. 2. Any holomorphic functional Φ on J ∞ satisfies r Φ = 0.
(1.19)
Proof. The differential dΦ defines a linear form on the tangent space; the second order differential defines a bilinear form on the tangent space, or equivalently a linear form on the tensor product of the tangent space by itself. Using these notations we have: ∂cos kθ cos kθ, dΦ = −k cos kθ sin kθ, dΦ + cos kθ ⊗ cos kθ, d 2 Φ ; ∂sin kθ sin kθ, dΦ = k sin kθ cos kθ, dΦ + sin kθ ⊗ sin kθ, d 2 Φ ; ∂cos kθ cos kθ, dΦ + ∂sin kθ sin kθ, dΦ = cos kθ ⊗ cos kθ, d 2 Φ + sin kθ ⊗ sin kθ, d 2 Φ . For example, the first of these equations is obtained as follows: with the identification (1.8), we take gε (θ ) = θ + ε cos(kθ ), then d cos kθ, dΦ = Φ θ + ε cos(kθ ) dε ε=0
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and d cos k θ + ε cos(kθ ) , dΦ θ + ε cos(kθ ) . ∂cos kθ cos kθ, dΦ = dε ε=0 In the same way, ∂sin kθ cos kθ, dΦ = −k sin2 kθ, dΦ + cos kθ ⊗ sin kθ, d 2 Φ , ∂cos kθ sin kθ, dΦ = k cos2 kθ, dΦ + cos kθ ⊗ sin kθ, d 2 Φ . According to (1.14) the ∂¯ operator on J ∞ (see [29]) corresponds to ϑ¯ p := ∂cos pθ +
√
−1∂J (cos pθ) = ∂cos pθ +
[ϑp , ϑ¯ p ] vanishes on J ∞ ,
√
−1∂sin pθ ;
(1.20)
p > 1,
(1.21)
where z denotes the real part of a complex number z; therefore 4 r =
p>1
rp rp (ϑp ϑ¯ p + ϑ¯ p ϑp ) = 2 (ϑp ϑ¯ p ). −p p3 − p
p3
2
p>1
The Brownian motion “on” M will be discussed in Section 4; a main feature is that it takes its values in the group Gh , thus getting us out of the C ∞ category where (1.9) has been established. 2. Douady–Earle infinitesimal extension Beurling and Ahlfors characterized the boundary values of quasi-conformal maps of the disk as the quasi-symmetric homeomorphisms of the circle; they gave a construction from a given quasi-symmetric boundary homeomorphism to the quasi-conformal extension. Their methodology is based on Fourier analysis. In [5] we extended the Beurling–Ahlfors construction to general infinitesimal transformations of the circle. In contrast to this, Douady and Earle constructed a canonical and conformally natural extension covariant under the action of the Möbius group. In this paper we shall use an infinitesimal version of the Douady–Earle extension which leads to a more transparent formalism than Fourier approach. Let ϕ be a quasi-symmetric homeomorphism of the circle; its Douady–Earle extension Φ is characterized (see [11, p. 28]) by the identity ∂D
ϕ(ζ ) − Φ(z)
|dζ | = 0. 1 − Φ(z)ϕ(ζ ) |z − ζ |2
In other words, if z ∈ D then Φ(z) is the unique point in D such that (2.1) holds. For instance, taking ϕ0 (ζ ) = ζ , then we get Φ0 (z) = z, which means that ∂D
ζ − z |dζ | = 0. 1 − z¯ ζ |z − ζ |2
(2.1)
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Indeed νz (dζ ) :=
(1 − |z|2 ) × |dζ | |z − ζ |2
is the Poisson kernel of the point z; therefore the holomorphic function h(ζ˜ ) :=
ζ˜ − z 1 − z¯ ζ˜
satisfies
h(ζ˜ ) νz (d ζ˜ ) = h(z) = 0.
∂D
Moreover if ϕ1 (ζ ) =
ζ −a 1 − aζ ¯
and u := Φ1 (z) =
z−a , 1 − az ¯
then 1 − a z¯ ζ −z ϕ1 (ζ ) − u = × . 1 − uϕ ¯ 1 (ζ ) 1 − az ¯ 1 − z¯ ζ Thus the defining relation (2.1) is satisfied for homographic transformations. We proceed now infinitesimally. Let ϕt be a family of diffeomorphisms of ∂D depending smoothly on the parameter t such that ϕ0 = Identity; setting Φt the corresponding Douady–Earle extensions, we have ϕt (ζ ) − Φt (z) |dζ | = 0. (2.2) 1 − Φt (z)ϕt (ζ ) |z − ζ |2 ∂D
As in Vasil’ev [35, Section 5], we extend vector fields on the circle to vector fields inside the disk. Using 1 ζ 1 = = (ζ − z)(1 − z¯ ζ ) (1 − ζ¯ z)(1 − z¯ ζ ) |ζ − z|2 and dζ =
√ −1 ζ |dζ |, we get dζ |dζ | . =i (ζ − z)(1 − z¯ ζ ) |ζ − z|2
Therefore (2.2) takes the form ∂D
ϕt (ζ ) − Φt (z) dζ = 0. 1 − Φt (z)ϕt (ζ ) (ζ − z)(1 − z¯ ζ )
(2.3)
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Differentiating (2.3) relatively to t and setting d v(ζ ) = ϕt (ζ ), dt t=0
d V (z) = Φt (z), dt t=0
(2.4)
gives
S1
v−V dζ + (1 − z¯ ζ )2 (ζ − z)
S1
z¯ v + ζ V¯ dζ = 0. (1 − z¯ ζ )3
(2.5)
Since ζ → is holomorphic, the integral V (z) =
(1 − |z|2 )2 2iπ
ζ V¯ S 1 (1−¯zζ )3
S1
ζ V¯ (1 − z¯ ζ )3
dζ is zero. With Cauchy’s integral formula we obtain
z¯ (1 − |z|2 )2 v(ζ ) dζ + 2iπ (1 − z¯ ζ )2 (ζ − z)
S1
v(ζ ) dζ. (1 − z¯ ζ )3
(2.6)
The ∂¯ derivative of the vector field V (z) in (2.6) has been calculated in Reich and Chen ¯ is obtained in a different manner. [32, formulas (2.1)–(2.3)]. With the following theorem ∂V Theorem 2.1. 1. If v(ζ ) = ζ p where p 0, then V (z) = zp .
(2.7)
p(p + 1) V (z) = z¯ p 1 + p(1 − z¯z) + (1 − z¯z)2 . 2
(2.8)
2. If v(ζ ) = ζ −p where p 1 then
3. Moreover, if V (z) is given by (2.8), then ∂V p(p + 1)(p + 2) = z¯ p−1 (1 − z¯z)2 , ∂ z¯ 2
(2.9)
and if V is given by (2.7) then ∂V = 0. ∂ z¯ For V given by (2.8) we also have
(2.10)
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
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∂V = −p z¯ p+1 1 + (p + 1)(1 − z¯z) . ∂z
(2.11)
Proof. We first prove (2.7); in this case the second integral of (2.6) vanishes and the first integral equals the residues at the point z. To prove (2.8) apply elementary residues to (2.6) as follows: V (z) ×
2iπ 1 = (1 − z¯z)2 (1 − z¯z)2 +
z¯ 1 − z¯z
S1
S1
z¯ v(ζ ) dζ + ζ −z (1 − z¯z)2 v(ζ ) dζ + z¯ (1 − z¯ ζ )2
S1
S1
v(ζ ) dζ 1 − z¯ ζ
v(ζ ) dζ. (1 − z¯ ζ )3
(2.12)
For v(ζ ) = ζ −k , the first integral in (2.12) cancels since
dζ =0 − z)
ζ k (ζ S1
if k 1.
Thus we obtain
1 V (z) z¯ = Resζ =0 (1 − z¯ ζ )ζ k (1 − z¯z)2 (1 − z¯z)2
1 1 z¯ + Resζ =0 + z¯ Resζ =0 . 1 − z¯z (1 − z¯ ζ )2 ζ k (1 − z¯ ζ )3 ζ k
(2.13)
Calculating the three residues gives V (z) z¯ z¯ k(k + 1) k−1 k z¯ k−1 + z¯ z¯ . = z¯ k−1 + (1 − z¯z) 2 (1 − z¯z)2 (1 − z¯z)2
2
3. Loewner type equation of a conformal welding flow Let t → Ct be a map from [0, 1] into the space g, which is assumed to be continuous for the C ∞ topology; assume furthermore that the Fourier coefficients of Ct on 1, cos θ , sin θ vanish.
(3.1)
To these data consider the flow of C ∞ diffeomorphisms of S 1 defined by d gt (θ ) = Ct gt (θ ) , dt
g0 = Identity .
(3.2)
¯ Because of (3.1) Let z → C˜ t (z) be the Douady–Earle extension of Ct to the closed disk D. ∞ ˜ and (2.7)–(2.8), we have Ct (0) = 0. Now consider the flow of C diffeomorphisms g˜ t of D¯ defined by the equation
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d g˜ t (z) = C˜ t g˜ t (z) , dt
g˜ 0 = Identity .
(3.3)
As gt ∈ C ∞ (S 1 ), the conformal welding (1.9) for gt exists; set gt (θ ) = ft−1 ◦ ht exp(iθ ) .
(3.4)
Next define a function Ft on the whole complex plane by Ft (z) :=
|z| > 1, |z| 1.
ht (z), (ft ◦ g˜ t )(z),
(3.5)
As the restriction of g˜ t to ∂D equals gt , we observe that Ft has a continuous extension to the whole complex plane.
(3.6)
Consider the infinitesimal increment
δt (F ) :=
d Ft ◦ Ft−1 . dt
(3.7)
ϕ
Given a univalent function ϕt let Ft := ϕt ◦ Ft . Then we have δt F ϕ = ϕt ϕt−1 × δt (F ) ◦ ϕt−1 + δt (ϕ).
(3.8)
¯ t (ϕ)) = 0 and from (3.8), Moreover, since ϕt is holomorphic, ∂(δ ∂¯ δt F ϕ = ∂¯ ϕt ϕt−1 × δt (F ) ◦ ϕt−1 . In the special case of an affine transformation ϕt (z) = αt z + βt , we find δt F ϕ = ϕt ◦ δt (F ) ◦ ϕt−1 + δt (ϕ) − βt .
(3.9)
Theorem 3.1 (Loewner equation along a conformal welding flow). On D we have d ft = δt (F ) ◦ ft − (∂ft ) × C˜ t dt
(3.10)
d ¯ ∂¯ δt (F ) ◦ ft − (∂ft ) × C˜ t = (∂f t ) = 0. dt
(3.11)
and
Thus δt (F ) satisfies on ft (D) the following identity:
∂¯ δt (F ) = At , We have
At :=
∂ft ∂ft
¯ ˜ × ∂ Ct ◦ ft−1 .
(3.12)
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¯ t At = ∂W
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(3.13)
where Wt is the image of the vector field C˜ t through the map ft , Wt (u) = ft ft−1 (u) C˜ t ft−1 (u) ,
(3.14)
denoting ft (u) = ∂ft (u). On the other hand,
∂¯ δt (F ) (z) = 0,
c z ∈ adherence ft (D) .
(3.15)
Proof. From ft = Ft ◦ g˜ t−1 ¯ t = 0, we arrive at (3.11). To obtain we get formula (3.10), and by taking into account that ∂f (3.12), recall the rule of change of variables for the holomorphic and antiholomorphic derivatives which can be found in [1, p. 8]: ¯ ¯ ◦ v) = (∂u) ¯ ◦ v ∂v + (∂u) ◦ v (∂v). ∂(u
(3.16)
By means of (3.16), taking u = exp(ηδt (F )) and v = ft , the vector fields being considered as infinitesimal transformations, we get ¯ t (F ) × ∂ft . ∂¯ δt (F ) ◦ ft = ∂δ
(3.17)
∂¯ ∂ft × C˜ t = ∂ft × ∂¯ C˜ t .
(3.18)
On the other hand, we have
Eqs. (3.17) and (3.18), along with (3.11), imply the claimed formula (3.12). To prove (3.14)–(3.15), we calculate again exploiting formula (3.16), the expression ∂¯ C˜ t ft−1 (u) = (∂¯ C˜ t ) ft−1 (u) × ∂ft−1 (u). We find ¯ t = ft ft−1 (u) × (∂¯ C˜ t ) ft−1 (u) × ∂ft−1 (u) ∂W which coincides with At since ft (ft−1 (u)) = 1/(ft−1 ) (u).
(3.19)
2
Remark 3.2. Identity (3.10) permits to obtain Kirillov vector fields, see [21], as well as [30] where identities like (3.10) are integrated via line integrals. Identities (3.14)–(3.15) can be deduced directly from (3.8); however they are delicate since the Taylor part in the expansion of Wt is different from f f −1 (u) × (Taylor part of C˜ t ) f −1 (u) . We are able to integrate (3.12) as follows.
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Theorem 3.3. The following identity is valid in the whole complex plane: 1 δt (Ft ) (z) − 2πi
ft (D)
1 At z dz ∧ d z¯ = αt z + βt , z − z
αt ∈ C, βt ∈ C.
(3.20)
Proof. Consider H1 (z) :=
1 2πi
ft (D)
1 At z dz ∧ d z¯ ,
z−z
z ∈ C.
(3.21)
Note that H1 is continuous in C. Using the fact that the Cauchy kernel is the elementary solution of the ∂¯ operator we get At (z), ¯ (∂H1 )(z) = 0,
z ∈ ft (D), z∈ / ft (D).
(3.22)
Therefore setting H2 := H1 − δt (F ), we have c H2 is holomorphic on ft (∂D) and continuous on C; thus by Morera’s theorem, H2 is holomorphic on C. As H2 is of order O(z) at infinity, by Liouville’s theorem, it is an affine function; we conclude by using the fact that H1 (∞) = 0. 2 Remark 3.4. The indeterminacy appearing in formula (3.20) through the choice of αt and βt relies on the fact that our construction is done for the space S of shapes, where objects are defined up to left multiplication by an affine transformation: indeed such a multiplication induces at the level of differentials, as shown in formula (3.9), the addition of an arbitrarily chosen infinitesimal affine transformation. Theorem 3.5 (Holomorphy of the welding functionals). Consider the functional Φ on M ∞ defined by Φ(g) = h,
(3.23)
where h is determined by the welding relation (1.9). Assuming the normalization Φ(g)(z) = z + o(1),
z → ∞,
(3.24)
then for any fixed zo ∈ C, |zo | > 1, the mapping g → Φ(g)(z0 )
(3.25)
is a holomorphic functional for the Kähler structure of M ∞ . Consequently with r being defined as in (1.18), we have r (Φ) = 0.
(3.26)
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Proof. We have to compute the differential of the functional (Φz0 )(g) := Φ(g)(z0 ). Given C ∈ g satisfying (3.1), fix g ∈ G∞ and consider the function hε = Φ exp(εC)g . Then h0 = Φ(g) = h and we have d C, dΦz0 g = hε (z0 ). dε ε=0 The derivative of the variation hε can be calculated by applying the results of Theorems 3.1 and 3.3. Using (3.20) and (3.24), we get according to (3.12) d 1 hε (z0 ) = dε ε=0 2πi
f (D)
1 AC z dz ∧ d z¯
h(z0 ) − z
(3.27)
where the univalent function f on D is given via the welding of g and
AC =
∂f ∂f
¯ ˜ × ∂ C ◦ f −1 .
According to (1.20), holomorphy on M ∞ is equivalent to √ ∂Ck + −1∂Ck Φ(g) (z0 ) = 0,
Ck = cos kθ, Ck = sin kθ.
This vanishing is assured through (3.27) if ∂¯ C˜ k +
√ −1 × ∂¯ C˜ k = 0.
(3.28)
The associated vector fields are
1 k+1 1 1 1 and Ck (ζ ) = Ck (ζ ) = ζ ζ k+1 − k−1 . + k−1 2 2i ζ ζ Taking the extension of Ck (ζ ) + iCk (ζ ) = ζ k+1 , we observe that (3.28) is true as a consequence of Eq. (2.10). Finally the second part of Theorem 1.4 gives (3.26). 2 II. Kählerian Brownian motion 4. Canonical Brownian on the diffeomorphism group of the disk We start by recalling the construction of the canonical Brownian motion “on” G∞ . The regularized canonical Brownian motion on the group of diffeomorphisms of the circle is the stochastic flow on the circle S 1 associated to the Stratonovich SDE r r r dψx,t ψx,t (θ ) (θ ) = dvx,t
(4.1)
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r (θ ) is the regularized g-valued Brownian motion defined by where vx,t
r vx,t (θ ) :=
rk x2k (t) cos kθ + x2k+1 (t) sin kθ , √ 3 k>1 k − k
θ ∈ S1.
(4.2)
Here {x∗ } is a sequence of independent scalar Brownian motions and r ∈ ]0, 1[. r (θ ) constitutes a C ∞ difIt results from Kunita’s theory of stochastic flows [23] that θ → ψx,t 1 r feomorphism of S . It can be proved that the limr→1 ψx,t = ψx,t exists uniformly in θ , defining a random homeomorphism ψx,t which is the so-called canonical Brownian motion “on” Diff(S 1 ); this random homeomorphism is furthermore Hölder continuous [28,13,6]. The corresponding infinitesimal generators r and of these processes are given by (1.18), resp. (1.17). The fact that the construction (4.1)–(4.2) gives Brownian motion with respect to the Levi-Civita connection on M ∞ has been proved in [4, p. 103]. r (θ) r (eiθ ) = eiψx,t r (eiθ ) = iΨ r (eiθ ) d ψ r (θ ). Thus in the variable Writing Ψx,t , then dt Ψx,t t x,t x,t ζ = exp(iθ ), we obtain from (4.2) the vector field r (ζ ) = χx,t
k>1
rk
iζ √ 3 k −k 2
1 1 k k ζ + k x2k (t) − i ζ − k x2k+1 (t) . ζ ζ
(4.3)
1 (ζ ) or equivalently When r = 1, we denote χx,t (ζ ) = χx,t
χx,t (ζ ) =
1 i 1 ζ k+1 x2k (t) − ix2k+1 (t) + k−1 x2k (t) + ix2k+1 (t) . (4.4) √ 2 ζ k3 − k k>1
According to Theorem 2.1 (items 1 and 2), the extension of the vector field (4.4) inside the unit disk is given by Vx,t (z) =
1 i k × Bx,t (z), √ 2 k3 − k
z ∈ D,
(4.5)
k>1
where k (z) := zk+1 x2k (t) − ix2k+1 (t) Bx,t
k(k − 1) k−1 2 1 + (k − 1)(1 − z¯z) + x2k (t) + ix2k+1 (t) . (1 − z¯z) + z¯ 2
(4.6)
Note that for fixed t, the function z → Vx,t (z)
(4.7)
is C ∞ on the open disk and vanishing at 0. As a consequence of its smoothness, Vx,t (∗) induces a flow of local diffeomorphism of D in the sense of Kunita [23], via the Stratonovich SDE dt Ψx,t (z) = (dt Vx,t ) Ψx,t (z) .
(4.8)
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r of V Extending the vector field (4.3) to the disk, we obtain the regularization Vx,t x,t defined
as r Vx,t (z) =
rk i k × Bx,t (z). √ 3 2 k −k
(4.9)
k>1
The corresponding regularized Stratonovich SDE r r r dt Ψx,t Ψx,t (z) (z) = dt Vx,t
(4.10)
defines a local C ∞ flow on the closure D¯ of the unit disk. In terms of polar coordinates z = eiθ e−y ,
resp.
Ψx,t (z) = eiθt e−yt ,
(4.11)
the Stratonovich SDE (4.10) becomes the following Stratonovich SDE ⎧ e−kyt r k ⎪ ⎪ 1 + Rk e2yt cos(kθt ) ◦ dx2k (t) + sin(kθt ) ◦ dx2k+1 (t) , √ ⎪ dθt = ⎪ ⎨ 2 k3 − k k>1
e−kyt r k ⎪ ⎪ ⎪ ⎪ dy 1 − Rk e2yt sin(kθt ) ◦ dx2k (t) − cos(kθt ) ◦ dx2k+1 (t) = √ t ⎩ 3 k>1 2 k − k
(4.12)
where e2y Rk (y) =
k(k + 1) 2y k(k − 1) −2y e − (k − 1)(k + 1) + e . 2 2
(4.13)
Remark 4.1. For y = 0, as it should be, we recover Eq. (4.1). Theorem 4.2. The Itô contraction of the Stratonovich system (4.12) is given by
d(θt + iyt ) ∗ d(θt + iyt ) =
1 −2yt 1 −4yt dt. e − e 2 4
Proof. The Itô contractions will be expressed essentially as a sum of geometric series or derivatives of geometric series. We write the Ito contractions for (4.12) when r = 1, dθt ∗ dθt =
e−2kyt 2 1 + Rk e2yt dt, 3 4(k − k) k>1
dθt ∗ dyt = 0, dyt ∗ dyt =
e−2kyt 2 1 − Rk e2yt dt. 3 4(k − k) k>1
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Thus d(θt + iyt ) ∗ d(θt + iyt ) =
e−2kyt Rk e2yt dt k3 − k k>1
and we have e−2ky 1 1 Rk e2y = e−2y − e−4y . 3 2 4 k −k
2
k>1
Theorem 4.3. For a given z0 ∈ D, let Ψx,t (z0 ) be solution of Eq. (4.8). Consider the stopping time Tz0 = inf t > 0: Ψx,t (z0 ) ∈ ∂D .
(4.14)
Then Tz0 = ∞. Proof. First remark that t → yt is a Markov process: indeed there exists an independent family of scalar Brownian motions ωk , independent of θ , such that ωk law cos(kθt )x2k+1 (t) − sin(kθt )x2k (t).
(4.15)
This allows to compare the Markov processes yt with the process having as infinitesimal generator the ODE d2 1 d q(y) 2 + w(y) , 2 dy dy
(4.16)
where q(y) =
k>1
2 e−2ky 1 − Rk e2y , 3 4(k − k)
1 1 w(y) = e−2y − e−4y . 2 4
(4.17)
We have w(y) > 0, and by (4.13) the estimation of q(y) at y = 0 gives the result. The comparison equation in Itô form reads as d y˜ = y˜ dbt where bt is an abstract Brownian motion; therefore
t y(t) ˜ = y(0) ˜ exp b(t) − 2 which never vanishes.
2
r takes values in the C ∞ orientation preserving diffeomorphisms Theorem 4.4. The process Ψx,t of the open disk D.
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r (z) is a C ∞ diffeomorphism on the open Proof. By Kunita’s theory of stochastic flows [23], Ψx,t random set {z: Tz > t}; we conclude as in Theorem 4.3. 2
Remark 4.5. The Ito contractions for a two points process governed by (4.12) are also expressed (j ) (j ) as sums of geometric series. Denote by (θt , yt ), j = 1, 2, solutions of Eqs. (4.12) with given initial conditions at t = 0. It is not difficult to see that (1)
dθt
(1)
∗ dθt
(1) (2) = Ly (1) ,y (2) θt − θt dt t
t
where Ly (1) ,y (2) (θ ) =
1 1 − cos(θ ) cosh y (1) + y (2) 2 (1) (2) (1) (2) × log 1 − 2e−(y +y ) cos(θ ) + e−(y +y ) + terms bounded in (θ, y)
(4.18)
with θ = θ (1) − θ (2) and y = (y (1) , y (2) ), y (1) 0, y (2) 0. We observe that the induced flow is isotropic in θ , see [26]; moreover it is log-Lipschitzian as in the case of its restriction to the circle, see [6,13,28]. 5. Regularized welding process, its holomorphy This whole section will be written for a fixed value of the regularization parameter r. We start by recalling the classical solution of the smooth welding problem. Define the complex modulus of quasi-conformality r (z) := μΨx,t
r ¯ x,t ∂Ψ r (z) ∂Ψx,t
r of the following Beltrami equation: and consider a solution Fx,t
r ¯ x,t ∂F r (z), μΨx,t (z) = r 0, ∂Fx,t
|z| 1, |z| > 1.
(5.1)
Normalizing the solution by the conditions r (z) = z + o(1), Fx,t
z → ∞,
(5.2)
r is analytically expressible by the Ahlfors–Bojarski series (see [1, Chapt. 5]; then Fx,t [7, Chapt. 5]). This solution is unique and therefore gives rise to a functional on the underlying probability space. Define
r −1 r r (z) = Fx,t ◦ Ψx,t (z), fx,t
z ∈ D;
r hrx,t (z) = Fx,t (z),
z∈ / D.
(5.3)
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Then r is holomorphic and univalent on D, and fx,t
hrx,t is holomorphic and univalent on D¯ c .
(5.4)
Differential calculus along the time variable will permit to use the results of Section 3 established in the case of C ∞ welding depending smoothly on time. Our tool for this purpose is the transfer principle; we proceed by smoothing the Brownian motion. To this end, we fix a mollifier, that is a positive C ∞ function a of compact support contained in the interval [0, 1] and integral equal to 1. To every ε > 0 we associate the smoothened Brownian motion defined as 1 xkε (t) =
xk (t + sε)a(s) ds.
(5.5)
0
Note that xkε (∗) are C ∞ functions and limε→0 xkε (∗) = xk (∗). Replacing in Eqs. (4.5) and (4.6) the Brownian motions x∗ by its smooth regularization x∗ε , we get a C ∞ vector field depending smoothly upon time: Vxrε ,t (z),
¯ z ∈ D,
(5.6)
to which we associate the following non-autonomous ODE: d r Ψ ε (z0 ) = Vxrε ,t Ψxrε ,t (z0 ) . dt x ,t
(5.7)
Theorem 5.1. We have r (z), lim Ψxrε ,t (z) = Ψx,t
ε→0
∀z ∈ C, uniformly on any compact.
(5.8)
Proof. The transfer principle (see for instance [27, Chapt. VIII]) states that the solution of the ODE driven by the regularized Brownian x ε converges locally uniformly towards the corresponding Stratonovich SDE driven by x. 2 Theorem 5.2. Let Fxrε ,t be defined by (5.1) and (5.2) with x replaced by x ε . The following identity is valid on the whole complex plane:
−1 d r 1 Fx ε ,t ◦ Fxrε ,t (z) = dt 2πi
fxrε ,t (D)
1 Arx ε ,t z dz ∧ d z¯
z−z
where
Arx ε ,t
:=
∂fxrε ,t ∂fxrε ,t
−1 r ¯ × ∂V x ε ,t ◦ fxrε ,t .
(5.9)
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Proof. Apply Loewner’s equation established in Theorem 3.1 along with the normalization (5.2). 2 r . Indeed, letting z = Letting ε → 0 in (5.9), we obtain a stochastic differential for Fx,t then
r )−1 (ζ ), (Fx,t
r dt Fx,t (z) =
1 2πi
r (D) fx,t
r r −1 ∂fx,t 1 r ¯ ◦ fx,t × d (ζ1 ) dζ1 ∧ dζ1 . ∂V t x,t r ζ − ζ1 ∂fx,t
(5.10)
Theorem 5.3. Fix a finite subset {zj }j ∈{1,2,...,d} of distinct points of D c , and define j
r wx,t := Fx,t (zj ).
There exist d complex Brownian motions bj such that
j wx,t
j − wx,0
t √ j = ( A)i dbi (t)
(5.11)
0
where the stochastic integrals are of Itô type and where the Hermitian matrix A is given by j
Aq :=
1 4π 2
r (D)2 fx,t
1 j (wx,t
q − z )(wx,t
− z
)
Cfr z , z
dz ∧ d z¯ ∧ dz
∧ d z¯
(5.12)
where r r −1 r −1
∂fx,t ¯ r r −1
r −1 ∂fx,t z , fx,t z f z fx,t z Cfr z , z
:= C r fx,t x,t r r ¯ x,t ∂fx,t ∂f
with ¯ tr (u) ∗ dt ∂V ¯ tr (v). C r (u, v) dt := dt ∂V
(5.13)
Proof. In finite dimension it is well known that the image of a Brownian motion on a Kähler manifold through a holomorphic function is a conformal martingale in C, equal in law to a timechanged complex Brownian motion; this fact extends to finite systems of holomorphic functions. In our case we apply the transfer principle together with the key fact of holomorphy of the conformal welding (Theorem 3.5). This implies the vanishing of Itô contractions induced by the passage from Stratonovich SDE to Itô SDE; only the martingale parts remain and (5.11) is established. Thus the computation of the martingale covariance matrix through Itô calculus involves only first order derivatives computed from (5.9) which finally establishes (5.12). 2
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6. Covariance for the ∂¯ of Douady–Earle extension The next step is the computation of Cfr in (5.12). We start from the definition of the regularized r (z) given in (4.9). According to (2.9)–(2.10) we have vector fields Vx,t i(1 − z¯ z)2 k 3 r (z) = r k − k x2k (t) + ix2k+1 (t) z¯ k−2 ∂¯z Vx,t 4
(6.1)
2 r ¯ x,t (z) = i(1 − z¯ z) r k k 3 − k dx2k (t) + i dx2k+1 (t) z¯ k−2 . dt ∂V 4
(6.2)
k>1
and
k>1
The covariance associated to this random vector field is r r (z ) ¯ x,t ¯ x,t C r (z1 , z2 ) dt = dt ∂V (z1 ) ∗ dt ∂V 2
(6.3)
where ∗ denotes the Itô contraction. Theorem 6.1. For any 0 < r 1, we have C r (z1 , z2 ) =
3r 4 (1 − |z1 |2 )2 (1 − |z2 |2 )2 4(1 − r 2 z¯ 1 z2 )4
(6.4)
and r C (z1 , z2 ) 12 exp −2dH (z1 , z2 ) ,
z1 , z2 ∈ D,
(6.5)
where dH is the Poincaré distance on the unit disk D. Proof. From (6.2) we get C r (z1 , z2 ) =
2 2 3 1 1 − |z1 |2 1 − |z2 |2 k − k r 2k (¯z1 z2 )k−2 . 8 k>1
Using the fact that k 3 − k X k−2 = k>1
6 , (1 − X)4
we obtain (6.4). Next for fixed z1 , z2 ∈ D, we verify that the function u(r) =
r4 , |1 − r 2 z1 z2 |4
0 r 1,
is increasing in r; thus it is enough to prove estimate (6.5) for r = 1.
(6.6)
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Since 1 − |z1 |2 1 − |z2 |2 = |1 − z¯ 1 z2 |2 − |z1 − z2 |2 , we have
|z1 − z2 |2 2 3 C(z1 , z2 ) = 1− . 4 |1 − z¯ 1 z2 |2 Note that the function φ(z1 , z2 ) :=
|z1 − z2 |2 |1 − z¯ 1 z2 |2
is invariant under homographic transformations T (z) =
z−a , 1 − az ¯
i.e. φ(T z1 , T z2 ) = φ(z1 , z2 ).
(6.7)
Hence it is sufficient to obtain the wanted upper bound when z1 = 0; then C(0, z) =
2 3 1 − |z|2 . 4
Denoting r = |z|, we have 1−r exp −dH (0, z) = , 1+r and finally since 0 < r 1, 1 − r 2 2(1 − r) 4 exp −dH (0, z) ; thus 2 1 − r 2 16 exp −2dH (0, z) .
2
7. A priori Hölderian estimates for the regularized welding process Granted to the normalization (5.2) we have hrx,t (z) = z + q(u) :=
k>0 ck z
−k .
Setting
u 1 = , hrx,t (u−1 ) 1 + k>0 ck uk+1
then q is a univalent function on the unit disk D satisfying q(0) = 0, q (0) = 1. Applying the Koebe 1/4-Theorem, we get 14 D ⊂ q(D).
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Lemma 7.1. Denoting for a Jordan curve Γ ⊂ C, |Γ |∞ := sup |z|: z ∈ Γ we have r h (∂D) 4, x,t ∞
∀t 0.
(7.1)
Proof. If z ∈ ∂D, then 1/z ∈ ∂D. We have q(1/z) = 1/ h(z). The two requirements D(0; 1/4) ⊂ q(D) and 1/z ∈ ∂D imply that q(1/z) 1/4; thus h(z) 4. 2 r where as in (6.3), Letting ε → 0 in (5.9), we obtained the stochastic differential (5.10) for Fx,t r r ¯ x,t is given by (6.2). Replacing dt ∂V ¯ x,t by expression (6.2), we the stochastic differential dt ∂V r −1 get with z = (Fx,t ) (ζ ),
r dt Fx,t (z) =
√ 1 k 3 r k − k dx2k (t) + −1 dx2k+1 (t) × Ik (ζ ) 8π
(7.2)
k>1
where we denote Ik (ζ ) = r (D) fx,t
r r −1 ∂fx,t 1 ◦ fx,t × u (ζ1 ) dζ1 ∧ dζ1 k r ζ − ζ1 ∂fx,t
(7.3)
and uk (z) = (1 − z¯z)2 z¯ k−2 . r (∂D), say ζ = F r (z ), ζ = F r (z ), our next objective is to evaluate Letting ζ0 , ζ0 ∈ Fx,t 0 x,t 0 x,t 0 0 the Itô contraction (see for example [24]) r r r r z
dt Fx,t z0 ∗ dt Fx,t (z0 ) − Fx,t (z0 ) − Fx,t 0 .
(7.4)
By means of (7.3), we obtain r r r r z
dt Fx,t z0 ∗ dt Fx,t (z0 ) − Fx,t (z0 ) − Fx,t 0 2 2 2k 3 = r k − k Ik (ζ0 ) − Ik ζ0 dt 64π 2
(7.5)
k>1
and Ik (ζ0 ) − Ik ζ0 = ζ0 − ζ0 × r (D) fx,t
r r −1 ∂fx,t 1 × uk ◦ fx,t (ζ1 ) dζ1 ∧ dζ1 . r (ζ0 − ζ1 )(ζ0 − ζ1 ) ∂fx,t (7.6)
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For the covariance of Brownian motion on the diffeomorphisms of the circle, log-Lipschitzian r = f r ◦ Ψ r restricted to estimates have been established in [28,6,13]. Here, we consider Fx,t x,t x,t the circle. r (z ) in (7.6), we get Carrying out the change of variable ζ1 = fx,t 1 Ik (ζ0 ) − Ik ζ 2 = ζ − ζ0 2 × Lk 0
0
with
1
Lk = D2
r (z ))(ζ
(ζ0 − fx,t 1 0
×
×
r ∂fx,t r ∂fx,t r ∂fx,t r ∂fx,t
r (z )) (ζ − fx,t 1 0
r (z ))(ζ − f r (z )) − fx,t 2 x,t 2 0
r 2 (z1 )uk (z1 )∂fx,t (z1 ) r 2 (z2 )uk (z2 )∂fx,t (z2 ) dz1 ∧ dz1 ∧ dz2 ∧ dz2 .
(7.7)
From (6.6) and (6.5) we have 2k 3 r k − k uk (z1 )uk (z2 ) 8 × 12 exp −2dH (z1 , z2 ) . k>1
Substituting in (7.5) we obtain 2 r r r r z
dt Fx,t z0 ∗ dt Fx,t = ζ0 − ζ0 × J dt (z0 ) − Fx,t (z0 ) − Fx,t 0
(7.8)
where 3 J 2 π
D2
exp(−2dH (z1 , z2 )) r (z ))(ζ − f r (z ))(ζ − f r (z ))(ζ − f r (z ))| |(ζ0 − fx,t 1 0 x,t 1 x,t 2 x,t 2 0 0
r 2 r 2 × ∂fx,t (z1 ) ∂fx,t (z2 ) dz1 ∧ dz1 ∧ dz2 ∧ dz2 .
(7.9)
r (z), this gives Transforming back to the ζ -variable, ζ = fx,t
3 J 2 π
r (D))2 (fx,t
r )−1 (ζ ), (f r )−1 (ζ ))) exp(−2dH ((fx,t 1 2 x,t dζ1 ∧ dζ1 ∧ dζ2 ∧ dζ2 |(ζ0 − ζ1 )(ζ0 − ζ1 )(ζ0 − ζ2 )(ζ0 − ζ2 )|
(7.10)
where as above dH denotes the Poincaré distance on D. Moreover, because of (5.3), the domain of integration for the integral (7.10) is r r Fx,t (D) = fx,t (D).
The following theorem is obtained by establishing an upper bound for J .
(7.11)
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Theorem 7.2. Denote r r r r z
dt Fx,t z0 ∗ dt Fx,t (z0 ) − Fx,t (z0 ) − Fx,t 0
2 r r r r (ζ0 ) − δt Fx,t ζ0 dt, ζ0 = Fx,t z0 . (z0 ), ζ0 = Fx,t =: EFt δt Fx,t r (∂D), Then there exists a numerical constant c, independent of r, such that for all ζ0 , ζ0 ∈ Fx,t
2 r r 2 (ζ0 ) − δt Fx,t ζ0 cζ0 − ζ0 log EFt δt Fx,t
16 . |ζ0 − ζ0 |
(7.12)
Proof. We already proved that 2 2 EFt δt F r (ζ0 ) − δt F r ζ0 cζ0 − ζ0 × J for some constant c where J satisfies estimate (7.10). Recall the definition of the approximate hyperbolic metric for an open subset Ω ⊂ D (see [31, p. 92], [15] and [9]): |dζ | (7.13) dΩ (η1 , η2 ) = inf γ dist(ζ, ∂Ω) γ
where γ is any rectifiable curve joining η1 to η2 . Then by [31, formula (17), p. 9 and formula (6), p. 92] we have for the Poincaré distance dH on D: 1 dH f −1 (η1 ), f −1 (η2 ) dΓ + (η1 , η2 ) 4
(7.14)
where f is a univalent function mapping the disk upon Γ + . As we used (6.5) to derive (7.10), we now have
r −1 r −1 1 (ζ1 ), fx,t (ζ2 ) exp − dΓ + (ζ1 , ζ2 ) . (7.15) exp −2dH fx,t 2 The proof of Theorem 7.2 will be completed after some preparatory lemmas.
2
r (∂D); then Denote by Ω the complement in the complex plane of the two points ζ0 , ζ0 ∈ Fx,t Γ + ⊂ Ω therefore
dΓ + (ζ1 , ζ2 ) dΩ (ζ1 , ζ2 ).
(7.16)
Set δ = δ(ζ0 , ζ0 ) := |ζ0 − ζ0 |, then up to a Euclidean motion of the complex plane, the distance dΩ is characterized by δ. Lemma 7.3. We have J Kδ Fx,t (D) Kδ (4D), where
(7.17)
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
Kδ (B) :=
3 π2
B2
exp(−dΩ (ζ1 , ζ2 )/2) dζ1 ∧ d ζ¯1 ∧ dζ2 ∧ d ζ¯2 . |(ζ0 − ζ1 )(ζ0 − ζ1 )(ζ0 − ζ2 )(ζ0 − ζ2 )|
3061
(7.18)
For all λ > 0, Kδ (B) = Kλδ Hλ (B)
(7.19)
where Hλ denotes the homothety of ratio λ and center (ζ0 + ζ0 )/2; in particular
J K1
8D . δ
(7.20)
Proof. The first inequality in (7.17) is a direct consequence of (7.13)–(7.16) and (7.10)–(7.11); the second inequality a consequence of Lemma 7.1. On the other hand, Hλ (ζ ) = λζ + (1 − λ)
ζ0 + ζ0
; 2
(7.21)
thus Hλ (ζ0 ) =
1+λ 1−λ
ζ0 + ζ , 2 2 0
1+λ 1−λ ζ + ζ0 Hλ ζ0 = 2 0 2
and Hλ ζ0 − Hλ (ζ0 ) = λ ζ0 − ζ0 ,
(7.22)
i.e., Hλ (u1 ) − Hλ (u2 ) = λ(u1 − u2 )
∀u1 , u2 .
With the change of variables ζ1 = Hλ (u1 ), we have φ(ζ1 ) φ(Hλ (u1 )) dζ1 ∧ dζ1 = du1 ∧ du1 .
|(Hλ (ζ0 ) − ζ1 )(Hλ (ζ0 ) − ζ1 )| |(ζ0 − u1 )(ζ0 − u1 )| B
Hλ (B)
In the same way, we carry out the change of variables in integrals of the type φ(ζ1 , ζ2 ) dζ1 ∧ dζ1 ∧ dζ2 ∧ dζ2 .
|(Hλ (ζ0 ) − ζ1 )(Hλ (ζ0 ) − ζ1 )(Hλ (ζ0 ) − ζ2 )(Hλ (ζ0 ) − ζ2 )| Hλ (B)2
Taking φ(ζ1 , ζ2 ) = exp(− 12 dHλ (B) (ζ1 , ζ2 )) and using dΩ (ζ1 , ζ2 ) = dHλ (Ω) Hλ (ζ1 ), Hλ (ζ2 ) , we see that integral in (7.18) is invariant under Hλ and we have (7.19).
(7.23)
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Finally to establish (7.20) we observe that
8D . K1 H1/δ (4D) K1 δ
2
(7.24)
Lemma 7.4. |ζ1 | . lim dΩ (λζ1 , λζ2 ) log λ→∞ |ζ |
(7.25)
2
Proof. Set ρ(ζ ) := |ζ − ζ0 |, r1 := ρ (ζ1 ),
r1 := ρ(ζ1 ),
ρ (ζ ) := ζ − ζ0 , r2 := ρ(ζ2 ),
r2 := ρ (ζ2 ).
Write C(η, a) for the circle of center η and radius a. Since ζ1 ∈ C(ζ0 , r1 ) and ζ2 ∈ C(ζ0 , r2 ), we have dΩ (ζ1 , ζ2 ) dΩ C(ζ0 , r1 ), C ζ0 , r2 .
(7.26)
The distance dΩ (C(ζ0 , r1 ), C(ζ0 , r2 )) vanishes if the two circles intersect which means that r1 − r δ r1 + r . 2
2
(7.27)
Assume that r1 r2 and that (7.27) fails (that is the two circles do not intersect). Then either r2 > δ + r1 or r1 + r2 < δ. In the first case, we have r − δ dΩ C(ζ0 , r1 ), C ζ0 , r2 = log 2 dΩ (ζ1 , ζ2 ), r1 and in the second case, δ − r2
|r − δ| dΩ C(ζ0 , r1 ), D¯ ζ0 , r2 = log = log 2 dΩ (ζ1 , ζ2 ). r1 r1 Thus log
|r2 − δ| dΩ (ζ1 , ζ2 ), r1
r1 r2 ,
log
|r1 − δ| dΩ (ζ1 , ζ2 ), r2
r1 r2 .
(7.28)
The lemma is proved by fixing δ, rewriting (7.28) for λζ1 , λζ2 , and letting λ → ∞ in (7.28).
2
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End of Proof of Theorem 7.2. We have K1 (2λD) − K1 (λD) =
...−
{0<|ζ1 |<2λ} {0<|ζ2 |<2λ}
=2
3063
...
{0<|ζ1 |<λ} {0<|ζ2 |<λ}
....
(7.29)
{λ<|ζ2 |<2λ} {0<|ζ1 |<|ζ2 |}
The second equality in (7.29) is easily obtained by passing to polar coordinates, |ζj | = ρj and integrating over the squares {0 < ρj < 2λ, j = 1, 2} and {0 < ρj < λ, j = 1, 2}. Expressing the volume element in polar coordinates |ζj | = ρj ,
1 4π 2
ζj = ρj exp(iψj ),
2λ ...
{λ<|ζ2 |<2λ} {0<|ζ1 |<|ζ2 |}
λ
ρ2 dρ2 dρ1 −1 1/2 ρ ρ1 c+3 , ρ2 ρ1 2
(7.30)
0
we obtain lim sup K1 (2λD) − K1 (λD) (c + 6) log 2 < ∞.
(7.31)
λ→∞
On the other hand, let φ(λ) be a real-valued, continuous and increasing function of the variable λ; assume that lim φ(2λ) − φ(λ) < +∞.
λ→∞
Then φ(λ) C log λ for λ near ∞. This can be seen as follows: we verify that 0 < A = limn→∞ (φ(2n+1 ) − φ(2n )) < +∞ implies asymptotically φ 2n = φ 2n − φ 2n−1 + φ 2n−1 − φ 2n−2 + · · · n × A =: C log 2n ; then we extend the proof by considering ψ(2λ ) = φ(λ). The combination of formula (7.31) along with (7.20) proves (7.12). It remains to justify (7.30); to this end we need an upper bound for the integrand exp(−dΩ (ζ1 , ζ2 )/2) . |(ζ0 − ζ1 )(ζ0 − ζ1 )(ζ0 − ζ2 )(ζ0 − ζ2 )|
(7.32)
r (∂D), by (7.1), we note that |ζ | 4 and |ζ | 4. For λ > 8 and ρ = |ζ | > λ, Since ζ0 , ζ0 ∈ Fx,t 0 2 2 0 we have
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|ζ0 − ζ2 | >
ρ2 and ζ0 − ζ2 > . 2
ρ2 2
(7.33)
If ρ1 < ρ2 , then log(ρ1 /ρ2 ) < 0. By (7.25), dΩ (λζ1 , λζ2 ) log(ρ2 /ρ1 ).
(7.34)
1/2 1 ρ1 exp − dΩ (λζ1 , λζ2 ) . 2 ρ2
(7.35)
Then, for any λ > 8,
Since in polar coordinates, the volume element is ρj dρj dψj , j = 1, 2, we see that we have to estimate 2λ λ
dρ2 ρ2
ρ2 0
1 ρ1 1/2 ρ1 dρ1 . ρ12 ρ2
(7.36)
2
This gives (7.30).
8. Moduli of continuity of regularized welding 8.1. Local moduli of continuity of regularized welding r is C ∞ , its restriction to ∂D is Hölderian. The purpose of this subsection and the As Fx,t following theorem is to obtain uniform estimates in r and t.
Theorem 8.1. Let r r ζ , η(t) ≡ ηx (t) := Fx,t (ζ ) − Fx,t
ζ, ζ ∈ ∂D,
and log η(s) , s∈[0,t] log η(0)
γ + = sup
log η(s) . s∈[0,t] log η(0)
γ − = inf
Let δ be a constant such that 0 < δ < 1. Then
γ+ δ +1
if and only if
γ− 1−δ
if and only if
inf
s∈[0,t]
r (ζ ) − F r (ζ )| |Fx,s x,s
|ζ − ζ |(1+δ)
2
1
and
sup s∈[0,t]
r (ζ ) − F r (ζ )| |Fx,s x,s
|ζ − ζ |(1−δ)
2
1.
There exists a constant σ (t) depending on t, but independent of r, such that
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
√
2 σ (t)c δ2 log η(0) , γ− 1−δ √ exp − × 1 2σ (t)c2 2π δ log η(0) √
2 σ (t)c δ2 log η(0) . exp − × Prob γ + > δ + 1 √ 1 2σ (t)c2 2πδ log η(0)
Prob
3065
(8.1)
(8.2)
The function σ (t) is independent of r, ζ, ζ and δ and tends to zero as t → 0. Proof. The first two assertions are straightforward. We confine ourselves to prove (8.1) and (8.2). Let log η(s) log 1/η(s) . = (8.3) γs = log η(0) log 1/η(0) r (ζ ) − F r (ζ ) = ζ − ζ . If we assume that η(0) = |ζ − ζ | < 1, By definition, γ0 = 1 and Fx,0 x,0 then log(1/η(0)) > 0. All Itô differentials below are well defined up to the stopping time T = inf s ∈ [0, ∞[: γs < 0 .
The subsequent computations allow to evaluate the probability of the event {T < t} which is of small order. We may limit ourselves to the case where η(0) < 1; these facts legitimate the change of variables in (8.3). We have 1 1 1 √ × (1 − γs ) = log − log , (8.4) log η(0) η(0) η(s) from where we deduce that
1 1 1 − × 1− γ − log . = sup log log η(0) η(0) η(s) s∈[0,t] Furthermore we notice that the condition 1−
γ− >δ
is equivalent to
sup s∈[0,t]
1 − log η(0)
1 log η(s)
> δ log
1 η(0)
which amounts to say
inf
s∈[0,t]
1 − log η(s)
1 log η(0)
< −δ log
1 . η(0)
(8.5)
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In the same way, from (8.4) we obtain
1 1 1 × γ + − 1 = sup − log log log η(0) η(s) η(0) s∈[0,t]
and we conclude that γ+ −1>δ is equivalent to
1 1 1 − log > δ log . sup log η(s) η(0) η(0) s∈[0,t]
(8.6)
The probabilities of the events (8.5) and (8.6) will be evaluated in Lemma F below after several intermediate results. 2 The following lemma serves as a key lemma. Lemma A. Let v(t) ≡ vx,ζ,ζ (t) = Fx,t (ζ ) − Fx,t ζ ,
ζ, ζ ∈ ∂D.
Then dv(t) = A t, v(t) dz(t) + B t, v(t) dt
(8.7)
where z(t) is a Brownian motion in the complex plane, and A(t) cv(t) logv(t) + 1 , B(t) cv(t) 1 + logv(t) .
(8.8)
Proof. If v(t) satisfies (8.7), then the Itô contraction takes the form dv(t) ∗ dv(t) = 2A t, v(t) A t, v(t) dt. By (7.12), we deduce the first inequality in (8.8). The remaining claims result from Itô calculus applied to (7.2). For inequality (8.8), let b(t) be a real Brownian motion, and compare Eq. (7.2) to the Stratonovich SDE
1 dw(t) = c1 w(t) + 1 ◦ db(t), log w(t)
w(0) = v(0) < 1.
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√ ˜ Let A(w) = c1 w( log |w| + 1). In this case, the drift obtained by passing to the Itô SDE is 1 ∂ A˜ ˜ B˜ dt = A dt, 2 ∂w ˜ c2 |w|(log |w| + 1) results. and the estimate |B|
2
The next lemma is obtained from Lemma A by Itô’s formula. r (ζ ) − F r (ζ )|2 = η(t)2 . Then λ(t) is solution of the Itô equation Lemma B. Set λ(t) := |Fx,t x,t
dλ(t) = A1 (t) db(t) + B1 (t) dt
(8.9)
where b(t) is a one-dimensional real Brownian motion and where A1 (t), B1 (t) satisfy
0 < A1 (t) 2cη(t)
2
1+
log
1 η(t)
and
B1 (t) 2c + 4c2 η(t)2 1 + log 1 . η(t) Moreover, for η(t) =
(8.10)
√ λ(t), we obtain dη(t) = α(t) db(t) + β(t) dt
(8.11)
with 0 < α(t) < 2cη(t) log
1 , η(t)
β(t) < 4 c + 2c2 η(t) log 1 . η(t)
(8.12)
The estimates (8.12) are valid for small values of η(t); more precisely up to the first hitting time of η(t) at 1/e. Proof. By Eq. (8.7), we have dv(t) = A(t) dx(t) + i dy(t) + B(t) dt where x(t) and y(t) are two independent real Brownian motions. By Itô calculus, dλ = v d v¯ + v¯ dv + dv ∗ d v¯ ¯ dy + (v B¯ + vB ¯ dt = (v A¯ + vA) ¯ dx + i(vA ¯ − v A) ¯ + 2AA) ¯ dt = C1 dx + C2 dy + (v B¯ + vB ¯ + 2AA)
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with C1 , C2 real-valued. Since C1 dx + C2 dy = b, we obtain
C12 + C22 db for some real Brownian motion
dλ = A1 db + B1 dt with A1 > 0. Moreover, 2 logvx,ζ,ζ (t) + 1 , 0 < A1 2cvx,ζ,ζ (t) 2 |B1 | 2c + 4c2 vx,ζ,ζ (t) logvx,ζ,ζ (t) + 1 . This proves (8.10). √ Now let η = λ. Then again by Itô calculus
A21 1 1 A1 B1 dη = √ dλ − √ dλ ∗ dλ = √ db + √ − √ dt 2 λ 8λ λ 2 λ 2 λ 8λ λ and (8.12) is a consequence of (8.10).
2
Lemma C. Introduce the function 1 1 φ(x) := log , c x
x > 0,
and consider the process η(t) = |v(t)| as in (8.11) and (8.12). Then u(t) = φ η(t) is solution of the following Itô equation: du(t) = α1 (t) db1 (t) + β1 (t) dt,
0 < α1 1, |β1 | c1 u, c1 := 4 c + c2 ,
(8.13)
where b1 (t) is a Brownian motion. Proof. As
1 −1/2 1 1 log ; 2c x x
1 −3/2 1 1 −1/2 1 1 1 φ
(x) = − log log + , 4c x x x 2 2c x2 φ (x) = −
we see that φ
(x) > 0 for x < e−1/2 . Thus 0 < φ
(x) <
1 −1/2 1 1 log . 2c x x2
(8.14)
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By Itô calculus, 1 du(t) = φ η(t) dη(t) + φ
η(t) α 2 (t) dt 2 1 = φ η(t) α(t) db(t) + φ η(t) β(t) + φ
η(t) α 2 (t) dt 2 = α1 (t) db1 (t) + β1 (t) dt where b1 (t) = −b(t) is a Brownian motion and α1 (t) = −φ η(t) α(t),
1 β1 (t) = φ η(t) β(t) + φ
η(t) α 2 (t). 2
The function α1 (t) = −φ (η(t))α(t) satisfies 0 < α1 (t) < 1. The upper bound for β1 is deduced from (8.12). 2 Lemma D. Let u(s) be the process given by (8.13). Consider the two comparison processes du± = db1 ± c1 u± dt,
u± (0) = u(0),
(8.15)
or equivalently
+
+
t
u (t) − u (0) =
exp c1 (t − s) db1 (s),
0 −
−
t
u (t) − u (0) =
exp −c1 (t − s) db1 (s).
(8.16)
0
Then inf u− (s) − u− (0) inf u(s) − u(0)
s∈[0,t]
s∈[0,t]
< sup u(s) − u(0) s∈[0,t]
sup u+ (s) − u+ (0) . s∈[0,t]
Proof. We use Ikeda–Watanabe’s comparison theorem [18].
2
Lemma E. Let m be a positive integer. For the two processes u+ (t) and u− (t), we have
2τ − (t) − m2 − Prob inf u (s) − u (0) < −m exp − − √ s∈[0,t] 2τ (t) m 2π
(8.17)
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with τ − (t) =
exp(2c1 t) − 1 , 2c1 exp(2c1 t)
and
2τ + (t) + m2 + Prob sup u (s) − u (0) > m exp − + √ 2τ (t) m 2π s∈[0,t]
(8.18)
with τ + (t) =
exp(2c1 t) − 1 . 2c1
Proof. We may write t
exp −c1 (t − s) db1 (s) = exp(−c1 t)Bτ (t)
0
where Bt is a Brownian motion and the rescaling τ (t) is given by t τ (t) =
exp(2c1 s) ds = 0
e2c1 t − 1 . 2c1
Let τ −1 (s) =
log(2c1 s + 1) 2c1
be the inverse to τ . Since inf e−c1 s Bτ (s) =
s∈[0,t]
inf
s∈[0,τ (t)]
e−c1 τ
−1 (s)
Bs =
inf
s∈[0,τ (t)]
Bs , √ 2c1 s + 1
we obtain with the reflection principle of Brownian motion: Prob
inf u− (s) − u− (0) < −m = Prob
s∈[0,t]
Prob
inf
s∈[0,τ (t)]
inf
s∈[0,τ (t)]
Bs < −m 2c1 s + 1 Bs < −m 2c1 τ (t) + 1
= Prob |Bτ (t) | m∗
where m∗ := m 2c1 τ (t) + 1
+∞
dx x2 exp − =2 √ 2τ (t) 2πτ (t) m∗
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
3071
√
2 τ (t) m2 √ exp − ∗ . 2τ (t) m∗ 2π
This proves (8.17) with t
−
τ (t) =
2c1
0t 0
exp(2c1 s) ds exp(2c1 s) ds + 1
=
1 exp(2c1 t) − 1 . 2c1 exp(2c1 t)
For inequality (8.18) we proceed in the same way; in this case we find +
τ (t) =
t
exp(−2c1 s) ds 1 1 − exp(−2c1 t) exp(2c1 t) − 1 = = . t 2c1 1 − 2c1 0 exp(−2c1 s) ds 2c1 exp(−2c1 t) 0
2
Lemma F. The two following estimates hold:
1 1 1 Prob inf log − log < −δ log s∈[0,t] η(s) η(0) η(0)
δ 2 log 1 2 τ − (t)c η(0) exp − − √ 2 1 2τ (t)c 2πδ log η(0)
(8.19)
and
1 1 1 − log > δ log Prob sup log η(s) η(0) η(0) s∈[0,t]
δ 2 log 1 2 τ + (t)c η(0) . exp − + √ 2 1 2τ (t)c 2πδ log η(0)
Proof. The inequality
inf
s∈[0,t]
1 − log η(s)
1 log η(0)
< −δ log
is equivalent to δ 1 inf u(s) − u(0) < − log s∈[0,t] c η(0) which implies that − δ 1 − inf u (s) − u (0) < − log . s∈[0,t] c η(0) We apply (8.17) of Lemma E to obtain (8.19).
1 η(0)
(8.20)
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Eq. (8.20) is proved in a similar way,
sup s∈[0,t]
1 − log η(s)
1 log η(0)
> δ log
1 η(0)
is equivalent to δ 1 sup u(s) − u(0) > log . c η(0) s∈[0,t] This implies + δ 1 + sup u (s) − u (0) > log c η(0) s∈[0,t] and we may use (8.18) of Lemma E to conclude.
2
End of Proof of Theorem 8.1. Taking into account that τ − (t) < τ + (t) and that the function √
δ 2 log 1 2 τc η(0) ϕ(τ ) = √ exp − 1 2τ c2 2πδ log η(0) is increasing in τ , we obtain (8.1)–(8.2) with σ (t) = τ + (t).
2
8.2. Moduli of continuity of regularized welding In the previous section, we derived local estimates of modulus of continuity; these are esr (ζ ) − F r (ζ )| for ζ , ζ fixed. Moduli of continuity are obtained from local timates of |Fx,t 0 0 1 x,t 1 estimates through estimates for r ωx,t (ε) =
r r sup Fx,t (ζ0 ) − Fx,t (ζ1 ).
|ζ0 −ζ1 |<ε
(8.21)
We shall implement in this section the classical Kolmogorov methodology of deriving estimates of moduli of continuity of a stochastic process in terms of estimates of its local moduli. Given a continuous function u(θ ) defined for θ ∈ [0, 2π], we consider small intervals where the function has a small variation and we prove Hölderianity on these intervals by means of the triangular inequality, then considering bigger intervals we obtain an estimate in probability of the Hölderian norm. Introduce the dyadic numbers: A=
! n1
An
H. Airault et al. / Journal of Functional Analysis 259 (2010) 3037–3079
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where An = θ : θ = π × p × 2−n , p = 1, . . . , 2n+1 ,
n ∈ N.
The reduced Hölder exponent is defined as γq++ := inf inf
nq θ∈An
log |u(θ ) − u(θ + 2−n π)| . log(2−n π)
(8.22)
Obviously, ++ γq++ γq+1 ,
and for any θ ∈ An , n q, ++ u(θ ) − u θ θ − θ γq
with θ = θ +
π . 2n
(8.23)
Lemma 8.2. For all θ and θ such that 2−q π < |θ − θ | 2−q+1 π , we have ++
−γq ++
γq u(θ ) − u θ 2 2 . ++ θ − θ 1 − 2−γq
(8.24)
Proof. Given 0 < θ < θ 2π , let q ∈ N be such that 2−q π < θ − θ < 2−q+1 π and k ∈ N such that (k − 1)π2−q θ < kπ2−q . The integers q and k then fulfill θ < 2−q kπ θ ,
2−q π < θ − θ < 2−q+1 π.
Consider the dyadic development θ = 2−q k − 2−l ε , π
θ
= 2−q k + 2−l ε , π
>q
ε∗ , ε∗ = 0, 1,
>q
and write u 2−q kπ − u(θ ) = u 2−q kπ − u 2−q kπ − εq+1 2−(q+1) kπ + u 2−q kπ − εq+1 2−(q+1) kπ − u 2−q kπ − εq+1 2−(q+1) kπ − εq+2 2−(q+2) kπ + · · · . Since γ ++ γq++ for q, we get by (8.23) ++ ++ u(θ ) − u 2−q kπ π γq++ 2− γq 2−qγq × >q
++
2−γq 1−2
−γq++
++
× π γq .
We proceed similarly for θ . Since 2−q π < θ − θ , we obtain estimate (8.24).
2
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On the other hand, |u(θ ) − u(θ )| 1 |θ − θ |α
∀θ, θ ∈ An , n q,
(8.25)
implies that π |u(θ ) − u(θ )| 1 for all θ = θ + n , θ ∈ An , n q. |θ − θ |α 2
(8.26)
Taking inequalities (8.23) and (8.24) into account, we see that (8.26) implies sup 2−q π<|θ−θ |2−q+1 π
|u(θ ) − u(θ )| 2 × 2−α . |θ − θ |α 1 − 2−α
(8.27)
Let c 21−α /(1 − 2−α ) be a constant, then by (8.25)–(8.27), the condition sup 2−q π<|θ−θ |2−q+1 π
|u(θ ) − u(θ )| c |θ − θ |α
(8.28)
assures existence of θ, θ ∈ An such that |u(θ ) − u(θ )| 1. |θ − θ |α
(8.29)
Let |u(θ ) − u(θ )| Bn = u: ∃θ, θ ∈ An such that 1 , |θ − θ |α then Bn ⊂ Bn+1
and
Prob
! nq
Bn sup Prob(Bn ),
(8.30)
nq
for any probability measure on the set of considered functions u. Consider the following Hölderian norms: uH α = sup θ,θ
|u(θ ) − u(θ )| , |θ − θ |α
respectively " r " "F "
x,t H α
r (ζ ) − F r (ζ )| |Fx,t x,t .
|α |ζ − ζ
ζ,ζ ∈∂D
= sup
With Theorem 8.1, we obtain uniform estimates in s, r for 0 < s t and 0 < r < 1.
(8.31)
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Theorem 8.3. We have for any 0 < s t and 0 < r < 1, " " Prob "F r "
x,s H (1−δ)2
>8×2
q0 (1−δ)2
√ 1 2 σ (t)c √ × √ q log 2 2πδ q>q0
× exp −
δ2 × q log 2 2σ (t)c2
(8.32)
which is a converging series. r (∂D)| 4 for t 0 and 0 < r < 1. Thus Proof. By (7.1) it holds that |Fx,t ∞
sup
|ζ −ζ |2−q0
r (ζ ) − F r (ζ )| |Fx,t x,t 8 × 2q0 α , |ζ − ζ |α
(8.33)
hence for |ζ − ζ | 2−q0 we have Hölderianity. Next we consider ζ, ζ such that |ζ − ζ | < 2−q0 . We have ! ζ − ζ < 2−q0 = 2−(q+1) ζ − ζ < 2−q .
(8.34)
qq0
For a positive constant K, we may estimate by means of (8.34) as follows: " r " " α > K Prob Prob "Fx,t H +
sup
|ζ −ζ |2−q0
r (ζ ) − F r (ζ )| |Fx,t x,t >K |ζ − ζ |α
Prob
sup 2−(q+1) |ζ −ζ |<2−q
qq0
r (ζ ) − F r (ζ )| |Fx,t x,t >K . |ζ − ζ |α
Because of (8.33), the first term on the right-hand side is zero for K = 8 × 2q0 α . On the other hand, let A=
sup 2−(q+1) |ζ −ζ |<2−q
r (ζ ) − F r (ζ )| |Fx,t x,t |ζ − ζ |α
and consider the event {A > 8 × 2q0 α }. Taking α = (1 − δ)2 , then by (8.1) and (8.28)–(8.30), we get an upper bound for Prob{A > 8 × 2q0 α }. In estimate (8.1) we have η(0) = |ζ − ζ |. For 2−(q+1) |ζ − ζ | < 2−q , observe that η(0) ∼ 2−q and − log η(0) ∼ q log 2. We conclude by using estimate (8.1) since σ (t) does not depend upon ζ and ζ . 2
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8.3. Moduli of continuity of the inverse of regularized welding Theorem 8.4. There exist a positive constant α and a function ϕ(M) independent of r, such that ϕ(M) → 0 as M → ∞, and such that " r −1 " " α > M < ϕ(M) Prob " Fˆx,t H
(8.35)
r denotes the restriction of F r to ∂D. where Fˆx,t x,t
Proof. We use Theorem 8.1 and Lemma 8.2.
2
9. Welding Brownian measures to Hölderian Jordan curves 9.1. Welding of random homeomorphisms Theorem 9.1. Fix δ such that 0 < δ < 1. Then " r " " (1−δ)2 = ∞ A := x: lim sup"Fx,t H
satisfies Prob(A) = 0.
(9.1)
r→1
Proof. Assume that Prob(A) = ε > 0.
(9.2)
Fix q0 such that the right-hand side of (8.32) is smaller than ε/3. Define Bm := x:
inf
r1−m−1
" r " "F "
x,t H
(1−δ)2
2 > 8 × 2q0 (1−δ) ;
then Bm is an increasing sequence of measurable sets and we have A⊂
!
Bm ;
therefore lim Prob(Bm ) ε. m
m
(9.3)
Fix m0 such that Prob(Bm0 ) > 2ε/3. As " 1−m−1 " −1 Bm0 ⊂ x: "Fx,t 0 "H δ−1 8 × 2q0 δ , we deduce by means of (8.32) that Prob(Bm0 ) ε/3. By (9.4) however this would imply that 2/3 1/3.
2
(9.4)
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Theorem 9.2 (Stochastic welding theorem). Almost surely there exist univalent functions hx,t , fx,t such that for some α ∈ ]0, (1 − δ)2 ], hx,t H α < ∞,
fx,t H α < ∞,
(9.5)
and −1 (ζ ), fx,t (ζ ) = hx,t ◦ ψx,t
ζ ∈ ∂D.
(9.6)
Proof. According to (9.1), for almost all x, we find a sequence depending upon x, say rk (x), such that rk (x) → 1, together with " r (x) " sup"Fx,tk "H α < ∞.
(9.7)
k
r (x)
We extract a subsequence rkq (x) such that Fx,tk converges for |z| > 8. Since the limit satisfies z nearby z = ∞, the limit will not be constant but a univalent function hx,t belonging to the space H α . We have rk −1 rk rk (ζ ). fx,tq (ζ ) = hx,tq ◦ ψx,tq rk
−1 in some Hölderian We conclude using [6] which assures that (ψx,tq )−1 converges towards ψx,t norm. 2
9.2. Hölderianity of h−1 x,t Estimate (8.35) has to be used. 9.3. Uniqueness of the welding We take the point of view of [20, p. 304]. The circle S 1 is the boundary of the two closed 1 be the North hemisphere and S 1 the South hemihemispheres of the Riemann sphere. Let S+ − sphere. 1 ⊕ S 1 an equivalence relation where the equivalence Given h ∈ Homeo(S 1 ), we define on S+ − 1 which are idenclasses are composed of single points with the exception of the boundaries ∂S± tified using h. The set of equivalence classes has the structure of a topological manifold h . A continuous function Φ on h is given by the data of a couple of continuous function Φ± defined on the closed hemispheres such that Φ+ (s) = Φ− (h(s)) on the equator. This family of functions forms an algebra Ah ; another equivalent definition is to define h as the Gelfand spectrum of Ah . The welding problem is equivalent to the following question. Question. Does there exist a conformal structure on h which restricted to each of the open hemispheres coincides with the given conformal structure on the hemisphere? For such a conformal structure C , we denote C h the corresponding Riemann surface.
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By Poincaré’s uniformization theorem, it is known that up to a homeomorphism, there is a unique conformal structure on the sphere; this means that there is a homeomorphism Θ carrying 1 ) is a Jordan curve Γ C . A Hölderian Jordan h onto Identity . The image of the equator Θ(∂S+ h curve is by definition a curve which is parametrizable by a univalent function ϕ such that ϕ is Hölderian together with its inverse. C0
Theorem 9.3. Assume that there exists a welding conformal structure C0 such that Γh Hölderian Jordan curve. Then every welding structure C coincides with C0 .
is a
Proof. Let Θ0 , Θ be the corresponding homeomorphisms of h ; then v := Θ ◦ Θ0−1 defines a C new conformal structure on the complement of Γh 0 . By [19, Cor. 2 and Cor. 4, pp. 267–268], there is a unique conformal structure which coincides with the trivial one on the complement C of Γh 0 ; therefore C = C0 . 2 Acknowledgments We thank I. Markina, Yu. Neretin, A. Vasiliev for helpful discussions on related topics. References [1] L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Math. Stud., vol. 10, D. Van Nostrand Co., Inc., Princeton, Toronto, New York, London, 1966. [2] H. Airault, V. Bogachev, Realization of Virasoro unitarizing measures on the set of Jordan curves, C. R. Math. Acad. Sci. Paris 336 (2003) 429–434. [3] H. Airault, P. Malliavin, Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl. (9) 80 (2001) 627–667. [4] H. Airault, P. Malliavin, Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry, J. Funct. Anal. 241 (2006) 99–142. [5] H. Airault, P. Malliavin, A. Thalmaier, Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows, J. Math. Pures Appl. (9) 83 (2004) 955–1018. [6] H. Airault, J. Ren, Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle”, J. Funct. Anal. 196 (2002) 395–426. [7] K. Astala, T. Iwaniec, G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser., vol. 48, Princeton University Press, Princeton, NJ, 2009. [8] K. Astala, P. Jones, A. Kupiainen, E. Saksman, Random curves by conformal welding (Courbes aléatoires par soudure conforme), C. R. Math. Acad. Sci. Paris 348 (2010) 257–262. [9] A.F. Beardon, C. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978) 475–483. [10] C. Carathéodory, Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis, Math. Ann. 73 (1913) 305–320. [11] A. Douady, C.J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986) 23–48. [12] B.K. Driver, M. Gordina, Heat kernel analysis on infinite-dimensional Heisenberg groups, J. Funct. Anal. 255 (2008) 2395–2461. [13] S. Fang, Canonical Brownian motion on the diffeomorphism group of the circle, J. Funct. Anal. 196 (2002) 162–179. [14] S. Fang, T. Zhang, A class of stochastic differential equations with non-Lipschitzian coefficients: pathwise uniqueness and no explosion, C. R. Math. Acad. Sci. Paris 337 (2003) 737–740. [15] F.W. Gehring, B.P. Palka, Quasiconformally homogeneous domains, J. Anal. Math. 30 (1976) 172–199. [16] M. Gordina, M. Wu, Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group, Commun. Stoch. Anal. 2 (2008) 71–95. [17] E. Grong, P. Gumenyuk, A. Vasil’ev, Matching univalent functions and conformal welding, Ann. Acad. Sci. Fenn. Math. 34 (2009) 303–314.
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[18] N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977) 619–633. [19] P.W. Jones, S.K. Smirnov, Removability theorems for Sobolev functions and quasiconformal maps, Ark. Mat. 38 (2000) 263–279. [20] Y. Katznelson, S. Nag, D.P. Sullivan, On conformal welding homeomorphisms associated to Jordan curves, Ann. Acad. Sci. Fenn. Math. 15 (1990) 293–306. [21] A.A. Kirillov, Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl. (9) 77 (1998) 735–746. [22] M. Kontsevich, Y. Suhov, On Malliavin measures, SLE, and CFT, Proc. Steklov Inst. Math. 258 (2007) 100–146. [23] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Stud. Adv. Math., vol. 24, Cambridge University Press, Cambridge, 1990. [24] H. Kunita, S. Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967) 209–245. [25] G.F. Lawler, O. Schramm, W. Werner, Analyticity of intersection exponents for planar Brownian motion, Acta Math. 189 (2002) 179–201. [26] Y. Le Jan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826–873. [27] P. Malliavin, Stochastic Analysis, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 313, Springer-Verlag, Berlin, 1997. [28] P. Malliavin, The canonic diffusion above the diffeomorphism group of the circle, C. R. Math. Acad. Sci. Paris 329 (1999) 325–329. [29] J. Morrow, K. Kodaira, Complex Manifolds, Holt, Rinehart and Winston, Inc., New York, 1971. [30] S. Nag, Singular Cauchy integrals and conformal welding on Jordan curves, Ann. Acad. Sci. Fenn. Math. 21 (1996) 81–88. [31] C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. (Fundamental Principles of Mathematical Sciences), vol. 299, Springer-Verlag, Berlin, 1992. [32] E. Reich, J. Chen, Extensions with bounded ∂-derivative, Ann. Acad. Sci. Fenn. Math. 16 (1991) 377–389. [33] J. Ren, X. Zhang, Stochastic flow for SDEs with non-Lipschitz coefficients, Bull. Sci. Math. 127 (2003) 739–754. [34] E. Sharon, D. Mumford, 2D-shape analysis using conformal mapping, Internat. J. Comput. Vision 70 (2006) 55–75. [35] A. Vasil’ev, Evolution of conformal maps with quasiconformal extensions, Bull. Sci. Math. 129 (2005) 831–859.
Journal of Functional Analysis 259 (2010) 3080–3114 www.elsevier.com/locate/jfa
Schatten–von Neumann properties in the Weyl calculus Ernesto Buzano a , Joachim Toft b,∗ a Department of Mathematics, University of Torino, Italy b Department of Computer Science, Physics and Mathematics, Linnaeus University, Växjö, Sweden
Received 9 April 2009; accepted 31 August 2010 Available online 16 September 2010 Communicated by C. Kenig
Abstract Let Opt (a), for t ∈ R, be the pseudo-differential operator f (x) → (2π )−n
a (1 − t)x + ty, ξ f (y)eix−y,ξ dy dξ
and let Ip be the set of Schatten–von Neumann operators of order p ∈ [1, ∞] on L2 . We are especially concerned with the Weyl case (i.e. when t = 1/2). We prove that if m and g are appropriate metrics and k/2 weight functions respectively, hg is the Planck’s function, hg m ∈ Lp for some k 0 and a ∈ S(m, g), r , then Op (a) is bounded on L2 , p then Opt (a) ∈ Ip , iff a ∈ L . Consequently, if 0 δ < ρ 1 and a ∈ Sρ,δ t ∞ iff a ∈ L . © 2010 Elsevier Inc. All rights reserved. Keywords: Hörmander symbols; Schatten–von Neumann classes; Embeddings; Necessary conditions; Sufficient conditions
0. Introduction The aim of the paper is to continue the discussions in [10,12,26] on general continuity and compactness properties for pseudo-differential operators, especially for Weyl operators, with smooth symbols which belong to certain Hörmander classes. We are especially focused on find* Corresponding author.
E-mail addresses:
[email protected] (E. Buzano),
[email protected] (J. Toft). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.021
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ing necessary and sufficient conditions on particular symbols in order for the corresponding pseudo-differential operators should be Schatten–von Neumann operators of certain degrees. If V is a real vector space of finite dimension n, V its dual space, t ∈ R is fixed and a ∈ S (V × V ) (we use the same notation for the usual functions and distribution spaces as in [18]), then the pseudo-differential operator Opt (a) of a is the continuous linear map from S (V ) to S (V ) defined by Opt (a)f (x) = (2π)−n a (1 − t)x + ty, ξ f (y)eix−y,ξ dy dξ. (0.1) V ×V
(In the case when a is not an integrable function, Opt (a) is interpreted as the operator with Schwartz kernel equal to (2π)−n/2 F2−1 a((1 − t)x + ty, x − y), where F2 U (x, ξ ) denotes the partial Fourier transform F on U (x, y) with respect to the second variable. Here F is the Fourier transform which takes the form F f (ξ ) = f(ξ ) = (2π)−n/2 f (x)e−ix,ξ dx, (0.2) when f ∈ S (V ). See also Section 18.5 in [18].) The operator Op1/2 (a) is the Weyl operator of a, and is denoted by Opw (a). (See (0.1) in Section 1.) r (R2n ), for A family of symbol classes, which appears in several situations, concerns Sρ,δ r, ρ, δ ∈ R, which consists of all smooth functions a on R2n such that α β ∂ ∂ a(x, ξ ) Cα,β ξ r+|α|δ−|β|ρ . x ξ
Here ξ = (1 + |ξ |2 )1/2 . By letting st,∞ be the set of all a ∈ S such that the definition of Opt (a) extends to a continuous operator on L2 , the following is a consequence of Theorem 18.1.11 and r ⊆s the comments on page 94 in [18]: Assume that 0 δ ρ 1 and δ < 1. Then Sρ,δ t,∞ if and only if r 0. The latter equivalence can also be formulated as r ⊆ st,∞ Sρ,δ
⇐⇒
r Sρ,δ ⊆ L∞ .
(0.3)
A similar property holds for any “reasonable” family of symbol classes. This is a consequence of the investigations in [2,3,16,18]. For example, in [16,18], Hörmander introduces a family of symbol classes, denoted by S(m, g), which is parameterized by the weight function m and the Riemannian metric g. (See Section 1 for strict definition.) By choosing m and g in approprir is ate ways, it follows that most of those reasonable symbol classes can be obtained, e.g. Sρ,δ obtained in such way. If m and g are appropriate, then (0.3) is generalized into: S(m, g) ⊆ st,∞
⇐⇒
S(m, g) ⊆ L∞ .
(0.3)
(Here we remark that important contributions for improving the calculus on S(m, g) can be found in [6–9]. For example in [7], Bony extends parts of the theory to a family of symbol classes which contains any S(m, g) when m and g are appropriate.) In [10,26], the equivalence (0.3) is extended in such way that it involves Schatten–von Neumann properties. More precisely, let st,p (V × V ) be the set of all a ∈ S (V × V ) such that Opt (a) belongs to Ip , the set of Schatten–von Neumann operators of order p ∈ [1, ∞] on
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L2 (Rn ). (Cf. Section 1 for a strict definition of Schatten–von Neumann classes.) Then in Theorem 1.1 in [10] equivalence (0.3) is generalized into S(m, g) ⊆ st,p
⇐⇒
S(m, g) ⊆ Lp .
(0.3)
provided certain additional conditions are imposed on g comparing to [16–18]. In [26], Theorem 1.1 in [10] is improved, in the sense that the equivalence (0.3) still holds also when these additional conditions on g are absent (cf. Theorem 4.4 in [26]). Obviously, (0.3) completely characterizes the symbol classes of the form S(m, g) that are contained in st,p . Consequently, a complete characterization of operator classes of the form Opt (S(m, g)) to be contained in Ip follows from (0.3) . On the other hand, (0.3) might give rather poor information about Schatten–von Neumann properties for a particular pseudodifferential operator Opt (a), when a belongs to a fixed but arbitrary symbol class S(m, g). For example, if a ∈ S(m, g) Lp , then (0.3) does not give any information whether Opt (a) belongs to Ip or not. In this context, Theorem 3.9 in [17] seems to be more adapted to particular pseudo-differential operators with symbols in S(m, g), instead of whole classes of such operators. The theorem involves conditions in terms of Planck’s function hg , and can be formulated as: N/2
Assume that hg
m ∈ Lp holds for some N 0 and a ∈ S(m, g),
(0.4)
for p = 1. Then a ∈ Lp
⇒
Opt (a) ∈ Ip ,
(0.5)
for p = 1 and t = 1/2. Equivalently, if (0.4) holds for p = 1, then a ∈ Lp
⇒
a ∈ st,p ,
(0.5)
for p = 1 and t = 1/2. Theorem 3.9 in [17] is extended in [26], where it is proved that if p ∈ [1, ∞], t ∈ R and (0.4) holds, then (0.5) and (0.5) hold. (Cf. Theorem 4.4 and Remark 6.4 in [26].) In Section 2 in the present paper we prove that if (0.4) holds, then (0.5) and (0.5) holds with the opposite implication. Consequently, if (0.4) holds, then a ∈ Lp
⇐⇒
Opt (a) ∈ Ip .
(0.6)
(See Theorem 2.1 and Theorem 2.9.) We remark that a different proof of (0.6) in the case p = ∞ can be found in [12]. In Section 3 we also give some further remarks on embeddings of the form (0.5) in the case p ∈ [1, 2] and t = 1/2 (the Weyl case). More precisely, Theorem 3.9 in [17] was generalized in Proposition 4.5 in [26] as remarked at the above. On the other hand, the proof of Theorem 3.9 in [17] contains some techniques which are not available in [26]. In Section 3 we combine these techniques with arguments in harmonic analysis to prove some stressed estimates of the st,p norm of compactly supported elements in C N . (See Lemmas 3.2–3.4, which might be useful in other problems in the future as well.) Thereafter we combine these estimates with arguments in the proofs of Theorem 4.4 and Proposition 4.5 in [26]. These investigations lead to Theorem 3.1, where slight different sufficiency conditions on the symbols comparing to Theorem 4.4
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and Proposition 4.5 in [26] are obtained in order for the corresponding pseudo-differential operators should be Schatten–von Neumann operators of certain degrees. Roughly speaking, the main differences between Proposition 4.5 (or Theorem 4.4 ) in [26] and Theorem 3.1 is that less regularity is imposed on the symbols in Theorem 3.1, while weaker assumptions are imposed on the parameterizing weight functions in Proposition 4.5 in [26]. r . In Section 4 we apply our results to symbol classes, which are related to Sρ,δ 1. Preliminaries In this section we recall some well-known facts which are needed. After a short review about integration over vector spaces, we continue with discussing certain facts on symplectic vector spaces. Thereafter we recall the definition of the symbol classes, and discuss appropriate conditions for the Riemannian metrics and weight functions which parameterize these classes. 1.1. Integration on vector spaces In order to formulate our problems in a coordinate invariant way, we consider, as in [23,25, 26], integration of densities on a real vector space V of finite dimension n. A volume form on V is positive homogeneous is a non-zero mapping μ : n V \ {0} → C which of order one, i.e. such that μ(tω) = |t|μ(ω), when t ∈ R \ {0} and ω ∈ n (V ) \ {0}. Since n V has dimension 1, the volume form μ is completely determined by μ(e1 ∧ · · · ∧ en ), where e1 , . . . , en is a basis of V . If we fix a volume form μ, it is possible to associate to each function f : V → C a density f μ and define f μ dx ≡ . . . f (x1 e1 + · · · + xn en )μ(e1 ∧ · · · ∧ en ) dx1 · · · dxn , (1.1) Rn
V
n where e1 , . . . , en is any basis of V and x = i=1 xi ei . In fact, it is easy to prove that the integral V f μ dx does not depend on the choice of the basis e1 , . . . , en of V , even though it depends on the volume form μ. If we consider only bases e1 , . . . , en for V such that μ(e1 ∧ · · · ∧ en ) = 1, (1.1) assumes the simpler form f μ dx = . . . f (x1 e1 + · · · + xn en ) dx1 · · · dxn , V
Rn
and therefore we can omit μ in the left-hand side, i.e. f dx = . . . f (x1 e1 + · · · + xn en ) dx1 · · · dxn . V
Rn
Definition (1.1) allows to consider invariant Lp (V ) spaces. Since invariant definition of spaces of differentiable functions like C0∞ (V ) and S (V ) is not a problem, we can also consider the dual spaces of distributions as D (V ) and S (V ).
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If f and g belong to S (V ), we consider the pairing f, g ≡
f gμ dx, V
which extends to the dual pairing between S (V ) and S (V ). We also let (f, g) = f, g for admissible f and g. The extension of (·,·) from S (V ) to L2 (V ) is then the usual scalar product. 1.2. Symplectic vector spaces Next we recall some facts about symplectic vector spaces. A real vector space W of finite dimension 2n is called symplectic if there exists a non-degenerate anti-symmetric bilinear form σ on W , i.e. σ (X, Y ) = −σ (Y, X),
for all X, Y ∈ W,
and σ (X, Y ) = 0,
for every Y ∈ W
⇒
X = 0.
The form σ is called the symplectic form of W . A basis e1 , . . . , en , ε1 , . . . , εn for W is called symplectic if it satisfies σ (ej , ek ) = σ (εj , εk ) = 0,
σ (ej , εk ) = −δj k ,
for j, k = 1, . . . , n. In some situations we use the notation en+1 , . . . , e2n for the vectors ε1 , . . . , εn . Then, with respect to this basis, σ is given by σ (X, Y ) =
n
(yj ξj − xj ηj ), j =1
when X=
n
(xj ej + ξj εj ) j =1
and Y =
n
(yj ej + ηj εj ). j =1
We refer to [18] for more facts about symplectic vector spaces. In order to have invariant measure and integration on the symplectic vector space W , we choose |σ ∧n |/n! as symplectic volume form. Since σ ∧n (e1 ∧ · · · ∧ en ∧ ε1 ∧ · · · ∧ εn ) = n!
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for a symplectic basis e1 , . . . , en , ε1 , . . . , εn (which we sometimes abbreviate as e1 , . . . , εn ), when we integrate on W , we can omit the symplectic volume form, and the integral for a ∈ L1 (W ) becomes
a(X) dX = W
a(x1 e1 + · · · + xn en + ξ1 ε1 + · · · + ξn εn ) dx dξ R2n
=
a(x1 e1 + · · · + ξn εn ) dx dξ. R2n
With this choice of volume form, the measure of subsets of W coincides with the standard Lebesgue measure |U | =
χU dX =
W
χU (x1 e1 + · · · + ξn εn ) dx dξ, R2n
where χU is the characteristic function of U ⊆ W . The symplectic Fourier transform Fσ on S (W ) is defined by the formula Fσ a(X) ≡ π −n
a(Y )e2iσ (X,Y ) dY, W
when a ∈ S (W ). Then Fσ is a homeomorphism on S (W ) which extends uniquely to a homeomorphism on S (W ), and to a unitary operator on L2 (W ). Moreover, (Fσ )2 is the identity operator. Also note that Fσ is defined without any reference of symplectic coordinates. By straight-forward computations it follows that Fσ (a ∗ b) = π n Fσ aFσ b,
Fσ (ab) = π −n Fσ a ∗ Fσ b,
when a ∈ S (W ), b ∈ S (W ), and ∗ denotes the usual convolution. We refer to [14,21–23] for more facts about the symplectic Fourier transform. Next we recall the definition of the Weyl quantization. Let V be a real vector space of finite dimension n, V its dual space and let W = V × V . The vector space W has a natural symplectic structure given by the symplectic form σ (X, Y ) = y, ξ − x, η,
(1.2)
where X = (x, ξ ) ∈ V × V ,
Y = (y, η) ∈ V × V ,
and ·,· is the duality pairing between V and V . Remark 1.1. Observe that when W = V × V , and σ is defined as in (1.2), then a symplectic basis for W is given by any basis e1 , . . . , en for V × {0} together with its dual basis ε1 , . . . , εn
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for {0} × V . We call such a symplectic basis split. Obviously, there are symplectic bases which are not split. On the other hand, assume that W is an n-dimensional symplectic vector space, e1 , . . . , en , ε1 , . . . , εn is a fix symplectic basis, and V and V are the vector spaces spanned by e1 , . . . , en and ε1 , . . . , εn respectively. Then V is the dual of V , with symplectic form as the dual form, and W can be identified with V × V , in which the symplectic basis e1 , . . . , εn is split. The Weyl quantization Opw (a) of a symbol a ∈ S (W ) is equal to Opt (a) for t = 1/2 (cf. the introduction). In particular, if a ∈ S (W ) and f ∈ S (V ), then Opw (a)f (x) = (2π)−n
a (x + y)/2, ξ f (y)eix−y,ξ dy dξ,
(0.1)
V ×V
where f ∈ S (V ) and the integration is performed with respect to a split symplectic basis for W = V × V . The definition of Opw (a) extends to each a ∈ S (W ), giving a continuous operator Opw (a): S (V ) → S (V ). (See [18,21–23].) We also note that Opw (a) = Op1/2 (a), when Opt (a) is given by (0.1). 1.3. Operators and symbol classes We recall the definition of symbol classes which are considered. (See [18].) Assume that a ∈ C N (W ), g is an arbitrary Riemannian metric on W , and that m > 0 is a measurable function on W . For each k = 0, . . . , N , let g |a|k (X) = supa (k) (X; Y1 , . . . , Yk ),
(1.3)
where the supremum is taken over all Y1 , . . . , Yk ∈ W such that gX (Yj ) 1 for j = 1, . . . , k. Also set g
am,N ≡
N
g sup |a|k (X)/m(X) ,
(1.4)
k=0 X∈W
g
let SN (m, g) be the set of all a ∈ C N (W ) such that am,N < ∞, and let S(m, g) ≡
SN (m, g).
N 0
Next we recall some properties for the metric g on W (cf. [25,26]). It follows from Section 18.6 in [18] that for each X ∈ W , there are symplectic coordinates Z = nj=1 (zj ej + ζj εj ) which diagonalize gX , i.e. gX takes the form gX (Z) =
n
j =1
λj (X) zj2 + ζj2 ,
(1.5)
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λ1 (X) λ2 (X) · · · λn (X) > 0,
(1.6)
where
only depend on gX and are independent of the choice of symplectic coordinates which diagonalize gX . The dual metric g σ and Planck’s function hg with respect to g and the symplectic form σ are defined by σ gX (Z) ≡
σ (Y, Z)2 sup Y =0 gX (Y )
gX (Z) and hg (X) = sup σ Z=0 gX (Z)
1/2
respectively. It follows that if (1.5) and (1.6) are fulfilled, then hg (X) = λ1 (X) and σ (Z) = gX
n
λj (X)−1 zj2 + ζj2 .
(1.5)
j =1
In most of the applications we have that hg (X) 1 everywhere, i.e. the uncertainly principle holds. σ for every X ∈ W . It follows that g is symplectic The metric g is called symplectic if gX = gX if and only if λ1 (X) = · · · = λn (X) = 1 in (1.5). We recall that parallel to g and g σ , there is also a canonical way to assign a corresponding symplectic metric g 0 . (See e.g. [26].) More precisely, let Mg = (g + g σ )/2 and define 0 = lim M k g. gX
k→∞
Then g 0 is a symplectic metric, defined in a symplectically invariant way and if Z = n j =1 (zj ei + ζj εj ) are symplectic coordinates such that (1.5) is fulfilled, then 0 gX (Z) =
n
2 zj + ζj2 . j =1
The Riemannian metric g on W is called slowly varying if there are positive constants c and C such that gX (Y − X) c
⇒
C −1 gY gX CgY .
(1.7)
More generally, assume that g and G are Riemannian metrics on W . Then G is called gcontinuous, if there are positive constants c and C such that gX (Y − X) c
⇒
C −1 GY GX CGY .
(1.7)
By duality it follows that g is slowly varying if and only if g σ is g-continuous, and that (1.7) is equivalent to (1.7) , when G = g σ .
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A positive function m on W is called g-continuous if there are constants c and C such that gX (Y − X) c
⇒
C −1 m(Y ) m(X) Cm(Y ).
(1.8)
We observe that if g is slowly varying, N 0 is an integer and m is g-continuous, then SN (m, g) is a Banach space when the topology is defined by the norm (1.4). Moreover, S(m, g) is a Fréchet space under the topology defined by the norms (1.4) for all N 0. The Riemannian metric g on W is called σ -temperate, if there is a constant C > 0 and an integer N 0 such that N gY (Z) CgX (Z) 1 + gYσ (X − Y ) ,
for all X, Y, Z ∈ W .
(1.9)
We observe that if (1.9) holds, then (1.9) still holds after the term gYσ (X − Y ) is replaced by σ (X − Y ), provided the constants C and N have been replaced by larger ones if necessary. (See gX also [18].) More generally, if g and G are Riemannian metrics on W , then G is called (σ, g)-temperate, if there is a constant C and an integer N 0 such that
σ (X − Y ) N , GX (Z) CGY (Z) 1 + gX N GX (Z) CGY (Z) 1 + gYσ (X − Y ) ,
for all X, Y, Z ∈ W.
(1.9)
By duality it follows that G is (σ, g)-temperate, if and only if Gσ is (σ, g)-temperate. In particular, g is σ -temperate, if and only if g σ is (σ, g)-temperate. We also note that if g is σ -temperate and one of the inequalities in (1.9) holds, then G is (σ, g)-temperate. The weight function m is called (σ, g)-temperate if (1.9) holds after GX (Z) and GY (Z) have been replaced by m(X) and m(Y ) respectively. In the following proposition we give examples on important functions related to the slowly varying metric g and which are symplectically invariantly defined. Here we set Λg (X) = λ1 (X) · · · λn (X),
(1.10)
when gX is given by (1.5). Proposition 1.2. Assume that g is a Riemannian metric on W , and that X ∈ W is fixed. Also assume that the symplectic coordinates are chosen such that (1.5) holds. Then the following are true: (1) λj for 1 j n and Λg are symplectically invariantly defined; (2) if in addition g is slowly varying, then λj for 1 j n and Λg are g-continuous; (3) if in addition g is σ -temperate, then λj for 1 j n and Λg are (σ, g)-temperate. Proof. The assertion follows immediately from the fact that
gX (Y ) 1/2 , λj (X) = inf sup σ (Y ) Wj Y ∈Wj \0 gX where the infimum is taken over all symplectic subspaces Wj of W of dimension 2(n − j + 1). 2
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We note that an alternative proof of (1) in Proposition 1.2 can be found in Section 18.5 in [18]. The following definition is motivated by the general theory of Weyl calculus. (See Sections 18.4–18.6 in [18].) Definition 1.3. Assume that g is a Riemannian metric on W . Then g is called (i) feasible if g is slowly varying and hg 1 everywhere; (ii) strongly feasible if g is feasible and σ -temperate. Note that feasible and strongly feasible metrics are not standard terminology. In the literature it is common to use the term “Hörmander metric” or “admissible metric” instead of “strongly feasible” for metrics which satisfy (ii) in Definition 1.3. (See [6–10].) An important reason for us to follow [25,26] concerning this terminology is that we permit metrics which are not admissible in the sense of [6–10], and that we prefer similar names for metrics which satisfy (i) or (ii) in Definition 1.3. Remark 1.4. We note that if g is strongly feasible, then g 0 is strongly feasible, and g and h−s g g g is strongly feasible which is are (σ, g 0 )-temperate when 0 s 1 (cf. [26]). In particular, h−s g also an immediate consequence of Proposition 18.5.6 in [18]. Remark 1.5. Assume that g is slowly varying on W and let c be the same as in (1.7). Then it follows from Theorem 1.4.10 in [18] that there is a constant ε0 > 0, an integer N0 0 and a sequence {Xj }j ∈N in W such that the following is true: (1) gXj (Xj − Xk ) ε0 for every j, k ∈ N such that j = k; (2) W = j ∈N Uj , where Uj is the gXj -ball {X; gXj (X − Xj ) < c}; (3) the intersection of more than N0 balls Uj is empty. Remark 1.6. It follows from Section 1.4 and Section 18.4 in [18] that if g is a slowly varying metric on W , and (1)–(3) in Remark 1.5 holds, then there is a sequence {ϕj }j ∈N in C0∞ (W ) such that the following is true: (1) 0 ϕj ∈ C0∞ (Uj ) for every j ∈ N; gX
(2) supj ∈N ϕj 1,Nj < ∞ for every integer N 0 (i.e. {ϕj }j ∈N is a bounded sequence in S(1, g)); (3) j ∈N ϕj = 1 on W . 1.4. Schatten–von Neumann operators Next we recall some facts about Schatten–von Neumann operators. (see [20].) Let ON0 (V ) be the set of all finite orthonormal sequences {fj }j ∈J in L2 (V ) such that fj ∈ S (V ) for every j ∈ J . Then the linear operator T from S (V ) to S (V ) is called a Schatten–von Neumann operator of order p ∈ [1, ∞] (on L2 (V )), if
p 1/p (Tfj , gj ) < ∞, (1.11) T Ip ≡ sup j ∈J
where the supremum is taken over all sequences {fj }j ∈J and {gj }j ∈J in ON0 (V ). The set of Schatten–von Neumann operators of order p is denoted by Ip . Then Ip is a Banach space
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under the norm · Ip , and I1 , I2 and I∞ are the spaces of trace-class, Hilbert–Schmidt, and continuous operators on L2 (V ) respectively. Moreover, Ip increases with p, · Ip decreases with p, and if T ∈ Ip for p < ∞, then T is compact on L2 (V ). We refer to [20] for more facts about Schatten–von Neumann spaces. For each p ∈ [1, ∞] and t ∈ R, we let st,p (W ) be the set of all a ∈ S (W ) such that Opt (a) ∈ Ip . We also let st, (W ) be the subspace of st,∞ (W ) consisting of all a such that Opt (a) is compact on L2 (V ). The spaces st,p (W ) and st, (W ) are equipped by the norms ast,p ≡ Opt (a)Ip and · st,∞ respectively. It follows that the map a → Opt (a) is an isometric homeomorphism from st,p (W ) to Ip , for every p ∈ [1, ∞] (see [21–23]). Since the Weyl case is particularily interesting we also use the notation spw and sw instead of st,p and st, when t = 1/2. In the following propositions, we recall some facts for the spw -spaces. The proofs are omitted since the results are restatements of certain results in [21–23]. Here and in what follows, p ∈ [1, ∞] denotes the conjugate exponent of p ∈ [1, ∞], i.e. 1/p + 1/p = 1. We also use the ∞ notation L∞ 0 (W ) for the set of all a ∈ L (W ) such that lim ess sup|X|R a(X) = 0, R→∞
where |X| is any Euclidean norm of X ∈ W . We refer to [24] for more facts about the st,p spaces for general t ∈ R. Proposition 1.7. Assume that p, p1 , p2 ∈ [1, ∞] are such that p1 p2 < ∞. Then spw (W ) and sw (W ) are Banach spaces with continuous embeddings w (W ) → S (W ). S (W ) → spw1 (W ) → spw2 (W ) → sw (W ) → s∞
Moreover, s2w (W ) = L2 (W ). If a ∈ S (W ) and T is an affine symplectic map, then Fσ and the pullback T ∗ are homeomorphisms on spw (W ) and on sw (W ), and aspw = T ∗ a s w = Fσ aspw , aspw aspw , 2
a
L∞
1
p
2 a , n
a
s1w
s2w
= (2π)−n/2 aL2 .
Proposition 1.8. Assume that p ∈ [1, ∞]. Then the following is true: (1) the bilinear form ·,· on S (W ) and the L2 -form (·,·) on S (W ) extend uniquely to the duality between spw (W ) and spw (W ), and for every a ∈ spw (W ) and b ∈ spw (W ) it holds (a, b) as w bs w a, b as w bs w , p p p
p
and aspw = supa, c = sup(a, c) where the suprema are taken over all c ∈ spw (W ) such that cs w 1; p (2) if p < ∞, then the dual space for spw can be identified with spw through the form ·,· or (·,·).
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In what follows we let Br (X) denote the open ball with center at X ∈ W and radius r, provided there is no confusion about the Euclidean structure in W . For future references we also set B(X) = B1 (X). The following result is an immediate consequence of [23, Corollary 2.12]. Proposition 1.9. Assume that p ∈ [1, ∞] and r ∈ (1, ∞). Then spw (W ) ∩ E (W ) = Fσ Lp (W ) ∩ E (W ), and for some constant C which only depends on r and n it holds C −1 Fσ aLp aspw CFσ aLp ,
(1.12)
for all a ∈ E (Br (0)). Here the open ball Br (0) is taken with respect to any choice of symplectic coordinates. The next proposition concerns interpolation properties. Here and in what follows we use similar notations as in [5] concerning interpolation spaces. Proposition 1.10. Assume that p, p1 , p2 ∈ [1, ∞] and 0 θ 1 such that 1/p = (1 − θ )/p1 + θ/p2 . Then the (complex) interpolation space (spw1 , spw2 )[θ] is equal to spw with equality in norms. 2. Necessary and sufficient conditions for symbols to define Schatten–von Neumann operators 2.1. Necessary and sufficient conditions for symbols in the Weyl calculus In this subsection we continue the discussion from [10,26] concerning Schatten–von Neumann properties for pseudo-differential operators. We discuss necessity for symbols in S(m, g) in order to the corresponding Weyl operators should be Schatten–von Neumann operators of certain degrees. We essentially prove that the sufficiency results in Section 6 in [26] are to some extent also necessary. More precisely we have the following result. Theorem 2.1. Assume that p ∈ [1, ∞], g is strongly feasible, m is g-continuous, and that a ∈ S(m, g). Then the following is true: k/2
(1) if hg m ∈ Lp (W ) for some k 0, then a ∈ spw (W ) if and only if a ∈ Lp (W ); w ∞ (2) if hg m ∈ L∞ 0 (W ) for some k 0, then a ∈ s (W ) if and only if a ∈ L0 (W ). k/2
Using completely different techniques, Theorem 2.1 has already been proved in [12] when p = ∞ and m is (σ, g)-temperate. For general p, Theorem 2.1 an immediate consequence of Proposition 2.2 and Proposition 2.3 below. It suffices to prove Proposition 2.3, since Proposition 2.2 is an immediate consequence of Proposition 4.5 in [26]. Proposition 2.2. Assume that g is feasible when p ∈ [1, 2] and strongly feasible when p ∈ (2, ∞], and that m is g-continuous. Then the following is true:
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(1) if hg m ∈ Lp (W ) for some k 0, then there exists an integer Np 0 such that if N Np , then SN (m, g) ∩ Lp (W ) ⊆ spw (W ),
(2.1)
k/2 g aspw C aLp + am,N hg mLp ,
(2.2)
and
for some constant C which is independent of a ∈ SN (m, g) ∩ Lp (W ); k/2 (2) if hg m ∈ L∞ 0 (W ) for some k 0, and N N∞ , then w SN (m, g) ∩ L∞ 0 (W ) ⊆ s (W ).
(2.3)
The next result is the needed converse of Proposition 2.2. Proposition 2.3. Assume that g is strongly feasible when p ∈ [1, 2) and feasible when p ∈ [2, ∞], and that m is g-continuous. Then the following is true: k/2
(1) if hg m ∈ Lp (W ) for some k 0, then S(m, g) ∩ spw (W ) ⊆ Lp (W );
(2.4)
(2) if hg m ∈ L∞ 0 (W ) for some k 0, then k/2
S(m, g) ∩ sw (W ) ⊆ L∞ 0 (W ).
(2.5)
The proof of Proposition 2.3 is based on certain estimates, and duality arguments in combination with Proposition 2.2. We need three preparing lemmas for the proof. In the second lemma we present important estimates for derivatives in S(m, g). The first lemma, which is needed for the second lemma is essentially the same as Lemma 3.1 of [19]. The third lemma which is related to Proposition 2.2 is convenient in the duality arguments. Lemma 2.4. Assume that f ∈ C 2 ([0, r]). Then f (0) 4 r −1 + 1 max f (t) + max f (t) . t∈[0,r]
t∈[0,r]
Proof. We may assume f (0) = 0. Set Mj = max f (j ) (t), t∈[0,r]
for j = 0, 2.
By the mean value theorem we have |f (t) − f (0)| M2 t, for all t ∈ [0, r]. Then f (0)/2 f (t)
for 2M2 t f (0) and 0 t r.
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Let δ = min{r, |f (0)|/(2M2 )}. Then using the mean value theorem again it follows that f (δ) − f (0) = f (s)δ for some s ∈ [0, δ]. This gives f (0)/2 f (s) f (δ) − f (0)/δ 2M0 /δ. Then either f (0)/2 2M0 /r
or
f (0)/2 4M0 M2 /f (0).
The result now follows by combining these inequalities.
2
Lemma 2.5. Assume that g is slowly varying, N ∈ N, and consider the open ball UX = Y ∈ W ; gX (Y − X) < c , where c is the same as in (1.7). Then there exists a positive constant C0 , depending only on N , n and the constants in (1.7) such that g
sup sup |a|k (Y ) C0
kN Y ∈U X
g sup a(Y ) + sup |a|N (Y ) ,
Y ∈U X
Y ∈U X
for all X ∈ W and all a ∈ C N (W ). Proof. By induction we may assume N = 2. Let X0 ∈ U X be fixed. We shall find an appropriate basis e1 , . . . , e2n , orthonormal with respect to gX , and such that (2.6) X0 + tej ∈ UX , for 0 t < c/2n. Let us first show that it is always possible to find e1 , . . . , e2n such that (2.6) is fulfilled. Since this is obviously true for X0 = X, we may assume X0 = X. Let g˜ X (Y, Z) = gX (Y + Z) − gX (Y − Z) /4 be the polarization of g and choose the basis e1 , . . . , e2n such that g˜ X (X0 − X, ej ) = − gX (X0 − X)/(2n), for j = 1, . . . , 2n. This is possible if we choose e1 , . . . , e2n in such way that X0 − X = −t0 (e1 + · · · + e2n ) for some t0 > 0. Then we have gX (X0 + tej − X) = gX (X0 − X) − 2t gX (X0 − X)/(2n) + t 2 2 = t − gX (X0 − X)/(2n) + 1 − (2n)−1 gX (X0 − X) < c, since it follows from the assumptions that − c/(2n) t − gX (X0 − X)/(2n) c/(2n)
and gX (X0 − X) < c.
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Since g˜ X (Z, ej ) 1, for gX (Z) = 1, we have 2n
(1) a (X0 ; ej ), sup a (1) (X0 ; Z) C
g
|a|1 (X0 ) C
gX (Z)=1
j =1
where C is the same as in (1.7). √ If we let f (t) = a(Z + tei ) and r = c/2n, Lemma 2.4 shows that 2n
(1) (2) a (X0 ; ei ) 4 2n/c + 1 max a(Y ) + max a (Y ; ej , ej ) , Y ∈U X
j =1 Y ∈U X
(2.7)
for i = 1, . . . , 2n. The result now follows from (2.7) and 2n
g max a (2) a(Y ; ej , ej ) 2n max |a|2 (Y ),
j =1 Y ∈U X
which completes the proof.
Y ∈U X
2
Lemma 2.6. Assume that g is feasible and set GX = hg (X)−1/2 gX . Then the following is true: (1) G is feasible. If in addition g is strongly feasible, then G is strongly feasible; (2) if g is strongly feasible when p ∈ [1, 2] and feasible when p ∈ (2, ∞], and m is g-continuous k/2 and satisfies hg m ∈ Lp for some k 0, then there is an integer N 0 such that N/4
hg
m ∈ Lp (W ),
N/4
hG mp−1 ∈ Lp (W ),
and SN mp−1 , G ∩ Lp ⊆ spw .
(2.8)
Furthermore, for some constant C it holds N/4 h m p , bs w C bLp + bG p−1 G L ,N m p
b ∈ S mp−1 , G .
(2.2)
Proof. We only prove the result in the case p < ∞. The case p = ∞ follows by similar arguments and is left for the reader. Since hg 1, it follows from Remark 1.4 that G is feasible (strongly feasible) when g is feasible (strongly feasible). The remaining part follows from Propo N/4 p−1 p ∈ Lp , sition 2.2 and the facts that hg 1, hN g m ∈ L for some N 0, if and only if hG m for some N 0. 2 Proof of Proposition 2.3. We only prove (1) in the case p < ∞. The case p = ∞ and assertion (2) follow by similar arguments and are left for the reader. k/2 / Lp (W ). We shall combine ProposiAssume that k 0, hg m ∈ Lp , a ∈ S(m, g) and a ∈ tion 2.2 with appropriate estimates, and duality to prove that a ∈ / spw , which will give the result. p We start to prove that a ∈ / L implies that |a| satisfies appropriate estimates from below in a union of convex sets in W with infinite measure. In fact, let GX = hg (X)−1/2 gX , and let Uj and Xj for j ∈ N be the same as in Remark 1.5 after g has been replaced by G. Also let N 0 be chosen such that (2.8) and (2.2) are fulfilled, let I0 be the set of all j ∈ N such that
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2hg (X)N/4 m(X) |a(X)| for some X ∈ U j , and set for each J ⊆ N, ΩJ = j ∈J Uj . For each j ∈ N we choose a point Yj ∈ U j such that |a(X)| |a(Yj )| when X ∈ U j . Then it follows that p N/4 I0 is an infinite set and that aLp (ΩI0 ) = +∞, since a ∈ Lloc (W ) \ Lp (W ) and hg m ∈ Lp . In a moment we shall prove that there are positive constants C and r0 , and a sequence {Xj0 }j ∈I0 such that for any j ∈ I0 it holds Uj0 = Y ∈ W ; GXj Y − Xj0 < 4r02 ⊂ Uj ,
(2.9)
hg (X)N/4 m(X) a(Yj )/2 a(X) a(Yj ), |a|G k (X) C a(X) ,
(2.10)
and
for all X ∈ Uj0 and k N . Admitting this for a while we may proceed as follows. Let Uj1 be the open ball with center at Xj0 and radius r0 (with respect to the metric GXj ), and choose a bounded sequence {ϕj }j ∈I0 in S(1, G) such that 0 ϕj 1, ϕj ∈ C0∞ (Uj0 ) and ϕj = 1 in Uj1 . Also let J be an arbitrary finite subset of I0 . Then it follows from (2.9) and (2.10) and the fact that there is a bound of overlapping Uj , that for some constant C which is independent of j ∈ I0 and J it holds 1 0 U U |Uj | C U 1 , j
j
j
j ∈ I0
and
1/p p a(Yj )p U 1 C a j
j ∈J
1/p
Lp (Uj1 )
j ∈J
C aLp (ΩJ ) C 2
3
a(Yj )p U 1 j
1/p
j ∈J
Now we let bJ (X) =
j ∈J
p−2 p−1 a(X)a(X) ϕj (X) aLp (ΩJ ) .
Since there is a bound of overlapping Uj , (2.11) gives bJ Lp
1 p−1
aLp (ΩJ )
C1
p
1/p a(X)p−1 ϕj (X) dX
1 p−1
aLp (ΩJ )
j ∈J
j ∈J U
j
a(X)p dX
1/p
.
(2.11)
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C2
1 p−1
aLp (ΩJ )
a(X)p dX
(p−1)/p = C2 ,
ΩJ
for some constants C1 and C2 . Hence sup bJ Lp (ΩJ ) < ∞. J
Furthermore, by (2.10) it follows that the set of all bJ is a bounded subset of SN (mp−1 , G). Hence (2.2) and (2.8) give sup bJ s w < ∞. p
J
By Proposition 1.8(1) and (2.11) it follows now that there are positive constants C1 and C2 which are independent of J such that aspw C1 (a, bJ ) C2 aLp (ΩJ ) . By letting J increase to I0 we therefore obtain aspw C2 aLp (ΩI0 ) = ∞, which proves the assertion. It remains to prove (2.9) and (2.10). By Lemma 2.5 we have sup |a|G k (X) C X∈U j
sup a(X) + sup |a|G (X) N Y ∈U j
X∈U j
(X) , C a(Yj ) + sup |a|G N
for all j ∈ I0 and k N.
(2.12)
Y ∈U j
On the other hand we have N/4
|a|G N (X) = hg
g
N/4
(X)|a|N (X) Chg
(X)m(X) C a(Yj ),
(2.13)
for all X ∈ Uj . From (2.12) and (2.13) we obtain sup |a|G k (X) C a(Yj ) ,
for all j ∈ I0 and k N.
X∈U j
Next we consider the Taylor expansion 1 a(X) = a(Yj ) +
a (1) Yj + t (X − Yj ); X − Yj dt,
0
which, together with (2.14), yields the estimate a(Yj ) a(X) + C a(Yj )GX (X − Yj )1/2 , j
(2.14)
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for all X ∈ Uj and j ∈ I0 . But then we can choose ε1 > 0 so small that a(Yj ) 2a(X), Let ε2 =
√
for X ∈ Uj , GXj (X − Yj ) < ε12 and j ∈ I0 .
c, with c as in Remark 1.5 and define Xj0 =
1 ε1 1 ε1 + 2ε2 Xj + Yj 2 ε1 + ε2 2 ε1 + ε2
and r0 =
1 ε1 ε2 . 4 ε1 + ε2
Then it is easy to check that (2.9) and (2.10) are satisfied and this completes the proof.
2
2.2. Necessary and sufficient conditions for symbols in other pseudo-differential calculi In this subsection we extend the results of Subsection 2.1 to other calculi of pseudo-differential operators, whose definition is a natural generalization of the Weyl quantization (0.1) . As in the definition of the Weyl quantization given in Subsection 1.2, let V be a real vector space of finite dimension n, V its dual space, and W = V × V the symplectic vector space with the symplectic form (1.2). Let t ∈ R be fixed, and assume that a ∈ S (W ). Then the pseudodifferential operator Opt (a) is defined by the formula (0.1) when f ∈ S (V ). We recall that the operator Opt (a) is continuous on S (V ), and the definition of Opt (a) extends to each a ∈ S (W ), and then Opt (a) is a continuous operator from S (V ) to S (V ). Moreover, the map a → Opt (a) from S (W ) to the set of linear and continuous operators from S (V ) to S (V ) is bijective. (See [18].) We note that a(x, D) = Op0 (a) is the standard representation (Kohn–Nirenberg representation) and Opw (a) = Op1/2 (a) is the Weyl quantization. We also recall that if s, t ∈ R and a, b ∈ S (W ) are arbitrary, then Ops (a) = Opt (b)
⇐⇒
a(X) = ei(s−t)Φ(D) b(X),
where Φ(X) = x, ξ
(2.15)
and X = (x, ξ ) ∈ V × V . We note that the right-hand side of (2.15) is equivalent to F a(X) = ei(s−t)Φ(X) F b(X). (See the introduction for the definition of the Fourier transform F .) In particular, eitΦ(D) is a bijective and continuous mapping on S (W ) which extends uniquely to bijective and continuous mapping on S (W ), and to a unitary operator on L2 (W ). The extension of the symbolic calculus to pseudo-differential operators of the kind (0.1) requires that the metric g has to be split (see [8]), i.e. g should satisfy the following identity gX (z, ζ ) = gX (z, −ζ ),
(2.16)
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for all X ∈ W , z ∈ V , and ζ ∈ V . (Cf. the discussion after Theorem 18.5.5 and before Theorem 18.5.10 in [18].) Observe that (2.16) is equivalent to gX (z, ζ ) = g1,X (z) + g2,X (ζ ),
(2.17)
where g1 and g2 are positive definite quadratic forms on V and V respectively. The diagonalization of the metric assume a special form when g is split. Recall that the definition of split symplectic basis is given in Remark 1.1. Lemma 2.7. Assume that g is split on W = V × V . Then for all X ∈ W there exists a split symplectic basis e1 , . . . , en , ε1 , . . . , εn such that gX (Z) =
n
λj (X) zj2 + ζj2
j =1
for all Z = (z, ζ ) =
n
j =1 (zj ej
+ ζj εj ).
Proof. Since it is well known that it is possible to diagonalize two quadratic forms, it follows from (2.17) that there exists a split symplectic basis e˜1 , . . . , e˜n , ε˜ 1 , . . . , ε˜ n such that n
2 gX (Z) = z˜ j + μj (X)ζ˜j2 , j =1
where Z =
n
μ1/4 e˜j , and
zj e˜j + ζ˜j ε˜ j ), and μ1 · · · μn j =1 (˜ −1/4 εj = μj ε˜ j , for j = 1, . . . , n. 2
1/2
> 0. Then it suffices to set λj = μj , ej =
The following proposition follows from Proposition 18.5.10 in [18]. Proposition 2.8. Assume that g is strongly feasible and split, and that m is g-continuous and (σ, g)-temperate. Also assume that t ∈ R. Then eitΦ(D) on S (W ) restricts to a homeomorphism on S(m, g). Furthermore, for every integer N 0 and a ∈ S(m, g) it holds eitΦ(D) a −
k itΦ(D) a/k! ∈ S hN g m, g .
(2.18)
k
Now we can state the extension of Theorem 2.1, when the involved metrics are split. Theorem 2.9. Assume that p ∈ [1, ∞], t ∈ R, g is strongly feasible and split, m is g-continuous and (σ, g)-temperate, and that a ∈ S(m, g). Then the following is true: k/2
(1) if hg m ∈ Lp (W ) for some k 0, then a ∈ st,p (W ) if and only if a ∈ Lp (W ); k/2 ∞ (2) if hg m ∈ L∞ 0 (W ) for some k 0, then a ∈ st, (W ) if and only if a ∈ L0 (W ). Theorem 2.9 is an immediate consequence of (2.15), Theorem 2.1 and the following result.
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Proposition 2.10. Assume that p ∈ [1, ∞], g is strongly feasible and split, and that m is gN/2 N/2 continuous, (σ, g)-temperate and satisfies hg m ∈ Lp (W ) (hg m ∈ L∞ 0 (W )) for some N 0. If t ∈ R is fixed, then ei tΦ(D) on S (W ) restricts to a continuous isomorphism on S(m, g) ∩ Lp (W ) and on S(m, g) ∩ L∞ 0 (W ). Proof. We only prove the result for p < ∞. The remaining cases follow by similar arguments and are left for the reader. We need to prove that b = eitΦ(D) a ∈ Lp (W ) whenever a ∈ S(m, g) ∩ Lp (W ). Let N0 , {Xj }j ∈N , {Uj }j ∈N and G be as in the proof of Proposition 2.3, and let {ϕj }j ∈N be as in Remark 1.6. Also let {ψj }j ∈N be a bounded set of non-negative functions in S(1, G) such that 1/2 supp ψj ⊆ Uj and ψj = 1 on supp ϕj . Then G is strongly feasible. Since hG = hg 1, it follows from Proposition 2.8 that
k p itΦ(D) a/k! ∈ S hN g m, g ⊆ L (W ).
eitΦ(D) a −
k
We therefore need to prove that Φ(D)k a ∈ Lp (W ) when k < N . By Lemma 2.7 there exists a split symplectic basis e1 , . . . , en , ε1 , . . . , εn such that GXj (Z) =
n
λi (Xj ) zi2 + ζi2
i=1
for all Z = (z, ζ ) =
n
i=1 (zi ei
+ ζi εi ). Let aj = ϕj a and j ∈ N and
Hj (z1 , . . . , zn , ζ1 , . . . , ζn ) = aj (Xj + z1 e1 + · · · + zn en + ζ1 ε1 + · · · + ζn εn ). By Theorem 4.13 in [1] or Lemma A.1 in Appendix A there exists a positive constant C depending only on N , n and p such that C Hj Lp +
α β ∂ ∂ Hj z ζ
Lp
γ ∂z ∂ δ Hj ζ
,
Lp
|γ +δ|=N
when |α + β| N . In particular we obtain Dz , Dζ k Hj
Lp
C Hj Lp +
, Lp
γ ∂z ∂ δ Hj ζ
|γ +δ|=N
for k < N . Since ∂zi Hj (z1 , . . . , zn , ζ1 , . . . , ζn ) = a (1) (X + Z; ei ) and ∂ζi Hj (z1 , . . . , zn , ζ1 , . . . , ζn ) = a (1) (X + Z; εi )
(2.19)
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for i = 1, . . . , n, we have Dz , Dζ k Hj (z1 , . . . , zn , ζ1 , . . . , ζn ) = Φ(D)k aj (Xj + Z) and α β ∂ ∂ Hj (z1 , . . . , zn , ζ1 , . . . , ζn ) Cλ(Xj )α+β m(Xj ) z ζ
ChG (Xj )k m(Xj ) = Chg (Xj )k/2 m(Xj ), where |α + β| = k and the constant C does not depend on (z, ζ ) nor on j ∈ N. From (2.19) it follows that Φ(D)k aj
Lp
N/2 C aj Lp + hg mψj Lp ,
where the constant C does not depend on j . Since there is a bound of overlapping Uj we therefore obtain Φ(D)k a
Lp
C1
Φ(D)k aj p p L
j ∈N
C2
1/p
N/2 p 1/p p aj Lp + hg mψj Lp
j ∈N
N/2 C3 aLp + hg mLp < ∞, for some constants C1 , C2 , C3 , and the result follows.
2
2.3. Some extensions In [8] Bony and Chemin introduce a broad family of Hilbert spaces H (m, g) of Sobolev type on V , parameterized with a strongly feasible metric g and a weight m which is strongly g-feasible. That is, m is g-continuous and (σ, g)-temperate on W . They also establish several important properties for these spaces. For example, if m0 is strongly g-feasible, then they prove: (1) if a ∈ S(m0 , g), then Opw (a) is continuous from H (m, g0 ) into H (m/m0 , g); (2) S (V ) ⊆ H (m, g) ⊆ S (V ) and H (1, g) = L2 (V ); (3) there exist a ∈ S(m0 , g) and b ∈ S(1/m0 , g) such that a#b = b#a = 1. In particular, Opw (a) : H (m, g0 ) → H (m/m0 , g0 )
and Opw (b) : H (m/m0 , g0 ) → H (m, g0 )
are continuous bijections and inverses to each others. The properties (1)–(3) here above can be used to extend the Schatten–von Neumann results in the present section to more general situations, involving the Bony–Chemin spaces H (m, g).
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In fact, for each pairs of strongly g-feasible weights m1 and m2 and each t ∈ R, the Banach space st,p (m1 , m2 , g) consists of all a ∈ S (W ) such that T = Opt (a) is a Schatten–von Neumann operator of order p ∈ [1, ∞] from H (m1 , g) to H (m2 , g). That is, a ∈ st,p (m1 , m2 , g), if and only if (1.11) holds, where (Tfj , gj ) = (Tfj , gj )H (m2 ,g) , and the supremum is taken over all orthonormal sets {fj }j ∈J in H (m1 , g) and {gj }j ∈J in H (m2 , g). We also let st, (m1 , m2 , g) be the set of all linear and compact operators from H (m1 , g) to H (m2 , g), and we set sw = st, and spw = st,p when t = 1/2. The most of the properties valid for st,p (W ) carry over to st,p (m1 , m2 , g). For example, if m1 , m2 , m3 are appropriate weights on W and p, q, r ∈ [1, ∞] satisfy 1/p + 1/q 1/r, then the map (a, b) → a#b on S (W ) extends uniquely to a continuous mapping from spw (m2 , m3 , g) × sqw (m1 , m2 , g) to srw (m1 , m3 , g) (cf. e.g. [27, Proposition 2.2]). The following results are now straight-forward consequences of the latter fact, [26, Theorem 4.4], (1)–(3) here above, and Theorems 2.1 and 2.9. Theorem 2.11. Assume that t = 1/2, p ∈ [1, ∞], g is strongly feasible, and m, m1 , m2 are gcontinuous and (σ, g)-temperate. Then the following is true: w (m , m , g), if and only if m m/m ∈ Lp (W ); (1) S(m, g) ⊆ st,p 1 2 2 1 w (m , m , g), if and only if m m/m ∈ L∞ (W ); (2) S(m, g) ⊆ st, 1 2 2 1 0 N/2
w (m , m , g) if and only if m a/m ∈ (3) if hg m2 m/m1 ∈ Lp (W ) for some N 0, then a ∈ st,p 1 2 2 1 p L (W ); N/2 w (4) if hg m2 m/m1 ∈ L∞ 0 (W ) for some N 0, then a ∈ st, (m1 , m2 , g) if and only if m2 a/m1 ∈ ∞ L0 (W ).
Furthermore, if in addition g is split, then (1)–(4) hold for general t ∈ R. 3. Further sufficient conditions for symbols to define Schatten–von Neumann operators in the Weyl calculus In this section we combine techniques in [17] with arguments in the proof of Theorem 4.4 in [26]. These investigations lead to Theorem 3.1 (2) below, where different sufficient conditions on N , m and g comparing to Proposition 4.5 (1) in [26] are presented in order for the embedding SN (m, g) ∩ Lp (W ) ⊆ spw (W ),
(3.1)
should hold. More precisely we prove have the following result, where the first part is essentially a restatement of Proposition 4.5 (1) in [26]. Recall (1.10) for the definition of Λg . Theorem 3.1. Assume that g is slowly varying on W , G = g + g 0 , p ∈ [1, 2], and let m ∈ p Lloc (W ). Then the following is true: N/2
(1) if N 2[2n(1/p − 1/2)] + 1 − δp,2 is an integer and hg m ∈ Lp (W ), then (3.1) holds; 1/p N/2 (2) if N [2n(1/p − 1/2)] + 1 − δp,2 is an integer, ΛG hg m ∈ Lp (W ), and in addition m is g-continuous, then (3.1) holds.
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We remark that both (1) and (2) in Theorem 3.1 generalize Theorem 3.9 in [17]. We also observe the differences between (1) and (2) in Theorem 3.1. In (1) we have stronger assumptions on N while the assumptions on m are relaxed, comparing to (2). We need some preparation for the proof of (2). The first result is a generalization of the estimate (3.9) in [17]. Lemma 3.2. Assume that p, q ∈ [1, ∞] are such that p q when p 2, and that N
[2n(1/p − 1/q )] + 1, if p < 2, 0 if p 2.
Then there is a constant C such that aspw C
2n
N D a q , j L
(3.2)
j =1
when a ∈ C0N (W ) is supported in a ball of radius one. For the proof we recall that for all 1 p ∞, R > 0, and multi-index α such that |α| N we have aLp (2R)|α| D α a Lp ,
for all a ∈ C0N BR (X) .
(3.3)
Proof. We may assume that a is supported in a ball with center at X = 0. First assume that 1 p < 2, and let a ∈ C0N (B(0)). By Hölder’s inequality it follows that it is no restriction to assume that q < 2, which in particular implies that q < p . Let Ω0 = B2 (0) and let Ωj = X ∈ R2n ; |X| 1, |Xj | > |X|/(4n) , 0 such that supp ϕ ⊆ Ω and for j = 1, . . . , 2n, and choose {ϕj }j =0,...,2n ⊆ S0,0 j j Then it follows from (1.12) that
aspw CFσ aLp C
2n
2n
j =0 ϕj
= 1.
ϕj Fσ aLp .
j =0
We have to estimate ϕj Fσ aLp for j = 0, . . . , 2n. First assume that j = 0. By (3.3), and Hausdorff–Young’s and Hölder’s inequalities it follows that for some constants C1 , C2 and C3 it holds ϕ0 Fσ aLp ϕ0 Lp Fσ aL∞ C1 ϕ0 Lp aL1 C2 aLq C3
2n
N D a q . j L j =1
The last inequality follows from (3.3).
(3.4)
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Next assume that j 1. Since q > p, it follows that we may choose r ∈ [1, ∞] such that 0 ∩ Lr . Hence by integration by 1/r = 1/p − 1/q . Then ψj (X) ≡ Xj−N ϕj (X) belongs to S0,0 parts and Hölder’s inequality it follows that ϕj Fσ aLp
2iσ (·,X) = ϕj a(X)e dX
Lp
R2n
=2
N 2iσ (·,X) Dj a (X)e dX ψj
−N
Lp
R2n
ψj Lr Fσ DjN a Lq . The fact that q > 2 and Hausdorff–Young’s inequality give ϕj Fσ aLp C DjN a Lq ,
for j = 1, . . . , 2n,
(3.5)
where C = ψj Lr is finite in view of the assumptions. The assertion now follows in this case by combining (3.4) and (3.5). Next assume that p 2. Then (1.12) and Hausdorff–Young’s and Hölder’s inequalities now give aspw C1 Fσ aLp C2 aLp C3 aLq . 2
The assertion now follows from (3.3) and the proof is complete.
Certain parts and ideas of the next result can be found in the proof of Lemma 3.8 in [17]. We set, as in [26], |a|W p (Ω) ≡ N
∂ α a |α|=N
Lp (Ω)
∂ α a
and aW p (Ω) ≡ N
|α|N
Lp (Ω)
.
Lemma 3.3. Assume that Ω ⊆ R2n is open, bounded and convex, p ∈ [1, ∞], N κp , and that ϕ ∈ C0N (Ω). Then there exists a positive constant C such that ϕaspw C
a (α) (Y ) + |a|W ∞ (Ω) , |α|N −1
N
(3.6)
for all Y ∈ Ω and all a ∈ C N (Ω). Furthermore, if q ∈ [1, ∞], and Ω0 ⊆ Ω is open and non-empty, then ϕaspw C aLq (Ω0 ) + |a|WN∞ (Ω) , for all a ∈ C N (Ω).
(3.7)
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Proof. By choosing a finite numbers of appropriate open balls Bk ⊂ Ω and ψk ∈ C0∞ (Bk ), k ∈ I such that supp ϕ ⊆
Bk ,
k∈I
and
ψk = 1,
on supp ϕ,
k∈I
we reduce ourself to the case that supp ϕ ⊆ B ⊆ Ω, for some open ball B. Let Y ∈ Ω be arbitrary. By Taylor expansion it follows that a = b + c, where b(X) = Ta,N −1 (X) ≡
|α|N −1
a (α) (Y ) (X − Y )α α!
is the Taylor polynomial of a at Y to the order N − 1, and
N c(X) = Ra,N −1 (X) ≡ (1 − t)N −1 a (α) Y + t (X − Y ) (X − Y )α dt α! 1
|α|=N
0
is the remainder term. The inequality (3.6) follows if we prove that ϕbspw C
a (α) (Y ),
(3.8)
|α|N −1
and ϕcspw C|a|WN∞ (Ω) ,
(3.9)
for some constant C which is independent of Y and a. First we prove (3.8). By straight-forward computations we get ϕbspw C1
a (α) (Y )(· − Y )α ϕ |α|N −1
= C1
spw
a (α) (Y )(· − Y )α ϕ
|α|N −1
spw
C2
a (α) (Y ), |α|N −1
for some constants C1 and C2 . This proves the assertion. Next we prove (3.9). We have that ∂ α c(Y ) = 0 when |α| N − 1, and that ∂ α c(X) = ∂ α a(X) when |α| = N , since c = a − Ta,N −1 and ∂Xα (Ta,N −1 ) = 0 for |α| = N . Hence for any multi-index β such that |β| < N , it follows that
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∂ β c(X) = Rc(β) ,N −|β|−1 (X)
N − |β| (1 − t)N −|β|−1 c(β+γ ) Y + t (X − Y ) (X − Y )γ dt γ! 1
=
|γ |=N −|β|
=
|γ |=N −|β|
0
N − |β| γ!
1
(1 − t)N −|β|−1 a (β+γ ) Y + t (X − Y ) (X − Y )γ dt.
0
Hence, there is a constant C which is independent of Y such that cWN∞ (Ω) C|a|WN∞ (Ω) , which implies that |ϕc|WN∞ (Ω) C|a|WN∞ (Ω) . An application of Lemma 3.2 with q = 2 and Hölder inequality now give ϕcspw C1 |ϕc|W 2 (Ω) C2 |ϕc|WN∞ (Ω) C3 |a|WN∞ (Ω) , N
which proves (3.9). It remains to prove (3.7). By Hölder’s inequality, it suffices to prove the result for q = 1, since Ω0 is bounded. Let Ω1 be a non-empty open ball such that Ω1 ⊆ Ω0 . By applying the L1 (Ω1 )norm with respect to the Y -variables in (3.6), and using Theorem 4.14 of [1] or Lemma A.1 in Appendix A, we get ϕaspw C1 aW 1 (Ω1 ) + |a|WN∞ (Ω) N−1 C2 aL1 (Ω0 ) + |a|W 1 (Ω0 ) + |a|WN∞ (Ω) N C3 aL1 (Ω0 ) + |a|WN∞ (Ω) , for some constants C1 , . . . , C3 . This proves (3.7).
2
In order to generalize Lemma 3.8 in [17], it is convenient to use particular classes of modulation spaces, introduced by Feichtinger in [13]. Assume that ϕ ∈ S (Rn ) \ 0 is fixed and that p ∈ [1, ∞]. Then the (classical) modulation space M p (Rn ) is the set of all f ∈ S (Rn ) such that f M p ≡
F f ϕ(· − x) (ξ )p dx dξ 1/p
is finite. (With obvious interpretation when p = ∞.) Here recall (0.2) for the definition of the Fourier transform F . We note that the definition of M p is independent of ϕ ∈ S (Rn ) \ 0 and that different ϕ gives rise to equivalent norms.
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The M p spaces fulfill the usual (complex) interpolation properties, i.e. p M 1 , M p2 [θ] = M p ,
1−θ θ 1 + = , p1 p2 p
1 p1 , p2 < ∞, 0 θ 1.
(Cf. [5,13,23].) The next result generalizes Lemma 3.8 in [17]. Here it is convenient to set sup D α a(X + Y ), |a|Ω,N (X) = Y ∈Ω,|α|=N
(3.10)
(3.11)
when Ω ⊆ R2n , N 0 is an integer and a ∈ C N (R2n ). Lemma 3.4. Assume that p ∈ [1, 2], and that N κp . Then there is a constant C such that aspw C aLp + |a|B(0),N Lp ,
(3.12)
for all a ∈ C N (R2n ). Proof. Let ϕ ∈ C0∞ (B(0)) be fixed such that ϕ dX = 1, and set aX (Y ) = ϕ(Y − X)a(Y ). By Lemma 3.3 it follows that (3.13) aX spw C aX Lp + |a|B(0),N (X) , for some constant C. We claim that aspw C
1/p
p
aX s w dX p
(3.14)
.
R2n
Admitting this for a while, we obtain aspw C1
p aX s w p
1/p dX
R2n
C2
p aX Lp + |a|B(0),N (X) dX
1/p
R2n
C3 aLp + |a|B(0),N Lp , for some constants C1 , C2 , C3 , and the result follows. It remains to prove (3.14). First we note that for some constant C we have C −1 Fσ aϕ(· − X) Lp aX spw C Fσ aϕ(· − X) Lp in view of Proposition 1.9, since ϕ has compact support. This implies that C −1 aM p
p
aX s w dX p
1/p CaM p .
(3.15)
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Hence it suffices to prove that aspw CaM p ,
when 1 p 2.
(3.16)
A proof of (3.16) can be found in [15,24]. In order to be self-contained we present an explicit proof here. First assume that p = 1. By (3.15) and Minkowski’s inequality we have as1w = aX dX aX s1w dX CaM 1 . s1w
R2n
R2n
This proves the result in this case. Next we consider the case p = 2. We have a2s w = (2π)−n a2L2 = (2π)−n
2
−n
R2n
aX 2L2
= (2π)
a(Y )ϕ(Y − X)2 dY dX
R2n −n
dX = (2π)
R2n
Fσ (aX )2 2 dX Ca2 2 , L M
R2n
for some constant C, and the result follows from this case as well. The inequality (3.16) now follows for general p ∈ [1, 2] by interpolation, using Theorem 5.1.2 of [5], Proposition 1.10 and (3.10). This proves the assertion. 2 Proof of Theorem 3.1. Let ϕj and Uj for j ∈ N be as in Remark 1.5 and Remark 1.6, and let {ψj }j ∈N be a bounded sequence in S(1, g) such that ψj ∈ C0∞ (Uj ) and ψj = 1 in the support of ϕj . Also let a ∈ SN (m, g) ∩ Lp (W ), and set aj = ϕj a. For each j ∈ N, we choose symplectic 0 and Gj ≡ GXj are coordinates such that gj ≡ gXj attains its diagonal form. Then g 0 ≡ gX j also given by their diagonal forms, and these coordinates form an orthonormal basis for W with respect to gj0 . Also set mj = m(Xj ) and Kj = X; gj (X + Y − Xj ) c, gj0 (Y ) 1 , where c is the same as in (1.7). Since ψj is equal to 1 on the support of aj , Lemma 3.4 gives p p aj s w C aj Lp + Ij , p
where Ij ≡
p g0 sup |aj |Nj (X + Y ) dX
gj0 (Y )1
R2n
C1
R2n
g
sup |aj |Nj (X + Y )hgj (Xj )N/2
p dX
gj0 (Y )1
p p C2 hgj (Xj )N/2 mj |Kj | C3 hgj (Xj )N/2 mj ΛGj (Xj )|Uj |, for some constants C, C1 , C2 and C3 which are independent of j ∈ N.
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Since there is a bound of overlapping Uj , and the fact that ΛGj is g-continuous in view of Proposition 1.2, it follows that
1/p N/2 p p mj hgj (Xj )N/2 ΛGj (Xj )|Uj | C ΛG hg mLp ,
j ∈N
and
p
p
aj Lp CaLp ,
j ∈N
for some constant C. The result is now a consequence of the estimate p
as w C p
p
ϕj as w
(see Corollary 4.2 in [26]). The proof is complete.
(3.17)
p
j ∈N
2
4. Consequences for a particular class of symbols In this section we apply the results from the previous sections on pseudo-differential operators, where the symbols belong to a certain types of symbol classes which are defined in a similar way r (cf. the introduction). as Sρ,δ r,s (R2n ) be the set of all a ∈ C ∞ (R2n ) such that For each r, s ∈ R and ρ, δ ∈ R2n , we let Sρ,δ α β ∂ ∂ a(x, ξ ) Cα,β xs−Π2 ρ,α+Π2 δ,β ξ r−Π1 ρ,β+Π1 δ,α , x ξ
for some constants Cα,β which are independent of x and ξ . Here Πj : R2n → Rn are the projections Π1 (ρ1 , . . . , ρn , ρn+1 , . . . , ρ2n ) = (ρ1 , . . . , ρn ) and Π2 (ρ1 , . . . , ρn , ρn+1 , . . . , ρ2n ) = (ρn+1 , . . . , ρ2n ). We note that if r, ρ0 , δ0 ∈ R, ρ0 , 1 j n, ρj = 0, n + 1 j 2n
and δj =
δ0 , 0,
1 j n, n + 1 j 2n,
r,s then Sρ,δ = Sρr 0 ,δ0 . r,s when A simple computation shows that S(m, g) = Sρ,δ
gx,ξ (y, η) =
n n
x−2ρn+j ξ 2δj yj2 + x2δn+j ξ −2ρj ηj2 , j =1
j =1
(4.1)
E. Buzano, J. Toft / Journal of Functional Analysis 259 (2010) 3080–3114
3109
m(x, ξ ) = xs ξ r ,
(4.2)
and
Here we recall that x = (1 + |x|2 )1/2 . The proof of the following lemma is omitted, since the result follows by similar arguments as in Section 18.4 in [18]. Here and in what follows it is convenient to use the following convention. Assume that μ, ν ∈ Rn and that r ∈ R. Then ν < μ and ν μ mean that νj < μj and νj μj respectively, for all j = 1, . . . , n. Moreover r < μ,
r μ,
μ
r μj ,
μj < r
and μ r
mean that r < μj ,
and μj r
respectively, for all j = 1, . . . , n. Lemma 4.1. Assume that m and g are given by (4.2) and (4.1), respectively. Then g is split, σ (y, η) = gx,ξ
n n
−2δn+j 2ρj 2 x ξ yj + x2ρn+j ξ −2δj ηj2 , j =1
j =1
and hg (x, ξ ) = max xδn+j −ρn+j ξ δj −ρj . 1j n
Moreover, (1) (2) (3) (4)
if ρ 1 and 0 δ, then g is slowly varying; if 0 δ ρ 1, then g is feasible; if 0 δ ρ 1 and δ < 1, then g is strongly feasible; if ρ 1 and 0 δ, then m is g-continuous.
The following result now follows from Theorem 2.9 and Lemma 4.1. Proposition 4.2. Assume that t ∈ R, p ∈ [1, ∞), r, s ∈ R, ρ, δ ∈ R2n are such that 0 δ ρ 1 r,s and δ < 1, and that a ∈ Sρ,δ (R2n ). Then the following is true: (1) if either r < −n/p or Π1 δ < Π1 ρ, and either s < −n/p or Π2 δ < Π2 ρ, then a ∈ st,p if and only if a ∈ Lp (R2n ); (2) if either r 0 or Π1 δ < Π1 ρ, and either s 0 or Π2 δ < Π2 ρ, then a ∈ st,∞ if and only if a ∈ L∞ (R2n ); (3) if either r < 0 or Π1 δ < Π1 ρ, and either s < 0 or Π2 δ < Π2 ρ, then a ∈ st, if and only if 2n a ∈ L∞ 0 (R ).
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Next we focus on the case when g in (4.1) is not necessarily feasible and illustrate the differences between (1) and (2) in Theorem 3.1. Set np = 2n(1/p − 1/2) . The following result is an immediate consequence of Theorem 3.1(1) and Lemma 4.1. Proposition 4.3. Assume that 1 p 2, ρ, δ ∈ R2n are such that ρ 1, 0 δ and ρ δ, and r,s that a ∈ Sρ,δ . Also assume that ⎧ max (δj − ρj ), ⎨ r < −n − p(np + 1/2) 1j n ⎩ s < −n − p(np + 1/2) max (δn+j − ρn+j ),
(4.3)
1j n
and that a ∈ Lp (R2n ). Then a ∈ spw . In order to apply Theorem 3.1(2) we need to analyze ΛG (cf. (1.10)) with G = g + g 0 . The symplectic transformation
yj = xδn+ j +ρn+j ξ −δj −ρj zj , ηj = x−δn+j −ρn+j ξ δj +ρj ζj ,
with j = 1, . . . , n, puts G in diagonal form Gx,ξ (z, ζ ) =
n
δ −ρ x n+j n+j ξ δj −ρj + 1 zj2 + ζj2 , j =1
so that n
ΛG (x, ξ ) =
δ −ρ x n+j n+j ξ δj −ρj + 1 .
(4.4)
j =1
The following result is an immediate consequence of Theorem 3.1, Lemma 4.1 and (4.4). Proposition 4.4. Assume that 1 p 2, ρ, δ ∈ R2n are such that ρ 1, 0 δ and ρ δ, and r,s that a ∈ Sρ,δ . Also assume that ⎧ r < −n − p(np + 1) max (δj − ρj )/2 − (δj − ρj ), ⎪ ⎨ 1j n 1j n ⎪ (δn+j − ρn+j ), ⎩ s < −n − p(np + 1) max (δn+j − ρn+j )/2 − 1j n
and that a ∈ Lp (R2n ). Then a ∈ spw .
1j n
(4.5)
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Remark 4.5. We note that Propositions 4.3 and 4.4 do not contain each others. Consequently, (1) and (2) in Theorem 3.1 do not contain each others as well. In fact, as simple examples show, the conditions imposed on r and s in (4.3) can be stronger, weaker or not comparable with those in (4.5). Appendix A In this appendix we consider some of the key estimates of Lp -type in Sections 2 and 3 again, and prove that they can be obtained by using similar techniques as in [11]. We start to consider n-sectors in Rn . An n-sector H in Rn is a set of the form H = HT = T (x); xj 0, j = 1, . . . , n , where T = TH is a linear and bijective map on Rn such that |T (e)| = 1 when e is a vector in the standard basis for Rn . We note that different T might give raise to the same sector, and hence, TH is not unique. On the other hand, if AT is the matrix for the linear map T , then Υ (H ) ≡ det(AT ) is independent of the choice of TH . For any set Ω ⊆ Rn , ε 0 and n-sector H , we set ΩH,ε ≡ x + y; x ∈ Ω, y ∈ H ∩ Bε (0) . The following lemma is, to some extent, a stressed version of Theorem 4.10 in Chapter 5 in [4]. Lemma A.1. Assume that Ω ⊆ Rn is a closed convex set, N 0 is an integer, H an n-sector, ε > 0, and that p ∈ [1, ∞]. Then there is a constant C, depending on n, N , ε and Υ (H ) only such that
α β ∂ f p ∂ f Lp (Ω ) , (A.1) C f Lp (ΩH,ε ) + L (Ω) H,ε
|β|=N
when |α| N and f ∈ C N (ΩH,ε ). For the proof we recall some facts on difference operators and B-splines in Section 5.4 in [4]. Let χ(0,1) be the characteristic function for the interval (0, 1). Then the function Hj , for j 1, defined inductively on the real line by H1 = χ(0,1) ,
Hj +1 = H1 ∗ Hj ,
j 1,
is called the B-spline of order j . j For any h ∈ Rn , let {Th }j 1 be a sequence of operators on C(Rn ) which is inductively defined by Th1 f (x) = f (x + h) − f (x),
j +1
Th
j
= Th ◦ Th1
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when f ∈ C(Rn ). An important relation for the B-splines and the operator Th when n = 1 is the relation j Th f (x) = f (j ) (x + th)hj Hj (t) dt. (A.2) Proof of Lemma A.1. We shall mainly follow the proof of Theorem 4.10 in Chapter 5 in [4]. We may assume that ε 1. Furthermore, by making a change of variables, we may assume that H = x ∈ Rn ; xj 0, j = 1, . . . , n . The result is obviously true when |α| = 0 and |α| = N . We may therefore assume that 0 < |α| < N . First we consider the case n = 1. Since the support of Hj is equal to [0, j ] and that the integral of Hj is equal to 1, the mean-value theorem and (A.2) give j Tε/N 2 f (x) = ε j N −2j
f (j ) x + εt/N 2 Hj (t) dt
= ε j N −2j f (j ) (x + θ ),
(A.3)
for some 0 θ ε/N . Furthermore, θ f (j ) (x) = f (j ) (x + θ ) −
f (j +1) (x + y) dy,
0
and combining this equality with (A.3) gives
f
(j )
(x) = ε
−j
N
2j
j Tε/N 2 f (x) −
θ
f (j +1) (x + y) dy.
0
By applying the Lp (Ω) norm on the latter equality, and using the fact that θ θ 1/p p (j +1) (j +1) f (x + y) dy (x + y) dy , f 0
0
by Hölder’s inequality, we get (j ) j f p 2N 2 /ε f Lp (ΩH,j ε/N 2 ) + ε f (j +1) Lp (Ω L (Ω)
H,ε/N )
/N.
Iteration of this result gives (A.1). Next assume that n 1 is arbitrary. From the first part of the proof we get j ∂ f k
Lp (Ω)
C f Lp (ΩH,ε/n ) + ∂kN f Lp (Ω
H,ε/n )
.
(A.4)
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3113
For any arbitrary multi-index α we also let gk = ∂kαk · · · ∂NαN f and rk = r − ni=k+1 αi . From (A.4) we get gk Lp (ΩH,(k−1)ε/n ) C gk+1 Lp (ΩH,kε/n ) + |f |W p (ΩH,kε/n ) . N
This gives α ∂ f
Lp (Ω)
= g1 Lp (Ω) C1 g2 Lp (ΩH,ε/n ) + |f |W p (ΩH,ε/n ) N · · · Cn−1 gn Lp (ΩH,(n−1)ε/n ) + |f |W p (ΩH,(n−1)ε/n ) N Cn f Lp (ΩH,ε ) + |f |W p (ΩH,ε ) , N
for some constants Ck which only depend on ε, n and N . The proof is complete.
2
Next we apply Lemma A.1 to a family of subsets of Rn which contains each convex sets. A subset Ω of Rn is called conistic (of order ε > 0) if for each x ∈ Ω, there is an n-sector H in Rn such that (A.5) Υ (H ) ε and x + H ∩ Bε (0) ⊆ Ω. By straight-forward computations it follows that any convex set is conistic. Consequently, since the Euclidean structure in Lemma 2.5 is completely determined by the Euclidean metric gX (note here that X is fixed), Lemma 2.5 is a consequence of the following result. Proposition A.2. Assume that Ω ⊆ Rn is bounded and conistic of order ε > 0, and that N ∈ N. Then there exists a positive constant C, depending on n, N and ε only such that
α β ∂ f ∞ ∞ ∂ , |α| N, f ∈ C N Rn . C f + f L (Ω) L (Ω) L∞ (Ω) |β|=N
Proof. We may assume that Ω is a closed set. Let x0 ∈ Ω be chosen such that α ∂ f (x0 ) = ∂ α f ∞ , L (Ω) and let the sector H be chosen such that (A.5) is fulfilled for x = x0 . If ω = x0 + (H ∩ Bε/2 (0)), then it follows that ωH,ε/2 = x0 + H ∩ Bε (0) ⊆ Ω. Hence Lemma A.1 gives α ∂ f ∞ = ∂ α f (x0 ) = ∂ α f L∞ (ω) L (Ω)
∂ β f ∞ C f L∞ (ωH,ε/2 ) + L (ω |β|=N
H,ε/2 )
β ∂ f L∞ (Ω) , × C f L∞ (Ω) +
|β|=N
when f ∈ C N , and the result follows.
2
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References [1] R.A. Adams, Sobolev Spaces, Pure Appl. Math., vol. 65, Academic Press, Orlando, 1975. [2] R. Beals, C. Feffermann, On local solvability of linear partial differential equations, Ann. of Math. 97 (1973) 482– 498. [3] R. Beals, C. Feffermann, Spatially inhomogeneous pseudo-differential operators I, Comm. Pure Appl. Math. 27 (1974) 1–24. [4] C. Bennett, R. Sharpley, Interpolation Operators, Pure Appl. Math., vol. 129, Academic Press, Boston, 1988. [5] J. Bergh, J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin–Heidelberg–New York, 1976. [6] J.M. Bony, Caractérisations des Opérateurs Pseudo-Différentiels, in: Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, Exp. No. XXIII, Sémin. École Polytech., Palaiseau, 1997. [7] J.M. Bony, Sur l’Inégalité de Fefferman–Phong, in: Séminaire sur les Équations aux Dérivées Partielles, 1998–1999, Exp. No. III, Sémin. École Polytech., Palaiseau, 1999. [8] J.M. Bony, J.Y. Chemin, Espaces Functionnels Associés au Calcul de Weyl–Hörmander, Bull. Soc. Math. France 122 (1994) 77–118. [9] J.M. Bony, N. Lerner, Quantification Asymptotique et Microlocalisations d’Ordre Su-pé-rieur I, Ann. Sci. École Norm. Sup. 22 (1989) 377–433. [10] E. Buzano, N. Nicola, Pseudo-differential operators and Schatten–von Neumann classes, in: P. Boggiatto, R. Ashino, M.W. Wong (Eds.), Advances in Pseudo-Differential Operators, Proceedings of the Fourth ISAAC Congress, in: Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2004. [11] E. Buzano, J. Toft, Schatten–von Neumann properties in the Weyl calculus, Research report 07052, Växjö University, 2007, also available on arXiv:0809.1207. [12] E. Buzano, J. Toft, Continuity and compactness properties of pseudo-differential operators, Fields Inst. Commun. 52 (2008). [13] H.G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical report, University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds.), Wavelets and Their Applications, Allied Publishers Private Limited, New Delhi, Mumbai, Kolkata, Chennai, Hagpur, Ahmedabad, Bangalore, Hyderbad, Lucknow, 2003, pp. 99–140. [14] G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, 1989. [15] K.H. Gröchenig, C. Heil, Modulation spaces and pseudo-differential operators, Integral Equations Operator Theory (4) 34 (1999) 439–457. [16] L. Hörmander, The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math. 32 (1979) 359–443. [17] L. Hörmander, On the asymptotic distributions of the eigenvalues of pseudodifferential operators in Rn , Ark. Mat. 17 (1979) 297–313. [18] L. Hörmander, The Analysis of Linear Partial Differential Operators, vols. I, III, Springer-Verlag, Berlin– Heidelberg–New York–Tokyo, 1983–1985. [19] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. [20] B. Simon, Trace Ideals and Their Applications I, London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge–London–New York–Melbourne, 1979. [21] J. Toft, Continuity and positivity problems in pseudo-differential calculus, Thesis, Department of Mathematics, University of Lund, Lund, 1996. [22] J. Toft, Regularizations, decompositions and lower bound problems in the Weyl calculus, Comm. Partial Differential Equations 25 (7 & 8) (2000) 1201–1234. [23] J. Toft, Continuity properties for non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math. (2) 126 (2002) 115–142. [24] J. Toft, Continuity properties for modulation spaces with applications to pseudo-differential calculus, I, J. Func. Anal. 207 (2004) 399–429. [25] J. Toft, Schatten–von Neumann properties in the Weyl calculus, and calculus of metrics on symplectic vector spaces, Research report, Växjö University, 2004. [26] J. Toft, Schatten–von Neumann properties in the Weyl calculus, and calculus of metrics on symplectic vector spaces, Ann. Global Anal. Geom. 30 (2006) 169–209. [27] J. Toft, Continuity and Schatten properties for pseudo-differential operators on modulation spaces, in: J. Toft, M.W. Wong, H. Zhu (Eds.), Modern Trends in Pseudo-Differential Operators, in: Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel, 2007, pp. 173–206.
Journal of Functional Analysis 259 (2010) 3115–3162 www.elsevier.com/locate/jfa
An infinite dimensional Schur–Horn Theorem and majorization theory Victor Kaftal ∗ , Gary Weiss University of Cincinnati, Department of Mathematics, Cincinnati, OH, 45221-0025, USA Received 10 May 2009; accepted 27 August 2010 Available online 16 September 2010 Communicated by D. Voiculescu
Abstract The main result of this paper is the extension of the Schur–Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0,there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if nj=1 ξj nj=1 ηj for every n if and only if ξ = Qη for some orthostochastic matrix Q. When ξ and η are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices. © 2010 Elsevier Inc. All rights reserved. Keywords: Schur–Horn Theorem; Majorization; Stochastic matrices
1. Introduction The Schur–Horn Theorem characterizes the diagonals of a (finite) self-adjoint matrix in terms of sequence that is, the order relation for ξ, η ∈ RN given by the condin N ξ η n majorization, ∗ ∗ ∗ ∗ tions j =1 ξj j =1 ηj for 1 n N and j =1 ξj = N j =1 ηj , where ξ , η denote the monotone nonincreasing rearrangement of ξ , η. The theory of majorization arose during the early part of the 20th century from a number of apparently unrelated topics: wealth distribution (Lorenz [26]), inequalities involving convex functions (Hardy, Littlewood and Pólya [11]), con* Corresponding author.
E-mail addresses:
[email protected] (V. Kaftal),
[email protected] (G. Weiss). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.018
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V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
vex combinations of permutation matrices (Birkhoff [3]), and more central to our interests herein, doubly stochastic matrices and the relation between eigenvalue lists and diagonals of self-adjoint matrices. Theorem 1.1. Let ξ, η ∈ RN . (i) Hardy, Littlewood and Pólya Theorem [11]. ξ η if and only if ξ = P η for some doubly stochastic matrix P . (ii) Horn Theorem [12, Theorem 4]. ξ η if and only if ξ = Qη for some orthostochastic matrix Q, i.e., the Schur-square of an orthogonal matrix (Qij = (Uij )2 ∀i, j for some unitary matrix U with real entries). (iii) Schur–Horn Theorem [33,12]. Given a self-adjoint N × N matrix A having eigenvalue list η, there is a basis in which A has diagonal entries ξ if and only if ξ η. The sufficiency part of the Schur–Horn Theorem is due to Schur and the necessity follows immediately from the Horn Theorem. The main goal of this paper is to extend to infinite dimension the Horn Theorem and hence the Schur–Horn Theorem. The notion of majorization extends seamlessly to infinite sequences that admit a monotone nonincreasing rearrangement. To avoid always having to pass to monotone rearrangements, in this paper we will focus on sequences decreasing monotonically to 0 and will denote by c∗o their positive cone and by (1 )∗ the subcone of summable decreasing sequences. (We note explicitly that c∗o and (1 )∗ do not denote herein the duals of co and 1 .) Even for finite sequences, the terminology and notations describing majorization vary considerably in the literature. In this paper, we will use the following notations: Definition 1.2. For ξ, η ∈ c∗o we say that • η majorizes ξ (ξ ≺ η) if nj=1 ξj nj=1 ηj for every n ∈ N; • η strongly majorizes ξ (ξ η) if ξ ≺ η and lim nj=1 (ηj − ξj ) = 0; nk nk ηj for some sequence of positive • η block majorizes ξ (ξ ≺b η) if ξ ≺ η and j =1 ξj = j =1 integers nk ↑ ∞. For ξ, η ∈ c+ o we say that one of the above relations holds for ξ and η if it holds for their monotone rearrangements ξ ∗ and η∗ . For nonsummable monotone decreasing sequences, the condition lim nj=1 (ηj − ξj ) = 0 retains many of the key properties of “equality at the end” for finite and for summable sequences (see [23]) and will prove crucial for our extension of the Schur–Horn Theorem. Block-majorization is both a natural way to bring the results of finite majorization theory to bear on infinite sequences and it also arises naturally in Section 4. This notion is further developed in [23] for its relevance in the study of operator ideals. In 1964, in two papers that are not nearly as well-known as they deserve and with two almost disjoint approaches, Markus [27] and Gohberg and Markus [9] found infinite dimensional versions of the Hardy, Littlewood and Pólya Theorem 1.1(i) and an extension to the summable case of the Horn Theorem [12, Theorem 4] (Theorem 1.1(ii)). More recently, Arveson and Kadison obtained other characterizations in [2] using still different methods.
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
3117
If ξ, η ∈ c∗o , then ξ ≺η
⇔
ξ = Qη for some substochastic Q (Q row-stochastic if ξn > 0 ∀n) [27, Lemma 3.1],
ξ ≺η
⇔
ξ = Qη
with Qij = |Wij |2 ,
for some co-isometry W [9, Proposition III, p. 205], ξ ≺η
⇔
ξ = Qη
with Qij = |Lij |2 ,
for some contraction L [2, Theorem 4.2].
(1)
If ξ, η ∈ (1 )+ , then ξ η
⇔
ξ = Qη
with Qij = |Uij |2 , for some unitary U [9, Theorem 1],
ξ η
⇔
ξ = Qη
with Qij = |Wij |2 , for some isometry W [2, Theorem 4.1].
(2)
By reformulating matricially the Markus construction in [27, Lemma 3.1] and slightly tightening it (see Remark 3.8), we can identify a sequence of orthogonal matrices (unitary matrices with real entries) underlying the construction. An analysis of their properties and infinite products permits us to obtain: • If ξ, η ∈ c∗o , ξn > 0 ∀n, and ξ ≺ η, then there is a canonical co-isometry with real entries W (ξ, η) for which ξ = Q(ξ, η)η with Q(ξ, η)ij = (W (ξ, η)ij )2 (Theorem 3.7). • If ξ, η ∈ c∗o , then ξ η ⇔ ξ = Qη with Qij = |Uij |2 for some orthogonal matrix U (Theorem 3.9). Not surprisingly, this construction applied to finite sequences provides another proof of the (finite) Horn Theorem 1.1(ii). The canonical matrix Q(ξ, η) is obtained as an infinite product of T-transforms (see Section 3 for details) and is therefore completely determined by the sequence {mk , tk } where mk is the matrix size and 0 < tk 1 is the “convex coefficient” of the k-th transform. In Section 4 we further analyze this double sequence and, more precisely, the set {tk | mk = 1}, in order to link properties of the majorization ξ ≺ η with properties of the corresponding canonical matrix Q(ξ, η): • ξ η if and only if Q(ξ, η) is orthostochastic (that is, if W (ξ, η) is orthogonal) if and only if {tk | mk = 1} = ∞ (Theorem 4.7). • ξ ≺b η if and only if Q(ξ, η) is the direct sum of finite orthostochastic matrices if and only if card{k | tk = mk = 1} = ∞ (Proposition 4.4). Notice that if ξ, η ∈ c∗o and ξ = Qη for some orthostochastic matrix Q, then by (1) it follows that ξ ≺ η, but in general it does not follow that ξ η. In fact, one of the main results of this paper is: / (1 )∗ and ξ ≺ η, then there is an orthostochastic matrix Q for which ξ = Qη • If ξ, η ∈ c∗o , ξ ∈ (Theorem 5.3).
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Section 5 is devoted to the proof of this theorem by showing that a pair of c∗o -sequences ξ ≺ η with ξ nonsummable and ξ η can be decomposed into “mutually orthogonal” pairs of infinite subsequences for which strong majorization holds (Lemmas 5.1–5.2) and then invoking Theorem 3.9 to obtain an orthostochastic matrix for each pair and taking their direct sum. Together, Theorems 3.9 and 5.3 provide the following infinite dimensional extension of the Horn Theorem (Corollary 5.4): • If
ξ, η ∈ c∗o
then ξ = Qη for some orthostochastic matrix Q ⇔
ξ ≺η ξ η
when ξ ∈ / 1 , when ξ ∈ 1 .
To apply the Horn Theorem to positive compact operators, notice first that the eigenvalue list with multiplicity (which is the sequence s(A) of s-numbers of A ∈ K(H )+ ) “ignores” the nullspace of A (e.g., see (32)) and hence it characterizes the partial isometry orbit of A V(A) := V AV ∗ V partial isometry, V ∗ V A = A rather than, as in the finite rank case, the unitary orbit U(A) of A. Then if we fix an orthonormal basis of the Hilbert space H and denote by E the canonical conditional expectation on the corresponding atomic masa D (i.e., the operation of “taking the main diagonal”), we obtain the following infinite dimension extension of the Schur–Horn Theorem for positive compact operators: {B ∈ D ∩ K(H )+ | s(B) ≺ s(A)} \ L1 if Tr(A) = ∞, • E(V(A)) = if Tr(A) < ∞ {B ∈ D ∩ K(H )+ | s(B) s(A)} (Proposition 6.4). If A has finite rank, then U(A) = V(A) and if A ∈ K(H )+ has dense range, i.e., its range projection RA is the identity, then • E(U(A)) =
{B ∈ D ∩ K(H )+ | s(B) ≺ s(A), RB = I } \ L1 {B ∈ D ∩ K(H )+ | s(B) s(A), RB = I }
if Tr(A) = ∞, if Tr(A) < ∞
(Proposition 6.6). For positive compact operators with infinite rank some sufficient conditions and some necessary conditions for membership in E(U(A)) are presented in Propositions 6.6 and 6.10. Our work extends some of the results of Gohberg and Marcus in [9] and Arveson and Kadison in [2]. There are only limited overlaps between our paper and those by A. Neumann [31] and by Antezana, Massey, Ruiz, and Stojanoff [1] as these authors characterize the closures of the conditional expectation of the unitary orbit of a self-adjoint not necessarily compact operator while we do not take closures. The connections with these three papers are further discussed in Section 6 where we also answer a couple of questions of Neumann. The following, in the case of sequences ξ, η ∈ c∗o , compares these different results to Proposition 6.4. If ξ ∈ / (1 )∗ , then ξ ≺η
⇔
⎧ ⎨ E(U(diag η)) . [31, Theorem 3.13], diag ξ ∈ E{L diag ηL∗ | L ∈ B(H )1 } [2, Theorem 4.2], ⎩ E(U(diag η)) (Proposition 6.4).
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
3119
If ξ ∈ (1 )∗ , then
ξ η
⇔
⎧ . ⎪ ⎨ E(U(diag η)) 1 diag ξ ∈ E(U(diag η) . 1 ) ⎪ ⎩ E(U(diag η))
[1, Proposition 3.13], [2, Theorem 4.2], ([9, Theorem 1], Proposition 6.4).
What first led us to investigate infinite majorization theory and the Schur–Horn Theorem was its natural connection to the theory of operator ideals, i.e., two-sided non-closed ideals of the algebra B(H ) for a separable Hilbert space H . It is well known since Calkin [4] that operator ideals are in one-to-one correspondence with certain positive subcones of c∗o (the characteristic sets) and for ξ, η ∈ c∗o , the order relation ξ ≺ η translates into the inequality ξa ηa between the (Cesaro) arithmetic mean sequences ξa and ηa (i.e., (ξa )n := n1 nj=1 ξj and similarly for ηa ). Arithmetic mean operations on sequences and hence on ideals are the key to the recent characterization of the commutator ideals of operator ideals (see [6] and the earlier partial results [34–36,24]). Part of our long-term project [17–23] investigating arithmetic mean ideals is [23] where we apply tools developed here to characterize the so-called arithmetic mean closed ideas in terms of invariance under various classes of stochastic and block stochastic matrices and in terms of the conditional expectation E. In that paper we develop also further majorization properties including: • If ξ, η ∈ c∗o and ξ ≺ η, then there are ζ, ρ ∈ c∗o for which ξ ζ η and ξ ρ η (existence of intermediate sequences). • Necessary and sufficient condition for the existence of ζ, ρ ∈ c∗o for which ξ ≺b ζ η and ξ ρ ≺b η. • Analogous results for “majorization at infinity” for summable sequences. 2. Notations and preliminaries on stochastic matrices Let c∗o denote the cone of nonnegative monotone nonincreasing sequences converging to 0 and (1 )∗ the cone of nonnegative monotone nonincreasing summable sequences. (Again, notice that c∗o and (1 )∗ here do not denote the duals of co and 1 .) If ξ ∈ (co )+ , denote by ξ ∗ its nonincreasing rearrangement. For every sequence ξ = ξ1 , ξ2 , . . . and every n = 0, 1, . . . , denote by ξ (n) the truncated sequence ξ (n) = ξn+1 , ξn+2 , . . . and by ξ χ[1, n] the sequence ξ1 , ξ2 , . . . , ξn , 0, . . . . We will of course identify ξ χ[1, n] with a vector in Rn and conversely, embed Rn into co by completing finite sequences with infinitely many zeros. When applying the majorization notations of Definition 1.2 to finite sequences, we caution the reader again that what we call majorization (ξ ≺ η, i.e., k1 ξj k1 ηj for all 1 k n) is called often nweak majorization, and what we call strong majorization (ξ η, i.e., ξ ≺ η with n ξ = j =1 j j =1 ηj ) is mostly called majorization, although with no universal agreement about notations or even about the direction of the inequalities (see [14, Remark, p. 198]). For the theory of majorization of finite sequences we refer the reader to Marshall and Olkin [28]. An immediate consequence of Definition 1.2 is that if ξ, η ∈ c∗o , then ξ ≺b η
⇒
ξ η
⇒
ξ ≺ η.
(3)
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Once we have fixed an orthonormal basis {ek } for a complex separable infinite dimensional Hilbert space H , i.e., once we have identified H with 2 , we will also identify infinite matrices with operators and will use these terms interchangeably. E.g., when we apply a Hilbert space operator to a sequence in c∗o , what we mean is that we apply the corresponding matrix to that sequence – which for substochastic matrices is always possible (e.g., see Remark 2.2). Also, for typographical reasons we are not going to distinguish between row and column vectors, e.g., if P is a matrix, we shall write P ξ1 , ξ2 , . . . in lieu the more precise P ξ1 , ξ2 , . . . T . K(H ) denotes the ideal of compact operators and L1 the trace class ideal, with Tr denoting the usual trace. Given a compact operator A ∈ K(H ), the sequence s(A) ∈ c∗o of its s-numbers (singular numbers) consists of the eigenvalues of (A∗ A)1/2 in monotone order, with repetition according to multiplicity, and with infinitely many zeros added in case A has finite rank. In particular, if A 0 has infinite rank, then s(A) is precisely the “eigenvalue list” of A. Given a sequence ρ ∈ ∞ , we denote by diag ρ the diagonal matrix having ρ as its main diagonal. Given an operator A ∈ B(H ), we denote by E(A) the diagonal matrix having as diagonal the main diagonal of A, i.e., E : B(H ) → D is the normal faithful and trace preserving conditional expectation from B(H ) onto the masa D of diagonal operators. Stochastic matrices play a key role in majorization theory of finite sequences (e.g., see Theorem 1.1(i)). A similar but necessarily more complex role is played in the case of infinite sequences. Definition 2.1. A matrix P with nonnegative entries is called • • • • •
substochastic if its row and column sums are bounded by 1; column-stochastic if it is substochastic with column sums equal to 1; row-stochastic if it is substochastic with row sums equal to 1; doubly stochastic if it is both column- and row-stochastic; block stochastic if it is the direct sum of finite doubly stochastic matrices.
Remark 2.2. (i) Contrary to the finite case a (square) matrix can be column-stochastic without being rowstochastic and vice versa. (ii) We can apply a substochastic matrix P to any sequence ρ ∈ ∞ , where by Pρ we just mean the sequence ∞ j =1 Pij ρj . (iii) If ρ ∈ co and P is substochastic, then Pρ ∈ co . (iv) By Schur’s test (e.g., see [10, Problem 45]) substochastic matrices viewed as Hilbert space operators are contractions. An important class of stochastic matrices is the one obtained as “Schur-squares” of contractions. To be more precise, we should call them the Schur product of a contraction by its complex conjugate matrix, but in most cases we consider only matrices with real entries. The Schur product is also called Hadamard product or entrywise product. The relevance of these stochastic matrices is clear from the following well-known lemma whose verification is straightforward. Lemma 2.3. Let ξ, η ∈ c∗o and let Qij = |Lij |2 for all i, j for some contraction L ∈ B(H ). Then ξ = Qη if and only if diag ξ = E(L diag ηL∗ ).
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
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Lemma 2.4. Let Qij = |Lij |2 for all i, j for some contraction L ∈ B(H ). Then (i) Q is substochastic. (ii) Q is column-stochastic if and only if L is an isometry. (ii ) Q is row-stochastic if and only if L is a co-isometry. Proof. Notice first that ∞
Qij =
j =1
∞
2 Lij L∗j i = L∗ ei 1 for every i,
(4)
j =1
and similarly ∞
Qij = Lej 2 1 for every j.
(5)
i=1
(i) Immediate from (4) and (5). (ii) Sufficiency is immediate from (i) and (5). Conversely, assume that Q is column-stochastic and hence Lej = 1 for all j by (5). Then (L∗ Lej , ej ) = 1 for all j and thus it follows that E(I − L∗ L) = 0. Since E is faithful and I − L∗ L 0 because L is a contraction, it follows that L∗ L = I . (ii ) Apply (ii) to L∗ . 2 Definition 2.5. If Qij = |Lij |2 for some contraction L, then we say that Q is isometry stochastic (resp. co-isometry stochastic, unistochastic, orthostochastic) if L is an isometry (resp. coisometry, unitary, orthogonal matrix, i.e., a unitary matrix with real entries). If L is the direct sum of finite unitary (resp. orthogonal) matrices, we say that Q is block unistochastic (resp. block orthostochastic). Remark 2.6. (i) The terminology “orthostochastic” goes back at least to Horn (cf. [12]). When the entries of the unitary matrix are not necessarily real, its Schur-square is called unitary stochastic in [28], Pythagorean in [16, Section 4], and orthostochastic in [13], although unistochastic appears to be the more common term now. (1) (2) (ii) Q can be the Schur-square of different contractions, e.g., Qij = |Lij |2 = |Lij |2 ; but L(1) is an isometry, a co-isometry, a unitary if and only if so is L(2) , respectively. Of course, L(1) may have real entries, while L(2) does not. (iii) L does not need to be a contraction for Q to be doubly stochastic, e.g., consider the 4 × 4 matrix L with constant entries 12 . (iv) As remarked by Horn [12], every 2 × 2 doubly stochastic matrix is necessarily orthostochastic but 1 2
1 1 0 1 0 1 0 1 1
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is doubly stochastic but not unistochastic (see [28, p. 39] for more examples). (v) Let P be a matrix and Π be a permutation matrix. Then ΠP is substochastic (resp. rowstochastic, column-stochastic, isometry stochastic, co-isometry stochastic, unistochastic, orthostochastic) precisely when P is. However, permutations do not preserve block stochasticity. An immediate consequence of Theorem 1.1(i) and (ii) is that if ξ, η ∈ c∗o ,
then ξ ≺b η
⇔
ξ = Qη
for a block orthostochastic matrix Q
⇔
ξ = Qη
for a block stochastic matrix Q.
(6)
Another simple application of the Horn Theorem, which we will need in Theorem 3.9, is to the case when η is a sequence with finite support. This result generalizes [16, Theorem 13] (see also [1, Theorem 4.7]). Lemma 2.7. If ξ, η ∈ c∗o , ξ η, and η has finite support, then ξ = Qη for some orthostochastic matrix Q. Proof. The case when η = 0 being trivial, let n be the largest integer for which ηn > 0. If n = 1, then let U be an orthogonal matrix that has as its first column the unit vector ( ηξ11 , ηξ21 , ηξ31 ,... )T and let Qij := Uij2 . Then Qη = ξ . In the case that n > 1, it follows that ∞ n n−1 m−1 n−1 m j =1 ξj = j =1 ηj > j =1 ξj j =1 ηj . Let m be the index for which j =1 ξj j =1 ηj < m n−1 ∞ and let α := j =1 ξj − j =1 ηj . Then m n, 0 < α = ηn − j =m+1 ξj ηn and ξ1 , ξ2 , . . . , ξm η1 , η2 , . . . , ηn−1 , α, 0, . . . , 0 . By applying the Horn Theorem (see Theorem 1.1(ii)) to the above two vectors of (R m )+ , we find an m × m orthostochastic matrix Qo for which Qo η1 , η2 , . . . , ηn−1 , α, 0, . . . , 0 = ξ1 , ξ2 , . . . , ξm and let U o be an orthogonal matrix for which Qoij = (Uijo )2 . In particular, the first n columns U1 , U2 , . . . , Un of U o are orthonormal. Denote by Q1 , Q2 , . . . , Qn their Schur-squares, i.e., the first n columns of Qo . Therefore the 2 vectors ⎛ α ⎞ η n Un ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ U2 Un−1 U1 ⎜ ξm+1 ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ηn ⎟ ⎜ ⎟,⎜ ⎟,...,⎜ ⎝ 0 ⎠ , ⎜ ξm+2 ⎟ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎜ ⎟ .. .. .. ηn ⎠ ⎝ . . . .. . are also orthonormal. Complete them to an orthonormal basis of 2 with real entry vectors and denote by U the orthogonal matrix having as columns these vectors and by Q the orthostochastic matrix Qij = Uij2 . Then
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
⎛
· · · Qn−1
Q1
⎜ 0 ⎜ Q=⎜ ⎜ 0 ⎝ .. .
···
0
··· .. .
0 .. .
α ηn Qn ξm+1 ηn ξm+2 ηn
.. .
∗
∗
∗
∗
∗ .. .
∗ .. .
···
3123
⎞
···⎟ ⎟ ⎟ ···⎟ ⎠ .. .
and hence ⎛ ⎛
Q1
⎜ 0 ⎜ Qη = ⎜ ⎜ 0 ⎝ .. .
⎜
⎞ α ⎜ (Q 1 ⎜ ηn Qn ⎛ η1 ⎞ ⎜ ξm+1 ⎟ ⎜ η n ⎟ ⎜ η2 ⎟ ⎜ ⎟ ⎜ ξm+2 ⎟ ⎟ ⎝ .. ⎠ = ⎜ ⎜ . ηn ⎠ ⎜
· · · Qn−1 ···
0
··· .. .
0 .. .
.. .
⎛
⎞⎞
⎛
⎜ ⎜ ⎜ ⎜ (Q 1 ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
· · · Qn−1
⎜ ⎜ ⎝
ηn
⎞⎞ η1 ⎜ η2 ⎟ ⎟ α ⎜ . ⎟⎟ Q ) n ηn ⎝ . ⎠⎟ . ⎟ ⎟ η n⎞⎟ ⎛ ⎟ ξm+1 ⎞ η 1 ⎟ ηn ⎟ ξm+2 ⎟ ⎜ η2 ⎟ ⎟ ⎜ ⎟ ηn ⎠ ⎝ . ⎠ ⎟ .. ⎠ .. . ηn ⎛
⎛
· · · Qn−1 ⎛0 ··· 0 ⎜0 ··· 0 ⎝ .. .. .. . . . ⎛ ⎞⎞ η 1
⎜ η2 ⎟ ⎟ ⎜ ⎜ . ⎟⎟ ⎜ ⎜ . ⎟⎟ ⎜ ⎜ η1 ⎟ ⎟ ⎜ . ⎟⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟⎟ ⎜ ⎜ η2 ⎟ ⎟ ⎜ ⎟ o ⎜ ηn−1 ⎟ ⎟ ⎟ Qn ) ⎜ ⎟ ⎜Q ⎜ ⎜ .. ⎟ ⎟ ⎜ α ⎟⎟ ⎜ ⎜ . ⎟⎟ ⎜ ⎟⎟ ⎜ ⎠⎟ ⎝ ⎜ ⎟ ⎟ = ξ. 0 = ⎜ ⎟ ηn−1 ⎟ ⎜ ⎜ . ⎟⎟ ⎟ ⎝ .. ⎠ ⎟ ⎜ ⎛ α ⎞⎟ ⎜ ⎟ ⎟ ξm+1 ⎟ ⎜ 0 ⎞⎟ ⎜ ⎟ ⎟ ⎛ ⎝ ξm+2 ⎠ ⎠ ⎜ ξm+1 ⎟ ⎜ ⎟ .. ⎝ ⎝ ξm+2 ⎠ ⎠ . .. .
2
The following lemma is a key “bridge” between properties of majorization and properties of stochastic matrices. Lemma 2.8. If P is a substochastic matrix for which lim n with ηn > 0 for all n, then P is column-stochastic.
n
∗ i=1 (ηi − (P η)i ) = 0 for some η ∈ co
Proof.
n n n ηi − (P η)i = Pij (ηj − ηn ) 1− i=1
j =1
+
i=1 ∞ n
j =n+1 i=1
1−
n i=1
Pij (ηn − ηj ) + n −
n ∞
Pij ηn
i=1 j =1
Pij (ηj − ηn ) 0 for all n j.
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Thus for all j ,
0 lim n 1 −
n
Pij (ηj − ηn ) = 1 −
i=1
hence
∞
i=1 Pij
= 0.
∞
Pij ηj 0,
i=1
2
Remark 2.9. (i) The first line of the proof is based on Ostrowski’s decomposition [32] and shows that n i=1 (ηi − (P η)i ) 0 whether P η is monotone or not. It was used by Markus to prove that P η ≺ η in [27, Lemma 3.1]. (ii) If P η is monotone, then the condition P η η is equivalent to lim n ni=1 (ηi − (P η)i ) = 0 and hence implies that P is column-stochastic. (P ζ )j = ∞ (iii) In the case that ζ ∈ (1 )+ and ∞ j =1 j =1 ζj , it is immediate to verify that ∞ =1 Pj n = 1 for every n which ζn = 0. The same conclusion can be obtained by the operator theoretic argument in [2, Theorem 4.1] in the case that Pij = |Lij |2 for some contraction L. For the reader’s convenience, a sketch of the argument is that then Tr(diag ζ ) = 1 1 Tr(E(L diag ζ L∗ )) and hence Tr(E((diag ζ ) 2 (I − L∗ L)(diag ζ ) 2 )) = 0. Since I − L∗ L 0 1 1 and Tr and E are faithful, it follows that (diag ζ ) 2 (I − L∗ L)(diag ζ ) 2 = 0 and hence Len = 1 for every n for which ζn = 0. (iv) Notice that if P is a substochastic matrix for which P η η for some η ∈ c∗o with ηn > 0 for all n, P can fail to be row-stochastic as is the case for ⎛
1/2 0 ⎜ 1/2 0 ⎜ ⎜ 0 1/2 ⎜ 0 1/2 P := ⎜ ⎜ 0 ⎜ 0 ⎜ 0 ⎝ 0 .. .. . .
0 0 0 0 1/2 1/2 .. .
0 0 0 0 0 0 .. .
⎞ ··· ···⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ···⎟ ⎟ ···⎠ .. .
and η any summable sequence with ηn > 0 for all n. For summable sequences the converse of Lemma 2.8 holds. Lemma 2.10. Let ξ, η ∈ c∗o and ξ = P η for some column-stochastic matrix P . If ξ ∈ (1 )∗ (resp. η ∈ (1 )∗ ), then η ∈ (1 )∗ (resp. ξ ∈ (1 )∗ ) and ξ η. Proof. We know from (1) that ξ ≺ η. Moreover, ∞ i=1
ξi =
∞ i=1
(P η)i =
∞ ∞
Pij ηj =
i=1 j =1
thus ξ ∈ (1 )∗ if and only if η ∈ (1 )∗ and ξ η.
∞ ∞ j =1 i=1
2
Pij ηj =
∞ j =1
ηj ,
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
3125
Without the condition of summability, the implication in Lemma 2.10 can fail. In fact, the following example shows that it can fail even for an orthostochastic matrix as seen by modifying the matrix in Remark 2.9(iv) as follows. Let ω denote the harmonic sequence, i.e., ω := n1 . Example 2.11. An example of an orthostochastic matrix Q for which Qω ω. Proof. Partition N into two infinite strictly increasing sequences {nk } and {mk } for which
lim
2k k 1 1 − nj mj
j =k+1
> 0.
j =1
For instance, this can be achieved by setting mk := (k + 1)2 and listing N \ {mk } as {nk }. Defining ⎧ √1 ⎪ if i = 2k − 1, j = nk , k ∈ N, ⎪ 2 ⎪ ⎪ ⎪ 1 ⎪ if i = 2k − 1, j = mk , k ∈ N, ⎪ ⎨ √2 1 Uij = √ if i = 2k, j = nk , k ∈ N, ⎪ 2 ⎪ ⎪ ⎪ ⎪ − √1 if i = 2k, j = mk , k ∈ N, ⎪ ⎪ 2 ⎩ 0 otherwise, it is easy to see that U is an orthogonal matrix. Let Q be the Schur-square of U , i.e., Qij := Uij2 . Then a simple computation shows that for every k ∈ N, (Qω)2k−1 = (Qω)2k =
1 1 1 + 2 nk mk
and hence (Qω)2k > (Qω)2k+1 , that is, Qω is monotone nonincreasing. Moreover, 2k j =1
2k 2k k 2k k 1 1 1 1 1 − ωj − (Qω)j = + − . j ni mi ni mi j =1
j =1
j =k+1
i=1
j =1
Similarly, 2k+1 j =1
ωj −
2k+1
2k+1
j =1
j =1
(Qω)j =
k 1 1 1 1 1 1 − − + + j ni mi 2 nk+1 mk+1
2k j =k+1
i=1
k 1 1 1 1 1 . − − + ni mi 2 nk+1 mk+1 j =1
Thus lim ( nj=1 ωj − nj=1 (Qω)j ) lim ( 2k j =k+1
1 nj
−
k
1 j =1 mj
) > 0, i.e., Qω ω.
2
We will see from Theorem 5.3 that for any nonsummable sequence η we can choose an orthostochastic matrix Q for which Qη = 12 η and hence Qη η.
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We know of no simple condition that characterizes substochastic matrices for which P η η for all η ∈ c∗o . Notice that since P η is not necessarily monotone, by the latter condition we mean (P η)∗ η for the monotone rearrangement (P η)∗ of P η. A sufficient condition is that P is block stochastic, i.e., the direct sum of doubly stochastic finite matrices. A more general sufficient condition is provided by the following lemma. Lemma 2.12. If P is a substochastic matrix and lim (n −
n
i,j =1 Pij ) = 0,
then
(i) P is doubly stochastic; (ii) P η η for every η ∈ c∗o . Proof. (i) n−
n
Pij =
i,j =1
n
1−
Pij
j =1
i=1
n
n
1−
∞
Pij
j =1
i=1
thus
0 = lim n −
∞
n
Pij
i,j =1
1−
∞
Pij .
j =1
i=1
∞ Then because P is substochastic, ∞ j =1 Pij = 1 for all i. Similarly, i=1 Pij = 1 for all j . n n ∗ (ii) Since i=1 (P η)i i=1 (P η)i for every n, n n ηi − (P η)∗i ηi − (P η)i i=1
i=1
=
n j =1
1−
Pij ηj −
i=1
n
n
1−
j =1
= n−
n i=1
n
n ∞
Pij ηj
i=1 j =n+1
Pij η ∞
Pij η ∞ ,
i,j =1
hence lim ni=1 (ηi − (P η)i )∗ lim (n − ni,j =1 Pij ) η ∞ = 0. Thus (P η)∗ η, i.e., by Definition 1.2, P η η. 2
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Remark 2.13. The condition lim (n − ni,j =1 Pij ) = 0 is not necessary for (ii). For instance, ∗ (ii) holds trivially for every permutation matrix Π since (Πη) = η for every η, but it is easy to find a permutation matrix Π for which lim (n − ni,j =1 Πij ) > 0. 3. The canonical co-isometry of a majorization We start with some historical notes about the link between majorization and stochastic matrices. Muirhead [30] for the case of integer-valued finite sequences and then Hardy, Littlewood and Pólya [11, p. 47] for the case of real-valued finite sequences proved that for all ξ, η ∈ RN with ξ η, there is a doubly stochastic matrix P with ξ = P η. P was obtained as a finite product of T-transforms, i.e. matrices of the form tI + (1 − t)Π with Π a transposition and 0 t < 1. The T-transforms were chosen so to reduce at each step the Hamming distance (i.e., the number of positions where two sequences differ) between the sequence ξ and the iterated transform of η. Notice that while individual T-transforms are orthostochastic, the product of two T-transforms can fail to be even unistochastic [28, Chapter II, Section G]. In 1952, Horn proved that the matrix P can be chosen to be orthostochastic by using a different method based on convexity arguments and a technically difficult proof [12, Theorem 4]. A proof based on properties of determinants was given a few years later by Mirsky [29, Theorem 2]. After a four decade hiatus, a proof based on composition of Givens rotations (special permutations of T-transforms) was obtained by Casazza and Leon in the Appendix of [5]. More recently, Arveson and Kadison [2, Theorem 2.1] gave an elegant proof of the Horn Theorem showing that P can be chosen to be unistochastic (see also [15, Lemma 5 and Theorem 6]). Reformulating their result in our terminology, they showed that ξ is obtained by applying to η a finite number of T-transforms and that by properly choosing unitary matrices whose Schur-squares are those T-transforms, the Schur-square of their product (a unistochastic matrix by definition) applied to η also yields ξ . Another recent proof was obtained by Kornelson and Larson [25, Theorem 2]. More precisely, they proved the equivalent statement that every positive finite rank operator B with eigenvalue list η can be decomposed as the linear combination B = kj =1 ξj Pj of rank-one projections (not required to be mutually orthogonal) with the given monotone nonincreasing coefficient sequence ξ := ξ1 , ξ2 , . . . , ξk , 0, . . . , if and only if (in our notations) ξ η. This link between majorization and stochastic matrices was partially extended to the infinite case by Markus in [27, Lemma 3.1] (see (1)) and the Schur–Horn Theorem was extended to summable sequences by Gohberg and Markus in [9, Lemma 1] (see (2)) based on [9, Theorem 1]. The latter proof depended crucially on the summability of the sequence, so we focus on the former proof. At the core of Markus’s proof, although he did not employ this terminology nor exhibit explicitly the matrices, is the construction, for every ξ, η ∈ c∗o with ξ ≺ η, of an infinite sequence of permutations of T-transforms whose product is a substochastic matrix Q for which ξ = Qη. Furthermore, a remark in his proof states that when ξn > 0 for all n, the matrix Q is row-stochastic. In this section we revisit and slightly tighten the Markus construction for the case when ξn > 0 for all n (see Remark 3.8) and prove that it provides a co-isometry stochastic matrix (Theorem 3.7) and in the strong majorization case, an orthostochastic matrix (Theorems 3.9, 4.7). Not surprisingly, this construction restricted to finite sequences yields another proof of the Horn Theorem (Remark 4.3).
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For every integer m 1 and 0 < t 1, define the m + 1 × m + 1 orthogonal matrix √ √ ⎛ ⎞ 0 0 ··· 0 t − 1−t ⎜1 0 ··· 0 ⎟ 0 0 ⎜ ⎟ ⎜0 1 ··· 0 ⎟ 0 0 ⎜ ⎟. V (m, t) := ⎜ .. .. .. .. .. .. ⎟ . . . . ⎜. . ⎟ ⎝0 0 ··· 1 ⎠ 0 0 √ √ 1−t t 0 0 ··· 0
(7)
Notice that the Schur-square of the matrix V (m, t) is the product of the permutation matrix Π1 that sends 1, 2, . . . , m, m + 1 to m, 1, 2, . . . , m − 1, m + 1 and of the so-called T-transform tI + (1 − t)Π2 where Π2 is the transposition that interchanges m and m + 1. Given a sequence {mk , tk } where mk ∈ N and 0 < tk 1, define for every n ∈ N W (n) := In−1 ⊕ V (mn , tn ) ⊕ I∞ · · · I1 ⊕ V (m2 , t2 ) ⊕ I∞ V (m1 , t1 ) ⊕ I∞
(8)
where In denotes the n × n identity matrix for 1 n ∞ and Io is simply dropped. Define also R (n) to be the Schur-square of V (mn , tn ) ⊕ I∞ , and let Q(n) := In−1 ⊕ R (n) In−2 ⊕ R (n−1) · · · I1 ⊕ R (2) R (1) .
(9)
Being a product of orthogonal matrices, all W (n) are also orthogonal. Denote by Pn := In ⊕ 0 the projection on span{e1 , . . . , en }. Proposition 3.1. Let {mk , tk } be a sequence with mk ∈ N and 0 < tk 1. Then: (i) The sequence of operators W (n) converges in the weak operator topology to a co-isometry W ({mk , tk }) and Pn W ({mk , tk }) = Pn W (n) for every n. The convergence is in the strong operator topology if and only if W ({mk , tk }) is orthogonal. (ii) The sequence of operators Q(n) converges in the weak operator topology to a row-stochastic operator Q({mk , tk }) and Pn Q({mk , tk }) = Pn Q(n) for every n. Proof. (i) From (8) we have for all integers j > n, W (j ) = Ij −1 ⊕ V (mj , tj ) ⊕ I∞ · · · In ⊕ V (mn+1 , tn+1 ) ⊕ I∞ W (n) and hence Pn W (j ) = Pn W (n)
for all j n.
(10)
As a consequence, ((W (j ) − W (i) )x, y) = ((W (j ) − W (i) )x, Pn⊥ y) for all x, y ∈ H and i, j n. Thus the sequence of orthogonal matrices {W (j ) } is weakly Cauchy and hence converges weakly to a contraction W ({mk , tk }) with real entries. Set W := W ({mk , tk }). From (10), it follows that (n) Pn W = Pn W (n) for all n, that is, the first n rows of the matrix W (n) stabilize: Wij = Wij for all n, j and all i n. Therefore ∗ W W ∗ = s-lim Pn W W ∗ Pn = s-lim Pn W (n) W (n) Pn = s-lim Pn = I,
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i.e., W is a co-isometry. Since (W − W (n) )x 2 → x 2 − W x 2 for all x ∈ H , the sequence W (n) converges strongly to W if and only if W is orthogonal. (ii) Same proof. 2 Remark 3.2. Proposition 3.1 holds even if we allow tk = 0. However, in order to obtain the uniqueness of the sequence {mk , tk } in the construction in Theorem 3.7, we will have to assume there that tk > 0 for all k. In addition, this assumption will simplify some of the proofs. A case where it is simple to find the form of W ({mk , tk }) and Q({mk , tk }) is when mk = 1 for all k. Example 3.3. ⎛
√
t1 t (1 2 − t1 ) ⎜ ⎜ t3 (1 − t2 )(1 − t1 ) ⎜ ⎜ . . W {1, tk } = ⎜ ⎜ . ⎜ tk k−1 (1 − ti ) i=1 ⎝ √
√ − 1 − t1 √ t2 t1 t3 (1 − t2 )t1 . .. k−1 tk t1 i=2 (1 − ti )
0 √ − 1 − t2 √ t3 t2 . .. k−1 tk t2 i=3 (1 − ti )
0 0 − 1 − t3 . .. k−1 tk t3 i=4 (1 − ti )
. ..
. ..
. ..
. ..
⎛
t1 ⎜ t2 (1 − t1 ) ⎜ t3 (1 − t2 )(1 − t1 ) ⎜ ⎜ .. Q {1, tk } = ⎜ ⎜ . ⎜ tk k−1 (1 − ti ) i=1 ⎝ .. .
1 − t1 t2 t1 t3 (1 − t2 )t1 .. . k−1 tk t1 i=2 (1 − ti )
0 1 − t2 t3 t2 .. . k−1 tk t2 i=3 (1 − ti )
0 0 1 − t3 .. . k−1 tk t3 i=4 (1 − ti )
.. .
.. .
.. .
⎞
···⎞ ···⎟ ···⎟ ⎟ . ⎟ .. ⎟ , ⎟ ···⎟ ⎠ . ..
··· ···⎟ ···⎟ ⎟ .. ⎟ . . ⎟ ⎟ ···⎟ ⎠ .. .
We see in this case that Q({1, tk }) is the Schur-square of W ({1, tk }). For the general case we first state a couple of elementary lemmas leaving their proof to the reader. Lemma 3.4. Let A and B be two bounded matrices with the property that for every i, j there is at most one index k for which Aik Bkj = 0 and let A and B be the Schur-square of A and B respectively. Then A B is the Schur-square of AB. If furthermore A and B are orthogonal (resp. unitary) then A B is orthostochastic (resp. unistochastic). In particular, if Q is orthostochastic (resp. unistochastic) and Π is a permutation, then ΠQ is orthostochastic (resp. unistochastic). Next, we consider a simple case where this sufficient condition is satisfied. Lemma 3.5. Let A and B be bounded matrices, let n ∈ N, and assume that for every j there is at most one index i > n for which Bij = 0. (i) For every i and j there is at most one index k for which (In ⊕ A)ik Bkj = 0. (ii) If for every j there is at most one index i > 1 (resp. i > 0) for which Aij = 0, then for every j there is at most one index i > n + 1 (resp. i > n) for which ((In ⊕ A)B)ij = 0. (iii) Let A(n) be a sequence of bounded matrices for which for every n and j , A(n)ij = 0 for at most one index i > 1 and let Q = (In−1 ⊕ A(n))(In−2 ⊕ A(n − 1)) · · · (I1 ⊕ A(2))A(1). Then for every j , Qij = 0 for at most one index i > n.
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Given a sequence {mk , tk } where mk ∈ N and 0 < tk 1, then for every n, j , at most one of (n) the entries Wij for i > n is nonzero. Proposition 3.6. Let {mk , tk } be a sequence with mk ∈ N and 0 < tk 1 and let W (n) and Q(n) be as in (8) and (9). Then (i) Q(n) is the Schur-square of W (n) for every n and for every n and j there is at most one index (n) i > n for which Qij = 0. (ii) Q({mk , tk }) is the Schur-square of W ({mk , tk }). Proof. (i) We reason by induction. Q(1) is by definition the Schur-square of W (1) . Assume that Q(n) is the Schur-square of W (n) . Now W (n+1) = (In ⊕ V (mn+1 , tn+1 ) ⊕ I∞ )W (n) . Since for every factor V (mk , tk ) ⊕ I∞ and every j there is at most one index i > 1 for which the i, j entry (V (mk , tk ) ⊕ I∞ )ij = 0, it follows from Lemma 3.5(iii) that for ev(n) ery n and j , Wij = 0 for at most one index i > n. Thus by Lemma 3.5(i), for every i (n) does not vanish for at most one index k. But then, and j , (In ⊕ V (mn+1 , tn+1 ) ⊕ I∞ )ik Wkj the product of Q(n+1) = (In ⊕ R (n+1) )Q(n) of the Schur-square of In ⊕ V (mn+1 , tn+1 ) ⊕ I∞ by the Schur-square of W (n) coincides by Lemma 3.4 with the Schur-square of W (n+1) = (In ⊕ V (mn+1 , tn+1 ) ⊕ I∞ )W (n) . (ii) Obvious since the first n rows of W ({mk , tk }) (resp. Q({mk , tk })) coincide with the first n rows of W (n) (resp. Q(n) ). 2
To every majorization ξ ≺ η with ξn > 0 for all n, the following construction associates a sequence {mk , tk } with mk ∈ N and 0 < tk 1 and hence associates the corresponding co-isometry W (ξ, η) := W ({mk , tk }). Theorem 3.7. Let ξ, η ∈ c∗o with ξn > 0 for every n. If ξ ≺ η, then there is a canonical co-isometry W (ξ, η) with real entries whose Schur-square Q(ξ, η) satisfies ξ = Q(ξ, η)η. Proof. We construct the following sequence {mk , tk } where mk ∈ N and 0 < tk 1. Set ρ(0) := η and choose m1 ∈ N for which ηm1 +1 < ξ1 ηm1 . Since ξ ≺ η and hence ξ1 η1 , and since ξ1 > 0 and ηj → 0, such an integer exists and by the monotonicity of η, it is unique. Express ξ1 as a convex combination of ηm1 and ηm1 +1 , that is, choose t1 for which ξ1 = t1 ηm1 + (1 − t1 )ηm1 +1 . Thus 0 < t1 1 and also t1 is uniquely determined. Set δ1 := (1 − t1 )ηm1 + t1 ηm1 +1 and hence δ1 = ηm1 + ηm1 +1 − ξ1 . Define the sequence ρ(1) := η1 , η2 , . . . , ηm1 −1 , δ1 , ηm1 +2 , ηm1 +3 , . . . where if m1 = 1, then the first entry of ρ(1) is δ1 . Since ηm1 +2 ηm1 +1 δ1 < ηm1 ηm1 −1 , we see that ρ(1) is monotone nonincreasing and ρ(1) η. Let R (1) be the Schur(1) square of V (m1 , t1 ) ⊕ I∞ , i.e., Rij = ((V (m1 , t1 ) ⊕ I∞ )ij )2 for all i, j . Then R (1) η = ξ1 , ρ(1)1 , ρ(1)2 , . . . . Moreover, ξ (1) ≺ ρ(1).
(11)
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(Recall the notation ξ (1) := ξ2 , ξ3 , . . . .) Indeed, for every 1 n < m1 , n
(1)
ρ(1)j − ξj
j =1
=
n
ηj −
j =1
n+1 j =2
ξj =
n (ηj − ξj ) + ξ1 − ξn+1 0, j =1
and for every n m1 m n+1 n+1 n+1 n 1 −1 (1) ρ(1)j − ξj = ηj + δ1 + ηj − ξj = (ηj − ξj ) 0. j =1
j =1
j =m1 +2
j =2
j =1
Repeat the construction applying it to the pair ξ (1) ≺ ρ(1), and so on. By the assumption that ξk > 0 for all k, the process can be iterated providing an infinite sequence of pairs {mk , tk } with mk ∈ N and 0 < tk 1 and from these, of sequences ρ(k) and scalars δk satisfying for all k the relations: ρ(k − 1)mk +1 < ξk ρ(k − 1)mk ,
ξk = tk ρ(k − 1)mk + (1 − tk )ρ(k − 1)mk +1 ,
δk := (1 − tk )ρ(k − 1)mk + tk ρ(k − 1)mk +1 = ρ(k − 1)mk + ρ(k − 1)mk +1 − ξk , ρ(k) := ρ(k − 1)1 , . . . , ρ(k − 1)mk −1 , δk , ρ(k − 1)mk +2 , . . . , i.e., ⎧ for all j < mk , ⎨ ρ(k − 1)j for j = mk , ρ(k)j = δk ⎩ ρ(k − 1) j +1 for all j > mk , ξ (k) ≺ ρ(k),
η = ρ(0) ρ(1) ρ(2) · · · ,
n n+1 (k) (k−1) ρ(k)j − ξj = ρ(k − 1)j − ξj j =1
for all n mk .
(12) (13)
(14) (15) (16)
j =1
Let R (n) be the Schur-square of V (mn , tn ) ⊕ I∞ , and let Q(n) := (In−1 ⊕ R (n) ) · · · (I1 ⊕ R (2) )R (1) as in (9). Then for all n, R (n) ρ(n − 1) = ξn , ρ(n)1 , ρ(n)2 , . . . ,
Q(n) η = ξ1 , ξ2 , . . . , ξn , ρ(n)1 , ρ(n)2 , . . . . (17)
Let W (ξ, η) := W {mk , tk } and Q(ξ, η) := Q {mk , tk } . Then by Propositions 3.1 and 3.6, W (ξ, η) is a co-isometry and Q(ξ, η) is its Schur-square. Finally, by Remark 2.2, Q(ξ, η)η is defined and is a sequence in co . In fact, by (17), Pn Q(ξ, η)η = Pn Q(n) η = ξ1 , ξ2 , . . . , ξn , 0, 0, . . . → ξ and hence Q(ξ, η)η = ξ .
2
pointwise,
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Remark 3.8. (i) The construction of the sequence {mk , tk } and the associated sequence of matrices Q(n) follows the Markus construction in [27, Lemma 3.1]. A minor difference is that while Markus chose mk to be an index for which ηmk +1 ξk ηmk so to treat at the same time also the case when ξ is finitely supported, here we consider only the case of infinitely supported ξ and then request that ηmk +1 < ξk ηmk , which makes the construction canonical. The main difference is that Markus’s analysis is at the level of the action of the matrices Q(n) on η, and thus yields only that their limit Q is row-stochastic. It is by introducing the underlying matrices W (n) and analyzing their properties that we can obtain that Q is co-isometry stochastic. (ii) A consequence of [9, Proposition III, p. 205] obtained by Gohberg and Markus with different methods, is that if ξ ≺ η, then ξ = Qη for some co-isometry stochastic matrix Q (see Remark 6.5(ii) for more details). When the majorization is strong, we obtain the following extension of the Horn Theorem [12, Theorem 4] (see Theorem 1.1(ii)). In the nonsummable case strong majorization will not be required, as we will see in Theorem 5.3. Theorem 3.9. If ξ, η ∈ c∗o and ξ η, then ξ = Qη for some orthostochastic matrix Q. Proof. If η has finite support, then the conclusion follows from Lemma 2.7. If η has infinite ξ toohas infinite support. Indeed, if otherwise ξn = 0 for some n, then ∞ support,then n−1 n−1 η = ξ j =1 j j =1 j j =1 ηj which implies that ηn = 0, a contradiction. But then, by Theorem 3.7, ξ = Q(ξ, η)η where Q(ξ, η) is the Schur-square of the co-isometry W (ξ, η). By Lemma 2.8 and Remark 2.9(ii), Q(ξ, η) is column-stochastic and hence by Lemma 2.4, W (ξ, η) is also an isometry and hence unitary. Since by construction W (ξ, η) has real entries, it is orthogonal, hence Q(ξ, η) is orthostochastic. 2 Remark 3.10. (i) The above proof shows that if ξ η and η has infinite support, then any co-isometry stochastic matrix Q for which ξ = Qη must be unistochastic, i.e., the Schur product of a unitary matrix by its complex conjugate. (ii) In the case when ξ η and ξ has infinite support but η does not, we cannot invoke Lemma 2.8 to conclude that W (ξ, η) is orthogonal, so for simplicity’s sake, we have chosen in lieu of Q(ξ, η) the orthostochastic matrix provided by Lemma 2.7. However, in the next section we will prove that if ξ η, then Q(ξ, η) itself is orthostochastic (Theorem 4.7). (iii) If ξ has finite support, say {1, . . . , N}, then the construction of Theorem 3.7 can still be carried on for the first N steps and it provides yet another proof of the Horn Theorem (see Remark 4.3). 4. Properties of the canonical matrix Q(ξ, η) of a majorization In this section, on which the following ones do not depend, we further analyze the construction in Theorem 3.7 to relate the properties of the majorization ξ ≺ η to those of the canonical coisometry stochastic matrix Q(ξ, η) via the properties of the set {tk | mk = 1}. In the next lemmas
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we collect the additional needed properties of the sequences mk , tk , δk , ρ(k), W (n) , etc. that were introduced in Theorem 3.7. Lemma 4.1. Let ξ, η ∈ c∗o , ξ ≺ η, and assume that ξn > 0 for all n. Then for every k ∈ N (i) mk mk−1 − 1; (ii) ρ(k)j = ηj +k for every j > mk ; (k) (iii) if n mk , then nj=1 (ρ(k)j − ξj ) = n+k j =1 (ηj − ξj ); k (iv) if mk = 1, then δk = ηk+1 + j =1 (ηj − ξj ); (v) if tk = 1, and mk = mk−1 − 1, then tk−1 = 1; (vi) if nj=1 (ηj − ξj ) = 0 and n−1 j =1 (ηj − ξj ) > 0 for some n > 1, then mn = tn = 1. Proof. (i) Assume by contradiction that mk < mk−1 − 1, then ξk > ρ(k − 1)mk +1 = ρ(k − 2)mk +1
by (12) by (14) since mk + 1 < mk−1
ρ(k − 2)mk−1 by the monotonicity of ρ(k − 2), since mk + 1 < mk−1 ξk−1 by (12) .
This is a contradiction because of the monotonicity of ξ . (ii) The proof is by induction on k. The property holds by (14) for k = 1 since by definition ρ(0) = η. Assume it holds for some k and let j > mk+1 . Then, ρ(k + 1)j = ρ(k)j +1 = ηj +1+k
by (14) by the induction hypothesis, since by (i), j + 1 > mk .
(iii) If n mk , then by (i), n + p mk−p for all 0 p < k. Thus iterating (16) n n+k n+k (k) (0) ρ(k)j − ξj = ρ(0)j − ξj = (ηj − ξj ). j =1
j =1
j =1
(iv) Since δk = ρ(k)1 by (14), setting n = 1 in (iii) we obtain δk = ξk+1 +
k+1 k (ηj − ξj ) = ηk+1 + (ηj − ξj ). j =1
j =1
(v) ξk−1 ρ(k − 2)mk−1 ρ(k − 2)mk = ρ(k − 1)mk
by (12)
by the monotonicity of ρ(k − 2), since mk < mk−1 by (14), since mk < mk−1
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by (12), since tk = 1
= ξk ξk−1
(by the monotonicity of ξ ).
But then, ξk−1 = ρ(k − 2)mk−1 and hence by (12), tk−1 = 1. (vi) We reason by induction on n and first prove the property for n = 2. If η1 + η2 = ξ1 + ξ2 and η1 > ξ1 , then η2 < ξ2 ξ1 < η1 . Thus m1 = 1, ρ(1)1 = δ1 = η1 + η2 − ξ1 = ξ2 and hence (vi) holds for some n 2 for every pair of sequences m2 = t2 = 1. Assume now that theproperty n (η − ξ ) = 0 and (η − ξj ) > 0. Then ηn+1 < ξn+1 ξ1 ηm1 implies that and that n+1 j j j j =1 j =1 n (1) n + 1 > m1 and from (16) we obtain that j =1 (ρ(1)j − ξj ) = n+1 j =1 (ηj − ξj ) = 0. We claim n−1 (1) that j =1 (ρ(1)j − ξj ) > 0. If n > m1 , the claim holds because by (14)
ρ(1)n = ηn+1 < ξn+1 = ξn(1)
(by definition)
n−1 = ρ(1)n + ρ(1)j − ξj(1)
since
j =1
n
ρ(1)j − ξj(1) = 0
.
j =1
If n = m1 , i.e., ηn+1 < ξ1 ηn , then n−1 ρ(1)j − ξj(1) = ξn(1) − ρ(1)n j =1
= ξn+1 − δ1
n ρ(1)j − ξj(1) = 0 since
by (14)
j =1
= ξn+1 − ηn+1 + ξ1 − ηn by (13)
n−1 n n+1 ηj − ξj (ηj − ξj ) = 0 . = since j =1
Thus, if ξ1 = ηn , then
n−1
j =2
j =1
(1)
− ξj ) = ξn+1 − ηn+1 > 0. If on the other hand ξ1 < ηn , then n−1 n by the monotonicity of η and ξ , we have n−1 j =2 ξj , thus completing the j =1 ηj > j =1 ξj (1) proof of the claim. Therefore the sequences ξ ≺ ρ(1) satisfy the hypotheses of (vi) for n and hence, by the induction hypothesis, satisfy the thesis of (vi). But by definition, the pair {mn , tn } for ξ (1) ≺ ρ(1) coincides with the pair {mn+1 , tn+1 } for ξ ≺ η, which concludes the induction proof. 2 j =1 (ρ(1)j
Without the assumption that n−1 j =1 (ηj − ξj ) > 0, the conclusion of (vi) may fail: consider for instance ξ = 1, 1, ∗, . . . and η = 1, 1, 1, 0, . . . where m2 = 2. Lemma 4.2. Let ξ, η ∈ c∗o with ξn > 0 for all n and ξ ≺ η. ⊥ for every n. (i) W (n) = Pn+mn W (n) Pn+mn + Pn+m n
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(iii) If mn = tn = 1, then Pn commutes
n
j =1 (ηj − ξj ) = 0. with W (n) and with W (ξ, η)
(ii) If Pn commutes with W (ξ, η), then
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and ρ(n) = η(n) .
Proof. (i) By Lemma 4.1(i), the sizes k +mk of the matrices Ik−1 ⊕V (mk , tk ) are nondecreasing. Thus for every 1 k n, ⊥ ⊥ ⊥ Pn+m Ik−1 ⊕ V (mk , tk ) ⊕ I∞ = Ik−1 ⊕ V (mk , tk ) ⊕ I∞ Pn+m = Pn+m . n n n ⊥ ⊥ ⊥ W (n) = W (n) Pn+m = Pn+m and hence the claim follows. By (8), Pn+m n n n (ii) If W (ξ, η) commutes with Pn , i.e., W (ξ, η)ij = 0 when 1 i n and j > n and when 1 j n and i > n, then so does its Schur-square Q(ξ, η). But then, the following n × n matrix Qn := Pn Q(ξ, η)Pn |Pn H is also orthostochastic. Since Q(ξ, η)η = ξ , it follows also that Qn η1 , . . . , ηn = ξ1 , . . . , ξn and hence nj=1 (ηj − ξj ) = 0. (iii) By Lemma 4.1(i), for every k, either mk−1 mk or mk−1 = mk + 1. Let j be the largest index i n for which mi−1 mi and if there is none, set j = 1. Then mi = n + 1 − i for all j i n. By applying recursively Lemma 4.1(v) we obtain that ti = 1 for all j i n. But then the size of all the matrices Ii−1 ⊕ V (n + 1 − i, 1) is constant and equal to n + 1, hence
W (n) =
In−1 ⊕ V (1, 1) In−2 ⊕ V (2, 1) · · · Ij −1 ⊕ V (n + 1 − j, 1) ⊕ I∞ W (j −1)
where we set W (0) = I if j = 1. All the matrices Ii−1 ⊕ V (n + 1 − i, 1) for j i n are permutation matrices of order n + 1 that leave the n + 1 position fixed and hence they commute with Pn . If j = 1, then W (0) = I commutes trivially with Pn , while if j > 1, then mj −1 = mj = n + 1 − j , hence n = j − 1 + mj −1 , and thus by (i), W (j −1) also commutes with Pn . Thus Pn W (n) = W (n) Pn . As Pn W (ξ, η) = Pn W (n) by Proposition 3.1, it follows that Pn W (ξ, η)Pn⊥ = 0. On the other hand, W (ξ, η) is a co-isometry and W (n) is unitary, hence ∗ ∗ Pn⊥ W (ξ, η)Pn = Pn⊥ W (ξ, η) W (n) W (n) Pn = Pn⊥ W (ξ, η) W (n) Pn W (n) = Pn⊥ W (ξ, η)W (ξ, η)∗ Pn W (n) = Pn⊥ Pn W (n) = 0 which proves that W (ξ, η) commutes with Pn . Moreover, ρ(n) = δn , ηn+2 , . . . by (14) and Lemma 4.1(ii), since mn = 1 n (ηj − ξj ), ηn+2 , . . . by Lemma 4.1(iv), since mn = 1 = ηn+1 + j =1
= η(n)
by (ii) .
2
Remark 4.3 (Proof of the Horn Theorem). If the sequence ξ has finite support, say {1, . . . , N}, then, as mentioned in Remark 3.10(iii), the construction in Theorem 3.7 can be carried out for the first N steps, and the properties obtained in Lemmas 4.1 and 4.2 hold for 1 n N . Thus Q(N ) is an infinite orthostochastic matrix and Q(N ) η = ξ1 , . . . , ξN , ρ(N )1 , ρ(N )2 , . . . . If furthermore ξ η, then also ηj = 0 for all j > N and ρ(N) ≡ 0. It is then easy to verify that then the upper left N × N block QN of Q(N ) is also orthostochastic and that ξ1 , . . . , ξN = QN η1 , . . . , ηN .
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Thus if we start with two finite (monotone) sequences ξ, η ∈ RN with ξ η we can obtain the required orthostochastic matrix QN by applying the construction in Theorem 3.7 to N × N matrices, thus providing an algorithmic proof of the Horn Theorem. For the reader’s convenience we summarize this adaptation. Proof. For every integer 1 m n − 1 < N and 0 < t 1, define the n × n orthogonal matrix ⎛0 0 ⎜1 0 ⎜0 1 ⎜ ⎜ .. .. ⎜. . ⎜ ⎜0 0 V (m, t, n) := ⎜ ⎜0 0 ⎜ ⎜0 0 ⎜ ⎜0 0 ⎜. . ⎝. . . .
··· 0 ··· 0 ··· 0 .. . ··· ··· ··· ···
√ t 0 0 .. .
1 √0 1−t 0 1 0 1 0 .. .. . . 0 0 ··· 1 0
√ − 1−t 0 0 .. . √0 t 0 0 .. .
0 0 0 0 0 0 .. .. . . 0 0 0 0 1 0 0 1 .. .. . .
··· 0⎞ ··· 0⎟ ··· 0⎟ ⎟ .. ⎟ .⎟ ⎟ ··· 0⎟ ⎟, ··· 0⎟ ⎟ ··· 0⎟ ⎟ ··· 0⎟ .. ⎟ ⎠ .
0
0 0
··· 1
where the first nonzero entry on the first row occurs in position m. Construct the sequence −1 {mk , tk }N with 1 mk N − k and 0 < tk 1 for which ξk = tk ρ(k) ˜ mk + (1 − tk )ρ(k) ˜ mk+1 1 where ρ(k) ˜ is defined inductively by ˜ − 1)mk−1 , (1 − tk )ρ(k ˜ − 1)mk ρ(k) ˜ := ρ(k ˜ − 1)1 , . . . , ρ(k + tk ρ(k ˜ − 1)mk+1 − ξk−1 , ρ(k ˜ − 1)mk+2 , . . . , ρ(k ˜ − 1)N −k
starting with ρ(0) ˜ := η = η1 , . . . , ηN . Then WN := IN −2 ⊕ V (mN −1 , tN −1 , 2) IN −3 ⊕ V (mN −2 , tN −2 , 3) · · · V (m1 , t1 , N ) is an orthogonal matrix and its Schur-square QN satisfies ξ = QN η.
2
Now we return to infinite sequences and apply Lemmas 4.1 and 4.2 to show that the set {tk | mk = 1} encodes key information about W (ξ, η) and Q(ξ, η). First, we characterize block-majorization (see Definition 1.2), both because it might be of independent interest and because it provides a key step in the characterization of strong majorization. Recall from (6) that an immediate consequence of the Horn Theorem is that ξ ≺b η if and only if ξ = Qη for some block orthostochastic matrix Q. The next proposition states that if ξ ≺b η, then Q(ξ, η) itself must be block orthostochastic (equivalently, W (ξ, η) is the direct sum of finite orthogonal matrices) and characterizes when this occurs in terms of the sequence {mk , tk }. Proposition 4.4. Let ξ ≺ η for some ξ, η ∈ c∗o with ξn > 0 for every n. Then the following conditions are equivalent.
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(i) ξ ≺b η. (ii) The set {k | mk = tk = 1} is infinite. (iii) Q(ξ, η) is block orthostochastic. nk Proof. (i) ⇒ (ii) By definition, j =1 (ηj − ξj ) = 0 for some strictly increasing sequence {nk }. pk Then either there is an infinite sequence of integers pk for which j =1 (ηj − ξj ) = 0 and n pk −1 j =1 (ηj − ξj ) = 0 for all n N and j =1 (ηj − ξj ) > 0 or there is some N ∈ N for which hence ηj = ξj for all j > N . In the first case mpk = tpk = 1 for all k by Lemma 4.1(vi). In the −1 second case, choose the smallest N with this property. If N > 1, then N j =1 (ηj − ξj ) > 0 and hence mN = tN = 1 by Lemma 4.1(vi) and by (14) and Lemma 4.1(ii) by Lemma 4.1(iv), since (ηj − ξj ) = 0 = ηN +1 , ηN +2 , . . .
ρ(N ) = δN , ηN +2 , . . .
j =1
=η
(N )
=ξ
(N )
.
If N = 1 then we see directly that ξ = η. It is easy to see now that whether N = 1 or N > 1, ρ(j ) = η(j ) for all j N . Since η → 0 and η has infinite support since by assumption and so has ξ , there is an infinite collection of indices j > N for which ρ(j − 1)2 = ηj +1 < ξj = ηj = ρ(j − 1)1 and thus for those indices mj = 1 = tj . (ii) ⇒ (iii) By Lemma 4.2(ii), W (ξ, η) commutes with every Pk for which mk = tk = 1. Thus W (ξ, η) is block diagonal with each (finite) block an orthogonal matrix and hence its Schursquare Q(ξ, η) is block orthostochastic. (iii) ⇒ (i) Obvious (see (6)). 2 Next, we proceed to characterize strong majorization ξ η. To do so, we will first need to further analyze the property obtained in Proposition 3.6(i) that for every n the orthogonal matrix W (n) has in each column at most one nonzero entry below row n. For a given j , define q(n, j ) = γ (n, j ) = 0
(n)
if all the entries of Wij for i > n are zero,
(n)
Wn+q(n,j ),j = γ (n, j ) is the unique nonzero entry.
(18)
Reformulating (18) in vector form, ⎛
(n)
⎞
⎠= 0 γ (n, j )eq(n,j ) n+2,j ···
Wn+1,j
⎝ W (n)
and thus we obtain the recurrence relation
if q(n, j ) = 0, if q(n, j ) = 0
(19)
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⎛ (n) ⎞ Wn+1,j ⎝ W (n+1) ⎠ = V (mn+1 , tn+1 ) ⊕ I∞ ⎝ W (n) ⎠ n+2,j n+2,j ··· ··· 0 = γ (n, j )(V (mn+1 , tn+1 ) ⊕ I∞ )eq(n,j ) ⎛
(n+1) ⎞
Wn+1,j
if q(n, j ) = 0, if q(n, j ) = 0.
We leave to the reader to verify the following lemma. Lemma 4.5. Given a sequence {mk , tk } where mk ∈ N and 0 < tk 1, and given the corresponding co-isometry W := W ({mk , tk }), let q(n, j ) and γ (n, j ) be the sequences defined for every n, j by (18) (i) ⎧ j ⎪ ⎨ j q(1, j ) = ⎪ ⎩0 j −1
for j for j for j for j
< m1 , = m1 , t1 = 1, = m1 , t1 = 1, > m1 ,
⎧ 1 ⎪ ⎨√ 1 − t1 γ (1, j ) = √ ⎪ ⎩ t1 1
for j for j for j for j
< m1 , = m1 , = m1 + 1, > m1 + 1,
and ⎧ q(n, j ) for q(n, j ) < mn+1 , ⎪ ⎨ q(n, j ) for q(n, j ) = mn+1 , tn+1 = 1, q(n + 1, j ) = for q(n, j ) = mn+1 , tn+1 = 1, ⎪ ⎩0 q(n, j ) − 1 for q(n, j ) > mn+1 , ⎧ γ (n, j ) for q(n, j ) < mn+1 , ⎪ ⎨√ 1 − tn+1 γ (n, j ) for q(n, j ) = mn+1 , γ (n + 1, j ) = √ for q(n, j ) = mn+1 + 1, ⎪ ⎩ tn+1 γ (n, j ) γ (n, j ) for q(n, j ) > mn+1 + 1. (ii) 0 q(n + 1, j ) q(n, j ) j and 0 γ (n + 1, j ) γ (n, j ) 1 for every n and j . (iii) Pn⊥ W (n) ej = γ (n, j ) for every n and j . (iv) For all n > 1 and all j , (n) Wnj = Wnj
=
0 γ (n − 1, j )(V (mn , tn ) ⊕ I∞ )1,q(n−1,j )
if q(n − 1, j ) = 0, if q(n − 1, j ) = 0.
In particular, all the√entries of W are either 0, 1, or products of a finite number of the factors √ √ tk , 1 − tk and − 1 − tk , but not more than one for each k. The case j = 1 is of special use. Lemma 4.6. Given a sequence {mk , tk } where mk ∈ N and 0 < tk 1, and given the corresponding co-isometry W := W ({mk , tk }), let gn := γ (n, 1)2 and set g∞ := lim gn . Then
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
3139
1 mk > 1 for all k, (i) g∞ = {(1 − tk ) | mk = 1} otherwise. √ tn gn−1 if mn = 1, (ii) Wn1 = 0 if mn > 1. 2 (iii) W e1 = 1 − g∞ . Proof. (i) It is straightforward to solve the recurrence relation in Lemma 4.5(i) for j = 1 and obtain q(n, 1) =
0 if mk = 1, tk = 1 for some 1 k n, 1 otherwise,
1 if mk > 1 for all 1 k n, γ (n, 1) = √ { 1 − tk | mk = 1, 1 k n} otherwise. Now (i) follows immediately. (ii) By the proof of (i), q(n, 1) ∈ {0, 1} for all n and thus Wn1 =
0, γ (n − 1, 1)(V (mn , tn ) ⊕ I∞ )1,1 ,
0, = √
q(n − 1, 1) = 0, q(n − 1, 1) = 1
by Lemma 4.5(iv)
q(n − 1, 1) = 0, gn−1 (V (mn , tn ))1,1 , q(n − 1, 1) = 1 √ √ tn , mn = 1, = gn−1 q(n − 1, 1) = 0 ⇔ gn−1 = 0 . 0, mn > 1 (iii) Assume first that the set {k | mk = 1} is non-empty and order it into a strictly increasing, possibly finite, sequence {kn }1nN ∞ . If N = ∞, then g∞ = limn gkn . If N < ∞, then we have gk = gkN for every k kN , and hence g∞ = gkN . Furthermore, for every 1 n N , gkn −1 = gkn−1 , where we set no = 0 and go = 1. Thus tkn gkn −1 = gkn−1 − gkn . But then, from (ii) we have
W e1 = 2
N
tkn gkn −1
n=1 N = (gkn−1 − gkn ) n=1
= gko − lim gk = 1 − g∞ . Finally, if the set {k | mk = 1} is empty, then gk = 1 for all k by (i) and hence g∞ = 1. By (ii), Wn1 = 0 for all n and hence W e1 2 = 0, also satisfying (iii). 2
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Theorem 4.7. Let ξ ≺ η for some ξ, η ∈ c∗o with ξn > 0 for every n. Then the following conditions are equivalent. (i) ξ η. (ii) {tk | mk = 1} = ∞. (iii) Q(ξ, η) is orthostochastic. Proof. Notice that by Lemma 2.4 (see also Remark 2.6(ii)), Q(ξ, η) is orthostochastic if and only if W (ξ, η) is unitary, in fact, orthogonal, since it has real entries. (iii) ⇒ (ii) In the case that there are infinitely many indices k for which mk = 1 and tk = 1, then the equality {tk | mk = 1} = ∞ holds trivially, thus assume that there is an integer N for which there are no k > N with mk = 1 and tk = 1. Then ∗ W (ξ, η) W (N ) = w- lim In−1 ⊕ V (mn , tn ) ⊕ I∞ · · · IN ⊕ V (mN +1 , tN +1 ) ⊕ I∞ n = IN ⊕ w- lim Ij −N −1 ⊕ V (mj , tj ) ⊕ I∞ · · · V (mN +1 , tN +1 ) ⊕ I∞ j
= IN ⊕ W {mk , tk }k>N . By construction, W ({mk , tk }k>N ) = W (ξ (N ) , ρ(N )) is a co-isometry, however, since W (N ) and W (ξ, η) are orthogonal, the former by construction, the latter by hypothesis, it follows that W ({mk , tk }k>N ) too is orthogonal. But then, by Lemma 4.6 applied to the majorization relation ξ (N ) ≺ ρ(N ), we have 2 1 = W ξ (N ) , ρ(N ) e1 = 1 − g∞ ! =1− (1 − tk ) mk = 1, k > N . It follows that {(1 − tk ) | mk = 1, k > N} = 0, hence {tk | mk = 1, k > N } = ∞ and therefore {tk | mk = 1} = ∞. (ii) ⇒ (iii) W (ξ, η) is unitary
⇔ ⇔ ⇔ ⇔ ⇔
W (ξ, η)ej = 1 for all j as W (ξ, η) is a co-isometry Pn W (ξ, η)ej → 1 for all j Pn W (n) ej → 1 for all j (by Proposition 3.1) ⊥ (n) P W ej → 0 for all j as W (n) is unitary n γ (n, j ) → 0 for all j by Lemma 4.5(iii) .
For a fixed j , by Lemma 4.5(ii) and (i), the integer sequence q(n, j ) is monotone nonincreasing in n and it decreases by 1 for every n for which mn+1 = 1 and q(n, j ) > 1. Since there are infinitely many integers k for which mk = 1, the sequence q(n, j ) must stabilize to either 0 or 1. If it is the former, since γ (n, j ) = 0 whenever q(n, j ) = 0 we are done. If q(n, j ) = 1 for all n N for some N ∈ N, then we obtain from the recurrence relation in Lemma 4.5(i)
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
γ (n, j ) =
3141
"! # { 1 − tk | mk = 1, n k > N } γ (N, j ).
Thus 2 "! # 2 (1 − tk ) mk = 1, k > N γ (N, j ) = 0 lim γ (n, j ) = n
because {tk | mk = 1, k > N} = ∞. (ii) ⇒ (i) If there are infinitely many indices k for which mk = tk = 1, then ξ ≺b η by Proposition 4.4 and hence ξ η. If there are only finitely many k for which mk = tk = 1 and K is the (K) ≺ η(K) . Thus it largest one, then by Lemma 4.2(ii) and (iii), N j =1 (ηJ − ξj ) = 0 and hence ξ is sufficient (and necessary) to prove that ξ (K) ρ(K). Since by Lemma 4.2(iii), ρ(K) = η(K) , and hence {mk , tk }k>K is the sequence generated by ξ (K) ≺ η(K) , we can assume without loss of generality that tk < 1 whenever mk = 1. Order the indices k for which mk = 1 into a strictly increasing sequence {kn } and set qn := nj=1 (1 − tkn ). By the assumption that 0 < tkn < 1, it fol lows that qn > 0 for all n and qn is strictly decreasing. The condition ∞ n=1 tkn = ∞ guarantees kn that qn → 0. Since δkn = ηkn +1 + j =1 (ηj − ξj ) by Lemma 4.1(iv), in order to show that ξ η it is sufficient to prove that limn δkn = 0. For every n > 1, δkn = (1 − tkn )ρ(kn − 1)1 + tkn ρ(kn − 1)2 by definition, see (13) (1 − tkn )ρ(kn−1 )1 + tkn ρ(kn−1 )2 by (15) = (1 − tkn )δkn−1 + tkn ρ(kn−1 )2 since mkn−1 = 1, see (14) = (1 − tkn )δkn−1 + tkn ηkn−1 +2 by Lemma 4.1(ii) (1 − tkn )δkn−1 + tkn ηkn−1
(by the monotonicity of η).
Also δk1 = (1 − tk1 )ρ(k1 − 1)1 + tk1 ρ(k1 − 1)2 (1 − tk1 )η1 + tk1 η2 . For convenience, set ko := 0 and ηko := η2 . Iterating, we obtain
δkn
=
=
n !
n n ! (1 − tkj ) (1 − tk1 )η1 + tk1 η2 + (1 − tki )ηkj −2 + tkn ηkn−1 tkj −1
j =2 n !
(1 − tki ) η1 +
i=1 n !
(1 − tki ) η1 +
n j =2
tkj −1
n !
i=j
(1 − tki )ηkj −2 + tkn ηkn−1
i=j
n n ! j =2
i=1
j =3
i=j
(1 − tki ) −
n !
(1 − tki ) ηkj −2 + 1 − (1 − tkn ) ηkn−1
i=j −1
n qn qn qn ηkj −2 + 1 − ηkn−1 = q n η1 + − qj −1 qj −2 qn−1 j =2
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= qn
n 1 1 1 ηkj −2 − − ηk η1 + + ηkn−1 qj −1 qj −2 qn−1 n−1 j =2
n 1 = qn (η1 − η2 ) + qn (ηk − ηkj −1 ) + ηkn−1 , qj −1 j −2 j =2
where the last equality is obtained by “summation by parts”. We know that qn → 0 and clearly, ηkn−1 → 0. We claim that also qn nj=2 qj1−1 (ηkj −2 − ηkj −1 ) → 0. Indeed, for every > 0, choose m for which ηkm < and choose N m + 2 so that for all n N
qn
m+1 j =2
1 (ηk − ηkj −1 ) < . qj −1 j −2
Then by the monotonicity of q and η
qn
n n 1 1 (ηkj −2 − ηkj −1 ) < + qn (ηk − ηkj −1 ) qj −1 qj −1 j −2 j =2
j =m+2
+
n
(ηkj −2 − ηkj −1 ) < + ηkm < 2 .
j =m+2
This proves that limn δkn = 0 and hence that ξ η. We split the proof of the implication (i) ⇒ (ii) or (iii) in two cases. If η has infinite support, then (i) ⇒ (iii). This follows immediately from Remark 3.10. If η has finite support, then (i) ⇒ (ii). Let ηN > 0 and ηN +1 = 0. First notice that if mh = 1 for some h N − 1, then ρ(h)2 = ηh+2 = 0 by Lemma 4.1(ii). For every k h, ρ(k)2 ρ(h)2 , hence ρ(k)2 = 0 and by the definition of mk we have mk = 1. Thus the sequence {mk } either eventually stabilizes at 1 or is bounded away from 1 from N − 1 on. We claim that the latter case is impossible. Reasoning by contradiction, assume that mk 2 for all k N − 1. Then for every k N − 1 and every n mk we have ξN ρ(N − 1)1 since ξ (N −1) ≺ ρ(N − 1) by (15) = ρ(k)1 by (14)
n
ρ(k)j
j =1
=
n+k
ηj −
j =1
=
∞ j =k+1
k
by Lemma 4.1(iii)
ξj
j =1
ξj → 0
since
N j =1
ηj =
∞ j =1
ξj
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
3143
which contradicts the assumption that ξ has infinite support and hence ξN > 0. Therefore there all k K. But then, if k > K, ρ(k − 1) = δk−1 , 0, . . . and is a K N such that mk = 1 for ∞ ξ = hence ξk = tk δk−1 , where δk−1 = kj =1 ηj − k−1 j =k ξj by Lemma 4.1(iv). Therefore j =1 j 1 − tk =
∞ j =k+1 ξj ∞ j =k ξj
∈ (0, 1) and hence for all M K + 1 M !
∞
j =M+1 ξj
(1 − tk ) = ∞
j =K+1 ξj
k=K+1
As a consequence,
∞
k=K+1 tk
= ∞ and hence
→ 0 for M → ∞.
{tk | mk = 1} = ∞.
2
Remark 4.8. Given a sequence {mk , tk } where mk ∈ N and 0 < tk 1, Proposition 3.1 constructs the co-isometry W ({mk , tk }) and Lemmas 4.5, and 4.6 provide further properties for that construction. It is easy to see that, if in lieu of Q(ξ, η) we consider the Schur-square of W ({mk , tk }), the implications (ii) ⇔ (iii) in Proposition 4.4 and Theorem 4.7 still hold for this more general setting. 5. An extension of the Horn Theorem to nonsummable sequences In Theorem 3.9 we proved that if ξ η, then ξ = Qη for some orthostochastic matrix Q. While strong majorization is necessary and sufficient in the summable case by Lemma 2.10, in the nonsummable case it is not, as seen in Example 2.11. In fact, we are going to prove that the condition ξ ≺ η will always suffice when ξ is nonsummable. Our strategy will be to decompose any pair of sequences ξ, η ∈ c∗o with ξ ≺ η and ξ nonsummable into “direct sums” of pairs of sequences ξ(r) η(r) (r = 1, 2, . . .). The key step in this process is the following “shift” lemma. Lemma 5.1. Let ξ, η ∈ c∗o \ (1 )∗ and assume that ξ ≺ η but ξ η. Then there are integers p and n with 0 p < n, for which ξ χ[1, n] ≺ ηχ[1, n − p] and ξ (n) ≺ η(n−p) . Proof. By hypothesis, α := lim
n
j =1 (ηj
− ξj ) > 0. If there is some n ∈ N for which
n m (ηj − ξj ) (ηj − ξj ) j =1
for all m n,
j =1
which is certainly the case if α = ∞, then ξ (n) ≺ η(n) and hence the pair p = 0 and n satisfies the requirement. Assume therefore that there is no such n and hence that nj=1 (ηj − ξj ) > α for every n. In particular, η1 > ξ1 + α > α. Let N1 be an integer for which ηN1 < α and for every n N1 , let p(n) be the largest integer in [1, n) for which n
ηj α.
(20)
j =n−p(n)+1
n By the monotonicity of η, n+1 j =n−p(n)+1 ηj α and hence by the maximality j =n−p(n)+2 ηj of p(n+1), it follows that p(n+1) p(n) for every n N1 , i.e., the sequence p(n) is monotone
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nondecreasing. Since ηj → 0, it follows that p(n) → ∞, and since η is nonsummable, it follows that n − p(n) → ∞. Then ηn−p(n) < α2 for all n N2 for some N2 N1 and hence by the maximality of p(n), n
ηj >
j =n−p(n)+1
α 2
for every n N2 .
(21)
Now n−p(n)
ηj −
j =1
n
ξj =
j =1
n (ηj − ξj ) − j =1
n
ηj > 0
j =n−p(n)+1
from which it follows that ξ χ[1, n] ≺ ηχ[1, n − p(n)] for every n N2 . It remains to prove that there is an n N2 for which ξ (n) ≺ η(n−p(n)) . Reasoning by contradiction, assume that q(n)−n (n−p(n)) (n) − ξj ) < 0, i.e., for every n N2 there is an integer q(n) > n for which j =1 (ηj q(n)−p(n) q(n) j =n+1 ξj > j =n−p(n)+1 ηj . Then q(n)
(ξj − ηj ) >
j =n+1
q(n)−p(n) j =n−p(n)+1
ηj −
q(n)
n
ηj =
j =n+1
q(n)
ηj −
j =n−p(n)+1
ηj 0, (22)
j =q(n)−p(n)+1
for every n N2 , where the last inequality follows form the monotonicity of η. From this inequality and (21), we have q(n)
j =q(n)−p(n)+1 ηj n j =n−p(n)+1 ηj
q(n)
j =n+1 (ξj
> 1 − n
− ηj )
j =n−p(n)+1 ηj
>1−
q(n) 2 (ξj − ηj ). α j =n+1
Set m1 = N2 and mk+1 := q(mk ). The sequence mk is strictly increasing and for every k 1, mk+1
j =mk+1 −p(mk )+1 ηj
mk
j =mk −p(mk )+1 ηj
>1−
2 α
mk+1
(ξj − ηj ).
j =mk +1
Given that p(mk ) is nondecreasing and hence mk+1 − p(mk+1 ) mk+1 − p(mk ), the average of the nonincreasing sequence η over the integer interval {mk+1 − p(mk+1 ) j mk+1 } must be at least as large as its average over the integer interval {mk+1 − p(mk ) j mk+1 } and hence mk+1 1 j =mk+1 −p(mk+1 )+1 ηj p(mk+1 ) 1 mk j =mk −p(mk )+1 ηj p(mk )
1 mk+1 j =mk+1 −p(mk )+1 ηj p(mk ) 1 mk j =mk −p(mk )+1 ηj p(mk )
2 >1− α
mk+1
(ξj − ηj ).
j =mk +1
(23)
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mk+1 Now by (22), j =mk +1 (ξj − ηj ) > 0 for every k 1 and by assumption we also have mk j =1 (ηj − ξj ) > α > 0. Thus for every h > 1, mh mh m1 m1 h−1 m k+1 (ηj − ξj ) = (ηj − ξj ) − (ξj − ηj ) = (ηj − ξj ) − (ξj − ηj ) > 0, j =1
j =1
j =m1 +1
j =1
k=1 j =mk +1
∞ mk+1 whence k=1 j =mk +1 (ξj − ηj ) < ∞. Choose an integer ko large enough so to obtain ∞ mk+1 α 2 mk+1 (ξ k=ko j =mk +1 j − ηj ) < 2 . In particular, for all k ko we have 0 < α j =mk +1 (ξj − ηj ) < 1 and hence from Eq. (23) we have for every K ko K !
0<
k=ko
2 1− α
mk+1
(ξj − ηj ) <
j =mk +1
mK+1 1 j =mK+1 −p(mK+1 )+1 ηj p(mK+1) mko 1 j =mko −p(mko )+1 ηj p(mko )
2
p(mko ) , p(mK+1 )
where the last inequality follows from the inequalities (20) and (21). Now, on the one p(mko ) hand, p(mk ) → ∞ and hence 2 p(mK+1 ) → 0 for K → ∞. On the other hand, the sequence ∞ mk+1 2 2 mk+1 k=ko (1 − α j =mk +1 (ξj − ηj ) ∈ (0, 1) and is summable, hence j =mk +1 (ξj − ηj )) > 0, α a contradiction. 2 Lemma 5.2. Let ξ, η ∈ c∗o \ (1 )∗ and assume that ξ ≺ η but ξ η. Then there are two partitions (1) ˙ (2) (1) (2) (1) (1) of N into sequences, N = {nj } ∪ {nj } and N = {mj } ∪˙ {mj } with n1 = m1 = 1 for which, if ξ := {ξn(1) }, η := {ηm(1) }, ξ := {ξn(2) }, and η := {ηm(2) } are the corresponding subsequences j
j
j
j
of ξ and η, then ξ η , ξ ≺ η , ξ η , and ξ ∈ (1 )∗ . Proof. By Lemma 5.1, ξ χ[1, N] ≺ ηχ[1, N − p] and ξ (N ) ≺ η(N −p) for some pair of integers p and N with 0 p < N . Let
α := lim
k
(ηj − ξj ) ,
j =1
β :=
N −p
ηj −
j =1
N
k (N −p) (N ) ηj . and γ := lim − ξj
ξj
j =1
j =1
By hypothesis, α > 0 and β 0, γ 0. Since for k > N − p k−N k+p k +p (N −p) (N ) ηj + (ηj − ξj ) = β + − ξj ξj j =1
j =1
j =k+1
k+p and j =k+1 ξj → 0 for k → ∞, it follows that 0 < α = β + γ , so β and γ cannot both vanish. Assume first that β > 0. The strategy for the construction of the sequences ξ , ξ , η , and η is to first move a finite number of entries from the infinite sequence ξ (N ) to the finite sequence ξ χ[1, N], i.e., delete them from the first sequence and insert them after the last nonzero term of the second one, and do so while controlling the sum and preserving the majorization by ηχ[1, N − p] of the new finite sequence. This will automatically preserve majorization of the new infinite sequence by η(N −p) . At the next step, move a single entry from the sequence η(N −p)
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to the sequence ηχ[1, N − p], so to preserve majorization of the two infinite sequences and still control the sums, while majorization of the two finite ones is automatically preserved. And then iterate the process. Now we make this strategy precise. We construct three strictly increasing sequences of integers kj , hi and qi with N < kqi < hi hi + p < kqi +1 < kqi +2 < · · · < kqi+1 so that i−1
β+
ηhj −
j =1
δi :=
qi
i 1 < ξkj < β + ηhj , i
q
i−1
j =1
j =1
(24)
kqi +qi
ξkj −
j =1
$
(25)
ξj > δi−1 ,
j =kqi +1
% i−1 1 ηhj , ηhi < min i , δi − 2
(26)
j =1
where for i = 1 we take 0 in place of i−1 j =1 ηhj and of δi−1 . To start the construction, use the fact that ξj → 0 and is nonsummable to choose q1 − 1 q −1 integers N < k1 < · · · < kq1 −1 for which β − 1 < j 1=1 ξkj < β. Since ξ has infinite support, it has an infinite subsequence for which ξpn > ξpn +1 . Choose kq1 ∈ {pn } large enough so that q1 q1 kq1 +q1 j =1 ξkj < β. By the monotonicity of ξ , it follows that δ1 := i=1 ξki − j =kq1 +1 ξj > 0 and conditions (24) and (25) are thus satisfied for i = 1. To satisfy also (26) it is enough to choose h1 > kq1 so that ηh1 < min{ 12 , δ1 }, which is always possible since ηj → 0 and δ1 > 0. Assume now the construction of the three integer sequences up to some i − 1 and choose positive integers hi−1 + p < kqi−1 +1 < kqi−1 +2 < · · · < kqi −1 for which β+
i−1
qi−1
ηhj −
j =1
j =1
1 ξkj − < i
q i −1 j =qi−1 +1
Choose kqi ∈ {pn }, kqi > kqi −1 large enough so that i.e., so to satisfy (24). Now
δi − δi−1 =
qi
=
j =kqi +1
j =kqi +qi−1 +1
qi
kqi +qi
ξkj −
ξkj .
j =1
<β+
i−1
j =1 ηhj
−
j =kqi +qi−1 +1
> 0 (because ξkqi > ξkqi +1 ).
ξj
j =kqi−1 +1 kqi−1 +qi−1
kqi +qi
ξkj −
ηhj −
kqi−1 +qi−1
ξj +
j =qi−1 +1
j =qi−1 +1
j =qi−1 +1 ξkj
qi−1
j =1
qi
kqi +qi
ξkj −
j =qi−1 +1 qi
ξkj < β +
i−1
ξj +
j =kqi−1 +1
ξj
kqi +qi−1
ξj −
ξj
j =kqi +1
(by the monotonicity of ξ )
qi−1
j =1 ξkj ,
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Thus (25) is satisfied. By the induction assumption that ηhn < min{ 21n , δn − n−1 j =1 ηhj } for all i−i 1 n i − 1, we see that δi > δi−1 > j =1 ηhj and since ηn → 0 we can choose hi > kqi so to satisfy also (26). Now define n(1) := 1, . . . , N, k1 , k2 , . . . and m(1) := 1, . . . , N − p, h1 , h2 , . . .
(27)
and n(2) , m(2) are the complementary sequences of n(1) , m(1) respectively. Explicitly, ξ := ξ1 , . . . , ξN , ξk1 , ξk2 , . . .
and η := η1 , . . . , ηN −p , ηh1 , ηh2 , . . . ,
ξ := ξN +1 , ξN +2 , . . . , ξk1 −1 , ξk1 +1 , . . . and η := ηN −p+1 , . . . , ηh1 −1 , ηh1 +1 , . . . .
(28)
First we verify that ξ η . m If m N − p, then m j =1 (ηj − ξj ) = j =1 (ηj − ξj ) 0. m N −p If N − p < m N , then j =1 (ηj − ξj ) j =1 ηj − N j =1 ξj = β > 0. Finally, if m > N , let qi−1 < m − N qi , where we set qo = 0 for convenience. Then m > N + qi−1 N + i − 1 N − p + i − 1 and hence m
ηj
N −p+i−1
j =1
ηj =
j =1
N −p
=
ηj +
j =1
ηj − β +
j =1 N
N −p
qi
i−1
ηhj
j =1
ξkj
by (26)
j =1
ξj +
j =1
qi
ξkj
j =1
N
ξj +
j =1
m−N
ξkj =
j =1
m
ξj .
j =1
Thus ξ ≺ η . For every i > 1, N −p+i−1
ηj
=
j =1
N −p
ηj +
j =1
N j =1
=
j =1
ηhj =
j =1
ξj +
N +qi
i−1
qi
ξj + β +
j =1
ξkj +
j =1
ξj +
N
1 i
i−1
ηhj
(by the definition of β)
j =1
by (26)
∞
1 1 < ξj + . i i j =1
∞ 1 ∗ 1 ∗ Therefore ∞ j =1 ηj j =1 ξj and since ξ ≺ η and by (26), η ∈ ( ) and hence ξ ∈ ( ) , equality follows, i.e., ξ η .
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Next, we verify that ξ ≺ η . We start with the following two inequalities. If hi + p N + m < hi+1 + p, then m
ηj =
m+i
j =1
(N −p)
ηj
−
j =1
i
ηhj
j =1
m
(N −p)
ηj
−
j =1
i
(29)
ηhj .
j =1
If kqi N + m < kqi+1 and N + m h1 + p, then m
ξj =
j =1
m+q i
(N )
ξj
−
qi
ξkj =
j =1
j =1
m
kqi +qi (N ) ξj
+
j =1
m
(N )
ξj
+
j =1
ξj −
j =kqi +1
N +m+q i
ξj −
j =N +m+1
qi
ξkj =
j =1
m
(N )
ξj
qi
ξkj
j =1
− δi ,
(30)
j =1
where the inequality follows from the monotonicity of ξ . mSinceN < h1 + p < kq2 < · · · < kqi < hi hi + p < kqi+1 < · · · , in order to prove that j =1 (ηj − ξj ) 0, we need to consider three cases: N + m < h1 + p, hi + p N + m < kqi+1 for some i 1, and kqi N + m < hi + p for some i 2. In the first case, since ξ (N ) ≺ η(N −p) , m m m (N −p) (N −p) (N ) ηj − ξj = ηj ηj 0. − ξj − ξj j =1
j =1
j =1
In the second case hi + p N + m < hi+1 + p and kqi N + m < kqi+1 while N + m h1 + p, hence combining (29) and (30) yields m m i (N −p) ηj − ξj ηj − ξj + δi − ηhj > 0 j =1
j =1
j =1
where the last inequality follows from ξ (N ) ≺ η(N −p) and (26). In the third case we have hi−1 + p N + m < hi + p and kqi N + m < kqi+1 while N + m kq2 > h1 + p, hence combining (29) and (30) yields m m i−1 (N −p) ηj − ξj ηj − ξj + δi − ηhj > 0. j =1
j =1
This proves that ξ ≺ η . Finally, since
lim
j =1
∞
j =1 ηj
=
∞
j =1 ξj
= ∞, it follows that
k k ηj − ξj = lim (ηj − ξj ) = α > 0, j =1
j =1
i.e., ξ η . This completes the proof of the lemma for the case β > 0.
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The proof for the case where γ > 0 is similar, but simpler. To further slightly simplify the N −p proof, assume without loss of generality that β = 0, i.e., N j =1 ξj = j =1 ηj . Choose an M ∈ N m (N −p) (N ) − ξj ) > γo for all m M and select a strictly increasing and γo > 0 so that j =1 (ηj infinite sequence of integers {hj } with h1 N − p + M for which ∞ j =1 ηhj γo . Then choose strictly increasing sequences of integers {kj } and {qi } so that for all i 1 i
i 1 ξkj < ηhj < i
ηhj −
j =1
q
i
j =1
j =1
(31)
which can be achieved as in the proof of the case β > 0, by using the fact that ξi → 0 and ξ is nonsummable. Define the sequences n(1) , m(1) and their complements n(2) , m(2) as in (27) and hence ξ , η , ξ , and η as in (28), i.e., by “moving” the entries ξkj (resp. ηkj ) from ξ (N ) to ξ χ[1, N] (resp. from η(N −p) to ηχ[1, N − p]). First we show that ξ η . If 1 n N , then from ξ χ[1, N] ≺ ηχ[1, N − p] we have n
ξj =
j =1
n
ξj
j =1
n n ηχ[1, N − p] j ηj . j =1
j =1
Set qo = 0. If qi−1 < m qi for some i 1, then m qi−1 + 1 i and by (31), N +m
ξj
j =1
N +qi
ξj
=
j =1
N
ξj +
j =1
qi
ξkj
j =1
N −p
ηj +
j =1
i j =1
ηhj =
N +i
j =1
ηj =
N −p j =1
ηj +
i j =1
ηhj <
j =1
which proves that ξ ≺ η . By construction, η is summable and hence the other hand, from (31), for every i, N −p+i
ηj
N +m
ηj ,
j =1
∞
j =1 ξ
∞
j =1 η
.
On
qi N +qi N ∞ 1 1 1 + ξj + ξkj = + ξj < + ξj . i i i j =1
j =1
j =1
j =1
∞ Thus ∞ j =1 ηj = j =1 ξj and hence ξ η . (N ) ≺ η , and hence, since ξ ξ (N ) , that ξ ≺ η . Indeed, if N − Next we prove that ξ m (N −p) (N ) m p + m < h1 , then j =1 ηj = m j =1 ηj j =1 ξj . If hi N − p + m < hi+1 for some m (N −p) − ij =1 ηhj . But ij =1 ηhj < γo and since we i 1, then, as in (29), m j =1 ηj j =1 ηj (N −p) (N ) − ξj ) > γo . Thus we conclude have m > h1 − N + p M it follows also that m j =1 (ηj m (N ) that m j =1 ηj > j =1 ξj . Finally, as in the case of β > 0 it is now immediate to see that ξ η . 2 Having thus prepared the groundwork, we can present an infinite dimensional extension of the Horn Theorem [12, Theorem 4] (see Theorem 1.1(ii)) for nonsummable sequences. / (1 )∗ , then ξ ≺ η if and only if ξ = Qη for some orthostochasTheorem 5.3. If ξ, η ∈ c∗o and ξ ∈ tic matrix Q.
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Proof. The sufficiency is well known since Markus (see (1) or the proof of Lemma 2.8). For the necessity, if ξ η, then the result follows from Theorem 3.9. If ξ η, then Lemma 5.2 (r) applied iteratively partitions N into the union of infinitely many sequences {nk }, and sequences & & ∞ ∞ (r) (r) (r) {mk }, i.e., N = ˙ k=1 {nk } = ˙ k=1 {mk }, so that {ξn(r) } {ηm(r) }, the exhaustion of N being k
k
(1) guaranteed by the condition in Lemma 5.2 that n(1) 1 = m1 = 1. Then by Theorem 3.9 we can find for each r an orthogonal matrix W (r) with Schur-square Q(r), for which {ξn(r) } = Q(r){ηm(r) }. k k “Direct summing” the matrices Q(r) with respect to the two decompositions of the basis yields the required orthostochastic matrix Q. Explicitly, let Rr , Sr ∈ B(H ) be the diagonal projections on the subspaces with bases {em(r) } k ∞ and {em(r) } respectively. Then ∞ r=1 Rr = r=1 Sr = I . We can identify the matrices Q(r) and k ∞ W (r) with ∞ corresponding operators in B(Rr H, Sr H ) and then define Q: r=1 Sr Q(r)Rr and W : r=1 Sr W (r)Rr . It is clear that W, Q ∈ B(H ), W is unitary and its matrix is
Wij := (W (r))hk 0
if i = nh , j = mk for some r, h, k, otherwise.
Qij := (Q(r))hk 0
if i = nh , j = mk for some r, h, k, otherwise.
(r)
(r)
(r)
(r)
Similarly,
Thus the matrix Q is the Schur-square of the matrix W and hence Q is orthostochastic. For every (r) i ∈ N, i = nh for a unique r and h and (Qη)i =
∞
Qij ηj =
j =1
i.e., ξ = Qη.
∞
Q(r)hk ηm(r) = ξn(r) = ξi ,
k=1
k
h
2
Thus combining Theorem 5.3 with Theorem 3.9, we have the following infinite dimensional extension of the Horn Theorem. Corollary 5.4. If ξ, η ∈ c∗o then ξ = Qη for some orthostochastic matrix Q
⇔
ξ ≺η ξ η
when ξ ∈ / 1 , when ξ ∈ 1 .
6. Diagonals of compact operators Recall that given a Hilbert space H , we denote by K(H ) and L1 the ideals of compact operators and of trace class operators respectively, given an orthonormal basis of H , we denote by D the masa of diagonal operators, by E : B(H ) → D the operation of taking the main diagonal, i.e., the normal faithful and trace preserving conditional expectation from B(H ) onto D, and by diag : ∞ → D the isometric isomorphism that maps a sequence η to the diagonal operator having diagonal η.
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For the readers’ convenience, let us first collect here some of the equivalent conditions of the majorization relations that we have obtained explicitly or that are immediate consequences of results obtained in the previous sections. Corollary 6.1. Let ξ, η ∈ c∗o . If ξ ∈ / (1 )∗ , then the following conditions are equivalent: NS(i) NS(ii) NS(ii ) NS(iiid) NS(iiis) NS(iii is) NS(iii cs) NS(iii cn)
ξ ≺ η. ξ = Qη for some orthostochastic matrix Q. diag ξ = E(U diag ηU ∗ ) for some orthogonal matrix U . ξ = Qη for some doubly stochastic matrix Q. ξ = Qη for some substochastic matrix Q. diag ξ = E(W diag ηW ∗ ) for some isometry W . diag ξ = E(W diag ηW ∗ ) for some co-isometry W . diag ξ = E(L diag ηL∗ ) for some contraction L.
If ξ ∈ (1 )∗ , then the following conditions are equivalent: S(i) S(ii) S(ii ) S(iiid) S(iiic) S(iii is)
ξ η. ξ = Qη for some orthostochastic matrix Q. diag ξ = E(U diag ηU ∗ ) for some orthogonal matrix U . ξ = Qη for some doubly stochastic matrix Q. ξ = Qη for some column-stochastic matrix Q. diag ξ = E(W diag ηW ∗ ) for some isometry W .
In general the following conditions are equivalent: (i) (ii) (iii) (ii ) (iii ) (iv)
ξ ≺ η. ξ = DQη for some orthostochastic matrix Q and some D ∈ D with 0 D I . ξ = QDη for some orthostochastic matrix Q and some D ∈ D with 0 D I . diag ξ = E(DU diag η(DU )∗ ) = for some orthogonal U and some D ∈ D with 0 D I . diag ξ = E(U D diag η(U D)∗ ) for some orthogonal U and some D ∈ D with 0 D I . diag ξ = E(L diag ηL∗ ) for some contraction L.
Remark 6.2. (i) In lieu of orthogonal matrices it suffices to take unitary matrices and conversely, we can always ask that the isometries, co-isometries, and contractions have real entries. (ii) The equivalence of (i) and (iv) was proven by Arveson and Kadison in [2, Theorem 4.2]. In the case of finite rank positive operators, the “sequence” formulation S(i) ⇔ S(iii) in the above corollary is obviously equivalent to the operator theory formulation of the classical Schur– Horn Theorem (see Theorem 1.1(iii)). For infinite rank positive compact operators however, the above corollary leads to a somewhat different operator theory reformulation. We find convenient to introduce the notion of partial isometry orbit of an operator.
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For every operator A ∈ B(H ), denote by U(A) := U AU ∗ U ∈ B(H ) unitary , the unitary orbit of A, V(A) := V AV ∗ V ∈ B(H ), V ∗ V = RA ∨ RA∗ , the partial isometry orbit of A. Lemma 6.3. U(A) ⊂ V(A) for every A ∈ B(H ) and U(A) = V(A) if and only if A has finite rank. Proof. The inclusion is obvious since if U is unitary, then V := U (RA ∨ RA∗ ) is a partial isometry with V ∗ V = RA ∨ RA∗ and U AU ∗ = V AV ∗ . If A has finite rank and V is a partial isometry with V ∗ V = RA ∨ RA∗ , then both (V ∗ V )⊥ and (V V ∗ )⊥ are infinite and hence equivalent projections. Thus V can be extended to a unitary U and V AV ∗ = U AU ∗ . Thus U(A) = V(A). If A has infinite rank, then choose a partial isometry V with V ∗ V = RA ∨ RA∗ but such that (V V ∗ )⊥ (V ∗ V )⊥ and let B = V AV ∗ . Since V RA V ∗ B = V RA V ∗ V AV ∗ = B, it follows that V V ∗ V RA V ∗ RB . Similarly, V RA∗ V ∗ B ∗ = B ∗ and hence V V ∗ V RA∗ V ∗ RB ∗ . Thus V V ∗ V (RA ∨ RA∗ )V ∗ = V RA V ∗ ∨ V RA∗ V ∗ RB ∨ RB ∗ . So far, we have only used the fact that V ∗ V RA ∨ RA∗ , and since A = V ∗ BV , the same argument shows that V ∗ (RB ∨ RB ∗ )V RA ∨ RA∗ . But then RB ∨ RB ∗ V V ∗ = V (RA ∨ RA∗ )V ∗ V V ∗ (RB ∨ RB ∗ )V V ∗ = RB ∨ RB ∗ whence V V ∗ = RB ∨ RB ∗ . But then RB ∨ RB ∗ is not unitarily equivalent to RA ∨ RA∗ , hence B∈ / U(A). 2 Denote by s(A) the sequence of s-numbers of A. In particular, if A is a positive compact operator, s(A) is the eigenvalue list of A in monotone nonincreasing order, with repetition according to multiplicity, and with infinitely many zeros added in case A has finite rank. Notice that the eigenvalue list of A “ignores” the null space of A, i.e., A and A ⊕ 0n where 0n denotes the zero operator on a space of dimension n ∈ N ∪ {∞} share the same eigenvalue list. Since for all A ∈ K(H )+ there is an isometry V for which A = V diag s(A)V ∗ (and V can be chosen unitary when A has finite rank or when RA = I ), we see that V(A) = V diag s(A) = B ∈ K(H )+ s(B) = s(A) =
'
U diag s(A) ⊕ 0n . (32)
0n∞ . 1
when A is In [2] the set V(A) is denoted by Os(A) and it was shown that V(A) = U(A) trace class [2, Proposition 3.1]. The infinite dimensional extensions of the Horn Theorem obtained in Theorems 3.9 and 5.3 provide a characterization of “the diagonals” of the partial isometry orbit of a positive compact operator.
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Proposition 6.4. Let A ∈ K(H )+ . Then {B ∈ D ∩ K(H )+ | s(B) ≺ s(A)} \ L1 E V(A) = {B ∈ D ∩ K(H )+ | s(B) s(A)}
if Tr(A) = ∞, if Tr(A) < ∞.
Proof. By (32), assume without loss of generality that A = diag s(A). To prove that the left-hand set is contained in the right-hand set, let V be a partial isometry for which V ∗ V = Rdiag s(A) and let B = E(V diag s(A)V ∗ ). Then Tr(B) = Tr V diag s(A)V ∗ = Tr diag s(A) = Tr(A) and we only need to prove that s(B) ≺ s(A). Let W be the isometry that rearranges the sequence s(B) into the sequence of the diagonal entries of B, i.e., B = W diag s(B)W ∗ . Then W commutes with the expectation E and hence diag s(B) = W ∗ BW = W ∗ E V diag s(A)V ∗ W = E W ∗ V diag s(A)V ∗ W . If diag s(A) has infinite rank then it must have zero kernel and hence V is an isometry, while if diag s(A) has finite rank, we can extend V to an isometry. Since W W ∗ RB RV diag s(A)V ∗ = V V ∗ , it follows that W ∗ V is an isometry. By Lemma 2.3, s(B) = Qs(A) where Qij = |(W ∗ V )ij |2 and hence by (1) and Lemma 2.4 it follows that s(B) ≺ s(A). To prove the opposite inclusion, let B ∈ D ∩ K(H )+ with s(B) ≺ s(A) and if Tr(A) = ∞ assume that Tr(B) = ∞ while if Tr(A) < ∞ assume also that Tr(B) = Tr(A) and hence s(B) s(A). As a consequence of Theorem 3.9 and Theorem 5.3, there is a unitary U for which diag s(B) = E(U diag s(A)U ∗ ). As above, let W be the isometry that rearranges the sequence s(B) into the sequence of the diagonal entries of B. Then W U is an isometry and hence B = W diag s(B)W ∗ = W E U diag s(A)U ∗ W ∗ = E W U diag s(A)U ∗ W ∗ ∈ E V diag s(A) = E V(A) .
2
Remark 6.5. (i) The proof that s(E(C)) ≺ s(C) for all C ∈ K(H )+ (i.e., the inclusion of the left-hand set into the right-hand set in Proposition 6.4) is usually attributed to Ky Fan [7]. See also [8] and see an elegant proof in [2, Theorem 4.2] of the more general fact that s(E(LAL∗ )) ≺ s(A) for every contraction L. (ii) Gohberg and Markus have proven in [9, Proposition III, p. 205] that if A ∈ K(H )+ , ξ ∈ c∗o , and ξ ≺ s(A), then there is an orthonormal sequence fn ∈ H for which (Afn , fn ) = ξn for all n. Thus setting W ∗ en = fn for the fixed orthonormal basis {en }, defines a co-isometry W for which diag ξ = E(W AW ∗ ). Applied to the case of A = diag η, their result proves that if ξ ≺ η, then ξ = Qη for some co-isometry stochastic matrix Q (cf. Theorem 3.7). In the case that A is of trace class and ξ s(A), Gohberg and Markus have furthermore proven in [9, Theorem 1] that A vanishes on span{fn }⊥ , i.e., that W ∗ W RA . As a consequence, V := W RA is a partial isometry and diag ξ ∈ E(V(A)), which proves the inclusion of the right-hand set into the left-hand set in Proposition 6.4.
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(iii) In the trace class case, Proposition 6.4 was derived from the (finite dimensional) Schur–Horn Theorem by Arveson and Kadison using compactness arguments (see [2, Theorem 4.1]). Since s(A ⊕ 0) = s(A) but in general E(U(A ⊕ 0)) = E(U(A)), it is clear that we cannot characterize E(U(A)) only in terms of the sequence s(A). Proposition 6.6. Let A ∈ K(H )+ . Then ⊥ )}. (i) E(U(A)) ⊂ E(V(A)) ∩ {B ∈ D | Tr(RB⊥ ) Tr(RA (ii) If RA = I , then
E U(A) = E V(A) ∩ {B ∈ D | RB = I } {B ∈ D ∩ K(H )+ | s(B) ≺ s(A), RB = I } \ L1 = {B ∈ D ∩ K(H )+ | s(B) s(A), RB = I }
if Tr(A) = ∞, if Tr(A) < ∞.
(iii) The inclusion in (i) is proper unless RA = I or A has finite rank. Proof. (i) That E(U(A)) ⊂ E(V(A)) follows from Lemma 6.3. Let B = E(U AU ∗ ) for some unitary U . Then E RB⊥ U AU ∗ RB⊥ = RB⊥ E U AU ∗ RB⊥ = 0 since RB ∈ D. By the faithfulness of the expectation, it follows that RB⊥ U AU ∗ RB⊥ = 0, hence ⊥ U ∗ , whence Tr(R ⊥ ) Tr(R ⊥ ). RB⊥ U AU ∗ = 0 and thus RB⊥ (RU AU ∗ )⊥ = U RA B A (ii) The second set equality is given by Proposition 6.4. By (i), E U(A) ⊂ E V(A) ∩ {B ∈ D | RB = I }. To prove the opposite inclusion, let B ∈ D ∩ K(H )+ with RB = I and s(B) ≺ s(A) and assume that s(B) s(A) if Tr(A) < ∞ and Tr(B) = ∞ if Tr(A) = ∞. By Theorems 3.9 and 5.3 (or see for convenience Corollary 6.1), diag s(B) ∈ E(U(diag s(A))). Since RB = I , there is a permutation matrix Π for which B = Π diag s(A)Π ∗ . As Π commutes with the expectation, B ∈ E(U(diag s(A))). But U(diag s(A)) = U(A) since RA = I , and thus B ∈ E(U(A)). (iii) If RA = I , the equality holds by (ii) and if A has finite rank, then U(A) = V(A) by Lemma 6.3 and hence E U(A) = E V(A) = E V(A) ∩ B ∈ D Tr RB⊥ ∞ . ⊥ ) ∈ N ∪ {∞} and η := s(A). Assume now that A has infinite rank but RA = I . Set n := Tr(RA Then ηj = 0 for all j and n = 0. Let Λ be an infinite subset of N with card N \ Λ = n, let π : N → Λ be a bijection,
η˜ k :=
0 ηj
if k ∈ / Λ, if k = π(j ).
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Then U(A) = U(diag η). ˜ Clearly, B := diag η ∈ E V(A) = E V(diag η)
and RB⊥ = 0.
We claim that B ∈E(U(A)). / Reasoning by contradiction, assume that diag η = E(U diag ηU ˜ ∗) for some unitary U . By Lemma 2.3, η = Qη˜ for the unistochastic and hence doubly stochastic matrix Q given by Qij = |Uij |2 . But then, for every i ∈ N ∞
Qiπ(j ) ηj =
j =1
∞
Qiπ(j ) η˜ π(j ) =
j =1
∞
Qik η˜ k = ηi =
k=1
∞
Qik ηi =
k ∈Λ /
k=1
Qik ηi +
∞
Qiπ(j ) ηi
j =1
since Q is column-stochastic. Hence for all i ∈ N,
Qik ηi +
k ∈Λ /
∞
Qiπ(j ) (ηi − ηj ) = 0.
(33)
j =1
Let np be the strictly increasing sequence of integers starting with no = 0 for which ηj = ηnp for all np−1 < j np . Applying the identity (33) to any 0 < i n1 we have
Qik ηi +
k ∈Λ /
∞
Qiπ(j ) (ηi − ηj ) =
Qik ηi +
k ∈Λ /
j =n1 +1
∞
Qiπ(j ) (ηi − ηj ) = 0.
j =1
Since ηi = ηn1 > ηj > 0 for all j > n1 , and Qik 0 for all k, we see that Qik = 0 for all 1 k∈ / {π(1), . . . , π(n1 )} and in particular for all k ∈ / Λ. But then nj =1 Qiπ(j ) = 1 since Q is n1 row-stochastic. Hence i,j =1 Qiπ(j ) = n1 . Since Q is also column-stochastic, it follows that n1 i=1 Qiπ(j ) = 1 for every 0 < j n1 . Thus Qiπ(j ) = 0 for every pair i > n1 and 0 < j n1 . Now applying the identity (33) to n1 < i n2 we obtain k ∈Λ /
Qik ηi +
∞
Qiπ(j ) (ηi − ηj ) =
j =n2 +1
Qik ηi +
k ∈Λ /
∞
Qiπ(j ) (ηi − ηj ) = 0.
j =1
Thus again we obtain for all n1 < i n2 that Qik = 0 for all k ∈ / {π(n1 + 1), . . . , π(n2 )} and in particular for all k ∈ / Λ. Iterating, we obtain that Qik = 0 for all k ∈ / Λ and all i. Since Λ = ∅ we conclude that Q is not column-stochastic, a contradiction. 2 Notice that if ξ, ζ ∈ (co )+ , then (ξ + ζ )∗ ≺ ξ ∗ + ζ ∗ . Thus the majorization condition in Proposition 6.4 is preserved by convex combinations, which yields the following simple conclusions. Corollary 6.7. Let A ∈ K(H )+ . Then (i) E(V(A)) is convex. (ii) If A has finite rank or RA = I , then E(U(A)) is convex.
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Proof. (i) For i = 1, 2, let Bi ∈ D ∩ K(H )+ with s(Bi ) ≺ s(A) and let B := tB1 + (1 − t)B2 for some t ∈ [0, 1]. Then B ∈ D ∩ K(H )+ and since and s (−t)B2 ≺ s (1 − t)A ,
s(tB1 ) ≺ s(tA) it follows that
s(B) ≺ s(tA) + s (1 − t)A = s(A). Furthermore, if Tr(A) = ∞ and we choose Bi ∈ / L1 , then B ∈ / L1 , while if Tr(A) < ∞ and we choose Bi with Tr(Bi ) = Tr(A), then also Tr(B) = Tr(A). Thus the right-hand set in Proposition 6.4 is convex and hence so is E(V(A)). (ii) If in the proof of (i) we choose the diagonal operators Bi so that RBi = I for i = 1, 2, i.e., all the diagonal entries of Bi don’t vanish, then RB = I and hence the conclusion follows from Proposition 6.6. 2 When A has infinite rank but RA = I , we can identify a distinguished subset of E(U(A)) in terms of the following stronger notion of sequence majorization. Definition 6.8. Let ξ ∈ c∗o , η ∈ (co )+ , p ∈ Z+ , and N ∈ N. Then we say that p
(i) ξ ≺ η if ξ ≺ η∗ and N p
n+p
j =1 ξj
n
j =1 ηj
for all n N .
p
(ii) ξ ≺ η if ξ ≺ η for some N ∈ N. N
p
Lemma 6.9. Let ξ, η ∈ c∗o and ξ ≺ η for some p ∈ N, and if ξ ∈ 1 assume also that ξ η. Then p
( )* + ξ = Q 0, 0, . . . , 0, η for some orthostochastic matrix Q. Proof. We start by disposing of the case where ξ or η or both have finite support. If ξ has finite support, by hypothesis ξ η and hence η too has finite support. If η has finite support, then p ( )* + there is a permutation matrix Π for which Π 0, 0, . . . , 0, η = η. By Lemma 2.7, there is an p ( )* + orthostochastic matrix Q for which ξ = Q η = Q Π 0, 0, . . . , 0, η . By Lemma 3.4, Q := Q Π is also orthostochastic. p Thus assume henceforth that both ξn and ηn never vanish. Let N be an integer for which ξ ≺ η p
N
( )* + and let η˜ := η1 , η2 , . . . , ηN , 0, 0, . . . , 0, η(N ) (recall the notation η(N ) = ηN +1 , ηN +2 , . . . ). As p ( )* + in the first part of the proof, since η˜ is a permutation of the original sequence 0, 0, . . . , 0, η , it suffices to find an orthostochastic matrix Q for which ξ = Qη. ˜ Consider first the case when ξ1 > ηN and apply the first step of the construction in the proof of Theorem 3.7 to the sequences ξ ≺ η. We have proven in (11) that ξ (1) ≺ ρ(1). The assump-
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tion that ξ1 > ηN guarantees that m1 N − 1 and hence for every n N − 1 we see from Lemma 4.1(iii) or from a direct elementary computation that n
ρ(1)j =
j =1
n+1
ηj − ξ1
j =1
n+1+p
ξj − ξ1 =
j =1
n+p
(1)
ξj .
j =1
p
This shows that ξ (1) ≺ ρ(1). Furthermore, if ξ ∈ 1 and thus ξ η, then ξ (1) ρ(1). N −1
Recall that R (1) is the Schur-square of V (m1 , t1 ) ⊕ I∞ , where V (m1 , t1 ) is the orthogonal m1 + 1 × m1 + 1 matrix defined in (7). Then R (1) is orthostochastic and for every j there is at (1) most one index i > 1 for which Rij = 0. Since R (1) η = ξ1 , ρ(1) and m1 + 1 N , we also have
R
(1)
p
( )* + η˜ = ξ1 , ρ(1)1 , ρ(1)2 , . . . , ρ(1)N −1 , 0, 0, . . . , 0, ρ(1)(N −1) .
Now, consider the case when ξ1 ηN and define t1 := ηξN1 , R˜ (1) to be the Schur-square of V (N, ρ(1) := η1 , η2 , . . . , ηN −1 , ηN − ξ1 , η(N ) ∗ . Then ρ(1)j = ηj if j N − 1 t1n) ⊕ I∞ , and and j =1 ρ(1)j nj=1 ηj − ξ1 for all n N . Thus n
$ ρ(1)j
j =1
=
n
j =1 ηj n j =1 ηj
n (1) j =1 ξj j =1 ξ n+p n+p−1 − ξ1 j =1 ξj − ξ1 = j =1 ξ (1)
n
if n N − 1, if n N.
p−1
This shows that ξ (1) ≺ ρ(1) and if ξ ∈ 1 , then clearly, ξ (1) ρ(1). Furthermore, since N
t1 ηN = ξ1 and (1 − t1 )ηN = ηN − ξ1 , we have p−1
( )* + R˜ (1) η˜ = ξ1 , η1 , η2 , . . . , ηN −1 , ηN − ξ1 , 0, 0, . . . , 0, η(N ) . Let Π1 be the permutation matrix for which p−1
p−1
( )* + ( )* + Π1 ηN − ξ1 , 0, 0, . . . , 0, η(N ) = ρ(1), 0, 0, . . . , 0, ρ(1)(N )
and let R (1) := (IN ⊕ Π1 )R˜ (1) . Then
R
(1)
p−1
( )* + η˜ = ξ1 , ρ(1)1 , ρ(1)2 , . . . , ρ(1), 0, 0, . . . , 0, ρ(1)(N ) .
Furthermore, by Lemma 3.4, R (1) is orthostochastic and by Lemma 3.5(ii) for every j there is at (1) most one index i > 1 for which Rij = 0. Iterate this construction. At every step we decrease by one unit either N or p. Thus we end the process when after k p + N − 1 steps we reach p = 0. Define as in (9)
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Q(k) := Ik−1 ⊕ R (k) Ik−2 ⊕ R (k−1) · · · I1 ⊕ R (2) R (1) (h)
where for every 1 h k and every j there is at most one index i > 1 for which Rij = 0. By Lemma 3.5(iii), as in the proof of Proposition 3.6, Q(k) is orthostochastic and for every j there is (k) at most one index i > k for which Qij = 0. Then Q(k) η˜ = ξ1 , ξ2 , . . . , ξk , ρ(k) and ξ (k) ≺ ρ(k), with ξ (k) ρ(k) when ξ ∈ 1 . By Theorem 3.9 there is an orthostochastic matrix Q for which ˜ Finally, by Lemmas 3.5(i) and 3.4, Q is ξ (k) = Q ρ(k). Let Q := (Ik ⊕ Q )Q(k) . Then ξ = Qη. also orthostochastic. 2 Proposition 6.10. Let A ∈ K(H )+ and assume that A has infinite rank. Then (i) '
p E V(A) ∩ B ∈ D ∩ K(H )+ s(B) ≺ s(A),
⊥) p∈Z+ , 0pTr(RA
⊥ ⊥ − p Tr RB⊥ Tr RA ⊂ E U(A) . Tr RA
(ii) If RA = I , then the inclusion in (i) is proper. ⊥ ) < ∞, by passing if Proof. (i) Let B ∈ D ∩ K(H )+ belong to the left-hand set and if Tr(RA ⊥ ) − p. necessary to a smaller integer, assume without loss of generality that Tr(RB⊥ ) = Tr(RA Furthermore, Tr(B) = Tr(A), so if Tr(A) < ∞ then s(B) s(A) (see also Proposition 6.4). But p ( )* + then by Lemma 6.9, there is an orthostochastic matrix Q for which s(B) = Q 0, 0, . . . , 0, s(A) . Thus in B(RB H ), diag s(B) = ERB (V (diag s(A) ⊕ 0p )V ∗ ) where V ∈ B(RB H ) is unitary and ERB denotes the conditional expectation with respect to the natural orthonormal basis of RB H inherited from the given basis on H . Extend V to a unitary U ∈ B(H ) that commutes with RB . Then
diag s(B) ⊕ 0Tr(R ⊥ ) = E V diag s(A) ⊕ 0p V ∗ ⊕ 0Tr(R ⊥ )−p B A = E U diag s(A) ⊕ 0Tr(RA )⊥ U ∗ ∈ E U(A) . Finally, B = Π(diag s(B) ⊕ 0Tr(R ⊥ ) )Π ∗ for some permutation matrix Π . But then B
B = ΠE U(A) Π ∗ = E Π U(A)Π ∗ = E U(A) . ⊥ ). Set η := s(A) and η˜ := 0, η , η , . . . . Then (ii) Assume that RA = I and let N := Tr(RA 1 2 A is unitarily equivalent to the diagonal operator
D :=
diag η˜ 0N −1 ⊕ diag η˜
if N = 1, if N > 1.
Let U be an orthogonal matrix for which Uij = 0 for all i > j > 1, all other entries being nonzero (such matrices exist, see Example 6.11 below) and let Q be the Schur-square of U . Let
V. Kaftal, G. Weiss / Journal of Functional Analysis 259 (2010) 3115–3162
U U˜ := IN −1 ⊕ U
if N = 1, if N > 1
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and B := E U˜ D U˜ ∗ ,
and let ξ := Qη. ˜ Then B=
E(U diag ηU ˜ ∗ ) = diag ξ ˜ ∗ ) = 0N −1 ⊕ diag ξ E(0N −1 ⊕ U diag ηU
if N = 1, if N > 1.
It is immediate to verify that ξn > 0 for all n and hence s(B) is the monotone rearrangement ξ ∗ p
1
⊥ ) − 1 but ξ ∗ ≺ η and hence s(B) ≺ s(A) of ξ . Now B ∈ E(U(A)) and Tr(RB⊥ ) = N − 1 = Tr(RA for all p > 0. Indeed, since ηj > 0 for all j ,
Qij
= 0 for i > j > 1, > 0 otherwise,
and Q is doubly stochastic and hence column-stochastic, we have for every n > 1 n
ξi∗
i=1
n
ξi =
n ∞
=
Qij η˜ j
i=1 j =1
i=1 n ∞
Qij ηj −1 =
i=1 j =2
=
n−1
Qij ηj −1 +
j =2 i=1
ηj +
j =1
n n
∞ n
∞ n
Qij ηj −1
j =n+1 i=1
Qij ηj −1 >
j =n+1 i=1
n−1
ηj .
2
j =1
Example 6.11. An orthogonal matrix U for which Uij = 0 for all i > j > 1, all other entries being nonzero is obtained as follows. Given a sequence {an } with an > 0 for all n and ∞ j =1 an = 1, n −1 set bn := ( j =1 aj ) and ⎛ √a
1
√ ⎜ a2 ⎜ √a ⎜ 3 ⎜ . ⎜ . U := ⎜ . ⎜ √a ⎜ n ⎜√ ⎝ an+1 .. .
√ a1 (b1 − b2 ) √ − 1 − a2 b2 0 .. . 0 0 .. .
a (b − b3 ) 1 2 a2 (b2 − b3 ) √ − 1 − a3 b3 .. . 0 0 .. .
··· ··· ··· .. . ··· ··· .. .
a (b − bn ) 1 n−1 a (b − bn ) 2 n−1 a3 (bn−1 − bn ) .. √ . − 1 − an bn 0 .. .
a (b − bn+1 ) 1 n a (b − bn+1 ) 2 n a3 (bn − bn+1 ) .. . an (bn − bn+1 ) − 1 − an+1 bn+1 .. .
···⎞ ···⎟ ⎟ ···⎟
⎟ ⎟ ⎟. ⎟ ···⎟ ⎟ ···⎠ .. .
A direct computation shows that U is indeed unitary. Thus our characterization of E(U(A)) is complete for the cases when RA = I or when A has finite rank and it points to an interesting and delicate role of RA in the general case. A test question is: Question 6.12. What is E(U(diag 0, 1, 12 , 13 , 14 , . . . ))?
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As seen from the work of A. Neumann [31], Arveson and Kadison [2], and Antezana, Massey Ruiz, and Stojanoff [1], the role of RA disappears when one takes the closure of E(U(A)) under the operator or the trace norm (in the trace class case). For the readers’ convenience we collect here below the relations between the various sets and their closures in the special case when RA = I or A has finite rank (see also the introduction). Let 0 = A ∈ K(H )+ . • If RA = I and Tr(A) = ∞, then E U(A) = E V(A) ∩ {B ∈ D | RB = I } (Proposition 6.6) = B ∈ D ∩ K(H )+ s(B) ≺ s(A), RB = I \ L1 (Proposition 6.4) E V(A) = B ∈ D ∩ K(H )+ s(B) ≺ s(A) \ L1 Propositions 6.6(iii) and 6.4 B ∈ D ∩ K(H )+ s(B) ≺ s(A) (obvious) . = E U(A) [31, Corollary 2.18, Theorem 3.13]. • If RA = I and Tr(A) < ∞ then E U(A) = E V(A) ∩ {B ∈ D | RB = I } (Proposition 6.6) (Proposition 6.4) = B ∈ D ∩ K(H )+ s(B) s(A), RB = I Propositions 6.6(iii) and 6.4 E V(A) = B ∈ D ∩ K(H )+ s(B) s(A) . 1 = E U(A) [1, Proposition 3.13] . 1 [2, Theorem 3.1] = E U(A) . E U(A) = B ∈ D ∩ K(H )+ s(B) ≺ s(A) [31, Corollary 2.18, Theorem 3.13]. • If A has finite rank, then E U(A) = E V(A) (Lemma 6.3) = B ∈ D ∩ K(H )+ s(B) s(A) (Proposition 6.4) . 1 . 1 = E U(A) [1, Proposition 3.13], [2, Theorem 3.1] = E U(A) . E U(A) = B ∈ D ∩ K(H )+ s(B) ≺ s(A) [31, Corollary 2.18, Theorem 3.13]. So, in particular, E(U(A)) is closed in the trace norm if and only if A has finite rank (see also [1, Remark 4.8]) and it is never closed in the operator norm but for the trivial case when A = 0. This answers in the negative the question by A. Neumann [31, p. 447] on whether E(U(A)) must
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be norm closed if A is of trace class or Hilbert–Schmidt class and provides an alternative to [31, Remark 3.7]. The inclusions in the finite rank case are illustrated by the following simple example. Example 6.13. Let η := 1, 0, . . . . Then U(diag η) = V(diag η) consists of all the rank-one projections, E(V (diag η)) consists of all trace class positive diagonal operators with trace = 1, while . E(U(diag η)) consists of all trace class positive diagonal operators with trace 1. That the latter set contains 0 is immediately clear since if Fn denotes the infinite matrix with the upper left n × n corner having entries all equal to n1 and all other entries zero, then Fn is a rank-one projections and hence Fn ∈ U(diag η) and E(Fn ) = n1 → 0. Finally, notice that in the summable case, if we pass from the unitary orbit to the bounded orbit, we obtain: ∗ Corollary 6.14. If 0 = η ∈ (1 )∗ , then L+ 1 ∩ D = E{T diag ηT | T ∈ B(H )}. ∗ Proof. Let B ∈ L+ 1 ∩ D. Then B = V diag s(B)V for some isometry V that commutes with E. Tr B Choose c η1 . Then s(B) ≺ cη and by Corollary 6.1, diag s(B) = E(DU diag cηU ∗ D) for some D ∈ D with 0 D I and some unitary U . Then
∗ B = V E DU diag cηU ∗ V ∗ = E c1/2 V DU diag η c1/2 V DU . The opposite inclusion is trivial since Tr(E(T diag ηT ∗ )) = Tr(T diag ηT ∗ ) < ∞ for every operator T ∈ B(H ). 2 References [1] J. Antezana, P. Massey, M. Ruiz, D. Stojanoff, The Schur–Horn Theorem for operators and frames with prescribed norms and frame operator, Illinois J. Math. 51 (2) (2007) 537–560. [2] W. Arveson, R.V. Kadison, Diagonals of self-adjoint operators, in: Operator Theory, Operator Algebras, and Applications, in: Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 247–263. [3] G. Birkhoff, Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev. Ser. A 5 (1946) 147–151. [4] J.W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. 42 (2) (1941) 839–873. [5] P. Casazza, M. Leon, Frames with a given frame operator, preprint, 2002. [6] K. Dykema, T. Figiel, G. Weiss, M. Wodzicki, The commutator structure of operator ideals, Adv. Math. 185 (1) (2004) 1–79. [7] K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. Natl. Acad. Sci. USA 35 (1949) 652–655. [8] K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766. [9] I.C. Gohberg, A.S. Markus, Some relations between eigenvalues and matrix elements of linear operators, Mat. Sb. 64 (106) (1964) 481–496 (in Russian); Amer. Math. Soc. Transl. Ser. 2, vol. 52, 1966, pp. 201–216 (in English). [10] P.R. Halmos, A Hilbert Space Problem Book, second ed., Grad. Texts in Math., vol. 19, Springer-Verlag, 1982, from first ed., D. Van Nostrand Co., Inc., Princeton, NJ/Toronto, ON/London, 1967. [11] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, second ed., Cambridge University Press, 1952. [12] A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954) 620–630. [13] A. Horn, C.R. Johnson, Topics in Matrix Analysis, corrected reprint of the 1991 original, Cambridge University Press, Cambridge, 1995.
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[14] A. Horn, C.R. Johnson, Matrix Analysis, corrected reprint of the 1985 original, Cambridge University Press, Cambridge, 1996. [15] R. Kadison, The Pythagorean Theorem I: the finite case, Proc. Natl. Acad. Sci. USA 99 (7) (2002) 4178–4184. [16] R. Kadison, The Pythagorean Theorem II: the infinite discrete case, Proc. Natl. Acad. Sci. USA 99 (8) (2002) 5217–5222. [17] V. Kaftal, G. Weiss, Traces, ideals, and arithmetic means, Proc. Natl. Acad. Sci. USA 99 (11) (2002) 7356–7360. [18] V. Kaftal, G. Weiss, Soft ideals and arithmetic mean ideals, Integral Equations Operator Theory 58 (2007) 363–405. [19] V. Kaftal, G. Weiss, Second order arithmetic means in operator ideals, Oper. Matrices 1 (2) (2007) 235–256. [20] V. Kaftal, G. Weiss, A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur–Horn majorization theorem, in: Hot Topics in Operator Theory, Theta, 2008, pp. 101–135. [21] V. Kaftal, G. Weiss, Traces on operator ideals and arithmetic means, J. Operator Theory 63 (1) (2010) 3–46. [22] V. Kaftal, G. Weiss, The B(H ) lattices, density and arithmetic mean ideals, Houston J. Math. 37 (1) (2011), in press. [23] V. Kaftal, G. Weiss, Majorization and arithmetic mean ideals, preprint. [24] N.J. Kalton, Trace-class operators and commutators, J. Funct. Anal. 86 (1989) 41–74. [25] K. Kornelson, D. Larson, Rank-one decompositions of operators and construction of frames, in: Wavelets, Frames and Operator Theory, in: Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 203–214. [26] M.O. Lorenz, Methods of measuring concentration of wealth, J. Amer. Statist. Assoc. 9 (1905) 209–219. [27] A.S. Markus, The eigen- and singular values of the sum and product of linear operators, Uspekhi Mat. Nauk 4 (118) (1964) 93–123. [28] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications, Math. Sci. Eng., vol. 143, Academic Press Inc. (Harcourt Brace Jovanovich Publishers), 1979. [29] L. Mirsky, Matrices with prescribed characteristic roots and diagonal elements, J. London Math. Soc. 33 (1958) 14–24. [30] R.F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinb. Math. Soc. 21 (1903) 144–157. [31] A. Neumann, An infinite-dimensional generalization of the Schur–Horn convexity theorem, J. Funct. Anal. 161 (2) (1999) 418–451. [32] A.M. Ostrowski, Sur quelques applications des fonctions convexes et concaves au sens de I. Schur, J. Math. Pures Appl. (9) 31 (1952) 253–292. [33] I. Schur, Über eine Klasse von Mittelbildungen mit Anwendungen auf der Determinantentheorie, Sitzungsber. Berliner Mat. Ges. 22 (1923) 9–29. [34] G. Weiss, Commutators and operators ideals, dissertation, University of Michigan Microfilm, 1975. [35] G. Weiss, Commutators of Hilbert–Schmidt operators. I, Integral Equations Operator Theory 3 (4) (1980) 574–600. [36] G. Weiss, Commutators of Hilbert–Schmidt operators. II, Integral Equations Operator Theory 9 (6) (1986) 877–892.
Journal of Functional Analysis 259 (2010) 3163–3204 www.elsevier.com/locate/jfa
Central limit theorem for the heat kernel measure on the unitary group Thierry Lévy a , Mylène Maïda b,∗ a Département de Mathématiques, Ecole Normale Supérieure, 45, rue d’Ulm, F-75230 Paris Cedex 05, France b Laboratoire de Mathématiques, Faculté des Sciences d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France
Received 22 May 2009; accepted 10 August 2010
Communicated by S. Vaes
Abstract We prove that for a finite collection of real-valued functions f1 , . . . , fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of (tr f1 , . . . , tr fn ) under the properly scaled heat kernel measure at a given time on the unitary group U(N ) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N −1 . In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results. © 2010 Elsevier Inc. All rights reserved. Keywords: Central limit theorem; Random matrices; Unitary matrices; Heat kernel; Free probability
1. Introduction In [8], P. Diaconis and S.N. Evans studied the fluctuations of the trace of functions of a unitary matrix picked uniformly at random. Let us recall briefly their main result. If U is a unitary matrix of size N 1 and f a real-valued function on the set U of complex numbers of modulus 1, then the eigenvalues λ1 , . . . , λN of U belong to U and tr f (U ) = N1 N i=1 f (λi ), where tr is the normalized trace (so that tr(IN ) = 1) and the matrix f (U ) is obtained from U and f by functional calculus. Using Weyl’s integration formula and the rotational invariance of the Haar * Corresponding author. Fax: +33 1 69 15 72 34.
E-mail addresses:
[email protected] (T. Lévy),
[email protected] (M. Maïda). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.005
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measure, it is easy to see that if f : U → R is defined almost everywhere, is integrable and has zero mean on U, then tr f (U ) is defined for almost every U , and seen as a random variable under the Haar measure, also has zero mean. The function f being fixed, tr f can be seen as a random variable on the unitary group U(N), endowed with the Haar measure, for all N 1. Thus, the single function f gives rise to a sequence of random variables indexed by the integer N, which is their main object of study. In order to understand the behavior of this sequence, a fundamental fact, which has been proved and used extensively in this context in [8], is the following: for all p, q ∈ Z, one has E[tr(U p )tr(U q )] = δp,q N −2 min(|p|, N ). Using this, one can easily check that, if f is squareintegrable on U, then the variance of tr f converges to 0 as N tends to infinity. Moreover, if f 1 belongs to the Sobolev space H 2 (U) (see Definition 9.1 below), then the series of the variances of tr f on U(N) converges, which gives a strong law of large numbers. The main result of [8] is that the fluctuations of tr f under the Haar measure are asymptotically 1 Gaussian. More precisely, they have proved that if f belongs to H 2 (U) and has zero mean on U, then N tr f converges in distribution to a centered Gaussian random variable with variance equal 1 to the square of the H 2 -norm of f (see Theorem 9.2 below for a precise statement). In this paper, we consider the fluctuations of tr f when the unitary matrix is picked not under the Haar measure, but rather under the heat kernel measure at a certain time. The heat kernel measure at time T is the distribution of UN (T ), where (UN (t))t0 is the Brownian motion on U(N ) issued from the identity matrix, that is, the Markov process whose generator is the Laplace–Beltrami operator associated to a certain Riemannian metric on U(N ). The choice of a Riemannian metric that we make is explicited at the beginning of Section 2. Apart from being one of the most natural stochastic processes with values in the unitary group, the Brownian motion arises for example in the context of two-dimensional U(N ) Yang–Mills theory [18,12,11]. Let f : U → R be a function, as above. Once a time T 0 is fixed, tr f is a random variable on U(N ) for each N 1, the unitary group being endowed with the heat kernel measure at time T . With our choice of Riemannian metric, it is known since the work of P. Biane [3] that if f is continuous, then tr f converges almost surely towards the integral of f against a probability measure νT on U, which is characterized by the formula (4) below. By this almost sure convergence, we mean that the expectations of these variables and the series of their variances converge. For all T > 0, the measure νT is absolutely continuous with respect to the uniform measure on U, with a density which unfortunately cannot be expressed in terms of usual functions. Its support is the full circle only for T 4. For T ∈ (0, 4), its support is an arc of circle containing 1, symmetric with respect to the horizontal axis, which grows continuously with T , and for the width of which a simple explicit formula exists. In fact, as N tends to infinity, not only the distribution of the eigenvalues of UN (T ) but the Brownian motion itself as a stochastic process converges in a certain sense towards a limiting object called the free multiplicative Brownian motion, which is defined in the language of free probability. The measure νT is the non-commutative distribution of this free process at time T and can be considered as a multiplicative analogue of the Wigner semi-circle law. The main result of this paper is that for any function f : U → R with Lipschitz continuous derivative, the fluctuations of N tr f are asymptotically Gaussian with variance σT (f, f ), where σT is the quadratic form defined in Definition 2.4. This definition of σT (f, f ) involves three free multiplicative Brownian motions which are mutually free and the functional calculus associated to f . It makes sense for functions of class C 1 , or at best for absolutely continuous functions. An alternative definition of σT (f, f ) is given by Definition 9.10 in terms of the Fourier co-
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efficients of f and the solution of an infinite triangular differential system (see Lemma 9.7). We prove that, when T is large enough, this second definition makes sense for functions in the 1 Sobolev space H 2 (U), which are not even necessarily continuous. Moreover, we prove that, as T tends to infinity, σT (f, f ) converges towards the square of the 1 H 2 -norm of f . This convergence is consistent, at a heuristic level, with the result of P. Diaconis and S.N. Evans, since the Haar measure is the invariant measure of the Brownian motion, and its limiting distribution as time tends to infinity. For small values of T , the analysis seems much harder to perform. We have no expression of the covariance other than Definition 2.4 and it seems plausible, considering the limiting support of the distribution of the eigenvalues of UN (T ) and some puzzling numerical simulations (see Fig. 1 in Section 9), that the largest space of functions f for which N tr f has Gaussian fluctuations might depend on T , say for T 4. Unfortunately, we have no precise conjecture to offer in this respect. The understanding of global fluctuations of random matrices has been widely developed in the literature using various techniques. By combinatorial methods applied to the computation of moments, Y. Sinai and A. Soshnikov [29] derived a central limit theorem (CLT) for moments of Wigner matrices growing as o(N 2/3 ). An important breakthrough is the work of K. Johansson [17] where he got, using techniques of orthogonal polynomials on the explicit joint density of eigenvalues, a CLT for Hermitian or real symmetric matrices whose entries have joint density eN tr V (M) , for a large class of potentials V . Recently, M. Shcherbina [28] has been able to lower, in the symmetric case, the regularity of those functions for which the CLT holds. The study of Stieltjes transform for this purpose, initiated by L.A. Pastur and others [25,26], has recently given some striking results, among which one can cite the works of G.W. Anderson and O. Zeitouni [1] or W. Hachem, P. Loubaton and J. Najim [15]. Recently S. Chatterjee [6] proposed “a soft approach” based on second-order Poincaré inequalities. The technique of proof that we have chosen is rather of the flavor of the one introduced in [5]. Therein, T. Cabanal-Duvillard proposed an approach based on matricial stochastic calculus to get a CLT for Hermitian and Wishart Brownian motions but also for several Gaussian Wigner matrices. In this direction we can also mention a CLT for band matrices obtained by A. Guionnet [13]. Some tools of free probability will play a key role in our analysis. The notion of second-order freeness was developed in a series of papers [24,23,7] in order to give a general framework to CLT’s for large random matrices. In particular, the second paper [23] of the series deals with unitary matrices and the results therein might be relevant to the problem under consideration (see Section 8 for more details). Let us mention the work of F. Benaych-Georges [2], which is closely related to ours. He also considers unitary matrices taken under the heat kernel measure, and he obtains a CLT for functions of the entries of these matrices, whereas we are rather considering functions of their empirical measure. The paper is organized as follows: Section 2 is devoted to defining the Brownian motion on the unitary group, recalling from [3] its asymptotics, defining the proper covariance functional and stating our main result (Theorem 2.6). In Section 3, we present the structure of the proof of our main theorem by introducing a family of martingales (see Eq. (6)) that will be the main object of study. The proof will in fact boil down to proving the convergence of the bracket of these martingales (Section 5) and to controlling the variance of this bracket (Section 6), relying on some technical results on the functional calculus on U(N ) gathered in Section 4. In Section 7, we extend our result to other Brownian motions on the unitary group and to the Brownian motion on
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the special unitary group. In Section 8, we deal with the fluctuations of unitary Brownian motions stopped at different times. Section 9 is devoted to the study of the covariance for large time, in connexion with the CLT for Haar unitaries [8]. Finally, in Section 10, we discuss a combinatorial approach to some of our previous results and we obtain, via representation theoretic arguments, an explicit formula (Theorem 10.2) for mixed moments of the heat kernel on SU(N ). 2. The Brownian motion on the unitary group 2.1. The stochastic differential equation Let N 1 be an integer. We denote by U(N ) the group of unitary N × N matrices and by u(N ) its Lie algebra, which is the space of anti-Hermitian N × N matrices. We denote by IN the identity matrix. We will use systematically the following convention for traces: we denote the usual trace by Tr and the normalized trace by tr, so that Tr(IN ) = N and tr(IN ) = 1. Let us endow u(N ) with the real scalar product X, Y u(N ) = N Tr(X ∗ Y ) = −N Tr(XY ). We denote by · u(N ) the corresponding norm. The scalar product ·,·u(N ) determines a Brownian motion with values in u(N ), namely the unique continuous Gaussian process (KN (t))t0 with values in u(N ) such that ∀s, t 0, ∀A, B ∈ u(N ),
E A, KN (s) u(N ) B, KN (t) u(N ) = min(s, t)A, Bu(N ) .
Equivalently, let (Bkl , Ckl , Dk )k,l1 be independent standard real Brownian motions. Then KN (t) has the same distribution as the anti-Hermitian matrix whose upper-diagonal coefficients are the √1 (Bkl (t) + iCkl (t)) and whose diagonal coefficients are the √i Dk (t). N 2N The linear stochastic differential equation 1 dUN (t) = UN (t) dKN (t) − UN (t) dt 2
(1)
admits a strong solution which is a process with values in MN (C). This process satisfies the identity d(UN UN∗ )(t) = 0, as one can check by using Itô’s formula. Hence, this equation defines a Markov process on the unitary group U(N ), which we call the unitary Brownian motion. The generator of this Markov process can be described as follows. Let (X1 , . . . , XN 2 ) be an orthonormal basis of u(N ). Each element X of u(N ) can be identified with the left-invariant first-order differential operator LX on U(N ) by setting, for all differentiable function F : U(N ) → R and all U ∈ U(N), (LX F )(U ) =
d F U etX . dt |t=0
The generator of the unitary Brownian motion is the second-order differential operator N2
1 1 2 = L Xk . 2 2 k=1
(2)
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This operator does not depend on the choice of the orthonormal basis of u(N ). We denote the associated semi-group by (Pt )t0 . From now on, we will always consider the Brownian motion issued from the identity matrix, so that UN (0) = IN . The stochastic differential equation satisfied by UN can be translated into an Itô formula, as follows. Proposition 2.1. Let F : R × U(N ) → R be a function of class C 2 . Then for all t 0, 2
N F t, UN (t) = F (0, IN ) +
t
(LXk F ) s, UN (s) dXk , KN u(N ) (s)
k=1 0
t +
1 F + ∂t F s, UN (s) ds, 2
(3)
0
and the processes {Xk , KN u(N ) : k ∈ {1, . . . , N 2 }} are independent standard real Brownian motions. This result is classical in the framework of stochastic analysis on manifolds (see for example [16]), but since our whole analysis relies on this formula and for the convenience of the reader, we offer a sketch of proof in this particular setting. Proof of Proposition 2.1. For all a, b ∈ {1, . . . , N}, let εab : MN (C) → C denote the coordinate mapping which to a matrix M associates the entry Mab . Let also ∂ab denote the partial derivation with respect to the ab-entry. The definition of LX given by (2) makes sense for any matrix X. One can check the following identities:
∀X ∈ MN (C),
N
LX =
εac Xcb ∂ab
and
a,b,c=1
L2X − LX2 =
N
εac Xcb εa c Xc b ∂ab ∂a b ,
a,b,c,a ,b ,c =1 2
= LC +
N
N
εac (Xk )cb εa c (Xk )c b ∂ab ∂a b ,
k=1 a,b,c,a ,b ,c =1
2 2 where C = N i=1 Xi . Moreover, C = −IN , regardless of the choice of the orthonormal basis (X1 , . . . , XN 2 ). Any smooth function F : R × U(N ) is the restriction of a smooth function defined on R × MN (C). Applying the usual Itô formula to this extended function and using the identities above leads immediately to (3). 2
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2.2. The free multiplicative Brownian motion We are interested in the large N behavior of the stochastic process UN issued from IN . P. Biane has described in [3] the limiting distribution of this process seen as a collection of elements of the non-commutative probability space (L∞ ⊗ MN (C), E ⊗ tr). We start by describing the limiting object. As a general reference on non-commutative probability and freeness, we recommend [30]. Definition 2.2. Let (A, τ ) be a (non-commutative) ∗-probability space. A collection of unitaries (ut )t0 in A is called a free multiplicative Brownian motion if the following properties hold. 1. For all 0 t1 · · · tn , the elements ut1 , ut2 u∗t1 , . . . , utn u∗tn−1 are free. 2. For all 0 s t, the element ut u∗s has the same distribution as ut−s . 3. For all t 0, the distribution of ut is the probability measure νt on U = {z ∈ C: |z| = 1} characterized by the identity
U
1 1−
z tz 2t z+1 e e ξ
dνt (ξ ) = 1 + z,
(4)
valid for z in a neighborhood of 0. The following result was proved by P. Biane. The second assertion follows from the first by a general result of D. Voiculescu. Theorem 2.3. The collection (UN (t))t0 of non-commutative random variables converges in distribution, as N tends to +∞, towards a free multiplicative Brownian motion. Moreover, if UN(1) , UN(2) , . . . , UN(n) are n independent sequences of unitary Brownian motions, (1) (2) (n) then the family ((UN (t))t0 , (UN (t))t0 , . . . , (UN (t))t0 ) converges in non-commutative (1) (n) (1) distribution, as N tends to infinity, towards ((ut )t0 , (u(2) t )t0 , . . . , (ut )t0 ) where u , (n) . . . , u are n free multiplicative Brownian motions which are mutually free. 2.3. Statement of the central limit theorem Recall that U denotes the group of complex numbers of modulus 1. Let f : U → R be a function. Then, by the functional calculus, f induces a function, still denoted by f , from U(N ) to MN (C). Moreover, for all unitary matrix U , the matrix f (U ) is Hermitian. We endow U with the usual length distance, that is, the distance d(eiα , eiβ ) = |α − β| for all α, β ∈ R such that |α − β| π . Accordingly, we define the Lipschitz norm of a function f : U → R as follows: f Lip =
|f (z) − f (w)| . d(z, w) z,w∈U,z =w sup
Note that if f is Lipschitz continuous and z, w belong to U, then the following inequalities hold: |f (z) − f (w)| f Lip d(z, w) π2 f Lip |z − w|.
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By the derivative of a differentiable function f : U → R, we mean the function f : U → R defined by ∀z ∈ U,
f (zeih ) − f (z) . h→0 h
f (z) = lim
We denote by L1 (U) the space of integrable functions on U, with respect to the Lebesgue measure. We denote by C 1 (U) the space of continuously differentiable functions and by C 1,1 (U) the subspace of C 1 (U) consisting of those functions whose derivative is Lipschitz continuous. We define a family of bilinear forms on C 1 (U) as follows. Definition 2.4. Let (A, τ ) be a C ∗ -probability space which carries three free multiplicative Brownian motions u, v, w which are mutually free. Let T 0 be a real number. Let f, g : U → R be two functions of C 1 (U). For all s ∈ [0, T ], we set σT ,s (f, g) = τ (f (us vT −s )g (us wT −s )). Then, we define
T σT (f, g) =
T σT ,s (f, g) ds =
0
τ f (us vT −s )g (us wT −s ) ds.
0
Lemma 2.5. For all T 0, σT is a symmetric non-negative bilinear form on C 1 (U). Proof. The symmetry of σT comes from the fact that the triples (u, v, w) and (u, w, v) have the same distribution. In order to prove the non-negativity, let us realize (u, v, w) on the free product of three non-commutative probability spaces. So, let (Au , τu ), (Av , τv ) and (Aw , τw ) be three non-commutative probability spaces which carry respectively u, v and w. We consider their free product, so we define A = Au ∗ Av ∗ Aw and τ = τu ∗ τv ∗ τw . We also use the notation τu , τv , τw for the partial traces on A. Then
T σT (f, f ) =
τu τv f (us vT −s ) τw f (us wT −s ) ds
0
T =
2 τu τv f (us vT −s ) ds 0,
0
the positivity coming from the fact that f (us vT −s ) is self-adjoint.
2
We will use the notation σT (f ) = σT (f, f ). Let us state our main result. Theorem 2.6. Let T 0 be a real number. Let n 1 be an integer. Let f1 , . . . , fn : U → R be n functions of C 1,1 (U). Let us define an n × n real non-negative symmetric matrix by setting ΣT (f1 , . . . , fn ) = (σT (fi , fj ))i,j ∈{1,...,n} . Then, as N tends to infinity, the following convergence of random vectors in Rn holds in distribution: (d) N tr fi UN (T ) − E tr fi UN (T ) i∈{1,...,n} −−−−→ N 0, ΣT (f1 , . . . , fn ) . N →∞
(5)
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3. Structure of the proof For T = 0, the result is straightforward. Let us choose once for all a real T > 0. In order to study the left-hand side of (5), we write each component of this random vector as the difference between the final and the initial value of a martingale. To do this, let (FN,t )t0 denote the filtration generated by the unitary Brownian motion UN . To each function f of L1 (U) we associate a f real-valued martingale (MN (t))t∈[0,T ] by setting f MN (t) = E tr f UN (T ) FN,t . f
(6)
f
The left-hand side of (5) is simply N (MNi (T )−MNi (0))i∈{1,...,n} and we are going to study the f quadratic variations and covariations of the martingales MNi . In order to state the main technical results, let us introduce some notation. Recall that the gradient of a differentiable function F : U(N ) → C is the vector field on U(N ) 2 defined by ∇F = N k=1 (LXk F )Xk , where (X1 , . . . , XN 2 ) is an orthonormal basis of u(N ). To f,g each pair of functions f, g ∈ L1 (U) we associate a function EN on [0, T ) × U(N ) by setting f,g EN (s, U ) = N 2 ∇ PT −s (tr f ) (U ), ∇ PT −s (tr g) (U ) u(N ) . Let us check that this function is well defined. By the Weyl integration formula, the fact that f is integrable on U implies that tr f is an integrable function on U(N ). Hence, for all s ∈ [0, T ), f,g PT −s (tr f ) is a function of class C ∞ on U(N ) and EN is well defined. Proposition 3.1. Consider f, g ∈ L1 (U). With the notation introduced above, the following properties hold. f
g
1. For all t ∈ [0, T ], the quadratic covariation of the martingales N MN and N MN is given by f g N MN , N M N t
t =
f,g
EN
s, UN (s) ds.
0
2. Assume that f and g are Lipschitz continuous. Then for all s ∈ [0, T ) and all U ∈ U(N ), f,g |EN (s, U )| (f Lip +gLip )2 . Moreover, if f and g belong to C 1 (U), then the following convergence holds: f,g E EN s, UN (s) −−−−→ σT ,s (f, g). N →∞
3. Assume that f and g belong to C 1,1 (U). Then the following estimate holds: f,g sup Var EN s, UN (s) = O N −2 .
s∈[0,T )
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Let us show that these results imply Theorem 2.6. Proof of Theorem 2.6. For all N 1, define an Rn -valued martingale QN = (Q1N , . . . , QnN ) f
f
by setting QN (t) = N (MNj (t) − MNj (0))j ∈{1,...,n} . It is a martingale indexed by [0, T ], issued f
from 0 and with the same bracket as N (MNj )j ∈{1,...,n} . For all ξ = (ξ1 , . . . , ξn ) ∈ Rn and all t ∈ [0, T ], set RN (t) = exp i
n
j ξj QN (t) +
j =1
t n 1 ξj ξk σT ,s (fj , fk ) ds . 2 j,k=1
0
Itô’s formula yields
n 1 f ,f E RN (t) = 1 + ξj ξk E RN (s) σT ,s (fj , fk ) − ENj k s, UN (s) ds. 2 t
j,k=1
0
Thus, 2 nT ξ 2
2
E RN (t) − 1 nξ e 2 maxj =1...n fj ∞ 2
t
f ,f × max E σT ,s (fj , fk ) − ENj k s, UN (s) ds. j,k=1...n
0
For fixed j and k, the last integral is smaller than
t
σT ,s (fj , fk ) − E E fj ,fk (s, UN )(s) ds N
0
t +E
fj ,fk f ,f
E (s, UN )(s) − E E j k (s, UN )(s) ds. N
N
0
By the second part of Proposition 3.1, and by the dominated convergence theorem, the first integral tends to 0 as N tends to infinity. The square of the second integral is smaller than t f ,f t 0 Var(ENj k (s, UN (s))) ds, which, thanks to the third part of Proposition 3.1 and by dominated convergence again, tends also to 0. Finally, we have proved that
∀ξ ∈ R , n
t n i n ξj Qj (t) 1 j =1 N = exp − lim E e ξj ξk σT ,s (fj , fk ) ds , N →∞ 2 j,k=1
which, for t = T , yields the expected result.
2
0
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In Section 4, we collect some technical results that we use in Sections 5 and 6 to prove Proposition 3.1. 4. Regularity of the functional calculus In this section, we relate the regularity of a function f : U → R to the regularity of the functional calculus mapping f : U(N ) → MN (C) and the function tr f : U(N ) → R. We start with a result which, logically speaking, is not necessary for our exposition, but which is the simplest instance of a crucial phenomenon. 4.1. Lipschitz norms The group U(N) becomes a metric space when it is endowed with the Riemannian distance, denoted by d, associated to the Riemannian metric induced by the scalar product ·,·u(N ) on u(N ). We denote by F Lip the corresponding Lipschitz norm of a function F : U(N ) → R, that is, |F (U ) − F (V )| : U, V ∈ U(N ), U = V . F Lip = sup d(U, V ) As a reference for the notions of Riemannian geometry that we use, we recommend [9]. Proposition 4.1. Let f : U → R be a Lipschitz continuous function. Then tr f : U(N ) → R is also Lipschitz continuous and tr f Lip =
1 f Lip . N
Note that this result can be compared to Lemma 1.2 in [14], where it was a key point towards the concentration results for Wigner and Wishart random matrices. In order to prove this proposition, we use the following lemma. Lemma 4.2. Let U and V be two elements of U(N ). Then there exist A, B ∈ U(N ) such that AU A−1 and BV B −1 are diagonal and d(AU A−1 , BV B −1 ) d(U, V ). Proof. Let O be the conjugacy class of V . It is a compact submanifold of U(N ). Let V be a point of O which minimizes the distance to U . Let γ : [0, 1] → U(N ) be a minimizing geodesic path from V to U parametrized at constant speed. It is thus of the form γ (t) = V etZ for some Z ∈ u(N ). Since V minimizes the distance to U , the vector γ˙ (0) is orthogonal to the tangent space TV O. This space TV O, identified with a subspace of u(N ) by a left translation, is the range of the linear mapping Ad(V −1 ) − Id. Hence, Z belongs to the kernel of the adjoint linear mapping, that is, to the kernel of Ad(V ) − Id. In other words, V ZV −1 = Z. It follows that Z and V can be simultaneously diagonalized, in an orthonormal basis, and the same is true for V and V eZ = U . Finally, V and U are conjugated by a same unitary matrix to two diagonal unitary matrices. The result follows easily from the fact that translations are isometries on U(N ). 2 Proof of Proposition 4.1. Let f : U → R be Lipschitz continuous. Consider U and V in U(N ). Thanks to Lemma 4.2, let us choose U and V which are both diagonal, conjugated respectively
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to U and V , and such that d(U , V ) d(U, V ). Let us write U = diag(eiα1 , . . . , eiαN ) and V = diag(eiβ1 , . . . , eiβN ) in such a way that |βj − αj | π for all j ∈ {1, . . . , N }. Let us compute d(U , V ). It is equal to d(IN , U −1 V ), hence to d IN , ei diag(β1 −α1 ,...,βN −αN ) = i diag(β1 − α1 , . . . , βN − αN )u(N ) N = N (βj − αj )2 . j =1
It follows that d(U, V )
N
j =1 |βj
− αj |. On the other hand,
N N
iβ
f e j − f eiαj 1 f Lip
tr f (V ) − tr f (U ) 1 |βj − αj | N N j =1
j =1
1 f Lip d(U, V ). N
This proves the inequality tr f Lip N1 f Lip . By choosing α, β such that |f (eiβ ) − f (eiα )| is close to f Lip |β − α| and by considering U = eiα IN , V = eiβ IN , one verifies that the opposite inequality holds. 2 Let us make a short heuristic comment on this result. The scalar product which we have chosen on u(N ) corresponds to a metric structure on U(N ) which gives this group the diameter d(IN , −IN ) = i diag(π, . . . , π)u(N ) = N π , of the order of N . The function f : U → R being fixed, the variations of the function tr f : U(N ) → R are of the same order of magnitude as those of f but occur on a space N times as large. This makes the equality that we have just proved plausible. In the same order of ideas, note that the distance to the origin at time T of a linear Brownian motion √ in a Euclidean space of large dimension d is, by the law of large numbers, of the order of dT . Assuming that the Brownian motion on the unitary group behaves in a comparable way, and considering the fact that the dimension of√U(N ) is N 2 , this indicates that the Brownian motion UN (T ) might be at a distance of order N T of IN , thus a fraction of the diameter of U(N ) which does not depend on N . This gives an intuitive justification for the choice of the normalization. 4.2. First derivatives We are now going to prove that the functional calculus induced by f is differentiable when f is differentiable, and to compute its differential. For this, we introduce some notation. Let f : U → C be a differentiable function. Let us define a function Df : U × U → C by setting f (z)−f (w) ∀z, w ∈ U,
Df (z, w) =
z−w − zi f (z)
if z = w, if z = w.
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The function Df is symmetric and, if f is C 1 (U), it is continuous and bounded by π2 f ∞ . Note that Df takes its values in C even if f is real-valued. If the function f is only Lipschitz continuous, then it is differentiable with bounded differential outside a negligible subset of U, and the definition of Df still makes sense outside the corresponding negligible subset of the diagonal of U × U. Moreover, outside this subset, the inequality |Df (z, w)| π2 f ∞ holds. If U is a unitary matrix, we denote by LU and RU the linear operators on MN (C) of left and right multiplication by U respectively. These operators commute and they are normal with respect to the scalar product A, B = N Tr(A∗ B) on MN (C). In fact, L∗U = LU −1 and RU∗ = RU −1 . Hence, if g is a function on U × U, then g(LU , RU ) is a well-defined endomorphism of MN (C). Even when f is only Lipschitz continuous, Df (LU , RU ) is well defined for almost all U ∈ U(N). Let us define a special orthonormal basis of u(N ). We use the notation (Ej k )j,k∈{1,...,N } for the canonical basis of MN (C). For all j, k with 1 j < k N , set Xj k = √1 (Ej k − Ekj ) and Yj k = √ i (Ej k + Ekj ). For all j ∈ {1, . . . , N}, set Hj = 2N orthonormal basis of u(N ).
2N
√i Ejj . N
These matrices form an
Proposition 4.3. Let f : U → C be a differentiable function. Let U be an element of U(N ). Let X be an element of u(N ). Then d f U etX = Df (LU , RU ) (U X). dt |t=0
(7)
In particular, when U is a diagonal matrix with diagonal coefficients (u1 , . . . , uN ), the following equalities hold. 1. For all j ∈ {1, . . . , N}, 2. For all j, k 3. For all j, k
d tHj ) = Df (u , u )U H . j j j dt |t=0 f (U e d tX j k ∈ {1, . . . , N} with j < k, dt |t=0 f (U e ) = Df (uj , uk )U Xj k . d tY j k ∈ {1, . . . , N} with j < k, dt |t=0 f (U e ) = Df (uj , uk )U Yj k .
If f is only Lipschitz continuous, then the same conclusions hold for almost all U ∈ U(N ) (with respect to Haar measure). Proof. We will give the proof under the assumption that f is differentiable. The extension to the Lipschitz continuous case is straightforward (we have to take into account that in this case the differential operators involved are only defined for almost all U with respect to Haar measure). Let us start by proving the part of the statement which concerns a diagonal matrix U . 1. Since U etHj is diagonal, this assertion is proved by an easy direct computation. 2. This case is less trivial. Let us assume that uj = uk . Then for small t, there is a unique pair of continuous functions (uj (t), uk (t)) such that the spectrum of U etXj k is deduced from that of U by replacing uj and uk respectively by uj (t) and uk (t). The functions uj and uk are in fact smooth and they satisfy uj (0) = uk (0) = 0, an equality which can be phrased by saying that the right multiplication by etXj k does not affect the spectrum of U at the first order. Let D(t) be the diagonal matrix obtained from U by replacing uj and uk by uj (t) and uk (t) respectively. By diagonalizing U etXj k for small t, one can find a unitary matrix P (t) which
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depends smoothly on t, such that P (0) = IN , such that the only non-zero off-diagonal terms of P (t) are P (t)j k and P (t)kj , and finally such that U etXj k = P (t)D(t)P (t)−1 .
(8)
By differentiating with respect to t at t = 0, one finds U Xj k = P (0), U , u
j k and P (0)kj = √1 uku−u . By applying f to from which one deduces that P (0)j k = √1 uk −u j j 2N 2N both sides of (8) and then differentiating again with respect to t at t = 0, we find
d f U etXj k = P (0), f (U ) . dt |t=0 Knowing the off-diagonal terms of P (0) is enough to compute this bracket and we find the expected result. The case where uj = uk is left to the reader, as well as the third assertion. Let us now turn to the first part of the statement, where no assumption is made on U . Let us first prove that (7) is true when U is a diagonal matrix with diagonal coefficients (u1 , . . . , uN ). In this case, for all j, k ∈ {1, . . . , N}, the matrix Ej k is an eigenvector for LU and RU , with the eigenvalues uj and uk respectively. Hence, by definition of Df , Ej k is an eigenvector of Df (LU , RU ) with the eigenvalue Df (uj , uk ). The validity of (7) in this case follows, because U Hj (resp. U Xj k , U Yj k ) has the same vanishing entries as Hj (resp. Xj k , Yj k ). Let us finally prove that (7) holds for any unitary matrix. Consider U ∈ U(N ). Choose P , D ∈ U(N ) such that D is diagonal and U = P DP −1 . Set Y = P −1 XP . Then U etX = P DetY P −1 . The result now follows easily. 2 Before we apply the last result in order to compute the differential of tr f , let us state a classical yet very useful lemma, of which a version can be found in [27]. Lemma 4.4. Let (Xk )k∈{1,...,N 2 } be an orthonormal basis of u(N ). Let A, B be elements of MN (C). Then the following equalities hold: 2
N
tr(AXk ) tr(BXk ) = −
k=1
1 tr(AB), N2
(9)
2
N
tr(AXk BXk ) = − tr(A) tr(B).
k=1
Proof. 1. For A, B ∈ u(N ), this equality multiplied by N 4 is indeed simply 2
N A, Xk u(N ) B, Xk u(N ) = A, Bu(N ) . k=1
(10)
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The general case follows thanks to the equality MN (C) = u(N ) ⊕ iu(N ) and the fact that the relations are C-bilinear in (A, B). 2. Choose i, j, l, m ∈ {1, . . . , N 2 }. By taking A = Ej i and B = Eml in the first relation, we find 2
N 1 (Xk )ij (Xk )lm = − δi,m δj,l . N k=1
The second relation follows by developing the trace.
2
Proposition 4.5. Let f : U → R be a differentiable function. Then tr f is differentiable and, for all U ∈ U(N ) and all Y ∈ u(N ), we have LY (tr f ) (U ) = −i tr f (U )Y . In particular, ∀U ∈ U(N ), ∇(tr f )(U )2 =
1 N2
(11)
tr(f (U )2 ).
Proof. Since tr f is invariant by conjugation, we have for all U, V ∈ U(N ) and all Y ∈ u(N ) the equality (LY (tr f ))(U ) = (LV Y V −1 (tr f ))(V U V −1 ). Hence, it suffices to check (11) for all Y when U is diagonal. In this case, the result is a direct consequence of Proposition 4.3. The second assertion follows from the definition of the gradient and the identity (9). 2 4.3. Lipschitz norms again At the end of the proof of Proposition 3.1 (see Section 6.2), we will need to estimate the Lipschitz norm of a function of a unitary matrix of a special form. We state and prove this estimation below, although the reader might want to skip it now and jump to Section 5. Proposition 4.6. Let f be an element of C 1,1 (U). Let V , W be two elements of U(N ). Define a function FV ,W : U(N ) → C by setting FV ,W (U ) = tr f (U V )f (U W ) . Then F is Lipschitz continuous and we have the estimate FV ,W Lip
π f ∞ f ∞ . L L N
Proof. We prove that FV ,W is differentiable almost everywhere on U(N ) and estimate the L∞ norm of its differential. According to Proposition 4.3, we have, for all X ∈ u(N ) and almost all U ∈ U(N), the equality (LX FV ,W )(U ) = tr V −1 Df (LV U , RV U )(V U X)Vf (U W ) + tr f (U V )W −1 Df (LW U , RW U )(W U X)W .
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d d We have used the fact that dt f (U etX V ) = V −1 dt f (V U etX )V . Let us focus on the first |t=0 |t=0 term of the right-hand side, the second being similar. By the Cauchy–Schwarz inequality,
−1
tr V Df (LV U , RV U )(V U X)Vf (U W ) 2 tr M ∗ M tr f (U W )∗ f (U W ) , where we have set M = Df (LV U , RV U )(V U X). Recall that MN (C) is endowed with the scalar product A, B = N Tr(A∗ B). We claim that the operator norm of the endomorphism Df (LV U , RV U ) of MN (C) with respect to this norm is bounded above by π2 f L∞ . Indeed, this operator is normal with respect to this scalar product, so that its operator norm equals its spectral radius, which is smaller than the L∞ norm of Df . Hence, we find 1 π 1 tr M ∗ M 2 f L∞ tr X ∗ X 2 . 2 It follows that π Xu(N ) f ∞ , LX FV ,W L∞ 2 f L∞ L 2 N from which the result follows easily.
2
5. Convergence of the bracket In this section, we prove the first two assertions of Proposition 3.1. Let us first prove a fundamental property of the generator of the Brownian motion on U(N ). The action of U(N ) on u(N ) by conjugation is an isometric action. Hence, for all V ∈ U(N ), the processes UN and V UN V −1 satisfy two stochastic differential equations (see (1)) driven by two processes in u(N ) with the same distribution, so that they have the same distribution. Lemma 5.1. Let F : U(N ) → R be a Lipschitz continuous function. Let Y be an element of u(N ). Let t 0 be a real number. Then LY (Pt F ) = Pt (LY F ). Proof. Since F is Lipschitz continuous, LY F is well defined as an element of L∞ (U(N )). The result amounts simply to the interversion of an integration and a derivation: for all U ∈ U(N ), d d E F U esY UN (t) = E F U UN (t)esY ds |s=0 ds |s=0 d =E F U UN (t)esY = Pt (LY F )(U ). ds |s=0
LY (Pt F )(U ) =
We have used the fact that UN (t) has the same distribution as e−sY UN (t)esY .
2
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5.1. Itô formula The following result summarizes the applications of Itô formula that we will use. The third assertion below implies, by polarization, the first assertion of Proposition 3.1. Proposition 5.2. Let F : U(N ) → R be an integrable function. Define a real-valued martingale LF indexed by [0, T ] by setting, for all t ∈ [0, T ], LF (t) = E[F (UN (T ))|FN,t ]. Let (Xk )k∈{1,...,N 2 } be an orthonormal basis of u(N ). Then the following equalities hold for all t ∈ [0, T ]. 1. LF (t) = (PT −t F )(UN (t)). t 2 2. LF (t) = LF (0) + 0 N k=1 LXk (PT −s F )(UN (s)) dXk , KN u(N ) (s). t 3. LF (t) = 0 (∇(PT −s F ))(UN (s))2 ds. t 2 2 4. If F is Lipschitz continuous, then LF (t) = 0 N k=1 [PT −s (LXk F )(UN (s))] ds. Proof. 1. Choose t ∈ [0, T ]. Since the unitary Brownian motion has independent multiplicative increments, LF (t) can be rewritten as LF (t) = E F UN (T ) FN,t = E F UN (t)UN∗ (t)UN (T ) FN,t = E F UN (t)VN (T − t) FN,t , where VN is a Brownian motion on U(N ) with the same distribution as UN and independent of UN . The result follows. 2. Let us apply (3) to the function G : [0, T ] × U(N ) → R defined by G(t, U ) = (PT −t F )(U ). It follows from the definition of the semigroup (Pt )t0 that G satisfies the time-reversed heat equation 12 G + ∂t G = 0. Hence, Itô’s formula reads N
LXk (PT −s F ) UN (s) dXk , KN u(N ) (s). L (t) = L (0) + 2
F
t
F
k=1 0
3. The equality follows immediately from the equality 2 and the fact that the processes {Xk , KN u(N ) : k ∈ {1, . . . , N 2 }} are independent standard real Brownian motions. 4. This equality follows from the previous one by applying Lemma 5.1. 2 5.2. Expectation of the bracket We can now prove the second assertion of Proposition 3.1. Recall that we use the notation = N 2 ∇(PT −s (tr f ))(U ), ∇(PT −s (tr g))(U )u(N ) . We will use the fact, which is a consequence of Jensen’s inequality, that for any square-integrable function G : U(N ) → R, and for all t 0, (Pt G)2 Pt (G2 ).
f,g EN (s, U )
Proof of the second assertion of Proposition 3.1. Let f : U → R be Lipschitz continuous. By definition and by Lemma 5.1
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2
f,f EN (s, U ) = N 2
N PT −s LXk (tr f ) (U )2 k=1 2
N
2
N
2 PT −s (LXk tr f )2 (U ) = N 2 PT −s ∇(tr f ) (U ).
k=1
By Proposition 4.5 and the fact that PT −s does not increase the uniform norm, this implies that
f,f
E (s, U ) f 2 ∞ . N
L
f,g
By polarization, the estimation of |EN (s, U )| follows. Now, let us consider two independent copies VN and WN of the unitary Brownian motion UN . Then, denoting by EVN ,WN the expectation with respect to VN and WN only, we have f,f
s, UN (s)
EN
2
N 2 PT −s LXk (tr f ) UN (s) =N 2
k=1 2
=N
2
N
EVN ,WN (LXk tr f ) UN (s)VN (T − s) (LXk tr f ) UN (s)WN (T − s) .
k=1
Using successively Proposition 4.5 and Lemma 4.4, we find f,f
EN
s, UN (s) = EVN ,WN tr f UN (s)VN (T − s) f UN (s)WN (T − s) .
Taking the expectation with respect to UN , we find finally f,f E EN s, UN (s) = E tr f UN (s)VN (T − s) f UN (s)WN (T − s) . Let (A, τ ) be a C ∗ -probability space which carries three free multiplicative Brownian motions u, v, w which are mutually free. According to Theorem 2.3, the family (UN (s), VN (t), WN (u))s,t,u0 , seen as a collection of non-commutative random variables in the non-commutative probability space (L∞ ⊗ MN (C), E ⊗ tr), converges in distribution to (us , vt , wu )s,t,u0 as N tends to infinity. This implies in particular that for all non-commutative polynomial p in three variables and their adjoints, and for all s, t, u 0 E tr p UN (s), VN (t), WN (u) −−−−→ τ p(us , vt , wu ) . N →∞
Let us fix s ∈ [0, T ). Since A is a C ∗ -algebra, there is a continuous functional calculus on normal elements, hence on unitary elements, and f (us vT −s )f (us wT −s ) is a well-defined element of A. On the other hand, choose ε > 0 and let q(z, w) be a polynomial function in z, w and their adjoints such that supz,w∈U |f (z)f (w) − q(z, w)| < ε. Then
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E tr f UN (s)VN (T − s) f UN (s)WN (T − s) − τ f (us vT −s )f (us wT −s )
E tr f UN (s)VN (T − s) f UN (s)WN (T − s) − tr q UN (s)VN (T − s), UN (s)WN (T − s)
+ E tr q UN (s)VN (T − s), UN (s)WN (T − s) − τ q(us vT −s , us wT −s )
+ τ q(us vT −s , us wT −s ) − τ f (us vT −s )f (us wT −s ) . The first and the third terms are smaller than the uniform distance between q(·,·) and f (·)f (·), hence smaller than ε. The middle term tends to 0 as N tends to infinity. Altogether, this proves that f,f E EN s, UN (s) −−−−→ τ f (us vT −s )f (us wT −s ) , N →∞
from which the expected result follows by polarization.
2
6. Convergence of the variance of the bracket This section is devoted to the proof of the third assertion of Proposition 3.1. 6.1. A weak concentration inequality Consider a function F : U(N ) → R. If F is Lipschitz continuous, then the equality F Lip = ∇F L∞ holds. The goal of this subsection is to prove the following inequality. Proposition 6.1. Let F : U(N ) → R be a Lipschitz continuous function. For all T 0, one has the following inequality: Var F UN (T ) T F 2Lip . Note that this inequality is preserved by rescaling of the Riemannian metric on U(N ), that is, by rescaling of the scalar product on u(N ). Indeed, let λ be a positive real and let us consider the scalar product ·,·u = λ·,·u on u(N ). Then, putting a tilde to the quantities associated with 1 −1 this new scalar product, we have on one hand d˜ = λ 2 d and F Lip = λ 2 F Lip , and on the N (T ) has the distribution of UN (λ−1 T ). = λ−1 and U other hand Proof of Proposition 6.1. Recall the definition of the martingale LF (see Proposition 5.2). The left-hand side is equal to E[LF (T )], thus, by the third assertion of Proposition 5.2, to
T E 0
∇(PT −s F ) UN (s) 2 ds T sup ∇(PT −s F )2 ∞ L s∈[0,T )
= T sup PT −s F 2Lip . s∈[0,T )
On the other hand, since F is Lipschitz continuous, for all t 0, Pt F Lip F Lip . The result follows. 2
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6.2. An estimate of a Lipschitz norm f,f
With Proposition 6.1 in mind, we are going to study the Lipschitz norm of U → EN (s, U ) f,f in order to estimate the variance of EN (s, UN (s)). Proposition 6.2. Assume that f is of class C 1,1 (U). Then f,f sup EN (s, ·)Lip = O N −1 .
s∈[0,T ]
Proof. The proof relies on the identity f,f
EN
s, UN (s) = EVN ,WN tr f UN (s)VN (T − s) f UN (s)WN (T − s) .
By Proposition 4.6, the expression between the brackets is a Lipschitz continuous function of UN (s) for all values of VN (T − s) and WN (T − s), with a Lipschitz norm which does not depend on VN (T − s) and WN (T − s) and is O(N −1 ). Hence, the same estimate holds for the expectation. 2 Proof of the third assertion of Proposition 3.1. It suffices to combine Proposition 6.2 and f,f f,g Proposition 6.1 to find that sups∈[0,T ) Var[EN (s, UN (s))] = O(N −2 ). The same result for EN follows easily. 2 This concludes the proof of Proposition 3.1 and thus of Theorem 2.6. 7. Other Brownian motions, on unitary and special unitary groups In this section, we explain how Theorem 2.6 can be extended to other Brownian motions on the unitary group and to the Brownian motion on the special unitary group. In this paper so far, we have considered the Brownian motion UN on U(N ) associated to the scalar product on u(N ) given by X, Y u(N ) = N Tr(X ∗ Y ), for any X, Y ∈ u(N ). The crucial property of this scalar product is its invariance under the action of U(N ) on u(N ) by conjugation. There is in fact a two-parameter family of scalar products with this invariance property, namely a Tr((X − tr X)∗ (Y − tr Y )) + b Tr(X ∗ ) Tr(Y ) with a, b > 0. Multiplying the two parameters a and b by the same constant simply affects the Brownian motion by a global rescaling of time, indeed dividing time by this constant, so that we may choose the value of one of them. We take a = N in order to have correct asymptotics as N tends to infinity. This choice being made, varying b really yields different Brownian motions. It turns out to be more convenient to take 1 α = b− 2 as the parameter: we define, for all α > 0, the scalar product 1 (α) X, Y u(N ) = N Tr (X − tr X)∗ (Y − tr Y ) + 2 Tr X ∗ Tr(Y ) α on u(N ). In particular, the scalar product considered in the rest of this paper corresponds to α = 1.
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In order to understand the Brownian motions associated to the scalar products ·,·u(N ) , we start by defining the Brownian motion on SU(N ), which corresponds to the limit where α tends to 0. Let us denote by su(N ) the hyperplane of u(N ) consisting of traceless matrices, which is also the Lie algebra of the special unitary group SU(N ), and let KN0 be the linear Brownian motion on su(N ) corresponding to the scalar product induced by ·,·u(N ) . Let VN be the solution of the stochastic differential equation dVN (t) = VN (t) dKN0 (t) −
1 1 1 − 2 VN (t) dt. 2 N
(12)
One can check that if the initial condition is in the special unitary group, then the process VN stays in it: the constant 1 − N12 is designed for that purpose. We call VN the Brownian motion on SU(N). Now, for all α 0, let us consider the following process with values in U(N ): (α)
VN (t) = e
iαBt N
VN (t),
where (Bt )t0 is a standard real Brownian motion independent of VN . Let (Y1 , . . . , YN 2 −1 ) be (α) an orthonormal basis of su(N ). For all α 0, the generator of VN is given by 1 (α) 1 = 2 2
N 2 −1
L2Yi
k=1
+α L i I 2 2
N N
,
(α)
and we call VN the α-Brownian motion on U(N ). (α) (α) For each α > 0, the process VN is naturally associated with the scalar product X, Y u(N ) (α)
on u(N ). Indeed, let KN be the linear Brownian motion on u(N ) corresponding to this scalar (α) (α) product. It can be expressed as KN = KN0 + iα N B. Then the process VN satisfies the stochastic differential equation (α)
(α)
(α)
dVN (t) = VN (t) dKN (t) − (1)
α2 − 1 1 (α) VN (t) dt. 1+ 2 N2
(13)
In particular, VN has the same distribution as UN . The main feature of the Brownian motion on U(N ) which we have used extensively in the proof of Theorem 2.6 is that its generator commutes with all Lie derivatives. Since the Lie derivative in the direction of iIN commutes with all Lie derivatives, this is also the case for (α) the generator of VN and of all the processes VN , α > 0. (α) Finally, following [3], one can check that for all α 0, the process VN converges as N tends to infinity to a free multiplicative Brownian motion. Let us now define a modified version of the covariance σT .
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Definition 7.1. With all the notation of Definition 2.4, we define, for all α 0, σT(α) (f, g) =
T
τ f (us vT −s )g (us wT −s ) + α 2 − 1 τ f (us vT −s ) τ g (us wT −s ) ds.
0
Following step by step the proof of Theorem 2.6, one finds the following result. Theorem 7.2. Let T 0 be a real number. Let n 1 be an integer. Let f1 , . . . , fn : U → R be n functions of C 1,1 (U). Let us define an n × n real non-negative symmetric matrix by setting (α) (α) ΣT (f1 , . . . , fn ) = (σT (fi , fj ))i,j ∈{1,...,n} . Then, as N tends to infinity, the following convergence of random vectors in Rn holds in distribution: (α) (α) (d) (α) N tr fi VN (T ) − E tr fi VN (T ) i∈{1,...,n} −−−−→ N 0, ΣT (f1 , . . . , fn ) . N →∞
(14)
We leave the details to the reader, since every step can be adapted in a straightforward way. The only substantial change is in Lemma 4.4, which now will take the following form. (α)
Lemma 7.3. Let (Xk )k∈{1,...,N 2 } be an orthonormal basis of (u(N ), ·,·u(N ) ). Let A, B be elements of MN (C). Then the following equality holds: 2
N
tr(AXk ) tr(BXk ) = −
k=1
1 tr(AB) + α 2 − 1 tr(A) tr(B) . 2 N
(15)
Assume that (X1 , . . . , XN 2 −1 ) forms an orthonormal basis of su(N ) endowed with the scalar product induced by ·,·u(N ) . Then 2 −1 N
tr(AXk ) tr(BXk ) = −
k=1
1 tr(AB) − tr(A) tr(B) . 2 N
(16)
It is this modification which gives rise to the new covariance introduced in Definition 7.1. 8. Joint fluctuations of the unitary Brownian motion at different times A natural generalization of our main result consists in considering several Brownian motions stopped at possibly different times. The goal of this section is to establish an analogue of Theorem 2.6 in this case. In order to state the result, we define a new covariance function. Definition 8.1. Let (A, τ ) be a C ∗ -probability space which carries three free multiplicative Brownian motions u, v, w which are mutually free. Let T1 , T2 0 be real numbers. Let f, g : U → R be two functions of C 1 (U). For all s ∈ [0, T1 ∧ T2 ], we set σT1 ,T2 ,s (f, g) = τ (f (us vT1 −s )g (us wT2 −s )). Then, we define T
1 ∧T2
σT1 ,T2 (f, g) =
T
1 ∧T2
τ f (us vT1 −s )g (us wT2 −s ) ds.
σT1 ,T2 ,s (f, g) ds = 0
0
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We have the following result. Theorem 8.2. Let n 1 be an integer. Let T1 , . . . , Tn 0 be real numbers. Let f1 , . . . , fn : U → R be n functions of C 1,1 (U). Let us define an n × n real non-negative symmetric matrix by setting ΣT1 ,...,Tn (f1 , . . . , fn ) = (σTi ,Tj (fi , fj ))i,j ∈{1,...,n} . Then, as N tends to infinity, the following convergence of random vectors in Rn holds in distribution: (d) N tr fi UN (Ti ) − E tr fi UN (Ti ) i∈{1,...,n} −−−−→ N 0, ΣT1 ,...,Tn (f1 , . . . , fn ) . N →∞
(17)
The proof of this result is very similar to the proof of Theorem 2.6 and, as in the previous section, we simply point out the small differences between the two. For the sake of convenience, let us assume T1 · · · Tn . Let f1 , . . . , fn : U → R be n functions of C 1,1 (U). We define for each i ∈ {1, . . . , n} a martingale indexed by [0, Tn ] by setting f MNi (t) = E tr fi UN (Ti ) FN,t . f
Observe that the martingale MNi is constant on the interval [Ti , Tn ]. Let us now define the vectorf f valued martingale QN (t) = N (MNi (t) − MNi (0))i∈{1,...,n} , so that the left hand-side of (17) is equal to QN (Tn ). The proof of Theorem 8.2 relies on an analogue of Proposition 3.1, for which we introduce the following notation: for all i, j ∈ {1, . . . , n} with i j and all s ∈ [0, Ti ), we set f ,fj
ENi
(s, U ) = N 2 ∇ PTi −s (tr fi ) (U ), ∇ PTj −s (tr fj ) (U ) u(N ) .
We state the following result for the two functions f1 and f2 . Proposition 8.3. With the notation introduced above, the following properties hold. f
f
1. For all t ∈ [0, T2 ], the quadratic covariation of the martingales N MN1 and N MN2 is given by f f N MN1 , N MN2 t
t∧T
1
=
f ,f2
EN1
s, UN (s) ds.
0
2. Assume that f1 and f2 are Lipschitz continuous. Then for all s ∈ [0, T1 ) and all U ∈ U(N ), f ,f |EN1 2 (s, U )| (f1 Lip + f2 Lip )2 . Moreover, if f1 and f2 belong to C 1 (U), then the following convergence holds: f ,f E EN1 2 s, UN (s) −−−−→ σT1 ,T2 ,s (f1 , f2 ). N →∞
3. Assume that f1 and f2 belong to C 1,1 (U). Then the following estimate holds: f ,f sup Var EN1 2 s, UN (s) = O N −2 .
s∈[0,T1 )
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The proof of this proposition is in no way different from the proof of Proposition 3.1. The f unique novelty is the fact that MN1 is constant on the interval [T1 , T2 ] so that the quadratic f f covariation MN1 , MN2 vanishes on this interval. Then, one deduces Theorem 8.2 from Proposition 8.3 just as one deduces Theorem 2.6 from Proposition 3.1. Let us mention that, in the case where the functions f1 , . . . , fn are polynomial, and given Theorem 2.6, the Gaussian character of the fluctuations in the case where the Brownian motions are stopped at different times is a consequence of the work of J.A. Mingo, R. Speicher and ´ P. Sniady [24,23] on the notion of second-order freeness and its specialization to the case of unitary matrices. Their work also provides one with a covariance function and it could be interesting to investigate the relation between our expression of what we call σT1 ,T2 and theirs. Another natural question which is answered by the theory of second-order freeness is that of the asymptotic fluctuations of random variables of the form tr p(UN (T1 ), . . . , UN (Tk )) where p is a non-commutative polynomial. It seems more difficult, although not hopeless, to apply our techniques to such functionals. 9. Behavior of the covariance for large time For any fixed N , the Markov process (UN (T ))T 0 converges in distribution, as T goes to infinity, to its invariant measure, which is the Haar measure on U(N ). In [8], P. Diaconis and S.N. Evans established a central limit theorem for Haar distributed unitary random matrices. In this section, we relate our result to theirs by comparing the limit as T tends to infinity of the covariance σT with the covariance which they have found. 9.1. Statement of the result of convergence In order to state the result of Diaconis and Evans, we need to introduce some notation. 1
Definition 9.1. Let H 2 (U) denote the space of functions that are square-integrable on U and such that 1 f 1 := 2 16π 2
|f (eiϕ ) − f (eiθ )|2
2
[0,2π]2
sin2 ( ϕ−θ 2 )
dϕ dθ < ∞.
We denote by ·,· 1 the inner product associated to this Hilbertian semi-norm. 2
For all f : U → C which is square-integrable and all j ∈ Z, we denote by aj (f ) = 1 (ξ )e−ij ξ dξ the j -th Fourier coefficient of f . One can check that f ∈ H 2 (U) if and U f only if j ∈Z |j ||aj (f )|2 is finite and that, in this case, 1 2π
f 21 = 2
2 |j | aj (f ) .
j ∈Z
The result of Diaconis and Evans states as follows.
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Theorem 9.2. (See 5.1 in [8].) For all N ∈ N, let MN be an N × N unitary matrix distributed 1 according to the Haar measure on U(N ). Let n 1 be an integer. For all f1 , . . . , fn ∈ H 2 (U), let Σ(f1 , . . . , fn ) be the n × n real non-negative symmetric matrix defined by Σ(f1 , . . . , fn ) = (fi , fj 1 )i,j =1,...,n . As N goes to infinity, the following convergence of random vectors in Rn 2 holds in distribution: (d) N tr fi (MN ) − E tr fi (MN ) i∈{1,...,n} −−−−→ N 0, Σ(f1 , . . . , fn ) . N →∞
In view of this result, it is natural to expect the covariance that we have introduced in Defi1 nition 2.4 to converge, as T tends to infinity, to the covariance given by the H 2 -scalar product. This is what the following result expresses. 1
Theorem 9.3. For all n 1 and all f1 , . . . , fn ∈ H 2 (U), ΣT (f1 , . . . , fn ) −−−−→ Σ(f1 , . . . , fn ). T →∞
Let us emphasize that ΣT (f1 , . . . , fn ) has only been defined so far for functions in C 1,1 (U). From this point on, we focus on extending the definition of the covariance to functions of the 1 space H 2 (U) and proving Theorem 9.3. 9.2. The main estimate In the sequel, (ut )t0 , (vt )t0 and (wt )t0 will be three multiplicative free Brownian motions, that are mutually free. For all T 0 and all k ∈ Z, let us denote by μk (T ) = τ (ukT ) the k-th moment of uT . Recall that, since uT has the same law as u∗T , one has, for all k ∈ Z, the equality μk (T ) = μ−k (T ). For each k 1, according to [3], μk (T ) is given by μk (T ) = e
− kT 2
k−1 k (−T )l k l−1 . l! l+1
(18)
l=0
Lemma 9.4. For all ε > 0, all T T0 (ε) = 2ε log(1 + 2ε ) and all k ∈ Z, one has
μk (T ) e−|k|T ( 12 −ε) . Proof. If k = 0 or ε 12 , the inequality is trivial. Moreover, since μk (T ) = μ−k (T ), it suffices to prove the inequality for k > 0. So, let us assume that ε 12 and k ∈ N∗ . It is easy to check that the expression (18) of μk (T ) is equivalent to the following: kT
e− 2 μk (T ) = 2ikπ
e
−kT z
1 1+ z
k dz,
where we integrate over a closed path of index 1 around the origin of the complex plane. If we choose as our contour the circle of radius 2ε centered at the origin, we get
T. Lévy, M. Maïda / Journal of Functional Analysis 259 (2010) 3163–3204 kT
e− 2 μk (T ) = 2ikπ
2π e
−kT 2ε eiθ
2 1 + iθ εe
k
3187
ε i eiθ dθ, 2
0
so that, provided T T0 (ε), k
μk (T ) ε e− kT2 ekT 2ε 1 + 2 e−kT ( 12 −ε) , 2k ε as expected.
2
We will denote by T0 a real large enough such that for all T T0 and all k ∈ Z, the inequality T |μk (T )| e−|k| 3 holds. One can check that 31 is large enough but we choose T0 = 32 for reasons which will soon become apparent. For all j, k ∈ Z and T > 0, we define
T τj,k (T ) =
τ (us vT −s )j (us wT −s )k ds.
(19)
0
Proposition 9.5. Set T0 = 32. For all T T0 and all (j, k) = (0, 0), the following inequality holds: |j +k|
− 4 T
|j |+|k|
τj,k (T ) 4 e + |j | + |k| T0 e− 4 (T −T0 ) . |j | + |k|
(20)
Moreover, if j = 0, then
− T4
|j |
τj,−j (T ) − 1 e + 2|j |T0 e− 2 (T −T0 ) .
|j | |j |
(21)
In particular, for all (j, k) = (0, 0), the following convergence holds: lim τj,k (T ) = δj +k,0
T →∞
1 . |j |
The proof of these estimates relies on a differential system satisfied by the functions τj,k . This differential system is a consequence of the free Itô calculus for free multiplicative Brownian motions. We state the form that we use, which is of interest on its own. Proposition 9.6. Let (ut )t0 be a free multiplicative Brownian motion on some non-commutative ∗-probability space (A, τ ). Let a1 , . . . , an ∈ A be random variables such that the two families {ut : t 0} and {a1 , . . . , an } are free. Finally, choose ε1 , . . . , εn ∈ {1, ∗}. Then
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d ε1 n τ ut a1 · · · uεt n an = − τ uεt 1 a1 · · · uεt n an dt 2 ε ε − 1εi =εj τ ai · · · aj −1 ut j τ aj · · · ai−1 ut i 1i<j n
+
1εi =εj τ (ai · · · aj −1 )τ (aj · · · ai−1 ),
1i<j n ε
ε
where for all 1 i < j n, we have used the shorthands ai · · · aj −1 for ai ut i+1 ai+1 · · · ut j −1 aj −1 ε ε and aj · · · ai−1 for aj ut j +1 aj +1 · · · uεt n an uεt 1 a1 · · · ut i−1 ai−1 . Proof. In [3], P. Biane showed that the free multiplicative Brownian motion (ut )t0 satisfies the free stochastic differential equation dut = iut dxt − 12 ut dt, where (xt )t0 is a free additive (Hermitian) Brownian motion. The identity above follows from this fact by free stochastic calculus, which has been developed by P. Biane and R. Speicher and is exposed in [4]. For the reader not familiar with free stochastic calculus, and without entering into the details, let us explain how the computation goes. The analogy with usual Itô calculus should be a helpful guide. The equation satisfied by ut implies that u∗t satisfies the equation du∗t = −i dxt u∗t − 12 u∗t dt. The time derivative of τ (uεt 1 a1 · · · uεt n an ) is computed formally by applying the formula n ε d τ uεt 1 a1 · · · uεt n an = τ uεt 1 a1 · · · dut i · · · uεt n an i=1
+
ε τ uεt 1 a1 · · · duεt i · · · dut j · · · uεt n an ,
1i<j n
together with the rules τ (a dt) = τ (a) dt,
τ (a dxt ) = 0,
τ (a dt b dt) = τ (a dt b dxt ) = 0,
and
τ (a dxt b dxt ) = τ (a)τ (b) dt valid for all a, b ∈ A, and using the invariance of τ under cyclic permutation of its arguments. 2 Lemma 9.7. The family (τj,k )(j,k)∈Z2 satisfies the following system of differential equations:
τ˙j,k (T ) = μj +k (T ) −
|j |−1 |j | + |k| |j | − l μl (T )τsgn(j )(|j |−l),k (T ) τj,k (T ) − 2 l=1
−
|k|−1
|k| − m μm (T )τj,sgn(k)(|k|−m) (T ),
m=1
where τ˙j,k is the derivative of the function T → τj,k (T ).
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Proof. This differential system follows easily from an application of Proposition 9.6 to the expression (19). 2 Before we turn to the proof of Proposition 9.5, let us state some elementary properties of the kT functions τj,k . For all k 0, define the polynomial Pk by the relation μk (T ) = e− 2 Pk (T ). For k < 0, define Pk = P−k . Lemma 9.8. For all j, k ∈ Z, the function τj,k is real-valued and satisfies τj,k = τk,j = τ−j,−k . Moreover, there exists a family of polynomials (Rj,k )j,k∈Z with rational coefficients such that the following equality holds: ∀j, k ∈ Z,
τj,k (T ) =
|j |+|k| 1j =0 δj +k,0 + e− 2 T Rj,k (T ). |j |
(22)
These polynomials are characterized by the fact that for all j, k ∈ Z, Rj,k (0) = 0 and R˙ j,k = 1j k0 Pj +k −
|j |−1
|k|−1 |j | − l Pl Rsgn(j )(|j |−l),k − |k| − m Pm Rj,sgn(k)(|k|−m) . (23)
l=1
m=1
Proof. The equalities τj,k = τ−j,−k = τk,j follow from the definition of τj,k , using the unitarity of u, v, w, the traciality of τ , and the fact that the families (u, v, w) and (u, w, v) have the same joint distribution. The fact that τj,k is real-valued can be proved by induction using the differential system stated in Lemma 9.7, or directly using the definition and the fact that (u, v, w) and (u∗ , v ∗ , w ∗ ) have the same distribution. The functions Rj,k defined by (22) are easily checked to satisfy the differential system (23) and, by induction, to be polynomial. 2 Proof of Proposition 9.5. Since the differential equation for τj,k expressed by Lemma 9.7 involves only indices (j , k ) such that |j | + |k | |j | + |k|, we will prove the conjunction of (20) and (21) by induction on |j | + |k|. It is understood that k = −j in (21). The symmetry properties of τj,k allow us to restrict ourselves to the two cases where j, k 0 and j > 0, k < 0. We may also assume that j + k 0. T The smallest possible value of |j | + |k| is 1. So, we start with τ1,0 (T ) = T μ1 (T ) = T e− 2 , T which is smaller than e− 4 for T larger than T0 . Hence, if |j | + |k| = 1 and T T0 , then T |τj,k (T )| e− 4 . This proves the result when |j | + |k| = 1. Let us consider now j and k and assume that (20) and (21) have been proved for all j , k such that |j | + |k | < |j | + |k|. Let us first assume that j + k = 0. In this case, define ρj,k (T ) = e
|j |+|k| 2 T
τj,k (T ).
Then Lemmas 9.4 and 9.7 and the induction hypothesis imply the inequality T |j |−1
T e−|sgn(j )(|j |−l)+k| 4 |j |+|k| |j +k| |j |+|k|
ρ˙j,k (T ) e 2 T e− 3 T + 4e 2 T |j | − l e−l 3 |j | − l + |k|
l=1
3190
T. Lévy, M. Maïda / Journal of Functional Analysis 259 (2010) 3163–3204 |j |−1 T T −T0 |j |+|k| |j | − 1 e−l 3 el 4 + |j | + |k| − 1 T0 e 4 (T +T0 ) l=1
+ 4e
|j |+|k| 2 T
|k|−1
m=1
T
−|j +sgn(k)(|k|−m)| 4 T e |k| − m e−m 3 |j | + |k| − m
|k|−1 T −T0 |j |+|k| T (T +T0 ) 4 + |j | + |k| − 1 T0 e |k| − 1 e−m 3 em 4 . m=1
Since |j | − l |j | − l + |k|, |k| − m |j | + |k| − m and e−l
T0 4
1, we find
|j |−1
|j |+|k| |j +k| |j |+|k| T T
ρ˙j,k (T ) e 2 T e− 3 T + 4e 2 T e−l 3 e−|sgn(j )(|j |−l)+k| 4 l=1
+ 4e
|j |+|k| 2 T
|k|−1
T
T
e−m 3 e−|j +sgn(k)(|k|−m)| 4
m=1 ∞ 2 |j |+|k| T (T +T0 ) 4 + 2 |j | + |k| − 1 T0 e e−l 12 .
(24)
l=1
If we are in the case where j, k 0, then we obtain immediately the estimate
|j |+|k|
ρ˙j,k (T ) e 2 T |j +k| × e− 3 T +
e− 12 T
T
1 − e− 12
− |j +k| T 2 |j |+|k| 8e 4 + 2 |j | + |k| − 1 T0 e− 4 (T −T0 ) . (25)
In the case where j > 0 and k < 0, the computation is slightly more complicated. In this case, let us also assume that j + k > 0, as we have indicated that it is possible to do. Then the estimation of the sum over m in (24) is the same as before, since j + sgn(k)(|k| − m) is positive for all values of m. However, the sign of sgn(j )(|j | − l) + k now depends on l. Thus, we bound the first sum over l by e−
|j +k| 4 T
j +k l=1
+∞
T
e−l 12 +
T
T
e−l 3 e−(l−(j +k)) 4 .
l=j +k+1 T
In the first term, we could actually have e−l 3 instead of e−l 12 but we are not seeking any optimality. In the second term, we write T
T
T
T
T
T
e−(l−(j +k)) 4 = e−(2l−(j +k)) 4 el 4 e−(j +k) 4 el 4 , and we find that the first sum over l in (24) is bounded by 2e− established that, when j > 0, k < 0 and j + k > 0,
|j +k| 4 T
e
T − 12
1−e
T − 12
. Finally, we have
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3191
|j |+|k|
ρ˙j,k (T ) e 2 T |j +k| × e− 3 T +
T
e− 12 T
1 − e− 12
− |j +k| T 2 |j |+|k| 12e 4 + 2 |j | + |k| − 1 T0 e− 4 (T −T0 ) .
In view of (25), the last estimate holds for all values of j and k. Our choice of T0 guarantees that for T T0 , the inequalities T
e
T − 12
+ 12
T
e− 12 1−e
T − 12
1
and
e− 12 1−e
T − 12
1 8
hold. Hence, we find
|j |+|k| |j +k| |j |+|k|
ρ˙j,k (T ) e 2 T e− 4 T + 1 |j | + |k| − 1 2 T0 e− 4 (T −T0 ) . 4 Integrating the last inequality from T0 on and using the fact that
|j |+|k| 2
− |j +k| 4
|j |+|k| 4 ,
we find
− |j +k| T
4 − |j |+|k| (T −T ) |j |+|k| |j |+|k| 0
ρj,k (T ) T0 e 2 T0 + e 2 T 4 e , + |j | + |k| − 1 T0 e 4 |j | + |k| from which it follows immediately that |j +k|
− 4 T
|j |+|k|
τj,k (T ) 4 e + |j | + |k| T0 e− 4 (T −T0 ) , |j | + |k|
which is the expected equality. Let us now treat the case where k = −j . As before, we can assume that j > 0. Setting ρj (T ) = e|j |T (τj,−j (T ) − |j1| ), we find, using the same estimates as before, that ∞ ∞
2 |j | T 1 −l 7T
ρ˙j (T ) 8e|j |T e 12 + 2 2|j | − 1 e 2 (T +T0 ) T0 e−l 12 . 2 l=1
l=1
It follows that
ρj (T ) T0 e|j |T0 + e|j |T
T
e− 2 2(|j | −
1 2)
|j | + 2|j | − 1 T0 e 2 (T +T0 ) ,
so that
− T2
|j |
τj,−j (T ) − 1 e + 2|j |T0 e− 2 (T −T0 ) ,
|j | |j | which is the expected inequality. This concludes the proof.
2
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9.3. Extension of the definition of the covariance 1
Proposition 9.9. Let f ∈ H 2 (U) be real-valued. The following properties hold. 1. For all T > T0 , j,k∈Z |j kaj (f )ak (f )τj,k (T )| < ∞. 2. limT →∞ j,k∈Z j kaj (f )ak (f )τj,k (T ) = −f 21 . 2
Proof. Choose an integer n 1. Then for all T T0 , Proposition 9.5 implies
j kaj (f )ak (f )τj,k (T ) |j |,|k|n
|j |,|k|n
+ T0
|j +k| 4|j k|
aj (f )ak (f ) e− 4 T |j | + |k|
|j |+|k| |j k| |j | + |k| aj (f )ak (f ) e− 4 (T −T0 )
|j |,|k|n
2
|j +k| |j k| aj (f )ak (f ) e− 4 T
|j |,|k|n
+2
|j |
|k|
|j |2 aj (f ) e− 4 (T −T0 ) |k|2 ak (f ) e− 4 (T −T0 )
|j |,|k|n
2
l∈Z
T
e−|l| 4
|j k| aj (f )ak (f ) + 2
|j |,|k|n,j +k=l
|j | T f 21 2 e−|l| 4 + 2 |j |3 e− 2 (T −T0 ) . 2
l∈Z
|j |
|j |2 aj (f ) e− 4 (T −T0 )
2
|j |n
j ∈Z
The first assertion follows. The second is a consequence of the second statement in Proposition 9.5 and the theorem of dominated convergence. 2 Proposition 9.9 above allows us to give a new definition of the covariance σT when T is large enough. 1
Definition 9.10. For all T > T0 and all f ∈ H 2 (U), we define σT (f, f ) = −
j kaj (f )ak (f )τj,k (T ).
j,k∈Z
Lemma 9.11. Let f be a function of C 1,1 (U). For all T > T0 , the two definitions (Definition 2.4 and Definition 9.10) of σT (f, f ) coincide. Proof. The series j ∈Z |aj (f )| is convergent, so that Sn (f )(eiξ ) = i |j |n j aj (f )eij ξ converges uniformly to f on U as n tends to infinity. Therefore, starting from Definition 2.4,
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T j k σT (f, f ) = − τ j kaj (f )ak (f )(us vT −s ) (us wT −s ) ds. j,k∈Z
0
As the processes are unitary and for all T 0,
j ∈Z |j ||aj (f )| < ∞,
σT (f, f ) = −
we get by dominated convergence that,
j kaj (f )ak (f )τj,k (T ),
j,k∈Z
as expected.
2
Theorem 9.3 is now a straightforward consequence of the polarization of Definition 9.10 and Proposition 9.9. Remark 9.12. Let us emphasize that Proposition 9.5 implies that, for all ε > 0 and all T > T0 , the following series converges:
1−ε
τj,k (T ) 2 < +∞. |j | + |k| j,k∈Z
Hence, for all T > T0 , the equality KT eiθ , eiϕ =
eij θ eikϕ τj,k (T )
(j,k)∈Z2 \{(0,0)}
defines KT as a square-integrable real-valued function on U2 and, for all ε > 0 and f, g ∈ 3 H 4 +ε (U), one has the equality
σT (f, g) =
dθ dϕ f eiθ KT eiθ , eiϕ g eiϕ . 4π 2
[0,2π]2
We conclude this study of the covariance by showing some puzzling numerical experiments (see Fig. 1). It is striking on these pictures that the behavior of the covariance σT (f, g) is complicated and interesting for small T , and much simpler for large T . It is thus not surprising that we have been only able to analyze σT for large T . 10. Combinatorial approaches 10.1. The differential system satisfied by the τj,k The differential system satisfied by the functions τj,k (Lemma 9.7) can be interpreted, at least when j and k have the same sign, in terms of enumeration of walks on the symmetric group, in the same vein as the computations made by one of us in [19]. This is what we explain in this section.
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Fig. 1. For all k 1, let us define sk (eiθ ) = sin(kθ) and ck (eiθ ) = cos(kθ). The pictures above are the graphs of the following functions of T for T ∈ [0, 6]. Top left: σT (sk , sk ) and σT (ck , ck ) for k ∈ {1, . . . , 8}. Bottom left: μk (T ) for k ∈ {1, . . . , 6}. Top center: σT (sk , sk+1 ) for k ∈ {1, . . . , 15}. Bottom center: σT (ck , ck+1 ) for k ∈ {1, . . . , 15}. Top right: σT (sk , sk+3 ) for k ∈ {1, 4, 7, 10, 13}. Bottom right: σT (sk , sk+2 ) for odd k ∈ {1, . . . , 13}.
Fix j 1. We consider the Cayley graph on the symmetric group Sj generated by all transpositions. The vertices of this graph are the elements of Sj and two permutations σ1 and σ2 are joined by an edge if and only if σ1 σ2−1 is a transposition. A finite sequence (σ0 , . . . , σn ) of permutations such that σi and σi+1 are joined by an edge for all i ∈ {0, . . . , n − 1} is called a path of length n. The distance between two permutations is the length of the shortest path that joins them. We call defect of a path the number of steps in the path which increase the distance to identity. Heuristically, one can understand the defect as follows: each time we compose a permutation with a transposition, either we cut a cycle into two pieces and this is a step which decreases the distance to identity, or we coalesce two cycles into a bigger one and this is a step which increases the distance to identity. The defect counts the number of steps of the second kind. For any σ ∈ Sj , and any two integers n, d 0, we denote by S(σ, n, d) the number of paths in the Cayley graph of Sj starting from σ, of length n and with defect d. The interested reader can find more details about those combinatorial objects in [19]. Let j, k 1. If σ ∈ Sj and τ ∈ Sk , we denote by σ × τ the concatenation of σ and τ, that is the permutation in Sj +k such that σ × τ (i) = σ (i) if 1 i j and σ × τ (i) = τ (i − j ) + j if j + 1 i j + k. From Theorem 3.3 in [19], it follows that for all T 0, E tr UN (T )j tr UN (T )k − E tr UN (T )j E tr UN (T )k ∞ (−T )n −(j +k) T2 =e S (1 · · · j ) × (1 · · · k), n, d n!N 2d n,d=0
−
∞ n1 ,n2 ,d1 ,d2 =0
(−T )n1 +n2 S (1 · · · j ), n1 , d1 S (1 · · · k), n2 , d2 . n1 !n2 !N 2(d1 +d2 )
(26)
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Moreover, for all T 0, we recall that all the expansions involved converge uniformly on (N, T ) ∈ N × [0, T ]. Using this equality, it is for example easy to check that lim E tr UN (T )j tr UN (T )k − E tr UN (T )j E tr UN (T )k
N →∞
=e
−(j +k) T2
∞ (−T )n n=0
n!
S (1 · · · j ) × (1 · · · k), n, 0
n n S (1 · · · j ), n1 , 0 S (1 · · · k), n − n1 , 0 = 0, − n1 n1 =0
where the last equality comes from Proposition 5.3 of [19]. Each term of the sum is indeed zero and heuristically, it means that a path without defect starting from (1 · · · j ) × (1 · · · k) is simply obtained by “shuffling” two paths without defect from each of the two cycles in their respective symmetric group. More interesting for us is the fact we can also deduce from (26) that
(def)
κj,k (T ) =
lim N 2 E tr UN (T )j tr UN (T )k − E tr UN (T )j E tr UN (T )k
N →∞
= e
−(j +k) T2
∞ (−T )n n=0
n!
S (1 · · · j ) × (1 · · · k), n, 1 ,
(27)
where, σ ∈ Sj , τ ∈ Sk and n 1 being given, we use the notation S (σ × τ, n, 1) = S(σ × τ, n, 1) n n S(σ, n1 , 1)S(τ, n − n1 , 0) + S(σ, n1 , 0)S(τ, n − n1 , 1) . − n1 n1 =0
Thus defined, S (σ × τ, n, 1) is the number of paths of length n starting from σ × τ such that the unique step which increases the distance to the identity is the multiplication by a transposition which exchanges an element of {1, . . . , j } with an element of {j + 1, . . . , j + k}. Thus, heuristically, the unique step which is a coalescence is a coalescence between σ and τ . Our goal is now to show the following combinatorial identity Proposition 10.1. For any integers j, k 1, and n 0, we have S (1 · · · j ) × (1 · · · k), n + 1, 1 = j kS (1 · · · j + k), n, 0
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+j
j −1 n n S (1 · · · l), p, 0 S (1 · · · j − l) × (1 · · · k), n − p, 1 p l=1 p=0
+k
k−1 n n S (1 · · · m), q, 0 S (1 · · · j ) × (1 · · · k − m), n − q, 1 . q
m=1 q=0
The combinatorial interpretation of this identity is the following: let us consider a path of length n + 1 from (1 · · · j ) × (1 · · · k) whose unique step increasing the distance to identity is a true coalescence between the two cycles. The first step of such a path can be of three kinds, corresponding respectively to the three terms of the right hand-side: • Either it coalesces the cycles, creating a (j + k)-cycle, and this can be done by choosing an element in each cycle. Then the path can be completed by any path of length n without defect from a (j + k)-cycle. • Either it cuts the cycle (1 · · · j ) into two cycles, one of length l that will then be cut p times without being affected by the coalescence and another of length j − l which contains the element which will be exchanged with an element of {j + 1, . . . , j + k} during the coalescing step. • Either, symmetrically, it cuts the cycle (1 · · · k). We will hereafter propose a rigorous proof of this identity through the free stochastic calculus tools introduced above in the paper. It should be noted that the combinatorics which we investigate here is related to that of annular noncrossing partitions introduced by J.A. Mingo and A. Nica [22]. Proof of Proposition 10.1. Let the integers j, k 1 and the real T 0 be fixed. If we consider the quantities κj,k (T ) as defined in (27), if we denote, for any r ∈ Z, by fr : U → C the function given by fr (z) = zr , then, from Definition 2.4 and Theorem 2.6, we get κj,k (T ) = σT (fj , fk ) and from (19), it can be reexpressed as κj,k (T ) = −j kτj,k (T ). Now, from Lemma 9.7, we get immediately κ˙ j,k (T ) = −j kμj +k (T ) − −j
j −1
j +k κj,k (T ) 2
μl (T )σj −l,k (T ) − k
l=1
k−1
μm (T )κj,k−m (T ),
m=1
so that we get immediately the announced result, as we know from [19] that, for any r ∈ N∗ , T
μr (T ) = e−r 2
∞ (−T )n S (1 · · · r), n, 0 n! n=0
and from (27) that κ˙ j,k (T ) = −
∞ n T (−T ) j +k κj,k (T ) − e−(j +k) 2 S (1 · · · j ) × (1 · · · k), n + 1, 1 . 2 n! n=0
2
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10.2. Mixed moments of special unitary matrices In principle, any computation involving functions invariant by conjugation on the unitary group can be performed by using harmonic analysis, that is, the representation theory of the unitary group. In this section, we use this approach to prove the following formula, which yields for each N 3 an explicit expression for the covariance of traces of powers of the Brownian motion on SU(N ). With the help of Section 7, it is easy to deduce the analogous result for the Brownian motion on U(N ). Theorem 10.2. Let N 3 be an integer. Consider, on SU(N ), the Brownian motion (VN (t))t0 associated with the scalar product X, Y su(N ) = N Tr(X ∗ Y ) on su(N ). Let n and m be positive integers. Assume that N n + m + 1. Then E Tr VN (t)n Tr VN (t)m −(n+m) t − n(n−1)+m(m−1) t − (n−m)
2 t
2 N 2 N2 2 = nδn,m + (−1)n+m e n−1 m−1 N + r1 −nr2 t m − 1 N + r2 r1 +r2 −nr1 2t n − 1 2 (−1) e × e r1 r2 n m r1 =0 r2 =0 (N + r1 + r2 + 1)(N − n − m + r1 + r2 + 1) . × (N − n + r1 + r2 + 1)(N − m + r1 + r2 + 1)
The basic strategy for the proof is to expand the heat kernel and the traces in the basis of Schur functions, and then to use the multiplication rules for Schur functions and their orthogonality properties. The multiplication rules are expressed by the Littlewood–Richardson formula and they are rather complicated. Fortunately, in the present situation, the Young diagrams which occur are simple enough for the computation to be tractable. Let us recall the fundamental facts about Schur functions. Details can be found in [10]. A Young diagram is a non-increasing sequence of non-negative integers. If λ = (λ1 · · · λk > 0) is such a sequence, we call k the length of λ and denote it by (λ). The set of Young diagrams of length at most k is denoted by Nk↓ . We draw Young diagrams downwards in rows, according to the convention illustrated by the left part of Fig. 2.
Fig. 2. The Young diagram on the left is (7, 6, 3, 3, 1), which we also denote by 7 6 32 1. The diagram on the right is ηn,r = (n − 1)1r .
The Schur function sλ is a symmetric function which, when evaluated on strictly less than (λ) variables, yields 0. Whenever (λ) N , the function sλ is well defined and non-zero on SU(N ). Its value sλ (IN ) at the identity matrix in particular is a positive integer, which is the dimension of the irreducible representation of SU(N ) of which sλ is the character. Another number attached
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to λ will play an important role for us, which is the non-negative real number c(λ) such that sλ = −c(λ)sλ . It happens that distinct Young diagrams yield the same function on SU(N ): if λ and μ are Young diagrams such that (λ), (μ) N , then sλ = sμ if and only if there exists l ∈ Z such that λ = μ + (l, . . . , l) = μ + l N . In fact, if ρλ and ρμ are the representations of U(N ) corresponding to λ and μ, then ρλ = ρμ ⊗ det⊗l and the restrictions of these representations to SU(N ) are equal. Finally, we need to use the decomposition of the heat kernel and the function U → Tr(U n ) in terms of Schur functions. For the latter, we introduce a class of Young diagrams called hooks. For all n 1 and all r ∈ {0, . . . , n − 1}, we define ηn,r = (n − r, 1, . . . , 1) = (n − r)1r , ! r
which is depicted on the right part of Fig. 2. The heat kernel at time t on SU(N ) is the density, denoted by Qt : SU(N ) → R, of the distribution of VN (t) with respect to the Haar measure. Proposition 10.3. Choose N 1 and U ∈ SU(N ). Then the following equalities hold. r 1. For all n 1, Tr(U n ) = n−1 r=0 (−1) sηn,r (U ). c(λ) 2. For all t 0, Qt (U ) = λ∈NN−1 e− 2 t sλ (IN )sλ (U ). ↓
The proof of the first equality can be found in [21], the proof of the second in [20]. The expectation that we want to compute in order to prove Theorem 10.2 is thus equal to n−1 m−1 c(λ) e− 2 t sλ (IN ) (−1)r1 +r2 E Tr VN (t)n Tr VN (t)m = r1 =0 r2 =0
λ∈NN−1 ↓
×
sλ (U )sηn,r1 (U )sηm,r2 (U ) dU.
SU(N )
The multiplication of Schur functions is governed by the Littlewood–Richardson formula, γ which describes a non-negative integer Nα,β for each triple of Young diagrams α, β, γ , in such a way that sα sβ =
γ
Nα,β sγ .
γ
Using these coefficients, the integral above can be rewritten as
sλ (U )sηn,r1 (U )sηm,r2 (U ) dU =
γ
SU(N )
=
γ
γ
Nλ,ηn,r
sγ (U )sηm,r2 (U ) dU
1
SU(N )
γ Nλ,ηn,r 1
1γ =ηm,r
2 +l
l0
N
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=
ηm,r +l N
Nλ,ηn,r2
1
l0
3199
.
Thus, we need to compute n−1 m−1 r1 =0 r2 =0
(−1)r1 +r2
ηm,r +l N
Nλ,ηn,r2
1
l0
(28)
.
It turns out that a slightly more general computation is simpler to perform: we compute the β Littlewood–Richardson coefficient Nα,ηn,r for all α, β and all n, r. Let us introduce some notation. Let α = (α1 , . . .) and β = (β1 , . . .) be two Young diagrams. Set |α| = i αi and |β| = i βi . We assume that α ⊂ β, that is, αi βi for all i. Then we denote by β/α the set of boxes of the graphical representation of β which are not contained in α. We say that a subset of β/α is connected if one can go from any box to any other inside this subset by a path which jumps from a box to another only when they share an edge. We denote by k(β/α) the number of connected components of β/α. Also, we define v(β/α) as the number of boxes of β/α which are such that the box located immediately above also belongs to β/α. Alternatively, this is the number of distinct occurrences of the motif formed by two consecutive boxes one above the other in β/α. Our main combinatorial result is the following. β
Proposition 10.4. Let α and β be two Young diagrams. Let ηn,r be a hook. Then Nα,ηn,r is nonzero if and only if the following conditions are satisfied: α ⊂ β, |β| = |α| + n, β/α contains no β 2 × 2 square, and v(β/α) r v(β/α) + k(β/α) − 1. In this case, Nα,ηn,r = k(β/α)−1 r−v(β/α) . β
Proof. According to the Littlewood–Richardson rule, Nα,ηn,r is the number of strict expansions of α by ηn,r which yield β, that is, the number of fillings of β/α with the boxes of ηn,r such that the following conditions are satisfied: 1. for all s 1, the union of α and the boxes of β/α filled by the first s rows of ηn,r is a Young diagram, 2. no two boxes of the first row of ηn,r are put in the same column of β/α, 3. if one goes through the boxes of β/α from right to left and from top to bottom, writing for each box the number of the row of ηn,r from which is issued the box which has been used to fill it, one obtains a sequence which starts with 1, and in which all other numbers 2, . . . , r appear, not necessarily consecutively, in this order. It is important to notice that, according to the third rule, a strict expansion of α by a hook which yields β is completely characterized by the set of boxes of β/α which are filled by boxes issued from the first row of the hook. We say for short that these boxes of β/α are filled by the first row. The first two conditions α ⊂ β and |β| = |α| + n are obviously implied by this rule. A less trivial implication is that there cannot exist a strict expansion if β/α contains a 2 × 2 square. Indeed, by the first two rules, the bottom-left box of the square cannot be filled by the first row
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Fig. 3. White boxes must be filled by boxes issued from the first row of ηn,r . Gray boxes cannot. The black box may or may not, except if this snake is the topmost connected component of β/α, in which case it must also be filled by a box issued from the first row of ηn,r .
and the bottom-right box must then be filled with a boxed issued from a strictly lower (in the graphical representation) row of ηn,r . This contradicts the third rule. Let us assume that β/α contains no 2 × 2 square. Then each connected component of β/α is a “snake” (see Fig. 3). Any box of such a snake which has a box on its right must be filled by the first row. These boxes are the white boxes in Fig. 3. Any box located below a white box cannot be filled by the first row. These boxes are the gray boxes in Fig. 3. Only one box is not in one of these two cases, the top-right box of the snake. In the topmost connected component of β/α the third rule implies that this box must be filled by the first row. Finally, if the first three conditions are satisfied, then β/α contains one box in each connected component, except the topmost one, which can either be filled by the first row or not. The minimal number of boxes which are not filled by the first row is the number of gray boxes, which we have denoted by v(β/α). This is the minimal value of r for which there exists a strict expansion of α by ηn,r which yields β. Moreover, for this value of r, the expansion is unique, since the boxes filled by the first row are completely determined. Similarly, the maximal value of r is v(β/α) + k(β/α) − 1. For r between these two bounds, there are exactly as many expansions as there are choices of which snakes have their top-right box filled by the first row. There are thus k(β/α)−1 r−v(β/α) such expansions. 2 Corollary 10.5. Let α and β be two Young diagrams. Choose n 1. Then n−1
β (−1)r Nα,η = (−1)v(β/α) n,r
r=0
if α ⊂ β, |β| = |α| + n, β/α contains no 2 × 2 square and is connected. Otherwise, it is equal to 0. β
Proof. If the first three conditions are not satisfied, then Nα,ηn,r = 0 for all r = 0 . . . n − 1. Let us assume that they are satisfied. Then, by the previous proposition, the sum above is equal to v(β/α)+k(β/α)−1
(−1)r
r=v(β/α)
k(β/α) − 1 , r − v(β/α)
which is equal to 0 unless k(β/α) = 1. In this case, only one term of the sum is non-zero, for r = v(β/α). 2
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Fig. 4. The diagram λN m,r2 ,n,r1 .
We apply now this result when β is of the sum of a hook and a rectangle. Lemma 10.6. Consider n m 1, r2 ∈ {0, . . . , m − 1}, and N m + n. For all r1 ∈ {0, . . . , n − 1}, define r2 N −r1 −r2 −2 λN (n − r1 − 1)r1 . m,r2 ,n,r1 = (n − r1 + m − r2 )(n − r1 + 1) (n − r1 ) −1 and all l 1, (See Fig. 4.) Then, for all λ ∈ NN ↓ n−1
ηm,r +l N
(−1)r1 Nλ,ηn,r2
r1 =0
1
=
(−1)n−l 0
if l ∈ {1, . . . , n} and λ = λN m,r2 ,n,n−l , otherwise.
Moreover, when l ∈ {1, . . . , n}, the only non-zero term of the sum is the term corresponding to r1 = n − l. ηm,r Finally, if n = m, then Nλ,ηn,r2 = 1 if r1 = r2 and λ is the empty diagram, and 0 otherwise. 1
Proof. Let us first consider the case n > m. In this case, according to Corollary 10.5, in order for the sum to be non-zero, λ must be a Young diagram of length at most N − 1, contained in ηm,r2 + l N , such that (ηm,r2 + l N )/λ contains no 2 × 2 square and is connected. Since n > m, l must be positive, so that the diagram ηm,r2 + l N has length N whereas λ has length at most N − 1. Thus, the N -th row of (ηm,r2 + l N )/λ is not empty, it has actually length l. In particular, |ηm,r2 + l N | − |λ| l. If l > n, all the Littlewood–Richardson coefficients appearing in the sum are zero. Otherwise, if l n, there is exactly one way to choose λ a subdiagram of ηm,r2 + l N such that all conditions are satisfied: it is λ = λN m,r2 ,n,n−l . When n = m, nothing changes for l 1. However, the sum may be non-zero even for l = 0. ηm,r The diagram λ must be the empty diagram and it is easy to check that N∅,ηn,r2 = δn,m δr1 ,r2 . 2 1
We can now go on to compute (28). We find the following result.
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Proposition 10.7. Let N , n and m be three positive integers. Assume that n m and N n + m + 1. Then n−1 m−1 c(λN m,r2 ,n,r1 ) n r1 +r2 − t m 2 E Tr VN (t) Tr VN (t) = nδn,m + (−1) e sλNm,r
2 ,n,r1
r1 =0 r2 =0
(IN ).
Proof. We have n−1 m−1
(−1)r1 +r2
r1 =0 r2 =0
ηm,r +l N
Nλ,ηn,r2
1
l0
= nδm,n 1λ=∅ +
m−1
(−1)r2
r2 =0
= nδm,n 1λ=∅ +
m,r2 ,n,n−l
l=1
n−1 m−1 r1 =0 r2 =0
The claimed equality follows easily.
n (−1)n−l 1λ=λN
(−1)r1 +r2 1λ=λNm,r
2 ,n,r1
.
2
In order to prove Theorem 10.2, there remains to compute c(λN (IN ). m,r2 ,n,r1 ) and sλN m,r2 ,n,r1 This is by no means complicated but slightly tedious. We recall the general formulae, give the results in this particular case and invite the reader to check them if she/he feels inclined to do so. Lemma 10.8. Consider n, m 1, r1 ∈ {0, . . . , n − 1} and r2 ∈ {0, . . . , m − 1}. Then the following identities hold. m(m − 2r2 − 1) (n − m)2 n(n − 2r1 − 1) + m + − = n + , c λN m,r2 ,n,r1 N N N2 (N − r1 − r2 − 1)(N + n + m − r1 − r2 − 1) sλNm,r ,n,r (IN ) = 2 1 (N + n − r1 − r2 − 1)(N + m − r1 − r2 − 1) n − 1 N + n − r1 − 1 m − 1 N + m − r2 − 1 . × n r2 m r1 Proof. The general formulae are the following: for all α ∈ NN ↓ , one has 1 c(α) = N
N i=1
αi2
+
1i<j N
on one hand and, using the notation (λ) = . . . , 1, 0), sα (IN ) = on the other hand.
1 (αi − αj ) − 2 N
"
1i<j N (λi
N
2 αi
i=1
− λj ) and δ = (N − 1, N − 2,
(α + δ) (α)
2
Theorem 10.2 now easily follows from Proposition 10.7 and Lemma 10.8.
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References [1] G.W. Anderson, O. Zeitouni, A CLT for a band matrix model, Probab. Theory Related Fields 134 (2) (2006) 283– 338. [2] F. Benaych-Georges, Central limit theorems for the Brownian motion on large unitary groups, preprint, arXiv:0904.1681, 2009. [3] P. Biane, Free Brownian motion, free stochastic calculus and random matrices, in: Free Probability Theory, Waterloo, ON, 1995, in: Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 1–19. [4] P. Biane, R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (3) (1998) 373–409. [5] T. Cabanal-Duvillard, Fluctuations de la loi empirique de grandes matrices aléatoires, Ann. Inst. H. Poincaré Probab. Statist. 37 (3) (2001) 373–402. [6] S. Chatterjee, Fluctuations of eigenvalues and second order Poincaré inequalities, Probab. Theory Related Fields 143 (1–2) (2009) 1–40. ´ [7] B. Collins, J.A. Mingo, P. Sniady, R. Speicher, Second order freeness and fluctuations of random matrices. III. Higher order freeness and free cumulants, Doc. Math. 12 (2007) 1–70 (electronic). [8] P. Diaconis, S.N. Evans, Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (7) (2001) 2615–2633 (electronic). [9] M.P. do Carmo, Riemannian Geometry, Math. Theory Appl., Birkhäuser Boston Inc., Boston, MA, 1992, translated from the second Portuguese edition by Francis Flaherty. [10] W. Fulton, J. Harris, Representation Theory. A First Course, Grad. Texts Math. Read. Math., vol. 129, SpringerVerlag, New York, 1991. [11] D.J. Gross, A. Matytsin, Some properties of large-N two-dimensional Yang–Mills theory, Nuclear Phys. B 437 (3) (1995) 541–584. [12] D.J. Gross, W. Taylor IV, Twists and Wilson loops in the string theory of two-dimensional QCD, Nuclear Phys. B 403 (1–2) (1993) 395–449. [13] A. Guionnet, Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices, Ann. Inst. H. Poincaré Probab. Statist. 38 (3) (2002) 341–384. [14] A. Guionnet, O. Zeitouni, Concentration of the spectral measure for large matrices, Electron. Comm. Probab. 5 (2000) 119–136 (electronic). [15] W. Hachem, P. Loubaton, J. Najim, A clt for information-theoretic statistics of gram random matrices with a given variance profile, Ann. Appl. Probab. 18 (6) (2008) 2071–2130. [16] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second ed., North-Holland Math. Library, vol. 24, North-Holland Publishing Co., Amsterdam, 1989. [17] K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1) (1998) 151–204. [18] V.A. Kazakov, I.K. Kostov, Nonlinear strings in two-dimensional U (∞) gauge theory, Nuclear Phys. B 176 (1) (1980) 199–215. [19] T. Lévy, Schur–Weyl duality and the heat kernel measure on the unitary group, Adv. Math. 218 (2) (2008) 537–575. [20] M. Liao, Lévy Processes in Lie Groups, Cambridge Tracts in Math., vol. 162, Cambridge University Press, Cambridge, 2004. [21] I.G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., Oxford Math. Monogr., The Clarendon Press/Oxford University Press, New York, 1995, with contributions by A. Zelevinsky, Oxford Science Publications. [22] J.A. Mingo, A. Nica, Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices, Int. Math. Res. Not. 28 (2004) 1413–1460. ´ [23] J.A. Mingo, P. Sniady, R. Speicher, Second order freeness and fluctuations of random matrices. II. Unitary random matrices, Adv. Math. 209 (1) (2007) 212–240. [24] J.A. Mingo, R. Speicher, Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces, J. Funct. Anal. 235 (1) (2006) 226–270. [25] L.A. Pastur, A simple approach to the global regime of Gaussian ensembles of random matrices, Ukraïn. Mat. Zh. 57 (6) (2005) 790–817. [26] L.A. Pastur, A. Lytova, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, preprint, http://arxiv.org/abs/0809.4698, 2008. [27] A.N. Sengupta, Traces in two-dimensional QCD: the large-N limit, in: Sergio Albeverio, Matilde Marcolli, Sylvie Paycha, Jorge Plazas (Eds.), Traces in Geometry, Number Theory and Quantum Fields, Vieweg, 2008. [28] M. Shcherbina, Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models, Zh. Mat. Fiz. Anal. Geom. 4 (1) (2008) 171–195, 204.
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[29] Y. Sinai, A. Soshnikov, Central limit theorem for traces of large random symmetric matrices with independent matrix elements, Bol. Soc. Brasil. Mat. (N.S.) 29 (1) (1998) 1–24. [30] D. Voiculescu, Lectures on free probability theory, in: Lectures on Probability Theory and Statistics, Saint-Flour, 1998, in: Lecture Notes in Math., vol. 1738, Springer-Verlag, Berlin, 2000, pp. 279–349.
Journal of Functional Analysis 259 (2010) 3205–3229 www.elsevier.com/locate/jfa
Endpoint Lp → Lq bounds for integration along certain polynomial curves Betsy Stovall 1 UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, United States Received 26 May 2009; accepted 11 August 2010 Available online 6 September 2010 Communicated by J. Bourgain
Abstract We establish strong-type endpoint Lp (Rd ) → Lq (Rd ) bounds for the operator given by convolution with affine arclength measure on polynomial curves for d 4. The bounds established depend only on the dimension d and the degree of the polynomial. © 2010 Elsevier Inc. All rights reserved. Keywords: Generalized Radon transform; Affine arclength; Polynomial curves
1. Introduction Let P : R → Rd be a polynomial and define the operator TP by TP f (x) :=
f x − P (s) dσP (s),
(1)
I
where I is an interval and dσP represents affine arclength measure along P , 2/d(d+1) ds. dσP (s) := det P (s), P (s), . . . , P (d) (s) E-mail address:
[email protected]. 1 The author was supported in part by NSF grants DMS-040126 and DMS-0901569.
0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.008
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The goal of this article is to establish Lp → Lq bounds for TP in the full conjectured range of exponents in dimensions d 4 (together with a slight improvement in Lorentz spaces). The conjectured range of (p, q) depends only on d, and our bounds on the operator norms depend only on p, q, d, and the degree of P . This result has already been established in dimension 2 by Oberlin [23] and in dimension 3 by Dendrinos, Laghi and Wright [9]. Theorem 1. Let d 4, let P : R → Rd be a polynomial of degree N , and let TP be the operator d+1 d p q defined in (1). Let pd := d+1 2 and qd := 2 d−1 . Then TP maps L → L if (p, q) = (pd , qd ) or (qd , pd ), with bounds depending only on d, N . Moreover, TP maps the Lorentz space Lpd ,u (Rd ) boundedly into Lqd ,v (Rd ) and Lqd ,v (Rd ) into Lpd ,u (Rd ) whenever u < qd , v > pd , and u < v. In the case when I has infinite length and dσP ≡ 0, this theorem is sharp up to Lorentz space endpoints. A proof of this in the case when P (t) = (t, t 2 , . . . , t d ) is given in [27]. If I has finite length, then TP is easily seen to be bounded from L1 to L1 and L∞ to L∞ (though finite Lp → Lq bounds depending only on d, N cannot hold when (p −1 , q −1 ) lies off the line segment joining (pd−1 , qd−1 ) and (1 − qd−1 , 1 − pd−1 )). By interpolation with the bounds established in Theorem 1, we obtain nearly sharp bounds in this case as well. We now give a little of the history of this problem. It was observed by Drury in [11] that the affine arclength dσP is in some ways a more natural choice than euclidean arclength for averages along P and for the restriction of the Fourier transform to P . For one, affine (like euclidean) arclength measure is easily seen to be parametrization independent; that is, if ψ : R → R is a diffeomorphism, then 2/d(d+1) ds. dσP ◦ψ (s) = ψ (s)det P ψ(s) , . . . , P (d) ψ(s) For two, affine (unlike euclidean) arclength measure compensates for degeneracies in the curve P by vanishing where the torsion vanishes (for instance at the cusp in the curve parametrized by (t 2 , t 3 )). This results in an Lp → Lq mapping theory in which p and q are independent of the curve P . Finally, as its name implies, affine (again unlike euclidean) arclength measure behaves nicely under affine transformations of Rd . This property will allow us to prove bounds on the operator norms of T that depend only on d and the degree of P . A general discussion of affine arclength may be found in [16]. Drury [11] established Lp → Lq bounds for the optimal range of p, q in dimension 2 for P (t) = (t, p(t)) satisfying certain regularity conditions (though not necessarily polynomial). Later, Oberlin [23] strengthened Drury’s result in the polynomial case by establishing optimal Lp → Lq bounds for arbitrary polynomials in dimension 2 and for polynomials of the form P (t) = (t, p1 (t), p2 (t)) in dimension 3. In dimension 2, Oberlin established bounds for the operator norms of TP depending only on p, q, and the degree of P ; he remarked that such invariant bounds were likely to hold in higher dimensions as well. Recently, Dendrinos, Laghi and Wright [9] used a geometric inequality established in [10] to treat the general three-dimensional case (also establishing invariant bounds). Other results in a similar vein are due to Choi in [4,3] and by Pan in [24]. A recent preprint of Oberlin [19] establishes bounds for TP in certain non-polynomial cases for d = 2, 3, 4. More broadly, there is a growing body of literature on the so-called generalized Radon transforms, operators defined by integration over families of submanifolds of Rd . We mention here a few articles which are particularly closely related to this one; by no means is this an exhaustive bibliography. In [6], Christ developed a combinatorial technique that he used to establish optimal
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restricted weak type estimates for convolution with affine arclength measure on the moment curve (t, t 2 , . . . , t d ) for d 4 (strong type bounds had been proved in lower dimensions by Littman [18] and Oberlin [20–22] via different techniques). Later on, Tao and Wright used ideas from [6] together with several new ideas to establish optimal (up to Lebesgue space endpoint) bounds for averages over families of smooth curves in [28]. This article is one of a few recent efforts toward establishing the endpoint bounds in special cases of the Tao–Wright theorem. In [15], Gressman proved the restricted weak-type estimate at the endpoint in the polynomial case of Tao and Wright’s theorem. In [7], Christ showed that it was possible to use arguments similar to those in [6] to prove strong-type estimates. We mention three subsequent applications of this technique (this article being a fourth), namely [9] wherein Dendrinos, Laghi and Wright consider TP in dimension 3, [17] in which Laghi proves endpoint bounds for a restricted X-ray transform, and the author’s [27] which establishes strong-type bounds for convolution with affine arclength measure along the moment curve in dimensions d 4 (as mentioned above, low-dimensional results were already known). In this article, we will use techniques from [6,7,9,27], mentioned above, as well as from [14], in which Gressman established restricted weak-type bounds for convolution with certain measures along polynomial curves whose entries are monomials. Related to our problem is the restriction of the Fourier transform to curves with affine arclength. In this case, it is conjectured that uniform Lp → Lq bounds hold, and though considerable progress has been made, the conjecture has not been completely resolved. A detailed history of this problem may be found in [10], for instance, along with some recent results. Other articles in this vein include the recent work of Bak, Oberlin and Seeger [2,1] and the less recent work of Drury and Marshall [12,13] and of Sjölin [26] (and of course Drury’s [11]). Notation. In this article, we will write A B when A CB with the constant C depending only on the ambient dimension and the degree of the polynomial P . We will also use the accompanying notation A ∼ B. We will occasionally use the ‘big O’ notation, writing A = O(B) instead of A B. In addition, if 1 p ∞, p refers to the exponent dual to p (which satisfies 1 = p −1 + p −1 ). 2. Initial simplifications and a key theorem As mentioned above, our problem is closely related to the problem of the restriction of the Fourier transform to curves with affine arclength. One of the main tools used here and in [9] is a theorem which was originally proved by Dendrinos and Wright in [10] as one step toward such a restriction theorem. Let LP (s) := det P (s), P (s), . . . , P (d) (s) , 2/d(d+1)
for s ∈ R. Hence dσP = LP
(3)
ds. For t = (t1 , t2 , . . . , td ) ∈ Rd , define
JP (t) := det P (t1 ), P (t2 ), . . . , P (td ) .
(4)
As indicated by the notation, this last term will arise as the jacobian of a certain map from Rd → Rd . Theorem 2 (Dendrinos and Wright). Let P : R → Rd be a polynomial of degree N such that N LP (s) ≡ 0. Then there exists a decomposition R = C j =1 Ij such that the Ij are pairwise disjoint
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open intervals, and for each j , there exist a positive constant Aj , a non-negative integer Kj N , and a real number bj ∈ R \ Ij such that (i) |LP (s)| ∼ Aj |s − bj |Kj for every s ∈ Ij . (ii) Whenever (t1 , . . . , td ) ∈ Ijd , d LP (tk )1/d JP (t1 , . . . , td ) C |tk − tl |. l
k=1
(iii) For any ε ∈ {−1, 1}d , define Φ ε (t1 , . . . , td ) := cardinality of (Φ ε )−1 {x} ∩ Ijd is at most d!.
d
j =1 εj P (tj ).
Then for each x ∈ Rd , the
Here C and the implicit constants depend only on N and d. We may assume in proving Theorem 1 that I has finite length. Theorem 2 allows us to make a few further simplifications. First, it suffices to prove the desired bounds for an operator as in (1), only with the integral restricted to one of the intervals Ij from the decomposition above. Next, after translating Ij and reflecting it across 0 if needed, we may assume that the real number bj in Theorem 2 is equal to 0 and that Ij ⊂ (0, ∞). The Lpd ,u → Lqd ,v bounds are invariant under scalings in s (P a (s) = P (as)) and multiplication of P by a constant. By scaling in the s variable, we may assume that |Ij | = 1. Finally, using the fact that LλP = λd LP , after multiplying P by an appropriate constant, we may assume that the constant Aj in Theorem 2 is equal to 1. In particular, |LP (s)| ∼ s K uniformly on I , where the implicit constants depend only on d, N . In summary, it suffices to prove uniform estimates for the operator Tf (x) :=
f x − P (s) dμ(s),
(5)
I
where dμ(s) = s 2K/d(d+1) ds and I ⊂ (0, ∞) has length 1. We note that Dendrinos, Folch-Gabayet and Wright have recently shown in [8] that Theorem 2 extends to d-tuples of rational functions of bounded degree. It therefore seems likely that Theorem 1 could be generalized to give bounds for convolution with affine arclength along curves parametrized by such functions, but the author has not investigated the extent to which the arguments in this article would need to be changed. 3. Reduction of Theorem 1 to two lemmas If E and F are subsets of Rd , then we will denote T χE , χF =: T (E, F ). We define two quantities, α :=
T (E, F ) , |F |
β :=
T (E, F ) , |E|
(6)
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which have played an important role in much of the recent literature on generalized Radon transforms and, in particular, appeared in most of the references mentioned in Section 1. Note that β, for instance, represents the average over x ∈ E of
T ∗ χF (x) = μ s ∈ I : x + P (s) ∈ F . The quantities α, β have mostly been used to prove restricted weak type bounds, i.e. T (E, F ) |E|1/p |F |1/q . One can easily check that in our case, the restricted weak-type bound at the endpoint would follow from |E| α d(d+1)/2 (β/α)d−1 . In [7], Christ proved that “trilinear” versions of such estimates could be used to establish strongtype bounds. This method was applied in [9] to prove strong-type estimates for the operator in (1) in the cases d = 2, 3 and in [27] to prove strong-type estimates for convolution with affine arclength measure along the moment curve. Using Christ’s techniques, one can show that the next two lemmas imply Theorem 1. See [27] for details. For notational convenience, we define the quantity n := d(d + 1)/ 2K + d(d + 1) .
(7)
μ 0, r n = nr.
(8)
Notice that for 0 r 1,
Lemma 3. Let E1 , E2 , F ⊂ Rd have finite positive measures. Assume that, for j = 1, 2, T (E ,F ) T χEj (y) αj for all y ∈ F and |Ejj | βj , where α2 α1 . Then d(d+1) 2
|E2 | α1
β1 α1
d−1 (α2 /α1 )(d+1)/2+n(d−1)/2 .
(9)
Lemma 4. Let η > 0, and let E, F1 , F2 ⊂ Rd have finite, nonzero measures. Assume that, for T (E,F ) j = 1, 2, T ∗ χFj (x) βj for all x ∈ E, where βj η |E| j and β2 β1 . Assume as well that T (E,Fj ) |Fj |
αj , where α2 α1 . Then |F2 | ηC α1r1 α2r2 β1s1 β2s2 ,
(10)
for some quadruple (r1 , r2 , s1 , s2 ) which is taken from a finite list that depends on d, N and which satisfies d(d − 1) = r1 + r2 , 2
d = s1 + s2 ,
The constant C is allowed to depend on N , d alone.
0<
s2 r2 − − 1. qd qd
(11)
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In some sense, Lemma 4 corresponds to a more general formulation. The relative simplicity of Lemma 3 is a matter of luck more than anything else. See [27] for more explanation. We remark that Lemma 4 is slightly different from the corresponding lemma in [27], but implies the strong-type bound (and accompanying Lorentz space improvement) by exactly the same proof. 4. Proof of Lemma 3 4.1. Setup For i 1, define i Φi (t1 , . . . , ti ) := (−1)j +1 P (ti ). j =1
Our first step will be to identify sets Ωi , satisfying certain helpful properties, such that for each i, Φi (Ωi ) is contained in F or one of the Ej . We assume the hypotheses of Lemma 3 and define γ1 := max{α1 , β1 }. Lemma 5. There exist a point x0 ∈ E1 , a constant c > 0, and measurable sets Ω1 ⊂ I , Ωi ⊂ Ωi−1 × I for 2 i 2d such that • x0 + Φi (Ωi ) is contained in: F if i is odd, E2 if i = 2d, and E1 otherwise. • μ(Ω1 ) = cβ1 , and if t ∈ Ωi−1 , then μ{ti ∈ I : (t, ti ) ∈ Ωi } equals: cβ1 if i > 1 is odd, cα2 if i = 2d, and cα1 otherwise. • If t ∈ Ωi , then ti is greater than or equal to: cγ1n if 1 i < 2d and cα2n if i = 2d. Further−2K/d(d+1) more, if j < i < 2d, then |ti − tj | is greater than or equal to: cβ1 tj if i is odd and −2K/d(d+1)
cα1 tj to
c t
i
if
tj α2n
if i < 2d is even. Finally, if j < i = 2d, then |ti − tj | is greater than or equal −2K/d(d+1)
and c α2 tj
otherwise.
Proof. This is quite similar to Lemma 1 of [6], but the last item involves changes inspired by the works [14] and [9]. First, suppose that we have proved the existence of sets Ωi as described in the lemma for 1 i 2d − 1. Then every point t ∈ Ω2d−1 corresponds to a point y(t) = x0 + Φ2d−1 (t) ∈ F . By the assumed lower bound T χE2 (y) α2 on F , there exists a set I (t) ⊂ I with μ(I (t)) α2 such that y(t) − P (I (t)) ∈ E2 . Next, we refine the sets I (t) to guarantee the third condition of the lemma. Since μ[0, cα2n ] ∼ 1/n c α2 , we can assume that s cα2n for s ∈ I (t) while maintaining μ(I (t)) α2 . With this assumption in place, let j < 2d. If tj < cα2n , then we excise the region [0, 2cα2n ] from I (t) if necessary. Having done that, if s ∈ I (t), then s − tj s/2. We claim that if tj > cα2n , then −2K/d(d+1) μ(B(tj )) := μ{s ∈ I (t): |s − tj | < c α2 tj } < cα2 , for c sufficiently small. To see this,
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note that if s ∈ B(tj ), then |s − tj | < c α2n < c s, so s ∼ tj and −2K/d(d+1)
μ B(tj ) =
tj +c α2 tj
2K/d(d+1)−2K/d(d+1)
s 2K/d(d+1) ds cα2 tj
.
−2K/d(d+1) tj −c α2 tj
Thus, by removing small “bad” portions of each I (t) if needed (this may be done while preserving measurability), we may set Ω2d = {(t, s) ∈ Ω2d−1 × I : s ∈ I (t)}, and satisfy the conclusions of the lemma. With some modifications similar to those indicated above, the proof of the existence of sets Ωi , 1 i 2d − 1 is essentially the same as the proof of Lemma 1 in [14]. The key observation is that Tγ1 χE1 , χF T (E1 , F ), where
f x − P (s) dμ(s).
Tγ1 f (x) =
(12)
I \[0,cγ1n ]
The proof then proceeds as in Lemma 1 of [6].
2
4.2. The proof when β1 α1 Let t0 ∈ Ωd , and let ω := {t ∈ I d : (t0 , t) ∈ Ω2d }. For consistency of notation, we will refer to elements of ω as t = (td+1 , . . . , t2d ). In this section, we will prove the following lemma. Lemma 6. Let rd = 1 + 3 + · · · + d − 1 if d is even and rd = 2 + 4 + · · · + d − 1 if d is odd. Then d(d+1)/2
|E2 | α1
(β1 /α1 )rd (α2 /α1 )(d+1)/2+n(d−1)/2 .
Note that rd d − 1, so when β1 α1 , the conclusion of Lemma 6 implies Lemma 3. Proof of Lemma 6. Let Φ(t) := x0 + Φ2d (t0 , t). Note that (iii) of Theorem 2 implies that |E2 |
det DΦ(t) dt =
ω
JP (t) dt.
ω
Hence, by (i)–(ii) of Theorem 2 and the fact that ω ⊂ I d , |E2 |
2d
K/d(d−1) K/d(d−1) tl |tk
tk
− tl | dt.
(13)
ω k=d+1 d
β1 (α2 /α1 ), and our next task will be to prove a lower bound We know that μd (ω) α1 for (13) which involves integrals with respect to μ. It will suffice to prove the following:
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Lemma 7. For every t ∈ ω and k 2d: • If k is odd and j < k, then: |tk − tj | β1 (tk · tj )−K/d(d+1) . • If k < 2d is even and j < k, then: |tk − tj | α1 (tk · tj )−K/d(d+1) . (1+n)/2 (1−n)/2 • Finally, if j < k = 2d, then: |tk − tj | α2 α1 (tk · tj )−K/d(d+1) . Separating the integrand in (13) into a product over even k times a product over odd k, and manipulating the inequalities in Lemma 7, we see that
r
|E2 | α1rd β1d−1
α2 α1
d(d+1)/2 α1
(n+1)(d−1)/2
β1 α1
rd
2d
2K/d(d+1)
tk
dt
ω k=d+1
α2 α1
The last inequality also requires a little algebra.
(d+1)/2+n(d−1)/2 . 2
Proof of Lemma 7. We will prove the lemma when k = 2d. The proof when k < 2d is similar, but a little simpler. Suppose that tj tk or tj tk . Then by the triangle inequality, |tk − tj | tk , and the lower 1+K/d(d+1) K/d(d+1) (1+n)/2 (1−n)/2 bounds in Lemma 5 imply that tk tj α2 α1 . −2K/d(d+1)
If tj ∼ tk , then |tj − tk | α2 tj follows from α2 α1 . 2
∼ α2 (tk tj )−K/d(d+1) and the claimed inequality
This completes the proof of Lemma 3 in the case β1 α1 . 4.3. A combinatorial argument To prove Lemma 3 in the remaining case, β1 α1 , we modify the “band structure” argument of Christ in [6] to accommodate the weight s 2K/d(d+1) in the measure μ. The modifications we make are inspired by the arguments of [9]. We will follow Christ’s construction as closely as possible. Definition. Let Γ be a finite set of positive integers, called indices. A band structure on Γ is a partition of Γ into subsets called “bands”. Given a band structure on Γ , we designate the indices in Γ as free, quasi-free, or bound as follows: • The least index of each band is free. • An index is quasi-free if it is the larger element of a two-element band; it is quasi-bound (not bound) to the smaller (free) element of that band. • An index is bound (to the free element of its band) if it is one of two or more non-free elements of some band. Lemma 8. Let ε > 0. Then there exist parameters δ, δ satisfying cd,ε < δ < εδ < εc, an integer d k < 2d, an element t0 ∈ Ω2d−k , a set ω ⊂ {t: (t0 , t) ∈ Ω2d } with μk (ω) ∼
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k/2 k/2
α1 β1 (α2 /α1 ), and a band structure on [2d − k + 1, 2d] such that the following properties hold: (i) (ii) (iii) (iv)
There are exactly d free or quasi-free indices. In particular, each even index is free. |ti − tj | > δα1 (ti · tj )−K/d(d+1) , unless i and j lie in the same band. c0 β1 (ti · tj )−K/d(d+1) < |ti − tj | < δα1 (ti · tj )−K/d(d+1) whenever i is quasi-bound to j . δ α1 (ti · tj )−K/d(d+1) > |ti − tj | whenever i is bound to j .
In our application of the lemma, the parameter ε will be the value whose existence is guaranteed by the forthcoming Lemma 9. Sketch of proof of Lemma 8. Because of the lower bounds from Lemma 7 and the fact that α2 α1 , the proof of this lemma follows from arguments similar to those in [27], which are in turn slight modifications of arguments in [6]. We sketch the beginnings of the modified argument. There exists ω0 ⊂ Ω2d with |ω0 | |Ω2d | and a permutation σ ∈ Perm2d (where Permj is the set of permutations on j indices) such that tσ (1) < tσ (2) < · · · < tσ (2d) c
N,d on ω0 . Initially set δ = 2d , where cN,d is some small constant to be determined below. Refining ω0 if necessary, but retaining the above lower bound on |ω0 |, there exists a sequence 1 = L1 < L2 < · · · < LR 2d such that tσ (i−1) < tσ (i) − δα1 (tσ (i−1) tσ (i) )−K/d(d+1) if and only if i = Lj for some 1 < j R. After this first step of the decomposition, our bands are
σ (L1 ) = σ (1), . . . , σ (L2 − 1) , σ (L2 ), . . . , σ (L3 − 1) , . . . , σ (LR ), . . . , σ (2d) . Assume that i < j and σ (i), σ (j ) are in the same band. Then by induction, tσ (j ) > tσ (i) tσ (j ) − δα1 (tσ (i) tσ (i+1) )−K/d(d+1) + · · · + (tσ (j −1) tσ (j ) )−K/d(d+1) −2K/d(d+1)
> tσ (j ) − 2dδα1 tσ (i)
−2K/d(d+1)
tσ (j ) − cN,d α1 tσ (i)
tσ (j ) − cN,d α1n .
The inequality on the second line follows from the monotonicity of the tσ (·) and the second inequality on the last line follows from tσ (i) α1n (recall α2 > α1 ) and a bit of algebra. For cN,d sufficiently small (depending only on N , d), the second inequality on that line implies that |tσ (i) − tσ (j ) | min{tσ (i) , tσ (j ) }, which in turn implies that tσ (i) ∼ tσ (j ) (we will use this fact several times in the coming pages). Returning to the first line of the sequence of inequalities, tσ (j ) > tσ (i) > tσ (j ) − cN,d α1 (tσ (i) tσ (j ) )−K/d(d+1) . From this and the lower bounds in Lemma 7, the maximum of σ (i) and σ (j ) cannot be even. Thus an even index is always the minimum element of a band containing it, and in particular, no band contains two or more even indices after the first step.
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Assume that i < j and σ (i), σ (j ) are in different bands. Say σ (i) lies in the band
σ (La−1 ), . . . , σ (La − 1) .
Then j La and tσ (j ) − tσ (i) =
j −1
tσ (k+1) − tσ (k) tσ (La ) − tσ (La −1)
k=i
δα1 (tσ (La ) tσ (La −1) )−K/d(d+1) δα1 (tσ (j ) tσ (i) )−K/d(d+1) . The first inequality on the second line follows from the definition of the sequence La . The second inequality follows from the monotonicity of the tσ (·) and the fact that tσ (i) ∼ tσ (La −1) because both lie in the same band (this was proved in the previous paragraph). Note that it is vital that α2 α1 , so that t2d α1n on Ω2d . We have explained the first step of an iterative procedure which terminates after O(d) steps. In each step of this procedure, the quantities δ, δ are decreased, but this is only done finitely many times and in a way which guarantees the lower bound cd,ε < δ . For the remainder of the algorithm, we refer the reader to [27] (the arguments there are adapted from [6]) with the caveat that adjustments as above will be necessary. 2 To state the next lemma, we need a little further notation. Suppose ω, k, and t0 are as in the conclusion of Lemma 8. With t ∈ Rk denoted by (t2d−k+1 , . . . , t2d ), define Φ(t) := x0 + Φ2d (t0 , t). Thus Φ(ω) ⊂ E2 . Let Λ ⊂ [2d − k + 1, 2d] be the set of free or quasi-free indices. Given i ∈ [2d − k + 1, 2d] \ Λ, let B(i) denote the index to which i is bound. We define three mappings: Rd τ (t), R
k−d
s(t),
Rk−d σ (t),
τi (t) = ti ,
for i ∈ Λ,
sj (t) = tj − tB(j ) , for j ∈ / Λ, 2K/d(d+1) σj (t) = sj (t) τB(j ) (t) .
The function t → (τ, σ ) is invertible for t ∈ (0, ∞)k . We let t (τ, σ ) denote its inverse, and define J (τ, σ ) := det Dτ Φ t (τ, σ ) . Lemma 9. There exists ε > 0 such that given any k, any band structure on [2d − k + 1, 2d], and any ω satisfying the conclusion of Lemma 8, the following lower bound holds for all (τ, σ ) ∈ Rk such that t (τ, σ ) ∈ ω:
M (1+n)(d−1)/2 d α2 2K/d(d+1) J (τ, σ ) cα d(d−1)/2 β1 τj , 1 α1 α1
(14)
j =1
for some c > 0, where M is the number of quasi-free indices in the band structure. Here the values of ε, c depend only on d, N , and the constant c0 in (iii) of Lemma 8 (which in turn depends on d, N ).
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Completion of proof of Lemma 3. The remainder of the proof is quite similar to the argument from [6]. Let k, ω, etc., be as in Lemma 8, and fix σ ∈ Rk−d . Let ωσ = {τ : t (τ, σ ) ∈ ω}. By Bezout’s theorem (as stated in [25], for a similar application see [6]), for each σ , under the map ω τ → Φ(t (τ, σ )) ∈ E2 , points x lying off a set of measure zero have at most CN,d preimages. Therefore |E2 |
J (τ, σ ) dτ α d(d−1)/2 (β1 /α1 )M (α2 /α1 )(1+n)(d−1)/2 μd (ωσ ). 1
(15)
ωσ
It is easy to check that if t (τ, σ ) ∈ ω, then |σ | α1 . Integrating both sides of (15) over |σ | α1 , d(d−1)/2 α1k−d |E2 | α1 (β1 /α1 )M (α2 /α1 )(1+n)(d−1)/2
dμd (τ ) dσ.
(16)
t (τ,σ )∈ω
We switch the order of integration and make the change of variables (τ, σ ) → t to show that the integral on the right of (16) equals
2K/d(d+1)
tB(j )
/ ω j ∈Λ
dtj
2K/d(d+1)
ti
(17)
dti .
i∈Λ
We showed above (in the proof of Lemma 8) that ti ∼ tj whenever i is bound to j and t ∈ ω. Approximating the tB(j ) in (17) by tj , we then have d(d−1)/2 α1k−d |E2 | α1 (β1 /α1 )M (α2 /α1 )(1+n)(d−1)/2
dμk (t).
(18)
ω
Next, applying (i) of Lemma 8 and (18), we have shown that
d(d+1)/2 |E2 | α1
β1 α1
M+k/2 (α2 /α1 )(d+1)/2+n(d−1)/2 .
Finally, at least k/2 + 1 indices are free (the even indices together with the first index), and M plus the number of free indices equals d. Therefore M + k/2 d − 1, and we have proved Lemma 3 in the remaining case, β1 α1 . 2 5. Proof of Lemma 9 Now we return to the lower bound on J (τ, σ ). We recall some important information about the interval I in (5). The interval came from the decomposition in Theorem 2 and is contained in (0, ∞) by the reductions in Section 2. In addition, the corresponding quantity b equals zero. Analysis similar to that in [6] can be used to write J (τ, s) in a more helpful form. We summarize the argument. Note that
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(−1)j +1 P (τj ) + Φ t (τ, σ ) = z0 + (−1)i+1 P (τj + si ) j ∈Λ
i⇒j
i+1 θj P (τj ) + P (τj + si ) − P (τj ) , (−1) = z0 + j ∈Λ
i⇒j
−2K/d(d+1)
where z0 is a constant, si = σi τj , and i ⇒ j means i is bound to j . In addition, the j +1 i+1 quantity θj := (−1) + i⇒j (−1) is never zero because bound indices are always odd and because an index j ∈ Λ has zero or at least two indices bound to it. See [6] for details. For fixed σ , we wish to compute the jacobian with respect to τ . Using the definition of si in the previous paragraph, we have ∂ 2K si P (τj + si ) P τj + si (τj , σi ) − P (τj ) = P (τj + si ) − P (τj ) − . ∂τj d(d + 1) τj Now, using multilinearity of the determinant, J (τ, σ ) = C0 JP (τ ) +
error terms,
(19)
where |C0 | = |± j ∈Λ θj | ∼ 1. The number of terms in the sum above is Cd and each error term is equal to a constant times a certain determinant, for instance
si(jl ) Ck,l det P (τj1 ), . . . , P (τjk + si(jk ) ) − P (τjk ), . . . , P (τjl + si(jl ) ), . . . . τ jl
(20)
Here Λ = {j1 , . . . , jd }, each i(j ) is bound to j , either k < l or l d (so not all entries of the matrix are of the form P (τj )), and k−1 d−l+1 −2K θ ji |Ck,l | = Cd,N . d(d + 1) i=1
The determinants in (20) can be viewed as a hybrid of two types of error terms. The first of these is det P (τj1 ), . . . , P (τjk−1 ), P (τjk + si(jk ) ) − P (τjk ), . . . , P (τjd + si(jd ) ) − P (τjd ) τjk +si(jk )
=
τjd +si(jd )
··· τjk
τjd
d ∂ ∂τj τj
=k
=tj
JP (τ ) dtjd · · · dtjk ,
(21)
where k d and i(j ) is bound to j . The second type is
si(j ) JP (ti(1) , . . . , ti(d) ). τj
(22)
j ∈Λ
In (22) t = t (τ, σ ) and ∅ = Λ ⊂ Λ, but now i(j ) equals j if j ∈ / Λ and is bound to j otherwise.
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For clarity of exposition, we will explicitly bound the quantities (21) and (22), but our analysis applies equally well to (20), the hybrid of these terms. 5.1. Aside: A geometric identity Before proceeding in our analysis of the error terms, we record an identity relating the jacobian JP defined in (4) to determinants of certain minors of the matrix (P (t), . . . , P (d) (t)). A proof of this identity may be found in [10]. For 1 j d, we define polynomials Lj = LP ,j by ⎛
P1 (s) ⎜ . Lj (s) := det ⎝ .. Pj (s)
... .. . ...
⎞ (j ) P1 (s) ⎟ .. ⎠. . (j ) Pj (s)
(23)
Note that Ld = LP , where LP is the polynomial defined in (3). Using this, we recursively define rational functions Jk : Rk → R by J1 (t1 ) :=
Ld−2 (t1 )Ld (t1 ) , Ld−1 (t1 )2
(24)
t2 tk k Ld−k−1 (tj )Ld−k+1 (tj ) ··· Jk−1 (s1 , . . . , sk−1 ) ds1 · · · dsk−1 . Jk (t1 , . . . , tk ) := Ld−k (tj )2 j =1
t1
(25)
tk−1
The convention L0 ≡ L−1 ≡ 1 is required to define Jd−1 and Jd . The algorithm in [10] begins with an initial decomposition CN,d
R=
Ij ,
j =1
where the Ij are disjoint open intervals, and on each Ij , the polynomials L1 , . . . , Ld are all single-signed. Then for t = (t1 , . . . , td ) ∈ Ijd , we have the identity JP (t) = Jd (t).
(26)
5.2. Back to the proof of Lemma 9 We examine a typical instance of (21). For ease of notation, let the τj be indexed by j ∈ {1, . . . , d} instead of Λ. We expand the identity (26) JP (τ ) =
d j =1
τ2
τd ···
L1 (τj ) τ1
τd−1
Jd−1 (s1 , . . . , sd−1 ) ds1 · · · dsd−1 .
(27)
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Let h(τ ) :=
d
H (τ ) := JP (τ )/ h(τ ).
L1 (τj ),
j =1
Then
∂ JP (τ ) = T1 · T2 := ∂τj j ∈Λ
Λ ⊂Λ
j ∈Λ \Λ
Λ ⊂Λ
∂ ∂ h(τ ) · H (τ ). ∂τj ∂τj
(28)
j ∈Λ
We will use the following lemmas to bound T1 and T2 in a typical term from the above sum. Lemma 10. With I ⊂ (0, ∞) and b = 0 coming from Theorem 2, whenever s ∈ I |L1 (s)| L (s) . 1 s
(29)
Lemma 11. With I , b as in Lemma 10, whenever τ ∈ I d
j ∈Λ
∂ H (τ ) ∂τj
|H (τ )|
δj j ∈Λ τj |τj
δ,i
− τi(j ) |1−δj
,
(30)
where the sum is taken over δ ∈ {0, 1}Λ and functions i : Λ → Λ with i(j ) = j , for all j ∈ Λ . We remark that the conditions on I and b, particularly the fact that I , b are determined by the algorithm in [10], are crucial hypotheses to these lemmas. We postpone the proofs of Lemmas 10 and 11 for a moment and finish proving Lemma 9. By Lemma 10 |T1 |
|h(τ )|
l∈Λ \Λ τl
.
(31)
Lemma 11 gives an upper bound for |T2 |, and combining that with (31) and (28), we obtain ∂ JP (tΛ , τΛc ) ∂t j j ∈Λ
Λ ⊂Λ ,δ,i
|JP (tΛ , τΛc )| , δl 1−δl ) l∈Λ \Λ tl j ∈Λ (tj |tj − ui(j ) |
where δ, i are as in Lemma 11 and ui = ti if i ∈ Λ and τi if i ∈ Λc . We return to (21), which is the term we want to estimate. Let j ∈ Λ, let i be bound to j , and let j ∈ Λ \ j . Suppose t (τ, s) ∈ ω. By condition (4) of Lemma 8 and the definition of si , −K/d(d+1) . |si | εδα1 τj · (τj + si )
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Since t (τ, s) ∈ ω, both τj = tj and τj + si = ti are α1n (recall α2 α1 ). Raising these lower bounds to the negative power −K/d(d + 1) gives an upper bound on |si | and using τj α1n again implies |si | εδα1n ετj .
(32)
This implies the inequality (already noted above) τj ∼ τj + si .
(33)
We next compare |τj − τj | and |si |. If τj τj or if τj τj , then |τj − τj | τj ε −1 |si |, by (32). Say τj ∼ τj . Since j and j must lie in different bands, approximating τj by τj in conclusion (iv) of Lemma 8, −2K/d(d+1)
|τj − τj | δα1 τj
−K/d(d+1) ∼ δα1 τj (τj + si ) ,
where the second estimate follows from (33). Thus, regardless of the relative sizes of τj , τj , |τj − τj | ε −1 |si |.
(34)
Therefore, we may estimate the integrand in (21) by the constant
|JP (τ )| , δl 1−δl ) j ∈Λ \Λ τj l∈Λ (τl |τl − τl |
which implies that the error term in (21) is
|JP (τ )| j ∈Λ |sj | ε |Λ | JP (τ ), δl 1−δl ) j ∈Λ \Λ τj l∈Λ (τl |τl − τl |
by (32) and (34). We now have everything we need to estimate (22) as well. By our bound on (21), JP (ti(1) , . . . , ti(d) ) ∼ JP (τ1 , . . . , τd ), and by (32), | j ∈Λ si(j ) /τj | ε |Λ | . Combining these two estimates, J (τ, σ ) = CJP (τ ) + O(ε)JP (τ ). It remains to show that JP (τ ) is bounded from below by the term on the right of (14). By our assumptions on I (in particular, the conclusions of Theorem 2), d K/d JP (τ ) τj |τj − τk |. j =1
k<j
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The first term on the right is in the form we want, but we need to work on the second term. By (iv)–(v) of Lemma 8 and the fact that each quasi-free index is quasi-bound to a unique free index, we have d k=1
|τk |K/d
d(d−1)/2
|τk − τl | α1
k
β1 α1
M
α2 α1
(n+1)(d−1)/2 d k=1
K/d
τk
(τk · τl )−K/d(d+1) .
k
Putting these two inequalities together and using the definition of n to perform a quick computation proves the lower bound claimed in Lemma 9. 6. Proofs of Lemmas 10 and 11 Our proofs of Lemmas 10 and 11 are not self-contained, but rather consist of some minor adjustments and additions to the arguments of Dendrinos and Wright in Sections 5, 7, 9 of [10]. For clarity, in this section we will provide a rough sketch of the decomposition procedure in [10] before explaining how Lemmas 10 and 11 follow. We recall the identity quoted in Section 5.1. It will be important in what follows to note that one can prove inductively from the definition that Jd−1 is an antisymmetric function. Two decomposition procedures, applied iteratively, constitute the main part of the algorithm in [10]. We describe them here. D1 : Let η1 , . . . , ηd be complex numbers and J an interval. This procedure decomposes J = di=1 j =i Iij , where the Iij are pairwise disjoint, and each Iij is a union of Od (1) open intervals. Each Iij is associated to the real number bi = Re ηi , and for each i, j and s ∈ Iij , 1 k d ,
|s − bi | |s − ηk |,
|s − ηk | ∼ Aik |s − bi |δik .
(35)
Here δik ∈ {0, 1}, and Aik ∈ R. D2 : Let J be an interval and b a real number. Let β1 , . . . , βd be (not necessarily distinct) complex numbers with |b + β1 | · · · |b + βd |. This procedure produces a decomposition J=
Cd j =1
Cd
Gj ∪
Dj ,
j =1
where the Gj , Dk are pairwise disjoint, and each Gj and each Dk is a union of O(1) open intervals. On each Gj (called a gap), |s − b − βk | ∼ |βk |1−εkj |s − b|εkj ,
|s − b − βk | |s − b|,
(36)
for some εkj ∈ {0, 1}, and on each Dj (called a dyadic interval), |s − b| ∼ Dj . We move now to the implementation of the procedures D1 and D2 . Fix an interval J coming from the initial decomposition above (thus (26) holds).
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Step 0: Apply D1 to J with respect to the zeros ηi of Ld . Fix an interval I0 with corresponding real number b0 from this decomposition. Step n + 1: Assume that step n has been completed (0 n d − 2), leaving us with an interval In and real number bn . Apply D2 to In with respect to bn and the zeros of Ln+1 (· + bn ). There are two possibilities for an interval J from this decomposition. Case I: J is a gap. In this case, step n+1 is complete and has produced an interval In+1 = J and real number bn+1 = bn . Case II: J is dyadic. We apply D1 to J with respect to bn and the zeros of LP ,n+1 and b0 , b1 , . . . , bn (these are the real numbers corresponding to In and its ancestor intervals). We are left with interval-number pairs. We choose one of these pairs (arbitrarily) and denote its elements In+1 , bn+1 to complete step n + 1. In either case, the decomposition has been performed so that on In+1 , Ld (s) ∼ Ad |s − bn+1 |kd , L1 (s) ∼ A1 |s − bn+1 |k1 , Ln+1 (s) ∼ An+1 |s − bn+1 |kn+1 .
..., (37)
Here, the constants Aj and the non-negative integer exponents kj are allowed to change from line to line to reflect the fact that if the interval J is dyadic, then |s − bn | is nearly constant. If |s − bn | is nearly constant, then by the analogue of (37) from step n, so are Ld , L1 , . . . , Ln . The final step (n = d) of the decomposition is to decompose each interval Id−1 coming from step d − 1 so that none of the subintervals Id contains any of the real numbers b0 , b1 , . . . , bd−1 associated to Id−1 and its ancestors. A detailed proof of (37) may be found in [10]. To simplify the exposition, we have omitted some crucial details in the sketch above. For example, linear transformations are used between steps to guarantee certain exponents kj do not arise which would prevent the deduction of (ii) of Theorem 2 from (37) and (26). We have also made a minor change in the algorithm of Dendrinos and Wright by using the real numbers bj determined in previous steps to perform the decomposition in Case II of step n + 1. This alteration is miniscule, and moreover, the modified algorithm is needed only to establish the upper bound on the error terms in (19). Having bounded those error terms, one can use the original algorithm in [10] to prove Theorem 2, so this change does not cause any technical issues to arise. In fact, the modified algorithm could also be employed in the proof of Theorem 2. Proof of Lemma 10. After d − 1 steps as above, we select an interval I with corresponding real number b. From (37), we know that for s ∈ I , L1 (s) ∼ A1 |s − b|k1 . To prove the lemma (wherein things have been arranged so b = 0, I ⊂ (0, ∞)), we must show that |L1 (s)| L (s) . 1 |s − b|
(38)
We let b0 , b1 , . . . , bd−1 = b be the real numbers corresponding to the ancestor intervals of I in the decomposition procedure.
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We begin our analysis after the application of D2 in step 1. Case I: the ancestor J of I arising after D2 is a gap. Then b1 = b0 and L1 (s) = B1
(s − b0 − βi )ni , i
where the βi are the complex numbers with respect to which D2 was performed. By the product rule and the triangle inequality, L (s) 1
i
ni |L1 (s)| . |s − b0 − βj |
Thus, by (36), L (s) CN,d |L1 (s)| = CN,d |L1 (s)| , 1 |s − b0 | |s − b1 | where CN,d = D1 ,
i
ni . Case II: the ancestor J of I arising after D2 is dyadic. In the notation of
L1 (s) = B1
(s − ηi )ni , i
and b1 satisfies (35) with the index i replaced by 1 on I . Hence, by the same arguments as in Case I, L (s) CN,d |L1 (s)| . 1 |s − b1 | We wish to prove that after step n, L (s) CN,d |L1 (s)| . 1 |s − bn |
(39)
We proceed by induction, assuming that (39) holds after step n. Step n + 1 begins with a D2 decomposition. If after this the ancestor J of I is a gap, we have bn+1 = bn . If the ancestor J of I is dyadic, by the way D1 is performed in the above version of the algorithm, |s − bn+1 | |s − bn |. In either case, the analogue of (39) holds. This completes the proof of Lemma 10. 2 Proof of Lemma 11. We will prove the lemma by rewriting the left side of (30) as an integral and then approximating the integral. It is easy to see that if Λ = {1, . . . , d}, then T2 = 0. Let i1 , . . . , ik be an increasing enumeration of the elements of Λ \ Λ . Using antisymmetry of Jd−1 ,
j ∈Λ
τi2 τik ∂ H (τ ) = ± · · · Jd−1 (s1 , . . . , sk−1 , τΛ ) ds1 · · · dsk−1 . ∂τj τi1
τik−1
(40)
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Next, we proceed to the estimation of Jd−1 (t) for t ∈ I d . Recall that we are assuming that b = 0 and that I ⊂ (0, ∞). Since JP is antisymmetric, by reordering indices if needed, we may assume that t1 < · · · < td . Thus the only points (s1 , . . . , sd−1 ) relevant to the integral in (26) are those with s1 < · · · < sd−1 . By induction, for each 1 r d, only those s ∈ I r−1 with s1 < · · · < sr−1 are relevant in (25). Because the si form a monotone sequence and since the Lj are all single-signed on I (by the initial decomposition), each integrand Jk−1 in (25) is single signed on the domain of integration. This fact will make valid the approximations below. We will proceed by induction. In [10, Section 9], the authors define certain integers σ1 , . . . , σd−1 such that if Sd−1 is defined inductively by
Sr (t1 , . . . , tr ) :=
r
t2
tr ···
tsσr
s=1
t1
Sr−1 (w1 , . . . , wr−1 ) dwr−1 · · · dw1 ,
tr−1
for 1 < r d − 1, then we have (at the last stage) that |Jd−1 | ∼ |Sd−1 |. We note in particular that the absolute value of the right-hand side of (40) is τik
τi2 ∼± τi1
···
Sd−1 (s1 , . . . , sk−1 , τΛ ) ds1 · · · dsk−1 .
(41)
τik−1
We establish inductively a formula for Sr . To do this, we will use the fact that the algorithm in [10] ensures that the σj are such that none of the Sj contains a ti−1 term (and hence none contains a log ti term). Suppose that
Sr−1 (t1 , . . . , tr−1 ) = Cr−1
k
k
ρ(r−1) sgn(ρ)t1 ρ(1) · · · tr−1 ,
(42)
ρ∈Permr−1
where Cr−1 = 0, and the ki are integers. (We recall that Permr−1 is the set of permutations on r − 1 indices.) The case r = 2 of this hypothesis is trivial. Then t2 Tr (t1 , . . . , tr ) :=
tr ···
t1
= C
Sr−1 (w1 , . . . , wr−1 ) dwr−1 · · · dw1
tr−1
ρ∈Permr−1
sgn(ρ)
r−1 j =1
kρ(j ) +1 kρ(j ) +1 tj . − tj +1
When we multiply out one of the summands in the last line above, we will have a sum of two types of monomials: those with no repeated indices, and those with repeated indices. For example, (t1 − t2 )(t2 − t3 ) = (t1 t2 − t1 t3 + t2 t3 ) − t2 t2 .
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The terms with repeated indices will cancel when we sum on ρ ∈ Permr−1 , and we obtain Tr (t1 , . . . , tr ) = C
r kρ(j −1) +1 kρ(j ) +1 k +1 k +1 sgn(ρ) (−1)r−j t1 ρ(1) · · · tj −1 tj +1 · · · tr ρ(r−1) j =1
ρ∈Permr−1
= C
sgn(τ )t1 τ (1)
· · · tr τ (r) ,
τ ∈Permr
where j = kj + 1 for 1 j r − 1 and r = 0. Therefore Sr (t1 , . . . , tr ) is also of the form (42). This implies that α d Td (t1 , . . . , td ) = det tjα1 , . . . , tj d j =1 . This is a Vandermonde-type determinant and is equal to Td (t1 , . . . , td ) = CP ,I P(t)
d
(tj − ti ),
j =1 i<j
with CP ,I ∼ 1 and P ≡ 0 a symmetric polynomial with non-negative coefficients. For a reference, see [29, pp. 200–201]; this was used in related contexts in [5,14]. The right side of (41) equals
j ∈Λ
∂ Td (t), ∂τj
whose absolute value is easily shown to be
|Td (t)|
δj j ∈Λ tj |tj
− tj |1−δj
∼
|H (t)|
δj j ∈Λ tj |tj
− tj |1−δj
,
where the sum is as in the statement of the lemma. This completes the proof of Lemma 11.
2
7. Proof of Lemma 4 In this section, we modify the techniques in the previous section (using arguments similar to those in [27]) to prove Lemma 4. Recall that γ1 := max{α1 , β1 }. We also define γ2 := max{α2 , β2 }. The proof of the following lemma is almost exactly the same as the proof of Lemma 5. We leave the details to the reader. Lemma 12. There exist x0 ∈ E, a constant c > 0, and measurable sets Ω1 ⊂ I , Ωi ⊂ Ωi−1 × I for 2 i 2d − 1 such that
B. Stovall / Journal of Functional Analysis 259 (2010) 3205–3229
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• x0 + Φi (Ωi ) is contained in E if i is even, F2 if i = 2d − 1 and F1 otherwise. • μ(Ω1 ) = cβ1 , and if t ∈ Ωi−1 , then μ{ti ∈ I : (t, ti ) ∈ Ωi } equals: cα1 if i is even, cβ2 if i = 2d − 1, and cβ1 otherwise. • t ∈ Ωi implies that ti is greater than or equal to: cγ1n if i < 2d − 1 and cβ2n if i = 2d − 1. −2K/d(d+1) Furthermore, if j < i < 2d − 1, then |ti − tj | is greater than or equal to: cβ1 tj if −2K/d(d+1)
i is odd and cα1 tj than or equal to
c t
i
if
if i is even. Finally, if j < i = 2d − 1, then |ti − tj | is greater
tj c β2n
−2K/d(d+1)
and c β2 tj
otherwise.
As in the proof of Lemma 3, the proof of Lemma 4 breaks into two cases. The first case is when β1 α1 (thus β2 β1 α1 α2 by assumption). Proceeding as in Lemma 6, one can show that
d(d+1)/2 |F2 | α1
β1 α1
rd+1
β2 β1
(d+1)/2+n(d−1)/2 .
(43)
Note that the analogue of Lemma 7 for our current setup is the same as the original except that the first two conditions hold when k < 2d − 1, and when j < k = 2d − 1, we have |tk − tj | (1+n)/2 (1−n)/2 β2 β1 (tk tj )−K/d(d+1) . The second case is when β1 α1 . To begin, we establish a lower bound for t2d−1 . A priori, the best we can do is t2d−1 β2n . We will show that we can assume that t2d−1 ηn α2n (η being the quantity in the statement of Lemma 4). To see this, we let EB ⊂ E be the set of x such that
μ s cηn α2n : x + P (s) ∈ F2 β2 . Here c n. By the hypothesis that T ∗ χF2 β2 on E, we have that TB∗ χF2 β2 on EB , where TB f (x) =
f x − P (s) dμ(s).
[0,cηn α2n ]
Supposing that |EB | |E|, this implies that TB (EB , F2 ) := χEB , TB∗ χF2 β2 |E| ηT (E, F2 ). On the other hand, TB χE (x) ηα2 for all x, so we must have TB (EB , F2 ) ηα2 |F2 | = ηT (E, F2 ). This is a contradiction, so we must actually have |EB | |E|. Let EG = E \ EB . Then |EG | ∼ |E|. Moreover, on EG , T χF1 α1 ,
Tcηn α2n χF2 α2
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(see (12)). Thus T (EG , F1 ) β1 |E| ηT (E, F1 ) and Tcηn α2n (EG , F2 ) β2 |E| ηT (E, F2 ). Thus with α˜ 1 :=
T (EG , F1 ) , |F1 |
α˜ 2 :=
Tcηn α2n (EG , F2 ) |F2 |
,
we have αj α˜ j ηαj . Henceforth, we will proceed as though η ∼ 1, and hence α˜ j ∼ αj . The form of the conclusion in (11) is such that the general case requires little change in the analysis. By refining Ω2d−1 if necessary, we may assume that either t2d−1 α1n or that t2d−1 α1n throughout Ω2d−1 . We deal with these cases separately. 7.1. If t2d−1 α1n Keeping in mind that we just proved that the lower bound t2d−1 γ2n holds, we perform a band decomposition precisely as in Section 4.3, except that we declare the index 2d − 1 to be free from the beginning (note that t2d−1 ti whenever i < 2d − 1). Since |tj − t2d−1 | ∼ tj whenever j < 2d − 1, and since by definition 2d − 1 can have no indices bound to it, the inequalities (32) and (34) hold whenever i is bound to j and j = j ∈ Λ. In addition, whenever j < 2d − 1, (n+1)/2 (1−n)/2 γ2 (tj t2d−1 )−2K/d(d+1) .
|t2d−1 − tj | ∼ tj α1n α1
Thus the analogue of Lemma 9 implies that
d(d−1)/2 J (τ, σ ) α1
β1 α1
M
γ2 α1
(1−n)(d−1)/2 ,
which implies that
d(d+1)/2 |F2 | α1
β1 α1
M+k/2
γ2 α1
(1−n)(d−1)/2
β2 , β1
by arguments similar to those in Section 4.3. Because 2d − 1, the first index (2d − k), and all of the even indices are free, there are at least k/2 + 1 free indices. As the number of free plus the number of quasi-free indices equals d, the exponent of αβ11 in the above inequality is d − 1. From that and the definition of γ2 ,
max{0,(1−n)(d−1)/2−1} β2 α2 β2 α1 α1 β1
d 2 max{0,(1−n)(d−1)/2−1} β2 α2 d(d+1)/2 β1 = α1 . α1 β1 α1 d(d+1)/2
|F2 | α1
β1 α1
d−1
By checking the cases when the above max is zero and nonzero separately, one can verify that (11) is satisfied, and Lemma 4 is proved in the case β1 α1 and t2d−1 α1n .
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7.2. If t2d−1 α1n We perform a band decomposition. The following lemma holds: Lemma 13. Let ε > 0. Then there exist cd,ε < δ < εδ < εc, an integer d k 2d − 1, an k/2 k/2 element t0 ∈ Ω2d−k−1 , a set ω ⊂ {t: (t0 , t) ∈ Ω2d−1 } with μk (ω) ∼ α1 β1 (β2 /β1 ), and a band structure on [2d − k, 2d − 1] such that properties (i)–(iv) of Lemma 8 hold. As with Lemma 8, this can be proved by making a few modifications to the proof of the analogous lemma in [6]. Say 2d − 1 is free. Note that if β2 α1 , we have that (1+n)/2 (1−n)/2 α1 (t2d−1 tj )−K/d(d+1) ,
|t2d−1 − tj | β2
as can be shown in a manner similar to the proof of Lemma 7. On the other hand, if β2 < α1 , then by (ii) of Lemma 8, |t2d−1 − tj | α1 (t2d−1 tj )−K/d(d+1) β2 (t2d−1 tj )−K/d(d+1) . In addition, because t2d−1 α1n , the arguments leading up to (32) and (34) apply, and we have that whenever i is bound to j and j = j ∈ Λ (Λ being the set of free and quasi-free indices) |ti − tj | < εtj ,
|ti − tj | < ε|tj − tj |.
Thus, for sufficiently small ε, the analogue of Lemma 9 implies that d 2K/d(d+1) J (τ, σ ) α d(d+1)/2 (β1 /α1 )M+k/2 (β2 /α1 )ρ (β2 /β1 ) τ , j
1
(44)
j =1
where ρ = (1 + n)/(d − 1)/2 if β2 α1 and 1 if β2 < α1 . Since d 4, ρ is at least 1 when β2 α1 , and (44) holds with ρ = 1 regardless of the relative magnitudes of β2 and α1 . Moreover, 2d − 1, all of the even indices, and the least index (2d − k) are free. Since M plus the number of free indices equals d, one can check that the exponent of β1 /α1 is at most d − 1. Hence d(d−1)/2 d β1 (β2 /β1 )2 .
|F2 | α1
One can verify that the inequalities and equalities needed for Lemma 4 are satisfied. Finally, suppose that 2d − 1 is not free (hence β2 < α1 ). Then one can show that if t ∈ ω and j < 2d − 1, then |tj − t2d−1 | β2 (tj t2d−1 )−K/d(d+1) . Indeed, if j and 2d − 1 are in different bands, |tj − t2d−1 | > δα1 (tj t2d−1 )−K/d(d+1) and if j and 2d − 1 are in the same band, tj ∼ t2d−1 as was shown in Section 4.3, and the inequality follows from the construction of Ω2d−1 .
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One can produce a two-stage band structure as in [27] by partitioning the band B containing 2d − 1. For completeness, we note that one obtains the following lemma: Lemma 14. Let ε > 0. Then there exist parameters δ, δ , ρ, ρ satisfying 0 < cd,ε < ρ < ερ,
ρ < δ,
cd,ε < δ < εδ,
a set ω ⊂ Ω2d−1 with μ2d−1 (ω) ∼ μ2d−1 (Ω2d−1 ), and a two-stage band structure on {1, . . . , 2d − 1} satisfying the following: The first stage is a band structure on {1, . . . , 2d − 1}. Each even index is free after the first stage. The second stage is a band structure on the band B containing 2d − 1. Let t ∈ ω. Consider the bands created in the first stage. • If i and j lie in different bands, then |ti − tj | δα1 (ti tj )−K/d(d+1) . • If i is quasi-bound to j , then cn β1 (ti tj )−K/d(d+1) |ti − tj | < δα1 (ti tj )−K/d(d+1) . • If i is bound to j , then |ti − tj | < δ α1 (ti tj )−K/d(d+1) . Now we let i, j ∈ B. • If i and j lie in different bands, then |ti − tj | ργ2 (ti tj )−K/d(d+1) . • If i is quasi-bound to j , then cn β1 (ti tj )−K/d(d+1) |ti − tj | < ργ2 (ti tj )−K/d(d+1) . If i = 2d − 1 is quasi-bound to j , the lower bound is cn β2 (ti tj )−K/d(d+1) < |ti − tj |. • If i is bound to j , then |ti − tj | ρ γ2 (ti tj )−K/d(d+1) . Here, γ2 = max{α2 , β2 }. The proof of Lemma 14 is similar to arguments in [27], with modifications as in previous sections to handle the measure μ. Here one uses the fact that β2 < α1 implies that γ2 < α1 (because α2 < α1 ), and hence ti > γ2n for i ∈ B. With Lemma 14 proved, the proof of Lemma 4 is exactly as in [27], with adaptations made to handle the measure μ as in Section 4.3. One needs an analogue of Lemma 9, but the adaptation is straightforward. In making this adaptation, it is important to note the following: letting τ , s, σ be as in Section 4.3, then (32) and (34) hold whenever i is bound to j and j = j is free or quasi-free. Now Lemma 4 has been proved in all possible cases. Acknowledgments This article was adapted from part of the author’s PhD thesis, and she would like to thank her advisor, Mike Christ, for initially drawing her attention to the article [9] and for his continuing support and advice. In particular, Prof. Christ’s kind proofreading of and comments on earlier versions of this manuscript significantly improved the exposition. The author would also like to thank the anonymous referee at the JFA for many valuable comments and suggestions.
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References [1] J.-G. Bak, D.M. Oberlin, A. Seeger, Restriction of Fourier transforms to curves. II. Some classes with vanishing torsion, J. Aust. Math. Soc. 85 (1) (2008) 1–28. [2] J.-G. Bak, D.M. Oberlin, A. Seeger, Restriction of Fourier transforms to curves and related oscillatory integrals, Amer. J. Math. 131 (2) (2009) 277–311. [3] Y. Choi, Convolution operators with the affine arclength measure on plane curves, J. Korean Math. Soc. 36 (1) (1999) 193–207. [4] Y. Choi, The Lp − Lq mapping properties of convolution operators with the affine arclength measure on space curves, J. Aust. Math. Soc. 75 (2) (2003) 247–261. [5] M. Christ, On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc. 287 (1) (1985) 223–238. [6] M. Christ, Convolution, curvature, and combinatorics. A case study, Int. Math. Res. Not. IMRN 19 (1998) 1033– 1048. [7] M. Christ, Quasi-extremals for a Radon-like transform, preprint. [8] S. Dendrinos, M. Folch-Gabayet, J. Wright, An affine invariant inequality for rational functions and applications in harmonic analysis, preprint. [9] S. Dendrinos, N. Laghi, J. Wright, Universal Lp improving for averages along polynomial curves in low dimensions, J. Funct. Anal. 257 (5) (2009) 1355–1378. [10] S. Dendrinos, J. Wright, Fourier restriction to polynomial curves I: a geometric inequality, preprint. [11] S.W. Drury, Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc. 108 (1) (1990) 89–96. [12] S.W. Drury, B.P. Marshall, Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc. 97 (1) (1985) 111–125. [13] S.W. Drury, B.P. Marshall, Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc. 101 (3) (1987) 541–553. [14] P.T. Gressman, Convolution and fractional integration with measures on homogeneous curves in Rn , Math. Res. Lett. 11 (5–6) (2004) 869–881. [15] P.T. Gressman, Lp -improving properties of averages on polynomial curves and related integral estimates, Math. Res. Lett. 16 (6) (2009) 971–989. [16] H.W. Guggenheimer, Differential Geometry, McGraw–Hill Book Co., Inc., New York, San Francisco, Toronto, London, 1963. [17] N. Laghi, A note on restricted X-ray transforms, Math. Proc. Cambridge Philos. Soc. 146 (3) (2009) 719–729. [18] W. Littman, Lp − Lq -estimates for singular integral operators arising from hyperbolic equations. Partial differential equations, in: Proc. Sympos. Pure Math., vol. XXIII, Univ. California, Berkeley, CA, 1971, Amer. Math. Soc., Providence, RI, 1973, pp. 479–481. [19] D.M. Oberlin, Convolution with measures on flat curves in low dimensions, preprint, arXiv:0911.1471. [20] D.M. Oberlin, Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1) (1987) 56–60. [21] D.M. Oberlin, A convolution estimate for a measure on a curve in R4 , Proc. Amer. Math. Soc. 125 (5) (1997) 1355–1361. [22] D.M. Oberlin, A convolution estimate for a measure on a curve in R4 . II, Proc. Amer. Math. Soc. 127 (1) (1999) 217–221. [23] D.M. Oberlin, Convolution with measures on polynomial curves, Math. Scand. 90 (1) (2002) 126–138. [24] Y. Pan, Lp -improving properties for some measures supported on curves, Math. Scand. 78 (1) (1996) 121–132. [25] I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, Berlin, 1977. [26] P. Sjölin, Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in R 2 , Studia Math. 51 (1974) 169–182. [27] B. Stovall, Endpoint bounds for a generalized Radon transform, J. Lond. Math. Soc. (2) 80 (2) (2009) 357–374. [28] T. Tao, J. Wright, Lp improving bounds for averages along curves, J. Amer. Math. Soc. 16 (3) (2003) 605–638. [29] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton NJ, 1946.
Journal of Functional Analysis 259 (2010) 3230–3264 www.elsevier.com/locate/jfa
Hardy space estimates for the wave equation on compact Lie groups ✩ Jiecheng Chen a , Dashan Fan b,∗ , Lijing Sun b a Department of Mathematics, Zhejiang University, Hangzhou, China b Department of Mathematics, University of Wisconsin–Milwaukee, Milwaukee, WI 53201, USA
Received 18 June 2009; accepted 31 August 2010 Available online 15 September 2010 Communicated by C. Kenig
Abstract Let −L be the Laplacian. In this paper, we prove that on a compact Lie group G of dimension n, the √ −β
multiplier operator eis L (1 + L) 2 , s ∈ (0, 1], extends to a bounded operator on the Hardy space H p (G), β 0 < p < ∞, if and only if | p1 − 12 | n−1 . The result is an analogue of a well-known theorem in Euclidean space. © 2010 Elsevier Inc. All rights reserved. Keywords: Oscillating multiplier; H p spaces; Compact Lie groups; Fourier series; Wave equation
1. Introduction Let G be a connected, simply connected, semisimple compact Lie group of dimension n. Any C ∞ function f on G has the Fourier expansion f (x) =
dλ χλ ∗ f (x)
λ∈Λ
✩
This work is partially supported by the NSF of China (Grant Nos. 10931001, 10871173).
* Corresponding author.
E-mail addresses:
[email protected] (J. Chen),
[email protected] (D. Fan),
[email protected] (L. Sun). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.020
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
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(the detailed definition can be found in the second section). In this paper, we will study the H p boundedness of the oscillating multiplier operator Tβ,s (f )(x) = Kβ,s ∗ f (x),
β 0.
Here Kβ,s is a central kernel defined by
Kβ,s (y) =
λ∈Λ, λ+δ=0
eisλ+δ dλ χλ (ξ ), λ + δβ
where s ∈ R+ , δ is half the sum of all positive roots (see Section 2), and y is conjugate to the element exp ξ in a fixed maximal torus of G. Thus, Tβ,s (f ) has the Fourier expansion
Tβ,s (f )(x) =
λ∈Λ, λ+δ=0
eisλ+δ dλ χλ ∗ f (x) λ + δβ
for any f ∈ C ∞ (G). The main purpose of this paper is to study the H p boundedness of Tβ,s , where H p denotes the Hardy space on G. It is well known that H p = Lp when p > 1. In order to describe the motivation of the problem, we introduce a multiplier operator mβ,s on a more general connected Lie group G of polynomial volume growth (see [1]). Fix a choice of left invariant vector fields Y1 , Y2 , . . . , Yn that, together with their successive Lie brackets Yi1 , Yi2 , . . . , [Yim−1 , Yim ] . . . , generate the Lie algebra of G. Let L be the sub-Laplacian L = − Y12 + · · · + Yn2 . Using the spectral theorem, one can define on G the operator √
mβ,s :=
eis L β
(1 + L) 2
.
When G = G is a compact Lie group, mβ,s (f ) coincides with the operator Sβ,s (f )(x) =
λ∈Λ
eisλ+δ dλ χλ ∗ f (x), (1 + λ + δ2 )β/2
and the H p boundedness of Sβ,s is equivalent to that of Tβ,s . Let G be a compact Lie group. Consider the following Cauchy problem for the wave equation on G × R+ associated with L: ∂ 2u + Lu = 0, ∂s 2
u(x, 0) = f (x),
∂u(x, 0) = g(x), ∂s
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where s ∈ R+ denotes time. The solution to this problem is formally given by u(x, s) =
sin(sλ + δ) cos sλ + δ dλ χλ ∗ f (x). dλ χλ ∗ g(x) + λ + δ
λ∈Λ
λ∈Λ
In the spectral sense, we can write √ √ sin(s L ) u(x, s) = g(x) + cos(s L )f (x). √ L If we measure smoothness of the above solution u(x, s) for fixed s in terms of Sobolev norms of the form 2 β/2 1 + λ + δ f H p (G) := d χ ∗ f , λ λ β
H p (G)
λ∈Λ
where β ∈ R, we are naturally led to study the mapping properties of the operators such as Sβ,s and √ sin(s L ) . √ L(1 + L)β/2 When s = 1, we denote mβ,1 = mβ ,
Sβ,1 (f ) = Sβ (f ),
Tβ,1 = Tβ ,
Kβ,1 = Kβ .
In the last three decades, much work has been published regarding the H p mapping properties of mβ,s and its related operators on various Lie groups (see [1,10,11,13], etc.). Below, we list a few results that are of interest to us in this paper. Theorem A. (See [1].) Let G be a connected Lie group of polynomial volume growth with local dimension d. Then mβ is bounded on Lp (G), 1 p ∞, if 1 1 β − < . 2 p d Theorem B. (See [13].) Let Hm be the Heisenberg group of Euclidean dimension d = 2m + 1. Then mβ,s is bounded on Lp (Hm ), 1 p ∞, if 1 1 − < β . 2 p d − 1 The above theorems are slightly weaker than the following well-known sharp result on Rd : Theorem C. (See [12,14].) mβ is bounded on H p (Rd ), 0 < p ∞, if and only if 1 1 − β . 2 p d − 1
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Based on the above observation, one naturally expects to obtain the optimal result on certain Lie groups or manifolds, analogous to the classical result on Euclidean space as in Theorem C. In this paper, we study this topic for a compact Lie group G. We are aware that when p > 1, local analogues of Theorem C hold true for solutions to strictly hyperbolic differential equations. Indeed, as previously shown in [15], the sharp estimates for p > 1 in Theorem C hold true locally for more general classes of Fourier integral operators. It is possible to derive the estimate for p > 1 on a compact Lie group from results in [15]. However, we notice that in the literature, all estimates on mβ are on H p (G) for p 1, except in Theorem C, in which the estimate is on the underlying space G = Rd . Thus the investigation of the case 0 < p < 1 becomes more interesting, even in the local case. Motivated by the above facts, we will establish the following theorem in this paper: Theorem 1. Let G be a connected, simply connected, semisimple compact Lie group of dimension n. Assume s ∈ (0, 1]. Then: (1) Tβp is bounded on H p (G) with βp = ( p1 − 12 )(n − 1); (2) For 0 < p < ∞, Tβ,s (f ) p Cs f H p (G) H (G) if and only if 1 1 − β , 2 p n − 1 where Cs is a constant depending on s. By the theorem and a known result in [6], we easily obtain the following corollary: Corollary 2. For s ∈ (0, 1], Tβ,s (f ) if and only if β >
L1 (G)
Cs f L1 (G)
n−1 2 .
By checking the proof in this paper, it is not difficult to see that the boundedness of Tβ,s is equivalent to the boundedness of mβ,s , and that for 0 < p < ∞, √ sin(s L ) √ g Cs gH p (G) p β/2 L(1 + L) H (G) p
p
if β − 1 (n − 1)|1/2 − 1/p|. Thus for initial conditions f ∈ Hβ and g ∈ Hβ−1 in the Cauchy problem for the wave equation, the solution u satisfies u(·, s) p Cs f H p + gH p H (G) β
for all 0 < p < ∞ provided β (n − 1)|1/2 − 1/p|.
β−1
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As we mentioned above, the case p > 1 of our main theorem might be a deduction of results in [15], because of the compactness of G. The proof of the wave equation estimates in this paper, however, is different and uses Lie group machinery. Thus, for self independence, we will present proofs for the whole range of p ∈ (0, ∞). For simplicity, in the proof of the sufficiency part of the theorem, we will show the boundedness properties on the operator Tβ . The restriction to time s = 1 is inessential in our proof. In order to prove the sufficiency part of the theorem, we follow a standard complex interpolation [2] and define the analytic family of operators Tz with kernels Kz =
λ∈Λ: λ+δ=0
eiλ+δ dλ χλ , λ + δz
z ∈ C.
Then the sufficiency part of the theorem follows from the two inequalities Tz (f )
L2
f L2
Tz (f ) p f H p H
if Re(z) = 0,
1 1 − if Re(z) = (n − 1). p 2
The first inequality follows easily from Plancherel’s theorem, while the second inequality follows from (1) of the theorem. Thus, one key ingredient of the proof is to obtain the sharp H p estimate on the operator. To this end, we must carefully estimate the derivatives of the kernel by using some Lie group machinery developed in [7]. The plan of this paper is as follows: in Section 2, we will recall some necessary notation and definitions for a compact Lie group; the kernel Kβ will be decomposed and carefully estimated in Sections 3 and 4. With these preparations, we will prove the sufficiency part of the theorem in Sections 5.1–5.2, and prove the necessity part and the corollary in Sections 5.3 and 5.4, respectively. In this paper, we use the notation A B to mean that there is a positive constant C independent of all essential variables such that A CB. The notation A B means that there are two positive constants c1 and c2 independent of all essential variables such that c1 A B c2 A. 2. Notation and definitions Let G be a connected, simply connected, compact semisimple Lie group of dimension n. Let g be the Lie algebra of G and b the Lie algebra of a fixed maximal torus T in G of dimension m. 1 Let + be a system of positive roots for (g, b), so that | + | = n−m , and let δ = a∈ + a. 2 2 C Let | · | be the norm of g induced by the negative of the Killing form B on g , the complexification of g. Then | · | induces a bi-invariant metric dG on G. Furthermore, since B|bC ×bC is nondegenerate, given λ ∈ homC (bC , C), there is a unique Hλ in bC such that λ(H ) = B(H, Hλ ) for each H ∈ bC . We let , and · denote the inner product and norm transferred from b to homC (b, iR) by means of this canonical isomorphism. Let N = {H ∈ b, exp H = I }, where I is the identity in G. The weight lattice ℘ is defined by ℘ = {λ ∈ b: λ, n ∈ 2πZ for any n ∈ N} with dominant weights defined by Λ = {λ ∈ ℘,
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λ, a 0 for any a ∈ + }. Λ provides a full set of parameters for the equivalence classes of unitary irreducible representation of G: for λ ∈ Λ, the representation Φλ has dimension λ + δ, a
, δ, a
+
dλ =
a∈
and its associated character is w∈W
χλ (ξ ) =
(w)ei w(λ+δ),ξ
, D(ξ )
where ξ ∈ b, W is the Weyl group, (w) is the signature of w ∈ W , and
D(ξ ) =
ei wδ,ξ =
sin
α∈ +
w∈W
α, ξ
2
is the Weyl denominator. Any function f ∈ L1 (G) has the Fourier series
dλ χλ ∗ f (x).
λ∈Λ
The oscillating multiplier
Tβ (f )(x) =
λ∈Λ: λ+δ=0
eiλ+δ dλ χλ ∗ f (x) λ + δβ
is initially defined on all f ∈ C ∞ , where λ + δ2 = λ + δ, λ + δ . Thus, Tβ is a convolution operator Tβ (f )(x) = Kβ ∗ f (x) and Kβ is a central kernel defined by Kβ (y) =
λ∈Λ: λ+δ=0
eiλ+δ dλ χλ (ξ ), λ + δβ
where exp ξ ∈ T is conjugate to y. Let Q be a fixed fundamental domain. Any element y ∈ G is conjugate to exactly one element in exp(Q) and 0 ∈ Q. We denote y ∼ exp ξ if y is conjugate to exp ξ . Let Qν = {ξ + ν: ξ ∈ Q},
ν ∈ N,
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and Q0 = Q,
Q∞ =
Qν .
ν=0
Without loss of generality, we may assume Q0 ⊂ θ ∈ Rm : |θ | 4 . It is known that there is a σ > 0 such that |θ | > 1 + σ
for any θ ∈ Qν if ν = 0.
It is also known that, up to sets of measure zero, {Qν } is a sequence of mutually disjoint subsets in the Lie algebra b. Denote Γm =
Qν .
ν∈N
Γm is an unbounded subset of Rm . For any subset E = ξ ∈ Rm , c1 < |ξ | < c2 one has measure(E) measure(E ∩ Γm ). Next, we briefly review the definition of the Hardy space on G and some of its properties. For the heat kernel (see [16]) Wt =
e−t{λ+δ
2 −δ2 }
dλ χλ ,
λ∈Λ
the Hardy space H p (G) is the collection of all distributions f ∈ S (G) such that f H p (G) = sup|Wt ∗ f | t>0
Lp (G)
< ∞.
Since the Hardy–Littlewood maximal function is a bounded operator on Lp (G), if 1 < p < ∞, and since it majorizes the function supt>0 |Wt ∗ f |, we see from [3] that H p (G) = Lp (G) if 1 < p < ∞. Since the Lp (G) boundedness of Tβ,s is known when p > 1, it is of interest to study the boundedness of Tβ,s on H p (G) when 0 < p 1. To this end, we need to introduce the atomic decomposition of H p (G) spaces. The following definition of atomic H p spaces on G was given in [7] (see also [8] and [4] for some equivalent definitions of H p on G). An exceptional atom is an L∞ function bounded by 1. In order to define a regular atom, one considers a faithful unitary representation Φ of G. Then G can be identified as a submanifold in
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
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a real vector space V underlying End(CL ). A regular p-atom for 0 < p 1 is a function a(x) supported in a small ball B(x0 , ρ) such that −n( 1 − 1 ) aL2 (G) ρ p 2 and a(x)P Φ(x) dx = 0, G
where P is any polynomial on V of degree less than or equal to for any fixed integer [n( p1 − 1)], the integer part of n( p1 − 1). p The space Ha (G), 0 < p 1, is the space of all f ∈ S (G) having the form f=
ck ak
with
|ck |p < ∞,
where each a(x) is either a regular p-atom, or an exceptional atom. The “norm” f Hap is the infimum of all expressions ( |ck |p )1/p for which we have a representation f = ck ak . Various characterizations of Hardy spaces on compact Lie groups G were studied in [4]. In particular, we have f Hap (G) f H p (G) . p
For this reason, in the sequel we only need to study the H p boundedness on the atomic Ha spaces. Another important characterization of H p is that the space H p can be defined by using the Riesz transforms. For an integer L 0, and a multi-index J = {j1 , . . . , jN } ∈ {1, 2, . . . , n}L , let RJ (f ) denote the generalized Riesz transform RJ (f ) = Rj1 · · · RjL f , where Rj (f ) is the n−1 j th Riesz transform of f if j = 0 and R0 (f ) = f . It is known (see [5]) that for p > n−1+L and ∞ p all f ∈ C (G) ∩ H (G), RJ (f )
Lp
RJ (f )
f H p ,
Hp
f H p .
J
We now have Tβ,s (f )
Hp
RJ Tβ,s (f )
Lp
J
=
Tβ,s RJ (f )
Lp
.
J
Thus, since C ∞ (G) ∩ H p (G) is dense in H p (G) (see [4]), to prove Tβ,s (f ) p f H p , H it suffices to show that for any f ∈ C ∞ (G) ∩ Ha (G), Tβ,s (f ) p f Hap (G) . L (G) p
Furthermore, since f ∈ C ∞ (G) ∩ H p (G) has the atomic decomposition, by a standard argument (see [5] or [7]), to prove the H p boundedness of Tβ,s , 0 < p 1, one only needs to
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J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
show that there is a constant C, independent of atoms, such that for all exceptional and regular p-atoms a(x), Tβ,s (a)
Lp (G)
C.
3. Decomposing the kernel Kβ The kernel Kβ is a distribution kernel for small β. We need to treat the cases β β > n−1 2 differently. 3.1. Decomposing Kβ if β
n−1 2
and
n−1 2
It is known that for any y ∼ exp ξ , in the distribution sense (see [9]), Kβ (y) = Kβ (exp ξ )
=
eiλ+δ β λ+δ
λ∈Λ: λ+δ=0
( a∈ + δ + λ, a ) w∈W (w)ei w(λ+δ),ξ
D(ξ ) α∈ + α, δ
( a∈ + μ, a )ei μ,ξ
= D(ξ ) a∈ + a, δ
iμ ∂ ( a∈ + ∂α ) μ∈℘\{0} e β ei μ,ξ
μ . D(ξ ) a∈ + a, δ
eiμ μ∈℘\{0} μβ
Noting that the kernel Kβ might be a distribution kernel for small β, we need to decompose the kernel. Choose a C ∞ function Z(t) on the real line with support in the interval [ 12 , 2] and satisfying ∞
Zk (t) = 1
k=−∞
for all t ∈ (0, ∞), where Zk (t) = Z(2−k t). Since the set of dominant weights is a countable set with at most finitely many elements of any given norm, without loss of generality, we may write
∞ eiμ i μ,ξ
∂ eiμ i μ,ξ
∂ μ e = Z e . k ∂α μβ ∂α μβ + + μ∈℘ a∈
μ∈℘\{0}
a∈
k=0
Now we have Tβ (f )(x) = Kβ ∗ f (x) =
∞ k=0
Hk ∗ f (x)
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
with Hk (y) =
( a∈ +
∂ μ∈℘ ∂α )
D(ξ )
iμ
e i μ,ξ
Zk (μ) μ βe
a∈ + a, δ
.
By the Poisson summation formula (see [7,9,17]), we can write
ν∈N (
Hk (y)
∂ a∈ + ∂α )Gk (ξ
D(ξ )
+ ν)
a∈ + a, δ
,
where Gk (ξ ) = Rm
−i ξ,H
eiH H e Z dH. k H β
Denote Z(H ) , Ψ H = H β where Ψ is a C ∞ function with support in {H : Gk (ξ ) = 2−k(β−m)
1 2
H 2}. By this notation, we have
k H −2k ξ,H )
ei(2
Ψ H dH.
Rm
We now write the kernel Kβ by Kβ ∗ f (x) = Kβ,0 ∗ f (x) + Kβ,∞ ∗ f (x), where Kβ,0 (y) =
∞ ∂ ( a∈ + ∂α )Gk (ξ ) D(ξ ) a∈ + a, δ
k=0
and Kβ,∞ (y) =
∞ ( a∈ + k=0
3.2. An integral formula for Kβ when β >
∂ ν=0 Gk (ξ ∂α )
D(ξ )
a∈ + a, δ
+ ν)
.
n−1 2
Choose a C ∞ radial function ψ on Rm such that ψ(t) = 0
if t ∈ (0, c1 )
and ψ(t) = 1 if t ∈ (c2 , ∞),
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J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
where c1 and c2 are two fixed positive numbers such that, for y ∼ exp ξ , Kβ (y) =
eiμ μ∈℘ μβ
ψ(μ)( a∈ + μ, a )ei μ,ξ
. D(ξ ) a∈ + a, δ
By a similar argument to that used in the previous section for the case β summation formula, we have
n−1 2 , using the Poisson
∂ a∈ + ∂α )ν (ξ ) , D(ξ ) a∈ + a, δ
ν∈N (
Kβ (y)
(3.2-1)
(3.2-2)
where 0 (ξ ) = Rm
eiH ψ H e−i ξ,H dH β H
and ν (ξ ) = 0 (ξ + ν).
We denote (ξ ) = 0 (ξ ). By [17, Ch. 4] (or see [7]), ∞ (ξ )
eit ψ(t)V m−2 t|ξ | t −β+m−1 dt, 2
0
where V m−2 (t) = 2
and J m−2 (t) is the Bessel function of order 2
m−2 2 .
J m−2 (t) t
2 m−2 2
An easy computation shows (or see [7]),
∂ V m−2 t|ξ | α, ξ V n−2 t|ξ | t n−m . 2 2 ∂α + + a∈
a∈
We obtain that, in the integral formula (3.2-2) for Kβ ,
∞ ∂ ξ α, ξ
eit ψ(t)V n−2 t|ξ | t −β+n−1 dt 2 ∂α + + a∈
a∈
= |ξ |
2−n 2
0
a∈ +
∞ n α, ξ
eit ψ(t)J n−2 t|ξ | t 2 −β dt. 2
0
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
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The Bessel function satisfies (see [18]) J n−2 (t) e∓it t −1/2 + O t −3/2 2
so the last improper integral exists, since β >
as t → ∞,
n−1 2 .
3.3. Decomposing the Weyl denominator D(θ ) We introduce some notation from [7]. Let Π be the set of all simple roots in + and let ΠL be the set of all largest roots. For θ ∈ Q, introduce the following sets 1 , I = Iθ = α ∈ Π: α(θ ) R 1 J = Jθ = β ∈ ΠL : β(θ ) 2π − . R Here, we assume that R is a fixed large number so that elements in Iθ and Jθ are independent. We define the facet (a terminology in [7]) FI,J = ξ ∈ Q: α(ξ ) = 0 for α ∈ Iθ , β(ξ ) = 2π for β ∈ Jθ ,
0 < α(ξ ) < 2π for α ∈ Π\Iθ , 0 < β(ξ ) < 2π for β ∈ ΠL \Jθ ,
and let FI,J be the affine subspace generated by FI,J so that FI,J = ξ ∈ b: α(ξ ) = 0 for α ∈ Iθ and β(ξ ) = 2π for β ∈ Jθ . A positive root γ is R-singular of type 1 at θ if the following equivalent conditions are satisfied: (i) γ {FI,J } = {0}, (ii) γ {FI,J } = {0}, (iii) γ can be written as γ=
nα α,
nα ∈ N.
α∈Iθ
A positive root γ is R-singular of type 2 at θ , if the following equivalent conditions are satisfied: (i) γ {FI,J } = {2π}, (ii) γ {FI,J } = {2π}, (iii) γ can be written as γ =β−
α∈Iθ
for some β ∈ Jθ .
nα α,
nα ∈ N,
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Both R-singular roots of types 1 and 2 are called singular roots. By this definition, it is easy to see that if γ is a positive non-singular root, then 1 1 < γ (θ ) < 2π − . R R For a singular root α of type 1, let α be the orthogonal symmetry with respect to the hyperplane α = 0. For a singular root β of type 2, let β be the orthogonal symmetry with respect to the hyperplane α = 2π . Let WI,J be the group generated by {α }α∈Iθ ∪ { β }β∈Jθ . This group is a finite subgroup of the affine Weyl group. Now we define (R)
Γθ = Γθ
= convex hull of {wθ }w∈WI,J .
Also, we write the root system + + = + s ∪ ns + where + s is the set of all singular (R-singular) roots and ns is the set of all non-singular roots. Denote by μR the number of singular roots and denote
D R (θ ) =
sin
a∈ + ns
α, θ
. 2
Thus the Weyl denominator can be written as D(θ ) = D R (θ )
sin
a∈ + s
α, θ
. 2
(R)
In addition, the above definitions of D(θ ) and Γθ can be defined on the torus T itself. In fact, if x ∈ T , then x = exp θ for some θ in Q, and we define d(exp θ ) = D(θ ),
d R (exp θ ) = D R (θ )
(R)
and Γx(R) = exp Γθ
.
Some properties of the domain Γx(R) can be found in [7]. In particular, it is known from Lemma (2.9) in [7] that there exists a constant c > 0 such that R D (ξ ) cD R (θ ) (R)
for all ξ ∈ Γθ
.
(3.3-1)
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
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3.4. Derivatives on central functions Fixing a vector basis of gC , say Y1 , Y2 , . . . , Yn , we denote the element Y1 1 Y2 2 · · · Yn n by Y J , where J = (j1 , . . . , jn ). As J varies over all possible n-tuples, the {Y J } forms a basis of the complex universal enveloping algebra U (g) of g. Similarly, we fix a basis Θ1 , . . . , Θm of bC and im use the notation Θ I for Θ1i1 Θ2i2 · · · Θm , with I = (i1 , . . . , im ). We can find the following two theorems in [7]. j
j
j
Theorem D. (See [7].) Let p, q be positive integers. If f is a C ∞ central function, then there exists a constant C such that for each I with |I | p and J with |J | q, p+q I J Θ Y f (y) C Rj j =0
sup Θ K f (v). (R)
|K|p+q−j v∈Γy
Theorem E. (See [7].) Let p be a positive integer. Assume f (y) = d(y)−1 g(y) and that g is a C ∞ central function which is skew-invariant by the Weyl group. There exists a constant C such that for each I with |I | p p I Θ f (y) C d R (y)−1 Rj
sup Θ K g(v). (R)
j =0
|K|p+μR −j v∈Γy
3.5. Fixing a small r > 0 Let x ∼ exp θ , y ∼ exp ξ and xy −1 ∼ exp ζ . Noting exp ζ → exp θ
as dG (y, I ) → 0,
by compactness and Lemma 6.4 in [7], we have the following lemma. Lemma 3. There exist a small r1 > 0 and a large number M depending only on r1 such that if |sin α,θ
2 | Mρ then 1 |sin α,θ
2 |
1
|sin α,ζ 2 |
whenever dG (y, I ) ρ < r1 . By continuity, it is also easy to see that there is an r2 > 0 such that for all x, y ∈ G and ρ < r2 , we have θ + ν ζ + ν uniformly for ν ∈ N, whenever dG (y, I ) ρ < r2 and dG (x, I ) 100ρ. Choose a small r3 > 0 such that the number R in Section 3.3 is smaller than 1/r3 . We now fix r = min{r1 , r2 , r3 } throughout the rest of this paper.
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4. Estimating the kernel Kβ 4.1. Estimating each ( a∈ +
∂ ∂α )Gk (ξ )
Recall that we may obtain
Gk (ξ ) 2
−k(β−m)
∞
k ei2 t Ψ (t)V m−2 2k t|ξ | t m−1 dt. 2
0
By [17, Ch. 4] (or see [7]), up to a constant independent of k,
∞ n 2−n n ∂ k −k(β− −1) 2 Gk (ξ ) |ξ | 2 α, ξ 2 ei2 t Ψ (t)J n−2 2k t|ξ | t 2 dt. 2 ∂α + + a∈
a∈
0
Lemma 4. If 2k |ξ | 100, then
∂ n−m Gk (ξ ) 2−k(β−n) |ξ | 2 . ∂α + a∈
Proof. It is easy to obtain the lemma by observing
|ξ | n−m 2 , α, ξ
a∈ +
the well-known formula n−2 J n−2 (t) = O t 2 , 2
and that the support of Ψ (t) lies in the interval [ 12 , 2].
as t → 0 2
Lemma 5. Suppose 2k |ξ | > 100. Then for any positive integers N and L, we have
L −N ∂ 1−m n G (ξ ) |ξ | 2 −j 2−k(β− 2 −1/2+j ) 2−N k 1 − |ξ | k ∂α + j =0
a∈
−m−1 n + O |ξ | 2 −L 2−k(β− 2 +1/2+L) uniformly on k, where β =
n−1 2 .
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
3245
Proof. Using the asymptotic expansion of J n−2 (t|ξ |) (see [17,18]): 2
L −it|ξ | J n−2 t|ξ | e 2
j =0
+e
it|ξ |
aj (t|ξ |)j +1/2
L j =0
bj (t|ξ |)j +1/2
−(L+1)−1/2 , + O t|ξ |
(4.1-1)
where a1 , b1 , . . . , aL , bL are constants, we have
∞ n 2−n ∂ k −k(β− n2 −1) 2 Gk (ξ ) = |ξ | α, ξ 2 ei2 t Ψ (t)J n−2 2k t|ξ | t 2 dt 2 ∂α + + a∈
a∈
0
∞ L 1−n n k −j −k(β− −1/2+j ) 2 α, ξ
|ξ | 2 2 ei2 t (1±|ξ |) hj (t) dt
a∈ +
+O
j =0
0
−n+1 −L−1 −k(β− n2 +1/2+L) 2 , α, ξ |ξ | 2
a∈ +
where each hj is a C ∞ function with support in the interval [ 12 , 2]. Integration by parts now yields ∞ −N i2k t (1±|ξ |) hj (t) dt 2−N k 1 − |ξ | e
(4.1-2)
0
for any positive integer N . So the lemma follows from (4.1-2) and the known formula n−m α, ξ |ξ | 2 . 2 a∈ +
4.2. The kernel Kβ,∞ is integrable For the kernel Kβ,∞ (y) =
∞ ( a∈ + k=0
∂ ν=0 Gk (ξ ∂α )
D(ξ )
a∈ + a, δ
we have the following estimate: Proposition 6. If 0 < β
n−1 2 ,
then Kβ,∞ L1 (G) 1.
+ ν)
,
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Proof. Recall
Q∞ =
Qν ,
ν∈N\{0}
and 1 − |ξ | σ for all ξ ∈ Q∞ . By the Weyl integral formula we have ∞ ∂ ( a∈ + ∂α ) ν=0 Gk (ξ + ν) 2 Kβ L1 (G) = D(ξ ) dξ D(ξ ) a∈ + a, δ
Q k=0
∞ ∂ Gk (ξ )D(ξ ) dξ ∂α + k=0Q ∞
a∈
= j1 + j2 . Here, j1 =
j2 =
∞
k=0
Q∞ ∩{|ξ |<100/2k }
∞
k=0
Q∞ ∩{|ξ |100/2k }
∂ D(ξ ) dξ, G (ξ ) k ∂α + a∈
∂ Gk (ξ )D(ξ ) dξ. ∂α + a∈
By Lemma 4, j1
∞ k=0
2
−k(β−m)
dξ
|ξ |<100/2k
∞
2−kβ 1.
k=0
Choosing suitable N and L in Lemma 5, we have j2
∞ k=0
+
n
2−k(β− 2 −1/2) 2−N k
∞
|1−|ξ ||>σ
2
−k(β− n2 +1/2+L)
k=0
This completes the proof.
1 − |ξ |−N |ξ | 1−m 2 D(ξ ) dξ
|ξ |>1
2
|ξ |
−n−1 2 −L
D(ξ ) dξ 1.
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
4.3. Derivatives on Kβ when β >
3247
n−1 2
The main purpose of this section is to decompose the kernel Kβ by writing Kβ (y) = K β,∞,0 (y) + K β,∞,∞ (y) + K β,0,0 (y) + K β,0,0 (y), and then to give different estimates for these four kernels. Recall that for y ∼ exp ξ and β > n−1 2 , we obtained in Section 3.2 that Kβ (y)
∂ a∈ + ∂α )ν (ξ ) , D(ξ ) a∈ + a, δ
ν∈N (
where
∞ ∂ it ν (ξ ) e ψ(t) α, ξ + ν V n−2 t|ξ + ν| t −β+n−1 dt. 2 ∂α + + a∈
a∈
0
Fix a positive number ρ < r, where r is the number chosen in Section 3.5. Let ϕ(t) be a C ∞ function on (0, ∞) that satisfies ϕ(t) = 0 if t < 0.05,
ϕ(t) = 1 if t > 0.1,
and let ϕ∞ (t) = ϕ(ρt). We set ϕ0 (t) = 1 − ϕ∞ (t). Now write Kβ (y) = K β,∞ (y) + K β,0 (y). In the definition of K β,∞ (y) we replace the factor ( a∈ +
∂ ∂α )(ξ )
in Kβ by
∞ ∂ ∞ (ξ ) = eit ϕ∞ (t) α, ξ V n−2 t|ξ | t −β+n−1 dt, 2 ∂α + + a∈
0
a∈
and in the definition of K β,0 (y) we replace the factor ( a∈ +
∂ ∂α )(ξ )
in Kβ by
∞ ∂ 0 α, ξ V n−2 t|ξ | t −β+n−1 dt. (ξ ) = eit ϕ0 (t)ψ(t) 2 ∂α + + a∈
0
a∈
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We further write
ν∈N\{0} (
K β,∞ (y)
∂ ∞ a∈ + ∂α )ν (ξ )
D(ξ )
+
∂ ( a∈ + ∂α )∞ (ξ ) D(ξ )
=K β,∞,∞ (y) + K β,∞,0 (y), where ∞ ∞ ν (ξ ) = (ξ + ν).
We have the following estimates for these two kernels: Proposition 7. Let r be the small number chosen in Section 3.5 and let 0 < ρ < r. Suppose y ∼ exp ξ . Then for any integers L and N we have N K β,∞,0 (y) j =0
−L n+1 α∈ + | α, ξ | 1 − |ξ | ρ (β− 2 +L+j ) n−1 +j 2
|ξ |
+ρ
|D(ξ )|
ρ (β− n−1 2 +N )
(β− n−1 2 +N ) α∈ + n+1 +N 2
|(1 − |ξ |)| K β,∞,0 (y)
|ξ |
β− n+1 2
α∈ +
| α, ξ |
|D(ξ )|
| α, ξ |
|D(ξ )|
if 1 − |ξ | ρ;
if 1 − |ξ | < 10ρ;
and
L K β,∞,∞ (y) dy ρ .
G
Proof. As in the argument in Section 4.1, we use the asymptotic expansion of the Bessel function [18] to obtain, for any integer N 0,
∞ N n−1 ∂ a∈ + α, ξ
∞ (ξ ) aj eit (1−|ξ |) ϕ∞ (t)t −β+ 2 −j dt n−1 ∂α |ξ | 2 +j + a∈
j =0
+
0
N
bj
|ξ |
j =0
+O
a∈ + α, ξ
n−1 2 +j
a∈ + α, ξ
n+1 2 +N
|ξ |
∞
e−it (1+|ξ |) ϕ∞ (t)t −β+
0
∞ ϕ∞ (t)t 0
n−1 2 −j
−β+ n−1 2 −N −1
dt .
dt
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
3249
It is easy to see that ∞
ϕ∞ (t)t −β+
n−1 2 −N −1
dt ρ β−
n−1 2 +N
.
0
On the other hand, using integration by parts in ∞
e−it (1±|ξ |) ϕ∞ (t)t −β+
n−1 2 −j
dt
0
as many times as necessary, by the choice of ϕ∞ , it is easy to see that ∞ −L n+1 −it (1±|ξ |) −β+ n−1 −j 2 ϕ∞ (t)t dt 1 − |ξ | ρ (β− 2 +L+j ) e
for j = 1, 2, . . . , N,
0
where the positive integer L can be obtained as large as necessary. If |(1 − |ξ |)| < 10ρ, by changing variables we obtain ∞ it (1−|ξ |) −β+ n−1 −j 2 ϕ∞ (t)t dt e 0
∞ β− n+1 +j β− n+1 +j n−1 it −β+ −j 2 2 e t 2 1 − |ξ | dt 1 − |ξ | . 0.1
∂ Taking these estimates into ( a∈ + ∂α )∞ (ξ )/D(ξ ), we obtain the estimate for K β,∞,0 (y). In addition, by the Weyl integral formula we have G
∂
∞ −1 K dx (y) (ξ )D(ξ ) β,∞,∞ ν dy ∂α + ν∈N\{0} G
|ξ |>1+σ
N
ρ
a∈
∂ ∞ (ξ )D(ξ ) dξ ∂α + a∈
(β− n+1 2 +L+j )
j =0
+ρ
β− n−1 2 +N
| a∈ + α, ξ ||D(ξ )|
|ξ |>1+σ
|ξ |>1+σ
|ξ |
n−1 2 +j
|(1 − |ξ |)|L
| a∈ + α, ξ | |ξ |
n+1 2 +N
dξ.
dξ
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Noting that N and L can be chosen arbitrarily, we easily obtain L K β,∞,∞ (y) dy ρ . G
This completes the proof.
2
We now study derivatives on the kernel K β,0 . Recall
∂ 0 a∈ + ∂α )ν (ξ ) , D(ξ ) a∈ + a, δ
ν∈N (
K β,0 (y) where
∞
∂ 0 α, ξ + ν
eit ϕ0 (t)ψ(t)V n−2 t|ξ + ν| t −β+n−1 dt. ν (ξ ) = 2 ∂α + + a∈
a∈
0
We continue to decompose the kernel K β,0 (y) by writing K β,0 (y) = K β,0,0 (y) + K β,0,∞ (y), where K β,0,∞ (y)
ν∈N\{0} (
D(ξ )
∂ 0 a∈ + ∂α )ν (ξ )
a∈ + a, δ
and ∂ ( a∈ + ∂α )0 (ξ ) K . (y) β,0,0 D(ξ ) a∈ + a, δ
Proposition 8. For any non-negative integer L and any multi-index M with |M| = q, M −1 −L q Y K sup d R (x) |ν| . β,0,∞ (z) R (R)
ν=0
x∈Γz
Proof. The kernel K β,0,∞ is skew-invariant by the Weyl group. By Theorems D and E, we obtain M Y K β,0,∞ (z) sup Θ J K β,0,∞ (x) (R)
|J |q x∈Γz
−1 R q sup d R (x) (R)
x∈Γz
|J |q |I ||J |+μR
I ∂
∂ 0ν (ξ ), sup ∂ξ ∂α (R) (R) + x∈Γ y∈Γ sup z
x
ν∈N\{0}
a∈
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3251
where exp ξ ∼ y. We have (see [7]) |I | ∂ I ∂
Pk+ n−m (ξ ) ∞ −β+ n +k 0 it 2 2 0 (ξ ) e ϕ0 (t)ψ(t)J n−2 +k t|ξ | t dt , n−2 2 ∂ξ ∂α |ξ | 2 +k + k=0
a∈
0
where Pk is a homogeneous polynomial of degree k. For each k, using the asymptotic expansion of the Bessel function (see (4.1-1)), we have Pk+ n−m (ξ )
∞
|ξ | =
2 n−2 2 +k
N
2
0
Pk+ n−m (ξ )
∞
aj
|ξ |
j =0
+
n eit ϕ0 (t)ψ(t)J n−2 +k t|ξ | t −β+ 2 +k dt
N
eit (1−|ξ |) ϕ0 (t)ψ(t)t −β+
n−1 2 −j +k
dt
0
Pk+ n−m (ξ )
∞
bj
j =0
+O
2 n−1 2 +j +k
|ξ |
2 n−1 2 +k+j
0
∞ Pk+ n−m (ξ ) 2
|ξ |
n+1 2 +N +k
e−it (1+|ξ |) ϕ0 (t)ψ(t)t −β+
n−1 2 −j +k
dt
ϕ0 (t)ψ(t)t
−β+ n−1 2 −N −1+k
(4.3-1)
dt .
0
Noting that we may use integration by parts on the t-variable as many times as necessary, we now obtain that when |ξ | 1 + σ ,
∂ ∂ξ
I
∂ 00 (ξ ) = O |ξ |−L ∂α + a∈
(R)
for any positive integer L. So we obtain the proposition by noting that for any y ∈ {Γx (R) Γz } and ν = 0, one has sup sup |ξ + ν|−L |ν|−L , ξ ∈ Q, (R)
x∈Γz
where y ∼ exp ξ .
: x∈
(R)
y∈Γx
2
For the kernel K β,0,0 , we have the following proposition. 1 . For any multi-index J with |J | = q, and any integer N Proposition 9. Suppose dG (z, I ) < 10 n−1 satisfying −β + 2 + q < N , we have
J Y K β,0,0 (z) sup (R)
y∈Γz
where y ∼ exp ξ .
1 |ξ |
n+1 2 +N
,
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Proof. By Theorem D, without loss of generality, we may write J ∂ sup Kβ,0,0 (y). ∂ξ (R)
J Y K β,0,0 (z)
|J |=q y∈Γz
Since dG (z, I ) <
1 10
(R)
and y ∈ Γz
, without loss of generality we may assume that + α, ξ
a∈ + a∈ sin α, ξ
is analytic. Using this to our advantage, without loss of generality, we may assume J J ∞ ∂ ∂ −β+n−1 it e ϕ0 (t)ψ(t)V n−2 t|ξ | t dt . ∂ξ Kβ,0,0 (y) ∂ξ 2 0
Thus by (4.3-1), we know that, for y ∼ exp ξ , ξ ∈ Q, J |J | N ∂ (y) K β,0,0 ∂ξ k=0 j =0
+
∞ n−1 it (1−|ξ |) −β+ 2 −j +k e ϕ (t)ψ(t)t dt 0 n−1 +j |ξ | 2 1
0
|J | N k=0 j =0
+O
∞ n−1 −it (1+|ξ |) −β+ 2 −j +k e ϕ (t)ψ(t)t dt 0 n−1 +j |ξ | 2 1
0
1 |ξ |
∞
n+1 2 +N
ϕ0 (t)ψ(t)t
−β+ n−1 2 −N −1+|J |
dt .
(4.3-2)
0
1 Noting dG (z, I ) < 10 implies that |ξ | 1/2 for any y ∈ Γz(R) , using integration by parts, we easily obtain the proposition. 2
√ + Proposition 10. Let x ∼ exp θ . Suppose |sin α,θ
2 | M ρ for all α ∈ , where M is sufficiently large. In addition, assume dG (z, I ) < ρ < r and dG (x, I ) > 100ρ. For any multi-index J with |J | = q and any non-negative integer L, if q2 > L + β − n+1 2 then we have −1 J 1 − |θ |−L ρ β− n+1 Y K 2 −q+L . β,0,0 (xz) D(θ ) Proof. By Theorem D in Section 3.4, we have q J Y K Rj β,0,0 (xz) j =0
I ∂ sup Kβ,0,0 (y) ∂ξ (R)
|I |q−j y∈Γxz
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3253
where y ∼ exp ξ . Since J
∂ α, ξ −|J | 1 sin 1 . max ∂ξ D(ξ ) |D(ξ )| α∈ + 2 Choosing R =
√1 , ρ
by the assumption of the proposition and (3.3-1), it is easy to see
sup (R)
y∈Γxz
α, ξ −|J | |J | 1 1 ρ− 2 max sin . |D(ξ )| α∈ + 2 |D(θ )|
Thus, by the Leibniz rule and checking the proof of Proposition 9, with the assumptions on x and z we only need to show ∞ −L n+1 dt it (1−|ξ |) −β+ n−1 +q 2 ϕ0 (t)ψ(t)t 1 − |ξ | ρ β− 2 −q+L , e t 0
since the estimates on other terms are exactly the same. Using integration by parts on the t-variable L times and noting the choice of the function ϕ0 (t)ψ(t), we may write ∞ ∞ −L n−1 it (1−|ξ |) −β+ n+1 +q it (1−|ξ |) −β+ +q−L 2 2 ϕ0 (t)ψ(t)t dt 1 − |ξ | e ϕ0 (t)ψ(t)t dt e 0
0
−L 1 − |ξ |
∞
ϕ0 (t)t −β+
n−1 2 +q−L
dt.
0
Thus, the proposition follows easily by the definition of ϕ0 .
2
5. Proof of the theorem 5.1. H p boundedness of Tβp , βp = ( p1 − 12 )(n − 1) In this section, we will prove (1) of the theorem. As mentioned in Section 2, to prove the H p boundedness of Tβp , it suffices to show Tβ (a) p 1 p L (G) uniformly for all atoms a. Since Tβp is bounded on L2 , we have Tβ (a) p Tβp (a)L2 (G) 1 p L (G) r , uniformly for all exceptional atoms and p-atoms a whose supports B(x0 , ρ) satisfy ρ 100 where the positive number r was chosen in Section 3.5. Thus, by the translation invariance, to
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complete the proof of the result we need to show that for any p-atom a of support in B(I, ρ) r , we have with ρ < 100 Tβ (a) p 1. p L (G)
(5.1-1)
By Hölder’s inequality, p a(y)Kβ xy −1 dy dx p
1
B(I,100ρ n )
G
p 2 2 a(y)Kβp xy −1 dy dx ρ (2−p)/2
G B(I,ρ)
a
p ρ (2−p)/2 L2−βp
ρ −(1−p/2) ρ (2−p)/2 = 1.
The last inequality is obtained by the Sobolev imbedding theorem. Let Cρ = x ∈ G: x ∼ exp θ with 1 − 1000ρ < |θ | < 1 + 1000ρ . For each α ∈ + , we define the set Bα,M
= x ∼ exp θ :
α, θ M √ρ , sin 2
where M is a large positive number. Clearly, the volumes of Cρ and Bα,M satisfy |Cρ | ρ
for each α ∈ + .
and |Bα,M | ρ
By Hölder’s inequality and the Sobolev imbedding theorem, we have p p a(y)Kβ xy −1 dy dx + a(y)Kβ xy −1 dy dx p p Cρ G
ρ
−(1−p/2)
α∈ + Bα,M G
|Cρ |
(2−p)/2
+
|Bα,M |
(2−p)/2
1.
α∈ +
Denote 1 C Eρ = x ∈ G: dG (x, I ) > 100ρ n ∩ CρC ∩α∈ + Bα,M , where E C denotes the complement of a set E. It remains to show −1 p a(y)K dy dx 1, βp ,0,0 xy Eρ G
(5.1-2)
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
−1 p a(y)K dx 1 xy dy βp ,0,∞
3255
(5.1-3)
Eρ G
and −1 p a(y)K dy dx 1, βp ,∞ xy Eρ G
where K βp ,0,0 , K βp ,0,∞ and K βp ,∞ are defined in Section 4.3. By Hölder’s inequality, Proposition 7 and Lemma 3, we have −1 p a(y)K dy dx βp ,∞ xy Eρ G
aL1 (G) K βp ,∞,∞ L1 (G) +
Eρ G
p
1 + aL1 (G) 1 + ρ −n+pn
−1 p dx K βp ,∞,0 xy
sup
Eρ
−1 p a(y)K dy dx βp ,∞,0 xy
dG (y,I )<ρ
n+1 1 − |θ | p(βp − 2 ) dθ
|1−|θ||<10ρ
+ρ
−n+np
N
ρ
(βp − n+1 2 +L+j )p
j =0
+ ρ −n+np
{|1−|θ||>ρ}∩Eρ
{|1−|θ||>ρ}∩Eρ
ρ
(βp − n−1 2 +N )p
|θ |p
n+1 2 +Np
−pL |θ |n−m 1 − |θ | dθ p n−1 +pj |θ | 2
|θ |n−m dθ.
Thus, it is easy to compute that −1 p a(y)K dy dx 1 βp ,∞ xy Eρ G
if we choose L p2 . This proves (5.1-4). Let n0 = 2[ pn − n] + 2. Using the cancellation condition of a, we have G
−1 dy = a(y)K βp ,0,∞ xy
−1 − Tnx0 (K a(y) K βp ,0,∞ xy βp ,0,∞ )(y) dy,
G
where Tnx0 (K βp ,0,∞ ) is the Taylor polynomial of K βp ,0,∞ at x. Hence,
(5.1-4)
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−1 a(y)K ρ n0 +1 xy dy β ,0,∞ p G
a(y) dy
B(I,ρ)
ρ
− pn +n+1+n0
sup
z∈B(x,ρ), |J |n0 +1
sup
z∈B(I,ρ), |J |n0 +1
J Y K β ,0,∞ (z) p
J Y K βp ,0,∞ (xz) .
Noting that we may fix any small r > 0 and assume ρ r/100 in our proof, by Proposition 8 we have −1 −L J n0 R Y K sup d (x) |ν| , βp ,0,∞ (xz) R z∈B(I,ρ), |J |n0 +1
ν=0
for any positive integer L, where we may let R =
√ ρ. This observation gives us
−1 p dy dx 1. a(y)K βp ,0,∞ xy
Eρ B(I,ρ)
It now remains to prove (5.1-2). We define Eρ,0 = x ∈ Eρ : x ∼ exp θ with |θ |
1 1000
and Eρ,1 = x ∈ Eρ : x ∼ exp θ with |θ | >
1 . 1000
Then, using the same argument as before, by the cancellation of a we obtain
−1 p dx xy dy a(y)K βp ,0,0
Eρ B(I,ρ)
ρ
−n+pn+p+pn0
sup
Eρ,0
+ρ
−n+pn+p+qp
p J Y K βp ,0,0 (xz) dx
z∈B(I,ρ), |J |n0 +1
sup
Eρ,1
z∈B(I,ρ), |J |q+1
p J Y K βp ,0,0 (xz) dx,
where q is sufficiently large. From Proposition 9, we choose N = [ p1 ] to obtain ρ −n+pn+p+pn0
sup
Eρ,0
z∈B(I,ρ), |J |n0 +1
p J Y K βp ,0,0 (xz) dx
ρ 1/n 2 100 <|θ|< 3
|D(θ )|2 |θ |p(
By Proposition 10, we choose a sufficiently large q and a suitable L to obtain
n+1 2 )+1
dθ 1.
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
ρ
−n+pn+p+qp
Eρ,1
sup
z∈B(I,ρ), |J |q+1
ρ −n+pn+p+qp ρ pβp −p
3257
p J Y K βp ,0,0 (xz) dx
n+1 2 −pq−p+pL
1 − |θ |−pL dθ
ρ 1000 |1−|θ||4
ρ −n+pn+pβp −p
n+1 2 +1
= 1.
The H p boundedness of Tβp is proved. 5.2. Proving the sufficiency part of (2) in Theorem 1 We use a standard complex interpolation. Define an analytic family of operators
Tz (f )(x) = Kz ∗ f (x) =
λ∈Λ: λ+δ=0
By the Plancherel formula, we have Tz (f )
L2 (G)
eiλ+δ dλ χλ ∗ f (x), λ + δz
f L2 (G)
z ∈ C.
if Re z = 0.
Checking the proofs in Section 5.1, it is easy to see Tz (f ) p f H p (G) H (G)
(5.2-1)
(5.2-2)
if Re z = ( p1 − 12 )(n − 1). Thus, by a complex interpolation (see [2]) on inequalities (5.1-1), (5.1-2), we have, for 0 < p 2, Tβ (f ) p f H p (G) H (G) if
1 p
−
1 2
β n−1 .
When p > 2 the theorem follows trivially by a dual argument.
5.3. Proving necessity part of Theorem 1 For simplicity, again we assume s = 1. We first show that the condition 1 1 − β p 2 n − 1 is necessary for the Lp boundedness of Tβ for p > 1. To this end, clearly we only need to consider 0 < β n−1 2 . Fix an orthonormal basis X1 , X2 , . . . , Xn of the Lie algebra g such that X1 , X2 , . . . , Xm form an orthonormal basis of b. Let r > 0 be the fixed small positive number as be the exponential mapping from g to G. We may choose r small such that in Section 3.5 and Φ is a C ∞ -diffeomorphism from the neighborhood Φ B(I, r) = y ∈ G: dG (y, I ) < r
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J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
of the origin in Rn , which satisfies dG (u, v) ≈ |Φ(u) to a neighborhood N − Φ(v)|, for all u, v ∈ B(I, r), where |Φ(u) − Φ(v)| is the Euclidean distance between Φ(u) and Φ(v). Without loss of generality, we may assume 1 − Φ(v) . − Φ(v) dG (u, v) 2Φ(u) Φ(u) 2 Let Ωr = {ξ ∈ b: exp ξ ∈ B(I, r)}. Ωr is a neighborhood of 0 in b. We can fix r sufficiently small such that all translations of Ωr by elements of N are disjoint. Let η(ξ ) be a C ∞ function supported in Ωr/10 and identically one on Ωr/20 . We also choose 0 η(ξ ) 1. Let fε be a ξ +n central function defined by fε (exp ξ ) = n∈N η( 0.001ε ). It is easy to check that for all ε in the interval (0, r), we have for any 1 p ∞,
fε (x)p dx ≈
2 ξ + n p D(ξ ) dξ η 0.001ε
Q n∈N
G
p ξ D(ξ )2 dξ ε n . = η 0.001ε Q
On the other hand, we may write Tβ (f )(x) =
λ∈Λ: λ+δ=0
eiλ+δ Ψ λ + δ dλ χλ ∗ f (x), β λ + δ
where Ψ (t) is a C ∞ function on the interval (0, ∞) and satisfies Ψ (t) ≡ 0 if 0 < t <
1 100
and Ψ (t) ≡ 1 if t >
1 . 50
We define the Abel mean Gu , u > 0, by Gu (f )(x) =
e−uλ+δ dλ χλ ∗ f (x).
λ∈Λ
It is known from [19] that for all 1 p ∞, Gu (f )
Lp (G)
f Lp (G)
and lim Gu (f )Lp (G) = f Lp (G) .
u→0+
J. Chen et al. / Journal of Functional Analysis 259 (2010) 3230–3264
3259
Thus, lim Gu Tβ (f ) Lp (G) = Tβ (f )Lp (G) .
u→0+
Now, Gu Tβ (f ) (x) = Gu (Kβ ) ∗ f (x), where Gu (Kβ )(y) =
λ∈Λ: λ+δ=0
eiλ+δ e−uλ+δ Ψ λ + δ dλ χλ (y) β λ + δ
is a C ∞ kernel for each u > 0. Arguing the same as in Section 3.1, by the Poisson summation formula we have
∞
Gu (Kβ )(y)
ν∈N 0
α, ξ + ν
V n−2 t|ξ + ν| t n−1−β dt eit−ut Ψ (t) α,ξ
2 sin 2 a∈ +
= Gu (Kβ,0 )(y) + Gu (Kβ,∞ )(y), where Gu (Kβ,0 )(y) =
a∈ +
α, ξ
sin
∞
α,ξ
2
eit−ut Ψ (t)V n−2 t|ξ | t n−1−β dt 2
0
and Gu (Kβ,∞ )(y)
∞ α, ξ + ν ν∈N\{0}
a∈ +
sin
α,ξ
2
eit−ut Ψ (t)V n−2 t|ξ + ν| t n−1−β dt. 2
0
Observe that the singularity of
∞ α, ξ + ν
a∈ +
sin
α,ξ
2
eit−ut Ψ (t)V n−2 t|ξ + ν| t n−1−β dt 2
0
is at |ξ + ν| = 1 and that for all ν ∈ N\{0}, |ξ + ν| > 1 + σ
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uniformly on ξ ∈ Q. By Proposition 6 and the Young inequality, we have Gu (Kβ,∞ ) ∗ f p Kβ,∞ ∗ f Lp (G) f Lp (G) . L (G) We construct the sets E1 , E2 by E1 = θ ∈ Q: |θ | > r , α, θ
2 D(θ ) r , and r E2 = θ ∈ E1 : 0 0 , sin α,θ
2 a∈ + where r0 is a fixed small positive number such that −k −k 1 − |θ | 1 − |θ | dθ ≈ dθ. E1
E2
We also let A = {x ∈ G: x ∼ exp θ for θ ∈ E2 }. Now, if Tβ is bounded on Lp , we must have fε Lp Tβ (fε )Lp Gu (Tβ )(fε )Lp Gu (Kβ,0 ) ∗ fε Lp − fε Lp . Thus, we have fε Lp Gu (Kβ,0 ) ∗ fε Lp p 1 p −1 Gu (Kβ,0 ) xy fε (y) dy dx . A G
Let x ∼ exp θ . Using the asymptotic expansion of the Bessel function, we have, for x ∈ A, − n−2 2
Gu (Kβ,0 )(x) = |θ |
a∈ +
=
a∈ +
+
α, θ
sin
+O
sin
α,θ
2
sin
α,θ
2
N
n eit−ut Ψ (t)J n−2 t|θ | t 2 −β dt 2
0 − n−1 2 −j
α,θ
2
∞
aj |θ |
j =0
α, θ
a∈ +
∞
α, θ
0
N
bj |θ |−
j =0
α, θ
a∈ +
sin α,θ
2
1
n
eit (1−|θ|)−ut Ψ (t)t 2 −β− 2 −j dt
n−1 2 −j
∞
n
1
eit (1+|θ|)−ut Ψ (t)t 2 −β− 2 −j dt
0 − n−2 2 −(N +1)−1/2
∞
|θ |
Ψ (t)t 0
n 2 −β−(N +1)−1/2
dt ,
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3261
where {aj } and {bj } are constants, and in particular, a0 = 0. Choose N for which n − β − (N + 1) − 1/2 < −1. 2 This inequality, along with an integration by parts on the integral ∞
1
eit (1±|θ|)−ut Ψ (t)t 2 −β− 2 −j dt, n
0
now yield that
Gu (Kβ,0 )(x) =
a∈ +
N
α, θ
sin
α,θ
2
∞
− n−1 2 −j
aj |θ |
j =0
1
eit (1−|θ|)−ut Ψ (t)t 2 −β− 2 −j dt + C(θ, u), n
0
where |C(θ, u)| 1 uniformly on θ ∈ E2 and u > 0. By Lemma 2 in [12], we know that for each ε > 0, ∞ lim
u→0
n 1 eit (1−|θ|)−ut Ψ (t)t −β+ 2 − 2 −j dt = Fμ 1 − |θ |
0
uniformly on |(1 − |θ |)| ε, where n 1 − − j, 2 2 −μ−1 Fμ 1 − |θ | = Aμ 1 − |θ | + i0 + F |θ | μ = −β +
if μ = −1, −2, . . .
and −μ−1 Fμ 1 − |θ | = 1 − |θ | Bμ + Cμ log 1 − |θ | + i0 + F |θ | if μ = −1, −2, . . . . Here, Aμ , Bμ , Cμ are non-zero constants depending only on μ, and F (|θ |) is a smooth function. We estimate the case for which n2 − β − 12 = −1, −2, . . . (the estimate of the other case is the same). If ε |(1 − |θ |)| 1, we have, in the distribution sense, lim Gu (Kβ,0 )(x) =
u→0
a∈ +
α, θ
sin
α,θ
2
|ξ |−
n−1 2
β− n − 1 2 2 a0 A n −β− 1 1 − |θ | + i0 2
2
β− n + 1 + O 1 − |θ | 2 2 h1 (θ ) + h2 |θ | , where h1 and h2 are continuous and bounded functions on the closure of the set E2 . Without loss of generality, we assume that the constant κ = a0 A n −β− 1 is positive. 2
2
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Returning to p 1 p Gu (Kβ,0 ) xy −1 fε (y) dy dx , A G
we note that the choice of fε implies p Gu (Kβ,0 ) xy −1 fε (y) dy dx = 0 A G
only if dG (y, I ) |ξ | < 0.0001rε. Also, we can choose ε as small as necessary. When x ∈ A, by continuity we may write κ lim Gu (Kβ,0 ) xy −1 u→0 100
a∈ +
β− n − 1 n−1 2 2 |ξ |− 2 1 − |θ | α,θ
α, θ
sin
2
β− n + 1 − 104 1 − |θ | 2 2 h1 (θ ) − 104 h2 |θ | , whenever ε 1 − |θ | 1. Combining all results, we obtain p 1 p Gu (Kβ,0 ) xy −1 fε (y) dy dx u→0
fε Lp lim
A G
βp− np − p 2 2 dθ 1 − |θ |
1/p fε L1 .
E2 ∩{ε1−|θ|}
Thus, the Lp boundedness of Tβ implies fε Lp 1, ε→0 u→0 Gu (Tβ )(fε )Lp lim lim
which is true if and only if lim ε −n+pn
ε→0
Since βp −
np 2
−
p 2
βp− np − p 2 2 dθ 1. 1 − |θ |
E2 ∩{ε1−|θ|}
< −1, the last inequality holds if and only if βp −
np p − + 1 n − pn. 2 2
Thus, we have proven the necessity condition 1 1 β − p 2 n−1
(5.3-1)
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in the case 1 < p 2. By an easy dual argument, we obtain the necessary condition 1 1 − β p 2 n − 1 for the Lp boundedness of Tβ for all 1 < p < ∞. Clearly, (5.3-1) is also a necessary condition for the H p boundedness of Tβ for 0 < p 1. Otherwise, a complex interpolation used in Section 5.2 would yield a contradiction to the proven case for p > 1. 5.4. Proof of Corollary 2 Again, we assume s = 1. The sufficiency part of the corollary is proved in [9]. To prove the necessity part, it suffices to show that the operator T n−1 is not bounded on L1 . Using the same 2
proof as in Section 5.3, we find that T n−1 is bounded on L1 only if 2
lim
ε→0 E2 ∩{ε1−|θ|}
−1 1 − |θ | dθ 1,
which is clearly not possible. Acknowledgment We are grateful to the referee for his very helpful suggestions and comments. References [1] G. Alexopoulos, Oscillating multipliers on Lie groups and Riemannian manifolds, Tohoku Math. J. 46 (1994) 457– 468. [2] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press, 1988. [3] B. Blank, Nontangential maximal functions over compact Riemannian manifolds, Proc. Amer. Math. Soc. 103 (1988) 999–1002. [4] B. Blank, D. Fan, H p spaces on compact Lie groups, Ann. Fac. Sci. Toulouse Math. (6) 6 (1997) 429–479. [5] W.R. Bloom, Z. Xu, Approximation of H p functions by Bochner–Riesz means on compact Lie groups, Math. Z. 216 (1994) 131–145. [6] J. Chen, D. Fan, Central oscillating multipliers on compact Lie groups, Math. Z., doi:10.1007/s00209-009-0618-4, in press. [7] J.L. Clerc, Bochner–Riesz means of H p functions (0 < p < 1) on compact Lie groups, in: Lecture Notes in Math., vol. 1234, Springer-Verlag, 1987, pp. 86–107. [8] R. Coifman, G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569–645. [9] M. Cowling, A.M. Mantero, F. Ricci, Pointwise estimates for some kernels on compact Lie groups, Rend. Circ. Mat. Palermo (2) XXXI (1982) 145–158. [10] S. Giulini, S. Meda, Oscillating multiplier on noncompact symmetric spaces, J. Reine Angew. Math. 409 (1990) 93–105. [11] M. Marias, Lp -boundedness of oscillating spectral multipliers on Riemannian manifolds, Ann. Math. 10 (2003) 133–160. [12] A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (2) (1981) 267–315. [13] D. Müller, E.M. Stein, Lp -estimates for the wave equation on the Heisenberg group, Rev. Mat. Iberoamericana 15 (1999) 297–334.
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[14] J. Peral, Lp estimates for the wave equation, J. Funct. Anal. 36 (1980) 114–145. [15] A. Seeger, C.D. Sogge, E.M. Stein, Regularity properties of Fourier integral operators, Ann. of Math. 134 (1991) 231–251. [16] E.M. Stein, Topics in Harmonic Analysis, Ann. of Math. Stud., vol. 63, Princeton University Press, Princeton, NJ, 1970. [17] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [18] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922. [19] Z. Xu, The generalized Abel means of H p functions on compact Lie groups, Chinese Ann. Math. Ser. A 13 (1) (1992) 101–110.
Journal of Functional Analysis 259 (2010) 3265–3327 www.elsevier.com/locate/jfa
Canonical conservative state/signal shift realizations of passive discrete time behaviors Damir Z. Arov a,1 , Olof J. Staffans b,∗ a Division of Applied Mathematics and Informatics, Institute of Physics and Mathematics, South-Ukrainian
Pedagogical University, 65020 Odessa, Ukraine b Åbo Akademi University, Department of Mathematics, FIN-20500 Åbo, Finland
Received 17 November 2009; accepted 29 May 2010
Communicated by J. Coron
Abstract A passive linear discrete time invariant s/s (state/signal) system Σ = (V ; X , W) consists of a Hilbert (state) space X , a Kre˘ın (signal) space W, a maximal nonnegative (generating) subspace V of the Kre˘ın space K := −X [] X [] W. The sets of trajectories (x(·); w(·)) generated by V on the discrete time intervals I ⊂ Z are defined by
x(n + 1); x(n); w(n) ∈ V ,
n ∈ I.
This system is forward conservative, or backward conservative, or conservative if V ⊂ V [⊥] , V [⊥] ⊂ V , or V [⊥] = V , respectively. The set WΣ + of all signal components w(·) of trajectories (x(·); w(·)) of Σ on I = Z+ with x(0) = 0 and w(·) ∈ 2 (Z+ ; W) is called the future time domain behavior of Σ. The Fourier Σ of WΣ is called the future frequency domain behavior of Σ. This set is a maximal nonnegatransform W + + tive right-shift invariant subspace in the Kre˘ın space K 2 (D; W) that as a topological vector space coincides with the usual Hardy space H 2 (D; W), but has the indefinite Kre˘ın space inner product inherited from W. A subspace of K 2 (D; W) with the above properties is called a passive future frequency domain behavior on W. It has been shown earlier by the present authors that every passive future frequency domain behavior + on W may be realized as the future frequency domain behavior of some passive s/s system Σ, and W * Corresponding author.
E-mail address:
[email protected] (O.J. Staffans). URL: http://users.abo.fi/staffans/ (O.J. Staffans). 1 Damir Z. Arov thanks Åbo Akademi for its hospitality and the Academy of Finland and the Magnus Ehrnrooth Foundation for their financial support during his visits to Åbo in 2003–2008. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.05.019
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that it is possible to require, in addition, that Σ is (a) controllable and forward conservative, (b) observable and backward conservative, or (c) simple and conservative. These three types of realizations are determined + up to unitary similarity. Canonical functional shift realizations of the types (a) and (b) have been by W obtained earlier by the present authors, and their connection to the classical de Branges–Rovnyak models have been discussed. Here we present analogous results for a realization of the type (c). © 2010 Elsevier Inc. All rights reserved. Keywords: Passive; Conservative; Behavior; State/signal system; De Branges–Rovnyak model; Input/state/output system; Transfer function; Scattering matrix; Krein space; Fundamental decomposition
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary notions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Kre˘ın spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Passive and conservative state/signal systems . . . . . . . . . . . . . . . . . . . . . 2.3. Future, past, and full behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Forward and backward conservative canonical models . . . . . . . . . . . . . . . 3. The simple conservative canonical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The full stable trajectories of a conservative state/signal system . . . . . . . . . . . . . . 5. Incoming and outgoing inner channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Alternative characterizations of L(W) and D(W) . . . . . . . . . . . . . . . . . . . . . . . 7. Forward and backward conservative compressions of the conservative model . . . . 8. Conservative dilations of the forward and backward conservative canonical models 9. Passive realizations of frequency domain behaviors . . . . . . . . . . . . . . . . . . . . . . 10. The de Branges–Rovnyak conservative i/s/o model . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction This article may be regarded as a continuation of [6] and [7], which in turn continued the development of a passive time-invariant linear s/s (state/signal) systems theory in discrete time that was begun in [2–5].Some further comments on the earlier history are given at the end of this introduction, and also in [7]. A linear discrete time invariant s/s (state/signal) system Σ consists of a Hilbert (state) space X , a Kre˘ın (signal) space W, and a family of trajectories (x(·), w(·)) of Σ on each discrete time interval I defined by an equation of the form x(n + 1) = F x(n), w(n) ,
n ∈ I,
(1.1)
where F is a bounded linear operator from a closed domain D(F ) ⊂ X [] W into X with the extra property that for every x ∈ X there exists at least one w ∈ W such that wx ∈ D(F ). The three most important cases are I = Z+ := {0, 1, 2, . . .}, I = Z = {0, ±1, ±2, . . .}, and I = Z− := {−1, −2, . . .}. By a past, full, and future trajectory of Σ we mean a trajectory of Σ on Z− , Z, and Z+ , respectively. The extra property of F mentioned above is equivalent to the requirement that for each x0 ∈ X and for each interval I with finite left end-point m there exists at least one
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trajectory (x(·), w(·)) of Σ on I satisfying x(m) = x0 . However, instead of working directly with equation (1.1) we shall use the graph form of (1.1), given by x(n + 1) ∈ V , n ∈ I, (1.2) x(n) w(n) where V is the graph of F defined by the formula
F (x, w) −X
x ∈ D(F ) . V := ∈ X x
w w W
(1.3)
The subspace V is called the generating subspace of Σ, since it generates the sets of all trajectories of Σ on the discrete intervals I by formula (1.2). We denote the system by Σ = (V ; X , W). The properties of F listed above can be rewritten in terms of conditions on V , as was done in [2]. If V is a maximal nonnegative subspace of the Kre˘ın node space K := −X [] X [] W then the conditions listed above are satisfied, and hence every maximal nonnegative subspace V is the generating subspace of a s/s system. By a passive s/s system we mean a system whose generating subspace V is maximal nonnegative. In addition to passivity we shall often assume that Σ is forward conservative, backward conservative, or conservative, which means that V ⊂ V [⊥] , V [⊥] ⊂ V , or V [⊥] = V , respectively, where V [⊥] is the orthogonal companion to V in K. Passivity implies that all trajectories (x(·), w(·)) of Σ on I satisfy 2 2 −x(n + 1)X + x(n)X + w(n), w(n) W 0, n ∈ I, (1.4) and forward conservativity means that (1.4) holds in form of an equality 2 2 −x(n + 1)X + x(n)X + w(n), w(n) W = 0, n ∈ I.
(1.5)
See Sections 2.1–2.2 for details. The future behavior WΣ + of a passive s/s system Σ = (V ; X ; W) consists of all the signal components w(·) of all trajectories (x(·), w(·)) of Σ on I = Z+ with x(0) = 0 and w(·) ∈ 2 (Z+ ; W). This set is a maximal nonnegative right-shift invariant subspace of the Kre˘ın space 2 (W). As a topological vector space this space coincides with the Hilbert space 2 (Z+ ; W), k+ 2 (W) but it has the indefinite Kre˘ın space inner product inherited from W. A subspace W+ of k+ with the above properties is called a passive future behavior on W. By replacing the interval Z+ by either Z− or Z we get two more behaviors induced by the 2 passive s/s system Σ , namely the past behavior WΣ − consisting of sequences in k− (W), and the Σ 2 2 2 full behavior W consisting of sequences in k (W), where k− (W) and k (W) are topologically equal to 2 (Z− ; W) and 2 (Z; W), respectively, but carry the inner products inherited from W. In the definitions of these behaviors we replace the condition x(0) = 0 in the definition of WΣ + by is a maximal nonnegative right-shift invariant the condition x(k) → 0 as k → −∞. The set WΣ − 2 (W). A subspace W of k 2 (W) with these properties is called a passive past subspace of k− − − behavior on W. The set WΣ is a maximal nonnegative subspace of k 2 (W) which is bilaterally shift-invariant, and it has an extra causality property which will be explained in Section 2.3. A subspace W of k 2 (W) with these properties is called passive full behavior on W. It turns
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Σ Σ out that any two of the three behaviors WΣ + , W , and W− can be recovered from the third by formulas (2.29)–(2.31). The same formulas can be used to uniquely define any two of the above types of passive behaviors W− , W, and W+ by means of the third. See Section 2.3 for more details. It was shown in [3] that every passive future behavior W+ on W may be realized as the future behavior of some passive s/s system Σ , and that it is possible to require, in addition, that Σ is (a) controllable and forward conservative, (b) observable and backward conservative, or (c) simple and conservative. These three types of realizations are determined by W+ (or equivalently, by W or W− ) up to unitary similarity. In [7] the present authors obtained canonical functional shift realizations of the types (a) and (b), and discussed their connections to the respective two classical de Branges–Rovnyak models. The main purpose of the present paper is to obtain analogous results for a realization of the type (c), and to further study the properties of simple conservative s/s systems. This paper is organized as follows. In Section 2.1 we review the notion of a Kre˘ın space and present some Kre˘ın space results that will be needed later. Some background on passive s/s systems is presented in Section 2.2. Passive future, past, and full behaviors on a Kre˘ın signal space and related results are presented in Section 2.3. Two crucial Hilbert spaces H(W[⊥] − ) and H(W+ ) are introduced in Section 2.4, constructed with the help of the passive past behavior W− and the corresponding passive future behavior W+ , as well as the past/future map ΓW , which is a linear contraction H(W[⊥] − ) → H(W+ ) with some special properties. The Hilbert spaces H(W[⊥] − ) and H(W+ ) and the past/future map ΓW are used in Section 3 in our construction of a canonical simple conservative realization of a given passive full behavior W. The state space D(W) of this realization is a certain subspace of the quotient space k 2 (W)/(W[⊥] − [] W+ ), and the dynamics of the system is defined in terms of a left-shift applied to sequences in the equivalence class defined by the initial state. Some new results on the dynamics of conservative s/s systems that follow from our canonical model are discussed in Section 4. The significant notion of incoming and outgoing inner channels of a simple conservative s/s system are discussed in Section 5. Alternative characterizations of the state space D(W) of our canonical simple conservative model are developed in Section 6, and at the same time we discuss the properties of the inverse image L(W) of D(W) under the quotient map k 2 (W) → k 2 (W)/(W[⊥] − [] W+ ). The connection of the new canonical simple conservative s/s model to the controllable forward conservative s/s model and the observable backward conservative s/s model constructed in [7] is explained in Sections 7 and 8. Finally, the connection between our canonical simple conservative s/s model and the simple conservative input/state/output de Branges–Rovnyak scattering model is discussed in Section 10. This model is formulated in frequency domain terms, and for this reason we explain in Section 9 how to convert the time domain results from Sections 2–8 into corresponding frequency domain results. As we mentioned earlier, this article may be regarded as a continuation of [6] and [7], which in turn continued the development of a passive time-invariant linear s/s (state/signal) systems theory in discrete time that was begun in [2–5]. Some preliminary steps in this direction were taken already in [9] by J. Ball and the second author. See, in particular, [9] for a discussion of the connection with the theory of passive and conservative behaviors presented in the papers [22–26] and the monograph [19]. As explained in [2], part of the motivation comes from classical passive time-invariant circuit theory, see, e.g., [10] and [27]. Continuous time passive s/s systems theory has been studied in [16] and [15].
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As we also mentioned above, as a corollary of our main result we recover the simple conservative input/state/output de Branges–Rovnyak model, that was originally presented in [12,13], and which can be found also, e.g., in [1] and in [17,18]. In [17,18] Nikolski˘ı and Vasyunin present a “coordinate free” model of a simple conservative i/s/o scattering system whose scattering matrix coincides with a given Schur function. The philosophy behind the work of Nikolski˘ı and Vasyunin is very different from the philosophy underlying our work. The coordinate free Nikolski˘ı–Vasyunin model contains a “free” parameter Π , and by the appropriate choice of this parameter it is possible to recover all the standard simple conservative shift models whose characteristic function is equal to a given Schur function ϕ, including the Sz.-Nagy–Foia¸s model, the de Branges–Rovnyak model, and the Pavlov model. In this sense the Nikolski˘ı–Vasyunin model is “universal”. On the other hand, our canonical simple conservative s/s shift model is completely determined by a given future behavior, and in particular, it is “coordinate free” in the sense that it does not depend on some arbitrarily chosen fundamental decomposition W = −Y [] U of the given signal space W. Different choices of such a decomposition give rise to different graph representations of the frequency domain version of the given future behavior as the graphs of the multiplication operators induced by different Schur functions ϕ (with varying input and output spaces), and the corresponding i/s/o representations of our canonical s/s model are equivalent to the i/s/o de Branges–Rovnyak realizations of ϕ. On a conceptual level our construction of a simple conservative realization is vaguely reminiscent of the abstract realization theory presented in [14, Part IV]. More precisely, the basic realization in [14, Section 10.5] whose state space is the set of past input sequences factored over the kernel of the Hankel operator of the given Schur functions is analogous to our controllable forward conservative realization presented in [7], whose state space is essentially the quotient of the past behavior over the kernel of the past/future map, and the realization mentioned in [14, pp. 262–263] whose state space is the range of the kernel of the Hankel operator is analogous to our observable backward conservative realization presented in [7], whose state space is essentially the range of the past/future map. However, the construction in [14] is completely algebraic as opposed to our construction which also make crucial use of topological properties (in the form of various indefinite inner products derived from the energy balance equations). Moreover, whereas the construction in [14] is based entirely on i/o considerations, our construction is completely i/o free. Notations. The following standard notations are used below. C is the complex plane, D+ := {z ∈ C | |z| < 1}, D− := {z ∈ C | |z| > 1} ∪ {∞}, T = {z ∈ C | |z| = 1}, Z = {0, ±1, ±2, . . .}, Z+ = {0, 1, 2, . . .}, and Z− = {−1, −2, −3, . . .}. For any set Ω, we denote the closure of Ω by Ω, and we denote the closed linear span of a collection {Ωα }α∈A of sets in a Hilbert or Kre˘ın space by α∈A Ωα . The space of bounded linear operators from one Kre˘ın space U to another Kre˘ın space Y is denoted by B(U; Y). The domain, range, and kernel of a linear operator A are denoted by D(A), R(A), and N (A), respectively. The restriction of A to some subspace Z ⊂ D(A) is denoted by A|Z . The identity operator on U is denoted by 1U . The inner product in a Hilbert space X is denoted by (·,·)X , and the inner product in a Kre˘ın space K is denoted by [·,·]K . The anti-space −K of a Kre˘ın space is algebraically the same space as K, but it has a different inner product [·,·]−K := −[·,·]K . An orthogonal (inner) direct sum decomposition of a Hilbert or Kre˘ın space W into two closed subspaces Y and U will be denoted by W = Y ⊕ U in the case of a Hilbert space and by W = Y [] U in the case of a Kre˘ın space. The subspaces Y and U become Hilbert or Kre˘ın spaces
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when we let them inherit the inner product from W. In this case we denote the (orthogonal) projections of W onto Y and U by PY and PU , respectively. Y We denote the orthogonal (external) direct sum of two Hilbert or Kre˘ın spaces Y and U by . By this we mean the Cartesian product of Y and U equipped with the standard algebraic U operations and the inner product (in the Kre˘ın space notation) y y , = [y, y]Y + [u, u]U . u u Y U
Clearly UY = Y0 [] U0 . After identifying Y0 with Y and U0 with U we can identify UY with Y [] U . Analogous notations are used for orthogonal sums with three or more components. If w(·) is a sequence with values in a Kre˘ın or Hilbert space W defined on Z, then Sw is the sequence w(·) shifted one step to the right. For sequences w(·) defined on Z+ we define (S+ w)(n) = w(n − 1), n 1, (S+ w)(0) = 0, and for sequences w(·) defined on Z− we define (S− w)(n) = w(n − 1), n ∈ Z− . Some additional notations will be introduced in Sections 2 and 3. 2. Preliminary notions and results 2.1. Kre˘ın spaces Throughout this work both the signal space W and the node space K will be a Kre˘ın space. We therefore begin with a review of the most important Kre˘ın space notions and results that will be needed here. A Kre˘ın space W is a (possibly infinite-dimensional) vector space with an inner product [·,·]W that satisfies all the standard properties required by an inner product, except for the condition [w, w]W > 0 for nonzero w, with the additional property that W can be decomposed into a direct sum W = −Y U in such a way that the following conditions are satisfied: 1) U and −Y are orthogonal to each other with respect to the inner product [·,·]W , i.e., [y, u]W = 0 for all u ∈ U and all y ∈ −Y. 2) U is a Hilbert space with the inner product (u, u )U := [u, u ]W , u, u ∈ U , inherited from W. 3) −Y is an anti-Hilbert space with the inner product [y, y ]−Y := [y, y ]W , y, y ∈ −Y, inherited from W. Here and later we shall use the notation −Y for the anti-space of a vector space Y equipped with a (possibly indefinite) inner product. This is algebraically the same space as Y, but the inner product [·,·]Y in Y has been replaced by the inner product [y, y ]−Y := −[y, y ]Y , y, y ∈ −Y. The condition that −Y is an anti-Hilbert space with the inner product inherited from W is equivalent to saying that Y is a Hilbert space with the inner product (y, y )Y := −[y, y ]W , y, y ∈ −Y, inherited from −W. Since Y and U are orthogonal to each other we shall denote the direct sum by W = −Y [] U . Any decomposition W = −Y [] U with properties 1)–3) is called a fundamental decomposition of W. If the space W itself is neither a Hilbert space nor an anti-Hilbert space, then it has
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infinite many fundamental decompositions. If W = −Y [] U is a fundamental decomposition of W, then [w, w]W = − y 2Y + u 2U ,
w = u + y, u ∈ U, y ∈ Y.
(2.1)
The dimensions of the positive space U and the negative space −Y do not depend on the particular decomposition. These dimensions are called the positive and negative indices of W, respectively, and they are denoted by ind+ W and ind− W. An arbitrary choice of fundamental decomposition W = −Y [] U determines a Hilbert space norm on W by w 2Y ⊕U = y 2Y + u 2U ,
w = u + y, u ∈ U, y ∈ Y.
(2.2)
While the norm · Y ⊕U itself depends on the choice of fundamental decomposition W = −Y [] U for W, all these norms are equivalent and the resulting strong and weak topologies are each independent of the choice of the fundamental decomposition. Thus, we can define topological notions, such as convergence, or closedness, with respect to any one of these norms. Any norm on W arising in this way from some choice of fundamental decomposition W = −Y [] U for W we shall call an admissible norm on W, and we shall refer to the corresponding positive inner product on Y ⊕ U as an admissible Hilbert space inner product on W. A subspace L of W is positive if every nonzero vector w ∈ L is positive ([w, w]W > 0), it is neutral if every vector w ∈ L is neutral ([w, w]W = 0), and negative if every nonzero vector w ∈ L is negative ([w, w]W < 0). Nonnegative and nonpositive subspaces are defined in the analogous way. A nonnegative subspace which is not strictly contained in any other nonnegative subspace is called maximal nonnegative, and the notion of a maximal nonpositive subspace is defined in an analogous way. Every nonnegative subspace is contained in some maximal nonnegative subspace, and every nonpositive subspace is contained in some maximal nonpositive subspace. Maximal nonnegative or nonpositive subspaces are always closed. The orthogonal companion L[⊥] of an arbitrary subset L ⊂ W with respect to the Kre˘ın space inner product [·,·]W consists of all vectors in W that are orthogonal to all vectors in L, i.e.,
L[⊥] = w ∈ W w , w W = 0 for all w ∈ L . This is always a closed subspace of W, and L = (L[⊥] )[⊥] if and only if L is a closed subspace. If W is a Hilbert space, then we write L⊥ instead of L[⊥] . Note that, by definition, a subspace L is neutral if and only if L ⊂ L[⊥] . A stronger notion than a neutral subspace is that of a Lagrangian subspace: a subspace L ⊂ W is called Lagrangian if L = L[⊥] . A direct sum decomposition W = F E of W where both F and E are neutral is called a Lagrangian decomposition of W. The subspaces F and E are automatically Lagrangian in this case. Such a decomposition exists if and only if ind+ W = ind− W (this index may be finite or infinite). If we fix a fundamental decomposition W = −Y [] U , then we may view elements of W as consisting of column vectors y −Y w= ∈ , u U where we view Y and U as Hilbert spaces, and the Kre˘ın space inner product on W is given by
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y y y y −1Y 0 ,
= , u u W u Y ⊕U 0 1U u = − y, y Y + u, u U .
(2.3)
In this representation, nonnegative, neutral, nonpositive, and Lagrangian subspaces are characterized as follows. Proposition 2.1. Let W be a Kre˘ın space represented in the form W =
−Y
with Kre˘ın space −1 0 inner product equal to the quadratic form [·,·]J induced by the operator J = 0Y 1 in the U Hilbert space inner product of UY as explained above, and let L be a subspace of W. Then the following claims are true: U
1) L is nonnegative if and only if there is a linear Hilbert space contraction K+ : D+ → Y from some domain D+ ⊂ U into Y such that L=
K+ K+ d+
d D+ = ∈ D
+ + . 1U d+
(2.4)
L is maximal nonnegative if and only if, in addition, D+ = U . 2) L is nonpositive if and only if there is a linear contraction K− : D− → U from some domain D− ⊂ Y into U such that
1Y d−
D− = L=
d− ∈ D− . K− K− d−
(2.5)
L is maximal nonpositive if and only if, in addition, D− = Y. 3) L is neutral if and only if there is an isometry K+ mapping a subspace D+ of U isometrically onto a subspace D− of Y, or equivalently, an isometry K− mapping D− ⊂ Y isometrically onto D+ ⊂ U , such that K+ 1Y D+ = D− . L= 1U K−
(2.6)
L is Lagrangian if and only if, in addition, D+ = U and D− = Y. 4) L is maximal nonnegative if and only if L is closed and L[⊥] is maximal nonpositive. More precisely, if L has the representation (2.4) with D+ = U , then L[⊥] has the representation L
[⊥]
1Y = ∗ Y, K+
(2.7)
∗ is computed with respect to the Hilbert space inner product in Y (instead of the where K+ anti-Hilbert space inner product in −Y inherited from W). 5) L is maximal nonnegative if and only if L is closed and nonnegative and L[⊥] is nonpositive. In particular, L is Lagrangian if and only if L is both maximal nonnegative and maximal nonpositive.
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Proof. See [8, Section 1.8, pp. 48–64] or [11, Theorem 11.7 on p. 54, Theorems 4.2 and 4.4 on pp. 105–106, and Lemma 4.5 on p. 106]. 2 The fundamental decompositions that we have considered above are a special case of orthogonal decompositions W = −Y [] U of W, where Y and U are orthogonal with respect to [·,·]W , and both Y and U are Kre˘ın spaces with the inner products inherited from −W and W, respectively. Thus, if w = y + u with y ∈ Y and u ∈ U , then [w, w]W = [y, y]W + [u, u]W = −[y, y]Y + [u, u]U .
(2.8)
This orthogonal decomposition is fundamental if and only if Y and U are Hilbert spaces, or equivalently, if they are both nonnegative. Lemma 2.2. Let X and Z be two Hilbert spaces and W a Kre˘ın space, and let K be the Kre˘ın space K =
−Z
X W
.
1) A nonnegative subspace V of K is maximal nonnegative if and only if conditions (a) and (b) below hold: (a) For each x ∈ X there exists some z ∈ Z and w ∈ W such that
z x w
∈V; z (b) The set of all w ∈ W for which there exists some z ∈ Z such that 0 ∈ V is maximal w
nonnegative in W. 2) A nonpositive subspace V of K is maximal nonpositive if and only if conditions (c) and (d) below hold: (c) For each z ∈ Z there exists some x ∈ X and w ∈ W such that
z x w
∈V; 0 (d) The set of all w ∈ W for which there exists some x ∈ X such that x ∈ V is maximal w
nonpositive in W. 3) A neutral subspace V of K is Lagrangian if and only if conditions (a)–(d) above hold. Proof. Proof of 1). Assume first that (a) and (b) hold. Let W = −Y [] U be a fundamental decomposition of W. Then −
Z 0 Y
0
[] X is a fundamental decomposition of K. By assertion 1) U
in Proposition 2.1 V has a representation ⎧⎡ ⎪ ⎨ ⎢ V= ⎣ ⎪ ⎩ K1
⎫ x ⎤ 0
K1 u ⎪
⎬ 0 ⎥ x0 x0 ∈ D
⎦ + ,
u0 ⎪ x ⎭ 0 K2 u + u0 0
(2.9)
Z is a contraction defined on some subspace D+ of X with values in Y . By PropoU sition 2.1, in order to show that V is maximal nonnegative it suffices to show that D+ = X . U where
K2
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x be an arbitrary vector in X . Then by (a), there exist z1 ∈ Z and w1 ∈ W such that zLet 1 x ∈ V . Since the set in (b) is maximal nonnegative it follows that for any u ∈ U there exist w1 z 2 z2 ∈ Z and w2 ∈ W such that PU w2 = u − PU w1 and 0 ∈ V . Since V is a subspace, also w2
z1 z2 z1 + z2 ∈ V, x + 0 = x w1 w2 w1 + w2 x x x with PU (w1 + w2 ) = u. Thus u ∈ D+ , z1 + z2 = K1 u , and PY (w1 + w2 ) = K2 u . Since x ∈ X and u ∈ U are arbitrary we find that D+ = X . This proves that V is maximal nonnegaU tive. Conversely, suppose that V maximal nonnegative. By Proposition 2.1, V has a representation X K1 X : U → Y . Clearly this implies that (a) holds. of the form (2.9) for some contraction K 2 0 Moreover, the set in (b) is given by {u0 + K2 u0 |u0 ∈ U}, and by Proposition 2.1 it is maximal nonnegative. Thus also (b) holds. Proof of 2). The proof of 2) is analogous to the proof of 1). Proof of 3). This follows from 1) and 2) together with assertion 5) in Proposition 2.1. 2 The Hilbert space H(Z). In [6] was constructed a Hilbert space H(Z), where Z is a maximal nonnegative subspace of a Kre˘ın space. Below we give a short review of this construction. Let Z be a maximal nonnegative subspace of the Kre˘ın space K, and let K/Z be the quotient of K modulo Z. We define H(Z) by
(2.10) H(Z) = h ∈ K/Z sup −[x, x]K x ∈ h < ∞ . It turns out that sup{−[x, x]K | x ∈ h} 0 for all h ∈ H(Z), that H(Z) is a subspace of K/Z, that H(Z) is a Hilbert space with the norm
1/2 , h H(Z ) = sup −[x, x]K x ∈ h
h ∈ H(Z),
(2.11)
and that H(Z) is continuously contained in X /Z. We denote the equivalence class h ∈ K/Z that contains a particular vector x ∈ K by h = x + Z. Thus, with this notation, (2.10) and (2.11) can be rewritten in the form
H(Z) = x + Z ∈ K/Z x + Z 2H(Z ) < ∞ ,
x + Z 2H(Z ) = sup −[x + z, x + z]K z ∈ Z , x + Z ∈ H(Z). A very important (and easily proved) fact is that if we define
H0 (Z) := z† + Z z† ∈ Z [⊥] ,
(2.12) (2.13)
(2.14)
then H0 (Z) is a subspace of H(Z). However, even more is true: H0 (Z) is a dense subspace of H(Z), and for every z† ∈ Z [⊥] it is true that † z + Z 2
H (Z )
= − z† , z† K ,
z† ∈ Z [⊥] .
(2.15)
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Furthermore, if we denote
K(Z) = x ∈ K x + Z ∈ H(Z) , then Z + Z [⊥] ⊂ K(Z) ⊂ Z + Z [⊥] and † z + Z, x + Z H(Z ) = − z† , x K ,
z† ∈ Z [⊥] , x ∈ K(Z).
(2.16)
(2.17)
See [6] for more details. Connection between H(Z) and de Branges complementary space. Let A ∈ B(U; Y) be a contractive operator between the Hilbert spaces U and Y. The de Branges complementary space H(A) is defined by the formulas
(2.18) H(A) = y ∈ Y y H(A) < ∞ , where
y H(A) = sup y − Au 2Y − u 2Y u ∈ U .
(2.19)
This is a Hilbert space continuously contained in Y. It was introduced and used in [12,13], + defined in formula (10.2) below, as the state space in the with A replaced by the operator D canonical de Branges–Rovnyak model of a scattering i/s/o observable backward conservative system with a given Schur class scattering matrix Φ. We shall derive this model from our s/s model in Section 10. Later it was observed that H(A) has another alternative characterization: 1/2 , H(A) = R 1 − AA∗ 1/2 [−1] y Y , y ∈ H(A), y H(A) = 1 − AA∗
(2.20)
where the upper index [−1] represents a pseudo-inverse, i.e., B [−1] : R(B) → (N (B))⊥ is the inverse of the injective operator B|(N (B))⊥ → R(B). The operator (1 − AA∗ )1/2 is usually called the defect operator of the contraction A∗ . See [1] and [20] for more details. In [6] it was explained how the space H(Z) defined earlier in this section is related to the space H(A), where A is the contraction appearing in the graph representation
Au
u ∈ U Z=
u of the maximal nonnegative subspace Z of K with respect to some fundamental decomposition K = −Y [] U . The connection is the following. There exists a unitary map T : H(Z) → H(A) with the property that the image of x + Z ∈ H(Z) under T is the unique vector y in this equivalence class whose projection onto U is zero. Explicitly this means that y y T + Z = y − Au, ∈ K(Z), u u y + Z, y ∈ H(A). T −1 y = 0
(2.21)
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The operator T maps H0 (Z) one-to-one onto the dense subspace R(1 − AA∗ ) of H(A). In the sequel we denote H0 (A) := R(1 − AA∗ ). 2.2. Passive and conservative state/signal systems A s/s system Σ = (V ; X , W) forward passive if the inequality (1.4) holds for every is called x(·) interval I and every trajectory w(·) of Σ on I . This is equivalent to the requirement that V is a nonnegative subspace of the node space K. We call Σ forward conservative if the equality (1.5) holds. This stronger notion is equivalent to the condition that V is neutral, i.e., V ⊂ V [⊥] . The notions of a passive and a conservative s/s system Σ = (V ; X , W) depend on the notion of the adjoint s/s system Σ∗ = (V∗ ; X , −W). This system has same state space X as Σ , its signal space is −W, its node space is K∗ = −X [] X [] −W, and its generating subspace V∗ is defined by 0 1X 0 V∗ := 1X (2.22) 0 0 V [⊥] , 0 0 I where I is the identity operator acting from W to −W. Proposition 2.3. (See [3, Proposition 4.6].) Let Σ = (V ; X , W) be a s/s system with the adjoint Σ∗ = (V∗ ; X , W∗ ). 1) A sequence (x(·), w(·)) is a trajectory of Σ on Z+ if and only if n % & ' w(k), I −1 w∗ (n − k) W = 0 − x(n + 1), x∗ (0) X + x(0), x∗ (n + 1) X +
(2.23)
k=0
for all trajectories of Σ∗ on Z+ and all n ∈ Z+ . 2) A sequence (x∗ (·), w∗ (·)) is a trajectory of Σ∗ on Z+ if and only if (2.23) holds for all trajectories of Σ on Z+ and all n ∈ Z+ . From this proposition follows that (Σ∗ )∗ = Σ . By a backward conservative s/s system Σ we mean a system whose dual system Σ∗ is forward conservative. A s/s system Σ = (V ; X , W) is called passive if both Σ and the dual system Σ∗ are forward passive, and it is called conservative if both Σ and the dual system Σ∗ are forward conservative, or in other words, Σ is both forward and backward conservative. The dual node space K∗ and the dual generating subspace V∗ have been defined in such a way that V∗ is nonnegative in K∗ if and only if V [⊥] is nonpositive in K, and therefore, by Proposition 2.1, Σ is passive if and only if V is a maximal nonnegative subspace of K. Likewise, V∗ is a neutral subspace of K∗ , i.e., V∗[⊥] ⊂ V∗ , if and only if V [⊥] ⊂ V , and hence Σ is conservative if and only if V = V [⊥] , i.e., V is a Lagrangian subspace of K. A trajectory (x(·), w(·)) defined on some interval I with a finite left end-point m is called externally generated on I if x(m) = 0. In the case where the left end-point of I is −∞ we replace this condition by limk→−∞ x(k) = 0. By (1.4), if Σ is passive, and if [x(·)w(·)] is an externally generated trajectory of Σ on some interval I with left end-point m −∞, then
D.Z. Arov, O.J. Staffans / Journal of Functional Analysis 259 (2010) 3265–3327 n % x(n + 1)2 w(k), w(k) W , X
n ∈ I.
3277
(2.24)
k=m 2 If (n w(·) ∈ (I ; W) with respect to some admissible norm in W, then the sum k=m [w(k), w(k)]W has an upper bound independent of n, and it follows from (2.24) that the sequence x(·) is bounded, i.e., x(·) ∈ ∞ (I ; X ). We call a trajectory (x(·), w(·)) of Σ on some interval I stable if w(·) ∈ 2 (I ; W) and x(·) ∈ ∞ (I ; X ). Thus, externally generated trajectories of a passive s/s system are stable on I whenever the signal part belongs to 2 (I ; W). Every passive s/s system Σ = (V ; X , W) is well-posed in the forward time direction in the following sense:
x 1 1) For every x0 ∈ X there exists x1 ∈ X and w0 ∈ W such that x0 ∈ V ; w0 x 1 2) For every x0 ∈ V there exists a stable future trajectory (x(·), w(·)) of Σ satisfying w0
x(0) = x0 , x(1) = x1 , and w(0) = w0 ; see, e.g., [3, Proposition 5.12] and [7, Lemma 2.3, assertion 7)]. If Σ is conservative then it is also well-posed in the backward time direction in the sense that the following two conditions hold: x 0 3) For every x0 ∈ X there exists x−1 ∈ X and w−1 ∈ W such that x−1 ∈ V ; w−1 x 0 4) For every x−1 ∈ V there exists a stable past trajectory (x(·), w(·)) of Σ satisfying x(0) = w−1
x0 , x(−1) = x−1 , and w(−1) = w−1 ; this follows from [7, Lemma 3.1] and the fact that the adjoint system Σ∗ is well-posed in the forward time direction. The subspace of X that we get by taking the closure in X of all states x(n) that appear in externally generated trajectories (x(·), w(·)) of Σ on Z+ is called the (approximately) reachable subspace, and we denote it by RΣ . If RΣ = X , then Σ is called controllable. The subspace of all x0 ∈ X with the property that there exists some trajectory (x(·), w(·)) of Σ on Z+ with x(0) = x0 for which w(·) vanishes identically is called the unobservable subspace, and it is denoted by UΣ . If UΣ = {0}, then Σ is called (approximately) observable. A s/s system Σ is ⊥ called simple if X = RΣ + U⊥ Σ , or equivalently, if UΣ ∩ RΣ = {0}, and it is minimal if it is both controllable and observable. Throughout the rest of this paper all s/s systems that we shall consider will be assumed to be passive. The main object of study in this paper is the subclass of simple conservative s/s systems. 2.3. Future, past, and full behaviors Let Σ = (V ; X , W) be a passive s/s system. By the (stable) behavior of Σ on the discrete time interval I ⊂ Z we mean the set of all the signal parts w(·) of all stable externally generated trajectories of Σ on I . We denote this set by WΣ (I ), and introduce the abbreviations − Σ WΣ − := W Z ,
WΣ := WΣ (Z),
+ Σ WΣ + := W Z .
(2.25)
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Σ Σ These three behaviors WΣ − , W , and W+ are called the past behavior, the full behavior, and the future behavior of Σ , respectively. These are the signal parts of all stable externally generated past, full, and future trajectories of Σ , respectively. By k 2 (I ; W) we denote the Kre˘ın space that coincides with the Hilbert space 2 (I, W) as a topological vector space, and is equipped with the indefinite inner product
% w1 (·), w2 (·) k 2 (I ;W ) = w1 (k), w2 (k) W ,
(2.26)
k∈I
and use the abbreviations 2 k− (W) := k 2 Z− ; W ,
k 2 (W) := k 2 (Z; W),
2 k+ (W) := k 2 Z+ ; W .
(2.27)
If W = −Y []U is a fundamental decomposition of W, then k 2 (I, W) = −2 (I, Y)[]2 (I, U) is a fundamental decomposition of k 2 (I, W). It follows from (2.24) that WΣ (I ) is a nonnegative subspace of k 2 (I ; W) for all intervals I . Actually, the maximal nonnegativity of V in K implies that these subspaces are even maximal Σ nonnegative. This was proved in the cases of WΣ ± and W in [7, Theorem 2.8], and the proof for a general interval I is similar. Σ Apart from being maximal nonnegative the three behaviors WΣ ± and W are shift-invariant 2 in the following sense. We denote the right-shift operator on k± (W) by S± and the right-shift Σ operator on k 2 (W) by S. It is easy to see that WΣ ± are S± -invariant, and that W is S-reducing Σ in the following way. Let π be the (SWΣ = WΣ ). In addition WΣ can be recovered from W ± ± 2 (W). Then orthogonal projection of k 2 (W) onto k±
Σ Σ
, WΣ − = π− W = π− w w ∈ W
Σ 2 Σ
π− w = 0 . WΣ + = W ∩ k+ (W) = w ∈ W
(2.28)
Σ It is also possible to recover WΣ from WΣ − and from W+ as described in Proposition 2.5 below. The above facts motivate the following definition.
Definition 2.4. Let W be a Kre˘ın space. 2 (W) is called a passive past behavior 1) A maximal nonnegative S− -invariant subspace of k− on the (signal) space W. 2 (W) is called a passive future behavior 2) A maximal nonnegative S+ -invariant subspace of k+ on the Kre˘ın (signal) space W. 3) A maximal nonnegative S-reducing subspace W of k 2 (W) is causal if W− := π− W and 2 (W) are maximal nonnegative subspaces of k 2 (W) and k 2 (W), respectively. W+ := W∩k+ − + 4) A maximal nonnegative S-reducing causal subspace of k 2 (W) is called a passive full behavior on the (signal) space W.
As the following proposition shows, the two additional conditions required of W in the above definition of causality are equivalent.
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Proposition 2.5. (See [7, Theorem 2.11].) Let W be a Kre˘ın space. 1) If W is a maximal nonnegative S-reducing subspace of k 2 (W), and if we define W− and W+ by W− := π− W,
2 W+ := W ∩ k+ (W),
(2.29)
then W− is a passive past behavior if and only if W+ is a passive future behavior (and in this case W is a passive full behavior). Moreover, W can be recovered from W+ and from W− by the formulas W=
)
w(·) ∈ k 2 (W) π− S −n w ∈ W− ,
n∈Z+
W=
*
(2.30)
S −n W+ .
(2.31)
n∈Z+
2) If W− is a passive past behavior on W, and if we define W by (2.30), then W is a passive full behavior on W and W− = π− W. 3) If W+ is a passive future behavior on W, and if we define W by (2.31), then W is a passive 2 (W). full behavior on W and W+ = W ∩ k+ This proposition combined with our earlier results on the behaviors induced by a passive s/s systems imply the following result. Proposition 2.6. (See [7, Theorem 2.8].) Let Σ = (V ; X , W) be a passive s/s system. Then the past, full, and future behaviors of Σ are passive past, full, and future behaviors, respectively, in the sense of Definition 2.4. Each one of these behaviors determine the two others uniquely through formulas (2.29)–(2.31). This proposition has the following converse. Proposition 2.7. Let W be a Kre˘ın space, and let W− , W, and W+ be past, full, and future behaviors on W connected to each other by Eqs. (2.29)–(2.31). Then there exists a passive s/s system Σ = (V ; X , W) whose past, full, and future behaviors are equal to W− , W, and W+ , respectively. Moreover, it is possible to require, in addition, that Σ is (a) controllable and forward conservative, (b) observable and backward conservative, or (c) simple and conservative. These three types of realizations are defined uniquely by the given behaviors up to unitary similarity. Proof. This follows from [7, Theorem 1.1] and Propositions 2.5 and 2.6.
2
Two canonical shift models of the type (a) and (b) were originally found in [7], and they will be recalled in Section 2.4. Graph representations of passive behaviors. Let W = −Y [] U be a fundamental de2 (W) = −2 (Y) [] 2 (U) are composition of W. Then k 2 (W) = −2 (Y) [] 2 (U) and k± ± ± 2 2 (W), respectively. By assertion fundamental decompositions of the Kre˘ın spaces k (W) and k± 1) and 4) of Proposition 2.1, every passive past, full, and future behavior W− , W, and W+
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on W and their orthogonal companions have a graph representation with respect to the above fundamental decompositions of the type
D± u
Du
2 2 , u ∈ (U) , W = u ∈
± u u
y y
2 [⊥] 2 y ∈ y ∈ W[⊥] = (Y) , W = (Y) ,
± ± D∗± y D∗ y
W± =
where D± and D are linear contractions between the respective 2 -spaces. Since S± W± ⊂ W± and SW = W we have S ± D± = D± S ±
and SD = DS.
(2.32)
Furthermore, if W± and W are related to each other by the relations (2.29)–(2.31), then D+ = D|2 (U ) , +
D− = π− D|2 (U ) , −
D∗+ = π+ D∗ |2 (U ) , +
D∗− = D∗ |2 (U ) . −
(2.33)
From (2.32) and (2.33) follow that D± and D are convolution operators of the type
(D+ u+ )(n) =
n %
D(n − k)u+ (k),
u+ ∈ 2+ (U), n ∈ Z+ ,
k=0 n %
(D− u− )(n) =
D(n − k)u− (k),
u− ∈ 2− (U), n ∈ Z− ,
k=−∞
(Du)(n) =
n %
D(n − k)u(k),
u ∈ 2 (U), n ∈ Z,
(2.34)
k=−∞
with the same sequence {D(k)}∞ k=0 of operators in B(U; Y) in the three formulas above. The contractivity of D+ implies, in particular, that D(0) is a contraction. The adjoint of these causal convolution operators are the anti-causal convolutions operators ∞ % ∗ D+ y+ (n) = D ∗ (n − k)y+ (k),
y+ ∈ 2+ (Y), n ∈ Z+ ,
k=n −1 % ∗ D− y− (n) = D ∗ (n − k)y− (k),
y− ∈ 2− (Y), n ∈ Z− ,
k=n ∞ % ∗ D y (n) = D ∗ (n − k)y(k), k=n
y ∈ 2 (Y), n ∈ Z.
(2.35)
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Lemma 2.8. Let W be a Kre˘ın space. 1) The zero section
W+ (0) := w(0) w ∈ W+ of every passive future behavior W+ on W is a maximal nonnegative subspace of W. Conversely, every maximal nonnegative subspace W0 of W is the zero section of some passive future behavior on W. 2) The (−1)-section
[⊥]
W[⊥] − (−1) := w(−1) w ∈ W− of the orthogonal companion of every passive past behavior W− on W is a maximal nonpositive subspace of W. Conversely, every maximal nonpositive subspace W−1 of W is the −1-section of the orthogonal companion of some passive past behavior W− . Proof. We only prove 1) below, and leave the analogous proof of 2) to the reader. Let W = −Y [] U be a fundamental decomposition of W. By (2.34), W+ (0) has the rep 0 resentation W+ (0) = D(0)u , where D(0) is a contraction U → Y. By Proposition 2.1, this u0 implies that W+ (0) is maximal nonnegative. Conversely, if W0 is maximal nonnegative in W, then by Proposition 2.1, W0 has a graph representation W0 = { Du00u0 | u0 ∈ U} with respect to the fundamental decomposition W = −Y [] U of W, where D0 ∈ B(U; Y) is a contraction. It is easy to see that W+ := {D0 u(·)] | 2 (W), and it follows from Proposition 2.1 that W u(·) ∈ 2+ (U)} is a S+ -invariant subspace of k+ + is maximal nonnegative since it is the graph of a contraction operator 2+ (U) → 2+ (Y). Thus, W0 is the zero section of the passive future behavior W+ . 2 2.4. Forward and backward conservative canonical models In this section we shall present two special Hilbert spaces that play a central role throughout the rest of this article. Among others, they were used in [7] as the state spaces of two of our canonical realizations of a passive behavior. These two spaces are special cases of the Hilbert space H(Z) described in the preceding section. The Hilbert space H (W+ ) and the observable backward conservative canonical model. Let W+ be a given passive future behavior on a Kre˘ın signal space W, i.e., W+ is a maximal 2 (W). We take K = k 2 (W) and Z = W in the discusnonnegative S+ -invariant subspace of k+ + + sion in Section 2.1, and adapting our earlier formulas for H(Z) and H0 (Z) to this case we get the following result. Theorem 2.9. (See [7, Theorem 4.1].) Let W+ be a passive future behavior on the Kre˘ın space 2 (W). Define k+
2 H(W+ ) = h+ ∈ k+ (W)/W+ sup −[w+ , w+ ]k 2 (W ) w+ ∈ h+ < ∞ , +
(2.36)
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and define ·H(W
+)
by
1/2 h+ H(W+ ) = sup −[w+ , w+ ]k 2 (W ) w+ ∈ h+ ,
h+ ∈ H(W+ ).
+
(2.37)
Then H(W+ ) is a Hilbert space with the norm ·H(W ) that is continuously contained in + 2 (W)/W . The set k+ +
† † + W+ w + ∈ W[⊥] H0 (W+ ) := w+ +
(2.38)
is a dense subspace of H(W+ ), and † w + W+ 2 + H(W
+)
† † = − w+ (·), w+ (·) k 2 (W ) , +
† w+ ∈ W[⊥] + .
(2.39)
The set
2 (W) w+ (·) + W+ ∈ H(W+ ) K(W+ ) = w+ (·) ∈ k+
(2.40)
2 (W), and is a subspace of k+
† w+ (·) + W+ , w+ (·) + W+ H(W
+)
if
† w+ (·) ∈ W[⊥] +
† = − w+ (·), w+ (·) k 2 (W ) , +
and w+ (·) ∈ K(W+ ).
(2.41)
Lemma 2.10. (See [7, Lemma 4.3].) If w+ (·) ∈ K(W+ ), where W+ is a passive future behavior ∗ w ∈ K(W ) and on the Kre˘ın space W, then S+ + + ∗ S w+ + W+ 2 + H(W
+)
w+ + W+ 2H(W+ ) + w+ (0), w+ (0) W .
(2.42)
If w+ (·) ∈ W[⊥] + , then w+ (·) ∈ K(W+ ) and (2.42) holds as an equality. Theorem 2.11. (See [7, Theorem 7.1].) Let W+ be a passive future behavior on the Kre˘ın space W, and let W Vobc+
=
∗w+W S+ + w + W+ w(0)
H(W+ )
∈ H(W+ ) w ∈ K(W+ ) , W
W
(2.43)
W
where K(W+ ) is the space defined in (2.40). Then Σobc+ = (Vobc+ ; H(W+ ), W) is a passive observable backward conservative s/s system whose future behavior is equal to W+ . Moreover, W (x(·), w(·)) is a stable future trajectory of Σobc+ if and only if w ∈ K(W+ )
∗ n and x(n) = S+ w + W+ ,
n ∈ Z+ .
(2.44)
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The Hilbert space H (W⊥ − ). Let W− be a given passive past behavior on a Kre˘ın signal 2 (W). Then W[⊥] is space W, i.e., W− is a maximal nonnegative S− -invariant subspace of k− − ∗ 2 a maximal nonpositive S− -invariant subspace of k− (W), and hence it can be interpreted as a ∗ -invariant subspace of the anti-space −k 2 (W). This time we take K = maximal nonnegative S− − [⊥] 2 −k− (W) and Z = W− in the definition of H(Z). Adapting our earlier formulas to this case we get the following result. Theorem 2.12. (See [7, Theorem 4.4].) Let W− be a passive past behavior on the Kre˘ın space 2 (W), and interpret W[⊥] as a maximal nonnegative S ∗ -invariant subspace of the anti-space k− − − 2 (W). Define −k−
2
sup w− (·), w− (·) 2
w− (·) ∈ h− < ∞ , (2.45) H W[⊥] = h− ∈ −k− (W)/W[⊥] − − k (W ) −
and define · H(W[⊥] ) by −
h− 2
H(W[⊥] − )
= sup w− (·), w− (·) k 2 (W ) w− (·) ∈ h− . −
(2.46)
Then H(W[⊥] − ) is a Hilbert space with the norm · H(W[⊥] ) that is continuously contained in −
2 (W)/W[⊥] . The set −k− −
w− (·) ∈ W− H0 W[⊥] = w− (·) + W[⊥] − −
(2.47)
is a dense subspace of H(W[⊥] − ), and w− + W[⊥] 2 −
H(W[⊥] − )
= w− (·), w− (·) k 2 (W ) , −
w− (·) ∈ W− .
(2.48)
The set
[⊥] 2 = w− (·) ∈ k− (W) w− (·) + W[⊥] K W[⊥] − − ∈ H W−
(2.49)
2 (W), and is a subspace of k−
[⊥] w− (·) + W[⊥] − , v− (·) + W− H(W[⊥] ) = w− (·), v− (·) k 2 (W ) , −
if w− (·) ∈ W−
and v− (·) ∈ K W[⊥] − .
−
(2.50)
[⊥] Lemma 2.13. (See [7, Lemma 4.6].) If w− (·) ∈ K(W[⊥] − ), then S− w− ∈ K(W− ) and
S− w− + W[⊥] 2 −
H(W[⊥] − )
2 w− + W[⊥] − H(W[⊥] ) − w− (−1), w− (−1) W . −
If w− (·) ∈ W− , then w− (·) ∈ K(W[⊥] − ) and (2.51) holds as an equality.
(2.51)
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The past/future map and the controllable forward conservative canonical model. The passive controllable forward conservative canonical model developed in [7, Section 8] used the past/future map of a passive full behavior, which will be defined below. Lemma 2.14. (See [7, Lemma 6.1].) Let W be a passive full behavior on W with the corre2 (W). sponding passive past behavior W− = π− W and passive future behavior W+ = W ∩ k+ [⊥] [⊥] Then π+ w + W+ ∈ H(W+ ) and π− w + W− ⊂ H(W− ) whenever w ∈ W, and there exists a unique contraction ΓW : H(W[⊥] − ) → H(W+ ) satisfying ΓW π− w + W[⊥] = π+ w + W+ , −
w ∈ W.
(2.52)
Definition 2.15. The contraction ΓW : H(W[⊥] − ) → H(W+ ) in Lemma 2.14 is called the past/future map of the full behavior W. If W is the full behavior of a passive s/s system Σ , then we also call ΓW the past/future map of Σ and denote it by ΓΣ . Theorem 2.16. (See [7, Theorems 8.1 and 8.6].) Let W be a passive full behavior on the Kre˘ın 2 (W) be the corresponding passive past and space W, and let W− = π− W and W+ = W ∩ k+ future behaviors. Let W V˚cfc −
⎧⎡ ⎫ ⎤ [⊥] ⎤ ⎡ H(W[⊥] ⎨ w − + W− ⎬ − )
⎦ ∈ ⎣ H(W[⊥] ) ⎦
w− ∈ W− = ⎣ S− w− + W[⊥] − − ⎩ ⎭ w− (−1) W
(2.53)
[⊥] and let Vcfc − be the closure of V˚cfc − in the Kre˘ın space K− := −H(W[⊥] − ) [] H(W− ) [] W. Then ⎧⎡ ⎫ [⊥] ⎤
−1
w = w + w , w ∈ KW[⊥] , w ∈ K(W ), ⎬ ⎨ π− S w + W− − + − + + W − ⎦
Vcfc − = ⎣ π− w + W[⊥] (2.54) −
and w+ + W+ = ΓW w− + W[⊥] , ⎩ ⎭ − w(0) W
W
and Σcfc − = (Vcfc − ; H(W[⊥] − ), W) is a passive controllable forward conservative s/s system with past behavior W− and full behavior W. W
W
The input and output maps of a passive state/signal system. In [7, Section 5] the input and output maps of a passive s/s system were defined in the following way. Lemma 2.17. (See [7, Lemma 5.10].) Let Σ = (V ; X ; W) be a passive s/s system with past behavior W− . Then there exists a unique linear contraction BΣ : H(W[⊥] − ) → X , called the ) is given by input map of Σ , whose restriction to H0 (W[⊥] − BΣ w− + W[⊥] = x(0), −
w− (·) ∈ W− ,
(2.55)
where (x(·), w− (·)) is the unique stable externally generated past trajectory of Σ whose signal part is w− (·).
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Lemma 2.18. (See [7, Lemma 5.2].) Let Σ = (V ; X ; W) be a passive s/s system with future behavior W+ . Then the formula
w+ (·) is the signal part of some stable future
C Σ x 0 = w + + W+
trajectory x(·), w+ (·) of Σ with x(0) = x0
(2.56)
defines a linear contraction CΣ : X → H(W+ ), called the output map of Σ. Lemma 2.19. (See [7, Lemma 5.12].) Let Σ = (V ; X ; W) be a passive s/s system with past behavior W− , full behavior W, future behavior W+ , input map BΣ , and output map CΣ . Then (x(·), w(·)) is an externally generated stable past trajectory of Σ if and only if w ∈ W−
and x(n) = BΣ π− S −n w + W[⊥] − ,
n 0,
(2.57)
and (x(·), w(·)) is an externally generated stable full trajectory of Σ if and only if w ∈ W and x(n) = BΣ π− S −n w + W[⊥] − ,
n ∈ Z.
(2.58)
In the latter case we have, in addition, CΣ x(n) = π+ S −n w + W+ ,
n ∈ Z.
(2.59)
Lemma 2.20. (See [7, Lemma 6.3].) The past/future map ΓΣ of a passive s/s system Σ = (V ; X , W) factors into the product ΓΣ = CΣ BΣ
(2.60)
of the input map BΣ and the output map CΣ of Σ . Lemma 2.21. (See [7, Lemma 5.15].) If Σ is a passive forward conservative s/s system, then the input map BΣ of Σ is an isometry with R(BΣ ) = RΣ . Lemma 2.22. (See [7, Lemma 5.20].) If Σ is a passive backward conservative s/s system, then the output map CΣ of Σ is a co-isometry with N (CΣ ) = UΣ . The null controllable and backward unobservable subspaces. As we mentioned earlier, every conservative s/s system Σ = (V ; X , W) is well-posed both in the forward and the backward time direction in the sense that to each x0 ∈ X there exists a stable full trajectory of Σ with x(0) = x0 . Much of what we have said earlier remains true if we interchange the roles played by Z+ and Z− , provided we at the same time replace the notion of an externally generated trajectory by the notion of a backward externally generated trajectory. This notion is defined in the natural way: A trajectory (x(·), w(·)) of Σ defined on an interval I with a finite right end-point n is backward externally generated if x(n) = 0, and if the right end-point of I is +∞ then we replace the condition x(n) = 0 by limk→∞ x(k) = 0. The anti-causal future, full, and past behaviors of Σ are the signal parts of all backward externally generated future, full, and past stable trajectories of Σ , respectively. Here the “past”, “full”, and “future” still refer to the same time intervals as before, i.e., “past” refers to Z− , “full”
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[⊥] , and to Z, and “future” to Z+ . By [7, Theorem 3.4], these behaviors are equal to W[⊥] + , W [⊥] W− , respectively, where W+ , W, and W− are the future, full, and past behaviors of Σ. The causal versions of the input, output, and past/future maps BΣ , CΣ , and ΓΣ defined above also have anti-causal counterparts B†Σ , C†Σ , and ΓΣ† , which we obtain by the same constructions as before, but interchange the roles of Z+ and Z− , and also interchange the roles of H(W+ ) and H(W[⊥] − ). Thus, if (x(·), w(·)) is a backward externally generated stable future trajectory of Σ , then x(0) = B†Σ (w + W+ ), C†Σ x(0) is the equivalence class of all the signal parts of stable past trajectories (x(·), w(·)) of Σ with x(0) = x0 , and
† ΓW π+ w † + W+ = π− w † + W[⊥] − ,
w † ∈ W[⊥] .
As shown in [7, Lemma 5.19], B†Σ = C∗Σ , and C†Σ = B∗Σ , and by [7, Lemma 6.8], ΓΣ† = ΓΣ∗ . Our earlier definition of the reachable and unobservable subspaces RΣ and UΣ also have a built-in direction of time. These two subspaces do not, in general, remain invariant under time reversal, and the subspaces RΣ and UΣ that we defined earlier are the causal versions of these subspaces. We denote the anti-causal counterparts of RΣ and UΣ by R†Σ and U†Σ , respectively. Thus, R†Σ is the closure in X of all states x(n) that appear in backward externally generated past trajectories (x(·), w(·)) of Σ, and U† consists of all x0 ∈ X with the property that there exists some past trajectory (x(·), w(·)) of Σ which with x(0) = x0 for which w(·) vanishes identically. We shall follow the control theory tradition and call R†Σ the (approximately) null controllable subspace. The space U†Σ does not have an established name in control theory, and here we shall use the name backward unobservable subspace. By a backward unobservable trajectory we mean a past trajectory (x(·), w(·)) of Σ for which w(·) vanishes identically. A full trajectory (x(·), w(·)) whose signal part w(·) vanishes identically will be called a bilaterally unobservable trajectory. The restriction of such a trajectory to Z+ is unobservable, and the restriction to Z− is backward unobservable. By [3, Proposition 4.7], R†Σ = R(B∗Σ ) = N (CΣ )⊥ = U⊥ Σ and U†Σ = N (C∗Σ ) = R(BΣ )⊥ = (RΣ )⊥ . 3. The simple conservative canonical model Let W be a Kre˘ın space, and let W be a passive full behavior on W, with corresponding past 2 (W). To shorten the notations we define and future behaviors W− = π− W and W+ = W ∩ k+ H− := H W[⊥] − , H+ := H(W+ ),
0 H− := H0 W[⊥] − , 0 H+ := H0 (W+ ).
(3.1)
Moreover, we denote Q− w := π− w + W[⊥] Q+ w := π+ w + W+ , − , [⊥] Qw := w + W− + W+ , w ∈ k 2 (W).
(3.2)
Below we shall encounter the quotient space k 2 (W)/(W+ W[⊥] − ). Each vector in this space [⊥] is an equivalence class of the type x := w + (W+ W− ) for some w ∈ k 2 (W). Above we 2 (W) [] k 2 (W), denoted the corresponding quotient map by Q, i.e., x = Qw. Since k 2 (W) = k+ −
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2 (W) and W[⊥] is a closed subspace of k 2 (W), it and since W+ is a closed subspace of k+ − − k+2 (W )/W+ [⊥] 2 follows that we can identify k (W)/(W+ W− ) with the product space 2 [⊥] . We
W[⊥] − )
k− (W )/W− 2 k− (W)/W[⊥] − by
denote the projections of onto and P+ + and P− , respectively. Thus, P± is the operator which for each w ∈ k 2 (W) maps x = Qw into 2 (W)/W and H is continuously contained in Q± w. Since H+ is continuously contained in k+ + − H [⊥] + 2 k− (W)/W− , this means that H can be interpreted as a continuously contained subspace of k 2 (W)/(W
2 (W)/W k+ +
−
k 2 (W)/(W+ W[⊥] − ). Let
AW :=
1H+ ∗ ΓW
ΓW . 1H−
(3.3)
This is a bounded linear operator on H+ ⊕ H− . It is nonnegative since ΓW is a contraction x H− → H+ , and by the Schwarz inequality, for all x−+ ∈ H+ ⊕ H− ,
x x+ , AW + = x+ 2H+ + 2(x+ , ΓW x− )H+ + x− 2H− x− x− H+ ⊕H− x+ 2H+ − 2 x+ H+ x− H− + x− 2H− 0. 1/2
We define D(W) to be the range of AW , with the range norm, i.e., x+ 1/2 [−1] x+ A = , x− W x− H+ ⊕H− D (W) x x 1/2 1/2 1/2 where (AW )[−1] is the pseudo-inverse of AW , i.e., x + := (AW )[−1] x−+ is the unique vector − x 1/2 x in R(AW ) which satisfies x−+ = AW x + . With respect to this inner product in the range space −
1/2
the operator AW |R(AW ) is a unitary operator mapping R(AW ) onto D(W). In particular, D(W) is a Hilbert space. Lemma 3.1. Define AW by (3.3). 1) R(AW ) is a dense subset of the Hilbert space D(W), D(W) is a dense subspace of R(AW ), and D(W) is continuously contained in H+ ⊕ H− . 2) AW is bounded as an operator H+ ⊕ H− → D(W). 3) If x ∈ D(W) and y = AW y , then y ∈ D(W), and (x, y)D(W) = (x, y )H− ⊕H+ . Γ 4) AW |H− = 1HW is an isometry H− → D(W). − 1H 5) AW |H+ = Γ ∗+ is an isometry H+ → D(W). W
1/2
1/2
Proof. Proof of 1). Clearly R(AW ) ⊂ R(AW ) = D(W). As is well known, R(AW ) = R(AW ), and thus D(W) is a dense subspace of R(AW ). Let U be the unitary map U := 1/2 AW |R(AW ) : R(AW ) → D(W). Since D(W) is a dense subspace of R(AW ), the image of
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D(W) under U is a dense subspace of D(W). But this image is equal to R(AW ). Thus, R(AW ) is dense in D(W). To show that D(W) is continuously contained in H+ ⊕ H− we take some x ∈ D(W). Then 1/2 x = AW y for some y ∈ R(AW ), and x D(W) = y H+ ⊕H− . Therefore 1/2 2 x 2H+ ⊕H− = AW y H
+ ⊕H−
= (y, AW y)H+ ⊕H− AW y 2H+ ⊕H−
= AW x 2D(W) . This shows that D(W) is continuously embedded in R(AW ), and hence continuously contained in H+ ⊕ H− . 1/2 Proof of 2). With the same notation as in the proof of 1), AW factors into AW = U AW , where 1/2 AW is a bounded linear operator in H+ ⊕ H− , and U is a unitary operator R(AW ) → D(W). Thus AW is bounded as an operator H+ ⊕ H− → D(W). Proof of 3). First assume that x = AW x for some x ∈ H+ ⊕ H− . Then 1/2 1/2 (x, y)D(W) = AW x , AW y D(W) = AW x , AW y H ⊕H + −
= AW x , y H ⊕H = x, y H ⊕H . +
−
+
−
If x is an arbitrary vector in D(W), then there exists a sequence xn ∈ R(AW ) such that xn → x in D(W) as n → ∞. Since D(W) is continuously contained in H+ ⊕ H− , it is also true that xn → x in H+ ⊕ H− . Consequently (x, y)D(W) = lim (xn , y)D(W) = lim xn , y H ⊕H = x, y H ⊕H . n→∞
n→∞
+
−
+
−
Proof of 4). This follows from 3) since we have for all x− ∈ H− , 2 2 ΓW ΓW ΓW x , x = = (AW x− , AW x− )D(W) 1H x− 1H− − 1H− − D(W) − D (W) ΓW x = (x− , AW x− )H+ ⊕H− = x− , 1H− − H+ ⊕H− = x− 2H− . The proof of 5) is analogous.
2
In the sequel we shall throughout interpret AW as a bounded linear operator H+ ⊕ H− → D(W), instead of interpreting AW as a self-adjoint operator in H+ ⊕ H− . In particular, in this setting the operator AW is not self-adjoint unless D(W) = H+ ⊕ H− , i.e., unless ΓW = 0. When the duality in the range space is taken with respect to the inner product in D(W) instead of the inner product in H+ ⊕ H− the operator A∗W becomes a bounded linear operator D(W) → H+ ⊕ H− . 2 Recall that we denoted the projections of k 2 (W)/(W+ W[⊥] − ) onto k+ (W)/W+ and H+ [⊥] 2 (W)/W k− − by P+ and P− , respectively. We denote the restrictions of P± to H− by Π± , x x H so that Π± x−+ = x± for all x−+ ∈ H + . −
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Lemma 3.2. Let AW be the operator defined in (3.3), interpreted as bounded linear operator H+ ⊕ H− → D(W), whose adjoint A∗W is a bounded linear operator D(W) → H+ ⊕ H− . H 1) A∗W is equal to the embedding operator D(W) → H + . − 2) (AW |H+ )∗ = Π+ |D(W) and (AW |H− )∗ = Π− |D(W) . (In the computation of these adjoints we interpret AW |H± as operators H± → D(W).) Proof. By Part 3) of Lemma 3.1, for all x ∈ D(W) and all y ∈ H+ ⊕ H− , x, AW y D(W) = x, y H
− ⊕H+
.
This proves Claim 1). If we in the same computation replace y ∈ H+ ⊕ H− by either y ∈ H+ or y ∈ H− we get Claim 2). 2 As the following lemma shows, the subspace D0 (W) defined by
D0 (W) := Q z + z† z ∈ W, z† ∈ W[⊥]
(3.4)
L(W) = w ∈ k 2 (W) Qw ∈ D(W) ,
L0 (W) = z + z† z ∈ W, z† ∈ W[⊥] ,
(3.5)
is dense in D(W). We define
and (w1 , w2 )L(W) = (Qw1 , Qw2 )D(W) , w L(W) = Qw D(W) ,
w1 , w2 ∈ L(W), w ∈ L(W).
(3.6) (3.7)
Then (·,·)L(W) is a semi-inner product in L(W) and · L(W) is a semi-norm in L(W). Lemma 3.3. 1) If z ∈ W and z† ∈ W[⊥] , then Q(z + z† ) = AW L0 (W) ⊂ L(W). 2) D0 (W) is a dense subspace of D(W). 3) If w ∈ L(W), z ∈ W, and z† ∈ W[⊥] , then
Q+ z † Q− z
. In particular, D0 (W) ⊂ R(AW ) and
(w, z)L(W) = (Q− w, Q− z)H− = [π− w, π− z]k 2 (W ) , − † † w, z L(W) = Q+ w, Q+ z H = − π+ w, π+ z† k 2 (W ) . +
In particular,
+
(3.8) (3.9)
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z 2L(W) = Q− z 2H− = [π− z, π− z]k 2 (W ) , −
† 2 z
L(W)
z ∈ W,
2 = Q+ z† H = − π+ z† , π+ z† k 2 (W ) , +
+
z† ∈ W[⊥] .
(3.10) (3.11)
Proof. Step 1: Proof of 1). Let z ∈ W. Then π+ z + W+ = ΓW (π− z + W[⊥] − ), and consequently Q+ z ΓW Q− z = AW Q− z. Qz = = 1H− Q− z
An analogous computation shows that Qz† = AW Q+ z† for all z† ∈ W[⊥] . Thus, Q(z + z† ) = + z† AW Q . Q z −
0 is dense in H , and since R(A ) Step 2: D0 (W) is a dense subspace of D(W). Since H± ± W 0 0 is dense in D(W), the image of H+ ⊕ H− under AW is dense in D(W). However, by Claim 1, this image is equal to D0 (W). Step 3: Proof of (3.8)–(3.11). By part 3) of Lemma 3.1 and Theorem 2.12
(w, z)L(W) = (Qw, Qz)D(W) = (Qw, AW Q− z)D(W) = (Qw, Q− z)H+ ⊕H− = (Q− w, Q− z)H− = [π− w, π− z]k 2 (W ) . −
This proves (3.8), and an analogous computation together with Theorem 2.9 can be used to prove (3.9). The equalities (3.10) and (3.11) follow directly from (3.8) and (3.9). 2 Lemma 3.4. 1) If w ∈ L(W), then S −1 w ∈ L(W), and −1 2 S w
L(W)
= w(0), w(0) W + w 2L(W) .
(3.12)
2) If w ∈ L(W), then Sw ∈ L(W), and Sw 2L(W) = − w(−1), w(−1) W + w 2L(W) .
(3.13)
3) If w1 , w2 ∈ L(W), then w1 , S −1 w2 L(W) = w1 (−1), w2 (0) W + (Sw1 , w2 )L(W) .
(3.14)
Proof. Step 1: Proof of 1). It follows from Lemma 3.3 that if w ∈ L0 (W), then S −1 w ∈ L0 (W) and (3.12) holds. Now let w ∈ L(W), and choose xm ∈ D0 (W) such that xm → Qw in D(W) as m → ∞. Let R be a bounded left-inverse of the quotient map Q, and define wm := w + R(xm − Qw). Then Qwm = xm → Qw in D(W), wm ∈ L0 (W), and wm → w in k 2 (W) as m → ∞. It then follows from (3.12) applied to wm ∈ L0 (W) that QS −1 wm is a Cauchy sequence
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in D(W), and hence it tends to a limit y in D(W) satisfying y 2D(W) = [w(0), w(0)]W + Qw 2D(W) . By the continuity of Q and S −1 , RQS −1 w = RQS −1 lim wm = R lim QS −1 wm = Ry, m→∞
m→∞
and hence y = QRy = QS −1 w. This proves Claim 1). Step 2: Proof of 2). This proof is analogous to the proof of 1). Step 3: Proof of (3.14). By polarizing (3.12) we get −1 S w1 , S −1 w2 L(W) = w1 (0), w2 (0) W + [w1 , w2 ]2L(W) for all w1 , w2 ∈ L(W). If we here replace w1 by Sw1 , then we get (3.14).
2
Theorem 3.5. Let W be a passive full behavior on the Kre˘ın space W, and let W− = π− W 2 (W) be the corresponding passive past and future behaviors. Let D(W) be and W+ = W ∩ k+ 1/2 the range space of the operator AW , where AW is the nonnegative self-adjoint operator on H+ ⊕ H− defined by (3.3), and define L(W) by (3.5). The subspace Vsc defined by Vsc :=
QS −1 w
Qw
w ∈ L(W) w(0)
(3.15)
is the generating subspace of a simple conservative s/s system Σsc = (Vsc ; D(W), W) whose full Γ behavior is W. The input map of Σsc is BΣsc = 1HW with B∗Σsc = Π− |D(W) , and the output − 1H map of Σsc is CΣsc = Π+ |D(W) with C∗Σsc = Γ ∗+ . Moreover, (x(·), w(·)) is a stable externally W generated full trajectory of Σsc if and only if w ∈ W and x(n) = QS −n w, n ∈ Z. Proof. Step 1: Vsc = Vsc[⊥] , and hence Vsc generates a conservative s/s system Σsc = (Vsc ; D(W), W). Our proof of Step 1 is based on Lemma 2.2 with Z = X = D(W). That Vsc is a neutral subspace of K follows from equality (3.12). Clearly condition (a) in that lemma holds because of the definition of L(W), and (c) holds because of Lemma 3.4. The set described in condition (b) is equal to the zero section W+ (0) = {w(0) ∈ W | w ∈ W+ }, which according to Lemma 2.8 is maximal nonnegative, and the set described in condition (d) is equal to the [⊥] −1-section W[⊥] − (−1) = {w(−1) ∈ W | w ∈ W− }, which according to Lemma 2.8 is maximal nonpositive. Thus, by Lemma 2.2, Vsc is Lagrangian, and hence it generates a conservative s/s system Σsc = (Vsc ; D(W), W). Step 2: The behavior of Σsc is equal to W. If w ∈ W+ , then Qw ∈ D(W), and it follows from (3.15) that (x(·), w(·)), where x(n) = QS −n w, n ∈ Z+ , is an externally generated stable sc future trajectory of Σsc . This implies that W+ ⊂ WΣ + . Since W+ is maximal nonnegative and Σsc Σsc sc W+ is nonnegative, this implies that W+ = W+ . From this follows that also W− = WΣ − and Σ sc W=W . Step 3: (x(·), w(·)) is an externally generated full trajectory of Σsc if and only if w ∈ W and x(n) = QS −n w, n ∈ Z. By the definition of W, if (x(·), w(·)) is an externally generated full trajectory of Σsc , then w ∈ W. Conversely, let w ∈ W. Then w ∈ L(W), and it follows from (3.15)
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that (x(·), w(·)) is a full trajectory of Σsc , where x(n) = QS −n w, n ∈ Z. This trajectory is externally generated, since, according to Lemma 3.3 −n −n −n x(n)2 D (W) = QS w L(W) = π− S w, π− S w k 2 (W ) → 0 −
as n → −∞.
As an externally generated full trajectory (x(·), w(·)) is determined uniquely by its signal part w (see Lemma 2.19), it follows that every externally generated trajectory (x(·), w(·)) of Σsc satisfies x(n) = QS −n w, n ∈ Z. Γ Step 4: The input map of Σ is 1HW . According to Lemma 2.17, the operator BΣsc is the − unique operator H− → D(W) which satisfies BΣsc Q− w = x(0) for every w ∈ W, where x(·) is the state component of the unique externally generated trajectory (x(·), w(·)) whose signal part is w. Let w ∈ W. By Step 3, ΓW Q+ w Q− w. = x(0) = Qw = 1H− Q− w
Γ Thus, BΣsc = 1HW . By Lemma 3.2, B∗Σsc = Π− |D(W) . − Step 5: The output map of Σ is Π+ |D(W) . According to Lemma 2.18, CΣsc is the operator which maps x0 ∈ D(W) into the equivalence class consisting of all the signal parts w(·) of all stable future trajectories (x(·), w(·)) of Σsc satisfying x(0) = x0 . Let x0 ∈ D(W), and choose some w0 ∈ L(W) such that Qw0 = x0 . It follows from (3.15) that (x(·), w0 (·)), where x(n) = QS −n w0 , n ∈ Z+ , is a stable future trajectory of Σsc satisfying x(0) = x0 . If (x1 (·), w1 (·)) is another stable future trajectory of Σsc satisfying x1 (0) = x(0) = x0 , then (x − x1 , w0 − w− ) is an externally generated stable future trajectory of Σsc , and hence w1 − w0 ∈ W+ . Thus, the equivalence class of all the signal parts w(·) of all stable future trajectories (x(·), w(·)) of Σsc satisfying x(0) = x0 is equal to Q+ w0 . Consequently, CΣsc = Π+ |D(W) . By Lemma 3.2, C∗Σsc = 1H+ . Γ W∗ Γ Step 6: Σsc is simple. According to Lemma 3.1, the linear span of the ranges of BΣsc = 1WW − 1W and C∗Σsc = Γ ∗+ is dense in the state space D(W), and hence Σsc is simple. 2 W
Let R be the reachable subspace, U the unobservable subspace, R† the null controllable subspace, and U† the backward unobservable subspace of Σsc . As we noticed earlier, R† = U⊥ and U† = R⊥ . By Lemma 3.2 and Theorem 3.5,
ΓW x−
x− ∈ H− , x−
∗ † 2 U = N BΣsc = N (Π− |D(W) ) = Qw w ∈ L(W) ∩ k+ (W) ,
∗ x+ 1H+
† = R = R CΣsc = R
x+ ∈ H+ , ∗ ∗ x ΓW ΓW +
2 U = N (CΣsc ) = N (Π+ |D(W) ) = Qw w ∈ L(W) ∩ k− (W) .
R = R(BΣsc ) = R
ΓW 1H−
=
The orthogonal projections onto these subspaces are given by
(3.16)
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PR = BΣsc B∗Σsc =
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ΓW Π− |D(W) , Π− |D(W)
PU† = 1D(W) − PR = Π+ |D(W) − ΓW Π− |D(W) , Π+ |D(W) ∗ , PR† = CΣsc CΣsc = ∗ Π | ΓW + D (W) ∗ PU = 1D(W) − PR† = Π− |D(W) − ΓW Π+ |D(W) .
(3.17)
4. The full stable trajectories of a conservative state/signal system The causal input and output maps BΣ and CΣ together with their anti-causal counterparts B†Σ = C∗Σ and C†Σ = B∗Σ of a conservative s/s system Σ can be used to describe the relationship between the state component x(·) and the signal component w(·) of an arbitrary stable full trajectory (x(·), w(·)) of Σ . Theorem 4.1. Let Σ = (V ; X , W) be a conservative s/s system with full behavior W, input map BΣ , output map CΣ , reachable subspace R = R(BΣ ), unobservable subspace UΣ = N (CΣ ), null controllable subspace R†Σ = R(C∗Σ ) and backward unobservable subspace U†Σ = N (B∗Σ ). 1) The operator Cfull Σ :=
CΣ B∗Σ
(4.1)
† is a co-isometry from X onto D(W), with kernel X0 := N (Cfull Σ ) = U ∩ U . Thus, Σ is simple full if and only if CΣ is injective, i.e., it is unitary. full full ∗ full 2) Denote the adjoint of Cfull Σ by BΣ := (CΣ ) . Then BΣ is an isometry D(W) → X with ⊥ range X0 = R + R† , which is uniquely determined by the fact that
BΣ = Bfull Σ
ΓW , 1H−
C∗Σ = Bfull Σ
1H+ . ∗ ΓW
(4.2)
In particular, Bfull Σ is surjective, i.e., it is unitary, if and only if Σ is simple. 3) A full trajectory (x(·), w(·)) of Σ is stable if and only if w ∈ k 2 (W). 4) If (x(·), w(·)) is a stable full trajectory of Σ, then w ∈ L(W), QS −n w = Cfull Σ x(n), and −n w for all n ∈ Z. PX ⊥ x(n) = Bfull QS Σ 0
−n 5) Conversely, let w ∈ L(W), and define x(n) := Bfull Σ QS w, n ∈ Z. Then (x(·), w(·)) is a stable full trajectory of Σ. 6) The state component x(·) of a stable full trajectory (x(·), w(·)) of Σ is determined uniquely by the signal component w(·) if and only if Σ is simple. W 7) If Σ is simple, then Σ is unitarily similar to the canonical simple conservative model Σsc full with unitary similarity operator CΣ .
Proof. Proof of 1). The claim about the kernel of Cfull Σ is trivial. Therefore, to prove 1) it suffices to show that the restriction of Cfull to a dense subspace of X0⊥ is isometric, and that the range of Σ
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this restriction is a dense subspace of D(W). We choose this dense subspace of X0⊥ to be the set of all vectors x0 ∈ X of the form x0 = BΣ Q− z + C∗Σ Q+ z† , where z ∈ W and z† ∈ W[⊥] . By Lemmas 2.21 and 2.22, both BΣ and C∗Σ are isometric, and since ΓW = CΣ BΣ , Q+ z = ∗ Q z† , we have ΓW Q− z and Q− z† = ΓW + 2 x0 2X = BΣ Q− z + C∗Σ Q+ z† X 2 = BΣ Q− z 2X + C∗Σ Q+ z† X + 2 BΣ Q− z, C∗Σ Q+ z† X 2 = Q− z 2H− + Q+ z† H + 2 ΓW Q− z, Q+ z† H + + † † Q+ (z + z ) Q+ z , = . Q− z Q− (z + z† ) H+ ⊕H− On the other hand, Q+ z† ΓW Q− z + Q+ z† CΣ ∗ † BΣ Q− z + CΣ Q+ z = = AW . = B∗Σ Q− z + Γ ∗ Q+ z† Q− z
Cfull Σ x0
Thus, by Lemma 3.1, Cfull Σ x0 ∈ D(W), and full 2 C x 0 Σ
Q+ z† Q+ z† , AW Q− z Q− z H+ ⊕H− † † Q+ (z + z ) Q+ z , = = x0 2X . Q− z Q− (z + z† ) H+ ⊕H−
D (W) =
⊥ This proves that the restriction of Cfull Σ to a dense subspace of X0 is an isometric map of this subspace into D(W). The image of the same subspace is equal to D0 (W), which is dense in ⊥ D(W). Thus, Cfull Σ is a unitary map from X0 onto D(W), and hence a co-isometric map from X onto D(W). full Proof of 2). We begin by observing that (4.2) defines Bfull Σ uniquely, since it defines BΣ on R(AW ), which is dense in D(W). Thus, it suffices to prove that (4.2) holds. Let z ∈ W, and let x0 = BΣ Q− z. Then by the proof of Step 1, Cfull Σ x0 = Qz. On the other full = 1 hand, since Cfull B we also have D (W) Σ Σ full full Cfull Σ BΣ Qz = Qz = CΣ BΣ Q− z. full ⊥ ⊥ We also know that both x0 = BΣ Q− z and Bfull Σ Qz lie in X0 , and that CΣ is injective on X0 . Thus,
full BΣ Q− z = Bfull Σ Qz = BΣ
ΓW Q− z 1H−
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ΓW 0 0 for all z ∈ W. Consequently, BΣ x− = Bfull Σ 1H− x− for all x− ∈ H− . The subspace H− is ΓW full 1H+ . ∗ dense in H− , and thus BΣ = Bfull ∗ Σ 1H− . An analogous argument shows that CΣ = BΣ ΓW Proof of 3). By definition of a stable trajectory, if (x(·), w(·)) is a stable full trajectory, then w ∈ k 2 (W). Conversely, if w ∈ k 2 (W), then it follows from the balance equation n % 2 2 w(n), w(n) W = 0 −x(n + 1)X + x(m)X + k=m
that x(n) 2X has a finite limit at ±∞, and so x ∈ ∞ (W). Proof of 4). If (x(·), w(·)) is a stable full trajectory of Σ , and if we shift this trajectory to the left or right, then the shifted trajectory is still a stable full trajectory of Σ . Thus, it suffices to prove Claim 4) with n = 0. If (x(·), w(·)) is a stable full trajectory of Σ , then the restriction of this trajectory to Z+ is a stable future trajectory of Σ, and by the definition of CΣ , this implies that Q+ w = CΣ x(0). The same argument applied to the anti-causal adjoint system implies that Q− w = B∗Σ x(0), and confull full full full sequently, Qw = Cfull Σ x(0). By applying BΣ to this identity we get BΣ Qw = BΣ CΣ x(0) = PX ⊥ x(0). 0
Proof of 5). We first claim that if w = z + z† , where z ∈ W and z† ∈ W[⊥] , and if we define −n x(n) := Bfull Σ QS w, n ∈ Z, then (x(·), w(·)) is a stable full trajectory of Σ . We first consider the case where w = z ∈ W. Then, by Claim 2), −n full x(n) = Bfull Σ QS z = BΣ
ΓW Q− S −n z = BΣ Q− S −n z, 1H−
and it follows from Lemma 2.19 that (x(·), w(·)) is a stable full trajectory of Σ . An analogous argument can be used in the case where w = z† ∈ W[⊥] . Let now w be an arbitrary vector in L(W). Choose some sequence ym ∈ D0 (W) such that ym → Qw in D(W) as m → ∞. Let R be a bounded left-inverse of the quotient map Q, and define wm := w + R(ym − Qw). Then Qwm = ym → Qw in D(W), wm ∈ L0 (W), and wm → w full −n −n in k 2 (W) as m → ∞. Define x(n) = Bfull Σ QS w and xm (n) = BΣ QS wm , n ∈ Z. Then (xm (·), wm (·)) is a stable full trajectory of Σ for all m, and xm (n) → x(n) for all n ∈ Z as m → ∞. Since V is closed, also (x(·), w(·)) is a full trajectory of Σ , and it is stable since w ∈ k 2 (W). Proof of 6). If Σ is simple, then X0⊥ = X , and it follows from 4) that the state component x(·) of a stable full trajectory (x(·), w(·)) of Σ is determined uniquely by w(·). On the other hand, if Σ is not simple, then X0 = {0}, and for each x0 ∈ X0 there exists a (unique) stable full trajectory (x(·), 0) of Σ with x(0) x0 and zero signal part. = Proof of 7). Let
x1 x0 w0
∈ V , and choose some arbitrary stable full trajectory (x(·), w(·))
such that x(0) = x0 , x(1) = x1 , and w(0) = w0 ; this is possible since Σ is well-posed both in the forward and in the backward time direction. By parts 5) and 6), the unique full trajectory (xsc (·), w(·)) of Σsc whose signal part is w(·) satisfies xsc (n) = QS −n w, n ∈ Z, and by the
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full −n same argument, x(n) = Bfull Σ QS w, n ∈ Z. Thus, in particular, x0 = x(0) = BΣ xsc (0) and
x1 = x(1) = Bfull Σ xsc (1), where
xsc (1) xsc (0) w0
∈ Vsc . This gives
⎡
Bfull Σ ⎣ V⊂ 0 0
0 Bfull Σ 0
⎤ 0 0 ⎦ Vsc . 1W
By interchanging the roles of Σ and Σsc we get the opposite inclusion. Thus, Σ is unitarily W with unitary similarity operator (Bfull )−1 = Cfull . 2 similar to Σsc Σ Σ Alternative proof of Theorem 3.5. Let Σ = (V ; X , W) be an arbitrary simple conservative s/s realization of W; that such a s/s system exists follows from [3, Theorem 8.6]. It follows from Theorem 4.1 that V has the image representation ⎫ ⎧⎡ full −1 ⎤ ⎬ ⎨ BΣ QS w
⎦ w ∈ L(W) . V = ⎣ Bfull Qw Σ ⎭ ⎩ w(0) full ∗ If we to this system apply a unitary similarity transform with similarity operator Cfull Σ = (BΣ ) , then we get another simple conservative s/s realization of W. The generating subspace that we get in this way is the same one which is given in (3.15). 2
The above proof of Theorem 3.5 is very short, but it is not fully self-contained in the sense that it is based on the knowledge that every passive full behavior has a simple conservative realization. The original proof given in Section 3 is complete in the sense that it does not rely on any a priori knowledge of the existence of a simple conservative realization of W. Corollary 4.2. Let W be a full behavior on the Kre˘ın space W. Then the sequence (x(·), w(·)) W if and only if is a stable full trajectory of Σsc w ∈ L(W)
and x(n) = QS −n w,
Proof. This follows from Theorem 4.1.
n ∈ Z.
2
full Definition 4.3. We call the operators Cfull Σ and BΣ defined in Theorem 4.1 the bilateral output and input maps, respectively, of the conservative s/s system Σ .
5. Incoming and outgoing inner channels In this section we throughout let Σ = (V ; X , W) be a conservative s/s system with full behavior W, input map BΣ , and output map CΣ . Let RΣ = R(BΣ ) be the reachable subspace, let UΣ = N (CΣ ) be the unobservable subspace, let R†Σ = R(C∗Σ ) be the null controllable subspace, and let U†Σ = N (B∗Σ ) be the backward unobservable subspace of Σ . Lemma 5.1. Let Σ = (V ; X , W) be a conservative s/s system.
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1) There exists a unique isometry A− on UΣ such that (x(·), 0) is an unobservable future trajectory of Σ if and only if x(n) = An− x(0). Every unobservable future trajectory (x(·), 0) is uniquely determined by the value of x(n) for any fixed n ∈ Z+ . 2) There exists a unique isometry A+ on U†Σ such that (x(·), 0) is a backward unobservable |n| past trajectory of Σ if and only if x(n) = A+ x(0), n ∈ Z− . Every backward unobservable past trajectory (x(·), 0) is uniquely determined by the value of x(n) for any fixed n ∈ Z− . Proof. By the definition of UΣ , for each x0 ∈ UΣ there exists a unique unobservable future trajectory (x(·), 0) of Σ with x(0) = x0 . Let A− be the mapping from x0 to x(1). That A− is an isometry follows from the conservativity of Σ which implies that x(1) 2X = x(0) 2X . If we left-shift an unobservable trajectory by n steps, then the shifted trajectory is still an unobservable trajectory, and hence x(n + 1) = A− x(n) for all n ∈ Z+ . Since A− is isometric, the condition x(n + 1) = A− x(n) implies x(n) = A∗− x(n + 1), and therefore x(n + 1) = A− x(n),
x(n) = A∗− x(n + 1),
n ∈ Z+ .
(5.1)
Clearly, if we know x(n) for any fixed n ∈ Z+ , then (5.1) determines the full future trajectory uniquely. The proof of Claim 2) is analogous. This time (5.1) is replaced by x(n) = A+ x(n + 1),
x(n + 1) = A∗+ x(n),
n ∈ Z− .
2
(5.2)
Definition 5.2. We call (A∗+ , U† ) the incoming inner channel and (A− ; U) the outgoing inner channel of the conservative s/s system Σ, where A+ and A− are the operators defined in Lemma 5.1. Theorem 5.3. Let Σ = (V ; X , W) be a conservative s/s system with bilateral input and output † full ∗ maps Bfull Σ and CΣ , respectively, and incoming and outgoing inner channels (A+ ; UΣ ) and (A− ; UΣ ), respectively. Then ) ) full ⊥ UΣ ∩ U†Σ = N Cfull = An− UΣ = An+ U†Σ . Σ = R BΣ n∈Z+
(5.3)
n∈Z+
Consequently, the following four conditions are equivalently: 1) 2) 3) 4)
Σ is simple. The operator A− is completely non-unitary. The operator A+ is completely non-unitary. Σ has no nontrivial bilaterally unobservable trajectory.
full )⊥ . This follows from Theorem 4.1. Proof. Step 1: UΣ ∩ U†Σ = N (Cfull Σ ) = R(BΣ + + † n Step 2: UΣ ∩ UΣ ⊂ ( n∈Z+ A− UΣ ) ∩ ( n∈Z+ An+ U†Σ ). Let x0 ∈ UΣ ∩ U†Σ . Then by the definitions of UΣ and U†Σ , Σ has a bilaterally unobservable full trajectory (x(·), 0) with x(0) = x0 . The set of all bilaterally unobservable full trajectories of Σ is invariant under both rightand left-shifts, and together with (5.1) and (5.2) this implies that for all n ∈ Z+ we have x(0) = + + An− x(−n) = An+ x(n). Consequently x0 ∈ ( n∈Z+ An− UΣ ) ∩ ( n∈Z+ An+ U†Σ ).
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Step 3: Clearly
+
n∈Z+
An− UΣ ⊂ UΣ ∩U†Σ and A− X0 ⊂ X0 =
+
)
n∈Z+
An+ U†Σ ⊂ UΣ ∩U†Σ . Denote X0 =
An− UΣ ⊂
n0
)
+
n∈Z+
An− UΣ .
An− UΣ = A− X0 .
n1
Thus, A− X0 = X0 , and hence A0 := A− |X0 maps X0 unitarily onto itself. Let x0 ∈ X0 , and define x(n) = An0 x0 , n ∈ Z. Then x(n + 1) = A− x(n), n ∈ Z. By the definition of A− ,
A− x(n) x(n) 0
∈V,
n ∈ Z, and hence (x(·), 0) is a bilaterally unobservable trajectory of Σ . By the definitions of UΣ + and U†Σ , this implies that x(0) = x0 ∈ UΣ ∩ U†Σ . Thus n∈Z+ An− UΣ ⊂ UΣ ∩ U†Σ . An analogous + argument shows that also n∈Z+ An+ U†Σ ⊂ UΣ ∩ U†Σ . 2 Lemma 5.4. Let Σ = (V ; X , W) be a conservative s/s system with incoming and outgoing inner channels (A∗+ ; U†Σ ) and (A− ; UΣ ), respectively. Then x1 PU† x1 = A∗+ PU† x0 and PUΣ x0 = A∗− PUΣ x1 whenever x0 ∈ V . (5.4) Σ Σ w0 Proof. Let z0 ∈ U†Σ and
x 1
x0 w0
0=
∈ V . By Lemma 5.1,
z0 A+ z0 0
∈ V and since V = V [⊥] we get
z0 x1 = −(z0 , x1 )X + (A+ z0 , x0 )X A+ z0 , x0 0 w0 K
= −(z0 , PU† x1 )X + (A+ z0 , PU† x0 )X Σ Σ ∗ = z0 , −PU† x1 + A+ PU† x0 X . Σ
Σ
This being true for all z0 ∈ U†Σ we have PU† x1 = A∗+ PU† x0 . An analogous computation shows Σ Σ that PUΣ x0 = A∗− PUΣ x1 . 2 Theorem 5.5. Let Σ = (V ; X , W) be a conservative s/s system. 1) If (x(·), w(·)) is a future trajectory of Σ , then n PU† x(n) = A∗+ PU† x(0), Σ
Σ
n 0.
(5.5)
In particular, PU† x(n)X is a nonincreasing function of n. Σ 2) If (x(·), w(·)) is a past trajectory of Σ , then |n| PUΣ x(n) = A∗− PUΣ x(0),
n 0.
In particular, PUΣ x(n) X is a nonincreasing function of |n|.
(5.6)
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3) The following three assertions are equivalent: (a) Σ is simple; (b) PU† x(n) → 0 in X as n → ∞ for every future trajectory (x(·), w(·)) of Σ ; Σ (c) PUΣ x(n) → 0 in X as n → −∞ for every past trajectory (x(·), w(·)) of Σ. Proof. That (5.5) and (5.6) hold follows from Lemma 5.4, and the monotonicity of the norm follows from the fact that A∗+ and A∗− are contractions. By the Wold decomposition (see, e.g., [21, Theorem 1.1, p. 3]), A± is completely non-unitary if and only if A∗± is strongly stable, and hence 3) follows from 1) and 2) combined with Theorem 5.3. 2 non-unitary Suppose that Σ is simple, and denote N− = N (A∗− ). Since A− is completely , n A it follows from the Wold decomposition [21, Theorem 1.1, p. 3] that UΣ = ∞ n=0 − N− . This 2 makes it possible to define a unitary map U− : UΣ → − (N− ) by −(k+1) −∞ U− x = PN− A∗− x k=−1 ,
x ∈ UΣ .
(5.7)
∗ on 2 (N ) in the sense that The operator U− intertwines A− with the outgoing shift S− − − ∗ U− A− = S− U− .
Analogously we define N+ = N (A∗+ ). Then U†Σ = define a unitary map U+ : U†Σ → 2+ (N+ ) by
(5.8)
,∞
n n=0 A+ N+ ,
k ∞ U+ x = PN+ A∗+ x k=0 ,
and this makes it possible to
x ∈ U†Σ .
(5.9)
The operator U+ intertwines A+ with the outgoing shift S+ on 2+ (N+ ) in the sense that U+ A+ = S+ U+ .
(5.10)
In the case of the system Σsc = (Vsc ; D(W), W) it is possible to give more explicit formulas for the operators A± defined in Lemma 5.1 and their adjoints. Lemma 5.6. In the case of the canonical simple conservative system Σsc = (Vsc ; D(W), W) with behavior W and past/future map ΓW the operators A∓ and their adjoints are given by the following formulas: A− Qw = QS −1 w, w ∈ L(W), π+ w = 0, 0 ∗ Sw, w ∈ L(W), π+ w = 0, A− Qw = ∗ Q Q− − ΓW + A+ Qw = QSw, w ∈ L(W), π− w = 0, Q+ − ΓW Q− −1 ∗ S w, w ∈ L(W), π− w = 0. A+ Qw = 0 Thus, the defect subspaces N− and N+ for the system Σsc are given by
(5.11)
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∗ N− = Qw w ∈ L(W) with π+ w = 0 and Q− Sw = ΓW Q+ Sw ,
N+ = Qw w ∈ L(W) with π− w = 0 and Q+ S −1 w = ΓW Q− S −1 w .
(5.12)
Proof. Below we only prove the formulas for A− and A∗− , and leave the proof of the formulas for A+ and A∗+ to the reader. By the definition of A− , for all x0 ∈ U we have A− x0 = x1 where
x1 x0 0
∈ Vsc . Since x0 ∈ U it
follows from (3.16) that x0 = Qw for some w ∈ L(W) satisfying π+ w = 0. By the definition of Vsc and the fact that the first component of Vsc is determined uniquely by the last two components we have x1 = QS −1 w. This proves the first equation in (5.11). To compute A∗− we let x0 ∈ U, and choose some representatives w such that x0 = Qw with ∗ ∗ w ∈ L(W) π+ w = 0. By Theorem 5.5 and (3.17), A− x0 = PU x −1 = (Π − − ΓW Π+ )x−1 , with where
x0 x−1 w−1
∈ Vsc for some w−1 ∈ W. One such vector is
x0 x−1 w−1
=
Qw QSw w(−1)
. Thus,
∗ ∗ A∗− x0 = Π− − ΓW Π+ QSw = Q− Sw − ΓW Q+ Sw. This proves the second equation in (5.11).
2
In the case of the system Σsc = (Vsc ; D(W), W) it is possible to give alternative descriptions of the unobservable and backward unobservable subspaces. W = (V ; D(W), W) be the canonical model of a simple conservative s/s Lemma 5.7. Let Σsc sc system with passive full behavior W and past/future map ΓW . Let U and U† be the unobservW . Finally, let H(Γ ) and H(Γ ∗ ) be the de able and backward unobservable subspaces of Σsc W W ∗ . Then the following Branges complementary spaces of the contractive operators ΓW and ΓW claims hold.
1) U is given by
U = x ∈ D(W) Π+ x = 0 =
∗ H+
0 x ∈ , ∈ H Γ
− W H− x−
(5.13)
and ∗ ), x D(W) = Π− x H(ΓW
x ∈ U.
∗ ). Thus, Π− |U is a unitary map from U onto H(ΓW 2) U† is given by
H+
x+ x ∈ ∈ H(Γ ) , U† = x ∈ D(W) Π− x = 0 =
+ W 0 H−
(5.14)
(5.15)
and ∗ ), x D(W) = Π+ x H(ΓW
Thus, Π+ |U† is a unitary map from U† onto H(ΓW ).
x ∈ U† .
(5.16)
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3) The s/s system Σsc is observable if and only if ΓW is isometric, or equivalently, if and only ∗ Π | if Π− |D(W) = ΓW + D (W) . 4) The s/s system Σsc is controllable if and only if ΓW is co-isometric, or equivalently, if and only if Π+ |D(W) = ΓW Π− |D(W) . 5) The s/s system Σsc is minimal if and only if ΓW is unitary. Proof. The first equalities in (5.13) and (5.15) follow from (3.16). Define ∗ ΓW , W = 1H− − ΓW
(5.17)
with x ∈ H . Then and let x− = W x− − −
0 1 0 =
= H∗+ ∗ Γ x
x− − ΓW x− ΓW W −
ΓW 1H−
where AW is the operator in (3.3). Consequently,
−ΓW x− −ΓW x− = AW ,
x− x−
0 x− ∈ R(AW ) ⊂ D(W),
and
2 0 x−
1/2 −ΓW x− 2 A = W x− D (W) H+ ⊕H− −ΓW x− −ΓW x− , AW = x− x− H+ ⊕H− 0 −ΓW x− , = x− H+ ⊕H− x−
2 = (x− , x− )H− = x− , W x− H = x− H(Γ ∗ ) . −
W
Thus,
0
x− ∈ R( W ) ⊂ D(W) ∩ (0 ⊕ H− ) = U, x−
∗ ), we find that and (5.14) holds for x− ∈ R( W ). Since R( W ) is a dense subspace of H(ΓW ∗ ), and that U := { 0 | x ∈ H(Γ ∗ )} is a closed subspace of U. (5.14) holds all x− ∈ H(ΓW 0 − W x− 0 To prove that U0 = U we let x− ∈ U be orthogonal to U0 in D(W). Then, for all x− ∈ H− ,
- 0=
.
0 −ΓW x−
, AW = x − , x−
H− , x− x− D (W)
which implies that x− = 0. This proves assertion 1). ∗ . Assertions 3) Assertion 2) may be proved in an analogous way, with ΓW replaced by ΓW and 4) follows from assertions 1) and 2) and (3.17), and assertion 5) follows from assertions 3) and 4). 2 For use in subsequent work we record the following fact.
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Proposition 5.8. Let Σ = (V ; X , W) be a simple conservative s/s system for which both the incoming inner channel (A∗+ ; U† ) and the outgoing inner channel (A1 ; U) are nontrivial. Let Γ ∈ B(UΣ ; U†Σ ) be the operator
Γ = PU† U .
(5.18)
Γ A− = A∗+ Γ
(5.19)
Σ
Σ
Then
and Γ ∗ Γ < 1UΣ ,
Γ Γ ∗ < 1U† .
(5.20)
Σ
Proof. That Γ has property (5.20) follows from its definition (5.18) and the fact that UΣ ∩ † U (5.19) we take some arbitrary x− ∈ UΣ and x+ ∈ U†Σ . Since both Σ = 0. To check the relation A− x− x− 0
∈ V and
x+ A+ x+ 0
∈ V and V = V [⊥] , we have
(A− x− , x+ )X = (x− , A+ x+ )X . Thus, (Γ A− x− , x+ )X = (PU† |UΣ A− x− , x+ )X = (A− x− , x+ )X = (x− , A+ x+ )X Σ = (x− , PUΣ |U† A+ x+ )X = x− , Γ ∗ A+ x+ X . Σ
This proves (5.19).
2
An operator Γ ∈ B(UΣ ; U†Σ ) satisfying the intertwining condition (5.19) with respect to the isometric completely non-unitary operator A− and the co-isometric completely non-unitary operator A∗+ is usually called a Hankel operator. By (5.20), the Hankel operator Γ defined in (5.19) is a contraction which does not have any singular numbers on the unit circle. 6. Alternative characterizations of L(W) and D(W) 2 (W) and Let W be a full passive behavior on the Kre˘ın signal space W, let W+ = W ∩ k+ W− = π− W be the corresponding future and past passive behaviors, and denote H+ = H(W+ ) and H− = H(W[⊥] − ). For each n ∈ Z+ we define
2 −n
L− n (W) := w(·) ∈ k (W) Q− S w ∈ H− , n−1 % 2 ρn− (w) = Q− S −n w H − w(k), w(k) W , −
k=0
w ∈ L− n (W).
(6.1) (6.2)
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If w ∈ L(W), then by Lemma 3.4, Q− S −n w ∈ H− for all n ∈ Z+ , and consequently L(W) ⊂ + L− n (W) for all n ∈ Z . Theorem 6.1. Let W be a passive full behavior on W. 1) A sequence w ∈ k 2 (W) belongs to L(W) if and only if Q− S −n w ∈ H− for all n ∈ Z+ and sup Q− S −n w H < ∞.
(6.3)
−
n∈Z+
2) If w ∈ L(W), then the sequence ρn− (w) is nonnegative, nondecreasing and bounded, and Qw 2D(W) = w 2L(W) = sup ρn− (w) = lim ρn− (w) n→∞
n∈Z+
2 = lim Q− S −n w H − [π+ w, π+ w]k 2 (W ) . −
n→∞
+
(6.4)
Γ Proof. Step 1: If (6.3) holds, then w ∈ L(W). For each n ∈ Z+ , the vector yn := 1HW Q− S −n w − belongs to D(W), and as the sequence Q− S −n w ∈ H− is assumed to be uniformly bounded in H− , the sequence yn is uniformly bounded in D(W). Let R+ be a right-inverse of the quotient map Q+ , and define wn , n 1, by wn = S n R+ ΓW Q− S −n w + π− S −n w = S n R+ ΓW Q− S −n w + Pk 2 ((−∞,n−1];W ) w. Then wn (k) = w(k) for k n − 1, wn is uniformly bounded in k 2 (W), and yn = QS −n wn . Finally, define xn = Qwn . Then, by Lemma 3.4, xn D(W) = Qwn D(W) 2
2
n−1 % wn (k), wn (k) W D (W) −
2 = QS −n wn
k=0
= yn 2D(W) −
n−1 %
w(k), w(k) W ,
k=0
and hence the sequence xn is uniformly bounded in D(W). Since the unit ball in D(W) is weakly compact, we can without loss of generality (by passing to a subsequence) suppose that xn tends weakly to a limit x ∈ D(W), and hence also in k 2 (W)/(W+ + W[⊥] − ). Since wn (k) = w(k) for k < n, it is also true that wn tends weakly to w in k 2 (W) as n → ∞. Let R : k 2 (W)/ 2 (W+ + W[⊥] − ) → k (W) be a bounded right-inverse of Q. Then, on one hand, RQwn tends weakly to RQw in k 2 (W), and on the other hand, RQwn = Rxn → Rx
weakly as n → ∞.
Therefore, RQw = Rx. Since R is injective, this implies that Qw = x ∈ D(W), and consequently, w ∈ L(W).
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Step 2: If w ∈ L(W), then (6.3) and (6.4) hold. It follows from Lemma 3.4 that if w ∈ L(W), then QS −n w ∈ L(W), and since D(W) is continuously contained in H+ ⊕ H− , this implies that Q− S −n w ∈ H− for all n ∈ Z+ . Let (x(·), w(·)) be the (unique) stable full trajectory of Σsc whose signal part is w, i.e., x(n) = QS −n w, n ∈ Z. By the conservativity of Σsc , n % x(0)2 w(n), w(n) W , = + D (W) D (W)
x(n)2
n ∈ Z+ .
n=0
Write x(n)2
PR x(n)2 P † x(n)2 = + U D (W) D (W) D (W) ,
where PR x(n)2
D (W)
2 ΓW 2 −n = Q− S −n w H . Π− QS w D (W) = 1H − − D (W)
2 = PR QS −n w
Thus, n−1 % 2 ρn− (w) = Q− S −n w H − w(k), w(k) W −
k=0 n−1 % 2 w(k), w(k) W = PR x(n)D(W) − k=0
2 = x(0)
2 D (W) − PU† x(n) D (W) 2 = w 2L(W) − PU† x(n)D(W) . Thus ρn− (w) 0 since PU† x(n) D(W) w L(W) . By Theorem 5.5, the sequence PU† x(n) 2D(W) is nonincreasing and tends to zero as n → ∞, and hence the sequence ρn− (w) is nondecreasing and tends to w 2L(W) as n → ∞. 2 Above we looked at the behavior of the sequence Q− S −n w in H− as n → ∞, and related this to the condition w ∈ L(W). It is also possible to instead look at how the sequence Q+ S n w behaves in H+ as n → ∞. For each n ∈ Z+ we define
2 n
L+ n (W) := w(·) ∈ k (W) Q+ S w ∈ H+ , −1 % 2 ρn+ (w) := Q+ S n w H − w(k), w(k) W , +
w ∈ L+ n (W).
(6.5) (6.6)
k=−n
If w ∈ L(W), then by Lemma 3.4, Q+ S n w ∈ H+ for all n ∈ Z+ , and consequently L(W) ⊂ + L+ n (W) for all n ∈ Z .
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Theorem 6.2. Let W be a passive full behavior on W. 1) A sequence w ∈ k 2 (W) belongs to L(W) if and only if Q+ S n w ∈ H+ for all n ∈ Z+ and sup Q+ S n w H < ∞. (6.7) +
n∈Z+
2) If w ∈ L(W), then the sequence ρn+ (w) is nonnegative, nondecreasing and bounded, and Qw 2D(W) = w 2L(W) = sup ρn+ (w) = lim ρn+ (w) n→∞
n∈Z+
2 = lim Q+ S n w H + [π− w, π− w]k 2 (W ) . n→∞
+
−
(6.8)
2 (W) Proof. This proof is analogous to the proof of Theorem 6.1. One throughout interchanges k− 2 −1 and k+ (W), π− and π+ , Q− and Q+ , S and S, and H− and H+ . (However, W and W[⊥] should not be interchanged.) 2
Lemma 6.3. Let w ∈ L(W). 1) The following conditions are equivalent: (a) w ∈ W; (b) w 2L(W) = [π− w, π− w]k 2 (W ) ; −
(c) limn→∞ Q− S −n w 2H− = [w, w]k 2 (W ) ;
(d) limn→∞ Q+ S n w 2H+ = 0. 2) The following conditions are equivalent: (e) w ∈ W[⊥] ; (f) w 2L(W) = −[π+ w, π+ w]k 2 (W ) ; +
(g) limn→∞ Q+ S n w 2H+ = [w, w]k 2 (W ) ;
(h) limn→∞ Q− S −n w 2H− = 0.
Proof. (a) ⇒ (b): This follows from Lemma 3.3. (b) ⇔ (c): This follows from Theorem 6.1. (b) ⇔ (d): This follows from Theorem 6.2. (d) ⇒ (a): Let (x(·), w(·)) be the unique stable full trajectory of Σsc with signal part w(·), i.e., x(n) = QS −n w for all n ∈ Z. We decompose x(n) in two orthogonal components, x(n) = PR† x(n) + PU x(n). By Theorem 5.5, PU x(n) → 0 as n → −∞, and by (3.17), PR† x(n) = Q+ −n S w, which tends to zero as n → −∞ if (d) holds. Thus, (x(·), w(·)) is an externally ∗ Q ΓW + generated trajectory of Σsc , and so w ∈ W. Proof of 2). This proof is analogous to the one above. 2 7. Forward and backward conservative compressions of the conservative model In Section 2.4 we presented two additional canonical models, namely the controllable forward W W conservative model Σcfc − , and the observable backward conservative model Σobc+ , which were originally obtained in [7]. Here we shall study the relationships between these two models and
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the model Σsc presented in Section 3. As we shall see, the two models in Section 2.4 can be obtained from Σsc by first performing an orthogonal compression, and then applying a unitary similarity transform. /, W) is an orthogonal outgoing dilation /; X / = (V We recall from [2,3] that the s/s system Σ /, if of the s/s system Σ = (V ; X , W) and Σ is an orthogonal outgoing compression onto X of Σ / and X ⊂X
0 1X 0
PX 0 0
0 0 1W
0
/ 1 X PX /∩ X V = 0 W 0
0 PX 0
0 0 1W
/. V
(7.1)
In (7.1) it is possible to take X to be the orthogonal complement to the unobservable subspace / and in this case Σ is observable. If instead of (7.1), we have of Σ,
PX 0 0
0 1X 0
0 0 1W
0
/ 1 / X X /∩ X /∩ X , V =V W W
(7.2)
/ is an orthogonal incoming dilation of Σ and Σ is an orthogonal incoming compresthen Σ / In (7.2) it is possible to take X to be the reachable subspace of Σ /, and in this case sion of Σ. Σ is controllable. These orthogonal compressions of a passive s/s system are passive, and they have the same past, full, and future behaviors as the dilated systems. By compressing Σsc orc = (V c ; R, W), thogonally onto R we get a controllable forward conservative s/s system Σsc sc † and by compressing Σsc orthogonally onto R we get an observable backward conservative s/s o = (V o ; R† , W), both of which have the same future, full, and past behaviors as Σ . Σsc sc sc By compressing Σsc orthogonally onto R we get a controllable forward conservative s/s sysc = (V c ; R, W), and by compressing Σ orthogonally onto R† we get an observable tem Σsc sc sc o = (V o ; R† , W), both of which have the same future, full, and backward conservative s/s Σsc sc past behaviors as Σsc . c of Σ to R is given by The generating subspace Vscc of the compression Σsc sc
Vscc
1 0 PR 0 D(W) D(W) 0 = 0 1R = Vsc ∩ Vsc ∩ R R 0 W W 0 0 1W
QS −1 w
= Qw
w ∈ L(W), Q+ w = ΓW Q− w . w(0)
(7.3)
Thus, this is an incoming compression of Σsc . That the two different formulas for Vscc given above are the same follows from the fact R is strongly invariant in the sense that x1 ∈ R whenever x1 x0 w0
W
∈ Vsc and x0 ∈ R. The subspace Vscc can be compared to the generating subspace Vcfc − of W
W
the canonical controllable and forward conservative model Σcfc − = (Vcfc − ; H− , W) described in Theorem 2.16. These two systems are unitarily similar, with the similarity operator BΣsc = ΓW : H− → R, whose inverse is Π− |R . 1H−
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3307
o of Σ to R† is given by The generating subspace Vsco of the compression Σsc sc
Vsco
1 0 PR† D(W) PR† 0 0 † = = Vsc ∩ R 0 0 1R† 0 W 0 0 1W 0 ⎫ ⎧ ⎡ 1H ⎤ −1 + ⎪ ⎪ ⎪ ⎪ ∗ Q+ S w
⎪ ⎪⎢ ΓW ⎬ ⎨ ⎥
⎢ ⎥ = ⎢ 1H+ ⎥ w ∈ L(W) . ⎪ ⎪ ⎣ ∗ Q+ w ⎦
⎪ ⎪ ⎪ ⎪ ΓW ⎭ ⎩ w(0)
0 PR† 0
0 0 1W
Vsc
(7.4)
Thus, this is an outgoing compression. That the two different formulas for Vsco are equivalent x 1 follows from the fact that x1 ∈ U = (R† )⊥ whenever x0 ∈ Vsc and x0 ∈ U; cf. Lemma 5.1. 0
Here the right-hand side depends only on the projection w+ := π+ w of w. By Theorem 6.2, w+ ∈ K(W+ ). Conversely, if w+ ∈ K(W+ ), the w+ can be written in the form w+ = π+ w 1H where w ∈ L(W) is an arbitrary sequence satisfying Qw = Γ ∗+ Q+ w+ and π+ w = w+ ; that 1H W such a sequence exists follows from the fact that C∗Σsc = Γ ∗+ maps H+ into D(W). Therefore W we can rewrite (7.4) in the form ⎧⎡ 1 ⎫ ⎤ H+ ∗ ⎪ ⎪ ⎪ ⎪ ∗ Q+ S+ w+
⎪ ⎪ Γ ⎨⎢ W ⎬ ⎥
⎢ ⎥
o Vsc = ⎢ 1H+ ⎥ w+ ∈ K(W+ ) . ⎪ ⎪ ⎣ Γ ∗ Q+ w+ ⎦
⎪ ⎪ ⎪ ⎪ W ⎩ ⎭ w+ (0)
(7.5)
W
This can be compared to the generating subspace Vobc+ of the canonical observable and backward W W conservative model Σobc+ = (Vobc+ ; H+ , W) presented in Theorem 2.11. These two systems 1H are unitarily similar, with the similarity operator C∗Σsc = Γ ∗+ : H+ → R† , whose inverse W is Π+ |R† . We now want to investigate the connections between Σsc and the two compressions defined above in more detail. Here the results described in Section 5 again become important. These results describe the part of the dynamics which stays in the unobservable subspace U or the backward unobservable subspace U† . To get a complete picture we also have to describe the part of the dynamics that crosses over between these subspaces and the reachable subspace R or the null controllable subspace R† , respectively. Here we only directly look at the simple conservative canonical model, but the results can easily be adapted to an arbitrary simple conservative system by using the unitary similarity described in part 7) of Theorem 4.1. Lemma 7.1. Define Vsco by (7.4) let N− = N (A∗− ). Then the formula Xo
PR† QS −1 w := PU QS −1 w, Qw w(0)
w ∈ L(W), Qw ∈ R† ,
(7.6)
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defines a bounded linear operator Xo from Vsco onto N− with
1R† N (Xo ) = 0 0
0 PR† 0
0 0 1W
0
R† Vsc ∩ D(W) W
1
R† = Vsc ∩ D(W) . W
(7.7)
This operator is isometric with respect to the inner product that Vsco inherits from Ko := −R† [] R† [] W. Proof. Let x0 ∈ R† , and choose some w ∈ L(W) such that Qw = x0 . By Theorem 3.5,
QS −1 w ∈ Vsc x0 w(0)
and hence by the conservativity of Σsc , 2 −QS −1 w D(W) + x0 2D(W) + w(0), w(0) W = 0. Here we split QS −1 w into two orthogonal components QS −1 w = x1 + z1 , where x1 := PR† QS −1 w ∈ R† ,
z1 := PU QS −1 w ∈ U.
This gives z1 2D(W) = x0 2D(W) + w(0), w(0) W − x1 2D(W) x1 x1 = . x0 , x0 w(0) w(0) −R† []R† []W By (7.4),
x1 x0 w(0)
∈ Vsco . This shows that (7.6) defines an isometric map Xo from Vsco into U
whose kernel is the maximal neutral subspace of Vsco . This subspace is equal to the orthogonal o is backward conservative, and it is not difficult to show that it complement to Vsco in Ko since Σsc is explicitly given by (7.7). That the two expressions for N (Xo ) are equivalent follows different from the fact that x0 ∈ R† whenever
x1 x0 w0
∈ Vsc and x1 ∈ R† .
It remains to show that R(Xo ) = N− . Let z 1 ∈ R(X o ). Then by the definition of Xo , there z1 +x1 † † x0 ∈ Vsc . By Lemma 5.6, exists w0 ∈ W, x0 ∈ R , and x1 ∈ R such that w0
0 = PU x0 = A∗− PU (z1 + x1 ) = A∗− z1 . Thus, z1 ∈ N (A∗− ) = N− . Conversely, suppose that z1 ∈ N− , i.e., that z1 ∈ U and A∗− z1 = 0. Since Σsc is well-posed in the backward time direction it is possible to find some x0 ∈ D(W) and w0 ∈ W such that
z1 x0 w0
∈ Vsc . By Lemma 5.6, PU x0 = A∗− PU z1 = A∗− z1 = 0, and so x0 ∈ R† .
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Thus, by the definition of Xo , z1 = Xo N− .
2
0 x0 w0
3309
], and so z1 ∈ R(Xo ). This proves that R(Xo ) =
Theorem 7.2. Let Σsc = (Vsc ; D(W), W) be the canonical model of a simple conservative s/s o = (V o ; R† , W) be the orthogonal compression system with passive full behavior W, and let Σsc sc † of Σsc onto the null controllable subspace R of Σsc . Let (A− ; U) be the outgoing inner channel of Σsc , and define the isometric operator Xo from Vsco onto N− = N (A∗− ) by (7.6). Then ⎧ ⎫ ⎡ x ⎤ 1 ⎪ ⎪ ⎪ ⎪ X A− z0 x1 ⎨ x1 ⎬ ⎢ o x0 ⎥
o w0 ⎥ + . Vsc = , z ∈ U ∈ V x0 + ⎢ z x
0 0 0 sc ⎦ ⎣ ⎪ ⎪ 0 ⎪ ⎪ w 0 w ⎩ 0 ⎭ 0 0 x Proof. Let
1 +z1 x0 +z0 w0
x1 x0 w0
∈ Vsc , where xi ∈ R† and zi ∈ U for i = 1, 2. Then
∈ Vsco ,
Moreover, z1 − A− z0 = Xo
x1 + z1 x0 + z0 w0
x 1
Conversely, if
x0 w0
(7.8)
A− z0 z0 0
∈ Vsc ,
and
x1 + z1 − A− z0 x0 w0
∈ Vsc .
x 1
as was shown in the proof of Lemma 7.1. Thus,
x0 w0
x1 = x0 w0
z1 + z0 0
x1 = x0 w0
x ⎤ 1 X A− z0 ⎢ o x0 ⎥ w0 ⎥ + +⎢ z0 ⎦ ⎣ 0 0 0 ⎡
is an arbitrary vector in Vsco and z0 is an arbitrary vector in U, then the above
sum belongs to Vsc , as can be seen from the proof of Lemma 7.1.
2
Lemma 7.3. Define Vscc by (7.3). Then the formula Xc
QS −1 w PR Qw := PU† Qw, w(0)
w ∈ L(W), QS −1 w ∈ R,
(7.9)
defines a bounded linear operator Xc from
[⊥] Vscc
PR = 0 0
0 PR 0
0 0 1W
Vsc =
PR QS −1 w
PR Qw
w ∈ L(W) , w(0)
onto N+ with N (Xc ) = Vscc . This operator is isometric with respect to the inner product that (Vscc )[⊥] inherits from −(−R [] R [] W).
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o Proof. The proof of this lemma is analogous to the proof of Lemma 7.1, where one replaces Σsc c. 2 by the anti-causal dual of Σsc
Corollary 7.4. Let Σsc = (Vsc ; D(W), W) be the canonical model of a simple conservative s/s c = (V c ; R, W) be the orthogonal compression system with passive full behavior W, and let Σsc sc of Σsc onto the reachable subspace R of Σsc . Let (A∗+ ; U† ) be the incoming inner channel of Σsc , and define the isometric operator Xc from −(Vscc )[⊥] to N+ = N (A∗+ ) by (7.9). Then ⎫ ⎧ ⎤ ⎡ 0 ⎪ ⎪ ∗ ⎪ ⎪ A+ z0 x1 ⎬ ⎨ x1 x1 ⎥ ⎢ c [⊥]
† ⎥ ⎢ Vsc = , z0 ∈ U . x0 + ⎣ X c x0 ⎦ + z0
x0 ∈ Vsc ⎪ ⎪ w0 ⎪ ⎪ 0 w0 ⎭ ⎩ w0 0
(7.10)
Proof. The proof is analogous to the proof of Theorem 7.2, where the past and the future have interchanged places, and Lemma 7.1 has been replaced by Lemma 7.3 2 Lemma 7.5. The subspaces N± = N (A∗± ) have the following representations: N− =
w ∈ L(W) and 0 −1
S w
, ∗ Q ∗ Q w Q− − ΓW + Q− w = ΓW +
N+ =
w ∈ L(W) and Q+ − ΓW Q−
Sw
. 0 Q+ w = ΓW Q− w
(7.11)
Proof. Formula (7.11) holds since N− = R(Xo ) and N+ = R(Xc ), where Xo and Xc are the operator defined in Lemmas 7.1 and 7.3. In this formula we have also used (3.16) and substituted the values of PR , PU , PR† and PU† given in (3.17). 2 Theorem 7.6. Let Σsc = (Vsc ; D(W), W) be the canonical model of a simple conservative s/s c = (V c ; R, W) be the orthogonal compression system with passive full behavior W, and let Σsc sc of Σsc onto the reachable subspace R of Σsc . Let (A∗+ ; U† ) be the incoming inner channel of Σsc , and define N+ := N (A∗+ ). Then Vsc = Vscc + V1 + V0 , where R V1 = Vsc ∩ N+ , W
z1
† V0 = A+ z1 z1 ∈ U . 0
(7.12)
All of Vscc , V0 , and V1 are subspaces of Vsc , and
Vscc
∩ V0 = {0},
V1 ∩ V0 = {0},
Vscc
D(W) ∩ V1 = Vsc ∩ . 0 W
(7.13)
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The interpretation of the above splitting is the following: The subspace Vscc represents the dynamics of the forward conservative controllable compression of Σsc , the subspace V0 represents the internal dynamics in U† , and the subspace V1 describes the part of the dynamics that crosses over from N+ ⊂ U† to R. c Proof x of Theorem 7.6. It is clear that each of Vsc , V0 , and V1 are subspaces of Vsc . Conversely, 1 let x0 be an arbitrary vector in Vsc . Let x01 = PR x0 and z1 = PU† x1 . By (7.3), it is possible to w0 x z 11 1 find x11 ∈ R and w01 ∈ W such that x01 ∈ Vscc ⊂ Vsc , and by Lemma 5.1, A+ z1 ∈ Vsc . Thus, 0 x x wx01 z 12 1 11 1 also the difference x02 := x0 − x01 − A+ z1 belongs to Vsc . We have now decomposed w02 w0 w01 0 x 1 x0 into w0
x1 x0 w0
x11 = x01 w01
x12 + x02 w02
z1 + A+ z1 , 0
where each term belongs to Vsc , the first term belongs to Vscc , and the last term belongs to V0 . The two top components of the middle term satisfies x12 ∈ R and x02 ∈ U† , and hence by Lemma 5.4, A∗+ x02 = PU† x12 = 0. Thus x02 ∈ N+ , and we conclude that the middle term belongs to V1 . The splitting of x0 into x0 = x01 + x02 + A+ z1 is orthogonal, since x01 ∈ R, x01 ∈ N+ , and A+ z1 ∈ R(A+ ) = U† N+ . This together with (7.3) implies (7.13). 2 8. Conservative dilations of the forward and backward conservative canonical models In the preceding section we described how to obtain the forward and backward conservative models from the simple conservative model Σsc by first performing an orthogonal compression, and then applying a unitary similarity transform. Here we shall proceed in the opposite direction and show how to construct a simple conservative model by a dilation from a forward or backward conservative model. We begin with a central lemma, which is related to Lemma 7.1. Lemma 8.1. Let V be a maximal nonnegative subspace of a Kre˘ın space K satisfying V [⊥] ⊂ V , V [⊥] = V . Let N0 be the quotient N0 = V /V [⊥] . Then N0 is a Hilbert space with the inner product inherited from K. Let Q be the quotient map V → V /V [⊥] , and define
κ K
Vext = ∈
κ ∈V . Qκ N0 Then Vext is a Lagrangian subspace of K [] −N0 . Proof. If κ ∈ V , then
κ κ = [κ, κ]K − [Qκ, Qκ]N0 = 0. , Qκ Qκ K[]−N 0
Thus, Vext is a neutral subspace of K [] −N0
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κ Next suppose that κ1 , κ2 ∈ V , and that Πκ12 is orthogonal to Vext in K [] −N0 . Then, for all κ ∈ V , κ κ1 , 0= = [κ1 , κ]K − [Qκ2 , Qκ]N0 Qκ K[]−N Qκ2 0
= [κ1 , κ]K − [κ2 , κ]K = [κ1 − κ2 , κ]K . Thus κ1 − κ2 ∈ V [⊥] ⊂ V . Since κ2 ∈ V , this implies that κ1 ∈ V , and that Qκ1 = Qκ2 , and κ κ consequently Qκ12 = Qκ11 ∈ Vext . 2 Theorem 8.2. Let Σ o = (V o ; X o , W) be an observable backward conservative passive s/s system which is not conservative. Then the quotient space No := V o /(V o )[⊥] is a Hilbert space −X o
X o . Let X o : V o → V o /(V o )[⊥] W o := X o ⊕ Z o . Z o := 2− (No ) and let Xext
with the inner product inherited from the node space Ko =
be the quotient map X o κ := κ + (V o )[⊥] , κ ∈ V o , let o by Define Vext ⎧⎡ ⎤ ⎡ 0 ⎤ ⎫ x1
⎪ ⎪ ∗z ⎪ z1 ⎪ S x x
⎨ 1 1 ⎬ ⎢ ⎥ ⎢ − 0 ⎥ o x0 ⎥ + ⎢ 0 ⎥
x0 ∈ V o , z0 ∈ Z o , z1 = d−1 (·)X o x0 , (8.1) Vext = ⎢ ⎣ ⎦
⎦ ⎣ z ⎪ ⎪ ⎪ 0 0 w0 w0 ⎪ ⎩ ⎭ w0 0
o where d−1 (·) is the scalar sequence defined by d−1 (−1) = 1, d−1 (k) = 0 for k < −1. Then Vext o = (V o ; X o , W) which is is the generating subspace of a simple conservative s/s system Σext ext ext an outgoing dilation of Σ o with outgoing inner channel (S− , Z o ).
Proof. Define
Z0o = w− ∈ 2− (N− ) w− (k) = 0 for k < −1 ,
Z1o = w− ∈ 2− (N− ) w− (−1) = 0 . o [] X o [] W can be written as the orthogonal sum of two Kre˘ın The node space Koext = −Xext ext o spaces Kext = K1 [] K0 , where
−Z1o o K0 = Z , 0
−(X o ⊕ Z0o ) o K1 = . X W
o in the same way to get V o = V [] V , where We decompose Vext 0 1 ext
V0 =
∗z S− 0 z0 0
o
z0 ∈ Z ,
⎧⎡ ⎤ ⎫ x1
⎪ ⎪ ⎪ x1 ⎪ ⎨⎢ z1 ⎥ x1 ⎬
o o ⎢ ⎥ . V1 = ⎣ x0 ⎦ x0 ∈ V , z1 = d−1 X x0
⎪ ⎪ ⎪ 0 w0 w0 ⎪ ⎩ ⎭ w0
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Here V0 is a Lagrangian subspace of K0 , and by Lemma 8.1, V1 is a Lagrangian subspace of K1 . o is a Lagrangian subspace of Ko = K [] K . This means that V o is the generating Thus, Vext 1 0 ext ext o = (V o ; X o , W). It can easily be seen that Σ o is an subspace of a conservative s/s system Σext ext ext ext o is simple follows from outgoing dilation of Σ o with outgoing inner channel (S− ; Z o ). That Σext Theorem 5.3 and the fact that S− is completely non-unitary. 2 Theorem 8.3. Let Σ c = (V c ; X c , W) be a controllable forward conservative passive s/s system. Then the quotient space Nc := (V c )[⊥] /V c is a Hilbert space with the inner product inherited let
Xc
−X c . Let X c : V c → (V c )[⊥] /V c be the quotient map X c κ −W c := X c ⊕ Z c , and define V c by Z c := 2+ (Nc ) and Xext ext
from −Kc =
:= κ +V c , κ ∈ (V c )⊥ ,
⎧⎡ ⎤ ⎡ 0 ⎤ ⎫ x1
⎪ ⎪ ⎪ ⎪ x ⎨⎢ 0 ⎥ ⎢ z1 ⎥ x1 1 ⎬ c x0 ⎥ + ⎢ 0 ⎥
x0 ∈ V c [⊥] , z1 ∈ Z c , z0 = d0 X c x0 , (8.2) Vext = ⎢ ⎣ ⎦ ⎣ S z ⎦
⎪ ⎪ ⎪ + 1 w0 w0 ⎪ ⎩ z0 ⎭ w0 0 c is the where d0 (·) is the scalar sequence defined by d0 (0) = 1, d0 (k) = 0 for k > 1. Then Vext c = (V c ; X c , W) which is an ingenerating subspace of a simple conservative s/s system Σext ext ext ∗ ; Z c ). coming dilation of Σ c with incoming inner channel (S+
Proof. The proof of this theorem is analogous to the proof of 8.2.
2
By applying Theorems 8.2 and 8.3 to the canonical backward conservative and forward conW W servative models Σobc+ and Σcfc − presented in Section 2.4 we can construct two additional non-symmetrical models of a simple conservative s/s system with a given passive behavior W. The state space in the model that we get by applying Theorem 8.2 to the observable backW W+ /(V W+ )[⊥] with the ward conservative model Σobc+ is H(W+ ) ⊕ 2− (No ), where No = Vobc obc inner product constructed in Lemma 8.1. This model is unitarily similar to the symmetrical model Σsc . If we decompose the state space D(W) of Σsc into D(W) = R† ⊕ U, then with respect to this decomposition the similarity operator Uo between these two models is block 1H(W
† −1 + diagonal, i.e., it is of the form Uo = R . Here UR† = and UR ∗ † = Π+ . To ΓW 0 UU o compute UU we first investigate the restriction ofUU |No , which maps N unitarily onto N− .
U
+
+
0
U R† 0 0 0 U R† 0 0 0 1W −R† [] R† [] W,
W /(V W )[⊥] . The operator Recall that No = Vobc obc
)
W
is a unitary map of Kobc+ :=
−H(W+ ) [] H(W+ ) [] W onto Kosc := and hence it induces a unitary + + [⊥] W W o map of Vobc /(Vobc ) onto Vsc /N (Xo ), where Xo is the operator defined in (7.6). By composing this unitary map with the operator Xo in (7.6) we get the unitary map UU |No : No → N− . The space 2− (N− ) can be mapped unitarily onto U by means of the inverse of the operator U− in (5.8), and by combining this map with the earlier described unitary similarity from No onto N− we get the full formula for UU . The state space in the model that we get by applying Theorem 8.3 to the controllable forward W W− [⊥] W 2 c c conservative model Σcfc − is H(W[⊥] /Vcfc − with the inner − ) ⊕ + (N ), where N = (Vcfc ) W product given in Lemma 8.1, with the node space Kcfc− replaced by its anti-space. This model is unitarily similar to the symmetrical model Σsc . If we decompose the state space D(W) of
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Σsc into D(W) = R ⊕ U† , then with respect to this decomposition the similarity operator Uc UR 0 between these two models is block diagonal, i.e., it is of the form Uc = 0 U † . Here UR = U ΓW −1 and UR = Π− . To compute UU† we first investigate the restriction of UU† |Nc , which 1 [⊥] H(W− ) U 0 0 R W− [⊥] W− c c maps N unitarily onto N+ . Recall that N = (Vcfc ) /Vcfc . The operator 0 UR 0 is 0
0 1W
[⊥] c a unitary map of Kcfc− := −H(W[⊥] − ) [] H(W− ) [] W onto Ksc := −R [] R [] W, and W− [⊥] W− hence it induces a unitary map of (Vcfc ) /Vcfc onto Vscc /N (Xc ), where Xc is the operator defined in (7.9). By composing this unitary map with the operator Xc in (7.9) we get the unitary map UU† |Nc : Nc → N+ . The space 2+ (N+ ) can be mapped unitarily onto U† by means of the inverse of the operator U+ in (5.10), and by combining this map with the earlier described unitary similarity from Nc onto N+ we get the full formula for UU† . W
9. Passive realizations of frequency domain behaviors The Fourier transform. Up to now we have throughout worked in the time domain, and formulated all our results in terms of sequences in k 2 (I ; W), where I is a discrete time interval. It is also possible to work in the frequency domain instead, replacing all the signal sequences w(·) by their Fourier transforms. In this section we assume that the signal space W is separable. As is well known,( for each Hilbert space X , the Fourier transform F , formally defined by n 2 (F w)(z) := w(z) ˆ = ∞ n=−∞ w(n)z is a unitary map from (X ) onto the Lebesgue space 2 2 L (X ) := L (T; X ), where T := {ξ ∈ C | |ξ | = 1}. The restrictions F± = F |2 (X ) of F to ±
2± (X ) are unitary maps from 2± (X ) onto the Hardy spaces H±2 (X ) := H 2 (D± ; X ), where
D+ := z ∈ Z |z| < 1 ,
D− := ζ ∈ Z |ζ | > 1} ∪ {∞}.
Functions in H±2 (X ) are analytic in D± , they have nontangential boundary values in the strong sense a.e. on T, and the boundary function belongs to L2 (X ). The norms in L2 (X ) and H±2 (X ) are given by the same formula 2 2 2 2 dξ 2 1 1 w(ξ w(ξ w(·) ˆ )X |dξ | = ˆ )X , ˆ L2 (X ) = (9.1) 2π 2πi ξ ξ ∈T
ξ ∈T
and L2 (X ) = H+2 (X ) ⊕ H−2 (X ). We denote the orthogonal projections of L2 (X ) onto H±2 (X ) by πˆ ± . They are explicitly given by (πˆ + w)(z) ˆ =
1 2πi
2 ξ ∈T
1 (πˆ − w)(ζ ˆ )=− 2πi
2
ξ ∈T
w(ξ ˆ ) dξ, ξ −z w(ξ ˆ ) dξ, ξ −ζ
wˆ ∈ L2 (W), z ∈ D+ ,
wˆ ∈ L2 (W), ζ ∈ D− .
(9.2)
If we denote the inverse Fourier transform of wˆ by w, then w(0) = (πˆ + w)(0), ˆ and it can be computed from the first formula in (9.2) with z = 0.
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Above we discussed the situation where X is a Hilbert space, and these considerations can be extended to the case where X is replaced by a Kre˘ın space W. We denote the images of k 2 (W) 2 (W) under the Fourier transform by K 2 (W) := K 2 (T; W) and K 2 (W) := K 2 (D ; W), and k± ± ± respectively, and define the indefinite inner products in these spaces so that the Fourier transform is a unitary operator in each case. This means that, if we fix some admissible Hilbert space 2 (W) coincide with L2 (W) and H 2 (W), inner product in W, then the spaces K 2 (W) and K± ± 2 (W) are given by the same formula respectively, and that the inner product in K 2 (W) and K± 2 1 wˆ 1 (·), wˆ 2 (·) K 2 (W ) = wˆ 1 (ξ ), wˆ 2 (ξ ) W |dξ | 2π ξ ∈T
1 = 2πi
2
ξ ∈T
dξ wˆ 1 (ξ ), wˆ 2 (ξ ) W . ξ
(9.3)
Every fundamental decomposition W = −Y [] U of the signal space gives rise to the fundamental decompositions K 2 (W) = −L2 (Y) [] L2 (U),
2 K± (W) = −H±2 (Y) [] H±2 (U).
Under the Fourier transform the three shift operators S+ , S, and S− and their adjoints are mapped into the frequency domain shift operators ∗ 1 2 ˆ − w(0) ˆ , w(·) ˆ ∈ K+ S+ wˆ (z) := w(z) (W), z −1 1 ˆ ), w(·) ˆ ∈ K 2 (W), ( S w)(ξ ˆ ) := ξ w(ξ ˆ ), S wˆ (ξ ) := w(ξ ξ ∗ 1 2 ˆ ), w(·) ˆ ∈ K− ˆ ) := ζ w(ζ ˆ ) − lim ζ w(ζ ˆ ), S− wˆ (ζ ) := w(ζ (W). ( S− w)(ζ ζ →∞ ζ ( S+ w)(z) ˆ := zw(z), ˆ
(9.4)
Frequency domain behaviors. Under the Fourier transform the class of all passive future S+ -invariant subspaces behaviors W+ on W is mapped onto the class of all maximal nonnegative 2 (W), the class of all passive past behaviors W on W is mapped onto the class of + of K+ W − 2 (W), and the class of all passive full − of K− all maximal nonnegative S− -invariant subspaces W behaviors W is mapped onto the class of all maximal nonnegative S-reducing causal subspaces of K 2 (W). The definition of causality in the frequency domain is analogous to the definition W is causal if it of causality in time domain, i.e., a S-reducing maximal nonnegative subspace W 2 (W), or equivalently, that − := πˆ − W is a maximal nonnegative subspace of K− is true that W 2 (W) is a maximal nonnegative subspace of K 2 (W). ∩ K+ + := W W + All our earlier results on passive realizations of passive (time domain) behaviors can be reformulated in frequency domain terms. In particular, the frequency domain analogues of (2.29)– (2.31) are * 2 ∩ K+ = + = W +, − = πˆ − W, (W), W S −n W W W n∈Z+
)
= − . w(·) ˆ ∈ K 2 (W) πˆ − W S −n wˆ ∈ W n∈Z+
(9.5)
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Σ The Fourier transforms of WΣ and WΣ ± give the respective frequency domain behaviors W Σ and W± . Frequency domain versions of the canonical models. The frequency domain analogue + ), where W + is a maximal nonnegative S+ of the space H(W+ ) is the Hilbert space H(W 2 (W), and the frequency domain analogue of the space H(W[⊥] ) is the invariant subspace of K+ − 2 (W). [⊥] Hilbert space H(W ), where W is a maximal nonnegative S -invariant subspace of K− − − − 2 2 (W) These spaces are defined in the same way as in Section 2.4, with k± (W) replaced by K± 2 (W) onto K 2 (W), and since ± . Since the F± is a unitary map of k± and with W± replaced by W ± the frequency domain constructions are identical to the time domain constructions, the Fourier ± ) which map H0 (W± ) isometrically onto transform induces two unitary maps H(W± ) → H(W 0 H (W± ). We shall use the same notation F± for these two unitary maps. The frequency domain analogues of the quotient maps Q and Q± are the quotient maps Q and Q± given by [⊥] + +W wˆ = wˆ + W Q − ,
+, + wˆ = πˆ + wˆ + W Q
[⊥] − wˆ = πˆ − wˆ + W Q − ,
for wˆ ∈ K 2 (W). We denote the frequency domain analogue of the past/future map ΓW by ΓW 3 , i.e., ΓW 3 := −1 F+ ΓW F− . Thus, if W is a passive full behavior on W with the corresponding passive future [⊥] and past behaviors W+ and W− , then ΓW 3 is the unique linear contraction H(W − ) → H(W+ ), which is defined by the relation ˆ + wˆ = Γ 3 Q Q W − w,
wˆ ∈ W,
[⊥] [⊥] [⊥] on the dense subspace H0 (W − ) := Q− W of H(W− ) and then extended to H(W− ) by continuity. Frequency domain versions of the two canonical backward and forward conservative models presented in Theorems 2.11 and 2.16 were given in [7, Section 10], and there it was also shown how to derive the respective de Branges–Rovnyak canonical scattering models from our backward and forward conservative canonical models. Below we shall carry out the same program for the simple conservative canonical model presented in Theorem 3.5. which is the The frequency domain analogue of the space D(W) is the Hilbert space D(W), 1/2 range space of the operator AW 3 in H+ ⊕ H− where
1H + AW 3 := ∗ ΓW 3
ΓW 3 , 1H −
+ ), + := H(W H
[⊥] − . − := H W H
(9.6)
of The frequency domain analogue of the subspace L(W) of k 2 (W) is the subspace L(W) K 2 (W) defined by
= wˆ ∈ K 2 (W) Q . wˆ ∈ D(W) L(W)
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The frequency domain analogue of the canonical model Σsc = (Vsc ; D(W), W) is the canon W), where sc ; D(W), sc = (V ical model Σ
−1 QS wˆ
. sc := wˆ V Q
wˆ ∈ L(W) w(0) ˆ sc is BΣ = The input map of Σ sc + | 3 with C∗ = is CΣsc = Π D (W) sc Σ onto H± .
ΓW 3 1H − 1 + H ΓW 3
(9.7)
− | 3 , and the output map of Σsc with B∗Σ = Π D (W) sc
+ ⊕ H − ± are the orthoprojections from H , where Π
10. The de Branges–Rovnyak conservative i/s/o model In this final section we shall use our simple conservative frequency domain canonical s/s sc to recover the classical de Branges–Rovnyak model of a simple conservative scattermodel Σ ing realization of a given Schur function, originally developed in [12,13]. Graph representations of frequency domain behaviors. Let W be a passive full behavior 2 (W) and W = π W be the corresponding on the Kre˘ın signal space W, let W+ = W ∩ k+ − − and W ± be the corresponding frequency domain passive future and past behaviors, and let W behaviors. Let W = −Y [] U be a fundamental decomposition of W, with the corresponding 2 (W) = −H 2 (Y) [] H 2 (U) fundamental decompositions K 2 (W) = −L2 (Y) [] L2 (U) and K± ± ± 2 2 of K (W) and K± (W), respectively. Since W and W± are maximal nonnegative subspaces of 2 (W), respectively, it follows from assertion 1) of Proposition 2.1 that they have K 2 (W) and K± the graph representations uˆ
D 2 W = wˆ =
uˆ ∈ L (U) , uˆ
± = wˆ ± = D± uˆ ±
uˆ ± ∈ H±2 (U) , W uˆ ±
(10.1)
∈ B(L2 (U); L2 (Y)) and D ± ∈ B(H±2 (U); H±2 (Y)) are contractions. It follows from where D 2 . As was shown in [7, Section 9], the operators (9.5) that D+ = D|H 2 (U ) and D− = πˆ − D| H − (U ) + and D ± are (Laurent) operators of the type D u)(ξ (D ˆ ) = Φ(ξ )u(ξ ˆ ),
uˆ ∈ L2 (U), ξ ∈ T,
+ uˆ + )(z) = Φ(z)uˆ + (z), uˆ + ∈ H+2 (U), z ∈ D+ , (D 2 Φ(ξ )uˆ − (ξ ) − uˆ − )(ζ ) = − 1 (D dξ, uˆ − ∈ H−2 (U), ζ ∈ D− , 2πi ξ −ζ
(10.2)
ξ ∈T
whose symbol Φ is a function in the Schur class S(U, Y), i.e., Φ is an analytic B(U; Y)-valued function in D+ satisfying Φ(z) B(U ,Y ) 1, z ∈ D+ . Such a function has a strong nontangential
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limit Φ(ζ ) for almost all ζ ∈ T. The boundary function belongs to L∞ (T), Φ(ζ ) B(U ,Y ) 1 for almost all ζ ∈ T, and Φ H ∞ (D+ ) = sup Φ(z)B(U ,Y ) = ess supΦ(ζ )B(U ,Y ) = Φ L∞ (T) . ζ ∈T
z∈D+
and W ± have the representations The orthogonal companions of W
yˆ
2 [⊥] = wˆ = y ˆ ∈ L (Y) , W
∗ yˆ D
yˆ±
[⊥] 2 W± = wˆ ± = ∗
yˆ± ∈ H± (Y) . D± yˆ±
(10.3)
is the Laurent (multiplication) operator whose symbol is Φ ∗ (ζ ), ζ ∈ T, and ∗ of D The adjoint D ∗ ∗ ∗ . More precisely, D+ and D+ are the appropriate compressions of D ∗ yˆ (ξ ) = Φ(ξ )∗ y(ξ D ˆ ), yˆ ∈ L2 (Y), ξ ∈ T, 2 ∗ Φ(ξ )∗ yˆ+ (ξ ) + yˆ+ (z) = 1 dξ, yˆ+ ∈ H+2 (Y), z ∈ D+ , D 2πi ξ −z ζ ∈T
∗ − yˆ− (ζ ) = Φ(1/ζ )∗ yˆ− (ζ ), D
yˆ− ∈ H−2 (Y), ζ ∈ D− .
(10.4)
+ ) and H (D ∗ ). The connection between the The de Branges complementary spaces H (D − Hilbert space H(Z), where Z is a maximal nonnegative subspace of a Kre˘ın space K, and the de Branges complementary space H(A), where A is a contraction between two Hilbert spaces, was explained in Section 2.1. We shall now use this connection with the following two sets of substitutions: 2 (W), U → H 2 (U), Y → H 2 (Y), A → D + , K → K+ + , and T → T+ , 1) Z → W + + [⊥] 2 2 2 ∗− , and T → T− . 2) Z → W− , K → −K− (W), U → H− (Y), Y → H− (U), A → D
+ ) → H(D + ) and T− : H(W [⊥] Here T+ : H(W − ) → H(D− ) are the unitary operators that we get by carrying out the above substitutions in (2.21), and they are explicitly given by yˆ+ yˆ+ + ), = yˆ+ − D+ uˆ + , ∈ K(W T+ Q+ uˆ + uˆ + [⊥] yˆ− yˆ− ∗ − , = uˆ − − D− yˆ− , ∈K W T− Q− uˆ − uˆ − + yˆ+ , yˆ+ ∈ H(W+ ), T+−1 yˆ+ = Q 0 − 0 , uˆ − ∈ H W[⊥] . T−−1 uˆ − = Q − uˆ −
(10.5)
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∗ ) to H (D + ). By using the unitary maps T− : H(W [⊥] The past/future map from H (D − )→ − ∗ H(D− ) and T+ : H(W+ ) → H(D+ ) we can define a version of the past/future map ΓW of a ∗− ) to H(D + ), namely passive full behavior which is a contraction from H(D + Γ 3 T−−1 = T+ F+ ΓW F−−1 T−−1 . Γ(D ∗ ,D + ) := T W −
This map is related to but not identical with the Hankel operator 2 2 2 ˆ + D| ΓD := π H (U ) : H− (U) → H+ (Y) −
induced by D. We recall the following results from [7]. and write wˆ in the form wˆ = Lemma 10.1. (See [7, Lemma 9.1].) Let wˆ ∈ W, 2 uˆ = PL2 (U ) wˆ ∈ L (U) (cf. (10.1)). Then
D uˆ uˆ
where ∈W
[⊥] ∗− D − uˆ − , = 1 H 2 (U ) − D T− πˆ − wˆ + W − −
+ ) = Γ uˆ − , T+ (πˆ + wˆ + W D
(10.6)
where uˆ − = πˆ − u ∈ H−2 (U). Lemma 10.2. (See [7, Lemma 9.2].) The operator Γ(D ∗ ,D + ) is the unique linear contraction − ∗ H(D− ) → H(D+ ), which is defined by the relation ∗− D − [−1] , Γ(D ∗ ,D + ) = ΓD 1 H 2 (U ) − D −
−
(10.7)
∗− ) = R(1 2 ∗ ∗ ∗ on the dense subspace H0 (D H− (U ) − D− D− ) of H(D− ) and then extended to H(D− ) by continuity. ∗− ) → H−2 (U) is Lemma 10.3. (See [7, Lemma 9.3].) The adjoint of the inclusion map I− : H(D ∗ ∗ 2 ∗ the operator I− = 1H 2 (U ) − D− D− : H− (U) → H(D− ). −
We shall also need the dual versions of Lemmas 10.1–10.3, which read as follows, and which may be proved in the same way as Lemmas 10.1–10.3. [⊥] , and write wˆ † in the form wˆ † = Lemma 10.4. Let wˆ † ∈ W L2 (Y) (cf. (10.3)). Then
yˆ ∗ yˆ D
∗ [⊥] = ΓD T− πˆ − wˆ † + W − yˆ+ , + = 1 2 ∗ T+ πˆ + wˆ † + W H (Y ) − D+ D+ yˆ+ , +
where yˆ+ = πˆ + yˆ ∈ H+2 (Y).
where yˆ = PL2 (Y ) wˆ † ∈
(10.8)
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Lemma 10.5. The operator Γ(∗D ∗ ,D is defined by the relation
−
+)
Γ(∗D ∗ ,D −
+)
+ ) → H(D ∗− ), which is the unique linear contraction H(D ∗ ∗ [−1] , = ΓD 1 H 2 (Y ) − D+ D+
(10.9)
+
+ ) = R(1 2 ∗ ∗ ∗ on the dense subspace H0 (D H+ (Y ) − D+ D+ ) of H(D+ ) and then extended to H(D+ ) by continuity. ∗ = + ) → H+2 (Y) is the operator I+ Lemma 10.6. The adjoint of the inclusion map I+ : H(D ∗ 2 1H 2 (Y ) − D+ D+ : H+ (Y) → H(D+ ). +
∗ ) as reproducing kernel Hilbert spaces. We begin by show + ) and H (D The spaces H (D − ing that our space H(D+ ) is equal to the standard de Branges reproducing kernel Hilbert space + , as defined in, e.g., [1, Definition 2.1.1]. H(Φ), where Φ is the symbol of D 2 Let EH 2 (Y ) (z) : H+ (Y) → Y be the point evaluation operator EH 2 (Y ) (z)yˆ+ = yˆ+ (z), z ∈ D+ . + + + ) is continuously contained in H+2 (Y), the restriction E+ (z) = E 2 (z)| is Since H(D H + (Y )
H (D + )
+ ) → Y given by the same formula E+ (z)yˆ+ = yˆ+ (z), z ∈ D+ . a bounded linear operator H(D + ) is determined Since each E+ (z) is a bounded linear operator, and since each yˆ+ ∈ H(D + ) is a reproducing uniquely by its values in D+ , it follows from [1, Theorem 1.1.2] that H(D ∗ kernel Hilbert space, with the reproducing kernel KD + (z, z∗ ) = E+ (z)E+ (z∗ ) on D+ × D+ . Let 2 ∗ ∗ ∗ + ) → H+ (Y). Then E+ (z∗ ) = I+ be the inclusion map H(D I+ EH 2 (Y ) (z∗ ) . By Lemma 10.6, + ∗ ∗ I+ = 1 2 − D+ D+ . A direct computation shows that H + (Y )
y0 , y0 ∈ Y, EH 2 (Y ) (z∗ ) y0 = z → + 1 − zz∗ ∗ + D∗+ E 2 (z∗ )∗ y0 I+ EH 2 (Y ) (z∗ )∗ y0 = 1H 2 (Y ) − D H + (Y ) + + y0 − Φ(z)Φ(z∗ )∗ y0 , y0 ∈ Y, = z → 1 − zz∗
∗
∗ KD I+ EH 2 (Y ) (z∗ )∗ + (z, z∗ ) = EH 2 (Y ) (z) +
=
+
1Y − Φ(z)Φ(z∗ )∗ , 1 − zz∗
(z, z∗ ) ∈ D+ × D+ .
This is the reproducing kernel of the standard de Branges space H(Φ) (see, e.g., [1, Defini + ) = H(Φ). tion 2.1.1]). Thus, we conclude that H(D ∗− ). This is a reproducing kernel Hilbert A similar result can be derived for our space H(D space of analytic U -valued functions defined on D− and continuously contained in H−2 (U). A computation similar to the one above shows that the reproducing kernel of this space is given by ∗ ∗ KD I− EH 2 (U ) (ζ∗ )∗ ∗ (ζ, ζ∗ ) = E− (ζ )E− (ζ∗ ) = EH 2 (U ) (ζ ) −
−
−
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∗− ) and H−2 (U), respecwhere E− (ζ ) and EH 2 (U ) (ζ ) are the point evaluation operator in H(D − ∗ 2 − ) → H− (U). A direct computation I− is the inclusion map H(D tively, at the point ζ ∈ D− , and shows that u0 ∗ EH 2 (U ) (ζ∗ ) u0 = ζ → , u0 ∈ U, − ζζ∗ − 1 1U − Φ(1/ζ )∗ Φ(1/ζ ∗ ) ∗ , I− EH 2 (U ) (ζ∗ )∗ = KD ∗ (ζ, ζ∗ ) = EH 2 (U ) (ζ ) − − − ζζ∗ − 1
(ζ, ζ∗ ) ∈ D− × D− .
Let R be the reflection operator which maps uˆ − ∈ H−2 (U) onto the function (R uˆ − )(z) = †+ = R D ∗− R −1 . Then D †+ is a causal convolution operator (1/z)uˆ − (1/z) ∈ H+2 (U), and define D † ∗ whose symbol is the Schur function Φ (z) = Φ(z) . The operator R is a unitary map from ∗− ) onto H(D †+ ), as H−2 (U) onto H+2 (U), and this implies that R|H(D ∗ ) is a unitary map of H(D − can easily be seen from the definition of these two spaces. By comparing the above reproducing kernel to the reproducing kernel of the de Branges space H(Φ † ) (see, e.g., [1, Definition 2.1.1]) ∗− ) = R −1 H(Φ † ). we conclude that H(D as a reproducing kernel Hilbert space.We let D(D) ⊂ H(D The space D (D) + ) ⊕ H(D− ) T+ 0 [⊥] . Since D(D) be the image of D(W) ⊂ H(W+ ) ⊕ H(W− ) under the unitary map 0 T− + ) ⊕ H(D − ) which is continuously contained in H+2 (Y) ⊕ is continuously contained in H(D yˆ (z) yˆ are continuous for all H−2 (U), it is true that the point evaluation operators uˆ + → + − uˆ − (ζ ) by interpreting each (z, ζ ) ∈ D+ × D− . We can apply [1, Theorem 1.1.2] to the space D(D) vector in D(D) as a function defined on Ω = D+ × D− , so that the point evaluation ED(D) is 1H 2 (Y) 0 yˆ+ yˆ+ (z) yˆ+ We claim that D(D) = + = uˆ (ζ ) , uˆ ∈ D(D). D(Φ), given by ED(D) (z, ζ ) uˆ −1 −
−
−
0
R
where R is the reflection operator defined above and D(Φ) is the standard de Branges space This space was introduced in [12] and [13] as the state space in induced by the symbol Φ of D. the de Branges–Rovnyak canonical model of a simple conservative i/s/o scattering system with scattering matrix Φ. The same space is characterized in [1] as a reproducing kernel Hilbert space. See the cited references for details, as well as [20]. Arguing as above we find that the reproducing kernel of the reproducing kernel Hilbert space of (Y × U)-valued functions defined on Ω = D+ × D− is given by D(D) ∗ K(z, ζ ; z∗ , ζ∗ ) = EH 2 (Y )⊕H 2 (U ) (z, ζ ) I ∗ EH 2 (Y )⊕H 2 (U ) (z∗ , ζ∗ ), +
−
+
−
where EH 2 (Y )⊕H 2 (U ) (z, ζ ) is the point evaluation operator in H+2 (Y) ⊕ H−2 (U) at the point + − → H+2 (Y) ⊕ H−2 (U). To compute the I is the inclusion map D(D) (z, ζ ) ∈ D+ × D− , and I 0 → ID(D) ID(D) adjoint of I we factor it into I= + , where is the inclusion map D(D) 0 I− + ) ⊕ H(D− ) → H+2 (Y ⊕ U). Thus, ∗− ), and I+ 0 is the inclusion map H(D + ) ⊕ H(D H(D − 0 I ∗ I+ 0 ∗ ∗ I∗ = ID . By Lemma 3.2, ID , where = AD (D) (D) 0 I−
1 H (D +) AD := Γ(∗D ) ∗ ,D −
+
Γ(D ∗ ,D +) − , 1 H (D [⊥] ) −
(10.10)
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and by Lemmas 10.3 and 10.6,
I+ 0
∗
0 I−
=
+D ∗+ 1 H 2 (Y ) − D
0
+
∗− D − . 1 H 2 (U ) − D
0
−
These identities together with (10.7) and (10.9) imply that I ∗ = BD :=
+D ∗+ 1 H 2 (Y ) − D +
∗ ΓD
ΓD ∗− D − . −D
1 H 2 (U ) −
(10.11)
A direct computation shows that ∗
EH 2 (Y ⊕U ) (ζ )
y0 = z → u0
y0 1−zz∗ u0 ζ ζ ∗ −1
,
y0 Y ∈ , U u0
∗ K(z, ζ ; z∗ , ζ∗ ) = EH 2 (Y )⊕H 2 (U ) (z, ζ ) I ∗ EH 2 (Y )⊕H 2 (U ) (z∗ , ζ∗ )
⎡
=⎣
+
−
1Y −Φ(z)Φ(z∗ 1−zz∗
+
)∗
Φ(1/ζ )∗ −Φ(z∗ )∗ 1−ζ z∗
−
Φ(z)−Φ(1/ζ ∗ ) zζ ∗ −1 1U −Φ(1/ζ )∗ Φ(1/ζ ∗ ) ζ ζ ∗ −1
⎤
⎦,
(z, ζ ; z∗ , ζ∗ ) ∈ (D+ × D+ ) × (D− × D− ). This differs from the reproducing kernel of the standard de Branges space D(Φ) (see, e.g., [1, Definition 2.1.1]) only by a reflection in the second component. Thus, we conclude that = D(D)
1H 2 (Y)
0
0
R −1
+
D(Φ).
→ H+2 (Y) ⊕ H−2 (U). Above we defined BD I : D(D) to be the adjoint of the inclusion map 2 2 We can also interpret BD as an operator mapping H+ (Y) ⊕ H− (U) into itself by multiplying ∗ A 3 T−1 , where BD to the left by I, after which it becomes equal to IAD I . Here AD =T W T T := + 0
[⊥] ∗ 0 +) ⊕ H W +) ⊕ H D − → H(D − . : H(W T−
(10.12)
+ ) ⊕ H(D ∗− ). Thus, The operator AD can be interpreted as a nonnegative operator on H(D 2 2 with this interpretation BD becomes a nonnegative operator on H+ (Y) ⊕ H− (U). Moreover, 1/2 = R(A1/2 ) is the unitary image under the oper D) A1/2 T−1 . Thus the range space D( = T AD 3 W D ator T| 3 of the range space D(W). D (W)
1/2
1/2
Lemma 10.7. With the above definitions, R(BD ) = R(AD ), with equality of range norms. 1/2 Thus, the Hilbert space D(D) is the range space of the operator BD in H+ (Y) ⊕ H− (U) as 1/2 ∗ well as the range space of the operator A in H(D+ ) ⊕ H(D− ). D
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Proof. Clearly R(BD ) ⊂ R(AD ) ⊂ R(AD ) is dense in R(BD ). Moreover, R(BD ) and 1/2 I ∗ ) is dense in R(AD R(AD y = ), so to prove the lemma it suffices to show that for all x = BD ∗ I y ∈ R(B ) we have A D
D
2 BD y
1/2
R(BD )
2 I ∗ y R(A1/2 ) . = AD D
But this follows from the fact that 2 BD y
1/2 R(BD )
1/2 2 = BD y)H 2 (Y )⊕H 2 (U ) y H 2 (Y )⊕H 2 (U ) = (y, BD +
+
−
−
∗ I ∗ y H 2 (Y )⊕H 2 (U ) = I ∗ y H (D = y, IAD I y, AD ∗ ) + − )⊕H− (D + − − 2 I ∗ y R(A1/2 ) . = AD 2 D
as the range of the operator A1/2 in Remark 10.8. The characterization in Lemma 10.7 of D(D) D + ) ⊕ H(D ∗− ) is equivalent to the one given in [1, Theorem 3.4.3]. The operator Λ the space H(D appearing in that theorem is given by Λ = R −1 Γ(∗D ). ∗ ,D −
+
Scattering i/s/o representations of a passive s/s system. Let Σ = (V ; X , W) be a passive s/s system with future, full, and past behaviors W+ , W, and W− , respectively. Let W = −Y [] U be a fundamental decomposition of W, and let D and D± be the operators in the graph representations of W and W± . The Kre˘ın node space K = −X [] X [] W has the fundamental decomposition K = −(X ⊕ Y) [] (X ⊕ U). By assertion 1) of Proposition 2.1, V has the graph representation V=
Axˆ0 + Bu0 xˆ0 C xˆ0 + Du0 + u0
∈ K x0 ∈ X , u0 ∈ U ,
(10.13)
A B where C is a contraction X ⊕ U → X ⊕ Y. This means that Σ has i/s/o representation D A B ; X , U, Y) where the state space X , input space U , and output space Y are Σi/s/o = ( C D Hilbert spaces. The set of trajectories of Σi/s/o on an interval I consists of triples (x(·), u(·), y(·)) satisfying x(n + 1) = Ax(n) + Bu(n), y(n) = Cx(n) + Du(n),
n ∈ I.
(10.14)
The i/s/o system Σi/s/o defined above is called a scattering representation of the passive s/s system Σ. The transfer function, which is also called the scattering matrix, of this i/s/o representation is given by Φ(z) = zC(1X − zA)−1 B + D, and it is a Schur function in D+ .
(10.15)
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A scattering representation of a s/s system is controllable, or observable, or simple, or minimal, or forward conservative, or backward conservative, or conservative if the corresponding s/s system has the corresponding property. More details about scattering representations of passive s/s systems can be found in, e.g., [3] and [7]. Scattering representations of the frequency domain versions of the canonical s/s model. sc corresponding to a We continue by developing a description of the i/s/o representation of Σ fundamental decomposition W = −Y [] U of the signal space W. This description contains the unitary operator T defined in (10.12). The operator T and its inverse are explicitly given by + πˆ + yˆ yˆ yˆ πˆ + −D , ∈ L(W), TQ = ∗− πˆ − uˆ uˆ uˆ −D πˆ − yˆ+ yˆ+ −1 yˆ+ =Q , ∈ D(W). T uˆ − uˆ − uˆ −
(10.16)
sc in order to replace the state We begin by applying the unitary similarity transform T to Σ D of the new system Σsc W) with of Σ sc by the state space D(D) = (VscD ; D(D), space D(W) generating subspace VscD
T 0 0 3 := 0 T 0 VscW . 0 0 1W
(10.17)
D , and the correThe fundamental decomposition W = −Y [] U of W is admissible for Σsc D sc Bsc sponding i/s/o representation Σi/s/o =( A Csc Dsc ; X , U, Y) is a simple conservative scattering system with scattering matrix Φ. Explicit formulas for the operators Asc , Bsc , Csc , and Dsc can be computed in the following yˆ yˆ D be the initial state of Σsc . Then T−1 + is the corresponding initial way. Let + ∈ D(D) uˆ − uˆ − sc . By (10.16), this initial state can be written in the form state of Σ
yˆ −1 yˆ+ + +W [⊥] = + +W T − , uˆ − uˆ − and hence T−1
yˆ+ w, =Q ˆ where uˆ − + uˆ − yˆ− yˆ+ D + + ∗ , wˆ = uˆ − uˆ − D− yˆ−
and uˆ + and yˆ− are free parameters in H+2 (U) and H−2 (Y), respectively. By (9.7) and (10.17), VscD
−1 T QS wˆ
= T Qwˆ
wˆ ∈ L(W) . w(0) ˆ
(10.18)
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wˆ = yˆ+ and Here TQ uˆ − w(0) ˆ =
+ uˆ − yˆ (0) Φ(0)uˆ + (0) yˆ+ (0) D (0) = + . + + uˆ − 0 0 uˆ + (0)
In order to compute TQ S −1 wˆ we apply TQ S −1 to each of the components in (10.18), and get πˆ + yˆ+ 0 S −1 −1 yˆ+ T QS = , ∗− πˆ − uˆ − πˆ − uˆ − −D S −1 S −1 ∗D ∗ + + − D S+ S+ D+ uˆ + = TQ S −1 ∗− πˆ − + uˆ + , uˆ + πˆ + S −1 − D S −1 D yˆ = 0. TQ S −1 ∗− D− yˆ−
(10.19)
The expressions above can be computed explicitly by means of (9.2), (9.4), (10.2), and (10.4), and they turn out to be
1 πˆ + 0 S −1 yˆ+ z (yˆ+ (z) − yˆ+ (0)) = (z, ζ ) → , 1 ∗− πˆ − uˆ − −D πˆ − S −1 S −1 ˆ − (ζ ) − Φ(1/ζ )∗ yˆ+ (0)) ζ (u 1 ∗D ∗ + + − D S+ S+ z (Φ(z) − Φ(0)) uˆ = (z, ζ ) → 1 uˆ + (0) . ∗− πˆ − + + πˆ + S −1 − D S −1 D (1U − Φ(1/ζ )∗ Φ(0)) ζ
Thus, we conclude that VscD has the representation VscD
=
Asc xˆ0 + Bsc u0 xˆ0 Csc xˆ0 + Dsc u0 + u0
D(D)
∈ D(D) xˆ0 ∈ D(D), u0 ∈ U , W
(10.20)
where 1 (yˆ+ (z) − yˆ+ (0)) yˆ+ z Asc (z, ζ ) = 1 , uˆ − ˆ − (ζ ) − Φ(1/ζ )∗ yˆ+ (0)) ζ (u 1 (Φ(z) − Φ(0)) u0 , (Bsc u0 )(z, ζ ) = (z, ζ ) → 1 z ∗ ζ (1U − Φ(1/ζ ) Φ(0)) yˆ Csc + = yˆ+ (0), uˆ − Dsc = Φ(0).
(10.21)
sc Bsc Comparing these coefficients A Csc Dsc to those given in, e.g., [1] we find that the scattering Asc Bsc D U, Y) of Σsc representation Σi/s/o = ( Csc Dsc , D(D), corresponding to the fundamental de 1 2 0 composition W = −Y [] U of W is unitarily similar with similarity operator H+ (Y) −1 to 0
R
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the canonical de Branges–Rovnyak model of a simple conservative i/s/o scattering system with scattering matrix Φ. Scattering representations of the two conservative dilations in Section 8. It is possible to apply the Fourier transform to convert also the two models at the end of Section 8 into frequency domain models. The scattering representations of the models that we obtain by applying the same method that have been used earlier in this section coincide with the corresponding models in [1, Section 2.4]. We leave the details to the reader. References [1] Daniel Alpay, Aad Dijksma, James Rovnyak, Henrik de Snoo, Schur Functions, Operator Colligations, and Reproducing Kernel Hilbert Spaces, Oper. Theory Adv. Appl., vol. 96, Birkhäuser-Verlag, Basel–Boston–Berlin, 1997. [2] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part I: Discrete time systems, in: The State Space Method, Generalizations and Applications, in: Oper. Theory Adv. Appl., vol. 161, BirkhäuserVerlag, Basel–Boston–Berlin, 2005, pp. 115–177. [3] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory: Passive discrete time systems, Internat. J. Robust Nonlinear Control 17 (2007) 497–548. [4] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems, in: Operator Theory, Structured Matrices, and Dilations, Tiberiu Constantinescu Memorial Volume, Theta Foundation, Bucharest, Romania, 2007, pp. 101–140, available from American Mathematical Society. [5] Damir Z. Arov, Olof J. Staffans, State/signal linear time-invariant systems theory. Part IV: Affine representations of discrete time systems, Complex Anal. Oper. Theory 1 (2007) 457–521. [6] Damir Z. Arov, Olof J. Staffans, A Kre˘ın space coordinate free version of the de Branges complementary space, J. Funct. Anal. 256 (2009) 3892–3915. [7] Damir Z. Arov, Olof J. Staffans, Two canonical passive state/signal shift realizations of passive discrete time behaviors, J. Funct. Anal. 257 (2009) 2573–2634. [8] Tomas Ya. Azizov, Iosif S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley, New York–London, 1989. [9] Joseph A. Ball, Olof J. Staffans, Conservative state-space realizations of dissipative system behaviors, Integral Equations Operator Theory 54 (2006) 151–213. [10] Vitold Belevitch, Classical Network Theory, Holden-Day, Cambridge, San Francisco–California–Amsterdam, 1968. [11] János Bognár, Indefinite Inner Product Spaces, Ergeb. Math. Grenzgebiete, vol. 78, Springer-Verlag, Berlin– Heidelberg–New York, 1974. [12] Louis de Branges, James Rovnyak, Canonical Models in Quantum Scattering Theory, in: Perturbation Theory and Its Applications in Quantum Mechanics, Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965, Wiley, New York, 1966, pp. 295–392. [13] Louis de Branges, James Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [14] Rudolf E. Kalman, Peter L. Falb, Michael A. Arbib, Topics in Mathematical System Theory, McGraw–Hill, New York, 1969. [15] Mikael Kurula, On passive and conservative state/signal systems in continuous time, Integral Equations Operator Theory 67 (2010) 377–424, 449. [16] Mikael Kurula, Olof J. Staffans, Well-posed state/signal systems in continuous time, Complex Anal. Oper. Theory 4 (2009) 319–390. [17] Nikola˘ı K. Nikolski˘ı, Vasily I. Vasyunin, A unified approach to function models, and the transcription problem, in: The Gohberg Anniversary Collection, vol. II, Calgary, AB, 1988, in: Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 405–434. [18] Nikola˘ı K. Nikolski˘ı, Vasily I. Vasyunin, Elements of spectral theory in terms of the free function model. I. Basic constructions, in: Holomorphic Spaces, Berkeley, CA, 1995, in: Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 211–302. [19] Jan Willem Polderman, Jan C. Willems, Introduction to Mathematical Systems Theory: A Behavioral Approach, Springer-Verlag, New York, 1998. [20] Donald Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Univ. Arkansas Lecture Notes in Math. Sci., vol. 10, John Wiley & Sons Inc., New York, 1994, a Wiley–Interscience Publication.
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[21] Béla Sz.-Nagy, Ciprian Foia¸s, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam– London, 1970. [22] Jan C. Willems, Dissipative dynamical systems Part I: General theory, Arch. Ration. Mech. Anal. 45 (1972) 321– 351. [23] Jan C. Willems, Dissipative dynamical systems Part II: Linear systems with quadratic supply rates, Arch. Ration. Mech. Anal. 45 (1972) 352–393. [24] J.C. Willems, H.L. Trentelman, On quadratic differential forms, SIAM J. Control Optim. 36 (5) (1998) 1703–1749 (electronic). [25] Jan C. Willems, Harry L. Trentelman, Synthesis of dissipative systems using quadratic differential forms: Part I, IEEE Trans. Autom. Control 47 (2002) 53–69. [26] Jan C. Willems, Harry L. Trentelman, Synthesis of dissipative systems using quadratic differential forms: Part II, IEEE Trans. Autom. Control 47 (2002) 70–86. [27] M. Ronald Wohlers, Lumped and Distributed Passive Networks, Academic Press, New York–London, 1969.
Journal of Functional Analysis 259 (2010) 3328–3359 www.elsevier.com/locate/jfa
On the Cauchy problem for the transport equation with random noise Jong Uhn Kim Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, United States Received 29 December 2009; accepted 26 August 2010 Available online 6 September 2010 Communicated by H. Brezis
Abstract We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in R d . When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case. © 2010 Elsevier Inc. All rights reserved. Keywords: Transport equation; Random noise; Time-periodic measure
0. Introduction In this paper, we study an initial value problem for the transport equation in R d with random noise
ut + b · ∇u + cu =
∞ j =1
fj (u)
dBj , dt
u(0, x) = u0 (x),
(t, x) ∈ (0, ∞) × R d x ∈ Rd
E-mail address:
[email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.08.017
(0.1) (0.2)
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where u is a scalar function, b = b(t, x) is a given vector field and {Bj }∞ j =1 is a sequence of mutually independent standard Brownian motions. For the deterministic case where fj ≡ 0 for all j , and b is a smooth vector field, the results for the Cauchy problem are well known by the clas1,p sical method of characteristics. For nonsmooth vector fields of the type Wloc (R d ), 1 p ∞, [5] provides comprehensive results and analysis in conjunction with the well-posedness of corresponding ordinary differential equations. The result was extended in [1] for L∞ (R d )-solutions to the case where the vector fields are locally of bounded variation. Here we study the transport equation perturbed by random noise of standard semilinear form defined in the sense of Ito. The goal in the first part of this paper is to establish the well-posedness of the Cauchy problem (0.1)–(0.2) in the function class Lp (R d ), 1 p < ∞, under the same assumptions on the vector field b and the scalar function c as in [5]. For 2 p < ∞, we can obtain results comparable to those for the deterministic equation under usual assumptions on fj ’s in (0.1). However, for 1 p < 2, we need more restriction on fj ’s. The case p = ∞ under the same assumptions on b and c as in [5] and with random noise of standard semilinear form defined in the sense of Ito is completely open. This is not surprising considering that the theory of Ito stochastic integrals in Banach spaces has been established only for M-type 2 Banach spaces, which do not include Lp (R d ), 1 p < 2, or L∞ (R d ). However, the recent work [7] established the well-posedness of the Cauchy problem in L∞ (R d ) when the vector field b is globally Hölder continuous, and random noise is a linear functional of the gradient of u defined in the sense of Stratonovich. The uniqueness result in [7] covers even the case where the uniqueness of a solution to the corresponding deterministic equation fails. The main feature of analysis in [7] is the use of stochastic flow maps. Along this line, related problems were also discussed in earlier works in [2,6,8,10,12]. However, there seems to be no work which addressed our specific issue discussed in this paper. Our approach is entirely different from that of the above references. We first consider an additive noise. By regularizing coefficients with respect to the space variables, the stochastic approximate equation can be reduced to essentially a deterministic equation with nonhomogeneous terms. By means of estimates uniform with respect to the regularizing parameter, we can obtain a solution of the original equation as a limit. For a multiplicative noise, we employ a standard iteration scheme. Obviously, the key component of the whole procedure is a set of various estimates in Lp (R d ). In particular, we need estimates involving Lp (R d )-valued stochastic integrals. Even though the general theory of stochastic integrals in M-type 2 Banach spaces is well established (see [3,13] and references therein), we could not find direct references for some of necessary energy identities. So we proceed from the bottom based on Hilbert space valued stochastic integrals in a self-contained manner. In the second part of our work, we will study the case where the given coefficients in (0.1) including the vector field are time-periodic. Under the assumption of the nondegenerate multiplicative noise, we will show the existence of a time-periodic measure which can include an invariant measure as a special case. The interesting feature is dissipation of energy due to the noise term. In other words, the corresponding deterministic equation can have solutions which can grow exponentially fast in time. This study is motivated by the well-known example of the stochastic differential equation dY dB = aY + bY , dt dt
Y (0) > 0
where a and b are constants, and B(t) is a standard Brownian motion.
(0.3)
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By means of the explicit form of a solution, it is known that: (i) If a > 12 b2 , then Y (t) → ∞ as t → ∞, almost surely. (ii) If a < 12 b2 , then Y (t) → 0 as t → ∞, almost surely. (iii) If a = 12 b2 , then Y (t) fluctuates between arbitrarily large and small values as t → ∞, almost surely. For a < 12 b2 , the random noise has stabilizing effect. One of the methods to prove the existence of an invariant measure is the energy method for dissipative equations; see [4] and [14]. However, the method discussed in [4] and [14] cannot be directly used, because the corresponding deterministic equation may not be dissipative. We will adapt the presentation in [14] by utilizing the stabilizing effect of nondegenerate multiplicative noise. The key idea is to apply Ito’s formula twice in an appropriate way to capture the stabilizing effect. It is interesting to discover the hidden mechanism of dissipation. However, if we define the noise term in the sense of Stratonovich, our procedure breaks down. This implies that the noise in the sense of Stratonovich does not provide dissipation. So there is a limitation on our result. In Section 1, we explain our notations and present some technical lemmas. In Section 2, we state our main results. The remaining sections consist of proofs. 1. Notation and preliminaries A version of the following fact was already used in [5]. Another version was proved in [11]. For a proof of the following fact, we could not find a direct reference, and thus, we provide the full details. q
Lemma 1.1. Let b be a time-dependent vector field in R d such that b ∈ L1 (0, T ; Lloc (R d )), 1 q < ∞, and ∇x · b ∈ L1 (0, T ; L∞ (R d )). Then, there is a sequence of time-dependent vector ∞ ∞ d fields {bk }∞ k=1 in C ([0, T ] × R ) and a sequence of positive numbers {Lk }k=1 such that (i) (ii) (iii) (iv)
Lk ↑ ∞, supp bk ⊂ [0, T ] × BLk , k 1, ∇x · bk L1 (0,T ;L∞ (R d )) K for all k 1, for some constant K > 0, for each 0 < L < ∞, bk → b in L1 (0, T ; Lq (BL )),
where BL = x ∈ R d |x| < L
(1.1)
Proof. Let ρ = ρ (t) and ρˆ = ρˆ (x) be the standard Friedrichs mollifiers in R and R d , respectively. Extend b so that b(t, ·) = 0 for t ∈ / [0, T ] and define ck = (b ∗ ρ 1 ) ∗ ρˆk k
We can choose a sequence {k }∞ k=1 of positive numbers decreasing to zero such that ck → b
q in L1 0, T ; Lloc R d
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Furthermore, ∇x · ck L1 (0,T ;L∞ (R d )) ∇x · b L1 (0,T ;L∞ (R d )) Let x1 hk (t, x1 , . . . , xd ) = (∇x · ck )(t, y, x2 , . . . , xd ) dy 0
The divergence-free vector field ck − (hk , 0, . . . , 0) corresponds to a (d − 1)-form αk ∈ C ∞ [0, T ] × R d ; Λd−1 R d defined by αk = (ck,1 − hk ) dx2 ∧ dx3 ∧ · · · ∧ dxd − ck,2 dx1 ∧ dx3 ∧ · · · ∧ dxd .. . (−1)d−1 ck,d dx1 ∧ · · · ∧ dxd−1 where ck = (ck,1 , . . . , ck,d ) and Λk (R d ) stands for the space of all skew-symmetric k-linear functions on R d . Since dαk = 0, it follows from the constructive proof of Poincare’s lemma in [15] that there is a (d − 2)-form βk ∈ C ∞ ([0, T ]; Λd−2 (R d )) such that αk = dβk . Define χk (x) = χ
x k
(1.2)
where χ is a function in C ∞ (R d ) such that χ(x) = 1 for |x| 1, 0 χ(x) 1, for 1 |x| 2, and χ(x) = 0 for |x| > 2. Then, we define αˆ k = d(χ2k βk ) Let γk = (γk,1 , . . . , γk,d ) be the vector field associated with αˆ k by αˆ k = γk,1 dx2 ∧ dx3 ∧ · · · ∧ dxd − γk,2 dx1 ∧ dx3 ∧ · · · ∧ dxd .. . (−1)d−1 γk,d dx1 ∧ · · · ∧ dxd−1 Also, αˆ k = (dχ2k ) ∧ βk + χ2k αk
(1.3)
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and hence, αˆ k = αk
for (t, x) ∈ [0, T ] × B2k
Define bk = γk + (χ2k hk , 0, . . . , 0) Then, bk ∈ C ∞ ([0, T ] × R d ) and supp bk ⊂ [0, T ] × B4k . It holds that ∇x · bk = ∇x · γk + χ2k
∂hk ∂χ2k + hk ∂x1 ∂x1
But ∇x · γk = 0, and ∂χ2k ⊂ [0, T ] × x 2k |x| 4k supp ∂x1 hk (t, x) 4k ∇x · ck (t) ∞ d , for |x| 4k L (R ) ∂χ2k C ∂x (x) k , for all x for some constant C > 0 1 Hence, ∂χ2k h C ∂x k 1 1 L (0,T ;L∞ (R d )) uniformly in k 1. Thus, ∇x · bk is bounded in L1 (0, T ; L∞ (R d )) uniformly in k 1. Next choose any 0 < L < ∞. For 2k > L, bk = ck − (hk , 0, . . . , 0) + (χ2k hk , 0, . . . , 0) = ck on [0, T ] × BL , and bk → b in L1 (0, T ; Lq (BL )).
2
1,q
Here we note that if b ∈ L1 (0, T ; Wloc (R d )) in addition to the above conditions, then bk → b
in L1 0, T ; W 1,q (BL ) , for each 0 < L < ∞
A stochastic basis (Ω, F , P ) is given throughout this paper. Let {Ft } be a filtration over (Ω, F , P ) such that it satisfies the usual condition, i.e., it is right continuous, and F0 contains all P -negligible sets in F . Let G = A ∈ FT ⊗ B [0, T ] A ∩ Ω × [0, t] ∈ Ft ⊗ B [0, t] for each t ∈ [0, T ] Then, (Ω × [0, T ], G, dP × dt) is a finite measure space. Let X be a Banach space. A function f : Ω × [0, T ] → X is said to be X -valued progressively measurable if f −1 (G) ∈ G for each G ∈ B(X ). B(X ) stands for the collection of all Borel subsets of X . For 1 p < ∞, 1 r < ∞,
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let Sp,r be the set of equivalent classes of X -valued progressively measurable functions f such that
p
T
r
f X dt r
Ω
dP < ∞
0
Next let Yr = Lr [0, T ], B [0, T ] , dt; X Each member of Sp,r belongs to an equivalent class in Lp (Ω, F , dP ; Yr ). By the usual convention, we regard an equivalent class in Lp (Ω, F , dP ; Yr ) as a member of Sp,r if it contains a representative from Sp,r . See [9, p. 131]. In this way, it is easy to see that Sp,r is a closed linear subspace of Lp (Ω, F , dP ; Yr ). Hence, it is also weakly closed. The following fact can be proved by the standard argument. Lemma 1.2. Let X be a reflexive Banach space. Let {fn }∞ n=1 be a sequence of X -valued progressively measurable functions such that for some 1 < p < ∞, p ess sup fn (t) X dP C, for all n 1 for some constant C > 0 Ω
0tT
Then, there is an X -valued progressively measurable function f and a subsequence {fnk }∞ k=1 such that p ess sup f (t) X dP C Ω
0tT
and fnk → f
weakly in Lp (Ω, F , dP ; Yr ), for every 1 r < ∞
Also, for each ψ ∈ L1 ([0, T ], B([0, T ]), dt; X ∗ ), and each A ∈ F , T
fnk (t), ψ(t) dt dP →
A 0
T
f (t), ψ(t) dt dP
A 0
as k → ∞, where ·,· denotes the duality pairing between X and X ∗ . We also need the following facts which can be proved in the same manner as in [5]. Lemma 1.3. Let b ∈ L1 (0, T ; Wloc (R d )) and w ∈ L∞ (0, T ; Lloc (R d )), 1 p < ∞, Then, as δ = 2 → 0, (b · ∇w) ∗ ρ − (b ∗ ρδ ) · ∇(w ∗ ρ ) → 0 in L1 0, T ; L1loc R d 1,q
p
where the convolution is taken with respect to the space variables only.
1 p
+ q1 = 1.
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1 ∞ d ∞ d Lemma √ 1.4. Let c ∈ L (0, T ; L (R )) and w ∈ L (0, T ; Lloc (R )), 1 p < ∞. Then, as ν = → 0, p
(c ∗ ρν )(w ∗ ρ ) − (cw) ∗ ρ → 0 in L1 0, T ; L1loc R d For 2 < p < ∞, we will show that a stochastic integral of an Lp (R d )-valued progressively measurable random variable can be defined as an Lp (R d )-valued continuous martingale based on the Ito stochastic integral in the Hilbert space setting. We assume gj is Lp R d -valued progressively measurable such that
∞
p
T
E
j =1 0
(1.4)
2
gj 2Lp (R d ) dt
<∞
(1.5)
Lemma 1.5. Under the assumptions (1.4)–(1.5), there is a unique Lp (R d )-valued continuous martingale M(t), 0 t T , such that for each ψ ∈ Lq (R d ),
M(t), ψ =
∞ t
gj (s), ψ dBj (s)
(1.6)
j =1 0
holds for all t ∈ [0, T ], for almost all ω ∈ Ω, where ·,· denotes the duality pairing between Lp (R d ) and Lq (R d ). Furthermore, it holds that E
∞ T
p 2 p 2 gj (s) Lp (R d ) ds sup M(t) Lp (R d ) CE
t∈[0,T ]
(1.7)
j =1 0
for some constant C > 0 independent of {gj }∞ j =1 . p
Proof. We first note that the mapping F : w → w Lp (R d ) is of class C 2 from H m (R d ) into R, m > d2 . Its first derivative DF is given by DF (w) (ψ) =
p|w|p−1 sgn(w)ψ dx,
∀ψ ∈ H m R d
Rd
and the second derivative D 2 F is given by 2 D F (w) (ψ1 , ψ2 ) =
Rd
p(p − 1)|w|p−2 ψ1 ψ2 dx,
∀ψ1 , ψ2 ∈ H m R d .
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Let χk be a function in C0∞ (R d ) defined by (1.2), and define Mk, (t) =
∞
t
gj,k, dBj
j =1 0
where gj,k, = (gj χk ) ∗ ρ . Then, Mk, (t) is an H m (R d )-valued continuous martingale. It follows from Ito’s formula that Mk, (t) p p
L (R d )
=
∞ t
p|Mk, |p−1 sgn(Mk, )gj,k, dx dBj
j =1 0 d R ∞
1 + 2
t p(p − 1)|Mk, |p−2 (gj,k, )2 dx ds
(1.8)
j =1 0 d R
By the Burkholder–Davis–Gundy inequality and Hölder’s inequality, we can derive from (1.8) E
∞ T
p 2 p sup Mk, (t) Lp (R d ) CE gj,k, 2Lp (R d ) dt
0tT
(1.9)
j =1 0
for some positive constant C independent of k and . Now for each k = 1, 2, . . . , there is k > 0 such that as k → ∞, E
∞
p
T
j =1 0
2
gj,k,k − gj 2Lp (R d ) ds
→0
(1.10)
The inequality (1.9) is valid for Mk,k − Ml,l on the left and gj,k,k − gj,l,l on the right. p p d Thus, {Mk,k }∞ k=1 is a Cauchy sequence in L (Ω; C([0, T ]; L (R ))). Let lim Mk,k = M
k→∞
in Lp Ω; C [0, T ]; Lp R d
(1.11)
For each k, (1.6) is valid with Mk,k on the left and gj,k,k on the right; see Lemma 2.4.1 of [14]. Hence, M(t) satisfies (1.6) with gj . By (1.10) and (1.11), (1.7) holds. Next suppose there is another Lp (R d )-valued continuous martingale Mˆ which satisfies (1.6). Let {ψk } be a countable ˜ = 0, and for each dense subset of Lq (R d ). Then, there is some Ω˜ ⊂ Ω such that P (Ω \ Ω) ˜ ω∈Ω M, Mˆ ∈ C [0, T ]; Lp R d ˆ ψk , for all t ∈ [0, T ] and all k = 1, 2, . . . M(t), ψk = M(t), ˜ Then, M = Mˆ for all ω ∈ Ω.
2
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Lemma 1.6. Assume (1.4)–(1.5) with 1 p < 2. Let m > d2 . There is a unique H −m (R d )-valued continuous martingale M(t), 0 t T , such that for each ψ ∈ H m (R d ),
M(t), ψ =
∞ t
(1.12)
gj ψ dx dBj
j =1 0 d R
for all t ∈ [0, T ], for almost all ω, where ·,· is the duality pairing between H m (R d ) and H −m (R d ). Proof. Since Lp (R d ) is continuously embedded into H −m (R d ), M(t) can be defined as an H −m (R d )-valued continuous martingale. (1.12) follows from Lemma 2.4.1 of [14]. 2 2. Statement of the main results Let (Ω, F , P ) be a given stochastic basis with a filtration {Ft } which satisfies the usual condition. Let 1 p < ∞, and p1 + q1 = 1. We assume that T > 0 is given, and 1,q b ∈ L1 0, T ; Wloc R d ∇ · b, c ∈ L1 0, T ; L∞ R d ∈ L1 0, T ; L∞ R d + L1 0, T ; L1 R d
(2.1) (2.2)
b 1 + |x| gj is Lp R d -valued progressively measurable such that ∞ T
p 2 E gj 2Lp (R d ) dt <∞
(2.4)
u0 is F0 -measurable and u0 ∈ Lp Ω; Lp R d
(2.5)
(2.3)
j =1 0
We first define a solution of ut + b · ∇u + cu =
∞
gj
j =1
dBj , dt
u(0, x) = u0 (x),
(t, x) ∈ (0, T ) × R d x ∈ Rd
(2.6) (2.7)
Definition 2.1. Let 2 p < ∞. An Lp (R d )-valued progressively measurable stochastic process u is a solution of (2.6)–(2.7) if u ∈ C([0, T ]; Lp (R d )) for almost all ω, and
u(t)φ dx = Rd
t u0 φ dx +
u(s)b · ∇φ dx ds 0 Rd
Rd
t
(∇ · b − c)uφ dx ds +
+ 0 Rd
Rd
M(t)φ dx
(2.8)
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holds for all φ ∈ C0∞ (R d ), for all t ∈ [0, T ], for almost all ω. Here M(t) is defined in Lemma 1.5. Theorem 2.2. Let 2 p < ∞, and T > 0 be given. Under the above assumptions (2.1)–(2.5), there exists a pathwise unique solution of (2.6)–(2.7) in Lp (Ω; C([0, T ]; Lp (R d ))). Furthermore, it holds that E
∞ T
p 2 p p 2 gj Lp (R d ) dt sup u(t) Lp (R d ) CE u0 Lp (R d ) + CE
t∈[0,T ]
(2.9)
j =1 0
for some constants C independent of u0 and gj ’s. Next we assume the following condition on fj ’s in Eq. (0.1). For each w ∈ Lp (R d ), fj (w) = gj + hj (w)
(2.10)
where gj ’s satisfy (2.4), and hj is a time-independent continuous map from Lp (R d ) into itself such that hj (0) = 0
(2.11)
and hj (w1 ) − hj (w2 ) p d λj w1 − w2 p d , L (R ) L (R )
∀w1 , w2 ∈ Lp R d
(2.12)
for some constant λj . We suppose that ∞
λ2j < ∞
(2.13)
j =1
Theorem 2.3. Let 2 p < ∞. Under the assumptions (2.1)–(2.5) and (2.10)–(2.13) there is a pathwise unique solution of (0.1)–(0.2) in Lp (Ω; C([0, T ]; Lp (R d ))). For 1 p < 2, we assume that fj (w) = aj w,
∀w ∈ Lp R d
(2.14)
where aj = aj (t, ω) is bounded and progressively measurable such that ∞
aj 2L∞ ((0,T )×Ω) < ∞
(2.15)
j =1
We also need to restrict the condition (2.3) to 2 b ∈ L1 0, T ; L∞ R d ∩ L1 0, T ; L 2−p R d 1 + |x|
(2.16)
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For 1 p < 2, a solution of (0.1)–(0.2) is defined as in Definition 2.1 with the last term in (2.8) is replaced by M(t), φ where M(t) is defined in Lemma 1.6 with gj = fj (u), and ·,· is the duality pairing between H m (R d ) and H −m (R d ) for m > d2 . Theorem 2.4. Let 1 p < 2. Under the assumption (2.1), (2.2), (2.5) and (2.14)–(2.16), there is a pathwise unique solution to (0.1)–(0.2) in Lp (Ω; C([0, T ]; Lp (R d ))). For the existence of a time-periodic measure, we consider the following equation ∞
ut + b · ∇u + cu = h
dBj dB0 + , fj (u) dt dt
(t, x) ∈ (0, ∞) × R d
(2.17)
j =1
where we assume that {Bj }∞ j =0 is a sequence of mutually independent standard Brownian motions, and 1,q b ∈ L1loc 0, ∞; Wloc R d h ∈ L∞ 0, ∞; Lp R d ∇ · b, c ∈ L1loc 0, ∞; L∞ R d , b(t, ·) = b(t + T , ·), b 1 + |x|
c(t, ·) = c(t + T , ·), h(t, ·) = h(t + T , ·), ∈ L1loc 0, ∞; L∞ R d + L1loc 0, ∞; L1 R d ∇ · b − pc κ,
∀t 0
(2.18) (2.19) (2.20) (2.21)
∀(t, x)
(2.22)
for some positive constant κ. We also need the following assumption on fj ’s. Each fj satisfies (2.10)–(2.13) with gj ≡ 0, and for all w1 , w2 ∈ Lp (R d ), ∞ j =1
|w1 − w2 |p−1 sgn(w1 − w2 ) fj (w1 ) − fj (w2 ) dx
2 η1
Rd
2 |w1 − w2 |p dx
Rd
(2.23) ∞
2 |w1 − w2 |p−2 fj (w1 ) − fj (w2 ) dx η2
j =1 d R
|w1 − w2 |p dx
(2.24)
Rd
where η1 and η2 are positive constants which satisfy
2 1 + 2κ η1 > η2 1 − p p In the special case, fj (w) = aj w for some constant aj , we may take η1 = η2 =
∞ j =1
aj2 < ∞
(2.25)
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Let 2 p < ∞. Under the conditions (2.18)–(2.25), a pathwise unique solution of (2.17) exists on the interval [s, η] for the given initial condition u(s) = y ∈ Lp (R d ), for any 0 s < η < ∞, which follows from Theorem 2.3. We denote the solution by Y (·; s, y). Following [4] and [14], we define for 0 s t < ∞, (Ps,t ϕ)(y) = E ϕ Y (t; s, y) for each bounded continuous function ϕ on Lp (R d ). Definition 2.5. A probability measure μ over (Lp (R d ), B(Lp (R d ))) is called a T -periodic measure to (2.17) if
(P0,t ϕ)(y) dμ(y) =
Lp (R d )
(P0,t+T ϕ)(y) dμ(y)
(2.26)
Lp (R d )
for all t 0 and all bounded continuous functions ϕ on Lp (R d ). By convention, (2.26) is written as ∗ ∗ P0,t μ = P0,t+T μ
Theorem 2.6. Let 2 p < ∞. Under the above assumptions (2.10)–(2.13) with gj ≡ 0, and (2.18)–(2.25), there is a unique T -periodic measure over Lp (R d ) to (2.17). If b, c, and h are time-independent, there is a unique invariant measure. 3. Proof of Theorems 2.2 and 2.3 Throughout this section, 2 p < ∞ and T > 0 are fixed. The proof consists of several steps. Lemma 3.1. Let m > d/2. Suppose that b ∈ C([0, T ]; H m+1 (R d )), c ∈ C([0, T ]; H m (R d )), M ∈ C([0, T ]; H m+1 (R d )), and u0 ∈ H m (R d ). Then, there is a unique solution u ∈ C([0, T ]; H m (R d )) of ∂M ∂u + (b · ∇)u + cu = ∂t ∂t u(0) = u0 Furthermore, the mapping (u0 , M) → u is a continuous linear map from H m (R d ) × C([0, t]; H m+1 (R d )) into C([0, t]; H m (R d )) for each t ∈ [0, T ].
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Proof. Set v = u − M. Consider the initial value problem ∂v + (b · ∇)v + cv = −(b · ∇)M − cM ∂t v(0) = u0 − M(0) Then, the result follows easily from the classical theory of linear hyperbolic equations.
2
By the continuous dependence on u0 and M, we have the following fact. Lemma 3.2. Suppose that u0 and M in the above lemma are random such that M : Ω → C([0, t]; H m+1 (R d )) is Ft -measurable for each t ∈ [0, T ], and u0 : Ω → H m (R d ) is F0 measurable. Then, u : Ω → C([0, t]; H m (R d )) is also Ft -measurable for each t ∈ [0, T ]. By Lemma 1.1 and the remark right below Lemma 1.1, we can find sequences {bk }∞ k=1 in m+1 d d C([0, T ]; H (R )) such that supp bk is a compact subset of [0, T ] × R , and 1,q in L1 0, T ; Wloc R d ∇ · bk is bounded in L1 0, T ; L∞ R d uniformly in k bk → b
(3.1) (3.2)
m d We can also find a sequence {ck }∞ k=1 in C([0, T ]; H (R )) such that
ck → c
q in L1 0, T ; Lloc R d
(3.3)
and ck is bounded in L1 0, T ; L∞ R d uniformly in k
(3.4)
Under the assumption (2.4), we can define an Lp (R d )-valued continuous martingale M(t) by Lemma 1.5. Let us write Mk (t) = Mk,k (t) where Mk,k was defined in the proof of Lemma 1.5. p m+1 (R d ))) such that Then, {Mk }∞ k=1 is a sequence in L (Ω; C([0, T ]; H Mk → M
in Lp Ω; C [0, T ]; Lp R d
(3.5)
p m d Next we choose a sequence {u0,k }∞ k=1 in L (Ω; H (R )) such that each u0,k is F0 -measurable, and
u0,k → u0
in Lp Ω; Lp R d
(3.6)
By virtue of Lemmas 3.1–3.2, we obtain the solution uk of ∂Mk ∂uk + (bk · ∇)uk + ck uk = ∂t ∂t uk (0) = u0,k By the regularity of uk , bk , ck , and Mk , we can proceed as in the proof of Lemma 1.5 to obtain
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
uk (t)p dx =
Rd
t |u0,k | dx + ∞ t
pck |uk |p dx ds
p
0 Rd
Rd
+
t (∇ · bk )|uk | dx ds −
p
3341
0 Rd
p|uk |p−1 sgn(uk )gj,k,k dx dBj (s)
j =1 0 d R ∞
1 + 2
t p(p − 1)|uk |p−2 (gj,k,k )2 dx ds
(3.7)
j =1 0 d R
for all t ∈ [0, T ], for almost all ω. By means of (1.10), (3.2) and (3.4), we use the Burkholder– Davis–Gundy inequality to derive E
p sup uk (t) Lp (R d ) C
(3.8)
t∈[0,T ]
for all k 1, for some constant C > 0. Thus, by Lemma 1.2, there is a progressively measurable u ∈ Lp Ω; L∞ 0, T ; Lp R d
(3.9)
such that for some subsequence still denoted by {uk }, T
T uk ψ dx dt dP → A 0 Rd
uψ dx dt dP
(3.10)
A 0 Rd
for all A ∈ F , and all ψ ∈ L1 (0, T ; Lq (R d )). It is easy to see that for each φ ∈ C0∞ ([0, T ) × R d ),
T uφt dx dt + 0
Rd
T u0 φ(0) dx +
Rd
T
T ub · ∇φ dx dt +
0
Rd
u(∇ · b − c)φ dx dt 0
Rd
=
Mφt dx dt 0 Rd
holds for almost all ω. Meanwhile, there is a countable subset S of C0∞ ([0, T ) × R d ) such that each φ ∈ C0∞ ([0, T ) × R d ) can be approximated by a sequence in S with respect to the norm of C 1 ([0, T ]; W 1,q (R d ) ∩ W 1,∞ (R d )). Thus, it follows that ∂M ∂u + (b · ∇)u + cu = ∂t ∂t holds in D ((0, T ) × R d ), for almost all ω, and
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∂ ∂t
(u − M)ψ dx = Rd
u(b · ∇ψ) dx +
Rd
u(∇ · b − c)ψ dx
(3.11)
Rd
holds in D ((0, T )), for all ψ ∈ C0∞ (R d ), for almost all ω. The right-hand side belongs to L1 (0, T ) for all ψ ∈ C0∞ (R d ), for almost all ω. Since M ∈ C([0, T ]; Lp (R d )) for almost all ω, it follows from (3.11) that u ∈ C [0, T ]; D R d ,
for almost all ω
which, combined with (3.9), implies that u(t) is Lp R d -weakly continuous in t ∈ [0, T ], for almost all ω
(3.12)
and
u(t)ψ dx =
Rd
t u0 ψ dx +
u(b · ∇ψ) dx ds 0 Rd
Rd
t +
u(∇ · b − c)ψ dx ds + 0
Rd
M(t)ψ dx
(3.13)
Rd
for all t ∈ [0, T ] and all ψ ∈ C0∞ (R d ), for almost all ω. Obviously, it follows that u(0) = u0 Next we prove continuity in t with respect to the norm of Lp (R d ). For each 0 < K < ∞ and > 0, the mapping w → w ∗ ρ is continuous from Lp (R d ) into H m (BK ), m > d2 , with respect to the weak topology of Lp (R d ). Thus, by (3.12), u ∗ ρ ∈ C([0, T ]; H m (BK )), for almost all ω, and it follows from (3.13) that d(u ∗ ρ ) = −(bδ · ∇)(u ∗ ρ ) dt − cν (u ∗ ρ ) dt + f dt + h dt + dM
(3.14)
holds in H m (BL ), m > d2 , for all = 1k , k = 1, 2, . . . , and L = 1, 2, . . . , for almost all ω, where δ = 2, bδ = b ∗ ρ δ ,
ν=
√
cν = c ∗ ρν
f = (bδ · ∇)(u ∗ ρ ) − (b · ∇u) ∗ ρ h = cν (u ∗ ρ ) − (cu) ∗ ρ
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3343
and M (t) =
∞
t
gj ∗ ρ dBj
j =1 0
Here the convolution is taken with respect to the space variables only. We define β ∈ C 2 (R) such that β(y) =
|y|p , 2p ,
|y| 1, |y| > 2,
β(y) 0
Let βL (y) = |L|p β(y/L). We also use χK (·) defined by (1.2). Since the mapping w → χK βL (w) dx Rd
is C 2 -mapping from H m (B2K ) into R, we can apply Ito’s formula to (3.14) to obtain
χK βL u(t) ∗ ρ dx
Rd
=
t χK βL (u0 ∗ ρ ) dx −
χK cν βL (u ∗ ρ )(u ∗ ρ ) dx ds
0 Rd
Rd
t
+
t χK (∇ · bδ )βL (u ∗ ρ ) dx ds +
0 Rd
t +
0 Rd
χK (f + h )βL (u ∗ ρ ) dx ds +
1 + 2
∞ t
χK βL (u ∗ ρ )(gj ∗ ρ ) dx dBj (s)
j =1 0 d R
0 Rd ∞
(bδ · ∇χK )βL (u ∗ ρ ) dx ds
t
χK βL (u ∗ ρ )(gj ∗ ρ )2 dx ds
(3.15)
j =1 0 d R
There is Ω1 ⊂ Ω, such that P (Ω \ Ω1 ) = 0, and for each ω ∈ Ω1 , (i) (3.15) holds for all t ∈ [0, T ], all = 1/k, k = 1, 2, . . . , all L = 1, 2, . . . , all K = 1, 2, . . . , and (ii) u(ω) ∈ L∞ (0, T ; Lp (R d )), and u(t, ω) ∈ Lp (R d ), for all t ∈ [0, T ], which follows from (3.12). Fix K and L, and pass = 1/k → 0 using Lemmas 1.3 and 1.4. There is Ω2 ⊂ Ω1 such that P (Ω \ Ω2 ) = 0, and for each ω ∈ Ω2 ,
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χK βL u(t) dx =
Rd
t χK βL (u0 ) dx −
χK cβL (u)u dx ds
0 Rd
Rd
t +
χK (∇ · b)βL (u) dx ds + 0
+
t (b · ∇χK )βL (u) dx ds 0
Rd
∞ t
χK βL (u)gj
j =1 0 d R
Rd ∞
1 dx dBj (s) + 2
t
χK βL (u)(gj )2 dx ds
j =1 0 d R
holds for all K = 1, 2, . . . , L = 1, 2, . . . , and all t ∈ [0, T ]. Here we used the fact that as → 0, 2 ∞ T χK β (u ∗ ρ )(gj ∗ ρ ) dx − χK β (u)gj dx dt → 0 E L L
j =1 0 Rd
Rd
Next fix L and pass K → ∞. By (2.3), we can proceed as in [5] to find t (b · ∇χK )βL (u) dx ds = 0
lim
K→∞ 0 Rd
for all t ∈ [0, T ], for almost all ω. There is Ω3 ⊂ Ω2 such that P (Ω \ Ω3 ) = 0, and for each ω ∈ Ω3 ,
βL u(t) dx =
Rd
t βL (u0 ) dx +
t (∇ · b)βL (u) dx ds −
0 Rd
Rd
+
∞ t
βL (u)gj
j =1 0 d R
cβL (u)u dx ds
0 Rd ∞
1 dx dBj (s) + 2
t
βL (u)(gj )2 dx ds
j =1 0 d R
holds for all L = 1, 2, . . . , and all t ∈ [0, T ]. Next pass L → ∞. Rd
u(t)p dx =
t |u0 | dx +
+
p
0 Rd
Rd ∞ t
t (∇ · b)|u| dx ds −
p
pc|u|p dx ds 0 Rd
p|u|p−1 sgn(u)gj dx dBj (s)
j =1 0 d R ∞
1 + 2
t
j =1 0 d R
p(p − 1)|u|p−2 (gj )2 dx ds
(3.16)
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3345
for all t ∈ [0, T ], for almost all ω. Hence, u(t) Lp (R d ) is continuous in t ∈ [0, T ], for almost all ω. Since Lp (R d ) is uniformly convex, u ∈ C([0, T ]; Lp (R d )) holds for almost all ω. In the meantime, by the Burkholder–Davis–Gundy inequality and Hölder’s inequality, we arrive at (2.9). For pathwise uniqueness of a solution, let u1 and u2 be two solutions of (2.6)–(2.7). By the same procedure as for (2.9),
p sup u1 (t) − u2 (t) Lp (R d ) 0
E
0tT
which yields u1 = u2 for almost all ω. Now the proof of Theorem 2.2 is complete. We now proceed to prove Theorem 2.3. Let u(0) ≡ u0 , and u(n) be the solution of (n) ut
+ b · ∇u
(n)
∞ dBj , gj + hj u(n−1) = dt
+ cu
(n)
(t, x) ∈ (0, ∞) × R d
(3.17)
j =1
u(n) (0, x) = u0 (x),
x ∈ Rd
(3.18)
for n = 1, 2, . . . . It follows from (2.9) and (2.12) that p sup u(n+1) (s) − u(n) (s) Lp (R d )
E
0st
CE
∞
t
2 λ2 u(n) (s) − u(n−1) (s) p j
L
j =1 0
Ct
p−2 2
t E
(n) u (s) − u(n−1) (s) p p
p 2
(R d )
L (R d )
ds
n = 1, 2, . . . ,
ds ,
(3.19)
0
where C denotes positive constants independent of t and n. Now suppose that E
p sup u(n) (s) − u(n−1) (s) Lp (R d ) C1 t α ,
∀t ∈ [0, T ]
0st
Then, (3.19) implies E
p t α+ 2 , sup u(n+1) (s) − u(n) (s) Lp (R d ) CC1 α+1 0st p
∀t ∈ [0, T ]
(3.20)
In the meantime, it follows from (2.9) that E
p sup u(1) (s) − u(0) (s) Lp (R d )
0st
2p E
p p sup u(1) (s) Lp (R d ) + 2p E u0 Lp (R d ) C˜
0st
(3.21)
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
for all t ∈ [0, T ], for some constant C˜ depending on u0 and T . By induction using (3.20)–(3.21), we derive E
p ˜ n sup u(n+1) (s) − u(n) (s) Lp (R d ) CC
0st
t
pn 2
( p2 + 1)(p + 1) · · · ( p(n−1) + 1) 2
˜ C independent of n and t. Hence, we have for all t ∈ [0, T ], and all n 1, for some constants C, ∞ ∞ n pn p1 1 p C T 2 p E sup u(n+1) (t) − u(n) (t) Lp (R d ) C˜ <∞ n! 0tT n=1
n=1
So {u(n) } is a Cauchy sequence in Lp (Ω; C([0, T ]; Lp (R d ))), and the limit is a solution of (0.1)–(0.2). Next suppose that u1 and u2 are two solutions of (0.1)–(0.2). As above, it follows from (2.9) and (2.12) that E
p sup u1 (s) − u2 (s) Lp (R d ) CE
0st
t
u1 (s) − u2 (s) p p
L (R d )
ds
0
for all t ∈ [0, T ], which yields u1 = u2 for almost all ω. This completes the proof of Theorem 2.3. 4. Proof of Theorem 2.4 Let 1 p < 2 and T > 0 be fixed. The idea is to obtain a solution as the limit of a sequence of L2 (R d )-valued solutions. Let {u0,n } be a sequence in Lp (Ω; Lp (R d )) ∩ L2 (Ω; L2 (R d )) such that each u0,n is F0 -measurable, and u0,n → u0 in Lp Ω; Lp R d (4.1) We can construct such a sequence as follows. Let 1, for |y| n ξn (y) = 0, for |y| > n Then, ξn ( u0 Lp (R d ) )u0 ∈ L2 (Ω; Lp (R d )), and ξn u0 Lp (R d ) u0 → u0
in Lp Ω; Lp R d
For each fixed n 1, ξn u0 Lp (R d ) u0 ∗ ρ1/k ∈ L2 Ω; L2 R d and, as k → ∞, ξn u0 Lp (R d ) (u0 ∗ ρ1/k ) = ξn u0 Lp (R d ) u0 ∗ ρ1/k → ξn u0 Lp (R d ) u0 in L2 (Ω; Lp (R d )). Thus, for each n, there is n > 0 such that
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3347
def u0,n = ξn u0 Lp (R d ) (u0 ∗ ρn ) ∈ L2 Ω; L2 R d ∩ Lp Ω; Lp R d and u0,n → u0
in Lp Ω; Lp R d
For 0 < η < 1, choose a nonnegative function γη ∈ C 2 (R) such that (η + |y|2 )p/2 , |y| 1, γ (y) C, ∀|y| 1 γη (y) = p η |y| 2, 2 , where C is a constant independent of η. For 1 L < ∞, set
y γL,η (y) = Lp γη L
(4.2)
(4.3)
Then, γ (y)y C|y|p L,η
(4.4)
γ (y)y 2 C |y|p + Lp ηp/2 L,η
(4.5)
and
for all 0 < η < 1, 1 L < ∞, and all y ∈ R, for some constant C > 0. According to Theorem 2.3 with p = 2, there is a solution un of (0.1)–(0.2) with un (0) = u0,n . Here we note that the conditions on b and c for 1 p < 2 also satisfy the corresponding conditions for p = 2. Choose any m > n 1 and set w = un − um Then, (3.14) is valid with u replaced by w and with M defined by M (t) =
∞
t
aj (w ∗ ρ ) dBj
j =1 0
By applying Ito’s formula to (3.14), we obtain as in (3.15) χK γL,η w(t) ∗ ρ dx Rd
=
χK γL,η w(0) ∗ ρ dx −
t
χK cν γL,η (w ∗ ρ )(w ∗ ρ ) dx ds
0 Rd
Rd
t +
t χK (∇ · bδ )γL,η (w ∗ ρ ) dx ds +
0 Rd
(bδ · ∇χK )γL,η (w ∗ ρ ) dx ds 0 Rd
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
t
χK (f + h )γL,η (w ∗ ρ ) dx ds
+ 0 Rd
+
∞ t
χK γL,η (w ∗ ρ )aj (w ∗ ρ ) dx dBj (s)
j =1 0 d R ∞
1 + 2
t
χK γL,η (w ∗ ρ )aj2 (w ∗ ρ )2 dx ds
(4.6)
j =1 0 d R
for all t ∈ [0, T ], = 1k , k = 1, 2, . . . , for almost all ω. By virtue of (2.15) and (4.2)–(4.3), we can pass = k1 → 0 to find χK γL,η w(t) dx Rd
=
χK γL,η w(0) dx −
t
χK cγL,η (w)w dx ds
0 Rd
Rd
t
+
t χK (∇ · b)γL,η (w) dx ds +
0 Rd
+
(b · ∇χK )γL,η (w) dx ds 0 Rd
∞ t
χK γL,η (w)aj w dx dBj (s) +
j =1 0 d R
∞
1 2
t
χK γL,η (w)aj2 w 2 dx ds
(4.7)
j =1 0 d R
for all t ∈ [0, T ], L = 1, 2, . . . , η = 1k , k = 1, 2, . . . , for almost all ω. Next by (4.4)–(4.5), we take η = 1/L4 , and pass L → ∞ to obtain p χK w(t) dx Rd
=
p χK w(0) dx −
t cχK p|w|p dx ds 0 Rd
Rd
t
t χK (∇ · b)|w| dx ds +
+ 0 Rd
+
(b · ∇χK )|w|p dx ds
p
∞ t
0 Rd
j =1 0 d R
∞
1 pχK |w| aj dx dBj (s) + 2
t
p
for all t ∈ [0, T ], K = 1, 2, . . . , for almost all ω.
j =1 0 d R
p(p − 1)χK |w|p aj2 dx ds
(4.8)
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3349
By the Burkholder–Davis–Gundy inequality, we can estimate the stochastic integral on the right-hand side. ∞ s
p √ p E sup pχK |w| aj dx dBj C tE sup χK w(s) dx 0st 0st
j =1 0 d R
(4.9)
Rd
for some constant C independent of K and t. We also see that E
∞ t
p(p − 1)χK |w|p aj2 dx ds
CtE
sup
p χK w(s) dx
(4.10)
0st Rd
j =1 0 d R
for some constant C independent of K and t. The second and the third integrals on the right-hand side can be estimated in the same way. To estimate the fourth integral, we use the fact w ∈ L2 Ω; C [0, T ]; L2 R d Let Ξ (t) =
b 2 ∈ L1 (0, T ) 1 + |x| L 2−p d (R )
Then, def
T
CK = E
|b · ∇χK ||w| dx ds p
0 Rd
T CE 0
Ξ (s)
w(s)2 dx
p 2
ds → 0 as K → ∞
K|x|2K
Let us write ZK (t) = E
sup
p χK w(s) dx
0st Rd
Θ(t) = ∇ · b(t) L∞ (R d ) + c(t) L∞ (R d ) ∈ L1 (0, T ) Combining (4.8)–(4.11), we find that t ZK (t) ZK (0) + C
√ Θ(s)ZK (s) ds + C( t + t)ZK (t) + CK
0
for some constant C > 0 independent of K and t.
(4.11)
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Hence, there is some positive integer N such that for all 0 < t T /N ,
p p χK w(s) dx CE χK w(0) dx + CCK
sup
E
0st Rd
(4.12)
Rd
where we can choose a positive constant C independent of K, w and t, but depending on Θ L1 (0,T ) . We now partition the interval [0, T ] 0 = t0 < t1 < · · · < tN = T ,
tj − tj −1 = T /N
By considering the interval [tj −1 , tj ] instead of [0, t] in (4.12), it holds that E
p p χK w(s) dx CE χK w(tj −1 ) dx + CCK ,
sup
tj −1 stj
Rd
j = 1, 2, . . . , N
Rd
for some constant C independent of w and K. Thus, E
N p χK w(t) dx E
sup 0tT
j =1
Rd
CE
sup
tj −1 ttj
p χK w(t) dx
Rd
w(0)p dx + CCK
(4.13)
Rd
for some constant C independent of K and w. Next we see that
p w(s)p dx = lim E sup E sup χK w(s) dx K→∞
0st Rd
It follows form (4.11), (4.13) and (4.14) that
p p w(t) dx CE w(0) dx E sup 0tT
Rd
(4.14)
0st Rd
(4.15)
Rd
This is also valid with w replaced by each un . Hence, by (4.1), {un } is a Cauchy sequence in Lp (Ω; L∞ (0, T ; Lp (R d ))). In fact, it is a Cauchy sequence in Lp (Ω; C([0, T ]; Lp (R d ))). We will show this. It is enough to show that each un ∈ C([0, T ]; Lp (R d )), for almost all ω. First of all, each un ∈ C([0, T ]; L2 (R d )), for almost all ω, and hence, un (t) ∈ Lp (R d ), for all t ∈ [0, T ], for almost all ω. Fix any n 1. With w replaced by un , (4.8) is valid for all t ∈ [0, T ], for almost all ω. By passing K → ∞, Rd
un (t)p dx =
Rd
un (0)p dx −
t cp|un |p dx ds 0 Rd
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
t +
(∇ · b)|un | dx ds + p
∞ t
3351
p|un |p aj dx dBj (s)
j =1 0 d R
0 Rd ∞
1 + 2
t p(p − 1)|un |p aj2 dx ds
(4.16)
j =1 0 d R
for all t ∈ [0, T ], for almost all ω. By subtracting (4.8) (with w replaced by un ) from (4.16), we have
p (1 − χK )un (t) dx =
Rd
p (1 − χK )un (0) dx −
t cp(1 − χK )|un |p dx ds 0 Rd
Rd
t
+
t (∇ · b)(1 − χK )|un | dx ds −
(b · ∇χK )|un |p dx ds
p
0 Rd
0 Rd
∞ t
+
p(1 − χK )|un |p aj dx dBj (s)
j =1 0 d R ∞
1 + 2
t p(p − 1)(1 − χK )|un |p aj2 dx ds
(4.17)
j =1 0 d R
for all t ∈ [0, T ], for almost all ω. By the Burkholder–Davis–Gundy inequality, it holds that ∞ t E sup p(1 − χK )|un |p aj dx dBj (s) 0tT
j =1 0 d R
T CE 0
p (1 − χK )un (t) dx
1
2
2
dt
→0
as K → ∞
(4.18)
Rd
and E
∞ T
p(p − 1)(1 − χK )|un |p aj2 dx dt
j =1 0 d R
T CE 0 Rd
p (1 − χK ) un (t) dx dt → 0
as K → ∞
(4.19)
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
Combining (4.11), (4.17)–(4.19), we find that for given > 0, there is 0 < K() < ∞ such that E sup (1 − χK() )un (t) Lp (R d ) < (4.20) 0tT
Fix any δ > 0 and choose sequences of positive numbers {mk } and {k } such that ∞
mk ↑ ∞, k ↓ 0,
mk k < δ
k=1
Let Kk = K(k ) be determined by (4.20) with = k . Define Qk = ω sup (I − χK )un (t) p k
0tT
L
> (R d )
1 mk
Then, P (Qk ) < mk k . Let Q=
∞
Qk
k=1
˜ = 0, and for each ω ∈ Ω, ˜ un (ω) ∈ C([0, T ]; L2 (R d )), and thus, Let Ω˜ ⊂ Ω such that P (Ω \ Ω) χKk un (ω) ∈ C([0, T ]; Lp (R d )), for each k. Then, P (Q) < δ and for each ω ∈ Ω˜ \ Q, un (ω) ∈ C [0, T ]; Lp R d Since δ > 0 is arbitrary, un (ω) ∈ C([0, T ]; Lp (R d )), for almost all ω, and {un } is a Cauchy sequence in Lp (Ω; C([0, T ]; Lp (R d ))). Let u be the limit. Then, u ∈ Lp Ω; C [0, T ]; Lp R d and it is easy to see that it is a solution of (0.1)–(0.2). Next we consider the pathwise uniqueness. Let u1 and u2 be two solutions. Set w = u1 − u2 ∈ Lp Ω; C [0, T ]; Lp R d We can repeat the same procedure as above to arrive at (4.8). But the fourth integral on the right-hand side of (4.8) is estimated differently. Let b ∈ L1 (0, T ) Ξ (t) = 1 + |x| L∞ (R d ) Then,
t
t
|b · ∇χK ||w| dx ds CE p
E 0 Rd
as K → ∞.
0
Ξ (s)
K|x|2K
w(s)p dx ds → 0
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3353
For other terms, we use the fact that w ∈ C([0, T ]; Lp (R d )) to pass K → ∞, and arrive at (4.15), which yields the uniqueness of a solution. The proof of Theorem 2.4 is complete. 5. Proof of Theorem 2.6 Let 2 p < ∞ be fixed. Under the assumptions (2.10)–(2.13) with gj ≡ 0, and (2.18)–(2.25), there is a unique solution on the interval [0, T ], for every 0 < T < ∞ according to Theorem 2.3. Hence, a solution exists in C([0, ∞); Lp (R d )) for almost all ω. Let us define Λ1 (u) =
(∇ · b − pc)|u|p dx
Rd
Λ2 (u, h) = p
h|u|p−1 sgn(u) dx
Rd
1 Λ3 (u, h) = p(p − 1) 2
|u|p−2 h2 dx
Rd
Λ4,j (u) = p
|u|p−1 sgn(u)fj (u) dx
Rd
1 Λ5,j (u) = p(p − 1) 2 X(t) =
2 |u|p−2 fj (u) dx
Rd
u(t)p dx
Rd
As in (3.16), we can obtain
dX = Λ1 (u) dt + Λ3 (u, h) dt +
∞
Λ5,j (u) dt + Λ2 (u, h) dB0 +
j =1
∞
Λ4,j (u) dBj
(5.1)
j =1
Let 0 < K < 1, 0 < r < 12 , and λ be a small positive number to be determined later. By Ito’s formula, it holds that r r e K + X(t) = K + X(0) +
t
λt
r λeλs K + X(s) ds
0
t +
r−1 e r K + X(s) Λ1 (u) ds +
t
λs
0
0
r−1 eλs r K + X(s) Λ3 (u, h) ds
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
+
∞
t
r−1 eλs r K + X(s) Λ5,j (u) ds
j =1 0
t +
r−1 eλs r K + X(s) Λ2 (u, h) dB0
0
+
∞
t
r−1 eλs r K + X(s) Λ4,j (u) dBj
j =1 0
+
r(r − 1) 2
t
r−2 2 Λ2 (u, h) ds eλs K + X(s)
0 ∞
r(r − 1) + 2
t
r−2 2 Λ4,j (u) ds eλs K + X(s)
(5.2)
j =1 0
for all t 0, for almost all ω. We will estimate the integrals on the right-hand side. t
r−1 e r K + X(s) Λ1 (u) ds rκ
t
λs
0
r−1 eλs K + X(s) X(s) ds
0
t rκ
r eλs K + X(s) ds
(5.3)
0
t
r−1 eλs r K + X(s) Λ3 (u, h) ds
0
t 1 r
r−1 e K + X(s) X(s) ds + Cp,1 r
t
λs
0
t 1 r
r−1 h(s) p p d ds eλs K + X(s) L (R )
0
p r eλs K + X(s) ds + Cp,1 rK r−1 sup h(s) Lp (R d ) 0s<∞
0
where we have used Λ3 u(t), h(t) 1 X(t) + Cp, h(t) p p d 1 L (R ) for some positive constants Cp,1 , for every 1 > 0.
t eλs ds 0
(5.4)
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
r(r − 1) 2
t
r−2 2 Λ2 (u, h) ds 0 eλs K + X(s)
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(5.5)
0 ∞ j =1 0
t
r−1 1 e r K + X(s) Λ5,j (u) ds p(p − 1)rη2 2
t
λs
r−1 eλs K + X(s) X(s) ds
0
1 p(p − 1)rη2 2
t
r eλs K + X(s) ds
(5.6)
0 ∞
1 − r(1 − r) 2
t
r−2 2 Λ4,j (u) ds eλs K + X(s)
j =1 0
1 − r(1 − r)p 2 η1 2
t
r−2 eλs K + X(s) X(s)2 ds
0
1 − r(1 − r)p 2 η1 (1 − 2 ) 2
t
r eλs K + X(s) ds
0
1 + r(1 − r)p 2 η1 CK,2 K r−2 2
t eλs ds
(5.7)
0
where we have used
1 K 2, (1 + δ)X 2 (K + X)2 − 1 + δ
∀δ > 0
which can be written as X 2 (1 − 2 )(K + X)2 − CK,2 ,
∀2 > 0
By virtue of (2.25), we can choose some small 0 < r < 12 , 1 > 0, 2 > 0, and λ > 0 such that 1 1 r(1 − r)p 2 (1 − 2 )η1 > p(p − 1)rη2 + r1 + λ + rκ 2 2
(5.8)
Then, it follows from (5.2)–(5.8) that r r 1 E eλt K + X(t) E K + X(0) + CK eλt − 1 λ
(5.9)
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
Next let u1 and u2 be the solutions of (2.17) with u1 (0) = u0,1 and u2 (0) = u0,2 , respectively. Then, u1 − u2 is a solution of ut + b · ∇u + cu =
∞ dBj , fj (u1 ) − fj (u2 ) dt
(t, x) ∈ (0, ∞) × R d
(5.10)
j =1
Let us set X1,2 (t) =
u1 (t) − u2 (t)p dx
Rd
By repetition of the above procedure as in (5.2), we arrive at r r e K + X1,2 (t) = K + X1,2 (0) +
t
λt
r λeλs K + X1,2 (s) ds
0
t +
r−1 eλs r K + X1,2 (s) Λ1 (u1 − u2 ) ds
0
+
∞
t
r−1 eλs r K + X1,2 (s) Λ5,j (u1 , u2 ) ds
t
r−1 eλs r K + X1,2 (s) Λ4,j (u1 , u2 ) dBj
j =1 0
+
∞ j =1 0
∞
r(r − 1) + 2
t
r−2 2 Λ4,j (u1 , u2 ) ds eλs K + X1,2 (s)
(5.11)
j =1 0
where Λ4,j (u1 , u2 ) = p
|u1 − u2 |p−1 sgn(u1 − u2 ) fj (u1 ) − fj (u2 ) dx
Rd
1 Λ5,j (u1 , u2 ) = p(p − 1) 2
2 |u1 − u2 |p−2 fj (u1 ) − fj (u2 ) dx
Rd
A salient feature of the estimate of the terms of (5.11) is that we can pass K → 0, which is impossible in the presence of h. t
t
r r−1 λs e r K + X1,2 (s) Λ1 (u1 − u2 ) ds rκE e X1,2 (s) ds λs
lim E
K→0
0
0
(5.12)
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
lim E
K→0
∞
t
r−1 eλs r K + X1,2 (s) Λ5,j (u1 , u2 ) ds
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j =1 0
1 p(p − 1)rη2 E 2
t
r eλs X1,2 (s) ds
(5.13)
0
∞
1 lim E − r(1 − r) K→0 2
t
r−2 2 Λ4,j (u1 , u2 ) ds e K + X1,2 (s)
λs
j =1 0
1 − r(1 − r)p 2 η1 E 2
t e
λs
r X1,2 (s) ds
(5.14)
0
Under the same condition as in (5.8), we have r r E eλt X1,2 (t) E X1,2 (0)
(5.15)
Following [4,14], we can introduce a sequence of mutually independent Brownian motions ∞ ˜ {B˜ j (t)}∞ j =0 and a filtration {Ft } which extend the original {Bj (t)}j =0 and {Ft } to the whole real line −∞ < t < ∞. We then denote by Y (t; s, z) the solution u(t) of (2.17) on the interval [s, ∞) with the initial condition u(s) = z ∈ Lp R d Let L{. . .} denote the probability law. Then, by virtue of (2.20) and the procedure to construct the solution, we have L Y (t; s, z) = L Y (t + kT ; s + kT , z)
(5.16)
for all integer k. Also, (5.9) and (5.15) are valid in the form r r 1 E eλ(t−s) K + X(t) E K + X(s) + CK eλ(t−s) − 1 λ
(5.17)
r r E eλ(t−s) X1,2 (t) E X1,2 (s)
(5.18)
and
for all −∞ < s < t < ∞. By using the same method as in [4,14], it can be shown that Y (·; ·, ·) satisfies the Markov property. It follows from (5.17) and (5.18) that {Y (0; −nT , 0)}∞ n=1 is a Cauchy sequence in Lpr (Ω; Lp (R d )), which is a complete metric space. Let us define Y∞ = lim Y (0; −nT , 0) in Lpr Ω; Lp R d n→∞ μn = L Y (0; −nT , 0) μ = L{Y∞ },
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J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
Then, μ and μn are probability measures on Lp (R d ), and μn → μ weakly We will show that ∗ ∗ P0,t μ = P0,t+T μ,
∀t 0
(5.19)
Let us choose any bounded continuous function ϕ on Lp (R d ). (P0,t ϕ)(y) dμ(y) Lp (R d )
E ϕ Y (t; 0, y) dμ(y)
= Lp (R d )
= lim
n→∞ Lp (R d )
= lim
n→∞
E ϕ Y (t; 0, y) dμn (y) = lim
E ϕ Y (t; 0, y) y=Y (0;−nT ,0) dP
n→∞ Ω
E ϕ Y (t; −nT , 0) Fˆ 0 dP = lim
E ϕ Y (t; −nT , 0) dP
n→∞
Ω
= lim
n→∞
Ω
E ϕ Y t + T ; −(n − 1)T , 0 dP
Ω
= lim
n→∞ Ω
= lim
E ϕ Y t + T ; −(n − 1)T , 0 Fˆ 0 dP
n→∞ Lp (R d )
E ϕ Y (t + T ; 0, y) dμn (y) =
(P0,t+T ϕ)(y) dμ(y)
Lp (R d )
which proves (5.19). Next we will show that lim (P0,kT ϕ)(ξ ) =
k→∞
ϕ(y) dμ(y)
(5.20)
Lp (R d )
for each ξ ∈ Lp (R d ), and each bounded Lipschitz continuous function ϕ on Lp (R d ). By means of (5.19), (P0,kT ϕ)(ξ ) − ϕ(y) dμ(y) = Lp (R d )
Lp (R d )
(P0,kT ϕ)(ξ ) − (P0,kT ϕ)(y) dμ(y)
J.U. Kim / Journal of Functional Analysis 259 (2010) 3328–3359
3359
E ϕ Y (kT ; 0, ξ ) − ϕ Y (kT ; 0, y) dμ(y)
Lp (R d )
It follows from (5.18) that pr pr E Y (kT ; 0, ξ ) − Y (kT ; 0, y) Lp (R d ) Ce−kT ξ − y Lp (R d ) → 0 as k → ∞. Thus, for each ξ , y, Y (kT ; 0, ξ ) − Y (kT ; 0, y) p d → 0 in probability L (R ) as k → ∞, and lim E ϕ Y (kT ; 0, ξ ) − ϕ Y (kT ; 0, y) = 0,
k→∞
for each ξ, y
This proves (5.20) for every bounded Lipschitz continuous function on Lp (R d ). This yields the uniqueness of μ among all probability measures over (Lp (R d ), B(Lp (R d ))) which satisfy (5.19). Now we consider the case where b, c, and h are time-independent. Then, we may take T > 0 arbitrary. By the uniqueness of T -periodic measure, the above measure μ is a unique invariant measure. This completes the proof of Theorem 2.6. References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004) 227–260. [2] L. Ambrosio, A. Figalli, On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna– Lions, J. Funct. Anal. 256 (2009) 179–214. [3] Z. Brzezniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995) 1–45. [4] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. [5] R. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511–547. [6] W. E, E. Vanden Eijnden, Generalized flows, intrinsic stochasticity and turbulent transport, Proc. Natl. Acad. Sci. 97 (2000) 8200–8205. [7] F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010) 1–53. [8] T. Funaki, Construction of a solution of random transport equation with boundary condition, J. Math. Soc. Japan 31 (1979) 719–744. [9] I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, second ed., Springer, New York, Berlin, Heidelberg, 1997. [10] Y. Le Jan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826–873. [11] P.L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1, Clarendon Press, Oxford, 1996. [12] S. Lototskii, B. Rozovskii, Passive scalar equation in a turbulent incompressible Gaussian velocity field, Russian Math. Surveys 59 (2004) 297–312. [13] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. 426 (2004) 1–63. [14] C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1905, Springer, Berlin, 2007. [15] I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1967.
